LMFDB - Embedded newform 43.7.b.b.42.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,7,Mod(42,43)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("43.42");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	43.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.89232559565$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1038 x^{18} + 455829 x^{16} + 110435384 x^{14} + 16133606976 x^{12} + 1458210485616 x^{10} + \cdots + 64\!\cdots\!40 $ x^20 + 1038x^18 + 455829x^16 + 110435384x^14 + 16133606976x^12 + 1458210485616x^10 + 80362197690736x^8 + 2545997652841536x^6 + 40210452531479040x^4 + 212033222410436608*x^2 + 64236717122519040
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			42.8
Root			$-6.90119i$ of defining polynomial
Character	$\chi$	$=$	43.42
Dual form			43.7.b.b.42.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.90119i q^{2} -11.0801i q^{3} +16.3736 q^{4} +107.518i q^{5} -76.4660 q^{6} -594.412i q^{7} -554.673i q^{8} +606.231 q^{9} +O(q^{10})$	\(q-6.90119i q^{2} -11.0801i q^{3} +16.3736 q^{4} +107.518i q^{5} -76.4660 q^{6} -594.412i q^{7} -554.673i q^{8} +606.231 q^{9} +742.004 q^{10} -326.752 q^{11} -181.422i q^{12} -3327.43 q^{13} -4102.15 q^{14} +1191.32 q^{15} -2779.99 q^{16} +1730.37 q^{17} -4183.71i q^{18} -6997.69i q^{19} +1760.47i q^{20} -6586.16 q^{21} +2254.98i q^{22} -7593.56 q^{23} -6145.85 q^{24} +4064.80 q^{25} +22963.2i q^{26} -14794.5i q^{27} -9732.68i q^{28} +10036.1i q^{29} -8221.51i q^{30} +25812.1 q^{31} -16313.8i q^{32} +3620.46i q^{33} -11941.6i q^{34} +63910.2 q^{35} +9926.19 q^{36} +37980.1i q^{37} -48292.4 q^{38} +36868.3i q^{39} +59637.6 q^{40} -18315.7 q^{41} +45452.3i q^{42} +(61156.1 - 50806.4i) q^{43} -5350.12 q^{44} +65180.9i q^{45} +52404.6i q^{46} +130402. q^{47} +30802.7i q^{48} -235677. q^{49} -28051.9i q^{50} -19172.7i q^{51} -54482.1 q^{52} +272103. q^{53} -102100. q^{54} -35131.9i q^{55} -329705. q^{56} -77535.3 q^{57} +69261.1 q^{58} +216781. q^{59} +19506.2 q^{60} +344144. i q^{61} -178134. i q^{62} -360351. i q^{63} -290504. q^{64} -357760. i q^{65} +24985.5 q^{66} -227534. q^{67} +28332.4 q^{68} +84137.6i q^{69} -441056. i q^{70} +281680. i q^{71} -336260. i q^{72} +294591. i q^{73} +262108. q^{74} -45038.5i q^{75} -114578. i q^{76} +194226. i q^{77} +254435. q^{78} -497124. q^{79} -298900. i q^{80} +278017. q^{81} +126400. i q^{82} -143881. q^{83} -107839. q^{84} +186046. i q^{85} +(-350625. - 422050. i) q^{86} +111201. q^{87} +181241. i q^{88} +523099. i q^{89} +449826. q^{90} +1.97786e6i q^{91} -124334. q^{92} -286002. i q^{93} -899926. i q^{94} +752380. q^{95} -180760. q^{96} -430866. q^{97} +1.62645e6i q^{98} -198087. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 796 q^{4} - 258 q^{6} - 2736 q^{9}+O(q^{10}) $ 20 * q - 796 * q^4 - 258 * q^6 - 2736 * q^9	$ 20 q - 796 q^{4} - 258 q^{6} - 2736 q^{9} - 1962 q^{10} + 2616 q^{11} - 5612 q^{13} - 3876 q^{14} + 4048 q^{15} + 14900 q^{16} + 3328 q^{17} + 10544 q^{21} - 836 q^{23} + 26966 q^{24} - 57904 q^{25} + 17116 q^{31} - 150472 q^{35} - 132110 q^{36} + 2266 q^{38} + 401286 q^{40} - 141936 q^{41} + 58160 q^{43} + 88544 q^{44} + 48452 q^{47} - 20644 q^{49} + 12292 q^{52} + 425212 q^{53} + 173740 q^{54} + 447864 q^{56} - 92904 q^{57} + 86502 q^{58} - 918856 q^{59} + 429848 q^{60} - 156892 q^{64} - 1246176 q^{66} + 1098496 q^{67} - 2589854 q^{68} - 1224106 q^{74} + 855716 q^{78} - 401492 q^{79} + 470644 q^{81} + 1354360 q^{83} - 637268 q^{84} + 3046356 q^{86} + 2116740 q^{87} - 683716 q^{90} - 485998 q^{92} - 2711660 q^{95} - 2480062 q^{96} + 7814560 q^{97} + 3744560 q^{99}+O(q^{100}) $ 20 * q - 796 * q^4 - 258 * q^6 - 2736 * q^9 - 1962 * q^10 + 2616 * q^11 - 5612 * q^13 - 3876 * q^14 + 4048 * q^15 + 14900 * q^16 + 3328 * q^17 + 10544 * q^21 - 836 * q^23 + 26966 * q^24 - 57904 * q^25 + 17116 * q^31 - 150472 * q^35 - 132110 * q^36 + 2266 * q^38 + 401286 * q^40 - 141936 * q^41 + 58160 * q^43 + 88544 * q^44 + 48452 * q^47 - 20644 * q^49 + 12292 * q^52 + 425212 * q^53 + 173740 * q^54 + 447864 * q^56 - 92904 * q^57 + 86502 * q^58 - 918856 * q^59 + 429848 * q^60 - 156892 * q^64 - 1246176 * q^66 + 1098496 * q^67 - 2589854 * q^68 - 1224106 * q^74 + 855716 * q^78 - 401492 * q^79 + 470644 * q^81 + 1354360 * q^83 - 637268 * q^84 + 3046356 * q^86 + 2116740 * q^87 - 683716 * q^90 - 485998 * q^92 - 2711660 * q^95 - 2480062 * q^96 + 7814560 * q^97 + 3744560 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	6.90119i		−	0.862648i	−0.902197	−	0.431324i	$-0.858047\pi$
$2$		−	6.90119i		−	0.862648i	0.902197	−	0.431324i	$-0.141953\pi$
$3$		−	11.0801i		−	0.410375i	−0.978723	−	0.205188i	$-0.934220\pi$
$3$		−	11.0801i		−	0.410375i	0.978723	−	0.205188i	$-0.0657804\pi$
$4$	16.3736			0.255838
$5$			107.518i			0.860147i	0.902794	+	0.430074i	$0.141512\pi$
$5$			107.518i			0.860147i	−0.902794	+	0.430074i	$0.858488\pi$
$6$	−76.4660			−0.354009
$7$		−	594.412i		−	1.73298i	−0.499194	−	0.866490i	$-0.666371\pi$
$7$		−	594.412i		−	1.73298i	0.499194	−	0.866490i	$-0.333629\pi$
$8$		−	554.673i		−	1.08335i
$9$	606.231			0.831592
$10$	742.004			0.742004
$11$	−326.752			−0.245494			−0.122747	−	0.992438i	$-0.539170\pi$
$11$	−326.752			−0.245494			−0.122747	+	0.992438i	$0.539170\pi$
$12$		−	181.422i		−	0.104989i
$13$	−3327.43			−1.51453			−0.757266	−	0.653106i	$-0.773466\pi$
$13$	−3327.43			−1.51453			−0.757266	+	0.653106i	$0.773466\pi$
$14$	−4102.15			−1.49495
$15$	1191.32			0.352983
$16$	−2779.99			−0.678709
$17$	1730.37			0.352202			0.176101	−	0.984372i	$-0.443651\pi$
$17$	1730.37			0.352202			0.176101	+	0.984372i	$0.443651\pi$
$18$		−	4183.71i		−	0.717372i
$19$		−	6997.69i		−	1.02022i	−0.860109	−	0.510110i	$-0.829605\pi$
$19$		−	6997.69i		−	1.02022i	0.860109	−	0.510110i	$-0.170395\pi$
$20$			1760.47i			0.220058i
$21$	−6586.16			−0.711172
$22$			2254.98i			0.211775i
$23$	−7593.56			−0.624111			−0.312055	−	0.950064i	$-0.601017\pi$
$23$	−7593.56			−0.624111			−0.312055	+	0.950064i	$0.601017\pi$
$24$	−6145.85			−0.444578
$25$	4064.80			0.260147
$26$			22963.2i			1.30651i
$27$		−	14794.5i		−	0.751640i
$28$		−	9732.68i		−	0.443362i
$29$			10036.1i			0.411502i	0.978604	+	0.205751i	$0.0659637\pi$
$29$			10036.1i			0.411502i	−0.978604	+	0.205751i	$0.934036\pi$
$30$		−	8221.51i		−	0.304500i
$31$	25812.1			0.866440			0.433220	−	0.901288i	$-0.357377\pi$
$31$	25812.1			0.866440			0.433220	+	0.901288i	$0.357377\pi$
$32$		−	16313.8i		−	0.497859i
$33$			3620.46i			0.100745i
$34$		−	11941.6i		−	0.303826i
$35$	63910.2			1.49062
$36$	9926.19			0.212753
$37$			37980.1i			0.749809i	0.927063	+	0.374905i	$0.122325\pi$
$37$			37980.1i			0.749809i	−0.927063	+	0.374905i	$0.877675\pi$
$38$	−48292.4			−0.880091
$39$			36868.3i			0.621527i
$40$	59637.6			0.931837
$41$	−18315.7			−0.265749			−0.132874	−	0.991133i	$-0.542421\pi$
$41$	−18315.7			−0.265749			−0.132874	+	0.991133i	$0.542421\pi$
$42$			45452.3i			0.613491i
$43$	61156.1	−	50806.4i	0.769191	−	0.639019i
$43$	61156.1	−	50806.4i	0.769191	−	0.639019i
$44$	−5350.12			−0.0628066
$45$			65180.9i			0.715292i
$46$			52404.6i			0.538388i
$47$	130402.			1.25600			0.627999	−	0.778214i	$-0.283874\pi$
$47$	130402.			1.25600			0.627999	+	0.778214i	$0.283874\pi$
$48$			30802.7i			0.278525i
$49$	−235677.			−2.00322
$50$		−	28051.9i		−	0.224415i
$51$		−	19172.7i		−	0.144535i
$52$	−54482.1			−0.387475
$53$	272103.			1.82771			0.913853	−	0.406045i	$-0.133092\pi$
$53$	272103.			1.82771			0.913853	+	0.406045i	$0.133092\pi$
$54$	−102100.			−0.648401
$55$		−	35131.9i		−	0.211161i
$56$	−329705.			−1.87742
$57$	−77535.3			−0.418673
$58$	69261.1			0.354981
$59$	216781.			1.05552			0.527759	−	0.849394i	$-0.323032\pi$
$59$	216781.			1.05552			0.527759	+	0.849394i	$0.323032\pi$
$60$	19506.2			0.0903064
$61$			344144.i			1.51618i	0.652151	+	0.758089i	$0.273867\pi$
$61$			344144.i			1.51618i	−0.652151	+	0.758089i	$0.726133\pi$
$62$		−	178134.i		−	0.747433i
$63$		−	360351.i		−	1.44113i
$64$	−290504.			−1.10819
$65$		−	357760.i		−	1.30272i
$66$	24985.5			0.0869071
$67$	−227534.			−0.756524			−0.378262	−	0.925699i	$-0.623478\pi$
$67$	−227534.			−0.756524			−0.378262	+	0.925699i	$0.623478\pi$
$68$	28332.4			0.0901065
$69$			84137.6i			0.256120i
$70$		−	441056.i		−	1.28588i
$71$			281680.i			0.787013i	0.919322	+	0.393506i	$0.128738\pi$
$71$			281680.i			0.787013i	−0.919322	+	0.393506i	$0.871262\pi$
$72$		−	336260.i		−	0.900902i
$73$			294591.i			0.757270i	0.925546	+	0.378635i	$0.123606\pi$
$73$			294591.i			0.757270i	−0.925546	+	0.378635i	$0.876394\pi$
$74$	262108.			0.646822
$75$		−	45038.5i		−	0.106758i
$76$		−	114578.i		−	0.261011i
$77$			194226.i			0.425436i
$78$	254435.			0.536159
$79$	−497124.			−1.00828			−0.504142	−	0.863621i	$-0.668191\pi$
$79$	−497124.			−1.00828			−0.504142	+	0.863621i	$0.668191\pi$
$80$		−	298900.i		−	0.583790i
$81$	278017.			0.523138
$82$			126400.i			0.229248i
$83$	−143881.			−0.251634			−0.125817	−	0.992053i	$-0.540155\pi$
$83$	−143881.			−0.251634			−0.125817	+	0.992053i	$0.540155\pi$
$84$	−107839.			−0.181945
$85$			186046.i			0.302945i
$86$	−350625.	−	422050.i	−0.551248	−	0.663542i
$87$	111201.			0.168870
$88$			181241.i			0.265955i
$89$			523099.i			0.742017i	0.928629	+	0.371009i	$0.120988\pi$
$89$			523099.i			0.742017i	−0.928629	+	0.371009i	$0.879012\pi$
$90$	449826.			0.617045
$91$			1.97786e6i			2.62466i
$92$	−124334.			−0.159671
$93$		−	286002.i		−	0.355565i
$94$		−	899926.i		−	1.08349i
$95$	752380.			0.877539
$96$	−180760.			−0.204309
$97$	−430866.			−0.472093			−0.236046	−	0.971742i	$-0.575852\pi$
$97$	−430866.			−0.472093			−0.236046	+	0.971742i	$0.575852\pi$
$98$			1.62645e6i			1.72807i
$99$	−198087.			−0.204151
$100$	66555.4			0.0665554
$101$	842018.			0.817254			0.408627	−	0.912701i	$-0.366008\pi$
$101$	842018.			0.817254			0.408627	+	0.912701i	$0.366008\pi$
$102$	−132314.			−0.124683
$103$	104065.			0.0952343			0.0476171	−	0.998866i	$-0.484837\pi$
$103$	104065.			0.0952343			0.0476171	+	0.998866i	$0.484837\pi$
$104$			1.84564e6i			1.64076i
$105$		−	708134.i		−	0.611713i
$106$		−	1.87784e6i		−	1.57667i
$107$	837014.			0.683253			0.341627	−	0.939836i	$-0.389022\pi$
$107$	837014.			0.683253			0.341627	+	0.939836i	$0.389022\pi$
$108$		−	242240.i		−	0.192298i
$109$	2.47342e6			1.90993			0.954967	−	0.296711i	$-0.0958899\pi$
$109$	2.47342e6			1.90993			0.954967	+	0.296711i	$0.0958899\pi$
$110$	−242452.			−0.182158
$111$	420824.			0.307703
$112$			1.65246e6i			1.17619i
$113$		−	2.68260e6i		−	1.85918i	−0.368597	−	0.929589i	$-0.620162\pi$
$113$		−	2.68260e6i		−	1.85918i	0.368597	−	0.929589i	$-0.379838\pi$
$114$			535086.i			0.361168i
$115$		−	816447.i		−	0.536827i
$116$			164328.i			0.105278i
$117$	−2.01719e6			−1.25947
$118$		−	1.49605e6i		−	0.910541i
$119$		−	1.02855e6i		−	0.610359i
$120$		−	660792.i		−	0.382403i
$121$	−1.66479e6			−0.939733
$122$	2.37500e6			1.30793
$123$			202940.i			0.109057i
$124$	422638.			0.221668
$125$			2.11702e6i			1.08391i
$126$	−2.48685e6			−1.24319
$127$	899817.			0.439281			0.219641	−	0.975581i	$-0.429512\pi$
$127$	899817.			0.439281			0.219641	+	0.975581i	$0.429512\pi$
$128$			960739.i			0.458116i
$129$	−562942.	−	677617.i	−0.262237	−	0.315657i
$130$	−2.46897e6			−1.12379
$131$			1.57065e6i			0.698662i	0.936999	+	0.349331i	$0.113591\pi$
$131$			1.57065e6i			0.698662i	−0.936999	+	0.349331i	$0.886409\pi$
$132$			59280.0i			0.0257743i
$133$	−4.15951e6			−1.76802
$134$			1.57026e6i			0.652614i
$135$	1.59068e6			0.646521
$136$		−	959789.i		−	0.381557i
$137$		−	2.54070e6i		−	0.988080i	−0.869439	−	0.494040i	$-0.835520\pi$
$137$		−	2.54070e6i		−	0.988080i	0.869439	−	0.494040i	$-0.164480\pi$
$138$	580649.			0.220941
$139$	−4.18653e6			−1.55887			−0.779434	−	0.626484i	$-0.784493\pi$
$139$	−4.18653e6			−1.55887			−0.779434	+	0.626484i	$0.784493\pi$
$140$	1.04644e6			0.381356
$141$		−	1.44487e6i		−	0.515431i
$142$	1.94393e6			0.678915
$143$	1.08724e6			0.371808
$144$	−1.68532e6			−0.564409
$145$	−1.07907e6			−0.353952
$146$	2.03303e6			0.653258
$147$			2.61133e6i			0.822072i
$148$			621872.i			0.191830i
$149$		−	3.45889e6i		−	1.04563i	−0.852446	−	0.522815i	$-0.824882\pi$
$149$		−	3.45889e6i		−	1.04563i	0.852446	−	0.522815i	$-0.175118\pi$
$150$	−310819.			−0.0920945
$151$		−	6.00850e6i		−	1.74516i	−0.488472	−	0.872579i	$-0.662446\pi$
$151$		−	6.00850e6i		−	1.74516i	0.488472	−	0.872579i	$-0.337554\pi$
$152$	−3.88143e6			−1.10525
$153$	1.04900e6			0.292888
$154$	1.34039e6			0.367002
$155$			2.77528e6i			0.745266i
$156$			603668.i			0.159010i
$157$			3.21178e6i			0.829940i	0.909835	+	0.414970i	$0.136208\pi$
$157$			3.21178e6i			0.829940i	−0.909835	+	0.414970i	$0.863792\pi$
$158$			3.43074e6i			0.869795i
$159$		−	3.01494e6i		−	0.750045i
$160$	1.75404e6			0.428232
$161$			4.51370e6i			1.08157i
$162$		−	1.91865e6i		−	0.451284i
$163$		−	2.40767e6i		−	0.555947i	−0.960589	−	0.277974i	$-0.910337\pi$
$163$		−	2.40767e6i		−	0.555947i	0.960589	−	0.277974i	$-0.0896628\pi$
$164$	−299894.			−0.0679886
$165$	−389266.			−0.0866551
$166$			992950.i			0.217072i
$167$	473244.			0.101610			0.0508049	−	0.998709i	$-0.483821\pi$
$167$	473244.			0.101610			0.0508049	+	0.998709i	$0.483821\pi$
$168$			3.65317e6i			0.770446i
$169$	6.24497e6			1.29381
$170$	1.28394e6			0.261335
$171$		−	4.24221e6i		−	0.848407i
$172$	1.00135e6	−	831885.i	0.196788	−	0.163485i
$173$	2.36439e6			0.456648			0.228324	−	0.973585i	$-0.426675\pi$
$173$	2.36439e6			0.456648			0.228324	+	0.973585i	$0.426675\pi$
$174$		−	767422.i		−	0.145676i
$175$		−	2.41616e6i		−	0.450829i
$176$	908369.			0.166619
$177$		−	2.40197e6i		−	0.433159i
$178$	3.61001e6			0.640100
$179$			5.36583e6i			0.935573i	0.883841	+	0.467787i	$0.154948\pi$
$179$			5.36583e6i			0.935573i	−0.883841	+	0.467787i	$0.845052\pi$
$180$			1.06725e6i			0.182999i
$181$	−1.32962e6			−0.224228			−0.112114	−	0.993695i	$-0.535762\pi$
$181$	−1.32962e6			−0.224228			−0.112114	+	0.993695i	$0.535762\pi$
$182$	1.36496e7			2.26415
$183$	3.81316e6			0.622202
$184$			4.21194e6i			0.676128i
$185$	−4.08356e6			−0.644946
$186$	−1.97375e6			−0.306728
$187$	−565402.			−0.0864634
$188$	2.13515e6			0.321332
$189$	−8.79405e6			−1.30258
$190$		−	5.19232e6i		−	0.757008i
$191$			1.27305e7i			1.82703i	0.406807	+	0.913514i	$0.366642\pi$
$191$			1.27305e7i			1.82703i	−0.406807	+	0.913514i	$0.633358\pi$
$192$			3.21883e6i			0.454772i
$193$	−3.87363e6			−0.538823			−0.269411	−	0.963025i	$-0.586829\pi$
$193$	−3.87363e6			−0.538823			−0.269411	+	0.963025i	$0.586829\pi$
$194$			2.97349e6i			0.407250i
$195$	−3.96402e6			−0.534604
$196$	−3.85888e6			−0.512499
$197$	−7.79358e6			−1.01939			−0.509693	−	0.860356i	$-0.670241\pi$
$197$	−7.79358e6			−1.01939			−0.509693	+	0.860356i	$0.670241\pi$
$198$			1.36704e6i			0.176110i
$199$			2.68474e6i			0.340678i	0.985386	+	0.170339i	$0.0544862\pi$
$199$			2.68474e6i			0.340678i	−0.985386	+	0.170339i	$0.945514\pi$
$200$		−	2.25463e6i		−	0.281829i
$201$			2.52111e6i			0.310459i
$202$		−	5.81092e6i		−	0.705003i
$203$	5.96559e6			0.713124
$204$		−	313926.i		−	0.0369775i
$205$		−	1.96927e6i		−	0.228583i
$206$		−	718172.i		−	0.0821537i
$207$	−4.60345e6			−0.519006
$208$	9.25023e6			1.02793
$209$			2.28651e6i			0.250458i
$210$	−4.88696e6			−0.527693
$211$			1.89548e6i			0.201777i	0.994898	+	0.100888i	$0.0321685\pi$
$211$			1.89548e6i			0.201777i	−0.994898	+	0.100888i	$0.967831\pi$
$212$	4.45532e6			0.467596
$213$	3.12106e6			0.322970
$214$		−	5.77639e6i		−	0.589407i
$215$	5.46263e6	+	6.57540e6i	0.549650	+	0.661618i
$216$	−8.20613e6			−0.814287
$217$		−	1.53430e7i		−	1.50152i
$218$		−	1.70695e7i		−	1.64760i
$219$	3.26411e6			0.310765
$220$		−	575236.i		−	0.0540229i
$221$	−5.75768e6			−0.533421
$222$		−	2.90419e6i		−	0.265440i
$223$		−	5.86386e6i		−	0.528772i	−0.964417	−	0.264386i	$-0.914831\pi$
$223$		−	5.86386e6i		−	0.528772i	0.964417	−	0.264386i	$-0.0851694\pi$
$224$	−9.69715e6			−0.862780
$225$	2.46420e6			0.216336
$226$	−1.85131e7			−1.60382
$227$			1.08262e7i			0.925551i	0.886476	+	0.462775i	$0.153146\pi$
$227$			1.08262e7i			0.925551i	−0.886476	+	0.462775i	$0.846854\pi$
$228$	−1.26953e6			−0.107112
$229$	−4.16761e6			−0.347041			−0.173520	−	0.984830i	$-0.555514\pi$
$229$	−4.16761e6			−0.347041			−0.173520	+	0.984830i	$0.555514\pi$
$230$	−5.63445e6			−0.463093
$231$	2.15204e6			0.174588
$232$	5.56677e6			0.445799
$233$		−	2.22514e7i		−	1.75909i	−0.475812	−	0.879547i	$-0.657846\pi$
$233$		−	2.22514e7i		−	1.75909i	0.475812	−	0.879547i	$-0.342154\pi$
$234$			1.39210e7i			1.08648i
$235$			1.40206e7i			1.08034i
$236$	3.54950e6			0.270042
$237$			5.50820e6i			0.413775i
$238$	−7.09823e6			−0.526525
$239$	2.26230e7			1.65713			0.828567	−	0.559891i	$-0.189157\pi$
$239$	2.26230e7			1.65713			0.828567	+	0.559891i	$0.189157\pi$
$240$	−3.31185e6			−0.239573
$241$		−	1.73057e7i		−	1.23634i	−0.786045	−	0.618170i	$-0.787874\pi$
$241$		−	1.73057e7i		−	1.23634i	0.786045	−	0.618170i	$-0.212126\pi$
$242$			1.14891e7i			0.810659i
$243$		−	1.38657e7i		−	0.966323i
$244$			5.63488e6i			0.387896i
$245$		−	2.53396e7i		−	1.72306i
$246$	1.40053e6			0.0940776
$247$			2.32843e7i			1.54516i
$248$		−	1.43173e7i		−	0.938654i
$249$			1.59422e6i			0.103264i
$250$	1.46099e7			0.935035
$251$	−2.40761e7			−1.52253			−0.761264	−	0.648442i	$-0.775421\pi$
$251$	−2.40761e7			−1.52253			−0.761264	+	0.648442i	$0.775421\pi$
$252$		−	5.90025e6i		−	0.368696i
$253$	2.48121e6			0.153215
$254$		−	6.20980e6i		−	0.378945i
$255$	2.06142e6			0.124321
$256$	−1.19620e7			−0.712993
$257$			9.76539e6i			0.575294i	0.957736	+	0.287647i	$0.0928730\pi$
$257$			9.76539e6i			0.575294i	−0.957736	+	0.287647i	$0.907127\pi$
$258$	−4.67636e6	+	3.88497e6i	−0.272301	+	0.226219i
$259$	2.25758e7			1.29940
$260$		−	5.85782e6i		−	0.333285i
$261$			6.08420e6i			0.342202i
$262$	1.08394e7			0.602699
$263$		−	7.57778e6i		−	0.416557i	−0.978070	−	0.208279i	$-0.933214\pi$
$263$		−	7.57778e6i		−	0.416557i	0.978070	−	0.208279i	$-0.0667861\pi$
$264$	2.00817e6			0.109141
$265$			2.92561e7i			1.57210i
$266$			2.87056e7i			1.52518i
$267$	5.79601e6			0.304506
$268$	−3.72556e6			−0.193547
$269$	−8.27452e6			−0.425095			−0.212548	−	0.977151i	$-0.568176\pi$
$269$	−8.27452e6			−0.425095			−0.212548	+	0.977151i	$0.568176\pi$
$270$		−	1.09776e7i		−	0.557720i
$271$	−1.77022e7			−0.889445			−0.444723	−	0.895668i	$-0.646698\pi$
$271$	−1.77022e7			−0.889445			−0.444723	+	0.895668i	$0.646698\pi$
$272$	−4.81041e6			−0.239043
$273$	2.19150e7			1.07709
$274$	−1.75339e7			−0.852365
$275$	−1.32818e6			−0.0638645
$276$			1.37764e6i			0.0655251i
$277$			3.35720e7i			1.57957i	0.613385	+	0.789784i	$0.289807\pi$
$277$			3.35720e7i			1.57957i	−0.613385	+	0.789784i	$0.710193\pi$
$278$			2.88920e7i			1.34475i
$279$	1.56481e7			0.720525
$280$		−	3.54493e7i		−	1.61486i
$281$	3.12675e7			1.40921			0.704604	−	0.709601i	$-0.251125\pi$
$281$	3.12675e7			1.40921			0.704604	+	0.709601i	$0.251125\pi$
$282$	−9.97129e6			−0.444635
$283$	4.43759e7			1.95789			0.978945	−	0.204124i	$-0.0654346\pi$
$283$	4.43759e7			1.95789			0.978945	+	0.204124i	$0.0654346\pi$
$284$			4.61213e6i			0.201348i
$285$		−	8.33647e6i		−	0.360120i
$286$		−	7.50328e6i		−	0.320740i
$287$			1.08871e7i			0.460537i
$288$		−	9.88995e6i		−	0.414016i
$289$	−2.11434e7			−0.875954
$290$			7.44684e6i			0.305336i
$291$			4.77405e6i			0.193735i
$292$			4.82352e6i			0.193738i
$293$	2.10928e6			0.0838553			0.0419276	−	0.999121i	$-0.486650\pi$
$293$	2.10928e6			0.0838553			0.0419276	+	0.999121i	$0.486650\pi$
$294$	1.80213e7			0.709159
$295$			2.33080e7i			0.907901i
$296$	2.10665e7			0.812303
$297$			4.83415e6i			0.184523i
$298$	−2.38705e7			−0.902011
$299$	2.52670e7			0.945236
$300$		−	737443.i		−	0.0273127i
$301$	−3.02000e7	−	3.63519e7i	−1.10741	−	1.33299i
$302$	−4.14658e7			−1.50546
$303$		−	9.32967e6i		−	0.335381i
$304$			1.94535e7i			0.692433i
$305$	−3.70018e7			−1.30414
$306$		−	7.23936e6i		−	0.252660i
$307$	1.34969e7			0.466466			0.233233	−	0.972421i	$-0.425070\pi$
$307$	1.34969e7			0.466466			0.233233	+	0.972421i	$0.425070\pi$
$308$			3.18017e6i			0.108843i
$309$		−	1.15305e6i		−	0.0390818i
$310$	1.91527e7			0.642902
$311$	−4.02057e7			−1.33662			−0.668308	−	0.743885i	$-0.732981\pi$
$311$	−4.02057e7			−1.33662			−0.668308	+	0.743885i	$0.732981\pi$
$312$	2.04499e7			0.673329
$313$			2.75678e6i			0.0899021i	0.998989	+	0.0449510i	$0.0143132\pi$
$313$			2.75678e6i			0.0899021i	−0.998989	+	0.0449510i	$0.985687\pi$
$314$	2.21651e7			0.715946
$315$	3.87443e7			1.23959
$316$	−8.13972e6			−0.257957
$317$	1.69320e7			0.531532			0.265766	−	0.964038i	$-0.414375\pi$
$317$	1.69320e7			0.531532			0.265766	+	0.964038i	$0.414375\pi$
$318$	−2.08067e7			−0.647025
$319$		−	3.27932e6i		−	0.101021i
$320$		−	3.12346e7i		−	0.953203i
$321$		−	9.27423e6i		−	0.280390i
$322$	3.11499e7			0.933016
$323$		−	1.21086e7i		−	0.359323i
$324$	4.55214e6			0.133838
$325$	−1.35253e7			−0.394001
$326$	−1.66158e7			−0.479587
$327$		−	2.74058e7i		−	0.783790i
$328$			1.01592e7i			0.287898i
$329$		−	7.75123e7i		−	2.17662i
$330$			2.68640e6i			0.0747529i
$331$			1.17055e6i			0.0322778i	0.999870	+	0.0161389i	$0.00513740\pi$
$331$			1.17055e6i			0.0322778i	−0.999870	+	0.0161389i	$0.994863\pi$
$332$	−2.35585e6			−0.0643775
$333$			2.30247e7i			0.623536i
$334$		−	3.26595e6i		−	0.0876536i
$335$		−	2.44641e7i		−	0.650722i
$336$	1.83095e7			0.482679
$337$	−1.71166e7			−0.447228			−0.223614	−	0.974678i	$-0.571785\pi$
$337$	−1.71166e7			−0.447228			−0.223614	+	0.974678i	$0.571785\pi$
$338$		−	4.30977e7i		−	1.11610i
$339$	−2.97236e7			−0.762961
$340$			3.04625e6i			0.0775049i
$341$	−8.43416e6			−0.212706
$342$	−2.92763e7			−0.731877
$343$			7.01572e7i			1.73856i
$344$	−2.81810e7	−	3.39217e7i	−0.692278	−	0.833301i
$345$	−9.04634e6			−0.220301
$346$		−	1.63171e7i		−	0.393927i
$347$			4.50028e7i			1.07709i	0.842597	+	0.538544i	$0.181026\pi$
$347$			4.50028e7i			1.07709i	−0.842597	+	0.538544i	$0.818974\pi$
$348$	1.82077e6			0.0432034
$349$			4.97760e7i			1.17097i	0.810685	+	0.585483i	$0.199095\pi$
$349$			4.97760e7i			1.17097i	−0.810685	+	0.585483i	$0.800905\pi$
$350$	−1.66744e7			−0.388907
$351$			4.92277e7i			1.13838i
$352$			5.33059e6i			0.122221i
$353$	−6.55699e7			−1.49067			−0.745333	−	0.666693i	$-0.767709\pi$
$353$	−6.55699e7			−1.49067			−0.745333	+	0.666693i	$0.767709\pi$
$354$	−1.65764e7			−0.373664
$355$	−3.02858e7			−0.676947
$356$			8.56503e6i			0.189836i
$357$	−1.13965e7			−0.250476
$358$	3.70306e7			0.807071
$359$	4.55309e7			0.984063			0.492031	−	0.870577i	$-0.336254\pi$
$359$	4.55309e7			0.984063			0.492031	+	0.870577i	$0.336254\pi$
$360$	3.61541e7			0.774909
$361$	−1.92179e6			−0.0408493
$362$			9.17593e6i			0.193430i
$363$			1.84461e7i			0.385643i
$364$			3.23848e7i			0.671486i
$365$	−3.16740e7			−0.651364
$366$		−	2.63153e7i		−	0.536742i
$367$	−6.30366e7			−1.27525			−0.637623	−	0.770348i	$-0.720082\pi$
$367$	−6.30366e7			−1.27525			−0.637623	+	0.770348i	$0.720082\pi$
$368$	2.11100e7			0.423590
$369$	−1.11035e7			−0.220995
$370$			2.81814e7i			0.556362i
$371$		−	1.61742e8i		−	3.16738i
$372$		−	4.68288e6i		−	0.0909671i
$373$		−	2.47114e7i		−	0.476179i	−0.971243	−	0.238089i	$-0.923479\pi$
$373$		−	2.47114e7i		−	0.476179i	0.971243	−	0.238089i	$-0.0765211\pi$
$374$			3.90194e6i			0.0745875i
$375$	2.34568e7			0.444810
$376$		−	7.23303e7i		−	1.36068i
$377$		−	3.33945e7i		−	0.623233i
$378$			6.06894e7i			1.12367i
$379$	5.77691e7			1.06115			0.530577	−	0.847637i	$-0.321975\pi$
$379$	5.77691e7			1.06115			0.530577	+	0.847637i	$0.321975\pi$
$380$	1.23192e7			0.224508
$381$		−	9.97008e6i		−	0.180270i
$382$	8.78556e7			1.57608
$383$			8.68330e7i			1.54557i	0.634668	+	0.772785i	$0.281137\pi$
$383$			8.68330e7i			1.54557i	−0.634668	+	0.772785i	$0.718863\pi$
$384$	1.06451e7			0.188000
$385$	−2.08828e7			−0.365937
$386$			2.67326e7i			0.464815i
$387$	3.70747e7	−	3.08004e7i	0.639653	−	0.531403i
$388$	−7.05484e6			−0.120779
$389$			6.43558e6i			0.109330i	0.998505	+	0.0546650i	$0.0174091\pi$
$389$			6.43558e6i			0.109330i	−0.998505	+	0.0546650i	$0.982591\pi$
$390$			2.73565e7i			0.461176i
$391$	−1.31396e7			−0.219813
$392$			1.30724e8i			2.17018i
$393$	1.74031e7			0.286713
$394$			5.37850e7i			0.879371i
$395$		−	5.34499e7i		−	0.867273i
$396$	−3.24341e6			−0.0522295
$397$	1.21619e8			1.94370			0.971850	−	0.235601i	$-0.0757060\pi$
$397$	1.21619e8			1.94370			0.971850	+	0.235601i	$0.0757060\pi$
$398$	1.85279e7			0.293885
$399$			4.60879e7i			0.725552i
$400$	−1.13001e7			−0.176564
$401$	1.77372e7			0.275075			0.137538	−	0.990497i	$-0.456081\pi$
$401$	1.77372e7			0.275075			0.137538	+	0.990497i	$0.456081\pi$
$402$	1.73987e7			0.267817
$403$	−8.58879e7			−1.31225
$404$	1.37869e7			0.209084
$405$			2.98919e7i			0.449975i
$406$		−	4.11697e7i		−	0.615176i
$407$		−	1.24101e7i		−	0.184074i
$408$	−1.06346e7			−0.156581
$409$		−	3.42591e7i		−	0.500732i	−0.968151	−	0.250366i	$-0.919449\pi$
$409$		−	3.42591e7i		−	0.500732i	0.968151	−	0.250366i	$-0.0805510\pi$
$410$	−1.35903e7			−0.197187
$411$	−2.81513e7			−0.405483
$412$	1.70392e6			0.0243645
$413$		−	1.28857e8i		−	1.82919i
$414$			3.17692e7i			0.447719i
$415$		−	1.54699e7i		−	0.216442i
$416$			5.42832e7i			0.754024i
$417$			4.63872e7i			0.639721i
$418$	1.57796e7			0.216057
$419$			7.43380e7i			1.01058i	0.862951	+	0.505288i	$0.168614\pi$
$419$			7.43380e7i			1.01058i	−0.862951	+	0.505288i	$0.831386\pi$
$420$		−	1.15947e7i		−	0.156499i
$421$		−	6.57042e7i		−	0.880535i	−0.897867	−	0.440268i	$-0.854884\pi$
$421$		−	6.57042e7i		−	0.880535i	0.897867	−	0.440268i	$-0.145116\pi$
$422$	1.30810e7			0.174062
$423$	7.90534e7			1.04448
$424$		−	1.50929e8i		−	1.98004i
$425$	7.03359e6			0.0916242
$426$		−	2.15390e7i		−	0.278610i
$427$	2.04563e8			2.62751
$428$	1.37050e7			0.174802
$429$		−	1.20468e7i		−	0.152581i
$430$	4.53781e7	−	3.76986e7i	0.570743	−	0.474155i
$431$	−7.65329e7			−0.955908			−0.477954	−	0.878385i	$-0.658621\pi$
$431$	−7.65329e7			−0.955908			−0.477954	+	0.878385i	$0.658621\pi$
$432$			4.11287e7i			0.510145i
$433$			1.00323e8i			1.23577i	0.786269	+	0.617884i	$0.212010\pi$
$433$			1.00323e8i			1.23577i	−0.786269	+	0.617884i	$0.787990\pi$
$434$	−1.05885e8			−1.29529
$435$			1.19562e7i			0.145253i
$436$	4.04988e7			0.488633
$437$			5.31374e7i			0.636730i
$438$		−	2.25262e7i		−	0.268081i
$439$	3.20358e7			0.378653			0.189327	−	0.981914i	$-0.439369\pi$
$439$	3.20358e7			0.378653			0.189327	+	0.981914i	$0.439369\pi$
$440$	−1.94867e7			−0.228760
$441$	−1.42875e8			−1.66586
$442$			3.97348e7i			0.460155i
$443$	−8.21035e7			−0.944388			−0.472194	−	0.881495i	$-0.656538\pi$
$443$	−8.21035e7			−0.944388			−0.472194	+	0.881495i	$0.656538\pi$
$444$	6.89042e6			0.0787221
$445$	−5.62428e7			−0.638244
$446$	−4.04676e7			−0.456145
$447$	−3.83250e7			−0.429101
$448$			1.72679e8i			1.92047i
$449$			4.47173e7i			0.494011i	0.969014	+	0.247005i	$0.0794465\pi$
$449$			4.47173e7i			0.494011i	−0.969014	+	0.247005i	$0.920553\pi$
$450$		−	1.70059e7i		−	0.186622i
$451$	5.98469e6			0.0652397
$452$		−	4.39239e7i		−	0.475648i
$453$	−6.65749e7			−0.716170
$454$	7.47139e7			0.798425
$455$	−2.12657e8			−2.25759
$456$			4.30068e7i			0.453568i
$457$		−	1.04216e8i		−	1.09191i	−0.837816	−	0.545953i	$-0.816168\pi$
$457$		−	1.04216e8i		−	1.09191i	0.837816	−	0.545953i	$-0.183832\pi$
$458$			2.87614e7i			0.299374i
$459$		−	2.56000e7i		−	0.264729i
$460$		−	1.33682e7i		−	0.137341i
$461$	5.88536e6			0.0600717			0.0300359	−	0.999549i	$-0.490438\pi$
$461$	5.88536e6			0.0600717			0.0300359	+	0.999549i	$0.490438\pi$
$462$		−	1.48517e7i		−	0.150608i
$463$		−	8.38526e7i		−	0.844838i	−0.906401	−	0.422419i	$-0.861181\pi$
$463$		−	8.38526e7i		−	0.844838i	0.906401	−	0.422419i	$-0.138819\pi$
$464$		−	2.79003e7i		−	0.279290i
$465$	3.07504e7			0.305839
$466$	−1.53561e8			−1.51748
$467$			5.43691e7i			0.533828i	0.963720	+	0.266914i	$0.0860040\pi$
$467$			5.43691e7i			0.533828i	−0.963720	+	0.266914i	$0.913996\pi$
$468$	−3.30287e7			−0.322221
$469$			1.35249e8i			1.31104i
$470$	9.67585e7			0.931957
$471$	3.55869e7			0.340587
$472$		−	1.20243e8i		−	1.14349i
$473$	−1.99829e7	+	1.66011e7i	−0.188832	+	0.156875i
$474$	3.80131e7			0.356942
$475$		−	2.84442e7i		−	0.265407i
$476$		−	1.68411e7i		−	0.156153i
$477$	1.64957e8			1.51991
$478$		−	1.56126e8i		−	1.42952i
$479$	−2.39565e7			−0.217980			−0.108990	−	0.994043i	$-0.534762\pi$
$479$	−2.39565e7			−0.217980			−0.108990	+	0.994043i	$0.534762\pi$
$480$		−	1.94350e7i		−	0.175736i
$481$		−	1.26376e8i		−	1.13561i
$482$	−1.19430e8			−1.06653
$483$	5.00124e7			0.443850
$484$	−2.72587e7			−0.240419
$485$		−	4.63260e7i		−	0.406069i
$486$	−9.56896e7			−0.833597
$487$	7.69776e6			0.0666465			0.0333233	−	0.999445i	$-0.489391\pi$
$487$	7.69776e6			0.0666465			0.0333233	+	0.999445i	$0.489391\pi$
$488$	1.90887e8			1.64255
$489$	−2.66773e7			−0.228147
$490$	−1.74873e8			−1.48640
$491$		−	1.38326e8i		−	1.16858i	−0.811544	−	0.584292i	$-0.801372\pi$
$491$		−	1.38326e8i		−	1.16858i	0.811544	−	0.584292i	$-0.198628\pi$
$492$			3.32286e6i			0.0279008i
$493$			1.73662e7i			0.144932i
$494$	1.60689e8			1.33293
$495$		−	2.12980e7i		−	0.175600i
$496$	−7.17575e7			−0.588061
$497$	1.67434e8			1.36388
$498$	1.10020e7			0.0890808
$499$		−	2.71530e7i		−	0.218532i	−0.994013	−	0.109266i	$-0.965150\pi$
$499$		−	2.71530e7i		−	0.218532i	0.994013	−	0.109266i	$-0.0348501\pi$
$500$			3.46632e7i			0.277306i
$501$		−	5.24361e6i		−	0.0416982i
$502$			1.66154e8i			1.31341i
$503$		−	4.39907e7i		−	0.345666i	−0.984951	−	0.172833i	$-0.944708\pi$
$503$		−	4.39907e7i		−	0.345666i	0.984951	−	0.172833i	$-0.0552921\pi$
$504$	−1.99877e8			−1.56125
$505$			9.05324e7i			0.702959i
$506$		−	1.71233e7i		−	0.132171i
$507$		−	6.91951e7i		−	0.530948i
$508$	1.47333e7			0.112385
$509$	9.94585e7			0.754203			0.377102	−	0.926172i	$-0.376921\pi$
$509$	9.94585e7			0.754203			0.377102	+	0.926172i	$0.376921\pi$
$510$		−	1.42262e7i		−	0.107246i
$511$	1.75108e8			1.31233
$512$			1.44040e8i			1.07318i
$513$	−1.03528e8			−0.766838
$514$	6.73927e7			0.496277
$515$			1.11889e7i			0.0819155i
$516$	−9.21740e6	−	1.10951e7i	−0.0670902	−	0.0807570i
$517$	−4.26090e7			−0.308340
$518$		−	1.55800e8i		−	1.12093i
$519$		−	2.61978e7i		−	0.187397i
$520$	−1.98440e8			−1.41130
$521$		−	1.64109e7i		−	0.116043i	−0.998315	−	0.0580216i	$-0.981521\pi$
$521$		−	1.64109e7i		−	0.116043i	0.998315	−	0.0580216i	$-0.0184792\pi$
$522$	4.19882e7			0.295200
$523$			1.31984e8i			0.922603i	0.887243	+	0.461302i	$0.152617\pi$
$523$			1.31984e8i			0.922603i	−0.887243	+	0.461302i	$0.847383\pi$
$524$			2.57173e7i			0.178744i
$525$	−2.67714e7			−0.185009
$526$	−5.22957e7			−0.359343
$527$	4.46644e7			0.305162
$528$		−	1.00648e7i		−	0.0683763i
$529$	−9.03738e7			−0.610486
$530$	2.01902e8			1.35617
$531$	1.31420e8			0.877761
$532$	−6.81063e7			−0.452327
$533$	6.09441e7			0.402485
$534$		−	3.99993e7i		−	0.262681i
$535$			8.99944e7i			0.587698i
$536$			1.26207e8i			0.819578i
$537$	5.94541e7			0.383936
$538$			5.71040e7i			0.366708i
$539$	7.70079e7			0.491778
$540$	2.60453e7			0.165404
$541$	1.85731e8			1.17299			0.586493	−	0.809954i	$-0.300508\pi$
$541$	1.85731e8			1.17299			0.586493	+	0.809954i	$0.300508\pi$
$542$			1.22166e8i			0.767279i
$543$			1.47323e7i			0.0920178i
$544$		−	2.82289e7i		−	0.175347i
$545$			2.65938e8i			1.64282i
$546$		−	1.51239e8i		−	0.929153i
$547$	−1.99821e8			−1.22090			−0.610450	−	0.792055i	$-0.709011\pi$
$547$	−1.99821e8			−1.22090			−0.610450	+	0.792055i	$0.709011\pi$
$548$		−	4.16005e7i		−	0.252788i
$549$			2.08631e8i			1.26084i
$550$			9.16603e6i			0.0550926i
$551$	7.02296e7			0.419822
$552$	4.66689e7			0.277466
$553$			2.95496e8i			1.74734i
$554$	2.31687e8			1.36261
$555$			4.52464e7i			0.264670i
$556$	−6.85486e7			−0.398817
$557$	−7.45441e7			−0.431368			−0.215684	−	0.976463i	$-0.569198\pi$
$557$	−7.45441e7			−0.431368			−0.215684	+	0.976463i	$0.569198\pi$
$558$		−	1.07990e8i		−	0.621559i
$559$	−2.03493e8	+	1.69055e8i	−1.16497	+	0.967815i
$560$	−1.77670e8			−1.01170
$561$			6.26472e6i			0.0354824i
$562$		−	2.15783e8i		−	1.21565i
$563$	−2.84802e8			−1.59595			−0.797973	−	0.602693i	$-0.794094\pi$
$563$	−2.84802e8			−1.59595			−0.797973	+	0.602693i	$0.794094\pi$
$564$		−	2.36577e7i		−	0.131867i
$565$	2.88429e8			1.59917
$566$		−	3.06247e8i		−	1.68897i
$567$		−	1.65257e8i		−	0.906587i
$568$	1.56241e8			0.852607
$569$	7.04148e7			0.382232			0.191116	−	0.981567i	$-0.438789\pi$
$569$	7.04148e7			0.382232			0.191116	+	0.981567i	$0.438789\pi$
$570$	−5.75316e7			−0.310657
$571$			1.43310e8i			0.769780i	0.922962	+	0.384890i	$0.125761\pi$
$571$			1.43310e8i			0.769780i	−0.922962	+	0.384890i	$0.874239\pi$
$572$	1.78021e7			0.0951227
$573$	1.41056e8			0.749767
$574$	7.51336e7			0.397282
$575$	−3.08663e7			−0.162361
$576$	−1.76113e8			−0.921559
$577$		−	3.52177e7i		−	0.183330i	−0.995790	−	0.0916651i	$-0.970781\pi$
$577$		−	3.52177e7i		−	0.183330i	0.995790	−	0.0916651i	$-0.0292189\pi$
$578$			1.45915e8i			0.755640i
$579$			4.29203e7i			0.221120i
$580$	−1.76682e7			−0.0905543
$581$			8.55247e7i			0.436077i
$582$	3.29466e7			0.167125
$583$	−8.89104e7			−0.448691
$584$	1.63402e8			0.820386
$585$		−	2.16885e8i		−	1.08333i
$586$		−	1.45565e7i		−	0.0723376i
$587$			1.60114e8i			0.791617i	0.918333	+	0.395809i	$0.129536\pi$
$587$			1.60114e8i			0.791617i	−0.918333	+	0.395809i	$0.870464\pi$
$588$			4.27569e7i			0.210317i
$589$		−	1.80625e8i		−	0.883959i
$590$	1.60853e8			0.783200
$591$			8.63539e7i			0.418331i
$592$		−	1.05584e8i		−	0.508903i
$593$			2.06554e8i			0.990533i	0.868741	+	0.495266i	$0.164930\pi$
$593$			2.06554e8i			0.990533i	−0.868741	+	0.495266i	$0.835070\pi$
$594$	3.33613e7			0.159178
$595$	1.10588e8			0.524998
$596$		−	5.66346e7i		−	0.267512i
$597$	2.97473e7			0.139806
$598$		−	1.74372e8i		−	0.815407i
$599$	1.21447e8			0.565075			0.282537	−	0.959256i	$-0.408824\pi$
$599$	1.21447e8			0.565075			0.282537	+	0.959256i	$0.408824\pi$
$600$	−2.49816e7			−0.115656
$601$		−	3.10614e8i		−	1.43086i	−0.698684	−	0.715430i	$-0.746231\pi$
$601$		−	3.10614e8i		−	1.43086i	0.698684	−	0.715430i	$-0.253769\pi$
$602$	−2.50871e8	+	2.08416e8i	−1.14990	+	0.955302i
$603$	−1.37938e8			−0.629119
$604$		−	9.83808e7i		−	0.446478i
$605$		−	1.78996e8i		−	0.808308i
$606$	−6.43858e7			−0.289316
$607$		−	4.20257e8i		−	1.87909i	−0.342421	−	0.939547i	$-0.611247\pi$
$607$		−	4.20257e8i		−	1.87909i	0.342421	−	0.939547i	$-0.388753\pi$
$608$	−1.14159e8			−0.507926
$609$		−	6.60995e7i		−	0.292649i
$610$			2.55356e8i			1.12501i
$611$	−4.33902e8			−1.90225
$612$	1.71760e7			0.0749319
$613$	−4.00813e8			−1.74004			−0.870022	−	0.493014i	$-0.835895\pi$
$613$	−4.00813e8			−1.74004			−0.870022	+	0.493014i	$0.835895\pi$
$614$		−	9.31449e7i		−	0.402396i
$615$	−2.18198e7			−0.0938048
$616$	1.07732e8			0.460894
$617$	2.59681e8			1.10556			0.552782	−	0.833326i	$-0.313566\pi$
$617$	2.59681e8			1.10556			0.552782	+	0.833326i	$0.313566\pi$
$618$	−7.95744e6			−0.0337138
$619$	−1.81356e8			−0.764644			−0.382322	−	0.924029i	$-0.624875\pi$
$619$	−1.81356e8			−0.764644			−0.382322	+	0.924029i	$0.624875\pi$
$620$			4.54413e7i			0.190667i
$621$			1.12343e8i			0.469107i
$622$			2.77467e8i			1.15303i
$623$	3.10937e8			1.28590
$624$		−	1.02494e8i		−	0.421836i
$625$	−1.64106e8			−0.672177
$626$	1.90251e7			0.0775539
$627$	2.53348e7			0.102782
$628$			5.25884e7i			0.212330i
$629$			6.57195e7i			0.264084i
$630$		−	2.67382e8i		−	1.06933i
$631$		−	2.84986e8i		−	1.13432i	−0.823608	−	0.567160i	$-0.808042\pi$
$631$		−	2.84986e8i		−	1.13432i	0.823608	−	0.567160i	$-0.191958\pi$
$632$			2.75741e8i			1.09232i
$633$	2.10021e7			0.0828042
$634$		−	1.16851e8i		−	0.458526i
$635$			9.67468e7i			0.377847i
$636$		−	4.93655e7i		−	0.191890i
$637$	7.84198e8			3.03394
$638$	−2.26312e7			−0.0871457
$639$			1.70763e8i			0.654473i
$640$	−1.03297e8			−0.394047
$641$		−	3.06450e8i		−	1.16355i	−0.813350	−	0.581775i	$-0.802359\pi$
$641$		−	3.06450e8i		−	1.16355i	0.813350	−	0.581775i	$-0.197641\pi$
$642$	−6.40032e7			−0.241878
$643$	−3.24019e8			−1.21882			−0.609408	−	0.792857i	$-0.708593\pi$
$643$	−3.24019e8			−1.21882			−0.609408	+	0.792857i	$0.708593\pi$
$644$			7.39056e7i			0.276707i
$645$	7.28563e7	−	6.05266e7i	0.271511	−	0.225563i
$646$	−8.35636e7			−0.309970
$647$		−	3.47049e8i		−	1.28138i	−0.767800	−	0.640690i	$-0.778648\pi$
$647$		−	3.47049e8i		−	1.28138i	0.767800	−	0.640690i	$-0.221352\pi$
$648$		−	1.54209e8i		−	0.566739i
$649$	−7.08338e7			−0.259123
$650$			9.33408e7i			0.339884i
$651$	−1.70003e8			−0.616188
$652$		−	3.94222e7i		−	0.142232i
$653$			1.64007e8i			0.589011i	0.955650	+	0.294505i	$0.0951549\pi$
$653$			1.64007e8i			0.589011i	−0.955650	+	0.294505i	$0.904845\pi$
$654$	−1.89133e8			−0.676135
$655$	−1.68874e8			−0.600952
$656$	5.09174e7			0.180366
$657$			1.78590e8i			0.629740i
$658$	−5.34927e8			−1.87766
$659$	8.42317e7			0.294320			0.147160	−	0.989113i	$-0.452987\pi$
$659$	8.42317e7			0.294320			0.147160	+	0.989113i	$0.452987\pi$
$660$	−6.37369e6			−0.0221697
$661$	3.01790e7			0.104496			0.0522481	−	0.998634i	$-0.483361\pi$
$661$	3.01790e7			0.104496			0.0522481	+	0.998634i	$0.483361\pi$
$662$	8.07816e6			0.0278444
$663$			6.37958e7i			0.218903i
$664$			7.98070e7i			0.272607i
$665$		−	4.47224e8i		−	1.52076i
$666$	1.58898e8			0.537892
$667$		−	7.62098e7i		−	0.256823i
$668$	7.74872e6			0.0259956
$669$	−6.49723e7			−0.216995
$670$	−1.68832e8			−0.561344
$671$		−	1.12450e8i		−	0.372212i
$672$			1.07446e8i			0.354063i
$673$			2.18792e8i			0.717772i	0.933381	+	0.358886i	$0.116843\pi$
$673$			2.18792e8i			0.717772i	−0.933381	+	0.358886i	$0.883157\pi$
$674$			1.18125e8i			0.385800i
$675$		−	6.01368e7i		−	0.195537i
$676$	1.02253e8			0.331006
$677$			5.58092e8i			1.79862i	0.437312	+	0.899310i	$0.355931\pi$
$677$			5.58092e8i			1.79862i	−0.437312	+	0.899310i	$0.644069\pi$
$678$			2.05128e8i			0.658167i
$679$			2.56112e8i			0.818127i
$680$	1.03195e8			0.328195
$681$	1.19956e8			0.379823
$682$			5.82057e7i			0.183490i
$683$	4.01935e8			1.26152			0.630759	−	0.775979i	$-0.282744\pi$
$683$	4.01935e8			1.26152			0.630759	+	0.775979i	$0.282744\pi$
$684$		−	6.94604e7i		−	0.217055i
$685$	2.73172e8			0.849894
$686$	4.84168e8			1.49977
$687$			4.61776e7i			0.142417i
$688$	−1.70014e8	+	1.41242e8i	−0.522057	+	0.433708i
$689$	−9.05405e8			−2.76812
$690$			6.24305e7i			0.190042i
$691$			3.35495e8i			1.01684i	0.861109	+	0.508420i	$0.169770\pi$
$691$			3.35495e8i			1.01684i	−0.861109	+	0.508420i	$0.830230\pi$
$692$	3.87137e7			0.116828
$693$			1.17745e8i			0.353789i
$694$	3.10573e8			0.929149
$695$		−	4.50128e8i		−	1.34086i
$696$		−	6.16805e7i		−	0.182945i
$697$	−3.16928e7			−0.0935972
$698$	3.43514e8			1.01013
$699$	−2.46548e8			−0.721889
$700$		−	3.95614e7i		−	0.115339i
$701$	−2.59344e8			−0.752873			−0.376436	−	0.926443i	$-0.622851\pi$
$701$	−2.59344e8			−0.752873			−0.376436	+	0.926443i	$0.622851\pi$
$702$	3.39730e8			0.982025
$703$	2.65773e8			0.764971
$704$	9.49230e7			0.272053
$705$	1.55350e8			0.443346
$706$			4.52510e8i			1.28592i
$707$		−	5.00506e8i		−	1.41629i
$708$		−	3.93289e7i		−	0.110818i
$709$	1.30606e8			0.366458			0.183229	−	0.983070i	$-0.441345\pi$
$709$	1.30606e8			0.366458			0.183229	+	0.983070i	$0.441345\pi$
$710$			2.09008e8i			0.583967i
$711$	−3.01372e8			−0.838482
$712$	2.90149e8			0.803862
$713$	−1.96006e8			−0.540754
$714$			7.86493e7i			0.216073i
$715$			1.16899e8i			0.319810i
$716$			8.78580e7i			0.239355i
$717$		−	2.50666e8i		−	0.680046i
$718$		−	3.14217e8i		−	0.848900i
$719$	4.57584e8			1.23107			0.615537	−	0.788108i	$-0.288939\pi$
$719$	4.57584e8			1.23107			0.615537	+	0.788108i	$0.288939\pi$
$720$		−	1.81203e8i		−	0.485475i
$721$		−	6.18575e7i		−	0.165039i
$722$			1.32626e7i			0.0352386i
$723$	−1.91749e8			−0.507363
$724$	−2.17706e7			−0.0573661
$725$			4.07948e7i			0.107051i
$726$	1.27300e8			0.332674
$727$		−	3.80141e8i		−	0.989330i	−0.869084	−	0.494665i	$-0.835291\pi$
$727$		−	3.80141e8i		−	0.989330i	0.869084	−	0.494665i	$-0.164709\pi$
$728$	1.09707e9			2.84341
$729$	4.90408e7			0.126583
$730$			2.18588e8i			0.561898i
$731$	1.05823e8	−	8.79138e7i	0.270911	−	0.225064i
$732$	6.24352e7			0.159183
$733$			3.54939e8i			0.901242i	0.892715	+	0.450621i	$0.148797\pi$
$733$			3.54939e8i			0.901242i	−0.892715	+	0.450621i	$0.851203\pi$
$734$			4.35027e8i			1.10009i
$735$	−2.80766e8			−0.707103
$736$			1.23880e8i			0.310719i
$737$	7.43474e7			0.185722
$738$			7.66275e7i			0.190641i
$739$			3.35280e7i			0.0830757i	0.999137	+	0.0415378i	$0.0132257\pi$
$739$			3.35280e7i			0.0830757i	−0.999137	+	0.0415378i	$0.986774\pi$
$740$	−6.68626e7			−0.165002
$741$	2.57993e8			0.634094
$742$	−1.11621e9			−2.73233
$743$		−	4.64197e8i		−	1.13171i	−0.824504	−	0.565856i	$-0.808546\pi$
$743$		−	4.64197e8i		−	1.13171i	0.824504	−	0.565856i	$-0.191454\pi$
$744$	−1.58637e8			−0.385200
$745$	3.71894e8			0.899396
$746$	−1.70538e8			−0.410775
$747$	−8.72251e7			−0.209257
$748$	−9.25767e6			−0.0221206
$749$		−	4.97532e8i		−	1.18406i
$750$		−	1.61880e8i		−	0.383715i
$751$			3.26025e8i			0.769716i	0.922976	+	0.384858i	$0.125750\pi$
$751$			3.26025e8i			0.769716i	−0.922976	+	0.384858i	$0.874250\pi$
$752$	−3.62515e8			−0.852458
$753$			2.66766e8i			0.624808i
$754$	−2.30461e8			−0.537631
$755$	6.46024e8			1.50109
$756$	−1.43990e8			−0.333248
$757$			6.61201e8i			1.52421i	0.647451	+	0.762107i	$0.275835\pi$
$757$			6.61201e8i			1.52421i	−0.647451	+	0.762107i	$0.724165\pi$
$758$		−	3.98675e8i		−	0.915402i
$759$		−	2.74921e7i		−	0.0628758i
$760$		−	4.17325e8i		−	0.950679i
$761$			8.45615e8i			1.91875i	0.282131	+	0.959376i	$0.408959\pi$
$761$			8.45615e8i			1.91875i	−0.282131	+	0.959376i	$0.591041\pi$
$762$	−6.88054e7			−0.155510
$763$		−	1.47023e9i		−	3.30988i
$764$			2.08444e8i			0.467423i
$765$			1.12787e8i			0.251927i
$766$	5.99251e8			1.33328
$767$	−7.21325e8			−1.59862
$768$			1.32541e8i			0.292595i
$769$	1.38008e8			0.303477			0.151738	−	0.988421i	$-0.451513\pi$
$769$	1.38008e8			0.303477			0.151738	+	0.988421i	$0.451513\pi$
$770$			1.44116e8i			0.315675i
$771$	1.08202e8			0.236086
$772$	−6.34253e7			−0.137851
$773$			6.00044e8i			1.29911i	0.760317	+	0.649553i	$0.225044\pi$
$773$			6.00044e8i			1.29911i	−0.760317	+	0.649553i	$0.774956\pi$
$774$	−2.12560e8	−	2.55859e8i	−0.458414	−	0.551796i
$775$	1.04921e8			0.225402
$776$			2.38990e8i			0.511440i
$777$		−	2.50143e8i		−	0.533244i
$778$	4.44132e7			0.0943133
$779$			1.28167e8i			0.271122i
$780$	−6.49054e7			−0.136772
$781$		−	9.20397e7i		−	0.193207i
$782$			9.06791e7i			0.189621i
$783$	1.48480e8			0.309301
$784$	6.55180e8			1.35960
$785$	−3.45325e8			−0.713870
$786$		−	1.20102e8i		−	0.247333i
$787$	−4.15603e7			−0.0852619			−0.0426309	−	0.999091i	$-0.513574\pi$
$787$	−4.15603e7			−0.0852619			−0.0426309	+	0.999091i	$0.513574\pi$
$788$	−1.27609e8			−0.260797
$789$	−8.39628e7			−0.170945
$790$	−3.68868e8			−0.748152
$791$	−1.59457e9			−3.22192
$792$			1.09874e8i			0.221166i
$793$		−	1.14511e9i		−	2.29630i
$794$		−	8.39314e8i		−	1.67673i
$795$	3.24162e8			0.645149
$796$			4.39590e7i			0.0871582i
$797$	−1.92759e8			−0.380749			−0.190375	−	0.981712i	$-0.560970\pi$
$797$	−1.92759e8			−0.380749			−0.190375	+	0.981712i	$0.560970\pi$
$798$	3.18061e8			0.625896
$799$	2.25643e8			0.442365
$800$		−	6.63125e7i		−	0.129517i
$801$			3.17119e8i			0.617056i
$802$		−	1.22408e8i		−	0.237293i
$803$		−	9.62583e7i		−	0.185905i
$804$			4.12797e7i			0.0794271i
$805$	−4.85306e8			−0.930311
$806$			5.92729e8i			1.13201i
$807$			9.16828e7i			0.174449i
$808$		−	4.67045e8i		−	0.885369i
$809$	1.20496e8			0.227577			0.113788	−	0.993505i	$-0.463701\pi$
$809$	1.20496e8			0.227577			0.113788	+	0.993505i	$0.463701\pi$
$810$	2.06290e8			0.388171
$811$		−	3.17658e7i		−	0.0595522i	−0.999557	−	0.0297761i	$-0.990521\pi$
$811$		−	3.17658e7i		−	0.0595522i	0.999557	−	0.0297761i	$-0.00947943\pi$
$812$	9.76783e7			0.182444
$813$			1.96143e8i			0.365006i
$814$	−8.56443e7			−0.158791
$815$	2.58868e8			0.478196
$816$			5.33000e7i			0.0980972i
$817$	−3.55528e8	−	4.27951e8i	−0.651940	−	0.784745i
$818$	−2.36428e8			−0.431956
$819$			1.19904e9i			2.18264i
$820$		−	3.22441e7i		−	0.0584802i
$821$	−9.23308e8			−1.66846			−0.834232	−	0.551413i	$-0.814089\pi$
$821$	−9.23308e8			−1.66846			−0.834232	+	0.551413i	$0.814089\pi$
$822$			1.94277e8i			0.349790i
$823$	5.55575e8			0.996650			0.498325	−	0.866990i	$-0.333949\pi$
$823$	5.55575e8			0.996650			0.498325	+	0.866990i	$0.333949\pi$
$824$		−	5.77221e7i		−	0.103172i
$825$			1.47164e7i			0.0262084i
$826$	−8.89270e8			−1.57795
$827$	−9.37783e8			−1.65800			−0.829002	−	0.559245i	$-0.811091\pi$
$827$	−9.37783e8			−1.65800			−0.829002	+	0.559245i	$0.811091\pi$
$828$	−7.53751e7			−0.132781
$829$			3.79853e8i			0.666732i	0.942797	+	0.333366i	$0.108185\pi$
$829$			3.79853e8i			0.666732i	−0.942797	+	0.333366i	$0.891815\pi$
$830$	−1.06760e8			−0.186714
$831$	3.71982e8			0.648215
$832$	9.66633e8			1.67838
$833$	−4.07808e8			−0.705538
$834$	3.20127e8			0.551854
$835$			5.08825e7i			0.0873994i
$836$			3.74385e7i			0.0640766i
$837$		−	3.81878e8i		−	0.651251i
$838$	5.13020e8			0.871771
$839$		−	2.41418e8i		−	0.408774i	−0.978890	−	0.204387i	$-0.934480\pi$
$839$		−	2.41418e8i		−	0.408774i	0.978890	−	0.204387i	$-0.0655201\pi$
$840$	−3.92783e8			−0.662697
$841$	4.94100e8			0.830666
$842$	−4.53437e8			−0.759592
$843$		−	3.46448e8i		−	0.578304i
$844$			3.10358e7i			0.0516222i
$845$			6.71450e8i			1.11287i
$846$		−	5.45563e8i		−	0.901018i
$847$			9.89574e8i			1.62854i
$848$	−7.56446e8			−1.24048
$849$		−	4.91691e8i		−	0.803470i
$850$		−	4.85401e7i		−	0.0790395i
$851$		−	2.88404e8i		−	0.467964i
$852$	5.11030e7			0.0826280
$853$	1.12096e9			1.80611			0.903056	−	0.429522i	$-0.141318\pi$
$853$	1.12096e9			1.80611			0.903056	+	0.429522i	$0.141318\pi$
$854$		−	1.41173e9i		−	2.26661i
$855$	4.56116e8			0.729755
$856$		−	4.64270e8i		−	0.740200i
$857$	−1.37829e8			−0.218977			−0.109488	−	0.993988i	$-0.534921\pi$
$857$	−1.37829e8			−0.218977			−0.109488	+	0.993988i	$0.534921\pi$
$858$	−8.31373e7			−0.131624
$859$			1.06257e8i			0.167640i	0.996481	+	0.0838201i	$0.0267121\pi$
$859$			1.06257e8i			0.167640i	−0.996481	+	0.0838201i	$0.973288\pi$
$860$	8.94430e7	+	1.07663e8i	0.140621	+	0.169267i
$861$	1.20630e8			0.188993
$862$			5.28168e8i			0.824613i
$863$			2.54275e8i			0.395613i	0.980241	+	0.197806i	$0.0633818\pi$
$863$			2.54275e8i			0.395613i	−0.980241	+	0.197806i	$0.936618\pi$
$864$	−2.41356e8			−0.374211
$865$			2.54216e8i			0.392785i
$866$	6.92348e8			1.06603
$867$			2.34272e8i			0.359470i
$868$		−	2.51221e8i		−	0.384146i
$869$	1.62436e8			0.247528
$870$	8.25120e7			0.125302
$871$	7.57105e8			1.14578
$872$		−	1.37194e9i		−	2.06912i
$873$	−2.61204e8			−0.392588
$874$	3.66711e8			0.549274
$875$	1.25838e9			1.87840
$876$	5.34452e7			0.0795054
$877$	−3.73053e8			−0.553059			−0.276530	−	0.961005i	$-0.589184\pi$
$877$	−3.73053e8			−0.553059			−0.276530	+	0.961005i	$0.589184\pi$
$878$		−	2.21085e8i		−	0.326645i
$879$		−	2.33710e7i		−	0.0344121i
$880$			9.76664e7i			0.143317i
$881$	5.36850e8			0.785100			0.392550	−	0.919731i	$-0.371593\pi$
$881$	5.36850e8			0.785100			0.392550	+	0.919731i	$0.371593\pi$
$882$			9.86004e8i			1.43705i
$883$	3.92225e8			0.569710			0.284855	−	0.958571i	$-0.408055\pi$
$883$	3.92225e8			0.569710			0.284855	+	0.958571i	$0.408055\pi$
$884$	−9.42740e7			−0.136469
$885$	2.58255e8			0.372580
$886$			5.66612e8i			0.814675i
$887$			7.67617e8i			1.09995i	0.835181	+	0.549976i	$0.185363\pi$
$887$			7.67617e8i			1.09995i	−0.835181	+	0.549976i	$0.814637\pi$
$888$		−	2.33420e8i		−	0.333349i
$889$		−	5.34862e8i		−	0.761266i
$890$			3.88142e8i			0.550580i
$891$	−9.08426e7			−0.128427
$892$		−	9.60126e7i		−	0.135280i
$893$		−	9.12510e8i		−	1.28140i
$894$			2.64488e8i			0.370163i
$895$	−5.76925e8			−0.804730
$896$	5.71075e8			0.793906
$897$		−	2.79962e8i		−	0.387902i
$898$	3.08602e8			0.426158
$899$			2.59053e8i			0.356542i
$900$	4.03479e7			0.0553470
$901$	4.70839e8			0.643721
$902$		−	4.13014e7i		−	0.0562789i
$903$	−4.02784e8	+	3.34620e8i	−0.547027	+	0.454452i
$904$	−1.48797e9			−2.01413
$905$		−	1.42958e8i		−	0.192869i
$906$			4.59446e8i			0.617803i
$907$	−7.67990e8			−1.02928			−0.514640	−	0.857406i	$-0.672074\pi$
$907$	−7.67990e8			−1.02928			−0.514640	+	0.857406i	$0.672074\pi$
$908$			1.77265e8i			0.236791i
$909$	5.10457e8			0.679622
$910$			1.46758e9i			1.94751i
$911$			1.28662e9i			1.70175i	0.525367	+	0.850876i	$0.323928\pi$
$911$			1.28662e9i			1.70175i	−0.525367	+	0.850876i	$0.676072\pi$
$912$	2.15548e8			0.284157
$913$	4.70135e7			0.0617746
$914$	−7.19214e8			−0.941931
$915$			4.09985e8i			0.535185i
$916$	−6.82388e7			−0.0887861
$917$	9.33616e8			1.21077
$918$	−1.76670e8			−0.228368
$919$	−2.49791e7			−0.0321832			−0.0160916	−	0.999871i	$-0.505122\pi$
$919$	−2.49791e7			−0.0321832			−0.0160916	+	0.999871i	$0.505122\pi$
$920$	−4.52861e8			−0.581570
$921$		−	1.49548e8i		−	0.191426i
$922$		−	4.06160e7i		−	0.0518208i
$923$		−	9.37272e8i		−	1.19196i
$924$	3.52367e7			0.0446663
$925$			1.54381e8i			0.195061i
$926$	−5.78683e8			−0.728798
$927$	6.30874e7			0.0791961
$928$	1.63728e8			0.204870
$929$		−	5.90013e7i		−	0.0735893i	−0.999323	−	0.0367947i	$-0.988285\pi$
$929$		−	5.90013e7i		−	0.0735893i	0.999323	−	0.0367947i	$-0.0117147\pi$
$930$		−	2.12214e8i		−	0.263831i
$931$			1.64919e9i			2.04373i
$932$		−	3.64336e8i		−	0.450043i
$933$			4.45484e8i			0.548514i
$934$	3.75211e8			0.460506
$935$		−	6.07911e7i		−	0.0743712i
$936$			1.11888e9i			1.36445i
$937$		−	1.48181e8i		−	0.180125i	−0.995936	−	0.0900623i	$-0.971293\pi$
$937$		−	1.48181e8i		−	0.180125i	0.995936	−	0.0900623i	$-0.0287066\pi$
$938$	9.33380e8			1.13097
$939$	3.05455e7			0.0368936
$940$			2.29567e8i			0.276393i
$941$	5.23299e8			0.628031			0.314015	−	0.949418i	$-0.398326\pi$
$941$	5.23299e8			0.628031			0.314015	+	0.949418i	$0.398326\pi$
$942$		−	2.45592e8i		−	0.293806i
$943$	1.39081e8			0.165857
$944$	−6.02651e8			−0.716390
$945$		−	9.45522e8i		−	1.12041i
$946$	1.14567e8	+	1.37906e8i	0.135328	+	0.162895i
$947$	5.80378e8			0.683378			0.341689	−	0.939813i	$-0.389001\pi$
$947$	5.80378e8			0.683378			0.341689	+	0.939813i	$0.389001\pi$
$948$			9.01891e7i			0.105859i
$949$		−	9.80231e8i		−	1.14691i
$950$	−1.96299e8			−0.228953
$951$		−	1.87608e8i		−	0.218128i
$952$	−5.70510e8			−0.661230
$953$		−	8.72300e8i		−	1.00783i	−0.863753	−	0.503915i	$-0.831893\pi$
$953$		−	8.72300e8i		−	1.00783i	0.863753	−	0.503915i	$-0.168107\pi$
$954$		−	1.13840e9i		−	1.31114i
$955$	−1.36876e9			−1.57151
$956$	3.70421e8			0.423957
$957$	−3.63353e7			−0.0414566
$958$			1.65328e8i			0.188040i
$959$	−1.51022e9			−1.71232
$960$	−3.46083e8			−0.391171
$961$	−2.21239e8			−0.249282
$962$	−8.72145e8			−0.979633
$963$	5.07424e8			0.568188
$964$		−	2.83357e8i		−	0.316302i
$965$		−	4.16486e8i		−	0.463467i
$966$		−	3.45145e8i		−	0.382887i
$967$	1.02817e9			1.13707			0.568533	−	0.822660i	$-0.307511\pi$
$967$	1.02817e9			1.13707			0.568533	+	0.822660i	$0.307511\pi$
$968$			9.23417e8i			1.01806i
$969$	−1.34165e8			−0.147457
$970$	−3.19705e8			−0.350295
$971$	1.64931e9			1.80155			0.900773	−	0.434290i	$-0.143001\pi$
$971$	1.64931e9			1.80155			0.900773	+	0.434290i	$0.143001\pi$
$972$		−	2.27031e8i		−	0.247222i
$973$			2.48852e9i			2.70149i
$974$		−	5.31237e7i		−	0.0574925i
$975$			1.49862e8i			0.161688i
$976$		−	9.56717e8i		−	1.02904i
$977$	−1.06836e9			−1.14561			−0.572804	−	0.819693i	$-0.694144\pi$
$977$	−1.06836e9			−1.14561			−0.572804	+	0.819693i	$0.694144\pi$
$978$			1.84105e8i			0.196811i
$979$		−	1.70924e8i		−	0.182161i
$980$		−	4.14901e8i		−	0.440825i
$981$	1.49946e9			1.58829
$982$	−9.54614e8			−1.00808
$983$		−	7.22940e8i		−	0.761100i	−0.924760	−	0.380550i	$-0.875735\pi$
$983$		−	7.22940e8i		−	0.761100i	0.924760	−	0.380550i	$-0.124265\pi$
$984$	1.12565e8			0.118146
$985$		−	8.37954e8i		−	0.876822i
$986$	1.19847e8			0.125025
$987$	−8.58846e8			−0.893231
$988$			3.81249e8i			0.395310i
$989$	−4.64392e8	+	3.85802e8i	−0.480061	+	0.398818i
$990$	−1.46982e8			−0.151481
$991$			2.66146e7i			0.0273463i	0.999907	+	0.0136732i	$0.00435243\pi$
$991$			2.66146e7i			0.0273463i	−0.999907	+	0.0136732i	$0.995648\pi$
$992$		−	4.21095e8i		−	0.431365i
$993$	1.29698e7			0.0132460
$994$		−	1.15550e9i		−	1.17655i
$995$	−2.88659e8			−0.293033
$996$			2.61032e7i			0.0264189i
$997$		−	2.79719e8i		−	0.282251i	−0.989992	−	0.141126i	$-0.954928\pi$
$997$		−	2.79719e8i		−	0.282251i	0.989992	−	0.141126i	$-0.0450722\pi$
$998$	−1.87388e8			−0.188517
$999$	5.61898e8			0.563587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.b.b.42.8	✓	20
43.42	odd	2	inner	43.7.b.b.42.13	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.b.b.42.8	✓	20	1.1	even	1	trivial
43.7.b.b.42.13	yes	20	43.42	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.90119i q^{2} -11.0801i q^{3} +16.3736 q^{4} +107.518i q^{5} -76.4660 q^{6} -594.412i q^{7} -554.673i q^{8} +606.231 q^{9} +O(q^{10})\)	\(q-6.90119i q^{2} -11.0801i q^{3} +16.3736 q^{4} +107.518i q^{5} -76.4660 q^{6} -594.412i q^{7} -554.673i q^{8} +606.231 q^{9} +742.004 q^{10} -326.752 q^{11} -181.422i q^{12} -3327.43 q^{13} -4102.15 q^{14} +1191.32 q^{15} -2779.99 q^{16} +1730.37 q^{17} -4183.71i q^{18} -6997.69i q^{19} +1760.47i q^{20} -6586.16 q^{21} +2254.98i q^{22} -7593.56 q^{23} -6145.85 q^{24} +4064.80 q^{25} +22963.2i q^{26} -14794.5i q^{27} -9732.68i q^{28} +10036.1i q^{29} -8221.51i q^{30} +25812.1 q^{31} -16313.8i q^{32} +3620.46i q^{33} -11941.6i q^{34} +63910.2 q^{35} +9926.19 q^{36} +37980.1i q^{37} -48292.4 q^{38} +36868.3i q^{39} +59637.6 q^{40} -18315.7 q^{41} +45452.3i q^{42} +(61156.1 - 50806.4i) q^{43} -5350.12 q^{44} +65180.9i q^{45} +52404.6i q^{46} +130402. q^{47} +30802.7i q^{48} -235677. q^{49} -28051.9i q^{50} -19172.7i q^{51} -54482.1 q^{52} +272103. q^{53} -102100. q^{54} -35131.9i q^{55} -329705. q^{56} -77535.3 q^{57} +69261.1 q^{58} +216781. q^{59} +19506.2 q^{60} +344144. i q^{61} -178134. i q^{62} -360351. i q^{63} -290504. q^{64} -357760. i q^{65} +24985.5 q^{66} -227534. q^{67} +28332.4 q^{68} +84137.6i q^{69} -441056. i q^{70} +281680. i q^{71} -336260. i q^{72} +294591. i q^{73} +262108. q^{74} -45038.5i q^{75} -114578. i q^{76} +194226. i q^{77} +254435. q^{78} -497124. q^{79} -298900. i q^{80} +278017. q^{81} +126400. i q^{82} -143881. q^{83} -107839. q^{84} +186046. i q^{85} +(-350625. - 422050. i) q^{86} +111201. q^{87} +181241. i q^{88} +523099. i q^{89} +449826. q^{90} +1.97786e6i q^{91} -124334. q^{92} -286002. i q^{93} -899926. i q^{94} +752380. q^{95} -180760. q^{96} -430866. q^{97} +1.62645e6i q^{98} -198087. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 796 q^{4} - 258 q^{6} - 2736 q^{9}+O(q^{10}) \) 20 * q - 796 * q^4 - 258 * q^6 - 2736 * q^9	\( 20 q - 796 q^{4} - 258 q^{6} - 2736 q^{9} - 1962 q^{10} + 2616 q^{11} - 5612 q^{13} - 3876 q^{14} + 4048 q^{15} + 14900 q^{16} + 3328 q^{17} + 10544 q^{21} - 836 q^{23} + 26966 q^{24} - 57904 q^{25} + 17116 q^{31} - 150472 q^{35} - 132110 q^{36} + 2266 q^{38} + 401286 q^{40} - 141936 q^{41} + 58160 q^{43} + 88544 q^{44} + 48452 q^{47} - 20644 q^{49} + 12292 q^{52} + 425212 q^{53} + 173740 q^{54} + 447864 q^{56} - 92904 q^{57} + 86502 q^{58} - 918856 q^{59} + 429848 q^{60} - 156892 q^{64} - 1246176 q^{66} + 1098496 q^{67} - 2589854 q^{68} - 1224106 q^{74} + 855716 q^{78} - 401492 q^{79} + 470644 q^{81} + 1354360 q^{83} - 637268 q^{84} + 3046356 q^{86} + 2116740 q^{87} - 683716 q^{90} - 485998 q^{92} - 2711660 q^{95} - 2480062 q^{96} + 7814560 q^{97} + 3744560 q^{99}+O(q^{100}) \) 20 * q - 796 * q^4 - 258 * q^6 - 2736 * q^9 - 1962 * q^10 + 2616 * q^11 - 5612 * q^13 - 3876 * q^14 + 4048 * q^15 + 14900 * q^16 + 3328 * q^17 + 10544 * q^21 - 836 * q^23 + 26966 * q^24 - 57904 * q^25 + 17116 * q^31 - 150472 * q^35 - 132110 * q^36 + 2266 * q^38 + 401286 * q^40 - 141936 * q^41 + 58160 * q^43 + 88544 * q^44 + 48452 * q^47 - 20644 * q^49 + 12292 * q^52 + 425212 * q^53 + 173740 * q^54 + 447864 * q^56 - 92904 * q^57 + 86502 * q^58 - 918856 * q^59 + 429848 * q^60 - 156892 * q^64 - 1246176 * q^66 + 1098496 * q^67 - 2589854 * q^68 - 1224106 * q^74 + 855716 * q^78 - 401492 * q^79 + 470644 * q^81 + 1354360 * q^83 - 637268 * q^84 + 3046356 * q^86 + 2116740 * q^87 - 683716 * q^90 - 485998 * q^92 - 2711660 * q^95 - 2480062 * q^96 + 7814560 * q^97 + 3744560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more