LMFDB - Embedded newform 138.3.b.a.91.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,3,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	138.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:	$3.76022764817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1358954496.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 $ x^8 + 8x^6 + 20x^4 + 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.1
Root			$-1.58671i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.3.b.a.91.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -1.69484i q^{5} +2.44949 q^{6} +4.18388i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -1.69484i q^{5} +2.44949 q^{6} +4.18388i q^{7} -2.82843 q^{8} +3.00000 q^{9} +2.39686i q^{10} -20.2970i q^{11} -3.46410 q^{12} -11.6410 q^{13} -5.91690i q^{14} +2.93555i q^{15} +4.00000 q^{16} -12.5377i q^{17} -4.24264 q^{18} -27.1971i q^{19} -3.38968i q^{20} -7.24670i q^{21} +28.7043i q^{22} +(4.46926 - 22.5616i) q^{23} +4.89898 q^{24} +22.1275 q^{25} +16.4629 q^{26} -5.19615 q^{27} +8.36776i q^{28} -50.9389 q^{29} -4.15149i q^{30} -11.5732 q^{31} -5.65685 q^{32} +35.1554i q^{33} +17.7310i q^{34} +7.09100 q^{35} +6.00000 q^{36} +52.7319i q^{37} +38.4625i q^{38} +20.1628 q^{39} +4.79373i q^{40} +42.4390 q^{41} +10.2484i q^{42} -2.10282i q^{43} -40.5940i q^{44} -5.08451i q^{45} +(-6.32049 + 31.9069i) q^{46} +20.0260 q^{47} -6.92820 q^{48} +31.4951 q^{49} -31.2930 q^{50} +21.7160i q^{51} -23.2820 q^{52} +49.6304i q^{53} +7.34847 q^{54} -34.4001 q^{55} -11.8338i q^{56} +47.1067i q^{57} +72.0384 q^{58} -39.9319 q^{59} +5.87109i q^{60} -70.9635i q^{61} +16.3669 q^{62} +12.5516i q^{63} +8.00000 q^{64} +19.7296i q^{65} -49.7173i q^{66} -4.38918i q^{67} -25.0755i q^{68} +(-7.74099 + 39.0778i) q^{69} -10.0282 q^{70} -46.5698 q^{71} -8.48528 q^{72} -3.71985 q^{73} -74.5741i q^{74} -38.3260 q^{75} -54.3941i q^{76} +84.9203 q^{77} -28.5145 q^{78} -26.4542i q^{79} -6.77935i q^{80} +9.00000 q^{81} -60.0178 q^{82} -106.806i q^{83} -14.4934i q^{84} -21.2494 q^{85} +2.97384i q^{86} +88.2287 q^{87} +57.4086i q^{88} +137.125i q^{89} +7.19059i q^{90} -48.7046i q^{91} +(8.93852 - 45.1232i) q^{92} +20.0453 q^{93} -28.3210 q^{94} -46.0946 q^{95} +9.79796 q^{96} -119.514i q^{97} -44.5408 q^{98} -60.8910i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 + 24 * q^9	$ 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) $ 8 * q + 16 * q^4 + 24 * q^9 + 16 * q^13 + 32 * q^16 + 16 * q^23 + 72 * q^25 + 32 * q^26 - 144 * q^29 - 128 * q^31 - 112 * q^35 + 48 * q^36 + 48 * q^39 - 16 * q^41 - 80 * q^46 - 112 * q^47 + 40 * q^49 - 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 + 80 * q^59 - 96 * q^62 + 64 * q^64 - 72 * q^69 - 144 * q^70 + 32 * q^71 + 64 * q^73 + 48 * q^75 + 224 * q^77 - 144 * q^78 + 72 * q^81 + 48 * q^85 + 96 * q^87 + 32 * q^92 + 192 * q^93 - 16 * q^94 + 112 * q^95 + 224 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	2.00000			0.500000
$5$		−	1.69484i		−	0.338968i	−0.985533	−	0.169484i	$-0.945790\pi$
$5$		−	1.69484i		−	0.338968i	0.985533	−	0.169484i	$-0.0542100\pi$
$6$	2.44949			0.408248
$7$			4.18388i			0.597697i	0.954300	+	0.298849i	$0.0966026\pi$
$7$			4.18388i			0.597697i	−0.954300	+	0.298849i	$0.903397\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$			2.39686i			0.239686i
$11$		−	20.2970i		−	1.84518i	−0.385779	−	0.922591i	$-0.626067\pi$
$11$		−	20.2970i		−	1.84518i	0.385779	−	0.922591i	$-0.373933\pi$
$12$	−3.46410			−0.288675
$13$	−11.6410			−0.895461			−0.447731	−	0.894168i	$-0.647768\pi$
$13$	−11.6410			−0.895461			−0.447731	+	0.894168i	$0.647768\pi$
$14$		−	5.91690i		−	0.422636i
$15$			2.93555i			0.195703i
$16$	4.00000			0.250000
$17$		−	12.5377i		−	0.737514i	−0.929526	−	0.368757i	$-0.879783\pi$
$17$		−	12.5377i		−	0.737514i	0.929526	−	0.368757i	$-0.120217\pi$
$18$	−4.24264			−0.235702
$19$		−	27.1971i		−	1.43142i	−0.698395	−	0.715712i	$-0.746102\pi$
$19$		−	27.1971i		−	1.43142i	0.698395	−	0.715712i	$-0.253898\pi$
$20$		−	3.38968i		−	0.169484i
$21$		−	7.24670i		−	0.345081i
$22$			28.7043i			1.30474i
$23$	4.46926	−	22.5616i	0.194316	−	0.980939i
$23$	4.46926	−	22.5616i	0.194316	−	0.980939i
$24$	4.89898			0.204124
$25$	22.1275			0.885101
$26$	16.4629			0.633187
$27$	−5.19615			−0.192450
$28$			8.36776i			0.298849i
$29$	−50.9389			−1.75651			−0.878256	−	0.478191i	$-0.841293\pi$
$29$	−50.9389			−1.75651			−0.878256	+	0.478191i	$0.841293\pi$
$30$		−	4.15149i		−	0.138383i
$31$	−11.5732			−0.373328			−0.186664	−	0.982424i	$-0.559768\pi$
$31$	−11.5732			−0.373328			−0.186664	+	0.982424i	$0.559768\pi$
$32$	−5.65685			−0.176777
$33$			35.1554i			1.06532i
$34$			17.7310i			0.521501i
$35$	7.09100			0.202600
$36$	6.00000			0.166667
$37$			52.7319i			1.42519i	0.701578	+	0.712593i	$0.252479\pi$
$37$			52.7319i			1.42519i	−0.701578	+	0.712593i	$0.747521\pi$
$38$			38.4625i			1.01217i
$39$	20.1628			0.516995
$40$			4.79373i			0.119843i
$41$	42.4390			1.03510			0.517548	−	0.855654i	$-0.326845\pi$
$41$	42.4390			1.03510			0.517548	+	0.855654i	$0.326845\pi$
$42$			10.2484i			0.244009i
$43$		−	2.10282i		−	0.0489029i	−0.999701	−	0.0244514i	$-0.992216\pi$
$43$		−	2.10282i		−	0.0489029i	0.999701	−	0.0244514i	$-0.00778391\pi$
$44$		−	40.5940i		−	0.922591i
$45$		−	5.08451i		−	0.112989i
$46$	−6.32049	+	31.9069i	−0.137402	+	0.693629i
$47$	20.0260			0.426085			0.213043	−	0.977043i	$-0.431663\pi$
$47$	20.0260			0.426085			0.213043	+	0.977043i	$0.431663\pi$
$48$	−6.92820			−0.144338
$49$	31.4951			0.642758
$50$	−31.2930			−0.625861
$51$			21.7160i			0.425804i
$52$	−23.2820			−0.447731
$53$			49.6304i			0.936423i	0.883616	+	0.468212i	$0.155102\pi$
$53$			49.6304i			0.936423i	−0.883616	+	0.468212i	$0.844898\pi$
$54$	7.34847			0.136083
$55$	−34.4001			−0.625457
$56$		−	11.8338i		−	0.211318i
$57$			47.1067i			0.826434i
$58$	72.0384			1.24204
$59$	−39.9319			−0.676811			−0.338406	−	0.941000i	$-0.609888\pi$
$59$	−39.9319			−0.676811			−0.338406	+	0.941000i	$0.609888\pi$
$60$			5.87109i			0.0978515i
$61$		−	70.9635i		−	1.16334i	−0.813426	−	0.581668i	$-0.802400\pi$
$61$		−	70.9635i		−	1.16334i	0.813426	−	0.581668i	$-0.197600\pi$
$62$	16.3669			0.263983
$63$			12.5516i			0.199232i
$64$	8.00000			0.125000
$65$			19.7296i			0.303532i
$66$		−	49.7173i		−	0.753292i
$67$		−	4.38918i		−	0.0655101i	−0.999463	−	0.0327551i	$-0.989572\pi$
$67$		−	4.38918i		−	0.0655101i	0.999463	−	0.0327551i	$-0.0104281\pi$
$68$		−	25.0755i		−	0.368757i
$69$	−7.74099	+	39.0778i	−0.112188	+	0.566345i
$70$	−10.0282			−0.143260
$71$	−46.5698			−0.655912			−0.327956	−	0.944693i	$-0.606360\pi$
$71$	−46.5698			−0.655912			−0.327956	+	0.944693i	$0.606360\pi$
$72$	−8.48528			−0.117851
$73$	−3.71985			−0.0509569			−0.0254784	−	0.999675i	$-0.508111\pi$
$73$	−3.71985			−0.0509569			−0.0254784	+	0.999675i	$0.508111\pi$
$74$		−	74.5741i		−	1.00776i
$75$	−38.3260			−0.511013
$76$		−	54.3941i		−	0.715712i
$77$	84.9203			1.10286
$78$	−28.5145			−0.365571
$79$		−	26.4542i		−	0.334864i	−0.985884	−	0.167432i	$-0.946453\pi$
$79$		−	26.4542i		−	0.334864i	0.985884	−	0.167432i	$-0.0535474\pi$
$80$		−	6.77935i		−	0.0847419i
$81$	9.00000			0.111111
$82$	−60.0178			−0.731924
$83$		−	106.806i		−	1.28681i	−0.765524	−	0.643407i	$-0.777520\pi$
$83$		−	106.806i		−	1.28681i	0.765524	−	0.643407i	$-0.222480\pi$
$84$		−	14.4934i		−	0.172540i
$85$	−21.2494			−0.249993
$86$			2.97384i			0.0345795i
$87$	88.2287			1.01412
$88$			57.4086i			0.652370i
$89$			137.125i			1.54073i	0.637602	+	0.770366i	$0.279926\pi$
$89$			137.125i			1.54073i	−0.637602	+	0.770366i	$0.720074\pi$
$90$			7.19059i			0.0798954i
$91$		−	48.7046i		−	0.535215i
$92$	8.93852	−	45.1232i	0.0971579	−	0.490470i
$93$	20.0453			0.215541
$94$	−28.3210			−0.301288
$95$	−46.0946			−0.485207
$96$	9.79796			0.102062
$97$		−	119.514i		−	1.23211i	−0.787705	−	0.616053i	$-0.788731\pi$
$97$		−	119.514i		−	1.23211i	0.787705	−	0.616053i	$-0.211269\pi$
$98$	−44.5408			−0.454498
$99$		−	60.8910i		−	0.615061i
$100$	44.2550			0.442550
$101$	24.5512			0.243082			0.121541	−	0.992586i	$-0.461216\pi$
$101$	24.5512			0.243082			0.121541	+	0.992586i	$0.461216\pi$
$102$		−	30.7111i		−	0.301089i
$103$			7.77997i			0.0755337i	0.999287	+	0.0377668i	$0.0120244\pi$
$103$			7.77997i			0.0755337i	−0.999287	+	0.0377668i	$0.987976\pi$
$104$	32.9257			0.316593
$105$	−12.2820			−0.116971
$106$		−	70.1880i		−	0.662151i
$107$			42.8558i			0.400522i	0.979743	+	0.200261i	$0.0641790\pi$
$107$			42.8558i			0.400522i	−0.979743	+	0.200261i	$0.935821\pi$
$108$	−10.3923			−0.0962250
$109$			1.36851i			0.0125552i	0.999980	+	0.00627759i	$0.00199823\pi$
$109$			1.36851i			0.0125552i	−0.999980	+	0.00627759i	$0.998002\pi$
$110$	48.6491			0.442265
$111$		−	91.3343i		−	0.822831i
$112$			16.7355i			0.149424i
$113$			121.497i			1.07520i	0.843201	+	0.537598i	$0.180668\pi$
$113$			121.497i			1.07520i	−0.843201	+	0.537598i	$0.819332\pi$
$114$		−	66.6190i		−	0.584377i
$115$	−38.2383	−	7.57467i	−0.332507	−	0.0658667i
$116$	−101.878			−0.878256
$117$	−34.9230			−0.298487
$118$	56.4722			0.478578
$119$	52.4564			0.440810
$120$		−	8.30298i		−	0.0691915i
$121$	−290.968			−2.40470
$122$			100.358i			0.822603i
$123$	−73.5065			−0.597614
$124$	−23.1463			−0.186664
$125$		−	79.8735i		−	0.638988i
$126$		−	17.7507i		−	0.140879i
$127$	223.757			1.76186			0.880932	−	0.473243i	$-0.156917\pi$
$127$	223.757			1.76186			0.880932	+	0.473243i	$0.156917\pi$
$128$	−11.3137			−0.0883883
$129$			3.64220i			0.0282341i
$130$		−	27.9019i		−	0.214630i
$131$	49.2839			0.376213			0.188106	−	0.982149i	$-0.439765\pi$
$131$	49.2839			0.376213			0.188106	+	0.982149i	$0.439765\pi$
$132$			70.3109i			0.532658i
$133$	113.789			0.855559
$134$			6.20724i			0.0463227i
$135$			8.80664i			0.0652343i
$136$			35.4621i			0.260751i
$137$			97.8172i			0.713994i	0.934105	+	0.356997i	$0.116199\pi$
$137$			97.8172i			0.713994i	−0.934105	+	0.356997i	$0.883801\pi$
$138$	10.9474	−	55.2644i	0.0793291	−	0.400467i
$139$	174.434			1.25492			0.627459	−	0.778650i	$-0.284095\pi$
$139$	174.434			1.25492			0.627459	+	0.778650i	$0.284095\pi$
$140$	14.1820			0.101300
$141$	−34.6861			−0.246000
$142$	65.8596			0.463800
$143$			236.277i			1.65229i
$144$	12.0000			0.0833333
$145$			86.3331i			0.595401i
$146$	5.26067			0.0360320
$147$	−54.5512			−0.371096
$148$			105.464i			0.712593i
$149$			8.72531i			0.0585591i	0.999571	+	0.0292796i	$0.00932131\pi$
$149$			8.72531i			0.0585591i	−0.999571	+	0.0292796i	$0.990679\pi$
$150$	54.2011			0.361341
$151$	−86.1741			−0.570689			−0.285345	−	0.958425i	$-0.592108\pi$
$151$	−86.1741			−0.570689			−0.285345	+	0.958425i	$0.592108\pi$
$152$			76.9249i			0.506085i
$153$		−	37.6132i		−	0.245838i
$154$	−120.095			−0.779840
$155$			19.6146i			0.126546i
$156$	40.3256			0.258497
$157$			177.857i			1.13285i	0.824114	+	0.566424i	$0.191673\pi$
$157$			177.857i			1.13285i	−0.824114	+	0.566424i	$0.808327\pi$
$158$			37.4119i			0.236784i
$159$		−	85.9625i		−	0.540644i
$160$			9.58745i			0.0599216i
$161$	94.3951	+	18.6989i	0.586305	+	0.116142i
$162$	−12.7279			−0.0785674
$163$	−116.103			−0.712291			−0.356146	−	0.934430i	$-0.615909\pi$
$163$	−116.103			−0.712291			−0.356146	+	0.934430i	$0.615909\pi$
$164$	84.8780			0.517548
$165$	59.5828			0.361108
$166$			151.046i			0.909915i
$167$	−281.959			−1.68838			−0.844189	−	0.536046i	$-0.819917\pi$
$167$	−281.959			−1.68838			−0.844189	+	0.536046i	$0.819917\pi$
$168$			20.4968i			0.122004i
$169$	−33.4872			−0.198149
$170$	30.0512			0.176772
$171$		−	81.5912i		−	0.477142i
$172$		−	4.20565i		−	0.0244514i
$173$	115.730			0.668962			0.334481	−	0.942403i	$-0.391439\pi$
$173$	115.730			0.668962			0.334481	+	0.942403i	$0.391439\pi$
$174$	−124.774			−0.717093
$175$			92.5790i			0.529023i
$176$		−	81.1880i		−	0.461296i
$177$	69.1640			0.390757
$178$		−	193.924i		−	1.08946i
$179$	184.904			1.03299			0.516493	−	0.856292i	$-0.327237\pi$
$179$	184.904			1.03299			0.516493	+	0.856292i	$0.327237\pi$
$180$		−	10.1690i		−	0.0564946i
$181$		−	349.157i		−	1.92904i	−0.264001	−	0.964522i	$-0.585042\pi$
$181$		−	349.157i		−	1.92904i	0.264001	−	0.964522i	$-0.414958\pi$
$182$			68.8787i			0.378454i
$183$			122.912i			0.671652i
$184$	−12.6410	+	63.8138i	−0.0687010	+	0.346814i
$185$	89.3720			0.483092
$186$	−28.3484			−0.152410
$187$	−254.478			−1.36085
$188$	40.0520			0.213043
$189$		−	21.7401i		−	0.115027i
$190$	65.1877			0.343093
$191$		−	319.163i		−	1.67101i	−0.549483	−	0.835505i	$-0.685175\pi$
$191$		−	319.163i		−	1.67101i	0.549483	−	0.835505i	$-0.314825\pi$
$192$	−13.8564			−0.0721688
$193$	371.496			1.92485			0.962425	−	0.271546i	$-0.0875349\pi$
$193$	371.496			1.92485			0.962425	+	0.271546i	$0.0875349\pi$
$194$			169.019i			0.871231i
$195$		−	34.1727i		−	0.175245i
$196$	62.9903			0.321379
$197$	−190.805			−0.968552			−0.484276	−	0.874915i	$-0.660917\pi$
$197$	−190.805			−0.968552			−0.484276	+	0.874915i	$0.660917\pi$
$198$			86.1129i			0.434914i
$199$			165.204i			0.830171i	0.909782	+	0.415085i	$0.136248\pi$
$199$			165.204i			0.830171i	−0.909782	+	0.415085i	$0.863752\pi$
$200$	−62.5861			−0.312930
$201$			7.60228i			0.0378223i
$202$	−34.7207			−0.171885
$203$		−	213.122i		−	1.04986i
$204$			43.4320i			0.212902i
$205$		−	71.9272i		−	0.350864i
$206$		−	11.0025i		−	0.0534104i
$207$	13.4078	−	67.6848i	0.0647719	−	0.326980i
$208$	−46.5640			−0.223865
$209$	−552.019			−2.64124
$210$	17.3693			0.0827111
$211$	217.886			1.03263			0.516317	−	0.856397i	$-0.327303\pi$
$211$	217.886			1.03263			0.516317	+	0.856397i	$0.327303\pi$
$212$			99.2609i			0.468212i
$213$	80.6612			0.378691
$214$		−	60.6073i		−	0.283212i
$215$	−3.56394			−0.0165765
$216$	14.6969			0.0680414
$217$		−	48.4208i		−	0.223137i
$218$		−	1.93537i		−	0.00887785i
$219$	6.44297			0.0294200
$220$	−68.8003			−0.312729
$221$			145.952i			0.660415i
$222$			129.166i			0.581830i
$223$	−229.084			−1.02728			−0.513641	−	0.858005i	$-0.671704\pi$
$223$	−229.084			−1.02728			−0.513641	+	0.858005i	$0.671704\pi$
$224$		−	23.6676i		−	0.105659i
$225$	66.3826			0.295034
$226$		−	171.823i		−	0.760279i
$227$		−	272.590i		−	1.20084i	−0.799686	−	0.600418i	$-0.795001\pi$
$227$		−	272.590i		−	1.20084i	0.799686	−	0.600418i	$-0.204999\pi$
$228$			94.2134i			0.413217i
$229$			155.398i			0.678593i	0.940679	+	0.339297i	$0.110189\pi$
$229$			155.398i			0.678593i	−0.940679	+	0.339297i	$0.889811\pi$
$230$	54.0771	+	10.7122i	0.235118	+	0.0465748i
$231$	−147.086			−0.636737
$232$	144.077			0.621021
$233$	200.272			0.859537			0.429768	−	0.902939i	$-0.358595\pi$
$233$	200.272			0.859537			0.429768	+	0.902939i	$0.358595\pi$
$234$	49.3886			0.211062
$235$		−	33.9408i		−	0.144429i
$236$	−79.8637			−0.338406
$237$			45.8201i			0.193334i
$238$	−74.1846			−0.311700
$239$	440.599			1.84351			0.921755	−	0.387772i	$-0.126755\pi$
$239$	440.599			1.84351			0.921755	+	0.387772i	$0.126755\pi$
$240$			11.7422i			0.0489258i
$241$		−	331.186i		−	1.37421i	−0.726556	−	0.687107i	$-0.758880\pi$
$241$		−	331.186i		−	1.37421i	0.726556	−	0.687107i	$-0.241120\pi$
$242$	411.491			1.70038
$243$	−15.5885			−0.0641500
$244$		−	141.927i		−	0.581668i
$245$		−	53.3791i		−	0.217874i
$246$	103.954			0.422577
$247$			316.601i			1.28179i
$248$	32.7339			0.131991
$249$			184.993i			0.742943i
$250$			112.958i			0.451833i
$251$		−	163.892i		−	0.652957i	−0.945205	−	0.326479i	$-0.894138\pi$
$251$		−	163.892i		−	0.652957i	0.945205	−	0.326479i	$-0.105862\pi$
$252$			25.1033i			0.0996162i
$253$	−457.933	−	90.7126i	−1.81001	−	0.358548i
$254$	−316.440			−1.24583
$255$	36.8051			0.144334
$256$	16.0000			0.0625000
$257$	−210.371			−0.818565			−0.409283	−	0.912408i	$-0.634221\pi$
$257$	−210.371			−0.818565			−0.409283	+	0.912408i	$0.634221\pi$
$258$		−	5.15084i		−	0.0199645i
$259$	−220.624			−0.851830
$260$			39.4592i			0.151766i
$261$	−152.817			−0.585504
$262$	−69.6979			−0.266023
$263$			257.455i			0.978916i	0.872027	+	0.489458i	$0.162805\pi$
$263$			257.455i			0.978916i	−0.872027	+	0.489458i	$0.837195\pi$
$264$		−	99.4346i		−	0.376646i
$265$	84.1156			0.317417
$266$	−160.922			−0.604972
$267$		−	237.508i		−	0.889542i
$268$		−	8.77836i		−	0.0327551i
$269$	126.614			0.470683			0.235342	−	0.971913i	$-0.424379\pi$
$269$	126.614			0.470683			0.235342	+	0.971913i	$0.424379\pi$
$270$		−	12.4545i		−	0.0461277i
$271$	452.242			1.66879			0.834394	−	0.551168i	$-0.185818\pi$
$271$	452.242			1.66879			0.834394	+	0.551168i	$0.185818\pi$
$272$		−	50.1509i		−	0.184378i
$273$			84.3588i			0.309007i
$274$		−	138.334i		−	0.504870i
$275$		−	449.122i		−	1.63317i
$276$	−15.4820	+	78.1557i	−0.0560941	+	0.283173i
$277$	97.8176			0.353132			0.176566	−	0.984289i	$-0.443501\pi$
$277$	97.8176			0.353132			0.176566	+	0.984289i	$0.443501\pi$
$278$	−246.686			−0.887361
$279$	−34.7195			−0.124443
$280$	−20.0564			−0.0716299
$281$		−	411.158i		−	1.46320i	−0.681736	−	0.731598i	$-0.738775\pi$
$281$		−	411.158i		−	1.46320i	0.681736	−	0.731598i	$-0.261225\pi$
$282$	49.0535			0.173949
$283$			167.168i			0.590701i	0.955389	+	0.295350i	$0.0954364\pi$
$283$			167.168i			0.590701i	−0.955389	+	0.295350i	$0.904564\pi$
$284$	−93.1395			−0.327956
$285$	79.8383			0.280134
$286$		−	334.147i		−	1.16835i
$287$			177.560i			0.618675i
$288$	−16.9706			−0.0589256
$289$	131.805			0.456073
$290$		−	122.093i		−	0.421012i
$291$			207.005i			0.711357i
$292$	−7.43970			−0.0254784
$293$		−	359.823i		−	1.22807i	−0.789280	−	0.614033i	$-0.789546\pi$
$293$		−	359.823i		−	1.22807i	0.789280	−	0.614033i	$-0.210454\pi$
$294$	77.1470			0.262405
$295$			67.6780i			0.229417i
$296$		−	149.148i		−	0.503879i
$297$			105.466i			0.355105i
$298$		−	12.3395i		−	0.0414076i
$299$	−52.0267	+	262.640i	−0.174002	+	0.878393i
$300$	−76.6520			−0.255507
$301$	8.79796			0.0292291
$302$	121.869			0.403538
$303$	−42.5240			−0.140343
$304$		−	108.788i		−	0.357856i
$305$	−120.272			−0.394333
$306$			53.1931i			0.173834i
$307$	159.380			0.519152			0.259576	−	0.965723i	$-0.416417\pi$
$307$	159.380			0.519152			0.259576	+	0.965723i	$0.416417\pi$
$308$	169.841			0.551430
$309$		−	13.4753i		−	0.0436094i
$310$		−	27.7393i		−	0.0894816i
$311$	−212.603			−0.683610			−0.341805	−	0.939771i	$-0.611038\pi$
$311$	−212.603			−0.683610			−0.341805	+	0.939771i	$0.611038\pi$
$312$	−57.0290			−0.182785
$313$			6.75351i			0.0215767i	0.999942	+	0.0107884i	$0.00343411\pi$
$313$			6.75351i			0.0215767i	−0.999942	+	0.0107884i	$0.996566\pi$
$314$		−	251.528i		−	0.801044i
$315$	21.2730			0.0675334
$316$		−	52.9085i		−	0.167432i
$317$	262.878			0.829268			0.414634	−	0.909988i	$-0.363910\pi$
$317$	262.878			0.829268			0.414634	+	0.909988i	$0.363910\pi$
$318$			121.569i			0.382293i
$319$			1033.91i			3.24108i
$320$		−	13.5587i		−	0.0423710i
$321$		−	74.2285i		−	0.231241i
$322$	−133.495	−	26.4442i	−0.414580	−	0.0821248i
$323$	−340.990			−1.05570
$324$	18.0000			0.0555556
$325$	−257.586			−0.792574
$326$	164.195			0.503666
$327$		−	2.37034i		−	0.00724873i
$328$	−120.036			−0.365962
$329$			83.7865i			0.254670i
$330$	−84.2628			−0.255342
$331$	353.071			1.06668			0.533339	−	0.845901i	$-0.320937\pi$
$331$	353.071			1.06668			0.533339	+	0.845901i	$0.320937\pi$
$332$		−	213.611i		−	0.643407i
$333$			158.196i			0.475062i
$334$	398.750			1.19386
$335$	−7.43895			−0.0222058
$336$		−	28.9868i		−	0.0862702i
$337$			95.3746i			0.283011i	0.989937	+	0.141505i	$0.0451942\pi$
$337$			95.3746i			0.283011i	−0.989937	+	0.141505i	$0.954806\pi$
$338$	47.3580			0.140112
$339$		−	210.439i		−	0.620765i
$340$	−42.4989			−0.124997
$341$			234.901i			0.688858i
$342$			115.387i			0.337390i
$343$			336.782i			0.981872i
$344$			5.94768i			0.0172898i
$345$	66.2306	+	13.1197i	0.191973	+	0.0380282i
$346$	−163.668			−0.473027
$347$	−348.533			−1.00442			−0.502209	−	0.864746i	$-0.667479\pi$
$347$	−348.533			−1.00442			−0.502209	+	0.864746i	$0.667479\pi$
$348$	176.457			0.507061
$349$	458.963			1.31508			0.657541	−	0.753419i	$-0.271597\pi$
$349$	458.963			1.31508			0.657541	+	0.753419i	$0.271597\pi$
$350$		−	130.926i		−	0.374075i
$351$	60.4884			0.172332
$352$			114.817i			0.326185i
$353$	225.020			0.637452			0.318726	−	0.947847i	$-0.396745\pi$
$353$	225.020			0.637452			0.318726	+	0.947847i	$0.396745\pi$
$354$	−97.8127			−0.276307
$355$			78.9282i			0.222333i
$356$			274.250i			0.770366i
$357$	−90.8572			−0.254502
$358$	−261.494			−0.730431
$359$			318.972i			0.888502i	0.895902	+	0.444251i	$0.146530\pi$
$359$			318.972i			0.888502i	−0.895902	+	0.444251i	$0.853470\pi$
$360$			14.3812i			0.0399477i
$361$	−378.681			−1.04898
$362$			493.783i			1.36404i
$363$	503.972			1.38835
$364$		−	97.4091i		−	0.267608i
$365$			6.30455i			0.0172727i
$366$		−	173.824i		−	0.474930i
$367$		−	10.8441i		−	0.0295479i	−0.999891	−	0.0147740i	$-0.995297\pi$
$367$		−	10.8441i		−	0.0295479i	0.999891	−	0.0147740i	$-0.00470287\pi$
$368$	17.8770	−	90.2464i	0.0485789	−	0.245235i
$369$	127.317			0.345032
$370$	−126.391			−0.341598
$371$	−207.648			−0.559698
$372$	40.0906			0.107770
$373$			252.337i			0.676507i	0.941055	+	0.338254i	$0.109836\pi$
$373$			252.337i			0.676507i	−0.941055	+	0.338254i	$0.890164\pi$
$374$	359.887			0.962264
$375$			138.345i			0.368920i
$376$	−56.6421			−0.150644
$377$	592.979			1.57289
$378$			30.7451i			0.0813363i
$379$			675.838i			1.78321i	0.452811	+	0.891606i	$0.350421\pi$
$379$			675.838i			1.78321i	−0.452811	+	0.891606i	$0.649579\pi$
$380$	−92.1893			−0.242603
$381$	−387.558			−1.01721
$382$			451.364i			1.18158i
$383$		−	285.533i		−	0.745516i	−0.927929	−	0.372758i	$-0.878412\pi$
$383$		−	285.533i		−	0.745516i	0.927929	−	0.372758i	$-0.121588\pi$
$384$	19.5959			0.0510310
$385$		−	143.926i		−	0.373834i
$386$	−525.375			−1.36108
$387$		−	6.30847i		−	0.0163010i
$388$		−	239.029i		−	0.616053i
$389$			119.403i			0.306949i	0.988153	+	0.153475i	$0.0490463\pi$
$389$			119.403i			0.306949i	−0.988153	+	0.153475i	$0.950954\pi$
$390$			48.3275i			0.123917i
$391$	−282.871	−	56.0344i	−0.723456	−	0.143311i
$392$	−89.0817			−0.227249
$393$	−85.3622			−0.217207
$394$	269.839			0.684869
$395$	−44.8356			−0.113508
$396$		−	121.782i		−	0.307530i
$397$	−135.547			−0.341429			−0.170715	−	0.985321i	$-0.554608\pi$
$397$	−135.547			−0.341429			−0.170715	+	0.985321i	$0.554608\pi$
$398$		−	233.634i		−	0.587020i
$399$	−197.089			−0.493957
$400$	88.5101			0.221275
$401$			135.478i			0.337850i	0.985629	+	0.168925i	$0.0540296\pi$
$401$			135.478i			0.337850i	−0.985629	+	0.168925i	$0.945970\pi$
$402$		−	10.7512i		−	0.0267444i
$403$	134.723			0.334301
$404$	49.1025			0.121541
$405$		−	15.2535i		−	0.0376631i
$406$			301.400i			0.742365i
$407$	1070.30			2.62973
$408$		−	61.4221i		−	0.150544i
$409$	−271.448			−0.663686			−0.331843	−	0.943335i	$-0.607670\pi$
$409$	−271.448			−0.663686			−0.331843	+	0.943335i	$0.607670\pi$
$410$			101.720i			0.248099i
$411$		−	169.424i		−	0.412225i
$412$			15.5599i			0.0377668i
$413$		−	167.070i		−	0.404528i
$414$	−18.9615	+	95.7208i	−0.0458007	+	0.231210i
$415$	−181.018			−0.436188
$416$	65.8514			0.158297
$417$	−302.128			−0.724527
$418$	780.673			1.86764
$419$			214.387i			0.511664i	0.966721	+	0.255832i	$0.0823494\pi$
$419$			214.387i			0.511664i	−0.966721	+	0.255832i	$0.917651\pi$
$420$	−24.5640			−0.0584856
$421$		−	569.380i		−	1.35245i	−0.736696	−	0.676224i	$-0.763615\pi$
$421$		−	569.380i		−	1.35245i	0.736696	−	0.676224i	$-0.236385\pi$
$422$	−308.137			−0.730183
$423$	60.0780			0.142028
$424$		−	140.376i		−	0.331076i
$425$		−	277.429i		−	0.652774i
$426$	−114.072			−0.267775
$427$	296.903			0.695323
$428$			85.7117i			0.200261i
$429$		−	409.244i		−	0.953950i
$430$	5.04018			0.0117213
$431$			183.203i			0.425064i	0.977154	+	0.212532i	$0.0681710\pi$
$431$			183.203i			0.425064i	−0.977154	+	0.212532i	$0.931829\pi$
$432$	−20.7846			−0.0481125
$433$		−	749.313i		−	1.73052i	−0.501327	−	0.865258i	$-0.667155\pi$
$433$		−	749.313i		−	1.73052i	0.501327	−	0.865258i	$-0.332845\pi$
$434$			68.4773i			0.157782i
$435$		−	149.533i		−	0.343755i
$436$			2.73703i			0.00627759i
$437$	−613.609	−	121.551i	−1.40414	−	0.278148i
$438$	−9.11174			−0.0208031
$439$	91.4979			0.208423			0.104212	−	0.994555i	$-0.466768\pi$
$439$	91.4979			0.208423			0.104212	+	0.994555i	$0.466768\pi$
$440$	97.2983			0.221132
$441$	94.4854			0.214253
$442$		−	206.407i		−	0.466984i
$443$	−366.672			−0.827701			−0.413851	−	0.910345i	$-0.635816\pi$
$443$	−366.672			−0.827701			−0.413851	+	0.910345i	$0.635816\pi$
$444$		−	182.669i		−	0.411416i
$445$	232.405			0.522258
$446$	323.974			0.726398
$447$		−	15.1127i		−	0.0338091i
$448$			33.4711i			0.0747122i
$449$	−410.543			−0.914349			−0.457175	−	0.889377i	$-0.651139\pi$
$449$	−410.543			−0.914349			−0.457175	+	0.889377i	$0.651139\pi$
$450$	−93.8791			−0.208620
$451$		−	861.384i		−	1.90994i
$452$			242.994i			0.537598i
$453$	149.258			0.329488
$454$			385.500i			0.849120i
$455$	−82.5464			−0.181421
$456$		−	133.238i		−	0.292188i
$457$		−	840.088i		−	1.83827i	−0.393947	−	0.919133i	$-0.628891\pi$
$457$		−	840.088i		−	1.83827i	0.393947	−	0.919133i	$-0.371109\pi$
$458$		−	219.766i		−	0.479838i
$459$			65.1480i			0.141935i
$460$	−76.4765	−	15.1493i	−0.166253	−	0.0329334i
$461$	35.9604			0.0780052			0.0390026	−	0.999239i	$-0.487582\pi$
$461$	35.9604			0.0780052			0.0390026	+	0.999239i	$0.487582\pi$
$462$	208.011			0.450241
$463$	−615.180			−1.32868			−0.664341	−	0.747430i	$-0.731288\pi$
$463$	−615.180			−1.32868			−0.664341	+	0.747430i	$0.731288\pi$
$464$	−203.755			−0.439128
$465$		−	33.9736i		−	0.0730614i
$466$	−283.228			−0.607784
$467$			504.679i			1.08068i	0.841446	+	0.540342i	$0.181705\pi$
$467$			504.679i			1.08068i	−0.841446	+	0.540342i	$0.818295\pi$
$468$	−69.8460			−0.149244
$469$	18.3638			0.0391552
$470$			47.9996i			0.102127i
$471$		−	308.058i		−	0.654050i
$472$	112.944			0.239289
$473$	−42.6810			−0.0902347
$474$		−	64.7994i		−	0.136708i
$475$		−	601.804i		−	1.26696i
$476$	104.913			0.220405
$477$			148.891i			0.312141i
$478$	−623.101			−1.30356
$479$			50.1143i			0.104623i	0.998631	+	0.0523114i	$0.0166588\pi$
$479$			50.1143i			0.104623i	−0.998631	+	0.0523114i	$0.983341\pi$
$480$		−	16.6060i		−	0.0345957i
$481$		−	613.852i		−	1.27620i
$482$			468.367i			0.971716i
$483$	−163.497	−	32.3874i	−0.338503	−	0.0670546i
$484$	−581.937			−1.20235
$485$	−202.557			−0.417644
$486$	22.0454			0.0453609
$487$	359.536			0.738268			0.369134	−	0.929376i	$-0.379654\pi$
$487$	359.536			0.738268			0.369134	+	0.929376i	$0.379654\pi$
$488$			200.715i			0.411301i
$489$	201.097			0.411242
$490$			75.4895i			0.154060i
$491$	516.399			1.05173			0.525864	−	0.850569i	$-0.323742\pi$
$491$	516.399			1.05173			0.525864	+	0.850569i	$0.323742\pi$
$492$	−147.013			−0.298807
$493$			638.658i			1.29545i
$494$		−	447.742i		−	0.906359i
$495$	−103.200			−0.208486
$496$	−46.2927			−0.0933320
$497$		−	194.842i		−	0.392037i
$498$		−	261.619i		−	0.525340i
$499$	−61.0247			−0.122294			−0.0611470	−	0.998129i	$-0.519476\pi$
$499$	−61.0247			−0.122294			−0.0611470	+	0.998129i	$0.519476\pi$
$500$		−	159.747i		−	0.319494i
$501$	488.367			0.974785
$502$			231.779i			0.461710i
$503$			54.2794i			0.107911i	0.998543	+	0.0539557i	$0.0171830\pi$
$503$			54.2794i			0.107911i	−0.998543	+	0.0539557i	$0.982817\pi$
$504$		−	35.5014i		−	0.0704393i
$505$		−	41.6104i		−	0.0823968i
$506$	647.615	+	128.287i	1.27987	+	0.253532i
$507$	58.0015			0.114401
$508$	447.513			0.880932
$509$	515.585			1.01294			0.506468	−	0.862258i	$-0.330951\pi$
$509$	515.585			1.01294			0.506468	+	0.862258i	$0.330951\pi$
$510$	−52.0503			−0.102059
$511$		−	15.5634i		−	0.0304568i
$512$	−22.6274			−0.0441942
$513$			141.320i			0.275478i
$514$	297.510			0.578813
$515$	13.1858			0.0256035
$516$			7.28439i			0.0141170i
$517$		−	406.468i		−	0.786205i
$518$	312.009			0.602335
$519$	−200.451			−0.386225
$520$		−	55.8038i		−	0.107315i
$521$			950.381i			1.82415i	0.410027	+	0.912073i	$0.365519\pi$
$521$			950.381i			1.82415i	−0.410027	+	0.912073i	$0.634481\pi$
$522$	216.115			0.414014
$523$		−	632.161i		−	1.20872i	−0.796711	−	0.604360i	$-0.793429\pi$
$523$		−	632.161i		−	1.20872i	0.796711	−	0.604360i	$-0.206571\pi$
$524$	98.5678			0.188106
$525$		−	160.351i		−	0.305431i
$526$		−	364.096i		−	0.692198i
$527$			145.101i			0.275335i
$528$			140.622i			0.266329i
$529$	−489.051	−	201.667i	−0.924483	−	0.381224i
$530$	−118.957			−0.224448
$531$	−119.796			−0.225604
$532$	227.579			0.427780
$533$	−494.032			−0.926889
$534$			335.887i			0.629001i
$535$	72.6337			0.135764
$536$			12.4145i			0.0231613i
$537$	−320.264			−0.596394
$538$	−179.059			−0.332823
$539$		−	639.257i		−	1.18601i
$540$			17.6133i			0.0326172i
$541$	42.8880			0.0792755			0.0396377	−	0.999214i	$-0.487380\pi$
$541$	42.8880			0.0792755			0.0396377	+	0.999214i	$0.487380\pi$
$542$	−639.566			−1.18001
$543$			604.758i			1.11373i
$544$			70.9241i			0.130375i
$545$	2.31941			0.00425580
$546$		−	119.301i		−	0.218501i
$547$	−597.669			−1.09263			−0.546315	−	0.837580i	$-0.683970\pi$
$547$	−597.669			−1.09263			−0.546315	+	0.837580i	$0.683970\pi$
$548$			195.634i			0.356997i
$549$		−	212.890i		−	0.387779i
$550$			635.155i			1.15483i
$551$			1385.39i			2.51432i
$552$	21.8948	−	110.529i	0.0396645	−	0.200233i
$553$	110.681			0.200147
$554$	−138.335			−0.249702
$555$	−154.797			−0.278913
$556$	348.867			0.627459
$557$			39.1372i			0.0702642i	0.999383	+	0.0351321i	$0.0111852\pi$
$557$			39.1372i			0.0702642i	−0.999383	+	0.0351321i	$0.988815\pi$
$558$	49.1008			0.0879942
$559$			24.4790i			0.0437906i
$560$	28.3640			0.0506500
$561$	440.770			0.785686
$562$			581.466i			1.03464i
$563$		−	420.979i		−	0.747743i	−0.927481	−	0.373871i	$-0.878030\pi$
$563$		−	420.979i		−	0.747743i	0.927481	−	0.373871i	$-0.121970\pi$
$564$	−69.3721			−0.123000
$565$	205.918			0.364457
$566$		−	236.412i		−	0.417688i
$567$			37.6549i			0.0664108i
$568$	131.719			0.231900
$569$			519.422i			0.912868i	0.889757	+	0.456434i	$0.150874\pi$
$569$			519.422i			0.912868i	−0.889757	+	0.456434i	$0.849126\pi$
$570$	−112.908			−0.198085
$571$			824.370i			1.44373i	0.692034	+	0.721865i	$0.256715\pi$
$571$			824.370i			1.44373i	−0.692034	+	0.721865i	$0.743285\pi$
$572$			472.555i			0.826145i
$573$			552.806i			0.964758i
$574$		−	251.107i		−	0.437469i
$575$	98.8937	−	499.232i	0.171989	−	0.868230i
$576$	24.0000			0.0416667
$577$	326.554			0.565951			0.282975	−	0.959127i	$-0.408679\pi$
$577$	326.554			0.565951			0.282975	+	0.959127i	$0.408679\pi$
$578$	−186.401			−0.322493
$579$	−643.450			−1.11131
$580$			172.666i			0.297700i
$581$	446.862			0.769126
$582$		−	292.749i		−	0.503005i
$583$	1007.35			1.72787
$584$	10.5213			0.0180160
$585$			59.1888i			0.101177i
$586$			508.867i			0.868374i
$587$	−142.994			−0.243602			−0.121801	−	0.992555i	$-0.538867\pi$
$587$	−142.994			−0.243602			−0.121801	+	0.992555i	$0.538867\pi$
$588$	−109.102			−0.185548
$589$			314.756i			0.534391i
$590$		−	95.7112i		−	0.162222i
$591$	330.483			0.559194
$592$			210.928i			0.356296i
$593$	340.100			0.573525			0.286763	−	0.958002i	$-0.407421\pi$
$593$	340.100			0.573525			0.286763	+	0.958002i	$0.407421\pi$
$594$		−	149.152i		−	0.251097i
$595$		−	88.9051i		−	0.149420i
$596$			17.4506i			0.0292796i
$597$		−	286.142i		−	0.479299i
$598$	73.5768	−	371.428i	0.123038	−	0.621118i
$599$	810.734			1.35348			0.676739	−	0.736223i	$-0.263392\pi$
$599$	810.734			1.35348			0.676739	+	0.736223i	$0.263392\pi$
$600$	108.402			0.180670
$601$	−917.012			−1.52581			−0.762906	−	0.646510i	$-0.776228\pi$
$601$	−917.012			−1.52581			−0.762906	+	0.646510i	$0.776228\pi$
$602$	−12.4422			−0.0206681
$603$		−	13.1675i		−	0.0218367i
$604$	−172.348			−0.285345
$605$			493.144i			0.815114i
$606$	60.1380			0.0992377
$607$	−891.833			−1.46925			−0.734624	−	0.678475i	$-0.762641\pi$
$607$	−891.833			−1.46925			−0.734624	+	0.678475i	$0.762641\pi$
$608$			153.850i			0.253043i
$609$			369.138i			0.606139i
$610$	170.090			0.278836
$611$	−233.123			−0.381543
$612$		−	75.2264i		−	0.122919i
$613$			451.763i			0.736970i	0.929634	+	0.368485i	$0.120123\pi$
$613$			451.763i			0.736970i	−0.929634	+	0.368485i	$0.879877\pi$
$614$	−225.397			−0.367096
$615$			124.582i			0.202572i
$616$	−240.191			−0.389920
$617$			123.296i			0.199832i	0.994996	+	0.0999161i	$0.0318574\pi$
$617$			123.296i			0.199832i	−0.994996	+	0.0999161i	$0.968143\pi$
$618$			19.0570i			0.0308365i
$619$		−	417.170i		−	0.673941i	−0.941515	−	0.336971i	$-0.890598\pi$
$619$		−	417.170i		−	0.673941i	0.941515	−	0.336971i	$-0.109402\pi$
$620$			39.2293i			0.0632730i
$621$	−23.2230	+	117.234i	−0.0373961	+	0.188782i
$622$	300.666			0.483385
$623$	−573.715			−0.920891
$624$	80.6512			0.129249
$625$	417.815			0.668505
$626$		−	9.55091i		−	0.0152570i
$627$	956.125			1.52492
$628$			355.714i			0.566424i
$629$	661.138			1.05109
$630$	−30.0846			−0.0477533
$631$		−	553.468i		−	0.877129i	−0.898700	−	0.438564i	$-0.855487\pi$
$631$		−	553.468i		−	0.877129i	0.898700	−	0.438564i	$-0.144513\pi$
$632$			74.8239i			0.118392i
$633$	−377.389			−0.596192
$634$	−371.766			−0.586381
$635$		−	379.231i		−	0.597215i
$636$		−	171.925i		−	0.270322i
$637$	−366.635			−0.575565
$638$		−	1462.16i		−	2.29179i
$639$	−139.709			−0.218637
$640$			19.1749i			0.0299608i
$641$			524.194i			0.817776i	0.912585	+	0.408888i	$0.134083\pi$
$641$			524.194i			0.817776i	−0.912585	+	0.408888i	$0.865917\pi$
$642$			104.975i			0.163512i
$643$		−	946.150i		−	1.47146i	−0.677274	−	0.735731i	$-0.736839\pi$
$643$		−	946.150i		−	1.47146i	0.677274	−	0.735731i	$-0.263161\pi$
$644$	188.790	+	37.3977i	0.293152	+	0.0580710i
$645$	6.17293			0.00957044
$646$	482.232			0.746490
$647$	−157.004			−0.242665			−0.121332	−	0.992612i	$-0.538717\pi$
$647$	−157.004			−0.242665			−0.121332	+	0.992612i	$0.538717\pi$
$648$	−25.4558			−0.0392837
$649$			810.497i			1.24884i
$650$	364.282			0.560434
$651$			83.8672i			0.128828i
$652$	−232.207			−0.356146
$653$	−111.109			−0.170152			−0.0850761	−	0.996374i	$-0.527113\pi$
$653$	−111.109			−0.170152			−0.0850761	+	0.996374i	$0.527113\pi$
$654$			3.35216i			0.00512563i
$655$		−	83.5282i		−	0.127524i
$656$	169.756			0.258774
$657$	−11.1596			−0.0169856
$658$		−	118.492i		−	0.180079i
$659$		−	397.086i		−	0.602559i	−0.953536	−	0.301279i	$-0.902586\pi$
$659$		−	397.086i		−	0.602559i	0.953536	−	0.301279i	$-0.0974137\pi$
$660$	119.166			0.180554
$661$			755.142i			1.14242i	0.820803	+	0.571212i	$0.193526\pi$
$661$			755.142i			1.14242i	−0.820803	+	0.571212i	$0.806474\pi$
$662$	−499.317			−0.754256
$663$		−	252.796i		−	0.381291i
$664$			302.092i			0.454958i
$665$		−	192.855i		−	0.290007i
$666$		−	223.722i		−	0.335920i
$667$	−227.659	+	1149.26i	−0.341318	+	1.72303i
$668$	−563.918			−0.844189
$669$	396.785			0.593102
$670$	10.5203			0.0157019
$671$	−1440.35			−2.14657
$672$			40.9935i			0.0610022i
$673$	394.477			0.586147			0.293073	−	0.956090i	$-0.405322\pi$
$673$	394.477			0.586147			0.293073	+	0.956090i	$0.405322\pi$
$674$		−	134.880i		−	0.200119i
$675$	−114.978			−0.170338
$676$	−66.9743			−0.0990745
$677$		−	776.614i		−	1.14714i	−0.819156	−	0.573571i	$-0.805558\pi$
$677$		−	776.614i		−	1.14714i	0.819156	−	0.573571i	$-0.194442\pi$
$678$			297.606i			0.438947i
$679$	500.034			0.736427
$680$	60.1025			0.0883860
$681$			472.140i			0.693303i
$682$		−	332.200i		−	0.487096i
$683$	809.349			1.18499			0.592496	−	0.805574i	$-0.298143\pi$
$683$	809.349			1.18499			0.592496	+	0.805574i	$0.298143\pi$
$684$		−	163.182i		−	0.238571i
$685$	165.784			0.242021
$686$		−	476.282i		−	0.694288i
$687$		−	269.157i		−	0.391786i
$688$		−	8.41129i		−	0.0122257i
$689$		−	577.748i		−	0.838531i
$690$	−93.6642	−	18.5541i	−0.135745	−	0.0268900i
$691$	−118.726			−0.171818			−0.0859088	−	0.996303i	$-0.527379\pi$
$691$	−118.726			−0.171818			−0.0859088	+	0.996303i	$0.527379\pi$
$692$	231.461			0.334481
$693$	254.761			0.367620
$694$	492.900			0.710231
$695$		−	295.637i		−	0.425376i
$696$	−249.548			−0.358547
$697$		−	532.089i		−	0.763398i
$698$	−649.072			−0.929903
$699$	−346.881			−0.496254
$700$			185.158i			0.264511i
$701$			751.269i			1.07171i	0.844310	+	0.535856i	$0.180011\pi$
$701$			751.269i			1.07171i	−0.844310	+	0.535856i	$0.819989\pi$
$702$	−85.5435			−0.121857
$703$	1434.15			2.04005
$704$		−	162.376i		−	0.230648i
$705$			58.7873i			0.0833862i
$706$	−318.227			−0.450746
$707$			102.720i			0.145289i
$708$	138.328			0.195379
$709$			80.3638i			0.113348i	0.998393	+	0.0566740i	$0.0180496\pi$
$709$			80.3638i			0.113348i	−0.998393	+	0.0566740i	$0.981950\pi$
$710$		−	111.621i		−	0.157213i
$711$		−	79.3627i		−	0.111621i
$712$		−	387.848i		−	0.544731i
$713$	−51.7235	+	261.109i	−0.0725435	+	0.366212i
$714$	128.491			0.179960
$715$	400.452			0.560073
$716$	369.809			0.516493
$717$	−763.140			−1.06435
$718$		−	451.095i		−	0.628266i
$719$	−211.128			−0.293641			−0.146821	−	0.989163i	$-0.546904\pi$
$719$	−211.128			−0.293641			−0.146821	+	0.989163i	$0.546904\pi$
$720$		−	20.3381i		−	0.0282473i
$721$	−32.5505			−0.0451463
$722$	535.536			0.741739
$723$			573.630i			0.793403i
$724$		−	698.314i		−	0.964522i
$725$	−1127.15			−1.55469
$726$	−712.724			−0.981713
$727$			870.940i			1.19799i	0.800752	+	0.598996i	$0.204433\pi$
$727$			870.940i			1.19799i	−0.800752	+	0.598996i	$0.795567\pi$
$728$			137.757i			0.189227i
$729$	27.0000			0.0370370
$730$		−	8.91598i		−	0.0122137i
$731$	−26.3646			−0.0360665
$732$			245.825i			0.335826i
$733$			1194.53i			1.62964i	0.579713	+	0.814821i	$0.303165\pi$
$733$			1194.53i			1.62964i	−0.579713	+	0.814821i	$0.696835\pi$
$734$			15.3359i			0.0208935i
$735$			92.4554i			0.125790i
$736$	−25.2820	+	127.628i	−0.0343505	+	0.173407i
$737$	−89.0872			−0.120878
$738$	−180.053			−0.243975
$739$	−1344.03			−1.81872			−0.909360	−	0.416011i	$-0.863428\pi$
$739$	−1344.03			−1.81872			−0.909360	+	0.416011i	$0.863428\pi$
$740$	178.744			0.241546
$741$		−	548.369i		−	0.740039i
$742$	293.659			0.395766
$743$		−	1070.72i		−	1.44107i	−0.693417	−	0.720537i	$-0.743896\pi$
$743$		−	1070.72i		−	1.44107i	0.693417	−	0.720537i	$-0.256104\pi$
$744$	−56.6967			−0.0762052
$745$	14.7880			0.0198497
$746$		−	356.859i		−	0.478363i
$747$		−	320.417i		−	0.428938i
$748$	−508.957			−0.680424
$749$	−179.304			−0.239391
$750$		−	195.649i		−	0.260866i
$751$		−	812.180i		−	1.08146i	−0.841195	−	0.540732i	$-0.818147\pi$
$751$		−	812.180i		−	1.08146i	0.841195	−	0.540732i	$-0.181853\pi$
$752$	80.1040			0.106521
$753$			283.870i			0.376985i
$754$	−838.599			−1.11220
$755$			146.051i			0.193445i
$756$		−	43.4802i		−	0.0575135i
$757$			624.896i			0.825491i	0.910846	+	0.412745i	$0.135430\pi$
$757$			624.896i			0.825491i	−0.910846	+	0.412745i	$0.864570\pi$
$758$		−	955.779i		−	1.26092i
$759$	793.163	+	157.119i	1.04501	+	0.207008i
$760$	130.375			0.171546
$761$	1119.42			1.47098			0.735491	−	0.677534i	$-0.236951\pi$
$761$	1119.42			1.47098			0.735491	+	0.677534i	$0.236951\pi$
$762$	548.090			0.719278
$763$	−5.72570			−0.00750419
$764$		−	638.326i		−	0.835505i
$765$	−63.7483			−0.0833311
$766$			403.804i			0.527160i
$767$	464.847			0.606058
$768$	−27.7128			−0.0360844
$769$		−	483.747i		−	0.629059i	−0.949248	−	0.314530i	$-0.898153\pi$
$769$		−	483.747i		−	0.629059i	0.949248	−	0.314530i	$-0.101847\pi$
$770$			203.542i			0.264341i
$771$	364.374			0.472599
$772$	742.992			0.962425
$773$		−	1101.62i		−	1.42512i	−0.701609	−	0.712562i	$-0.747535\pi$
$773$		−	1101.62i		−	1.42512i	0.701609	−	0.712562i	$-0.252465\pi$
$774$			8.92152i			0.0115265i
$775$	−256.086			−0.330433
$776$			338.038i			0.435615i
$777$	382.132			0.491804
$778$		−	168.862i		−	0.217046i
$779$		−	1154.22i		−	1.48166i
$780$		−	68.3454i		−	0.0876223i
$781$			945.227i			1.21028i
$782$	400.041	+	79.2446i	0.511561	+	0.101336i
$783$	264.686			0.338041
$784$	125.981			0.160689
$785$	301.439			0.383999
$786$	120.720			0.153588
$787$		−	1186.68i		−	1.50785i	−0.656961	−	0.753925i	$-0.728158\pi$
$787$		−	1186.68i		−	1.50785i	0.656961	−	0.753925i	$-0.271842\pi$
$788$	−381.609			−0.484276
$789$		−	445.925i		−	0.565178i
$790$	63.4072			0.0802623
$791$	−508.330			−0.642642
$792$			172.226i			0.217457i
$793$			826.086i			1.04172i
$794$	191.693			0.241427
$795$	−145.692			−0.183261
$796$			330.408i			0.415085i
$797$		−	89.5551i		−	0.112365i	−0.998421	−	0.0561826i	$-0.982107\pi$
$797$		−	89.5551i		−	0.112365i	0.998421	−	0.0561826i	$-0.0178929\pi$
$798$	278.726			0.349281
$799$		−	251.081i		−	0.314244i
$800$	−125.172			−0.156465
$801$			411.375i			0.513577i
$802$		−	191.595i		−	0.238896i
$803$			75.5019i			0.0940247i
$804$			15.2046i			0.0189111i
$805$	31.6915	−	159.984i	0.0393684	−	0.198738i
$806$	−190.527			−0.236386
$807$	−219.302			−0.271749
$808$	−69.4414			−0.0859423
$809$	−1189.39			−1.47020			−0.735099	−	0.677960i	$-0.762864\pi$
$809$	−1189.39			−1.47020			−0.735099	+	0.677960i	$0.762864\pi$
$810$			21.5718i			0.0266318i
$811$	−1569.97			−1.93585			−0.967923	−	0.251247i	$-0.919159\pi$
$811$	−1569.97			−1.93585			−0.967923	+	0.251247i	$0.919159\pi$
$812$		−	426.244i		−	0.524931i
$813$	−783.306			−0.963476
$814$	−1513.63			−1.85950
$815$			196.777i			0.241444i
$816$			86.8640i			0.106451i
$817$	−57.1906			−0.0700008
$818$	383.885			0.469297
$819$		−	146.114i		−	0.178405i
$820$		−	143.854i		−	0.175432i
$821$	49.4062			0.0601780			0.0300890	−	0.999547i	$-0.490421\pi$
$821$	49.4062			0.0601780			0.0300890	+	0.999547i	$0.490421\pi$
$822$			239.602i			0.291487i
$823$	−41.3668			−0.0502635			−0.0251317	−	0.999684i	$-0.508001\pi$
$823$	−41.3668			−0.0502635			−0.0251317	+	0.999684i	$0.508001\pi$
$824$		−	22.0051i		−	0.0267052i
$825$			777.903i			0.942913i
$826$			236.273i			0.286045i
$827$			101.706i			0.122982i	0.998108	+	0.0614912i	$0.0195856\pi$
$827$			101.706i			0.122982i	−0.998108	+	0.0614912i	$0.980414\pi$
$828$	26.8156	−	135.370i	0.0323860	−	0.163490i
$829$	1215.98			1.46680			0.733401	−	0.679796i	$-0.237932\pi$
$829$	1215.98			1.46680			0.733401	+	0.679796i	$0.237932\pi$
$830$	255.998			0.308432
$831$	−169.425			−0.203881
$832$	−93.1280			−0.111933
$833$		−	394.878i		−	0.474043i
$834$	427.273			0.512318
$835$			477.875i			0.572305i
$836$	−1104.04			−1.32062
$837$	60.1359			0.0718470
$838$		−	303.189i		−	0.361801i
$839$		−	593.795i		−	0.707741i	−0.935294	−	0.353871i	$-0.884865\pi$
$839$		−	593.795i		−	0.707741i	0.935294	−	0.353871i	$-0.115135\pi$
$840$	34.7387			0.0413556
$841$	1753.77			2.08533
$842$			805.225i			0.956325i
$843$			712.147i			0.844777i
$844$	435.772			0.516317
$845$			56.7553i			0.0671661i
$846$	−84.9631			−0.100429
$847$		−	1217.38i		−	1.43728i
$848$			198.522i			0.234106i
$849$		−	289.544i		−	0.341041i
$850$			392.344i			0.461581i
$851$	1189.72	+	235.673i	1.39802	+	0.276936i
$852$	161.322			0.189346
$853$	−708.023			−0.830038			−0.415019	−	0.909813i	$-0.636225\pi$
$853$	−708.023			−0.830038			−0.415019	+	0.909813i	$0.636225\pi$
$854$	−419.884			−0.491668
$855$	−138.284			−0.161736
$856$		−	121.215i		−	0.141606i
$857$	1406.05			1.64067			0.820335	−	0.571884i	$-0.193787\pi$
$857$	1406.05			1.64067			0.820335	+	0.571884i	$0.193787\pi$
$858$			578.759i			0.674544i
$859$	848.615			0.987911			0.493955	−	0.869487i	$-0.335551\pi$
$859$	848.615			0.987911			0.493955	+	0.869487i	$0.335551\pi$
$860$	−7.12789			−0.00828824
$861$		−	307.542i		−	0.357192i
$862$		−	259.088i		−	0.300566i
$863$	−869.000			−1.00695			−0.503476	−	0.864009i	$-0.667946\pi$
$863$	−869.000			−1.00695			−0.503476	+	0.864009i	$0.667946\pi$
$864$	29.3939			0.0340207
$865$		−	196.144i		−	0.226756i
$866$			1059.69i			1.22366i
$867$	−228.293			−0.263314
$868$		−	96.8415i		−	0.111569i
$869$	−536.942			−0.617885
$870$			211.472i			0.243071i
$871$			51.0944i			0.0586618i
$872$		−	3.87074i		−	0.00443892i
$873$		−	358.543i		−	0.410702i
$874$	867.775	+	171.899i	0.992877	+	0.196681i
$875$	334.181			0.381922
$876$	12.8859			0.0147100
$877$	223.104			0.254394			0.127197	−	0.991877i	$-0.459402\pi$
$877$	223.104			0.254394			0.127197	+	0.991877i	$0.459402\pi$
$878$	−129.398			−0.147378
$879$			623.232i			0.709024i
$880$	−137.601			−0.156364
$881$		−	151.919i		−	0.172439i	−0.996276	−	0.0862194i	$-0.972521\pi$
$881$		−	151.919i		−	0.172439i	0.996276	−	0.0862194i	$-0.0274786\pi$
$882$	−133.623			−0.151499
$883$	−658.614			−0.745882			−0.372941	−	0.927855i	$-0.621651\pi$
$883$	−658.614			−0.745882			−0.372941	+	0.927855i	$0.621651\pi$
$884$			291.904i			0.330208i
$885$		−	117.222i		−	0.132454i
$886$	518.552			0.585273
$887$	−623.832			−0.703306			−0.351653	−	0.936130i	$-0.614380\pi$
$887$	−623.832			−0.703306			−0.351653	+	0.936130i	$0.614380\pi$
$888$			258.332i			0.290915i
$889$			936.172i			1.05306i
$890$	−328.670			−0.369292
$891$		−	182.673i		−	0.205020i
$892$	−458.168			−0.513641
$893$		−	544.649i		−	0.609909i
$894$			21.3726i			0.0239067i
$895$		−	313.383i		−	0.350149i
$896$		−	47.3352i		−	0.0528295i
$897$	90.1128	−	454.905i	0.100460	−	0.507140i
$898$	580.595			0.646542
$899$	589.524			0.655755
$900$	132.765			0.147517
$901$	622.253			0.690625
$902$			1218.18i			1.35053i
$903$	−15.2385			−0.0168754
$904$		−	343.646i		−	0.380139i
$905$	−591.765			−0.653884
$906$	−211.083			−0.232983
$907$		−	111.606i		−	0.123050i	−0.998106	−	0.0615249i	$-0.980404\pi$
$907$		−	111.606i		−	0.123050i	0.998106	−	0.0615249i	$-0.0195964\pi$
$908$		−	545.180i		−	0.600418i
$909$	73.6537			0.0810272
$910$	116.738			0.128284
$911$			1401.85i			1.53880i	0.638767	+	0.769400i	$0.279445\pi$
$911$			1401.85i			1.53880i	−0.638767	+	0.769400i	$0.720555\pi$
$912$			188.427i			0.206608i
$913$	−2167.83			−2.37441
$914$			1188.06i			1.29985i
$915$	208.317			0.227668
$916$			310.796i			0.339297i
$917$			206.198i			0.224861i
$918$		−	92.1332i		−	0.100363i
$919$			952.820i			1.03680i	0.855138	+	0.518401i	$0.173472\pi$
$919$			952.820i			1.03680i	−0.855138	+	0.518401i	$0.826528\pi$
$920$	108.154	+	21.4244i	0.117559	+	0.0232874i
$921$	−276.054			−0.299733
$922$	−50.8557			−0.0551580
$923$	542.118			0.587344
$924$	−294.172			−0.318368
$925$			1166.83i			1.26143i
$926$	869.995			0.939520
$927$			23.3399i			0.0251779i
$928$	288.154			0.310510
$929$	−977.171			−1.05185			−0.525926	−	0.850530i	$-0.676281\pi$
$929$	−977.171			−1.05185			−0.525926	+	0.850530i	$0.676281\pi$
$930$			48.0459i			0.0516622i
$931$		−	856.575i		−	0.920059i
$932$	400.544			0.429768
$933$	368.239			0.394682
$934$		−	713.724i		−	0.764159i
$935$			431.300i			0.461283i
$936$	98.7771			0.105531
$937$		−	1381.30i		−	1.47418i	−0.675796	−	0.737089i	$-0.736200\pi$
$937$		−	1381.30i		−	1.47418i	0.675796	−	0.737089i	$-0.263800\pi$
$938$	−25.9703			−0.0276869
$939$		−	11.6974i		−	0.0124573i
$940$		−	67.8817i		−	0.0722145i
$941$		−	918.677i		−	0.976277i	−0.872766	−	0.488138i	$-0.837676\pi$
$941$		−	918.677i		−	0.976277i	0.872766	−	0.488138i	$-0.162324\pi$
$942$			435.659i			0.462483i
$943$	189.671	−	957.491i	0.201136	−	1.01537i
$944$	−159.727			−0.169203
$945$	−36.8459			−0.0389904
$946$	60.3601			0.0638056
$947$	346.985			0.366404			0.183202	−	0.983075i	$-0.441354\pi$
$947$	346.985			0.366404			0.183202	+	0.983075i	$0.441354\pi$
$948$			91.6402i			0.0966668i
$949$	43.3028			0.0456299
$950$			851.079i			0.895873i
$951$	−455.318			−0.478778
$952$	−148.369			−0.155850
$953$			1718.04i			1.80277i	0.433023	+	0.901383i	$0.357447\pi$
$953$			1718.04i			1.80277i	−0.433023	+	0.901383i	$0.642553\pi$
$954$		−	210.564i		−	0.220717i
$955$	−540.929			−0.566418
$956$	881.198			0.921755
$957$		−	1790.78i		−	1.87124i
$958$		−	70.8724i		−	0.0739795i
$959$	−409.256			−0.426753
$960$			23.4844i			0.0244629i
$961$	−827.062			−0.860626
$962$			868.117i			0.902409i
$963$			128.568i			0.133507i
$964$		−	662.371i		−	0.687107i
$965$		−	629.626i		−	0.652462i
$966$	231.220	+	45.8027i	0.239358	+	0.0474148i
$967$	−1277.14			−1.32073			−0.660363	−	0.750946i	$-0.729598\pi$
$967$	−1277.14			−1.32073			−0.660363	+	0.750946i	$0.729598\pi$
$968$	822.983			0.850189
$969$	590.612			0.609506
$970$	286.459			0.295319
$971$			1001.34i			1.03125i	0.856814	+	0.515626i	$0.172440\pi$
$971$			1001.34i			1.03125i	−0.856814	+	0.515626i	$0.827560\pi$
$972$	−31.1769			−0.0320750
$973$			729.809i			0.750061i
$974$	−508.461			−0.522034
$975$	446.153			0.457593
$976$		−	283.854i		−	0.290834i
$977$			980.126i			1.00320i	0.865100	+	0.501600i	$0.167255\pi$
$977$			980.126i			1.00320i	−0.865100	+	0.501600i	$0.832745\pi$
$978$	−284.394			−0.290792
$979$	2783.23			2.84293
$980$		−	106.758i		−	0.108937i
$981$			4.10554i			0.00418506i
$982$	−730.298			−0.743684
$983$		−	1017.15i		−	1.03474i	−0.855761	−	0.517371i	$-0.826911\pi$
$983$		−	1017.15i		−	1.03474i	0.855761	−	0.517371i	$-0.173089\pi$
$984$	207.908			0.211288
$985$			323.383i			0.328308i
$986$		−	903.199i		−	0.916023i
$987$		−	145.122i		−	0.147034i
$988$			633.202i			0.640893i
$989$	−47.4430	−	9.39807i	−0.0479707	−	0.00950259i
$990$	145.947			0.147422
$991$	1094.77			1.10471			0.552356	−	0.833608i	$-0.313729\pi$
$991$	1094.77			1.10471			0.552356	+	0.833608i	$0.313729\pi$
$992$	65.4677			0.0659957
$993$	−611.536			−0.615847
$994$			275.549i			0.277212i
$995$	279.994			0.281401
$996$			369.985i			0.371471i
$997$	385.578			0.386738			0.193369	−	0.981126i	$-0.438059\pi$
$997$	385.578			0.386738			0.193369	+	0.981126i	$0.438059\pi$
$998$	86.3020			0.0864749
$999$		−	274.003i		−	0.274277i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.b.a.91.1	✓	8
3.2	odd	2		414.3.b.c.91.8		8
4.3	odd	2		1104.3.c.c.1057.6		8
23.22	odd	2	inner	138.3.b.a.91.2	yes	8
69.68	even	2		414.3.b.c.91.5		8
92.91	even	2		1104.3.c.c.1057.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.b.a.91.1	✓	8	1.1	even	1	trivial
138.3.b.a.91.2	yes	8	23.22	odd	2	inner
414.3.b.c.91.5		8	69.68	even	2
414.3.b.c.91.8		8	3.2	odd	2
1104.3.c.c.1057.6		8	4.3	odd	2
1104.3.c.c.1057.7		8	92.91	even	2

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -1.69484i q^{5} +2.44949 q^{6} +4.18388i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -1.69484i q^{5} +2.44949 q^{6} +4.18388i q^{7} -2.82843 q^{8} +3.00000 q^{9} +2.39686i q^{10} -20.2970i q^{11} -3.46410 q^{12} -11.6410 q^{13} -5.91690i q^{14} +2.93555i q^{15} +4.00000 q^{16} -12.5377i q^{17} -4.24264 q^{18} -27.1971i q^{19} -3.38968i q^{20} -7.24670i q^{21} +28.7043i q^{22} +(4.46926 - 22.5616i) q^{23} +4.89898 q^{24} +22.1275 q^{25} +16.4629 q^{26} -5.19615 q^{27} +8.36776i q^{28} -50.9389 q^{29} -4.15149i q^{30} -11.5732 q^{31} -5.65685 q^{32} +35.1554i q^{33} +17.7310i q^{34} +7.09100 q^{35} +6.00000 q^{36} +52.7319i q^{37} +38.4625i q^{38} +20.1628 q^{39} +4.79373i q^{40} +42.4390 q^{41} +10.2484i q^{42} -2.10282i q^{43} -40.5940i q^{44} -5.08451i q^{45} +(-6.32049 + 31.9069i) q^{46} +20.0260 q^{47} -6.92820 q^{48} +31.4951 q^{49} -31.2930 q^{50} +21.7160i q^{51} -23.2820 q^{52} +49.6304i q^{53} +7.34847 q^{54} -34.4001 q^{55} -11.8338i q^{56} +47.1067i q^{57} +72.0384 q^{58} -39.9319 q^{59} +5.87109i q^{60} -70.9635i q^{61} +16.3669 q^{62} +12.5516i q^{63} +8.00000 q^{64} +19.7296i q^{65} -49.7173i q^{66} -4.38918i q^{67} -25.0755i q^{68} +(-7.74099 + 39.0778i) q^{69} -10.0282 q^{70} -46.5698 q^{71} -8.48528 q^{72} -3.71985 q^{73} -74.5741i q^{74} -38.3260 q^{75} -54.3941i q^{76} +84.9203 q^{77} -28.5145 q^{78} -26.4542i q^{79} -6.77935i q^{80} +9.00000 q^{81} -60.0178 q^{82} -106.806i q^{83} -14.4934i q^{84} -21.2494 q^{85} +2.97384i q^{86} +88.2287 q^{87} +57.4086i q^{88} +137.125i q^{89} +7.19059i q^{90} -48.7046i q^{91} +(8.93852 - 45.1232i) q^{92} +20.0453 q^{93} -28.3210 q^{94} -46.0946 q^{95} +9.79796 q^{96} -119.514i q^{97} -44.5408 q^{98} -60.8910i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 + 24 * q^9	\( 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) \) 8 * q + 16 * q^4 + 24 * q^9 + 16 * q^13 + 32 * q^16 + 16 * q^23 + 72 * q^25 + 32 * q^26 - 144 * q^29 - 128 * q^31 - 112 * q^35 + 48 * q^36 + 48 * q^39 - 16 * q^41 - 80 * q^46 - 112 * q^47 + 40 * q^49 - 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 + 80 * q^59 - 96 * q^62 + 64 * q^64 - 72 * q^69 - 144 * q^70 + 32 * q^71 + 64 * q^73 + 48 * q^75 + 224 * q^77 - 144 * q^78 + 72 * q^81 + 48 * q^85 + 96 * q^87 + 32 * q^92 + 192 * q^93 - 16 * q^94 + 112 * q^95 + 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more