LMFDB - Embedded newform 99.5.c.c.10.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,5,Mod(10,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("99.10");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	99.c (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:	$10.2336263453$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 $ x^8 + 102x^6 + 2913x^4 + 23292*x^2 + 41364
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			10.5
Root			$1.57474i$ of defining polynomial
Character	$\chi$	$=$	99.10
Dual form			99.5.c.c.10.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57474i q^{2} +13.5202 q^{4} -15.5526 q^{5} -93.8006i q^{7} +46.4867i q^{8} +O(q^{10})$	\(q+1.57474i q^{2} +13.5202 q^{4} -15.5526 q^{5} -93.8006i q^{7} +46.4867i q^{8} -24.4913i q^{10} +(-60.9369 - 104.536i) q^{11} +29.4527i q^{13} +147.712 q^{14} +143.119 q^{16} -251.915i q^{17} -80.2497i q^{19} -210.274 q^{20} +(164.617 - 95.9599i) q^{22} +702.557 q^{23} -383.118 q^{25} -46.3804 q^{26} -1268.20i q^{28} -1449.00i q^{29} +1279.46 q^{31} +969.161i q^{32} +396.701 q^{34} +1458.84i q^{35} +115.552 q^{37} +126.373 q^{38} -722.987i q^{40} +1076.75i q^{41} +1887.47i q^{43} +(-823.879 - 1413.34i) q^{44} +1106.35i q^{46} -1591.75 q^{47} -6397.55 q^{49} -603.311i q^{50} +398.206i q^{52} -1190.69 q^{53} +(947.726 + 1625.80i) q^{55} +4360.48 q^{56} +2281.80 q^{58} -1895.56 q^{59} +3776.15i q^{61} +2014.82i q^{62} +763.719 q^{64} -458.065i q^{65} -1253.00 q^{67} -3405.94i q^{68} -2297.30 q^{70} +3591.39 q^{71} +1750.75i q^{73} +181.965i q^{74} -1084.99i q^{76} +(-9805.50 + 5715.92i) q^{77} -6749.08i q^{79} -2225.86 q^{80} -1695.60 q^{82} +8616.78i q^{83} +3917.92i q^{85} -2972.27 q^{86} +(4859.51 - 2832.75i) q^{88} +9334.05 q^{89} +2762.68 q^{91} +9498.71 q^{92} -2506.59i q^{94} +1248.09i q^{95} +11068.8 q^{97} -10074.5i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) $ 8 * q - 76 * q^4 + 36 * q^5	$ 8 q - 76 q^{4} + 36 q^{5} - 36 q^{11} + 1140 q^{14} + 1412 q^{16} - 2532 q^{20} - 780 q^{22} - 516 q^{23} - 2280 q^{25} + 1524 q^{26} + 2752 q^{31} - 4920 q^{34} + 5296 q^{37} - 696 q^{38} + 6540 q^{44} - 420 q^{47} - 6832 q^{49} - 3540 q^{53} + 3784 q^{55} - 17964 q^{56} + 21624 q^{58} + 16632 q^{59} - 27508 q^{64} - 3656 q^{67} + 3312 q^{70} + 13212 q^{71} - 23268 q^{77} + 4476 q^{80} + 17088 q^{82} - 19896 q^{86} - 12516 q^{88} - 15528 q^{89} - 19752 q^{91} + 81180 q^{92} + 7624 q^{97}+O(q^{100}) $ 8 * q - 76 * q^4 + 36 * q^5 - 36 * q^11 + 1140 * q^14 + 1412 * q^16 - 2532 * q^20 - 780 * q^22 - 516 * q^23 - 2280 * q^25 + 1524 * q^26 + 2752 * q^31 - 4920 * q^34 + 5296 * q^37 - 696 * q^38 + 6540 * q^44 - 420 * q^47 - 6832 * q^49 - 3540 * q^53 + 3784 * q^55 - 17964 * q^56 + 21624 * q^58 + 16632 * q^59 - 27508 * q^64 - 3656 * q^67 + 3312 * q^70 + 13212 * q^71 - 23268 * q^77 + 4476 * q^80 + 17088 * q^82 - 19896 * q^86 - 12516 * q^88 - 15528 * q^89 - 19752 * q^91 + 81180 * q^92 + 7624 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.57474i			0.393685i	0.980435	+	0.196843i	$0.0630688\pi$
$2$			1.57474i			0.393685i	−0.980435	+	0.196843i	$0.936931\pi$
$3$		0			0
$3$		0			0
$4$	13.5202			0.845012
$5$	−15.5526			−0.622103			−0.311051	−	0.950393i	$-0.600681\pi$
$5$	−15.5526			−0.622103			−0.311051	+	0.950393i	$0.600681\pi$
$6$		0			0
$7$		−	93.8006i		−	1.91430i	−0.289596	−	0.957149i	$-0.593521\pi$
$7$		−	93.8006i		−	1.91430i	0.289596	−	0.957149i	$-0.406479\pi$
$8$			46.4867i			0.726354i
$9$		0			0
$10$		−	24.4913i		−	0.244913i
$11$	−60.9369	−	104.536i	−0.503611	−	0.863931i
$11$	−60.9369	−	104.536i	−0.503611	−	0.863931i
$12$		0			0
$13$			29.4527i			0.174276i	0.996196	+	0.0871382i	$0.0277722\pi$
$13$			29.4527i			0.174276i	−0.996196	+	0.0871382i	$0.972228\pi$
$14$	147.712			0.753631
$15$		0			0
$16$	143.119			0.559057
$17$		−	251.915i		−	0.871677i	−0.900025	−	0.435839i	$-0.856452\pi$
$17$		−	251.915i		−	0.871677i	0.900025	−	0.435839i	$-0.143548\pi$
$18$		0			0
$19$		−	80.2497i		−	0.222298i	−0.993804	−	0.111149i	$-0.964547\pi$
$19$		−	80.2497i		−	0.222298i	0.993804	−	0.111149i	$-0.0354531\pi$
$20$	−210.274			−0.525684
$21$		0			0
$22$	164.617	−	95.9599i	0.340117	−	0.198264i
$23$	702.557			1.32809			0.664043	−	0.747695i	$-0.268839\pi$
$23$	702.557			1.32809			0.664043	+	0.747695i	$0.268839\pi$
$24$		0			0
$25$	−383.118			−0.612988
$26$	−46.3804			−0.0686100
$27$		0			0
$28$		−	1268.20i		−	1.61760i
$29$		−	1449.00i		−	1.72295i	−0.507798	−	0.861476i	$-0.669540\pi$
$29$		−	1449.00i		−	1.72295i	0.507798	−	0.861476i	$-0.330460\pi$
$30$		0			0
$31$	1279.46			1.33138			0.665692	−	0.746227i	$-0.268137\pi$
$31$	1279.46			1.33138			0.665692	+	0.746227i	$0.268137\pi$
$32$			969.161i			0.946447i
$33$		0			0
$34$	396.701			0.343167
$35$			1458.84i			1.19089i
$36$		0			0
$37$	115.552			0.0844063			0.0422031	−	0.999109i	$-0.486562\pi$
$37$	115.552			0.0844063			0.0422031	+	0.999109i	$0.486562\pi$
$38$	126.373			0.0875156
$39$		0			0
$40$		−	722.987i		−	0.451867i
$41$			1076.75i			0.640541i	0.947326	+	0.320270i	$0.103774\pi$
$41$			1076.75i			0.640541i	−0.947326	+	0.320270i	$0.896226\pi$
$42$		0			0
$43$			1887.47i			1.02080i	0.859936	+	0.510402i	$0.170503\pi$
$43$			1887.47i			1.02080i	−0.859936	+	0.510402i	$0.829497\pi$
$44$	−823.879	−	1413.34i	−0.425557	−	0.730032i
$45$		0			0
$46$			1106.35i			0.522848i
$47$	−1591.75			−0.720575			−0.360287	−	0.932841i	$-0.617321\pi$
$47$	−1591.75			−0.720575			−0.360287	+	0.932841i	$0.617321\pi$
$48$		0			0
$49$	−6397.55			−2.66454
$50$		−	603.311i		−	0.241324i
$51$		0			0
$52$			398.206i			0.147266i
$53$	−1190.69			−0.423886			−0.211943	−	0.977282i	$-0.567979\pi$
$53$	−1190.69			−0.423886			−0.211943	+	0.977282i	$0.567979\pi$
$54$		0			0
$55$	947.726	+	1625.80i	0.313298	+	0.537454i
$56$	4360.48			1.39046
$57$		0			0
$58$	2281.80			0.678301
$59$	−1895.56			−0.544544			−0.272272	−	0.962220i	$-0.587775\pi$
$59$	−1895.56			−0.544544			−0.272272	+	0.962220i	$0.587775\pi$
$60$		0			0
$61$			3776.15i			1.01482i	0.861704	+	0.507411i	$0.169397\pi$
$61$			3776.15i			1.01482i	−0.861704	+	0.507411i	$0.830603\pi$
$62$			2014.82i			0.524146i
$63$		0			0
$64$	763.719			0.186455
$65$		−	458.065i		−	0.108418i
$66$		0			0
$67$	−1253.00			−0.279127			−0.139564	−	0.990213i	$-0.544570\pi$
$67$	−1253.00			−0.279127			−0.139564	+	0.990213i	$0.544570\pi$
$68$		−	3405.94i		−	0.736578i
$69$		0			0
$70$	−2297.30			−0.468836
$71$	3591.39			0.712437			0.356218	−	0.934403i	$-0.384066\pi$
$71$	3591.39			0.712437			0.356218	+	0.934403i	$0.384066\pi$
$72$		0			0
$73$			1750.75i			0.328533i	0.986416	+	0.164267i	$0.0525257\pi$
$73$			1750.75i			0.328533i	−0.986416	+	0.164267i	$0.947474\pi$
$74$			181.965i			0.0332295i
$75$		0			0
$76$		−	1084.99i		−	0.187845i
$77$	−9805.50	+	5715.92i	−1.65382	+	0.964061i
$78$		0			0
$79$		−	6749.08i		−	1.08141i	−0.841212	−	0.540705i	$-0.818157\pi$
$79$		−	6749.08i		−	1.08141i	0.841212	−	0.540705i	$-0.181843\pi$
$80$	−2225.86			−0.347791
$81$		0			0
$82$	−1695.60			−0.252171
$83$			8616.78i			1.25080i	0.780303	+	0.625401i	$0.215065\pi$
$83$			8616.78i			1.25080i	−0.780303	+	0.625401i	$0.784935\pi$
$84$		0			0
$85$			3917.92i			0.542273i
$86$	−2972.27			−0.401876
$87$		0			0
$88$	4859.51	−	2832.75i	0.627520	−	0.365800i
$89$	9334.05			1.17839			0.589196	−	0.807990i	$-0.299444\pi$
$89$	9334.05			1.17839			0.589196	+	0.807990i	$0.299444\pi$
$90$		0			0
$91$	2762.68			0.333617
$92$	9498.71			1.12225
$93$		0			0
$94$		−	2506.59i		−	0.283680i
$95$			1248.09i			0.138292i
$96$		0			0
$97$	11068.8			1.17640			0.588202	−	0.808714i	$-0.299836\pi$
$97$	11068.8			1.17640			0.588202	+	0.808714i	$0.299836\pi$
$98$		−	10074.5i		−	1.04899i
$99$		0			0
$100$	−5179.82			−0.517982
$101$		−	9732.30i		−	0.954053i	−0.878889	−	0.477027i	$-0.841715\pi$
$101$		−	9732.30i		−	0.954053i	0.878889	−	0.477027i	$-0.158285\pi$
$102$		0			0
$103$	2229.35			0.210138			0.105069	−	0.994465i	$-0.466494\pi$
$103$	2229.35			0.210138			0.105069	+	0.994465i	$0.466494\pi$
$104$	−1369.16			−0.126586
$105$		0			0
$106$		−	1875.04i		−	0.166878i
$107$		−	8911.94i		−	0.778404i	−0.921152	−	0.389202i	$-0.872751\pi$
$107$		−	8911.94i		−	0.778404i	0.921152	−	0.389202i	$-0.127249\pi$
$108$		0			0
$109$		−	8075.42i		−	0.679692i	−0.940481	−	0.339846i	$-0.889625\pi$
$109$		−	8075.42i		−	0.679692i	0.940481	−	0.339846i	$-0.110375\pi$
$110$	−2560.21	+	1492.42i	−0.211588	+	0.123341i
$111$		0			0
$112$		−	13424.6i		−	1.07020i
$113$	16548.3			1.29597			0.647987	−	0.761652i	$-0.275611\pi$
$113$	16548.3			1.29597			0.647987	+	0.761652i	$0.275611\pi$
$114$		0			0
$115$	−10926.6			−0.826206
$116$		−	19590.8i		−	1.45592i
$117$		0			0
$118$		−	2985.01i		−	0.214379i
$119$	−23629.8			−1.66865
$120$		0			0
$121$	−7214.38	+	12740.2i	−0.492752	+	0.870170i
$122$	−5946.46			−0.399520
$123$		0			0
$124$	17298.5			1.12503
$125$	15678.8			1.00344
$126$		0			0
$127$			2165.25i			0.134246i	0.997745	+	0.0671228i	$0.0213819\pi$
$127$			2165.25i			0.134246i	−0.997745	+	0.0671228i	$0.978618\pi$
$128$			16709.2i			1.01985i
$129$		0			0
$130$	721.334			0.0426825
$131$			15267.4i			0.889658i	0.895616	+	0.444829i	$0.146736\pi$
$131$			15267.4i			0.889658i	−0.895616	+	0.444829i	$0.853264\pi$
$132$		0			0
$133$	−7527.47			−0.425545
$134$		−	1973.16i		−	0.109888i
$135$		0			0
$136$	11710.7			0.633146
$137$	−10391.2			−0.553636			−0.276818	−	0.960922i	$-0.589280\pi$
$137$	−10391.2			−0.553636			−0.276818	+	0.960922i	$0.589280\pi$
$138$		0			0
$139$		−	21983.9i		−	1.13782i	−0.822399	−	0.568911i	$-0.807365\pi$
$139$		−	21983.9i		−	1.13782i	0.822399	−	0.568911i	$-0.192635\pi$
$140$			19723.8i			1.00632i
$141$		0			0
$142$			5655.52i			0.280476i
$143$	3078.86	−	1794.76i	0.150563	−	0.0877675i
$144$		0			0
$145$			22535.7i			1.07185i
$146$	−2756.98			−0.129339
$147$		0			0
$148$	1562.29			0.0713243
$149$			10316.9i			0.464705i	0.972632	+	0.232353i	$0.0746423\pi$
$149$			10316.9i			0.464705i	−0.972632	+	0.232353i	$0.925358\pi$
$150$		0			0
$151$			20057.9i			0.879694i	0.898073	+	0.439847i	$0.144967\pi$
$151$			20057.9i			0.879694i	−0.898073	+	0.439847i	$0.855033\pi$
$152$	3730.54			0.161467
$153$		0			0
$154$	−9001.10	−	15441.1i	−0.379537	−	0.651085i
$155$	−19898.9			−0.828258
$156$		0			0
$157$	−5742.18			−0.232958			−0.116479	−	0.993193i	$-0.537161\pi$
$157$	−5742.18			−0.232958			−0.116479	+	0.993193i	$0.537161\pi$
$158$	10628.1			0.425735
$159$		0			0
$160$		−	15073.0i		−	0.588787i
$161$		−	65900.3i		−	2.54235i
$162$		0			0
$163$	−24660.6			−0.928174			−0.464087	−	0.885790i	$-0.653617\pi$
$163$	−24660.6			−0.928174			−0.464087	+	0.885790i	$0.653617\pi$
$164$			14557.8i			0.541265i
$165$		0			0
$166$	−13569.2			−0.492423
$167$			47391.6i			1.69930i	0.527351	+	0.849648i	$0.323185\pi$
$167$			47391.6i			1.69930i	−0.527351	+	0.849648i	$0.676815\pi$
$168$		0			0
$169$	27693.5			0.969628
$170$	−6169.71			−0.213485
$171$		0			0
$172$			25518.9i			0.862591i
$173$			1326.29i			0.0443147i	0.999754	+	0.0221573i	$0.00705347\pi$
$173$			1326.29i			0.0443147i	−0.999754	+	0.0221573i	$0.992947\pi$
$174$		0			0
$175$			35936.7i			1.17344i
$176$	−8721.21	−	14961.0i	−0.281547	−	0.482986i
$177$		0			0
$178$			14698.7i			0.463916i
$179$	53822.7			1.67981			0.839903	−	0.542736i	$-0.182612\pi$
$179$	53822.7			1.67981			0.839903	+	0.542736i	$0.182612\pi$
$180$		0			0
$181$	9961.35			0.304061			0.152031	−	0.988376i	$-0.451419\pi$
$181$	9961.35			0.304061			0.152031	+	0.988376i	$0.451419\pi$
$182$			4350.51i			0.131340i
$183$		0			0
$184$			32659.5i			0.964660i
$185$	−1797.13			−0.0525094
$186$		0			0
$187$	−26334.1	+	15350.9i	−0.753069	+	0.438986i
$188$	−21520.8			−0.608894
$189$		0			0
$190$	−1965.42			−0.0544437
$191$	54964.8			1.50667			0.753335	−	0.657637i	$-0.228444\pi$
$191$	54964.8			1.50667			0.753335	+	0.657637i	$0.228444\pi$
$192$		0			0
$193$		−	1722.76i		−	0.0462498i	−0.999733	−	0.0231249i	$-0.992638\pi$
$193$		−	1722.76i		−	0.0462498i	0.999733	−	0.0231249i	$-0.00736154\pi$
$194$			17430.5i			0.463133i
$195$		0			0
$196$	−86496.1			−2.25156
$197$			10387.5i			0.267656i	0.991005	+	0.133828i	$0.0427270\pi$
$197$			10387.5i			0.267656i	−0.991005	+	0.133828i	$0.957273\pi$
$198$		0			0
$199$	50964.3			1.28695			0.643473	−	0.765469i	$-0.277493\pi$
$199$	50964.3			1.28695			0.643473	+	0.765469i	$0.277493\pi$
$200$		−	17809.9i		−	0.445246i
$201$		0			0
$202$	15325.9			0.375597
$203$	−135917.			−3.29824
$204$		0			0
$205$		−	16746.2i		−	0.398482i
$206$			3510.66i			0.0827283i
$207$		0			0
$208$			4215.23i			0.0974304i
$209$	−8388.95	+	4890.17i	−0.192050	+	0.111952i
$210$		0			0
$211$		−	68681.5i		−	1.54268i	−0.636426	−	0.771338i	$-0.719588\pi$
$211$		−	68681.5i		−	1.54268i	0.636426	−	0.771338i	$-0.280412\pi$
$212$	−16098.4			−0.358188
$213$		0			0
$214$	14034.0			0.306446
$215$		−	29355.0i		−	0.635045i
$216$		0			0
$217$		−	120014.i		−	2.54866i
$218$	12716.7			0.267585
$219$		0			0
$220$	12813.4	+	21981.1i	0.264740	+	0.454155i
$221$	7419.57			0.151913
$222$		0			0
$223$	−1628.89			−0.0327553			−0.0163777	−	0.999866i	$-0.505213\pi$
$223$	−1628.89			−0.0327553			−0.0163777	+	0.999866i	$0.505213\pi$
$224$	90907.9			1.81178
$225$		0			0
$226$			26059.3i			0.510206i
$227$			42599.1i			0.826701i	0.910572	+	0.413351i	$0.135642\pi$
$227$			42599.1i			0.826701i	−0.910572	+	0.413351i	$0.864358\pi$
$228$		0			0
$229$	32387.8			0.617605			0.308802	−	0.951126i	$-0.400072\pi$
$229$	32387.8			0.617605			0.308802	+	0.951126i	$0.400072\pi$
$230$		−	17206.5i		−	0.325265i
$231$		0			0
$232$	67359.3			1.25147
$233$		−	57867.1i		−	1.06591i	−0.846144	−	0.532954i	$-0.821082\pi$
$233$		−	57867.1i		−	1.06591i	0.846144	−	0.532954i	$-0.178918\pi$
$234$		0			0
$235$	24755.8			0.448272
$236$	−25628.3			−0.460146
$237$		0			0
$238$		−	37210.8i		−	0.656923i
$239$		−	3758.46i		−	0.0657982i	−0.999459	−	0.0328991i	$-0.989526\pi$
$239$		−	3758.46i		−	0.0657982i	0.999459	−	0.0328991i	$-0.0104740\pi$
$240$		0			0
$241$		−	53393.2i		−	0.919288i	−0.888103	−	0.459644i	$-0.847977\pi$
$241$		−	53393.2i		−	0.919288i	0.888103	−	0.459644i	$-0.152023\pi$
$242$	−20062.5	−	11360.8i	−0.342573	−	0.193989i
$243$		0			0
$244$			51054.3i			0.857536i
$245$	99498.4			1.65762
$246$		0			0
$247$	2363.57			0.0387414
$248$			59477.8i			0.967056i
$249$		0			0
$250$			24690.1i			0.395041i
$251$	−1857.31			−0.0294807			−0.0147403	−	0.999891i	$-0.504692\pi$
$251$	−1857.31			−0.0294807			−0.0147403	+	0.999891i	$0.504692\pi$
$252$		0			0
$253$	−42811.7	−	73442.2i	−0.668838	−	1.14737i
$254$	−3409.70			−0.0528505
$255$		0			0
$256$	−14093.2			−0.215046
$257$	1262.77			0.0191187			0.00955933	−	0.999954i	$-0.496957\pi$
$257$	1262.77			0.0191187			0.00955933	+	0.999954i	$0.496957\pi$
$258$		0			0
$259$		−	10838.9i		−	0.161579i
$260$		−	6193.13i		−	0.0916143i
$261$		0			0
$262$	−24042.2			−0.350245
$263$		−	118343.i		−	1.71092i	−0.517870	−	0.855459i	$-0.673275\pi$
$263$		−	118343.i		−	1.71092i	0.517870	−	0.855459i	$-0.326725\pi$
$264$		0			0
$265$	18518.4			0.263700
$266$		−	11853.8i		−	0.167531i
$267$		0			0
$268$	−16940.8			−0.235866
$269$	−100514.			−1.38906			−0.694528	−	0.719465i	$-0.744387\pi$
$269$	−100514.			−1.38906			−0.694528	+	0.719465i	$0.744387\pi$
$270$		0			0
$271$			138354.i			1.88389i	0.335773	+	0.941943i	$0.391003\pi$
$271$			138354.i			1.88389i	−0.335773	+	0.941943i	$0.608997\pi$
$272$		−	36053.7i		−	0.487317i
$273$		0			0
$274$		−	16363.5i		−	0.217958i
$275$	23346.0	+	40049.4i	0.308707	+	0.529579i
$276$		0			0
$277$		−	40696.2i		−	0.530388i	−0.964195	−	0.265194i	$-0.914564\pi$
$277$		−	40696.2i		−	0.530388i	0.964195	−	0.265194i	$-0.0854360\pi$
$278$	34618.9			0.447944
$279$		0			0
$280$	−67816.6			−0.865008
$281$			17381.6i			0.220129i	0.993924	+	0.110065i	$0.0351058\pi$
$281$			17381.6i			0.220129i	−0.993924	+	0.110065i	$0.964894\pi$
$282$		0			0
$283$			102799.i			1.28355i	0.766891	+	0.641777i	$0.221802\pi$
$283$			102799.i			1.28355i	−0.766891	+	0.641777i	$0.778198\pi$
$284$	48556.3			0.602018
$285$		0			0
$286$	2826.28	+	4848.40i	0.0345528	+	0.0592743i
$287$	101000.			1.22619
$288$		0			0
$289$	20059.9			0.240178
$290$	−35487.9			−0.421973
$291$		0			0
$292$			23670.5i			0.277614i
$293$			106779.i			1.24380i	0.783098	+	0.621898i	$0.213638\pi$
$293$			106779.i			1.24380i	−0.783098	+	0.621898i	$0.786362\pi$
$294$		0			0
$295$	29480.8			0.338762
$296$			5371.64i			0.0613088i
$297$		0			0
$298$	−16246.5			−0.182948
$299$			20692.2i			0.231454i
$300$		0			0
$301$	177045.			1.95412
$302$	−31586.0			−0.346322
$303$		0			0
$304$		−	11485.2i		−	0.124277i
$305$		−	58728.8i		−	0.631323i
$306$		0			0
$307$		−	78416.1i		−	0.832010i	−0.909362	−	0.416005i	$-0.863430\pi$
$307$		−	78416.1i		−	0.832010i	0.909362	−	0.416005i	$-0.136570\pi$
$308$	−132572.	+	77280.3i	−1.39750	+	0.814643i
$309$		0			0
$310$		−	31335.6i		−	0.326073i
$311$	60684.2			0.627415			0.313707	−	0.949520i	$-0.398429\pi$
$311$	60684.2			0.627415			0.313707	+	0.949520i	$0.398429\pi$
$312$		0			0
$313$	−85739.0			−0.875165			−0.437582	−	0.899178i	$-0.644165\pi$
$313$	−85739.0			−0.875165			−0.437582	+	0.899178i	$0.644165\pi$
$314$		−	9042.45i		−	0.0917121i
$315$		0			0
$316$		−	91248.9i		−	0.913804i
$317$	−117644.			−1.17071			−0.585357	−	0.810776i	$-0.699045\pi$
$317$	−117644.			−1.17071			−0.585357	+	0.810776i	$0.699045\pi$
$318$		0			0
$319$	−151472.	+	88297.8i	−1.48851	+	0.867698i
$320$	−11877.8			−0.115994
$321$		0			0
$322$	103776.			1.00089
$323$	−20216.1			−0.193773
$324$		0			0
$325$		−	11283.8i		−	0.106829i
$326$		−	38834.1i		−	0.365408i
$327$		0			0
$328$	−50054.5			−0.465259
$329$			149307.i			1.37940i
$330$		0			0
$331$	−109718.			−1.00143			−0.500717	−	0.865611i	$-0.666930\pi$
$331$	−109718.			−1.00143			−0.500717	+	0.865611i	$0.666930\pi$
$332$			116501.i			1.05694i
$333$		0			0
$334$	−74629.6			−0.668988
$335$	19487.4			0.173646
$336$		0			0
$337$		−	73221.5i		−	0.644731i	−0.946615	−	0.322365i	$-0.895522\pi$
$337$		−	73221.5i		−	0.644731i	0.946615	−	0.322365i	$-0.104478\pi$
$338$			43610.2i			0.381728i
$339$		0			0
$340$			52971.1i			0.458227i
$341$	−77966.3	−	133749.i	−0.670499	−	1.15022i
$342$		0			0
$343$			374879.i			3.18642i
$344$	−87742.0			−0.741465
$345$		0			0
$346$	−2088.57			−0.0174460
$347$		−	160256.i		−	1.33093i	−0.746428	−	0.665466i	$-0.768233\pi$
$347$		−	160256.i		−	1.33093i	0.746428	−	0.665466i	$-0.231767\pi$
$348$		0			0
$349$		−	24791.6i		−	0.203542i	−0.994808	−	0.101771i	$-0.967549\pi$
$349$		−	24791.6i		−	0.203542i	0.994808	−	0.101771i	$-0.0324509\pi$
$350$	−56590.9			−0.461967
$351$		0			0
$352$	101312.	−	59057.7i	0.817664	−	0.476641i
$353$	140695.			1.12909			0.564547	−	0.825401i	$-0.309051\pi$
$353$	140695.			1.12909			0.564547	+	0.825401i	$0.309051\pi$
$354$		0			0
$355$	−55855.4			−0.443209
$356$	126198.			0.995756
$357$		0			0
$358$			84756.8i			0.661315i
$359$		−	14603.5i		−	0.113310i	−0.998394	−	0.0566552i	$-0.981956\pi$
$359$		−	14603.5i		−	0.113310i	0.998394	−	0.0566552i	$-0.0180436\pi$
$360$		0			0
$361$	123881.			0.950583
$362$			15686.5i			0.119704i
$363$		0			0
$364$	37352.0			0.281910
$365$		−	27228.7i		−	0.204381i
$366$		0			0
$367$	−168034.			−1.24757			−0.623785	−	0.781596i	$-0.714406\pi$
$367$	−168034.			−1.24757			−0.623785	+	0.781596i	$0.714406\pi$
$368$	100549.			0.742475
$369$		0			0
$370$		−	2830.02i		−	0.0206722i
$371$			111688.i			0.811443i
$372$		0			0
$373$			179739.i			1.29189i	0.763385	+	0.645944i	$0.223536\pi$
$373$			179739.i			1.29189i	−0.763385	+	0.645944i	$0.776464\pi$
$374$	−24173.7	−	41469.3i	−0.172822	−	0.296472i
$375$		0			0
$376$		−	73995.2i		−	0.523393i
$377$	42677.0			0.300270
$378$		0			0
$379$	−88424.0			−0.615590			−0.307795	−	0.951453i	$-0.599591\pi$
$379$	−88424.0			−0.615590			−0.307795	+	0.951453i	$0.599591\pi$
$380$			16874.4i			0.116859i
$381$		0			0
$382$			86555.4i			0.593154i
$383$	−95956.7			−0.654151			−0.327075	−	0.944998i	$-0.606063\pi$
$383$	−95956.7			−0.654151			−0.327075	+	0.944998i	$0.606063\pi$
$384$		0			0
$385$	152501.	−	88897.3i	1.02885	−	0.599745i
$386$	2712.90			0.0182079
$387$		0			0
$388$	149652.			0.994075
$389$	118749.			0.784747			0.392373	−	0.919806i	$-0.371654\pi$
$389$	118749.			0.784747			0.392373	+	0.919806i	$0.371654\pi$
$390$		0			0
$391$		−	176985.i		−	1.15766i
$392$		−	297401.i		−	1.93540i
$393$		0			0
$394$	−16357.6			−0.105372
$395$			104966.i			0.672748i
$396$		0			0
$397$	−159427.			−1.01154			−0.505768	−	0.862670i	$-0.668791\pi$
$397$	−159427.			−1.01154			−0.505768	+	0.862670i	$0.668791\pi$
$398$			80255.6i			0.506652i
$399$		0			0
$400$	−54831.2			−0.342695
$401$	41646.5			0.258994			0.129497	−	0.991580i	$-0.458664\pi$
$401$	41646.5			0.258994			0.129497	+	0.991580i	$0.458664\pi$
$402$		0			0
$403$			37683.5i			0.232029i
$404$		−	131583.i		−	0.806186i
$405$		0			0
$406$		−	214035.i		−	1.29847i
$407$	−7041.40	−	12079.3i	−0.0425079	−	0.0729212i
$408$		0			0
$409$		−	265054.i		−	1.58449i	−0.610206	−	0.792243i	$-0.708913\pi$
$409$		−	265054.i		−	1.58449i	0.610206	−	0.792243i	$-0.291087\pi$
$410$	26371.0			0.156877
$411$		0			0
$412$	30141.3			0.177569
$413$			177804.i			1.04242i
$414$		0			0
$415$		−	134013.i		−	0.778128i
$416$	−28544.4			−0.164943
$417$		0			0
$418$	−7700.76	−	13210.4i	−0.0440738	−	0.0756074i
$419$	−255711.			−1.45654			−0.728270	−	0.685291i	$-0.759675\pi$
$419$	−255711.			−1.45654			−0.728270	+	0.685291i	$0.759675\pi$
$420$		0			0
$421$	189209.			1.06753			0.533763	−	0.845634i	$-0.320778\pi$
$421$	189209.			1.06753			0.533763	+	0.845634i	$0.320778\pi$
$422$	108156.			0.607329
$423$		0			0
$424$		−	55351.4i		−	0.307891i
$425$			96513.0i			0.534328i
$426$		0			0
$427$	354205.			1.94267
$428$		−	120491.i		−	0.657760i
$429$		0			0
$430$	46226.5			0.250008
$431$			110749.i			0.596190i	0.954536	+	0.298095i	$0.0963512\pi$
$431$			110749.i			0.596190i	−0.954536	+	0.298095i	$0.903649\pi$
$432$		0			0
$433$	172877.			0.922062			0.461031	−	0.887384i	$-0.347480\pi$
$433$	172877.			0.922062			0.461031	+	0.887384i	$0.347480\pi$
$434$	188991.			1.00337
$435$		0			0
$436$		−	109181.i		−	0.574348i
$437$		−	56380.0i		−	0.295231i
$438$		0			0
$439$			155214.i			0.805383i	0.915336	+	0.402691i	$0.131925\pi$
$439$			155214.i			0.805383i	−0.915336	+	0.402691i	$0.868075\pi$
$440$	−75577.9	+	44056.6i	−0.390382	+	0.227565i
$441$		0			0
$442$			11683.9i			0.0598058i
$443$	−104175.			−0.530828			−0.265414	−	0.964134i	$-0.585509\pi$
$443$	−104175.			−0.530828			−0.265414	+	0.964134i	$0.585509\pi$
$444$		0			0
$445$	−145168.			−0.733082
$446$		−	2565.08i		−	0.0128953i
$447$		0			0
$448$		−	71637.3i		−	0.356930i
$449$	−88086.4			−0.436934			−0.218467	−	0.975844i	$-0.570106\pi$
$449$	−88086.4			−0.436934			−0.218467	+	0.975844i	$0.570106\pi$
$450$		0			0
$451$	112559.	−	65613.8i	0.553383	−	0.322583i
$452$	223736.			1.09511
$453$		0			0
$454$	−67082.5			−0.325460
$455$	−42966.8			−0.207544
$456$		0			0
$457$			77946.3i			0.373219i	0.982434	+	0.186609i	$0.0597499\pi$
$457$			77946.3i			0.373219i	−0.982434	+	0.186609i	$0.940250\pi$
$458$			51002.4i			0.243142i
$459$		0			0
$460$	−147729.			−0.698154
$461$		−	115963.i		−	0.545653i	−0.962063	−	0.272827i	$-0.912041\pi$
$461$		−	115963.i		−	0.545653i	0.962063	−	0.272827i	$-0.0879586\pi$
$462$		0			0
$463$	−313839.			−1.46401			−0.732007	−	0.681297i	$-0.761416\pi$
$463$	−313839.			−1.46401			−0.732007	+	0.681297i	$0.761416\pi$
$464$		−	207379.i		−	0.963228i
$465$		0			0
$466$	91125.7			0.419632
$467$	96033.3			0.440340			0.220170	−	0.975462i	$-0.429339\pi$
$467$	96033.3			0.440340			0.220170	+	0.975462i	$0.429339\pi$
$468$		0			0
$469$			117532.i			0.534333i
$470$			38984.0i			0.176478i
$471$		0			0
$472$		−	88118.1i		−	0.395532i
$473$	197307.	−	115016.i	0.881904	−	0.514088i
$474$		0			0
$475$			30745.1i			0.136266i
$476$	−319479.			−1.41003
$477$		0			0
$478$	5918.60			0.0259038
$479$		−	215693.i		−	0.940082i	−0.882645	−	0.470041i	$-0.844239\pi$
$479$		−	215693.i		−	0.940082i	0.882645	−	0.470041i	$-0.155761\pi$
$480$		0			0
$481$			3403.32i			0.0147100i
$482$	84080.5			0.361910
$483$		0			0
$484$	−97539.8	+	172249.i	−0.416381	+	0.735304i
$485$	−172148.			−0.731844
$486$		0			0
$487$	−281250.			−1.18587			−0.592933	−	0.805252i	$-0.702030\pi$
$487$	−281250.			−1.18587			−0.592933	+	0.805252i	$0.702030\pi$
$488$	−175541.			−0.737120
$489$		0			0
$490$			156684.i			0.652579i
$491$			229094.i			0.950279i	0.879910	+	0.475139i	$0.157602\pi$
$491$			229094.i			0.950279i	−0.879910	+	0.475139i	$0.842398\pi$
$492$		0			0
$493$	−365025.			−1.50186
$494$			3722.01i			0.0152519i
$495$		0			0
$496$	183114.			0.744319
$497$		−	336875.i		−	1.36382i
$498$		0			0
$499$	37652.6			0.151215			0.0756073	−	0.997138i	$-0.475910\pi$
$499$	37652.6			0.151215			0.0756073	+	0.997138i	$0.475910\pi$
$500$	211981.			0.847922
$501$		0			0
$502$		−	2924.79i		−	0.0116061i
$503$			353544.i			1.39736i	0.715435	+	0.698679i	$0.246228\pi$
$503$			353544.i			1.39736i	−0.715435	+	0.698679i	$0.753772\pi$
$504$		0			0
$505$			151362.i			0.593519i
$506$	115653.	−	67417.3i	0.451704	−	0.263312i
$507$		0			0
$508$			29274.6i			0.113439i
$509$	240356.			0.927727			0.463864	−	0.885907i	$-0.346463\pi$
$509$	240356.			0.927727			0.463864	+	0.885907i	$0.346463\pi$
$510$		0			0
$511$	164222.			0.628910
$512$			245155.i			0.935191i
$513$		0			0
$514$			1988.53i			0.00752673i
$515$	−34672.2			−0.130727
$516$		0			0
$517$	96996.4	+	166395.i	0.362889	+	0.622527i
$518$	17068.4			0.0636112
$519$		0			0
$520$	21293.9			0.0787497
$521$	−484627.			−1.78539			−0.892693	−	0.450666i	$-0.851187\pi$
$521$	−484627.			−1.78539			−0.892693	+	0.450666i	$0.851187\pi$
$522$		0			0
$523$			188211.i			0.688084i	0.938954	+	0.344042i	$0.111796\pi$
$523$			188211.i			0.688084i	−0.938954	+	0.344042i	$0.888204\pi$
$524$			206418.i			0.751771i
$525$		0			0
$526$	186359.			0.673564
$527$		−	322315.i		−	1.16054i
$528$		0			0
$529$	213746.			0.763811
$530$			29161.6i			0.103815i
$531$		0			0
$532$	−101773.			−0.359591
$533$	−31713.2			−0.111631
$534$		0			0
$535$			138604.i			0.484247i
$536$		−	58247.9i		−	0.202745i
$537$		0			0
$538$		−	158283.i		−	0.546851i
$539$	389847.	+	668772.i	1.34189	+	2.30197i
$540$		0			0
$541$		−	329143.i		−	1.12458i	−0.826940	−	0.562290i	$-0.809920\pi$
$541$		−	329143.i		−	1.12458i	0.826940	−	0.562290i	$-0.190080\pi$
$542$	−217873.			−0.741658
$543$		0			0
$544$	244146.			0.824996
$545$			125594.i			0.422838i
$546$		0			0
$547$			201214.i			0.672486i	0.941775	+	0.336243i	$0.109156\pi$
$547$			201214.i			0.672486i	−0.941775	+	0.336243i	$0.890844\pi$
$548$	−140491.			−0.467829
$549$		0			0
$550$	−63067.5	+	36763.9i	−0.208488	+	0.121534i
$551$	−116282.			−0.383010
$552$		0			0
$553$	−633068.			−2.07014
$554$	64085.9			0.208806
$555$		0			0
$556$		−	297226.i		−	0.961474i
$557$		−	52859.7i		−	0.170378i	−0.996365	−	0.0851892i	$-0.972851\pi$
$557$		−	52859.7i		−	0.170378i	0.996365	−	0.0851892i	$-0.0271495\pi$
$558$		0			0
$559$	−55591.0			−0.177902
$560$			208787.i			0.665775i
$561$		0			0
$562$	−27371.5			−0.0866616
$563$			143681.i			0.453295i	0.973977	+	0.226648i	$0.0727766\pi$
$563$			143681.i			0.453295i	−0.973977	+	0.226648i	$0.927223\pi$
$564$		0			0
$565$	−257368.			−0.806229
$566$	−161881.			−0.505316
$567$		0			0
$568$			166952.i			0.517481i
$569$		−	568025.i		−	1.75446i	−0.480071	−	0.877230i	$-0.659389\pi$
$569$		−	568025.i		−	1.75446i	0.480071	−	0.877230i	$-0.340611\pi$
$570$		0			0
$571$		−	165896.i		−	0.508818i	−0.967097	−	0.254409i	$-0.918119\pi$
$571$		−	165896.i		−	0.508818i	0.967097	−	0.254409i	$-0.0818810\pi$
$572$	41626.7	−	24265.5i	0.127227	−	0.0741645i
$573$		0			0
$574$			159048.i			0.482731i
$575$	−269162.			−0.814100
$576$		0			0
$577$	157238.			0.472286			0.236143	−	0.971718i	$-0.424117\pi$
$577$	157238.			0.472286			0.236143	+	0.971718i	$0.424117\pi$
$578$			31589.2i			0.0945547i
$579$		0			0
$580$			304687.i			0.905729i
$581$	808259.			2.39441
$582$		0			0
$583$	72557.3	+	124470.i	0.213473	+	0.366208i
$584$	−81386.6			−0.238631
$585$		0			0
$586$	−168149.			−0.489664
$587$	−9552.74			−0.0277237			−0.0138619	−	0.999904i	$-0.504413\pi$
$587$	−9552.74			−0.0277237			−0.0138619	+	0.999904i	$0.504413\pi$
$588$		0			0
$589$		−	102676.i		−	0.295964i
$590$			46424.6i			0.133366i
$591$		0			0
$592$	16537.7			0.0471879
$593$			164511.i			0.467827i	0.972257	+	0.233913i	$0.0751532\pi$
$593$			164511.i			0.467827i	−0.972257	+	0.233913i	$0.924847\pi$
$594$		0			0
$595$	367503.			1.03807
$596$			139487.i			0.392681i
$597$		0			0
$598$	−32584.9			−0.0911200
$599$	−318614.			−0.887996			−0.443998	−	0.896028i	$-0.646440\pi$
$599$	−318614.			−0.887996			−0.443998	+	0.896028i	$0.646440\pi$
$600$		0			0
$601$		−	385872.i		−	1.06830i	−0.845389	−	0.534151i	$-0.820631\pi$
$601$		−	385872.i		−	1.06830i	0.845389	−	0.534151i	$-0.179369\pi$
$602$			278801.i			0.769310i
$603$		0			0
$604$			271187.i			0.743351i
$605$	112202.	−	198142.i	0.306542	−	0.541335i
$606$		0			0
$607$			218753.i			0.593712i	0.954922	+	0.296856i	$0.0959381\pi$
$607$			218753.i			0.593712i	−0.954922	+	0.296856i	$0.904062\pi$
$608$	77774.9			0.210394
$609$		0			0
$610$	92482.7			0.248543
$611$		−	46881.3i		−	0.125579i
$612$		0			0
$613$			348766.i			0.928140i	0.885799	+	0.464070i	$0.153611\pi$
$613$			348766.i			0.928140i	−0.885799	+	0.464070i	$0.846389\pi$
$614$	123485.			0.327550
$615$		0			0
$616$	−265714.	−	455825.i	−0.700250	−	1.20126i
$617$	−222234.			−0.583767			−0.291884	−	0.956454i	$-0.594282\pi$
$617$	−222234.			−0.583767			−0.291884	+	0.956454i	$0.594282\pi$
$618$		0			0
$619$	478523.			1.24888			0.624441	−	0.781072i	$-0.285327\pi$
$619$	478523.			1.24888			0.624441	+	0.781072i	$0.285327\pi$
$620$	−269037.			−0.699888
$621$		0			0
$622$			95561.9i			0.247004i
$623$		−	875539.i		−	2.25579i
$624$		0			0
$625$	−4397.54			−0.0112577
$626$		−	135017.i		−	0.344539i
$627$		0			0
$628$	−77635.4			−0.196852
$629$		−	29109.3i		−	0.0735750i
$630$		0			0
$631$	−332157.			−0.834229			−0.417115	−	0.908854i	$-0.636959\pi$
$631$	−332157.			−0.834229			−0.417115	+	0.908854i	$0.636959\pi$
$632$	313742.			0.785487
$633$		0			0
$634$		−	185259.i		−	0.460893i
$635$		−	33675.2i		−	0.0835146i
$636$		0			0
$637$		−	188425.i		−	0.464366i
$638$	−139046.	−	238530.i	−0.341600	−	0.586005i
$639$		0			0
$640$		−	259872.i		−	0.634452i
$641$	194677.			0.473803			0.236902	−	0.971534i	$-0.423868\pi$
$641$	194677.			0.473803			0.236902	+	0.971534i	$0.423868\pi$
$642$		0			0
$643$	598433.			1.44742			0.723709	−	0.690106i	$-0.242436\pi$
$643$	598433.			1.44742			0.723709	+	0.690106i	$0.242436\pi$
$644$		−	890984.i		−	2.14832i
$645$		0			0
$646$		−	31835.1i		−	0.0762854i
$647$	789504.			1.88602			0.943008	−	0.332769i	$-0.107983\pi$
$647$	789504.			1.88602			0.943008	+	0.332769i	$0.107983\pi$
$648$		0			0
$649$	115509.	+	198153.i	0.274238	+	0.470448i
$650$	17769.1			0.0420571
$651$		0			0
$652$	−333417.			−0.784318
$653$	−704147.			−1.65134			−0.825671	−	0.564151i	$-0.809203\pi$
$653$	−704147.			−1.65134			−0.825671	+	0.564151i	$0.809203\pi$
$654$		0			0
$655$		−	237448.i		−	0.553459i
$656$			154103.i			0.358099i
$657$		0			0
$658$	−235120.			−0.543048
$659$			603395.i			1.38941i	0.719294	+	0.694706i	$0.244465\pi$
$659$			603395.i			1.38941i	−0.719294	+	0.694706i	$0.755535\pi$
$660$		0			0
$661$	257502.			0.589356			0.294678	−	0.955597i	$-0.404788\pi$
$661$	257502.			0.589356			0.294678	+	0.955597i	$0.404788\pi$
$662$		−	172778.i		−	0.394250i
$663$		0			0
$664$	−400565.			−0.908526
$665$	117072.			0.264733
$666$		0			0
$667$		−	1.01801e6i		−	2.28823i
$668$			640744.i			1.43592i
$669$		0			0
$670$			30687.6i			0.0683619i
$671$	394742.	−	230107.i	0.876735	−	0.511075i
$672$		0			0
$673$		−	10491.1i		−	0.0231629i	−0.999933	−	0.0115814i	$-0.996313\pi$
$673$		−	10491.1i		−	0.0231629i	0.999933	−	0.0115814i	$-0.00368656\pi$
$674$	115305.			0.253821
$675$		0			0
$676$	374422.			0.819347
$677$			243377.i			0.531008i	0.964110	+	0.265504i	$0.0855384\pi$
$677$			243377.i			0.531008i	−0.964110	+	0.265504i	$0.914462\pi$
$678$		0			0
$679$		−	1.03826e6i		−	2.25199i
$680$	−182131.			−0.393882
$681$		0			0
$682$	210620.	−	122777.i	0.452826	−	0.263966i
$683$	−94210.5			−0.201957			−0.100978	−	0.994889i	$-0.532197\pi$
$683$	−94210.5			−0.201957			−0.100978	+	0.994889i	$0.532197\pi$
$684$		0			0
$685$	161610.			0.344419
$686$	−590337.			−1.25445
$687$		0			0
$688$			270131.i			0.570687i
$689$		−	35069.2i		−	0.0738732i
$690$		0			0
$691$	−49743.2			−0.104178			−0.0520892	−	0.998642i	$-0.516588\pi$
$691$	−49743.2			−0.104178			−0.0520892	+	0.998642i	$0.516588\pi$
$692$			17931.7i			0.0374464i
$693$		0			0
$694$	252362.			0.523968
$695$			341906.i			0.707843i
$696$		0			0
$697$	271249.			0.558345
$698$	39040.3			0.0801314
$699$		0			0
$700$			485870.i			0.991572i
$701$		−	337972.i		−	0.687772i	−0.939011	−	0.343886i	$-0.888257\pi$
$701$		−	337972.i		−	0.687772i	0.939011	−	0.343886i	$-0.111743\pi$
$702$		0			0
$703$		−	9273.03i		−	0.0187634i
$704$	−46538.7	−	79835.8i	−0.0939006	−	0.161084i
$705$		0			0
$706$			221558.i			0.444507i
$707$	−912895.			−1.82634
$708$		0			0
$709$	209306.			0.416379			0.208189	−	0.978089i	$-0.433243\pi$
$709$	209306.			0.416379			0.208189	+	0.978089i	$0.433243\pi$
$710$		−	87957.8i		−	0.174485i
$711$		0			0
$712$			433909.i			0.855930i
$713$	898894.			1.76819
$714$		0			0
$715$	−47884.1	+	27913.1i	−0.0936655	+	0.0546004i
$716$	727693.			1.41946
$717$		0			0
$718$	22996.8			0.0446086
$719$	−382242.			−0.739403			−0.369701	−	0.929151i	$-0.620540\pi$
$719$	−382242.			−0.739403			−0.369701	+	0.929151i	$0.620540\pi$
$720$		0			0
$721$		−	209115.i		−	0.402267i
$722$			195081.i			0.374231i
$723$		0			0
$724$	134679.			0.256935
$725$			555138.i			1.05615i
$726$		0			0
$727$	−573664.			−1.08540			−0.542698	−	0.839928i	$-0.682597\pi$
$727$	−573664.			−1.08540			−0.542698	+	0.839928i	$0.682597\pi$
$728$			128428.i			0.242324i
$729$		0			0
$730$	42878.2			0.0804619
$731$	475481.			0.889812
$732$		0			0
$733$			898874.i			1.67298i	0.547982	+	0.836490i	$0.315396\pi$
$733$			898874.i			1.67298i	−0.547982	+	0.836490i	$0.684604\pi$
$734$		−	264610.i		−	0.491150i
$735$		0			0
$736$			680891.i			1.25696i
$737$	76354.1	+	130983.i	0.140572	+	0.241147i
$738$		0			0
$739$		−	716712.i		−	1.31237i	−0.754601	−	0.656184i	$-0.772170\pi$
$739$		−	716712.i		−	1.31237i	0.754601	−	0.656184i	$-0.227830\pi$
$740$	−24297.6			−0.0443711
$741$		0			0
$742$	−175879.			−0.319453
$743$			327330.i			0.592937i	0.955043	+	0.296468i	$0.0958089\pi$
$743$			327330.i			0.592937i	−0.955043	+	0.296468i	$0.904191\pi$
$744$		0			0
$745$		−	160455.i		−	0.289094i
$746$	−283043.			−0.508597
$747$		0			0
$748$	−356042.	+	207547.i	−0.636352	+	0.370949i
$749$	−835946.			−1.49010
$750$		0			0
$751$	473695.			0.839883			0.419941	−	0.907551i	$-0.362051\pi$
$751$	473695.			0.839883			0.419941	+	0.907551i	$0.362051\pi$
$752$	−227809.			−0.402842
$753$		0			0
$754$			67205.3i			0.118212i
$755$		−	311952.i		−	0.547260i
$756$		0			0
$757$	168385.			0.293841			0.146920	−	0.989148i	$-0.453064\pi$
$757$	168385.			0.293841			0.146920	+	0.989148i	$0.453064\pi$
$758$		−	139245.i		−	0.242349i
$759$		0			0
$760$	−58019.5			−0.100449
$761$		−	216244.i		−	0.373400i	−0.982417	−	0.186700i	$-0.940221\pi$
$761$		−	216244.i		−	0.373400i	0.982417	−	0.186700i	$-0.0597792\pi$
$762$		0			0
$763$	−757479.			−1.30113
$764$	743135.			1.27315
$765$		0			0
$766$		−	151107.i		−	0.257530i
$767$		−	55829.3i		−	0.0949011i
$768$		0			0
$769$			356784.i			0.603328i	0.953414	+	0.301664i	$0.0975420\pi$
$769$			356784.i			0.603328i	−0.953414	+	0.301664i	$0.902458\pi$
$770$	139990.	+	240149.i	0.236111	+	0.405042i
$771$		0			0
$772$		−	23292.0i		−	0.0390816i
$773$	91587.2			0.153277			0.0766383	−	0.997059i	$-0.475581\pi$
$773$	91587.2			0.153277			0.0766383	+	0.997059i	$0.475581\pi$
$774$		0			0
$775$	−490183.			−0.816122
$776$			514551.i			0.854486i
$777$		0			0
$778$			186998.i			0.308943i
$779$	86408.8			0.142391
$780$		0			0
$781$	−218849.	−	375429.i	−0.358791	−	0.615496i
$782$	278705.			0.455755
$783$		0			0
$784$	−915608.			−1.48963
$785$	89305.7			0.144924
$786$		0			0
$787$			77739.6i			0.125514i	0.998029	+	0.0627571i	$0.0199894\pi$
$787$			77739.6i			0.125514i	−0.998029	+	0.0627571i	$0.980011\pi$
$788$			140441.i			0.226173i
$789$		0			0
$790$	−165294.			−0.264851
$791$		−	1.55224e6i		−	2.48088i
$792$		0			0
$793$	−111218.			−0.176859
$794$		−	251056.i		−	0.398227i
$795$		0			0
$796$	689047.			1.08748
$797$	−125376.			−0.197377			−0.0986887	−	0.995118i	$-0.531465\pi$
$797$	−125376.			−0.197377			−0.0986887	+	0.995118i	$0.531465\pi$
$798$		0			0
$799$			400985.i			0.628109i
$800$		−	371303.i		−	0.580160i
$801$		0			0
$802$			65582.5i			0.101962i
$803$	183016.	−	106685.i	0.283830	−	0.165453i
$804$		0			0
$805$			1.02492e6i			1.58160i
$806$	−59341.8			−0.0913463
$807$		0			0
$808$	452422.			0.692981
$809$			466047.i			0.712087i	0.934469	+	0.356044i	$0.115875\pi$
$809$			466047.i			0.712087i	−0.934469	+	0.356044i	$0.884125\pi$
$810$		0			0
$811$			1.07207e6i			1.62998i	0.579473	+	0.814992i	$0.303259\pi$
$811$			1.07207e6i			1.62998i	−0.579473	+	0.814992i	$0.696741\pi$
$812$	−1.83763e6			−2.78706
$813$		0			0
$814$	19021.8	−	11088.4i	0.0287080	−	0.0167347i
$815$	383536.			0.577420
$816$		0			0
$817$	151469.			0.226923
$818$	417392.			0.623789
$819$		0			0
$820$		−	226412.i		−	0.336722i
$821$			423155.i			0.627789i	0.949458	+	0.313894i	$0.101634\pi$
$821$			423155.i			0.627789i	−0.949458	+	0.313894i	$0.898366\pi$
$822$		0			0
$823$	1.01630e6			1.50045			0.750227	−	0.661180i	$-0.229944\pi$
$823$	1.01630e6			1.50045			0.750227	+	0.661180i	$0.229944\pi$
$824$			103635.i			0.152635i
$825$		0			0
$826$	−279996.			−0.410385
$827$			83099.6i			0.121503i	0.998153	+	0.0607517i	$0.0193498\pi$
$827$			83099.6i			0.121503i	−0.998153	+	0.0607517i	$0.980650\pi$
$828$		0			0
$829$	−293001.			−0.426343			−0.213172	−	0.977015i	$-0.568379\pi$
$829$	−293001.			−0.426343			−0.213172	+	0.977015i	$0.568379\pi$
$830$	211036.			0.306338
$831$		0			0
$832$			22493.6i			0.0324946i
$833$			1.61164e6i			2.32262i
$834$		0			0
$835$		−	737062.i		−	1.05714i
$836$	−113420.	+	66116.1i	−0.162285	+	0.0946007i
$837$		0			0
$838$		−	402679.i		−	0.573418i
$839$	−426202.			−0.605468			−0.302734	−	0.953075i	$-0.597899\pi$
$839$	−426202.			−0.605468			−0.302734	+	0.953075i	$0.597899\pi$
$840$		0			0
$841$	−1.39233e6			−1.96856
$842$			297956.i			0.420269i
$843$		0			0
$844$		−	928587.i		−	1.30358i
$845$	−430706.			−0.603208
$846$		0			0
$847$	1.19503e6	+	676713.i	1.66576	+	0.943274i
$848$	−170410.			−0.236976
$849$		0			0
$850$	−151983.			−0.210357
$851$	81182.0			0.112099
$852$		0			0
$853$			1.33250e6i			1.83134i	0.401926	+	0.915672i	$0.368341\pi$
$853$			1.33250e6i			1.83134i	−0.401926	+	0.915672i	$0.631659\pi$
$854$			557781.i			0.764801i
$855$		0			0
$856$	414287.			0.565397
$857$		−	295654.i		−	0.402552i	−0.979535	−	0.201276i	$-0.935491\pi$
$857$		−	295654.i		−	0.402552i	0.979535	−	0.201276i	$-0.0645088\pi$
$858$		0			0
$859$	−931988.			−1.26306			−0.631530	−	0.775352i	$-0.717573\pi$
$859$	−931988.			−1.26306			−0.631530	+	0.775352i	$0.717573\pi$
$860$		−	396885.i		−	0.536621i
$861$		0			0
$862$	−174401.			−0.234711
$863$	−159744.			−0.214488			−0.107244	−	0.994233i	$-0.534203\pi$
$863$	−159744.			−0.214488			−0.107244	+	0.994233i	$0.534203\pi$
$864$		0			0
$865$		−	20627.3i		−	0.0275683i
$866$			272236.i			0.363002i
$867$		0			0
$868$		−	1.62261e6i		−	2.15365i
$869$	−705519.	+	411268.i	−0.934263	+	0.544610i
$870$		0			0
$871$		−	36904.3i		−	0.0486453i
$872$	375399.			0.493697
$873$		0			0
$874$	88784.0			0.116228
$875$		−	1.47068e6i		−	1.92089i
$876$		0			0
$877$		−	1.43711e6i		−	1.86849i	−0.356636	−	0.934244i	$-0.616076\pi$
$877$		−	1.43711e6i		−	1.86849i	0.356636	−	0.934244i	$-0.383924\pi$
$878$	−244422.			−0.317067
$879$		0			0
$880$	135637.	+	232682.i	0.175151	+	0.300467i
$881$	−478465.			−0.616451			−0.308225	−	0.951313i	$-0.599735\pi$
$881$	−478465.			−0.616451			−0.308225	+	0.951313i	$0.599735\pi$
$882$		0			0
$883$	280446.			0.359689			0.179845	−	0.983695i	$-0.442440\pi$
$883$	280446.			0.359689			0.179845	+	0.983695i	$0.442440\pi$
$884$	100314.			0.128368
$885$		0			0
$886$		−	164048.i		−	0.208979i
$887$			1.20382e6i			1.53008i	0.643982	+	0.765041i	$0.277281\pi$
$887$			1.20382e6i			1.53008i	−0.643982	+	0.765041i	$0.722719\pi$
$888$		0			0
$889$	203101.			0.256986
$890$		−	228603.i		−	0.288603i
$891$		0			0
$892$	−22022.9			−0.0276786
$893$			127738.i			0.160183i
$894$		0			0
$895$	−837081.			−1.04501
$896$	1.56734e6			1.95230
$897$		0			0
$898$		−	138713.i		−	0.172015i
$899$		−	1.85394e6i		−	2.29391i
$900$		0			0
$901$			299954.i			0.369491i
$902$	103325.	+	177251.i	0.126996	+	0.217859i
$903$		0			0
$904$			769274.i			0.941335i
$905$	−154925.			−0.189157
$906$		0			0
$907$	224182.			0.272512			0.136256	−	0.990674i	$-0.456493\pi$
$907$	224182.			0.272512			0.136256	+	0.990674i	$0.456493\pi$
$908$			575948.i			0.698572i
$909$		0			0
$910$		−	67661.6i		−	0.0817070i
$911$	815416.			0.982522			0.491261	−	0.871012i	$-0.336536\pi$
$911$	815416.			0.982522			0.491261	+	0.871012i	$0.336536\pi$
$912$		0			0
$913$	900760.	−	525080.i	1.08061	−	0.629918i
$914$	−122745.			−0.146931
$915$		0			0
$916$	437889.			0.521883
$917$	1.43209e6			1.70307
$918$		0			0
$919$		−	858763.i		−	1.01682i	−0.861116	−	0.508408i	$-0.830234\pi$
$919$		−	858763.i		−	1.01682i	0.861116	−	0.508408i	$-0.169766\pi$
$920$		−	507940.i		−	0.600118i
$921$		0			0
$922$	182611.			0.214816
$923$			105776.i			0.124161i
$924$		0			0
$925$	−44270.1			−0.0517400
$926$		−	494215.i		−	0.576361i
$927$		0			0
$928$	1.40432e6			1.63068
$929$	−489087.			−0.566702			−0.283351	−	0.959016i	$-0.591446\pi$
$929$	−489087.			−0.566702			−0.283351	+	0.959016i	$0.591446\pi$
$930$		0			0
$931$			513402.i			0.592322i
$932$		−	782374.i		−	0.900705i
$933$		0			0
$934$			151228.i			0.173355i
$935$	409562.	−	238746.i	0.468486	−	0.273095i
$936$		0			0
$937$		−	276854.i		−	0.315335i	−0.987492	−	0.157667i	$-0.949603\pi$
$937$		−	276854.i		−	0.315335i	0.987492	−	0.157667i	$-0.0503974\pi$
$938$	−185083.			−0.210359
$939$		0			0
$940$	334703.			0.378795
$941$		−	61582.8i		−	0.0695472i	−0.999395	−	0.0347736i	$-0.988929\pi$
$941$		−	61582.8i		−	0.0695472i	0.999395	−	0.0347736i	$-0.0110710\pi$
$942$		0			0
$943$			756478.i			0.850693i
$944$	−271289.			−0.304431
$945$		0			0
$946$	181121.	+	310708.i	0.202389	+	0.347193i
$947$	1.23913e6			1.38171			0.690856	−	0.722992i	$-0.257234\pi$
$947$	1.23913e6			1.38171			0.690856	+	0.722992i	$0.257234\pi$
$948$		0			0
$949$	−51564.4			−0.0572555
$950$	−48415.5			−0.0536460
$951$		0			0
$952$		−	1.09847e6i		−	1.21203i
$953$		−	1.44690e6i		−	1.59314i	−0.604548	−	0.796569i	$-0.706646\pi$
$953$		−	1.44690e6i		−	1.59314i	0.604548	−	0.796569i	$-0.293354\pi$
$954$		0			0
$955$	−854844.			−0.937303
$956$		−	50815.1i		−	0.0556002i
$957$		0			0
$958$	339661.			0.370096
$959$			974701.i			1.05982i
$960$		0			0
$961$	713496.			0.772582
$962$	−5359.35			−0.00579112
$963$		0			0
$964$		−	721886.i		−	0.776810i
$965$			26793.3i			0.0287721i
$966$		0			0
$967$			382759.i			0.409328i	0.978832	+	0.204664i	$0.0656102\pi$
$967$			382759.i			0.409328i	−0.978832	+	0.204664i	$0.934390\pi$
$968$	−592247.	−	335373.i	−0.632051	−	0.357912i
$969$		0			0
$970$		−	271089.i		−	0.288116i
$971$	−571623.			−0.606277			−0.303138	−	0.952947i	$-0.598034\pi$
$971$	−571623.			−0.606277			−0.303138	+	0.952947i	$0.598034\pi$
$972$		0			0
$973$	−2.06210e6			−2.17813
$974$		−	442897.i		−	0.466858i
$975$		0			0
$976$			540437.i			0.567343i
$977$	−1.22248e6			−1.28072			−0.640360	−	0.768075i	$-0.721215\pi$
$977$	−1.22248e6			−1.28072			−0.640360	+	0.768075i	$0.721215\pi$
$978$		0			0
$979$	−568788.	−	975740.i	−0.593451	−	1.01805i
$980$	1.34524e6			1.40071
$981$		0			0
$982$	−360764.			−0.374111
$983$	1.01427e6			1.04966			0.524829	−	0.851208i	$-0.324129\pi$
$983$	1.01427e6			1.04966			0.524829	+	0.851208i	$0.324129\pi$
$984$		0			0
$985$		−	161552.i		−	0.166510i
$986$		−	574820.i		−	0.591260i
$987$		0			0
$988$	31955.9			0.0327369
$989$			1.32605e6i			1.35571i
$990$		0			0
$991$	609717.			0.620842			0.310421	−	0.950599i	$-0.399530\pi$
$991$	609717.			0.620842			0.310421	+	0.950599i	$0.399530\pi$
$992$			1.24000e6i			1.26008i
$993$		0			0
$994$	530491.			0.536914
$995$	−792626.			−0.800612
$996$		0			0
$997$			134059.i			0.134867i	0.997724	+	0.0674335i	$0.0214811\pi$
$997$			134059.i			0.134867i	−0.997724	+	0.0674335i	$0.978519\pi$
$998$			59293.1i			0.0595310i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.5.c.c.10.5		8
3.2	odd	2		33.5.c.a.10.4	✓	8
11.10	odd	2	inner	99.5.c.c.10.4		8
12.11	even	2		528.5.j.a.241.8		8
33.32	even	2		33.5.c.a.10.5	yes	8
132.131	odd	2		528.5.j.a.241.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.c.a.10.4	✓	8	3.2	odd	2
33.5.c.a.10.5	yes	8	33.32	even	2
99.5.c.c.10.4		8	11.10	odd	2	inner
99.5.c.c.10.5		8	1.1	even	1	trivial
528.5.j.a.241.7		8	132.131	odd	2
528.5.j.a.241.8		8	12.11	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	99.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.57474i q^{2} +13.5202 q^{4} -15.5526 q^{5} -93.8006i q^{7} +46.4867i q^{8} +O(q^{10})\)	\(q+1.57474i q^{2} +13.5202 q^{4} -15.5526 q^{5} -93.8006i q^{7} +46.4867i q^{8} -24.4913i q^{10} +(-60.9369 - 104.536i) q^{11} +29.4527i q^{13} +147.712 q^{14} +143.119 q^{16} -251.915i q^{17} -80.2497i q^{19} -210.274 q^{20} +(164.617 - 95.9599i) q^{22} +702.557 q^{23} -383.118 q^{25} -46.3804 q^{26} -1268.20i q^{28} -1449.00i q^{29} +1279.46 q^{31} +969.161i q^{32} +396.701 q^{34} +1458.84i q^{35} +115.552 q^{37} +126.373 q^{38} -722.987i q^{40} +1076.75i q^{41} +1887.47i q^{43} +(-823.879 - 1413.34i) q^{44} +1106.35i q^{46} -1591.75 q^{47} -6397.55 q^{49} -603.311i q^{50} +398.206i q^{52} -1190.69 q^{53} +(947.726 + 1625.80i) q^{55} +4360.48 q^{56} +2281.80 q^{58} -1895.56 q^{59} +3776.15i q^{61} +2014.82i q^{62} +763.719 q^{64} -458.065i q^{65} -1253.00 q^{67} -3405.94i q^{68} -2297.30 q^{70} +3591.39 q^{71} +1750.75i q^{73} +181.965i q^{74} -1084.99i q^{76} +(-9805.50 + 5715.92i) q^{77} -6749.08i q^{79} -2225.86 q^{80} -1695.60 q^{82} +8616.78i q^{83} +3917.92i q^{85} -2972.27 q^{86} +(4859.51 - 2832.75i) q^{88} +9334.05 q^{89} +2762.68 q^{91} +9498.71 q^{92} -2506.59i q^{94} +1248.09i q^{95} +11068.8 q^{97} -10074.5i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) \) 8 * q - 76 * q^4 + 36 * q^5	\( 8 q - 76 q^{4} + 36 q^{5} - 36 q^{11} + 1140 q^{14} + 1412 q^{16} - 2532 q^{20} - 780 q^{22} - 516 q^{23} - 2280 q^{25} + 1524 q^{26} + 2752 q^{31} - 4920 q^{34} + 5296 q^{37} - 696 q^{38} + 6540 q^{44} - 420 q^{47} - 6832 q^{49} - 3540 q^{53} + 3784 q^{55} - 17964 q^{56} + 21624 q^{58} + 16632 q^{59} - 27508 q^{64} - 3656 q^{67} + 3312 q^{70} + 13212 q^{71} - 23268 q^{77} + 4476 q^{80} + 17088 q^{82} - 19896 q^{86} - 12516 q^{88} - 15528 q^{89} - 19752 q^{91} + 81180 q^{92} + 7624 q^{97}+O(q^{100}) \) 8 * q - 76 * q^4 + 36 * q^5 - 36 * q^11 + 1140 * q^14 + 1412 * q^16 - 2532 * q^20 - 780 * q^22 - 516 * q^23 - 2280 * q^25 + 1524 * q^26 + 2752 * q^31 - 4920 * q^34 + 5296 * q^37 - 696 * q^38 + 6540 * q^44 - 420 * q^47 - 6832 * q^49 - 3540 * q^53 + 3784 * q^55 - 17964 * q^56 + 21624 * q^58 + 16632 * q^59 - 27508 * q^64 - 3656 * q^67 + 3312 * q^70 + 13212 * q^71 - 23268 * q^77 + 4476 * q^80 + 17088 * q^82 - 19896 * q^86 - 12516 * q^88 - 15528 * q^89 - 19752 * q^91 + 81180 * q^92 + 7624 * q^97

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