LMFDB - Embedded newform 1225.2.b.n.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.n.99.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.381966i q^{2} +0.874032i q^{3} +1.85410 q^{4} +0.333851 q^{6} -1.47214i q^{8} +2.23607 q^{9} +2.23607 q^{11} +1.62054i q^{12} -4.03631i q^{13} +3.14590 q^{16} -7.40492i q^{17} -0.854102i q^{18} -4.24264 q^{19} -0.854102i q^{22} +3.76393i q^{23} +1.28669 q^{24} -1.54173 q^{26} +4.57649i q^{27} -2.23607 q^{29} +6.86474 q^{31} -4.14590i q^{32} +1.95440i q^{33} -2.82843 q^{34} +4.14590 q^{36} -10.7082i q^{37} +1.62054i q^{38} +3.52786 q^{39} -4.78282 q^{41} +5.00000i q^{43} +4.14590 q^{44} +1.43769 q^{46} +9.48683i q^{47} +2.74962i q^{48} +6.47214 q^{51} -7.48373i q^{52} +9.70820i q^{53} +1.74806 q^{54} -3.70820i q^{57} +0.854102i q^{58} +13.1893 q^{59} -3.03476 q^{61} -2.62210i q^{62} +4.70820 q^{64} +0.746512 q^{66} -8.70820i q^{67} -13.7295i q^{68} -3.28980 q^{69} -7.47214 q^{71} -3.29180i q^{72} +2.62210i q^{73} -4.09017 q^{74} -7.86629 q^{76} -1.34752i q^{78} +4.70820 q^{79} +2.70820 q^{81} +1.82688i q^{82} +6.86474i q^{83} +1.90983 q^{86} -1.95440i q^{87} -3.29180i q^{88} -7.94510 q^{89} +6.97871i q^{92} +6.00000i q^{93} +3.62365 q^{94} +3.62365 q^{96} +10.1058i q^{97} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4} + 52 q^{16} + 60 q^{36} + 64 q^{39} + 60 q^{44} + 92 q^{46} + 16 q^{51} - 16 q^{64} - 24 q^{71} + 12 q^{74} - 16 q^{79} - 32 q^{81} + 60 q^{86} + 40 q^{99}+O(q^{100}) $ 8 * q - 12 * q^4 + 52 * q^16 + 60 * q^36 + 64 * q^39 + 60 * q^44 + 92 * q^46 + 16 * q^51 - 16 * q^64 - 24 * q^71 + 12 * q^74 - 16 * q^79 - 32 * q^81 + 60 * q^86 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.381966i	− 0.270091i	−0.990839	−	0.135045i	$-0.956882\pi$
$2$	− 0.381966i	− 0.270091i	0.990839	−	0.135045i	$-0.0431180\pi$
$3$	0.874032i	0.504623i	0.967646	+	0.252311i	$0.0811907\pi$
$3$	0.874032i	0.504623i	−0.967646	+	0.252311i	$0.918809\pi$
$4$	1.85410	0.927051
$5$	0	0
$5$	0	0
$6$	0.333851	0.136294
$7$	0	0
$7$	0	0
$8$	− 1.47214i	− 0.520479i
$9$	2.23607	0.745356
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	1.62054i	0.467811i
$13$	− 4.03631i	− 1.11947i	−0.828671	−	0.559735i	$-0.810903\pi$
$13$	− 4.03631i	− 1.11947i	0.828671	−	0.559735i	$-0.189097\pi$
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	− 7.40492i	− 1.79596i	−0.440040	−	0.897978i	$-0.645036\pi$
$17$	− 7.40492i	− 1.79596i	0.440040	−	0.897978i	$-0.354964\pi$
$18$	− 0.854102i	− 0.201314i
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	0	0
$22$	− 0.854102i	− 0.182095i
$23$	3.76393i	0.784834i	0.919787	+	0.392417i	$0.128361\pi$
$23$	3.76393i	0.784834i	−0.919787	+	0.392417i	$0.871639\pi$
$24$	1.28669	0.262645
$25$	0	0
$26$	−1.54173	−0.302359
$27$	4.57649i	0.880746i
$28$	0	0
$29$	−2.23607	−0.415227	−0.207614	−	0.978211i	$-0.566570\pi$
$29$	−2.23607	−0.415227	−0.207614	+	0.978211i	$0.566570\pi$
$30$	0	0
$31$	6.86474	1.23294	0.616472	−	0.787377i	$-0.288562\pi$
$31$	6.86474	1.23294	0.616472	+	0.787377i	$0.288562\pi$
$32$	− 4.14590i	− 0.732898i
$33$	1.95440i	0.340217i
$34$	−2.82843	−0.485071
$35$	0	0
$36$	4.14590	0.690983
$37$	− 10.7082i	− 1.76042i	−0.474586	−	0.880209i	$-0.657402\pi$
$37$	− 10.7082i	− 1.76042i	0.474586	−	0.880209i	$-0.342598\pi$
$38$	1.62054i	0.262887i
$39$	3.52786	0.564910
$40$	0	0
$41$	−4.78282	−0.746951	−0.373476	−	0.927640i	$-0.621834\pi$
$41$	−4.78282	−0.746951	−0.373476	+	0.927640i	$0.621834\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	4.14590	0.625018
$45$	0	0
$46$	1.43769	0.211976
$47$	9.48683i	1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$	9.48683i	1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$	2.74962i	0.396873i
$49$	0	0
$50$	0	0
$51$	6.47214	0.906280
$52$	− 7.48373i	− 1.03781i
$53$	9.70820i	1.33352i	0.745271	+	0.666762i	$0.232320\pi$
$53$	9.70820i	1.33352i	−0.745271	+	0.666762i	$0.767680\pi$
$54$	1.74806	0.237881
$55$	0	0
$56$	0	0
$57$	− 3.70820i	− 0.491164i
$58$	0.854102i	0.112149i
$59$	13.1893	1.71710	0.858550	−	0.512730i	$-0.171366\pi$
$59$	13.1893	1.71710	0.858550	+	0.512730i	$0.171366\pi$
$60$	0	0
$61$	−3.03476	−0.388561	−0.194280	−	0.980946i	$-0.562237\pi$
$61$	−3.03476	−0.388561	−0.194280	+	0.980946i	$0.562237\pi$
$62$	− 2.62210i	− 0.333007i
$63$	0	0
$64$	4.70820	0.588525
$65$	0	0
$66$	0.746512	0.0918893
$67$	− 8.70820i	− 1.06388i	−0.846783	−	0.531938i	$-0.821464\pi$
$67$	− 8.70820i	− 1.06388i	0.846783	−	0.531938i	$-0.178536\pi$
$68$	− 13.7295i	− 1.66494i
$69$	−3.28980	−0.396045
$70$	0	0
$71$	−7.47214	−0.886779	−0.443390	−	0.896329i	$-0.646224\pi$
$71$	−7.47214	−0.886779	−0.443390	+	0.896329i	$0.646224\pi$
$72$	− 3.29180i	− 0.387942i
$73$	2.62210i	0.306893i	0.988157	+	0.153447i	$0.0490373\pi$
$73$	2.62210i	0.306893i	−0.988157	+	0.153447i	$0.950963\pi$
$74$	−4.09017	−0.475473
$75$	0	0
$76$	−7.86629	−0.902325
$77$	0	0
$78$	− 1.34752i	− 0.152577i
$79$	4.70820	0.529714	0.264857	−	0.964288i	$-0.414675\pi$
$79$	4.70820	0.529714	0.264857	+	0.964288i	$0.414675\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	1.82688i	0.201745i
$83$	6.86474i	0.753503i	0.926314	+	0.376751i	$0.122959\pi$
$83$	6.86474i	0.753503i	−0.926314	+	0.376751i	$0.877041\pi$
$84$	0	0
$85$	0	0
$86$	1.90983	0.205942
$87$	− 1.95440i	− 0.209533i
$88$	− 3.29180i	− 0.350907i
$89$	−7.94510	−0.842179	−0.421089	−	0.907019i	$-0.638352\pi$
$89$	−7.94510	−0.842179	−0.421089	+	0.907019i	$0.638352\pi$
$90$	0	0
$91$	0	0
$92$	6.97871i	0.727581i
$93$	6.00000i	0.622171i
$94$	3.62365	0.373751
$95$	0	0
$96$	3.62365	0.369837
$97$	10.1058i	1.02609i	0.858361	+	0.513046i	$0.171483\pi$
$97$	10.1058i	1.02609i	−0.858361	+	0.513046i	$0.828517\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	11.1074	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$101$	11.1074	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$102$	− 2.47214i	− 0.244778i
$103$	9.48683i	0.934765i	0.884055	+	0.467383i	$0.154803\pi$
$103$	9.48683i	0.934765i	−0.884055	+	0.467383i	$0.845197\pi$
$104$	−5.94200	−0.582661
$105$	0	0
$106$	3.70820	0.360173
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	8.48528i	0.816497i
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	9.35931	0.888347
$112$	0	0
$113$	9.76393i	0.918513i	0.888304	+	0.459257i	$0.151884\pi$
$113$	9.76393i	0.918513i	−0.888304	+	0.459257i	$0.848116\pi$
$114$	−1.41641	−0.132659
$115$	0	0
$116$	−4.14590	−0.384937
$117$	− 9.02546i	− 0.834404i
$118$	− 5.03786i	− 0.463773i
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	1.15917i	0.104947i
$123$	− 4.18034i	− 0.376929i
$124$	12.7279	1.14300
$125$	0	0
$126$	0	0
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	− 10.0902i	− 0.891853i
$129$	−4.37016	−0.384771
$130$	0	0
$131$	−14.8098	−1.29394	−0.646971	−	0.762515i	$-0.723964\pi$
$131$	−14.8098	−1.29394	−0.646971	+	0.762515i	$0.723964\pi$
$132$	3.62365i	0.315398i
$133$	0	0
$134$	−3.32624	−0.287343
$135$	0	0
$136$	−10.9010	−0.934757
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	1.25659i	0.106968i
$139$	19.5927	1.66183	0.830914	−	0.556401i	$-0.187818\pi$
$139$	19.5927	1.66183	0.830914	+	0.556401i	$0.187818\pi$
$140$	0	0
$141$	−8.29180	−0.698295
$142$	2.85410i	0.239511i
$143$	− 9.02546i	− 0.754747i
$144$	7.03444	0.586203
$145$	0	0
$146$	1.00155	0.0828890
$147$	0	0
$148$	− 19.8541i	− 1.63200i
$149$	−9.65248	−0.790762	−0.395381	−	0.918517i	$-0.629387\pi$
$149$	−9.65248	−0.790762	−0.395381	+	0.918517i	$0.629387\pi$
$150$	0	0
$151$	8.41641	0.684918	0.342459	−	0.939533i	$-0.388740\pi$
$151$	8.41641	0.684918	0.342459	+	0.939533i	$0.388740\pi$
$152$	6.24574i	0.506597i
$153$	− 16.5579i	− 1.33863i
$154$	0	0
$155$	0	0
$156$	6.54102	0.523701
$157$	5.45052i	0.434999i	0.976060	+	0.217500i	$0.0697901\pi$
$157$	5.45052i	0.434999i	−0.976060	+	0.217500i	$0.930210\pi$
$158$	− 1.79837i	− 0.143071i
$159$	−8.48528	−0.672927
$160$	0	0
$161$	0	0
$162$	− 1.03444i	− 0.0812734i
$163$	− 2.29180i	− 0.179507i	−0.995964	−	0.0897537i	$-0.971392\pi$
$163$	− 2.29180i	− 0.179507i	0.995964	−	0.0897537i	$-0.0286080\pi$
$164$	−8.86784	−0.692462
$165$	0	0
$166$	2.62210	0.203514
$167$	− 14.2697i	− 1.10422i	−0.833772	−	0.552110i	$-0.813823\pi$
$167$	− 14.2697i	− 1.10422i	0.833772	−	0.552110i	$-0.186177\pi$
$168$	0	0
$169$	−3.29180	−0.253215
$170$	0	0
$171$	−9.48683	−0.725476
$172$	9.27051i	0.706870i
$173$	4.24264i	0.322562i	0.986909	+	0.161281i	$0.0515625\pi$
$173$	4.24264i	0.322562i	−0.986909	+	0.161281i	$0.948437\pi$
$174$	−0.746512	−0.0565930
$175$	0	0
$176$	7.03444	0.530241
$177$	11.5279i	0.866487i
$178$	3.03476i	0.227465i
$179$	−3.70820	−0.277164	−0.138582	−	0.990351i	$-0.544254\pi$
$179$	−3.70820	−0.277164	−0.138582	+	0.990351i	$0.544254\pi$
$180$	0	0
$181$	11.1074	0.825605	0.412802	−	0.910821i	$-0.364550\pi$
$181$	11.1074	0.825605	0.412802	+	0.910821i	$0.364550\pi$
$182$	0	0
$183$	− 2.65248i	− 0.196077i
$184$	5.54102	0.408489
$185$	0	0
$186$	2.29180	0.168043
$187$	− 16.5579i	− 1.21083i
$188$	17.5896i	1.28285i
$189$	0	0
$190$	0	0
$191$	−23.1246	−1.67324	−0.836619	−	0.547785i	$-0.815471\pi$
$191$	−23.1246	−1.67324	−0.836619	+	0.547785i	$0.815471\pi$
$192$	4.11512i	0.296983i
$193$	− 10.7082i	− 0.770793i	−0.922751	−	0.385397i	$-0.874065\pi$
$193$	− 10.7082i	− 0.770793i	0.922751	−	0.385397i	$-0.125935\pi$
$194$	3.86008	0.277138
$195$	0	0
$196$	0	0
$197$	5.94427i	0.423512i	0.977323	+	0.211756i	$0.0679182\pi$
$197$	5.94427i	0.423512i	−0.977323	+	0.211756i	$0.932082\pi$
$198$	− 1.90983i	− 0.135726i
$199$	−25.2495	−1.78989	−0.894945	−	0.446176i	$-0.852786\pi$
$199$	−25.2495	−1.78989	−0.894945	+	0.446176i	$0.852786\pi$
$200$	0	0
$201$	7.61125	0.536856
$202$	− 4.24264i	− 0.298511i
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	3.62365	0.252472
$207$	8.41641i	0.584981i
$208$	− 12.6978i	− 0.880435i
$209$	−9.48683	−0.656218
$210$	0	0
$211$	7.41641	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$212$	18.0000i	1.23625i
$213$	− 6.53089i	− 0.447489i
$214$	−2.29180	−0.156664
$215$	0	0
$216$	6.73722	0.458410
$217$	0	0
$218$	− 0.381966i	− 0.0258700i
$219$	−2.29180	−0.154865
$220$	0	0
$221$	−29.8885	−2.01052
$222$	− 3.57494i	− 0.239934i
$223$	− 5.86319i	− 0.392628i	−0.980541	−	0.196314i	$-0.937103\pi$
$223$	− 5.86319i	− 0.392628i	0.980541	−	0.196314i	$-0.0628972\pi$
$224$	0	0
$225$	0	0
$226$	3.72949	0.248082
$227$	26.4574i	1.75604i	0.478625	+	0.878020i	$0.341135\pi$
$227$	26.4574i	1.75604i	−0.478625	+	0.878020i	$0.658865\pi$
$228$	− 6.87539i	− 0.455334i
$229$	7.07107	0.467269	0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107	0.467269	0.233635	+	0.972324i	$0.424938\pi$
$230$	0	0
$231$	0	0
$232$	3.29180i	0.216117i
$233$	4.52786i	0.296630i	0.988940	+	0.148315i	$0.0473850\pi$
$233$	4.52786i	0.296630i	−0.988940	+	0.148315i	$0.952615\pi$
$234$	−3.44742	−0.225365
$235$	0	0
$236$	24.4543	1.59184
$237$	4.11512i	0.267306i
$238$	0	0
$239$	−29.1246	−1.88391	−0.941957	−	0.335733i	$-0.891016\pi$
$239$	−29.1246	−1.88391	−0.941957	+	0.335733i	$0.891016\pi$
$240$	0	0
$241$	−14.7310	−0.948909	−0.474454	−	0.880280i	$-0.657355\pi$
$241$	−14.7310	−0.948909	−0.474454	+	0.880280i	$0.657355\pi$
$242$	2.29180i	0.147322i
$243$	16.0965i	1.03259i
$244$	−5.62675	−0.360216
$245$	0	0
$246$	−1.59675	−0.101805
$247$	17.1246i	1.08961i
$248$	− 10.1058i	− 0.641721i
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−11.5687	−0.730213	−0.365106	−	0.930966i	$-0.618967\pi$
$251$	−11.5687	−0.730213	−0.365106	+	0.930966i	$0.618967\pi$
$252$	0	0
$253$	8.41641i	0.529135i
$254$	−2.67376	−0.167767
$255$	0	0
$256$	5.56231	0.347644
$257$	− 12.1089i	− 0.755334i	−0.925941	−	0.377667i	$-0.876726\pi$
$257$	− 12.1089i	− 0.755334i	0.925941	−	0.377667i	$-0.123274\pi$
$258$	1.66925i	0.103923i
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	5.65685i	0.349482i
$263$	29.9443i	1.84644i	0.384268	+	0.923221i	$0.374454\pi$
$263$	29.9443i	1.84644i	−0.384268	+	0.923221i	$0.625546\pi$
$264$	2.87714	0.177075
$265$	0	0
$266$	0	0
$267$	− 6.94427i	− 0.424983i
$268$	− 16.1459i	− 0.986268i
$269$	15.3500	0.935907	0.467954	−	0.883753i	$-0.344991\pi$
$269$	15.3500	0.935907	0.467954	+	0.883753i	$0.344991\pi$
$270$	0	0
$271$	−4.44897	−0.270256	−0.135128	−	0.990828i	$-0.543145\pi$
$271$	−4.44897	−0.270256	−0.135128	+	0.990828i	$0.543145\pi$
$272$	− 23.2951i	− 1.41247i
$273$	0	0
$274$	2.29180	0.138452
$275$	0	0
$276$	−6.09962	−0.367154
$277$	24.0000i	1.44202i	0.692925	+	0.721010i	$0.256322\pi$
$277$	24.0000i	1.44202i	−0.692925	+	0.721010i	$0.743678\pi$
$278$	− 7.48373i	− 0.448844i
$279$	15.3500	0.918982
$280$	0	0
$281$	−24.5967	−1.46732	−0.733659	−	0.679517i	$-0.762189\pi$
$281$	−24.5967	−1.46732	−0.733659	+	0.679517i	$0.762189\pi$
$282$	3.16718i	0.188603i
$283$	− 20.8005i	− 1.23646i	−0.785996	−	0.618232i	$-0.787849\pi$
$283$	− 20.8005i	− 1.23646i	0.785996	−	0.618232i	$-0.212151\pi$
$284$	−13.8541	−0.822090
$285$	0	0
$286$	−3.44742	−0.203850
$287$	0	0
$288$	− 9.27051i	− 0.546270i
$289$	−37.8328	−2.22546
$290$	0	0
$291$	−8.83282	−0.517789
$292$	4.86163i	0.284506i
$293$	18.4335i	1.07690i	0.842659	+	0.538448i	$0.180989\pi$
$293$	18.4335i	1.07690i	−0.842659	+	0.538448i	$0.819011\pi$
$294$	0	0
$295$	0	0
$296$	−15.7639	−0.916260
$297$	10.2333i	0.593799i
$298$	3.68692i	0.213577i
$299$	15.1924	0.878599
$300$	0	0
$301$	0	0
$302$	− 3.21478i	− 0.184990i
$303$	9.70820i	0.557722i
$304$	−13.3469	−0.765498
$305$	0	0
$306$	−6.32456	−0.361551
$307$	− 14.5548i	− 0.830686i	−0.909665	−	0.415343i	$-0.863662\pi$
$307$	− 14.5548i	− 0.830686i	0.909665	−	0.415343i	$-0.136338\pi$
$308$	0	0
$309$	−8.29180	−0.471704
$310$	0	0
$311$	−9.56564	−0.542418	−0.271209	−	0.962520i	$-0.587423\pi$
$311$	−9.56564	−0.542418	−0.271209	+	0.962520i	$0.587423\pi$
$312$	− 5.19350i	− 0.294024i
$313$	21.2132i	1.19904i	0.800359	+	0.599521i	$0.204642\pi$
$313$	21.2132i	1.19904i	−0.800359	+	0.599521i	$0.795358\pi$
$314$	2.08191	0.117489
$315$	0	0
$316$	8.72949	0.491072
$317$	11.9443i	0.670857i	0.942066	+	0.335429i	$0.108881\pi$
$317$	11.9443i	0.670857i	−0.942066	+	0.335429i	$0.891119\pi$
$318$	3.24109i	0.181751i
$319$	−5.00000	−0.279946
$320$	0	0
$321$	5.24419	0.292702
$322$	0	0
$323$	31.4164i	1.74806i
$324$	5.02129	0.278960
$325$	0	0
$326$	−0.875388	−0.0484833
$327$	0.874032i	0.0483341i
$328$	7.04096i	0.388772i
$329$	0	0
$330$	0	0
$331$	−1.29180	−0.0710035	−0.0355018	−	0.999370i	$-0.511303\pi$
$331$	−1.29180	−0.0710035	−0.0355018	+	0.999370i	$0.511303\pi$
$332$	12.7279i	0.698535i
$333$	− 23.9443i	− 1.31214i
$334$	−5.45052	−0.298239
$335$	0	0
$336$	0	0
$337$	23.1246i	1.25968i	0.776726	+	0.629839i	$0.216879\pi$
$337$	23.1246i	1.25968i	−0.776726	+	0.629839i	$0.783121\pi$
$338$	1.25735i	0.0683911i
$339$	−8.53399	−0.463503
$340$	0	0
$341$	15.3500	0.831250
$342$	3.62365i	0.195944i
$343$	0	0
$344$	7.36068	0.396861
$345$	0	0
$346$	1.62054	0.0871210
$347$	− 27.6525i	− 1.48446i	−0.670144	−	0.742231i	$-0.733768\pi$
$347$	− 27.6525i	− 1.48446i	0.670144	−	0.742231i	$-0.266232\pi$
$348$	− 3.62365i	− 0.194248i
$349$	1.00155	0.0536118	0.0268059	−	0.999641i	$-0.491466\pi$
$349$	1.00155	0.0536118	0.0268059	+	0.999641i	$0.491466\pi$
$350$	0	0
$351$	18.4721	0.985970
$352$	− 9.27051i	− 0.494120i
$353$	− 12.1089i	− 0.644493i	−0.946656	−	0.322247i	$-0.895562\pi$
$353$	− 12.1089i	− 0.644493i	0.946656	−	0.322247i	$-0.104438\pi$
$354$	4.40325	0.234030
$355$	0	0
$356$	−14.7310	−0.780743
$357$	0	0
$358$	1.41641i	0.0748595i
$359$	−0.0557281	−0.00294122	−0.00147061	−	0.999999i	$-0.500468\pi$
$359$	−0.0557281	−0.00294122	−0.00147061	+	0.999999i	$0.500468\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	− 4.24264i	− 0.222988i
$363$	− 5.24419i	− 0.275249i
$364$	0	0
$365$	0	0
$366$	−1.01316	−0.0529585
$367$	8.69161i	0.453698i	0.973930	+	0.226849i	$0.0728425\pi$
$367$	8.69161i	0.453698i	−0.973930	+	0.226849i	$0.927158\pi$
$368$	11.8409i	0.617252i
$369$	−10.6947	−0.556745
$370$	0	0
$371$	0	0
$372$	11.1246i	0.576784i
$373$	− 16.1246i	− 0.834901i	−0.908700	−	0.417450i	$-0.862924\pi$
$373$	− 16.1246i	− 0.834901i	0.908700	−	0.417450i	$-0.137076\pi$
$374$	−6.32456	−0.327035
$375$	0	0
$376$	13.9659	0.720237
$377$	9.02546i	0.464835i
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	6.11822	0.313446
$382$	8.83282i	0.451926i
$383$	− 13.1893i	− 0.673941i	−0.941515	−	0.336971i	$-0.890598\pi$
$383$	− 13.1893i	− 0.673941i	0.941515	−	0.336971i	$-0.109402\pi$
$384$	8.81913	0.450049
$385$	0	0
$386$	−4.09017	−0.208184
$387$	11.1803i	0.568329i
$388$	18.7372i	0.951239i
$389$	−27.7639	−1.40769	−0.703844	−	0.710355i	$-0.748534\pi$
$389$	−27.7639	−1.40769	−0.703844	+	0.710355i	$0.748534\pi$
$390$	0	0
$391$	27.8716	1.40953
$392$	0	0
$393$	− 12.9443i	− 0.652952i
$394$	2.27051	0.114387
$395$	0	0
$396$	9.27051	0.465861
$397$	− 9.89949i	− 0.496841i	−0.968652	−	0.248421i	$-0.920088\pi$
$397$	− 9.89949i	− 0.496841i	0.968652	−	0.248421i	$-0.0799115\pi$
$398$	9.64446i	0.483433i
$399$	0	0
$400$	0	0
$401$	23.0689	1.15201	0.576003	−	0.817448i	$-0.304612\pi$
$401$	23.0689	1.15201	0.576003	+	0.817448i	$0.304612\pi$
$402$	− 2.90724i	− 0.145000i
$403$	− 27.7082i	− 1.38024i
$404$	20.5942	1.02460
$405$	0	0
$406$	0	0
$407$	− 23.9443i	− 1.18687i
$408$	− 9.52786i	− 0.471700i
$409$	14.7310	0.728402	0.364201	−	0.931320i	$-0.381342\pi$
$409$	14.7310	0.728402	0.364201	+	0.931320i	$0.381342\pi$
$410$	0	0
$411$	−5.24419	−0.258677
$412$	17.5896i	0.866575i
$413$	0	0
$414$	3.21478	0.157998
$415$	0	0
$416$	−16.7341	−0.820458
$417$	17.1246i	0.838596i
$418$	3.62365i	0.177238i
$419$	−4.86163	−0.237506	−0.118753	−	0.992924i	$-0.537890\pi$
$419$	−4.86163	−0.237506	−0.118753	+	0.992924i	$0.537890\pi$
$420$	0	0
$421$	−9.29180	−0.452854	−0.226427	−	0.974028i	$-0.572705\pi$
$421$	−9.29180	−0.452854	−0.226427	+	0.974028i	$0.572705\pi$
$422$	− 2.83282i	− 0.137899i
$423$	21.2132i	1.03142i
$424$	14.2918	0.694071
$425$	0	0
$426$	−2.49458	−0.120863
$427$	0	0
$428$	− 11.1246i	− 0.537728i
$429$	7.88854	0.380862
$430$	0	0
$431$	6.76393	0.325807	0.162904	−	0.986642i	$-0.447914\pi$
$431$	6.76393	0.325807	0.162904	+	0.986642i	$0.447914\pi$
$432$	14.3972i	0.692684i
$433$	− 32.7031i	− 1.57161i	−0.618473	−	0.785806i	$-0.712248\pi$
$433$	− 32.7031i	− 1.57161i	0.618473	−	0.785806i	$-0.287752\pi$
$434$	0	0
$435$	0	0
$436$	1.85410	0.0887954
$437$	− 15.9690i	− 0.763901i
$438$	0.875388i	0.0418277i
$439$	−17.7658	−0.847915	−0.423957	−	0.905682i	$-0.639359\pi$
$439$	−17.7658	−0.847915	−0.423957	+	0.905682i	$0.639359\pi$
$440$	0	0
$441$	0	0
$442$	11.4164i	0.543023i
$443$	20.1803i	0.958797i	0.877597	+	0.479398i	$0.159145\pi$
$443$	20.1803i	0.958797i	−0.877597	+	0.479398i	$0.840855\pi$
$444$	17.3531	0.823543
$445$	0	0
$446$	−2.23954	−0.106045
$447$	− 8.43657i	− 0.399036i
$448$	0	0
$449$	12.5967	0.594477	0.297239	−	0.954803i	$-0.403934\pi$
$449$	12.5967	0.594477	0.297239	+	0.954803i	$0.403934\pi$
$450$	0	0
$451$	−10.6947	−0.503594
$452$	18.1033i	0.851509i
$453$	7.35621i	0.345625i
$454$	10.1058	0.474290
$455$	0	0
$456$	−5.45898	−0.255640
$457$	− 7.29180i	− 0.341096i	−0.985349	−	0.170548i	$-0.945446\pi$
$457$	− 7.29180i	− 0.341096i	0.985349	−	0.170548i	$-0.0545538\pi$
$458$	− 2.70091i	− 0.126205i
$459$	33.8885	1.58178
$460$	0	0
$461$	−33.4009	−1.55564	−0.777819	−	0.628489i	$-0.783674\pi$
$461$	−33.4009	−1.55564	−0.777819	+	0.628489i	$0.783674\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	−7.03444	−0.326566
$465$	0	0
$466$	1.72949	0.0801171
$467$	9.64446i	0.446292i	0.974785	+	0.223146i	$0.0716327\pi$
$467$	9.64446i	0.446292i	−0.974785	+	0.223146i	$0.928367\pi$
$468$	− 16.7341i	− 0.773535i
$469$	0	0
$470$	0	0
$471$	−4.76393	−0.219510
$472$	− 19.4164i	− 0.893714i
$473$	11.1803i	0.514073i
$474$	1.57184	0.0721968
$475$	0	0
$476$	0	0
$477$	21.7082i	0.993950i
$478$	11.1246i	0.508828i
$479$	9.02546	0.412384	0.206192	−	0.978512i	$-0.433893\pi$
$479$	9.02546	0.412384	0.206192	+	0.978512i	$0.433893\pi$
$480$	0	0
$481$	−43.2216	−1.97074
$482$	5.62675i	0.256291i
$483$	0	0
$484$	−11.1246	−0.505664
$485$	0	0
$486$	6.14833	0.278894
$487$	− 28.7082i	− 1.30089i	−0.759552	−	0.650446i	$-0.774582\pi$
$487$	− 28.7082i	− 1.30089i	0.759552	−	0.650446i	$-0.225418\pi$
$488$	4.46758i	0.202238i
$489$	2.00310	0.0905835
$490$	0	0
$491$	7.47214	0.337213	0.168606	−	0.985683i	$-0.446073\pi$
$491$	7.47214	0.337213	0.168606	+	0.985683i	$0.446073\pi$
$492$	− 7.75078i	− 0.349432i
$493$	16.5579i	0.745730i
$494$	6.54102	0.294294
$495$	0	0
$496$	21.5958	0.969678
$497$	0	0
$498$	2.29180i	0.102698i
$499$	25.4164	1.13779	0.568897	−	0.822409i	$-0.307370\pi$
$499$	25.4164	1.13779	0.568897	+	0.822409i	$0.307370\pi$
$500$	0	0
$501$	12.4721	0.557214
$502$	4.41887i	0.197224i
$503$	16.8918i	0.753166i	0.926383	+	0.376583i	$0.122901\pi$
$503$	16.8918i	0.753166i	−0.926383	+	0.376583i	$0.877099\pi$
$504$	0	0
$505$	0	0
$506$	3.21478	0.142914
$507$	− 2.87714i	− 0.127778i
$508$	− 12.9787i	− 0.575837i
$509$	−5.78437	−0.256388	−0.128194	−	0.991749i	$-0.540918\pi$
$509$	−5.78437	−0.256388	−0.128194	+	0.991749i	$0.540918\pi$
$510$	0	0
$511$	0	0
$512$	− 22.3050i	− 0.985749i
$513$	− 19.4164i	− 0.857255i
$514$	−4.62520	−0.204009
$515$	0	0
$516$	−8.10272	−0.356702
$517$	21.2132i	0.932956i
$518$	0	0
$519$	−3.70820	−0.162772
$520$	0	0
$521$	36.0230	1.57820	0.789099	−	0.614266i	$-0.210548\pi$
$521$	36.0230	1.57820	0.789099	+	0.614266i	$0.210548\pi$
$522$	1.90983i	0.0835910i
$523$	− 3.24109i	− 0.141723i	−0.997486	−	0.0708615i	$-0.977425\pi$
$523$	− 3.24109i	− 0.141723i	0.997486	−	0.0708615i	$-0.0225748\pi$
$524$	−27.4589	−1.19955
$525$	0	0
$526$	11.4377	0.498707
$527$	− 50.8328i	− 2.21431i
$528$	6.14833i	0.267572i
$529$	8.83282	0.384035
$530$	0	0
$531$	29.4922	1.27985
$532$	0	0
$533$	19.3050i	0.836190i
$534$	−2.65248	−0.114784
$535$	0	0
$536$	−12.8197	−0.553725
$537$	− 3.24109i	− 0.139863i
$538$	− 5.86319i	− 0.252780i
$539$	0	0
$540$	0	0
$541$	−8.70820	−0.374395	−0.187197	−	0.982322i	$-0.559940\pi$
$541$	−8.70820	−0.374395	−0.187197	+	0.982322i	$0.559940\pi$
$542$	1.69936i	0.0729936i
$543$	9.70820i	0.416619i
$544$	−30.7000	−1.31625
$545$	0	0
$546$	0	0
$547$	− 29.0000i	− 1.23995i	−0.784621	−	0.619975i	$-0.787143\pi$
$547$	− 29.0000i	− 1.23995i	0.784621	−	0.619975i	$-0.212857\pi$
$548$	11.1246i	0.475220i
$549$	−6.78593	−0.289616
$550$	0	0
$551$	9.48683	0.404153
$552$	4.84303i	0.206133i
$553$	0	0
$554$	9.16718	0.389476
$555$	0	0
$556$	36.3268	1.54060
$557$	− 24.5967i	− 1.04220i	−0.853496	−	0.521099i	$-0.825522\pi$
$557$	− 24.5967i	− 1.04220i	0.853496	−	0.521099i	$-0.174478\pi$
$558$	− 5.86319i	− 0.248208i
$559$	20.1815	0.853589
$560$	0	0
$561$	14.4721	0.611014
$562$	9.39512i	0.396309i
$563$	− 13.8083i	− 0.581950i	−0.956731	−	0.290975i	$-0.906020\pi$
$563$	− 13.8083i	− 0.581950i	0.956731	−	0.290975i	$-0.0939796\pi$
$564$	−15.3738	−0.647355
$565$	0	0
$566$	−7.94510	−0.333957
$567$	0	0
$568$	11.0000i	0.461550i
$569$	29.9443	1.25533	0.627665	−	0.778484i	$-0.284011\pi$
$569$	29.9443	1.25533	0.627665	+	0.778484i	$0.284011\pi$
$570$	0	0
$571$	26.4164	1.10549	0.552746	−	0.833350i	$-0.313580\pi$
$571$	26.4164	1.10549	0.552746	+	0.833350i	$0.313580\pi$
$572$	− 16.7341i	− 0.699689i
$573$	− 20.2117i	− 0.844354i
$574$	0	0
$575$	0	0
$576$	10.5279	0.438661
$577$	− 42.1900i	− 1.75639i	−0.478301	−	0.878196i	$-0.658747\pi$
$577$	− 42.1900i	− 1.75639i	0.478301	−	0.878196i	$-0.341253\pi$
$578$	14.4508i	0.601076i
$579$	9.35931	0.388960
$580$	0	0
$581$	0	0
$582$	3.37384i	0.139850i
$583$	21.7082i	0.899062i
$584$	3.86008	0.159731
$585$	0	0
$586$	7.04096	0.290860
$587$	− 5.86319i	− 0.242000i	−0.992653	−	0.121000i	$-0.961390\pi$
$587$	− 5.86319i	− 0.242000i	0.992653	−	0.121000i	$-0.0386100\pi$
$588$	0	0
$589$	−29.1246	−1.20006
$590$	0	0
$591$	−5.19548	−0.213714
$592$	− 33.6869i	− 1.38452i
$593$	− 30.7000i	− 1.26070i	−0.776311	−	0.630350i	$-0.782912\pi$
$593$	− 30.7000i	− 1.26070i	0.776311	−	0.630350i	$-0.217088\pi$
$594$	3.90879	0.160380
$595$	0	0
$596$	−17.8967	−0.733076
$597$	− 22.0689i	− 0.903219i
$598$	− 5.80298i	− 0.237301i
$599$	6.81966	0.278644	0.139322	−	0.990247i	$-0.455508\pi$
$599$	6.81966	0.278644	0.139322	+	0.990247i	$0.455508\pi$
$600$	0	0
$601$	−29.4621	−1.20178	−0.600891	−	0.799331i	$-0.705187\pi$
$601$	−29.4621	−1.20178	−0.600891	+	0.799331i	$0.705187\pi$
$602$	0	0
$603$	− 19.4721i	− 0.792967i
$604$	15.6049	0.634953
$605$	0	0
$606$	3.70820	0.150635
$607$	45.2247i	1.83562i	0.397025	+	0.917808i	$0.370042\pi$
$607$	45.2247i	1.83562i	−0.397025	+	0.917808i	$0.629958\pi$
$608$	17.5896i	0.713351i
$609$	0	0
$610$	0	0
$611$	38.2918	1.54912
$612$	− 30.7000i	− 1.24098i
$613$	14.4164i	0.582273i	0.956681	+	0.291137i	$0.0940334\pi$
$613$	14.4164i	0.582273i	−0.956681	+	0.291137i	$0.905967\pi$
$614$	−5.55944	−0.224361
$615$	0	0
$616$	0	0
$617$	− 19.3607i	− 0.779432i	−0.920935	−	0.389716i	$-0.872573\pi$
$617$	− 19.3607i	− 0.779432i	0.920935	−	0.389716i	$-0.127427\pi$
$618$	3.16718i	0.127403i
$619$	9.28050	0.373015	0.186507	−	0.982454i	$-0.440283\pi$
$619$	9.28050	0.373015	0.186507	+	0.982454i	$0.440283\pi$
$620$	0	0
$621$	−17.2256	−0.691240
$622$	3.65375i	0.146502i
$623$	0	0
$624$	11.0983	0.444288
$625$	0	0
$626$	8.10272	0.323850
$627$	− 8.29180i	− 0.331142i
$628$	10.1058i	0.403266i
$629$	−79.2934	−3.16163
$630$	0	0
$631$	−28.1246	−1.11962	−0.559812	−	0.828620i	$-0.689126\pi$
$631$	−28.1246	−1.11962	−0.559812	+	0.828620i	$0.689126\pi$
$632$	− 6.93112i	− 0.275705i
$633$	6.48218i	0.257643i
$634$	4.56231	0.181192
$635$	0	0
$636$	−15.7326	−0.623837
$637$	0	0
$638$	1.90983i	0.0756109i
$639$	−16.7082	−0.660966
$640$	0	0
$641$	−16.5279	−0.652811	−0.326406	−	0.945230i	$-0.605838\pi$
$641$	−16.5279	−0.652811	−0.326406	+	0.945230i	$0.605838\pi$
$642$	− 2.00310i	− 0.0790562i
$643$	22.2148i	0.876064i	0.898959	+	0.438032i	$0.144324\pi$
$643$	22.2148i	0.876064i	−0.898959	+	0.438032i	$0.855676\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	− 19.0525i	− 0.749030i	−0.927221	−	0.374515i	$-0.877809\pi$
$647$	− 19.0525i	− 0.749030i	0.927221	−	0.374515i	$-0.122191\pi$
$648$	− 3.98684i	− 0.156618i
$649$	29.4922	1.15767
$650$	0	0
$651$	0	0
$652$	− 4.24922i	− 0.166412i
$653$	1.41641i	0.0554283i	0.999616	+	0.0277142i	$0.00882282\pi$
$653$	1.41641i	0.0554283i	−0.999616	+	0.0277142i	$0.991177\pi$
$654$	0.333851	0.0130546
$655$	0	0
$656$	−15.0463	−0.587458
$657$	5.86319i	0.228745i
$658$	0	0
$659$	8.83282	0.344078	0.172039	−	0.985090i	$-0.444965\pi$
$659$	8.83282	0.344078	0.172039	+	0.985090i	$0.444965\pi$
$660$	0	0
$661$	19.9752	0.776946	0.388473	−	0.921460i	$-0.373003\pi$
$661$	19.9752	0.776946	0.388473	+	0.921460i	$0.373003\pi$
$662$	0.493422i	0.0191774i
$663$	− 26.1235i	− 1.01455i
$664$	10.1058	0.392182
$665$	0	0
$666$	−9.14590	−0.354396
$667$	− 8.41641i	− 0.325885i
$668$	− 26.4574i	− 1.02367i
$669$	5.12461	0.198129
$670$	0	0
$671$	−6.78593	−0.261968
$672$	0	0
$673$	− 41.1246i	− 1.58524i	−0.609718	−	0.792619i	$-0.708717\pi$
$673$	− 41.1246i	− 1.58524i	0.609718	−	0.792619i	$-0.291283\pi$
$674$	8.83282	0.340227
$675$	0	0
$676$	−6.10333	−0.234743
$677$	13.8083i	0.530696i	0.964153	+	0.265348i	$0.0854868\pi$
$677$	13.8083i	0.530696i	−0.964153	+	0.265348i	$0.914513\pi$
$678$	3.25969i	0.125188i
$679$	0	0
$680$	0	0
$681$	−23.1246	−0.886137
$682$	− 5.86319i	− 0.224513i
$683$	41.0689i	1.57146i	0.618571	+	0.785729i	$0.287712\pi$
$683$	41.0689i	1.57146i	−0.618571	+	0.785729i	$0.712288\pi$
$684$	−17.5896	−0.672553
$685$	0	0
$686$	0	0
$687$	6.18034i	0.235795i
$688$	15.7295i	0.599681i
$689$	39.1853	1.49284
$690$	0	0
$691$	1.79677	0.0683524	0.0341762	−	0.999416i	$-0.489119\pi$
$691$	1.79677	0.0683524	0.0341762	+	0.999416i	$0.489119\pi$
$692$	7.86629i	0.299031i
$693$	0	0
$694$	−10.5623	−0.400940
$695$	0	0
$696$	−2.87714	−0.109058
$697$	35.4164i	1.34149i
$698$	− 0.382559i	− 0.0144801i
$699$	−3.95750	−0.149686
$700$	0	0
$701$	31.4164	1.18658	0.593291	−	0.804988i	$-0.297828\pi$
$701$	31.4164	1.18658	0.593291	+	0.804988i	$0.297828\pi$
$702$	− 7.05573i	− 0.266301i
$703$	45.4311i	1.71346i
$704$	10.5279	0.396784
$705$	0	0
$706$	−4.62520	−0.174072
$707$	0	0
$708$	21.3738i	0.803278i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	10.5279	0.394826
$712$	11.6963i	0.438336i
$713$	25.8384i	0.967656i
$714$	0	0
$715$	0	0
$716$	−6.87539	−0.256945
$717$	− 25.4558i	− 0.950666i
$718$	0.0212862i	0 0.000794395i
$719$	−1.54173	−0.0574969	−0.0287485	−	0.999587i	$-0.509152\pi$
$719$	−1.54173	−0.0574969	−0.0287485	+	0.999587i	$0.509152\pi$
$720$	0	0
$721$	0	0
$722$	0.381966i	0.0142153i
$723$	− 12.8754i	− 0.478841i
$724$	20.5942	0.765378
$725$	0	0
$726$	−2.00310	−0.0743421
$727$	− 5.65685i	− 0.209801i	−0.994483	−	0.104901i	$-0.966548\pi$
$727$	− 5.65685i	− 0.209801i	0.994483	−	0.104901i	$-0.0334524\pi$
$728$	0	0
$729$	−5.94427	−0.220158
$730$	0	0
$731$	37.0246	1.36940
$732$	− 4.91796i	− 0.181773i
$733$	− 6.48218i	− 0.239425i	−0.992809	−	0.119712i	$-0.961803\pi$
$733$	− 6.48218i	− 0.239425i	0.992809	−	0.119712i	$-0.0381972\pi$
$734$	3.31990	0.122540
$735$	0	0
$736$	15.6049	0.575203
$737$	− 19.4721i	− 0.717265i
$738$	4.08502i	0.150372i
$739$	7.29180	0.268233	0.134117	−	0.990966i	$-0.457180\pi$
$739$	7.29180	0.268233	0.134117	+	0.990966i	$0.457180\pi$
$740$	0	0
$741$	−14.9675	−0.549843
$742$	0	0
$743$	33.7082i	1.23663i	0.785929	+	0.618317i	$0.212185\pi$
$743$	33.7082i	1.23663i	−0.785929	+	0.618317i	$0.787815\pi$
$744$	8.83282	0.323827
$745$	0	0
$746$	−6.15905	−0.225499
$747$	15.3500i	0.561628i
$748$	− 30.7000i	− 1.12250i
$749$	0	0
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	29.8446i	1.08832i
$753$	− 10.1115i	− 0.368482i
$754$	3.44742	0.125548
$755$	0	0
$756$	0	0
$757$	12.4164i	0.451282i	0.974211	+	0.225641i	$0.0724476\pi$
$757$	12.4164i	0.451282i	−0.974211	+	0.225641i	$0.927552\pi$
$758$	− 7.25735i	− 0.263599i
$759$	−7.35621	−0.267014
$760$	0	0
$761$	52.9148	1.91816	0.959080	−	0.283136i	$-0.0913747\pi$
$761$	52.9148	1.91816	0.959080	+	0.283136i	$0.0913747\pi$
$762$	− 2.33695i	− 0.0846589i
$763$	0	0
$764$	−42.8754	−1.55118
$765$	0	0
$766$	−5.03786	−0.182025
$767$	− 53.2361i	− 1.92224i
$768$	4.86163i	0.175429i
$769$	−8.69161	−0.313428	−0.156714	−	0.987644i	$-0.550090\pi$
$769$	−8.69161	−0.313428	−0.156714	+	0.987644i	$0.550090\pi$
$770$	0	0
$771$	10.5836	0.381159
$772$	− 19.8541i	− 0.714565i
$773$	15.3500i	0.552102i	0.961143	+	0.276051i	$0.0890258\pi$
$773$	15.3500i	0.552102i	−0.961143	+	0.276051i	$0.910974\pi$
$774$	4.27051	0.153500
$775$	0	0
$776$	14.8771	0.534059
$777$	0	0
$778$	10.6049i	0.380203i
$779$	20.2918	0.727029
$780$	0	0
$781$	−16.7082	−0.597867
$782$	− 10.6460i	− 0.380700i
$783$	− 10.2333i	− 0.365710i
$784$	0	0
$785$	0	0
$786$	−4.94427	−0.176356
$787$	16.3516i	0.582871i	0.956591	+	0.291435i	$0.0941328\pi$
$787$	16.3516i	0.582871i	−0.956591	+	0.291435i	$0.905867\pi$
$788$	11.0213i	0.392617i
$789$	−26.1723	−0.931757
$790$	0	0
$791$	0	0
$792$	− 7.36068i	− 0.261550i
$793$	12.2492i	0.434983i
$794$	−3.78127	−0.134192
$795$	0	0
$796$	−46.8152	−1.65932
$797$	39.0277i	1.38243i	0.722648	+	0.691216i	$0.242925\pi$
$797$	39.0277i	1.38243i	−0.722648	+	0.691216i	$0.757075\pi$
$798$	0	0
$799$	70.2492	2.48524
$800$	0	0
$801$	−17.7658	−0.627723
$802$	− 8.81153i	− 0.311146i
$803$	5.86319i	0.206907i
$804$	14.1120	0.497693
$805$	0	0
$806$	−10.5836	−0.372791
$807$	13.4164i	0.472280i
$808$	− 16.3516i	− 0.575246i
$809$	−29.0689	−1.02201	−0.511004	−	0.859578i	$-0.670726\pi$
$809$	−29.0689	−1.02201	−0.511004	+	0.859578i	$0.670726\pi$
$810$	0	0
$811$	7.07107	0.248299	0.124149	−	0.992264i	$-0.460380\pi$
$811$	7.07107	0.248299	0.124149	+	0.992264i	$0.460380\pi$
$812$	0	0
$813$	− 3.88854i	− 0.136377i
$814$	−9.14590	−0.320564
$815$	0	0
$816$	20.3607	0.712766
$817$	− 21.2132i	− 0.742156i
$818$	− 5.62675i	− 0.196735i
$819$	0	0
$820$	0	0
$821$	10.5836	0.369370	0.184685	−	0.982798i	$-0.440874\pi$
$821$	10.5836	0.369370	0.184685	+	0.982798i	$0.440874\pi$
$822$	2.00310i	0.0698662i
$823$	− 29.5410i	− 1.02974i	−0.857270	−	0.514868i	$-0.827841\pi$
$823$	− 29.5410i	− 1.02974i	0.857270	−	0.514868i	$-0.172159\pi$
$824$	13.9659	0.486525
$825$	0	0
$826$	0	0
$827$	1.47214i	0.0511912i	0.999672	+	0.0255956i	$0.00814822\pi$
$827$	1.47214i	0.0511912i	−0.999672	+	0.0255956i	$0.991852\pi$
$828$	15.6049i	0.542307i
$829$	−10.0757	−0.349944	−0.174972	−	0.984573i	$-0.555984\pi$
$829$	−10.0757	−0.349944	−0.174972	+	0.984573i	$0.555984\pi$
$830$	0	0
$831$	−20.9768	−0.727676
$832$	− 19.0038i	− 0.658837i
$833$	0	0
$834$	6.54102	0.226497
$835$	0	0
$836$	−17.5896	−0.608348
$837$	31.4164i	1.08591i
$838$	1.85698i	0.0641483i
$839$	31.3190	1.08125	0.540626	−	0.841263i	$-0.318187\pi$
$839$	31.3190	1.08125	0.540626	+	0.841263i	$0.318187\pi$
$840$	0	0
$841$	−24.0000	−0.827586
$842$	3.54915i	0.122312i
$843$	− 21.4983i	− 0.740442i
$844$	13.7508	0.473321
$845$	0	0
$846$	8.10272	0.278577
$847$	0	0
$848$	30.5410i	1.04878i
$849$	18.1803	0.623948
$850$	0	0
$851$	40.3050	1.38164
$852$	− 12.1089i	− 0.414845i
$853$	− 22.2148i	− 0.760619i	−0.924859	−	0.380309i	$-0.875818\pi$
$853$	− 22.2148i	− 0.760619i	0.924859	−	0.380309i	$-0.124182\pi$
$854$	0	0
$855$	0	0
$856$	−8.83282	−0.301899
$857$	− 20.6730i	− 0.706177i	−0.935590	−	0.353088i	$-0.885131\pi$
$857$	− 20.6730i	− 0.706177i	0.935590	−	0.353088i	$-0.114869\pi$
$858$	− 3.01316i	− 0.102867i
$859$	−39.7742	−1.35708	−0.678539	−	0.734564i	$-0.737387\pi$
$859$	−39.7742	−1.35708	−0.678539	+	0.734564i	$0.737387\pi$
$860$	0	0
$861$	0	0
$862$	− 2.58359i	− 0.0879975i
$863$	− 1.36068i	− 0.0463181i	−0.999732	−	0.0231590i	$-0.992628\pi$
$863$	− 1.36068i	− 0.0463181i	0.999732	−	0.0231590i	$-0.00737241\pi$
$864$	18.9737	0.645497
$865$	0	0
$866$	−12.4915	−0.424478
$867$	− 33.0671i	− 1.12302i
$868$	0	0
$869$	10.5279	0.357133
$870$	0	0
$871$	−35.1490	−1.19098
$872$	− 1.47214i	− 0.0498528i
$873$	22.5973i	0.764803i
$874$	−6.09962	−0.206323
$875$	0	0
$876$	−4.24922	−0.143568
$877$	38.2918i	1.29302i	0.762905	+	0.646511i	$0.223773\pi$
$877$	38.2918i	1.29302i	−0.762905	+	0.646511i	$0.776227\pi$
$878$	6.78593i	0.229014i
$879$	−16.1115	−0.543426
$880$	0	0
$881$	−1.62054	−0.0545975	−0.0272988	−	0.999627i	$-0.508691\pi$
$881$	−1.62054	−0.0545975	−0.0272988	+	0.999627i	$0.508691\pi$
$882$	0	0
$883$	− 32.7082i	− 1.10072i	−0.834928	−	0.550359i	$-0.814491\pi$
$883$	− 32.7082i	− 1.10072i	0.834928	−	0.550359i	$-0.185509\pi$
$884$	−55.4164	−1.86386
$885$	0	0
$886$	7.70820	0.258962
$887$	24.9945i	0.839232i	0.907702	+	0.419616i	$0.137835\pi$
$887$	24.9945i	0.839232i	−0.907702	+	0.419616i	$0.862165\pi$
$888$	− 13.7782i	− 0.462366i
$889$	0	0
$890$	0	0
$891$	6.05573	0.202875
$892$	− 10.8709i	− 0.363986i
$893$	− 40.2492i	− 1.34689i
$894$	−3.22248	−0.107776
$895$	0	0
$896$	0	0
$897$	13.2786i	0.443361i
$898$	− 4.81153i	− 0.160563i
$899$	−15.3500	−0.511952
$900$	0	0
$901$	71.8885	2.39495
$902$	4.08502i	0.136016i
$903$	0	0
$904$	14.3738	0.478067
$905$	0	0
$906$	2.80982	0.0933501
$907$	14.8328i	0.492516i	0.969204	+	0.246258i	$0.0792010\pi$
$907$	14.8328i	0.492516i	−0.969204	+	0.246258i	$0.920799\pi$
$908$	49.0547i	1.62794i
$909$	24.8369	0.823786
$910$	0	0
$911$	21.7639	0.721071	0.360536	−	0.932745i	$-0.382594\pi$
$911$	21.7639	0.721071	0.360536	+	0.932745i	$0.382594\pi$
$912$	− 11.6656i	− 0.386288i
$913$	15.3500i	0.508011i
$914$	−2.78522	−0.0921268
$915$	0	0
$916$	13.1105	0.433182
$917$	0	0
$918$	− 12.9443i	− 0.427225i
$919$	−31.8328	−1.05007	−0.525034	−	0.851081i	$-0.675947\pi$
$919$	−31.8328	−1.05007	−0.525034	+	0.851081i	$0.675947\pi$
$920$	0	0
$921$	12.7214	0.419183
$922$	12.7580i	0.420163i
$923$	30.1599i	0.992724i
$924$	0	0
$925$	0	0
$926$	−5.34752	−0.175731
$927$	21.2132i	0.696733i
$928$	9.27051i	0.304319i
$929$	47.0516	1.54371	0.771857	−	0.635797i	$-0.219328\pi$
$929$	47.0516	1.54371	0.771857	+	0.635797i	$0.219328\pi$
$930$	0	0
$931$	0	0
$932$	8.39512i	0.274991i
$933$	− 8.36068i	− 0.273716i
$934$	3.68385	0.120539
$935$	0	0
$936$	−13.2867	−0.434290
$937$	28.4906i	0.930747i	0.885114	+	0.465374i	$0.154080\pi$
$937$	28.4906i	0.930747i	−0.885114	+	0.465374i	$0.845920\pi$
$938$	0	0
$939$	−18.5410	−0.605063
$940$	0	0
$941$	3.62365	0.118128	0.0590638	−	0.998254i	$-0.481188\pi$
$941$	3.62365	0.118128	0.0590638	+	0.998254i	$0.481188\pi$
$942$	1.81966i	0.0592877i
$943$	− 18.0022i	− 0.586233i
$944$	41.4922	1.35046
$945$	0	0
$946$	4.27051	0.138846
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	7.62985i	0.247806i
$949$	10.5836	0.343558
$950$	0	0
$951$	−10.4397	−0.338530
$952$	0	0
$953$	− 36.5967i	− 1.18548i	−0.805392	−	0.592742i	$-0.798045\pi$
$953$	− 36.5967i	− 1.18548i	0.805392	−	0.592742i	$-0.201955\pi$
$954$	8.29180	0.268457
$955$	0	0
$956$	−54.0000	−1.74648
$957$	− 4.37016i	− 0.141267i
$958$	− 3.44742i	− 0.111381i
$959$	0	0
$960$	0	0
$961$	16.1246	0.520149
$962$	16.5092i	0.532278i
$963$	− 13.4164i	− 0.432338i
$964$	−27.3128	−0.879687
$965$	0	0
$966$	0	0
$967$	46.2492i	1.48727i	0.668583	+	0.743637i	$0.266901\pi$
$967$	46.2492i	1.48727i	−0.668583	+	0.743637i	$0.733099\pi$
$968$	8.83282i	0.283897i
$969$	−27.4589	−0.882108
$970$	0	0
$971$	−12.7279	−0.408458	−0.204229	−	0.978923i	$-0.565469\pi$
$971$	−12.7279	−0.408458	−0.204229	+	0.978923i	$0.565469\pi$
$972$	29.8446i	0.957266i
$973$	0	0
$974$	−10.9656	−0.351359
$975$	0	0
$976$	−9.54704	−0.305593
$977$	27.7639i	0.888247i	0.895966	+	0.444123i	$0.146485\pi$
$977$	27.7639i	0.888247i	−0.895966	+	0.444123i	$0.853515\pi$
$978$	− 0.765117i	− 0.0244658i
$979$	−17.7658	−0.567797
$980$	0	0
$981$	2.23607	0.0713922
$982$	− 2.85410i	− 0.0910781i
$983$	8.10272i	0.258437i	0.991616	+	0.129218i	$0.0412468\pi$
$983$	8.10272i	0.258437i	−0.991616	+	0.129218i	$0.958753\pi$
$984$	−6.15403	−0.196183
$985$	0	0
$986$	6.32456	0.201415
$987$	0	0
$988$	31.7508i	1.01013i
$989$	−18.8197	−0.598430
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	− 28.4605i	− 0.903622i
$993$	− 1.12907i	− 0.0358300i
$994$	0	0
$995$	0	0
$996$	−11.1246	−0.352497
$997$	48.6722i	1.54146i	0.637160	+	0.770731i	$0.280109\pi$
$997$	48.6722i	1.54146i	−0.637160	+	0.770731i	$0.719891\pi$
$998$	− 9.70820i	− 0.307308i
$999$	49.0060	1.55048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.n.99.4		8
5.2	odd	4		1225.2.a.bc.1.2	yes	4
5.3	odd	4		1225.2.a.ba.1.3	✓	4
5.4	even	2	inner	1225.2.b.n.99.5		8
7.6	odd	2	inner	1225.2.b.n.99.3		8
35.13	even	4		1225.2.a.ba.1.4	yes	4
35.27	even	4		1225.2.a.bc.1.1	yes	4
35.34	odd	2	inner	1225.2.b.n.99.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.ba.1.3	✓	4	5.3	odd	4
1225.2.a.ba.1.4	yes	4	35.13	even	4
1225.2.a.bc.1.1	yes	4	35.27	even	4
1225.2.a.bc.1.2	yes	4	5.2	odd	4
1225.2.b.n.99.3		8	7.6	odd	2	inner
1225.2.b.n.99.4		8	1.1	even	1	trivial
1225.2.b.n.99.5		8	5.4	even	2	inner
1225.2.b.n.99.6		8	35.34	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.381966i q^{2} +0.874032i q^{3} +1.85410 q^{4} +0.333851 q^{6} -1.47214i q^{8} +2.23607 q^{9} +2.23607 q^{11} +1.62054i q^{12} -4.03631i q^{13} +3.14590 q^{16} -7.40492i q^{17} -0.854102i q^{18} -4.24264 q^{19} -0.854102i q^{22} +3.76393i q^{23} +1.28669 q^{24} -1.54173 q^{26} +4.57649i q^{27} -2.23607 q^{29} +6.86474 q^{31} -4.14590i q^{32} +1.95440i q^{33} -2.82843 q^{34} +4.14590 q^{36} -10.7082i q^{37} +1.62054i q^{38} +3.52786 q^{39} -4.78282 q^{41} +5.00000i q^{43} +4.14590 q^{44} +1.43769 q^{46} +9.48683i q^{47} +2.74962i q^{48} +6.47214 q^{51} -7.48373i q^{52} +9.70820i q^{53} +1.74806 q^{54} -3.70820i q^{57} +0.854102i q^{58} +13.1893 q^{59} -3.03476 q^{61} -2.62210i q^{62} +4.70820 q^{64} +0.746512 q^{66} -8.70820i q^{67} -13.7295i q^{68} -3.28980 q^{69} -7.47214 q^{71} -3.29180i q^{72} +2.62210i q^{73} -4.09017 q^{74} -7.86629 q^{76} -1.34752i q^{78} +4.70820 q^{79} +2.70820 q^{81} +1.82688i q^{82} +6.86474i q^{83} +1.90983 q^{86} -1.95440i q^{87} -3.29180i q^{88} -7.94510 q^{89} +6.97871i q^{92} +6.00000i q^{93} +3.62365 q^{94} +3.62365 q^{96} +10.1058i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4} + 52 q^{16} + 60 q^{36} + 64 q^{39} + 60 q^{44} + 92 q^{46} + 16 q^{51} - 16 q^{64} - 24 q^{71} + 12 q^{74} - 16 q^{79} - 32 q^{81} + 60 q^{86} + 40 q^{99}+O(q^{100}) \) 8 * q - 12 * q^4 + 52 * q^16 + 60 * q^36 + 64 * q^39 + 60 * q^44 + 92 * q^46 + 16 * q^51 - 16 * q^64 - 24 * q^71 + 12 * q^74 - 16 * q^79 - 32 * q^81 + 60 * q^86 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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