LMFDB - Embedded newform 5625.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

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:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.741379$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.741379 q^{2} -1.45036 q^{4} +1.03586 q^{7} +2.55802 q^{8} +O(q^{10})$	$q-0.741379 q^{2} -1.45036 q^{4} +1.03586 q^{7} +2.55802 q^{8} +0.513903 q^{11} +3.54391 q^{13} -0.767968 q^{14} +1.00425 q^{16} +1.36691 q^{17} +0.894573 q^{19} -0.380997 q^{22} -5.45465 q^{23} -2.62738 q^{26} -1.50237 q^{28} +9.65038 q^{29} +10.4630 q^{31} -5.86057 q^{32} -1.01340 q^{34} -2.19473 q^{37} -0.663218 q^{38} -3.12460 q^{41} +10.2866 q^{43} -0.745343 q^{44} +4.04396 q^{46} -7.65509 q^{47} -5.92699 q^{49} -5.13993 q^{52} +10.5524 q^{53} +2.64976 q^{56} -7.15459 q^{58} -11.9238 q^{59} +7.85170 q^{61} -7.75704 q^{62} +2.33641 q^{64} +8.80731 q^{67} -1.98251 q^{68} -5.00948 q^{71} +5.82505 q^{73} +1.62713 q^{74} -1.29745 q^{76} +0.532334 q^{77} -6.74036 q^{79} +2.31651 q^{82} +7.99598 q^{83} -7.62626 q^{86} +1.31458 q^{88} -12.9917 q^{89} +3.67101 q^{91} +7.91118 q^{92} +5.67533 q^{94} -7.27747 q^{97} +4.39414 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−0.741379	−0.524234	−0.262117	−	0.965036i	$-0.584421\pi$
$2$	−0.741379	−0.524234	−0.262117	+	0.965036i	$0.584421\pi$
$3$	0	0
$3$	0	0
$4$	−1.45036	−0.725178
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.03586	0.391520	0.195760	−	0.980652i	$-0.437283\pi$
$7$	1.03586	0.391520	0.195760	+	0.980652i	$0.437283\pi$
$8$	2.55802	0.904398
$9$	0	0
$10$	0	0
$11$	0.513903	0.154948	0.0774738	−	0.996994i	$-0.475315\pi$
$11$	0.513903	0.154948	0.0774738	+	0.996994i	$0.475315\pi$
$12$	0	0
$13$	3.54391	0.982904	0.491452	−	0.870905i	$-0.336466\pi$
$13$	3.54391	0.982904	0.491452	+	0.870905i	$0.336466\pi$
$14$	−0.767968	−0.205248
$15$	0	0
$16$	1.00425	0.251062
$17$	1.36691	0.331525	0.165762	−	0.986166i	$-0.446992\pi$
$17$	1.36691	0.331525	0.165762	+	0.986166i	$0.446992\pi$
$18$	0	0
$19$	0.894573	0.205229	0.102615	−	0.994721i	$-0.467279\pi$
$19$	0.894573	0.205229	0.102615	+	0.994721i	$0.467279\pi$
$20$	0	0
$21$	0	0
$22$	−0.380997	−0.0812289
$23$	−5.45465	−1.13737	−0.568686	−	0.822554i	$-0.692548\pi$
$23$	−5.45465	−1.13737	−0.568686	+	0.822554i	$0.692548\pi$
$24$	0	0
$25$	0	0
$26$	−2.62738	−0.515272
$27$	0	0
$28$	−1.50237	−0.283922
$29$	9.65038	1.79203	0.896015	−	0.444024i	$-0.146450\pi$
$29$	9.65038	1.79203	0.896015	+	0.444024i	$0.146450\pi$
$30$	0	0
$31$	10.4630	1.87921	0.939604	−	0.342265i	$-0.111194\pi$
$31$	10.4630	1.87921	0.939604	+	0.342265i	$0.111194\pi$
$32$	−5.86057	−1.03601
$33$	0	0
$34$	−1.01340	−0.173797
$35$	0	0
$36$	0	0
$37$	−2.19473	−0.360812	−0.180406	−	0.983592i	$-0.557741\pi$
$37$	−2.19473	−0.360812	−0.180406	+	0.983592i	$0.557741\pi$
$38$	−0.663218	−0.107588
$39$	0	0
$40$	0	0
$41$	−3.12460	−0.487980	−0.243990	−	0.969778i	$-0.578456\pi$
$41$	−3.12460	−0.487980	−0.243990	+	0.969778i	$0.578456\pi$
$42$	0	0
$43$	10.2866	1.56869	0.784345	−	0.620325i	$-0.212999\pi$
$43$	10.2866	1.56869	0.784345	+	0.620325i	$0.212999\pi$
$44$	−0.745343	−0.112365
$45$	0	0
$46$	4.04396	0.596250
$47$	−7.65509	−1.11661	−0.558305	−	0.829636i	$-0.688548\pi$
$47$	−7.65509	−1.11661	−0.558305	+	0.829636i	$0.688548\pi$
$48$	0	0
$49$	−5.92699	−0.846712
$50$	0	0
$51$	0	0
$52$	−5.13993	−0.712780
$53$	10.5524	1.44948	0.724742	−	0.689020i	$-0.241959\pi$
$53$	10.5524	1.44948	0.724742	+	0.689020i	$0.241959\pi$
$54$	0	0
$55$	0	0
$56$	2.64976	0.354090
$57$	0	0
$58$	−7.15459	−0.939443
$59$	−11.9238	−1.55234	−0.776172	−	0.630521i	$-0.782841\pi$
$59$	−11.9238	−1.55234	−0.776172	+	0.630521i	$0.782841\pi$
$60$	0	0
$61$	7.85170	1.00531	0.502653	−	0.864488i	$-0.332357\pi$
$61$	7.85170	1.00531	0.502653	+	0.864488i	$0.332357\pi$
$62$	−7.75704	−0.985145
$63$	0	0
$64$	2.33641	0.292052
$65$	0	0
$66$	0	0
$67$	8.80731	1.07598	0.537992	−	0.842950i	$-0.319183\pi$
$67$	8.80731	1.07598	0.537992	+	0.842950i	$0.319183\pi$
$68$	−1.98251	−0.240414
$69$	0	0
$70$	0	0
$71$	−5.00948	−0.594516	−0.297258	−	0.954797i	$-0.596072\pi$
$71$	−5.00948	−0.594516	−0.297258	+	0.954797i	$0.596072\pi$
$72$	0	0
$73$	5.82505	0.681770	0.340885	−	0.940105i	$-0.389273\pi$
$73$	5.82505	0.681770	0.340885	+	0.940105i	$0.389273\pi$
$74$	1.62713	0.189150
$75$	0	0
$76$	−1.29745	−0.148828
$77$	0.532334	0.0606651
$78$	0	0
$79$	−6.74036	−0.758350	−0.379175	−	0.925325i	$-0.623792\pi$
$79$	−6.74036	−0.758350	−0.379175	+	0.925325i	$0.623792\pi$
$80$	0	0
$81$	0	0
$82$	2.31651	0.255816
$83$	7.99598	0.877673	0.438837	−	0.898567i	$-0.355391\pi$
$83$	7.99598	0.877673	0.438837	+	0.898567i	$0.355391\pi$
$84$	0	0
$85$	0	0
$86$	−7.62626	−0.822361
$87$	0	0
$88$	1.31458	0.140134
$89$	−12.9917	−1.37711	−0.688557	−	0.725182i	$-0.741756\pi$
$89$	−12.9917	−1.37711	−0.688557	+	0.725182i	$0.741756\pi$
$90$	0	0
$91$	3.67101	0.384826
$92$	7.91118	0.824798
$93$	0	0
$94$	5.67533	0.585365
$95$	0	0
$96$	0	0
$97$	−7.27747	−0.738916	−0.369458	−	0.929248i	$-0.620457\pi$
$97$	−7.27747	−0.738916	−0.369458	+	0.929248i	$0.620457\pi$
$98$	4.39414	0.443876
$99$	0	0
$100$	0	0
$101$	−11.5323	−1.14751	−0.573753	−	0.819028i	$-0.694513\pi$
$101$	−11.5323	−1.14751	−0.573753	+	0.819028i	$0.694513\pi$
$102$	0	0
$103$	−1.95586	−0.192717	−0.0963585	−	0.995347i	$-0.530720\pi$
$103$	−1.95586	−0.192717	−0.0963585	+	0.995347i	$0.530720\pi$
$104$	9.06540	0.888936
$105$	0	0
$106$	−7.82333	−0.759869
$107$	15.2874	1.47789	0.738945	−	0.673766i	$-0.235324\pi$
$107$	15.2874	1.47789	0.738945	+	0.673766i	$0.235324\pi$
$108$	0	0
$109$	−0.346596	−0.0331979	−0.0165989	−	0.999862i	$-0.505284\pi$
$109$	−0.346596	−0.0331979	−0.0165989	+	0.999862i	$0.505284\pi$
$110$	0	0
$111$	0	0
$112$	1.04026	0.0982958
$113$	3.91425	0.368222	0.184111	−	0.982905i	$-0.441059\pi$
$113$	3.91425	0.368222	0.184111	+	0.982905i	$0.441059\pi$
$114$	0	0
$115$	0	0
$116$	−13.9965	−1.29954
$117$	0	0
$118$	8.84004	0.813792
$119$	1.41593	0.129798
$120$	0	0
$121$	−10.7359	−0.975991
$122$	−5.82108	−0.527016
$123$	0	0
$124$	−15.1751	−1.36276
$125$	0	0
$126$	0	0
$127$	−6.02350	−0.534499	−0.267249	−	0.963627i	$-0.586115\pi$
$127$	−6.02350	−0.534499	−0.267249	+	0.963627i	$0.586115\pi$
$128$	9.98898	0.882910
$129$	0	0
$130$	0	0
$131$	−1.26444	−0.110474	−0.0552372	−	0.998473i	$-0.517592\pi$
$131$	−1.26444	−0.110474	−0.0552372	+	0.998473i	$0.517592\pi$
$132$	0	0
$133$	0.926656	0.0803513
$134$	−6.52956	−0.564068
$135$	0	0
$136$	3.49659	0.299830
$137$	−1.85695	−0.158650	−0.0793249	−	0.996849i	$-0.525276\pi$
$137$	−1.85695	−0.158650	−0.0793249	+	0.996849i	$0.525276\pi$
$138$	0	0
$139$	17.4734	1.48207	0.741037	−	0.671464i	$-0.234334\pi$
$139$	17.4734	1.48207	0.741037	+	0.671464i	$0.234334\pi$
$140$	0	0
$141$	0	0
$142$	3.71392	0.311666
$143$	1.82123	0.152299
$144$	0	0
$145$	0	0
$146$	−4.31857	−0.357407
$147$	0	0
$148$	3.18314	0.261653
$149$	−20.5555	−1.68397	−0.841985	−	0.539502i	$-0.818613\pi$
$149$	−20.5555	−1.68397	−0.841985	+	0.539502i	$0.818613\pi$
$150$	0	0
$151$	2.86944	0.233512	0.116756	−	0.993161i	$-0.462750\pi$
$151$	2.86944	0.233512	0.116756	+	0.993161i	$0.462750\pi$
$152$	2.28834	0.185609
$153$	0	0
$154$	−0.394661	−0.0318027
$155$	0	0
$156$	0	0
$157$	−14.3887	−1.14834	−0.574172	−	0.818735i	$-0.694676\pi$
$157$	−14.3887	−1.14834	−0.574172	+	0.818735i	$0.694676\pi$
$158$	4.99716	0.397553
$159$	0	0
$160$	0	0
$161$	−5.65027	−0.445304
$162$	0	0
$163$	−7.90445	−0.619125	−0.309562	−	0.950879i	$-0.600183\pi$
$163$	−7.90445	−0.619125	−0.309562	+	0.950879i	$0.600183\pi$
$164$	4.53178	0.353873
$165$	0	0
$166$	−5.92806	−0.460106
$167$	18.9057	1.46297	0.731483	−	0.681859i	$-0.238828\pi$
$167$	18.9057	1.46297	0.731483	+	0.681859i	$0.238828\pi$
$168$	0	0
$169$	−0.440707	−0.0339006
$170$	0	0
$171$	0	0
$172$	−14.9192	−1.13758
$173$	4.41184	0.335426	0.167713	−	0.985836i	$-0.446362\pi$
$173$	4.41184	0.335426	0.167713	+	0.985836i	$0.446362\pi$
$174$	0	0
$175$	0	0
$176$	0.516086	0.0389015
$177$	0	0
$178$	9.63175	0.721931
$179$	11.6515	0.870876	0.435438	−	0.900219i	$-0.356593\pi$
$179$	11.6515	0.870876	0.435438	+	0.900219i	$0.356593\pi$
$180$	0	0
$181$	15.5625	1.15675	0.578375	−	0.815771i	$-0.303687\pi$
$181$	15.5625	1.15675	0.578375	+	0.815771i	$0.303687\pi$
$182$	−2.72161	−0.201739
$183$	0	0
$184$	−13.9531	−1.02864
$185$	0	0
$186$	0	0
$187$	0.702460	0.0513690
$188$	11.1026	0.809741
$189$	0	0
$190$	0	0
$191$	21.4213	1.54999	0.774996	−	0.631966i	$-0.217752\pi$
$191$	21.4213	1.54999	0.774996	+	0.631966i	$0.217752\pi$
$192$	0	0
$193$	−3.32583	−0.239398	−0.119699	−	0.992810i	$-0.538193\pi$
$193$	−3.32583	−0.239398	−0.119699	+	0.992810i	$0.538193\pi$
$194$	5.39537	0.387365
$195$	0	0
$196$	8.59624	0.614017
$197$	−14.8102	−1.05518	−0.527591	−	0.849498i	$-0.676905\pi$
$197$	−14.8102	−1.05518	−0.527591	+	0.849498i	$0.676905\pi$
$198$	0	0
$199$	−15.3279	−1.08657	−0.543285	−	0.839549i	$-0.682820\pi$
$199$	−15.3279	−1.08657	−0.543285	+	0.839549i	$0.682820\pi$
$200$	0	0
$201$	0	0
$202$	8.54981	0.601562
$203$	9.99648	0.701615
$204$	0	0
$205$	0	0
$206$	1.45004	0.101029
$207$	0	0
$208$	3.55896	0.246770
$209$	0.459724	0.0317998
$210$	0	0
$211$	−0.377050	−0.0259572	−0.0129786	−	0.999916i	$-0.504131\pi$
$211$	−0.377050	−0.0259572	−0.0129786	+	0.999916i	$0.504131\pi$
$212$	−15.3047	−1.05113
$213$	0	0
$214$	−11.3338	−0.774761
$215$	0	0
$216$	0	0
$217$	10.8382	0.735747
$218$	0.256959	0.0174035
$219$	0	0
$220$	0	0
$221$	4.84421	0.325857
$222$	0	0
$223$	0.270144	0.0180902	0.00904509	−	0.999959i	$-0.497121\pi$
$223$	0.270144	0.0180902	0.00904509	+	0.999959i	$0.497121\pi$
$224$	−6.07076	−0.405620
$225$	0	0
$226$	−2.90194	−0.193034
$227$	−7.14071	−0.473946	−0.236973	−	0.971516i	$-0.576155\pi$
$227$	−7.14071	−0.473946	−0.236973	+	0.971516i	$0.576155\pi$
$228$	0	0
$229$	24.8881	1.64465	0.822325	−	0.569018i	$-0.192677\pi$
$229$	24.8881	1.64465	0.822325	+	0.569018i	$0.192677\pi$
$230$	0	0
$231$	0	0
$232$	24.6859	1.62071
$233$	16.9547	1.11074	0.555368	−	0.831604i	$-0.312577\pi$
$233$	16.9547	1.11074	0.555368	+	0.831604i	$0.312577\pi$
$234$	0	0
$235$	0	0
$236$	17.2937	1.12573
$237$	0	0
$238$	−1.04974	−0.0680448
$239$	4.12456	0.266796	0.133398	−	0.991063i	$-0.457411\pi$
$239$	4.12456	0.266796	0.133398	+	0.991063i	$0.457411\pi$
$240$	0	0
$241$	−5.94241	−0.382784	−0.191392	−	0.981514i	$-0.561300\pi$
$241$	−5.94241	−0.382784	−0.191392	+	0.981514i	$0.561300\pi$
$242$	7.95938	0.511648
$243$	0	0
$244$	−11.3878	−0.729027
$245$	0	0
$246$	0	0
$247$	3.17029	0.201721
$248$	26.7646	1.69955
$249$	0	0
$250$	0	0
$251$	18.1052	1.14279	0.571394	−	0.820676i	$-0.306403\pi$
$251$	18.1052	1.14279	0.571394	+	0.820676i	$0.306403\pi$
$252$	0	0
$253$	−2.80316	−0.176233
$254$	4.46570	0.280203
$255$	0	0
$256$	−12.0784	−0.754903
$257$	23.9627	1.49475	0.747375	−	0.664402i	$-0.231314\pi$
$257$	23.9627	1.49475	0.747375	+	0.664402i	$0.231314\pi$
$258$	0	0
$259$	−2.27344	−0.141265
$260$	0	0
$261$	0	0
$262$	0.937428	0.0579145
$263$	13.2525	0.817184	0.408592	−	0.912717i	$-0.366020\pi$
$263$	13.2525	0.817184	0.408592	+	0.912717i	$0.366020\pi$
$264$	0	0
$265$	0	0
$266$	−0.687004	−0.0421229
$267$	0	0
$268$	−12.7737	−0.780281
$269$	0.728075	0.0443915	0.0221958	−	0.999754i	$-0.492934\pi$
$269$	0.728075	0.0443915	0.0221958	+	0.999754i	$0.492934\pi$
$270$	0	0
$271$	−16.9085	−1.02712	−0.513558	−	0.858055i	$-0.671673\pi$
$271$	−16.9085	−1.02712	−0.513558	+	0.858055i	$0.671673\pi$
$272$	1.37272	0.0832332
$273$	0	0
$274$	1.37670	0.0831697
$275$	0	0
$276$	0	0
$277$	10.0386	0.603160	0.301580	−	0.953441i	$-0.402486\pi$
$277$	10.0386	0.603160	0.301580	+	0.953441i	$0.402486\pi$
$278$	−12.9544	−0.776954
$279$	0	0
$280$	0	0
$281$	−20.5358	−1.22507	−0.612533	−	0.790445i	$-0.709849\pi$
$281$	−20.5358	−1.22507	−0.612533	+	0.790445i	$0.709849\pi$
$282$	0	0
$283$	21.2483	1.26308	0.631541	−	0.775343i	$-0.282423\pi$
$283$	21.2483	1.26308	0.631541	+	0.775343i	$0.282423\pi$
$284$	7.26553	0.431130
$285$	0	0
$286$	−1.35022	−0.0798402
$287$	−3.23666	−0.191054
$288$	0	0
$289$	−15.1316	−0.890091
$290$	0	0
$291$	0	0
$292$	−8.44840	−0.494405
$293$	29.8828	1.74577	0.872885	−	0.487927i	$-0.162247\pi$
$293$	29.8828	1.74577	0.872885	+	0.487927i	$0.162247\pi$
$294$	0	0
$295$	0	0
$296$	−5.61417	−0.326317
$297$	0	0
$298$	15.2394	0.882794
$299$	−19.3308	−1.11793
$300$	0	0
$301$	10.6555	0.614173
$302$	−2.12735	−0.122415
$303$	0	0
$304$	0.898374	0.0515253
$305$	0	0
$306$	0	0
$307$	−0.531050	−0.0303086	−0.0151543	−	0.999885i	$-0.504824\pi$
$307$	−0.531050	−0.0303086	−0.0151543	+	0.999885i	$0.504824\pi$
$308$	−0.772074	−0.0439930
$309$	0	0
$310$	0	0
$311$	18.2322	1.03385	0.516927	−	0.856030i	$-0.327076\pi$
$311$	18.2322	1.03385	0.516927	+	0.856030i	$0.327076\pi$
$312$	0	0
$313$	29.3523	1.65909	0.829546	−	0.558439i	$-0.188599\pi$
$313$	29.3523	1.65909	0.829546	+	0.558439i	$0.188599\pi$
$314$	10.6675	0.602001
$315$	0	0
$316$	9.77593	0.549939
$317$	32.8989	1.84778	0.923892	−	0.382653i	$-0.124989\pi$
$317$	32.8989	1.84778	0.923892	+	0.382653i	$0.124989\pi$
$318$	0	0
$319$	4.95936	0.277671
$320$	0	0
$321$	0	0
$322$	4.18899	0.233444
$323$	1.22280	0.0680385
$324$	0	0
$325$	0	0
$326$	5.86020	0.324566
$327$	0	0
$328$	−7.99279	−0.441328
$329$	−7.92963	−0.437175
$330$	0	0
$331$	−3.75461	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$331$	−3.75461	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$332$	−11.5970	−0.636470
$333$	0	0
$334$	−14.0163	−0.766937
$335$	0	0
$336$	0	0
$337$	−14.3206	−0.780095	−0.390047	−	0.920795i	$-0.627541\pi$
$337$	−14.3206	−0.780095	−0.390047	+	0.920795i	$0.627541\pi$
$338$	0.326731	0.0177718
$339$	0	0
$340$	0	0
$341$	5.37696	0.291179
$342$	0	0
$343$	−13.3906	−0.723024
$344$	26.3133	1.41872
$345$	0	0
$346$	−3.27085	−0.175842
$347$	−5.91568	−0.317571	−0.158785	−	0.987313i	$-0.550758\pi$
$347$	−5.91568	−0.317571	−0.158785	+	0.987313i	$0.550758\pi$
$348$	0	0
$349$	26.2468	1.40496	0.702479	−	0.711705i	$-0.252077\pi$
$349$	26.2468	1.40496	0.702479	+	0.711705i	$0.252077\pi$
$350$	0	0
$351$	0	0
$352$	−3.01177	−0.160528
$353$	−14.3082	−0.761546	−0.380773	−	0.924669i	$-0.624342\pi$
$353$	−14.3082	−0.761546	−0.380773	+	0.924669i	$0.624342\pi$
$354$	0	0
$355$	0	0
$356$	18.8426	0.998653
$357$	0	0
$358$	−8.63820	−0.456543
$359$	−36.3183	−1.91681	−0.958404	−	0.285416i	$-0.907868\pi$
$359$	−36.3183	−1.91681	−0.958404	+	0.285416i	$0.907868\pi$
$360$	0	0
$361$	−18.1997	−0.957881
$362$	−11.5377	−0.606408
$363$	0	0
$364$	−5.32427	−0.279068
$365$	0	0
$366$	0	0
$367$	27.3604	1.42820	0.714100	−	0.700043i	$-0.246836\pi$
$367$	27.3604	1.42820	0.714100	+	0.700043i	$0.246836\pi$
$368$	−5.47782	−0.285551
$369$	0	0
$370$	0	0
$371$	10.9309	0.567502
$372$	0	0
$373$	−30.3763	−1.57283	−0.786413	−	0.617702i	$-0.788064\pi$
$373$	−30.3763	−1.57283	−0.786413	+	0.617702i	$0.788064\pi$
$374$	−0.520789	−0.0269294
$375$	0	0
$376$	−19.5819	−1.00986
$377$	34.2001	1.76139
$378$	0	0
$379$	27.5797	1.41668	0.708338	−	0.705873i	$-0.249445\pi$
$379$	27.5797	1.41668	0.708338	+	0.705873i	$0.249445\pi$
$380$	0	0
$381$	0	0
$382$	−15.8813	−0.812559
$383$	−14.6341	−0.747765	−0.373883	−	0.927476i	$-0.621974\pi$
$383$	−14.6341	−0.747765	−0.373883	+	0.927476i	$0.621974\pi$
$384$	0	0
$385$	0	0
$386$	2.46570	0.125501
$387$	0	0
$388$	10.5549	0.535846
$389$	22.0844	1.11972	0.559862	−	0.828586i	$-0.310854\pi$
$389$	22.0844	1.11972	0.559862	+	0.828586i	$0.310854\pi$
$390$	0	0
$391$	−7.45602	−0.377067
$392$	−15.1614	−0.765765
$393$	0	0
$394$	10.9800	0.553163
$395$	0	0
$396$	0	0
$397$	10.3527	0.519589	0.259794	−	0.965664i	$-0.416345\pi$
$397$	10.3527	0.519589	0.259794	+	0.965664i	$0.416345\pi$
$398$	11.3638	0.569617
$399$	0	0
$400$	0	0
$401$	19.8832	0.992919	0.496460	−	0.868060i	$-0.334633\pi$
$401$	19.8832	0.992919	0.496460	+	0.868060i	$0.334633\pi$
$402$	0	0
$403$	37.0799	1.84708
$404$	16.7259	0.832147
$405$	0	0
$406$	−7.41118	−0.367811
$407$	−1.12788	−0.0559069
$408$	0	0
$409$	−7.05843	−0.349017	−0.174509	−	0.984656i	$-0.555834\pi$
$409$	−7.05843	−0.349017	−0.174509	+	0.984656i	$0.555834\pi$
$410$	0	0
$411$	0	0
$412$	2.83670	0.139754
$413$	−12.3514	−0.607773
$414$	0	0
$415$	0	0
$416$	−20.7693	−1.01830
$417$	0	0
$418$	−0.340830	−0.0166705
$419$	9.67801	0.472802	0.236401	−	0.971656i	$-0.424032\pi$
$419$	9.67801	0.472802	0.236401	+	0.971656i	$0.424032\pi$
$420$	0	0
$421$	4.30464	0.209795	0.104898	−	0.994483i	$-0.466548\pi$
$421$	4.30464	0.209795	0.104898	+	0.994483i	$0.466548\pi$
$422$	0.279537	0.0136076
$423$	0	0
$424$	26.9933	1.31091
$425$	0	0
$426$	0	0
$427$	8.13329	0.393597
$428$	−22.1722	−1.07173
$429$	0	0
$430$	0	0
$431$	5.53549	0.266635	0.133318	−	0.991073i	$-0.457437\pi$
$431$	5.53549	0.266635	0.133318	+	0.991073i	$0.457437\pi$
$432$	0	0
$433$	28.1770	1.35410	0.677051	−	0.735936i	$-0.263258\pi$
$433$	28.1770	1.35410	0.677051	+	0.735936i	$0.263258\pi$
$434$	−8.03524	−0.385704
$435$	0	0
$436$	0.502688	0.0240744
$437$	−4.87958	−0.233422
$438$	0	0
$439$	−10.5674	−0.504353	−0.252176	−	0.967681i	$-0.581146\pi$
$439$	−10.5674	−0.504353	−0.252176	+	0.967681i	$0.581146\pi$
$440$	0	0
$441$	0	0
$442$	−3.59140	−0.170825
$443$	36.7964	1.74825	0.874126	−	0.485699i	$-0.161435\pi$
$443$	36.7964	1.74825	0.874126	+	0.485699i	$0.161435\pi$
$444$	0	0
$445$	0	0
$446$	−0.200279	−0.00948350
$447$	0	0
$448$	2.42021	0.114344
$449$	−26.9341	−1.27110	−0.635549	−	0.772060i	$-0.719226\pi$
$449$	−26.9341	−1.27110	−0.635549	+	0.772060i	$0.719226\pi$
$450$	0	0
$451$	−1.60574	−0.0756114
$452$	−5.67706	−0.267026
$453$	0	0
$454$	5.29398	0.248459
$455$	0	0
$456$	0	0
$457$	24.6289	1.15209	0.576046	−	0.817417i	$-0.304595\pi$
$457$	24.6289	1.15209	0.576046	+	0.817417i	$0.304595\pi$
$458$	−18.4515	−0.862182
$459$	0	0
$460$	0	0
$461$	−24.6914	−1.14999	−0.574996	−	0.818156i	$-0.694996\pi$
$461$	−24.6914	−1.14999	−0.574996	+	0.818156i	$0.694996\pi$
$462$	0	0
$463$	−22.7190	−1.05584	−0.527921	−	0.849293i	$-0.677028\pi$
$463$	−22.7190	−1.05584	−0.527921	+	0.849293i	$0.677028\pi$
$464$	9.69137	0.449911
$465$	0	0
$466$	−12.5698	−0.582286
$467$	−10.8747	−0.503222	−0.251611	−	0.967829i	$-0.580960\pi$
$467$	−10.8747	−0.503222	−0.251611	+	0.967829i	$0.580960\pi$
$468$	0	0
$469$	9.12318	0.421269
$470$	0	0
$471$	0	0
$472$	−30.5013	−1.40394
$473$	5.28631	0.243065
$474$	0	0
$475$	0	0
$476$	−2.05361	−0.0941270
$477$	0	0
$478$	−3.05786	−0.139863
$479$	−0.739002	−0.0337659	−0.0168829	−	0.999857i	$-0.505374\pi$
$479$	−0.739002	−0.0337659	−0.0168829	+	0.999857i	$0.505374\pi$
$480$	0	0
$481$	−7.77793	−0.354643
$482$	4.40558	0.200669
$483$	0	0
$484$	15.5709	0.707768
$485$	0	0
$486$	0	0
$487$	−29.1699	−1.32181	−0.660907	−	0.750468i	$-0.729828\pi$
$487$	−29.1699	−1.32181	−0.660907	+	0.750468i	$0.729828\pi$
$488$	20.0848	0.909197
$489$	0	0
$490$	0	0
$491$	31.6811	1.42975	0.714875	−	0.699252i	$-0.246484\pi$
$491$	31.6811	1.42975	0.714875	+	0.699252i	$0.246484\pi$
$492$	0	0
$493$	13.1912	0.594102
$494$	−2.35038	−0.105749
$495$	0	0
$496$	10.5074	0.471798
$497$	−5.18914	−0.232765
$498$	0	0
$499$	−8.89261	−0.398088	−0.199044	−	0.979991i	$-0.563784\pi$
$499$	−8.89261	−0.398088	−0.199044	+	0.979991i	$0.563784\pi$
$500$	0	0
$501$	0	0
$502$	−13.4228	−0.599088
$503$	17.0282	0.759248	0.379624	−	0.925141i	$-0.376053\pi$
$503$	17.0282	0.759248	0.379624	+	0.925141i	$0.376053\pi$
$504$	0	0
$505$	0	0
$506$	2.07821	0.0923875
$507$	0	0
$508$	8.73622	0.387607
$509$	18.6709	0.827571	0.413786	−	0.910374i	$-0.364206\pi$
$509$	18.6709	0.827571	0.413786	+	0.910374i	$0.364206\pi$
$510$	0	0
$511$	6.03396	0.266927
$512$	−11.0233	−0.487164
$513$	0	0
$514$	−17.7654	−0.783600
$515$	0	0
$516$	0	0
$517$	−3.93398	−0.173016
$518$	1.68548	0.0740559
$519$	0	0
$520$	0	0
$521$	−4.44863	−0.194898	−0.0974489	−	0.995241i	$-0.531068\pi$
$521$	−4.44863	−0.194898	−0.0974489	+	0.995241i	$0.531068\pi$
$522$	0	0
$523$	20.4791	0.895490	0.447745	−	0.894161i	$-0.352227\pi$
$523$	20.4791	0.895490	0.447745	+	0.894161i	$0.352227\pi$
$524$	1.83389	0.0801137
$525$	0	0
$526$	−9.82513	−0.428396
$527$	14.3020	0.623003
$528$	0	0
$529$	6.75318	0.293616
$530$	0	0
$531$	0	0
$532$	−1.34398	−0.0582690
$533$	−11.0733	−0.479637
$534$	0	0
$535$	0	0
$536$	22.5293	0.973118
$537$	0	0
$538$	−0.539780	−0.0232716
$539$	−3.04590	−0.131196
$540$	0	0
$541$	19.4764	0.837355	0.418678	−	0.908135i	$-0.362494\pi$
$541$	19.4764	0.837355	0.418678	+	0.908135i	$0.362494\pi$
$542$	12.5356	0.538450
$543$	0	0
$544$	−8.01088	−0.343464
$545$	0	0
$546$	0	0
$547$	40.3751	1.72631	0.863157	−	0.504936i	$-0.168484\pi$
$547$	40.3751	1.72631	0.863157	+	0.504936i	$0.168484\pi$
$548$	2.69324	0.115049
$549$	0	0
$550$	0	0
$551$	8.63297	0.367777
$552$	0	0
$553$	−6.98210	−0.296909
$554$	−7.44240	−0.316197
$555$	0	0
$556$	−25.3427	−1.07477
$557$	1.34777	0.0571067	0.0285534	−	0.999592i	$-0.490910\pi$
$557$	1.34777	0.0571067	0.0285534	+	0.999592i	$0.490910\pi$
$558$	0	0
$559$	36.4547	1.54187
$560$	0	0
$561$	0	0
$562$	15.2249	0.642222
$563$	9.88833	0.416744	0.208372	−	0.978050i	$-0.433184\pi$
$563$	9.88833	0.416744	0.208372	+	0.978050i	$0.433184\pi$
$564$	0	0
$565$	0	0
$566$	−15.7531	−0.662150
$567$	0	0
$568$	−12.8144	−0.537679
$569$	−9.70429	−0.406825	−0.203413	−	0.979093i	$-0.565203\pi$
$569$	−9.70429	−0.406825	−0.203413	+	0.979093i	$0.565203\pi$
$570$	0	0
$571$	4.61768	0.193244	0.0966219	−	0.995321i	$-0.469196\pi$
$571$	4.61768	0.193244	0.0966219	+	0.995321i	$0.469196\pi$
$572$	−2.64143	−0.110444
$573$	0	0
$574$	2.39959	0.100157
$575$	0	0
$576$	0	0
$577$	2.05700	0.0856338	0.0428169	−	0.999083i	$-0.486367\pi$
$577$	2.05700	0.0856338	0.0428169	+	0.999083i	$0.486367\pi$
$578$	11.2182	0.466616
$579$	0	0
$580$	0	0
$581$	8.28275	0.343626
$582$	0	0
$583$	5.42291	0.224594
$584$	14.9006	0.616592
$585$	0	0
$586$	−22.1545	−0.915192
$587$	18.7955	0.775774	0.387887	−	0.921707i	$-0.373205\pi$
$587$	18.7955	0.775774	0.387887	+	0.921707i	$0.373205\pi$
$588$	0	0
$589$	9.35990	0.385668
$590$	0	0
$591$	0	0
$592$	−2.20405	−0.0905861
$593$	−23.8873	−0.980934	−0.490467	−	0.871460i	$-0.663174\pi$
$593$	−23.8873	−0.980934	−0.490467	+	0.871460i	$0.663174\pi$
$594$	0	0
$595$	0	0
$596$	29.8128	1.22118
$597$	0	0
$598$	14.3314	0.586056
$599$	0.641179	0.0261979	0.0130989	−	0.999914i	$-0.495830\pi$
$599$	0.641179	0.0261979	0.0130989	+	0.999914i	$0.495830\pi$
$600$	0	0
$601$	28.1316	1.14751	0.573755	−	0.819027i	$-0.305486\pi$
$601$	28.1316	1.14751	0.573755	+	0.819027i	$0.305486\pi$
$602$	−7.89977	−0.321971
$603$	0	0
$604$	−4.16172	−0.169338
$605$	0	0
$606$	0	0
$607$	0.0458025	0.00185907	0.000929533	−	1.00000i	$-0.499704\pi$
$607$	0.0458025	0.00185907	0.000929533		1.00000i	$0.499704\pi$
$608$	−5.24271	−0.212620
$609$	0	0
$610$	0	0
$611$	−27.1289	−1.09752
$612$	0	0
$613$	−23.3986	−0.945062	−0.472531	−	0.881314i	$-0.656660\pi$
$613$	−23.3986	−0.945062	−0.472531	+	0.881314i	$0.656660\pi$
$614$	0.393710	0.0158888
$615$	0	0
$616$	1.36172	0.0548654
$617$	−5.62099	−0.226292	−0.113146	−	0.993578i	$-0.536093\pi$
$617$	−5.62099	−0.226292	−0.113146	+	0.993578i	$0.536093\pi$
$618$	0	0
$619$	−9.38706	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$619$	−9.38706	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$620$	0	0
$621$	0	0
$622$	−13.5170	−0.541981
$623$	−13.4576	−0.539167
$624$	0	0
$625$	0	0
$626$	−21.7612	−0.869753
$627$	0	0
$628$	20.8687	0.832754
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−0.494043	−0.0196675	−0.00983376	−	0.999952i	$-0.503130\pi$
$631$	−0.494043	−0.0196675	−0.00983376	+	0.999952i	$0.503130\pi$
$632$	−17.2420	−0.685850
$633$	0	0
$634$	−24.3905	−0.968672
$635$	0	0
$636$	0	0
$637$	−21.0047	−0.832237
$638$	−3.67677	−0.145565
$639$	0	0
$640$	0	0
$641$	29.9045	1.18116	0.590578	−	0.806981i	$-0.298900\pi$
$641$	29.9045	1.18116	0.590578	+	0.806981i	$0.298900\pi$
$642$	0	0
$643$	−17.8273	−0.703039	−0.351519	−	0.936181i	$-0.614335\pi$
$643$	−17.8273	−0.703039	−0.351519	+	0.936181i	$0.614335\pi$
$644$	8.19491	0.322925
$645$	0	0
$646$	−0.906560	−0.0356681
$647$	6.25123	0.245761	0.122881	−	0.992421i	$-0.460787\pi$
$647$	6.25123	0.245761	0.122881	+	0.992421i	$0.460787\pi$
$648$	0	0
$649$	−6.12767	−0.240532
$650$	0	0
$651$	0	0
$652$	11.4643	0.448976
$653$	3.40836	0.133379	0.0666897	−	0.997774i	$-0.478756\pi$
$653$	3.40836	0.133379	0.0666897	+	0.997774i	$0.478756\pi$
$654$	0	0
$655$	0	0
$656$	−3.13787	−0.122513
$657$	0	0
$658$	5.87886	0.229182
$659$	8.85617	0.344988	0.172494	−	0.985011i	$-0.444818\pi$
$659$	8.85617	0.344988	0.172494	+	0.985011i	$0.444818\pi$
$660$	0	0
$661$	−16.5504	−0.643735	−0.321867	−	0.946785i	$-0.604311\pi$
$661$	−16.5504	−0.643735	−0.321867	+	0.946785i	$0.604311\pi$
$662$	2.78359	0.108187
$663$	0	0
$664$	20.4539	0.793766
$665$	0	0
$666$	0	0
$667$	−52.6394	−2.03821
$668$	−27.4200	−1.06091
$669$	0	0
$670$	0	0
$671$	4.03501	0.155770
$672$	0	0
$673$	−50.3337	−1.94022	−0.970110	−	0.242665i	$-0.921979\pi$
$673$	−50.3337	−1.94022	−0.970110	+	0.242665i	$0.921979\pi$
$674$	10.6170	0.408952
$675$	0	0
$676$	0.639183	0.0245840
$677$	0.624178	0.0239891	0.0119946	−	0.999928i	$-0.496182\pi$
$677$	0.624178	0.0239891	0.0119946	+	0.999928i	$0.496182\pi$
$678$	0	0
$679$	−7.53847	−0.289300
$680$	0	0
$681$	0	0
$682$	−3.98637	−0.152646
$683$	41.3608	1.58263	0.791313	−	0.611411i	$-0.209398\pi$
$683$	41.3608	1.58263	0.791313	+	0.611411i	$0.209398\pi$
$684$	0	0
$685$	0	0
$686$	9.92751	0.379034
$687$	0	0
$688$	10.3303	0.393838
$689$	37.3968	1.42470
$690$	0	0
$691$	−40.0489	−1.52353	−0.761767	−	0.647851i	$-0.775668\pi$
$691$	−40.0489	−1.52353	−0.761767	+	0.647851i	$0.775668\pi$
$692$	−6.39874	−0.243244
$693$	0	0
$694$	4.38577	0.166481
$695$	0	0
$696$	0	0
$697$	−4.27105	−0.161777
$698$	−19.4588	−0.736527
$699$	0	0
$700$	0	0
$701$	−39.8261	−1.50421	−0.752106	−	0.659043i	$-0.770962\pi$
$701$	−39.8261	−1.50421	−0.752106	+	0.659043i	$0.770962\pi$
$702$	0	0
$703$	−1.96335	−0.0740491
$704$	1.20069	0.0452527
$705$	0	0
$706$	10.6078	0.399229
$707$	−11.9459	−0.449272
$708$	0	0
$709$	25.1649	0.945087	0.472543	−	0.881307i	$-0.343336\pi$
$709$	25.1649	0.945087	0.472543	+	0.881307i	$0.343336\pi$
$710$	0	0
$711$	0	0
$712$	−33.2330	−1.24546
$713$	−57.0719	−2.13736
$714$	0	0
$715$	0	0
$716$	−16.8989	−0.631541
$717$	0	0
$718$	26.9256	1.00486
$719$	−19.1076	−0.712593	−0.356297	−	0.934373i	$-0.615961\pi$
$719$	−19.1076	−0.712593	−0.356297	+	0.934373i	$0.615961\pi$
$720$	0	0
$721$	−2.02601	−0.0754525
$722$	13.4929	0.502154
$723$	0	0
$724$	−22.5711	−0.838850
$725$	0	0
$726$	0	0
$727$	22.4025	0.830862	0.415431	−	0.909625i	$-0.363631\pi$
$727$	22.4025	0.830862	0.415431	+	0.909625i	$0.363631\pi$
$728$	9.39052	0.348036
$729$	0	0
$730$	0	0
$731$	14.0608	0.520059
$732$	0	0
$733$	−45.4058	−1.67710	−0.838551	−	0.544823i	$-0.816597\pi$
$733$	−45.4058	−1.67710	−0.838551	+	0.544823i	$0.816597\pi$
$734$	−20.2844	−0.748712
$735$	0	0
$736$	31.9674	1.17833
$737$	4.52611	0.166721
$738$	0	0
$739$	41.4109	1.52332	0.761662	−	0.647974i	$-0.224384\pi$
$739$	41.4109	1.52332	0.761662	+	0.647974i	$0.224384\pi$
$740$	0	0
$741$	0	0
$742$	−8.10391	−0.297504
$743$	12.1759	0.446689	0.223344	−	0.974740i	$-0.428303\pi$
$743$	12.1759	0.446689	0.223344	+	0.974740i	$0.428303\pi$
$744$	0	0
$745$	0	0
$746$	22.5204	0.824529
$747$	0	0
$748$	−1.01882	−0.0372517
$749$	15.8357	0.578623
$750$	0	0
$751$	−31.1833	−1.13789	−0.568947	−	0.822374i	$-0.692649\pi$
$751$	−31.1833	−1.13789	−0.568947	+	0.822374i	$0.692649\pi$
$752$	−7.68761	−0.280338
$753$	0	0
$754$	−25.3552	−0.923382
$755$	0	0
$756$	0	0
$757$	−25.8685	−0.940207	−0.470104	−	0.882611i	$-0.655783\pi$
$757$	−25.8685	−0.940207	−0.470104	+	0.882611i	$0.655783\pi$
$758$	−20.4471	−0.742670
$759$	0	0
$760$	0	0
$761$	−31.2246	−1.13189	−0.565945	−	0.824443i	$-0.691489\pi$
$761$	−31.2246	−1.13189	−0.565945	+	0.824443i	$0.691489\pi$
$762$	0	0
$763$	−0.359026	−0.0129976
$764$	−31.0686	−1.12402
$765$	0	0
$766$	10.8494	0.392004
$767$	−42.2568	−1.52580
$768$	0	0
$769$	−1.72556	−0.0622253	−0.0311127	−	0.999516i	$-0.509905\pi$
$769$	−1.72556	−0.0622253	−0.0311127	+	0.999516i	$0.509905\pi$
$770$	0	0
$771$	0	0
$772$	4.82363	0.173606
$773$	−39.2992	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$773$	−39.2992	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$774$	0	0
$775$	0	0
$776$	−18.6159	−0.668274
$777$	0	0
$778$	−16.3729	−0.586998
$779$	−2.79518	−0.100148
$780$	0	0
$781$	−2.57439	−0.0921188
$782$	5.52774	0.197671
$783$	0	0
$784$	−5.95216	−0.212577
$785$	0	0
$786$	0	0
$787$	50.7384	1.80863	0.904315	−	0.426866i	$-0.140383\pi$
$787$	50.7384	1.80863	0.904315	+	0.426866i	$0.140383\pi$
$788$	21.4801	0.765195
$789$	0	0
$790$	0	0
$791$	4.05463	0.144166
$792$	0	0
$793$	27.8257	0.988120
$794$	−7.67530	−0.272386
$795$	0	0
$796$	22.2310	0.787957
$797$	−1.77689	−0.0629405	−0.0314703	−	0.999505i	$-0.510019\pi$
$797$	−1.77689	−0.0629405	−0.0314703	+	0.999505i	$0.510019\pi$
$798$	0	0
$799$	−10.4638	−0.370184
$800$	0	0
$801$	0	0
$802$	−14.7410	−0.520522
$803$	2.99351	0.105639
$804$	0	0
$805$	0	0
$806$	−27.4902	−0.968302
$807$	0	0
$808$	−29.4999	−1.03780
$809$	50.7926	1.78577	0.892887	−	0.450281i	$-0.148676\pi$
$809$	50.7926	1.78577	0.892887	+	0.450281i	$0.148676\pi$
$810$	0	0
$811$	−26.4748	−0.929655	−0.464827	−	0.885401i	$-0.653884\pi$
$811$	−26.4748	−0.929655	−0.464827	+	0.885401i	$0.653884\pi$
$812$	−14.4985	−0.508796
$813$	0	0
$814$	0.836186	0.0293083
$815$	0	0
$816$	0	0
$817$	9.20210	0.321941
$818$	5.23298	0.182967
$819$	0	0
$820$	0	0
$821$	−30.1888	−1.05359	−0.526797	−	0.849991i	$-0.676607\pi$
$821$	−30.1888	−1.05359	−0.526797	+	0.849991i	$0.676607\pi$
$822$	0	0
$823$	10.5256	0.366901	0.183450	−	0.983029i	$-0.441273\pi$
$823$	10.5256	0.366901	0.183450	+	0.983029i	$0.441273\pi$
$824$	−5.00314	−0.174293
$825$	0	0
$826$	9.15708	0.318616
$827$	−48.0149	−1.66964	−0.834820	−	0.550523i	$-0.814428\pi$
$827$	−48.0149	−1.66964	−0.834820	+	0.550523i	$0.814428\pi$
$828$	0	0
$829$	−27.0527	−0.939581	−0.469790	−	0.882778i	$-0.655671\pi$
$829$	−27.0527	−0.939581	−0.469790	+	0.882778i	$0.655671\pi$
$830$	0	0
$831$	0	0
$832$	8.28003	0.287059
$833$	−8.10166	−0.280706
$834$	0	0
$835$	0	0
$836$	−0.666764	−0.0230605
$837$	0	0
$838$	−7.17508	−0.247859
$839$	−48.1448	−1.66214	−0.831072	−	0.556165i	$-0.812272\pi$
$839$	−48.1448	−1.66214	−0.831072	+	0.556165i	$0.812272\pi$
$840$	0	0
$841$	64.1297	2.21137
$842$	−3.19137	−0.109982
$843$	0	0
$844$	0.546856	0.0188236
$845$	0	0
$846$	0	0
$847$	−11.1209	−0.382120
$848$	10.5972	0.363910
$849$	0	0
$850$	0	0
$851$	11.9715	0.410377
$852$	0	0
$853$	6.56698	0.224849	0.112425	−	0.993660i	$-0.464138\pi$
$853$	6.56698	0.224849	0.112425	+	0.993660i	$0.464138\pi$
$854$	−6.02985	−0.206337
$855$	0	0
$856$	39.1056	1.33660
$857$	26.7026	0.912143	0.456072	−	0.889943i	$-0.349256\pi$
$857$	26.7026	0.912143	0.456072	+	0.889943i	$0.349256\pi$
$858$	0	0
$859$	−12.6095	−0.430231	−0.215116	−	0.976589i	$-0.569013\pi$
$859$	−12.6095	−0.430231	−0.215116	+	0.976589i	$0.569013\pi$
$860$	0	0
$861$	0	0
$862$	−4.10390	−0.139779
$863$	7.76101	0.264188	0.132094	−	0.991237i	$-0.457830\pi$
$863$	7.76101	0.264188	0.132094	+	0.991237i	$0.457830\pi$
$864$	0	0
$865$	0	0
$866$	−20.8899	−0.709867
$867$	0	0
$868$	−15.7193	−0.533548
$869$	−3.46389	−0.117505
$870$	0	0
$871$	31.2123	1.05759
$872$	−0.886601	−0.0300241
$873$	0	0
$874$	3.61762	0.122368
$875$	0	0
$876$	0	0
$877$	15.3303	0.517667	0.258834	−	0.965922i	$-0.416662\pi$
$877$	15.3303	0.517667	0.258834	+	0.965922i	$0.416662\pi$
$878$	7.83443	0.264399
$879$	0	0
$880$	0	0
$881$	−37.4961	−1.26328	−0.631638	−	0.775263i	$-0.717617\pi$
$881$	−37.4961	−1.26328	−0.631638	+	0.775263i	$0.717617\pi$
$882$	0	0
$883$	−1.99657	−0.0671899	−0.0335950	−	0.999436i	$-0.510696\pi$
$883$	−1.99657	−0.0671899	−0.0335950	+	0.999436i	$0.510696\pi$
$884$	−7.02583	−0.236304
$885$	0	0
$886$	−27.2801	−0.916494
$887$	13.0003	0.436506	0.218253	−	0.975892i	$-0.429964\pi$
$887$	13.0003	0.436506	0.218253	+	0.975892i	$0.429964\pi$
$888$	0	0
$889$	−6.23952	−0.209267
$890$	0	0
$891$	0	0
$892$	−0.391805	−0.0131186
$893$	−6.84804	−0.229161
$894$	0	0
$895$	0	0
$896$	10.3472	0.345677
$897$	0	0
$898$	19.9684	0.666354
$899$	100.972	3.36760
$900$	0	0
$901$	14.4242	0.480539
$902$	1.19046	0.0396381
$903$	0	0
$904$	10.0127	0.333019
$905$	0	0
$906$	0	0
$907$	15.3852	0.510857	0.255429	−	0.966828i	$-0.417783\pi$
$907$	15.3852	0.510857	0.255429	+	0.966828i	$0.417783\pi$
$908$	10.3566	0.343695
$909$	0	0
$910$	0	0
$911$	0.397650	0.0131747	0.00658736	−	0.999978i	$-0.497903\pi$
$911$	0.397650	0.0131747	0.00658736	+	0.999978i	$0.497903\pi$
$912$	0	0
$913$	4.10916	0.135993
$914$	−18.2594	−0.603966
$915$	0	0
$916$	−36.0966	−1.19266
$917$	−1.30979	−0.0432529
$918$	0	0
$919$	4.87620	0.160851	0.0804255	−	0.996761i	$-0.474372\pi$
$919$	4.87620	0.160851	0.0804255	+	0.996761i	$0.474372\pi$
$920$	0	0
$921$	0	0
$922$	18.3057	0.602865
$923$	−17.7531	−0.584352
$924$	0	0
$925$	0	0
$926$	16.8434	0.553509
$927$	0	0
$928$	−56.5567	−1.85657
$929$	−25.0861	−0.823048	−0.411524	−	0.911399i	$-0.635003\pi$
$929$	−25.0861	−0.823048	−0.411524	+	0.911399i	$0.635003\pi$
$930$	0	0
$931$	−5.30212	−0.173770
$932$	−24.5903	−0.805482
$933$	0	0
$934$	8.06229	0.263806
$935$	0	0
$936$	0	0
$937$	39.6519	1.29537	0.647686	−	0.761907i	$-0.275737\pi$
$937$	39.6519	1.29537	0.647686	+	0.761907i	$0.275737\pi$
$938$	−6.76373	−0.220844
$939$	0	0
$940$	0	0
$941$	−10.8852	−0.354847	−0.177424	−	0.984135i	$-0.556776\pi$
$941$	−10.8852	−0.354847	−0.177424	+	0.984135i	$0.556776\pi$
$942$	0	0
$943$	17.0436	0.555015
$944$	−11.9744	−0.389735
$945$	0	0
$946$	−3.91916	−0.127423
$947$	34.0211	1.10554	0.552768	−	0.833335i	$-0.313571\pi$
$947$	34.0211	1.10554	0.552768	+	0.833335i	$0.313571\pi$
$948$	0	0
$949$	20.6434	0.670115
$950$	0	0
$951$	0	0
$952$	3.62199	0.117389
$953$	40.0497	1.29734	0.648669	−	0.761071i	$-0.275326\pi$
$953$	40.0497	1.29734	0.648669	+	0.761071i	$0.275326\pi$
$954$	0	0
$955$	0	0
$956$	−5.98208	−0.193474
$957$	0	0
$958$	0.547881	0.0177012
$959$	−1.92355	−0.0621146
$960$	0	0
$961$	78.4740	2.53142
$962$	5.76639	0.185916
$963$	0	0
$964$	8.61862	0.277587
$965$	0	0
$966$	0	0
$967$	−1.82683	−0.0587469	−0.0293734	−	0.999569i	$-0.509351\pi$
$967$	−1.82683	−0.0587469	−0.0293734	+	0.999569i	$0.509351\pi$
$968$	−27.4627	−0.882684
$969$	0	0
$970$	0	0
$971$	−40.4775	−1.29899	−0.649493	−	0.760368i	$-0.725019\pi$
$971$	−40.4775	−1.29899	−0.649493	+	0.760368i	$0.725019\pi$
$972$	0	0
$973$	18.1001	0.580261
$974$	21.6259	0.692940
$975$	0	0
$976$	7.88505	0.252394
$977$	20.8782	0.667954	0.333977	−	0.942581i	$-0.391609\pi$
$977$	20.8782	0.667954	0.333977	+	0.942581i	$0.391609\pi$
$978$	0	0
$979$	−6.67646	−0.213381
$980$	0	0
$981$	0	0
$982$	−23.4877	−0.749524
$983$	21.0491	0.671361	0.335680	−	0.941976i	$-0.391034\pi$
$983$	21.0491	0.671361	0.335680	+	0.941976i	$0.391034\pi$
$984$	0	0
$985$	0	0
$986$	−9.77968	−0.311449
$987$	0	0
$988$	−4.59805	−0.146283
$989$	−56.1097	−1.78418
$990$	0	0
$991$	42.2138	1.34097	0.670483	−	0.741925i	$-0.266087\pi$
$991$	42.2138	1.34097	0.670483	+	0.741925i	$0.266087\pi$
$992$	−61.3191	−1.94688
$993$	0	0
$994$	3.84712	0.122023
$995$	0	0
$996$	0	0
$997$	32.3301	1.02390	0.511952	−	0.859014i	$-0.328923\pi$
$997$	32.3301	1.02390	0.511952	+	0.859014i	$0.328923\pi$
$998$	6.59280	0.208691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.u.1.4		8
3.2	odd	2		1875.2.a.o.1.5	yes	8
5.4	even	2		5625.2.a.bc.1.5		8
15.2	even	4		1875.2.b.g.1249.9		16
15.8	even	4		1875.2.b.g.1249.8		16
15.14	odd	2		1875.2.a.n.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.4	✓	8	15.14	odd	2
1875.2.a.o.1.5	yes	8	3.2	odd	2
1875.2.b.g.1249.8		16	15.8	even	4
1875.2.b.g.1249.9		16	15.2	even	4
5625.2.a.u.1.4		8	1.1	even	1	trivial
5625.2.a.bc.1.5		8	5.4	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.741379 q^{2} -1.45036 q^{4} +1.03586 q^{7} +2.55802 q^{8} +O(q^{10})\)	\(q-0.741379 q^{2} -1.45036 q^{4} +1.03586 q^{7} +2.55802 q^{8} +0.513903 q^{11} +3.54391 q^{13} -0.767968 q^{14} +1.00425 q^{16} +1.36691 q^{17} +0.894573 q^{19} -0.380997 q^{22} -5.45465 q^{23} -2.62738 q^{26} -1.50237 q^{28} +9.65038 q^{29} +10.4630 q^{31} -5.86057 q^{32} -1.01340 q^{34} -2.19473 q^{37} -0.663218 q^{38} -3.12460 q^{41} +10.2866 q^{43} -0.745343 q^{44} +4.04396 q^{46} -7.65509 q^{47} -5.92699 q^{49} -5.13993 q^{52} +10.5524 q^{53} +2.64976 q^{56} -7.15459 q^{58} -11.9238 q^{59} +7.85170 q^{61} -7.75704 q^{62} +2.33641 q^{64} +8.80731 q^{67} -1.98251 q^{68} -5.00948 q^{71} +5.82505 q^{73} +1.62713 q^{74} -1.29745 q^{76} +0.532334 q^{77} -6.74036 q^{79} +2.31651 q^{82} +7.99598 q^{83} -7.62626 q^{86} +1.31458 q^{88} -12.9917 q^{89} +3.67101 q^{91} +7.91118 q^{92} +5.67533 q^{94} -7.27747 q^{97} +4.39414 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

Introduction

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