LMFDB - Embedded newform 3025.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35567 q^{2} -0.575493 q^{3} -0.162147 q^{4} -0.780181 q^{6} -3.64941 q^{7} -2.93117 q^{8} -2.66881 q^{9} +O(q^{10})$	\(q+1.35567 q^{2} -0.575493 q^{3} -0.162147 q^{4} -0.780181 q^{6} -3.64941 q^{7} -2.93117 q^{8} -2.66881 q^{9} +0.0933146 q^{12} -2.83095 q^{13} -4.94742 q^{14} -3.64941 q^{16} +3.69195 q^{17} -3.61803 q^{18} +0.0951243 q^{19} +2.10021 q^{21} -1.16215 q^{23} +1.68687 q^{24} -3.83785 q^{26} +3.26236 q^{27} +0.591742 q^{28} +6.75389 q^{29} +6.77837 q^{31} +0.914918 q^{32} +5.00509 q^{34} +0.432740 q^{36} -9.83980 q^{37} +0.128958 q^{38} +1.62920 q^{39} +8.31822 q^{41} +2.84720 q^{42} -2.96862 q^{43} -1.57549 q^{46} +2.22491 q^{47} +2.10021 q^{48} +6.31822 q^{49} -2.12469 q^{51} +0.459031 q^{52} -2.99393 q^{53} +4.42270 q^{54} +10.6970 q^{56} -0.0547434 q^{57} +9.15607 q^{58} -8.50860 q^{59} +8.48037 q^{61} +9.18926 q^{62} +9.73958 q^{63} +8.53916 q^{64} +13.4153 q^{67} -0.598640 q^{68} +0.668808 q^{69} +8.30309 q^{71} +7.82272 q^{72} -1.32003 q^{73} -13.3396 q^{74} -0.0154241 q^{76} +2.20866 q^{78} +13.8661 q^{79} +6.12896 q^{81} +11.2768 q^{82} -10.6445 q^{83} -0.340544 q^{84} -4.02448 q^{86} -3.88682 q^{87} -12.1612 q^{89} +10.3313 q^{91} +0.188439 q^{92} -3.90091 q^{93} +3.01625 q^{94} -0.526529 q^{96} +4.33133 q^{97} +8.56545 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 - q^4 + q^6 - 3 * q^7 - 3 * q^8	$ 4 q - q^{2} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8} - 8 q^{12} - q^{13} + 2 q^{14} - 3 q^{16} + q^{17} - 10 q^{18} + 20 q^{19} + 10 q^{21} - 5 q^{23} + 11 q^{24} - 15 q^{26} + 15 q^{27} - 13 q^{28} + 12 q^{29} - 5 q^{31} + 8 q^{32} + 2 q^{34} - 7 q^{37} - 20 q^{38} + 7 q^{39} + 11 q^{41} - 12 q^{42} - 19 q^{43} - 4 q^{46} - 5 q^{47} + 10 q^{48} + 3 q^{49} + 7 q^{51} + 11 q^{52} + 11 q^{53} - 8 q^{54} + 11 q^{56} + 5 q^{57} + 14 q^{58} + 9 q^{59} + 12 q^{61} + 35 q^{62} + 5 q^{63} - 3 q^{64} + 19 q^{67} + 3 q^{68} - 8 q^{69} + 5 q^{71} + 25 q^{72} - 11 q^{73} + 8 q^{78} + 34 q^{79} + 4 q^{81} + 6 q^{82} + 11 q^{83} + 11 q^{84} + q^{86} + 19 q^{87} - 8 q^{89} - 8 q^{91} + 12 q^{92} + 5 q^{93} - q^{94} - 34 q^{96} - 32 q^{97} + 8 q^{98}+O(q^{100}) $ 4 * q - q^2 - q^4 + q^6 - 3 * q^7 - 3 * q^8 - 8 * q^12 - q^13 + 2 * q^14 - 3 * q^16 + q^17 - 10 * q^18 + 20 * q^19 + 10 * q^21 - 5 * q^23 + 11 * q^24 - 15 * q^26 + 15 * q^27 - 13 * q^28 + 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 - 7 * q^37 - 20 * q^38 + 7 * q^39 + 11 * q^41 - 12 * q^42 - 19 * q^43 - 4 * q^46 - 5 * q^47 + 10 * q^48 + 3 * q^49 + 7 * q^51 + 11 * q^52 + 11 * q^53 - 8 * q^54 + 11 * q^56 + 5 * q^57 + 14 * q^58 + 9 * q^59 + 12 * q^61 + 35 * q^62 + 5 * q^63 - 3 * q^64 + 19 * q^67 + 3 * q^68 - 8 * q^69 + 5 * q^71 + 25 * q^72 - 11 * q^73 + 8 * q^78 + 34 * q^79 + 4 * q^81 + 6 * q^82 + 11 * q^83 + 11 * q^84 + q^86 + 19 * q^87 - 8 * q^89 - 8 * q^91 + 12 * q^92 + 5 * q^93 - q^94 - 34 * q^96 - 32 * q^97 + 8 * 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 $

$2$	1.35567	0.958606	0.479303	−	0.877649i	$-0.340889\pi$
$2$	1.35567	0.958606	0.479303	+	0.877649i	$0.340889\pi$
$3$	−0.575493	−0.332261	−0.166131	−	0.986104i	$-0.553127\pi$
$3$	−0.575493	−0.332261	−0.166131	+	0.986104i	$0.553127\pi$
$4$	−0.162147	−0.0810736
$5$	0	0
$5$	0	0
$6$	−0.780181	−0.318508
$7$	−3.64941	−1.37935	−0.689674	−	0.724120i	$-0.742246\pi$
$7$	−3.64941	−1.37935	−0.689674	+	0.724120i	$0.742246\pi$
$8$	−2.93117	−1.03632
$9$	−2.66881	−0.889603
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.0933146	0.0269376
$13$	−2.83095	−0.785166	−0.392583	−	0.919717i	$-0.628418\pi$
$13$	−2.83095	−0.785166	−0.392583	+	0.919717i	$0.628418\pi$
$14$	−4.94742	−1.32225
$15$	0	0
$16$	−3.64941	−0.912353
$17$	3.69195	0.895431	0.447715	−	0.894176i	$-0.352238\pi$
$17$	3.69195	0.895431	0.447715	+	0.894176i	$0.352238\pi$
$18$	−3.61803	−0.852779
$19$	0.0951243	0.0218230	0.0109115	−	0.999940i	$-0.496527\pi$
$19$	0.0951243	0.0218230	0.0109115	+	0.999940i	$0.496527\pi$
$20$	0	0
$21$	2.10021	0.458304
$22$	0	0
$23$	−1.16215	−0.242324	−0.121162	−	0.992633i	$-0.538662\pi$
$23$	−1.16215	−0.242324	−0.121162	+	0.992633i	$0.538662\pi$
$24$	1.68687	0.344330
$25$	0	0
$26$	−3.83785	−0.752665
$27$	3.26236	0.627841
$28$	0.591742	0.111829
$29$	6.75389	1.25417	0.627083	−	0.778953i	$-0.284249\pi$
$29$	6.75389	1.25417	0.627083	+	0.778953i	$0.284249\pi$
$30$	0	0
$31$	6.77837	1.21743	0.608716	−	0.793388i	$-0.291685\pi$
$31$	6.77837	1.21743	0.608716	+	0.793388i	$0.291685\pi$
$32$	0.914918	0.161736
$33$	0	0
$34$	5.00509	0.858366
$35$	0	0
$36$	0.432740	0.0721233
$37$	−9.83980	−1.61765	−0.808826	−	0.588048i	$-0.799897\pi$
$37$	−9.83980	−1.61765	−0.808826	+	0.588048i	$0.799897\pi$
$38$	0.128958	0.0209197
$39$	1.62920	0.260880
$40$	0	0
$41$	8.31822	1.29909	0.649544	−	0.760324i	$-0.274960\pi$
$41$	8.31822	1.29909	0.649544	+	0.760324i	$0.274960\pi$
$42$	2.84720	0.439333
$43$	−2.96862	−0.452710	−0.226355	−	0.974045i	$-0.572681\pi$
$43$	−2.96862	−0.452710	−0.226355	+	0.974045i	$0.572681\pi$
$44$	0	0
$45$	0	0
$46$	−1.57549	−0.232294
$47$	2.22491	0.324536	0.162268	−	0.986747i	$-0.448119\pi$
$47$	2.22491	0.324536	0.162268	+	0.986747i	$0.448119\pi$
$48$	2.10021	0.303140
$49$	6.31822	0.902603
$50$	0	0
$51$	−2.12469	−0.297517
$52$	0.459031	0.0636562
$53$	−2.99393	−0.411248	−0.205624	−	0.978631i	$-0.565922\pi$
$53$	−2.99393	−0.411248	−0.205624	+	0.978631i	$0.565922\pi$
$54$	4.42270	0.601853
$55$	0	0
$56$	10.6970	1.42945
$57$	−0.0547434	−0.00725094
$58$	9.15607	1.20225
$59$	−8.50860	−1.10773	−0.553863	−	0.832608i	$-0.686847\pi$
$59$	−8.50860	−1.10773	−0.553863	+	0.832608i	$0.686847\pi$
$60$	0	0
$61$	8.48037	1.08580	0.542900	−	0.839797i	$-0.317326\pi$
$61$	8.48037	1.08580	0.542900	+	0.839797i	$0.317326\pi$
$62$	9.18926	1.16704
$63$	9.73958	1.22707
$64$	8.53916	1.06739
$65$	0	0
$66$	0	0
$67$	13.4153	1.63894	0.819469	−	0.573123i	$-0.194268\pi$
$67$	13.4153	1.63894	0.819469	+	0.573123i	$0.194268\pi$
$68$	−0.598640	−0.0725958
$69$	0.668808	0.0805150
$70$	0	0
$71$	8.30309	0.985396	0.492698	−	0.870200i	$-0.336011\pi$
$71$	8.30309	0.985396	0.492698	+	0.870200i	$0.336011\pi$
$72$	7.82272	0.921917
$73$	−1.32003	−0.154498	−0.0772490	−	0.997012i	$-0.524614\pi$
$73$	−1.32003	−0.154498	−0.0772490	+	0.997012i	$0.524614\pi$
$74$	−13.3396	−1.55069
$75$	0	0
$76$	−0.0154241	−0.00176927
$77$	0	0
$78$	2.20866	0.250081
$79$	13.8661	1.56006	0.780028	−	0.625744i	$-0.215205\pi$
$79$	13.8661	1.56006	0.780028	+	0.625744i	$0.215205\pi$
$80$	0	0
$81$	6.12896	0.680995
$82$	11.2768	1.24531
$83$	−10.6445	−1.16838	−0.584191	−	0.811616i	$-0.698588\pi$
$83$	−10.6445	−1.16838	−0.584191	+	0.811616i	$0.698588\pi$
$84$	−0.340544	−0.0371564
$85$	0	0
$86$	−4.02448	−0.433971
$87$	−3.88682	−0.416710
$88$	0	0
$89$	−12.1612	−1.28908	−0.644540	−	0.764570i	$-0.722951\pi$
$89$	−12.1612	−1.28908	−0.644540	+	0.764570i	$0.722951\pi$
$90$	0	0
$91$	10.3313	1.08302
$92$	0.188439	0.0196461
$93$	−3.90091	−0.404505
$94$	3.01625	0.311102
$95$	0	0
$96$	−0.526529	−0.0537387
$97$	4.33133	0.439780	0.219890	−	0.975525i	$-0.429430\pi$
$97$	4.33133	0.439780	0.219890	+	0.975525i	$0.429430\pi$
$98$	8.56545	0.865241
$99$	0	0
$100$	0	0
$101$	−9.90570	−0.985654	−0.492827	−	0.870127i	$-0.664036\pi$
$101$	−9.90570	−0.985654	−0.492827	+	0.870127i	$0.664036\pi$
$102$	−2.88039	−0.285201
$103$	−4.06590	−0.400625	−0.200313	−	0.979732i	$-0.564196\pi$
$103$	−4.06590	−0.400625	−0.200313	+	0.979732i	$0.564196\pi$
$104$	8.29800	0.813686
$105$	0	0
$106$	−4.05879	−0.394225
$107$	1.93858	0.187409	0.0937046	−	0.995600i	$-0.470129\pi$
$107$	1.93858	0.187409	0.0937046	+	0.995600i	$0.470129\pi$
$108$	−0.528983	−0.0509014
$109$	−6.12664	−0.586825	−0.293413	−	0.955986i	$-0.594791\pi$
$109$	−6.12664	−0.586825	−0.293413	+	0.955986i	$0.594791\pi$
$110$	0	0
$111$	5.66273	0.537483
$112$	13.3182	1.25845
$113$	−5.78527	−0.544232	−0.272116	−	0.962264i	$-0.587724\pi$
$113$	−5.78527	−0.544232	−0.272116	+	0.962264i	$0.587724\pi$
$114$	−0.0742142	−0.00695080
$115$	0	0
$116$	−1.09512	−0.101680
$117$	7.55527	0.698485
$118$	−11.5349	−1.06187
$119$	−13.4735	−1.23511
$120$	0	0
$121$	0	0
$122$	11.4966	1.04085
$123$	−4.78708	−0.431636
$124$	−1.09909	−0.0987016
$125$	0	0
$126$	13.2037	1.17628
$127$	2.43783	0.216322	0.108161	−	0.994133i	$-0.465504\pi$
$127$	2.43783	0.216322	0.108161	+	0.994133i	$0.465504\pi$
$128$	9.74648	0.861475
$129$	1.70842	0.150418
$130$	0	0
$131$	−7.04156	−0.615224	−0.307612	−	0.951512i	$-0.599530\pi$
$131$	−7.04156	−0.615224	−0.307612	+	0.951512i	$0.599530\pi$
$132$	0	0
$133$	−0.347148	−0.0301016
$134$	18.1868	1.57110
$135$	0	0
$136$	−10.8217	−0.927956
$137$	9.57286	0.817864	0.408932	−	0.912565i	$-0.365901\pi$
$137$	9.57286	0.817864	0.408932	+	0.912565i	$0.365901\pi$
$138$	0.906685	0.0771822
$139$	−0.515502	−0.0437243	−0.0218621	−	0.999761i	$-0.506959\pi$
$139$	−0.515502	−0.0437243	−0.0218621	+	0.999761i	$0.506959\pi$
$140$	0	0
$141$	−1.28042	−0.107831
$142$	11.2563	0.944607
$143$	0	0
$144$	9.73958	0.811632
$145$	0	0
$146$	−1.78953	−0.148103
$147$	−3.63609	−0.299900
$148$	1.59550	0.131149
$149$	8.15983	0.668479	0.334240	−	0.942488i	$-0.391521\pi$
$149$	8.15983	0.668479	0.334240	+	0.942488i	$0.391521\pi$
$150$	0	0
$151$	−1.94023	−0.157893	−0.0789466	−	0.996879i	$-0.525156\pi$
$151$	−1.94023	−0.157893	−0.0789466	+	0.996879i	$0.525156\pi$
$152$	−0.278825	−0.0226157
$153$	−9.85312	−0.796577
$154$	0	0
$155$	0	0
$156$	−0.264169	−0.0211505
$157$	−21.2745	−1.69789	−0.848944	−	0.528483i	$-0.822761\pi$
$157$	−21.2745	−1.69789	−0.848944	+	0.528483i	$0.822761\pi$
$158$	18.7979	1.49548
$159$	1.72298	0.136642
$160$	0	0
$161$	4.24116	0.334250
$162$	8.30887	0.652807
$163$	15.9810	1.25173	0.625863	−	0.779933i	$-0.284747\pi$
$163$	15.9810	1.25173	0.625863	+	0.779933i	$0.284747\pi$
$164$	−1.34878	−0.105322
$165$	0	0
$166$	−14.4304	−1.12002
$167$	17.7090	1.37037	0.685183	−	0.728371i	$-0.259722\pi$
$167$	17.7090	1.37037	0.685183	+	0.728371i	$0.259722\pi$
$168$	−6.15607	−0.474951
$169$	−4.98569	−0.383515
$170$	0	0
$171$	−0.253869	−0.0194138
$172$	0.481353	0.0367029
$173$	15.8855	1.20775	0.603875	−	0.797079i	$-0.293622\pi$
$173$	15.8855	1.20775	0.603875	+	0.797079i	$0.293622\pi$
$174$	−5.26926	−0.399461
$175$	0	0
$176$	0	0
$177$	4.89664	0.368054
$178$	−16.4866	−1.23572
$179$	−16.8810	−1.26175	−0.630874	−	0.775885i	$-0.717304\pi$
$179$	−16.8810	−1.26175	−0.630874	+	0.775885i	$0.717304\pi$
$180$	0	0
$181$	−24.0874	−1.79040	−0.895200	−	0.445664i	$-0.852968\pi$
$181$	−24.0874	−1.79040	−0.895200	+	0.445664i	$0.852968\pi$
$182$	14.0059	1.03819
$183$	−4.88039	−0.360769
$184$	3.40645	0.251127
$185$	0	0
$186$	−5.28836	−0.387761
$187$	0	0
$188$	−0.360762	−0.0263113
$189$	−11.9057	−0.866012
$190$	0	0
$191$	5.38279	0.389485	0.194743	−	0.980854i	$-0.437613\pi$
$191$	5.38279	0.389485	0.194743	+	0.980854i	$0.437613\pi$
$192$	−4.91423	−0.354654
$193$	18.2840	1.31611	0.658057	−	0.752968i	$-0.271378\pi$
$193$	18.2840	1.31611	0.658057	+	0.752968i	$0.271378\pi$
$194$	5.87187	0.421576
$195$	0	0
$196$	−1.02448	−0.0731773
$197$	2.64566	0.188496	0.0942478	−	0.995549i	$-0.469955\pi$
$197$	2.64566	0.188496	0.0942478	+	0.995549i	$0.469955\pi$
$198$	0	0
$199$	6.52800	0.462757	0.231379	−	0.972864i	$-0.425676\pi$
$199$	6.52800	0.462757	0.231379	+	0.972864i	$0.425676\pi$
$200$	0	0
$201$	−7.72041	−0.544555
$202$	−13.4289	−0.944854
$203$	−24.6477	−1.72993
$204$	0.344513	0.0241208
$205$	0	0
$206$	−5.51204	−0.384042
$207$	3.10155	0.215572
$208$	10.3313	0.716349
$209$	0	0
$210$	0	0
$211$	27.4478	1.88958	0.944792	−	0.327671i	$-0.106264\pi$
$211$	27.4478	1.88958	0.944792	+	0.327671i	$0.106264\pi$
$212$	0.485457	0.0333413
$213$	−4.77837	−0.327409
$214$	2.62808	0.179652
$215$	0	0
$216$	−9.56252	−0.650647
$217$	−24.7371	−1.67926
$218$	−8.30573	−0.562535
$219$	0.759669	0.0513337
$220$	0	0
$221$	−10.4518	−0.703061
$222$	7.67682	0.515235
$223$	5.08194	0.340312	0.170156	−	0.985417i	$-0.445573\pi$
$223$	5.08194	0.340312	0.170156	+	0.985417i	$0.445573\pi$
$224$	−3.33892	−0.223091
$225$	0	0
$226$	−7.84294	−0.521705
$227$	3.73980	0.248219	0.124110	−	0.992269i	$-0.460393\pi$
$227$	3.73980	0.248219	0.124110	+	0.992269i	$0.460393\pi$
$228$	0.00887649	0.000587860 0
$229$	26.9241	1.77920	0.889598	−	0.456745i	$-0.150985\pi$
$229$	26.9241	1.77920	0.889598	+	0.456745i	$0.150985\pi$
$230$	0	0
$231$	0	0
$232$	−19.7968	−1.29972
$233$	18.3167	1.19997	0.599984	−	0.800012i	$-0.295174\pi$
$233$	18.3167	1.19997	0.599984	+	0.800012i	$0.295174\pi$
$234$	10.2425	0.669573
$235$	0	0
$236$	1.37965	0.0898073
$237$	−7.97984	−0.518346
$238$	−18.2656	−1.18399
$239$	10.9559	0.708676	0.354338	−	0.935117i	$-0.384706\pi$
$239$	10.9559	0.708676	0.354338	+	0.935117i	$0.384706\pi$
$240$	0	0
$241$	9.99444	0.643798	0.321899	−	0.946774i	$-0.395679\pi$
$241$	9.99444	0.643798	0.321899	+	0.946774i	$0.395679\pi$
$242$	0	0
$243$	−13.3143	−0.854110
$244$	−1.37507	−0.0880297
$245$	0	0
$246$	−6.48972	−0.413769
$247$	−0.269293	−0.0171347
$248$	−19.8685	−1.26165
$249$	6.12581	0.388208
$250$	0	0
$251$	9.65743	0.609572	0.304786	−	0.952421i	$-0.401415\pi$
$251$	9.65743	0.609572	0.304786	+	0.952421i	$0.401415\pi$
$252$	−1.57925	−0.0994832
$253$	0	0
$254$	3.30490	0.207368
$255$	0	0
$256$	−3.86526	−0.241579
$257$	−10.4276	−0.650454	−0.325227	−	0.945636i	$-0.605441\pi$
$257$	−10.4276	−0.650454	−0.325227	+	0.945636i	$0.605441\pi$
$258$	2.31606	0.144192
$259$	35.9095	2.23131
$260$	0	0
$261$	−18.0248	−1.11571
$262$	−9.54606	−0.589757
$263$	−10.9619	−0.675937	−0.337968	−	0.941157i	$-0.609740\pi$
$263$	−10.9619	−0.675937	−0.337968	+	0.941157i	$0.609740\pi$
$264$	0	0
$265$	0	0
$266$	−0.470620	−0.0288555
$267$	6.99867	0.428311
$268$	−2.17525	−0.132875
$269$	−0.0893449	−0.00544746	−0.00272373	−	0.999996i	$-0.500867\pi$
$269$	−0.0893449	−0.00544746	−0.00272373	+	0.999996i	$0.500867\pi$
$270$	0	0
$271$	13.3996	0.813965	0.406982	−	0.913436i	$-0.366581\pi$
$271$	13.3996	0.813965	0.406982	+	0.913436i	$0.366581\pi$
$272$	−13.4735	−0.816949
$273$	−5.94561	−0.359844
$274$	12.9777	0.784010
$275$	0	0
$276$	−0.108445	−0.00652764
$277$	−3.90669	−0.234730	−0.117365	−	0.993089i	$-0.537445\pi$
$277$	−3.90669	−0.234730	−0.117365	+	0.993089i	$0.537445\pi$
$278$	−0.698853	−0.0419144
$279$	−18.0902	−1.08303
$280$	0	0
$281$	1.53743	0.0917155	0.0458577	−	0.998948i	$-0.485398\pi$
$281$	1.53743	0.0917155	0.0458577	+	0.998948i	$0.485398\pi$
$282$	−1.73583	−0.103367
$283$	−5.41170	−0.321692	−0.160846	−	0.986980i	$-0.551422\pi$
$283$	−5.41170	−0.321692	−0.160846	+	0.986980i	$0.551422\pi$
$284$	−1.34632	−0.0798896
$285$	0	0
$286$	0	0
$287$	−30.3566	−1.79190
$288$	−2.44174	−0.143881
$289$	−3.36947	−0.198204
$290$	0	0
$291$	−2.49265	−0.146122
$292$	0.214039	0.0125257
$293$	11.4165	0.666958	0.333479	−	0.942757i	$-0.391777\pi$
$293$	11.4165	0.666958	0.333479	+	0.942757i	$0.391777\pi$
$294$	−4.92936	−0.287486
$295$	0	0
$296$	28.8421	1.67641
$297$	0	0
$298$	11.0621	0.640808
$299$	3.28999	0.190265
$300$	0	0
$301$	10.8337	0.624445
$302$	−2.63031	−0.151358
$303$	5.70066	0.327494
$304$	−0.347148	−0.0199103
$305$	0	0
$306$	−13.3576	−0.763604
$307$	−4.25008	−0.242565	−0.121282	−	0.992618i	$-0.538701\pi$
$307$	−4.25008	−0.242565	−0.121282	+	0.992618i	$0.538701\pi$
$308$	0	0
$309$	2.33990	0.133112
$310$	0	0
$311$	−16.6195	−0.942404	−0.471202	−	0.882025i	$-0.656180\pi$
$311$	−16.6195	−0.942404	−0.471202	+	0.882025i	$0.656180\pi$
$312$	−4.77544	−0.270356
$313$	26.5770	1.50222	0.751109	−	0.660178i	$-0.229519\pi$
$313$	26.5770	1.50222	0.751109	+	0.660178i	$0.229519\pi$
$314$	−28.8413	−1.62761
$315$	0	0
$316$	−2.24835	−0.126479
$317$	−5.79694	−0.325589	−0.162794	−	0.986660i	$-0.552051\pi$
$317$	−5.79694	−0.325589	−0.162794	+	0.986660i	$0.552051\pi$
$318$	2.33581	0.130985
$319$	0	0
$320$	0	0
$321$	−1.11564	−0.0622688
$322$	5.74963	0.320414
$323$	0.351195	0.0195410
$324$	−0.993793	−0.0552107
$325$	0	0
$326$	21.6650	1.19991
$327$	3.52584	0.194979
$328$	−24.3821	−1.34628
$329$	−8.11961	−0.447648
$330$	0	0
$331$	−12.9230	−0.710311	−0.355155	−	0.934807i	$-0.615572\pi$
$331$	−12.9230	−0.710311	−0.355155	+	0.934807i	$0.615572\pi$
$332$	1.72597	0.0947249
$333$	26.2605	1.43907
$334$	24.0077	1.31364
$335$	0	0
$336$	−7.66454	−0.418135
$337$	13.3854	0.729148	0.364574	−	0.931174i	$-0.381215\pi$
$337$	13.3854	0.729148	0.364574	+	0.931174i	$0.381215\pi$
$338$	−6.75898	−0.367640
$339$	3.32938	0.180827
$340$	0	0
$341$	0	0
$342$	−0.344163	−0.0186102
$343$	2.48809	0.134344
$344$	8.70152	0.469155
$345$	0	0
$346$	21.5355	1.15776
$347$	−8.44899	−0.453565	−0.226783	−	0.973945i	$-0.572821\pi$
$347$	−8.44899	−0.453565	−0.226783	+	0.973945i	$0.572821\pi$
$348$	0.630236	0.0337842
$349$	10.3988	0.556636	0.278318	−	0.960489i	$-0.410223\pi$
$349$	10.3988	0.556636	0.278318	+	0.960489i	$0.410223\pi$
$350$	0	0
$351$	−9.23559	−0.492959
$352$	0	0
$353$	−19.1073	−1.01698	−0.508489	−	0.861069i	$-0.669796\pi$
$353$	−19.1073	−1.01698	−0.508489	+	0.861069i	$0.669796\pi$
$354$	6.63825	0.352819
$355$	0	0
$356$	1.97190	0.104510
$357$	7.75389	0.410379
$358$	−22.8852	−1.20952
$359$	4.41417	0.232971	0.116486	−	0.993192i	$-0.462837\pi$
$359$	4.41417	0.232971	0.116486	+	0.993192i	$0.462837\pi$
$360$	0	0
$361$	−18.9910	−0.999524
$362$	−32.6546	−1.71629
$363$	0	0
$364$	−1.67520	−0.0878041
$365$	0	0
$366$	−6.61622	−0.345836
$367$	29.3617	1.53267	0.766335	−	0.642442i	$-0.222078\pi$
$367$	29.3617	1.53267	0.766335	+	0.642442i	$0.222078\pi$
$368$	4.24116	0.221086
$369$	−22.1997	−1.15567
$370$	0	0
$371$	10.9261	0.567254
$372$	0.632521	0.0327947
$373$	−4.96478	−0.257067	−0.128533	−	0.991705i	$-0.541027\pi$
$373$	−4.96478	−0.257067	−0.128533	+	0.991705i	$0.541027\pi$
$374$	0	0
$375$	0	0
$376$	−6.52157	−0.336325
$377$	−19.1200	−0.984728
$378$	−16.1403	−0.830165
$379$	7.92315	0.406985	0.203492	−	0.979077i	$-0.434771\pi$
$379$	7.92315	0.406985	0.203492	+	0.979077i	$0.434771\pi$
$380$	0	0
$381$	−1.40295	−0.0718755
$382$	7.29731	0.373363
$383$	24.5155	1.25268	0.626342	−	0.779549i	$-0.284551\pi$
$383$	24.5155	1.25268	0.626342	+	0.779549i	$0.284551\pi$
$384$	−5.60903	−0.286235
$385$	0	0
$386$	24.7872	1.26164
$387$	7.92268	0.402732
$388$	−0.702312	−0.0356545
$389$	5.46094	0.276881	0.138440	−	0.990371i	$-0.455791\pi$
$389$	5.46094	0.276881	0.138440	+	0.990371i	$0.455791\pi$
$390$	0	0
$391$	−4.29059	−0.216985
$392$	−18.5198	−0.935389
$393$	4.05237	0.204415
$394$	3.58665	0.180693
$395$	0	0
$396$	0	0
$397$	6.43455	0.322941	0.161470	−	0.986878i	$-0.448376\pi$
$397$	6.43455	0.322941	0.161470	+	0.986878i	$0.448376\pi$
$398$	8.84984	0.443602
$399$	0.199781	0.0100016
$400$	0	0
$401$	−14.7026	−0.734213	−0.367107	−	0.930179i	$-0.619652\pi$
$401$	−14.7026	−0.734213	−0.367107	+	0.930179i	$0.619652\pi$
$402$	−10.4664	−0.522014
$403$	−19.1893	−0.955885
$404$	1.60618	0.0799105
$405$	0	0
$406$	−33.4143	−1.65832
$407$	0	0
$408$	6.22784	0.308324
$409$	−4.39576	−0.217356	−0.108678	−	0.994077i	$-0.534662\pi$
$409$	−4.39576	−0.217356	−0.108678	+	0.994077i	$0.534662\pi$
$410$	0	0
$411$	−5.50911	−0.271745
$412$	0.659275	0.0324802
$413$	31.0514	1.52794
$414$	4.20469	0.206649
$415$	0	0
$416$	−2.59009	−0.126990
$417$	0.296668	0.0145279
$418$	0	0
$419$	17.8526	0.872159	0.436079	−	0.899908i	$-0.356367\pi$
$419$	17.8526	0.872159	0.436079	+	0.899908i	$0.356367\pi$
$420$	0	0
$421$	−4.82854	−0.235328	−0.117664	−	0.993053i	$-0.537541\pi$
$421$	−4.82854	−0.235328	−0.117664	+	0.993053i	$0.537541\pi$
$422$	37.2103	1.81137
$423$	−5.93785	−0.288708
$424$	8.77570	0.426186
$425$	0	0
$426$	−6.47792	−0.313856
$427$	−30.9484	−1.49770
$428$	−0.314335	−0.0151939
$429$	0	0
$430$	0	0
$431$	24.8739	1.19814	0.599068	−	0.800698i	$-0.295538\pi$
$431$	24.8739	1.19814	0.599068	+	0.800698i	$0.295538\pi$
$432$	−11.9057	−0.572813
$433$	21.2502	1.02122	0.510611	−	0.859812i	$-0.329419\pi$
$433$	21.2502	1.02122	0.510611	+	0.859812i	$0.329419\pi$
$434$	−33.5354	−1.60975
$435$	0	0
$436$	0.993417	0.0475761
$437$	−0.110548	−0.00528825
$438$	1.02986	0.0492088
$439$	15.9119	0.759434	0.379717	−	0.925103i	$-0.376021\pi$
$439$	15.9119	0.759434	0.379717	+	0.925103i	$0.376021\pi$
$440$	0	0
$441$	−16.8621	−0.802958
$442$	−14.1692	−0.673959
$443$	−26.2876	−1.24896	−0.624481	−	0.781040i	$-0.714690\pi$
$443$	−26.2876	−1.24896	−0.624481	+	0.781040i	$0.714690\pi$
$444$	−0.918197	−0.0435757
$445$	0	0
$446$	6.88945	0.326225
$447$	−4.69592	−0.222110
$448$	−31.1629	−1.47231
$449$	−8.18961	−0.386492	−0.193246	−	0.981150i	$-0.561902\pi$
$449$	−8.18961	−0.386492	−0.193246	+	0.981150i	$0.561902\pi$
$450$	0	0
$451$	0	0
$452$	0.938065	0.0441229
$453$	1.11659	0.0524618
$454$	5.06995	0.237945
$455$	0	0
$456$	0.160462	0.00751432
$457$	11.9164	0.557425	0.278713	−	0.960375i	$-0.410092\pi$
$457$	11.9164	0.557425	0.278713	+	0.960375i	$0.410092\pi$
$458$	36.5003	1.70555
$459$	12.0445	0.562188
$460$	0	0
$461$	−6.96172	−0.324240	−0.162120	−	0.986771i	$-0.551833\pi$
$461$	−6.96172	−0.324240	−0.162120	+	0.986771i	$0.551833\pi$
$462$	0	0
$463$	−12.4762	−0.579817	−0.289909	−	0.957054i	$-0.593625\pi$
$463$	−12.4762	−0.579817	−0.289909	+	0.957054i	$0.593625\pi$
$464$	−24.6477	−1.14424
$465$	0	0
$466$	24.8315	1.15030
$467$	6.14617	0.284411	0.142205	−	0.989837i	$-0.454581\pi$
$467$	6.14617	0.284411	0.142205	+	0.989837i	$0.454581\pi$
$468$	−1.22507	−0.0566287
$469$	−48.9579	−2.26067
$470$	0	0
$471$	12.2433	0.564142
$472$	24.9401	1.14796
$473$	0	0
$474$	−10.8181	−0.496890
$475$	0	0
$476$	2.18469	0.100135
$477$	7.99022	0.365847
$478$	14.8526	0.679341
$479$	−22.1942	−1.01408	−0.507039	−	0.861923i	$-0.669260\pi$
$479$	−22.1942	−1.01408	−0.507039	+	0.861923i	$0.669260\pi$
$480$	0	0
$481$	27.8560	1.27013
$482$	13.5492	0.617149
$483$	−2.44076	−0.111058
$484$	0	0
$485$	0	0
$486$	−18.0498	−0.818755
$487$	34.2306	1.55114	0.775569	−	0.631263i	$-0.217463\pi$
$487$	34.2306	1.55114	0.775569	+	0.631263i	$0.217463\pi$
$488$	−24.8574	−1.12524
$489$	−9.19693	−0.415900
$490$	0	0
$491$	16.9957	0.767007	0.383503	−	0.923539i	$-0.374717\pi$
$491$	16.9957	0.767007	0.383503	+	0.923539i	$0.374717\pi$
$492$	0.776212	0.0349943
$493$	24.9351	1.12302
$494$	−0.365073	−0.0164254
$495$	0	0
$496$	−24.7371	−1.11073
$497$	−30.3014	−1.35920
$498$	8.30461	0.372138
$499$	5.22946	0.234103	0.117051	−	0.993126i	$-0.462656\pi$
$499$	5.22946	0.234103	0.117051	+	0.993126i	$0.462656\pi$
$500$	0	0
$501$	−10.1914	−0.455319
$502$	13.0923	0.584339
$503$	41.9448	1.87023	0.935113	−	0.354350i	$-0.115298\pi$
$503$	41.9448	1.87023	0.935113	+	0.354350i	$0.115298\pi$
$504$	−28.5484	−1.27164
$505$	0	0
$506$	0	0
$507$	2.86923	0.127427
$508$	−0.395287	−0.0175380
$509$	20.3678	0.902787	0.451393	−	0.892325i	$-0.350927\pi$
$509$	20.3678	0.902787	0.451393	+	0.892325i	$0.350927\pi$
$510$	0	0
$511$	4.81734	0.213107
$512$	−24.7330	−1.09305
$513$	0.310330	0.0137014
$514$	−14.1364	−0.623529
$515$	0	0
$516$	−0.277016	−0.0121949
$517$	0	0
$518$	48.6816	2.13895
$519$	−9.14199	−0.401289
$520$	0	0
$521$	14.4779	0.634287	0.317143	−	0.948378i	$-0.397276\pi$
$521$	14.4779	0.634287	0.317143	+	0.948378i	$0.397276\pi$
$522$	−24.4358	−1.06953
$523$	11.1601	0.487998	0.243999	−	0.969775i	$-0.421541\pi$
$523$	11.1601	0.487998	0.243999	+	0.969775i	$0.421541\pi$
$524$	1.14177	0.0498784
$525$	0	0
$526$	−14.8607	−0.647958
$527$	25.0254	1.09013
$528$	0	0
$529$	−21.6494	−0.941279
$530$	0	0
$531$	22.7078	0.985436
$532$	0.0562891	0.00244044
$533$	−23.5485	−1.02000
$534$	9.48791	0.410582
$535$	0	0
$536$	−39.3225	−1.69847
$537$	9.71492	0.419230
$538$	−0.121123	−0.00522197
$539$	0	0
$540$	0	0
$541$	−10.6808	−0.459203	−0.229602	−	0.973285i	$-0.573742\pi$
$541$	−10.6808	−0.459203	−0.229602	+	0.973285i	$0.573742\pi$
$542$	18.1654	0.780272
$543$	13.8621	0.594880
$544$	3.37784	0.144824
$545$	0	0
$546$	−8.06031	−0.344949
$547$	1.74760	0.0747220	0.0373610	−	0.999302i	$-0.488105\pi$
$547$	1.74760	0.0747220	0.0373610	+	0.999302i	$0.488105\pi$
$548$	−1.55221	−0.0663072
$549$	−22.6325	−0.965930
$550$	0	0
$551$	0.642459	0.0273697
$552$	−1.96039	−0.0834396
$553$	−50.6031	−2.15186
$554$	−5.29619	−0.225014
$555$	0	0
$556$	0.0835872	0.00354489
$557$	−19.4844	−0.825579	−0.412790	−	0.910826i	$-0.635446\pi$
$557$	−19.4844	−0.825579	−0.412790	+	0.910826i	$0.635446\pi$
$558$	−24.5244	−1.03820
$559$	8.40403	0.355453
$560$	0	0
$561$	0	0
$562$	2.08426	0.0879191
$563$	−14.6892	−0.619074	−0.309537	−	0.950887i	$-0.600174\pi$
$563$	−14.6892	−0.619074	−0.309537	+	0.950887i	$0.600174\pi$
$564$	0.207616	0.00874222
$565$	0	0
$566$	−7.33650	−0.308376
$567$	−22.3671	−0.939330
$568$	−24.3377	−1.02119
$569$	−19.9335	−0.835658	−0.417829	−	0.908526i	$-0.637209\pi$
$569$	−19.9335	−0.835658	−0.417829	+	0.908526i	$0.637209\pi$
$570$	0	0
$571$	−5.24422	−0.219464	−0.109732	−	0.993961i	$-0.534999\pi$
$571$	−5.24422	−0.219464	−0.109732	+	0.993961i	$0.534999\pi$
$572$	0	0
$573$	−3.09776	−0.129411
$574$	−41.1537	−1.71772
$575$	0	0
$576$	−22.7894	−0.949557
$577$	−37.6004	−1.56533	−0.782663	−	0.622446i	$-0.786139\pi$
$577$	−37.6004	−1.56533	−0.782663	+	0.622446i	$0.786139\pi$
$578$	−4.56790	−0.190000
$579$	−10.5223	−0.437294
$580$	0	0
$581$	38.8460	1.61161
$582$	−3.37922	−0.140073
$583$	0	0
$584$	3.86923	0.160110
$585$	0	0
$586$	15.4770	0.639351
$587$	−25.5711	−1.05543	−0.527716	−	0.849421i	$-0.676952\pi$
$587$	−25.5711	−1.05543	−0.527716	+	0.849421i	$0.676952\pi$
$588$	0.589582	0.0243140
$589$	0.644788	0.0265680
$590$	0	0
$591$	−1.52256	−0.0626297
$592$	35.9095	1.47587
$593$	40.2260	1.65188	0.825942	−	0.563754i	$-0.190644\pi$
$593$	40.2260	1.65188	0.825942	+	0.563754i	$0.190644\pi$
$594$	0	0
$595$	0	0
$596$	−1.32309	−0.0541960
$597$	−3.75682	−0.153756
$598$	4.46015	0.182389
$599$	4.92997	0.201433	0.100716	−	0.994915i	$-0.467886\pi$
$599$	4.92997	0.201433	0.100716	+	0.994915i	$0.467886\pi$
$600$	0	0
$601$	−46.0896	−1.88003	−0.940017	−	0.341127i	$-0.889191\pi$
$601$	−46.0896	−1.88003	−0.940017	+	0.341127i	$0.889191\pi$
$602$	14.6870	0.598597
$603$	−35.8028	−1.45800
$604$	0.314602	0.0128010
$605$	0	0
$606$	7.72824	0.313938
$607$	−45.1365	−1.83203	−0.916016	−	0.401141i	$-0.868614\pi$
$607$	−45.1365	−1.83203	−0.916016	+	0.401141i	$0.868614\pi$
$608$	0.0870310	0.00352957
$609$	14.1846	0.574789
$610$	0	0
$611$	−6.29861	−0.254815
$612$	1.59766	0.0645814
$613$	4.73418	0.191212	0.0956059	−	0.995419i	$-0.469521\pi$
$613$	4.73418	0.191212	0.0956059	+	0.995419i	$0.469521\pi$
$614$	−5.76172	−0.232524
$615$	0	0
$616$	0	0
$617$	−17.8468	−0.718486	−0.359243	−	0.933244i	$-0.616965\pi$
$617$	−17.8468	−0.718486	−0.359243	+	0.933244i	$0.616965\pi$
$618$	3.17214	0.127602
$619$	0.356952	0.0143471	0.00717356	−	0.999974i	$-0.497717\pi$
$619$	0.356952	0.0143471	0.00717356	+	0.999974i	$0.497717\pi$
$620$	0	0
$621$	−3.79134	−0.152141
$622$	−22.5306	−0.903394
$623$	44.3811	1.77809
$624$	−5.94561	−0.238015
$625$	0	0
$626$	36.0297	1.44004
$627$	0	0
$628$	3.44960	0.137654
$629$	−36.3281	−1.44850
$630$	0	0
$631$	−31.9922	−1.27359	−0.636795	−	0.771033i	$-0.719740\pi$
$631$	−31.9922	−1.27359	−0.636795	+	0.771033i	$0.719740\pi$
$632$	−40.6438	−1.61672
$633$	−15.7960	−0.627835
$634$	−7.85876	−0.312111
$635$	0	0
$636$	−0.279377	−0.0110780
$637$	−17.8866	−0.708693
$638$	0	0
$639$	−22.1594	−0.876610
$640$	0	0
$641$	1.01285	0.0400050	0.0200025	−	0.999800i	$-0.493633\pi$
$641$	1.01285	0.0400050	0.0200025	+	0.999800i	$0.493633\pi$
$642$	−1.51244	−0.0596912
$643$	14.9724	0.590455	0.295228	−	0.955427i	$-0.404605\pi$
$643$	14.9724	0.590455	0.295228	+	0.955427i	$0.404605\pi$
$644$	−0.687692	−0.0270988
$645$	0	0
$646$	0.476106	0.0187321
$647$	−17.8873	−0.703224	−0.351612	−	0.936146i	$-0.614366\pi$
$647$	−17.8873	−0.703224	−0.351612	+	0.936146i	$0.614366\pi$
$648$	−17.9650	−0.705732
$649$	0	0
$650$	0	0
$651$	14.2360	0.557954
$652$	−2.59127	−0.101482
$653$	−45.7642	−1.79089	−0.895446	−	0.445169i	$-0.853144\pi$
$653$	−45.7642	−1.79089	−0.895446	+	0.445169i	$0.853144\pi$
$654$	4.77989	0.186908
$655$	0	0
$656$	−30.3566	−1.18523
$657$	3.52291	0.137442
$658$	−11.0075	−0.429119
$659$	9.54036	0.371640	0.185820	−	0.982584i	$-0.440506\pi$
$659$	9.54036	0.371640	0.185820	+	0.982584i	$0.440506\pi$
$660$	0	0
$661$	15.7769	0.613651	0.306825	−	0.951766i	$-0.400733\pi$
$661$	15.7769	0.613651	0.306825	+	0.951766i	$0.400733\pi$
$662$	−17.5193	−0.680908
$663$	6.01491	0.233600
$664$	31.2007	1.21082
$665$	0	0
$666$	35.6007	1.37950
$667$	−7.84901	−0.303915
$668$	−2.87147	−0.111100
$669$	−2.92462	−0.113072
$670$	0	0
$671$	0	0
$672$	1.92152	0.0741243
$673$	47.3031	1.82340	0.911700	−	0.410856i	$-0.134770\pi$
$673$	47.3031	1.82340	0.911700	+	0.410856i	$0.134770\pi$
$674$	18.1462	0.698966
$675$	0	0
$676$	0.808416	0.0310929
$677$	27.5431	1.05857	0.529284	−	0.848445i	$-0.322461\pi$
$677$	27.5431	1.05857	0.529284	+	0.848445i	$0.322461\pi$
$678$	4.51356	0.173342
$679$	−15.8068	−0.606609
$680$	0	0
$681$	−2.15223	−0.0824736
$682$	0	0
$683$	27.1617	1.03931	0.519656	−	0.854375i	$-0.326060\pi$
$683$	27.1617	1.03931	0.519656	+	0.854375i	$0.326060\pi$
$684$	0.0411641	0.00157395
$685$	0	0
$686$	3.37304	0.128783
$687$	−15.4946	−0.591157
$688$	10.8337	0.413032
$689$	8.47567	0.322897
$690$	0	0
$691$	−7.52680	−0.286333	−0.143166	−	0.989699i	$-0.545728\pi$
$691$	−7.52680	−0.286333	−0.143166	+	0.989699i	$0.545728\pi$
$692$	−2.57579	−0.0979167
$693$	0	0
$694$	−11.4541	−0.434791
$695$	0	0
$696$	11.3929	0.431847
$697$	30.7105	1.16324
$698$	14.0974	0.533595
$699$	−10.5411	−0.398702
$700$	0	0
$701$	31.8207	1.20185	0.600926	−	0.799305i	$-0.294799\pi$
$701$	31.8207	1.20185	0.600926	+	0.799305i	$0.294799\pi$
$702$	−12.5205	−0.472554
$703$	−0.936004	−0.0353021
$704$	0	0
$705$	0	0
$706$	−25.9032	−0.974882
$707$	36.1500	1.35956
$708$	−0.793977	−0.0298395
$709$	−14.4381	−0.542235	−0.271118	−	0.962546i	$-0.587393\pi$
$709$	−14.4381	−0.542235	−0.271118	+	0.962546i	$0.587393\pi$
$710$	0	0
$711$	−37.0059	−1.38783
$712$	35.6464	1.33591
$713$	−7.87747	−0.295013
$714$	10.5117	0.393392
$715$	0	0
$716$	2.73721	0.102294
$717$	−6.30503	−0.235465
$718$	5.98418	0.223328
$719$	5.41004	0.201761	0.100880	−	0.994899i	$-0.467834\pi$
$719$	5.41004	0.201761	0.100880	+	0.994899i	$0.467834\pi$
$720$	0	0
$721$	14.8382	0.552602
$722$	−25.7455	−0.958150
$723$	−5.75173	−0.213909
$724$	3.90570	0.145154
$725$	0	0
$726$	0	0
$727$	16.7753	0.622161	0.311080	−	0.950384i	$-0.399309\pi$
$727$	16.7753	0.622161	0.311080	+	0.950384i	$0.399309\pi$
$728$	−30.2828	−1.12236
$729$	−10.7246	−0.397208
$730$	0	0
$731$	−10.9600	−0.405371
$732$	0.791342	0.0292488
$733$	−14.0851	−0.520243	−0.260122	−	0.965576i	$-0.583763\pi$
$733$	−14.0851	−0.520243	−0.260122	+	0.965576i	$0.583763\pi$
$734$	39.8049	1.46923
$735$	0	0
$736$	−1.06327	−0.0391926
$737$	0	0
$738$	−30.0956	−1.10783
$739$	36.3457	1.33700	0.668499	−	0.743713i	$-0.266937\pi$
$739$	36.3457	1.33700	0.668499	+	0.743713i	$0.266937\pi$
$740$	0	0
$741$	0.154976	0.00569319
$742$	14.8122	0.543773
$743$	−1.95716	−0.0718012	−0.0359006	−	0.999355i	$-0.511430\pi$
$743$	−1.95716	−0.0718012	−0.0359006	+	0.999355i	$0.511430\pi$
$744$	11.4342	0.419198
$745$	0	0
$746$	−6.73063	−0.246426
$747$	28.4080	1.03939
$748$	0	0
$749$	−7.07466	−0.258503
$750$	0	0
$751$	18.7106	0.682759	0.341379	−	0.939926i	$-0.389106\pi$
$751$	18.7106	0.682759	0.341379	+	0.939926i	$0.389106\pi$
$752$	−8.11961	−0.296092
$753$	−5.55778	−0.202537
$754$	−25.9204	−0.943967
$755$	0	0
$756$	1.93048	0.0702107
$757$	14.5470	0.528721	0.264361	−	0.964424i	$-0.414839\pi$
$757$	14.5470	0.528721	0.264361	+	0.964424i	$0.414839\pi$
$758$	10.7412	0.390138
$759$	0	0
$760$	0	0
$761$	−13.1406	−0.476345	−0.238173	−	0.971223i	$-0.576548\pi$
$761$	−13.1406	−0.476345	−0.238173	+	0.971223i	$0.576548\pi$
$762$	−1.90195	−0.0689003
$763$	22.3586	0.809437
$764$	−0.872805	−0.0315770
$765$	0	0
$766$	33.2350	1.20083
$767$	24.0875	0.869748
$768$	2.22443	0.0802673
$769$	−38.9767	−1.40554	−0.702768	−	0.711419i	$-0.748053\pi$
$769$	−38.9767	−1.40554	−0.702768	+	0.711419i	$0.748053\pi$
$770$	0	0
$771$	6.00099	0.216121
$772$	−2.96471	−0.106702
$773$	38.7539	1.39388	0.696940	−	0.717129i	$-0.254544\pi$
$773$	38.7539	1.39388	0.696940	+	0.717129i	$0.254544\pi$
$774$	10.7406	0.386062
$775$	0	0
$776$	−12.6958	−0.455754
$777$	−20.6657	−0.741377
$778$	7.40326	0.265420
$779$	0.791265	0.0283500
$780$	0	0
$781$	0	0
$782$	−5.81665	−0.208003
$783$	22.0336	0.787417
$784$	−23.0578	−0.823493
$785$	0	0
$786$	5.49369	0.195953
$787$	−21.3842	−0.762265	−0.381132	−	0.924521i	$-0.624466\pi$
$787$	−21.3842	−0.762265	−0.381132	+	0.924521i	$0.624466\pi$
$788$	−0.428986	−0.0152820
$789$	6.30847	0.224588
$790$	0	0
$791$	21.1128	0.750686
$792$	0	0
$793$	−24.0075	−0.852533
$794$	8.72315	0.309573
$795$	0	0
$796$	−1.05850	−0.0375174
$797$	2.22456	0.0787978	0.0393989	−	0.999224i	$-0.487456\pi$
$797$	2.22456	0.0787978	0.0393989	+	0.999224i	$0.487456\pi$
$798$	0.270838	0.00958757
$799$	8.21426	0.290599
$800$	0	0
$801$	32.4558	1.14677
$802$	−19.9319	−0.703821
$803$	0	0
$804$	1.25184	0.0441491
$805$	0	0
$806$	−26.0144	−0.916318
$807$	0.0514174	0.00180998
$808$	29.0353	1.02146
$809$	−21.1682	−0.744234	−0.372117	−	0.928186i	$-0.621368\pi$
$809$	−21.1682	−0.744234	−0.372117	+	0.928186i	$0.621368\pi$
$810$	0	0
$811$	36.7172	1.28932	0.644658	−	0.764471i	$-0.277000\pi$
$811$	36.7172	1.28932	0.644658	+	0.764471i	$0.277000\pi$
$812$	3.99656	0.140252
$813$	−7.71135	−0.270449
$814$	0	0
$815$	0	0
$816$	7.75389	0.271440
$817$	−0.282388	−0.00987951
$818$	−5.95922	−0.208359
$819$	−27.5723	−0.963455
$820$	0	0
$821$	39.6693	1.38447	0.692235	−	0.721673i	$-0.256626\pi$
$821$	39.6693	1.38447	0.692235	+	0.721673i	$0.256626\pi$
$822$	−7.46856	−0.260496
$823$	45.9283	1.60096	0.800480	−	0.599359i	$-0.204578\pi$
$823$	45.9283	1.60096	0.800480	+	0.599359i	$0.204578\pi$
$824$	11.9178	0.415178
$825$	0	0
$826$	42.0956	1.46469
$827$	−39.6949	−1.38033	−0.690164	−	0.723653i	$-0.742462\pi$
$827$	−39.6949	−1.38033	−0.690164	+	0.723653i	$0.742462\pi$
$828$	−0.502907	−0.0174772
$829$	7.65584	0.265898	0.132949	−	0.991123i	$-0.457555\pi$
$829$	7.65584	0.265898	0.132949	+	0.991123i	$0.457555\pi$
$830$	0	0
$831$	2.24827	0.0779916
$832$	−24.1740	−0.838082
$833$	23.3266	0.808218
$834$	0.402185	0.0139265
$835$	0	0
$836$	0	0
$837$	22.1135	0.764354
$838$	24.2024	0.836057
$839$	27.5886	0.952465	0.476233	−	0.879319i	$-0.342002\pi$
$839$	27.5886	0.952465	0.476233	+	0.879319i	$0.342002\pi$
$840$	0	0
$841$	16.6150	0.572932
$842$	−6.54592	−0.225587
$843$	−0.884781	−0.0304735
$844$	−4.45058	−0.153195
$845$	0	0
$846$	−8.04979	−0.276757
$847$	0	0
$848$	10.9261	0.375203
$849$	3.11439	0.106886
$850$	0	0
$851$	11.4353	0.391997
$852$	0.774800	0.0265442
$853$	42.1496	1.44318	0.721588	−	0.692323i	$-0.243413\pi$
$853$	42.1496	1.44318	0.721588	+	0.692323i	$0.243413\pi$
$854$	−41.9559	−1.43570
$855$	0	0
$856$	−5.68229	−0.194217
$857$	−45.0850	−1.54008	−0.770038	−	0.637998i	$-0.779763\pi$
$857$	−45.0850	−1.54008	−0.770038	+	0.637998i	$0.779763\pi$
$858$	0	0
$859$	−11.8257	−0.403488	−0.201744	−	0.979438i	$-0.564661\pi$
$859$	−11.8257	−0.403488	−0.201744	+	0.979438i	$0.564661\pi$
$860$	0	0
$861$	17.4700	0.595377
$862$	33.7210	1.14854
$863$	−27.8713	−0.948750	−0.474375	−	0.880323i	$-0.657326\pi$
$863$	−27.8713	−0.948750	−0.474375	+	0.880323i	$0.657326\pi$
$864$	2.98479	0.101545
$865$	0	0
$866$	28.8084	0.978950
$867$	1.93911	0.0658555
$868$	4.01105	0.136144
$869$	0	0
$870$	0	0
$871$	−37.9781	−1.28684
$872$	17.9582	0.608141
$873$	−11.5595	−0.391229
$874$	−0.149868	−0.00506935
$875$	0	0
$876$	−0.123178	−0.00416181
$877$	−11.4471	−0.386543	−0.193271	−	0.981145i	$-0.561910\pi$
$877$	−11.4471	−0.386543	−0.193271	+	0.981145i	$0.561910\pi$
$878$	21.5714	0.727998
$879$	−6.57011	−0.221604
$880$	0	0
$881$	47.0037	1.58360	0.791798	−	0.610783i	$-0.209145\pi$
$881$	47.0037	1.58360	0.791798	+	0.610783i	$0.209145\pi$
$882$	−22.8595	−0.769721
$883$	46.9146	1.57880	0.789401	−	0.613877i	$-0.210391\pi$
$883$	46.9146	1.57880	0.789401	+	0.613877i	$0.210391\pi$
$884$	1.69472	0.0569997
$885$	0	0
$886$	−35.6374	−1.19726
$887$	−27.8427	−0.934866	−0.467433	−	0.884029i	$-0.654821\pi$
$887$	−27.8427	−0.934866	−0.467433	+	0.884029i	$0.654821\pi$
$888$	−16.5984	−0.557007
$889$	−8.89664	−0.298384
$890$	0	0
$891$	0	0
$892$	−0.824022	−0.0275903
$893$	0.211643	0.00708236
$894$	−6.36614	−0.212916
$895$	0	0
$896$	−35.5689	−1.18828
$897$	−1.89336	−0.0632176
$898$	−11.1024	−0.370494
$899$	45.7804	1.52686
$900$	0	0
$901$	−11.0534	−0.368244
$902$	0	0
$903$	−6.23473	−0.207479
$904$	16.9576	0.564001
$905$	0	0
$906$	1.51373	0.0502902
$907$	28.6233	0.950421	0.475210	−	0.879872i	$-0.342372\pi$
$907$	28.6233	0.950421	0.475210	+	0.879872i	$0.342372\pi$
$908$	−0.606398	−0.0201240
$909$	26.4364	0.876840
$910$	0	0
$911$	5.12823	0.169906	0.0849529	−	0.996385i	$-0.472926\pi$
$911$	5.12823	0.169906	0.0849529	+	0.996385i	$0.472926\pi$
$912$	0.199781	0.00661542
$913$	0	0
$914$	16.1547	0.534351
$915$	0	0
$916$	−4.36567	−0.144246
$917$	25.6976	0.848608
$918$	16.3284	0.538917
$919$	−35.2810	−1.16381	−0.581906	−	0.813256i	$-0.697693\pi$
$919$	−35.2810	−1.16381	−0.581906	+	0.813256i	$0.697693\pi$
$920$	0	0
$921$	2.44589	0.0805949
$922$	−9.43783	−0.310818
$923$	−23.5057	−0.773699
$924$	0	0
$925$	0	0
$926$	−16.9136	−0.555817
$927$	10.8511	0.356397
$928$	6.17926	0.202844
$929$	−59.1427	−1.94041	−0.970204	−	0.242289i	$-0.922102\pi$
$929$	−59.1427	−1.94041	−0.970204	+	0.242289i	$0.922102\pi$
$930$	0	0
$931$	0.601017	0.0196975
$932$	−2.97000	−0.0972857
$933$	9.56439	0.313124
$934$	8.33220	0.272638
$935$	0	0
$936$	−22.1458	−0.723857
$937$	14.4425	0.471817	0.235909	−	0.971775i	$-0.424193\pi$
$937$	14.4425	0.471817	0.235909	+	0.971775i	$0.424193\pi$
$938$	−66.3710	−2.16709
$939$	−15.2949	−0.499129
$940$	0	0
$941$	18.6591	0.608269	0.304135	−	0.952629i	$-0.401633\pi$
$941$	18.6591	0.608269	0.304135	+	0.952629i	$0.401633\pi$
$942$	16.5979	0.540790
$943$	−9.66700	−0.314801
$944$	31.0514	1.01064
$945$	0	0
$946$	0	0
$947$	0.991391	0.0322159	0.0161079	−	0.999870i	$-0.494872\pi$
$947$	0.991391	0.0322159	0.0161079	+	0.999870i	$0.494872\pi$
$948$	1.29391	0.0420242
$949$	3.73695	0.121306
$950$	0	0
$951$	3.33610	0.108180
$952$	39.4930	1.27998
$953$	−8.26404	−0.267699	−0.133849	−	0.991002i	$-0.542734\pi$
$953$	−8.26404	−0.267699	−0.133849	+	0.991002i	$0.542734\pi$
$954$	10.8321	0.350703
$955$	0	0
$956$	−1.77646	−0.0574549
$957$	0	0
$958$	−30.0881	−0.972101
$959$	−34.9353	−1.12812
$960$	0	0
$961$	14.9463	0.482139
$962$	37.7637	1.21755
$963$	−5.17368	−0.166720
$964$	−1.62057	−0.0521950
$965$	0	0
$966$	−3.30887	−0.106461
$967$	7.36029	0.236691	0.118345	−	0.992972i	$-0.462241\pi$
$967$	7.36029	0.236691	0.118345	+	0.992972i	$0.462241\pi$
$968$	0	0
$969$	−0.202110	−0.00649271
$970$	0	0
$971$	−4.97733	−0.159730	−0.0798650	−	0.996806i	$-0.525449\pi$
$971$	−4.97733	−0.159730	−0.0798650	+	0.996806i	$0.525449\pi$
$972$	2.15887	0.0692457
$973$	1.88128	0.0603111
$974$	46.4056	1.48693
$975$	0	0
$976$	−30.9484	−0.990633
$977$	−10.3368	−0.330704	−0.165352	−	0.986235i	$-0.552876\pi$
$977$	−10.3368	−0.330704	−0.165352	+	0.986235i	$0.552876\pi$
$978$	−12.4680	−0.398684
$979$	0	0
$980$	0	0
$981$	16.3508	0.522041
$982$	23.0407	0.735258
$983$	29.0614	0.926913	0.463457	−	0.886120i	$-0.346609\pi$
$983$	29.0614	0.926913	0.463457	+	0.886120i	$0.346609\pi$
$984$	14.0317	0.447315
$985$	0	0
$986$	33.8038	1.07653
$987$	4.67278	0.148736
$988$	0.0436651	0.00138917
$989$	3.44997	0.109703
$990$	0	0
$991$	7.70381	0.244719	0.122360	−	0.992486i	$-0.460954\pi$
$991$	7.70381	0.244719	0.122360	+	0.992486i	$0.460954\pi$
$992$	6.20166	0.196903
$993$	7.43708	0.236009
$994$	−41.0788	−1.30294
$995$	0	0
$996$	−0.993283	−0.0314734
$997$	−2.86418	−0.0907095	−0.0453547	−	0.998971i	$-0.514442\pi$
$997$	−2.86418	−0.0907095	−0.0453547	+	0.998971i	$0.514442\pi$
$998$	7.08945	0.224413
$999$	−32.1010	−1.01563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.w.1.4		4
5.4	even	2		605.2.a.k.1.1		4
11.2	odd	10		275.2.h.a.26.2		8
11.6	odd	10		275.2.h.a.201.2		8
11.10	odd	2		3025.2.a.bd.1.1		4
15.14	odd	2		5445.2.a.bi.1.4		4
20.19	odd	2		9680.2.a.cm.1.2		4
55.2	even	20		275.2.z.a.224.1		16
55.4	even	10		605.2.g.e.511.1		8
55.9	even	10		605.2.g.k.81.2		8
55.13	even	20		275.2.z.a.224.4		16
55.14	even	10		605.2.g.e.251.1		8
55.17	even	20		275.2.z.a.124.4		16
55.19	odd	10		605.2.g.m.251.2		8
55.24	odd	10		55.2.g.b.26.1	✓	8
55.28	even	20		275.2.z.a.124.1		16
55.29	odd	10		605.2.g.m.511.2		8
55.39	odd	10		55.2.g.b.36.1	yes	8
55.49	even	10		605.2.g.k.366.2		8
55.54	odd	2		605.2.a.j.1.4		4
165.134	even	10		495.2.n.e.136.2		8
165.149	even	10		495.2.n.e.91.2		8
165.164	even	2		5445.2.a.bp.1.1		4
220.39	even	10		880.2.bo.h.641.1		8
220.79	even	10		880.2.bo.h.81.1		8
220.219	even	2		9680.2.a.cn.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.26.1	✓	8	55.24	odd	10
55.2.g.b.36.1	yes	8	55.39	odd	10
275.2.h.a.26.2		8	11.2	odd	10
275.2.h.a.201.2		8	11.6	odd	10
275.2.z.a.124.1		16	55.28	even	20
275.2.z.a.124.4		16	55.17	even	20
275.2.z.a.224.1		16	55.2	even	20
275.2.z.a.224.4		16	55.13	even	20
495.2.n.e.91.2		8	165.149	even	10
495.2.n.e.136.2		8	165.134	even	10
605.2.a.j.1.4		4	55.54	odd	2
605.2.a.k.1.1		4	5.4	even	2
605.2.g.e.251.1		8	55.14	even	10
605.2.g.e.511.1		8	55.4	even	10
605.2.g.k.81.2		8	55.9	even	10
605.2.g.k.366.2		8	55.49	even	10
605.2.g.m.251.2		8	55.19	odd	10
605.2.g.m.511.2		8	55.29	odd	10
880.2.bo.h.81.1		8	220.79	even	10
880.2.bo.h.641.1		8	220.39	even	10
3025.2.a.w.1.4		4	1.1	even	1	trivial
3025.2.a.bd.1.1		4	11.10	odd	2
5445.2.a.bi.1.4		4	15.14	odd	2
5445.2.a.bp.1.1		4	165.164	even	2
9680.2.a.cm.1.2		4	20.19	odd	2
9680.2.a.cn.1.2		4	220.219	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35567 q^{2} -0.575493 q^{3} -0.162147 q^{4} -0.780181 q^{6} -3.64941 q^{7} -2.93117 q^{8} -2.66881 q^{9} +O(q^{10})\)	\(q+1.35567 q^{2} -0.575493 q^{3} -0.162147 q^{4} -0.780181 q^{6} -3.64941 q^{7} -2.93117 q^{8} -2.66881 q^{9} +0.0933146 q^{12} -2.83095 q^{13} -4.94742 q^{14} -3.64941 q^{16} +3.69195 q^{17} -3.61803 q^{18} +0.0951243 q^{19} +2.10021 q^{21} -1.16215 q^{23} +1.68687 q^{24} -3.83785 q^{26} +3.26236 q^{27} +0.591742 q^{28} +6.75389 q^{29} +6.77837 q^{31} +0.914918 q^{32} +5.00509 q^{34} +0.432740 q^{36} -9.83980 q^{37} +0.128958 q^{38} +1.62920 q^{39} +8.31822 q^{41} +2.84720 q^{42} -2.96862 q^{43} -1.57549 q^{46} +2.22491 q^{47} +2.10021 q^{48} +6.31822 q^{49} -2.12469 q^{51} +0.459031 q^{52} -2.99393 q^{53} +4.42270 q^{54} +10.6970 q^{56} -0.0547434 q^{57} +9.15607 q^{58} -8.50860 q^{59} +8.48037 q^{61} +9.18926 q^{62} +9.73958 q^{63} +8.53916 q^{64} +13.4153 q^{67} -0.598640 q^{68} +0.668808 q^{69} +8.30309 q^{71} +7.82272 q^{72} -1.32003 q^{73} -13.3396 q^{74} -0.0154241 q^{76} +2.20866 q^{78} +13.8661 q^{79} +6.12896 q^{81} +11.2768 q^{82} -10.6445 q^{83} -0.340544 q^{84} -4.02448 q^{86} -3.88682 q^{87} -12.1612 q^{89} +10.3313 q^{91} +0.188439 q^{92} -3.90091 q^{93} +3.01625 q^{94} -0.526529 q^{96} +4.33133 q^{97} +8.56545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 - q^4 + q^6 - 3 * q^7 - 3 * q^8	\( 4 q - q^{2} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8} - 8 q^{12} - q^{13} + 2 q^{14} - 3 q^{16} + q^{17} - 10 q^{18} + 20 q^{19} + 10 q^{21} - 5 q^{23} + 11 q^{24} - 15 q^{26} + 15 q^{27} - 13 q^{28} + 12 q^{29} - 5 q^{31} + 8 q^{32} + 2 q^{34} - 7 q^{37} - 20 q^{38} + 7 q^{39} + 11 q^{41} - 12 q^{42} - 19 q^{43} - 4 q^{46} - 5 q^{47} + 10 q^{48} + 3 q^{49} + 7 q^{51} + 11 q^{52} + 11 q^{53} - 8 q^{54} + 11 q^{56} + 5 q^{57} + 14 q^{58} + 9 q^{59} + 12 q^{61} + 35 q^{62} + 5 q^{63} - 3 q^{64} + 19 q^{67} + 3 q^{68} - 8 q^{69} + 5 q^{71} + 25 q^{72} - 11 q^{73} + 8 q^{78} + 34 q^{79} + 4 q^{81} + 6 q^{82} + 11 q^{83} + 11 q^{84} + q^{86} + 19 q^{87} - 8 q^{89} - 8 q^{91} + 12 q^{92} + 5 q^{93} - q^{94} - 34 q^{96} - 32 q^{97} + 8 q^{98}+O(q^{100}) \) 4 * q - q^2 - q^4 + q^6 - 3 * q^7 - 3 * q^8 - 8 * q^12 - q^13 + 2 * q^14 - 3 * q^16 + q^17 - 10 * q^18 + 20 * q^19 + 10 * q^21 - 5 * q^23 + 11 * q^24 - 15 * q^26 + 15 * q^27 - 13 * q^28 + 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 - 7 * q^37 - 20 * q^38 + 7 * q^39 + 11 * q^41 - 12 * q^42 - 19 * q^43 - 4 * q^46 - 5 * q^47 + 10 * q^48 + 3 * q^49 + 7 * q^51 + 11 * q^52 + 11 * q^53 - 8 * q^54 + 11 * q^56 + 5 * q^57 + 14 * q^58 + 9 * q^59 + 12 * q^61 + 35 * q^62 + 5 * q^63 - 3 * q^64 + 19 * q^67 + 3 * q^68 - 8 * q^69 + 5 * q^71 + 25 * q^72 - 11 * q^73 + 8 * q^78 + 34 * q^79 + 4 * q^81 + 6 * q^82 + 11 * q^83 + 11 * q^84 + q^86 + 19 * q^87 - 8 * q^89 - 8 * q^91 + 12 * q^92 + 5 * q^93 - q^94 - 34 * q^96 - 32 * q^97 + 8 * q^98

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