LMFDB - Embedded newform 3025.2.a.w.1.2

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			1.2
Root			$0.737640$ 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)

$f(q)$	$=$	$q-0.737640 q^{2} +2.81156 q^{3} -1.45589 q^{4} -2.07392 q^{6} +1.03138 q^{7} +2.54920 q^{8} +4.90488 q^{9} +O(q^{10})$	\(q-0.737640 q^{2} +2.81156 q^{3} -1.45589 q^{4} -2.07392 q^{6} +1.03138 q^{7} +2.54920 q^{8} +4.90488 q^{9} -4.09331 q^{12} +3.44899 q^{13} -0.760787 q^{14} +1.03138 q^{16} +2.39822 q^{17} -3.61803 q^{18} +7.66881 q^{19} +2.89979 q^{21} -2.45589 q^{23} +7.16724 q^{24} -2.54411 q^{26} +5.35567 q^{27} -1.50157 q^{28} +5.95431 q^{29} -3.68820 q^{31} -5.85919 q^{32} -1.76902 q^{34} -7.14094 q^{36} -5.95858 q^{37} -5.65682 q^{38} +9.69704 q^{39} -3.93626 q^{41} -2.13900 q^{42} -7.64941 q^{43} +1.81156 q^{46} -5.84294 q^{47} +2.89979 q^{48} -5.93626 q^{49} +6.74273 q^{51} -5.02134 q^{52} +11.8480 q^{53} -3.95056 q^{54} +2.62920 q^{56} +21.5613 q^{57} -4.39214 q^{58} +2.94630 q^{59} -2.48037 q^{61} +2.72057 q^{62} +5.05879 q^{63} +2.25922 q^{64} +6.14702 q^{67} -3.49153 q^{68} -6.90488 q^{69} +2.02315 q^{71} +12.5035 q^{72} -0.825867 q^{73} +4.39529 q^{74} -11.1649 q^{76} -7.15293 q^{78} +12.0782 q^{79} +0.343178 q^{81} +2.90354 q^{82} +1.61002 q^{83} -4.22176 q^{84} +5.64252 q^{86} +16.7409 q^{87} +8.16116 q^{89} +3.55722 q^{91} +3.57549 q^{92} -10.3696 q^{93} +4.30999 q^{94} -16.4735 q^{96} -2.44278 q^{97} +4.37882 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.737640	−0.521590	−0.260795	−	0.965394i	$-0.583985\pi$
$2$	−0.737640	−0.521590	−0.260795	+	0.965394i	$0.583985\pi$
$3$	2.81156	1.62326	0.811628	−	0.584175i	$-0.198582\pi$
$3$	2.81156	1.62326	0.811628	+	0.584175i	$0.198582\pi$
$4$	−1.45589	−0.727943
$5$	0	0
$5$	0	0
$6$	−2.07392	−0.846675
$7$	1.03138	0.389825	0.194912	−	0.980821i	$-0.437558\pi$
$7$	1.03138	0.389825	0.194912	+	0.980821i	$0.437558\pi$
$8$	2.54920	0.901279
$9$	4.90488	1.63496
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−4.09331	−1.18164
$13$	3.44899	0.956577	0.478289	−	0.878203i	$-0.341257\pi$
$13$	3.44899	0.956577	0.478289	+	0.878203i	$0.341257\pi$
$14$	−0.760787	−0.203329
$15$	0	0
$16$	1.03138	0.257845
$17$	2.39822	0.581653	0.290826	−	0.956776i	$-0.406070\pi$
$17$	2.39822	0.581653	0.290826	+	0.956776i	$0.406070\pi$
$18$	−3.61803	−0.852779
$19$	7.66881	1.75935	0.879673	−	0.475580i	$-0.157762\pi$
$19$	7.66881	1.75935	0.879673	+	0.475580i	$0.157762\pi$
$20$	0	0
$21$	2.89979	0.632786
$22$	0	0
$23$	−2.45589	−0.512088	−0.256044	−	0.966665i	$-0.582419\pi$
$23$	−2.45589	−0.512088	−0.256044	+	0.966665i	$0.582419\pi$
$24$	7.16724	1.46301
$25$	0	0
$26$	−2.54411	−0.498942
$27$	5.35567	1.03070
$28$	−1.50157	−0.283770
$29$	5.95431	1.10569	0.552844	−	0.833285i	$-0.313542\pi$
$29$	5.95431	1.10569	0.552844	+	0.833285i	$0.313542\pi$
$30$	0	0
$31$	−3.68820	−0.662421	−0.331210	−	0.943557i	$-0.607457\pi$
$31$	−3.68820	−0.662421	−0.331210	+	0.943557i	$0.607457\pi$
$32$	−5.85919	−1.03577
$33$	0	0
$34$	−1.76902	−0.303384
$35$	0	0
$36$	−7.14094	−1.19016
$37$	−5.95858	−0.979584	−0.489792	−	0.871839i	$-0.662927\pi$
$37$	−5.95858	−0.979584	−0.489792	+	0.871839i	$0.662927\pi$
$38$	−5.65682	−0.917658
$39$	9.69704	1.55277
$40$	0	0
$41$	−3.93626	−0.614740	−0.307370	−	0.951590i	$-0.599449\pi$
$41$	−3.93626	−0.614740	−0.307370	+	0.951590i	$0.599449\pi$
$42$	−2.13900	−0.330055
$43$	−7.64941	−1.16652	−0.583262	−	0.812284i	$-0.698224\pi$
$43$	−7.64941	−1.16652	−0.583262	+	0.812284i	$0.698224\pi$
$44$	0	0
$45$	0	0
$46$	1.81156	0.267100
$47$	−5.84294	−0.852281	−0.426140	−	0.904657i	$-0.640127\pi$
$47$	−5.84294	−0.852281	−0.426140	+	0.904657i	$0.640127\pi$
$48$	2.89979	0.418548
$49$	−5.93626	−0.848037
$50$	0	0
$51$	6.74273	0.944171
$52$	−5.02134	−0.696334
$53$	11.8480	1.62745	0.813726	−	0.581249i	$-0.197436\pi$
$53$	11.8480	1.62745	0.813726	+	0.581249i	$0.197436\pi$
$54$	−3.95056	−0.537603
$55$	0	0
$56$	2.62920	0.351341
$57$	21.5613	2.85587
$58$	−4.39214	−0.576717
$59$	2.94630	0.383575	0.191788	−	0.981436i	$-0.438572\pi$
$59$	2.94630	0.383575	0.191788	+	0.981436i	$0.438572\pi$
$60$	0	0
$61$	−2.48037	−0.317579	−0.158789	−	0.987312i	$-0.550759\pi$
$61$	−2.48037	−0.317579	−0.158789	+	0.987312i	$0.550759\pi$
$62$	2.72057	0.345512
$63$	5.05879	0.637348
$64$	2.25922	0.282402
$65$	0	0
$66$	0	0
$67$	6.14702	0.750978	0.375489	−	0.926827i	$-0.377475\pi$
$67$	6.14702	0.750978	0.375489	+	0.926827i	$0.377475\pi$
$68$	−3.49153	−0.423410
$69$	−6.90488	−0.831249
$70$	0	0
$71$	2.02315	0.240103	0.120052	−	0.992768i	$-0.461694\pi$
$71$	2.02315	0.240103	0.120052	+	0.992768i	$0.461694\pi$
$72$	12.5035	1.47355
$73$	−0.825867	−0.0966604	−0.0483302	−	0.998831i	$-0.515390\pi$
$73$	−0.825867	−0.0966604	−0.0483302	+	0.998831i	$0.515390\pi$
$74$	4.39529	0.510942
$75$	0	0
$76$	−11.1649	−1.28070
$77$	0	0
$78$	−7.15293	−0.809910
$79$	12.0782	1.35890	0.679451	−	0.733721i	$-0.262218\pi$
$79$	12.0782	1.35890	0.679451	+	0.733721i	$0.262218\pi$
$80$	0	0
$81$	0.343178	0.0381309
$82$	2.90354	0.320642
$83$	1.61002	0.176722	0.0883612	−	0.996088i	$-0.471837\pi$
$83$	1.61002	0.176722	0.0883612	+	0.996088i	$0.471837\pi$
$84$	−4.22176	−0.460632
$85$	0	0
$86$	5.64252	0.608448
$87$	16.7409	1.79481
$88$	0	0
$89$	8.16116	0.865081	0.432541	−	0.901614i	$-0.357617\pi$
$89$	8.16116	0.865081	0.432541	+	0.901614i	$0.357617\pi$
$90$	0	0
$91$	3.55722	0.372898
$92$	3.57549	0.372771
$93$	−10.3696	−1.07528
$94$	4.30999	0.444541
$95$	0	0
$96$	−16.4735	−1.68132
$97$	−2.44278	−0.248027	−0.124013	−	0.992281i	$-0.539577\pi$
$97$	−2.44278	−0.248027	−0.124013	+	0.992281i	$0.539577\pi$
$98$	4.37882	0.442328
$99$	0	0
$100$	0	0
$101$	7.52373	0.748640	0.374320	−	0.927300i	$-0.377876\pi$
$101$	7.52373	0.748640	0.374320	+	0.927300i	$0.377876\pi$
$102$	−4.97371	−0.492471
$103$	9.48231	0.934320	0.467160	−	0.884173i	$-0.345277\pi$
$103$	9.48231	0.934320	0.467160	+	0.884173i	$0.345277\pi$
$104$	8.79217	0.862143
$105$	0	0
$106$	−8.73958	−0.848863
$107$	−4.64678	−0.449221	−0.224611	−	0.974449i	$-0.572111\pi$
$107$	−4.64678	−0.449221	−0.224611	+	0.974449i	$0.572111\pi$
$108$	−7.79726	−0.750291
$109$	5.32826	0.510355	0.255178	−	0.966894i	$-0.417866\pi$
$109$	5.32826	0.510355	0.255178	+	0.966894i	$0.417866\pi$
$110$	0	0
$111$	−16.7529	−1.59012
$112$	1.06374	0.100514
$113$	−0.304901	−0.0286826	−0.0143413	−	0.999897i	$-0.504565\pi$
$113$	−0.304901	−0.0286826	−0.0143413	+	0.999897i	$0.504565\pi$
$114$	−15.9045	−1.48959
$115$	0	0
$116$	−8.66881	−0.804879
$117$	16.9169	1.56396
$118$	−2.17331	−0.200069
$119$	2.47347	0.226743
$120$	0	0
$121$	0	0
$122$	1.82962	0.165646
$123$	−11.0670	−0.997880
$124$	5.36960	0.482205
$125$	0	0
$126$	−3.73157	−0.332434
$127$	−11.9100	−1.05684	−0.528419	−	0.848984i	$-0.677215\pi$
$127$	−11.9100	−1.05684	−0.528419	+	0.848984i	$0.677215\pi$
$128$	10.0519	0.888470
$129$	−21.5068	−1.89357
$130$	0	0
$131$	11.1875	0.977452	0.488726	−	0.872437i	$-0.337462\pi$
$131$	11.1875	0.977452	0.488726	+	0.872437i	$0.337462\pi$
$132$	0	0
$133$	7.90945	0.685837
$134$	−4.53429	−0.391703
$135$	0	0
$136$	6.11353	0.524231
$137$	4.28124	0.365771	0.182886	−	0.983134i	$-0.441456\pi$
$137$	4.28124	0.365771	0.182886	+	0.983134i	$0.441456\pi$
$138$	5.09331	0.433572
$139$	5.95320	0.504943	0.252472	−	0.967604i	$-0.418757\pi$
$139$	5.95320	0.504943	0.252472	+	0.967604i	$0.418757\pi$
$140$	0	0
$141$	−16.4278	−1.38347
$142$	−1.49235	−0.125236
$143$	0	0
$144$	5.05879	0.421566
$145$	0	0
$146$	0.609193	0.0504171
$147$	−16.6901	−1.37658
$148$	8.67501	0.713082
$149$	3.78444	0.310034	0.155017	−	0.987912i	$-0.450457\pi$
$149$	3.78444	0.310034	0.155017	+	0.987912i	$0.450457\pi$
$150$	0	0
$151$	24.3566	1.98211	0.991057	−	0.133437i	$-0.0426013\pi$
$151$	24.3566	1.98211	0.991057	+	0.133437i	$0.0426013\pi$
$152$	19.5493	1.58566
$153$	11.7629	0.950978
$154$	0	0
$155$	0	0
$156$	−14.1178	−1.13033
$157$	−7.23210	−0.577184	−0.288592	−	0.957452i	$-0.593187\pi$
$157$	−7.23210	−0.577184	−0.288592	+	0.957452i	$0.593187\pi$
$158$	−8.90936	−0.708790
$159$	33.3115	2.64177
$160$	0	0
$161$	−2.53295	−0.199625
$162$	−0.253142	−0.0198887
$163$	−18.6892	−1.46385	−0.731924	−	0.681386i	$-0.761377\pi$
$163$	−18.6892	−1.46385	−0.731924	+	0.681386i	$0.761377\pi$
$164$	5.73074	0.447496
$165$	0	0
$166$	−1.18761	−0.0921767
$167$	7.85328	0.607705	0.303852	−	0.952719i	$-0.401727\pi$
$167$	7.85328	0.607705	0.303852	+	0.952719i	$0.401727\pi$
$168$	7.39214	0.570316
$169$	−1.10448	−0.0849597
$170$	0	0
$171$	37.6145	2.87646
$172$	11.1367	0.849164
$173$	11.2047	0.851877	0.425938	−	0.904752i	$-0.359944\pi$
$173$	11.2047	0.851877	0.425938	+	0.904752i	$0.359944\pi$
$174$	−12.3488	−0.936158
$175$	0	0
$176$	0	0
$177$	8.28370	0.622641
$178$	−6.02000	−0.451218
$179$	1.46463	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$179$	1.46463	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$180$	0	0
$181$	9.28900	0.690446	0.345223	−	0.938521i	$-0.387803\pi$
$181$	9.28900	0.690446	0.345223	+	0.938521i	$0.387803\pi$
$182$	−2.62395	−0.194500
$183$	−6.97371	−0.515511
$184$	−6.26055	−0.461534
$185$	0	0
$186$	7.64904	0.560855
$187$	0	0
$188$	8.50666	0.620412
$189$	5.52373	0.401793
$190$	0	0
$191$	−4.47296	−0.323652	−0.161826	−	0.986819i	$-0.551738\pi$
$191$	−4.47296	−0.323652	−0.161826	+	0.986819i	$0.551738\pi$
$192$	6.35192	0.458410
$193$	−22.6660	−1.63154	−0.815768	−	0.578380i	$-0.803685\pi$
$193$	−22.6660	−1.63154	−0.815768	+	0.578380i	$0.803685\pi$
$194$	1.80189	0.129368
$195$	0	0
$196$	8.64252	0.617323
$197$	−11.2080	−0.798535	−0.399267	−	0.916835i	$-0.630735\pi$
$197$	−11.2080	−0.798535	−0.399267	+	0.916835i	$0.630735\pi$
$198$	0	0
$199$	−7.81979	−0.554330	−0.277165	−	0.960822i	$-0.589395\pi$
$199$	−7.81979	−0.554330	−0.277165	+	0.960822i	$0.589395\pi$
$200$	0	0
$201$	17.2827	1.21903
$202$	−5.54981	−0.390483
$203$	6.14116	0.431025
$204$	−9.81665	−0.687303
$205$	0	0
$206$	−6.99454	−0.487332
$207$	−12.0458	−0.837242
$208$	3.55722	0.246649
$209$	0	0
$210$	0	0
$211$	22.7670	1.56734	0.783672	−	0.621175i	$-0.213344\pi$
$211$	22.7670	1.56734	0.783672	+	0.621175i	$0.213344\pi$
$212$	−17.2494	−1.18469
$213$	5.68820	0.389749
$214$	3.42765	0.234309
$215$	0	0
$216$	13.6527	0.928948
$217$	−3.80394	−0.258228
$218$	−3.93034	−0.266196
$219$	−2.32197	−0.156905
$220$	0	0
$221$	8.27142	0.556396
$222$	12.3576	0.829389
$223$	16.0427	1.07430	0.537148	−	0.843488i	$-0.319501\pi$
$223$	16.0427	1.07430	0.537148	+	0.843488i	$0.319501\pi$
$224$	−6.04305	−0.403768
$225$	0	0
$226$	0.224907	0.0149606
$227$	−24.1562	−1.60330	−0.801652	−	0.597791i	$-0.796045\pi$
$227$	−24.1562	−1.60330	−0.801652	+	0.597791i	$0.796045\pi$
$228$	−31.3908	−2.07891
$229$	−15.0143	−0.992172	−0.496086	−	0.868274i	$-0.665230\pi$
$229$	−15.0143	−0.992172	−0.496086	+	0.868274i	$0.665230\pi$
$230$	0	0
$231$	0	0
$232$	15.1787	0.996534
$233$	11.4259	0.748538	0.374269	−	0.927320i	$-0.377894\pi$
$233$	11.4259	0.748538	0.374269	+	0.927320i	$0.377894\pi$
$234$	−12.4786	−0.815749
$235$	0	0
$236$	−4.28948	−0.279221
$237$	33.9586	2.20584
$238$	−1.82453	−0.118267
$239$	−27.4067	−1.77279	−0.886397	−	0.462927i	$-0.846799\pi$
$239$	−27.4067	−1.77279	−0.886397	+	0.462927i	$0.846799\pi$
$240$	0	0
$241$	−10.9387	−0.704624	−0.352312	−	0.935883i	$-0.614604\pi$
$241$	−10.9387	−0.704624	−0.352312	+	0.935883i	$0.614604\pi$
$242$	0	0
$243$	−15.1022	−0.968804
$244$	3.61114	0.231179
$245$	0	0
$246$	8.16348	0.520485
$247$	26.4496	1.68295
$248$	−9.40197	−0.597026
$249$	4.52666	0.286866
$250$	0	0
$251$	17.2311	1.08762	0.543809	−	0.839209i	$-0.316982\pi$
$251$	17.2311	1.08762	0.543809	+	0.839209i	$0.316982\pi$
$252$	−7.36503	−0.463953
$253$	0	0
$254$	8.78527	0.551237
$255$	0	0
$256$	−11.9331	−0.745819
$257$	−18.4954	−1.15371	−0.576856	−	0.816846i	$-0.695721\pi$
$257$	−18.4954	−1.15371	−0.576856	+	0.816846i	$0.695721\pi$
$258$	15.8643	0.987667
$259$	−6.14556	−0.381866
$260$	0	0
$261$	29.2052	1.80775
$262$	−8.25232	−0.509830
$263$	3.69135	0.227618	0.113809	−	0.993503i	$-0.463695\pi$
$263$	3.69135	0.227618	0.113809	+	0.993503i	$0.463695\pi$
$264$	0	0
$265$	0	0
$266$	−5.83433	−0.357726
$267$	22.9456	1.40425
$268$	−8.94936	−0.546669
$269$	−9.94510	−0.606363	−0.303182	−	0.952933i	$-0.598049\pi$
$269$	−9.94510	−0.606363	−0.303182	+	0.952933i	$0.598049\pi$
$270$	0	0
$271$	−1.25365	−0.0761539	−0.0380770	−	0.999275i	$-0.512123\pi$
$271$	−1.25365	−0.0761539	−0.0380770	+	0.999275i	$0.512123\pi$
$272$	2.47347	0.149976
$273$	10.0013	0.605308
$274$	−3.15802	−0.190783
$275$	0	0
$276$	10.0527	0.605102
$277$	−8.09331	−0.486280	−0.243140	−	0.969991i	$-0.578177\pi$
$277$	−8.09331	−0.486280	−0.243140	+	0.969991i	$0.578177\pi$
$278$	−4.39132	−0.263374
$279$	−18.0902	−1.08303
$280$	0	0
$281$	−25.3702	−1.51346	−0.756731	−	0.653726i	$-0.773205\pi$
$281$	−25.3702	−1.51346	−0.756731	+	0.653726i	$0.773205\pi$
$282$	12.1178	0.721604
$283$	−20.4424	−1.21517	−0.607587	−	0.794253i	$-0.707863\pi$
$283$	−20.4424	−1.21517	−0.607587	+	0.794253i	$0.707863\pi$
$284$	−2.94547	−0.174782
$285$	0	0
$286$	0	0
$287$	−4.05977	−0.239641
$288$	−28.7386	−1.69344
$289$	−11.2486	−0.661680
$290$	0	0
$291$	−6.86803	−0.402611
$292$	1.20237	0.0703633
$293$	2.54907	0.148918	0.0744591	−	0.997224i	$-0.476277\pi$
$293$	2.54907	0.148918	0.0744591	+	0.997224i	$0.476277\pi$
$294$	12.3113	0.718011
$295$	0	0
$296$	−15.1896	−0.882878
$297$	0	0
$298$	−2.79156	−0.161711
$299$	−8.47033	−0.489852
$300$	0	0
$301$	−7.88945	−0.454740
$302$	−17.9664	−1.03385
$303$	21.1534	1.21523
$304$	7.90945	0.453638
$305$	0	0
$306$	−8.67682	−0.496021
$307$	8.99273	0.513242	0.256621	−	0.966512i	$-0.417391\pi$
$307$	8.99273	0.513242	0.256621	+	0.966512i	$0.417391\pi$
$308$	0	0
$309$	26.6601	1.51664
$310$	0	0
$311$	−20.1232	−1.14108	−0.570540	−	0.821270i	$-0.693266\pi$
$311$	−20.1232	−1.14108	−0.570540	+	0.821270i	$0.693266\pi$
$312$	24.7197	1.39948
$313$	−7.10483	−0.401589	−0.200794	−	0.979633i	$-0.564352\pi$
$313$	−7.10483	−0.401589	−0.200794	+	0.979633i	$0.564352\pi$
$314$	5.33469	0.301054
$315$	0	0
$316$	−17.5845	−0.989204
$317$	−2.29323	−0.128801	−0.0644003	−	0.997924i	$-0.520513\pi$
$317$	−2.29323	−0.128801	−0.0644003	+	0.997924i	$0.520513\pi$
$318$	−24.5719	−1.37792
$319$	0	0
$320$	0	0
$321$	−13.0647	−0.729201
$322$	1.86841	0.104122
$323$	18.3915	1.02333
$324$	−0.499629	−0.0277572
$325$	0	0
$326$	13.7859	0.763529
$327$	14.9807	0.828437
$328$	−10.0343	−0.554052
$329$	−6.02629	−0.332240
$330$	0	0
$331$	15.3951	0.846192	0.423096	−	0.906085i	$-0.360943\pi$
$331$	15.3951	0.846192	0.423096	+	0.906085i	$0.360943\pi$
$332$	−2.34400	−0.128644
$333$	−29.2261	−1.60158
$334$	−5.79289	−0.316973
$335$	0	0
$336$	2.99078	0.163161
$337$	−19.4968	−1.06206	−0.531030	−	0.847353i	$-0.678195\pi$
$337$	−19.4968	−1.06206	−0.531030	+	0.847353i	$0.678195\pi$
$338$	0.814706	0.0443141
$339$	−0.857247	−0.0465592
$340$	0	0
$341$	0	0
$342$	−27.7460	−1.50033
$343$	−13.3422	−0.720411
$344$	−19.4999	−1.05136
$345$	0	0
$346$	−8.26503	−0.444331
$347$	−2.16905	−0.116440	−0.0582202	−	0.998304i	$-0.518543\pi$
$347$	−2.16905	−0.116440	−0.0582202	+	0.998304i	$0.518543\pi$
$348$	−24.3729	−1.30652
$349$	25.0520	1.34100	0.670502	−	0.741908i	$-0.266079\pi$
$349$	25.0520	1.34100	0.670502	+	0.741908i	$0.266079\pi$
$350$	0	0
$351$	18.4717	0.985944
$352$	0	0
$353$	23.2532	1.23764	0.618821	−	0.785532i	$-0.287611\pi$
$353$	23.2532	1.23764	0.618821	+	0.785532i	$0.287611\pi$
$354$	−6.11039	−0.324764
$355$	0	0
$356$	−11.8817	−0.629730
$357$	6.95431	0.368061
$358$	−1.08037	−0.0570993
$359$	−10.1224	−0.534239	−0.267119	−	0.963663i	$-0.586072\pi$
$359$	−10.1224	−0.534239	−0.267119	+	0.963663i	$0.586072\pi$
$360$	0	0
$361$	39.8106	2.09530
$362$	−6.85194	−0.360130
$363$	0	0
$364$	−5.17891	−0.271448
$365$	0	0
$366$	5.14409	0.268886
$367$	−3.70925	−0.193621	−0.0968105	−	0.995303i	$-0.530864\pi$
$367$	−3.70925	−0.193621	−0.0968105	+	0.995303i	$0.530864\pi$
$368$	−2.53295	−0.132039
$369$	−19.3068	−1.00507
$370$	0	0
$371$	12.2198	0.634421
$372$	15.0970	0.782741
$373$	−9.34017	−0.483616	−0.241808	−	0.970324i	$-0.577740\pi$
$373$	−9.34017	−0.483616	−0.241808	+	0.970324i	$0.577740\pi$
$374$	0	0
$375$	0	0
$376$	−14.8948	−0.768142
$377$	20.5364	1.05768
$378$	−4.07453	−0.209571
$379$	−9.81169	−0.503993	−0.251996	−	0.967728i	$-0.581087\pi$
$379$	−9.81169	−0.503993	−0.251996	+	0.967728i	$0.581087\pi$
$380$	0	0
$381$	−33.4856	−1.71552
$382$	3.29944	0.168814
$383$	18.0468	0.922149	0.461074	−	0.887362i	$-0.347464\pi$
$383$	18.0468	0.922149	0.461074	+	0.887362i	$0.347464\pi$
$384$	28.2615	1.44221
$385$	0	0
$386$	16.7194	0.850993
$387$	−37.5194	−1.90722
$388$	3.55641	0.180550
$389$	31.1915	1.58147	0.790737	−	0.612156i	$-0.209698\pi$
$389$	31.1915	1.58147	0.790737	+	0.612156i	$0.209698\pi$
$390$	0	0
$391$	−5.88974	−0.297857
$392$	−15.1327	−0.764317
$393$	31.4542	1.58666
$394$	8.26745	0.416508
$395$	0	0
$396$	0	0
$397$	10.6212	0.533062	0.266531	−	0.963826i	$-0.414123\pi$
$397$	10.6212	0.533062	0.266531	+	0.963826i	$0.414123\pi$
$398$	5.76820	0.289133
$399$	22.2379	1.11329
$400$	0	0
$401$	−27.5679	−1.37668	−0.688338	−	0.725390i	$-0.741659\pi$
$401$	−27.5679	−1.37668	−0.688338	+	0.725390i	$0.741659\pi$
$402$	−12.7484	−0.635834
$403$	−12.7206	−0.633657
$404$	−10.9537	−0.544967
$405$	0	0
$406$	−4.52997	−0.224818
$407$	0	0
$408$	17.1886	0.850961
$409$	−14.3682	−0.710460	−0.355230	−	0.934779i	$-0.615598\pi$
$409$	−14.3682	−0.710460	−0.355230	+	0.934779i	$0.615598\pi$
$410$	0	0
$411$	12.0370	0.593740
$412$	−13.8052	−0.680132
$413$	3.03875	0.149527
$414$	8.88548	0.436698
$415$	0	0
$416$	−20.2083	−0.990793
$417$	16.7378	0.819652
$418$	0	0
$419$	−31.4707	−1.53744	−0.768722	−	0.639584i	$-0.779107\pi$
$419$	−31.4707	−1.53744	−0.768722	+	0.639584i	$0.779107\pi$
$420$	0	0
$421$	26.5712	1.29500	0.647500	−	0.762065i	$-0.275815\pi$
$421$	26.5712	1.29500	0.647500	+	0.762065i	$0.275815\pi$
$422$	−16.7939	−0.817512
$423$	−28.6589	−1.39344
$424$	30.2030	1.46679
$425$	0	0
$426$	−4.19585	−0.203289
$427$	−2.55820	−0.123800
$428$	6.76518	0.327008
$429$	0	0
$430$	0	0
$431$	3.86870	0.186349	0.0931744	−	0.995650i	$-0.470299\pi$
$431$	3.86870	0.186349	0.0931744	+	0.995650i	$0.470299\pi$
$432$	5.52373	0.265761
$433$	−40.1388	−1.92895	−0.964474	−	0.264179i	$-0.914899\pi$
$433$	−40.1388	−1.92895	−0.964474	+	0.264179i	$0.914899\pi$
$434$	2.80594	0.134689
$435$	0	0
$436$	−7.75735	−0.371510
$437$	−18.8337	−0.900939
$438$	1.71278	0.0818399
$439$	−1.02336	−0.0488425	−0.0244212	−	0.999702i	$-0.507774\pi$
$439$	−1.02336	−0.0488425	−0.0244212	+	0.999702i	$0.507774\pi$
$440$	0	0
$441$	−29.1166	−1.38650
$442$	−6.10133	−0.290211
$443$	16.0728	0.763644	0.381822	−	0.924236i	$-0.375297\pi$
$443$	16.0728	0.763644	0.381822	+	0.924236i	$0.375297\pi$
$444$	24.3903	1.15751
$445$	0	0
$446$	−11.8337	−0.560343
$447$	10.6402	0.503264
$448$	2.33011	0.110087
$449$	35.8421	1.69149	0.845746	−	0.533585i	$-0.179156\pi$
$449$	35.8421	1.69149	0.845746	+	0.533585i	$0.179156\pi$
$450$	0	0
$451$	0	0
$452$	0.443901	0.0208793
$453$	68.4802	3.21748
$454$	17.8186	0.836268
$455$	0	0
$456$	54.9641	2.57393
$457$	−25.1525	−1.17658	−0.588291	−	0.808649i	$-0.700199\pi$
$457$	−25.1525	−1.17658	−0.588291	+	0.808649i	$0.700199\pi$
$458$	11.0751	0.517507
$459$	12.8441	0.599509
$460$	0	0
$461$	−6.65631	−0.310015	−0.155008	−	0.987913i	$-0.549540\pi$
$461$	−6.65631	−0.310015	−0.155008	+	0.987913i	$0.549540\pi$
$462$	0	0
$463$	−38.7730	−1.80194	−0.900968	−	0.433886i	$-0.857142\pi$
$463$	−38.7730	−1.80194	−0.900968	+	0.433886i	$0.857142\pi$
$464$	6.14116	0.285096
$465$	0	0
$466$	−8.42823	−0.390430
$467$	−22.5494	−1.04346	−0.521731	−	0.853110i	$-0.674714\pi$
$467$	−22.5494	−1.04346	−0.521731	+	0.853110i	$0.674714\pi$
$468$	−24.6290	−1.13848
$469$	6.33991	0.292750
$470$	0	0
$471$	−20.3335	−0.936918
$472$	7.51071	0.345708
$473$	0	0
$474$	−25.0492	−1.15055
$475$	0	0
$476$	−3.60109	−0.165056
$477$	58.1131	2.66082
$478$	20.2163	0.924672
$479$	1.63186	0.0745618	0.0372809	−	0.999305i	$-0.488130\pi$
$479$	1.63186	0.0745618	0.0372809	+	0.999305i	$0.488130\pi$
$480$	0	0
$481$	−20.5511	−0.937048
$482$	8.06883	0.367525
$483$	−7.12155	−0.324042
$484$	0	0
$485$	0	0
$486$	11.1400	0.505319
$487$	−1.05030	−0.0475936	−0.0237968	−	0.999717i	$-0.507575\pi$
$487$	−1.05030	−0.0475936	−0.0237968	+	0.999717i	$0.507575\pi$
$488$	−6.32296	−0.286227
$489$	−52.5457	−2.37620
$490$	0	0
$491$	13.9141	0.627934	0.313967	−	0.949434i	$-0.398342\pi$
$491$	13.9141	0.627934	0.313967	+	0.949434i	$0.398342\pi$
$492$	16.1123	0.726400
$493$	14.2797	0.643127
$494$	−19.5103	−0.877811
$495$	0	0
$496$	−3.80394	−0.170802
$497$	2.08663	0.0935983
$498$	−3.33905	−0.149626
$499$	17.3673	0.777466	0.388733	−	0.921350i	$-0.372913\pi$
$499$	17.3673	0.777466	0.388733	+	0.921350i	$0.372913\pi$
$500$	0	0
$501$	22.0800	0.986460
$502$	−12.7104	−0.567291
$503$	35.8536	1.59863	0.799316	−	0.600910i	$-0.205195\pi$
$503$	35.8536	1.59863	0.799316	+	0.600910i	$0.205195\pi$
$504$	12.8959	0.574428
$505$	0	0
$506$	0	0
$507$	−3.10530	−0.137911
$508$	17.3396	0.769319
$509$	2.13878	0.0947999	0.0474000	−	0.998876i	$-0.484906\pi$
$509$	2.13878	0.0947999	0.0474000	+	0.998876i	$0.484906\pi$
$510$	0	0
$511$	−0.851782	−0.0376806
$512$	−11.3014	−0.499458
$513$	41.0716	1.81336
$514$	13.6430	0.601765
$515$	0	0
$516$	31.3115	1.37841
$517$	0	0
$518$	4.53321	0.199178
$519$	31.5027	1.38281
$520$	0	0
$521$	−12.7352	−0.557940	−0.278970	−	0.960300i	$-0.589993\pi$
$521$	−12.7352	−0.557940	−0.278970	+	0.960300i	$0.589993\pi$
$522$	−21.5429	−0.942908
$523$	23.9088	1.04546	0.522729	−	0.852499i	$-0.324914\pi$
$523$	23.9088	1.04546	0.522729	+	0.852499i	$0.324914\pi$
$524$	−16.2877	−0.711530
$525$	0	0
$526$	−2.72289	−0.118723
$527$	−8.84510	−0.385299
$528$	0	0
$529$	−16.9686	−0.737766
$530$	0	0
$531$	14.4512	0.627130
$532$	−11.5153	−0.499250
$533$	−13.5761	−0.588046
$534$	−16.9256	−0.732443
$535$	0	0
$536$	15.6700	0.676840
$537$	4.11790	0.177700
$538$	7.33590	0.316273
$539$	0	0
$540$	0	0
$541$	−1.31921	−0.0567171	−0.0283586	−	0.999598i	$-0.509028\pi$
$541$	−1.31921	−0.0567171	−0.0283586	+	0.999598i	$0.509028\pi$
$542$	0.924744	0.0397212
$543$	26.1166	1.12077
$544$	−14.0516	−0.602457
$545$	0	0
$546$	−7.37739	−0.315723
$547$	9.32128	0.398549	0.199275	−	0.979944i	$-0.436141\pi$
$547$	9.32128	0.398549	0.199275	+	0.979944i	$0.436141\pi$
$548$	−6.23301	−0.266261
$549$	−12.1659	−0.519228
$550$	0	0
$551$	45.6625	1.94529
$552$	−17.6019	−0.749187
$553$	12.4572	0.529734
$554$	5.96996	0.253639
$555$	0	0
$556$	−8.66718	−0.367570
$557$	39.3172	1.66592	0.832961	−	0.553331i	$-0.186644\pi$
$557$	39.3172	1.66592	0.832961	+	0.553331i	$0.186644\pi$
$558$	13.3440	0.564898
$559$	−26.3827	−1.11587
$560$	0	0
$561$	0	0
$562$	18.7141	0.789407
$563$	19.9810	0.842097	0.421048	−	0.907038i	$-0.361662\pi$
$563$	19.9810	0.842097	0.421048	+	0.907038i	$0.361662\pi$
$564$	23.9170	1.00709
$565$	0	0
$566$	15.0791	0.633824
$567$	0.353947	0.0148644
$568$	5.15741	0.216400
$569$	34.5647	1.44903	0.724515	−	0.689260i	$-0.242064\pi$
$569$	34.5647	1.44903	0.724515	+	0.689260i	$0.242064\pi$
$570$	0	0
$571$	−3.15090	−0.131861	−0.0659306	−	0.997824i	$-0.521002\pi$
$571$	−3.15090	−0.131861	−0.0659306	+	0.997824i	$0.521002\pi$
$572$	0	0
$573$	−12.5760	−0.525370
$574$	2.99465	0.124994
$575$	0	0
$576$	11.0812	0.461715
$577$	−27.3226	−1.13745	−0.568727	−	0.822526i	$-0.692564\pi$
$577$	−27.3226	−1.13745	−0.568727	+	0.822526i	$0.692564\pi$
$578$	8.29739	0.345126
$579$	−63.7269	−2.64840
$580$	0	0
$581$	1.66054	0.0688908
$582$	5.06614	0.209998
$583$	0	0
$584$	−2.10530	−0.0871180
$585$	0	0
$586$	−1.88030	−0.0776743
$587$	46.1679	1.90555	0.952776	−	0.303675i	$-0.0982138\pi$
$587$	46.1679	1.90555	0.952776	+	0.303675i	$0.0982138\pi$
$588$	24.2990	1.00207
$589$	−28.2841	−1.16543
$590$	0	0
$591$	−31.5119	−1.29623
$592$	−6.14556	−0.252581
$593$	39.4265	1.61905	0.809525	−	0.587085i	$-0.199725\pi$
$593$	39.4265	1.61905	0.809525	+	0.587085i	$0.199725\pi$
$594$	0	0
$595$	0	0
$596$	−5.50972	−0.225687
$597$	−21.9858	−0.899820
$598$	6.24805	0.255502
$599$	1.04875	0.0428507	0.0214253	−	0.999770i	$-0.493180\pi$
$599$	1.04875	0.0428507	0.0214253	+	0.999770i	$0.493180\pi$
$600$	0	0
$601$	−27.2498	−1.11154	−0.555771	−	0.831336i	$-0.687577\pi$
$601$	−27.2498	−1.11154	−0.555771	+	0.831336i	$0.687577\pi$
$602$	5.81958	0.237188
$603$	30.1504	1.22782
$604$	−35.4605	−1.44287
$605$	0	0
$606$	−15.6036	−0.633854
$607$	21.9217	0.889774	0.444887	−	0.895587i	$-0.353244\pi$
$607$	21.9217	0.889774	0.444887	+	0.895587i	$0.353244\pi$
$608$	−44.9330	−1.82227
$609$	17.2662	0.699664
$610$	0	0
$611$	−20.1522	−0.815272
$612$	−17.1255	−0.692258
$613$	10.5921	0.427809	0.213905	−	0.976855i	$-0.431382\pi$
$613$	10.5921	0.427809	0.213905	+	0.976855i	$0.431382\pi$
$614$	−6.63340	−0.267702
$615$	0	0
$616$	0	0
$617$	−4.60402	−0.185351	−0.0926755	−	0.995696i	$-0.529542\pi$
$617$	−4.60402	−0.185351	−0.0926755	+	0.995696i	$0.529542\pi$
$618$	−19.6656	−0.791065
$619$	37.0037	1.48731	0.743653	−	0.668566i	$-0.233092\pi$
$619$	37.0037	1.48731	0.743653	+	0.668566i	$0.233092\pi$
$620$	0	0
$621$	−13.1529	−0.527809
$622$	14.8437	0.595177
$623$	8.41726	0.337230
$624$	10.0013	0.400374
$625$	0	0
$626$	5.24081	0.209465
$627$	0	0
$628$	10.5291	0.420157
$629$	−14.2900	−0.569778
$630$	0	0
$631$	−24.8406	−0.988889	−0.494444	−	0.869209i	$-0.664628\pi$
$631$	−24.8406	−0.988889	−0.494444	+	0.869209i	$0.664628\pi$
$632$	30.7897	1.22475
$633$	64.0108	2.54420
$634$	1.69158	0.0671812
$635$	0	0
$636$	−48.4977	−1.92306
$637$	−20.4741	−0.811213
$638$	0	0
$639$	9.92328	0.392559
$640$	0	0
$641$	−44.4293	−1.75485	−0.877425	−	0.479714i	$-0.840740\pi$
$641$	−44.4293	−1.75485	−0.877425	+	0.479714i	$0.840740\pi$
$642$	9.63705	0.380344
$643$	−25.8610	−1.01986	−0.509929	−	0.860217i	$-0.670328\pi$
$643$	−25.8610	−1.01986	−0.509929	+	0.860217i	$0.670328\pi$
$644$	3.68769	0.145315
$645$	0	0
$646$	−13.5663	−0.533758
$647$	−19.4865	−0.766093	−0.383047	−	0.923729i	$-0.625125\pi$
$647$	−19.4865	−0.766093	−0.383047	+	0.923729i	$0.625125\pi$
$648$	0.874831	0.0343666
$649$	0	0
$650$	0	0
$651$	−10.6950	−0.419170
$652$	27.2093	1.06560
$653$	16.7298	0.654687	0.327344	−	0.944905i	$-0.393847\pi$
$653$	16.7298	0.654687	0.327344	+	0.944905i	$0.393847\pi$
$654$	−11.0504	−0.432105
$655$	0	0
$656$	−4.05977	−0.158508
$657$	−4.05077	−0.158036
$658$	4.44524	0.173293
$659$	1.66127	0.0647137	0.0323569	−	0.999476i	$-0.489699\pi$
$659$	1.66127	0.0647137	0.0323569	+	0.999476i	$0.489699\pi$
$660$	0	0
$661$	−44.0130	−1.71191	−0.855953	−	0.517053i	$-0.827029\pi$
$661$	−44.0130	−1.71191	−0.855953	+	0.517053i	$0.827029\pi$
$662$	−11.3561	−0.441365
$663$	23.2556	0.903172
$664$	4.10426	0.159276
$665$	0	0
$666$	21.5583	0.835369
$667$	−14.6231	−0.566210
$668$	−11.4335	−0.442375
$669$	45.1050	1.74386
$670$	0	0
$671$	0	0
$672$	−16.9904	−0.655419
$673$	−38.5949	−1.48772	−0.743862	−	0.668333i	$-0.767008\pi$
$673$	−38.5949	−1.48772	−0.743862	+	0.668333i	$0.767008\pi$
$674$	14.3817	0.553960
$675$	0	0
$676$	1.60799	0.0618458
$677$	−38.5988	−1.48347	−0.741737	−	0.670691i	$-0.765998\pi$
$677$	−38.5988	−1.48347	−0.741737	+	0.670691i	$0.765998\pi$
$678$	0.632340	0.0242849
$679$	−2.51944	−0.0966871
$680$	0	0
$681$	−67.9167	−2.60257
$682$	0	0
$683$	0.748158	0.0286275	0.0143137	−	0.999898i	$-0.495444\pi$
$683$	0.748158	0.0286275	0.0143137	+	0.999898i	$0.495444\pi$
$684$	−54.7625	−2.09390
$685$	0	0
$686$	9.84174	0.375759
$687$	−42.2136	−1.61055
$688$	−7.88945	−0.300783
$689$	40.8637	1.55678
$690$	0	0
$691$	5.22184	0.198648	0.0993242	−	0.995055i	$-0.468332\pi$
$691$	5.22184	0.198648	0.0993242	+	0.995055i	$0.468332\pi$
$692$	−16.3128	−0.620118
$693$	0	0
$694$	1.59998	0.0607342
$695$	0	0
$696$	42.6760	1.61763
$697$	−9.43999	−0.357565
$698$	−18.4794	−0.699455
$699$	32.1247	1.21507
$700$	0	0
$701$	14.1580	0.534740	0.267370	−	0.963594i	$-0.413845\pi$
$701$	14.1580	0.534740	0.267370	+	0.963594i	$0.413845\pi$
$702$	−13.6254	−0.514259
$703$	−45.6952	−1.72343
$704$	0	0
$705$	0	0
$706$	−17.1525	−0.645542
$707$	7.75983	0.291838
$708$	−12.0601	−0.453247
$709$	−17.2144	−0.646499	−0.323249	−	0.946314i	$-0.604775\pi$
$709$	−17.2144	−0.646499	−0.323249	+	0.946314i	$0.604775\pi$
$710$	0	0
$711$	59.2420	2.22175
$712$	20.8044	0.779680
$713$	9.05781	0.339217
$714$	−5.12978	−0.191977
$715$	0	0
$716$	−2.13233	−0.0796891
$717$	−77.0557	−2.87770
$718$	7.46667	0.278654
$719$	−26.5559	−0.990369	−0.495185	−	0.868788i	$-0.664900\pi$
$719$	−26.5559	−0.990369	−0.495185	+	0.868788i	$0.664900\pi$
$720$	0	0
$721$	9.77987	0.364221
$722$	−29.3659	−1.09289
$723$	−30.7548	−1.14379
$724$	−13.5237	−0.502606
$725$	0	0
$726$	0	0
$727$	−44.1917	−1.63898	−0.819490	−	0.573094i	$-0.805743\pi$
$727$	−44.1917	−1.63898	−0.819490	+	0.573094i	$0.805743\pi$
$728$	9.06806	0.336085
$729$	−43.4902	−1.61075
$730$	0	0
$731$	−18.3449	−0.678512
$732$	10.1529	0.375263
$733$	24.9604	0.921935	0.460968	−	0.887417i	$-0.347502\pi$
$733$	24.9604	0.921935	0.460968	+	0.887417i	$0.347502\pi$
$734$	2.73609	0.100991
$735$	0	0
$736$	14.3895	0.530404
$737$	0	0
$738$	14.2415	0.524237
$739$	−21.2342	−0.781114	−0.390557	−	0.920579i	$-0.627718\pi$
$739$	−21.2342	−0.781114	−0.390557	+	0.920579i	$0.627718\pi$
$740$	0	0
$741$	74.3648	2.73186
$742$	−9.01383	−0.330908
$743$	−30.6527	−1.12454	−0.562270	−	0.826954i	$-0.690072\pi$
$743$	−30.6527	−1.12454	−0.562270	+	0.826954i	$0.690072\pi$
$744$	−26.4342	−0.969125
$745$	0	0
$746$	6.88968	0.252249
$747$	7.89694	0.288934
$748$	0	0
$749$	−4.79259	−0.175118
$750$	0	0
$751$	−30.3073	−1.10593	−0.552965	−	0.833204i	$-0.686504\pi$
$751$	−30.3073	−1.10593	−0.552965	+	0.833204i	$0.686504\pi$
$752$	−6.02629	−0.219756
$753$	48.4463	1.76548
$754$	−15.1485	−0.551674
$755$	0	0
$756$	−8.04193	−0.292482
$757$	34.8694	1.26735	0.633674	−	0.773600i	$-0.281546\pi$
$757$	34.8694	1.26735	0.633674	+	0.773600i	$0.281546\pi$
$758$	7.23750	0.262878
$759$	0	0
$760$	0	0
$761$	2.68972	0.0975022	0.0487511	−	0.998811i	$-0.484476\pi$
$761$	2.68972	0.0975022	0.0487511	+	0.998811i	$0.484476\pi$
$762$	24.7003	0.894798
$763$	5.49546	0.198949
$764$	6.51212	0.235600
$765$	0	0
$766$	−13.3121	−0.480984
$767$	10.1617	0.366920
$768$	−33.5507	−1.21066
$769$	32.5735	1.17463	0.587315	−	0.809359i	$-0.300185\pi$
$769$	32.5735	1.17463	0.587315	+	0.809359i	$0.300185\pi$
$770$	0	0
$771$	−52.0010	−1.87277
$772$	32.9991	1.18767
$773$	−41.6637	−1.49854	−0.749270	−	0.662265i	$-0.769595\pi$
$773$	−41.6637	−1.49854	−0.749270	+	0.662265i	$0.769595\pi$
$774$	27.6758	0.994788
$775$	0	0
$776$	−6.22714	−0.223541
$777$	−17.2786	−0.619867
$778$	−23.0081	−0.824882
$779$	−30.1864	−1.08154
$780$	0	0
$781$	0	0
$782$	4.34451	0.155359
$783$	31.8894	1.13963
$784$	−6.12253	−0.218662
$785$	0	0
$786$	−23.2019	−0.827584
$787$	−35.9207	−1.28044	−0.640218	−	0.768193i	$-0.721156\pi$
$787$	−35.9207	−1.28044	−0.640218	+	0.768193i	$0.721156\pi$
$788$	16.3175	0.581288
$789$	10.3784	0.369482
$790$	0	0
$791$	−0.314468	−0.0111812
$792$	0	0
$793$	−8.55476	−0.303789
$794$	−7.83461	−0.278040
$795$	0	0
$796$	11.3847	0.403521
$797$	31.7197	1.12357	0.561785	−	0.827283i	$-0.310115\pi$
$797$	31.7197	1.12357	0.561785	+	0.827283i	$0.310115\pi$
$798$	−16.4036	−0.580680
$799$	−14.0126	−0.495731
$800$	0	0
$801$	40.0295	1.41437
$802$	20.3352	0.718061
$803$	0	0
$804$	−25.1617	−0.887384
$805$	0	0
$806$	9.38320	0.330509
$807$	−27.9612	−0.984283
$808$	19.1795	0.674733
$809$	8.51572	0.299397	0.149698	−	0.988732i	$-0.452170\pi$
$809$	8.51572	0.299397	0.149698	+	0.988732i	$0.452170\pi$
$810$	0	0
$811$	−9.03030	−0.317097	−0.158548	−	0.987351i	$-0.550681\pi$
$811$	−9.03030	−0.317097	−0.158548	+	0.987351i	$0.550681\pi$
$812$	−8.94083	−0.313762
$813$	−3.52472	−0.123617
$814$	0	0
$815$	0	0
$816$	6.95431	0.243450
$817$	−58.6619	−2.05232
$818$	10.5985	0.370569
$819$	17.4477	0.609672
$820$	0	0
$821$	54.3225	1.89587	0.947935	−	0.318465i	$-0.103167\pi$
$821$	54.3225	1.89587	0.947935	+	0.318465i	$0.103167\pi$
$822$	−8.87896	−0.309689
$823$	−17.8594	−0.622541	−0.311271	−	0.950321i	$-0.600755\pi$
$823$	−17.8594	−0.622541	−0.311271	+	0.950321i	$0.600755\pi$
$824$	24.1723	0.842083
$825$	0	0
$826$	−2.24151	−0.0779920
$827$	−51.9494	−1.80646	−0.903229	−	0.429159i	$-0.858810\pi$
$827$	−51.9494	−1.80646	−0.903229	+	0.429159i	$0.858810\pi$
$828$	17.5373	0.609465
$829$	−19.7460	−0.685807	−0.342904	−	0.939371i	$-0.611410\pi$
$829$	−19.7460	−0.685807	−0.342904	+	0.939371i	$0.611410\pi$
$830$	0	0
$831$	−22.7548	−0.789357
$832$	7.79201	0.270139
$833$	−14.2364	−0.493263
$834$	−12.3465	−0.427523
$835$	0	0
$836$	0	0
$837$	−19.7528	−0.682757
$838$	23.2140	0.801916
$839$	−4.11650	−0.142117	−0.0710586	−	0.997472i	$-0.522638\pi$
$839$	−4.11650	−0.142117	−0.0710586	+	0.997472i	$0.522638\pi$
$840$	0	0
$841$	6.45386	0.222547
$842$	−19.6000	−0.675460
$843$	−71.3300	−2.45674
$844$	−33.1462	−1.14094
$845$	0	0
$846$	21.1400	0.726807
$847$	0	0
$848$	12.2198	0.419630
$849$	−57.4751	−1.97254
$850$	0	0
$851$	14.6336	0.501633
$852$	−8.28138	−0.283715
$853$	5.50285	0.188414	0.0942070	−	0.995553i	$-0.469968\pi$
$853$	5.50285	0.188414	0.0942070	+	0.995553i	$0.469968\pi$
$854$	1.88703	0.0645729
$855$	0	0
$856$	−11.8456	−0.404873
$857$	−26.9281	−0.919847	−0.459924	−	0.887959i	$-0.652123\pi$
$857$	−26.9281	−0.919847	−0.459924	+	0.887959i	$0.652123\pi$
$858$	0	0
$859$	19.1519	0.653456	0.326728	−	0.945118i	$-0.394054\pi$
$859$	19.1519	0.653456	0.326728	+	0.945118i	$0.394054\pi$
$860$	0	0
$861$	−11.4143	−0.388998
$862$	−2.85371	−0.0971977
$863$	−4.96151	−0.168892	−0.0844458	−	0.996428i	$-0.526912\pi$
$863$	−4.96151	−0.168892	−0.0844458	+	0.996428i	$0.526912\pi$
$864$	−31.3799	−1.06757
$865$	0	0
$866$	29.6080	1.00612
$867$	−31.6260	−1.07408
$868$	5.53810	0.187975
$869$	0	0
$870$	0	0
$871$	21.2010	0.718368
$872$	13.5828	0.459972
$873$	−11.9815	−0.405514
$874$	13.8925	0.469921
$875$	0	0
$876$	3.38053	0.114218
$877$	−27.2053	−0.918659	−0.459329	−	0.888266i	$-0.651910\pi$
$877$	−27.2053	−0.918659	−0.459329	+	0.888266i	$0.651910\pi$
$878$	0.754874	0.0254758
$879$	7.16686	0.241732
$880$	0	0
$881$	10.3570	0.348935	0.174467	−	0.984663i	$-0.444180\pi$
$881$	10.3570	0.348935	0.174467	+	0.984663i	$0.444180\pi$
$882$	21.4776	0.723188
$883$	−7.39489	−0.248858	−0.124429	−	0.992229i	$-0.539710\pi$
$883$	−7.39489	−0.248858	−0.124429	+	0.992229i	$0.539710\pi$
$884$	−12.0422	−0.405025
$885$	0	0
$886$	−11.8560	−0.398309
$887$	−18.0590	−0.606363	−0.303181	−	0.952933i	$-0.598049\pi$
$887$	−18.0590	−0.606363	−0.303181	+	0.952933i	$0.598049\pi$
$888$	−42.7065	−1.43314
$889$	−12.2837	−0.411982
$890$	0	0
$891$	0	0
$892$	−23.3563	−0.782027
$893$	−44.8084	−1.49946
$894$	−7.84864	−0.262498
$895$	0	0
$896$	10.3673	0.346348
$897$	−23.8148	−0.795154
$898$	−26.4386	−0.882267
$899$	−21.9607	−0.732431
$900$	0	0
$901$	28.4141	0.946612
$902$	0	0
$903$	−22.1817	−0.738160
$904$	−0.777253	−0.0258511
$905$	0	0
$906$	−50.5137	−1.67821
$907$	−26.2971	−0.873179	−0.436590	−	0.899661i	$-0.643814\pi$
$907$	−26.2971	−0.873179	−0.436590	+	0.899661i	$0.643814\pi$
$908$	35.1687	1.16711
$909$	36.9030	1.22399
$910$	0	0
$911$	28.6489	0.949179	0.474589	−	0.880207i	$-0.342597\pi$
$911$	28.6489	0.949179	0.474589	+	0.880207i	$0.342597\pi$
$912$	22.2379	0.736371
$913$	0	0
$914$	18.5535	0.613694
$915$	0	0
$916$	21.8591	0.722245
$917$	11.5385	0.381035
$918$	−9.47430	−0.312698
$919$	−38.8568	−1.28177	−0.640884	−	0.767638i	$-0.721432\pi$
$919$	−38.8568	−1.28177	−0.640884	+	0.767638i	$0.721432\pi$
$920$	0	0
$921$	25.2836	0.833123
$922$	4.90996	0.161701
$923$	6.97781	0.229677
$924$	0	0
$925$	0	0
$926$	28.6006	0.939873
$927$	46.5096	1.52757
$928$	−34.8875	−1.14524
$929$	7.42132	0.243486	0.121743	−	0.992562i	$-0.461152\pi$
$929$	7.42132	0.243486	0.121743	+	0.992562i	$0.461152\pi$
$930$	0	0
$931$	−45.5240	−1.49199
$932$	−16.6349	−0.544893
$933$	−56.5775	−1.85227
$934$	16.6334	0.544260
$935$	0	0
$936$	43.1245	1.40957
$937$	−14.9360	−0.487937	−0.243968	−	0.969783i	$-0.578449\pi$
$937$	−14.9360	−0.487937	−0.243968	+	0.969783i	$0.578449\pi$
$938$	−4.67657	−0.152696
$939$	−19.9757	−0.651881
$940$	0	0
$941$	52.3409	1.70626	0.853132	−	0.521695i	$-0.174700\pi$
$941$	52.3409	1.70626	0.853132	+	0.521695i	$0.174700\pi$
$942$	14.9988	0.488687
$943$	9.66700	0.314801
$944$	3.03875	0.0989030
$945$	0	0
$946$	0	0
$947$	3.69553	0.120088	0.0600442	−	0.998196i	$-0.480876\pi$
$947$	3.69553	0.120088	0.0600442	+	0.998196i	$0.480876\pi$
$948$	−49.4398	−1.60573
$949$	−2.84841	−0.0924631
$950$	0	0
$951$	−6.44756	−0.209076
$952$	6.30538	0.204358
$953$	43.1526	1.39785	0.698925	−	0.715195i	$-0.253662\pi$
$953$	43.1526	1.39785	0.698925	+	0.715195i	$0.253662\pi$
$954$	−42.8666	−1.38786
$955$	0	0
$956$	39.9011	1.29049
$957$	0	0
$958$	−1.20373	−0.0388907
$959$	4.41559	0.142587
$960$	0	0
$961$	−17.3972	−0.561199
$962$	15.1593	0.488755
$963$	−22.7919	−0.734458
$964$	15.9255	0.512927
$965$	0	0
$966$	5.25314	0.169017
$967$	−29.2144	−0.939471	−0.469736	−	0.882807i	$-0.655651\pi$
$967$	−29.2144	−0.939471	−0.469736	+	0.882807i	$0.655651\pi$
$968$	0	0
$969$	51.7087	1.66112
$970$	0	0
$971$	−26.4046	−0.847365	−0.423683	−	0.905811i	$-0.639263\pi$
$971$	−26.4046	−0.847365	−0.423683	+	0.905811i	$0.639263\pi$
$972$	21.9870	0.705234
$973$	6.14001	0.196840
$974$	0.774743	0.0248244
$975$	0	0
$976$	−2.55820	−0.0818861
$977$	15.8434	0.506875	0.253437	−	0.967352i	$-0.418439\pi$
$977$	15.8434	0.506875	0.253437	+	0.967352i	$0.418439\pi$
$978$	38.7598	1.23940
$979$	0	0
$980$	0	0
$981$	26.1345	0.834410
$982$	−10.2636	−0.327525
$983$	−35.1039	−1.11964	−0.559821	−	0.828614i	$-0.689130\pi$
$983$	−35.1039	−1.11964	−0.559821	+	0.828614i	$0.689130\pi$
$984$	−28.2121	−0.899368
$985$	0	0
$986$	−10.5333	−0.335449
$987$	−16.9433	−0.539311
$988$	−38.5077	−1.22509
$989$	18.7861	0.597363
$990$	0	0
$991$	18.9700	0.602600	0.301300	−	0.953529i	$-0.402579\pi$
$991$	18.9700	0.602600	0.301300	+	0.953529i	$0.402579\pi$
$992$	21.6099	0.686114
$993$	43.2843	1.37359
$994$	−1.53918	−0.0488200
$995$	0	0
$996$	−6.59031	−0.208822
$997$	30.1347	0.954375	0.477188	−	0.878801i	$-0.341656\pi$
$997$	30.1347	0.954375	0.477188	+	0.878801i	$0.341656\pi$
$998$	−12.8108	−0.405519
$999$	−31.9122	−1.00966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.w.1.2		4
5.4	even	2		605.2.a.k.1.3		4
11.2	odd	10		275.2.h.a.26.1		8
11.6	odd	10		275.2.h.a.201.1		8
11.10	odd	2		3025.2.a.bd.1.3		4
15.14	odd	2		5445.2.a.bi.1.2		4
20.19	odd	2		9680.2.a.cm.1.4		4
55.2	even	20		275.2.z.a.224.3		16
55.4	even	10		605.2.g.e.511.2		8
55.9	even	10		605.2.g.k.81.1		8
55.13	even	20		275.2.z.a.224.2		16
55.14	even	10		605.2.g.e.251.2		8
55.17	even	20		275.2.z.a.124.2		16
55.19	odd	10		605.2.g.m.251.1		8
55.24	odd	10		55.2.g.b.26.2	✓	8
55.28	even	20		275.2.z.a.124.3		16
55.29	odd	10		605.2.g.m.511.1		8
55.39	odd	10		55.2.g.b.36.2	yes	8
55.49	even	10		605.2.g.k.366.1		8
55.54	odd	2		605.2.a.j.1.2		4
165.134	even	10		495.2.n.e.136.1		8
165.149	even	10		495.2.n.e.91.1		8
165.164	even	2		5445.2.a.bp.1.3		4
220.39	even	10		880.2.bo.h.641.2		8
220.79	even	10		880.2.bo.h.81.2		8
220.219	even	2		9680.2.a.cn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.26.2	✓	8	55.24	odd	10
55.2.g.b.36.2	yes	8	55.39	odd	10
275.2.h.a.26.1		8	11.2	odd	10
275.2.h.a.201.1		8	11.6	odd	10
275.2.z.a.124.2		16	55.17	even	20
275.2.z.a.124.3		16	55.28	even	20
275.2.z.a.224.2		16	55.13	even	20
275.2.z.a.224.3		16	55.2	even	20
495.2.n.e.91.1		8	165.149	even	10
495.2.n.e.136.1		8	165.134	even	10
605.2.a.j.1.2		4	55.54	odd	2
605.2.a.k.1.3		4	5.4	even	2
605.2.g.e.251.2		8	55.14	even	10
605.2.g.e.511.2		8	55.4	even	10
605.2.g.k.81.1		8	55.9	even	10
605.2.g.k.366.1		8	55.49	even	10
605.2.g.m.251.1		8	55.19	odd	10
605.2.g.m.511.1		8	55.29	odd	10
880.2.bo.h.81.2		8	220.79	even	10
880.2.bo.h.641.2		8	220.39	even	10
3025.2.a.w.1.2		4	1.1	even	1	trivial
3025.2.a.bd.1.3		4	11.10	odd	2
5445.2.a.bi.1.2		4	15.14	odd	2
5445.2.a.bp.1.3		4	165.164	even	2
9680.2.a.cm.1.4		4	20.19	odd	2
9680.2.a.cn.1.4		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)

\(f(q)\)	\(=\)	\(q-0.737640 q^{2} +2.81156 q^{3} -1.45589 q^{4} -2.07392 q^{6} +1.03138 q^{7} +2.54920 q^{8} +4.90488 q^{9} +O(q^{10})\)	\(q-0.737640 q^{2} +2.81156 q^{3} -1.45589 q^{4} -2.07392 q^{6} +1.03138 q^{7} +2.54920 q^{8} +4.90488 q^{9} -4.09331 q^{12} +3.44899 q^{13} -0.760787 q^{14} +1.03138 q^{16} +2.39822 q^{17} -3.61803 q^{18} +7.66881 q^{19} +2.89979 q^{21} -2.45589 q^{23} +7.16724 q^{24} -2.54411 q^{26} +5.35567 q^{27} -1.50157 q^{28} +5.95431 q^{29} -3.68820 q^{31} -5.85919 q^{32} -1.76902 q^{34} -7.14094 q^{36} -5.95858 q^{37} -5.65682 q^{38} +9.69704 q^{39} -3.93626 q^{41} -2.13900 q^{42} -7.64941 q^{43} +1.81156 q^{46} -5.84294 q^{47} +2.89979 q^{48} -5.93626 q^{49} +6.74273 q^{51} -5.02134 q^{52} +11.8480 q^{53} -3.95056 q^{54} +2.62920 q^{56} +21.5613 q^{57} -4.39214 q^{58} +2.94630 q^{59} -2.48037 q^{61} +2.72057 q^{62} +5.05879 q^{63} +2.25922 q^{64} +6.14702 q^{67} -3.49153 q^{68} -6.90488 q^{69} +2.02315 q^{71} +12.5035 q^{72} -0.825867 q^{73} +4.39529 q^{74} -11.1649 q^{76} -7.15293 q^{78} +12.0782 q^{79} +0.343178 q^{81} +2.90354 q^{82} +1.61002 q^{83} -4.22176 q^{84} +5.64252 q^{86} +16.7409 q^{87} +8.16116 q^{89} +3.55722 q^{91} +3.57549 q^{92} -10.3696 q^{93} +4.30999 q^{94} -16.4735 q^{96} -2.44278 q^{97} +4.37882 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 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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