LMFDB - Embedded newform 6025.2.a.q.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47177 q^{2} -1.07414 q^{3} +0.166115 q^{4} +1.58089 q^{6} -3.64779 q^{7} +2.69906 q^{8} -1.84623 q^{9} +O(q^{10})$	\(q-1.47177 q^{2} -1.07414 q^{3} +0.166115 q^{4} +1.58089 q^{6} -3.64779 q^{7} +2.69906 q^{8} -1.84623 q^{9} +5.15794 q^{11} -0.178431 q^{12} -0.639676 q^{13} +5.36872 q^{14} -4.30464 q^{16} +0.722806 q^{17} +2.71723 q^{18} +4.10252 q^{19} +3.91823 q^{21} -7.59132 q^{22} -1.02481 q^{23} -2.89917 q^{24} +0.941458 q^{26} +5.20552 q^{27} -0.605954 q^{28} +2.08345 q^{29} +1.88445 q^{31} +0.937323 q^{32} -5.54035 q^{33} -1.06381 q^{34} -0.306686 q^{36} +8.66520 q^{37} -6.03797 q^{38} +0.687101 q^{39} +2.67828 q^{41} -5.76675 q^{42} -7.04494 q^{43} +0.856813 q^{44} +1.50829 q^{46} +1.75154 q^{47} +4.62378 q^{48} +6.30638 q^{49} -0.776394 q^{51} -0.106260 q^{52} -10.6700 q^{53} -7.66134 q^{54} -9.84561 q^{56} -4.40667 q^{57} -3.06636 q^{58} +4.03176 q^{59} +0.491373 q^{61} -2.77348 q^{62} +6.73465 q^{63} +7.22975 q^{64} +8.15413 q^{66} -15.9873 q^{67} +0.120069 q^{68} +1.10079 q^{69} +2.40999 q^{71} -4.98308 q^{72} +11.1626 q^{73} -12.7532 q^{74} +0.681491 q^{76} -18.8151 q^{77} -1.01126 q^{78} -0.140139 q^{79} -0.0527699 q^{81} -3.94182 q^{82} +4.61199 q^{83} +0.650878 q^{84} +10.3686 q^{86} -2.23791 q^{87} +13.9216 q^{88} -3.21821 q^{89} +2.33341 q^{91} -0.170237 q^{92} -2.02416 q^{93} -2.57786 q^{94} -1.00682 q^{96} +5.91955 q^{97} -9.28156 q^{98} -9.52273 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

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 $

$2$	−1.47177	−1.04070	−0.520350	−	0.853953i	$-0.674199\pi$
$2$	−1.47177	−1.04070	−0.520350	+	0.853953i	$0.674199\pi$
$3$	−1.07414	−0.620154	−0.310077	−	0.950711i	$-0.600355\pi$
$3$	−1.07414	−0.620154	−0.310077	+	0.950711i	$0.600355\pi$
$4$	0.166115	0.0830576
$5$	0	0
$5$	0	0
$6$	1.58089	0.645395
$7$	−3.64779	−1.37874	−0.689368	−	0.724412i	$-0.742112\pi$
$7$	−3.64779	−1.37874	−0.689368	+	0.724412i	$0.742112\pi$
$8$	2.69906	0.954262
$9$	−1.84623	−0.615409
$10$	0	0
$11$	5.15794	1.55518	0.777589	−	0.628772i	$-0.216442\pi$
$11$	5.15794	1.55518	0.777589	+	0.628772i	$0.216442\pi$
$12$	−0.178431	−0.0515085
$13$	−0.639676	−0.177414	−0.0887071	−	0.996058i	$-0.528274\pi$
$13$	−0.639676	−0.177414	−0.0887071	+	0.996058i	$0.528274\pi$
$14$	5.36872	1.43485
$15$	0	0
$16$	−4.30464	−1.07616
$17$	0.722806	0.175306	0.0876532	−	0.996151i	$-0.472063\pi$
$17$	0.722806	0.175306	0.0876532	+	0.996151i	$0.472063\pi$
$18$	2.71723	0.640456
$19$	4.10252	0.941182	0.470591	−	0.882352i	$-0.344041\pi$
$19$	4.10252	0.941182	0.470591	+	0.882352i	$0.344041\pi$
$20$	0	0
$21$	3.91823	0.855029
$22$	−7.59132	−1.61848
$23$	−1.02481	−0.213688	−0.106844	−	0.994276i	$-0.534074\pi$
$23$	−1.02481	−0.213688	−0.106844	+	0.994276i	$0.534074\pi$
$24$	−2.89917	−0.591790
$25$	0	0
$26$	0.941458	0.184635
$27$	5.20552	1.00180
$28$	−0.605954	−0.114515
$29$	2.08345	0.386887	0.193443	−	0.981111i	$-0.438034\pi$
$29$	2.08345	0.386887	0.193443	+	0.981111i	$0.438034\pi$
$30$	0	0
$31$	1.88445	0.338457	0.169229	−	0.985577i	$-0.445872\pi$
$31$	1.88445	0.338457	0.169229	+	0.985577i	$0.445872\pi$
$32$	0.937323	0.165697
$33$	−5.54035	−0.964451
$34$	−1.06381	−0.182441
$35$	0	0
$36$	−0.306686	−0.0511144
$37$	8.66520	1.42455	0.712275	−	0.701900i	$-0.247665\pi$
$37$	8.66520	1.42455	0.712275	+	0.701900i	$0.247665\pi$
$38$	−6.03797	−0.979488
$39$	0.687101	0.110024
$40$	0	0
$41$	2.67828	0.418277	0.209139	−	0.977886i	$-0.432934\pi$
$41$	2.67828	0.418277	0.209139	+	0.977886i	$0.432934\pi$
$42$	−5.76675	−0.889829
$43$	−7.04494	−1.07434	−0.537172	−	0.843473i	$-0.680507\pi$
$43$	−7.04494	−1.07434	−0.537172	+	0.843473i	$0.680507\pi$
$44$	0.856813	0.129169
$45$	0	0
$46$	1.50829	0.222385
$47$	1.75154	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$47$	1.75154	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$48$	4.62378	0.667385
$49$	6.30638	0.900911
$50$	0	0
$51$	−0.776394	−0.108717
$52$	−0.106260	−0.0147356
$53$	−10.6700	−1.46564	−0.732819	−	0.680423i	$-0.761796\pi$
$53$	−10.6700	−1.46564	−0.732819	+	0.680423i	$0.761796\pi$
$54$	−7.66134	−1.04258
$55$	0	0
$56$	−9.84561	−1.31568
$57$	−4.40667	−0.583678
$58$	−3.06636	−0.402633
$59$	4.03176	0.524890	0.262445	−	0.964947i	$-0.415471\pi$
$59$	4.03176	0.524890	0.262445	+	0.964947i	$0.415471\pi$
$60$	0	0
$61$	0.491373	0.0629138	0.0314569	−	0.999505i	$-0.489985\pi$
$61$	0.491373	0.0629138	0.0314569	+	0.999505i	$0.489985\pi$
$62$	−2.77348	−0.352233
$63$	6.73465	0.848486
$64$	7.22975	0.903718
$65$	0	0
$66$	8.15413	1.00370
$67$	−15.9873	−1.95316	−0.976579	−	0.215159i	$-0.930973\pi$
$67$	−15.9873	−1.95316	−0.976579	+	0.215159i	$0.930973\pi$
$68$	0.120069	0.0145605
$69$	1.10079	0.132519
$70$	0	0
$71$	2.40999	0.286013	0.143007	−	0.989722i	$-0.454323\pi$
$71$	2.40999	0.286013	0.143007	+	0.989722i	$0.454323\pi$
$72$	−4.98308	−0.587261
$73$	11.1626	1.30648	0.653240	−	0.757151i	$-0.273409\pi$
$73$	11.1626	1.30648	0.653240	+	0.757151i	$0.273409\pi$
$74$	−12.7532	−1.48253
$75$	0	0
$76$	0.681491	0.0781723
$77$	−18.8151	−2.14418
$78$	−1.01126	−0.114502
$79$	−0.140139	−0.0157669	−0.00788343	−	0.999969i	$-0.502509\pi$
$79$	−0.140139	−0.0157669	−0.00788343	+	0.999969i	$0.502509\pi$
$80$	0	0
$81$	−0.0527699	−0.00586332
$82$	−3.94182	−0.435301
$83$	4.61199	0.506232	0.253116	−	0.967436i	$-0.418545\pi$
$83$	4.61199	0.506232	0.253116	+	0.967436i	$0.418545\pi$
$84$	0.650878	0.0710167
$85$	0	0
$86$	10.3686	1.11807
$87$	−2.23791	−0.239929
$88$	13.9216	1.48405
$89$	−3.21821	−0.341130	−0.170565	−	0.985346i	$-0.554559\pi$
$89$	−3.21821	−0.341130	−0.170565	+	0.985346i	$0.554559\pi$
$90$	0	0
$91$	2.33341	0.244607
$92$	−0.170237	−0.0177484
$93$	−2.02416	−0.209896
$94$	−2.57786	−0.265886
$95$	0	0
$96$	−1.00682	−0.102758
$97$	5.91955	0.601039	0.300519	−	0.953776i	$-0.402840\pi$
$97$	5.91955	0.601039	0.300519	+	0.953776i	$0.402840\pi$
$98$	−9.28156	−0.937579
$99$	−9.52273	−0.957071
$100$	0	0
$101$	−12.7292	−1.26660	−0.633302	−	0.773904i	$-0.718301\pi$
$101$	−12.7292	−1.26660	−0.633302	+	0.773904i	$0.718301\pi$
$102$	1.14268	0.113142
$103$	−4.49787	−0.443188	−0.221594	−	0.975139i	$-0.571126\pi$
$103$	−4.49787	−0.443188	−0.221594	+	0.975139i	$0.571126\pi$
$104$	−1.72653	−0.169300
$105$	0	0
$106$	15.7038	1.52529
$107$	−0.717660	−0.0693788	−0.0346894	−	0.999398i	$-0.511044\pi$
$107$	−0.717660	−0.0693788	−0.0346894	+	0.999398i	$0.511044\pi$
$108$	0.864716	0.0832074
$109$	8.40103	0.804673	0.402337	−	0.915492i	$-0.368198\pi$
$109$	8.40103	0.804673	0.402337	+	0.915492i	$0.368198\pi$
$110$	0	0
$111$	−9.30763	−0.883441
$112$	15.7024	1.48374
$113$	−4.77126	−0.448842	−0.224421	−	0.974492i	$-0.572049\pi$
$113$	−4.77126	−0.448842	−0.224421	+	0.974492i	$0.572049\pi$
$114$	6.48562	0.607434
$115$	0	0
$116$	0.346093	0.0321339
$117$	1.18099	0.109182
$118$	−5.93383	−0.546253
$119$	−2.63665	−0.241701
$120$	0	0
$121$	15.6044	1.41858
$122$	−0.723189	−0.0654744
$123$	−2.87685	−0.259396
$124$	0.313036	0.0281115
$125$	0	0
$126$	−9.91187	−0.883020
$127$	−3.97373	−0.352611	−0.176306	−	0.984335i	$-0.556415\pi$
$127$	−3.97373	−0.352611	−0.176306	+	0.984335i	$0.556415\pi$
$128$	−12.5152	−1.10620
$129$	7.56724	0.666259
$130$	0	0
$131$	−8.58625	−0.750184	−0.375092	−	0.926988i	$-0.622389\pi$
$131$	−8.58625	−0.750184	−0.375092	+	0.926988i	$0.622389\pi$
$132$	−0.920336	−0.0801050
$133$	−14.9651	−1.29764
$134$	23.5297	2.03265
$135$	0	0
$136$	1.95090	0.167288
$137$	−19.3080	−1.64959	−0.824795	−	0.565432i	$-0.808710\pi$
$137$	−19.3080	−1.64959	−0.824795	+	0.565432i	$0.808710\pi$
$138$	−1.62011	−0.137913
$139$	−17.9309	−1.52088	−0.760439	−	0.649410i	$-0.775016\pi$
$139$	−17.9309	−1.52088	−0.760439	+	0.649410i	$0.775016\pi$
$140$	0	0
$141$	−1.88139	−0.158442
$142$	−3.54696	−0.297654
$143$	−3.29941	−0.275911
$144$	7.94733	0.662278
$145$	0	0
$146$	−16.4288	−1.35965
$147$	−6.77393	−0.558704
$148$	1.43942	0.118320
$149$	−12.5996	−1.03220	−0.516098	−	0.856529i	$-0.672616\pi$
$149$	−12.5996	−1.03220	−0.516098	+	0.856529i	$0.672616\pi$
$150$	0	0
$151$	18.4897	1.50467	0.752337	−	0.658778i	$-0.228926\pi$
$151$	18.4897	1.50467	0.752337	+	0.658778i	$0.228926\pi$
$152$	11.0729	0.898134
$153$	−1.33446	−0.107885
$154$	27.6916	2.23145
$155$	0	0
$156$	0.114138	0.00913835
$157$	19.1795	1.53069	0.765346	−	0.643619i	$-0.222568\pi$
$157$	19.1795	1.53069	0.765346	+	0.643619i	$0.222568\pi$
$158$	0.206253	0.0164086
$159$	11.4611	0.908922
$160$	0	0
$161$	3.73829	0.294619
$162$	0.0776653	0.00610196
$163$	3.07077	0.240521	0.120261	−	0.992742i	$-0.461627\pi$
$163$	3.07077	0.240521	0.120261	+	0.992742i	$0.461627\pi$
$164$	0.444903	0.0347411
$165$	0	0
$166$	−6.78781	−0.526836
$167$	0.691075	0.0534770	0.0267385	−	0.999642i	$-0.491488\pi$
$167$	0.691075	0.0534770	0.0267385	+	0.999642i	$0.491488\pi$
$168$	10.5756	0.815922
$169$	−12.5908	−0.968524
$170$	0	0
$171$	−7.57417	−0.579211
$172$	−1.17027	−0.0892324
$173$	−14.5364	−1.10518	−0.552591	−	0.833453i	$-0.686361\pi$
$173$	−14.5364	−1.10518	−0.552591	+	0.833453i	$0.686361\pi$
$174$	3.29370	0.249695
$175$	0	0
$176$	−22.2031	−1.67362
$177$	−4.33066	−0.325513
$178$	4.73648	0.355014
$179$	24.5031	1.83145	0.915723	−	0.401810i	$-0.131619\pi$
$179$	24.5031	1.83145	0.915723	+	0.401810i	$0.131619\pi$
$180$	0	0
$181$	14.6604	1.08970	0.544851	−	0.838533i	$-0.316586\pi$
$181$	14.6604	1.08970	0.544851	+	0.838533i	$0.316586\pi$
$182$	−3.43424	−0.254563
$183$	−0.527802	−0.0390163
$184$	−2.76603	−0.203914
$185$	0	0
$186$	2.97911	0.218439
$187$	3.72820	0.272633
$188$	0.290957	0.0212202
$189$	−18.9886	−1.38122
$190$	0	0
$191$	1.86222	0.134745	0.0673726	−	0.997728i	$-0.478538\pi$
$191$	1.86222	0.134745	0.0673726	+	0.997728i	$0.478538\pi$
$192$	−7.76575	−0.560445
$193$	−12.8038	−0.921635	−0.460818	−	0.887495i	$-0.652444\pi$
$193$	−12.8038	−0.921635	−0.460818	+	0.887495i	$0.652444\pi$
$194$	−8.71223	−0.625502
$195$	0	0
$196$	1.04759	0.0748276
$197$	5.81726	0.414462	0.207231	−	0.978292i	$-0.433555\pi$
$197$	5.81726	0.414462	0.207231	+	0.978292i	$0.433555\pi$
$198$	14.0153	0.996024
$199$	5.66555	0.401620	0.200810	−	0.979630i	$-0.435643\pi$
$199$	5.66555	0.401620	0.200810	+	0.979630i	$0.435643\pi$
$200$	0	0
$201$	17.1726	1.21126
$202$	18.7345	1.31816
$203$	−7.59999	−0.533414
$204$	−0.128971	−0.00902977
$205$	0	0
$206$	6.61985	0.461227
$207$	1.89203	0.131505
$208$	2.75357	0.190926
$209$	21.1605	1.46371
$210$	0	0
$211$	−5.10215	−0.351247	−0.175623	−	0.984457i	$-0.556194\pi$
$211$	−5.10215	−0.351247	−0.175623	+	0.984457i	$0.556194\pi$
$212$	−1.77245	−0.121733
$213$	−2.58866	−0.177372
$214$	1.05623	0.0722025
$215$	0	0
$216$	14.0500	0.955983
$217$	−6.87409	−0.466643
$218$	−12.3644	−0.837424
$219$	−11.9902	−0.810219
$220$	0	0
$221$	−0.462362	−0.0311018
$222$	13.6987	0.919398
$223$	−16.4515	−1.10167	−0.550835	−	0.834614i	$-0.685691\pi$
$223$	−16.4515	−1.10167	−0.550835	+	0.834614i	$0.685691\pi$
$224$	−3.41916	−0.228452
$225$	0	0
$226$	7.02221	0.467111
$227$	18.3505	1.21797	0.608984	−	0.793183i	$-0.291577\pi$
$227$	18.3505	1.21797	0.608984	+	0.793183i	$0.291577\pi$
$228$	−0.732015	−0.0484789
$229$	19.4658	1.28634	0.643169	−	0.765724i	$-0.277619\pi$
$229$	19.4658	1.28634	0.643169	+	0.765724i	$0.277619\pi$
$230$	0	0
$231$	20.2100	1.32972
$232$	5.62336	0.369191
$233$	−4.26877	−0.279657	−0.139828	−	0.990176i	$-0.544655\pi$
$233$	−4.26877	−0.279657	−0.139828	+	0.990176i	$0.544655\pi$
$234$	−1.73814	−0.113626
$235$	0	0
$236$	0.669736	0.0435961
$237$	0.150529	0.00977788
$238$	3.88055	0.251538
$239$	−10.4338	−0.674907	−0.337453	−	0.941342i	$-0.609566\pi$
$239$	−10.4338	−0.674907	−0.337453	+	0.941342i	$0.609566\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−22.9661	−1.47632
$243$	−15.5599	−0.998166
$244$	0.0816245	0.00522547
$245$	0	0
$246$	4.23406	0.269954
$247$	−2.62428	−0.166979
$248$	5.08625	0.322977
$249$	−4.95392	−0.313942
$250$	0	0
$251$	6.54496	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$251$	6.54496	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$252$	1.11873	0.0704732
$253$	−5.28591	−0.332322
$254$	5.84842	0.366963
$255$	0	0
$256$	3.96003	0.247502
$257$	9.04873	0.564444	0.282222	−	0.959349i	$-0.408929\pi$
$257$	9.04873	0.564444	0.282222	+	0.959349i	$0.408929\pi$
$258$	−11.1373	−0.693376
$259$	−31.6089	−1.96408
$260$	0	0
$261$	−3.84652	−0.238093
$262$	12.6370	0.780717
$263$	26.8876	1.65796	0.828980	−	0.559279i	$-0.188922\pi$
$263$	26.8876	1.65796	0.828980	+	0.559279i	$0.188922\pi$
$264$	−14.9537	−0.920339
$265$	0	0
$266$	22.0253	1.35046
$267$	3.45681	0.211553
$268$	−2.65573	−0.162225
$269$	16.7557	1.02162	0.510808	−	0.859695i	$-0.329346\pi$
$269$	16.7557	1.02162	0.510808	+	0.859695i	$0.329346\pi$
$270$	0	0
$271$	9.01529	0.547640	0.273820	−	0.961781i	$-0.411713\pi$
$271$	9.01529	0.547640	0.273820	+	0.961781i	$0.411713\pi$
$272$	−3.11142	−0.188657
$273$	−2.50640	−0.151694
$274$	28.4169	1.71673
$275$	0	0
$276$	0.182858	0.0110067
$277$	−8.62442	−0.518191	−0.259096	−	0.965852i	$-0.583424\pi$
$277$	−8.62442	−0.518191	−0.259096	+	0.965852i	$0.583424\pi$
$278$	26.3902	1.58278
$279$	−3.47912	−0.208290
$280$	0	0
$281$	6.99214	0.417116	0.208558	−	0.978010i	$-0.433123\pi$
$281$	6.99214	0.417116	0.208558	+	0.978010i	$0.433123\pi$
$282$	2.76898	0.164891
$283$	26.3583	1.56684	0.783420	−	0.621492i	$-0.213473\pi$
$283$	26.3583	1.56684	0.783420	+	0.621492i	$0.213473\pi$
$284$	0.400336	0.0237556
$285$	0	0
$286$	4.85599	0.287141
$287$	−9.76981	−0.576694
$288$	−1.73051	−0.101971
$289$	−16.4776	−0.969268
$290$	0	0
$291$	−6.35841	−0.372737
$292$	1.85427	0.108513
$293$	−3.45835	−0.202039	−0.101019	−	0.994884i	$-0.532210\pi$
$293$	−3.45835	−0.202039	−0.101019	+	0.994884i	$0.532210\pi$
$294$	9.96968	0.581443
$295$	0	0
$296$	23.3879	1.35940
$297$	26.8498	1.55798
$298$	18.5437	1.07421
$299$	0.655547	0.0379112
$300$	0	0
$301$	25.6985	1.48124
$302$	−27.2127	−1.56592
$303$	13.6729	0.785490
$304$	−17.6598	−1.01286
$305$	0	0
$306$	1.96403	0.112276
$307$	−8.30367	−0.473915	−0.236958	−	0.971520i	$-0.576150\pi$
$307$	−8.30367	−0.473915	−0.236958	+	0.971520i	$0.576150\pi$
$308$	−3.12548	−0.178091
$309$	4.83134	0.274845
$310$	0	0
$311$	−30.4373	−1.72594	−0.862969	−	0.505256i	$-0.831398\pi$
$311$	−30.4373	−1.72594	−0.862969	+	0.505256i	$0.831398\pi$
$312$	1.85453	0.104992
$313$	−8.14895	−0.460606	−0.230303	−	0.973119i	$-0.573972\pi$
$313$	−8.14895	−0.460606	−0.230303	+	0.973119i	$0.573972\pi$
$314$	−28.2279	−1.59299
$315$	0	0
$316$	−0.0232792	−0.00130956
$317$	9.97145	0.560053	0.280026	−	0.959992i	$-0.409657\pi$
$317$	9.97145	0.560053	0.280026	+	0.959992i	$0.409657\pi$
$318$	−16.8681	−0.945916
$319$	10.7463	0.601678
$320$	0	0
$321$	0.770866	0.0430255
$322$	−5.50192	−0.306610
$323$	2.96533	0.164995
$324$	−0.00876588	−0.000486994 0
$325$	0	0
$326$	−4.51947	−0.250310
$327$	−9.02387	−0.499021
$328$	7.22885	0.399146
$329$	−6.38924	−0.352250
$330$	0	0
$331$	6.74088	0.370513	0.185256	−	0.982690i	$-0.440688\pi$
$331$	6.74088	0.370513	0.185256	+	0.982690i	$0.440688\pi$
$332$	0.766122	0.0420464
$333$	−15.9979	−0.876681
$334$	−1.01711	−0.0556536
$335$	0	0
$336$	−16.8666	−0.920147
$337$	24.3161	1.32458	0.662291	−	0.749246i	$-0.269584\pi$
$337$	24.3161	1.32458	0.662291	+	0.749246i	$0.269584\pi$
$338$	18.5308	1.00794
$339$	5.12500	0.278352
$340$	0	0
$341$	9.71990	0.526362
$342$	11.1475	0.602786
$343$	2.53018	0.136617
$344$	−19.0147	−1.02521
$345$	0	0
$346$	21.3943	1.15016
$347$	−6.98090	−0.374754	−0.187377	−	0.982288i	$-0.559999\pi$
$347$	−6.98090	−0.374754	−0.187377	+	0.982288i	$0.559999\pi$
$348$	−0.371751	−0.0199280
$349$	5.68571	0.304349	0.152175	−	0.988354i	$-0.451372\pi$
$349$	5.68571	0.304349	0.152175	+	0.988354i	$0.451372\pi$
$350$	0	0
$351$	−3.32985	−0.177734
$352$	4.83466	0.257688
$353$	−32.8357	−1.74767	−0.873834	−	0.486225i	$-0.838374\pi$
$353$	−32.8357	−1.74767	−0.873834	+	0.486225i	$0.838374\pi$
$354$	6.37375	0.338761
$355$	0	0
$356$	−0.534594	−0.0283334
$357$	2.83212	0.149892
$358$	−36.0630	−1.90599
$359$	−23.6423	−1.24779	−0.623897	−	0.781507i	$-0.714451\pi$
$359$	−23.6423	−1.24779	−0.623897	+	0.781507i	$0.714451\pi$
$360$	0	0
$361$	−2.16936	−0.114177
$362$	−21.5768	−1.13405
$363$	−16.7613	−0.879739
$364$	0.387614	0.0203165
$365$	0	0
$366$	0.776805	0.0406042
$367$	5.24220	0.273641	0.136820	−	0.990596i	$-0.456312\pi$
$367$	5.24220	0.273641	0.136820	+	0.990596i	$0.456312\pi$
$368$	4.41143	0.229962
$369$	−4.94471	−0.257411
$370$	0	0
$371$	38.9220	2.02073
$372$	−0.336244	−0.0174335
$373$	30.3804	1.57304	0.786519	−	0.617566i	$-0.211881\pi$
$373$	30.3804	1.57304	0.786519	+	0.617566i	$0.211881\pi$
$374$	−5.48706	−0.283729
$375$	0	0
$376$	4.72751	0.243803
$377$	−1.33273	−0.0686392
$378$	27.9470	1.43744
$379$	−27.9316	−1.43475	−0.717376	−	0.696687i	$-0.754657\pi$
$379$	−27.9316	−1.43475	−0.717376	+	0.696687i	$0.754657\pi$
$380$	0	0
$381$	4.26833	0.218673
$382$	−2.74076	−0.140229
$383$	25.0588	1.28045	0.640223	−	0.768189i	$-0.278842\pi$
$383$	25.0588	1.28045	0.640223	+	0.768189i	$0.278842\pi$
$384$	13.4430	0.686013
$385$	0	0
$386$	18.8442	0.959147
$387$	13.0066	0.661160
$388$	0.983327	0.0499209
$389$	35.3045	1.79001	0.895004	−	0.446058i	$-0.147172\pi$
$389$	35.3045	1.79001	0.895004	+	0.446058i	$0.147172\pi$
$390$	0	0
$391$	−0.740739	−0.0374608
$392$	17.0213	0.859706
$393$	9.22282	0.465230
$394$	−8.56168	−0.431331
$395$	0	0
$396$	−1.58187	−0.0794920
$397$	−5.77731	−0.289955	−0.144977	−	0.989435i	$-0.546311\pi$
$397$	−5.77731	−0.289955	−0.144977	+	0.989435i	$0.546311\pi$
$398$	−8.33841	−0.417967
$399$	16.0746	0.804737
$400$	0	0
$401$	−14.3217	−0.715193	−0.357596	−	0.933876i	$-0.616404\pi$
$401$	−14.3217	−0.715193	−0.357596	+	0.933876i	$0.616404\pi$
$402$	−25.2741	−1.26056
$403$	−1.20544	−0.0600472
$404$	−2.11452	−0.105201
$405$	0	0
$406$	11.1855	0.555125
$407$	44.6946	2.21543
$408$	−2.09554	−0.103745
$409$	−14.9869	−0.741055	−0.370527	−	0.928822i	$-0.620823\pi$
$409$	−14.9869	−0.741055	−0.370527	+	0.928822i	$0.620823\pi$
$410$	0	0
$411$	20.7394	1.02300
$412$	−0.747165	−0.0368102
$413$	−14.7070	−0.723684
$414$	−2.78464	−0.136858
$415$	0	0
$416$	−0.599583	−0.0293970
$417$	19.2602	0.943178
$418$	−31.1435	−1.52328
$419$	17.3996	0.850026	0.425013	−	0.905187i	$-0.360269\pi$
$419$	17.3996	0.850026	0.425013	+	0.905187i	$0.360269\pi$
$420$	0	0
$421$	15.5810	0.759372	0.379686	−	0.925115i	$-0.376032\pi$
$421$	15.5810	0.759372	0.379686	+	0.925115i	$0.376032\pi$
$422$	7.50921	0.365543
$423$	−3.23373	−0.157230
$424$	−28.7990	−1.39860
$425$	0	0
$426$	3.80992	0.184591
$427$	−1.79242	−0.0867415
$428$	−0.119214	−0.00576244
$429$	3.54403	0.171107
$430$	0	0
$431$	−17.8974	−0.862088	−0.431044	−	0.902331i	$-0.641855\pi$
$431$	−17.8974	−0.862088	−0.431044	+	0.902331i	$0.641855\pi$
$432$	−22.4079	−1.07810
$433$	−3.99742	−0.192104	−0.0960519	−	0.995376i	$-0.530621\pi$
$433$	−3.99742	−0.192104	−0.0960519	+	0.995376i	$0.530621\pi$
$434$	10.1171	0.485636
$435$	0	0
$436$	1.39554	0.0668342
$437$	−4.20430	−0.201119
$438$	17.6468	0.843196
$439$	−8.96731	−0.427986	−0.213993	−	0.976835i	$-0.568647\pi$
$439$	−8.96731	−0.427986	−0.213993	+	0.976835i	$0.568647\pi$
$440$	0	0
$441$	−11.6430	−0.554429
$442$	0.680492	0.0323677
$443$	11.1014	0.527444	0.263722	−	0.964599i	$-0.415050\pi$
$443$	11.1014	0.527444	0.263722	+	0.964599i	$0.415050\pi$
$444$	−1.54614	−0.0733765
$445$	0	0
$446$	24.2128	1.14651
$447$	13.5337	0.640121
$448$	−26.3726	−1.24599
$449$	−26.1449	−1.23386	−0.616928	−	0.787020i	$-0.711623\pi$
$449$	−26.1449	−1.23386	−0.616928	+	0.787020i	$0.711623\pi$
$450$	0	0
$451$	13.8144	0.650496
$452$	−0.792579	−0.0372798
$453$	−19.8605	−0.933130
$454$	−27.0078	−1.26754
$455$	0	0
$456$	−11.8939	−0.556982
$457$	19.2894	0.902320	0.451160	−	0.892443i	$-0.351010\pi$
$457$	19.2894	0.902320	0.451160	+	0.892443i	$0.351010\pi$
$458$	−28.6493	−1.33869
$459$	3.76258	0.175622
$460$	0	0
$461$	−34.7496	−1.61845	−0.809226	−	0.587498i	$-0.800113\pi$
$461$	−34.7496	−1.61845	−0.809226	+	0.587498i	$0.800113\pi$
$462$	−29.7446	−1.38384
$463$	32.2638	1.49943	0.749714	−	0.661762i	$-0.230191\pi$
$463$	32.2638	1.49943	0.749714	+	0.661762i	$0.230191\pi$
$464$	−8.96849	−0.416352
$465$	0	0
$466$	6.28266	0.291039
$467$	−5.13683	−0.237704	−0.118852	−	0.992912i	$-0.537921\pi$
$467$	−5.13683	−0.237704	−0.118852	+	0.992912i	$0.537921\pi$
$468$	0.196180	0.00906842
$469$	58.3183	2.69289
$470$	0	0
$471$	−20.6015	−0.949265
$472$	10.8820	0.500883
$473$	−36.3374	−1.67080
$474$	−0.221544	−0.0101758
$475$	0	0
$476$	−0.437987	−0.0200751
$477$	19.6993	0.901967
$478$	15.3562	0.702376
$479$	42.8634	1.95848	0.979240	−	0.202702i	$-0.0649723\pi$
$479$	42.8634	1.95848	0.979240	+	0.202702i	$0.0649723\pi$
$480$	0	0
$481$	−5.54293	−0.252736
$482$	1.47177	0.0670374
$483$	−4.01544	−0.182709
$484$	2.59213	0.117824
$485$	0	0
$486$	22.9006	1.03879
$487$	−16.1377	−0.731270	−0.365635	−	0.930758i	$-0.619148\pi$
$487$	−16.1377	−0.731270	−0.365635	+	0.930758i	$0.619148\pi$
$488$	1.32624	0.0600363
$489$	−3.29843	−0.149160
$490$	0	0
$491$	−6.34829	−0.286494	−0.143247	−	0.989687i	$-0.545754\pi$
$491$	−6.34829	−0.286494	−0.143247	+	0.989687i	$0.545754\pi$
$492$	−0.477888	−0.0215449
$493$	1.50593	0.0678237
$494$	3.86235	0.173775
$495$	0	0
$496$	−8.11188	−0.364234
$497$	−8.79114	−0.394337
$498$	7.29104	0.326719
$499$	−15.1873	−0.679876	−0.339938	−	0.940448i	$-0.610406\pi$
$499$	−15.1873	−0.679876	−0.339938	+	0.940448i	$0.610406\pi$
$500$	0	0
$501$	−0.742311	−0.0331640
$502$	−9.63269	−0.429928
$503$	26.1449	1.16574	0.582872	−	0.812564i	$-0.301929\pi$
$503$	26.1449	1.16574	0.582872	+	0.812564i	$0.301929\pi$
$504$	18.1772	0.809678
$505$	0	0
$506$	7.77966	0.345848
$507$	13.5243	0.600634
$508$	−0.660097	−0.0292871
$509$	14.0110	0.621025	0.310512	−	0.950569i	$-0.399499\pi$
$509$	14.0110	0.621025	0.310512	+	0.950569i	$0.399499\pi$
$510$	0	0
$511$	−40.7187	−1.80129
$512$	19.2021	0.848622
$513$	21.3557	0.942878
$514$	−13.3177	−0.587417
$515$	0	0
$516$	1.25703	0.0553379
$517$	9.03433	0.397329
$518$	46.5211	2.04402
$519$	15.6141	0.685383
$520$	0	0
$521$	44.4005	1.94522	0.972610	−	0.232445i	$-0.0746725\pi$
$521$	44.4005	1.94522	0.972610	+	0.232445i	$0.0746725\pi$
$522$	5.66120	0.247784
$523$	38.9673	1.70392	0.851961	−	0.523604i	$-0.175413\pi$
$523$	38.9673	1.70392	0.851961	+	0.523604i	$0.175413\pi$
$524$	−1.42631	−0.0623085
$525$	0	0
$526$	−39.5724	−1.72544
$527$	1.36209	0.0593337
$528$	23.8492	1.03790
$529$	−21.9498	−0.954338
$530$	0	0
$531$	−7.44353	−0.323022
$532$	−2.48594	−0.107779
$533$	−1.71323	−0.0742084
$534$	−5.08763	−0.220163
$535$	0	0
$536$	−43.1507	−1.86383
$537$	−26.3197	−1.13578
$538$	−24.6606	−1.06320
$539$	32.5280	1.40108
$540$	0	0
$541$	12.8548	0.552672	0.276336	−	0.961061i	$-0.410880\pi$
$541$	12.8548	0.552672	0.276336	+	0.961061i	$0.410880\pi$
$542$	−13.2685	−0.569929
$543$	−15.7473	−0.675783
$544$	0.677503	0.0290477
$545$	0	0
$546$	3.68885	0.157868
$547$	11.4502	0.489577	0.244788	−	0.969577i	$-0.421281\pi$
$547$	11.4502	0.489577	0.244788	+	0.969577i	$0.421281\pi$
$548$	−3.20735	−0.137011
$549$	−0.907185	−0.0387177
$550$	0	0
$551$	8.54738	0.364131
$552$	2.97109	0.126458
$553$	0.511197	0.0217383
$554$	12.6932	0.539282
$555$	0	0
$556$	−2.97859	−0.126320
$557$	17.1929	0.728486	0.364243	−	0.931304i	$-0.381328\pi$
$557$	17.1929	0.728486	0.364243	+	0.931304i	$0.381328\pi$
$558$	5.12048	0.216767
$559$	4.50648	0.190604
$560$	0	0
$561$	−4.00460	−0.169074
$562$	−10.2908	−0.434093
$563$	29.2212	1.23153	0.615764	−	0.787931i	$-0.288847\pi$
$563$	29.2212	1.23153	0.615764	+	0.787931i	$0.288847\pi$
$564$	−0.312528	−0.0131598
$565$	0	0
$566$	−38.7935	−1.63061
$567$	0.192494	0.00808397
$568$	6.50471	0.272932
$569$	21.2086	0.889112	0.444556	−	0.895751i	$-0.353362\pi$
$569$	21.2086	0.889112	0.444556	+	0.895751i	$0.353362\pi$
$570$	0	0
$571$	23.5590	0.985913	0.492957	−	0.870054i	$-0.335916\pi$
$571$	23.5590	0.985913	0.492957	+	0.870054i	$0.335916\pi$
$572$	−0.548083	−0.0229165
$573$	−2.00028	−0.0835628
$574$	14.3789	0.600165
$575$	0	0
$576$	−13.3477	−0.556156
$577$	44.0330	1.83312	0.916560	−	0.399898i	$-0.130954\pi$
$577$	44.0330	1.83312	0.916560	+	0.399898i	$0.130954\pi$
$578$	24.2512	1.00872
$579$	13.7530	0.571556
$580$	0	0
$581$	−16.8236	−0.697960
$582$	9.35814	0.387907
$583$	−55.0353	−2.27933
$584$	30.1285	1.24673
$585$	0	0
$586$	5.08990	0.210262
$587$	6.14338	0.253564	0.126782	−	0.991931i	$-0.459535\pi$
$587$	6.14338	0.253564	0.126782	+	0.991931i	$0.459535\pi$
$588$	−1.12525	−0.0464046
$589$	7.73099	0.318550
$590$	0	0
$591$	−6.24854	−0.257031
$592$	−37.3006	−1.53304
$593$	40.5705	1.66603	0.833015	−	0.553251i	$-0.186613\pi$
$593$	40.5705	1.66603	0.833015	+	0.553251i	$0.186613\pi$
$594$	−39.5168	−1.62139
$595$	0	0
$596$	−2.09298	−0.0857318
$597$	−6.08559	−0.249067
$598$	−0.964816	−0.0394543
$599$	−13.1336	−0.536625	−0.268313	−	0.963332i	$-0.586466\pi$
$599$	−13.1336	−0.536625	−0.268313	+	0.963332i	$0.586466\pi$
$600$	0	0
$601$	−43.0533	−1.75618	−0.878090	−	0.478496i	$-0.841182\pi$
$601$	−43.0533	−1.75618	−0.878090	+	0.478496i	$0.841182\pi$
$602$	−37.8223	−1.54152
$603$	29.5161	1.20199
$604$	3.07143	0.124975
$605$	0	0
$606$	−20.1235	−0.817460
$607$	8.55193	0.347112	0.173556	−	0.984824i	$-0.444474\pi$
$607$	8.55193	0.347112	0.173556	+	0.984824i	$0.444474\pi$
$608$	3.84538	0.155951
$609$	8.16344	0.330799
$610$	0	0
$611$	−1.12042	−0.0453272
$612$	−0.221675	−0.00896068
$613$	−16.6073	−0.670764	−0.335382	−	0.942082i	$-0.608865\pi$
$613$	−16.6073	−0.670764	−0.335382	+	0.942082i	$0.608865\pi$
$614$	12.2211	0.493204
$615$	0	0
$616$	−50.7831	−2.04611
$617$	−2.31997	−0.0933987	−0.0466993	−	0.998909i	$-0.514870\pi$
$617$	−2.31997	−0.0933987	−0.0466993	+	0.998909i	$0.514870\pi$
$618$	−7.11063	−0.286032
$619$	33.0310	1.32763	0.663814	−	0.747898i	$-0.268937\pi$
$619$	33.0310	1.32763	0.663814	+	0.747898i	$0.268937\pi$
$620$	0	0
$621$	−5.33467	−0.214073
$622$	44.7967	1.79619
$623$	11.7394	0.470328
$624$	−2.95772	−0.118404
$625$	0	0
$626$	11.9934	0.479353
$627$	−22.7294	−0.907723
$628$	3.18601	0.127136
$629$	6.26327	0.249733
$630$	0	0
$631$	0.120776	0.00480800	0.00240400	−	0.999997i	$-0.499235\pi$
$631$	0.120776	0.00480800	0.00240400	+	0.999997i	$0.499235\pi$
$632$	−0.378244	−0.0150457
$633$	5.48042	0.217827
$634$	−14.6757	−0.582847
$635$	0	0
$636$	1.90386	0.0754929
$637$	−4.03404	−0.159835
$638$	−15.8161	−0.626167
$639$	−4.44939	−0.176015
$640$	0	0
$641$	43.9369	1.73540	0.867702	−	0.497084i	$-0.165596\pi$
$641$	43.9369	1.73540	0.867702	+	0.497084i	$0.165596\pi$
$642$	−1.13454	−0.0447767
$643$	−0.294166	−0.0116008	−0.00580040	−	0.999983i	$-0.501846\pi$
$643$	−0.294166	−0.0116008	−0.00580040	+	0.999983i	$0.501846\pi$
$644$	0.620988	0.0244703
$645$	0	0
$646$	−4.36429	−0.171710
$647$	40.0093	1.57293	0.786464	−	0.617637i	$-0.211910\pi$
$647$	40.0093	1.57293	0.786464	+	0.617637i	$0.211910\pi$
$648$	−0.142429	−0.00559515
$649$	20.7956	0.816297
$650$	0	0
$651$	7.38372	0.289391
$652$	0.510101	0.0199771
$653$	−44.2141	−1.73023	−0.865116	−	0.501571i	$-0.832755\pi$
$653$	−44.2141	−1.73023	−0.865116	+	0.501571i	$0.832755\pi$
$654$	13.2811	0.519332
$655$	0	0
$656$	−11.5290	−0.450133
$657$	−20.6086	−0.804020
$658$	9.40351	0.366587
$659$	32.9525	1.28365	0.641824	−	0.766852i	$-0.278178\pi$
$659$	32.9525	1.28365	0.641824	+	0.766852i	$0.278178\pi$
$660$	0	0
$661$	−16.1861	−0.629567	−0.314784	−	0.949163i	$-0.601932\pi$
$661$	−16.1861	−0.629567	−0.314784	+	0.949163i	$0.601932\pi$
$662$	−9.92105	−0.385593
$663$	0.496641	0.0192879
$664$	12.4481	0.483078
$665$	0	0
$666$	23.5453	0.912362
$667$	−2.13514	−0.0826729
$668$	0.114798	0.00444167
$669$	17.6711	0.683206
$670$	0	0
$671$	2.53447	0.0978422
$672$	3.67265	0.141676
$673$	24.7488	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$673$	24.7488	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$674$	−35.7878	−1.37849
$675$	0	0
$676$	−2.09153	−0.0804433
$677$	23.4140	0.899873	0.449937	−	0.893060i	$-0.351447\pi$
$677$	23.4140	0.899873	0.449937	+	0.893060i	$0.351447\pi$
$678$	−7.54283	−0.289681
$679$	−21.5933	−0.828674
$680$	0	0
$681$	−19.7110	−0.755328
$682$	−14.3055	−0.547785
$683$	20.4270	0.781617	0.390809	−	0.920472i	$-0.372195\pi$
$683$	20.4270	0.781617	0.390809	+	0.920472i	$0.372195\pi$
$684$	−1.25819	−0.0481079
$685$	0	0
$686$	−3.72386	−0.142178
$687$	−20.9090	−0.797728
$688$	30.3259	1.15616
$689$	6.82535	0.260025
$690$	0	0
$691$	25.1700	0.957512	0.478756	−	0.877948i	$-0.341088\pi$
$691$	25.1700	0.957512	0.478756	+	0.877948i	$0.341088\pi$
$692$	−2.41472	−0.0917938
$693$	34.7369	1.31955
$694$	10.2743	0.390007
$695$	0	0
$696$	−6.04026	−0.228956
$697$	1.93588	0.0733267
$698$	−8.36807	−0.316736
$699$	4.58525	0.173430
$700$	0	0
$701$	46.4292	1.75361	0.876803	−	0.480850i	$-0.159672\pi$
$701$	46.4292	1.75361	0.876803	+	0.480850i	$0.159672\pi$
$702$	4.90078	0.184968
$703$	35.5491	1.34076
$704$	37.2906	1.40544
$705$	0	0
$706$	48.3267	1.81880
$707$	46.4335	1.74631
$708$	−0.719389	−0.0270363
$709$	23.5588	0.884768	0.442384	−	0.896826i	$-0.354133\pi$
$709$	23.5588	0.884768	0.442384	+	0.896826i	$0.354133\pi$
$710$	0	0
$711$	0.258728	0.00970306
$712$	−8.68615	−0.325527
$713$	−1.93120	−0.0723242
$714$	−4.16824	−0.155993
$715$	0	0
$716$	4.07034	0.152116
$717$	11.2074	0.418546
$718$	34.7961	1.29858
$719$	−30.8865	−1.15187	−0.575936	−	0.817495i	$-0.695362\pi$
$719$	−30.8865	−1.15187	−0.575936	+	0.817495i	$0.695362\pi$
$720$	0	0
$721$	16.4073	0.611040
$722$	3.19281	0.118824
$723$	1.07414	0.0399476
$724$	2.43532	0.0905080
$725$	0	0
$726$	24.6688	0.915545
$727$	7.06911	0.262179	0.131089	−	0.991371i	$-0.458153\pi$
$727$	7.06911	0.262179	0.131089	+	0.991371i	$0.458153\pi$
$728$	6.29801	0.233420
$729$	16.8718	0.624880
$730$	0	0
$731$	−5.09213	−0.188339
$732$	−0.0876760	−0.00324060
$733$	33.3816	1.23298	0.616488	−	0.787364i	$-0.288555\pi$
$733$	33.3816	1.23298	0.616488	+	0.787364i	$0.288555\pi$
$734$	−7.71533	−0.284778
$735$	0	0
$736$	−0.960578	−0.0354074
$737$	−82.4615	−3.03751
$738$	7.27749	0.267888
$739$	−5.57262	−0.204992	−0.102496	−	0.994733i	$-0.532683\pi$
$739$	−5.57262	−0.204992	−0.102496	+	0.994733i	$0.532683\pi$
$740$	0	0
$741$	2.81884	0.103553
$742$	−57.2843	−2.10297
$743$	−25.9781	−0.953045	−0.476523	−	0.879162i	$-0.658103\pi$
$743$	−25.9781	−0.953045	−0.476523	+	0.879162i	$0.658103\pi$
$744$	−5.46334	−0.200296
$745$	0	0
$746$	−44.7130	−1.63706
$747$	−8.51478	−0.311540
$748$	0.619310	0.0226442
$749$	2.61787	0.0956550
$750$	0	0
$751$	−43.9694	−1.60446	−0.802232	−	0.597012i	$-0.796354\pi$
$751$	−43.9694	−1.60446	−0.802232	+	0.597012i	$0.796354\pi$
$752$	−7.53973	−0.274946
$753$	−7.03019	−0.256194
$754$	1.96148	0.0714329
$755$	0	0
$756$	−3.15430	−0.114721
$757$	−37.9194	−1.37821	−0.689103	−	0.724664i	$-0.741995\pi$
$757$	−37.9194	−1.37821	−0.689103	+	0.724664i	$0.741995\pi$
$758$	41.1090	1.49315
$759$	5.67780	0.206091
$760$	0	0
$761$	47.1567	1.70943	0.854714	−	0.519099i	$-0.173732\pi$
$761$	47.1567	1.70943	0.854714	+	0.519099i	$0.173732\pi$
$762$	−6.28202	−0.227573
$763$	−30.6452	−1.10943
$764$	0.309342	0.0111916
$765$	0	0
$766$	−36.8809	−1.33256
$767$	−2.57902	−0.0931230
$768$	−4.25362	−0.153489
$769$	−33.0196	−1.19072	−0.595358	−	0.803460i	$-0.702990\pi$
$769$	−33.0196	−1.19072	−0.595358	+	0.803460i	$0.702990\pi$
$770$	0	0
$771$	−9.71959	−0.350042
$772$	−2.12690	−0.0765489
$773$	6.41462	0.230718	0.115359	−	0.993324i	$-0.463198\pi$
$773$	6.41462	0.230718	0.115359	+	0.993324i	$0.463198\pi$
$774$	−19.1427	−0.688070
$775$	0	0
$776$	15.9772	0.573549
$777$	33.9523	1.21803
$778$	−51.9602	−1.86286
$779$	10.9877	0.393675
$780$	0	0
$781$	12.4306	0.444802
$782$	1.09020	0.0389855
$783$	10.8454	0.387584
$784$	−27.1467	−0.969524
$785$	0	0
$786$	−13.5739	−0.484165
$787$	−39.4086	−1.40477	−0.702383	−	0.711799i	$-0.747881\pi$
$787$	−39.4086	−1.40477	−0.702383	+	0.711799i	$0.747881\pi$
$788$	0.966335	0.0344243
$789$	−28.8810	−1.02819
$790$	0	0
$791$	17.4046	0.618835
$792$	−25.7024	−0.913297
$793$	−0.314319	−0.0111618
$794$	8.50288	0.301756
$795$	0	0
$796$	0.941135	0.0333576
$797$	49.4557	1.75181	0.875905	−	0.482484i	$-0.160265\pi$
$797$	49.4557	1.75181	0.875905	+	0.482484i	$0.160265\pi$
$798$	−23.6582	−0.837491
$799$	1.26602	0.0447887
$800$	0	0
$801$	5.94155	0.209934
$802$	21.0783	0.744302
$803$	57.5759	2.03181
$804$	2.85262	0.100604
$805$	0	0
$806$	1.77413	0.0624911
$807$	−17.9980	−0.633559
$808$	−34.3569	−1.20867
$809$	−9.70167	−0.341092	−0.170546	−	0.985350i	$-0.554553\pi$
$809$	−9.70167	−0.341092	−0.170546	+	0.985350i	$0.554553\pi$
$810$	0	0
$811$	−35.0724	−1.23156	−0.615779	−	0.787919i	$-0.711158\pi$
$811$	−35.0724	−1.23156	−0.615779	+	0.787919i	$0.711158\pi$
$812$	−1.26247	−0.0443041
$813$	−9.68367	−0.339621
$814$	−65.7804	−2.30560
$815$	0	0
$816$	3.34210	0.116997
$817$	−28.9020	−1.01115
$818$	22.0573	0.771216
$819$	−4.30799	−0.150534
$820$	0	0
$821$	28.5076	0.994921	0.497461	−	0.867487i	$-0.334266\pi$
$821$	28.5076	0.994921	0.497461	+	0.867487i	$0.334266\pi$
$822$	−30.5237	−1.06464
$823$	49.8334	1.73708	0.868541	−	0.495617i	$-0.165058\pi$
$823$	49.8334	1.73708	0.868541	+	0.495617i	$0.165058\pi$
$824$	−12.1400	−0.422918
$825$	0	0
$826$	21.6454	0.753139
$827$	47.2249	1.64217	0.821085	−	0.570806i	$-0.193369\pi$
$827$	47.2249	1.64217	0.821085	+	0.570806i	$0.193369\pi$
$828$	0.314295	0.0109225
$829$	26.8324	0.931926	0.465963	−	0.884804i	$-0.345708\pi$
$829$	26.8324	0.931926	0.465963	+	0.884804i	$0.345708\pi$
$830$	0	0
$831$	9.26382	0.321358
$832$	−4.62470	−0.160333
$833$	4.55829	0.157935
$834$	−28.3467	−0.981566
$835$	0	0
$836$	3.51509	0.121572
$837$	9.80955	0.339068
$838$	−25.6083	−0.884623
$839$	16.5300	0.570678	0.285339	−	0.958427i	$-0.407894\pi$
$839$	16.5300	0.570678	0.285339	+	0.958427i	$0.407894\pi$
$840$	0	0
$841$	−24.6592	−0.850319
$842$	−22.9317	−0.790279
$843$	−7.51052	−0.258676
$844$	−0.847546	−0.0291737
$845$	0	0
$846$	4.75932	0.163629
$847$	−56.9216	−1.95585
$848$	45.9305	1.57726
$849$	−28.3125	−0.971683
$850$	0	0
$851$	−8.88019	−0.304409
$852$	−0.430017	−0.0147321
$853$	−45.5823	−1.56071	−0.780354	−	0.625338i	$-0.784961\pi$
$853$	−45.5823	−1.56071	−0.780354	+	0.625338i	$0.784961\pi$
$854$	2.63804	0.0902719
$855$	0	0
$856$	−1.93701	−0.0662056
$857$	−53.6009	−1.83097	−0.915486	−	0.402349i	$-0.868194\pi$
$857$	−53.6009	−1.83097	−0.915486	+	0.402349i	$0.868194\pi$
$858$	−5.21601	−0.178071
$859$	−53.2011	−1.81520	−0.907598	−	0.419840i	$-0.862086\pi$
$859$	−53.2011	−1.81520	−0.907598	+	0.419840i	$0.862086\pi$
$860$	0	0
$861$	10.4941	0.357639
$862$	26.3409	0.897175
$863$	43.7556	1.48946	0.744730	−	0.667366i	$-0.232578\pi$
$863$	43.7556	1.48946	0.744730	+	0.667366i	$0.232578\pi$
$864$	4.87925	0.165996
$865$	0	0
$866$	5.88330	0.199923
$867$	17.6992	0.601095
$868$	−1.14189	−0.0387583
$869$	−0.722829	−0.0245203
$870$	0	0
$871$	10.2267	0.346518
$872$	22.6749	0.767869
$873$	−10.9288	−0.369885
$874$	6.18777	0.209305
$875$	0	0
$876$	−1.99175	−0.0672949
$877$	−17.5582	−0.592897	−0.296448	−	0.955049i	$-0.595802\pi$
$877$	−17.5582	−0.592897	−0.296448	+	0.955049i	$0.595802\pi$
$878$	13.1978	0.445406
$879$	3.71475	0.125295
$880$	0	0
$881$	−31.6488	−1.06628	−0.533138	−	0.846028i	$-0.678987\pi$
$881$	−31.6488	−1.06628	−0.533138	+	0.846028i	$0.678987\pi$
$882$	17.1359	0.576994
$883$	−26.1244	−0.879157	−0.439579	−	0.898204i	$-0.644872\pi$
$883$	−26.1244	−0.879157	−0.439579	+	0.898204i	$0.644872\pi$
$884$	−0.0768054	−0.00258325
$885$	0	0
$886$	−16.3388	−0.548912
$887$	24.4007	0.819294	0.409647	−	0.912244i	$-0.365652\pi$
$887$	24.4007	0.819294	0.409647	+	0.912244i	$0.365652\pi$
$888$	−25.1219	−0.843035
$889$	14.4953	0.486158
$890$	0	0
$891$	−0.272184	−0.00911851
$892$	−2.73284	−0.0915022
$893$	7.18571	0.240461
$894$	−19.9185	−0.666175
$895$	0	0
$896$	45.6528	1.52515
$897$	−0.704148	−0.0235108
$898$	38.4794	1.28407
$899$	3.92616	0.130945
$900$	0	0
$901$	−7.71235	−0.256936
$902$	−20.3317	−0.676971
$903$	−27.6037	−0.918594
$904$	−12.8779	−0.428314
$905$	0	0
$906$	29.2302	0.971109
$907$	42.3274	1.40546	0.702728	−	0.711458i	$-0.251965\pi$
$907$	42.3274	1.40546	0.702728	+	0.711458i	$0.251965\pi$
$908$	3.04830	0.101162
$909$	23.5010	0.779480
$910$	0	0
$911$	37.6349	1.24690	0.623450	−	0.781863i	$-0.285730\pi$
$911$	37.6349	1.24690	0.623450	+	0.781863i	$0.285730\pi$
$912$	18.9691	0.628130
$913$	23.7884	0.787281
$914$	−28.3896	−0.939045
$915$	0	0
$916$	3.23357	0.106840
$917$	31.3209	1.03431
$918$	−5.53767	−0.182770
$919$	−46.8857	−1.54662	−0.773308	−	0.634031i	$-0.781399\pi$
$919$	−46.8857	−1.54662	−0.773308	+	0.634031i	$0.781399\pi$
$920$	0	0
$921$	8.91929	0.293901
$922$	51.1435	1.68432
$923$	−1.54161	−0.0507428
$924$	3.35719	0.110444
$925$	0	0
$926$	−47.4850	−1.56045
$927$	8.30409	0.272742
$928$	1.95287	0.0641059
$929$	−27.0258	−0.886687	−0.443343	−	0.896352i	$-0.646208\pi$
$929$	−27.0258	−0.886687	−0.443343	+	0.896352i	$0.646208\pi$
$930$	0	0
$931$	25.8720	0.847921
$932$	−0.709108	−0.0232276
$933$	32.6938	1.07035
$934$	7.56025	0.247379
$935$	0	0
$936$	3.18756	0.104189
$937$	−1.25183	−0.0408954	−0.0204477	−	0.999791i	$-0.506509\pi$
$937$	−1.25183	−0.0408954	−0.0204477	+	0.999791i	$0.506509\pi$
$938$	−85.8313	−2.80249
$939$	8.75311	0.285647
$940$	0	0
$941$	−5.72639	−0.186675	−0.0933374	−	0.995635i	$-0.529754\pi$
$941$	−5.72639	−0.186675	−0.0933374	+	0.995635i	$0.529754\pi$
$942$	30.3207	0.987901
$943$	−2.74473	−0.0893807
$944$	−17.3552	−0.564865
$945$	0	0
$946$	53.4804	1.73880
$947$	0.468304	0.0152178	0.00760892	−	0.999971i	$-0.497578\pi$
$947$	0.468304	0.0152178	0.00760892	+	0.999971i	$0.497578\pi$
$948$	0.0250051	0.000812128 0
$949$	−7.14043	−0.231788
$950$	0	0
$951$	−10.7107	−0.347319
$952$	−7.11647	−0.230646
$953$	0.239737	0.00776586	0.00388293	−	0.999992i	$-0.498764\pi$
$953$	0.239737	0.00776586	0.00388293	+	0.999992i	$0.498764\pi$
$954$	−28.9928	−0.938678
$955$	0	0
$956$	−1.73321	−0.0560562
$957$	−11.5430	−0.373133
$958$	−63.0853	−2.03819
$959$	70.4314	2.27435
$960$	0	0
$961$	−27.4488	−0.885447
$962$	8.15793	0.263022
$963$	1.32496	0.0426963
$964$	−0.166115	−0.00535021
$965$	0	0
$966$	5.90982	0.190145
$967$	17.0034	0.546793	0.273396	−	0.961901i	$-0.411853\pi$
$967$	17.0034	0.546793	0.273396	+	0.961901i	$0.411853\pi$
$968$	42.1172	1.35370
$969$	−3.18517	−0.102322
$970$	0	0
$971$	−2.90090	−0.0930943	−0.0465471	−	0.998916i	$-0.514822\pi$
$971$	−2.90090	−0.0930943	−0.0465471	+	0.998916i	$0.514822\pi$
$972$	−2.58473	−0.0829053
$973$	65.4081	2.09689
$974$	23.7510	0.761033
$975$	0	0
$976$	−2.11518	−0.0677053
$977$	−23.7482	−0.759773	−0.379886	−	0.925033i	$-0.624037\pi$
$977$	−23.7482	−0.759773	−0.379886	+	0.925033i	$0.624037\pi$
$978$	4.85454	0.155231
$979$	−16.5994	−0.530518
$980$	0	0
$981$	−15.5102	−0.495203
$982$	9.34325	0.298155
$983$	10.3096	0.328827	0.164413	−	0.986392i	$-0.447427\pi$
$983$	10.3096	0.328827	0.164413	+	0.986392i	$0.447427\pi$
$984$	−7.76478	−0.247532
$985$	0	0
$986$	−2.21639	−0.0705842
$987$	6.86293	0.218449
$988$	−0.435933	−0.0138689
$989$	7.21973	0.229574
$990$	0	0
$991$	39.5286	1.25567	0.627834	−	0.778347i	$-0.283941\pi$
$991$	39.5286	1.25567	0.627834	+	0.778347i	$0.283941\pi$
$992$	1.76634	0.0560814
$993$	−7.24064	−0.229775
$994$	12.9386	0.410386
$995$	0	0
$996$	−0.822922	−0.0260753
$997$	38.5135	1.21974	0.609868	−	0.792503i	$-0.291222\pi$
$997$	38.5135	1.21974	0.609868	+	0.792503i	$0.291222\pi$
$998$	22.3522	0.707547
$999$	45.1069	1.42712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.19		66
5.2	odd	4		1205.2.b.d.724.19	✓	66
5.3	odd	4		1205.2.b.d.724.48	yes	66
5.4	even	2	inner	6025.2.a.q.1.48		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.19	✓	66	5.2	odd	4
1205.2.b.d.724.48	yes	66	5.3	odd	4
6025.2.a.q.1.19		66	1.1	even	1	trivial
6025.2.a.q.1.48		66	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47177 q^{2} -1.07414 q^{3} +0.166115 q^{4} +1.58089 q^{6} -3.64779 q^{7} +2.69906 q^{8} -1.84623 q^{9} +O(q^{10})\)	\(q-1.47177 q^{2} -1.07414 q^{3} +0.166115 q^{4} +1.58089 q^{6} -3.64779 q^{7} +2.69906 q^{8} -1.84623 q^{9} +5.15794 q^{11} -0.178431 q^{12} -0.639676 q^{13} +5.36872 q^{14} -4.30464 q^{16} +0.722806 q^{17} +2.71723 q^{18} +4.10252 q^{19} +3.91823 q^{21} -7.59132 q^{22} -1.02481 q^{23} -2.89917 q^{24} +0.941458 q^{26} +5.20552 q^{27} -0.605954 q^{28} +2.08345 q^{29} +1.88445 q^{31} +0.937323 q^{32} -5.54035 q^{33} -1.06381 q^{34} -0.306686 q^{36} +8.66520 q^{37} -6.03797 q^{38} +0.687101 q^{39} +2.67828 q^{41} -5.76675 q^{42} -7.04494 q^{43} +0.856813 q^{44} +1.50829 q^{46} +1.75154 q^{47} +4.62378 q^{48} +6.30638 q^{49} -0.776394 q^{51} -0.106260 q^{52} -10.6700 q^{53} -7.66134 q^{54} -9.84561 q^{56} -4.40667 q^{57} -3.06636 q^{58} +4.03176 q^{59} +0.491373 q^{61} -2.77348 q^{62} +6.73465 q^{63} +7.22975 q^{64} +8.15413 q^{66} -15.9873 q^{67} +0.120069 q^{68} +1.10079 q^{69} +2.40999 q^{71} -4.98308 q^{72} +11.1626 q^{73} -12.7532 q^{74} +0.681491 q^{76} -18.8151 q^{77} -1.01126 q^{78} -0.140139 q^{79} -0.0527699 q^{81} -3.94182 q^{82} +4.61199 q^{83} +0.650878 q^{84} +10.3686 q^{86} -2.23791 q^{87} +13.9216 q^{88} -3.21821 q^{89} +2.33341 q^{91} -0.170237 q^{92} -2.02416 q^{93} -2.57786 q^{94} -1.00682 q^{96} +5.91955 q^{97} -9.28156 q^{98} -9.52273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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