LMFDB - Embedded newform 6025.2.a.p.1.31

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:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.31
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)

$f(q)$	$=$	$q+0.930197 q^{2} -1.32621 q^{3} -1.13473 q^{4} -1.23364 q^{6} +4.33497 q^{7} -2.91592 q^{8} -1.24116 q^{9} +O(q^{10})$	\(q+0.930197 q^{2} -1.32621 q^{3} -1.13473 q^{4} -1.23364 q^{6} +4.33497 q^{7} -2.91592 q^{8} -1.24116 q^{9} +0.882357 q^{11} +1.50490 q^{12} -2.68232 q^{13} +4.03238 q^{14} -0.442916 q^{16} -0.304406 q^{17} -1.15452 q^{18} -4.13564 q^{19} -5.74909 q^{21} +0.820766 q^{22} +2.19664 q^{23} +3.86713 q^{24} -2.49509 q^{26} +5.62468 q^{27} -4.91903 q^{28} +8.97537 q^{29} +5.91415 q^{31} +5.41984 q^{32} -1.17019 q^{33} -0.283158 q^{34} +1.40839 q^{36} -10.9828 q^{37} -3.84696 q^{38} +3.55733 q^{39} -6.07092 q^{41} -5.34779 q^{42} +2.71940 q^{43} -1.00124 q^{44} +2.04331 q^{46} -9.50435 q^{47} +0.587400 q^{48} +11.7920 q^{49} +0.403707 q^{51} +3.04372 q^{52} -11.0767 q^{53} +5.23206 q^{54} -12.6404 q^{56} +5.48474 q^{57} +8.34887 q^{58} -7.93694 q^{59} -1.60387 q^{61} +5.50133 q^{62} -5.38040 q^{63} +5.92735 q^{64} -1.08851 q^{66} +15.5753 q^{67} +0.345420 q^{68} -2.91321 q^{69} -9.40214 q^{71} +3.61913 q^{72} +4.11094 q^{73} -10.2162 q^{74} +4.69285 q^{76} +3.82499 q^{77} +3.30902 q^{78} +9.90696 q^{79} -3.73604 q^{81} -5.64716 q^{82} +10.7285 q^{83} +6.52368 q^{84} +2.52958 q^{86} -11.9033 q^{87} -2.57288 q^{88} -9.48120 q^{89} -11.6278 q^{91} -2.49260 q^{92} -7.84342 q^{93} -8.84092 q^{94} -7.18786 q^{96} +1.41584 q^{97} +10.9689 q^{98} -1.09515 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * 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 $

$2$	0.930197	0.657749	0.328874	−	0.944374i	$-0.393331\pi$
$2$	0.930197	0.657749	0.328874	+	0.944374i	$0.393331\pi$
$3$	−1.32621	−0.765689	−0.382845	−	0.923813i	$-0.625056\pi$
$3$	−1.32621	−0.765689	−0.382845	+	0.923813i	$0.625056\pi$
$4$	−1.13473	−0.567366
$5$	0	0
$5$	0	0
$6$	−1.23364	−0.503631
$7$	4.33497	1.63847	0.819233	−	0.573461i	$-0.194400\pi$
$7$	4.33497	1.63847	0.819233	+	0.573461i	$0.194400\pi$
$8$	−2.91592	−1.03093
$9$	−1.24116	−0.413720
$10$	0	0
$11$	0.882357	0.266041	0.133020	−	0.991113i	$-0.457532\pi$
$11$	0.882357	0.266041	0.133020	+	0.991113i	$0.457532\pi$
$12$	1.50490	0.434426
$13$	−2.68232	−0.743943	−0.371971	−	0.928244i	$-0.621318\pi$
$13$	−2.68232	−0.743943	−0.371971	+	0.928244i	$0.621318\pi$
$14$	4.03238	1.07770
$15$	0	0
$16$	−0.442916	−0.110729
$17$	−0.304406	−0.0738294	−0.0369147	−	0.999318i	$-0.511753\pi$
$17$	−0.304406	−0.0738294	−0.0369147	+	0.999318i	$0.511753\pi$
$18$	−1.15452	−0.272124
$19$	−4.13564	−0.948781	−0.474390	−	0.880315i	$-0.657331\pi$
$19$	−4.13564	−0.948781	−0.474390	+	0.880315i	$0.657331\pi$
$20$	0	0
$21$	−5.74909	−1.25455
$22$	0.820766	0.174988
$23$	2.19664	0.458031	0.229015	−	0.973423i	$-0.426449\pi$
$23$	2.19664	0.458031	0.229015	+	0.973423i	$0.426449\pi$
$24$	3.86713	0.789375
$25$	0	0
$26$	−2.49509	−0.489328
$27$	5.62468	1.08247
$28$	−4.91903	−0.929610
$29$	8.97537	1.66669	0.833343	−	0.552757i	$-0.186424\pi$
$29$	8.97537	1.66669	0.833343	+	0.552757i	$0.186424\pi$
$30$	0	0
$31$	5.91415	1.06221	0.531107	−	0.847305i	$-0.321776\pi$
$31$	5.91415	1.06221	0.531107	+	0.847305i	$0.321776\pi$
$32$	5.41984	0.958102
$33$	−1.17019	−0.203704
$34$	−0.283158	−0.0485612
$35$	0	0
$36$	1.40839	0.234731
$37$	−10.9828	−1.80556	−0.902779	−	0.430105i	$-0.858477\pi$
$37$	−10.9828	−1.80556	−0.902779	+	0.430105i	$0.858477\pi$
$38$	−3.84696	−0.624059
$39$	3.55733	0.569629
$40$	0	0
$41$	−6.07092	−0.948119	−0.474059	−	0.880493i	$-0.657212\pi$
$41$	−6.07092	−0.948119	−0.474059	+	0.880493i	$0.657212\pi$
$42$	−5.34779	−0.825182
$43$	2.71940	0.414705	0.207352	−	0.978266i	$-0.433515\pi$
$43$	2.71940	0.414705	0.207352	+	0.978266i	$0.433515\pi$
$44$	−1.00124	−0.150943
$45$	0	0
$46$	2.04331	0.301269
$47$	−9.50435	−1.38635	−0.693176	−	0.720769i	$-0.743789\pi$
$47$	−9.50435	−1.38635	−0.693176	+	0.720769i	$0.743789\pi$
$48$	0.587400	0.0847839
$49$	11.7920	1.68457
$50$	0	0
$51$	0.403707	0.0565303
$52$	3.04372	0.422088
$53$	−11.0767	−1.52150	−0.760748	−	0.649048i	$-0.775168\pi$
$53$	−11.0767	−1.52150	−0.760748	+	0.649048i	$0.775168\pi$
$54$	5.23206	0.711994
$55$	0	0
$56$	−12.6404	−1.68915
$57$	5.48474	0.726471
$58$	8.34887	1.09626
$59$	−7.93694	−1.03330	−0.516651	−	0.856196i	$-0.672821\pi$
$59$	−7.93694	−1.03330	−0.516651	+	0.856196i	$0.672821\pi$
$60$	0	0
$61$	−1.60387	−0.205355	−0.102678	−	0.994715i	$-0.532741\pi$
$61$	−1.60387	−0.205355	−0.102678	+	0.994715i	$0.532741\pi$
$62$	5.50133	0.698670
$63$	−5.38040	−0.677866
$64$	5.92735	0.740919
$65$	0	0
$66$	−1.08851	−0.133986
$67$	15.5753	1.90282	0.951410	−	0.307926i	$-0.0996349\pi$
$67$	15.5753	1.90282	0.951410	+	0.307926i	$0.0996349\pi$
$68$	0.345420	0.0418883
$69$	−2.91321	−0.350709
$70$	0	0
$71$	−9.40214	−1.11583	−0.557915	−	0.829898i	$-0.688398\pi$
$71$	−9.40214	−1.11583	−0.557915	+	0.829898i	$0.688398\pi$
$72$	3.61913	0.426518
$73$	4.11094	0.481150	0.240575	−	0.970631i	$-0.422664\pi$
$73$	4.11094	0.481150	0.240575	+	0.970631i	$0.422664\pi$
$74$	−10.2162	−1.18760
$75$	0	0
$76$	4.69285	0.538306
$77$	3.82499	0.435898
$78$	3.30902	0.374673
$79$	9.90696	1.11462	0.557310	−	0.830304i	$-0.311833\pi$
$79$	9.90696	1.11462	0.557310	+	0.830304i	$0.311833\pi$
$80$	0	0
$81$	−3.73604	−0.415115
$82$	−5.64716	−0.623624
$83$	10.7285	1.17761	0.588803	−	0.808277i	$-0.299599\pi$
$83$	10.7285	1.17761	0.588803	+	0.808277i	$0.299599\pi$
$84$	6.52368	0.711792
$85$	0	0
$86$	2.52958	0.272772
$87$	−11.9033	−1.27616
$88$	−2.57288	−0.274270
$89$	−9.48120	−1.00501	−0.502503	−	0.864576i	$-0.667587\pi$
$89$	−9.48120	−1.00501	−0.502503	+	0.864576i	$0.667587\pi$
$90$	0	0
$91$	−11.6278	−1.21892
$92$	−2.49260	−0.259871
$93$	−7.84342	−0.813325
$94$	−8.84092	−0.911871
$95$	0	0
$96$	−7.18786	−0.733608
$97$	1.41584	0.143757	0.0718784	−	0.997413i	$-0.477101\pi$
$97$	1.41584	0.143757	0.0718784	+	0.997413i	$0.477101\pi$
$98$	10.9689	1.10802
$99$	−1.09515	−0.110066
$100$	0	0
$101$	16.9462	1.68621	0.843104	−	0.537750i	$-0.180726\pi$
$101$	16.9462	1.68621	0.843104	+	0.537750i	$0.180726\pi$
$102$	0.375528	0.0371828
$103$	1.90525	0.187729	0.0938647	−	0.995585i	$-0.470078\pi$
$103$	1.90525	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$104$	7.82144	0.766956
$105$	0	0
$106$	−10.3035	−1.00076
$107$	−8.61469	−0.832814	−0.416407	−	0.909178i	$-0.636711\pi$
$107$	−8.61469	−0.832814	−0.416407	+	0.909178i	$0.636711\pi$
$108$	−6.38251	−0.614157
$109$	−8.16711	−0.782267	−0.391134	−	0.920334i	$-0.627917\pi$
$109$	−8.16711	−0.782267	−0.391134	+	0.920334i	$0.627917\pi$
$110$	0	0
$111$	14.5655	1.38250
$112$	−1.92003	−0.181426
$113$	−4.19717	−0.394837	−0.197418	−	0.980319i	$-0.563256\pi$
$113$	−4.19717	−0.394837	−0.197418	+	0.980319i	$0.563256\pi$
$114$	5.10189	0.477835
$115$	0	0
$116$	−10.1847	−0.945621
$117$	3.32920	0.307784
$118$	−7.38292	−0.679653
$119$	−1.31959	−0.120967
$120$	0	0
$121$	−10.2214	−0.929222
$122$	−1.49192	−0.135072
$123$	8.05133	0.725964
$124$	−6.71098	−0.602664
$125$	0	0
$126$	−5.00483	−0.445866
$127$	4.94642	0.438924	0.219462	−	0.975621i	$-0.429570\pi$
$127$	4.94642	0.438924	0.219462	+	0.975621i	$0.429570\pi$
$128$	−5.32607	−0.470763
$129$	−3.60650	−0.317535
$130$	0	0
$131$	13.6851	1.19567	0.597836	−	0.801618i	$-0.296027\pi$
$131$	13.6851	1.19567	0.597836	+	0.801618i	$0.296027\pi$
$132$	1.32786	0.115575
$133$	−17.9279	−1.55454
$134$	14.4881	1.25158
$135$	0	0
$136$	0.887624	0.0761132
$137$	−19.0577	−1.62821	−0.814105	−	0.580717i	$-0.802772\pi$
$137$	−19.0577	−1.62821	−0.814105	+	0.580717i	$0.802772\pi$
$138$	−2.70986	−0.230678
$139$	−21.7352	−1.84356	−0.921778	−	0.387717i	$-0.873264\pi$
$139$	−21.7352	−1.84356	−0.921778	+	0.387717i	$0.873264\pi$
$140$	0	0
$141$	12.6048	1.06151
$142$	−8.74585	−0.733935
$143$	−2.36677	−0.197919
$144$	0.549730	0.0458108
$145$	0	0
$146$	3.82399	0.316476
$147$	−15.6387	−1.28986
$148$	12.4625	1.02441
$149$	−6.03822	−0.494671	−0.247335	−	0.968930i	$-0.579555\pi$
$149$	−6.03822	−0.494671	−0.247335	+	0.968930i	$0.579555\pi$
$150$	0	0
$151$	−15.0115	−1.22162	−0.610808	−	0.791779i	$-0.709155\pi$
$151$	−15.0115	−1.22162	−0.610808	+	0.791779i	$0.709155\pi$
$152$	12.0592	0.978130
$153$	0.377817	0.0305447
$154$	3.55800	0.286712
$155$	0	0
$156$	−4.03662	−0.323188
$157$	−4.38682	−0.350107	−0.175053	−	0.984559i	$-0.556010\pi$
$157$	−4.38682	−0.350107	−0.175053	+	0.984559i	$0.556010\pi$
$158$	9.21543	0.733140
$159$	14.6900	1.16499
$160$	0	0
$161$	9.52236	0.750467
$162$	−3.47525	−0.273042
$163$	5.32658	0.417210	0.208605	−	0.978000i	$-0.433108\pi$
$163$	5.32658	0.417210	0.208605	+	0.978000i	$0.433108\pi$
$164$	6.88888	0.537931
$165$	0	0
$166$	9.97963	0.774569
$167$	8.59533	0.665127	0.332563	−	0.943081i	$-0.392086\pi$
$167$	8.59533	0.665127	0.332563	+	0.943081i	$0.392086\pi$
$168$	16.7639	1.29336
$169$	−5.80514	−0.446549
$170$	0	0
$171$	5.13299	0.392530
$172$	−3.08579	−0.235290
$173$	−3.91350	−0.297538	−0.148769	−	0.988872i	$-0.547531\pi$
$173$	−3.91350	−0.297538	−0.148769	+	0.988872i	$0.547531\pi$
$174$	−11.0724	−0.839395
$175$	0	0
$176$	−0.390810	−0.0294584
$177$	10.5261	0.791187
$178$	−8.81939	−0.661041
$179$	−2.52352	−0.188617	−0.0943084	−	0.995543i	$-0.530064\pi$
$179$	−2.52352	−0.188617	−0.0943084	+	0.995543i	$0.530064\pi$
$180$	0	0
$181$	−16.5056	−1.22685	−0.613426	−	0.789752i	$-0.710209\pi$
$181$	−16.5056	−1.22685	−0.613426	+	0.789752i	$0.710209\pi$
$182$	−10.8161	−0.801746
$183$	2.12708	0.157238
$184$	−6.40522	−0.472199
$185$	0	0
$186$	−7.29593	−0.534964
$187$	−0.268595	−0.0196416
$188$	10.7849	0.786569
$189$	24.3828	1.77359
$190$	0	0
$191$	−22.6006	−1.63532	−0.817660	−	0.575701i	$-0.804729\pi$
$191$	−22.6006	−1.63532	−0.817660	+	0.575701i	$0.804729\pi$
$192$	−7.86093	−0.567314
$193$	−6.58611	−0.474079	−0.237039	−	0.971500i	$-0.576177\pi$
$193$	−6.58611	−0.474079	−0.237039	+	0.971500i	$0.576177\pi$
$194$	1.31701	0.0945559
$195$	0	0
$196$	−13.3807	−0.955767
$197$	3.59184	0.255908	0.127954	−	0.991780i	$-0.459159\pi$
$197$	3.59184	0.255908	0.127954	+	0.991780i	$0.459159\pi$
$198$	−1.01870	−0.0723961
$199$	−21.0736	−1.49387	−0.746934	−	0.664899i	$-0.768475\pi$
$199$	−21.0736	−1.49387	−0.746934	+	0.664899i	$0.768475\pi$
$200$	0	0
$201$	−20.6561	−1.45697
$202$	15.7633	1.10910
$203$	38.9080	2.73081
$204$	−0.458100	−0.0320734
$205$	0	0
$206$	1.77225	0.123479
$207$	−2.72638	−0.189496
$208$	1.18804	0.0823760
$209$	−3.64911	−0.252414
$210$	0	0
$211$	28.9947	1.99607	0.998037	−	0.0626191i	$-0.0199453\pi$
$211$	28.9947	1.99607	0.998037	+	0.0626191i	$0.0199453\pi$
$212$	12.5690	0.863245
$213$	12.4692	0.854378
$214$	−8.01336	−0.547782
$215$	0	0
$216$	−16.4011	−1.11595
$217$	25.6377	1.74040
$218$	−7.59702	−0.514535
$219$	−5.45199	−0.368411
$220$	0	0
$221$	0.816516	0.0549248
$222$	13.5488	0.909335
$223$	7.57709	0.507399	0.253700	−	0.967283i	$-0.418353\pi$
$223$	7.57709	0.507399	0.253700	+	0.967283i	$0.418353\pi$
$224$	23.4949	1.56982
$225$	0	0
$226$	−3.90420	−0.259704
$227$	9.73925	0.646416	0.323208	−	0.946328i	$-0.395239\pi$
$227$	9.73925	0.646416	0.323208	+	0.946328i	$0.395239\pi$
$228$	−6.22371	−0.412175
$229$	−24.7729	−1.63704	−0.818520	−	0.574478i	$-0.805205\pi$
$229$	−24.7729	−1.63704	−0.818520	+	0.574478i	$0.805205\pi$
$230$	0	0
$231$	−5.07275	−0.333763
$232$	−26.1715	−1.71824
$233$	−20.9912	−1.37518	−0.687589	−	0.726100i	$-0.741331\pi$
$233$	−20.9912	−1.37518	−0.687589	+	0.726100i	$0.741331\pi$
$234$	3.09681	0.202445
$235$	0	0
$236$	9.00630	0.586260
$237$	−13.1387	−0.853453
$238$	−1.22748	−0.0795658
$239$	−0.936020	−0.0605461	−0.0302730	−	0.999542i	$-0.509638\pi$
$239$	−0.936020	−0.0605461	−0.0302730	+	0.999542i	$0.509638\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−9.50796	−0.611195
$243$	−11.9193	−0.764621
$244$	1.81997	0.116512
$245$	0	0
$246$	7.48933	0.477502
$247$	11.0931	0.705839
$248$	−17.2452	−1.09507
$249$	−14.2283	−0.901680
$250$	0	0
$251$	11.8204	0.746099	0.373050	−	0.927811i	$-0.378312\pi$
$251$	11.8204	0.746099	0.373050	+	0.927811i	$0.378312\pi$
$252$	6.10531	0.384598
$253$	1.93822	0.121855
$254$	4.60115	0.288702
$255$	0	0
$256$	−16.8090	−1.05056
$257$	14.9566	0.932964	0.466482	−	0.884531i	$-0.345521\pi$
$257$	14.9566	0.932964	0.466482	+	0.884531i	$0.345521\pi$
$258$	−3.35476	−0.208858
$259$	−47.6100	−2.95834
$260$	0	0
$261$	−11.1399	−0.689541
$262$	12.7298	0.786453
$263$	12.3331	0.760494	0.380247	−	0.924885i	$-0.375839\pi$
$263$	12.3331	0.760494	0.380247	+	0.924885i	$0.375839\pi$
$264$	3.41219	0.210006
$265$	0	0
$266$	−16.6765	−1.02250
$267$	12.5741	0.769522
$268$	−17.6738	−1.07960
$269$	−2.61251	−0.159287	−0.0796437	−	0.996823i	$-0.525378\pi$
$269$	−2.61251	−0.159287	−0.0796437	+	0.996823i	$0.525378\pi$
$270$	0	0
$271$	−9.75632	−0.592654	−0.296327	−	0.955086i	$-0.595762\pi$
$271$	−9.75632	−0.592654	−0.296327	+	0.955086i	$0.595762\pi$
$272$	0.134826	0.00817505
$273$	15.4209	0.933317
$274$	−17.7274	−1.07095
$275$	0	0
$276$	3.30571	0.198980
$277$	−0.508682	−0.0305637	−0.0152819	−	0.999883i	$-0.504865\pi$
$277$	−0.508682	−0.0305637	−0.0152819	+	0.999883i	$0.504865\pi$
$278$	−20.2180	−1.21260
$279$	−7.34042	−0.439459
$280$	0	0
$281$	−25.0618	−1.49506	−0.747530	−	0.664228i	$-0.768760\pi$
$281$	−25.0618	−1.49506	−0.747530	+	0.664228i	$0.768760\pi$
$282$	11.7249	0.698210
$283$	−1.51258	−0.0899134	−0.0449567	−	0.998989i	$-0.514315\pi$
$283$	−1.51258	−0.0899134	−0.0449567	+	0.998989i	$0.514315\pi$
$284$	10.6689	0.633084
$285$	0	0
$286$	−2.20156	−0.130181
$287$	−26.3173	−1.55346
$288$	−6.72689	−0.396386
$289$	−16.9073	−0.994549
$290$	0	0
$291$	−1.87771	−0.110073
$292$	−4.66482	−0.272988
$293$	−27.7625	−1.62190	−0.810950	−	0.585115i	$-0.801049\pi$
$293$	−27.7625	−1.62190	−0.810950	+	0.585115i	$0.801049\pi$
$294$	−14.5470	−0.848401
$295$	0	0
$296$	32.0249	1.86141
$297$	4.96298	0.287981
$298$	−5.61674	−0.325369
$299$	−5.89209	−0.340749
$300$	0	0
$301$	11.7885	0.679479
$302$	−13.9636	−0.803516
$303$	−22.4742	−1.29111
$304$	1.83174	0.105057
$305$	0	0
$306$	0.351444	0.0200907
$307$	19.1572	1.09336	0.546679	−	0.837342i	$-0.315892\pi$
$307$	19.1572	1.09336	0.546679	+	0.837342i	$0.315892\pi$
$308$	−4.34034	−0.247314
$309$	−2.52676	−0.143742
$310$	0	0
$311$	−4.09651	−0.232292	−0.116146	−	0.993232i	$-0.537054\pi$
$311$	−4.09651	−0.232292	−0.116146	+	0.993232i	$0.537054\pi$
$312$	−10.3729	−0.587250
$313$	−27.8074	−1.57177	−0.785883	−	0.618375i	$-0.787791\pi$
$313$	−27.8074	−1.57177	−0.785883	+	0.618375i	$0.787791\pi$
$314$	−4.08061	−0.230282
$315$	0	0
$316$	−11.2418	−0.632398
$317$	−14.6718	−0.824048	−0.412024	−	0.911173i	$-0.635178\pi$
$317$	−14.6718	−0.824048	−0.412024	+	0.911173i	$0.635178\pi$
$318$	13.6646	0.766273
$319$	7.91949	0.443406
$320$	0	0
$321$	11.4249	0.637676
$322$	8.85767	0.493619
$323$	1.25891	0.0700479
$324$	4.23941	0.235523
$325$	0	0
$326$	4.95477	0.274420
$327$	10.8313	0.598974
$328$	17.7023	0.977448
$329$	−41.2011	−2.27149
$330$	0	0
$331$	−14.8331	−0.815303	−0.407651	−	0.913138i	$-0.633652\pi$
$331$	−14.8331	−0.815303	−0.407651	+	0.913138i	$0.633652\pi$
$332$	−12.1740	−0.668134
$333$	13.6314	0.746996
$334$	7.99536	0.437486
$335$	0	0
$336$	2.54636	0.138916
$337$	3.92047	0.213562	0.106781	−	0.994283i	$-0.465946\pi$
$337$	3.92047	0.213562	0.106781	+	0.994283i	$0.465946\pi$
$338$	−5.39992	−0.293717
$339$	5.56634	0.302322
$340$	0	0
$341$	5.21840	0.282592
$342$	4.77470	0.258186
$343$	20.7731	1.12164
$344$	−7.92956	−0.427533
$345$	0	0
$346$	−3.64033	−0.195705
$347$	1.88740	0.101321	0.0506606	−	0.998716i	$-0.483867\pi$
$347$	1.88740	0.101321	0.0506606	+	0.998716i	$0.483867\pi$
$348$	13.5070	0.724052
$349$	−5.79692	−0.310302	−0.155151	−	0.987891i	$-0.549586\pi$
$349$	−5.79692	−0.310302	−0.155151	+	0.987891i	$0.549586\pi$
$350$	0	0
$351$	−15.0872	−0.805296
$352$	4.78224	0.254894
$353$	20.8216	1.10822	0.554111	−	0.832443i	$-0.313058\pi$
$353$	20.8216	1.10822	0.554111	+	0.832443i	$0.313058\pi$
$354$	9.79132	0.520403
$355$	0	0
$356$	10.7586	0.570206
$357$	1.75006	0.0926230
$358$	−2.34737	−0.124062
$359$	−15.4514	−0.815492	−0.407746	−	0.913095i	$-0.633685\pi$
$359$	−15.4514	−0.815492	−0.407746	+	0.913095i	$0.633685\pi$
$360$	0	0
$361$	−1.89649	−0.0998153
$362$	−15.3535	−0.806961
$363$	13.5558	0.711495
$364$	13.1944	0.691577
$365$	0	0
$366$	1.97860	0.103423
$367$	−38.0380	−1.98557	−0.992785	−	0.119912i	$-0.961739\pi$
$367$	−38.0380	−1.98557	−0.992785	+	0.119912i	$0.961739\pi$
$368$	−0.972925	−0.0507172
$369$	7.53499	0.392256
$370$	0	0
$371$	−48.0170	−2.49292
$372$	8.90019	0.461453
$373$	−15.6921	−0.812505	−0.406253	−	0.913761i	$-0.633165\pi$
$373$	−15.6921	−0.812505	−0.406253	+	0.913761i	$0.633165\pi$
$374$	−0.249846	−0.0129192
$375$	0	0
$376$	27.7139	1.42924
$377$	−24.0749	−1.23992
$378$	22.6808	1.16658
$379$	−4.85553	−0.249412	−0.124706	−	0.992194i	$-0.539799\pi$
$379$	−4.85553	−0.249412	−0.124706	+	0.992194i	$0.539799\pi$
$380$	0	0
$381$	−6.56001	−0.336079
$382$	−21.0230	−1.07563
$383$	18.3938	0.939878	0.469939	−	0.882699i	$-0.344276\pi$
$383$	18.3938	0.939878	0.469939	+	0.882699i	$0.344276\pi$
$384$	7.06351	0.360458
$385$	0	0
$386$	−6.12638	−0.311825
$387$	−3.37521	−0.171572
$388$	−1.60660	−0.0815628
$389$	−30.6318	−1.55310	−0.776548	−	0.630058i	$-0.783031\pi$
$389$	−30.6318	−1.55310	−0.776548	+	0.630058i	$0.783031\pi$
$390$	0	0
$391$	−0.668670	−0.0338161
$392$	−34.3845	−1.73668
$393$	−18.1494	−0.915514
$394$	3.34112	0.168323
$395$	0	0
$396$	1.24270	0.0624480
$397$	4.67514	0.234639	0.117319	−	0.993094i	$-0.462570\pi$
$397$	4.67514	0.234639	0.117319	+	0.993094i	$0.462570\pi$
$398$	−19.6026	−0.982590
$399$	23.7762	1.19030
$400$	0	0
$401$	−7.50388	−0.374726	−0.187363	−	0.982291i	$-0.559994\pi$
$401$	−7.50388	−0.374726	−0.187363	+	0.982291i	$0.559994\pi$
$402$	−19.2143	−0.958320
$403$	−15.8637	−0.790226
$404$	−19.2294	−0.956698
$405$	0	0
$406$	36.1921	1.79618
$407$	−9.69073	−0.480352
$408$	−1.17718	−0.0582790
$409$	−7.22831	−0.357417	−0.178709	−	0.983902i	$-0.557192\pi$
$409$	−7.22831	−0.357417	−0.178709	+	0.983902i	$0.557192\pi$
$410$	0	0
$411$	25.2746	1.24670
$412$	−2.16195	−0.106511
$413$	−34.4064	−1.69303
$414$	−2.53607	−0.124641
$415$	0	0
$416$	−14.5378	−0.712773
$417$	28.8255	1.41159
$418$	−3.39439	−0.166025
$419$	12.9158	0.630979	0.315490	−	0.948929i	$-0.397831\pi$
$419$	12.9158	0.630979	0.315490	+	0.948929i	$0.397831\pi$
$420$	0	0
$421$	25.6611	1.25064	0.625322	−	0.780367i	$-0.284967\pi$
$421$	25.6611	1.25064	0.625322	+	0.780367i	$0.284967\pi$
$422$	26.9708	1.31292
$423$	11.7964	0.573562
$424$	32.2986	1.56856
$425$	0	0
$426$	11.5989	0.561966
$427$	−6.95275	−0.336467
$428$	9.77537	0.472510
$429$	3.13884	0.151544
$430$	0	0
$431$	−4.38626	−0.211279	−0.105639	−	0.994404i	$-0.533689\pi$
$431$	−4.38626	−0.211279	−0.105639	+	0.994404i	$0.533689\pi$
$432$	−2.49126	−0.119861
$433$	10.6818	0.513335	0.256667	−	0.966500i	$-0.417376\pi$
$433$	10.6818	0.513335	0.256667	+	0.966500i	$0.417376\pi$
$434$	23.8481	1.14475
$435$	0	0
$436$	9.26749	0.443832
$437$	−9.08450	−0.434571
$438$	−5.07142	−0.242322
$439$	6.07260	0.289829	0.144915	−	0.989444i	$-0.453709\pi$
$439$	6.07260	0.289829	0.144915	+	0.989444i	$0.453709\pi$
$440$	0	0
$441$	−14.6357	−0.696940
$442$	0.759521	0.0361267
$443$	23.3422	1.10902	0.554512	−	0.832176i	$-0.312905\pi$
$443$	23.3422	1.10902	0.554512	+	0.832176i	$0.312905\pi$
$444$	−16.5280	−0.784382
$445$	0	0
$446$	7.04819	0.333741
$447$	8.00797	0.378764
$448$	25.6949	1.21397
$449$	14.5863	0.688371	0.344185	−	0.938902i	$-0.388155\pi$
$449$	14.5863	0.688371	0.344185	+	0.938902i	$0.388155\pi$
$450$	0	0
$451$	−5.35672	−0.252238
$452$	4.76267	0.224017
$453$	19.9084	0.935378
$454$	9.05942	0.425180
$455$	0	0
$456$	−15.9931	−0.748943
$457$	21.6065	1.01071	0.505355	−	0.862912i	$-0.331362\pi$
$457$	21.6065	1.01071	0.505355	+	0.862912i	$0.331362\pi$
$458$	−23.0437	−1.07676
$459$	−1.71219	−0.0799181
$460$	0	0
$461$	−8.71776	−0.406026	−0.203013	−	0.979176i	$-0.565073\pi$
$461$	−8.71776	−0.406026	−0.203013	+	0.979176i	$0.565073\pi$
$462$	−4.71866	−0.219532
$463$	−16.2752	−0.756375	−0.378187	−	0.925729i	$-0.623452\pi$
$463$	−16.2752	−0.756375	−0.378187	+	0.925729i	$0.623452\pi$
$464$	−3.97534	−0.184550
$465$	0	0
$466$	−19.5260	−0.904522
$467$	6.86128	0.317502	0.158751	−	0.987319i	$-0.449253\pi$
$467$	6.86128	0.317502	0.158751	+	0.987319i	$0.449253\pi$
$468$	−3.77775	−0.174626
$469$	67.5183	3.11771
$470$	0	0
$471$	5.81786	0.268073
$472$	23.1435	1.06526
$473$	2.39948	0.110328
$474$	−12.2216	−0.561358
$475$	0	0
$476$	1.49738	0.0686325
$477$	13.7479	0.629473
$478$	−0.870683	−0.0398241
$479$	17.6985	0.808664	0.404332	−	0.914612i	$-0.367504\pi$
$479$	17.6985	0.808664	0.404332	+	0.914612i	$0.367504\pi$
$480$	0	0
$481$	29.4594	1.34323
$482$	0.930197	0.0423693
$483$	−12.6287	−0.574624
$484$	11.5986	0.527210
$485$	0	0
$486$	−11.0873	−0.502928
$487$	9.06661	0.410847	0.205424	−	0.978673i	$-0.434143\pi$
$487$	9.06661	0.410847	0.205424	+	0.978673i	$0.434143\pi$
$488$	4.67677	0.211707
$489$	−7.06418	−0.319453
$490$	0	0
$491$	−16.2481	−0.733269	−0.366634	−	0.930365i	$-0.619490\pi$
$491$	−16.2481	−0.733269	−0.366634	+	0.930365i	$0.619490\pi$
$492$	−9.13611	−0.411888
$493$	−2.73216	−0.123050
$494$	10.3188	0.464265
$495$	0	0
$496$	−2.61947	−0.117618
$497$	−40.7580	−1.82825
$498$	−13.2351	−0.593079
$499$	31.2373	1.39837	0.699186	−	0.714940i	$-0.253546\pi$
$499$	31.2373	1.39837	0.699186	+	0.714940i	$0.253546\pi$
$500$	0	0
$501$	−11.3992	−0.509280
$502$	10.9953	0.490746
$503$	−36.0521	−1.60749	−0.803743	−	0.594977i	$-0.797161\pi$
$503$	−36.0521	−1.60749	−0.803743	+	0.594977i	$0.797161\pi$
$504$	15.6888	0.698835
$505$	0	0
$506$	1.80293	0.0801498
$507$	7.69884	0.341918
$508$	−5.61287	−0.249031
$509$	6.06492	0.268823	0.134412	−	0.990926i	$-0.457086\pi$
$509$	6.06492	0.268823	0.134412	+	0.990926i	$0.457086\pi$
$510$	0	0
$511$	17.8208	0.788347
$512$	−4.98355	−0.220244
$513$	−23.2616	−1.02703
$514$	13.9125	0.613656
$515$	0	0
$516$	4.09242	0.180159
$517$	−8.38623	−0.368826
$518$	−44.2867	−1.94585
$519$	5.19013	0.227822
$520$	0	0
$521$	−33.8087	−1.48119	−0.740593	−	0.671954i	$-0.765455\pi$
$521$	−33.8087	−1.48119	−0.740593	+	0.671954i	$0.765455\pi$
$522$	−10.3623	−0.453545
$523$	23.5838	1.03125	0.515625	−	0.856815i	$-0.327560\pi$
$523$	23.5838	1.03125	0.515625	+	0.856815i	$0.327560\pi$
$524$	−15.5289	−0.678385
$525$	0	0
$526$	11.4723	0.500214
$527$	−1.80031	−0.0784225
$528$	0.518297	0.0225560
$529$	−18.1748	−0.790208
$530$	0	0
$531$	9.85101	0.427498
$532$	20.3433	0.881996
$533$	16.2842	0.705346
$534$	11.6964	0.506152
$535$	0	0
$536$	−45.4162	−1.96168
$537$	3.34672	0.144422
$538$	−2.43015	−0.104771
$539$	10.4047	0.448164
$540$	0	0
$541$	−12.4241	−0.534152	−0.267076	−	0.963675i	$-0.586058\pi$
$541$	−12.4241	−0.534152	−0.267076	+	0.963675i	$0.586058\pi$
$542$	−9.07531	−0.389818
$543$	21.8899	0.939388
$544$	−1.64983	−0.0707360
$545$	0	0
$546$	14.3445	0.613888
$547$	12.3673	0.528786	0.264393	−	0.964415i	$-0.414828\pi$
$547$	12.3673	0.528786	0.264393	+	0.964415i	$0.414828\pi$
$548$	21.6254	0.923792
$549$	1.99067	0.0849596
$550$	0	0
$551$	−37.1189	−1.58132
$552$	8.49468	0.361558
$553$	42.9464	1.82627
$554$	−0.473174	−0.0201033
$555$	0	0
$556$	24.6637	1.04597
$557$	−14.4719	−0.613195	−0.306598	−	0.951839i	$-0.599191\pi$
$557$	−14.4719	−0.613195	−0.306598	+	0.951839i	$0.599191\pi$
$558$	−6.82804	−0.289054
$559$	−7.29432	−0.308517
$560$	0	0
$561$	0.356214	0.0150394
$562$	−23.3124	−0.983374
$563$	−2.71732	−0.114521	−0.0572607	−	0.998359i	$-0.518237\pi$
$563$	−2.71732	−0.114521	−0.0572607	+	0.998359i	$0.518237\pi$
$564$	−14.3031	−0.602268
$565$	0	0
$566$	−1.40700	−0.0591404
$567$	−16.1956	−0.680152
$568$	27.4159	1.15035
$569$	−35.2838	−1.47917	−0.739586	−	0.673062i	$-0.764979\pi$
$569$	−35.2838	−1.47917	−0.739586	+	0.673062i	$0.764979\pi$
$570$	0	0
$571$	−34.4873	−1.44325	−0.721625	−	0.692285i	$-0.756604\pi$
$571$	−34.4873	−1.44325	−0.721625	+	0.692285i	$0.756604\pi$
$572$	2.68565	0.112293
$573$	29.9732	1.25215
$574$	−24.4803	−1.02179
$575$	0	0
$576$	−7.35680	−0.306533
$577$	6.57108	0.273558	0.136779	−	0.990602i	$-0.456325\pi$
$577$	6.57108	0.273558	0.136779	+	0.990602i	$0.456325\pi$
$578$	−15.7272	−0.654164
$579$	8.73458	0.362997
$580$	0	0
$581$	46.5078	1.92947
$582$	−1.74664	−0.0724004
$583$	−9.77357	−0.404780
$584$	−11.9872	−0.496033
$585$	0	0
$586$	−25.8246	−1.06680
$587$	−3.17806	−0.131173	−0.0655863	−	0.997847i	$-0.520892\pi$
$587$	−3.17806	−0.131173	−0.0655863	+	0.997847i	$0.520892\pi$
$588$	17.7457	0.731821
$589$	−24.4588	−1.00781
$590$	0	0
$591$	−4.76354	−0.195946
$592$	4.86445	0.199928
$593$	35.0450	1.43913	0.719563	−	0.694428i	$-0.244342\pi$
$593$	35.0450	1.43913	0.719563	+	0.694428i	$0.244342\pi$
$594$	4.61655	0.189419
$595$	0	0
$596$	6.85177	0.280660
$597$	27.9481	1.14384
$598$	−5.48081	−0.224127
$599$	−18.6149	−0.760583	−0.380292	−	0.924867i	$-0.624176\pi$
$599$	−18.6149	−0.760583	−0.380292	+	0.924867i	$0.624176\pi$
$600$	0	0
$601$	8.79674	0.358827	0.179413	−	0.983774i	$-0.442580\pi$
$601$	8.79674	0.358827	0.179413	+	0.983774i	$0.442580\pi$
$602$	10.9657	0.446927
$603$	−19.3314	−0.787235
$604$	17.0340	0.693104
$605$	0	0
$606$	−20.9055	−0.849227
$607$	−44.5048	−1.80639	−0.903197	−	0.429226i	$-0.858787\pi$
$607$	−44.5048	−1.80639	−0.903197	+	0.429226i	$0.858787\pi$
$608$	−22.4145	−0.909028
$609$	−51.6003	−2.09095
$610$	0	0
$611$	25.4937	1.03137
$612$	−0.428721	−0.0173300
$613$	47.1882	1.90591	0.952956	−	0.303109i	$-0.0980246\pi$
$613$	47.1882	1.90591	0.952956	+	0.303109i	$0.0980246\pi$
$614$	17.8200	0.719155
$615$	0	0
$616$	−11.1534	−0.449382
$617$	−3.11780	−0.125518	−0.0627590	−	0.998029i	$-0.519990\pi$
$617$	−3.11780	−0.125518	−0.0627590	+	0.998029i	$0.519990\pi$
$618$	−2.35039	−0.0945464
$619$	23.7548	0.954787	0.477393	−	0.878690i	$-0.341582\pi$
$619$	23.7548	0.954787	0.477393	+	0.878690i	$0.341582\pi$
$620$	0	0
$621$	12.3554	0.495804
$622$	−3.81056	−0.152789
$623$	−41.1007	−1.64667
$624$	−1.57560	−0.0630744
$625$	0	0
$626$	−25.8664	−1.03383
$627$	4.83950	0.193271
$628$	4.97787	0.198639
$629$	3.34323	0.133303
$630$	0	0
$631$	−47.0576	−1.87334	−0.936668	−	0.350220i	$-0.886107\pi$
$631$	−47.0576	−1.87334	−0.936668	+	0.350220i	$0.886107\pi$
$632$	−28.8879	−1.14910
$633$	−38.4531	−1.52837
$634$	−13.6476	−0.542016
$635$	0	0
$636$	−16.6692	−0.660978
$637$	−31.6299	−1.25322
$638$	7.36668	0.291650
$639$	11.6696	0.461641
$640$	0	0
$641$	25.2677	0.998015	0.499007	−	0.866598i	$-0.333698\pi$
$641$	25.2677	0.998015	0.499007	+	0.866598i	$0.333698\pi$
$642$	10.6274	0.419431
$643$	1.36761	0.0539332	0.0269666	−	0.999636i	$-0.491415\pi$
$643$	1.36761	0.0539332	0.0269666	+	0.999636i	$0.491415\pi$
$644$	−10.8053	−0.425790
$645$	0	0
$646$	1.17104	0.0460739
$647$	47.9634	1.88563	0.942817	−	0.333310i	$-0.108166\pi$
$647$	47.9634	1.88563	0.942817	+	0.333310i	$0.108166\pi$
$648$	10.8940	0.427956
$649$	−7.00321	−0.274900
$650$	0	0
$651$	−34.0010	−1.33261
$652$	−6.04425	−0.236711
$653$	18.9603	0.741975	0.370987	−	0.928638i	$-0.379019\pi$
$653$	18.9603	0.741975	0.370987	+	0.928638i	$0.379019\pi$
$654$	10.0753	0.393974
$655$	0	0
$656$	2.68891	0.104984
$657$	−5.10234	−0.199061
$658$	−38.3251	−1.49407
$659$	−8.62976	−0.336168	−0.168084	−	0.985773i	$-0.553758\pi$
$659$	−8.62976	−0.336168	−0.168084	+	0.985773i	$0.553758\pi$
$660$	0	0
$661$	−16.6323	−0.646920	−0.323460	−	0.946242i	$-0.604846\pi$
$661$	−16.6323	−0.646920	−0.323460	+	0.946242i	$0.604846\pi$
$662$	−13.7977	−0.536265
$663$	−1.08287	−0.0420553
$664$	−31.2835	−1.21403
$665$	0	0
$666$	12.6799	0.491336
$667$	19.7156	0.763393
$668$	−9.75341	−0.377371
$669$	−10.0488	−0.388510
$670$	0	0
$671$	−1.41519	−0.0546328
$672$	−31.1592	−1.20199
$673$	10.4577	0.403113	0.201557	−	0.979477i	$-0.435400\pi$
$673$	10.4577	0.403113	0.201557	+	0.979477i	$0.435400\pi$
$674$	3.64681	0.140470
$675$	0	0
$676$	6.58728	0.253357
$677$	−41.7207	−1.60346	−0.801728	−	0.597689i	$-0.796086\pi$
$677$	−41.7207	−1.60346	−0.801728	+	0.597689i	$0.796086\pi$
$678$	5.17780	0.198852
$679$	6.13763	0.235541
$680$	0	0
$681$	−12.9163	−0.494954
$682$	4.85414	0.185875
$683$	32.9092	1.25924	0.629618	−	0.776905i	$-0.283211\pi$
$683$	32.9092	1.25924	0.629618	+	0.776905i	$0.283211\pi$
$684$	−5.82457	−0.222708
$685$	0	0
$686$	19.3231	0.737758
$687$	32.8541	1.25346
$688$	−1.20447	−0.0459198
$689$	29.7112	1.13191
$690$	0	0
$691$	23.1023	0.878855	0.439427	−	0.898278i	$-0.355181\pi$
$691$	23.1023	0.878855	0.439427	+	0.898278i	$0.355181\pi$
$692$	4.44078	0.168813
$693$	−4.74743	−0.180340
$694$	1.75566	0.0666438
$695$	0	0
$696$	34.7089	1.31564
$697$	1.84803	0.0699990
$698$	−5.39228	−0.204101
$699$	27.8388	1.05296
$700$	0	0
$701$	16.5438	0.624852	0.312426	−	0.949942i	$-0.398858\pi$
$701$	16.5438	0.624852	0.312426	+	0.949942i	$0.398858\pi$
$702$	−14.0341	−0.529683
$703$	45.4208	1.71308
$704$	5.23004	0.197115
$705$	0	0
$706$	19.3682	0.728932
$707$	73.4612	2.76279
$708$	−11.9443	−0.448893
$709$	31.5902	1.18639	0.593197	−	0.805058i	$-0.297866\pi$
$709$	31.5902	1.18639	0.593197	+	0.805058i	$0.297866\pi$
$710$	0	0
$711$	−12.2961	−0.461141
$712$	27.6464	1.03609
$713$	12.9913	0.486526
$714$	1.62790	0.0609227
$715$	0	0
$716$	2.86352	0.107015
$717$	1.24136	0.0463595
$718$	−14.3728	−0.536389
$719$	−17.6028	−0.656474	−0.328237	−	0.944595i	$-0.606454\pi$
$719$	−17.6028	−0.656474	−0.328237	+	0.944595i	$0.606454\pi$
$720$	0	0
$721$	8.25919	0.307588
$722$	−1.76411	−0.0656534
$723$	−1.32621	−0.0493224
$724$	18.7295	0.696075
$725$	0	0
$726$	12.6096	0.467985
$727$	−34.7819	−1.28999	−0.644995	−	0.764187i	$-0.723141\pi$
$727$	−34.7819	−1.28999	−0.644995	+	0.764187i	$0.723141\pi$
$728$	33.9057	1.25663
$729$	27.0156	1.00058
$730$	0	0
$731$	−0.827803	−0.0306174
$732$	−2.41367	−0.0892117
$733$	35.3016	1.30389	0.651947	−	0.758265i	$-0.273952\pi$
$733$	35.3016	1.30389	0.651947	+	0.758265i	$0.273952\pi$
$734$	−35.3829	−1.30601
$735$	0	0
$736$	11.9054	0.438840
$737$	13.7429	0.506228
$738$	7.00903	0.258006
$739$	−14.8693	−0.546975	−0.273487	−	0.961876i	$-0.588177\pi$
$739$	−14.8693	−0.546975	−0.273487	+	0.961876i	$0.588177\pi$
$740$	0	0
$741$	−14.7118	−0.540453
$742$	−44.6653	−1.63971
$743$	−28.8502	−1.05841	−0.529206	−	0.848494i	$-0.677510\pi$
$743$	−28.8502	−1.05841	−0.529206	+	0.848494i	$0.677510\pi$
$744$	22.8708	0.838484
$745$	0	0
$746$	−14.5967	−0.534424
$747$	−13.3158	−0.487199
$748$	0.304784	0.0111440
$749$	−37.3444	−1.36454
$750$	0	0
$751$	26.4978	0.966918	0.483459	−	0.875367i	$-0.339380\pi$
$751$	26.4978	0.966918	0.483459	+	0.875367i	$0.339380\pi$
$752$	4.20963	0.153509
$753$	−15.6764	−0.571280
$754$	−22.3944	−0.815555
$755$	0	0
$756$	−27.6680	−1.00628
$757$	34.8758	1.26758	0.633790	−	0.773505i	$-0.281498\pi$
$757$	34.8758	1.26758	0.633790	+	0.773505i	$0.281498\pi$
$758$	−4.51660	−0.164050
$759$	−2.57049	−0.0933029
$760$	0	0
$761$	43.4881	1.57644	0.788221	−	0.615392i	$-0.211002\pi$
$761$	43.4881	1.57644	0.788221	+	0.615392i	$0.211002\pi$
$762$	−6.10210	−0.221056
$763$	−35.4042	−1.28172
$764$	25.6456	0.927826
$765$	0	0
$766$	17.1098	0.618203
$767$	21.2894	0.768717
$768$	22.2923	0.804405
$769$	49.0482	1.76872	0.884362	−	0.466801i	$-0.154594\pi$
$769$	49.0482	1.76872	0.884362	+	0.466801i	$0.154594\pi$
$770$	0	0
$771$	−19.8356	−0.714361
$772$	7.47348	0.268976
$773$	28.0305	1.00819	0.504093	−	0.863649i	$-0.331827\pi$
$773$	28.0305	1.00819	0.504093	+	0.863649i	$0.331827\pi$
$774$	−3.13962	−0.112851
$775$	0	0
$776$	−4.12848	−0.148204
$777$	63.1410	2.26517
$778$	−28.4937	−1.02155
$779$	25.1071	0.899557
$780$	0	0
$781$	−8.29605	−0.296856
$782$	−0.621995	−0.0222425
$783$	50.4836	1.80414
$784$	−5.22285	−0.186530
$785$	0	0
$786$	−16.8825	−0.602178
$787$	−14.4420	−0.514801	−0.257400	−	0.966305i	$-0.582866\pi$
$787$	−14.4420	−0.514801	−0.257400	+	0.966305i	$0.582866\pi$
$788$	−4.07578	−0.145194
$789$	−16.3564	−0.582302
$790$	0	0
$791$	−18.1946	−0.646926
$792$	3.19336	0.113471
$793$	4.30211	0.152772
$794$	4.34880	0.154333
$795$	0	0
$796$	23.9129	0.847570
$797$	10.9710	0.388612	0.194306	−	0.980941i	$-0.437755\pi$
$797$	10.9710	0.388612	0.194306	+	0.980941i	$0.437755\pi$
$798$	22.1165	0.782917
$799$	2.89318	0.102353
$800$	0	0
$801$	11.7677	0.415791
$802$	−6.98009	−0.246475
$803$	3.62732	0.128005
$804$	23.4392	0.826635
$805$	0	0
$806$	−14.7564	−0.519770
$807$	3.46474	0.121965
$808$	−49.4137	−1.73837
$809$	−39.0394	−1.37255	−0.686276	−	0.727342i	$-0.740756\pi$
$809$	−39.0394	−1.37255	−0.686276	+	0.727342i	$0.740756\pi$
$810$	0	0
$811$	−22.6660	−0.795912	−0.397956	−	0.917405i	$-0.630280\pi$
$811$	−22.6660	−0.795912	−0.397956	+	0.917405i	$0.630280\pi$
$812$	−44.1502	−1.54937
$813$	12.9390	0.453789
$814$	−9.01430	−0.315951
$815$	0	0
$816$	−0.178808	−0.00625954
$817$	−11.2465	−0.393464
$818$	−6.72376	−0.235091
$819$	14.4320	0.504294
$820$	0	0
$821$	24.9834	0.871926	0.435963	−	0.899965i	$-0.356408\pi$
$821$	24.9834	0.871926	0.435963	+	0.899965i	$0.356408\pi$
$822$	23.5104	0.820018
$823$	35.0663	1.22233	0.611167	−	0.791502i	$-0.290700\pi$
$823$	35.0663	1.22233	0.611167	+	0.791502i	$0.290700\pi$
$824$	−5.55555	−0.193537
$825$	0	0
$826$	−32.0047	−1.11359
$827$	−5.28536	−0.183790	−0.0918949	−	0.995769i	$-0.529292\pi$
$827$	−5.28536	−0.183790	−0.0918949	+	0.995769i	$0.529292\pi$
$828$	3.09371	0.107514
$829$	0.699646	0.0242997	0.0121499	−	0.999926i	$-0.496132\pi$
$829$	0.699646	0.0242997	0.0121499	+	0.999926i	$0.496132\pi$
$830$	0	0
$831$	0.674620	0.0234023
$832$	−15.8991	−0.551202
$833$	−3.58955	−0.124371
$834$	26.8134	0.928473
$835$	0	0
$836$	4.14077	0.143211
$837$	33.2652	1.14981
$838$	12.0143	0.415026
$839$	44.8426	1.54814	0.774069	−	0.633101i	$-0.218218\pi$
$839$	44.8426	1.54814	0.774069	+	0.633101i	$0.218218\pi$
$840$	0	0
$841$	51.5573	1.77784
$842$	23.8699	0.822610
$843$	33.2372	1.14475
$844$	−32.9012	−1.13251
$845$	0	0
$846$	10.9730	0.377260
$847$	−44.3097	−1.52250
$848$	4.90603	0.168474
$849$	2.00600	0.0688457
$850$	0	0
$851$	−24.1252	−0.827001
$852$	−14.1493	−0.484746
$853$	−10.7624	−0.368497	−0.184249	−	0.982880i	$-0.558985\pi$
$853$	−10.7624	−0.368497	−0.184249	+	0.982880i	$0.558985\pi$
$854$	−6.46743	−0.221311
$855$	0	0
$856$	25.1197	0.858575
$857$	21.5155	0.734956	0.367478	−	0.930032i	$-0.380221\pi$
$857$	21.5155	0.734956	0.367478	+	0.930032i	$0.380221\pi$
$858$	2.91974	0.0996782
$859$	2.12944	0.0726556	0.0363278	−	0.999340i	$-0.488434\pi$
$859$	2.12944	0.0726556	0.0363278	+	0.999340i	$0.488434\pi$
$860$	0	0
$861$	34.9023	1.18947
$862$	−4.08009	−0.138968
$863$	10.2693	0.349571	0.174785	−	0.984607i	$-0.444077\pi$
$863$	10.2693	0.349571	0.174785	+	0.984607i	$0.444077\pi$
$864$	30.4849	1.03712
$865$	0	0
$866$	9.93618	0.337645
$867$	22.4227	0.761515
$868$	−29.0919	−0.987444
$869$	8.74148	0.296534
$870$	0	0
$871$	−41.7779	−1.41559
$872$	23.8146	0.806466
$873$	−1.75729	−0.0594751
$874$	−8.45038	−0.285838
$875$	0	0
$876$	6.18655	0.209024
$877$	5.83603	0.197069	0.0985343	−	0.995134i	$-0.468585\pi$
$877$	5.83603	0.197069	0.0985343	+	0.995134i	$0.468585\pi$
$878$	5.64871	0.190635
$879$	36.8189	1.24187
$880$	0	0
$881$	23.6510	0.796823	0.398412	−	0.917207i	$-0.369562\pi$
$881$	23.6510	0.796823	0.398412	+	0.917207i	$0.369562\pi$
$882$	−13.6141	−0.458411
$883$	−21.4740	−0.722658	−0.361329	−	0.932438i	$-0.617677\pi$
$883$	−21.4740	−0.722658	−0.361329	+	0.932438i	$0.617677\pi$
$884$	−0.926528	−0.0311625
$885$	0	0
$886$	21.7129	0.729459
$887$	−19.7851	−0.664320	−0.332160	−	0.943223i	$-0.607777\pi$
$887$	−19.7851	−0.664320	−0.332160	+	0.943223i	$0.607777\pi$
$888$	−42.4718	−1.42526
$889$	21.4426	0.719162
$890$	0	0
$891$	−3.29652	−0.110438
$892$	−8.59797	−0.287881
$893$	39.3066	1.31534
$894$	7.44899	0.249132
$895$	0	0
$896$	−23.0884	−0.771329
$897$	7.81417	0.260907
$898$	13.5681	0.452775
$899$	53.0817	1.77038
$900$	0	0
$901$	3.37180	0.112331
$902$	−4.98281	−0.165909
$903$	−15.6341	−0.520270
$904$	12.2386	0.407051
$905$	0	0
$906$	18.5187	0.615244
$907$	0.445947	0.0148074	0.00740371	−	0.999973i	$-0.497643\pi$
$907$	0.445947	0.0148074	0.00740371	+	0.999973i	$0.497643\pi$
$908$	−11.0514	−0.366755
$909$	−21.0329	−0.697619
$910$	0	0
$911$	3.07240	0.101793	0.0508965	−	0.998704i	$-0.483792\pi$
$911$	3.07240	0.101793	0.0508965	+	0.998704i	$0.483792\pi$
$912$	−2.42928	−0.0804414
$913$	9.46637	0.313291
$914$	20.0983	0.664793
$915$	0	0
$916$	28.1106	0.928802
$917$	59.3245	1.95907
$918$	−1.59267	−0.0525660
$919$	9.98217	0.329281	0.164641	−	0.986354i	$-0.447354\pi$
$919$	9.98217	0.329281	0.164641	+	0.986354i	$0.447354\pi$
$920$	0	0
$921$	−25.4065	−0.837172
$922$	−8.10924	−0.267063
$923$	25.2196	0.830113
$924$	5.75622	0.189366
$925$	0	0
$926$	−15.1392	−0.497505
$927$	−2.36472	−0.0776675
$928$	48.6451	1.59685
$929$	2.23534	0.0733392	0.0366696	−	0.999327i	$-0.488325\pi$
$929$	2.23534	0.0733392	0.0366696	+	0.999327i	$0.488325\pi$
$930$	0	0
$931$	−48.7674	−1.59829
$932$	23.8194	0.780230
$933$	5.43284	0.177863
$934$	6.38235	0.208837
$935$	0	0
$936$	−9.70767	−0.317305
$937$	−46.5278	−1.52000	−0.759999	−	0.649924i	$-0.774801\pi$
$937$	−46.5278	−1.52000	−0.759999	+	0.649924i	$0.774801\pi$
$938$	62.8053	2.05067
$939$	36.8785	1.20348
$940$	0	0
$941$	37.9006	1.23552	0.617761	−	0.786366i	$-0.288040\pi$
$941$	37.9006	1.23552	0.617761	+	0.786366i	$0.288040\pi$
$942$	5.41176	0.176325
$943$	−13.3356	−0.434267
$944$	3.51539	0.114416
$945$	0	0
$946$	2.23199	0.0725684
$947$	−22.0207	−0.715576	−0.357788	−	0.933803i	$-0.616469\pi$
$947$	−22.0207	−0.715576	−0.357788	+	0.933803i	$0.616469\pi$
$948$	14.9090	0.484220
$949$	−11.0269	−0.357948
$950$	0	0
$951$	19.4579	0.630964
$952$	3.84783	0.124709
$953$	−49.8117	−1.61356	−0.806779	−	0.590853i	$-0.798791\pi$
$953$	−49.8117	−1.61356	−0.806779	+	0.590853i	$0.798791\pi$
$954$	12.7883	0.414035
$955$	0	0
$956$	1.06213	0.0343518
$957$	−10.5029	−0.339511
$958$	16.4631	0.531898
$959$	−82.6147	−2.66777
$960$	0	0
$961$	3.97722	0.128297
$962$	27.4030	0.883509
$963$	10.6922	0.344552
$964$	−1.13473	−0.0365473
$965$	0	0
$966$	−11.7472	−0.377959
$967$	12.7604	0.410348	0.205174	−	0.978726i	$-0.434224\pi$
$967$	12.7604	0.410348	0.205174	+	0.978726i	$0.434224\pi$
$968$	29.8049	0.957966
$969$	−1.66959	−0.0536349
$970$	0	0
$971$	6.51886	0.209200	0.104600	−	0.994514i	$-0.466644\pi$
$971$	6.51886	0.209200	0.104600	+	0.994514i	$0.466644\pi$
$972$	13.5252	0.433820
$973$	−94.2215	−3.02060
$974$	8.43374	0.270234
$975$	0	0
$976$	0.710381	0.0227388
$977$	−21.9764	−0.703089	−0.351544	−	0.936171i	$-0.614343\pi$
$977$	−21.9764	−0.703089	−0.351544	+	0.936171i	$0.614343\pi$
$978$	−6.57108	−0.210120
$979$	−8.36580	−0.267372
$980$	0	0
$981$	10.1367	0.323640
$982$	−15.1140	−0.482307
$983$	9.07592	0.289477	0.144738	−	0.989470i	$-0.453766\pi$
$983$	9.07592	0.289477	0.144738	+	0.989470i	$0.453766\pi$
$984$	−23.4770	−0.748421
$985$	0	0
$986$	−2.54145	−0.0809362
$987$	54.6414	1.73925
$988$	−12.5877	−0.400469
$989$	5.97354	0.189947
$990$	0	0
$991$	19.4358	0.617400	0.308700	−	0.951159i	$-0.400106\pi$
$991$	19.4358	0.617400	0.308700	+	0.951159i	$0.400106\pi$
$992$	32.0538	1.01771
$993$	19.6719	0.624269
$994$	−37.9130	−1.20253
$995$	0	0
$996$	16.1453	0.511583
$997$	35.1276	1.11250	0.556250	−	0.831015i	$-0.312240\pi$
$997$	35.1276	1.11250	0.556250	+	0.831015i	$0.312240\pi$
$998$	29.0568	0.919777
$999$	−61.7746	−1.95446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.31		46
5.2	odd	4		1205.2.b.c.724.31	yes	46
5.3	odd	4		1205.2.b.c.724.16	✓	46
5.4	even	2	inner	6025.2.a.p.1.16		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.16	✓	46	5.3	odd	4
1205.2.b.c.724.31	yes	46	5.2	odd	4
6025.2.a.p.1.16		46	5.4	even	2	inner
6025.2.a.p.1.31		46	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.930197 q^{2} -1.32621 q^{3} -1.13473 q^{4} -1.23364 q^{6} +4.33497 q^{7} -2.91592 q^{8} -1.24116 q^{9} +O(q^{10})\)	\(q+0.930197 q^{2} -1.32621 q^{3} -1.13473 q^{4} -1.23364 q^{6} +4.33497 q^{7} -2.91592 q^{8} -1.24116 q^{9} +0.882357 q^{11} +1.50490 q^{12} -2.68232 q^{13} +4.03238 q^{14} -0.442916 q^{16} -0.304406 q^{17} -1.15452 q^{18} -4.13564 q^{19} -5.74909 q^{21} +0.820766 q^{22} +2.19664 q^{23} +3.86713 q^{24} -2.49509 q^{26} +5.62468 q^{27} -4.91903 q^{28} +8.97537 q^{29} +5.91415 q^{31} +5.41984 q^{32} -1.17019 q^{33} -0.283158 q^{34} +1.40839 q^{36} -10.9828 q^{37} -3.84696 q^{38} +3.55733 q^{39} -6.07092 q^{41} -5.34779 q^{42} +2.71940 q^{43} -1.00124 q^{44} +2.04331 q^{46} -9.50435 q^{47} +0.587400 q^{48} +11.7920 q^{49} +0.403707 q^{51} +3.04372 q^{52} -11.0767 q^{53} +5.23206 q^{54} -12.6404 q^{56} +5.48474 q^{57} +8.34887 q^{58} -7.93694 q^{59} -1.60387 q^{61} +5.50133 q^{62} -5.38040 q^{63} +5.92735 q^{64} -1.08851 q^{66} +15.5753 q^{67} +0.345420 q^{68} -2.91321 q^{69} -9.40214 q^{71} +3.61913 q^{72} +4.11094 q^{73} -10.2162 q^{74} +4.69285 q^{76} +3.82499 q^{77} +3.30902 q^{78} +9.90696 q^{79} -3.73604 q^{81} -5.64716 q^{82} +10.7285 q^{83} +6.52368 q^{84} +2.52958 q^{86} -11.9033 q^{87} -2.57288 q^{88} -9.48120 q^{89} -11.6278 q^{91} -2.49260 q^{92} -7.84342 q^{93} -8.84092 q^{94} -7.18786 q^{96} +1.41584 q^{97} +10.9689 q^{98} -1.09515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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