LMFDB - Embedded newform 6025.2.a.p.1.16

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			1.16
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	$-0.606669\pi$
$2$	−0.930197	−0.657749	−0.328874	+	0.944374i	$0.606669\pi$
$3$	1.32621	0.765689	0.382845	−	0.923813i	$-0.374944\pi$
$3$	1.32621	0.765689	0.382845	+	0.923813i	$0.374944\pi$
$4$	−1.13473	−0.567366
$5$	0	0
$5$	0	0
$6$	−1.23364	−0.503631
$7$	−4.33497	−1.63847	−0.819233	−	0.573461i	$-0.805600\pi$
$7$	−4.33497	−1.63847	−0.819233	+	0.573461i	$0.805600\pi$
$8$	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	$-0.378682\pi$
$13$	2.68232	0.743943	0.371971	+	0.928244i	$0.378682\pi$
$14$	4.03238	1.07770
$15$	0	0
$16$	−0.442916	−0.110729
$17$	0.304406	0.0738294	0.0369147	−	0.999318i	$-0.488247\pi$
$17$	0.304406	0.0738294	0.0369147	+	0.999318i	$0.488247\pi$
$18$	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	$-0.573551\pi$
$23$	−2.19664	−0.458031	−0.229015	+	0.973423i	$0.573551\pi$
$24$	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	$-0.141523\pi$
$37$	10.9828	1.80556	0.902779	+	0.430105i	$0.141523\pi$
$38$	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	$-0.566485\pi$
$43$	−2.71940	−0.414705	−0.207352	+	0.978266i	$0.566485\pi$
$44$	−1.00124	−0.150943
$45$	0	0
$46$	2.04331	0.301269
$47$	9.50435	1.38635	0.693176	−	0.720769i	$-0.256211\pi$
$47$	9.50435	1.38635	0.693176	+	0.720769i	$0.256211\pi$
$48$	−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	$-0.224832\pi$
$53$	11.0767	1.52150	0.760748	+	0.649048i	$0.224832\pi$
$54$	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	$-0.900365\pi$
$67$	−15.5753	−1.90282	−0.951410	+	0.307926i	$0.900365\pi$
$68$	−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	$-0.577336\pi$
$73$	−4.11094	−0.481150	−0.240575	+	0.970631i	$0.577336\pi$
$74$	−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	$-0.700401\pi$
$83$	−10.7285	−1.17761	−0.588803	+	0.808277i	$0.700401\pi$
$84$	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	$-0.522899\pi$
$97$	−1.41584	−0.143757	−0.0718784	+	0.997413i	$0.522899\pi$
$98$	−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	$-0.529922\pi$
$103$	−1.90525	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$104$	7.82144	0.766956
$105$	0	0
$106$	−10.3035	−1.00076
$107$	8.61469	0.832814	0.416407	−	0.909178i	$-0.363289\pi$
$107$	8.61469	0.832814	0.416407	+	0.909178i	$0.363289\pi$
$108$	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	$-0.436744\pi$
$113$	4.19717	0.394837	0.197418	+	0.980319i	$0.436744\pi$
$114$	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	$-0.570430\pi$
$127$	−4.94642	−0.438924	−0.219462	+	0.975621i	$0.570430\pi$
$128$	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	$-0.197228\pi$
$137$	19.0577	1.62821	0.814105	+	0.580717i	$0.197228\pi$
$138$	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	$-0.443990\pi$
$157$	4.38682	0.350107	0.175053	+	0.984559i	$0.443990\pi$
$158$	−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	$-0.566892\pi$
$163$	−5.32658	−0.417210	−0.208605	+	0.978000i	$0.566892\pi$
$164$	6.88888	0.537931
$165$	0	0
$166$	9.97963	0.774569
$167$	−8.59533	−0.665127	−0.332563	−	0.943081i	$-0.607914\pi$
$167$	−8.59533	−0.665127	−0.332563	+	0.943081i	$0.607914\pi$
$168$	−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	$-0.452469\pi$
$173$	3.91350	0.297538	0.148769	+	0.988872i	$0.452469\pi$
$174$	−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	$-0.423823\pi$
$193$	6.58611	0.474079	0.237039	+	0.971500i	$0.423823\pi$
$194$	1.31701	0.0945559
$195$	0	0
$196$	−13.3807	−0.955767
$197$	−3.59184	−0.255908	−0.127954	−	0.991780i	$-0.540841\pi$
$197$	−3.59184	−0.255908	−0.127954	+	0.991780i	$0.540841\pi$
$198$	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	$-0.581647\pi$
$223$	−7.57709	−0.507399	−0.253700	+	0.967283i	$0.581647\pi$
$224$	23.4949	1.56982
$225$	0	0
$226$	−3.90420	−0.259704
$227$	−9.73925	−0.646416	−0.323208	−	0.946328i	$-0.604761\pi$
$227$	−9.73925	−0.646416	−0.323208	+	0.946328i	$0.604761\pi$
$228$	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	$-0.258669\pi$
$233$	20.9912	1.37518	0.687589	+	0.726100i	$0.258669\pi$
$234$	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	$-0.654479\pi$
$257$	−14.9566	−0.932964	−0.466482	+	0.884531i	$0.654479\pi$
$258$	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	$-0.624161\pi$
$263$	−12.3331	−0.760494	−0.380247	+	0.924885i	$0.624161\pi$
$264$	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	$-0.495135\pi$
$277$	0.508682	0.0305637	0.0152819	+	0.999883i	$0.495135\pi$
$278$	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	$-0.485685\pi$
$283$	1.51258	0.0899134	0.0449567	+	0.998989i	$0.485685\pi$
$284$	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	$-0.198951\pi$
$293$	27.7625	1.62190	0.810950	+	0.585115i	$0.198951\pi$
$294$	−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	$-0.684108\pi$
$307$	−19.1572	−1.09336	−0.546679	+	0.837342i	$0.684108\pi$
$308$	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	$-0.212209\pi$
$313$	27.8074	1.57177	0.785883	+	0.618375i	$0.212209\pi$
$314$	−4.08061	−0.230282
$315$	0	0
$316$	−11.2418	−0.632398
$317$	14.6718	0.824048	0.412024	−	0.911173i	$-0.364822\pi$
$317$	14.6718	0.824048	0.412024	+	0.911173i	$0.364822\pi$
$318$	−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	$-0.534054\pi$
$337$	−3.92047	−0.213562	−0.106781	+	0.994283i	$0.534054\pi$
$338$	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	$-0.516133\pi$
$347$	−1.88740	−0.101321	−0.0506606	+	0.998716i	$0.516133\pi$
$348$	−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	$-0.686942\pi$
$353$	−20.8216	−1.10822	−0.554111	+	0.832443i	$0.686942\pi$
$354$	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	$-0.0382613\pi$
$367$	38.0380	1.98557	0.992785	+	0.119912i	$0.0382613\pi$
$368$	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	$-0.366835\pi$
$373$	15.6921	0.812505	0.406253	+	0.913761i	$0.366835\pi$
$374$	−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	$-0.655724\pi$
$383$	−18.3938	−0.939878	−0.469939	+	0.882699i	$0.655724\pi$
$384$	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	$-0.537430\pi$
$397$	−4.67514	−0.234639	−0.117319	+	0.993094i	$0.537430\pi$
$398$	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	$-0.582624\pi$
$433$	−10.6818	−0.513335	−0.256667	+	0.966500i	$0.582624\pi$
$434$	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	$-0.687095\pi$
$443$	−23.3422	−1.10902	−0.554512	+	0.832176i	$0.687095\pi$
$444$	−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	$-0.668638\pi$
$457$	−21.6065	−1.01071	−0.505355	+	0.862912i	$0.668638\pi$
$458$	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	$-0.376548\pi$
$463$	16.2752	0.756375	0.378187	+	0.925729i	$0.376548\pi$
$464$	−3.97534	−0.184550
$465$	0	0
$466$	−19.5260	−0.904522
$467$	−6.86128	−0.317502	−0.158751	−	0.987319i	$-0.550747\pi$
$467$	−6.86128	−0.317502	−0.158751	+	0.987319i	$0.550747\pi$
$468$	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	$-0.565857\pi$
$487$	−9.06661	−0.410847	−0.205424	+	0.978673i	$0.565857\pi$
$488$	−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	$-0.202839\pi$
$503$	36.0521	1.60749	0.803743	+	0.594977i	$0.202839\pi$
$504$	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	$-0.672440\pi$
$523$	−23.5838	−1.03125	−0.515625	+	0.856815i	$0.672440\pi$
$524$	−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	$-0.585172\pi$
$547$	−12.3673	−0.528786	−0.264393	+	0.964415i	$0.585172\pi$
$548$	−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	$-0.400809\pi$
$557$	14.4719	0.613195	0.306598	+	0.951839i	$0.400809\pi$
$558$	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	$-0.481763\pi$
$563$	2.71732	0.114521	0.0572607	+	0.998359i	$0.481763\pi$
$564$	−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	$-0.543675\pi$
$577$	−6.57108	−0.273558	−0.136779	+	0.990602i	$0.543675\pi$
$578$	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	$-0.479108\pi$
$587$	3.17806	0.131173	0.0655863	+	0.997847i	$0.479108\pi$
$588$	−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	$-0.755658\pi$
$593$	−35.0450	−1.43913	−0.719563	+	0.694428i	$0.755658\pi$
$594$	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	$-0.141213\pi$
$607$	44.5048	1.80639	0.903197	+	0.429226i	$0.141213\pi$
$608$	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	$-0.901975\pi$
$613$	−47.1882	−1.90591	−0.952956	+	0.303109i	$0.901975\pi$
$614$	17.8200	0.719155
$615$	0	0
$616$	−11.1534	−0.449382
$617$	3.11780	0.125518	0.0627590	−	0.998029i	$-0.480010\pi$
$617$	3.11780	0.125518	0.0627590	+	0.998029i	$0.480010\pi$
$618$	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	$-0.508585\pi$
$643$	−1.36761	−0.0539332	−0.0269666	+	0.999636i	$0.508585\pi$
$644$	−10.8053	−0.425790
$645$	0	0
$646$	1.17104	0.0460739
$647$	−47.9634	−1.88563	−0.942817	−	0.333310i	$-0.891834\pi$
$647$	−47.9634	−1.88563	−0.942817	+	0.333310i	$0.891834\pi$
$648$	−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	$-0.620981\pi$
$653$	−18.9603	−0.741975	−0.370987	+	0.928638i	$0.620981\pi$
$654$	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	$-0.564600\pi$
$673$	−10.4577	−0.403113	−0.201557	+	0.979477i	$0.564600\pi$
$674$	3.64681	0.140470
$675$	0	0
$676$	6.58728	0.253357
$677$	41.7207	1.60346	0.801728	−	0.597689i	$-0.203914\pi$
$677$	41.7207	1.60346	0.801728	+	0.597689i	$0.203914\pi$
$678$	−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	$-0.716789\pi$
$683$	−32.9092	−1.25924	−0.629618	+	0.776905i	$0.716789\pi$
$684$	−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	$-0.276859\pi$
$727$	34.7819	1.28999	0.644995	+	0.764187i	$0.276859\pi$
$728$	−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	$-0.726048\pi$
$733$	−35.3016	−1.30389	−0.651947	+	0.758265i	$0.726048\pi$
$734$	−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	$-0.322490\pi$
$743$	28.8502	1.05841	0.529206	+	0.848494i	$0.322490\pi$
$744$	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	$-0.718502\pi$
$757$	−34.8758	−1.26758	−0.633790	+	0.773505i	$0.718502\pi$
$758$	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	$-0.668173\pi$
$773$	−28.0305	−1.00819	−0.504093	+	0.863649i	$0.668173\pi$
$774$	−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	$-0.417134\pi$
$787$	14.4420	0.514801	0.257400	+	0.966305i	$0.417134\pi$
$788$	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	$-0.562245\pi$
$797$	−10.9710	−0.388612	−0.194306	+	0.980941i	$0.562245\pi$
$798$	−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	$-0.709300\pi$
$823$	−35.0663	−1.22233	−0.611167	+	0.791502i	$0.709300\pi$
$824$	−5.55555	−0.193537
$825$	0	0
$826$	−32.0047	−1.11359
$827$	5.28536	0.183790	0.0918949	−	0.995769i	$-0.470708\pi$
$827$	5.28536	0.183790	0.0918949	+	0.995769i	$0.470708\pi$
$828$	−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	$-0.441015\pi$
$853$	10.7624	0.368497	0.184249	+	0.982880i	$0.441015\pi$
$854$	−6.46743	−0.221311
$855$	0	0
$856$	25.1197	0.858575
$857$	−21.5155	−0.734956	−0.367478	−	0.930032i	$-0.619779\pi$
$857$	−21.5155	−0.734956	−0.367478	+	0.930032i	$0.619779\pi$
$858$	−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	$-0.555923\pi$
$863$	−10.2693	−0.349571	−0.174785	+	0.984607i	$0.555923\pi$
$864$	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	$-0.531415\pi$
$877$	−5.83603	−0.197069	−0.0985343	+	0.995134i	$0.531415\pi$
$878$	−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	$-0.382323\pi$
$883$	21.4740	0.722658	0.361329	+	0.932438i	$0.382323\pi$
$884$	−0.926528	−0.0311625
$885$	0	0
$886$	21.7129	0.729459
$887$	19.7851	0.664320	0.332160	−	0.943223i	$-0.392223\pi$
$887$	19.7851	0.664320	0.332160	+	0.943223i	$0.392223\pi$
$888$	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	$-0.502357\pi$
$907$	−0.445947	−0.0148074	−0.00740371	+	0.999973i	$0.502357\pi$
$908$	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	$-0.225199\pi$
$937$	46.5278	1.52000	0.759999	+	0.649924i	$0.225199\pi$
$938$	−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	$-0.383531\pi$
$947$	22.0207	0.715576	0.357788	+	0.933803i	$0.383531\pi$
$948$	−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	$-0.201209\pi$
$953$	49.8117	1.61356	0.806779	+	0.590853i	$0.201209\pi$
$954$	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	$-0.565776\pi$
$967$	−12.7604	−0.410348	−0.205174	+	0.978726i	$0.565776\pi$
$968$	−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	$-0.385657\pi$
$977$	21.9764	0.703089	0.351544	+	0.936171i	$0.385657\pi$
$978$	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	$-0.546234\pi$
$983$	−9.07592	−0.289477	−0.144738	+	0.989470i	$0.546234\pi$
$984$	−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	$-0.687760\pi$
$997$	−35.1276	−1.11250	−0.556250	+	0.831015i	$0.687760\pi$
$998$	−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.1.16		46
5.2	odd	4		1205.2.b.c.724.16	✓	46
5.3	odd	4		1205.2.b.c.724.31	yes	46
5.4	even	2	inner	6025.2.a.p.1.31		46

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

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