LMFDB - Embedded newform 6025.2.a.q.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.33352 q^{2} +1.59052 q^{3} +3.44531 q^{4} -3.71151 q^{6} -3.91727 q^{7} -3.37267 q^{8} -0.470249 q^{9} +O(q^{10})$	\(q-2.33352 q^{2} +1.59052 q^{3} +3.44531 q^{4} -3.71151 q^{6} -3.91727 q^{7} -3.37267 q^{8} -0.470249 q^{9} +0.470102 q^{11} +5.47984 q^{12} +6.58515 q^{13} +9.14102 q^{14} +0.979554 q^{16} +4.71018 q^{17} +1.09734 q^{18} +5.29136 q^{19} -6.23049 q^{21} -1.09699 q^{22} -3.10917 q^{23} -5.36429 q^{24} -15.3666 q^{26} -5.51950 q^{27} -13.4962 q^{28} +8.03181 q^{29} +2.48847 q^{31} +4.45952 q^{32} +0.747705 q^{33} -10.9913 q^{34} -1.62015 q^{36} -6.41257 q^{37} -12.3475 q^{38} +10.4738 q^{39} +4.02545 q^{41} +14.5390 q^{42} +6.67474 q^{43} +1.61965 q^{44} +7.25531 q^{46} -0.431389 q^{47} +1.55800 q^{48} +8.34498 q^{49} +7.49164 q^{51} +22.6879 q^{52} -13.5727 q^{53} +12.8799 q^{54} +13.2116 q^{56} +8.41601 q^{57} -18.7424 q^{58} +5.05909 q^{59} +9.90032 q^{61} -5.80688 q^{62} +1.84209 q^{63} -12.3655 q^{64} -1.74479 q^{66} -7.78824 q^{67} +16.2281 q^{68} -4.94519 q^{69} +7.90125 q^{71} +1.58599 q^{72} -11.9926 q^{73} +14.9639 q^{74} +18.2304 q^{76} -1.84151 q^{77} -24.4408 q^{78} -5.95673 q^{79} -7.36812 q^{81} -9.39346 q^{82} -2.79875 q^{83} -21.4660 q^{84} -15.5756 q^{86} +12.7748 q^{87} -1.58550 q^{88} +1.23195 q^{89} -25.7958 q^{91} -10.7121 q^{92} +3.95795 q^{93} +1.00665 q^{94} +7.09295 q^{96} -7.82518 q^{97} -19.4732 q^{98} -0.221065 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.33352	−1.65005	−0.825024	−	0.565098i	$-0.808838\pi$
$2$	−2.33352	−1.65005	−0.825024	+	0.565098i	$0.808838\pi$
$3$	1.59052	0.918287	0.459143	−	0.888362i	$-0.348156\pi$
$3$	1.59052	0.918287	0.459143	+	0.888362i	$0.348156\pi$
$4$	3.44531	1.72266
$5$	0	0
$5$	0	0
$6$	−3.71151	−1.51522
$7$	−3.91727	−1.48059	−0.740294	−	0.672283i	$-0.765314\pi$
$7$	−3.91727	−1.48059	−0.740294	+	0.672283i	$0.765314\pi$
$8$	−3.37267	−1.19242
$9$	−0.470249	−0.156750
$10$	0	0
$11$	0.470102	0.141741	0.0708705	−	0.997486i	$-0.477422\pi$
$11$	0.470102	0.141741	0.0708705	+	0.997486i	$0.477422\pi$
$12$	5.47984	1.58189
$13$	6.58515	1.82639	0.913196	−	0.407521i	$-0.133607\pi$
$13$	6.58515	1.82639	0.913196	+	0.407521i	$0.133607\pi$
$14$	9.14102	2.44304
$15$	0	0
$16$	0.979554	0.244889
$17$	4.71018	1.14239	0.571194	−	0.820815i	$-0.306480\pi$
$17$	4.71018	1.14239	0.571194	+	0.820815i	$0.306480\pi$
$18$	1.09734	0.258644
$19$	5.29136	1.21392	0.606961	−	0.794732i	$-0.292388\pi$
$19$	5.29136	1.21392	0.606961	+	0.794732i	$0.292388\pi$
$20$	0	0
$21$	−6.23049	−1.35960
$22$	−1.09699	−0.233879
$23$	−3.10917	−0.648307	−0.324153	−	0.946005i	$-0.605079\pi$
$23$	−3.10917	−0.648307	−0.324153	+	0.946005i	$0.605079\pi$
$24$	−5.36429	−1.09498
$25$	0	0
$26$	−15.3666	−3.01363
$27$	−5.51950	−1.06223
$28$	−13.4962	−2.55054
$29$	8.03181	1.49147	0.745735	−	0.666243i	$-0.232098\pi$
$29$	8.03181	1.49147	0.745735	+	0.666243i	$0.232098\pi$
$30$	0	0
$31$	2.48847	0.446942	0.223471	−	0.974711i	$-0.428261\pi$
$31$	2.48847	0.446942	0.223471	+	0.974711i	$0.428261\pi$
$32$	4.45952	0.788339
$33$	0.747705	0.130159
$34$	−10.9913	−1.88499
$35$	0	0
$36$	−1.62015	−0.270026
$37$	−6.41257	−1.05422	−0.527110	−	0.849797i	$-0.676724\pi$
$37$	−6.41257	−1.05422	−0.527110	+	0.849797i	$0.676724\pi$
$38$	−12.3475	−2.00303
$39$	10.4738	1.67715
$40$	0	0
$41$	4.02545	0.628669	0.314335	−	0.949312i	$-0.398219\pi$
$41$	4.02545	0.628669	0.314335	+	0.949312i	$0.398219\pi$
$42$	14.5390	2.24341
$43$	6.67474	1.01789	0.508944	−	0.860799i	$-0.330036\pi$
$43$	6.67474	1.01789	0.508944	+	0.860799i	$0.330036\pi$
$44$	1.61965	0.244171
$45$	0	0
$46$	7.25531	1.06974
$47$	−0.431389	−0.0629245	−0.0314623	−	0.999505i	$-0.510016\pi$
$47$	−0.431389	−0.0629245	−0.0314623	+	0.999505i	$0.510016\pi$
$48$	1.55800	0.224878
$49$	8.34498	1.19214
$50$	0	0
$51$	7.49164	1.04904
$52$	22.6879	3.14625
$53$	−13.5727	−1.86435	−0.932175	−	0.362007i	$-0.882092\pi$
$53$	−13.5727	−1.86435	−0.932175	+	0.362007i	$0.882092\pi$
$54$	12.8799	1.75273
$55$	0	0
$56$	13.2116	1.76548
$57$	8.41601	1.11473
$58$	−18.7424	−2.46100
$59$	5.05909	0.658637	0.329319	−	0.944219i	$-0.393181\pi$
$59$	5.05909	0.658637	0.329319	+	0.944219i	$0.393181\pi$
$60$	0	0
$61$	9.90032	1.26761	0.633803	−	0.773495i	$-0.281493\pi$
$61$	9.90032	1.26761	0.633803	+	0.773495i	$0.281493\pi$
$62$	−5.80688	−0.737475
$63$	1.84209	0.232082
$64$	−12.3655	−1.54569
$65$	0	0
$66$	−1.74479	−0.214768
$67$	−7.78824	−0.951485	−0.475743	−	0.879585i	$-0.657821\pi$
$67$	−7.78824	−0.951485	−0.475743	+	0.879585i	$0.657821\pi$
$68$	16.2281	1.96794
$69$	−4.94519	−0.595331
$70$	0	0
$71$	7.90125	0.937705	0.468853	−	0.883276i	$-0.344668\pi$
$71$	7.90125	0.937705	0.468853	+	0.883276i	$0.344668\pi$
$72$	1.58599	0.186911
$73$	−11.9926	−1.40363	−0.701817	−	0.712358i	$-0.747627\pi$
$73$	−11.9926	−1.40363	−0.701817	+	0.712358i	$0.747627\pi$
$74$	14.9639	1.73951
$75$	0	0
$76$	18.2304	2.09117
$77$	−1.84151	−0.209860
$78$	−24.4408	−2.76738
$79$	−5.95673	−0.670185	−0.335092	−	0.942185i	$-0.608768\pi$
$79$	−5.95673	−0.670185	−0.335092	+	0.942185i	$0.608768\pi$
$80$	0	0
$81$	−7.36812	−0.818680
$82$	−9.39346	−1.03733
$83$	−2.79875	−0.307202	−0.153601	−	0.988133i	$-0.549087\pi$
$83$	−2.79875	−0.307202	−0.153601	+	0.988133i	$0.549087\pi$
$84$	−21.4660	−2.34213
$85$	0	0
$86$	−15.5756	−1.67956
$87$	12.7748	1.36960
$88$	−1.58550	−0.169014
$89$	1.23195	0.130586	0.0652930	−	0.997866i	$-0.479202\pi$
$89$	1.23195	0.130586	0.0652930	+	0.997866i	$0.479202\pi$
$90$	0	0
$91$	−25.7958	−2.70413
$92$	−10.7121	−1.11681
$93$	3.95795	0.410421
$94$	1.00665	0.103828
$95$	0	0
$96$	7.09295	0.723922
$97$	−7.82518	−0.794527	−0.397263	−	0.917705i	$-0.630040\pi$
$97$	−7.82518	−0.794527	−0.397263	+	0.917705i	$0.630040\pi$
$98$	−19.4732	−1.96709
$99$	−0.221065	−0.0222178
$100$	0	0
$101$	−15.2668	−1.51910	−0.759551	−	0.650448i	$-0.774581\pi$
$101$	−15.2668	−1.51910	−0.759551	+	0.650448i	$0.774581\pi$
$102$	−17.4819	−1.73096
$103$	−0.895158	−0.0882025	−0.0441012	−	0.999027i	$-0.514042\pi$
$103$	−0.895158	−0.0882025	−0.0441012	+	0.999027i	$0.514042\pi$
$104$	−22.2095	−2.17782
$105$	0	0
$106$	31.6721	3.07627
$107$	1.57330	0.152097	0.0760483	−	0.997104i	$-0.475770\pi$
$107$	1.57330	0.152097	0.0760483	+	0.997104i	$0.475770\pi$
$108$	−19.0164	−1.82985
$109$	5.24868	0.502732	0.251366	−	0.967892i	$-0.419120\pi$
$109$	5.24868	0.502732	0.251366	+	0.967892i	$0.419120\pi$
$110$	0	0
$111$	−10.1993	−0.968076
$112$	−3.83718	−0.362579
$113$	19.6197	1.84567	0.922834	−	0.385197i	$-0.125867\pi$
$113$	19.6197	1.84567	0.922834	+	0.385197i	$0.125867\pi$
$114$	−19.6389	−1.83935
$115$	0	0
$116$	27.6721	2.56929
$117$	−3.09666	−0.286286
$118$	−11.8055	−1.08678
$119$	−18.4510	−1.69140
$120$	0	0
$121$	−10.7790	−0.979910
$122$	−23.1026	−2.09161
$123$	6.40255	0.577298
$124$	8.57355	0.769927
$125$	0	0
$126$	−4.29855	−0.382946
$127$	9.78587	0.868355	0.434178	−	0.900827i	$-0.357039\pi$
$127$	9.78587	0.868355	0.434178	+	0.900827i	$0.357039\pi$
$128$	19.9361	1.76212
$129$	10.6163	0.934714
$130$	0	0
$131$	−5.45842	−0.476904	−0.238452	−	0.971154i	$-0.576640\pi$
$131$	−5.45842	−0.476904	−0.238452	+	0.971154i	$0.576640\pi$
$132$	2.57608	0.224219
$133$	−20.7277	−1.79732
$134$	18.1740	1.57000
$135$	0	0
$136$	−15.8859	−1.36220
$137$	−13.4327	−1.14763	−0.573817	−	0.818984i	$-0.694538\pi$
$137$	−13.4327	−1.14763	−0.573817	+	0.818984i	$0.694538\pi$
$138$	11.5397	0.982325
$139$	−18.1503	−1.53949	−0.769744	−	0.638352i	$-0.779616\pi$
$139$	−18.1503	−1.53949	−0.769744	+	0.638352i	$0.779616\pi$
$140$	0	0
$141$	−0.686132	−0.0577827
$142$	−18.4377	−1.54726
$143$	3.09569	0.258874
$144$	−0.460634	−0.0383862
$145$	0	0
$146$	27.9851	2.31606
$147$	13.2728	1.09473
$148$	−22.0933	−1.81606
$149$	17.0894	1.40002	0.700010	−	0.714133i	$-0.253179\pi$
$149$	17.0894	1.40002	0.700010	+	0.714133i	$0.253179\pi$
$150$	0	0
$151$	16.7927	1.36657	0.683283	−	0.730153i	$-0.260551\pi$
$151$	16.7927	1.36657	0.683283	+	0.730153i	$0.260551\pi$
$152$	−17.8460	−1.44750
$153$	−2.21496	−0.179069
$154$	4.29721	0.346279
$155$	0	0
$156$	36.0855	2.88915
$157$	−4.23355	−0.337874	−0.168937	−	0.985627i	$-0.554033\pi$
$157$	−4.23355	−0.337874	−0.168937	+	0.985627i	$0.554033\pi$
$158$	13.9001	1.10584
$159$	−21.5876	−1.71201
$160$	0	0
$161$	12.1794	0.959875
$162$	17.1936	1.35086
$163$	22.0283	1.72539	0.862695	−	0.505725i	$-0.168775\pi$
$163$	22.0283	1.72539	0.862695	+	0.505725i	$0.168775\pi$
$164$	13.8689	1.08298
$165$	0	0
$166$	6.53093	0.506898
$167$	9.58474	0.741689	0.370845	−	0.928695i	$-0.379068\pi$
$167$	9.58474	0.741689	0.370845	+	0.928695i	$0.379068\pi$
$168$	21.0133	1.62122
$169$	30.3642	2.33571
$170$	0	0
$171$	−2.48826	−0.190282
$172$	22.9966	1.75347
$173$	12.3388	0.938100	0.469050	−	0.883171i	$-0.344596\pi$
$173$	12.3388	0.938100	0.469050	+	0.883171i	$0.344596\pi$
$174$	−29.8101	−2.25990
$175$	0	0
$176$	0.460490	0.0347107
$177$	8.04658	0.604818
$178$	−2.87477	−0.215473
$179$	0.704547	0.0526603	0.0263302	−	0.999653i	$-0.491618\pi$
$179$	0.704547	0.0526603	0.0263302	+	0.999653i	$0.491618\pi$
$180$	0	0
$181$	−12.6190	−0.937965	−0.468982	−	0.883207i	$-0.655379\pi$
$181$	−12.6190	−0.937965	−0.468982	+	0.883207i	$0.655379\pi$
$182$	60.1950	4.46195
$183$	15.7466	1.16403
$184$	10.4862	0.773052
$185$	0	0
$186$	−9.23596	−0.677214
$187$	2.21426	0.161923
$188$	−1.48627	−0.108397
$189$	21.6213	1.57272
$190$	0	0
$191$	16.1302	1.16714	0.583572	−	0.812062i	$-0.301655\pi$
$191$	16.1302	1.16714	0.583572	+	0.812062i	$0.301655\pi$
$192$	−19.6675	−1.41938
$193$	−7.38527	−0.531603	−0.265802	−	0.964028i	$-0.585637\pi$
$193$	−7.38527	−0.531603	−0.265802	+	0.964028i	$0.585637\pi$
$194$	18.2602	1.31101
$195$	0	0
$196$	28.7511	2.05365
$197$	1.92571	0.137201	0.0686006	−	0.997644i	$-0.478147\pi$
$197$	1.92571	0.137201	0.0686006	+	0.997644i	$0.478147\pi$
$198$	0.515859	0.0366605
$199$	5.80835	0.411743	0.205871	−	0.978579i	$-0.433997\pi$
$199$	5.80835	0.411743	0.205871	+	0.978579i	$0.433997\pi$
$200$	0	0
$201$	−12.3873	−0.873736
$202$	35.6253	2.50659
$203$	−31.4628	−2.20825
$204$	25.8110	1.80713
$205$	0	0
$206$	2.08887	0.145538
$207$	1.46208	0.101622
$208$	6.45051	0.447262
$209$	2.48748	0.172062
$210$	0	0
$211$	22.7032	1.56295	0.781475	−	0.623936i	$-0.214467\pi$
$211$	22.7032	1.56295	0.781475	+	0.623936i	$0.214467\pi$
$212$	−46.7621	−3.21164
$213$	12.5671	0.861082
$214$	−3.67132	−0.250966
$215$	0	0
$216$	18.6154	1.26662
$217$	−9.74799	−0.661736
$218$	−12.2479	−0.829532
$219$	−19.0745	−1.28894
$220$	0	0
$221$	31.0173	2.08645
$222$	23.8003	1.59737
$223$	26.9690	1.80598	0.902988	−	0.429665i	$-0.141368\pi$
$223$	26.9690	1.80598	0.902988	+	0.429665i	$0.141368\pi$
$224$	−17.4691	−1.16721
$225$	0	0
$226$	−45.7830	−3.04544
$227$	−1.56332	−0.103761	−0.0518807	−	0.998653i	$-0.516522\pi$
$227$	−1.56332	−0.103761	−0.0518807	+	0.998653i	$0.516522\pi$
$228$	28.9958	1.92029
$229$	11.7583	0.777009	0.388505	−	0.921447i	$-0.372992\pi$
$229$	11.7583	0.777009	0.388505	+	0.921447i	$0.372992\pi$
$230$	0	0
$231$	−2.92896	−0.192712
$232$	−27.0886	−1.77845
$233$	3.15348	0.206592	0.103296	−	0.994651i	$-0.467061\pi$
$233$	3.15348	0.206592	0.103296	+	0.994651i	$0.467061\pi$
$234$	7.22611	0.472386
$235$	0	0
$236$	17.4301	1.13461
$237$	−9.47429	−0.615422
$238$	43.0559	2.79090
$239$	21.0101	1.35903	0.679516	−	0.733661i	$-0.262190\pi$
$239$	21.0101	1.35903	0.679516	+	0.733661i	$0.262190\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	25.1530	1.61690
$243$	4.83936	0.310445
$244$	34.1097	2.18365
$245$	0	0
$246$	−14.9405	−0.952570
$247$	34.8444	2.21710
$248$	−8.39276	−0.532941
$249$	−4.45146	−0.282100
$250$	0	0
$251$	−8.98642	−0.567218	−0.283609	−	0.958940i	$-0.591532\pi$
$251$	−8.98642	−0.567218	−0.283609	+	0.958940i	$0.591532\pi$
$252$	6.34658	0.399797
$253$	−1.46163	−0.0918916
$254$	−22.8355	−1.43283
$255$	0	0
$256$	−21.7902	−1.36189
$257$	−10.2473	−0.639207	−0.319604	−	0.947551i	$-0.603550\pi$
$257$	−10.2473	−0.639207	−0.319604	+	0.947551i	$0.603550\pi$
$258$	−24.7734	−1.54232
$259$	25.1197	1.56086
$260$	0	0
$261$	−3.77695	−0.233787
$262$	12.7373	0.786915
$263$	1.27093	0.0783692	0.0391846	−	0.999232i	$-0.487524\pi$
$263$	1.27093	0.0783692	0.0391846	+	0.999232i	$0.487524\pi$
$264$	−2.52176	−0.155204
$265$	0	0
$266$	48.3684	2.96566
$267$	1.95943	0.119915
$268$	−26.8329	−1.63908
$269$	−29.4318	−1.79449	−0.897244	−	0.441535i	$-0.854434\pi$
$269$	−29.4318	−1.79449	−0.897244	+	0.441535i	$0.854434\pi$
$270$	0	0
$271$	−8.30266	−0.504351	−0.252175	−	0.967682i	$-0.581146\pi$
$271$	−8.30266	−0.504351	−0.252175	+	0.967682i	$0.581146\pi$
$272$	4.61388	0.279758
$273$	−41.0287	−2.48317
$274$	31.3455	1.89365
$275$	0	0
$276$	−17.0377	−1.02555
$277$	−1.24737	−0.0749474	−0.0374737	−	0.999298i	$-0.511931\pi$
$277$	−1.24737	−0.0749474	−0.0374737	+	0.999298i	$0.511931\pi$
$278$	42.3541	2.54023
$279$	−1.17020	−0.0700580
$280$	0	0
$281$	15.8283	0.944238	0.472119	−	0.881535i	$-0.343489\pi$
$281$	15.8283	0.944238	0.472119	+	0.881535i	$0.343489\pi$
$282$	1.60110	0.0953442
$283$	14.0158	0.833150	0.416575	−	0.909101i	$-0.363230\pi$
$283$	14.0158	0.833150	0.416575	+	0.909101i	$0.363230\pi$
$284$	27.2223	1.61534
$285$	0	0
$286$	−7.22385	−0.427155
$287$	−15.7687	−0.930800
$288$	−2.09709	−0.123572
$289$	5.18584	0.305049
$290$	0	0
$291$	−12.4461	−0.729603
$292$	−41.3184	−2.41798
$293$	18.4033	1.07513	0.537566	−	0.843222i	$-0.319344\pi$
$293$	18.4033	1.07513	0.537566	+	0.843222i	$0.319344\pi$
$294$	−30.9724	−1.80635
$295$	0	0
$296$	21.6274	1.25707
$297$	−2.59472	−0.150561
$298$	−39.8785	−2.31010
$299$	−20.4743	−1.18406
$300$	0	0
$301$	−26.1467	−1.50707
$302$	−39.1860	−2.25490
$303$	−24.2821	−1.39497
$304$	5.18318	0.297276
$305$	0	0
$306$	5.16865	0.295472
$307$	−18.1605	−1.03647	−0.518237	−	0.855237i	$-0.673411\pi$
$307$	−18.1605	−1.03647	−0.518237	+	0.855237i	$0.673411\pi$
$308$	−6.34459	−0.361516
$309$	−1.42377	−0.0809952
$310$	0	0
$311$	26.5122	1.50337	0.751684	−	0.659524i	$-0.229242\pi$
$311$	26.5122	1.50337	0.751684	+	0.659524i	$0.229242\pi$
$312$	−35.3246	−1.99986
$313$	−13.4467	−0.760050	−0.380025	−	0.924976i	$-0.624085\pi$
$313$	−13.4467	−0.760050	−0.380025	+	0.924976i	$0.624085\pi$
$314$	9.87906	0.557508
$315$	0	0
$316$	−20.5228	−1.15450
$317$	−0.363420	−0.0204117	−0.0102058	−	0.999948i	$-0.503249\pi$
$317$	−0.363420	−0.0204117	−0.0102058	+	0.999948i	$0.503249\pi$
$318$	50.3751	2.82490
$319$	3.77577	0.211402
$320$	0	0
$321$	2.50236	0.139668
$322$	−28.4210	−1.58384
$323$	24.9233	1.38677
$324$	−25.3855	−1.41030
$325$	0	0
$326$	−51.4035	−2.84698
$327$	8.34813	0.461652
$328$	−13.5765	−0.749636
$329$	1.68986	0.0931652
$330$	0	0
$331$	33.6960	1.85210	0.926050	−	0.377400i	$-0.123182\pi$
$331$	33.6960	1.85210	0.926050	+	0.377400i	$0.123182\pi$
$332$	−9.64256	−0.529204
$333$	3.01550	0.165249
$334$	−22.3662	−1.22382
$335$	0	0
$336$	−6.10310	−0.332951
$337$	−7.43800	−0.405174	−0.202587	−	0.979264i	$-0.564935\pi$
$337$	−7.43800	−0.405174	−0.202587	+	0.979264i	$0.564935\pi$
$338$	−70.8554	−3.85403
$339$	31.2055	1.69485
$340$	0	0
$341$	1.16983	0.0633499
$342$	5.80640	0.313974
$343$	−5.26864	−0.284480
$344$	−22.5117	−1.21375
$345$	0	0
$346$	−28.7928	−1.54791
$347$	10.7653	0.577913	0.288956	−	0.957342i	$-0.406692\pi$
$347$	10.7653	0.577913	0.288956	+	0.957342i	$0.406692\pi$
$348$	44.0130	2.35935
$349$	−33.3506	−1.78521	−0.892607	−	0.450835i	$-0.851126\pi$
$349$	−33.3506	−1.78521	−0.892607	+	0.450835i	$0.851126\pi$
$350$	0	0
$351$	−36.3467	−1.94004
$352$	2.09643	0.111740
$353$	22.2683	1.18522	0.592611	−	0.805489i	$-0.298097\pi$
$353$	22.2683	1.18522	0.592611	+	0.805489i	$0.298097\pi$
$354$	−18.7768	−0.997978
$355$	0	0
$356$	4.24444	0.224955
$357$	−29.3467	−1.55319
$358$	−1.64408	−0.0868921
$359$	21.8499	1.15319	0.576596	−	0.817030i	$-0.304381\pi$
$359$	21.8499	1.15319	0.576596	+	0.817030i	$0.304381\pi$
$360$	0	0
$361$	8.99851	0.473606
$362$	29.4467	1.54769
$363$	−17.1442	−0.899838
$364$	−88.8745	−4.65829
$365$	0	0
$366$	−36.7451	−1.92070
$367$	28.6234	1.49413	0.747066	−	0.664750i	$-0.231462\pi$
$367$	28.6234	1.49413	0.747066	+	0.664750i	$0.231462\pi$
$368$	−3.04560	−0.158763
$369$	−1.89296	−0.0985437
$370$	0	0
$371$	53.1678	2.76034
$372$	13.6364	0.707014
$373$	8.59652	0.445111	0.222555	−	0.974920i	$-0.428560\pi$
$373$	8.59652	0.445111	0.222555	+	0.974920i	$0.428560\pi$
$374$	−5.16703	−0.267181
$375$	0	0
$376$	1.45493	0.0750323
$377$	52.8907	2.72401
$378$	−50.4538	−2.59506
$379$	10.5322	0.541003	0.270501	−	0.962720i	$-0.412811\pi$
$379$	10.5322	0.541003	0.270501	+	0.962720i	$0.412811\pi$
$380$	0	0
$381$	15.5646	0.797399
$382$	−37.6402	−1.92584
$383$	20.5706	1.05111	0.525553	−	0.850761i	$-0.323858\pi$
$383$	20.5706	1.05111	0.525553	+	0.850761i	$0.323858\pi$
$384$	31.7087	1.61813
$385$	0	0
$386$	17.2337	0.877170
$387$	−3.13879	−0.159554
$388$	−26.9602	−1.36870
$389$	−5.11767	−0.259476	−0.129738	−	0.991548i	$-0.541414\pi$
$389$	−5.11767	−0.259476	−0.129738	+	0.991548i	$0.541414\pi$
$390$	0	0
$391$	−14.6448	−0.740618
$392$	−28.1448	−1.42153
$393$	−8.68172	−0.437935
$394$	−4.49368	−0.226388
$395$	0	0
$396$	−0.761637	−0.0382737
$397$	−34.5170	−1.73236	−0.866179	−	0.499734i	$-0.833431\pi$
$397$	−34.5170	−1.73236	−0.866179	+	0.499734i	$0.833431\pi$
$398$	−13.5539	−0.679395
$399$	−32.9678	−1.65045
$400$	0	0
$401$	−14.3533	−0.716768	−0.358384	−	0.933574i	$-0.616672\pi$
$401$	−14.3533	−0.716768	−0.358384	+	0.933574i	$0.616672\pi$
$402$	28.9061	1.44171
$403$	16.3869	0.816291
$404$	−52.5989	−2.61689
$405$	0	0
$406$	73.4189	3.64372
$407$	−3.01456	−0.149426
$408$	−25.2668	−1.25089
$409$	11.4961	0.568448	0.284224	−	0.958758i	$-0.408264\pi$
$409$	11.4961	0.568448	0.284224	+	0.958758i	$0.408264\pi$
$410$	0	0
$411$	−21.3650	−1.05386
$412$	−3.08410	−0.151943
$413$	−19.8178	−0.975170
$414$	−3.41180	−0.167681
$415$	0	0
$416$	29.3666	1.43982
$417$	−28.8684	−1.41369
$418$	−5.80458	−0.283911
$419$	6.37230	0.311307	0.155654	−	0.987812i	$-0.450252\pi$
$419$	6.37230	0.311307	0.155654	+	0.987812i	$0.450252\pi$
$420$	0	0
$421$	−4.55939	−0.222211	−0.111105	−	0.993809i	$-0.535439\pi$
$421$	−4.55939	−0.222211	−0.111105	+	0.993809i	$0.535439\pi$
$422$	−52.9783	−2.57894
$423$	0.202860	0.00986339
$424$	45.7761	2.22308
$425$	0	0
$426$	−29.3255	−1.42083
$427$	−38.7822	−1.87680
$428$	5.42051	0.262010
$429$	4.92375	0.237721
$430$	0	0
$431$	18.0127	0.867642	0.433821	−	0.900999i	$-0.357165\pi$
$431$	18.0127	0.867642	0.433821	+	0.900999i	$0.357165\pi$
$432$	−5.40665	−0.260127
$433$	−21.8765	−1.05132	−0.525658	−	0.850696i	$-0.676181\pi$
$433$	−21.8765	−1.05132	−0.525658	+	0.850696i	$0.676181\pi$
$434$	22.7471	1.09190
$435$	0	0
$436$	18.0833	0.866035
$437$	−16.4517	−0.786994
$438$	44.5108	2.12681
$439$	−28.5395	−1.36212	−0.681058	−	0.732229i	$-0.738480\pi$
$439$	−28.5395	−1.36212	−0.681058	+	0.732229i	$0.738480\pi$
$440$	0	0
$441$	−3.92422	−0.186868
$442$	−72.3794	−3.44274
$443$	11.9389	0.567234	0.283617	−	0.958938i	$-0.408466\pi$
$443$	11.9389	0.567234	0.283617	+	0.958938i	$0.408466\pi$
$444$	−35.1398	−1.66766
$445$	0	0
$446$	−62.9326	−2.97995
$447$	27.1811	1.28562
$448$	48.4389	2.28852
$449$	−0.361654	−0.0170675	−0.00853375	−	0.999964i	$-0.502716\pi$
$449$	−0.361654	−0.0170675	−0.00853375	+	0.999964i	$0.502716\pi$
$450$	0	0
$451$	1.89237	0.0891082
$452$	67.5961	3.17945
$453$	26.7090	1.25490
$454$	3.64804	0.171211
$455$	0	0
$456$	−28.3844	−1.32922
$457$	14.9179	0.697831	0.348915	−	0.937154i	$-0.386550\pi$
$457$	14.9179	0.697831	0.348915	+	0.937154i	$0.386550\pi$
$458$	−27.4382	−1.28210
$459$	−25.9979	−1.21348
$460$	0	0
$461$	20.5687	0.957980	0.478990	−	0.877820i	$-0.341003\pi$
$461$	20.5687	0.957980	0.478990	+	0.877820i	$0.341003\pi$
$462$	6.83479	0.317983
$463$	−3.01029	−0.139900	−0.0699501	−	0.997550i	$-0.522284\pi$
$463$	−3.01029	−0.139900	−0.0699501	+	0.997550i	$0.522284\pi$
$464$	7.86760	0.365244
$465$	0	0
$466$	−7.35872	−0.340886
$467$	−0.252192	−0.0116701	−0.00583504	−	0.999983i	$-0.501857\pi$
$467$	−0.252192	−0.0116701	−0.00583504	+	0.999983i	$0.501857\pi$
$468$	−10.6690	−0.493173
$469$	30.5086	1.40876
$470$	0	0
$471$	−6.73354	−0.310265
$472$	−17.0626	−0.785370
$473$	3.13781	0.144276
$474$	22.1084	1.01547
$475$	0	0
$476$	−63.5696	−2.91371
$477$	6.38254	0.292236
$478$	−49.0275	−2.24247
$479$	7.16728	0.327481	0.163741	−	0.986503i	$-0.447644\pi$
$479$	7.16728	0.327481	0.163741	+	0.986503i	$0.447644\pi$
$480$	0	0
$481$	−42.2277	−1.92542
$482$	2.33352	0.106289
$483$	19.3716	0.881440
$484$	−37.1370	−1.68805
$485$	0	0
$486$	−11.2927	−0.512249
$487$	−15.1944	−0.688526	−0.344263	−	0.938873i	$-0.611871\pi$
$487$	−15.1944	−0.688526	−0.344263	+	0.938873i	$0.611871\pi$
$488$	−33.3905	−1.51152
$489$	35.0364	1.58440
$490$	0	0
$491$	−8.26090	−0.372809	−0.186405	−	0.982473i	$-0.559684\pi$
$491$	−8.26090	−0.372809	−0.186405	+	0.982473i	$0.559684\pi$
$492$	22.0588	0.994487
$493$	37.8313	1.70384
$494$	−81.3101	−3.65831
$495$	0	0
$496$	2.43759	0.109451
$497$	−30.9513	−1.38835
$498$	10.3876	0.465478
$499$	−12.9549	−0.579941	−0.289970	−	0.957036i	$-0.593645\pi$
$499$	−12.9549	−0.579941	−0.289970	+	0.957036i	$0.593645\pi$
$500$	0	0
$501$	15.2447	0.681083
$502$	20.9700	0.935936
$503$	20.9327	0.933345	0.466672	−	0.884430i	$-0.345453\pi$
$503$	20.9327	0.933345	0.466672	+	0.884430i	$0.345453\pi$
$504$	−6.21276	−0.276738
$505$	0	0
$506$	3.41073	0.151626
$507$	48.2948	2.14485
$508$	33.7154	1.49588
$509$	16.7485	0.742364	0.371182	−	0.928560i	$-0.378953\pi$
$509$	16.7485	0.742364	0.371182	+	0.928560i	$0.378953\pi$
$510$	0	0
$511$	46.9784	2.07820
$512$	10.9758	0.485065
$513$	−29.2057	−1.28946
$514$	23.9122	1.05472
$515$	0	0
$516$	36.5765	1.61019
$517$	−0.202796	−0.00891898
$518$	−58.6174	−2.57550
$519$	19.6251	0.861445
$520$	0	0
$521$	17.6831	0.774712	0.387356	−	0.921930i	$-0.373388\pi$
$521$	17.6831	0.774712	0.387356	+	0.921930i	$0.373388\pi$
$522$	8.81359	0.385760
$523$	−15.8399	−0.692628	−0.346314	−	0.938119i	$-0.612567\pi$
$523$	−15.8399	−0.692628	−0.346314	+	0.938119i	$0.612567\pi$
$524$	−18.8060	−0.821542
$525$	0	0
$526$	−2.96575	−0.129313
$527$	11.7211	0.510581
$528$	0.732418	0.0318744
$529$	−13.3331	−0.579698
$530$	0	0
$531$	−2.37903	−0.103241
$532$	−71.4133	−3.09616
$533$	26.5082	1.14820
$534$	−4.57238	−0.197866
$535$	0	0
$536$	26.2671	1.13457
$537$	1.12060	0.0483573
$538$	68.6797	2.96099
$539$	3.92299	0.168975
$540$	0	0
$541$	6.93852	0.298310	0.149155	−	0.988814i	$-0.452345\pi$
$541$	6.93852	0.298310	0.149155	+	0.988814i	$0.452345\pi$
$542$	19.3744	0.832202
$543$	−20.0708	−0.861321
$544$	21.0052	0.900589
$545$	0	0
$546$	95.7412	4.09735
$547$	−24.1685	−1.03337	−0.516686	−	0.856175i	$-0.672834\pi$
$547$	−24.1685	−1.03337	−0.516686	+	0.856175i	$0.672834\pi$
$548$	−46.2799	−1.97698
$549$	−4.65561	−0.198697
$550$	0	0
$551$	42.4992	1.81053
$552$	16.6785	0.709883
$553$	23.3341	0.992267
$554$	2.91077	0.123667
$555$	0	0
$556$	−62.5335	−2.65201
$557$	4.70378	0.199306	0.0996528	−	0.995022i	$-0.468227\pi$
$557$	4.70378	0.199306	0.0996528	+	0.995022i	$0.468227\pi$
$558$	2.73068	0.115599
$559$	43.9542	1.85906
$560$	0	0
$561$	3.52183	0.148692
$562$	−36.9357	−1.55804
$563$	−22.3338	−0.941257	−0.470629	−	0.882331i	$-0.655973\pi$
$563$	−22.3338	−0.941257	−0.470629	+	0.882331i	$0.655973\pi$
$564$	−2.36394	−0.0995398
$565$	0	0
$566$	−32.7060	−1.37474
$567$	28.8629	1.21213
$568$	−26.6483	−1.11814
$569$	3.23670	0.135690	0.0678448	−	0.997696i	$-0.478388\pi$
$569$	3.23670	0.135690	0.0678448	+	0.997696i	$0.478388\pi$
$570$	0	0
$571$	−24.3476	−1.01892	−0.509458	−	0.860496i	$-0.670154\pi$
$571$	−24.3476	−1.01892	−0.509458	+	0.860496i	$0.670154\pi$
$572$	10.6656	0.445952
$573$	25.6555	1.07177
$574$	36.7967	1.53586
$575$	0	0
$576$	5.81486	0.242286
$577$	−24.3182	−1.01238	−0.506191	−	0.862421i	$-0.668947\pi$
$577$	−24.3182	−1.01238	−0.506191	+	0.862421i	$0.668947\pi$
$578$	−12.1013	−0.503346
$579$	−11.7464	−0.488164
$580$	0	0
$581$	10.9634	0.454840
$582$	29.0432	1.20388
$583$	−6.38054	−0.264255
$584$	40.4472	1.67372
$585$	0	0
$586$	−42.9445	−1.77402
$587$	35.1463	1.45064	0.725321	−	0.688411i	$-0.241691\pi$
$587$	35.1463	1.45064	0.725321	+	0.688411i	$0.241691\pi$
$588$	45.7291	1.88584
$589$	13.1674	0.542552
$590$	0	0
$591$	3.06288	0.125990
$592$	−6.28146	−0.258166
$593$	−25.3555	−1.04123	−0.520614	−	0.853792i	$-0.674297\pi$
$593$	−25.3555	−1.04123	−0.520614	+	0.853792i	$0.674297\pi$
$594$	6.05484	0.248433
$595$	0	0
$596$	58.8784	2.41175
$597$	9.23829	0.378098
$598$	47.7773	1.95376
$599$	10.8537	0.443469	0.221734	−	0.975107i	$-0.428828\pi$
$599$	10.8537	0.443469	0.221734	+	0.975107i	$0.428828\pi$
$600$	0	0
$601$	1.20127	0.0490009	0.0245004	−	0.999700i	$-0.492200\pi$
$601$	1.20127	0.0490009	0.0245004	+	0.999700i	$0.492200\pi$
$602$	61.0139	2.48674
$603$	3.66241	0.149145
$604$	57.8559	2.35413
$605$	0	0
$606$	56.6628	2.30177
$607$	32.1366	1.30438	0.652191	−	0.758054i	$-0.273850\pi$
$607$	32.1366	1.30438	0.652191	+	0.758054i	$0.273850\pi$
$608$	23.5969	0.956982
$609$	−50.0421	−2.02781
$610$	0	0
$611$	−2.84076	−0.114925
$612$	−7.63123	−0.308474
$613$	−32.7866	−1.32424	−0.662120	−	0.749398i	$-0.730343\pi$
$613$	−32.7866	−1.32424	−0.662120	+	0.749398i	$0.730343\pi$
$614$	42.3778	1.71023
$615$	0	0
$616$	6.21081	0.250241
$617$	15.5099	0.624405	0.312202	−	0.950016i	$-0.398933\pi$
$617$	15.5099	0.624405	0.312202	+	0.950016i	$0.398933\pi$
$618$	3.32238	0.133646
$619$	−40.0353	−1.60915	−0.804577	−	0.593848i	$-0.797608\pi$
$619$	−40.0353	−1.60915	−0.804577	+	0.593848i	$0.797608\pi$
$620$	0	0
$621$	17.1611	0.688649
$622$	−61.8667	−2.48063
$623$	−4.82586	−0.193344
$624$	10.2597	0.410715
$625$	0	0
$626$	31.3780	1.25412
$627$	3.95638	0.158003
$628$	−14.5859	−0.582041
$629$	−30.2044	−1.20433
$630$	0	0
$631$	−8.93103	−0.355539	−0.177769	−	0.984072i	$-0.556888\pi$
$631$	−8.93103	−0.355539	−0.177769	+	0.984072i	$0.556888\pi$
$632$	20.0901	0.799140
$633$	36.1098	1.43524
$634$	0.848047	0.0336803
$635$	0	0
$636$	−74.3761	−2.94920
$637$	54.9529	2.17731
$638$	−8.81083	−0.348824
$639$	−3.71555	−0.146985
$640$	0	0
$641$	−25.8585	−1.02135	−0.510674	−	0.859775i	$-0.670604\pi$
$641$	−25.8585	−1.02135	−0.510674	+	0.859775i	$0.670604\pi$
$642$	−5.83931	−0.230459
$643$	38.0613	1.50099	0.750496	−	0.660875i	$-0.229815\pi$
$643$	38.0613	1.50099	0.750496	+	0.660875i	$0.229815\pi$
$644$	41.9620	1.65353
$645$	0	0
$646$	−58.1590	−2.28823
$647$	35.3972	1.39161	0.695804	−	0.718232i	$-0.255048\pi$
$647$	35.3972	1.39161	0.695804	+	0.718232i	$0.255048\pi$
$648$	24.8502	0.976208
$649$	2.37829	0.0933559
$650$	0	0
$651$	−15.5044	−0.607664
$652$	75.8944	2.97225
$653$	0.363598	0.0142287	0.00711434	−	0.999975i	$-0.497735\pi$
$653$	0.363598	0.0142287	0.00711434	+	0.999975i	$0.497735\pi$
$654$	−19.4805	−0.761748
$655$	0	0
$656$	3.94314	0.153954
$657$	5.63953	0.220019
$658$	−3.94333	−0.153727
$659$	38.1286	1.48528	0.742639	−	0.669691i	$-0.233574\pi$
$659$	38.1286	1.48528	0.742639	+	0.669691i	$0.233574\pi$
$660$	0	0
$661$	3.88199	0.150992	0.0754960	−	0.997146i	$-0.475946\pi$
$661$	3.88199	0.150992	0.0754960	+	0.997146i	$0.475946\pi$
$662$	−78.6303	−3.05605
$663$	49.3336	1.91596
$664$	9.43924	0.366313
$665$	0	0
$666$	−7.03674	−0.272668
$667$	−24.9723	−0.966930
$668$	33.0224	1.27768
$669$	42.8947	1.65840
$670$	0	0
$671$	4.65415	0.179672
$672$	−27.7850	−1.07183
$673$	−7.18749	−0.277057	−0.138529	−	0.990358i	$-0.544237\pi$
$673$	−7.18749	−0.277057	−0.138529	+	0.990358i	$0.544237\pi$
$674$	17.3567	0.668556
$675$	0	0
$676$	104.614	4.02362
$677$	38.4175	1.47650	0.738252	−	0.674525i	$-0.235651\pi$
$677$	38.4175	1.47650	0.738252	+	0.674525i	$0.235651\pi$
$678$	−72.8187	−2.79659
$679$	30.6533	1.17637
$680$	0	0
$681$	−2.48650	−0.0952827
$682$	−2.72983	−0.104530
$683$	11.5291	0.441147	0.220573	−	0.975370i	$-0.429207\pi$
$683$	11.5291	0.441147	0.220573	+	0.975370i	$0.429207\pi$
$684$	−8.57282	−0.327790
$685$	0	0
$686$	12.2945	0.469405
$687$	18.7018	0.713517
$688$	6.53827	0.249269
$689$	−89.3781	−3.40503
$690$	0	0
$691$	−0.312537	−0.0118895	−0.00594473	−	0.999982i	$-0.501892\pi$
$691$	−0.312537	−0.0118895	−0.00594473	+	0.999982i	$0.501892\pi$
$692$	42.5110	1.61602
$693$	0.865970	0.0328955
$694$	−25.1211	−0.953583
$695$	0	0
$696$	−43.0850	−1.63313
$697$	18.9606	0.718184
$698$	77.8242	2.94569
$699$	5.01568	0.189710
$700$	0	0
$701$	−28.8002	−1.08777	−0.543885	−	0.839160i	$-0.683047\pi$
$701$	−28.8002	−1.08777	−0.543885	+	0.839160i	$0.683047\pi$
$702$	84.8158	3.20116
$703$	−33.9312	−1.27974
$704$	−5.81304	−0.219087
$705$	0	0
$706$	−51.9635	−1.95567
$707$	59.8041	2.24916
$708$	27.7230	1.04189
$709$	0.935652	0.0351392	0.0175696	−	0.999846i	$-0.494407\pi$
$709$	0.935652	0.0351392	0.0175696	+	0.999846i	$0.494407\pi$
$710$	0	0
$711$	2.80115	0.105051
$712$	−4.15494	−0.155713
$713$	−7.73707	−0.289755
$714$	68.4812	2.56284
$715$	0	0
$716$	2.42739	0.0907157
$717$	33.4170	1.24798
$718$	−50.9871	−1.90282
$719$	−0.493382	−0.0184000	−0.00920002	−	0.999958i	$-0.502928\pi$
$719$	−0.493382	−0.0184000	−0.00920002	+	0.999958i	$0.502928\pi$
$720$	0	0
$721$	3.50657	0.130592
$722$	−20.9982	−0.781472
$723$	−1.59052	−0.0591520
$724$	−43.4765	−1.61579
$725$	0	0
$726$	40.0064	1.48478
$727$	8.52980	0.316353	0.158176	−	0.987411i	$-0.449439\pi$
$727$	8.52980	0.316353	0.158176	+	0.987411i	$0.449439\pi$
$728$	87.0005	3.22445
$729$	29.8014	1.10376
$730$	0	0
$731$	31.4393	1.16282
$732$	54.2521	2.00522
$733$	−23.2369	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$733$	−23.2369	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$734$	−66.7934	−2.46539
$735$	0	0
$736$	−13.8654	−0.511086
$737$	−3.66126	−0.134864
$738$	4.41726	0.162602
$739$	−52.3706	−1.92648	−0.963242	−	0.268636i	$-0.913427\pi$
$739$	−52.3706	−1.92648	−0.963242	+	0.268636i	$0.913427\pi$
$740$	0	0
$741$	55.4207	2.03593
$742$	−124.068	−4.55468
$743$	−23.8100	−0.873503	−0.436752	−	0.899582i	$-0.643871\pi$
$743$	−23.8100	−0.873503	−0.436752	+	0.899582i	$0.643871\pi$
$744$	−13.3489	−0.489393
$745$	0	0
$746$	−20.0601	−0.734454
$747$	1.31611	0.0481539
$748$	7.62884	0.278938
$749$	−6.16303	−0.225192
$750$	0	0
$751$	−42.9807	−1.56839	−0.784193	−	0.620517i	$-0.786923\pi$
$751$	−42.9807	−1.56839	−0.784193	+	0.620517i	$0.786923\pi$
$752$	−0.422569	−0.0154095
$753$	−14.2931	−0.520869
$754$	−123.421	−4.49474
$755$	0	0
$756$	74.4923	2.70926
$757$	38.3109	1.39243	0.696216	−	0.717833i	$-0.254866\pi$
$757$	38.3109	1.39243	0.696216	+	0.717833i	$0.254866\pi$
$758$	−24.5771	−0.892680
$759$	−2.32474	−0.0843828
$760$	0	0
$761$	15.1540	0.549331	0.274666	−	0.961540i	$-0.411433\pi$
$761$	15.1540	0.549331	0.274666	+	0.961540i	$0.411433\pi$
$762$	−36.3203	−1.31575
$763$	−20.5605	−0.744339
$764$	55.5737	2.01059
$765$	0	0
$766$	−48.0018	−1.73438
$767$	33.3149	1.20293
$768$	−34.6578	−1.25060
$769$	−51.9569	−1.87361	−0.936806	−	0.349849i	$-0.886233\pi$
$769$	−51.9569	−1.87361	−0.936806	+	0.349849i	$0.886233\pi$
$770$	0	0
$771$	−16.2985	−0.586975
$772$	−25.4445	−0.915769
$773$	28.0084	1.00739	0.503697	−	0.863880i	$-0.331973\pi$
$773$	28.0084	1.00739	0.503697	+	0.863880i	$0.331973\pi$
$774$	7.32443	0.263271
$775$	0	0
$776$	26.3917	0.947408
$777$	39.9534	1.43332
$778$	11.9422	0.428148
$779$	21.3001	0.763155
$780$	0	0
$781$	3.71439	0.132911
$782$	34.1738	1.22205
$783$	−44.3316	−1.58428
$784$	8.17436	0.291941
$785$	0	0
$786$	20.2590	0.722613
$787$	8.64287	0.308085	0.154043	−	0.988064i	$-0.450771\pi$
$787$	8.64287	0.308085	0.154043	+	0.988064i	$0.450771\pi$
$788$	6.63467	0.236350
$789$	2.02145	0.0719654
$790$	0	0
$791$	−76.8557	−2.73267
$792$	0.745577	0.0264929
$793$	65.1951	2.31514
$794$	80.5461	2.85847
$795$	0	0
$796$	20.0116	0.709292
$797$	−36.0159	−1.27575	−0.637874	−	0.770141i	$-0.720186\pi$
$797$	−36.0159	−1.27575	−0.637874	+	0.770141i	$0.720186\pi$
$798$	76.9309	2.72332
$799$	−2.03192	−0.0718842
$800$	0	0
$801$	−0.579322	−0.0204693
$802$	33.4936	1.18270
$803$	−5.63776	−0.198952
$804$	−42.6783	−1.50515
$805$	0	0
$806$	−38.2392	−1.34692
$807$	−46.8118	−1.64785
$808$	51.4898	1.81140
$809$	45.4691	1.59861	0.799305	−	0.600926i	$-0.205201\pi$
$809$	45.4691	1.59861	0.799305	+	0.600926i	$0.205201\pi$
$810$	0	0
$811$	26.9051	0.944767	0.472383	−	0.881393i	$-0.343394\pi$
$811$	26.9051	0.944767	0.472383	+	0.881393i	$0.343394\pi$
$812$	−108.399	−3.80406
$813$	−13.2055	−0.463138
$814$	7.03453	0.246560
$815$	0	0
$816$	7.33847	0.256898
$817$	35.3185	1.23564
$818$	−26.8265	−0.937966
$819$	12.1304	0.423872
$820$	0	0
$821$	−32.2980	−1.12721	−0.563605	−	0.826045i	$-0.690586\pi$
$821$	−32.2980	−1.12721	−0.563605	+	0.826045i	$0.690586\pi$
$822$	49.8556	1.73891
$823$	7.44846	0.259637	0.129818	−	0.991538i	$-0.458561\pi$
$823$	7.44846	0.259637	0.129818	+	0.991538i	$0.458561\pi$
$824$	3.01907	0.105174
$825$	0	0
$826$	46.2452	1.60908
$827$	42.8805	1.49110	0.745551	−	0.666449i	$-0.232186\pi$
$827$	42.8805	1.49110	0.745551	+	0.666449i	$0.232186\pi$
$828$	5.03734	0.175060
$829$	−26.8911	−0.933965	−0.466983	−	0.884267i	$-0.654659\pi$
$829$	−26.8911	−0.933965	−0.466983	+	0.884267i	$0.654659\pi$
$830$	0	0
$831$	−1.98397	−0.0688232
$832$	−81.4286	−2.82303
$833$	39.3064	1.36189
$834$	67.3650	2.33266
$835$	0	0
$836$	8.57014	0.296404
$837$	−13.7351	−0.474754
$838$	−14.8699	−0.513671
$839$	40.1098	1.38474	0.692372	−	0.721540i	$-0.256566\pi$
$839$	40.1098	1.38474	0.692372	+	0.721540i	$0.256566\pi$
$840$	0	0
$841$	35.5100	1.22448
$842$	10.6394	0.366659
$843$	25.1752	0.867081
$844$	78.2195	2.69243
$845$	0	0
$846$	−0.473378	−0.0162751
$847$	42.2242	1.45084
$848$	−13.2952	−0.456558
$849$	22.2923	0.765070
$850$	0	0
$851$	19.9378	0.683458
$852$	43.2975	1.48335
$853$	−12.8732	−0.440769	−0.220384	−	0.975413i	$-0.570731\pi$
$853$	−12.8732	−0.440769	−0.220384	+	0.975413i	$0.570731\pi$
$854$	90.4990	3.09681
$855$	0	0
$856$	−5.30621	−0.181363
$857$	−35.7049	−1.21966	−0.609828	−	0.792534i	$-0.708762\pi$
$857$	−35.7049	−1.21966	−0.609828	+	0.792534i	$0.708762\pi$
$858$	−11.4897	−0.392251
$859$	−44.6693	−1.52410	−0.762049	−	0.647520i	$-0.775806\pi$
$859$	−44.6693	−1.52410	−0.762049	+	0.647520i	$0.775806\pi$
$860$	0	0
$861$	−25.0805	−0.854741
$862$	−42.0330	−1.43165
$863$	45.2951	1.54186	0.770932	−	0.636918i	$-0.219791\pi$
$863$	45.2951	1.54186	0.770932	+	0.636918i	$0.219791\pi$
$864$	−24.6143	−0.837396
$865$	0	0
$866$	51.0491	1.73472
$867$	8.24817	0.280123
$868$	−33.5849	−1.13994
$869$	−2.80027	−0.0949926
$870$	0	0
$871$	−51.2867	−1.73778
$872$	−17.7020	−0.599467
$873$	3.67978	0.124542
$874$	38.3905	1.29858
$875$	0	0
$876$	−65.7177	−2.22040
$877$	52.1350	1.76047	0.880236	−	0.474535i	$-0.157384\pi$
$877$	52.1350	1.76047	0.880236	+	0.474535i	$0.157384\pi$
$878$	66.5975	2.24756
$879$	29.2708	0.987280
$880$	0	0
$881$	50.2368	1.69252	0.846261	−	0.532769i	$-0.178849\pi$
$881$	50.2368	1.69252	0.846261	+	0.532769i	$0.178849\pi$
$882$	9.15724	0.308340
$883$	13.3320	0.448657	0.224328	−	0.974514i	$-0.427981\pi$
$883$	13.3320	0.448657	0.224328	+	0.974514i	$0.427981\pi$
$884$	106.864	3.59423
$885$	0	0
$886$	−27.8597	−0.935964
$887$	−42.9656	−1.44265	−0.721323	−	0.692599i	$-0.756465\pi$
$887$	−42.9656	−1.44265	−0.721323	+	0.692599i	$0.756465\pi$
$888$	34.3989	1.15435
$889$	−38.3338	−1.28568
$890$	0	0
$891$	−3.46376	−0.116040
$892$	92.9166	3.11108
$893$	−2.28263	−0.0763854
$894$	−63.4275	−2.12133
$895$	0	0
$896$	−78.0949	−2.60897
$897$	−32.5648	−1.08731
$898$	0.843927	0.0281622
$899$	19.9869	0.666600
$900$	0	0
$901$	−63.9298	−2.12981
$902$	−4.41588	−0.147033
$903$	−41.5869	−1.38393
$904$	−66.1708	−2.20081
$905$	0	0
$906$	−62.3261	−2.07064
$907$	−16.8356	−0.559015	−0.279508	−	0.960143i	$-0.590171\pi$
$907$	−16.8356	−0.559015	−0.279508	+	0.960143i	$0.590171\pi$
$908$	−5.38614	−0.178745
$909$	7.17919	0.238119
$910$	0	0
$911$	−3.06013	−0.101387	−0.0506933	−	0.998714i	$-0.516143\pi$
$911$	−3.06013	−0.101387	−0.0506933	+	0.998714i	$0.516143\pi$
$912$	8.24394	0.272984
$913$	−1.31570	−0.0435432
$914$	−34.8113	−1.15145
$915$	0	0
$916$	40.5110	1.33852
$917$	21.3821	0.706099
$918$	60.6665	2.00229
$919$	25.6211	0.845163	0.422582	−	0.906325i	$-0.361124\pi$
$919$	25.6211	0.845163	0.422582	+	0.906325i	$0.361124\pi$
$920$	0	0
$921$	−28.8846	−0.951780
$922$	−47.9975	−1.58071
$923$	52.0309	1.71262
$924$	−10.0912	−0.331976
$925$	0	0
$926$	7.02458	0.230842
$927$	0.420947	0.0138257
$928$	35.8180	1.17578
$929$	−23.9187	−0.784747	−0.392374	−	0.919806i	$-0.628346\pi$
$929$	−23.9187	−0.784747	−0.392374	+	0.919806i	$0.628346\pi$
$930$	0	0
$931$	44.1563	1.44716
$932$	10.8647	0.355886
$933$	42.1681	1.38052
$934$	0.588496	0.0192562
$935$	0	0
$936$	10.4440	0.341373
$937$	−0.136233	−0.00445054	−0.00222527	−	0.999998i	$-0.500708\pi$
$937$	−0.136233	−0.00445054	−0.00222527	+	0.999998i	$0.500708\pi$
$938$	−71.1925	−2.32452
$939$	−21.3872	−0.697944
$940$	0	0
$941$	19.6643	0.641036	0.320518	−	0.947242i	$-0.396143\pi$
$941$	19.6643	0.641036	0.320518	+	0.947242i	$0.396143\pi$
$942$	15.7128	0.511952
$943$	−12.5158	−0.407570
$944$	4.95565	0.161293
$945$	0	0
$946$	−7.32213	−0.238063
$947$	−48.5859	−1.57883	−0.789415	−	0.613859i	$-0.789616\pi$
$947$	−48.5859	−1.57883	−0.789415	+	0.613859i	$0.789616\pi$
$948$	−32.6419	−1.06016
$949$	−78.9734	−2.56358
$950$	0	0
$951$	−0.578026	−0.0187438
$952$	62.2292	2.01686
$953$	−18.7715	−0.608067	−0.304034	−	0.952661i	$-0.598333\pi$
$953$	−18.7715	−0.608067	−0.304034	+	0.952661i	$0.598333\pi$
$954$	−14.8938	−0.482204
$955$	0	0
$956$	72.3864	2.34114
$957$	6.00543	0.194128
$958$	−16.7250	−0.540359
$959$	52.6195	1.69917
$960$	0	0
$961$	−24.8075	−0.800243
$962$	98.5392	3.17703
$963$	−0.739842	−0.0238411
$964$	−3.44531	−0.110966
$965$	0	0
$966$	−45.2041	−1.45442
$967$	22.1290	0.711622	0.355811	−	0.934558i	$-0.384205\pi$
$967$	22.1290	0.711622	0.355811	+	0.934558i	$0.384205\pi$
$968$	36.3540	1.16846
$969$	39.6410	1.27345
$970$	0	0
$971$	27.0585	0.868349	0.434174	−	0.900829i	$-0.357040\pi$
$971$	27.0585	0.868349	0.434174	+	0.900829i	$0.357040\pi$
$972$	16.6731	0.534790
$973$	71.0996	2.27935
$974$	35.4565	1.13610
$975$	0	0
$976$	9.69790	0.310422
$977$	−51.8655	−1.65933	−0.829663	−	0.558265i	$-0.811467\pi$
$977$	−51.8655	−1.65933	−0.829663	+	0.558265i	$0.811467\pi$
$978$	−81.7582	−2.61434
$979$	0.579140	0.0185094
$980$	0	0
$981$	−2.46819	−0.0788031
$982$	19.2770	0.615153
$983$	−14.1055	−0.449897	−0.224948	−	0.974371i	$-0.572221\pi$
$983$	−14.1055	−0.449897	−0.224948	+	0.974371i	$0.572221\pi$
$984$	−21.5937	−0.688381
$985$	0	0
$986$	−88.2801	−2.81141
$987$	2.68776	0.0855524
$988$	120.050	3.81929
$989$	−20.7529	−0.659904
$990$	0	0
$991$	−10.3220	−0.327889	−0.163944	−	0.986470i	$-0.552422\pi$
$991$	−10.3220	−0.327889	−0.163944	+	0.986470i	$0.552422\pi$
$992$	11.0974	0.352342
$993$	53.5942	1.70076
$994$	72.2254	2.29085
$995$	0	0
$996$	−15.3367	−0.485961
$997$	51.8688	1.64270	0.821351	−	0.570423i	$-0.193221\pi$
$997$	51.8688	1.64270	0.821351	+	0.570423i	$0.193221\pi$
$998$	30.2305	0.956930
$999$	35.3942	1.11982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.9		66
5.2	odd	4		1205.2.b.d.724.9	✓	66
5.3	odd	4		1205.2.b.d.724.58	yes	66
5.4	even	2	inner	6025.2.a.q.1.58		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.9	✓	66	5.2	odd	4
1205.2.b.d.724.58	yes	66	5.3	odd	4
6025.2.a.q.1.9		66	1.1	even	1	trivial
6025.2.a.q.1.58		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)

\(f(q)\)	\(=\)	\(q-2.33352 q^{2} +1.59052 q^{3} +3.44531 q^{4} -3.71151 q^{6} -3.91727 q^{7} -3.37267 q^{8} -0.470249 q^{9} +O(q^{10})\)	\(q-2.33352 q^{2} +1.59052 q^{3} +3.44531 q^{4} -3.71151 q^{6} -3.91727 q^{7} -3.37267 q^{8} -0.470249 q^{9} +0.470102 q^{11} +5.47984 q^{12} +6.58515 q^{13} +9.14102 q^{14} +0.979554 q^{16} +4.71018 q^{17} +1.09734 q^{18} +5.29136 q^{19} -6.23049 q^{21} -1.09699 q^{22} -3.10917 q^{23} -5.36429 q^{24} -15.3666 q^{26} -5.51950 q^{27} -13.4962 q^{28} +8.03181 q^{29} +2.48847 q^{31} +4.45952 q^{32} +0.747705 q^{33} -10.9913 q^{34} -1.62015 q^{36} -6.41257 q^{37} -12.3475 q^{38} +10.4738 q^{39} +4.02545 q^{41} +14.5390 q^{42} +6.67474 q^{43} +1.61965 q^{44} +7.25531 q^{46} -0.431389 q^{47} +1.55800 q^{48} +8.34498 q^{49} +7.49164 q^{51} +22.6879 q^{52} -13.5727 q^{53} +12.8799 q^{54} +13.2116 q^{56} +8.41601 q^{57} -18.7424 q^{58} +5.05909 q^{59} +9.90032 q^{61} -5.80688 q^{62} +1.84209 q^{63} -12.3655 q^{64} -1.74479 q^{66} -7.78824 q^{67} +16.2281 q^{68} -4.94519 q^{69} +7.90125 q^{71} +1.58599 q^{72} -11.9926 q^{73} +14.9639 q^{74} +18.2304 q^{76} -1.84151 q^{77} -24.4408 q^{78} -5.95673 q^{79} -7.36812 q^{81} -9.39346 q^{82} -2.79875 q^{83} -21.4660 q^{84} -15.5756 q^{86} +12.7748 q^{87} -1.58550 q^{88} +1.23195 q^{89} -25.7958 q^{91} -10.7121 q^{92} +3.95795 q^{93} +1.00665 q^{94} +7.09295 q^{96} -7.82518 q^{97} -19.4732 q^{98} -0.221065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 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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