LMFDB - Embedded newform 6025.2.a.p.1.38

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			1.38
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84209 q^{2} +1.69681 q^{3} +1.39329 q^{4} +3.12567 q^{6} -1.54055 q^{7} -1.11761 q^{8} -0.120844 q^{9} +O(q^{10})$	\(q+1.84209 q^{2} +1.69681 q^{3} +1.39329 q^{4} +3.12567 q^{6} -1.54055 q^{7} -1.11761 q^{8} -0.120844 q^{9} +2.53578 q^{11} +2.36415 q^{12} -6.36889 q^{13} -2.83783 q^{14} -4.84532 q^{16} +3.54355 q^{17} -0.222605 q^{18} +0.713569 q^{19} -2.61402 q^{21} +4.67113 q^{22} -1.66453 q^{23} -1.89637 q^{24} -11.7321 q^{26} -5.29547 q^{27} -2.14644 q^{28} -8.23983 q^{29} +10.5697 q^{31} -6.69029 q^{32} +4.30273 q^{33} +6.52753 q^{34} -0.168370 q^{36} +0.496832 q^{37} +1.31446 q^{38} -10.8068 q^{39} -8.92984 q^{41} -4.81525 q^{42} -1.50676 q^{43} +3.53308 q^{44} -3.06622 q^{46} -6.94849 q^{47} -8.22158 q^{48} -4.62670 q^{49} +6.01272 q^{51} -8.87372 q^{52} +9.87593 q^{53} -9.75473 q^{54} +1.72174 q^{56} +1.21079 q^{57} -15.1785 q^{58} -10.1744 q^{59} -9.13781 q^{61} +19.4703 q^{62} +0.186166 q^{63} -2.63346 q^{64} +7.92601 q^{66} +1.85057 q^{67} +4.93720 q^{68} -2.82439 q^{69} -9.47485 q^{71} +0.135056 q^{72} +8.88186 q^{73} +0.915209 q^{74} +0.994210 q^{76} -3.90650 q^{77} -19.9071 q^{78} -6.12948 q^{79} -8.62287 q^{81} -16.4496 q^{82} +0.835537 q^{83} -3.64209 q^{84} -2.77558 q^{86} -13.9814 q^{87} -2.83402 q^{88} +2.27480 q^{89} +9.81161 q^{91} -2.31918 q^{92} +17.9348 q^{93} -12.7997 q^{94} -11.3521 q^{96} +7.56572 q^{97} -8.52280 q^{98} -0.306433 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.84209	1.30255	0.651277	−	0.758840i	$-0.274234\pi$
$2$	1.84209	1.30255	0.651277	+	0.758840i	$0.274234\pi$
$3$	1.69681	0.979652	0.489826	−	0.871820i	$-0.337060\pi$
$3$	1.69681	0.979652	0.489826	+	0.871820i	$0.337060\pi$
$4$	1.39329	0.696645
$5$	0	0
$5$	0	0
$6$	3.12567	1.27605
$7$	−1.54055	−0.582274	−0.291137	−	0.956681i	$-0.594033\pi$
$7$	−1.54055	−0.582274	−0.291137	+	0.956681i	$0.594033\pi$
$8$	−1.11761	−0.395136
$9$	−0.120844	−0.0402812
$10$	0	0
$11$	2.53578	0.764566	0.382283	−	0.924045i	$-0.375138\pi$
$11$	2.53578	0.764566	0.382283	+	0.924045i	$0.375138\pi$
$12$	2.36415	0.682470
$13$	−6.36889	−1.76641	−0.883207	−	0.468984i	$-0.844620\pi$
$13$	−6.36889	−1.76641	−0.883207	+	0.468984i	$0.844620\pi$
$14$	−2.83783	−0.758442
$15$	0	0
$16$	−4.84532	−1.21133
$17$	3.54355	0.859437	0.429719	−	0.902963i	$-0.358613\pi$
$17$	3.54355	0.859437	0.429719	+	0.902963i	$0.358613\pi$
$18$	−0.222605	−0.0524684
$19$	0.713569	0.163704	0.0818520	−	0.996644i	$-0.473917\pi$
$19$	0.713569	0.163704	0.0818520	+	0.996644i	$0.473917\pi$
$20$	0	0
$21$	−2.61402	−0.570426
$22$	4.67113	0.995889
$23$	−1.66453	−0.347079	−0.173540	−	0.984827i	$-0.555520\pi$
$23$	−1.66453	−0.347079	−0.173540	+	0.984827i	$0.555520\pi$
$24$	−1.89637	−0.387095
$25$	0	0
$26$	−11.7321	−2.30085
$27$	−5.29547	−1.01911
$28$	−2.14644	−0.405638
$29$	−8.23983	−1.53010	−0.765049	−	0.643972i	$-0.777285\pi$
$29$	−8.23983	−1.53010	−0.765049	+	0.643972i	$0.777285\pi$
$30$	0	0
$31$	10.5697	1.89838	0.949188	−	0.314711i	$-0.101908\pi$
$31$	10.5697	1.89838	0.949188	+	0.314711i	$0.101908\pi$
$32$	−6.69029	−1.18269
$33$	4.30273	0.749009
$34$	6.52753	1.11946
$35$	0	0
$36$	−0.168370	−0.0280617
$37$	0.496832	0.0816787	0.0408393	−	0.999166i	$-0.486997\pi$
$37$	0.496832	0.0816787	0.0408393	+	0.999166i	$0.486997\pi$
$38$	1.31446	0.213233
$39$	−10.8068	−1.73047
$40$	0	0
$41$	−8.92984	−1.39461	−0.697303	−	0.716776i	$-0.745617\pi$
$41$	−8.92984	−1.39461	−0.697303	+	0.716776i	$0.745617\pi$
$42$	−4.81525	−0.743010
$43$	−1.50676	−0.229778	−0.114889	−	0.993378i	$-0.536651\pi$
$43$	−1.50676	−0.229778	−0.114889	+	0.993378i	$0.536651\pi$
$44$	3.53308	0.532632
$45$	0	0
$46$	−3.06622	−0.452089
$47$	−6.94849	−1.01354	−0.506771	−	0.862081i	$-0.669161\pi$
$47$	−6.94849	−1.01354	−0.506771	+	0.862081i	$0.669161\pi$
$48$	−8.22158	−1.18668
$49$	−4.62670	−0.660958
$50$	0	0
$51$	6.01272	0.841950
$52$	−8.87372	−1.23056
$53$	9.87593	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$53$	9.87593	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$54$	−9.75473	−1.32745
$55$	0	0
$56$	1.72174	0.230077
$57$	1.21079	0.160373
$58$	−15.1785	−1.99303
$59$	−10.1744	−1.32459	−0.662296	−	0.749242i	$-0.730418\pi$
$59$	−10.1744	−1.32459	−0.662296	+	0.749242i	$0.730418\pi$
$60$	0	0
$61$	−9.13781	−1.16998	−0.584988	−	0.811042i	$-0.698901\pi$
$61$	−9.13781	−1.16998	−0.584988	+	0.811042i	$0.698901\pi$
$62$	19.4703	2.47273
$63$	0.186166	0.0234547
$64$	−2.63346	−0.329183
$65$	0	0
$66$	7.92601	0.975625
$67$	1.85057	0.226083	0.113041	−	0.993590i	$-0.463941\pi$
$67$	1.85057	0.226083	0.113041	+	0.993590i	$0.463941\pi$
$68$	4.93720	0.598723
$69$	−2.82439	−0.340017
$70$	0	0
$71$	−9.47485	−1.12446	−0.562229	−	0.826981i	$-0.690056\pi$
$71$	−9.47485	−1.12446	−0.562229	+	0.826981i	$0.690056\pi$
$72$	0.135056	0.0159165
$73$	8.88186	1.03954	0.519771	−	0.854305i	$-0.326017\pi$
$73$	8.88186	1.03954	0.519771	+	0.854305i	$0.326017\pi$
$74$	0.915209	0.106391
$75$	0	0
$76$	0.994210	0.114044
$77$	−3.90650	−0.445187
$78$	−19.9071	−2.25403
$79$	−6.12948	−0.689621	−0.344810	−	0.938672i	$-0.612057\pi$
$79$	−6.12948	−0.689621	−0.344810	+	0.938672i	$0.612057\pi$
$80$	0	0
$81$	−8.62287	−0.958096
$82$	−16.4496	−1.81655
$83$	0.835537	0.0917122	0.0458561	−	0.998948i	$-0.485398\pi$
$83$	0.835537	0.0917122	0.0458561	+	0.998948i	$0.485398\pi$
$84$	−3.64209	−0.397384
$85$	0	0
$86$	−2.77558	−0.299299
$87$	−13.9814	−1.49896
$88$	−2.83402	−0.302107
$89$	2.27480	0.241129	0.120564	−	0.992706i	$-0.461530\pi$
$89$	2.27480	0.241129	0.120564	+	0.992706i	$0.461530\pi$
$90$	0	0
$91$	9.81161	1.02854
$92$	−2.31918	−0.241791
$93$	17.9348	1.85975
$94$	−12.7997	−1.32019
$95$	0	0
$96$	−11.3521	−1.15862
$97$	7.56572	0.768182	0.384091	−	0.923295i	$-0.374515\pi$
$97$	7.56572	0.768182	0.384091	+	0.923295i	$0.374515\pi$
$98$	−8.52280	−0.860933
$99$	−0.306433	−0.0307976
$100$	0	0
$101$	−16.9324	−1.68483	−0.842417	−	0.538826i	$-0.818868\pi$
$101$	−16.9324	−1.68483	−0.842417	+	0.538826i	$0.818868\pi$
$102$	11.0760	1.09668
$103$	−2.83252	−0.279096	−0.139548	−	0.990215i	$-0.544565\pi$
$103$	−2.83252	−0.279096	−0.139548	+	0.990215i	$0.544565\pi$
$104$	7.11795	0.697973
$105$	0	0
$106$	18.1923	1.76700
$107$	−4.86488	−0.470306	−0.235153	−	0.971958i	$-0.575559\pi$
$107$	−4.86488	−0.470306	−0.235153	+	0.971958i	$0.575559\pi$
$108$	−7.37813	−0.709961
$109$	10.2870	0.985317	0.492658	−	0.870223i	$-0.336025\pi$
$109$	10.2870	0.985317	0.492658	+	0.870223i	$0.336025\pi$
$110$	0	0
$111$	0.843029	0.0800167
$112$	7.46447	0.705326
$113$	−12.6981	−1.19454	−0.597270	−	0.802041i	$-0.703748\pi$
$113$	−12.6981	−1.19454	−0.597270	+	0.802041i	$0.703748\pi$
$114$	2.23038	0.208894
$115$	0	0
$116$	−11.4805	−1.06594
$117$	0.769640	0.0711532
$118$	−18.7421	−1.72535
$119$	−5.45902	−0.500427
$120$	0	0
$121$	−4.56982	−0.415438
$122$	−16.8326	−1.52396
$123$	−15.1522	−1.36623
$124$	14.7267	1.32249
$125$	0	0
$126$	0.342934	0.0305510
$127$	−1.17595	−0.104349	−0.0521743	−	0.998638i	$-0.516615\pi$
$127$	−1.17595	−0.104349	−0.0521743	+	0.998638i	$0.516615\pi$
$128$	8.52951	0.753909
$129$	−2.55668	−0.225103
$130$	0	0
$131$	18.7573	1.63884	0.819418	−	0.573196i	$-0.194297\pi$
$131$	18.7573	1.63884	0.819418	+	0.573196i	$0.194297\pi$
$132$	5.99496	0.521794
$133$	−1.09929	−0.0953205
$134$	3.40891	0.294485
$135$	0	0
$136$	−3.96031	−0.339594
$137$	5.04016	0.430610	0.215305	−	0.976547i	$-0.430925\pi$
$137$	5.04016	0.430610	0.215305	+	0.976547i	$0.430925\pi$
$138$	−5.20278	−0.442890
$139$	−0.226430	−0.0192056	−0.00960278	−	0.999954i	$-0.503057\pi$
$139$	−0.226430	−0.0192056	−0.00960278	+	0.999954i	$0.503057\pi$
$140$	0	0
$141$	−11.7902	−0.992918
$142$	−17.4535	−1.46467
$143$	−16.1501	−1.35054
$144$	0.585526	0.0487938
$145$	0	0
$146$	16.3612	1.35406
$147$	−7.85063	−0.647509
$148$	0.692232	0.0569011
$149$	−4.00539	−0.328134	−0.164067	−	0.986449i	$-0.552461\pi$
$149$	−4.00539	−0.328134	−0.164067	+	0.986449i	$0.552461\pi$
$150$	0	0
$151$	−5.73080	−0.466366	−0.233183	−	0.972433i	$-0.574914\pi$
$151$	−5.73080	−0.466366	−0.233183	+	0.972433i	$0.574914\pi$
$152$	−0.797494	−0.0646853
$153$	−0.428215	−0.0346191
$154$	−7.19612	−0.579880
$155$	0	0
$156$	−15.0570	−1.20552
$157$	−4.02238	−0.321021	−0.160510	−	0.987034i	$-0.551314\pi$
$157$	−4.02238	−0.321021	−0.160510	+	0.987034i	$0.551314\pi$
$158$	−11.2911	−0.898268
$159$	16.7575	1.32896
$160$	0	0
$161$	2.56430	0.202095
$162$	−15.8841	−1.24797
$163$	2.51003	0.196601	0.0983004	−	0.995157i	$-0.468659\pi$
$163$	2.51003	0.196601	0.0983004	+	0.995157i	$0.468659\pi$
$164$	−12.4419	−0.971546
$165$	0	0
$166$	1.53913	0.119460
$167$	25.3075	1.95835	0.979177	−	0.203008i	$-0.0650718\pi$
$167$	25.3075	1.95835	0.979177	+	0.203008i	$0.0650718\pi$
$168$	2.92146	0.225395
$169$	27.5628	2.12022
$170$	0	0
$171$	−0.0862302	−0.00659419
$172$	−2.09935	−0.160074
$173$	−19.0394	−1.44754	−0.723769	−	0.690043i	$-0.757592\pi$
$173$	−19.0394	−1.44754	−0.723769	+	0.690043i	$0.757592\pi$
$174$	−25.7550	−1.95248
$175$	0	0
$176$	−12.2867	−0.926143
$177$	−17.2640	−1.29764
$178$	4.19039	0.314083
$179$	12.3757	0.925007	0.462503	−	0.886618i	$-0.346951\pi$
$179$	12.3757	0.925007	0.462503	+	0.886618i	$0.346951\pi$
$180$	0	0
$181$	5.62774	0.418307	0.209153	−	0.977883i	$-0.432929\pi$
$181$	5.62774	0.418307	0.209153	+	0.977883i	$0.432929\pi$
$182$	18.0738	1.33972
$183$	−15.5051	−1.14617
$184$	1.86030	0.137143
$185$	0	0
$186$	33.0374	2.42242
$187$	8.98566	0.657097
$188$	−9.68127	−0.706079
$189$	8.15794	0.593403
$190$	0	0
$191$	−16.0108	−1.15850	−0.579251	−	0.815149i	$-0.696655\pi$
$191$	−16.0108	−1.15850	−0.579251	+	0.815149i	$0.696655\pi$
$192$	−4.46848	−0.322485
$193$	6.92078	0.498169	0.249084	−	0.968482i	$-0.419870\pi$
$193$	6.92078	0.498169	0.249084	+	0.968482i	$0.419870\pi$
$194$	13.9367	1.00060
$195$	0	0
$196$	−6.44634	−0.460453
$197$	−4.08208	−0.290836	−0.145418	−	0.989370i	$-0.546453\pi$
$197$	−4.08208	−0.290836	−0.145418	+	0.989370i	$0.546453\pi$
$198$	−0.564476	−0.0401156
$199$	7.24353	0.513481	0.256740	−	0.966480i	$-0.417352\pi$
$199$	7.24353	0.513481	0.256740	+	0.966480i	$0.417352\pi$
$200$	0	0
$201$	3.14006	0.221483
$202$	−31.1909	−2.19459
$203$	12.6939	0.890935
$204$	8.37747	0.586540
$205$	0	0
$206$	−5.21775	−0.363538
$207$	0.201148	0.0139808
$208$	30.8593	2.13971
$209$	1.80945	0.125163
$210$	0	0
$211$	0.0986890	0.00679403	0.00339701	−	0.999994i	$-0.498919\pi$
$211$	0.0986890	0.00679403	0.00339701	+	0.999994i	$0.498919\pi$
$212$	13.7600	0.945043
$213$	−16.0770	−1.10158
$214$	−8.96154	−0.612598
$215$	0	0
$216$	5.91828	0.402688
$217$	−16.2832	−1.10537
$218$	18.9496	1.28343
$219$	15.0708	1.01839
$220$	0	0
$221$	−22.5685	−1.51812
$222$	1.55293	0.104226
$223$	15.5911	1.04406	0.522028	−	0.852928i	$-0.325176\pi$
$223$	15.5911	1.04406	0.522028	+	0.852928i	$0.325176\pi$
$224$	10.3067	0.688647
$225$	0	0
$226$	−23.3911	−1.55595
$227$	24.7410	1.64212	0.821059	−	0.570843i	$-0.193384\pi$
$227$	24.7410	1.64212	0.821059	+	0.570843i	$0.193384\pi$
$228$	1.68698	0.111723
$229$	−21.0262	−1.38945	−0.694725	−	0.719275i	$-0.744474\pi$
$229$	−21.0262	−1.38945	−0.694725	+	0.719275i	$0.744474\pi$
$230$	0	0
$231$	−6.62858	−0.436128
$232$	9.20893	0.604596
$233$	−8.57165	−0.561548	−0.280774	−	0.959774i	$-0.590591\pi$
$233$	−8.57165	−0.561548	−0.280774	+	0.959774i	$0.590591\pi$
$234$	1.41774	0.0926809
$235$	0	0
$236$	−14.1759	−0.922771
$237$	−10.4006	−0.675589
$238$	−10.0560	−0.651833
$239$	22.7072	1.46880	0.734402	−	0.678715i	$-0.237463\pi$
$239$	22.7072	1.46880	0.734402	+	0.678715i	$0.237463\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−8.41801	−0.541130
$243$	1.25507	0.0805127
$244$	−12.7316	−0.815059
$245$	0	0
$246$	−27.9117	−1.77959
$247$	−4.54465	−0.289169
$248$	−11.8128	−0.750115
$249$	1.41775	0.0898460
$250$	0	0
$251$	−12.9356	−0.816489	−0.408244	−	0.912873i	$-0.633859\pi$
$251$	−12.9356	−0.816489	−0.408244	+	0.912873i	$0.633859\pi$
$252$	0.259383	0.0163396
$253$	−4.22089	−0.265365
$254$	−2.16620	−0.135920
$255$	0	0
$256$	20.9790	1.31119
$257$	−10.8510	−0.676865	−0.338432	−	0.940991i	$-0.609897\pi$
$257$	−10.8510	−0.676865	−0.338432	+	0.940991i	$0.609897\pi$
$258$	−4.70963	−0.293209
$259$	−0.765395	−0.0475593
$260$	0	0
$261$	0.995730	0.0616341
$262$	34.5527	2.13467
$263$	14.1032	0.869638	0.434819	−	0.900518i	$-0.356812\pi$
$263$	14.1032	0.869638	0.434819	+	0.900518i	$0.356812\pi$
$264$	−4.80878	−0.295960
$265$	0	0
$266$	−2.02499	−0.124160
$267$	3.85991	0.236222
$268$	2.57838	0.157500
$269$	−7.55668	−0.460739	−0.230370	−	0.973103i	$-0.573993\pi$
$269$	−7.55668	−0.460739	−0.230370	+	0.973103i	$0.573993\pi$
$270$	0	0
$271$	20.9578	1.27309	0.636546	−	0.771238i	$-0.280362\pi$
$271$	20.9578	1.27309	0.636546	+	0.771238i	$0.280362\pi$
$272$	−17.1696	−1.04106
$273$	16.6484	1.00761
$274$	9.28443	0.560893
$275$	0	0
$276$	−3.93520	−0.236871
$277$	−14.9632	−0.899052	−0.449526	−	0.893267i	$-0.648407\pi$
$277$	−14.9632	−0.899052	−0.449526	+	0.893267i	$0.648407\pi$
$278$	−0.417105	−0.0250163
$279$	−1.27728	−0.0764688
$280$	0	0
$281$	30.7490	1.83433	0.917165	−	0.398507i	$-0.130471\pi$
$281$	30.7490	1.83433	0.917165	+	0.398507i	$0.130471\pi$
$282$	−21.7187	−1.29333
$283$	4.06104	0.241404	0.120702	−	0.992689i	$-0.461485\pi$
$283$	4.06104	0.241404	0.120702	+	0.992689i	$0.461485\pi$
$284$	−13.2012	−0.783349
$285$	0	0
$286$	−29.7499	−1.75915
$287$	13.7569	0.812043
$288$	0.808478	0.0476400
$289$	−4.44326	−0.261368
$290$	0	0
$291$	12.8376	0.752551
$292$	12.3750	0.724193
$293$	−18.7249	−1.09392	−0.546961	−	0.837158i	$-0.684215\pi$
$293$	−18.7249	−1.09392	−0.546961	+	0.837158i	$0.684215\pi$
$294$	−14.4615	−0.843415
$295$	0	0
$296$	−0.555266	−0.0322742
$297$	−13.4282	−0.779180
$298$	−7.37828	−0.427412
$299$	10.6012	0.613085
$300$	0	0
$301$	2.32124	0.133794
$302$	−10.5566	−0.607467
$303$	−28.7310	−1.65055
$304$	−3.45747	−0.198300
$305$	0	0
$306$	−0.788810	−0.0450933
$307$	−2.41046	−0.137572	−0.0687860	−	0.997631i	$-0.521913\pi$
$307$	−2.41046	−0.137572	−0.0687860	+	0.997631i	$0.521913\pi$
$308$	−5.44289	−0.310137
$309$	−4.80624	−0.273418
$310$	0	0
$311$	−16.0513	−0.910183	−0.455092	−	0.890445i	$-0.650394\pi$
$311$	−16.0513	−0.910183	−0.455092	+	0.890445i	$0.650394\pi$
$312$	12.0778	0.683771
$313$	6.01435	0.339951	0.169976	−	0.985448i	$-0.445631\pi$
$313$	6.01435	0.339951	0.169976	+	0.985448i	$0.445631\pi$
$314$	−7.40958	−0.418147
$315$	0	0
$316$	−8.54015	−0.480421
$317$	27.5411	1.54686	0.773430	−	0.633881i	$-0.218539\pi$
$317$	27.5411	1.54686	0.773430	+	0.633881i	$0.218539\pi$
$318$	30.8689	1.73104
$319$	−20.8944	−1.16986
$320$	0	0
$321$	−8.25476	−0.460736
$322$	4.72366	0.263240
$323$	2.52857	0.140693
$324$	−12.0142	−0.667453
$325$	0	0
$326$	4.62370	0.256083
$327$	17.4551	0.965268
$328$	9.98010	0.551059
$329$	10.7045	0.590158
$330$	0	0
$331$	0.248029	0.0136329	0.00681645	−	0.999977i	$-0.497830\pi$
$331$	0.248029	0.0136329	0.00681645	+	0.999977i	$0.497830\pi$
$332$	1.16415	0.0638909
$333$	−0.0600390	−0.00329011
$334$	46.6187	2.55086
$335$	0	0
$336$	12.6658	0.690974
$337$	−21.1908	−1.15433	−0.577167	−	0.816626i	$-0.695842\pi$
$337$	−21.1908	−1.15433	−0.577167	+	0.816626i	$0.695842\pi$
$338$	50.7731	2.76169
$339$	−21.5463	−1.17023
$340$	0	0
$341$	26.8024	1.45143
$342$	−0.158844	−0.00858929
$343$	17.9115	0.967132
$344$	1.68397	0.0907936
$345$	0	0
$346$	−35.0722	−1.88549
$347$	17.9531	0.963775	0.481887	−	0.876233i	$-0.339951\pi$
$347$	17.9531	0.963775	0.481887	+	0.876233i	$0.339951\pi$
$348$	−19.4802	−1.04425
$349$	7.68481	0.411358	0.205679	−	0.978619i	$-0.434060\pi$
$349$	7.68481	0.411358	0.205679	+	0.978619i	$0.434060\pi$
$350$	0	0
$351$	33.7263	1.80018
$352$	−16.9651	−0.904243
$353$	25.0052	1.33089	0.665446	−	0.746446i	$-0.268241\pi$
$353$	25.0052	1.33089	0.665446	+	0.746446i	$0.268241\pi$
$354$	−31.8018	−1.69025
$355$	0	0
$356$	3.16946	0.167981
$357$	−9.26291	−0.490245
$358$	22.7972	1.20487
$359$	7.21328	0.380703	0.190351	−	0.981716i	$-0.439037\pi$
$359$	7.21328	0.380703	0.190351	+	0.981716i	$0.439037\pi$
$360$	0	0
$361$	−18.4908	−0.973201
$362$	10.3668	0.544867
$363$	−7.75410	−0.406985
$364$	13.6704	0.716525
$365$	0	0
$366$	−28.5618	−1.49295
$367$	10.0531	0.524767	0.262383	−	0.964964i	$-0.415491\pi$
$367$	10.0531	0.524767	0.262383	+	0.964964i	$0.415491\pi$
$368$	8.06520	0.420428
$369$	1.07911	0.0561764
$370$	0	0
$371$	−15.2144	−0.789891
$372$	24.9883	1.29558
$373$	31.7029	1.64151	0.820756	−	0.571279i	$-0.193553\pi$
$373$	31.7029	1.64151	0.820756	+	0.571279i	$0.193553\pi$
$374$	16.5524	0.855904
$375$	0	0
$376$	7.76571	0.400486
$377$	52.4786	2.70278
$378$	15.0277	0.772939
$379$	−8.44277	−0.433676	−0.216838	−	0.976208i	$-0.569574\pi$
$379$	−8.44277	−0.433676	−0.216838	+	0.976208i	$0.569574\pi$
$380$	0	0
$381$	−1.99536	−0.102225
$382$	−29.4933	−1.50901
$383$	−9.65154	−0.493171	−0.246585	−	0.969121i	$-0.579309\pi$
$383$	−9.65154	−0.493171	−0.246585	+	0.969121i	$0.579309\pi$
$384$	14.4729	0.738569
$385$	0	0
$386$	12.7487	0.648891
$387$	0.182082	0.00925574
$388$	10.5412	0.535151
$389$	23.9535	1.21449	0.607246	−	0.794514i	$-0.292274\pi$
$389$	23.9535	1.21449	0.607246	+	0.794514i	$0.292274\pi$
$390$	0	0
$391$	−5.89836	−0.298293
$392$	5.17086	0.261168
$393$	31.8276	1.60549
$394$	−7.51955	−0.378829
$395$	0	0
$396$	−0.426950	−0.0214550
$397$	−26.7766	−1.34388	−0.671940	−	0.740606i	$-0.734539\pi$
$397$	−26.7766	−1.34388	−0.671940	+	0.740606i	$0.734539\pi$
$398$	13.3432	0.668836
$399$	−1.86528	−0.0933810
$400$	0	0
$401$	−25.2168	−1.25927	−0.629635	−	0.776891i	$-0.716795\pi$
$401$	−25.2168	−1.25927	−0.629635	+	0.776891i	$0.716795\pi$
$402$	5.78426	0.288493
$403$	−67.3173	−3.35331
$404$	−23.5917	−1.17373
$405$	0	0
$406$	23.3832	1.16049
$407$	1.25986	0.0624488
$408$	−6.71989	−0.332684
$409$	−13.2389	−0.654619	−0.327310	−	0.944917i	$-0.606142\pi$
$409$	−13.2389	−0.654619	−0.327310	+	0.944917i	$0.606142\pi$
$410$	0	0
$411$	8.55219	0.421848
$412$	−3.94652	−0.194431
$413$	15.6742	0.771275
$414$	0.370533	0.0182107
$415$	0	0
$416$	42.6097	2.08911
$417$	−0.384209	−0.0188148
$418$	3.33318	0.163031
$419$	−18.5429	−0.905880	−0.452940	−	0.891541i	$-0.649625\pi$
$419$	−18.5429	−0.905880	−0.452940	+	0.891541i	$0.649625\pi$
$420$	0	0
$421$	−25.9339	−1.26394	−0.631972	−	0.774992i	$-0.717754\pi$
$421$	−25.9339	−1.26394	−0.631972	+	0.774992i	$0.717754\pi$
$422$	0.181794	0.00884958
$423$	0.839680	0.0408266
$424$	−11.0375	−0.536026
$425$	0	0
$426$	−29.6153	−1.43486
$427$	14.0773	0.681246
$428$	−6.77819	−0.327636
$429$	−27.4036	−1.32306
$430$	0	0
$431$	22.5328	1.08537	0.542683	−	0.839937i	$-0.317408\pi$
$431$	22.5328	1.08537	0.542683	+	0.839937i	$0.317408\pi$
$432$	25.6583	1.23448
$433$	−33.5776	−1.61364	−0.806818	−	0.590801i	$-0.798812\pi$
$433$	−33.5776	−1.61364	−0.806818	+	0.590801i	$0.798812\pi$
$434$	−29.9950	−1.43981
$435$	0	0
$436$	14.3328	0.686416
$437$	−1.18776	−0.0568182
$438$	27.7618	1.32651
$439$	−3.64200	−0.173823	−0.0869116	−	0.996216i	$-0.527700\pi$
$439$	−3.64200	−0.173823	−0.0869116	+	0.996216i	$0.527700\pi$
$440$	0	0
$441$	0.559107	0.0266242
$442$	−41.5732	−1.97743
$443$	27.0746	1.28635	0.643177	−	0.765718i	$-0.277616\pi$
$443$	27.0746	1.28635	0.643177	+	0.765718i	$0.277616\pi$
$444$	1.17458	0.0557433
$445$	0	0
$446$	28.7202	1.35994
$447$	−6.79637	−0.321457
$448$	4.05698	0.191674
$449$	40.1427	1.89445	0.947224	−	0.320571i	$-0.103875\pi$
$449$	40.1427	1.89445	0.947224	+	0.320571i	$0.103875\pi$
$450$	0	0
$451$	−22.6441	−1.06627
$452$	−17.6922	−0.832170
$453$	−9.72407	−0.456877
$454$	45.5751	2.13895
$455$	0	0
$456$	−1.35319	−0.0633691
$457$	−22.3710	−1.04647	−0.523236	−	0.852188i	$-0.675275\pi$
$457$	−22.3710	−1.04647	−0.523236	+	0.852188i	$0.675275\pi$
$458$	−38.7321	−1.80983
$459$	−18.7648	−0.875864
$460$	0	0
$461$	−31.3342	−1.45938	−0.729689	−	0.683780i	$-0.760335\pi$
$461$	−31.3342	−1.45938	−0.729689	+	0.683780i	$0.760335\pi$
$462$	−12.2104	−0.568080
$463$	−9.21061	−0.428053	−0.214027	−	0.976828i	$-0.568658\pi$
$463$	−9.21061	−0.428053	−0.214027	+	0.976828i	$0.568658\pi$
$464$	39.9246	1.85345
$465$	0	0
$466$	−15.7897	−0.731446
$467$	−1.38103	−0.0639065	−0.0319532	−	0.999489i	$-0.510173\pi$
$467$	−1.38103	−0.0639065	−0.0319532	+	0.999489i	$0.510173\pi$
$468$	1.07233	0.0495686
$469$	−2.85089	−0.131642
$470$	0	0
$471$	−6.82520	−0.314489
$472$	11.3710	0.523394
$473$	−3.82081	−0.175681
$474$	−19.1587	−0.879990
$475$	0	0
$476$	−7.60600	−0.348620
$477$	−1.19344	−0.0546440
$478$	41.8286	1.91320
$479$	1.68759	0.0771080	0.0385540	−	0.999257i	$-0.487725\pi$
$479$	1.68759	0.0771080	0.0385540	+	0.999257i	$0.487725\pi$
$480$	0	0
$481$	−3.16427	−0.144278
$482$	1.84209	0.0839048
$483$	4.35112	0.197983
$484$	−6.36709	−0.289413
$485$	0	0
$486$	2.31195	0.104872
$487$	−32.1394	−1.45638	−0.728188	−	0.685377i	$-0.759637\pi$
$487$	−32.1394	−1.45638	−0.728188	+	0.685377i	$0.759637\pi$
$488$	10.2125	0.462299
$489$	4.25904	0.192600
$490$	0	0
$491$	−36.6379	−1.65345	−0.826723	−	0.562608i	$-0.809798\pi$
$491$	−36.6379	−1.65345	−0.826723	+	0.562608i	$0.809798\pi$
$492$	−21.1115	−0.951778
$493$	−29.1982	−1.31502
$494$	−8.37164	−0.376658
$495$	0	0
$496$	−51.2136	−2.29956
$497$	14.5965	0.654742
$498$	2.61161	0.117029
$499$	38.2943	1.71429	0.857145	−	0.515076i	$-0.172236\pi$
$499$	38.2943	1.71429	0.857145	+	0.515076i	$0.172236\pi$
$500$	0	0
$501$	42.9420	1.91851
$502$	−23.8285	−1.06352
$503$	−36.2174	−1.61485	−0.807427	−	0.589967i	$-0.799141\pi$
$503$	−36.2174	−1.61485	−0.807427	+	0.589967i	$0.799141\pi$
$504$	−0.208061	−0.00926777
$505$	0	0
$506$	−7.77525	−0.345652
$507$	46.7688	2.07707
$508$	−1.63844	−0.0726940
$509$	18.3681	0.814153	0.407076	−	0.913394i	$-0.366548\pi$
$509$	18.3681	0.814153	0.407076	+	0.913394i	$0.366548\pi$
$510$	0	0
$511$	−13.6830	−0.605298
$512$	21.5862	0.953985
$513$	−3.77869	−0.166833
$514$	−19.9885	−0.881653
$515$	0	0
$516$	−3.56220	−0.156817
$517$	−17.6198	−0.774920
$518$	−1.40993	−0.0619486
$519$	−32.3062	−1.41808
$520$	0	0
$521$	−37.0378	−1.62265	−0.811327	−	0.584593i	$-0.801254\pi$
$521$	−37.0378	−1.62265	−0.811327	+	0.584593i	$0.801254\pi$
$522$	1.83422	0.0802817
$523$	20.9663	0.916791	0.458395	−	0.888748i	$-0.348424\pi$
$523$	20.9663	0.916791	0.458395	+	0.888748i	$0.348424\pi$
$524$	26.1344	1.14169
$525$	0	0
$526$	25.9793	1.13275
$527$	37.4543	1.63153
$528$	−20.8481	−0.907298
$529$	−20.2293	−0.879536
$530$	0	0
$531$	1.22951	0.0533561
$532$	−1.53163	−0.0664046
$533$	56.8732	2.46345
$534$	7.11029	0.307692
$535$	0	0
$536$	−2.06822	−0.0893334
$537$	20.9993	0.906185
$538$	−13.9201	−0.600137
$539$	−11.7323	−0.505346
$540$	0	0
$541$	9.75202	0.419272	0.209636	−	0.977779i	$-0.432772\pi$
$541$	9.75202	0.419272	0.209636	+	0.977779i	$0.432772\pi$
$542$	38.6060	1.65827
$543$	9.54920	0.409795
$544$	−23.7074	−1.01645
$545$	0	0
$546$	30.6678	1.31246
$547$	30.8874	1.32065	0.660326	−	0.750979i	$-0.270418\pi$
$547$	30.8874	1.32065	0.660326	+	0.750979i	$0.270418\pi$
$548$	7.02241	0.299983
$549$	1.10424	0.0471280
$550$	0	0
$551$	−5.87969	−0.250483
$552$	3.15658	0.134353
$553$	9.44278	0.401548
$554$	−27.5636	−1.17106
$555$	0	0
$556$	−0.315483	−0.0133795
$557$	25.1804	1.06693	0.533463	−	0.845823i	$-0.320890\pi$
$557$	25.1804	1.06693	0.533463	+	0.845823i	$0.320890\pi$
$558$	−2.35286	−0.0996047
$559$	9.59638	0.405884
$560$	0	0
$561$	15.2469	0.643726
$562$	56.6424	2.38931
$563$	37.8072	1.59338	0.796692	−	0.604386i	$-0.206581\pi$
$563$	37.8072	1.59338	0.796692	+	0.604386i	$0.206581\pi$
$564$	−16.4272	−0.691712
$565$	0	0
$566$	7.48080	0.314441
$567$	13.2840	0.557874
$568$	10.5892	0.444313
$569$	−9.89011	−0.414615	−0.207307	−	0.978276i	$-0.566470\pi$
$569$	−9.89011	−0.414615	−0.207307	+	0.978276i	$0.566470\pi$
$570$	0	0
$571$	−20.8124	−0.870972	−0.435486	−	0.900196i	$-0.643423\pi$
$571$	−20.8124	−0.870972	−0.435486	+	0.900196i	$0.643423\pi$
$572$	−22.5018	−0.940848
$573$	−27.1673	−1.13493
$574$	25.3414	1.05773
$575$	0	0
$576$	0.318237	0.0132599
$577$	22.7214	0.945906	0.472953	−	0.881088i	$-0.343188\pi$
$577$	22.7214	0.945906	0.472953	+	0.881088i	$0.343188\pi$
$578$	−8.18487	−0.340446
$579$	11.7432	0.488032
$580$	0	0
$581$	−1.28719	−0.0534016
$582$	23.6479	0.980238
$583$	25.0432	1.03718
$584$	−9.92647	−0.410760
$585$	0	0
$586$	−34.4930	−1.42489
$587$	26.2568	1.08373	0.541866	−	0.840465i	$-0.317718\pi$
$587$	26.2568	1.08373	0.541866	+	0.840465i	$0.317718\pi$
$588$	−10.9382	−0.451084
$589$	7.54222	0.310772
$590$	0	0
$591$	−6.92650	−0.284918
$592$	−2.40731	−0.0989399
$593$	−16.3255	−0.670406	−0.335203	−	0.942146i	$-0.608805\pi$
$593$	−16.3255	−0.670406	−0.335203	+	0.942146i	$0.608805\pi$
$594$	−24.7358	−1.01492
$595$	0	0
$596$	−5.58067	−0.228593
$597$	12.2909	0.503032
$598$	19.5284	0.798576
$599$	−45.7396	−1.86887	−0.934436	−	0.356132i	$-0.884095\pi$
$599$	−45.7396	−1.86887	−0.934436	+	0.356132i	$0.884095\pi$
$600$	0	0
$601$	−17.3734	−0.708676	−0.354338	−	0.935117i	$-0.615294\pi$
$601$	−17.3734	−0.708676	−0.354338	+	0.935117i	$0.615294\pi$
$602$	4.27592	0.174274
$603$	−0.223629	−0.00910688
$604$	−7.98467	−0.324892
$605$	0	0
$606$	−52.9250	−2.14993
$607$	4.48838	0.182178	0.0910890	−	0.995843i	$-0.470965\pi$
$607$	4.48838	0.182178	0.0910890	+	0.995843i	$0.470965\pi$
$608$	−4.77399	−0.193611
$609$	21.5391	0.872807
$610$	0	0
$611$	44.2542	1.79033
$612$	−0.596628	−0.0241173
$613$	−29.9448	−1.20946	−0.604730	−	0.796430i	$-0.706719\pi$
$613$	−29.9448	−1.20946	−0.604730	+	0.796430i	$0.706719\pi$
$614$	−4.44028	−0.179195
$615$	0	0
$616$	4.36595	0.175909
$617$	−48.0028	−1.93252	−0.966261	−	0.257566i	$-0.917079\pi$
$617$	−48.0028	−1.93252	−0.966261	+	0.257566i	$0.917079\pi$
$618$	−8.85352	−0.356141
$619$	−6.70266	−0.269403	−0.134701	−	0.990886i	$-0.543008\pi$
$619$	−6.70266	−0.269403	−0.134701	+	0.990886i	$0.543008\pi$
$620$	0	0
$621$	8.81449	0.353713
$622$	−29.5678	−1.18556
$623$	−3.50445	−0.140403
$624$	52.3624	2.09617
$625$	0	0
$626$	11.0790	0.442805
$627$	3.07030	0.122616
$628$	−5.60434	−0.223638
$629$	1.76055	0.0701977
$630$	0	0
$631$	−7.36072	−0.293025	−0.146513	−	0.989209i	$-0.546805\pi$
$631$	−7.36072	−0.293025	−0.146513	+	0.989209i	$0.546805\pi$
$632$	6.85039	0.272494
$633$	0.167456	0.00665579
$634$	50.7331	2.01487
$635$	0	0
$636$	23.3481	0.925814
$637$	29.4670	1.16752
$638$	−38.4893	−1.52381
$639$	1.14497	0.0452945
$640$	0	0
$641$	−37.1888	−1.46887	−0.734434	−	0.678680i	$-0.762552\pi$
$641$	−37.1888	−1.46887	−0.734434	+	0.678680i	$0.762552\pi$
$642$	−15.2060	−0.600133
$643$	27.0935	1.06846	0.534232	−	0.845338i	$-0.320601\pi$
$643$	27.0935	1.06846	0.534232	+	0.845338i	$0.320601\pi$
$644$	3.57281	0.140789
$645$	0	0
$646$	4.65785	0.183261
$647$	16.1175	0.633646	0.316823	−	0.948485i	$-0.397384\pi$
$647$	16.1175	0.633646	0.316823	+	0.948485i	$0.397384\pi$
$648$	9.63702	0.378578
$649$	−25.8000	−1.01274
$650$	0	0
$651$	−27.6294	−1.08288
$652$	3.49720	0.136961
$653$	−11.7529	−0.459928	−0.229964	−	0.973199i	$-0.573861\pi$
$653$	−11.7529	−0.459928	−0.229964	+	0.973199i	$0.573861\pi$
$654$	32.1538	1.25731
$655$	0	0
$656$	43.2680	1.68933
$657$	−1.07331	−0.0418740
$658$	19.7186	0.768713
$659$	−12.7347	−0.496075	−0.248038	−	0.968750i	$-0.579786\pi$
$659$	−12.7347	−0.496075	−0.248038	+	0.968750i	$0.579786\pi$
$660$	0	0
$661$	−2.32682	−0.0905028	−0.0452514	−	0.998976i	$-0.514409\pi$
$661$	−2.32682	−0.0905028	−0.0452514	+	0.998976i	$0.514409\pi$
$662$	0.456891	0.0177576
$663$	−38.2944	−1.48723
$664$	−0.933807	−0.0362387
$665$	0	0
$666$	−0.110597	−0.00428555
$667$	13.7155	0.531065
$668$	35.2607	1.36428
$669$	26.4551	1.02281
$670$	0	0
$671$	−23.1715	−0.894525
$672$	17.4885	0.674635
$673$	21.2017	0.817267	0.408633	−	0.912699i	$-0.366005\pi$
$673$	21.2017	0.817267	0.408633	+	0.912699i	$0.366005\pi$
$674$	−39.0353	−1.50358
$675$	0	0
$676$	38.4030	1.47704
$677$	51.8603	1.99315	0.996576	−	0.0826794i	$-0.0263477\pi$
$677$	51.8603	1.99315	0.996576	+	0.0826794i	$0.0263477\pi$
$678$	−39.6902	−1.52429
$679$	−11.6554	−0.447292
$680$	0	0
$681$	41.9807	1.60870
$682$	49.3725	1.89057
$683$	22.4656	0.859621	0.429811	−	0.902919i	$-0.358580\pi$
$683$	22.4656	0.859621	0.429811	+	0.902919i	$0.358580\pi$
$684$	−0.120144	−0.00459381
$685$	0	0
$686$	32.9946	1.25974
$687$	−35.6774	−1.36118
$688$	7.30073	0.278338
$689$	−62.8987	−2.39625
$690$	0	0
$691$	51.2504	1.94966	0.974829	−	0.222953i	$-0.0715698\pi$
$691$	51.2504	1.94966	0.974829	+	0.222953i	$0.0715698\pi$
$692$	−26.5274	−1.00842
$693$	0.472075	0.0179327
$694$	33.0713	1.25537
$695$	0	0
$696$	15.6258	0.592294
$697$	−31.6433	−1.19858
$698$	14.1561	0.535816
$699$	−14.5444	−0.550121
$700$	0	0
$701$	0.425393	0.0160669	0.00803344	−	0.999968i	$-0.497443\pi$
$701$	0.425393	0.0160669	0.00803344	+	0.999968i	$0.497443\pi$
$702$	62.1268	2.34483
$703$	0.354524	0.0133711
$704$	−6.67788	−0.251682
$705$	0	0
$706$	46.0618	1.73356
$707$	26.0852	0.981035
$708$	−24.0537	−0.903995
$709$	−15.9902	−0.600523	−0.300261	−	0.953857i	$-0.597074\pi$
$709$	−15.9902	−0.600523	−0.300261	+	0.953857i	$0.597074\pi$
$710$	0	0
$711$	0.740709	0.0277787
$712$	−2.54235	−0.0952785
$713$	−17.5936	−0.658886
$714$	−17.0631	−0.638570
$715$	0	0
$716$	17.2430	0.644402
$717$	38.5297	1.43892
$718$	13.2875	0.495885
$719$	−26.2235	−0.977973	−0.488986	−	0.872291i	$-0.662633\pi$
$719$	−26.2235	−0.977973	−0.488986	+	0.872291i	$0.662633\pi$
$720$	0	0
$721$	4.36364	0.162510
$722$	−34.0617	−1.26765
$723$	1.69681	0.0631050
$724$	7.84108	0.291412
$725$	0	0
$726$	−14.2837	−0.530120
$727$	−25.2570	−0.936730	−0.468365	−	0.883535i	$-0.655157\pi$
$727$	−25.2570	−0.936730	−0.468365	+	0.883535i	$0.655157\pi$
$728$	−10.9656	−0.406411
$729$	27.9982	1.03697
$730$	0	0
$731$	−5.33927	−0.197480
$732$	−21.6031	−0.798474
$733$	8.92252	0.329561	0.164780	−	0.986330i	$-0.447308\pi$
$733$	8.92252	0.329561	0.164780	+	0.986330i	$0.447308\pi$
$734$	18.5187	0.683537
$735$	0	0
$736$	11.1362	0.410486
$737$	4.69263	0.172855
$738$	1.98782	0.0731728
$739$	−14.2439	−0.523970	−0.261985	−	0.965072i	$-0.584377\pi$
$739$	−14.2439	−0.523970	−0.261985	+	0.965072i	$0.584377\pi$
$740$	0	0
$741$	−7.71139	−0.283285
$742$	−28.0262	−1.02887
$743$	14.9854	0.549762	0.274881	−	0.961478i	$-0.411362\pi$
$743$	14.9854	0.549762	0.274881	+	0.961478i	$0.411362\pi$
$744$	−20.0441	−0.734852
$745$	0	0
$746$	58.3995	2.13816
$747$	−0.100969	−0.00369427
$748$	12.5196	0.457763
$749$	7.49459	0.273847
$750$	0	0
$751$	−29.1216	−1.06266	−0.531330	−	0.847165i	$-0.678308\pi$
$751$	−29.1216	−1.06266	−0.531330	+	0.847165i	$0.678308\pi$
$752$	33.6677	1.22773
$753$	−21.9492	−0.799875
$754$	96.6702	3.52052
$755$	0	0
$756$	11.3664	0.413392
$757$	−29.9865	−1.08988	−0.544940	−	0.838475i	$-0.683447\pi$
$757$	−29.9865	−1.08988	−0.544940	+	0.838475i	$0.683447\pi$
$758$	−15.5523	−0.564886
$759$	−7.16204	−0.259966
$760$	0	0
$761$	2.01232	0.0729466	0.0364733	−	0.999335i	$-0.488388\pi$
$761$	2.01232	0.0729466	0.0364733	+	0.999335i	$0.488388\pi$
$762$	−3.67563	−0.133154
$763$	−15.8477	−0.573724
$764$	−22.3077	−0.807065
$765$	0	0
$766$	−17.7790	−0.642381
$767$	64.7996	2.33978
$768$	35.5974	1.28451
$769$	46.1301	1.66350	0.831748	−	0.555154i	$-0.187341\pi$
$769$	46.1301	1.66350	0.831748	+	0.555154i	$0.187341\pi$
$770$	0	0
$771$	−18.4120	−0.663092
$772$	9.64266	0.347047
$773$	−48.0505	−1.72826	−0.864128	−	0.503272i	$-0.832130\pi$
$773$	−48.0505	−1.72826	−0.864128	+	0.503272i	$0.832130\pi$
$774$	0.335411	0.0120561
$775$	0	0
$776$	−8.45554	−0.303536
$777$	−1.29873	−0.0465916
$778$	44.1245	1.58194
$779$	−6.37206	−0.228303
$780$	0	0
$781$	−24.0261	−0.859723
$782$	−10.8653	−0.388542
$783$	43.6338	1.55934
$784$	22.4179	0.800638
$785$	0	0
$786$	58.6292	2.09124
$787$	−36.3244	−1.29482	−0.647412	−	0.762140i	$-0.724149\pi$
$787$	−36.3244	−1.29482	−0.647412	+	0.762140i	$0.724149\pi$
$788$	−5.68752	−0.202610
$789$	23.9303	0.851943
$790$	0	0
$791$	19.5621	0.695548
$792$	0.342473	0.0121692
$793$	58.1977	2.06666
$794$	−49.3249	−1.75048
$795$	0	0
$796$	10.0923	0.357714
$797$	−36.6218	−1.29721	−0.648606	−	0.761124i	$-0.724648\pi$
$797$	−36.6218	−1.29721	−0.648606	+	0.761124i	$0.724648\pi$
$798$	−3.43602	−0.121634
$799$	−24.6223	−0.871075
$800$	0	0
$801$	−0.274895	−0.00971295
$802$	−46.4517	−1.64027
$803$	22.5224	0.794799
$804$	4.37501	0.154295
$805$	0	0
$806$	−124.004	−4.36787
$807$	−12.8222	−0.451364
$808$	18.9238	0.665738
$809$	−34.5599	−1.21506	−0.607530	−	0.794297i	$-0.707840\pi$
$809$	−34.5599	−1.21506	−0.607530	+	0.794297i	$0.707840\pi$
$810$	0	0
$811$	−17.3270	−0.608435	−0.304217	−	0.952603i	$-0.598395\pi$
$811$	−17.3270	−0.608435	−0.304217	+	0.952603i	$0.598395\pi$
$812$	17.6863	0.620666
$813$	35.5613	1.24719
$814$	2.32077	0.0813429
$815$	0	0
$816$	−29.1336	−1.01988
$817$	−1.07518	−0.0376156
$818$	−24.3871	−0.852677
$819$	−1.18567	−0.0414306
$820$	0	0
$821$	13.5004	0.471169	0.235584	−	0.971854i	$-0.424300\pi$
$821$	13.5004	0.471169	0.235584	+	0.971854i	$0.424300\pi$
$822$	15.7539	0.549480
$823$	−46.7783	−1.63059	−0.815295	−	0.579046i	$-0.803425\pi$
$823$	−46.7783	−1.63059	−0.815295	+	0.579046i	$0.803425\pi$
$824$	3.16566	0.110281
$825$	0	0
$826$	28.8732	1.00463
$827$	−44.0656	−1.53231	−0.766155	−	0.642656i	$-0.777833\pi$
$827$	−44.0656	−1.53231	−0.766155	+	0.642656i	$0.777833\pi$
$828$	0.280258	0.00973963
$829$	−4.27440	−0.148456	−0.0742280	−	0.997241i	$-0.523649\pi$
$829$	−4.27440	−0.148456	−0.0742280	+	0.997241i	$0.523649\pi$
$830$	0	0
$831$	−25.3897	−0.880759
$832$	16.7722	0.581473
$833$	−16.3950	−0.568051
$834$	−0.707746	−0.0245073
$835$	0	0
$836$	2.52110	0.0871940
$837$	−55.9716	−1.93466
$838$	−34.1577	−1.17996
$839$	−31.6115	−1.09135	−0.545675	−	0.837997i	$-0.683727\pi$
$839$	−31.6115	−1.09135	−0.545675	+	0.837997i	$0.683727\pi$
$840$	0	0
$841$	38.8947	1.34120
$842$	−47.7726	−1.64635
$843$	52.1751	1.79701
$844$	0.137502	0.00473303
$845$	0	0
$846$	1.54677	0.0531789
$847$	7.04004	0.241899
$848$	−47.8521	−1.64325
$849$	6.89080	0.236492
$850$	0	0
$851$	−0.826993	−0.0283490
$852$	−22.3999	−0.767409
$853$	−5.03330	−0.172337	−0.0861684	−	0.996281i	$-0.527462\pi$
$853$	−5.03330	−0.172337	−0.0861684	+	0.996281i	$0.527462\pi$
$854$	25.9316	0.887359
$855$	0	0
$856$	5.43705	0.185834
$857$	−17.8339	−0.609195	−0.304598	−	0.952481i	$-0.598522\pi$
$857$	−17.8339	−0.609195	−0.304598	+	0.952481i	$0.598522\pi$
$858$	−50.4799	−1.72336
$859$	29.9247	1.02102	0.510509	−	0.859873i	$-0.329457\pi$
$859$	29.9247	1.02102	0.510509	+	0.859873i	$0.329457\pi$
$860$	0	0
$861$	23.3428	0.795519
$862$	41.5074	1.41375
$863$	−21.1041	−0.718392	−0.359196	−	0.933262i	$-0.616949\pi$
$863$	−21.1041	−0.718392	−0.359196	+	0.933262i	$0.616949\pi$
$864$	35.4282	1.20529
$865$	0	0
$866$	−61.8529	−2.10185
$867$	−7.53935	−0.256050
$868$	−22.6872	−0.770053
$869$	−15.5430	−0.527261
$870$	0	0
$871$	−11.7861	−0.399356
$872$	−11.4969	−0.389334
$873$	−0.914268	−0.0309433
$874$	−2.18796	−0.0740088
$875$	0	0
$876$	20.9980	0.709457
$877$	12.0202	0.405892	0.202946	−	0.979190i	$-0.434948\pi$
$877$	12.0202	0.405892	0.202946	+	0.979190i	$0.434948\pi$
$878$	−6.70889	−0.226414
$879$	−31.7726	−1.07166
$880$	0	0
$881$	−15.0651	−0.507556	−0.253778	−	0.967263i	$-0.581673\pi$
$881$	−15.0651	−0.507556	−0.253778	+	0.967263i	$0.581673\pi$
$882$	1.02993	0.0346794
$883$	−3.17468	−0.106836	−0.0534182	−	0.998572i	$-0.517012\pi$
$883$	−3.17468	−0.106836	−0.0534182	+	0.998572i	$0.517012\pi$
$884$	−31.4445	−1.05759
$885$	0	0
$886$	49.8738	1.67554
$887$	−33.6952	−1.13137	−0.565687	−	0.824620i	$-0.691389\pi$
$887$	−33.6952	−1.13137	−0.565687	+	0.824620i	$0.691389\pi$
$888$	−0.942179	−0.0316175
$889$	1.81161	0.0607594
$890$	0	0
$891$	−21.8657	−0.732528
$892$	21.7229	0.727337
$893$	−4.95823	−0.165921
$894$	−12.5195	−0.418715
$895$	0	0
$896$	−13.1401	−0.438981
$897$	17.9883	0.600610
$898$	73.9463	2.46762
$899$	−87.0925	−2.90470
$900$	0	0
$901$	34.9958	1.16588
$902$	−41.7125	−1.38887
$903$	3.93869	0.131071
$904$	14.1916	0.472005
$905$	0	0
$906$	−17.9126	−0.595106
$907$	15.5495	0.516311	0.258156	−	0.966103i	$-0.416885\pi$
$907$	15.5495	0.516311	0.258156	+	0.966103i	$0.416885\pi$
$908$	34.4714	1.14397
$909$	2.04617	0.0678671
$910$	0	0
$911$	47.4791	1.57305	0.786527	−	0.617556i	$-0.211877\pi$
$911$	47.4791	1.57305	0.786527	+	0.617556i	$0.211877\pi$
$912$	−5.86667	−0.194265
$913$	2.11874	0.0701200
$914$	−41.2094	−1.36309
$915$	0	0
$916$	−29.2956	−0.967954
$917$	−28.8966	−0.954251
$918$	−34.5664	−1.14086
$919$	−43.8093	−1.44514	−0.722568	−	0.691299i	$-0.757039\pi$
$919$	−43.8093	−1.44514	−0.722568	+	0.691299i	$0.757039\pi$
$920$	0	0
$921$	−4.09008	−0.134773
$922$	−57.7203	−1.90092
$923$	60.3443	1.98626
$924$	−9.23554	−0.303827
$925$	0	0
$926$	−16.9668	−0.557562
$927$	0.342292	0.0112423
$928$	55.1268	1.80963
$929$	−23.2341	−0.762285	−0.381143	−	0.924516i	$-0.624469\pi$
$929$	−23.2341	−0.762285	−0.381143	+	0.924516i	$0.624469\pi$
$930$	0	0
$931$	−3.30147	−0.108201
$932$	−11.9428	−0.391200
$933$	−27.2359	−0.891663
$934$	−2.54398	−0.0832416
$935$	0	0
$936$	−0.860159	−0.0281152
$937$	26.0794	0.851975	0.425988	−	0.904729i	$-0.359927\pi$
$937$	26.0794	0.851975	0.425988	+	0.904729i	$0.359927\pi$
$938$	−5.25160	−0.171471
$939$	10.2052	0.333034
$940$	0	0
$941$	−23.0611	−0.751771	−0.375885	−	0.926666i	$-0.622661\pi$
$941$	−23.0611	−0.751771	−0.375885	+	0.926666i	$0.622661\pi$
$942$	−12.5726	−0.409638
$943$	14.8640	0.484039
$944$	49.2982	1.60452
$945$	0	0
$946$	−7.03826	−0.228834
$947$	−37.2266	−1.20970	−0.604851	−	0.796339i	$-0.706767\pi$
$947$	−37.2266	−1.20970	−0.604851	+	0.796339i	$0.706767\pi$
$948$	−14.4910	−0.470646
$949$	−56.5676	−1.83626
$950$	0	0
$951$	46.7319	1.51539
$952$	6.10107	0.197737
$953$	23.0528	0.746753	0.373377	−	0.927680i	$-0.378200\pi$
$953$	23.0528	0.746753	0.373377	+	0.927680i	$0.378200\pi$
$954$	−2.19843	−0.0711767
$955$	0	0
$956$	31.6377	1.02324
$957$	−35.4538	−1.14606
$958$	3.10869	0.100437
$959$	−7.76463	−0.250733
$960$	0	0
$961$	80.7187	2.60383
$962$	−5.82887	−0.187930
$963$	0.587889	0.0189445
$964$	1.39329	0.0448749
$965$	0	0
$966$	8.01515	0.257883
$967$	−29.7340	−0.956181	−0.478090	−	0.878311i	$-0.658671\pi$
$967$	−29.7340	−0.956181	−0.478090	+	0.878311i	$0.658671\pi$
$968$	5.10729	0.164154
$969$	4.29049	0.137831
$970$	0	0
$971$	−16.0126	−0.513868	−0.256934	−	0.966429i	$-0.582712\pi$
$971$	−16.0126	−0.513868	−0.256934	+	0.966429i	$0.582712\pi$
$972$	1.74868	0.0560888
$973$	0.348827	0.0111829
$974$	−59.2037	−1.89701
$975$	0	0
$976$	44.2756	1.41723
$977$	−16.8004	−0.537493	−0.268747	−	0.963211i	$-0.586609\pi$
$977$	−16.8004	−0.537493	−0.268747	+	0.963211i	$0.586609\pi$
$978$	7.84553	0.250872
$979$	5.76840	0.184359
$980$	0	0
$981$	−1.24312	−0.0396897
$982$	−67.4903	−2.15370
$983$	19.1223	0.609908	0.304954	−	0.952367i	$-0.401359\pi$
$983$	19.1223	0.609908	0.304954	+	0.952367i	$0.401359\pi$
$984$	16.9343	0.539846
$985$	0	0
$986$	−53.7857	−1.71289
$987$	18.1635	0.578150
$988$	−6.33202	−0.201448
$989$	2.50805	0.0797513
$990$	0	0
$991$	−10.1892	−0.323670	−0.161835	−	0.986818i	$-0.551741\pi$
$991$	−10.1892	−0.323670	−0.161835	+	0.986818i	$0.551741\pi$
$992$	−70.7144	−2.24518
$993$	0.420857	0.0133555
$994$	26.8880	0.852837
$995$	0	0
$996$	1.97533	0.0625908
$997$	16.0609	0.508652	0.254326	−	0.967118i	$-0.418146\pi$
$997$	16.0609	0.508652	0.254326	+	0.967118i	$0.418146\pi$
$998$	70.5415	2.23295
$999$	−2.63096	−0.0832399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.38		46
5.2	odd	4		1205.2.b.c.724.38	yes	46
5.3	odd	4		1205.2.b.c.724.9	✓	46
5.4	even	2	inner	6025.2.a.p.1.9		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.9	✓	46	5.3	odd	4
1205.2.b.c.724.38	yes	46	5.2	odd	4
6025.2.a.p.1.9		46	5.4	even	2	inner
6025.2.a.p.1.38		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)

\(f(q)\)	\(=\)	\(q+1.84209 q^{2} +1.69681 q^{3} +1.39329 q^{4} +3.12567 q^{6} -1.54055 q^{7} -1.11761 q^{8} -0.120844 q^{9} +O(q^{10})\)	\(q+1.84209 q^{2} +1.69681 q^{3} +1.39329 q^{4} +3.12567 q^{6} -1.54055 q^{7} -1.11761 q^{8} -0.120844 q^{9} +2.53578 q^{11} +2.36415 q^{12} -6.36889 q^{13} -2.83783 q^{14} -4.84532 q^{16} +3.54355 q^{17} -0.222605 q^{18} +0.713569 q^{19} -2.61402 q^{21} +4.67113 q^{22} -1.66453 q^{23} -1.89637 q^{24} -11.7321 q^{26} -5.29547 q^{27} -2.14644 q^{28} -8.23983 q^{29} +10.5697 q^{31} -6.69029 q^{32} +4.30273 q^{33} +6.52753 q^{34} -0.168370 q^{36} +0.496832 q^{37} +1.31446 q^{38} -10.8068 q^{39} -8.92984 q^{41} -4.81525 q^{42} -1.50676 q^{43} +3.53308 q^{44} -3.06622 q^{46} -6.94849 q^{47} -8.22158 q^{48} -4.62670 q^{49} +6.01272 q^{51} -8.87372 q^{52} +9.87593 q^{53} -9.75473 q^{54} +1.72174 q^{56} +1.21079 q^{57} -15.1785 q^{58} -10.1744 q^{59} -9.13781 q^{61} +19.4703 q^{62} +0.186166 q^{63} -2.63346 q^{64} +7.92601 q^{66} +1.85057 q^{67} +4.93720 q^{68} -2.82439 q^{69} -9.47485 q^{71} +0.135056 q^{72} +8.88186 q^{73} +0.915209 q^{74} +0.994210 q^{76} -3.90650 q^{77} -19.9071 q^{78} -6.12948 q^{79} -8.62287 q^{81} -16.4496 q^{82} +0.835537 q^{83} -3.64209 q^{84} -2.77558 q^{86} -13.9814 q^{87} -2.83402 q^{88} +2.27480 q^{89} +9.81161 q^{91} -2.31918 q^{92} +17.9348 q^{93} -12.7997 q^{94} -11.3521 q^{96} +7.56572 q^{97} -8.52280 q^{98} -0.306433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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