LMFDB - Embedded newform 6025.2.a.p.1.36

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			1.36
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)

$f(q)$	$=$	$q+1.70963 q^{2} +1.38336 q^{3} +0.922819 q^{4} +2.36503 q^{6} +5.05937 q^{7} -1.84158 q^{8} -1.08631 q^{9} +O(q^{10})$	\(q+1.70963 q^{2} +1.38336 q^{3} +0.922819 q^{4} +2.36503 q^{6} +5.05937 q^{7} -1.84158 q^{8} -1.08631 q^{9} -5.25074 q^{11} +1.27659 q^{12} -2.84467 q^{13} +8.64964 q^{14} -4.99404 q^{16} -6.16813 q^{17} -1.85719 q^{18} +1.47178 q^{19} +6.99894 q^{21} -8.97680 q^{22} +2.54443 q^{23} -2.54756 q^{24} -4.86332 q^{26} -5.65285 q^{27} +4.66889 q^{28} -9.53193 q^{29} -7.94968 q^{31} -4.85479 q^{32} -7.26367 q^{33} -10.5452 q^{34} -1.00247 q^{36} +8.85694 q^{37} +2.51619 q^{38} -3.93520 q^{39} +1.60908 q^{41} +11.9656 q^{42} +0.0179130 q^{43} -4.84548 q^{44} +4.35003 q^{46} -5.95134 q^{47} -6.90856 q^{48} +18.5973 q^{49} -8.53275 q^{51} -2.62511 q^{52} -8.07317 q^{53} -9.66425 q^{54} -9.31723 q^{56} +2.03600 q^{57} -16.2960 q^{58} -3.28732 q^{59} +7.19911 q^{61} -13.5910 q^{62} -5.49606 q^{63} +1.68822 q^{64} -12.4182 q^{66} -2.92491 q^{67} -5.69207 q^{68} +3.51987 q^{69} -4.49618 q^{71} +2.00053 q^{72} +4.29619 q^{73} +15.1421 q^{74} +1.35819 q^{76} -26.5655 q^{77} -6.72772 q^{78} +2.34077 q^{79} -4.56099 q^{81} +2.75093 q^{82} -9.27436 q^{83} +6.45875 q^{84} +0.0306245 q^{86} -13.1861 q^{87} +9.66964 q^{88} +13.8608 q^{89} -14.3922 q^{91} +2.34805 q^{92} -10.9973 q^{93} -10.1746 q^{94} -6.71593 q^{96} +11.0653 q^{97} +31.7944 q^{98} +5.70395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.70963	1.20889	0.604444	−	0.796648i	$-0.293395\pi$
$2$	1.70963	1.20889	0.604444	+	0.796648i	$0.293395\pi$
$3$	1.38336	0.798684	0.399342	−	0.916802i	$-0.369239\pi$
$3$	1.38336	0.798684	0.399342	+	0.916802i	$0.369239\pi$
$4$	0.922819	0.461409
$5$	0	0
$5$	0	0
$6$	2.36503	0.965519
$7$	5.05937	1.91226	0.956132	−	0.292936i	$-0.0946324\pi$
$7$	5.05937	1.91226	0.956132	+	0.292936i	$0.0946324\pi$
$8$	−1.84158	−0.651096
$9$	−1.08631	−0.362104
$10$	0	0
$11$	−5.25074	−1.58316	−0.791579	−	0.611066i	$-0.790741\pi$
$11$	−5.25074	−1.58316	−0.791579	+	0.611066i	$0.790741\pi$
$12$	1.27659	0.368520
$13$	−2.84467	−0.788969	−0.394485	−	0.918903i	$-0.629077\pi$
$13$	−2.84467	−0.788969	−0.394485	+	0.918903i	$0.629077\pi$
$14$	8.64964	2.31171
$15$	0	0
$16$	−4.99404	−1.24851
$17$	−6.16813	−1.49599	−0.747996	−	0.663704i	$-0.768984\pi$
$17$	−6.16813	−1.49599	−0.747996	+	0.663704i	$0.768984\pi$
$18$	−1.85719	−0.437743
$19$	1.47178	0.337649	0.168825	−	0.985646i	$-0.446003\pi$
$19$	1.47178	0.337649	0.168825	+	0.985646i	$0.446003\pi$
$20$	0	0
$21$	6.99894	1.52729
$22$	−8.97680	−1.91386
$23$	2.54443	0.530551	0.265275	−	0.964173i	$-0.414537\pi$
$23$	2.54443	0.530551	0.265275	+	0.964173i	$0.414537\pi$
$24$	−2.54756	−0.520019
$25$	0	0
$26$	−4.86332	−0.953775
$27$	−5.65285	−1.08789
$28$	4.66889	0.882336
$29$	−9.53193	−1.77003	−0.885017	−	0.465558i	$-0.845854\pi$
$29$	−9.53193	−1.77003	−0.885017	+	0.465558i	$0.845854\pi$
$30$	0	0
$31$	−7.94968	−1.42780	−0.713902	−	0.700245i	$-0.753074\pi$
$31$	−7.94968	−1.42780	−0.713902	+	0.700245i	$0.753074\pi$
$32$	−4.85479	−0.858214
$33$	−7.26367	−1.26444
$34$	−10.5452	−1.80849
$35$	0	0
$36$	−1.00247	−0.167078
$37$	8.85694	1.45607	0.728036	−	0.685539i	$-0.240433\pi$
$37$	8.85694	1.45607	0.728036	+	0.685539i	$0.240433\pi$
$38$	2.51619	0.408180
$39$	−3.93520	−0.630137
$40$	0	0
$41$	1.60908	0.251297	0.125648	−	0.992075i	$-0.459899\pi$
$41$	1.60908	0.251297	0.125648	+	0.992075i	$0.459899\pi$
$42$	11.9656	1.84633
$43$	0.0179130	0.00273171	0.00136585	−	0.999999i	$-0.499565\pi$
$43$	0.0179130	0.00273171	0.00136585	+	0.999999i	$0.499565\pi$
$44$	−4.84548	−0.730484
$45$	0	0
$46$	4.35003	0.641376
$47$	−5.95134	−0.868092	−0.434046	−	0.900891i	$-0.642915\pi$
$47$	−5.95134	−0.868092	−0.434046	+	0.900891i	$0.642915\pi$
$48$	−6.90856	−0.997165
$49$	18.5973	2.65675
$50$	0	0
$51$	−8.53275	−1.19482
$52$	−2.62511	−0.364038
$53$	−8.07317	−1.10894	−0.554468	−	0.832205i	$-0.687078\pi$
$53$	−8.07317	−1.10894	−0.554468	+	0.832205i	$0.687078\pi$
$54$	−9.66425	−1.31514
$55$	0	0
$56$	−9.31723	−1.24507
$57$	2.03600	0.269675
$58$	−16.2960	−2.13977
$59$	−3.28732	−0.427973	−0.213986	−	0.976837i	$-0.568645\pi$
$59$	−3.28732	−0.427973	−0.213986	+	0.976837i	$0.568645\pi$
$60$	0	0
$61$	7.19911	0.921751	0.460876	−	0.887465i	$-0.347535\pi$
$61$	7.19911	0.921751	0.460876	+	0.887465i	$0.347535\pi$
$62$	−13.5910	−1.72606
$63$	−5.49606	−0.692439
$64$	1.68822	0.211027
$65$	0	0
$66$	−12.4182	−1.52857
$67$	−2.92491	−0.357335	−0.178667	−	0.983910i	$-0.557179\pi$
$67$	−2.92491	−0.357335	−0.178667	+	0.983910i	$0.557179\pi$
$68$	−5.69207	−0.690264
$69$	3.51987	0.423742
$70$	0	0
$71$	−4.49618	−0.533599	−0.266799	−	0.963752i	$-0.585966\pi$
$71$	−4.49618	−0.533599	−0.266799	+	0.963752i	$0.585966\pi$
$72$	2.00053	0.235765
$73$	4.29619	0.502831	0.251416	−	0.967879i	$-0.419104\pi$
$73$	4.29619	0.502831	0.251416	+	0.967879i	$0.419104\pi$
$74$	15.1421	1.76023
$75$	0	0
$76$	1.35819	0.155795
$77$	−26.5655	−3.02742
$78$	−6.72772	−0.761765
$79$	2.34077	0.263357	0.131679	−	0.991292i	$-0.457963\pi$
$79$	2.34077	0.263357	0.131679	+	0.991292i	$0.457963\pi$
$80$	0	0
$81$	−4.56099	−0.506776
$82$	2.75093	0.303790
$83$	−9.27436	−1.01799	−0.508997	−	0.860768i	$-0.669984\pi$
$83$	−9.27436	−1.01799	−0.508997	+	0.860768i	$0.669984\pi$
$84$	6.45875	0.704708
$85$	0	0
$86$	0.0306245	0.00330232
$87$	−13.1861	−1.41370
$88$	9.66964	1.03079
$89$	13.8608	1.46924	0.734619	−	0.678479i	$-0.237361\pi$
$89$	13.8608	1.46924	0.734619	+	0.678479i	$0.237361\pi$
$90$	0	0
$91$	−14.3922	−1.50872
$92$	2.34805	0.244801
$93$	−10.9973	−1.14036
$94$	−10.1746	−1.04943
$95$	0	0
$96$	−6.71593	−0.685441
$97$	11.0653	1.12351	0.561756	−	0.827303i	$-0.310126\pi$
$97$	11.0653	1.12351	0.561756	+	0.827303i	$0.310126\pi$
$98$	31.7944	3.21172
$99$	5.70395	0.573269
$100$	0	0
$101$	0.249085	0.0247849	0.0123924	−	0.999923i	$-0.496055\pi$
$101$	0.249085	0.0247849	0.0123924	+	0.999923i	$0.496055\pi$
$102$	−14.5878	−1.44441
$103$	−13.4549	−1.32575	−0.662875	−	0.748730i	$-0.730664\pi$
$103$	−13.4549	−1.32575	−0.662875	+	0.748730i	$0.730664\pi$
$104$	5.23867	0.513694
$105$	0	0
$106$	−13.8021	−1.34058
$107$	−1.30787	−0.126437	−0.0632185	−	0.998000i	$-0.520136\pi$
$107$	−1.30787	−0.126437	−0.0632185	+	0.998000i	$0.520136\pi$
$108$	−5.21655	−0.501963
$109$	19.2547	1.84426	0.922131	−	0.386878i	$-0.126446\pi$
$109$	19.2547	1.84426	0.922131	+	0.386878i	$0.126446\pi$
$110$	0	0
$111$	12.2523	1.16294
$112$	−25.2667	−2.38748
$113$	−17.0383	−1.60283	−0.801415	−	0.598108i	$-0.795919\pi$
$113$	−17.0383	−1.60283	−0.801415	+	0.598108i	$0.795919\pi$
$114$	3.48080	0.326007
$115$	0	0
$116$	−8.79624	−0.816710
$117$	3.09020	0.285689
$118$	−5.62009	−0.517371
$119$	−31.2069	−2.86073
$120$	0	0
$121$	16.5703	1.50639
$122$	12.3078	1.11429
$123$	2.22594	0.200707
$124$	−7.33611	−0.658802
$125$	0	0
$126$	−9.39621	−0.837081
$127$	10.0749	0.894007	0.447004	−	0.894532i	$-0.352491\pi$
$127$	10.0749	0.894007	0.447004	+	0.894532i	$0.352491\pi$
$128$	12.5958	1.11332
$129$	0.0247801	0.00218177
$130$	0	0
$131$	5.37122	0.469285	0.234643	−	0.972082i	$-0.424608\pi$
$131$	5.37122	0.469285	0.234643	+	0.972082i	$0.424608\pi$
$132$	−6.70305	−0.583426
$133$	7.44628	0.645675
$134$	−5.00050	−0.431978
$135$	0	0
$136$	11.3591	0.974034
$137$	10.2584	0.876436	0.438218	−	0.898869i	$-0.355610\pi$
$137$	10.2584	0.876436	0.438218	+	0.898869i	$0.355610\pi$
$138$	6.01766	0.512257
$139$	14.6133	1.23949	0.619743	−	0.784805i	$-0.287237\pi$
$139$	14.6133	1.23949	0.619743	+	0.784805i	$0.287237\pi$
$140$	0	0
$141$	−8.23285	−0.693331
$142$	−7.68679	−0.645061
$143$	14.9366	1.24906
$144$	5.42509	0.452091
$145$	0	0
$146$	7.34488	0.607866
$147$	25.7267	2.12191
$148$	8.17335	0.671845
$149$	−15.8135	−1.29549	−0.647747	−	0.761856i	$-0.724289\pi$
$149$	−15.8135	−1.29549	−0.647747	+	0.761856i	$0.724289\pi$
$150$	0	0
$151$	−15.2888	−1.24419	−0.622093	−	0.782944i	$-0.713717\pi$
$151$	−15.2888	−1.24419	−0.622093	+	0.782944i	$0.713717\pi$
$152$	−2.71039	−0.219842
$153$	6.70052	0.541705
$154$	−45.4170	−3.65981
$155$	0	0
$156$	−3.63148	−0.290751
$157$	0.857546	0.0684396	0.0342198	−	0.999414i	$-0.489105\pi$
$157$	0.857546	0.0684396	0.0342198	+	0.999414i	$0.489105\pi$
$158$	4.00184	0.318369
$159$	−11.1681	−0.885689
$160$	0	0
$161$	12.8732	1.01455
$162$	−7.79758	−0.612635
$163$	9.36765	0.733731	0.366866	−	0.930274i	$-0.380431\pi$
$163$	9.36765	0.733731	0.366866	+	0.930274i	$0.380431\pi$
$164$	1.48489	0.115951
$165$	0	0
$166$	−15.8557	−1.23064
$167$	2.74053	0.212069	0.106034	−	0.994362i	$-0.466185\pi$
$167$	2.74053	0.212069	0.106034	+	0.994362i	$0.466185\pi$
$168$	−12.8891	−0.994415
$169$	−4.90786	−0.377528
$170$	0	0
$171$	−1.59881	−0.122264
$172$	0.0165304	0.00126043
$173$	1.88911	0.143627	0.0718134	−	0.997418i	$-0.477121\pi$
$173$	1.88911	0.143627	0.0718134	+	0.997418i	$0.477121\pi$
$174$	−22.5433	−1.70900
$175$	0	0
$176$	26.2224	1.97659
$177$	−4.54755	−0.341815
$178$	23.6967	1.77614
$179$	−0.490228	−0.0366413	−0.0183207	−	0.999832i	$-0.505832\pi$
$179$	−0.490228	−0.0366413	−0.0183207	+	0.999832i	$0.505832\pi$
$180$	0	0
$181$	0.282628	0.0210076	0.0105038	−	0.999945i	$-0.496656\pi$
$181$	0.282628	0.0210076	0.0105038	+	0.999945i	$0.496656\pi$
$182$	−24.6053	−1.82387
$183$	9.95896	0.736188
$184$	−4.68577	−0.345439
$185$	0	0
$186$	−18.8012	−1.37857
$187$	32.3873	2.36839
$188$	−5.49201	−0.400546
$189$	−28.5999	−2.08033
$190$	0	0
$191$	18.5700	1.34368	0.671840	−	0.740696i	$-0.265504\pi$
$191$	18.5700	1.34368	0.671840	+	0.740696i	$0.265504\pi$
$192$	2.33541	0.168544
$193$	−25.6312	−1.84498	−0.922488	−	0.386026i	$-0.873847\pi$
$193$	−25.6312	−1.84498	−0.922488	+	0.386026i	$0.873847\pi$
$194$	18.9175	1.35820
$195$	0	0
$196$	17.1619	1.22585
$197$	−12.1900	−0.868499	−0.434250	−	0.900793i	$-0.642986\pi$
$197$	−12.1900	−0.868499	−0.434250	+	0.900793i	$0.642986\pi$
$198$	9.75162	0.693017
$199$	6.70955	0.475628	0.237814	−	0.971311i	$-0.423569\pi$
$199$	6.70955	0.475628	0.237814	+	0.971311i	$0.423569\pi$
$200$	0	0
$201$	−4.04621	−0.285398
$202$	0.425842	0.0299621
$203$	−48.2256	−3.38477
$204$	−7.87418	−0.551303
$205$	0	0
$206$	−23.0028	−1.60268
$207$	−2.76405	−0.192115
$208$	14.2064	0.985036
$209$	−7.72794	−0.534552
$210$	0	0
$211$	7.92574	0.545630	0.272815	−	0.962066i	$-0.412045\pi$
$211$	7.92574	0.545630	0.272815	+	0.962066i	$0.412045\pi$
$212$	−7.45007	−0.511673
$213$	−6.21984	−0.426177
$214$	−2.23597	−0.152848
$215$	0	0
$216$	10.4101	0.708321
$217$	−40.2204	−2.73034
$218$	32.9183	2.22951
$219$	5.94318	0.401603
$220$	0	0
$221$	17.5463	1.18029
$222$	20.9469	1.40587
$223$	−18.7959	−1.25867	−0.629335	−	0.777134i	$-0.716673\pi$
$223$	−18.7959	−1.25867	−0.629335	+	0.777134i	$0.716673\pi$
$224$	−24.5622	−1.64113
$225$	0	0
$226$	−29.1292	−1.93764
$227$	1.76519	0.117160	0.0585799	−	0.998283i	$-0.481343\pi$
$227$	1.76519	0.117160	0.0585799	+	0.998283i	$0.481343\pi$
$228$	1.87886	0.124431
$229$	−21.1776	−1.39945	−0.699726	−	0.714411i	$-0.746695\pi$
$229$	−21.1776	−1.39945	−0.699726	+	0.714411i	$0.746695\pi$
$230$	0	0
$231$	−36.7496	−2.41795
$232$	17.5538	1.15246
$233$	−6.09009	−0.398975	−0.199488	−	0.979900i	$-0.563928\pi$
$233$	−6.09009	−0.398975	−0.199488	+	0.979900i	$0.563928\pi$
$234$	5.28308	0.345366
$235$	0	0
$236$	−3.03360	−0.197471
$237$	3.23813	0.210339
$238$	−53.3521	−3.45830
$239$	−1.39490	−0.0902286	−0.0451143	−	0.998982i	$-0.514365\pi$
$239$	−1.39490	−0.0902286	−0.0451143	+	0.998982i	$0.514365\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	28.3290	1.82106
$243$	10.6490	0.683137
$244$	6.64347	0.425305
$245$	0	0
$246$	3.80553	0.242632
$247$	−4.18672	−0.266395
$248$	14.6399	0.929637
$249$	−12.8298	−0.813055
$250$	0	0
$251$	10.4868	0.661920	0.330960	−	0.943645i	$-0.392627\pi$
$251$	10.4868	0.661920	0.330960	+	0.943645i	$0.392627\pi$
$252$	−5.07187	−0.319498
$253$	−13.3602	−0.839946
$254$	17.2244	1.08075
$255$	0	0
$256$	18.1577	1.13485
$257$	−14.8521	−0.926448	−0.463224	−	0.886241i	$-0.653308\pi$
$257$	−14.8521	−0.926448	−0.463224	+	0.886241i	$0.653308\pi$
$258$	0.0423647	0.00263751
$259$	44.8106	2.78439
$260$	0	0
$261$	10.3547	0.640937
$262$	9.18277	0.567313
$263$	1.20965	0.0745901	0.0372951	−	0.999304i	$-0.488126\pi$
$263$	1.20965	0.0745901	0.0372951	+	0.999304i	$0.488126\pi$
$264$	13.3766	0.823273
$265$	0	0
$266$	12.7304	0.780548
$267$	19.1744	1.17346
$268$	−2.69916	−0.164878
$269$	15.6437	0.953811	0.476905	−	0.878955i	$-0.341758\pi$
$269$	15.6437	0.953811	0.476905	+	0.878955i	$0.341758\pi$
$270$	0	0
$271$	−25.9732	−1.57776	−0.788880	−	0.614547i	$-0.789339\pi$
$271$	−25.9732	−1.57776	−0.788880	+	0.614547i	$0.789339\pi$
$272$	30.8039	1.86776
$273$	−19.9097	−1.20499
$274$	17.5380	1.05951
$275$	0	0
$276$	3.24820	0.195519
$277$	11.7554	0.706312	0.353156	−	0.935564i	$-0.385108\pi$
$277$	11.7554	0.706312	0.353156	+	0.935564i	$0.385108\pi$
$278$	24.9833	1.49840
$279$	8.63584	0.517014
$280$	0	0
$281$	−6.68164	−0.398593	−0.199297	−	0.979939i	$-0.563866\pi$
$281$	−6.68164	−0.398593	−0.199297	+	0.979939i	$0.563866\pi$
$282$	−14.0751	−0.838159
$283$	7.36154	0.437598	0.218799	−	0.975770i	$-0.429786\pi$
$283$	7.36154	0.437598	0.218799	+	0.975770i	$0.429786\pi$
$284$	−4.14916	−0.246207
$285$	0	0
$286$	25.5360	1.50998
$287$	8.14096	0.480546
$288$	5.27382	0.310763
$289$	21.0458	1.23799
$290$	0	0
$291$	15.3073	0.897331
$292$	3.96461	0.232011
$293$	12.0504	0.703992	0.351996	−	0.936001i	$-0.385503\pi$
$293$	12.0504	0.703992	0.351996	+	0.936001i	$0.385503\pi$
$294$	43.9831	2.56515
$295$	0	0
$296$	−16.3107	−0.948042
$297$	29.6816	1.72230
$298$	−27.0352	−1.56611
$299$	−7.23807	−0.418588
$300$	0	0
$301$	0.0906285	0.00522374
$302$	−26.1381	−1.50408
$303$	0.344574	0.0197953
$304$	−7.35013	−0.421559
$305$	0	0
$306$	11.4554	0.654860
$307$	−25.7906	−1.47195	−0.735974	−	0.677010i	$-0.763275\pi$
$307$	−25.7906	−1.47195	−0.735974	+	0.677010i	$0.763275\pi$
$308$	−24.5151	−1.39688
$309$	−18.6130	−1.05886
$310$	0	0
$311$	−19.1825	−1.08774	−0.543869	−	0.839170i	$-0.683041\pi$
$311$	−19.1825	−1.08774	−0.543869	+	0.839170i	$0.683041\pi$
$312$	7.24698	0.410279
$313$	4.33448	0.245000	0.122500	−	0.992469i	$-0.460909\pi$
$313$	4.33448	0.245000	0.122500	+	0.992469i	$0.460909\pi$
$314$	1.46608	0.0827358
$315$	0	0
$316$	2.16011	0.121516
$317$	−7.30667	−0.410383	−0.205192	−	0.978722i	$-0.565782\pi$
$317$	−7.30667	−0.410383	−0.205192	+	0.978722i	$0.565782\pi$
$318$	−19.0933	−1.07070
$319$	50.0497	2.80225
$320$	0	0
$321$	−1.80926	−0.100983
$322$	22.0084	1.22648
$323$	−9.07813	−0.505121
$324$	−4.20896	−0.233831
$325$	0	0
$326$	16.0152	0.886998
$327$	26.6361	1.47298
$328$	−2.96325	−0.163618
$329$	−30.1101	−1.66002
$330$	0	0
$331$	1.02032	0.0560821	0.0280411	−	0.999607i	$-0.491073\pi$
$331$	1.02032	0.0560821	0.0280411	+	0.999607i	$0.491073\pi$
$332$	−8.55856	−0.469712
$333$	−9.62141	−0.527250
$334$	4.68529	0.256367
$335$	0	0
$336$	−34.9530	−1.90684
$337$	−18.9655	−1.03312	−0.516559	−	0.856251i	$-0.672787\pi$
$337$	−18.9655	−1.03312	−0.516559	+	0.856251i	$0.672787\pi$
$338$	−8.39060	−0.456389
$339$	−23.5702	−1.28015
$340$	0	0
$341$	41.7417	2.26044
$342$	−2.73337	−0.147804
$343$	58.6750	3.16815
$344$	−0.0329881	−0.00177860
$345$	0	0
$346$	3.22968	0.173629
$347$	31.3900	1.68510	0.842550	−	0.538617i	$-0.181053\pi$
$347$	31.3900	1.68510	0.842550	+	0.538617i	$0.181053\pi$
$348$	−12.1684	−0.652293
$349$	−12.0895	−0.647139	−0.323569	−	0.946204i	$-0.604883\pi$
$349$	−12.0895	−0.647139	−0.323569	+	0.946204i	$0.604883\pi$
$350$	0	0
$351$	16.0805	0.858312
$352$	25.4913	1.35869
$353$	4.93700	0.262770	0.131385	−	0.991331i	$-0.458058\pi$
$353$	4.93700	0.262770	0.131385	+	0.991331i	$0.458058\pi$
$354$	−7.77461	−0.413216
$355$	0	0
$356$	12.7910	0.677921
$357$	−43.1704	−2.28482
$358$	−0.838106	−0.0442953
$359$	0.458022	0.0241735	0.0120867	−	0.999927i	$-0.496153\pi$
$359$	0.458022	0.0241735	0.0120867	+	0.999927i	$0.496153\pi$
$360$	0	0
$361$	−16.8339	−0.885993
$362$	0.483189	0.0253958
$363$	22.9227	1.20313
$364$	−13.2814	−0.696136
$365$	0	0
$366$	17.0261	0.889968
$367$	16.2642	0.848987	0.424493	−	0.905431i	$-0.360452\pi$
$367$	16.2642	0.848987	0.424493	+	0.905431i	$0.360452\pi$
$368$	−12.7070	−0.662398
$369$	−1.74797	−0.0909957
$370$	0	0
$371$	−40.8452	−2.12058
$372$	−10.1485	−0.526175
$373$	−19.6620	−1.01806	−0.509029	−	0.860749i	$-0.669995\pi$
$373$	−19.6620	−1.01806	−0.509029	+	0.860749i	$0.669995\pi$
$374$	55.3701	2.86312
$375$	0	0
$376$	10.9598	0.565211
$377$	27.1152	1.39650
$378$	−48.8950	−2.51489
$379$	−38.4555	−1.97533	−0.987664	−	0.156587i	$-0.949951\pi$
$379$	−38.4555	−1.97533	−0.987664	+	0.156587i	$0.949951\pi$
$380$	0	0
$381$	13.9373	0.714029
$382$	31.7478	1.62436
$383$	25.5844	1.30730	0.653652	−	0.756796i	$-0.273236\pi$
$383$	25.5844	1.30730	0.653652	+	0.756796i	$0.273236\pi$
$384$	17.4245	0.889192
$385$	0	0
$386$	−43.8198	−2.23037
$387$	−0.0194591	−0.000989162 0
$388$	10.2113	0.518399
$389$	1.54506	0.0783376	0.0391688	−	0.999233i	$-0.487529\pi$
$389$	1.54506	0.0783376	0.0391688	+	0.999233i	$0.487529\pi$
$390$	0	0
$391$	−15.6944	−0.793699
$392$	−34.2483	−1.72980
$393$	7.43033	0.374811
$394$	−20.8403	−1.04992
$395$	0	0
$396$	5.26371	0.264511
$397$	15.9215	0.799078	0.399539	−	0.916716i	$-0.369170\pi$
$397$	15.9215	0.799078	0.399539	+	0.916716i	$0.369170\pi$
$398$	11.4708	0.574980
$399$	10.3009	0.515690
$400$	0	0
$401$	−16.9817	−0.848024	−0.424012	−	0.905657i	$-0.639379\pi$
$401$	−16.9817	−0.848024	−0.424012	+	0.905657i	$0.639379\pi$
$402$	−6.91750	−0.345014
$403$	22.6142	1.12649
$404$	0.229860	0.0114360
$405$	0	0
$406$	−82.4477	−4.09181
$407$	−46.5055	−2.30519
$408$	15.7137	0.777945
$409$	−26.2104	−1.29602	−0.648010	−	0.761631i	$-0.724399\pi$
$409$	−26.2104	−1.29602	−0.648010	+	0.761631i	$0.724399\pi$
$410$	0	0
$411$	14.1911	0.699995
$412$	−12.4164	−0.611714
$413$	−16.6318	−0.818397
$414$	−4.72549	−0.232245
$415$	0	0
$416$	13.8103	0.677104
$417$	20.2155	0.989957
$418$	−13.2119	−0.646214
$419$	−14.5100	−0.708862	−0.354431	−	0.935082i	$-0.615325\pi$
$419$	−14.5100	−0.708862	−0.354431	+	0.935082i	$0.615325\pi$
$420$	0	0
$421$	14.9113	0.726732	0.363366	−	0.931646i	$-0.381627\pi$
$421$	14.9113	0.726732	0.363366	+	0.931646i	$0.381627\pi$
$422$	13.5500	0.659606
$423$	6.46502	0.314340
$424$	14.8674	0.722023
$425$	0	0
$426$	−10.6336	−0.515200
$427$	36.4230	1.76263
$428$	−1.20693	−0.0583392
$429$	20.6627	0.997606
$430$	0	0
$431$	−27.0648	−1.30367	−0.651833	−	0.758363i	$-0.726000\pi$
$431$	−27.0648	−1.30367	−0.651833	+	0.758363i	$0.726000\pi$
$432$	28.2306	1.35824
$433$	−32.9065	−1.58139	−0.790693	−	0.612212i	$-0.790280\pi$
$433$	−32.9065	−1.58139	−0.790693	+	0.612212i	$0.790280\pi$
$434$	−68.7618	−3.30067
$435$	0	0
$436$	17.7686	0.850960
$437$	3.74484	0.179140
$438$	10.1606	0.485493
$439$	−18.6369	−0.889490	−0.444745	−	0.895657i	$-0.646706\pi$
$439$	−18.6369	−0.889490	−0.444745	+	0.895657i	$0.646706\pi$
$440$	0	0
$441$	−20.2025	−0.962022
$442$	29.9976	1.42684
$443$	−0.950840	−0.0451758	−0.0225879	−	0.999745i	$-0.507191\pi$
$443$	−0.950840	−0.0451758	−0.0225879	+	0.999745i	$0.507191\pi$
$444$	11.3067	0.536592
$445$	0	0
$446$	−32.1340	−1.52159
$447$	−21.8758	−1.03469
$448$	8.54132	0.403539
$449$	12.1598	0.573858	0.286929	−	0.957952i	$-0.407366\pi$
$449$	12.1598	0.573858	0.286929	+	0.957952i	$0.407366\pi$
$450$	0	0
$451$	−8.44889	−0.397843
$452$	−15.7233	−0.739561
$453$	−21.1499	−0.993710
$454$	3.01781	0.141633
$455$	0	0
$456$	−3.74945	−0.175584
$457$	−34.9685	−1.63576	−0.817880	−	0.575389i	$-0.804851\pi$
$457$	−34.9685	−1.63576	−0.817880	+	0.575389i	$0.804851\pi$
$458$	−36.2057	−1.69178
$459$	34.8675	1.62747
$460$	0	0
$461$	−10.2125	−0.475646	−0.237823	−	0.971309i	$-0.576434\pi$
$461$	−10.2125	−0.475646	−0.237823	+	0.971309i	$0.576434\pi$
$462$	−62.8281	−2.92303
$463$	−17.4281	−0.809951	−0.404976	−	0.914328i	$-0.632720\pi$
$463$	−17.4281	−0.809951	−0.404976	+	0.914328i	$0.632720\pi$
$464$	47.6029	2.20991
$465$	0	0
$466$	−10.4118	−0.482316
$467$	−18.9150	−0.875281	−0.437640	−	0.899150i	$-0.644186\pi$
$467$	−18.9150	−0.875281	−0.437640	+	0.899150i	$0.644186\pi$
$468$	2.85169	0.131820
$469$	−14.7982	−0.683318
$470$	0	0
$471$	1.18630	0.0546616
$472$	6.05385	0.278651
$473$	−0.0940565	−0.00432472
$474$	5.53599	0.254277
$475$	0	0
$476$	−28.7983	−1.31997
$477$	8.76999	0.401550
$478$	−2.38476	−0.109076
$479$	−19.0476	−0.870306	−0.435153	−	0.900357i	$-0.643306\pi$
$479$	−19.0476	−0.870306	−0.435153	+	0.900357i	$0.643306\pi$
$480$	0	0
$481$	−25.1951	−1.14880
$482$	1.70963	0.0778713
$483$	17.8083	0.810307
$484$	15.2914	0.695063
$485$	0	0
$486$	18.2059	0.825835
$487$	22.4723	1.01832	0.509159	−	0.860673i	$-0.329957\pi$
$487$	22.4723	1.01832	0.509159	+	0.860673i	$0.329957\pi$
$488$	−13.2577	−0.600148
$489$	12.9588	0.586019
$490$	0	0
$491$	−7.02401	−0.316989	−0.158495	−	0.987360i	$-0.550664\pi$
$491$	−7.02401	−0.316989	−0.158495	+	0.987360i	$0.550664\pi$
$492$	2.05414	0.0926079
$493$	58.7942	2.64796
$494$	−7.15773	−0.322041
$495$	0	0
$496$	39.7010	1.78263
$497$	−22.7479	−1.02038
$498$	−21.9341	−0.982892
$499$	21.8733	0.979183	0.489592	−	0.871952i	$-0.337146\pi$
$499$	21.8733	0.979183	0.489592	+	0.871952i	$0.337146\pi$
$500$	0	0
$501$	3.79115	0.169376
$502$	17.9285	0.800187
$503$	−30.0983	−1.34202	−0.671009	−	0.741449i	$-0.734139\pi$
$503$	−30.0983	−1.34202	−0.671009	+	0.741449i	$0.734139\pi$
$504$	10.1214	0.450844
$505$	0	0
$506$	−22.8409	−1.01540
$507$	−6.78934	−0.301525
$508$	9.29735	0.412503
$509$	7.77815	0.344761	0.172380	−	0.985030i	$-0.444854\pi$
$509$	7.77815	0.344761	0.172380	+	0.985030i	$0.444854\pi$
$510$	0	0
$511$	21.7360	0.961546
$512$	5.85120	0.258589
$513$	−8.31974	−0.367326
$514$	−25.3915	−1.11997
$515$	0	0
$516$	0.0228676	0.00100669
$517$	31.2490	1.37433
$518$	76.6093	3.36602
$519$	2.61333	0.114712
$520$	0	0
$521$	−14.0318	−0.614746	−0.307373	−	0.951589i	$-0.599450\pi$
$521$	−14.0318	−0.614746	−0.307373	+	0.951589i	$0.599450\pi$
$522$	17.7026	0.774821
$523$	2.80012	0.122440	0.0612202	−	0.998124i	$-0.480501\pi$
$523$	2.80012	0.122440	0.0612202	+	0.998124i	$0.480501\pi$
$524$	4.95666	0.216533
$525$	0	0
$526$	2.06805	0.0901711
$527$	49.0347	2.13598
$528$	36.2751	1.57867
$529$	−16.5259	−0.718516
$530$	0	0
$531$	3.57106	0.154971
$532$	6.87157	0.297920
$533$	−4.57731	−0.198265
$534$	32.7811	1.41858
$535$	0	0
$536$	5.38645	0.232659
$537$	−0.678162	−0.0292648
$538$	26.7448	1.15305
$539$	−97.6495	−4.20606
$540$	0	0
$541$	31.2317	1.34276	0.671379	−	0.741114i	$-0.265702\pi$
$541$	31.2317	1.34276	0.671379	+	0.741114i	$0.265702\pi$
$542$	−44.4045	−1.90734
$543$	0.390977	0.0167784
$544$	29.9450	1.28388
$545$	0	0
$546$	−34.0381	−1.45669
$547$	42.9144	1.83489	0.917444	−	0.397865i	$-0.130249\pi$
$547$	42.9144	1.83489	0.917444	+	0.397865i	$0.130249\pi$
$548$	9.46666	0.404396
$549$	−7.82048	−0.333770
$550$	0	0
$551$	−14.0289	−0.597651
$552$	−6.48211	−0.275897
$553$	11.8428	0.503609
$554$	20.0973	0.853852
$555$	0	0
$556$	13.4854	0.571910
$557$	39.2314	1.66229	0.831145	−	0.556056i	$-0.187686\pi$
$557$	39.2314	1.66229	0.831145	+	0.556056i	$0.187686\pi$
$558$	14.7641	0.625012
$559$	−0.0509565	−0.00215523
$560$	0	0
$561$	44.8033	1.89160
$562$	−11.4231	−0.481855
$563$	9.68487	0.408169	0.204084	−	0.978953i	$-0.434578\pi$
$563$	9.68487	0.408169	0.204084	+	0.978953i	$0.434578\pi$
$564$	−7.59743	−0.319909
$565$	0	0
$566$	12.5855	0.529007
$567$	−23.0757	−0.969090
$568$	8.28006	0.347424
$569$	−12.8438	−0.538441	−0.269221	−	0.963079i	$-0.586766\pi$
$569$	−12.8438	−0.538441	−0.269221	+	0.963079i	$0.586766\pi$
$570$	0	0
$571$	8.76755	0.366910	0.183455	−	0.983028i	$-0.441272\pi$
$571$	8.76755	0.366910	0.183455	+	0.983028i	$0.441272\pi$
$572$	13.7838	0.576329
$573$	25.6890	1.07318
$574$	13.9180	0.580926
$575$	0	0
$576$	−1.83393	−0.0764138
$577$	−42.9318	−1.78727	−0.893637	−	0.448791i	$-0.851855\pi$
$577$	−42.9318	−1.78727	−0.893637	+	0.448791i	$0.851855\pi$
$578$	35.9805	1.49659
$579$	−35.4572	−1.47355
$580$	0	0
$581$	−46.9225	−1.94667
$582$	26.1698	1.08477
$583$	42.3901	1.75562
$584$	−7.91177	−0.327391
$585$	0	0
$586$	20.6017	0.851047
$587$	−39.1573	−1.61619	−0.808097	−	0.589049i	$-0.799503\pi$
$587$	−39.1573	−1.61619	−0.808097	+	0.589049i	$0.799503\pi$
$588$	23.7411	0.979067
$589$	−11.7002	−0.482097
$590$	0	0
$591$	−16.8631	−0.693656
$592$	−44.2320	−1.81792
$593$	21.3089	0.875050	0.437525	−	0.899206i	$-0.355855\pi$
$593$	21.3089	0.875050	0.437525	+	0.899206i	$0.355855\pi$
$594$	50.7445	2.08207
$595$	0	0
$596$	−14.5930	−0.597753
$597$	9.28173	0.379876
$598$	−12.3744	−0.506026
$599$	−7.76975	−0.317463	−0.158732	−	0.987322i	$-0.550740\pi$
$599$	−7.76975	−0.317463	−0.158732	+	0.987322i	$0.550740\pi$
$600$	0	0
$601$	16.9993	0.693415	0.346707	−	0.937973i	$-0.387300\pi$
$601$	16.9993	0.693415	0.346707	+	0.937973i	$0.387300\pi$
$602$	0.154941	0.00631492
$603$	3.17737	0.129392
$604$	−14.1088	−0.574079
$605$	0	0
$606$	0.589093	0.0239303
$607$	−20.7512	−0.842267	−0.421133	−	0.906999i	$-0.638368\pi$
$607$	−20.7512	−0.842267	−0.421133	+	0.906999i	$0.638368\pi$
$608$	−7.14518	−0.289775
$609$	−66.7134	−2.70336
$610$	0	0
$611$	16.9296	0.684898
$612$	6.18337	0.249948
$613$	−17.5567	−0.709108	−0.354554	−	0.935035i	$-0.615367\pi$
$613$	−17.5567	−0.709108	−0.354554	+	0.935035i	$0.615367\pi$
$614$	−44.0923	−1.77942
$615$	0	0
$616$	48.9224	1.97114
$617$	6.03640	0.243016	0.121508	−	0.992590i	$-0.461227\pi$
$617$	6.03640	0.243016	0.121508	+	0.992590i	$0.461227\pi$
$618$	−31.8212	−1.28004
$619$	−21.4216	−0.861008	−0.430504	−	0.902589i	$-0.641664\pi$
$619$	−21.4216	−0.861008	−0.430504	+	0.902589i	$0.641664\pi$
$620$	0	0
$621$	−14.3833	−0.577181
$622$	−32.7948	−1.31495
$623$	70.1268	2.80957
$624$	19.6526	0.786733
$625$	0	0
$626$	7.41034	0.296177
$627$	−10.6905	−0.426938
$628$	0.791359	0.0315787
$629$	−54.6308	−2.17827
$630$	0	0
$631$	17.0323	0.678044	0.339022	−	0.940778i	$-0.389904\pi$
$631$	17.0323	0.678044	0.339022	+	0.940778i	$0.389904\pi$
$632$	−4.31071	−0.171471
$633$	10.9642	0.435786
$634$	−12.4917	−0.496107
$635$	0	0
$636$	−10.3061	−0.408665
$637$	−52.9031	−2.09610
$638$	85.5662	3.38760
$639$	4.88426	0.193218
$640$	0	0
$641$	−5.88960	−0.232625	−0.116313	−	0.993213i	$-0.537107\pi$
$641$	−5.88960	−0.232625	−0.116313	+	0.993213i	$0.537107\pi$
$642$	−3.09316	−0.122077
$643$	−27.5753	−1.08746	−0.543731	−	0.839259i	$-0.682989\pi$
$643$	−27.5753	−1.08746	−0.543731	+	0.839259i	$0.682989\pi$
$644$	11.8797	0.468124
$645$	0	0
$646$	−15.5202	−0.610634
$647$	26.2229	1.03093	0.515465	−	0.856911i	$-0.327619\pi$
$647$	26.2229	1.03093	0.515465	+	0.856911i	$0.327619\pi$
$648$	8.39940	0.329960
$649$	17.2609	0.677549
$650$	0	0
$651$	−55.6393	−2.18068
$652$	8.64464	0.338550
$653$	−5.20569	−0.203714	−0.101857	−	0.994799i	$-0.532478\pi$
$653$	−5.20569	−0.203714	−0.101857	+	0.994799i	$0.532478\pi$
$654$	45.5378	1.78067
$655$	0	0
$656$	−8.03584	−0.313747
$657$	−4.66701	−0.182077
$658$	−51.4769	−2.00678
$659$	−7.97812	−0.310783	−0.155392	−	0.987853i	$-0.549664\pi$
$659$	−7.97812	−0.310783	−0.155392	+	0.987853i	$0.549664\pi$
$660$	0	0
$661$	17.0900	0.664724	0.332362	−	0.943152i	$-0.392154\pi$
$661$	17.0900	0.664724	0.332362	+	0.943152i	$0.392154\pi$
$662$	1.74437	0.0677970
$663$	24.2728	0.942679
$664$	17.0795	0.662811
$665$	0	0
$666$	−16.4490	−0.637386
$667$	−24.2533	−0.939093
$668$	2.52902	0.0978506
$669$	−26.0016	−1.00528
$670$	0	0
$671$	−37.8007	−1.45928
$672$	−33.9784	−1.31074
$673$	−40.1751	−1.54864	−0.774319	−	0.632796i	$-0.781907\pi$
$673$	−40.1751	−1.54864	−0.774319	+	0.632796i	$0.781907\pi$
$674$	−32.4240	−1.24892
$675$	0	0
$676$	−4.52907	−0.174195
$677$	9.73259	0.374054	0.187027	−	0.982355i	$-0.440115\pi$
$677$	9.73259	0.374054	0.187027	+	0.982355i	$0.440115\pi$
$678$	−40.2961	−1.54756
$679$	55.9836	2.14845
$680$	0	0
$681$	2.44190	0.0935736
$682$	71.3627	2.73262
$683$	9.78489	0.374408	0.187204	−	0.982321i	$-0.440057\pi$
$683$	9.78489	0.374408	0.187204	+	0.982321i	$0.440057\pi$
$684$	−1.47541	−0.0564139
$685$	0	0
$686$	100.312	3.82994
$687$	−29.2962	−1.11772
$688$	−0.0894582	−0.00341056
$689$	22.9655	0.874916
$690$	0	0
$691$	3.54792	0.134969	0.0674846	−	0.997720i	$-0.478503\pi$
$691$	3.54792	0.134969	0.0674846	+	0.997720i	$0.478503\pi$
$692$	1.74331	0.0662707
$693$	28.8584	1.09624
$694$	53.6651	2.03710
$695$	0	0
$696$	24.2832	0.920452
$697$	−9.92505	−0.375938
$698$	−20.6686	−0.782318
$699$	−8.42479	−0.318655
$700$	0	0
$701$	21.2003	0.800724	0.400362	−	0.916357i	$-0.368884\pi$
$701$	21.2003	0.800724	0.400362	+	0.916357i	$0.368884\pi$
$702$	27.4916	1.03760
$703$	13.0355	0.491642
$704$	−8.86439	−0.334089
$705$	0	0
$706$	8.44042	0.317659
$707$	1.26021	0.0473952
$708$	−4.19656	−0.157717
$709$	35.8330	1.34574	0.672869	−	0.739762i	$-0.265062\pi$
$709$	35.8330	1.34574	0.672869	+	0.739762i	$0.265062\pi$
$710$	0	0
$711$	−2.54281	−0.0953628
$712$	−25.5257	−0.956615
$713$	−20.2274	−0.757523
$714$	−73.8052	−2.76209
$715$	0	0
$716$	−0.452391	−0.0169067
$717$	−1.92965	−0.0720641
$718$	0.783046	0.0292230
$719$	−17.4449	−0.650586	−0.325293	−	0.945613i	$-0.605463\pi$
$719$	−17.4449	−0.650586	−0.325293	+	0.945613i	$0.605463\pi$
$720$	0	0
$721$	−68.0734	−2.53518
$722$	−28.7796	−1.07107
$723$	1.38336	0.0514477
$724$	0.260815	0.00969310
$725$	0	0
$726$	39.1892	1.45445
$727$	−2.83872	−0.105282	−0.0526411	−	0.998613i	$-0.516764\pi$
$727$	−2.83872	−0.105282	−0.0526411	+	0.998613i	$0.516764\pi$
$728$	26.5044	0.982319
$729$	28.4144	1.05239
$730$	0	0
$731$	−0.110490	−0.00408661
$732$	9.19032	0.339684
$733$	−19.1179	−0.706135	−0.353067	−	0.935598i	$-0.614861\pi$
$733$	−19.1179	−0.706135	−0.353067	+	0.935598i	$0.614861\pi$
$734$	27.8058	1.02633
$735$	0	0
$736$	−12.3527	−0.455326
$737$	15.3580	0.565718
$738$	−2.98837	−0.110004
$739$	−10.7460	−0.395299	−0.197650	−	0.980273i	$-0.563331\pi$
$739$	−10.7460	−0.395299	−0.197650	+	0.980273i	$0.563331\pi$
$740$	0	0
$741$	−5.79175	−0.212765
$742$	−69.8300	−2.56354
$743$	−21.9385	−0.804846	−0.402423	−	0.915454i	$-0.631832\pi$
$743$	−21.9385	−0.804846	−0.402423	+	0.915454i	$0.631832\pi$
$744$	20.2523	0.742486
$745$	0	0
$746$	−33.6146	−1.23072
$747$	10.0749	0.368620
$748$	29.8876	1.09280
$749$	−6.61702	−0.241781
$750$	0	0
$751$	−24.7399	−0.902773	−0.451386	−	0.892329i	$-0.649070\pi$
$751$	−24.7399	−0.902773	−0.451386	+	0.892329i	$0.649070\pi$
$752$	29.7212	1.08382
$753$	14.5070	0.528665
$754$	46.3568	1.68821
$755$	0	0
$756$	−26.3925	−0.959885
$757$	15.0100	0.545546	0.272773	−	0.962078i	$-0.412059\pi$
$757$	15.0100	0.545546	0.272773	+	0.962078i	$0.412059\pi$
$758$	−65.7446	−2.38795
$759$	−18.4819	−0.670851
$760$	0	0
$761$	−26.0521	−0.944389	−0.472194	−	0.881495i	$-0.656538\pi$
$761$	−26.0521	−0.944389	−0.472194	+	0.881495i	$0.656538\pi$
$762$	23.8275	0.863181
$763$	97.4166	3.52672
$764$	17.1368	0.619987
$765$	0	0
$766$	43.7398	1.58038
$767$	9.35134	0.337657
$768$	25.1186	0.906389
$769$	−24.4144	−0.880407	−0.440203	−	0.897898i	$-0.645094\pi$
$769$	−24.4144	−0.880407	−0.440203	+	0.897898i	$0.645094\pi$
$770$	0	0
$771$	−20.5458	−0.739939
$772$	−23.6530	−0.851289
$773$	41.9248	1.50793	0.753966	−	0.656914i	$-0.228138\pi$
$773$	41.9248	1.50793	0.753966	+	0.656914i	$0.228138\pi$
$774$	−0.0332678	−0.00119579
$775$	0	0
$776$	−20.3776	−0.731514
$777$	61.9892	2.22385
$778$	2.64147	0.0947014
$779$	2.36822	0.0848502
$780$	0	0
$781$	23.6083	0.844771
$782$	−26.8315	−0.959494
$783$	53.8825	1.92560
$784$	−92.8756	−3.31699
$785$	0	0
$786$	12.7031	0.453104
$787$	42.1295	1.50176	0.750878	−	0.660441i	$-0.229631\pi$
$787$	42.1295	1.50176	0.750878	+	0.660441i	$0.229631\pi$
$788$	−11.2491	−0.400734
$789$	1.67338	0.0595739
$790$	0	0
$791$	−86.2033	−3.06504
$792$	−10.5043	−0.373253
$793$	−20.4791	−0.727233
$794$	27.2198	0.965995
$795$	0	0
$796$	6.19170	0.219459
$797$	32.8495	1.16359	0.581795	−	0.813335i	$-0.302351\pi$
$797$	32.8495	1.16359	0.581795	+	0.813335i	$0.302351\pi$
$798$	17.6107	0.623411
$799$	36.7086	1.29866
$800$	0	0
$801$	−15.0571	−0.532018
$802$	−29.0323	−1.02517
$803$	−22.5582	−0.796062
$804$	−3.73392	−0.131685
$805$	0	0
$806$	38.6618	1.36180
$807$	21.6408	0.761793
$808$	−0.458709	−0.0161373
$809$	21.0371	0.739625	0.369812	−	0.929106i	$-0.379422\pi$
$809$	21.0371	0.739625	0.369812	+	0.929106i	$0.379422\pi$
$810$	0	0
$811$	38.1121	1.33830	0.669149	−	0.743128i	$-0.266659\pi$
$811$	38.1121	1.33830	0.669149	+	0.743128i	$0.266659\pi$
$812$	−44.5035	−1.56177
$813$	−35.9303	−1.26013
$814$	−79.5070	−2.78672
$815$	0	0
$816$	42.6129	1.49175
$817$	0.0263640	0.000922359 0
$818$	−44.8100	−1.56674
$819$	15.6345	0.546313
$820$	0	0
$821$	−13.3082	−0.464459	−0.232230	−	0.972661i	$-0.574602\pi$
$821$	−13.3082	−0.464459	−0.232230	+	0.972661i	$0.574602\pi$
$822$	24.2614	0.846215
$823$	9.58043	0.333953	0.166976	−	0.985961i	$-0.446600\pi$
$823$	9.58043	0.333953	0.166976	+	0.985961i	$0.446600\pi$
$824$	24.7782	0.863190
$825$	0	0
$826$	−28.4341	−0.989350
$827$	5.10249	0.177431	0.0887154	−	0.996057i	$-0.471724\pi$
$827$	5.10249	0.177431	0.0887154	+	0.996057i	$0.471724\pi$
$828$	−2.55072	−0.0886435
$829$	−38.5370	−1.33845	−0.669223	−	0.743061i	$-0.733373\pi$
$829$	−38.5370	−1.33845	−0.669223	+	0.743061i	$0.733373\pi$
$830$	0	0
$831$	16.2619	0.564120
$832$	−4.80241	−0.166494
$833$	−114.710	−3.97448
$834$	34.5609	1.19675
$835$	0	0
$836$	−7.13148	−0.246647
$837$	44.9383	1.55330
$838$	−24.8067	−0.856934
$839$	9.97929	0.344524	0.172262	−	0.985051i	$-0.444893\pi$
$839$	9.97929	0.344524	0.172262	+	0.985051i	$0.444893\pi$
$840$	0	0
$841$	61.8576	2.13302
$842$	25.4927	0.878537
$843$	−9.24312	−0.318350
$844$	7.31402	0.251759
$845$	0	0
$846$	11.0528	0.380002
$847$	83.8354	2.88062
$848$	40.3178	1.38452
$849$	10.1837	0.349503
$850$	0	0
$851$	22.5359	0.772520
$852$	−5.73979	−0.196642
$853$	−0.528777	−0.0181050	−0.00905250	−	0.999959i	$-0.502882\pi$
$853$	−0.528777	−0.0181050	−0.00905250	+	0.999959i	$0.502882\pi$
$854$	62.2697	2.13082
$855$	0	0
$856$	2.40855	0.0823225
$857$	13.2208	0.451615	0.225808	−	0.974172i	$-0.427498\pi$
$857$	13.2208	0.451615	0.225808	+	0.974172i	$0.427498\pi$
$858$	35.3255	1.20599
$859$	15.8466	0.540679	0.270339	−	0.962765i	$-0.412864\pi$
$859$	15.8466	0.540679	0.270339	+	0.962765i	$0.412864\pi$
$860$	0	0
$861$	11.2619	0.383804
$862$	−46.2707	−1.57599
$863$	42.9425	1.46178	0.730890	−	0.682495i	$-0.239105\pi$
$863$	42.9425	1.46178	0.730890	+	0.682495i	$0.239105\pi$
$864$	27.4434	0.933642
$865$	0	0
$866$	−56.2578	−1.91172
$867$	29.1140	0.988763
$868$	−37.1161	−1.25980
$869$	−12.2908	−0.416937
$870$	0	0
$871$	8.32040	0.281926
$872$	−35.4589	−1.20079
$873$	−12.0204	−0.406829
$874$	6.40228	0.216560
$875$	0	0
$876$	5.48448	0.185303
$877$	13.9351	0.470554	0.235277	−	0.971928i	$-0.424400\pi$
$877$	13.9351	0.470554	0.235277	+	0.971928i	$0.424400\pi$
$878$	−31.8621	−1.07529
$879$	16.6701	0.562267
$880$	0	0
$881$	38.0645	1.28243	0.641213	−	0.767363i	$-0.278431\pi$
$881$	38.0645	1.28243	0.641213	+	0.767363i	$0.278431\pi$
$882$	−34.5386	−1.16298
$883$	−50.9419	−1.71433	−0.857165	−	0.515042i	$-0.827776\pi$
$883$	−50.9419	−1.71433	−0.857165	+	0.515042i	$0.827776\pi$
$884$	16.1920	0.544597
$885$	0	0
$886$	−1.62558	−0.0546124
$887$	51.0399	1.71375	0.856875	−	0.515523i	$-0.172403\pi$
$887$	51.0399	1.71375	0.856875	+	0.515523i	$0.172403\pi$
$888$	−22.5636	−0.757186
$889$	50.9729	1.70958
$890$	0	0
$891$	23.9486	0.802307
$892$	−17.3452	−0.580762
$893$	−8.75906	−0.293111
$894$	−37.3994	−1.25082
$895$	0	0
$896$	63.7268	2.12896
$897$	−10.0129	−0.334320
$898$	20.7887	0.693729
$899$	75.7758	2.52726
$900$	0	0
$901$	49.7964	1.65896
$902$	−14.4444	−0.480947
$903$	0.125372	0.00417212
$904$	31.3774	1.04360
$905$	0	0
$906$	−36.1585	−1.20128
$907$	−20.9409	−0.695333	−0.347666	−	0.937618i	$-0.613026\pi$
$907$	−20.9409	−0.695333	−0.347666	+	0.937618i	$0.613026\pi$
$908$	1.62895	0.0540586
$909$	−0.270584	−0.00897471
$910$	0	0
$911$	−28.0321	−0.928744	−0.464372	−	0.885640i	$-0.653720\pi$
$911$	−28.0321	−0.928744	−0.464372	+	0.885640i	$0.653720\pi$
$912$	−10.1679	−0.336692
$913$	48.6973	1.61165
$914$	−59.7831	−1.97745
$915$	0	0
$916$	−19.5430	−0.645720
$917$	27.1750	0.897398
$918$	59.6103	1.96743
$919$	−21.5713	−0.711572	−0.355786	−	0.934567i	$-0.615787\pi$
$919$	−21.5713	−0.711572	−0.355786	+	0.934567i	$0.615787\pi$
$920$	0	0
$921$	−35.6777	−1.17562
$922$	−17.4596	−0.575002
$923$	12.7901	0.420993
$924$	−33.9133	−1.11566
$925$	0	0
$926$	−29.7955	−0.979140
$927$	14.6162	0.480060
$928$	46.2755	1.51907
$929$	12.2984	0.403499	0.201749	−	0.979437i	$-0.435337\pi$
$929$	12.2984	0.403499	0.201749	+	0.979437i	$0.435337\pi$
$930$	0	0
$931$	27.3711	0.897051
$932$	−5.62005	−0.184091
$933$	−26.5363	−0.868758
$934$	−32.3375	−1.05812
$935$	0	0
$936$	−5.69084	−0.186011
$937$	−0.227516	−0.00743264	−0.00371632	−	0.999993i	$-0.501183\pi$
$937$	−0.227516	−0.00743264	−0.00371632	+	0.999993i	$0.501183\pi$
$938$	−25.2994	−0.826055
$939$	5.99615	0.195677
$940$	0	0
$941$	31.0857	1.01336	0.506682	−	0.862133i	$-0.330872\pi$
$941$	31.0857	1.01336	0.506682	+	0.862133i	$0.330872\pi$
$942$	2.02812	0.0660797
$943$	4.09421	0.133326
$944$	16.4170	0.534329
$945$	0	0
$946$	−0.160801	−0.00522810
$947$	−28.5537	−0.927871	−0.463935	−	0.885869i	$-0.653563\pi$
$947$	−28.5537	−0.927871	−0.463935	+	0.885869i	$0.653563\pi$
$948$	2.98821	0.0970525
$949$	−12.2212	−0.396718
$950$	0	0
$951$	−10.1078	−0.327767
$952$	57.4699	1.86261
$953$	21.7444	0.704372	0.352186	−	0.935930i	$-0.385439\pi$
$953$	21.7444	0.704372	0.352186	+	0.935930i	$0.385439\pi$
$954$	14.9934	0.485429
$955$	0	0
$956$	−1.28724	−0.0416323
$957$	69.2368	2.23811
$958$	−32.5642	−1.05210
$959$	51.9012	1.67598
$960$	0	0
$961$	32.1974	1.03863
$962$	−43.0741	−1.38877
$963$	1.42076	0.0457834
$964$	0.922819	0.0297220
$965$	0	0
$966$	30.4456	0.979570
$967$	−11.6791	−0.375576	−0.187788	−	0.982210i	$-0.560132\pi$
$967$	−11.6791	−0.375576	−0.187788	+	0.982210i	$0.560132\pi$
$968$	−30.5155	−0.980805
$969$	−12.5583	−0.403432
$970$	0	0
$971$	−8.70745	−0.279435	−0.139718	−	0.990191i	$-0.544619\pi$
$971$	−8.70745	−0.279435	−0.139718	+	0.990191i	$0.544619\pi$
$972$	9.82714	0.315206
$973$	73.9342	2.37022
$974$	38.4192	1.23103
$975$	0	0
$976$	−35.9527	−1.15082
$977$	44.5056	1.42386	0.711930	−	0.702250i	$-0.247821\pi$
$977$	44.5056	1.42386	0.711930	+	0.702250i	$0.247821\pi$
$978$	22.1548	0.708431
$979$	−72.7794	−2.32604
$980$	0	0
$981$	−20.9166	−0.667815
$982$	−12.0084	−0.383204
$983$	43.9573	1.40202	0.701010	−	0.713152i	$-0.252733\pi$
$983$	43.9573	1.40202	0.701010	+	0.713152i	$0.252733\pi$
$984$	−4.09925	−0.130679
$985$	0	0
$986$	100.516	3.20108
$987$	−41.6531	−1.32583
$988$	−3.86359	−0.122917
$989$	0.0455784	0.00144931
$990$	0	0
$991$	19.5023	0.619510	0.309755	−	0.950816i	$-0.399753\pi$
$991$	19.5023	0.619510	0.309755	+	0.950816i	$0.399753\pi$
$992$	38.5940	1.22536
$993$	1.41148	0.0447919
$994$	−38.8903	−1.23353
$995$	0	0
$996$	−11.8396	−0.375151
$997$	−53.0288	−1.67944	−0.839719	−	0.543021i	$-0.817280\pi$
$997$	−53.0288	−1.67944	−0.839719	+	0.543021i	$0.817280\pi$
$998$	37.3951	1.18372
$999$	−50.0669	−1.58405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.36		46
5.2	odd	4		1205.2.b.c.724.36	yes	46
5.3	odd	4		1205.2.b.c.724.11	✓	46
5.4	even	2	inner	6025.2.a.p.1.11		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.11	✓	46	5.3	odd	4
1205.2.b.c.724.36	yes	46	5.2	odd	4
6025.2.a.p.1.11		46	5.4	even	2	inner
6025.2.a.p.1.36		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)

\(f(q)\)	\(=\)	\(q+1.70963 q^{2} +1.38336 q^{3} +0.922819 q^{4} +2.36503 q^{6} +5.05937 q^{7} -1.84158 q^{8} -1.08631 q^{9} +O(q^{10})\)	\(q+1.70963 q^{2} +1.38336 q^{3} +0.922819 q^{4} +2.36503 q^{6} +5.05937 q^{7} -1.84158 q^{8} -1.08631 q^{9} -5.25074 q^{11} +1.27659 q^{12} -2.84467 q^{13} +8.64964 q^{14} -4.99404 q^{16} -6.16813 q^{17} -1.85719 q^{18} +1.47178 q^{19} +6.99894 q^{21} -8.97680 q^{22} +2.54443 q^{23} -2.54756 q^{24} -4.86332 q^{26} -5.65285 q^{27} +4.66889 q^{28} -9.53193 q^{29} -7.94968 q^{31} -4.85479 q^{32} -7.26367 q^{33} -10.5452 q^{34} -1.00247 q^{36} +8.85694 q^{37} +2.51619 q^{38} -3.93520 q^{39} +1.60908 q^{41} +11.9656 q^{42} +0.0179130 q^{43} -4.84548 q^{44} +4.35003 q^{46} -5.95134 q^{47} -6.90856 q^{48} +18.5973 q^{49} -8.53275 q^{51} -2.62511 q^{52} -8.07317 q^{53} -9.66425 q^{54} -9.31723 q^{56} +2.03600 q^{57} -16.2960 q^{58} -3.28732 q^{59} +7.19911 q^{61} -13.5910 q^{62} -5.49606 q^{63} +1.68822 q^{64} -12.4182 q^{66} -2.92491 q^{67} -5.69207 q^{68} +3.51987 q^{69} -4.49618 q^{71} +2.00053 q^{72} +4.29619 q^{73} +15.1421 q^{74} +1.35819 q^{76} -26.5655 q^{77} -6.72772 q^{78} +2.34077 q^{79} -4.56099 q^{81} +2.75093 q^{82} -9.27436 q^{83} +6.45875 q^{84} +0.0306245 q^{86} -13.1861 q^{87} +9.66964 q^{88} +13.8608 q^{89} -14.3922 q^{91} +2.34805 q^{92} -10.9973 q^{93} -10.1746 q^{94} -6.71593 q^{96} +11.0653 q^{97} +31.7944 q^{98} +5.70395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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