LMFDB - Embedded newform 9025.2.a.ck.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 $ x^16 - 22x^14 + 190x^12 - 820x^10 + 1862x^8 - 2154x^6 + 1163x^4 - 256*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.93600$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93600 q^{2} -2.41423 q^{3} +1.74811 q^{4} +4.67396 q^{6} +1.46100 q^{7} +0.487655 q^{8} +2.82851 q^{9} +O(q^{10})$	\(q-1.93600 q^{2} -2.41423 q^{3} +1.74811 q^{4} +4.67396 q^{6} +1.46100 q^{7} +0.487655 q^{8} +2.82851 q^{9} -2.89488 q^{11} -4.22035 q^{12} +6.12528 q^{13} -2.82851 q^{14} -4.44033 q^{16} -6.29889 q^{17} -5.47600 q^{18} -3.52719 q^{21} +5.60450 q^{22} +0.508852 q^{23} -1.17731 q^{24} -11.8586 q^{26} +0.414027 q^{27} +2.55400 q^{28} -1.72159 q^{29} +8.44484 q^{31} +7.62118 q^{32} +6.98890 q^{33} +12.1947 q^{34} +4.94455 q^{36} +3.13569 q^{37} -14.7878 q^{39} -7.44400 q^{41} +6.82866 q^{42} +6.90372 q^{43} -5.06057 q^{44} -0.985139 q^{46} -0.316851 q^{47} +10.7200 q^{48} -4.86547 q^{49} +15.2070 q^{51} +10.7077 q^{52} +4.34236 q^{53} -0.801557 q^{54} +0.712465 q^{56} +3.33300 q^{58} -2.25084 q^{59} -6.29374 q^{61} -16.3492 q^{62} +4.13245 q^{63} -5.87399 q^{64} -13.5305 q^{66} -10.0324 q^{67} -11.0112 q^{68} -1.22849 q^{69} +6.63567 q^{71} +1.37933 q^{72} +12.9644 q^{73} -6.07070 q^{74} -4.22942 q^{77} +28.6293 q^{78} -14.6730 q^{79} -9.48507 q^{81} +14.4116 q^{82} -4.94202 q^{83} -6.16593 q^{84} -13.3656 q^{86} +4.15631 q^{87} -1.41170 q^{88} +5.35408 q^{89} +8.94904 q^{91} +0.889530 q^{92} -20.3878 q^{93} +0.613425 q^{94} -18.3993 q^{96} +15.9482 q^{97} +9.41958 q^{98} -8.18818 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * 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 $

$2$	−1.93600	−1.36896	−0.684481	−	0.729031i	$-0.739971\pi$
$2$	−1.93600	−1.36896	−0.684481	+	0.729031i	$0.739971\pi$
$3$	−2.41423	−1.39386	−0.696928	−	0.717141i	$-0.745450\pi$
$3$	−2.41423	−1.39386	−0.696928	+	0.717141i	$0.745450\pi$
$4$	1.74811	0.874056
$5$	0	0
$5$	0	0
$6$	4.67396	1.90814
$7$	1.46100	0.552207	0.276103	−	0.961128i	$-0.410957\pi$
$7$	1.46100	0.552207	0.276103	+	0.961128i	$0.410957\pi$
$8$	0.487655	0.172412
$9$	2.82851	0.942835
$10$	0	0
$11$	−2.89488	−0.872839	−0.436419	−	0.899743i	$-0.643754\pi$
$11$	−2.89488	−0.872839	−0.436419	+	0.899743i	$0.643754\pi$
$12$	−4.22035	−1.21831
$13$	6.12528	1.69885	0.849423	−	0.527713i	$-0.176950\pi$
$13$	6.12528	1.69885	0.849423	+	0.527713i	$0.176950\pi$
$14$	−2.82851	−0.755950
$15$	0	0
$16$	−4.44033	−1.11008
$17$	−6.29889	−1.52771	−0.763853	−	0.645391i	$-0.776695\pi$
$17$	−6.29889	−1.52771	−0.763853	+	0.645391i	$0.776695\pi$
$18$	−5.47600	−1.29071
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−3.52719	−0.769697
$22$	5.60450	1.19488
$23$	0.508852	0.106103	0.0530515	−	0.998592i	$-0.483105\pi$
$23$	0.508852	0.106103	0.0530515	+	0.998592i	$0.483105\pi$
$24$	−1.17731	−0.240318
$25$	0	0
$26$	−11.8586	−2.32565
$27$	0.414027	0.0796795
$28$	2.55400	0.482660
$29$	−1.72159	−0.319691	−0.159845	−	0.987142i	$-0.551100\pi$
$29$	−1.72159	−0.319691	−0.159845	+	0.987142i	$0.551100\pi$
$30$	0	0
$31$	8.44484	1.51674	0.758369	−	0.651826i	$-0.225997\pi$
$31$	8.44484	1.51674	0.758369	+	0.651826i	$0.225997\pi$
$32$	7.62118	1.34725
$33$	6.98890	1.21661
$34$	12.1947	2.09137
$35$	0	0
$36$	4.94455	0.824091
$37$	3.13569	0.515504	0.257752	−	0.966211i	$-0.417018\pi$
$37$	3.13569	0.515504	0.257752	+	0.966211i	$0.417018\pi$
$38$	0	0
$39$	−14.7878	−2.36795
$40$	0	0
$41$	−7.44400	−1.16256	−0.581279	−	0.813704i	$-0.697447\pi$
$41$	−7.44400	−1.16256	−0.581279	+	0.813704i	$0.697447\pi$
$42$	6.82866	1.05369
$43$	6.90372	1.05281	0.526404	−	0.850235i	$-0.323540\pi$
$43$	6.90372	1.05281	0.526404	+	0.850235i	$0.323540\pi$
$44$	−5.06057	−0.762910
$45$	0	0
$46$	−0.985139	−0.145251
$47$	−0.316851	−0.0462175	−0.0231087	−	0.999733i	$-0.507356\pi$
$47$	−0.316851	−0.0462175	−0.0231087	+	0.999733i	$0.507356\pi$
$48$	10.7200	1.54729
$49$	−4.86547	−0.695068
$50$	0	0
$51$	15.2070	2.12940
$52$	10.7077	1.48489
$53$	4.34236	0.596468	0.298234	−	0.954493i	$-0.403602\pi$
$53$	4.34236	0.596468	0.298234	+	0.954493i	$0.403602\pi$
$54$	−0.801557	−0.109078
$55$	0	0
$56$	0.712465	0.0952071
$57$	0	0
$58$	3.33300	0.437644
$59$	−2.25084	−0.293035	−0.146517	−	0.989208i	$-0.546806\pi$
$59$	−2.25084	−0.293035	−0.146517	+	0.989208i	$0.546806\pi$
$60$	0	0
$61$	−6.29374	−0.805831	−0.402916	−	0.915237i	$-0.632003\pi$
$61$	−6.29374	−0.805831	−0.402916	+	0.915237i	$0.632003\pi$
$62$	−16.3492	−2.07636
$63$	4.13245	0.520640
$64$	−5.87399	−0.734249
$65$	0	0
$66$	−13.5305	−1.66549
$67$	−10.0324	−1.22565	−0.612827	−	0.790217i	$-0.709968\pi$
$67$	−10.0324	−1.22565	−0.612827	+	0.790217i	$0.709968\pi$
$68$	−11.0112	−1.33530
$69$	−1.22849	−0.147892
$70$	0	0
$71$	6.63567	0.787509	0.393754	−	0.919216i	$-0.371176\pi$
$71$	6.63567	0.787509	0.393754	+	0.919216i	$0.371176\pi$
$72$	1.37933	0.162556
$73$	12.9644	1.51737	0.758685	−	0.651458i	$-0.225842\pi$
$73$	12.9644	1.51737	0.758685	+	0.651458i	$0.225842\pi$
$74$	−6.07070	−0.705705
$75$	0	0
$76$	0	0
$77$	−4.22942	−0.481987
$78$	28.6293	3.24163
$79$	−14.6730	−1.65084	−0.825420	−	0.564519i	$-0.809062\pi$
$79$	−14.6730	−1.65084	−0.825420	+	0.564519i	$0.809062\pi$
$80$	0	0
$81$	−9.48507	−1.05390
$82$	14.4116	1.59150
$83$	−4.94202	−0.542458	−0.271229	−	0.962515i	$-0.587430\pi$
$83$	−4.94202	−0.542458	−0.271229	+	0.962515i	$0.587430\pi$
$84$	−6.16593	−0.672758
$85$	0	0
$86$	−13.3656	−1.44125
$87$	4.15631	0.445603
$88$	−1.41170	−0.150488
$89$	5.35408	0.567532	0.283766	−	0.958894i	$-0.408416\pi$
$89$	5.35408	0.567532	0.283766	+	0.958894i	$0.408416\pi$
$90$	0	0
$91$	8.94904	0.938114
$92$	0.889530	0.0927400
$93$	−20.3878	−2.11411
$94$	0.613425	0.0632700
$95$	0	0
$96$	−18.3993	−1.87787
$97$	15.9482	1.61929	0.809646	−	0.586918i	$-0.199659\pi$
$97$	15.9482	1.61929	0.809646	+	0.586918i	$0.199659\pi$
$98$	9.41958	0.951521
$99$	−8.18818	−0.822943
$100$	0	0
$101$	−6.24001	−0.620904	−0.310452	−	0.950589i	$-0.600480\pi$
$101$	−6.24001	−0.620904	−0.310452	+	0.950589i	$0.600480\pi$
$102$	−29.4408	−2.91507
$103$	7.93296	0.781658	0.390829	−	0.920463i	$-0.372188\pi$
$103$	7.93296	0.781658	0.390829	+	0.920463i	$0.372188\pi$
$104$	2.98702	0.292902
$105$	0	0
$106$	−8.40682	−0.816543
$107$	−17.3832	−1.68050	−0.840250	−	0.542200i	$-0.817592\pi$
$107$	−17.3832	−1.68050	−0.840250	+	0.542200i	$0.817592\pi$
$108$	0.723765	0.0696443
$109$	−8.66544	−0.829999	−0.414999	−	0.909822i	$-0.636218\pi$
$109$	−8.66544	−0.829999	−0.414999	+	0.909822i	$0.636218\pi$
$110$	0	0
$111$	−7.57027	−0.718538
$112$	−6.48733	−0.612995
$113$	−9.86985	−0.928477	−0.464239	−	0.885710i	$-0.653672\pi$
$113$	−9.86985	−0.928477	−0.464239	+	0.885710i	$0.653672\pi$
$114$	0	0
$115$	0	0
$116$	−3.00953	−0.279428
$117$	17.3254	1.60173
$118$	4.35764	0.401153
$119$	−9.20269	−0.843609
$120$	0	0
$121$	−2.61968	−0.238153
$122$	12.1847	1.10315
$123$	17.9715	1.62044
$124$	14.7625	1.32571
$125$	0	0
$126$	−8.00044	−0.712736
$127$	−11.9797	−1.06303	−0.531515	−	0.847049i	$-0.678377\pi$
$127$	−11.9797	−1.06303	−0.531515	+	0.847049i	$0.678377\pi$
$128$	−3.87030	−0.342089
$129$	−16.6672	−1.46746
$130$	0	0
$131$	−8.04755	−0.703118	−0.351559	−	0.936166i	$-0.614348\pi$
$131$	−8.04755	−0.703118	−0.351559	+	0.936166i	$0.614348\pi$
$132$	12.2174	1.06339
$133$	0	0
$134$	19.4228	1.67787
$135$	0	0
$136$	−3.07169	−0.263395
$137$	−3.13952	−0.268227	−0.134113	−	0.990966i	$-0.542819\pi$
$137$	−3.13952	−0.268227	−0.134113	+	0.990966i	$0.542819\pi$
$138$	2.37835	0.202459
$139$	18.7439	1.58983	0.794917	−	0.606719i	$-0.207515\pi$
$139$	18.7439	1.58983	0.794917	+	0.606719i	$0.207515\pi$
$140$	0	0
$141$	0.764951	0.0644205
$142$	−12.8467	−1.07807
$143$	−17.7319	−1.48282
$144$	−12.5595	−1.04662
$145$	0	0
$146$	−25.0992	−2.07722
$147$	11.7464	0.968824
$148$	5.48153	0.450579
$149$	−8.91189	−0.730090	−0.365045	−	0.930990i	$-0.618946\pi$
$149$	−8.91189	−0.730090	−0.365045	+	0.930990i	$0.618946\pi$
$150$	0	0
$151$	1.41610	0.115241	0.0576204	−	0.998339i	$-0.481649\pi$
$151$	1.41610	0.115241	0.0576204	+	0.998339i	$0.481649\pi$
$152$	0	0
$153$	−17.8164	−1.44037
$154$	8.18818	0.659822
$155$	0	0
$156$	−25.8508	−2.06972
$157$	−4.37921	−0.349499	−0.174750	−	0.984613i	$-0.555912\pi$
$157$	−4.37921	−0.349499	−0.174750	+	0.984613i	$0.555912\pi$
$158$	28.4070	2.25994
$159$	−10.4834	−0.831391
$160$	0	0
$161$	0.743434	0.0585908
$162$	18.3631	1.44274
$163$	−3.09759	−0.242622	−0.121311	−	0.992615i	$-0.538710\pi$
$163$	−3.09759	−0.242622	−0.121311	+	0.992615i	$0.538710\pi$
$164$	−13.0130	−1.01614
$165$	0	0
$166$	9.56778	0.742604
$167$	−3.25702	−0.252036	−0.126018	−	0.992028i	$-0.540220\pi$
$167$	−3.25702	−0.252036	−0.126018	+	0.992028i	$0.540220\pi$
$168$	−1.72005	−0.132705
$169$	24.5190	1.88608
$170$	0	0
$171$	0	0
$172$	12.0685	0.920213
$173$	10.5204	0.799848	0.399924	−	0.916548i	$-0.369036\pi$
$173$	10.5204	0.799848	0.399924	+	0.916548i	$0.369036\pi$
$174$	−8.04663	−0.610013
$175$	0	0
$176$	12.8542	0.968922
$177$	5.43405	0.408448
$178$	−10.3655	−0.776929
$179$	14.3853	1.07521	0.537605	−	0.843197i	$-0.319329\pi$
$179$	14.3853	1.07521	0.537605	+	0.843197i	$0.319329\pi$
$180$	0	0
$181$	−8.13512	−0.604679	−0.302339	−	0.953200i	$-0.597768\pi$
$181$	−8.13512	−0.604679	−0.302339	+	0.953200i	$0.597768\pi$
$182$	−17.3254	−1.28424
$183$	15.1945	1.12321
$184$	0.248144	0.0182934
$185$	0	0
$186$	39.4708	2.89414
$187$	18.2345	1.33344
$188$	−0.553891	−0.0403967
$189$	0.604894	0.0439995
$190$	0	0
$191$	−6.19452	−0.448220	−0.224110	−	0.974564i	$-0.571947\pi$
$191$	−6.19452	−0.448220	−0.224110	+	0.974564i	$0.571947\pi$
$192$	14.1812	1.02344
$193$	12.5560	0.903798	0.451899	−	0.892069i	$-0.350747\pi$
$193$	12.5560	0.903798	0.451899	+	0.892069i	$0.350747\pi$
$194$	−30.8758	−2.21675
$195$	0	0
$196$	−8.50540	−0.607528
$197$	3.43662	0.244849	0.122425	−	0.992478i	$-0.460933\pi$
$197$	3.43662	0.244849	0.122425	+	0.992478i	$0.460933\pi$
$198$	15.8523	1.12658
$199$	−0.866118	−0.0613975	−0.0306987	−	0.999529i	$-0.509773\pi$
$199$	−0.866118	−0.0613975	−0.0306987	+	0.999529i	$0.509773\pi$
$200$	0	0
$201$	24.2205	1.70839
$202$	12.0807	0.849994
$203$	−2.51524	−0.176535
$204$	26.5835	1.86122
$205$	0	0
$206$	−15.3582	−1.07006
$207$	1.43929	0.100038
$208$	−27.1982	−1.88586
$209$	0	0
$210$	0	0
$211$	7.97947	0.549329	0.274665	−	0.961540i	$-0.411433\pi$
$211$	7.97947	0.549329	0.274665	+	0.961540i	$0.411433\pi$
$212$	7.59093	0.521347
$213$	−16.0200	−1.09767
$214$	33.6540	2.30054
$215$	0	0
$216$	0.201902	0.0137377
$217$	12.3379	0.837553
$218$	16.7763	1.13624
$219$	−31.2991	−2.11500
$220$	0	0
$221$	−38.5824	−2.59534
$222$	14.6561	0.983651
$223$	15.2911	1.02397	0.511985	−	0.858995i	$-0.328911\pi$
$223$	15.2911	1.02397	0.511985	+	0.858995i	$0.328911\pi$
$224$	11.1346	0.743959
$225$	0	0
$226$	19.1081	1.27105
$227$	−7.39075	−0.490541	−0.245271	−	0.969455i	$-0.578877\pi$
$227$	−7.39075	−0.490541	−0.245271	+	0.969455i	$0.578877\pi$
$228$	0	0
$229$	−1.00545	−0.0664417	−0.0332209	−	0.999448i	$-0.510576\pi$
$229$	−1.00545	−0.0664417	−0.0332209	+	0.999448i	$0.510576\pi$
$230$	0	0
$231$	10.2108	0.671821
$232$	−0.839541	−0.0551185
$233$	17.0150	1.11469	0.557344	−	0.830281i	$-0.311820\pi$
$233$	17.0150	1.11469	0.557344	+	0.830281i	$0.311820\pi$
$234$	−33.5420	−2.19271
$235$	0	0
$236$	−3.93473	−0.256129
$237$	35.4240	2.30103
$238$	17.8164	1.15487
$239$	8.40908	0.543938	0.271969	−	0.962306i	$-0.412325\pi$
$239$	8.40908	0.543938	0.271969	+	0.962306i	$0.412325\pi$
$240$	0	0
$241$	13.0814	0.842648	0.421324	−	0.906910i	$-0.361566\pi$
$241$	13.0814	0.842648	0.421324	+	0.906910i	$0.361566\pi$
$242$	5.07171	0.326022
$243$	21.6571	1.38930
$244$	−11.0022	−0.704342
$245$	0	0
$246$	−34.7930	−2.21832
$247$	0	0
$248$	4.11817	0.261504
$249$	11.9312	0.756108
$250$	0	0
$251$	13.6747	0.863137	0.431568	−	0.902080i	$-0.357960\pi$
$251$	13.6747	0.863137	0.431568	+	0.902080i	$0.357960\pi$
$252$	7.22399	0.455069
$253$	−1.47306	−0.0926107
$254$	23.1928	1.45525
$255$	0	0
$256$	19.2409	1.20256
$257$	2.94446	0.183670	0.0918352	−	0.995774i	$-0.470727\pi$
$257$	2.94446	0.183670	0.0918352	+	0.995774i	$0.470727\pi$
$258$	32.2677	2.00890
$259$	4.58124	0.284665
$260$	0	0
$261$	−4.86952	−0.301416
$262$	15.5801	0.962541
$263$	−13.1907	−0.813370	−0.406685	−	0.913568i	$-0.633315\pi$
$263$	−13.1907	−0.813370	−0.406685	+	0.913568i	$0.633315\pi$
$264$	3.40817	0.209759
$265$	0	0
$266$	0	0
$267$	−12.9260	−0.791058
$268$	−17.5378	−1.07129
$269$	30.9906	1.88953	0.944765	−	0.327749i	$-0.106290\pi$
$269$	30.9906	1.88953	0.944765	+	0.327749i	$0.106290\pi$
$270$	0	0
$271$	−7.50772	−0.456062	−0.228031	−	0.973654i	$-0.573229\pi$
$271$	−7.50772	−0.456062	−0.228031	+	0.973654i	$0.573229\pi$
$272$	27.9691	1.69588
$273$	−21.6050	−1.30760
$274$	6.07812	0.367192
$275$	0	0
$276$	−2.14753	−0.129266
$277$	9.58988	0.576200	0.288100	−	0.957600i	$-0.406976\pi$
$277$	9.58988	0.576200	0.288100	+	0.957600i	$0.406976\pi$
$278$	−36.2882	−2.17642
$279$	23.8863	1.43003
$280$	0	0
$281$	−16.4874	−0.983555	−0.491778	−	0.870721i	$-0.663653\pi$
$281$	−16.4874	−0.983555	−0.491778	+	0.870721i	$0.663653\pi$
$282$	−1.48095	−0.0881892
$283$	−29.3679	−1.74574	−0.872871	−	0.487951i	$-0.837744\pi$
$283$	−29.3679	−1.74574	−0.872871	+	0.487951i	$0.837744\pi$
$284$	11.5999	0.688327
$285$	0	0
$286$	34.3291	2.02992
$287$	−10.8757	−0.641972
$288$	21.5566	1.27023
$289$	22.6760	1.33388
$290$	0	0
$291$	−38.5026	−2.25706
$292$	22.6633	1.32627
$293$	0.995738	0.0581717	0.0290858	−	0.999577i	$-0.490740\pi$
$293$	0.995738	0.0581717	0.0290858	+	0.999577i	$0.490740\pi$
$294$	−22.7410	−1.32628
$295$	0	0
$296$	1.52913	0.0888790
$297$	−1.19856	−0.0695473
$298$	17.2535	0.999466
$299$	3.11686	0.180253
$300$	0	0
$301$	10.0864	0.581368
$302$	−2.74158	−0.157760
$303$	15.0648	0.865451
$304$	0	0
$305$	0	0
$306$	34.4927	1.97182
$307$	−9.86517	−0.563035	−0.281518	−	0.959556i	$-0.590838\pi$
$307$	−9.86517	−0.563035	−0.281518	+	0.959556i	$0.590838\pi$
$308$	−7.39351	−0.421284
$309$	−19.1520	−1.08952
$310$	0	0
$311$	0.418188	0.0237133	0.0118566	−	0.999930i	$-0.496226\pi$
$311$	0.418188	0.0237133	0.0118566	+	0.999930i	$0.496226\pi$
$312$	−7.21136	−0.408263
$313$	−6.86188	−0.387856	−0.193928	−	0.981016i	$-0.562123\pi$
$313$	−6.86188	−0.387856	−0.193928	+	0.981016i	$0.562123\pi$
$314$	8.47817	0.478451
$315$	0	0
$316$	−25.6500	−1.44293
$317$	11.9255	0.669804	0.334902	−	0.942253i	$-0.391297\pi$
$317$	11.9255	0.669804	0.334902	+	0.942253i	$0.391297\pi$
$318$	20.2960	1.13814
$319$	4.98378	0.279038
$320$	0	0
$321$	41.9671	2.34237
$322$	−1.43929	−0.0802085
$323$	0	0
$324$	−16.5810	−0.921165
$325$	0	0
$326$	5.99694	0.332140
$327$	20.9204	1.15690
$328$	−3.63010	−0.200439
$329$	−0.462920	−0.0255216
$330$	0	0
$331$	7.96898	0.438015	0.219007	−	0.975723i	$-0.429718\pi$
$331$	7.96898	0.438015	0.219007	+	0.975723i	$0.429718\pi$
$332$	−8.63922	−0.474139
$333$	8.86930	0.486035
$334$	6.30561	0.345028
$335$	0	0
$336$	15.6619	0.854427
$337$	−4.04620	−0.220410	−0.110205	−	0.993909i	$-0.535151\pi$
$337$	−4.04620	−0.220410	−0.110205	+	0.993909i	$0.535151\pi$
$338$	−47.4689	−2.58197
$339$	23.8281	1.29416
$340$	0	0
$341$	−24.4468	−1.32387
$342$	0	0
$343$	−17.3355	−0.936028
$344$	3.36664	0.181517
$345$	0	0
$346$	−20.3675	−1.09496
$347$	3.26228	0.175129	0.0875643	−	0.996159i	$-0.472092\pi$
$347$	3.26228	0.175129	0.0875643	+	0.996159i	$0.472092\pi$
$348$	7.26569	0.389482
$349$	−6.32235	−0.338428	−0.169214	−	0.985579i	$-0.554123\pi$
$349$	−6.32235	−0.338428	−0.169214	+	0.985579i	$0.554123\pi$
$350$	0	0
$351$	2.53603	0.135363
$352$	−22.0624	−1.17593
$353$	6.35500	0.338242	0.169121	−	0.985595i	$-0.445907\pi$
$353$	6.35500	0.338242	0.169121	+	0.985595i	$0.445907\pi$
$354$	−10.5203	−0.559150
$355$	0	0
$356$	9.35954	0.496055
$357$	22.2174	1.17587
$358$	−27.8500	−1.47192
$359$	−2.70761	−0.142902	−0.0714512	−	0.997444i	$-0.522763\pi$
$359$	−2.70761	−0.142902	−0.0714512	+	0.997444i	$0.522763\pi$
$360$	0	0
$361$	0	0
$362$	15.7496	0.827782
$363$	6.32451	0.331951
$364$	15.6439	0.819965
$365$	0	0
$366$	−29.4167	−1.53764
$367$	−23.2443	−1.21334	−0.606671	−	0.794953i	$-0.707495\pi$
$367$	−23.2443	−1.21334	−0.606671	+	0.794953i	$0.707495\pi$
$368$	−2.25947	−0.117783
$369$	−21.0554	−1.09610
$370$	0	0
$371$	6.34419	0.329374
$372$	−35.6401	−1.84785
$373$	6.24784	0.323501	0.161750	−	0.986832i	$-0.448286\pi$
$373$	6.24784	0.323501	0.161750	+	0.986832i	$0.448286\pi$
$374$	−35.3021	−1.82543
$375$	0	0
$376$	−0.154514	−0.00796845
$377$	−10.5452	−0.543105
$378$	−1.17108	−0.0602337
$379$	−13.0469	−0.670176	−0.335088	−	0.942187i	$-0.608766\pi$
$379$	−13.0469	−0.670176	−0.335088	+	0.942187i	$0.608766\pi$
$380$	0	0
$381$	28.9219	1.48171
$382$	11.9926	0.613596
$383$	1.83744	0.0938889	0.0469445	−	0.998898i	$-0.485052\pi$
$383$	1.83744	0.0938889	0.0469445	+	0.998898i	$0.485052\pi$
$384$	9.34379	0.476823
$385$	0	0
$386$	−24.3084	−1.23726
$387$	19.5272	0.992624
$388$	27.8792	1.41535
$389$	18.6941	0.947828	0.473914	−	0.880571i	$-0.342841\pi$
$389$	18.6941	0.947828	0.473914	+	0.880571i	$0.342841\pi$
$390$	0	0
$391$	−3.20520	−0.162094
$392$	−2.37267	−0.119838
$393$	19.4286	0.980045
$394$	−6.65332	−0.335189
$395$	0	0
$396$	−14.3139	−0.719298
$397$	−24.9631	−1.25286	−0.626431	−	0.779477i	$-0.715485\pi$
$397$	−24.9631	−1.25286	−0.626431	+	0.779477i	$0.715485\pi$
$398$	1.67681	0.0840508
$399$	0	0
$400$	0	0
$401$	25.4399	1.27041	0.635203	−	0.772345i	$-0.280916\pi$
$401$	25.4399	1.27041	0.635203	+	0.772345i	$0.280916\pi$
$402$	−46.8911	−2.33871
$403$	51.7270	2.57670
$404$	−10.9082	−0.542705
$405$	0	0
$406$	4.86952	0.241670
$407$	−9.07743	−0.449951
$408$	7.41575	0.367135
$409$	−26.8692	−1.32859	−0.664297	−	0.747468i	$-0.731269\pi$
$409$	−26.8692	−1.32859	−0.664297	+	0.747468i	$0.731269\pi$
$410$	0	0
$411$	7.57951	0.373870
$412$	13.8677	0.683213
$413$	−3.28849	−0.161816
$414$	−2.78647	−0.136948
$415$	0	0
$416$	46.6818	2.28877
$417$	−45.2520	−2.21600
$418$	0	0
$419$	21.0428	1.02801	0.514004	−	0.857788i	$-0.328162\pi$
$419$	21.0428	1.02801	0.514004	+	0.857788i	$0.328162\pi$
$420$	0	0
$421$	2.52939	0.123275	0.0616376	−	0.998099i	$-0.480368\pi$
$421$	2.52939	0.123275	0.0616376	+	0.998099i	$0.480368\pi$
$422$	−15.4483	−0.752011
$423$	−0.896215	−0.0435755
$424$	2.11757	0.102838
$425$	0	0
$426$	31.0148	1.50267
$427$	−9.19517	−0.444985
$428$	−30.3878	−1.46885
$429$	42.8089	2.06684
$430$	0	0
$431$	−30.2212	−1.45570	−0.727852	−	0.685734i	$-0.759481\pi$
$431$	−30.2212	−1.45570	−0.727852	+	0.685734i	$0.759481\pi$
$432$	−1.83841	−0.0884507
$433$	13.0760	0.628392	0.314196	−	0.949358i	$-0.398265\pi$
$433$	13.0760	0.628392	0.314196	+	0.949358i	$0.398265\pi$
$434$	−23.8863	−1.14658
$435$	0	0
$436$	−15.1482	−0.725466
$437$	0	0
$438$	60.5952	2.89535
$439$	−38.3642	−1.83102	−0.915512	−	0.402290i	$-0.868214\pi$
$439$	−38.3642	−1.83102	−0.915512	+	0.402290i	$0.868214\pi$
$440$	0	0
$441$	−13.7620	−0.655334
$442$	74.6958	3.55292
$443$	11.5357	0.548079	0.274039	−	0.961718i	$-0.411640\pi$
$443$	11.5357	0.548079	0.274039	+	0.961718i	$0.411640\pi$
$444$	−13.2337	−0.628043
$445$	0	0
$446$	−29.6037	−1.40177
$447$	21.5153	1.01764
$448$	−8.58191	−0.405457
$449$	20.1918	0.952909	0.476455	−	0.879199i	$-0.341922\pi$
$449$	20.1918	0.952909	0.476455	+	0.879199i	$0.341922\pi$
$450$	0	0
$451$	21.5495	1.01473
$452$	−17.2536	−0.811541
$453$	−3.41880	−0.160629
$454$	14.3085	0.671533
$455$	0	0
$456$	0	0
$457$	−17.6887	−0.827442	−0.413721	−	0.910404i	$-0.635771\pi$
$457$	−17.6887	−0.827442	−0.413721	+	0.910404i	$0.635771\pi$
$458$	1.94655	0.0909562
$459$	−2.60791	−0.121727
$460$	0	0
$461$	24.9640	1.16269	0.581345	−	0.813657i	$-0.302527\pi$
$461$	24.9640	1.16269	0.581345	+	0.813657i	$0.302527\pi$
$462$	−19.7681	−0.919697
$463$	17.4046	0.808860	0.404430	−	0.914569i	$-0.367470\pi$
$463$	17.4046	0.808860	0.404430	+	0.914569i	$0.367470\pi$
$464$	7.64441	0.354883
$465$	0	0
$466$	−32.9411	−1.52597
$467$	14.6807	0.679342	0.339671	−	0.940544i	$-0.389684\pi$
$467$	14.6807	0.679342	0.339671	+	0.940544i	$0.389684\pi$
$468$	30.2867	1.40000
$469$	−14.6574	−0.676814
$470$	0	0
$471$	10.5724	0.487151
$472$	−1.09763	−0.0505227
$473$	−19.9854	−0.918931
$474$	−68.5810	−3.15003
$475$	0	0
$476$	−16.0873	−0.737362
$477$	12.2824	0.562372
$478$	−16.2800	−0.744631
$479$	2.88202	0.131683	0.0658415	−	0.997830i	$-0.479027\pi$
$479$	2.88202	0.131683	0.0658415	+	0.997830i	$0.479027\pi$
$480$	0	0
$481$	19.2069	0.875761
$482$	−25.3257	−1.15355
$483$	−1.79482	−0.0816671
$484$	−4.57950	−0.208159
$485$	0	0
$486$	−41.9282	−1.90190
$487$	−22.9093	−1.03812	−0.519061	−	0.854737i	$-0.673718\pi$
$487$	−22.9093	−1.03812	−0.519061	+	0.854737i	$0.673718\pi$
$488$	−3.06918	−0.138935
$489$	7.47828	0.338180
$490$	0	0
$491$	−21.2033	−0.956889	−0.478445	−	0.878118i	$-0.658799\pi$
$491$	−21.2033	−0.956889	−0.478445	+	0.878118i	$0.658799\pi$
$492$	31.4163	1.41635
$493$	10.8441	0.488393
$494$	0	0
$495$	0	0
$496$	−37.4978	−1.68370
$497$	9.69472	0.434868
$498$	−23.0988	−1.03508
$499$	−36.5201	−1.63487	−0.817433	−	0.576024i	$-0.804604\pi$
$499$	−36.5201	−1.63487	−0.817433	+	0.576024i	$0.804604\pi$
$500$	0	0
$501$	7.86320	0.351302
$502$	−26.4742	−1.18160
$503$	−32.6315	−1.45497	−0.727483	−	0.686125i	$-0.759310\pi$
$503$	−32.6315	−1.45497	−0.727483	+	0.686125i	$0.759310\pi$
$504$	2.01521	0.0897646
$505$	0	0
$506$	2.85186	0.126781
$507$	−59.1945	−2.62892
$508$	−20.9419	−0.929148
$509$	43.2651	1.91769	0.958846	−	0.283927i	$-0.0916374\pi$
$509$	43.2651	1.91769	0.958846	+	0.283927i	$0.0916374\pi$
$510$	0	0
$511$	18.9410	0.837902
$512$	−29.5099	−1.30416
$513$	0	0
$514$	−5.70049	−0.251438
$515$	0	0
$516$	−29.1361	−1.28265
$517$	0.917245	0.0403404
$518$	−8.86930	−0.389695
$519$	−25.3986	−1.11487
$520$	0	0
$521$	38.7647	1.69831	0.849156	−	0.528142i	$-0.177111\pi$
$521$	38.7647	1.69831	0.849156	+	0.528142i	$0.177111\pi$
$522$	9.42741	0.412627
$523$	−27.6586	−1.20943	−0.604714	−	0.796443i	$-0.706712\pi$
$523$	−27.6586	−1.20943	−0.604714	+	0.796443i	$0.706712\pi$
$524$	−14.0680	−0.614564
$525$	0	0
$526$	25.5372	1.11347
$527$	−53.1931	−2.31713
$528$	−31.0330	−1.35054
$529$	−22.7411	−0.988742
$530$	0	0
$531$	−6.36652	−0.276284
$532$	0	0
$533$	−45.5966	−1.97501
$534$	25.0248	1.08293
$535$	0	0
$536$	−4.89235	−0.211318
$537$	−34.7295	−1.49869
$538$	−59.9979	−2.58669
$539$	14.0850	0.606682
$540$	0	0
$541$	−4.19755	−0.180467	−0.0902335	−	0.995921i	$-0.528761\pi$
$541$	−4.19755	−0.180467	−0.0902335	+	0.995921i	$0.528761\pi$
$542$	14.5350	0.624331
$543$	19.6400	0.842835
$544$	−48.0050	−2.05820
$545$	0	0
$546$	41.8274	1.79005
$547$	14.4970	0.619848	0.309924	−	0.950761i	$-0.399696\pi$
$547$	14.4970	0.619848	0.309924	+	0.950761i	$0.399696\pi$
$548$	−5.48823	−0.234445
$549$	−17.8019	−0.759766
$550$	0	0
$551$	0	0
$552$	−0.599077	−0.0254984
$553$	−21.4373	−0.911605
$554$	−18.5661	−0.788796
$555$	0	0
$556$	32.7664	1.38960
$557$	−28.9192	−1.22535	−0.612673	−	0.790337i	$-0.709906\pi$
$557$	−28.9192	−1.22535	−0.612673	+	0.790337i	$0.709906\pi$
$558$	−46.2439	−1.95766
$559$	42.2872	1.78856
$560$	0	0
$561$	−44.0223	−1.85862
$562$	31.9197	1.34645
$563$	24.5426	1.03435	0.517174	−	0.855880i	$-0.326984\pi$
$563$	24.5426	1.03435	0.517174	+	0.855880i	$0.326984\pi$
$564$	1.33722	0.0563072
$565$	0	0
$566$	56.8564	2.38985
$567$	−13.8577	−0.581969
$568$	3.23592	0.135776
$569$	0.454757	0.0190644	0.00953221	−	0.999955i	$-0.496966\pi$
$569$	0.454757	0.0190644	0.00953221	+	0.999955i	$0.496966\pi$
$570$	0	0
$571$	−24.0492	−1.00643	−0.503213	−	0.864162i	$-0.667849\pi$
$571$	−24.0492	−1.00643	−0.503213	+	0.864162i	$0.667849\pi$
$572$	−30.9974	−1.29607
$573$	14.9550	0.624754
$574$	21.0554	0.878835
$575$	0	0
$576$	−16.6146	−0.692275
$577$	31.5181	1.31211	0.656057	−	0.754711i	$-0.272223\pi$
$577$	31.5181	1.31211	0.656057	+	0.754711i	$0.272223\pi$
$578$	−43.9009	−1.82604
$579$	−30.3130	−1.25976
$580$	0	0
$581$	−7.22031	−0.299549
$582$	74.5412	3.08983
$583$	−12.5706	−0.520621
$584$	6.32216	0.261613
$585$	0	0
$586$	−1.92775	−0.0796348
$587$	−0.243299	−0.0100420	−0.00502101	−	0.999987i	$-0.501598\pi$
$587$	−0.243299	−0.0100420	−0.00502101	+	0.999987i	$0.501598\pi$
$588$	20.5340	0.846807
$589$	0	0
$590$	0	0
$591$	−8.29680	−0.341285
$592$	−13.9235	−0.572251
$593$	−38.5061	−1.58126	−0.790628	−	0.612296i	$-0.790246\pi$
$593$	−38.5061	−1.58126	−0.790628	+	0.612296i	$0.790246\pi$
$594$	2.32041	0.0952076
$595$	0	0
$596$	−15.5790	−0.638140
$597$	2.09101	0.0855792
$598$	−6.03425	−0.246759
$599$	14.3109	0.584728	0.292364	−	0.956307i	$-0.405558\pi$
$599$	14.3109	0.584728	0.292364	+	0.956307i	$0.405558\pi$
$600$	0	0
$601$	36.8537	1.50329	0.751647	−	0.659566i	$-0.229260\pi$
$601$	36.8537	1.50329	0.751647	+	0.659566i	$0.229260\pi$
$602$	−19.5272	−0.795870
$603$	−28.3767	−1.15559
$604$	2.47551	0.100727
$605$	0	0
$606$	−29.1656	−1.18477
$607$	45.1585	1.83293	0.916463	−	0.400119i	$-0.131031\pi$
$607$	45.1585	1.83293	0.916463	+	0.400119i	$0.131031\pi$
$608$	0	0
$609$	6.07237	0.246065
$610$	0	0
$611$	−1.94080	−0.0785164
$612$	−31.1452	−1.25897
$613$	12.8540	0.519167	0.259583	−	0.965721i	$-0.416415\pi$
$613$	12.8540	0.519167	0.259583	+	0.965721i	$0.416415\pi$
$614$	19.0990	0.770774
$615$	0	0
$616$	−2.06250	−0.0831004
$617$	42.2758	1.70196	0.850979	−	0.525199i	$-0.176009\pi$
$617$	42.2758	1.70196	0.850979	+	0.525199i	$0.176009\pi$
$618$	37.0783	1.49151
$619$	25.2048	1.01307	0.506533	−	0.862221i	$-0.330927\pi$
$619$	25.2048	1.01307	0.506533	+	0.862221i	$0.330927\pi$
$620$	0	0
$621$	0.210678	0.00845423
$622$	−0.809614	−0.0324626
$623$	7.82233	0.313395
$624$	65.6628	2.62861
$625$	0	0
$626$	13.2846	0.530960
$627$	0	0
$628$	−7.65535	−0.305482
$629$	−19.7513	−0.787538
$630$	0	0
$631$	1.49432	0.0594880	0.0297440	−	0.999558i	$-0.490531\pi$
$631$	1.49432	0.0594880	0.0297440	+	0.999558i	$0.490531\pi$
$632$	−7.15536	−0.284625
$633$	−19.2643	−0.765686
$634$	−23.0878	−0.916936
$635$	0	0
$636$	−18.3262	−0.726683
$637$	−29.8024	−1.18081
$638$	−9.64863	−0.381993
$639$	18.7690	0.742491
$640$	0	0
$641$	−6.57587	−0.259731	−0.129866	−	0.991532i	$-0.541455\pi$
$641$	−6.57587	−0.259731	−0.129866	+	0.991532i	$0.541455\pi$
$642$	−81.2485	−3.20662
$643$	4.60673	0.181672	0.0908359	−	0.995866i	$-0.471046\pi$
$643$	4.60673	0.181672	0.0908359	+	0.995866i	$0.471046\pi$
$644$	1.29961	0.0512116
$645$	0	0
$646$	0	0
$647$	3.53706	0.139056	0.0695281	−	0.997580i	$-0.477851\pi$
$647$	3.53706	0.139056	0.0695281	+	0.997580i	$0.477851\pi$
$648$	−4.62544	−0.181705
$649$	6.51592	0.255772
$650$	0	0
$651$	−29.7866	−1.16743
$652$	−5.41493	−0.212065
$653$	48.6489	1.90378	0.951889	−	0.306444i	$-0.0991391\pi$
$653$	48.6489	1.90378	0.951889	+	0.306444i	$0.0991391\pi$
$654$	−40.5019	−1.58375
$655$	0	0
$656$	33.0538	1.29053
$657$	36.6699	1.43063
$658$	0.896215	0.0349381
$659$	−21.2450	−0.827588	−0.413794	−	0.910371i	$-0.635797\pi$
$659$	−21.2450	−0.827588	−0.413794	+	0.910371i	$0.635797\pi$
$660$	0	0
$661$	14.7872	0.575154	0.287577	−	0.957757i	$-0.407150\pi$
$661$	14.7872	0.575154	0.287577	+	0.957757i	$0.407150\pi$
$662$	−15.4280	−0.599625
$663$	93.1469	3.61753
$664$	−2.41000	−0.0935262
$665$	0	0
$666$	−17.1710	−0.665363
$667$	−0.876033	−0.0339201
$668$	−5.69364	−0.220294
$669$	−36.9163	−1.42727
$670$	0	0
$671$	18.2196	0.703361
$672$	−26.8814	−1.03697
$673$	6.36313	0.245280	0.122640	−	0.992451i	$-0.460864\pi$
$673$	6.36313	0.245280	0.122640	+	0.992451i	$0.460864\pi$
$674$	7.83345	0.301733
$675$	0	0
$676$	42.8620	1.64854
$677$	45.6873	1.75591	0.877953	−	0.478746i	$-0.158909\pi$
$677$	45.6873	1.75591	0.877953	+	0.478746i	$0.158909\pi$
$678$	−46.1313	−1.77166
$679$	23.3003	0.894184
$680$	0	0
$681$	17.8430	0.683744
$682$	47.3291	1.81232
$683$	−15.8141	−0.605109	−0.302554	−	0.953132i	$-0.597839\pi$
$683$	−15.8141	−0.605109	−0.302554	+	0.953132i	$0.597839\pi$
$684$	0	0
$685$	0	0
$686$	33.5616	1.28139
$687$	2.42738	0.0926102
$688$	−30.6548	−1.16870
$689$	26.5981	1.01331
$690$	0	0
$691$	−35.2478	−1.34089	−0.670444	−	0.741960i	$-0.733896\pi$
$691$	−35.2478	−1.34089	−0.670444	+	0.741960i	$0.733896\pi$
$692$	18.3908	0.699113
$693$	−11.9629	−0.454435
$694$	−6.31580	−0.239744
$695$	0	0
$696$	2.02684	0.0768273
$697$	46.8890	1.77605
$698$	12.2401	0.463295
$699$	−41.0781	−1.55372
$700$	0	0
$701$	−51.0439	−1.92790	−0.963951	−	0.266079i	$-0.914272\pi$
$701$	−51.0439	−1.92790	−0.963951	+	0.266079i	$0.914272\pi$
$702$	−4.90976	−0.185307
$703$	0	0
$704$	17.0045	0.640880
$705$	0	0
$706$	−12.3033	−0.463041
$707$	−9.11667	−0.342867
$708$	9.49934	0.357007
$709$	1.76418	0.0662552	0.0331276	−	0.999451i	$-0.489453\pi$
$709$	1.76418	0.0662552	0.0331276	+	0.999451i	$0.489453\pi$
$710$	0	0
$711$	−41.5026	−1.55647
$712$	2.61095	0.0978493
$713$	4.29717	0.160930
$714$	−43.0130	−1.60972
$715$	0	0
$716$	25.1472	0.939793
$717$	−20.3015	−0.758172
$718$	5.24195	0.195628
$719$	−51.5725	−1.92333	−0.961665	−	0.274227i	$-0.911578\pi$
$719$	−51.5725	−1.92333	−0.961665	+	0.274227i	$0.911578\pi$
$720$	0	0
$721$	11.5901	0.431637
$722$	0	0
$723$	−31.5815	−1.17453
$724$	−14.2211	−0.528523
$725$	0	0
$726$	−12.2443	−0.454428
$727$	39.7695	1.47497	0.737484	−	0.675364i	$-0.236014\pi$
$727$	39.7695	1.47497	0.737484	+	0.675364i	$0.236014\pi$
$728$	4.36404	0.161742
$729$	−23.8299	−0.882589
$730$	0	0
$731$	−43.4858	−1.60838
$732$	26.5618	0.981751
$733$	−32.2476	−1.19109	−0.595547	−	0.803321i	$-0.703065\pi$
$733$	−32.2476	−1.19109	−0.595547	+	0.803321i	$0.703065\pi$
$734$	45.0010	1.66102
$735$	0	0
$736$	3.87805	0.142947
$737$	29.0426	1.06980
$738$	40.7633	1.50052
$739$	−33.2127	−1.22175	−0.610875	−	0.791727i	$-0.709182\pi$
$739$	−33.2127	−1.22175	−0.610875	+	0.791727i	$0.709182\pi$
$740$	0	0
$741$	0	0
$742$	−12.2824	−0.450900
$743$	−21.0511	−0.772290	−0.386145	−	0.922438i	$-0.626194\pi$
$743$	−21.0511	−0.772290	−0.386145	+	0.922438i	$0.626194\pi$
$744$	−9.94220	−0.364499
$745$	0	0
$746$	−12.0958	−0.442860
$747$	−13.9785	−0.511448
$748$	31.8760	1.16550
$749$	−25.3969	−0.927983
$750$	0	0
$751$	−9.54033	−0.348132	−0.174066	−	0.984734i	$-0.555691\pi$
$751$	−9.54033	−0.348132	−0.174066	+	0.984734i	$0.555691\pi$
$752$	1.40692	0.0513052
$753$	−33.0138	−1.20309
$754$	20.4155	0.743490
$755$	0	0
$756$	1.05742	0.0384581
$757$	36.0653	1.31082	0.655408	−	0.755275i	$-0.272497\pi$
$757$	36.0653	1.31082	0.655408	+	0.755275i	$0.272497\pi$
$758$	25.2589	0.917445
$759$	3.55632	0.129086
$760$	0	0
$761$	−13.0993	−0.474848	−0.237424	−	0.971406i	$-0.576303\pi$
$761$	−13.0993	−0.474848	−0.237424	+	0.971406i	$0.576303\pi$
$762$	−55.9928	−2.02841
$763$	−12.6602	−0.458331
$764$	−10.8287	−0.391769
$765$	0	0
$766$	−3.55730	−0.128530
$767$	−13.7870	−0.497821
$768$	−46.4519	−1.67619
$769$	−43.3213	−1.56221	−0.781103	−	0.624402i	$-0.785343\pi$
$769$	−43.3213	−1.56221	−0.781103	+	0.624402i	$0.785343\pi$
$770$	0	0
$771$	−7.10861	−0.256010
$772$	21.9492	0.789970
$773$	−2.58122	−0.0928401	−0.0464200	−	0.998922i	$-0.514781\pi$
$773$	−2.58122	−0.0928401	−0.0464200	+	0.998922i	$0.514781\pi$
$774$	−37.8048	−1.35886
$775$	0	0
$776$	7.77721	0.279186
$777$	−11.0602	−0.396781
$778$	−36.1918	−1.29754
$779$	0	0
$780$	0	0
$781$	−19.2094	−0.687368
$782$	6.20529	0.221901
$783$	−0.712783	−0.0254728
$784$	21.6043	0.771582
$785$	0	0
$786$	−37.6139	−1.34164
$787$	−32.3874	−1.15449	−0.577243	−	0.816572i	$-0.695872\pi$
$787$	−32.3874	−1.15449	−0.577243	+	0.816572i	$0.695872\pi$
$788$	6.00761	0.214012
$789$	31.8453	1.13372
$790$	0	0
$791$	−14.4199	−0.512711
$792$	−3.99301	−0.141885
$793$	−38.5509	−1.36898
$794$	48.3286	1.71512
$795$	0	0
$796$	−1.51407	−0.0536648
$797$	−0.660265	−0.0233878	−0.0116939	−	0.999932i	$-0.503722\pi$
$797$	−0.660265	−0.0233878	−0.0116939	+	0.999932i	$0.503722\pi$
$798$	0	0
$799$	1.99581	0.0706067
$800$	0	0
$801$	15.1441	0.535089
$802$	−49.2517	−1.73914
$803$	−37.5304	−1.32442
$804$	42.3402	1.49322
$805$	0	0
$806$	−100.144	−3.52741
$807$	−74.8184	−2.63373
$808$	−3.04297	−0.107051
$809$	−39.0366	−1.37246	−0.686228	−	0.727387i	$-0.740735\pi$
$809$	−39.0366	−1.37246	−0.686228	+	0.727387i	$0.740735\pi$
$810$	0	0
$811$	−29.9982	−1.05338	−0.526690	−	0.850057i	$-0.676567\pi$
$811$	−29.9982	−1.05338	−0.526690	+	0.850057i	$0.676567\pi$
$812$	−4.39693	−0.154302
$813$	18.1254	0.635684
$814$	17.5739	0.615966
$815$	0	0
$816$	−67.5239	−2.36381
$817$	0	0
$818$	52.0188	1.81880
$819$	25.3124	0.884487
$820$	0	0
$821$	−36.5733	−1.27642	−0.638208	−	0.769864i	$-0.720324\pi$
$821$	−36.5733	−1.27642	−0.638208	+	0.769864i	$0.720324\pi$
$822$	−14.6740	−0.511814
$823$	1.19784	0.0417542	0.0208771	−	0.999782i	$-0.493354\pi$
$823$	1.19784	0.0417542	0.0208771	+	0.999782i	$0.493354\pi$
$824$	3.86855	0.134767
$825$	0	0
$826$	6.36652	0.221520
$827$	0.0989534	0.00344095	0.00172047	−	0.999999i	$-0.499452\pi$
$827$	0.0989534	0.00344095	0.00172047	+	0.999999i	$0.499452\pi$
$828$	2.51604	0.0874385
$829$	3.38031	0.117403	0.0587015	−	0.998276i	$-0.481304\pi$
$829$	3.38031	0.117403	0.0587015	+	0.998276i	$0.481304\pi$
$830$	0	0
$831$	−23.1522	−0.803140
$832$	−35.9798	−1.24738
$833$	30.6471	1.06186
$834$	87.6080	3.03362
$835$	0	0
$836$	0	0
$837$	3.49639	0.120853
$838$	−40.7389	−1.40730
$839$	−11.7241	−0.404759	−0.202380	−	0.979307i	$-0.564867\pi$
$839$	−11.7241	−0.404759	−0.202380	+	0.979307i	$0.564867\pi$
$840$	0	0
$841$	−26.0361	−0.897798
$842$	−4.89692	−0.168759
$843$	39.8043	1.37093
$844$	13.9490	0.480145
$845$	0	0
$846$	1.73508	0.0596532
$847$	−3.82736	−0.131510
$848$	−19.2815	−0.662129
$849$	70.9009	2.43331
$850$	0	0
$851$	1.59560	0.0546964
$852$	−28.0048	−0.959429
$853$	−11.8398	−0.405389	−0.202694	−	0.979242i	$-0.564970\pi$
$853$	−11.8398	−0.405389	−0.202694	+	0.979242i	$0.564970\pi$
$854$	17.8019	0.609168
$855$	0	0
$856$	−8.47701	−0.289738
$857$	−52.7265	−1.80110	−0.900552	−	0.434749i	$-0.856837\pi$
$857$	−52.7265	−1.80110	−0.900552	+	0.434749i	$0.856837\pi$
$858$	−82.8783	−2.82942
$859$	−5.72563	−0.195356	−0.0976779	−	0.995218i	$-0.531142\pi$
$859$	−5.72563	−0.195356	−0.0976779	+	0.995218i	$0.531142\pi$
$860$	0	0
$861$	26.2564	0.894817
$862$	58.5084	1.99280
$863$	−10.4374	−0.355293	−0.177646	−	0.984094i	$-0.556848\pi$
$863$	−10.4374	−0.355293	−0.177646	+	0.984094i	$0.556848\pi$
$864$	3.15537	0.107348
$865$	0	0
$866$	−25.3152	−0.860244
$867$	−54.7451	−1.85924
$868$	21.5681	0.732068
$869$	42.4765	1.44092
$870$	0	0
$871$	−61.4513	−2.08220
$872$	−4.22575	−0.143102
$873$	45.1095	1.52673
$874$	0	0
$875$	0	0
$876$	−54.7143	−1.84863
$877$	16.9211	0.571386	0.285693	−	0.958321i	$-0.407776\pi$
$877$	16.9211	0.571386	0.285693	+	0.958321i	$0.407776\pi$
$878$	74.2733	2.50660
$879$	−2.40394	−0.0810829
$880$	0	0
$881$	−28.2082	−0.950360	−0.475180	−	0.879889i	$-0.657617\pi$
$881$	−28.2082	−0.950360	−0.475180	+	0.879889i	$0.657617\pi$
$882$	26.6433	0.897128
$883$	2.89071	0.0972800	0.0486400	−	0.998816i	$-0.484511\pi$
$883$	2.89071	0.0972800	0.0486400	+	0.998816i	$0.484511\pi$
$884$	−67.4465	−2.26847
$885$	0	0
$886$	−22.3332	−0.750299
$887$	−32.9315	−1.10573	−0.552866	−	0.833270i	$-0.686466\pi$
$887$	−32.9315	−1.10573	−0.552866	+	0.833270i	$0.686466\pi$
$888$	−3.69168	−0.123885
$889$	−17.5024	−0.587013
$890$	0	0
$891$	27.4581	0.919882
$892$	26.7306	0.895007
$893$	0	0
$894$	−41.6538	−1.39311
$895$	0	0
$896$	−5.65451	−0.188904
$897$	−7.52481	−0.251246
$898$	−39.0914	−1.30450
$899$	−14.5385	−0.484887
$900$	0	0
$901$	−27.3520	−0.911228
$902$	−41.7199	−1.38912
$903$	−24.3508	−0.810343
$904$	−4.81308	−0.160081
$905$	0	0
$906$	6.61881	0.219895
$907$	22.8800	0.759717	0.379859	−	0.925045i	$-0.375973\pi$
$907$	22.8800	0.759717	0.379859	+	0.925045i	$0.375973\pi$
$908$	−12.9199	−0.428761
$909$	−17.6499	−0.585410
$910$	0	0
$911$	11.6167	0.384880	0.192440	−	0.981309i	$-0.438360\pi$
$911$	11.6167	0.384880	0.192440	+	0.981309i	$0.438360\pi$
$912$	0	0
$913$	14.3066	0.473478
$914$	34.2454	1.13274
$915$	0	0
$916$	−1.75763	−0.0580738
$917$	−11.7575	−0.388266
$918$	5.04892	0.166639
$919$	−12.4353	−0.410203	−0.205101	−	0.978741i	$-0.565752\pi$
$919$	−12.4353	−0.410203	−0.205101	+	0.978741i	$0.565752\pi$
$920$	0	0
$921$	23.8168	0.784790
$922$	−48.3305	−1.59168
$923$	40.6453	1.33786
$924$	17.8496	0.587209
$925$	0	0
$926$	−33.6954	−1.10730
$927$	22.4384	0.736974
$928$	−13.1205	−0.430703
$929$	−15.7247	−0.515910	−0.257955	−	0.966157i	$-0.583049\pi$
$929$	−15.7247	−0.515910	−0.257955	+	0.966157i	$0.583049\pi$
$930$	0	0
$931$	0	0
$932$	29.7441	0.974301
$933$	−1.00960	−0.0330529
$934$	−28.4219	−0.929993
$935$	0	0
$936$	8.44881	0.276158
$937$	−16.4857	−0.538563	−0.269282	−	0.963061i	$-0.586786\pi$
$937$	−16.4857	−0.538563	−0.269282	+	0.963061i	$0.586786\pi$
$938$	28.3767	0.926533
$939$	16.5661	0.540616
$940$	0	0
$941$	−13.7308	−0.447611	−0.223806	−	0.974634i	$-0.571848\pi$
$941$	−13.7308	−0.447611	−0.223806	+	0.974634i	$0.571848\pi$
$942$	−20.4683	−0.666892
$943$	−3.78789	−0.123351
$944$	9.99448	0.325293
$945$	0	0
$946$	38.6919	1.25798
$947$	−45.7355	−1.48620	−0.743101	−	0.669179i	$-0.766646\pi$
$947$	−45.7355	−1.48620	−0.743101	+	0.669179i	$0.766646\pi$
$948$	61.9251	2.01123
$949$	79.4106	2.57778
$950$	0	0
$951$	−28.7909	−0.933610
$952$	−4.48774	−0.145448
$953$	−32.7211	−1.05994	−0.529971	−	0.848016i	$-0.677797\pi$
$953$	−32.7211	−1.05994	−0.529971	+	0.848016i	$0.677797\pi$
$954$	−23.7787	−0.769865
$955$	0	0
$956$	14.7000	0.475433
$957$	−12.0320	−0.388939
$958$	−5.57961	−0.180269
$959$	−4.58684	−0.148117
$960$	0	0
$961$	40.3153	1.30049
$962$	−37.1847	−1.19888
$963$	−49.1685	−1.58443
$964$	22.8678	0.736522
$965$	0	0
$966$	3.47478	0.111799
$967$	−11.4909	−0.369523	−0.184761	−	0.982783i	$-0.559151\pi$
$967$	−11.4909	−0.369523	−0.184761	+	0.982783i	$0.559151\pi$
$968$	−1.27750	−0.0410604
$969$	0	0
$970$	0	0
$971$	−54.4334	−1.74685	−0.873425	−	0.486959i	$-0.838106\pi$
$971$	−54.4334	−1.74685	−0.873425	+	0.486959i	$0.838106\pi$
$972$	37.8590	1.21433
$973$	27.3848	0.877917
$974$	44.3526	1.42115
$975$	0	0
$976$	27.9463	0.894539
$977$	−23.6834	−0.757698	−0.378849	−	0.925459i	$-0.623680\pi$
$977$	−23.6834	−0.757698	−0.378849	+	0.925459i	$0.623680\pi$
$978$	−14.4780	−0.462955
$979$	−15.4994	−0.495364
$980$	0	0
$981$	−24.5103	−0.782552
$982$	41.0496	1.30994
$983$	−4.93140	−0.157287	−0.0786436	−	0.996903i	$-0.525059\pi$
$983$	−4.93140	−0.157287	−0.0786436	+	0.996903i	$0.525059\pi$
$984$	8.76391	0.279383
$985$	0	0
$986$	−20.9942	−0.668592
$987$	1.11760	0.0355735
$988$	0	0
$989$	3.51297	0.111706
$990$	0	0
$991$	−12.3542	−0.392445	−0.196223	−	0.980559i	$-0.562867\pi$
$991$	−12.3542	−0.392445	−0.196223	+	0.980559i	$0.562867\pi$
$992$	64.3596	2.04342
$993$	−19.2389	−0.610530
$994$	−18.7690	−0.595317
$995$	0	0
$996$	20.8571	0.660881
$997$	33.3639	1.05664	0.528322	−	0.849044i	$-0.322821\pi$
$997$	33.3639	1.05664	0.528322	+	0.849044i	$0.322821\pi$
$998$	70.7032	2.23807
$999$	1.29826	0.0410750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.3		16
5.2	odd	4		1805.2.b.j.1084.3	yes	16
5.3	odd	4		1805.2.b.j.1084.14	yes	16
5.4	even	2	inner	9025.2.a.ck.1.14		16
19.18	odd	2		9025.2.a.cl.1.14		16
95.18	even	4		1805.2.b.i.1084.3	✓	16
95.37	even	4		1805.2.b.i.1084.14	yes	16
95.94	odd	2		9025.2.a.cl.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.3	✓	16	95.18	even	4
1805.2.b.i.1084.14	yes	16	95.37	even	4
1805.2.b.j.1084.3	yes	16	5.2	odd	4
1805.2.b.j.1084.14	yes	16	5.3	odd	4
9025.2.a.ck.1.3		16	1.1	even	1	trivial
9025.2.a.ck.1.14		16	5.4	even	2	inner
9025.2.a.cl.1.3		16	95.94	odd	2
9025.2.a.cl.1.14		16	19.18	odd	2

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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93600 q^{2} -2.41423 q^{3} +1.74811 q^{4} +4.67396 q^{6} +1.46100 q^{7} +0.487655 q^{8} +2.82851 q^{9} +O(q^{10})\)	\(q-1.93600 q^{2} -2.41423 q^{3} +1.74811 q^{4} +4.67396 q^{6} +1.46100 q^{7} +0.487655 q^{8} +2.82851 q^{9} -2.89488 q^{11} -4.22035 q^{12} +6.12528 q^{13} -2.82851 q^{14} -4.44033 q^{16} -6.29889 q^{17} -5.47600 q^{18} -3.52719 q^{21} +5.60450 q^{22} +0.508852 q^{23} -1.17731 q^{24} -11.8586 q^{26} +0.414027 q^{27} +2.55400 q^{28} -1.72159 q^{29} +8.44484 q^{31} +7.62118 q^{32} +6.98890 q^{33} +12.1947 q^{34} +4.94455 q^{36} +3.13569 q^{37} -14.7878 q^{39} -7.44400 q^{41} +6.82866 q^{42} +6.90372 q^{43} -5.06057 q^{44} -0.985139 q^{46} -0.316851 q^{47} +10.7200 q^{48} -4.86547 q^{49} +15.2070 q^{51} +10.7077 q^{52} +4.34236 q^{53} -0.801557 q^{54} +0.712465 q^{56} +3.33300 q^{58} -2.25084 q^{59} -6.29374 q^{61} -16.3492 q^{62} +4.13245 q^{63} -5.87399 q^{64} -13.5305 q^{66} -10.0324 q^{67} -11.0112 q^{68} -1.22849 q^{69} +6.63567 q^{71} +1.37933 q^{72} +12.9644 q^{73} -6.07070 q^{74} -4.22942 q^{77} +28.6293 q^{78} -14.6730 q^{79} -9.48507 q^{81} +14.4116 q^{82} -4.94202 q^{83} -6.16593 q^{84} -13.3656 q^{86} +4.15631 q^{87} -1.41170 q^{88} +5.35408 q^{89} +8.94904 q^{91} +0.889530 q^{92} -20.3878 q^{93} +0.613425 q^{94} -18.3993 q^{96} +15.9482 q^{97} +9.41958 q^{98} -8.18818 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

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