LMFDB - Embedded newform 9025.2.a.ck.1.4

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			1.4
Root			$-1.85244$ 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)

$f(q)$	$=$	$q-1.85244 q^{2} +1.45440 q^{3} +1.43152 q^{4} -2.69418 q^{6} -0.477604 q^{7} +1.05306 q^{8} -0.884731 q^{9} +O(q^{10})$	\(q-1.85244 q^{2} +1.45440 q^{3} +1.43152 q^{4} -2.69418 q^{6} -0.477604 q^{7} +1.05306 q^{8} -0.884731 q^{9} +1.95728 q^{11} +2.08200 q^{12} -3.06043 q^{13} +0.884731 q^{14} -4.81379 q^{16} -0.973502 q^{17} +1.63891 q^{18} -0.694625 q^{21} -3.62574 q^{22} +5.59328 q^{23} +1.53157 q^{24} +5.66926 q^{26} -5.64994 q^{27} -0.683702 q^{28} +10.1756 q^{29} -5.60367 q^{31} +6.81111 q^{32} +2.84666 q^{33} +1.80335 q^{34} -1.26651 q^{36} +5.65754 q^{37} -4.45108 q^{39} -10.4679 q^{41} +1.28675 q^{42} -6.20983 q^{43} +2.80189 q^{44} -10.3612 q^{46} -2.47845 q^{47} -7.00115 q^{48} -6.77189 q^{49} -1.41586 q^{51} -4.38109 q^{52} +10.7711 q^{53} +10.4662 q^{54} -0.502948 q^{56} -18.8496 q^{58} -7.32031 q^{59} +8.76766 q^{61} +10.3805 q^{62} +0.422551 q^{63} -2.98958 q^{64} -5.27326 q^{66} -8.82042 q^{67} -1.39359 q^{68} +8.13485 q^{69} +13.1846 q^{71} -0.931679 q^{72} -4.97121 q^{73} -10.4802 q^{74} -0.934804 q^{77} +8.24535 q^{78} +0.707984 q^{79} -5.56306 q^{81} +19.3911 q^{82} +11.8783 q^{83} -0.994373 q^{84} +11.5033 q^{86} +14.7993 q^{87} +2.06114 q^{88} -2.14855 q^{89} +1.46167 q^{91} +8.00692 q^{92} -8.14996 q^{93} +4.59117 q^{94} +9.90605 q^{96} -5.89541 q^{97} +12.5445 q^{98} -1.73167 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.85244	−1.30987	−0.654936	−	0.755685i	$-0.727304\pi$
$2$	−1.85244	−1.30987	−0.654936	+	0.755685i	$0.727304\pi$
$3$	1.45440	0.839696	0.419848	−	0.907594i	$-0.362083\pi$
$3$	1.45440	0.839696	0.419848	+	0.907594i	$0.362083\pi$
$4$	1.43152	0.715762
$5$	0	0
$5$	0	0
$6$	−2.69418	−1.09989
$7$	−0.477604	−0.180517	−0.0902586	−	0.995918i	$-0.528769\pi$
$7$	−0.477604	−0.180517	−0.0902586	+	0.995918i	$0.528769\pi$
$8$	1.05306	0.372315
$9$	−0.884731	−0.294910
$10$	0	0
$11$	1.95728	0.590142	0.295071	−	0.955475i	$-0.404657\pi$
$11$	1.95728	0.590142	0.295071	+	0.955475i	$0.404657\pi$
$12$	2.08200	0.601023
$13$	−3.06043	−0.848812	−0.424406	−	0.905472i	$-0.639517\pi$
$13$	−3.06043	−0.848812	−0.424406	+	0.905472i	$0.639517\pi$
$14$	0.884731	0.236454
$15$	0	0
$16$	−4.81379	−1.20345
$17$	−0.973502	−0.236109	−0.118054	−	0.993007i	$-0.537666\pi$
$17$	−0.973502	−0.236109	−0.118054	+	0.993007i	$0.537666\pi$
$18$	1.63891	0.386295
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.694625	−0.151580
$22$	−3.62574	−0.773010
$23$	5.59328	1.16628	0.583140	−	0.812372i	$-0.301824\pi$
$23$	5.59328	1.16628	0.583140	+	0.812372i	$0.301824\pi$
$24$	1.53157	0.312631
$25$	0	0
$26$	5.66926	1.11183
$27$	−5.64994	−1.08733
$28$	−0.683702	−0.129207
$29$	10.1756	1.88955	0.944777	−	0.327714i	$-0.106278\pi$
$29$	10.1756	1.88955	0.944777	+	0.327714i	$0.106278\pi$
$30$	0	0
$31$	−5.60367	−1.00645	−0.503225	−	0.864156i	$-0.667853\pi$
$31$	−5.60367	−1.00645	−0.503225	+	0.864156i	$0.667853\pi$
$32$	6.81111	1.20405
$33$	2.84666	0.495540
$34$	1.80335	0.309272
$35$	0	0
$36$	−1.26651	−0.211086
$37$	5.65754	0.930094	0.465047	−	0.885286i	$-0.346037\pi$
$37$	5.65754	0.930094	0.465047	+	0.885286i	$0.346037\pi$
$38$	0	0
$39$	−4.45108	−0.712744
$40$	0	0
$41$	−10.4679	−1.63481	−0.817404	−	0.576065i	$-0.804587\pi$
$41$	−10.4679	−1.63481	−0.817404	+	0.576065i	$0.804587\pi$
$42$	1.28675	0.198550
$43$	−6.20983	−0.946990	−0.473495	−	0.880796i	$-0.657008\pi$
$43$	−6.20983	−0.946990	−0.473495	+	0.880796i	$0.657008\pi$
$44$	2.80189	0.422401
$45$	0	0
$46$	−10.3612	−1.52768
$47$	−2.47845	−0.361519	−0.180759	−	0.983527i	$-0.557856\pi$
$47$	−2.47845	−0.361519	−0.180759	+	0.983527i	$0.557856\pi$
$48$	−7.00115	−1.01053
$49$	−6.77189	−0.967414
$50$	0	0
$51$	−1.41586	−0.198260
$52$	−4.38109	−0.607547
$53$	10.7711	1.47953	0.739764	−	0.672866i	$-0.234937\pi$
$53$	10.7711	1.47953	0.739764	+	0.672866i	$0.234937\pi$
$54$	10.4662	1.42426
$55$	0	0
$56$	−0.502948	−0.0672092
$57$	0	0
$58$	−18.8496	−2.47507
$59$	−7.32031	−0.953024	−0.476512	−	0.879168i	$-0.658099\pi$
$59$	−7.32031	−0.953024	−0.476512	+	0.879168i	$0.658099\pi$
$60$	0	0
$61$	8.76766	1.12258	0.561292	−	0.827618i	$-0.310305\pi$
$61$	8.76766	1.12258	0.561292	+	0.827618i	$0.310305\pi$
$62$	10.3805	1.31832
$63$	0.422551	0.0532364
$64$	−2.98958	−0.373698
$65$	0	0
$66$	−5.27326	−0.649093
$67$	−8.82042	−1.07759	−0.538793	−	0.842438i	$-0.681120\pi$
$67$	−8.82042	−1.07759	−0.538793	+	0.842438i	$0.681120\pi$
$68$	−1.39359	−0.168998
$69$	8.13485	0.979321
$70$	0	0
$71$	13.1846	1.56472	0.782362	−	0.622825i	$-0.214015\pi$
$71$	13.1846	1.56472	0.782362	+	0.622825i	$0.214015\pi$
$72$	−0.931679	−0.109799
$73$	−4.97121	−0.581836	−0.290918	−	0.956748i	$-0.593961\pi$
$73$	−4.97121	−0.581836	−0.290918	+	0.956748i	$0.593961\pi$
$74$	−10.4802	−1.21830
$75$	0	0
$76$	0	0
$77$	−0.934804	−0.106531
$78$	8.24535	0.933603
$79$	0.707984	0.0796544	0.0398272	−	0.999207i	$-0.487319\pi$
$79$	0.707984	0.0796544	0.0398272	+	0.999207i	$0.487319\pi$
$80$	0	0
$81$	−5.56306	−0.618118
$82$	19.3911	2.14139
$83$	11.8783	1.30382	0.651909	−	0.758297i	$-0.273968\pi$
$83$	11.8783	1.30382	0.651909	+	0.758297i	$0.273968\pi$
$84$	−0.994373	−0.108495
$85$	0	0
$86$	11.5033	1.24043
$87$	14.7993	1.58665
$88$	2.06114	0.219718
$89$	−2.14855	−0.227746	−0.113873	−	0.993495i	$-0.536326\pi$
$89$	−2.14855	−0.227746	−0.113873	+	0.993495i	$0.536326\pi$
$90$	0	0
$91$	1.46167	0.153225
$92$	8.00692	0.834779
$93$	−8.14996	−0.845112
$94$	4.59117	0.473543
$95$	0	0
$96$	9.90605	1.01103
$97$	−5.89541	−0.598588	−0.299294	−	0.954161i	$-0.596751\pi$
$97$	−5.89541	−0.598588	−0.299294	+	0.954161i	$0.596751\pi$
$98$	12.5445	1.26719
$99$	−1.73167	−0.174039
$100$	0	0
$101$	0.258211	0.0256930	0.0128465	−	0.999917i	$-0.495911\pi$
$101$	0.258211	0.0256930	0.0128465	+	0.999917i	$0.495911\pi$
$102$	2.62279	0.259695
$103$	−14.3714	−1.41605	−0.708026	−	0.706186i	$-0.750414\pi$
$103$	−14.3714	−1.41605	−0.708026	+	0.706186i	$0.750414\pi$
$104$	−3.22283	−0.316025
$105$	0	0
$106$	−19.9529	−1.93799
$107$	5.73115	0.554051	0.277026	−	0.960863i	$-0.410651\pi$
$107$	5.73115	0.554051	0.277026	+	0.960863i	$0.410651\pi$
$108$	−8.08803	−0.778271
$109$	5.77812	0.553444	0.276722	−	0.960950i	$-0.410752\pi$
$109$	5.77812	0.553444	0.276722	+	0.960950i	$0.410752\pi$
$110$	0	0
$111$	8.22830	0.780996
$112$	2.29908	0.217243
$113$	13.9762	1.31477	0.657386	−	0.753554i	$-0.271662\pi$
$113$	13.9762	1.31477	0.657386	+	0.753554i	$0.271662\pi$
$114$	0	0
$115$	0	0
$116$	14.5666	1.35247
$117$	2.70766	0.250323
$118$	13.5604	1.24834
$119$	0.464948	0.0426217
$120$	0	0
$121$	−7.16906	−0.651733
$122$	−16.2415	−1.47044
$123$	−15.2244	−1.37274
$124$	−8.02180	−0.720379
$125$	0	0
$126$	−0.782749	−0.0697328
$127$	9.23319	0.819313	0.409657	−	0.912240i	$-0.365649\pi$
$127$	9.23319	0.819313	0.409657	+	0.912240i	$0.365649\pi$
$128$	−8.08421	−0.714550
$129$	−9.03155	−0.795184
$130$	0	0
$131$	−13.9316	−1.21721	−0.608603	−	0.793475i	$-0.708270\pi$
$131$	−13.9316	−1.21721	−0.608603	+	0.793475i	$0.708270\pi$
$132$	4.07506	0.354689
$133$	0	0
$134$	16.3393	1.41150
$135$	0	0
$136$	−1.02516	−0.0879068
$137$	−3.84336	−0.328360	−0.164180	−	0.986430i	$-0.552498\pi$
$137$	−3.84336	−0.328360	−0.164180	+	0.986430i	$0.552498\pi$
$138$	−15.0693	−1.28278
$139$	−18.8730	−1.60079	−0.800395	−	0.599473i	$-0.795377\pi$
$139$	−18.8730	−1.60079	−0.800395	+	0.599473i	$0.795377\pi$
$140$	0	0
$141$	−3.60464	−0.303566
$142$	−24.4236	−2.04959
$143$	−5.99012	−0.500919
$144$	4.25891	0.354909
$145$	0	0
$146$	9.20885	0.762130
$147$	−9.84902	−0.812333
$148$	8.09891	0.665726
$149$	−23.9268	−1.96016	−0.980082	−	0.198594i	$-0.936363\pi$
$149$	−23.9268	−1.96016	−0.980082	+	0.198594i	$0.936363\pi$
$150$	0	0
$151$	5.90301	0.480380	0.240190	−	0.970726i	$-0.422790\pi$
$151$	5.90301	0.480380	0.240190	+	0.970726i	$0.422790\pi$
$152$	0	0
$153$	0.861288	0.0696310
$154$	1.73167	0.139542
$155$	0	0
$156$	−6.37184	−0.510155
$157$	−10.5707	−0.843631	−0.421815	−	0.906682i	$-0.638607\pi$
$157$	−10.5707	−0.843631	−0.421815	+	0.906682i	$0.638607\pi$
$158$	−1.31150	−0.104337
$159$	15.6655	1.24235
$160$	0	0
$161$	−2.67137	−0.210534
$162$	10.3052	0.809654
$163$	18.6090	1.45757	0.728785	−	0.684743i	$-0.240085\pi$
$163$	18.6090	1.45757	0.728785	+	0.684743i	$0.240085\pi$
$164$	−14.9850	−1.17013
$165$	0	0
$166$	−22.0039	−1.70783
$167$	−2.77731	−0.214914	−0.107457	−	0.994210i	$-0.534271\pi$
$167$	−2.77731	−0.214914	−0.107457	+	0.994210i	$0.534271\pi$
$168$	−0.731485	−0.0564353
$169$	−3.63375	−0.279519
$170$	0	0
$171$	0	0
$172$	−8.88952	−0.677820
$173$	8.92075	0.678232	0.339116	−	0.940745i	$-0.389872\pi$
$173$	8.92075	0.678232	0.339116	+	0.940745i	$0.389872\pi$
$174$	−27.4148	−2.07831
$175$	0	0
$176$	−9.42192	−0.710204
$177$	−10.6466	−0.800250
$178$	3.98006	0.298318
$179$	−15.2646	−1.14093	−0.570467	−	0.821321i	$-0.693238\pi$
$179$	−15.2646	−1.14093	−0.570467	+	0.821321i	$0.693238\pi$
$180$	0	0
$181$	8.54101	0.634848	0.317424	−	0.948284i	$-0.397182\pi$
$181$	8.54101	0.634848	0.317424	+	0.948284i	$0.397182\pi$
$182$	−2.70766	−0.200705
$183$	12.7516	0.942629
$184$	5.89009	0.434223
$185$	0	0
$186$	15.0973	1.10699
$187$	−1.90542	−0.139338
$188$	−3.54796	−0.258761
$189$	2.69843	0.196282
$190$	0	0
$191$	−16.2967	−1.17919	−0.589594	−	0.807700i	$-0.700712\pi$
$191$	−16.2967	−1.17919	−0.589594	+	0.807700i	$0.700712\pi$
$192$	−4.34804	−0.313792
$193$	16.4692	1.18548	0.592739	−	0.805394i	$-0.298046\pi$
$193$	16.4692	1.18548	0.592739	+	0.805394i	$0.298046\pi$
$194$	10.9209	0.784074
$195$	0	0
$196$	−9.69414	−0.692438
$197$	10.0229	0.714102	0.357051	−	0.934085i	$-0.383782\pi$
$197$	10.0229	0.714102	0.357051	+	0.934085i	$0.383782\pi$
$198$	3.20780	0.227969
$199$	−12.6643	−0.897748	−0.448874	−	0.893595i	$-0.648175\pi$
$199$	−12.6643	−0.897748	−0.448874	+	0.893595i	$0.648175\pi$
$200$	0	0
$201$	−12.8284	−0.904845
$202$	−0.478321	−0.0336545
$203$	−4.85988	−0.341097
$204$	−2.02684	−0.141907
$205$	0	0
$206$	26.6221	1.85485
$207$	−4.94855	−0.343948
$208$	14.7323	1.02150
$209$	0	0
$210$	0	0
$211$	−5.73609	−0.394888	−0.197444	−	0.980314i	$-0.563264\pi$
$211$	−5.73609	−0.394888	−0.197444	+	0.980314i	$0.563264\pi$
$212$	15.4191	1.05899
$213$	19.1756	1.31389
$214$	−10.6166	−0.725736
$215$	0	0
$216$	−5.94975	−0.404829
$217$	2.67634	0.181682
$218$	−10.7036	−0.724940
$219$	−7.23010	−0.488565
$220$	0	0
$221$	2.97934	0.200412
$222$	−15.2424	−1.02300
$223$	12.8478	0.860353	0.430176	−	0.902745i	$-0.358451\pi$
$223$	12.8478	0.860353	0.430176	+	0.902745i	$0.358451\pi$
$224$	−3.25301	−0.217351
$225$	0	0
$226$	−25.8901	−1.72218
$227$	15.5808	1.03413	0.517067	−	0.855945i	$-0.327024\pi$
$227$	15.5808	1.03413	0.517067	+	0.855945i	$0.327024\pi$
$228$	0	0
$229$	−3.77930	−0.249743	−0.124871	−	0.992173i	$-0.539852\pi$
$229$	−3.77930	−0.249743	−0.124871	+	0.992173i	$0.539852\pi$
$230$	0	0
$231$	−1.35958	−0.0894535
$232$	10.7155	0.703508
$233$	11.0745	0.725512	0.362756	−	0.931884i	$-0.381836\pi$
$233$	11.0745	0.725512	0.362756	+	0.931884i	$0.381836\pi$
$234$	−5.01577	−0.327891
$235$	0	0
$236$	−10.4792	−0.682139
$237$	1.02969	0.0668855
$238$	−0.861288	−0.0558290
$239$	−10.6496	−0.688865	−0.344433	−	0.938811i	$-0.611929\pi$
$239$	−10.6496	−0.688865	−0.344433	+	0.938811i	$0.611929\pi$
$240$	0	0
$241$	−5.01271	−0.322897	−0.161448	−	0.986881i	$-0.551617\pi$
$241$	−5.01271	−0.322897	−0.161448	+	0.986881i	$0.551617\pi$
$242$	13.2802	0.853686
$243$	8.85892	0.568300
$244$	12.5511	0.803503
$245$	0	0
$246$	28.2023	1.79811
$247$	0	0
$248$	−5.90103	−0.374716
$249$	17.2758	1.09481
$250$	0	0
$251$	−26.1721	−1.65197	−0.825986	−	0.563691i	$-0.809381\pi$
$251$	−26.1721	−1.65197	−0.825986	+	0.563691i	$0.809381\pi$
$252$	0.604892	0.0381046
$253$	10.9476	0.688271
$254$	−17.1039	−1.07319
$255$	0	0
$256$	20.9546	1.30967
$257$	−22.4024	−1.39742	−0.698712	−	0.715403i	$-0.746243\pi$
$257$	−22.4024	−1.39742	−0.698712	+	0.715403i	$0.746243\pi$
$258$	16.7304	1.04159
$259$	−2.70206	−0.167898
$260$	0	0
$261$	−9.00263	−0.557249
$262$	25.8074	1.59438
$263$	−25.9338	−1.59915	−0.799573	−	0.600569i	$-0.794941\pi$
$263$	−25.9338	−1.59915	−0.799573	+	0.600569i	$0.794941\pi$
$264$	2.99772	0.184497
$265$	0	0
$266$	0	0
$267$	−3.12485	−0.191238
$268$	−12.6267	−0.771296
$269$	9.24978	0.563969	0.281985	−	0.959419i	$-0.409007\pi$
$269$	9.24978	0.563969	0.281985	+	0.959419i	$0.409007\pi$
$270$	0	0
$271$	−22.4688	−1.36488	−0.682441	−	0.730941i	$-0.739082\pi$
$271$	−22.4688	−1.36488	−0.682441	+	0.730941i	$0.739082\pi$
$272$	4.68623	0.284145
$273$	2.12585	0.128663
$274$	7.11958	0.430110
$275$	0	0
$276$	11.6452	0.700961
$277$	−9.16598	−0.550730	−0.275365	−	0.961340i	$-0.588799\pi$
$277$	−9.16598	−0.550730	−0.275365	+	0.961340i	$0.588799\pi$
$278$	34.9611	2.09683
$279$	4.95774	0.296812
$280$	0	0
$281$	19.3577	1.15479	0.577393	−	0.816467i	$-0.304070\pi$
$281$	19.3577	1.15479	0.577393	+	0.816467i	$0.304070\pi$
$282$	6.67738	0.397632
$283$	−8.72514	−0.518656	−0.259328	−	0.965789i	$-0.583501\pi$
$283$	−8.72514	−0.518656	−0.259328	+	0.965789i	$0.583501\pi$
$284$	18.8741	1.11997
$285$	0	0
$286$	11.0963	0.656140
$287$	4.99950	0.295111
$288$	−6.02600	−0.355085
$289$	−16.0523	−0.944253
$290$	0	0
$291$	−8.57427	−0.502632
$292$	−7.11641	−0.416456
$293$	−16.7749	−0.980002	−0.490001	−	0.871722i	$-0.663004\pi$
$293$	−16.7749	−0.980002	−0.490001	+	0.871722i	$0.663004\pi$
$294$	18.2447	1.06405
$295$	0	0
$296$	5.95776	0.346287
$297$	−11.0585	−0.641680
$298$	44.3230	2.56756
$299$	−17.1179	−0.989952
$300$	0	0
$301$	2.96584	0.170948
$302$	−10.9350	−0.629236
$303$	0.375542	0.0215743
$304$	0	0
$305$	0	0
$306$	−1.59548	−0.0912076
$307$	−6.40893	−0.365777	−0.182889	−	0.983134i	$-0.558545\pi$
$307$	−6.40893	−0.365777	−0.182889	+	0.983134i	$0.558545\pi$
$308$	−1.33819	−0.0762507
$309$	−20.9017	−1.18905
$310$	0	0
$311$	10.6248	0.602477	0.301238	−	0.953549i	$-0.402600\pi$
$311$	10.6248	0.602477	0.301238	+	0.953549i	$0.402600\pi$
$312$	−4.68728	−0.265365
$313$	4.75656	0.268857	0.134428	−	0.990923i	$-0.457080\pi$
$313$	4.75656	0.268857	0.134428	+	0.990923i	$0.457080\pi$
$314$	19.5815	1.10505
$315$	0	0
$316$	1.01350	0.0570136
$317$	−0.0612218	−0.00343856	−0.00171928	−	0.999999i	$-0.500547\pi$
$317$	−0.0612218	−0.00343856	−0.00171928	+	0.999999i	$0.500547\pi$
$318$	−29.0194	−1.62732
$319$	19.9164	1.11510
$320$	0	0
$321$	8.33536	0.465235
$322$	4.94855	0.275772
$323$	0	0
$324$	−7.96366	−0.442425
$325$	0	0
$326$	−34.4720	−1.90923
$327$	8.40368	0.464724
$328$	−11.0234	−0.608663
$329$	1.18372	0.0652603
$330$	0	0
$331$	−17.6067	−0.967752	−0.483876	−	0.875137i	$-0.660771\pi$
$331$	−17.6067	−0.967752	−0.483876	+	0.875137i	$0.660771\pi$
$332$	17.0041	0.933224
$333$	−5.00540	−0.274294
$334$	5.14478	0.281510
$335$	0	0
$336$	3.34378	0.182418
$337$	−6.90874	−0.376343	−0.188171	−	0.982136i	$-0.560256\pi$
$337$	−6.90874	−0.376343	−0.188171	+	0.982136i	$0.560256\pi$
$338$	6.73129	0.366134
$339$	20.3270	1.10401
$340$	0	0
$341$	−10.9680	−0.593948
$342$	0	0
$343$	6.57751	0.355152
$344$	−6.53935	−0.352578
$345$	0	0
$346$	−16.5251	−0.888396
$347$	−7.32576	−0.393267	−0.196634	−	0.980477i	$-0.563001\pi$
$347$	−7.32576	−0.393267	−0.196634	+	0.980477i	$0.563001\pi$
$348$	21.1856	1.13567
$349$	−2.16127	−0.115690	−0.0578452	−	0.998326i	$-0.518423\pi$
$349$	−2.16127	−0.115690	−0.0578452	+	0.998326i	$0.518423\pi$
$350$	0	0
$351$	17.2913	0.922939
$352$	13.3312	0.710557
$353$	16.6022	0.883644	0.441822	−	0.897103i	$-0.354332\pi$
$353$	16.6022	0.883644	0.441822	+	0.897103i	$0.354332\pi$
$354$	19.7222	1.04822
$355$	0	0
$356$	−3.07571	−0.163012
$357$	0.676219	0.0357893
$358$	28.2768	1.49448
$359$	17.0830	0.901608	0.450804	−	0.892623i	$-0.351137\pi$
$359$	17.0830	0.901608	0.450804	+	0.892623i	$0.351137\pi$
$360$	0	0
$361$	0	0
$362$	−15.8217	−0.831569
$363$	−10.4267	−0.547257
$364$	2.09242	0.109673
$365$	0	0
$366$	−23.6216	−1.23472
$367$	−19.5631	−1.02119	−0.510594	−	0.859822i	$-0.670574\pi$
$367$	−19.5631	−1.02119	−0.510594	+	0.859822i	$0.670574\pi$
$368$	−26.9249	−1.40356
$369$	9.26125	0.482122
$370$	0	0
$371$	−5.14433	−0.267080
$372$	−11.6669	−0.604899
$373$	−6.59700	−0.341580	−0.170790	−	0.985307i	$-0.554632\pi$
$373$	−6.59700	−0.341580	−0.170790	+	0.985307i	$0.554632\pi$
$374$	3.52966	0.182515
$375$	0	0
$376$	−2.60997	−0.134599
$377$	−31.1416	−1.60387
$378$	−4.99868	−0.257104
$379$	−21.6574	−1.11246	−0.556232	−	0.831027i	$-0.687753\pi$
$379$	−21.6574	−1.11246	−0.556232	+	0.831027i	$0.687753\pi$
$380$	0	0
$381$	13.4287	0.687974
$382$	30.1886	1.54458
$383$	−24.3214	−1.24277	−0.621383	−	0.783507i	$-0.713429\pi$
$383$	−24.3214	−1.24277	−0.621383	+	0.783507i	$0.713429\pi$
$384$	−11.7576	−0.600005
$385$	0	0
$386$	−30.5082	−1.55282
$387$	5.49403	0.279277
$388$	−8.43943	−0.428447
$389$	15.8227	0.802241	0.401120	−	0.916025i	$-0.368621\pi$
$389$	15.8227	0.802241	0.401120	+	0.916025i	$0.368621\pi$
$390$	0	0
$391$	−5.44507	−0.275369
$392$	−7.13124	−0.360182
$393$	−20.2620	−1.02208
$394$	−18.5668	−0.935381
$395$	0	0
$396$	−2.47892	−0.124571
$397$	35.4753	1.78046	0.890228	−	0.455515i	$-0.150545\pi$
$397$	35.4753	1.78046	0.890228	+	0.455515i	$0.150545\pi$
$398$	23.4598	1.17593
$399$	0	0
$400$	0	0
$401$	11.2107	0.559837	0.279919	−	0.960024i	$-0.409693\pi$
$401$	11.2107	0.559837	0.279919	+	0.960024i	$0.409693\pi$
$402$	23.7638	1.18523
$403$	17.1497	0.854286
$404$	0.369636	0.0183901
$405$	0	0
$406$	9.00263	0.446793
$407$	11.0734	0.548887
$408$	−1.49099	−0.0738150
$409$	−1.91450	−0.0946660	−0.0473330	−	0.998879i	$-0.515072\pi$
$409$	−1.91450	−0.0946660	−0.0473330	+	0.998879i	$0.515072\pi$
$410$	0	0
$411$	−5.58977	−0.275723
$412$	−20.5730	−1.01356
$413$	3.49621	0.172037
$414$	9.16688	0.450528
$415$	0	0
$416$	−20.8449	−1.02201
$417$	−27.4489	−1.34418
$418$	0	0
$419$	−22.1856	−1.08384	−0.541918	−	0.840431i	$-0.682302\pi$
$419$	−22.1856	−1.08384	−0.541918	+	0.840431i	$0.682302\pi$
$420$	0	0
$421$	−13.3793	−0.652069	−0.326034	−	0.945358i	$-0.605713\pi$
$421$	−13.3793	−0.652069	−0.326034	+	0.945358i	$0.605713\pi$
$422$	10.6257	0.517253
$423$	2.19276	0.106616
$424$	11.3427	0.550850
$425$	0	0
$426$	−35.5216	−1.72103
$427$	−4.18747	−0.202646
$428$	8.20428	0.396569
$429$	−8.71201	−0.420620
$430$	0	0
$431$	37.1687	1.79035	0.895177	−	0.445711i	$-0.147049\pi$
$431$	37.1687	1.79035	0.895177	+	0.445711i	$0.147049\pi$
$432$	27.1976	1.30855
$433$	−8.32630	−0.400137	−0.200068	−	0.979782i	$-0.564116\pi$
$433$	−8.32630	−0.400137	−0.200068	+	0.979782i	$0.564116\pi$
$434$	−4.95774	−0.237979
$435$	0	0
$436$	8.27152	0.396134
$437$	0	0
$438$	13.3933	0.639957
$439$	17.0746	0.814928	0.407464	−	0.913221i	$-0.366413\pi$
$439$	17.0746	0.814928	0.407464	+	0.913221i	$0.366413\pi$
$440$	0	0
$441$	5.99131	0.285300
$442$	−5.51904	−0.262514
$443$	7.18135	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$443$	7.18135	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$444$	11.7790	0.559008
$445$	0	0
$446$	−23.7998	−1.12695
$447$	−34.7991	−1.64594
$448$	1.42784	0.0674589
$449$	−23.1024	−1.09027	−0.545134	−	0.838349i	$-0.683521\pi$
$449$	−23.1024	−1.09027	−0.545134	+	0.838349i	$0.683521\pi$
$450$	0	0
$451$	−20.4886	−0.964768
$452$	20.0073	0.941065
$453$	8.58532	0.403373
$454$	−28.8625	−1.35458
$455$	0	0
$456$	0	0
$457$	−13.7430	−0.642872	−0.321436	−	0.946931i	$-0.604166\pi$
$457$	−13.7430	−0.642872	−0.321436	+	0.946931i	$0.604166\pi$
$458$	7.00091	0.327131
$459$	5.50023	0.256729
$460$	0	0
$461$	−26.0522	−1.21337	−0.606686	−	0.794942i	$-0.707501\pi$
$461$	−26.0522	−1.21337	−0.606686	+	0.794942i	$0.707501\pi$
$462$	2.51853	0.117173
$463$	4.24698	0.197374	0.0986869	−	0.995119i	$-0.468536\pi$
$463$	4.24698	0.197374	0.0986869	+	0.995119i	$0.468536\pi$
$464$	−48.9830	−2.27398
$465$	0	0
$466$	−20.5147	−0.950327
$467$	27.7571	1.28444	0.642222	−	0.766519i	$-0.278013\pi$
$467$	27.7571	1.28444	0.642222	+	0.766519i	$0.278013\pi$
$468$	3.87608	0.179172
$469$	4.21267	0.194523
$470$	0	0
$471$	−15.3739	−0.708394
$472$	−7.70877	−0.354825
$473$	−12.1544	−0.558858
$474$	−1.90744	−0.0876114
$475$	0	0
$476$	0.665585	0.0305070
$477$	−9.52956	−0.436328
$478$	19.7277	0.902325
$479$	−7.11883	−0.325268	−0.162634	−	0.986686i	$-0.551999\pi$
$479$	−7.11883	−0.325268	−0.162634	+	0.986686i	$0.551999\pi$
$480$	0	0
$481$	−17.3145	−0.789474
$482$	9.28573	0.422953
$483$	−3.88523	−0.176784
$484$	−10.2627	−0.466486
$485$	0	0
$486$	−16.4106	−0.744400
$487$	3.54621	0.160694	0.0803470	−	0.996767i	$-0.474397\pi$
$487$	3.54621	0.160694	0.0803470	+	0.996767i	$0.474397\pi$
$488$	9.23291	0.417954
$489$	27.0649	1.22392
$490$	0	0
$491$	−32.1308	−1.45004	−0.725020	−	0.688727i	$-0.758170\pi$
$491$	−32.1308	−1.45004	−0.725020	+	0.688727i	$0.758170\pi$
$492$	−21.7942	−0.982557
$493$	−9.90593	−0.446141
$494$	0	0
$495$	0	0
$496$	26.9749	1.21121
$497$	−6.29701	−0.282459
$498$	−32.0024	−1.43406
$499$	−11.9147	−0.533376	−0.266688	−	0.963783i	$-0.585929\pi$
$499$	−11.9147	−0.533376	−0.266688	+	0.963783i	$0.585929\pi$
$500$	0	0
$501$	−4.03930	−0.180463
$502$	48.4823	2.16387
$503$	−0.239744	−0.0106896	−0.00534482	−	0.999986i	$-0.501701\pi$
$503$	−0.239744	−0.0106896	−0.00534482	+	0.999986i	$0.501701\pi$
$504$	0.444973	0.0198207
$505$	0	0
$506$	−20.2798	−0.901546
$507$	−5.28491	−0.234711
$508$	13.2175	0.586434
$509$	29.3663	1.30164	0.650819	−	0.759233i	$-0.274426\pi$
$509$	29.3663	1.30164	0.650819	+	0.759233i	$0.274426\pi$
$510$	0	0
$511$	2.37427	0.105031
$512$	−22.6488	−1.00094
$513$	0	0
$514$	41.4991	1.83045
$515$	0	0
$516$	−12.9289	−0.569163
$517$	−4.85101	−0.213347
$518$	5.00540	0.219925
$519$	12.9743	0.569509
$520$	0	0
$521$	−12.1069	−0.530414	−0.265207	−	0.964191i	$-0.585440\pi$
$521$	−12.1069	−0.530414	−0.265207	+	0.964191i	$0.585440\pi$
$522$	16.6768	0.729924
$523$	6.43779	0.281505	0.140752	−	0.990045i	$-0.455048\pi$
$523$	6.43779	0.281505	0.140752	+	0.990045i	$0.455048\pi$
$524$	−19.9434	−0.871231
$525$	0	0
$526$	48.0407	2.09467
$527$	5.45519	0.237632
$528$	−13.7032	−0.596356
$529$	8.28480	0.360209
$530$	0	0
$531$	6.47651	0.281057
$532$	0	0
$533$	32.0362	1.38764
$534$	5.78859	0.250497
$535$	0	0
$536$	−9.28848	−0.401201
$537$	−22.2008	−0.958037
$538$	−17.1346	−0.738727
$539$	−13.2545	−0.570911
$540$	0	0
$541$	−15.1179	−0.649969	−0.324985	−	0.945719i	$-0.605359\pi$
$541$	−15.1179	−0.649969	−0.324985	+	0.945719i	$0.605359\pi$
$542$	41.6220	1.78782
$543$	12.4220	0.533079
$544$	−6.63063	−0.284286
$545$	0	0
$546$	−3.93801	−0.168531
$547$	−36.1694	−1.54649	−0.773246	−	0.634106i	$-0.781368\pi$
$547$	−36.1694	−1.54649	−0.773246	+	0.634106i	$0.781368\pi$
$548$	−5.50186	−0.235028
$549$	−7.75702	−0.331061
$550$	0	0
$551$	0	0
$552$	8.56652	0.364615
$553$	−0.338136	−0.0143790
$554$	16.9794	0.721386
$555$	0	0
$556$	−27.0172	−1.14579
$557$	−37.0585	−1.57022	−0.785109	−	0.619358i	$-0.787393\pi$
$557$	−37.0585	−1.57022	−0.785109	+	0.619358i	$0.787393\pi$
$558$	−9.18391	−0.388786
$559$	19.0048	0.803816
$560$	0	0
$561$	−2.77123	−0.117001
$562$	−35.8590	−1.51262
$563$	12.9283	0.544863	0.272432	−	0.962175i	$-0.412172\pi$
$563$	12.9283	0.544863	0.272432	+	0.962175i	$0.412172\pi$
$564$	−5.16014	−0.217281
$565$	0	0
$566$	16.1628	0.679372
$567$	2.65694	0.111581
$568$	13.8842	0.582569
$569$	−18.5114	−0.776038	−0.388019	−	0.921651i	$-0.626841\pi$
$569$	−18.5114	−0.776038	−0.388019	+	0.921651i	$0.626841\pi$
$570$	0	0
$571$	16.2207	0.678816	0.339408	−	0.940639i	$-0.389773\pi$
$571$	16.2207	0.678816	0.339408	+	0.940639i	$0.389773\pi$
$572$	−8.57501	−0.358539
$573$	−23.7019	−0.990160
$574$	−9.26125	−0.386557
$575$	0	0
$576$	2.64498	0.110207
$577$	−17.7980	−0.740939	−0.370470	−	0.928845i	$-0.620803\pi$
$577$	−17.7980	−0.740939	−0.370470	+	0.928845i	$0.620803\pi$
$578$	29.7359	1.23685
$579$	23.9527	0.995442
$580$	0	0
$581$	−5.67314	−0.235362
$582$	15.8833	0.658384
$583$	21.0821	0.873132
$584$	−5.23500	−0.216626
$585$	0	0
$586$	31.0745	1.28368
$587$	−35.8593	−1.48007	−0.740036	−	0.672567i	$-0.765191\pi$
$587$	−35.8593	−1.48007	−0.740036	+	0.672567i	$0.765191\pi$
$588$	−14.0991	−0.581438
$589$	0	0
$590$	0	0
$591$	14.5773	0.599628
$592$	−27.2342	−1.11932
$593$	−18.2848	−0.750868	−0.375434	−	0.926849i	$-0.622506\pi$
$593$	−18.2848	−0.750868	−0.375434	+	0.926849i	$0.622506\pi$
$594$	20.4852	0.840518
$595$	0	0
$596$	−34.2519	−1.40301
$597$	−18.4189	−0.753836
$598$	31.7098	1.29671
$599$	7.85141	0.320800	0.160400	−	0.987052i	$-0.448722\pi$
$599$	7.85141	0.320800	0.160400	+	0.987052i	$0.448722\pi$
$600$	0	0
$601$	−17.3527	−0.707833	−0.353917	−	0.935277i	$-0.615150\pi$
$601$	−17.3527	−0.707833	−0.353917	+	0.935277i	$0.615150\pi$
$602$	−5.49403	−0.223920
$603$	7.80370	0.317791
$604$	8.45030	0.343838
$605$	0	0
$606$	−0.695668	−0.0282596
$607$	40.7364	1.65344	0.826720	−	0.562614i	$-0.190204\pi$
$607$	40.7364	1.65344	0.826720	+	0.562614i	$0.190204\pi$
$608$	0	0
$609$	−7.06820	−0.286418
$610$	0	0
$611$	7.58512	0.306861
$612$	1.23295	0.0498392
$613$	27.0535	1.09268	0.546340	−	0.837564i	$-0.316021\pi$
$613$	27.0535	1.09268	0.546340	+	0.837564i	$0.316021\pi$
$614$	11.8721	0.479121
$615$	0	0
$616$	−0.984409	−0.0396630
$617$	19.8932	0.800871	0.400436	−	0.916325i	$-0.368859\pi$
$617$	19.8932	0.800871	0.400436	+	0.916325i	$0.368859\pi$
$618$	38.7190	1.55751
$619$	16.8447	0.677044	0.338522	−	0.940958i	$-0.390073\pi$
$619$	16.8447	0.677044	0.338522	+	0.940958i	$0.390073\pi$
$620$	0	0
$621$	−31.6017	−1.26813
$622$	−19.6818	−0.789167
$623$	1.02616	0.0411121
$624$	21.4266	0.857749
$625$	0	0
$626$	−8.81123	−0.352168
$627$	0	0
$628$	−15.1322	−0.603839
$629$	−5.50763	−0.219603
$630$	0	0
$631$	27.2013	1.08287	0.541433	−	0.840744i	$-0.317882\pi$
$631$	27.2013	1.08287	0.541433	+	0.840744i	$0.317882\pi$
$632$	0.745553	0.0296565
$633$	−8.34254	−0.331586
$634$	0.113410	0.00450407
$635$	0	0
$636$	22.4255	0.889231
$637$	20.7249	0.821152
$638$	−36.8939	−1.46064
$639$	−11.6648	−0.461453
$640$	0	0
$641$	17.6090	0.695514	0.347757	−	0.937585i	$-0.386943\pi$
$641$	17.6090	0.695514	0.347757	+	0.937585i	$0.386943\pi$
$642$	−15.4407	−0.609398
$643$	−8.95765	−0.353255	−0.176628	−	0.984278i	$-0.556519\pi$
$643$	−8.95765	−0.353255	−0.176628	+	0.984278i	$0.556519\pi$
$644$	−3.82414	−0.150692
$645$	0	0
$646$	0	0
$647$	49.1850	1.93366	0.966831	−	0.255416i	$-0.0822124\pi$
$647$	49.1850	1.93366	0.966831	+	0.255416i	$0.0822124\pi$
$648$	−5.85826	−0.230134
$649$	−14.3279	−0.562419
$650$	0	0
$651$	3.89245	0.152557
$652$	26.6393	1.04327
$653$	11.7035	0.457993	0.228997	−	0.973427i	$-0.426456\pi$
$653$	11.7035	0.457993	0.228997	+	0.973427i	$0.426456\pi$
$654$	−15.5673	−0.608729
$655$	0	0
$656$	50.3901	1.96740
$657$	4.39818	0.171589
$658$	−2.19276	−0.0854826
$659$	−34.7573	−1.35395	−0.676976	−	0.736005i	$-0.736710\pi$
$659$	−34.7573	−1.35395	−0.676976	+	0.736005i	$0.736710\pi$
$660$	0	0
$661$	17.9064	0.696477	0.348239	−	0.937406i	$-0.386780\pi$
$661$	17.9064	0.696477	0.348239	+	0.937406i	$0.386780\pi$
$662$	32.6153	1.26763
$663$	4.33314	0.168285
$664$	12.5087	0.485431
$665$	0	0
$666$	9.27219	0.359290
$667$	56.9148	2.20375
$668$	−3.97578	−0.153828
$669$	18.6858	0.722435
$670$	0	0
$671$	17.1607	0.662483
$672$	−4.73117	−0.182509
$673$	10.9828	0.423355	0.211677	−	0.977340i	$-0.432107\pi$
$673$	10.9828	0.423355	0.211677	+	0.977340i	$0.432107\pi$
$674$	12.7980	0.492961
$675$	0	0
$676$	−5.20180	−0.200069
$677$	4.77520	0.183526	0.0917630	−	0.995781i	$-0.470750\pi$
$677$	4.77520	0.183526	0.0917630	+	0.995781i	$0.470750\pi$
$678$	−37.6544	−1.44611
$679$	2.81567	0.108056
$680$	0	0
$681$	22.6607	0.868359
$682$	20.3175	0.777995
$683$	−46.0853	−1.76341	−0.881703	−	0.471805i	$-0.843603\pi$
$683$	−46.0853	−1.76341	−0.881703	+	0.471805i	$0.843603\pi$
$684$	0	0
$685$	0	0
$686$	−12.1844	−0.465203
$687$	−5.49660	−0.209708
$688$	29.8928	1.13965
$689$	−32.9643	−1.25584
$690$	0	0
$691$	15.2699	0.580893	0.290447	−	0.956891i	$-0.406196\pi$
$691$	15.2699	0.580893	0.290447	+	0.956891i	$0.406196\pi$
$692$	12.7703	0.485453
$693$	0.827050	0.0314170
$694$	13.5705	0.515130
$695$	0	0
$696$	15.5846	0.590733
$697$	10.1905	0.385993
$698$	4.00362	0.151539
$699$	16.1067	0.609210
$700$	0	0
$701$	−16.6852	−0.630191	−0.315096	−	0.949060i	$-0.602037\pi$
$701$	−16.6852	−0.630191	−0.315096	+	0.949060i	$0.602037\pi$
$702$	−32.0310	−1.20893
$703$	0	0
$704$	−5.85144	−0.220535
$705$	0	0
$706$	−30.7545	−1.15746
$707$	−0.123323	−0.00463803
$708$	−15.2409	−0.572789
$709$	1.95870	0.0735604	0.0367802	−	0.999323i	$-0.488290\pi$
$709$	1.95870	0.0735604	0.0367802	+	0.999323i	$0.488290\pi$
$710$	0	0
$711$	−0.626375	−0.0234909
$712$	−2.26257	−0.0847933
$713$	−31.3429	−1.17380
$714$	−1.25265	−0.0468794
$715$	0	0
$716$	−21.8517	−0.816637
$717$	−15.4887	−0.578438
$718$	−31.6452	−1.18099
$719$	−23.0165	−0.858370	−0.429185	−	0.903217i	$-0.641199\pi$
$719$	−23.0165	−0.858370	−0.429185	+	0.903217i	$0.641199\pi$
$720$	0	0
$721$	6.86382	0.255622
$722$	0	0
$723$	−7.29046	−0.271135
$724$	12.2267	0.454400
$725$	0	0
$726$	19.3147	0.716837
$727$	10.3883	0.385279	0.192640	−	0.981270i	$-0.438295\pi$
$727$	10.3883	0.385279	0.192640	+	0.981270i	$0.438295\pi$
$728$	1.53924	0.0570479
$729$	29.5736	1.09532
$730$	0	0
$731$	6.04528	0.223593
$732$	18.2543	0.674698
$733$	−40.7900	−1.50661	−0.753306	−	0.657670i	$-0.771542\pi$
$733$	−40.7900	−1.50661	−0.753306	+	0.657670i	$0.771542\pi$
$734$	36.2395	1.33762
$735$	0	0
$736$	38.0965	1.40425
$737$	−17.2640	−0.635929
$738$	−17.1559	−0.631517
$739$	−7.92965	−0.291697	−0.145848	−	0.989307i	$-0.546591\pi$
$739$	−7.92965	−0.291697	−0.145848	+	0.989307i	$0.546591\pi$
$740$	0	0
$741$	0	0
$742$	9.52956	0.349841
$743$	−7.27911	−0.267045	−0.133522	−	0.991046i	$-0.542629\pi$
$743$	−7.27911	−0.267045	−0.133522	+	0.991046i	$0.542629\pi$
$744$	−8.58244	−0.314648
$745$	0	0
$746$	12.2205	0.447426
$747$	−10.5091	−0.384509
$748$	−2.72765	−0.0997327
$749$	−2.73722	−0.100016
$750$	0	0
$751$	37.8815	1.38232	0.691158	−	0.722704i	$-0.257101\pi$
$751$	37.8815	1.38232	0.691158	+	0.722704i	$0.257101\pi$
$752$	11.9307	0.435068
$753$	−38.0647	−1.38715
$754$	57.6879	2.10087
$755$	0	0
$756$	3.86287	0.140491
$757$	−51.1527	−1.85917	−0.929587	−	0.368602i	$-0.879837\pi$
$757$	−51.1527	−1.85917	−0.929587	+	0.368602i	$0.879837\pi$
$758$	40.1189	1.45718
$759$	15.9222	0.577938
$760$	0	0
$761$	−10.8749	−0.394213	−0.197107	−	0.980382i	$-0.563155\pi$
$761$	−10.8749	−0.394213	−0.197107	+	0.980382i	$0.563155\pi$
$762$	−24.8759	−0.901158
$763$	−2.75965	−0.0999061
$764$	−23.3291	−0.844019
$765$	0	0
$766$	45.0539	1.62786
$767$	22.4033	0.808938
$768$	30.4764	1.09972
$769$	−18.8020	−0.678016	−0.339008	−	0.940783i	$-0.610091\pi$
$769$	−18.8020	−0.678016	−0.339008	+	0.940783i	$0.610091\pi$
$770$	0	0
$771$	−32.5820	−1.17341
$772$	23.5761	0.848521
$773$	−24.3856	−0.877090	−0.438545	−	0.898709i	$-0.644506\pi$
$773$	−24.3856	−0.877090	−0.438545	+	0.898709i	$0.644506\pi$
$774$	−10.1773	−0.365817
$775$	0	0
$776$	−6.20825	−0.222863
$777$	−3.92987	−0.140983
$778$	−29.3105	−1.05083
$779$	0	0
$780$	0	0
$781$	25.8059	0.923408
$782$	10.0867	0.360698
$783$	−57.4913	−2.05457
$784$	32.5985	1.16423
$785$	0	0
$786$	37.5341	1.33880
$787$	9.17617	0.327095	0.163547	−	0.986535i	$-0.447706\pi$
$787$	9.17617	0.327095	0.163547	+	0.986535i	$0.447706\pi$
$788$	14.3480	0.511127
$789$	−37.7180	−1.34280
$790$	0	0
$791$	−6.67510	−0.237339
$792$	−1.82356	−0.0647972
$793$	−26.8328	−0.952862
$794$	−65.7158	−2.33217
$795$	0	0
$796$	−18.1293	−0.642574
$797$	−46.8095	−1.65808	−0.829039	−	0.559191i	$-0.811112\pi$
$797$	−46.8095	−1.65808	−0.829039	+	0.559191i	$0.811112\pi$
$798$	0	0
$799$	2.41277	0.0853578
$800$	0	0
$801$	1.90089	0.0671647
$802$	−20.7672	−0.733314
$803$	−9.73004	−0.343366
$804$	−18.3642	−0.647654
$805$	0	0
$806$	−31.7687	−1.11900
$807$	13.4529	0.473563
$808$	0.271913	0.00956588
$809$	11.7440	0.412898	0.206449	−	0.978457i	$-0.433809\pi$
$809$	11.7440	0.412898	0.206449	+	0.978457i	$0.433809\pi$
$810$	0	0
$811$	−41.3856	−1.45324	−0.726622	−	0.687038i	$-0.758911\pi$
$811$	−41.3856	−1.45324	−0.726622	+	0.687038i	$0.758911\pi$
$812$	−6.95704	−0.244144
$813$	−32.6785	−1.14609
$814$	−20.5127	−0.718971
$815$	0	0
$816$	6.81564	0.238595
$817$	0	0
$818$	3.54650	0.124000
$819$	−1.29319	−0.0451877
$820$	0	0
$821$	−15.6567	−0.546424	−0.273212	−	0.961954i	$-0.588086\pi$
$821$	−15.6567	−0.546424	−0.273212	+	0.961954i	$0.588086\pi$
$822$	10.3547	0.361161
$823$	−42.0847	−1.46698	−0.733490	−	0.679700i	$-0.762110\pi$
$823$	−42.0847	−1.46698	−0.733490	+	0.679700i	$0.762110\pi$
$824$	−15.1340	−0.527217
$825$	0	0
$826$	−6.47651	−0.225347
$827$	−23.0974	−0.803175	−0.401588	−	0.915821i	$-0.631541\pi$
$827$	−23.0974	−0.803175	−0.401588	+	0.915821i	$0.631541\pi$
$828$	−7.08397	−0.246185
$829$	−44.6607	−1.55113	−0.775566	−	0.631267i	$-0.782535\pi$
$829$	−44.6607	−1.55113	−0.775566	+	0.631267i	$0.782535\pi$
$830$	0	0
$831$	−13.3310	−0.462446
$832$	9.14941	0.317199
$833$	6.59245	0.228415
$834$	50.8473	1.76070
$835$	0	0
$836$	0	0
$837$	31.6604	1.09434
$838$	41.0974	1.41969
$839$	−27.2641	−0.941260	−0.470630	−	0.882331i	$-0.655973\pi$
$839$	−27.2641	−0.941260	−0.470630	+	0.882331i	$0.655973\pi$
$840$	0	0
$841$	74.5420	2.57041
$842$	24.7844	0.854126
$843$	28.1538	0.969669
$844$	−8.21135	−0.282646
$845$	0	0
$846$	−4.06195	−0.139653
$847$	3.42397	0.117649
$848$	−51.8499	−1.78053
$849$	−12.6898	−0.435513
$850$	0	0
$851$	31.6442	1.08475
$852$	27.4504	0.940434
$853$	6.64990	0.227688	0.113844	−	0.993499i	$-0.463684\pi$
$853$	6.64990	0.227688	0.113844	+	0.993499i	$0.463684\pi$
$854$	7.75702	0.265440
$855$	0	0
$856$	6.03527	0.206281
$857$	−47.5875	−1.62556	−0.812780	−	0.582571i	$-0.802047\pi$
$857$	−47.5875	−1.62556	−0.812780	+	0.582571i	$0.802047\pi$
$858$	16.1385	0.550958
$859$	47.5779	1.62334	0.811669	−	0.584118i	$-0.198559\pi$
$859$	47.5779	1.62334	0.811669	+	0.584118i	$0.198559\pi$
$860$	0	0
$861$	7.27125	0.247804
$862$	−68.8527	−2.34513
$863$	−0.387407	−0.0131875	−0.00659374	−	0.999978i	$-0.502099\pi$
$863$	−0.387407	−0.0131875	−0.00659374	+	0.999978i	$0.502099\pi$
$864$	−38.4823	−1.30920
$865$	0	0
$866$	15.4240	0.524127
$867$	−23.3464	−0.792885
$868$	3.83124	0.130041
$869$	1.38572	0.0470074
$870$	0	0
$871$	26.9943	0.914667
$872$	6.08473	0.206055
$873$	5.21585	0.176530
$874$	0	0
$875$	0	0
$876$	−10.3501	−0.349697
$877$	−39.0509	−1.31865	−0.659327	−	0.751856i	$-0.729159\pi$
$877$	−39.0509	−1.31865	−0.659327	+	0.751856i	$0.729159\pi$
$878$	−31.6297	−1.06745
$879$	−24.3974	−0.822904
$880$	0	0
$881$	22.9067	0.771747	0.385873	−	0.922552i	$-0.373900\pi$
$881$	22.9067	0.771747	0.385873	+	0.922552i	$0.373900\pi$
$882$	−11.0985	−0.373707
$883$	28.5966	0.962353	0.481176	−	0.876624i	$-0.340210\pi$
$883$	28.5966	0.962353	0.481176	+	0.876624i	$0.340210\pi$
$884$	4.26500	0.143447
$885$	0	0
$886$	−13.3030	−0.446923
$887$	8.79919	0.295448	0.147724	−	0.989029i	$-0.452805\pi$
$887$	8.79919	0.295448	0.147724	+	0.989029i	$0.452805\pi$
$888$	8.66494	0.290776
$889$	−4.40981	−0.147900
$890$	0	0
$891$	−10.8885	−0.364777
$892$	18.3919	0.615808
$893$	0	0
$894$	64.4632	2.15597
$895$	0	0
$896$	3.86105	0.128989
$897$	−24.8962	−0.831259
$898$	42.7957	1.42811
$899$	−57.0205	−1.90174
$900$	0	0
$901$	−10.4857	−0.349330
$902$	37.9538	1.26372
$903$	4.31350	0.143544
$904$	14.7179	0.489509
$905$	0	0
$906$	−15.9038	−0.528367
$907$	23.7448	0.788432	0.394216	−	0.919018i	$-0.371016\pi$
$907$	23.7448	0.788432	0.394216	+	0.919018i	$0.371016\pi$
$908$	22.3043	0.740194
$909$	−0.228448	−0.00757713
$910$	0	0
$911$	8.14760	0.269942	0.134971	−	0.990850i	$-0.456906\pi$
$911$	8.14760	0.269942	0.134971	+	0.990850i	$0.456906\pi$
$912$	0	0
$913$	23.2492	0.769438
$914$	25.4581	0.842080
$915$	0	0
$916$	−5.41016	−0.178757
$917$	6.65377	0.219727
$918$	−10.1888	−0.336281
$919$	−4.38700	−0.144714	−0.0723569	−	0.997379i	$-0.523052\pi$
$919$	−4.38700	−0.144714	−0.0723569	+	0.997379i	$0.523052\pi$
$920$	0	0
$921$	−9.32113	−0.307142
$922$	48.2601	1.58936
$923$	−40.3506	−1.32816
$924$	−1.94627	−0.0640274
$925$	0	0
$926$	−7.86726	−0.258534
$927$	12.7148	0.417609
$928$	69.3068	2.27511
$929$	−0.394075	−0.0129292	−0.00646459	−	0.999979i	$-0.502058\pi$
$929$	−0.394075	−0.0129292	−0.00646459	+	0.999979i	$0.502058\pi$
$930$	0	0
$931$	0	0
$932$	15.8534	0.519294
$933$	15.4527	0.505898
$934$	−51.4182	−1.68246
$935$	0	0
$936$	2.85134	0.0931990
$937$	28.6351	0.935468	0.467734	−	0.883869i	$-0.345070\pi$
$937$	28.6351	0.935468	0.467734	+	0.883869i	$0.345070\pi$
$938$	−7.80370	−0.254800
$939$	6.91792	0.225758
$940$	0	0
$941$	26.6636	0.869208	0.434604	−	0.900622i	$-0.356888\pi$
$941$	26.6636	0.869208	0.434604	+	0.900622i	$0.356888\pi$
$942$	28.4793	0.927904
$943$	−58.5498	−1.90664
$944$	35.2384	1.14691
$945$	0	0
$946$	22.5152	0.732032
$947$	18.0730	0.587295	0.293648	−	0.955914i	$-0.405131\pi$
$947$	18.0730	0.587295	0.293648	+	0.955914i	$0.405131\pi$
$948$	1.47403	0.0478741
$949$	15.2140	0.493869
$950$	0	0
$951$	−0.0890408	−0.00288735
$952$	0.489621	0.0158687
$953$	−23.2504	−0.753155	−0.376577	−	0.926385i	$-0.622899\pi$
$953$	−23.2504	−0.753155	−0.376577	+	0.926385i	$0.622899\pi$
$954$	17.6529	0.571534
$955$	0	0
$956$	−15.2452	−0.493064
$957$	28.9663	0.936349
$958$	13.1872	0.426059
$959$	1.83560	0.0592747
$960$	0	0
$961$	0.401173	0.0129411
$962$	32.0741	1.03411
$963$	−5.07053	−0.163395
$964$	−7.17581	−0.231117
$965$	0	0
$966$	7.19715	0.231565
$967$	38.9509	1.25258	0.626289	−	0.779591i	$-0.284573\pi$
$967$	38.9509	1.25258	0.626289	+	0.779591i	$0.284573\pi$
$968$	−7.54948	−0.242650
$969$	0	0
$970$	0	0
$971$	−11.0320	−0.354034	−0.177017	−	0.984208i	$-0.556645\pi$
$971$	−11.0320	−0.354034	−0.177017	+	0.984208i	$0.556645\pi$
$972$	12.6818	0.406768
$973$	9.01384	0.288970
$974$	−6.56913	−0.210488
$975$	0	0
$976$	−42.2056	−1.35097
$977$	−26.3934	−0.844401	−0.422201	−	0.906502i	$-0.638742\pi$
$977$	−26.3934	−0.844401	−0.422201	+	0.906502i	$0.638742\pi$
$978$	−50.1360	−1.60317
$979$	−4.20532	−0.134403
$980$	0	0
$981$	−5.11208	−0.163216
$982$	59.5202	1.89937
$983$	47.2339	1.50653	0.753264	−	0.657719i	$-0.228478\pi$
$983$	47.2339	1.50653	0.753264	+	0.657719i	$0.228478\pi$
$984$	−16.0323	−0.511092
$985$	0	0
$986$	18.3501	0.584387
$987$	1.72159	0.0547989
$988$	0	0
$989$	−34.7333	−1.10446
$990$	0	0
$991$	45.6843	1.45121	0.725604	−	0.688112i	$-0.241560\pi$
$991$	45.6843	1.45121	0.725604	+	0.688112i	$0.241560\pi$
$992$	−38.1672	−1.21181
$993$	−25.6071	−0.812618
$994$	11.6648	0.369986
$995$	0	0
$996$	24.7308	0.783625
$997$	−28.9871	−0.918031	−0.459016	−	0.888428i	$-0.651798\pi$
$997$	−28.9871	−0.918031	−0.459016	+	0.888428i	$0.651798\pi$
$998$	22.0713	0.698654
$999$	−31.9648	−1.01132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.4		16
5.2	odd	4		1805.2.b.j.1084.4	yes	16
5.3	odd	4		1805.2.b.j.1084.13	yes	16
5.4	even	2	inner	9025.2.a.ck.1.13		16
19.18	odd	2		9025.2.a.cl.1.13		16
95.18	even	4		1805.2.b.i.1084.4	✓	16
95.37	even	4		1805.2.b.i.1084.13	yes	16
95.94	odd	2		9025.2.a.cl.1.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.4	✓	16	95.18	even	4
1805.2.b.i.1084.13	yes	16	95.37	even	4
1805.2.b.j.1084.4	yes	16	5.2	odd	4
1805.2.b.j.1084.13	yes	16	5.3	odd	4
9025.2.a.ck.1.4		16	1.1	even	1	trivial
9025.2.a.ck.1.13		16	5.4	even	2	inner
9025.2.a.cl.1.4		16	95.94	odd	2
9025.2.a.cl.1.13		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)

\(f(q)\)	\(=\)	\(q-1.85244 q^{2} +1.45440 q^{3} +1.43152 q^{4} -2.69418 q^{6} -0.477604 q^{7} +1.05306 q^{8} -0.884731 q^{9} +O(q^{10})\)	\(q-1.85244 q^{2} +1.45440 q^{3} +1.43152 q^{4} -2.69418 q^{6} -0.477604 q^{7} +1.05306 q^{8} -0.884731 q^{9} +1.95728 q^{11} +2.08200 q^{12} -3.06043 q^{13} +0.884731 q^{14} -4.81379 q^{16} -0.973502 q^{17} +1.63891 q^{18} -0.694625 q^{21} -3.62574 q^{22} +5.59328 q^{23} +1.53157 q^{24} +5.66926 q^{26} -5.64994 q^{27} -0.683702 q^{28} +10.1756 q^{29} -5.60367 q^{31} +6.81111 q^{32} +2.84666 q^{33} +1.80335 q^{34} -1.26651 q^{36} +5.65754 q^{37} -4.45108 q^{39} -10.4679 q^{41} +1.28675 q^{42} -6.20983 q^{43} +2.80189 q^{44} -10.3612 q^{46} -2.47845 q^{47} -7.00115 q^{48} -6.77189 q^{49} -1.41586 q^{51} -4.38109 q^{52} +10.7711 q^{53} +10.4662 q^{54} -0.502948 q^{56} -18.8496 q^{58} -7.32031 q^{59} +8.76766 q^{61} +10.3805 q^{62} +0.422551 q^{63} -2.98958 q^{64} -5.27326 q^{66} -8.82042 q^{67} -1.39359 q^{68} +8.13485 q^{69} +13.1846 q^{71} -0.931679 q^{72} -4.97121 q^{73} -10.4802 q^{74} -0.934804 q^{77} +8.24535 q^{78} +0.707984 q^{79} -5.56306 q^{81} +19.3911 q^{82} +11.8783 q^{83} -0.994373 q^{84} +11.5033 q^{86} +14.7993 q^{87} +2.06114 q^{88} -2.14855 q^{89} +1.46167 q^{91} +8.00692 q^{92} -8.14996 q^{93} +4.59117 q^{94} +9.90605 q^{96} -5.89541 q^{97} +12.5445 q^{98} -1.73167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 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

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