LMFDB - Embedded newform 7865.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7865 = 5 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7865.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,1,-5,7,-8,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.8023411897$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 13x^{3} - 35x^{2} - 3x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 13x^3 - 35x^2 - 3x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.769771$ of defining polynomial
Character	$\chi$	$=$	7865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.769771 q^{2} -1.78335 q^{3} -1.40745 q^{4} -1.00000 q^{5} +1.37277 q^{6} +4.81089 q^{7} +2.62296 q^{8} +0.180351 q^{9} +0.769771 q^{10} +2.50999 q^{12} -1.00000 q^{13} -3.70329 q^{14} +1.78335 q^{15} +0.795827 q^{16} +5.95944 q^{17} -0.138829 q^{18} -6.43825 q^{19} +1.40745 q^{20} -8.57952 q^{21} -2.25396 q^{23} -4.67766 q^{24} +1.00000 q^{25} +0.769771 q^{26} +5.02843 q^{27} -6.77110 q^{28} +7.17182 q^{29} -1.37277 q^{30} +8.29957 q^{31} -5.85852 q^{32} -4.58740 q^{34} -4.81089 q^{35} -0.253835 q^{36} -8.81816 q^{37} +4.95598 q^{38} +1.78335 q^{39} -2.62296 q^{40} +6.20968 q^{41} +6.60427 q^{42} +0.808116 q^{43} -0.180351 q^{45} +1.73503 q^{46} +1.10118 q^{47} -1.41924 q^{48} +16.1447 q^{49} -0.769771 q^{50} -10.6278 q^{51} +1.40745 q^{52} +8.25370 q^{53} -3.87074 q^{54} +12.6188 q^{56} +11.4817 q^{57} -5.52066 q^{58} +2.21329 q^{59} -2.50999 q^{60} -4.19995 q^{61} -6.38877 q^{62} +0.867648 q^{63} +2.91807 q^{64} +1.00000 q^{65} +9.52010 q^{67} -8.38762 q^{68} +4.01960 q^{69} +3.70329 q^{70} +4.50728 q^{71} +0.473052 q^{72} -13.8253 q^{73} +6.78796 q^{74} -1.78335 q^{75} +9.06152 q^{76} -1.37277 q^{78} +6.73319 q^{79} -0.795827 q^{80} -9.50853 q^{81} -4.78003 q^{82} +5.58735 q^{83} +12.0753 q^{84} -5.95944 q^{85} -0.622065 q^{86} -12.7899 q^{87} +13.3953 q^{89} +0.138829 q^{90} -4.81089 q^{91} +3.17234 q^{92} -14.8011 q^{93} -0.847655 q^{94} +6.43825 q^{95} +10.4478 q^{96} +4.75201 q^{97} -12.4277 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 5 q^{3} + 7 q^{4} - 8 q^{5} - 7 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} - 15 q^{12} - 8 q^{13} - 7 q^{14} + 5 q^{15} + 9 q^{16} + 4 q^{17} + 8 q^{18} + 16 q^{19} - 7 q^{20} - 23 q^{21}+ \cdots + 41 q^{98}+O(q^{100}) $ 8 * q + q^2 - 5 * q^3 + 7 * q^4 - 8 * q^5 - 7 * q^6 + 9 * q^7 + 3 * q^8 + 3 * q^9 - q^10 - 15 * q^12 - 8 * q^13 - 7 * q^14 + 5 * q^15 + 9 * q^16 + 4 * q^17 + 8 * q^18 + 16 * q^19 - 7 * q^20 - 23 * q^21 - 9 * q^23 + 5 * q^24 + 8 * q^25 - q^26 - 11 * q^27 + 11 * q^28 - 13 * q^29 + 7 * q^30 - 28 * q^32 + 13 * q^34 - 9 * q^35 + 12 * q^36 - 5 * q^37 + 55 * q^38 + 5 * q^39 - 3 * q^40 + 13 * q^41 - 5 * q^42 + 5 * q^43 - 3 * q^45 + 27 * q^46 + 10 * q^47 - 36 * q^48 + 23 * q^49 + q^50 - 37 * q^51 - 7 * q^52 + 14 * q^53 - 45 * q^54 - 18 * q^56 - 21 * q^57 + 3 * q^58 - 20 * q^59 + 15 * q^60 + 6 * q^61 + 25 * q^62 + 26 * q^63 - 3 * q^64 + 8 * q^65 + 4 * q^67 - 9 * q^69 + 7 * q^70 - 9 * q^71 + 35 * q^72 + 10 * q^73 + 42 * q^74 - 5 * q^75 + 19 * q^76 + 7 * q^78 + 13 * q^79 - 9 * q^80 + 16 * q^82 + 26 * q^83 - 58 * q^84 - 4 * q^85 + 26 * q^86 + 11 * q^87 - 6 * q^89 - 8 * q^90 - 9 * q^91 - 16 * q^92 + 3 * q^93 + 23 * q^94 - 16 * q^95 + 21 * q^96 + 26 * q^97 + 41 * q^98

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 $

$2$	−0.769771	−0.544310	−0.272155	−	0.962253i	$-0.587736\pi$
$2$	−0.769771	−0.544310	−0.272155	+	0.962253i	$0.587736\pi$
$3$	−1.78335	−1.02962	−0.514810	−	0.857304i	$-0.672137\pi$
$3$	−1.78335	−1.02962	−0.514810	+	0.857304i	$0.672137\pi$
$4$	−1.40745	−0.703726
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.37277	0.560433
$7$	4.81089	1.81835	0.909173	−	0.416418i	$-0.136715\pi$
$7$	4.81089	1.81835	0.909173	+	0.416418i	$0.136715\pi$
$8$	2.62296	0.927356
$9$	0.180351	0.0601169
$10$	0.769771	0.243423
$11$	0	0
$11$	0	0
$12$	2.50999	0.724570
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.70329	−0.989745
$15$	1.78335	0.460460
$16$	0.795827	0.198957
$17$	5.95944	1.44538	0.722688	−	0.691174i	$-0.242906\pi$
$17$	5.95944	1.44538	0.722688	+	0.691174i	$0.242906\pi$
$18$	−0.138829	−0.0327222
$19$	−6.43825	−1.47703	−0.738517	−	0.674234i	$-0.764474\pi$
$19$	−6.43825	−1.47703	−0.738517	+	0.674234i	$0.764474\pi$
$20$	1.40745	0.314716
$21$	−8.57952	−1.87221
$22$	0	0
$23$	−2.25396	−0.469982	−0.234991	−	0.971997i	$-0.575506\pi$
$23$	−2.25396	−0.469982	−0.234991	+	0.971997i	$0.575506\pi$
$24$	−4.67766	−0.954824
$25$	1.00000	0.200000
$26$	0.769771	0.150965
$27$	5.02843	0.967722
$28$	−6.77110	−1.27962
$29$	7.17182	1.33177	0.665886	−	0.746053i	$-0.268054\pi$
$29$	7.17182	1.33177	0.665886	+	0.746053i	$0.268054\pi$
$30$	−1.37277	−0.250633
$31$	8.29957	1.49065	0.745324	−	0.666703i	$-0.232295\pi$
$31$	8.29957	1.49065	0.745324	+	0.666703i	$0.232295\pi$
$32$	−5.85852	−1.03565
$33$	0	0
$34$	−4.58740	−0.786733
$35$	−4.81089	−0.813189
$36$	−0.253835	−0.0423058
$37$	−8.81816	−1.44970	−0.724848	−	0.688909i	$-0.758090\pi$
$37$	−8.81816	−1.44970	−0.724848	+	0.688909i	$0.758090\pi$
$38$	4.95598	0.803965
$39$	1.78335	0.285565
$40$	−2.62296	−0.414726
$41$	6.20968	0.969789	0.484894	−	0.874573i	$-0.338858\pi$
$41$	6.20968	0.969789	0.484894	+	0.874573i	$0.338858\pi$
$42$	6.60427	1.01906
$43$	0.808116	0.123237	0.0616183	−	0.998100i	$-0.480374\pi$
$43$	0.808116	0.123237	0.0616183	+	0.998100i	$0.480374\pi$
$44$	0	0
$45$	−0.180351	−0.0268851
$46$	1.73503	0.255816
$47$	1.10118	0.160623	0.0803116	−	0.996770i	$-0.474408\pi$
$47$	1.10118	0.160623	0.0803116	+	0.996770i	$0.474408\pi$
$48$	−1.41924	−0.204850
$49$	16.1447	2.30638
$50$	−0.769771	−0.108862
$51$	−10.6278	−1.48819
$52$	1.40745	0.195179
$53$	8.25370	1.13373	0.566867	−	0.823810i	$-0.308156\pi$
$53$	8.25370	1.13373	0.566867	+	0.823810i	$0.308156\pi$
$54$	−3.87074	−0.526741
$55$	0	0
$56$	12.6188	1.68625
$57$	11.4817	1.52078
$58$	−5.52066	−0.724898
$59$	2.21329	0.288146	0.144073	−	0.989567i	$-0.453980\pi$
$59$	2.21329	0.288146	0.144073	+	0.989567i	$0.453980\pi$
$60$	−2.50999	−0.324038
$61$	−4.19995	−0.537749	−0.268874	−	0.963175i	$-0.586652\pi$
$61$	−4.19995	−0.537749	−0.268874	+	0.963175i	$0.586652\pi$
$62$	−6.38877	−0.811375
$63$	0.867648	0.109313
$64$	2.91807	0.364758
$65$	1.00000	0.124035
$66$	0	0
$67$	9.52010	1.16307	0.581533	−	0.813523i	$-0.302453\pi$
$67$	9.52010	1.16307	0.581533	+	0.813523i	$0.302453\pi$
$68$	−8.38762	−1.01715
$69$	4.01960	0.483903
$70$	3.70329	0.442627
$71$	4.50728	0.534916	0.267458	−	0.963570i	$-0.413816\pi$
$71$	4.50728	0.534916	0.267458	+	0.963570i	$0.413816\pi$
$72$	0.473052	0.0557497
$73$	−13.8253	−1.61813	−0.809064	−	0.587720i	$-0.800026\pi$
$73$	−13.8253	−1.61813	−0.809064	+	0.587720i	$0.800026\pi$
$74$	6.78796	0.789085
$75$	−1.78335	−0.205924
$76$	9.06152	1.03943
$77$	0	0
$78$	−1.37277	−0.155436
$79$	6.73319	0.757543	0.378771	−	0.925490i	$-0.376347\pi$
$79$	6.73319	0.757543	0.378771	+	0.925490i	$0.376347\pi$
$80$	−0.795827	−0.0889761
$81$	−9.50853	−1.05650
$82$	−4.78003	−0.527866
$83$	5.58735	0.613292	0.306646	−	0.951824i	$-0.400793\pi$
$83$	5.58735	0.613292	0.306646	+	0.951824i	$0.400793\pi$
$84$	12.0753	1.31752
$85$	−5.95944	−0.646392
$86$	−0.622065	−0.0670790
$87$	−12.7899	−1.37122
$88$	0	0
$89$	13.3953	1.41990	0.709951	−	0.704251i	$-0.248717\pi$
$89$	13.3953	1.41990	0.709951	+	0.704251i	$0.248717\pi$
$90$	0.138829	0.0146338
$91$	−4.81089	−0.504319
$92$	3.17234	0.330739
$93$	−14.8011	−1.53480
$94$	−0.847655	−0.0874289
$95$	6.43825	0.660550
$96$	10.4478	1.06633
$97$	4.75201	0.482494	0.241247	−	0.970464i	$-0.422444\pi$
$97$	4.75201	0.482494	0.241247	+	0.970464i	$0.422444\pi$
$98$	−12.4277	−1.25539
$99$	0	0
$100$	−1.40745	−0.140745
$101$	−10.7491	−1.06958	−0.534788	−	0.844986i	$-0.679609\pi$
$101$	−10.7491	−1.06958	−0.534788	+	0.844986i	$0.679609\pi$
$102$	8.18096	0.810036
$103$	−9.97862	−0.983223	−0.491611	−	0.870815i	$-0.663592\pi$
$103$	−9.97862	−0.983223	−0.491611	+	0.870815i	$0.663592\pi$
$104$	−2.62296	−0.257202
$105$	8.57952	0.837276
$106$	−6.35346	−0.617103
$107$	−8.74273	−0.845191	−0.422596	−	0.906318i	$-0.638881\pi$
$107$	−8.74273	−0.845191	−0.422596	+	0.906318i	$0.638881\pi$
$108$	−7.07728	−0.681011
$109$	10.4009	0.996229	0.498114	−	0.867111i	$-0.334026\pi$
$109$	10.4009	0.996229	0.498114	+	0.867111i	$0.334026\pi$
$110$	0	0
$111$	15.7259	1.49264
$112$	3.82864	0.361772
$113$	3.78844	0.356387	0.178193	−	0.983995i	$-0.442975\pi$
$113$	3.78844	0.356387	0.178193	+	0.983995i	$0.442975\pi$
$114$	−8.83826	−0.827779
$115$	2.25396	0.210183
$116$	−10.0940	−0.937203
$117$	−0.180351	−0.0166734
$118$	−1.70373	−0.156841
$119$	28.6702	2.62819
$120$	4.67766	0.427010
$121$	0	0
$122$	3.23300	0.292702
$123$	−11.0741	−0.998514
$124$	−11.6813	−1.04901
$125$	−1.00000	−0.0894427
$126$	−0.667890	−0.0595004
$127$	15.4486	1.37084	0.685422	−	0.728146i	$-0.259618\pi$
$127$	15.4486	1.37084	0.685422	+	0.728146i	$0.259618\pi$
$128$	9.47080	0.837108
$129$	−1.44116	−0.126887
$130$	−0.769771	−0.0675134
$131$	−19.1810	−1.67585	−0.837926	−	0.545784i	$-0.816232\pi$
$131$	−19.1810	−1.67585	−0.837926	+	0.545784i	$0.816232\pi$
$132$	0	0
$133$	−30.9737	−2.68576
$134$	−7.32830	−0.633069
$135$	−5.02843	−0.432779
$136$	15.6314	1.34038
$137$	−15.0138	−1.28272	−0.641358	−	0.767242i	$-0.721629\pi$
$137$	−15.0138	−1.28272	−0.641358	+	0.767242i	$0.721629\pi$
$138$	−3.09417	−0.263394
$139$	−0.809467	−0.0686581	−0.0343291	−	0.999411i	$-0.510929\pi$
$139$	−0.809467	−0.0686581	−0.0343291	+	0.999411i	$0.510929\pi$
$140$	6.77110	0.572263
$141$	−1.96379	−0.165381
$142$	−3.46958	−0.291160
$143$	0	0
$144$	0.143528	0.0119607
$145$	−7.17182	−0.595587
$146$	10.6423	0.880764
$147$	−28.7917	−2.37470
$148$	12.4111	1.02019
$149$	−12.9714	−1.06266	−0.531328	−	0.847166i	$-0.678307\pi$
$149$	−12.9714	−1.06266	−0.531328	+	0.847166i	$0.678307\pi$
$150$	1.37277	0.112087
$151$	−3.01030	−0.244975	−0.122487	−	0.992470i	$-0.539087\pi$
$151$	−3.01030	−0.244975	−0.122487	+	0.992470i	$0.539087\pi$
$152$	−16.8873	−1.36974
$153$	1.07479	0.0868915
$154$	0	0
$155$	−8.29957	−0.666638
$156$	−2.50999	−0.200960
$157$	−6.42928	−0.513113	−0.256556	−	0.966529i	$-0.582588\pi$
$157$	−6.42928	−0.513113	−0.256556	+	0.966529i	$0.582588\pi$
$158$	−5.18301	−0.412338
$159$	−14.7193	−1.16731
$160$	5.85852	0.463157
$161$	−10.8435	−0.854591
$162$	7.31939	0.575065
$163$	−13.0458	−1.02182	−0.510912	−	0.859633i	$-0.670692\pi$
$163$	−13.0458	−1.02182	−0.510912	+	0.859633i	$0.670692\pi$
$164$	−8.73982	−0.682466
$165$	0	0
$166$	−4.30098	−0.333821
$167$	18.0745	1.39865	0.699325	−	0.714804i	$-0.253484\pi$
$167$	18.0745	1.39865	0.699325	+	0.714804i	$0.253484\pi$
$168$	−22.5037	−1.73620
$169$	1.00000	0.0769231
$170$	4.58740	0.351838
$171$	−1.16114	−0.0887947
$172$	−1.13739	−0.0867248
$173$	−9.81302	−0.746070	−0.373035	−	0.927817i	$-0.621683\pi$
$173$	−9.81302	−0.746070	−0.373035	+	0.927817i	$0.621683\pi$
$174$	9.84529	0.746369
$175$	4.81089	0.363669
$176$	0	0
$177$	−3.94708	−0.296680
$178$	−10.3113	−0.772867
$179$	−7.42273	−0.554801	−0.277401	−	0.960754i	$-0.589473\pi$
$179$	−7.42273	−0.554801	−0.277401	+	0.960754i	$0.589473\pi$
$180$	0.253835	0.0189197
$181$	0.644667	0.0479177	0.0239589	−	0.999713i	$-0.492373\pi$
$181$	0.644667	0.0479177	0.0239589	+	0.999713i	$0.492373\pi$
$182$	3.70329	0.274506
$183$	7.49000	0.553677
$184$	−5.91203	−0.435841
$185$	8.81816	0.648324
$186$	11.3934	0.835408
$187$	0	0
$188$	−1.54985	−0.113035
$189$	24.1912	1.75965
$190$	−4.95598	−0.359544
$191$	−17.4590	−1.26329	−0.631643	−	0.775259i	$-0.717619\pi$
$191$	−17.4590	−1.26329	−0.631643	+	0.775259i	$0.717619\pi$
$192$	−5.20395	−0.375562
$193$	4.85709	0.349621	0.174811	−	0.984602i	$-0.444069\pi$
$193$	4.85709	0.349621	0.174811	+	0.984602i	$0.444069\pi$
$194$	−3.65796	−0.262626
$195$	−1.78335	−0.127709
$196$	−22.7229	−1.62306
$197$	20.1931	1.43870	0.719351	−	0.694647i	$-0.244440\pi$
$197$	20.1931	1.43870	0.719351	+	0.694647i	$0.244440\pi$
$198$	0	0
$199$	3.67299	0.260372	0.130186	−	0.991490i	$-0.458443\pi$
$199$	3.67299	0.260372	0.130186	+	0.991490i	$0.458443\pi$
$200$	2.62296	0.185471
$201$	−16.9777	−1.19752
$202$	8.27436	0.582182
$203$	34.5028	2.42162
$204$	14.9581	1.04728
$205$	−6.20968	−0.433703
$206$	7.68126	0.535178
$207$	−0.406502	−0.0282539
$208$	−0.795827	−0.0551807
$209$	0	0
$210$	−6.60427	−0.455738
$211$	14.9622	1.03004	0.515021	−	0.857178i	$-0.327784\pi$
$211$	14.9622	1.03004	0.515021	+	0.857178i	$0.327784\pi$
$212$	−11.6167	−0.797838
$213$	−8.03808	−0.550760
$214$	6.72990	0.460046
$215$	−0.808116	−0.0551131
$216$	13.1894	0.897423
$217$	39.9284	2.71051
$218$	−8.00634	−0.542258
$219$	24.6554	1.66606
$220$	0	0
$221$	−5.95944	−0.400875
$222$	−12.1053	−0.812457
$223$	−17.5869	−1.17770	−0.588852	−	0.808241i	$-0.700420\pi$
$223$	−17.5869	−1.17770	−0.588852	+	0.808241i	$0.700420\pi$
$224$	−28.1847	−1.88317
$225$	0.180351	0.0120234
$226$	−2.91623	−0.193985
$227$	−9.77823	−0.649004	−0.324502	−	0.945885i	$-0.605197\pi$
$227$	−9.77823	−0.649004	−0.324502	+	0.945885i	$0.605197\pi$
$228$	−16.1599	−1.07022
$229$	−23.3186	−1.54094	−0.770469	−	0.637477i	$-0.779978\pi$
$229$	−23.3186	−1.54094	−0.770469	+	0.637477i	$0.779978\pi$
$230$	−1.73503	−0.114405
$231$	0	0
$232$	18.8114	1.23503
$233$	−10.2178	−0.669390	−0.334695	−	0.942326i	$-0.608633\pi$
$233$	−10.2178	−0.669390	−0.334695	+	0.942326i	$0.608633\pi$
$234$	0.138829	0.00907552
$235$	−1.10118	−0.0718329
$236$	−3.11510	−0.202776
$237$	−12.0077	−0.779981
$238$	−22.0695	−1.43055
$239$	−15.7709	−1.02014	−0.510068	−	0.860134i	$-0.670380\pi$
$239$	−15.7709	−1.02014	−0.510068	+	0.860134i	$0.670380\pi$
$240$	1.41924	0.0916116
$241$	17.6971	1.13997	0.569987	−	0.821654i	$-0.306948\pi$
$241$	17.6971	1.13997	0.569987	+	0.821654i	$0.306948\pi$
$242$	0	0
$243$	1.87177	0.120074
$244$	5.91123	0.378428
$245$	−16.1447	−1.03145
$246$	8.52448	0.543501
$247$	6.43825	0.409656
$248$	21.7694	1.38236
$249$	−9.96423	−0.631457
$250$	0.769771	0.0486846
$251$	19.7062	1.24385	0.621923	−	0.783079i	$-0.286352\pi$
$251$	19.7062	1.24385	0.621923	+	0.783079i	$0.286352\pi$
$252$	−1.22117	−0.0769266
$253$	0	0
$254$	−11.8919	−0.746165
$255$	10.6278	0.665538
$256$	−13.1265	−0.820405
$257$	10.5662	0.659103	0.329552	−	0.944138i	$-0.393102\pi$
$257$	10.5662	0.659103	0.329552	+	0.944138i	$0.393102\pi$
$258$	1.10936	0.0690658
$259$	−42.4232	−2.63605
$260$	−1.40745	−0.0872865
$261$	1.29344	0.0800620
$262$	14.7650	0.912184
$263$	−1.89308	−0.116732	−0.0583662	−	0.998295i	$-0.518589\pi$
$263$	−1.89308	−0.116732	−0.0583662	+	0.998295i	$0.518589\pi$
$264$	0	0
$265$	−8.25370	−0.507021
$266$	23.8427	1.46189
$267$	−23.8886	−1.46196
$268$	−13.3991	−0.818480
$269$	12.0263	0.733254	0.366627	−	0.930368i	$-0.380513\pi$
$269$	12.0263	0.733254	0.366627	+	0.930368i	$0.380513\pi$
$270$	3.87074	0.235566
$271$	23.5822	1.43252	0.716259	−	0.697835i	$-0.245853\pi$
$271$	23.5822	1.43252	0.716259	+	0.697835i	$0.245853\pi$
$272$	4.74268	0.287567
$273$	8.57952	0.519256
$274$	11.5572	0.698195
$275$	0	0
$276$	−5.65740	−0.340535
$277$	−0.275729	−0.0165669	−0.00828346	−	0.999966i	$-0.502637\pi$
$277$	−0.275729	−0.0165669	−0.00828346	+	0.999966i	$0.502637\pi$
$278$	0.623105	0.0373713
$279$	1.49683	0.0896131
$280$	−12.6188	−0.754116
$281$	−19.7025	−1.17535	−0.587677	−	0.809096i	$-0.699957\pi$
$281$	−19.7025	−1.17535	−0.587677	+	0.809096i	$0.699957\pi$
$282$	1.51167	0.0900185
$283$	−16.1023	−0.957180	−0.478590	−	0.878039i	$-0.658852\pi$
$283$	−16.1023	−0.957180	−0.478590	+	0.878039i	$0.658852\pi$
$284$	−6.34379	−0.376434
$285$	−11.4817	−0.680115
$286$	0	0
$287$	29.8741	1.76341
$288$	−1.05659	−0.0622600
$289$	18.5149	1.08911
$290$	5.52066	0.324184
$291$	−8.47451	−0.496785
$292$	19.4584	1.13872
$293$	−7.53151	−0.439996	−0.219998	−	0.975500i	$-0.570605\pi$
$293$	−7.53151	−0.439996	−0.219998	+	0.975500i	$0.570605\pi$
$294$	22.1630	1.29257
$295$	−2.21329	−0.128863
$296$	−23.1297	−1.34438
$297$	0	0
$298$	9.98499	0.578415
$299$	2.25396	0.130350
$300$	2.50999	0.144914
$301$	3.88776	0.224087
$302$	2.31724	0.133342
$303$	19.1695	1.10126
$304$	−5.12373	−0.293866
$305$	4.19995	0.240489
$306$	−0.827341	−0.0472959
$307$	15.0655	0.859836	0.429918	−	0.902868i	$-0.358542\pi$
$307$	15.0655	0.859836	0.429918	+	0.902868i	$0.358542\pi$
$308$	0	0
$309$	17.7954	1.01235
$310$	6.38877	0.362858
$311$	1.91004	0.108308	0.0541541	−	0.998533i	$-0.482754\pi$
$311$	1.91004	0.108308	0.0541541	+	0.998533i	$0.482754\pi$
$312$	4.67766	0.264821
$313$	33.8248	1.91189	0.955946	−	0.293541i	$-0.0948338\pi$
$313$	33.8248	1.91189	0.955946	+	0.293541i	$0.0948338\pi$
$314$	4.94908	0.279293
$315$	−0.867648	−0.0488864
$316$	−9.47664	−0.533103
$317$	−16.5773	−0.931075	−0.465537	−	0.885028i	$-0.654139\pi$
$317$	−16.5773	−0.931075	−0.465537	+	0.885028i	$0.654139\pi$
$318$	11.3305	0.635381
$319$	0	0
$320$	−2.91807	−0.163125
$321$	15.5914	0.870226
$322$	8.34705	0.465163
$323$	−38.3683	−2.13487
$324$	13.3828	0.743489
$325$	−1.00000	−0.0554700
$326$	10.0423	0.556190
$327$	−18.5485	−1.02574
$328$	16.2877	0.899339
$329$	5.29765	0.292069
$330$	0	0
$331$	−3.63821	−0.199974	−0.0999869	−	0.994989i	$-0.531880\pi$
$331$	−3.63821	−0.199974	−0.0999869	+	0.994989i	$0.531880\pi$
$332$	−7.86394	−0.431590
$333$	−1.59036	−0.0871512
$334$	−13.9133	−0.761299
$335$	−9.52010	−0.520139
$336$	−6.82782	−0.372488
$337$	−27.7343	−1.51079	−0.755393	−	0.655272i	$-0.772554\pi$
$337$	−27.7343	−1.51079	−0.755393	+	0.655272i	$0.772554\pi$
$338$	−0.769771	−0.0418700
$339$	−6.75613	−0.366943
$340$	8.38762	0.454883
$341$	0	0
$342$	0.893813	0.0483319
$343$	43.9941	2.37546
$344$	2.11966	0.114284
$345$	−4.01960	−0.216408
$346$	7.55378	0.406094
$347$	27.0546	1.45237	0.726183	−	0.687502i	$-0.241293\pi$
$347$	27.0546	1.45237	0.726183	+	0.687502i	$0.241293\pi$
$348$	18.0012	0.964963
$349$	32.8638	1.75916	0.879580	−	0.475750i	$-0.157823\pi$
$349$	32.8638	1.75916	0.879580	+	0.475750i	$0.157823\pi$
$350$	−3.70329	−0.197949
$351$	−5.02843	−0.268398
$352$	0	0
$353$	−7.74548	−0.412250	−0.206125	−	0.978526i	$-0.566085\pi$
$353$	−7.74548	−0.412250	−0.206125	+	0.978526i	$0.566085\pi$
$354$	3.03835	0.161486
$355$	−4.50728	−0.239222
$356$	−18.8533	−0.999222
$357$	−51.1291	−2.70604
$358$	5.71381	0.301984
$359$	−28.2495	−1.49095	−0.745476	−	0.666532i	$-0.767778\pi$
$359$	−28.2495	−1.49095	−0.745476	+	0.666532i	$0.767778\pi$
$360$	−0.473052	−0.0249320
$361$	22.4510	1.18163
$362$	−0.496246	−0.0260821
$363$	0	0
$364$	6.77110	0.354902
$365$	13.8253	0.723649
$366$	−5.76559	−0.301372
$367$	12.2235	0.638059	0.319030	−	0.947745i	$-0.396643\pi$
$367$	12.2235	0.638059	0.319030	+	0.947745i	$0.396643\pi$
$368$	−1.79376	−0.0935062
$369$	1.11992	0.0583007
$370$	−6.78796	−0.352889
$371$	39.7077	2.06152
$372$	20.8318	1.08008
$373$	10.0129	0.518448	0.259224	−	0.965817i	$-0.416533\pi$
$373$	10.0129	0.518448	0.259224	+	0.965817i	$0.416533\pi$
$374$	0	0
$375$	1.78335	0.0920920
$376$	2.88834	0.148955
$377$	−7.17182	−0.369367
$378$	−18.6217	−0.957798
$379$	0.466922	0.0239842	0.0119921	−	0.999928i	$-0.496183\pi$
$379$	0.466922	0.0239842	0.0119921	+	0.999928i	$0.496183\pi$
$380$	−9.06152	−0.464846
$381$	−27.5504	−1.41145
$382$	13.4394	0.687620
$383$	30.7149	1.56946	0.784729	−	0.619838i	$-0.212802\pi$
$383$	30.7149	1.56946	0.784729	+	0.619838i	$0.212802\pi$
$384$	−16.8898	−0.861903
$385$	0	0
$386$	−3.73885	−0.190303
$387$	0.145744	0.00740860
$388$	−6.68823	−0.339543
$389$	12.8144	0.649718	0.324859	−	0.945762i	$-0.394683\pi$
$389$	12.8144	0.649718	0.324859	+	0.945762i	$0.394683\pi$
$390$	1.37277	0.0695131
$391$	−13.4323	−0.679301
$392$	42.3469	2.13884
$393$	34.2065	1.72549
$394$	−15.5441	−0.783100
$395$	−6.73319	−0.338783
$396$	0	0
$397$	5.19520	0.260740	0.130370	−	0.991465i	$-0.458384\pi$
$397$	5.19520	0.260740	0.130370	+	0.991465i	$0.458384\pi$
$398$	−2.82737	−0.141723
$399$	55.2371	2.76531
$400$	0.795827	0.0397913
$401$	5.73954	0.286619	0.143309	−	0.989678i	$-0.454226\pi$
$401$	5.73954	0.286619	0.143309	+	0.989678i	$0.454226\pi$
$402$	13.0689	0.651820
$403$	−8.29957	−0.413431
$404$	15.1289	0.752689
$405$	9.50853	0.472482
$406$	−26.5593	−1.31812
$407$	0	0
$408$	−27.8762	−1.38008
$409$	−13.8462	−0.684649	−0.342324	−	0.939582i	$-0.611214\pi$
$409$	−13.8462	−0.684649	−0.342324	+	0.939582i	$0.611214\pi$
$410$	4.78003	0.236069
$411$	26.7749	1.32071
$412$	14.0444	0.691920
$413$	10.6479	0.523949
$414$	0.312914	0.0153789
$415$	−5.58735	−0.274272
$416$	5.85852	0.287238
$417$	1.44357	0.0706918
$418$	0	0
$419$	−17.7580	−0.867537	−0.433769	−	0.901024i	$-0.642816\pi$
$419$	−17.7580	−0.867537	−0.433769	+	0.901024i	$0.642816\pi$
$420$	−12.0753	−0.589213
$421$	−26.7851	−1.30542	−0.652712	−	0.757606i	$-0.726369\pi$
$421$	−26.7851	−1.30542	−0.652712	+	0.757606i	$0.726369\pi$
$422$	−11.5175	−0.560662
$423$	0.198598	0.00965617
$424$	21.6491	1.05137
$425$	5.95944	0.289075
$426$	6.18748	0.299785
$427$	−20.2055	−0.977814
$428$	12.3050	0.594783
$429$	0	0
$430$	0.622065	0.0299986
$431$	−1.26832	−0.0610929	−0.0305464	−	0.999533i	$-0.509725\pi$
$431$	−1.26832	−0.0610929	−0.0305464	+	0.999533i	$0.509725\pi$
$432$	4.00176	0.192535
$433$	36.1333	1.73645	0.868227	−	0.496167i	$-0.165259\pi$
$433$	36.1333	1.73645	0.868227	+	0.496167i	$0.165259\pi$
$434$	−30.7357	−1.47536
$435$	12.7899	0.613228
$436$	−14.6388	−0.701072
$437$	14.5115	0.694180
$438$	−18.9790	−0.906852
$439$	14.9325	0.712688	0.356344	−	0.934355i	$-0.384023\pi$
$439$	14.9325	0.712688	0.356344	+	0.934355i	$0.384023\pi$
$440$	0	0
$441$	2.91171	0.138653
$442$	4.58740	0.218201
$443$	10.7791	0.512132	0.256066	−	0.966659i	$-0.417573\pi$
$443$	10.7791	0.512132	0.256066	+	0.966659i	$0.417573\pi$
$444$	−22.1335	−1.05041
$445$	−13.3953	−0.634999
$446$	13.5379	0.641036
$447$	23.1325	1.09413
$448$	14.0385	0.663257
$449$	32.6453	1.54063	0.770313	−	0.637666i	$-0.220100\pi$
$449$	32.6453	1.54063	0.770313	+	0.637666i	$0.220100\pi$
$450$	−0.138829	−0.00654445
$451$	0	0
$452$	−5.33205	−0.250799
$453$	5.36843	0.252231
$454$	7.52700	0.353260
$455$	4.81089	0.225538
$456$	30.1159	1.41031
$457$	36.5686	1.71061	0.855303	−	0.518128i	$-0.173371\pi$
$457$	36.5686	1.71061	0.855303	+	0.518128i	$0.173371\pi$
$458$	17.9500	0.838749
$459$	29.9666	1.39872
$460$	−3.17234	−0.147911
$461$	−22.3491	−1.04090	−0.520451	−	0.853892i	$-0.674236\pi$
$461$	−22.3491	−1.04090	−0.520451	+	0.853892i	$0.674236\pi$
$462$	0	0
$463$	−13.2068	−0.613771	−0.306885	−	0.951746i	$-0.599287\pi$
$463$	−13.2068	−0.613771	−0.306885	+	0.951746i	$0.599287\pi$
$464$	5.70752	0.264965
$465$	14.8011	0.686384
$466$	7.86537	0.364356
$467$	−2.37755	−0.110020	−0.0550100	−	0.998486i	$-0.517519\pi$
$467$	−2.37755	−0.110020	−0.0550100	+	0.998486i	$0.517519\pi$
$468$	0.253835	0.0117335
$469$	45.8002	2.11486
$470$	0.847655	0.0390994
$471$	11.4657	0.528311
$472$	5.80536	0.267214
$473$	0	0
$474$	9.24315	0.424552
$475$	−6.43825	−0.295407
$476$	−40.3520	−1.84953
$477$	1.48856	0.0681565
$478$	12.1400	0.555271
$479$	−33.3935	−1.52579	−0.762894	−	0.646524i	$-0.776222\pi$
$479$	−33.3935	−1.52579	−0.762894	+	0.646524i	$0.776222\pi$
$480$	−10.4478	−0.476875
$481$	8.81816	0.402073
$482$	−13.6228	−0.620499
$483$	19.3379	0.879904
$484$	0	0
$485$	−4.75201	−0.215778
$486$	−1.44083	−0.0653575
$487$	25.9415	1.17552	0.587761	−	0.809034i	$-0.300009\pi$
$487$	25.9415	1.17552	0.587761	+	0.809034i	$0.300009\pi$
$488$	−11.0163	−0.498684
$489$	23.2652	1.05209
$490$	12.4277	0.561427
$491$	−21.9490	−0.990544	−0.495272	−	0.868738i	$-0.664932\pi$
$491$	−21.9490	−0.990544	−0.495272	+	0.868738i	$0.664932\pi$
$492$	15.5862	0.702680
$493$	42.7400	1.92491
$494$	−4.95598	−0.222980
$495$	0	0
$496$	6.60502	0.296574
$497$	21.6841	0.972663
$498$	7.67018	0.343709
$499$	24.2606	1.08605	0.543027	−	0.839715i	$-0.317278\pi$
$499$	24.2606	1.08605	0.543027	+	0.839715i	$0.317278\pi$
$500$	1.40745	0.0629432
$501$	−32.2333	−1.44008
$502$	−15.1693	−0.677038
$503$	−29.8823	−1.33238	−0.666192	−	0.745780i	$-0.732077\pi$
$503$	−29.8823	−1.33238	−0.666192	+	0.745780i	$0.732077\pi$
$504$	2.27580	0.101372
$505$	10.7491	0.478329
$506$	0	0
$507$	−1.78335	−0.0792015
$508$	−21.7432	−0.964699
$509$	24.2781	1.07611	0.538055	−	0.842910i	$-0.319159\pi$
$509$	24.2781	1.07611	0.538055	+	0.842910i	$0.319159\pi$
$510$	−8.18096	−0.362259
$511$	−66.5120	−2.94232
$512$	−8.83721	−0.390553
$513$	−32.3743	−1.42936
$514$	−8.13358	−0.358757
$515$	9.97862	0.439711
$516$	2.02836	0.0892936
$517$	0	0
$518$	32.6562	1.43483
$519$	17.5001	0.768169
$520$	2.62296	0.115024
$521$	7.59574	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$521$	7.59574	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$522$	−0.995654	−0.0435786
$523$	41.9952	1.83632	0.918161	−	0.396208i	$-0.129674\pi$
$523$	41.9952	1.83632	0.918161	+	0.396208i	$0.129674\pi$
$524$	26.9964	1.17934
$525$	−8.57952	−0.374441
$526$	1.45724	0.0635387
$527$	49.4608	2.15455
$528$	0	0
$529$	−17.9197	−0.779117
$530$	6.35346	0.275977
$531$	0.399168	0.0173224
$532$	43.5940	1.89004
$533$	−6.20968	−0.268971
$534$	18.3888	0.795759
$535$	8.74273	0.377981
$536$	24.9708	1.07858
$537$	13.2374	0.571234
$538$	−9.25746	−0.399118
$539$	0	0
$540$	7.07728	0.304558
$541$	38.3298	1.64792	0.823962	−	0.566644i	$-0.191759\pi$
$541$	38.3298	1.64792	0.823962	+	0.566644i	$0.191759\pi$
$542$	−18.1529	−0.779734
$543$	−1.14967	−0.0493370
$544$	−34.9135	−1.49690
$545$	−10.4009	−0.445527
$546$	−6.60427	−0.282637
$547$	−31.5937	−1.35085	−0.675424	−	0.737430i	$-0.736039\pi$
$547$	−31.5937	−1.35085	−0.675424	+	0.737430i	$0.736039\pi$
$548$	21.1312	0.902680
$549$	−0.757464	−0.0323278
$550$	0	0
$551$	−46.1739	−1.96707
$552$	10.5432	0.448750
$553$	32.3926	1.37748
$554$	0.212248	0.00901755
$555$	−15.7259	−0.667527
$556$	1.13929	0.0483165
$557$	31.6007	1.33896	0.669482	−	0.742828i	$-0.266516\pi$
$557$	31.6007	1.33896	0.669482	+	0.742828i	$0.266516\pi$
$558$	−1.15222	−0.0487773
$559$	−0.808116	−0.0341797
$560$	−3.82864	−0.161789
$561$	0	0
$562$	15.1664	0.639757
$563$	−12.6455	−0.532945	−0.266473	−	0.963842i	$-0.585858\pi$
$563$	−12.6455	−0.532945	−0.266473	+	0.963842i	$0.585858\pi$
$564$	2.76394	0.116383
$565$	−3.78844	−0.159381
$566$	12.3951	0.521003
$567$	−45.7445	−1.92109
$568$	11.8224	0.496058
$569$	16.5839	0.695233	0.347617	−	0.937637i	$-0.386991\pi$
$569$	16.5839	0.695233	0.347617	+	0.937637i	$0.386991\pi$
$570$	8.83826	0.370194
$571$	38.2935	1.60253	0.801266	−	0.598308i	$-0.204160\pi$
$571$	38.2935	1.60253	0.801266	+	0.598308i	$0.204160\pi$
$572$	0	0
$573$	31.1355	1.30071
$574$	−22.9962	−0.959843
$575$	−2.25396	−0.0939965
$576$	0.526275	0.0219281
$577$	21.5547	0.897333	0.448666	−	0.893699i	$-0.351899\pi$
$577$	21.5547	0.897333	0.448666	+	0.893699i	$0.351899\pi$
$578$	−14.2522	−0.592815
$579$	−8.66192	−0.359977
$580$	10.0940	0.419130
$581$	26.8802	1.11518
$582$	6.52344	0.270405
$583$	0	0
$584$	−36.2632	−1.50058
$585$	0.180351	0.00745658
$586$	5.79754	0.239494
$587$	−3.46105	−0.142853	−0.0714265	−	0.997446i	$-0.522755\pi$
$587$	−3.46105	−0.142853	−0.0714265	+	0.997446i	$0.522755\pi$
$588$	40.5229	1.67114
$589$	−53.4347	−2.20174
$590$	1.70373	0.0701413
$591$	−36.0115	−1.48132
$592$	−7.01773	−0.288427
$593$	−23.4891	−0.964580	−0.482290	−	0.876012i	$-0.660195\pi$
$593$	−23.4891	−0.964580	−0.482290	+	0.876012i	$0.660195\pi$
$594$	0	0
$595$	−28.6702	−1.17536
$596$	18.2566	0.747819
$597$	−6.55025	−0.268084
$598$	−1.73503	−0.0709507
$599$	15.3588	0.627542	0.313771	−	0.949499i	$-0.398408\pi$
$599$	15.3588	0.627542	0.313771	+	0.949499i	$0.398408\pi$
$600$	−4.67766	−0.190965
$601$	18.8174	0.767577	0.383789	−	0.923421i	$-0.374619\pi$
$601$	18.8174	0.767577	0.383789	+	0.923421i	$0.374619\pi$
$602$	−2.99269	−0.121973
$603$	1.71696	0.0699199
$604$	4.23685	0.172395
$605$	0	0
$606$	−14.7561	−0.599426
$607$	3.94448	0.160102	0.0800508	−	0.996791i	$-0.474492\pi$
$607$	3.94448	0.160102	0.0800508	+	0.996791i	$0.474492\pi$
$608$	37.7186	1.52969
$609$	−61.5308	−2.49335
$610$	−3.23300	−0.130900
$611$	−1.10118	−0.0445489
$612$	−1.51271	−0.0611478
$613$	38.3168	1.54760	0.773800	−	0.633430i	$-0.218353\pi$
$613$	38.3168	1.54760	0.773800	+	0.633430i	$0.218353\pi$
$614$	−11.5970	−0.468018
$615$	11.0741	0.446549
$616$	0	0
$617$	−36.4383	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$617$	−36.4383	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$618$	−13.6984	−0.551030
$619$	−35.2585	−1.41716	−0.708580	−	0.705631i	$-0.750664\pi$
$619$	−35.2585	−1.41716	−0.708580	+	0.705631i	$0.750664\pi$
$620$	11.6813	0.469131
$621$	−11.3339	−0.454812
$622$	−1.47029	−0.0589533
$623$	64.4435	2.58187
$624$	1.41924	0.0568151
$625$	1.00000	0.0400000
$626$	−26.0374	−1.04066
$627$	0	0
$628$	9.04891	0.361091
$629$	−52.5513	−2.09536
$630$	0.667890	0.0266094
$631$	27.3796	1.08996	0.544982	−	0.838448i	$-0.316536\pi$
$631$	27.3796	1.08996	0.544982	+	0.838448i	$0.316536\pi$
$632$	17.6609	0.702512
$633$	−26.6829	−1.06055
$634$	12.7607	0.506794
$635$	−15.4486	−0.613060
$636$	20.7167	0.821469
$637$	−16.1447	−0.639676
$638$	0	0
$639$	0.812891	0.0321575
$640$	−9.47080	−0.374366
$641$	−23.5701	−0.930964	−0.465482	−	0.885057i	$-0.654119\pi$
$641$	−23.5701	−0.930964	−0.465482	+	0.885057i	$0.654119\pi$
$642$	−12.0018	−0.473673
$643$	21.0883	0.831642	0.415821	−	0.909447i	$-0.363494\pi$
$643$	21.0883	0.831642	0.415821	+	0.909447i	$0.363494\pi$
$644$	15.2618	0.601398
$645$	1.44116	0.0567455
$646$	29.5348	1.16203
$647$	−10.0611	−0.395543	−0.197771	−	0.980248i	$-0.563370\pi$
$647$	−10.0611	−0.395543	−0.197771	+	0.980248i	$0.563370\pi$
$648$	−24.9405	−0.979754
$649$	0	0
$650$	0.769771	0.0301929
$651$	−71.2064	−2.79080
$652$	18.3613	0.719084
$653$	47.4073	1.85519	0.927595	−	0.373587i	$-0.121872\pi$
$653$	47.4073	1.85519	0.927595	+	0.373587i	$0.121872\pi$
$654$	14.2781	0.558319
$655$	19.1810	0.749464
$656$	4.94183	0.192946
$657$	−2.49340	−0.0972768
$658$	−4.07798	−0.158976
$659$	−3.04156	−0.118482	−0.0592412	−	0.998244i	$-0.518868\pi$
$659$	−3.04156	−0.118482	−0.0592412	+	0.998244i	$0.518868\pi$
$660$	0	0
$661$	6.50350	0.252957	0.126478	−	0.991969i	$-0.459633\pi$
$661$	6.50350	0.252957	0.126478	+	0.991969i	$0.459633\pi$
$662$	2.80059	0.108848
$663$	10.6278	0.412749
$664$	14.6554	0.568740
$665$	30.9737	1.20111
$666$	1.22421	0.0474373
$667$	−16.1650	−0.625910
$668$	−25.4390	−0.984266
$669$	31.3636	1.21259
$670$	7.32830	0.283117
$671$	0	0
$672$	50.2633	1.93895
$673$	−40.1794	−1.54880	−0.774400	−	0.632696i	$-0.781948\pi$
$673$	−40.1794	−1.54880	−0.774400	+	0.632696i	$0.781948\pi$
$674$	21.3491	0.822337
$675$	5.02843	0.193544
$676$	−1.40745	−0.0541328
$677$	29.1600	1.12071	0.560355	−	0.828252i	$-0.310665\pi$
$677$	29.1600	1.12071	0.560355	+	0.828252i	$0.310665\pi$
$678$	5.20068	0.199731
$679$	22.8614	0.877340
$680$	−15.6314	−0.599435
$681$	17.4381	0.668228
$682$	0	0
$683$	9.80230	0.375075	0.187537	−	0.982257i	$-0.439949\pi$
$683$	9.80230	0.375075	0.187537	+	0.982257i	$0.439949\pi$
$684$	1.63425	0.0624872
$685$	15.0138	0.573648
$686$	−33.8654	−1.29299
$687$	41.5854	1.58658
$688$	0.643121	0.0245187
$689$	−8.25370	−0.314441
$690$	3.09417	0.117793
$691$	−2.41002	−0.0916815	−0.0458407	−	0.998949i	$-0.514597\pi$
$691$	−2.41002	−0.0916815	−0.0458407	+	0.998949i	$0.514597\pi$
$692$	13.8114	0.525029
$693$	0	0
$694$	−20.8258	−0.790538
$695$	0.809467	0.0307048
$696$	−33.5473	−1.27161
$697$	37.0062	1.40171
$698$	−25.2976	−0.957529
$699$	18.2219	0.689217
$700$	−6.77110	−0.255924
$701$	−24.3491	−0.919654	−0.459827	−	0.888009i	$-0.652089\pi$
$701$	−24.3491	−0.919654	−0.459827	+	0.888009i	$0.652089\pi$
$702$	3.87074	0.146092
$703$	56.7735	2.14125
$704$	0	0
$705$	1.96379	0.0739606
$706$	5.96225	0.224392
$707$	−51.7128	−1.94486
$708$	5.55532	0.208782
$709$	49.7851	1.86972	0.934858	−	0.355021i	$-0.115526\pi$
$709$	49.7851	1.86972	0.934858	+	0.355021i	$0.115526\pi$
$710$	3.46958	0.130211
$711$	1.21433	0.0455411
$712$	35.1354	1.31675
$713$	−18.7069	−0.700578
$714$	39.3577	1.47293
$715$	0	0
$716$	10.4471	0.390428
$717$	28.1251	1.05035
$718$	21.7457	0.811541
$719$	−40.9306	−1.52645	−0.763227	−	0.646131i	$-0.776386\pi$
$719$	−40.9306	−1.52645	−0.763227	+	0.646131i	$0.776386\pi$
$720$	−0.143528	−0.00534897
$721$	−48.0061	−1.78784
$722$	−17.2821	−0.643175
$723$	−31.5603	−1.17374
$724$	−0.907338	−0.0337210
$725$	7.17182	0.266355
$726$	0	0
$727$	0.403973	0.0149825	0.00749127	−	0.999972i	$-0.497615\pi$
$727$	0.403973	0.0149825	0.00749127	+	0.999972i	$0.497615\pi$
$728$	−12.6188	−0.467683
$729$	25.1876	0.932872
$730$	−10.6423	−0.393890
$731$	4.81592	0.178123
$732$	−10.5418	−0.389637
$733$	−9.15898	−0.338295	−0.169147	−	0.985591i	$-0.554101\pi$
$733$	−9.15898	−0.338295	−0.169147	+	0.985591i	$0.554101\pi$
$734$	−9.40927	−0.347302
$735$	28.7917	1.06200
$736$	13.2049	0.486737
$737$	0	0
$738$	−0.862081	−0.0317337
$739$	−9.62485	−0.354056	−0.177028	−	0.984206i	$-0.556648\pi$
$739$	−9.62485	−0.354056	−0.177028	+	0.984206i	$0.556648\pi$
$740$	−12.4111	−0.456243
$741$	−11.4817	−0.421790
$742$	−30.5658	−1.12211
$743$	52.8702	1.93962	0.969810	−	0.243863i	$-0.0784147\pi$
$743$	52.8702	1.93962	0.969810	+	0.243863i	$0.0784147\pi$
$744$	−38.8226	−1.42331
$745$	12.9714	0.475234
$746$	−7.70763	−0.282196
$747$	1.00768	0.0368692
$748$	0	0
$749$	−42.0603	−1.53685
$750$	−1.37277	−0.0501266
$751$	21.8046	0.795660	0.397830	−	0.917459i	$-0.369763\pi$
$751$	21.8046	0.795660	0.397830	+	0.917459i	$0.369763\pi$
$752$	0.876347	0.0319571
$753$	−35.1431	−1.28069
$754$	5.52066	0.201050
$755$	3.01030	0.109556
$756$	−34.0480	−1.23831
$757$	44.3404	1.61158	0.805789	−	0.592203i	$-0.201741\pi$
$757$	44.3404	1.61158	0.805789	+	0.592203i	$0.201741\pi$
$758$	−0.359423	−0.0130548
$759$	0	0
$760$	16.8873	0.612565
$761$	−13.9311	−0.505001	−0.252500	−	0.967597i	$-0.581253\pi$
$761$	−13.9311	−0.505001	−0.252500	+	0.967597i	$0.581253\pi$
$762$	21.2075	0.768266
$763$	50.0378	1.81149
$764$	24.5727	0.889008
$765$	−1.07479	−0.0388591
$766$	−23.6435	−0.854273
$767$	−2.21329	−0.0799172
$768$	23.4092	0.844705
$769$	6.09027	0.219621	0.109810	−	0.993953i	$-0.464976\pi$
$769$	6.09027	0.219621	0.109810	+	0.993953i	$0.464976\pi$
$770$	0	0
$771$	−18.8433	−0.678626
$772$	−6.83613	−0.246038
$773$	−9.22521	−0.331808	−0.165904	−	0.986142i	$-0.553054\pi$
$773$	−9.22521	−0.331808	−0.165904	+	0.986142i	$0.553054\pi$
$774$	−0.112190	−0.00403258
$775$	8.29957	0.298130
$776$	12.4643	0.447443
$777$	75.6556	2.71413
$778$	−9.86418	−0.353648
$779$	−39.9794	−1.43241
$780$	2.50999	0.0898719
$781$	0	0
$782$	10.3398	0.369751
$783$	36.0630	1.28879
$784$	12.8484	0.458871
$785$	6.42928	0.229471
$786$	−26.3312	−0.939203
$787$	21.7516	0.775361	0.387681	−	0.921794i	$-0.373276\pi$
$787$	21.7516	0.775361	0.387681	+	0.921794i	$0.373276\pi$
$788$	−28.4209	−1.01245
$789$	3.37604	0.120190
$790$	5.18301	0.184403
$791$	18.2258	0.648034
$792$	0	0
$793$	4.19995	0.149145
$794$	−3.99911	−0.141923
$795$	14.7193	0.522039
$796$	−5.16956	−0.183230
$797$	−4.56935	−0.161855	−0.0809274	−	0.996720i	$-0.525788\pi$
$797$	−4.56935	−0.161855	−0.0809274	+	0.996720i	$0.525788\pi$
$798$	−42.5199	−1.50519
$799$	6.56240	0.232161
$800$	−5.85852	−0.207130
$801$	2.41585	0.0853600
$802$	−4.41813	−0.156010
$803$	0	0
$804$	23.8953	0.842723
$805$	10.8435	0.382185
$806$	6.38877	0.225035
$807$	−21.4471	−0.754972
$808$	−28.1945	−0.991878
$809$	20.7922	0.731013	0.365507	−	0.930809i	$-0.380896\pi$
$809$	20.7922	0.731013	0.365507	+	0.930809i	$0.380896\pi$
$810$	−7.31939	−0.257177
$811$	−25.6999	−0.902444	−0.451222	−	0.892412i	$-0.649012\pi$
$811$	−25.6999	−0.902444	−0.451222	+	0.892412i	$0.649012\pi$
$812$	−48.5611	−1.70416
$813$	−42.0554	−1.47495
$814$	0	0
$815$	13.0458	0.456974
$816$	−8.45788	−0.296085
$817$	−5.20285	−0.182025
$818$	10.6584	0.372661
$819$	−0.867648	−0.0303181
$820$	8.73982	0.305208
$821$	−21.0166	−0.733483	−0.366742	−	0.930323i	$-0.619527\pi$
$821$	−21.0166	−0.733483	−0.366742	+	0.930323i	$0.619527\pi$
$822$	−20.6106	−0.718876
$823$	−27.2129	−0.948581	−0.474291	−	0.880368i	$-0.657295\pi$
$823$	−27.2129	−0.948581	−0.474291	+	0.880368i	$0.657295\pi$
$824$	−26.1735	−0.911798
$825$	0	0
$826$	−8.19644	−0.285191
$827$	48.8435	1.69845	0.849227	−	0.528027i	$-0.177068\pi$
$827$	48.8435	1.69845	0.849227	+	0.528027i	$0.177068\pi$
$828$	0.572133	0.0198830
$829$	8.83979	0.307019	0.153509	−	0.988147i	$-0.450942\pi$
$829$	8.83979	0.307019	0.153509	+	0.988147i	$0.450942\pi$
$830$	4.30098	0.149289
$831$	0.491722	0.0170576
$832$	−2.91807	−0.101166
$833$	96.2133	3.33359
$834$	−1.11122	−0.0384783
$835$	−18.0745	−0.625495
$836$	0	0
$837$	41.7338	1.44253
$838$	13.6696	0.472209
$839$	−3.79104	−0.130881	−0.0654407	−	0.997856i	$-0.520845\pi$
$839$	−3.79104	−0.130881	−0.0654407	+	0.997856i	$0.520845\pi$
$840$	22.5037	0.776453
$841$	22.4349	0.773619
$842$	20.6184	0.710556
$843$	35.1365	1.21017
$844$	−21.0586	−0.724867
$845$	−1.00000	−0.0344010
$846$	−0.152875	−0.00525595
$847$	0	0
$848$	6.56852	0.225564
$849$	28.7160	0.985531
$850$	−4.58740	−0.157347
$851$	19.8757	0.681332
$852$	11.3132	0.387584
$853$	−22.7928	−0.780412	−0.390206	−	0.920728i	$-0.627596\pi$
$853$	−22.7928	−0.780412	−0.390206	+	0.920728i	$0.627596\pi$
$854$	15.5536	0.532234
$855$	1.16114	0.0397102
$856$	−22.9318	−0.783793
$857$	−1.99257	−0.0680649	−0.0340324	−	0.999421i	$-0.510835\pi$
$857$	−1.99257	−0.0680649	−0.0340324	+	0.999421i	$0.510835\pi$
$858$	0	0
$859$	−36.1048	−1.23188	−0.615941	−	0.787793i	$-0.711224\pi$
$859$	−36.1048	−1.23188	−0.615941	+	0.787793i	$0.711224\pi$
$860$	1.13739	0.0387845
$861$	−53.2761	−1.81564
$862$	0.976317	0.0332535
$863$	40.2320	1.36951	0.684757	−	0.728771i	$-0.259908\pi$
$863$	40.2320	1.36951	0.684757	+	0.728771i	$0.259908\pi$
$864$	−29.4592	−1.00222
$865$	9.81302	0.333653
$866$	−27.8144	−0.945170
$867$	−33.0186	−1.12137
$868$	−56.1973	−1.90746
$869$	0	0
$870$	−9.84529	−0.333786
$871$	−9.52010	−0.322576
$872$	27.2812	0.923859
$873$	0.857028	0.0290060
$874$	−11.1706	−0.377850
$875$	−4.81089	−0.162638
$876$	−34.7013	−1.17245
$877$	−24.2151	−0.817686	−0.408843	−	0.912605i	$-0.634068\pi$
$877$	−24.2151	−0.817686	−0.408843	+	0.912605i	$0.634068\pi$
$878$	−11.4946	−0.387923
$879$	13.4314	0.453028
$880$	0	0
$881$	4.32854	0.145832	0.0729162	−	0.997338i	$-0.476769\pi$
$881$	4.32854	0.145832	0.0729162	+	0.997338i	$0.476769\pi$
$882$	−2.24135	−0.0754701
$883$	5.54148	0.186486	0.0932428	−	0.995643i	$-0.470277\pi$
$883$	5.54148	0.186486	0.0932428	+	0.995643i	$0.470277\pi$
$884$	8.38762	0.282106
$885$	3.94708	0.132680
$886$	−8.29747	−0.278759
$887$	15.4014	0.517129	0.258564	−	0.965994i	$-0.416751\pi$
$887$	15.4014	0.517129	0.258564	+	0.965994i	$0.416751\pi$
$888$	41.2484	1.38420
$889$	74.3217	2.49267
$890$	10.3113	0.345637
$891$	0	0
$892$	24.7527	0.828781
$893$	−7.08965	−0.237246
$894$	−17.8068	−0.595547
$895$	7.42273	0.248115
$896$	45.5630	1.52215
$897$	−4.01960	−0.134211
$898$	−25.1294	−0.838579
$899$	59.5230	1.98520
$900$	−0.253835	−0.00846116
$901$	49.1874	1.63867
$902$	0	0
$903$	−6.93325	−0.230724
$904$	9.93693	0.330497
$905$	−0.644667	−0.0214295
$906$	−4.13246	−0.137292
$907$	−54.7316	−1.81733	−0.908666	−	0.417524i	$-0.862898\pi$
$907$	−54.7316	−1.81733	−0.908666	+	0.417524i	$0.862898\pi$
$908$	13.7624	0.456721
$909$	−1.93861	−0.0642996
$910$	−3.70329	−0.122763
$911$	−1.26545	−0.0419263	−0.0209632	−	0.999780i	$-0.506673\pi$
$911$	−1.26545	−0.0419263	−0.0209632	+	0.999780i	$0.506673\pi$
$912$	9.13742	0.302570
$913$	0	0
$914$	−28.1494	−0.931101
$915$	−7.49000	−0.247612
$916$	32.8199	1.08440
$917$	−92.2778	−3.04728
$918$	−23.0674	−0.761339
$919$	30.2491	0.997824	0.498912	−	0.866653i	$-0.333733\pi$
$919$	30.2491	0.997824	0.498912	+	0.866653i	$0.333733\pi$
$920$	5.91203	0.194914
$921$	−26.8672	−0.885304
$922$	17.2037	0.566573
$923$	−4.50728	−0.148359
$924$	0	0
$925$	−8.81816	−0.289939
$926$	10.1662	0.334082
$927$	−1.79965	−0.0591083
$928$	−42.0162	−1.37925
$929$	−54.3209	−1.78221	−0.891105	−	0.453798i	$-0.850069\pi$
$929$	−54.3209	−1.78221	−0.891105	+	0.453798i	$0.850069\pi$
$930$	−11.3934	−0.373606
$931$	−103.943	−3.40661
$932$	14.3811	0.471067
$933$	−3.40627	−0.111516
$934$	1.83017	0.0598850
$935$	0	0
$936$	−0.473052	−0.0154622
$937$	−20.4888	−0.669340	−0.334670	−	0.942335i	$-0.608625\pi$
$937$	−20.4888	−0.669340	−0.334670	+	0.942335i	$0.608625\pi$
$938$	−35.2557	−1.15114
$939$	−60.3217	−1.96852
$940$	1.54985	0.0505507
$941$	−12.9168	−0.421076	−0.210538	−	0.977586i	$-0.567522\pi$
$941$	−12.9168	−0.421076	−0.210538	+	0.977586i	$0.567522\pi$
$942$	−8.82595	−0.287565
$943$	−13.9963	−0.455784
$944$	1.76139	0.0573285
$945$	−24.1912	−0.786941
$946$	0	0
$947$	−13.1625	−0.427723	−0.213862	−	0.976864i	$-0.568604\pi$
$947$	−13.1625	−0.427723	−0.213862	+	0.976864i	$0.568604\pi$
$948$	16.9002	0.548893
$949$	13.8253	0.448788
$950$	4.95598	0.160793
$951$	29.5632	0.958653
$952$	75.2008	2.43727
$953$	−53.8137	−1.74320	−0.871598	−	0.490222i	$-0.836916\pi$
$953$	−53.8137	−1.74320	−0.871598	+	0.490222i	$0.836916\pi$
$954$	−1.14585	−0.0370983
$955$	17.4590	0.564959
$956$	22.1968	0.717897
$957$	0	0
$958$	25.7054	0.830502
$959$	−72.2298	−2.33242
$960$	5.20395	0.167957
$961$	37.8829	1.22203
$962$	−6.78796	−0.218853
$963$	−1.57676	−0.0508103
$964$	−24.9079	−0.802229
$965$	−4.85709	−0.156355
$966$	−14.8857	−0.478941
$967$	−18.8913	−0.607503	−0.303751	−	0.952751i	$-0.598239\pi$
$967$	−18.8913	−0.607503	−0.303751	+	0.952751i	$0.598239\pi$
$968$	0	0
$969$	68.4243	2.19810
$970$	3.65796	0.117450
$971$	27.1116	0.870054	0.435027	−	0.900417i	$-0.356739\pi$
$971$	27.1116	0.870054	0.435027	+	0.900417i	$0.356739\pi$
$972$	−2.63442	−0.0844992
$973$	−3.89426	−0.124844
$974$	−19.9690	−0.639849
$975$	1.78335	0.0571130
$976$	−3.34243	−0.106989
$977$	1.49840	0.0479379	0.0239690	−	0.999713i	$-0.492370\pi$
$977$	1.49840	0.0479379	0.0239690	+	0.999713i	$0.492370\pi$
$978$	−17.9089	−0.572664
$979$	0	0
$980$	22.7229	0.725856
$981$	1.87582	0.0598902
$982$	16.8957	0.539163
$983$	11.5045	0.366936	0.183468	−	0.983026i	$-0.441268\pi$
$983$	11.5045	0.366936	0.183468	+	0.983026i	$0.441268\pi$
$984$	−29.0468	−0.925977
$985$	−20.1931	−0.643407
$986$	−32.9000	−1.04775
$987$	−9.44758	−0.300720
$988$	−9.06152	−0.288285
$989$	−1.82146	−0.0579190
$990$	0	0
$991$	−40.3643	−1.28221	−0.641107	−	0.767452i	$-0.721525\pi$
$991$	−40.3643	−1.28221	−0.641107	+	0.767452i	$0.721525\pi$
$992$	−48.6232	−1.54379
$993$	6.48821	0.205897
$994$	−16.6918	−0.529431
$995$	−3.67299	−0.116442
$996$	14.0242	0.444373
$997$	36.1786	1.14579	0.572893	−	0.819630i	$-0.305821\pi$
$997$	36.1786	1.14579	0.572893	+	0.819630i	$0.305821\pi$
$998$	−18.6751	−0.591150
$999$	−44.3415	−1.40290

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7865.2.a.u.1.3	yes	8
11.10	odd	2		7865.2.a.t.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7865.2.a.t.1.6	✓	8	11.10	odd	2
7865.2.a.u.1.3	yes	8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7865.a (trivial)

\(f(q)\)	\(=\)	\(q-0.769771 q^{2} -1.78335 q^{3} -1.40745 q^{4} -1.00000 q^{5} +1.37277 q^{6} +4.81089 q^{7} +2.62296 q^{8} +0.180351 q^{9} +0.769771 q^{10} +2.50999 q^{12} -1.00000 q^{13} -3.70329 q^{14} +1.78335 q^{15} +0.795827 q^{16} +5.95944 q^{17} -0.138829 q^{18} -6.43825 q^{19} +1.40745 q^{20} -8.57952 q^{21} -2.25396 q^{23} -4.67766 q^{24} +1.00000 q^{25} +0.769771 q^{26} +5.02843 q^{27} -6.77110 q^{28} +7.17182 q^{29} -1.37277 q^{30} +8.29957 q^{31} -5.85852 q^{32} -4.58740 q^{34} -4.81089 q^{35} -0.253835 q^{36} -8.81816 q^{37} +4.95598 q^{38} +1.78335 q^{39} -2.62296 q^{40} +6.20968 q^{41} +6.60427 q^{42} +0.808116 q^{43} -0.180351 q^{45} +1.73503 q^{46} +1.10118 q^{47} -1.41924 q^{48} +16.1447 q^{49} -0.769771 q^{50} -10.6278 q^{51} +1.40745 q^{52} +8.25370 q^{53} -3.87074 q^{54} +12.6188 q^{56} +11.4817 q^{57} -5.52066 q^{58} +2.21329 q^{59} -2.50999 q^{60} -4.19995 q^{61} -6.38877 q^{62} +0.867648 q^{63} +2.91807 q^{64} +1.00000 q^{65} +9.52010 q^{67} -8.38762 q^{68} +4.01960 q^{69} +3.70329 q^{70} +4.50728 q^{71} +0.473052 q^{72} -13.8253 q^{73} +6.78796 q^{74} -1.78335 q^{75} +9.06152 q^{76} -1.37277 q^{78} +6.73319 q^{79} -0.795827 q^{80} -9.50853 q^{81} -4.78003 q^{82} +5.58735 q^{83} +12.0753 q^{84} -5.95944 q^{85} -0.622065 q^{86} -12.7899 q^{87} +13.3953 q^{89} +0.138829 q^{90} -4.81089 q^{91} +3.17234 q^{92} -14.8011 q^{93} -0.847655 q^{94} +6.43825 q^{95} +10.4478 q^{96} +4.75201 q^{97} -12.4277 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 5 q^{3} + 7 q^{4} - 8 q^{5} - 7 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} - 15 q^{12} - 8 q^{13} - 7 q^{14} + 5 q^{15} + 9 q^{16} + 4 q^{17} + 8 q^{18} + 16 q^{19} - 7 q^{20} - 23 q^{21}+ \cdots + 41 q^{98}+O(q^{100}) \) 8 * q + q^2 - 5 * q^3 + 7 * q^4 - 8 * q^5 - 7 * q^6 + 9 * q^7 + 3 * q^8 + 3 * q^9 - q^10 - 15 * q^12 - 8 * q^13 - 7 * q^14 + 5 * q^15 + 9 * q^16 + 4 * q^17 + 8 * q^18 + 16 * q^19 - 7 * q^20 - 23 * q^21 - 9 * q^23 + 5 * q^24 + 8 * q^25 - q^26 - 11 * q^27 + 11 * q^28 - 13 * q^29 + 7 * q^30 - 28 * q^32 + 13 * q^34 - 9 * q^35 + 12 * q^36 - 5 * q^37 + 55 * q^38 + 5 * q^39 - 3 * q^40 + 13 * q^41 - 5 * q^42 + 5 * q^43 - 3 * q^45 + 27 * q^46 + 10 * q^47 - 36 * q^48 + 23 * q^49 + q^50 - 37 * q^51 - 7 * q^52 + 14 * q^53 - 45 * q^54 - 18 * q^56 - 21 * q^57 + 3 * q^58 - 20 * q^59 + 15 * q^60 + 6 * q^61 + 25 * q^62 + 26 * q^63 - 3 * q^64 + 8 * q^65 + 4 * q^67 - 9 * q^69 + 7 * q^70 - 9 * q^71 + 35 * q^72 + 10 * q^73 + 42 * q^74 - 5 * q^75 + 19 * q^76 + 7 * q^78 + 13 * q^79 - 9 * q^80 + 16 * q^82 + 26 * q^83 - 58 * q^84 - 4 * q^85 + 26 * q^86 + 11 * q^87 - 6 * q^89 - 8 * q^90 - 9 * q^91 - 16 * q^92 + 3 * q^93 + 23 * q^94 - 16 * q^95 + 21 * q^96 + 26 * q^97 + 41 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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