LMFDB - Embedded newform 8330.2.a.cf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, 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("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,-2,4,4,-2,0,4,6,4,0,-2,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.87996$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.87996 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.87996 q^{6} +1.00000 q^{8} +0.534253 q^{9} +1.00000 q^{10} -2.10126 q^{11} +1.87996 q^{12} -7.07288 q^{13} +1.87996 q^{15} +1.00000 q^{16} -1.00000 q^{17} +0.534253 q^{18} -1.77871 q^{19} +1.00000 q^{20} -2.10126 q^{22} -7.53863 q^{23} +1.87996 q^{24} +1.00000 q^{25} -7.07288 q^{26} -4.63551 q^{27} +1.24445 q^{29} +1.87996 q^{30} +1.24445 q^{31} +1.00000 q^{32} -3.95028 q^{33} -1.00000 q^{34} +0.534253 q^{36} +1.85158 q^{37} -1.77871 q^{38} -13.2967 q^{39} +1.00000 q^{40} -4.60713 q^{41} -3.55741 q^{43} -2.10126 q^{44} +0.534253 q^{45} -7.53863 q^{46} +11.2684 q^{47} +1.87996 q^{48} +1.00000 q^{50} -1.87996 q^{51} -7.07288 q^{52} +11.5582 q^{53} -4.63551 q^{54} -2.10126 q^{55} -3.34390 q^{57} +1.24445 q^{58} +6.48891 q^{59} +1.87996 q^{60} -8.90131 q^{61} +1.24445 q^{62} +1.00000 q^{64} -7.07288 q^{65} -3.95028 q^{66} +1.27102 q^{67} -1.00000 q^{68} -14.1723 q^{69} -14.0986 q^{71} +0.534253 q^{72} -2.75555 q^{73} +1.85158 q^{74} +1.87996 q^{75} -1.77871 q^{76} -13.2967 q^{78} -1.98303 q^{79} +1.00000 q^{80} -10.3173 q^{81} -4.60713 q^{82} -17.4897 q^{83} -1.00000 q^{85} -3.55741 q^{86} +2.33952 q^{87} -2.10126 q^{88} +4.12260 q^{89} +0.534253 q^{90} -7.53863 q^{92} +2.33952 q^{93} +11.2684 q^{94} -1.77871 q^{95} +1.87996 q^{96} -6.93149 q^{97} -1.12260 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{8} + 6 q^{9} + 4 q^{10} - 2 q^{12} - 12 q^{13} - 2 q^{15} + 4 q^{16} - 4 q^{17} + 6 q^{18} - 6 q^{19} + 4 q^{20} - 10 q^{23} - 2 q^{24} + 4 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^8 + 6 * q^9 + 4 * q^10 - 2 * q^12 - 12 * q^13 - 2 * q^15 + 4 * q^16 - 4 * q^17 + 6 * q^18 - 6 * q^19 + 4 * q^20 - 10 * q^23 - 2 * q^24 + 4 * q^25 - 12 * q^26 - 14 * q^27 - 2 * q^30 + 4 * q^32 - 26 * q^33 - 4 * q^34 + 6 * q^36 - 10 * q^37 - 6 * q^38 - 2 * q^39 + 4 * q^40 - 6 * q^41 - 12 * q^43 + 6 * q^45 - 10 * q^46 - 14 * q^47 - 2 * q^48 + 4 * q^50 + 2 * q^51 - 12 * q^52 - 4 * q^53 - 14 * q^54 + 12 * q^57 + 16 * q^59 - 2 * q^60 - 8 * q^61 + 4 * q^64 - 12 * q^65 - 26 * q^66 - 4 * q^67 - 4 * q^68 - 20 * q^69 + 2 * q^71 + 6 * q^72 - 16 * q^73 - 10 * q^74 - 2 * q^75 - 6 * q^76 - 2 * q^78 - 2 * q^79 + 4 * q^80 - 20 * q^81 - 6 * q^82 - 12 * q^83 - 4 * q^85 - 12 * q^86 + 8 * q^87 - 10 * q^89 + 6 * q^90 - 10 * q^92 + 8 * q^93 - 14 * q^94 - 6 * q^95 - 2 * q^96 - 20 * q^97 + 22 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.87996	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$3$	1.87996	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.87996	0.767491
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0.534253	0.178084
$10$	1.00000	0.316228
$11$	−2.10126	−0.633552	−0.316776	−	0.948500i	$-0.602600\pi$
$11$	−2.10126	−0.633552	−0.316776	+	0.948500i	$0.602600\pi$
$12$	1.87996	0.542698
$13$	−7.07288	−1.96166	−0.980832	−	0.194856i	$-0.937576\pi$
$13$	−7.07288	−1.96166	−0.980832	+	0.194856i	$0.937576\pi$
$14$	0	0
$15$	1.87996	0.485404
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.534253	0.125925
$19$	−1.77871	−0.408063	−0.204031	−	0.978964i	$-0.565404\pi$
$19$	−1.77871	−0.408063	−0.204031	+	0.978964i	$0.565404\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.10126	−0.447989
$23$	−7.53863	−1.57191	−0.785956	−	0.618282i	$-0.787829\pi$
$23$	−7.53863	−1.57191	−0.785956	+	0.618282i	$0.787829\pi$
$24$	1.87996	0.383745
$25$	1.00000	0.200000
$26$	−7.07288	−1.38711
$27$	−4.63551	−0.892104
$28$	0	0
$29$	1.24445	0.231089	0.115545	−	0.993302i	$-0.463139\pi$
$29$	1.24445	0.231089	0.115545	+	0.993302i	$0.463139\pi$
$30$	1.87996	0.343232
$31$	1.24445	0.223510	0.111755	−	0.993736i	$-0.464353\pi$
$31$	1.24445	0.223510	0.111755	+	0.993736i	$0.464353\pi$
$32$	1.00000	0.176777
$33$	−3.95028	−0.687655
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0.534253	0.0890421
$37$	1.85158	0.304399	0.152199	−	0.988350i	$-0.451364\pi$
$37$	1.85158	0.304399	0.152199	+	0.988350i	$0.451364\pi$
$38$	−1.77871	−0.288544
$39$	−13.2967	−2.12918
$40$	1.00000	0.158114
$41$	−4.60713	−0.719513	−0.359757	−	0.933046i	$-0.617140\pi$
$41$	−4.60713	−0.719513	−0.359757	+	0.933046i	$0.617140\pi$
$42$	0	0
$43$	−3.55741	−0.542500	−0.271250	−	0.962509i	$-0.587437\pi$
$43$	−3.55741	−0.542500	−0.271250	+	0.962509i	$0.587437\pi$
$44$	−2.10126	−0.316776
$45$	0.534253	0.0796417
$46$	−7.53863	−1.11151
$47$	11.2684	1.64366	0.821830	−	0.569733i	$-0.192953\pi$
$47$	11.2684	1.64366	0.821830	+	0.569733i	$0.192953\pi$
$48$	1.87996	0.271349
$49$	0	0
$50$	1.00000	0.141421
$51$	−1.87996	−0.263247
$52$	−7.07288	−0.980832
$53$	11.5582	1.58764	0.793818	−	0.608156i	$-0.208090\pi$
$53$	11.5582	1.58764	0.793818	+	0.608156i	$0.208090\pi$
$54$	−4.63551	−0.630813
$55$	−2.10126	−0.283333
$56$	0	0
$57$	−3.34390	−0.442910
$58$	1.24445	0.163405
$59$	6.48891	0.844783	0.422392	−	0.906413i	$-0.361191\pi$
$59$	6.48891	0.844783	0.422392	+	0.906413i	$0.361191\pi$
$60$	1.87996	0.242702
$61$	−8.90131	−1.13970	−0.569848	−	0.821750i	$-0.692998\pi$
$61$	−8.90131	−1.13970	−0.569848	+	0.821750i	$0.692998\pi$
$62$	1.24445	0.158046
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.07288	−0.877283
$66$	−3.95028	−0.486246
$67$	1.27102	0.155279	0.0776397	−	0.996981i	$-0.475262\pi$
$67$	1.27102	0.155279	0.0776397	+	0.996981i	$0.475262\pi$
$68$	−1.00000	−0.121268
$69$	−14.1723	−1.70615
$70$	0	0
$71$	−14.0986	−1.67320	−0.836598	−	0.547817i	$-0.815459\pi$
$71$	−14.0986	−1.67320	−0.836598	+	0.547817i	$0.815459\pi$
$72$	0.534253	0.0629623
$73$	−2.75555	−0.322512	−0.161256	−	0.986913i	$-0.551555\pi$
$73$	−2.75555	−0.322512	−0.161256	+	0.986913i	$0.551555\pi$
$74$	1.85158	0.215242
$75$	1.87996	0.217079
$76$	−1.77871	−0.204031
$77$	0	0
$78$	−13.2967	−1.50556
$79$	−1.98303	−0.223108	−0.111554	−	0.993758i	$-0.535583\pi$
$79$	−1.98303	−0.223108	−0.111554	+	0.993758i	$0.535583\pi$
$80$	1.00000	0.111803
$81$	−10.3173	−1.14637
$82$	−4.60713	−0.508773
$83$	−17.4897	−1.91974	−0.959870	−	0.280447i	$-0.909517\pi$
$83$	−17.4897	−1.91974	−0.959870	+	0.280447i	$0.909517\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−3.55741	−0.383605
$87$	2.33952	0.250823
$88$	−2.10126	−0.223995
$89$	4.12260	0.436995	0.218497	−	0.975838i	$-0.429884\pi$
$89$	4.12260	0.436995	0.218497	+	0.975838i	$0.429884\pi$
$90$	0.534253	0.0563152
$91$	0	0
$92$	−7.53863	−0.785956
$93$	2.33952	0.242597
$94$	11.2684	1.16224
$95$	−1.77871	−0.182491
$96$	1.87996	0.191873
$97$	−6.93149	−0.703787	−0.351893	−	0.936040i	$-0.614462\pi$
$97$	−6.93149	−0.703787	−0.351893	+	0.936040i	$0.614462\pi$
$98$	0	0
$99$	−1.12260	−0.112826
$100$	1.00000	0.100000
$101$	−7.73336	−0.769498	−0.384749	−	0.923021i	$-0.625712\pi$
$101$	−7.73336	−0.769498	−0.384749	+	0.923021i	$0.625712\pi$
$102$	−1.87996	−0.186144
$103$	4.97419	0.490121	0.245061	−	0.969508i	$-0.421192\pi$
$103$	4.97419	0.490121	0.245061	+	0.969508i	$0.421192\pi$
$104$	−7.07288	−0.693553
$105$	0	0
$106$	11.5582	1.12263
$107$	−5.80708	−0.561392	−0.280696	−	0.959797i	$-0.590565\pi$
$107$	−5.80708	−0.561392	−0.280696	+	0.959797i	$0.590565\pi$
$108$	−4.63551	−0.446052
$109$	2.48891	0.238394	0.119197	−	0.992871i	$-0.461968\pi$
$109$	2.48891	0.238394	0.119197	+	0.992871i	$0.461968\pi$
$110$	−2.10126	−0.200347
$111$	3.48091	0.330393
$112$	0	0
$113$	−15.3137	−1.44059	−0.720296	−	0.693667i	$-0.755994\pi$
$113$	−15.3137	−1.44059	−0.720296	+	0.693667i	$0.755994\pi$
$114$	−3.34390	−0.313185
$115$	−7.53863	−0.702981
$116$	1.24445	0.115545
$117$	−3.77871	−0.349341
$118$	6.48891	0.597352
$119$	0	0
$120$	1.87996	0.171616
$121$	−6.58473	−0.598611
$122$	−8.90131	−0.805886
$123$	−8.66123	−0.780957
$124$	1.24445	0.111755
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.97684	0.264152	0.132076	−	0.991240i	$-0.457836\pi$
$127$	2.97684	0.264152	0.132076	+	0.991240i	$0.457836\pi$
$128$	1.00000	0.0883883
$129$	−6.68779	−0.588827
$130$	−7.07288	−0.620333
$131$	7.22567	0.631310	0.315655	−	0.948874i	$-0.397776\pi$
$131$	7.22567	0.631310	0.315655	+	0.948874i	$0.397776\pi$
$132$	−3.95028	−0.343828
$133$	0	0
$134$	1.27102	0.109799
$135$	−4.63551	−0.398961
$136$	−1.00000	−0.0857493
$137$	−3.81702	−0.326110	−0.163055	−	0.986617i	$-0.552135\pi$
$137$	−3.81702	−0.326110	−0.163055	+	0.986617i	$0.552135\pi$
$138$	−14.1723	−1.20643
$139$	−2.05591	−0.174380	−0.0871899	−	0.996192i	$-0.527789\pi$
$139$	−2.05591	−0.174380	−0.0871899	+	0.996192i	$0.527789\pi$
$140$	0	0
$141$	21.1841	1.78402
$142$	−14.0986	−1.18313
$143$	14.8619	1.24282
$144$	0.534253	0.0445211
$145$	1.24445	0.103346
$146$	−2.75555	−0.228051
$147$	0	0
$148$	1.85158	0.152199
$149$	−7.12698	−0.583865	−0.291932	−	0.956439i	$-0.594298\pi$
$149$	−7.12698	−0.583865	−0.291932	+	0.956439i	$0.594298\pi$
$150$	1.87996	0.153498
$151$	13.7297	1.11731	0.558655	−	0.829400i	$-0.311318\pi$
$151$	13.7297	1.11731	0.558655	+	0.829400i	$0.311318\pi$
$152$	−1.77871	−0.144272
$153$	−0.534253	−0.0431918
$154$	0	0
$155$	1.24445	0.0999568
$156$	−13.2967	−1.06459
$157$	−9.71798	−0.775579	−0.387790	−	0.921748i	$-0.626761\pi$
$157$	−9.71798	−0.775579	−0.387790	+	0.921748i	$0.626761\pi$
$158$	−1.98303	−0.157761
$159$	21.7289	1.72321
$160$	1.00000	0.0790569
$161$	0	0
$162$	−10.3173	−0.810606
$163$	10.4587	0.819190	0.409595	−	0.912267i	$-0.365670\pi$
$163$	10.4587	0.819190	0.409595	+	0.912267i	$0.365670\pi$
$164$	−4.60713	−0.359757
$165$	−3.95028	−0.307529
$166$	−17.4897	−1.35746
$167$	−21.9304	−1.69703	−0.848514	−	0.529173i	$-0.822502\pi$
$167$	−21.9304	−1.69703	−0.848514	+	0.529173i	$0.822502\pi$
$168$	0	0
$169$	37.0256	2.84813
$170$	−1.00000	−0.0766965
$171$	−0.950278	−0.0726696
$172$	−3.55741	−0.271250
$173$	−16.7831	−1.27599	−0.637997	−	0.770039i	$-0.720237\pi$
$173$	−16.7831	−1.27599	−0.637997	+	0.770039i	$0.720237\pi$
$174$	2.33952	0.177359
$175$	0	0
$176$	−2.10126	−0.158388
$177$	12.1989	0.916925
$178$	4.12260	0.309002
$179$	18.7153	1.39885	0.699425	−	0.714706i	$-0.253440\pi$
$179$	18.7153	1.39885	0.699425	+	0.714706i	$0.253440\pi$
$180$	0.534253	0.0398208
$181$	19.7724	1.46967	0.734836	−	0.678244i	$-0.237259\pi$
$181$	19.7724	1.46967	0.734836	+	0.678244i	$0.237259\pi$
$182$	0	0
$183$	−16.7341	−1.23702
$184$	−7.53863	−0.555755
$185$	1.85158	0.136131
$186$	2.33952	0.171542
$187$	2.10126	0.153659
$188$	11.2684	0.821830
$189$	0	0
$190$	−1.77871	−0.129041
$191$	14.4124	1.04284	0.521422	−	0.853299i	$-0.325402\pi$
$191$	14.4124	1.04284	0.521422	+	0.853299i	$0.325402\pi$
$192$	1.87996	0.135674
$193$	0.977810	0.0703843	0.0351922	−	0.999381i	$-0.488796\pi$
$193$	0.977810	0.0703843	0.0351922	+	0.999381i	$0.488796\pi$
$194$	−6.93149	−0.497652
$195$	−13.2967	−0.952199
$196$	0	0
$197$	−0.294174	−0.0209591	−0.0104795	−	0.999945i	$-0.503336\pi$
$197$	−0.294174	−0.0209591	−0.0104795	+	0.999945i	$0.503336\pi$
$198$	−1.12260	−0.0797798
$199$	−14.9572	−1.06029	−0.530144	−	0.847907i	$-0.677862\pi$
$199$	−14.9572	−1.06029	−0.530144	+	0.847907i	$0.677862\pi$
$200$	1.00000	0.0707107
$201$	2.38946	0.168540
$202$	−7.73336	−0.544117
$203$	0	0
$204$	−1.87996	−0.131624
$205$	−4.60713	−0.321776
$206$	4.97419	0.346568
$207$	−4.02753	−0.279933
$208$	−7.07288	−0.490416
$209$	3.73751	0.258529
$210$	0	0
$211$	−8.47750	−0.583615	−0.291808	−	0.956477i	$-0.594257\pi$
$211$	−8.47750	−0.583615	−0.291808	+	0.956477i	$0.594257\pi$
$212$	11.5582	0.793818
$213$	−26.5048	−1.81608
$214$	−5.80708	−0.396964
$215$	−3.55741	−0.242613
$216$	−4.63551	−0.315406
$217$	0	0
$218$	2.48891	0.168570
$219$	−5.18032	−0.350054
$220$	−2.10126	−0.141667
$221$	7.07288	0.475773
$222$	3.48091	0.233623
$223$	17.2143	1.15275	0.576376	−	0.817184i	$-0.304466\pi$
$223$	17.2143	1.15275	0.576376	+	0.817184i	$0.304466\pi$
$224$	0	0
$225$	0.534253	0.0356168
$226$	−15.3137	−1.01865
$227$	−2.05228	−0.136215	−0.0681074	−	0.997678i	$-0.521696\pi$
$227$	−2.05228	−0.136215	−0.0681074	+	0.997678i	$0.521696\pi$
$228$	−3.34390	−0.221455
$229$	−12.1921	−0.805675	−0.402837	−	0.915272i	$-0.631976\pi$
$229$	−12.1921	−0.805675	−0.402837	+	0.915272i	$0.631976\pi$
$230$	−7.53863	−0.497082
$231$	0	0
$232$	1.24445	0.0817023
$233$	3.64947	0.239085	0.119543	−	0.992829i	$-0.461857\pi$
$233$	3.64947	0.239085	0.119543	+	0.992829i	$0.461857\pi$
$234$	−3.77871	−0.247022
$235$	11.2684	0.735067
$236$	6.48891	0.422392
$237$	−3.72802	−0.242161
$238$	0	0
$239$	25.0860	1.62268	0.811339	−	0.584576i	$-0.198739\pi$
$239$	25.0860	1.62268	0.811339	+	0.584576i	$0.198739\pi$
$240$	1.87996	0.121351
$241$	−12.9040	−0.831217	−0.415609	−	0.909544i	$-0.636431\pi$
$241$	−12.9040	−0.831217	−0.415609	+	0.909544i	$0.636431\pi$
$242$	−6.58473	−0.423282
$243$	−5.48966	−0.352162
$244$	−8.90131	−0.569848
$245$	0	0
$246$	−8.66123	−0.552220
$247$	12.5806	0.800482
$248$	1.24445	0.0790228
$249$	−32.8799	−2.08368
$250$	1.00000	0.0632456
$251$	−17.4443	−1.10107	−0.550537	−	0.834810i	$-0.685577\pi$
$251$	−17.4443	−1.10107	−0.550537	+	0.834810i	$0.685577\pi$
$252$	0	0
$253$	15.8406	0.995889
$254$	2.97684	0.186784
$255$	−1.87996	−0.117728
$256$	1.00000	0.0625000
$257$	−11.5128	−0.718150	−0.359075	−	0.933309i	$-0.616908\pi$
$257$	−11.5128	−0.718150	−0.359075	+	0.933309i	$0.616908\pi$
$258$	−6.68779	−0.416364
$259$	0	0
$260$	−7.07288	−0.438641
$261$	0.664852	0.0411533
$262$	7.22567	0.446403
$263$	−27.6620	−1.70571	−0.852855	−	0.522147i	$-0.825131\pi$
$263$	−27.6620	−1.70571	−0.852855	+	0.522147i	$0.825131\pi$
$264$	−3.95028	−0.243123
$265$	11.5582	0.710012
$266$	0	0
$267$	7.75033	0.474312
$268$	1.27102	0.0776397
$269$	26.8034	1.63423	0.817115	−	0.576475i	$-0.195572\pi$
$269$	26.8034	1.63423	0.817115	+	0.576475i	$0.195572\pi$
$270$	−4.63551	−0.282108
$271$	−23.1841	−1.40833	−0.704166	−	0.710035i	$-0.748679\pi$
$271$	−23.1841	−1.40833	−0.704166	+	0.710035i	$0.748679\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−3.81702	−0.230595
$275$	−2.10126	−0.126710
$276$	−14.1723	−0.853074
$277$	4.10819	0.246837	0.123419	−	0.992355i	$-0.460614\pi$
$277$	4.10819	0.246837	0.123419	+	0.992355i	$0.460614\pi$
$278$	−2.05591	−0.123305
$279$	0.664852	0.0398037
$280$	0	0
$281$	30.7047	1.83169	0.915844	−	0.401535i	$-0.131523\pi$
$281$	30.7047	1.83169	0.915844	+	0.401535i	$0.131523\pi$
$282$	21.1841	1.26149
$283$	−1.26880	−0.0754226	−0.0377113	−	0.999289i	$-0.512007\pi$
$283$	−1.26880	−0.0754226	−0.0377113	+	0.999289i	$0.512007\pi$
$284$	−14.0986	−0.836598
$285$	−3.34390	−0.198075
$286$	14.8619	0.878804
$287$	0	0
$288$	0.534253	0.0314811
$289$	1.00000	0.0588235
$290$	1.24445	0.0730768
$291$	−13.0309	−0.763887
$292$	−2.75555	−0.161256
$293$	3.86374	0.225722	0.112861	−	0.993611i	$-0.463999\pi$
$293$	3.86374	0.225722	0.112861	+	0.993611i	$0.463999\pi$
$294$	0	0
$295$	6.48891	0.377799
$296$	1.85158	0.107621
$297$	9.74039	0.565195
$298$	−7.12698	−0.412855
$299$	53.3198	3.08356
$300$	1.87996	0.108540
$301$	0	0
$302$	13.7297	0.790057
$303$	−14.5384	−0.835210
$304$	−1.77871	−0.102016
$305$	−8.90131	−0.509687
$306$	−0.534253	−0.0305412
$307$	−22.6914	−1.29507	−0.647534	−	0.762037i	$-0.724200\pi$
$307$	−22.6914	−1.29507	−0.647534	+	0.762037i	$0.724200\pi$
$308$	0	0
$309$	9.35128	0.531975
$310$	1.24445	0.0706802
$311$	9.71276	0.550760	0.275380	−	0.961335i	$-0.411196\pi$
$311$	9.71276	0.550760	0.275380	+	0.961335i	$0.411196\pi$
$312$	−13.2967	−0.752779
$313$	23.3386	1.31917	0.659587	−	0.751628i	$-0.270731\pi$
$313$	23.3386	1.31917	0.659587	+	0.751628i	$0.270731\pi$
$314$	−9.71798	−0.548417
$315$	0	0
$316$	−1.98303	−0.111554
$317$	2.89393	0.162539	0.0812696	−	0.996692i	$-0.474103\pi$
$317$	2.89393	0.162539	0.0812696	+	0.996692i	$0.474103\pi$
$318$	21.7289	1.21850
$319$	−2.61491	−0.146407
$320$	1.00000	0.0559017
$321$	−10.9171	−0.609332
$322$	0	0
$323$	1.77871	0.0989698
$324$	−10.3173	−0.573185
$325$	−7.07288	−0.392333
$326$	10.4587	0.579255
$327$	4.67904	0.258752
$328$	−4.60713	−0.254386
$329$	0	0
$330$	−3.95028	−0.217456
$331$	19.6381	1.07941	0.539703	−	0.841855i	$-0.318537\pi$
$331$	19.6381	1.07941	0.539703	+	0.841855i	$0.318537\pi$
$332$	−17.4897	−0.959870
$333$	0.989214	0.0542086
$334$	−21.9304	−1.19998
$335$	1.27102	0.0694431
$336$	0	0
$337$	22.9846	1.25205	0.626026	−	0.779802i	$-0.284680\pi$
$337$	22.9846	1.25205	0.626026	+	0.779802i	$0.284680\pi$
$338$	37.0256	2.01393
$339$	−28.7892	−1.56361
$340$	−1.00000	−0.0542326
$341$	−2.61491	−0.141605
$342$	−0.950278	−0.0513852
$343$	0	0
$344$	−3.55741	−0.191803
$345$	−14.1723	−0.763012
$346$	−16.7831	−0.902264
$347$	27.3010	1.46560	0.732798	−	0.680447i	$-0.238214\pi$
$347$	27.3010	1.46560	0.732798	+	0.680447i	$0.238214\pi$
$348$	2.33952	0.125412
$349$	−9.37408	−0.501783	−0.250892	−	0.968015i	$-0.580724\pi$
$349$	−9.37408	−0.501783	−0.250892	+	0.968015i	$0.580724\pi$
$350$	0	0
$351$	32.7864	1.75001
$352$	−2.10126	−0.111997
$353$	34.1203	1.81604	0.908021	−	0.418924i	$-0.137593\pi$
$353$	34.1203	1.81604	0.908021	+	0.418924i	$0.137593\pi$
$354$	12.1989	0.648364
$355$	−14.0986	−0.748276
$356$	4.12260	0.218497
$357$	0	0
$358$	18.7153	0.989136
$359$	1.95368	0.103112	0.0515558	−	0.998670i	$-0.483582\pi$
$359$	1.95368	0.103112	0.0515558	+	0.998670i	$0.483582\pi$
$360$	0.534253	0.0281576
$361$	−15.8362	−0.833485
$362$	19.7724	1.03922
$363$	−12.3790	−0.649730
$364$	0	0
$365$	−2.75555	−0.144232
$366$	−16.7341	−0.874706
$367$	8.31115	0.433838	0.216919	−	0.976190i	$-0.430399\pi$
$367$	8.31115	0.433838	0.216919	+	0.976190i	$0.430399\pi$
$368$	−7.53863	−0.392978
$369$	−2.46137	−0.128134
$370$	1.85158	0.0962593
$371$	0	0
$372$	2.33952	0.121299
$373$	32.0850	1.66130	0.830650	−	0.556795i	$-0.187969\pi$
$373$	32.0850	1.66130	0.830650	+	0.556795i	$0.187969\pi$
$374$	2.10126	0.108653
$375$	1.87996	0.0970808
$376$	11.2684	0.581121
$377$	−8.80186	−0.453319
$378$	0	0
$379$	0.195575	0.0100460	0.00502300	−	0.999987i	$-0.498401\pi$
$379$	0.195575	0.0100460	0.00502300	+	0.999987i	$0.498401\pi$
$380$	−1.77871	−0.0912457
$381$	5.59635	0.286710
$382$	14.4124	0.737403
$383$	−16.7603	−0.856410	−0.428205	−	0.903682i	$-0.640854\pi$
$383$	−16.7603	−0.856410	−0.428205	+	0.903682i	$0.640854\pi$
$384$	1.87996	0.0959363
$385$	0	0
$386$	0.977810	0.0497692
$387$	−1.90056	−0.0966107
$388$	−6.93149	−0.351893
$389$	−8.58132	−0.435090	−0.217545	−	0.976050i	$-0.569805\pi$
$389$	−8.58132	−0.435090	−0.217545	+	0.976050i	$0.569805\pi$
$390$	−13.2967	−0.673306
$391$	7.53863	0.381245
$392$	0	0
$393$	13.5840	0.685221
$394$	−0.294174	−0.0148203
$395$	−1.98303	−0.0997770
$396$	−1.12260	−0.0564128
$397$	−26.9793	−1.35405	−0.677026	−	0.735959i	$-0.736732\pi$
$397$	−26.9793	−1.35405	−0.677026	+	0.735959i	$0.736732\pi$
$398$	−14.9572	−0.749737
$399$	0	0
$400$	1.00000	0.0500000
$401$	27.9296	1.39474	0.697369	−	0.716713i	$-0.254354\pi$
$401$	27.9296	1.39474	0.697369	+	0.716713i	$0.254354\pi$
$402$	2.38946	0.119176
$403$	−8.80186	−0.438452
$404$	−7.73336	−0.384749
$405$	−10.3173	−0.512672
$406$	0	0
$407$	−3.89065	−0.192853
$408$	−1.87996	−0.0930719
$409$	−34.8618	−1.72381	−0.861903	−	0.507073i	$-0.830728\pi$
$409$	−34.8618	−1.72381	−0.861903	+	0.507073i	$0.830728\pi$
$410$	−4.60713	−0.227530
$411$	−7.17585	−0.353959
$412$	4.97419	0.245061
$413$	0	0
$414$	−4.02753	−0.197942
$415$	−17.4897	−0.858533
$416$	−7.07288	−0.346776
$417$	−3.86503	−0.189271
$418$	3.73751	0.182808
$419$	−28.4922	−1.39194	−0.695968	−	0.718073i	$-0.745024\pi$
$419$	−28.4922	−1.39194	−0.695968	+	0.718073i	$0.745024\pi$
$420$	0	0
$421$	2.79311	0.136128	0.0680640	−	0.997681i	$-0.478318\pi$
$421$	2.79311	0.136128	0.0680640	+	0.997681i	$0.478318\pi$
$422$	−8.47750	−0.412678
$423$	6.02015	0.292710
$424$	11.5582	0.561314
$425$	−1.00000	−0.0485071
$426$	−26.5048	−1.28416
$427$	0	0
$428$	−5.80708	−0.280696
$429$	27.9398	1.34895
$430$	−3.55741	−0.171554
$431$	−1.05516	−0.0508251	−0.0254126	−	0.999677i	$-0.508090\pi$
$431$	−1.05516	−0.0508251	−0.0254126	+	0.999677i	$0.508090\pi$
$432$	−4.63551	−0.223026
$433$	10.1243	0.486544	0.243272	−	0.969958i	$-0.421779\pi$
$433$	10.1243	0.486544	0.243272	+	0.969958i	$0.421779\pi$
$434$	0	0
$435$	2.33952	0.112171
$436$	2.48891	0.119197
$437$	13.4090	0.641439
$438$	−5.18032	−0.247525
$439$	4.49850	0.214702	0.107351	−	0.994221i	$-0.465763\pi$
$439$	4.49850	0.214702	0.107351	+	0.994221i	$0.465763\pi$
$440$	−2.10126	−0.100173
$441$	0	0
$442$	7.07288	0.336423
$443$	6.41377	0.304727	0.152364	−	0.988324i	$-0.451311\pi$
$443$	6.41377	0.304727	0.152364	+	0.988324i	$0.451311\pi$
$444$	3.48091	0.165197
$445$	4.12260	0.195430
$446$	17.2143	0.815119
$447$	−13.3984	−0.633724
$448$	0	0
$449$	−10.0047	−0.472152	−0.236076	−	0.971735i	$-0.575861\pi$
$449$	−10.0047	−0.472152	−0.236076	+	0.971735i	$0.575861\pi$
$450$	0.534253	0.0251849
$451$	9.68076	0.455849
$452$	−15.3137	−0.720296
$453$	25.8114	1.21272
$454$	−2.05228	−0.0963185
$455$	0	0
$456$	−3.34390	−0.156592
$457$	−15.3211	−0.716690	−0.358345	−	0.933589i	$-0.616659\pi$
$457$	−15.3211	−0.716690	−0.358345	+	0.933589i	$0.616659\pi$
$458$	−12.1921	−0.569698
$459$	4.63551	0.216367
$460$	−7.53863	−0.351490
$461$	21.3029	0.992176	0.496088	−	0.868272i	$-0.334769\pi$
$461$	21.3029	0.992176	0.496088	+	0.868272i	$0.334769\pi$
$462$	0	0
$463$	39.9493	1.85660	0.928302	−	0.371828i	$-0.121269\pi$
$463$	39.9493	1.85660	0.928302	+	0.371828i	$0.121269\pi$
$464$	1.24445	0.0577723
$465$	2.33952	0.108493
$466$	3.64947	0.169059
$467$	−3.26364	−0.151023	−0.0755116	−	0.997145i	$-0.524059\pi$
$467$	−3.26364	−0.151023	−0.0755116	+	0.997145i	$0.524059\pi$
$468$	−3.77871	−0.174671
$469$	0	0
$470$	11.2684	0.519771
$471$	−18.2694	−0.841811
$472$	6.48891	0.298676
$473$	7.47503	0.343702
$474$	−3.72802	−0.171233
$475$	−1.77871	−0.0816126
$476$	0	0
$477$	6.17498	0.282733
$478$	25.0860	1.14741
$479$	−32.3373	−1.47753	−0.738764	−	0.673965i	$-0.764590\pi$
$479$	−32.3373	−1.47753	−0.738764	+	0.673965i	$0.764590\pi$
$480$	1.87996	0.0858081
$481$	−13.0960	−0.597128
$482$	−12.9040	−0.587759
$483$	0	0
$484$	−6.58473	−0.299306
$485$	−6.93149	−0.314743
$486$	−5.48966	−0.249016
$487$	−2.50212	−0.113382	−0.0566910	−	0.998392i	$-0.518055\pi$
$487$	−2.50212	−0.113382	−0.0566910	+	0.998392i	$0.518055\pi$
$488$	−8.90131	−0.402943
$489$	19.6620	0.889145
$490$	0	0
$491$	−24.4712	−1.10437	−0.552185	−	0.833721i	$-0.686206\pi$
$491$	−24.4712	−1.10437	−0.552185	+	0.833721i	$0.686206\pi$
$492$	−8.66123	−0.390478
$493$	−1.24445	−0.0560473
$494$	12.5806	0.566027
$495$	−1.12260	−0.0504572
$496$	1.24445	0.0558776
$497$	0	0
$498$	−32.8799	−1.47338
$499$	−8.40587	−0.376298	−0.188149	−	0.982140i	$-0.560249\pi$
$499$	−8.40587	−0.376298	−0.188149	+	0.982140i	$0.560249\pi$
$500$	1.00000	0.0447214
$501$	−41.2284	−1.84195
$502$	−17.4443	−0.778578
$503$	24.1677	1.07759	0.538793	−	0.842438i	$-0.318881\pi$
$503$	24.1677	1.07759	0.538793	+	0.842438i	$0.318881\pi$
$504$	0	0
$505$	−7.73336	−0.344130
$506$	15.8406	0.704200
$507$	69.6067	3.09134
$508$	2.97684	0.132076
$509$	−15.0846	−0.668615	−0.334307	−	0.942464i	$-0.608502\pi$
$509$	−15.0846	−0.668615	−0.334307	+	0.942464i	$0.608502\pi$
$510$	−1.87996	−0.0832461
$511$	0	0
$512$	1.00000	0.0441942
$513$	8.24520	0.364035
$514$	−11.5128	−0.507808
$515$	4.97419	0.219189
$516$	−6.68779	−0.294414
$517$	−23.6777	−1.04134
$518$	0	0
$519$	−31.5515	−1.38496
$520$	−7.07288	−0.310166
$521$	38.6314	1.69247	0.846237	−	0.532807i	$-0.178863\pi$
$521$	38.6314	1.69247	0.846237	+	0.532807i	$0.178863\pi$
$522$	0.664852	0.0290998
$523$	26.5278	1.15998	0.579991	−	0.814623i	$-0.303056\pi$
$523$	26.5278	1.15998	0.579991	+	0.814623i	$0.303056\pi$
$524$	7.22567	0.315655
$525$	0	0
$526$	−27.6620	−1.20612
$527$	−1.24445	−0.0542092
$528$	−3.95028	−0.171914
$529$	33.8309	1.47091
$530$	11.5582	0.502054
$531$	3.46672	0.150443
$532$	0	0
$533$	32.5857	1.41144
$534$	7.75033	0.335390
$535$	−5.80708	−0.251062
$536$	1.27102	0.0548996
$537$	35.1841	1.51830
$538$	26.8034	1.15558
$539$	0	0
$540$	−4.63551	−0.199481
$541$	8.62728	0.370916	0.185458	−	0.982652i	$-0.440623\pi$
$541$	8.62728	0.370916	0.185458	+	0.982652i	$0.440623\pi$
$542$	−23.1841	−0.995841
$543$	37.1714	1.59518
$544$	−1.00000	−0.0428746
$545$	2.48891	0.106613
$546$	0	0
$547$	−23.4691	−1.00346	−0.501732	−	0.865023i	$-0.667304\pi$
$547$	−23.4691	−1.00346	−0.501732	+	0.865023i	$0.667304\pi$
$548$	−3.81702	−0.163055
$549$	−4.75555	−0.202962
$550$	−2.10126	−0.0895978
$551$	−2.21351	−0.0942989
$552$	−14.1723	−0.603214
$553$	0	0
$554$	4.10819	0.174540
$555$	3.48091	0.147756
$556$	−2.05591	−0.0871899
$557$	−24.3475	−1.03164	−0.515819	−	0.856698i	$-0.672512\pi$
$557$	−24.3475	−1.03164	−0.515819	+	0.856698i	$0.672512\pi$
$558$	0.664852	0.0281454
$559$	25.1611	1.06420
$560$	0	0
$561$	3.95028	0.166781
$562$	30.7047	1.29520
$563$	25.8402	1.08903	0.544517	−	0.838750i	$-0.316713\pi$
$563$	25.8402	1.08903	0.544517	+	0.838750i	$0.316713\pi$
$564$	21.1841	0.892011
$565$	−15.3137	−0.644253
$566$	−1.26880	−0.0533318
$567$	0	0
$568$	−14.0986	−0.591564
$569$	33.4050	1.40041	0.700206	−	0.713941i	$-0.253092\pi$
$569$	33.4050	1.40041	0.700206	+	0.713941i	$0.253092\pi$
$570$	−3.34390	−0.140060
$571$	35.7347	1.49545	0.747726	−	0.664008i	$-0.231146\pi$
$571$	35.7347	1.49545	0.747726	+	0.664008i	$0.231146\pi$
$572$	14.8619	0.621408
$573$	27.0948	1.13190
$574$	0	0
$575$	−7.53863	−0.314382
$576$	0.534253	0.0222605
$577$	28.3537	1.18038	0.590191	−	0.807264i	$-0.299052\pi$
$577$	28.3537	1.18038	0.590191	+	0.807264i	$0.299052\pi$
$578$	1.00000	0.0415945
$579$	1.83824	0.0763948
$580$	1.24445	0.0516731
$581$	0	0
$582$	−13.0309	−0.540150
$583$	−24.2867	−1.00585
$584$	−2.75555	−0.114025
$585$	−3.77871	−0.156230
$586$	3.86374	0.159610
$587$	−0.0197516	−0.000815236 0	−0.000407618	−	1.00000i	$-0.500130\pi$
$587$	−0.0197516	−0.000815236 0	−0.000407618		1.00000i	$0.500130\pi$
$588$	0	0
$589$	−2.21351	−0.0912063
$590$	6.48891	0.267144
$591$	−0.553036	−0.0227489
$592$	1.85158	0.0760997
$593$	−14.6064	−0.599812	−0.299906	−	0.953969i	$-0.596955\pi$
$593$	−14.6064	−0.599812	−0.299906	+	0.953969i	$0.596955\pi$
$594$	9.74039	0.399653
$595$	0	0
$596$	−7.12698	−0.291932
$597$	−28.1190	−1.15083
$598$	53.3198	2.18041
$599$	−30.4071	−1.24240	−0.621200	−	0.783652i	$-0.713355\pi$
$599$	−30.4071	−1.24240	−0.621200	+	0.783652i	$0.713355\pi$
$600$	1.87996	0.0767491
$601$	−22.9330	−0.935456	−0.467728	−	0.883872i	$-0.654927\pi$
$601$	−22.9330	−0.935456	−0.467728	+	0.883872i	$0.654927\pi$
$602$	0	0
$603$	0.679044	0.0276528
$604$	13.7297	0.558655
$605$	−6.58473	−0.267707
$606$	−14.5384	−0.590583
$607$	−34.1741	−1.38709	−0.693543	−	0.720416i	$-0.743951\pi$
$607$	−34.1741	−1.38709	−0.693543	+	0.720416i	$0.743951\pi$
$608$	−1.77871	−0.0721360
$609$	0	0
$610$	−8.90131	−0.360403
$611$	−79.6998	−3.22431
$612$	−0.534253	−0.0215959
$613$	−14.5648	−0.588266	−0.294133	−	0.955764i	$-0.595031\pi$
$613$	−14.5648	−0.588266	−0.294133	+	0.955764i	$0.595031\pi$
$614$	−22.6914	−0.915751
$615$	−8.66123	−0.349254
$616$	0	0
$617$	−6.57222	−0.264588	−0.132294	−	0.991211i	$-0.542234\pi$
$617$	−6.57222	−0.264588	−0.132294	+	0.991211i	$0.542234\pi$
$618$	9.35128	0.376163
$619$	12.0409	0.483964	0.241982	−	0.970281i	$-0.422203\pi$
$619$	12.0409	0.483964	0.241982	+	0.970281i	$0.422203\pi$
$620$	1.24445	0.0499784
$621$	34.9454	1.40231
$622$	9.71276	0.389446
$623$	0	0
$624$	−13.2967	−0.532295
$625$	1.00000	0.0400000
$626$	23.3386	0.932797
$627$	7.02638	0.280607
$628$	−9.71798	−0.387790
$629$	−1.85158	−0.0738275
$630$	0	0
$631$	16.5669	0.659518	0.329759	−	0.944065i	$-0.393032\pi$
$631$	16.5669	0.659518	0.329759	+	0.944065i	$0.393032\pi$
$632$	−1.98303	−0.0788806
$633$	−15.9374	−0.633454
$634$	2.89393	0.114933
$635$	2.97684	0.118132
$636$	21.7289	0.861607
$637$	0	0
$638$	−2.61491	−0.103525
$639$	−7.53222	−0.297970
$640$	1.00000	0.0395285
$641$	−20.2780	−0.800935	−0.400467	−	0.916311i	$-0.631152\pi$
$641$	−20.2780	−0.800935	−0.400467	+	0.916311i	$0.631152\pi$
$642$	−10.9171	−0.430863
$643$	18.7171	0.738132	0.369066	−	0.929403i	$-0.379678\pi$
$643$	18.7171	0.738132	0.369066	+	0.929403i	$0.379678\pi$
$644$	0	0
$645$	−6.68779	−0.263332
$646$	1.77871	0.0699822
$647$	−35.2219	−1.38471	−0.692357	−	0.721555i	$-0.743428\pi$
$647$	−35.2219	−1.38471	−0.692357	+	0.721555i	$0.743428\pi$
$648$	−10.3173	−0.405303
$649$	−13.6348	−0.535215
$650$	−7.07288	−0.277421
$651$	0	0
$652$	10.4587	0.409595
$653$	20.1040	0.786732	0.393366	−	0.919382i	$-0.371311\pi$
$653$	20.1040	0.786732	0.393366	+	0.919382i	$0.371311\pi$
$654$	4.67904	0.182965
$655$	7.22567	0.282330
$656$	−4.60713	−0.179878
$657$	−1.47216	−0.0574344
$658$	0	0
$659$	15.3668	0.598607	0.299303	−	0.954158i	$-0.403246\pi$
$659$	15.3668	0.598607	0.299303	+	0.954158i	$0.403246\pi$
$660$	−3.95028	−0.153764
$661$	−40.6360	−1.58056	−0.790279	−	0.612747i	$-0.790065\pi$
$661$	−40.6360	−1.58056	−0.790279	+	0.612747i	$0.790065\pi$
$662$	19.6381	0.763255
$663$	13.2967	0.516402
$664$	−17.4897	−0.678730
$665$	0	0
$666$	0.989214	0.0383313
$667$	−9.38146	−0.363252
$668$	−21.9304	−0.848514
$669$	32.3621	1.25119
$670$	1.27102	0.0491037
$671$	18.7039	0.722057
$672$	0	0
$673$	9.77243	0.376699	0.188350	−	0.982102i	$-0.439686\pi$
$673$	9.77243	0.376699	0.188350	+	0.982102i	$0.439686\pi$
$674$	22.9846	0.885335
$675$	−4.63551	−0.178421
$676$	37.0256	1.42406
$677$	12.3868	0.476064	0.238032	−	0.971257i	$-0.423498\pi$
$677$	12.3868	0.476064	0.238032	+	0.971257i	$0.423498\pi$
$678$	−28.7892	−1.10564
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	−3.85821	−0.147847
$682$	−2.61491	−0.100130
$683$	−41.1332	−1.57392	−0.786958	−	0.617006i	$-0.788345\pi$
$683$	−41.1332	−1.57392	−0.786958	+	0.617006i	$0.788345\pi$
$684$	−0.950278	−0.0363348
$685$	−3.81702	−0.145841
$686$	0	0
$687$	−22.9206	−0.874476
$688$	−3.55741	−0.135625
$689$	−81.7495	−3.11441
$690$	−14.1723	−0.539531
$691$	41.3084	1.57145	0.785723	−	0.618579i	$-0.212291\pi$
$691$	41.3084	1.57145	0.785723	+	0.618579i	$0.212291\pi$
$692$	−16.7831	−0.637997
$693$	0	0
$694$	27.3010	1.03633
$695$	−2.05591	−0.0779850
$696$	2.33952	0.0886794
$697$	4.60713	0.174508
$698$	−9.37408	−0.354814
$699$	6.86087	0.259502
$700$	0	0
$701$	−45.1019	−1.70348	−0.851738	−	0.523968i	$-0.824451\pi$
$701$	−45.1019	−1.70348	−0.851738	+	0.523968i	$0.824451\pi$
$702$	32.7864	1.23744
$703$	−3.29342	−0.124214
$704$	−2.10126	−0.0791940
$705$	21.1841	0.797838
$706$	34.1203	1.28414
$707$	0	0
$708$	12.1989	0.458462
$709$	19.4151	0.729149	0.364574	−	0.931174i	$-0.381214\pi$
$709$	19.4151	0.729149	0.364574	+	0.931174i	$0.381214\pi$
$710$	−14.0986	−0.529111
$711$	−1.05944	−0.0397320
$712$	4.12260	0.154501
$713$	−9.38146	−0.351339
$714$	0	0
$715$	14.8619	0.555805
$716$	18.7153	0.699425
$717$	47.1607	1.76125
$718$	1.95368	0.0729109
$719$	−6.65902	−0.248339	−0.124170	−	0.992261i	$-0.539627\pi$
$719$	−6.65902	−0.248339	−0.124170	+	0.992261i	$0.539627\pi$
$720$	0.534253	0.0199104
$721$	0	0
$722$	−15.8362	−0.589363
$723$	−24.2589	−0.902200
$724$	19.7724	0.734836
$725$	1.24445	0.0462178
$726$	−12.3790	−0.459429
$727$	−28.5622	−1.05931	−0.529656	−	0.848212i	$-0.677679\pi$
$727$	−28.5622	−1.05931	−0.529656	+	0.848212i	$0.677679\pi$
$728$	0	0
$729$	20.6317	0.764136
$730$	−2.75555	−0.101987
$731$	3.55741	0.131576
$732$	−16.7341	−0.618510
$733$	−22.5891	−0.834347	−0.417174	−	0.908827i	$-0.636979\pi$
$733$	−22.5891	−0.834347	−0.417174	+	0.908827i	$0.636979\pi$
$734$	8.31115	0.306770
$735$	0	0
$736$	−7.53863	−0.277877
$737$	−2.67073	−0.0983776
$738$	−2.46137	−0.0906044
$739$	32.9503	1.21210	0.606048	−	0.795428i	$-0.292754\pi$
$739$	32.9503	1.21210	0.606048	+	0.795428i	$0.292754\pi$
$740$	1.85158	0.0680656
$741$	23.6510	0.868840
$742$	0	0
$743$	−25.3045	−0.928333	−0.464166	−	0.885748i	$-0.653646\pi$
$743$	−25.3045	−0.928333	−0.464166	+	0.885748i	$0.653646\pi$
$744$	2.33952	0.0857710
$745$	−7.12698	−0.261112
$746$	32.0850	1.17472
$747$	−9.34390	−0.341875
$748$	2.10126	0.0768295
$749$	0	0
$750$	1.87996	0.0686465
$751$	−2.70839	−0.0988305	−0.0494152	−	0.998778i	$-0.515736\pi$
$751$	−2.70839	−0.0988305	−0.0494152	+	0.998778i	$0.515736\pi$
$752$	11.2684	0.410915
$753$	−32.7946	−1.19510
$754$	−8.80186	−0.320545
$755$	13.7297	0.499676
$756$	0	0
$757$	−30.5364	−1.10986	−0.554932	−	0.831896i	$-0.687256\pi$
$757$	−30.5364	−1.10986	−0.554932	+	0.831896i	$0.687256\pi$
$758$	0.195575	0.00710359
$759$	29.7797	1.08093
$760$	−1.77871	−0.0645204
$761$	6.84380	0.248088	0.124044	−	0.992277i	$-0.460414\pi$
$761$	6.84380	0.248088	0.124044	+	0.992277i	$0.460414\pi$
$762$	5.59635	0.202734
$763$	0	0
$764$	14.4124	0.521422
$765$	−0.534253	−0.0193159
$766$	−16.7603	−0.605573
$767$	−45.8952	−1.65718
$768$	1.87996	0.0678372
$769$	−34.3724	−1.23950	−0.619750	−	0.784799i	$-0.712766\pi$
$769$	−34.3724	−1.23950	−0.619750	+	0.784799i	$0.712766\pi$
$770$	0	0
$771$	−21.6436	−0.779477
$772$	0.977810	0.0351922
$773$	−51.5189	−1.85301	−0.926503	−	0.376287i	$-0.877201\pi$
$773$	−51.5189	−1.85301	−0.926503	+	0.376287i	$0.877201\pi$
$774$	−1.90056	−0.0683141
$775$	1.24445	0.0447021
$776$	−6.93149	−0.248826
$777$	0	0
$778$	−8.58132	−0.307655
$779$	8.19473	0.293607
$780$	−13.2967	−0.476100
$781$	29.6248	1.06006
$782$	7.53863	0.269581
$783$	−5.76867	−0.206155
$784$	0	0
$785$	−9.71798	−0.346850
$786$	13.5840	0.484524
$787$	−32.4460	−1.15658	−0.578288	−	0.815833i	$-0.696279\pi$
$787$	−32.4460	−1.15658	−0.578288	+	0.815833i	$0.696279\pi$
$788$	−0.294174	−0.0104795
$789$	−52.0034	−1.85137
$790$	−1.98303	−0.0705530
$791$	0	0
$792$	−1.12260	−0.0398899
$793$	62.9579	2.23570
$794$	−26.9793	−0.957460
$795$	21.7289	0.770644
$796$	−14.9572	−0.530144
$797$	32.4481	1.14937	0.574685	−	0.818375i	$-0.305125\pi$
$797$	32.4481	1.14937	0.574685	+	0.818375i	$0.305125\pi$
$798$	0	0
$799$	−11.2684	−0.398646
$800$	1.00000	0.0353553
$801$	2.20251	0.0778219
$802$	27.9296	0.986228
$803$	5.79011	0.204329
$804$	2.38946	0.0842698
$805$	0	0
$806$	−8.80186	−0.310032
$807$	50.3893	1.77379
$808$	−7.73336	−0.272059
$809$	34.3574	1.20794	0.603971	−	0.797006i	$-0.293584\pi$
$809$	34.3574	1.20794	0.603971	+	0.797006i	$0.293584\pi$
$810$	−10.3173	−0.362514
$811$	−11.5607	−0.405952	−0.202976	−	0.979184i	$-0.565061\pi$
$811$	−11.5607	−0.405952	−0.202976	+	0.979184i	$0.565061\pi$
$812$	0	0
$813$	−43.5852	−1.52860
$814$	−3.89065	−0.136367
$815$	10.4587	0.366353
$816$	−1.87996	−0.0658118
$817$	6.32758	0.221374
$818$	−34.8618	−1.21892
$819$	0	0
$820$	−4.60713	−0.160888
$821$	28.4762	0.993827	0.496913	−	0.867800i	$-0.334467\pi$
$821$	28.4762	0.993827	0.496913	+	0.867800i	$0.334467\pi$
$822$	−7.17585	−0.250287
$823$	−28.2339	−0.984173	−0.492086	−	0.870546i	$-0.663766\pi$
$823$	−28.2339	−0.984173	−0.492086	+	0.870546i	$0.663766\pi$
$824$	4.97419	0.173284
$825$	−3.95028	−0.137531
$826$	0	0
$827$	8.74958	0.304253	0.152126	−	0.988361i	$-0.451388\pi$
$827$	8.74958	0.304253	0.152126	+	0.988361i	$0.451388\pi$
$828$	−4.02753	−0.139966
$829$	−18.7214	−0.650221	−0.325110	−	0.945676i	$-0.605401\pi$
$829$	−18.7214	−0.650221	−0.325110	+	0.945676i	$0.605401\pi$
$830$	−17.4897	−0.607075
$831$	7.72324	0.267916
$832$	−7.07288	−0.245208
$833$	0	0
$834$	−3.86503	−0.133835
$835$	−21.9304	−0.758934
$836$	3.73751	0.129265
$837$	−5.76867	−0.199394
$838$	−28.4922	−0.984247
$839$	−46.0492	−1.58979	−0.794897	−	0.606744i	$-0.792475\pi$
$839$	−46.0492	−1.58979	−0.794897	+	0.606744i	$0.792475\pi$
$840$	0	0
$841$	−27.4513	−0.946598
$842$	2.79311	0.0962571
$843$	57.7236	1.98811
$844$	−8.47750	−0.291808
$845$	37.0256	1.27372
$846$	6.02015	0.206977
$847$	0	0
$848$	11.5582	0.396909
$849$	−2.38530	−0.0818633
$850$	−1.00000	−0.0342997
$851$	−13.9584	−0.478488
$852$	−26.5048	−0.908040
$853$	31.6455	1.08352	0.541760	−	0.840533i	$-0.317758\pi$
$853$	31.6455	1.08352	0.541760	+	0.840533i	$0.317758\pi$
$854$	0	0
$855$	−0.950278	−0.0324988
$856$	−5.80708	−0.198482
$857$	16.7555	0.572359	0.286179	−	0.958176i	$-0.407615\pi$
$857$	16.7555	0.572359	0.286179	+	0.958176i	$0.407615\pi$
$858$	27.9398	0.953850
$859$	34.5726	1.17960	0.589801	−	0.807549i	$-0.299206\pi$
$859$	34.5726	1.17960	0.589801	+	0.807549i	$0.299206\pi$
$860$	−3.55741	−0.121307
$861$	0	0
$862$	−1.05516	−0.0359388
$863$	14.6783	0.499656	0.249828	−	0.968290i	$-0.419626\pi$
$863$	14.6783	0.499656	0.249828	+	0.968290i	$0.419626\pi$
$864$	−4.63551	−0.157703
$865$	−16.7831	−0.570642
$866$	10.1243	0.344038
$867$	1.87996	0.0638468
$868$	0	0
$869$	4.16685	0.141351
$870$	2.33952	0.0793172
$871$	−8.98975	−0.304606
$872$	2.48891	0.0842850
$873$	−3.70317	−0.125333
$874$	13.4090	0.453566
$875$	0	0
$876$	−5.18032	−0.175027
$877$	−40.6335	−1.37210	−0.686048	−	0.727556i	$-0.740656\pi$
$877$	−40.6335	−1.37210	−0.686048	+	0.727556i	$0.740656\pi$
$878$	4.49850	0.151817
$879$	7.26368	0.244998
$880$	−2.10126	−0.0708333
$881$	−20.4629	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$881$	−20.4629	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$882$	0	0
$883$	47.3526	1.59354	0.796772	−	0.604280i	$-0.206539\pi$
$883$	47.3526	1.59354	0.796772	+	0.604280i	$0.206539\pi$
$884$	7.07288	0.237887
$885$	12.1989	0.410061
$886$	6.41377	0.215475
$887$	−46.2119	−1.55164	−0.775822	−	0.630952i	$-0.782665\pi$
$887$	−46.2119	−1.55164	−0.775822	+	0.630952i	$0.782665\pi$
$888$	3.48091	0.116812
$889$	0	0
$890$	4.12260	0.138190
$891$	21.6794	0.726286
$892$	17.2143	0.576376
$893$	−20.0431	−0.670717
$894$	−13.3984	−0.448111
$895$	18.7153	0.625584
$896$	0	0
$897$	100.239	3.34689
$898$	−10.0047	−0.333862
$899$	1.54866	0.0516508
$900$	0.534253	0.0178084
$901$	−11.5582	−0.385058
$902$	9.68076	0.322334
$903$	0	0
$904$	−15.3137	−0.509326
$905$	19.7724	0.657258
$906$	25.8114	0.857525
$907$	−52.8510	−1.75489	−0.877445	−	0.479677i	$-0.840754\pi$
$907$	−52.8510	−1.75489	−0.877445	+	0.479677i	$0.840754\pi$
$908$	−2.05228	−0.0681074
$909$	−4.13157	−0.137035
$910$	0	0
$911$	−10.3519	−0.342974	−0.171487	−	0.985186i	$-0.554857\pi$
$911$	−10.3519	−0.342974	−0.171487	+	0.985186i	$0.554857\pi$
$912$	−3.34390	−0.110727
$913$	36.7502	1.21626
$914$	−15.3211	−0.506777
$915$	−16.7341	−0.553213
$916$	−12.1921	−0.402837
$917$	0	0
$918$	4.63551	0.152995
$919$	12.1099	0.399468	0.199734	−	0.979850i	$-0.435992\pi$
$919$	12.1099	0.399468	0.199734	+	0.979850i	$0.435992\pi$
$920$	−7.53863	−0.248541
$921$	−42.6590	−1.40566
$922$	21.3029	0.701574
$923$	99.7177	3.28225
$924$	0	0
$925$	1.85158	0.0608797
$926$	39.9493	1.31282
$927$	2.65747	0.0872829
$928$	1.24445	0.0408512
$929$	−38.0450	−1.24822	−0.624108	−	0.781338i	$-0.714538\pi$
$929$	−38.0450	−1.24822	−0.624108	+	0.781338i	$0.714538\pi$
$930$	2.33952	0.0767159
$931$	0	0
$932$	3.64947	0.119543
$933$	18.2596	0.597793
$934$	−3.26364	−0.106789
$935$	2.10126	0.0687184
$936$	−3.77871	−0.123511
$937$	29.7496	0.971878	0.485939	−	0.873993i	$-0.338478\pi$
$937$	29.7496	0.971878	0.485939	+	0.873993i	$0.338478\pi$
$938$	0	0
$939$	43.8756	1.43183
$940$	11.2684	0.367533
$941$	7.34253	0.239360	0.119680	−	0.992813i	$-0.461813\pi$
$941$	7.34253	0.239360	0.119680	+	0.992813i	$0.461813\pi$
$942$	−18.2694	−0.595250
$943$	34.7315	1.13101
$944$	6.48891	0.211196
$945$	0	0
$946$	7.47503	0.243034
$947$	−7.28061	−0.236588	−0.118294	−	0.992979i	$-0.537743\pi$
$947$	−7.28061	−0.236588	−0.118294	+	0.992979i	$0.537743\pi$
$948$	−3.72802	−0.121080
$949$	19.4897	0.632661
$950$	−1.77871	−0.0577088
$951$	5.44047	0.176419
$952$	0	0
$953$	−40.4457	−1.31016	−0.655082	−	0.755558i	$-0.727366\pi$
$953$	−40.4457	−1.31016	−0.655082	+	0.755558i	$0.727366\pi$
$954$	6.17498	0.199922
$955$	14.4124	0.466374
$956$	25.0860	0.811339
$957$	−4.91593	−0.158910
$958$	−32.3373	−1.04477
$959$	0	0
$960$	1.87996	0.0606755
$961$	−29.4513	−0.950043
$962$	−13.0960	−0.422233
$963$	−3.10245	−0.0999750
$964$	−12.9040	−0.415609
$965$	0.977810	0.0314768
$966$	0	0
$967$	4.43365	0.142577	0.0712884	−	0.997456i	$-0.477289\pi$
$967$	4.43365	0.142577	0.0712884	+	0.997456i	$0.477289\pi$
$968$	−6.58473	−0.211641
$969$	3.34390	0.107421
$970$	−6.93149	−0.222557
$971$	41.9723	1.34695	0.673477	−	0.739208i	$-0.264800\pi$
$971$	41.9723	1.34695	0.673477	+	0.739208i	$0.264800\pi$
$972$	−5.48966	−0.176081
$973$	0	0
$974$	−2.50212	−0.0801731
$975$	−13.2967	−0.425836
$976$	−8.90131	−0.284924
$977$	−27.5656	−0.881901	−0.440951	−	0.897531i	$-0.645359\pi$
$977$	−27.5656	−0.881901	−0.440951	+	0.897531i	$0.645359\pi$
$978$	19.6620	0.628721
$979$	−8.66264	−0.276859
$980$	0	0
$981$	1.32970	0.0424542
$982$	−24.4712	−0.780908
$983$	−5.21594	−0.166363	−0.0831814	−	0.996534i	$-0.526508\pi$
$983$	−5.21594	−0.166363	−0.0831814	+	0.996534i	$0.526508\pi$
$984$	−8.66123	−0.276110
$985$	−0.294174	−0.00937318
$986$	−1.24445	−0.0396314
$987$	0	0
$988$	12.5806	0.400241
$989$	26.8180	0.852763
$990$	−1.12260	−0.0356786
$991$	11.2187	0.356375	0.178187	−	0.983997i	$-0.442977\pi$
$991$	11.2187	0.356375	0.178187	+	0.983997i	$0.442977\pi$
$992$	1.24445	0.0395114
$993$	36.9188	1.17158
$994$	0	0
$995$	−14.9572	−0.474176
$996$	−32.8799	−1.04184
$997$	44.5220	1.41002	0.705012	−	0.709195i	$-0.250941\pi$
$997$	44.5220	1.41002	0.705012	+	0.709195i	$0.250941\pi$
$998$	−8.40587	−0.266083
$999$	−8.58304	−0.271555

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cf.1.4		4
7.2	even	3		1190.2.i.j.851.1	yes	8
7.4	even	3		1190.2.i.j.681.1	✓	8
7.6	odd	2		8330.2.a.ck.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.i.j.681.1	✓	8	7.4	even	3
1190.2.i.j.851.1	yes	8	7.2	even	3
8330.2.a.cf.1.4		4	1.1	even	1	trivial
8330.2.a.ck.1.1		4	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.87996 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.87996 q^{6} +1.00000 q^{8} +0.534253 q^{9} +1.00000 q^{10} -2.10126 q^{11} +1.87996 q^{12} -7.07288 q^{13} +1.87996 q^{15} +1.00000 q^{16} -1.00000 q^{17} +0.534253 q^{18} -1.77871 q^{19} +1.00000 q^{20} -2.10126 q^{22} -7.53863 q^{23} +1.87996 q^{24} +1.00000 q^{25} -7.07288 q^{26} -4.63551 q^{27} +1.24445 q^{29} +1.87996 q^{30} +1.24445 q^{31} +1.00000 q^{32} -3.95028 q^{33} -1.00000 q^{34} +0.534253 q^{36} +1.85158 q^{37} -1.77871 q^{38} -13.2967 q^{39} +1.00000 q^{40} -4.60713 q^{41} -3.55741 q^{43} -2.10126 q^{44} +0.534253 q^{45} -7.53863 q^{46} +11.2684 q^{47} +1.87996 q^{48} +1.00000 q^{50} -1.87996 q^{51} -7.07288 q^{52} +11.5582 q^{53} -4.63551 q^{54} -2.10126 q^{55} -3.34390 q^{57} +1.24445 q^{58} +6.48891 q^{59} +1.87996 q^{60} -8.90131 q^{61} +1.24445 q^{62} +1.00000 q^{64} -7.07288 q^{65} -3.95028 q^{66} +1.27102 q^{67} -1.00000 q^{68} -14.1723 q^{69} -14.0986 q^{71} +0.534253 q^{72} -2.75555 q^{73} +1.85158 q^{74} +1.87996 q^{75} -1.77871 q^{76} -13.2967 q^{78} -1.98303 q^{79} +1.00000 q^{80} -10.3173 q^{81} -4.60713 q^{82} -17.4897 q^{83} -1.00000 q^{85} -3.55741 q^{86} +2.33952 q^{87} -2.10126 q^{88} +4.12260 q^{89} +0.534253 q^{90} -7.53863 q^{92} +2.33952 q^{93} +11.2684 q^{94} -1.77871 q^{95} +1.87996 q^{96} -6.93149 q^{97} -1.12260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{8} + 6 q^{9} + 4 q^{10} - 2 q^{12} - 12 q^{13} - 2 q^{15} + 4 q^{16} - 4 q^{17} + 6 q^{18} - 6 q^{19} + 4 q^{20} - 10 q^{23} - 2 q^{24} + 4 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^8 + 6 * q^9 + 4 * q^10 - 2 * q^12 - 12 * q^13 - 2 * q^15 + 4 * q^16 - 4 * q^17 + 6 * q^18 - 6 * q^19 + 4 * q^20 - 10 * q^23 - 2 * q^24 + 4 * q^25 - 12 * q^26 - 14 * q^27 - 2 * q^30 + 4 * q^32 - 26 * q^33 - 4 * q^34 + 6 * q^36 - 10 * q^37 - 6 * q^38 - 2 * q^39 + 4 * q^40 - 6 * q^41 - 12 * q^43 + 6 * q^45 - 10 * q^46 - 14 * q^47 - 2 * q^48 + 4 * q^50 + 2 * q^51 - 12 * q^52 - 4 * q^53 - 14 * q^54 + 12 * q^57 + 16 * q^59 - 2 * q^60 - 8 * q^61 + 4 * q^64 - 12 * q^65 - 26 * q^66 - 4 * q^67 - 4 * q^68 - 20 * q^69 + 2 * q^71 + 6 * q^72 - 16 * q^73 - 10 * q^74 - 2 * q^75 - 6 * q^76 - 2 * q^78 - 2 * q^79 + 4 * q^80 - 20 * q^81 - 6 * q^82 - 12 * q^83 - 4 * q^85 - 12 * q^86 + 8 * q^87 - 10 * q^89 + 6 * q^90 - 10 * q^92 + 8 * q^93 - 14 * q^94 - 6 * q^95 - 2 * q^96 - 20 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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