LMFDB - Embedded newform 8330.2.a.cd.1.3

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,-2,0,4,4,-4,2,-2,-14] 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:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
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			1.3
Root			$-1.51658$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.816594 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.816594 q^{6} +1.00000 q^{8} -2.33317 q^{9} -1.00000 q^{10} +5.84975 q^{11} +0.816594 q^{12} -4.53844 q^{13} -0.816594 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.33317 q^{18} -5.03316 q^{19} -1.00000 q^{20} +5.84975 q^{22} +6.57160 q^{23} +0.816594 q^{24} +1.00000 q^{25} -4.53844 q^{26} -4.35504 q^{27} -5.96636 q^{29} -0.816594 q^{30} -6.09474 q^{31} +1.00000 q^{32} +4.77687 q^{33} +1.00000 q^{34} -2.33317 q^{36} -3.25628 q^{37} -5.03316 q^{38} -3.70607 q^{39} -1.00000 q^{40} -5.07689 q^{41} -4.20479 q^{43} +5.84975 q^{44} +2.33317 q^{45} +6.57160 q^{46} -3.95627 q^{47} +0.816594 q^{48} +1.00000 q^{50} +0.816594 q^{51} -4.53844 q^{52} -13.9048 q^{53} -4.35504 q^{54} -5.84975 q^{55} -4.11005 q^{57} -5.96636 q^{58} +10.6817 q^{59} -0.816594 q^{60} +0.0437295 q^{61} -6.09474 q^{62} +1.00000 q^{64} +4.53844 q^{65} +4.77687 q^{66} +12.0226 q^{67} +1.00000 q^{68} +5.36633 q^{69} -4.49873 q^{71} -2.33317 q^{72} +7.73315 q^{73} -3.25628 q^{74} +0.816594 q^{75} -5.03316 q^{76} -3.70607 q^{78} -7.67036 q^{79} -1.00000 q^{80} +3.44322 q^{81} -5.07689 q^{82} -13.7943 q^{83} -1.00000 q^{85} -4.20479 q^{86} -4.87210 q^{87} +5.84975 q^{88} -7.85376 q^{89} +2.33317 q^{90} +6.57160 q^{92} -4.97693 q^{93} -3.95627 q^{94} +5.03316 q^{95} +0.816594 q^{96} +3.51002 q^{97} -13.6485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} - 2 q^{12} - 14 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} - 4 q^{19} - 4 q^{20} + 2 q^{22} + 6 q^{23}+ \cdots - 26 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 + 4 * q^9 - 4 * q^10 + 2 * q^11 - 2 * q^12 - 14 * q^13 + 2 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 - 4 * q^19 - 4 * q^20 + 2 * q^22 + 6 * q^23 - 2 * q^24 + 4 * q^25 - 14 * q^26 - 8 * q^27 + 2 * q^30 - 18 * q^31 + 4 * q^32 + 8 * q^33 + 4 * q^34 + 4 * q^36 - 8 * q^37 - 4 * q^38 + 6 * q^39 - 4 * q^40 - 12 * q^41 + 14 * q^43 + 2 * q^44 - 4 * q^45 + 6 * q^46 - 8 * q^47 - 2 * q^48 + 4 * q^50 - 2 * q^51 - 14 * q^52 - 22 * q^53 - 8 * q^54 - 2 * q^55 + 8 * q^57 - 2 * q^59 + 2 * q^60 + 8 * q^61 - 18 * q^62 + 4 * q^64 + 14 * q^65 + 8 * q^66 + 8 * q^67 + 4 * q^68 - 8 * q^69 - 8 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 2 * q^75 - 4 * q^76 + 6 * q^78 - 6 * q^79 - 4 * q^80 - 24 * q^81 - 12 * q^82 - 6 * q^83 - 4 * q^85 + 14 * q^86 - 42 * q^87 + 2 * q^88 - 12 * q^89 - 4 * q^90 + 6 * q^92 - 20 * q^93 - 8 * q^94 + 4 * q^95 - 2 * q^96 - 16 * q^97 - 26 * 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
$3$	0.816594	0.471461	0.235730	−	0.971818i	$-0.424252\pi$
$3$	0.816594	0.471461	0.235730	+	0.971818i	$0.424252\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.816594	0.333373
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.33317	−0.777725
$10$	−1.00000	−0.316228
$11$	5.84975	1.76377	0.881884	−	0.471467i	$-0.156275\pi$
$11$	5.84975	1.76377	0.881884	+	0.471467i	$0.156275\pi$
$12$	0.816594	0.235730
$13$	−4.53844	−1.25874	−0.629369	−	0.777107i	$-0.716687\pi$
$13$	−4.53844	−1.25874	−0.629369	+	0.777107i	$0.716687\pi$
$14$	0	0
$15$	−0.816594	−0.210844
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.33317	−0.549934
$19$	−5.03316	−1.15469	−0.577343	−	0.816502i	$-0.695910\pi$
$19$	−5.03316	−1.15469	−0.577343	+	0.816502i	$0.695910\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.84975	1.24717
$23$	6.57160	1.37027	0.685137	−	0.728414i	$-0.259742\pi$
$23$	6.57160	1.37027	0.685137	+	0.728414i	$0.259742\pi$
$24$	0.816594	0.166687
$25$	1.00000	0.200000
$26$	−4.53844	−0.890062
$27$	−4.35504	−0.838128
$28$	0	0
$29$	−5.96636	−1.10793	−0.553963	−	0.832541i	$-0.686885\pi$
$29$	−5.96636	−1.10793	−0.553963	+	0.832541i	$0.686885\pi$
$30$	−0.816594	−0.149089
$31$	−6.09474	−1.09465	−0.547324	−	0.836921i	$-0.684354\pi$
$31$	−6.09474	−1.09465	−0.547324	+	0.836921i	$0.684354\pi$
$32$	1.00000	0.176777
$33$	4.77687	0.831547
$34$	1.00000	0.171499
$35$	0	0
$36$	−2.33317	−0.388862
$37$	−3.25628	−0.535330	−0.267665	−	0.963512i	$-0.586252\pi$
$37$	−3.25628	−0.535330	−0.267665	+	0.963512i	$0.586252\pi$
$38$	−5.03316	−0.816486
$39$	−3.70607	−0.593446
$40$	−1.00000	−0.158114
$41$	−5.07689	−0.792877	−0.396438	−	0.918061i	$-0.629754\pi$
$41$	−5.07689	−0.792877	−0.396438	+	0.918061i	$0.629754\pi$
$42$	0	0
$43$	−4.20479	−0.641225	−0.320612	−	0.947210i	$-0.603889\pi$
$43$	−4.20479	−0.641225	−0.320612	+	0.947210i	$0.603889\pi$
$44$	5.84975	0.881884
$45$	2.33317	0.347809
$46$	6.57160	0.968930
$47$	−3.95627	−0.577081	−0.288541	−	0.957468i	$-0.593170\pi$
$47$	−3.95627	−0.577081	−0.288541	+	0.957468i	$0.593170\pi$
$48$	0.816594	0.117865
$49$	0	0
$50$	1.00000	0.141421
$51$	0.816594	0.114346
$52$	−4.53844	−0.629369
$53$	−13.9048	−1.90997	−0.954984	−	0.296657i	$-0.904128\pi$
$53$	−13.9048	−1.90997	−0.954984	+	0.296657i	$0.904128\pi$
$54$	−4.35504	−0.592646
$55$	−5.84975	−0.788781
$56$	0	0
$57$	−4.11005	−0.544389
$58$	−5.96636	−0.783422
$59$	10.6817	1.39063	0.695316	−	0.718704i	$-0.255264\pi$
$59$	10.6817	1.39063	0.695316	+	0.718704i	$0.255264\pi$
$60$	−0.816594	−0.105422
$61$	0.0437295	0.00559899	0.00279949	−	0.999996i	$-0.499109\pi$
$61$	0.0437295	0.00559899	0.00279949	+	0.999996i	$0.499109\pi$
$62$	−6.09474	−0.774033
$63$	0	0
$64$	1.00000	0.125000
$65$	4.53844	0.562925
$66$	4.77687	0.587993
$67$	12.0226	1.46879	0.734397	−	0.678721i	$-0.237465\pi$
$67$	12.0226	1.46879	0.734397	+	0.678721i	$0.237465\pi$
$68$	1.00000	0.121268
$69$	5.36633	0.646031
$70$	0	0
$71$	−4.49873	−0.533900	−0.266950	−	0.963710i	$-0.586016\pi$
$71$	−4.49873	−0.533900	−0.266950	+	0.963710i	$0.586016\pi$
$72$	−2.33317	−0.274967
$73$	7.73315	0.905096	0.452548	−	0.891740i	$-0.350515\pi$
$73$	7.73315	0.905096	0.452548	+	0.891740i	$0.350515\pi$
$74$	−3.25628	−0.378535
$75$	0.816594	0.0942922
$76$	−5.03316	−0.577343
$77$	0	0
$78$	−3.70607	−0.419629
$79$	−7.67036	−0.862983	−0.431491	−	0.902117i	$-0.642012\pi$
$79$	−7.67036	−0.862983	−0.431491	+	0.902117i	$0.642012\pi$
$80$	−1.00000	−0.111803
$81$	3.44322	0.382580
$82$	−5.07689	−0.560649
$83$	−13.7943	−1.51412	−0.757058	−	0.653348i	$-0.773364\pi$
$83$	−13.7943	−1.51412	−0.757058	+	0.653348i	$0.773364\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−4.20479	−0.453414
$87$	−4.87210	−0.522344
$88$	5.84975	0.623586
$89$	−7.85376	−0.832497	−0.416249	−	0.909251i	$-0.636655\pi$
$89$	−7.85376	−0.832497	−0.416249	+	0.909251i	$0.636655\pi$
$90$	2.33317	0.245938
$91$	0	0
$92$	6.57160	0.685137
$93$	−4.97693	−0.516084
$94$	−3.95627	−0.408058
$95$	5.03316	0.516391
$96$	0.816594	0.0833433
$97$	3.51002	0.356389	0.178194	−	0.983995i	$-0.442974\pi$
$97$	3.51002	0.356389	0.178194	+	0.983995i	$0.442974\pi$
$98$	0	0
$99$	−13.6485	−1.37173
$100$	1.00000	0.100000
$101$	11.0995	1.10444	0.552220	−	0.833699i	$-0.313781\pi$
$101$	11.0995	1.10444	0.552220	+	0.833699i	$0.313781\pi$
$102$	0.816594	0.0808549
$103$	0.651043	0.0641492	0.0320746	−	0.999485i	$-0.489789\pi$
$103$	0.651043	0.0641492	0.0320746	+	0.999485i	$0.489789\pi$
$104$	−4.53844	−0.445031
$105$	0	0
$106$	−13.9048	−1.35055
$107$	−14.2593	−1.37850	−0.689251	−	0.724522i	$-0.742060\pi$
$107$	−14.2593	−1.37850	−0.689251	+	0.724522i	$0.742060\pi$
$108$	−4.35504	−0.419064
$109$	−4.39949	−0.421395	−0.210698	−	0.977551i	$-0.567574\pi$
$109$	−4.39949	−0.421395	−0.210698	+	0.977551i	$0.567574\pi$
$110$	−5.84975	−0.557752
$111$	−2.65906	−0.252387
$112$	0	0
$113$	7.45379	0.701194	0.350597	−	0.936526i	$-0.385979\pi$
$113$	7.45379	0.701194	0.350597	+	0.936526i	$0.385979\pi$
$114$	−4.11005	−0.384941
$115$	−6.57160	−0.612805
$116$	−5.96636	−0.553963
$117$	10.5890	0.978952
$118$	10.6817	0.983326
$119$	0	0
$120$	−0.816594	−0.0745445
$121$	23.2196	2.11087
$122$	0.0437295	0.00395908
$123$	−4.14576	−0.373810
$124$	−6.09474	−0.547324
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.4249	1.81242	0.906208	−	0.422832i	$-0.138964\pi$
$127$	20.4249	1.81242	0.906208	+	0.422832i	$0.138964\pi$
$128$	1.00000	0.0883883
$129$	−3.43361	−0.302312
$130$	4.53844	0.398048
$131$	5.72793	0.500452	0.250226	−	0.968187i	$-0.419495\pi$
$131$	5.72793	0.500452	0.250226	+	0.968187i	$0.419495\pi$
$132$	4.77687	0.415774
$133$	0	0
$134$	12.0226	1.03859
$135$	4.35504	0.374822
$136$	1.00000	0.0857493
$137$	−7.11005	−0.607452	−0.303726	−	0.952759i	$-0.598231\pi$
$137$	−7.11005	−0.607452	−0.303726	+	0.952759i	$0.598231\pi$
$138$	5.36633	0.456813
$139$	3.89348	0.330241	0.165120	−	0.986273i	$-0.447199\pi$
$139$	3.89348	0.330241	0.165120	+	0.986273i	$0.447199\pi$
$140$	0	0
$141$	−3.23067	−0.272071
$142$	−4.49873	−0.377525
$143$	−26.5488	−2.22012
$144$	−2.33317	−0.194431
$145$	5.96636	0.495479
$146$	7.73315	0.640000
$147$	0	0
$148$	−3.25628	−0.267665
$149$	−6.86153	−0.562118	−0.281059	−	0.959690i	$-0.590686\pi$
$149$	−6.86153	−0.562118	−0.281059	+	0.959690i	$0.590686\pi$
$150$	0.816594	0.0666746
$151$	4.07120	0.331309	0.165655	−	0.986184i	$-0.447026\pi$
$151$	4.07120	0.331309	0.165655	+	0.986184i	$0.447026\pi$
$152$	−5.03316	−0.408243
$153$	−2.33317	−0.188626
$154$	0	0
$155$	6.09474	0.489542
$156$	−3.70607	−0.296723
$157$	−8.20527	−0.654852	−0.327426	−	0.944877i	$-0.606181\pi$
$157$	−8.20527	−0.654852	−0.327426	+	0.944877i	$0.606181\pi$
$158$	−7.67036	−0.610221
$159$	−11.3546	−0.900475
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	3.44322	0.270525
$163$	−17.9480	−1.40580	−0.702899	−	0.711290i	$-0.748111\pi$
$163$	−17.9480	−1.40580	−0.702899	+	0.711290i	$0.748111\pi$
$164$	−5.07689	−0.396438
$165$	−4.77687	−0.371879
$166$	−13.7943	−1.07064
$167$	−22.4978	−1.74093	−0.870465	−	0.492231i	$-0.836182\pi$
$167$	−22.4978	−1.74093	−0.870465	+	0.492231i	$0.836182\pi$
$168$	0	0
$169$	7.59748	0.584421
$170$	−1.00000	−0.0766965
$171$	11.7432	0.898028
$172$	−4.20479	−0.320612
$173$	−3.21256	−0.244246	−0.122123	−	0.992515i	$-0.538970\pi$
$173$	−3.21256	−0.244246	−0.122123	+	0.992515i	$0.538970\pi$
$174$	−4.87210	−0.369353
$175$	0	0
$176$	5.84975	0.440942
$177$	8.72258	0.655629
$178$	−7.85376	−0.588664
$179$	8.83543	0.660391	0.330196	−	0.943913i	$-0.392885\pi$
$179$	8.83543	0.660391	0.330196	+	0.943913i	$0.392885\pi$
$180$	2.33317	0.173905
$181$	−8.18949	−0.608720	−0.304360	−	0.952557i	$-0.598443\pi$
$181$	−8.18949	−0.608720	−0.304360	+	0.952557i	$0.598443\pi$
$182$	0	0
$183$	0.0357093	0.00263970
$184$	6.57160	0.484465
$185$	3.25628	0.239407
$186$	−4.97693	−0.364926
$187$	5.84975	0.427776
$188$	−3.95627	−0.288541
$189$	0	0
$190$	5.03316	0.365144
$191$	−23.9990	−1.73651	−0.868255	−	0.496118i	$-0.834758\pi$
$191$	−23.9990	−1.73651	−0.868255	+	0.496118i	$0.834758\pi$
$192$	0.816594	0.0589326
$193$	−15.3969	−1.10830	−0.554148	−	0.832418i	$-0.686956\pi$
$193$	−15.3969	−1.10830	−0.554148	+	0.832418i	$0.686956\pi$
$194$	3.51002	0.252005
$195$	3.70607	0.265397
$196$	0	0
$197$	24.6654	1.75734	0.878668	−	0.477433i	$-0.158433\pi$
$197$	24.6654	1.75734	0.878668	+	0.477433i	$0.158433\pi$
$198$	−13.6485	−0.969956
$199$	−24.8930	−1.76462	−0.882309	−	0.470670i	$-0.844012\pi$
$199$	−24.8930	−1.76462	−0.882309	+	0.470670i	$0.844012\pi$
$200$	1.00000	0.0707107
$201$	9.81758	0.692478
$202$	11.0995	0.780957
$203$	0	0
$204$	0.816594	0.0571730
$205$	5.07689	0.354585
$206$	0.651043	0.0453603
$207$	−15.3327	−1.06570
$208$	−4.53844	−0.314685
$209$	−29.4427	−2.03660
$210$	0	0
$211$	22.8711	1.57451	0.787257	−	0.616625i	$-0.211500\pi$
$211$	22.8711	1.57451	0.787257	+	0.616625i	$0.211500\pi$
$212$	−13.9048	−0.954984
$213$	−3.67363	−0.251713
$214$	−14.2593	−0.974748
$215$	4.20479	0.286764
$216$	−4.35504	−0.296323
$217$	0	0
$218$	−4.39949	−0.297971
$219$	6.31484	0.426717
$220$	−5.84975	−0.394390
$221$	−4.53844	−0.305289
$222$	−2.65906	−0.178465
$223$	−26.0275	−1.74293	−0.871464	−	0.490459i	$-0.836829\pi$
$223$	−26.0275	−1.74293	−0.871464	+	0.490459i	$0.836829\pi$
$224$	0	0
$225$	−2.33317	−0.155545
$226$	7.45379	0.495819
$227$	−3.11125	−0.206501	−0.103251	−	0.994655i	$-0.532924\pi$
$227$	−3.11125	−0.206501	−0.103251	+	0.994655i	$0.532924\pi$
$228$	−4.11005	−0.272195
$229$	6.62262	0.437635	0.218817	−	0.975766i	$-0.429780\pi$
$229$	6.62262	0.437635	0.218817	+	0.975766i	$0.429780\pi$
$230$	−6.57160	−0.433319
$231$	0	0
$232$	−5.96636	−0.391711
$233$	−27.6885	−1.81393	−0.906966	−	0.421205i	$-0.861607\pi$
$233$	−27.6885	−1.81393	−0.906966	+	0.421205i	$0.861607\pi$
$234$	10.5890	0.692223
$235$	3.95627	0.258079
$236$	10.6817	0.695316
$237$	−6.26357	−0.406863
$238$	0	0
$239$	−13.7717	−0.890815	−0.445407	−	0.895328i	$-0.646941\pi$
$239$	−13.7717	−0.890815	−0.445407	+	0.895328i	$0.646941\pi$
$240$	−0.816594	−0.0527109
$241$	4.36888	0.281425	0.140712	−	0.990051i	$-0.455061\pi$
$241$	4.36888	0.281425	0.140712	+	0.990051i	$0.455061\pi$
$242$	23.2196	1.49261
$243$	15.8768	1.01850
$244$	0.0437295	0.00279949
$245$	0	0
$246$	−4.14576	−0.264324
$247$	22.8427	1.45345
$248$	−6.09474	−0.387017
$249$	−11.2643	−0.713846
$250$	−1.00000	−0.0632456
$251$	−28.5196	−1.80014	−0.900072	−	0.435742i	$-0.856486\pi$
$251$	−28.5196	−1.80014	−0.900072	+	0.435742i	$0.856486\pi$
$252$	0	0
$253$	38.4423	2.41684
$254$	20.4249	1.28157
$255$	−0.816594	−0.0511371
$256$	1.00000	0.0625000
$257$	0.743237	0.0463619	0.0231809	−	0.999731i	$-0.492621\pi$
$257$	0.743237	0.0463619	0.0231809	+	0.999731i	$0.492621\pi$
$258$	−3.43361	−0.213767
$259$	0	0
$260$	4.53844	0.281462
$261$	13.9206	0.861661
$262$	5.72793	0.353873
$263$	4.36586	0.269210	0.134605	−	0.990899i	$-0.457023\pi$
$263$	4.36586	0.269210	0.134605	+	0.990899i	$0.457023\pi$
$264$	4.77687	0.293996
$265$	13.9048	0.854164
$266$	0	0
$267$	−6.41334	−0.392490
$268$	12.0226	0.734397
$269$	−15.1432	−0.923298	−0.461649	−	0.887063i	$-0.652742\pi$
$269$	−15.1432	−0.923298	−0.461649	+	0.887063i	$0.652742\pi$
$270$	4.35504	0.265039
$271$	−2.60003	−0.157940	−0.0789702	−	0.996877i	$-0.525163\pi$
$271$	−2.60003	−0.157940	−0.0789702	+	0.996877i	$0.525163\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−7.11005	−0.429534
$275$	5.84975	0.352753
$276$	5.36633	0.323015
$277$	−15.8789	−0.954071	−0.477035	−	0.878884i	$-0.658289\pi$
$277$	−15.8789	−0.954071	−0.477035	+	0.878884i	$0.658289\pi$
$278$	3.89348	0.233516
$279$	14.2201	0.851335
$280$	0	0
$281$	16.2226	0.967762	0.483881	−	0.875134i	$-0.339227\pi$
$281$	16.2226	0.967762	0.483881	+	0.875134i	$0.339227\pi$
$282$	−3.23067	−0.192383
$283$	−2.17043	−0.129019	−0.0645093	−	0.997917i	$-0.520548\pi$
$283$	−2.17043	−0.129019	−0.0645093	+	0.997917i	$0.520548\pi$
$284$	−4.49873	−0.266950
$285$	4.11005	0.243458
$286$	−26.5488	−1.56986
$287$	0	0
$288$	−2.33317	−0.137484
$289$	1.00000	0.0588235
$290$	5.96636	0.350357
$291$	2.86626	0.168023
$292$	7.73315	0.452548
$293$	15.0043	0.876558	0.438279	−	0.898839i	$-0.355588\pi$
$293$	15.0043	0.876558	0.438279	+	0.898839i	$0.355588\pi$
$294$	0	0
$295$	−10.6817	−0.621910
$296$	−3.25628	−0.189268
$297$	−25.4759	−1.47826
$298$	−6.86153	−0.397478
$299$	−29.8249	−1.72482
$300$	0.816594	0.0471461
$301$	0	0
$302$	4.07120	0.234271
$303$	9.06377	0.520700
$304$	−5.03316	−0.288672
$305$	−0.0437295	−0.00250394
$306$	−2.33317	−0.133379
$307$	27.9407	1.59466	0.797331	−	0.603542i	$-0.206244\pi$
$307$	27.9407	1.59466	0.797331	+	0.603542i	$0.206244\pi$
$308$	0	0
$309$	0.531638	0.0302438
$310$	6.09474	0.346158
$311$	22.3520	1.26747	0.633733	−	0.773552i	$-0.281522\pi$
$311$	22.3520	1.26747	0.633733	+	0.773552i	$0.281522\pi$
$312$	−3.70607	−0.209815
$313$	−12.9513	−0.732052	−0.366026	−	0.930605i	$-0.619282\pi$
$313$	−12.9513	−0.732052	−0.366026	+	0.930605i	$0.619282\pi$
$314$	−8.20527	−0.463050
$315$	0	0
$316$	−7.67036	−0.431491
$317$	18.2222	1.02346	0.511729	−	0.859147i	$-0.329005\pi$
$317$	18.2222	1.02346	0.511729	+	0.859147i	$0.329005\pi$
$318$	−11.3546	−0.636732
$319$	−34.9018	−1.95412
$320$	−1.00000	−0.0559017
$321$	−11.6441	−0.649910
$322$	0	0
$323$	−5.03316	−0.280052
$324$	3.44322	0.191290
$325$	−4.53844	−0.251748
$326$	−17.9480	−0.994049
$327$	−3.59260	−0.198671
$328$	−5.07689	−0.280324
$329$	0	0
$330$	−4.77687	−0.262958
$331$	−11.8307	−0.650274	−0.325137	−	0.945667i	$-0.605410\pi$
$331$	−11.8307	−0.650274	−0.325137	+	0.945667i	$0.605410\pi$
$332$	−13.7943	−0.757058
$333$	7.59748	0.416339
$334$	−22.4978	−1.23102
$335$	−12.0226	−0.656864
$336$	0	0
$337$	−1.49993	−0.0817063	−0.0408532	−	0.999165i	$-0.513008\pi$
$337$	−1.49993	−0.0817063	−0.0408532	+	0.999165i	$0.513008\pi$
$338$	7.59748	0.413248
$339$	6.08672	0.330585
$340$	−1.00000	−0.0542326
$341$	−35.6528	−1.93070
$342$	11.7432	0.635002
$343$	0	0
$344$	−4.20479	−0.226707
$345$	−5.36633	−0.288914
$346$	−3.21256	−0.172708
$347$	7.98374	0.428590	0.214295	−	0.976769i	$-0.431255\pi$
$347$	7.98374	0.428590	0.214295	+	0.976769i	$0.431255\pi$
$348$	−4.87210	−0.261172
$349$	−0.190444	−0.0101942	−0.00509711	−	0.999987i	$-0.501622\pi$
$349$	−0.190444	−0.0101942	−0.00509711	+	0.999987i	$0.501622\pi$
$350$	0	0
$351$	19.7651	1.05498
$352$	5.84975	0.311793
$353$	8.13312	0.432882	0.216441	−	0.976296i	$-0.430555\pi$
$353$	8.13312	0.432882	0.216441	+	0.976296i	$0.430555\pi$
$354$	8.72258	0.463600
$355$	4.49873	0.238768
$356$	−7.85376	−0.416249
$357$	0	0
$358$	8.83543	0.466967
$359$	−32.6758	−1.72456	−0.862282	−	0.506429i	$-0.830965\pi$
$359$	−32.6758	−1.72456	−0.862282	+	0.506429i	$0.830965\pi$
$360$	2.33317	0.122969
$361$	6.33270	0.333300
$362$	−8.18949	−0.430430
$363$	18.9610	0.995195
$364$	0	0
$365$	−7.73315	−0.404771
$366$	0.0357093	0.00186655
$367$	3.88827	0.202966	0.101483	−	0.994837i	$-0.467641\pi$
$367$	3.88827	0.202966	0.101483	+	0.994837i	$0.467641\pi$
$368$	6.57160	0.342569
$369$	11.8453	0.616640
$370$	3.25628	0.169286
$371$	0	0
$372$	−4.97693	−0.258042
$373$	25.9270	1.34245	0.671225	−	0.741253i	$-0.265768\pi$
$373$	25.9270	1.34245	0.671225	+	0.741253i	$0.265768\pi$
$374$	5.84975	0.302484
$375$	−0.816594	−0.0421687
$376$	−3.95627	−0.204029
$377$	27.0780	1.39459
$378$	0	0
$379$	−28.1128	−1.44406	−0.722030	−	0.691862i	$-0.756791\pi$
$379$	−28.1128	−1.44406	−0.722030	+	0.691862i	$0.756791\pi$
$380$	5.03316	0.258196
$381$	16.6788	0.854483
$382$	−23.9990	−1.22790
$383$	10.5359	0.538359	0.269180	−	0.963090i	$-0.413247\pi$
$383$	10.5359	0.538359	0.269180	+	0.963090i	$0.413247\pi$
$384$	0.816594	0.0416716
$385$	0	0
$386$	−15.3969	−0.783684
$387$	9.81051	0.498696
$388$	3.51002	0.178194
$389$	−21.0052	−1.06501	−0.532503	−	0.846428i	$-0.678749\pi$
$389$	−21.0052	−1.06501	−0.532503	+	0.846428i	$0.678749\pi$
$390$	3.70607	0.187664
$391$	6.57160	0.332340
$392$	0	0
$393$	4.67740	0.235943
$394$	24.6654	1.24262
$395$	7.67036	0.385938
$396$	−13.6485	−0.685863
$397$	11.0638	0.555275	0.277637	−	0.960686i	$-0.410449\pi$
$397$	11.0638	0.555275	0.277637	+	0.960686i	$0.410449\pi$
$398$	−24.8930	−1.24777
$399$	0	0
$400$	1.00000	0.0500000
$401$	3.09914	0.154764	0.0773819	−	0.997002i	$-0.475344\pi$
$401$	3.09914	0.154764	0.0773819	+	0.997002i	$0.475344\pi$
$402$	9.81758	0.489656
$403$	27.6607	1.37788
$404$	11.0995	0.552220
$405$	−3.44322	−0.171095
$406$	0	0
$407$	−19.0485	−0.944197
$408$	0.816594	0.0404274
$409$	17.5765	0.869101	0.434551	−	0.900647i	$-0.356907\pi$
$409$	17.5765	0.869101	0.434551	+	0.900647i	$0.356907\pi$
$410$	5.07689	0.250730
$411$	−5.80602	−0.286390
$412$	0.651043	0.0320746
$413$	0	0
$414$	−15.3327	−0.753561
$415$	13.7943	0.677133
$416$	−4.53844	−0.222516
$417$	3.17940	0.155696
$418$	−29.4427	−1.44009
$419$	26.3257	1.28609	0.643046	−	0.765827i	$-0.277670\pi$
$419$	26.3257	1.28609	0.643046	+	0.765827i	$0.277670\pi$
$420$	0	0
$421$	14.5206	0.707690	0.353845	−	0.935304i	$-0.384874\pi$
$421$	14.5206	0.707690	0.353845	+	0.935304i	$0.384874\pi$
$422$	22.8711	1.11335
$423$	9.23067	0.448810
$424$	−13.9048	−0.675276
$425$	1.00000	0.0485071
$426$	−3.67363	−0.177988
$427$	0	0
$428$	−14.2593	−0.689251
$429$	−21.6796	−1.04670
$430$	4.20479	0.202773
$431$	4.51891	0.217668	0.108834	−	0.994060i	$-0.465288\pi$
$431$	4.51891	0.217668	0.108834	+	0.994060i	$0.465288\pi$
$432$	−4.35504	−0.209532
$433$	21.5608	1.03615	0.518073	−	0.855336i	$-0.326649\pi$
$433$	21.5608	1.03615	0.518073	+	0.855336i	$0.326649\pi$
$434$	0	0
$435$	4.87210	0.233599
$436$	−4.39949	−0.210698
$437$	−33.0759	−1.58224
$438$	6.31484	0.301735
$439$	−16.7239	−0.798189	−0.399095	−	0.916910i	$-0.630676\pi$
$439$	−16.7239	−0.798189	−0.399095	+	0.916910i	$0.630676\pi$
$440$	−5.84975	−0.278876
$441$	0	0
$442$	−4.53844	−0.215872
$443$	24.9741	1.18656	0.593278	−	0.804997i	$-0.297833\pi$
$443$	24.9741	1.18656	0.593278	+	0.804997i	$0.297833\pi$
$444$	−2.65906	−0.126194
$445$	7.85376	0.372304
$446$	−26.0275	−1.23244
$447$	−5.60308	−0.265017
$448$	0	0
$449$	30.0889	1.41998	0.709992	−	0.704210i	$-0.248699\pi$
$449$	30.0889	1.41998	0.709992	+	0.704210i	$0.248699\pi$
$450$	−2.33317	−0.109987
$451$	−29.6986	−1.39845
$452$	7.45379	0.350597
$453$	3.32452	0.156199
$454$	−3.11125	−0.146018
$455$	0	0
$456$	−4.11005	−0.192471
$457$	6.14576	0.287486	0.143743	−	0.989615i	$-0.454086\pi$
$457$	6.14576	0.287486	0.143743	+	0.989615i	$0.454086\pi$
$458$	6.62262	0.309455
$459$	−4.35504	−0.203276
$460$	−6.57160	−0.306403
$461$	−23.4047	−1.09007	−0.545033	−	0.838415i	$-0.683483\pi$
$461$	−23.4047	−1.09007	−0.545033	+	0.838415i	$0.683483\pi$
$462$	0	0
$463$	3.32782	0.154657	0.0773284	−	0.997006i	$-0.475361\pi$
$463$	3.32782	0.154657	0.0773284	+	0.997006i	$0.475361\pi$
$464$	−5.96636	−0.276981
$465$	4.97693	0.230800
$466$	−27.6885	−1.28264
$467$	−20.2070	−0.935067	−0.467534	−	0.883975i	$-0.654857\pi$
$467$	−20.2070	−0.935067	−0.467534	+	0.883975i	$0.654857\pi$
$468$	10.5890	0.489476
$469$	0	0
$470$	3.95627	0.182489
$471$	−6.70038	−0.308737
$472$	10.6817	0.491663
$473$	−24.5970	−1.13097
$474$	−6.26357	−0.287695
$475$	−5.03316	−0.230937
$476$	0	0
$477$	32.4423	1.48543
$478$	−13.7717	−0.629901
$479$	−6.87355	−0.314060	−0.157030	−	0.987594i	$-0.550192\pi$
$479$	−6.87355	−0.314060	−0.157030	+	0.987594i	$0.550192\pi$
$480$	−0.816594	−0.0372723
$481$	14.7785	0.673840
$482$	4.36888	0.198997
$483$	0	0
$484$	23.2196	1.05544
$485$	−3.51002	−0.159382
$486$	15.8768	0.720188
$487$	11.3757	0.515482	0.257741	−	0.966214i	$-0.417022\pi$
$487$	11.3757	0.515482	0.257741	+	0.966214i	$0.417022\pi$
$488$	0.0437295	0.00197954
$489$	−14.6563	−0.662779
$490$	0	0
$491$	−13.2890	−0.599723	−0.299861	−	0.953983i	$-0.596940\pi$
$491$	−13.2890	−0.599723	−0.299861	+	0.953983i	$0.596940\pi$
$492$	−4.14576	−0.186905
$493$	−5.96636	−0.268711
$494$	22.8427	1.02774
$495$	13.6485	0.613454
$496$	−6.09474	−0.273662
$497$	0	0
$498$	−11.2643	−0.504766
$499$	−20.5112	−0.918208	−0.459104	−	0.888382i	$-0.651830\pi$
$499$	−20.5112	−0.918208	−0.459104	+	0.888382i	$0.651830\pi$
$500$	−1.00000	−0.0447214
$501$	−18.3715	−0.820780
$502$	−28.5196	−1.27289
$503$	−30.2906	−1.35059	−0.675295	−	0.737548i	$-0.735984\pi$
$503$	−30.2906	−1.35059	−0.675295	+	0.737548i	$0.735984\pi$
$504$	0	0
$505$	−11.0995	−0.493920
$506$	38.4423	1.70897
$507$	6.20406	0.275532
$508$	20.4249	0.906208
$509$	19.1432	0.848508	0.424254	−	0.905543i	$-0.360536\pi$
$509$	19.1432	0.848508	0.424254	+	0.905543i	$0.360536\pi$
$510$	−0.816594	−0.0361594
$511$	0	0
$512$	1.00000	0.0441942
$513$	21.9196	0.967774
$514$	0.743237	0.0327828
$515$	−0.651043	−0.0286884
$516$	−3.43361	−0.151156
$517$	−23.1432	−1.01784
$518$	0	0
$519$	−2.62335	−0.115152
$520$	4.53844	0.199024
$521$	−25.5629	−1.11993	−0.559965	−	0.828516i	$-0.689185\pi$
$521$	−25.5629	−1.11993	−0.559965	+	0.828516i	$0.689185\pi$
$522$	13.9206	0.609286
$523$	−33.0344	−1.44449	−0.722246	−	0.691636i	$-0.756890\pi$
$523$	−33.0344	−1.44449	−0.722246	+	0.691636i	$0.756890\pi$
$524$	5.72793	0.250226
$525$	0	0
$526$	4.36586	0.190360
$527$	−6.09474	−0.265491
$528$	4.77687	0.207887
$529$	20.1860	0.877651
$530$	13.9048	0.603985
$531$	−24.9222	−1.08153
$532$	0	0
$533$	23.0412	0.998024
$534$	−6.41334	−0.277532
$535$	14.2593	0.616485
$536$	12.0226	0.519297
$537$	7.21496	0.311349
$538$	−15.1432	−0.652870
$539$	0	0
$540$	4.35504	0.187411
$541$	11.6412	0.500495	0.250247	−	0.968182i	$-0.419488\pi$
$541$	11.6412	0.500495	0.250247	+	0.968182i	$0.419488\pi$
$542$	−2.60003	−0.111681
$543$	−6.68749	−0.286988
$544$	1.00000	0.0428746
$545$	4.39949	0.188454
$546$	0	0
$547$	7.14696	0.305582	0.152791	−	0.988259i	$-0.451174\pi$
$547$	7.14696	0.305582	0.152791	+	0.988259i	$0.451174\pi$
$548$	−7.11005	−0.303726
$549$	−0.102029	−0.00435447
$550$	5.84975	0.249434
$551$	30.0297	1.27931
$552$	5.36633	0.228406
$553$	0	0
$554$	−15.8789	−0.674630
$555$	2.65906	0.112871
$556$	3.89348	0.165120
$557$	−1.48510	−0.0629258	−0.0314629	−	0.999505i	$-0.510017\pi$
$557$	−1.48510	−0.0629258	−0.0314629	+	0.999505i	$0.510017\pi$
$558$	14.2201	0.601985
$559$	19.0832	0.807134
$560$	0	0
$561$	4.77687	0.201680
$562$	16.2226	0.684311
$563$	40.3479	1.70046	0.850230	−	0.526412i	$-0.176463\pi$
$563$	40.3479	1.70046	0.850230	+	0.526412i	$0.176463\pi$
$564$	−3.23067	−0.136036
$565$	−7.45379	−0.313583
$566$	−2.17043	−0.0912300
$567$	0	0
$568$	−4.49873	−0.188762
$569$	−4.45413	−0.186727	−0.0933634	−	0.995632i	$-0.529762\pi$
$569$	−4.45413	−0.186727	−0.0933634	+	0.995632i	$0.529762\pi$
$570$	4.11005	0.172151
$571$	−18.2048	−0.761847	−0.380923	−	0.924607i	$-0.624394\pi$
$571$	−18.2048	−0.761847	−0.380923	+	0.924607i	$0.624394\pi$
$572$	−26.5488	−1.11006
$573$	−19.5975	−0.818696
$574$	0	0
$575$	6.57160	0.274055
$576$	−2.33317	−0.0972156
$577$	−26.4000	−1.09905	−0.549523	−	0.835479i	$-0.685190\pi$
$577$	−26.4000	−1.09905	−0.549523	+	0.835479i	$0.685190\pi$
$578$	1.00000	0.0415945
$579$	−12.5731	−0.522518
$580$	5.96636	0.247740
$581$	0	0
$582$	2.86626	0.118810
$583$	−81.3395	−3.36874
$584$	7.73315	0.320000
$585$	−10.5890	−0.437800
$586$	15.0043	0.619820
$587$	20.2638	0.836378	0.418189	−	0.908360i	$-0.362665\pi$
$587$	20.2638	0.836378	0.418189	+	0.908360i	$0.362665\pi$
$588$	0	0
$589$	30.6758	1.26398
$590$	−10.6817	−0.439757
$591$	20.1416	0.828515
$592$	−3.25628	−0.133832
$593$	−14.6659	−0.602255	−0.301128	−	0.953584i	$-0.597363\pi$
$593$	−14.6659	−0.602255	−0.301128	+	0.953584i	$0.597363\pi$
$594$	−25.4759	−1.04529
$595$	0	0
$596$	−6.86153	−0.281059
$597$	−20.3275	−0.831948
$598$	−29.8249	−1.21963
$599$	−36.0073	−1.47122	−0.735609	−	0.677407i	$-0.763104\pi$
$599$	−36.0073	−1.47122	−0.735609	+	0.677407i	$0.763104\pi$
$600$	0.816594	0.0333373
$601$	37.4519	1.52769	0.763847	−	0.645397i	$-0.223308\pi$
$601$	37.4519	1.52769	0.763847	+	0.645397i	$0.223308\pi$
$602$	0	0
$603$	−28.0508	−1.14232
$604$	4.07120	0.165655
$605$	−23.2196	−0.944012
$606$	9.06377	0.368190
$607$	−2.63742	−0.107050	−0.0535248	−	0.998567i	$-0.517046\pi$
$607$	−2.63742	−0.107050	−0.0535248	+	0.998567i	$0.517046\pi$
$608$	−5.03316	−0.204122
$609$	0	0
$610$	−0.0437295	−0.00177056
$611$	17.9553	0.726394
$612$	−2.33317	−0.0943130
$613$	40.9887	1.65552	0.827760	−	0.561083i	$-0.189615\pi$
$613$	40.9887	1.65552	0.827760	+	0.561083i	$0.189615\pi$
$614$	27.9407	1.12760
$615$	4.14576	0.167173
$616$	0	0
$617$	−30.6838	−1.23528	−0.617642	−	0.786459i	$-0.711912\pi$
$617$	−30.6838	−1.23528	−0.617642	+	0.786459i	$0.711912\pi$
$618$	0.531638	0.0213856
$619$	−6.78817	−0.272840	−0.136420	−	0.990651i	$-0.543560\pi$
$619$	−6.78817	−0.272840	−0.136420	+	0.990651i	$0.543560\pi$
$620$	6.09474	0.244771
$621$	−28.6196	−1.14846
$622$	22.3520	0.896234
$623$	0	0
$624$	−3.70607	−0.148361
$625$	1.00000	0.0400000
$626$	−12.9513	−0.517639
$627$	−24.0428	−0.960176
$628$	−8.20527	−0.327426
$629$	−3.25628	−0.129837
$630$	0	0
$631$	−13.4306	−0.534663	−0.267332	−	0.963605i	$-0.586142\pi$
$631$	−13.4306	−0.534663	−0.267332	+	0.963605i	$0.586142\pi$
$632$	−7.67036	−0.305110
$633$	18.6764	0.742322
$634$	18.2222	0.723695
$635$	−20.4249	−0.810537
$636$	−11.3546	−0.450238
$637$	0	0
$638$	−34.9018	−1.38177
$639$	10.4963	0.415228
$640$	−1.00000	−0.0395285
$641$	−15.5305	−0.613419	−0.306710	−	0.951803i	$-0.599228\pi$
$641$	−15.5305	−0.613419	−0.306710	+	0.951803i	$0.599228\pi$
$642$	−11.6441	−0.459556
$643$	−11.8772	−0.468392	−0.234196	−	0.972189i	$-0.575246\pi$
$643$	−11.8772	−0.468392	−0.234196	+	0.972189i	$0.575246\pi$
$644$	0	0
$645$	3.43361	0.135198
$646$	−5.03316	−0.198027
$647$	42.1244	1.65608	0.828040	−	0.560668i	$-0.189456\pi$
$647$	42.1244	1.65608	0.828040	+	0.560668i	$0.189456\pi$
$648$	3.44322	0.135263
$649$	62.4850	2.45275
$650$	−4.53844	−0.178012
$651$	0	0
$652$	−17.9480	−0.702899
$653$	21.2015	0.829679	0.414840	−	0.909895i	$-0.363838\pi$
$653$	21.2015	0.829679	0.414840	+	0.909895i	$0.363838\pi$
$654$	−3.59260	−0.140482
$655$	−5.72793	−0.223809
$656$	−5.07689	−0.198219
$657$	−18.0428	−0.703916
$658$	0	0
$659$	41.7673	1.62702	0.813511	−	0.581549i	$-0.197553\pi$
$659$	41.7673	1.62702	0.813511	+	0.581549i	$0.197553\pi$
$660$	−4.77687	−0.185940
$661$	37.2533	1.44898	0.724492	−	0.689283i	$-0.242074\pi$
$661$	37.2533	1.44898	0.724492	+	0.689283i	$0.242074\pi$
$662$	−11.8307	−0.459813
$663$	−3.70607	−0.143932
$664$	−13.7943	−0.535321
$665$	0	0
$666$	7.59748	0.294396
$667$	−39.2086	−1.51816
$668$	−22.4978	−0.870465
$669$	−21.2539	−0.821722
$670$	−12.0226	−0.464473
$671$	0.255807	0.00987531
$672$	0	0
$673$	37.5956	1.44920	0.724601	−	0.689168i	$-0.242024\pi$
$673$	37.5956	1.44920	0.724601	+	0.689168i	$0.242024\pi$
$674$	−1.49993	−0.0577751
$675$	−4.35504	−0.167626
$676$	7.59748	0.292211
$677$	10.1091	0.388524	0.194262	−	0.980950i	$-0.437769\pi$
$677$	10.1091	0.388524	0.194262	+	0.980950i	$0.437769\pi$
$678$	6.08672	0.233759
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	−2.54063	−0.0973572
$682$	−35.6528	−1.36521
$683$	18.9419	0.724793	0.362397	−	0.932024i	$-0.381959\pi$
$683$	18.9419	0.724793	0.362397	+	0.932024i	$0.381959\pi$
$684$	11.7432	0.449014
$685$	7.11005	0.271661
$686$	0	0
$687$	5.40799	0.206328
$688$	−4.20479	−0.160306
$689$	63.1061	2.40415
$690$	−5.36633	−0.204293
$691$	−7.05648	−0.268441	−0.134221	−	0.990951i	$-0.542853\pi$
$691$	−7.05648	−0.268441	−0.134221	+	0.990951i	$0.542853\pi$
$692$	−3.21256	−0.122123
$693$	0	0
$694$	7.98374	0.303059
$695$	−3.89348	−0.147688
$696$	−4.87210	−0.184676
$697$	−5.07689	−0.192301
$698$	−0.190444	−0.00720840
$699$	−22.6102	−0.855198
$700$	0	0
$701$	−1.81491	−0.0685483	−0.0342742	−	0.999412i	$-0.510912\pi$
$701$	−1.81491	−0.0685483	−0.0342742	+	0.999412i	$0.510912\pi$
$702$	19.7651	0.745986
$703$	16.3894	0.618138
$704$	5.84975	0.220471
$705$	3.23067	0.121674
$706$	8.13312	0.306094
$707$	0	0
$708$	8.72258	0.327814
$709$	10.4477	0.392372	0.196186	−	0.980567i	$-0.437144\pi$
$709$	10.4477	0.392372	0.196186	+	0.980567i	$0.437144\pi$
$710$	4.49873	0.168834
$711$	17.8963	0.671163
$712$	−7.85376	−0.294332
$713$	−40.0522	−1.49997
$714$	0	0
$715$	26.5488	0.992868
$716$	8.83543	0.330196
$717$	−11.2459	−0.419984
$718$	−32.6758	−1.21945
$719$	27.7418	1.03459	0.517297	−	0.855806i	$-0.326938\pi$
$719$	27.7418	1.03459	0.517297	+	0.855806i	$0.326938\pi$
$720$	2.33317	0.0869523
$721$	0	0
$722$	6.33270	0.235679
$723$	3.56760	0.132681
$724$	−8.18949	−0.304360
$725$	−5.96636	−0.221585
$726$	18.9610	0.703709
$727$	41.2137	1.52853	0.764265	−	0.644902i	$-0.223102\pi$
$727$	41.2137	1.52853	0.764265	+	0.644902i	$0.223102\pi$
$728$	0	0
$729$	2.63526	0.0976022
$730$	−7.73315	−0.286217
$731$	−4.20479	−0.155520
$732$	0.0357093	0.00131985
$733$	32.2400	1.19081	0.595406	−	0.803425i	$-0.296991\pi$
$733$	32.2400	1.19081	0.595406	+	0.803425i	$0.296991\pi$
$734$	3.88827	0.143519
$735$	0	0
$736$	6.57160	0.242233
$737$	70.3292	2.59061
$738$	11.8453	0.436030
$739$	33.5067	1.23256	0.616281	−	0.787526i	$-0.288638\pi$
$739$	33.5067	1.23256	0.616281	+	0.787526i	$0.288638\pi$
$740$	3.25628	0.119703
$741$	18.6532	0.685243
$742$	0	0
$743$	10.8697	0.398770	0.199385	−	0.979921i	$-0.436106\pi$
$743$	10.8697	0.398770	0.199385	+	0.979921i	$0.436106\pi$
$744$	−4.97693	−0.182463
$745$	6.86153	0.251387
$746$	25.9270	0.949256
$747$	32.1844	1.17757
$748$	5.84975	0.213888
$749$	0	0
$750$	−0.816594	−0.0298178
$751$	18.1771	0.663292	0.331646	−	0.943404i	$-0.392396\pi$
$751$	18.1771	0.663292	0.331646	+	0.943404i	$0.392396\pi$
$752$	−3.95627	−0.144270
$753$	−23.2890	−0.848697
$754$	27.0780	0.986123
$755$	−4.07120	−0.148166
$756$	0	0
$757$	−37.9958	−1.38098	−0.690490	−	0.723342i	$-0.742605\pi$
$757$	−37.9958	−1.38098	−0.690490	+	0.723342i	$0.742605\pi$
$758$	−28.1128	−1.02110
$759$	31.3917	1.13945
$760$	5.03316	0.182572
$761$	−45.7314	−1.65776	−0.828881	−	0.559426i	$-0.811022\pi$
$761$	−45.7314	−1.65776	−0.828881	+	0.559426i	$0.811022\pi$
$762$	16.6788	0.604211
$763$	0	0
$764$	−23.9990	−0.868255
$765$	2.33317	0.0843561
$766$	10.5359	0.380677
$767$	−48.4781	−1.75044
$768$	0.816594	0.0294663
$769$	37.2625	1.34372	0.671861	−	0.740678i	$-0.265495\pi$
$769$	37.2625	1.34372	0.671861	+	0.740678i	$0.265495\pi$
$770$	0	0
$771$	0.606923	0.0218578
$772$	−15.3969	−0.554148
$773$	27.8021	0.999973	0.499987	−	0.866033i	$-0.333338\pi$
$773$	27.8021	0.999973	0.499987	+	0.866033i	$0.333338\pi$
$774$	9.81051	0.352632
$775$	−6.09474	−0.218930
$776$	3.51002	0.126002
$777$	0	0
$778$	−21.0052	−0.753073
$779$	25.5528	0.915524
$780$	3.70607	0.132698
$781$	−26.3164	−0.941676
$782$	6.57160	0.235000
$783$	25.9837	0.928583
$784$	0	0
$785$	8.20527	0.292859
$786$	4.67740	0.166837
$787$	−14.1271	−0.503576	−0.251788	−	0.967782i	$-0.581019\pi$
$787$	−14.1271	−0.503576	−0.251788	+	0.967782i	$0.581019\pi$
$788$	24.6654	0.878668
$789$	3.56513	0.126922
$790$	7.67036	0.272899
$791$	0	0
$792$	−13.6485	−0.484978
$793$	−0.198464	−0.00704766
$794$	11.0638	0.392638
$795$	11.3546	0.402705
$796$	−24.8930	−0.882309
$797$	22.6562	0.802523	0.401262	−	0.915964i	$-0.368572\pi$
$797$	22.6562	0.802523	0.401262	+	0.915964i	$0.368572\pi$
$798$	0	0
$799$	−3.95627	−0.139963
$800$	1.00000	0.0353553
$801$	18.3242	0.647454
$802$	3.09914	0.109435
$803$	45.2370	1.59638
$804$	9.81758	0.346239
$805$	0	0
$806$	27.6607	0.974305
$807$	−12.3659	−0.435299
$808$	11.0995	0.390478
$809$	22.8477	0.803283	0.401641	−	0.915797i	$-0.368440\pi$
$809$	22.8477	0.803283	0.401641	+	0.915797i	$0.368440\pi$
$810$	−3.44322	−0.120983
$811$	30.9767	1.08774	0.543870	−	0.839170i	$-0.316959\pi$
$811$	30.9767	1.08774	0.543870	+	0.839170i	$0.316959\pi$
$812$	0	0
$813$	−2.12317	−0.0744628
$814$	−19.0485	−0.667648
$815$	17.9480	0.628692
$816$	0.816594	0.0285865
$817$	21.1634	0.740413
$818$	17.5765	0.614547
$819$	0	0
$820$	5.07689	0.177293
$821$	3.83406	0.133810	0.0669048	−	0.997759i	$-0.478688\pi$
$821$	3.83406	0.133810	0.0669048	+	0.997759i	$0.478688\pi$
$822$	−5.80602	−0.202508
$823$	−5.57247	−0.194244	−0.0971221	−	0.995272i	$-0.530964\pi$
$823$	−5.57247	−0.194244	−0.0971221	+	0.995272i	$0.530964\pi$
$824$	0.651043	0.0226802
$825$	4.77687	0.166309
$826$	0	0
$827$	2.74540	0.0954668	0.0477334	−	0.998860i	$-0.484800\pi$
$827$	2.74540	0.0954668	0.0477334	+	0.998860i	$0.484800\pi$
$828$	−15.3327	−0.532848
$829$	12.2389	0.425075	0.212537	−	0.977153i	$-0.431827\pi$
$829$	12.2389	0.425075	0.212537	+	0.977153i	$0.431827\pi$
$830$	13.7943	0.478805
$831$	−12.9666	−0.449807
$832$	−4.53844	−0.157342
$833$	0	0
$834$	3.17940	0.110093
$835$	22.4978	0.778567
$836$	−29.4427	−1.01830
$837$	26.5428	0.917455
$838$	26.3257	0.909405
$839$	−10.1766	−0.351335	−0.175667	−	0.984450i	$-0.556208\pi$
$839$	−10.1766	−0.351335	−0.175667	+	0.984450i	$0.556208\pi$
$840$	0	0
$841$	6.59748	0.227499
$842$	14.5206	0.500413
$843$	13.2473	0.456262
$844$	22.8711	0.787257
$845$	−7.59748	−0.261361
$846$	9.23067	0.317357
$847$	0	0
$848$	−13.9048	−0.477492
$849$	−1.77236	−0.0608272
$850$	1.00000	0.0342997
$851$	−21.3990	−0.733549
$852$	−3.67363	−0.125857
$853$	−27.3969	−0.938053	−0.469027	−	0.883184i	$-0.655395\pi$
$853$	−27.3969	−0.938053	−0.469027	+	0.883184i	$0.655395\pi$
$854$	0	0
$855$	−11.7432	−0.401610
$856$	−14.2593	−0.487374
$857$	18.0461	0.616444	0.308222	−	0.951314i	$-0.400266\pi$
$857$	18.0461	0.616444	0.308222	+	0.951314i	$0.400266\pi$
$858$	−21.6796	−0.740129
$859$	16.2225	0.553505	0.276752	−	0.960941i	$-0.410742\pi$
$859$	16.2225	0.553505	0.276752	+	0.960941i	$0.410742\pi$
$860$	4.20479	0.143382
$861$	0	0
$862$	4.51891	0.153915
$863$	−16.5634	−0.563823	−0.281912	−	0.959440i	$-0.590969\pi$
$863$	−16.5634	−0.563823	−0.281912	+	0.959440i	$0.590969\pi$
$864$	−4.35504	−0.148161
$865$	3.21256	0.109230
$866$	21.5608	0.732666
$867$	0.816594	0.0277330
$868$	0	0
$869$	−44.8697	−1.52210
$870$	4.87210	0.165180
$871$	−54.5639	−1.84883
$872$	−4.39949	−0.148986
$873$	−8.18949	−0.277172
$874$	−33.0759	−1.11881
$875$	0	0
$876$	6.31484	0.213359
$877$	−23.5091	−0.793845	−0.396922	−	0.917852i	$-0.629922\pi$
$877$	−23.5091	−0.793845	−0.396922	+	0.917852i	$0.629922\pi$
$878$	−16.7239	−0.564405
$879$	12.2524	0.413263
$880$	−5.84975	−0.197195
$881$	−39.7518	−1.33927	−0.669635	−	0.742690i	$-0.733550\pi$
$881$	−39.7518	−1.33927	−0.669635	+	0.742690i	$0.733550\pi$
$882$	0	0
$883$	−12.6768	−0.426607	−0.213304	−	0.976986i	$-0.568422\pi$
$883$	−12.6768	−0.426607	−0.213304	+	0.976986i	$0.568422\pi$
$884$	−4.53844	−0.152644
$885$	−8.72258	−0.293206
$886$	24.9741	0.839022
$887$	−4.03229	−0.135391	−0.0676955	−	0.997706i	$-0.521565\pi$
$887$	−4.03229	−0.135391	−0.0676955	+	0.997706i	$0.521565\pi$
$888$	−2.65906	−0.0892323
$889$	0	0
$890$	7.85376	0.263259
$891$	20.1420	0.674783
$892$	−26.0275	−0.871464
$893$	19.9125	0.666348
$894$	−5.60308	−0.187395
$895$	−8.83543	−0.295336
$896$	0	0
$897$	−24.3548	−0.813183
$898$	30.0889	1.00408
$899$	36.3634	1.21279
$900$	−2.33317	−0.0777725
$901$	−13.9048	−0.463235
$902$	−29.6986	−0.988854
$903$	0	0
$904$	7.45379	0.247909
$905$	8.18949	0.272228
$906$	3.32452	0.110450
$907$	−47.3978	−1.57382	−0.786910	−	0.617068i	$-0.788320\pi$
$907$	−47.3978	−1.57382	−0.786910	+	0.617068i	$0.788320\pi$
$908$	−3.11125	−0.103251
$909$	−25.8970	−0.858950
$910$	0	0
$911$	27.8023	0.921130	0.460565	−	0.887626i	$-0.347647\pi$
$911$	27.8023	0.921130	0.460565	+	0.887626i	$0.347647\pi$
$912$	−4.11005	−0.136097
$913$	−80.6930	−2.67055
$914$	6.14576	0.203284
$915$	−0.0357093	−0.00118051
$916$	6.62262	0.218817
$917$	0	0
$918$	−4.35504	−0.143738
$919$	−35.9165	−1.18477	−0.592387	−	0.805653i	$-0.701814\pi$
$919$	−35.9165	−1.18477	−0.592387	+	0.805653i	$0.701814\pi$
$920$	−6.57160	−0.216659
$921$	22.8162	0.751821
$922$	−23.4047	−0.770793
$923$	20.4172	0.672041
$924$	0	0
$925$	−3.25628	−0.107066
$926$	3.32782	0.109359
$927$	−1.51900	−0.0498904
$928$	−5.96636	−0.195855
$929$	19.5763	0.642279	0.321139	−	0.947032i	$-0.395934\pi$
$929$	19.5763	0.642279	0.321139	+	0.947032i	$0.395934\pi$
$930$	4.97693	0.163200
$931$	0	0
$932$	−27.6885	−0.906966
$933$	18.2525	0.597561
$934$	−20.2070	−0.661193
$935$	−5.84975	−0.191307
$936$	10.5890	0.346112
$937$	−50.2723	−1.64233	−0.821163	−	0.570694i	$-0.806674\pi$
$937$	−50.2723	−1.64233	−0.821163	+	0.570694i	$0.806674\pi$
$938$	0	0
$939$	−10.5760	−0.345134
$940$	3.95627	0.129039
$941$	−18.8397	−0.614156	−0.307078	−	0.951684i	$-0.599351\pi$
$941$	−18.8397	−0.614156	−0.307078	+	0.951684i	$0.599351\pi$
$942$	−6.70038	−0.218310
$943$	−33.3633	−1.08646
$944$	10.6817	0.347658
$945$	0	0
$946$	−24.5970	−0.799717
$947$	54.5904	1.77395	0.886975	−	0.461816i	$-0.152802\pi$
$947$	54.5904	1.77395	0.886975	+	0.461816i	$0.152802\pi$
$948$	−6.26357	−0.203431
$949$	−35.0965	−1.13928
$950$	−5.03316	−0.163297
$951$	14.8801	0.482521
$952$	0	0
$953$	13.0559	0.422922	0.211461	−	0.977386i	$-0.432178\pi$
$953$	13.0559	0.422922	0.211461	+	0.977386i	$0.432178\pi$
$954$	32.4423	1.05036
$955$	23.9990	0.776591
$956$	−13.7717	−0.445407
$957$	−28.5006	−0.921292
$958$	−6.87355	−0.222074
$959$	0	0
$960$	−0.816594	−0.0263555
$961$	6.14590	0.198255
$962$	14.7785	0.476477
$963$	33.2695	1.07210
$964$	4.36888	0.140712
$965$	15.3969	0.495645
$966$	0	0
$967$	−23.6607	−0.760875	−0.380438	−	0.924807i	$-0.624227\pi$
$967$	−23.6607	−0.760875	−0.380438	+	0.924807i	$0.624227\pi$
$968$	23.2196	0.746307
$969$	−4.11005	−0.132034
$970$	−3.51002	−0.112700
$971$	44.9081	1.44117	0.720584	−	0.693367i	$-0.243874\pi$
$971$	44.9081	1.44117	0.720584	+	0.693367i	$0.243874\pi$
$972$	15.8768	0.509250
$973$	0	0
$974$	11.3757	0.364501
$975$	−3.70607	−0.118689
$976$	0.0437295	0.00139975
$977$	48.5719	1.55395	0.776976	−	0.629530i	$-0.216753\pi$
$977$	48.5719	1.55395	0.776976	+	0.629530i	$0.216753\pi$
$978$	−14.6563	−0.468655
$979$	−45.9426	−1.46833
$980$	0	0
$981$	10.2648	0.327729
$982$	−13.2890	−0.424068
$983$	−35.9187	−1.14563	−0.572814	−	0.819686i	$-0.694148\pi$
$983$	−35.9187	−1.14563	−0.572814	+	0.819686i	$0.694148\pi$
$984$	−4.14576	−0.132162
$985$	−24.6654	−0.785905
$986$	−5.96636	−0.190008
$987$	0	0
$988$	22.8427	0.726724
$989$	−27.6322	−0.878654
$990$	13.6485	0.433778
$991$	29.2039	0.927693	0.463847	−	0.885916i	$-0.346469\pi$
$991$	29.2039	0.927693	0.463847	+	0.885916i	$0.346469\pi$
$992$	−6.09474	−0.193508
$993$	−9.66088	−0.306579
$994$	0	0
$995$	24.8930	0.789161
$996$	−11.2643	−0.356923
$997$	17.2940	0.547705	0.273853	−	0.961772i	$-0.411702\pi$
$997$	17.2940	0.547705	0.273853	+	0.961772i	$0.411702\pi$
$998$	−20.5112	−0.649271
$999$	14.1812	0.448675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cd.1.3		4
7.2	even	3		1190.2.i.k.851.2	yes	8
7.4	even	3		1190.2.i.k.681.2	✓	8
7.6	odd	2		8330.2.a.cl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.i.k.681.2	✓	8	7.4	even	3
1190.2.i.k.851.2	yes	8	7.2	even	3
8330.2.a.cd.1.3		4	1.1	even	1	trivial
8330.2.a.cl.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.816594 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.816594 q^{6} +1.00000 q^{8} -2.33317 q^{9} -1.00000 q^{10} +5.84975 q^{11} +0.816594 q^{12} -4.53844 q^{13} -0.816594 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.33317 q^{18} -5.03316 q^{19} -1.00000 q^{20} +5.84975 q^{22} +6.57160 q^{23} +0.816594 q^{24} +1.00000 q^{25} -4.53844 q^{26} -4.35504 q^{27} -5.96636 q^{29} -0.816594 q^{30} -6.09474 q^{31} +1.00000 q^{32} +4.77687 q^{33} +1.00000 q^{34} -2.33317 q^{36} -3.25628 q^{37} -5.03316 q^{38} -3.70607 q^{39} -1.00000 q^{40} -5.07689 q^{41} -4.20479 q^{43} +5.84975 q^{44} +2.33317 q^{45} +6.57160 q^{46} -3.95627 q^{47} +0.816594 q^{48} +1.00000 q^{50} +0.816594 q^{51} -4.53844 q^{52} -13.9048 q^{53} -4.35504 q^{54} -5.84975 q^{55} -4.11005 q^{57} -5.96636 q^{58} +10.6817 q^{59} -0.816594 q^{60} +0.0437295 q^{61} -6.09474 q^{62} +1.00000 q^{64} +4.53844 q^{65} +4.77687 q^{66} +12.0226 q^{67} +1.00000 q^{68} +5.36633 q^{69} -4.49873 q^{71} -2.33317 q^{72} +7.73315 q^{73} -3.25628 q^{74} +0.816594 q^{75} -5.03316 q^{76} -3.70607 q^{78} -7.67036 q^{79} -1.00000 q^{80} +3.44322 q^{81} -5.07689 q^{82} -13.7943 q^{83} -1.00000 q^{85} -4.20479 q^{86} -4.87210 q^{87} +5.84975 q^{88} -7.85376 q^{89} +2.33317 q^{90} +6.57160 q^{92} -4.97693 q^{93} -3.95627 q^{94} +5.03316 q^{95} +0.816594 q^{96} +3.51002 q^{97} -13.6485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} - 2 q^{12} - 14 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} - 4 q^{19} - 4 q^{20} + 2 q^{22} + 6 q^{23}+ \cdots - 26 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^10 + 2 * q^11 - 2 * q^12 - 14 * q^13 + 2 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 - 4 * q^19 - 4 * q^20 + 2 * q^22 + 6 * q^23 - 2 * q^24 + 4 * q^25 - 14 * q^26 - 8 * q^27 + 2 * q^30 - 18 * q^31 + 4 * q^32 + 8 * q^33 + 4 * q^34 + 4 * q^36 - 8 * q^37 - 4 * q^38 + 6 * q^39 - 4 * q^40 - 12 * q^41 + 14 * q^43 + 2 * q^44 - 4 * q^45 + 6 * q^46 - 8 * q^47 - 2 * q^48 + 4 * q^50 - 2 * q^51 - 14 * q^52 - 22 * q^53 - 8 * q^54 - 2 * q^55 + 8 * q^57 - 2 * q^59 + 2 * q^60 + 8 * q^61 - 18 * q^62 + 4 * q^64 + 14 * q^65 + 8 * q^66 + 8 * q^67 + 4 * q^68 - 8 * q^69 - 8 * q^71 + 4 * q^72 + 12 * q^73 - 8 * q^74 - 2 * q^75 - 4 * q^76 + 6 * q^78 - 6 * q^79 - 4 * q^80 - 24 * q^81 - 12 * q^82 - 6 * q^83 - 4 * q^85 + 14 * q^86 - 42 * q^87 + 2 * q^88 - 12 * q^89 - 4 * q^90 + 6 * q^92 - 20 * q^93 - 8 * q^94 + 4 * q^95 - 2 * q^96 - 16 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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