LMFDB - Embedded newform 650.4.b.h.599.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,4,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-16,0,-32,0,0,44,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{454})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 51529 $ x^4 + 51529
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.3
Root			$-10.6536 - 10.6536i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.4.b.h.599.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +4.00000i q^{3} -4.00000 q^{4} -8.00000 q^{6} -16.3073i q^{7} -8.00000i q^{8} +11.0000 q^{9} -10.3073 q^{11} -16.0000i q^{12} +13.0000i q^{13} +32.6146 q^{14} +16.0000 q^{16} +95.6146i q^{17} +22.0000i q^{18} +59.2291 q^{19} +65.2291 q^{21} -20.6146i q^{22} -57.8437i q^{23} +32.0000 q^{24} -26.0000 q^{26} +152.000i q^{27} +65.2291i q^{28} -272.687 q^{29} -313.995 q^{31} +32.0000i q^{32} -41.2291i q^{33} -191.229 q^{34} -44.0000 q^{36} +366.000i q^{37} +118.458i q^{38} -52.0000 q^{39} +172.916 q^{41} +130.458i q^{42} +76.3019i q^{43} +41.2291 q^{44} +115.687 q^{46} +101.078i q^{47} +64.0000i q^{48} +77.0728 q^{49} -382.458 q^{51} -52.0000i q^{52} -181.156i q^{53} -304.000 q^{54} -130.458 q^{56} +236.916i q^{57} -545.375i q^{58} -834.453 q^{59} -556.073 q^{61} -627.989i q^{62} -179.380i q^{63} -64.0000 q^{64} +82.4582 q^{66} +220.922i q^{67} -382.458i q^{68} +231.375 q^{69} +702.916 q^{71} -88.0000i q^{72} +76.1455i q^{73} -732.000 q^{74} -236.916 q^{76} +168.084i q^{77} -104.000i q^{78} -100.927 q^{79} -311.000 q^{81} +345.833i q^{82} -318.464i q^{83} -260.916 q^{84} -152.604 q^{86} -1090.75i q^{87} +82.4582i q^{88} +161.833 q^{89} +211.995 q^{91} +231.375i q^{92} -1255.98i q^{93} -202.156 q^{94} -128.000 q^{96} +1329.98i q^{97} +154.146i q^{98} -113.380 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 32 q^{6} + 44 q^{9} + 44 q^{11} - 40 q^{14} + 64 q^{16} - 104 q^{19} - 80 q^{21} + 128 q^{24} - 104 q^{26} - 68 q^{29} - 148 q^{31} - 424 q^{34} - 176 q^{36} - 208 q^{39} - 672 q^{41} - 176 q^{44}+ \cdots + 484 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 32 * q^6 + 44 * q^9 + 44 * q^11 - 40 * q^14 + 64 * q^16 - 104 * q^19 - 80 * q^21 + 128 * q^24 - 104 * q^26 - 68 * q^29 - 148 * q^31 - 424 * q^34 - 176 * q^36 - 208 * q^39 - 672 * q^41 - 176 * q^44 - 560 * q^46 - 544 * q^49 - 848 * q^51 - 1216 * q^54 + 160 * q^56 - 1548 * q^59 - 1372 * q^61 - 256 * q^64 - 352 * q^66 - 1120 * q^69 + 1448 * q^71 - 2928 * q^74 + 416 * q^76 - 1256 * q^79 - 1244 * q^81 + 320 * q^84 + 1776 * q^86 - 2080 * q^89 - 260 * q^91 - 1320 * q^94 - 512 * q^96 + 484 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−8.00000	−0.544331
$7$	− 16.3073i	− 0.880510i	−0.897873	−	0.440255i	$-0.854888\pi$
$7$	− 16.3073i	− 0.880510i	0.897873	−	0.440255i	$-0.145112\pi$
$8$	− 8.00000i	− 0.353553i
$9$	11.0000	0.407407
$10$	0	0
$11$	−10.3073	−0.282524	−0.141262	−	0.989972i	$-0.545116\pi$
$11$	−10.3073	−0.282524	−0.141262	+	0.989972i	$0.545116\pi$
$12$	− 16.0000i	− 0.384900i
$13$	13.0000i	0.277350i
$13$	13.0000i	0.277350i
$14$	32.6146	0.622615
$15$	0	0
$16$	16.0000	0.250000
$17$	95.6146i	1.36411i	0.731299	+	0.682057i	$0.238914\pi$
$17$	95.6146i	1.36411i	−0.731299	+	0.682057i	$0.761086\pi$
$18$	22.0000i	0.288081i
$19$	59.2291	0.715163	0.357581	−	0.933882i	$-0.383602\pi$
$19$	59.2291	0.715163	0.357581	+	0.933882i	$0.383602\pi$
$20$	0	0
$21$	65.2291	0.677817
$22$	− 20.6146i	− 0.199774i
$23$	− 57.8437i	− 0.524402i	−0.965013	−	0.262201i	$-0.915552\pi$
$23$	− 57.8437i	− 0.524402i	0.965013	−	0.262201i	$-0.0844483\pi$
$24$	32.0000	0.272166
$25$	0	0
$26$	−26.0000	−0.196116
$27$	152.000i	1.08342i
$28$	65.2291i	0.440255i
$29$	−272.687	−1.74610	−0.873048	−	0.487635i	$-0.837860\pi$
$29$	−272.687	−1.74610	−0.873048	+	0.487635i	$0.837860\pi$
$30$	0	0
$31$	−313.995	−1.81920	−0.909598	−	0.415489i	$-0.863611\pi$
$31$	−313.995	−1.81920	−0.909598	+	0.415489i	$0.863611\pi$
$32$	32.0000i	0.176777i
$33$	− 41.2291i	− 0.217487i
$34$	−191.229	−0.964574
$35$	0	0
$36$	−44.0000	−0.203704
$37$	366.000i	1.62622i	0.582112	+	0.813109i	$0.302226\pi$
$37$	366.000i	1.62622i	−0.582112	+	0.813109i	$0.697774\pi$
$38$	118.458i	0.505696i
$39$	−52.0000	−0.213504
$40$	0	0
$41$	172.916	0.658659	0.329329	−	0.944215i	$-0.393177\pi$
$41$	172.916	0.658659	0.329329	+	0.944215i	$0.393177\pi$
$42$	130.458i	0.479289i
$43$	76.3019i	0.270603i	0.990804	+	0.135301i	$0.0432003\pi$
$43$	76.3019i	0.270603i	−0.990804	+	0.135301i	$0.956800\pi$
$44$	41.2291	0.141262
$45$	0	0
$46$	115.687	0.370808
$47$	101.078i	0.313697i	0.987623	+	0.156849i	$0.0501335\pi$
$47$	101.078i	0.313697i	−0.987623	+	0.156849i	$0.949867\pi$
$48$	64.0000i	0.192450i
$49$	77.0728	0.224702
$50$	0	0
$51$	−382.458	−1.05010
$52$	− 52.0000i	− 0.138675i
$53$	− 181.156i	− 0.469504i	−0.972055	−	0.234752i	$-0.924572\pi$
$53$	− 181.156i	− 0.469504i	0.972055	−	0.234752i	$-0.0754279\pi$
$54$	−304.000	−0.766096
$55$	0	0
$56$	−130.458	−0.311307
$57$	236.916i	0.550532i
$58$	− 545.375i	− 1.23468i
$59$	−834.453	−1.84130	−0.920648	−	0.390393i	$-0.872339\pi$
$59$	−834.453	−1.84130	−0.920648	+	0.390393i	$0.872339\pi$
$60$	0	0
$61$	−556.073	−1.16718	−0.583589	−	0.812049i	$-0.698352\pi$
$61$	−556.073	−1.16718	−0.583589	+	0.812049i	$0.698352\pi$
$62$	− 627.989i	− 1.28637i
$63$	− 179.380i	− 0.358726i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	82.4582	0.153786
$67$	220.922i	0.402834i	0.979506	+	0.201417i	$0.0645547\pi$
$67$	220.922i	0.402834i	−0.979506	+	0.201417i	$0.935445\pi$
$68$	− 382.458i	− 0.682057i
$69$	231.375	0.403685
$70$	0	0
$71$	702.916	1.17494	0.587471	−	0.809245i	$-0.300124\pi$
$71$	702.916	1.17494	0.587471	+	0.809245i	$0.300124\pi$
$72$	− 88.0000i	− 0.144040i
$73$	76.1455i	0.122084i	0.998135	+	0.0610422i	$0.0194424\pi$
$73$	76.1455i	0.122084i	−0.998135	+	0.0610422i	$0.980558\pi$
$74$	−732.000	−1.14991
$75$	0	0
$76$	−236.916	−0.357581
$77$	168.084i	0.248765i
$78$	− 104.000i	− 0.150970i
$79$	−100.927	−0.143737	−0.0718684	−	0.997414i	$-0.522896\pi$
$79$	−100.927	−0.143737	−0.0718684	+	0.997414i	$0.522896\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	345.833i	0.465742i
$83$	− 318.464i	− 0.421156i	−0.977577	−	0.210578i	$-0.932465\pi$
$83$	− 318.464i	− 0.421156i	0.977577	−	0.210578i	$-0.0675345\pi$
$84$	−260.916	−0.338909
$85$	0	0
$86$	−152.604	−0.191345
$87$	− 1090.75i	− 1.34414i
$88$	82.4582i	0.0998872i
$89$	161.833	0.192744	0.0963722	−	0.995345i	$-0.469276\pi$
$89$	161.833	0.192744	0.0963722	+	0.995345i	$0.469276\pi$
$90$	0	0
$91$	211.995	0.244210
$92$	231.375i	0.262201i
$93$	− 1255.98i	− 1.40042i
$94$	−202.156	−0.221817
$95$	0	0
$96$	−128.000	−0.136083
$97$	1329.98i	1.39215i	0.717968	+	0.696076i	$0.245073\pi$
$97$	1329.98i	1.39215i	−0.717968	+	0.696076i	$0.754927\pi$
$98$	154.146i	0.158888i
$99$	−113.380	−0.115102
$100$	0	0
$101$	−107.615	−0.106020	−0.0530101	−	0.998594i	$-0.516882\pi$
$101$	−107.615	−0.106020	−0.0530101	+	0.998594i	$0.516882\pi$
$102$	− 764.916i	− 0.742530i
$103$	− 882.771i	− 0.844485i	−0.906483	−	0.422243i	$-0.861243\pi$
$103$	− 882.771i	− 0.844485i	0.906483	−	0.422243i	$-0.138757\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	362.313	0.331990
$107$	− 583.051i	− 0.526782i	−0.964689	−	0.263391i	$-0.915159\pi$
$107$	− 583.051i	− 0.526782i	0.964689	−	0.263391i	$-0.0848409\pi$
$108$	− 608.000i	− 0.541711i
$109$	−495.062	−0.435031	−0.217515	−	0.976057i	$-0.569795\pi$
$109$	−495.062	−0.435031	−0.217515	+	0.976057i	$0.569795\pi$
$110$	0	0
$111$	−1464.00	−1.25186
$112$	− 260.916i	− 0.220128i
$113$	− 392.458i	− 0.326720i	−0.986567	−	0.163360i	$-0.947767\pi$
$113$	− 392.458i	− 0.326720i	0.986567	−	0.163360i	$-0.0522332\pi$
$114$	−473.833	−0.389285
$115$	0	0
$116$	1090.75	0.873048
$117$	143.000i	0.112994i
$118$	− 1668.91i	− 1.30199i
$119$	1559.21	1.20112
$120$	0	0
$121$	−1224.76	−0.920180
$122$	− 1112.15i	− 0.825319i
$123$	691.666i	0.507036i
$124$	1255.98	0.909598
$125$	0	0
$126$	358.760	0.253658
$127$	954.135i	0.666660i	0.942810	+	0.333330i	$0.108172\pi$
$127$	954.135i	0.666660i	−0.942810	+	0.333330i	$0.891828\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−305.207	−0.208310
$130$	0	0
$131$	56.9164	0.0379604	0.0189802	−	0.999820i	$-0.493958\pi$
$131$	56.9164	0.0379604	0.0189802	+	0.999820i	$0.493958\pi$
$132$	164.916i	0.108743i
$133$	− 965.865i	− 0.629708i
$134$	−441.844	−0.284847
$135$	0	0
$136$	764.916	0.482287
$137$	2568.27i	1.60162i	0.598918	+	0.800810i	$0.295598\pi$
$137$	2568.27i	1.60162i	−0.598918	+	0.800810i	$0.704402\pi$
$138$	462.749i	0.285448i
$139$	−1603.05	−0.978194	−0.489097	−	0.872229i	$-0.662674\pi$
$139$	−1603.05	−0.978194	−0.489097	+	0.872229i	$0.662674\pi$
$140$	0	0
$141$	−404.313	−0.241484
$142$	1405.83i	0.830809i
$143$	− 133.995i	− 0.0783580i
$144$	176.000	0.101852
$145$	0	0
$146$	−152.291	−0.0863267
$147$	308.291i	0.172976i
$148$	− 1464.00i	− 0.813109i
$149$	−1585.08	−0.871511	−0.435755	−	0.900065i	$-0.643519\pi$
$149$	−1585.08	−0.871511	−0.435755	+	0.900065i	$0.643519\pi$
$150$	0	0
$151$	−2077.20	−1.11947	−0.559736	−	0.828671i	$-0.689097\pi$
$151$	−2077.20	−1.11947	−0.559736	+	0.828671i	$0.689097\pi$
$152$	− 473.833i	− 0.252848i
$153$	1051.76i	0.555750i
$154$	−336.167	−0.175903
$155$	0	0
$156$	208.000	0.106752
$157$	− 1605.73i	− 0.816248i	−0.912926	−	0.408124i	$-0.866183\pi$
$157$	− 1605.73i	− 0.816248i	0.912926	−	0.408124i	$-0.133817\pi$
$158$	− 201.854i	− 0.101637i
$159$	724.625	0.361425
$160$	0	0
$161$	−943.272	−0.461741
$162$	− 622.000i	− 0.301660i
$163$	497.105i	0.238873i	0.992842	+	0.119436i	$0.0381088\pi$
$163$	497.105i	0.238873i	−0.992842	+	0.119436i	$0.961891\pi$
$164$	−691.666	−0.329329
$165$	0	0
$166$	636.927	0.297802
$167$	1889.06i	0.875329i	0.899138	+	0.437665i	$0.144194\pi$
$167$	1889.06i	0.875329i	−0.899138	+	0.437665i	$0.855806\pi$
$168$	− 521.833i	− 0.239645i
$169$	−169.000	−0.0769231
$170$	0	0
$171$	651.520	0.291363
$172$	− 305.207i	− 0.135301i
$173$	− 3257.86i	− 1.43174i	−0.698235	−	0.715869i	$-0.746031\pi$
$173$	− 3257.86i	− 1.43174i	0.698235	−	0.715869i	$-0.253969\pi$
$174$	2181.50	0.950454
$175$	0	0
$176$	−164.916	−0.0706309
$177$	− 3337.81i	− 1.41743i
$178$	323.666i	0.136291i
$179$	4249.23	1.77431	0.887157	−	0.461467i	$-0.152677\pi$
$179$	4249.23	1.77431	0.887157	+	0.461467i	$0.152677\pi$
$180$	0	0
$181$	−3803.12	−1.56179	−0.780895	−	0.624662i	$-0.785237\pi$
$181$	−3803.12	−1.56179	−0.780895	+	0.624662i	$0.785237\pi$
$182$	423.989i	0.172682i
$183$	− 2224.29i	− 0.898494i
$184$	−462.749	−0.185404
$185$	0	0
$186$	2511.96	0.990245
$187$	− 985.526i	− 0.385394i
$188$	− 404.313i	− 0.156849i
$189$	2478.71	0.953965
$190$	0	0
$191$	−3338.38	−1.26470	−0.632348	−	0.774685i	$-0.717909\pi$
$191$	−3338.38	−1.26470	−0.632348	+	0.774685i	$0.717909\pi$
$192$	− 256.000i	− 0.0962250i
$193$	2879.98i	1.07412i	0.843544	+	0.537061i	$0.180465\pi$
$193$	2879.98i	1.07412i	−0.843544	+	0.537061i	$0.819535\pi$
$194$	−2659.96	−0.984401
$195$	0	0
$196$	−308.291	−0.112351
$197$	1038.73i	0.375666i	0.982201	+	0.187833i	$0.0601464\pi$
$197$	1038.73i	0.375666i	−0.982201	+	0.187833i	$0.939854\pi$
$198$	− 226.760i	− 0.0813896i
$199$	3322.13	1.18342	0.591708	−	0.806152i	$-0.298454\pi$
$199$	3322.13	1.18342	0.591708	+	0.806152i	$0.298454\pi$
$200$	0	0
$201$	−883.687	−0.310102
$202$	− 215.229i	− 0.0749677i
$203$	4446.79i	1.53745i
$204$	1529.83	0.525048
$205$	0	0
$206$	1765.54	0.597141
$207$	− 636.280i	− 0.213645i
$208$	208.000i	0.0693375i
$209$	−610.491	−0.202050
$210$	0	0
$211$	−649.240	−0.211827	−0.105914	−	0.994375i	$-0.533777\pi$
$211$	−649.240	−0.211827	−0.105914	+	0.994375i	$0.533777\pi$
$212$	724.625i	0.234752i
$213$	2811.67i	0.904470i
$214$	1166.10	0.372491
$215$	0	0
$216$	1216.00	0.383048
$217$	5120.40i	1.60182i
$218$	− 990.124i	− 0.307613i
$219$	−304.582	−0.0939806
$220$	0	0
$221$	−1242.99	−0.378337
$222$	− 2928.00i	− 0.885200i
$223$	4168.41i	1.25174i	0.779928	+	0.625869i	$0.215256\pi$
$223$	4168.41i	1.25174i	−0.779928	+	0.625869i	$0.784744\pi$
$224$	521.833	0.155654
$225$	0	0
$226$	784.916	0.231026
$227$	− 3076.15i	− 0.899433i	−0.893171	−	0.449717i	$-0.851525\pi$
$227$	− 3076.15i	− 0.899433i	0.893171	−	0.449717i	$-0.148475\pi$
$228$	− 947.666i	− 0.275266i
$229$	−2960.10	−0.854188	−0.427094	−	0.904207i	$-0.640463\pi$
$229$	−2960.10	−0.854188	−0.427094	+	0.904207i	$0.640463\pi$
$230$	0	0
$231$	−672.334	−0.191499
$232$	2181.50i	0.617338i
$233$	1049.67i	0.295133i	0.989052	+	0.147566i	$0.0471440\pi$
$233$	1049.67i	0.295133i	−0.989052	+	0.147566i	$0.952856\pi$
$234$	−286.000	−0.0798992
$235$	0	0
$236$	3337.81	0.920648
$237$	− 403.709i	− 0.110649i
$238$	3118.43i	0.849317i
$239$	−3746.54	−1.01399	−0.506995	−	0.861949i	$-0.669244\pi$
$239$	−3746.54	−1.01399	−0.506995	+	0.861949i	$0.669244\pi$
$240$	0	0
$241$	5009.14	1.33887	0.669434	−	0.742871i	$-0.266537\pi$
$241$	5009.14	1.33887	0.669434	+	0.742871i	$0.266537\pi$
$242$	− 2449.52i	− 0.650666i
$243$	2860.00i	0.755017i
$244$	2224.29	0.583589
$245$	0	0
$246$	−1383.33	−0.358528
$247$	769.978i	0.198350i
$248$	2511.96i	0.643183i
$249$	1273.85	0.324206
$250$	0	0
$251$	4368.23	1.09849	0.549243	−	0.835663i	$-0.314916\pi$
$251$	4368.23	1.09849	0.549243	+	0.835663i	$0.314916\pi$
$252$	717.520i	0.179363i
$253$	596.210i	0.148156i
$254$	−1908.27	−0.471399
$255$	0	0
$256$	256.000	0.0625000
$257$	− 5543.85i	− 1.34559i	−0.739830	−	0.672794i	$-0.765094\pi$
$257$	− 5543.85i	− 1.34559i	0.739830	−	0.672794i	$-0.234906\pi$
$258$	− 610.415i	− 0.147298i
$259$	5968.46	1.43190
$260$	0	0
$261$	−2999.56	−0.711372
$262$	113.833i	0.0268420i
$263$	4154.73i	0.974112i	0.873371	+	0.487056i	$0.161929\pi$
$263$	4154.73i	0.974112i	−0.873371	+	0.487056i	$0.838071\pi$
$264$	−329.833	−0.0768932
$265$	0	0
$266$	1931.73	0.445271
$267$	647.331i	0.148375i
$268$	− 883.687i	− 0.201417i
$269$	−669.625	−0.151776	−0.0758881	−	0.997116i	$-0.524179\pi$
$269$	−669.625	−0.151776	−0.0758881	+	0.997116i	$0.524179\pi$
$270$	0	0
$271$	2284.27	0.512029	0.256014	−	0.966673i	$-0.417591\pi$
$271$	2284.27	0.512029	0.256014	+	0.966673i	$0.417591\pi$
$272$	1529.83i	0.341028i
$273$	847.978i	0.187993i
$274$	−5136.54	−1.13252
$275$	0	0
$276$	−925.498	−0.201842
$277$	7405.67i	1.60637i	0.595733	+	0.803183i	$0.296862\pi$
$277$	7405.67i	1.60637i	−0.595733	+	0.803183i	$0.703138\pi$
$278$	− 3206.10i	− 0.691688i
$279$	−3453.94	−0.741154
$280$	0	0
$281$	7166.92	1.52150	0.760752	−	0.649043i	$-0.224831\pi$
$281$	7166.92	1.52150	0.760752	+	0.649043i	$0.224831\pi$
$282$	− 808.625i	− 0.170755i
$283$	4612.32i	0.968813i	0.874843	+	0.484407i	$0.160964\pi$
$283$	4612.32i	0.968813i	−0.874843	+	0.484407i	$0.839036\pi$
$284$	−2811.67	−0.587471
$285$	0	0
$286$	267.989	0.0554075
$287$	− 2819.80i	− 0.579956i
$288$	352.000i	0.0720201i
$289$	−4229.14	−0.860807
$290$	0	0
$291$	−5319.91	−1.07168
$292$	− 304.582i	− 0.0610422i
$293$	6504.41i	1.29690i	0.761257	+	0.648450i	$0.224583\pi$
$293$	6504.41i	1.29690i	−0.761257	+	0.648450i	$0.775417\pi$
$294$	−616.582	−0.122312
$295$	0	0
$296$	2928.00	0.574955
$297$	− 1566.71i	− 0.306093i
$298$	− 3170.17i	− 0.616251i
$299$	751.968	0.145443
$300$	0	0
$301$	1244.28	0.238269
$302$	− 4154.40i	− 0.791586i
$303$	− 430.458i	− 0.0816144i
$304$	947.666	0.178791
$305$	0	0
$306$	−2103.52	−0.392975
$307$	− 4407.10i	− 0.819304i	−0.912242	−	0.409652i	$-0.865650\pi$
$307$	− 4407.10i	− 0.819304i	0.912242	−	0.409652i	$-0.134350\pi$
$308$	− 672.334i	− 0.124382i
$309$	3531.08	0.650085
$310$	0	0
$311$	6594.09	1.20230	0.601152	−	0.799135i	$-0.294709\pi$
$311$	6594.09	1.20230	0.601152	+	0.799135i	$0.294709\pi$
$312$	416.000i	0.0754851i
$313$	− 907.782i	− 0.163932i	−0.996635	−	0.0819662i	$-0.973880\pi$
$313$	− 907.782i	− 0.163932i	0.996635	−	0.0819662i	$-0.0261200\pi$
$314$	3211.46	0.577175
$315$	0	0
$316$	403.709	0.0718684
$317$	1802.54i	0.319371i	0.987168	+	0.159686i	$0.0510480\pi$
$317$	1802.54i	0.319371i	−0.987168	+	0.159686i	$0.948952\pi$
$318$	1449.25i	0.255566i
$319$	2810.66	0.493313
$320$	0	0
$321$	2332.20	0.405517
$322$	− 1886.54i	− 0.326500i
$323$	5663.16i	0.975563i
$324$	1244.00	0.213306
$325$	0	0
$326$	−994.210	−0.168909
$327$	− 1980.25i	− 0.334887i
$328$	− 1383.33i	− 0.232871i
$329$	1648.31	0.276214
$330$	0	0
$331$	9001.00	1.49468	0.747341	−	0.664441i	$-0.231330\pi$
$331$	9001.00	1.49468	0.747341	+	0.664441i	$0.231330\pi$
$332$	1273.85i	0.210578i
$333$	4026.00i	0.662533i
$334$	−3778.12	−0.618951
$335$	0	0
$336$	1043.67	0.169454
$337$	4012.30i	0.648558i	0.945961	+	0.324279i	$0.105122\pi$
$337$	4012.30i	0.648558i	−0.945961	+	0.324279i	$0.894878\pi$
$338$	− 338.000i	− 0.0543928i
$339$	1569.83	0.251509
$340$	0	0
$341$	3236.43	0.513966
$342$	1303.04i	0.206024i
$343$	− 6850.24i	− 1.07836i
$344$	610.415	0.0956726
$345$	0	0
$346$	6515.72	1.01239
$347$	1447.68i	0.223964i	0.993710	+	0.111982i	$0.0357198\pi$
$347$	1447.68i	0.223964i	−0.993710	+	0.111982i	$0.964280\pi$
$348$	4363.00i	0.672072i
$349$	−6963.83	−1.06810	−0.534048	−	0.845454i	$-0.679330\pi$
$349$	−6963.83	−1.06810	−0.534048	+	0.845454i	$0.679330\pi$
$350$	0	0
$351$	−1976.00	−0.300487
$352$	− 329.833i	− 0.0499436i
$353$	− 10576.5i	− 1.59470i	−0.603518	−	0.797350i	$-0.706235\pi$
$353$	− 10576.5i	− 1.59470i	0.603518	−	0.797350i	$-0.293765\pi$
$354$	6675.62	1.00227
$355$	0	0
$356$	−647.331	−0.0963722
$357$	6236.85i	0.924620i
$358$	8498.46i	1.25463i
$359$	8925.48	1.31217	0.656084	−	0.754688i	$-0.272212\pi$
$359$	8925.48	1.31217	0.656084	+	0.754688i	$0.272212\pi$
$360$	0	0
$361$	−3350.91	−0.488543
$362$	− 7606.25i	− 1.10435i
$363$	− 4899.04i	− 0.708355i
$364$	−847.978	−0.122105
$365$	0	0
$366$	4448.58	0.635331
$367$	9449.06i	1.34397i	0.740565	+	0.671985i	$0.234558\pi$
$367$	9449.06i	1.34397i	−0.740565	+	0.671985i	$0.765442\pi$
$368$	− 925.498i	− 0.131100i
$369$	1902.08	0.268342
$370$	0	0
$371$	−2954.17	−0.413403
$372$	5023.91i	0.700209i
$373$	− 1298.21i	− 0.180211i	−0.995932	−	0.0901054i	$-0.971280\pi$
$373$	− 1298.21i	− 0.180211i	0.995932	−	0.0901054i	$-0.0287204\pi$
$374$	1971.05	0.272515
$375$	0	0
$376$	808.625	0.110909
$377$	− 3544.94i	− 0.484280i
$378$	4957.41i	0.674555i
$379$	13044.6	1.76796	0.883980	−	0.467524i	$-0.154854\pi$
$379$	13044.6	1.76796	0.883980	+	0.467524i	$0.154854\pi$
$380$	0	0
$381$	−3816.54	−0.513195
$382$	− 6676.76i	− 0.894275i
$383$	− 1304.12i	− 0.173989i	−0.996209	−	0.0869943i	$-0.972274\pi$
$383$	− 1304.12i	− 0.173989i	0.996209	−	0.0869943i	$-0.0277262\pi$
$384$	512.000	0.0680414
$385$	0	0
$386$	−5759.96	−0.759518
$387$	839.320i	0.110246i
$388$	− 5319.91i	− 0.696076i
$389$	−12390.4	−1.61496	−0.807479	−	0.589896i	$-0.799169\pi$
$389$	−12390.4	−1.61496	−0.807479	+	0.589896i	$0.799169\pi$
$390$	0	0
$391$	5530.70	0.715343
$392$	− 616.582i	− 0.0794441i
$393$	227.666i	0.0292219i
$394$	−2077.46	−0.265636
$395$	0	0
$396$	453.520	0.0575511
$397$	− 12332.6i	− 1.55909i	−0.626348	−	0.779543i	$-0.715451\pi$
$397$	− 12332.6i	− 1.55909i	0.626348	−	0.779543i	$-0.284549\pi$
$398$	6644.27i	0.836802i
$399$	3863.46	0.484749
$400$	0	0
$401$	2213.06	0.275599	0.137799	−	0.990460i	$-0.455997\pi$
$401$	2213.06	0.275599	0.137799	+	0.990460i	$0.455997\pi$
$402$	− 1767.37i	− 0.219275i
$403$	− 4081.93i	− 0.504554i
$404$	430.458	0.0530101
$405$	0	0
$406$	−8893.57	−1.08714
$407$	− 3772.46i	− 0.459445i
$408$	3059.67i	0.371265i
$409$	−2838.39	−0.343153	−0.171576	−	0.985171i	$-0.554886\pi$
$409$	−2838.39	−0.343153	−0.171576	+	0.985171i	$0.554886\pi$
$410$	0	0
$411$	−10273.1	−1.23293
$412$	3531.08i	0.422243i
$413$	13607.7i	1.62128i
$414$	1272.56	0.151070
$415$	0	0
$416$	−416.000	−0.0490290
$417$	− 6412.20i	− 0.753014i
$418$	− 1220.98i	− 0.142871i
$419$	9560.58	1.11471	0.557357	−	0.830273i	$-0.311816\pi$
$419$	9560.58	1.11471	0.557357	+	0.830273i	$0.311816\pi$
$420$	0	0
$421$	−2405.00	−0.278415	−0.139207	−	0.990263i	$-0.544455\pi$
$421$	−2405.00	−0.278415	−0.139207	+	0.990263i	$0.544455\pi$
$422$	− 1298.48i	− 0.149784i
$423$	1111.86i	0.127803i
$424$	−1449.25	−0.165995
$425$	0	0
$426$	−5623.33	−0.639557
$427$	9068.03i	1.02771i
$428$	2332.20i	0.263391i
$429$	535.978	0.0603200
$430$	0	0
$431$	5319.46	0.594499	0.297250	−	0.954800i	$-0.403931\pi$
$431$	5319.46	0.594499	0.297250	+	0.954800i	$0.403931\pi$
$432$	2432.00i	0.270856i
$433$	15949.8i	1.77021i	0.465394	+	0.885104i	$0.345913\pi$
$433$	15949.8i	1.77021i	−0.465394	+	0.885104i	$0.654087\pi$
$434$	−10240.8	−1.13266
$435$	0	0
$436$	1980.25	0.217515
$437$	− 3426.03i	− 0.375032i
$438$	− 609.164i	− 0.0664543i
$439$	9047.43	0.983622	0.491811	−	0.870702i	$-0.336335\pi$
$439$	9047.43	0.983622	0.491811	+	0.870702i	$0.336335\pi$
$440$	0	0
$441$	847.800	0.0915452
$442$	− 2485.98i	− 0.267525i
$443$	− 16553.2i	− 1.77531i	−0.460505	−	0.887657i	$-0.652332\pi$
$443$	− 16553.2i	− 1.77531i	0.460505	−	0.887657i	$-0.347668\pi$
$444$	5856.00	0.625931
$445$	0	0
$446$	−8336.83	−0.885113
$447$	− 6340.33i	− 0.670889i
$448$	1043.67i	0.110064i
$449$	−8102.66	−0.851644	−0.425822	−	0.904807i	$-0.640015\pi$
$449$	−8102.66	−0.851644	−0.425822	+	0.904807i	$0.640015\pi$
$450$	0	0
$451$	−1782.30	−0.186087
$452$	1569.83i	0.163360i
$453$	− 8308.81i	− 0.861770i
$454$	6152.30	0.635995
$455$	0	0
$456$	1895.33	0.194643
$457$	18327.3i	1.87596i	0.346687	+	0.937981i	$0.387306\pi$
$457$	18327.3i	1.87596i	−0.346687	+	0.937981i	$0.612694\pi$
$458$	− 5920.20i	− 0.604002i
$459$	−14533.4	−1.47791
$460$	0	0
$461$	−5131.83	−0.518467	−0.259233	−	0.965815i	$-0.583470\pi$
$461$	−5131.83	−0.518467	−0.259233	+	0.965815i	$0.583470\pi$
$462$	− 1344.67i	− 0.135410i
$463$	6659.01i	0.668403i	0.942502	+	0.334201i	$0.108467\pi$
$463$	6659.01i	0.668403i	−0.942502	+	0.334201i	$0.891533\pi$
$464$	−4363.00	−0.436524
$465$	0	0
$466$	−2099.33	−0.208690
$467$	7754.58i	0.768392i	0.923252	+	0.384196i	$0.125521\pi$
$467$	7754.58i	0.768392i	−0.923252	+	0.384196i	$0.874479\pi$
$468$	− 572.000i	− 0.0564972i
$469$	3602.63	0.354700
$470$	0	0
$471$	6422.91	0.628348
$472$	6675.62i	0.650997i
$473$	− 786.464i	− 0.0764517i
$474$	807.418	0.0782404
$475$	0	0
$476$	−6236.85	−0.600558
$477$	− 1992.72i	− 0.191280i
$478$	− 7493.09i	− 0.717000i
$479$	−4029.17	−0.384337	−0.192169	−	0.981362i	$-0.561552\pi$
$479$	−4029.17	−0.384337	−0.192169	+	0.981362i	$0.561552\pi$
$480$	0	0
$481$	−4758.00	−0.451031
$482$	10018.3i	0.946723i
$483$	− 3773.09i	− 0.355448i
$484$	4899.04	0.460090
$485$	0	0
$486$	−5720.00	−0.533878
$487$	− 9544.26i	− 0.888073i	−0.896009	−	0.444037i	$-0.853546\pi$
$487$	− 9544.26i	− 0.888073i	0.896009	−	0.444037i	$-0.146454\pi$
$488$	4448.58i	0.412660i
$489$	−1988.42	−0.183884
$490$	0	0
$491$	528.593	0.0485847	0.0242923	−	0.999705i	$-0.492267\pi$
$491$	528.593	0.0485847	0.0242923	+	0.999705i	$0.492267\pi$
$492$	− 2766.66i	− 0.253518i
$493$	− 26072.9i	− 2.38187i
$494$	−1539.96	−0.140255
$495$	0	0
$496$	−5023.91	−0.454799
$497$	− 11462.7i	− 1.03455i
$498$	2547.71i	0.229248i
$499$	−10891.2	−0.977072	−0.488536	−	0.872544i	$-0.662469\pi$
$499$	−10891.2	−0.977072	−0.488536	+	0.872544i	$0.662469\pi$
$500$	0	0
$501$	−7556.25	−0.673829
$502$	8736.45i	0.776747i
$503$	− 12648.0i	− 1.12117i	−0.828098	−	0.560583i	$-0.810577\pi$
$503$	− 12648.0i	− 1.12117i	0.828098	−	0.560583i	$-0.189423\pi$
$504$	−1435.04	−0.126829
$505$	0	0
$506$	−1192.42	−0.104762
$507$	− 676.000i	− 0.0592154i
$508$	− 3816.54i	− 0.333330i
$509$	−5593.39	−0.487078	−0.243539	−	0.969891i	$-0.578308\pi$
$509$	−5593.39	−0.487078	−0.243539	+	0.969891i	$0.578308\pi$
$510$	0	0
$511$	1241.73	0.107497
$512$	512.000i	0.0441942i
$513$	9002.82i	0.774823i
$514$	11087.7	0.951474
$515$	0	0
$516$	1220.83	0.104155
$517$	− 1041.84i	− 0.0886269i
$518$	11936.9i	1.01251i
$519$	13031.4	1.10215
$520$	0	0
$521$	−3285.58	−0.276284	−0.138142	−	0.990412i	$-0.544113\pi$
$521$	−3285.58	−0.276284	−0.138142	+	0.990412i	$0.544113\pi$
$522$	− 5999.12i	− 0.503016i
$523$	− 8833.62i	− 0.738560i	−0.929318	−	0.369280i	$-0.879604\pi$
$523$	− 8833.62i	− 0.738560i	0.929318	−	0.369280i	$-0.120396\pi$
$524$	−227.666	−0.0189802
$525$	0	0
$526$	−8309.46	−0.688801
$527$	− 30022.5i	− 2.48159i
$528$	− 659.666i	− 0.0543717i
$529$	8821.11	0.725003
$530$	0	0
$531$	−9178.98	−0.750158
$532$	3863.46i	0.314854i
$533$	2247.91i	0.182679i
$534$	−1294.66	−0.104917
$535$	0	0
$536$	1767.37	0.142423
$537$	16996.9i	1.36587i
$538$	− 1339.25i	− 0.107322i
$539$	−794.410	−0.0634836
$540$	0	0
$541$	22151.3	1.76036	0.880182	−	0.474636i	$-0.157420\pi$
$541$	22151.3	1.76036	0.880182	+	0.474636i	$0.157420\pi$
$542$	4568.55i	0.362059i
$543$	− 15212.5i	− 1.20227i
$544$	−3059.67	−0.241144
$545$	0	0
$546$	−1695.96	−0.132931
$547$	− 16355.9i	− 1.27848i	−0.769008	−	0.639239i	$-0.779249\pi$
$547$	− 16355.9i	− 1.27848i	0.769008	−	0.639239i	$-0.220751\pi$
$548$	− 10273.1i	− 0.800810i
$549$	−6116.80	−0.475517
$550$	0	0
$551$	−16151.0	−1.24874
$552$	− 1851.00i	− 0.142724i
$553$	1645.85i	0.126562i
$554$	−14811.3	−1.13587
$555$	0	0
$556$	6412.20	0.489097
$557$	17659.8i	1.34340i	0.740825	+	0.671698i	$0.234435\pi$
$557$	17659.8i	1.34340i	−0.740825	+	0.671698i	$0.765565\pi$
$558$	− 6907.88i	− 0.524075i
$559$	−991.924	−0.0750517
$560$	0	0
$561$	3942.10	0.296677
$562$	14333.8i	1.07587i
$563$	− 10559.9i	− 0.790493i	−0.918575	−	0.395247i	$-0.870659\pi$
$563$	− 10559.9i	− 0.790493i	0.918575	−	0.395247i	$-0.129341\pi$
$564$	1617.25	0.120742
$565$	0	0
$566$	−9224.65	−0.685054
$567$	5071.56i	0.375636i
$568$	− 5623.33i	− 0.415404i
$569$	10050.9	0.740519	0.370259	−	0.928928i	$-0.379269\pi$
$569$	10050.9	0.740519	0.370259	+	0.928928i	$0.379269\pi$
$570$	0	0
$571$	−4946.83	−0.362554	−0.181277	−	0.983432i	$-0.558023\pi$
$571$	−4946.83	−0.362554	−0.181277	+	0.983432i	$0.558023\pi$
$572$	535.978i	0.0391790i
$573$	− 13353.5i	− 0.973563i
$574$	5639.59	0.410091
$575$	0	0
$576$	−704.000	−0.0509259
$577$	7370.89i	0.531810i	0.963999	+	0.265905i	$0.0856707\pi$
$577$	7370.89i	0.531810i	−0.963999	+	0.265905i	$0.914329\pi$
$578$	− 8458.28i	− 0.608682i
$579$	−11519.9	−0.826859
$580$	0	0
$581$	−5193.27	−0.370832
$582$	− 10639.8i	− 0.757792i
$583$	1867.23i	0.132646i
$584$	609.164	0.0431633
$585$	0	0
$586$	−13008.8	−0.917047
$587$	− 12477.5i	− 0.877343i	−0.898647	−	0.438672i	$-0.855449\pi$
$587$	− 12477.5i	− 0.877343i	0.898647	−	0.438672i	$-0.144551\pi$
$588$	− 1233.16i	− 0.0864878i
$589$	−18597.6	−1.30102
$590$	0	0
$591$	−4154.91	−0.289188
$592$	5856.00i	0.406554i
$593$	− 1045.45i	− 0.0723970i	−0.999345	−	0.0361985i	$-0.988475\pi$
$593$	− 1045.45i	− 0.0723970i	0.999345	−	0.0361985i	$-0.0115249\pi$
$594$	3133.41	0.216440
$595$	0	0
$596$	6340.33	0.435755
$597$	13288.5i	0.910994i
$598$	1503.94i	0.102844i
$599$	−7835.13	−0.534448	−0.267224	−	0.963634i	$-0.586106\pi$
$599$	−7835.13	−0.534448	−0.267224	+	0.963634i	$0.586106\pi$
$600$	0	0
$601$	18645.8	1.26552	0.632760	−	0.774348i	$-0.281922\pi$
$601$	18645.8	1.26552	0.632760	+	0.774348i	$0.281922\pi$
$602$	2488.55i	0.168481i
$603$	2430.14i	0.164118i
$604$	8308.81	0.559736
$605$	0	0
$606$	860.916	0.0577101
$607$	8342.46i	0.557842i	0.960314	+	0.278921i	$0.0899767\pi$
$607$	8342.46i	0.557842i	−0.960314	+	0.278921i	$0.910023\pi$
$608$	1895.33i	0.126424i
$609$	−17787.1	−1.18353
$610$	0	0
$611$	−1314.02	−0.0870039
$612$	− 4207.04i	− 0.277875i
$613$	3323.14i	0.218957i	0.993989	+	0.109478i	$0.0349180\pi$
$613$	3323.14i	0.218957i	−0.993989	+	0.109478i	$0.965082\pi$
$614$	8814.20	0.579336
$615$	0	0
$616$	1344.67	0.0879517
$617$	2447.10i	0.159670i	0.996808	+	0.0798351i	$0.0254394\pi$
$617$	2447.10i	0.159670i	−0.996808	+	0.0798351i	$0.974561\pi$
$618$	7062.17i	0.459680i
$619$	−7493.38	−0.486566	−0.243283	−	0.969955i	$-0.578224\pi$
$619$	−7493.38	−0.486566	−0.243283	+	0.969955i	$0.578224\pi$
$620$	0	0
$621$	8792.24	0.568149
$622$	13188.2i	0.850157i
$623$	− 2639.05i	− 0.169713i
$624$	−832.000	−0.0533761
$625$	0	0
$626$	1815.56	0.115918
$627$	− 2441.96i	− 0.155538i
$628$	6422.91i	0.408124i
$629$	−34994.9	−2.21835
$630$	0	0
$631$	15601.8	0.984306	0.492153	−	0.870509i	$-0.336210\pi$
$631$	15601.8	0.984306	0.492153	+	0.870509i	$0.336210\pi$
$632$	807.418i	0.0508186i
$633$	− 2596.96i	− 0.163065i
$634$	−3605.08	−0.225830
$635$	0	0
$636$	−2898.50	−0.180712
$637$	1001.95i	0.0623211i
$638$	5621.33i	0.348825i
$639$	7732.08	0.478680
$640$	0	0
$641$	20944.9	1.29060	0.645300	−	0.763929i	$-0.276732\pi$
$641$	20944.9	1.29060	0.645300	+	0.763929i	$0.276732\pi$
$642$	4664.41i	0.286744i
$643$	− 10633.1i	− 0.652147i	−0.945344	−	0.326073i	$-0.894274\pi$
$643$	− 10633.1i	− 0.652147i	0.945344	−	0.326073i	$-0.105726\pi$
$644$	3773.09	0.230870
$645$	0	0
$646$	−11326.3	−0.689827
$647$	− 21552.9i	− 1.30963i	−0.755789	−	0.654815i	$-0.772746\pi$
$647$	− 21552.9i	− 1.30963i	0.755789	−	0.654815i	$-0.227254\pi$
$648$	2488.00i	0.150830i
$649$	8600.94	0.520210
$650$	0	0
$651$	−20481.6	−1.23308
$652$	− 1988.42i	− 0.119436i
$653$	− 6840.02i	− 0.409909i	−0.978771	−	0.204955i	$-0.934295\pi$
$653$	− 6840.02i	− 0.409909i	0.978771	−	0.204955i	$-0.0657047\pi$
$654$	3960.50	0.236801
$655$	0	0
$656$	2766.66	0.164665
$657$	837.601i	0.0497381i
$658$	3296.62i	0.195312i
$659$	1443.68	0.0853379	0.0426689	−	0.999089i	$-0.486414\pi$
$659$	1443.68	0.0853379	0.0426689	+	0.999089i	$0.486414\pi$
$660$	0	0
$661$	1676.98	0.0986793	0.0493396	−	0.998782i	$-0.484288\pi$
$661$	1676.98	0.0986793	0.0493396	+	0.998782i	$0.484288\pi$
$662$	18002.0i	1.05690i
$663$	− 4971.96i	− 0.291244i
$664$	−2547.71	−0.148901
$665$	0	0
$666$	−8052.00	−0.468481
$667$	15773.2i	0.915655i
$668$	− 7556.25i	− 0.437665i
$669$	−16673.7	−0.963589
$670$	0	0
$671$	5731.60	0.329755
$672$	2087.33i	0.119822i
$673$	− 10856.9i	− 0.621845i	−0.950435	−	0.310923i	$-0.899362\pi$
$673$	− 10856.9i	− 0.621845i	0.950435	−	0.310923i	$-0.100638\pi$
$674$	−8024.61	−0.458600
$675$	0	0
$676$	676.000	0.0384615
$677$	− 32301.0i	− 1.83372i	−0.399212	−	0.916859i	$-0.630716\pi$
$677$	− 32301.0i	− 1.83372i	0.399212	−	0.916859i	$-0.369284\pi$
$678$	3139.67i	0.177844i
$679$	21688.3	1.22580
$680$	0	0
$681$	12304.6	0.692384
$682$	6472.86i	0.363429i
$683$	− 24244.8i	− 1.35827i	−0.734012	−	0.679137i	$-0.762354\pi$
$683$	− 24244.8i	− 1.35827i	0.734012	−	0.679137i	$-0.237646\pi$
$684$	−2606.08	−0.145681
$685$	0	0
$686$	13700.5	0.762517
$687$	− 11840.4i	− 0.657554i
$688$	1220.83i	0.0676507i
$689$	2355.03	0.130217
$690$	0	0
$691$	436.970	0.0240566	0.0120283	−	0.999928i	$-0.496171\pi$
$691$	436.970	0.0240566	0.0120283	+	0.999928i	$0.496171\pi$
$692$	13031.4i	0.715869i
$693$	1848.92i	0.101349i
$694$	−2895.35	−0.158366
$695$	0	0
$696$	−8725.99	−0.475227
$697$	16533.3i	0.898485i
$698$	− 13927.7i	− 0.755258i
$699$	−4198.66	−0.227193
$700$	0	0
$701$	26635.7	1.43512	0.717558	−	0.696499i	$-0.245260\pi$
$701$	26635.7	1.43512	0.717558	+	0.696499i	$0.245260\pi$
$702$	− 3952.00i	− 0.212477i
$703$	21677.9i	1.16301i
$704$	659.666	0.0353155
$705$	0	0
$706$	21152.9	1.12762
$707$	1754.90i	0.0933519i
$708$	13351.2i	0.708715i
$709$	964.139	0.0510705	0.0255353	−	0.999674i	$-0.491871\pi$
$709$	964.139	0.0510705	0.0255353	+	0.999674i	$0.491871\pi$
$710$	0	0
$711$	−1110.20	−0.0585594
$712$	− 1294.66i	− 0.0681454i
$713$	18162.6i	0.953989i
$714$	−12473.7	−0.653805
$715$	0	0
$716$	−16996.9	−0.887157
$717$	− 14986.2i	− 0.780570i
$718$	17851.0i	0.927843i
$719$	−28916.1	−1.49984	−0.749921	−	0.661527i	$-0.769909\pi$
$719$	−28916.1	−1.49984	−0.749921	+	0.661527i	$0.769909\pi$
$720$	0	0
$721$	−14395.6	−0.743578
$722$	− 6701.83i	− 0.345452i
$723$	20036.6i	1.03066i
$724$	15212.5	0.780895
$725$	0	0
$726$	9798.08	0.500883
$727$	13826.5i	0.705359i	0.935744	+	0.352679i	$0.114729\pi$
$727$	13826.5i	0.705359i	−0.935744	+	0.352679i	$0.885271\pi$
$728$	− 1695.96i	− 0.0863411i
$729$	−19837.0	−1.00782
$730$	0	0
$731$	−7295.57	−0.369133
$732$	8897.16i	0.449247i
$733$	14930.0i	0.752325i	0.926554	+	0.376162i	$0.122756\pi$
$733$	14930.0i	0.752325i	−0.926554	+	0.376162i	$0.877244\pi$
$734$	−18898.1	−0.950330
$735$	0	0
$736$	1851.00	0.0927020
$737$	− 2277.10i	− 0.113810i
$738$	3804.16i	0.189747i
$739$	438.798	0.0218423	0.0109211	−	0.999940i	$-0.496524\pi$
$739$	438.798	0.0218423	0.0109211	+	0.999940i	$0.496524\pi$
$740$	0	0
$741$	−3079.91	−0.152690
$742$	− 5908.33i	− 0.292320i
$743$	− 25656.0i	− 1.26679i	−0.773828	−	0.633395i	$-0.781661\pi$
$743$	− 25656.0i	− 1.26679i	0.773828	−	0.633395i	$-0.218339\pi$
$744$	−10047.8	−0.495123
$745$	0	0
$746$	2596.41	0.127428
$747$	− 3503.10i	− 0.171582i
$748$	3942.10i	0.192697i
$749$	−9507.98	−0.463837
$750$	0	0
$751$	−19032.2	−0.924759	−0.462380	−	0.886682i	$-0.653004\pi$
$751$	−19032.2	−0.924759	−0.462380	+	0.886682i	$0.653004\pi$
$752$	1617.25i	0.0784243i
$753$	17472.9i	0.845615i
$754$	7089.87	0.342437
$755$	0	0
$756$	−9914.82	−0.476982
$757$	5549.07i	0.266426i	0.991087	+	0.133213i	$0.0425294\pi$
$757$	5549.07i	0.266426i	−0.991087	+	0.133213i	$0.957471\pi$
$758$	26089.2i	1.25014i
$759$	−2384.84	−0.114050
$760$	0	0
$761$	395.056	0.0188183	0.00940917	−	0.999956i	$-0.497005\pi$
$761$	395.056	0.0188183	0.00940917	+	0.999956i	$0.497005\pi$
$762$	− 7633.08i	− 0.362883i
$763$	8073.11i	0.383049i
$764$	13353.5	0.632348
$765$	0	0
$766$	2608.25	0.123028
$767$	− 10847.9i	− 0.510684i
$768$	1024.00i	0.0481125i
$769$	34896.6	1.63641	0.818207	−	0.574924i	$-0.194968\pi$
$769$	34896.6	1.63641	0.818207	+	0.574924i	$0.194968\pi$
$770$	0	0
$771$	22175.4	1.03583
$772$	− 11519.9i	− 0.537061i
$773$	10103.9i	0.470131i	0.971980	+	0.235065i	$0.0755304\pi$
$773$	10103.9i	0.470131i	−0.971980	+	0.235065i	$0.924470\pi$
$774$	−1678.64	−0.0779554
$775$	0	0
$776$	10639.8	0.492200
$777$	23873.9i	1.10228i
$778$	− 24780.8i	− 1.14195i
$779$	10241.7	0.471048
$780$	0	0
$781$	−7245.15	−0.331949
$782$	11061.4i	0.505824i
$783$	− 41448.5i	− 1.89176i
$784$	1233.16	0.0561755
$785$	0	0
$786$	−455.331	−0.0206630
$787$	17858.8i	0.808894i	0.914562	+	0.404447i	$0.132536\pi$
$787$	17858.8i	0.808894i	−0.914562	+	0.404447i	$0.867464\pi$
$788$	− 4154.91i	− 0.187833i
$789$	−16618.9	−0.749872
$790$	0	0
$791$	−6399.92	−0.287680
$792$	907.040i	0.0406948i
$793$	− 7228.95i	− 0.323717i
$794$	24665.3	1.10244
$795$	0	0
$796$	−13288.5	−0.591708
$797$	29321.3i	1.30315i	0.758583	+	0.651577i	$0.225892\pi$
$797$	29321.3i	1.30315i	−0.758583	+	0.651577i	$0.774108\pi$
$798$	7726.92i	0.342770i
$799$	−9664.54	−0.427919
$800$	0	0
$801$	1780.16	0.0785255
$802$	4426.12i	0.194878i
$803$	− 784.853i	− 0.0344917i
$804$	3534.75	0.155051
$805$	0	0
$806$	8163.86	0.356774
$807$	− 2678.50i	− 0.116837i
$808$	860.916i	0.0374838i
$809$	17552.7	0.762817	0.381408	−	0.924407i	$-0.375439\pi$
$809$	17552.7	0.762817	0.381408	+	0.924407i	$0.375439\pi$
$810$	0	0
$811$	−8239.17	−0.356740	−0.178370	−	0.983963i	$-0.557082\pi$
$811$	−8239.17	−0.356740	−0.178370	+	0.983963i	$0.557082\pi$
$812$	− 17787.1i	− 0.768727i
$813$	9137.10i	0.394160i
$814$	7544.93	0.324877
$815$	0	0
$816$	−6119.33	−0.262524
$817$	4519.29i	0.193525i
$818$	− 5676.79i	− 0.242646i
$819$	2331.94	0.0994928
$820$	0	0
$821$	−41252.6	−1.75362	−0.876812	−	0.480833i	$-0.840334\pi$
$821$	−41252.6	−1.75362	−0.876812	+	0.480833i	$0.840334\pi$
$822$	− 20546.2i	− 0.871812i
$823$	− 20332.6i	− 0.861179i	−0.902548	−	0.430590i	$-0.858306\pi$
$823$	− 20332.6i	− 0.861179i	0.902548	−	0.430590i	$-0.141694\pi$
$824$	−7062.17	−0.298571
$825$	0	0
$826$	−27215.3	−1.14642
$827$	− 19919.9i	− 0.837583i	−0.908082	−	0.418792i	$-0.862454\pi$
$827$	− 19919.9i	− 0.837583i	0.908082	−	0.418792i	$-0.137546\pi$
$828$	2545.12i	0.106823i
$829$	38514.9	1.61360	0.806801	−	0.590823i	$-0.201197\pi$
$829$	38514.9	1.61360	0.806801	+	0.590823i	$0.201197\pi$
$830$	0	0
$831$	−29622.7	−1.23658
$832$	− 832.000i	− 0.0346688i
$833$	7369.28i	0.306519i
$834$	12824.4	0.532462
$835$	0	0
$836$	2441.96	0.101025
$837$	− 47727.2i	− 1.97096i
$838$	19121.2i	0.788222i
$839$	12174.7	0.500976	0.250488	−	0.968120i	$-0.419409\pi$
$839$	12174.7	0.500976	0.250488	+	0.968120i	$0.419409\pi$
$840$	0	0
$841$	49969.4	2.04885
$842$	− 4810.01i	− 0.196869i
$843$	28667.7i	1.17125i
$844$	2596.96	0.105914
$845$	0	0
$846$	−2223.72	−0.0903700
$847$	19972.5i	0.810228i
$848$	− 2898.50i	− 0.117376i
$849$	−18449.3	−0.745793
$850$	0	0
$851$	21170.8	0.852791
$852$	− 11246.7i	− 0.452235i
$853$	47252.5i	1.89671i	0.317211	+	0.948355i	$0.397254\pi$
$853$	47252.5i	1.89671i	−0.317211	+	0.948355i	$0.602746\pi$
$854$	−18136.1	−0.726702
$855$	0	0
$856$	−4664.41	−0.186246
$857$	− 25523.4i	− 1.01734i	−0.860961	−	0.508671i	$-0.830137\pi$
$857$	− 25523.4i	− 1.01734i	0.860961	−	0.508671i	$-0.169863\pi$
$858$	1071.96i	0.0426527i
$859$	36484.2	1.44916	0.724578	−	0.689193i	$-0.242035\pi$
$859$	36484.2	1.44916	0.724578	+	0.689193i	$0.242035\pi$
$860$	0	0
$861$	11279.2	0.446450
$862$	10638.9i	0.420374i
$863$	13762.9i	0.542868i	0.962457	+	0.271434i	$0.0874979\pi$
$863$	13762.9i	0.542868i	−0.962457	+	0.271434i	$0.912502\pi$
$864$	−4864.00	−0.191524
$865$	0	0
$866$	−31899.7	−1.25173
$867$	− 16916.6i	− 0.662649i
$868$	− 20481.6i	− 0.800910i
$869$	1040.28	0.0406090
$870$	0	0
$871$	−2871.98	−0.111726
$872$	3960.50i	0.153807i
$873$	14629.8i	0.567173i
$874$	6852.06	0.265188
$875$	0	0
$876$	1218.33	0.0469903
$877$	27068.8i	1.04224i	0.853482	+	0.521122i	$0.174486\pi$
$877$	27068.8i	1.04224i	−0.853482	+	0.521122i	$0.825514\pi$
$878$	18094.9i	0.695526i
$879$	−26017.7	−0.998355
$880$	0	0
$881$	−28471.1	−1.08878	−0.544390	−	0.838833i	$-0.683239\pi$
$881$	−28471.1	−1.08878	−0.544390	+	0.838833i	$0.683239\pi$
$882$	1695.60i	0.0647322i
$883$	− 2832.32i	− 0.107945i	−0.998542	−	0.0539724i	$-0.982812\pi$
$883$	− 2832.32i	− 0.107945i	0.998542	−	0.0539724i	$-0.0171883\pi$
$884$	4971.96	0.189169
$885$	0	0
$886$	33106.3	1.25534
$887$	− 22466.8i	− 0.850463i	−0.905085	−	0.425232i	$-0.860193\pi$
$887$	− 22466.8i	− 0.850463i	0.905085	−	0.425232i	$-0.139807\pi$
$888$	11712.0i	0.442600i
$889$	15559.3	0.587000
$890$	0	0
$891$	3205.56	0.120528
$892$	− 16673.7i	− 0.625869i
$893$	5986.77i	0.224344i
$894$	12680.7	0.474390
$895$	0	0
$896$	−2087.33	−0.0778268
$897$	3007.87i	0.111962i
$898$	− 16205.3i	− 0.602203i
$899$	85622.3	3.17649
$900$	0	0
$901$	17321.2	0.640458
$902$	− 3564.59i	− 0.131583i
$903$	4977.10i	0.183419i
$904$	−3139.67	−0.115513
$905$	0	0
$906$	16617.6	0.609363
$907$	− 12037.4i	− 0.440677i	−0.975423	−	0.220339i	$-0.929284\pi$
$907$	− 12037.4i	− 0.440677i	0.975423	−	0.220339i	$-0.0707162\pi$
$908$	12304.6i	0.449717i
$909$	−1183.76	−0.0431934
$910$	0	0
$911$	12445.6	0.452626	0.226313	−	0.974055i	$-0.427333\pi$
$911$	12445.6	0.452626	0.226313	+	0.974055i	$0.427333\pi$
$912$	3790.66i	0.137633i
$913$	3282.49i	0.118986i
$914$	−36654.6	−1.32651
$915$	0	0
$916$	11840.4	0.427094
$917$	− 928.152i	− 0.0334245i
$918$	− 29066.8i	− 1.04504i
$919$	39392.6	1.41397	0.706987	−	0.707227i	$-0.250054\pi$
$919$	39392.6	1.41397	0.706987	+	0.707227i	$0.250054\pi$
$920$	0	0
$921$	17628.4	0.630701
$922$	− 10263.7i	− 0.366612i
$923$	9137.91i	0.325870i
$924$	2689.34	0.0957497
$925$	0	0
$926$	−13318.0	−0.472632
$927$	− 9710.48i	− 0.344050i
$928$	− 8725.99i	− 0.308669i
$929$	9475.86	0.334653	0.167327	−	0.985902i	$-0.446487\pi$
$929$	9475.86	0.334653	0.167327	+	0.985902i	$0.446487\pi$
$930$	0	0
$931$	4564.95	0.160698
$932$	− 4198.66i	− 0.147566i
$933$	26376.4i	0.925534i
$934$	−15509.2	−0.543335
$935$	0	0
$936$	1144.00	0.0399496
$937$	27681.1i	0.965103i	0.875868	+	0.482551i	$0.160290\pi$
$937$	27681.1i	0.965103i	−0.875868	+	0.482551i	$0.839710\pi$
$938$	7205.27i	0.250811i
$939$	3631.13	0.126195
$940$	0	0
$941$	−21404.0	−0.741501	−0.370750	−	0.928733i	$-0.620899\pi$
$941$	−21404.0	−0.741501	−0.370750	+	0.928733i	$0.620899\pi$
$942$	12845.8i	0.444309i
$943$	− 10002.1i	− 0.345402i
$944$	−13351.2	−0.460324
$945$	0	0
$946$	1572.93	0.0540595
$947$	14611.5i	0.501384i	0.968067	+	0.250692i	$0.0806582\pi$
$947$	14611.5i	0.501384i	−0.968067	+	0.250692i	$0.919342\pi$
$948$	1614.84i	0.0553243i
$949$	−989.892	−0.0338601
$950$	0	0
$951$	−7210.15	−0.245852
$952$	− 12473.7i	− 0.424659i
$953$	40724.9i	1.38427i	0.721768	+	0.692135i	$0.243330\pi$
$953$	40724.9i	1.38427i	−0.721768	+	0.692135i	$0.756670\pi$
$954$	3985.44	0.135255
$955$	0	0
$956$	14986.2	0.506995
$957$	11242.7i	0.379753i
$958$	− 8058.34i	− 0.271767i
$959$	41881.5	1.41024
$960$	0	0
$961$	68801.6	2.30948
$962$	− 9516.00i	− 0.318927i
$963$	− 6413.56i	− 0.214615i
$964$	−20036.6	−0.669434
$965$	0	0
$966$	7546.18	0.251340
$967$	48565.2i	1.61505i	0.589836	+	0.807523i	$0.299193\pi$
$967$	48565.2i	1.61505i	−0.589836	+	0.807523i	$0.700807\pi$
$968$	9798.08i	0.325333i
$969$	−22652.7	−0.750989
$970$	0	0
$971$	−7681.71	−0.253880	−0.126940	−	0.991910i	$-0.540516\pi$
$971$	−7681.71	−0.253880	−0.126940	+	0.991910i	$0.540516\pi$
$972$	− 11440.0i	− 0.377508i
$973$	26141.4i	0.861310i
$974$	19088.5	0.627963
$975$	0	0
$976$	−8897.16	−0.291794
$977$	19001.3i	0.622218i	0.950374	+	0.311109i	$0.100700\pi$
$977$	19001.3i	0.622218i	−0.950374	+	0.311109i	$0.899300\pi$
$978$	− 3976.84i	− 0.130026i
$979$	−1668.06	−0.0544548
$980$	0	0
$981$	−5445.68	−0.177235
$982$	1057.19i	0.0343545i
$983$	34310.5i	1.11326i	0.830761	+	0.556630i	$0.187906\pi$
$983$	34310.5i	1.11326i	−0.830761	+	0.556630i	$0.812094\pi$
$984$	5533.33	0.179264
$985$	0	0
$986$	52145.7	1.68424
$987$	6593.24i	0.212629i
$988$	− 3079.91i	− 0.0991752i
$989$	4413.58	0.141905
$990$	0	0
$991$	16266.7	0.521422	0.260711	−	0.965417i	$-0.416043\pi$
$991$	16266.7	0.521422	0.260711	+	0.965417i	$0.416043\pi$
$992$	− 10047.8i	− 0.321592i
$993$	36004.0i	1.15061i
$994$	22925.3	0.731536
$995$	0	0
$996$	−5095.42	−0.162103
$997$	− 56708.8i	− 1.80139i	−0.434453	−	0.900694i	$-0.643058\pi$
$997$	− 56708.8i	− 1.80139i	0.434453	−	0.900694i	$-0.356942\pi$
$998$	− 21782.5i	− 0.690894i
$999$	−55632.0	−1.76188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.b.h.599.3		4
5.2	odd	4		650.4.a.n.1.2	✓	2
5.3	odd	4		650.4.a.o.1.1	yes	2
5.4	even	2	inner	650.4.b.h.599.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.4.a.n.1.2	✓	2	5.2	odd	4
650.4.a.o.1.1	yes	2	5.3	odd	4
650.4.b.h.599.2		4	5.4	even	2	inner
650.4.b.h.599.3		4	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +4.00000i q^{3} -4.00000 q^{4} -8.00000 q^{6} -16.3073i q^{7} -8.00000i q^{8} +11.0000 q^{9} -10.3073 q^{11} -16.0000i q^{12} +13.0000i q^{13} +32.6146 q^{14} +16.0000 q^{16} +95.6146i q^{17} +22.0000i q^{18} +59.2291 q^{19} +65.2291 q^{21} -20.6146i q^{22} -57.8437i q^{23} +32.0000 q^{24} -26.0000 q^{26} +152.000i q^{27} +65.2291i q^{28} -272.687 q^{29} -313.995 q^{31} +32.0000i q^{32} -41.2291i q^{33} -191.229 q^{34} -44.0000 q^{36} +366.000i q^{37} +118.458i q^{38} -52.0000 q^{39} +172.916 q^{41} +130.458i q^{42} +76.3019i q^{43} +41.2291 q^{44} +115.687 q^{46} +101.078i q^{47} +64.0000i q^{48} +77.0728 q^{49} -382.458 q^{51} -52.0000i q^{52} -181.156i q^{53} -304.000 q^{54} -130.458 q^{56} +236.916i q^{57} -545.375i q^{58} -834.453 q^{59} -556.073 q^{61} -627.989i q^{62} -179.380i q^{63} -64.0000 q^{64} +82.4582 q^{66} +220.922i q^{67} -382.458i q^{68} +231.375 q^{69} +702.916 q^{71} -88.0000i q^{72} +76.1455i q^{73} -732.000 q^{74} -236.916 q^{76} +168.084i q^{77} -104.000i q^{78} -100.927 q^{79} -311.000 q^{81} +345.833i q^{82} -318.464i q^{83} -260.916 q^{84} -152.604 q^{86} -1090.75i q^{87} +82.4582i q^{88} +161.833 q^{89} +211.995 q^{91} +231.375i q^{92} -1255.98i q^{93} -202.156 q^{94} -128.000 q^{96} +1329.98i q^{97} +154.146i q^{98} -113.380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 32 q^{6} + 44 q^{9} + 44 q^{11} - 40 q^{14} + 64 q^{16} - 104 q^{19} - 80 q^{21} + 128 q^{24} - 104 q^{26} - 68 q^{29} - 148 q^{31} - 424 q^{34} - 176 q^{36} - 208 q^{39} - 672 q^{41} - 176 q^{44}+ \cdots + 484 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 32 * q^6 + 44 * q^9 + 44 * q^11 - 40 * q^14 + 64 * q^16 - 104 * q^19 - 80 * q^21 + 128 * q^24 - 104 * q^26 - 68 * q^29 - 148 * q^31 - 424 * q^34 - 176 * q^36 - 208 * q^39 - 672 * q^41 - 176 * q^44 - 560 * q^46 - 544 * q^49 - 848 * q^51 - 1216 * q^54 + 160 * q^56 - 1548 * q^59 - 1372 * q^61 - 256 * q^64 - 352 * q^66 - 1120 * q^69 + 1448 * q^71 - 2928 * q^74 + 416 * q^76 - 1256 * q^79 - 1244 * q^81 + 320 * q^84 + 1776 * q^86 - 2080 * q^89 - 260 * q^91 - 1320 * q^94 - 512 * q^96 + 484 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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