LMFDB - Embedded newform 600.6.f.k.49.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,6,Mod(49,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.49"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-324,0,128] 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:	$96.2302918878$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1489})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 745x^{2} + 138384 $ x^4 + 745*x^2 + 138384
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-18.7938i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.6.f.k.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +146.350i q^{7} -81.0000 q^{9} +649.401 q^{11} +118.949i q^{13} -1853.75i q^{17} -2400.45 q^{19} -1317.15 q^{21} -4764.86i q^{23} -729.000i q^{27} -5909.21 q^{29} +5170.55 q^{31} +5844.61i q^{33} +3636.26i q^{37} -1070.54 q^{39} -15159.1 q^{41} +22138.0i q^{43} -22666.1i q^{47} -4611.40 q^{49} +16683.8 q^{51} -6420.69i q^{53} -21604.1i q^{57} +2964.61 q^{59} +1113.75 q^{61} -11854.4i q^{63} -33997.2i q^{67} +42883.7 q^{69} -3671.15 q^{71} -44104.9i q^{73} +95040.0i q^{77} -41037.9 q^{79} +6561.00 q^{81} -101143. i q^{83} -53182.9i q^{87} +63379.3 q^{89} -17408.3 q^{91} +46535.0i q^{93} +70291.7i q^{97} -52601.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{9} + 128 q^{11} - 5280 q^{19} + 288 q^{21} - 3880 q^{29} + 12656 q^{31} - 20952 q^{39} + 4808 q^{41} - 28324 q^{49} + 38952 q^{51} - 10368 q^{59} - 107912 q^{61} - 720 q^{69} - 143104 q^{71}+ \cdots - 10368 q^{99}+O(q^{100}) $ 4 * q - 324 * q^9 + 128 * q^11 - 5280 * q^19 + 288 * q^21 - 3880 * q^29 + 12656 * q^31 - 20952 * q^39 + 4808 * q^41 - 28324 * q^49 + 38952 * q^51 - 10368 * q^59 - 107912 * q^61 - 720 * q^69 - 143104 * q^71 - 58576 * q^79 + 26244 * q^81 + 279448 * q^89 + 304512 * q^91 - 10368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	146.350i	1.12888i	0.825474	+	0.564441i	$0.190908\pi$
$7$	146.350i	1.12888i	−0.825474	+	0.564441i	$0.809092\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	649.401	1.61820	0.809098	−	0.587673i	$-0.199956\pi$
$11$	649.401	1.61820	0.809098	+	0.587673i	$0.199956\pi$
$12$	0	0
$13$	118.949i	0.195211i	0.995225	+	0.0976053i	$0.0311183\pi$
$13$	118.949i	0.195211i	−0.995225	+	0.0976053i	$0.968882\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1853.75i	− 1.55571i	−0.628443	−	0.777856i	$-0.716307\pi$
$17$	− 1853.75i	− 1.55571i	0.628443	−	0.777856i	$-0.283693\pi$
$18$	0	0
$19$	−2400.45	−1.52549	−0.762744	−	0.646700i	$-0.776149\pi$
$19$	−2400.45	−1.52549	−0.762744	+	0.646700i	$0.776149\pi$
$20$	0	0
$21$	−1317.15	−0.651760
$22$	0	0
$23$	− 4764.86i	− 1.87815i	−0.343712	−	0.939075i	$-0.611684\pi$
$23$	− 4764.86i	− 1.87815i	0.343712	−	0.939075i	$-0.388316\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	−5909.21	−1.30477	−0.652385	−	0.757887i	$-0.726232\pi$
$29$	−5909.21	−1.30477	−0.652385	+	0.757887i	$0.726232\pi$
$30$	0	0
$31$	5170.55	0.966346	0.483173	−	0.875525i	$-0.339484\pi$
$31$	5170.55	0.966346	0.483173	+	0.875525i	$0.339484\pi$
$32$	0	0
$33$	5844.61i	0.934266i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3636.26i	0.436667i	0.975874	+	0.218334i	$0.0700621\pi$
$37$	3636.26i	0.436667i	−0.975874	+	0.218334i	$0.929938\pi$
$38$	0	0
$39$	−1070.54	−0.112705
$40$	0	0
$41$	−15159.1	−1.40836	−0.704181	−	0.710020i	$-0.748686\pi$
$41$	−15159.1	−1.40836	−0.704181	+	0.710020i	$0.748686\pi$
$42$	0	0
$43$	22138.0i	1.82586i	0.408116	+	0.912930i	$0.366186\pi$
$43$	22138.0i	1.82586i	−0.408116	+	0.912930i	$0.633814\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 22666.1i	− 1.49669i	−0.663311	−	0.748344i	$-0.730849\pi$
$47$	− 22666.1i	− 1.49669i	0.663311	−	0.748344i	$-0.269151\pi$
$48$	0	0
$49$	−4611.40	−0.274374
$50$	0	0
$51$	16683.8	0.898191
$52$	0	0
$53$	− 6420.69i	− 0.313973i	−0.987601	−	0.156986i	$-0.949822\pi$
$53$	− 6420.69i	− 0.313973i	0.987601	−	0.156986i	$-0.0501779\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 21604.1i	− 0.880741i
$58$	0	0
$59$	2964.61	0.110876	0.0554380	−	0.998462i	$-0.482344\pi$
$59$	2964.61	0.110876	0.0554380	+	0.998462i	$0.482344\pi$
$60$	0	0
$61$	1113.75	0.0383232	0.0191616	−	0.999816i	$-0.493900\pi$
$61$	1113.75	0.0383232	0.0191616	+	0.999816i	$0.493900\pi$
$62$	0	0
$63$	− 11854.4i	− 0.376294i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 33997.2i	− 0.925243i	−0.886556	−	0.462622i	$-0.846909\pi$
$67$	− 33997.2i	− 0.925243i	0.886556	−	0.462622i	$-0.153091\pi$
$68$	0	0
$69$	42883.7	1.08435
$70$	0	0
$71$	−3671.15	−0.0864283	−0.0432142	−	0.999066i	$-0.513760\pi$
$71$	−3671.15	−0.0864283	−0.0432142	+	0.999066i	$0.513760\pi$
$72$	0	0
$73$	− 44104.9i	− 0.968678i	−0.874880	−	0.484339i	$-0.839060\pi$
$73$	− 44104.9i	− 0.968678i	0.874880	−	0.484339i	$-0.160940\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	95040.0i	1.82675i
$78$	0	0
$79$	−41037.9	−0.739805	−0.369903	−	0.929071i	$-0.620609\pi$
$79$	−41037.9	−0.739805	−0.369903	+	0.929071i	$0.620609\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	− 101143.i	− 1.61154i	−0.592227	−	0.805771i	$-0.701751\pi$
$83$	− 101143.i	− 1.61154i	0.592227	−	0.805771i	$-0.298249\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 53182.9i	− 0.753310i
$88$	0	0
$89$	63379.3	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$89$	63379.3	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$90$	0	0
$91$	−17408.3	−0.220370
$92$	0	0
$93$	46535.0i	0.557920i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	70291.7i	0.758533i	0.925287	+	0.379267i	$0.123824\pi$
$97$	70291.7i	0.758533i	−0.925287	+	0.379267i	$0.876176\pi$
$98$	0	0
$99$	−52601.5	−0.539399
$100$	0	0
$101$	−44145.4	−0.430608	−0.215304	−	0.976547i	$-0.569074\pi$
$101$	−44145.4	−0.430608	−0.215304	+	0.976547i	$0.569074\pi$
$102$	0	0
$103$	37009.4i	0.343731i	0.985120	+	0.171866i	$0.0549795\pi$
$103$	37009.4i	0.343731i	−0.985120	+	0.171866i	$0.945020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 54879.8i	− 0.463397i	−0.972788	−	0.231698i	$-0.925572\pi$
$107$	− 54879.8i	− 0.463397i	0.972788	−	0.231698i	$-0.0744282\pi$
$108$	0	0
$109$	−112407.	−0.906206	−0.453103	−	0.891458i	$-0.649683\pi$
$109$	−112407.	−0.906206	−0.453103	+	0.891458i	$0.649683\pi$
$110$	0	0
$111$	−32726.3	−0.252110
$112$	0	0
$113$	− 206991.i	− 1.52495i	−0.647018	−	0.762474i	$-0.723984\pi$
$113$	− 206991.i	− 1.52495i	0.647018	−	0.762474i	$-0.276016\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9634.89i	− 0.0650702i
$118$	0	0
$119$	271297.	1.75621
$120$	0	0
$121$	260671.	1.61856
$122$	0	0
$123$	− 136432.i	− 0.813119i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	22883.3i	0.125895i	0.998017	+	0.0629475i	$0.0200501\pi$
$127$	22883.3i	0.125895i	−0.998017	+	0.0629475i	$0.979950\pi$
$128$	0	0
$129$	−199242.	−1.05416
$130$	0	0
$131$	49165.6	0.250313	0.125156	−	0.992137i	$-0.460057\pi$
$131$	49165.6	0.250313	0.125156	+	0.992137i	$0.460057\pi$
$132$	0	0
$133$	− 351307.i	− 1.72210i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 189263.i	− 0.861518i	−0.902467	−	0.430759i	$-0.858246\pi$
$137$	− 189263.i	− 0.861518i	0.902467	−	0.430759i	$-0.141754\pi$
$138$	0	0
$139$	−247755.	−1.08764	−0.543820	−	0.839202i	$-0.683023\pi$
$139$	−247755.	−1.08764	−0.543820	+	0.839202i	$0.683023\pi$
$140$	0	0
$141$	203995.	0.864113
$142$	0	0
$143$	77245.8i	0.315889i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 41502.6i	− 0.158410i
$148$	0	0
$149$	−95894.1	−0.353856	−0.176928	−	0.984224i	$-0.556616\pi$
$149$	−95894.1	−0.353856	−0.176928	+	0.984224i	$0.556616\pi$
$150$	0	0
$151$	414806.	1.48048	0.740241	−	0.672342i	$-0.234711\pi$
$151$	414806.	1.48048	0.740241	+	0.672342i	$0.234711\pi$
$152$	0	0
$153$	150154.i	0.518571i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 218866.i	− 0.708645i	−0.935123	−	0.354322i	$-0.884712\pi$
$157$	− 218866.i	− 0.708645i	0.935123	−	0.354322i	$-0.115288\pi$
$158$	0	0
$159$	57786.2	0.181272
$160$	0	0
$161$	697338.	2.12021
$162$	0	0
$163$	− 411608.i	− 1.21343i	−0.794920	−	0.606715i	$-0.792487\pi$
$163$	− 411608.i	− 1.21343i	0.794920	−	0.606715i	$-0.207513\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	189476.i	0.525730i	0.964833	+	0.262865i	$0.0846675\pi$
$167$	189476.i	0.525730i	−0.964833	+	0.262865i	$0.915333\pi$
$168$	0	0
$169$	357144.	0.961893
$170$	0	0
$171$	194437.	0.508496
$172$	0	0
$173$	− 150896.i	− 0.383320i	−0.981461	−	0.191660i	$-0.938613\pi$
$173$	− 150896.i	− 0.383320i	0.981461	−	0.191660i	$-0.0613871\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26681.5i	0.0640143i
$178$	0	0
$179$	409591.	0.955473	0.477736	−	0.878503i	$-0.341457\pi$
$179$	409591.	0.955473	0.477736	+	0.878503i	$0.341457\pi$
$180$	0	0
$181$	321345.	0.729080	0.364540	−	0.931188i	$-0.381226\pi$
$181$	321345.	0.729080	0.364540	+	0.931188i	$0.381226\pi$
$182$	0	0
$183$	10023.7i	0.0221259i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.20383e6i	− 2.51745i
$188$	0	0
$189$	106689.	0.217253
$190$	0	0
$191$	204362.	0.405338	0.202669	−	0.979247i	$-0.435039\pi$
$191$	204362.	0.405338	0.202669	+	0.979247i	$0.435039\pi$
$192$	0	0
$193$	420929.i	0.813422i	0.913557	+	0.406711i	$0.133324\pi$
$193$	420929.i	0.813422i	−0.913557	+	0.406711i	$0.866676\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	492903.i	0.904891i	0.891792	+	0.452445i	$0.149448\pi$
$197$	492903.i	0.904891i	−0.891792	+	0.452445i	$0.850552\pi$
$198$	0	0
$199$	−312259.	−0.558962	−0.279481	−	0.960151i	$-0.590162\pi$
$199$	−312259.	−0.558962	−0.279481	+	0.960151i	$0.590162\pi$
$200$	0	0
$201$	305975.	0.534189
$202$	0	0
$203$	− 864814.i	− 1.47293i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	385953.i	0.626050i
$208$	0	0
$209$	−1.55886e6	−2.46854
$210$	0	0
$211$	−991009.	−1.53240	−0.766199	−	0.642604i	$-0.777854\pi$
$211$	−991009.	−1.53240	−0.766199	+	0.642604i	$0.777854\pi$
$212$	0	0
$213$	− 33040.3i	− 0.0498994i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	756712.i	1.09089i
$218$	0	0
$219$	396944.	0.559267
$220$	0	0
$221$	220502.	0.303691
$222$	0	0
$223$	− 1.25333e6i	− 1.68773i	−0.536557	−	0.843864i	$-0.680275\pi$
$223$	− 1.25333e6i	− 1.68773i	0.536557	−	0.843864i	$-0.319725\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 479319.i	− 0.617390i	−0.951161	−	0.308695i	$-0.900108\pi$
$227$	− 479319.i	− 0.617390i	0.951161	−	0.308695i	$-0.0998923\pi$
$228$	0	0
$229$	1.09794e6	1.38353	0.691767	−	0.722120i	$-0.256832\pi$
$229$	1.09794e6	1.38353	0.691767	+	0.722120i	$0.256832\pi$
$230$	0	0
$231$	−855360.	−1.05468
$232$	0	0
$233$	− 539999.i	− 0.651633i	−0.945433	−	0.325817i	$-0.894361\pi$
$233$	− 539999.i	− 0.651633i	0.945433	−	0.325817i	$-0.105639\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 369341.i	− 0.427127i
$238$	0	0
$239$	−1.25769e6	−1.42422	−0.712112	−	0.702065i	$-0.752261\pi$
$239$	−1.25769e6	−1.42422	−0.712112	+	0.702065i	$0.752261\pi$
$240$	0	0
$241$	110008.	0.122006	0.0610028	−	0.998138i	$-0.480570\pi$
$241$	110008.	0.122006	0.0610028	+	0.998138i	$0.480570\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 285532.i	− 0.297791i
$248$	0	0
$249$	910289.	0.930424
$250$	0	0
$251$	−373992.	−0.374695	−0.187347	−	0.982294i	$-0.559989\pi$
$251$	−373992.	−0.374695	−0.187347	+	0.982294i	$0.559989\pi$
$252$	0	0
$253$	− 3.09430e6i	− 3.03922i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.26750e6i	− 1.19706i	−0.801100	−	0.598531i	$-0.795751\pi$
$257$	− 1.26750e6i	− 1.19706i	0.801100	−	0.598531i	$-0.204249\pi$
$258$	0	0
$259$	−532167.	−0.492945
$260$	0	0
$261$	478646.	0.434924
$262$	0	0
$263$	− 42575.9i	− 0.0379555i	−0.999820	−	0.0189778i	$-0.993959\pi$
$263$	− 42575.9i	− 0.0379555i	0.999820	−	0.0189778i	$-0.00604117\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	570414.i	0.489679i
$268$	0	0
$269$	2.01104e6	1.69449	0.847247	−	0.531198i	$-0.178258\pi$
$269$	2.01104e6	1.69449	0.847247	+	0.531198i	$0.178258\pi$
$270$	0	0
$271$	1.81113e6	1.49805	0.749023	−	0.662544i	$-0.230523\pi$
$271$	1.81113e6	1.49805	0.749023	+	0.662544i	$0.230523\pi$
$272$	0	0
$273$	− 156674.i	− 0.127230i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.15345e6i	0.903231i	0.892213	+	0.451616i	$0.149152\pi$
$277$	1.15345e6i	0.903231i	−0.892213	+	0.451616i	$0.850848\pi$
$278$	0	0
$279$	−418815.	−0.322115
$280$	0	0
$281$	1.67992e6	1.26918	0.634589	−	0.772849i	$-0.281169\pi$
$281$	1.67992e6	1.26918	0.634589	+	0.772849i	$0.281169\pi$
$282$	0	0
$283$	756853.i	0.561753i	0.959744	+	0.280877i	$0.0906252\pi$
$283$	756853.i	0.561753i	−0.959744	+	0.280877i	$0.909375\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.21854e6i	− 1.58987i
$288$	0	0
$289$	−2.01654e6	−1.42024
$290$	0	0
$291$	−632625.	−0.437939
$292$	0	0
$293$	2.22662e6i	1.51523i	0.652704	+	0.757613i	$0.273634\pi$
$293$	2.22662e6i	1.51523i	−0.652704	+	0.757613i	$0.726366\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 473413.i	− 0.311422i
$298$	0	0
$299$	566776.	0.366635
$300$	0	0
$301$	−3.23990e6	−2.06118
$302$	0	0
$303$	− 397309.i	− 0.248612i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	270015.i	0.163509i	0.996652	+	0.0817545i	$0.0260524\pi$
$307$	270015.i	0.163509i	−0.996652	+	0.0817545i	$0.973948\pi$
$308$	0	0
$309$	−333085.	−0.198453
$310$	0	0
$311$	−1.24052e6	−0.727280	−0.363640	−	0.931539i	$-0.618466\pi$
$311$	−1.24052e6	−0.727280	−0.363640	+	0.931539i	$0.618466\pi$
$312$	0	0
$313$	316675.i	0.182706i	0.995819	+	0.0913529i	$0.0291191\pi$
$313$	316675.i	0.182706i	−0.995819	+	0.0913529i	$0.970881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	983045.i	0.549446i	0.961523	+	0.274723i	$0.0885862\pi$
$317$	983045.i	0.549446i	−0.961523	+	0.274723i	$0.911414\pi$
$318$	0	0
$319$	−3.83745e6	−2.11138
$320$	0	0
$321$	493918.	0.267542
$322$	0	0
$323$	4.44984e6i	2.37322i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.01166e6i	− 0.523198i
$328$	0	0
$329$	3.31718e6	1.68958
$330$	0	0
$331$	−1.01843e6	−0.510927	−0.255464	−	0.966819i	$-0.582228\pi$
$331$	−1.01843e6	−0.510927	−0.255464	+	0.966819i	$0.582228\pi$
$332$	0	0
$333$	− 294537.i	− 0.145556i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.23120e6i	0.590544i	0.955413	+	0.295272i	$0.0954102\pi$
$337$	1.23120e6i	0.590544i	−0.955413	+	0.295272i	$0.904590\pi$
$338$	0	0
$339$	1.86292e6	0.880430
$340$	0	0
$341$	3.35776e6	1.56374
$342$	0	0
$343$	1.78483e6i	0.819146i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 727325.i	− 0.324269i	−0.986769	−	0.162134i	$-0.948162\pi$
$347$	− 727325.i	− 0.324269i	0.986769	−	0.162134i	$-0.0518378\pi$
$348$	0	0
$349$	166838.	0.0733214	0.0366607	−	0.999328i	$-0.488328\pi$
$349$	166838.	0.0733214	0.0366607	+	0.999328i	$0.488328\pi$
$350$	0	0
$351$	86714.0	0.0375683
$352$	0	0
$353$	826694.i	0.353108i	0.984291	+	0.176554i	$0.0564951\pi$
$353$	826694.i	0.353108i	−0.984291	+	0.176554i	$0.943505\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.44167e6i	1.01395i
$358$	0	0
$359$	−911164.	−0.373130	−0.186565	−	0.982443i	$-0.559735\pi$
$359$	−911164.	−0.373130	−0.186565	+	0.982443i	$0.559735\pi$
$360$	0	0
$361$	3.28607e6	1.32712
$362$	0	0
$363$	2.34604e6i	0.934476i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	417688.i	0.161878i	0.996719	+	0.0809389i	$0.0257919\pi$
$367$	417688.i	0.161878i	−0.996719	+	0.0809389i	$0.974208\pi$
$368$	0	0
$369$	1.22789e6	0.469454
$370$	0	0
$371$	939670.	0.354438
$372$	0	0
$373$	− 1.82262e6i	− 0.678303i	−0.940732	−	0.339151i	$-0.889860\pi$
$373$	− 1.82262e6i	− 0.678303i	0.940732	−	0.339151i	$-0.110140\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 702896.i	− 0.254705i
$378$	0	0
$379$	−1.13976e6	−0.407583	−0.203792	−	0.979014i	$-0.565326\pi$
$379$	−1.13976e6	−0.407583	−0.203792	+	0.979014i	$0.565326\pi$
$380$	0	0
$381$	−205949.	−0.0726855
$382$	0	0
$383$	− 2.37135e6i	− 0.826036i	−0.910723	−	0.413018i	$-0.864475\pi$
$383$	− 2.37135e6i	− 0.826036i	0.910723	−	0.413018i	$-0.135525\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.79318e6i	− 0.608620i
$388$	0	0
$389$	1.44160e6	0.483025	0.241513	−	0.970398i	$-0.422356\pi$
$389$	1.44160e6	0.483025	0.241513	+	0.970398i	$0.422356\pi$
$390$	0	0
$391$	−8.83286e6	−2.92186
$392$	0	0
$393$	442490.i	0.144518i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	843782.i	0.268691i	0.990935	+	0.134346i	$0.0428933\pi$
$397$	843782.i	0.268691i	−0.990935	+	0.134346i	$0.957107\pi$
$398$	0	0
$399$	3.16176e6	0.994253
$400$	0	0
$401$	−3.08091e6	−0.956795	−0.478397	−	0.878143i	$-0.658782\pi$
$401$	−3.08091e6	−0.956795	−0.478397	+	0.878143i	$0.658782\pi$
$402$	0	0
$403$	615033.i	0.188641i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.36139e6i	0.706613i
$408$	0	0
$409$	−5.59642e6	−1.65425	−0.827126	−	0.562016i	$-0.810026\pi$
$409$	−5.59642e6	−1.65425	−0.827126	+	0.562016i	$0.810026\pi$
$410$	0	0
$411$	1.70337e6	0.497398
$412$	0	0
$413$	433871.i	0.125166i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.22979e6i	− 0.627949i
$418$	0	0
$419$	−2.45597e6	−0.683420	−0.341710	−	0.939805i	$-0.611006\pi$
$419$	−2.45597e6	−0.683420	−0.341710	+	0.939805i	$0.611006\pi$
$420$	0	0
$421$	6.36693e6	1.75075	0.875376	−	0.483443i	$-0.160614\pi$
$421$	6.36693e6	1.75075	0.875376	+	0.483443i	$0.160614\pi$
$422$	0	0
$423$	1.83595e6i	0.498896i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	162997.i	0.0432623i
$428$	0	0
$429$	−695212.	−0.182379
$430$	0	0
$431$	−4.71463e6	−1.22252	−0.611258	−	0.791432i	$-0.709336\pi$
$431$	−4.71463e6	−1.22252	−0.611258	+	0.791432i	$0.709336\pi$
$432$	0	0
$433$	− 3.99653e6i	− 1.02438i	−0.858871	−	0.512192i	$-0.828834\pi$
$433$	− 3.99653e6i	− 1.02438i	0.858871	−	0.512192i	$-0.171166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.14378e7i	2.86510i
$438$	0	0
$439$	−328891.	−0.0814500	−0.0407250	−	0.999170i	$-0.512967\pi$
$439$	−328891.	−0.0814500	−0.0407250	+	0.999170i	$0.512967\pi$
$440$	0	0
$441$	373523.	0.0914578
$442$	0	0
$443$	− 2.41671e6i	− 0.585081i	−0.956253	−	0.292540i	$-0.905499\pi$
$443$	− 2.41671e6i	− 0.585081i	0.956253	−	0.292540i	$-0.0945006\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 863047.i	− 0.204299i
$448$	0	0
$449$	4.00121e6	0.936647	0.468323	−	0.883557i	$-0.344858\pi$
$449$	4.00121e6	0.936647	0.468323	+	0.883557i	$0.344858\pi$
$450$	0	0
$451$	−9.84435e6	−2.27901
$452$	0	0
$453$	3.73326e6i	0.854756i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.86212e6i	1.76096i	0.474084	+	0.880480i	$0.342779\pi$
$457$	7.86212e6i	1.76096i	−0.474084	+	0.880480i	$0.657221\pi$
$458$	0	0
$459$	−1.35138e6	−0.299397
$460$	0	0
$461$	−4.64530e6	−1.01803	−0.509016	−	0.860757i	$-0.669991\pi$
$461$	−4.64530e6	−1.01803	−0.509016	+	0.860757i	$0.669991\pi$
$462$	0	0
$463$	− 1.10561e6i	− 0.239690i	−0.992793	−	0.119845i	$-0.961760\pi$
$463$	− 1.10561e6i	− 0.239690i	0.992793	−	0.119845i	$-0.0382397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.73047e6i	0.367175i	0.983003	+	0.183587i	$0.0587710\pi$
$467$	1.73047e6i	0.367175i	−0.983003	+	0.183587i	$0.941229\pi$
$468$	0	0
$469$	4.97550e6	1.04449
$470$	0	0
$471$	1.96979e6	0.409136
$472$	0	0
$473$	1.43764e7i	2.95460i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	520076.i	0.104658i
$478$	0	0
$479$	9.07136e6	1.80648	0.903241	−	0.429134i	$-0.141181\pi$
$479$	9.07136e6	1.80648	0.903241	+	0.429134i	$0.141181\pi$
$480$	0	0
$481$	−432530.	−0.0852420
$482$	0	0
$483$	6.27604e6i	1.22410i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.55244e6i	− 0.678741i	−0.940653	−	0.339370i	$-0.889786\pi$
$487$	− 3.55244e6i	− 0.678741i	0.940653	−	0.339370i	$-0.110214\pi$
$488$	0	0
$489$	3.70447e6	0.700574
$490$	0	0
$491$	−8.51045e6	−1.59312	−0.796560	−	0.604559i	$-0.793349\pi$
$491$	−8.51045e6	−1.59312	−0.796560	+	0.604559i	$0.793349\pi$
$492$	0	0
$493$	1.09542e7i	2.02985i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 537273.i	− 0.0975673i
$498$	0	0
$499$	566565.	0.101859	0.0509294	−	0.998702i	$-0.483782\pi$
$499$	566565.	0.101859	0.0509294	+	0.998702i	$0.483782\pi$
$500$	0	0
$501$	−1.70528e6	−0.303531
$502$	0	0
$503$	− 3.80262e6i	− 0.670135i	−0.942194	−	0.335068i	$-0.891241\pi$
$503$	− 3.80262e6i	− 0.670135i	0.942194	−	0.335068i	$-0.108759\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.21430e6i	0.555349i
$508$	0	0
$509$	2.73265e6	0.467508	0.233754	−	0.972296i	$-0.424899\pi$
$509$	2.73265e6	0.467508	0.233754	+	0.972296i	$0.424899\pi$
$510$	0	0
$511$	6.45476e6	1.09352
$512$	0	0
$513$	1.74993e6i	0.293580i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.47194e7i	− 2.42194i
$518$	0	0
$519$	1.35806e6	0.221310
$520$	0	0
$521$	−6.32941e6	−1.02157	−0.510786	−	0.859708i	$-0.670645\pi$
$521$	−6.32941e6	−1.02157	−0.510786	+	0.859708i	$0.670645\pi$
$522$	0	0
$523$	− 4.73315e6i	− 0.756651i	−0.925673	−	0.378326i	$-0.876500\pi$
$523$	− 4.73315e6i	− 0.756651i	0.925673	−	0.378326i	$-0.123500\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.58492e6i	− 1.50336i
$528$	0	0
$529$	−1.62675e7	−2.52745
$530$	0	0
$531$	−240133.	−0.0369587
$532$	0	0
$533$	− 1.80317e6i	− 0.274927i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.68632e6i	0.551642i
$538$	0	0
$539$	−2.99465e6	−0.443990
$540$	0	0
$541$	−7.95329e6	−1.16830	−0.584149	−	0.811647i	$-0.698572\pi$
$541$	−7.95329e6	−1.16830	−0.584149	+	0.811647i	$0.698572\pi$
$542$	0	0
$543$	2.89211e6i	0.420935i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.47929e6i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$547$	8.47929e6i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$548$	0	0
$549$	−90213.4	−0.0127744
$550$	0	0
$551$	1.41848e7	1.99041
$552$	0	0
$553$	− 6.00591e6i	− 0.835152i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.41411e6i	0.466272i	0.972444	+	0.233136i	$0.0748988\pi$
$557$	3.41411e6i	0.466272i	−0.972444	+	0.233136i	$0.925101\pi$
$558$	0	0
$559$	−2.63330e6	−0.356427
$560$	0	0
$561$	1.08345e7	1.45345
$562$	0	0
$563$	1.20757e7i	1.60561i	0.596243	+	0.802804i	$0.296660\pi$
$563$	1.20757e7i	1.60561i	−0.596243	+	0.802804i	$0.703340\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	960204.i	0.125431i
$568$	0	0
$569$	−3.01621e6	−0.390554	−0.195277	−	0.980748i	$-0.562561\pi$
$569$	−3.01621e6	−0.390554	−0.195277	+	0.980748i	$0.562561\pi$
$570$	0	0
$571$	−1.38871e7	−1.78247	−0.891235	−	0.453542i	$-0.850160\pi$
$571$	−1.38871e7	−1.78247	−0.891235	+	0.453542i	$0.850160\pi$
$572$	0	0
$573$	1.83926e6i	0.234022i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.84090e6i	− 0.355235i	−0.984100	−	0.177618i	$-0.943161\pi$
$577$	− 2.84090e6i	− 0.355235i	0.984100	−	0.177618i	$-0.0568390\pi$
$578$	0	0
$579$	−3.78836e6	−0.469629
$580$	0	0
$581$	1.48023e7	1.81924
$582$	0	0
$583$	− 4.16960e6i	− 0.508070i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.94501e6i	− 0.831912i	−0.909385	−	0.415956i	$-0.863447\pi$
$587$	− 6.94501e6i	− 0.831912i	0.909385	−	0.415956i	$-0.136553\pi$
$588$	0	0
$589$	−1.24117e7	−1.47415
$590$	0	0
$591$	−4.43613e6	−0.522439
$592$	0	0
$593$	− 3.67971e6i	− 0.429712i	−0.976646	−	0.214856i	$-0.931072\pi$
$593$	− 3.67971e6i	− 0.429712i	0.976646	−	0.214856i	$-0.0689282\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.81033e6i	− 0.322717i
$598$	0	0
$599$	1.13529e7	1.29282	0.646411	−	0.762990i	$-0.276269\pi$
$599$	1.13529e7	1.29282	0.646411	+	0.762990i	$0.276269\pi$
$600$	0	0
$601$	−127107.	−0.0143543	−0.00717715	−	0.999974i	$-0.502285\pi$
$601$	−127107.	−0.0143543	−0.00717715	+	0.999974i	$0.502285\pi$
$602$	0	0
$603$	2.75377e6i	0.308414i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.39012e7i	− 1.53138i	−0.643213	−	0.765688i	$-0.722399\pi$
$607$	− 1.39012e7i	− 1.53138i	0.643213	−	0.765688i	$-0.277601\pi$
$608$	0	0
$609$	7.78333e6	0.850398
$610$	0	0
$611$	2.69611e6	0.292169
$612$	0	0
$613$	− 9.80675e6i	− 1.05408i	−0.849840	−	0.527040i	$-0.823302\pi$
$613$	− 9.80675e6i	− 1.05408i	0.849840	−	0.527040i	$-0.176698\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.11020e6i	0.857667i	0.903384	+	0.428833i	$0.141075\pi$
$617$	8.11020e6i	0.857667i	−0.903384	+	0.428833i	$0.858925\pi$
$618$	0	0
$619$	−1.01637e7	−1.06617	−0.533083	−	0.846063i	$-0.678967\pi$
$619$	−1.01637e7	−1.06617	−0.533083	+	0.846063i	$0.678967\pi$
$620$	0	0
$621$	−3.47358e6	−0.361450
$622$	0	0
$623$	9.27557e6i	0.957460i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.40297e7i	− 1.42521i
$628$	0	0
$629$	6.74072e6	0.679328
$630$	0	0
$631$	−9.29395e6	−0.929238	−0.464619	−	0.885511i	$-0.653809\pi$
$631$	−9.29395e6	−0.929238	−0.464619	+	0.885511i	$0.653809\pi$
$632$	0	0
$633$	− 8.91908e6i	− 0.884730i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 548522.i	− 0.0535606i
$638$	0	0
$639$	297363.	0.0288094
$640$	0	0
$641$	−1.40969e7	−1.35512	−0.677559	−	0.735468i	$-0.736962\pi$
$641$	−1.40969e7	−1.35512	−0.677559	+	0.735468i	$0.736962\pi$
$642$	0	0
$643$	− 8.27016e6i	− 0.788836i	−0.918931	−	0.394418i	$-0.870946\pi$
$643$	− 8.27016e6i	− 0.788836i	0.918931	−	0.394418i	$-0.129054\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.31832e6i	0.405559i	0.979224	+	0.202779i	$0.0649974\pi$
$647$	4.31832e6i	0.405559i	−0.979224	+	0.202779i	$0.935003\pi$
$648$	0	0
$649$	1.92522e6	0.179419
$650$	0	0
$651$	−6.81041e6	−0.629826
$652$	0	0
$653$	1.89476e7i	1.73888i	0.494037	+	0.869441i	$0.335521\pi$
$653$	1.89476e7i	1.73888i	−0.494037	+	0.869441i	$0.664479\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.57250e6i	0.322893i
$658$	0	0
$659$	−6.91261e6	−0.620053	−0.310026	−	0.950728i	$-0.600338\pi$
$659$	−6.91261e6	−0.620053	−0.310026	+	0.950728i	$0.600338\pi$
$660$	0	0
$661$	5.82317e6	0.518389	0.259195	−	0.965825i	$-0.416543\pi$
$661$	5.82317e6	0.518389	0.259195	+	0.965825i	$0.416543\pi$
$662$	0	0
$663$	1.98452e6i	0.175336i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.81565e7i	2.45056i
$668$	0	0
$669$	1.12800e7	0.974410
$670$	0	0
$671$	723268.	0.0620144
$672$	0	0
$673$	− 1.75215e7i	− 1.49119i	−0.666400	−	0.745594i	$-0.732166\pi$
$673$	− 1.75215e7i	− 1.49119i	0.666400	−	0.745594i	$-0.267834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.18408e7i	− 0.992908i	−0.868063	−	0.496454i	$-0.834635\pi$
$677$	− 1.18408e7i	− 0.992908i	0.868063	−	0.496454i	$-0.165365\pi$
$678$	0	0
$679$	−1.02872e7	−0.856294
$680$	0	0
$681$	4.31387e6	0.356450
$682$	0	0
$683$	− 430258.i	− 0.0352921i	−0.999844	−	0.0176460i	$-0.994383\pi$
$683$	− 430258.i	− 0.0352921i	0.999844	−	0.0176460i	$-0.00561720\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.88146e6i	0.798784i
$688$	0	0
$689$	763736.	0.0612908
$690$	0	0
$691$	2.02121e7	1.61034	0.805169	−	0.593046i	$-0.202075\pi$
$691$	2.02121e7	1.61034	0.805169	+	0.593046i	$0.202075\pi$
$692$	0	0
$693$	− 7.69824e6i	− 0.608917i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.81012e7i	2.19101i
$698$	0	0
$699$	4.85999e6	0.376221
$700$	0	0
$701$	1.11172e7	0.854475	0.427237	−	0.904139i	$-0.359487\pi$
$701$	1.11172e7	0.854475	0.427237	+	0.904139i	$0.359487\pi$
$702$	0	0
$703$	− 8.72866e6i	− 0.666131i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.46069e6i	− 0.486106i
$708$	0	0
$709$	−4.24313e6	−0.317008	−0.158504	−	0.987358i	$-0.550667\pi$
$709$	−4.24313e6	−0.317008	−0.158504	+	0.987358i	$0.550667\pi$
$710$	0	0
$711$	3.32407e6	0.246602
$712$	0	0
$713$	− 2.46370e7i	− 1.81494i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.13192e7i	− 0.822277i
$718$	0	0
$719$	−1.14107e7	−0.823172	−0.411586	−	0.911371i	$-0.635025\pi$
$719$	−1.14107e7	−0.823172	−0.411586	+	0.911371i	$0.635025\pi$
$720$	0	0
$721$	−5.41634e6	−0.388032
$722$	0	0
$723$	990068.i	0.0704400i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.19426e7i	− 0.838038i	−0.907977	−	0.419019i	$-0.862374\pi$
$727$	− 1.19426e7i	− 0.838038i	0.907977	−	0.419019i	$-0.137626\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	4.10384e7	2.84051
$732$	0	0
$733$	− 3.09667e6i	− 0.212880i	−0.994319	−	0.106440i	$-0.966055\pi$
$733$	− 3.09667e6i	− 0.212880i	0.994319	−	0.106440i	$-0.0339452\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.20778e7i	− 1.49723i
$738$	0	0
$739$	1.27952e7	0.861861	0.430930	−	0.902385i	$-0.358185\pi$
$739$	1.27952e7	0.861861	0.430930	+	0.902385i	$0.358185\pi$
$740$	0	0
$741$	2.56979e6	0.171930
$742$	0	0
$743$	− 1.33759e7i	− 0.888897i	−0.895805	−	0.444448i	$-0.853400\pi$
$743$	− 1.33759e7i	− 0.888897i	0.895805	−	0.444448i	$-0.146600\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.19260e6i	0.537181i
$748$	0	0
$749$	8.03167e6	0.523120
$750$	0	0
$751$	1.73785e7	1.12438	0.562188	−	0.827009i	$-0.309960\pi$
$751$	1.73785e7	1.12438	0.562188	+	0.827009i	$0.309960\pi$
$752$	0	0
$753$	− 3.36593e6i	− 0.216330i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.53992e6i	0.541644i	0.962629	+	0.270822i	$0.0872955\pi$
$757$	8.53992e6i	0.541644i	−0.962629	+	0.270822i	$0.912705\pi$
$758$	0	0
$759$	2.78487e7	1.75469
$760$	0	0
$761$	−2.72355e7	−1.70480	−0.852401	−	0.522888i	$-0.824855\pi$
$761$	−2.72355e7	−1.70480	−0.852401	+	0.522888i	$0.824855\pi$
$762$	0	0
$763$	− 1.64508e7i	− 1.02300i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	352638.i	0.0216442i
$768$	0	0
$769$	1.97556e7	1.20469	0.602343	−	0.798237i	$-0.294234\pi$
$769$	1.97556e7	1.20469	0.602343	+	0.798237i	$0.294234\pi$
$770$	0	0
$771$	1.14075e7	0.691124
$772$	0	0
$773$	1.13923e6i	0.0685746i	0.999412	+	0.0342873i	$0.0109161\pi$
$773$	1.13923e6i	0.0685746i	−0.999412	+	0.0342873i	$0.989084\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.78951e6i	− 0.284602i
$778$	0	0
$779$	3.63888e7	2.14844
$780$	0	0
$781$	−2.38405e6	−0.139858
$782$	0	0
$783$	4.30781e6i	0.251103i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.80161e6i	0.333897i	0.985966	+	0.166948i	$0.0533913\pi$
$787$	5.80161e6i	0.333897i	−0.985966	+	0.166948i	$0.946609\pi$
$788$	0	0
$789$	383183.	0.0219136
$790$	0	0
$791$	3.02932e7	1.72149
$792$	0	0
$793$	132479.i	0.00748109i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.44645e7i	− 0.806597i	−0.915068	−	0.403299i	$-0.867864\pi$
$797$	− 1.44645e7i	− 0.806597i	0.915068	−	0.403299i	$-0.132136\pi$
$798$	0	0
$799$	−4.20172e7	−2.32842
$800$	0	0
$801$	−5.13372e6	−0.282716
$802$	0	0
$803$	− 2.86418e7i	− 1.56751i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.80994e7i	0.978317i
$808$	0	0
$809$	−2.19617e7	−1.17976	−0.589880	−	0.807491i	$-0.700825\pi$
$809$	−2.19617e7	−1.17976	−0.589880	+	0.807491i	$0.700825\pi$
$810$	0	0
$811$	−1.33812e7	−0.714402	−0.357201	−	0.934028i	$-0.616269\pi$
$811$	−1.33812e7	−0.714402	−0.357201	+	0.934028i	$0.616269\pi$
$812$	0	0
$813$	1.63001e7i	0.864897i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.31412e7i	− 2.78533i
$818$	0	0
$819$	1.41007e6	0.0734565
$820$	0	0
$821$	9.25548e6	0.479227	0.239613	−	0.970868i	$-0.422979\pi$
$821$	9.25548e6	0.479227	0.239613	+	0.970868i	$0.422979\pi$
$822$	0	0
$823$	− 4.88939e6i	− 0.251626i	−0.992054	−	0.125813i	$-0.959846\pi$
$823$	− 4.88939e6i	− 0.251626i	0.992054	−	0.125813i	$-0.0401539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.68085e6i	0.0854606i	0.999087	+	0.0427303i	$0.0136056\pi$
$827$	1.68085e6i	0.0854606i	−0.999087	+	0.0427303i	$0.986394\pi$
$828$	0	0
$829$	−2.21062e7	−1.11719	−0.558596	−	0.829440i	$-0.688660\pi$
$829$	−2.21062e7	−1.11719	−0.558596	+	0.829440i	$0.688660\pi$
$830$	0	0
$831$	−1.03810e7	−0.521481
$832$	0	0
$833$	8.54838e6i	0.426846i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.76933e6i	− 0.185973i
$838$	0	0
$839$	−9.64088e6	−0.472838	−0.236419	−	0.971651i	$-0.575974\pi$
$839$	−9.64088e6	−0.472838	−0.236419	+	0.971651i	$0.575974\pi$
$840$	0	0
$841$	1.44076e7	0.702427
$842$	0	0
$843$	1.51193e7i	0.732761i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.81492e7i	1.82716i
$848$	0	0
$849$	−6.81168e6	−0.324328
$850$	0	0
$851$	1.73263e7	0.820126
$852$	0	0
$853$	− 4.10226e6i	− 0.193042i	−0.995331	−	0.0965208i	$-0.969229\pi$
$853$	− 4.10226e6i	− 0.193042i	0.995331	−	0.0965208i	$-0.0307714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.30835e6i	0.293402i	0.989181	+	0.146701i	$0.0468656\pi$
$857$	6.30835e6i	0.293402i	−0.989181	+	0.146701i	$0.953134\pi$
$858$	0	0
$859$	8.57588e6	0.396548	0.198274	−	0.980147i	$-0.436466\pi$
$859$	8.57588e6	0.396548	0.198274	+	0.980147i	$0.436466\pi$
$860$	0	0
$861$	1.99669e7	0.917915
$862$	0	0
$863$	− 3.66361e7i	− 1.67449i	−0.546828	−	0.837245i	$-0.684165\pi$
$863$	− 3.66361e7i	− 1.67449i	0.546828	−	0.837245i	$-0.315835\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.81488e7i	− 0.819976i
$868$	0	0
$869$	−2.66500e7	−1.19715
$870$	0	0
$871$	4.04394e6	0.180617
$872$	0	0
$873$	− 5.69363e6i	− 0.252844i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.24157e7i	− 0.984132i	−0.870558	−	0.492066i	$-0.836242\pi$
$877$	− 2.24157e7i	− 0.984132i	0.870558	−	0.492066i	$-0.163758\pi$
$878$	0	0
$879$	−2.00396e7	−0.874816
$880$	0	0
$881$	−8.55161e6	−0.371200	−0.185600	−	0.982625i	$-0.559423\pi$
$881$	−8.55161e6	−0.371200	−0.185600	+	0.982625i	$0.559423\pi$
$882$	0	0
$883$	7.16620e6i	0.309305i	0.987969	+	0.154653i	$0.0494258\pi$
$883$	7.16620e6i	0.309305i	−0.987969	+	0.154653i	$0.950574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.43138e7i	1.03763i	0.854885	+	0.518817i	$0.173628\pi$
$887$	2.43138e7i	1.03763i	−0.854885	+	0.518817i	$0.826372\pi$
$888$	0	0
$889$	−3.34897e6	−0.142121
$890$	0	0
$891$	4.26072e6	0.179800
$892$	0	0
$893$	5.44088e7i	2.28318i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.10099e6i	0.211677i
$898$	0	0
$899$	−3.05539e7	−1.26086
$900$	0	0
$901$	−1.19024e7	−0.488451
$902$	0	0
$903$	− 2.91591e7i	− 1.19002i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.63820e7i	− 1.46848i	−0.678890	−	0.734240i	$-0.737539\pi$
$907$	− 3.63820e7i	− 1.46848i	0.678890	−	0.734240i	$-0.262461\pi$
$908$	0	0
$909$	3.57578e6	0.143536
$910$	0	0
$911$	−5.45897e6	−0.217929	−0.108964	−	0.994046i	$-0.534753\pi$
$911$	−5.45897e6	−0.217929	−0.108964	+	0.994046i	$0.534753\pi$
$912$	0	0
$913$	− 6.56825e7i	− 2.60779i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.19540e6i	0.282573i
$918$	0	0
$919$	1.27369e7	0.497477	0.248739	−	0.968571i	$-0.419984\pi$
$919$	1.27369e7	0.497477	0.248739	+	0.968571i	$0.419984\pi$
$920$	0	0
$921$	−2.43013e6	−0.0944020
$922$	0	0
$923$	− 436680.i	− 0.0168717i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.99776e6i	− 0.114577i
$928$	0	0
$929$	−1.00615e7	−0.382492	−0.191246	−	0.981542i	$-0.561253\pi$
$929$	−1.00615e7	−0.382492	−0.191246	+	0.981542i	$0.561253\pi$
$930$	0	0
$931$	1.10694e7	0.418554
$932$	0	0
$933$	− 1.11647e7i	− 0.419896i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.63556e7i	− 1.35276i	−0.736551	−	0.676382i	$-0.763547\pi$
$937$	− 3.63556e7i	− 1.35276i	0.736551	−	0.676382i	$-0.236453\pi$
$938$	0	0
$939$	−2.85007e6	−0.105485
$940$	0	0
$941$	−3.12726e7	−1.15130	−0.575652	−	0.817695i	$-0.695252\pi$
$941$	−3.12726e7	−1.15130	−0.575652	+	0.817695i	$0.695252\pi$
$942$	0	0
$943$	7.22311e7i	2.64512i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.71083e7i	− 0.982261i	−0.871086	−	0.491131i	$-0.836584\pi$
$947$	− 2.71083e7i	− 0.982261i	0.871086	−	0.491131i	$-0.163416\pi$
$948$	0	0
$949$	5.24624e6	0.189096
$950$	0	0
$951$	−8.84740e6	−0.317223
$952$	0	0
$953$	3.57296e7i	1.27437i	0.770710	+	0.637186i	$0.219902\pi$
$953$	3.57296e7i	1.27437i	−0.770710	+	0.637186i	$0.780098\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.45370e7i	− 1.21900i
$958$	0	0
$959$	2.76987e7	0.972552
$960$	0	0
$961$	−1.89453e6	−0.0661749
$962$	0	0
$963$	4.44526e6i	0.154466i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.32985e7i	0.801239i	0.916245	+	0.400619i	$0.131205\pi$
$967$	2.32985e7i	0.801239i	−0.916245	+	0.400619i	$0.868795\pi$
$968$	0	0
$969$	−4.00486e7	−1.37018
$970$	0	0
$971$	−1.00397e7	−0.341722	−0.170861	−	0.985295i	$-0.554655\pi$
$971$	−1.00397e7	−0.341722	−0.170861	+	0.985295i	$0.554655\pi$
$972$	0	0
$973$	− 3.62590e7i	− 1.22782i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.37438e7i	1.13099i	0.824753	+	0.565493i	$0.191314\pi$
$977$	3.37438e7i	1.13099i	−0.824753	+	0.565493i	$0.808686\pi$
$978$	0	0
$979$	4.11586e7	1.37247
$980$	0	0
$981$	9.10496e6	0.302069
$982$	0	0
$983$	− 5.65485e7i	− 1.86654i	−0.359175	−	0.933270i	$-0.616942\pi$
$983$	− 5.65485e7i	− 1.86654i	0.359175	−	0.933270i	$-0.383058\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.98546e7i	0.975482i
$988$	0	0
$989$	1.05484e8	3.42924
$990$	0	0
$991$	5.32533e7	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$991$	5.32533e7	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$992$	0	0
$993$	− 9.16583e6i	− 0.294984i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.28334e7i	− 0.408886i	−0.978878	−	0.204443i	$-0.934462\pi$
$997$	− 1.28334e7i	− 0.408886i	0.978878	−	0.204443i	$-0.0655383\pi$
$998$	0	0
$999$	2.65083e6	0.0840366

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.f.k.49.4		4
5.2	odd	4		120.6.a.g.1.1	✓	2
5.3	odd	4		600.6.a.k.1.2		2
5.4	even	2	inner	600.6.f.k.49.1		4
15.2	even	4		360.6.a.n.1.1		2
20.7	even	4		240.6.a.o.1.2		2
40.27	even	4		960.6.a.bn.1.2		2
40.37	odd	4		960.6.a.bi.1.1		2
60.47	odd	4		720.6.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.g.1.1	✓	2	5.2	odd	4
240.6.a.o.1.2		2	20.7	even	4
360.6.a.n.1.1		2	15.2	even	4
600.6.a.k.1.2		2	5.3	odd	4
600.6.f.k.49.1		4	5.4	even	2	inner
600.6.f.k.49.4		4	1.1	even	1	trivial
720.6.a.bf.1.2		2	60.47	odd	4
960.6.a.bi.1.1		2	40.37	odd	4
960.6.a.bn.1.2		2	40.27	even	4

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +146.350i q^{7} -81.0000 q^{9} +649.401 q^{11} +118.949i q^{13} -1853.75i q^{17} -2400.45 q^{19} -1317.15 q^{21} -4764.86i q^{23} -729.000i q^{27} -5909.21 q^{29} +5170.55 q^{31} +5844.61i q^{33} +3636.26i q^{37} -1070.54 q^{39} -15159.1 q^{41} +22138.0i q^{43} -22666.1i q^{47} -4611.40 q^{49} +16683.8 q^{51} -6420.69i q^{53} -21604.1i q^{57} +2964.61 q^{59} +1113.75 q^{61} -11854.4i q^{63} -33997.2i q^{67} +42883.7 q^{69} -3671.15 q^{71} -44104.9i q^{73} +95040.0i q^{77} -41037.9 q^{79} +6561.00 q^{81} -101143. i q^{83} -53182.9i q^{87} +63379.3 q^{89} -17408.3 q^{91} +46535.0i q^{93} +70291.7i q^{97} -52601.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{9} + 128 q^{11} - 5280 q^{19} + 288 q^{21} - 3880 q^{29} + 12656 q^{31} - 20952 q^{39} + 4808 q^{41} - 28324 q^{49} + 38952 q^{51} - 10368 q^{59} - 107912 q^{61} - 720 q^{69} - 143104 q^{71}+ \cdots - 10368 q^{99}+O(q^{100}) \) 4 * q - 324 * q^9 + 128 * q^11 - 5280 * q^19 + 288 * q^21 - 3880 * q^29 + 12656 * q^31 - 20952 * q^39 + 4808 * q^41 - 28324 * q^49 + 38952 * q^51 - 10368 * q^59 - 107912 * q^61 - 720 * q^69 - 143104 * q^71 - 58576 * q^79 + 26244 * q^81 + 279448 * q^89 + 304512 * q^91 - 10368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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