LMFDB - Embedded newform 600.4.f.j.49.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,4,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, 4); 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, 4, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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,-36,0,-32,0,0,0,0,0,0,0,420] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 55x^{2} + 729 $ x^4 + 55*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-5.72015i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.4.f.j.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +21.8806i q^{7} -9.00000 q^{9} +54.6418 q^{11} -82.7612i q^{13} -100.403i q^{17} +84.1194 q^{19} -65.6418 q^{21} +0.880613i q^{23} -27.0000i q^{27} +99.1194 q^{29} -78.9255 q^{31} +163.926i q^{33} +390.806i q^{37} +248.284 q^{39} +104.642 q^{41} -241.403i q^{43} +512.567i q^{47} -135.761 q^{49} +301.209 q^{51} -284.642i q^{53} +252.358i q^{57} +709.045 q^{59} +470.522 q^{61} -196.926i q^{63} -667.583i q^{67} -2.64184 q^{69} -51.5378 q^{71} -371.284i q^{73} +1195.60i q^{77} +79.3428 q^{79} +81.0000 q^{81} +682.180i q^{83} +297.358i q^{87} -628.478 q^{89} +1810.87 q^{91} -236.777i q^{93} +1519.09i q^{97} -491.777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} - 32 q^{11} + 420 q^{19} - 12 q^{21} + 480 q^{29} + 436 q^{31} + 492 q^{39} + 168 q^{41} - 376 q^{49} - 48 q^{51} + 2168 q^{59} + 1548 q^{61} + 240 q^{69} + 2216 q^{71} + 2656 q^{79} + 324 q^{81}+ \cdots + 288 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 - 32 * q^11 + 420 * q^19 - 12 * q^21 + 480 * q^29 + 436 * q^31 + 492 * q^39 + 168 * q^41 - 376 * q^49 - 48 * q^51 + 2168 * q^59 + 1548 * q^61 + 240 * q^69 + 2216 * q^71 + 2656 * q^79 + 324 * q^81 - 2848 * q^89 + 3652 * q^91 + 288 * 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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.8806i	1.18144i	0.806876	+	0.590721i	$0.201157\pi$
$7$	21.8806i	1.18144i	−0.806876	+	0.590721i	$0.798843\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	54.6418	1.49774	0.748870	−	0.662717i	$-0.230597\pi$
$11$	54.6418	1.49774	0.748870	+	0.662717i	$0.230597\pi$
$12$	0	0
$13$	− 82.7612i	− 1.76568i	−0.469674	−	0.882840i	$-0.655629\pi$
$13$	− 82.7612i	− 1.76568i	0.469674	−	0.882840i	$-0.344371\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 100.403i	− 1.43243i	−0.697879	−	0.716215i	$-0.745873\pi$
$17$	− 100.403i	− 1.43243i	0.697879	−	0.716215i	$-0.254127\pi$
$18$	0	0
$19$	84.1194	1.01570	0.507850	−	0.861445i	$-0.330440\pi$
$19$	84.1194	1.01570	0.507850	+	0.861445i	$0.330440\pi$
$20$	0	0
$21$	−65.6418	−0.682106
$22$	0	0
$23$	0.880613i	0.00798350i	0.999992	+	0.00399175i	$0.00127062\pi$
$23$	0.880613i	0.00798350i	−0.999992	+	0.00399175i	$0.998729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	99.1194	0.634690	0.317345	−	0.948310i	$-0.397209\pi$
$29$	99.1194	0.634690	0.317345	+	0.948310i	$0.397209\pi$
$30$	0	0
$31$	−78.9255	−0.457272	−0.228636	−	0.973512i	$-0.573427\pi$
$31$	−78.9255	−0.457272	−0.228636	+	0.973512i	$0.573427\pi$
$32$	0	0
$33$	163.926i	0.864720i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	390.806i	1.73644i	0.496183	+	0.868218i	$0.334735\pi$
$37$	390.806i	1.73644i	−0.496183	+	0.868218i	$0.665265\pi$
$38$	0	0
$39$	248.284	1.01942
$40$	0	0
$41$	104.642	0.398593	0.199296	−	0.979939i	$-0.436134\pi$
$41$	104.642	0.398593	0.199296	+	0.979939i	$0.436134\pi$
$42$	0	0
$43$	− 241.403i	− 0.856131i	−0.903748	−	0.428065i	$-0.859195\pi$
$43$	− 241.403i	− 0.856131i	0.903748	−	0.428065i	$-0.140805\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	512.567i	1.59076i	0.606112	+	0.795379i	$0.292728\pi$
$47$	512.567i	1.59076i	−0.606112	+	0.795379i	$0.707272\pi$
$48$	0	0
$49$	−135.761	−0.395805
$50$	0	0
$51$	301.209	0.827014
$52$	0	0
$53$	− 284.642i	− 0.737709i	−0.929487	−	0.368854i	$-0.879750\pi$
$53$	− 284.642i	− 0.737709i	0.929487	−	0.368854i	$-0.120250\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	252.358i	0.586415i
$58$	0	0
$59$	709.045	1.56457	0.782286	−	0.622919i	$-0.214053\pi$
$59$	709.045	1.56457	0.782286	+	0.622919i	$0.214053\pi$
$60$	0	0
$61$	470.522	0.987610	0.493805	−	0.869573i	$-0.335606\pi$
$61$	470.522	0.987610	0.493805	+	0.869573i	$0.335606\pi$
$62$	0	0
$63$	− 196.926i	− 0.393814i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 667.583i	− 1.21729i	−0.793444	−	0.608643i	$-0.791714\pi$
$67$	− 667.583i	− 1.21729i	0.793444	−	0.608643i	$-0.208286\pi$
$68$	0	0
$69$	−2.64184	−0.00460928
$70$	0	0
$71$	−51.5378	−0.0861466	−0.0430733	−	0.999072i	$-0.513715\pi$
$71$	−51.5378	−0.0861466	−0.0430733	+	0.999072i	$0.513715\pi$
$72$	0	0
$73$	− 371.284i	− 0.595280i	−0.954678	−	0.297640i	$-0.903800\pi$
$73$	− 371.284i	− 0.595280i	0.954678	−	0.297640i	$-0.0961996\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1195.60i	1.76949i
$78$	0	0
$79$	79.3428	0.112997	0.0564985	−	0.998403i	$-0.482006\pi$
$79$	79.3428	0.112997	0.0564985	+	0.998403i	$0.482006\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	682.180i	0.902156i	0.892485	+	0.451078i	$0.148960\pi$
$83$	682.180i	0.902156i	−0.892485	+	0.451078i	$0.851040\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	297.358i	0.366438i
$88$	0	0
$89$	−628.478	−0.748522	−0.374261	−	0.927323i	$-0.622104\pi$
$89$	−628.478	−0.748522	−0.374261	+	0.927323i	$0.622104\pi$
$90$	0	0
$91$	1810.87	2.08605
$92$	0	0
$93$	− 236.777i	− 0.264006i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1519.09i	1.59011i	0.606541	+	0.795053i	$0.292557\pi$
$97$	1519.09i	1.59011i	−0.606541	+	0.795053i	$0.707443\pi$
$98$	0	0
$99$	−491.777	−0.499246
$100$	0	0
$101$	977.777	0.963291	0.481646	−	0.876366i	$-0.340039\pi$
$101$	977.777	0.963291	0.481646	+	0.876366i	$0.340039\pi$
$102$	0	0
$103$	− 759.194i	− 0.726268i	−0.931737	−	0.363134i	$-0.881707\pi$
$103$	− 759.194i	− 0.726268i	0.931737	−	0.363134i	$-0.118293\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	945.448i	0.854205i	0.904203	+	0.427102i	$0.140466\pi$
$107$	945.448i	0.854205i	−0.904203	+	0.427102i	$0.859534\pi$
$108$	0	0
$109$	688.314	0.604849	0.302425	−	0.953173i	$-0.402204\pi$
$109$	688.314	0.604849	0.302425	+	0.953173i	$0.402204\pi$
$110$	0	0
$111$	−1172.42	−1.00253
$112$	0	0
$113$	350.865i	0.292094i	0.989278	+	0.146047i	$0.0466551\pi$
$113$	350.865i	0.292094i	−0.989278	+	0.146047i	$0.953345\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	744.851i	0.588560i
$118$	0	0
$119$	2196.88	1.69233
$120$	0	0
$121$	1654.73	1.24322
$122$	0	0
$123$	313.926i	0.230128i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1588.21i	1.10969i	0.831953	+	0.554846i	$0.187223\pi$
$127$	1588.21i	1.10969i	−0.831953	+	0.554846i	$0.812777\pi$
$128$	0	0
$129$	724.209	0.494287
$130$	0	0
$131$	−2156.02	−1.43795	−0.718977	−	0.695034i	$-0.755389\pi$
$131$	−2156.02	−1.43795	−0.718977	+	0.695034i	$0.755389\pi$
$132$	0	0
$133$	1840.58i	1.19999i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	316.716i	0.197510i	0.995112	+	0.0987551i	$0.0314860\pi$
$137$	316.716i	0.197510i	−0.995112	+	0.0987551i	$0.968514\pi$
$138$	0	0
$139$	2624.87	1.60171	0.800857	−	0.598855i	$-0.204377\pi$
$139$	2624.87	1.60171	0.800857	+	0.598855i	$0.204377\pi$
$140$	0	0
$141$	−1537.70	−0.918425
$142$	0	0
$143$	− 4522.23i	− 2.64453i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 407.284i	− 0.228518i
$148$	0	0
$149$	−2782.87	−1.53008	−0.765038	−	0.643985i	$-0.777280\pi$
$149$	−2782.87	−1.53008	−0.765038	+	0.643985i	$0.777280\pi$
$150$	0	0
$151$	1227.25	0.661407	0.330704	−	0.943735i	$-0.392714\pi$
$151$	1227.25	0.661407	0.330704	+	0.943735i	$0.392714\pi$
$152$	0	0
$153$	903.628i	0.477477i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3187.33i	− 1.62023i	−0.586268	−	0.810117i	$-0.699404\pi$
$157$	− 3187.33i	− 1.62023i	0.586268	−	0.810117i	$-0.300596\pi$
$158$	0	0
$159$	853.926	0.425916
$160$	0	0
$161$	−19.2684	−0.00943204
$162$	0	0
$163$	− 2481.64i	− 1.19250i	−0.802800	−	0.596249i	$-0.796657\pi$
$163$	− 2481.64i	− 1.19250i	0.802800	−	0.596249i	$-0.203343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1904.55i	− 0.882508i	−0.897382	−	0.441254i	$-0.854534\pi$
$167$	− 1904.55i	− 0.882508i	0.897382	−	0.441254i	$-0.145466\pi$
$168$	0	0
$169$	−4652.42	−2.11762
$170$	0	0
$171$	−757.074	−0.338567
$172$	0	0
$173$	3788.88i	1.66511i	0.553946	+	0.832553i	$0.313121\pi$
$173$	3788.88i	1.66511i	−0.553946	+	0.832553i	$0.686879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2127.13i	0.903306i
$178$	0	0
$179$	1390.63	0.580672	0.290336	−	0.956925i	$-0.406233\pi$
$179$	1390.63	0.580672	0.290336	+	0.956925i	$0.406233\pi$
$180$	0	0
$181$	2512.55	1.03180	0.515902	−	0.856647i	$-0.327457\pi$
$181$	2512.55	1.03180	0.515902	+	0.856647i	$0.327457\pi$
$182$	0	0
$183$	1411.57i	0.570197i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5486.21i	− 2.14541i
$188$	0	0
$189$	590.777	0.227369
$190$	0	0
$191$	2872.82	1.08833	0.544163	−	0.838980i	$-0.316847\pi$
$191$	2872.82	1.08833	0.544163	+	0.838980i	$0.316847\pi$
$192$	0	0
$193$	− 4130.79i	− 1.54063i	−0.637665	−	0.770314i	$-0.720100\pi$
$193$	− 4130.79i	− 1.54063i	0.637665	−	0.770314i	$-0.279900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2243.52i	0.811393i	0.914008	+	0.405697i	$0.132971\pi$
$197$	2243.52i	0.811393i	−0.914008	+	0.405697i	$0.867029\pi$
$198$	0	0
$199$	−1111.28	−0.395864	−0.197932	−	0.980216i	$-0.563422\pi$
$199$	−1111.28	−0.395864	−0.197932	+	0.980216i	$0.563422\pi$
$200$	0	0
$201$	2002.75	0.702801
$202$	0	0
$203$	2168.79i	0.749849i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.92552i	− 0.00266117i
$208$	0	0
$209$	4596.44	1.52125
$210$	0	0
$211$	−4546.63	−1.48343	−0.741713	−	0.670717i	$-0.765986\pi$
$211$	−4546.63	−1.48343	−0.741713	+	0.670717i	$0.765986\pi$
$212$	0	0
$213$	− 154.613i	− 0.0497368i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1726.94i	− 0.540241i
$218$	0	0
$219$	1113.85	0.343685
$220$	0	0
$221$	−8309.48	−2.52921
$222$	0	0
$223$	− 3012.93i	− 0.904755i	−0.891826	−	0.452378i	$-0.850576\pi$
$223$	− 3012.93i	− 0.904755i	0.891826	−	0.452378i	$-0.149424\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1515.27i	0.443049i	0.975155	+	0.221524i	$0.0711032\pi$
$227$	1515.27i	0.443049i	−0.975155	+	0.221524i	$0.928897\pi$
$228$	0	0
$229$	−2539.98	−0.732956	−0.366478	−	0.930427i	$-0.619436\pi$
$229$	−2539.98	−0.732956	−0.366478	+	0.930427i	$0.619436\pi$
$230$	0	0
$231$	−3586.79	−1.02162
$232$	0	0
$233$	4772.91i	1.34199i	0.741461	+	0.670996i	$0.234133\pi$
$233$	4772.91i	1.34199i	−0.741461	+	0.670996i	$0.765867\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	238.029i	0.0652389i
$238$	0	0
$239$	−244.344	−0.0661309	−0.0330655	−	0.999453i	$-0.510527\pi$
$239$	−244.344	−0.0661309	−0.0330655	+	0.999453i	$0.510527\pi$
$240$	0	0
$241$	5573.18	1.48963	0.744813	−	0.667273i	$-0.232538\pi$
$241$	5573.18	1.48963	0.744813	+	0.667273i	$0.232538\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6961.82i	− 1.79340i
$248$	0	0
$249$	−2046.54	−0.520860
$250$	0	0
$251$	−5786.78	−1.45521	−0.727606	−	0.685995i	$-0.759367\pi$
$251$	−5786.78	−1.45521	−0.727606	+	0.685995i	$0.759367\pi$
$252$	0	0
$253$	48.1183i	0.0119572i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7780.75i	− 1.88852i	−0.329200	−	0.944260i	$-0.606779\pi$
$257$	− 7780.75i	− 1.88852i	0.329200	−	0.944260i	$-0.393221\pi$
$258$	0	0
$259$	−8551.08	−2.05150
$260$	0	0
$261$	−892.074	−0.211563
$262$	0	0
$263$	5122.61i	1.20104i	0.799609	+	0.600521i	$0.205040\pi$
$263$	5122.61i	1.20104i	−0.799609	+	0.600521i	$0.794960\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1885.43i	− 0.432160i
$268$	0	0
$269$	−7125.12	−1.61497	−0.807484	−	0.589890i	$-0.799171\pi$
$269$	−7125.12	−1.61497	−0.807484	+	0.589890i	$0.799171\pi$
$270$	0	0
$271$	1761.58	0.394865	0.197433	−	0.980316i	$-0.436740\pi$
$271$	1761.58	0.394865	0.197433	+	0.980316i	$0.436740\pi$
$272$	0	0
$273$	5432.60i	1.20438i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5810.41i	1.26034i	0.776458	+	0.630169i	$0.217014\pi$
$277$	5810.41i	1.26034i	−0.776458	+	0.630169i	$0.782986\pi$
$278$	0	0
$279$	710.330	0.152424
$280$	0	0
$281$	−4126.19	−0.875972	−0.437986	−	0.898982i	$-0.644308\pi$
$281$	−4126.19	−0.875972	−0.437986	+	0.898982i	$0.644308\pi$
$282$	0	0
$283$	− 718.330i	− 0.150884i	−0.997150	−	0.0754422i	$-0.975963\pi$
$283$	− 718.330i	− 0.150884i	0.997150	−	0.0754422i	$-0.0240368\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2289.63i	0.470914i
$288$	0	0
$289$	−5167.78	−1.05186
$290$	0	0
$291$	−4557.27	−0.918048
$292$	0	0
$293$	3756.21i	0.748942i	0.927239	+	0.374471i	$0.122176\pi$
$293$	3756.21i	0.748942i	−0.927239	+	0.374471i	$0.877824\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1475.33i	− 0.288240i
$298$	0	0
$299$	72.8806	0.0140963
$300$	0	0
$301$	5282.05	1.01147
$302$	0	0
$303$	2933.33i	0.556156i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7599.52i	− 1.41279i	−0.707816	−	0.706397i	$-0.750319\pi$
$307$	− 7599.52i	− 1.41279i	0.707816	−	0.706397i	$-0.249681\pi$
$308$	0	0
$309$	2277.58	0.419311
$310$	0	0
$311$	2850.03	0.519648	0.259824	−	0.965656i	$-0.416336\pi$
$311$	2850.03	0.519648	0.259824	+	0.965656i	$0.416336\pi$
$312$	0	0
$313$	− 799.684i	− 0.144411i	−0.997390	−	0.0722057i	$-0.976996\pi$
$313$	− 799.684i	− 0.144411i	0.997390	−	0.0722057i	$-0.0230038\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1003.79i	− 0.177850i	−0.996038	−	0.0889250i	$-0.971657\pi$
$317$	− 1003.79i	− 0.177850i	0.996038	−	0.0889250i	$-0.0283431\pi$
$318$	0	0
$319$	5416.07	0.950600
$320$	0	0
$321$	−2836.34	−0.493175
$322$	0	0
$323$	− 8445.84i	− 1.45492i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2064.94i	0.349210i
$328$	0	0
$329$	−11215.3	−1.87939
$330$	0	0
$331$	8367.68	1.38951	0.694757	−	0.719245i	$-0.255512\pi$
$331$	8367.68	1.38951	0.694757	+	0.719245i	$0.255512\pi$
$332$	0	0
$333$	− 3517.26i	− 0.578812i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5034.79i	− 0.813835i	−0.913465	−	0.406918i	$-0.866604\pi$
$337$	− 5034.79i	− 0.813835i	0.913465	−	0.406918i	$-0.133396\pi$
$338$	0	0
$339$	−1052.60	−0.168641
$340$	0	0
$341$	−4312.64	−0.684875
$342$	0	0
$343$	4534.51i	0.713821i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2105.53i	− 0.325737i	−0.986648	−	0.162868i	$-0.947925\pi$
$347$	− 2105.53i	− 0.325737i	0.986648	−	0.162868i	$-0.0520746\pi$
$348$	0	0
$349$	8813.47	1.35179	0.675894	−	0.736999i	$-0.263758\pi$
$349$	8813.47	1.35179	0.675894	+	0.736999i	$0.263758\pi$
$350$	0	0
$351$	−2234.55	−0.339805
$352$	0	0
$353$	− 9348.98i	− 1.40962i	−0.709396	−	0.704810i	$-0.751032\pi$
$353$	− 9348.98i	− 1.40962i	0.709396	−	0.704810i	$-0.248968\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6590.64i	0.977069i
$358$	0	0
$359$	9233.33	1.35743	0.678714	−	0.734403i	$-0.262538\pi$
$359$	9233.33	1.35743	0.678714	+	0.734403i	$0.262538\pi$
$360$	0	0
$361$	217.071	0.0316477
$362$	0	0
$363$	4964.19i	0.717775i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6996.60i	− 0.995148i	−0.867421	−	0.497574i	$-0.834224\pi$
$367$	− 6996.60i	− 0.995148i	0.867421	−	0.497574i	$-0.165776\pi$
$368$	0	0
$369$	−941.777	−0.132864
$370$	0	0
$371$	6228.14	0.871560
$372$	0	0
$373$	− 2945.48i	− 0.408877i	−0.978879	−	0.204438i	$-0.934463\pi$
$373$	− 2945.48i	− 0.408877i	0.978879	−	0.204438i	$-0.0655368\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8203.24i	− 1.12066i
$378$	0	0
$379$	−8499.83	−1.15200	−0.575998	−	0.817451i	$-0.695387\pi$
$379$	−8499.83	−1.15200	−0.575998	+	0.817451i	$0.695387\pi$
$380$	0	0
$381$	−4764.63	−0.640681
$382$	0	0
$383$	6426.01i	0.857320i	0.903466	+	0.428660i	$0.141014\pi$
$383$	6426.01i	0.857320i	−0.903466	+	0.428660i	$0.858986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2172.63i	0.285377i
$388$	0	0
$389$	675.805	0.0880840	0.0440420	−	0.999030i	$-0.485976\pi$
$389$	675.805	0.0880840	0.0440420	+	0.999030i	$0.485976\pi$
$390$	0	0
$391$	88.4162	0.0114358
$392$	0	0
$393$	− 6468.05i	− 0.830203i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 9331.43i	− 1.17967i	−0.807522	−	0.589837i	$-0.799192\pi$
$397$	− 9331.43i	− 1.17967i	0.807522	−	0.589837i	$-0.200808\pi$
$398$	0	0
$399$	−5521.75	−0.692815
$400$	0	0
$401$	−427.213	−0.0532021	−0.0266010	−	0.999646i	$-0.508468\pi$
$401$	−427.213	−0.0532021	−0.0266010	+	0.999646i	$0.508468\pi$
$402$	0	0
$403$	6531.97i	0.807396i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21354.4i	2.60073i
$408$	0	0
$409$	−1243.18	−0.150297	−0.0751484	−	0.997172i	$-0.523943\pi$
$409$	−1243.18	−0.150297	−0.0751484	+	0.997172i	$0.523943\pi$
$410$	0	0
$411$	−950.149	−0.114033
$412$	0	0
$413$	15514.3i	1.84845i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7874.60i	0.924750i
$418$	0	0
$419$	3802.17	0.443313	0.221656	−	0.975125i	$-0.428854\pi$
$419$	3802.17	0.443313	0.221656	+	0.975125i	$0.428854\pi$
$420$	0	0
$421$	−12785.6	−1.48012	−0.740059	−	0.672542i	$-0.765203\pi$
$421$	−12785.6	−1.48012	−0.740059	+	0.672542i	$0.765203\pi$
$422$	0	0
$423$	− 4613.11i	− 0.530253i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10295.3i	1.16680i
$428$	0	0
$429$	13566.7	1.52682
$430$	0	0
$431$	−9588.11	−1.07156	−0.535781	−	0.844357i	$-0.679983\pi$
$431$	−9588.11	−1.07156	−0.535781	+	0.844357i	$0.679983\pi$
$432$	0	0
$433$	− 8493.83i	− 0.942697i	−0.881947	−	0.471348i	$-0.843767\pi$
$433$	− 8493.83i	− 0.942697i	0.881947	−	0.471348i	$-0.156233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	74.0766i	0.00810885i
$438$	0	0
$439$	−5167.40	−0.561792	−0.280896	−	0.959738i	$-0.590632\pi$
$439$	−5167.40	−0.561792	−0.280896	+	0.959738i	$0.590632\pi$
$440$	0	0
$441$	1221.85	0.131935
$442$	0	0
$443$	− 8192.86i	− 0.878679i	−0.898321	−	0.439339i	$-0.855213\pi$
$443$	− 8192.86i	− 0.878679i	0.898321	−	0.439339i	$-0.144787\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8348.60i	− 0.883390i
$448$	0	0
$449$	18252.3	1.91844	0.959218	−	0.282666i	$-0.0912188\pi$
$449$	18252.3	1.91844	0.959218	+	0.282666i	$0.0912188\pi$
$450$	0	0
$451$	5717.82	0.596988
$452$	0	0
$453$	3681.76i	0.381864i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 330.123i	− 0.0337910i	−0.999857	−	0.0168955i	$-0.994622\pi$
$457$	− 330.123i	− 0.0337910i	0.999857	−	0.0168955i	$-0.00537826\pi$
$458$	0	0
$459$	−2710.88	−0.275671
$460$	0	0
$461$	−4773.18	−0.482232	−0.241116	−	0.970496i	$-0.577513\pi$
$461$	−4773.18	−0.482232	−0.241116	+	0.970496i	$0.577513\pi$
$462$	0	0
$463$	− 7860.46i	− 0.788999i	−0.918896	−	0.394499i	$-0.870918\pi$
$463$	− 7860.46i	− 0.788999i	0.918896	−	0.394499i	$-0.129082\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10520.8i	1.04250i	0.853404	+	0.521249i	$0.174534\pi$
$467$	10520.8i	1.04250i	−0.853404	+	0.521249i	$0.825466\pi$
$468$	0	0
$469$	14607.1	1.43815
$470$	0	0
$471$	9561.99	0.935442
$472$	0	0
$473$	− 13190.7i	− 1.28226i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2561.78i	0.245903i
$478$	0	0
$479$	3013.64	0.287467	0.143734	−	0.989616i	$-0.454089\pi$
$479$	3013.64	0.287467	0.143734	+	0.989616i	$0.454089\pi$
$480$	0	0
$481$	32343.6	3.06599
$482$	0	0
$483$	− 57.8051i	− 0.00544559i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7048.81i	0.655876i	0.944699	+	0.327938i	$0.106354\pi$
$487$	7048.81i	0.655876i	−0.944699	+	0.327938i	$0.893646\pi$
$488$	0	0
$489$	7444.93	0.688489
$490$	0	0
$491$	6338.68	0.582608	0.291304	−	0.956630i	$-0.405911\pi$
$491$	6338.68	0.582608	0.291304	+	0.956630i	$0.405911\pi$
$492$	0	0
$493$	− 9951.89i	− 0.909149i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1127.68i	− 0.101777i
$498$	0	0
$499$	5402.66	0.484682	0.242341	−	0.970191i	$-0.422085\pi$
$499$	5402.66	0.484682	0.242341	+	0.970191i	$0.422085\pi$
$500$	0	0
$501$	5713.66	0.509516
$502$	0	0
$503$	7770.85i	0.688837i	0.938816	+	0.344419i	$0.111924\pi$
$503$	7770.85i	0.688837i	−0.938816	+	0.344419i	$0.888076\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 13957.3i	− 1.22261i
$508$	0	0
$509$	−729.019	−0.0634837	−0.0317418	−	0.999496i	$-0.510105\pi$
$509$	−729.019	−0.0634837	−0.0317418	+	0.999496i	$0.510105\pi$
$510$	0	0
$511$	8123.91	0.703289
$512$	0	0
$513$	− 2271.22i	− 0.195472i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28007.6i	2.38254i
$518$	0	0
$519$	−11366.6	−0.961349
$520$	0	0
$521$	14783.8	1.24317	0.621585	−	0.783346i	$-0.286489\pi$
$521$	14783.8	1.24317	0.621585	+	0.783346i	$0.286489\pi$
$522$	0	0
$523$	− 395.434i	− 0.0330614i	−0.999863	−	0.0165307i	$-0.994738\pi$
$523$	− 395.434i	− 0.0330614i	0.999863	−	0.0165307i	$-0.00526212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7924.36i	0.655011i
$528$	0	0
$529$	12166.2	0.999936
$530$	0	0
$531$	−6381.40	−0.521524
$532$	0	0
$533$	− 8660.29i	− 0.703787i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4171.88i	0.335251i
$538$	0	0
$539$	−7418.24	−0.592813
$540$	0	0
$541$	−16507.9	−1.31188	−0.655942	−	0.754811i	$-0.727728\pi$
$541$	−16507.9	−1.31188	−0.655942	+	0.754811i	$0.727728\pi$
$542$	0	0
$543$	7537.66i	0.595713i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9242.35i	0.722440i	0.932481	+	0.361220i	$0.117640\pi$
$547$	9242.35i	0.722440i	−0.932481	+	0.361220i	$0.882360\pi$
$548$	0	0
$549$	−4234.70	−0.329203
$550$	0	0
$551$	8337.86	0.644655
$552$	0	0
$553$	1736.07i	0.133499i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10449.5i	− 0.794898i	−0.917624	−	0.397449i	$-0.869896\pi$
$557$	− 10449.5i	− 0.794898i	0.917624	−	0.397449i	$-0.130104\pi$
$558$	0	0
$559$	−19978.8	−1.51165
$560$	0	0
$561$	16458.6	1.23865
$562$	0	0
$563$	13670.1i	1.02331i	0.859191	+	0.511656i	$0.170968\pi$
$563$	13670.1i	1.02331i	−0.859191	+	0.511656i	$0.829032\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1772.33i	0.131271i
$568$	0	0
$569$	−13616.7	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$569$	−13616.7	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$570$	0	0
$571$	−11833.4	−0.867270	−0.433635	−	0.901089i	$-0.642769\pi$
$571$	−11833.4	−0.867270	−0.433635	+	0.901089i	$0.642769\pi$
$572$	0	0
$573$	8618.47i	0.628345i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7131.30i	0.514523i	0.966342	+	0.257262i	$0.0828202\pi$
$577$	7131.30i	0.514523i	−0.966342	+	0.257262i	$0.917180\pi$
$578$	0	0
$579$	12392.4	0.889482
$580$	0	0
$581$	−14926.5	−1.06584
$582$	0	0
$583$	− 15553.4i	− 1.10490i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25214.3i	1.77292i	0.462804	+	0.886461i	$0.346843\pi$
$587$	25214.3i	1.77292i	−0.462804	+	0.886461i	$0.653157\pi$
$588$	0	0
$589$	−6639.17	−0.464452
$590$	0	0
$591$	−6730.57	−0.468458
$592$	0	0
$593$	− 4218.04i	− 0.292098i	−0.989277	−	0.146049i	$-0.953344\pi$
$593$	− 4218.04i	− 0.292098i	0.989277	−	0.146049i	$-0.0466557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3333.85i	− 0.228552i
$598$	0	0
$599$	−5956.57	−0.406308	−0.203154	−	0.979147i	$-0.565119\pi$
$599$	−5956.57	−0.406308	−0.203154	+	0.979147i	$0.565119\pi$
$600$	0	0
$601$	−5182.29	−0.351731	−0.175865	−	0.984414i	$-0.556272\pi$
$601$	−5182.29	−0.351731	−0.175865	+	0.984414i	$0.556272\pi$
$602$	0	0
$603$	6008.24i	0.405762i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10142.0i	0.678171i	0.940755	+	0.339086i	$0.110118\pi$
$607$	10142.0i	0.678171i	−0.940755	+	0.339086i	$0.889882\pi$
$608$	0	0
$609$	−6506.38	−0.432926
$610$	0	0
$611$	42420.7	2.80877
$612$	0	0
$613$	4022.95i	0.265066i	0.991179	+	0.132533i	$0.0423110\pi$
$613$	4022.95i	0.265066i	−0.991179	+	0.132533i	$0.957689\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27074.4i	− 1.76657i	−0.468837	−	0.883285i	$-0.655327\pi$
$617$	− 27074.4i	− 1.76657i	0.468837	−	0.883285i	$-0.344673\pi$
$618$	0	0
$619$	17245.7	1.11981	0.559905	−	0.828557i	$-0.310838\pi$
$619$	17245.7	1.11981	0.559905	+	0.828557i	$0.310838\pi$
$620$	0	0
$621$	23.7766	0.00153643
$622$	0	0
$623$	− 13751.5i	− 0.884336i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	13789.3i	0.878297i
$628$	0	0
$629$	39238.1	2.48732
$630$	0	0
$631$	−14254.4	−0.899302	−0.449651	−	0.893204i	$-0.648452\pi$
$631$	−14254.4	−0.899302	−0.449651	+	0.893204i	$0.648452\pi$
$632$	0	0
$633$	− 13639.9i	− 0.856456i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11235.8i	0.698865i
$638$	0	0
$639$	463.840	0.0287155
$640$	0	0
$641$	2611.61	0.160924	0.0804621	−	0.996758i	$-0.474360\pi$
$641$	2611.61	0.160924	0.0804621	+	0.996758i	$0.474360\pi$
$642$	0	0
$643$	27414.4i	1.68136i	0.541529	+	0.840682i	$0.317846\pi$
$643$	27414.4i	1.68136i	−0.541529	+	0.840682i	$0.682154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12820.8i	0.779041i	0.921018	+	0.389521i	$0.127359\pi$
$647$	12820.8i	0.779041i	−0.921018	+	0.389521i	$0.872641\pi$
$648$	0	0
$649$	38743.5	2.34332
$650$	0	0
$651$	5180.82	0.311908
$652$	0	0
$653$	15786.4i	0.946049i	0.881049	+	0.473025i	$0.156838\pi$
$653$	15786.4i	0.946049i	−0.881049	+	0.473025i	$0.843162\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3341.55i	0.198427i
$658$	0	0
$659$	−4822.61	−0.285072	−0.142536	−	0.989790i	$-0.545526\pi$
$659$	−4822.61	−0.285072	−0.142536	+	0.989790i	$0.545526\pi$
$660$	0	0
$661$	−23624.9	−1.39017	−0.695084	−	0.718929i	$-0.744633\pi$
$661$	−23624.9	−1.39017	−0.695084	+	0.718929i	$0.744633\pi$
$662$	0	0
$663$	− 24928.4i	− 1.46024i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	87.2858i	0.00506705i
$668$	0	0
$669$	9038.78	0.522361
$670$	0	0
$671$	25710.2	1.47918
$672$	0	0
$673$	8903.65i	0.509971i	0.966945	+	0.254985i	$0.0820707\pi$
$673$	8903.65i	0.509971i	−0.966945	+	0.254985i	$0.917929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9034.56i	− 0.512890i	−0.966559	−	0.256445i	$-0.917449\pi$
$677$	− 9034.56i	− 0.512890i	0.966559	−	0.256445i	$-0.0825512\pi$
$678$	0	0
$679$	−33238.6	−1.87862
$680$	0	0
$681$	−4545.81	−0.255794
$682$	0	0
$683$	− 28676.8i	− 1.60657i	−0.595596	−	0.803284i	$-0.703084\pi$
$683$	− 28676.8i	− 1.60657i	0.595596	−	0.803284i	$-0.296916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7619.95i	− 0.423172i
$688$	0	0
$689$	−23557.3	−1.30256
$690$	0	0
$691$	−20653.1	−1.13702	−0.568509	−	0.822677i	$-0.692480\pi$
$691$	−20653.1	−1.13702	−0.568509	+	0.822677i	$0.692480\pi$
$692$	0	0
$693$	− 10760.4i	− 0.589831i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10506.4i	− 0.570957i
$698$	0	0
$699$	−14318.7	−0.774799
$700$	0	0
$701$	−11000.1	−0.592678	−0.296339	−	0.955083i	$-0.595766\pi$
$701$	−11000.1	−0.592678	−0.296339	+	0.955083i	$0.595766\pi$
$702$	0	0
$703$	32874.4i	1.76370i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21394.4i	1.13807i
$708$	0	0
$709$	372.560	0.0197345	0.00986725	−	0.999951i	$-0.496859\pi$
$709$	372.560	0.0197345	0.00986725	+	0.999951i	$0.496859\pi$
$710$	0	0
$711$	−714.086	−0.0376657
$712$	0	0
$713$	− 69.5028i	− 0.00365063i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 733.032i	− 0.0381807i
$718$	0	0
$719$	7207.32	0.373835	0.186918	−	0.982376i	$-0.440150\pi$
$719$	7207.32	0.373835	0.186918	+	0.982376i	$0.440150\pi$
$720$	0	0
$721$	16611.6	0.858043
$722$	0	0
$723$	16719.5i	0.860036i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23423.6i	− 1.19496i	−0.801885	−	0.597479i	$-0.796169\pi$
$727$	− 23423.6i	− 1.19496i	0.801885	−	0.597479i	$-0.203831\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−24237.6	−1.22635
$732$	0	0
$733$	20368.0i	1.02634i	0.858286	+	0.513172i	$0.171530\pi$
$733$	20368.0i	1.02634i	−0.858286	+	0.513172i	$0.828470\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36477.9i	− 1.82318i
$738$	0	0
$739$	−2603.64	−0.129603	−0.0648014	−	0.997898i	$-0.520641\pi$
$739$	−2603.64	−0.129603	−0.0648014	+	0.997898i	$0.520641\pi$
$740$	0	0
$741$	20885.5	1.03542
$742$	0	0
$743$	− 8627.83i	− 0.426009i	−0.977051	−	0.213004i	$-0.931675\pi$
$743$	− 8627.83i	− 0.426009i	0.977051	−	0.213004i	$-0.0683249\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6139.62i	− 0.300719i
$748$	0	0
$749$	−20687.0	−1.00919
$750$	0	0
$751$	−18735.5	−0.910343	−0.455172	−	0.890404i	$-0.650422\pi$
$751$	−18735.5	−0.910343	−0.455172	+	0.890404i	$0.650422\pi$
$752$	0	0
$753$	− 17360.3i	− 0.840167i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10787.3i	0.517926i	0.965887	+	0.258963i	$0.0833807\pi$
$757$	10787.3i	0.517926i	−0.965887	+	0.258963i	$0.916619\pi$
$758$	0	0
$759$	−144.355	−0.00690349
$760$	0	0
$761$	−2175.95	−0.103651	−0.0518253	−	0.998656i	$-0.516504\pi$
$761$	−2175.95	−0.103651	−0.0518253	+	0.998656i	$0.516504\pi$
$762$	0	0
$763$	15060.7i	0.714594i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 58681.4i	− 2.76253i
$768$	0	0
$769$	−18705.8	−0.877176	−0.438588	−	0.898688i	$-0.644521\pi$
$769$	−18705.8	−0.877176	−0.438588	+	0.898688i	$0.644521\pi$
$770$	0	0
$771$	23342.2	1.09034
$772$	0	0
$773$	− 8243.60i	− 0.383573i	−0.981437	−	0.191786i	$-0.938572\pi$
$773$	− 8243.60i	− 0.383573i	0.981437	−	0.191786i	$-0.0614281\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 25653.2i	− 1.18443i
$778$	0	0
$779$	8802.41	0.404851
$780$	0	0
$781$	−2816.12	−0.129025
$782$	0	0
$783$	− 2676.22i	− 0.122146i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29442.1i	− 1.33354i	−0.745263	−	0.666770i	$-0.767676\pi$
$787$	− 29442.1i	− 1.33354i	0.745263	−	0.666770i	$-0.232324\pi$
$788$	0	0
$789$	−15367.8	−0.693422
$790$	0	0
$791$	−7677.15	−0.345092
$792$	0	0
$793$	− 38941.0i	− 1.74380i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42045.4i	1.86866i	0.356403	+	0.934332i	$0.384003\pi$
$797$	42045.4i	1.86866i	−0.356403	+	0.934332i	$0.615997\pi$
$798$	0	0
$799$	51463.3	2.27865
$800$	0	0
$801$	5656.30	0.249507
$802$	0	0
$803$	− 20287.6i	− 0.891575i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21375.4i	− 0.932402i
$808$	0	0
$809$	−43757.6	−1.90165	−0.950825	−	0.309727i	$-0.899762\pi$
$809$	−43757.6	−1.90165	−0.950825	+	0.309727i	$0.899762\pi$
$810$	0	0
$811$	3273.00	0.141715	0.0708574	−	0.997486i	$-0.477426\pi$
$811$	3273.00	0.141715	0.0708574	+	0.997486i	$0.477426\pi$
$812$	0	0
$813$	5284.74i	0.227976i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20306.7i	− 0.869572i
$818$	0	0
$819$	−16297.8	−0.695349
$820$	0	0
$821$	−40442.6	−1.71919	−0.859595	−	0.510976i	$-0.829284\pi$
$821$	−40442.6	−1.71919	−0.859595	+	0.510976i	$0.829284\pi$
$822$	0	0
$823$	27236.8i	1.15360i	0.816884	+	0.576802i	$0.195700\pi$
$823$	27236.8i	1.15360i	−0.816884	+	0.576802i	$0.804300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33344.5i	− 1.40206i	−0.713133	−	0.701028i	$-0.752725\pi$
$827$	− 33344.5i	− 1.40206i	0.713133	−	0.701028i	$-0.247275\pi$
$828$	0	0
$829$	32085.0	1.34422	0.672110	−	0.740452i	$-0.265388\pi$
$829$	32085.0	1.34422	0.672110	+	0.740452i	$0.265388\pi$
$830$	0	0
$831$	−17431.2	−0.727656
$832$	0	0
$833$	13630.8i	0.566964i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2130.99i	0.0880021i
$838$	0	0
$839$	7072.24	0.291014	0.145507	−	0.989357i	$-0.453519\pi$
$839$	7072.24	0.291014	0.145507	+	0.989357i	$0.453519\pi$
$840$	0	0
$841$	−14564.3	−0.597169
$842$	0	0
$843$	− 12378.6i	− 0.505743i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36206.5i	1.46880i
$848$	0	0
$849$	2154.99	0.0871131
$850$	0	0
$851$	−344.149	−0.0138628
$852$	0	0
$853$	10225.0i	0.410431i	0.978717	+	0.205215i	$0.0657895\pi$
$853$	10225.0i	0.410431i	−0.978717	+	0.205215i	$0.934211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9783.87i	0.389977i	0.980805	+	0.194989i	$0.0624670\pi$
$857$	9783.87i	0.389977i	−0.980805	+	0.194989i	$0.937533\pi$
$858$	0	0
$859$	22931.8	0.910853	0.455427	−	0.890273i	$-0.349487\pi$
$859$	22931.8	0.910853	0.455427	+	0.890273i	$0.349487\pi$
$860$	0	0
$861$	−6868.88	−0.271883
$862$	0	0
$863$	23506.0i	0.927175i	0.886051	+	0.463588i	$0.153438\pi$
$863$	23506.0i	0.927175i	−0.886051	+	0.463588i	$0.846562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 15503.3i	− 0.607290i
$868$	0	0
$869$	4335.44	0.169240
$870$	0	0
$871$	−55250.0	−2.14934
$872$	0	0
$873$	− 13671.8i	− 0.530035i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34672.0i	1.33499i	0.744613	+	0.667497i	$0.232634\pi$
$877$	34672.0i	1.33499i	−0.744613	+	0.667497i	$0.767366\pi$
$878$	0	0
$879$	−11268.6	−0.432402
$880$	0	0
$881$	−2920.92	−0.111701	−0.0558504	−	0.998439i	$-0.517787\pi$
$881$	−2920.92	−0.111701	−0.0558504	+	0.998439i	$0.517787\pi$
$882$	0	0
$883$	− 7123.74i	− 0.271498i	−0.990743	−	0.135749i	$-0.956656\pi$
$883$	− 7123.74i	− 0.271498i	0.990743	−	0.135749i	$-0.0433441\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7923.42i	− 0.299935i	−0.988691	−	0.149968i	$-0.952083\pi$
$887$	− 7923.42i	− 0.299935i	0.988691	−	0.149968i	$-0.0479169\pi$
$888$	0	0
$889$	−34751.0	−1.31104
$890$	0	0
$891$	4425.99	0.166415
$892$	0	0
$893$	43116.9i	1.61573i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	218.642i	0.00813850i
$898$	0	0
$899$	−7823.05	−0.290226
$900$	0	0
$901$	−28578.9	−1.05672
$902$	0	0
$903$	15846.1i	0.583972i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19771.7i	0.723825i	0.932212	+	0.361912i	$0.117876\pi$
$907$	19771.7i	0.723825i	−0.932212	+	0.361912i	$0.882124\pi$
$908$	0	0
$909$	−8799.99	−0.321097
$910$	0	0
$911$	38222.7	1.39009	0.695046	−	0.718966i	$-0.255384\pi$
$911$	38222.7	1.39009	0.695046	+	0.718966i	$0.255384\pi$
$912$	0	0
$913$	37275.5i	1.35119i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 47174.9i	− 1.69886i
$918$	0	0
$919$	−39237.3	−1.40840	−0.704199	−	0.710003i	$-0.748694\pi$
$919$	−39237.3	−1.40840	−0.704199	+	0.710003i	$0.748694\pi$
$920$	0	0
$921$	22798.6	0.815677
$922$	0	0
$923$	4265.33i	0.152107i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6832.74i	0.242089i
$928$	0	0
$929$	−20024.5	−0.707194	−0.353597	−	0.935398i	$-0.615042\pi$
$929$	−20024.5	−0.707194	−0.353597	+	0.935398i	$0.615042\pi$
$930$	0	0
$931$	−11420.2	−0.402020
$932$	0	0
$933$	8550.09i	0.300019i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8717.10i	− 0.303922i	−0.988386	−	0.151961i	$-0.951441\pi$
$937$	− 8717.10i	− 0.303922i	0.988386	−	0.151961i	$-0.0485588\pi$
$938$	0	0
$939$	2399.05	0.0833760
$940$	0	0
$941$	39276.2	1.36065	0.680323	−	0.732913i	$-0.261840\pi$
$941$	39276.2	1.36065	0.680323	+	0.732913i	$0.261840\pi$
$942$	0	0
$943$	92.1490i	0.00318217i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20080.8i	− 0.689060i	−0.938775	−	0.344530i	$-0.888038\pi$
$947$	− 20080.8i	− 0.689060i	0.938775	−	0.344530i	$-0.111962\pi$
$948$	0	0
$949$	−30727.9	−1.05107
$950$	0	0
$951$	3011.37	0.102682
$952$	0	0
$953$	− 38549.9i	− 1.31034i	−0.755481	−	0.655170i	$-0.772597\pi$
$953$	− 38549.9i	− 1.31034i	0.755481	−	0.655170i	$-0.227403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16248.2i	0.548829i
$958$	0	0
$959$	−6929.95	−0.233347
$960$	0	0
$961$	−23561.8	−0.790902
$962$	0	0
$963$	− 8509.03i	− 0.284735i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 24108.2i	− 0.801724i	−0.916138	−	0.400862i	$-0.868711\pi$
$967$	− 24108.2i	− 0.801724i	0.916138	−	0.400862i	$-0.131289\pi$
$968$	0	0
$969$	25337.5	0.839999
$970$	0	0
$971$	−41760.1	−1.38017	−0.690084	−	0.723729i	$-0.742427\pi$
$971$	−41760.1	−1.38017	−0.690084	+	0.723729i	$0.742427\pi$
$972$	0	0
$973$	57433.7i	1.89233i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4532.52i	0.148422i	0.997243	+	0.0742109i	$0.0236438\pi$
$977$	4532.52i	0.148422i	−0.997243	+	0.0742109i	$0.976356\pi$
$978$	0	0
$979$	−34341.2	−1.12109
$980$	0	0
$981$	−6194.83	−0.201616
$982$	0	0
$983$	48822.8i	1.58414i	0.610432	+	0.792068i	$0.290996\pi$
$983$	48822.8i	1.58414i	−0.610432	+	0.792068i	$0.709004\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 33645.9i	− 1.08507i
$988$	0	0
$989$	212.583	0.00683492
$990$	0	0
$991$	−26937.6	−0.863473	−0.431737	−	0.902000i	$-0.642099\pi$
$991$	−26937.6	−0.863473	−0.431737	+	0.902000i	$0.642099\pi$
$992$	0	0
$993$	25103.0i	0.802236i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9191.03i	− 0.291959i	−0.989288	−	0.145979i	$-0.953367\pi$
$997$	− 9191.03i	− 0.291959i	0.989288	−	0.145979i	$-0.0466333\pi$
$998$	0	0
$999$	10551.8	0.334177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.f.j.49.4		4
3.2	odd	2		1800.4.f.z.649.4		4
4.3	odd	2		1200.4.f.x.49.1		4
5.2	odd	4		600.4.a.u.1.1	yes	2
5.3	odd	4		600.4.a.s.1.2	✓	2
5.4	even	2	inner	600.4.f.j.49.1		4
15.2	even	4		1800.4.a.bm.1.1		2
15.8	even	4		1800.4.a.bo.1.2		2
15.14	odd	2		1800.4.f.z.649.1		4
20.3	even	4		1200.4.a.br.1.1		2
20.7	even	4		1200.4.a.bp.1.2		2
20.19	odd	2		1200.4.f.x.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.s.1.2	✓	2	5.3	odd	4
600.4.a.u.1.1	yes	2	5.2	odd	4
600.4.f.j.49.1		4	5.4	even	2	inner
600.4.f.j.49.4		4	1.1	even	1	trivial
1200.4.a.bp.1.2		2	20.7	even	4
1200.4.a.br.1.1		2	20.3	even	4
1200.4.f.x.49.1		4	4.3	odd	2
1200.4.f.x.49.4		4	20.19	odd	2
1800.4.a.bm.1.1		2	15.2	even	4
1800.4.a.bo.1.2		2	15.8	even	4
1800.4.f.z.649.1		4	15.14	odd	2
1800.4.f.z.649.4		4	3.2	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +21.8806i q^{7} -9.00000 q^{9} +54.6418 q^{11} -82.7612i q^{13} -100.403i q^{17} +84.1194 q^{19} -65.6418 q^{21} +0.880613i q^{23} -27.0000i q^{27} +99.1194 q^{29} -78.9255 q^{31} +163.926i q^{33} +390.806i q^{37} +248.284 q^{39} +104.642 q^{41} -241.403i q^{43} +512.567i q^{47} -135.761 q^{49} +301.209 q^{51} -284.642i q^{53} +252.358i q^{57} +709.045 q^{59} +470.522 q^{61} -196.926i q^{63} -667.583i q^{67} -2.64184 q^{69} -51.5378 q^{71} -371.284i q^{73} +1195.60i q^{77} +79.3428 q^{79} +81.0000 q^{81} +682.180i q^{83} +297.358i q^{87} -628.478 q^{89} +1810.87 q^{91} -236.777i q^{93} +1519.09i q^{97} -491.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} - 32 q^{11} + 420 q^{19} - 12 q^{21} + 480 q^{29} + 436 q^{31} + 492 q^{39} + 168 q^{41} - 376 q^{49} - 48 q^{51} + 2168 q^{59} + 1548 q^{61} + 240 q^{69} + 2216 q^{71} + 2656 q^{79} + 324 q^{81}+ \cdots + 288 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 - 32 * q^11 + 420 * q^19 - 12 * q^21 + 480 * q^29 + 436 * q^31 + 492 * q^39 + 168 * q^41 - 376 * q^49 - 48 * q^51 + 2168 * q^59 + 1548 * q^61 + 240 * q^69 + 2216 * q^71 + 2656 * q^79 + 324 * q^81 - 2848 * q^89 + 3652 * q^91 + 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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