LMFDB - Embedded newform 4900.2.e.r.2549.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] 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("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,-4, 0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 980)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.r.2549.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{3} -2.82843 q^{9} +1.82843 q^{11} +6.41421i q^{13} -3.58579i q^{17} +7.65685 q^{19} +3.41421i q^{23} +0.414214i q^{27} +4.65685 q^{29} +7.41421 q^{31} +4.41421i q^{33} +0.585786i q^{37} -15.4853 q^{39} +3.41421 q^{41} +0.343146i q^{43} -10.8995i q^{47} +8.65685 q^{51} -12.2426i q^{53} +18.4853i q^{57} -0.585786 q^{59} +10.8284 q^{61} -3.07107i q^{67} -8.24264 q^{69} -10.4853 q^{71} +10.8284i q^{73} +15.1421 q^{79} -9.48528 q^{81} -8.00000i q^{83} +11.2426i q^{87} +16.9706 q^{89} +17.8995i q^{93} +9.72792i q^{97} -5.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} + 8 q^{19} - 4 q^{29} + 24 q^{31} - 28 q^{39} + 8 q^{41} + 12 q^{51} - 8 q^{59} + 32 q^{61} - 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^11 + 8 * q^19 - 4 * q^29 + 24 * q^31 - 28 * q^39 + 8 * q^41 + 12 * q^51 - 8 * q^59 + 32 * q^61 - 16 * q^69 - 8 * q^71 + 4 * q^79 - 4 * q^81 - 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.41421i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	2.41421i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	1.82843	0.551292	0.275646	−	0.961259i	$-0.411108\pi$
$11$	1.82843	0.551292	0.275646	+	0.961259i	$0.411108\pi$
$12$	0	0
$13$	6.41421i	1.77898i	0.456952	+	0.889491i	$0.348941\pi$
$13$	6.41421i	1.77898i	−0.456952	+	0.889491i	$0.651059\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.58579i	− 0.869681i	−0.900508	−	0.434840i	$-0.856805\pi$
$17$	− 3.58579i	− 0.869681i	0.900508	−	0.434840i	$-0.143195\pi$
$18$	0	0
$19$	7.65685	1.75660	0.878301	−	0.478107i	$-0.158677\pi$
$19$	7.65685	1.75660	0.878301	+	0.478107i	$0.158677\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.41421i	0.711913i	0.934503	+	0.355956i	$0.115845\pi$
$23$	3.41421i	0.711913i	−0.934503	+	0.355956i	$0.884155\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.414214i	0.0797154i
$28$	0	0
$29$	4.65685	0.864756	0.432378	−	0.901692i	$-0.357675\pi$
$29$	4.65685	0.864756	0.432378	+	0.901692i	$0.357675\pi$
$30$	0	0
$31$	7.41421	1.33163	0.665816	−	0.746116i	$-0.268084\pi$
$31$	7.41421	1.33163	0.665816	+	0.746116i	$0.268084\pi$
$32$	0	0
$33$	4.41421i	0.768416i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.585786i	0.0963027i	0.998840	+	0.0481513i	$0.0153330\pi$
$37$	0.585786i	0.0963027i	−0.998840	+	0.0481513i	$0.984667\pi$
$38$	0	0
$39$	−15.4853	−2.47963
$40$	0	0
$41$	3.41421	0.533211	0.266605	−	0.963806i	$-0.414098\pi$
$41$	3.41421	0.533211	0.266605	+	0.963806i	$0.414098\pi$
$42$	0	0
$43$	0.343146i	0.0523292i	0.999658	+	0.0261646i	$0.00832941\pi$
$43$	0.343146i	0.0523292i	−0.999658	+	0.0261646i	$0.991671\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.8995i	− 1.58985i	−0.606705	−	0.794927i	$-0.707509\pi$
$47$	− 10.8995i	− 1.58985i	0.606705	−	0.794927i	$-0.292491\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.65685	1.21220
$52$	0	0
$53$	− 12.2426i	− 1.68166i	−0.541302	−	0.840828i	$-0.682069\pi$
$53$	− 12.2426i	− 1.68166i	0.541302	−	0.840828i	$-0.317931\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.4853i	2.44844i
$58$	0	0
$59$	−0.585786	−0.0762629	−0.0381314	−	0.999273i	$-0.512141\pi$
$59$	−0.585786	−0.0762629	−0.0381314	+	0.999273i	$0.512141\pi$
$60$	0	0
$61$	10.8284	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$61$	10.8284	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.07107i	− 0.375191i	−0.982246	−	0.187595i	$-0.939931\pi$
$67$	− 3.07107i	− 0.375191i	0.982246	−	0.187595i	$-0.0600694\pi$
$68$	0	0
$69$	−8.24264	−0.992297
$70$	0	0
$71$	−10.4853	−1.24437	−0.622187	−	0.782869i	$-0.713756\pi$
$71$	−10.4853	−1.24437	−0.622187	+	0.782869i	$0.713756\pi$
$72$	0	0
$73$	10.8284i	1.26737i	0.773591	+	0.633686i	$0.218459\pi$
$73$	10.8284i	1.26737i	−0.773591	+	0.633686i	$0.781541\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	15.1421	1.70362	0.851812	−	0.523848i	$-0.175504\pi$
$79$	15.1421	1.70362	0.851812	+	0.523848i	$0.175504\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.2426i	1.20534i
$88$	0	0
$89$	16.9706	1.79888	0.899438	−	0.437048i	$-0.143976\pi$
$89$	16.9706	1.79888	0.899438	+	0.437048i	$0.143976\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.8995i	1.85609i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.72792i	0.987721i	0.869541	+	0.493860i	$0.164415\pi$
$97$	9.72792i	0.987721i	−0.869541	+	0.493860i	$0.835585\pi$
$98$	0	0
$99$	−5.17157	−0.519763
$100$	0	0
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	0	0
$103$	4.41421i	0.434945i	0.976066	+	0.217473i	$0.0697813\pi$
$103$	4.41421i	0.434945i	−0.976066	+	0.217473i	$0.930219\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.1716i	1.08000i	0.841666	+	0.539998i	$0.181575\pi$
$107$	11.1716i	1.08000i	−0.841666	+	0.539998i	$0.818425\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	−1.41421	−0.134231
$112$	0	0
$113$	9.07107i	0.853334i	0.904409	+	0.426667i	$0.140312\pi$
$113$	9.07107i	0.853334i	−0.904409	+	0.426667i	$0.859688\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 18.1421i	− 1.67724i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.65685	−0.696078
$122$	0	0
$123$	8.24264i	0.743214i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.10051i	0.186390i	0.995648	+	0.0931948i	$0.0297079\pi$
$127$	2.10051i	0.186390i	−0.995648	+	0.0931948i	$0.970292\pi$
$128$	0	0
$129$	−0.828427	−0.0729389
$130$	0	0
$131$	−20.2426	−1.76861	−0.884304	−	0.466912i	$-0.845367\pi$
$131$	−20.2426	−1.76861	−0.884304	+	0.466912i	$0.845367\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.9706i	1.79164i	0.444421	+	0.895818i	$0.353409\pi$
$137$	20.9706i	1.79164i	−0.444421	+	0.895818i	$0.646591\pi$
$138$	0	0
$139$	−5.89949	−0.500389	−0.250194	−	0.968196i	$-0.580495\pi$
$139$	−5.89949	−0.500389	−0.250194	+	0.968196i	$0.580495\pi$
$140$	0	0
$141$	26.3137	2.21601
$142$	0	0
$143$	11.7279i	0.980738i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.4853	−1.67822	−0.839110	−	0.543962i	$-0.816924\pi$
$149$	−20.4853	−1.67822	−0.839110	+	0.543962i	$0.816924\pi$
$150$	0	0
$151$	−6.17157	−0.502235	−0.251118	−	0.967957i	$-0.580798\pi$
$151$	−6.17157	−0.502235	−0.251118	+	0.967957i	$0.580798\pi$
$152$	0	0
$153$	10.1421i	0.819943i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.48528i	0.517582i	0.965933	+	0.258791i	$0.0833241\pi$
$157$	6.48528i	0.517582i	−0.965933	+	0.258791i	$0.916676\pi$
$158$	0	0
$159$	29.5563	2.34397
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 9.55635i	− 0.748511i	−0.927326	−	0.374256i	$-0.877898\pi$
$163$	− 9.55635i	− 0.748511i	0.927326	−	0.374256i	$-0.122102\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.414214i	0.0320528i	0.999872	+	0.0160264i	$0.00510158\pi$
$167$	0.414214i	0.0320528i	−0.999872	+	0.0160264i	$0.994898\pi$
$168$	0	0
$169$	−28.1421	−2.16478
$170$	0	0
$171$	−21.6569	−1.65614
$172$	0	0
$173$	− 12.5563i	− 0.954642i	−0.878729	−	0.477321i	$-0.841608\pi$
$173$	− 12.5563i	− 0.954642i	0.878729	−	0.477321i	$-0.158392\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.41421i	− 0.106299i
$178$	0	0
$179$	10.4853	0.783707	0.391853	−	0.920028i	$-0.371834\pi$
$179$	10.4853	0.783707	0.391853	+	0.920028i	$0.371834\pi$
$180$	0	0
$181$	10.2426	0.761329	0.380665	−	0.924713i	$-0.375695\pi$
$181$	10.2426	0.761329	0.380665	+	0.924713i	$0.375695\pi$
$182$	0	0
$183$	26.1421i	1.93248i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.55635i	− 0.479448i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.34315	0.386616	0.193308	−	0.981138i	$-0.438078\pi$
$191$	5.34315	0.386616	0.193308	+	0.981138i	$0.438078\pi$
$192$	0	0
$193$	− 5.65685i	− 0.407189i	−0.979055	−	0.203595i	$-0.934738\pi$
$193$	− 5.65685i	− 0.407189i	0.979055	−	0.203595i	$-0.0652625\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.55635i	− 0.253379i	−0.991942	−	0.126690i	$-0.959565\pi$
$197$	− 3.55635i	− 0.253379i	0.991942	−	0.126690i	$-0.0404352\pi$
$198$	0	0
$199$	1.27208	0.0901752	0.0450876	−	0.998983i	$-0.485643\pi$
$199$	1.27208	0.0901752	0.0450876	+	0.998983i	$0.485643\pi$
$200$	0	0
$201$	7.41421	0.522958
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 9.65685i	− 0.671198i
$208$	0	0
$209$	14.0000	0.968400
$210$	0	0
$211$	−1.68629	−0.116089	−0.0580445	−	0.998314i	$-0.518487\pi$
$211$	−1.68629	−0.116089	−0.0580445	+	0.998314i	$0.518487\pi$
$212$	0	0
$213$	− 25.3137i	− 1.73447i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−26.1421	−1.76652
$220$	0	0
$221$	23.0000	1.54715
$222$	0	0
$223$	9.92893i	0.664890i	0.943123	+	0.332445i	$0.107874\pi$
$223$	9.92893i	0.664890i	−0.943123	+	0.332445i	$0.892126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.7279i	1.17664i	0.808627	+	0.588322i	$0.200211\pi$
$227$	17.7279i	1.17664i	−0.808627	+	0.588322i	$0.799789\pi$
$228$	0	0
$229$	−4.10051	−0.270969	−0.135485	−	0.990779i	$-0.543259\pi$
$229$	−4.10051	−0.270969	−0.135485	+	0.990779i	$0.543259\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.8284i	− 0.971443i	−0.874114	−	0.485721i	$-0.838557\pi$
$233$	− 14.8284i	− 0.971443i	0.874114	−	0.485721i	$-0.161443\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	36.5563i	2.37459i
$238$	0	0
$239$	−19.8284	−1.28259	−0.641297	−	0.767293i	$-0.721603\pi$
$239$	−19.8284	−1.28259	−0.641297	+	0.767293i	$0.721603\pi$
$240$	0	0
$241$	27.5563	1.77506	0.887530	−	0.460749i	$-0.152419\pi$
$241$	27.5563	1.77506	0.887530	+	0.460749i	$0.152419\pi$
$242$	0	0
$243$	− 21.6569i	− 1.38929i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	49.1127i	3.12497i
$248$	0	0
$249$	19.3137	1.22396
$250$	0	0
$251$	−12.9289	−0.816067	−0.408033	−	0.912967i	$-0.633785\pi$
$251$	−12.9289	−0.816067	−0.408033	+	0.912967i	$0.633785\pi$
$252$	0	0
$253$	6.24264i	0.392471i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.7990i	− 0.860757i	−0.902648	−	0.430379i	$-0.858380\pi$
$257$	− 13.7990i	− 0.860757i	0.902648	−	0.430379i	$-0.141620\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−13.1716	−0.815300
$262$	0	0
$263$	10.0000i	0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$	10.0000i	0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	40.9706i	2.50736i
$268$	0	0
$269$	−8.72792	−0.532151	−0.266075	−	0.963952i	$-0.585727\pi$
$269$	−8.72792	−0.532151	−0.266075	+	0.963952i	$0.585727\pi$
$270$	0	0
$271$	−4.97056	−0.301940	−0.150970	−	0.988538i	$-0.548240\pi$
$271$	−4.97056	−0.301940	−0.150970	+	0.988538i	$0.548240\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.58579i	0.275533i	0.990465	+	0.137767i	$0.0439924\pi$
$277$	4.58579i	0.275533i	−0.990465	+	0.137767i	$0.956008\pi$
$278$	0	0
$279$	−20.9706	−1.25547
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	− 27.2426i	− 1.61941i	−0.586839	−	0.809703i	$-0.699628\pi$
$283$	− 27.2426i	− 1.61941i	0.586839	−	0.809703i	$-0.300372\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.14214	0.243655
$290$	0	0
$291$	−23.4853	−1.37673
$292$	0	0
$293$	− 11.5858i	− 0.676849i	−0.940994	−	0.338424i	$-0.890106\pi$
$293$	− 11.5858i	− 0.676849i	0.940994	−	0.338424i	$-0.109894\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.757359i	0.0439464i
$298$	0	0
$299$	−21.8995	−1.26648
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.41421i	− 0.366079i	−0.983106	−	0.183039i	$-0.941406\pi$
$307$	− 6.41421i	− 0.366079i	0.983106	−	0.183039i	$-0.0585935\pi$
$308$	0	0
$309$	−10.6569	−0.606247
$310$	0	0
$311$	−26.9706	−1.52936	−0.764680	−	0.644410i	$-0.777103\pi$
$311$	−26.9706	−1.52936	−0.764680	+	0.644410i	$0.777103\pi$
$312$	0	0
$313$	− 14.5563i	− 0.822774i	−0.911461	−	0.411387i	$-0.865045\pi$
$313$	− 14.5563i	− 0.822774i	0.911461	−	0.411387i	$-0.134955\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.9706i	− 1.06549i	−0.846275	−	0.532746i	$-0.821160\pi$
$317$	− 18.9706i	− 1.06549i	0.846275	−	0.532746i	$-0.178840\pi$
$318$	0	0
$319$	8.51472	0.476733
$320$	0	0
$321$	−26.9706	−1.50535
$322$	0	0
$323$	− 27.4558i	− 1.52768i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 21.7279i	− 1.20156i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.828427	−0.0455345	−0.0227672	−	0.999741i	$-0.507248\pi$
$331$	−0.828427	−0.0455345	−0.0227672	+	0.999741i	$0.507248\pi$
$332$	0	0
$333$	− 1.65685i	− 0.0907951i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6.72792i	− 0.366493i	−0.983067	−	0.183247i	$-0.941339\pi$
$337$	− 6.72792i	− 0.366493i	0.983067	−	0.183247i	$-0.0586607\pi$
$338$	0	0
$339$	−21.8995	−1.18942
$340$	0	0
$341$	13.5563	0.734117
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.58579i	0.246178i	0.992396	+	0.123089i	$0.0392800\pi$
$347$	4.58579i	0.246178i	−0.992396	+	0.123089i	$0.960720\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	−2.65685	−0.141812
$352$	0	0
$353$	12.0711i	0.642478i	0.946998	+	0.321239i	$0.104099\pi$
$353$	12.0711i	0.642478i	−0.946998	+	0.321239i	$0.895901\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	39.6274	2.08565
$362$	0	0
$363$	− 18.4853i	− 0.970226i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.72792i	0.298995i	0.988762	+	0.149498i	$0.0477656\pi$
$367$	5.72792i	0.298995i	−0.988762	+	0.149498i	$0.952234\pi$
$368$	0	0
$369$	−9.65685	−0.502716
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 0.485281i	− 0.0251269i	−0.999921	−	0.0125635i	$-0.996001\pi$
$373$	− 0.485281i	− 0.0251269i	0.999921	−	0.0125635i	$-0.00399918\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.8701i	1.53839i
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	−5.07107	−0.259799
$382$	0	0
$383$	− 20.4853i	− 1.04675i	−0.852103	−	0.523374i	$-0.824673\pi$
$383$	− 20.4853i	− 1.04675i	0.852103	−	0.523374i	$-0.175327\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 0.970563i	− 0.0493365i
$388$	0	0
$389$	15.1421	0.767737	0.383868	−	0.923388i	$-0.374592\pi$
$389$	15.1421	0.767737	0.383868	+	0.923388i	$0.374592\pi$
$390$	0	0
$391$	12.2426	0.619137
$392$	0	0
$393$	− 48.8701i	− 2.46517i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11.5858i	− 0.581474i	−0.956803	−	0.290737i	$-0.906100\pi$
$397$	− 11.5858i	− 0.581474i	0.956803	−	0.290737i	$-0.0939005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.4853	0.873173	0.436587	−	0.899662i	$-0.356187\pi$
$401$	17.4853	0.873173	0.436587	+	0.899662i	$0.356187\pi$
$402$	0	0
$403$	47.5563i	2.36895i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.07107i	0.0530909i
$408$	0	0
$409$	−27.1716	−1.34355	−0.671774	−	0.740756i	$-0.734467\pi$
$409$	−27.1716	−1.34355	−0.671774	+	0.740756i	$0.734467\pi$
$410$	0	0
$411$	−50.6274	−2.49727
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.2426i	− 0.697465i
$418$	0	0
$419$	−21.5563	−1.05310	−0.526548	−	0.850145i	$-0.676514\pi$
$419$	−21.5563	−1.05310	−0.526548	+	0.850145i	$0.676514\pi$
$420$	0	0
$421$	−2.31371	−0.112763	−0.0563816	−	0.998409i	$-0.517956\pi$
$421$	−2.31371	−0.112763	−0.0563816	+	0.998409i	$0.517956\pi$
$422$	0	0
$423$	30.8284i	1.49893i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−28.3137	−1.36700
$430$	0	0
$431$	−4.17157	−0.200938	−0.100469	−	0.994940i	$-0.532034\pi$
$431$	−4.17157	−0.200938	−0.100469	+	0.994940i	$0.532034\pi$
$432$	0	0
$433$	34.2843i	1.64760i	0.566883	+	0.823798i	$0.308149\pi$
$433$	34.2843i	1.64760i	−0.566883	+	0.823798i	$0.691851\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.1421i	1.25055i
$438$	0	0
$439$	−15.7574	−0.752058	−0.376029	−	0.926608i	$-0.622711\pi$
$439$	−15.7574	−0.752058	−0.376029	+	0.926608i	$0.622711\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.4558i	0.734329i	0.930156	+	0.367165i	$0.119671\pi$
$443$	15.4558i	0.734329i	−0.930156	+	0.367165i	$0.880329\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 49.4558i	− 2.33918i
$448$	0	0
$449$	−18.1716	−0.857570	−0.428785	−	0.903407i	$-0.641058\pi$
$449$	−18.1716	−0.857570	−0.428785	+	0.903407i	$0.641058\pi$
$450$	0	0
$451$	6.24264	0.293954
$452$	0	0
$453$	− 14.8995i	− 0.700039i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.928932i	0.0434536i	0.999764	+	0.0217268i	$0.00691640\pi$
$457$	0.928932i	0.0434536i	−0.999764	+	0.0217268i	$0.993084\pi$
$458$	0	0
$459$	1.48528	0.0693270
$460$	0	0
$461$	33.6569	1.56756	0.783778	−	0.621041i	$-0.213290\pi$
$461$	33.6569	1.56756	0.783778	+	0.621041i	$0.213290\pi$
$462$	0	0
$463$	37.4558i	1.74072i	0.492415	+	0.870360i	$0.336114\pi$
$463$	37.4558i	1.74072i	−0.492415	+	0.870360i	$0.663886\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.8995i	0.874564i	0.899324	+	0.437282i	$0.144059\pi$
$467$	18.8995i	0.874564i	−0.899324	+	0.437282i	$0.855941\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.6569	−0.721430
$472$	0	0
$473$	0.627417i	0.0288487i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	34.6274i	1.58548i
$478$	0	0
$479$	−6.58579	−0.300912	−0.150456	−	0.988617i	$-0.548074\pi$
$479$	−6.58579	−0.300912	−0.150456	+	0.988617i	$0.548074\pi$
$480$	0	0
$481$	−3.75736	−0.171321
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.58579i	0.207802i	0.994588	+	0.103901i	$0.0331325\pi$
$487$	4.58579i	0.207802i	−0.994588	+	0.103901i	$0.966868\pi$
$488$	0	0
$489$	23.0711	1.04331
$490$	0	0
$491$	−39.2843	−1.77287	−0.886437	−	0.462849i	$-0.846827\pi$
$491$	−39.2843	−1.77287	−0.886437	+	0.462849i	$0.846827\pi$
$492$	0	0
$493$	− 16.6985i	− 0.752062i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.6274	−1.41584	−0.707919	−	0.706294i	$-0.750366\pi$
$499$	−31.6274	−1.41584	−0.707919	+	0.706294i	$0.750366\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	− 9.04163i	− 0.403146i	−0.979473	−	0.201573i	$-0.935395\pi$
$503$	− 9.04163i	− 0.403146i	0.979473	−	0.201573i	$-0.0646054\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 67.9411i	− 3.01737i
$508$	0	0
$509$	20.9289	0.927659	0.463829	−	0.885925i	$-0.346475\pi$
$509$	20.9289	0.927659	0.463829	+	0.885925i	$0.346475\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.17157i	0.140028i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 19.9289i	− 0.876473i
$518$	0	0
$519$	30.3137	1.33062
$520$	0	0
$521$	29.3137	1.28426	0.642128	−	0.766597i	$-0.278052\pi$
$521$	29.3137	1.28426	0.642128	+	0.766597i	$0.278052\pi$
$522$	0	0
$523$	30.1421i	1.31802i	0.752133	+	0.659012i	$0.229025\pi$
$523$	30.1421i	1.31802i	−0.752133	+	0.659012i	$0.770975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 26.5858i	− 1.15810i
$528$	0	0
$529$	11.3431	0.493180
$530$	0	0
$531$	1.65685	0.0719014
$532$	0	0
$533$	21.8995i	0.948572i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25.3137i	1.09237i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.9706	−0.772615	−0.386307	−	0.922370i	$-0.626250\pi$
$541$	−17.9706	−0.772615	−0.386307	+	0.922370i	$0.626250\pi$
$542$	0	0
$543$	24.7279i	1.06118i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.4853i	− 1.04692i	−0.852052	−	0.523458i	$-0.824642\pi$
$547$	− 24.4853i	− 1.04692i	0.852052	−	0.523458i	$-0.175358\pi$
$548$	0	0
$549$	−30.6274	−1.30715
$550$	0	0
$551$	35.6569	1.51903
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.7990i	− 1.34737i	−0.739020	−	0.673683i	$-0.764711\pi$
$557$	− 31.7990i	− 1.34737i	0.739020	−	0.673683i	$-0.235289\pi$
$558$	0	0
$559$	−2.20101	−0.0930928
$560$	0	0
$561$	15.8284	0.668277
$562$	0	0
$563$	24.6274i	1.03792i	0.854798	+	0.518961i	$0.173681\pi$
$563$	24.6274i	1.03792i	−0.854798	+	0.518961i	$0.826319\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.79899	−0.326951	−0.163475	−	0.986547i	$-0.552270\pi$
$569$	−7.79899	−0.326951	−0.163475	+	0.986547i	$0.552270\pi$
$570$	0	0
$571$	−0.828427	−0.0346686	−0.0173343	−	0.999850i	$-0.505518\pi$
$571$	−0.828427	−0.0346686	−0.0173343	+	0.999850i	$0.505518\pi$
$572$	0	0
$573$	12.8995i	0.538884i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 0.0710678i	− 0.00295859i	−0.999999	−	0.00147930i	$-0.999529\pi$
$577$	− 0.0710678i	− 0.00295859i	0.999999	−	0.00147930i	$-0.000470875\pi$
$578$	0	0
$579$	13.6569	0.567559
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 22.3848i	− 0.927083i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.1716i	1.03894i	0.854488	+	0.519471i	$0.173871\pi$
$587$	25.1716i	1.03894i	−0.854488	+	0.519471i	$0.826129\pi$
$588$	0	0
$589$	56.7696	2.33915
$590$	0	0
$591$	8.58579	0.353172
$592$	0	0
$593$	− 7.72792i	− 0.317348i	−0.987331	−	0.158674i	$-0.949278\pi$
$593$	− 7.72792i	− 0.317348i	0.987331	−	0.158674i	$-0.0507218\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.07107i	0.125690i
$598$	0	0
$599$	−26.7990	−1.09498	−0.547489	−	0.836813i	$-0.684416\pi$
$599$	−26.7990	−1.09498	−0.547489	+	0.836813i	$0.684416\pi$
$600$	0	0
$601$	28.3431	1.15614	0.578071	−	0.815987i	$-0.303806\pi$
$601$	28.3431	1.15614	0.578071	+	0.815987i	$0.303806\pi$
$602$	0	0
$603$	8.68629i	0.353733i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.0711i	− 1.13937i	−0.821863	−	0.569685i	$-0.807065\pi$
$607$	− 28.0711i	− 1.13937i	0.821863	−	0.569685i	$-0.192935\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	69.9117	2.82832
$612$	0	0
$613$	− 18.6863i	− 0.754732i	−0.926064	−	0.377366i	$-0.876830\pi$
$613$	− 18.6863i	− 0.754732i	0.926064	−	0.377366i	$-0.123170\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.8701i	0.518129i	0.965860	+	0.259065i	$0.0834142\pi$
$617$	12.8701i	0.518129i	−0.965860	+	0.259065i	$0.916586\pi$
$618$	0	0
$619$	42.8701	1.72309	0.861547	−	0.507679i	$-0.169496\pi$
$619$	42.8701	1.72309	0.861547	+	0.507679i	$0.169496\pi$
$620$	0	0
$621$	−1.41421	−0.0567504
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	33.7990i	1.34980i
$628$	0	0
$629$	2.10051	0.0837526
$630$	0	0
$631$	−22.5147	−0.896297	−0.448148	−	0.893959i	$-0.647916\pi$
$631$	−22.5147	−0.896297	−0.448148	+	0.893959i	$0.647916\pi$
$632$	0	0
$633$	− 4.07107i	− 0.161810i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	29.6569	1.17321
$640$	0	0
$641$	16.6863	0.659069	0.329534	−	0.944144i	$-0.393108\pi$
$641$	16.6863	0.659069	0.329534	+	0.944144i	$0.393108\pi$
$642$	0	0
$643$	− 38.2132i	− 1.50698i	−0.657458	−	0.753491i	$-0.728368\pi$
$643$	− 38.2132i	− 1.50698i	0.657458	−	0.753491i	$-0.271632\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.1127i	1.06591i	0.846144	+	0.532955i	$0.178919\pi$
$647$	27.1127i	1.06591i	−0.846144	+	0.532955i	$0.821081\pi$
$648$	0	0
$649$	−1.07107	−0.0420431
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 49.3137i	− 1.92979i	−0.262628	−	0.964897i	$-0.584589\pi$
$653$	− 49.3137i	− 1.92979i	0.262628	−	0.964897i	$-0.415411\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 30.6274i	− 1.19489i
$658$	0	0
$659$	9.34315	0.363957	0.181979	−	0.983302i	$-0.441750\pi$
$659$	9.34315	0.363957	0.181979	+	0.983302i	$0.441750\pi$
$660$	0	0
$661$	−5.85786	−0.227845	−0.113922	−	0.993490i	$-0.536341\pi$
$661$	−5.85786	−0.227845	−0.113922	+	0.993490i	$0.536341\pi$
$662$	0	0
$663$	55.5269i	2.15649i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.8995i	0.615631i
$668$	0	0
$669$	−23.9706	−0.926755
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	− 44.4853i	− 1.71478i	−0.514666	−	0.857391i	$-0.672084\pi$
$673$	− 44.4853i	− 1.71478i	0.514666	−	0.857391i	$-0.327916\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.7279i	1.37314i	0.727066	+	0.686568i	$0.240884\pi$
$677$	35.7279i	1.37314i	−0.727066	+	0.686568i	$0.759116\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−42.7990	−1.64006
$682$	0	0
$683$	− 27.1127i	− 1.03744i	−0.854945	−	0.518719i	$-0.826409\pi$
$683$	− 27.1127i	− 1.03744i	0.854945	−	0.518719i	$-0.173591\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 9.89949i	− 0.377689i
$688$	0	0
$689$	78.5269	2.99164
$690$	0	0
$691$	20.4853	0.779297	0.389648	−	0.920964i	$-0.372597\pi$
$691$	20.4853	0.779297	0.389648	+	0.920964i	$0.372597\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.2426i	− 0.463723i
$698$	0	0
$699$	35.7990	1.35404
$700$	0	0
$701$	17.4853	0.660410	0.330205	−	0.943909i	$-0.392882\pi$
$701$	17.4853	0.660410	0.330205	+	0.943909i	$0.392882\pi$
$702$	0	0
$703$	4.48528i	0.169166i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	41.6274	1.56335	0.781675	−	0.623686i	$-0.214365\pi$
$709$	41.6274	1.56335	0.781675	+	0.623686i	$0.214365\pi$
$710$	0	0
$711$	−42.8284	−1.60619
$712$	0	0
$713$	25.3137i	0.948006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 47.8701i	− 1.78774i
$718$	0	0
$719$	−24.2426	−0.904098	−0.452049	−	0.891993i	$-0.649307\pi$
$719$	−24.2426	−0.904098	−0.452049	+	0.891993i	$0.649307\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	66.5269i	2.47416i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	0	0
$729$	23.8284	0.882534
$730$	0	0
$731$	1.23045	0.0455097
$732$	0	0
$733$	28.6985i	1.06000i	0.847997	+	0.530001i	$0.177809\pi$
$733$	28.6985i	1.06000i	−0.847997	+	0.530001i	$0.822191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.61522i	− 0.206839i
$738$	0	0
$739$	−9.34315	−0.343693	−0.171847	−	0.985124i	$-0.554973\pi$
$739$	−9.34315	−0.343693	−0.171847	+	0.985124i	$0.554973\pi$
$740$	0	0
$741$	−118.569	−4.35572
$742$	0	0
$743$	− 23.0711i	− 0.846395i	−0.906037	−	0.423198i	$-0.860908\pi$
$743$	− 23.0711i	− 0.846395i	0.906037	−	0.423198i	$-0.139092\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.6274i	0.827894i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.1421	−0.990431	−0.495215	−	0.868770i	$-0.664911\pi$
$751$	−27.1421	−0.990431	−0.495215	+	0.868770i	$0.664911\pi$
$752$	0	0
$753$	− 31.2132i	− 1.13747i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 42.8284i	− 1.55663i	−0.627877	−	0.778313i	$-0.716076\pi$
$757$	− 42.8284i	− 1.55663i	0.627877	−	0.778313i	$-0.283924\pi$
$758$	0	0
$759$	−15.0711	−0.547045
$760$	0	0
$761$	34.8701	1.26404	0.632019	−	0.774953i	$-0.282226\pi$
$761$	34.8701	1.26404	0.632019	+	0.774953i	$0.282226\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.75736i	− 0.135670i
$768$	0	0
$769$	0.142136	0.00512554	0.00256277	−	0.999997i	$-0.499184\pi$
$769$	0.142136	0.00512554	0.00256277	+	0.999997i	$0.499184\pi$
$770$	0	0
$771$	33.3137	1.19976
$772$	0	0
$773$	− 18.2721i	− 0.657201i	−0.944469	−	0.328600i	$-0.893423\pi$
$773$	− 18.2721i	− 0.657201i	0.944469	−	0.328600i	$-0.106577\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.1421	0.936639
$780$	0	0
$781$	−19.1716	−0.686013
$782$	0	0
$783$	1.92893i	0.0689344i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.2721i	0.651329i	0.945485	+	0.325665i	$0.105588\pi$
$787$	18.2721i	0.651329i	−0.945485	+	0.325665i	$0.894412\pi$
$788$	0	0
$789$	−24.1421	−0.859483
$790$	0	0
$791$	0	0
$792$	0	0
$793$	69.4558i	2.46645i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.38478i	− 0.0490513i	−0.999699	−	0.0245256i	$-0.992192\pi$
$797$	− 1.38478i	− 0.0490513i	0.999699	−	0.0245256i	$-0.00780754\pi$
$798$	0	0
$799$	−39.0833	−1.38267
$800$	0	0
$801$	−48.0000	−1.69600
$802$	0	0
$803$	19.7990i	0.698691i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21.0711i	− 0.741737i
$808$	0	0
$809$	−39.9706	−1.40529	−0.702645	−	0.711541i	$-0.747998\pi$
$809$	−39.9706	−1.40529	−0.702645	+	0.711541i	$0.747998\pi$
$810$	0	0
$811$	11.5563	0.405798	0.202899	−	0.979200i	$-0.434964\pi$
$811$	11.5563	0.405798	0.202899	+	0.979200i	$0.434964\pi$
$812$	0	0
$813$	− 12.0000i	− 0.420858i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.62742i	0.0919217i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.1716	1.47180	0.735899	−	0.677091i	$-0.236760\pi$
$821$	42.1716	1.47180	0.735899	+	0.677091i	$0.236760\pi$
$822$	0	0
$823$	− 28.5269i	− 0.994386i	−0.867640	−	0.497193i	$-0.834364\pi$
$823$	− 28.5269i	− 0.994386i	0.867640	−	0.497193i	$-0.165636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 38.0416i	− 1.32284i	−0.750017	−	0.661419i	$-0.769955\pi$
$827$	− 38.0416i	− 1.32284i	0.750017	−	0.661419i	$-0.230045\pi$
$828$	0	0
$829$	2.72792	0.0947446	0.0473723	−	0.998877i	$-0.484915\pi$
$829$	2.72792	0.0947446	0.0473723	+	0.998877i	$0.484915\pi$
$830$	0	0
$831$	−11.0711	−0.384051
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.07107i	0.106152i
$838$	0	0
$839$	14.3848	0.496618	0.248309	−	0.968681i	$-0.420125\pi$
$839$	14.3848	0.496618	0.248309	+	0.968681i	$0.420125\pi$
$840$	0	0
$841$	−7.31371	−0.252197
$842$	0	0
$843$	− 2.41421i	− 0.0831499i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	65.7696	2.25721
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	26.2843i	0.899956i	0.893040	+	0.449978i	$0.148568\pi$
$853$	26.2843i	0.899956i	−0.893040	+	0.449978i	$0.851432\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.1716i	0.518251i	0.965844	+	0.259126i	$0.0834343\pi$
$857$	15.1716i	0.518251i	−0.965844	+	0.259126i	$0.916566\pi$
$858$	0	0
$859$	−5.45584	−0.186151	−0.0930755	−	0.995659i	$-0.529670\pi$
$859$	−5.45584	−0.186151	−0.0930755	+	0.995659i	$0.529670\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 28.2426i	− 0.961391i	−0.876888	−	0.480695i	$-0.840384\pi$
$863$	− 28.2426i	− 0.961391i	0.876888	−	0.480695i	$-0.159616\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.0000i	0.339618i
$868$	0	0
$869$	27.6863	0.939193
$870$	0	0
$871$	19.6985	0.667458
$872$	0	0
$873$	− 27.5147i	− 0.931232i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.3137i	− 1.12492i	−0.826823	−	0.562462i	$-0.809854\pi$
$877$	− 33.3137i	− 1.12492i	0.826823	−	0.562462i	$-0.190146\pi$
$878$	0	0
$879$	27.9706	0.943424
$880$	0	0
$881$	0.284271	0.00957734	0.00478867	−	0.999989i	$-0.498476\pi$
$881$	0.284271	0.00957734	0.00478867	+	0.999989i	$0.498476\pi$
$882$	0	0
$883$	55.1127i	1.85469i	0.374208	+	0.927345i	$0.377915\pi$
$883$	55.1127i	1.85469i	−0.374208	+	0.927345i	$0.622085\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.5980i	0.725189i	0.931947	+	0.362595i	$0.118109\pi$
$887$	21.5980i	0.725189i	−0.931947	+	0.362595i	$0.881891\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−17.3431	−0.581017
$892$	0	0
$893$	− 83.4558i	− 2.79274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 52.8701i	− 1.76528i
$898$	0	0
$899$	34.5269	1.15154
$900$	0	0
$901$	−43.8995	−1.46250
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42.5269i	1.41208i	0.708170	+	0.706041i	$0.249521\pi$
$907$	42.5269i	1.41208i	−0.708170	+	0.706041i	$0.750479\pi$
$908$	0	0
$909$	−2.34315	−0.0777172
$910$	0	0
$911$	23.5980	0.781836	0.390918	−	0.920426i	$-0.372158\pi$
$911$	23.5980	0.781836	0.390918	+	0.920426i	$0.372158\pi$
$912$	0	0
$913$	− 14.6274i	− 0.484097i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35.4853	1.17055	0.585276	−	0.810834i	$-0.300986\pi$
$919$	35.4853	1.17055	0.585276	+	0.810834i	$0.300986\pi$
$920$	0	0
$921$	15.4853	0.510257
$922$	0	0
$923$	− 67.2548i	− 2.21372i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.4853i	− 0.410070i
$928$	0	0
$929$	46.7279	1.53309	0.766547	−	0.642189i	$-0.221973\pi$
$929$	46.7279	1.53309	0.766547	+	0.642189i	$0.221973\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 65.1127i	− 2.13169i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 17.7279i	− 0.579146i	−0.957156	−	0.289573i	$-0.906487\pi$
$937$	− 17.7279i	− 0.579146i	0.957156	−	0.289573i	$-0.0935133\pi$
$938$	0	0
$939$	35.1421	1.14682
$940$	0	0
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	11.6569i	0.379599i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.0416i	− 0.521283i	−0.965436	−	0.260641i	$-0.916066\pi$
$947$	− 16.0416i	− 0.521283i	0.965436	−	0.260641i	$-0.0839340\pi$
$948$	0	0
$949$	−69.4558	−2.25463
$950$	0	0
$951$	45.7990	1.48513
$952$	0	0
$953$	14.8701i	0.481688i	0.970564	+	0.240844i	$0.0774243\pi$
$953$	14.8701i	0.481688i	−0.970564	+	0.240844i	$0.922576\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.5563i	0.664492i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	23.9706	0.773244
$962$	0	0
$963$	− 31.5980i	− 1.01823i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43.4558i	− 1.39745i	−0.715392	−	0.698723i	$-0.753752\pi$
$967$	− 43.4558i	− 1.39745i	0.715392	−	0.698723i	$-0.246248\pi$
$968$	0	0
$969$	66.2843	2.12936
$970$	0	0
$971$	42.5269	1.36475	0.682377	−	0.731001i	$-0.260946\pi$
$971$	42.5269	1.36475	0.682377	+	0.731001i	$0.260946\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 20.1421i	− 0.644404i	−0.946671	−	0.322202i	$-0.895577\pi$
$977$	− 20.1421i	− 0.644404i	0.946671	−	0.322202i	$-0.104423\pi$
$978$	0	0
$979$	31.0294	0.991705
$980$	0	0
$981$	25.4558	0.812743
$982$	0	0
$983$	40.0122i	1.27619i	0.769957	+	0.638095i	$0.220277\pi$
$983$	40.0122i	1.27619i	−0.769957	+	0.638095i	$0.779723\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.17157	−0.0372539
$990$	0	0
$991$	2.68629	0.0853329	0.0426664	−	0.999089i	$-0.486415\pi$
$991$	2.68629	0.0853329	0.0426664	+	0.999089i	$0.486415\pi$
$992$	0	0
$993$	− 2.00000i	− 0.0634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 36.8406i	− 1.16675i	−0.812201	−	0.583377i	$-0.801731\pi$
$997$	− 36.8406i	− 1.16675i	0.812201	−	0.583377i	$-0.198269\pi$
$998$	0	0
$999$	−0.242641	−0.00767681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.r.2549.4		4
5.2	odd	4		980.2.a.k.1.2	yes	2
5.3	odd	4		4900.2.a.x.1.1		2
5.4	even	2	inner	4900.2.e.r.2549.1		4
7.6	odd	2		4900.2.e.q.2549.1		4
15.2	even	4		8820.2.a.bl.1.1		2
20.7	even	4		3920.2.a.bo.1.1		2
35.2	odd	12		980.2.i.k.361.1		4
35.12	even	12		980.2.i.l.361.2		4
35.13	even	4		4900.2.a.z.1.2		2
35.17	even	12		980.2.i.l.961.2		4
35.27	even	4		980.2.a.j.1.1	✓	2
35.32	odd	12		980.2.i.k.961.1		4
35.34	odd	2		4900.2.e.q.2549.4		4
105.62	odd	4		8820.2.a.bg.1.1		2
140.27	odd	4		3920.2.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.j.1.1	✓	2	35.27	even	4
980.2.a.k.1.2	yes	2	5.2	odd	4
980.2.i.k.361.1		4	35.2	odd	12
980.2.i.k.961.1		4	35.32	odd	12
980.2.i.l.361.2		4	35.12	even	12
980.2.i.l.961.2		4	35.17	even	12
3920.2.a.bo.1.1		2	20.7	even	4
3920.2.a.bx.1.2		2	140.27	odd	4
4900.2.a.x.1.1		2	5.3	odd	4
4900.2.a.z.1.2		2	35.13	even	4
4900.2.e.q.2549.1		4	7.6	odd	2
4900.2.e.q.2549.4		4	35.34	odd	2
4900.2.e.r.2549.1		4	5.4	even	2	inner
4900.2.e.r.2549.4		4	1.1	even	1	trivial
8820.2.a.bg.1.1		2	105.62	odd	4
8820.2.a.bl.1.1		2	15.2	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{3} -2.82843 q^{9} +1.82843 q^{11} +6.41421i q^{13} -3.58579i q^{17} +7.65685 q^{19} +3.41421i q^{23} +0.414214i q^{27} +4.65685 q^{29} +7.41421 q^{31} +4.41421i q^{33} +0.585786i q^{37} -15.4853 q^{39} +3.41421 q^{41} +0.343146i q^{43} -10.8995i q^{47} +8.65685 q^{51} -12.2426i q^{53} +18.4853i q^{57} -0.585786 q^{59} +10.8284 q^{61} -3.07107i q^{67} -8.24264 q^{69} -10.4853 q^{71} +10.8284i q^{73} +15.1421 q^{79} -9.48528 q^{81} -8.00000i q^{83} +11.2426i q^{87} +16.9706 q^{89} +17.8995i q^{93} +9.72792i q^{97} -5.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} + 8 q^{19} - 4 q^{29} + 24 q^{31} - 28 q^{39} + 8 q^{41} + 12 q^{51} - 8 q^{59} + 32 q^{61} - 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^11 + 8 * q^19 - 4 * q^29 + 24 * q^31 - 28 * q^39 + 8 * q^41 + 12 * q^51 - 8 * q^59 + 32 * q^61 - 16 * q^69 - 8 * q^71 + 4 * q^79 - 4 * q^81 - 32 * q^99

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