LMFDB - Embedded newform 552.4.m.a.137.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [552,4,Mod(137,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.137"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	552.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$32.5690543232$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.15
Character	$\chi$	$=$	552.137
Dual form			552.4.m.a.137.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.28443 + 2.94001i) q^{3} -12.3977 q^{5} +18.3617i q^{7} +(9.71264 - 25.1926i) q^{9} -9.38725 q^{11} +81.1567 q^{13} +(53.1169 - 36.4493i) q^{15} +92.4031 q^{17} +109.758i q^{19} +(-53.9836 - 78.6694i) q^{21} +(98.5466 - 49.5538i) q^{23} +28.7019 q^{25} +(32.4533 + 136.491i) q^{27} +102.963i q^{29} -264.903 q^{31} +(40.2190 - 27.5986i) q^{33} -227.642i q^{35} +127.694i q^{37} +(-347.710 + 238.602i) q^{39} -166.636i q^{41} -14.5807i q^{43} +(-120.414 + 312.329i) q^{45} -177.957i q^{47} +5.84812 q^{49} +(-395.894 + 271.666i) q^{51} -109.326 q^{53} +116.380 q^{55} +(-322.691 - 470.251i) q^{57} +108.660i q^{59} -114.919i q^{61} +(462.578 + 178.341i) q^{63} -1006.15 q^{65} +646.548i q^{67} +(-276.527 + 502.038i) q^{69} +235.609i q^{71} -218.735 q^{73} +(-122.971 + 84.3839i) q^{75} -172.366i q^{77} +1338.31i q^{79} +(-540.329 - 489.372i) q^{81} -126.330 q^{83} -1145.58 q^{85} +(-302.713 - 441.139i) q^{87} -1177.54 q^{89} +1490.18i q^{91} +(1134.96 - 778.817i) q^{93} -1360.75i q^{95} +358.360i q^{97} +(-91.1750 + 236.489i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 28 q^{9} + 1872 q^{25} - 444 q^{27} - 216 q^{31} - 68 q^{39} - 4200 q^{49} - 576 q^{55} + 1376 q^{69} - 384 q^{73} - 3592 q^{75} - 252 q^{81} + 3480 q^{85} - 412 q^{87} + 780 q^{93}+O(q^{100}) $ 72 * q + 28 * q^9 + 1872 * q^25 - 444 * q^27 - 216 * q^31 - 68 * q^39 - 4200 * q^49 - 576 * q^55 + 1376 * q^69 - 384 * q^73 - 3592 * q^75 - 252 * q^81 + 3480 * q^85 - 412 * q^87 + 780 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−4.28443	+	2.94001i	−0.824538	+	0.565806i
$3$	−4.28443	+	2.94001i	−0.824538	+	0.565806i
$4$		0			0
$5$	−12.3977			−1.10888			−0.554440	−	0.832224i	$-0.687068\pi$
$5$	−12.3977			−1.10888			−0.554440	+	0.832224i	$0.687068\pi$
$6$		0			0
$7$			18.3617i			0.991438i	0.868483	+	0.495719i	$0.165095\pi$
$7$			18.3617i			0.991438i	−0.868483	+	0.495719i	$0.834905\pi$
$8$		0			0
$9$	9.71264	−	25.1926i	0.359727	−	0.933057i
$10$		0			0
$11$	−9.38725			−0.257306			−0.128653	−	0.991690i	$-0.541065\pi$
$11$	−9.38725			−0.257306			−0.128653	+	0.991690i	$0.541065\pi$
$12$		0			0
$13$	81.1567			1.73145			0.865724	−	0.500521i	$-0.166858\pi$
$13$	81.1567			1.73145			0.865724	+	0.500521i	$0.166858\pi$
$14$		0			0
$15$	53.1169	−	36.4493i	0.914314	−	0.627411i
$16$		0			0
$17$	92.4031			1.31830			0.659148	−	0.752013i	$-0.270917\pi$
$17$	92.4031			1.31830			0.659148	+	0.752013i	$0.270917\pi$
$18$		0			0
$19$			109.758i			1.32528i	0.748939	+	0.662639i	$0.230563\pi$
$19$			109.758i			1.32528i	−0.748939	+	0.662639i	$0.769437\pi$
$20$		0			0
$21$	−53.9836	−	78.6694i	−0.560962	−	0.817479i
$22$		0			0
$23$	98.5466	−	49.5538i	0.893408	−	0.449247i
$23$	98.5466	−	49.5538i	0.893408	−	0.449247i
$24$		0			0
$25$	28.7019			0.229615
$26$		0			0
$27$	32.4533	+	136.491i	0.231320	+	0.972878i
$28$		0			0
$29$			102.963i			0.659304i	0.944103	+	0.329652i	$0.106931\pi$
$29$			102.963i			0.659304i	−0.944103	+	0.329652i	$0.893069\pi$
$30$		0			0
$31$	−264.903			−1.53477			−0.767386	−	0.641186i	$-0.778443\pi$
$31$	−264.903			−1.53477			−0.767386	+	0.641186i	$0.778443\pi$
$32$		0			0
$33$	40.2190	−	27.5986i	0.212158	−	0.145585i
$34$		0			0
$35$		−	227.642i		−	1.09939i
$36$		0			0
$37$			127.694i			0.567370i	0.958917	+	0.283685i	$0.0915571\pi$
$37$			127.694i			0.567370i	−0.958917	+	0.283685i	$0.908443\pi$
$38$		0			0
$39$	−347.710	+	238.602i	−1.42765	+	0.979664i
$40$		0			0
$41$		−	166.636i		−	0.634736i	−0.948303	−	0.317368i	$-0.897201\pi$
$41$		−	166.636i		−	0.634736i	0.948303	−	0.317368i	$-0.102799\pi$
$42$		0			0
$43$		−	14.5807i		−	0.0517100i	−0.999666	−	0.0258550i	$-0.991769\pi$
$43$		−	14.5807i		−	0.0517100i	0.999666	−	0.0258550i	$-0.00823082\pi$
$44$		0			0
$45$	−120.414	+	312.329i	−0.398895	+	1.03465i
$46$		0			0
$47$		−	177.957i		−	0.552292i	−0.961116	−	0.276146i	$-0.910943\pi$
$47$		−	177.957i		−	0.552292i	0.961116	−	0.276146i	$-0.0890574\pi$
$48$		0			0
$49$	5.84812			0.0170499
$50$		0			0
$51$	−395.894	+	271.666i	−1.08699	+	0.745900i
$52$		0			0
$53$	−109.326			−0.283342			−0.141671	−	0.989914i	$-0.545248\pi$
$53$	−109.326			−0.283342			−0.141671	+	0.989914i	$0.545248\pi$
$54$		0			0
$55$	116.380			0.285321
$56$		0			0
$57$	−322.691	−	470.251i	−0.749850	−	1.09274i
$58$		0			0
$59$			108.660i			0.239767i	0.992788	+	0.119884i	$0.0382521\pi$
$59$			108.660i			0.239767i	−0.992788	+	0.119884i	$0.961748\pi$
$60$		0			0
$61$		−	114.919i		−	0.241210i	−0.992701	−	0.120605i	$-0.961517\pi$
$61$		−	114.919i		−	0.241210i	0.992701	−	0.120605i	$-0.0384835\pi$
$62$		0			0
$63$	462.578	+	178.341i	0.925069	+	0.356648i
$64$		0			0
$65$	−1006.15			−1.91997
$66$		0			0
$67$			646.548i			1.17893i	0.807793	+	0.589466i	$0.200662\pi$
$67$			646.548i			1.17893i	−0.807793	+	0.589466i	$0.799338\pi$
$68$		0			0
$69$	−276.527	+	502.038i	−0.482463	+	0.875917i
$70$		0			0
$71$			235.609i			0.393826i	0.980421	+	0.196913i	$0.0630916\pi$
$71$			235.609i			0.393826i	−0.980421	+	0.196913i	$0.936908\pi$
$72$		0			0
$73$	−218.735			−0.350699			−0.175349	−	0.984506i	$-0.556105\pi$
$73$	−218.735			−0.350699			−0.175349	+	0.984506i	$0.556105\pi$
$74$		0			0
$75$	−122.971	+	84.3839i	−0.189327	+	0.129918i
$76$		0			0
$77$		−	172.366i		−	0.255103i
$78$		0			0
$79$			1338.31i			1.90596i	0.303028	+	0.952982i	$0.402003\pi$
$79$			1338.31i			1.90596i	−0.303028	+	0.952982i	$0.597997\pi$
$80$		0			0
$81$	−540.329	−	489.372i	−0.741192	−	0.671293i
$82$		0			0
$83$	−126.330			−0.167066			−0.0835332	−	0.996505i	$-0.526620\pi$
$83$	−126.330			−0.167066			−0.0835332	+	0.996505i	$0.526620\pi$
$84$		0			0
$85$	−1145.58			−1.46183
$86$		0			0
$87$	−302.713	−	441.139i	−0.373038	−	0.543621i
$88$		0			0
$89$	−1177.54			−1.40246			−0.701228	−	0.712937i	$-0.747365\pi$
$89$	−1177.54			−1.40246			−0.701228	+	0.712937i	$0.747365\pi$
$90$		0			0
$91$			1490.18i			1.71662i
$92$		0			0
$93$	1134.96	−	778.817i	1.26548	−	0.868383i
$94$		0			0
$95$		−	1360.75i		−	1.46957i
$96$		0			0
$97$			358.360i			0.375113i	0.982254	+	0.187556i	$0.0600567\pi$
$97$			358.360i			0.375113i	−0.982254	+	0.187556i	$0.939943\pi$
$98$		0			0
$99$	−91.1750	+	236.489i	−0.0925599	+	0.240081i
$100$		0			0
$101$		−	1543.12i		−	1.52026i	−0.649770	−	0.760130i	$-0.725135\pi$
$101$		−	1543.12i		−	1.52026i	0.649770	−	0.760130i	$-0.274865\pi$
$102$		0			0
$103$			30.4240i			0.0291045i	0.999894	+	0.0145522i	$0.00463228\pi$
$103$			30.4240i			0.0291045i	−0.999894	+	0.0145522i	$0.995368\pi$
$104$		0			0
$105$	669.271	+	975.316i	0.622039	+	0.906486i
$106$		0			0
$107$	−35.0488			−0.0316663			−0.0158332	−	0.999875i	$-0.505040\pi$
$107$	−35.0488			−0.0316663			−0.0158332	+	0.999875i	$0.505040\pi$
$108$		0			0
$109$		−	35.0649i		−	0.0308130i	−0.999881	−	0.0154065i	$-0.995096\pi$
$109$		−	35.0649i		−	0.0308130i	0.999881	−	0.0154065i	$-0.00490423\pi$
$110$		0			0
$111$	−375.421	−	547.094i	−0.321021	−	0.467819i
$112$		0			0
$113$	788.978			0.656822			0.328411	−	0.944535i	$-0.393487\pi$
$113$	788.978			0.656822			0.328411	+	0.944535i	$0.393487\pi$
$114$		0			0
$115$	−1221.75	+	614.351i	−0.990682	+	0.498161i
$116$		0			0
$117$	788.246	−	2044.55i	0.622850	−	1.61554i
$118$		0			0
$119$			1696.68i			1.30701i
$120$		0			0
$121$	−1242.88			−0.933794
$122$		0			0
$123$	489.912	+	713.940i	0.359137	+	0.523364i
$124$		0			0
$125$	1193.87			0.854264
$126$		0			0
$127$	1138.68			0.795604			0.397802	−	0.917471i	$-0.369773\pi$
$127$	1138.68			0.795604			0.397802	+	0.917471i	$0.369773\pi$
$128$		0			0
$129$	42.8674	+	62.4698i	0.0292578	+	0.0426369i
$130$		0			0
$131$			1140.46i			0.760627i	0.924858	+	0.380314i	$0.124184\pi$
$131$			1140.46i			0.760627i	−0.924858	+	0.380314i	$0.875816\pi$
$132$		0			0
$133$	−2015.35			−1.31393
$134$		0			0
$135$	−402.345	−	1692.17i	−0.256506	−	1.07880i
$136$		0			0
$137$	−2927.94			−1.82592			−0.912959	−	0.408051i	$-0.866208\pi$
$137$	−2927.94			−1.82592			−0.912959	+	0.408051i	$0.866208\pi$
$138$		0			0
$139$	−644.552			−0.393311			−0.196655	−	0.980473i	$-0.563008\pi$
$139$	−644.552			−0.393311			−0.196655	+	0.980473i	$0.563008\pi$
$140$		0			0
$141$	523.197	+	762.445i	0.312490	+	0.455386i
$142$		0			0
$143$	−761.839			−0.445512
$144$		0			0
$145$		−	1276.50i		−	0.731089i
$146$		0			0
$147$	−25.0558	+	17.1935i	−0.0140583	+	0.00964693i
$148$		0			0
$149$	−1130.40			−0.621515			−0.310758	−	0.950489i	$-0.600583\pi$
$149$	−1130.40			−0.621515			−0.310758	+	0.950489i	$0.600583\pi$
$150$		0			0
$151$	−2578.93			−1.38987			−0.694936	−	0.719072i	$-0.744567\pi$
$151$	−2578.93			−1.38987			−0.694936	+	0.719072i	$0.744567\pi$
$152$		0			0
$153$	897.478	−	2327.87i	0.474228	−	1.23005i
$154$		0			0
$155$	3284.17			1.70188
$156$		0			0
$157$			3457.15i			1.75739i	0.477384	+	0.878695i	$0.341585\pi$
$157$			3457.15i			1.75739i	−0.477384	+	0.878695i	$0.658415\pi$
$158$		0			0
$159$	468.401	−	321.421i	0.233626	−	0.160317i
$160$		0			0
$161$	909.891	+	1809.48i	0.445401	+	0.885759i
$162$		0			0
$163$	2659.70			1.27806			0.639029	−	0.769182i	$-0.279336\pi$
$163$	2659.70			1.27806			0.639029	+	0.769182i	$0.279336\pi$
$164$		0			0
$165$	−498.621	+	342.159i	−0.235258	+	0.161436i
$166$		0			0
$167$		−	1148.16i		−	0.532021i	−0.963970	−	0.266010i	$-0.914294\pi$
$167$		−	1148.16i		−	0.532021i	0.963970	−	0.266010i	$-0.0857055\pi$
$168$		0			0
$169$	4389.42			1.99791
$170$		0			0
$171$	2765.09	+	1066.04i	1.23656	+	0.476739i
$172$		0			0
$173$			3089.66i			1.35782i	0.734223	+	0.678908i	$0.237547\pi$
$173$			3089.66i			1.35782i	−0.734223	+	0.678908i	$0.762453\pi$
$174$		0			0
$175$			527.015i			0.227649i
$176$		0			0
$177$	−319.460	−	465.544i	−0.135662	−	0.197697i
$178$		0			0
$179$			3862.31i			1.61275i	0.591403	+	0.806376i	$0.298574\pi$
$179$			3862.31i			1.61275i	−0.591403	+	0.806376i	$0.701426\pi$
$180$		0			0
$181$			702.667i			0.288557i	0.989537	+	0.144278i	$0.0460861\pi$
$181$			702.667i			0.288557i	−0.989537	+	0.144278i	$0.953914\pi$
$182$		0			0
$183$	337.862	+	492.360i	0.136478	+	0.198887i
$184$		0			0
$185$		−	1583.10i		−	0.629146i
$186$		0			0
$187$	−867.411			−0.339205
$188$		0			0
$189$	−2506.21	+	595.898i	−0.964548	+	0.229340i
$190$		0			0
$191$	1894.71			0.717781			0.358890	−	0.933380i	$-0.383155\pi$
$191$	1894.71			0.717781			0.358890	+	0.933380i	$0.383155\pi$
$192$		0			0
$193$	2282.29			0.851205			0.425602	−	0.904910i	$-0.360062\pi$
$193$	2282.29			0.851205			0.425602	+	0.904910i	$0.360062\pi$
$194$		0			0
$195$	4310.79	−	2958.10i	1.58309	−	1.08633i
$196$		0			0
$197$			1778.90i			0.643357i	0.946849	+	0.321678i	$0.104247\pi$
$197$			1778.90i			0.643357i	−0.946849	+	0.321678i	$0.895753\pi$
$198$		0			0
$199$		−	4622.24i		−	1.64654i	−0.567649	−	0.823271i	$-0.692147\pi$
$199$		−	4622.24i		−	1.64654i	0.567649	−	0.823271i	$-0.307853\pi$
$200$		0			0
$201$	−1900.86	−	2770.09i	−0.667047	−	0.972075i
$202$		0			0
$203$	−1890.58			−0.653659
$204$		0			0
$205$			2065.90i			0.703846i
$206$		0			0
$207$	−291.239	−	2963.94i	−0.0977898	−	0.995207i
$208$		0			0
$209$		−	1030.33i		−	0.341001i
$210$		0			0
$211$	2658.39			0.867352			0.433676	−	0.901069i	$-0.357216\pi$
$211$	2658.39			0.867352			0.433676	+	0.901069i	$0.357216\pi$
$212$		0			0
$213$	−692.693	−	1009.45i	−0.222829	−	0.324724i
$214$		0			0
$215$			180.766i			0.0573402i
$216$		0			0
$217$		−	4864.06i		−	1.52163i
$218$		0			0
$219$	937.154	−	643.084i	0.289165	−	0.198427i
$220$		0			0
$221$	7499.13			2.28256
$222$		0			0
$223$	−4495.95			−1.35010			−0.675048	−	0.737774i	$-0.735877\pi$
$223$	−4495.95			−1.35010			−0.675048	+	0.737774i	$0.735877\pi$
$224$		0			0
$225$	278.771	−	723.074i	0.0825989	−	0.214244i
$226$		0			0
$227$	1640.38			0.479630			0.239815	−	0.970819i	$-0.422913\pi$
$227$	1640.38			0.479630			0.239815	+	0.970819i	$0.422913\pi$
$228$		0			0
$229$		−	4875.55i		−	1.40692i	−0.710734	−	0.703461i	$-0.751637\pi$
$229$		−	4875.55i		−	1.40692i	0.710734	−	0.703461i	$-0.248363\pi$
$230$		0			0
$231$	506.758	+	738.489i	0.144339	+	0.210342i
$232$		0			0
$233$		−	5182.79i		−	1.45724i	−0.684920	−	0.728618i	$-0.740163\pi$
$233$		−	5182.79i		−	1.45724i	0.684920	−	0.728618i	$-0.259837\pi$
$234$		0			0
$235$			2206.25i			0.612426i
$236$		0			0
$237$	−3934.64	−	5733.87i	−1.07841	−	1.57154i
$238$		0			0
$239$		−	5644.76i		−	1.52774i	−0.645371	−	0.763869i	$-0.723297\pi$
$239$		−	5644.76i		−	1.52774i	0.645371	−	0.763869i	$-0.276703\pi$
$240$		0			0
$241$			7241.71i			1.93560i	0.251719	+	0.967800i	$0.419004\pi$
$241$			7241.71i			1.93560i	−0.251719	+	0.967800i	$0.580996\pi$
$242$		0			0
$243$	3753.76	+	508.106i	0.990963	+	0.134136i
$244$		0			0
$245$	−72.5029			−0.0189063
$246$		0			0
$247$			8907.62i			2.29465i
$248$		0			0
$249$	541.251	−	371.412i	0.137753	−	0.0945271i
$250$		0			0
$251$	−7405.04			−1.86216			−0.931080	−	0.364815i	$-0.881132\pi$
$251$	−7405.04			−1.86216			−0.931080	+	0.364815i	$0.881132\pi$
$252$		0			0
$253$	−925.081	+	465.174i	−0.229879	+	0.115594i
$254$		0			0
$255$	4908.16	−	3368.03i	1.20534	−	0.827114i
$256$		0			0
$257$		−	251.362i		−	0.0610098i	−0.999535	−	0.0305049i	$-0.990288\pi$
$257$		−	251.362i		−	0.0610098i	0.999535	−	0.0305049i	$-0.00971152\pi$
$258$		0			0
$259$	−2344.67			−0.562513
$260$		0			0
$261$	2593.91	+	1000.05i	0.615168	+	0.237170i
$262$		0			0
$263$	−2544.88			−0.596669			−0.298335	−	0.954461i	$-0.596431\pi$
$263$	−2544.88			−0.596669			−0.298335	+	0.954461i	$0.596431\pi$
$264$		0			0
$265$	1355.39			0.314192
$266$		0			0
$267$	5045.07	−	3461.97i	1.15638	−	0.793518i
$268$		0			0
$269$			2725.67i			0.617795i	0.951095	+	0.308898i	$0.0999601\pi$
$269$			2725.67i			0.617795i	−0.951095	+	0.308898i	$0.900040\pi$
$270$		0			0
$271$	2763.77			0.619509			0.309754	−	0.950817i	$-0.399753\pi$
$271$	2763.77			0.619509			0.309754	+	0.950817i	$0.399753\pi$
$272$		0			0
$273$	−4381.14	−	6384.55i	−0.971276	−	1.41542i
$274$		0			0
$275$	−269.432			−0.0590813
$276$		0			0
$277$	−6405.08			−1.38933			−0.694664	−	0.719334i	$-0.744447\pi$
$277$	−6405.08			−1.38933			−0.694664	+	0.719334i	$0.744447\pi$
$278$		0			0
$279$	−2572.90	+	6673.57i	−0.552100	+	1.43203i
$280$		0			0
$281$	4327.11			0.918626			0.459313	−	0.888275i	$-0.348096\pi$
$281$	4327.11			0.918626			0.459313	+	0.888275i	$0.348096\pi$
$282$		0			0
$283$			876.639i			0.184137i	0.995753	+	0.0920685i	$0.0293479\pi$
$283$			876.639i			0.184137i	−0.995753	+	0.0920685i	$0.970652\pi$
$284$		0			0
$285$	4000.61	+	5830.01i	0.831493	+	1.21172i
$286$		0			0
$287$	3059.72			0.629301
$288$		0			0
$289$	3625.33			0.737906
$290$		0			0
$291$	−1053.58	−	1535.37i	−0.212241	−	0.309295i
$292$		0			0
$293$	−5416.90			−1.08006			−0.540032	−	0.841645i	$-0.681588\pi$
$293$	−5416.90			−1.08006			−0.540032	+	0.841645i	$0.681588\pi$
$294$		0			0
$295$		−	1347.12i		−	0.265873i
$296$		0			0
$297$	−304.648	−	1281.28i	−0.0595200	−	0.250327i
$298$		0			0
$299$	7997.72	−	4021.62i	1.54689	−	0.777848i
$300$		0			0
$301$	267.726			0.0512673
$302$		0			0
$303$	4536.80	+	6611.39i	0.860173	+	1.25351i
$304$		0			0
$305$			1424.72i			0.267473i
$306$		0			0
$307$	−5827.01			−1.08327			−0.541637	−	0.840613i	$-0.682195\pi$
$307$	−5827.01			−1.08327			−0.541637	+	0.840613i	$0.682195\pi$
$308$		0			0
$309$	−89.4468	−	130.349i	−0.0164675	−	0.0239978i
$310$		0			0
$311$			3293.72i			0.600546i	0.953853	+	0.300273i	$0.0970778\pi$
$311$			3293.72i			0.600546i	−0.953853	+	0.300273i	$0.902922\pi$
$312$		0			0
$313$		−	4905.94i		−	0.885943i	−0.896536	−	0.442971i	$-0.853924\pi$
$313$		−	4905.94i		−	0.885943i	0.896536	−	0.442971i	$-0.146076\pi$
$314$		0			0
$315$	−5734.88	−	2211.01i	−1.02579	−	0.395479i
$316$		0			0
$317$		−	6513.58i		−	1.15407i	−0.816721	−	0.577033i	$-0.804210\pi$
$317$		−	6513.58i		−	1.15407i	0.816721	−	0.577033i	$-0.195790\pi$
$318$		0			0
$319$		−	966.542i		−	0.169643i
$320$		0			0
$321$	150.164	−	103.044i	0.0261101	−	0.0179170i
$322$		0			0
$323$			10142.0i			1.74711i
$324$		0			0
$325$	2329.35			0.397567
$326$		0			0
$327$	103.091	+	150.233i	0.0174342	+	0.0254065i
$328$		0			0
$329$	3267.60			0.547564
$330$		0			0
$331$	−1331.12			−0.221042			−0.110521	−	0.993874i	$-0.535252\pi$
$331$	−1331.12			−0.221042			−0.110521	+	0.993874i	$0.535252\pi$
$332$		0			0
$333$	3216.93	+	1240.24i	0.529389	+	0.204099i
$334$		0			0
$335$		−	8015.68i		−	1.30729i
$336$		0			0
$337$		−	4198.23i		−	0.678612i	−0.940676	−	0.339306i	$-0.889808\pi$
$337$		−	4198.23i		−	0.678612i	0.940676	−	0.339306i	$-0.110192\pi$
$338$		0			0
$339$	−3380.32	+	2319.61i	−0.541575	+	0.371633i
$340$		0			0
$341$	2486.71			0.394906
$342$		0			0
$343$			6405.44i			1.00834i
$344$		0			0
$345$	3428.28	−	6224.09i	0.534993	−	0.971286i
$346$		0			0
$347$			9399.49i			1.45415i	0.686556	+	0.727077i	$0.259122\pi$
$347$			9399.49i			1.45415i	−0.686556	+	0.727077i	$0.740878\pi$
$348$		0			0
$349$	3399.24			0.521367			0.260683	−	0.965424i	$-0.416052\pi$
$349$	3399.24			0.521367			0.260683	+	0.965424i	$0.416052\pi$
$350$		0			0
$351$	2633.81	+	11077.2i	0.400519	+	1.68449i
$352$		0			0
$353$			376.541i			0.0567741i	0.999597	+	0.0283870i	$0.00903709\pi$
$353$			376.541i			0.0567741i	−0.999597	+	0.0283870i	$0.990963\pi$
$354$		0			0
$355$		−	2921.00i		−	0.436705i
$356$		0			0
$357$	−4988.26	−	7269.29i	−0.739514	−	1.07768i
$358$		0			0
$359$	−1497.46			−0.220148			−0.110074	−	0.993923i	$-0.535109\pi$
$359$	−1497.46			−0.220148			−0.110074	+	0.993923i	$0.535109\pi$
$360$		0			0
$361$	−5187.87			−0.756360
$362$		0			0
$363$	5325.03	−	3654.08i	0.769949	−	0.528346i
$364$		0			0
$365$	2711.80			0.388883
$366$		0			0
$367$		−	13581.0i		−	1.93167i	−0.259156	−	0.965835i	$-0.583444\pi$
$367$		−	13581.0i		−	1.93167i	0.259156	−	0.965835i	$-0.416556\pi$
$368$		0			0
$369$	−4197.98	−	1618.48i	−0.592245	−	0.228332i
$370$		0			0
$371$		−	2007.42i		−	0.280916i
$372$		0			0
$373$			10774.5i			1.49567i	0.663887	+	0.747833i	$0.268906\pi$
$373$			10774.5i			1.49567i	−0.663887	+	0.747833i	$0.731094\pi$
$374$		0			0
$375$	−5115.05	+	3510.00i	−0.704374	+	0.483348i
$376$		0			0
$377$			8356.17i			1.14155i
$378$		0			0
$379$			11553.2i			1.56582i	0.622135	+	0.782910i	$0.286265\pi$
$379$			11553.2i			1.56582i	−0.622135	+	0.782910i	$0.713735\pi$
$380$		0			0
$381$	−4878.60	+	3347.74i	−0.656006	+	0.450158i
$382$		0			0
$383$	−11918.1			−1.59004			−0.795019	−	0.606584i	$-0.792539\pi$
$383$	−11918.1			−1.59004			−0.795019	+	0.606584i	$0.792539\pi$
$384$		0			0
$385$			2136.93i			0.282878i
$386$		0			0
$387$	−367.324	−	141.617i	−0.0482484	−	0.0186015i
$388$		0			0
$389$	−3563.37			−0.464447			−0.232223	−	0.972662i	$-0.574600\pi$
$389$	−3563.37			−0.464447			−0.232223	+	0.972662i	$0.574600\pi$
$390$		0			0
$391$	9106.01	−	4578.92i	1.17778	−	0.592241i
$392$		0			0
$393$	−3352.96	−	4886.20i	−0.430367	−	0.627166i
$394$		0			0
$395$		−	16591.9i		−	2.11349i
$396$		0			0
$397$	−9061.44			−1.14554			−0.572771	−	0.819715i	$-0.694132\pi$
$397$	−9061.44			−1.14554			−0.572771	+	0.819715i	$0.694132\pi$
$398$		0			0
$399$	8634.61	−	5925.15i	1.08339	−	0.743430i
$400$		0			0
$401$	10202.5			1.27054			0.635271	−	0.772289i	$-0.280888\pi$
$401$	10202.5			1.27054			0.635271	+	0.772289i	$0.280888\pi$
$402$		0			0
$403$	−21498.6			−2.65738
$404$		0			0
$405$	6698.82	+	6067.07i	0.821893	+	0.744383i
$406$		0			0
$407$		−	1198.69i		−	0.145988i
$408$		0			0
$409$	1518.58			0.183592			0.0917958	−	0.995778i	$-0.470739\pi$
$409$	1518.58			0.183592			0.0917958	+	0.995778i	$0.470739\pi$
$410$		0			0
$411$	12544.5	−	8608.18i	1.50554	−	1.03312i
$412$		0			0
$413$	−1995.17			−0.237714
$414$		0			0
$415$	1566.19			0.185257
$416$		0			0
$417$	2761.54	−	1894.99i	0.324300	−	0.222538i
$418$		0			0
$419$	4942.95			0.576322			0.288161	−	0.957582i	$-0.406956\pi$
$419$	4942.95			0.576322			0.288161	+	0.957582i	$0.406956\pi$
$420$		0			0
$421$		−	4158.33i		−	0.481389i	−0.970601	−	0.240694i	$-0.922625\pi$
$421$		−	4158.33i		−	0.481389i	0.970601	−	0.240694i	$-0.0773751\pi$
$422$		0			0
$423$	−4483.20	−	1728.43i	−0.515320	−	0.198675i
$424$		0			0
$425$	2652.14			0.302701
$426$		0			0
$427$	2110.10			0.239145
$428$		0			0
$429$	3264.04	−	2239.82i	0.367342	−	0.252073i
$430$		0			0
$431$	−9325.32			−1.04219			−0.521096	−	0.853498i	$-0.674477\pi$
$431$	−9325.32			−1.04219			−0.521096	+	0.853498i	$0.674477\pi$
$432$		0			0
$433$			13944.4i			1.54763i	0.633411	+	0.773815i	$0.281654\pi$
$433$			13944.4i			1.54763i	−0.633411	+	0.773815i	$0.718346\pi$
$434$		0			0
$435$	3752.94	+	5469.09i	0.413654	+	0.602811i
$436$		0			0
$437$	5438.94	+	10816.3i	0.595377	+	1.18401i
$438$		0			0
$439$	10433.9			1.13436			0.567178	−	0.823595i	$-0.308035\pi$
$439$	10433.9			1.13436			0.567178	+	0.823595i	$0.308035\pi$
$440$		0			0
$441$	56.8007	−	147.329i	0.00613332	−	0.0159085i
$442$		0			0
$443$		−	5374.83i		−	0.576447i	−0.957563	−	0.288223i	$-0.906935\pi$
$443$		−	5374.83i		−	0.576447i	0.957563	−	0.288223i	$-0.0930646\pi$
$444$		0			0
$445$	14598.7			1.55516
$446$		0			0
$447$	4843.11	−	3323.39i	0.512463	−	0.351657i
$448$		0			0
$449$			17791.1i			1.86996i	0.354698	+	0.934981i	$0.384584\pi$
$449$			17791.1i			1.86996i	−0.354698	+	0.934981i	$0.615416\pi$
$450$		0			0
$451$			1564.25i			0.163321i
$452$		0			0
$453$	11049.3	−	7582.10i	1.14600	−	0.786397i
$454$		0			0
$455$		−	18474.7i		−	1.90353i
$456$		0			0
$457$			3807.81i			0.389763i	0.980827	+	0.194882i	$0.0624322\pi$
$457$			3807.81i			0.389763i	−0.980827	+	0.194882i	$0.937568\pi$
$458$		0			0
$459$	2998.79	+	12612.2i	0.304949	+	1.28254i
$460$		0			0
$461$			9661.96i			0.976144i	0.872803	+	0.488072i	$0.162300\pi$
$461$			9661.96i			0.976144i	−0.872803	+	0.488072i	$0.837700\pi$
$462$		0			0
$463$	−3899.22			−0.391386			−0.195693	−	0.980665i	$-0.562696\pi$
$463$	−3899.22			−0.391386			−0.195693	+	0.980665i	$0.562696\pi$
$464$		0			0
$465$	−14070.8	+	9655.51i	−1.40326	+	0.962933i
$466$		0			0
$467$	−4231.91			−0.419335			−0.209668	−	0.977773i	$-0.567238\pi$
$467$	−4231.91			−0.419335			−0.209668	+	0.977773i	$0.567238\pi$
$468$		0			0
$469$	−11871.7			−1.16884
$470$		0			0
$471$	−10164.1	−	14811.9i	−0.994342	−	1.44904i
$472$		0			0
$473$			136.872i			0.0133053i
$474$		0			0
$475$			3150.27i			0.304304i
$476$		0			0
$477$	−1061.85	+	2754.21i	−0.101926	+	0.264374i
$478$		0			0
$479$	−19702.3			−1.87938			−0.939689	−	0.342030i	$-0.888886\pi$
$479$	−19702.3			−1.87938			−0.939689	+	0.342030i	$0.888886\pi$
$480$		0			0
$481$			10363.2i			0.982373i
$482$		0			0
$483$	−9218.26	−	5077.50i	−0.868417	−	0.478332i
$484$		0			0
$485$		−	4442.82i		−	0.415955i
$486$		0			0
$487$	−2563.01			−0.238483			−0.119241	−	0.992865i	$-0.538046\pi$
$487$	−2563.01			−0.238483			−0.119241	+	0.992865i	$0.538046\pi$
$488$		0			0
$489$	−11395.3	+	7819.54i	−1.05381	+	0.723133i
$490$		0			0
$491$			4514.55i			0.414946i	0.978241	+	0.207473i	$0.0665240\pi$
$491$			4514.55i			0.414946i	−0.978241	+	0.207473i	$0.933476\pi$
$492$		0			0
$493$			9514.13i			0.869158i
$494$		0			0
$495$	1130.36	−	2931.91i	0.102638	−	0.266221i
$496$		0			0
$497$	−4326.18			−0.390454
$498$		0			0
$499$	4927.41			0.442047			0.221023	−	0.975269i	$-0.429060\pi$
$499$	4927.41			0.442047			0.221023	+	0.975269i	$0.429060\pi$
$500$		0			0
$501$	3375.61	+	4919.22i	0.301020	+	0.438671i
$502$		0			0
$503$	−8329.96			−0.738399			−0.369199	−	0.929350i	$-0.620368\pi$
$503$	−8329.96			−0.738399			−0.369199	+	0.929350i	$0.620368\pi$
$504$		0			0
$505$			19131.1i			1.68579i
$506$		0			0
$507$	−18806.1	+	12904.9i	−1.64736	+	1.13043i
$508$		0			0
$509$		−	2654.64i		−	0.231169i	−0.993298	−	0.115584i	$-0.963126\pi$
$509$		−	2654.64i		−	0.231169i	0.993298	−	0.115584i	$-0.0368740\pi$
$510$		0			0
$511$		−	4016.35i		−	0.347696i
$512$		0			0
$513$	−14981.0	+	3562.02i	−1.28933	+	0.306563i
$514$		0			0
$515$		−	377.186i		−	0.0322734i
$516$		0			0
$517$			1670.53i			0.142108i
$518$		0			0
$519$	−9083.63	−	13237.4i	−0.768261	−	1.11957i
$520$		0			0
$521$	15201.4			1.27828			0.639141	−	0.769090i	$-0.279290\pi$
$521$	15201.4			1.27828			0.639141	+	0.769090i	$0.279290\pi$
$522$		0			0
$523$			16925.8i			1.41513i	0.706646	+	0.707567i	$0.250207\pi$
$523$			16925.8i			1.41513i	−0.706646	+	0.707567i	$0.749793\pi$
$524$		0			0
$525$	−1549.43	−	2257.96i	−0.128805	−	0.187706i
$526$		0			0
$527$	−24477.8			−2.02328
$528$		0			0
$529$	7255.85	−	9766.71i	0.596355	−	0.802721i
$530$		0			0
$531$	2737.41	+	1055.37i	0.223717	+	0.0862508i
$532$		0			0
$533$		−	13523.6i		−	1.09901i
$534$		0			0
$535$	434.523			0.0351142
$536$		0			0
$537$	−11355.2	−	16547.8i	−0.912504	−	1.32978i
$538$		0			0
$539$	−54.8977			−0.00438704
$540$		0			0
$541$	−16485.5			−1.31011			−0.655053	−	0.755583i	$-0.727354\pi$
$541$	−16485.5			−1.31011			−0.655053	+	0.755583i	$0.727354\pi$
$542$		0			0
$543$	−2065.85	−	3010.52i	−0.163267	−	0.237926i
$544$		0			0
$545$			434.723i			0.0341679i
$546$		0			0
$547$	17417.0			1.36142			0.680710	−	0.732553i	$-0.261671\pi$
$547$	17417.0			1.36142			0.680710	+	0.732553i	$0.261671\pi$
$548$		0			0
$549$	−2895.09	−	1116.16i	−0.225063	−	0.0867699i
$550$		0			0
$551$	−11301.1			−0.873760
$552$		0			0
$553$	−24573.6			−1.88965
$554$		0			0
$555$	4654.34	+	6782.69i	0.355974	+	0.518755i
$556$		0			0
$557$	−19632.3			−1.49344			−0.746720	−	0.665139i	$-0.768372\pi$
$557$	−19632.3			−1.49344			−0.746720	+	0.665139i	$0.768372\pi$
$558$		0			0
$559$		−	1183.32i		−	0.0895333i
$560$		0			0
$561$	3716.36	−	2550.20i	0.279688	−	0.191924i
$562$		0			0
$563$	−18598.0			−1.39221			−0.696104	−	0.717941i	$-0.745085\pi$
$563$	−18598.0			−1.39221			−0.696104	+	0.717941i	$0.745085\pi$
$564$		0			0
$565$	−9781.48			−0.728336
$566$		0			0
$567$	8985.71	−	9921.36i	0.665545	−	0.734847i
$568$		0			0
$569$	16101.0			1.18627			0.593135	−	0.805103i	$-0.297890\pi$
$569$	16101.0			1.18627			0.593135	+	0.805103i	$0.297890\pi$
$570$		0			0
$571$		−	7527.11i		−	0.551663i	−0.961206	−	0.275832i	$-0.911047\pi$
$571$		−	7527.11i		−	0.551663i	0.961206	−	0.275832i	$-0.0889532\pi$
$572$		0			0
$573$	−8117.73	+	5570.46i	−0.591838	+	0.406125i
$574$		0			0
$575$	2828.47	−	1422.29i	0.205140	−	0.103154i
$576$		0			0
$577$	15693.5			1.13228			0.566141	−	0.824308i	$-0.308436\pi$
$577$	15693.5			1.13228			0.566141	+	0.824308i	$0.308436\pi$
$578$		0			0
$579$	−9778.29	+	6709.95i	−0.701851	+	0.481617i
$580$		0			0
$581$		−	2319.63i		−	0.165636i
$582$		0			0
$583$	1026.27			0.0729055
$584$		0			0
$585$	−9772.41	+	25347.6i	−0.690666	+	1.79144i
$586$		0			0
$587$			7032.47i			0.494483i	0.968954	+	0.247241i	$0.0795240\pi$
$587$			7032.47i			0.494483i	−0.968954	+	0.247241i	$0.920476\pi$
$588$		0			0
$589$		−	29075.3i		−	2.03400i
$590$		0			0
$591$	−5229.99	−	7621.56i	−0.364015	−	0.530472i
$592$		0			0
$593$			14778.1i			1.02338i	0.859171	+	0.511688i	$0.170980\pi$
$593$			14778.1i			1.02338i	−0.859171	+	0.511688i	$0.829020\pi$
$594$		0			0
$595$		−	21034.8i		−	1.44932i
$596$		0			0
$597$	13589.4	+	19803.6i	0.931623	+	1.35764i
$598$		0			0
$599$			10705.3i			0.730226i	0.930963	+	0.365113i	$0.118970\pi$
$599$			10705.3i			0.730226i	−0.930963	+	0.365113i	$0.881030\pi$
$600$		0			0
$601$	−6917.27			−0.469486			−0.234743	−	0.972057i	$-0.575425\pi$
$601$	−6917.27			−0.469486			−0.234743	+	0.972057i	$0.575425\pi$
$602$		0			0
$603$	16288.2	+	6279.69i	1.10001	+	0.424094i
$604$		0			0
$605$	15408.8			1.03547
$606$		0			0
$607$	28683.4			1.91800			0.958998	−	0.283413i	$-0.0914667\pi$
$607$	28683.4			1.91800			0.958998	+	0.283413i	$0.0914667\pi$
$608$		0			0
$609$	8100.06	−	5558.33i	0.538967	−	0.369844i
$610$		0			0
$611$		−	14442.4i		−	0.956265i
$612$		0			0
$613$			13259.0i			0.873614i	0.899555	+	0.436807i	$0.143891\pi$
$613$			13259.0i			0.873614i	−0.899555	+	0.436807i	$0.856109\pi$
$614$		0			0
$615$	−6073.76	−	8851.18i	−0.398240	−	0.580348i
$616$		0			0
$617$	23587.3			1.53904			0.769522	−	0.638621i	$-0.220495\pi$
$617$	23587.3			1.53904			0.769522	+	0.638621i	$0.220495\pi$
$618$		0			0
$619$			274.662i			0.0178345i	0.999960	+	0.00891727i	$0.00283849\pi$
$619$			274.662i			0.0178345i	−0.999960	+	0.00891727i	$0.997162\pi$
$620$		0			0
$621$	9961.81	+	11842.5i	0.643725	+	0.765257i
$622$		0			0
$623$		−	21621.6i		−	1.39045i
$624$		0			0
$625$	−18388.9			−1.17689
$626$		0			0
$627$	3029.18	+	4414.37i	0.192941	+	0.281169i
$628$		0			0
$629$			11799.3i			0.747962i
$630$		0			0
$631$		−	18688.8i		−	1.17907i	−0.807744	−	0.589533i	$-0.799312\pi$
$631$		−	18688.8i		−	1.17907i	0.807744	−	0.589533i	$-0.200688\pi$
$632$		0			0
$633$	−11389.7	+	7815.71i	−0.715165	+	0.490753i
$634$		0			0
$635$	−14117.0			−0.882230
$636$		0			0
$637$	474.614			0.0295210
$638$		0			0
$639$	5935.59	+	2288.38i	0.367462	+	0.141670i
$640$		0			0
$641$	1053.54			0.0649179			0.0324590	−	0.999473i	$-0.489666\pi$
$641$	1053.54			0.0649179			0.0324590	+	0.999473i	$0.489666\pi$
$642$		0			0
$643$		−	11628.4i		−	0.713186i	−0.934260	−	0.356593i	$-0.883938\pi$
$643$		−	11628.4i		−	0.713186i	0.934260	−	0.356593i	$-0.116062\pi$
$644$		0			0
$645$	−531.455	−	774.479i	−0.0324434	−	0.0472792i
$646$		0			0
$647$			2835.47i			0.172293i	0.996282	+	0.0861467i	$0.0274554\pi$
$647$			2835.47i			0.172293i	−0.996282	+	0.0861467i	$0.972545\pi$
$648$		0			0
$649$		−	1020.01i		−	0.0616935i
$650$		0			0
$651$	14300.4	+	20839.7i	0.860948	+	1.25464i
$652$		0			0
$653$		−	29050.2i		−	1.74093i	−0.492235	−	0.870463i	$-0.663820\pi$
$653$		−	29050.2i		−	1.74093i	0.492235	−	0.870463i	$-0.336180\pi$
$654$		0			0
$655$		−	14139.0i		−	0.843444i
$656$		0			0
$657$	−2124.50	+	5510.49i	−0.126156	+	0.327222i
$658$		0			0
$659$	13975.6			0.826120			0.413060	−	0.910704i	$-0.364460\pi$
$659$	13975.6			0.826120			0.413060	+	0.910704i	$0.364460\pi$
$660$		0			0
$661$		−	23391.1i		−	1.37641i	−0.725517	−	0.688205i	$-0.758399\pi$
$661$		−	23391.1i		−	1.37641i	0.725517	−	0.688205i	$-0.241601\pi$
$662$		0			0
$663$	−32129.5	+	22047.6i	−1.88206	+	1.29149i
$664$		0			0
$665$	24985.6			1.45699
$666$		0			0
$667$	5102.22	+	10146.7i	0.296190	+	0.589027i
$668$		0			0
$669$	19262.6	−	13218.2i	1.11321	−	0.763892i
$670$		0			0
$671$			1078.77i			0.0620647i
$672$		0			0
$673$	23103.0			1.32326			0.661631	−	0.749830i	$-0.269865\pi$
$673$	23103.0			1.32326			0.661631	+	0.749830i	$0.269865\pi$
$674$		0			0
$675$	931.472	+	3917.55i	0.0531146	+	0.223387i
$676$		0			0
$677$	25309.1			1.43679			0.718394	−	0.695636i	$-0.244877\pi$
$677$	25309.1			1.43679			0.718394	+	0.695636i	$0.244877\pi$
$678$		0			0
$679$	−6580.09			−0.371901
$680$		0			0
$681$	−7028.10	+	4822.75i	−0.395474	+	0.271378i
$682$		0			0
$683$			8912.66i			0.499317i	0.968334	+	0.249659i	$0.0803184\pi$
$683$			8912.66i			0.499317i	−0.968334	+	0.249659i	$0.919682\pi$
$684$		0			0
$685$	36299.6			2.02472
$686$		0			0
$687$	14334.2	+	20888.9i	0.796045	+	1.16006i
$688$		0			0
$689$	−8872.57			−0.490592
$690$		0			0
$691$	−11850.1			−0.652387			−0.326193	−	0.945303i	$-0.605766\pi$
$691$	−11850.1			−0.652387			−0.326193	+	0.945303i	$0.605766\pi$
$692$		0			0
$693$	−4342.34	−	1674.13i	−0.238026	−	0.0917675i
$694$		0			0
$695$	7990.94			0.436135
$696$		0			0
$697$		−	15397.7i		−	0.836770i
$698$		0			0
$699$	15237.5	+	22205.3i	0.824513	+	1.20155i
$700$		0			0
$701$	−2889.72			−0.155697			−0.0778483	−	0.996965i	$-0.524805\pi$
$701$	−2889.72			−0.155697			−0.0778483	+	0.996965i	$0.524805\pi$
$702$		0			0
$703$	−14015.4			−0.751923
$704$		0			0
$705$	−6486.41	−	9452.53i	−0.346514	−	0.504969i
$706$		0			0
$707$	28334.3			1.50725
$708$		0			0
$709$		−	22383.1i		−	1.18563i	−0.805337	−	0.592817i	$-0.798016\pi$
$709$		−	22383.1i		−	1.18563i	0.805337	−	0.592817i	$-0.201984\pi$
$710$		0			0
$711$	33715.3	+	12998.5i	1.77837	+	0.685627i
$712$		0			0
$713$	−26105.2	+	13126.9i	−1.37118	+	0.689491i
$714$		0			0
$715$	9445.02			0.494019
$716$		0			0
$717$	16595.7	+	24184.6i	0.864403	+	1.25968i
$718$		0			0
$719$			4058.05i			0.210486i	0.994447	+	0.105243i	$0.0335621\pi$
$719$			4058.05i			0.210486i	−0.994447	+	0.105243i	$0.966438\pi$
$720$		0			0
$721$	−558.635			−0.0288553
$722$		0			0
$723$	−21290.7	−	31026.6i	−1.09517	−	1.59598i
$724$		0			0
$725$			2955.24i			0.151386i
$726$		0			0
$727$			13565.8i			0.692062i	0.938223	+	0.346031i	$0.112471\pi$
$727$			13565.8i			0.692062i	−0.938223	+	0.346031i	$0.887529\pi$
$728$		0			0
$729$	−17576.6	+	8859.17i	−0.892982	+	0.450093i
$730$		0			0
$731$		−	1347.30i		−	0.0681692i
$732$		0			0
$733$			4576.75i			0.230622i	0.993329	+	0.115311i	$0.0367865\pi$
$733$			4576.75i			0.230622i	−0.993329	+	0.115311i	$0.963214\pi$
$734$		0			0
$735$	310.634	−	213.160i	0.0155890	−	0.0106973i
$736$		0			0
$737$		−	6069.31i		−	0.303346i
$738$		0			0
$739$	30960.5			1.54114			0.770569	−	0.637356i	$-0.219972\pi$
$739$	30960.5			1.54114			0.770569	+	0.637356i	$0.219972\pi$
$740$		0			0
$741$	−26188.5	−	38164.1i	−1.29833	−	1.89203i
$742$		0			0
$743$	16597.1			0.819498			0.409749	−	0.912198i	$-0.365616\pi$
$743$	16597.1			0.819498			0.409749	+	0.912198i	$0.365616\pi$
$744$		0			0
$745$	14014.3			0.689186
$746$		0			0
$747$	−1227.00	+	3182.57i	−0.0600983	+	0.155882i
$748$		0			0
$749$		−	643.556i		−	0.0313952i
$750$		0			0
$751$		−	21526.7i		−	1.04597i	−0.852343	−	0.522983i	$-0.824819\pi$
$751$		−	21526.7i		−	1.04597i	0.852343	−	0.522983i	$-0.175181\pi$
$752$		0			0
$753$	31726.4	−	21770.9i	1.53542	−	1.05362i
$754$		0			0
$755$	31972.7			1.54120
$756$		0			0
$757$		−	30152.8i		−	1.44772i	−0.689949	−	0.723858i	$-0.742367\pi$
$757$		−	30152.8i		−	1.44772i	0.689949	−	0.723858i	$-0.257633\pi$
$758$		0			0
$759$	2595.83	−	4712.76i	0.124140	−	0.225378i
$760$		0			0
$761$		−	25726.3i		−	1.22546i	−0.790291	−	0.612732i	$-0.790070\pi$
$761$		−	25726.3i		−	1.22546i	0.790291	−	0.612732i	$-0.209930\pi$
$762$		0			0
$763$	643.852			0.0305492
$764$		0			0
$765$	−11126.6	+	28860.1i	−0.525861	+	1.36397i
$766$		0			0
$767$			8818.45i			0.415145i
$768$		0			0
$769$		−	12938.2i		−	0.606716i	−0.952877	−	0.303358i	$-0.901892\pi$
$769$		−	12938.2i		−	0.606716i	0.952877	−	0.303358i	$-0.0981078\pi$
$770$		0			0
$771$	739.008	+	1076.94i	0.0345197	+	0.0503050i
$772$		0			0
$773$	−18029.6			−0.838911			−0.419455	−	0.907776i	$-0.637779\pi$
$773$	−18029.6			−0.838911			−0.419455	+	0.907776i	$0.637779\pi$
$774$		0			0
$775$	−7603.21			−0.352407
$776$		0			0
$777$	10045.6	−	6893.37i	0.463813	−	0.318273i
$778$		0			0
$779$	18289.7			0.841201
$780$		0			0
$781$		−	2211.72i		−	0.101334i
$782$		0			0
$783$	−14053.6	+	3341.50i	−0.641422	+	0.152510i
$784$		0			0
$785$		−	42860.5i		−	1.94873i
$786$		0			0
$787$			7131.49i			0.323012i	0.986872	+	0.161506i	$0.0516351\pi$
$787$			7131.49i			0.323012i	−0.986872	+	0.161506i	$0.948365\pi$
$788$		0			0
$789$	10903.4	−	7481.98i	0.491977	−	0.337599i
$790$		0			0
$791$			14487.0i			0.651198i
$792$		0			0
$793$		−	9326.42i		−	0.417643i
$794$		0			0
$795$	−5807.07	+	3984.87i	−0.259064	+	0.177772i
$796$		0			0
$797$	26586.0			1.18159			0.590794	−	0.806822i	$-0.298815\pi$
$797$	26586.0			1.18159			0.590794	+	0.806822i	$0.298815\pi$
$798$		0			0
$799$		−	16443.8i		−	0.728085i
$800$		0			0
$801$	−11437.0	+	29665.1i	−0.504502	+	1.30857i
$802$		0			0
$803$	2053.32			0.0902368
$804$		0			0
$805$	−11280.5	−	22433.3i	−0.493896	−	0.982200i
$806$		0			0
$807$	−8013.50	−	11677.9i	−0.349552	−	0.509396i
$808$		0			0
$809$			44879.9i			1.95042i	0.221275	+	0.975212i	$0.428978\pi$
$809$			44879.9i			1.95042i	−0.221275	+	0.975212i	$0.571022\pi$
$810$		0			0
$811$	−36002.8			−1.55885			−0.779425	−	0.626495i	$-0.784489\pi$
$811$	−36002.8			−1.55885			−0.779425	+	0.626495i	$0.784489\pi$
$812$		0			0
$813$	−11841.2	+	8125.51i	−0.510809	+	0.350522i
$814$		0			0
$815$	−32974.0			−1.41721
$816$		0			0
$817$	1600.35			0.0685301
$818$		0			0
$819$	37541.3	+	14473.5i	1.60171	+	0.617517i
$820$		0			0
$821$			35228.8i			1.49756i	0.662820	+	0.748779i	$0.269360\pi$
$821$			35228.8i			1.49756i	−0.662820	+	0.748779i	$0.730640\pi$
$822$		0			0
$823$	−13778.8			−0.583596			−0.291798	−	0.956480i	$-0.594253\pi$
$823$	−13778.8			−0.583596			−0.291798	+	0.956480i	$0.594253\pi$
$824$		0			0
$825$	1154.36	−	792.133i	0.0487148	−	0.0334285i
$826$		0			0
$827$	4860.54			0.204374			0.102187	−	0.994765i	$-0.467416\pi$
$827$	4860.54			0.204374			0.102187	+	0.994765i	$0.467416\pi$
$828$		0			0
$829$	4879.34			0.204423			0.102211	−	0.994763i	$-0.467408\pi$
$829$	4879.34			0.204423			0.102211	+	0.994763i	$0.467408\pi$
$830$		0			0
$831$	27442.1	−	18831.0i	1.14555	−	0.786090i
$832$		0			0
$833$	540.384			0.0224768
$834$		0			0
$835$			14234.5i			0.589947i
$836$		0			0
$837$	−8596.97	−	36156.8i	−0.355024	−	1.49315i
$838$		0			0
$839$	−2364.12			−0.0972808			−0.0486404	−	0.998816i	$-0.515489\pi$
$839$	−2364.12			−0.0972808			−0.0486404	+	0.998816i	$0.515489\pi$
$840$		0			0
$841$	13787.6			0.565319
$842$		0			0
$843$	−18539.2	+	12721.8i	−0.757442	+	0.519764i
$844$		0			0
$845$	−54418.5			−2.21545
$846$		0			0
$847$		−	22821.4i		−	0.925799i
$848$		0			0
$849$	−2577.33	−	3755.90i	−0.104186	−	0.151828i
$850$		0			0
$851$	6327.70	+	12583.8i	0.254889	+	0.506893i
$852$		0			0
$853$	36712.9			1.47365			0.736827	−	0.676082i	$-0.236323\pi$
$853$	36712.9			1.47365			0.736827	+	0.676082i	$0.236323\pi$
$854$		0			0
$855$	−34280.6	−	13216.4i	−1.37120	−	0.528646i
$856$		0			0
$857$			29902.0i			1.19187i	0.803032	+	0.595935i	$0.203219\pi$
$857$			29902.0i			1.19187i	−0.803032	+	0.595935i	$0.796781\pi$
$858$		0			0
$859$	−840.472			−0.0333836			−0.0166918	−	0.999861i	$-0.505313\pi$
$859$	−840.472			−0.0333836			−0.0166918	+	0.999861i	$0.505313\pi$
$860$		0			0
$861$	−13109.1	+	8995.61i	−0.518883	+	0.356062i
$862$		0			0
$863$		−	5568.66i		−	0.219652i	−0.993951	−	0.109826i	$-0.964971\pi$
$863$		−	5568.66i		−	0.219652i	0.993951	−	0.109826i	$-0.0350293\pi$
$864$		0			0
$865$		−	38304.5i		−	1.50566i
$866$		0			0
$867$	−15532.5	+	10658.5i	−0.608432	+	0.417512i
$868$		0			0
$869$		−	12563.0i		−	0.490415i
$870$		0			0
$871$			52471.7i			2.04126i
$872$		0			0
$873$	9028.00	+	3480.62i	0.350002	+	0.134938i
$874$		0			0
$875$			21921.5i			0.846951i
$876$		0			0
$877$	30311.8			1.16711			0.583556	−	0.812073i	$-0.301661\pi$
$877$	30311.8			1.16711			0.583556	+	0.812073i	$0.301661\pi$
$878$		0			0
$879$	23208.3	−	15925.8i	0.890554	−	0.611106i
$880$		0			0
$881$	442.674			0.0169286			0.00846428	−	0.999964i	$-0.497306\pi$
$881$	442.674			0.0169286			0.00846428	+	0.999964i	$0.497306\pi$
$882$		0			0
$883$	33297.2			1.26901			0.634507	−	0.772917i	$-0.281203\pi$
$883$	33297.2			1.26901			0.634507	+	0.772917i	$0.281203\pi$
$884$		0			0
$885$	3960.56	+	5771.65i	0.150433	+	0.219223i
$886$		0			0
$887$			579.492i			0.0219362i	0.999940	+	0.0109681i	$0.00349133\pi$
$887$			579.492i			0.0219362i	−0.999940	+	0.0109681i	$0.996509\pi$
$888$		0			0
$889$			20908.1i			0.788793i
$890$		0			0
$891$	5072.21	+	4593.86i	0.190713	+	0.172727i
$892$		0			0
$893$	19532.3			0.731940
$894$		0			0
$895$		−	47883.6i		−	1.78835i
$896$		0			0
$897$	−22442.0	+	40743.8i	−0.835359	+	1.51660i
$898$		0			0
$899$		−	27275.3i		−	1.01188i
$900$		0			0
$901$	−10102.1			−0.373529
$902$		0			0
$903$	−1147.05	+	787.118i	−0.0422719	+	0.0290073i
$904$		0			0
$905$		−	8711.42i		−	0.319975i
$906$		0			0
$907$		−	2559.33i		−	0.0936947i	−0.998902	−	0.0468474i	$-0.985083\pi$
$907$		−	2559.33i		−	0.0936947i	0.998902	−	0.0468474i	$-0.0149174\pi$
$908$		0			0
$909$	−38875.2	−	14987.8i	−1.41849	−	0.546880i
$910$		0			0
$911$	−10981.3			−0.399371			−0.199685	−	0.979860i	$-0.563992\pi$
$911$	−10981.3			−0.399371			−0.199685	+	0.979860i	$0.563992\pi$
$912$		0			0
$913$	1185.89			0.0429871
$914$		0			0
$915$	−4188.70	−	6104.11i	−0.151338	−	0.220542i
$916$		0			0
$917$	−20940.7			−0.754115
$918$		0			0
$919$		−	42227.2i		−	1.51572i	−0.652416	−	0.757861i	$-0.726245\pi$
$919$		−	42227.2i		−	1.51572i	0.652416	−	0.757861i	$-0.273755\pi$
$920$		0			0
$921$	24965.4	−	17131.5i	0.893201	−	0.612923i
$922$		0			0
$923$			19121.2i			0.681889i
$924$		0			0
$925$			3665.05i			0.130277i
$926$		0			0
$927$	766.457	+	295.497i	0.0271561	+	0.0104697i
$928$		0			0
$929$		−	19797.1i		−	0.699162i	−0.936906	−	0.349581i	$-0.886324\pi$
$929$		−	19797.1i		−	0.699162i	0.936906	−	0.349581i	$-0.113676\pi$
$930$		0			0
$931$			641.879i			0.0225958i
$932$		0			0
$933$	−9683.59	−	14111.7i	−0.339793	−	0.495174i
$934$		0			0
$935$	10753.9			0.376138
$936$		0			0
$937$			36899.2i			1.28649i	0.765659	+	0.643246i	$0.222413\pi$
$937$			36899.2i			1.28649i	−0.765659	+	0.643246i	$0.777587\pi$
$938$		0			0
$939$	14423.5	+	21019.1i	0.501272	+	0.730494i
$940$		0			0
$941$	−4129.78			−0.143068			−0.0715340	−	0.997438i	$-0.522789\pi$
$941$	−4129.78			−0.143068			−0.0715340	+	0.997438i	$0.522789\pi$
$942$		0			0
$943$	−8257.44	−	16421.4i	−0.285153	−	0.567078i
$944$		0			0
$945$	31071.1	−	7387.74i	1.06957	−	0.254310i
$946$		0			0
$947$			6391.30i			0.219313i	0.993970	+	0.109656i	$0.0349751\pi$
$947$			6391.30i			0.219313i	−0.993970	+	0.109656i	$0.965025\pi$
$948$		0			0
$949$	−17751.8			−0.607217
$950$		0			0
$951$	19150.0	+	27907.0i	0.652978	+	0.951572i
$952$		0			0
$953$	3659.59			0.124392			0.0621961	−	0.998064i	$-0.480190\pi$
$953$	3659.59			0.124392			0.0621961	+	0.998064i	$0.480190\pi$
$954$		0			0
$955$	−23489.9			−0.795933
$956$		0			0
$957$	2841.65	+	4141.08i	0.0959848	+	0.139877i
$958$		0			0
$959$		−	53761.9i		−	1.81029i
$960$		0			0
$961$	40382.4			1.35552
$962$		0			0
$963$	−340.417	+	882.969i	−0.0113913	+	0.0295465i
$964$		0			0
$965$	−28295.0			−0.943884
$966$		0			0
$967$	14635.7			0.486713			0.243356	−	0.969937i	$-0.421752\pi$
$967$	14635.7			0.486713			0.243356	+	0.969937i	$0.421752\pi$
$968$		0			0
$969$	−29817.6	−	43452.7i	−0.988524	−	1.44056i
$970$		0			0
$971$	2016.09			0.0666316			0.0333158	−	0.999445i	$-0.489393\pi$
$971$	2016.09			0.0666316			0.0333158	+	0.999445i	$0.489393\pi$
$972$		0			0
$973$		−	11835.1i		−	0.389943i
$974$		0			0
$975$	−9979.94	+	6848.33i	−0.327809	+	0.224946i
$976$		0			0
$977$	684.955			0.0224295			0.0112148	−	0.999937i	$-0.496430\pi$
$977$	684.955			0.0224295			0.0112148	+	0.999937i	$0.496430\pi$
$978$		0			0
$979$	11053.8			0.360860
$980$		0			0
$981$	−883.376	−	340.573i	−0.0287503	−	0.0110843i
$982$		0			0
$983$	−37483.0			−1.21620			−0.608099	−	0.793861i	$-0.708068\pi$
$983$	−37483.0			−1.21620			−0.608099	+	0.793861i	$0.708068\pi$
$984$		0			0
$985$		−	22054.2i		−	0.713405i
$986$		0			0
$987$	−13999.8	+	9606.78i	−0.451487	+	0.309815i
$988$		0			0
$989$	−722.527	−	1436.87i	−0.0232306	−	0.0461981i
$990$		0			0
$991$	32662.5			1.04698			0.523490	−	0.852032i	$-0.324630\pi$
$991$	32662.5			1.04698			0.523490	+	0.852032i	$0.324630\pi$
$992$		0			0
$993$	5703.08	−	3913.51i	0.182258	−	0.125067i
$994$		0			0
$995$			57304.9i			1.82582i
$996$		0			0
$997$	28008.9			0.889719			0.444860	−	0.895600i	$-0.353254\pi$
$997$	28008.9			0.889719			0.444860	+	0.895600i	$0.353254\pi$
$998$		0			0
$999$	−17429.0	+	4144.08i	−0.551982	+	0.131244i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.m.a.137.15	yes	72
3.2	odd	2	inner	552.4.m.a.137.14	yes	72
23.22	odd	2	inner	552.4.m.a.137.16	yes	72
69.68	even	2	inner	552.4.m.a.137.13	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.m.a.137.13	✓	72	69.68	even	2	inner
552.4.m.a.137.14	yes	72	3.2	odd	2	inner
552.4.m.a.137.15	yes	72	1.1	even	1	trivial
552.4.m.a.137.16	yes	72	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.28443 + 2.94001i) q^{3} -12.3977 q^{5} +18.3617i q^{7} +(9.71264 - 25.1926i) q^{9} -9.38725 q^{11} +81.1567 q^{13} +(53.1169 - 36.4493i) q^{15} +92.4031 q^{17} +109.758i q^{19} +(-53.9836 - 78.6694i) q^{21} +(98.5466 - 49.5538i) q^{23} +28.7019 q^{25} +(32.4533 + 136.491i) q^{27} +102.963i q^{29} -264.903 q^{31} +(40.2190 - 27.5986i) q^{33} -227.642i q^{35} +127.694i q^{37} +(-347.710 + 238.602i) q^{39} -166.636i q^{41} -14.5807i q^{43} +(-120.414 + 312.329i) q^{45} -177.957i q^{47} +5.84812 q^{49} +(-395.894 + 271.666i) q^{51} -109.326 q^{53} +116.380 q^{55} +(-322.691 - 470.251i) q^{57} +108.660i q^{59} -114.919i q^{61} +(462.578 + 178.341i) q^{63} -1006.15 q^{65} +646.548i q^{67} +(-276.527 + 502.038i) q^{69} +235.609i q^{71} -218.735 q^{73} +(-122.971 + 84.3839i) q^{75} -172.366i q^{77} +1338.31i q^{79} +(-540.329 - 489.372i) q^{81} -126.330 q^{83} -1145.58 q^{85} +(-302.713 - 441.139i) q^{87} -1177.54 q^{89} +1490.18i q^{91} +(1134.96 - 778.817i) q^{93} -1360.75i q^{95} +358.360i q^{97} +(-91.1750 + 236.489i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 28 q^{9} + 1872 q^{25} - 444 q^{27} - 216 q^{31} - 68 q^{39} - 4200 q^{49} - 576 q^{55} + 1376 q^{69} - 384 q^{73} - 3592 q^{75} - 252 q^{81} + 3480 q^{85} - 412 q^{87} + 780 q^{93}+O(q^{100}) \) 72 * q + 28 * q^9 + 1872 * q^25 - 444 * q^27 - 216 * q^31 - 68 * q^39 - 4200 * q^49 - 576 * q^55 + 1376 * q^69 - 384 * q^73 - 3592 * q^75 - 252 * q^81 + 3480 * q^85 - 412 * q^87 + 780 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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