LMFDB - Embedded newform 624.4.n.d.623.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(623,624)] 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("624.623"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [56,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			623.13
Character	$\chi$	$=$	624.623
Dual form			624.4.n.d.623.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.53056 - 3.81250i) q^{3} -9.28291 q^{5} -16.8768 q^{7} +(-2.07031 + 26.9205i) q^{9} +43.0842i q^{11} +(-42.8921 + 18.9015i) q^{13} +(32.7738 + 35.3911i) q^{15} +4.06776i q^{17} +76.7301 q^{19} +(59.5845 + 64.3428i) q^{21} +124.564 q^{23} -38.8276 q^{25} +(109.944 - 87.1514i) q^{27} -19.4422i q^{29} -219.497 q^{31} +(164.259 - 152.111i) q^{33} +156.666 q^{35} +69.3757i q^{37} +(223.495 + 96.7932i) q^{39} -252.460 q^{41} -246.736i q^{43} +(19.2185 - 249.901i) q^{45} +455.407i q^{47} -58.1740 q^{49} +(15.5083 - 14.3615i) q^{51} -302.539i q^{53} -399.947i q^{55} +(-270.900 - 292.534i) q^{57} -422.412i q^{59} -703.925 q^{61} +(34.9403 - 454.332i) q^{63} +(398.163 - 175.461i) q^{65} +839.371 q^{67} +(-439.781 - 474.901i) q^{69} +698.752i q^{71} -601.513i q^{73} +(137.083 + 148.030i) q^{75} -727.124i q^{77} -182.752i q^{79} +(-720.428 - 111.468i) q^{81} +66.1415i q^{83} -37.7606i q^{85} +(-74.1234 + 68.6419i) q^{87} +888.131 q^{89} +(723.881 - 318.997i) q^{91} +(774.948 + 836.833i) q^{93} -712.279 q^{95} +874.867i q^{97} +(-1159.85 - 89.1979i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 64 q^{9} + 8 q^{13} + 1208 q^{25} + 4680 q^{49} + 1616 q^{61} + 480 q^{69} + 1568 q^{81}+O(q^{100}) $ 56 * q - 64 * q^9 + 8 * q^13 + 1208 * q^25 + 4680 * q^49 + 1616 * q^61 + 480 * q^69 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−3.53056	−	3.81250i	−0.679456	−	0.733716i
$3$	−3.53056	−	3.81250i	−0.679456	−	0.733716i
$4$		0			0
$5$	−9.28291			−0.830289			−0.415144	−	0.909756i	$-0.636269\pi$
$5$	−9.28291			−0.830289			−0.415144	+	0.909756i	$0.636269\pi$
$6$		0			0
$7$	−16.8768			−0.911261			−0.455630	−	0.890169i	$-0.650586\pi$
$7$	−16.8768			−0.911261			−0.455630	+	0.890169i	$0.650586\pi$
$8$		0			0
$9$	−2.07031	+	26.9205i	−0.0766783	+	0.997056i
$10$		0			0
$11$			43.0842i			1.18094i	0.807058	+	0.590472i	$0.201058\pi$
$11$			43.0842i			1.18094i	−0.807058	+	0.590472i	$0.798942\pi$
$12$		0			0
$13$	−42.8921	+	18.9015i	−0.915087	+	0.403257i
$13$	−42.8921	+	18.9015i	−0.915087	+	0.403257i
$14$		0			0
$15$	32.7738	+	35.3911i	0.564145	+	0.609196i
$16$		0			0
$17$			4.06776i			0.0580339i	0.999579	+	0.0290170i	$0.00923768\pi$
$17$			4.06776i			0.0580339i	−0.999579	+	0.0290170i	$0.990762\pi$
$18$		0			0
$19$	76.7301			0.926479			0.463239	−	0.886233i	$-0.346687\pi$
$19$	76.7301			0.926479			0.463239	+	0.886233i	$0.346687\pi$
$20$		0			0
$21$	59.5845	+	64.3428i	0.619162	+	0.668607i
$22$		0			0
$23$	124.564			1.12928			0.564640	−	0.825337i	$-0.309015\pi$
$23$	124.564			1.12928			0.564640	+	0.825337i	$0.309015\pi$
$24$		0			0
$25$	−38.8276			−0.310621
$26$		0			0
$27$	109.944	−	87.1514i	0.783655	−	0.621196i
$28$		0			0
$29$		−	19.4422i		−	0.124494i	−0.998061	−	0.0622470i	$-0.980173\pi$
$29$		−	19.4422i		−	0.124494i	0.998061	−	0.0622470i	$-0.0198267\pi$
$30$		0			0
$31$	−219.497			−1.27171			−0.635853	−	0.771810i	$-0.719351\pi$
$31$	−219.497			−1.27171			−0.635853	+	0.771810i	$0.719351\pi$
$32$		0			0
$33$	164.259	−	152.111i	0.866478	−	0.802400i
$34$		0			0
$35$	156.666			0.756610
$36$		0			0
$37$			69.3757i			0.308251i	0.988051	+	0.154126i	$0.0492560\pi$
$37$			69.3757i			0.308251i	−0.988051	+	0.154126i	$0.950744\pi$
$38$		0			0
$39$	223.495	+	96.7932i	0.917638	+	0.397418i
$40$		0			0
$41$	−252.460			−0.961651			−0.480826	−	0.876816i	$-0.659663\pi$
$41$	−252.460			−0.961651			−0.480826	+	0.876816i	$0.659663\pi$
$42$		0			0
$43$		−	246.736i		−	0.875044i	−0.899208	−	0.437522i	$-0.855856\pi$
$43$		−	246.736i		−	0.875044i	0.899208	−	0.437522i	$-0.144144\pi$
$44$		0			0
$45$	19.2185	−	249.901i	0.0636651	−	0.827844i
$46$		0			0
$47$			455.407i			1.41336i	0.707533	+	0.706681i	$0.249808\pi$
$47$			455.407i			1.41336i	−0.707533	+	0.706681i	$0.750192\pi$
$48$		0			0
$49$	−58.1740			−0.169603
$50$		0			0
$51$	15.5083	−	14.3615i	0.0425804	−	0.0394315i
$52$		0			0
$53$		−	302.539i		−	0.784094i	−0.919945	−	0.392047i	$-0.871767\pi$
$53$		−	302.539i		−	0.784094i	0.919945	−	0.392047i	$-0.128233\pi$
$54$		0			0
$55$		−	399.947i		−	0.980524i
$56$		0			0
$57$	−270.900	−	292.534i	−0.629502	−	0.679772i
$58$		0			0
$59$		−	422.412i		−	0.932090i	−0.884761	−	0.466045i	$-0.845678\pi$
$59$		−	422.412i		−	0.932090i	0.884761	−	0.466045i	$-0.154322\pi$
$60$		0			0
$61$	−703.925			−1.47751			−0.738757	−	0.673972i	$-0.764587\pi$
$61$	−703.925			−1.47751			−0.738757	+	0.673972i	$0.764587\pi$
$62$		0			0
$63$	34.9403	−	454.332i	0.0698740	−	0.908578i
$64$		0			0
$65$	398.163	−	175.461i	0.759786	−	0.334820i
$66$		0			0
$67$	839.371			1.53053			0.765265	−	0.643715i	$-0.222608\pi$
$67$	839.371			1.53053			0.765265	+	0.643715i	$0.222608\pi$
$68$		0			0
$69$	−439.781	−	474.901i	−0.767297	−	0.828571i
$70$		0			0
$71$			698.752i			1.16798i	0.811761	+	0.583990i	$0.198509\pi$
$71$			698.752i			1.16798i	−0.811761	+	0.583990i	$0.801491\pi$
$72$		0			0
$73$		−	601.513i		−	0.964408i	−0.876059	−	0.482204i	$-0.839836\pi$
$73$		−	601.513i		−	0.964408i	0.876059	−	0.482204i	$-0.160164\pi$
$74$		0			0
$75$	137.083	+	148.030i	0.211053	+	0.227908i
$76$		0			0
$77$		−	727.124i		−	1.07615i
$78$		0			0
$79$		−	182.752i		−	0.260269i	−0.991496	−	0.130135i	$-0.958459\pi$
$79$		−	182.752i		−	0.260269i	0.991496	−	0.130135i	$-0.0415409\pi$
$80$		0			0
$81$	−720.428	−	111.468i	−0.988241	−	0.152905i
$82$		0			0
$83$			66.1415i			0.0874695i	0.999043	+	0.0437347i	$0.0139256\pi$
$83$			66.1415i			0.0874695i	−0.999043	+	0.0437347i	$0.986074\pi$
$84$		0			0
$85$		−	37.7606i		−	0.0481849i
$86$		0			0
$87$	−74.1234	+	68.6419i	−0.0913433	+	0.0845883i
$88$		0			0
$89$	888.131			1.05777			0.528886	−	0.848693i	$-0.322610\pi$
$89$	888.131			1.05777			0.528886	+	0.848693i	$0.322610\pi$
$90$		0			0
$91$	723.881	−	318.997i	0.833883	−	0.367472i
$92$		0			0
$93$	774.948	+	836.833i	0.864068	+	0.933071i
$94$		0			0
$95$	−712.279			−0.769245
$96$		0			0
$97$			874.867i			0.915766i	0.889013	+	0.457883i	$0.151392\pi$
$97$			874.867i			0.915766i	−0.889013	+	0.457883i	$0.848608\pi$
$98$		0			0
$99$	−1159.85	−	89.1979i	−1.17747	−	0.0905528i
$100$		0			0
$101$		−	1927.31i		−	1.89876i	−0.314129	−	0.949380i	$-0.601713\pi$
$101$		−	1927.31i		−	1.89876i	0.314129	−	0.949380i	$-0.398287\pi$
$102$		0			0
$103$		−	613.904i		−	0.587279i	−0.955916	−	0.293640i	$-0.905133\pi$
$103$		−	613.904i		−	0.587279i	0.955916	−	0.293640i	$-0.0948666\pi$
$104$		0			0
$105$	−553.117	−	597.288i	−0.514083	−	0.555137i
$106$		0			0
$107$	−204.617			−0.184870			−0.0924349	−	0.995719i	$-0.529465\pi$
$107$	−204.617			−0.184870			−0.0924349	+	0.995719i	$0.529465\pi$
$108$		0			0
$109$		−	1671.02i		−	1.46839i	−0.678937	−	0.734197i	$-0.737559\pi$
$109$		−	1671.02i		−	1.46839i	0.678937	−	0.734197i	$-0.262441\pi$
$110$		0			0
$111$	264.495	−	244.935i	0.226169	−	0.209443i
$112$		0			0
$113$			760.259i			0.632913i	0.948607	+	0.316456i	$0.102493\pi$
$113$			760.259i			0.632913i	−0.948607	+	0.316456i	$0.897507\pi$
$114$		0			0
$115$	−1156.32			−0.937628
$116$		0			0
$117$	−420.039	−	1193.81i	−0.331902	−	0.943314i
$118$		0			0
$119$		−	68.6507i		−	0.0528841i
$120$		0			0
$121$	−525.251			−0.394629
$122$		0			0
$123$	891.326	+	962.505i	0.653400	+	0.705579i
$124$		0			0
$125$	1520.80			1.08819
$126$		0			0
$127$		−	437.334i		−	0.305568i	−0.988260	−	0.152784i	$-0.951176\pi$
$127$		−	437.334i		−	0.305568i	0.988260	−	0.152784i	$-0.0488238\pi$
$128$		0			0
$129$	−940.682	+	871.116i	−0.642034	+	0.594554i
$130$		0			0
$131$	1941.72			1.29503			0.647514	−	0.762053i	$-0.275809\pi$
$131$	1941.72			1.29503			0.647514	+	0.762053i	$0.275809\pi$
$132$		0			0
$133$	−1294.96			−0.844264
$134$		0			0
$135$	−1020.60	+	809.018i	−0.650660	+	0.515772i
$136$		0			0
$137$	91.7739			0.0572319			0.0286160	−	0.999590i	$-0.490890\pi$
$137$	91.7739			0.0572319			0.0286160	+	0.999590i	$0.490890\pi$
$138$		0			0
$139$		−	2394.64i		−	1.46123i	−0.682791	−	0.730614i	$-0.739234\pi$
$139$		−	2394.64i		−	1.46123i	0.682791	−	0.730614i	$-0.260766\pi$
$140$		0			0
$141$	1736.24	−	1607.84i	1.03701	−	0.960317i
$142$		0			0
$143$	−814.358	−	1847.97i	−0.476224	−	1.08067i
$144$		0			0
$145$			180.480i			0.103366i
$146$		0			0
$147$	205.387	+	221.788i	0.115238	+	0.124441i
$148$		0			0
$149$	−1308.92			−0.719669			−0.359835	−	0.933016i	$-0.617167\pi$
$149$	−1308.92			−0.719669			−0.359835	+	0.933016i	$0.617167\pi$
$150$		0			0
$151$	−130.432			−0.0702940			−0.0351470	−	0.999382i	$-0.511190\pi$
$151$	−130.432			−0.0702940			−0.0351470	+	0.999382i	$0.511190\pi$
$152$		0			0
$153$	−109.506	−	8.42154i	−0.0578631	−	0.00444994i
$154$		0			0
$155$	2037.57			1.05588
$156$		0			0
$157$	699.252			0.355455			0.177727	−	0.984080i	$-0.443125\pi$
$157$	699.252			0.355455			0.177727	+	0.984080i	$0.443125\pi$
$158$		0			0
$159$	−1153.43	+	1068.13i	−0.575302	+	0.532758i
$160$		0			0
$161$	−2102.24			−1.02907
$162$		0			0
$163$	3824.04			1.83756			0.918780	−	0.394770i	$-0.129176\pi$
$163$	3824.04			1.83756			0.918780	+	0.394770i	$0.129176\pi$
$164$		0			0
$165$	−1524.80	+	1412.04i	−0.719426	+	0.666223i
$166$		0			0
$167$		−	304.441i		−	0.141068i	−0.997509	−	0.0705339i	$-0.977530\pi$
$167$		−	304.441i		−	0.141068i	0.997509	−	0.0705339i	$-0.0224703\pi$
$168$		0			0
$169$	1482.46	−	1621.45i	0.674768	−	0.738030i
$170$		0			0
$171$	−158.856	+	2065.61i	−0.0710408	+	0.923751i
$172$		0			0
$173$			527.690i			0.231905i	0.993255	+	0.115952i	$0.0369920\pi$
$173$			527.690i			0.231905i	−0.993255	+	0.115952i	$0.963008\pi$
$174$		0			0
$175$	655.286			0.283057
$176$		0			0
$177$	−1610.44	+	1491.35i	−0.683889	+	0.633314i
$178$		0			0
$179$	2143.41			0.895007			0.447504	−	0.894282i	$-0.352313\pi$
$179$	2143.41			0.895007			0.447504	+	0.894282i	$0.352313\pi$
$180$		0			0
$181$	1860.14			0.763885			0.381942	−	0.924186i	$-0.375255\pi$
$181$	1860.14			0.763885			0.381942	+	0.924186i	$0.375255\pi$
$182$		0			0
$183$	2485.25	+	2683.71i	1.00391	+	1.08408i
$184$		0			0
$185$		−	644.008i		−	0.255937i
$186$		0			0
$187$	−175.256			−0.0685348
$188$		0			0
$189$	−1855.50	+	1470.84i	−0.714115	+	0.566071i
$190$		0			0
$191$	240.884			0.0912551			0.0456276	−	0.998959i	$-0.485471\pi$
$191$	240.884			0.0912551			0.0456276	+	0.998959i	$0.485471\pi$
$192$		0			0
$193$			4404.98i			1.64289i	0.570290	+	0.821443i	$0.306831\pi$
$193$			4404.98i			1.64289i	−0.570290	+	0.821443i	$0.693169\pi$
$194$		0			0
$195$	−2074.68	−	898.522i	−0.761904	−	0.329972i
$196$		0			0
$197$	844.824			0.305539			0.152770	−	0.988262i	$-0.451181\pi$
$197$	844.824			0.305539			0.152770	+	0.988262i	$0.451181\pi$
$198$		0			0
$199$		−	263.619i		−	0.0939070i	−0.998897	−	0.0469535i	$-0.985049\pi$
$199$		−	263.619i		−	0.0939070i	0.998897	−	0.0469535i	$-0.0149513\pi$
$200$		0			0
$201$	−2963.45	−	3200.10i	−1.03993	−	1.12297i
$202$		0			0
$203$			328.122i			0.113447i
$204$		0			0
$205$	2343.57			0.798448
$206$		0			0
$207$	−257.887	+	3353.33i	−0.0865913	+	1.12596i
$208$		0			0
$209$			3305.86i			1.09412i
$210$		0			0
$211$		−	3550.81i		−	1.15852i	−0.815143	−	0.579260i	$-0.803341\pi$
$211$		−	3550.81i		−	1.15852i	0.815143	−	0.579260i	$-0.196659\pi$
$212$		0			0
$213$	2663.99	−	2466.98i	0.856966	−	0.793591i
$214$		0			0
$215$			2290.43i			0.726539i
$216$		0			0
$217$	3704.41			1.15886
$218$		0			0
$219$	−2293.27	+	2123.68i	−0.707602	+	0.655273i
$220$		0			0
$221$	−76.8869	−	174.475i	−0.0234026	−	0.0531061i
$222$		0			0
$223$	1532.84			0.460298			0.230149	−	0.973155i	$-0.426079\pi$
$223$	1532.84			0.460298			0.230149	+	0.973155i	$0.426079\pi$
$224$		0			0
$225$	80.3854	−	1045.26i	0.0238179	−	0.309706i
$226$		0			0
$227$			3596.28i			1.05151i	0.850635	+	0.525757i	$0.176218\pi$
$227$			3596.28i			1.05151i	−0.850635	+	0.525757i	$0.823782\pi$
$228$		0			0
$229$			3334.26i			0.962158i	0.876677	+	0.481079i	$0.159755\pi$
$229$			3334.26i			0.962158i	−0.876677	+	0.481079i	$0.840245\pi$
$230$		0			0
$231$	−2772.16	+	2567.15i	−0.789587	+	0.731196i
$232$		0			0
$233$			2980.15i			0.837923i	0.908004	+	0.418961i	$0.137606\pi$
$233$			2980.15i			0.837923i	−0.908004	+	0.418961i	$0.862394\pi$
$234$		0			0
$235$		−	4227.50i		−	1.17350i
$236$		0			0
$237$	−696.744	+	645.218i	−0.190964	+	0.176841i
$238$		0			0
$239$		−	6919.06i		−	1.87262i	−0.351172	−	0.936311i	$-0.614217\pi$
$239$		−	6919.06i		−	1.87262i	0.351172	−	0.936311i	$-0.385783\pi$
$240$		0			0
$241$		−	2963.76i		−	0.792168i	−0.918214	−	0.396084i	$-0.870369\pi$
$241$		−	2963.76i		−	0.792168i	0.918214	−	0.396084i	$-0.129631\pi$
$242$		0			0
$243$	2118.54	+	3140.17i	0.559278	+	0.828980i
$244$		0			0
$245$	540.024			0.140820
$246$		0			0
$247$	−3291.12	+	1450.32i	−0.847809	+	0.373609i
$248$		0			0
$249$	252.164	−	233.516i	0.0641778	−	0.0594317i
$250$		0			0
$251$	−7220.24			−1.81569			−0.907843	−	0.419309i	$-0.862272\pi$
$251$	−7220.24			−1.81569			−0.907843	+	0.419309i	$0.862272\pi$
$252$		0			0
$253$			5366.76i			1.33362i
$254$		0			0
$255$	−143.962	+	133.316i	−0.0353540	+	0.0327395i
$256$		0			0
$257$			690.995i			0.167716i	0.996478	+	0.0838582i	$0.0267243\pi$
$257$			690.995i			0.167716i	−0.996478	+	0.0838582i	$0.973276\pi$
$258$		0			0
$259$		−	1170.84i		−	0.280897i
$260$		0			0
$261$	523.394	+	40.2515i	0.124128	+	0.00954599i
$262$		0			0
$263$	−4959.65			−1.16283			−0.581416	−	0.813606i	$-0.697501\pi$
$263$	−4959.65			−1.16283			−0.581416	+	0.813606i	$0.697501\pi$
$264$		0			0
$265$			2808.45i			0.651024i
$266$		0			0
$267$	−3135.60	−	3386.00i	−0.718710	−	0.776104i
$268$		0			0
$269$			6904.63i			1.56499i	0.622656	+	0.782496i	$0.286054\pi$
$269$			6904.63i			1.56499i	−0.622656	+	0.782496i	$0.713946\pi$
$270$		0			0
$271$	−3639.53			−0.815814			−0.407907	−	0.913024i	$-0.633741\pi$
$271$	−3639.53			−0.815814			−0.407907	+	0.913024i	$0.633741\pi$
$272$		0			0
$273$	−3771.88	−	1633.56i	−0.836207	−	0.362152i
$274$		0			0
$275$		−	1672.86i		−	0.366826i
$276$		0			0
$277$	3926.59			0.851719			0.425859	−	0.904789i	$-0.359972\pi$
$277$	3926.59			0.851719			0.425859	+	0.904789i	$0.359972\pi$
$278$		0			0
$279$	454.428	−	5908.98i	0.0975122	−	1.26796i
$280$		0			0
$281$	6738.55			1.43056			0.715282	−	0.698836i	$-0.246298\pi$
$281$	6738.55			1.43056			0.715282	+	0.698836i	$0.246298\pi$
$282$		0			0
$283$		−	1725.89i		−	0.362522i	−0.983435	−	0.181261i	$-0.941982\pi$
$283$		−	1725.89i		−	0.362522i	0.983435	−	0.181261i	$-0.0580179\pi$
$284$		0			0
$285$	2514.74	+	2715.56i	0.522668	+	0.564407i
$286$		0			0
$287$	4260.72			0.876315
$288$		0			0
$289$	4896.45			0.996632
$290$		0			0
$291$	3335.43	−	3088.77i	0.671912	−	0.622223i
$292$		0			0
$293$	−8083.08			−1.61167			−0.805833	−	0.592142i	$-0.798282\pi$
$293$	−8083.08			−1.61167			−0.805833	+	0.592142i	$0.798282\pi$
$294$		0			0
$295$			3921.21i			0.773904i
$296$		0			0
$297$	3754.85	+	4736.84i	0.733598	+	0.925453i
$298$		0			0
$299$	−5342.82	+	2354.45i	−1.03339	+	0.455390i
$300$		0			0
$301$			4164.11i			0.797394i
$302$		0			0
$303$	−7347.88	+	6804.49i	−1.39315	+	1.29012i
$304$		0			0
$305$	6534.47			1.22676
$306$		0			0
$307$	2965.37			0.551278			0.275639	−	0.961261i	$-0.411111\pi$
$307$	2965.37			0.551278			0.275639	+	0.961261i	$0.411111\pi$
$308$		0			0
$309$	−2340.51	+	2167.42i	−0.430896	+	0.399031i
$310$		0			0
$311$	−5347.16			−0.974950			−0.487475	−	0.873137i	$-0.662082\pi$
$311$	−5347.16			−0.974950			−0.487475	+	0.873137i	$0.662082\pi$
$312$		0			0
$313$	1308.54			0.236303			0.118151	−	0.992996i	$-0.462303\pi$
$313$	1308.54			0.236303			0.118151	+	0.992996i	$0.462303\pi$
$314$		0			0
$315$	−324.347	+	4217.52i	−0.0580155	+	0.754382i
$316$		0			0
$317$	−8882.61			−1.57381			−0.786904	−	0.617076i	$-0.788317\pi$
$317$	−8882.61			−1.57381			−0.786904	+	0.617076i	$0.788317\pi$
$318$		0			0
$319$	837.653			0.147021
$320$		0			0
$321$	722.412	+	780.102i	0.125611	+	0.135642i
$322$		0			0
$323$			312.120i			0.0537672i
$324$		0			0
$325$	1665.40	−	733.901i	0.284245	−	0.125260i
$326$		0			0
$327$	−6370.77	+	5899.64i	−1.07738	+	0.997709i
$328$		0			0
$329$		−	7685.81i		−	1.28794i
$330$		0			0
$331$	−114.766			−0.0190578			−0.00952890	−	0.999955i	$-0.503033\pi$
$331$	−114.766			−0.0190578			−0.00952890	+	0.999955i	$0.503033\pi$
$332$		0			0
$333$	−1867.63	−	143.629i	−0.307344	−	0.0236362i
$334$		0			0
$335$	−7791.81			−1.27078
$336$		0			0
$337$	6600.09			1.06685			0.533427	−	0.845846i	$-0.320904\pi$
$337$	6600.09			1.06685			0.533427	+	0.845846i	$0.320904\pi$
$338$		0			0
$339$	2898.49	−	2684.14i	0.464378	−	0.430037i
$340$		0			0
$341$		−	9456.87i		−	1.50181i
$342$		0			0
$343$	6770.53			1.06581
$344$		0			0
$345$	4082.45	+	4408.46i	0.637077	+	0.687953i
$346$		0			0
$347$	11146.5			1.72443			0.862215	−	0.506543i	$-0.169077\pi$
$347$	11146.5			1.72443			0.862215	+	0.506543i	$0.169077\pi$
$348$		0			0
$349$		−	7940.69i		−	1.21792i	−0.793200	−	0.608962i	$-0.791586\pi$
$349$		−	7940.69i		−	1.21792i	0.793200	−	0.608962i	$-0.208414\pi$
$350$		0			0
$351$	−3068.43	+	5816.21i	−0.466611	+	0.884463i
$352$		0			0
$353$	−12526.0			−1.88864			−0.944320	−	0.329028i	$-0.893279\pi$
$353$	−12526.0			−1.88864			−0.944320	+	0.329028i	$0.893279\pi$
$354$		0			0
$355$		−	6486.45i		−	0.969760i
$356$		0			0
$357$	−261.731	+	242.375i	−0.0388019	+	0.0359324i
$358$		0			0
$359$			446.881i			0.0656977i	0.999460	+	0.0328488i	$0.0104580\pi$
$359$			446.881i			0.0656977i	−0.999460	+	0.0328488i	$0.989542\pi$
$360$		0			0
$361$	−971.486			−0.141637
$362$		0			0
$363$	1854.43	+	2002.52i	0.268133	+	0.289546i
$364$		0			0
$365$			5583.79i			0.800737i
$366$		0			0
$367$			824.173i			0.117225i	0.998281	+	0.0586124i	$0.0186676\pi$
$367$			824.173i			0.117225i	−0.998281	+	0.0586124i	$0.981332\pi$
$368$		0			0
$369$	522.672	−	6796.36i	0.0737378	−	0.958820i
$370$		0			0
$371$			5105.89i			0.714514i
$372$		0			0
$373$	7900.66			1.09673			0.548365	−	0.836239i	$-0.315250\pi$
$373$	7900.66			1.09673			0.548365	+	0.836239i	$0.315250\pi$
$374$		0			0
$375$	−5369.26	−	5798.04i	−0.739380	−	0.798425i
$376$		0			0
$377$	367.487	+	833.917i	0.0502031	+	0.113923i
$378$		0			0
$379$	−2866.75			−0.388536			−0.194268	−	0.980948i	$-0.562233\pi$
$379$	−2866.75			−0.388536			−0.194268	+	0.980948i	$0.562233\pi$
$380$		0			0
$381$	−1667.33	+	1544.03i	−0.224200	+	0.207620i
$382$		0			0
$383$		−	5271.61i		−	0.703307i	−0.936130	−	0.351654i	$-0.885619\pi$
$383$		−	5271.61i		−	0.703307i	0.936130	−	0.351654i	$-0.114381\pi$
$384$		0			0
$385$			6749.82i			0.893514i
$386$		0			0
$387$	6642.26	+	510.821i	0.872468	+	0.0670969i
$388$		0			0
$389$			9006.25i			1.17387i	0.809635	+	0.586934i	$0.199665\pi$
$389$			9006.25i			1.17387i	−0.809635	+	0.586934i	$0.800335\pi$
$390$		0			0
$391$			506.698i			0.0655366i
$392$		0			0
$393$	−6855.35	−	7402.80i	−0.879915	−	0.950183i
$394$		0			0
$395$			1696.47i			0.216098i
$396$		0			0
$397$		−	13005.2i		−	1.64412i	−0.569403	−	0.822058i	$-0.692826\pi$
$397$		−	13005.2i		−	1.64412i	0.569403	−	0.822058i	$-0.307174\pi$
$398$		0			0
$399$	4571.93	+	4937.03i	0.573641	+	0.619450i
$400$		0			0
$401$	−3443.96			−0.428886			−0.214443	−	0.976737i	$-0.568794\pi$
$401$	−3443.96			−0.428886			−0.214443	+	0.976737i	$0.568794\pi$
$402$		0			0
$403$	9414.70	−	4148.83i	1.16372	−	0.512824i
$404$		0			0
$405$	6687.66	+	1034.75i	0.820525	+	0.126955i
$406$		0			0
$407$	−2989.00			−0.364027
$408$		0			0
$409$		−	514.697i		−	0.0622252i	−0.999516	−	0.0311126i	$-0.990095\pi$
$409$		−	514.697i		−	0.0622252i	0.999516	−	0.0311126i	$-0.00990505\pi$
$410$		0			0
$411$	−324.013	−	349.888i	−0.0388866	−	0.0419920i
$412$		0			0
$413$			7128.95i			0.849377i
$414$		0			0
$415$		−	613.985i		−	0.0726249i
$416$		0			0
$417$	−9129.56	+	8454.41i	−1.07213	+	0.992840i
$418$		0			0
$419$	5248.91			0.611996			0.305998	−	0.952032i	$-0.401010\pi$
$419$	5248.91			0.611996			0.305998	+	0.952032i	$0.401010\pi$
$420$		0			0
$421$			11487.1i			1.32981i	0.746929	+	0.664904i	$0.231528\pi$
$421$			11487.1i			1.32981i	−0.746929	+	0.664904i	$0.768472\pi$
$422$		0			0
$423$	−12259.8	−	942.836i	−1.40920	−	0.108374i
$424$		0			0
$425$		−	157.941i		−	0.0180266i
$426$		0			0
$427$	11880.0			1.34640
$428$		0			0
$429$	−4170.26	+	9629.12i	−0.469329	+	1.08368i
$430$		0			0
$431$		−	8461.46i		−	0.945648i	−0.881157	−	0.472824i	$-0.843235\pi$
$431$		−	8461.46i		−	0.945648i	0.881157	−	0.472824i	$-0.156765\pi$
$432$		0			0
$433$	5640.82			0.626052			0.313026	−	0.949745i	$-0.398657\pi$
$433$	5640.82			0.626052			0.313026	+	0.949745i	$0.398657\pi$
$434$		0			0
$435$	688.081	−	637.196i	0.0758413	−	0.0702327i
$436$		0			0
$437$	9557.83			1.04625
$438$		0			0
$439$		−	7198.90i		−	0.782654i	−0.920252	−	0.391327i	$-0.872016\pi$
$439$		−	7198.90i		−	0.782654i	0.920252	−	0.391327i	$-0.127984\pi$
$440$		0			0
$441$	120.438	−	1566.07i	0.0130049	−	0.169104i
$442$		0			0
$443$	17156.9			1.84007			0.920035	−	0.391836i	$-0.128160\pi$
$443$	17156.9			1.84007			0.920035	+	0.391836i	$0.128160\pi$
$444$		0			0
$445$	−8244.44			−0.878256
$446$		0			0
$447$	4621.21	+	4990.25i	0.488984	+	0.528033i
$448$		0			0
$449$	4902.95			0.515333			0.257666	−	0.966234i	$-0.417046\pi$
$449$	4902.95			0.515333			0.257666	+	0.966234i	$0.417046\pi$
$450$		0			0
$451$		−	10877.1i		−	1.13566i
$452$		0			0
$453$	460.497	+	497.271i	0.0477617	+	0.0515758i
$454$		0			0
$455$	−6719.72	+	2961.22i	−0.692363	+	0.305108i
$456$		0			0
$457$		−	9773.10i		−	1.00036i	−0.865920	−	0.500182i	$-0.833266\pi$
$457$		−	9773.10i		−	1.00036i	0.865920	−	0.500182i	$-0.166734\pi$
$458$		0			0
$459$	354.511	+	447.225i	0.0360504	+	0.0454786i
$460$		0			0
$461$	−3178.87			−0.321160			−0.160580	−	0.987023i	$-0.551337\pi$
$461$	−3178.87			−0.321160			−0.160580	+	0.987023i	$0.551337\pi$
$462$		0			0
$463$	−11372.6			−1.14153			−0.570767	−	0.821112i	$-0.693354\pi$
$463$	−11372.6			−1.14153			−0.570767	+	0.821112i	$0.693354\pi$
$464$		0			0
$465$	−7193.77	−	7768.25i	−0.717426	−	0.774718i
$466$		0			0
$467$	12139.3			1.20287			0.601435	−	0.798922i	$-0.294596\pi$
$467$	12139.3			1.20287			0.601435	+	0.798922i	$0.294596\pi$
$468$		0			0
$469$	−14165.9			−1.39471
$470$		0			0
$471$	−2468.75	−	2665.90i	−0.241516	−	0.260803i
$472$		0			0
$473$	10630.4			1.03338
$474$		0			0
$475$	−2979.25			−0.287784
$476$		0			0
$477$	8144.52	+	626.352i	0.781786	+	0.0601230i
$478$		0			0
$479$			17422.1i			1.66187i	0.556370	+	0.830935i	$0.312194\pi$
$479$			17422.1i			1.66187i	−0.556370	+	0.830935i	$0.687806\pi$
$480$		0			0
$481$	−1311.31	−	2975.67i	−0.124304	−	0.282076i
$482$		0			0
$483$	7422.10	+	8014.81i	0.699207	+	0.755044i
$484$		0			0
$485$		−	8121.31i		−	0.760350i
$486$		0			0
$487$	283.738			0.0264012			0.0132006	−	0.999913i	$-0.495798\pi$
$487$	283.738			0.0264012			0.0132006	+	0.999913i	$0.495798\pi$
$488$		0			0
$489$	−13501.0	−	14579.2i	−1.24854	−	1.34825i
$490$		0			0
$491$	2027.77			0.186379			0.0931893	−	0.995648i	$-0.470294\pi$
$491$	2027.77			0.186379			0.0931893	+	0.995648i	$0.470294\pi$
$492$		0			0
$493$	79.0863			0.00722488
$494$		0			0
$495$	10766.8	+	828.016i	0.977638	+	0.0751850i
$496$		0			0
$497$		−	11792.7i		−	1.06433i
$498$		0			0
$499$	−10231.8			−0.917908			−0.458954	−	0.888460i	$-0.651776\pi$
$499$	−10231.8			−0.917908			−0.458954	+	0.888460i	$0.651776\pi$
$500$		0			0
$501$	−1160.68	+	1074.85i	−0.103504	+	0.0958494i
$502$		0			0
$503$	3177.73			0.281686			0.140843	−	0.990032i	$-0.455019\pi$
$503$	3177.73			0.281686			0.140843	+	0.990032i	$0.455019\pi$
$504$		0			0
$505$			17891.1i			1.57652i
$506$		0			0
$507$	−11415.7	+	72.7359i	−0.999980	+	0.00637143i
$508$		0			0
$509$	18102.5			1.57638			0.788191	−	0.615431i	$-0.211018\pi$
$509$	18102.5			1.57638			0.788191	+	0.615431i	$0.211018\pi$
$510$		0			0
$511$			10151.6i			0.878828i
$512$		0			0
$513$	8436.00	−	6687.14i	0.726040	−	0.575525i
$514$		0			0
$515$			5698.82i			0.487611i
$516$		0			0
$517$	−19620.9			−1.66910
$518$		0			0
$519$	2011.82	−	1863.04i	0.170152	−	0.157569i
$520$		0			0
$521$		−	13744.0i		−	1.15573i	−0.816133	−	0.577864i	$-0.803887\pi$
$521$		−	13744.0i		−	1.15573i	0.816133	−	0.577864i	$-0.196113\pi$
$522$		0			0
$523$		−	10574.1i		−	0.884076i	−0.896996	−	0.442038i	$-0.854256\pi$
$523$		−	10574.1i		−	0.884076i	0.896996	−	0.442038i	$-0.145744\pi$
$524$		0			0
$525$	−2313.52	−	2498.28i	−0.192325	−	0.207683i
$526$		0			0
$527$		−	892.862i		−	0.0738021i
$528$		0			0
$529$	3349.26			0.275274
$530$		0			0
$531$	11371.5	+	874.525i	0.929346	+	0.0714711i
$532$		0			0
$533$	10828.6	−	4771.89i	0.879994	−	0.387792i
$534$		0			0
$535$	1899.44			0.153495
$536$		0			0
$537$	−7567.45	−	8171.77i	−0.608118	−	0.656681i
$538$		0			0
$539$		−	2506.38i		−	0.200292i
$540$		0			0
$541$		−	2835.35i		−	0.225326i	−0.993633	−	0.112663i	$-0.964062\pi$
$541$		−	2835.35i		−	0.225326i	0.993633	−	0.112663i	$-0.0359380\pi$
$542$		0			0
$543$	−6567.33	−	7091.79i	−0.519026	−	0.560475i
$544$		0			0
$545$			15511.9i			1.21919i
$546$		0			0
$547$		−	16130.5i		−	1.26086i	−0.776247	−	0.630428i	$-0.782879\pi$
$547$		−	16130.5i		−	1.26086i	0.776247	−	0.630428i	$-0.217121\pi$
$548$		0			0
$549$	1457.35	−	18950.0i	0.113293	−	1.47316i
$550$		0			0
$551$		−	1491.80i		−	0.115341i
$552$		0			0
$553$			3084.27i			0.237173i
$554$		0			0
$555$	−2455.28	+	2273.71i	−0.187785	+	0.173898i
$556$		0			0
$557$	−8262.29			−0.628518			−0.314259	−	0.949337i	$-0.601756\pi$
$557$	−8262.29			−0.628518			−0.314259	+	0.949337i	$0.601756\pi$
$558$		0			0
$559$	4663.69	+	10583.0i	0.352868	+	0.800742i
$560$		0			0
$561$	618.753	+	668.165i	0.0465664	+	0.0502851i
$562$		0			0
$563$	−13178.8			−0.986537			−0.493269	−	0.869877i	$-0.664198\pi$
$563$	−13178.8			−0.986537			−0.493269	+	0.869877i	$0.664198\pi$
$564$		0			0
$565$		−	7057.42i		−	0.525500i
$566$		0			0
$567$	12158.5	+	1881.22i	0.900545	+	0.139336i
$568$		0			0
$569$		−	15552.6i		−	1.14587i	−0.819602	−	0.572934i	$-0.805805\pi$
$569$		−	15552.6i		−	1.14587i	0.819602	−	0.572934i	$-0.194195\pi$
$570$		0			0
$571$			14416.3i			1.05658i	0.849065	+	0.528288i	$0.177166\pi$
$571$			14416.3i			1.05658i	−0.849065	+	0.528288i	$0.822834\pi$
$572$		0			0
$573$	−850.454	−	918.369i	−0.0620039	−	0.0669553i
$574$		0			0
$575$	−4836.53			−0.350778
$576$		0			0
$577$		−	2926.84i		−	0.211171i	−0.994410	−	0.105586i	$-0.966328\pi$
$577$		−	2926.84i		−	0.211171i	0.994410	−	0.105586i	$-0.0336717\pi$
$578$		0			0
$579$	16794.0	−	15552.0i	1.20541	−	1.11627i
$580$		0			0
$581$		−	1116.26i		−	0.0797075i
$582$		0			0
$583$	13034.7			0.925971
$584$		0			0
$585$	3899.18	+	11082.0i	0.275575	+	0.783223i
$586$		0			0
$587$		−	23204.1i		−	1.63158i	−0.578351	−	0.815788i	$-0.696304\pi$
$587$		−	23204.1i		−	1.63158i	0.578351	−	0.815788i	$-0.303696\pi$
$588$		0			0
$589$	−16842.1			−1.17821
$590$		0			0
$591$	−2982.70	−	3220.89i	−0.207601	−	0.224179i
$592$		0			0
$593$	−19130.9			−1.32481			−0.662404	−	0.749147i	$-0.730464\pi$
$593$	−19130.9			−1.32481			−0.662404	+	0.749147i	$0.730464\pi$
$594$		0			0
$595$			637.278i			0.0439090i
$596$		0			0
$597$	−1005.05	+	930.724i	−0.0689011	+	0.0638057i
$598$		0			0
$599$	22189.1			1.51356			0.756778	−	0.653672i	$-0.226772\pi$
$599$	22189.1			1.51356			0.756778	+	0.653672i	$0.226772\pi$
$600$		0			0
$601$	−25181.6			−1.70912			−0.854559	−	0.519354i	$-0.826173\pi$
$601$	−25181.6			−1.70912			−0.854559	+	0.519354i	$0.826173\pi$
$602$		0			0
$603$	−1737.76	+	22596.3i	−0.117358	+	1.52602i
$604$		0			0
$605$	4875.86			0.327656
$606$		0			0
$607$			5820.53i			0.389206i	0.980882	+	0.194603i	$0.0623418\pi$
$607$			5820.53i			0.389206i	−0.980882	+	0.194603i	$0.937658\pi$
$608$		0			0
$609$	1250.97	−	1158.45i	0.0832376	−	0.0770820i
$610$		0			0
$611$	−8607.89	−	19533.4i	−0.569948	−	1.29335i
$612$		0			0
$613$		−	18698.4i		−	1.23201i	−0.787742	−	0.616005i	$-0.788750\pi$
$613$		−	18698.4i		−	1.23201i	0.787742	−	0.616005i	$-0.211250\pi$
$614$		0			0
$615$	−8274.10	−	8934.85i	−0.542510	−	0.585834i
$616$		0			0
$617$	−524.183			−0.0342023			−0.0171011	−	0.999854i	$-0.505444\pi$
$617$	−524.183			−0.0342023			−0.0171011	+	0.999854i	$0.505444\pi$
$618$		0			0
$619$	4171.79			0.270886			0.135443	−	0.990785i	$-0.456754\pi$
$619$	4171.79			0.270886			0.135443	+	0.990785i	$0.456754\pi$
$620$		0			0
$621$	13695.1	−	10855.9i	0.884967	−	0.701504i
$622$		0			0
$623$	−14988.8			−0.963906
$624$		0			0
$625$	−9263.96			−0.592894
$626$		0			0
$627$	12603.6	−	11671.5i	0.802773	−	0.743407i
$628$		0			0
$629$	−282.204			−0.0178890
$630$		0			0
$631$	6128.07			0.386616			0.193308	−	0.981138i	$-0.438078\pi$
$631$	6128.07			0.386616			0.193308	+	0.981138i	$0.438078\pi$
$632$		0			0
$633$	−13537.5	+	12536.3i	−0.850025	+	0.787164i
$634$		0			0
$635$			4059.73i			0.253709i
$636$		0			0
$637$	2495.20	−	1099.58i	0.155202	−	0.0683938i
$638$		0			0
$639$	−18810.8	−	1446.64i	−1.16454	−	0.0895587i
$640$		0			0
$641$		−	4201.54i		−	0.258894i	−0.991586	−	0.129447i	$-0.958680\pi$
$641$		−	4201.54i		−	0.258894i	0.991586	−	0.129447i	$-0.0413201\pi$
$642$		0			0
$643$	−25108.7			−1.53995			−0.769976	−	0.638072i	$-0.779732\pi$
$643$	−25108.7			−1.53995			−0.769976	+	0.638072i	$0.779732\pi$
$644$		0			0
$645$	8732.26	−	8086.49i	0.533073	−	0.493652i
$646$		0			0
$647$	3133.13			0.190380			0.0951900	−	0.995459i	$-0.469654\pi$
$647$	3133.13			0.190380			0.0951900	+	0.995459i	$0.469654\pi$
$648$		0			0
$649$	18199.3			1.10075
$650$		0			0
$651$	−13078.6	−	14123.1i	−0.787392	−	0.850271i
$652$		0			0
$653$		−	22671.8i		−	1.35868i	−0.733825	−	0.679339i	$-0.762267\pi$
$653$		−	22671.8i		−	1.35868i	0.733825	−	0.679339i	$-0.237733\pi$
$654$		0			0
$655$	−18024.8			−1.07525
$656$		0			0
$657$	16193.0	+	1245.32i	0.961569	+	0.0739492i
$658$		0			0
$659$	3368.74			0.199131			0.0995656	−	0.995031i	$-0.468255\pi$
$659$	3368.74			0.199131			0.0995656	+	0.995031i	$0.468255\pi$
$660$		0			0
$661$		−	32396.1i		−	1.90630i	−0.302505	−	0.953148i	$-0.597823\pi$
$661$		−	32396.1i		−	1.90630i	0.302505	−	0.953148i	$-0.402177\pi$
$662$		0			0
$663$	−393.731	+	909.125i	−0.0230637	+	0.0532541i
$664$		0			0
$665$	12021.0			0.700983
$666$		0			0
$667$		−	2421.80i		−	0.140589i
$668$		0			0
$669$	−5411.78	−	5843.95i	−0.312752	−	0.337728i
$670$		0			0
$671$		−	30328.1i		−	1.74486i
$672$		0			0
$673$	3243.21			0.185760			0.0928802	−	0.995677i	$-0.470393\pi$
$673$	3243.21			0.185760			0.0928802	+	0.995677i	$0.470393\pi$
$674$		0			0
$675$	−4268.86	+	3383.88i	−0.243420	+	0.192956i
$676$		0			0
$677$		−	20223.6i		−	1.14809i	−0.818823	−	0.574046i	$-0.805373\pi$
$677$		−	20223.6i		−	1.14809i	0.818823	−	0.574046i	$-0.194627\pi$
$678$		0			0
$679$		−	14764.9i		−	0.834501i
$680$		0			0
$681$	13710.8	−	12696.9i	0.771513	−	0.714458i
$682$		0			0
$683$			13278.6i			0.743913i	0.928250	+	0.371957i	$0.121313\pi$
$683$			13278.6i			0.743913i	−0.928250	+	0.371957i	$0.878687\pi$
$684$		0			0
$685$	−851.929			−0.0475190
$686$		0			0
$687$	12711.9	−	11771.8i	0.705951	−	0.653744i
$688$		0			0
$689$	5718.46	+	12976.6i	0.316191	+	0.717514i
$690$		0			0
$691$	5397.48			0.297149			0.148574	−	0.988901i	$-0.452532\pi$
$691$	5397.48			0.297149			0.148574	+	0.988901i	$0.452532\pi$
$692$		0			0
$693$	19574.5	+	1505.37i	1.07298	+	0.0825172i
$694$		0			0
$695$			22229.2i			1.21324i
$696$		0			0
$697$		−	1026.95i		−	0.0558084i
$698$		0			0
$699$	11361.8	−	10521.6i	0.614797	−	0.569332i
$700$		0			0
$701$		−	4506.62i		−	0.242814i	−0.992603	−	0.121407i	$-0.961259\pi$
$701$		−	4506.62i		−	0.242814i	0.992603	−	0.121407i	$-0.0387407\pi$
$702$		0			0
$703$			5323.20i			0.285588i
$704$		0			0
$705$	−16117.4	+	14925.4i	−0.861014	+	0.797340i
$706$		0			0
$707$			32526.9i			1.73027i
$708$		0			0
$709$			27832.3i			1.47428i	0.675739	+	0.737141i	$0.263825\pi$
$709$			27832.3i			1.47428i	−0.675739	+	0.737141i	$0.736175\pi$
$710$		0			0
$711$	4919.79	+	378.355i	0.259503	+	0.0199570i
$712$		0			0
$713$	−27341.5			−1.43611
$714$		0			0
$715$	7559.61	+	17154.6i	0.395403	+	0.897265i
$716$		0			0
$717$	−26378.9	+	24428.1i	−1.37397	+	1.27236i
$718$		0			0
$719$	9535.06			0.494573			0.247286	−	0.968942i	$-0.420461\pi$
$719$	9535.06			0.494573			0.247286	+	0.968942i	$0.420461\pi$
$720$		0			0
$721$			10360.7i			0.535165i
$722$		0			0
$723$	−11299.3	+	10463.7i	−0.581226	+	0.538244i
$724$		0			0
$725$			754.895i			0.0386705i
$726$		0			0
$727$			12962.8i			0.661298i	0.943754	+	0.330649i	$0.107268\pi$
$727$			12962.8i			0.661298i	−0.943754	+	0.330649i	$0.892732\pi$
$728$		0			0
$729$	4492.28	−	19163.5i	0.228232	−	0.973607i
$730$		0			0
$731$	1003.66			0.0507823
$732$		0			0
$733$			28525.8i			1.43741i	0.695314	+	0.718706i	$0.255265\pi$
$733$			28525.8i			1.43741i	−0.695314	+	0.718706i	$0.744735\pi$
$734$		0			0
$735$	−1906.59	−	2058.84i	−0.0956809	−	0.103322i
$736$		0			0
$737$			36163.7i			1.80747i
$738$		0			0
$739$	−13883.8			−0.691100			−0.345550	−	0.938400i	$-0.612308\pi$
$739$	−13883.8			−0.691100			−0.345550	+	0.938400i	$0.612308\pi$
$740$		0			0
$741$	17148.8	+	7426.96i	0.850172	+	0.368200i
$742$		0			0
$743$			1055.64i			0.0521233i	0.999660	+	0.0260616i	$0.00829662\pi$
$743$			1055.64i			0.0521233i	−0.999660	+	0.0260616i	$0.991703\pi$
$744$		0			0
$745$	12150.6			0.597533
$746$		0			0
$747$	−1780.56	−	136.934i	−0.0872120	−	0.00670701i
$748$		0			0
$749$	3453.28			0.168465
$750$		0			0
$751$			21421.1i			1.04084i	0.853912	+	0.520418i	$0.174224\pi$
$751$			21421.1i			1.04084i	−0.853912	+	0.520418i	$0.825776\pi$
$752$		0			0
$753$	25491.5	+	27527.2i	1.23368	+	1.33220i
$754$		0			0
$755$	1210.79			0.0583643
$756$		0			0
$757$	−24778.3			−1.18967			−0.594837	−	0.803846i	$-0.702783\pi$
$757$	−24778.3			−1.18967			−0.594837	+	0.803846i	$0.702783\pi$
$758$		0			0
$759$	20460.8	−	18947.6i	0.978496	−	0.906134i
$760$		0			0
$761$	−136.102			−0.00648319			−0.00324160	−	0.999995i	$-0.501032\pi$
$761$	−136.102			−0.00648319			−0.00324160	+	0.999995i	$0.501032\pi$
$762$		0			0
$763$			28201.5i			1.33809i
$764$		0			0
$765$	1016.54	+	78.1764i	0.0480430	+	0.00369474i
$766$		0			0
$767$	7984.22	+	18118.1i	0.375872	+	0.852943i
$768$		0			0
$769$		−	1618.85i		−	0.0759130i	−0.999279	−	0.0379565i	$-0.987915\pi$
$769$		−	1618.85i		−	0.0759130i	0.999279	−	0.0379565i	$-0.0120848\pi$
$770$		0			0
$771$	2634.42	−	2439.60i	0.123056	−	0.113956i
$772$		0			0
$773$	6005.83			0.279450			0.139725	−	0.990190i	$-0.455378\pi$
$773$	6005.83			0.279450			0.139725	+	0.990190i	$0.455378\pi$
$774$		0			0
$775$	8522.56			0.395018
$776$		0			0
$777$	−4463.82	+	4133.71i	−0.206099	+	0.190857i
$778$		0			0
$779$	−19371.3			−0.890949
$780$		0			0
$781$	−30105.2			−1.37932
$782$		0			0
$783$	−1694.42	−	2137.55i	−0.0773352	−	0.0975604i
$784$		0			0
$785$	−6491.10			−0.295130
$786$		0			0
$787$	4785.90			0.216771			0.108386	−	0.994109i	$-0.465432\pi$
$787$	4785.90			0.216771			0.108386	+	0.994109i	$0.465432\pi$
$788$		0			0
$789$	17510.3	+	18908.7i	0.790094	+	0.853189i
$790$		0			0
$791$		−	12830.7i		−	0.576749i
$792$		0			0
$793$	30192.8	−	13305.3i	1.35205	−	0.595818i
$794$		0			0
$795$	10707.2	−	9915.38i	0.477667	−	0.442343i
$796$		0			0
$797$			37086.1i			1.64825i	0.566408	+	0.824125i	$0.308333\pi$
$797$			37086.1i			1.64825i	−0.566408	+	0.824125i	$0.691667\pi$
$798$		0			0
$799$	−1852.49			−0.0820229
$800$		0			0
$801$	−1838.71	+	23908.9i	−0.0811082	+	1.05466i
$802$		0			0
$803$	25915.7			1.13891
$804$		0			0
$805$	19514.9			0.854424
$806$		0			0
$807$	26323.9	−	24377.2i	1.14826	−	1.06334i
$808$		0			0
$809$			16614.0i			0.722022i	0.932562	+	0.361011i	$0.117568\pi$
$809$			16614.0i			0.722022i	−0.932562	+	0.361011i	$0.882432\pi$
$810$		0			0
$811$	−8890.17			−0.384927			−0.192464	−	0.981304i	$-0.561648\pi$
$811$	−8890.17			−0.384927			−0.192464	+	0.981304i	$0.561648\pi$
$812$		0			0
$813$	12849.6	+	13875.7i	0.554310	+	0.598576i
$814$		0			0
$815$	−35498.3			−1.52571
$816$		0			0
$817$		−	18932.1i		−	0.810710i
$818$		0			0
$819$	7088.90	+	20147.7i	0.302450	+	0.859605i
$820$		0			0
$821$	−5346.72			−0.227286			−0.113643	−	0.993522i	$-0.536252\pi$
$821$	−5346.72			−0.227286			−0.113643	+	0.993522i	$0.536252\pi$
$822$		0			0
$823$			702.708i			0.0297629i	0.999889	+	0.0148815i	$0.00473709\pi$
$823$			702.708i			0.0297629i	−0.999889	+	0.0148815i	$0.995263\pi$
$824$		0			0
$825$	−6377.77	+	5906.12i	−0.269146	+	0.249242i
$826$		0			0
$827$			7134.36i			0.299983i	0.988687	+	0.149991i	$0.0479246\pi$
$827$			7134.36i			0.299983i	−0.988687	+	0.149991i	$0.952075\pi$
$828$		0			0
$829$	−6453.83			−0.270387			−0.135193	−	0.990819i	$-0.543166\pi$
$829$	−6453.83			−0.270387			−0.135193	+	0.990819i	$0.543166\pi$
$830$		0			0
$831$	−13863.1	−	14970.1i	−0.578706	−	0.624920i
$832$		0			0
$833$		−	236.638i		−	0.00984275i
$834$		0			0
$835$			2826.09i			0.117127i
$836$		0			0
$837$	−24132.4	+	19129.5i	−0.996579	+	0.789978i
$838$		0			0
$839$			21446.2i			0.882486i	0.897388	+	0.441243i	$0.145462\pi$
$839$			21446.2i			0.882486i	−0.897388	+	0.441243i	$0.854538\pi$
$840$		0			0
$841$	24011.0			0.984501
$842$		0			0
$843$	−23790.8	−	25690.7i	−0.972005	−	1.04963i
$844$		0			0
$845$	−13761.6	+	15051.8i	−0.560252	+	0.612778i
$846$		0			0
$847$	8864.56			0.359610
$848$		0			0
$849$	−6579.97	+	6093.37i	−0.265988	+	0.246318i
$850$		0			0
$851$			8641.73i			0.348102i
$852$		0			0
$853$		−	30530.3i		−	1.22548i	−0.790283	−	0.612742i	$-0.790066\pi$
$853$		−	30530.3i		−	1.22548i	0.790283	−	0.612742i	$-0.209934\pi$
$854$		0			0
$855$	1474.64	−	19174.9i	0.0589844	−	0.766980i
$856$		0			0
$857$			37294.0i			1.48651i	0.669008	+	0.743256i	$0.266719\pi$
$857$			37294.0i			1.48651i	−0.669008	+	0.743256i	$0.733281\pi$
$858$		0			0
$859$			35599.1i			1.41400i	0.707213	+	0.707000i	$0.249952\pi$
$859$			35599.1i			1.41400i	−0.707213	+	0.707000i	$0.750048\pi$
$860$		0			0
$861$	−15042.7	−	16244.0i	−0.595418	−	0.642966i
$862$		0			0
$863$		−	8099.09i		−	0.319463i	−0.987161	−	0.159731i	$-0.948937\pi$
$863$		−	8099.09i		−	0.319463i	0.987161	−	0.159731i	$-0.0510628\pi$
$864$		0			0
$865$		−	4898.50i		−	0.192548i
$866$		0			0
$867$	−17287.2	−	18667.7i	−0.677168	−	0.731245i
$868$		0			0
$869$	7873.75			0.307363
$870$		0			0
$871$	−36002.4	+	15865.4i	−1.40057	+	0.617197i
$872$		0			0
$873$	−23551.9	−	1811.25i	−0.913069	−	0.0702194i
$874$		0			0
$875$	−25666.2			−0.991628
$876$		0			0
$877$		−	10329.6i		−	0.397725i	−0.980027	−	0.198863i	$-0.936275\pi$
$877$		−	10329.6i		−	0.397725i	0.980027	−	0.198863i	$-0.0637248\pi$
$878$		0			0
$879$	28537.8	+	30816.7i	1.09506	+	1.18251i
$880$		0			0
$881$		−	41904.8i		−	1.60251i	−0.598325	−	0.801253i	$-0.704167\pi$
$881$		−	41904.8i		−	1.60251i	0.598325	−	0.801253i	$-0.295833\pi$
$882$		0			0
$883$			25236.1i			0.961793i	0.876777	+	0.480896i	$0.159689\pi$
$883$			25236.1i			0.961793i	−0.876777	+	0.480896i	$0.840311\pi$
$884$		0			0
$885$	14949.6	−	13844.1i	0.567825	−	0.525834i
$886$		0			0
$887$	44523.9			1.68542			0.842709	−	0.538370i	$-0.180960\pi$
$887$	44523.9			1.68542			0.842709	+	0.538370i	$0.180960\pi$
$888$		0			0
$889$			7380.79i			0.278452i
$890$		0			0
$891$	4802.51	−	31039.1i	0.180572	−	1.16706i
$892$		0			0
$893$			34943.5i			1.30945i
$894$		0			0
$895$	−19897.1			−0.743114
$896$		0			0
$897$	27839.5	+	12057.0i	1.03627	+	0.448797i
$898$		0			0
$899$			4267.51i			0.158320i
$900$		0			0
$901$	1230.66			0.0455041
$902$		0			0
$903$	15875.7	−	14701.6i	0.585061	−	0.541794i
$904$		0			0
$905$	−17267.5			−0.634245
$906$		0			0
$907$			43646.4i			1.59786i	0.601426	+	0.798928i	$0.294599\pi$
$907$			43646.4i			1.59786i	−0.601426	+	0.798928i	$0.705401\pi$
$908$		0			0
$909$	51884.2	+	3990.14i	1.89317	+	0.145594i
$910$		0			0
$911$	−27333.6			−0.994076			−0.497038	−	0.867729i	$-0.665579\pi$
$911$	−27333.6			−0.994076			−0.497038	+	0.867729i	$0.665579\pi$
$912$		0			0
$913$	−2849.65			−0.103297
$914$		0			0
$915$	−23070.3	−	24912.7i	−0.833532	−	0.900095i
$916$		0			0
$917$	−32770.0			−1.18011
$918$		0			0
$919$		−	8688.66i		−	0.311874i	−0.987767	−	0.155937i	$-0.950160\pi$
$919$		−	8688.66i		−	0.311874i	0.987767	−	0.155937i	$-0.0498397\pi$
$920$		0			0
$921$	−10469.4	−	11305.5i	−0.374569	−	0.404482i
$922$		0			0
$923$	−13207.5	−	29970.9i	−0.470996	−	1.06880i
$924$		0			0
$925$		−	2693.69i		−	0.0957492i
$926$		0			0
$927$	16526.6	+	1270.97i	0.585550	+	0.0450316i
$928$		0			0
$929$	−29884.3			−1.05541			−0.527703	−	0.849429i	$-0.676946\pi$
$929$	−29884.3			−1.05541			−0.527703	+	0.849429i	$0.676946\pi$
$930$		0			0
$931$	−4463.70			−0.157134
$932$		0			0
$933$	18878.5	+	20386.0i	0.662436	+	0.715337i
$934$		0			0
$935$	1626.89			0.0569037
$936$		0			0
$937$	−3468.12			−0.120916			−0.0604581	−	0.998171i	$-0.519256\pi$
$937$	−3468.12			−0.120916			−0.0604581	+	0.998171i	$0.519256\pi$
$938$		0			0
$939$	−4619.86	−	4988.79i	−0.160557	−	0.173379i
$940$		0			0
$941$	47466.8			1.64439			0.822196	−	0.569204i	$-0.192749\pi$
$941$	47466.8			1.64439			0.822196	+	0.569204i	$0.192749\pi$
$942$		0			0
$943$	−31447.5			−1.08597
$944$		0			0
$945$	17224.4	−	13653.6i	0.592921	−	0.470003i
$946$		0			0
$947$		−	16570.8i		−	0.568615i	−0.958733	−	0.284307i	$-0.908236\pi$
$947$		−	16570.8i		−	0.568615i	0.958733	−	0.284307i	$-0.0917636\pi$
$948$		0			0
$949$	11369.5	+	25800.2i	0.388904	+	0.882517i
$950$		0			0
$951$	31360.6	+	33865.0i	1.06933	+	1.15473i
$952$		0			0
$953$		−	2230.85i		−	0.0758284i	−0.999281	−	0.0379142i	$-0.987929\pi$
$953$		−	2230.85i		−	0.0758284i	0.999281	−	0.0379142i	$-0.0120714\pi$
$954$		0			0
$955$	−2236.10			−0.0757681
$956$		0			0
$957$	−2957.38	−	3193.55i	−0.0998940	−	0.107871i
$958$		0			0
$959$	−1548.85			−0.0521532
$960$		0			0
$961$	18388.1			0.617235
$962$		0			0
$963$	423.621	−	5508.39i	0.0141755	−	0.184326i
$964$		0			0
$965$		−	40891.0i		−	1.36407i
$966$		0			0
$967$	1058.32			0.0351946			0.0175973	−	0.999845i	$-0.494398\pi$
$967$	1058.32			0.0351946			0.0175973	+	0.999845i	$0.494398\pi$
$968$		0			0
$969$	1189.96	−	1101.96i	0.0394499	−	0.0365325i
$970$		0			0
$971$	−50203.6			−1.65923			−0.829614	−	0.558337i	$-0.811440\pi$
$971$	−50203.6			−1.65923			−0.829614	+	0.558337i	$0.811440\pi$
$972$		0			0
$973$			40413.8i			1.33156i
$974$		0			0
$975$	−8677.78	−	3758.25i	−0.285037	−	0.123446i
$976$		0			0
$977$	1194.58			0.0391177			0.0195589	−	0.999809i	$-0.493774\pi$
$977$	1194.58			0.0391177			0.0195589	+	0.999809i	$0.493774\pi$
$978$		0			0
$979$			38264.4i			1.24917i
$980$		0			0
$981$	44984.8	+	3459.54i	1.46407	+	0.112594i
$982$		0			0
$983$		−	48036.1i		−	1.55861i	−0.626645	−	0.779305i	$-0.715573\pi$
$983$		−	48036.1i		−	1.55861i	0.626645	−	0.779305i	$-0.284427\pi$
$984$		0			0
$985$	−7842.42			−0.253686
$986$		0			0
$987$	−29302.2	+	27135.2i	−0.944983	+	0.875099i
$988$		0			0
$989$		−	30734.5i		−	0.988170i
$990$		0			0
$991$		−	37626.2i		−	1.20609i	−0.797707	−	0.603046i	$-0.793954\pi$
$991$		−	37626.2i		−	1.20609i	0.797707	−	0.603046i	$-0.206046\pi$
$992$		0			0
$993$	405.190	+	437.547i	0.0129489	+	0.0139830i
$994$		0			0
$995$			2447.16i			0.0779699i
$996$		0			0
$997$	10770.7			0.342139			0.171070	−	0.985259i	$-0.445278\pi$
$997$	10770.7			0.342139			0.171070	+	0.985259i	$0.445278\pi$
$998$		0			0
$999$	6046.18	+	7627.42i	0.191484	+	0.241563i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.n.d.623.13	✓	56
3.2	odd	2	inner	624.4.n.d.623.42	yes	56
4.3	odd	2	inner	624.4.n.d.623.43	yes	56
12.11	even	2	inner	624.4.n.d.623.16	yes	56
13.12	even	2	inner	624.4.n.d.623.14	yes	56
39.38	odd	2	inner	624.4.n.d.623.41	yes	56
52.51	odd	2	inner	624.4.n.d.623.44	yes	56
156.155	even	2	inner	624.4.n.d.623.15	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.n.d.623.13	✓	56	1.1	even	1	trivial
624.4.n.d.623.14	yes	56	13.12	even	2	inner
624.4.n.d.623.15	yes	56	156.155	even	2	inner
624.4.n.d.623.16	yes	56	12.11	even	2	inner
624.4.n.d.623.41	yes	56	39.38	odd	2	inner
624.4.n.d.623.42	yes	56	3.2	odd	2	inner
624.4.n.d.623.43	yes	56	4.3	odd	2	inner
624.4.n.d.623.44	yes	56	52.51	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.53056 - 3.81250i) q^{3} -9.28291 q^{5} -16.8768 q^{7} +(-2.07031 + 26.9205i) q^{9} +43.0842i q^{11} +(-42.8921 + 18.9015i) q^{13} +(32.7738 + 35.3911i) q^{15} +4.06776i q^{17} +76.7301 q^{19} +(59.5845 + 64.3428i) q^{21} +124.564 q^{23} -38.8276 q^{25} +(109.944 - 87.1514i) q^{27} -19.4422i q^{29} -219.497 q^{31} +(164.259 - 152.111i) q^{33} +156.666 q^{35} +69.3757i q^{37} +(223.495 + 96.7932i) q^{39} -252.460 q^{41} -246.736i q^{43} +(19.2185 - 249.901i) q^{45} +455.407i q^{47} -58.1740 q^{49} +(15.5083 - 14.3615i) q^{51} -302.539i q^{53} -399.947i q^{55} +(-270.900 - 292.534i) q^{57} -422.412i q^{59} -703.925 q^{61} +(34.9403 - 454.332i) q^{63} +(398.163 - 175.461i) q^{65} +839.371 q^{67} +(-439.781 - 474.901i) q^{69} +698.752i q^{71} -601.513i q^{73} +(137.083 + 148.030i) q^{75} -727.124i q^{77} -182.752i q^{79} +(-720.428 - 111.468i) q^{81} +66.1415i q^{83} -37.7606i q^{85} +(-74.1234 + 68.6419i) q^{87} +888.131 q^{89} +(723.881 - 318.997i) q^{91} +(774.948 + 836.833i) q^{93} -712.279 q^{95} +874.867i q^{97} +(-1159.85 - 89.1979i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 64 q^{9} + 8 q^{13} + 1208 q^{25} + 4680 q^{49} + 1616 q^{61} + 480 q^{69} + 1568 q^{81}+O(q^{100}) \) 56 * q - 64 * q^9 + 8 * q^13 + 1208 * q^25 + 4680 * q^49 + 1616 * q^61 + 480 * q^69 + 1568 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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