LMFDB - Embedded newform 624.4.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(1,624)] 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("624.1"); 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([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,24,0,0] 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:	yes
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +19.4833 q^{5} -7.48331 q^{7} +9.00000 q^{9} -22.8999 q^{11} -13.0000 q^{13} +58.4499 q^{15} +67.0334 q^{17} -16.5167 q^{19} -22.4499 q^{21} +175.600 q^{23} +254.600 q^{25} +27.0000 q^{27} +291.800 q^{29} -117.283 q^{31} -68.6997 q^{33} -145.800 q^{35} -154.766 q^{37} -39.0000 q^{39} -251.716 q^{41} +502.566 q^{43} +175.350 q^{45} +281.733 q^{47} -287.000 q^{49} +201.100 q^{51} +366.999 q^{53} -446.166 q^{55} -49.5501 q^{57} +79.6663 q^{59} -194.865 q^{61} -67.3498 q^{63} -253.283 q^{65} -400.082 q^{67} +526.799 q^{69} -528.299 q^{71} -734.366 q^{73} +763.799 q^{75} +171.367 q^{77} -113.266 q^{79} +81.0000 q^{81} +933.466 q^{83} +1306.03 q^{85} +875.399 q^{87} +1190.91 q^{89} +97.2831 q^{91} -351.849 q^{93} -321.800 q^{95} +557.165 q^{97} -206.099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39}+ \cdots + 396 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 396 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	19.4833	1.74264	0.871320	−	0.490715i	$-0.163264\pi$
$5$	19.4833	1.74264	0.871320	+	0.490715i	$0.163264\pi$
$6$	0	0
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−22.8999	−0.627689	−0.313844	−	0.949474i	$-0.601617\pi$
$11$	−22.8999	−0.627689	−0.313844	+	0.949474i	$0.601617\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	58.4499	1.00611
$16$	0	0
$17$	67.0334	0.956352	0.478176	−	0.878264i	$-0.341298\pi$
$17$	67.0334	0.956352	0.478176	+	0.878264i	$0.341298\pi$
$18$	0	0
$19$	−16.5167	−0.199431	−0.0997155	−	0.995016i	$-0.531793\pi$
$19$	−16.5167	−0.199431	−0.0997155	+	0.995016i	$0.531793\pi$
$20$	0	0
$21$	−22.4499	−0.233285
$22$	0	0
$23$	175.600	1.59196	0.795979	−	0.605324i	$-0.206956\pi$
$23$	175.600	1.59196	0.795979	+	0.605324i	$0.206956\pi$
$24$	0	0
$25$	254.600	2.03680
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	291.800	1.86848	0.934239	−	0.356648i	$-0.116080\pi$
$29$	291.800	1.86848	0.934239	+	0.356648i	$0.116080\pi$
$30$	0	0
$31$	−117.283	−0.679505	−0.339753	−	0.940515i	$-0.610343\pi$
$31$	−117.283	−0.679505	−0.339753	+	0.940515i	$0.610343\pi$
$32$	0	0
$33$	−68.6997	−0.362396
$34$	0	0
$35$	−145.800	−0.704133
$36$	0	0
$37$	−154.766	−0.687661	−0.343830	−	0.939032i	$-0.611724\pi$
$37$	−154.766	−0.687661	−0.343830	+	0.939032i	$0.611724\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−251.716	−0.958815	−0.479407	−	0.877592i	$-0.659148\pi$
$41$	−251.716	−0.958815	−0.479407	+	0.877592i	$0.659148\pi$
$42$	0	0
$43$	502.566	1.78234	0.891170	−	0.453669i	$-0.149885\pi$
$43$	502.566	1.78234	0.891170	+	0.453669i	$0.149885\pi$
$44$	0	0
$45$	175.350	0.580880
$46$	0	0
$47$	281.733	0.874361	0.437181	−	0.899374i	$-0.355977\pi$
$47$	281.733	0.874361	0.437181	+	0.899374i	$0.355977\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	0	0
$51$	201.100	0.552150
$52$	0	0
$53$	366.999	0.951154	0.475577	−	0.879674i	$-0.342239\pi$
$53$	366.999	0.951154	0.475577	+	0.879674i	$0.342239\pi$
$54$	0	0
$55$	−446.166	−1.09384
$56$	0	0
$57$	−49.5501	−0.115141
$58$	0	0
$59$	79.6663	0.175791	0.0878955	−	0.996130i	$-0.471986\pi$
$59$	79.6663	0.175791	0.0878955	+	0.996130i	$0.471986\pi$
$60$	0	0
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	0	0
$63$	−67.3498	−0.134687
$64$	0	0
$65$	−253.283	−0.483322
$66$	0	0
$67$	−400.082	−0.729519	−0.364759	−	0.931102i	$-0.618849\pi$
$67$	−400.082	−0.729519	−0.364759	+	0.931102i	$0.618849\pi$
$68$	0	0
$69$	526.799	0.919117
$70$	0	0
$71$	−528.299	−0.883065	−0.441532	−	0.897245i	$-0.645565\pi$
$71$	−528.299	−0.883065	−0.441532	+	0.897245i	$0.645565\pi$
$72$	0	0
$73$	−734.366	−1.17741	−0.588706	−	0.808347i	$-0.700362\pi$
$73$	−734.366	−1.17741	−0.588706	+	0.808347i	$0.700362\pi$
$74$	0	0
$75$	763.799	1.17594
$76$	0	0
$77$	171.367	0.253625
$78$	0	0
$79$	−113.266	−0.161309	−0.0806545	−	0.996742i	$-0.525701\pi$
$79$	−113.266	−0.161309	−0.0806545	+	0.996742i	$0.525701\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	933.466	1.23447	0.617236	−	0.786778i	$-0.288252\pi$
$83$	933.466	1.23447	0.617236	+	0.786778i	$0.288252\pi$
$84$	0	0
$85$	1306.03	1.66658
$86$	0	0
$87$	875.399	1.07877
$88$	0	0
$89$	1190.91	1.41839	0.709195	−	0.705012i	$-0.249059\pi$
$89$	1190.91	1.41839	0.709195	+	0.705012i	$0.249059\pi$
$90$	0	0
$91$	97.2831	0.112066
$92$	0	0
$93$	−351.849	−0.392313
$94$	0	0
$95$	−321.800	−0.347536
$96$	0	0
$97$	557.165	0.583211	0.291606	−	0.956539i	$-0.405811\pi$
$97$	557.165	0.583211	0.291606	+	0.956539i	$0.405811\pi$
$98$	0	0
$99$	−206.099	−0.209230
$100$	0	0
$101$	−286.766	−0.282518	−0.141259	−	0.989973i	$-0.545115\pi$
$101$	−286.766	−0.282518	−0.141259	+	0.989973i	$0.545115\pi$
$102$	0	0
$103$	1911.36	1.82847	0.914234	−	0.405187i	$-0.132794\pi$
$103$	1911.36	1.82847	0.914234	+	0.405187i	$0.132794\pi$
$104$	0	0
$105$	−437.399	−0.406531
$106$	0	0
$107$	−834.334	−0.753814	−0.376907	−	0.926251i	$-0.623012\pi$
$107$	−834.334	−0.753814	−0.376907	+	0.926251i	$0.623012\pi$
$108$	0	0
$109$	−1077.66	−0.946986	−0.473493	−	0.880798i	$-0.657007\pi$
$109$	−1077.66	−0.946986	−0.473493	+	0.880798i	$0.657007\pi$
$110$	0	0
$111$	−464.299	−0.397021
$112$	0	0
$113$	−166.065	−0.138248	−0.0691241	−	0.997608i	$-0.522020\pi$
$113$	−166.065	−0.138248	−0.0691241	+	0.997608i	$0.522020\pi$
$114$	0	0
$115$	3421.26	2.77421
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−501.632	−0.386424
$120$	0	0
$121$	−806.595	−0.606007
$122$	0	0
$123$	−755.147	−0.553572
$124$	0	0
$125$	2525.03	1.80676
$126$	0	0
$127$	−1296.16	−0.905637	−0.452819	−	0.891603i	$-0.649581\pi$
$127$	−1296.16	−0.905637	−0.452819	+	0.891603i	$0.649581\pi$
$128$	0	0
$129$	1507.70	1.02903
$130$	0	0
$131$	197.201	0.131523	0.0657617	−	0.997835i	$-0.479052\pi$
$131$	197.201	0.131523	0.0657617	+	0.997835i	$0.479052\pi$
$132$	0	0
$133$	123.600	0.0805823
$134$	0	0
$135$	526.049	0.335371
$136$	0	0
$137$	−546.915	−0.341066	−0.170533	−	0.985352i	$-0.554549\pi$
$137$	−546.915	−0.341066	−0.170533	+	0.985352i	$0.554549\pi$
$138$	0	0
$139$	−609.666	−0.372023	−0.186012	−	0.982548i	$-0.559556\pi$
$139$	−609.666	−0.372023	−0.186012	+	0.982548i	$0.559556\pi$
$140$	0	0
$141$	845.199	0.504813
$142$	0	0
$143$	297.699	0.174090
$144$	0	0
$145$	5685.23	3.25609
$146$	0	0
$147$	−861.000	−0.483089
$148$	0	0
$149$	−2165.08	−1.19040	−0.595202	−	0.803576i	$-0.702928\pi$
$149$	−2165.08	−1.19040	−0.595202	+	0.803576i	$0.702928\pi$
$150$	0	0
$151$	846.549	0.456233	0.228116	−	0.973634i	$-0.426743\pi$
$151$	846.549	0.456233	0.228116	+	0.973634i	$0.426743\pi$
$152$	0	0
$153$	603.300	0.318784
$154$	0	0
$155$	−2285.06	−1.18413
$156$	0	0
$157$	1653.60	0.840581	0.420291	−	0.907390i	$-0.361928\pi$
$157$	1653.60	0.840581	0.420291	+	0.907390i	$0.361928\pi$
$158$	0	0
$159$	1101.00	0.549149
$160$	0	0
$161$	−1314.07	−0.643248
$162$	0	0
$163$	2866.51	1.37744	0.688720	−	0.725027i	$-0.258173\pi$
$163$	2866.51	1.37744	0.688720	+	0.725027i	$0.258173\pi$
$164$	0	0
$165$	−1338.50	−0.631526
$166$	0	0
$167$	−729.066	−0.337825	−0.168913	−	0.985631i	$-0.554026\pi$
$167$	−729.066	−0.337825	−0.168913	+	0.985631i	$0.554026\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−148.650	−0.0664770
$172$	0	0
$173$	−3834.83	−1.68530	−0.842650	−	0.538462i	$-0.819005\pi$
$173$	−3834.83	−1.68530	−0.842650	+	0.538462i	$0.819005\pi$
$174$	0	0
$175$	−1905.25	−0.822990
$176$	0	0
$177$	238.999	0.101493
$178$	0	0
$179$	283.862	0.118530	0.0592649	−	0.998242i	$-0.481124\pi$
$179$	283.862	0.118530	0.0592649	+	0.998242i	$0.481124\pi$
$180$	0	0
$181$	2363.60	0.970634	0.485317	−	0.874338i	$-0.338704\pi$
$181$	2363.60	0.970634	0.485317	+	0.874338i	$0.338704\pi$
$182$	0	0
$183$	−584.596	−0.236145
$184$	0	0
$185$	−3015.36	−1.19835
$186$	0	0
$187$	−1535.06	−0.600291
$188$	0	0
$189$	−202.049	−0.0777616
$190$	0	0
$191$	−2514.26	−0.952491	−0.476246	−	0.879312i	$-0.658003\pi$
$191$	−2514.26	−0.952491	−0.476246	+	0.879312i	$0.658003\pi$
$192$	0	0
$193$	2420.73	0.902839	0.451420	−	0.892312i	$-0.350918\pi$
$193$	2420.73	0.902839	0.451420	+	0.892312i	$0.350918\pi$
$194$	0	0
$195$	−759.849	−0.279046
$196$	0	0
$197$	−4633.65	−1.67581	−0.837903	−	0.545819i	$-0.816219\pi$
$197$	−4633.65	−1.67581	−0.837903	+	0.545819i	$0.816219\pi$
$198$	0	0
$199$	−3054.17	−1.08796	−0.543980	−	0.839098i	$-0.683083\pi$
$199$	−3054.17	−1.08796	−0.543980	+	0.839098i	$0.683083\pi$
$200$	0	0
$201$	−1200.25	−0.421188
$202$	0	0
$203$	−2183.63	−0.754979
$204$	0	0
$205$	−4904.26	−1.67087
$206$	0	0
$207$	1580.40	0.530653
$208$	0	0
$209$	378.230	0.125181
$210$	0	0
$211$	4031.60	1.31539	0.657694	−	0.753285i	$-0.271532\pi$
$211$	4031.60	1.31539	0.657694	+	0.753285i	$0.271532\pi$
$212$	0	0
$213$	−1584.90	−0.509838
$214$	0	0
$215$	9791.66	3.10598
$216$	0	0
$217$	877.666	0.274562
$218$	0	0
$219$	−2203.10	−0.679779
$220$	0	0
$221$	−871.434	−0.265244
$222$	0	0
$223$	−3784.95	−1.13659	−0.568294	−	0.822826i	$-0.692396\pi$
$223$	−3784.95	−1.13659	−0.568294	+	0.822826i	$0.692396\pi$
$224$	0	0
$225$	2291.40	0.678932
$226$	0	0
$227$	−2013.83	−0.588821	−0.294411	−	0.955679i	$-0.595123\pi$
$227$	−2013.83	−0.588821	−0.294411	+	0.955679i	$0.595123\pi$
$228$	0	0
$229$	−3050.73	−0.880340	−0.440170	−	0.897915i	$-0.645082\pi$
$229$	−3050.73	−0.880340	−0.440170	+	0.897915i	$0.645082\pi$
$230$	0	0
$231$	514.101	0.146430
$232$	0	0
$233$	5587.49	1.57103	0.785513	−	0.618846i	$-0.212399\pi$
$233$	5587.49	1.57103	0.785513	+	0.618846i	$0.212399\pi$
$234$	0	0
$235$	5489.09	1.52370
$236$	0	0
$237$	−339.798	−0.0931317
$238$	0	0
$239$	1335.69	0.361501	0.180750	−	0.983529i	$-0.442147\pi$
$239$	1335.69	0.361501	0.180750	+	0.983529i	$0.442147\pi$
$240$	0	0
$241$	−571.558	−0.152769	−0.0763845	−	0.997078i	$-0.524338\pi$
$241$	−571.558	−0.152769	−0.0763845	+	0.997078i	$0.524338\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−5591.71	−1.45813
$246$	0	0
$247$	214.717	0.0553122
$248$	0	0
$249$	2800.40	0.712723
$250$	0	0
$251$	−4088.60	−1.02817	−0.514084	−	0.857740i	$-0.671868\pi$
$251$	−4088.60	−1.02817	−0.514084	+	0.857740i	$0.671868\pi$
$252$	0	0
$253$	−4021.21	−0.999254
$254$	0	0
$255$	3918.10	0.962199
$256$	0	0
$257$	3050.23	0.740342	0.370171	−	0.928964i	$-0.379299\pi$
$257$	3050.23	0.740342	0.370171	+	0.928964i	$0.379299\pi$
$258$	0	0
$259$	1158.17	0.277857
$260$	0	0
$261$	2626.20	0.622826
$262$	0	0
$263$	−5770.99	−1.35306	−0.676530	−	0.736415i	$-0.736517\pi$
$263$	−5770.99	−1.35306	−0.676530	+	0.736415i	$0.736517\pi$
$264$	0	0
$265$	7150.35	1.65752
$266$	0	0
$267$	3572.74	0.818908
$268$	0	0
$269$	−2079.40	−0.471314	−0.235657	−	0.971836i	$-0.575724\pi$
$269$	−2079.40	−0.471314	−0.235657	+	0.971836i	$0.575724\pi$
$270$	0	0
$271$	−6012.00	−1.34761	−0.673807	−	0.738908i	$-0.735342\pi$
$271$	−6012.00	−1.34761	−0.673807	+	0.738908i	$0.735342\pi$
$272$	0	0
$273$	291.849	0.0647015
$274$	0	0
$275$	−5830.30	−1.27847
$276$	0	0
$277$	−735.201	−0.159473	−0.0797364	−	0.996816i	$-0.525408\pi$
$277$	−735.201	−0.159473	−0.0797364	+	0.996816i	$0.525408\pi$
$278$	0	0
$279$	−1055.55	−0.226502
$280$	0	0
$281$	−1902.92	−0.403981	−0.201990	−	0.979387i	$-0.564741\pi$
$281$	−1902.92	−0.403981	−0.201990	+	0.979387i	$0.564741\pi$
$282$	0	0
$283$	−2125.71	−0.446502	−0.223251	−	0.974761i	$-0.571667\pi$
$283$	−2125.71	−0.446502	−0.223251	+	0.974761i	$0.571667\pi$
$284$	0	0
$285$	−965.399	−0.200650
$286$	0	0
$287$	1883.67	0.387420
$288$	0	0
$289$	−419.527	−0.0853913
$290$	0	0
$291$	1671.49	0.336717
$292$	0	0
$293$	−1641.03	−0.327200	−0.163600	−	0.986527i	$-0.552311\pi$
$293$	−1641.03	−0.327200	−0.163600	+	0.986527i	$0.552311\pi$
$294$	0	0
$295$	1552.16	0.306341
$296$	0	0
$297$	−618.297	−0.120799
$298$	0	0
$299$	−2282.79	−0.441530
$300$	0	0
$301$	−3760.86	−0.720174
$302$	0	0
$303$	−860.299	−0.163112
$304$	0	0
$305$	−3796.62	−0.712767
$306$	0	0
$307$	3373.27	0.627111	0.313555	−	0.949570i	$-0.398480\pi$
$307$	3373.27	0.627111	0.313555	+	0.949570i	$0.398480\pi$
$308$	0	0
$309$	5734.09	1.05567
$310$	0	0
$311$	868.525	0.158359	0.0791793	−	0.996860i	$-0.474770\pi$
$311$	868.525	0.158359	0.0791793	+	0.996860i	$0.474770\pi$
$312$	0	0
$313$	−4343.19	−0.784319	−0.392159	−	0.919897i	$-0.628272\pi$
$313$	−4343.19	−0.784319	−0.392159	+	0.919897i	$0.628272\pi$
$314$	0	0
$315$	−1312.20	−0.234711
$316$	0	0
$317$	−3277.65	−0.580730	−0.290365	−	0.956916i	$-0.593777\pi$
$317$	−3277.65	−0.580730	−0.290365	+	0.956916i	$0.593777\pi$
$318$	0	0
$319$	−6682.18	−1.17282
$320$	0	0
$321$	−2503.00	−0.435215
$322$	0	0
$323$	−1107.17	−0.190726
$324$	0	0
$325$	−3309.79	−0.564906
$326$	0	0
$327$	−3232.99	−0.546743
$328$	0	0
$329$	−2108.30	−0.353295
$330$	0	0
$331$	−5589.62	−0.928197	−0.464099	−	0.885784i	$-0.653622\pi$
$331$	−5589.62	−0.928197	−0.464099	+	0.885784i	$0.653622\pi$
$332$	0	0
$333$	−1392.90	−0.229220
$334$	0	0
$335$	−7794.92	−1.27129
$336$	0	0
$337$	901.544	0.145728	0.0728638	−	0.997342i	$-0.476786\pi$
$337$	901.544	0.145728	0.0728638	+	0.997342i	$0.476786\pi$
$338$	0	0
$339$	−498.194	−0.0798176
$340$	0	0
$341$	2685.77	0.426518
$342$	0	0
$343$	4714.49	0.742153
$344$	0	0
$345$	10263.8	1.60169
$346$	0	0
$347$	812.318	0.125670	0.0628350	−	0.998024i	$-0.479986\pi$
$347$	812.318	0.125670	0.0628350	+	0.998024i	$0.479986\pi$
$348$	0	0
$349$	4437.96	0.680683	0.340342	−	0.940302i	$-0.389457\pi$
$349$	4437.96	0.680683	0.340342	+	0.940302i	$0.389457\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	7115.35	1.07284	0.536419	−	0.843952i	$-0.319777\pi$
$353$	7115.35	1.07284	0.536419	+	0.843952i	$0.319777\pi$
$354$	0	0
$355$	−10293.0	−1.53886
$356$	0	0
$357$	−1504.90	−0.223102
$358$	0	0
$359$	−4693.98	−0.690081	−0.345040	−	0.938588i	$-0.612135\pi$
$359$	−4693.98	−0.690081	−0.345040	+	0.938588i	$0.612135\pi$
$360$	0	0
$361$	−6586.20	−0.960227
$362$	0	0
$363$	−2419.79	−0.349878
$364$	0	0
$365$	−14307.9	−2.05181
$366$	0	0
$367$	−9243.98	−1.31480	−0.657400	−	0.753542i	$-0.728344\pi$
$367$	−9243.98	−1.31480	−0.657400	+	0.753542i	$0.728344\pi$
$368$	0	0
$369$	−2265.44	−0.319605
$370$	0	0
$371$	−2746.37	−0.384324
$372$	0	0
$373$	−4311.99	−0.598569	−0.299285	−	0.954164i	$-0.596748\pi$
$373$	−4311.99	−0.598569	−0.299285	+	0.954164i	$0.596748\pi$
$374$	0	0
$375$	7575.09	1.04314
$376$	0	0
$377$	−3793.40	−0.518223
$378$	0	0
$379$	2382.73	0.322936	0.161468	−	0.986878i	$-0.448377\pi$
$379$	2382.73	0.322936	0.161468	+	0.986878i	$0.448377\pi$
$380$	0	0
$381$	−3888.49	−0.522870
$382$	0	0
$383$	−4845.81	−0.646499	−0.323250	−	0.946314i	$-0.604775\pi$
$383$	−4845.81	−0.646499	−0.323250	+	0.946314i	$0.604775\pi$
$384$	0	0
$385$	3338.80	0.441976
$386$	0	0
$387$	4523.10	0.594113
$388$	0	0
$389$	9561.50	1.24624	0.623120	−	0.782127i	$-0.285865\pi$
$389$	9561.50	1.24624	0.623120	+	0.782127i	$0.285865\pi$
$390$	0	0
$391$	11771.0	1.52247
$392$	0	0
$393$	591.604	0.0759350
$394$	0	0
$395$	−2206.79	−0.281103
$396$	0	0
$397$	−7440.11	−0.940575	−0.470287	−	0.882513i	$-0.655850\pi$
$397$	−7440.11	−0.940575	−0.470287	+	0.882513i	$0.655850\pi$
$398$	0	0
$399$	370.799	0.0465242
$400$	0	0
$401$	−8687.80	−1.08192	−0.540958	−	0.841050i	$-0.681938\pi$
$401$	−8687.80	−1.08192	−0.540958	+	0.841050i	$0.681938\pi$
$402$	0	0
$403$	1524.68	0.188461
$404$	0	0
$405$	1578.15	0.193627
$406$	0	0
$407$	3544.13	0.431637
$408$	0	0
$409$	2556.10	0.309024	0.154512	−	0.987991i	$-0.450619\pi$
$409$	2556.10	0.309024	0.154512	+	0.987991i	$0.450619\pi$
$410$	0	0
$411$	−1640.74	−0.196915
$412$	0	0
$413$	−596.168	−0.0710303
$414$	0	0
$415$	18187.0	2.15124
$416$	0	0
$417$	−1829.00	−0.214788
$418$	0	0
$419$	3347.46	0.390296	0.195148	−	0.980774i	$-0.437481\pi$
$419$	3347.46	0.390296	0.195148	+	0.980774i	$0.437481\pi$
$420$	0	0
$421$	−1854.48	−0.214684	−0.107342	−	0.994222i	$-0.534234\pi$
$421$	−1854.48	−0.214684	−0.107342	+	0.994222i	$0.534234\pi$
$422$	0	0
$423$	2535.60	0.291454
$424$	0	0
$425$	17066.7	1.94789
$426$	0	0
$427$	1458.24	0.165267
$428$	0	0
$429$	893.096	0.100511
$430$	0	0
$431$	14043.1	1.56945	0.784725	−	0.619844i	$-0.212804\pi$
$431$	14043.1	1.56945	0.784725	+	0.619844i	$0.212804\pi$
$432$	0	0
$433$	3086.47	0.342555	0.171278	−	0.985223i	$-0.445210\pi$
$433$	3086.47	0.342555	0.171278	+	0.985223i	$0.445210\pi$
$434$	0	0
$435$	17055.7	1.87990
$436$	0	0
$437$	−2900.32	−0.317486
$438$	0	0
$439$	−2837.68	−0.308508	−0.154254	−	0.988031i	$-0.549297\pi$
$439$	−2837.68	−0.308508	−0.154254	+	0.988031i	$0.549297\pi$
$440$	0	0
$441$	−2583.00	−0.278912
$442$	0	0
$443$	−18309.4	−1.96367	−0.981834	−	0.189744i	$-0.939234\pi$
$443$	−18309.4	−1.96367	−0.981834	+	0.189744i	$0.939234\pi$
$444$	0	0
$445$	23203.0	2.47174
$446$	0	0
$447$	−6495.24	−0.687281
$448$	0	0
$449$	13861.2	1.45690	0.728451	−	0.685098i	$-0.240241\pi$
$449$	13861.2	1.45690	0.728451	+	0.685098i	$0.240241\pi$
$450$	0	0
$451$	5764.26	0.601837
$452$	0	0
$453$	2539.65	0.263406
$454$	0	0
$455$	1895.40	0.195291
$456$	0	0
$457$	−8990.36	−0.920243	−0.460122	−	0.887856i	$-0.652194\pi$
$457$	−8990.36	−0.920243	−0.460122	+	0.887856i	$0.652194\pi$
$458$	0	0
$459$	1809.90	0.184050
$460$	0	0
$461$	−3406.90	−0.344198	−0.172099	−	0.985080i	$-0.555055\pi$
$461$	−3406.90	−0.344198	−0.172099	+	0.985080i	$0.555055\pi$
$462$	0	0
$463$	7498.45	0.752662	0.376331	−	0.926485i	$-0.377186\pi$
$463$	7498.45	0.752662	0.376331	+	0.926485i	$0.377186\pi$
$464$	0	0
$465$	−6855.19	−0.683660
$466$	0	0
$467$	−7711.38	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$467$	−7711.38	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$468$	0	0
$469$	2993.94	0.294770
$470$	0	0
$471$	4960.79	0.485310
$472$	0	0
$473$	−11508.7	−1.11875
$474$	0	0
$475$	−4205.14	−0.406200
$476$	0	0
$477$	3302.99	0.317051
$478$	0	0
$479$	9439.82	0.900451	0.450226	−	0.892915i	$-0.351344\pi$
$479$	9439.82	0.900451	0.450226	+	0.892915i	$0.351344\pi$
$480$	0	0
$481$	2011.96	0.190723
$482$	0	0
$483$	−3942.20	−0.371380
$484$	0	0
$485$	10855.4	1.01633
$486$	0	0
$487$	6156.20	0.572821	0.286411	−	0.958107i	$-0.407538\pi$
$487$	6156.20	0.572821	0.286411	+	0.958107i	$0.407538\pi$
$488$	0	0
$489$	8599.54	0.795265
$490$	0	0
$491$	−3842.74	−0.353198	−0.176599	−	0.984283i	$-0.556510\pi$
$491$	−3842.74	−0.353198	−0.176599	+	0.984283i	$0.556510\pi$
$492$	0	0
$493$	19560.3	1.78692
$494$	0	0
$495$	−4015.49	−0.364612
$496$	0	0
$497$	3953.43	0.356812
$498$	0	0
$499$	12842.4	1.15211	0.576056	−	0.817410i	$-0.304591\pi$
$499$	12842.4	1.15211	0.576056	+	0.817410i	$0.304591\pi$
$500$	0	0
$501$	−2187.20	−0.195043
$502$	0	0
$503$	−8580.11	−0.760573	−0.380287	−	0.924869i	$-0.624175\pi$
$503$	−8580.11	−0.760573	−0.380287	+	0.924869i	$0.624175\pi$
$504$	0	0
$505$	−5587.16	−0.492327
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−43.5957	−0.00379635	−0.00189818	−	0.999998i	$-0.500604\pi$
$509$	−43.5957	−0.00379635	−0.00189818	+	0.999998i	$0.500604\pi$
$510$	0	0
$511$	5495.49	0.475746
$512$	0	0
$513$	−445.951	−0.0383805
$514$	0	0
$515$	37239.7	3.18636
$516$	0	0
$517$	−6451.66	−0.548827
$518$	0	0
$519$	−11504.5	−0.973008
$520$	0	0
$521$	11368.1	0.955939	0.477969	−	0.878377i	$-0.341373\pi$
$521$	11368.1	0.955939	0.477969	+	0.878377i	$0.341373\pi$
$522$	0	0
$523$	5229.53	0.437230	0.218615	−	0.975811i	$-0.429846\pi$
$523$	5229.53	0.437230	0.218615	+	0.975811i	$0.429846\pi$
$524$	0	0
$525$	−5715.75	−0.475154
$526$	0	0
$527$	−7861.88	−0.649846
$528$	0	0
$529$	18668.2	1.53433
$530$	0	0
$531$	716.997	0.0585970
$532$	0	0
$533$	3272.31	0.265927
$534$	0	0
$535$	−16255.6	−1.31363
$536$	0	0
$537$	851.586	0.0684333
$538$	0	0
$539$	6572.27	0.525209
$540$	0	0
$541$	−6567.99	−0.521959	−0.260980	−	0.965344i	$-0.584046\pi$
$541$	−6567.99	−0.521959	−0.260980	+	0.965344i	$0.584046\pi$
$542$	0	0
$543$	7090.79	0.560396
$544$	0	0
$545$	−20996.5	−1.65026
$546$	0	0
$547$	13675.7	1.06897	0.534487	−	0.845177i	$-0.320505\pi$
$547$	13675.7	1.06897	0.534487	+	0.845177i	$0.320505\pi$
$548$	0	0
$549$	−1753.79	−0.136339
$550$	0	0
$551$	−4819.57	−0.372632
$552$	0	0
$553$	847.604	0.0651786
$554$	0	0
$555$	−9046.09	−0.691865
$556$	0	0
$557$	4527.96	0.344445	0.172222	−	0.985058i	$-0.444905\pi$
$557$	4527.96	0.344445	0.172222	+	0.985058i	$0.444905\pi$
$558$	0	0
$559$	−6533.36	−0.494332
$560$	0	0
$561$	−4605.17	−0.346578
$562$	0	0
$563$	−18441.8	−1.38051	−0.690256	−	0.723566i	$-0.742502\pi$
$563$	−18441.8	−1.38051	−0.690256	+	0.723566i	$0.742502\pi$
$564$	0	0
$565$	−3235.49	−0.240917
$566$	0	0
$567$	−606.148	−0.0448957
$568$	0	0
$569$	−13553.5	−0.998578	−0.499289	−	0.866436i	$-0.666405\pi$
$569$	−13553.5	−0.998578	−0.499289	+	0.866436i	$0.666405\pi$
$570$	0	0
$571$	−14815.5	−1.08583	−0.542915	−	0.839788i	$-0.682679\pi$
$571$	−14815.5	−1.08583	−0.542915	+	0.839788i	$0.682679\pi$
$572$	0	0
$573$	−7542.79	−0.549921
$574$	0	0
$575$	44707.6	3.24249
$576$	0	0
$577$	21596.2	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$577$	21596.2	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$578$	0	0
$579$	7262.19	0.521254
$580$	0	0
$581$	−6985.42	−0.498802
$582$	0	0
$583$	−8404.23	−0.597029
$584$	0	0
$585$	−2279.55	−0.161107
$586$	0	0
$587$	918.801	0.0646047	0.0323024	−	0.999478i	$-0.489716\pi$
$587$	918.801	0.0646047	0.0323024	+	0.999478i	$0.489716\pi$
$588$	0	0
$589$	1937.13	0.135514
$590$	0	0
$591$	−13900.9	−0.967527
$592$	0	0
$593$	19816.0	1.37226	0.686128	−	0.727481i	$-0.259309\pi$
$593$	19816.0	1.37226	0.686128	+	0.727481i	$0.259309\pi$
$594$	0	0
$595$	−9773.45	−0.673399
$596$	0	0
$597$	−9162.50	−0.628134
$598$	0	0
$599$	5141.86	0.350736	0.175368	−	0.984503i	$-0.443889\pi$
$599$	5141.86	0.350736	0.175368	+	0.984503i	$0.443889\pi$
$600$	0	0
$601$	12380.9	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$601$	12380.9	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$602$	0	0
$603$	−3600.74	−0.243173
$604$	0	0
$605$	−15715.1	−1.05605
$606$	0	0
$607$	23717.0	1.58590	0.792951	−	0.609286i	$-0.208544\pi$
$607$	23717.0	1.58590	0.792951	+	0.609286i	$0.208544\pi$
$608$	0	0
$609$	−6550.89	−0.435887
$610$	0	0
$611$	−3662.53	−0.242504
$612$	0	0
$613$	−26157.1	−1.72345	−0.861726	−	0.507373i	$-0.830617\pi$
$613$	−26157.1	−1.72345	−0.861726	+	0.507373i	$0.830617\pi$
$614$	0	0
$615$	−14712.8	−0.964677
$616$	0	0
$617$	23613.9	1.54077	0.770387	−	0.637576i	$-0.220063\pi$
$617$	23613.9	1.54077	0.770387	+	0.637576i	$0.220063\pi$
$618$	0	0
$619$	−23345.4	−1.51588	−0.757940	−	0.652324i	$-0.773794\pi$
$619$	−23345.4	−1.51588	−0.757940	+	0.652324i	$0.773794\pi$
$620$	0	0
$621$	4741.19	0.306372
$622$	0	0
$623$	−8911.99	−0.573116
$624$	0	0
$625$	17371.0	1.11174
$626$	0	0
$627$	1134.69	0.0722730
$628$	0	0
$629$	−10374.5	−0.657645
$630$	0	0
$631$	−15245.7	−0.961841	−0.480921	−	0.876764i	$-0.659698\pi$
$631$	−15245.7	−0.961841	−0.480921	+	0.876764i	$0.659698\pi$
$632$	0	0
$633$	12094.8	0.759439
$634$	0	0
$635$	−25253.6	−1.57820
$636$	0	0
$637$	3731.00	0.232068
$638$	0	0
$639$	−4754.69	−0.294355
$640$	0	0
$641$	10192.7	0.628063	0.314032	−	0.949413i	$-0.398320\pi$
$641$	10192.7	0.628063	0.314032	+	0.949413i	$0.398320\pi$
$642$	0	0
$643$	5506.31	0.337710	0.168855	−	0.985641i	$-0.445993\pi$
$643$	5506.31	0.337710	0.168855	+	0.985641i	$0.445993\pi$
$644$	0	0
$645$	29375.0	1.79324
$646$	0	0
$647$	13297.5	0.808005	0.404003	−	0.914758i	$-0.367619\pi$
$647$	13297.5	0.808005	0.404003	+	0.914758i	$0.367619\pi$
$648$	0	0
$649$	−1824.35	−0.110342
$650$	0	0
$651$	2633.00	0.158518
$652$	0	0
$653$	−12440.2	−0.745519	−0.372760	−	0.927928i	$-0.621588\pi$
$653$	−12440.2	−0.745519	−0.372760	+	0.927928i	$0.621588\pi$
$654$	0	0
$655$	3842.14	0.229198
$656$	0	0
$657$	−6609.29	−0.392470
$658$	0	0
$659$	9562.87	0.565276	0.282638	−	0.959227i	$-0.408791\pi$
$659$	9562.87	0.565276	0.282638	+	0.959227i	$0.408791\pi$
$660$	0	0
$661$	2409.69	0.141795	0.0708973	−	0.997484i	$-0.477414\pi$
$661$	2409.69	0.141795	0.0708973	+	0.997484i	$0.477414\pi$
$662$	0	0
$663$	−2614.30	−0.153139
$664$	0	0
$665$	2408.13	0.140426
$666$	0	0
$667$	51239.9	2.97454
$668$	0	0
$669$	−11354.9	−0.656209
$670$	0	0
$671$	4462.40	0.256735
$672$	0	0
$673$	7929.02	0.454147	0.227074	−	0.973878i	$-0.427084\pi$
$673$	7929.02	0.454147	0.227074	+	0.973878i	$0.427084\pi$
$674$	0	0
$675$	6874.19	0.391982
$676$	0	0
$677$	−2628.26	−0.149206	−0.0746030	−	0.997213i	$-0.523769\pi$
$677$	−2628.26	−0.149206	−0.0746030	+	0.997213i	$0.523769\pi$
$678$	0	0
$679$	−4169.44	−0.235653
$680$	0	0
$681$	−6041.48	−0.339956
$682$	0	0
$683$	−10021.5	−0.561437	−0.280719	−	0.959790i	$-0.590573\pi$
$683$	−10021.5	−0.561437	−0.280719	+	0.959790i	$0.590573\pi$
$684$	0	0
$685$	−10655.7	−0.594356
$686$	0	0
$687$	−9152.18	−0.508264
$688$	0	0
$689$	−4770.99	−0.263803
$690$	0	0
$691$	−23987.2	−1.32057	−0.660286	−	0.751014i	$-0.729565\pi$
$691$	−23987.2	−1.32057	−0.660286	+	0.751014i	$0.729565\pi$
$692$	0	0
$693$	1542.30	0.0845415
$694$	0	0
$695$	−11878.3	−0.648303
$696$	0	0
$697$	−16873.4	−0.916964
$698$	0	0
$699$	16762.5	0.907032
$700$	0	0
$701$	−3763.71	−0.202787	−0.101393	−	0.994846i	$-0.532330\pi$
$701$	−3763.71	−0.202787	−0.101393	+	0.994846i	$0.532330\pi$
$702$	0	0
$703$	2556.23	0.137141
$704$	0	0
$705$	16467.3	0.879707
$706$	0	0
$707$	2145.96	0.114155
$708$	0	0
$709$	−36047.8	−1.90946	−0.954728	−	0.297479i	$-0.903854\pi$
$709$	−36047.8	−1.90946	−0.954728	+	0.297479i	$0.903854\pi$
$710$	0	0
$711$	−1019.39	−0.0537696
$712$	0	0
$713$	−20594.9	−1.08174
$714$	0	0
$715$	5800.15	0.303376
$716$	0	0
$717$	4007.08	0.208713
$718$	0	0
$719$	3944.18	0.204580	0.102290	−	0.994755i	$-0.467383\pi$
$719$	3944.18	0.204580	0.102290	+	0.994755i	$0.467383\pi$
$720$	0	0
$721$	−14303.3	−0.738812
$722$	0	0
$723$	−1714.68	−0.0882012
$724$	0	0
$725$	74292.1	3.80571
$726$	0	0
$727$	20447.8	1.04315	0.521573	−	0.853206i	$-0.325345\pi$
$727$	20447.8	1.04315	0.521573	+	0.853206i	$0.325345\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	33688.7	1.70454
$732$	0	0
$733$	−13536.2	−0.682089	−0.341045	−	0.940047i	$-0.610781\pi$
$733$	−13536.2	−0.682089	−0.341045	+	0.940047i	$0.610781\pi$
$734$	0	0
$735$	−16775.1	−0.841851
$736$	0	0
$737$	9161.83	0.457911
$738$	0	0
$739$	−15839.1	−0.788433	−0.394217	−	0.919018i	$-0.628984\pi$
$739$	−15839.1	−0.788433	−0.394217	+	0.919018i	$0.628984\pi$
$740$	0	0
$741$	644.151	0.0319345
$742$	0	0
$743$	1664.92	0.0822075	0.0411037	−	0.999155i	$-0.486913\pi$
$743$	1664.92	0.0822075	0.0411037	+	0.999155i	$0.486913\pi$
$744$	0	0
$745$	−42182.9	−2.07445
$746$	0	0
$747$	8401.19	0.411491
$748$	0	0
$749$	6243.58	0.304587
$750$	0	0
$751$	−22399.1	−1.08835	−0.544177	−	0.838970i	$-0.683158\pi$
$751$	−22399.1	−1.08835	−0.544177	+	0.838970i	$0.683158\pi$
$752$	0	0
$753$	−12265.8	−0.593613
$754$	0	0
$755$	16493.6	0.795050
$756$	0	0
$757$	23798.9	1.14265	0.571326	−	0.820723i	$-0.306429\pi$
$757$	23798.9	1.14265	0.571326	+	0.820723i	$0.306429\pi$
$758$	0	0
$759$	−12063.6	−0.576920
$760$	0	0
$761$	13693.5	0.652285	0.326142	−	0.945321i	$-0.394251\pi$
$761$	13693.5	0.652285	0.326142	+	0.945321i	$0.394251\pi$
$762$	0	0
$763$	8064.50	0.382640
$764$	0	0
$765$	11754.3	0.555526
$766$	0	0
$767$	−1035.66	−0.0487556
$768$	0	0
$769$	16299.9	0.764358	0.382179	−	0.924088i	$-0.375174\pi$
$769$	16299.9	0.764358	0.382179	+	0.924088i	$0.375174\pi$
$770$	0	0
$771$	9150.68	0.427437
$772$	0	0
$773$	33532.2	1.56024	0.780122	−	0.625628i	$-0.215157\pi$
$773$	33532.2	1.56024	0.780122	+	0.625628i	$0.215157\pi$
$774$	0	0
$775$	−29860.2	−1.38401
$776$	0	0
$777$	3474.50	0.160421
$778$	0	0
$779$	4157.51	0.191217
$780$	0	0
$781$	12098.0	0.554290
$782$	0	0
$783$	7878.59	0.359589
$784$	0	0
$785$	32217.5	1.46483
$786$	0	0
$787$	−16163.3	−0.732097	−0.366049	−	0.930596i	$-0.619290\pi$
$787$	−16163.3	−0.732097	−0.366049	+	0.930596i	$0.619290\pi$
$788$	0	0
$789$	−17313.0	−0.781189
$790$	0	0
$791$	1242.71	0.0558607
$792$	0	0
$793$	2533.25	0.113441
$794$	0	0
$795$	21451.1	0.956970
$796$	0	0
$797$	−39636.4	−1.76160	−0.880798	−	0.473492i	$-0.842993\pi$
$797$	−39636.4	−1.76160	−0.880798	+	0.473492i	$0.842993\pi$
$798$	0	0
$799$	18885.5	0.836197
$800$	0	0
$801$	10718.2	0.472797
$802$	0	0
$803$	16816.9	0.739048
$804$	0	0
$805$	−25602.4	−1.12095
$806$	0	0
$807$	−6238.21	−0.272113
$808$	0	0
$809$	−23811.2	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$809$	−23811.2	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$810$	0	0
$811$	−27218.6	−1.17851	−0.589256	−	0.807946i	$-0.700579\pi$
$811$	−27218.6	−1.17851	−0.589256	+	0.807946i	$0.700579\pi$
$812$	0	0
$813$	−18036.0	−0.778045
$814$	0	0
$815$	55849.2	2.40038
$816$	0	0
$817$	−8300.73	−0.355454
$818$	0	0
$819$	875.548	0.0373555
$820$	0	0
$821$	−43094.8	−1.83193	−0.915967	−	0.401253i	$-0.868575\pi$
$821$	−43094.8	−1.83193	−0.915967	+	0.401253i	$0.868575\pi$
$822$	0	0
$823$	−26541.1	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$823$	−26541.1	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$824$	0	0
$825$	−17490.9	−0.738127
$826$	0	0
$827$	44898.7	1.88788	0.943942	−	0.330112i	$-0.107087\pi$
$827$	44898.7	1.88788	0.943942	+	0.330112i	$0.107087\pi$
$828$	0	0
$829$	−7137.48	−0.299029	−0.149514	−	0.988760i	$-0.547771\pi$
$829$	−7137.48	−0.299029	−0.149514	+	0.988760i	$0.547771\pi$
$830$	0	0
$831$	−2205.60	−0.0920717
$832$	0	0
$833$	−19238.6	−0.800213
$834$	0	0
$835$	−14204.6	−0.588708
$836$	0	0
$837$	−3166.64	−0.130771
$838$	0	0
$839$	4387.17	0.180527	0.0902634	−	0.995918i	$-0.471229\pi$
$839$	4387.17	0.180527	0.0902634	+	0.995918i	$0.471229\pi$
$840$	0	0
$841$	60758.1	2.49121
$842$	0	0
$843$	−5708.76	−0.233239
$844$	0	0
$845$	3292.68	0.134049
$846$	0	0
$847$	6036.01	0.244864
$848$	0	0
$849$	−6377.12	−0.257788
$850$	0	0
$851$	−27176.9	−1.09473
$852$	0	0
$853$	−9328.85	−0.374459	−0.187230	−	0.982316i	$-0.559951\pi$
$853$	−9328.85	−0.374459	−0.187230	+	0.982316i	$0.559951\pi$
$854$	0	0
$855$	−2896.20	−0.115845
$856$	0	0
$857$	−5010.39	−0.199710	−0.0998552	−	0.995002i	$-0.531838\pi$
$857$	−5010.39	−0.199710	−0.0998552	+	0.995002i	$0.531838\pi$
$858$	0	0
$859$	−30233.4	−1.20088	−0.600438	−	0.799672i	$-0.705007\pi$
$859$	−30233.4	−1.20088	−0.600438	+	0.799672i	$0.705007\pi$
$860$	0	0
$861$	5651.01	0.223677
$862$	0	0
$863$	−4334.93	−0.170988	−0.0854940	−	0.996339i	$-0.527247\pi$
$863$	−4334.93	−0.170988	−0.0854940	+	0.996339i	$0.527247\pi$
$864$	0	0
$865$	−74715.2	−2.93687
$866$	0	0
$867$	−1258.58	−0.0493007
$868$	0	0
$869$	2593.78	0.101252
$870$	0	0
$871$	5201.06	0.202332
$872$	0	0
$873$	5014.48	0.194404
$874$	0	0
$875$	−18895.6	−0.730043
$876$	0	0
$877$	34683.3	1.33543	0.667716	−	0.744416i	$-0.267272\pi$
$877$	34683.3	1.33543	0.667716	+	0.744416i	$0.267272\pi$
$878$	0	0
$879$	−4923.08	−0.188909
$880$	0	0
$881$	−18269.2	−0.698642	−0.349321	−	0.937003i	$-0.613588\pi$
$881$	−18269.2	−0.698642	−0.349321	+	0.937003i	$0.613588\pi$
$882$	0	0
$883$	14592.0	0.556128	0.278064	−	0.960563i	$-0.410307\pi$
$883$	14592.0	0.556128	0.278064	+	0.960563i	$0.410307\pi$
$884$	0	0
$885$	4656.49	0.176866
$886$	0	0
$887$	−30459.3	−1.15301	−0.576507	−	0.817092i	$-0.695585\pi$
$887$	−30459.3	−1.15301	−0.576507	+	0.817092i	$0.695585\pi$
$888$	0	0
$889$	9699.60	0.365933
$890$	0	0
$891$	−1854.89	−0.0697432
$892$	0	0
$893$	−4653.30	−0.174375
$894$	0	0
$895$	5530.57	0.206555
$896$	0	0
$897$	−6848.38	−0.254917
$898$	0	0
$899$	−34223.2	−1.26964
$900$	0	0
$901$	24601.2	0.909638
$902$	0	0
$903$	−11282.6	−0.415793
$904$	0	0
$905$	46050.7	1.69147
$906$	0	0
$907$	9364.89	0.342840	0.171420	−	0.985198i	$-0.445164\pi$
$907$	9364.89	0.342840	0.171420	+	0.985198i	$0.445164\pi$
$908$	0	0
$909$	−2580.90	−0.0941727
$910$	0	0
$911$	−32479.8	−1.18123	−0.590616	−	0.806952i	$-0.701115\pi$
$911$	−32479.8	−1.18123	−0.590616	+	0.806952i	$0.701115\pi$
$912$	0	0
$913$	−21376.3	−0.774864
$914$	0	0
$915$	−11389.9	−0.411516
$916$	0	0
$917$	−1475.72	−0.0531435
$918$	0	0
$919$	−295.958	−0.0106232	−0.00531161	−	0.999986i	$-0.501691\pi$
$919$	−295.958	−0.0106232	−0.00531161	+	0.999986i	$0.501691\pi$
$920$	0	0
$921$	10119.8	0.362062
$922$	0	0
$923$	6867.89	0.244918
$924$	0	0
$925$	−39403.5	−1.40062
$926$	0	0
$927$	17202.3	0.609489
$928$	0	0
$929$	−5620.38	−0.198492	−0.0992458	−	0.995063i	$-0.531643\pi$
$929$	−5620.38	−0.198492	−0.0992458	+	0.995063i	$0.531643\pi$
$930$	0	0
$931$	4740.29	0.166871
$932$	0	0
$933$	2605.58	0.0914284
$934$	0	0
$935$	−29908.0	−1.04609
$936$	0	0
$937$	−32583.1	−1.13601	−0.568006	−	0.823024i	$-0.692285\pi$
$937$	−32583.1	−1.13601	−0.568006	+	0.823024i	$0.692285\pi$
$938$	0	0
$939$	−13029.6	−0.452827
$940$	0	0
$941$	8812.99	0.305308	0.152654	−	0.988280i	$-0.451218\pi$
$941$	8812.99	0.305308	0.152654	+	0.988280i	$0.451218\pi$
$942$	0	0
$943$	−44201.2	−1.52639
$944$	0	0
$945$	−3936.59	−0.135510
$946$	0	0
$947$	−13426.8	−0.460732	−0.230366	−	0.973104i	$-0.573992\pi$
$947$	−13426.8	−0.460732	−0.230366	+	0.973104i	$0.573992\pi$
$948$	0	0
$949$	9546.76	0.326555
$950$	0	0
$951$	−9832.96	−0.335285
$952$	0	0
$953$	−13394.6	−0.455293	−0.227647	−	0.973744i	$-0.573103\pi$
$953$	−13394.6	−0.455293	−0.227647	+	0.973744i	$0.573103\pi$
$954$	0	0
$955$	−48986.2	−1.65985
$956$	0	0
$957$	−20046.5	−0.677129
$958$	0	0
$959$	4092.74	0.137812
$960$	0	0
$961$	−16035.7	−0.538273
$962$	0	0
$963$	−7509.00	−0.251271
$964$	0	0
$965$	47163.8	1.57332
$966$	0	0
$967$	−45590.8	−1.51613	−0.758066	−	0.652178i	$-0.773856\pi$
$967$	−45590.8	−1.51613	−0.758066	+	0.652178i	$0.773856\pi$
$968$	0	0
$969$	−3321.51	−0.110116
$970$	0	0
$971$	−264.763	−0.00875041	−0.00437521	−	0.999990i	$-0.501393\pi$
$971$	−264.763	−0.00875041	−0.00437521	+	0.999990i	$0.501393\pi$
$972$	0	0
$973$	4562.32	0.150320
$974$	0	0
$975$	−9929.38	−0.326148
$976$	0	0
$977$	610.521	0.0199921	0.00999606	−	0.999950i	$-0.496818\pi$
$977$	610.521	0.0199921	0.00999606	+	0.999950i	$0.496818\pi$
$978$	0	0
$979$	−27271.8	−0.890308
$980$	0	0
$981$	−9698.98	−0.315662
$982$	0	0
$983$	57829.7	1.87638	0.938190	−	0.346121i	$-0.112501\pi$
$983$	57829.7	1.87638	0.938190	+	0.346121i	$0.112501\pi$
$984$	0	0
$985$	−90278.8	−2.92033
$986$	0	0
$987$	−6324.89	−0.203975
$988$	0	0
$989$	88250.4	2.83741
$990$	0	0
$991$	56780.7	1.82008	0.910039	−	0.414522i	$-0.136051\pi$
$991$	56780.7	1.82008	0.910039	+	0.414522i	$0.136051\pi$
$992$	0	0
$993$	−16768.9	−0.535895
$994$	0	0
$995$	−59505.3	−1.89592
$996$	0	0
$997$	18616.6	0.591369	0.295684	−	0.955286i	$-0.404452\pi$
$997$	18616.6	0.591369	0.295684	+	0.955286i	$0.404452\pi$
$998$	0	0
$999$	−4178.69	−0.132340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.r.1.2		2
3.2	odd	2		1872.4.a.t.1.1		2
4.3	odd	2		39.4.a.b.1.1	✓	2
8.3	odd	2		2496.4.a.bc.1.1		2
8.5	even	2		2496.4.a.s.1.1		2
12.11	even	2		117.4.a.c.1.2		2
20.19	odd	2		975.4.a.j.1.2		2
28.27	even	2		1911.4.a.h.1.1		2
52.31	even	4		507.4.b.f.337.3		4
52.47	even	4		507.4.b.f.337.2		4
52.51	odd	2		507.4.a.f.1.2		2
156.155	even	2		1521.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	4.3	odd	2
117.4.a.c.1.2		2	12.11	even	2
507.4.a.f.1.2		2	52.51	odd	2
507.4.b.f.337.2		4	52.47	even	4
507.4.b.f.337.3		4	52.31	even	4
624.4.a.r.1.2		2	1.1	even	1	trivial
975.4.a.j.1.2		2	20.19	odd	2
1521.4.a.s.1.1		2	156.155	even	2
1872.4.a.t.1.1		2	3.2	odd	2
1911.4.a.h.1.1		2	28.27	even	2
2496.4.a.s.1.1		2	8.5	even	2
2496.4.a.bc.1.1		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +19.4833 q^{5} -7.48331 q^{7} +9.00000 q^{9} -22.8999 q^{11} -13.0000 q^{13} +58.4499 q^{15} +67.0334 q^{17} -16.5167 q^{19} -22.4499 q^{21} +175.600 q^{23} +254.600 q^{25} +27.0000 q^{27} +291.800 q^{29} -117.283 q^{31} -68.6997 q^{33} -145.800 q^{35} -154.766 q^{37} -39.0000 q^{39} -251.716 q^{41} +502.566 q^{43} +175.350 q^{45} +281.733 q^{47} -287.000 q^{49} +201.100 q^{51} +366.999 q^{53} -446.166 q^{55} -49.5501 q^{57} +79.6663 q^{59} -194.865 q^{61} -67.3498 q^{63} -253.283 q^{65} -400.082 q^{67} +526.799 q^{69} -528.299 q^{71} -734.366 q^{73} +763.799 q^{75} +171.367 q^{77} -113.266 q^{79} +81.0000 q^{81} +933.466 q^{83} +1306.03 q^{85} +875.399 q^{87} +1190.91 q^{89} +97.2831 q^{91} -351.849 q^{93} -321.800 q^{95} +557.165 q^{97} -206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39}+ \cdots + 396 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 396 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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