LMFDB - Embedded newform 2312.4.a.r.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$136.412415933$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57629 q^{3} -11.2335 q^{5} -14.3042 q^{7} -24.5153 q^{9} +O(q^{10})$	$q+1.57629 q^{3} -11.2335 q^{5} -14.3042 q^{7} -24.5153 q^{9} -42.8514 q^{11} +70.4740 q^{13} -17.7073 q^{15} -36.6315 q^{19} -22.5476 q^{21} +182.516 q^{23} +1.19183 q^{25} -81.2033 q^{27} +264.558 q^{29} +341.852 q^{31} -67.5465 q^{33} +160.687 q^{35} +65.5704 q^{37} +111.088 q^{39} +15.1328 q^{41} +149.301 q^{43} +275.393 q^{45} -4.93068 q^{47} -138.390 q^{49} -658.873 q^{53} +481.372 q^{55} -57.7420 q^{57} +105.529 q^{59} +588.185 q^{61} +350.672 q^{63} -791.671 q^{65} -952.688 q^{67} +287.700 q^{69} +333.247 q^{71} -673.669 q^{73} +1.87868 q^{75} +612.956 q^{77} -525.620 q^{79} +533.913 q^{81} +369.953 q^{83} +417.021 q^{87} -234.437 q^{89} -1008.08 q^{91} +538.860 q^{93} +411.500 q^{95} -134.987 q^{97} +1050.52 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Display 100 coefficients 10 coefficients

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
$3$	1.57629	0.303358	0.151679	−	0.988430i	$-0.451532\pi$
$3$	1.57629	0.303358	0.151679	+	0.988430i	$0.451532\pi$
$4$	0	0
$5$	−11.2335	−1.00476	−0.502378	−	0.864648i	$-0.667541\pi$
$5$	−11.2335	−1.00476	−0.502378	+	0.864648i	$0.667541\pi$
$6$	0	0
$7$	−14.3042	−0.772355	−0.386177	−	0.922425i	$-0.626205\pi$
$7$	−14.3042	−0.772355	−0.386177	+	0.922425i	$0.626205\pi$
$8$	0	0
$9$	−24.5153	−0.907974
$10$	0	0
$11$	−42.8514	−1.17456	−0.587282	−	0.809383i	$-0.699802\pi$
$11$	−42.8514	−1.17456	−0.587282	+	0.809383i	$0.699802\pi$
$12$	0	0
$13$	70.4740	1.50354	0.751769	−	0.659427i	$-0.229201\pi$
$13$	70.4740	1.50354	0.751769	+	0.659427i	$0.229201\pi$
$14$	0	0
$15$	−17.7073	−0.304801
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−36.6315	−0.442307	−0.221154	−	0.975239i	$-0.570982\pi$
$19$	−36.6315	−0.442307	−0.221154	+	0.975239i	$0.570982\pi$
$20$	0	0
$21$	−22.5476	−0.234300
$22$	0	0
$23$	182.516	1.65467	0.827333	−	0.561712i	$-0.189857\pi$
$23$	182.516	1.65467	0.827333	+	0.561712i	$0.189857\pi$
$24$	0	0
$25$	1.19183	0.00953467
$26$	0	0
$27$	−81.2033	−0.578799
$28$	0	0
$29$	264.558	1.69404	0.847021	−	0.531560i	$-0.178394\pi$
$29$	264.558	1.69404	0.847021	+	0.531560i	$0.178394\pi$
$30$	0	0
$31$	341.852	1.98060	0.990298	−	0.138961i	$-0.0443764\pi$
$31$	341.852	1.98060	0.990298	+	0.138961i	$0.0443764\pi$
$32$	0	0
$33$	−67.5465	−0.356313
$34$	0	0
$35$	160.687	0.776028
$36$	0	0
$37$	65.5704	0.291344	0.145672	−	0.989333i	$-0.453466\pi$
$37$	65.5704	0.291344	0.145672	+	0.989333i	$0.453466\pi$
$38$	0	0
$39$	111.088	0.456110
$40$	0	0
$41$	15.1328	0.0576426	0.0288213	−	0.999585i	$-0.490825\pi$
$41$	15.1328	0.0576426	0.0288213	+	0.999585i	$0.490825\pi$
$42$	0	0
$43$	149.301	0.529492	0.264746	−	0.964318i	$-0.414712\pi$
$43$	149.301	0.529492	0.264746	+	0.964318i	$0.414712\pi$
$44$	0	0
$45$	275.393	0.912292
$46$	0	0
$47$	−4.93068	−0.0153024	−0.00765120	−	0.999971i	$-0.502435\pi$
$47$	−4.93068	−0.0153024	−0.00765120	+	0.999971i	$0.502435\pi$
$48$	0	0
$49$	−138.390	−0.403468
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−658.873	−1.70761	−0.853803	−	0.520596i	$-0.825710\pi$
$53$	−658.873	−1.70761	−0.853803	+	0.520596i	$0.825710\pi$
$54$	0	0
$55$	481.372	1.18015
$56$	0	0
$57$	−57.7420	−0.134178
$58$	0	0
$59$	105.529	0.232858	0.116429	−	0.993199i	$-0.462855\pi$
$59$	105.529	0.232858	0.116429	+	0.993199i	$0.462855\pi$
$60$	0	0
$61$	588.185	1.23458	0.617290	−	0.786736i	$-0.288230\pi$
$61$	588.185	1.23458	0.617290	+	0.786736i	$0.288230\pi$
$62$	0	0
$63$	350.672	0.701278
$64$	0	0
$65$	−791.671	−1.51069
$66$	0	0
$67$	−952.688	−1.73716	−0.868578	−	0.495553i	$-0.834965\pi$
$67$	−952.688	−1.73716	−0.868578	+	0.495553i	$0.834965\pi$
$68$	0	0
$69$	287.700	0.501956
$70$	0	0
$71$	333.247	0.557030	0.278515	−	0.960432i	$-0.410158\pi$
$71$	333.247	0.557030	0.278515	+	0.960432i	$0.410158\pi$
$72$	0	0
$73$	−673.669	−1.08010	−0.540048	−	0.841634i	$-0.681594\pi$
$73$	−673.669	−1.08010	−0.540048	+	0.841634i	$0.681594\pi$
$74$	0	0
$75$	1.87868	0.00289242
$76$	0	0
$77$	612.956	0.907179
$78$	0	0
$79$	−525.620	−0.748568	−0.374284	−	0.927314i	$-0.622112\pi$
$79$	−525.620	−0.748568	−0.374284	+	0.927314i	$0.622112\pi$
$80$	0	0
$81$	533.913	0.732391
$82$	0	0
$83$	369.953	0.489248	0.244624	−	0.969618i	$-0.421335\pi$
$83$	369.953	0.489248	0.244624	+	0.969618i	$0.421335\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	417.021	0.513901
$88$	0	0
$89$	−234.437	−0.279217	−0.139608	−	0.990207i	$-0.544584\pi$
$89$	−234.437	−0.279217	−0.139608	+	0.990207i	$0.544584\pi$
$90$	0	0
$91$	−1008.08	−1.16126
$92$	0	0
$93$	538.860	0.600830
$94$	0	0
$95$	411.500	0.444411
$96$	0	0
$97$	−134.987	−0.141297	−0.0706487	−	0.997501i	$-0.522507\pi$
$97$	−134.987	−0.141297	−0.0706487	+	0.997501i	$0.522507\pi$
$98$	0	0
$99$	1050.52	1.06647
$100$	0	0
$101$	−491.153	−0.483877	−0.241938	−	0.970292i	$-0.577783\pi$
$101$	−491.153	−0.483877	−0.241938	+	0.970292i	$0.577783\pi$
$102$	0	0
$103$	−1673.02	−1.60046	−0.800229	−	0.599695i	$-0.795289\pi$
$103$	−1673.02	−1.60046	−0.800229	+	0.599695i	$0.795289\pi$
$104$	0	0
$105$	253.289	0.235414
$106$	0	0
$107$	36.3413	0.0328341	0.0164171	−	0.999865i	$-0.494774\pi$
$107$	36.3413	0.0328341	0.0164171	+	0.999865i	$0.494774\pi$
$108$	0	0
$109$	485.516	0.426642	0.213321	−	0.976982i	$-0.431572\pi$
$109$	485.516	0.426642	0.213321	+	0.976982i	$0.431572\pi$
$110$	0	0
$111$	103.358	0.0883814
$112$	0	0
$113$	−1866.06	−1.55349	−0.776745	−	0.629815i	$-0.783130\pi$
$113$	−1866.06	−1.55349	−0.776745	+	0.629815i	$0.783130\pi$
$114$	0	0
$115$	−2050.30	−1.66254
$116$	0	0
$117$	−1727.69	−1.36517
$118$	0	0
$119$	0	0
$120$	0	0
$121$	505.247	0.379599
$122$	0	0
$123$	23.8537	0.0174863
$124$	0	0
$125$	1390.80	0.995176
$126$	0	0
$127$	1197.60	0.836768	0.418384	−	0.908270i	$-0.362597\pi$
$127$	1197.60	0.836768	0.418384	+	0.908270i	$0.362597\pi$
$128$	0	0
$129$	235.342	0.160626
$130$	0	0
$131$	−329.385	−0.219683	−0.109842	−	0.993949i	$-0.535034\pi$
$131$	−329.385	−0.219683	−0.109842	+	0.993949i	$0.535034\pi$
$132$	0	0
$133$	523.984	0.341618
$134$	0	0
$135$	912.198	0.581552
$136$	0	0
$137$	1656.76	1.03318	0.516592	−	0.856231i	$-0.327200\pi$
$137$	1656.76	1.03318	0.516592	+	0.856231i	$0.327200\pi$
$138$	0	0
$139$	−1068.15	−0.651797	−0.325898	−	0.945405i	$-0.605667\pi$
$139$	−1068.15	−0.651797	−0.325898	+	0.945405i	$0.605667\pi$
$140$	0	0
$141$	−7.77220	−0.00464211
$142$	0	0
$143$	−3019.91	−1.76600
$144$	0	0
$145$	−2971.92	−1.70210
$146$	0	0
$147$	−218.143	−0.122395
$148$	0	0
$149$	−1746.65	−0.960344	−0.480172	−	0.877174i	$-0.659426\pi$
$149$	−1746.65	−0.960344	−0.480172	+	0.877174i	$0.659426\pi$
$150$	0	0
$151$	−2331.41	−1.25647	−0.628236	−	0.778023i	$-0.716223\pi$
$151$	−2331.41	−1.25647	−0.628236	+	0.778023i	$0.716223\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3840.20	−1.99002
$156$	0	0
$157$	−289.904	−0.147369	−0.0736843	−	0.997282i	$-0.523476\pi$
$157$	−289.904	−0.147369	−0.0736843	+	0.997282i	$0.523476\pi$
$158$	0	0
$159$	−1038.58	−0.518016
$160$	0	0
$161$	−2610.75	−1.27799
$162$	0	0
$163$	−532.716	−0.255985	−0.127992	−	0.991775i	$-0.540853\pi$
$163$	−532.716	−0.255985	−0.127992	+	0.991775i	$0.540853\pi$
$164$	0	0
$165$	758.785	0.358008
$166$	0	0
$167$	−2528.48	−1.17161	−0.585807	−	0.810451i	$-0.699222\pi$
$167$	−2528.48	−1.17161	−0.585807	+	0.810451i	$0.699222\pi$
$168$	0	0
$169$	2769.59	1.26062
$170$	0	0
$171$	898.032	0.401604
$172$	0	0
$173$	2905.96	1.27709	0.638543	−	0.769586i	$-0.279537\pi$
$173$	2905.96	1.27709	0.638543	+	0.769586i	$0.279537\pi$
$174$	0	0
$175$	−17.0482	−0.00736415
$176$	0	0
$177$	166.344	0.0706395
$178$	0	0
$179$	−936.886	−0.391208	−0.195604	−	0.980683i	$-0.562667\pi$
$179$	−936.886	−0.391208	−0.195604	+	0.980683i	$0.562667\pi$
$180$	0	0
$181$	−2005.28	−0.823487	−0.411744	−	0.911300i	$-0.635080\pi$
$181$	−2005.28	−0.823487	−0.411744	+	0.911300i	$0.635080\pi$
$182$	0	0
$183$	927.153	0.374520
$184$	0	0
$185$	−736.587	−0.292729
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1161.55	0.447038
$190$	0	0
$191$	1488.35	0.563841	0.281920	−	0.959438i	$-0.409029\pi$
$191$	1488.35	0.563841	0.281920	+	0.959438i	$0.409029\pi$
$192$	0	0
$193$	−67.7080	−0.0252525	−0.0126262	−	0.999920i	$-0.504019\pi$
$193$	−67.7080	−0.0252525	−0.0126262	+	0.999920i	$0.504019\pi$
$194$	0	0
$195$	−1247.91	−0.458279
$196$	0	0
$197$	−274.106	−0.0991332	−0.0495666	−	0.998771i	$-0.515784\pi$
$197$	−274.106	−0.0991332	−0.0495666	+	0.998771i	$0.515784\pi$
$198$	0	0
$199$	2649.00	0.943632	0.471816	−	0.881697i	$-0.343599\pi$
$199$	2649.00	0.943632	0.471816	+	0.881697i	$0.343599\pi$
$200$	0	0
$201$	−1501.72	−0.526980
$202$	0	0
$203$	−3784.29	−1.30840
$204$	0	0
$205$	−169.994	−0.0579167
$206$	0	0
$207$	−4474.45	−1.50239
$208$	0	0
$209$	1569.71	0.519518
$210$	0	0
$211$	2804.76	0.915106	0.457553	−	0.889182i	$-0.348726\pi$
$211$	2804.76	0.915106	0.457553	+	0.889182i	$0.348726\pi$
$212$	0	0
$213$	525.295	0.168980
$214$	0	0
$215$	−1677.17	−0.532010
$216$	0	0
$217$	−4889.92	−1.52972
$218$	0	0
$219$	−1061.90	−0.327656
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4641.76	1.39388	0.696941	−	0.717129i	$-0.254544\pi$
$223$	4641.76	1.39388	0.696941	+	0.717129i	$0.254544\pi$
$224$	0	0
$225$	−29.2182	−0.00865723
$226$	0	0
$227$	−6736.62	−1.96971	−0.984857	−	0.173370i	$-0.944534\pi$
$227$	−6736.62	−1.96971	−0.984857	+	0.173370i	$0.944534\pi$
$228$	0	0
$229$	−186.607	−0.0538485	−0.0269243	−	0.999637i	$-0.508571\pi$
$229$	−186.607	−0.0538485	−0.0269243	+	0.999637i	$0.508571\pi$
$230$	0	0
$231$	966.199	0.275200
$232$	0	0
$233$	4965.47	1.39613	0.698066	−	0.716033i	$-0.254044\pi$
$233$	4965.47	1.39613	0.698066	+	0.716033i	$0.254044\pi$
$234$	0	0
$235$	55.3888	0.0153752
$236$	0	0
$237$	−828.532	−0.227084
$238$	0	0
$239$	5838.84	1.58026	0.790132	−	0.612936i	$-0.210012\pi$
$239$	5838.84	1.58026	0.790132	+	0.612936i	$0.210012\pi$
$240$	0	0
$241$	−1000.86	−0.267513	−0.133757	−	0.991014i	$-0.542704\pi$
$241$	−1000.86	−0.267513	−0.133757	+	0.991014i	$0.542704\pi$
$242$	0	0
$243$	3034.09	0.800976
$244$	0	0
$245$	1554.60	0.405387
$246$	0	0
$247$	−2581.57	−0.665026
$248$	0	0
$249$	583.154	0.148417
$250$	0	0
$251$	5379.45	1.35278	0.676390	−	0.736543i	$-0.263543\pi$
$251$	5379.45	1.35278	0.676390	+	0.736543i	$0.263543\pi$
$252$	0	0
$253$	−7821.10	−1.94351
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2908.12	−0.705849	−0.352925	−	0.935652i	$-0.614813\pi$
$257$	−2908.12	−0.705849	−0.352925	+	0.935652i	$0.614813\pi$
$258$	0	0
$259$	−937.933	−0.225021
$260$	0	0
$261$	−6485.72	−1.53815
$262$	0	0
$263$	5319.69	1.24725	0.623624	−	0.781725i	$-0.285660\pi$
$263$	5319.69	1.24725	0.623624	+	0.781725i	$0.285660\pi$
$264$	0	0
$265$	7401.46	1.71573
$266$	0	0
$267$	−369.542	−0.0847026
$268$	0	0
$269$	3173.93	0.719397	0.359699	−	0.933069i	$-0.382879\pi$
$269$	3173.93	0.719397	0.359699	+	0.933069i	$0.382879\pi$
$270$	0	0
$271$	−184.445	−0.0413441	−0.0206720	−	0.999786i	$-0.506581\pi$
$271$	−184.445	−0.0413441	−0.0206720	+	0.999786i	$0.506581\pi$
$272$	0	0
$273$	−1589.02	−0.352279
$274$	0	0
$275$	−51.0718	−0.0111991
$276$	0	0
$277$	−1624.92	−0.352463	−0.176231	−	0.984349i	$-0.556391\pi$
$277$	−1624.92	−0.352463	−0.176231	+	0.984349i	$0.556391\pi$
$278$	0	0
$279$	−8380.61	−1.79833
$280$	0	0
$281$	−7606.47	−1.61482	−0.807409	−	0.589992i	$-0.799131\pi$
$281$	−7606.47	−1.61482	−0.807409	+	0.589992i	$0.799131\pi$
$282$	0	0
$283$	−4654.04	−0.977575	−0.488788	−	0.872403i	$-0.662561\pi$
$283$	−4654.04	−0.977575	−0.488788	+	0.872403i	$0.662561\pi$
$284$	0	0
$285$	648.646	0.134816
$286$	0	0
$287$	−216.463	−0.0445205
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−212.779	−0.0428637
$292$	0	0
$293$	6710.00	1.33789	0.668946	−	0.743311i	$-0.266746\pi$
$293$	6710.00	1.33789	0.668946	+	0.743311i	$0.266746\pi$
$294$	0	0
$295$	−1185.46	−0.233966
$296$	0	0
$297$	3479.68	0.679836
$298$	0	0
$299$	12862.7	2.48785
$300$	0	0
$301$	−2135.63	−0.408956
$302$	0	0
$303$	−774.201	−0.146788
$304$	0	0
$305$	−6607.39	−1.24045
$306$	0	0
$307$	7628.87	1.41825	0.709125	−	0.705083i	$-0.249090\pi$
$307$	7628.87	1.41825	0.709125	+	0.705083i	$0.249090\pi$
$308$	0	0
$309$	−2637.17	−0.485512
$310$	0	0
$311$	4889.33	0.891475	0.445737	−	0.895164i	$-0.352942\pi$
$311$	4889.33	0.891475	0.445737	+	0.895164i	$0.352942\pi$
$312$	0	0
$313$	582.059	0.105112	0.0525558	−	0.998618i	$-0.483263\pi$
$313$	582.059	0.105112	0.0525558	+	0.998618i	$0.483263\pi$
$314$	0	0
$315$	−3939.28	−0.704613
$316$	0	0
$317$	−3318.90	−0.588038	−0.294019	−	0.955800i	$-0.594993\pi$
$317$	−3318.90	−0.588038	−0.294019	+	0.955800i	$0.594993\pi$
$318$	0	0
$319$	−11336.7	−1.98976
$320$	0	0
$321$	57.2846	0.00996049
$322$	0	0
$323$	0	0
$324$	0	0
$325$	83.9933	0.0143357
$326$	0	0
$327$	765.316	0.129425
$328$	0	0
$329$	70.5294	0.0118189
$330$	0	0
$331$	−8438.98	−1.40136	−0.700678	−	0.713478i	$-0.747119\pi$
$331$	−8438.98	−1.40136	−0.700678	+	0.713478i	$0.747119\pi$
$332$	0	0
$333$	−1607.48	−0.264532
$334$	0	0
$335$	10702.0	1.74542
$336$	0	0
$337$	−1586.90	−0.256509	−0.128255	−	0.991741i	$-0.540938\pi$
$337$	−1586.90	−0.256509	−0.128255	+	0.991741i	$0.540938\pi$
$338$	0	0
$339$	−2941.46	−0.471264
$340$	0	0
$341$	−14648.9	−2.32634
$342$	0	0
$343$	6885.90	1.08398
$344$	0	0
$345$	−3231.88	−0.504343
$346$	0	0
$347$	843.572	0.130505	0.0652526	−	0.997869i	$-0.479215\pi$
$347$	843.572	0.130505	0.0652526	+	0.997869i	$0.479215\pi$
$348$	0	0
$349$	−8943.23	−1.37169	−0.685845	−	0.727747i	$-0.740567\pi$
$349$	−8943.23	−1.37169	−0.685845	+	0.727747i	$0.740567\pi$
$350$	0	0
$351$	−5722.72	−0.870246
$352$	0	0
$353$	−5269.98	−0.794597	−0.397298	−	0.917689i	$-0.630052\pi$
$353$	−5269.98	−0.794597	−0.397298	+	0.917689i	$0.630052\pi$
$354$	0	0
$355$	−3743.53	−0.559679
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5611.15	−0.824917	−0.412458	−	0.910976i	$-0.635330\pi$
$359$	−5611.15	−0.824917	−0.412458	+	0.910976i	$0.635330\pi$
$360$	0	0
$361$	−5517.13	−0.804364
$362$	0	0
$363$	796.418	0.115154
$364$	0	0
$365$	7567.67	1.08523
$366$	0	0
$367$	−6413.60	−0.912227	−0.456113	−	0.889922i	$-0.650759\pi$
$367$	−6413.60	−0.912227	−0.456113	+	0.889922i	$0.650759\pi$
$368$	0	0
$369$	−370.985	−0.0523379
$370$	0	0
$371$	9424.65	1.31888
$372$	0	0
$373$	−7997.74	−1.11021	−0.555104	−	0.831781i	$-0.687321\pi$
$373$	−7997.74	−1.11021	−0.555104	+	0.831781i	$0.687321\pi$
$374$	0	0
$375$	2192.31	0.301895
$376$	0	0
$377$	18644.5	2.54705
$378$	0	0
$379$	−1278.51	−0.173279	−0.0866397	−	0.996240i	$-0.527613\pi$
$379$	−1278.51	−0.173279	−0.0866397	+	0.996240i	$0.527613\pi$
$380$	0	0
$381$	1887.77	0.253840
$382$	0	0
$383$	6778.38	0.904331	0.452166	−	0.891934i	$-0.350652\pi$
$383$	6778.38	0.904331	0.452166	+	0.891934i	$0.350652\pi$
$384$	0	0
$385$	−6885.65	−0.911494
$386$	0	0
$387$	−3660.15	−0.480765
$388$	0	0
$389$	−8833.21	−1.15132	−0.575658	−	0.817691i	$-0.695254\pi$
$389$	−8833.21	−1.15132	−0.575658	+	0.817691i	$0.695254\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−519.208	−0.0666427
$394$	0	0
$395$	5904.56	0.752128
$396$	0	0
$397$	−3367.66	−0.425738	−0.212869	−	0.977081i	$-0.568281\pi$
$397$	−3367.66	−0.425738	−0.212869	+	0.977081i	$0.568281\pi$
$398$	0	0
$399$	825.954	0.103633
$400$	0	0
$401$	12259.7	1.52674	0.763369	−	0.645962i	$-0.223544\pi$
$401$	12259.7	1.52674	0.763369	+	0.645962i	$0.223544\pi$
$402$	0	0
$403$	24091.7	2.97790
$404$	0	0
$405$	−5997.72	−0.735874
$406$	0	0
$407$	−2809.79	−0.342202
$408$	0	0
$409$	1574.44	0.190344	0.0951722	−	0.995461i	$-0.469660\pi$
$409$	1574.44	0.190344	0.0951722	+	0.995461i	$0.469660\pi$
$410$	0	0
$411$	2611.54	0.313425
$412$	0	0
$413$	−1509.50	−0.179849
$414$	0	0
$415$	−4155.87	−0.491575
$416$	0	0
$417$	−1683.73	−0.197728
$418$	0	0
$419$	−2224.33	−0.259346	−0.129673	−	0.991557i	$-0.541393\pi$
$419$	−2224.33	−0.259346	−0.129673	+	0.991557i	$0.541393\pi$
$420$	0	0
$421$	−16480.9	−1.90791	−0.953954	−	0.299952i	$-0.903029\pi$
$421$	−16480.9	−1.90791	−0.953954	+	0.299952i	$0.903029\pi$
$422$	0	0
$423$	120.877	0.0138942
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8413.52	−0.953534
$428$	0	0
$429$	−4760.28	−0.535730
$430$	0	0
$431$	7364.96	0.823104	0.411552	−	0.911386i	$-0.364987\pi$
$431$	7364.96	0.823104	0.411552	+	0.911386i	$0.364987\pi$
$432$	0	0
$433$	10646.9	1.18166	0.590829	−	0.806796i	$-0.298801\pi$
$433$	10646.9	1.18166	0.590829	+	0.806796i	$0.298801\pi$
$434$	0	0
$435$	−4684.62	−0.516345
$436$	0	0
$437$	−6685.85	−0.731871
$438$	0	0
$439$	−9757.79	−1.06085	−0.530426	−	0.847731i	$-0.677968\pi$
$439$	−9757.79	−1.06085	−0.530426	+	0.847731i	$0.677968\pi$
$440$	0	0
$441$	3392.66	0.366339
$442$	0	0
$443$	−154.772	−0.0165992	−0.00829960	−	0.999966i	$-0.502642\pi$
$443$	−154.772	−0.0165992	−0.00829960	+	0.999966i	$0.502642\pi$
$444$	0	0
$445$	2633.55	0.280545
$446$	0	0
$447$	−2753.24	−0.291328
$448$	0	0
$449$	−7207.73	−0.757581	−0.378791	−	0.925482i	$-0.623660\pi$
$449$	−7207.73	−0.757581	−0.378791	+	0.925482i	$0.623660\pi$
$450$	0	0
$451$	−648.462	−0.0677048
$452$	0	0
$453$	−3674.99	−0.381161
$454$	0	0
$455$	11324.2	1.16679
$456$	0	0
$457$	−628.393	−0.0643216	−0.0321608	−	0.999483i	$-0.510239\pi$
$457$	−628.393	−0.0643216	−0.0321608	+	0.999483i	$0.510239\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6495.41	−0.656228	−0.328114	−	0.944638i	$-0.606413\pi$
$461$	−6495.41	−0.656228	−0.328114	+	0.944638i	$0.606413\pi$
$462$	0	0
$463$	−11349.3	−1.13919	−0.569595	−	0.821925i	$-0.692900\pi$
$463$	−11349.3	−1.13919	−0.569595	+	0.821925i	$0.692900\pi$
$464$	0	0
$465$	−6053.29	−0.603687
$466$	0	0
$467$	2600.03	0.257634	0.128817	−	0.991668i	$-0.458882\pi$
$467$	2600.03	0.257634	0.128817	+	0.991668i	$0.458882\pi$
$468$	0	0
$469$	13627.5	1.34170
$470$	0	0
$471$	−456.974	−0.0447054
$472$	0	0
$473$	−6397.75	−0.621922
$474$	0	0
$475$	−43.6586	−0.00421726
$476$	0	0
$477$	16152.5	1.55046
$478$	0	0
$479$	−5384.27	−0.513599	−0.256799	−	0.966465i	$-0.582668\pi$
$479$	−5384.27	−0.513599	−0.256799	+	0.966465i	$0.582668\pi$
$480$	0	0
$481$	4621.01	0.438046
$482$	0	0
$483$	−4115.32	−0.387688
$484$	0	0
$485$	1516.38	0.141969
$486$	0	0
$487$	1577.04	0.146740	0.0733702	−	0.997305i	$-0.476625\pi$
$487$	1577.04	0.146740	0.0733702	+	0.997305i	$0.476625\pi$
$488$	0	0
$489$	−839.717	−0.0776550
$490$	0	0
$491$	−14506.3	−1.33332	−0.666661	−	0.745361i	$-0.732277\pi$
$491$	−14506.3	−1.33332	−0.666661	+	0.745361i	$0.732277\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−11801.0	−1.07155
$496$	0	0
$497$	−4766.83	−0.430225
$498$	0	0
$499$	−1347.85	−0.120918	−0.0604588	−	0.998171i	$-0.519256\pi$
$499$	−1347.85	−0.120918	−0.0604588	+	0.998171i	$0.519256\pi$
$500$	0	0
$501$	−3985.63	−0.355418
$502$	0	0
$503$	15458.0	1.37025	0.685126	−	0.728424i	$-0.259747\pi$
$503$	15458.0	1.37025	0.685126	+	0.728424i	$0.259747\pi$
$504$	0	0
$505$	5517.37	0.486178
$506$	0	0
$507$	4365.69	0.382420
$508$	0	0
$509$	12625.0	1.09940	0.549700	−	0.835362i	$-0.314742\pi$
$509$	12625.0	1.09940	0.549700	+	0.835362i	$0.314742\pi$
$510$	0	0
$511$	9636.30	0.834217
$512$	0	0
$513$	2974.60	0.256007
$514$	0	0
$515$	18793.8	1.60807
$516$	0	0
$517$	211.287	0.0179736
$518$	0	0
$519$	4580.64	0.387414
$520$	0	0
$521$	−10028.0	−0.843252	−0.421626	−	0.906770i	$-0.638540\pi$
$521$	−10028.0	−0.843252	−0.421626	+	0.906770i	$0.638540\pi$
$522$	0	0
$523$	14574.0	1.21851	0.609253	−	0.792976i	$-0.291470\pi$
$523$	14574.0	1.21851	0.609253	+	0.792976i	$0.291470\pi$
$524$	0	0
$525$	−26.8730	−0.00223397
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21145.3	1.73792
$530$	0	0
$531$	−2587.06	−0.211429
$532$	0	0
$533$	1066.47	0.0866677
$534$	0	0
$535$	−408.241	−0.0329903
$536$	0	0
$537$	−1476.81	−0.118676
$538$	0	0
$539$	5930.20	0.473899
$540$	0	0
$541$	10357.9	0.823142	0.411571	−	0.911378i	$-0.364980\pi$
$541$	10357.9	0.823142	0.411571	+	0.911378i	$0.364980\pi$
$542$	0	0
$543$	−3160.91	−0.249812
$544$	0	0
$545$	−5454.05	−0.428671
$546$	0	0
$547$	−22491.5	−1.75808	−0.879039	−	0.476750i	$-0.841815\pi$
$547$	−22491.5	−1.75808	−0.879039	+	0.476750i	$0.841815\pi$
$548$	0	0
$549$	−14419.5	−1.12097
$550$	0	0
$551$	−9691.16	−0.749287
$552$	0	0
$553$	7518.58	0.578160
$554$	0	0
$555$	−1161.08	−0.0888018
$556$	0	0
$557$	−9170.37	−0.697596	−0.348798	−	0.937198i	$-0.613410\pi$
$557$	−9170.37	−0.697596	−0.348798	+	0.937198i	$0.613410\pi$
$558$	0	0
$559$	10521.8	0.796111
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15721.3	1.17686	0.588431	−	0.808548i	$-0.299746\pi$
$563$	15721.3	1.17686	0.588431	+	0.808548i	$0.299746\pi$
$564$	0	0
$565$	20962.4	1.56088
$566$	0	0
$567$	−7637.20	−0.565665
$568$	0	0
$569$	−15528.5	−1.14409	−0.572044	−	0.820223i	$-0.693850\pi$
$569$	−15528.5	−1.14409	−0.572044	+	0.820223i	$0.693850\pi$
$570$	0	0
$571$	−8877.35	−0.650623	−0.325311	−	0.945607i	$-0.605469\pi$
$571$	−8877.35	−0.650623	−0.325311	+	0.945607i	$0.605469\pi$
$572$	0	0
$573$	2346.08	0.171046
$574$	0	0
$575$	217.529	0.0157767
$576$	0	0
$577$	−23810.5	−1.71793	−0.858965	−	0.512035i	$-0.828892\pi$
$577$	−23810.5	−1.71793	−0.858965	+	0.512035i	$0.828892\pi$
$578$	0	0
$579$	−106.728	−0.00766054
$580$	0	0
$581$	−5291.88	−0.377873
$582$	0	0
$583$	28233.7	2.00569
$584$	0	0
$585$	19408.1	1.37167
$586$	0	0
$587$	−327.420	−0.0230223	−0.0115111	−	0.999934i	$-0.503664\pi$
$587$	−327.420	−0.0230223	−0.0115111	+	0.999934i	$0.503664\pi$
$588$	0	0
$589$	−12522.6	−0.876032
$590$	0	0
$591$	−432.072	−0.0300728
$592$	0	0
$593$	−17984.7	−1.24543	−0.622716	−	0.782448i	$-0.713971\pi$
$593$	−17984.7	−1.24543	−0.622716	+	0.782448i	$0.713971\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4175.61	0.286258
$598$	0	0
$599$	−749.825	−0.0511469	−0.0255735	−	0.999673i	$-0.508141\pi$
$599$	−749.825	−0.0511469	−0.0255735	+	0.999673i	$0.508141\pi$
$600$	0	0
$601$	−16786.1	−1.13930	−0.569651	−	0.821887i	$-0.692922\pi$
$601$	−16786.1	−1.13930	−0.569651	+	0.821887i	$0.692922\pi$
$602$	0	0
$603$	23355.4	1.57729
$604$	0	0
$605$	−5675.70	−0.381405
$606$	0	0
$607$	−15003.1	−1.00322	−0.501612	−	0.865093i	$-0.667259\pi$
$607$	−15003.1	−1.00322	−0.501612	+	0.865093i	$0.667259\pi$
$608$	0	0
$609$	−5965.16	−0.396914
$610$	0	0
$611$	−347.485	−0.0230077
$612$	0	0
$613$	−12349.4	−0.813685	−0.406843	−	0.913498i	$-0.633370\pi$
$613$	−12349.4	−0.813685	−0.406843	+	0.913498i	$0.633370\pi$
$614$	0	0
$615$	−267.961	−0.0175695
$616$	0	0
$617$	−21078.1	−1.37532	−0.687658	−	0.726034i	$-0.741361\pi$
$617$	−21078.1	−1.37532	−0.687658	+	0.726034i	$0.741361\pi$
$618$	0	0
$619$	−16150.7	−1.04871	−0.524356	−	0.851499i	$-0.675694\pi$
$619$	−16150.7	−1.04871	−0.524356	+	0.851499i	$0.675694\pi$
$620$	0	0
$621$	−14820.9	−0.957719
$622$	0	0
$623$	3353.44	0.215654
$624$	0	0
$625$	−15772.6	−1.00944
$626$	0	0
$627$	2474.33	0.157600
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−18792.5	−1.18561	−0.592803	−	0.805347i	$-0.701979\pi$
$631$	−18792.5	−1.18561	−0.592803	+	0.805347i	$0.701979\pi$
$632$	0	0
$633$	4421.12	0.277605
$634$	0	0
$635$	−13453.2	−0.840748
$636$	0	0
$637$	−9752.88	−0.606630
$638$	0	0
$639$	−8169.65	−0.505769
$640$	0	0
$641$	−21193.4	−1.30591	−0.652956	−	0.757395i	$-0.726472\pi$
$641$	−21193.4	−1.30591	−0.652956	+	0.757395i	$0.726472\pi$
$642$	0	0
$643$	8190.97	0.502364	0.251182	−	0.967940i	$-0.419181\pi$
$643$	8190.97	0.502364	0.251182	+	0.967940i	$0.419181\pi$
$644$	0	0
$645$	−2643.72	−0.161390
$646$	0	0
$647$	−10226.2	−0.621382	−0.310691	−	0.950511i	$-0.600560\pi$
$647$	−10226.2	−0.621382	−0.310691	+	0.950511i	$0.600560\pi$
$648$	0	0
$649$	−4522.05	−0.273507
$650$	0	0
$651$	−7707.96	−0.464053
$652$	0	0
$653$	4458.37	0.267181	0.133591	−	0.991037i	$-0.457349\pi$
$653$	4458.37	0.267181	0.133591	+	0.991037i	$0.457349\pi$
$654$	0	0
$655$	3700.15	0.220728
$656$	0	0
$657$	16515.2	0.980699
$658$	0	0
$659$	25446.7	1.50419	0.752097	−	0.659052i	$-0.229042\pi$
$659$	25446.7	1.50419	0.752097	+	0.659052i	$0.229042\pi$
$660$	0	0
$661$	15004.3	0.882903	0.441452	−	0.897285i	$-0.354464\pi$
$661$	15004.3	0.882903	0.441452	+	0.897285i	$0.354464\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5886.19	−0.343243
$666$	0	0
$667$	48286.2	2.80307
$668$	0	0
$669$	7316.79	0.422845
$670$	0	0
$671$	−25204.6	−1.45009
$672$	0	0
$673$	−29358.1	−1.68154	−0.840768	−	0.541396i	$-0.817896\pi$
$673$	−29358.1	−1.68154	−0.840768	+	0.541396i	$0.817896\pi$
$674$	0	0
$675$	−96.7808	−0.00551866
$676$	0	0
$677$	−19476.3	−1.10567	−0.552833	−	0.833292i	$-0.686453\pi$
$677$	−19476.3	−1.10567	−0.552833	+	0.833292i	$0.686453\pi$
$678$	0	0
$679$	1930.88	0.109132
$680$	0	0
$681$	−10618.9	−0.597528
$682$	0	0
$683$	2181.28	0.122202	0.0611012	−	0.998132i	$-0.480539\pi$
$683$	2181.28	0.122202	0.0611012	+	0.998132i	$0.480539\pi$
$684$	0	0
$685$	−18611.2	−1.03810
$686$	0	0
$687$	−294.147	−0.0163354
$688$	0	0
$689$	−46433.4	−2.56745
$690$	0	0
$691$	16698.7	0.919317	0.459658	−	0.888096i	$-0.347972\pi$
$691$	16698.7	0.919317	0.459658	+	0.888096i	$0.347972\pi$
$692$	0	0
$693$	−15026.8	−0.823695
$694$	0	0
$695$	11999.1	0.654896
$696$	0	0
$697$	0	0
$698$	0	0
$699$	7827.05	0.423528
$700$	0	0
$701$	−20210.8	−1.08895	−0.544473	−	0.838778i	$-0.683270\pi$
$701$	−20210.8	−1.08895	−0.544473	+	0.838778i	$0.683270\pi$
$702$	0	0
$703$	−2401.94	−0.128863
$704$	0	0
$705$	87.3091	0.00466418
$706$	0	0
$707$	7025.55	0.373724
$708$	0	0
$709$	5445.77	0.288463	0.144231	−	0.989544i	$-0.453929\pi$
$709$	5445.77	0.288463	0.144231	+	0.989544i	$0.453929\pi$
$710$	0	0
$711$	12885.7	0.679680
$712$	0	0
$713$	62393.7	3.27722
$714$	0	0
$715$	33924.3	1.77440
$716$	0	0
$717$	9203.73	0.479386
$718$	0	0
$719$	−11433.5	−0.593045	−0.296522	−	0.955026i	$-0.595827\pi$
$719$	−11433.5	−0.593045	−0.296522	+	0.955026i	$0.595827\pi$
$720$	0	0
$721$	23931.2	1.23612
$722$	0	0
$723$	−1577.64	−0.0811523
$724$	0	0
$725$	315.309	0.0161521
$726$	0	0
$727$	−1229.45	−0.0627207	−0.0313603	−	0.999508i	$-0.509984\pi$
$727$	−1229.45	−0.0627207	−0.0313603	+	0.999508i	$0.509984\pi$
$728$	0	0
$729$	−9633.02	−0.489408
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−8494.44	−0.428035	−0.214017	−	0.976830i	$-0.568655\pi$
$733$	−8494.44	−0.428035	−0.214017	+	0.976830i	$0.568655\pi$
$734$	0	0
$735$	2450.51	0.122978
$736$	0	0
$737$	40824.1	2.04040
$738$	0	0
$739$	−22206.2	−1.10537	−0.552684	−	0.833391i	$-0.686396\pi$
$739$	−22206.2	−1.10537	−0.552684	+	0.833391i	$0.686396\pi$
$740$	0	0
$741$	−4069.31	−0.201741
$742$	0	0
$743$	−5721.55	−0.282508	−0.141254	−	0.989973i	$-0.545113\pi$
$743$	−5721.55	−0.282508	−0.141254	+	0.989973i	$0.545113\pi$
$744$	0	0
$745$	19621.0	0.964911
$746$	0	0
$747$	−9069.50	−0.444224
$748$	0	0
$749$	−519.834	−0.0253596
$750$	0	0
$751$	−36393.3	−1.76832	−0.884161	−	0.467182i	$-0.845269\pi$
$751$	−36393.3	−1.76832	−0.884161	+	0.467182i	$0.845269\pi$
$752$	0	0
$753$	8479.60	0.410377
$754$	0	0
$755$	26189.9	1.26245
$756$	0	0
$757$	8039.34	0.385991	0.192995	−	0.981200i	$-0.438180\pi$
$757$	8039.34	0.385991	0.192995	+	0.981200i	$0.438180\pi$
$758$	0	0
$759$	−12328.3	−0.589579
$760$	0	0
$761$	−10731.1	−0.511173	−0.255586	−	0.966786i	$-0.582269\pi$
$761$	−10731.1	−0.511173	−0.255586	+	0.966786i	$0.582269\pi$
$762$	0	0
$763$	−6944.92	−0.329519
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7437.02	0.350111
$768$	0	0
$769$	−34154.2	−1.60160	−0.800802	−	0.598929i	$-0.795593\pi$
$769$	−34154.2	−1.60160	−0.800802	+	0.598929i	$0.795593\pi$
$770$	0	0
$771$	−4584.05	−0.214125
$772$	0	0
$773$	5508.21	0.256296	0.128148	−	0.991755i	$-0.459097\pi$
$773$	5508.21	0.256296	0.128148	+	0.991755i	$0.459097\pi$
$774$	0	0
$775$	407.431	0.0188843
$776$	0	0
$777$	−1478.46	−0.0682618
$778$	0	0
$779$	−554.337	−0.0254957
$780$	0	0
$781$	−14280.1	−0.654267
$782$	0	0
$783$	−21483.0	−0.980510
$784$	0	0
$785$	3256.64	0.148069
$786$	0	0
$787$	15900.2	0.720179	0.360089	−	0.932918i	$-0.382746\pi$
$787$	15900.2	0.720179	0.360089	+	0.932918i	$0.382746\pi$
$788$	0	0
$789$	8385.40	0.378362
$790$	0	0
$791$	26692.5	1.19984
$792$	0	0
$793$	41451.8	1.85624
$794$	0	0
$795$	11666.9	0.520480
$796$	0	0
$797$	−2917.79	−0.129678	−0.0648391	−	0.997896i	$-0.520653\pi$
$797$	−2917.79	−0.129678	−0.0648391	+	0.997896i	$0.520653\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5747.30	0.253521
$802$	0	0
$803$	28867.7	1.26864
$804$	0	0
$805$	29327.9	1.28407
$806$	0	0
$807$	5003.05	0.218235
$808$	0	0
$809$	1427.09	0.0620194	0.0310097	−	0.999519i	$-0.490128\pi$
$809$	1427.09	0.0620194	0.0310097	+	0.999519i	$0.490128\pi$
$810$	0	0
$811$	−2722.96	−0.117899	−0.0589495	−	0.998261i	$-0.518775\pi$
$811$	−2722.96	−0.117899	−0.0589495	+	0.998261i	$0.518775\pi$
$812$	0	0
$813$	−290.740	−0.0125421
$814$	0	0
$815$	5984.27	0.257202
$816$	0	0
$817$	−5469.11	−0.234198
$818$	0	0
$819$	24713.3	1.05440
$820$	0	0
$821$	−12544.7	−0.533269	−0.266634	−	0.963798i	$-0.585912\pi$
$821$	−12544.7	−0.533269	−0.266634	+	0.963798i	$0.585912\pi$
$822$	0	0
$823$	−12532.6	−0.530812	−0.265406	−	0.964137i	$-0.585506\pi$
$823$	−12532.6	−0.530812	−0.265406	+	0.964137i	$0.585506\pi$
$824$	0	0
$825$	−80.5042	−0.00339733
$826$	0	0
$827$	1557.22	0.0654775	0.0327388	−	0.999464i	$-0.489577\pi$
$827$	1557.22	0.0654775	0.0327388	+	0.999464i	$0.489577\pi$
$828$	0	0
$829$	30476.5	1.27683	0.638416	−	0.769691i	$-0.279590\pi$
$829$	30476.5	1.27683	0.638416	+	0.769691i	$0.279590\pi$
$830$	0	0
$831$	−2561.36	−0.106922
$832$	0	0
$833$	0	0
$834$	0	0
$835$	28403.7	1.17719
$836$	0	0
$837$	−27759.5	−1.14637
$838$	0	0
$839$	−6306.23	−0.259494	−0.129747	−	0.991547i	$-0.541416\pi$
$839$	−6306.23	−0.259494	−0.129747	+	0.991547i	$0.541416\pi$
$840$	0	0
$841$	45602.0	1.86978
$842$	0	0
$843$	−11990.0	−0.489868
$844$	0	0
$845$	−31112.2	−1.26662
$846$	0	0
$847$	−7227.15	−0.293185
$848$	0	0
$849$	−7336.13	−0.296555
$850$	0	0
$851$	11967.7	0.482076
$852$	0	0
$853$	18731.1	0.751866	0.375933	−	0.926647i	$-0.377322\pi$
$853$	18731.1	0.751866	0.375933	+	0.926647i	$0.377322\pi$
$854$	0	0
$855$	−10088.1	−0.403514
$856$	0	0
$857$	22174.4	0.883852	0.441926	−	0.897051i	$-0.354295\pi$
$857$	22174.4	0.883852	0.441926	+	0.897051i	$0.354295\pi$
$858$	0	0
$859$	48642.6	1.93209	0.966045	−	0.258375i	$-0.0831870\pi$
$859$	48642.6	1.93209	0.966045	+	0.258375i	$0.0831870\pi$
$860$	0	0
$861$	−341.209	−0.0135056
$862$	0	0
$863$	33726.6	1.33032	0.665161	−	0.746700i	$-0.268363\pi$
$863$	33726.6	1.33032	0.665161	+	0.746700i	$0.268363\pi$
$864$	0	0
$865$	−32644.1	−1.28316
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22523.6	0.879241
$870$	0	0
$871$	−67139.8	−2.61188
$872$	0	0
$873$	3309.24	0.128294
$874$	0	0
$875$	−19894.3	−0.768629
$876$	0	0
$877$	1761.20	0.0678126	0.0339063	−	0.999425i	$-0.489205\pi$
$877$	1761.20	0.0678126	0.0339063	+	0.999425i	$0.489205\pi$
$878$	0	0
$879$	10576.9	0.405860
$880$	0	0
$881$	−8516.74	−0.325694	−0.162847	−	0.986651i	$-0.552068\pi$
$881$	−8516.74	−0.325694	−0.162847	+	0.986651i	$0.552068\pi$
$882$	0	0
$883$	14160.2	0.539669	0.269834	−	0.962907i	$-0.413031\pi$
$883$	14160.2	0.539669	0.269834	+	0.962907i	$0.413031\pi$
$884$	0	0
$885$	−1868.63	−0.0709754
$886$	0	0
$887$	−23486.3	−0.889054	−0.444527	−	0.895765i	$-0.646628\pi$
$887$	−23486.3	−0.889054	−0.444527	+	0.895765i	$0.646628\pi$
$888$	0	0
$889$	−17130.7	−0.646282
$890$	0	0
$891$	−22878.9	−0.860239
$892$	0	0
$893$	180.618	0.00676837
$894$	0	0
$895$	10524.5	0.393068
$896$	0	0
$897$	20275.4	0.754710
$898$	0	0
$899$	90439.8	3.35521
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3366.38	−0.124060
$904$	0	0
$905$	22526.3	0.827404
$906$	0	0
$907$	41862.9	1.53256	0.766282	−	0.642504i	$-0.222104\pi$
$907$	41862.9	1.53256	0.766282	+	0.642504i	$0.222104\pi$
$908$	0	0
$909$	12040.8	0.439347
$910$	0	0
$911$	−16997.9	−0.618184	−0.309092	−	0.951032i	$-0.600025\pi$
$911$	−16997.9	−0.618184	−0.309092	+	0.951032i	$0.600025\pi$
$912$	0	0
$913$	−15853.0	−0.574653
$914$	0	0
$915$	−10415.2	−0.376301
$916$	0	0
$917$	4711.59	0.169673
$918$	0	0
$919$	−2596.93	−0.0932152	−0.0466076	−	0.998913i	$-0.514841\pi$
$919$	−2596.93	−0.0932152	−0.0466076	+	0.998913i	$0.514841\pi$
$920$	0	0
$921$	12025.3	0.430237
$922$	0	0
$923$	23485.3	0.837516
$924$	0	0
$925$	78.1491	0.00277787
$926$	0	0
$927$	41014.5	1.45317
$928$	0	0
$929$	−1026.66	−0.0362579	−0.0181289	−	0.999836i	$-0.505771\pi$
$929$	−1026.66	−0.0362579	−0.0181289	+	0.999836i	$0.505771\pi$
$930$	0	0
$931$	5069.42	0.178457
$932$	0	0
$933$	7707.03	0.270436
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25346.1	−0.883695	−0.441848	−	0.897090i	$-0.645677\pi$
$937$	−25346.1	−0.883695	−0.441848	+	0.897090i	$0.645677\pi$
$938$	0	0
$939$	917.496	0.0318864
$940$	0	0
$941$	−35728.1	−1.23773	−0.618864	−	0.785498i	$-0.712407\pi$
$941$	−35728.1	−1.23773	−0.618864	+	0.785498i	$0.712407\pi$
$942$	0	0
$943$	2761.98	0.0953792
$944$	0	0
$945$	−13048.3	−0.449164
$946$	0	0
$947$	−27215.5	−0.933882	−0.466941	−	0.884288i	$-0.654644\pi$
$947$	−27215.5	−0.933882	−0.466941	+	0.884288i	$0.654644\pi$
$948$	0	0
$949$	−47476.2	−1.62396
$950$	0	0
$951$	−5231.56	−0.178386
$952$	0	0
$953$	2334.16	0.0793400	0.0396700	−	0.999213i	$-0.487369\pi$
$953$	2334.16	0.0793400	0.0396700	+	0.999213i	$0.487369\pi$
$954$	0	0
$955$	−16719.4	−0.566522
$956$	0	0
$957$	−17870.0	−0.603609
$958$	0	0
$959$	−23698.6	−0.797985
$960$	0	0
$961$	87071.9	2.92276
$962$	0	0
$963$	−890.919	−0.0298125
$964$	0	0
$965$	760.599	0.0253726
$966$	0	0
$967$	43408.1	1.44355	0.721774	−	0.692129i	$-0.243327\pi$
$967$	43408.1	1.44355	0.721774	+	0.692129i	$0.243327\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42830.6	−1.41555	−0.707775	−	0.706438i	$-0.750301\pi$
$971$	−42830.6	−1.41555	−0.707775	+	0.706438i	$0.750301\pi$
$972$	0	0
$973$	15279.1	0.503418
$974$	0	0
$975$	132.398	0.00434886
$976$	0	0
$977$	−31967.7	−1.04681	−0.523407	−	0.852083i	$-0.675339\pi$
$977$	−31967.7	−1.04681	−0.523407	+	0.852083i	$0.675339\pi$
$978$	0	0
$979$	10046.0	0.327958
$980$	0	0
$981$	−11902.6	−0.387380
$982$	0	0
$983$	11631.5	0.377403	0.188702	−	0.982034i	$-0.439572\pi$
$983$	11631.5	0.377403	0.188702	+	0.982034i	$0.439572\pi$
$984$	0	0
$985$	3079.17	0.0996046
$986$	0	0
$987$	111.175	0.00358535
$988$	0	0
$989$	27249.9	0.876132
$990$	0	0
$991$	53074.7	1.70129	0.850643	−	0.525744i	$-0.176213\pi$
$991$	53074.7	1.70129	0.850643	+	0.525744i	$0.176213\pi$
$992$	0	0
$993$	−13302.3	−0.425112
$994$	0	0
$995$	−29757.6	−0.948120
$996$	0	0
$997$	13729.4	0.436122	0.218061	−	0.975935i	$-0.430027\pi$
$997$	13729.4	0.436122	0.218061	+	0.975935i	$0.430027\pi$
$998$	0	0
$999$	−5324.54	−0.168629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.r.1.14		24
17.10	odd	16		136.4.n.a.49.4	yes	24
17.12	odd	16		136.4.n.a.25.4	✓	24
17.16	even	2	inner	2312.4.a.r.1.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.a.25.4	✓	24	17.12	odd	16
136.4.n.a.49.4	yes	24	17.10	odd	16
2312.4.a.r.1.11		24	17.16	even	2	inner
2312.4.a.r.1.14		24	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57629 q^{3} -11.2335 q^{5} -14.3042 q^{7} -24.5153 q^{9} +O(q^{10})\)	\(q+1.57629 q^{3} -11.2335 q^{5} -14.3042 q^{7} -24.5153 q^{9} -42.8514 q^{11} +70.4740 q^{13} -17.7073 q^{15} -36.6315 q^{19} -22.5476 q^{21} +182.516 q^{23} +1.19183 q^{25} -81.2033 q^{27} +264.558 q^{29} +341.852 q^{31} -67.5465 q^{33} +160.687 q^{35} +65.5704 q^{37} +111.088 q^{39} +15.1328 q^{41} +149.301 q^{43} +275.393 q^{45} -4.93068 q^{47} -138.390 q^{49} -658.873 q^{53} +481.372 q^{55} -57.7420 q^{57} +105.529 q^{59} +588.185 q^{61} +350.672 q^{63} -791.671 q^{65} -952.688 q^{67} +287.700 q^{69} +333.247 q^{71} -673.669 q^{73} +1.87868 q^{75} +612.956 q^{77} -525.620 q^{79} +533.913 q^{81} +369.953 q^{83} +417.021 q^{87} -234.437 q^{89} -1008.08 q^{91} +538.860 q^{93} +411.500 q^{95} -134.987 q^{97} +1050.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Introduction

L-functions

Modular forms

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