LMFDB - Embedded newform 2312.4.a.r.1.1

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			1.1
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)

$f(q)$	$=$	$q-8.95975 q^{3} +3.69237 q^{5} -1.47268 q^{7} +53.2771 q^{9} +O(q^{10})$	$q-8.95975 q^{3} +3.69237 q^{5} -1.47268 q^{7} +53.2771 q^{9} +59.1821 q^{11} +12.0896 q^{13} -33.0827 q^{15} -42.1697 q^{19} +13.1949 q^{21} +99.0510 q^{23} -111.366 q^{25} -235.436 q^{27} -254.868 q^{29} -168.785 q^{31} -530.257 q^{33} -5.43769 q^{35} +76.8147 q^{37} -108.320 q^{39} +47.1276 q^{41} -227.653 q^{43} +196.719 q^{45} +279.896 q^{47} -340.831 q^{49} -211.131 q^{53} +218.522 q^{55} +377.830 q^{57} +105.482 q^{59} +826.485 q^{61} -78.4603 q^{63} +44.6393 q^{65} +638.294 q^{67} -887.472 q^{69} +1081.04 q^{71} -142.492 q^{73} +997.815 q^{75} -87.1564 q^{77} +218.429 q^{79} +670.968 q^{81} -1249.33 q^{83} +2283.56 q^{87} -1223.82 q^{89} -17.8041 q^{91} +1512.27 q^{93} -155.706 q^{95} -172.553 q^{97} +3153.05 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.95975	−1.72430	−0.862152	−	0.506649i	$-0.830884\pi$
$3$	−8.95975	−1.72430	−0.862152	+	0.506649i	$0.830884\pi$
$4$	0	0
$5$	3.69237	0.330256	0.165128	−	0.986272i	$-0.447196\pi$
$5$	3.69237	0.330256	0.165128	+	0.986272i	$0.447196\pi$
$6$	0	0
$7$	−1.47268	−0.0795174	−0.0397587	−	0.999209i	$-0.512659\pi$
$7$	−1.47268	−0.0795174	−0.0397587	+	0.999209i	$0.512659\pi$
$8$	0	0
$9$	53.2771	1.97323
$10$	0	0
$11$	59.1821	1.62219	0.811094	−	0.584916i	$-0.198872\pi$
$11$	59.1821	1.62219	0.811094	+	0.584916i	$0.198872\pi$
$12$	0	0
$13$	12.0896	0.257927	0.128964	−	0.991649i	$-0.458835\pi$
$13$	12.0896	0.257927	0.128964	+	0.991649i	$0.458835\pi$
$14$	0	0
$15$	−33.0827	−0.569461
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−42.1697	−0.509178	−0.254589	−	0.967049i	$-0.581940\pi$
$19$	−42.1697	−0.509178	−0.254589	+	0.967049i	$0.581940\pi$
$20$	0	0
$21$	13.1949	0.137112
$22$	0	0
$23$	99.0510	0.897981	0.448990	−	0.893537i	$-0.351784\pi$
$23$	99.0510	0.897981	0.448990	+	0.893537i	$0.351784\pi$
$24$	0	0
$25$	−111.366	−0.890931
$26$	0	0
$27$	−235.436	−1.67814
$28$	0	0
$29$	−254.868	−1.63200	−0.815998	−	0.578055i	$-0.803812\pi$
$29$	−254.868	−1.63200	−0.815998	+	0.578055i	$0.803812\pi$
$30$	0	0
$31$	−168.785	−0.977894	−0.488947	−	0.872313i	$-0.662619\pi$
$31$	−168.785	−0.977894	−0.488947	+	0.872313i	$0.662619\pi$
$32$	0	0
$33$	−530.257	−2.79715
$34$	0	0
$35$	−5.43769	−0.0262611
$36$	0	0
$37$	76.8147	0.341304	0.170652	−	0.985331i	$-0.445413\pi$
$37$	76.8147	0.341304	0.170652	+	0.985331i	$0.445413\pi$
$38$	0	0
$39$	−108.320	−0.444745
$40$	0	0
$41$	47.1276	0.179515	0.0897573	−	0.995964i	$-0.471391\pi$
$41$	47.1276	0.179515	0.0897573	+	0.995964i	$0.471391\pi$
$42$	0	0
$43$	−227.653	−0.807367	−0.403684	−	0.914899i	$-0.632270\pi$
$43$	−227.653	−0.807367	−0.403684	+	0.914899i	$0.632270\pi$
$44$	0	0
$45$	196.719	0.651669
$46$	0	0
$47$	279.896	0.868660	0.434330	−	0.900754i	$-0.356985\pi$
$47$	279.896	0.868660	0.434330	+	0.900754i	$0.356985\pi$
$48$	0	0
$49$	−340.831	−0.993677
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−211.131	−0.547191	−0.273596	−	0.961845i	$-0.588213\pi$
$53$	−211.131	−0.547191	−0.273596	+	0.961845i	$0.588213\pi$
$54$	0	0
$55$	218.522	0.535737
$56$	0	0
$57$	377.830	0.877978
$58$	0	0
$59$	105.482	0.232756	0.116378	−	0.993205i	$-0.462872\pi$
$59$	105.482	0.232756	0.116378	+	0.993205i	$0.462872\pi$
$60$	0	0
$61$	826.485	1.73476	0.867381	−	0.497644i	$-0.165801\pi$
$61$	826.485	1.73476	0.867381	+	0.497644i	$0.165801\pi$
$62$	0	0
$63$	−78.4603	−0.156906
$64$	0	0
$65$	44.6393	0.0851819
$66$	0	0
$67$	638.294	1.16388	0.581940	−	0.813232i	$-0.302294\pi$
$67$	638.294	1.16388	0.581940	+	0.813232i	$0.302294\pi$
$68$	0	0
$69$	−887.472	−1.54839
$70$	0	0
$71$	1081.04	1.80698	0.903489	−	0.428612i	$-0.140997\pi$
$71$	1081.04	1.80698	0.903489	+	0.428612i	$0.140997\pi$
$72$	0	0
$73$	−142.492	−0.228457	−0.114229	−	0.993454i	$-0.536440\pi$
$73$	−142.492	−0.228457	−0.114229	+	0.993454i	$0.536440\pi$
$74$	0	0
$75$	997.815	1.53624
$76$	0	0
$77$	−87.1564	−0.128992
$78$	0	0
$79$	218.429	0.311078	0.155539	−	0.987830i	$-0.450289\pi$
$79$	218.429	0.311078	0.155539	+	0.987830i	$0.450289\pi$
$80$	0	0
$81$	670.968	0.920395
$82$	0	0
$83$	−1249.33	−1.65219	−0.826095	−	0.563531i	$-0.809443\pi$
$83$	−1249.33	−1.65219	−0.826095	+	0.563531i	$0.809443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2283.56	2.81406
$88$	0	0
$89$	−1223.82	−1.45758	−0.728792	−	0.684735i	$-0.759918\pi$
$89$	−1223.82	−1.45758	−0.728792	+	0.684735i	$0.759918\pi$
$90$	0	0
$91$	−17.8041	−0.0205097
$92$	0	0
$93$	1512.27	1.68619
$94$	0	0
$95$	−155.706	−0.168159
$96$	0	0
$97$	−172.553	−0.180619	−0.0903096	−	0.995914i	$-0.528786\pi$
$97$	−172.553	−0.180619	−0.0903096	+	0.995914i	$0.528786\pi$
$98$	0	0
$99$	3153.05	3.20094
$100$	0	0
$101$	−1975.79	−1.94652	−0.973260	−	0.229708i	$-0.926223\pi$
$101$	−1975.79	−1.94652	−0.973260	+	0.229708i	$0.926223\pi$
$102$	0	0
$103$	−463.108	−0.443023	−0.221511	−	0.975158i	$-0.571099\pi$
$103$	−463.108	−0.443023	−0.221511	+	0.975158i	$0.571099\pi$
$104$	0	0
$105$	48.7204	0.0452821
$106$	0	0
$107$	−1549.67	−1.40011	−0.700057	−	0.714087i	$-0.746842\pi$
$107$	−1549.67	−1.40011	−0.700057	+	0.714087i	$0.746842\pi$
$108$	0	0
$109$	1471.04	1.29266	0.646332	−	0.763056i	$-0.276302\pi$
$109$	1471.04	1.29266	0.646332	+	0.763056i	$0.276302\pi$
$110$	0	0
$111$	−688.241	−0.588513
$112$	0	0
$113$	−1214.12	−1.01075	−0.505374	−	0.862900i	$-0.668645\pi$
$113$	−1214.12	−1.01075	−0.505374	+	0.862900i	$0.668645\pi$
$114$	0	0
$115$	365.733	0.296563
$116$	0	0
$117$	644.099	0.508948
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2171.52	1.63149
$122$	0	0
$123$	−422.251	−0.309538
$124$	0	0
$125$	−872.753	−0.624491
$126$	0	0
$127$	2058.56	1.43833	0.719165	−	0.694840i	$-0.244525\pi$
$127$	2058.56	1.43833	0.719165	+	0.694840i	$0.244525\pi$
$128$	0	0
$129$	2039.72	1.39215
$130$	0	0
$131$	1641.73	1.09495	0.547476	−	0.836821i	$-0.315589\pi$
$131$	1641.73	1.09495	0.547476	+	0.836821i	$0.315589\pi$
$132$	0	0
$133$	62.1025	0.0404885
$134$	0	0
$135$	−869.318	−0.554215
$136$	0	0
$137$	370.553	0.231084	0.115542	−	0.993303i	$-0.463140\pi$
$137$	370.553	0.231084	0.115542	+	0.993303i	$0.463140\pi$
$138$	0	0
$139$	−254.184	−0.155105	−0.0775526	−	0.996988i	$-0.524711\pi$
$139$	−254.184	−0.155105	−0.0775526	+	0.996988i	$0.524711\pi$
$140$	0	0
$141$	−2507.80	−1.49783
$142$	0	0
$143$	715.488	0.418406
$144$	0	0
$145$	−941.069	−0.538976
$146$	0	0
$147$	3053.76	1.71340
$148$	0	0
$149$	2296.11	1.26245	0.631224	−	0.775601i	$-0.282553\pi$
$149$	2296.11	1.26245	0.631224	+	0.775601i	$0.282553\pi$
$150$	0	0
$151$	−1359.78	−0.732831	−0.366415	−	0.930451i	$-0.619415\pi$
$151$	−1359.78	−0.732831	−0.366415	+	0.930451i	$0.619415\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−623.218	−0.322955
$156$	0	0
$157$	−1388.62	−0.705884	−0.352942	−	0.935645i	$-0.614819\pi$
$157$	−1388.62	−0.705884	−0.352942	+	0.935645i	$0.614819\pi$
$158$	0	0
$159$	1891.68	0.943524
$160$	0	0
$161$	−145.871	−0.0714051
$162$	0	0
$163$	2126.69	1.02193	0.510967	−	0.859600i	$-0.329287\pi$
$163$	2126.69	1.02193	0.510967	+	0.859600i	$0.329287\pi$
$164$	0	0
$165$	−1957.90	−0.923773
$166$	0	0
$167$	3411.22	1.58065	0.790323	−	0.612690i	$-0.209912\pi$
$167$	3411.22	1.58065	0.790323	+	0.612690i	$0.209912\pi$
$168$	0	0
$169$	−2050.84	−0.933474
$170$	0	0
$171$	−2246.68	−1.00472
$172$	0	0
$173$	529.657	0.232769	0.116385	−	0.993204i	$-0.462870\pi$
$173$	529.657	0.232769	0.116385	+	0.993204i	$0.462870\pi$
$174$	0	0
$175$	164.007	0.0708445
$176$	0	0
$177$	−945.095	−0.401343
$178$	0	0
$179$	3795.28	1.58476	0.792382	−	0.610025i	$-0.208841\pi$
$179$	3795.28	1.58476	0.792382	+	0.610025i	$0.208841\pi$
$180$	0	0
$181$	2851.95	1.17118	0.585590	−	0.810607i	$-0.300863\pi$
$181$	2851.95	1.17118	0.585590	+	0.810607i	$0.300863\pi$
$182$	0	0
$183$	−7405.10	−2.99126
$184$	0	0
$185$	283.628	0.112718
$186$	0	0
$187$	0	0
$188$	0	0
$189$	346.723	0.133441
$190$	0	0
$191$	−521.292	−0.197484	−0.0987419	−	0.995113i	$-0.531482\pi$
$191$	−521.292	−0.197484	−0.0987419	+	0.995113i	$0.531482\pi$
$192$	0	0
$193$	1658.75	0.618651	0.309326	−	0.950956i	$-0.399897\pi$
$193$	1658.75	0.618651	0.309326	+	0.950956i	$0.399897\pi$
$194$	0	0
$195$	−399.957	−0.146880
$196$	0	0
$197$	−4502.66	−1.62843	−0.814216	−	0.580562i	$-0.802833\pi$
$197$	−4502.66	−1.62843	−0.814216	+	0.580562i	$0.802833\pi$
$198$	0	0
$199$	−1501.31	−0.534798	−0.267399	−	0.963586i	$-0.586164\pi$
$199$	−1501.31	−0.534798	−0.267399	+	0.963586i	$0.586164\pi$
$200$	0	0
$201$	−5718.95	−2.00688
$202$	0	0
$203$	375.340	0.129772
$204$	0	0
$205$	174.013	0.0592857
$206$	0	0
$207$	5277.15	1.77192
$208$	0	0
$209$	−2495.69	−0.825983
$210$	0	0
$211$	−4124.58	−1.34573	−0.672863	−	0.739767i	$-0.734936\pi$
$211$	−4124.58	−1.34573	−0.672863	+	0.739767i	$0.734936\pi$
$212$	0	0
$213$	−9685.81	−3.11578
$214$	0	0
$215$	−840.580	−0.266638
$216$	0	0
$217$	248.567	0.0777596
$218$	0	0
$219$	1276.69	0.393930
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4822.90	−1.44827	−0.724137	−	0.689656i	$-0.757762\pi$
$223$	−4822.90	−1.44827	−0.724137	+	0.689656i	$0.757762\pi$
$224$	0	0
$225$	−5933.28	−1.75801
$226$	0	0
$227$	−2058.80	−0.601970	−0.300985	−	0.953629i	$-0.597315\pi$
$227$	−2058.80	−0.601970	−0.300985	+	0.953629i	$0.597315\pi$
$228$	0	0
$229$	−4122.51	−1.18962	−0.594811	−	0.803866i	$-0.702773\pi$
$229$	−4122.51	−1.18962	−0.594811	+	0.803866i	$0.702773\pi$
$230$	0	0
$231$	780.900	0.222422
$232$	0	0
$233$	−5933.87	−1.66842	−0.834208	−	0.551450i	$-0.814075\pi$
$233$	−5933.87	−1.66842	−0.834208	+	0.551450i	$0.814075\pi$
$234$	0	0
$235$	1033.48	0.286880
$236$	0	0
$237$	−1957.07	−0.536393
$238$	0	0
$239$	−4929.63	−1.33419	−0.667094	−	0.744973i	$-0.732462\pi$
$239$	−4929.63	−1.33419	−0.667094	+	0.744973i	$0.732462\pi$
$240$	0	0
$241$	−1879.06	−0.502245	−0.251123	−	0.967955i	$-0.580800\pi$
$241$	−1879.06	−0.502245	−0.251123	+	0.967955i	$0.580800\pi$
$242$	0	0
$243$	345.076	0.0910972
$244$	0	0
$245$	−1258.48	−0.328168
$246$	0	0
$247$	−509.814	−0.131331
$248$	0	0
$249$	11193.7	2.84888
$250$	0	0
$251$	4701.75	1.18236	0.591179	−	0.806540i	$-0.298663\pi$
$251$	4701.75	1.18236	0.591179	+	0.806540i	$0.298663\pi$
$252$	0	0
$253$	5862.04	1.45669
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1288.15	0.312655	0.156328	−	0.987705i	$-0.450034\pi$
$257$	1288.15	0.312655	0.156328	+	0.987705i	$0.450034\pi$
$258$	0	0
$259$	−113.124	−0.0271396
$260$	0	0
$261$	−13578.7	−3.22030
$262$	0	0
$263$	−5452.62	−1.27841	−0.639207	−	0.769035i	$-0.720737\pi$
$263$	−5452.62	−1.27841	−0.639207	+	0.769035i	$0.720737\pi$
$264$	0	0
$265$	−779.575	−0.180713
$266$	0	0
$267$	10965.1	2.51332
$268$	0	0
$269$	−1179.04	−0.267239	−0.133619	−	0.991033i	$-0.542660\pi$
$269$	−1179.04	−0.267239	−0.133619	+	0.991033i	$0.542660\pi$
$270$	0	0
$271$	2402.65	0.538563	0.269281	−	0.963062i	$-0.413214\pi$
$271$	2402.65	0.538563	0.269281	+	0.963062i	$0.413214\pi$
$272$	0	0
$273$	159.521	0.0353649
$274$	0	0
$275$	−6590.89	−1.44526
$276$	0	0
$277$	6062.59	1.31504	0.657520	−	0.753437i	$-0.271606\pi$
$277$	6062.59	1.31504	0.657520	+	0.753437i	$0.271606\pi$
$278$	0	0
$279$	−8992.39	−1.92961
$280$	0	0
$281$	2482.24	0.526968	0.263484	−	0.964664i	$-0.415128\pi$
$281$	2482.24	0.526968	0.263484	+	0.964664i	$0.415128\pi$
$282$	0	0
$283$	3280.39	0.689043	0.344521	−	0.938778i	$-0.388041\pi$
$283$	3280.39	0.689043	0.344521	+	0.938778i	$0.388041\pi$
$284$	0	0
$285$	1395.09	0.289957
$286$	0	0
$287$	−69.4040	−0.0142745
$288$	0	0
$289$	0	0
$290$	0	0
$291$	1546.03	0.311443
$292$	0	0
$293$	−2945.92	−0.587381	−0.293690	−	0.955901i	$-0.594883\pi$
$293$	−2945.92	−0.587381	−0.293690	+	0.955901i	$0.594883\pi$
$294$	0	0
$295$	389.480	0.0768691
$296$	0	0
$297$	−13933.6	−2.72225
$298$	0	0
$299$	1197.49	0.231614
$300$	0	0
$301$	335.261	0.0641998
$302$	0	0
$303$	17702.6	3.35639
$304$	0	0
$305$	3051.69	0.572915
$306$	0	0
$307$	904.415	0.168136	0.0840679	−	0.996460i	$-0.473209\pi$
$307$	904.415	0.168136	0.0840679	+	0.996460i	$0.473209\pi$
$308$	0	0
$309$	4149.33	0.763906
$310$	0	0
$311$	626.879	0.114299	0.0571496	−	0.998366i	$-0.481799\pi$
$311$	626.879	0.114299	0.0571496	+	0.998366i	$0.481799\pi$
$312$	0	0
$313$	−2934.20	−0.529875	−0.264938	−	0.964266i	$-0.585351\pi$
$313$	−2934.20	−0.529875	−0.264938	+	0.964266i	$0.585351\pi$
$314$	0	0
$315$	−289.704	−0.0518190
$316$	0	0
$317$	−8256.02	−1.46279	−0.731395	−	0.681954i	$-0.761130\pi$
$317$	−8256.02	−1.46279	−0.731395	+	0.681954i	$0.761130\pi$
$318$	0	0
$319$	−15083.6	−2.64740
$320$	0	0
$321$	13884.6	2.41422
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1346.38	−0.229795
$326$	0	0
$327$	−13180.2	−2.22895
$328$	0	0
$329$	−412.198	−0.0690736
$330$	0	0
$331$	8230.61	1.36675	0.683376	−	0.730066i	$-0.260511\pi$
$331$	8230.61	1.36675	0.683376	+	0.730066i	$0.260511\pi$
$332$	0	0
$333$	4092.47	0.673471
$334$	0	0
$335$	2356.82	0.384378
$336$	0	0
$337$	1373.76	0.222057	0.111029	−	0.993817i	$-0.464585\pi$
$337$	1373.76	0.222057	0.111029	+	0.993817i	$0.464585\pi$
$338$	0	0
$339$	10878.2	1.74284
$340$	0	0
$341$	−9989.06	−1.58633
$342$	0	0
$343$	1007.07	0.158532
$344$	0	0
$345$	−3276.88	−0.511365
$346$	0	0
$347$	−11221.3	−1.73600	−0.867998	−	0.496568i	$-0.834593\pi$
$347$	−11221.3	−1.73600	−0.867998	+	0.496568i	$0.834593\pi$
$348$	0	0
$349$	−3294.18	−0.505253	−0.252627	−	0.967564i	$-0.581294\pi$
$349$	−3294.18	−0.505253	−0.252627	+	0.967564i	$0.581294\pi$
$350$	0	0
$351$	−2846.33	−0.432837
$352$	0	0
$353$	−4675.80	−0.705008	−0.352504	−	0.935810i	$-0.614670\pi$
$353$	−4675.80	−0.705008	−0.352504	+	0.935810i	$0.614670\pi$
$354$	0	0
$355$	3991.59	0.596765
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8491.22	1.24833	0.624163	−	0.781294i	$-0.285440\pi$
$359$	8491.22	1.24833	0.624163	+	0.781294i	$0.285440\pi$
$360$	0	0
$361$	−5080.72	−0.740738
$362$	0	0
$363$	−19456.3	−2.81319
$364$	0	0
$365$	−526.132	−0.0754493
$366$	0	0
$367$	−11329.7	−1.61146	−0.805729	−	0.592284i	$-0.798226\pi$
$367$	−11329.7	−1.61146	−0.805729	+	0.592284i	$0.798226\pi$
$368$	0	0
$369$	2510.82	0.354223
$370$	0	0
$371$	310.930	0.0435112
$372$	0	0
$373$	3133.52	0.434980	0.217490	−	0.976063i	$-0.430213\pi$
$373$	3133.52	0.434980	0.217490	+	0.976063i	$0.430213\pi$
$374$	0	0
$375$	7819.64	1.07681
$376$	0	0
$377$	−3081.26	−0.420936
$378$	0	0
$379$	−8813.73	−1.19454	−0.597271	−	0.802040i	$-0.703748\pi$
$379$	−8813.73	−1.19454	−0.597271	+	0.802040i	$0.703748\pi$
$380$	0	0
$381$	−18444.2	−2.48012
$382$	0	0
$383$	1695.79	0.226242	0.113121	−	0.993581i	$-0.463915\pi$
$383$	1695.79	0.226242	0.113121	+	0.993581i	$0.463915\pi$
$384$	0	0
$385$	−321.814	−0.0426004
$386$	0	0
$387$	−12128.7	−1.59312
$388$	0	0
$389$	1092.80	0.142435	0.0712175	−	0.997461i	$-0.477312\pi$
$389$	1092.80	0.142435	0.0712175	+	0.997461i	$0.477312\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14709.5	−1.88803
$394$	0	0
$395$	806.520	0.102735
$396$	0	0
$397$	−3928.12	−0.496591	−0.248295	−	0.968684i	$-0.579870\pi$
$397$	−3928.12	−0.496591	−0.248295	+	0.968684i	$0.579870\pi$
$398$	0	0
$399$	−556.423	−0.0698145
$400$	0	0
$401$	9146.08	1.13899	0.569493	−	0.821996i	$-0.307140\pi$
$401$	9146.08	1.13899	0.569493	+	0.821996i	$0.307140\pi$
$402$	0	0
$403$	−2040.55	−0.252225
$404$	0	0
$405$	2477.46	0.303966
$406$	0	0
$407$	4546.05	0.553660
$408$	0	0
$409$	−3945.09	−0.476949	−0.238475	−	0.971149i	$-0.576647\pi$
$409$	−3945.09	−0.476949	−0.238475	+	0.971149i	$0.576647\pi$
$410$	0	0
$411$	−3320.07	−0.398459
$412$	0	0
$413$	−155.342	−0.0185082
$414$	0	0
$415$	−4612.99	−0.545645
$416$	0	0
$417$	2277.43	0.267449
$418$	0	0
$419$	−124.587	−0.0145262	−0.00726309	−	0.999974i	$-0.502312\pi$
$419$	−124.587	−0.0145262	−0.00726309	+	0.999974i	$0.502312\pi$
$420$	0	0
$421$	8468.69	0.980377	0.490188	−	0.871617i	$-0.336928\pi$
$421$	8468.69	0.980377	0.490188	+	0.871617i	$0.336928\pi$
$422$	0	0
$423$	14912.0	1.71406
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1217.15	−0.137944
$428$	0	0
$429$	−6410.59	−0.721460
$430$	0	0
$431$	−6572.89	−0.734582	−0.367291	−	0.930106i	$-0.619715\pi$
$431$	−6572.89	−0.734582	−0.367291	+	0.930106i	$0.619715\pi$
$432$	0	0
$433$	1367.07	0.151725	0.0758627	−	0.997118i	$-0.475829\pi$
$433$	1367.07	0.151725	0.0758627	+	0.997118i	$0.475829\pi$
$434$	0	0
$435$	8431.74	0.929359
$436$	0	0
$437$	−4176.95	−0.457232
$438$	0	0
$439$	−7194.82	−0.782209	−0.391105	−	0.920346i	$-0.627907\pi$
$439$	−7194.82	−0.782209	−0.391105	+	0.920346i	$0.627907\pi$
$440$	0	0
$441$	−18158.5	−1.96075
$442$	0	0
$443$	4911.14	0.526716	0.263358	−	0.964698i	$-0.415170\pi$
$443$	4911.14	0.526716	0.263358	+	0.964698i	$0.415170\pi$
$444$	0	0
$445$	−4518.81	−0.481376
$446$	0	0
$447$	−20572.6	−2.17684
$448$	0	0
$449$	−14040.7	−1.47578	−0.737888	−	0.674924i	$-0.764177\pi$
$449$	−14040.7	−1.47578	−0.737888	+	0.674924i	$0.764177\pi$
$450$	0	0
$451$	2789.11	0.291206
$452$	0	0
$453$	12183.3	1.26362
$454$	0	0
$455$	−65.7395	−0.00677344
$456$	0	0
$457$	−8693.27	−0.889834	−0.444917	−	0.895572i	$-0.646767\pi$
$457$	−8693.27	−0.889834	−0.444917	+	0.895572i	$0.646767\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3632.76	−0.367016	−0.183508	−	0.983018i	$-0.558745\pi$
$461$	−3632.76	−0.367016	−0.183508	+	0.983018i	$0.558745\pi$
$462$	0	0
$463$	−4321.99	−0.433823	−0.216911	−	0.976191i	$-0.569598\pi$
$463$	−4321.99	−0.433823	−0.216911	+	0.976191i	$0.569598\pi$
$464$	0	0
$465$	5583.87	0.556873
$466$	0	0
$467$	−6617.01	−0.655672	−0.327836	−	0.944735i	$-0.606319\pi$
$467$	−6617.01	−0.655672	−0.327836	+	0.944735i	$0.606319\pi$
$468$	0	0
$469$	−940.004	−0.0925487
$470$	0	0
$471$	12441.7	1.21716
$472$	0	0
$473$	−13473.0	−1.30970
$474$	0	0
$475$	4696.28	0.453643
$476$	0	0
$477$	−11248.5	−1.07973
$478$	0	0
$479$	11683.3	1.11445	0.557225	−	0.830361i	$-0.311866\pi$
$479$	11683.3	1.11445	0.557225	+	0.830361i	$0.311866\pi$
$480$	0	0
$481$	928.659	0.0880316
$482$	0	0
$483$	1306.96	0.123124
$484$	0	0
$485$	−637.129	−0.0596506
$486$	0	0
$487$	3970.36	0.369434	0.184717	−	0.982792i	$-0.440863\pi$
$487$	3970.36	0.369434	0.184717	+	0.982792i	$0.440863\pi$
$488$	0	0
$489$	−19054.6	−1.76213
$490$	0	0
$491$	−3096.59	−0.284618	−0.142309	−	0.989822i	$-0.545453\pi$
$491$	−3096.59	−0.284618	−0.142309	+	0.989822i	$0.545453\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	11642.2	1.05713
$496$	0	0
$497$	−1592.02	−0.143686
$498$	0	0
$499$	7685.17	0.689450	0.344725	−	0.938704i	$-0.387972\pi$
$499$	7685.17	0.689450	0.344725	+	0.938704i	$0.387972\pi$
$500$	0	0
$501$	−30563.7	−2.72552
$502$	0	0
$503$	−7824.80	−0.693619	−0.346810	−	0.937936i	$-0.612735\pi$
$503$	−7824.80	−0.693619	−0.346810	+	0.937936i	$0.612735\pi$
$504$	0	0
$505$	−7295.35	−0.642849
$506$	0	0
$507$	18375.0	1.60959
$508$	0	0
$509$	2753.24	0.239755	0.119877	−	0.992789i	$-0.461750\pi$
$509$	2753.24	0.239755	0.119877	+	0.992789i	$0.461750\pi$
$510$	0	0
$511$	209.845	0.0181663
$512$	0	0
$513$	9928.27	0.854471
$514$	0	0
$515$	−1709.97	−0.146311
$516$	0	0
$517$	16564.8	1.40913
$518$	0	0
$519$	−4745.59	−0.401365
$520$	0	0
$521$	20660.0	1.73729	0.868646	−	0.495433i	$-0.164991\pi$
$521$	20660.0	1.73729	0.868646	+	0.495433i	$0.164991\pi$
$522$	0	0
$523$	2877.90	0.240615	0.120307	−	0.992737i	$-0.461612\pi$
$523$	2877.90	0.240615	0.120307	+	0.992737i	$0.461612\pi$
$524$	0	0
$525$	−1469.47	−0.122158
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−2355.90	−0.193631
$530$	0	0
$531$	5619.79	0.459281
$532$	0	0
$533$	569.754	0.0463016
$534$	0	0
$535$	−5721.95	−0.462395
$536$	0	0
$537$	−34004.8	−2.73262
$538$	0	0
$539$	−20171.1	−1.61193
$540$	0	0
$541$	20912.6	1.66193	0.830963	−	0.556328i	$-0.187790\pi$
$541$	20912.6	1.66193	0.830963	+	0.556328i	$0.187790\pi$
$542$	0	0
$543$	−25552.7	−2.01947
$544$	0	0
$545$	5431.64	0.426910
$546$	0	0
$547$	−4203.83	−0.328597	−0.164299	−	0.986411i	$-0.552536\pi$
$547$	−4203.83	−0.328597	−0.164299	+	0.986411i	$0.552536\pi$
$548$	0	0
$549$	44032.7	3.42308
$550$	0	0
$551$	10747.7	0.830977
$552$	0	0
$553$	−321.676	−0.0247361
$554$	0	0
$555$	−2541.24	−0.194360
$556$	0	0
$557$	12948.1	0.984971	0.492486	−	0.870321i	$-0.336088\pi$
$557$	12948.1	0.984971	0.492486	+	0.870321i	$0.336088\pi$
$558$	0	0
$559$	−2752.24	−0.208242
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2892.35	0.216515	0.108258	−	0.994123i	$-0.465473\pi$
$563$	2892.35	0.216515	0.108258	+	0.994123i	$0.465473\pi$
$564$	0	0
$565$	−4482.97	−0.333805
$566$	0	0
$567$	−988.123	−0.0731874
$568$	0	0
$569$	9936.96	0.732125	0.366062	−	0.930590i	$-0.380706\pi$
$569$	9936.96	0.732125	0.366062	+	0.930590i	$0.380706\pi$
$570$	0	0
$571$	16000.2	1.17266	0.586330	−	0.810072i	$-0.300572\pi$
$571$	16000.2	1.17266	0.586330	+	0.810072i	$0.300572\pi$
$572$	0	0
$573$	4670.65	0.340522
$574$	0	0
$575$	−11030.9	−0.800039
$576$	0	0
$577$	−1663.43	−0.120016	−0.0600081	−	0.998198i	$-0.519113\pi$
$577$	−1663.43	−0.120016	−0.0600081	+	0.998198i	$0.519113\pi$
$578$	0	0
$579$	−14862.0	−1.06674
$580$	0	0
$581$	1839.87	0.131378
$582$	0	0
$583$	−12495.2	−0.887647
$584$	0	0
$585$	2378.25	0.168083
$586$	0	0
$587$	−18366.1	−1.29140	−0.645699	−	0.763592i	$-0.723434\pi$
$587$	−18366.1	−1.29140	−0.645699	+	0.763592i	$0.723434\pi$
$588$	0	0
$589$	7117.61	0.497922
$590$	0	0
$591$	40342.7	2.80791
$592$	0	0
$593$	−18886.5	−1.30788	−0.653941	−	0.756546i	$-0.726886\pi$
$593$	−18886.5	−1.30788	−0.653941	+	0.756546i	$0.726886\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13451.3	0.922154
$598$	0	0
$599$	−21164.2	−1.44365	−0.721823	−	0.692078i	$-0.756695\pi$
$599$	−21164.2	−1.44365	−0.721823	+	0.692078i	$0.756695\pi$
$600$	0	0
$601$	−11491.5	−0.779946	−0.389973	−	0.920826i	$-0.627516\pi$
$601$	−11491.5	−0.779946	−0.389973	+	0.920826i	$0.627516\pi$
$602$	0	0
$603$	34006.4	2.29660
$604$	0	0
$605$	8018.05	0.538810
$606$	0	0
$607$	−10835.5	−0.724548	−0.362274	−	0.932072i	$-0.618000\pi$
$607$	−10835.5	−0.724548	−0.362274	+	0.932072i	$0.618000\pi$
$608$	0	0
$609$	−3362.96	−0.223767
$610$	0	0
$611$	3383.83	0.224051
$612$	0	0
$613$	−26334.1	−1.73511	−0.867556	−	0.497340i	$-0.834310\pi$
$613$	−26334.1	−1.73511	−0.867556	+	0.497340i	$0.834310\pi$
$614$	0	0
$615$	−1559.11	−0.102227
$616$	0	0
$617$	−13781.0	−0.899196	−0.449598	−	0.893231i	$-0.648433\pi$
$617$	−13781.0	−0.899196	−0.449598	+	0.893231i	$0.648433\pi$
$618$	0	0
$619$	15328.9	0.995347	0.497673	−	0.867364i	$-0.334188\pi$
$619$	15328.9	0.995347	0.497673	+	0.867364i	$0.334188\pi$
$620$	0	0
$621$	−23320.2	−1.50694
$622$	0	0
$623$	1802.30	0.115903
$624$	0	0
$625$	10698.3	0.684689
$626$	0	0
$627$	22360.7	1.42425
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−10666.5	−0.672943	−0.336471	−	0.941694i	$-0.609234\pi$
$631$	−10666.5	−0.672943	−0.336471	+	0.941694i	$0.609234\pi$
$632$	0	0
$633$	36955.2	2.32044
$634$	0	0
$635$	7600.98	0.475017
$636$	0	0
$637$	−4120.51	−0.256296
$638$	0	0
$639$	57594.5	3.56558
$640$	0	0
$641$	−8010.60	−0.493603	−0.246802	−	0.969066i	$-0.579380\pi$
$641$	−8010.60	−0.493603	−0.246802	+	0.969066i	$0.579380\pi$
$642$	0	0
$643$	−4013.14	−0.246132	−0.123066	−	0.992399i	$-0.539273\pi$
$643$	−4013.14	−0.246132	−0.123066	+	0.992399i	$0.539273\pi$
$644$	0	0
$645$	7531.39	0.459765
$646$	0	0
$647$	−7414.08	−0.450506	−0.225253	−	0.974300i	$-0.572321\pi$
$647$	−7414.08	−0.450506	−0.225253	+	0.974300i	$0.572321\pi$
$648$	0	0
$649$	6242.66	0.377574
$650$	0	0
$651$	−2227.10	−0.134081
$652$	0	0
$653$	11839.8	0.709536	0.354768	−	0.934954i	$-0.384560\pi$
$653$	11839.8	0.709536	0.354768	+	0.934954i	$0.384560\pi$
$654$	0	0
$655$	6061.88	0.361614
$656$	0	0
$657$	−7591.54	−0.450798
$658$	0	0
$659$	−7958.77	−0.470455	−0.235227	−	0.971940i	$-0.575583\pi$
$659$	−7958.77	−0.470455	−0.235227	+	0.971940i	$0.575583\pi$
$660$	0	0
$661$	−32464.8	−1.91034	−0.955170	−	0.296058i	$-0.904328\pi$
$661$	−32464.8	−1.91034	−0.955170	+	0.296058i	$0.904328\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	229.306	0.0133716
$666$	0	0
$667$	−25245.0	−1.46550
$668$	0	0
$669$	43212.0	2.49727
$670$	0	0
$671$	48913.1	2.81411
$672$	0	0
$673$	−22356.1	−1.28048	−0.640241	−	0.768174i	$-0.721166\pi$
$673$	−22356.1	−1.28048	−0.640241	+	0.768174i	$0.721166\pi$
$674$	0	0
$675$	26219.7	1.49511
$676$	0	0
$677$	13031.8	0.739812	0.369906	−	0.929069i	$-0.379390\pi$
$677$	13031.8	0.739812	0.369906	+	0.929069i	$0.379390\pi$
$678$	0	0
$679$	254.115	0.0143624
$680$	0	0
$681$	18446.3	1.03798
$682$	0	0
$683$	1994.90	0.111761	0.0558805	−	0.998437i	$-0.482203\pi$
$683$	1994.90	0.111761	0.0558805	+	0.998437i	$0.482203\pi$
$684$	0	0
$685$	1368.22	0.0763168
$686$	0	0
$687$	36936.7	2.05127
$688$	0	0
$689$	−2552.49	−0.141135
$690$	0	0
$691$	−18711.0	−1.03010	−0.515050	−	0.857160i	$-0.672227\pi$
$691$	−18711.0	−1.03010	−0.515050	+	0.857160i	$0.672227\pi$
$692$	0	0
$693$	−4643.44	−0.254531
$694$	0	0
$695$	−938.543	−0.0512244
$696$	0	0
$697$	0	0
$698$	0	0
$699$	53166.0	2.87686
$700$	0	0
$701$	3487.04	0.187880	0.0939399	−	0.995578i	$-0.470054\pi$
$701$	3487.04	0.187880	0.0939399	+	0.995578i	$0.470054\pi$
$702$	0	0
$703$	−3239.25	−0.173785
$704$	0	0
$705$	−9259.72	−0.494668
$706$	0	0
$707$	2909.71	0.154782
$708$	0	0
$709$	9834.71	0.520946	0.260473	−	0.965481i	$-0.416122\pi$
$709$	9834.71	0.520946	0.260473	+	0.965481i	$0.416122\pi$
$710$	0	0
$711$	11637.3	0.613827
$712$	0	0
$713$	−16718.3	−0.878130
$714$	0	0
$715$	2641.85	0.138181
$716$	0	0
$717$	44168.2	2.30055
$718$	0	0
$719$	1760.43	0.0913116	0.0456558	−	0.998957i	$-0.485462\pi$
$719$	1760.43	0.0913116	0.0456558	+	0.998957i	$0.485462\pi$
$720$	0	0
$721$	682.011	0.0352280
$722$	0	0
$723$	16835.9	0.866024
$724$	0	0
$725$	28383.8	1.45400
$726$	0	0
$727$	−16586.7	−0.846173	−0.423086	−	0.906089i	$-0.639053\pi$
$727$	−16586.7	−0.846173	−0.423086	+	0.906089i	$0.639053\pi$
$728$	0	0
$729$	−21207.9	−1.07747
$730$	0	0
$731$	0	0
$732$	0	0
$733$	20221.6	1.01897	0.509484	−	0.860480i	$-0.329836\pi$
$733$	20221.6	1.01897	0.509484	+	0.860480i	$0.329836\pi$
$734$	0	0
$735$	11275.6	0.565861
$736$	0	0
$737$	37775.5	1.88803
$738$	0	0
$739$	8176.50	0.407006	0.203503	−	0.979074i	$-0.434767\pi$
$739$	8176.50	0.407006	0.203503	+	0.979074i	$0.434767\pi$
$740$	0	0
$741$	4567.81	0.226454
$742$	0	0
$743$	5841.82	0.288446	0.144223	−	0.989545i	$-0.453932\pi$
$743$	5841.82	0.288446	0.144223	+	0.989545i	$0.453932\pi$
$744$	0	0
$745$	8478.09	0.416930
$746$	0	0
$747$	−66560.7	−3.26014
$748$	0	0
$749$	2282.17	0.111333
$750$	0	0
$751$	−32642.8	−1.58609	−0.793044	−	0.609164i	$-0.791505\pi$
$751$	−32642.8	−1.58609	−0.793044	+	0.609164i	$0.791505\pi$
$752$	0	0
$753$	−42126.5	−2.03875
$754$	0	0
$755$	−5020.82	−0.242022
$756$	0	0
$757$	6800.98	0.326533	0.163267	−	0.986582i	$-0.447797\pi$
$757$	6800.98	0.326533	0.163267	+	0.986582i	$0.447797\pi$
$758$	0	0
$759$	−52522.4	−2.51178
$760$	0	0
$761$	−17756.1	−0.845805	−0.422902	−	0.906175i	$-0.638989\pi$
$761$	−17756.1	−0.845805	−0.422902	+	0.906175i	$0.638989\pi$
$762$	0	0
$763$	−2166.38	−0.102789
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1275.24	0.0600341
$768$	0	0
$769$	16008.1	0.750673	0.375337	−	0.926889i	$-0.377527\pi$
$769$	16008.1	0.750673	0.375337	+	0.926889i	$0.377527\pi$
$770$	0	0
$771$	−11541.5	−0.539113
$772$	0	0
$773$	31660.1	1.47314	0.736569	−	0.676362i	$-0.236445\pi$
$773$	31660.1	1.47314	0.736569	+	0.676362i	$0.236445\pi$
$774$	0	0
$775$	18797.0	0.871236
$776$	0	0
$777$	1013.56	0.0467970
$778$	0	0
$779$	−1987.36	−0.0914049
$780$	0	0
$781$	63978.0	2.93126
$782$	0	0
$783$	60005.3	2.73871
$784$	0	0
$785$	−5127.29	−0.233122
$786$	0	0
$787$	−18765.4	−0.849956	−0.424978	−	0.905204i	$-0.639718\pi$
$787$	−18765.4	−0.849956	−0.424978	+	0.905204i	$0.639718\pi$
$788$	0	0
$789$	48854.1	2.20438
$790$	0	0
$791$	1788.01	0.0803721
$792$	0	0
$793$	9991.87	0.447442
$794$	0	0
$795$	6984.80	0.311604
$796$	0	0
$797$	14341.6	0.637396	0.318698	−	0.947856i	$-0.396754\pi$
$797$	14341.6	0.637396	0.318698	+	0.947856i	$0.396754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−65201.8	−2.87614
$802$	0	0
$803$	−8432.95	−0.370601
$804$	0	0
$805$	−538.609	−0.0235819
$806$	0	0
$807$	10563.9	0.460801
$808$	0	0
$809$	−26729.8	−1.16164	−0.580822	−	0.814031i	$-0.697269\pi$
$809$	−26729.8	−1.16164	−0.580822	+	0.814031i	$0.697269\pi$
$810$	0	0
$811$	−13970.4	−0.604893	−0.302446	−	0.953166i	$-0.597803\pi$
$811$	−13970.4	−0.604893	−0.302446	+	0.953166i	$0.597803\pi$
$812$	0	0
$813$	−21527.1	−0.928646
$814$	0	0
$815$	7852.53	0.337500
$816$	0	0
$817$	9600.06	0.411094
$818$	0	0
$819$	−948.553	−0.0404702
$820$	0	0
$821$	14495.7	0.616203	0.308101	−	0.951354i	$-0.400306\pi$
$821$	14495.7	0.616203	0.308101	+	0.951354i	$0.400306\pi$
$822$	0	0
$823$	−32684.8	−1.38435	−0.692175	−	0.721729i	$-0.743348\pi$
$823$	−32684.8	−1.38435	−0.692175	+	0.721729i	$0.743348\pi$
$824$	0	0
$825$	59052.8	2.49206
$826$	0	0
$827$	−17445.1	−0.733527	−0.366763	−	0.930314i	$-0.619534\pi$
$827$	−17445.1	−0.733527	−0.366763	+	0.930314i	$0.619534\pi$
$828$	0	0
$829$	28086.3	1.17669	0.588346	−	0.808609i	$-0.299779\pi$
$829$	28086.3	1.17669	0.588346	+	0.808609i	$0.299779\pi$
$830$	0	0
$831$	−54319.3	−2.26753
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12595.5	0.522018
$836$	0	0
$837$	39738.1	1.64104
$838$	0	0
$839$	−21207.4	−0.872658	−0.436329	−	0.899787i	$-0.643722\pi$
$839$	−21207.4	−0.872658	−0.436329	+	0.899787i	$0.643722\pi$
$840$	0	0
$841$	40568.9	1.66341
$842$	0	0
$843$	−22240.2	−0.908653
$844$	0	0
$845$	−7572.47	−0.308285
$846$	0	0
$847$	−3197.96	−0.129732
$848$	0	0
$849$	−29391.5	−1.18812
$850$	0	0
$851$	7608.57	0.306485
$852$	0	0
$853$	−7922.65	−0.318015	−0.159007	−	0.987277i	$-0.550829\pi$
$853$	−7922.65	−0.318015	−0.159007	+	0.987277i	$0.550829\pi$
$854$	0	0
$855$	−8295.57	−0.331816
$856$	0	0
$857$	13278.7	0.529280	0.264640	−	0.964347i	$-0.414747\pi$
$857$	13278.7	0.529280	0.264640	+	0.964347i	$0.414747\pi$
$858$	0	0
$859$	9239.88	0.367009	0.183504	−	0.983019i	$-0.441256\pi$
$859$	9239.88	0.367009	0.183504	+	0.983019i	$0.441256\pi$
$860$	0	0
$861$	621.843	0.0246136
$862$	0	0
$863$	27602.0	1.08874	0.544370	−	0.838845i	$-0.316769\pi$
$863$	27602.0	1.08874	0.544370	+	0.838845i	$0.316769\pi$
$864$	0	0
$865$	1955.69	0.0768733
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12927.1	0.504627
$870$	0	0
$871$	7716.71	0.300196
$872$	0	0
$873$	−9193.11	−0.356403
$874$	0	0
$875$	1285.29	0.0496579
$876$	0	0
$877$	10577.4	0.407267	0.203633	−	0.979047i	$-0.434725\pi$
$877$	10577.4	0.407267	0.203633	+	0.979047i	$0.434725\pi$
$878$	0	0
$879$	26394.7	1.01282
$880$	0	0
$881$	−21241.2	−0.812298	−0.406149	−	0.913807i	$-0.633129\pi$
$881$	−21241.2	−0.812298	−0.406149	+	0.913807i	$0.633129\pi$
$882$	0	0
$883$	−10705.8	−0.408016	−0.204008	−	0.978969i	$-0.565397\pi$
$883$	−10705.8	−0.408016	−0.204008	+	0.978969i	$0.565397\pi$
$884$	0	0
$885$	−3489.64	−0.132546
$886$	0	0
$887$	−32061.4	−1.21366	−0.606830	−	0.794831i	$-0.707559\pi$
$887$	−32061.4	−1.21366	−0.606830	+	0.794831i	$0.707559\pi$
$888$	0	0
$889$	−3031.61	−0.114372
$890$	0	0
$891$	39709.3	1.49305
$892$	0	0
$893$	−11803.1	−0.442303
$894$	0	0
$895$	14013.6	0.523378
$896$	0	0
$897$	−10729.2	−0.399372
$898$	0	0
$899$	43018.0	1.59592
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3003.85	−0.110700
$904$	0	0
$905$	10530.5	0.386789
$906$	0	0
$907$	50916.9	1.86402	0.932010	−	0.362432i	$-0.118053\pi$
$907$	50916.9	1.86402	0.932010	+	0.362432i	$0.118053\pi$
$908$	0	0
$909$	−105264.	−3.84092
$910$	0	0
$911$	1626.60	0.0591566	0.0295783	−	0.999562i	$-0.490584\pi$
$911$	1626.60	0.0591566	0.0295783	+	0.999562i	$0.490584\pi$
$912$	0	0
$913$	−73937.9	−2.68016
$914$	0	0
$915$	−27342.4	−0.987880
$916$	0	0
$917$	−2417.75	−0.0870677
$918$	0	0
$919$	3069.68	0.110184	0.0550922	−	0.998481i	$-0.482455\pi$
$919$	3069.68	0.110184	0.0550922	+	0.998481i	$0.482455\pi$
$920$	0	0
$921$	−8103.33	−0.289917
$922$	0	0
$923$	13069.3	0.466068
$924$	0	0
$925$	−8554.58	−0.304079
$926$	0	0
$927$	−24673.0	−0.874184
$928$	0	0
$929$	27405.6	0.967869	0.483934	−	0.875104i	$-0.339207\pi$
$929$	27405.6	0.967869	0.483934	+	0.875104i	$0.339207\pi$
$930$	0	0
$931$	14372.7	0.505959
$932$	0	0
$933$	−5616.68	−0.197087
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44275.2	−1.54366	−0.771829	−	0.635830i	$-0.780658\pi$
$937$	−44275.2	−1.54366	−0.771829	+	0.635830i	$0.780658\pi$
$938$	0	0
$939$	26289.7	0.913666
$940$	0	0
$941$	937.676	0.0324839	0.0162420	−	0.999868i	$-0.494830\pi$
$941$	937.676	0.0324839	0.0162420	+	0.999868i	$0.494830\pi$
$942$	0	0
$943$	4668.04	0.161201
$944$	0	0
$945$	1280.23	0.0440697
$946$	0	0
$947$	−33289.6	−1.14231	−0.571154	−	0.820843i	$-0.693504\pi$
$947$	−33289.6	−1.14231	−0.571154	+	0.820843i	$0.693504\pi$
$948$	0	0
$949$	−1722.67	−0.0589253
$950$	0	0
$951$	73971.9	2.52229
$952$	0	0
$953$	−25936.5	−0.881601	−0.440800	−	0.897605i	$-0.645305\pi$
$953$	−25936.5	−0.881601	−0.440800	+	0.897605i	$0.645305\pi$
$954$	0	0
$955$	−1924.80	−0.0652201
$956$	0	0
$957$	135146.	4.56493
$958$	0	0
$959$	−545.708	−0.0183752
$960$	0	0
$961$	−1302.56	−0.0437231
$962$	0	0
$963$	−82561.9	−2.76274
$964$	0	0
$965$	6124.73	0.204313
$966$	0	0
$967$	−7494.43	−0.249229	−0.124615	−	0.992205i	$-0.539769\pi$
$967$	−7494.43	−0.249229	−0.124615	+	0.992205i	$0.539769\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7191.65	−0.237684	−0.118842	−	0.992913i	$-0.537918\pi$
$971$	−7191.65	−0.237684	−0.118842	+	0.992913i	$0.537918\pi$
$972$	0	0
$973$	374.333	0.0123336
$974$	0	0
$975$	12063.2	0.396237
$976$	0	0
$977$	−36160.3	−1.18411	−0.592053	−	0.805899i	$-0.701682\pi$
$977$	−36160.3	−1.18411	−0.592053	+	0.805899i	$0.701682\pi$
$978$	0	0
$979$	−72428.4	−2.36448
$980$	0	0
$981$	78373.0	2.55072
$982$	0	0
$983$	−50237.0	−1.63002	−0.815011	−	0.579445i	$-0.803269\pi$
$983$	−50237.0	−1.63002	−0.815011	+	0.579445i	$0.803269\pi$
$984$	0	0
$985$	−16625.5	−0.537799
$986$	0	0
$987$	3693.19	0.119104
$988$	0	0
$989$	−22549.3	−0.725000
$990$	0	0
$991$	39841.8	1.27711	0.638556	−	0.769575i	$-0.279532\pi$
$991$	39841.8	1.27711	0.638556	+	0.769575i	$0.279532\pi$
$992$	0	0
$993$	−73744.2	−2.35670
$994$	0	0
$995$	−5543.38	−0.176620
$996$	0	0
$997$	−58983.2	−1.87364	−0.936818	−	0.349817i	$-0.886244\pi$
$997$	−58983.2	−1.87364	−0.936818	+	0.349817i	$0.886244\pi$
$998$	0	0
$999$	−18085.0	−0.572756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.r.1.1		24
17.11	odd	16		136.4.n.a.121.6	yes	24
17.14	odd	16		136.4.n.a.9.6	✓	24
17.16	even	2	inner	2312.4.a.r.1.24		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.a.9.6	✓	24	17.14	odd	16
136.4.n.a.121.6	yes	24	17.11	odd	16
2312.4.a.r.1.1		24	1.1	even	1	trivial
2312.4.a.r.1.24		24	17.16	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-8.95975 q^{3} +3.69237 q^{5} -1.47268 q^{7} +53.2771 q^{9} +O(q^{10})\)	\(q-8.95975 q^{3} +3.69237 q^{5} -1.47268 q^{7} +53.2771 q^{9} +59.1821 q^{11} +12.0896 q^{13} -33.0827 q^{15} -42.1697 q^{19} +13.1949 q^{21} +99.0510 q^{23} -111.366 q^{25} -235.436 q^{27} -254.868 q^{29} -168.785 q^{31} -530.257 q^{33} -5.43769 q^{35} +76.8147 q^{37} -108.320 q^{39} +47.1276 q^{41} -227.653 q^{43} +196.719 q^{45} +279.896 q^{47} -340.831 q^{49} -211.131 q^{53} +218.522 q^{55} +377.830 q^{57} +105.482 q^{59} +826.485 q^{61} -78.4603 q^{63} +44.6393 q^{65} +638.294 q^{67} -887.472 q^{69} +1081.04 q^{71} -142.492 q^{73} +997.815 q^{75} -87.1564 q^{77} +218.429 q^{79} +670.968 q^{81} -1249.33 q^{83} +2283.56 q^{87} -1223.82 q^{89} -17.8041 q^{91} +1512.27 q^{93} -155.706 q^{95} -172.553 q^{97} +3153.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 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

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