LMFDB - Embedded newform 2352.4.a.co.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.391168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 40x^{2} + 382 $ x^4 - 40*x^2 + 382
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.92368$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -14.5121 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -14.5121 q^{5} +9.00000 q^{9} -10.1116 q^{11} +0.623659 q^{13} -43.5362 q^{15} +16.8120 q^{17} +75.7927 q^{19} -46.7115 q^{23} +85.5999 q^{25} +27.0000 q^{27} +73.5290 q^{29} -135.435 q^{31} -30.3348 q^{33} +133.220 q^{37} +1.87098 q^{39} +69.3010 q^{41} -279.261 q^{43} -130.609 q^{45} +372.176 q^{47} +50.4361 q^{51} +656.872 q^{53} +146.740 q^{55} +227.378 q^{57} -730.411 q^{59} -39.5121 q^{61} -9.05057 q^{65} -417.962 q^{67} -140.135 q^{69} +831.408 q^{71} -1192.72 q^{73} +256.800 q^{75} +1034.49 q^{79} +81.0000 q^{81} +117.618 q^{83} -243.977 q^{85} +220.587 q^{87} +678.390 q^{89} -406.305 q^{93} -1099.91 q^{95} -1659.31 q^{97} -91.0044 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9} - 40 q^{11} - 48 q^{13} - 24 q^{15} - 72 q^{17} + 32 q^{19} - 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} - 120 q^{33} + 48 q^{37} - 144 q^{39} - 72 q^{41} - 512 q^{43} - 72 q^{45} + 160 q^{47} - 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} - 896 q^{61} - 136 q^{65} - 1088 q^{67} - 24 q^{69} - 1288 q^{71} - 1488 q^{73} + 492 q^{75} - 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} - 3160 q^{89} + 144 q^{93} + 240 q^{95} - 2384 q^{97} - 360 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9 - 40 * q^11 - 48 * q^13 - 24 * q^15 - 72 * q^17 + 32 * q^19 - 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 - 120 * q^33 + 48 * q^37 - 144 * q^39 - 72 * q^41 - 512 * q^43 - 72 * q^45 + 160 * q^47 - 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 - 896 * q^61 - 136 * q^65 - 1088 * q^67 - 24 * q^69 - 1288 * q^71 - 1488 * q^73 + 492 * q^75 - 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 - 3160 * q^89 + 144 * q^93 + 240 * q^95 - 2384 * q^97 - 360 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−14.5121	−1.29800	−0.648999	−	0.760789i	$-0.724812\pi$
$5$	−14.5121	−1.29800	−0.648999	+	0.760789i	$0.724812\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−10.1116	−0.277160	−0.138580	−	0.990351i	$-0.544254\pi$
$11$	−10.1116	−0.277160	−0.138580	+	0.990351i	$0.544254\pi$
$12$	0	0
$13$	0.623659	0.0133055	0.00665276	−	0.999978i	$-0.497882\pi$
$13$	0.623659	0.0133055	0.00665276	+	0.999978i	$0.497882\pi$
$14$	0	0
$15$	−43.5362	−0.749400
$16$	0	0
$17$	16.8120	0.239854	0.119927	−	0.992783i	$-0.461734\pi$
$17$	16.8120	0.239854	0.119927	+	0.992783i	$0.461734\pi$
$18$	0	0
$19$	75.7927	0.915159	0.457580	−	0.889169i	$-0.348716\pi$
$19$	75.7927	0.915159	0.457580	+	0.889169i	$0.348716\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−46.7115	−0.423479	−0.211740	−	0.977326i	$-0.567913\pi$
$23$	−46.7115	−0.423479	−0.211740	+	0.977326i	$0.567913\pi$
$24$	0	0
$25$	85.5999	0.684799
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	73.5290	0.470827	0.235414	−	0.971895i	$-0.424356\pi$
$29$	73.5290	0.470827	0.235414	+	0.971895i	$0.424356\pi$
$30$	0	0
$31$	−135.435	−0.784672	−0.392336	−	0.919822i	$-0.628333\pi$
$31$	−135.435	−0.784672	−0.392336	+	0.919822i	$0.628333\pi$
$32$	0	0
$33$	−30.3348	−0.160018
$34$	0	0
$35$	0	0
$36$	0	0
$37$	133.220	0.591926	0.295963	−	0.955199i	$-0.404359\pi$
$37$	133.220	0.591926	0.295963	+	0.955199i	$0.404359\pi$
$38$	0	0
$39$	1.87098	0.00768195
$40$	0	0
$41$	69.3010	0.263975	0.131988	−	0.991251i	$-0.457864\pi$
$41$	69.3010	0.263975	0.131988	+	0.991251i	$0.457864\pi$
$42$	0	0
$43$	−279.261	−0.990392	−0.495196	−	0.868781i	$-0.664904\pi$
$43$	−279.261	−0.990392	−0.495196	+	0.868781i	$0.664904\pi$
$44$	0	0
$45$	−130.609	−0.432666
$46$	0	0
$47$	372.176	1.15505	0.577526	−	0.816372i	$-0.304018\pi$
$47$	372.176	1.15505	0.577526	+	0.816372i	$0.304018\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	50.4361	0.138480
$52$	0	0
$53$	656.872	1.70242	0.851210	−	0.524825i	$-0.175869\pi$
$53$	656.872	1.70242	0.851210	+	0.524825i	$0.175869\pi$
$54$	0	0
$55$	146.740	0.359753
$56$	0	0
$57$	227.378	0.528368
$58$	0	0
$59$	−730.411	−1.61172	−0.805859	−	0.592107i	$-0.798296\pi$
$59$	−730.411	−1.61172	−0.805859	+	0.592107i	$0.798296\pi$
$60$	0	0
$61$	−39.5121	−0.0829345	−0.0414673	−	0.999140i	$-0.513203\pi$
$61$	−39.5121	−0.0829345	−0.0414673	+	0.999140i	$0.513203\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.05057	−0.0172705
$66$	0	0
$67$	−417.962	−0.762122	−0.381061	−	0.924550i	$-0.624441\pi$
$67$	−417.962	−0.762122	−0.381061	+	0.924550i	$0.624441\pi$
$68$	0	0
$69$	−140.135	−0.244496
$70$	0	0
$71$	831.408	1.38972	0.694859	−	0.719146i	$-0.255467\pi$
$71$	831.408	1.38972	0.694859	+	0.719146i	$0.255467\pi$
$72$	0	0
$73$	−1192.72	−1.91229	−0.956145	−	0.292894i	$-0.905382\pi$
$73$	−1192.72	−1.91229	−0.956145	+	0.292894i	$0.905382\pi$
$74$	0	0
$75$	256.800	0.395369
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1034.49	1.47327	0.736637	−	0.676288i	$-0.236413\pi$
$79$	1034.49	1.47327	0.736637	+	0.676288i	$0.236413\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	117.618	0.155545	0.0777726	−	0.996971i	$-0.475219\pi$
$83$	117.618	0.155545	0.0777726	+	0.996971i	$0.475219\pi$
$84$	0	0
$85$	−243.977	−0.311330
$86$	0	0
$87$	220.587	0.271832
$88$	0	0
$89$	678.390	0.807969	0.403984	−	0.914766i	$-0.367625\pi$
$89$	678.390	0.807969	0.403984	+	0.914766i	$0.367625\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−406.305	−0.453031
$94$	0	0
$95$	−1099.91	−1.18788
$96$	0	0
$97$	−1659.31	−1.73688	−0.868439	−	0.495795i	$-0.834877\pi$
$97$	−1659.31	−1.73688	−0.868439	+	0.495795i	$0.834877\pi$
$98$	0	0
$99$	−91.0044	−0.0923867
$100$	0	0
$101$	−485.839	−0.478641	−0.239321	−	0.970941i	$-0.576925\pi$
$101$	−485.839	−0.478641	−0.239321	+	0.970941i	$0.576925\pi$
$102$	0	0
$103$	276.792	0.264787	0.132394	−	0.991197i	$-0.457734\pi$
$103$	276.792	0.264787	0.132394	+	0.991197i	$0.457734\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−103.639	−0.0936373	−0.0468187	−	0.998903i	$-0.514908\pi$
$107$	−103.639	−0.0936373	−0.0468187	+	0.998903i	$0.514908\pi$
$108$	0	0
$109$	−50.1571	−0.0440751	−0.0220375	−	0.999757i	$-0.507015\pi$
$109$	−50.1571	−0.0440751	−0.0220375	+	0.999757i	$0.507015\pi$
$110$	0	0
$111$	399.661	0.341749
$112$	0	0
$113$	−1954.66	−1.62725	−0.813623	−	0.581393i	$-0.802508\pi$
$113$	−1954.66	−1.62725	−0.813623	+	0.581393i	$0.802508\pi$
$114$	0	0
$115$	677.880	0.549675
$116$	0	0
$117$	5.61293	0.00443517
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1228.76	−0.923182
$122$	0	0
$123$	207.903	0.152406
$124$	0	0
$125$	571.776	0.409130
$126$	0	0
$127$	1123.20	0.784783	0.392392	−	0.919798i	$-0.371648\pi$
$127$	1123.20	0.784783	0.392392	+	0.919798i	$0.371648\pi$
$128$	0	0
$129$	−837.782	−0.571803
$130$	0	0
$131$	1562.85	1.04234	0.521171	−	0.853453i	$-0.325496\pi$
$131$	1562.85	1.04234	0.521171	+	0.853453i	$0.325496\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−391.826	−0.249800
$136$	0	0
$137$	1283.91	0.800673	0.400337	−	0.916368i	$-0.368893\pi$
$137$	1283.91	0.800673	0.400337	+	0.916368i	$0.368893\pi$
$138$	0	0
$139$	−674.069	−0.411322	−0.205661	−	0.978623i	$-0.565934\pi$
$139$	−674.069	−0.411322	−0.205661	+	0.978623i	$0.565934\pi$
$140$	0	0
$141$	1116.53	0.666869
$142$	0	0
$143$	−6.30619	−0.00368776
$144$	0	0
$145$	−1067.06	−0.611133
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1293.72	−0.711314	−0.355657	−	0.934617i	$-0.615743\pi$
$149$	−1293.72	−0.711314	−0.355657	+	0.934617i	$0.615743\pi$
$150$	0	0
$151$	−2266.65	−1.22157	−0.610786	−	0.791795i	$-0.709147\pi$
$151$	−2266.65	−1.22157	−0.610786	+	0.791795i	$0.709147\pi$
$152$	0	0
$153$	151.308	0.0799512
$154$	0	0
$155$	1965.44	1.01850
$156$	0	0
$157$	−1868.03	−0.949585	−0.474792	−	0.880098i	$-0.657477\pi$
$157$	−1868.03	−0.949585	−0.474792	+	0.880098i	$0.657477\pi$
$158$	0	0
$159$	1970.61	0.982892
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1511.45	0.726293	0.363147	−	0.931732i	$-0.381702\pi$
$163$	1511.45	0.726293	0.363147	+	0.931732i	$0.381702\pi$
$164$	0	0
$165$	440.220	0.207704
$166$	0	0
$167$	−2371.40	−1.09883	−0.549415	−	0.835550i	$-0.685149\pi$
$167$	−2371.40	−1.09883	−0.549415	+	0.835550i	$0.685149\pi$
$168$	0	0
$169$	−2196.61	−0.999823
$170$	0	0
$171$	682.134	0.305053
$172$	0	0
$173$	−4124.67	−1.81267	−0.906337	−	0.422555i	$-0.861133\pi$
$173$	−4124.67	−1.81267	−0.906337	+	0.422555i	$0.861133\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2191.23	−0.930526
$178$	0	0
$179$	−3194.53	−1.33391	−0.666955	−	0.745098i	$-0.732403\pi$
$179$	−3194.53	−1.33391	−0.666955	+	0.745098i	$0.732403\pi$
$180$	0	0
$181$	−1389.99	−0.570815	−0.285407	−	0.958406i	$-0.592129\pi$
$181$	−1389.99	−0.570815	−0.285407	+	0.958406i	$0.592129\pi$
$182$	0	0
$183$	−118.536	−0.0478823
$184$	0	0
$185$	−1933.30	−0.768319
$186$	0	0
$187$	−169.996	−0.0664779
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1915.99	−0.725844	−0.362922	−	0.931820i	$-0.618221\pi$
$191$	−1915.99	−0.725844	−0.362922	+	0.931820i	$0.618221\pi$
$192$	0	0
$193$	−1092.29	−0.407381	−0.203690	−	0.979035i	$-0.565294\pi$
$193$	−1092.29	−0.407381	−0.203690	+	0.979035i	$0.565294\pi$
$194$	0	0
$195$	−27.1517	−0.00997115
$196$	0	0
$197$	4727.17	1.70963	0.854815	−	0.518933i	$-0.173671\pi$
$197$	4727.17	1.70963	0.854815	+	0.518933i	$0.173671\pi$
$198$	0	0
$199$	634.583	0.226052	0.113026	−	0.993592i	$-0.463946\pi$
$199$	634.583	0.226052	0.113026	+	0.993592i	$0.463946\pi$
$200$	0	0
$201$	−1253.89	−0.440011
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1005.70	−0.342640
$206$	0	0
$207$	−420.404	−0.141160
$208$	0	0
$209$	−766.385	−0.253646
$210$	0	0
$211$	1782.00	0.581413	0.290707	−	0.956812i	$-0.406110\pi$
$211$	1782.00	0.581413	0.290707	+	0.956812i	$0.406110\pi$
$212$	0	0
$213$	2494.22	0.802354
$214$	0	0
$215$	4052.65	1.28553
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3578.16	−1.10406
$220$	0	0
$221$	10.4850	0.00319138
$222$	0	0
$223$	−5362.96	−1.61045	−0.805225	−	0.592970i	$-0.797955\pi$
$223$	−5362.96	−1.61045	−0.805225	+	0.592970i	$0.797955\pi$
$224$	0	0
$225$	770.399	0.228266
$226$	0	0
$227$	−734.718	−0.214824	−0.107412	−	0.994215i	$-0.534256\pi$
$227$	−734.718	−0.214824	−0.107412	+	0.994215i	$0.534256\pi$
$228$	0	0
$229$	−5160.26	−1.48908	−0.744541	−	0.667577i	$-0.767332\pi$
$229$	−5160.26	−1.48908	−0.744541	+	0.667577i	$0.767332\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4068.43	−1.14391	−0.571956	−	0.820284i	$-0.693815\pi$
$233$	−4068.43	−1.14391	−0.571956	+	0.820284i	$0.693815\pi$
$234$	0	0
$235$	−5401.04	−1.49925
$236$	0	0
$237$	3103.46	0.850595
$238$	0	0
$239$	650.039	0.175931	0.0879655	−	0.996124i	$-0.471963\pi$
$239$	650.039	0.175931	0.0879655	+	0.996124i	$0.471963\pi$
$240$	0	0
$241$	−3233.14	−0.864170	−0.432085	−	0.901833i	$-0.642222\pi$
$241$	−3233.14	−0.864170	−0.432085	+	0.901833i	$0.642222\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	47.2687	0.0121767
$248$	0	0
$249$	352.854	0.0898041
$250$	0	0
$251$	1139.75	0.286615	0.143308	−	0.989678i	$-0.454226\pi$
$251$	1139.75	0.286615	0.143308	+	0.989678i	$0.454226\pi$
$252$	0	0
$253$	472.328	0.117372
$254$	0	0
$255$	−731.931	−0.179746
$256$	0	0
$257$	−2036.97	−0.494408	−0.247204	−	0.968963i	$-0.579512\pi$
$257$	−2036.97	−0.494408	−0.247204	+	0.968963i	$0.579512\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	661.761	0.156942
$262$	0	0
$263$	836.215	0.196058	0.0980289	−	0.995184i	$-0.468746\pi$
$263$	836.215	0.196058	0.0980289	+	0.995184i	$0.468746\pi$
$264$	0	0
$265$	−9532.56	−2.20974
$266$	0	0
$267$	2035.17	0.466481
$268$	0	0
$269$	−2682.99	−0.608122	−0.304061	−	0.952652i	$-0.598343\pi$
$269$	−2682.99	−0.608122	−0.304061	+	0.952652i	$0.598343\pi$
$270$	0	0
$271$	−6541.42	−1.46628	−0.733142	−	0.680076i	$-0.761947\pi$
$271$	−6541.42	−1.46628	−0.733142	+	0.680076i	$0.761947\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−865.552	−0.189799
$276$	0	0
$277$	−1293.11	−0.280490	−0.140245	−	0.990117i	$-0.544789\pi$
$277$	−1293.11	−0.280490	−0.140245	+	0.990117i	$0.544789\pi$
$278$	0	0
$279$	−1218.92	−0.261557
$280$	0	0
$281$	4684.69	0.994538	0.497269	−	0.867596i	$-0.334336\pi$
$281$	4684.69	0.994538	0.497269	+	0.867596i	$0.334336\pi$
$282$	0	0
$283$	565.240	0.118728	0.0593640	−	0.998236i	$-0.481093\pi$
$283$	565.240	0.118728	0.0593640	+	0.998236i	$0.481093\pi$
$284$	0	0
$285$	−3299.72	−0.685820
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4630.36	−0.942470
$290$	0	0
$291$	−4977.93	−1.00279
$292$	0	0
$293$	−809.451	−0.161395	−0.0806973	−	0.996739i	$-0.525715\pi$
$293$	−809.451	−0.161395	−0.0806973	+	0.996739i	$0.525715\pi$
$294$	0	0
$295$	10599.8	2.09201
$296$	0	0
$297$	−273.013	−0.0533395
$298$	0	0
$299$	−29.1320	−0.00563461
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1457.52	−0.276344
$304$	0	0
$305$	573.402	0.107649
$306$	0	0
$307$	3667.95	0.681892	0.340946	−	0.940083i	$-0.389253\pi$
$307$	3667.95	0.681892	0.340946	+	0.940083i	$0.389253\pi$
$308$	0	0
$309$	830.375	0.152875
$310$	0	0
$311$	−73.3047	−0.0133657	−0.00668284	−	0.999978i	$-0.502127\pi$
$311$	−73.3047	−0.0133657	−0.00668284	+	0.999978i	$0.502127\pi$
$312$	0	0
$313$	8300.73	1.49899	0.749497	−	0.662008i	$-0.230295\pi$
$313$	8300.73	1.49899	0.749497	+	0.662008i	$0.230295\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1288.89	0.228364	0.114182	−	0.993460i	$-0.463575\pi$
$317$	1288.89	0.228364	0.114182	+	0.993460i	$0.463575\pi$
$318$	0	0
$319$	−743.495	−0.130494
$320$	0	0
$321$	−310.918	−0.0540615
$322$	0	0
$323$	1274.23	0.219504
$324$	0	0
$325$	53.3851	0.00911161
$326$	0	0
$327$	−150.471	−0.0254468
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6517.90	1.08234	0.541172	−	0.840912i	$-0.317981\pi$
$331$	6517.90	1.08234	0.541172	+	0.840912i	$0.317981\pi$
$332$	0	0
$333$	1198.98	0.197309
$334$	0	0
$335$	6065.49	0.989233
$336$	0	0
$337$	−7477.45	−1.20867	−0.604337	−	0.796729i	$-0.706562\pi$
$337$	−7477.45	−1.20867	−0.604337	+	0.796729i	$0.706562\pi$
$338$	0	0
$339$	−5863.97	−0.939491
$340$	0	0
$341$	1369.46	0.217480
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2033.64	0.317355
$346$	0	0
$347$	1228.91	0.190119	0.0950596	−	0.995472i	$-0.469696\pi$
$347$	1228.91	0.190119	0.0950596	+	0.995472i	$0.469696\pi$
$348$	0	0
$349$	−1660.55	−0.254691	−0.127345	−	0.991858i	$-0.540646\pi$
$349$	−1660.55	−0.254691	−0.127345	+	0.991858i	$0.540646\pi$
$350$	0	0
$351$	16.8388	0.00256065
$352$	0	0
$353$	−10483.4	−1.58066	−0.790330	−	0.612681i	$-0.790091\pi$
$353$	−10483.4	−1.58066	−0.790330	+	0.612681i	$0.790091\pi$
$354$	0	0
$355$	−12065.4	−1.80385
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5329.76	0.783549	0.391775	−	0.920061i	$-0.371861\pi$
$359$	5329.76	0.783549	0.391775	+	0.920061i	$0.371861\pi$
$360$	0	0
$361$	−1114.47	−0.162483
$362$	0	0
$363$	−3686.27	−0.533000
$364$	0	0
$365$	17308.8	2.48215
$366$	0	0
$367$	−6608.43	−0.939937	−0.469969	−	0.882683i	$-0.655735\pi$
$367$	−6608.43	−0.939937	−0.469969	+	0.882683i	$0.655735\pi$
$368$	0	0
$369$	623.709	0.0879918
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6969.99	0.967540	0.483770	−	0.875195i	$-0.339267\pi$
$373$	6969.99	0.967540	0.483770	+	0.875195i	$0.339267\pi$
$374$	0	0
$375$	1715.33	0.236211
$376$	0	0
$377$	45.8570	0.00626460
$378$	0	0
$379$	7383.56	1.00071	0.500353	−	0.865821i	$-0.333203\pi$
$379$	7383.56	1.00071	0.500353	+	0.865821i	$0.333203\pi$
$380$	0	0
$381$	3369.59	0.453095
$382$	0	0
$383$	845.491	0.112800	0.0564002	−	0.998408i	$-0.482038\pi$
$383$	845.491	0.112800	0.0564002	+	0.998408i	$0.482038\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2513.35	−0.330131
$388$	0	0
$389$	−5897.87	−0.768725	−0.384362	−	0.923182i	$-0.625579\pi$
$389$	−5897.87	−0.768725	−0.384362	+	0.923182i	$0.625579\pi$
$390$	0	0
$391$	−785.315	−0.101573
$392$	0	0
$393$	4688.55	0.601796
$394$	0	0
$395$	−15012.5	−1.91231
$396$	0	0
$397$	10274.5	1.29890	0.649448	−	0.760406i	$-0.275000\pi$
$397$	10274.5	1.29890	0.649448	+	0.760406i	$0.275000\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4261.85	0.530740	0.265370	−	0.964147i	$-0.414506\pi$
$401$	4261.85	0.530740	0.265370	+	0.964147i	$0.414506\pi$
$402$	0	0
$403$	−84.4652	−0.0104405
$404$	0	0
$405$	−1175.48	−0.144222
$406$	0	0
$407$	−1347.07	−0.164058
$408$	0	0
$409$	−6389.21	−0.772436	−0.386218	−	0.922408i	$-0.626219\pi$
$409$	−6389.21	−0.772436	−0.386218	+	0.922408i	$0.626219\pi$
$410$	0	0
$411$	3851.74	0.462269
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1706.88	−0.201898
$416$	0	0
$417$	−2022.21	−0.237477
$418$	0	0
$419$	−7.77360	−0.000906361 0	−0.000453180	−	1.00000i	$-0.500144\pi$
$419$	−7.77360	−0.000906361 0	−0.000453180		1.00000i	$0.500144\pi$
$420$	0	0
$421$	8500.33	0.984040	0.492020	−	0.870584i	$-0.336259\pi$
$421$	8500.33	0.984040	0.492020	+	0.870584i	$0.336259\pi$
$422$	0	0
$423$	3349.58	0.385017
$424$	0	0
$425$	1439.11	0.164252
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.9186	−0.00212913
$430$	0	0
$431$	7264.32	0.811856	0.405928	−	0.913905i	$-0.366948\pi$
$431$	7264.32	0.811856	0.405928	+	0.913905i	$0.366948\pi$
$432$	0	0
$433$	−1023.35	−0.113578	−0.0567890	−	0.998386i	$-0.518086\pi$
$433$	−1023.35	−0.113578	−0.0567890	+	0.998386i	$0.518086\pi$
$434$	0	0
$435$	−3201.17	−0.352838
$436$	0	0
$437$	−3540.39	−0.387551
$438$	0	0
$439$	−8081.20	−0.878576	−0.439288	−	0.898346i	$-0.644769\pi$
$439$	−8081.20	−0.878576	−0.439288	+	0.898346i	$0.644769\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4900.66	−0.525592	−0.262796	−	0.964851i	$-0.584645\pi$
$443$	−4900.66	−0.525592	−0.262796	+	0.964851i	$0.584645\pi$
$444$	0	0
$445$	−9844.84	−1.04874
$446$	0	0
$447$	−3881.16	−0.410677
$448$	0	0
$449$	8493.30	0.892703	0.446351	−	0.894858i	$-0.352723\pi$
$449$	8493.30	0.892703	0.446351	+	0.894858i	$0.352723\pi$
$450$	0	0
$451$	−700.744	−0.0731635
$452$	0	0
$453$	−6799.96	−0.705275
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13042.6	−1.33503	−0.667514	−	0.744597i	$-0.732642\pi$
$457$	−13042.6	−1.33503	−0.667514	+	0.744597i	$0.732642\pi$
$458$	0	0
$459$	453.925	0.0461599
$460$	0	0
$461$	8900.01	0.899164	0.449582	−	0.893239i	$-0.351573\pi$
$461$	8900.01	0.899164	0.449582	+	0.893239i	$0.351573\pi$
$462$	0	0
$463$	17336.4	1.74015	0.870076	−	0.492917i	$-0.164070\pi$
$463$	17336.4	1.74015	0.870076	+	0.492917i	$0.164070\pi$
$464$	0	0
$465$	5896.32	0.588033
$466$	0	0
$467$	3071.51	0.304352	0.152176	−	0.988353i	$-0.451372\pi$
$467$	3071.51	0.304352	0.152176	+	0.988353i	$0.451372\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5604.08	−0.548243
$472$	0	0
$473$	2823.77	0.274497
$474$	0	0
$475$	6487.85	0.626701
$476$	0	0
$477$	5911.84	0.567473
$478$	0	0
$479$	−16272.9	−1.55225	−0.776125	−	0.630580i	$-0.782817\pi$
$479$	−16272.9	−1.55225	−0.776125	+	0.630580i	$0.782817\pi$
$480$	0	0
$481$	83.0839	0.00787589
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24080.0	2.25447
$486$	0	0
$487$	10851.9	1.00975	0.504874	−	0.863193i	$-0.331539\pi$
$487$	10851.9	1.00975	0.504874	+	0.863193i	$0.331539\pi$
$488$	0	0
$489$	4534.35	0.419326
$490$	0	0
$491$	−744.455	−0.0684252	−0.0342126	−	0.999415i	$-0.510892\pi$
$491$	−744.455	−0.0684252	−0.0342126	+	0.999415i	$0.510892\pi$
$492$	0	0
$493$	1236.17	0.112930
$494$	0	0
$495$	1320.66	0.119918
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13272.2	−1.19067	−0.595334	−	0.803478i	$-0.702980\pi$
$499$	−13272.2	−1.19067	−0.595334	+	0.803478i	$0.702980\pi$
$500$	0	0
$501$	−7114.21	−0.634410
$502$	0	0
$503$	−19485.3	−1.72725	−0.863626	−	0.504133i	$-0.831812\pi$
$503$	−19485.3	−1.72725	−0.863626	+	0.504133i	$0.831812\pi$
$504$	0	0
$505$	7050.52	0.621275
$506$	0	0
$507$	−6589.83	−0.577248
$508$	0	0
$509$	−1348.71	−0.117447	−0.0587235	−	0.998274i	$-0.518703\pi$
$509$	−1348.71	−0.117447	−0.0587235	+	0.998274i	$0.518703\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2046.40	0.176123
$514$	0	0
$515$	−4016.82	−0.343693
$516$	0	0
$517$	−3763.29	−0.320134
$518$	0	0
$519$	−12374.0	−1.04655
$520$	0	0
$521$	−3349.98	−0.281699	−0.140849	−	0.990031i	$-0.544983\pi$
$521$	−3349.98	−0.281699	−0.140849	+	0.990031i	$0.544983\pi$
$522$	0	0
$523$	−7588.42	−0.634452	−0.317226	−	0.948350i	$-0.602751\pi$
$523$	−7588.42	−0.634452	−0.317226	+	0.948350i	$0.602751\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2276.94	−0.188207
$528$	0	0
$529$	−9985.03	−0.820665
$530$	0	0
$531$	−6573.70	−0.537239
$532$	0	0
$533$	43.2202	0.00351233
$534$	0	0
$535$	1504.02	0.121541
$536$	0	0
$537$	−9583.58	−0.770134
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13690.4	−1.08797	−0.543987	−	0.839093i	$-0.683086\pi$
$541$	−13690.4	−1.08797	−0.543987	+	0.839093i	$0.683086\pi$
$542$	0	0
$543$	−4169.98	−0.329560
$544$	0	0
$545$	727.884	0.0572094
$546$	0	0
$547$	−15250.2	−1.19205	−0.596024	−	0.802967i	$-0.703254\pi$
$547$	−15250.2	−1.19205	−0.596024	+	0.802967i	$0.703254\pi$
$548$	0	0
$549$	−355.609	−0.0276448
$550$	0	0
$551$	5572.96	0.430882
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5799.90	−0.443589
$556$	0	0
$557$	−14231.6	−1.08261	−0.541305	−	0.840826i	$-0.682070\pi$
$557$	−14231.6	−1.08261	−0.541305	+	0.840826i	$0.682070\pi$
$558$	0	0
$559$	−174.163	−0.0131777
$560$	0	0
$561$	−509.989	−0.0383810
$562$	0	0
$563$	−23001.7	−1.72186	−0.860928	−	0.508727i	$-0.830116\pi$
$563$	−23001.7	−1.72186	−0.860928	+	0.508727i	$0.830116\pi$
$564$	0	0
$565$	28366.1	2.11216
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11996.5	0.883866	0.441933	−	0.897048i	$-0.354293\pi$
$569$	11996.5	0.883866	0.441933	+	0.897048i	$0.354293\pi$
$570$	0	0
$571$	25436.0	1.86421	0.932106	−	0.362186i	$-0.117970\pi$
$571$	25436.0	1.86421	0.932106	+	0.362186i	$0.117970\pi$
$572$	0	0
$573$	−5747.97	−0.419066
$574$	0	0
$575$	−3998.50	−0.289998
$576$	0	0
$577$	−1605.24	−0.115818	−0.0579090	−	0.998322i	$-0.518443\pi$
$577$	−1605.24	−0.115818	−0.0579090	+	0.998322i	$0.518443\pi$
$578$	0	0
$579$	−3276.86	−0.235201
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6642.02	−0.471843
$584$	0	0
$585$	−81.4552	−0.00575685
$586$	0	0
$587$	−20510.8	−1.44220	−0.721101	−	0.692829i	$-0.756364\pi$
$587$	−20510.8	−1.44220	−0.721101	+	0.692829i	$0.756364\pi$
$588$	0	0
$589$	−10265.0	−0.718100
$590$	0	0
$591$	14181.5	0.987055
$592$	0	0
$593$	−899.970	−0.0623227	−0.0311613	−	0.999514i	$-0.509921\pi$
$593$	−899.970	−0.0623227	−0.0311613	+	0.999514i	$0.509921\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1903.75	0.130511
$598$	0	0
$599$	17326.2	1.18185	0.590926	−	0.806726i	$-0.298762\pi$
$599$	17326.2	1.18185	0.590926	+	0.806726i	$0.298762\pi$
$600$	0	0
$601$	10193.7	0.691860	0.345930	−	0.938260i	$-0.387563\pi$
$601$	10193.7	0.691860	0.345930	+	0.938260i	$0.387563\pi$
$602$	0	0
$603$	−3761.66	−0.254041
$604$	0	0
$605$	17831.8	1.19829
$606$	0	0
$607$	−10674.8	−0.713803	−0.356901	−	0.934142i	$-0.616167\pi$
$607$	−10674.8	−0.713803	−0.356901	+	0.934142i	$0.616167\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	232.111	0.0153686
$612$	0	0
$613$	−20442.8	−1.34694	−0.673471	−	0.739214i	$-0.735197\pi$
$613$	−20442.8	−1.34694	−0.673471	+	0.739214i	$0.735197\pi$
$614$	0	0
$615$	−3017.10	−0.197823
$616$	0	0
$617$	−12084.9	−0.788523	−0.394261	−	0.918998i	$-0.629000\pi$
$617$	−12084.9	−0.788523	−0.394261	+	0.918998i	$0.629000\pi$
$618$	0	0
$619$	23057.3	1.49718	0.748589	−	0.663035i	$-0.230732\pi$
$619$	23057.3	1.49718	0.748589	+	0.663035i	$0.230732\pi$
$620$	0	0
$621$	−1261.21	−0.0814986
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18997.6	−1.21585
$626$	0	0
$627$	−2299.15	−0.146442
$628$	0	0
$629$	2239.70	0.141976
$630$	0	0
$631$	19201.2	1.21139	0.605694	−	0.795697i	$-0.292895\pi$
$631$	19201.2	1.21139	0.605694	+	0.795697i	$0.292895\pi$
$632$	0	0
$633$	5346.01	0.335679
$634$	0	0
$635$	−16299.9	−1.01865
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7482.67	0.463239
$640$	0	0
$641$	10581.2	0.651998	0.325999	−	0.945370i	$-0.394299\pi$
$641$	10581.2	0.651998	0.325999	+	0.945370i	$0.394299\pi$
$642$	0	0
$643$	29823.2	1.82910	0.914550	−	0.404474i	$-0.132545\pi$
$643$	29823.2	1.82910	0.914550	+	0.404474i	$0.132545\pi$
$644$	0	0
$645$	12157.9	0.742199
$646$	0	0
$647$	143.283	0.00870639	0.00435320	−	0.999991i	$-0.498614\pi$
$647$	143.283	0.00870639	0.00435320	+	0.999991i	$0.498614\pi$
$648$	0	0
$649$	7385.62	0.446704
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7516.26	0.450435	0.225217	−	0.974309i	$-0.427691\pi$
$653$	7516.26	0.450435	0.225217	+	0.974309i	$0.427691\pi$
$654$	0	0
$655$	−22680.2	−1.35296
$656$	0	0
$657$	−10734.5	−0.637430
$658$	0	0
$659$	4791.03	0.283205	0.141603	−	0.989924i	$-0.454774\pi$
$659$	4791.03	0.283205	0.141603	+	0.989924i	$0.454774\pi$
$660$	0	0
$661$	−4711.63	−0.277249	−0.138624	−	0.990345i	$-0.544268\pi$
$661$	−4711.63	−0.277249	−0.138624	+	0.990345i	$0.544268\pi$
$662$	0	0
$663$	31.4549	0.00184254
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3434.65	−0.199386
$668$	0	0
$669$	−16088.9	−0.929793
$670$	0	0
$671$	399.530	0.0229861
$672$	0	0
$673$	−10953.3	−0.627367	−0.313683	−	0.949528i	$-0.601563\pi$
$673$	−10953.3	−0.627367	−0.313683	+	0.949528i	$0.601563\pi$
$674$	0	0
$675$	2311.20	0.131790
$676$	0	0
$677$	27537.8	1.56331	0.781656	−	0.623710i	$-0.214375\pi$
$677$	27537.8	1.56331	0.781656	+	0.623710i	$0.214375\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2204.15	−0.124028
$682$	0	0
$683$	−3280.93	−0.183808	−0.0919042	−	0.995768i	$-0.529295\pi$
$683$	−3280.93	−0.183808	−0.0919042	+	0.995768i	$0.529295\pi$
$684$	0	0
$685$	−18632.2	−1.03927
$686$	0	0
$687$	−15480.8	−0.859722
$688$	0	0
$689$	409.664	0.0226516
$690$	0	0
$691$	−20420.4	−1.12421	−0.562106	−	0.827065i	$-0.690009\pi$
$691$	−20420.4	−1.12421	−0.562106	+	0.827065i	$0.690009\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9782.13	0.533895
$696$	0	0
$697$	1165.09	0.0633155
$698$	0	0
$699$	−12205.3	−0.660438
$700$	0	0
$701$	−19238.0	−1.03653	−0.518267	−	0.855219i	$-0.673423\pi$
$701$	−19238.0	−1.03653	−0.518267	+	0.855219i	$0.673423\pi$
$702$	0	0
$703$	10097.1	0.541707
$704$	0	0
$705$	−16203.1	−0.865595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−34685.9	−1.83731	−0.918657	−	0.395057i	$-0.870725\pi$
$709$	−34685.9	−1.83731	−0.918657	+	0.395057i	$0.870725\pi$
$710$	0	0
$711$	9310.37	0.491091
$712$	0	0
$713$	6326.37	0.332293
$714$	0	0
$715$	91.5157	0.00478671
$716$	0	0
$717$	1950.12	0.101574
$718$	0	0
$719$	21896.6	1.13575	0.567877	−	0.823114i	$-0.307765\pi$
$719$	21896.6	1.13575	0.567877	+	0.823114i	$0.307765\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9699.42	−0.498929
$724$	0	0
$725$	6294.07	0.322422
$726$	0	0
$727$	−22962.5	−1.17143	−0.585717	−	0.810516i	$-0.699187\pi$
$727$	−22962.5	−1.17143	−0.585717	+	0.810516i	$0.699187\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4694.94	−0.237549
$732$	0	0
$733$	22990.5	1.15849	0.579245	−	0.815153i	$-0.303347\pi$
$733$	22990.5	1.15849	0.579245	+	0.815153i	$0.303347\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4226.26	0.211230
$738$	0	0
$739$	−5010.14	−0.249392	−0.124696	−	0.992195i	$-0.539796\pi$
$739$	−5010.14	−0.249392	−0.124696	+	0.992195i	$0.539796\pi$
$740$	0	0
$741$	141.806	0.00703021
$742$	0	0
$743$	−14389.0	−0.710472	−0.355236	−	0.934777i	$-0.615599\pi$
$743$	−14389.0	−0.710472	−0.355236	+	0.934777i	$0.615599\pi$
$744$	0	0
$745$	18774.6	0.923284
$746$	0	0
$747$	1058.56	0.0518484
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−35586.5	−1.72912	−0.864560	−	0.502529i	$-0.832403\pi$
$751$	−35586.5	−1.72912	−0.864560	+	0.502529i	$0.832403\pi$
$752$	0	0
$753$	3419.25	0.165477
$754$	0	0
$755$	32893.8	1.58560
$756$	0	0
$757$	−12140.6	−0.582905	−0.291453	−	0.956585i	$-0.594139\pi$
$757$	−12140.6	−0.582905	−0.291453	+	0.956585i	$0.594139\pi$
$758$	0	0
$759$	1416.98	0.0677645
$760$	0	0
$761$	−22998.0	−1.09550	−0.547750	−	0.836642i	$-0.684516\pi$
$761$	−22998.0	−1.09550	−0.547750	+	0.836642i	$0.684516\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2195.79	−0.103777
$766$	0	0
$767$	−455.527	−0.0214448
$768$	0	0
$769$	−24004.6	−1.12566	−0.562828	−	0.826574i	$-0.690287\pi$
$769$	−24004.6	−1.12566	−0.562828	+	0.826574i	$0.690287\pi$
$770$	0	0
$771$	−6110.92	−0.285447
$772$	0	0
$773$	−2777.97	−0.129258	−0.0646292	−	0.997909i	$-0.520586\pi$
$773$	−2777.97	−0.129258	−0.0646292	+	0.997909i	$0.520586\pi$
$774$	0	0
$775$	−11593.2	−0.537343
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5252.50	0.241580
$780$	0	0
$781$	−8406.87	−0.385175
$782$	0	0
$783$	1985.28	0.0906107
$784$	0	0
$785$	27108.9	1.23256
$786$	0	0
$787$	25824.5	1.16969	0.584843	−	0.811147i	$-0.301156\pi$
$787$	25824.5	1.16969	0.584843	+	0.811147i	$0.301156\pi$
$788$	0	0
$789$	2508.64	0.113194
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.6421	−0.00110349
$794$	0	0
$795$	−28597.7	−1.27579
$796$	0	0
$797$	5363.77	0.238387	0.119193	−	0.992871i	$-0.461969\pi$
$797$	5363.77	0.238387	0.119193	+	0.992871i	$0.461969\pi$
$798$	0	0
$799$	6257.03	0.277043
$800$	0	0
$801$	6105.51	0.269323
$802$	0	0
$803$	12060.3	0.530010
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8048.98	−0.351100
$808$	0	0
$809$	15270.5	0.663638	0.331819	−	0.943343i	$-0.392338\pi$
$809$	15270.5	0.663638	0.331819	+	0.943343i	$0.392338\pi$
$810$	0	0
$811$	2993.47	0.129612	0.0648058	−	0.997898i	$-0.479357\pi$
$811$	2993.47	0.129612	0.0648058	+	0.997898i	$0.479357\pi$
$812$	0	0
$813$	−19624.3	−0.846559
$814$	0	0
$815$	−21934.2	−0.942728
$816$	0	0
$817$	−21165.9	−0.906367
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16002.9	0.680272	0.340136	−	0.940376i	$-0.389527\pi$
$821$	16002.9	0.680272	0.340136	+	0.940376i	$0.389527\pi$
$822$	0	0
$823$	11836.7	0.501340	0.250670	−	0.968073i	$-0.419349\pi$
$823$	11836.7	0.501340	0.250670	+	0.968073i	$0.419349\pi$
$824$	0	0
$825$	−2596.66	−0.109581
$826$	0	0
$827$	5675.18	0.238628	0.119314	−	0.992857i	$-0.461931\pi$
$827$	5675.18	0.238628	0.119314	+	0.992857i	$0.461931\pi$
$828$	0	0
$829$	28989.5	1.21453	0.607265	−	0.794499i	$-0.292266\pi$
$829$	28989.5	1.21453	0.607265	+	0.794499i	$0.292266\pi$
$830$	0	0
$831$	−3879.34	−0.161941
$832$	0	0
$833$	0	0
$834$	0	0
$835$	34413.9	1.42628
$836$	0	0
$837$	−3656.75	−0.151010
$838$	0	0
$839$	10021.9	0.412390	0.206195	−	0.978511i	$-0.433892\pi$
$839$	10021.9	0.412390	0.206195	+	0.978511i	$0.433892\pi$
$840$	0	0
$841$	−18982.5	−0.778322
$842$	0	0
$843$	14054.1	0.574197
$844$	0	0
$845$	31877.4	1.29777
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1695.72	0.0685477
$850$	0	0
$851$	−6222.92	−0.250668
$852$	0	0
$853$	19446.0	0.780562	0.390281	−	0.920696i	$-0.372378\pi$
$853$	19446.0	0.780562	0.390281	+	0.920696i	$0.372378\pi$
$854$	0	0
$855$	−9899.17	−0.395958
$856$	0	0
$857$	−39813.3	−1.58693	−0.793464	−	0.608617i	$-0.791725\pi$
$857$	−39813.3	−1.58693	−0.793464	+	0.608617i	$0.791725\pi$
$858$	0	0
$859$	46968.9	1.86561	0.932804	−	0.360384i	$-0.117354\pi$
$859$	46968.9	1.86561	0.932804	+	0.360384i	$0.117354\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13859.8	0.546689	0.273344	−	0.961916i	$-0.411870\pi$
$863$	13859.8	0.546689	0.273344	+	0.961916i	$0.411870\pi$
$864$	0	0
$865$	59857.4	2.35285
$866$	0	0
$867$	−13891.1	−0.544135
$868$	0	0
$869$	−10460.3	−0.408333
$870$	0	0
$871$	−260.665	−0.0101404
$872$	0	0
$873$	−14933.8	−0.578960
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27382.5	1.05432	0.527162	−	0.849765i	$-0.323256\pi$
$877$	27382.5	1.05432	0.527162	+	0.849765i	$0.323256\pi$
$878$	0	0
$879$	−2428.35	−0.0931812
$880$	0	0
$881$	17720.1	0.677647	0.338823	−	0.940850i	$-0.389971\pi$
$881$	17720.1	0.677647	0.338823	+	0.940850i	$0.389971\pi$
$882$	0	0
$883$	−45478.3	−1.73326	−0.866629	−	0.498953i	$-0.833718\pi$
$883$	−45478.3	−1.73326	−0.866629	+	0.498953i	$0.833718\pi$
$884$	0	0
$885$	31799.3	1.20782
$886$	0	0
$887$	35483.6	1.34321	0.671603	−	0.740911i	$-0.265606\pi$
$887$	35483.6	1.34321	0.671603	+	0.740911i	$0.265606\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−819.039	−0.0307956
$892$	0	0
$893$	28208.2	1.05706
$894$	0	0
$895$	46359.1	1.73141
$896$	0	0
$897$	−87.3961	−0.00325315
$898$	0	0
$899$	−9958.40	−0.369445
$900$	0	0
$901$	11043.3	0.408332
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20171.7	0.740917
$906$	0	0
$907$	4269.33	0.156296	0.0781481	−	0.996942i	$-0.475099\pi$
$907$	4269.33	0.156296	0.0781481	+	0.996942i	$0.475099\pi$
$908$	0	0
$909$	−4372.55	−0.159547
$910$	0	0
$911$	−45428.0	−1.65214	−0.826069	−	0.563569i	$-0.809428\pi$
$911$	−45428.0	−1.65214	−0.826069	+	0.563569i	$0.809428\pi$
$912$	0	0
$913$	−1189.31	−0.0431110
$914$	0	0
$915$	1720.21	0.0621511
$916$	0	0
$917$	0	0
$918$	0	0
$919$	28035.2	1.00631	0.503153	−	0.864198i	$-0.332173\pi$
$919$	28035.2	1.00631	0.503153	+	0.864198i	$0.332173\pi$
$920$	0	0
$921$	11003.8	0.393690
$922$	0	0
$923$	518.515	0.0184909
$924$	0	0
$925$	11403.6	0.405351
$926$	0	0
$927$	2491.12	0.0882624
$928$	0	0
$929$	−45234.1	−1.59751	−0.798754	−	0.601658i	$-0.794507\pi$
$929$	−45234.1	−1.59751	−0.798754	+	0.601658i	$0.794507\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−219.914	−0.00771668
$934$	0	0
$935$	2467.00	0.0862882
$936$	0	0
$937$	26573.2	0.926476	0.463238	−	0.886234i	$-0.346688\pi$
$937$	26573.2	0.926476	0.463238	+	0.886234i	$0.346688\pi$
$938$	0	0
$939$	24902.2	0.865445
$940$	0	0
$941$	−37470.6	−1.29809	−0.649047	−	0.760748i	$-0.724832\pi$
$941$	−37470.6	−1.29809	−0.649047	+	0.760748i	$0.724832\pi$
$942$	0	0
$943$	−3237.15	−0.111788
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46525.9	1.59650	0.798252	−	0.602323i	$-0.205758\pi$
$947$	46525.9	1.59650	0.798252	+	0.602323i	$0.205758\pi$
$948$	0	0
$949$	−743.849	−0.0254440
$950$	0	0
$951$	3866.68	0.131846
$952$	0	0
$953$	7066.68	0.240202	0.120101	−	0.992762i	$-0.461678\pi$
$953$	7066.68	0.240202	0.120101	+	0.992762i	$0.461678\pi$
$954$	0	0
$955$	27805.0	0.942144
$956$	0	0
$957$	−2230.49	−0.0753410
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−11448.4	−0.384289
$962$	0	0
$963$	−932.754	−0.0312124
$964$	0	0
$965$	15851.3	0.528779
$966$	0	0
$967$	31681.8	1.05359	0.526793	−	0.849994i	$-0.323394\pi$
$967$	31681.8	1.05359	0.526793	+	0.849994i	$0.323394\pi$
$968$	0	0
$969$	3822.68	0.126731
$970$	0	0
$971$	53411.5	1.76525	0.882625	−	0.470078i	$-0.155774\pi$
$971$	53411.5	1.76525	0.882625	+	0.470078i	$0.155774\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	160.155	0.00526059
$976$	0	0
$977$	56743.2	1.85811	0.929056	−	0.369939i	$-0.120622\pi$
$977$	56743.2	1.85811	0.929056	+	0.369939i	$0.120622\pi$
$978$	0	0
$979$	−6859.61	−0.223937
$980$	0	0
$981$	−451.414	−0.0146917
$982$	0	0
$983$	40797.4	1.32374	0.661869	−	0.749620i	$-0.269764\pi$
$983$	40797.4	1.32374	0.661869	+	0.749620i	$0.269764\pi$
$984$	0	0
$985$	−68601.0	−2.21910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13044.7	0.419411
$990$	0	0
$991$	−37456.6	−1.20066	−0.600328	−	0.799754i	$-0.704963\pi$
$991$	−37456.6	−1.20066	−0.600328	+	0.799754i	$0.704963\pi$
$992$	0	0
$993$	19553.7	0.624892
$994$	0	0
$995$	−9209.10	−0.293415
$996$	0	0
$997$	9679.23	0.307467	0.153733	−	0.988112i	$-0.450870\pi$
$997$	9679.23	0.307467	0.153733	+	0.988112i	$0.450870\pi$
$998$	0	0
$999$	3596.95	0.113916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.co.1.2		4
4.3	odd	2		1176.4.a.z.1.2	✓	4
7.6	odd	2		2352.4.a.cn.1.3		4
28.27	even	2		1176.4.a.be.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.z.1.2	✓	4	4.3	odd	2
1176.4.a.be.1.3	yes	4	28.27	even	2
2352.4.a.cn.1.3		4	7.6	odd	2
2352.4.a.co.1.2		4	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -14.5121 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -14.5121 q^{5} +9.00000 q^{9} -10.1116 q^{11} +0.623659 q^{13} -43.5362 q^{15} +16.8120 q^{17} +75.7927 q^{19} -46.7115 q^{23} +85.5999 q^{25} +27.0000 q^{27} +73.5290 q^{29} -135.435 q^{31} -30.3348 q^{33} +133.220 q^{37} +1.87098 q^{39} +69.3010 q^{41} -279.261 q^{43} -130.609 q^{45} +372.176 q^{47} +50.4361 q^{51} +656.872 q^{53} +146.740 q^{55} +227.378 q^{57} -730.411 q^{59} -39.5121 q^{61} -9.05057 q^{65} -417.962 q^{67} -140.135 q^{69} +831.408 q^{71} -1192.72 q^{73} +256.800 q^{75} +1034.49 q^{79} +81.0000 q^{81} +117.618 q^{83} -243.977 q^{85} +220.587 q^{87} +678.390 q^{89} -406.305 q^{93} -1099.91 q^{95} -1659.31 q^{97} -91.0044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9} - 40 q^{11} - 48 q^{13} - 24 q^{15} - 72 q^{17} + 32 q^{19} - 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} - 120 q^{33} + 48 q^{37} - 144 q^{39} - 72 q^{41} - 512 q^{43} - 72 q^{45} + 160 q^{47} - 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} - 896 q^{61} - 136 q^{65} - 1088 q^{67} - 24 q^{69} - 1288 q^{71} - 1488 q^{73} + 492 q^{75} - 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} - 3160 q^{89} + 144 q^{93} + 240 q^{95} - 2384 q^{97} - 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9 - 40 * q^11 - 48 * q^13 - 24 * q^15 - 72 * q^17 + 32 * q^19 - 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 - 120 * q^33 + 48 * q^37 - 144 * q^39 - 72 * q^41 - 512 * q^43 - 72 * q^45 + 160 * q^47 - 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 - 896 * q^61 - 136 * q^65 - 1088 * q^67 - 24 * q^69 - 1288 * q^71 - 1488 * q^73 + 492 * q^75 - 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 - 3160 * q^89 + 144 * q^93 + 240 * q^95 - 2384 * q^97 - 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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