LMFDB - Embedded newform 1323.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 42x^{4} + 369x^{2} - 112 $ x^6 - 42x^4 + 369x^2 - 112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.46131$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46131 q^{2} +3.98067 q^{4} +6.55599 q^{5} +13.9122 q^{8} +O(q^{10})$	$q-3.46131 q^{2} +3.98067 q^{4} +6.55599 q^{5} +13.9122 q^{8} -22.6923 q^{10} +2.29451 q^{11} -28.6343 q^{13} -79.9996 q^{16} -46.6643 q^{17} +67.5183 q^{19} +26.0972 q^{20} -7.94200 q^{22} -30.1188 q^{23} -82.0190 q^{25} +99.1122 q^{26} +24.0840 q^{29} +193.940 q^{31} +165.606 q^{32} +161.520 q^{34} +208.767 q^{37} -233.702 q^{38} +91.2080 q^{40} -234.347 q^{41} +46.0576 q^{43} +9.13366 q^{44} +104.251 q^{46} +194.239 q^{47} +283.893 q^{50} -113.984 q^{52} +221.635 q^{53} +15.0428 q^{55} -83.3623 q^{58} -710.748 q^{59} +634.111 q^{61} -671.285 q^{62} +66.7827 q^{64} -187.726 q^{65} +269.671 q^{67} -185.755 q^{68} -234.593 q^{71} -213.368 q^{73} -722.608 q^{74} +268.768 q^{76} +242.405 q^{79} -524.477 q^{80} +811.149 q^{82} -1337.82 q^{83} -305.931 q^{85} -159.420 q^{86} +31.9215 q^{88} +1105.39 q^{89} -119.893 q^{92} -672.322 q^{94} +442.650 q^{95} +1139.50 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 36 q^{4}+O(q^{10}) $ 6 * q + 36 * q^4	$ 6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58} + 1152 q^{61} + 2964 q^{64} + 924 q^{67} + 1260 q^{73} + 5868 q^{76} - 1500 q^{79} + 4500 q^{82} - 2232 q^{85} - 2460 q^{88} - 4968 q^{94} + 3312 q^{97}+O(q^{100}) $ 6 * q + 36 * q^4 + 180 * q^10 + 108 * q^13 + 420 * q^16 + 198 * q^19 - 84 * q^22 + 420 * q^25 - 90 * q^31 - 648 * q^34 - 402 * q^37 + 2844 * q^40 - 660 * q^43 - 1332 * q^46 + 1224 * q^52 + 846 * q^55 - 1800 * q^58 + 1152 * q^61 + 2964 * q^64 + 924 * q^67 + 1260 * q^73 + 5868 * q^76 - 1500 * q^79 + 4500 * q^82 - 2232 * q^85 - 2460 * q^88 - 4968 * q^94 + 3312 * q^97

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 $

$2$	−3.46131	−1.22376	−0.611879	−	0.790951i	$-0.709586\pi$
$2$	−3.46131	−1.22376	−0.611879	+	0.790951i	$0.709586\pi$
$3$	0	0
$3$	0	0
$4$	3.98067	0.497583
$5$	6.55599	0.586386	0.293193	−	0.956053i	$-0.405282\pi$
$5$	6.55599	0.586386	0.293193	+	0.956053i	$0.405282\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	13.9122	0.614837
$9$	0	0
$10$	−22.6923	−0.717594
$11$	2.29451	0.0628927	0.0314463	−	0.999505i	$-0.489989\pi$
$11$	2.29451	0.0628927	0.0314463	+	0.999505i	$0.489989\pi$
$12$	0	0
$13$	−28.6343	−0.610902	−0.305451	−	0.952208i	$-0.598807\pi$
$13$	−28.6343	−0.610902	−0.305451	+	0.952208i	$0.598807\pi$
$14$	0	0
$15$	0	0
$16$	−79.9996	−1.24999
$17$	−46.6643	−0.665750	−0.332875	−	0.942971i	$-0.608019\pi$
$17$	−46.6643	−0.665750	−0.332875	+	0.942971i	$0.608019\pi$
$18$	0	0
$19$	67.5183	0.815251	0.407625	−	0.913149i	$-0.366357\pi$
$19$	67.5183	0.815251	0.407625	+	0.913149i	$0.366357\pi$
$20$	26.0972	0.291776
$21$	0	0
$22$	−7.94200	−0.0769654
$23$	−30.1188	−0.273053	−0.136526	−	0.990636i	$-0.543594\pi$
$23$	−30.1188	−0.273053	−0.136526	+	0.990636i	$0.543594\pi$
$24$	0	0
$25$	−82.0190	−0.656152
$26$	99.1122	0.747597
$27$	0	0
$28$	0	0
$29$	24.0840	0.154217	0.0771085	−	0.997023i	$-0.475431\pi$
$29$	24.0840	0.154217	0.0771085	+	0.997023i	$0.475431\pi$
$30$	0	0
$31$	193.940	1.12363	0.561816	−	0.827262i	$-0.310103\pi$
$31$	193.940	1.12363	0.561816	+	0.827262i	$0.310103\pi$
$32$	165.606	0.914854
$33$	0	0
$34$	161.520	0.814717
$35$	0	0
$36$	0	0
$37$	208.767	0.927598	0.463799	−	0.885940i	$-0.346486\pi$
$37$	208.767	0.927598	0.463799	+	0.885940i	$0.346486\pi$
$38$	−233.702	−0.997669
$39$	0	0
$40$	91.2080	0.360531
$41$	−234.347	−0.892656	−0.446328	−	0.894869i	$-0.647269\pi$
$41$	−234.347	−0.892656	−0.446328	+	0.894869i	$0.647269\pi$
$42$	0	0
$43$	46.0576	0.163342	0.0816712	−	0.996659i	$-0.473974\pi$
$43$	46.0576	0.163342	0.0816712	+	0.996659i	$0.473974\pi$
$44$	9.13366	0.0312943
$45$	0	0
$46$	104.251	0.334150
$47$	194.239	0.602823	0.301412	−	0.953494i	$-0.402542\pi$
$47$	194.239	0.602823	0.301412	+	0.953494i	$0.402542\pi$
$48$	0	0
$49$	0	0
$50$	283.893	0.802971
$51$	0	0
$52$	−113.984	−0.303975
$53$	221.635	0.574414	0.287207	−	0.957869i	$-0.407273\pi$
$53$	221.635	0.574414	0.287207	+	0.957869i	$0.407273\pi$
$54$	0	0
$55$	15.0428	0.0368794
$56$	0	0
$57$	0	0
$58$	−83.3623	−0.188724
$59$	−710.748	−1.56833	−0.784165	−	0.620552i	$-0.786908\pi$
$59$	−710.748	−1.56833	−0.784165	+	0.620552i	$0.786908\pi$
$60$	0	0
$61$	634.111	1.33098	0.665489	−	0.746408i	$-0.268223\pi$
$61$	634.111	1.33098	0.665489	+	0.746408i	$0.268223\pi$
$62$	−671.285	−1.37505
$63$	0	0
$64$	66.7827	0.130435
$65$	−187.726	−0.358224
$66$	0	0
$67$	269.671	0.491725	0.245862	−	0.969305i	$-0.420929\pi$
$67$	269.671	0.491725	0.245862	+	0.969305i	$0.420929\pi$
$68$	−185.755	−0.331266
$69$	0	0
$70$	0	0
$71$	−234.593	−0.392127	−0.196063	−	0.980591i	$-0.562816\pi$
$71$	−234.593	−0.392127	−0.196063	+	0.980591i	$0.562816\pi$
$72$	0	0
$73$	−213.368	−0.342094	−0.171047	−	0.985263i	$-0.554715\pi$
$73$	−213.368	−0.342094	−0.171047	+	0.985263i	$0.554715\pi$
$74$	−722.608	−1.13516
$75$	0	0
$76$	268.768	0.405655
$77$	0	0
$78$	0	0
$79$	242.405	0.345224	0.172612	−	0.984990i	$-0.444779\pi$
$79$	242.405	0.345224	0.172612	+	0.984990i	$0.444779\pi$
$80$	−524.477	−0.732979
$81$	0	0
$82$	811.149	1.09240
$83$	−1337.82	−1.76922	−0.884610	−	0.466332i	$-0.845575\pi$
$83$	−1337.82	−1.76922	−0.884610	+	0.466332i	$0.845575\pi$
$84$	0	0
$85$	−305.931	−0.390387
$86$	−159.420	−0.199892
$87$	0	0
$88$	31.9215	0.0386687
$89$	1105.39	1.31653	0.658263	−	0.752788i	$-0.271292\pi$
$89$	1105.39	1.31653	0.658263	+	0.752788i	$0.271292\pi$
$90$	0	0
$91$	0	0
$92$	−119.893	−0.135866
$93$	0	0
$94$	−672.322	−0.737709
$95$	442.650	0.478051
$96$	0	0
$97$	1139.50	1.19277	0.596387	−	0.802697i	$-0.296602\pi$
$97$	1139.50	1.19277	0.596387	+	0.802697i	$0.296602\pi$
$98$	0	0
$99$	0	0
$100$	−326.490	−0.326490
$101$	−1351.19	−1.33117	−0.665586	−	0.746321i	$-0.731818\pi$
$101$	−1351.19	−1.33117	−0.665586	+	0.746321i	$0.731818\pi$
$102$	0	0
$103$	28.3780	0.0271472	0.0135736	−	0.999908i	$-0.495679\pi$
$103$	28.3780	0.0271472	0.0135736	+	0.999908i	$0.495679\pi$
$104$	−398.365	−0.375605
$105$	0	0
$106$	−767.148	−0.702943
$107$	1551.30	1.40159	0.700794	−	0.713364i	$-0.252829\pi$
$107$	1551.30	1.40159	0.700794	+	0.713364i	$0.252829\pi$
$108$	0	0
$109$	−1265.70	−1.11222	−0.556111	−	0.831108i	$-0.687707\pi$
$109$	−1265.70	−1.11222	−0.556111	+	0.831108i	$0.687707\pi$
$110$	−52.0677	−0.0451314
$111$	0	0
$112$	0	0
$113$	−480.406	−0.399937	−0.199968	−	0.979802i	$-0.564084\pi$
$113$	−480.406	−0.399937	−0.199968	+	0.979802i	$0.564084\pi$
$114$	0	0
$115$	−197.459	−0.160114
$116$	95.8705	0.0767358
$117$	0	0
$118$	2460.12	1.91926
$119$	0	0
$120$	0	0
$121$	−1325.74	−0.996045
$122$	−2194.86	−1.62879
$123$	0	0
$124$	772.009	0.559100
$125$	−1357.21	−0.971144
$126$	0	0
$127$	1837.72	1.28403	0.642014	−	0.766693i	$-0.278099\pi$
$127$	1837.72	1.28403	0.642014	+	0.766693i	$0.278099\pi$
$128$	−1556.01	−1.07447
$129$	0	0
$130$	649.779	0.438380
$131$	1198.23	0.799157	0.399578	−	0.916699i	$-0.369157\pi$
$131$	1198.23	0.799157	0.399578	+	0.916699i	$0.369157\pi$
$132$	0	0
$133$	0	0
$134$	−933.415	−0.601752
$135$	0	0
$136$	−649.201	−0.409328
$137$	−2019.83	−1.25960	−0.629802	−	0.776756i	$-0.716864\pi$
$137$	−2019.83	−1.25960	−0.629802	+	0.776756i	$0.716864\pi$
$138$	0	0
$139$	2320.01	1.41569	0.707843	−	0.706370i	$-0.249669\pi$
$139$	2320.01	1.41569	0.707843	+	0.706370i	$0.249669\pi$
$140$	0	0
$141$	0	0
$142$	811.997	0.479868
$143$	−65.7016	−0.0384213
$144$	0	0
$145$	157.895	0.0904307
$146$	738.534	0.418640
$147$	0	0
$148$	831.033	0.461557
$149$	1230.57	0.676592	0.338296	−	0.941040i	$-0.390149\pi$
$149$	1230.57	0.676592	0.338296	+	0.941040i	$0.390149\pi$
$150$	0	0
$151$	30.8369	0.0166190	0.00830952	−	0.999965i	$-0.497355\pi$
$151$	30.8369	0.0166190	0.00830952	+	0.999965i	$0.497355\pi$
$152$	939.326	0.501246
$153$	0	0
$154$	0	0
$155$	1271.47	0.658882
$156$	0	0
$157$	−2340.62	−1.18982	−0.594910	−	0.803792i	$-0.702812\pi$
$157$	−2340.62	−1.18982	−0.594910	+	0.803792i	$0.702812\pi$
$158$	−839.038	−0.422470
$159$	0	0
$160$	1085.71	0.536457
$161$	0	0
$162$	0	0
$163$	2129.76	1.02341	0.511705	−	0.859161i	$-0.329014\pi$
$163$	2129.76	1.02341	0.511705	+	0.859161i	$0.329014\pi$
$164$	−932.859	−0.444171
$165$	0	0
$166$	4630.62	2.16510
$167$	−158.641	−0.0735092	−0.0367546	−	0.999324i	$-0.511702\pi$
$167$	−158.641	−0.0735092	−0.0367546	+	0.999324i	$0.511702\pi$
$168$	0	0
$169$	−1377.08	−0.626798
$170$	1058.92	0.477739
$171$	0	0
$172$	183.340	0.0812764
$173$	1116.85	0.490823	0.245412	−	0.969419i	$-0.421077\pi$
$173$	1116.85	0.490823	0.245412	+	0.969419i	$0.421077\pi$
$174$	0	0
$175$	0	0
$176$	−183.560	−0.0786155
$177$	0	0
$178$	−3826.09	−1.61111
$179$	3503.57	1.46295	0.731477	−	0.681866i	$-0.238831\pi$
$179$	3503.57	1.46295	0.731477	+	0.681866i	$0.238831\pi$
$180$	0	0
$181$	1591.32	0.653492	0.326746	−	0.945112i	$-0.394048\pi$
$181$	1591.32	0.653492	0.326746	+	0.945112i	$0.394048\pi$
$182$	0	0
$183$	0	0
$184$	−419.018	−0.167883
$185$	1368.68	0.543930
$186$	0	0
$187$	−107.072	−0.0418708
$188$	773.201	0.299955
$189$	0	0
$190$	−1532.15	−0.585019
$191$	−4924.24	−1.86547	−0.932737	−	0.360558i	$-0.882586\pi$
$191$	−4924.24	−1.86547	−0.932737	+	0.360558i	$0.882586\pi$
$192$	0	0
$193$	3314.07	1.23602	0.618011	−	0.786170i	$-0.287939\pi$
$193$	3314.07	1.23602	0.618011	+	0.786170i	$0.287939\pi$
$194$	−3944.17	−1.45967
$195$	0	0
$196$	0	0
$197$	−1024.59	−0.370554	−0.185277	−	0.982686i	$-0.559318\pi$
$197$	−1024.59	−0.370554	−0.185277	+	0.982686i	$0.559318\pi$
$198$	0	0
$199$	−795.124	−0.283240	−0.141620	−	0.989921i	$-0.545231\pi$
$199$	−795.124	−0.283240	−0.141620	+	0.989921i	$0.545231\pi$
$200$	−1141.06	−0.403426
$201$	0	0
$202$	4676.89	1.62903
$203$	0	0
$204$	0	0
$205$	−1536.38	−0.523441
$206$	−98.2249	−0.0332216
$207$	0	0
$208$	2290.73	0.763624
$209$	154.921	0.0512733
$210$	0	0
$211$	−850.964	−0.277644	−0.138822	−	0.990317i	$-0.544332\pi$
$211$	−850.964	−0.277644	−0.138822	+	0.990317i	$0.544332\pi$
$212$	882.255	0.285819
$213$	0	0
$214$	−5369.53	−1.71520
$215$	301.953	0.0957817
$216$	0	0
$217$	0	0
$218$	4380.98	1.36109
$219$	0	0
$220$	59.8802	0.0183506
$221$	1336.20	0.406708
$222$	0	0
$223$	4204.86	1.26268	0.631342	−	0.775505i	$-0.282505\pi$
$223$	4204.86	1.26268	0.631342	+	0.775505i	$0.282505\pi$
$224$	0	0
$225$	0	0
$226$	1662.84	0.489426
$227$	5199.29	1.52022	0.760108	−	0.649796i	$-0.225146\pi$
$227$	5199.29	1.52022	0.760108	+	0.649796i	$0.225146\pi$
$228$	0	0
$229$	1357.28	0.391666	0.195833	−	0.980637i	$-0.437259\pi$
$229$	1357.28	0.391666	0.195833	+	0.980637i	$0.437259\pi$
$230$	683.466	0.195941
$231$	0	0
$232$	335.061	0.0948183
$233$	6711.98	1.88720	0.943598	−	0.331093i	$-0.107417\pi$
$233$	6711.98	1.88720	0.943598	+	0.331093i	$0.107417\pi$
$234$	0	0
$235$	1273.43	0.353487
$236$	−2829.25	−0.780375
$237$	0	0
$238$	0	0
$239$	−1850.76	−0.500902	−0.250451	−	0.968129i	$-0.580579\pi$
$239$	−1850.76	−0.500902	−0.250451	+	0.968129i	$0.580579\pi$
$240$	0	0
$241$	4864.74	1.30027	0.650136	−	0.759818i	$-0.274712\pi$
$241$	4864.74	1.30027	0.650136	+	0.759818i	$0.274712\pi$
$242$	4588.78	1.21892
$243$	0	0
$244$	2524.18	0.662272
$245$	0	0
$246$	0	0
$247$	−1933.34	−0.498039
$248$	2698.12	0.690850
$249$	0	0
$250$	4697.74	1.18844
$251$	1901.00	0.478049	0.239024	−	0.971014i	$-0.423172\pi$
$251$	1901.00	0.478049	0.239024	+	0.971014i	$0.423172\pi$
$252$	0	0
$253$	−69.1078	−0.0171730
$254$	−6360.93	−1.57134
$255$	0	0
$256$	4851.55	1.18446
$257$	1670.33	0.405417	0.202709	−	0.979239i	$-0.435026\pi$
$257$	1670.33	0.405417	0.202709	+	0.979239i	$0.435026\pi$
$258$	0	0
$259$	0	0
$260$	−747.276	−0.178246
$261$	0	0
$262$	−4147.43	−0.977974
$263$	−3792.54	−0.889195	−0.444597	−	0.895731i	$-0.646653\pi$
$263$	−3792.54	−0.889195	−0.444597	+	0.895731i	$0.646653\pi$
$264$	0	0
$265$	1453.04	0.336828
$266$	0	0
$267$	0	0
$268$	1073.47	0.244674
$269$	7923.59	1.79595	0.897973	−	0.440050i	$-0.145039\pi$
$269$	7923.59	1.79595	0.897973	+	0.440050i	$0.145039\pi$
$270$	0	0
$271$	6340.76	1.42130	0.710652	−	0.703543i	$-0.248400\pi$
$271$	6340.76	1.42130	0.710652	+	0.703543i	$0.248400\pi$
$272$	3733.13	0.832184
$273$	0	0
$274$	6991.25	1.54145
$275$	−188.193	−0.0412671
$276$	0	0
$277$	−3169.14	−0.687420	−0.343710	−	0.939076i	$-0.611684\pi$
$277$	−3169.14	−0.687420	−0.343710	+	0.939076i	$0.611684\pi$
$278$	−8030.26	−1.73246
$279$	0	0
$280$	0	0
$281$	3492.87	0.741520	0.370760	−	0.928729i	$-0.379097\pi$
$281$	3492.87	0.741520	0.370760	+	0.928729i	$0.379097\pi$
$282$	0	0
$283$	1904.23	0.399982	0.199991	−	0.979798i	$-0.435909\pi$
$283$	1904.23	0.399982	0.199991	+	0.979798i	$0.435909\pi$
$284$	−933.834	−0.195116
$285$	0	0
$286$	227.414	0.0470184
$287$	0	0
$288$	0	0
$289$	−2735.44	−0.556776
$290$	−546.523	−0.110665
$291$	0	0
$292$	−849.348	−0.170220
$293$	3988.81	0.795320	0.397660	−	0.917533i	$-0.369822\pi$
$293$	3988.81	0.795320	0.397660	+	0.917533i	$0.369822\pi$
$294$	0	0
$295$	−4659.66	−0.919646
$296$	2904.40	0.570321
$297$	0	0
$298$	−4259.38	−0.827985
$299$	862.432	0.166808
$300$	0	0
$301$	0	0
$302$	−106.736	−0.0203377
$303$	0	0
$304$	−5401.44	−1.01906
$305$	4157.23	0.780466
$306$	0	0
$307$	1415.59	0.263165	0.131583	−	0.991305i	$-0.457994\pi$
$307$	1415.59	0.263165	0.131583	+	0.991305i	$0.457994\pi$
$308$	0	0
$309$	0	0
$310$	−4400.94	−0.806311
$311$	8808.57	1.60607	0.803036	−	0.595931i	$-0.203217\pi$
$311$	8808.57	1.60607	0.803036	+	0.595931i	$0.203217\pi$
$312$	0	0
$313$	4338.32	0.783439	0.391719	−	0.920085i	$-0.371880\pi$
$313$	4338.32	0.783439	0.391719	+	0.920085i	$0.371880\pi$
$314$	8101.61	1.45605
$315$	0	0
$316$	964.933	0.171778
$317$	−8911.96	−1.57901	−0.789504	−	0.613745i	$-0.789662\pi$
$317$	−8911.96	−1.57901	−0.789504	+	0.613745i	$0.789662\pi$
$318$	0	0
$319$	55.2610	0.00969913
$320$	437.827	0.0764852
$321$	0	0
$322$	0	0
$323$	−3150.69	−0.542753
$324$	0	0
$325$	2348.56	0.400845
$326$	−7371.76	−1.25240
$327$	0	0
$328$	−3260.28	−0.548838
$329$	0	0
$330$	0	0
$331$	8405.16	1.39574	0.697869	−	0.716226i	$-0.254132\pi$
$331$	8405.16	1.39574	0.697869	+	0.716226i	$0.254132\pi$
$332$	−5325.43	−0.880334
$333$	0	0
$334$	549.107	0.0899574
$335$	1767.96	0.288340
$336$	0	0
$337$	7481.10	1.20926	0.604631	−	0.796506i	$-0.293321\pi$
$337$	7481.10	1.20926	0.604631	+	0.796506i	$0.293321\pi$
$338$	4766.49	0.767049
$339$	0	0
$340$	−1217.81	−0.194250
$341$	444.996	0.0706682
$342$	0	0
$343$	0	0
$344$	640.761	0.100429
$345$	0	0
$346$	−3865.76	−0.600649
$347$	1603.13	0.248013	0.124006	−	0.992281i	$-0.460426\pi$
$347$	1603.13	0.248013	0.124006	+	0.992281i	$0.460426\pi$
$348$	0	0
$349$	2238.34	0.343311	0.171656	−	0.985157i	$-0.445088\pi$
$349$	2238.34	0.343311	0.171656	+	0.985157i	$0.445088\pi$
$350$	0	0
$351$	0	0
$352$	379.984	0.0575376
$353$	9507.71	1.43355	0.716777	−	0.697303i	$-0.245617\pi$
$353$	9507.71	1.43355	0.716777	+	0.697303i	$0.245617\pi$
$354$	0	0
$355$	−1537.99	−0.229938
$356$	4400.17	0.655081
$357$	0	0
$358$	−12126.9	−1.79030
$359$	7856.96	1.15508	0.577541	−	0.816361i	$-0.304012\pi$
$359$	7856.96	1.15508	0.577541	+	0.816361i	$0.304012\pi$
$360$	0	0
$361$	−2300.28	−0.335366
$362$	−5508.06	−0.799716
$363$	0	0
$364$	0	0
$365$	−1398.84	−0.200599
$366$	0	0
$367$	2258.91	0.321292	0.160646	−	0.987012i	$-0.448642\pi$
$367$	2258.91	0.321292	0.160646	+	0.987012i	$0.448642\pi$
$368$	2409.50	0.341314
$369$	0	0
$370$	−4737.41	−0.665639
$371$	0	0
$372$	0	0
$373$	1789.23	0.248372	0.124186	−	0.992259i	$-0.460368\pi$
$373$	1789.23	0.248372	0.124186	+	0.992259i	$0.460368\pi$
$374$	370.608	0.0512398
$375$	0	0
$376$	2702.29	0.370638
$377$	−689.630	−0.0942116
$378$	0	0
$379$	−7021.61	−0.951651	−0.475826	−	0.879540i	$-0.657851\pi$
$379$	−7021.61	−0.951651	−0.475826	+	0.879540i	$0.657851\pi$
$380$	1762.04	0.237870
$381$	0	0
$382$	17044.3	2.28289
$383$	9614.42	1.28270	0.641350	−	0.767248i	$-0.278375\pi$
$383$	9614.42	1.28270	0.641350	+	0.767248i	$0.278375\pi$
$384$	0	0
$385$	0	0
$386$	−11471.0	−1.51259
$387$	0	0
$388$	4535.98	0.593504
$389$	−12211.5	−1.59164	−0.795822	−	0.605530i	$-0.792961\pi$
$389$	−12211.5	−1.59164	−0.795822	+	0.605530i	$0.792961\pi$
$390$	0	0
$391$	1405.47	0.181785
$392$	0	0
$393$	0	0
$394$	3546.43	0.453468
$395$	1589.20	0.202434
$396$	0	0
$397$	−10404.7	−1.31535	−0.657677	−	0.753300i	$-0.728461\pi$
$397$	−10404.7	−1.31535	−0.657677	+	0.753300i	$0.728461\pi$
$398$	2752.17	0.346618
$399$	0	0
$400$	6561.49	0.820186
$401$	−789.887	−0.0983667	−0.0491834	−	0.998790i	$-0.515662\pi$
$401$	−789.887	−0.0983667	−0.0491834	+	0.998790i	$0.515662\pi$
$402$	0	0
$403$	−5553.33	−0.686429
$404$	−5378.63	−0.662369
$405$	0	0
$406$	0	0
$407$	479.018	0.0583391
$408$	0	0
$409$	8685.05	1.05000	0.524998	−	0.851104i	$-0.324066\pi$
$409$	8685.05	1.05000	0.524998	+	0.851104i	$0.324066\pi$
$410$	5317.89	0.640565
$411$	0	0
$412$	112.963	0.0135080
$413$	0	0
$414$	0	0
$415$	−8770.76	−1.03745
$416$	−4742.02	−0.558886
$417$	0	0
$418$	−536.230	−0.0627461
$419$	−1296.36	−0.151148	−0.0755741	−	0.997140i	$-0.524079\pi$
$419$	−1296.36	−0.151148	−0.0755741	+	0.997140i	$0.524079\pi$
$420$	0	0
$421$	1303.36	0.150884	0.0754418	−	0.997150i	$-0.475963\pi$
$421$	1303.36	0.150884	0.0754418	+	0.997150i	$0.475963\pi$
$422$	2945.45	0.339768
$423$	0	0
$424$	3083.42	0.353170
$425$	3827.36	0.436833
$426$	0	0
$427$	0	0
$428$	6175.21	0.697407
$429$	0	0
$430$	−1045.15	−0.117214
$431$	−70.6768	−0.00789880	−0.00394940	−	0.999992i	$-0.501257\pi$
$431$	−70.6768	−0.00789880	−0.00394940	+	0.999992i	$0.501257\pi$
$432$	0	0
$433$	6842.92	0.759468	0.379734	−	0.925096i	$-0.376016\pi$
$433$	6842.92	0.759468	0.379734	+	0.925096i	$0.376016\pi$
$434$	0	0
$435$	0	0
$436$	−5038.33	−0.553423
$437$	−2033.57	−0.222606
$438$	0	0
$439$	−18342.5	−1.99417	−0.997084	−	0.0763102i	$-0.975686\pi$
$439$	−18342.5	−1.99417	−0.997084	+	0.0763102i	$0.975686\pi$
$440$	209.277	0.0226748
$441$	0	0
$442$	−4625.00	−0.497713
$443$	−10938.1	−1.17310	−0.586552	−	0.809911i	$-0.699515\pi$
$443$	−10938.1	−1.17310	−0.586552	+	0.809911i	$0.699515\pi$
$444$	0	0
$445$	7246.91	0.771992
$446$	−14554.3	−1.54522
$447$	0	0
$448$	0	0
$449$	10816.8	1.13692	0.568461	−	0.822710i	$-0.307539\pi$
$449$	10816.8	1.13692	0.568461	+	0.822710i	$0.307539\pi$
$450$	0	0
$451$	−537.712	−0.0561416
$452$	−1912.34	−0.199002
$453$	0	0
$454$	−17996.4	−1.86038
$455$	0	0
$456$	0	0
$457$	4525.45	0.463220	0.231610	−	0.972809i	$-0.425601\pi$
$457$	4525.45	0.463220	0.231610	+	0.972809i	$0.425601\pi$
$458$	−4697.96	−0.479304
$459$	0	0
$460$	−786.018	−0.0796701
$461$	−11088.4	−1.12025	−0.560126	−	0.828407i	$-0.689247\pi$
$461$	−11088.4	−1.12025	−0.560126	+	0.828407i	$0.689247\pi$
$462$	0	0
$463$	17567.2	1.76332	0.881660	−	0.471884i	$-0.156426\pi$
$463$	17567.2	1.76332	0.881660	+	0.471884i	$0.156426\pi$
$464$	−1926.71	−0.192770
$465$	0	0
$466$	−23232.3	−2.30947
$467$	6326.26	0.626862	0.313431	−	0.949611i	$-0.398522\pi$
$467$	6326.26	0.626862	0.313431	+	0.949611i	$0.398522\pi$
$468$	0	0
$469$	0	0
$470$	−4407.74	−0.432582
$471$	0	0
$472$	−9888.04	−0.964266
$473$	105.680	0.0102730
$474$	0	0
$475$	−5537.78	−0.534928
$476$	0	0
$477$	0	0
$478$	6406.05	0.612983
$479$	375.365	0.0358055	0.0179028	−	0.999840i	$-0.494301\pi$
$479$	375.365	0.0358055	0.0179028	+	0.999840i	$0.494301\pi$
$480$	0	0
$481$	−5977.91	−0.566672
$482$	−16838.4	−1.59122
$483$	0	0
$484$	−5277.31	−0.495615
$485$	7470.57	0.699425
$486$	0	0
$487$	25.9467	0.00241429	0.00120714	−	0.999999i	$-0.499616\pi$
$487$	25.9467	0.00241429	0.00120714	+	0.999999i	$0.499616\pi$
$488$	8821.86	0.818333
$489$	0	0
$490$	0	0
$491$	−6282.99	−0.577490	−0.288745	−	0.957406i	$-0.593238\pi$
$491$	−6282.99	−0.577490	−0.288745	+	0.957406i	$0.593238\pi$
$492$	0	0
$493$	−1123.87	−0.102670
$494$	6691.89	0.609479
$495$	0	0
$496$	−15515.1	−1.40453
$497$	0	0
$498$	0	0
$499$	−2371.78	−0.212776	−0.106388	−	0.994325i	$-0.533929\pi$
$499$	−2371.78	−0.212776	−0.106388	+	0.994325i	$0.533929\pi$
$500$	−5402.62	−0.483225
$501$	0	0
$502$	−6579.96	−0.585016
$503$	−11080.3	−0.982201	−0.491100	−	0.871103i	$-0.663405\pi$
$503$	−11080.3	−0.982201	−0.491100	+	0.871103i	$0.663405\pi$
$504$	0	0
$505$	−8858.39	−0.780580
$506$	239.204	0.0210156
$507$	0	0
$508$	7315.36	0.638910
$509$	10063.7	0.876355	0.438178	−	0.898888i	$-0.355624\pi$
$509$	10063.7	0.876355	0.438178	+	0.898888i	$0.355624\pi$
$510$	0	0
$511$	0	0
$512$	−4344.69	−0.375020
$513$	0	0
$514$	−5781.52	−0.496132
$515$	186.046	0.0159187
$516$	0	0
$517$	445.683	0.0379132
$518$	0	0
$519$	0	0
$520$	−2611.68	−0.220249
$521$	−9181.24	−0.772048	−0.386024	−	0.922489i	$-0.626152\pi$
$521$	−9181.24	−0.772048	−0.386024	+	0.922489i	$0.626152\pi$
$522$	0	0
$523$	−9392.06	−0.785250	−0.392625	−	0.919699i	$-0.628433\pi$
$523$	−9392.06	−0.785250	−0.392625	+	0.919699i	$0.628433\pi$
$524$	4769.74	0.397647
$525$	0	0
$526$	13127.2	1.08816
$527$	−9050.06	−0.748058
$528$	0	0
$529$	−11259.9	−0.925442
$530$	−5029.42	−0.412196
$531$	0	0
$532$	0	0
$533$	6710.38	0.545326
$534$	0	0
$535$	10170.3	0.821871
$536$	3751.71	0.302330
$537$	0	0
$538$	−27426.0	−2.19780
$539$	0	0
$540$	0	0
$541$	4114.52	0.326982	0.163491	−	0.986545i	$-0.447725\pi$
$541$	4114.52	0.326982	0.163491	+	0.986545i	$0.447725\pi$
$542$	−21947.3	−1.73933
$543$	0	0
$544$	−7727.90	−0.609064
$545$	−8297.92	−0.652191
$546$	0	0
$547$	−9796.38	−0.765746	−0.382873	−	0.923801i	$-0.625065\pi$
$547$	−9796.38	−0.765746	−0.382873	+	0.923801i	$0.625065\pi$
$548$	−8040.26	−0.626757
$549$	0	0
$550$	651.394	0.0505010
$551$	1626.11	0.125726
$552$	0	0
$553$	0	0
$554$	10969.4	0.841236
$555$	0	0
$556$	9235.16	0.704421
$557$	−4197.53	−0.319309	−0.159655	−	0.987173i	$-0.551038\pi$
$557$	−4197.53	−0.319309	−0.159655	+	0.987173i	$0.551038\pi$
$558$	0	0
$559$	−1318.83	−0.0997863
$560$	0	0
$561$	0	0
$562$	−12089.9	−0.907441
$563$	−10742.4	−0.804151	−0.402075	−	0.915607i	$-0.631711\pi$
$563$	−10742.4	−0.804151	−0.402075	+	0.915607i	$0.631711\pi$
$564$	0	0
$565$	−3149.54	−0.234517
$566$	−6591.15	−0.489482
$567$	0	0
$568$	−3263.69	−0.241094
$569$	7915.86	0.583216	0.291608	−	0.956538i	$-0.405810\pi$
$569$	7915.86	0.583216	0.291608	+	0.956538i	$0.405810\pi$
$570$	0	0
$571$	16923.1	1.24030	0.620149	−	0.784484i	$-0.287072\pi$
$571$	16923.1	1.24030	0.620149	+	0.784484i	$0.287072\pi$
$572$	−261.536	−0.0191178
$573$	0	0
$574$	0	0
$575$	2470.32	0.179164
$576$	0	0
$577$	24108.2	1.73941	0.869704	−	0.493573i	$-0.164310\pi$
$577$	24108.2	1.73941	0.869704	+	0.493573i	$0.164310\pi$
$578$	9468.21	0.681360
$579$	0	0
$580$	628.526	0.0449968
$581$	0	0
$582$	0	0
$583$	508.543	0.0361264
$584$	−2968.41	−0.210332
$585$	0	0
$586$	−13806.5	−0.973279
$587$	−13259.2	−0.932308	−0.466154	−	0.884704i	$-0.654361\pi$
$587$	−13259.2	−0.932308	−0.466154	+	0.884704i	$0.654361\pi$
$588$	0	0
$589$	13094.5	0.916041
$590$	16128.5	1.12542
$591$	0	0
$592$	−16701.3	−1.15949
$593$	18789.8	1.30119	0.650596	−	0.759424i	$-0.274519\pi$
$593$	18789.8	1.30119	0.650596	+	0.759424i	$0.274519\pi$
$594$	0	0
$595$	0	0
$596$	4898.49	0.336661
$597$	0	0
$598$	−2985.14	−0.204133
$599$	7105.41	0.484673	0.242337	−	0.970192i	$-0.422086\pi$
$599$	7105.41	0.484673	0.242337	+	0.970192i	$0.422086\pi$
$600$	0	0
$601$	−800.460	−0.0543286	−0.0271643	−	0.999631i	$-0.508648\pi$
$601$	−800.460	−0.0543286	−0.0271643	+	0.999631i	$0.508648\pi$
$602$	0	0
$603$	0	0
$604$	122.752	0.00826935
$605$	−8691.51	−0.584066
$606$	0	0
$607$	−8771.51	−0.586532	−0.293266	−	0.956031i	$-0.594742\pi$
$607$	−8771.51	−0.586532	−0.293266	+	0.956031i	$0.594742\pi$
$608$	11181.5	0.745835
$609$	0	0
$610$	−14389.5	−0.955102
$611$	−5561.90	−0.368266
$612$	0	0
$613$	−2533.30	−0.166915	−0.0834577	−	0.996511i	$-0.526596\pi$
$613$	−2533.30	−0.166915	−0.0834577	+	0.996511i	$0.526596\pi$
$614$	−4899.78	−0.322050
$615$	0	0
$616$	0	0
$617$	−1464.17	−0.0955350	−0.0477675	−	0.998858i	$-0.515211\pi$
$617$	−1464.17	−0.0955350	−0.0477675	+	0.998858i	$0.515211\pi$
$618$	0	0
$619$	18519.8	1.20254	0.601271	−	0.799045i	$-0.294661\pi$
$619$	18519.8	1.20254	0.601271	+	0.799045i	$0.294661\pi$
$620$	5061.28	0.327848
$621$	0	0
$622$	−30489.2	−1.96544
$623$	0	0
$624$	0	0
$625$	1354.48	0.0866867
$626$	−15016.3	−0.958739
$627$	0	0
$628$	−9317.23	−0.592035
$629$	−9741.98	−0.617549
$630$	0	0
$631$	5128.62	0.323562	0.161781	−	0.986827i	$-0.448276\pi$
$631$	5128.62	0.323562	0.161781	+	0.986827i	$0.448276\pi$
$632$	3372.38	0.212256
$633$	0	0
$634$	30847.1	1.93232
$635$	12048.1	0.752935
$636$	0	0
$637$	0	0
$638$	−191.275	−0.0118694
$639$	0	0
$640$	−10201.2	−0.630057
$641$	20864.3	1.28563	0.642817	−	0.766020i	$-0.277766\pi$
$641$	20864.3	1.28563	0.642817	+	0.766020i	$0.277766\pi$
$642$	0	0
$643$	9799.87	0.601040	0.300520	−	0.953775i	$-0.402840\pi$
$643$	9799.87	0.601040	0.300520	+	0.953775i	$0.402840\pi$
$644$	0	0
$645$	0	0
$646$	10905.5	0.664199
$647$	−8180.42	−0.497072	−0.248536	−	0.968623i	$-0.579949\pi$
$647$	−8180.42	−0.497072	−0.248536	+	0.968623i	$0.579949\pi$
$648$	0	0
$649$	−1630.81	−0.0986365
$650$	−8129.08	−0.490537
$651$	0	0
$652$	8477.86	0.509231
$653$	−13509.9	−0.809619	−0.404810	−	0.914401i	$-0.632662\pi$
$653$	−13509.9	−0.809619	−0.404810	+	0.914401i	$0.632662\pi$
$654$	0	0
$655$	7855.56	0.468614
$656$	18747.7	1.11582
$657$	0	0
$658$	0	0
$659$	−18426.1	−1.08919	−0.544597	−	0.838698i	$-0.683317\pi$
$659$	−18426.1	−1.08919	−0.544597	+	0.838698i	$0.683317\pi$
$660$	0	0
$661$	12328.0	0.725424	0.362712	−	0.931901i	$-0.381851\pi$
$661$	12328.0	0.725424	0.362712	+	0.931901i	$0.381851\pi$
$662$	−29092.8	−1.70804
$663$	0	0
$664$	−18612.0	−1.08778
$665$	0	0
$666$	0	0
$667$	−725.383	−0.0421094
$668$	−631.498	−0.0365769
$669$	0	0
$670$	−6119.46	−0.352859
$671$	1454.97	0.0837087
$672$	0	0
$673$	−20866.3	−1.19515	−0.597575	−	0.801813i	$-0.703869\pi$
$673$	−20866.3	−1.19515	−0.597575	+	0.801813i	$0.703869\pi$
$674$	−25894.4	−1.47984
$675$	0	0
$676$	−5481.68	−0.311884
$677$	638.005	0.0362194	0.0181097	−	0.999836i	$-0.494235\pi$
$677$	638.005	0.0362194	0.0181097	+	0.999836i	$0.494235\pi$
$678$	0	0
$679$	0	0
$680$	−4256.16	−0.240024
$681$	0	0
$682$	−1540.27	−0.0864808
$683$	−25045.2	−1.40312	−0.701558	−	0.712612i	$-0.747512\pi$
$683$	−25045.2	−1.40312	−0.701558	+	0.712612i	$0.747512\pi$
$684$	0	0
$685$	−13242.0	−0.738613
$686$	0	0
$687$	0	0
$688$	−3684.59	−0.204177
$689$	−6346.37	−0.350911
$690$	0	0
$691$	−3371.26	−0.185599	−0.0927994	−	0.995685i	$-0.529582\pi$
$691$	−3371.26	−0.185599	−0.0927994	+	0.995685i	$0.529582\pi$
$692$	4445.80	0.244225
$693$	0	0
$694$	−5548.93	−0.303508
$695$	15209.9	0.830138
$696$	0	0
$697$	10935.7	0.594286
$698$	−7747.59	−0.420130
$699$	0	0
$700$	0	0
$701$	24430.3	1.31629	0.658146	−	0.752890i	$-0.271341\pi$
$701$	24430.3	1.31629	0.658146	+	0.752890i	$0.271341\pi$
$702$	0	0
$703$	14095.6	0.756225
$704$	153.233	0.00820340
$705$	0	0
$706$	−32909.1	−1.75432
$707$	0	0
$708$	0	0
$709$	24270.6	1.28562	0.642809	−	0.766026i	$-0.277769\pi$
$709$	24270.6	1.28562	0.642809	+	0.766026i	$0.277769\pi$
$710$	5323.45	0.281388
$711$	0	0
$712$	15378.3	0.809448
$713$	−5841.23	−0.306811
$714$	0	0
$715$	−430.739	−0.0225297
$716$	13946.5	0.727941
$717$	0	0
$718$	−27195.4	−1.41354
$719$	−6891.17	−0.357437	−0.178718	−	0.983900i	$-0.557195\pi$
$719$	−6891.17	−0.357437	−0.178718	+	0.983900i	$0.557195\pi$
$720$	0	0
$721$	0	0
$722$	7961.97	0.410407
$723$	0	0
$724$	6334.53	0.325167
$725$	−1975.35	−0.101190
$726$	0	0
$727$	4904.38	0.250197	0.125099	−	0.992144i	$-0.460075\pi$
$727$	4904.38	0.250197	0.125099	+	0.992144i	$0.460075\pi$
$728$	0	0
$729$	0	0
$730$	4841.82	0.245485
$731$	−2149.25	−0.108745
$732$	0	0
$733$	6610.71	0.333114	0.166557	−	0.986032i	$-0.446735\pi$
$733$	6610.71	0.333114	0.166557	+	0.986032i	$0.446735\pi$
$734$	−7818.78	−0.393183
$735$	0	0
$736$	−4987.86	−0.249803
$737$	618.762	0.0309259
$738$	0	0
$739$	−27941.3	−1.39085	−0.695425	−	0.718599i	$-0.744784\pi$
$739$	−27941.3	−1.39085	−0.695425	+	0.718599i	$0.744784\pi$
$740$	5448.24	0.270651
$741$	0	0
$742$	0	0
$743$	−29535.5	−1.45835	−0.729174	−	0.684328i	$-0.760095\pi$
$743$	−29535.5	−1.45835	−0.729174	+	0.684328i	$0.760095\pi$
$744$	0	0
$745$	8067.61	0.396744
$746$	−6193.07	−0.303947
$747$	0	0
$748$	−426.216	−0.0208342
$749$	0	0
$750$	0	0
$751$	−35047.9	−1.70295	−0.851475	−	0.524395i	$-0.824291\pi$
$751$	−35047.9	−1.70295	−0.851475	+	0.524395i	$0.824291\pi$
$752$	−15539.1	−0.753525
$753$	0	0
$754$	2387.02	0.115292
$755$	202.167	0.00974517
$756$	0	0
$757$	19501.6	0.936326	0.468163	−	0.883642i	$-0.344916\pi$
$757$	19501.6	0.936326	0.468163	+	0.883642i	$0.344916\pi$
$758$	24304.0	1.16459
$759$	0	0
$760$	6158.21	0.293923
$761$	−22813.5	−1.08672	−0.543358	−	0.839501i	$-0.682847\pi$
$761$	−22813.5	−1.08672	−0.543358	+	0.839501i	$0.682847\pi$
$762$	0	0
$763$	0	0
$764$	−19601.7	−0.928228
$765$	0	0
$766$	−33278.5	−1.56971
$767$	20351.8	0.958096
$768$	0	0
$769$	−23272.0	−1.09130	−0.545650	−	0.838013i	$-0.683717\pi$
$769$	−23272.0	−1.09130	−0.545650	+	0.838013i	$0.683717\pi$
$770$	0	0
$771$	0	0
$772$	13192.2	0.615023
$773$	29648.9	1.37956	0.689778	−	0.724021i	$-0.257708\pi$
$773$	29648.9	1.37956	0.689778	+	0.724021i	$0.257708\pi$
$774$	0	0
$775$	−15906.7	−0.737273
$776$	15853.0	0.733361
$777$	0	0
$778$	42267.9	1.94779
$779$	−15822.7	−0.727739
$780$	0	0
$781$	−538.274	−0.0246619
$782$	−4864.78	−0.222461
$783$	0	0
$784$	0	0
$785$	−15345.1	−0.697694
$786$	0	0
$787$	19003.2	0.860726	0.430363	−	0.902656i	$-0.358386\pi$
$787$	19003.2	0.860726	0.430363	+	0.902656i	$0.358386\pi$
$788$	−4078.56	−0.184381
$789$	0	0
$790$	−5500.73	−0.247731
$791$	0	0
$792$	0	0
$793$	−18157.3	−0.813097
$794$	36013.8	1.60968
$795$	0	0
$796$	−3165.12	−0.140936
$797$	19286.8	0.857182	0.428591	−	0.903499i	$-0.359010\pi$
$797$	19286.8	0.857182	0.428591	+	0.903499i	$0.359010\pi$
$798$	0	0
$799$	−9064.03	−0.401330
$800$	−13582.8	−0.600283
$801$	0	0
$802$	2734.04	0.120377
$803$	−489.575	−0.0215152
$804$	0	0
$805$	0	0
$806$	19221.8	0.840023
$807$	0	0
$808$	−18798.0	−0.818453
$809$	−26463.5	−1.15007	−0.575034	−	0.818129i	$-0.695011\pi$
$809$	−26463.5	−1.15007	−0.575034	+	0.818129i	$0.695011\pi$
$810$	0	0
$811$	28814.1	1.24759	0.623797	−	0.781586i	$-0.285589\pi$
$811$	28814.1	1.24759	0.623797	+	0.781586i	$0.285589\pi$
$812$	0	0
$813$	0	0
$814$	−1658.03	−0.0713930
$815$	13962.7	0.600113
$816$	0	0
$817$	3109.73	0.133165
$818$	−30061.6	−1.28494
$819$	0	0
$820$	−6115.81	−0.260455
$821$	34564.5	1.46932	0.734658	−	0.678437i	$-0.237342\pi$
$821$	34564.5	1.46932	0.734658	+	0.678437i	$0.237342\pi$
$822$	0	0
$823$	10429.7	0.441746	0.220873	−	0.975303i	$-0.429109\pi$
$823$	10429.7	0.441746	0.220873	+	0.975303i	$0.429109\pi$
$824$	394.799	0.0166911
$825$	0	0
$826$	0	0
$827$	33622.9	1.41376	0.706881	−	0.707332i	$-0.250101\pi$
$827$	33622.9	1.41376	0.706881	+	0.707332i	$0.250101\pi$
$828$	0	0
$829$	37426.6	1.56801	0.784004	−	0.620756i	$-0.213174\pi$
$829$	37426.6	1.56801	0.784004	+	0.620756i	$0.213174\pi$
$830$	30358.3	1.26958
$831$	0	0
$832$	−1912.28	−0.0796830
$833$	0	0
$834$	0	0
$835$	−1040.05	−0.0431047
$836$	616.689	0.0255127
$837$	0	0
$838$	4487.09	0.184969
$839$	20073.3	0.825993	0.412996	−	0.910733i	$-0.364482\pi$
$839$	20073.3	0.825993	0.412996	+	0.910733i	$0.364482\pi$
$840$	0	0
$841$	−23809.0	−0.976217
$842$	−4511.34	−0.184645
$843$	0	0
$844$	−3387.40	−0.138151
$845$	−9028.10	−0.367546
$846$	0	0
$847$	0	0
$848$	−17730.7	−0.718014
$849$	0	0
$850$	−13247.7	−0.534578
$851$	−6287.82	−0.253283
$852$	0	0
$853$	31621.1	1.26927	0.634635	−	0.772812i	$-0.281151\pi$
$853$	31621.1	1.26927	0.634635	+	0.772812i	$0.281151\pi$
$854$	0	0
$855$	0	0
$856$	21581.9	0.861747
$857$	−5655.26	−0.225414	−0.112707	−	0.993628i	$-0.535952\pi$
$857$	−5655.26	−0.225414	−0.112707	+	0.993628i	$0.535952\pi$
$858$	0	0
$859$	−25859.7	−1.02715	−0.513574	−	0.858045i	$-0.671679\pi$
$859$	−25859.7	−1.02715	−0.513574	+	0.858045i	$0.671679\pi$
$860$	1201.98	0.0476593
$861$	0	0
$862$	244.634	0.00966621
$863$	−12328.5	−0.486288	−0.243144	−	0.969990i	$-0.578179\pi$
$863$	−12328.5	−0.486288	−0.243144	+	0.969990i	$0.578179\pi$
$864$	0	0
$865$	7322.05	0.287812
$866$	−23685.5	−0.929405
$867$	0	0
$868$	0	0
$869$	556.200	0.0217121
$870$	0	0
$871$	−7721.84	−0.300396
$872$	−17608.6	−0.683834
$873$	0	0
$874$	7038.82	0.272416
$875$	0	0
$876$	0	0
$877$	−18980.7	−0.730825	−0.365412	−	0.930846i	$-0.619072\pi$
$877$	−18980.7	−0.730825	−0.365412	+	0.930846i	$0.619072\pi$
$878$	63489.1	2.44038
$879$	0	0
$880$	−1203.42	−0.0460990
$881$	38081.2	1.45629	0.728144	−	0.685424i	$-0.240383\pi$
$881$	38081.2	1.45629	0.728144	+	0.685424i	$0.240383\pi$
$882$	0	0
$883$	6338.38	0.241567	0.120783	−	0.992679i	$-0.461459\pi$
$883$	6338.38	0.241567	0.120783	+	0.992679i	$0.461459\pi$
$884$	5318.97	0.202371
$885$	0	0
$886$	37860.2	1.43560
$887$	−22874.3	−0.865889	−0.432944	−	0.901421i	$-0.642525\pi$
$887$	−22874.3	−0.865889	−0.432944	+	0.901421i	$0.642525\pi$
$888$	0	0
$889$	0	0
$890$	−25083.8	−0.944731
$891$	0	0
$892$	16738.1	0.628290
$893$	13114.7	0.491452
$894$	0	0
$895$	22969.4	0.857856
$896$	0	0
$897$	0	0
$898$	−37440.4	−1.39132
$899$	4670.85	0.173283
$900$	0	0
$901$	−10342.4	−0.382416
$902$	1861.19	0.0687037
$903$	0	0
$904$	−6683.49	−0.245896
$905$	10432.7	0.383199
$906$	0	0
$907$	−33888.9	−1.24064	−0.620320	−	0.784349i	$-0.712997\pi$
$907$	−33888.9	−1.24064	−0.620320	+	0.784349i	$0.712997\pi$
$908$	20696.6	0.756434
$909$	0	0
$910$	0	0
$911$	−10145.9	−0.368990	−0.184495	−	0.982833i	$-0.559065\pi$
$911$	−10145.9	−0.368990	−0.184495	+	0.982833i	$0.559065\pi$
$912$	0	0
$913$	−3069.65	−0.111271
$914$	−15664.0	−0.566870
$915$	0	0
$916$	5402.87	0.194886
$917$	0	0
$918$	0	0
$919$	30423.3	1.09203	0.546013	−	0.837777i	$-0.316145\pi$
$919$	30423.3	1.09203	0.546013	+	0.837777i	$0.316145\pi$
$920$	−2747.08	−0.0984440
$921$	0	0
$922$	38380.3	1.37092
$923$	6717.40	0.239551
$924$	0	0
$925$	−17122.9	−0.608645
$926$	−60805.6	−2.15788
$927$	0	0
$928$	3988.47	0.141086
$929$	−18206.1	−0.642976	−0.321488	−	0.946914i	$-0.604183\pi$
$929$	−18206.1	−0.642976	−0.321488	+	0.946914i	$0.604183\pi$
$930$	0	0
$931$	0	0
$932$	26718.2	0.939037
$933$	0	0
$934$	−21897.2	−0.767127
$935$	−701.960	−0.0245525
$936$	0	0
$937$	15511.4	0.540806	0.270403	−	0.962747i	$-0.412843\pi$
$937$	15511.4	0.540806	0.270403	+	0.962747i	$0.412843\pi$
$938$	0	0
$939$	0	0
$940$	5069.10	0.175889
$941$	90.6984	0.00314206	0.00157103	−	0.999999i	$-0.499500\pi$
$941$	90.6984	0.00314206	0.00157103	+	0.999999i	$0.499500\pi$
$942$	0	0
$943$	7058.27	0.243742
$944$	56859.5	1.96040
$945$	0	0
$946$	−365.790	−0.0125717
$947$	35516.0	1.21871	0.609354	−	0.792899i	$-0.291429\pi$
$947$	35516.0	1.21871	0.609354	+	0.792899i	$0.291429\pi$
$948$	0	0
$949$	6109.66	0.208986
$950$	19168.0	0.654622
$951$	0	0
$952$	0	0
$953$	22433.4	0.762526	0.381263	−	0.924467i	$-0.375489\pi$
$953$	22433.4	0.762526	0.381263	+	0.924467i	$0.375489\pi$
$954$	0	0
$955$	−32283.3	−1.09389
$956$	−7367.25	−0.249241
$957$	0	0
$958$	−1299.25	−0.0438173
$959$	0	0
$960$	0	0
$961$	7821.56	0.262548
$962$	20691.4	0.693469
$963$	0	0
$964$	19364.9	0.646994
$965$	21727.0	0.724785
$966$	0	0
$967$	2753.37	0.0915639	0.0457820	−	0.998951i	$-0.485422\pi$
$967$	2753.37	0.0915639	0.0457820	+	0.998951i	$0.485422\pi$
$968$	−18443.8	−0.612405
$969$	0	0
$970$	−25858.0	−0.855927
$971$	−46754.1	−1.54522	−0.772611	−	0.634880i	$-0.781050\pi$
$971$	−46754.1	−1.54522	−0.772611	+	0.634880i	$0.781050\pi$
$972$	0	0
$973$	0	0
$974$	−89.8096	−0.00295450
$975$	0	0
$976$	−50728.7	−1.66371
$977$	34781.1	1.13894	0.569471	−	0.822011i	$-0.307148\pi$
$977$	34781.1	1.13894	0.569471	+	0.822011i	$0.307148\pi$
$978$	0	0
$979$	2536.32	0.0827998
$980$	0	0
$981$	0	0
$982$	21747.4	0.706708
$983$	29661.0	0.962401	0.481200	−	0.876611i	$-0.340201\pi$
$983$	29661.0	0.962401	0.481200	+	0.876611i	$0.340201\pi$
$984$	0	0
$985$	−6717.22	−0.217288
$986$	3890.05	0.125643
$987$	0	0
$988$	−7695.98	−0.247816
$989$	−1387.20	−0.0446011
$990$	0	0
$991$	−13621.6	−0.436636	−0.218318	−	0.975878i	$-0.570057\pi$
$991$	−13621.6	−0.436636	−0.218318	+	0.975878i	$0.570057\pi$
$992$	32117.6	1.02796
$993$	0	0
$994$	0	0
$995$	−5212.83	−0.166088
$996$	0	0
$997$	9439.71	0.299858	0.149929	−	0.988697i	$-0.452095\pi$
$997$	9439.71	0.299858	0.149929	+	0.988697i	$0.452095\pi$
$998$	8209.46	0.260387
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bg.1.2	yes	6
3.2	odd	2	inner	1323.4.a.bg.1.5	yes	6
7.6	odd	2		1323.4.a.bf.1.2	✓	6
21.20	even	2		1323.4.a.bf.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bf.1.2	✓	6	7.6	odd	2
1323.4.a.bf.1.5	yes	6	21.20	even	2
1323.4.a.bg.1.2	yes	6	1.1	even	1	trivial
1323.4.a.bg.1.5	yes	6	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-3.46131 q^{2} +3.98067 q^{4} +6.55599 q^{5} +13.9122 q^{8} +O(q^{10})\)	\(q-3.46131 q^{2} +3.98067 q^{4} +6.55599 q^{5} +13.9122 q^{8} -22.6923 q^{10} +2.29451 q^{11} -28.6343 q^{13} -79.9996 q^{16} -46.6643 q^{17} +67.5183 q^{19} +26.0972 q^{20} -7.94200 q^{22} -30.1188 q^{23} -82.0190 q^{25} +99.1122 q^{26} +24.0840 q^{29} +193.940 q^{31} +165.606 q^{32} +161.520 q^{34} +208.767 q^{37} -233.702 q^{38} +91.2080 q^{40} -234.347 q^{41} +46.0576 q^{43} +9.13366 q^{44} +104.251 q^{46} +194.239 q^{47} +283.893 q^{50} -113.984 q^{52} +221.635 q^{53} +15.0428 q^{55} -83.3623 q^{58} -710.748 q^{59} +634.111 q^{61} -671.285 q^{62} +66.7827 q^{64} -187.726 q^{65} +269.671 q^{67} -185.755 q^{68} -234.593 q^{71} -213.368 q^{73} -722.608 q^{74} +268.768 q^{76} +242.405 q^{79} -524.477 q^{80} +811.149 q^{82} -1337.82 q^{83} -305.931 q^{85} -159.420 q^{86} +31.9215 q^{88} +1105.39 q^{89} -119.893 q^{92} -672.322 q^{94} +442.650 q^{95} +1139.50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 36 q^{4}+O(q^{10}) \) 6 * q + 36 * q^4	\( 6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58} + 1152 q^{61} + 2964 q^{64} + 924 q^{67} + 1260 q^{73} + 5868 q^{76} - 1500 q^{79} + 4500 q^{82} - 2232 q^{85} - 2460 q^{88} - 4968 q^{94} + 3312 q^{97}+O(q^{100}) \) 6 * q + 36 * q^4 + 180 * q^10 + 108 * q^13 + 420 * q^16 + 198 * q^19 - 84 * q^22 + 420 * q^25 - 90 * q^31 - 648 * q^34 - 402 * q^37 + 2844 * q^40 - 660 * q^43 - 1332 * q^46 + 1224 * q^52 + 846 * q^55 - 1800 * q^58 + 1152 * q^61 + 2964 * q^64 + 924 * q^67 + 1260 * q^73 + 5868 * q^76 - 1500 * q^79 + 4500 * q^82 - 2232 * q^85 - 2460 * q^88 - 4968 * q^94 + 3312 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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