LMFDB - Embedded newform 1323.4.a.bp.1.4

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 72x^{10} + 1809x^{8} - 19062x^{6} + 78324x^{4} - 73368x^{2} + 19600 $ x^12 - 72x^10 + 1809x^8 - 19062x^6 + 78324x^4 - 73368*x^2 + 19600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.82234$ 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)

$f(q)$	$=$	$q-2.82234 q^{2} -0.0343991 q^{4} +15.1035 q^{5} +22.6758 q^{8} -42.6273 q^{10} +32.1724 q^{11} -33.5373 q^{13} -63.7236 q^{16} -41.0684 q^{17} +35.9540 q^{19} -0.519549 q^{20} -90.8015 q^{22} +92.4825 q^{23} +103.117 q^{25} +94.6535 q^{26} +12.3752 q^{29} -250.230 q^{31} -1.55672 q^{32} +115.909 q^{34} -345.832 q^{37} -101.474 q^{38} +342.485 q^{40} -318.201 q^{41} -363.945 q^{43} -1.10670 q^{44} -261.017 q^{46} -405.225 q^{47} -291.030 q^{50} +1.15365 q^{52} -258.677 q^{53} +485.917 q^{55} -34.9269 q^{58} +258.265 q^{59} +578.762 q^{61} +706.233 q^{62} +514.183 q^{64} -506.531 q^{65} +726.896 q^{67} +1.41272 q^{68} -1074.68 q^{71} +942.519 q^{73} +976.054 q^{74} -1.23679 q^{76} -708.532 q^{79} -962.452 q^{80} +898.072 q^{82} -957.080 q^{83} -620.278 q^{85} +1027.18 q^{86} +729.535 q^{88} +485.451 q^{89} -3.18132 q^{92} +1143.68 q^{94} +543.032 q^{95} -313.998 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{4} - 40 q^{5} + 444 q^{16} - 292 q^{17} - 524 q^{20} - 600 q^{22} + 324 q^{25} - 144 q^{26} + 216 q^{37} - 808 q^{38} - 1768 q^{41} + 180 q^{43} - 984 q^{47} + 1164 q^{58} - 1324 q^{59} - 3640 q^{62}+ \cdots - 1036 q^{89}+O(q^{100}) $ 12 * q + 48 * q^4 - 40 * q^5 + 444 * q^16 - 292 * q^17 - 524 * q^20 - 600 * q^22 + 324 * q^25 - 144 * q^26 + 216 * q^37 - 808 * q^38 - 1768 * q^41 + 180 * q^43 - 984 * q^47 + 1164 * q^58 - 1324 * q^59 - 3640 * q^62 + 3252 * q^64 + 972 * q^67 - 4528 * q^68 - 1752 * q^79 - 10536 * q^80 - 2872 * q^83 + 2664 * q^85 - 8160 * q^88 - 1036 * q^89

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 $

$2$	−2.82234	−0.997848	−0.498924	−	0.866646i	$-0.666271\pi$
$2$	−2.82234	−0.997848	−0.498924	+	0.866646i	$0.666271\pi$
$3$	0	0
$3$	0	0
$4$	−0.0343991	−0.00429989
$5$	15.1035	1.35090	0.675450	−	0.737405i	$-0.263949\pi$
$5$	15.1035	1.35090	0.675450	+	0.737405i	$0.263949\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6758	1.00214
$9$	0	0
$10$	−42.6273	−1.34799
$11$	32.1724	0.881850	0.440925	−	0.897544i	$-0.354650\pi$
$11$	32.1724	0.881850	0.440925	+	0.897544i	$0.354650\pi$
$12$	0	0
$13$	−33.5373	−0.715505	−0.357752	−	0.933817i	$-0.616457\pi$
$13$	−33.5373	−0.715505	−0.357752	+	0.933817i	$0.616457\pi$
$14$	0	0
$15$	0	0
$16$	−63.7236	−0.995682
$17$	−41.0684	−0.585915	−0.292958	−	0.956125i	$-0.594639\pi$
$17$	−41.0684	−0.585915	−0.292958	+	0.956125i	$0.594639\pi$
$18$	0	0
$19$	35.9540	0.434127	0.217063	−	0.976158i	$-0.430352\pi$
$19$	35.9540	0.434127	0.217063	+	0.976158i	$0.430352\pi$
$20$	−0.519549	−0.00580873
$21$	0	0
$22$	−90.8015	−0.879952
$23$	92.4825	0.838432	0.419216	−	0.907887i	$-0.362305\pi$
$23$	92.4825	0.838432	0.419216	+	0.907887i	$0.362305\pi$
$24$	0	0
$25$	103.117	0.824933
$26$	94.6535	0.713965
$27$	0	0
$28$	0	0
$29$	12.3752	0.0792417	0.0396209	−	0.999215i	$-0.487385\pi$
$29$	12.3752	0.0792417	0.0396209	+	0.999215i	$0.487385\pi$
$30$	0	0
$31$	−250.230	−1.44976	−0.724880	−	0.688875i	$-0.758105\pi$
$31$	−250.230	−1.44976	−0.724880	+	0.688875i	$0.758105\pi$
$32$	−1.55672	−0.00859973
$33$	0	0
$34$	115.909	0.584654
$35$	0	0
$36$	0	0
$37$	−345.832	−1.53660	−0.768302	−	0.640087i	$-0.778898\pi$
$37$	−345.832	−1.53660	−0.768302	+	0.640087i	$0.778898\pi$
$38$	−101.474	−0.433192
$39$	0	0
$40$	342.485	1.35379
$41$	−318.201	−1.21207	−0.606033	−	0.795440i	$-0.707240\pi$
$41$	−318.201	−1.21207	−0.606033	+	0.795440i	$0.707240\pi$
$42$	0	0
$43$	−363.945	−1.29072	−0.645362	−	0.763877i	$-0.723294\pi$
$43$	−363.945	−1.29072	−0.645362	+	0.763877i	$0.723294\pi$
$44$	−1.10670	−0.00379186
$45$	0	0
$46$	−261.017	−0.836628
$47$	−405.225	−1.25762	−0.628809	−	0.777560i	$-0.716457\pi$
$47$	−405.225	−1.25762	−0.628809	+	0.777560i	$0.716457\pi$
$48$	0	0
$49$	0	0
$50$	−291.030	−0.823158
$51$	0	0
$52$	1.15365	0.00307659
$53$	−258.677	−0.670414	−0.335207	−	0.942144i	$-0.608806\pi$
$53$	−258.677	−0.670414	−0.335207	+	0.942144i	$0.608806\pi$
$54$	0	0
$55$	485.917	1.19129
$56$	0	0
$57$	0	0
$58$	−34.9269	−0.0790712
$59$	258.265	0.569886	0.284943	−	0.958544i	$-0.408025\pi$
$59$	258.265	0.569886	0.284943	+	0.958544i	$0.408025\pi$
$60$	0	0
$61$	578.762	1.21480	0.607400	−	0.794396i	$-0.292212\pi$
$61$	578.762	1.21480	0.607400	+	0.794396i	$0.292212\pi$
$62$	706.233	1.44664
$63$	0	0
$64$	514.183	1.00426
$65$	−506.531	−0.966576
$66$	0	0
$67$	726.896	1.32544	0.662720	−	0.748867i	$-0.269402\pi$
$67$	726.896	1.32544	0.662720	+	0.748867i	$0.269402\pi$
$68$	1.41272	0.00251937
$69$	0	0
$70$	0	0
$71$	−1074.68	−1.79635	−0.898174	−	0.439639i	$-0.855106\pi$
$71$	−1074.68	−1.79635	−0.898174	+	0.439639i	$0.855106\pi$
$72$	0	0
$73$	942.519	1.51114	0.755572	−	0.655065i	$-0.227359\pi$
$73$	942.519	1.51114	0.755572	+	0.655065i	$0.227359\pi$
$74$	976.054	1.53330
$75$	0	0
$76$	−1.23679	−0.00186670
$77$	0	0
$78$	0	0
$79$	−708.532	−1.00906	−0.504532	−	0.863393i	$-0.668335\pi$
$79$	−708.532	−1.00906	−0.504532	+	0.863393i	$0.668335\pi$
$80$	−962.452	−1.34507
$81$	0	0
$82$	898.072	1.20946
$83$	−957.080	−1.26570	−0.632851	−	0.774274i	$-0.718115\pi$
$83$	−957.080	−1.26570	−0.632851	+	0.774274i	$0.718115\pi$
$84$	0	0
$85$	−620.278	−0.791513
$86$	1027.18	1.28795
$87$	0	0
$88$	729.535	0.883736
$89$	485.451	0.578177	0.289088	−	0.957302i	$-0.406648\pi$
$89$	485.451	0.578177	0.289088	+	0.957302i	$0.406648\pi$
$90$	0	0
$91$	0	0
$92$	−3.18132	−0.00360517
$93$	0	0
$94$	1143.68	1.25491
$95$	543.032	0.586462
$96$	0	0
$97$	−313.998	−0.328677	−0.164338	−	0.986404i	$-0.552549\pi$
$97$	−313.998	−0.328677	−0.164338	+	0.986404i	$0.552549\pi$
$98$	0	0
$99$	0	0
$100$	−3.54713	−0.00354713
$101$	−854.698	−0.842035	−0.421018	−	0.907052i	$-0.638327\pi$
$101$	−854.698	−0.842035	−0.421018	+	0.907052i	$0.638327\pi$
$102$	0	0
$103$	−846.014	−0.809323	−0.404662	−	0.914467i	$-0.632611\pi$
$103$	−846.014	−0.809323	−0.404662	+	0.914467i	$0.632611\pi$
$104$	−760.484	−0.717035
$105$	0	0
$106$	730.073	0.668972
$107$	1493.04	1.34895	0.674473	−	0.738299i	$-0.264371\pi$
$107$	1493.04	1.34895	0.674473	+	0.738299i	$0.264371\pi$
$108$	0	0
$109$	1825.31	1.60397	0.801984	−	0.597345i	$-0.203778\pi$
$109$	1825.31	1.60397	0.801984	+	0.597345i	$0.203778\pi$
$110$	−1371.42	−1.18873
$111$	0	0
$112$	0	0
$113$	545.012	0.453720	0.226860	−	0.973927i	$-0.427154\pi$
$113$	545.012	0.453720	0.226860	+	0.973927i	$0.427154\pi$
$114$	0	0
$115$	1396.81	1.13264
$116$	−0.425695	−0.000340731 0
$117$	0	0
$118$	−728.912	−0.568659
$119$	0	0
$120$	0	0
$121$	−295.935	−0.222341
$122$	−1633.46	−1.21219
$123$	0	0
$124$	8.60768	0.00623381
$125$	−330.515	−0.236498
$126$	0	0
$127$	−1993.09	−1.39259	−0.696293	−	0.717758i	$-0.745169\pi$
$127$	−1993.09	−1.39259	−0.696293	+	0.717758i	$0.745169\pi$
$128$	−1438.74	−0.993502
$129$	0	0
$130$	1429.60	0.964496
$131$	1668.82	1.11302	0.556510	−	0.830841i	$-0.312140\pi$
$131$	1668.82	1.11302	0.556510	+	0.830841i	$0.312140\pi$
$132$	0	0
$133$	0	0
$134$	−2051.55	−1.32259
$135$	0	0
$136$	−931.260	−0.587168
$137$	−18.8810	−0.0117746	−0.00588728	−	0.999983i	$-0.501874\pi$
$137$	−18.8810	−0.0117746	−0.00588728	+	0.999983i	$0.501874\pi$
$138$	0	0
$139$	−662.600	−0.404324	−0.202162	−	0.979352i	$-0.564797\pi$
$139$	−662.600	−0.404324	−0.202162	+	0.979352i	$0.564797\pi$
$140$	0	0
$141$	0	0
$142$	3033.10	1.79248
$143$	−1078.97	−0.630968
$144$	0	0
$145$	186.909	0.107048
$146$	−2660.11	−1.50789
$147$	0	0
$148$	11.8963	0.00660723
$149$	2344.27	1.28892	0.644462	−	0.764636i	$-0.277081\pi$
$149$	2344.27	1.28892	0.644462	+	0.764636i	$0.277081\pi$
$150$	0	0
$151$	1248.19	0.672693	0.336347	−	0.941738i	$-0.390809\pi$
$151$	1248.19	0.672693	0.336347	+	0.941738i	$0.390809\pi$
$152$	815.285	0.435055
$153$	0	0
$154$	0	0
$155$	−3779.35	−1.95848
$156$	0	0
$157$	2441.08	1.24089	0.620443	−	0.784252i	$-0.286953\pi$
$157$	2441.08	1.24089	0.620443	+	0.784252i	$0.286953\pi$
$158$	1999.72	1.00689
$159$	0	0
$160$	−23.5119	−0.0116174
$161$	0	0
$162$	0	0
$163$	−1496.26	−0.718993	−0.359496	−	0.933147i	$-0.617051\pi$
$163$	−1496.26	−0.718993	−0.359496	+	0.933147i	$0.617051\pi$
$164$	10.9459	0.00521175
$165$	0	0
$166$	2701.21	1.26298
$167$	−183.542	−0.0850472	−0.0425236	−	0.999095i	$-0.513540\pi$
$167$	−183.542	−0.0850472	−0.0425236	+	0.999095i	$0.513540\pi$
$168$	0	0
$169$	−1072.25	−0.488053
$170$	1750.64	0.789810
$171$	0	0
$172$	12.5194	0.00554998
$173$	−3807.75	−1.67340	−0.836700	−	0.547662i	$-0.815518\pi$
$173$	−3807.75	−1.67340	−0.836700	+	0.547662i	$0.815518\pi$
$174$	0	0
$175$	0	0
$176$	−2050.14	−0.878042
$177$	0	0
$178$	−1370.11	−0.576932
$179$	1467.27	0.612678	0.306339	−	0.951923i	$-0.400896\pi$
$179$	1467.27	0.612678	0.306339	+	0.951923i	$0.400896\pi$
$180$	0	0
$181$	1127.05	0.462833	0.231417	−	0.972855i	$-0.425664\pi$
$181$	1127.05	0.462833	0.231417	+	0.972855i	$0.425664\pi$
$182$	0	0
$183$	0	0
$184$	2097.12	0.840225
$185$	−5223.28	−2.07580
$186$	0	0
$187$	−1321.27	−0.516689
$188$	13.9394	0.00540763
$189$	0	0
$190$	−1532.62	−0.585200
$191$	−2930.52	−1.11018	−0.555092	−	0.831789i	$-0.687317\pi$
$191$	−2930.52	−1.11018	−0.555092	+	0.831789i	$0.687317\pi$
$192$	0	0
$193$	−2250.48	−0.839343	−0.419671	−	0.907676i	$-0.637855\pi$
$193$	−2250.48	−0.839343	−0.419671	+	0.907676i	$0.637855\pi$
$194$	886.208	0.327969
$195$	0	0
$196$	0	0
$197$	−3349.31	−1.21131	−0.605655	−	0.795727i	$-0.707089\pi$
$197$	−3349.31	−1.21131	−0.605655	+	0.795727i	$0.707089\pi$
$198$	0	0
$199$	−5149.60	−1.83440	−0.917200	−	0.398427i	$-0.869556\pi$
$199$	−5149.60	−1.83440	−0.917200	+	0.398427i	$0.869556\pi$
$200$	2338.25	0.826697
$201$	0	0
$202$	2412.25	0.840223
$203$	0	0
$204$	0	0
$205$	−4805.96	−1.63738
$206$	2387.74	0.807581
$207$	0	0
$208$	2137.12	0.712415
$209$	1156.73	0.382835
$210$	0	0
$211$	4907.47	1.60116	0.800579	−	0.599228i	$-0.204526\pi$
$211$	4907.47	1.60116	0.800579	+	0.599228i	$0.204526\pi$
$212$	8.89826	0.00288271
$213$	0	0
$214$	−4213.86	−1.34604
$215$	−5496.86	−1.74364
$216$	0	0
$217$	0	0
$218$	−5151.63	−1.60052
$219$	0	0
$220$	−16.7151	−0.00512243
$221$	1377.32	0.419225
$222$	0	0
$223$	592.442	0.177905	0.0889526	−	0.996036i	$-0.471648\pi$
$223$	592.442	0.177905	0.0889526	+	0.996036i	$0.471648\pi$
$224$	0	0
$225$	0	0
$226$	−1538.21	−0.452744
$227$	−2883.54	−0.843117	−0.421558	−	0.906801i	$-0.638517\pi$
$227$	−2883.54	−0.843117	−0.421558	+	0.906801i	$0.638517\pi$
$228$	0	0
$229$	−5150.21	−1.48618	−0.743090	−	0.669192i	$-0.766641\pi$
$229$	−5150.21	−1.48618	−0.743090	+	0.669192i	$0.766641\pi$
$230$	−3942.28	−1.13020
$231$	0	0
$232$	280.617	0.0794112
$233$	3799.78	1.06838	0.534188	−	0.845366i	$-0.320617\pi$
$233$	3799.78	1.06838	0.534188	+	0.845366i	$0.320617\pi$
$234$	0	0
$235$	−6120.32	−1.69892
$236$	−8.88411	−0.00245045
$237$	0	0
$238$	0	0
$239$	4159.65	1.12580	0.562899	−	0.826526i	$-0.309686\pi$
$239$	4159.65	1.12580	0.562899	+	0.826526i	$0.309686\pi$
$240$	0	0
$241$	1235.40	0.330203	0.165102	−	0.986277i	$-0.447205\pi$
$241$	1235.40	0.330203	0.165102	+	0.986277i	$0.447205\pi$
$242$	835.230	0.221862
$243$	0	0
$244$	−19.9089	−0.00522351
$245$	0	0
$246$	0	0
$247$	−1205.80	−0.310620
$248$	−5674.16	−1.45286
$249$	0	0
$250$	932.827	0.235989
$251$	−2831.44	−0.712027	−0.356013	−	0.934481i	$-0.615864\pi$
$251$	−2831.44	−0.712027	−0.356013	+	0.934481i	$0.615864\pi$
$252$	0	0
$253$	2975.39	0.739371
$254$	5625.19	1.38959
$255$	0	0
$256$	−52.8361	−0.0128994
$257$	6641.20	1.61193	0.805966	−	0.591962i	$-0.201646\pi$
$257$	6641.20	1.61193	0.805966	+	0.591962i	$0.201646\pi$
$258$	0	0
$259$	0	0
$260$	17.4242	0.00415617
$261$	0	0
$262$	−4709.98	−1.11062
$263$	−3206.48	−0.751788	−0.375894	−	0.926663i	$-0.622664\pi$
$263$	−3206.48	−0.751788	−0.375894	+	0.926663i	$0.622664\pi$
$264$	0	0
$265$	−3906.93	−0.905664
$266$	0	0
$267$	0	0
$268$	−25.0046	−0.00569925
$269$	−3786.52	−0.858245	−0.429123	−	0.903246i	$-0.641177\pi$
$269$	−3786.52	−0.858245	−0.429123	+	0.903246i	$0.641177\pi$
$270$	0	0
$271$	−3624.05	−0.812345	−0.406173	−	0.913796i	$-0.633137\pi$
$271$	−3624.05	−0.812345	−0.406173	+	0.913796i	$0.633137\pi$
$272$	2617.03	0.583385
$273$	0	0
$274$	53.2886	0.0117492
$275$	3317.51	0.727468
$276$	0	0
$277$	123.201	0.0267235	0.0133618	−	0.999911i	$-0.495747\pi$
$277$	123.201	0.0267235	0.0133618	+	0.999911i	$0.495747\pi$
$278$	1870.08	0.403453
$279$	0	0
$280$	0	0
$281$	−3441.06	−0.730522	−0.365261	−	0.930905i	$-0.619020\pi$
$281$	−3441.06	−0.730522	−0.365261	+	0.930905i	$0.619020\pi$
$282$	0	0
$283$	−3475.98	−0.730125	−0.365063	−	0.930983i	$-0.618952\pi$
$283$	−3475.98	−0.730125	−0.365063	+	0.930983i	$0.618952\pi$
$284$	36.9680	0.00772411
$285$	0	0
$286$	3045.23	0.629610
$287$	0	0
$288$	0	0
$289$	−3226.38	−0.656703
$290$	−527.520	−0.106817
$291$	0	0
$292$	−32.4219	−0.00649776
$293$	−8075.93	−1.61024	−0.805121	−	0.593111i	$-0.797900\pi$
$293$	−8075.93	−1.61024	−0.805121	+	0.593111i	$0.797900\pi$
$294$	0	0
$295$	3900.72	0.769860
$296$	−7842.01	−1.53989
$297$	0	0
$298$	−6616.31	−1.28615
$299$	−3101.61	−0.599902
$300$	0	0
$301$	0	0
$302$	−3522.83	−0.671245
$303$	0	0
$304$	−2291.12	−0.432252
$305$	8741.35	1.64108
$306$	0	0
$307$	2140.32	0.397898	0.198949	−	0.980010i	$-0.436247\pi$
$307$	2140.32	0.397898	0.198949	+	0.980010i	$0.436247\pi$
$308$	0	0
$309$	0	0
$310$	10666.6	1.95427
$311$	154.865	0.0282367	0.0141184	−	0.999900i	$-0.495506\pi$
$311$	154.865	0.0282367	0.0141184	+	0.999900i	$0.495506\pi$
$312$	0	0
$313$	−2025.62	−0.365799	−0.182899	−	0.983132i	$-0.558548\pi$
$313$	−2025.62	−0.365799	−0.182899	+	0.983132i	$0.558548\pi$
$314$	−6889.54	−1.23821
$315$	0	0
$316$	24.3729	0.00433887
$317$	838.305	0.148530	0.0742648	−	0.997239i	$-0.476339\pi$
$317$	838.305	0.148530	0.0742648	+	0.997239i	$0.476339\pi$
$318$	0	0
$319$	398.139	0.0698793
$320$	7765.97	1.35666
$321$	0	0
$322$	0	0
$323$	−1476.57	−0.254361
$324$	0	0
$325$	−3458.25	−0.590244
$326$	4222.94	0.717445
$327$	0	0
$328$	−7215.47	−1.21466
$329$	0	0
$330$	0	0
$331$	10516.3	1.74631	0.873157	−	0.487439i	$-0.162069\pi$
$331$	10516.3	1.74631	0.873157	+	0.487439i	$0.162069\pi$
$332$	32.9227	0.00544238
$333$	0	0
$334$	518.017	0.0848642
$335$	10978.7	1.79054
$336$	0	0
$337$	−5791.21	−0.936104	−0.468052	−	0.883701i	$-0.655044\pi$
$337$	−5791.21	−0.936104	−0.468052	+	0.883701i	$0.655044\pi$
$338$	3026.26	0.487003
$339$	0	0
$340$	21.3370	0.00340342
$341$	−8050.49	−1.27847
$342$	0	0
$343$	0	0
$344$	−8252.75	−1.29348
$345$	0	0
$346$	10746.8	1.66980
$347$	−4407.82	−0.681913	−0.340957	−	0.940079i	$-0.610751\pi$
$347$	−4407.82	−0.681913	−0.340957	+	0.940079i	$0.610751\pi$
$348$	0	0
$349$	−11551.9	−1.77181	−0.885905	−	0.463867i	$-0.846462\pi$
$349$	−11551.9	−1.77181	−0.885905	+	0.463867i	$0.846462\pi$
$350$	0	0
$351$	0	0
$352$	−50.0833	−0.00758367
$353$	8304.23	1.25210	0.626048	−	0.779785i	$-0.284671\pi$
$353$	8304.23	1.25210	0.626048	+	0.779785i	$0.284671\pi$
$354$	0	0
$355$	−16231.4	−2.42669
$356$	−16.6991	−0.00248610
$357$	0	0
$358$	−4141.15	−0.611359
$359$	−2188.61	−0.321756	−0.160878	−	0.986974i	$-0.551433\pi$
$359$	−2188.61	−0.321756	−0.160878	+	0.986974i	$0.551433\pi$
$360$	0	0
$361$	−5566.31	−0.811534
$362$	−3180.91	−0.461837
$363$	0	0
$364$	0	0
$365$	14235.4	2.04141
$366$	0	0
$367$	−3866.97	−0.550011	−0.275005	−	0.961443i	$-0.588680\pi$
$367$	−3866.97	−0.550011	−0.275005	+	0.961443i	$0.588680\pi$
$368$	−5893.32	−0.834811
$369$	0	0
$370$	14741.9	2.07133
$371$	0	0
$372$	0	0
$373$	11154.2	1.54837	0.774183	−	0.632962i	$-0.218161\pi$
$373$	11154.2	1.54837	0.774183	+	0.632962i	$0.218161\pi$
$374$	3729.07	0.515577
$375$	0	0
$376$	−9188.79	−1.26031
$377$	−415.029	−0.0566978
$378$	0	0
$379$	7752.17	1.05067	0.525333	−	0.850897i	$-0.323941\pi$
$379$	7752.17	1.05067	0.525333	+	0.850897i	$0.323941\pi$
$380$	−18.6798	−0.00252172
$381$	0	0
$382$	8270.93	1.10780
$383$	−3591.50	−0.479157	−0.239578	−	0.970877i	$-0.577009\pi$
$383$	−3591.50	−0.479157	−0.239578	+	0.970877i	$0.577009\pi$
$384$	0	0
$385$	0	0
$386$	6351.62	0.837536
$387$	0	0
$388$	10.8013	0.00141327
$389$	−10233.3	−1.33380	−0.666902	−	0.745145i	$-0.732380\pi$
$389$	−10233.3	−1.33380	−0.666902	+	0.745145i	$0.732380\pi$
$390$	0	0
$391$	−3798.11	−0.491250
$392$	0	0
$393$	0	0
$394$	9452.88	1.20870
$395$	−10701.3	−1.36315
$396$	0	0
$397$	13763.8	1.74001	0.870007	−	0.493039i	$-0.164114\pi$
$397$	13763.8	1.74001	0.870007	+	0.493039i	$0.164114\pi$
$398$	14533.9	1.83045
$399$	0	0
$400$	−6570.97	−0.821371
$401$	1357.25	0.169022	0.0845108	−	0.996423i	$-0.473067\pi$
$401$	1357.25	0.169022	0.0845108	+	0.996423i	$0.473067\pi$
$402$	0	0
$403$	8392.01	1.03731
$404$	29.4009	0.00362066
$405$	0	0
$406$	0	0
$407$	−11126.2	−1.35505
$408$	0	0
$409$	6114.11	0.739176	0.369588	−	0.929196i	$-0.379499\pi$
$409$	6114.11	0.739176	0.369588	+	0.929196i	$0.379499\pi$
$410$	13564.1	1.63386
$411$	0	0
$412$	29.1022	0.00348000
$413$	0	0
$414$	0	0
$415$	−14455.3	−1.70984
$416$	52.2080	0.00615314
$417$	0	0
$418$	−3264.67	−0.382011
$419$	−4387.27	−0.511533	−0.255766	−	0.966739i	$-0.582328\pi$
$419$	−4387.27	−0.511533	−0.255766	+	0.966739i	$0.582328\pi$
$420$	0	0
$421$	6052.81	0.700703	0.350352	−	0.936618i	$-0.386062\pi$
$421$	6052.81	0.700703	0.350352	+	0.936618i	$0.386062\pi$
$422$	−13850.5	−1.59771
$423$	0	0
$424$	−5865.70	−0.671848
$425$	−4234.84	−0.483341
$426$	0	0
$427$	0	0
$428$	−51.3592	−0.00580033
$429$	0	0
$430$	15514.0	1.73989
$431$	−5590.63	−0.624806	−0.312403	−	0.949950i	$-0.601134\pi$
$431$	−5590.63	−0.624806	−0.312403	+	0.949950i	$0.601134\pi$
$432$	0	0
$433$	−16225.8	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$433$	−16225.8	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$434$	0	0
$435$	0	0
$436$	−62.7890	−0.00689689
$437$	3325.11	0.363986
$438$	0	0
$439$	437.018	0.0475119	0.0237560	−	0.999718i	$-0.492438\pi$
$439$	437.018	0.0475119	0.0237560	+	0.999718i	$0.492438\pi$
$440$	11018.6	1.19384
$441$	0	0
$442$	−3887.27	−0.418323
$443$	7169.02	0.768872	0.384436	−	0.923152i	$-0.374396\pi$
$443$	7169.02	0.768872	0.384436	+	0.923152i	$0.374396\pi$
$444$	0	0
$445$	7332.03	0.781059
$446$	−1672.07	−0.177522
$447$	0	0
$448$	0	0
$449$	−1150.18	−0.120892	−0.0604459	−	0.998171i	$-0.519252\pi$
$449$	−1150.18	−0.120892	−0.0604459	+	0.998171i	$0.519252\pi$
$450$	0	0
$451$	−10237.3	−1.06886
$452$	−18.7479	−0.00195095
$453$	0	0
$454$	8138.34	0.841302
$455$	0	0
$456$	0	0
$457$	−8788.54	−0.899586	−0.449793	−	0.893133i	$-0.648502\pi$
$457$	−8788.54	−0.899586	−0.449793	+	0.893133i	$0.648502\pi$
$458$	14535.6	1.48298
$459$	0	0
$460$	−48.0492	−0.00487023
$461$	−12093.8	−1.22183	−0.610916	−	0.791695i	$-0.709199\pi$
$461$	−12093.8	−1.22183	−0.610916	+	0.791695i	$0.709199\pi$
$462$	0	0
$463$	17846.6	1.79137	0.895684	−	0.444690i	$-0.146686\pi$
$463$	17846.6	1.79137	0.895684	+	0.444690i	$0.146686\pi$
$464$	−788.590	−0.0788995
$465$	0	0
$466$	−10724.3	−1.06608
$467$	7435.85	0.736809	0.368405	−	0.929666i	$-0.379904\pi$
$467$	7435.85	0.736809	0.368405	+	0.929666i	$0.379904\pi$
$468$	0	0
$469$	0	0
$470$	17273.6	1.69526
$471$	0	0
$472$	5856.37	0.571105
$473$	−11709.0	−1.13823
$474$	0	0
$475$	3707.45	0.358126
$476$	0	0
$477$	0	0
$478$	−11739.9	−1.12337
$479$	10630.3	1.01401	0.507006	−	0.861942i	$-0.330752\pi$
$479$	10630.3	1.01401	0.507006	+	0.861942i	$0.330752\pi$
$480$	0	0
$481$	11598.2	1.09945
$482$	−3486.71	−0.329493
$483$	0	0
$484$	10.1799	0.000956041 0
$485$	−4742.47	−0.444010
$486$	0	0
$487$	13619.0	1.26722	0.633610	−	0.773653i	$-0.281572\pi$
$487$	13619.0	1.26722	0.633610	+	0.773653i	$0.281572\pi$
$488$	13123.9	1.21740
$489$	0	0
$490$	0	0
$491$	17456.7	1.60450	0.802251	−	0.596987i	$-0.203636\pi$
$491$	17456.7	1.60450	0.802251	+	0.596987i	$0.203636\pi$
$492$	0	0
$493$	−508.229	−0.0464289
$494$	3403.17	0.309951
$495$	0	0
$496$	15945.5	1.44350
$497$	0	0
$498$	0	0
$499$	4099.09	0.367736	0.183868	−	0.982951i	$-0.441138\pi$
$499$	4099.09	0.367736	0.183868	+	0.982951i	$0.441138\pi$
$500$	11.3694	0.00101691
$501$	0	0
$502$	7991.28	0.710494
$503$	−3326.13	−0.294841	−0.147420	−	0.989074i	$-0.547097\pi$
$503$	−3326.13	−0.294841	−0.147420	+	0.989074i	$0.547097\pi$
$504$	0	0
$505$	−12909.0	−1.13751
$506$	−8397.55	−0.737780
$507$	0	0
$508$	68.5607	0.00598797
$509$	10155.2	0.884322	0.442161	−	0.896936i	$-0.354212\pi$
$509$	10155.2	0.884322	0.442161	+	0.896936i	$0.354212\pi$
$510$	0	0
$511$	0	0
$512$	11659.1	1.00637
$513$	0	0
$514$	−18743.7	−1.60846
$515$	−12777.8	−1.09332
$516$	0	0
$517$	−13037.1	−1.10903
$518$	0	0
$519$	0	0
$520$	−11486.0	−0.968643
$521$	−1425.90	−0.119903	−0.0599517	−	0.998201i	$-0.519095\pi$
$521$	−1425.90	−0.119903	−0.0599517	+	0.998201i	$0.519095\pi$
$522$	0	0
$523$	9431.47	0.788545	0.394273	−	0.918993i	$-0.370997\pi$
$523$	9431.47	0.788545	0.394273	+	0.918993i	$0.370997\pi$
$524$	−57.4060	−0.00478586
$525$	0	0
$526$	9049.79	0.750170
$527$	10276.5	0.849436
$528$	0	0
$529$	−3613.98	−0.297032
$530$	11026.7	0.903714
$531$	0	0
$532$	0	0
$533$	10671.6	0.867239
$534$	0	0
$535$	22550.1	1.82229
$536$	16483.0	1.32827
$537$	0	0
$538$	10686.8	0.856398
$539$	0	0
$540$	0	0
$541$	−20808.6	−1.65366	−0.826831	−	0.562451i	$-0.809858\pi$
$541$	−20808.6	−1.65366	−0.826831	+	0.562451i	$0.809858\pi$
$542$	10228.3	0.810597
$543$	0	0
$544$	63.9319	0.00503871
$545$	27568.6	2.16680
$546$	0	0
$547$	−2854.36	−0.223114	−0.111557	−	0.993758i	$-0.535584\pi$
$547$	−2854.36	−0.223114	−0.111557	+	0.993758i	$0.535584\pi$
$548$	0.649491	5.06293e−5 0
$549$	0	0
$550$	−9363.15	−0.725902
$551$	444.936	0.0344009
$552$	0	0
$553$	0	0
$554$	−347.715	−0.0266660
$555$	0	0
$556$	22.7929	0.00173855
$557$	−23816.4	−1.81173	−0.905863	−	0.423571i	$-0.860777\pi$
$557$	−23816.4	−1.81173	−0.905863	+	0.423571i	$0.860777\pi$
$558$	0	0
$559$	12205.7	0.923519
$560$	0	0
$561$	0	0
$562$	9711.85	0.728950
$563$	10225.7	0.765477	0.382739	−	0.923857i	$-0.374981\pi$
$563$	10225.7	0.765477	0.382739	+	0.923857i	$0.374981\pi$
$564$	0	0
$565$	8231.60	0.612931
$566$	9810.39	0.728554
$567$	0	0
$568$	−24369.2	−1.80019
$569$	−11486.3	−0.846278	−0.423139	−	0.906065i	$-0.639072\pi$
$569$	−11486.3	−0.846278	−0.423139	+	0.906065i	$0.639072\pi$
$570$	0	0
$571$	17536.4	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$571$	17536.4	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$572$	37.1158	0.00271309
$573$	0	0
$574$	0	0
$575$	9536.49	0.691651
$576$	0	0
$577$	−19543.8	−1.41009	−0.705044	−	0.709163i	$-0.749073\pi$
$577$	−19543.8	−1.41009	−0.705044	+	0.709163i	$0.749073\pi$
$578$	9105.95	0.655290
$579$	0	0
$580$	−6.42950	−0.000460294 0
$581$	0	0
$582$	0	0
$583$	−8322.25	−0.591205
$584$	21372.4	1.51438
$585$	0	0
$586$	22793.0	1.60678
$587$	−5132.74	−0.360904	−0.180452	−	0.983584i	$-0.557756\pi$
$587$	−5132.74	−0.360904	−0.180452	+	0.983584i	$0.557756\pi$
$588$	0	0
$589$	−8996.74	−0.629379
$590$	−11009.2	−0.768203
$591$	0	0
$592$	22037.6	1.52997
$593$	−20882.4	−1.44610	−0.723049	−	0.690796i	$-0.757260\pi$
$593$	−20882.4	−1.44610	−0.723049	+	0.690796i	$0.757260\pi$
$594$	0	0
$595$	0	0
$596$	−80.6407	−0.00554224
$597$	0	0
$598$	8753.80	0.598611
$599$	−18584.0	−1.26765	−0.633823	−	0.773478i	$-0.718515\pi$
$599$	−18584.0	−1.26765	−0.633823	+	0.773478i	$0.718515\pi$
$600$	0	0
$601$	6118.34	0.415262	0.207631	−	0.978207i	$-0.433425\pi$
$601$	6118.34	0.415262	0.207631	+	0.978207i	$0.433425\pi$
$602$	0	0
$603$	0	0
$604$	−42.9368	−0.00289251
$605$	−4469.67	−0.300360
$606$	0	0
$607$	16078.1	1.07511	0.537555	−	0.843229i	$-0.319348\pi$
$607$	16078.1	1.07511	0.537555	+	0.843229i	$0.319348\pi$
$608$	−55.9701	−0.00373337
$609$	0	0
$610$	−24671.0	−1.63754
$611$	13590.1	0.899832
$612$	0	0
$613$	−11377.2	−0.749624	−0.374812	−	0.927101i	$-0.622293\pi$
$613$	−11377.2	−0.749624	−0.374812	+	0.927101i	$0.622293\pi$
$614$	−6040.72	−0.397042
$615$	0	0
$616$	0	0
$617$	8554.73	0.558185	0.279093	−	0.960264i	$-0.409966\pi$
$617$	8554.73	0.558185	0.279093	+	0.960264i	$0.409966\pi$
$618$	0	0
$619$	−10517.5	−0.682928	−0.341464	−	0.939895i	$-0.610923\pi$
$619$	−10517.5	−0.682928	−0.341464	+	0.939895i	$0.610923\pi$
$620$	130.006	0.00842126
$621$	0	0
$622$	−437.083	−0.0281759
$623$	0	0
$624$	0	0
$625$	−17881.5	−1.14442
$626$	5716.99	0.365011
$627$	0	0
$628$	−83.9709	−0.00533568
$629$	14202.8	0.900320
$630$	0	0
$631$	−1077.85	−0.0680010	−0.0340005	−	0.999422i	$-0.510825\pi$
$631$	−1077.85	−0.0680010	−0.0340005	+	0.999422i	$0.510825\pi$
$632$	−16066.5	−1.01122
$633$	0	0
$634$	−2365.98	−0.148210
$635$	−30102.7	−1.88125
$636$	0	0
$637$	0	0
$638$	−1123.68	−0.0697289
$639$	0	0
$640$	−21730.1	−1.34212
$641$	−8552.43	−0.526990	−0.263495	−	0.964661i	$-0.584875\pi$
$641$	−8552.43	−0.526990	−0.263495	+	0.964661i	$0.584875\pi$
$642$	0	0
$643$	11371.5	0.697429	0.348714	−	0.937229i	$-0.386618\pi$
$643$	11371.5	0.697429	0.348714	+	0.937229i	$0.386618\pi$
$644$	0	0
$645$	0	0
$646$	4167.39	0.253814
$647$	−9268.32	−0.563176	−0.281588	−	0.959535i	$-0.590861\pi$
$647$	−9268.32	−0.563176	−0.281588	+	0.959535i	$0.590861\pi$
$648$	0	0
$649$	8309.02	0.502554
$650$	9760.36	0.588973
$651$	0	0
$652$	51.4699	0.00309159
$653$	−29751.5	−1.78295	−0.891476	−	0.453068i	$-0.850329\pi$
$653$	−29751.5	−1.78295	−0.891476	+	0.453068i	$0.850329\pi$
$654$	0	0
$655$	25205.1	1.50358
$656$	20276.9	1.20683
$657$	0	0
$658$	0	0
$659$	32496.7	1.92093	0.960465	−	0.278402i	$-0.0898048\pi$
$659$	32496.7	1.92093	0.960465	+	0.278402i	$0.0898048\pi$
$660$	0	0
$661$	21778.6	1.28153	0.640764	−	0.767738i	$-0.278618\pi$
$661$	21778.6	1.28153	0.640764	+	0.767738i	$0.278618\pi$
$662$	−29680.7	−1.74256
$663$	0	0
$664$	−21702.6	−1.26841
$665$	0	0
$666$	0	0
$667$	1144.49	0.0664388
$668$	6.31368	0.000365694 0
$669$	0	0
$670$	−30985.6	−1.78668
$671$	18620.2	1.07127
$672$	0	0
$673$	20220.5	1.15816	0.579081	−	0.815270i	$-0.303412\pi$
$673$	20220.5	1.15816	0.579081	+	0.815270i	$0.303412\pi$
$674$	16344.7	0.934089
$675$	0	0
$676$	36.8846	0.00209858
$677$	−9696.26	−0.550454	−0.275227	−	0.961379i	$-0.588753\pi$
$677$	−9696.26	−0.550454	−0.275227	+	0.961379i	$0.588753\pi$
$678$	0	0
$679$	0	0
$680$	−14065.3	−0.793206
$681$	0	0
$682$	22721.2	1.27572
$683$	26698.2	1.49572	0.747860	−	0.663856i	$-0.231081\pi$
$683$	26698.2	1.49572	0.747860	+	0.663856i	$0.231081\pi$
$684$	0	0
$685$	−285.170	−0.0159063
$686$	0	0
$687$	0	0
$688$	23191.9	1.28515
$689$	8675.30	0.479685
$690$	0	0
$691$	5236.43	0.288282	0.144141	−	0.989557i	$-0.453958\pi$
$691$	5236.43	0.288282	0.144141	+	0.989557i	$0.453958\pi$
$692$	130.983	0.00719544
$693$	0	0
$694$	12440.4	0.680446
$695$	−10007.6	−0.546201
$696$	0	0
$697$	13068.0	0.710168
$698$	32603.5	1.76800
$699$	0	0
$700$	0	0
$701$	−4272.04	−0.230175	−0.115088	−	0.993355i	$-0.536715\pi$
$701$	−4272.04	−0.230175	−0.115088	+	0.993355i	$0.536715\pi$
$702$	0	0
$703$	−12434.0	−0.667081
$704$	16542.5	0.885609
$705$	0	0
$706$	−23437.4	−1.24940
$707$	0	0
$708$	0	0
$709$	29484.1	1.56178	0.780888	−	0.624671i	$-0.214767\pi$
$709$	29484.1	1.56178	0.780888	+	0.624671i	$0.214767\pi$
$710$	45810.6	2.42147
$711$	0	0
$712$	11008.0	0.579413
$713$	−23141.9	−1.21553
$714$	0	0
$715$	−16296.3	−0.852375
$716$	−50.4730	−0.00263445
$717$	0	0
$718$	6177.00	0.321063
$719$	9780.80	0.507319	0.253659	−	0.967294i	$-0.418366\pi$
$719$	9780.80	0.507319	0.253659	+	0.967294i	$0.418366\pi$
$720$	0	0
$721$	0	0
$722$	15710.0	0.809787
$723$	0	0
$724$	−38.7695	−0.00199013
$725$	1276.09	0.0653692
$726$	0	0
$727$	−16135.6	−0.823157	−0.411578	−	0.911374i	$-0.635022\pi$
$727$	−16135.6	−0.823157	−0.411578	+	0.911374i	$0.635022\pi$
$728$	0	0
$729$	0	0
$730$	−40177.1	−2.03701
$731$	14946.7	0.756255
$732$	0	0
$733$	−25327.5	−1.27625	−0.638126	−	0.769932i	$-0.720290\pi$
$733$	−25327.5	−1.27625	−0.638126	+	0.769932i	$0.720290\pi$
$734$	10913.9	0.548827
$735$	0	0
$736$	−143.969	−0.00721029
$737$	23386.0	1.16884
$738$	0	0
$739$	−3069.76	−0.152805	−0.0764024	−	0.997077i	$-0.524343\pi$
$739$	−3069.76	−0.152805	−0.0764024	+	0.997077i	$0.524343\pi$
$740$	179.676	0.00892572
$741$	0	0
$742$	0	0
$743$	−5480.22	−0.270592	−0.135296	−	0.990805i	$-0.543198\pi$
$743$	−5480.22	−0.270592	−0.135296	+	0.990805i	$0.543198\pi$
$744$	0	0
$745$	35406.7	1.74121
$746$	−31480.8	−1.54503
$747$	0	0
$748$	45.4506	0.00222171
$749$	0	0
$750$	0	0
$751$	19052.2	0.925730	0.462865	−	0.886429i	$-0.346821\pi$
$751$	19052.2	0.925730	0.462865	+	0.886429i	$0.346821\pi$
$752$	25822.4	1.25219
$753$	0	0
$754$	1171.35	0.0565758
$755$	18852.2	0.908742
$756$	0	0
$757$	9417.77	0.452173	0.226086	−	0.974107i	$-0.427407\pi$
$757$	9417.77	0.452173	0.226086	+	0.974107i	$0.427407\pi$
$758$	−21879.3	−1.04840
$759$	0	0
$760$	12313.7	0.587716
$761$	−11853.8	−0.564651	−0.282326	−	0.959319i	$-0.591106\pi$
$761$	−11853.8	−0.564651	−0.282326	+	0.959319i	$0.591106\pi$
$762$	0	0
$763$	0	0
$764$	100.808	0.00477367
$765$	0	0
$766$	10136.4	0.478126
$767$	−8661.51	−0.407756
$768$	0	0
$769$	−4643.71	−0.217759	−0.108879	−	0.994055i	$-0.534726\pi$
$769$	−4643.71	−0.217759	−0.108879	+	0.994055i	$0.534726\pi$
$770$	0	0
$771$	0	0
$772$	77.4146	0.00360908
$773$	−16703.5	−0.777211	−0.388606	−	0.921404i	$-0.627043\pi$
$773$	−16703.5	−0.777211	−0.388606	+	0.921404i	$0.627043\pi$
$774$	0	0
$775$	−25802.8	−1.19596
$776$	−7120.15	−0.329379
$777$	0	0
$778$	28881.9	1.33093
$779$	−11440.6	−0.526190
$780$	0	0
$781$	−34575.0	−1.58411
$782$	10719.6	0.490193
$783$	0	0
$784$	0	0
$785$	36868.9	1.67631
$786$	0	0
$787$	21278.9	0.963799	0.481899	−	0.876227i	$-0.339947\pi$
$787$	21278.9	0.963799	0.481899	+	0.876227i	$0.339947\pi$
$788$	115.213	0.00520851
$789$	0	0
$790$	30202.8	1.36021
$791$	0	0
$792$	0	0
$793$	−19410.1	−0.869195
$794$	−38846.1	−1.73627
$795$	0	0
$796$	177.142	0.00788772
$797$	−15624.4	−0.694410	−0.347205	−	0.937789i	$-0.612869\pi$
$797$	−15624.4	−0.694410	−0.347205	+	0.937789i	$0.612869\pi$
$798$	0	0
$799$	16641.9	0.736858
$800$	−160.523	−0.00709420
$801$	0	0
$802$	−3830.61	−0.168658
$803$	30323.1	1.33260
$804$	0	0
$805$	0	0
$806$	−23685.1	−1.03508
$807$	0	0
$808$	−19381.0	−0.843836
$809$	−11337.0	−0.492692	−0.246346	−	0.969182i	$-0.579230\pi$
$809$	−11337.0	−0.492692	−0.246346	+	0.969182i	$0.579230\pi$
$810$	0	0
$811$	29136.4	1.26155	0.630776	−	0.775965i	$-0.282737\pi$
$811$	29136.4	1.26155	0.630776	+	0.775965i	$0.282737\pi$
$812$	0	0
$813$	0	0
$814$	31402.0	1.35214
$815$	−22598.7	−0.971288
$816$	0	0
$817$	−13085.3	−0.560338
$818$	−17256.1	−0.737585
$819$	0	0
$820$	165.321	0.00704056
$821$	40165.6	1.70742	0.853709	−	0.520750i	$-0.174348\pi$
$821$	40165.6	1.70742	0.853709	+	0.520750i	$0.174348\pi$
$822$	0	0
$823$	24736.2	1.04769	0.523846	−	0.851813i	$-0.324497\pi$
$823$	24736.2	1.04769	0.523846	+	0.851813i	$0.324497\pi$
$824$	−19184.1	−0.811054
$825$	0	0
$826$	0	0
$827$	25802.8	1.08495	0.542473	−	0.840073i	$-0.317488\pi$
$827$	25802.8	1.08495	0.542473	+	0.840073i	$0.317488\pi$
$828$	0	0
$829$	19715.1	0.825976	0.412988	−	0.910736i	$-0.364485\pi$
$829$	19715.1	0.825976	0.412988	+	0.910736i	$0.364485\pi$
$830$	40797.7	1.70616
$831$	0	0
$832$	−17244.3	−0.718555
$833$	0	0
$834$	0	0
$835$	−2772.13	−0.114890
$836$	−39.7904	−0.00164615
$837$	0	0
$838$	12382.4	0.510432
$839$	−27462.9	−1.13007	−0.565033	−	0.825068i	$-0.691137\pi$
$839$	−27462.9	−1.13007	−0.565033	+	0.825068i	$0.691137\pi$
$840$	0	0
$841$	−24235.9	−0.993721
$842$	−17083.1	−0.699195
$843$	0	0
$844$	−168.813	−0.00688480
$845$	−16194.8	−0.659311
$846$	0	0
$847$	0	0
$848$	16483.8	0.667519
$849$	0	0
$850$	11952.2	0.482301
$851$	−31983.4	−1.28834
$852$	0	0
$853$	−26182.1	−1.05095	−0.525474	−	0.850810i	$-0.676112\pi$
$853$	−26182.1	−1.05095	−0.525474	+	0.850810i	$0.676112\pi$
$854$	0	0
$855$	0	0
$856$	33855.8	1.35183
$857$	−28940.2	−1.15353	−0.576767	−	0.816909i	$-0.695686\pi$
$857$	−28940.2	−1.15353	−0.576767	+	0.816909i	$0.695686\pi$
$858$	0	0
$859$	−20356.6	−0.808567	−0.404284	−	0.914634i	$-0.632479\pi$
$859$	−20356.6	−0.808567	−0.404284	+	0.914634i	$0.632479\pi$
$860$	189.087	0.00749747
$861$	0	0
$862$	15778.7	0.623461
$863$	25157.3	0.992313	0.496156	−	0.868233i	$-0.334744\pi$
$863$	25157.3	0.992313	0.496156	+	0.868233i	$0.334744\pi$
$864$	0	0
$865$	−57510.5	−2.26060
$866$	45794.8	1.79696
$867$	0	0
$868$	0	0
$869$	−22795.2	−0.889844
$870$	0	0
$871$	−24378.1	−0.948359
$872$	41390.3	1.60740
$873$	0	0
$874$	−9384.60	−0.363202
$875$	0	0
$876$	0	0
$877$	20273.6	0.780606	0.390303	−	0.920686i	$-0.372370\pi$
$877$	20273.6	0.780606	0.390303	+	0.920686i	$0.372370\pi$
$878$	−1233.41	−0.0474097
$879$	0	0
$880$	−30964.4	−1.18615
$881$	14440.1	0.552213	0.276106	−	0.961127i	$-0.410956\pi$
$881$	14440.1	0.552213	0.276106	+	0.961127i	$0.410956\pi$
$882$	0	0
$883$	17201.5	0.655581	0.327790	−	0.944751i	$-0.393696\pi$
$883$	17201.5	0.655581	0.327790	+	0.944751i	$0.393696\pi$
$884$	−47.3787	−0.00180262
$885$	0	0
$886$	−20233.4	−0.767217
$887$	−37287.0	−1.41147	−0.705736	−	0.708475i	$-0.749384\pi$
$887$	−37287.0	−1.41147	−0.705736	+	0.708475i	$0.749384\pi$
$888$	0	0
$889$	0	0
$890$	−20693.5	−0.779378
$891$	0	0
$892$	−20.3795	−0.000764974 0
$893$	−14569.4	−0.545966
$894$	0	0
$895$	22161.0	0.827667
$896$	0	0
$897$	0	0
$898$	3246.20	0.120632
$899$	−3096.63	−0.114881
$900$	0	0
$901$	10623.4	0.392806
$902$	28893.2	1.06656
$903$	0	0
$904$	12358.6	0.454691
$905$	17022.4	0.625242
$906$	0	0
$907$	−33447.8	−1.22450	−0.612248	−	0.790666i	$-0.709734\pi$
$907$	−33447.8	−1.22450	−0.612248	+	0.790666i	$0.709734\pi$
$908$	99.1914	0.00362531
$909$	0	0
$910$	0	0
$911$	−14250.2	−0.518254	−0.259127	−	0.965843i	$-0.583435\pi$
$911$	−14250.2	−0.518254	−0.259127	+	0.965843i	$0.583435\pi$
$912$	0	0
$913$	−30791.6	−1.11616
$914$	24804.3	0.897650
$915$	0	0
$916$	177.163	0.00639042
$917$	0	0
$918$	0	0
$919$	−1515.14	−0.0543850	−0.0271925	−	0.999630i	$-0.508657\pi$
$919$	−1515.14	−0.0543850	−0.0271925	+	0.999630i	$0.508657\pi$
$920$	31673.8	1.13506
$921$	0	0
$922$	34132.8	1.21920
$923$	36041.7	1.28530
$924$	0	0
$925$	−35661.0	−1.26760
$926$	−50369.3	−1.78751
$927$	0	0
$928$	−19.2646	−0.000681457 0
$929$	−33779.4	−1.19297	−0.596483	−	0.802626i	$-0.703436\pi$
$929$	−33779.4	−1.19297	−0.596483	+	0.802626i	$0.703436\pi$
$930$	0	0
$931$	0	0
$932$	−130.709	−0.00459390
$933$	0	0
$934$	−20986.5	−0.735223
$935$	−19955.9	−0.697996
$936$	0	0
$937$	25335.2	0.883312	0.441656	−	0.897184i	$-0.354391\pi$
$937$	25335.2	0.883312	0.441656	+	0.897184i	$0.354391\pi$
$938$	0	0
$939$	0	0
$940$	210.534	0.00730517
$941$	27536.8	0.953956	0.476978	−	0.878915i	$-0.341732\pi$
$941$	27536.8	0.953956	0.476978	+	0.878915i	$0.341732\pi$
$942$	0	0
$943$	−29428.1	−1.01623
$944$	−16457.6	−0.567425
$945$	0	0
$946$	33046.8	1.13578
$947$	29425.4	1.00971	0.504856	−	0.863203i	$-0.331545\pi$
$947$	29425.4	1.00971	0.504856	+	0.863203i	$0.331545\pi$
$948$	0	0
$949$	−31609.5	−1.08123
$950$	−10463.7	−0.357355
$951$	0	0
$952$	0	0
$953$	10346.9	0.351700	0.175850	−	0.984417i	$-0.443733\pi$
$953$	10346.9	0.351700	0.175850	+	0.984417i	$0.443733\pi$
$954$	0	0
$955$	−44261.3	−1.49975
$956$	−143.088	−0.00484081
$957$	0	0
$958$	−30002.4	−1.01183
$959$	0	0
$960$	0	0
$961$	32823.8	1.10180
$962$	−32734.2	−1.09708
$963$	0	0
$964$	−42.4966	−0.00141984
$965$	−33990.2	−1.13387
$966$	0	0
$967$	−17872.3	−0.594348	−0.297174	−	0.954823i	$-0.596044\pi$
$967$	−17872.3	−0.594348	−0.297174	+	0.954823i	$0.596044\pi$
$968$	−6710.57	−0.222816
$969$	0	0
$970$	13384.9	0.443054
$971$	−5739.41	−0.189687	−0.0948437	−	0.995492i	$-0.530235\pi$
$971$	−5739.41	−0.189687	−0.0948437	+	0.995492i	$0.530235\pi$
$972$	0	0
$973$	0	0
$974$	−38437.5	−1.26449
$975$	0	0
$976$	−36880.8	−1.20955
$977$	39025.6	1.27793	0.638965	−	0.769236i	$-0.279363\pi$
$977$	39025.6	1.27793	0.638965	+	0.769236i	$0.279363\pi$
$978$	0	0
$979$	15618.1	0.509865
$980$	0	0
$981$	0	0
$982$	−49268.8	−1.60105
$983$	−8168.04	−0.265025	−0.132513	−	0.991181i	$-0.542305\pi$
$983$	−8168.04	−0.265025	−0.132513	+	0.991181i	$0.542305\pi$
$984$	0	0
$985$	−50586.3	−1.63636
$986$	1434.39	0.0463290
$987$	0	0
$988$	41.4784	0.00133563
$989$	−33658.6	−1.08218
$990$	0	0
$991$	9879.16	0.316672	0.158336	−	0.987385i	$-0.449387\pi$
$991$	9879.16	0.316672	0.158336	+	0.987385i	$0.449387\pi$
$992$	389.537	0.0124675
$993$	0	0
$994$	0	0
$995$	−77777.2	−2.47809
$996$	0	0
$997$	−23600.4	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$997$	−23600.4	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$998$	−11569.0	−0.366945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bp.1.4	✓	12
3.2	odd	2		1323.4.a.bq.1.9	yes	12
7.6	odd	2		1323.4.a.bq.1.4	yes	12
21.20	even	2	inner	1323.4.a.bp.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bp.1.4	✓	12	1.1	even	1	trivial
1323.4.a.bp.1.9	yes	12	21.20	even	2	inner
1323.4.a.bq.1.4	yes	12	7.6	odd	2
1323.4.a.bq.1.9	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.82234 q^{2} -0.0343991 q^{4} +15.1035 q^{5} +22.6758 q^{8} -42.6273 q^{10} +32.1724 q^{11} -33.5373 q^{13} -63.7236 q^{16} -41.0684 q^{17} +35.9540 q^{19} -0.519549 q^{20} -90.8015 q^{22} +92.4825 q^{23} +103.117 q^{25} +94.6535 q^{26} +12.3752 q^{29} -250.230 q^{31} -1.55672 q^{32} +115.909 q^{34} -345.832 q^{37} -101.474 q^{38} +342.485 q^{40} -318.201 q^{41} -363.945 q^{43} -1.10670 q^{44} -261.017 q^{46} -405.225 q^{47} -291.030 q^{50} +1.15365 q^{52} -258.677 q^{53} +485.917 q^{55} -34.9269 q^{58} +258.265 q^{59} +578.762 q^{61} +706.233 q^{62} +514.183 q^{64} -506.531 q^{65} +726.896 q^{67} +1.41272 q^{68} -1074.68 q^{71} +942.519 q^{73} +976.054 q^{74} -1.23679 q^{76} -708.532 q^{79} -962.452 q^{80} +898.072 q^{82} -957.080 q^{83} -620.278 q^{85} +1027.18 q^{86} +729.535 q^{88} +485.451 q^{89} -3.18132 q^{92} +1143.68 q^{94} +543.032 q^{95} -313.998 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{4} - 40 q^{5} + 444 q^{16} - 292 q^{17} - 524 q^{20} - 600 q^{22} + 324 q^{25} - 144 q^{26} + 216 q^{37} - 808 q^{38} - 1768 q^{41} + 180 q^{43} - 984 q^{47} + 1164 q^{58} - 1324 q^{59} - 3640 q^{62}+ \cdots - 1036 q^{89}+O(q^{100}) \) 12 * q + 48 * q^4 - 40 * q^5 + 444 * q^16 - 292 * q^17 - 524 * q^20 - 600 * q^22 + 324 * q^25 - 144 * q^26 + 216 * q^37 - 808 * q^38 - 1768 * q^41 + 180 * q^43 - 984 * q^47 + 1164 * q^58 - 1324 * q^59 - 3640 * q^62 + 3252 * q^64 + 972 * q^67 - 4528 * q^68 - 1752 * q^79 - 10536 * q^80 - 2872 * q^83 + 2664 * q^85 - 8160 * q^88 - 1036 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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