LMFDB - Embedded newform 1323.4.a.be.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1323,4,Mod(1,1323)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,32,0,0,0,0,0,20,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$1$
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} - 40x^{4} + 453x^{2} - 1278 $ x^6 - 40x^4 + 453x^2 - 1278
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.70753$ 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)

$f(q)$	$=$	$q+4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} +29.0024 q^{8} -3.90938 q^{10} -68.2503 q^{11} +35.7721 q^{13} +23.2429 q^{16} -14.6779 q^{17} -86.7693 q^{19} -11.7599 q^{20} -321.290 q^{22} +58.4087 q^{23} -124.310 q^{25} +168.398 q^{26} -67.1398 q^{29} +69.5002 q^{31} -122.603 q^{32} -69.0965 q^{34} -289.240 q^{37} -408.469 q^{38} -24.0851 q^{40} +357.750 q^{41} -235.801 q^{43} -966.482 q^{44} +274.961 q^{46} -450.067 q^{47} -585.195 q^{50} +506.563 q^{52} +172.464 q^{53} +56.6787 q^{55} -316.063 q^{58} +445.856 q^{59} -476.314 q^{61} +327.174 q^{62} -763.099 q^{64} -29.7070 q^{65} -804.320 q^{67} -207.851 q^{68} +353.412 q^{71} +778.045 q^{73} -1361.61 q^{74} -1228.73 q^{76} -74.1996 q^{79} -19.3021 q^{80} +1684.12 q^{82} +1168.49 q^{83} +12.1893 q^{85} -1110.04 q^{86} -1979.42 q^{88} -1514.72 q^{89} +827.116 q^{92} -2118.70 q^{94} +72.0579 q^{95} -1631.46 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58}+ \cdots + 682 q^{97}+O(q^{100}) $ 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * q^97

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 $

$2$	4.70753	1.66436	0.832182	−	0.554503i	$-0.187092\pi$
$2$	4.70753	1.66436	0.832182	+	0.554503i	$0.187092\pi$
$3$	0	0
$3$	0	0
$4$	14.1608	1.77011
$5$	−0.830453	−0.0742780	−0.0371390	−	0.999310i	$-0.511824\pi$
$5$	−0.830453	−0.0742780	−0.0371390	+	0.999310i	$0.511824\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	29.0024	1.28174
$9$	0	0
$10$	−3.90938	−0.123626
$11$	−68.2503	−1.87075	−0.935374	−	0.353659i	$-0.884937\pi$
$11$	−68.2503	−1.87075	−0.935374	+	0.353659i	$0.884937\pi$
$12$	0	0
$13$	35.7721	0.763184	0.381592	−	0.924331i	$-0.375376\pi$
$13$	35.7721	0.763184	0.381592	+	0.924331i	$0.375376\pi$
$14$	0	0
$15$	0	0
$16$	23.2429	0.363170
$17$	−14.6779	−0.209406	−0.104703	−	0.994504i	$-0.533389\pi$
$17$	−14.6779	−0.209406	−0.104703	+	0.994504i	$0.533389\pi$
$18$	0	0
$19$	−86.7693	−1.04770	−0.523849	−	0.851811i	$-0.675504\pi$
$19$	−86.7693	−1.04770	−0.523849	+	0.851811i	$0.675504\pi$
$20$	−11.7599	−0.131480
$21$	0	0
$22$	−321.290	−3.11361
$23$	58.4087	0.529524	0.264762	−	0.964314i	$-0.414707\pi$
$23$	58.4087	0.529524	0.264762	+	0.964314i	$0.414707\pi$
$24$	0	0
$25$	−124.310	−0.994483
$26$	168.398	1.27022
$27$	0	0
$28$	0	0
$29$	−67.1398	−0.429915	−0.214958	−	0.976623i	$-0.568961\pi$
$29$	−67.1398	−0.429915	−0.214958	+	0.976623i	$0.568961\pi$
$30$	0	0
$31$	69.5002	0.402665	0.201332	−	0.979523i	$-0.435473\pi$
$31$	69.5002	0.402665	0.201332	+	0.979523i	$0.435473\pi$
$32$	−122.603	−0.677290
$33$	0	0
$34$	−69.0965	−0.348528
$35$	0	0
$36$	0	0
$37$	−289.240	−1.28516	−0.642578	−	0.766220i	$-0.722135\pi$
$37$	−289.240	−1.28516	−0.642578	+	0.766220i	$0.722135\pi$
$38$	−408.469	−1.74375
$39$	0	0
$40$	−24.0851	−0.0952048
$41$	357.750	1.36271	0.681355	−	0.731953i	$-0.261391\pi$
$41$	357.750	1.36271	0.681355	+	0.731953i	$0.261391\pi$
$42$	0	0
$43$	−235.801	−0.836263	−0.418132	−	0.908386i	$-0.637315\pi$
$43$	−235.801	−0.836263	−0.418132	+	0.908386i	$0.637315\pi$
$44$	−966.482	−3.31142
$45$	0	0
$46$	274.961	0.881320
$47$	−450.067	−1.39679	−0.698394	−	0.715714i	$-0.746102\pi$
$47$	−450.067	−1.39679	−0.698394	+	0.715714i	$0.746102\pi$
$48$	0	0
$49$	0	0
$50$	−585.195	−1.65518
$51$	0	0
$52$	506.563	1.35092
$53$	172.464	0.446977	0.223489	−	0.974707i	$-0.428255\pi$
$53$	172.464	0.446977	0.223489	+	0.974707i	$0.428255\pi$
$54$	0	0
$55$	56.6787	0.138955
$56$	0	0
$57$	0	0
$58$	−316.063	−0.715535
$59$	445.856	0.983823	0.491912	−	0.870645i	$-0.336298\pi$
$59$	445.856	0.983823	0.491912	+	0.870645i	$0.336298\pi$
$60$	0	0
$61$	−476.314	−0.999767	−0.499883	−	0.866093i	$-0.666624\pi$
$61$	−476.314	−0.999767	−0.499883	+	0.866093i	$0.666624\pi$
$62$	327.174	0.670180
$63$	0	0
$64$	−763.099	−1.49043
$65$	−29.7070	−0.0566878
$66$	0	0
$67$	−804.320	−1.46662	−0.733309	−	0.679896i	$-0.762025\pi$
$67$	−804.320	−1.46662	−0.733309	+	0.679896i	$0.762025\pi$
$68$	−207.851	−0.370671
$69$	0	0
$70$	0	0
$71$	353.412	0.590737	0.295368	−	0.955383i	$-0.404558\pi$
$71$	353.412	0.590737	0.295368	+	0.955383i	$0.404558\pi$
$72$	0	0
$73$	778.045	1.24744	0.623721	−	0.781647i	$-0.285620\pi$
$73$	778.045	1.24744	0.623721	+	0.781647i	$0.285620\pi$
$74$	−1361.61	−2.13897
$75$	0	0
$76$	−1228.73	−1.85454
$77$	0	0
$78$	0	0
$79$	−74.1996	−0.105672	−0.0528361	−	0.998603i	$-0.516826\pi$
$79$	−74.1996	−0.105672	−0.0528361	+	0.998603i	$0.516826\pi$
$80$	−19.3021	−0.0269755
$81$	0	0
$82$	1684.12	2.26804
$83$	1168.49	1.54529	0.772643	−	0.634840i	$-0.218934\pi$
$83$	1168.49	1.54529	0.772643	+	0.634840i	$0.218934\pi$
$84$	0	0
$85$	12.1893	0.0155543
$86$	−1110.04	−1.39185
$87$	0	0
$88$	−1979.42	−2.39781
$89$	−1514.72	−1.80405	−0.902023	−	0.431687i	$-0.857918\pi$
$89$	−1514.72	−1.80405	−0.902023	+	0.431687i	$0.857918\pi$
$90$	0	0
$91$	0	0
$92$	827.116	0.937313
$93$	0	0
$94$	−2118.70	−2.32476
$95$	72.0579	0.0778209
$96$	0	0
$97$	−1631.46	−1.70772	−0.853862	−	0.520499i	$-0.825746\pi$
$97$	−1631.46	−1.70772	−0.853862	+	0.520499i	$0.825746\pi$
$98$	0	0
$99$	0	0
$100$	−1760.34	−1.76034
$101$	−1205.73	−1.18787	−0.593934	−	0.804514i	$-0.702426\pi$
$101$	−1205.73	−1.18787	−0.593934	+	0.804514i	$0.702426\pi$
$102$	0	0
$103$	485.363	0.464313	0.232156	−	0.972679i	$-0.425422\pi$
$103$	485.363	0.464313	0.232156	+	0.972679i	$0.425422\pi$
$104$	1037.48	0.978201
$105$	0	0
$106$	811.881	0.743933
$107$	−258.101	−0.233192	−0.116596	−	0.993179i	$-0.537198\pi$
$107$	−258.101	−0.233192	−0.116596	+	0.993179i	$0.537198\pi$
$108$	0	0
$109$	581.246	0.510764	0.255382	−	0.966840i	$-0.417799\pi$
$109$	581.246	0.510764	0.255382	+	0.966840i	$0.417799\pi$
$110$	266.817	0.231272
$111$	0	0
$112$	0	0
$113$	708.244	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$113$	708.244	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$114$	0	0
$115$	−48.5057	−0.0393320
$116$	−950.756	−0.760996
$117$	0	0
$118$	2098.88	1.63744
$119$	0	0
$120$	0	0
$121$	3327.10	2.49970
$122$	−2242.26	−1.66398
$123$	0	0
$124$	984.182	0.712759
$125$	207.041	0.148146
$126$	0	0
$127$	2348.04	1.64059	0.820296	−	0.571939i	$-0.193809\pi$
$127$	2348.04	1.64059	0.820296	+	0.571939i	$0.193809\pi$
$128$	−2611.49	−1.80332
$129$	0	0
$130$	−139.847	−0.0943491
$131$	2405.82	1.60456	0.802279	−	0.596949i	$-0.203621\pi$
$131$	2405.82	1.60456	0.802279	+	0.596949i	$0.203621\pi$
$132$	0	0
$133$	0	0
$134$	−3786.36	−2.44098
$135$	0	0
$136$	−425.693	−0.268403
$137$	−1187.55	−0.740579	−0.370289	−	0.928916i	$-0.620742\pi$
$137$	−1187.55	−0.740579	−0.370289	+	0.928916i	$0.620742\pi$
$138$	0	0
$139$	712.439	0.434736	0.217368	−	0.976090i	$-0.430253\pi$
$139$	712.439	0.434736	0.217368	+	0.976090i	$0.430253\pi$
$140$	0	0
$141$	0	0
$142$	1663.70	0.983201
$143$	−2441.46	−1.42773
$144$	0	0
$145$	55.7564	0.0319332
$146$	3662.67	2.07620
$147$	0	0
$148$	−4095.89	−2.27486
$149$	−2209.24	−1.21468	−0.607342	−	0.794441i	$-0.707764\pi$
$149$	−2209.24	−1.21468	−0.607342	+	0.794441i	$0.707764\pi$
$150$	0	0
$151$	−546.983	−0.294787	−0.147394	−	0.989078i	$-0.547088\pi$
$151$	−546.983	−0.294787	−0.147394	+	0.989078i	$0.547088\pi$
$152$	−2516.52	−1.34287
$153$	0	0
$154$	0	0
$155$	−57.7166	−0.0299091
$156$	0	0
$157$	−1870.19	−0.950686	−0.475343	−	0.879801i	$-0.657676\pi$
$157$	−1870.19	−0.950686	−0.475343	+	0.879801i	$0.657676\pi$
$158$	−349.297	−0.175877
$159$	0	0
$160$	101.816	0.0503078
$161$	0	0
$162$	0	0
$163$	−3037.40	−1.45955	−0.729777	−	0.683685i	$-0.760376\pi$
$163$	−3037.40	−1.45955	−0.729777	+	0.683685i	$0.760376\pi$
$164$	5066.04	2.41214
$165$	0	0
$166$	5500.72	2.57192
$167$	−2967.09	−1.37485	−0.687426	−	0.726255i	$-0.741259\pi$
$167$	−2967.09	−1.37485	−0.687426	+	0.726255i	$0.741259\pi$
$168$	0	0
$169$	−917.358	−0.417550
$170$	57.3814	0.0258879
$171$	0	0
$172$	−3339.14	−1.48028
$173$	3866.81	1.69935	0.849677	−	0.527303i	$-0.176797\pi$
$173$	3866.81	1.69935	0.849677	+	0.527303i	$0.176797\pi$
$174$	0	0
$175$	0	0
$176$	−1586.33	−0.679399
$177$	0	0
$178$	−7130.60	−3.00259
$179$	334.726	0.139769	0.0698844	−	0.997555i	$-0.477737\pi$
$179$	334.726	0.139769	0.0698844	+	0.997555i	$0.477737\pi$
$180$	0	0
$181$	1564.56	0.642502	0.321251	−	0.946994i	$-0.395897\pi$
$181$	1564.56	0.642502	0.321251	+	0.946994i	$0.395897\pi$
$182$	0	0
$183$	0	0
$184$	1693.99	0.678710
$185$	240.200	0.0954588
$186$	0	0
$187$	1001.77	0.391746
$188$	−6373.33	−2.47246
$189$	0	0
$190$	339.215	0.129522
$191$	3067.14	1.16194	0.580971	−	0.813925i	$-0.302673\pi$
$191$	3067.14	1.16194	0.580971	+	0.813925i	$0.302673\pi$
$192$	0	0
$193$	−2223.60	−0.829316	−0.414658	−	0.909977i	$-0.636099\pi$
$193$	−2223.60	−0.829316	−0.414658	+	0.909977i	$0.636099\pi$
$194$	−7680.13	−2.84227
$195$	0	0
$196$	0	0
$197$	1856.18	0.671306	0.335653	−	0.941986i	$-0.391043\pi$
$197$	1856.18	0.671306	0.335653	+	0.941986i	$0.391043\pi$
$198$	0	0
$199$	993.602	0.353943	0.176971	−	0.984216i	$-0.443370\pi$
$199$	993.602	0.353943	0.176971	+	0.984216i	$0.443370\pi$
$200$	−3605.30	−1.27467
$201$	0	0
$202$	−5676.02	−1.97704
$203$	0	0
$204$	0	0
$205$	−297.094	−0.101219
$206$	2284.86	0.772785
$207$	0	0
$208$	831.446	0.277165
$209$	5922.03	1.95998
$210$	0	0
$211$	−545.079	−0.177843	−0.0889213	−	0.996039i	$-0.528342\pi$
$211$	−545.079	−0.177843	−0.0889213	+	0.996039i	$0.528342\pi$
$212$	2442.24	0.791197
$213$	0	0
$214$	−1215.02	−0.388116
$215$	195.822	0.0621160
$216$	0	0
$217$	0	0
$218$	2736.23	0.850097
$219$	0	0
$220$	802.618	0.245966
$221$	−525.058	−0.159815
$222$	0	0
$223$	3446.72	1.03502	0.517510	−	0.855677i	$-0.326859\pi$
$223$	3446.72	1.03502	0.517510	+	0.855677i	$0.326859\pi$
$224$	0	0
$225$	0	0
$226$	3334.08	0.981326
$227$	486.297	0.142188	0.0710940	−	0.997470i	$-0.477351\pi$
$227$	486.297	0.142188	0.0710940	+	0.997470i	$0.477351\pi$
$228$	0	0
$229$	−4214.68	−1.21622	−0.608109	−	0.793853i	$-0.708072\pi$
$229$	−4214.68	−1.21622	−0.608109	+	0.793853i	$0.708072\pi$
$230$	−228.342	−0.0654627
$231$	0	0
$232$	−1947.21	−0.551038
$233$	−1099.34	−0.309100	−0.154550	−	0.987985i	$-0.549393\pi$
$233$	−1099.34	−0.309100	−0.154550	+	0.987985i	$0.549393\pi$
$234$	0	0
$235$	373.760	0.103751
$236$	6313.71	1.74147
$237$	0	0
$238$	0	0
$239$	4558.99	1.23388	0.616939	−	0.787011i	$-0.288373\pi$
$239$	4558.99	1.23388	0.616939	+	0.787011i	$0.288373\pi$
$240$	0	0
$241$	1089.40	0.291181	0.145591	−	0.989345i	$-0.453492\pi$
$241$	1089.40	0.291181	0.145591	+	0.989345i	$0.453492\pi$
$242$	15662.4	4.16041
$243$	0	0
$244$	−6745.01	−1.76969
$245$	0	0
$246$	0	0
$247$	−3103.92	−0.799586
$248$	2015.67	0.516110
$249$	0	0
$250$	974.650	0.246569
$251$	−6356.75	−1.59855	−0.799273	−	0.600969i	$-0.794782\pi$
$251$	−6356.75	−1.59855	−0.799273	+	0.600969i	$0.794782\pi$
$252$	0	0
$253$	−3986.41	−0.990606
$254$	11053.5	2.73054
$255$	0	0
$256$	−6188.88	−1.51096
$257$	1063.43	0.258113	0.129056	−	0.991637i	$-0.458805\pi$
$257$	1063.43	0.258113	0.129056	+	0.991637i	$0.458805\pi$
$258$	0	0
$259$	0	0
$260$	−420.677	−0.100343
$261$	0	0
$262$	11325.5	2.67057
$263$	3674.59	0.861539	0.430770	−	0.902462i	$-0.358242\pi$
$263$	3674.59	0.861539	0.430770	+	0.902462i	$0.358242\pi$
$264$	0	0
$265$	−143.224	−0.0332006
$266$	0	0
$267$	0	0
$268$	−11389.9	−2.59607
$269$	−6959.94	−1.57753	−0.788764	−	0.614696i	$-0.789279\pi$
$269$	−6959.94	−1.57753	−0.788764	+	0.614696i	$0.789279\pi$
$270$	0	0
$271$	6595.78	1.47847	0.739235	−	0.673448i	$-0.235187\pi$
$271$	6595.78	1.47847	0.739235	+	0.673448i	$0.235187\pi$
$272$	−341.155	−0.0760499
$273$	0	0
$274$	−5590.43	−1.23259
$275$	8484.22	1.86043
$276$	0	0
$277$	−3121.06	−0.676990	−0.338495	−	0.940968i	$-0.609918\pi$
$277$	−3121.06	−0.676990	−0.338495	+	0.940968i	$0.609918\pi$
$278$	3353.83	0.723559
$279$	0	0
$280$	0	0
$281$	1662.38	0.352915	0.176457	−	0.984308i	$-0.443536\pi$
$281$	1662.38	0.352915	0.176457	+	0.984308i	$0.443536\pi$
$282$	0	0
$283$	−453.279	−0.0952107	−0.0476053	−	0.998866i	$-0.515159\pi$
$283$	−453.279	−0.0952107	−0.0476053	+	0.998866i	$0.515159\pi$
$284$	5004.62	1.04567
$285$	0	0
$286$	−11493.2	−2.37625
$287$	0	0
$288$	0	0
$289$	−4697.56	−0.956149
$290$	262.475	0.0531485
$291$	0	0
$292$	11017.8	2.20811
$293$	772.774	0.154082	0.0770408	−	0.997028i	$-0.475453\pi$
$293$	772.774	0.154082	0.0770408	+	0.997028i	$0.475453\pi$
$294$	0	0
$295$	−370.263	−0.0730764
$296$	−8388.65	−1.64723
$297$	0	0
$298$	−10400.1	−2.02168
$299$	2089.40	0.404124
$300$	0	0
$301$	0	0
$302$	−2574.94	−0.490633
$303$	0	0
$304$	−2016.77	−0.380492
$305$	395.557	0.0742607
$306$	0	0
$307$	−4193.64	−0.779621	−0.389810	−	0.920895i	$-0.627459\pi$
$307$	−4193.64	−0.779621	−0.389810	+	0.920895i	$0.627459\pi$
$308$	0	0
$309$	0	0
$310$	−271.703	−0.0497796
$311$	399.740	0.0728848	0.0364424	−	0.999336i	$-0.488397\pi$
$311$	399.740	0.0728848	0.0364424	+	0.999336i	$0.488397\pi$
$312$	0	0
$313$	−3730.48	−0.673672	−0.336836	−	0.941563i	$-0.609357\pi$
$313$	−3730.48	−0.673672	−0.336836	+	0.941563i	$0.609357\pi$
$314$	−8803.99	−1.58229
$315$	0	0
$316$	−1050.73	−0.187051
$317$	8373.47	1.48360	0.741800	−	0.670622i	$-0.233973\pi$
$317$	8373.47	1.48360	0.741800	+	0.670622i	$0.233973\pi$
$318$	0	0
$319$	4582.31	0.804263
$320$	633.718	0.110706
$321$	0	0
$322$	0	0
$323$	1273.59	0.219394
$324$	0	0
$325$	−4446.84	−0.758973
$326$	−14298.6	−2.42923
$327$	0	0
$328$	10375.6	1.74664
$329$	0	0
$330$	0	0
$331$	−1349.72	−0.224130	−0.112065	−	0.993701i	$-0.535747\pi$
$331$	−1349.72	−0.224130	−0.112065	+	0.993701i	$0.535747\pi$
$332$	16546.9	2.73532
$333$	0	0
$334$	−13967.7	−2.28825
$335$	667.950	0.108937
$336$	0	0
$337$	−6629.72	−1.07164	−0.535822	−	0.844331i	$-0.679998\pi$
$337$	−6629.72	−1.07164	−0.535822	+	0.844331i	$0.679998\pi$
$338$	−4318.49	−0.694955
$339$	0	0
$340$	172.610	0.0275327
$341$	−4743.41	−0.753284
$342$	0	0
$343$	0	0
$344$	−6838.79	−1.07187
$345$	0	0
$346$	18203.1	2.82834
$347$	4952.94	0.766246	0.383123	−	0.923697i	$-0.374848\pi$
$347$	4952.94	0.766246	0.383123	+	0.923697i	$0.374848\pi$
$348$	0	0
$349$	4498.12	0.689911	0.344955	−	0.938619i	$-0.387894\pi$
$349$	4498.12	0.689911	0.344955	+	0.938619i	$0.387894\pi$
$350$	0	0
$351$	0	0
$352$	8367.67	1.26704
$353$	1613.35	0.243258	0.121629	−	0.992576i	$-0.461188\pi$
$353$	1613.35	0.243258	0.121629	+	0.992576i	$0.461188\pi$
$354$	0	0
$355$	−293.492	−0.0438787
$356$	−21449.7	−3.19335
$357$	0	0
$358$	1575.73	0.232626
$359$	13186.9	1.93866	0.969328	−	0.245770i	$-0.0790409\pi$
$359$	13186.9	1.93866	0.969328	+	0.245770i	$0.0790409\pi$
$360$	0	0
$361$	669.919	0.0976700
$362$	7365.22	1.06936
$363$	0	0
$364$	0	0
$365$	−646.130	−0.0926575
$366$	0	0
$367$	−5103.39	−0.725871	−0.362936	−	0.931814i	$-0.618226\pi$
$367$	−5103.39	−0.725871	−0.362936	+	0.931814i	$0.618226\pi$
$368$	1357.58	0.192307
$369$	0	0
$370$	1130.75	0.158878
$371$	0	0
$372$	0	0
$373$	−1459.19	−0.202557	−0.101279	−	0.994858i	$-0.532293\pi$
$373$	−1459.19	−0.202557	−0.101279	+	0.994858i	$0.532293\pi$
$374$	4715.85	0.652008
$375$	0	0
$376$	−13053.0	−1.79031
$377$	−2401.73	−0.328104
$378$	0	0
$379$	2332.75	0.316162	0.158081	−	0.987426i	$-0.449469\pi$
$379$	2332.75	0.316162	0.158081	+	0.987426i	$0.449469\pi$
$380$	1020.40	0.137751
$381$	0	0
$382$	14438.7	1.93389
$383$	5651.11	0.753938	0.376969	−	0.926226i	$-0.376966\pi$
$383$	5651.11	0.753938	0.376969	+	0.926226i	$0.376966\pi$
$384$	0	0
$385$	0	0
$386$	−10467.7	−1.38028
$387$	0	0
$388$	−23102.8	−3.02285
$389$	−9040.82	−1.17838	−0.589188	−	0.807996i	$-0.700552\pi$
$389$	−9040.82	−1.17838	−0.589188	+	0.807996i	$0.700552\pi$
$390$	0	0
$391$	−857.314	−0.110886
$392$	0	0
$393$	0	0
$394$	8738.02	1.11730
$395$	61.6193	0.00784912
$396$	0	0
$397$	−3524.35	−0.445547	−0.222774	−	0.974870i	$-0.571511\pi$
$397$	−3524.35	−0.445547	−0.222774	+	0.974870i	$0.571511\pi$
$398$	4677.41	0.589089
$399$	0	0
$400$	−2889.33	−0.361166
$401$	−4469.81	−0.556638	−0.278319	−	0.960489i	$-0.589777\pi$
$401$	−4469.81	−0.556638	−0.278319	+	0.960489i	$0.589777\pi$
$402$	0	0
$403$	2486.17	0.307307
$404$	−17074.2	−2.10265
$405$	0	0
$406$	0	0
$407$	19740.7	2.40420
$408$	0	0
$409$	7532.36	0.910639	0.455320	−	0.890328i	$-0.349525\pi$
$409$	7532.36	0.910639	0.455320	+	0.890328i	$0.349525\pi$
$410$	−1398.58	−0.168466
$411$	0	0
$412$	6873.15	0.821882
$413$	0	0
$414$	0	0
$415$	−970.379	−0.114781
$416$	−4385.75	−0.516897
$417$	0	0
$418$	27878.2	3.26212
$419$	15486.4	1.80563	0.902814	−	0.430031i	$-0.141498\pi$
$419$	15486.4	1.80563	0.902814	+	0.430031i	$0.141498\pi$
$420$	0	0
$421$	4408.87	0.510393	0.255196	−	0.966889i	$-0.417860\pi$
$421$	4408.87	0.510393	0.255196	+	0.966889i	$0.417860\pi$
$422$	−2565.98	−0.295995
$423$	0	0
$424$	5001.88	0.572907
$425$	1824.61	0.208251
$426$	0	0
$427$	0	0
$428$	−3654.93	−0.412774
$429$	0	0
$430$	921.837	0.103384
$431$	10356.6	1.15745	0.578726	−	0.815522i	$-0.303550\pi$
$431$	10356.6	1.15745	0.578726	+	0.815522i	$0.303550\pi$
$432$	0	0
$433$	1398.15	0.155175	0.0775875	−	0.996986i	$-0.475278\pi$
$433$	1398.15	0.155175	0.0775875	+	0.996986i	$0.475278\pi$
$434$	0	0
$435$	0	0
$436$	8230.94	0.904107
$437$	−5068.08	−0.554781
$438$	0	0
$439$	3449.32	0.375005	0.187502	−	0.982264i	$-0.439961\pi$
$439$	3449.32	0.375005	0.187502	+	0.982264i	$0.439961\pi$
$440$	1643.82	0.178104
$441$	0	0
$442$	−2471.73	−0.265991
$443$	−2244.65	−0.240737	−0.120369	−	0.992729i	$-0.538408\pi$
$443$	−2244.65	−0.240737	−0.120369	+	0.992729i	$0.538408\pi$
$444$	0	0
$445$	1257.91	0.134001
$446$	16225.6	1.72265
$447$	0	0
$448$	0	0
$449$	7639.18	0.802929	0.401465	−	0.915875i	$-0.368501\pi$
$449$	7639.18	0.802929	0.401465	+	0.915875i	$0.368501\pi$
$450$	0	0
$451$	−24416.5	−2.54929
$452$	10029.3	1.04367
$453$	0	0
$454$	2289.26	0.236652
$455$	0	0
$456$	0	0
$457$	540.823	0.0553581	0.0276790	−	0.999617i	$-0.491188\pi$
$457$	540.823	0.0553581	0.0276790	+	0.999617i	$0.491188\pi$
$458$	−19840.8	−2.02423
$459$	0	0
$460$	−686.881	−0.0696217
$461$	8627.25	0.871608	0.435804	−	0.900042i	$-0.356464\pi$
$461$	8627.25	0.871608	0.435804	+	0.900042i	$0.356464\pi$
$462$	0	0
$463$	4497.71	0.451461	0.225730	−	0.974190i	$-0.427523\pi$
$463$	4497.71	0.451461	0.225730	+	0.974190i	$0.427523\pi$
$464$	−1560.52	−0.156132
$465$	0	0
$466$	−5175.19	−0.514455
$467$	−2493.07	−0.247035	−0.123518	−	0.992342i	$-0.539418\pi$
$467$	−2493.07	−0.247035	−0.123518	+	0.992342i	$0.539418\pi$
$468$	0	0
$469$	0	0
$470$	1759.48	0.172679
$471$	0	0
$472$	12930.9	1.26100
$473$	16093.5	1.56444
$474$	0	0
$475$	10786.3	1.04192
$476$	0	0
$477$	0	0
$478$	21461.6	2.05362
$479$	−8339.27	−0.795472	−0.397736	−	0.917500i	$-0.630204\pi$
$479$	−8339.27	−0.795472	−0.397736	+	0.917500i	$0.630204\pi$
$480$	0	0
$481$	−10346.7	−0.980811
$482$	5128.41	0.484632
$483$	0	0
$484$	47114.6	4.42474
$485$	1354.85	0.126846
$486$	0	0
$487$	−3104.85	−0.288899	−0.144450	−	0.989512i	$-0.546141\pi$
$487$	−3104.85	−0.288899	−0.144450	+	0.989512i	$0.546141\pi$
$488$	−13814.2	−1.28144
$489$	0	0
$490$	0	0
$491$	496.579	0.0456421	0.0228211	−	0.999740i	$-0.492735\pi$
$491$	496.579	0.0456421	0.0228211	+	0.999740i	$0.492735\pi$
$492$	0	0
$493$	985.468	0.0900269
$494$	−14611.8	−1.33080
$495$	0	0
$496$	1615.38	0.146236
$497$	0	0
$498$	0	0
$499$	−13253.6	−1.18900	−0.594501	−	0.804095i	$-0.702650\pi$
$499$	−13253.6	−1.18900	−0.594501	+	0.804095i	$0.702650\pi$
$500$	2931.87	0.262234
$501$	0	0
$502$	−29924.6	−2.66056
$503$	−1625.59	−0.144099	−0.0720493	−	0.997401i	$-0.522954\pi$
$503$	−1625.59	−0.144099	−0.0720493	+	0.997401i	$0.522954\pi$
$504$	0	0
$505$	1001.30	0.0882325
$506$	−18766.1	−1.64873
$507$	0	0
$508$	33250.3	2.90402
$509$	−7083.05	−0.616799	−0.308399	−	0.951257i	$-0.599793\pi$
$509$	−7083.05	−0.616799	−0.308399	+	0.951257i	$0.599793\pi$
$510$	0	0
$511$	0	0
$512$	−8242.42	−0.711459
$513$	0	0
$514$	5006.14	0.429594
$515$	−403.071	−0.0344882
$516$	0	0
$517$	30717.2	2.61304
$518$	0	0
$519$	0	0
$520$	−861.575	−0.0726588
$521$	−16565.3	−1.39298	−0.696489	−	0.717568i	$-0.745255\pi$
$521$	−16565.3	−1.39298	−0.696489	+	0.717568i	$0.745255\pi$
$522$	0	0
$523$	6472.69	0.541168	0.270584	−	0.962696i	$-0.412783\pi$
$523$	6472.69	0.541168	0.270584	+	0.962696i	$0.412783\pi$
$524$	34068.4	2.84024
$525$	0	0
$526$	17298.2	1.43391
$527$	−1020.11	−0.0843204
$528$	0	0
$529$	−8755.43	−0.719605
$530$	−674.229	−0.0552578
$531$	0	0
$532$	0	0
$533$	12797.5	1.04000
$534$	0	0
$535$	214.341	0.0173210
$536$	−23327.2	−1.87982
$537$	0	0
$538$	−32764.1	−2.62558
$539$	0	0
$540$	0	0
$541$	−3388.92	−0.269318	−0.134659	−	0.990892i	$-0.542994\pi$
$541$	−3388.92	−0.269318	−0.134659	+	0.990892i	$0.542994\pi$
$542$	31049.8	2.46071
$543$	0	0
$544$	1799.54	0.141829
$545$	−482.698	−0.0379385
$546$	0	0
$547$	10055.2	0.785980	0.392990	−	0.919543i	$-0.371441\pi$
$547$	10055.2	0.785980	0.392990	+	0.919543i	$0.371441\pi$
$548$	−16816.7	−1.31090
$549$	0	0
$550$	39939.7	3.09643
$551$	5825.67	0.450421
$552$	0	0
$553$	0	0
$554$	−14692.5	−1.12676
$555$	0	0
$556$	10088.7	0.769529
$557$	−5632.51	−0.428469	−0.214234	−	0.976782i	$-0.568726\pi$
$557$	−5632.51	−0.428469	−0.214234	+	0.976782i	$0.568726\pi$
$558$	0	0
$559$	−8435.10	−0.638223
$560$	0	0
$561$	0	0
$562$	7825.68	0.587378
$563$	−1450.41	−0.108575	−0.0542874	−	0.998525i	$-0.517289\pi$
$563$	−1450.41	−0.108575	−0.0542874	+	0.998525i	$0.517289\pi$
$564$	0	0
$565$	−588.163	−0.0437951
$566$	−2133.82	−0.158465
$567$	0	0
$568$	10249.8	0.757169
$569$	−16030.4	−1.18107	−0.590536	−	0.807012i	$-0.701084\pi$
$569$	−16030.4	−1.18107	−0.590536	+	0.807012i	$0.701084\pi$
$570$	0	0
$571$	−16540.8	−1.21228	−0.606139	−	0.795359i	$-0.707282\pi$
$571$	−16540.8	−1.21228	−0.606139	+	0.795359i	$0.707282\pi$
$572$	−34573.1	−2.52723
$573$	0	0
$574$	0	0
$575$	−7260.80	−0.526602
$576$	0	0
$577$	2039.27	0.147133	0.0735666	−	0.997290i	$-0.476562\pi$
$577$	2039.27	0.147133	0.0735666	+	0.997290i	$0.476562\pi$
$578$	−22113.9	−1.59138
$579$	0	0
$580$	789.558	0.0565252
$581$	0	0
$582$	0	0
$583$	−11770.7	−0.836182
$584$	22565.2	1.59889
$585$	0	0
$586$	3637.86	0.256448
$587$	−20392.1	−1.43385	−0.716927	−	0.697148i	$-0.754452\pi$
$587$	−20392.1	−1.43385	−0.716927	+	0.697148i	$0.754452\pi$
$588$	0	0
$589$	−6030.49	−0.421871
$590$	−1743.02	−0.121626
$591$	0	0
$592$	−6722.77	−0.466730
$593$	20269.0	1.40362	0.701810	−	0.712364i	$-0.252375\pi$
$593$	20269.0	1.40362	0.701810	+	0.712364i	$0.252375\pi$
$594$	0	0
$595$	0	0
$596$	−31284.7	−2.15012
$597$	0	0
$598$	9835.91	0.672609
$599$	−22317.8	−1.52234	−0.761169	−	0.648554i	$-0.775374\pi$
$599$	−22317.8	−1.52234	−0.761169	+	0.648554i	$0.775374\pi$
$600$	0	0
$601$	−1880.78	−0.127652	−0.0638259	−	0.997961i	$-0.520330\pi$
$601$	−1880.78	−0.127652	−0.0638259	+	0.997961i	$0.520330\pi$
$602$	0	0
$603$	0	0
$604$	−7745.74	−0.521804
$605$	−2763.00	−0.185673
$606$	0	0
$607$	16361.6	1.09407	0.547033	−	0.837111i	$-0.315757\pi$
$607$	16361.6	1.09407	0.547033	+	0.837111i	$0.315757\pi$
$608$	10638.2	0.709595
$609$	0	0
$610$	1862.09	0.123597
$611$	−16099.8	−1.06601
$612$	0	0
$613$	−15921.4	−1.04904	−0.524518	−	0.851399i	$-0.675755\pi$
$613$	−15921.4	−1.04904	−0.524518	+	0.851399i	$0.675755\pi$
$614$	−19741.7	−1.29757
$615$	0	0
$616$	0	0
$617$	7690.80	0.501815	0.250908	−	0.968011i	$-0.419271\pi$
$617$	7690.80	0.501815	0.250908	+	0.968011i	$0.419271\pi$
$618$	0	0
$619$	604.012	0.0392202	0.0196101	−	0.999808i	$-0.493758\pi$
$619$	604.012	0.0392202	0.0196101	+	0.999808i	$0.493758\pi$
$620$	−817.317	−0.0529423
$621$	0	0
$622$	1881.79	0.121307
$623$	0	0
$624$	0	0
$625$	15366.9	0.983479
$626$	−17561.3	−1.12123
$627$	0	0
$628$	−26483.5	−1.68281
$629$	4245.43	0.269120
$630$	0	0
$631$	−10806.1	−0.681752	−0.340876	−	0.940108i	$-0.610724\pi$
$631$	−10806.1	−0.681752	−0.340876	+	0.940108i	$0.610724\pi$
$632$	−2151.97	−0.135444
$633$	0	0
$634$	39418.4	2.46925
$635$	−1949.94	−0.121860
$636$	0	0
$637$	0	0
$638$	21571.4	1.33859
$639$	0	0
$640$	2168.72	0.133947
$641$	−4447.62	−0.274057	−0.137029	−	0.990567i	$-0.543755\pi$
$641$	−4447.62	−0.274057	−0.137029	+	0.990567i	$0.543755\pi$
$642$	0	0
$643$	−13531.3	−0.829897	−0.414949	−	0.909845i	$-0.636200\pi$
$643$	−13531.3	−0.829897	−0.414949	+	0.909845i	$0.636200\pi$
$644$	0	0
$645$	0	0
$646$	5995.46	0.365152
$647$	−18976.9	−1.15311	−0.576553	−	0.817059i	$-0.695603\pi$
$647$	−18976.9	−1.15311	−0.576553	+	0.817059i	$0.695603\pi$
$648$	0	0
$649$	−30429.8	−1.84049
$650$	−20933.6	−1.26321
$651$	0	0
$652$	−43012.1	−2.58357
$653$	−8796.72	−0.527171	−0.263585	−	0.964636i	$-0.584905\pi$
$653$	−8796.72	−0.527171	−0.263585	+	0.964636i	$0.584905\pi$
$654$	0	0
$655$	−1997.92	−0.119183
$656$	8315.13	0.494895
$657$	0	0
$658$	0	0
$659$	−24279.6	−1.43520	−0.717601	−	0.696455i	$-0.754760\pi$
$659$	−24279.6	−1.43520	−0.717601	+	0.696455i	$0.754760\pi$
$660$	0	0
$661$	25635.3	1.50847	0.754234	−	0.656605i	$-0.228008\pi$
$661$	25635.3	1.50847	0.754234	+	0.656605i	$0.228008\pi$
$662$	−6353.83	−0.373034
$663$	0	0
$664$	33889.1	1.98065
$665$	0	0
$666$	0	0
$667$	−3921.54	−0.227650
$668$	−42016.5	−2.43363
$669$	0	0
$670$	3144.40	0.181311
$671$	32508.6	1.87031
$672$	0	0
$673$	−23355.1	−1.33770	−0.668852	−	0.743396i	$-0.733214\pi$
$673$	−23355.1	−1.33770	−0.668852	+	0.743396i	$0.733214\pi$
$674$	−31209.6	−1.78361
$675$	0	0
$676$	−12990.6	−0.739108
$677$	17993.7	1.02150	0.510750	−	0.859730i	$-0.329368\pi$
$677$	17993.7	1.02150	0.510750	+	0.859730i	$0.329368\pi$
$678$	0	0
$679$	0	0
$680$	353.518	0.0199365
$681$	0	0
$682$	−22329.7	−1.25374
$683$	−17930.9	−1.00455	−0.502275	−	0.864708i	$-0.667504\pi$
$683$	−17930.9	−1.00455	−0.502275	+	0.864708i	$0.667504\pi$
$684$	0	0
$685$	986.205	0.0550087
$686$	0	0
$687$	0	0
$688$	−5480.69	−0.303706
$689$	6169.41	0.341126
$690$	0	0
$691$	−25818.0	−1.42137	−0.710683	−	0.703512i	$-0.751614\pi$
$691$	−25818.0	−1.42137	−0.710683	+	0.703512i	$0.751614\pi$
$692$	54757.4	3.00804
$693$	0	0
$694$	23316.1	1.27531
$695$	−591.647	−0.0322913
$696$	0	0
$697$	−5251.00	−0.285360
$698$	21175.0	1.14826
$699$	0	0
$700$	0	0
$701$	−16727.2	−0.901250	−0.450625	−	0.892713i	$-0.648799\pi$
$701$	−16727.2	−0.901250	−0.450625	+	0.892713i	$0.648799\pi$
$702$	0	0
$703$	25097.2	1.34646
$704$	52081.7	2.78821
$705$	0	0
$706$	7594.92	0.404870
$707$	0	0
$708$	0	0
$709$	−6891.41	−0.365039	−0.182519	−	0.983202i	$-0.558425\pi$
$709$	−6891.41	−0.365039	−0.182519	+	0.983202i	$0.558425\pi$
$710$	−1381.62	−0.0730302
$711$	0	0
$712$	−43930.5	−2.31231
$713$	4059.41	0.213220
$714$	0	0
$715$	2027.51	0.106049
$716$	4740.01	0.247406
$717$	0	0
$718$	62077.7	3.22663
$719$	10406.4	0.539767	0.269884	−	0.962893i	$-0.413015\pi$
$719$	10406.4	0.539767	0.269884	+	0.962893i	$0.413015\pi$
$720$	0	0
$721$	0	0
$722$	3153.66	0.162558
$723$	0	0
$724$	22155.5	1.13730
$725$	8346.17	0.427543
$726$	0	0
$727$	−2203.46	−0.112410	−0.0562049	−	0.998419i	$-0.517900\pi$
$727$	−2203.46	−0.112410	−0.0562049	+	0.998419i	$0.517900\pi$
$728$	0	0
$729$	0	0
$730$	−3041.68	−0.154216
$731$	3461.05	0.175119
$732$	0	0
$733$	8773.70	0.442106	0.221053	−	0.975262i	$-0.429051\pi$
$733$	8773.70	0.442106	0.221053	+	0.975262i	$0.429051\pi$
$734$	−24024.4	−1.20811
$735$	0	0
$736$	−7161.06	−0.358641
$737$	54895.1	2.74367
$738$	0	0
$739$	27385.9	1.36320	0.681601	−	0.731724i	$-0.261284\pi$
$739$	27385.9	1.36320	0.681601	+	0.731724i	$0.261284\pi$
$740$	3401.44	0.168972
$741$	0	0
$742$	0	0
$743$	28890.0	1.42648	0.713239	−	0.700921i	$-0.247228\pi$
$743$	28890.0	1.42648	0.713239	+	0.700921i	$0.247228\pi$
$744$	0	0
$745$	1834.67	0.0902243
$746$	−6869.17	−0.337129
$747$	0	0
$748$	14185.9	0.693432
$749$	0	0
$750$	0	0
$751$	−20807.7	−1.01103	−0.505515	−	0.862818i	$-0.668697\pi$
$751$	−20807.7	−1.01103	−0.505515	+	0.862818i	$0.668697\pi$
$752$	−10460.8	−0.507271
$753$	0	0
$754$	−11306.2	−0.546085
$755$	454.244	0.0218962
$756$	0	0
$757$	14372.8	0.690078	0.345039	−	0.938588i	$-0.387866\pi$
$757$	14372.8	0.690078	0.345039	+	0.938588i	$0.387866\pi$
$758$	10981.5	0.526208
$759$	0	0
$760$	2089.85	0.0997459
$761$	−26973.6	−1.28488	−0.642439	−	0.766337i	$-0.722077\pi$
$761$	−26973.6	−1.28488	−0.642439	+	0.766337i	$0.722077\pi$
$762$	0	0
$763$	0	0
$764$	43433.4	2.05676
$765$	0	0
$766$	26602.8	1.25483
$767$	15949.2	0.750838
$768$	0	0
$769$	3264.38	0.153077	0.0765387	−	0.997067i	$-0.475613\pi$
$769$	3264.38	0.153077	0.0765387	+	0.997067i	$0.475613\pi$
$770$	0	0
$771$	0	0
$772$	−31488.0	−1.46798
$773$	22135.4	1.02995	0.514977	−	0.857204i	$-0.327800\pi$
$773$	22135.4	1.02995	0.514977	+	0.857204i	$0.327800\pi$
$774$	0	0
$775$	−8639.59	−0.400443
$776$	−47316.1	−2.18885
$777$	0	0
$778$	−42560.0	−1.96124
$779$	−31041.7	−1.42771
$780$	0	0
$781$	−24120.5	−1.10512
$782$	−4035.83	−0.184554
$783$	0	0
$784$	0	0
$785$	1553.11	0.0706150
$786$	0	0
$787$	−29347.2	−1.32924	−0.664622	−	0.747180i	$-0.731407\pi$
$787$	−29347.2	−1.32924	−0.664622	+	0.747180i	$0.731407\pi$
$788$	26285.1	1.18828
$789$	0	0
$790$	290.075	0.0130638
$791$	0	0
$792$	0	0
$793$	−17038.7	−0.763006
$794$	−16591.0	−0.741553
$795$	0	0
$796$	14070.3	0.626516
$797$	−26622.6	−1.18321	−0.591606	−	0.806227i	$-0.701506\pi$
$797$	−26622.6	−1.18321	−0.591606	+	0.806227i	$0.701506\pi$
$798$	0	0
$799$	6606.02	0.292496
$800$	15240.8	0.673554
$801$	0	0
$802$	−21041.8	−0.926448
$803$	−53101.8	−2.33365
$804$	0	0
$805$	0	0
$806$	11703.7	0.511471
$807$	0	0
$808$	−34969.1	−1.52253
$809$	−21385.7	−0.929395	−0.464697	−	0.885470i	$-0.653837\pi$
$809$	−21385.7	−0.929395	−0.464697	+	0.885470i	$0.653837\pi$
$810$	0	0
$811$	26846.2	1.16239	0.581195	−	0.813764i	$-0.302585\pi$
$811$	26846.2	1.16239	0.581195	+	0.813764i	$0.302585\pi$
$812$	0	0
$813$	0	0
$814$	92930.1	4.00147
$815$	2522.42	0.108413
$816$	0	0
$817$	20460.3	0.876151
$818$	35458.8	1.51563
$819$	0	0
$820$	−4207.11	−0.179169
$821$	23679.3	1.00659	0.503296	−	0.864114i	$-0.332120\pi$
$821$	23679.3	1.00659	0.503296	+	0.864114i	$0.332120\pi$
$822$	0	0
$823$	−20546.6	−0.870242	−0.435121	−	0.900372i	$-0.643294\pi$
$823$	−20546.6	−0.870242	−0.435121	+	0.900372i	$0.643294\pi$
$824$	14076.7	0.595126
$825$	0	0
$826$	0	0
$827$	−24006.6	−1.00942	−0.504712	−	0.863288i	$-0.668401\pi$
$827$	−24006.6	−1.00942	−0.504712	+	0.863288i	$0.668401\pi$
$828$	0	0
$829$	20544.1	0.860706	0.430353	−	0.902661i	$-0.358389\pi$
$829$	20544.1	0.860706	0.430353	+	0.902661i	$0.358389\pi$
$830$	−4568.09	−0.191037
$831$	0	0
$832$	−27297.6	−1.13747
$833$	0	0
$834$	0	0
$835$	2464.03	0.102121
$836$	83861.0	3.46937
$837$	0	0
$838$	72902.5	3.00522
$839$	−8213.78	−0.337987	−0.168993	−	0.985617i	$-0.554052\pi$
$839$	−8213.78	−0.337987	−0.168993	+	0.985617i	$0.554052\pi$
$840$	0	0
$841$	−19881.3	−0.815173
$842$	20754.9	0.849479
$843$	0	0
$844$	−7718.79	−0.314800
$845$	761.823	0.0310148
$846$	0	0
$847$	0	0
$848$	4008.56	0.162329
$849$	0	0
$850$	8589.41	0.346605
$851$	−16894.1	−0.680521
$852$	0	0
$853$	−9158.34	−0.367615	−0.183808	−	0.982962i	$-0.558842\pi$
$853$	−9158.34	−0.367615	−0.183808	+	0.982962i	$0.558842\pi$
$854$	0	0
$855$	0	0
$856$	−7485.54	−0.298891
$857$	25763.8	1.02692	0.513462	−	0.858112i	$-0.328363\pi$
$857$	25763.8	1.02692	0.513462	+	0.858112i	$0.328363\pi$
$858$	0	0
$859$	−41844.0	−1.66205	−0.831024	−	0.556236i	$-0.812245\pi$
$859$	−41844.0	−1.66205	−0.831024	+	0.556236i	$0.812245\pi$
$860$	2773.00	0.109952
$861$	0	0
$862$	48754.2	1.92642
$863$	43821.4	1.72850	0.864250	−	0.503062i	$-0.167793\pi$
$863$	43821.4	1.72850	0.864250	+	0.503062i	$0.167793\pi$
$864$	0	0
$865$	−3211.21	−0.126225
$866$	6581.83	0.258268
$867$	0	0
$868$	0	0
$869$	5064.15	0.197686
$870$	0	0
$871$	−28772.2	−1.11930
$872$	16857.5	0.654665
$873$	0	0
$874$	−23858.1	−0.923357
$875$	0	0
$876$	0	0
$877$	−6690.44	−0.257606	−0.128803	−	0.991670i	$-0.541113\pi$
$877$	−6690.44	−0.257606	−0.128803	+	0.991670i	$0.541113\pi$
$878$	16237.8	0.624144
$879$	0	0
$880$	1317.37	0.0504644
$881$	−11946.0	−0.456835	−0.228417	−	0.973563i	$-0.573355\pi$
$881$	−11946.0	−0.456835	−0.228417	+	0.973563i	$0.573355\pi$
$882$	0	0
$883$	−4878.25	−0.185919	−0.0929593	−	0.995670i	$-0.529633\pi$
$883$	−4878.25	−0.185919	−0.0929593	+	0.995670i	$0.529633\pi$
$884$	−7435.26	−0.282890
$885$	0	0
$886$	−10566.8	−0.400674
$887$	30111.7	1.13986	0.569928	−	0.821695i	$-0.306971\pi$
$887$	30111.7	1.13986	0.569928	+	0.821695i	$0.306971\pi$
$888$	0	0
$889$	0	0
$890$	5921.63	0.223026
$891$	0	0
$892$	48808.5	1.83210
$893$	39052.0	1.46341
$894$	0	0
$895$	−277.974	−0.0103817
$896$	0	0
$897$	0	0
$898$	35961.7	1.33637
$899$	−4666.23	−0.173112
$900$	0	0
$901$	−2531.41	−0.0935998
$902$	−114942.	−4.24294
$903$	0	0
$904$	20540.8	0.755725
$905$	−1299.30	−0.0477238
$906$	0	0
$907$	−1986.84	−0.0727364	−0.0363682	−	0.999338i	$-0.511579\pi$
$907$	−1986.84	−0.0727364	−0.0363682	+	0.999338i	$0.511579\pi$
$908$	6886.38	0.251688
$909$	0	0
$910$	0	0
$911$	−31604.7	−1.14941	−0.574703	−	0.818362i	$-0.694882\pi$
$911$	−31604.7	−1.14941	−0.574703	+	0.818362i	$0.694882\pi$
$912$	0	0
$913$	−79750.0	−2.89084
$914$	2545.94	0.0921360
$915$	0	0
$916$	−59683.5	−2.15284
$917$	0	0
$918$	0	0
$919$	−26999.8	−0.969140	−0.484570	−	0.874752i	$-0.661024\pi$
$919$	−26999.8	−0.969140	−0.484570	+	0.874752i	$0.661024\pi$
$920$	−1406.78	−0.0504132
$921$	0	0
$922$	40613.1	1.45067
$923$	12642.3	0.450841
$924$	0	0
$925$	35955.5	1.27807
$926$	21173.1	0.751395
$927$	0	0
$928$	8231.51	0.291177
$929$	−17312.9	−0.611431	−0.305715	−	0.952123i	$-0.598896\pi$
$929$	−17312.9	−0.611431	−0.305715	+	0.952123i	$0.598896\pi$
$930$	0	0
$931$	0	0
$932$	−15567.6	−0.547140
$933$	0	0
$934$	−11736.2	−0.411156
$935$	−831.922	−0.0290981
$936$	0	0
$937$	31294.1	1.09107	0.545535	−	0.838088i	$-0.316327\pi$
$937$	31294.1	1.09107	0.545535	+	0.838088i	$0.316327\pi$
$938$	0	0
$939$	0	0
$940$	5292.75	0.183650
$941$	44549.5	1.54333	0.771665	−	0.636029i	$-0.219424\pi$
$941$	44549.5	1.54333	0.771665	+	0.636029i	$0.219424\pi$
$942$	0	0
$943$	20895.7	0.721587
$944$	10363.0	0.357295
$945$	0	0
$946$	75760.6	2.60380
$947$	−54541.4	−1.87155	−0.935775	−	0.352596i	$-0.885299\pi$
$947$	−54541.4	−1.87155	−0.935775	+	0.352596i	$0.885299\pi$
$948$	0	0
$949$	27832.3	0.952028
$950$	50777.0	1.73413
$951$	0	0
$952$	0	0
$953$	−19349.7	−0.657710	−0.328855	−	0.944380i	$-0.606663\pi$
$953$	−19349.7	−0.657710	−0.328855	+	0.944380i	$0.606663\pi$
$954$	0	0
$955$	−2547.12	−0.0863066
$956$	64559.2	2.18409
$957$	0	0
$958$	−39257.4	−1.32395
$959$	0	0
$960$	0	0
$961$	−24960.7	−0.837861
$962$	−48707.5	−1.63243
$963$	0	0
$964$	15426.9	0.515422
$965$	1846.59	0.0615999
$966$	0	0
$967$	52450.4	1.74425	0.872125	−	0.489283i	$-0.162742\pi$
$967$	52450.4	1.74425	0.872125	+	0.489283i	$0.162742\pi$
$968$	96493.9	3.20396
$969$	0	0
$970$	6377.99	0.211118
$971$	−13635.8	−0.450662	−0.225331	−	0.974282i	$-0.572346\pi$
$971$	−13635.8	−0.450662	−0.225331	+	0.974282i	$0.572346\pi$
$972$	0	0
$973$	0	0
$974$	−14616.2	−0.480834
$975$	0	0
$976$	−11070.9	−0.363085
$977$	30018.6	0.982989	0.491495	−	0.870881i	$-0.336451\pi$
$977$	30018.6	0.982989	0.491495	+	0.870881i	$0.336451\pi$
$978$	0	0
$979$	103380.	3.37492
$980$	0	0
$981$	0	0
$982$	2337.66	0.0759651
$983$	−6470.42	−0.209943	−0.104972	−	0.994475i	$-0.533475\pi$
$983$	−6470.42	−0.209943	−0.104972	+	0.994475i	$0.533475\pi$
$984$	0	0
$985$	−1541.47	−0.0498633
$986$	4639.12	0.149837
$987$	0	0
$988$	−43954.2	−1.41535
$989$	−13772.8	−0.442821
$990$	0	0
$991$	−23884.4	−0.765605	−0.382802	−	0.923830i	$-0.625041\pi$
$991$	−23884.4	−0.765605	−0.382802	+	0.923830i	$0.625041\pi$
$992$	−8520.91	−0.272721
$993$	0	0
$994$	0	0
$995$	−825.140	−0.0262902
$996$	0	0
$997$	−7790.90	−0.247483	−0.123741	−	0.992315i	$-0.539489\pi$
$997$	−7790.90	−0.247483	−0.123741	+	0.992315i	$0.539489\pi$
$998$	−62391.6	−1.97893
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.be.1.6		6
3.2	odd	2	inner	1323.4.a.be.1.1		6
7.3	odd	6		189.4.e.e.163.1	yes	12
7.5	odd	6		189.4.e.e.109.1	✓	12
7.6	odd	2		1323.4.a.bd.1.6		6
21.5	even	6		189.4.e.e.109.6	yes	12
21.17	even	6		189.4.e.e.163.6	yes	12
21.20	even	2		1323.4.a.bd.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.e.109.1	✓	12	7.5	odd	6
189.4.e.e.109.6	yes	12	21.5	even	6
189.4.e.e.163.1	yes	12	7.3	odd	6
189.4.e.e.163.6	yes	12	21.17	even	6
1323.4.a.bd.1.1		6	21.20	even	2
1323.4.a.bd.1.6		6	7.6	odd	2
1323.4.a.be.1.1		6	3.2	odd	2	inner
1323.4.a.be.1.6		6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} +29.0024 q^{8} -3.90938 q^{10} -68.2503 q^{11} +35.7721 q^{13} +23.2429 q^{16} -14.6779 q^{17} -86.7693 q^{19} -11.7599 q^{20} -321.290 q^{22} +58.4087 q^{23} -124.310 q^{25} +168.398 q^{26} -67.1398 q^{29} +69.5002 q^{31} -122.603 q^{32} -69.0965 q^{34} -289.240 q^{37} -408.469 q^{38} -24.0851 q^{40} +357.750 q^{41} -235.801 q^{43} -966.482 q^{44} +274.961 q^{46} -450.067 q^{47} -585.195 q^{50} +506.563 q^{52} +172.464 q^{53} +56.6787 q^{55} -316.063 q^{58} +445.856 q^{59} -476.314 q^{61} +327.174 q^{62} -763.099 q^{64} -29.7070 q^{65} -804.320 q^{67} -207.851 q^{68} +353.412 q^{71} +778.045 q^{73} -1361.61 q^{74} -1228.73 q^{76} -74.1996 q^{79} -19.3021 q^{80} +1684.12 q^{82} +1168.49 q^{83} +12.1893 q^{85} -1110.04 q^{86} -1979.42 q^{88} -1514.72 q^{89} +827.116 q^{92} -2118.70 q^{94} +72.0579 q^{95} -1631.46 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58}+ \cdots + 682 q^{97}+O(q^{100}) \) 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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