LMFDB - Embedded newform 1323.4.a.bg.1.1

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 = [6,0,0,36,0,0,0,0,0,180,0,0,108] 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:	$0$
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			1.1
Root			$-5.45019$ 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)

$f(q)$	$=$	$q-5.45019 q^{2} +21.7046 q^{4} -19.3432 q^{5} -74.6929 q^{8} +105.424 q^{10} +11.2132 q^{11} +46.3102 q^{13} +233.454 q^{16} +97.6541 q^{17} +98.9176 q^{19} -419.836 q^{20} -61.1139 q^{22} +138.173 q^{23} +249.158 q^{25} -252.400 q^{26} +180.123 q^{29} +31.9046 q^{31} -674.825 q^{32} -532.234 q^{34} -205.452 q^{37} -539.120 q^{38} +1444.80 q^{40} -234.404 q^{41} -320.568 q^{43} +243.377 q^{44} -753.067 q^{46} +312.715 q^{47} -1357.96 q^{50} +1005.15 q^{52} +53.7942 q^{53} -216.898 q^{55} -981.706 q^{58} -400.291 q^{59} +97.3536 q^{61} -173.886 q^{62} +1810.30 q^{64} -895.786 q^{65} +257.525 q^{67} +2119.55 q^{68} +253.601 q^{71} +1161.99 q^{73} +1119.75 q^{74} +2146.97 q^{76} -1070.16 q^{79} -4515.73 q^{80} +1277.55 q^{82} +889.495 q^{83} -1888.94 q^{85} +1747.16 q^{86} -837.543 q^{88} +647.303 q^{89} +2998.98 q^{92} -1704.36 q^{94} -1913.38 q^{95} -673.379 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	−5.45019	−1.92693	−0.963467	−	0.267826i	$-0.913695\pi$
$2$	−5.45019	−1.92693	−0.963467	+	0.267826i	$0.913695\pi$
$3$	0	0
$3$	0	0
$4$	21.7046	2.71308
$5$	−19.3432	−1.73011	−0.865053	−	0.501681i	$-0.832715\pi$
$5$	−19.3432	−1.73011	−0.865053	+	0.501681i	$0.832715\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−74.6929	−3.30099
$9$	0	0
$10$	105.424	3.33380
$11$	11.2132	0.307354	0.153677	−	0.988121i	$-0.450888\pi$
$11$	11.2132	0.307354	0.153677	+	0.988121i	$0.450888\pi$
$12$	0	0
$13$	46.3102	0.988010	0.494005	−	0.869459i	$-0.335532\pi$
$13$	46.3102	0.988010	0.494005	+	0.869459i	$0.335532\pi$
$14$	0	0
$15$	0	0
$16$	233.454	3.64771
$17$	97.6541	1.39321	0.696606	−	0.717454i	$-0.254693\pi$
$17$	97.6541	1.39321	0.696606	+	0.717454i	$0.254693\pi$
$18$	0	0
$19$	98.9176	1.19438	0.597191	−	0.802099i	$-0.296284\pi$
$19$	98.9176	1.19438	0.597191	+	0.802099i	$0.296284\pi$
$20$	−419.836	−4.69391
$21$	0	0
$22$	−61.1139	−0.592251
$23$	138.173	1.25265	0.626325	−	0.779562i	$-0.284558\pi$
$23$	138.173	1.25265	0.626325	+	0.779562i	$0.284558\pi$
$24$	0	0
$25$	249.158	1.99327
$26$	−252.400	−1.90383
$27$	0	0
$28$	0	0
$29$	180.123	1.15338	0.576690	−	0.816963i	$-0.304344\pi$
$29$	180.123	1.15338	0.576690	+	0.816963i	$0.304344\pi$
$30$	0	0
$31$	31.9046	0.184846	0.0924230	−	0.995720i	$-0.470539\pi$
$31$	31.9046	0.184846	0.0924230	+	0.995720i	$0.470539\pi$
$32$	−674.825	−3.72792
$33$	0	0
$34$	−532.234	−2.68463
$35$	0	0
$36$	0	0
$37$	−205.452	−0.912867	−0.456433	−	0.889758i	$-0.650873\pi$
$37$	−205.452	−0.912867	−0.456433	+	0.889758i	$0.650873\pi$
$38$	−539.120	−2.30149
$39$	0	0
$40$	1444.80	5.71106
$41$	−234.404	−0.892874	−0.446437	−	0.894815i	$-0.647307\pi$
$41$	−234.404	−0.892874	−0.446437	+	0.894815i	$0.647307\pi$
$42$	0	0
$43$	−320.568	−1.13689	−0.568443	−	0.822723i	$-0.692454\pi$
$43$	−320.568	−1.13689	−0.568443	+	0.822723i	$0.692454\pi$
$44$	243.377	0.833875
$45$	0	0
$46$	−753.067	−2.41378
$47$	312.715	0.970516	0.485258	−	0.874371i	$-0.338726\pi$
$47$	312.715	0.970516	0.485258	+	0.874371i	$0.338726\pi$
$48$	0	0
$49$	0	0
$50$	−1357.96	−3.84089
$51$	0	0
$52$	1005.15	2.68055
$53$	53.7942	0.139419	0.0697094	−	0.997567i	$-0.477793\pi$
$53$	53.7942	0.139419	0.0697094	+	0.997567i	$0.477793\pi$
$54$	0	0
$55$	−216.898	−0.531755
$56$	0	0
$57$	0	0
$58$	−981.706	−2.22249
$59$	−400.291	−0.883278	−0.441639	−	0.897193i	$-0.645603\pi$
$59$	−400.291	−0.883278	−0.441639	+	0.897193i	$0.645603\pi$
$60$	0	0
$61$	97.3536	0.204342	0.102171	−	0.994767i	$-0.467421\pi$
$61$	97.3536	0.204342	0.102171	+	0.994767i	$0.467421\pi$
$62$	−173.886	−0.356186
$63$	0	0
$64$	1810.30	3.53574
$65$	−895.786	−1.70936
$66$	0	0
$67$	257.525	0.469577	0.234789	−	0.972046i	$-0.424560\pi$
$67$	257.525	0.469577	0.234789	+	0.972046i	$0.424560\pi$
$68$	2119.55	3.77989
$69$	0	0
$70$	0	0
$71$	253.601	0.423900	0.211950	−	0.977280i	$-0.432019\pi$
$71$	253.601	0.423900	0.211950	+	0.977280i	$0.432019\pi$
$72$	0	0
$73$	1161.99	1.86303	0.931514	−	0.363706i	$-0.118489\pi$
$73$	1161.99	1.86303	0.931514	+	0.363706i	$0.118489\pi$
$74$	1119.75	1.75904
$75$	0	0
$76$	2146.97	3.24045
$77$	0	0
$78$	0	0
$79$	−1070.16	−1.52408	−0.762039	−	0.647531i	$-0.775802\pi$
$79$	−1070.16	−1.52408	−0.762039	+	0.647531i	$0.775802\pi$
$80$	−4515.73	−6.31093
$81$	0	0
$82$	1277.55	1.72051
$83$	889.495	1.17632	0.588161	−	0.808744i	$-0.299852\pi$
$83$	889.495	1.17632	0.588161	+	0.808744i	$0.299852\pi$
$84$	0	0
$85$	−1888.94	−2.41040
$86$	1747.16	2.19070
$87$	0	0
$88$	−837.543	−1.01457
$89$	647.303	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$89$	647.303	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$90$	0	0
$91$	0	0
$92$	2998.98	3.39854
$93$	0	0
$94$	−1704.36	−1.87012
$95$	−1913.38	−2.06641
$96$	0	0
$97$	−673.379	−0.704859	−0.352429	−	0.935838i	$-0.614644\pi$
$97$	−673.379	−0.704859	−0.352429	+	0.935838i	$0.614644\pi$
$98$	0	0
$99$	0	0
$100$	5407.89	5.40789
$101$	243.254	0.239650	0.119825	−	0.992795i	$-0.461767\pi$
$101$	243.254	0.239650	0.119825	+	0.992795i	$0.461767\pi$
$102$	0	0
$103$	−486.156	−0.465072	−0.232536	−	0.972588i	$-0.574702\pi$
$103$	−486.156	−0.465072	−0.232536	+	0.972588i	$0.574702\pi$
$104$	−3459.04	−3.26141
$105$	0	0
$106$	−293.189	−0.268651
$107$	−384.679	−0.347555	−0.173777	−	0.984785i	$-0.555597\pi$
$107$	−384.679	−0.347555	−0.173777	+	0.984785i	$0.555597\pi$
$108$	0	0
$109$	994.781	0.874153	0.437077	−	0.899424i	$-0.356014\pi$
$109$	994.781	0.874153	0.437077	+	0.899424i	$0.356014\pi$
$110$	1182.14	1.02466
$111$	0	0
$112$	0	0
$113$	189.932	0.158118	0.0790589	−	0.996870i	$-0.474808\pi$
$113$	189.932	0.158118	0.0790589	+	0.996870i	$0.474808\pi$
$114$	0	0
$115$	−2672.70	−2.16722
$116$	3909.50	3.12921
$117$	0	0
$118$	2181.66	1.70202
$119$	0	0
$120$	0	0
$121$	−1205.27	−0.905534
$122$	−530.596	−0.393754
$123$	0	0
$124$	692.476	0.501502
$125$	−2401.62	−1.71846
$126$	0	0
$127$	1679.82	1.17370	0.586851	−	0.809695i	$-0.300367\pi$
$127$	1679.82	1.17370	0.586851	+	0.809695i	$0.300367\pi$
$128$	−4467.88	−3.08522
$129$	0	0
$130$	4882.21	3.29383
$131$	−237.457	−0.158372	−0.0791859	−	0.996860i	$-0.525232\pi$
$131$	−237.457	−0.158372	−0.0791859	+	0.996860i	$0.525232\pi$
$132$	0	0
$133$	0	0
$134$	−1403.56	−0.904845
$135$	0	0
$136$	−7294.07	−4.59898
$137$	−526.657	−0.328433	−0.164217	−	0.986424i	$-0.552510\pi$
$137$	−526.657	−0.328433	−0.164217	+	0.986424i	$0.552510\pi$
$138$	0	0
$139$	−580.607	−0.354291	−0.177145	−	0.984185i	$-0.556686\pi$
$139$	−580.607	−0.354291	−0.177145	+	0.984185i	$0.556686\pi$
$140$	0	0
$141$	0	0
$142$	−1382.18	−0.816828
$143$	519.283	0.303669
$144$	0	0
$145$	−3484.15	−1.99547
$146$	−6333.09	−3.58993
$147$	0	0
$148$	−4459.26	−2.47668
$149$	−2781.48	−1.52931	−0.764657	−	0.644438i	$-0.777091\pi$
$149$	−2781.48	−1.52931	−0.764657	+	0.644438i	$0.777091\pi$
$150$	0	0
$151$	−1413.35	−0.761702	−0.380851	−	0.924636i	$-0.624369\pi$
$151$	−1413.35	−0.761702	−0.380851	+	0.924636i	$0.624369\pi$
$152$	−7388.44	−3.94264
$153$	0	0
$154$	0	0
$155$	−617.135	−0.319803
$156$	0	0
$157$	2356.59	1.19794	0.598970	−	0.800772i	$-0.295577\pi$
$157$	2356.59	1.19794	0.598970	+	0.800772i	$0.295577\pi$
$158$	5832.57	2.93680
$159$	0	0
$160$	13053.3	6.44969
$161$	0	0
$162$	0	0
$163$	1309.51	0.629255	0.314627	−	0.949215i	$-0.398120\pi$
$163$	1309.51	0.629255	0.314627	+	0.949215i	$0.398120\pi$
$164$	−5087.66	−2.42244
$165$	0	0
$166$	−4847.92	−2.26670
$167$	−1623.77	−0.752401	−0.376201	−	0.926538i	$-0.622770\pi$
$167$	−1623.77	−0.752401	−0.376201	+	0.926538i	$0.622770\pi$
$168$	0	0
$169$	−52.3668	−0.0238356
$170$	10295.1	4.64469
$171$	0	0
$172$	−6957.80	−3.08446
$173$	2219.78	0.975529	0.487765	−	0.872975i	$-0.337812\pi$
$173$	2219.78	0.975529	0.487765	+	0.872975i	$0.337812\pi$
$174$	0	0
$175$	0	0
$176$	2617.75	1.12114
$177$	0	0
$178$	−3527.92	−1.48556
$179$	−1124.59	−0.469584	−0.234792	−	0.972046i	$-0.575441\pi$
$179$	−1124.59	−0.469584	−0.234792	+	0.972046i	$0.575441\pi$
$180$	0	0
$181$	3951.04	1.62253	0.811267	−	0.584676i	$-0.198778\pi$
$181$	3951.04	1.62253	0.811267	+	0.584676i	$0.198778\pi$
$182$	0	0
$183$	0	0
$184$	−10320.5	−4.13499
$185$	3974.09	1.57936
$186$	0	0
$187$	1095.01	0.428209
$188$	6787.37	2.63308
$189$	0	0
$190$	10428.3	3.98183
$191$	−1177.40	−0.446039	−0.223019	−	0.974814i	$-0.571591\pi$
$191$	−1177.40	−0.446039	−0.223019	+	0.974814i	$0.571591\pi$
$192$	0	0
$193$	−1772.08	−0.660917	−0.330458	−	0.943821i	$-0.607203\pi$
$193$	−1772.08	−0.660917	−0.330458	+	0.943821i	$0.607203\pi$
$194$	3670.05	1.35822
$195$	0	0
$196$	0	0
$197$	794.274	0.287257	0.143629	−	0.989632i	$-0.454123\pi$
$197$	794.274	0.287257	0.143629	+	0.989632i	$0.454123\pi$
$198$	0	0
$199$	3662.14	1.30453	0.652267	−	0.757990i	$-0.273818\pi$
$199$	3662.14	1.30453	0.652267	+	0.757990i	$0.273818\pi$
$200$	−18610.3	−6.57975
$201$	0	0
$202$	−1325.78	−0.461791
$203$	0	0
$204$	0	0
$205$	4534.12	1.54477
$206$	2649.64	0.896162
$207$	0	0
$208$	10811.3	3.60398
$209$	1109.18	0.367098
$210$	0	0
$211$	2063.69	0.673318	0.336659	−	0.941627i	$-0.390703\pi$
$211$	2063.69	0.673318	0.336659	+	0.941627i	$0.390703\pi$
$212$	1167.58	0.378254
$213$	0	0
$214$	2096.58	0.669715
$215$	6200.79	1.96693
$216$	0	0
$217$	0	0
$218$	−5421.75	−1.68444
$219$	0	0
$220$	−4707.69	−1.44269
$221$	4522.38	1.37651
$222$	0	0
$223$	−3728.28	−1.11957	−0.559784	−	0.828638i	$-0.689116\pi$
$223$	−3728.28	−1.11957	−0.559784	+	0.828638i	$0.689116\pi$
$224$	0	0
$225$	0	0
$226$	−1035.17	−0.304683
$227$	4558.22	1.33277	0.666387	−	0.745606i	$-0.267840\pi$
$227$	4558.22	1.33277	0.666387	+	0.745606i	$0.267840\pi$
$228$	0	0
$229$	2068.22	0.596819	0.298410	−	0.954438i	$-0.403544\pi$
$229$	2068.22	0.596819	0.298410	+	0.954438i	$0.403544\pi$
$230$	14566.7	4.17609
$231$	0	0
$232$	−13453.9	−3.80730
$233$	−1459.56	−0.410383	−0.205191	−	0.978722i	$-0.565782\pi$
$233$	−1459.56	−0.410383	−0.205191	+	0.978722i	$0.565782\pi$
$234$	0	0
$235$	−6048.91	−1.67909
$236$	−8688.16	−2.39640
$237$	0	0
$238$	0	0
$239$	2220.74	0.601035	0.300518	−	0.953776i	$-0.402841\pi$
$239$	2220.74	0.601035	0.300518	+	0.953776i	$0.402841\pi$
$240$	0	0
$241$	−5064.37	−1.35363	−0.676815	−	0.736153i	$-0.736640\pi$
$241$	−5064.37	−1.35363	−0.676815	+	0.736153i	$0.736640\pi$
$242$	6568.93	1.74490
$243$	0	0
$244$	2113.02	0.554395
$245$	0	0
$246$	0	0
$247$	4580.89	1.18006
$248$	−2383.04	−0.610175
$249$	0	0
$250$	13089.3	3.31135
$251$	1797.86	0.452112	0.226056	−	0.974114i	$-0.427417\pi$
$251$	1797.86	0.452112	0.226056	+	0.974114i	$0.427417\pi$
$252$	0	0
$253$	1549.35	0.385007
$254$	−9155.37	−2.26165
$255$	0	0
$256$	9868.42	2.40928
$257$	2734.28	0.663656	0.331828	−	0.943340i	$-0.392335\pi$
$257$	2734.28	0.663656	0.331828	+	0.943340i	$0.392335\pi$
$258$	0	0
$259$	0	0
$260$	−19442.7	−4.63763
$261$	0	0
$262$	1294.19	0.305172
$263$	−8197.71	−1.92203	−0.961013	−	0.276504i	$-0.910824\pi$
$263$	−8197.71	−1.92203	−0.961013	+	0.276504i	$0.910824\pi$
$264$	0	0
$265$	−1040.55	−0.241209
$266$	0	0
$267$	0	0
$268$	5589.48	1.27400
$269$	−237.512	−0.0538342	−0.0269171	−	0.999638i	$-0.508569\pi$
$269$	−237.512	−0.0538342	−0.0269171	+	0.999638i	$0.508569\pi$
$270$	0	0
$271$	−4349.95	−0.975057	−0.487528	−	0.873107i	$-0.662101\pi$
$271$	−4349.95	−0.975057	−0.487528	+	0.873107i	$0.662101\pi$
$272$	22797.7	5.08204
$273$	0	0
$274$	2870.38	0.632869
$275$	2793.85	0.612638
$276$	0	0
$277$	6228.51	1.35103	0.675515	−	0.737347i	$-0.263921\pi$
$277$	6228.51	1.35103	0.675515	+	0.737347i	$0.263921\pi$
$278$	3164.42	0.682695
$279$	0	0
$280$	0	0
$281$	3360.95	0.713515	0.356757	−	0.934197i	$-0.383882\pi$
$281$	3360.95	0.713515	0.356757	+	0.934197i	$0.383882\pi$
$282$	0	0
$283$	8009.37	1.68236	0.841180	−	0.540755i	$-0.181862\pi$
$283$	8009.37	1.68236	0.841180	+	0.540755i	$0.181862\pi$
$284$	5504.32	1.15007
$285$	0	0
$286$	−2830.19	−0.585150
$287$	0	0
$288$	0	0
$289$	4623.33	0.941041
$290$	18989.3	3.84514
$291$	0	0
$292$	25220.6	5.05454
$293$	−9247.12	−1.84376	−0.921881	−	0.387472i	$-0.873348\pi$
$293$	−9247.12	−1.84376	−0.921881	+	0.387472i	$0.873348\pi$
$294$	0	0
$295$	7742.89	1.52816
$296$	15345.8	3.01336
$297$	0	0
$298$	15159.6	2.94689
$299$	6398.80	1.23763
$300$	0	0
$301$	0	0
$302$	7703.04	1.46775
$303$	0	0
$304$	23092.7	4.35676
$305$	−1883.13	−0.353533
$306$	0	0
$307$	3367.11	0.625965	0.312983	−	0.949759i	$-0.398672\pi$
$307$	3367.11	0.625965	0.312983	+	0.949759i	$0.398672\pi$
$308$	0	0
$309$	0	0
$310$	3363.51	0.616240
$311$	6555.92	1.19534	0.597672	−	0.801740i	$-0.296092\pi$
$311$	6555.92	1.19534	0.597672	+	0.801740i	$0.296092\pi$
$312$	0	0
$313$	−616.028	−0.111246	−0.0556229	−	0.998452i	$-0.517714\pi$
$313$	−616.028	−0.111246	−0.0556229	+	0.998452i	$0.517714\pi$
$314$	−12843.9	−2.30835
$315$	0	0
$316$	−23227.4	−4.13494
$317$	−3481.49	−0.616845	−0.308422	−	0.951250i	$-0.599801\pi$
$317$	−3481.49	−0.616845	−0.308422	+	0.951250i	$0.599801\pi$
$318$	0	0
$319$	2019.75	0.354496
$320$	−35016.9	−6.11720
$321$	0	0
$322$	0	0
$323$	9659.71	1.66403
$324$	0	0
$325$	11538.6	1.96937
$326$	−7137.07	−1.21253
$327$	0	0
$328$	17508.3	2.94737
$329$	0	0
$330$	0	0
$331$	−7187.44	−1.19353	−0.596763	−	0.802417i	$-0.703547\pi$
$331$	−7187.44	−1.19353	−0.596763	+	0.802417i	$0.703547\pi$
$332$	19306.2	3.19145
$333$	0	0
$334$	8849.86	1.44983
$335$	−4981.35	−0.812418
$336$	0	0
$337$	3575.89	0.578015	0.289007	−	0.957327i	$-0.406675\pi$
$337$	3575.89	0.578015	0.289007	+	0.957327i	$0.406675\pi$
$338$	285.410	0.0459297
$339$	0	0
$340$	−40998.8	−6.53962
$341$	357.751	0.0568131
$342$	0	0
$343$	0	0
$344$	23944.1	3.75285
$345$	0	0
$346$	−12098.2	−1.87978
$347$	574.458	0.0888719	0.0444359	−	0.999012i	$-0.485851\pi$
$347$	574.458	0.0888719	0.0444359	+	0.999012i	$0.485851\pi$
$348$	0	0
$349$	−11491.0	−1.76246	−0.881229	−	0.472689i	$-0.843283\pi$
$349$	−11491.0	−1.76246	−0.881229	+	0.472689i	$0.843283\pi$
$350$	0	0
$351$	0	0
$352$	−7566.92	−1.14579
$353$	1369.32	0.206464	0.103232	−	0.994657i	$-0.467082\pi$
$353$	1369.32	0.206464	0.103232	+	0.994657i	$0.467082\pi$
$354$	0	0
$355$	−4905.45	−0.733392
$356$	14049.5	2.09163
$357$	0	0
$358$	6129.21	0.904857
$359$	12915.0	1.89868	0.949339	−	0.314254i	$-0.101754\pi$
$359$	12915.0	1.89868	0.949339	+	0.314254i	$0.101754\pi$
$360$	0	0
$361$	2925.68	0.426546
$362$	−21533.9	−3.12652
$363$	0	0
$364$	0	0
$365$	−22476.6	−3.22323
$366$	0	0
$367$	1186.30	0.168731	0.0843653	−	0.996435i	$-0.473114\pi$
$367$	1186.30	0.168731	0.0843653	+	0.996435i	$0.473114\pi$
$368$	32256.9	4.56931
$369$	0	0
$370$	−21659.6	−3.04332
$371$	0	0
$372$	0	0
$373$	−6462.68	−0.897117	−0.448559	−	0.893753i	$-0.648062\pi$
$373$	−6462.68	−0.897117	−0.448559	+	0.893753i	$0.648062\pi$
$374$	−5968.02	−0.825131
$375$	0	0
$376$	−23357.6	−3.20366
$377$	8341.54	1.13955
$378$	0	0
$379$	−8555.31	−1.15952	−0.579758	−	0.814789i	$-0.696853\pi$
$379$	−8555.31	−1.15952	−0.579758	+	0.814789i	$0.696853\pi$
$380$	−41529.2	−5.60632
$381$	0	0
$382$	6417.04	0.859488
$383$	−570.982	−0.0761770	−0.0380885	−	0.999274i	$-0.512127\pi$
$383$	−570.982	−0.0761770	−0.0380885	+	0.999274i	$0.512127\pi$
$384$	0	0
$385$	0	0
$386$	9658.17	1.27354
$387$	0	0
$388$	−14615.4	−1.91234
$389$	14437.2	1.88174	0.940869	−	0.338771i	$-0.110011\pi$
$389$	14437.2	1.88174	0.940869	+	0.338771i	$0.110011\pi$
$390$	0	0
$391$	13493.1	1.74521
$392$	0	0
$393$	0	0
$394$	−4328.95	−0.553526
$395$	20700.3	2.63682
$396$	0	0
$397$	9384.81	1.18642	0.593212	−	0.805047i	$-0.297860\pi$
$397$	9384.81	1.18642	0.593212	+	0.805047i	$0.297860\pi$
$398$	−19959.4	−2.51375
$399$	0	0
$400$	58166.9	7.27086
$401$	−8167.50	−1.01712	−0.508560	−	0.861026i	$-0.669822\pi$
$401$	−8167.50	−1.01712	−0.508560	+	0.861026i	$0.669822\pi$
$402$	0	0
$403$	1477.51	0.182630
$404$	5279.74	0.650190
$405$	0	0
$406$	0	0
$407$	−2303.76	−0.280573
$408$	0	0
$409$	−11655.7	−1.40913	−0.704567	−	0.709637i	$-0.748859\pi$
$409$	−11655.7	−1.40913	−0.704567	+	0.709637i	$0.748859\pi$
$410$	−24711.9	−2.97666
$411$	0	0
$412$	−10551.8	−1.26178
$413$	0	0
$414$	0	0
$415$	−17205.7	−2.03516
$416$	−31251.3	−3.68322
$417$	0	0
$418$	−6045.23	−0.707373
$419$	−10257.6	−1.19598	−0.597992	−	0.801502i	$-0.704035\pi$
$419$	−10257.6	−1.19598	−0.597992	+	0.801502i	$0.704035\pi$
$420$	0	0
$421$	−8160.70	−0.944723	−0.472361	−	0.881405i	$-0.656598\pi$
$421$	−8160.70	−0.944723	−0.472361	+	0.881405i	$0.656598\pi$
$422$	−11247.5	−1.29744
$423$	0	0
$424$	−4018.04	−0.460220
$425$	24331.3	2.77704
$426$	0	0
$427$	0	0
$428$	−8349.32	−0.942943
$429$	0	0
$430$	−33795.5	−3.79015
$431$	8722.62	0.974835	0.487417	−	0.873169i	$-0.337939\pi$
$431$	8722.62	0.974835	0.487417	+	0.873169i	$0.337939\pi$
$432$	0	0
$433$	12628.4	1.40157	0.700787	−	0.713371i	$-0.252832\pi$
$433$	12628.4	1.40157	0.700787	+	0.713371i	$0.252832\pi$
$434$	0	0
$435$	0	0
$436$	21591.3	2.37165
$437$	13667.7	1.49614
$438$	0	0
$439$	−3487.47	−0.379153	−0.189576	−	0.981866i	$-0.560711\pi$
$439$	−3487.47	−0.379153	−0.189576	+	0.981866i	$0.560711\pi$
$440$	16200.7	1.75532
$441$	0	0
$442$	−24647.9	−2.65244
$443$	14138.0	1.51629	0.758144	−	0.652087i	$-0.226106\pi$
$443$	14138.0	1.51629	0.758144	+	0.652087i	$0.226106\pi$
$444$	0	0
$445$	−12520.9	−1.33381
$446$	20319.8	2.15734
$447$	0	0
$448$	0	0
$449$	−14007.0	−1.47223	−0.736114	−	0.676858i	$-0.763341\pi$
$449$	−14007.0	−1.47223	−0.736114	+	0.676858i	$0.763341\pi$
$450$	0	0
$451$	−2628.41	−0.274428
$452$	4122.40	0.428986
$453$	0	0
$454$	−24843.2	−2.56817
$455$	0	0
$456$	0	0
$457$	4837.29	0.495140	0.247570	−	0.968870i	$-0.420368\pi$
$457$	4837.29	0.495140	0.247570	+	0.968870i	$0.420368\pi$
$458$	−11272.2	−1.15003
$459$	0	0
$460$	−58009.9	−5.87983
$461$	10034.5	1.01378	0.506890	−	0.862011i	$-0.330795\pi$
$461$	10034.5	1.01378	0.506890	+	0.862011i	$0.330795\pi$
$462$	0	0
$463$	−15501.9	−1.55602	−0.778008	−	0.628254i	$-0.783770\pi$
$463$	−15501.9	−1.55602	−0.778008	+	0.628254i	$0.783770\pi$
$464$	42050.4	4.20720
$465$	0	0
$466$	7954.91	0.790781
$467$	9616.13	0.952851	0.476426	−	0.879215i	$-0.341932\pi$
$467$	9616.13	0.952851	0.476426	+	0.879215i	$0.341932\pi$
$468$	0	0
$469$	0	0
$470$	32967.7	3.23551
$471$	0	0
$472$	29898.8	2.91569
$473$	−3594.57	−0.349426
$474$	0	0
$475$	24646.1	2.38072
$476$	0	0
$477$	0	0
$478$	−12103.4	−1.15816
$479$	−14159.7	−1.35067	−0.675337	−	0.737509i	$-0.736002\pi$
$479$	−14159.7	−1.35067	−0.675337	+	0.737509i	$0.736002\pi$
$480$	0	0
$481$	−9514.51	−0.901922
$482$	27601.8	2.60836
$483$	0	0
$484$	−26159.8	−2.45678
$485$	13025.3	1.21948
$486$	0	0
$487$	15584.6	1.45012	0.725058	−	0.688688i	$-0.241813\pi$
$487$	15584.6	1.45012	0.725058	+	0.688688i	$0.241813\pi$
$488$	−7271.62	−0.674530
$489$	0	0
$490$	0	0
$491$	12898.0	1.18549	0.592747	−	0.805389i	$-0.298043\pi$
$491$	12898.0	1.18549	0.592747	+	0.805389i	$0.298043\pi$
$492$	0	0
$493$	17589.8	1.60690
$494$	−24966.7	−2.27390
$495$	0	0
$496$	7448.23	0.674265
$497$	0	0
$498$	0	0
$499$	6707.11	0.601707	0.300853	−	0.953670i	$-0.402729\pi$
$499$	6707.11	0.601707	0.300853	+	0.953670i	$0.402729\pi$
$500$	−52126.1	−4.66230
$501$	0	0
$502$	−9798.70	−0.871190
$503$	−14186.3	−1.25753	−0.628763	−	0.777597i	$-0.716438\pi$
$503$	−14186.3	−1.25753	−0.628763	+	0.777597i	$0.716438\pi$
$504$	0	0
$505$	−4705.31	−0.414620
$506$	−8444.26	−0.741884
$507$	0	0
$508$	36459.9	3.18435
$509$	12086.8	1.05253	0.526263	−	0.850322i	$-0.323593\pi$
$509$	12086.8	1.05253	0.526263	+	0.850322i	$0.323593\pi$
$510$	0	0
$511$	0	0
$512$	−18041.8	−1.55731
$513$	0	0
$514$	−14902.3	−1.27882
$515$	9403.80	0.804623
$516$	0	0
$517$	3506.53	0.298292
$518$	0	0
$519$	0	0
$520$	66908.8	5.64259
$521$	−12142.6	−1.02107	−0.510534	−	0.859857i	$-0.670552\pi$
$521$	−12142.6	−1.02107	−0.510534	+	0.859857i	$0.670552\pi$
$522$	0	0
$523$	−3408.38	−0.284967	−0.142484	−	0.989797i	$-0.545509\pi$
$523$	−3408.38	−0.284967	−0.142484	+	0.989797i	$0.545509\pi$
$524$	−5153.91	−0.429675
$525$	0	0
$526$	44679.1	3.70362
$527$	3115.61	0.257530
$528$	0	0
$529$	6924.66	0.569135
$530$	5671.20	0.464795
$531$	0	0
$532$	0	0
$533$	−10855.3	−0.882168
$534$	0	0
$535$	7440.92	0.601306
$536$	−19235.3	−1.55007
$537$	0	0
$538$	1294.49	0.103735
$539$	0	0
$540$	0	0
$541$	−16737.5	−1.33013	−0.665066	−	0.746785i	$-0.731597\pi$
$541$	−16737.5	−1.33013	−0.665066	+	0.746785i	$0.731597\pi$
$542$	23708.1	1.87887
$543$	0	0
$544$	−65899.5	−5.19378
$545$	−19242.2	−1.51238
$546$	0	0
$547$	−13940.6	−1.08968	−0.544840	−	0.838540i	$-0.683410\pi$
$547$	−13940.6	−1.08968	−0.544840	+	0.838540i	$0.683410\pi$
$548$	−11430.9	−0.891065
$549$	0	0
$550$	−15227.0	−1.18051
$551$	17817.3	1.37758
$552$	0	0
$553$	0	0
$554$	−33946.6	−2.60335
$555$	0	0
$556$	−12601.8	−0.961218
$557$	−818.376	−0.0622544	−0.0311272	−	0.999515i	$-0.509910\pi$
$557$	−818.376	−0.0622544	−0.0311272	+	0.999515i	$0.509910\pi$
$558$	0	0
$559$	−14845.5	−1.12325
$560$	0	0
$561$	0	0
$562$	−18317.9	−1.37490
$563$	8957.98	0.670575	0.335288	−	0.942116i	$-0.391166\pi$
$563$	8957.98	0.670575	0.335288	+	0.942116i	$0.391166\pi$
$564$	0	0
$565$	−3673.89	−0.273560
$566$	−43652.6	−3.24180
$567$	0	0
$568$	−18942.2	−1.39929
$569$	−19997.8	−1.47338	−0.736688	−	0.676233i	$-0.763611\pi$
$569$	−19997.8	−1.47338	−0.736688	+	0.676233i	$0.763611\pi$
$570$	0	0
$571$	−3391.26	−0.248546	−0.124273	−	0.992248i	$-0.539660\pi$
$571$	−3391.26	−0.248546	−0.124273	+	0.992248i	$0.539660\pi$
$572$	11270.8	0.823877
$573$	0	0
$574$	0	0
$575$	34426.8	2.49687
$576$	0	0
$577$	−3957.72	−0.285549	−0.142775	−	0.989755i	$-0.545602\pi$
$577$	−3957.72	−0.285549	−0.142775	+	0.989755i	$0.545602\pi$
$578$	−25198.1	−1.81332
$579$	0	0
$580$	−75622.2	−5.41387
$581$	0	0
$582$	0	0
$583$	603.202	0.0428509
$584$	−86792.6	−6.14983
$585$	0	0
$586$	50398.6	3.55281
$587$	12647.3	0.889283	0.444642	−	0.895709i	$-0.353331\pi$
$587$	12647.3	0.889283	0.444642	+	0.895709i	$0.353331\pi$
$588$	0	0
$589$	3155.92	0.220777
$590$	−42200.3	−2.94467
$591$	0	0
$592$	−47963.5	−3.32988
$593$	3910.60	0.270808	0.135404	−	0.990790i	$-0.456767\pi$
$593$	3910.60	0.270808	0.135404	+	0.990790i	$0.456767\pi$
$594$	0	0
$595$	0	0
$596$	−60371.0	−4.14915
$597$	0	0
$598$	−34874.7	−2.38484
$599$	13933.9	0.950455	0.475228	−	0.879863i	$-0.342366\pi$
$599$	13933.9	0.950455	0.475228	+	0.879863i	$0.342366\pi$
$600$	0	0
$601$	16095.1	1.09240	0.546202	−	0.837654i	$-0.316073\pi$
$601$	16095.1	1.09240	0.546202	+	0.837654i	$0.316073\pi$
$602$	0	0
$603$	0	0
$604$	−30676.3	−2.06656
$605$	23313.7	1.56667
$606$	0	0
$607$	−3050.91	−0.204008	−0.102004	−	0.994784i	$-0.532525\pi$
$607$	−3050.91	−0.204008	−0.102004	+	0.994784i	$0.532525\pi$
$608$	−66752.0	−4.45255
$609$	0	0
$610$	10263.4	0.681235
$611$	14481.9	0.958879
$612$	0	0
$613$	−1512.24	−0.0996394	−0.0498197	−	0.998758i	$-0.515865\pi$
$613$	−1512.24	−0.0996394	−0.0498197	+	0.998758i	$0.515865\pi$
$614$	−18351.4	−1.20619
$615$	0	0
$616$	0	0
$617$	−23466.1	−1.53114	−0.765569	−	0.643354i	$-0.777542\pi$
$617$	−23466.1	−1.53114	−0.765569	+	0.643354i	$0.777542\pi$
$618$	0	0
$619$	20736.1	1.34645	0.673226	−	0.739437i	$-0.264908\pi$
$619$	20736.1	1.34645	0.673226	+	0.739437i	$0.264908\pi$
$620$	−13394.7	−0.867651
$621$	0	0
$622$	−35731.1	−2.30335
$623$	0	0
$624$	0	0
$625$	15310.1	0.979844
$626$	3357.47	0.214364
$627$	0	0
$628$	51148.9	3.25010
$629$	−20063.2	−1.27182
$630$	0	0
$631$	15623.5	0.985673	0.492837	−	0.870122i	$-0.335960\pi$
$631$	15623.5	0.985673	0.492837	+	0.870122i	$0.335960\pi$
$632$	79933.2	5.03097
$633$	0	0
$634$	18974.8	1.18862
$635$	−32493.1	−2.03063
$636$	0	0
$637$	0	0
$638$	−11008.0	−0.683091
$639$	0	0
$640$	86423.0	5.33776
$641$	−12284.6	−0.756961	−0.378481	−	0.925609i	$-0.623553\pi$
$641$	−12284.6	−0.756961	−0.378481	+	0.925609i	$0.623553\pi$
$642$	0	0
$643$	−924.041	−0.0566728	−0.0283364	−	0.999598i	$-0.509021\pi$
$643$	−924.041	−0.0566728	−0.0283364	+	0.999598i	$0.509021\pi$
$644$	0	0
$645$	0	0
$646$	−52647.3	−3.20647
$647$	10330.2	0.627701	0.313850	−	0.949472i	$-0.398381\pi$
$647$	10330.2	0.627701	0.313850	+	0.949472i	$0.398381\pi$
$648$	0	0
$649$	−4488.52	−0.271479
$650$	−62887.4	−3.79484
$651$	0	0
$652$	28422.4	1.70722
$653$	−3064.12	−0.183627	−0.0918134	−	0.995776i	$-0.529266\pi$
$653$	−3064.12	−0.183627	−0.0918134	+	0.995776i	$0.529266\pi$
$654$	0	0
$655$	4593.17	0.274000
$656$	−54722.6	−3.25695
$657$	0	0
$658$	0	0
$659$	9157.78	0.541330	0.270665	−	0.962674i	$-0.412756\pi$
$659$	9157.78	0.541330	0.270665	+	0.962674i	$0.412756\pi$
$660$	0	0
$661$	23367.4	1.37502	0.687508	−	0.726177i	$-0.258705\pi$
$661$	23367.4	1.37502	0.687508	+	0.726177i	$0.258705\pi$
$662$	39172.9	2.29985
$663$	0	0
$664$	−66438.9	−3.88303
$665$	0	0
$666$	0	0
$667$	24888.1	1.44478
$668$	−35243.3	−2.04132
$669$	0	0
$670$	27149.3	1.56548
$671$	1091.64	0.0628053
$672$	0	0
$673$	34214.8	1.95971	0.979853	−	0.199719i	$-0.0640029\pi$
$673$	34214.8	1.95971	0.979853	+	0.199719i	$0.0640029\pi$
$674$	−19489.3	−1.11380
$675$	0	0
$676$	−1136.60	−0.0646679
$677$	−19288.8	−1.09502	−0.547511	−	0.836799i	$-0.684424\pi$
$677$	−19288.8	−1.09502	−0.547511	+	0.836799i	$0.684424\pi$
$678$	0	0
$679$	0	0
$680$	141090.	7.95672
$681$	0	0
$682$	−1949.81	−0.109475
$683$	−31119.7	−1.74343	−0.871714	−	0.490015i	$-0.836991\pi$
$683$	−31119.7	−1.74343	−0.871714	+	0.490015i	$0.836991\pi$
$684$	0	0
$685$	10187.2	0.568224
$686$	0	0
$687$	0	0
$688$	−74837.7	−4.14703
$689$	2491.22	0.137747
$690$	0	0
$691$	12732.5	0.700967	0.350483	−	0.936569i	$-0.386017\pi$
$691$	12732.5	0.700967	0.350483	+	0.936569i	$0.386017\pi$
$692$	48179.4	2.64669
$693$	0	0
$694$	−3130.91	−0.171250
$695$	11230.8	0.612961
$696$	0	0
$697$	−22890.6	−1.24396
$698$	62628.1	3.39614
$699$	0	0
$700$	0	0
$701$	9506.11	0.512184	0.256092	−	0.966652i	$-0.417565\pi$
$701$	9506.11	0.512184	0.256092	+	0.966652i	$0.417565\pi$
$702$	0	0
$703$	−20322.8	−1.09031
$704$	20299.2	1.08672
$705$	0	0
$706$	−7463.08	−0.397842
$707$	0	0
$708$	0	0
$709$	2500.48	0.132450	0.0662252	−	0.997805i	$-0.478904\pi$
$709$	2500.48	0.132450	0.0662252	+	0.997805i	$0.478904\pi$
$710$	26735.7	1.41320
$711$	0	0
$712$	−48348.9	−2.54487
$713$	4408.33	0.231548
$714$	0	0
$715$	−10044.6	−0.525379
$716$	−24408.7	−1.27402
$717$	0	0
$718$	−70389.0	−3.65863
$719$	19519.3	1.01245	0.506223	−	0.862403i	$-0.331041\pi$
$719$	19519.3	1.01245	0.506223	+	0.862403i	$0.331041\pi$
$720$	0	0
$721$	0	0
$722$	−15945.5	−0.821927
$723$	0	0
$724$	85755.9	4.40206
$725$	44879.2	2.29899
$726$	0	0
$727$	35008.3	1.78595	0.892975	−	0.450105i	$-0.148614\pi$
$727$	35008.3	1.78595	0.892975	+	0.450105i	$0.148614\pi$
$728$	0	0
$729$	0	0
$730$	122502.	6.21096
$731$	−31304.7	−1.58392
$732$	0	0
$733$	33284.0	1.67718	0.838590	−	0.544763i	$-0.183381\pi$
$733$	33284.0	1.67718	0.838590	+	0.544763i	$0.183381\pi$
$734$	−6465.54	−0.325133
$735$	0	0
$736$	−93242.3	−4.66978
$737$	2887.67	0.144326
$738$	0	0
$739$	−25554.2	−1.27203	−0.636013	−	0.771678i	$-0.719418\pi$
$739$	−25554.2	−1.27203	−0.636013	+	0.771678i	$0.719418\pi$
$740$	86256.1	4.28492
$741$	0	0
$742$	0	0
$743$	−3926.32	−0.193867	−0.0969333	−	0.995291i	$-0.530903\pi$
$743$	−3926.32	−0.193867	−0.0969333	+	0.995291i	$0.530903\pi$
$744$	0	0
$745$	53802.6	2.64587
$746$	35222.8	1.72869
$747$	0	0
$748$	23766.8	1.16177
$749$	0	0
$750$	0	0
$751$	2496.80	0.121318	0.0606589	−	0.998159i	$-0.480680\pi$
$751$	2496.80	0.121318	0.0606589	+	0.998159i	$0.480680\pi$
$752$	73004.6	3.54016
$753$	0	0
$754$	−45463.0	−2.19584
$755$	27338.7	1.31782
$756$	0	0
$757$	9014.06	0.432789	0.216395	−	0.976306i	$-0.430570\pi$
$757$	9014.06	0.432789	0.216395	+	0.976306i	$0.430570\pi$
$758$	46628.1	2.23431
$759$	0	0
$760$	142916.	6.82118
$761$	−2393.59	−0.114018	−0.0570089	−	0.998374i	$-0.518156\pi$
$761$	−2393.59	−0.114018	−0.0570089	+	0.998374i	$0.518156\pi$
$762$	0	0
$763$	0	0
$764$	−25554.9	−1.21014
$765$	0	0
$766$	3111.96	0.146788
$767$	−18537.5	−0.872688
$768$	0	0
$769$	9474.06	0.444270	0.222135	−	0.975016i	$-0.428697\pi$
$769$	9474.06	0.444270	0.222135	+	0.975016i	$0.428697\pi$
$770$	0	0
$771$	0	0
$772$	−38462.3	−1.79312
$773$	−12878.4	−0.599231	−0.299615	−	0.954060i	$-0.596858\pi$
$773$	−12878.4	−0.599231	−0.299615	+	0.954060i	$0.596858\pi$
$774$	0	0
$775$	7949.28	0.368447
$776$	50296.6	2.32673
$777$	0	0
$778$	−78685.6	−3.62599
$779$	−23186.7	−1.06643
$780$	0	0
$781$	2843.67	0.130287
$782$	−73540.2	−3.36290
$783$	0	0
$784$	0	0
$785$	−45584.0	−2.07256
$786$	0	0
$787$	−16966.5	−0.768477	−0.384239	−	0.923234i	$-0.625536\pi$
$787$	−16966.5	−0.768477	−0.384239	+	0.923234i	$0.625536\pi$
$788$	17239.4	0.779351
$789$	0	0
$790$	−112820.	−5.08097
$791$	0	0
$792$	0	0
$793$	4508.46	0.201892
$794$	−51149.0	−2.28616
$795$	0	0
$796$	79485.3	3.53930
$797$	−21.6935	−0.000964143 0	−0.000482072	−	1.00000i	$-0.500153\pi$
$797$	−21.6935	−0.000964143 0	−0.000482072		1.00000i	$0.500153\pi$
$798$	0	0
$799$	30538.0	1.35213
$800$	−168138.	−7.43073
$801$	0	0
$802$	44514.5	1.95993
$803$	13029.6	0.572609
$804$	0	0
$805$	0	0
$806$	−8052.69	−0.351916
$807$	0	0
$808$	−18169.3	−0.791083
$809$	−31766.3	−1.38052	−0.690262	−	0.723560i	$-0.742505\pi$
$809$	−31766.3	−1.38052	−0.690262	+	0.723560i	$0.742505\pi$
$810$	0	0
$811$	−31081.1	−1.34575	−0.672875	−	0.739756i	$-0.734941\pi$
$811$	−31081.1	−1.34575	−0.672875	+	0.739756i	$0.734941\pi$
$812$	0	0
$813$	0	0
$814$	12556.0	0.540646
$815$	−25330.0	−1.08868
$816$	0	0
$817$	−31709.8	−1.35788
$818$	63525.7	2.71531
$819$	0	0
$820$	98411.5	4.19107
$821$	−5021.74	−0.213472	−0.106736	−	0.994287i	$-0.534040\pi$
$821$	−5021.74	−0.213472	−0.106736	+	0.994287i	$0.534040\pi$
$822$	0	0
$823$	13443.5	0.569394	0.284697	−	0.958617i	$-0.408107\pi$
$823$	13443.5	0.569394	0.284697	+	0.958617i	$0.408107\pi$
$824$	36312.4	1.53520
$825$	0	0
$826$	0	0
$827$	37439.2	1.57423	0.787116	−	0.616805i	$-0.211573\pi$
$827$	37439.2	1.57423	0.787116	+	0.616805i	$0.211573\pi$
$828$	0	0
$829$	3890.21	0.162982	0.0814912	−	0.996674i	$-0.474032\pi$
$829$	3890.21	0.162982	0.0814912	+	0.996674i	$0.474032\pi$
$830$	93774.2	3.92162
$831$	0	0
$832$	83835.3	3.49335
$833$	0	0
$834$	0	0
$835$	31408.9	1.30173
$836$	24074.3	0.995965
$837$	0	0
$838$	55906.0	2.30458
$839$	−15062.0	−0.619783	−0.309891	−	0.950772i	$-0.600293\pi$
$839$	−15062.0	−0.619783	−0.309891	+	0.950772i	$0.600293\pi$
$840$	0	0
$841$	8055.35	0.330286
$842$	44477.4	1.82042
$843$	0	0
$844$	44791.5	1.82676
$845$	1012.94	0.0412381
$846$	0	0
$847$	0	0
$848$	12558.4	0.508560
$849$	0	0
$850$	−132611.	−5.35118
$851$	−28387.8	−1.14350
$852$	0	0
$853$	39207.9	1.57380	0.786900	−	0.617081i	$-0.211685\pi$
$853$	39207.9	1.57380	0.786900	+	0.617081i	$0.211685\pi$
$854$	0	0
$855$	0	0
$856$	28732.8	1.14727
$857$	37029.6	1.47597	0.737986	−	0.674816i	$-0.235777\pi$
$857$	37029.6	1.47597	0.737986	+	0.674816i	$0.235777\pi$
$858$	0	0
$859$	−37646.3	−1.49531	−0.747657	−	0.664085i	$-0.768821\pi$
$859$	−37646.3	−1.49531	−0.747657	+	0.664085i	$0.768821\pi$
$860$	134586.	5.33644
$861$	0	0
$862$	−47540.0	−1.87844
$863$	26698.1	1.05309	0.526543	−	0.850149i	$-0.323488\pi$
$863$	26698.1	1.05309	0.526543	+	0.850149i	$0.323488\pi$
$864$	0	0
$865$	−42937.5	−1.68777
$866$	−68827.2	−2.70074
$867$	0	0
$868$	0	0
$869$	−11999.8	−0.468431
$870$	0	0
$871$	11926.0	0.463947
$872$	−74303.0	−2.88557
$873$	0	0
$874$	−74491.6	−2.88297
$875$	0	0
$876$	0	0
$877$	−22177.5	−0.853912	−0.426956	−	0.904272i	$-0.640414\pi$
$877$	−22177.5	−0.853912	−0.426956	+	0.904272i	$0.640414\pi$
$878$	19007.4	0.730602
$879$	0	0
$880$	−50635.6	−1.93969
$881$	11503.1	0.439898	0.219949	−	0.975511i	$-0.429411\pi$
$881$	11503.1	0.439898	0.219949	+	0.975511i	$0.429411\pi$
$882$	0	0
$883$	41751.6	1.59123	0.795613	−	0.605806i	$-0.207149\pi$
$883$	41751.6	1.59123	0.795613	+	0.605806i	$0.207149\pi$
$884$	98156.6	3.73457
$885$	0	0
$886$	−77054.7	−2.92179
$887$	4670.18	0.176786	0.0883930	−	0.996086i	$-0.471827\pi$
$887$	4670.18	0.176786	0.0883930	+	0.996086i	$0.471827\pi$
$888$	0	0
$889$	0	0
$890$	68241.3	2.57017
$891$	0	0
$892$	−80920.8	−3.03748
$893$	30933.0	1.15917
$894$	0	0
$895$	21753.1	0.812430
$896$	0	0
$897$	0	0
$898$	76340.8	2.83689
$899$	5746.75	0.213198
$900$	0	0
$901$	5253.22	0.194240
$902$	14325.4	0.528805
$903$	0	0
$904$	−14186.6	−0.521945
$905$	−76425.7	−2.80715
$906$	0	0
$907$	33783.8	1.23679	0.618397	−	0.785866i	$-0.287782\pi$
$907$	33783.8	1.23679	0.618397	+	0.785866i	$0.287782\pi$
$908$	98934.5	3.61592
$909$	0	0
$910$	0	0
$911$	34880.5	1.26854	0.634271	−	0.773111i	$-0.281300\pi$
$911$	34880.5	1.26854	0.634271	+	0.773111i	$0.281300\pi$
$912$	0	0
$913$	9974.04	0.361547
$914$	−26364.2	−0.954102
$915$	0	0
$916$	44889.9	1.61922
$917$	0	0
$918$	0	0
$919$	−5964.51	−0.214092	−0.107046	−	0.994254i	$-0.534139\pi$
$919$	−5964.51	−0.214092	−0.107046	+	0.994254i	$0.534139\pi$
$920$	199631.	7.15397
$921$	0	0
$922$	−54689.9	−1.95349
$923$	11744.3	0.418818
$924$	0	0
$925$	−51190.0	−1.81959
$926$	84488.5	2.99834
$927$	0	0
$928$	−121552.	−4.29971
$929$	−54065.1	−1.90939	−0.954693	−	0.297594i	$-0.903816\pi$
$929$	−54065.1	−1.90939	−0.954693	+	0.297594i	$0.903816\pi$
$930$	0	0
$931$	0	0
$932$	−31679.3	−1.11340
$933$	0	0
$934$	−52409.8	−1.83608
$935$	−21181.0	−0.740847
$936$	0	0
$937$	14678.5	0.511766	0.255883	−	0.966708i	$-0.417634\pi$
$937$	14678.5	0.511766	0.255883	+	0.966708i	$0.417634\pi$
$938$	0	0
$939$	0	0
$940$	−131289.	−4.55551
$941$	−22198.8	−0.769032	−0.384516	−	0.923118i	$-0.625632\pi$
$941$	−22198.8	−0.769032	−0.384516	+	0.923118i	$0.625632\pi$
$942$	0	0
$943$	−32388.3	−1.11846
$944$	−93449.3	−3.22194
$945$	0	0
$946$	19591.1	0.673322
$947$	−3314.30	−0.113728	−0.0568639	−	0.998382i	$-0.518110\pi$
$947$	−3314.30	−0.113728	−0.0568639	+	0.998382i	$0.518110\pi$
$948$	0	0
$949$	53812.1	1.84069
$950$	−134326.	−4.58749
$951$	0	0
$952$	0	0
$953$	45137.3	1.53425	0.767125	−	0.641498i	$-0.221687\pi$
$953$	45137.3	1.53425	0.767125	+	0.641498i	$0.221687\pi$
$954$	0	0
$955$	22774.6	0.771694
$956$	48200.2	1.63066
$957$	0	0
$958$	77173.1	2.60266
$959$	0	0
$960$	0	0
$961$	−28773.1	−0.965832
$962$	51856.0	1.73794
$963$	0	0
$964$	−109920.	−3.67250
$965$	34277.6	1.14346
$966$	0	0
$967$	13423.9	0.446416	0.223208	−	0.974771i	$-0.428347\pi$
$967$	13423.9	0.446416	0.223208	+	0.974771i	$0.428347\pi$
$968$	90024.7	2.98916
$969$	0	0
$970$	−70990.4	−2.34986
$971$	−50282.5	−1.66183	−0.830917	−	0.556396i	$-0.812184\pi$
$971$	−50282.5	−1.66183	−0.830917	+	0.556396i	$0.812184\pi$
$972$	0	0
$973$	0	0
$974$	−84939.2	−2.79428
$975$	0	0
$976$	22727.6	0.745381
$977$	−24918.5	−0.815980	−0.407990	−	0.912986i	$-0.633770\pi$
$977$	−24918.5	−0.815980	−0.407990	+	0.912986i	$0.633770\pi$
$978$	0	0
$979$	7258.30	0.236952
$980$	0	0
$981$	0	0
$982$	−70296.5	−2.28437
$983$	5830.48	0.189179	0.0945897	−	0.995516i	$-0.469846\pi$
$983$	5830.48	0.189179	0.0945897	+	0.995516i	$0.469846\pi$
$984$	0	0
$985$	−15363.8	−0.496986
$986$	−95867.7	−3.09640
$987$	0	0
$988$	99426.5	3.20160
$989$	−44293.6	−1.42412
$990$	0	0
$991$	15669.9	0.502293	0.251146	−	0.967949i	$-0.419192\pi$
$991$	15669.9	0.502293	0.251146	+	0.967949i	$0.419192\pi$
$992$	−21530.0	−0.689091
$993$	0	0
$994$	0	0
$995$	−70837.4	−2.25698
$996$	0	0
$997$	−11367.5	−0.361095	−0.180548	−	0.983566i	$-0.557787\pi$
$997$	−11367.5	−0.361095	−0.180548	+	0.983566i	$0.557787\pi$
$998$	−36555.1	−1.15945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bg.1.1	yes	6
3.2	odd	2	inner	1323.4.a.bg.1.6	yes	6
7.6	odd	2		1323.4.a.bf.1.1	✓	6
21.20	even	2		1323.4.a.bf.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bf.1.1	✓	6	7.6	odd	2
1323.4.a.bf.1.6	yes	6	21.20	even	2
1323.4.a.bg.1.1	yes	6	1.1	even	1	trivial
1323.4.a.bg.1.6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-5.45019 q^{2} +21.7046 q^{4} -19.3432 q^{5} -74.6929 q^{8} +105.424 q^{10} +11.2132 q^{11} +46.3102 q^{13} +233.454 q^{16} +97.6541 q^{17} +98.9176 q^{19} -419.836 q^{20} -61.1139 q^{22} +138.173 q^{23} +249.158 q^{25} -252.400 q^{26} +180.123 q^{29} +31.9046 q^{31} -674.825 q^{32} -532.234 q^{34} -205.452 q^{37} -539.120 q^{38} +1444.80 q^{40} -234.404 q^{41} -320.568 q^{43} +243.377 q^{44} -753.067 q^{46} +312.715 q^{47} -1357.96 q^{50} +1005.15 q^{52} +53.7942 q^{53} -216.898 q^{55} -981.706 q^{58} -400.291 q^{59} +97.3536 q^{61} -173.886 q^{62} +1810.30 q^{64} -895.786 q^{65} +257.525 q^{67} +2119.55 q^{68} +253.601 q^{71} +1161.99 q^{73} +1119.75 q^{74} +2146.97 q^{76} -1070.16 q^{79} -4515.73 q^{80} +1277.55 q^{82} +889.495 q^{83} -1888.94 q^{85} +1747.16 q^{86} -837.543 q^{88} +647.303 q^{89} +2998.98 q^{92} -1704.36 q^{94} -1913.38 q^{95} -673.379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58}+ \cdots + 3312 q^{97}+O(q^{100}) \) 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

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Representations

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Database

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