LMFDB - Embedded newform 1323.4.a.bf.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:	$1$
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} - 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	$-0.167285\pi$
$5$	19.3432	1.73011	0.865053	+	0.501681i	$0.167285\pi$
$6$	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	$-0.664468\pi$
$13$	−46.3102	−0.988010	−0.494005	+	0.869459i	$0.664468\pi$
$14$	0	0
$15$	0	0
$16$	233.454	3.64771
$17$	−97.6541	−1.39321	−0.696606	−	0.717454i	$-0.745307\pi$
$17$	−97.6541	−1.39321	−0.696606	+	0.717454i	$0.745307\pi$
$18$	0	0
$19$	−98.9176	−1.19438	−0.597191	−	0.802099i	$-0.703716\pi$
$19$	−98.9176	−1.19438	−0.597191	+	0.802099i	$0.703716\pi$
$20$	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	$-0.529461\pi$
$31$	−31.9046	−0.184846	−0.0924230	+	0.995720i	$0.529461\pi$
$32$	−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	$-0.352693\pi$
$41$	234.404	0.892874	0.446437	+	0.894815i	$0.352693\pi$
$42$	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	$-0.661274\pi$
$47$	−312.715	−0.970516	−0.485258	+	0.874371i	$0.661274\pi$
$48$	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	$-0.354397\pi$
$59$	400.291	0.883278	0.441639	+	0.897193i	$0.354397\pi$
$60$	0	0
$61$	−97.3536	−0.204342	−0.102171	−	0.994767i	$-0.532579\pi$
$61$	−97.3536	−0.204342	−0.102171	+	0.994767i	$0.532579\pi$
$62$	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	$-0.881511\pi$
$73$	−1161.99	−1.86303	−0.931514	+	0.363706i	$0.881511\pi$
$74$	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	$-0.700148\pi$
$83$	−889.495	−1.17632	−0.588161	+	0.808744i	$0.700148\pi$
$84$	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	$-0.625961\pi$
$89$	−647.303	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$90$	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	$-0.385356\pi$
$97$	673.379	0.704859	0.352429	+	0.935838i	$0.385356\pi$
$98$	0	0
$99$	0	0
$100$	5407.89	5.40789
$101$	−243.254	−0.239650	−0.119825	−	0.992795i	$-0.538233\pi$
$101$	−243.254	−0.239650	−0.119825	+	0.992795i	$0.538233\pi$
$102$	0	0
$103$	486.156	0.465072	0.232536	−	0.972588i	$-0.425298\pi$
$103$	486.156	0.465072	0.232536	+	0.972588i	$0.425298\pi$
$104$	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	$-0.474768\pi$
$131$	237.457	0.158372	0.0791859	+	0.996860i	$0.474768\pi$
$132$	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	$-0.443314\pi$
$139$	580.607	0.354291	0.177145	+	0.984185i	$0.443314\pi$
$140$	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	$-0.704423\pi$
$157$	−2356.59	−1.19794	−0.598970	+	0.800772i	$0.704423\pi$
$158$	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	$-0.377230\pi$
$167$	1623.77	0.752401	0.376201	+	0.926538i	$0.377230\pi$
$168$	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	$-0.662188\pi$
$173$	−2219.78	−0.975529	−0.487765	+	0.872975i	$0.662188\pi$
$174$	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	$-0.801222\pi$
$181$	−3951.04	−1.62253	−0.811267	+	0.584676i	$0.801222\pi$
$182$	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	$-0.726182\pi$
$199$	−3662.14	−1.30453	−0.652267	+	0.757990i	$0.726182\pi$
$200$	−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	$-0.310884\pi$
$223$	3728.28	1.11957	0.559784	+	0.828638i	$0.310884\pi$
$224$	0	0
$225$	0	0
$226$	−1035.17	−0.304683
$227$	−4558.22	−1.33277	−0.666387	−	0.745606i	$-0.732160\pi$
$227$	−4558.22	−1.33277	−0.666387	+	0.745606i	$0.732160\pi$
$228$	0	0
$229$	−2068.22	−0.596819	−0.298410	−	0.954438i	$-0.596456\pi$
$229$	−2068.22	−0.596819	−0.298410	+	0.954438i	$0.596456\pi$
$230$	−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	$-0.263360\pi$
$241$	5064.37	1.35363	0.676815	+	0.736153i	$0.263360\pi$
$242$	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	$-0.572583\pi$
$251$	−1797.86	−0.452112	−0.226056	+	0.974114i	$0.572583\pi$
$252$	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	$-0.607665\pi$
$257$	−2734.28	−0.663656	−0.331828	+	0.943340i	$0.607665\pi$
$258$	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	$-0.491431\pi$
$269$	237.512	0.0538342	0.0269171	+	0.999638i	$0.491431\pi$
$270$	0	0
$271$	4349.95	0.975057	0.487528	−	0.873107i	$-0.337899\pi$
$271$	4349.95	0.975057	0.487528	+	0.873107i	$0.337899\pi$
$272$	−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	$-0.818138\pi$
$283$	−8009.37	−1.68236	−0.841180	+	0.540755i	$0.818138\pi$
$284$	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	$-0.126652\pi$
$293$	9247.12	1.84376	0.921881	+	0.387472i	$0.126652\pi$
$294$	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	$-0.601328\pi$
$307$	−3367.11	−0.625965	−0.312983	+	0.949759i	$0.601328\pi$
$308$	0	0
$309$	0	0
$310$	3363.51	0.616240
$311$	−6555.92	−1.19534	−0.597672	−	0.801740i	$-0.703908\pi$
$311$	−6555.92	−1.19534	−0.597672	+	0.801740i	$0.703908\pi$
$312$	0	0
$313$	616.028	0.111246	0.0556229	−	0.998452i	$-0.482286\pi$
$313$	616.028	0.111246	0.0556229	+	0.998452i	$0.482286\pi$
$314$	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	$-0.156717\pi$
$349$	11491.0	1.76246	0.881229	+	0.472689i	$0.156717\pi$
$350$	0	0
$351$	0	0
$352$	−7566.92	−1.14579
$353$	−1369.32	−0.206464	−0.103232	−	0.994657i	$-0.532918\pi$
$353$	−1369.32	−0.206464	−0.103232	+	0.994657i	$0.532918\pi$
$354$	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	$-0.526886\pi$
$367$	−1186.30	−0.168731	−0.0843653	+	0.996435i	$0.526886\pi$
$368$	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	$-0.487873\pi$
$383$	570.982	0.0761770	0.0380885	+	0.999274i	$0.487873\pi$
$384$	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	$-0.702140\pi$
$397$	−9384.81	−1.18642	−0.593212	+	0.805047i	$0.702140\pi$
$398$	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	$-0.251141\pi$
$409$	11655.7	1.40913	0.704567	+	0.709637i	$0.251141\pi$
$410$	−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	$-0.295965\pi$
$419$	10257.6	1.19598	0.597992	+	0.801502i	$0.295965\pi$
$420$	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	$-0.747168\pi$
$433$	−12628.4	−1.40157	−0.700787	+	0.713371i	$0.747168\pi$
$434$	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	$-0.439289\pi$
$439$	3487.47	0.379153	0.189576	+	0.981866i	$0.439289\pi$
$440$	−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	$-0.669205\pi$
$461$	−10034.5	−1.01378	−0.506890	+	0.862011i	$0.669205\pi$
$462$	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	$-0.658068\pi$
$467$	−9616.13	−0.952851	−0.476426	+	0.879215i	$0.658068\pi$
$468$	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	$-0.263998\pi$
$479$	14159.7	1.35067	0.675337	+	0.737509i	$0.263998\pi$
$480$	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	$-0.283562\pi$
$503$	14186.3	1.25753	0.628763	+	0.777597i	$0.283562\pi$
$504$	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	$-0.676407\pi$
$509$	−12086.8	−1.05253	−0.526263	+	0.850322i	$0.676407\pi$
$510$	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	$-0.329448\pi$
$521$	12142.6	1.02107	0.510534	+	0.859857i	$0.329448\pi$
$522$	0	0
$523$	3408.38	0.284967	0.142484	−	0.989797i	$-0.454491\pi$
$523$	3408.38	0.284967	0.142484	+	0.989797i	$0.454491\pi$
$524$	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	$-0.608834\pi$
$563$	−8957.98	−0.670575	−0.335288	+	0.942116i	$0.608834\pi$
$564$	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	$-0.454398\pi$
$577$	3957.72	0.285549	0.142775	+	0.989755i	$0.454398\pi$
$578$	−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	$-0.646669\pi$
$587$	−12647.3	−0.889283	−0.444642	+	0.895709i	$0.646669\pi$
$588$	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	$-0.543233\pi$
$593$	−3910.60	−0.270808	−0.135404	+	0.990790i	$0.543233\pi$
$594$	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	$-0.683927\pi$
$601$	−16095.1	−1.09240	−0.546202	+	0.837654i	$0.683927\pi$
$602$	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	$-0.467475\pi$
$607$	3050.91	0.204008	0.102004	+	0.994784i	$0.467475\pi$
$608$	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	$-0.735092\pi$
$619$	−20736.1	−1.34645	−0.673226	+	0.739437i	$0.735092\pi$
$620$	−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	$-0.490979\pi$
$643$	924.041	0.0566728	0.0283364	+	0.999598i	$0.490979\pi$
$644$	0	0
$645$	0	0
$646$	−52647.3	−3.20647
$647$	−10330.2	−0.627701	−0.313850	−	0.949472i	$-0.601619\pi$
$647$	−10330.2	−0.627701	−0.313850	+	0.949472i	$0.601619\pi$
$648$	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	$-0.741295\pi$
$661$	−23367.4	−1.37502	−0.687508	+	0.726177i	$0.741295\pi$
$662$	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	$-0.315576\pi$
$677$	19288.8	1.09502	0.547511	+	0.836799i	$0.315576\pi$
$678$	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	$-0.613983\pi$
$691$	−12732.5	−0.700967	−0.350483	+	0.936569i	$0.613983\pi$
$692$	−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	$-0.668959\pi$
$719$	−19519.3	−1.01245	−0.506223	+	0.862403i	$0.668959\pi$
$720$	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	$-0.851386\pi$
$727$	−35008.3	−1.78595	−0.892975	+	0.450105i	$0.851386\pi$
$728$	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	$-0.816619\pi$
$733$	−33284.0	−1.67718	−0.838590	+	0.544763i	$0.816619\pi$
$734$	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	$-0.481844\pi$
$761$	2393.59	0.114018	0.0570089	+	0.998374i	$0.481844\pi$
$762$	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	$-0.571303\pi$
$769$	−9474.06	−0.444270	−0.222135	+	0.975016i	$0.571303\pi$
$770$	0	0
$771$	0	0
$772$	−38462.3	−1.79312
$773$	12878.4	0.599231	0.299615	−	0.954060i	$-0.403142\pi$
$773$	12878.4	0.599231	0.299615	+	0.954060i	$0.403142\pi$
$774$	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	$-0.374464\pi$
$787$	16966.5	0.768477	0.384239	+	0.923234i	$0.374464\pi$
$788$	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	$-0.499847\pi$
$797$	21.6935	0.000964143 0	0.000482072		1.00000i	$0.499847\pi$
$798$	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	$-0.265059\pi$
$811$	31081.1	1.34575	0.672875	+	0.739756i	$0.265059\pi$
$812$	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	$-0.525968\pi$
$829$	−3890.21	−0.162982	−0.0814912	+	0.996674i	$0.525968\pi$
$830$	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	$-0.399707\pi$
$839$	15062.0	0.619783	0.309891	+	0.950772i	$0.399707\pi$
$840$	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	$-0.788315\pi$
$853$	−39207.9	−1.57380	−0.786900	+	0.617081i	$0.788315\pi$
$854$	0	0
$855$	0	0
$856$	28732.8	1.14727
$857$	−37029.6	−1.47597	−0.737986	−	0.674816i	$-0.764223\pi$
$857$	−37029.6	−1.47597	−0.737986	+	0.674816i	$0.764223\pi$
$858$	0	0
$859$	37646.3	1.49531	0.747657	−	0.664085i	$-0.231179\pi$
$859$	37646.3	1.49531	0.747657	+	0.664085i	$0.231179\pi$
$860$	−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	$-0.570589\pi$
$881$	−11503.1	−0.439898	−0.219949	+	0.975511i	$0.570589\pi$
$882$	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	$-0.528173\pi$
$887$	−4670.18	−0.176786	−0.0883930	+	0.996086i	$0.528173\pi$
$888$	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	$-0.0961841\pi$
$929$	54065.1	1.90939	0.954693	+	0.297594i	$0.0961841\pi$
$930$	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	$-0.582366\pi$
$937$	−14678.5	−0.511766	−0.255883	+	0.966708i	$0.582366\pi$
$938$	0	0
$939$	0	0
$940$	−131289.	−4.55551
$941$	22198.8	0.769032	0.384516	−	0.923118i	$-0.374368\pi$
$941$	22198.8	0.769032	0.384516	+	0.923118i	$0.374368\pi$
$942$	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	$-0.187816\pi$
$971$	50282.5	1.66183	0.830917	+	0.556396i	$0.187816\pi$
$972$	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	$-0.530154\pi$
$983$	−5830.48	−0.189179	−0.0945897	+	0.995516i	$0.530154\pi$
$984$	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	$-0.442213\pi$
$997$	11367.5	0.361095	0.180548	+	0.983566i	$0.442213\pi$
$998$	−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.bf.1.1	✓	6
3.2	odd	2	inner	1323.4.a.bf.1.6	yes	6
7.6	odd	2		1323.4.a.bg.1.1	yes	6
21.20	even	2		1323.4.a.bg.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bf.1.1	✓	6	1.1	even	1	trivial
1323.4.a.bf.1.6	yes	6	3.2	odd	2	inner
1323.4.a.bg.1.1	yes	6	7.6	odd	2
1323.4.a.bg.1.6	yes	6	21.20	even	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 $\F_{q}$
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

Fields

Representations

Groups

Database

Properties

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