LMFDB - Embedded newform 324.6.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$51.9643576194$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 110x^{3} + 39x^{2} + 2214x - 1944 $ x^5 - 2x^4 - 110x^3 + 39x^2 + 2214x - 1944
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$10.4099$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.1436 q^{5} -151.408 q^{7} +O(q^{10})$	$q+28.1436 q^{5} -151.408 q^{7} -277.747 q^{11} +583.858 q^{13} -1612.01 q^{17} +1368.76 q^{19} +856.028 q^{23} -2332.94 q^{25} +8534.97 q^{29} +2938.38 q^{31} -4261.17 q^{35} +4036.80 q^{37} +18899.6 q^{41} -20317.1 q^{43} -295.779 q^{47} +6117.35 q^{49} +3039.13 q^{53} -7816.81 q^{55} -17236.6 q^{59} +25652.5 q^{61} +16431.9 q^{65} -26280.3 q^{67} +76665.7 q^{71} +1496.33 q^{73} +42053.1 q^{77} +99274.2 q^{79} +50051.3 q^{83} -45367.7 q^{85} +136635. q^{89} -88400.8 q^{91} +38522.0 q^{95} -66649.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 21 q^{5} - 29 q^{7}+O(q^{10}) $ 5 * q + 21 * q^5 - 29 * q^7	$ 5 q + 21 q^{5} - 29 q^{7} - 177 q^{11} + 181 q^{13} + 1140 q^{17} - 416 q^{19} - 399 q^{23} + 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 18573 q^{35} - 7586 q^{37} + 18435 q^{41} - 1469 q^{43} + 25155 q^{47} + 4056 q^{49} + 58422 q^{53} + 7389 q^{55} + 90537 q^{59} - 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 114684 q^{71} + 7600 q^{73} + 211983 q^{77} - 29993 q^{79} + 228951 q^{83} + 49662 q^{85} + 299166 q^{89} + 62465 q^{91} + 394764 q^{95} - 40541 q^{97}+O(q^{100}) $ 5 * q + 21 * q^5 - 29 * q^7 - 177 * q^11 + 181 * q^13 + 1140 * q^17 - 416 * q^19 - 399 * q^23 + 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 18573 * q^35 - 7586 * q^37 + 18435 * q^41 - 1469 * q^43 + 25155 * q^47 + 4056 * q^49 + 58422 * q^53 + 7389 * q^55 + 90537 * q^59 - 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 114684 * q^71 + 7600 * q^73 + 211983 * q^77 - 29993 * q^79 + 228951 * q^83 + 49662 * q^85 + 299166 * q^89 + 62465 * q^91 + 394764 * q^95 - 40541 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	28.1436	0.503449	0.251724	−	0.967799i	$-0.419002\pi$
$5$	28.1436	0.503449	0.251724	+	0.967799i	$0.419002\pi$
$6$	0	0
$7$	−151.408	−1.16789	−0.583947	−	0.811792i	$-0.698492\pi$
$7$	−151.408	−1.16789	−0.583947	+	0.811792i	$0.698492\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−277.747	−0.692098	−0.346049	−	0.938216i	$-0.612477\pi$
$11$	−277.747	−0.692098	−0.346049	+	0.938216i	$0.612477\pi$
$12$	0	0
$13$	583.858	0.958185	0.479092	−	0.877765i	$-0.340966\pi$
$13$	583.858	0.958185	0.479092	+	0.877765i	$0.340966\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1612.01	−1.35283	−0.676417	−	0.736519i	$-0.736468\pi$
$17$	−1612.01	−1.35283	−0.676417	+	0.736519i	$0.736468\pi$
$18$	0	0
$19$	1368.76	0.869851	0.434925	−	0.900467i	$-0.356775\pi$
$19$	1368.76	0.869851	0.434925	+	0.900467i	$0.356775\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	856.028	0.337418	0.168709	−	0.985666i	$-0.446040\pi$
$23$	856.028	0.337418	0.168709	+	0.985666i	$0.446040\pi$
$24$	0	0
$25$	−2332.94	−0.746539
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8534.97	1.88455	0.942274	−	0.334844i	$-0.108683\pi$
$29$	8534.97	1.88455	0.942274	+	0.334844i	$0.108683\pi$
$30$	0	0
$31$	2938.38	0.549166	0.274583	−	0.961563i	$-0.411460\pi$
$31$	2938.38	0.549166	0.274583	+	0.961563i	$0.411460\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4261.17	−0.587975
$36$	0	0
$37$	4036.80	0.484767	0.242383	−	0.970181i	$-0.422071\pi$
$37$	4036.80	0.484767	0.242383	+	0.970181i	$0.422071\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	18899.6	1.75587	0.877937	−	0.478776i	$-0.158919\pi$
$41$	18899.6	1.75587	0.877937	+	0.478776i	$0.158919\pi$
$42$	0	0
$43$	−20317.1	−1.67568	−0.837840	−	0.545916i	$-0.816182\pi$
$43$	−20317.1	−1.67568	−0.837840	+	0.545916i	$0.816182\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−295.779	−0.0195309	−0.00976546	−	0.999952i	$-0.503108\pi$
$47$	−295.779	−0.0195309	−0.00976546	+	0.999952i	$0.503108\pi$
$48$	0	0
$49$	6117.35	0.363976
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3039.13	0.148614	0.0743069	−	0.997235i	$-0.476326\pi$
$53$	3039.13	0.148614	0.0743069	+	0.997235i	$0.476326\pi$
$54$	0	0
$55$	−7816.81	−0.348436
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−17236.6	−0.644647	−0.322324	−	0.946630i	$-0.604464\pi$
$59$	−17236.6	−0.644647	−0.322324	+	0.946630i	$0.604464\pi$
$60$	0	0
$61$	25652.5	0.882683	0.441342	−	0.897339i	$-0.354503\pi$
$61$	25652.5	0.882683	0.441342	+	0.897339i	$0.354503\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16431.9	0.482397
$66$	0	0
$67$	−26280.3	−0.715225	−0.357613	−	0.933870i	$-0.616409\pi$
$67$	−26280.3	−0.715225	−0.357613	+	0.933870i	$0.616409\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	76665.7	1.80491	0.902454	−	0.430786i	$-0.141764\pi$
$71$	76665.7	1.80491	0.902454	+	0.430786i	$0.141764\pi$
$72$	0	0
$73$	1496.33	0.0328640	0.0164320	−	0.999865i	$-0.494769\pi$
$73$	1496.33	0.0328640	0.0164320	+	0.999865i	$0.494769\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42053.1	0.808297
$78$	0	0
$79$	99274.2	1.78965	0.894826	−	0.446414i	$-0.147299\pi$
$79$	99274.2	1.78965	0.894826	+	0.446414i	$0.147299\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	50051.3	0.797481	0.398740	−	0.917064i	$-0.369447\pi$
$83$	50051.3	0.797481	0.398740	+	0.917064i	$0.369447\pi$
$84$	0	0
$85$	−45367.7	−0.681082
$86$	0	0
$87$	0	0
$88$	0	0
$89$	136635.	1.82847	0.914235	−	0.405185i	$-0.132793\pi$
$89$	136635.	1.82847	0.914235	+	0.405185i	$0.132793\pi$
$90$	0	0
$91$	−88400.8	−1.11906
$92$	0	0
$93$	0	0
$94$	0	0
$95$	38522.0	0.437925
$96$	0	0
$97$	−66649.9	−0.719234	−0.359617	−	0.933100i	$-0.617093\pi$
$97$	−66649.9	−0.719234	−0.359617	+	0.933100i	$0.617093\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−24677.4	−0.240711	−0.120355	−	0.992731i	$-0.538403\pi$
$101$	−24677.4	−0.240711	−0.120355	+	0.992731i	$0.538403\pi$
$102$	0	0
$103$	115767.	1.07521	0.537603	−	0.843198i	$-0.319330\pi$
$103$	115767.	1.07521	0.537603	+	0.843198i	$0.319330\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	84364.3	0.712360	0.356180	−	0.934417i	$-0.384079\pi$
$107$	84364.3	0.712360	0.356180	+	0.934417i	$0.384079\pi$
$108$	0	0
$109$	198400.	1.59947	0.799735	−	0.600354i	$-0.204973\pi$
$109$	198400.	1.59947	0.799735	+	0.600354i	$0.204973\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	228905.	1.68640	0.843199	−	0.537602i	$-0.180670\pi$
$113$	228905.	1.68640	0.843199	+	0.537602i	$0.180670\pi$
$114$	0	0
$115$	24091.8	0.169873
$116$	0	0
$117$	0	0
$118$	0	0
$119$	244070.	1.57997
$120$	0	0
$121$	−83907.7	−0.521001
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−153606.	−0.879293
$126$	0	0
$127$	−246629.	−1.35686	−0.678430	−	0.734665i	$-0.737339\pi$
$127$	−246629.	−1.35686	−0.678430	+	0.734665i	$0.737339\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−169225.	−0.861563	−0.430781	−	0.902456i	$-0.641762\pi$
$131$	−169225.	−0.861563	−0.430781	+	0.902456i	$0.641762\pi$
$132$	0	0
$133$	−207242.	−1.01589
$134$	0	0
$135$	0	0
$136$	0	0
$137$	99970.6	0.455062	0.227531	−	0.973771i	$-0.426935\pi$
$137$	99970.6	0.455062	0.227531	+	0.973771i	$0.426935\pi$
$138$	0	0
$139$	−37398.7	−0.164180	−0.0820899	−	0.996625i	$-0.526159\pi$
$139$	−37398.7	−0.164180	−0.0820899	+	0.996625i	$0.526159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−162165.	−0.663158
$144$	0	0
$145$	240205.	0.948773
$146$	0	0
$147$	0	0
$148$	0	0
$149$	254534.	0.939246	0.469623	−	0.882867i	$-0.344390\pi$
$149$	254534.	0.939246	0.469623	+	0.882867i	$0.344390\pi$
$150$	0	0
$151$	−236726.	−0.844895	−0.422448	−	0.906387i	$-0.638829\pi$
$151$	−236726.	−0.844895	−0.422448	+	0.906387i	$0.638829\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	82696.6	0.276477
$156$	0	0
$157$	254812.	0.825031	0.412515	−	0.910951i	$-0.364650\pi$
$157$	254812.	0.825031	0.412515	+	0.910951i	$0.364650\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−129609.	−0.394069
$162$	0	0
$163$	215050.	0.633973	0.316987	−	0.948430i	$-0.397329\pi$
$163$	215050.	0.633973	0.316987	+	0.948430i	$0.397329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	192337.	0.533669	0.266834	−	0.963742i	$-0.414022\pi$
$167$	192337.	0.533669	0.266834	+	0.963742i	$0.414022\pi$
$168$	0	0
$169$	−30402.3	−0.0818823
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8301.15	−0.0210874	−0.0105437	−	0.999944i	$-0.503356\pi$
$173$	−8301.15	−0.0210874	−0.0105437	+	0.999944i	$0.503356\pi$
$174$	0	0
$175$	353225.	0.871879
$176$	0	0
$177$	0	0
$178$	0	0
$179$	574496.	1.34015	0.670077	−	0.742292i	$-0.266261\pi$
$179$	574496.	1.34015	0.670077	+	0.742292i	$0.266261\pi$
$180$	0	0
$181$	−224707.	−0.509823	−0.254912	−	0.966964i	$-0.582046\pi$
$181$	−224707.	−0.509823	−0.254912	+	0.966964i	$0.582046\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	113610.	0.244055
$186$	0	0
$187$	447730.	0.936293
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−478082.	−0.948241	−0.474121	−	0.880460i	$-0.657234\pi$
$191$	−478082.	−0.948241	−0.474121	+	0.880460i	$0.657234\pi$
$192$	0	0
$193$	−526050.	−1.01656	−0.508281	−	0.861191i	$-0.669719\pi$
$193$	−526050.	−1.01656	−0.508281	+	0.861191i	$0.669719\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−222278.	−0.408067	−0.204034	−	0.978964i	$-0.565405\pi$
$197$	−222278.	−0.408067	−0.204034	+	0.978964i	$0.565405\pi$
$198$	0	0
$199$	−109696.	−0.196363	−0.0981813	−	0.995169i	$-0.531303\pi$
$199$	−109696.	−0.196363	−0.0981813	+	0.995169i	$0.531303\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.29226e6	−2.20095
$204$	0	0
$205$	531904.	0.883992
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−380170.	−0.602022
$210$	0	0
$211$	−618873.	−0.956963	−0.478482	−	0.878098i	$-0.658813\pi$
$211$	−618873.	−0.956963	−0.478482	+	0.878098i	$0.658813\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−571797.	−0.843618
$216$	0	0
$217$	−444893.	−0.641367
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−941183.	−1.29626
$222$	0	0
$223$	463306.	0.623886	0.311943	−	0.950101i	$-0.399020\pi$
$223$	463306.	0.623886	0.311943	+	0.950101i	$0.399020\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.14665e6	−1.47695	−0.738477	−	0.674279i	$-0.764455\pi$
$227$	−1.14665e6	−1.47695	−0.738477	+	0.674279i	$0.764455\pi$
$228$	0	0
$229$	−351055.	−0.442371	−0.221185	−	0.975232i	$-0.570993\pi$
$229$	−351055.	−0.442371	−0.221185	+	0.975232i	$0.570993\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	814275.	0.982610	0.491305	−	0.870988i	$-0.336520\pi$
$233$	814275.	0.982610	0.491305	+	0.870988i	$0.336520\pi$
$234$	0	0
$235$	−8324.30	−0.00983281
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−714903.	−0.809566	−0.404783	−	0.914413i	$-0.632653\pi$
$239$	−714903.	−0.809566	−0.404783	+	0.914413i	$0.632653\pi$
$240$	0	0
$241$	45296.9	0.0502373	0.0251186	−	0.999684i	$-0.492004\pi$
$241$	45296.9	0.0502373	0.0251186	+	0.999684i	$0.492004\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	172164.	0.183243
$246$	0	0
$247$	799165.	0.833477
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.35110e6	−1.35364	−0.676822	−	0.736147i	$-0.736643\pi$
$251$	−1.35110e6	−1.35364	−0.676822	+	0.736147i	$0.736643\pi$
$252$	0	0
$253$	−237759.	−0.233526
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.20019e6	1.13349	0.566747	−	0.823892i	$-0.308202\pi$
$257$	1.20019e6	1.13349	0.566747	+	0.823892i	$0.308202\pi$
$258$	0	0
$259$	−611203.	−0.566156
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.31360e6	−1.17105	−0.585525	−	0.810654i	$-0.699112\pi$
$263$	−1.31360e6	−1.17105	−0.585525	+	0.810654i	$0.699112\pi$
$264$	0	0
$265$	85532.1	0.0748195
$266$	0	0
$267$	0	0
$268$	0	0
$269$	321363.	0.270779	0.135389	−	0.990792i	$-0.456771\pi$
$269$	321363.	0.270779	0.135389	+	0.990792i	$0.456771\pi$
$270$	0	0
$271$	384928.	0.318388	0.159194	−	0.987247i	$-0.449110\pi$
$271$	384928.	0.318388	0.159194	+	0.987247i	$0.449110\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	647966.	0.516678
$276$	0	0
$277$	−1.69593e6	−1.32803	−0.664015	−	0.747719i	$-0.731149\pi$
$277$	−1.69593e6	−1.32803	−0.664015	+	0.747719i	$0.731149\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.06134e6	0.801841	0.400920	−	0.916113i	$-0.368690\pi$
$281$	1.06134e6	0.801841	0.400920	+	0.916113i	$0.368690\pi$
$282$	0	0
$283$	−384626.	−0.285478	−0.142739	−	0.989760i	$-0.545591\pi$
$283$	−384626.	−0.285478	−0.142739	+	0.989760i	$0.545591\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.86155e6	−2.05067
$288$	0	0
$289$	1.17871e6	0.830158
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.33529e6	−0.908667	−0.454334	−	0.890832i	$-0.650123\pi$
$293$	−1.33529e6	−0.908667	−0.454334	+	0.890832i	$0.650123\pi$
$294$	0	0
$295$	−485101.	−0.324547
$296$	0	0
$297$	0	0
$298$	0	0
$299$	499799.	0.323309
$300$	0	0
$301$	3.07617e6	1.95702
$302$	0	0
$303$	0	0
$304$	0	0
$305$	721954.	0.444386
$306$	0	0
$307$	636269.	0.385296	0.192648	−	0.981268i	$-0.438292\pi$
$307$	636269.	0.385296	0.192648	+	0.981268i	$0.438292\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−539781.	−0.316458	−0.158229	−	0.987402i	$-0.550578\pi$
$311$	−539781.	−0.316458	−0.158229	+	0.987402i	$0.550578\pi$
$312$	0	0
$313$	1.95321e6	1.12691	0.563454	−	0.826148i	$-0.309472\pi$
$313$	1.95321e6	1.12691	0.563454	+	0.826148i	$0.309472\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.56397e6	0.874136	0.437068	−	0.899428i	$-0.356017\pi$
$317$	1.56397e6	0.874136	0.437068	+	0.899428i	$0.356017\pi$
$318$	0	0
$319$	−2.37056e6	−1.30429
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.20646e6	−1.17676
$324$	0	0
$325$	−1.36210e6	−0.715323
$326$	0	0
$327$	0	0
$328$	0	0
$329$	44783.3	0.0228100
$330$	0	0
$331$	−2.56000e6	−1.28431	−0.642154	−	0.766575i	$-0.721959\pi$
$331$	−2.56000e6	−1.28431	−0.642154	+	0.766575i	$0.721959\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−739622.	−0.360079
$336$	0	0
$337$	−599292.	−0.287451	−0.143725	−	0.989618i	$-0.545908\pi$
$337$	−599292.	−0.287451	−0.143725	+	0.989618i	$0.545908\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−816125.	−0.380076
$342$	0	0
$343$	1.61850e6	0.742808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.68773e6	1.19829	0.599144	−	0.800641i	$-0.295508\pi$
$347$	2.68773e6	1.19829	0.599144	+	0.800641i	$0.295508\pi$
$348$	0	0
$349$	671614.	0.295159	0.147580	−	0.989050i	$-0.452852\pi$
$349$	671614.	0.295159	0.147580	+	0.989050i	$0.452852\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.52976e6	−0.653412	−0.326706	−	0.945126i	$-0.605939\pi$
$353$	−1.52976e6	−0.653412	−0.326706	+	0.945126i	$0.605939\pi$
$354$	0	0
$355$	2.15765e6	0.908678
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.27210e6	1.33996	0.669978	−	0.742381i	$-0.266303\pi$
$359$	3.27210e6	1.33996	0.669978	+	0.742381i	$0.266303\pi$
$360$	0	0
$361$	−602583.	−0.243360
$362$	0	0
$363$	0	0
$364$	0	0
$365$	42112.2	0.0165453
$366$	0	0
$367$	627142.	0.243053	0.121526	−	0.992588i	$-0.461221\pi$
$367$	627142.	0.243053	0.121526	+	0.992588i	$0.461221\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−460148.	−0.173565
$372$	0	0
$373$	132374.	0.0492639	0.0246320	−	0.999697i	$-0.492159\pi$
$373$	132374.	0.0492639	0.0246320	+	0.999697i	$0.492159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.98322e6	1.80574
$378$	0	0
$379$	−163225.	−0.0583700	−0.0291850	−	0.999574i	$-0.509291\pi$
$379$	−163225.	−0.0583700	−0.0291850	+	0.999574i	$0.509291\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−796601.	−0.277488	−0.138744	−	0.990328i	$-0.544307\pi$
$383$	−796601.	−0.277488	−0.138744	+	0.990328i	$0.544307\pi$
$384$	0	0
$385$	1.18353e6	0.406936
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.09490e6	1.37205	0.686024	−	0.727579i	$-0.259354\pi$
$389$	4.09490e6	1.37205	0.686024	+	0.727579i	$0.259354\pi$
$390$	0	0
$391$	−1.37992e6	−0.456471
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.79394e6	0.900998
$396$	0	0
$397$	−5.58867e6	−1.77964	−0.889820	−	0.456312i	$-0.849170\pi$
$397$	−5.58867e6	−1.77964	−0.889820	+	0.456312i	$0.849170\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.23353e6	0.693634	0.346817	−	0.937933i	$-0.387262\pi$
$401$	2.23353e6	0.693634	0.346817	+	0.937933i	$0.387262\pi$
$402$	0	0
$403$	1.71560e6	0.526202
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.12121e6	−0.335506
$408$	0	0
$409$	−4.60512e6	−1.36123	−0.680617	−	0.732640i	$-0.738288\pi$
$409$	−4.60512e6	−1.36123	−0.680617	+	0.732640i	$0.738288\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.60976e6	0.752880
$414$	0	0
$415$	1.40863e6	0.401491
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.14213e6	0.317819	0.158909	−	0.987293i	$-0.449202\pi$
$419$	1.14213e6	0.317819	0.158909	+	0.987293i	$0.449202\pi$
$420$	0	0
$421$	1.92477e6	0.529264	0.264632	−	0.964349i	$-0.414749\pi$
$421$	1.92477e6	0.529264	0.264632	+	0.964349i	$0.414749\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.76071e6	1.00994
$426$	0	0
$427$	−3.88399e6	−1.03088
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.21303e6	0.573843	0.286922	−	0.957954i	$-0.407368\pi$
$431$	2.21303e6	0.573843	0.286922	+	0.957954i	$0.407368\pi$
$432$	0	0
$433$	3.00235e6	0.769558	0.384779	−	0.923009i	$-0.374278\pi$
$433$	3.00235e6	0.769558	0.384779	+	0.923009i	$0.374278\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.17170e6	0.293503
$438$	0	0
$439$	1.92171e6	0.475913	0.237956	−	0.971276i	$-0.423522\pi$
$439$	1.92171e6	0.475913	0.237956	+	0.971276i	$0.423522\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.84852e6	−0.447521	−0.223761	−	0.974644i	$-0.571833\pi$
$443$	−1.84852e6	−0.447521	−0.223761	+	0.974644i	$0.571833\pi$
$444$	0	0
$445$	3.84541e6	0.920540
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.45640e6	0.809110	0.404555	−	0.914514i	$-0.367426\pi$
$449$	3.45640e6	0.809110	0.404555	+	0.914514i	$0.367426\pi$
$450$	0	0
$451$	−5.24931e6	−1.21524
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.48792e6	−0.563388
$456$	0	0
$457$	3.58637e6	0.803276	0.401638	−	0.915798i	$-0.368441\pi$
$457$	3.58637e6	0.803276	0.401638	+	0.915798i	$0.368441\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.43550e6	1.84867	0.924333	−	0.381586i	$-0.124622\pi$
$461$	8.43550e6	1.84867	0.924333	+	0.381586i	$0.124622\pi$
$462$	0	0
$463$	4.94474e6	1.07199	0.535995	−	0.844221i	$-0.319937\pi$
$463$	4.94474e6	1.07199	0.535995	+	0.844221i	$0.319937\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.12589e6	−0.875438	−0.437719	−	0.899112i	$-0.644213\pi$
$467$	−4.12589e6	−0.875438	−0.437719	+	0.899112i	$0.644213\pi$
$468$	0	0
$469$	3.97904e6	0.835307
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.64302e6	1.15973
$474$	0	0
$475$	−3.19324e6	−0.649378
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.12260e6	−0.223557	−0.111778	−	0.993733i	$-0.535655\pi$
$479$	−1.12260e6	−0.223557	−0.111778	+	0.993733i	$0.535655\pi$
$480$	0	0
$481$	2.35692e6	0.464496
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.87577e6	−0.362097
$486$	0	0
$487$	8.11380e6	1.55025	0.775126	−	0.631807i	$-0.217687\pi$
$487$	8.11380e6	1.55025	0.775126	+	0.631807i	$0.217687\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.32962e6	−0.623291	−0.311645	−	0.950199i	$-0.600880\pi$
$491$	−3.32962e6	−0.623291	−0.311645	+	0.950199i	$0.600880\pi$
$492$	0	0
$493$	−1.37584e7	−2.54948
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.16078e7	−2.10794
$498$	0	0
$499$	1.88126e6	0.338218	0.169109	−	0.985597i	$-0.445911\pi$
$499$	1.88126e6	0.338218	0.169109	+	0.985597i	$0.445911\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.96749e6	−1.40411	−0.702056	−	0.712122i	$-0.747734\pi$
$503$	−7.96749e6	−1.40411	−0.702056	+	0.712122i	$0.747734\pi$
$504$	0	0
$505$	−694511.	−0.121186
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.08112e7	1.84961	0.924805	−	0.380441i	$-0.124228\pi$
$509$	1.08112e7	1.84961	0.924805	+	0.380441i	$0.124228\pi$
$510$	0	0
$511$	−226556.	−0.0383817
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.25810e6	0.541311
$516$	0	0
$517$	82151.7	0.0135173
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.19670e6	−0.193148	−0.0965740	−	0.995326i	$-0.530788\pi$
$521$	−1.19670e6	−0.193148	−0.0965740	+	0.995326i	$0.530788\pi$
$522$	0	0
$523$	−6.43371e6	−1.02851	−0.514254	−	0.857638i	$-0.671931\pi$
$523$	−6.43371e6	−1.02851	−0.514254	+	0.857638i	$0.671931\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.73668e6	−0.742929
$528$	0	0
$529$	−5.70356e6	−0.886149
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.10347e7	1.68245
$534$	0	0
$535$	2.37432e6	0.358637
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.69907e6	−0.251907
$540$	0	0
$541$	5.85989e6	0.860788	0.430394	−	0.902641i	$-0.358375\pi$
$541$	5.85989e6	0.860788	0.430394	+	0.902641i	$0.358375\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.58370e6	0.805251
$546$	0	0
$547$	5.18872e6	0.741468	0.370734	−	0.928739i	$-0.379106\pi$
$547$	5.18872e6	0.741468	0.370734	+	0.928739i	$0.379106\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.16824e7	1.63927
$552$	0	0
$553$	−1.50309e7	−2.09012
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.07405e6	0.692973	0.346487	−	0.938055i	$-0.387375\pi$
$557$	5.07405e6	0.692973	0.346487	+	0.938055i	$0.387375\pi$
$558$	0	0
$559$	−1.18623e7	−1.60561
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.10878e6	−0.812239	−0.406119	−	0.913820i	$-0.633118\pi$
$563$	−6.10878e6	−0.812239	−0.406119	+	0.913820i	$0.633118\pi$
$564$	0	0
$565$	6.44223e6	0.849014
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.08097e7	−1.39969	−0.699844	−	0.714296i	$-0.746747\pi$
$569$	−1.08097e7	−1.39969	−0.699844	+	0.714296i	$0.746747\pi$
$570$	0	0
$571$	1.20559e7	1.54743	0.773714	−	0.633535i	$-0.218397\pi$
$571$	1.20559e7	1.54743	0.773714	+	0.633535i	$0.218397\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.99706e6	−0.251896
$576$	0	0
$577$	1.22323e7	1.52957	0.764786	−	0.644285i	$-0.222845\pi$
$577$	1.22323e7	1.52957	0.764786	+	0.644285i	$0.222845\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.57816e6	−0.931373
$582$	0	0
$583$	−844109.	−0.102855
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.59634e6	−1.02972	−0.514859	−	0.857275i	$-0.672156\pi$
$587$	−8.59634e6	−1.02972	−0.514859	+	0.857275i	$0.672156\pi$
$588$	0	0
$589$	4.02195e6	0.477692
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.78455e6	−0.675511	−0.337756	−	0.941234i	$-0.609668\pi$
$593$	−5.78455e6	−0.675511	−0.337756	+	0.941234i	$0.609668\pi$
$594$	0	0
$595$	6.86903e6	0.795432
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.25113e6	−0.597979	−0.298989	−	0.954256i	$-0.596650\pi$
$599$	−5.25113e6	−0.597979	−0.298989	+	0.954256i	$0.596650\pi$
$600$	0	0
$601$	−6.22338e6	−0.702813	−0.351407	−	0.936223i	$-0.614297\pi$
$601$	−6.22338e6	−0.702813	−0.351407	+	0.936223i	$0.614297\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.36147e6	−0.262297
$606$	0	0
$607$	−1.58810e7	−1.74947	−0.874733	−	0.484604i	$-0.838964\pi$
$607$	−1.58810e7	−1.74947	−0.874733	+	0.484604i	$0.838964\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−172693.	−0.0187142
$612$	0	0
$613$	−1.28661e7	−1.38292	−0.691458	−	0.722417i	$-0.743031\pi$
$613$	−1.28661e7	−1.38292	−0.691458	+	0.722417i	$0.743031\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.87594e6	−0.727141	−0.363571	−	0.931567i	$-0.618442\pi$
$617$	−6.87594e6	−0.727141	−0.363571	+	0.931567i	$0.618442\pi$
$618$	0	0
$619$	1.30482e7	1.36875	0.684373	−	0.729132i	$-0.260076\pi$
$619$	1.30482e7	1.36875	0.684373	+	0.729132i	$0.260076\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.06877e7	−2.13546
$624$	0	0
$625$	2.96739e6	0.303861
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.50734e6	−0.655809
$630$	0	0
$631$	−1.52784e7	−1.52758	−0.763790	−	0.645464i	$-0.776664\pi$
$631$	−1.52784e7	−1.52758	−0.763790	+	0.645464i	$0.776664\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.94103e6	−0.683109
$636$	0	0
$637$	3.57166e6	0.348756
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.19231e7	−1.14616	−0.573080	−	0.819499i	$-0.694252\pi$
$641$	−1.19231e7	−1.14616	−0.573080	+	0.819499i	$0.694252\pi$
$642$	0	0
$643$	−6.00603e6	−0.572875	−0.286438	−	0.958099i	$-0.592471\pi$
$643$	−6.00603e6	−0.572875	−0.286438	+	0.958099i	$0.592471\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.19440e7	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$647$	1.19440e7	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$648$	0	0
$649$	4.78742e6	0.446159
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.06593e6	−0.281371	−0.140686	−	0.990054i	$-0.544931\pi$
$653$	−3.06593e6	−0.281371	−0.140686	+	0.990054i	$0.544931\pi$
$654$	0	0
$655$	−4.76261e6	−0.433753
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.10130e6	−0.278183	−0.139092	−	0.990280i	$-0.544418\pi$
$659$	−3.10130e6	−0.278183	−0.139092	+	0.990280i	$0.544418\pi$
$660$	0	0
$661$	1.29771e7	1.15524	0.577621	−	0.816305i	$-0.303981\pi$
$661$	1.29771e7	1.15524	0.577621	+	0.816305i	$0.303981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.83253e6	−0.511450
$666$	0	0
$667$	7.30618e6	0.635881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.12490e6	−0.610903
$672$	0	0
$673$	2.21406e7	1.88431	0.942154	−	0.335182i	$-0.108798\pi$
$673$	2.21406e7	1.88431	0.942154	+	0.335182i	$0.108798\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.59970e6	0.134143	0.0670713	−	0.997748i	$-0.478634\pi$
$677$	1.59970e6	0.134143	0.0670713	+	0.997748i	$0.478634\pi$
$678$	0	0
$679$	1.00913e7	0.839989
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.36986e7	1.12363	0.561816	−	0.827262i	$-0.310103\pi$
$683$	1.36986e7	1.12363	0.561816	+	0.827262i	$0.310103\pi$
$684$	0	0
$685$	2.81353e6	0.229100
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.77442e6	0.142400
$690$	0	0
$691$	−4.18213e6	−0.333198	−0.166599	−	0.986025i	$-0.553279\pi$
$691$	−4.18213e6	−0.333198	−0.166599	+	0.986025i	$0.553279\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.05254e6	−0.0826561
$696$	0	0
$697$	−3.04663e7	−2.37540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.47134e7	−1.13089	−0.565443	−	0.824788i	$-0.691295\pi$
$701$	−1.47134e7	−1.13089	−0.565443	+	0.824788i	$0.691295\pi$
$702$	0	0
$703$	5.52543e6	0.421675
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.73635e6	0.281125
$708$	0	0
$709$	−8.51816e6	−0.636400	−0.318200	−	0.948024i	$-0.603078\pi$
$709$	−8.51816e6	−0.636400	−0.318200	+	0.948024i	$0.603078\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.51533e6	0.185299
$714$	0	0
$715$	−4.56391e6	−0.333866
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.20183e7	−0.867003	−0.433502	−	0.901153i	$-0.642722\pi$
$719$	−1.20183e7	−0.867003	−0.433502	+	0.901153i	$0.642722\pi$
$720$	0	0
$721$	−1.75280e7	−1.25573
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.99115e7	−1.40689
$726$	0	0
$727$	2.98514e6	0.209473	0.104737	−	0.994500i	$-0.466600\pi$
$727$	2.98514e6	0.209473	0.104737	+	0.994500i	$0.466600\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.27513e7	2.26691
$732$	0	0
$733$	2.11351e7	1.45293	0.726464	−	0.687205i	$-0.241162\pi$
$733$	2.11351e7	1.45293	0.726464	+	0.687205i	$0.241162\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.29926e6	0.495006
$738$	0	0
$739$	1.33961e7	0.902337	0.451168	−	0.892439i	$-0.351007\pi$
$739$	1.33961e7	0.902337	0.451168	+	0.892439i	$0.351007\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.19084e7	0.791371	0.395685	−	0.918386i	$-0.370507\pi$
$743$	1.19084e7	0.791371	0.395685	+	0.918386i	$0.370507\pi$
$744$	0	0
$745$	7.16350e6	0.472862
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.27734e7	−0.831961
$750$	0	0
$751$	4.66367e6	0.301737	0.150868	−	0.988554i	$-0.451793\pi$
$751$	4.66367e6	0.301737	0.150868	+	0.988554i	$0.451793\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.66232e6	−0.425361
$756$	0	0
$757$	138599.	0.00879065	0.00439532	−	0.999990i	$-0.498601\pi$
$757$	138599.	0.00879065	0.00439532	+	0.999990i	$0.498601\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.39712e6	0.150047	0.0750237	−	0.997182i	$-0.476097\pi$
$761$	2.39712e6	0.150047	0.0750237	+	0.997182i	$0.476097\pi$
$762$	0	0
$763$	−3.00394e7	−1.86801
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.00638e7	−0.617691
$768$	0	0
$769$	799479.	0.0487519	0.0243759	−	0.999703i	$-0.492240\pi$
$769$	799479.	0.0487519	0.0243759	+	0.999703i	$0.492240\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.07743e6	0.0648546	0.0324273	−	0.999474i	$-0.489676\pi$
$773$	1.07743e6	0.0648546	0.0324273	+	0.999474i	$0.489676\pi$
$774$	0	0
$775$	−6.85505e6	−0.409974
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.58691e7	1.52735
$780$	0	0
$781$	−2.12937e7	−1.24917
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.17132e6	0.415361
$786$	0	0
$787$	3.07091e6	0.176738	0.0883692	−	0.996088i	$-0.471834\pi$
$787$	3.07091e6	0.176738	0.0883692	+	0.996088i	$0.471834\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.46581e7	−1.96953
$792$	0	0
$793$	1.49774e7	0.845774
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.49848e7	0.835615	0.417808	−	0.908536i	$-0.362799\pi$
$797$	1.49848e7	0.835615	0.417808	+	0.908536i	$0.362799\pi$
$798$	0	0
$799$	476797.	0.0264221
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−415601.	−0.0227451
$804$	0	0
$805$	−3.64768e6	−0.198393
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.11069e6	−0.167103	−0.0835516	−	0.996503i	$-0.526626\pi$
$809$	−3.11069e6	−0.167103	−0.0835516	+	0.996503i	$0.526626\pi$
$810$	0	0
$811$	1.12694e7	0.601659	0.300830	−	0.953678i	$-0.402736\pi$
$811$	1.12694e7	0.601659	0.300830	+	0.953678i	$0.402736\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.05229e6	0.319173
$816$	0	0
$817$	−2.78093e7	−1.45759
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.09537e7	1.08493	0.542467	−	0.840077i	$-0.317490\pi$
$821$	2.09537e7	1.08493	0.542467	+	0.840077i	$0.317490\pi$
$822$	0	0
$823$	1.94816e6	0.100259	0.0501296	−	0.998743i	$-0.484037\pi$
$823$	1.94816e6	0.100259	0.0501296	+	0.998743i	$0.484037\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.11807e7	1.58534	0.792671	−	0.609650i	$-0.208690\pi$
$827$	3.11807e7	1.58534	0.792671	+	0.609650i	$0.208690\pi$
$828$	0	0
$829$	−1.67293e7	−0.845455	−0.422727	−	0.906257i	$-0.638927\pi$
$829$	−1.67293e7	−0.845455	−0.422727	+	0.906257i	$0.638927\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.86120e6	−0.492399
$834$	0	0
$835$	5.41306e6	0.268675
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.27321e6	−0.405760	−0.202880	−	0.979204i	$-0.565030\pi$
$839$	−8.27321e6	−0.405760	−0.202880	+	0.979204i	$0.565030\pi$
$840$	0	0
$841$	5.23346e7	2.55152
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−855632.	−0.0412235
$846$	0	0
$847$	1.27043e7	0.608473
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.45562e6	0.163569
$852$	0	0
$853$	3.46938e7	1.63260	0.816300	−	0.577628i	$-0.196022\pi$
$853$	3.46938e7	1.63260	0.816300	+	0.577628i	$0.196022\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.03725e6	0.234284	0.117142	−	0.993115i	$-0.462627\pi$
$857$	5.03725e6	0.234284	0.117142	+	0.993115i	$0.462627\pi$
$858$	0	0
$859$	−4.27819e7	−1.97823	−0.989116	−	0.147139i	$-0.952994\pi$
$859$	−4.27819e7	−1.97823	−0.989116	+	0.147139i	$0.952994\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.50904e6	0.0689721	0.0344861	−	0.999405i	$-0.489021\pi$
$863$	1.50904e6	0.0689721	0.0344861	+	0.999405i	$0.489021\pi$
$864$	0	0
$865$	−233624.	−0.0106164
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.75731e7	−1.23861
$870$	0	0
$871$	−1.53440e7	−0.685318
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.32572e7	1.02692
$876$	0	0
$877$	−3.29399e7	−1.44618	−0.723092	−	0.690752i	$-0.757280\pi$
$877$	−3.29399e7	−1.44618	−0.723092	+	0.690752i	$0.757280\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.62885e7	1.14110	0.570552	−	0.821261i	$-0.306729\pi$
$881$	2.62885e7	1.14110	0.570552	+	0.821261i	$0.306729\pi$
$882$	0	0
$883$	−2.07499e7	−0.895602	−0.447801	−	0.894133i	$-0.647793\pi$
$883$	−2.07499e7	−0.895602	−0.447801	+	0.894133i	$0.647793\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.28582e7	1.40228	0.701140	−	0.713023i	$-0.252675\pi$
$887$	3.28582e7	1.40228	0.701140	+	0.713023i	$0.252675\pi$
$888$	0	0
$889$	3.73416e7	1.58467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−404852.	−0.0169890
$894$	0	0
$895$	1.61684e7	0.674698
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.50790e7	1.03493
$900$	0	0
$901$	−4.89909e6	−0.201050
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.32407e6	−0.256670
$906$	0	0
$907$	4.71672e7	1.90380	0.951900	−	0.306408i	$-0.0991270\pi$
$907$	4.71672e7	1.90380	0.951900	+	0.306408i	$0.0991270\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.80492e7	1.51897	0.759486	−	0.650524i	$-0.225451\pi$
$911$	3.80492e7	1.51897	0.759486	+	0.650524i	$0.225451\pi$
$912$	0	0
$913$	−1.39016e7	−0.551935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.56220e7	1.00621
$918$	0	0
$919$	2.54292e7	0.993215	0.496607	−	0.867975i	$-0.334579\pi$
$919$	2.54292e7	0.993215	0.496607	+	0.867975i	$0.334579\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.47619e7	1.72943
$924$	0	0
$925$	−9.41759e6	−0.361898
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.40094e7	−0.532576	−0.266288	−	0.963894i	$-0.585797\pi$
$929$	−1.40094e7	−0.532576	−0.266288	+	0.963894i	$0.585797\pi$
$930$	0	0
$931$	8.37320e6	0.316605
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.26007e7	0.471375
$936$	0	0
$937$	−1.29313e7	−0.481164	−0.240582	−	0.970629i	$-0.577338\pi$
$937$	−1.29313e7	−0.481164	−0.240582	+	0.970629i	$0.577338\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9.59945e6	−0.353405	−0.176702	−	0.984264i	$-0.556543\pi$
$941$	−9.59945e6	−0.353405	−0.176702	+	0.984264i	$0.556543\pi$
$942$	0	0
$943$	1.61786e7	0.592464
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.43337e7	−0.519379	−0.259690	−	0.965692i	$-0.583620\pi$
$947$	−1.43337e7	−0.519379	−0.259690	+	0.965692i	$0.583620\pi$
$948$	0	0
$949$	873646.	0.0314898
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.14184e7	1.12060	0.560302	−	0.828289i	$-0.310685\pi$
$953$	3.14184e7	1.12060	0.560302	+	0.828289i	$0.310685\pi$
$954$	0	0
$955$	−1.34550e7	−0.477391
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.51363e7	−0.531464
$960$	0	0
$961$	−1.99951e7	−0.698417
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.48050e7	−0.511786
$966$	0	0
$967$	−1.35264e7	−0.465174	−0.232587	−	0.972576i	$-0.574719\pi$
$967$	−1.35264e7	−0.465174	−0.232587	+	0.972576i	$0.574719\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.90534e7	1.32926	0.664632	−	0.747171i	$-0.268588\pi$
$971$	3.90534e7	1.32926	0.664632	+	0.747171i	$0.268588\pi$
$972$	0	0
$973$	5.66246e6	0.191744
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.89049e7	0.633632	0.316816	−	0.948487i	$-0.397386\pi$
$977$	1.89049e7	0.633632	0.316816	+	0.948487i	$0.397386\pi$
$978$	0	0
$979$	−3.79500e7	−1.26548
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.90025e6	−0.194754	−0.0973771	−	0.995248i	$-0.531045\pi$
$983$	−5.90025e6	−0.194754	−0.0973771	+	0.995248i	$0.531045\pi$
$984$	0	0
$985$	−6.25572e6	−0.205441
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.73920e7	−0.565405
$990$	0	0
$991$	−3.67515e7	−1.18875	−0.594375	−	0.804188i	$-0.702601\pi$
$991$	−3.67515e7	−1.18875	−0.594375	+	0.804188i	$0.702601\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.08725e6	−0.0988585
$996$	0	0
$997$	2.92149e7	0.930820	0.465410	−	0.885095i	$-0.345907\pi$
$997$	2.92149e7	0.930820	0.465410	+	0.885095i	$0.345907\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.a.e.1.4		5
3.2	odd	2		324.6.a.d.1.2		5
9.2	odd	6		108.6.e.a.37.4		10
9.4	even	3		36.6.e.a.25.1	yes	10
9.5	odd	6		108.6.e.a.73.4		10
9.7	even	3		36.6.e.a.13.1	✓	10
36.7	odd	6		144.6.i.d.49.5		10
36.11	even	6		432.6.i.d.145.4		10
36.23	even	6		432.6.i.d.289.4		10
36.31	odd	6		144.6.i.d.97.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.1	✓	10	9.7	even	3
36.6.e.a.25.1	yes	10	9.4	even	3
108.6.e.a.37.4		10	9.2	odd	6
108.6.e.a.73.4		10	9.5	odd	6
144.6.i.d.49.5		10	36.7	odd	6
144.6.i.d.97.5		10	36.31	odd	6
324.6.a.d.1.2		5	3.2	odd	2
324.6.a.e.1.4		5	1.1	even	1	trivial
432.6.i.d.145.4		10	36.11	even	6
432.6.i.d.289.4		10	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+28.1436 q^{5} -151.408 q^{7} +O(q^{10})\)	\(q+28.1436 q^{5} -151.408 q^{7} -277.747 q^{11} +583.858 q^{13} -1612.01 q^{17} +1368.76 q^{19} +856.028 q^{23} -2332.94 q^{25} +8534.97 q^{29} +2938.38 q^{31} -4261.17 q^{35} +4036.80 q^{37} +18899.6 q^{41} -20317.1 q^{43} -295.779 q^{47} +6117.35 q^{49} +3039.13 q^{53} -7816.81 q^{55} -17236.6 q^{59} +25652.5 q^{61} +16431.9 q^{65} -26280.3 q^{67} +76665.7 q^{71} +1496.33 q^{73} +42053.1 q^{77} +99274.2 q^{79} +50051.3 q^{83} -45367.7 q^{85} +136635. q^{89} -88400.8 q^{91} +38522.0 q^{95} -66649.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 21 q^{5} - 29 q^{7}+O(q^{10}) \) 5 * q + 21 * q^5 - 29 * q^7	\( 5 q + 21 q^{5} - 29 q^{7} - 177 q^{11} + 181 q^{13} + 1140 q^{17} - 416 q^{19} - 399 q^{23} + 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 18573 q^{35} - 7586 q^{37} + 18435 q^{41} - 1469 q^{43} + 25155 q^{47} + 4056 q^{49} + 58422 q^{53} + 7389 q^{55} + 90537 q^{59} - 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 114684 q^{71} + 7600 q^{73} + 211983 q^{77} - 29993 q^{79} + 228951 q^{83} + 49662 q^{85} + 299166 q^{89} + 62465 q^{91} + 394764 q^{95} - 40541 q^{97}+O(q^{100}) \) 5 * q + 21 * q^5 - 29 * q^7 - 177 * q^11 + 181 * q^13 + 1140 * q^17 - 416 * q^19 - 399 * q^23 + 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 18573 * q^35 - 7586 * q^37 + 18435 * q^41 - 1469 * q^43 + 25155 * q^47 + 4056 * q^49 + 58422 * q^53 + 7389 * q^55 + 90537 * q^59 - 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 114684 * q^71 + 7600 * q^73 + 211983 * q^77 - 29993 * q^79 + 228951 * q^83 + 49662 * q^85 + 299166 * q^89 + 62465 * q^91 + 394764 * q^95 - 40541 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more