LMFDB - Embedded newform 6240.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.69353$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.819031 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -0.819031 q^{7} +1.00000 q^{9} -1.08298 q^{11} +1.00000 q^{13} -1.00000 q^{15} +1.18097 q^{17} +6.51016 q^{19} +0.819031 q^{21} -0.819031 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.51016 q^{29} +1.08298 q^{33} -0.819031 q^{35} +0.917024 q^{37} -1.00000 q^{39} +3.08298 q^{41} -8.41216 q^{43} +1.00000 q^{45} -6.32919 q^{49} -1.18097 q^{51} -7.49514 q^{53} -1.08298 q^{55} -6.51016 q^{57} -0.195985 q^{59} +2.55509 q^{61} -0.819031 q^{63} +1.00000 q^{65} -10.2462 q^{67} +0.819031 q^{69} -7.85708 q^{71} +4.09799 q^{73} -1.00000 q^{75} +0.886992 q^{77} +9.69112 q^{79} +1.00000 q^{81} -6.24621 q^{83} +1.18097 q^{85} -8.51016 q^{87} +11.6911 q^{89} -0.819031 q^{91} +6.51016 q^{95} +14.2013 q^{97} -1.08298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.819031	−0.309565	−0.154782	−	0.987949i	$-0.549468\pi$
$7$	−0.819031	−0.309565	−0.154782	+	0.987949i	$0.549468\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.08298	−0.326530	−0.163265	−	0.986582i	$-0.552202\pi$
$11$	−1.08298	−0.326530	−0.163265	+	0.986582i	$0.552202\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.18097	0.286427	0.143213	−	0.989692i	$-0.454256\pi$
$17$	1.18097	0.286427	0.143213	+	0.989692i	$0.454256\pi$
$18$	0	0
$19$	6.51016	1.49353	0.746766	−	0.665087i	$-0.231605\pi$
$19$	6.51016	1.49353	0.746766	+	0.665087i	$0.231605\pi$
$20$	0	0
$21$	0.819031	0.178727
$22$	0	0
$23$	−0.819031	−0.170780	−0.0853899	−	0.996348i	$-0.527214\pi$
$23$	−0.819031	−0.170780	−0.0853899	+	0.996348i	$0.527214\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.51016	1.58030	0.790148	−	0.612916i	$-0.210004\pi$
$29$	8.51016	1.58030	0.790148	+	0.612916i	$0.210004\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.08298	0.188522
$34$	0	0
$35$	−0.819031	−0.138442
$36$	0	0
$37$	0.917024	0.150758	0.0753789	−	0.997155i	$-0.475983\pi$
$37$	0.917024	0.150758	0.0753789	+	0.997155i	$0.475983\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.08298	0.481480	0.240740	−	0.970590i	$-0.422610\pi$
$41$	3.08298	0.481480	0.240740	+	0.970590i	$0.422610\pi$
$42$	0	0
$43$	−8.41216	−1.28284	−0.641421	−	0.767189i	$-0.721655\pi$
$43$	−8.41216	−1.28284	−0.641421	+	0.767189i	$0.721655\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.32919	−0.904170
$50$	0	0
$51$	−1.18097	−0.165369
$52$	0	0
$53$	−7.49514	−1.02954	−0.514768	−	0.857329i	$-0.672122\pi$
$53$	−7.49514	−1.02954	−0.514768	+	0.857329i	$0.672122\pi$
$54$	0	0
$55$	−1.08298	−0.146028
$56$	0	0
$57$	−6.51016	−0.862291
$58$	0	0
$59$	−0.195985	−0.0255150	−0.0127575	−	0.999919i	$-0.504061\pi$
$59$	−0.195985	−0.0255150	−0.0127575	+	0.999919i	$0.504061\pi$
$60$	0	0
$61$	2.55509	0.327145	0.163573	−	0.986531i	$-0.447698\pi$
$61$	2.55509	0.327145	0.163573	+	0.986531i	$0.447698\pi$
$62$	0	0
$63$	−0.819031	−0.103188
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	0.819031	0.0985998
$70$	0	0
$71$	−7.85708	−0.932463	−0.466232	−	0.884663i	$-0.654389\pi$
$71$	−7.85708	−0.932463	−0.466232	+	0.884663i	$0.654389\pi$
$72$	0	0
$73$	4.09799	0.479634	0.239817	−	0.970818i	$-0.422913\pi$
$73$	4.09799	0.479634	0.239817	+	0.970818i	$0.422913\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.886992	0.101082
$78$	0	0
$79$	9.69112	1.09034	0.545168	−	0.838327i	$-0.316466\pi$
$79$	9.69112	1.09034	0.545168	+	0.838327i	$0.316466\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.24621	−0.685611	−0.342805	−	0.939406i	$-0.611377\pi$
$83$	−6.24621	−0.685611	−0.342805	+	0.939406i	$0.611377\pi$
$84$	0	0
$85$	1.18097	0.128094
$86$	0	0
$87$	−8.51016	−0.912384
$88$	0	0
$89$	11.6911	1.23926	0.619628	−	0.784895i	$-0.287283\pi$
$89$	11.6911	1.23926	0.619628	+	0.784895i	$0.287283\pi$
$90$	0	0
$91$	−0.819031	−0.0858578
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.51016	0.667928
$96$	0	0
$97$	14.2013	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$97$	14.2013	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$98$	0	0
$99$	−1.08298	−0.108843
$100$	0	0
$101$	9.03805	0.899319	0.449660	−	0.893200i	$-0.351545\pi$
$101$	9.03805	0.899319	0.449660	+	0.893200i	$0.351545\pi$
$102$	0	0
$103$	10.5781	1.04229	0.521146	−	0.853467i	$-0.325505\pi$
$103$	10.5781	1.04229	0.521146	+	0.853467i	$0.325505\pi$
$104$	0	0
$105$	0.819031	0.0799293
$106$	0	0
$107$	14.1033	1.36342	0.681708	−	0.731624i	$-0.261237\pi$
$107$	14.1033	1.36342	0.681708	+	0.731624i	$0.261237\pi$
$108$	0	0
$109$	1.73606	0.166284	0.0831420	−	0.996538i	$-0.473505\pi$
$109$	1.73606	0.166284	0.0831420	+	0.996538i	$0.473505\pi$
$110$	0	0
$111$	−0.917024	−0.0870400
$112$	0	0
$113$	17.6462	1.66001	0.830007	−	0.557753i	$-0.188336\pi$
$113$	17.6462	1.66001	0.830007	+	0.557753i	$0.188336\pi$
$114$	0	0
$115$	−0.819031	−0.0763751
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−0.967250	−0.0886677
$120$	0	0
$121$	−9.82716	−0.893378
$122$	0	0
$123$	−3.08298	−0.277983
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.9701	−0.973438	−0.486719	−	0.873559i	$-0.661806\pi$
$127$	−10.9701	−0.973438	−0.486719	+	0.873559i	$0.661806\pi$
$128$	0	0
$129$	8.41216	0.740650
$130$	0	0
$131$	1.90201	0.166179	0.0830896	−	0.996542i	$-0.473521\pi$
$131$	1.90201	0.166179	0.0830896	+	0.996542i	$0.473521\pi$
$132$	0	0
$133$	−5.33202	−0.462345
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−4.77410	−0.407879	−0.203939	−	0.978983i	$-0.565375\pi$
$137$	−4.77410	−0.407879	−0.203939	+	0.978983i	$0.565375\pi$
$138$	0	0
$139$	−20.4652	−1.73584	−0.867919	−	0.496706i	$-0.834543\pi$
$139$	−20.4652	−1.73584	−0.867919	+	0.496706i	$0.834543\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.08298	−0.0905630
$144$	0	0
$145$	8.51016	0.706730
$146$	0	0
$147$	6.32919	0.522023
$148$	0	0
$149$	−15.9073	−1.30318	−0.651589	−	0.758573i	$-0.725897\pi$
$149$	−15.9073	−1.30318	−0.651589	+	0.758573i	$0.725897\pi$
$150$	0	0
$151$	3.27613	0.266607	0.133304	−	0.991075i	$-0.457441\pi$
$151$	3.27613	0.266607	0.133304	+	0.991075i	$0.457441\pi$
$152$	0	0
$153$	1.18097	0.0954756
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0502	0.961713	0.480856	−	0.876799i	$-0.340326\pi$
$157$	12.0502	0.961713	0.480856	+	0.876799i	$0.340326\pi$
$158$	0	0
$159$	7.49514	0.594403
$160$	0	0
$161$	0.670813	0.0528674
$162$	0	0
$163$	4.05306	0.317460	0.158730	−	0.987322i	$-0.449260\pi$
$163$	4.05306	0.317460	0.158730	+	0.987322i	$0.449260\pi$
$164$	0	0
$165$	1.08298	0.0843096
$166$	0	0
$167$	20.6283	1.59627	0.798135	−	0.602479i	$-0.205820\pi$
$167$	20.6283	1.59627	0.798135	+	0.602479i	$0.205820\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.51016	0.497844
$172$	0	0
$173$	−12.0502	−0.916162	−0.458081	−	0.888911i	$-0.651463\pi$
$173$	−12.0502	−0.916162	−0.458081	+	0.888911i	$0.651463\pi$
$174$	0	0
$175$	−0.819031	−0.0619130
$176$	0	0
$177$	0.195985	0.0147311
$178$	0	0
$179$	17.4802	1.30653	0.653267	−	0.757127i	$-0.273398\pi$
$179$	17.4802	1.30653	0.653267	+	0.757127i	$0.273398\pi$
$180$	0	0
$181$	3.47495	0.258291	0.129145	−	0.991626i	$-0.458777\pi$
$181$	3.47495	0.258291	0.129145	+	0.991626i	$0.458777\pi$
$182$	0	0
$183$	−2.55509	−0.188877
$184$	0	0
$185$	0.917024	0.0674209
$186$	0	0
$187$	−1.27896	−0.0935269
$188$	0	0
$189$	0.819031	0.0595758
$190$	0	0
$191$	1.11017	0.0803293	0.0401647	−	0.999193i	$-0.487212\pi$
$191$	1.11017	0.0803293	0.0401647	+	0.999193i	$0.487212\pi$
$192$	0	0
$193$	3.34692	0.240917	0.120458	−	0.992718i	$-0.461564\pi$
$193$	3.34692	0.240917	0.120458	+	0.992718i	$0.461564\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−8.05023	−0.573555	−0.286777	−	0.957997i	$-0.592584\pi$
$197$	−8.05023	−0.573555	−0.286777	+	0.957997i	$0.592584\pi$
$198$	0	0
$199$	17.5223	1.24213	0.621063	−	0.783761i	$-0.286701\pi$
$199$	17.5223	1.24213	0.621063	+	0.783761i	$0.286701\pi$
$200$	0	0
$201$	10.2462	0.722712
$202$	0	0
$203$	−6.97009	−0.489204
$204$	0	0
$205$	3.08298	0.215324
$206$	0	0
$207$	−0.819031	−0.0569266
$208$	0	0
$209$	−7.05034	−0.487683
$210$	0	0
$211$	−8.33190	−0.573592	−0.286796	−	0.957992i	$-0.592590\pi$
$211$	−8.33190	−0.573592	−0.286796	+	0.957992i	$0.592590\pi$
$212$	0	0
$213$	7.85708	0.538358
$214$	0	0
$215$	−8.41216	−0.573705
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.09799	−0.276917
$220$	0	0
$221$	1.18097	0.0794405
$222$	0	0
$223$	−14.7263	−0.986148	−0.493074	−	0.869987i	$-0.664127\pi$
$223$	−14.7263	−0.986148	−0.493074	+	0.869987i	$0.664127\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−15.3822	−1.02096	−0.510478	−	0.859891i	$-0.670531\pi$
$227$	−15.3822	−1.02096	−0.510478	+	0.859891i	$0.670531\pi$
$228$	0	0
$229$	10.7918	0.713144	0.356572	−	0.934268i	$-0.383945\pi$
$229$	10.7918	0.713144	0.356572	+	0.934268i	$0.383945\pi$
$230$	0	0
$231$	−0.886992	−0.0583598
$232$	0	0
$233$	2.29114	0.150098	0.0750488	−	0.997180i	$-0.476089\pi$
$233$	2.29114	0.150098	0.0750488	+	0.997180i	$0.476089\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.69112	−0.629506
$238$	0	0
$239$	−2.10329	−0.136050	−0.0680252	−	0.997684i	$-0.521670\pi$
$239$	−2.10329	−0.136050	−0.0680252	+	0.997684i	$0.521670\pi$
$240$	0	0
$241$	24.6584	1.58839	0.794193	−	0.607666i	$-0.207894\pi$
$241$	24.6584	1.58839	0.794193	+	0.607666i	$0.207894\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.32919	−0.404357
$246$	0	0
$247$	6.51016	0.414231
$248$	0	0
$249$	6.24621	0.395838
$250$	0	0
$251$	1.37412	0.0867336	0.0433668	−	0.999059i	$-0.486192\pi$
$251$	1.37412	0.0867336	0.0433668	+	0.999059i	$0.486192\pi$
$252$	0	0
$253$	0.886992	0.0557647
$254$	0	0
$255$	−1.18097	−0.0739551
$256$	0	0
$257$	−12.7263	−0.793847	−0.396923	−	0.917852i	$-0.629922\pi$
$257$	−12.7263	−0.793847	−0.396923	+	0.917852i	$0.629922\pi$
$258$	0	0
$259$	−0.751071	−0.0466693
$260$	0	0
$261$	8.51016	0.526765
$262$	0	0
$263$	6.82187	0.420654	0.210327	−	0.977631i	$-0.432547\pi$
$263$	6.82187	0.420654	0.210327	+	0.977631i	$0.432547\pi$
$264$	0	0
$265$	−7.49514	−0.460423
$266$	0	0
$267$	−11.6911	−0.715485
$268$	0	0
$269$	7.59030	0.462789	0.231394	−	0.972860i	$-0.425671\pi$
$269$	7.59030	0.462789	0.231394	+	0.972860i	$0.425671\pi$
$270$	0	0
$271$	5.44208	0.330583	0.165291	−	0.986245i	$-0.447144\pi$
$271$	5.44208	0.330583	0.165291	+	0.986245i	$0.447144\pi$
$272$	0	0
$273$	0.819031	0.0495700
$274$	0	0
$275$	−1.08298	−0.0653059
$276$	0	0
$277$	23.4325	1.40792	0.703961	−	0.710239i	$-0.251413\pi$
$277$	23.4325	1.40792	0.703961	+	0.710239i	$0.251413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.9701	1.72821	0.864105	−	0.503311i	$-0.167885\pi$
$281$	28.9701	1.72821	0.864105	+	0.503311i	$0.167885\pi$
$282$	0	0
$283$	24.5183	1.45746	0.728730	−	0.684801i	$-0.240111\pi$
$283$	24.5183	1.45746	0.728730	+	0.684801i	$0.240111\pi$
$284$	0	0
$285$	−6.51016	−0.385628
$286$	0	0
$287$	−2.52505	−0.149049
$288$	0	0
$289$	−15.6053	−0.917960
$290$	0	0
$291$	−14.2013	−0.832494
$292$	0	0
$293$	23.4325	1.36894	0.684470	−	0.729041i	$-0.260034\pi$
$293$	23.4325	1.36894	0.684470	+	0.729041i	$0.260034\pi$
$294$	0	0
$295$	−0.195985	−0.0114107
$296$	0	0
$297$	1.08298	0.0628407
$298$	0	0
$299$	−0.819031	−0.0473658
$300$	0	0
$301$	6.88983	0.397123
$302$	0	0
$303$	−9.03805	−0.519222
$304$	0	0
$305$	2.55509	0.146304
$306$	0	0
$307$	−1.19870	−0.0684135	−0.0342068	−	0.999415i	$-0.510890\pi$
$307$	−1.19870	−0.0684135	−0.0342068	+	0.999415i	$0.510890\pi$
$308$	0	0
$309$	−10.5781	−0.601768
$310$	0	0
$311$	27.3822	1.55270	0.776352	−	0.630299i	$-0.217068\pi$
$311$	27.3822	1.55270	0.776352	+	0.630299i	$0.217068\pi$
$312$	0	0
$313$	26.2964	1.48636	0.743181	−	0.669090i	$-0.233316\pi$
$313$	26.2964	1.48636	0.743181	+	0.669090i	$0.233316\pi$
$314$	0	0
$315$	−0.819031	−0.0461472
$316$	0	0
$317$	14.8243	0.832617	0.416309	−	0.909223i	$-0.363324\pi$
$317$	14.8243	0.832617	0.416309	+	0.909223i	$0.363324\pi$
$318$	0	0
$319$	−9.21630	−0.516014
$320$	0	0
$321$	−14.1033	−0.787169
$322$	0	0
$323$	7.68829	0.427788
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−1.73606	−0.0960041
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.78628	0.0981829	0.0490915	−	0.998794i	$-0.484367\pi$
$331$	1.78628	0.0981829	0.0490915	+	0.998794i	$0.484367\pi$
$332$	0	0
$333$	0.917024	0.0502526
$334$	0	0
$335$	−10.2462	−0.559810
$336$	0	0
$337$	−31.3522	−1.70786	−0.853932	−	0.520385i	$-0.825789\pi$
$337$	−31.3522	−1.70786	−0.853932	+	0.520385i	$0.825789\pi$
$338$	0	0
$339$	−17.6462	−0.958410
$340$	0	0
$341$	0	0
$342$	0	0
$343$	10.9170	0.589464
$344$	0	0
$345$	0.819031	0.0440952
$346$	0	0
$347$	−12.0733	−0.648126	−0.324063	−	0.946035i	$-0.605049\pi$
$347$	−12.0733	−0.648126	−0.324063	+	0.946035i	$0.605049\pi$
$348$	0	0
$349$	4.29398	0.229851	0.114926	−	0.993374i	$-0.463337\pi$
$349$	4.29398	0.229851	0.114926	+	0.993374i	$0.463337\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−19.4325	−1.03429	−0.517143	−	0.855899i	$-0.673005\pi$
$353$	−19.4325	−1.03429	−0.517143	+	0.855899i	$0.673005\pi$
$354$	0	0
$355$	−7.85708	−0.417010
$356$	0	0
$357$	0.967250	0.0511923
$358$	0	0
$359$	13.7538	0.725897	0.362949	−	0.931809i	$-0.381770\pi$
$359$	13.7538	0.725897	0.362949	+	0.931809i	$0.381770\pi$
$360$	0	0
$361$	23.3821	1.23064
$362$	0	0
$363$	9.82716	0.515792
$364$	0	0
$365$	4.09799	0.214499
$366$	0	0
$367$	0.804133	0.0419754	0.0209877	−	0.999780i	$-0.493319\pi$
$367$	0.804133	0.0419754	0.0209877	+	0.999780i	$0.493319\pi$
$368$	0	0
$369$	3.08298	0.160493
$370$	0	0
$371$	6.13876	0.318708
$372$	0	0
$373$	−13.5984	−0.704100	−0.352050	−	0.935981i	$-0.614515\pi$
$373$	−13.5984	−0.704100	−0.352050	+	0.935981i	$0.614515\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	8.51016	0.438295
$378$	0	0
$379$	−32.4447	−1.66657	−0.833285	−	0.552844i	$-0.813543\pi$
$379$	−32.4447	−1.66657	−0.833285	+	0.552844i	$0.813543\pi$
$380$	0	0
$381$	10.9701	0.562015
$382$	0	0
$383$	32.4026	1.65569	0.827847	−	0.560954i	$-0.189566\pi$
$383$	32.4026	1.65569	0.827847	+	0.560954i	$0.189566\pi$
$384$	0	0
$385$	0.886992	0.0452053
$386$	0	0
$387$	−8.41216	−0.427614
$388$	0	0
$389$	12.3142	0.624353	0.312177	−	0.950024i	$-0.398942\pi$
$389$	12.3142	0.624353	0.312177	+	0.950024i	$0.398942\pi$
$390$	0	0
$391$	−0.967250	−0.0489160
$392$	0	0
$393$	−1.90201	−0.0959436
$394$	0	0
$395$	9.69112	0.487613
$396$	0	0
$397$	−32.0678	−1.60944	−0.804719	−	0.593656i	$-0.797684\pi$
$397$	−32.0678	−1.60944	−0.804719	+	0.593656i	$0.797684\pi$
$398$	0	0
$399$	5.33202	0.266935
$400$	0	0
$401$	13.0801	0.653191	0.326596	−	0.945164i	$-0.394099\pi$
$401$	13.0801	0.653191	0.326596	+	0.945164i	$0.394099\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−0.993115	−0.0492269
$408$	0	0
$409$	−16.8544	−0.833394	−0.416697	−	0.909045i	$-0.636812\pi$
$409$	−16.8544	−0.833394	−0.416697	+	0.909045i	$0.636812\pi$
$410$	0	0
$411$	4.77410	0.235489
$412$	0	0
$413$	0.160518	0.00789855
$414$	0	0
$415$	−6.24621	−0.306614
$416$	0	0
$417$	20.4652	1.00219
$418$	0	0
$419$	−5.81215	−0.283942	−0.141971	−	0.989871i	$-0.545344\pi$
$419$	−5.81215	−0.283942	−0.141971	+	0.989871i	$0.545344\pi$
$420$	0	0
$421$	18.2639	0.890130	0.445065	−	0.895498i	$-0.353181\pi$
$421$	18.2639	0.890130	0.445065	+	0.895498i	$0.353181\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.18097	0.0572854
$426$	0	0
$427$	−2.09270	−0.101273
$428$	0	0
$429$	1.08298	0.0522866
$430$	0	0
$431$	29.1562	1.40441	0.702203	−	0.711977i	$-0.252200\pi$
$431$	29.1562	1.40441	0.702203	+	0.711977i	$0.252200\pi$
$432$	0	0
$433$	−2.19598	−0.105532	−0.0527661	−	0.998607i	$-0.516804\pi$
$433$	−2.19598	−0.105532	−0.0527661	+	0.998607i	$0.516804\pi$
$434$	0	0
$435$	−8.51016	−0.408031
$436$	0	0
$437$	−5.33202	−0.255065
$438$	0	0
$439$	22.4953	1.07364	0.536820	−	0.843697i	$-0.319625\pi$
$439$	22.4953	1.07364	0.536820	+	0.843697i	$0.319625\pi$
$440$	0	0
$441$	−6.32919	−0.301390
$442$	0	0
$443$	1.47495	0.0700768	0.0350384	−	0.999386i	$-0.488845\pi$
$443$	1.47495	0.0700768	0.0350384	+	0.999386i	$0.488845\pi$
$444$	0	0
$445$	11.6911	0.554212
$446$	0	0
$447$	15.9073	0.752390
$448$	0	0
$449$	25.2937	1.19368	0.596842	−	0.802359i	$-0.296422\pi$
$449$	25.2937	1.19368	0.596842	+	0.802359i	$0.296422\pi$
$450$	0	0
$451$	−3.33879	−0.157217
$452$	0	0
$453$	−3.27613	−0.153926
$454$	0	0
$455$	−0.819031	−0.0383968
$456$	0	0
$457$	9.47741	0.443334	0.221667	−	0.975122i	$-0.428850\pi$
$457$	9.47741	0.443334	0.221667	+	0.975122i	$0.428850\pi$
$458$	0	0
$459$	−1.18097	−0.0551229
$460$	0	0
$461$	−8.88156	−0.413655	−0.206828	−	0.978377i	$-0.566314\pi$
$461$	−8.88156	−0.413655	−0.206828	+	0.978377i	$0.566314\pi$
$462$	0	0
$463$	10.9850	0.510515	0.255258	−	0.966873i	$-0.417840\pi$
$463$	10.9850	0.510515	0.255258	+	0.966873i	$0.417840\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.6054	−0.722133	−0.361067	−	0.932540i	$-0.617587\pi$
$467$	−15.6054	−0.722133	−0.361067	+	0.932540i	$0.617587\pi$
$468$	0	0
$469$	8.39197	0.387505
$470$	0	0
$471$	−12.0502	−0.555245
$472$	0	0
$473$	9.11017	0.418886
$474$	0	0
$475$	6.51016	0.298706
$476$	0	0
$477$	−7.49514	−0.343179
$478$	0	0
$479$	17.2790	0.789496	0.394748	−	0.918789i	$-0.370832\pi$
$479$	17.2790	0.789496	0.394748	+	0.918789i	$0.370832\pi$
$480$	0	0
$481$	0.917024	0.0418127
$482$	0	0
$483$	−0.670813	−0.0305230
$484$	0	0
$485$	14.2013	0.644847
$486$	0	0
$487$	−19.8093	−0.897646	−0.448823	−	0.893621i	$-0.648157\pi$
$487$	−19.8093	−0.897646	−0.448823	+	0.893621i	$0.648157\pi$
$488$	0	0
$489$	−4.05306	−0.183286
$490$	0	0
$491$	−12.7564	−0.575687	−0.287843	−	0.957677i	$-0.592938\pi$
$491$	−12.7564	−0.575687	−0.287843	+	0.957677i	$0.592938\pi$
$492$	0	0
$493$	10.0502	0.452639
$494$	0	0
$495$	−1.08298	−0.0486762
$496$	0	0
$497$	6.43519	0.288658
$498$	0	0
$499$	16.2543	0.727644	0.363822	−	0.931468i	$-0.381472\pi$
$499$	16.2543	0.727644	0.363822	+	0.931468i	$0.381472\pi$
$500$	0	0
$501$	−20.6283	−0.921606
$502$	0	0
$503$	6.42990	0.286695	0.143347	−	0.989672i	$-0.454213\pi$
$503$	6.42990	0.286695	0.143347	+	0.989672i	$0.454213\pi$
$504$	0	0
$505$	9.03805	0.402188
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−7.61087	−0.337346	−0.168673	−	0.985672i	$-0.553948\pi$
$509$	−7.61087	−0.337346	−0.168673	+	0.985672i	$0.553948\pi$
$510$	0	0
$511$	−3.35638	−0.148478
$512$	0	0
$513$	−6.51016	−0.287430
$514$	0	0
$515$	10.5781	0.466127
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.0502	0.528946
$520$	0	0
$521$	12.9797	0.568650	0.284325	−	0.958728i	$-0.408230\pi$
$521$	12.9797	0.568650	0.284325	+	0.958728i	$0.408230\pi$
$522$	0	0
$523$	−22.3265	−0.976268	−0.488134	−	0.872769i	$-0.662322\pi$
$523$	−22.3265	−0.976268	−0.488134	+	0.872769i	$0.662322\pi$
$524$	0	0
$525$	0.819031	0.0357455
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.3292	−0.970834
$530$	0	0
$531$	−0.195985	−0.00850501
$532$	0	0
$533$	3.08298	0.133539
$534$	0	0
$535$	14.1033	0.609738
$536$	0	0
$537$	−17.4802	−0.754328
$538$	0	0
$539$	6.85436	0.295238
$540$	0	0
$541$	20.4299	0.878350	0.439175	−	0.898402i	$-0.355271\pi$
$541$	20.4299	0.878350	0.439175	+	0.898402i	$0.355271\pi$
$542$	0	0
$543$	−3.47495	−0.149124
$544$	0	0
$545$	1.73606	0.0743644
$546$	0	0
$547$	14.2721	0.610230	0.305115	−	0.952316i	$-0.401305\pi$
$547$	14.2721	0.610230	0.305115	+	0.952316i	$0.401305\pi$
$548$	0	0
$549$	2.55509	0.109048
$550$	0	0
$551$	55.4024	2.36022
$552$	0	0
$553$	−7.93734	−0.337530
$554$	0	0
$555$	−0.917024	−0.0389255
$556$	0	0
$557$	−25.7142	−1.08954	−0.544772	−	0.838584i	$-0.683384\pi$
$557$	−25.7142	−1.08954	−0.544772	+	0.838584i	$0.683384\pi$
$558$	0	0
$559$	−8.41216	−0.355797
$560$	0	0
$561$	1.27896	0.0539978
$562$	0	0
$563$	24.8515	1.04737	0.523683	−	0.851913i	$-0.324557\pi$
$563$	24.8515	1.04737	0.523683	+	0.851913i	$0.324557\pi$
$564$	0	0
$565$	17.6462	0.742381
$566$	0	0
$567$	−0.819031	−0.0343961
$568$	0	0
$569$	28.5685	1.19765	0.598827	−	0.800878i	$-0.295634\pi$
$569$	28.5685	1.19765	0.598827	+	0.800878i	$0.295634\pi$
$570$	0	0
$571$	−35.1236	−1.46988	−0.734938	−	0.678134i	$-0.762789\pi$
$571$	−35.1236	−1.46988	−0.734938	+	0.678134i	$0.762789\pi$
$572$	0	0
$573$	−1.11017	−0.0463782
$574$	0	0
$575$	−0.819031	−0.0341560
$576$	0	0
$577$	−10.0952	−0.420267	−0.210133	−	0.977673i	$-0.567390\pi$
$577$	−10.0952	−0.420267	−0.210133	+	0.977673i	$0.567390\pi$
$578$	0	0
$579$	−3.34692	−0.139093
$580$	0	0
$581$	5.11584	0.212241
$582$	0	0
$583$	8.11706	0.336174
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	32.5985	1.34549	0.672743	−	0.739876i	$-0.265116\pi$
$587$	32.5985	1.34549	0.672743	+	0.739876i	$0.265116\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8.05023	0.331142
$592$	0	0
$593$	1.58784	0.0652046	0.0326023	−	0.999468i	$-0.489621\pi$
$593$	1.58784	0.0652046	0.0326023	+	0.999468i	$0.489621\pi$
$594$	0	0
$595$	−0.967250	−0.0396534
$596$	0	0
$597$	−17.5223	−0.717142
$598$	0	0
$599$	20.3319	0.830739	0.415370	−	0.909653i	$-0.363652\pi$
$599$	20.3319	0.830739	0.415370	+	0.909653i	$0.363652\pi$
$600$	0	0
$601$	−6.38497	−0.260448	−0.130224	−	0.991485i	$-0.541570\pi$
$601$	−6.38497	−0.260448	−0.130224	+	0.991485i	$0.541570\pi$
$602$	0	0
$603$	−10.2462	−0.417258
$604$	0	0
$605$	−9.82716	−0.399531
$606$	0	0
$607$	−37.1360	−1.50731	−0.753653	−	0.657273i	$-0.771710\pi$
$607$	−37.1360	−1.50731	−0.753653	+	0.657273i	$0.771710\pi$
$608$	0	0
$609$	6.97009	0.282442
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.91702	−0.198597	−0.0992984	−	0.995058i	$-0.531660\pi$
$613$	−4.91702	−0.198597	−0.0992984	+	0.995058i	$0.531660\pi$
$614$	0	0
$615$	−3.08298	−0.124318
$616$	0	0
$617$	−7.22590	−0.290904	−0.145452	−	0.989365i	$-0.546464\pi$
$617$	−7.22590	−0.290904	−0.145452	+	0.989365i	$0.546464\pi$
$618$	0	0
$619$	−4.57566	−0.183911	−0.0919556	−	0.995763i	$-0.529312\pi$
$619$	−4.57566	−0.183911	−0.0919556	+	0.995763i	$0.529312\pi$
$620$	0	0
$621$	0.819031	0.0328666
$622$	0	0
$623$	−9.57540	−0.383630
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.05034	0.281564
$628$	0	0
$629$	1.08298	0.0431811
$630$	0	0
$631$	14.0761	0.560360	0.280180	−	0.959947i	$-0.409606\pi$
$631$	14.0761	0.560360	0.280180	+	0.959947i	$0.409606\pi$
$632$	0	0
$633$	8.33190	0.331163
$634$	0	0
$635$	−10.9701	−0.435335
$636$	0	0
$637$	−6.32919	−0.250772
$638$	0	0
$639$	−7.85708	−0.310821
$640$	0	0
$641$	40.3725	1.59462	0.797310	−	0.603571i	$-0.206256\pi$
$641$	40.3725	1.59462	0.797310	+	0.603571i	$0.206256\pi$
$642$	0	0
$643$	−10.0230	−0.395270	−0.197635	−	0.980276i	$-0.563326\pi$
$643$	−10.0230	−0.395270	−0.197635	+	0.980276i	$0.563326\pi$
$644$	0	0
$645$	8.41216	0.331229
$646$	0	0
$647$	9.87481	0.388219	0.194109	−	0.980980i	$-0.437818\pi$
$647$	9.87481	0.388219	0.194109	+	0.980980i	$0.437818\pi$
$648$	0	0
$649$	0.212247	0.00833141
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.5781	−0.805284	−0.402642	−	0.915358i	$-0.631908\pi$
$653$	−20.5781	−0.805284	−0.402642	+	0.915358i	$0.631908\pi$
$654$	0	0
$655$	1.90201	0.0743176
$656$	0	0
$657$	4.09799	0.159878
$658$	0	0
$659$	15.4502	0.601855	0.300927	−	0.953647i	$-0.402704\pi$
$659$	15.4502	0.601855	0.300927	+	0.953647i	$0.402704\pi$
$660$	0	0
$661$	14.6502	0.569828	0.284914	−	0.958553i	$-0.408035\pi$
$661$	14.6502	0.569828	0.284914	+	0.958553i	$0.408035\pi$
$662$	0	0
$663$	−1.18097	−0.0458650
$664$	0	0
$665$	−5.33202	−0.206767
$666$	0	0
$667$	−6.97009	−0.269883
$668$	0	0
$669$	14.7263	0.569353
$670$	0	0
$671$	−2.76710	−0.106823
$672$	0	0
$673$	−21.0503	−0.811431	−0.405716	−	0.913999i	$-0.632978\pi$
$673$	−21.0503	−0.811431	−0.405716	+	0.913999i	$0.632978\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	21.1332	0.812215	0.406107	−	0.913825i	$-0.366886\pi$
$677$	21.1332	0.812215	0.406107	+	0.913825i	$0.366886\pi$
$678$	0	0
$679$	−11.6313	−0.446368
$680$	0	0
$681$	15.3822	0.589449
$682$	0	0
$683$	−37.9604	−1.45251	−0.726257	−	0.687424i	$-0.758742\pi$
$683$	−37.9604	−1.45251	−0.726257	+	0.687424i	$0.758742\pi$
$684$	0	0
$685$	−4.77410	−0.182409
$686$	0	0
$687$	−10.7918	−0.411734
$688$	0	0
$689$	−7.49514	−0.285542
$690$	0	0
$691$	−18.7061	−0.711615	−0.355808	−	0.934559i	$-0.615794\pi$
$691$	−18.7061	−0.711615	−0.355808	+	0.934559i	$0.615794\pi$
$692$	0	0
$693$	0.886992	0.0336940
$694$	0	0
$695$	−20.4652	−0.776290
$696$	0	0
$697$	3.64090	0.137909
$698$	0	0
$699$	−2.29114	−0.0866589
$700$	0	0
$701$	−5.95790	−0.225027	−0.112513	−	0.993650i	$-0.535890\pi$
$701$	−5.95790	−0.225027	−0.112513	+	0.993650i	$0.535890\pi$
$702$	0	0
$703$	5.96997	0.225162
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.40244	−0.278398
$708$	0	0
$709$	10.7918	0.405296	0.202648	−	0.979252i	$-0.435045\pi$
$709$	10.7918	0.405296	0.202648	+	0.979252i	$0.435045\pi$
$710$	0	0
$711$	9.69112	0.363446
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.08298	−0.0405010
$716$	0	0
$717$	2.10329	0.0785487
$718$	0	0
$719$	26.7482	0.997541	0.498770	−	0.866734i	$-0.333785\pi$
$719$	26.7482	0.997541	0.498770	+	0.866734i	$0.333785\pi$
$720$	0	0
$721$	−8.66381	−0.322657
$722$	0	0
$723$	−24.6584	−0.917055
$724$	0	0
$725$	8.51016	0.316059
$726$	0	0
$727$	−16.9401	−0.628272	−0.314136	−	0.949378i	$-0.601715\pi$
$727$	−16.9401	−0.628272	−0.314136	+	0.949378i	$0.601715\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.93450	−0.367441
$732$	0	0
$733$	21.8014	0.805254	0.402627	−	0.915364i	$-0.368097\pi$
$733$	21.8014	0.805254	0.402627	+	0.915364i	$0.368097\pi$
$734$	0	0
$735$	6.32919	0.233456
$736$	0	0
$737$	11.0964	0.408741
$738$	0	0
$739$	−38.2787	−1.40810	−0.704052	−	0.710148i	$-0.748628\pi$
$739$	−38.2787	−1.40810	−0.704052	+	0.710148i	$0.748628\pi$
$740$	0	0
$741$	−6.51016	−0.239157
$742$	0	0
$743$	0.813735	0.0298531	0.0149265	−	0.999889i	$-0.495249\pi$
$743$	0.813735	0.0298531	0.0149265	+	0.999889i	$0.495249\pi$
$744$	0	0
$745$	−15.9073	−0.582799
$746$	0	0
$747$	−6.24621	−0.228537
$748$	0	0
$749$	−11.5510	−0.422066
$750$	0	0
$751$	−13.9972	−0.510764	−0.255382	−	0.966840i	$-0.582201\pi$
$751$	−13.9972	−0.510764	−0.255382	+	0.966840i	$0.582201\pi$
$752$	0	0
$753$	−1.37412	−0.0500756
$754$	0	0
$755$	3.27613	0.119230
$756$	0	0
$757$	25.1508	0.914121	0.457061	−	0.889436i	$-0.348902\pi$
$757$	25.1508	0.914121	0.457061	+	0.889436i	$0.348902\pi$
$758$	0	0
$759$	−0.886992	−0.0321958
$760$	0	0
$761$	29.0801	1.05415	0.527077	−	0.849817i	$-0.323288\pi$
$761$	29.0801	1.05415	0.527077	+	0.849817i	$0.323288\pi$
$762$	0	0
$763$	−1.42188	−0.0514757
$764$	0	0
$765$	1.18097	0.0426980
$766$	0	0
$767$	−0.195985	−0.00707660
$768$	0	0
$769$	−47.2163	−1.70266	−0.851332	−	0.524628i	$-0.824204\pi$
$769$	−47.2163	−1.70266	−0.851332	+	0.524628i	$0.824204\pi$
$770$	0	0
$771$	12.7263	0.458328
$772$	0	0
$773$	−26.3767	−0.948704	−0.474352	−	0.880335i	$-0.657318\pi$
$773$	−26.3767	−0.948704	−0.474352	+	0.880335i	$0.657318\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.751071	0.0269445
$778$	0	0
$779$	20.0707	0.719106
$780$	0	0
$781$	8.50903	0.304477
$782$	0	0
$783$	−8.51016	−0.304128
$784$	0	0
$785$	12.0502	0.430091
$786$	0	0
$787$	−9.94977	−0.354671	−0.177336	−	0.984150i	$-0.556748\pi$
$787$	−9.94977	−0.354671	−0.177336	+	0.984150i	$0.556748\pi$
$788$	0	0
$789$	−6.82187	−0.242865
$790$	0	0
$791$	−14.4528	−0.513882
$792$	0	0
$793$	2.55509	0.0907338
$794$	0	0
$795$	7.49514	0.265825
$796$	0	0
$797$	−32.6053	−1.15494	−0.577470	−	0.816412i	$-0.695960\pi$
$797$	−32.6053	−1.15494	−0.577470	+	0.816412i	$0.695960\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	11.6911	0.413086
$802$	0	0
$803$	−4.43803	−0.156615
$804$	0	0
$805$	0.670813	0.0236430
$806$	0	0
$807$	−7.59030	−0.267191
$808$	0	0
$809$	−46.9848	−1.65190	−0.825950	−	0.563744i	$-0.809360\pi$
$809$	−46.9848	−1.65190	−0.825950	+	0.563744i	$0.809360\pi$
$810$	0	0
$811$	10.6106	0.372589	0.186294	−	0.982494i	$-0.440352\pi$
$811$	10.6106	0.372589	0.186294	+	0.982494i	$0.440352\pi$
$812$	0	0
$813$	−5.44208	−0.190862
$814$	0	0
$815$	4.05306	0.141973
$816$	0	0
$817$	−54.7645	−1.91597
$818$	0	0
$819$	−0.819031	−0.0286193
$820$	0	0
$821$	−14.3648	−0.501334	−0.250667	−	0.968073i	$-0.580650\pi$
$821$	−14.3648	−0.501334	−0.250667	+	0.968073i	$0.580650\pi$
$822$	0	0
$823$	34.7386	1.21091	0.605456	−	0.795879i	$-0.292991\pi$
$823$	34.7386	1.21091	0.605456	+	0.795879i	$0.292991\pi$
$824$	0	0
$825$	1.08298	0.0377044
$826$	0	0
$827$	23.4625	0.815871	0.407936	−	0.913011i	$-0.366249\pi$
$827$	23.4625	0.815871	0.407936	+	0.913011i	$0.366249\pi$
$828$	0	0
$829$	6.82433	0.237019	0.118509	−	0.992953i	$-0.462188\pi$
$829$	6.82433	0.237019	0.118509	+	0.992953i	$0.462188\pi$
$830$	0	0
$831$	−23.4325	−0.812864
$832$	0	0
$833$	−7.47457	−0.258979
$834$	0	0
$835$	20.6283	0.713873
$836$	0	0
$837$	0	0
$838$	0	0
$839$	55.3139	1.90965	0.954824	−	0.297171i	$-0.0960432\pi$
$839$	55.3139	1.90965	0.954824	+	0.297171i	$0.0960432\pi$
$840$	0	0
$841$	43.4228	1.49734
$842$	0	0
$843$	−28.9701	−0.997783
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	8.04876	0.276558
$848$	0	0
$849$	−24.5183	−0.841465
$850$	0	0
$851$	−0.751071	−0.0257464
$852$	0	0
$853$	26.4596	0.905958	0.452979	−	0.891521i	$-0.350361\pi$
$853$	26.4596	0.905958	0.452979	+	0.891521i	$0.350361\pi$
$854$	0	0
$855$	6.51016	0.222643
$856$	0	0
$857$	−33.6734	−1.15026	−0.575131	−	0.818062i	$-0.695049\pi$
$857$	−33.6734	−1.15026	−0.575131	+	0.818062i	$0.695049\pi$
$858$	0	0
$859$	−2.95249	−0.100738	−0.0503688	−	0.998731i	$-0.516040\pi$
$859$	−2.95249	−0.100738	−0.0503688	+	0.998731i	$0.516040\pi$
$860$	0	0
$861$	2.52505	0.0860536
$862$	0	0
$863$	33.4066	1.13717	0.568587	−	0.822623i	$-0.307490\pi$
$863$	33.4066	1.13717	0.568587	+	0.822623i	$0.307490\pi$
$864$	0	0
$865$	−12.0502	−0.409720
$866$	0	0
$867$	15.6053	0.529984
$868$	0	0
$869$	−10.4953	−0.356027
$870$	0	0
$871$	−10.2462	−0.347180
$872$	0	0
$873$	14.2013	0.480641
$874$	0	0
$875$	−0.819031	−0.0276883
$876$	0	0
$877$	−48.3725	−1.63342	−0.816712	−	0.577045i	$-0.804206\pi$
$877$	−48.3725	−1.63342	−0.816712	+	0.577045i	$0.804206\pi$
$878$	0	0
$879$	−23.4325	−0.790358
$880$	0	0
$881$	9.36599	0.315548	0.157774	−	0.987475i	$-0.449568\pi$
$881$	9.36599	0.315548	0.157774	+	0.987475i	$0.449568\pi$
$882$	0	0
$883$	−6.27208	−0.211072	−0.105536	−	0.994415i	$-0.533656\pi$
$883$	−6.27208	−0.211072	−0.105536	+	0.994415i	$0.533656\pi$
$884$	0	0
$885$	0.195985	0.00658795
$886$	0	0
$887$	21.8393	0.733293	0.366647	−	0.930360i	$-0.380506\pi$
$887$	21.8393	0.733293	0.366647	+	0.930360i	$0.380506\pi$
$888$	0	0
$889$	8.98485	0.301342
$890$	0	0
$891$	−1.08298	−0.0362811
$892$	0	0
$893$	0	0
$894$	0	0
$895$	17.4802	0.584300
$896$	0	0
$897$	0.819031	0.0273467
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−8.85152	−0.294887
$902$	0	0
$903$	−6.88983	−0.229279
$904$	0	0
$905$	3.47495	0.115511
$906$	0	0
$907$	−35.2771	−1.17136	−0.585679	−	0.810543i	$-0.699172\pi$
$907$	−35.2771	−1.17136	−0.585679	+	0.810543i	$0.699172\pi$
$908$	0	0
$909$	9.03805	0.299773
$910$	0	0
$911$	10.8544	0.359621	0.179810	−	0.983701i	$-0.442452\pi$
$911$	10.8544	0.359621	0.179810	+	0.983701i	$0.442452\pi$
$912$	0	0
$913$	6.76450	0.223872
$914$	0	0
$915$	−2.55509	−0.0844686
$916$	0	0
$917$	−1.55780	−0.0514432
$918$	0	0
$919$	28.0979	0.926863	0.463431	−	0.886133i	$-0.346618\pi$
$919$	28.0979	0.926863	0.463431	+	0.886133i	$0.346618\pi$
$920$	0	0
$921$	1.19870	0.0394986
$922$	0	0
$923$	−7.85708	−0.258619
$924$	0	0
$925$	0.917024	0.0301516
$926$	0	0
$927$	10.5781	0.347431
$928$	0	0
$929$	−42.6814	−1.40033	−0.700166	−	0.713980i	$-0.746891\pi$
$929$	−42.6814	−1.40033	−0.700166	+	0.713980i	$0.746891\pi$
$930$	0	0
$931$	−41.2040	−1.35041
$932$	0	0
$933$	−27.3822	−0.896455
$934$	0	0
$935$	−1.27896	−0.0418265
$936$	0	0
$937$	−45.4827	−1.48586	−0.742928	−	0.669372i	$-0.766563\pi$
$937$	−45.4827	−1.48586	−0.742928	+	0.669372i	$0.766563\pi$
$938$	0	0
$939$	−26.2964	−0.858152
$940$	0	0
$941$	23.5210	0.766763	0.383381	−	0.923590i	$-0.374759\pi$
$941$	23.5210	0.766763	0.383381	+	0.923590i	$0.374759\pi$
$942$	0	0
$943$	−2.52505	−0.0822271
$944$	0	0
$945$	0.819031	0.0266431
$946$	0	0
$947$	−6.91002	−0.224546	−0.112273	−	0.993677i	$-0.535813\pi$
$947$	−6.91002	−0.224546	−0.112273	+	0.993677i	$0.535813\pi$
$948$	0	0
$949$	4.09799	0.133026
$950$	0	0
$951$	−14.8243	−0.480712
$952$	0	0
$953$	2.39727	0.0776550	0.0388275	−	0.999246i	$-0.487638\pi$
$953$	2.39727	0.0776550	0.0388275	+	0.999246i	$0.487638\pi$
$954$	0	0
$955$	1.11017	0.0359244
$956$	0	0
$957$	9.21630	0.297921
$958$	0	0
$959$	3.91014	0.126265
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	14.1033	0.454472
$964$	0	0
$965$	3.34692	0.107741
$966$	0	0
$967$	−39.0228	−1.25489	−0.627444	−	0.778662i	$-0.715899\pi$
$967$	−39.0228	−1.25489	−0.627444	+	0.778662i	$0.715899\pi$
$968$	0	0
$969$	−7.68829	−0.246983
$970$	0	0
$971$	2.92799	0.0939637	0.0469818	−	0.998896i	$-0.485040\pi$
$971$	2.92799	0.0939637	0.0469818	+	0.998896i	$0.485040\pi$
$972$	0	0
$973$	16.7617	0.537354
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−29.3020	−0.937454	−0.468727	−	0.883343i	$-0.655287\pi$
$977$	−29.3020	−0.937454	−0.468727	+	0.883343i	$0.655287\pi$
$978$	0	0
$979$	−12.6612	−0.404654
$980$	0	0
$981$	1.73606	0.0554280
$982$	0	0
$983$	54.5084	1.73855	0.869275	−	0.494329i	$-0.164586\pi$
$983$	54.5084	1.73855	0.869275	+	0.494329i	$0.164586\pi$
$984$	0	0
$985$	−8.05023	−0.256502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.88983	0.219084
$990$	0	0
$991$	−7.41905	−0.235674	−0.117837	−	0.993033i	$-0.537596\pi$
$991$	−7.41905	−0.235674	−0.117837	+	0.993033i	$0.537596\pi$
$992$	0	0
$993$	−1.78628	−0.0566859
$994$	0	0
$995$	17.5223	0.555495
$996$	0	0
$997$	−8.38780	−0.265644	−0.132822	−	0.991140i	$-0.542404\pi$
$997$	−8.38780	−0.265644	−0.132822	+	0.991140i	$0.542404\pi$
$998$	0	0
$999$	−0.917024	−0.0290133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.ce.1.3	✓	4
4.3	odd	2		6240.2.a.cf.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.ce.1.3	✓	4	1.1	even	1	trivial
6240.2.a.cf.1.2	yes	4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.819031 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.819031 q^{7} +1.00000 q^{9} -1.08298 q^{11} +1.00000 q^{13} -1.00000 q^{15} +1.18097 q^{17} +6.51016 q^{19} +0.819031 q^{21} -0.819031 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.51016 q^{29} +1.08298 q^{33} -0.819031 q^{35} +0.917024 q^{37} -1.00000 q^{39} +3.08298 q^{41} -8.41216 q^{43} +1.00000 q^{45} -6.32919 q^{49} -1.18097 q^{51} -7.49514 q^{53} -1.08298 q^{55} -6.51016 q^{57} -0.195985 q^{59} +2.55509 q^{61} -0.819031 q^{63} +1.00000 q^{65} -10.2462 q^{67} +0.819031 q^{69} -7.85708 q^{71} +4.09799 q^{73} -1.00000 q^{75} +0.886992 q^{77} +9.69112 q^{79} +1.00000 q^{81} -6.24621 q^{83} +1.18097 q^{85} -8.51016 q^{87} +11.6911 q^{89} -0.819031 q^{91} +6.51016 q^{95} +14.2013 q^{97} -1.08298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * q^99

Introduction

L-functions

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