LMFDB - Embedded newform 6240.2.a.bo.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} -1.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.37228 q^{17} -2.00000 q^{19} +1.37228 q^{21} +3.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -1.37228 q^{33} -1.37228 q^{35} -1.37228 q^{37} -1.00000 q^{39} +5.37228 q^{41} -6.74456 q^{43} -1.00000 q^{45} -8.74456 q^{47} -5.11684 q^{49} -5.37228 q^{51} -13.3723 q^{53} +1.37228 q^{55} -2.00000 q^{57} +8.74456 q^{59} +8.11684 q^{61} +1.37228 q^{63} +1.00000 q^{65} +12.7446 q^{67} +3.37228 q^{69} -1.37228 q^{71} -6.00000 q^{73} +1.00000 q^{75} -1.88316 q^{77} -15.3723 q^{79} +1.00000 q^{81} -3.25544 q^{83} +5.37228 q^{85} +8.74456 q^{87} +4.11684 q^{89} -1.37228 q^{91} -4.74456 q^{93} +2.00000 q^{95} -8.11684 q^{97} -1.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} + 3 q^{33} + 3 q^{35} + 3 q^{37} - 2 q^{39} + 5 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{47} + 7 q^{49} - 5 q^{51} - 21 q^{53} - 3 q^{55} - 4 q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 2 q^{65} + 14 q^{67} + q^{69} + 3 q^{71} - 12 q^{73} + 2 q^{75} - 21 q^{77} - 25 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{85} + 6 q^{87} - 9 q^{89} + 3 q^{91} + 2 q^{93} + 4 q^{95} + q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 4 * q^19 - 3 * q^21 + q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 + 3 * q^33 + 3 * q^35 + 3 * q^37 - 2 * q^39 + 5 * q^41 - 2 * q^43 - 2 * q^45 - 6 * q^47 + 7 * q^49 - 5 * q^51 - 21 * q^53 - 3 * q^55 - 4 * q^57 + 6 * q^59 - q^61 - 3 * q^63 + 2 * q^65 + 14 * q^67 + q^69 + 3 * q^71 - 12 * q^73 + 2 * q^75 - 21 * q^77 - 25 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^85 + 6 * q^87 - 9 * q^89 + 3 * q^91 + 2 * q^93 + 4 * q^95 + 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
$7$	1.37228	0.518674	0.259337	−	0.965787i	$-0.416496\pi$
$7$	1.37228	0.518674	0.259337	+	0.965787i	$0.416496\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.37228	−1.30297	−0.651485	−	0.758662i	$-0.725854\pi$
$17$	−5.37228	−1.30297	−0.651485	+	0.758662i	$0.725854\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.37228	0.299456
$22$	0	0
$23$	3.37228	0.703169	0.351585	−	0.936156i	$-0.385643\pi$
$23$	3.37228	0.703169	0.351585	+	0.936156i	$0.385643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	0	0
$33$	−1.37228	−0.238884
$34$	0	0
$35$	−1.37228	−0.231958
$36$	0	0
$37$	−1.37228	−0.225602	−0.112801	−	0.993618i	$-0.535982\pi$
$37$	−1.37228	−0.225602	−0.112801	+	0.993618i	$0.535982\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.37228	0.839009	0.419505	−	0.907753i	$-0.362204\pi$
$41$	5.37228	0.839009	0.419505	+	0.907753i	$0.362204\pi$
$42$	0	0
$43$	−6.74456	−1.02854	−0.514268	−	0.857629i	$-0.671936\pi$
$43$	−6.74456	−1.02854	−0.514268	+	0.857629i	$0.671936\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.74456	−1.27553	−0.637763	−	0.770233i	$-0.720140\pi$
$47$	−8.74456	−1.27553	−0.637763	+	0.770233i	$0.720140\pi$
$48$	0	0
$49$	−5.11684	−0.730978
$50$	0	0
$51$	−5.37228	−0.752270
$52$	0	0
$53$	−13.3723	−1.83682	−0.918412	−	0.395625i	$-0.870528\pi$
$53$	−13.3723	−1.83682	−0.918412	+	0.395625i	$0.870528\pi$
$54$	0	0
$55$	1.37228	0.185038
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	8.74456	1.13845	0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456	1.13845	0.569223	+	0.822183i	$0.307244\pi$
$60$	0	0
$61$	8.11684	1.03926	0.519628	−	0.854393i	$-0.326071\pi$
$61$	8.11684	1.03926	0.519628	+	0.854393i	$0.326071\pi$
$62$	0	0
$63$	1.37228	0.172891
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	12.7446	1.55700	0.778498	−	0.627647i	$-0.215982\pi$
$67$	12.7446	1.55700	0.778498	+	0.627647i	$0.215982\pi$
$68$	0	0
$69$	3.37228	0.405975
$70$	0	0
$71$	−1.37228	−0.162860	−0.0814299	−	0.996679i	$-0.525949\pi$
$71$	−1.37228	−0.162860	−0.0814299	+	0.996679i	$0.525949\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.88316	−0.214606
$78$	0	0
$79$	−15.3723	−1.72952	−0.864758	−	0.502188i	$-0.832528\pi$
$79$	−15.3723	−1.72952	−0.864758	+	0.502188i	$0.832528\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.25544	−0.357331	−0.178665	−	0.983910i	$-0.557178\pi$
$83$	−3.25544	−0.357331	−0.178665	+	0.983910i	$0.557178\pi$
$84$	0	0
$85$	5.37228	0.582706
$86$	0	0
$87$	8.74456	0.937516
$88$	0	0
$89$	4.11684	0.436385	0.218192	−	0.975906i	$-0.429984\pi$
$89$	4.11684	0.436385	0.218192	+	0.975906i	$0.429984\pi$
$90$	0	0
$91$	−1.37228	−0.143854
$92$	0	0
$93$	−4.74456	−0.491988
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−8.11684	−0.824141	−0.412070	−	0.911152i	$-0.635194\pi$
$97$	−8.11684	−0.824141	−0.412070	+	0.911152i	$0.635194\pi$
$98$	0	0
$99$	−1.37228	−0.137919
$100$	0	0
$101$	−16.7446	−1.66615	−0.833073	−	0.553163i	$-0.813421\pi$
$101$	−16.7446	−1.66615	−0.833073	+	0.553163i	$0.813421\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−1.37228	−0.133921
$106$	0	0
$107$	7.37228	0.712705	0.356353	−	0.934352i	$-0.384020\pi$
$107$	7.37228	0.712705	0.356353	+	0.934352i	$0.384020\pi$
$108$	0	0
$109$	−20.7446	−1.98697	−0.993484	−	0.113968i	$-0.963644\pi$
$109$	−20.7446	−1.98697	−0.993484	+	0.113968i	$0.963644\pi$
$110$	0	0
$111$	−1.37228	−0.130251
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−3.37228	−0.314467
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−7.37228	−0.675816
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	5.37228	0.484402
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−6.74456	−0.593826
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−2.74456	−0.237984
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	15.4891	1.32333	0.661663	−	0.749802i	$-0.269851\pi$
$137$	15.4891	1.32333	0.661663	+	0.749802i	$0.269851\pi$
$138$	0	0
$139$	−14.1168	−1.19738	−0.598688	−	0.800983i	$-0.704311\pi$
$139$	−14.1168	−1.19738	−0.598688	+	0.800983i	$0.704311\pi$
$140$	0	0
$141$	−8.74456	−0.736425
$142$	0	0
$143$	1.37228	0.114756
$144$	0	0
$145$	−8.74456	−0.726196
$146$	0	0
$147$	−5.11684	−0.422030
$148$	0	0
$149$	4.11684	0.337265	0.168632	−	0.985679i	$-0.446065\pi$
$149$	4.11684	0.337265	0.168632	+	0.985679i	$0.446065\pi$
$150$	0	0
$151$	−14.2337	−1.15832	−0.579161	−	0.815214i	$-0.696620\pi$
$151$	−14.2337	−1.15832	−0.579161	+	0.815214i	$0.696620\pi$
$152$	0	0
$153$	−5.37228	−0.434323
$154$	0	0
$155$	4.74456	0.381092
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−13.3723	−1.06049
$160$	0	0
$161$	4.62772	0.364715
$162$	0	0
$163$	10.8614	0.850731	0.425366	−	0.905022i	$-0.360146\pi$
$163$	10.8614	0.850731	0.425366	+	0.905022i	$0.360146\pi$
$164$	0	0
$165$	1.37228	0.106832
$166$	0	0
$167$	−8.74456	−0.676675	−0.338337	−	0.941025i	$-0.609864\pi$
$167$	−8.74456	−0.676675	−0.338337	+	0.941025i	$0.609864\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	11.4891	0.873502	0.436751	−	0.899582i	$-0.356129\pi$
$173$	11.4891	0.873502	0.436751	+	0.899582i	$0.356129\pi$
$174$	0	0
$175$	1.37228	0.103735
$176$	0	0
$177$	8.74456	0.657282
$178$	0	0
$179$	14.7446	1.10206	0.551030	−	0.834485i	$-0.314235\pi$
$179$	14.7446	1.10206	0.551030	+	0.834485i	$0.314235\pi$
$180$	0	0
$181$	12.1168	0.900638	0.450319	−	0.892868i	$-0.351310\pi$
$181$	12.1168	0.900638	0.450319	+	0.892868i	$0.351310\pi$
$182$	0	0
$183$	8.11684	0.600014
$184$	0	0
$185$	1.37228	0.100892
$186$	0	0
$187$	7.37228	0.539115
$188$	0	0
$189$	1.37228	0.0998188
$190$	0	0
$191$	−2.74456	−0.198590	−0.0992948	−	0.995058i	$-0.531659\pi$
$191$	−2.74456	−0.198590	−0.0992948	+	0.995058i	$0.531659\pi$
$192$	0	0
$193$	−3.88316	−0.279516	−0.139758	−	0.990186i	$-0.544632\pi$
$193$	−3.88316	−0.279516	−0.139758	+	0.990186i	$0.544632\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−19.4891	−1.38854	−0.694271	−	0.719713i	$-0.744273\pi$
$197$	−19.4891	−1.38854	−0.694271	+	0.719713i	$0.744273\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	12.7446	0.898932
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−5.37228	−0.375216
$206$	0	0
$207$	3.37228	0.234390
$208$	0	0
$209$	2.74456	0.189845
$210$	0	0
$211$	1.48913	0.102516	0.0512578	−	0.998685i	$-0.483677\pi$
$211$	1.48913	0.102516	0.0512578	+	0.998685i	$0.483677\pi$
$212$	0	0
$213$	−1.37228	−0.0940272
$214$	0	0
$215$	6.74456	0.459975
$216$	0	0
$217$	−6.51087	−0.441987
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	5.37228	0.361379
$222$	0	0
$223$	14.2337	0.953158	0.476579	−	0.879132i	$-0.341877\pi$
$223$	14.2337	0.953158	0.476579	+	0.879132i	$0.341877\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	−10.2337	−0.676261	−0.338131	−	0.941099i	$-0.609795\pi$
$229$	−10.2337	−0.676261	−0.338131	+	0.941099i	$0.609795\pi$
$230$	0	0
$231$	−1.88316	−0.123903
$232$	0	0
$233$	−10.6277	−0.696245	−0.348122	−	0.937449i	$-0.613181\pi$
$233$	−10.6277	−0.696245	−0.348122	+	0.937449i	$0.613181\pi$
$234$	0	0
$235$	8.74456	0.570432
$236$	0	0
$237$	−15.3723	−0.998537
$238$	0	0
$239$	25.6060	1.65631	0.828156	−	0.560497i	$-0.189390\pi$
$239$	25.6060	1.65631	0.828156	+	0.560497i	$0.189390\pi$
$240$	0	0
$241$	−4.74456	−0.305624	−0.152812	−	0.988255i	$-0.548833\pi$
$241$	−4.74456	−0.305624	−0.152812	+	0.988255i	$0.548833\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.11684	0.326903
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−3.25544	−0.206305
$250$	0	0
$251$	−26.9783	−1.70285	−0.851426	−	0.524475i	$-0.824262\pi$
$251$	−26.9783	−1.70285	−0.851426	+	0.524475i	$0.824262\pi$
$252$	0	0
$253$	−4.62772	−0.290942
$254$	0	0
$255$	5.37228	0.336425
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−1.88316	−0.117014
$260$	0	0
$261$	8.74456	0.541275
$262$	0	0
$263$	−25.4891	−1.57173	−0.785863	−	0.618400i	$-0.787781\pi$
$263$	−25.4891	−1.57173	−0.785863	+	0.618400i	$0.787781\pi$
$264$	0	0
$265$	13.3723	0.821453
$266$	0	0
$267$	4.11684	0.251947
$268$	0	0
$269$	−31.7228	−1.93417	−0.967087	−	0.254446i	$-0.918107\pi$
$269$	−31.7228	−1.93417	−0.967087	+	0.254446i	$0.918107\pi$
$270$	0	0
$271$	19.4891	1.18388	0.591940	−	0.805982i	$-0.298362\pi$
$271$	19.4891	1.18388	0.591940	+	0.805982i	$0.298362\pi$
$272$	0	0
$273$	−1.37228	−0.0830542
$274$	0	0
$275$	−1.37228	−0.0827517
$276$	0	0
$277$	−32.7446	−1.96743	−0.983715	−	0.179735i	$-0.942476\pi$
$277$	−32.7446	−1.96743	−0.983715	+	0.179735i	$0.942476\pi$
$278$	0	0
$279$	−4.74456	−0.284050
$280$	0	0
$281$	−19.4891	−1.16262	−0.581312	−	0.813681i	$-0.697460\pi$
$281$	−19.4891	−1.16262	−0.581312	+	0.813681i	$0.697460\pi$
$282$	0	0
$283$	−2.51087	−0.149256	−0.0746280	−	0.997211i	$-0.523777\pi$
$283$	−2.51087	−0.149256	−0.0746280	+	0.997211i	$0.523777\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	7.37228	0.435172
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	−8.11684	−0.475818
$292$	0	0
$293$	8.74456	0.510863	0.255431	−	0.966827i	$-0.417783\pi$
$293$	8.74456	0.510863	0.255431	+	0.966827i	$0.417783\pi$
$294$	0	0
$295$	−8.74456	−0.509128
$296$	0	0
$297$	−1.37228	−0.0796278
$298$	0	0
$299$	−3.37228	−0.195024
$300$	0	0
$301$	−9.25544	−0.533475
$302$	0	0
$303$	−16.7446	−0.961950
$304$	0	0
$305$	−8.11684	−0.464769
$306$	0	0
$307$	−8.11684	−0.463253	−0.231626	−	0.972805i	$-0.574405\pi$
$307$	−8.11684	−0.463253	−0.231626	+	0.972805i	$0.574405\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	25.7228	1.45861	0.729303	−	0.684190i	$-0.239844\pi$
$311$	25.7228	1.45861	0.729303	+	0.684190i	$0.239844\pi$
$312$	0	0
$313$	16.9783	0.959667	0.479834	−	0.877359i	$-0.340697\pi$
$313$	16.9783	0.959667	0.479834	+	0.877359i	$0.340697\pi$
$314$	0	0
$315$	−1.37228	−0.0773193
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	7.37228	0.411481
$322$	0	0
$323$	10.7446	0.597843
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−20.7446	−1.14718
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	22.0000	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$332$	0	0
$333$	−1.37228	−0.0752006
$334$	0	0
$335$	−12.7446	−0.696310
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	6.51087	0.352584
$342$	0	0
$343$	−16.6277	−0.897812
$344$	0	0
$345$	−3.37228	−0.181558
$346$	0	0
$347$	17.8832	0.960018	0.480009	−	0.877264i	$-0.340633\pi$
$347$	17.8832	0.960018	0.480009	+	0.877264i	$0.340633\pi$
$348$	0	0
$349$	−24.9783	−1.33706	−0.668528	−	0.743687i	$-0.733075\pi$
$349$	−24.9783	−1.33706	−0.668528	+	0.743687i	$0.733075\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−20.7446	−1.10412	−0.552061	−	0.833804i	$-0.686158\pi$
$353$	−20.7446	−1.10412	−0.552061	+	0.833804i	$0.686158\pi$
$354$	0	0
$355$	1.37228	0.0728331
$356$	0	0
$357$	−7.37228	−0.390183
$358$	0	0
$359$	20.7446	1.09486	0.547428	−	0.836853i	$-0.315607\pi$
$359$	20.7446	1.09486	0.547428	+	0.836853i	$0.315607\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−9.11684	−0.478510
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	13.2554	0.691928	0.345964	−	0.938248i	$-0.387552\pi$
$367$	13.2554	0.691928	0.345964	+	0.938248i	$0.387552\pi$
$368$	0	0
$369$	5.37228	0.279670
$370$	0	0
$371$	−18.3505	−0.952712
$372$	0	0
$373$	7.48913	0.387772	0.193886	−	0.981024i	$-0.437891\pi$
$373$	7.48913	0.387772	0.193886	+	0.981024i	$0.437891\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.74456	−0.450368
$378$	0	0
$379$	26.2337	1.34753	0.673767	−	0.738944i	$-0.264675\pi$
$379$	26.2337	1.34753	0.673767	+	0.738944i	$0.264675\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	4.51087	0.230495	0.115247	−	0.993337i	$-0.463234\pi$
$383$	4.51087	0.230495	0.115247	+	0.993337i	$0.463234\pi$
$384$	0	0
$385$	1.88316	0.0959745
$386$	0	0
$387$	−6.74456	−0.342845
$388$	0	0
$389$	−28.7446	−1.45741	−0.728704	−	0.684829i	$-0.759877\pi$
$389$	−28.7446	−1.45741	−0.728704	+	0.684829i	$0.759877\pi$
$390$	0	0
$391$	−18.1168	−0.916208
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	15.3723	0.773463
$396$	0	0
$397$	25.6060	1.28513	0.642563	−	0.766233i	$-0.277871\pi$
$397$	25.6060	1.28513	0.642563	+	0.766233i	$0.277871\pi$
$398$	0	0
$399$	−2.74456	−0.137400
$400$	0	0
$401$	−11.4891	−0.573740	−0.286870	−	0.957970i	$-0.592615\pi$
$401$	−11.4891	−0.573740	−0.286870	+	0.957970i	$0.592615\pi$
$402$	0	0
$403$	4.74456	0.236343
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	1.88316	0.0933446
$408$	0	0
$409$	6.23369	0.308236	0.154118	−	0.988052i	$-0.450746\pi$
$409$	6.23369	0.308236	0.154118	+	0.988052i	$0.450746\pi$
$410$	0	0
$411$	15.4891	0.764022
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	3.25544	0.159803
$416$	0	0
$417$	−14.1168	−0.691305
$418$	0	0
$419$	10.7446	0.524906	0.262453	−	0.964945i	$-0.415468\pi$
$419$	10.7446	0.524906	0.262453	+	0.964945i	$0.415468\pi$
$420$	0	0
$421$	24.7446	1.20598	0.602988	−	0.797750i	$-0.293977\pi$
$421$	24.7446	1.20598	0.602988	+	0.797750i	$0.293977\pi$
$422$	0	0
$423$	−8.74456	−0.425175
$424$	0	0
$425$	−5.37228	−0.260594
$426$	0	0
$427$	11.1386	0.539034
$428$	0	0
$429$	1.37228	0.0662544
$430$	0	0
$431$	14.2337	0.685613	0.342806	−	0.939406i	$-0.388623\pi$
$431$	14.2337	0.685613	0.342806	+	0.939406i	$0.388623\pi$
$432$	0	0
$433$	−0.510875	−0.0245511	−0.0122755	−	0.999925i	$-0.503908\pi$
$433$	−0.510875	−0.0245511	−0.0122755	+	0.999925i	$0.503908\pi$
$434$	0	0
$435$	−8.74456	−0.419270
$436$	0	0
$437$	−6.74456	−0.322636
$438$	0	0
$439$	−10.1168	−0.482851	−0.241425	−	0.970419i	$-0.577615\pi$
$439$	−10.1168	−0.482851	−0.241425	+	0.970419i	$0.577615\pi$
$440$	0	0
$441$	−5.11684	−0.243659
$442$	0	0
$443$	12.8614	0.611064	0.305532	−	0.952182i	$-0.401166\pi$
$443$	12.8614	0.611064	0.305532	+	0.952182i	$0.401166\pi$
$444$	0	0
$445$	−4.11684	−0.195157
$446$	0	0
$447$	4.11684	0.194720
$448$	0	0
$449$	−16.1168	−0.760601	−0.380300	−	0.924863i	$-0.624179\pi$
$449$	−16.1168	−0.760601	−0.380300	+	0.924863i	$0.624179\pi$
$450$	0	0
$451$	−7.37228	−0.347147
$452$	0	0
$453$	−14.2337	−0.668757
$454$	0	0
$455$	1.37228	0.0643335
$456$	0	0
$457$	−2.62772	−0.122919	−0.0614597	−	0.998110i	$-0.519576\pi$
$457$	−2.62772	−0.122919	−0.0614597	+	0.998110i	$0.519576\pi$
$458$	0	0
$459$	−5.37228	−0.250757
$460$	0	0
$461$	18.8614	0.878463	0.439232	−	0.898374i	$-0.355251\pi$
$461$	18.8614	0.878463	0.439232	+	0.898374i	$0.355251\pi$
$462$	0	0
$463$	−41.6060	−1.93359	−0.966797	−	0.255547i	$-0.917745\pi$
$463$	−41.6060	−1.93359	−0.966797	+	0.255547i	$0.917745\pi$
$464$	0	0
$465$	4.74456	0.220024
$466$	0	0
$467$	3.37228	0.156051	0.0780253	−	0.996951i	$-0.475139\pi$
$467$	3.37228	0.156051	0.0780253	+	0.996951i	$0.475139\pi$
$468$	0	0
$469$	17.4891	0.807573
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	9.25544	0.425565
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−13.3723	−0.612275
$478$	0	0
$479$	−23.0951	−1.05524	−0.527621	−	0.849480i	$-0.676916\pi$
$479$	−23.0951	−1.05524	−0.527621	+	0.849480i	$0.676916\pi$
$480$	0	0
$481$	1.37228	0.0625706
$482$	0	0
$483$	4.62772	0.210568
$484$	0	0
$485$	8.11684	0.368567
$486$	0	0
$487$	−41.8397	−1.89594	−0.947968	−	0.318366i	$-0.896866\pi$
$487$	−41.8397	−1.89594	−0.947968	+	0.318366i	$0.896866\pi$
$488$	0	0
$489$	10.8614	0.491170
$490$	0	0
$491$	−6.74456	−0.304378	−0.152189	−	0.988351i	$-0.548632\pi$
$491$	−6.74456	−0.304378	−0.152189	+	0.988351i	$0.548632\pi$
$492$	0	0
$493$	−46.9783	−2.11579
$494$	0	0
$495$	1.37228	0.0616795
$496$	0	0
$497$	−1.88316	−0.0844711
$498$	0	0
$499$	−36.9783	−1.65537	−0.827687	−	0.561190i	$-0.810344\pi$
$499$	−36.9783	−1.65537	−0.827687	+	0.561190i	$0.810344\pi$
$500$	0	0
$501$	−8.74456	−0.390678
$502$	0	0
$503$	41.4891	1.84991	0.924954	−	0.380078i	$-0.124103\pi$
$503$	41.4891	1.84991	0.924954	+	0.380078i	$0.124103\pi$
$504$	0	0
$505$	16.7446	0.745123
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−19.8832	−0.881305	−0.440653	−	0.897678i	$-0.645253\pi$
$509$	−19.8832	−0.881305	−0.440653	+	0.897678i	$0.645253\pi$
$510$	0	0
$511$	−8.23369	−0.364237
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	11.4891	0.504317
$520$	0	0
$521$	8.51087	0.372868	0.186434	−	0.982467i	$-0.440307\pi$
$521$	8.51087	0.372868	0.186434	+	0.982467i	$0.440307\pi$
$522$	0	0
$523$	−13.7228	−0.600057	−0.300028	−	0.953930i	$-0.596996\pi$
$523$	−13.7228	−0.600057	−0.300028	+	0.953930i	$0.596996\pi$
$524$	0	0
$525$	1.37228	0.0598913
$526$	0	0
$527$	25.4891	1.11032
$528$	0	0
$529$	−11.6277	−0.505553
$530$	0	0
$531$	8.74456	0.379482
$532$	0	0
$533$	−5.37228	−0.232699
$534$	0	0
$535$	−7.37228	−0.318732
$536$	0	0
$537$	14.7446	0.636275
$538$	0	0
$539$	7.02175	0.302448
$540$	0	0
$541$	7.48913	0.321983	0.160991	−	0.986956i	$-0.448531\pi$
$541$	7.48913	0.321983	0.160991	+	0.986956i	$0.448531\pi$
$542$	0	0
$543$	12.1168	0.519984
$544$	0	0
$545$	20.7446	0.888599
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	8.11684	0.346418
$550$	0	0
$551$	−17.4891	−0.745062
$552$	0	0
$553$	−21.0951	−0.897055
$554$	0	0
$555$	1.37228	0.0582501
$556$	0	0
$557$	3.25544	0.137937	0.0689687	−	0.997619i	$-0.478029\pi$
$557$	3.25544	0.137937	0.0689687	+	0.997619i	$0.478029\pi$
$558$	0	0
$559$	6.74456	0.285265
$560$	0	0
$561$	7.37228	0.311258
$562$	0	0
$563$	13.8832	0.585105	0.292553	−	0.956249i	$-0.405495\pi$
$563$	13.8832	0.585105	0.292553	+	0.956249i	$0.405495\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	1.37228	0.0576304
$568$	0	0
$569$	36.9783	1.55021	0.775104	−	0.631833i	$-0.217697\pi$
$569$	36.9783	1.55021	0.775104	+	0.631833i	$0.217697\pi$
$570$	0	0
$571$	23.6060	0.987879	0.493940	−	0.869496i	$-0.335556\pi$
$571$	23.6060	0.987879	0.493940	+	0.869496i	$0.335556\pi$
$572$	0	0
$573$	−2.74456	−0.114656
$574$	0	0
$575$	3.37228	0.140634
$576$	0	0
$577$	18.8614	0.785211	0.392605	−	0.919707i	$-0.371574\pi$
$577$	18.8614	0.785211	0.392605	+	0.919707i	$0.371574\pi$
$578$	0	0
$579$	−3.88316	−0.161378
$580$	0	0
$581$	−4.46738	−0.185338
$582$	0	0
$583$	18.3505	0.760001
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	43.4891	1.79499	0.897494	−	0.441026i	$-0.145385\pi$
$587$	43.4891	1.79499	0.897494	+	0.441026i	$0.145385\pi$
$588$	0	0
$589$	9.48913	0.390993
$590$	0	0
$591$	−19.4891	−0.801675
$592$	0	0
$593$	27.7228	1.13844	0.569220	−	0.822185i	$-0.307245\pi$
$593$	27.7228	1.13844	0.569220	+	0.822185i	$0.307245\pi$
$594$	0	0
$595$	7.37228	0.302234
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	14.8614	0.606209	0.303105	−	0.952957i	$-0.401977\pi$
$601$	14.8614	0.606209	0.303105	+	0.952957i	$0.401977\pi$
$602$	0	0
$603$	12.7446	0.518999
$604$	0	0
$605$	9.11684	0.370652
$606$	0	0
$607$	13.7228	0.556992	0.278496	−	0.960437i	$-0.410164\pi$
$607$	13.7228	0.556992	0.278496	+	0.960437i	$0.410164\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	8.74456	0.353767
$612$	0	0
$613$	48.3505	1.95286	0.976430	−	0.215835i	$-0.0692474\pi$
$613$	48.3505	1.95286	0.976430	+	0.215835i	$0.0692474\pi$
$614$	0	0
$615$	−5.37228	−0.216631
$616$	0	0
$617$	−46.4674	−1.87071	−0.935353	−	0.353716i	$-0.884918\pi$
$617$	−46.4674	−1.87071	−0.935353	+	0.353716i	$0.884918\pi$
$618$	0	0
$619$	−12.7446	−0.512247	−0.256124	−	0.966644i	$-0.582445\pi$
$619$	−12.7446	−0.512247	−0.256124	+	0.966644i	$0.582445\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	0	0
$623$	5.64947	0.226341
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.74456	0.109607
$628$	0	0
$629$	7.37228	0.293952
$630$	0	0
$631$	1.76631	0.0703158	0.0351579	−	0.999382i	$-0.488807\pi$
$631$	1.76631	0.0703158	0.0351579	+	0.999382i	$0.488807\pi$
$632$	0	0
$633$	1.48913	0.0591874
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	5.11684	0.202737
$638$	0	0
$639$	−1.37228	−0.0542866
$640$	0	0
$641$	−4.97825	−0.196629	−0.0983145	−	0.995155i	$-0.531345\pi$
$641$	−4.97825	−0.196629	−0.0983145	+	0.995155i	$0.531345\pi$
$642$	0	0
$643$	−20.1168	−0.793331	−0.396665	−	0.917963i	$-0.629833\pi$
$643$	−20.1168	−0.793331	−0.396665	+	0.917963i	$0.629833\pi$
$644$	0	0
$645$	6.74456	0.265567
$646$	0	0
$647$	15.6060	0.613534	0.306767	−	0.951785i	$-0.400753\pi$
$647$	15.6060	0.613534	0.306767	+	0.951785i	$0.400753\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−6.51087	−0.255181
$652$	0	0
$653$	−27.4891	−1.07573	−0.537866	−	0.843030i	$-0.680769\pi$
$653$	−27.4891	−1.07573	−0.537866	+	0.843030i	$0.680769\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	22.9783	0.895106	0.447553	−	0.894258i	$-0.352296\pi$
$659$	22.9783	0.895106	0.447553	+	0.894258i	$0.352296\pi$
$660$	0	0
$661$	39.4891	1.53595	0.767974	−	0.640480i	$-0.221265\pi$
$661$	39.4891	1.53595	0.767974	+	0.640480i	$0.221265\pi$
$662$	0	0
$663$	5.37228	0.208642
$664$	0	0
$665$	2.74456	0.106430
$666$	0	0
$667$	29.4891	1.14182
$668$	0	0
$669$	14.2337	0.550306
$670$	0	0
$671$	−11.1386	−0.430001
$672$	0	0
$673$	−30.2337	−1.16542	−0.582712	−	0.812679i	$-0.698008\pi$
$673$	−30.2337	−1.16542	−0.582712	+	0.812679i	$0.698008\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−26.8614	−1.03237	−0.516184	−	0.856478i	$-0.672648\pi$
$677$	−26.8614	−1.03237	−0.516184	+	0.856478i	$0.672648\pi$
$678$	0	0
$679$	−11.1386	−0.427460
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	0	0
$685$	−15.4891	−0.591809
$686$	0	0
$687$	−10.2337	−0.390440
$688$	0	0
$689$	13.3723	0.509443
$690$	0	0
$691$	16.9783	0.645883	0.322942	−	0.946419i	$-0.395328\pi$
$691$	16.9783	0.645883	0.322942	+	0.946419i	$0.395328\pi$
$692$	0	0
$693$	−1.88316	−0.0715352
$694$	0	0
$695$	14.1168	0.535482
$696$	0	0
$697$	−28.8614	−1.09320
$698$	0	0
$699$	−10.6277	−0.401977
$700$	0	0
$701$	6.23369	0.235443	0.117722	−	0.993047i	$-0.462441\pi$
$701$	6.23369	0.235443	0.117722	+	0.993047i	$0.462441\pi$
$702$	0	0
$703$	2.74456	0.103513
$704$	0	0
$705$	8.74456	0.329339
$706$	0	0
$707$	−22.9783	−0.864186
$708$	0	0
$709$	−45.2119	−1.69797	−0.848985	−	0.528417i	$-0.822786\pi$
$709$	−45.2119	−1.69797	−0.848985	+	0.528417i	$0.822786\pi$
$710$	0	0
$711$	−15.3723	−0.576506
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−1.37228	−0.0513204
$716$	0	0
$717$	25.6060	0.956272
$718$	0	0
$719$	−48.4674	−1.80753	−0.903764	−	0.428031i	$-0.859207\pi$
$719$	−48.4674	−1.80753	−0.903764	+	0.428031i	$0.859207\pi$
$720$	0	0
$721$	−10.9783	−0.408851
$722$	0	0
$723$	−4.74456	−0.176452
$724$	0	0
$725$	8.74456	0.324765
$726$	0	0
$727$	4.23369	0.157019	0.0785094	−	0.996913i	$-0.474984\pi$
$727$	4.23369	0.157019	0.0785094	+	0.996913i	$0.474984\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.2337	1.34015
$732$	0	0
$733$	33.6060	1.24126	0.620632	−	0.784102i	$-0.286876\pi$
$733$	33.6060	1.24126	0.620632	+	0.784102i	$0.286876\pi$
$734$	0	0
$735$	5.11684	0.188738
$736$	0	0
$737$	−17.4891	−0.644220
$738$	0	0
$739$	−43.9565	−1.61697	−0.808483	−	0.588520i	$-0.799711\pi$
$739$	−43.9565	−1.61697	−0.808483	+	0.588520i	$0.799711\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	0	0
$745$	−4.11684	−0.150829
$746$	0	0
$747$	−3.25544	−0.119110
$748$	0	0
$749$	10.1168	0.369661
$750$	0	0
$751$	35.8397	1.30781	0.653904	−	0.756578i	$-0.273130\pi$
$751$	35.8397	1.30781	0.653904	+	0.756578i	$0.273130\pi$
$752$	0	0
$753$	−26.9783	−0.983142
$754$	0	0
$755$	14.2337	0.518017
$756$	0	0
$757$	−8.74456	−0.317827	−0.158913	−	0.987293i	$-0.550799\pi$
$757$	−8.74456	−0.317827	−0.158913	+	0.987293i	$0.550799\pi$
$758$	0	0
$759$	−4.62772	−0.167976
$760$	0	0
$761$	31.4891	1.14148	0.570740	−	0.821131i	$-0.306656\pi$
$761$	31.4891	1.14148	0.570740	+	0.821131i	$0.306656\pi$
$762$	0	0
$763$	−28.4674	−1.03059
$764$	0	0
$765$	5.37228	0.194235
$766$	0	0
$767$	−8.74456	−0.315748
$768$	0	0
$769$	−51.9565	−1.87360	−0.936800	−	0.349866i	$-0.886227\pi$
$769$	−51.9565	−1.87360	−0.936800	+	0.349866i	$0.886227\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−7.72281	−0.277770	−0.138885	−	0.990308i	$-0.544352\pi$
$773$	−7.72281	−0.277770	−0.138885	+	0.990308i	$0.544352\pi$
$774$	0	0
$775$	−4.74456	−0.170430
$776$	0	0
$777$	−1.88316	−0.0675578
$778$	0	0
$779$	−10.7446	−0.384964
$780$	0	0
$781$	1.88316	0.0673846
$782$	0	0
$783$	8.74456	0.312505
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	11.2554	0.401213	0.200607	−	0.979672i	$-0.435709\pi$
$787$	11.2554	0.401213	0.200607	+	0.979672i	$0.435709\pi$
$788$	0	0
$789$	−25.4891	−0.907437
$790$	0	0
$791$	−2.74456	−0.0975854
$792$	0	0
$793$	−8.11684	−0.288238
$794$	0	0
$795$	13.3723	0.474266
$796$	0	0
$797$	−27.0951	−0.959757	−0.479879	−	0.877335i	$-0.659319\pi$
$797$	−27.0951	−0.959757	−0.479879	+	0.877335i	$0.659319\pi$
$798$	0	0
$799$	46.9783	1.66197
$800$	0	0
$801$	4.11684	0.145462
$802$	0	0
$803$	8.23369	0.290561
$804$	0	0
$805$	−4.62772	−0.163106
$806$	0	0
$807$	−31.7228	−1.11670
$808$	0	0
$809$	−4.97825	−0.175026	−0.0875130	−	0.996163i	$-0.527892\pi$
$809$	−4.97825	−0.175026	−0.0875130	+	0.996163i	$0.527892\pi$
$810$	0	0
$811$	0.978251	0.0343510	0.0171755	−	0.999852i	$-0.494533\pi$
$811$	0.978251	0.0343510	0.0171755	+	0.999852i	$0.494533\pi$
$812$	0	0
$813$	19.4891	0.683513
$814$	0	0
$815$	−10.8614	−0.380458
$816$	0	0
$817$	13.4891	0.471925
$818$	0	0
$819$	−1.37228	−0.0479514
$820$	0	0
$821$	21.3723	0.745898	0.372949	−	0.927852i	$-0.378347\pi$
$821$	21.3723	0.745898	0.372949	+	0.927852i	$0.378347\pi$
$822$	0	0
$823$	−34.9783	−1.21927	−0.609633	−	0.792684i	$-0.708683\pi$
$823$	−34.9783	−1.21927	−0.609633	+	0.792684i	$0.708683\pi$
$824$	0	0
$825$	−1.37228	−0.0477767
$826$	0	0
$827$	48.7446	1.69501	0.847507	−	0.530784i	$-0.178102\pi$
$827$	48.7446	1.69501	0.847507	+	0.530784i	$0.178102\pi$
$828$	0	0
$829$	27.4891	0.954737	0.477368	−	0.878703i	$-0.341591\pi$
$829$	27.4891	0.954737	0.477368	+	0.878703i	$0.341591\pi$
$830$	0	0
$831$	−32.7446	−1.13590
$832$	0	0
$833$	27.4891	0.952442
$834$	0	0
$835$	8.74456	0.302618
$836$	0	0
$837$	−4.74456	−0.163996
$838$	0	0
$839$	32.5842	1.12493	0.562466	−	0.826820i	$-0.309853\pi$
$839$	32.5842	1.12493	0.562466	+	0.826820i	$0.309853\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	−19.4891	−0.671241
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−12.5109	−0.429879
$848$	0	0
$849$	−2.51087	−0.0861730
$850$	0	0
$851$	−4.62772	−0.158636
$852$	0	0
$853$	31.8832	1.09166	0.545829	−	0.837896i	$-0.316215\pi$
$853$	31.8832	1.09166	0.545829	+	0.837896i	$0.316215\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	45.8397	1.56585	0.782926	−	0.622114i	$-0.213726\pi$
$857$	45.8397	1.56585	0.782926	+	0.622114i	$0.213726\pi$
$858$	0	0
$859$	50.3505	1.71794	0.858969	−	0.512028i	$-0.171105\pi$
$859$	50.3505	1.71794	0.858969	+	0.512028i	$0.171105\pi$
$860$	0	0
$861$	7.37228	0.251247
$862$	0	0
$863$	−11.4891	−0.391094	−0.195547	−	0.980694i	$-0.562648\pi$
$863$	−11.4891	−0.391094	−0.195547	+	0.980694i	$0.562648\pi$
$864$	0	0
$865$	−11.4891	−0.390642
$866$	0	0
$867$	11.8614	0.402834
$868$	0	0
$869$	21.0951	0.715602
$870$	0	0
$871$	−12.7446	−0.431833
$872$	0	0
$873$	−8.11684	−0.274714
$874$	0	0
$875$	−1.37228	−0.0463916
$876$	0	0
$877$	7.48913	0.252890	0.126445	−	0.991974i	$-0.459643\pi$
$877$	7.48913	0.252890	0.126445	+	0.991974i	$0.459643\pi$
$878$	0	0
$879$	8.74456	0.294947
$880$	0	0
$881$	35.7228	1.20353	0.601766	−	0.798672i	$-0.294464\pi$
$881$	35.7228	1.20353	0.601766	+	0.798672i	$0.294464\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	−8.74456	−0.293945
$886$	0	0
$887$	−12.3940	−0.416151	−0.208075	−	0.978113i	$-0.566720\pi$
$887$	−12.3940	−0.416151	−0.208075	+	0.978113i	$0.566720\pi$
$888$	0	0
$889$	−21.9565	−0.736397
$890$	0	0
$891$	−1.37228	−0.0459732
$892$	0	0
$893$	17.4891	0.585251
$894$	0	0
$895$	−14.7446	−0.492856
$896$	0	0
$897$	−3.37228	−0.112597
$898$	0	0
$899$	−41.4891	−1.38374
$900$	0	0
$901$	71.8397	2.39333
$902$	0	0
$903$	−9.25544	−0.308002
$904$	0	0
$905$	−12.1168	−0.402778
$906$	0	0
$907$	22.7446	0.755221	0.377610	−	0.925965i	$-0.376746\pi$
$907$	22.7446	0.755221	0.377610	+	0.925965i	$0.376746\pi$
$908$	0	0
$909$	−16.7446	−0.555382
$910$	0	0
$911$	29.7228	0.984761	0.492380	−	0.870380i	$-0.336127\pi$
$911$	29.7228	0.984761	0.492380	+	0.870380i	$0.336127\pi$
$912$	0	0
$913$	4.46738	0.147849
$914$	0	0
$915$	−8.11684	−0.268335
$916$	0	0
$917$	16.4674	0.543801
$918$	0	0
$919$	14.1168	0.465672	0.232836	−	0.972516i	$-0.425200\pi$
$919$	14.1168	0.465672	0.232836	+	0.972516i	$0.425200\pi$
$920$	0	0
$921$	−8.11684	−0.267459
$922$	0	0
$923$	1.37228	0.0451692
$924$	0	0
$925$	−1.37228	−0.0451203
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−4.35053	−0.142736	−0.0713682	−	0.997450i	$-0.522737\pi$
$929$	−4.35053	−0.142736	−0.0713682	+	0.997450i	$0.522737\pi$
$930$	0	0
$931$	10.2337	0.335396
$932$	0	0
$933$	25.7228	0.842127
$934$	0	0
$935$	−7.37228	−0.241099
$936$	0	0
$937$	−28.7446	−0.939044	−0.469522	−	0.882921i	$-0.655574\pi$
$937$	−28.7446	−0.939044	−0.469522	+	0.882921i	$0.655574\pi$
$938$	0	0
$939$	16.9783	0.554064
$940$	0	0
$941$	49.6060	1.61711	0.808554	−	0.588422i	$-0.200250\pi$
$941$	49.6060	1.61711	0.808554	+	0.588422i	$0.200250\pi$
$942$	0	0
$943$	18.1168	0.589966
$944$	0	0
$945$	−1.37228	−0.0446403
$946$	0	0
$947$	3.72281	0.120975	0.0604876	−	0.998169i	$-0.480734\pi$
$947$	3.72281	0.120975	0.0604876	+	0.998169i	$0.480734\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−39.3288	−1.27398	−0.636992	−	0.770870i	$-0.719822\pi$
$953$	−39.3288	−1.27398	−0.636992	+	0.770870i	$0.719822\pi$
$954$	0	0
$955$	2.74456	0.0888120
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	21.2554	0.686374
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	7.37228	0.237568
$964$	0	0
$965$	3.88316	0.125003
$966$	0	0
$967$	−28.7446	−0.924363	−0.462181	−	0.886785i	$-0.652933\pi$
$967$	−28.7446	−0.924363	−0.462181	+	0.886785i	$0.652933\pi$
$968$	0	0
$969$	10.7446	0.345165
$970$	0	0
$971$	51.2119	1.64347	0.821735	−	0.569870i	$-0.193007\pi$
$971$	51.2119	1.64347	0.821735	+	0.569870i	$0.193007\pi$
$972$	0	0
$973$	−19.3723	−0.621047
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	−5.64947	−0.180558
$980$	0	0
$981$	−20.7446	−0.662323
$982$	0	0
$983$	17.7663	0.566657	0.283329	−	0.959023i	$-0.408561\pi$
$983$	17.7663	0.566657	0.283329	+	0.959023i	$0.408561\pi$
$984$	0	0
$985$	19.4891	0.620975
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−22.7446	−0.723235
$990$	0	0
$991$	−11.1386	−0.353829	−0.176915	−	0.984226i	$-0.556612\pi$
$991$	−11.1386	−0.353829	−0.176915	+	0.984226i	$0.556612\pi$
$992$	0	0
$993$	22.0000	0.698149
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−3.72281	−0.117903	−0.0589513	−	0.998261i	$-0.518776\pi$
$997$	−3.72281	−0.117903	−0.0589513	+	0.998261i	$0.518776\pi$
$998$	0	0
$999$	−1.37228	−0.0434171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bo.1.2	yes	2
4.3	odd	2		6240.2.a.bk.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bk.1.1	✓	2	4.3	odd	2
6240.2.a.bo.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} -1.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.37228 q^{17} -2.00000 q^{19} +1.37228 q^{21} +3.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -1.37228 q^{33} -1.37228 q^{35} -1.37228 q^{37} -1.00000 q^{39} +5.37228 q^{41} -6.74456 q^{43} -1.00000 q^{45} -8.74456 q^{47} -5.11684 q^{49} -5.37228 q^{51} -13.3723 q^{53} +1.37228 q^{55} -2.00000 q^{57} +8.74456 q^{59} +8.11684 q^{61} +1.37228 q^{63} +1.00000 q^{65} +12.7446 q^{67} +3.37228 q^{69} -1.37228 q^{71} -6.00000 q^{73} +1.00000 q^{75} -1.88316 q^{77} -15.3723 q^{79} +1.00000 q^{81} -3.25544 q^{83} +5.37228 q^{85} +8.74456 q^{87} +4.11684 q^{89} -1.37228 q^{91} -4.74456 q^{93} +2.00000 q^{95} -8.11684 q^{97} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} + 3 q^{33} + 3 q^{35} + 3 q^{37} - 2 q^{39} + 5 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{47} + 7 q^{49} - 5 q^{51} - 21 q^{53} - 3 q^{55} - 4 q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 2 q^{65} + 14 q^{67} + q^{69} + 3 q^{71} - 12 q^{73} + 2 q^{75} - 21 q^{77} - 25 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{85} + 6 q^{87} - 9 q^{89} + 3 q^{91} + 2 q^{93} + 4 q^{95} + q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 4 * q^19 - 3 * q^21 + q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 + 3 * q^33 + 3 * q^35 + 3 * q^37 - 2 * q^39 + 5 * q^41 - 2 * q^43 - 2 * q^45 - 6 * q^47 + 7 * q^49 - 5 * q^51 - 21 * q^53 - 3 * q^55 - 4 * q^57 + 6 * q^59 - q^61 - 3 * q^63 + 2 * q^65 + 14 * q^67 + q^69 + 3 * q^71 - 12 * q^73 + 2 * q^75 - 21 * q^77 - 25 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^85 + 6 * q^87 - 9 * q^89 + 3 * q^91 + 2 * q^93 + 4 * q^95 + q^97 + 3 * q^99

Introduction

L-functions

Modular forms

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