LMFDB - Embedded newform 6240.2.a.bk.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			$3.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +4.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +4.37228 q^{7} +1.00000 q^{9} -4.37228 q^{11} -1.00000 q^{13} +1.00000 q^{15} +0.372281 q^{17} +2.00000 q^{19} -4.37228 q^{21} +2.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.74456 q^{29} -6.74456 q^{31} +4.37228 q^{33} -4.37228 q^{35} +4.37228 q^{37} +1.00000 q^{39} -0.372281 q^{41} -4.74456 q^{43} -1.00000 q^{45} -2.74456 q^{47} +12.1168 q^{49} -0.372281 q^{51} -7.62772 q^{53} +4.37228 q^{55} -2.00000 q^{57} +2.74456 q^{59} -9.11684 q^{61} +4.37228 q^{63} +1.00000 q^{65} -1.25544 q^{67} -2.37228 q^{69} -4.37228 q^{71} -6.00000 q^{73} -1.00000 q^{75} -19.1168 q^{77} +9.62772 q^{79} +1.00000 q^{81} +14.7446 q^{83} -0.372281 q^{85} +2.74456 q^{87} -13.1168 q^{89} -4.37228 q^{91} +6.74456 q^{93} -2.00000 q^{95} +9.11684 q^{97} -4.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.37228	1.65257	0.826284	−	0.563254i	$-0.190451\pi$
$7$	4.37228	1.65257	0.826284	+	0.563254i	$0.190451\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.37228	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$11$	−4.37228	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.372281	0.0902915	0.0451457	−	0.998980i	$-0.485625\pi$
$17$	0.372281	0.0902915	0.0451457	+	0.998980i	$0.485625\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−4.37228	−0.954110
$22$	0	0
$23$	2.37228	0.494655	0.247327	−	0.968932i	$-0.420448\pi$
$23$	2.37228	0.494655	0.247327	+	0.968932i	$0.420448\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	0	0
$33$	4.37228	0.761116
$34$	0	0
$35$	−4.37228	−0.739050
$36$	0	0
$37$	4.37228	0.718799	0.359399	−	0.933184i	$-0.382982\pi$
$37$	4.37228	0.718799	0.359399	+	0.933184i	$0.382982\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.372281	−0.0581406	−0.0290703	−	0.999577i	$-0.509255\pi$
$41$	−0.372281	−0.0581406	−0.0290703	+	0.999577i	$0.509255\pi$
$42$	0	0
$43$	−4.74456	−0.723539	−0.361770	−	0.932268i	$-0.617827\pi$
$43$	−4.74456	−0.723539	−0.361770	+	0.932268i	$0.617827\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	0	0
$49$	12.1168	1.73098
$50$	0	0
$51$	−0.372281	−0.0521298
$52$	0	0
$53$	−7.62772	−1.04775	−0.523874	−	0.851796i	$-0.675514\pi$
$53$	−7.62772	−1.04775	−0.523874	+	0.851796i	$0.675514\pi$
$54$	0	0
$55$	4.37228	0.589558
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	−9.11684	−1.16729	−0.583646	−	0.812008i	$-0.698374\pi$
$61$	−9.11684	−1.16729	−0.583646	+	0.812008i	$0.698374\pi$
$62$	0	0
$63$	4.37228	0.550856
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.25544	−0.153376	−0.0766880	−	0.997055i	$-0.524435\pi$
$67$	−1.25544	−0.153376	−0.0766880	+	0.997055i	$0.524435\pi$
$68$	0	0
$69$	−2.37228	−0.285589
$70$	0	0
$71$	−4.37228	−0.518894	−0.259447	−	0.965757i	$-0.583540\pi$
$71$	−4.37228	−0.518894	−0.259447	+	0.965757i	$0.583540\pi$
$72$	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
$77$	−19.1168	−2.17857
$78$	0	0
$79$	9.62772	1.08320	0.541601	−	0.840635i	$-0.317818\pi$
$79$	9.62772	1.08320	0.541601	+	0.840635i	$0.317818\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.7446	1.61843	0.809213	−	0.587515i	$-0.199894\pi$
$83$	14.7446	1.61843	0.809213	+	0.587515i	$0.199894\pi$
$84$	0	0
$85$	−0.372281	−0.0403796
$86$	0	0
$87$	2.74456	0.294248
$88$	0	0
$89$	−13.1168	−1.39038	−0.695191	−	0.718825i	$-0.744680\pi$
$89$	−13.1168	−1.39038	−0.695191	+	0.718825i	$0.744680\pi$
$90$	0	0
$91$	−4.37228	−0.458340
$92$	0	0
$93$	6.74456	0.699379
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	9.11684	0.925675	0.462838	−	0.886443i	$-0.346831\pi$
$97$	9.11684	0.925675	0.462838	+	0.886443i	$0.346831\pi$
$98$	0	0
$99$	−4.37228	−0.439431
$100$	0	0
$101$	−5.25544	−0.522936	−0.261468	−	0.965212i	$-0.584207\pi$
$101$	−5.25544	−0.522936	−0.261468	+	0.965212i	$0.584207\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	4.37228	0.426691
$106$	0	0
$107$	−1.62772	−0.157358	−0.0786788	−	0.996900i	$-0.525070\pi$
$107$	−1.62772	−0.157358	−0.0786788	+	0.996900i	$0.525070\pi$
$108$	0	0
$109$	−9.25544	−0.886510	−0.443255	−	0.896396i	$-0.646176\pi$
$109$	−9.25544	−0.886510	−0.443255	+	0.896396i	$0.646176\pi$
$110$	0	0
$111$	−4.37228	−0.414999
$112$	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
$115$	−2.37228	−0.221216
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	1.62772	0.149213
$120$	0	0
$121$	8.11684	0.737895
$122$	0	0
$123$	0.372281	0.0335675
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	4.74456	0.417735
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	8.74456	0.758250
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.48913	−0.639839	−0.319920	−	0.947445i	$-0.603656\pi$
$137$	−7.48913	−0.639839	−0.319920	+	0.947445i	$0.603656\pi$
$138$	0	0
$139$	−3.11684	−0.264367	−0.132184	−	0.991225i	$-0.542199\pi$
$139$	−3.11684	−0.264367	−0.132184	+	0.991225i	$0.542199\pi$
$140$	0	0
$141$	2.74456	0.231134
$142$	0	0
$143$	4.37228	0.365629
$144$	0	0
$145$	2.74456	0.227924
$146$	0	0
$147$	−12.1168	−0.999380
$148$	0	0
$149$	−13.1168	−1.07457	−0.537287	−	0.843400i	$-0.680551\pi$
$149$	−13.1168	−1.07457	−0.537287	+	0.843400i	$0.680551\pi$
$150$	0	0
$151$	−20.2337	−1.64659	−0.823297	−	0.567611i	$-0.807868\pi$
$151$	−20.2337	−1.64659	−0.823297	+	0.567611i	$0.807868\pi$
$152$	0	0
$153$	0.372281	0.0300972
$154$	0	0
$155$	6.74456	0.541736
$156$	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
$159$	7.62772	0.604917
$160$	0	0
$161$	10.3723	0.817450
$162$	0	0
$163$	17.8614	1.39901	0.699507	−	0.714626i	$-0.253403\pi$
$163$	17.8614	1.39901	0.699507	+	0.714626i	$0.253403\pi$
$164$	0	0
$165$	−4.37228	−0.340382
$166$	0	0
$167$	−2.74456	−0.212381	−0.106190	−	0.994346i	$-0.533865\pi$
$167$	−2.74456	−0.212381	−0.106190	+	0.994346i	$0.533865\pi$
$168$	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	$-0.643871\pi$
$173$	−11.4891	−0.873502	−0.436751	+	0.899582i	$0.643871\pi$
$174$	0	0
$175$	4.37228	0.330513
$176$	0	0
$177$	−2.74456	−0.206294
$178$	0	0
$179$	−3.25544	−0.243323	−0.121661	−	0.992572i	$-0.538822\pi$
$179$	−3.25544	−0.243323	−0.121661	+	0.992572i	$0.538822\pi$
$180$	0	0
$181$	−5.11684	−0.380332	−0.190166	−	0.981752i	$-0.560903\pi$
$181$	−5.11684	−0.380332	−0.190166	+	0.981752i	$0.560903\pi$
$182$	0	0
$183$	9.11684	0.673936
$184$	0	0
$185$	−4.37228	−0.321457
$186$	0	0
$187$	−1.62772	−0.119031
$188$	0	0
$189$	−4.37228	−0.318037
$190$	0	0
$191$	−8.74456	−0.632734	−0.316367	−	0.948637i	$-0.602463\pi$
$191$	−8.74456	−0.632734	−0.316367	+	0.948637i	$0.602463\pi$
$192$	0	0
$193$	−21.1168	−1.52002	−0.760012	−	0.649909i	$-0.774807\pi$
$193$	−21.1168	−1.52002	−0.760012	+	0.649909i	$0.774807\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	3.48913	0.248590	0.124295	−	0.992245i	$-0.460333\pi$
$197$	3.48913	0.248590	0.124295	+	0.992245i	$0.460333\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	1.25544	0.0885517
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0.372281	0.0260013
$206$	0	0
$207$	2.37228	0.164885
$208$	0	0
$209$	−8.74456	−0.604874
$210$	0	0
$211$	21.4891	1.47937	0.739686	−	0.672952i	$-0.234974\pi$
$211$	21.4891	1.47937	0.739686	+	0.672952i	$0.234974\pi$
$212$	0	0
$213$	4.37228	0.299584
$214$	0	0
$215$	4.74456	0.323576
$216$	0	0
$217$	−29.4891	−2.00185
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−0.372281	−0.0250424
$222$	0	0
$223$	20.2337	1.35495	0.677474	−	0.735547i	$-0.263075\pi$
$223$	20.2337	1.35495	0.677474	+	0.735547i	$0.263075\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	24.2337	1.60141	0.800704	−	0.599061i	$-0.204459\pi$
$229$	24.2337	1.60141	0.800704	+	0.599061i	$0.204459\pi$
$230$	0	0
$231$	19.1168	1.25780
$232$	0	0
$233$	−16.3723	−1.07258	−0.536292	−	0.844033i	$-0.680175\pi$
$233$	−16.3723	−1.07258	−0.536292	+	0.844033i	$0.680175\pi$
$234$	0	0
$235$	2.74456	0.179036
$236$	0	0
$237$	−9.62772	−0.625388
$238$	0	0
$239$	14.6060	0.944782	0.472391	−	0.881389i	$-0.343391\pi$
$239$	14.6060	0.944782	0.472391	+	0.881389i	$0.343391\pi$
$240$	0	0
$241$	6.74456	0.434455	0.217228	−	0.976121i	$-0.430299\pi$
$241$	6.74456	0.434455	0.217228	+	0.976121i	$0.430299\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.1168	−0.774117
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−14.7446	−0.934399
$250$	0	0
$251$	−18.9783	−1.19790	−0.598948	−	0.800788i	$-0.704415\pi$
$251$	−18.9783	−1.19790	−0.598948	+	0.800788i	$0.704415\pi$
$252$	0	0
$253$	−10.3723	−0.652100
$254$	0	0
$255$	0.372281	0.0233132
$256$	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
$259$	19.1168	1.18786
$260$	0	0
$261$	−2.74456	−0.169884
$262$	0	0
$263$	2.51087	0.154827	0.0774136	−	0.996999i	$-0.475334\pi$
$263$	2.51087	0.154827	0.0774136	+	0.996999i	$0.475334\pi$
$264$	0	0
$265$	7.62772	0.468567
$266$	0	0
$267$	13.1168	0.802738
$268$	0	0
$269$	25.7228	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$269$	25.7228	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$270$	0	0
$271$	3.48913	0.211949	0.105975	−	0.994369i	$-0.466204\pi$
$271$	3.48913	0.211949	0.105975	+	0.994369i	$0.466204\pi$
$272$	0	0
$273$	4.37228	0.264623
$274$	0	0
$275$	−4.37228	−0.263658
$276$	0	0
$277$	−21.2554	−1.27712	−0.638558	−	0.769574i	$-0.720469\pi$
$277$	−21.2554	−1.27712	−0.638558	+	0.769574i	$0.720469\pi$
$278$	0	0
$279$	−6.74456	−0.403786
$280$	0	0
$281$	3.48913	0.208144	0.104072	−	0.994570i	$-0.466813\pi$
$281$	3.48913	0.208144	0.104072	+	0.994570i	$0.466813\pi$
$282$	0	0
$283$	25.4891	1.51517	0.757586	−	0.652736i	$-0.226379\pi$
$283$	25.4891	1.51517	0.757586	+	0.652736i	$0.226379\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−1.62772	−0.0960812
$288$	0	0
$289$	−16.8614	−0.991847
$290$	0	0
$291$	−9.11684	−0.534439
$292$	0	0
$293$	−2.74456	−0.160339	−0.0801695	−	0.996781i	$-0.525546\pi$
$293$	−2.74456	−0.160339	−0.0801695	+	0.996781i	$0.525546\pi$
$294$	0	0
$295$	−2.74456	−0.159795
$296$	0	0
$297$	4.37228	0.253705
$298$	0	0
$299$	−2.37228	−0.137193
$300$	0	0
$301$	−20.7446	−1.19570
$302$	0	0
$303$	5.25544	0.301917
$304$	0	0
$305$	9.11684	0.522029
$306$	0	0
$307$	−9.11684	−0.520326	−0.260163	−	0.965565i	$-0.583776\pi$
$307$	−9.11684	−0.520326	−0.260163	+	0.965565i	$0.583776\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	31.7228	1.79884	0.899418	−	0.437090i	$-0.143991\pi$
$311$	31.7228	1.79884	0.899418	+	0.437090i	$0.143991\pi$
$312$	0	0
$313$	−28.9783	−1.63795	−0.818974	−	0.573831i	$-0.805457\pi$
$313$	−28.9783	−1.63795	−0.818974	+	0.573831i	$0.805457\pi$
$314$	0	0
$315$	−4.37228	−0.246350
$316$	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
$321$	1.62772	0.0908504
$322$	0	0
$323$	0.744563	0.0414286
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	9.25544	0.511827
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	4.37228	0.239600
$334$	0	0
$335$	1.25544	0.0685919
$336$	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
$341$	29.4891	1.59693
$342$	0	0
$343$	22.3723	1.20799
$344$	0	0
$345$	2.37228	0.127719
$346$	0	0
$347$	−35.1168	−1.88517	−0.942585	−	0.333965i	$-0.891613\pi$
$347$	−35.1168	−1.88517	−0.942585	+	0.333965i	$0.891613\pi$
$348$	0	0
$349$	20.9783	1.12294	0.561470	−	0.827497i	$-0.310236\pi$
$349$	20.9783	1.12294	0.561470	+	0.827497i	$0.310236\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−9.25544	−0.492617	−0.246309	−	0.969191i	$-0.579218\pi$
$353$	−9.25544	−0.492617	−0.246309	+	0.969191i	$0.579218\pi$
$354$	0	0
$355$	4.37228	0.232057
$356$	0	0
$357$	−1.62772	−0.0861480
$358$	0	0
$359$	−9.25544	−0.488483	−0.244242	−	0.969714i	$-0.578539\pi$
$359$	−9.25544	−0.488483	−0.244242	+	0.969714i	$0.578539\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−8.11684	−0.426024
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−24.7446	−1.29166	−0.645828	−	0.763483i	$-0.723488\pi$
$367$	−24.7446	−1.29166	−0.645828	+	0.763483i	$0.723488\pi$
$368$	0	0
$369$	−0.372281	−0.0193802
$370$	0	0
$371$	−33.3505	−1.73147
$372$	0	0
$373$	−15.4891	−0.801997	−0.400998	−	0.916079i	$-0.631337\pi$
$373$	−15.4891	−0.801997	−0.400998	+	0.916079i	$0.631337\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	2.74456	0.141352
$378$	0	0
$379$	8.23369	0.422936	0.211468	−	0.977385i	$-0.432176\pi$
$379$	8.23369	0.422936	0.211468	+	0.977385i	$0.432176\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−27.4891	−1.40463	−0.702314	−	0.711867i	$-0.747850\pi$
$383$	−27.4891	−1.40463	−0.702314	+	0.711867i	$0.747850\pi$
$384$	0	0
$385$	19.1168	0.974285
$386$	0	0
$387$	−4.74456	−0.241180
$388$	0	0
$389$	−17.2554	−0.874885	−0.437443	−	0.899246i	$-0.644116\pi$
$389$	−17.2554	−0.874885	−0.437443	+	0.899246i	$0.644116\pi$
$390$	0	0
$391$	0.883156	0.0446631
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	−9.62772	−0.484423
$396$	0	0
$397$	−14.6060	−0.733053	−0.366526	−	0.930408i	$-0.619453\pi$
$397$	−14.6060	−0.733053	−0.366526	+	0.930408i	$0.619453\pi$
$398$	0	0
$399$	−8.74456	−0.437776
$400$	0	0
$401$	11.4891	0.573740	0.286870	−	0.957970i	$-0.407385\pi$
$401$	11.4891	0.573740	0.286870	+	0.957970i	$0.407385\pi$
$402$	0	0
$403$	6.74456	0.335971
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−19.1168	−0.947587
$408$	0	0
$409$	−28.2337	−1.39607	−0.698033	−	0.716066i	$-0.745941\pi$
$409$	−28.2337	−1.39607	−0.698033	+	0.716066i	$0.745941\pi$
$410$	0	0
$411$	7.48913	0.369411
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−14.7446	−0.723782
$416$	0	0
$417$	3.11684	0.152633
$418$	0	0
$419$	0.744563	0.0363743	0.0181871	−	0.999835i	$-0.494211\pi$
$419$	0.744563	0.0363743	0.0181871	+	0.999835i	$0.494211\pi$
$420$	0	0
$421$	13.2554	0.646030	0.323015	−	0.946394i	$-0.395303\pi$
$421$	13.2554	0.646030	0.323015	+	0.946394i	$0.395303\pi$
$422$	0	0
$423$	−2.74456	−0.133445
$424$	0	0
$425$	0.372281	0.0180583
$426$	0	0
$427$	−39.8614	−1.92903
$428$	0	0
$429$	−4.37228	−0.211096
$430$	0	0
$431$	20.2337	0.974622	0.487311	−	0.873228i	$-0.337978\pi$
$431$	20.2337	0.974622	0.487311	+	0.873228i	$0.337978\pi$
$432$	0	0
$433$	−23.4891	−1.12882	−0.564408	−	0.825496i	$-0.690895\pi$
$433$	−23.4891	−1.12882	−0.564408	+	0.825496i	$0.690895\pi$
$434$	0	0
$435$	−2.74456	−0.131592
$436$	0	0
$437$	4.74456	0.226963
$438$	0	0
$439$	−7.11684	−0.339668	−0.169834	−	0.985473i	$-0.554323\pi$
$439$	−7.11684	−0.339668	−0.169834	+	0.985473i	$0.554323\pi$
$440$	0	0
$441$	12.1168	0.576993
$442$	0	0
$443$	15.8614	0.753598	0.376799	−	0.926295i	$-0.377025\pi$
$443$	15.8614	0.753598	0.376799	+	0.926295i	$0.377025\pi$
$444$	0	0
$445$	13.1168	0.621798
$446$	0	0
$447$	13.1168	0.620405
$448$	0	0
$449$	1.11684	0.0527071	0.0263536	−	0.999653i	$-0.491610\pi$
$449$	1.11684	0.0527071	0.0263536	+	0.999653i	$0.491610\pi$
$450$	0	0
$451$	1.62772	0.0766463
$452$	0	0
$453$	20.2337	0.950662
$454$	0	0
$455$	4.37228	0.204976
$456$	0	0
$457$	−8.37228	−0.391639	−0.195819	−	0.980640i	$-0.562737\pi$
$457$	−8.37228	−0.391639	−0.195819	+	0.980640i	$0.562737\pi$
$458$	0	0
$459$	−0.372281	−0.0173766
$460$	0	0
$461$	−9.86141	−0.459291	−0.229646	−	0.973274i	$-0.573757\pi$
$461$	−9.86141	−0.459291	−0.229646	+	0.973274i	$0.573757\pi$
$462$	0	0
$463$	1.39403	0.0647861	0.0323931	−	0.999475i	$-0.489687\pi$
$463$	1.39403	0.0647861	0.0323931	+	0.999475i	$0.489687\pi$
$464$	0	0
$465$	−6.74456	−0.312772
$466$	0	0
$467$	2.37228	0.109776	0.0548880	−	0.998493i	$-0.482520\pi$
$467$	2.37228	0.109776	0.0548880	+	0.998493i	$0.482520\pi$
$468$	0	0
$469$	−5.48913	−0.253464
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	20.7446	0.953836
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	−7.62772	−0.349249
$478$	0	0
$479$	−40.0951	−1.83199	−0.915996	−	0.401188i	$-0.868597\pi$
$479$	−40.0951	−1.83199	−0.915996	+	0.401188i	$0.868597\pi$
$480$	0	0
$481$	−4.37228	−0.199359
$482$	0	0
$483$	−10.3723	−0.471955
$484$	0	0
$485$	−9.11684	−0.413975
$486$	0	0
$487$	−32.8397	−1.48811	−0.744053	−	0.668120i	$-0.767099\pi$
$487$	−32.8397	−1.48811	−0.744053	+	0.668120i	$0.767099\pi$
$488$	0	0
$489$	−17.8614	−0.807721
$490$	0	0
$491$	−4.74456	−0.214119	−0.107060	−	0.994253i	$-0.534144\pi$
$491$	−4.74456	−0.214119	−0.107060	+	0.994253i	$0.534144\pi$
$492$	0	0
$493$	−1.02175	−0.0460173
$494$	0	0
$495$	4.37228	0.196519
$496$	0	0
$497$	−19.1168	−0.857508
$498$	0	0
$499$	−8.97825	−0.401922	−0.200961	−	0.979599i	$-0.564406\pi$
$499$	−8.97825	−0.401922	−0.200961	+	0.979599i	$0.564406\pi$
$500$	0	0
$501$	2.74456	0.122618
$502$	0	0
$503$	−18.5109	−0.825359	−0.412680	−	0.910876i	$-0.635407\pi$
$503$	−18.5109	−0.825359	−0.412680	+	0.910876i	$0.635407\pi$
$504$	0	0
$505$	5.25544	0.233864
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−37.1168	−1.64518	−0.822588	−	0.568638i	$-0.807470\pi$
$509$	−37.1168	−1.64518	−0.822588	+	0.568638i	$0.807470\pi$
$510$	0	0
$511$	−26.2337	−1.16051
$512$	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
$521$	31.4891	1.37956	0.689782	−	0.724017i	$-0.257706\pi$
$521$	31.4891	1.37956	0.689782	+	0.724017i	$0.257706\pi$
$522$	0	0
$523$	−43.7228	−1.91187	−0.955933	−	0.293586i	$-0.905151\pi$
$523$	−43.7228	−1.91187	−0.955933	+	0.293586i	$0.905151\pi$
$524$	0	0
$525$	−4.37228	−0.190822
$526$	0	0
$527$	−2.51087	−0.109375
$528$	0	0
$529$	−17.3723	−0.755317
$530$	0	0
$531$	2.74456	0.119104
$532$	0	0
$533$	0.372281	0.0161253
$534$	0	0
$535$	1.62772	0.0703724
$536$	0	0
$537$	3.25544	0.140482
$538$	0	0
$539$	−52.9783	−2.28193
$540$	0	0
$541$	−15.4891	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$541$	−15.4891	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$542$	0	0
$543$	5.11684	0.219585
$544$	0	0
$545$	9.25544	0.396459
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−9.11684	−0.389097
$550$	0	0
$551$	−5.48913	−0.233845
$552$	0	0
$553$	42.0951	1.79007
$554$	0	0
$555$	4.37228	0.185593
$556$	0	0
$557$	14.7446	0.624747	0.312374	−	0.949959i	$-0.398876\pi$
$557$	14.7446	0.624747	0.312374	+	0.949959i	$0.398876\pi$
$558$	0	0
$559$	4.74456	0.200674
$560$	0	0
$561$	1.62772	0.0687223
$562$	0	0
$563$	−31.1168	−1.31142	−0.655709	−	0.755013i	$-0.727630\pi$
$563$	−31.1168	−1.31142	−0.655709	+	0.755013i	$0.727630\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	4.37228	0.183619
$568$	0	0
$569$	−8.97825	−0.376388	−0.188194	−	0.982132i	$-0.560263\pi$
$569$	−8.97825	−0.376388	−0.188194	+	0.982132i	$0.560263\pi$
$570$	0	0
$571$	16.6060	0.694938	0.347469	−	0.937691i	$-0.387041\pi$
$571$	16.6060	0.694938	0.347469	+	0.937691i	$0.387041\pi$
$572$	0	0
$573$	8.74456	0.365309
$574$	0	0
$575$	2.37228	0.0989310
$576$	0	0
$577$	−9.86141	−0.410536	−0.205268	−	0.978706i	$-0.565807\pi$
$577$	−9.86141	−0.410536	−0.205268	+	0.978706i	$0.565807\pi$
$578$	0	0
$579$	21.1168	0.877586
$580$	0	0
$581$	64.4674	2.67456
$582$	0	0
$583$	33.3505	1.38124
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−20.5109	−0.846574	−0.423287	−	0.905996i	$-0.639124\pi$
$587$	−20.5109	−0.846574	−0.423287	+	0.905996i	$0.639124\pi$
$588$	0	0
$589$	−13.4891	−0.555810
$590$	0	0
$591$	−3.48913	−0.143523
$592$	0	0
$593$	−29.7228	−1.22057	−0.610285	−	0.792182i	$-0.708945\pi$
$593$	−29.7228	−1.22057	−0.610285	+	0.792182i	$0.708945\pi$
$594$	0	0
$595$	−1.62772	−0.0667300
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−13.8614	−0.565419	−0.282709	−	0.959206i	$-0.591233\pi$
$601$	−13.8614	−0.565419	−0.282709	+	0.959206i	$0.591233\pi$
$602$	0	0
$603$	−1.25544	−0.0511254
$604$	0	0
$605$	−8.11684	−0.329997
$606$	0	0
$607$	43.7228	1.77465	0.887327	−	0.461141i	$-0.152560\pi$
$607$	43.7228	1.77465	0.887327	+	0.461141i	$0.152560\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	2.74456	0.111033
$612$	0	0
$613$	−3.35053	−0.135327	−0.0676634	−	0.997708i	$-0.521554\pi$
$613$	−3.35053	−0.135327	−0.0676634	+	0.997708i	$0.521554\pi$
$614$	0	0
$615$	−0.372281	−0.0150118
$616$	0	0
$617$	22.4674	0.904502	0.452251	−	0.891891i	$-0.350621\pi$
$617$	22.4674	0.904502	0.452251	+	0.891891i	$0.350621\pi$
$618$	0	0
$619$	1.25544	0.0504603	0.0252301	−	0.999682i	$-0.491968\pi$
$619$	1.25544	0.0504603	0.0252301	+	0.999682i	$0.491968\pi$
$620$	0	0
$621$	−2.37228	−0.0951964
$622$	0	0
$623$	−57.3505	−2.29770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.74456	0.349224
$628$	0	0
$629$	1.62772	0.0649014
$630$	0	0
$631$	−36.2337	−1.44244	−0.721220	−	0.692706i	$-0.756418\pi$
$631$	−36.2337	−1.44244	−0.721220	+	0.692706i	$0.756418\pi$
$632$	0	0
$633$	−21.4891	−0.854116
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−12.1168	−0.480087
$638$	0	0
$639$	−4.37228	−0.172965
$640$	0	0
$641$	40.9783	1.61854	0.809272	−	0.587434i	$-0.199862\pi$
$641$	40.9783	1.61854	0.809272	+	0.587434i	$0.199862\pi$
$642$	0	0
$643$	2.88316	0.113701	0.0568503	−	0.998383i	$-0.481894\pi$
$643$	2.88316	0.113701	0.0568503	+	0.998383i	$0.481894\pi$
$644$	0	0
$645$	−4.74456	−0.186817
$646$	0	0
$647$	24.6060	0.967360	0.483680	−	0.875245i	$-0.339300\pi$
$647$	24.6060	0.967360	0.483680	+	0.875245i	$0.339300\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	29.4891	1.15577
$652$	0	0
$653$	−4.51087	−0.176524	−0.0882621	−	0.996097i	$-0.528131\pi$
$653$	−4.51087	−0.176524	−0.0882621	+	0.996097i	$0.528131\pi$
$654$	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
$661$	16.5109	0.642199	0.321099	−	0.947046i	$-0.395948\pi$
$661$	16.5109	0.642199	0.321099	+	0.947046i	$0.395948\pi$
$662$	0	0
$663$	0.372281	0.0144582
$664$	0	0
$665$	−8.74456	−0.339100
$666$	0	0
$667$	−6.51087	−0.252102
$668$	0	0
$669$	−20.2337	−0.782280
$670$	0	0
$671$	39.8614	1.53883
$672$	0	0
$673$	4.23369	0.163197	0.0815983	−	0.996665i	$-0.473998\pi$
$673$	4.23369	0.163197	0.0815983	+	0.996665i	$0.473998\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	1.86141	0.0715397	0.0357698	−	0.999360i	$-0.488612\pi$
$677$	1.86141	0.0715397	0.0357698	+	0.999360i	$0.488612\pi$
$678$	0	0
$679$	39.8614	1.52974
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	0	0
$685$	7.48913	0.286145
$686$	0	0
$687$	−24.2337	−0.924573
$688$	0	0
$689$	7.62772	0.290593
$690$	0	0
$691$	28.9783	1.10238	0.551192	−	0.834378i	$-0.314173\pi$
$691$	28.9783	1.10238	0.551192	+	0.834378i	$0.314173\pi$
$692$	0	0
$693$	−19.1168	−0.726189
$694$	0	0
$695$	3.11684	0.118229
$696$	0	0
$697$	−0.138593	−0.00524960
$698$	0	0
$699$	16.3723	0.619257
$700$	0	0
$701$	−28.2337	−1.06637	−0.533186	−	0.845998i	$-0.679005\pi$
$701$	−28.2337	−1.06637	−0.533186	+	0.845998i	$0.679005\pi$
$702$	0	0
$703$	8.74456	0.329807
$704$	0	0
$705$	−2.74456	−0.103366
$706$	0	0
$707$	−22.9783	−0.864186
$708$	0	0
$709$	35.2119	1.32241	0.661206	−	0.750204i	$-0.270045\pi$
$709$	35.2119	1.32241	0.661206	+	0.750204i	$0.270045\pi$
$710$	0	0
$711$	9.62772	0.361068
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−4.37228	−0.163514
$716$	0	0
$717$	−14.6060	−0.545470
$718$	0	0
$719$	−20.4674	−0.763304	−0.381652	−	0.924306i	$-0.624645\pi$
$719$	−20.4674	−0.763304	−0.381652	+	0.924306i	$0.624645\pi$
$720$	0	0
$721$	34.9783	1.30266
$722$	0	0
$723$	−6.74456	−0.250833
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	30.2337	1.12131	0.560653	−	0.828051i	$-0.310550\pi$
$727$	30.2337	1.12131	0.560653	+	0.828051i	$0.310550\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.76631	−0.0653294
$732$	0	0
$733$	−6.60597	−0.243997	−0.121999	−	0.992530i	$-0.538930\pi$
$733$	−6.60597	−0.243997	−0.121999	+	0.992530i	$0.538930\pi$
$734$	0	0
$735$	12.1168	0.446937
$736$	0	0
$737$	5.48913	0.202195
$738$	0	0
$739$	−47.9565	−1.76411	−0.882054	−	0.471148i	$-0.843840\pi$
$739$	−47.9565	−1.76411	−0.882054	+	0.471148i	$0.843840\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	38.0000	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$744$	0	0
$745$	13.1168	0.480564
$746$	0	0
$747$	14.7446	0.539475
$748$	0	0
$749$	−7.11684	−0.260044
$750$	0	0
$751$	38.8397	1.41728	0.708640	−	0.705571i	$-0.249309\pi$
$751$	38.8397	1.41728	0.708640	+	0.705571i	$0.249309\pi$
$752$	0	0
$753$	18.9783	0.691606
$754$	0	0
$755$	20.2337	0.736379
$756$	0	0
$757$	2.74456	0.0997528	0.0498764	−	0.998755i	$-0.484117\pi$
$757$	2.74456	0.0997528	0.0498764	+	0.998755i	$0.484117\pi$
$758$	0	0
$759$	10.3723	0.376490
$760$	0	0
$761$	8.51087	0.308519	0.154259	−	0.988030i	$-0.450701\pi$
$761$	8.51087	0.308519	0.154259	+	0.988030i	$0.450701\pi$
$762$	0	0
$763$	−40.4674	−1.46502
$764$	0	0
$765$	−0.372281	−0.0134599
$766$	0	0
$767$	−2.74456	−0.0991004
$768$	0	0
$769$	39.9565	1.44087	0.720434	−	0.693523i	$-0.243943\pi$
$769$	39.9565	1.44087	0.720434	+	0.693523i	$0.243943\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	49.7228	1.78841	0.894203	−	0.447662i	$-0.147743\pi$
$773$	49.7228	1.78841	0.894203	+	0.447662i	$0.147743\pi$
$774$	0	0
$775$	−6.74456	−0.242272
$776$	0	0
$777$	−19.1168	−0.685813
$778$	0	0
$779$	−0.744563	−0.0266767
$780$	0	0
$781$	19.1168	0.684054
$782$	0	0
$783$	2.74456	0.0980827
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−22.7446	−0.810756	−0.405378	−	0.914149i	$-0.632860\pi$
$787$	−22.7446	−0.810756	−0.405378	+	0.914149i	$0.632860\pi$
$788$	0	0
$789$	−2.51087	−0.0893895
$790$	0	0
$791$	−8.74456	−0.310921
$792$	0	0
$793$	9.11684	0.323749
$794$	0	0
$795$	−7.62772	−0.270527
$796$	0	0
$797$	36.0951	1.27855	0.639277	−	0.768977i	$-0.279234\pi$
$797$	36.0951	1.27855	0.639277	+	0.768977i	$0.279234\pi$
$798$	0	0
$799$	−1.02175	−0.0361469
$800$	0	0
$801$	−13.1168	−0.463461
$802$	0	0
$803$	26.2337	0.925767
$804$	0	0
$805$	−10.3723	−0.365575
$806$	0	0
$807$	−25.7228	−0.905486
$808$	0	0
$809$	40.9783	1.44072	0.720359	−	0.693601i	$-0.243977\pi$
$809$	40.9783	1.44072	0.720359	+	0.693601i	$0.243977\pi$
$810$	0	0
$811$	44.9783	1.57940	0.789700	−	0.613493i	$-0.210236\pi$
$811$	44.9783	1.57940	0.789700	+	0.613493i	$0.210236\pi$
$812$	0	0
$813$	−3.48913	−0.122369
$814$	0	0
$815$	−17.8614	−0.625658
$816$	0	0
$817$	−9.48913	−0.331982
$818$	0	0
$819$	−4.37228	−0.152780
$820$	0	0
$821$	15.6277	0.545411	0.272706	−	0.962098i	$-0.412082\pi$
$821$	15.6277	0.545411	0.272706	+	0.962098i	$0.412082\pi$
$822$	0	0
$823$	−10.9783	−0.382678	−0.191339	−	0.981524i	$-0.561283\pi$
$823$	−10.9783	−0.382678	−0.191339	+	0.981524i	$0.561283\pi$
$824$	0	0
$825$	4.37228	0.152223
$826$	0	0
$827$	−37.2554	−1.29550	−0.647749	−	0.761854i	$-0.724290\pi$
$827$	−37.2554	−1.29550	−0.647749	+	0.761854i	$0.724290\pi$
$828$	0	0
$829$	4.51087	0.156669	0.0783346	−	0.996927i	$-0.475040\pi$
$829$	4.51087	0.156669	0.0783346	+	0.996927i	$0.475040\pi$
$830$	0	0
$831$	21.2554	0.737343
$832$	0	0
$833$	4.51087	0.156293
$834$	0	0
$835$	2.74456	0.0949795
$836$	0	0
$837$	6.74456	0.233126
$838$	0	0
$839$	53.5842	1.84993	0.924966	−	0.380049i	$-0.124093\pi$
$839$	53.5842	1.84993	0.924966	+	0.380049i	$0.124093\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−3.48913	−0.120172
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	35.4891	1.21942
$848$	0	0
$849$	−25.4891	−0.874785
$850$	0	0
$851$	10.3723	0.355557
$852$	0	0
$853$	49.1168	1.68173	0.840864	−	0.541246i	$-0.182047\pi$
$853$	49.1168	1.68173	0.840864	+	0.541246i	$0.182047\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	−28.8397	−0.985144	−0.492572	−	0.870272i	$-0.663943\pi$
$857$	−28.8397	−0.985144	−0.492572	+	0.870272i	$0.663943\pi$
$858$	0	0
$859$	1.35053	0.0460796	0.0230398	−	0.999735i	$-0.492666\pi$
$859$	1.35053	0.0460796	0.0230398	+	0.999735i	$0.492666\pi$
$860$	0	0
$861$	1.62772	0.0554725
$862$	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
$867$	16.8614	0.572643
$868$	0	0
$869$	−42.0951	−1.42798
$870$	0	0
$871$	1.25544	0.0425389
$872$	0	0
$873$	9.11684	0.308558
$874$	0	0
$875$	−4.37228	−0.147810
$876$	0	0
$877$	−15.4891	−0.523031	−0.261515	−	0.965199i	$-0.584222\pi$
$877$	−15.4891	−0.523031	−0.261515	+	0.965199i	$0.584222\pi$
$878$	0	0
$879$	2.74456	0.0925718
$880$	0	0
$881$	−21.7228	−0.731860	−0.365930	−	0.930642i	$-0.619249\pi$
$881$	−21.7228	−0.731860	−0.365930	+	0.930642i	$0.619249\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	2.74456	0.0922575
$886$	0	0
$887$	52.6060	1.76634	0.883168	−	0.469057i	$-0.155406\pi$
$887$	52.6060	1.76634	0.883168	+	0.469057i	$0.155406\pi$
$888$	0	0
$889$	69.9565	2.34627
$890$	0	0
$891$	−4.37228	−0.146477
$892$	0	0
$893$	−5.48913	−0.183687
$894$	0	0
$895$	3.25544	0.108817
$896$	0	0
$897$	2.37228	0.0792082
$898$	0	0
$899$	18.5109	0.617372
$900$	0	0
$901$	−2.83966	−0.0946027
$902$	0	0
$903$	20.7446	0.690336
$904$	0	0
$905$	5.11684	0.170090
$906$	0	0
$907$	−11.2554	−0.373731	−0.186865	−	0.982386i	$-0.559833\pi$
$907$	−11.2554	−0.373731	−0.186865	+	0.982386i	$0.559833\pi$
$908$	0	0
$909$	−5.25544	−0.174312
$910$	0	0
$911$	27.7228	0.918498	0.459249	−	0.888308i	$-0.348119\pi$
$911$	27.7228	0.918498	0.459249	+	0.888308i	$0.348119\pi$
$912$	0	0
$913$	−64.4674	−2.13356
$914$	0	0
$915$	−9.11684	−0.301394
$916$	0	0
$917$	−52.4674	−1.73263
$918$	0	0
$919$	3.11684	0.102815	0.0514076	−	0.998678i	$-0.483629\pi$
$919$	3.11684	0.102815	0.0514076	+	0.998678i	$0.483629\pi$
$920$	0	0
$921$	9.11684	0.300410
$922$	0	0
$923$	4.37228	0.143915
$924$	0	0
$925$	4.37228	0.143760
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	47.3505	1.55352	0.776760	−	0.629796i	$-0.216862\pi$
$929$	47.3505	1.55352	0.776760	+	0.629796i	$0.216862\pi$
$930$	0	0
$931$	24.2337	0.794227
$932$	0	0
$933$	−31.7228	−1.03856
$934$	0	0
$935$	1.62772	0.0532321
$936$	0	0
$937$	−17.2554	−0.563711	−0.281855	−	0.959457i	$-0.590950\pi$
$937$	−17.2554	−0.563711	−0.281855	+	0.959457i	$0.590950\pi$
$938$	0	0
$939$	28.9783	0.945669
$940$	0	0
$941$	9.39403	0.306237	0.153118	−	0.988208i	$-0.451068\pi$
$941$	9.39403	0.306237	0.153118	+	0.988208i	$0.451068\pi$
$942$	0	0
$943$	−0.883156	−0.0287595
$944$	0	0
$945$	4.37228	0.142230
$946$	0	0
$947$	53.7228	1.74576	0.872878	−	0.487938i	$-0.162251\pi$
$947$	53.7228	1.74576	0.872878	+	0.487938i	$0.162251\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	58.3288	1.88945	0.944727	−	0.327857i	$-0.106326\pi$
$953$	58.3288	1.88945	0.944727	+	0.327857i	$0.106326\pi$
$954$	0	0
$955$	8.74456	0.282967
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−32.7446	−1.05738
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	−1.62772	−0.0524525
$964$	0	0
$965$	21.1168	0.679775
$966$	0	0
$967$	17.2554	0.554897	0.277449	−	0.960740i	$-0.410511\pi$
$967$	17.2554	0.554897	0.277449	+	0.960740i	$0.410511\pi$
$968$	0	0
$969$	−0.744563	−0.0239188
$970$	0	0
$971$	29.2119	0.937456	0.468728	−	0.883343i	$-0.344712\pi$
$971$	29.2119	0.937456	0.468728	+	0.883343i	$0.344712\pi$
$972$	0	0
$973$	−13.6277	−0.436885
$974$	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
$979$	57.3505	1.83293
$980$	0	0
$981$	−9.25544	−0.295503
$982$	0	0
$983$	−52.2337	−1.66600	−0.832998	−	0.553276i	$-0.813377\pi$
$983$	−52.2337	−1.66600	−0.832998	+	0.553276i	$0.813377\pi$
$984$	0	0
$985$	−3.48913	−0.111173
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−11.2554	−0.357902
$990$	0	0
$991$	39.8614	1.26624	0.633120	−	0.774054i	$-0.281774\pi$
$991$	39.8614	1.26624	0.633120	+	0.774054i	$0.281774\pi$
$992$	0	0
$993$	22.0000	0.698149
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	53.7228	1.70142	0.850709	−	0.525636i	$-0.176173\pi$
$997$	53.7228	1.70142	0.850709	+	0.525636i	$0.176173\pi$
$998$	0	0
$999$	−4.37228	−0.138333

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.37228 q^{7} +1.00000 q^{9} -4.37228 q^{11} -1.00000 q^{13} +1.00000 q^{15} +0.372281 q^{17} +2.00000 q^{19} -4.37228 q^{21} +2.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.74456 q^{29} -6.74456 q^{31} +4.37228 q^{33} -4.37228 q^{35} +4.37228 q^{37} +1.00000 q^{39} -0.372281 q^{41} -4.74456 q^{43} -1.00000 q^{45} -2.74456 q^{47} +12.1168 q^{49} -0.372281 q^{51} -7.62772 q^{53} +4.37228 q^{55} -2.00000 q^{57} +2.74456 q^{59} -9.11684 q^{61} +4.37228 q^{63} +1.00000 q^{65} -1.25544 q^{67} -2.37228 q^{69} -4.37228 q^{71} -6.00000 q^{73} -1.00000 q^{75} -19.1168 q^{77} +9.62772 q^{79} +1.00000 q^{81} +14.7446 q^{83} -0.372281 q^{85} +2.74456 q^{87} -13.1168 q^{89} -4.37228 q^{91} +6.74456 q^{93} -2.00000 q^{95} +9.11684 q^{97} -4.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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