LMFDB - Embedded newform 6240.2.w.f.3121.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(3121,6240)]

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, 1, 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("6240.3121");

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.w (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 $ x^20 - 2x^19 + 2x^18 - 5x^16 + 10x^15 - 12x^14 + 16x^13 - 2x^12 - 40x^11 + 72x^10 - 80x^9 - 8x^8 + 128x^7 - 192x^6 + 320x^5 - 320x^4 + 512x^2 - 1024*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{23} $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3121.8
Root			$0.288202 + 1.38454i$ of defining polynomial
Character	$\chi$	$=$	6240.3121
Dual form			6240.2.w.f.3121.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +3.39492 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} +3.39492 q^{7} -1.00000 q^{9} +0.396748i q^{11} -1.00000i q^{13} -1.00000 q^{15} +4.14322 q^{17} +6.40325i q^{19} -3.39492i q^{21} -7.72529 q^{23} -1.00000 q^{25} +1.00000i q^{27} -5.67448i q^{29} -4.53959 q^{31} +0.396748 q^{33} -3.39492i q^{35} +0.138729i q^{37} -1.00000 q^{39} +10.8070 q^{41} +0.00574917i q^{43} +1.00000i q^{45} +12.3072 q^{47} +4.52551 q^{49} -4.14322i q^{51} -6.40926i q^{53} +0.396748 q^{55} +6.40325 q^{57} -6.11036i q^{59} -10.7552i q^{61} -3.39492 q^{63} -1.00000 q^{65} -12.1232i q^{67} +7.72529i q^{69} +16.2844 q^{71} -4.82153 q^{73} +1.00000i q^{75} +1.34693i q^{77} +7.23539 q^{79} +1.00000 q^{81} -11.7504i q^{83} -4.14322i q^{85} -5.67448 q^{87} -8.53201 q^{89} -3.39492i q^{91} +4.53959i q^{93} +6.40325 q^{95} -1.60886 q^{97} -0.396748i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7} - 20 q^{9}+O(q^{10}) $ 20 * q + 16 * q^7 - 20 * q^9	$ 20 q + 16 q^{7} - 20 q^{9} - 20 q^{15} + 16 q^{17} - 48 q^{23} - 20 q^{25} + 8 q^{31} - 16 q^{33} - 20 q^{39} + 16 q^{41} + 8 q^{47} + 36 q^{49} - 16 q^{55} + 16 q^{57} - 16 q^{63} - 20 q^{65} + 8 q^{71} + 40 q^{73} + 8 q^{79} + 20 q^{81} - 8 q^{87} + 56 q^{89} + 16 q^{95} - 48 q^{97}+O(q^{100}) $ 20 * q + 16 * q^7 - 20 * q^9 - 20 * q^15 + 16 * q^17 - 48 * q^23 - 20 * q^25 + 8 * q^31 - 16 * q^33 - 20 * q^39 + 16 * q^41 + 8 * q^47 + 36 * q^49 - 16 * q^55 + 16 * q^57 - 16 * q^63 - 20 * q^65 + 8 * q^71 + 40 * q^73 + 8 * q^79 + 20 * q^81 - 8 * q^87 + 56 * q^89 + 16 * q^95 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6240\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$5761$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	3.39492	1.28316	0.641580	−	0.767056i	$-0.278279\pi$
$7$	3.39492	1.28316	0.641580	+	0.767056i	$0.278279\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.396748i	0.119624i	0.998210	+	0.0598120i	$0.0190501\pi$
$11$	0.396748i	0.119624i	−0.998210	+	0.0598120i	$0.980950\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.14322	1.00488	0.502439	−	0.864613i	$-0.332436\pi$
$17$	4.14322	1.00488	0.502439	+	0.864613i	$0.332436\pi$
$18$	0	0
$19$	6.40325i	1.46901i	0.678605	+	0.734504i	$0.262585\pi$
$19$	6.40325i	1.46901i	−0.678605	+	0.734504i	$0.737415\pi$
$20$	0	0
$21$	− 3.39492i	− 0.740833i
$22$	0	0
$23$	−7.72529	−1.61083	−0.805417	−	0.592709i	$-0.798059\pi$
$23$	−7.72529	−1.61083	−0.805417	+	0.592709i	$0.798059\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 5.67448i	− 1.05372i	−0.849951	−	0.526862i	$-0.823369\pi$
$29$	− 5.67448i	− 1.05372i	0.849951	−	0.526862i	$-0.176631\pi$
$30$	0	0
$31$	−4.53959	−0.815334	−0.407667	−	0.913131i	$-0.633658\pi$
$31$	−4.53959	−0.815334	−0.407667	+	0.913131i	$0.633658\pi$
$32$	0	0
$33$	0.396748	0.0690650
$34$	0	0
$35$	− 3.39492i	− 0.573847i
$36$	0	0
$37$	0.138729i	0.0228068i	0.999935	+	0.0114034i	$0.00362990\pi$
$37$	0.138729i	0.0228068i	−0.999935	+	0.0114034i	$0.996370\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	10.8070	1.68777	0.843883	−	0.536527i	$-0.180264\pi$
$41$	10.8070	1.68777	0.843883	+	0.536527i	$0.180264\pi$
$42$	0	0
$43$	0.00574917i	0 0.000876740i	1.00000		0.000438370i	$0.000139538\pi$
$43$	0.00574917i	0 0.000876740i	−1.00000		0.000438370i	$0.999860\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	12.3072	1.79519	0.897594	−	0.440823i	$-0.145313\pi$
$47$	12.3072	1.79519	0.897594	+	0.440823i	$0.145313\pi$
$48$	0	0
$49$	4.52551	0.646501
$50$	0	0
$51$	− 4.14322i	− 0.580167i
$52$	0	0
$53$	− 6.40926i	− 0.880380i	−0.897905	−	0.440190i	$-0.854911\pi$
$53$	− 6.40926i	− 0.880380i	0.897905	−	0.440190i	$-0.145089\pi$
$54$	0	0
$55$	0.396748	0.0534975
$56$	0	0
$57$	6.40325	0.848132
$58$	0	0
$59$	− 6.11036i	− 0.795502i	−0.917493	−	0.397751i	$-0.869791\pi$
$59$	− 6.11036i	− 0.795502i	0.917493	−	0.397751i	$-0.130209\pi$
$60$	0	0
$61$	− 10.7552i	− 1.37706i	−0.725207	−	0.688531i	$-0.758256\pi$
$61$	− 10.7552i	− 1.37706i	0.725207	−	0.688531i	$-0.241744\pi$
$62$	0	0
$63$	−3.39492	−0.427720
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	− 12.1232i	− 1.48109i	−0.672009	−	0.740543i	$-0.734568\pi$
$67$	− 12.1232i	− 1.48109i	0.672009	−	0.740543i	$-0.265432\pi$
$68$	0	0
$69$	7.72529i	0.930015i
$70$	0	0
$71$	16.2844	1.93260	0.966298	−	0.257424i	$-0.0828738\pi$
$71$	16.2844	1.93260	0.966298	+	0.257424i	$0.0828738\pi$
$72$	0	0
$73$	−4.82153	−0.564318	−0.282159	−	0.959368i	$-0.591051\pi$
$73$	−4.82153	−0.564318	−0.282159	+	0.959368i	$0.591051\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	1.34693i	0.153497i
$78$	0	0
$79$	7.23539	0.814045	0.407023	−	0.913418i	$-0.366567\pi$
$79$	7.23539	0.814045	0.407023	+	0.913418i	$0.366567\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 11.7504i	− 1.28977i	−0.764278	−	0.644886i	$-0.776905\pi$
$83$	− 11.7504i	− 1.28977i	0.764278	−	0.644886i	$-0.223095\pi$
$84$	0	0
$85$	− 4.14322i	− 0.449395i
$86$	0	0
$87$	−5.67448	−0.608368
$88$	0	0
$89$	−8.53201	−0.904391	−0.452196	−	0.891919i	$-0.649359\pi$
$89$	−8.53201	−0.904391	−0.452196	+	0.891919i	$0.649359\pi$
$90$	0	0
$91$	− 3.39492i	− 0.355885i
$92$	0	0
$93$	4.53959i	0.470733i
$94$	0	0
$95$	6.40325	0.656960
$96$	0	0
$97$	−1.60886	−0.163355	−0.0816777	−	0.996659i	$-0.526028\pi$
$97$	−1.60886	−0.163355	−0.0816777	+	0.996659i	$0.526028\pi$
$98$	0	0
$99$	− 0.396748i	− 0.0398747i
$100$	0	0
$101$	14.9155i	1.48415i	0.670319	+	0.742073i	$0.266157\pi$
$101$	14.9155i	1.48415i	−0.670319	+	0.742073i	$0.733843\pi$
$102$	0	0
$103$	−10.1548	−1.00058	−0.500290	−	0.865858i	$-0.666773\pi$
$103$	−10.1548	−1.00058	−0.500290	+	0.865858i	$0.666773\pi$
$104$	0	0
$105$	−3.39492	−0.331311
$106$	0	0
$107$	1.82095i	0.176038i	0.996119	+	0.0880190i	$0.0280536\pi$
$107$	1.82095i	0.176038i	−0.996119	+	0.0880190i	$0.971946\pi$
$108$	0	0
$109$	10.4131i	0.997390i	0.866777	+	0.498695i	$0.166187\pi$
$109$	10.4131i	0.997390i	−0.866777	+	0.498695i	$0.833813\pi$
$110$	0	0
$111$	0.138729	0.0131675
$112$	0	0
$113$	7.22500	0.679671	0.339835	−	0.940485i	$-0.389629\pi$
$113$	7.22500	0.679671	0.339835	+	0.940485i	$0.389629\pi$
$114$	0	0
$115$	7.72529i	0.720387i
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	14.0659	1.28942
$120$	0	0
$121$	10.8426	0.985690
$122$	0	0
$123$	− 10.8070i	− 0.974433i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−10.0781	−0.894284	−0.447142	−	0.894463i	$-0.647558\pi$
$127$	−10.0781	−0.894284	−0.447142	+	0.894463i	$0.647558\pi$
$128$	0	0
$129$	0.00574917	0.000506186 0
$130$	0	0
$131$	15.3388i	1.34015i	0.742292	+	0.670077i	$0.233739\pi$
$131$	15.3388i	1.34015i	−0.742292	+	0.670077i	$0.766261\pi$
$132$	0	0
$133$	21.7386i	1.88497i
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	16.8339	1.43822	0.719110	−	0.694897i	$-0.244550\pi$
$137$	16.8339	1.43822	0.719110	+	0.694897i	$0.244550\pi$
$138$	0	0
$139$	− 19.4538i	− 1.65005i	−0.565096	−	0.825025i	$-0.691161\pi$
$139$	− 19.4538i	− 1.65005i	0.565096	−	0.825025i	$-0.308839\pi$
$140$	0	0
$141$	− 12.3072i	− 1.03645i
$142$	0	0
$143$	0.396748	0.0331777
$144$	0	0
$145$	−5.67448	−0.471240
$146$	0	0
$147$	− 4.52551i	− 0.373257i
$148$	0	0
$149$	− 0.699116i	− 0.0572738i	−0.999590	−	0.0286369i	$-0.990883\pi$
$149$	− 0.699116i	− 0.0572738i	0.999590	−	0.0286369i	$-0.00911666\pi$
$150$	0	0
$151$	2.67552	0.217731	0.108865	−	0.994056i	$-0.465278\pi$
$151$	2.67552	0.217731	0.108865	+	0.994056i	$0.465278\pi$
$152$	0	0
$153$	−4.14322	−0.334959
$154$	0	0
$155$	4.53959i	0.364628i
$156$	0	0
$157$	− 11.7388i	− 0.936859i	−0.883501	−	0.468429i	$-0.844820\pi$
$157$	− 11.7388i	− 0.936859i	0.883501	−	0.468429i	$-0.155180\pi$
$158$	0	0
$159$	−6.40926	−0.508287
$160$	0	0
$161$	−26.2268	−2.06696
$162$	0	0
$163$	− 14.9680i	− 1.17239i	−0.810172	−	0.586193i	$-0.800626\pi$
$163$	− 14.9680i	− 1.17239i	0.810172	−	0.586193i	$-0.199374\pi$
$164$	0	0
$165$	− 0.396748i	− 0.0308868i
$166$	0	0
$167$	22.4410	1.73654	0.868269	−	0.496094i	$-0.165233\pi$
$167$	22.4410	1.73654	0.868269	+	0.496094i	$0.165233\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 6.40325i	− 0.489669i
$172$	0	0
$173$	7.75619i	0.589692i	0.955545	+	0.294846i	$0.0952684\pi$
$173$	7.75619i	0.589692i	−0.955545	+	0.294846i	$0.904732\pi$
$174$	0	0
$175$	−3.39492	−0.256632
$176$	0	0
$177$	−6.11036	−0.459283
$178$	0	0
$179$	17.2528i	1.28954i	0.764378	+	0.644768i	$0.223046\pi$
$179$	17.2528i	1.28954i	−0.764378	+	0.644768i	$0.776954\pi$
$180$	0	0
$181$	− 7.78316i	− 0.578518i	−0.957251	−	0.289259i	$-0.906591\pi$
$181$	− 7.78316i	− 0.578518i	0.957251	−	0.289259i	$-0.0934089\pi$
$182$	0	0
$183$	−10.7552	−0.795047
$184$	0	0
$185$	0.138729	0.0101995
$186$	0	0
$187$	1.64381i	0.120208i
$188$	0	0
$189$	3.39492i	0.246944i
$190$	0	0
$191$	−6.45160	−0.466821	−0.233411	−	0.972378i	$-0.574989\pi$
$191$	−6.45160	−0.466821	−0.233411	+	0.972378i	$0.574989\pi$
$192$	0	0
$193$	19.4499	1.40003	0.700017	−	0.714126i	$-0.253176\pi$
$193$	19.4499	1.40003	0.700017	+	0.714126i	$0.253176\pi$
$194$	0	0
$195$	1.00000i	0.0716115i
$196$	0	0
$197$	− 15.9730i	− 1.13803i	−0.822329	−	0.569013i	$-0.807325\pi$
$197$	− 15.9730i	− 1.13803i	0.822329	−	0.569013i	$-0.192675\pi$
$198$	0	0
$199$	8.58841	0.608816	0.304408	−	0.952542i	$-0.401541\pi$
$199$	8.58841	0.608816	0.304408	+	0.952542i	$0.401541\pi$
$200$	0	0
$201$	−12.1232	−0.855106
$202$	0	0
$203$	− 19.2644i	− 1.35210i
$204$	0	0
$205$	− 10.8070i	− 0.754792i
$206$	0	0
$207$	7.72529	0.536945
$208$	0	0
$209$	−2.54048	−0.175729
$210$	0	0
$211$	1.87864i	0.129331i	0.997907	+	0.0646656i	$0.0205981\pi$
$211$	1.87864i	0.129331i	−0.997907	+	0.0646656i	$0.979402\pi$
$212$	0	0
$213$	− 16.2844i	− 1.11579i
$214$	0	0
$215$	0.00574917	0.000392090 0
$216$	0	0
$217$	−15.4115	−1.04620
$218$	0	0
$219$	4.82153i	0.325809i
$220$	0	0
$221$	− 4.14322i	− 0.278703i
$222$	0	0
$223$	−17.7336	−1.18753	−0.593764	−	0.804639i	$-0.702359\pi$
$223$	−17.7336	−1.18753	−0.593764	+	0.804639i	$0.702359\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 21.6331i	− 1.43584i	−0.696126	−	0.717919i	$-0.745095\pi$
$227$	− 21.6331i	− 1.43584i	0.696126	−	0.717919i	$-0.254905\pi$
$228$	0	0
$229$	17.0379i	1.12590i	0.826492	+	0.562948i	$0.190333\pi$
$229$	17.0379i	1.12590i	−0.826492	+	0.562948i	$0.809667\pi$
$230$	0	0
$231$	1.34693	0.0886214
$232$	0	0
$233$	13.8709	0.908714	0.454357	−	0.890820i	$-0.349869\pi$
$233$	13.8709	0.908714	0.454357	+	0.890820i	$0.349869\pi$
$234$	0	0
$235$	− 12.3072i	− 0.802832i
$236$	0	0
$237$	− 7.23539i	− 0.469989i
$238$	0	0
$239$	18.3021	1.18386	0.591932	−	0.805988i	$-0.298365\pi$
$239$	18.3021	1.18386	0.591932	+	0.805988i	$0.298365\pi$
$240$	0	0
$241$	13.8153	0.889922	0.444961	−	0.895550i	$-0.353218\pi$
$241$	13.8153	0.889922	0.444961	+	0.895550i	$0.353218\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 4.52551i	− 0.289124i
$246$	0	0
$247$	6.40325	0.407429
$248$	0	0
$249$	−11.7504	−0.744651
$250$	0	0
$251$	− 14.0408i	− 0.886249i	−0.896460	−	0.443125i	$-0.853870\pi$
$251$	− 14.0408i	− 0.886249i	0.896460	−	0.443125i	$-0.146130\pi$
$252$	0	0
$253$	− 3.06499i	− 0.192694i
$254$	0	0
$255$	−4.14322	−0.259458
$256$	0	0
$257$	−17.8821	−1.11545	−0.557727	−	0.830025i	$-0.688326\pi$
$257$	−17.8821	−1.11545	−0.557727	+	0.830025i	$0.688326\pi$
$258$	0	0
$259$	0.470973i	0.0292648i
$260$	0	0
$261$	5.67448i	0.351241i
$262$	0	0
$263$	−26.7434	−1.64907	−0.824534	−	0.565812i	$-0.808563\pi$
$263$	−26.7434	−1.64907	−0.824534	+	0.565812i	$0.808563\pi$
$264$	0	0
$265$	−6.40926	−0.393718
$266$	0	0
$267$	8.53201i	0.522150i
$268$	0	0
$269$	− 28.9143i	− 1.76294i	−0.472244	−	0.881468i	$-0.656556\pi$
$269$	− 28.9143i	− 1.76294i	0.472244	−	0.881468i	$-0.343444\pi$
$270$	0	0
$271$	−6.44150	−0.391293	−0.195647	−	0.980674i	$-0.562681\pi$
$271$	−6.44150	−0.391293	−0.195647	+	0.980674i	$0.562681\pi$
$272$	0	0
$273$	−3.39492	−0.205470
$274$	0	0
$275$	− 0.396748i	− 0.0239248i
$276$	0	0
$277$	17.9769i	1.08013i	0.841625	+	0.540063i	$0.181599\pi$
$277$	17.9769i	1.08013i	−0.841625	+	0.540063i	$0.818401\pi$
$278$	0	0
$279$	4.53959	0.271778
$280$	0	0
$281$	3.10865	0.185447	0.0927233	−	0.995692i	$-0.470443\pi$
$281$	3.10865	0.185447	0.0927233	+	0.995692i	$0.470443\pi$
$282$	0	0
$283$	− 21.3067i	− 1.26655i	−0.773925	−	0.633277i	$-0.781709\pi$
$283$	− 21.3067i	− 1.26655i	0.773925	−	0.633277i	$-0.218291\pi$
$284$	0	0
$285$	− 6.40325i	− 0.379296i
$286$	0	0
$287$	36.6889	2.16568
$288$	0	0
$289$	0.166272	0.00978068
$290$	0	0
$291$	1.60886i	0.0943133i
$292$	0	0
$293$	32.3271i	1.88857i	0.329126	+	0.944286i	$0.393246\pi$
$293$	32.3271i	1.88857i	−0.329126	+	0.944286i	$0.606754\pi$
$294$	0	0
$295$	−6.11036	−0.355759
$296$	0	0
$297$	−0.396748	−0.0230217
$298$	0	0
$299$	7.72529i	0.446765i
$300$	0	0
$301$	0.0195180i	0.00112500i
$302$	0	0
$303$	14.9155	0.856872
$304$	0	0
$305$	−10.7552	−0.615841
$306$	0	0
$307$	12.9514i	0.739173i	0.929196	+	0.369587i	$0.120501\pi$
$307$	12.9514i	0.739173i	−0.929196	+	0.369587i	$0.879499\pi$
$308$	0	0
$309$	10.1548i	0.577685i
$310$	0	0
$311$	16.6519	0.944240	0.472120	−	0.881534i	$-0.343489\pi$
$311$	16.6519	0.944240	0.472120	+	0.881534i	$0.343489\pi$
$312$	0	0
$313$	−19.1587	−1.08292	−0.541458	−	0.840728i	$-0.682127\pi$
$313$	−19.1587	−1.08292	−0.541458	+	0.840728i	$0.682127\pi$
$314$	0	0
$315$	3.39492i	0.191282i
$316$	0	0
$317$	− 0.550475i	− 0.0309178i	−0.999881	−	0.0154589i	$-0.995079\pi$
$317$	− 0.550475i	− 0.0309178i	0.999881	−	0.0154589i	$-0.00492091\pi$
$318$	0	0
$319$	2.25134	0.126051
$320$	0	0
$321$	1.82095	0.101636
$322$	0	0
$323$	26.5301i	1.47617i
$324$	0	0
$325$	1.00000i	0.0554700i
$326$	0	0
$327$	10.4131	0.575844
$328$	0	0
$329$	41.7820	2.30351
$330$	0	0
$331$	20.2879i	1.11513i	0.830135	+	0.557563i	$0.188263\pi$
$331$	20.2879i	1.11513i	−0.830135	+	0.557563i	$0.811737\pi$
$332$	0	0
$333$	− 0.138729i	− 0.00760228i
$334$	0	0
$335$	−12.1232	−0.662362
$336$	0	0
$337$	0.423022	0.0230435	0.0115217	−	0.999934i	$-0.496332\pi$
$337$	0.423022	0.0230435	0.0115217	+	0.999934i	$0.496332\pi$
$338$	0	0
$339$	− 7.22500i	− 0.392408i
$340$	0	0
$341$	− 1.80107i	− 0.0975335i
$342$	0	0
$343$	−8.40072	−0.453596
$344$	0	0
$345$	7.72529	0.415915
$346$	0	0
$347$	− 8.60456i	− 0.461917i	−0.972964	−	0.230959i	$-0.925814\pi$
$347$	− 8.60456i	− 0.461917i	0.972964	−	0.230959i	$-0.0741862\pi$
$348$	0	0
$349$	− 22.2219i	− 1.18951i	−0.803908	−	0.594754i	$-0.797249\pi$
$349$	− 22.2219i	− 1.18951i	0.803908	−	0.594754i	$-0.202751\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−16.7816	−0.893196	−0.446598	−	0.894735i	$-0.647365\pi$
$353$	−16.7816	−0.893196	−0.446598	+	0.894735i	$0.647365\pi$
$354$	0	0
$355$	− 16.2844i	− 0.864284i
$356$	0	0
$357$	− 14.0659i	− 0.744447i
$358$	0	0
$359$	−4.32450	−0.228239	−0.114119	−	0.993467i	$-0.536405\pi$
$359$	−4.32450	−0.228239	−0.114119	+	0.993467i	$0.536405\pi$
$360$	0	0
$361$	−22.0017	−1.15798
$362$	0	0
$363$	− 10.8426i	− 0.569088i
$364$	0	0
$365$	4.82153i	0.252371i
$366$	0	0
$367$	3.26276	0.170315	0.0851574	−	0.996368i	$-0.472861\pi$
$367$	3.26276	0.170315	0.0851574	+	0.996368i	$0.472861\pi$
$368$	0	0
$369$	−10.8070	−0.562589
$370$	0	0
$371$	− 21.7590i	− 1.12967i
$372$	0	0
$373$	16.9512i	0.877701i	0.898560	+	0.438851i	$0.144614\pi$
$373$	16.9512i	0.877701i	−0.898560	+	0.438851i	$0.855386\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−5.67448	−0.292250
$378$	0	0
$379$	− 33.6603i	− 1.72902i	−0.502620	−	0.864508i	$-0.667630\pi$
$379$	− 33.6603i	− 1.72902i	0.502620	−	0.864508i	$-0.332370\pi$
$380$	0	0
$381$	10.0781i	0.516315i
$382$	0	0
$383$	30.2431	1.54535	0.772676	−	0.634801i	$-0.218918\pi$
$383$	30.2431	1.54535	0.772676	+	0.634801i	$0.218918\pi$
$384$	0	0
$385$	1.34693	0.0686459
$386$	0	0
$387$	− 0.00574917i	0 0.000292247i
$388$	0	0
$389$	− 1.13392i	− 0.0574919i	−0.999587	−	0.0287459i	$-0.990849\pi$
$389$	− 1.13392i	− 0.0574919i	0.999587	−	0.0287459i	$-0.00915138\pi$
$390$	0	0
$391$	−32.0076	−1.61869
$392$	0	0
$393$	15.3388	0.773738
$394$	0	0
$395$	− 7.23539i	− 0.364052i
$396$	0	0
$397$	3.71437i	0.186419i	0.995647	+	0.0932094i	$0.0297126\pi$
$397$	3.71437i	0.186419i	−0.995647	+	0.0932094i	$0.970287\pi$
$398$	0	0
$399$	21.7386	1.08829
$400$	0	0
$401$	−14.2660	−0.712411	−0.356206	−	0.934408i	$-0.615930\pi$
$401$	−14.2660	−0.712411	−0.356206	+	0.934408i	$0.615930\pi$
$402$	0	0
$403$	4.53959i	0.226133i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	−0.0550403	−0.00272824
$408$	0	0
$409$	7.42941	0.367361	0.183680	−	0.982986i	$-0.441199\pi$
$409$	7.42941	0.367361	0.183680	+	0.982986i	$0.441199\pi$
$410$	0	0
$411$	− 16.8339i	− 0.830356i
$412$	0	0
$413$	− 20.7442i	− 1.02076i
$414$	0	0
$415$	−11.7504	−0.576804
$416$	0	0
$417$	−19.4538	−0.952657
$418$	0	0
$419$	4.77450i	0.233250i	0.993176	+	0.116625i	$0.0372075\pi$
$419$	4.77450i	0.233250i	−0.993176	+	0.116625i	$0.962792\pi$
$420$	0	0
$421$	14.3349i	0.698639i	0.937004	+	0.349319i	$0.113587\pi$
$421$	14.3349i	0.698639i	−0.937004	+	0.349319i	$0.886413\pi$
$422$	0	0
$423$	−12.3072	−0.598396
$424$	0	0
$425$	−4.14322	−0.200976
$426$	0	0
$427$	− 36.5131i	− 1.76699i
$428$	0	0
$429$	− 0.396748i	− 0.0191552i
$430$	0	0
$431$	−6.99223	−0.336804	−0.168402	−	0.985718i	$-0.553861\pi$
$431$	−6.99223	−0.336804	−0.168402	+	0.985718i	$0.553861\pi$
$432$	0	0
$433$	29.2064	1.40357	0.701784	−	0.712390i	$-0.252387\pi$
$433$	29.2064	1.40357	0.701784	+	0.712390i	$0.252387\pi$
$434$	0	0
$435$	5.67448i	0.272070i
$436$	0	0
$437$	− 49.4670i	− 2.36633i
$438$	0	0
$439$	−26.0530	−1.24344	−0.621721	−	0.783239i	$-0.713566\pi$
$439$	−26.0530	−1.24344	−0.621721	+	0.783239i	$0.713566\pi$
$440$	0	0
$441$	−4.52551	−0.215500
$442$	0	0
$443$	3.98911i	0.189528i	0.995500	+	0.0947641i	$0.0302097\pi$
$443$	3.98911i	0.189528i	−0.995500	+	0.0947641i	$0.969790\pi$
$444$	0	0
$445$	8.53201i	0.404456i
$446$	0	0
$447$	−0.699116	−0.0330671
$448$	0	0
$449$	−28.6490	−1.35203	−0.676015	−	0.736888i	$-0.736295\pi$
$449$	−28.6490	−1.35203	−0.676015	+	0.736888i	$0.736295\pi$
$450$	0	0
$451$	4.28765i	0.201897i
$452$	0	0
$453$	− 2.67552i	− 0.125707i
$454$	0	0
$455$	−3.39492	−0.159156
$456$	0	0
$457$	6.75071	0.315785	0.157892	−	0.987456i	$-0.449530\pi$
$457$	6.75071	0.315785	0.157892	+	0.987456i	$0.449530\pi$
$458$	0	0
$459$	4.14322i	0.193389i
$460$	0	0
$461$	− 14.4044i	− 0.670882i	−0.942061	−	0.335441i	$-0.891115\pi$
$461$	− 14.4044i	− 0.670882i	0.942061	−	0.335441i	$-0.108885\pi$
$462$	0	0
$463$	−16.1305	−0.749646	−0.374823	−	0.927096i	$-0.622297\pi$
$463$	−16.1305	−0.749646	−0.374823	+	0.927096i	$0.622297\pi$
$464$	0	0
$465$	4.53959	0.210518
$466$	0	0
$467$	− 2.95857i	− 0.136906i	−0.997654	−	0.0684530i	$-0.978194\pi$
$467$	− 2.95857i	− 0.136906i	0.997654	−	0.0684530i	$-0.0218063\pi$
$468$	0	0
$469$	− 41.1574i	− 1.90047i
$470$	0	0
$471$	−11.7388	−0.540896
$472$	0	0
$473$	−0.00228097	−0.000104879 0
$474$	0	0
$475$	− 6.40325i	− 0.293801i
$476$	0	0
$477$	6.40926i	0.293460i
$478$	0	0
$479$	15.4082	0.704017	0.352008	−	0.935997i	$-0.385499\pi$
$479$	15.4082	0.704017	0.352008	+	0.935997i	$0.385499\pi$
$480$	0	0
$481$	0.138729	0.00632548
$482$	0	0
$483$	26.2268i	1.19336i
$484$	0	0
$485$	1.60886i	0.0730548i
$486$	0	0
$487$	−35.2141	−1.59570	−0.797851	−	0.602855i	$-0.794030\pi$
$487$	−35.2141	−1.59570	−0.797851	+	0.602855i	$0.794030\pi$
$488$	0	0
$489$	−14.9680	−0.676877
$490$	0	0
$491$	− 11.8979i	− 0.536946i	−0.963287	−	0.268473i	$-0.913481\pi$
$491$	− 11.8979i	− 0.536946i	0.963287	−	0.268473i	$-0.0865190\pi$
$492$	0	0
$493$	− 23.5106i	− 1.05886i
$494$	0	0
$495$	−0.396748	−0.0178325
$496$	0	0
$497$	55.2841	2.47983
$498$	0	0
$499$	21.1824i	0.948255i	0.880456	+	0.474127i	$0.157236\pi$
$499$	21.1824i	0.948255i	−0.880456	+	0.474127i	$0.842764\pi$
$500$	0	0
$501$	− 22.4410i	− 1.00259i
$502$	0	0
$503$	9.48031	0.422706	0.211353	−	0.977410i	$-0.432213\pi$
$503$	9.48031	0.422706	0.211353	+	0.977410i	$0.432213\pi$
$504$	0	0
$505$	14.9155	0.663730
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	2.37660i	0.105341i	0.998612	+	0.0526705i	$0.0167733\pi$
$509$	2.37660i	0.105341i	−0.998612	+	0.0526705i	$0.983227\pi$
$510$	0	0
$511$	−16.3687	−0.724110
$512$	0	0
$513$	−6.40325	−0.282711
$514$	0	0
$515$	10.1548i	0.447473i
$516$	0	0
$517$	4.88285i	0.214748i
$518$	0	0
$519$	7.75619	0.340459
$520$	0	0
$521$	−34.5802	−1.51499	−0.757494	−	0.652842i	$-0.773576\pi$
$521$	−34.5802	−1.51499	−0.757494	+	0.652842i	$0.773576\pi$
$522$	0	0
$523$	1.83692i	0.0803228i	0.999193	+	0.0401614i	$0.0127872\pi$
$523$	1.83692i	0.0803228i	−0.999193	+	0.0401614i	$0.987213\pi$
$524$	0	0
$525$	3.39492i	0.148167i
$526$	0	0
$527$	−18.8085	−0.819311
$528$	0	0
$529$	36.6801	1.59479
$530$	0	0
$531$	6.11036i	0.265167i
$532$	0	0
$533$	− 10.8070i	− 0.468102i
$534$	0	0
$535$	1.82095	0.0787266
$536$	0	0
$537$	17.2528	0.744514
$538$	0	0
$539$	1.79549i	0.0773371i
$540$	0	0
$541$	19.5442i	0.840269i	0.907462	+	0.420135i	$0.138017\pi$
$541$	19.5442i	0.840269i	−0.907462	+	0.420135i	$0.861983\pi$
$542$	0	0
$543$	−7.78316	−0.334008
$544$	0	0
$545$	10.4131	0.446047
$546$	0	0
$547$	− 36.9621i	− 1.58038i	−0.612860	−	0.790192i	$-0.709981\pi$
$547$	− 36.9621i	− 1.58038i	0.612860	−	0.790192i	$-0.290019\pi$
$548$	0	0
$549$	10.7552i	0.459021i
$550$	0	0
$551$	36.3351	1.54793
$552$	0	0
$553$	24.5636	1.04455
$554$	0	0
$555$	− 0.138729i	− 0.00588870i
$556$	0	0
$557$	30.1003i	1.27539i	0.770289	+	0.637695i	$0.220112\pi$
$557$	30.1003i	1.27539i	−0.770289	+	0.637695i	$0.779888\pi$
$558$	0	0
$559$	0.00574917	0.000243164 0
$560$	0	0
$561$	1.64381	0.0694019
$562$	0	0
$563$	− 5.42571i	− 0.228666i	−0.993442	−	0.114333i	$-0.963527\pi$
$563$	− 5.42571i	− 0.228666i	0.993442	−	0.114333i	$-0.0364731\pi$
$564$	0	0
$565$	− 7.22500i	− 0.303958i
$566$	0	0
$567$	3.39492	0.142573
$568$	0	0
$569$	11.5792	0.485426	0.242713	−	0.970098i	$-0.421963\pi$
$569$	11.5792	0.485426	0.242713	+	0.970098i	$0.421963\pi$
$570$	0	0
$571$	− 1.46060i	− 0.0611243i	−0.999533	−	0.0305621i	$-0.990270\pi$
$571$	− 1.46060i	− 0.0611243i	0.999533	−	0.0305621i	$-0.00972975\pi$
$572$	0	0
$573$	6.45160i	0.269519i
$574$	0	0
$575$	7.72529	0.322167
$576$	0	0
$577$	−4.03172	−0.167843	−0.0839214	−	0.996472i	$-0.526744\pi$
$577$	−4.03172	−0.167843	−0.0839214	+	0.996472i	$0.526744\pi$
$578$	0	0
$579$	− 19.4499i	− 0.808310i
$580$	0	0
$581$	− 39.8917i	− 1.65499i
$582$	0	0
$583$	2.54286	0.105315
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	5.86983i	0.242274i	0.992636	+	0.121137i	$0.0386540\pi$
$587$	5.86983i	0.242274i	−0.992636	+	0.121137i	$0.961346\pi$
$588$	0	0
$589$	− 29.0681i	− 1.19773i
$590$	0	0
$591$	−15.9730	−0.657040
$592$	0	0
$593$	28.9942	1.19065	0.595324	−	0.803486i	$-0.297024\pi$
$593$	28.9942	1.19065	0.595324	+	0.803486i	$0.297024\pi$
$594$	0	0
$595$	− 14.0659i	− 0.576646i
$596$	0	0
$597$	− 8.58841i	− 0.351500i
$598$	0	0
$599$	−23.1526	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$599$	−23.1526	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$600$	0	0
$601$	−12.0385	−0.491060	−0.245530	−	0.969389i	$-0.578962\pi$
$601$	−12.0385	−0.491060	−0.245530	+	0.969389i	$0.578962\pi$
$602$	0	0
$603$	12.1232i	0.493695i
$604$	0	0
$605$	− 10.8426i	− 0.440814i
$606$	0	0
$607$	35.4047	1.43703	0.718517	−	0.695509i	$-0.244821\pi$
$607$	35.4047	1.43703	0.718517	+	0.695509i	$0.244821\pi$
$608$	0	0
$609$	−19.2644	−0.780633
$610$	0	0
$611$	− 12.3072i	− 0.497896i
$612$	0	0
$613$	20.4475i	0.825868i	0.910761	+	0.412934i	$0.135496\pi$
$613$	20.4475i	0.825868i	−0.910761	+	0.412934i	$0.864504\pi$
$614$	0	0
$615$	−10.8070	−0.435779
$616$	0	0
$617$	6.54022	0.263299	0.131650	−	0.991296i	$-0.457973\pi$
$617$	6.54022	0.263299	0.131650	+	0.991296i	$0.457973\pi$
$618$	0	0
$619$	− 8.86767i	− 0.356422i	−0.983992	−	0.178211i	$-0.942969\pi$
$619$	− 8.86767i	− 0.356422i	0.983992	−	0.178211i	$-0.0570309\pi$
$620$	0	0
$621$	− 7.72529i	− 0.310005i
$622$	0	0
$623$	−28.9655	−1.16048
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.54048i	0.101457i
$628$	0	0
$629$	0.574783i	0.0229181i
$630$	0	0
$631$	−23.0301	−0.916815	−0.458407	−	0.888742i	$-0.651580\pi$
$631$	−23.0301	−0.916815	−0.458407	+	0.888742i	$0.651580\pi$
$632$	0	0
$633$	1.87864	0.0746694
$634$	0	0
$635$	10.0781i	0.399936i
$636$	0	0
$637$	− 4.52551i	− 0.179307i
$638$	0	0
$639$	−16.2844	−0.644199
$640$	0	0
$641$	−15.5602	−0.614590	−0.307295	−	0.951614i	$-0.599424\pi$
$641$	−15.5602	−0.614590	−0.307295	+	0.951614i	$0.599424\pi$
$642$	0	0
$643$	1.70224i	0.0671299i	0.999437	+	0.0335649i	$0.0106861\pi$
$643$	1.70224i	0.0671299i	−0.999437	+	0.0335649i	$0.989314\pi$
$644$	0	0
$645$	− 0.00574917i	0 0.000226373i
$646$	0	0
$647$	−26.5457	−1.04362	−0.521810	−	0.853062i	$-0.674743\pi$
$647$	−26.5457	−1.04362	−0.521810	+	0.853062i	$0.674743\pi$
$648$	0	0
$649$	2.42428	0.0951611
$650$	0	0
$651$	15.4115i	0.604026i
$652$	0	0
$653$	− 27.0146i	− 1.05716i	−0.848882	−	0.528582i	$-0.822724\pi$
$653$	− 27.0146i	− 1.05716i	0.848882	−	0.528582i	$-0.177276\pi$
$654$	0	0
$655$	15.3388	0.599335
$656$	0	0
$657$	4.82153	0.188106
$658$	0	0
$659$	28.6287i	1.11522i	0.830104	+	0.557608i	$0.188281\pi$
$659$	28.6287i	1.11522i	−0.830104	+	0.557608i	$0.811719\pi$
$660$	0	0
$661$	− 8.50842i	− 0.330939i	−0.986215	−	0.165470i	$-0.947086\pi$
$661$	− 8.50842i	− 0.330939i	0.986215	−	0.165470i	$-0.0529139\pi$
$662$	0	0
$663$	−4.14322	−0.160909
$664$	0	0
$665$	21.7386	0.842985
$666$	0	0
$667$	43.8370i	1.69737i
$668$	0	0
$669$	17.7336i	0.685620i
$670$	0	0
$671$	4.26710	0.164730
$672$	0	0
$673$	35.9798	1.38692	0.693459	−	0.720496i	$-0.256086\pi$
$673$	35.9798	1.38692	0.693459	+	0.720496i	$0.256086\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 1.98195i	− 0.0761725i	−0.999274	−	0.0380862i	$-0.987874\pi$
$677$	− 1.98195i	− 0.0761725i	0.999274	−	0.0380862i	$-0.0121262\pi$
$678$	0	0
$679$	−5.46197	−0.209611
$680$	0	0
$681$	−21.6331	−0.828982
$682$	0	0
$683$	− 8.18346i	− 0.313132i	−0.987668	−	0.156566i	$-0.949958\pi$
$683$	− 8.18346i	− 0.313132i	0.987668	−	0.156566i	$-0.0500423\pi$
$684$	0	0
$685$	− 16.8339i	− 0.643191i
$686$	0	0
$687$	17.0379	0.650036
$688$	0	0
$689$	−6.40926	−0.244173
$690$	0	0
$691$	18.6051i	0.707772i	0.935289	+	0.353886i	$0.115140\pi$
$691$	18.6051i	0.707772i	−0.935289	+	0.353886i	$0.884860\pi$
$692$	0	0
$693$	− 1.34693i	− 0.0511656i
$694$	0	0
$695$	−19.4538	−0.737925
$696$	0	0
$697$	44.7757	1.69600
$698$	0	0
$699$	− 13.8709i	− 0.524646i
$700$	0	0
$701$	− 5.69331i	− 0.215033i	−0.994203	−	0.107517i	$-0.965710\pi$
$701$	− 5.69331i	− 0.215033i	0.994203	−	0.107517i	$-0.0342899\pi$
$702$	0	0
$703$	−0.888314	−0.0335034
$704$	0	0
$705$	−12.3072	−0.463516
$706$	0	0
$707$	50.6369i	1.90440i
$708$	0	0
$709$	− 6.41637i	− 0.240972i	−0.992715	−	0.120486i	$-0.961555\pi$
$709$	− 6.41637i	− 0.240972i	0.992715	−	0.120486i	$-0.0384453\pi$
$710$	0	0
$711$	−7.23539	−0.271348
$712$	0	0
$713$	35.0696	1.31337
$714$	0	0
$715$	− 0.396748i	− 0.0148375i
$716$	0	0
$717$	− 18.3021i	− 0.683504i
$718$	0	0
$719$	21.2275	0.791652	0.395826	−	0.918326i	$-0.370458\pi$
$719$	21.2275	0.791652	0.395826	+	0.918326i	$0.370458\pi$
$720$	0	0
$721$	−34.4747	−1.28390
$722$	0	0
$723$	− 13.8153i	− 0.513797i
$724$	0	0
$725$	5.67448i	0.210745i
$726$	0	0
$727$	48.7583	1.80835	0.904173	−	0.427167i	$-0.140488\pi$
$727$	48.7583	1.80835	0.904173	+	0.427167i	$0.140488\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0.0238201i	0 0.000881017i
$732$	0	0
$733$	− 21.4416i	− 0.791965i	−0.918258	−	0.395982i	$-0.870404\pi$
$733$	− 21.4416i	− 0.791965i	0.918258	−	0.395982i	$-0.129596\pi$
$734$	0	0
$735$	−4.52551	−0.166926
$736$	0	0
$737$	4.80986	0.177174
$738$	0	0
$739$	30.7738i	1.13203i	0.824394	+	0.566016i	$0.191516\pi$
$739$	30.7738i	1.13203i	−0.824394	+	0.566016i	$0.808484\pi$
$740$	0	0
$741$	− 6.40325i	− 0.235229i
$742$	0	0
$743$	44.7693	1.64243	0.821214	−	0.570621i	$-0.193297\pi$
$743$	44.7693	1.64243	0.821214	+	0.570621i	$0.193297\pi$
$744$	0	0
$745$	−0.699116	−0.0256136
$746$	0	0
$747$	11.7504i	0.429924i
$748$	0	0
$749$	6.18199i	0.225885i
$750$	0	0
$751$	14.1338	0.515750	0.257875	−	0.966178i	$-0.416978\pi$
$751$	14.1338	0.515750	0.257875	+	0.966178i	$0.416978\pi$
$752$	0	0
$753$	−14.0408	−0.511676
$754$	0	0
$755$	− 2.67552i	− 0.0973722i
$756$	0	0
$757$	37.9994i	1.38111i	0.723279	+	0.690556i	$0.242634\pi$
$757$	37.9994i	1.38111i	−0.723279	+	0.690556i	$0.757366\pi$
$758$	0	0
$759$	−3.06499	−0.111252
$760$	0	0
$761$	−11.7193	−0.424824	−0.212412	−	0.977180i	$-0.568132\pi$
$761$	−11.7193	−0.424824	−0.212412	+	0.977180i	$0.568132\pi$
$762$	0	0
$763$	35.3515i	1.27981i
$764$	0	0
$765$	4.14322i	0.149798i
$766$	0	0
$767$	−6.11036	−0.220632
$768$	0	0
$769$	−48.3612	−1.74395	−0.871974	−	0.489552i	$-0.837160\pi$
$769$	−48.3612	−1.74395	−0.871974	+	0.489552i	$0.837160\pi$
$770$	0	0
$771$	17.8821i	0.644007i
$772$	0	0
$773$	45.4992i	1.63649i	0.574867	+	0.818247i	$0.305054\pi$
$773$	45.4992i	1.63649i	−0.574867	+	0.818247i	$0.694946\pi$
$774$	0	0
$775$	4.53959	0.163067
$776$	0	0
$777$	0.470973	0.0168961
$778$	0	0
$779$	69.1998i	2.47934i
$780$	0	0
$781$	6.46078i	0.231185i
$782$	0	0
$783$	5.67448	0.202789
$784$	0	0
$785$	−11.7388	−0.418976
$786$	0	0
$787$	− 26.3647i	− 0.939801i	−0.882719	−	0.469901i	$-0.844290\pi$
$787$	− 26.3647i	− 0.939801i	0.882719	−	0.469901i	$-0.155710\pi$
$788$	0	0
$789$	26.7434i	0.952090i
$790$	0	0
$791$	24.5283	0.872127
$792$	0	0
$793$	−10.7552	−0.381928
$794$	0	0
$795$	6.40926i	0.227313i
$796$	0	0
$797$	− 1.70781i	− 0.0604937i	−0.999542	−	0.0302469i	$-0.990371\pi$
$797$	− 1.70781i	− 0.0604937i	0.999542	−	0.0302469i	$-0.00962934\pi$
$798$	0	0
$799$	50.9914	1.80395
$800$	0	0
$801$	8.53201	0.301464
$802$	0	0
$803$	− 1.91293i	− 0.0675060i
$804$	0	0
$805$	26.2268i	0.924372i
$806$	0	0
$807$	−28.9143	−1.01783
$808$	0	0
$809$	0.618133	0.0217324	0.0108662	−	0.999941i	$-0.496541\pi$
$809$	0.618133	0.0217324	0.0108662	+	0.999941i	$0.496541\pi$
$810$	0	0
$811$	− 9.02126i	− 0.316779i	−0.987377	−	0.158390i	$-0.949370\pi$
$811$	− 9.02126i	− 0.316779i	0.987377	−	0.158390i	$-0.0506302\pi$
$812$	0	0
$813$	6.44150i	0.225913i
$814$	0	0
$815$	−14.9680	−0.524307
$816$	0	0
$817$	−0.0368134	−0.00128794
$818$	0	0
$819$	3.39492i	0.118628i
$820$	0	0
$821$	9.69790i	0.338459i	0.985577	+	0.169229i	$0.0541279\pi$
$821$	9.69790i	0.338459i	−0.985577	+	0.169229i	$0.945872\pi$
$822$	0	0
$823$	−6.83024	−0.238087	−0.119044	−	0.992889i	$-0.537983\pi$
$823$	−6.83024	−0.238087	−0.119044	+	0.992889i	$0.537983\pi$
$824$	0	0
$825$	−0.396748	−0.0138130
$826$	0	0
$827$	26.9443i	0.936945i	0.883478	+	0.468472i	$0.155195\pi$
$827$	26.9443i	0.936945i	−0.883478	+	0.468472i	$0.844805\pi$
$828$	0	0
$829$	28.2417i	0.980876i	0.871476	+	0.490438i	$0.163163\pi$
$829$	28.2417i	0.980876i	−0.871476	+	0.490438i	$0.836837\pi$
$830$	0	0
$831$	17.9769	0.623611
$832$	0	0
$833$	18.7502	0.649655
$834$	0	0
$835$	− 22.4410i	− 0.776603i
$836$	0	0
$837$	− 4.53959i	− 0.156911i
$838$	0	0
$839$	−45.0999	−1.55702	−0.778511	−	0.627631i	$-0.784025\pi$
$839$	−45.0999	−1.55702	−0.778511	+	0.627631i	$0.784025\pi$
$840$	0	0
$841$	−3.19967	−0.110333
$842$	0	0
$843$	− 3.10865i	− 0.107068i
$844$	0	0
$845$	1.00000i	0.0344010i
$846$	0	0
$847$	36.8098	1.26480
$848$	0	0
$849$	−21.3067	−0.731246
$850$	0	0
$851$	− 1.07172i	− 0.0367380i
$852$	0	0
$853$	− 27.8658i	− 0.954107i	−0.878874	−	0.477054i	$-0.841705\pi$
$853$	− 27.8658i	− 0.954107i	0.878874	−	0.477054i	$-0.158295\pi$
$854$	0	0
$855$	−6.40325	−0.218987
$856$	0	0
$857$	−6.54295	−0.223503	−0.111751	−	0.993736i	$-0.535646\pi$
$857$	−6.54295	−0.223503	−0.111751	+	0.993736i	$0.535646\pi$
$858$	0	0
$859$	22.5237i	0.768499i	0.923229	+	0.384250i	$0.125540\pi$
$859$	22.5237i	0.768499i	−0.923229	+	0.384250i	$0.874460\pi$
$860$	0	0
$861$	− 36.6889i	− 1.25035i
$862$	0	0
$863$	−39.6856	−1.35092	−0.675458	−	0.737399i	$-0.736054\pi$
$863$	−39.6856	−1.35092	−0.675458	+	0.737399i	$0.736054\pi$
$864$	0	0
$865$	7.75619	0.263718
$866$	0	0
$867$	− 0.166272i	− 0.00564688i
$868$	0	0
$869$	2.87063i	0.0973794i
$870$	0	0
$871$	−12.1232	−0.410779
$872$	0	0
$873$	1.60886	0.0544518
$874$	0	0
$875$	3.39492i	0.114769i
$876$	0	0
$877$	− 39.9136i	− 1.34779i	−0.738829	−	0.673893i	$-0.764621\pi$
$877$	− 39.9136i	− 1.34779i	0.738829	−	0.673893i	$-0.235379\pi$
$878$	0	0
$879$	32.3271	1.09037
$880$	0	0
$881$	−17.2027	−0.579572	−0.289786	−	0.957091i	$-0.593584\pi$
$881$	−17.2027	−0.579572	−0.289786	+	0.957091i	$0.593584\pi$
$882$	0	0
$883$	22.1032i	0.743830i	0.928267	+	0.371915i	$0.121299\pi$
$883$	22.1032i	0.743830i	−0.928267	+	0.371915i	$0.878701\pi$
$884$	0	0
$885$	6.11036i	0.205398i
$886$	0	0
$887$	13.3067	0.446797	0.223398	−	0.974727i	$-0.428285\pi$
$887$	13.3067	0.446797	0.223398	+	0.974727i	$0.428285\pi$
$888$	0	0
$889$	−34.2143	−1.14751
$890$	0	0
$891$	0.396748i	0.0132916i
$892$	0	0
$893$	78.8061i	2.63714i
$894$	0	0
$895$	17.2528	0.576698
$896$	0	0
$897$	7.72529	0.257940
$898$	0	0
$899$	25.7598i	0.859136i
$900$	0	0
$901$	− 26.5550i	− 0.884675i
$902$	0	0
$903$	0.0195180	0.000649518 0
$904$	0	0
$905$	−7.78316	−0.258721
$906$	0	0
$907$	55.0721i	1.82864i	0.404992	+	0.914320i	$0.367274\pi$
$907$	55.0721i	1.82864i	−0.404992	+	0.914320i	$0.632726\pi$
$908$	0	0
$909$	− 14.9155i	− 0.494715i
$910$	0	0
$911$	−25.4432	−0.842970	−0.421485	−	0.906835i	$-0.638491\pi$
$911$	−25.4432	−0.842970	−0.421485	+	0.906835i	$0.638491\pi$
$912$	0	0
$913$	4.66194	0.154288
$914$	0	0
$915$	10.7552i	0.355556i
$916$	0	0
$917$	52.0739i	1.71963i
$918$	0	0
$919$	−49.5245	−1.63366	−0.816831	−	0.576878i	$-0.804271\pi$
$919$	−49.5245	−1.63366	−0.816831	+	0.576878i	$0.804271\pi$
$920$	0	0
$921$	12.9514	0.426762
$922$	0	0
$923$	− 16.2844i	− 0.536006i
$924$	0	0
$925$	− 0.138729i	− 0.00456137i
$926$	0	0
$927$	10.1548	0.333527
$928$	0	0
$929$	33.9679	1.11445	0.557226	−	0.830361i	$-0.311866\pi$
$929$	33.9679	1.11445	0.557226	+	0.830361i	$0.311866\pi$
$930$	0	0
$931$	28.9780i	0.949714i
$932$	0	0
$933$	− 16.6519i	− 0.545157i
$934$	0	0
$935$	1.64381	0.0537585
$936$	0	0
$937$	−24.4041	−0.797248	−0.398624	−	0.917114i	$-0.630512\pi$
$937$	−24.4041	−0.797248	−0.398624	+	0.917114i	$0.630512\pi$
$938$	0	0
$939$	19.1587i	0.625222i
$940$	0	0
$941$	30.4557i	0.992828i	0.868086	+	0.496414i	$0.165350\pi$
$941$	30.4557i	0.992828i	−0.868086	+	0.496414i	$0.834650\pi$
$942$	0	0
$943$	−83.4870	−2.71871
$944$	0	0
$945$	3.39492	0.110437
$946$	0	0
$947$	− 10.4726i	− 0.340313i	−0.985417	−	0.170156i	$-0.945573\pi$
$947$	− 10.4726i	− 0.340313i	0.985417	−	0.170156i	$-0.0544273\pi$
$948$	0	0
$949$	4.82153i	0.156514i
$950$	0	0
$951$	−0.550475	−0.0178504
$952$	0	0
$953$	0.413661	0.0133998	0.00669989	−	0.999978i	$-0.497867\pi$
$953$	0.413661	0.0133998	0.00669989	+	0.999978i	$0.497867\pi$
$954$	0	0
$955$	6.45160i	0.208769i
$956$	0	0
$957$	− 2.25134i	− 0.0727754i
$958$	0	0
$959$	57.1499	1.84547
$960$	0	0
$961$	−10.3922	−0.335231
$962$	0	0
$963$	− 1.82095i	− 0.0586793i
$964$	0	0
$965$	− 19.4499i	− 0.626114i
$966$	0	0
$967$	−16.9499	−0.545073	−0.272537	−	0.962145i	$-0.587863\pi$
$967$	−16.9499	−0.545073	−0.272537	+	0.962145i	$0.587863\pi$
$968$	0	0
$969$	26.5301	0.852269
$970$	0	0
$971$	− 28.8227i	− 0.924966i	−0.886628	−	0.462483i	$-0.846959\pi$
$971$	− 28.8227i	− 0.924966i	0.886628	−	0.462483i	$-0.153041\pi$
$972$	0	0
$973$	− 66.0442i	− 2.11728i
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	56.6390	1.81204	0.906020	−	0.423234i	$-0.139105\pi$
$977$	56.6390	1.81204	0.906020	+	0.423234i	$0.139105\pi$
$978$	0	0
$979$	− 3.38506i	− 0.108187i
$980$	0	0
$981$	− 10.4131i	− 0.332463i
$982$	0	0
$983$	33.4141	1.06574	0.532872	−	0.846196i	$-0.321113\pi$
$983$	33.4141	1.06574	0.532872	+	0.846196i	$0.321113\pi$
$984$	0	0
$985$	−15.9730	−0.508941
$986$	0	0
$987$	− 41.7820i	− 1.32993i
$988$	0	0
$989$	− 0.0444140i	− 0.00141228i
$990$	0	0
$991$	−32.9225	−1.04582	−0.522909	−	0.852389i	$-0.675153\pi$
$991$	−32.9225	−1.04582	−0.522909	+	0.852389i	$0.675153\pi$
$992$	0	0
$993$	20.2879	0.643818
$994$	0	0
$995$	− 8.58841i	− 0.272271i
$996$	0	0
$997$	32.4298i	1.02706i	0.858071	+	0.513531i	$0.171663\pi$
$997$	32.4298i	1.02706i	−0.858071	+	0.513531i	$0.828337\pi$
$998$	0	0
$999$	−0.138729	−0.00438918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.w.f.3121.8		20
4.3	odd	2		1560.2.w.f.781.19	✓	20
8.3	odd	2		1560.2.w.f.781.20	yes	20
8.5	even	2	inner	6240.2.w.f.3121.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.w.f.781.19	✓	20	4.3	odd	2
1560.2.w.f.781.20	yes	20	8.3	odd	2
6240.2.w.f.3121.8		20	1.1	even	1	trivial
6240.2.w.f.3121.18		20	8.5	even	2	inner

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.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +3.39492 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} +3.39492 q^{7} -1.00000 q^{9} +0.396748i q^{11} -1.00000i q^{13} -1.00000 q^{15} +4.14322 q^{17} +6.40325i q^{19} -3.39492i q^{21} -7.72529 q^{23} -1.00000 q^{25} +1.00000i q^{27} -5.67448i q^{29} -4.53959 q^{31} +0.396748 q^{33} -3.39492i q^{35} +0.138729i q^{37} -1.00000 q^{39} +10.8070 q^{41} +0.00574917i q^{43} +1.00000i q^{45} +12.3072 q^{47} +4.52551 q^{49} -4.14322i q^{51} -6.40926i q^{53} +0.396748 q^{55} +6.40325 q^{57} -6.11036i q^{59} -10.7552i q^{61} -3.39492 q^{63} -1.00000 q^{65} -12.1232i q^{67} +7.72529i q^{69} +16.2844 q^{71} -4.82153 q^{73} +1.00000i q^{75} +1.34693i q^{77} +7.23539 q^{79} +1.00000 q^{81} -11.7504i q^{83} -4.14322i q^{85} -5.67448 q^{87} -8.53201 q^{89} -3.39492i q^{91} +4.53959i q^{93} +6.40325 q^{95} -1.60886 q^{97} -0.396748i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7} - 20 q^{9}+O(q^{10}) \) 20 * q + 16 * q^7 - 20 * q^9	\( 20 q + 16 q^{7} - 20 q^{9} - 20 q^{15} + 16 q^{17} - 48 q^{23} - 20 q^{25} + 8 q^{31} - 16 q^{33} - 20 q^{39} + 16 q^{41} + 8 q^{47} + 36 q^{49} - 16 q^{55} + 16 q^{57} - 16 q^{63} - 20 q^{65} + 8 q^{71} + 40 q^{73} + 8 q^{79} + 20 q^{81} - 8 q^{87} + 56 q^{89} + 16 q^{95} - 48 q^{97}+O(q^{100}) \) 20 * q + 16 * q^7 - 20 * q^9 - 20 * q^15 + 16 * q^17 - 48 * q^23 - 20 * q^25 + 8 * q^31 - 16 * q^33 - 20 * q^39 + 16 * q^41 + 8 * q^47 + 36 * q^49 - 16 * q^55 + 16 * q^57 - 16 * q^63 - 20 * q^65 + 8 * q^71 + 40 * q^73 + 8 * q^79 + 20 * q^81 - 8 * q^87 + 56 * q^89 + 16 * q^95 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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