LMFDB - Embedded newform 3360.2.g.a.1681.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1681,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.5
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3360.1681
Dual form			3360.2.g.a.1681.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -2.61313i q^{11} +5.44155i q^{13} +1.00000 q^{15} -1.53073 q^{17} -6.73925i q^{19} -1.00000i q^{21} +5.53073 q^{23} -1.00000 q^{25} -1.00000i q^{27} +4.52395i q^{29} -10.3086 q^{31} +2.61313 q^{33} +1.00000i q^{35} +1.53073i q^{37} -5.44155 q^{39} -2.39782 q^{41} +5.69552i q^{43} +1.00000i q^{45} -0.469266 q^{47} +1.00000 q^{49} -1.53073i q^{51} +4.44834i q^{53} -2.61313 q^{55} +6.73925 q^{57} +5.06147i q^{59} -10.2581i q^{61} +1.00000 q^{63} +5.44155 q^{65} -15.9150i q^{67} +5.53073i q^{69} -13.4962 q^{71} -4.80750 q^{73} -1.00000i q^{75} +2.61313i q^{77} -14.3842 q^{79} +1.00000 q^{81} -15.1580i q^{83} +1.53073i q^{85} -4.52395 q^{87} -9.55582 q^{89} -5.44155i q^{91} -10.3086i q^{93} -6.73925 q^{95} -10.8713 q^{97} +2.61313i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} + 32 q^{23} - 8 q^{25} - 32 q^{31} - 16 q^{47} + 8 q^{49} + 8 q^{63} - 16 q^{79} + 8 q^{81} + 16 q^{87}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 + 32 * q^23 - 8 * q^25 - 32 * q^31 - 16 * q^47 + 8 * q^49 + 8 * q^63 - 16 * q^79 + 8 * q^81 + 16 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.61313i	− 0.787887i	−0.919135	−	0.393944i	$-0.871111\pi$
$11$	− 2.61313i	− 0.787887i	0.919135	−	0.393944i	$-0.128889\pi$
$12$	0	0
$13$	5.44155i	1.50922i	0.656176	+	0.754608i	$0.272173\pi$
$13$	5.44155i	1.50922i	−0.656176	+	0.754608i	$0.727827\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.53073	−0.371257	−0.185629	−	0.982620i	$-0.559432\pi$
$17$	−1.53073	−0.371257	−0.185629	+	0.982620i	$0.559432\pi$
$18$	0	0
$19$	− 6.73925i	− 1.54609i	−0.634352	−	0.773045i	$-0.718733\pi$
$19$	− 6.73925i	− 1.54609i	0.634352	−	0.773045i	$-0.281267\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	5.53073	1.15324	0.576619	−	0.817013i	$-0.304372\pi$
$23$	5.53073	1.15324	0.576619	+	0.817013i	$0.304372\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.52395i	0.840076i	0.907507	+	0.420038i	$0.137983\pi$
$29$	4.52395i	0.840076i	−0.907507	+	0.420038i	$0.862017\pi$
$30$	0	0
$31$	−10.3086	−1.85149	−0.925744	−	0.378152i	$-0.876560\pi$
$31$	−10.3086	−1.85149	−0.925744	+	0.378152i	$0.876560\pi$
$32$	0	0
$33$	2.61313	0.454887
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	1.53073i	0.251651i	0.992052	+	0.125826i	$0.0401580\pi$
$37$	1.53073i	0.251651i	−0.992052	+	0.125826i	$0.959842\pi$
$38$	0	0
$39$	−5.44155	−0.871346
$40$	0	0
$41$	−2.39782	−0.374477	−0.187239	−	0.982314i	$-0.559954\pi$
$41$	−2.39782	−0.374477	−0.187239	+	0.982314i	$0.559954\pi$
$42$	0	0
$43$	5.69552i	0.868558i	0.900778	+	0.434279i	$0.142997\pi$
$43$	5.69552i	0.868558i	−0.900778	+	0.434279i	$0.857003\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−0.469266	−0.0684495	−0.0342248	−	0.999414i	$-0.510896\pi$
$47$	−0.469266	−0.0684495	−0.0342248	+	0.999414i	$0.510896\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 1.53073i	− 0.214346i
$52$	0	0
$53$	4.44834i	0.611027i	0.952188	+	0.305513i	$0.0988281\pi$
$53$	4.44834i	0.611027i	−0.952188	+	0.305513i	$0.901172\pi$
$54$	0	0
$55$	−2.61313	−0.352354
$56$	0	0
$57$	6.73925	0.892635
$58$	0	0
$59$	5.06147i	0.658947i	0.944165	+	0.329473i	$0.106871\pi$
$59$	5.06147i	0.658947i	−0.944165	+	0.329473i	$0.893129\pi$
$60$	0	0
$61$	− 10.2581i	− 1.31342i	−0.754144	−	0.656709i	$-0.771948\pi$
$61$	− 10.2581i	− 1.31342i	0.754144	−	0.656709i	$-0.228052\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.44155	0.674942
$66$	0	0
$67$	− 15.9150i	− 1.94432i	−0.234309	−	0.972162i	$-0.575283\pi$
$67$	− 15.9150i	− 1.94432i	0.234309	−	0.972162i	$-0.424717\pi$
$68$	0	0
$69$	5.53073i	0.665822i
$70$	0	0
$71$	−13.4962	−1.60171	−0.800854	−	0.598860i	$-0.795621\pi$
$71$	−13.4962	−1.60171	−0.800854	+	0.598860i	$0.795621\pi$
$72$	0	0
$73$	−4.80750	−0.562676	−0.281338	−	0.959609i	$-0.590778\pi$
$73$	−4.80750	−0.562676	−0.281338	+	0.959609i	$0.590778\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	2.61313i	0.297793i
$78$	0	0
$79$	−14.3842	−1.61835	−0.809177	−	0.587565i	$-0.800087\pi$
$79$	−14.3842	−1.61835	−0.809177	+	0.587565i	$0.800087\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 15.1580i	− 1.66381i	−0.554921	−	0.831903i	$-0.687251\pi$
$83$	− 15.1580i	− 1.66381i	0.554921	−	0.831903i	$-0.312749\pi$
$84$	0	0
$85$	1.53073i	0.166031i
$86$	0	0
$87$	−4.52395	−0.485018
$88$	0	0
$89$	−9.55582	−1.01291	−0.506457	−	0.862265i	$-0.669045\pi$
$89$	−9.55582	−1.01291	−0.506457	+	0.862265i	$0.669045\pi$
$90$	0	0
$91$	− 5.44155i	− 0.570430i
$92$	0	0
$93$	− 10.3086i	− 1.06896i
$94$	0	0
$95$	−6.73925	−0.691432
$96$	0	0
$97$	−10.8713	−1.10381	−0.551904	−	0.833907i	$-0.686099\pi$
$97$	−10.8713	−1.10381	−0.551904	+	0.833907i	$0.686099\pi$
$98$	0	0
$99$	2.61313i	0.262629i
$100$	0	0
$101$	6.23304i	0.620211i	0.950702	+	0.310105i	$0.100364\pi$
$101$	6.23304i	0.620211i	−0.950702	+	0.310105i	$0.899636\pi$
$102$	0	0
$103$	8.76017	0.863165	0.431583	−	0.902073i	$-0.357955\pi$
$103$	8.76017	0.863165	0.431583	+	0.902073i	$0.357955\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	12.0843i	1.16823i	0.811671	+	0.584115i	$0.198558\pi$
$107$	12.0843i	1.16823i	−0.811671	+	0.584115i	$0.801442\pi$
$108$	0	0
$109$	8.15571i	0.781176i	0.920566	+	0.390588i	$0.127728\pi$
$109$	8.15571i	0.781176i	−0.920566	+	0.390588i	$0.872272\pi$
$110$	0	0
$111$	−1.53073	−0.145291
$112$	0	0
$113$	−6.81657	−0.641249	−0.320625	−	0.947206i	$-0.603893\pi$
$113$	−6.81657	−0.641249	−0.320625	+	0.947206i	$0.603893\pi$
$114$	0	0
$115$	− 5.53073i	− 0.515744i
$116$	0	0
$117$	− 5.44155i	− 0.503072i
$118$	0	0
$119$	1.53073	0.140322
$120$	0	0
$121$	4.17157	0.379234
$122$	0	0
$123$	− 2.39782i	− 0.216205i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−8.15571	−0.723702	−0.361851	−	0.932236i	$-0.617855\pi$
$127$	−8.15571	−0.723702	−0.361851	+	0.932236i	$0.617855\pi$
$128$	0	0
$129$	−5.69552	−0.501462
$130$	0	0
$131$	3.22625i	0.281879i	0.990018	+	0.140939i	$0.0450123\pi$
$131$	3.22625i	0.281879i	−0.990018	+	0.140939i	$0.954988\pi$
$132$	0	0
$133$	6.73925i	0.584367i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	0.286743	0.0244981	0.0122490	−	0.999925i	$-0.496101\pi$
$137$	0.286743	0.0244981	0.0122490	+	0.999925i	$0.496101\pi$
$138$	0	0
$139$	− 4.61631i	− 0.391550i	−0.980649	−	0.195775i	$-0.937278\pi$
$139$	− 4.61631i	− 0.391550i	0.980649	−	0.195775i	$-0.0627223\pi$
$140$	0	0
$141$	− 0.469266i	− 0.0395193i
$142$	0	0
$143$	14.2195	1.18909
$144$	0	0
$145$	4.52395	0.375693
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	13.1412i	1.07657i	0.842762	+	0.538286i	$0.180928\pi$
$149$	13.1412i	1.07657i	−0.842762	+	0.538286i	$0.819072\pi$
$150$	0	0
$151$	−4.68554	−0.381304	−0.190652	−	0.981658i	$-0.561060\pi$
$151$	−4.68554	−0.381304	−0.190652	+	0.981658i	$0.561060\pi$
$152$	0	0
$153$	1.53073	0.123752
$154$	0	0
$155$	10.3086i	0.828010i
$156$	0	0
$157$	− 3.27677i	− 0.261515i	−0.991414	−	0.130757i	$-0.958259\pi$
$157$	− 3.27677i	− 0.261515i	0.991414	−	0.130757i	$-0.0417409\pi$
$158$	0	0
$159$	−4.44834	−0.352776
$160$	0	0
$161$	−5.53073	−0.435883
$162$	0	0
$163$	6.63405i	0.519619i	0.965660	+	0.259809i	$0.0836598\pi$
$163$	6.63405i	0.519619i	−0.965660	+	0.259809i	$0.916340\pi$
$164$	0	0
$165$	− 2.61313i	− 0.203432i
$166$	0	0
$167$	−22.7981	−1.76417	−0.882084	−	0.471091i	$-0.843860\pi$
$167$	−22.7981	−1.76417	−0.882084	+	0.471091i	$0.843860\pi$
$168$	0	0
$169$	−16.6105	−1.27773
$170$	0	0
$171$	6.73925i	0.515363i
$172$	0	0
$173$	− 23.0060i	− 1.74912i	−0.484920	−	0.874558i	$-0.661151\pi$
$173$	− 23.0060i	− 1.74912i	0.484920	−	0.874558i	$-0.338849\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−5.06147	−0.380443
$178$	0	0
$179$	− 3.47433i	− 0.259684i	−0.991535	−	0.129842i	$-0.958553\pi$
$179$	− 3.47433i	− 0.259684i	0.991535	−	0.129842i	$-0.0414470\pi$
$180$	0	0
$181$	− 12.7525i	− 0.947884i	−0.880556	−	0.473942i	$-0.842831\pi$
$181$	− 12.7525i	− 0.947884i	0.880556	−	0.473942i	$-0.157169\pi$
$182$	0	0
$183$	10.2581	0.758303
$184$	0	0
$185$	1.53073	0.112542
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−12.9563	−0.937483	−0.468741	−	0.883335i	$-0.655292\pi$
$191$	−12.9563	−0.937483	−0.468741	+	0.883335i	$0.655292\pi$
$192$	0	0
$193$	2.65914	0.191409	0.0957045	−	0.995410i	$-0.469490\pi$
$193$	2.65914	0.191409	0.0957045	+	0.995410i	$0.469490\pi$
$194$	0	0
$195$	5.44155i	0.389678i
$196$	0	0
$197$	− 13.4962i	− 0.961567i	−0.876839	−	0.480783i	$-0.840352\pi$
$197$	− 13.4962i	− 0.961567i	0.876839	−	0.480783i	$-0.159648\pi$
$198$	0	0
$199$	21.1144	1.49676	0.748381	−	0.663269i	$-0.230832\pi$
$199$	21.1144	1.49676	0.748381	+	0.663269i	$0.230832\pi$
$200$	0	0
$201$	15.9150	1.12256
$202$	0	0
$203$	− 4.52395i	− 0.317519i
$204$	0	0
$205$	2.39782i	0.167471i
$206$	0	0
$207$	−5.53073	−0.384413
$208$	0	0
$209$	−17.6105	−1.21814
$210$	0	0
$211$	5.12972i	0.353145i	0.984288	+	0.176572i	$0.0565010\pi$
$211$	5.12972i	0.353145i	−0.984288	+	0.176572i	$0.943499\pi$
$212$	0	0
$213$	− 13.4962i	− 0.924747i
$214$	0	0
$215$	5.69552	0.388431
$216$	0	0
$217$	10.3086	0.699796
$218$	0	0
$219$	− 4.80750i	− 0.324861i
$220$	0	0
$221$	− 8.32957i	− 0.560307i
$222$	0	0
$223$	−10.9604	−0.733965	−0.366982	−	0.930228i	$-0.619609\pi$
$223$	−10.9604	−0.733965	−0.366982	+	0.930228i	$0.619609\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	8.40795i	0.558055i	0.960283	+	0.279028i	$0.0900121\pi$
$227$	8.40795i	0.558055i	−0.960283	+	0.279028i	$0.909988\pi$
$228$	0	0
$229$	− 2.43649i	− 0.161008i	−0.996754	−	0.0805038i	$-0.974347\pi$
$229$	− 2.43649i	− 0.161008i	0.996754	−	0.0805038i	$-0.0256529\pi$
$230$	0	0
$231$	−2.61313	−0.171931
$232$	0	0
$233$	−13.3346	−0.873581	−0.436790	−	0.899563i	$-0.643885\pi$
$233$	−13.3346	−0.873581	−0.436790	+	0.899563i	$0.643885\pi$
$234$	0	0
$235$	0.469266i	0.0306116i
$236$	0	0
$237$	− 14.3842i	− 0.934357i
$238$	0	0
$239$	−9.10748	−0.589114	−0.294557	−	0.955634i	$-0.595172\pi$
$239$	−9.10748	−0.589114	−0.294557	+	0.955634i	$0.595172\pi$
$240$	0	0
$241$	−15.9091	−1.02479	−0.512397	−	0.858748i	$-0.671243\pi$
$241$	−15.9091	−1.02479	−0.512397	+	0.858748i	$0.671243\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	36.6720	2.33338
$248$	0	0
$249$	15.1580	0.960599
$250$	0	0
$251$	− 13.6433i	− 0.861156i	−0.902553	−	0.430578i	$-0.858310\pi$
$251$	− 13.6433i	− 0.861156i	0.902553	−	0.430578i	$-0.141690\pi$
$252$	0	0
$253$	− 14.4525i	− 0.908621i
$254$	0	0
$255$	−1.53073	−0.0958583
$256$	0	0
$257$	−20.1344	−1.25595	−0.627976	−	0.778233i	$-0.716116\pi$
$257$	−20.1344	−1.25595	−0.627976	+	0.778233i	$0.716116\pi$
$258$	0	0
$259$	− 1.53073i	− 0.0951152i
$260$	0	0
$261$	− 4.52395i	− 0.280025i
$262$	0	0
$263$	−26.5786	−1.63891	−0.819454	−	0.573145i	$-0.805723\pi$
$263$	−26.5786	−1.63891	−0.819454	+	0.573145i	$0.805723\pi$
$264$	0	0
$265$	4.44834	0.273259
$266$	0	0
$267$	− 9.55582i	− 0.584807i
$268$	0	0
$269$	− 6.70549i	− 0.408841i	−0.978883	−	0.204421i	$-0.934469\pi$
$269$	− 6.70549i	− 0.408841i	0.978883	−	0.204421i	$-0.0655310\pi$
$270$	0	0
$271$	31.2200	1.89648	0.948241	−	0.317553i	$-0.102861\pi$
$271$	31.2200	1.89648	0.948241	+	0.317553i	$0.102861\pi$
$272$	0	0
$273$	5.44155	0.329338
$274$	0	0
$275$	2.61313i	0.157577i
$276$	0	0
$277$	− 11.6537i	− 0.700201i	−0.936712	−	0.350101i	$-0.886147\pi$
$277$	− 11.6537i	− 0.700201i	0.936712	−	0.350101i	$-0.113853\pi$
$278$	0	0
$279$	10.3086	0.617162
$280$	0	0
$281$	30.8277	1.83903	0.919513	−	0.393061i	$-0.128584\pi$
$281$	30.8277	1.83903	0.919513	+	0.393061i	$0.128584\pi$
$282$	0	0
$283$	− 30.4608i	− 1.81071i	−0.424657	−	0.905354i	$-0.639605\pi$
$283$	− 30.4608i	− 1.81071i	0.424657	−	0.905354i	$-0.360395\pi$
$284$	0	0
$285$	− 6.73925i	− 0.399199i
$286$	0	0
$287$	2.39782	0.141539
$288$	0	0
$289$	−14.6569	−0.862168
$290$	0	0
$291$	− 10.8713i	− 0.637284i
$292$	0	0
$293$	24.8831i	1.45369i	0.686803	+	0.726843i	$0.259013\pi$
$293$	24.8831i	1.45369i	−0.686803	+	0.726843i	$0.740987\pi$
$294$	0	0
$295$	5.06147	0.294690
$296$	0	0
$297$	−2.61313	−0.151629
$298$	0	0
$299$	30.0958i	1.74048i
$300$	0	0
$301$	− 5.69552i	− 0.328284i
$302$	0	0
$303$	−6.23304	−0.358079
$304$	0	0
$305$	−10.2581	−0.587379
$306$	0	0
$307$	3.82164i	0.218112i	0.994036	+	0.109056i	$0.0347829\pi$
$307$	3.82164i	0.218112i	−0.994036	+	0.109056i	$0.965217\pi$
$308$	0	0
$309$	8.76017i	0.498349i
$310$	0	0
$311$	8.60896	0.488170	0.244085	−	0.969754i	$-0.421512\pi$
$311$	8.60896	0.488170	0.244085	+	0.969754i	$0.421512\pi$
$312$	0	0
$313$	−26.2087	−1.48140	−0.740701	−	0.671835i	$-0.765506\pi$
$313$	−26.2087	−1.48140	−0.740701	+	0.671835i	$0.765506\pi$
$314$	0	0
$315$	− 1.00000i	− 0.0563436i
$316$	0	0
$317$	9.32132i	0.523537i	0.965131	+	0.261769i	$0.0843058\pi$
$317$	9.32132i	0.523537i	−0.965131	+	0.261769i	$0.915694\pi$
$318$	0	0
$319$	11.8216	0.661885
$320$	0	0
$321$	−12.0843	−0.674478
$322$	0	0
$323$	10.3160i	0.573997i
$324$	0	0
$325$	− 5.44155i	− 0.301843i
$326$	0	0
$327$	−8.15571	−0.451012
$328$	0	0
$329$	0.469266	0.0258715
$330$	0	0
$331$	0.527131i	0.0289737i	0.999895	+	0.0144869i	$0.00461147\pi$
$331$	0.527131i	0.0289737i	−0.999895	+	0.0144869i	$0.995389\pi$
$332$	0	0
$333$	− 1.53073i	− 0.0838837i
$334$	0	0
$335$	−15.9150	−0.869528
$336$	0	0
$337$	20.0958	1.09469	0.547343	−	0.836908i	$-0.315639\pi$
$337$	20.0958	1.09469	0.547343	+	0.836908i	$0.315639\pi$
$338$	0	0
$339$	− 6.81657i	− 0.370225i
$340$	0	0
$341$	26.9378i	1.45876i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.53073	0.297765
$346$	0	0
$347$	16.3365i	0.876990i	0.898734	+	0.438495i	$0.144488\pi$
$347$	16.3365i	0.876990i	−0.898734	+	0.438495i	$0.855512\pi$
$348$	0	0
$349$	3.76377i	0.201470i	0.994913	+	0.100735i	$0.0321195\pi$
$349$	3.76377i	0.201470i	−0.994913	+	0.100735i	$0.967881\pi$
$350$	0	0
$351$	5.44155	0.290449
$352$	0	0
$353$	29.2513	1.55689	0.778446	−	0.627712i	$-0.216008\pi$
$353$	29.2513	1.55689	0.778446	+	0.627712i	$0.216008\pi$
$354$	0	0
$355$	13.4962i	0.716306i
$356$	0	0
$357$	1.53073i	0.0810150i
$358$	0	0
$359$	28.1508	1.48574	0.742871	−	0.669434i	$-0.233463\pi$
$359$	28.1508	1.48574	0.742871	+	0.669434i	$0.233463\pi$
$360$	0	0
$361$	−26.4174	−1.39039
$362$	0	0
$363$	4.17157i	0.218951i
$364$	0	0
$365$	4.80750i	0.251636i
$366$	0	0
$367$	−16.4661	−0.859522	−0.429761	−	0.902943i	$-0.641402\pi$
$367$	−16.4661	−0.859522	−0.429761	+	0.902943i	$0.641402\pi$
$368$	0	0
$369$	2.39782	0.124826
$370$	0	0
$371$	− 4.44834i	− 0.230946i
$372$	0	0
$373$	19.4161i	1.00533i	0.864482	+	0.502664i	$0.167647\pi$
$373$	19.4161i	1.00533i	−0.864482	+	0.502664i	$0.832353\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−24.6173	−1.26785
$378$	0	0
$379$	− 35.1900i	− 1.80759i	−0.427965	−	0.903795i	$-0.640769\pi$
$379$	− 35.1900i	− 1.80759i	0.427965	−	0.903795i	$-0.359231\pi$
$380$	0	0
$381$	− 8.15571i	− 0.417830i
$382$	0	0
$383$	7.73662	0.395323	0.197661	−	0.980270i	$-0.436665\pi$
$383$	7.73662	0.395323	0.197661	+	0.980270i	$0.436665\pi$
$384$	0	0
$385$	2.61313	0.133177
$386$	0	0
$387$	− 5.69552i	− 0.289519i
$388$	0	0
$389$	25.0994i	1.27259i	0.771446	+	0.636295i	$0.219534\pi$
$389$	25.0994i	1.27259i	−0.771446	+	0.636295i	$0.780466\pi$
$390$	0	0
$391$	−8.46608	−0.428148
$392$	0	0
$393$	−3.22625	−0.162743
$394$	0	0
$395$	14.3842i	0.723750i
$396$	0	0
$397$	− 7.56449i	− 0.379651i	−0.981818	−	0.189825i	$-0.939208\pi$
$397$	− 7.56449i	− 0.379651i	0.981818	−	0.189825i	$-0.0607922\pi$
$398$	0	0
$399$	−6.73925	−0.337384
$400$	0	0
$401$	6.50793	0.324991	0.162495	−	0.986709i	$-0.448046\pi$
$401$	6.50793	0.324991	0.162495	+	0.986709i	$0.448046\pi$
$402$	0	0
$403$	− 56.0950i	− 2.79429i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−26.6493	−1.31772	−0.658862	−	0.752264i	$-0.728962\pi$
$409$	−26.6493	−1.31772	−0.658862	+	0.752264i	$0.728962\pi$
$410$	0	0
$411$	0.286743i	0.0141440i
$412$	0	0
$413$	− 5.06147i	− 0.249059i
$414$	0	0
$415$	−15.1580	−0.744077
$416$	0	0
$417$	4.61631	0.226062
$418$	0	0
$419$	34.0675i	1.66431i	0.554546	+	0.832153i	$0.312892\pi$
$419$	34.0675i	1.66431i	−0.554546	+	0.832153i	$0.687108\pi$
$420$	0	0
$421$	− 2.12293i	− 0.103466i	−0.998661	−	0.0517328i	$-0.983526\pi$
$421$	− 2.12293i	− 0.103466i	0.998661	−	0.0517328i	$-0.0164744\pi$
$422$	0	0
$423$	0.469266	0.0228165
$424$	0	0
$425$	1.53073	0.0742515
$426$	0	0
$427$	10.2581i	0.496426i
$428$	0	0
$429$	14.2195i	0.686522i
$430$	0	0
$431$	−25.9770	−1.25127	−0.625634	−	0.780117i	$-0.715160\pi$
$431$	−25.9770	−1.25127	−0.625634	+	0.780117i	$0.715160\pi$
$432$	0	0
$433$	23.3437	1.12183	0.560913	−	0.827874i	$-0.310450\pi$
$433$	23.3437	1.12183	0.560913	+	0.827874i	$0.310450\pi$
$434$	0	0
$435$	4.52395i	0.216907i
$436$	0	0
$437$	− 37.2730i	− 1.78301i
$438$	0	0
$439$	−6.20761	−0.296273	−0.148137	−	0.988967i	$-0.547328\pi$
$439$	−6.20761	−0.296273	−0.148137	+	0.988967i	$0.547328\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	7.88220i	0.374495i	0.982313	+	0.187247i	$0.0599566\pi$
$443$	7.88220i	0.374495i	−0.982313	+	0.187247i	$0.940043\pi$
$444$	0	0
$445$	9.55582i	0.452989i
$446$	0	0
$447$	−13.1412	−0.621559
$448$	0	0
$449$	2.83113	0.133609	0.0668046	−	0.997766i	$-0.478720\pi$
$449$	2.83113	0.133609	0.0668046	+	0.997766i	$0.478720\pi$
$450$	0	0
$451$	6.26582i	0.295046i
$452$	0	0
$453$	− 4.68554i	− 0.220146i
$454$	0	0
$455$	−5.44155	−0.255104
$456$	0	0
$457$	24.3835	1.14061	0.570306	−	0.821432i	$-0.306825\pi$
$457$	24.3835	1.14061	0.570306	+	0.821432i	$0.306825\pi$
$458$	0	0
$459$	1.53073i	0.0714485i
$460$	0	0
$461$	− 3.62482i	− 0.168825i	−0.996431	−	0.0844124i	$-0.973099\pi$
$461$	− 3.62482i	− 0.168825i	0.996431	−	0.0844124i	$-0.0269013\pi$
$462$	0	0
$463$	16.1995	0.752855	0.376428	−	0.926446i	$-0.377152\pi$
$463$	16.1995	0.752855	0.376428	+	0.926446i	$0.377152\pi$
$464$	0	0
$465$	−10.3086	−0.478052
$466$	0	0
$467$	8.34240i	0.386040i	0.981195	+	0.193020i	$0.0618283\pi$
$467$	8.34240i	0.386040i	−0.981195	+	0.193020i	$0.938172\pi$
$468$	0	0
$469$	15.9150i	0.734886i
$470$	0	0
$471$	3.27677	0.150986
$472$	0	0
$473$	14.8831	0.684326
$474$	0	0
$475$	6.73925i	0.309218i
$476$	0	0
$477$	− 4.44834i	− 0.203676i
$478$	0	0
$479$	−6.42047	−0.293359	−0.146679	−	0.989184i	$-0.546859\pi$
$479$	−6.42047	−0.293359	−0.146679	+	0.989184i	$0.546859\pi$
$480$	0	0
$481$	−8.32957	−0.379796
$482$	0	0
$483$	− 5.53073i	− 0.251657i
$484$	0	0
$485$	10.8713i	0.493638i
$486$	0	0
$487$	42.2633	1.91513	0.957566	−	0.288215i	$-0.0930618\pi$
$487$	42.2633	1.91513	0.957566	+	0.288215i	$0.0930618\pi$
$488$	0	0
$489$	−6.63405	−0.300002
$490$	0	0
$491$	− 13.7756i	− 0.621685i	−0.950461	−	0.310843i	$-0.899389\pi$
$491$	− 13.7756i	− 0.621685i	0.950461	−	0.310843i	$-0.100611\pi$
$492$	0	0
$493$	− 6.92496i	− 0.311884i
$494$	0	0
$495$	2.61313	0.117451
$496$	0	0
$497$	13.4962	0.605389
$498$	0	0
$499$	17.2036i	0.770139i	0.922888	+	0.385070i	$0.125823\pi$
$499$	17.2036i	0.770139i	−0.922888	+	0.385070i	$0.874177\pi$
$500$	0	0
$501$	− 22.7981i	− 1.01854i
$502$	0	0
$503$	24.3456	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$503$	24.3456	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$504$	0	0
$505$	6.23304	0.277367
$506$	0	0
$507$	− 16.6105i	− 0.737698i
$508$	0	0
$509$	0.983083i	0.0435744i	0.999763	+	0.0217872i	$0.00693563\pi$
$509$	0.983083i	0.0435744i	−0.999763	+	0.0217872i	$0.993064\pi$
$510$	0	0
$511$	4.80750	0.212671
$512$	0	0
$513$	−6.73925	−0.297545
$514$	0	0
$515$	− 8.76017i	− 0.386019i
$516$	0	0
$517$	1.22625i	0.0539305i
$518$	0	0
$519$	23.0060	1.00985
$520$	0	0
$521$	−28.6636	−1.25578	−0.627888	−	0.778303i	$-0.716081\pi$
$521$	−28.6636	−1.25578	−0.627888	+	0.778303i	$0.716081\pi$
$522$	0	0
$523$	− 29.1927i	− 1.27651i	−0.769826	−	0.638254i	$-0.779657\pi$
$523$	− 29.1927i	− 1.27651i	0.769826	−	0.638254i	$-0.220343\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	15.7798	0.687378
$528$	0	0
$529$	7.58902	0.329957
$530$	0	0
$531$	− 5.06147i	− 0.219649i
$532$	0	0
$533$	− 13.0479i	− 0.565167i
$534$	0	0
$535$	12.0843	0.522449
$536$	0	0
$537$	3.47433	0.149928
$538$	0	0
$539$	− 2.61313i	− 0.112555i
$540$	0	0
$541$	− 30.1550i	− 1.29646i	−0.761443	−	0.648232i	$-0.775509\pi$
$541$	− 30.1550i	− 1.29646i	0.761443	−	0.648232i	$-0.224491\pi$
$542$	0	0
$543$	12.7525	0.547261
$544$	0	0
$545$	8.15571	0.349352
$546$	0	0
$547$	25.5127i	1.09084i	0.838162	+	0.545421i	$0.183630\pi$
$547$	25.5127i	1.09084i	−0.838162	+	0.545421i	$0.816370\pi$
$548$	0	0
$549$	10.2581i	0.437806i
$550$	0	0
$551$	30.4880	1.29883
$552$	0	0
$553$	14.3842	0.611680
$554$	0	0
$555$	1.53073i	0.0649760i
$556$	0	0
$557$	25.5046i	1.08066i	0.841452	+	0.540331i	$0.181701\pi$
$557$	25.5046i	1.08066i	−0.841452	+	0.540331i	$0.818299\pi$
$558$	0	0
$559$	−30.9925	−1.31084
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	23.8955i	1.00708i	0.863973	+	0.503538i	$0.167969\pi$
$563$	23.8955i	1.00708i	−0.863973	+	0.503538i	$0.832031\pi$
$564$	0	0
$565$	6.81657i	0.286775i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.36276	0.308663	0.154332	−	0.988019i	$-0.450678\pi$
$569$	7.36276	0.308663	0.154332	+	0.988019i	$0.450678\pi$
$570$	0	0
$571$	2.82843i	0.118366i	0.998247	+	0.0591830i	$0.0188495\pi$
$571$	2.82843i	0.118366i	−0.998247	+	0.0591830i	$0.981150\pi$
$572$	0	0
$573$	− 12.9563i	− 0.541256i
$574$	0	0
$575$	−5.53073	−0.230648
$576$	0	0
$577$	15.4802	0.644450	0.322225	−	0.946663i	$-0.395569\pi$
$577$	15.4802	0.644450	0.322225	+	0.946663i	$0.395569\pi$
$578$	0	0
$579$	2.65914i	0.110510i
$580$	0	0
$581$	15.1580i	0.628860i
$582$	0	0
$583$	11.6241	0.481420
$584$	0	0
$585$	−5.44155	−0.224981
$586$	0	0
$587$	7.28168i	0.300547i	0.988644	+	0.150274i	$0.0480154\pi$
$587$	7.28168i	0.300547i	−0.988644	+	0.150274i	$0.951985\pi$
$588$	0	0
$589$	69.4725i	2.86256i
$590$	0	0
$591$	13.4962	0.555161
$592$	0	0
$593$	28.1699	1.15680	0.578400	−	0.815753i	$-0.303677\pi$
$593$	28.1699	1.15680	0.578400	+	0.815753i	$0.303677\pi$
$594$	0	0
$595$	− 1.53073i	− 0.0627540i
$596$	0	0
$597$	21.1144i	0.864156i
$598$	0	0
$599$	−15.6373	−0.638923	−0.319462	−	0.947599i	$-0.603502\pi$
$599$	−15.6373	−0.638923	−0.319462	+	0.947599i	$0.603502\pi$
$600$	0	0
$601$	−10.0083	−0.408248	−0.204124	−	0.978945i	$-0.565435\pi$
$601$	−10.0083	−0.408248	−0.204124	+	0.978945i	$0.565435\pi$
$602$	0	0
$603$	15.9150i	0.648108i
$604$	0	0
$605$	− 4.17157i	− 0.169599i
$606$	0	0
$607$	−14.1293	−0.573491	−0.286745	−	0.958007i	$-0.592573\pi$
$607$	−14.1293	−0.573491	−0.286745	+	0.958007i	$0.592573\pi$
$608$	0	0
$609$	4.52395	0.183320
$610$	0	0
$611$	− 2.55354i	− 0.103305i
$612$	0	0
$613$	− 35.7812i	− 1.44519i	−0.691273	−	0.722594i	$-0.742950\pi$
$613$	− 35.7812i	− 1.44519i	0.691273	−	0.722594i	$-0.257050\pi$
$614$	0	0
$615$	−2.39782	−0.0966896
$616$	0	0
$617$	36.2197	1.45815	0.729075	−	0.684434i	$-0.239951\pi$
$617$	36.2197	1.45815	0.729075	+	0.684434i	$0.239951\pi$
$618$	0	0
$619$	38.5628i	1.54997i	0.631979	+	0.774986i	$0.282243\pi$
$619$	38.5628i	1.54997i	−0.631979	+	0.774986i	$0.717757\pi$
$620$	0	0
$621$	− 5.53073i	− 0.221941i
$622$	0	0
$623$	9.55582	0.382846
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 17.6105i	− 0.703296i
$628$	0	0
$629$	− 2.34315i	− 0.0934273i
$630$	0	0
$631$	−19.3046	−0.768505	−0.384253	−	0.923228i	$-0.625541\pi$
$631$	−19.3046	−0.768505	−0.384253	+	0.923228i	$0.625541\pi$
$632$	0	0
$633$	−5.12972	−0.203888
$634$	0	0
$635$	8.15571i	0.323650i
$636$	0	0
$637$	5.44155i	0.215602i
$638$	0	0
$639$	13.4962	0.533903
$640$	0	0
$641$	28.4344	1.12309	0.561545	−	0.827446i	$-0.310207\pi$
$641$	28.4344	1.12309	0.561545	+	0.827446i	$0.310207\pi$
$642$	0	0
$643$	10.1146i	0.398881i	0.979910	+	0.199441i	$0.0639125\pi$
$643$	10.1146i	0.398881i	−0.979910	+	0.199441i	$0.936087\pi$
$644$	0	0
$645$	5.69552i	0.224261i
$646$	0	0
$647$	−33.7940	−1.32858	−0.664290	−	0.747475i	$-0.731266\pi$
$647$	−33.7940	−1.32858	−0.664290	+	0.747475i	$0.731266\pi$
$648$	0	0
$649$	13.2263	0.519176
$650$	0	0
$651$	10.3086i	0.404028i
$652$	0	0
$653$	25.1395i	0.983785i	0.870656	+	0.491893i	$0.163695\pi$
$653$	25.1395i	0.983785i	−0.870656	+	0.491893i	$0.836305\pi$
$654$	0	0
$655$	3.22625	0.126060
$656$	0	0
$657$	4.80750	0.187559
$658$	0	0
$659$	36.7625i	1.43206i	0.698067	+	0.716032i	$0.254044\pi$
$659$	36.7625i	1.43206i	−0.698067	+	0.716032i	$0.745956\pi$
$660$	0	0
$661$	− 18.2400i	− 0.709453i	−0.934970	−	0.354727i	$-0.884574\pi$
$661$	− 18.2400i	− 0.709453i	0.934970	−	0.354727i	$-0.115426\pi$
$662$	0	0
$663$	8.32957	0.323494
$664$	0	0
$665$	6.73925	0.261337
$666$	0	0
$667$	25.0207i	0.968807i
$668$	0	0
$669$	− 10.9604i	− 0.423755i
$670$	0	0
$671$	−26.8058	−1.03483
$672$	0	0
$673$	41.2447	1.58987	0.794933	−	0.606697i	$-0.207506\pi$
$673$	41.2447	1.58987	0.794933	+	0.606697i	$0.207506\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	20.6810i	0.794837i	0.917637	+	0.397418i	$0.130094\pi$
$677$	20.6810i	0.794837i	−0.917637	+	0.397418i	$0.869906\pi$
$678$	0	0
$679$	10.8713	0.417200
$680$	0	0
$681$	−8.40795	−0.322193
$682$	0	0
$683$	8.28078i	0.316855i	0.987371	+	0.158428i	$0.0506424\pi$
$683$	8.28078i	0.316855i	−0.987371	+	0.158428i	$0.949358\pi$
$684$	0	0
$685$	− 0.286743i	− 0.0109559i
$686$	0	0
$687$	2.43649	0.0929578
$688$	0	0
$689$	−24.2059	−0.922171
$690$	0	0
$691$	21.8864i	0.832597i	0.909228	+	0.416298i	$0.136673\pi$
$691$	21.8864i	0.832597i	−0.909228	+	0.416298i	$0.863327\pi$
$692$	0	0
$693$	− 2.61313i	− 0.0992644i
$694$	0	0
$695$	−4.61631	−0.175107
$696$	0	0
$697$	3.67043	0.139027
$698$	0	0
$699$	− 13.3346i	− 0.504362i
$700$	0	0
$701$	40.3776i	1.52504i	0.646964	+	0.762521i	$0.276038\pi$
$701$	40.3776i	1.52504i	−0.646964	+	0.762521i	$0.723962\pi$
$702$	0	0
$703$	10.3160	0.389075
$704$	0	0
$705$	−0.469266	−0.0176736
$706$	0	0
$707$	− 6.23304i	− 0.234418i
$708$	0	0
$709$	− 29.5524i	− 1.10986i	−0.831896	−	0.554931i	$-0.812745\pi$
$709$	− 29.5524i	− 1.10986i	0.831896	−	0.554931i	$-0.187255\pi$
$710$	0	0
$711$	14.3842	0.539451
$712$	0	0
$713$	−57.0144	−2.13520
$714$	0	0
$715$	− 14.2195i	− 0.531778i
$716$	0	0
$717$	− 9.10748i	− 0.340125i
$718$	0	0
$719$	−21.9509	−0.818632	−0.409316	−	0.912393i	$-0.634233\pi$
$719$	−21.9509	−0.818632	−0.409316	+	0.912393i	$0.634233\pi$
$720$	0	0
$721$	−8.76017	−0.326246
$722$	0	0
$723$	− 15.9091i	− 0.591666i
$724$	0	0
$725$	− 4.52395i	− 0.168015i
$726$	0	0
$727$	−41.6972	−1.54646	−0.773232	−	0.634123i	$-0.781361\pi$
$727$	−41.6972	−1.54646	−0.773232	+	0.634123i	$0.781361\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.71832i	− 0.322459i
$732$	0	0
$733$	13.2549i	0.489580i	0.969576	+	0.244790i	$0.0787190\pi$
$733$	13.2549i	0.489580i	−0.969576	+	0.244790i	$0.921281\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−41.5879	−1.53191
$738$	0	0
$739$	15.2353i	0.560440i	0.959936	+	0.280220i	$0.0904075\pi$
$739$	15.2353i	0.560440i	−0.959936	+	0.280220i	$0.909593\pi$
$740$	0	0
$741$	36.6720i	1.34718i
$742$	0	0
$743$	12.2453	0.449236	0.224618	−	0.974447i	$-0.427887\pi$
$743$	12.2453	0.449236	0.224618	+	0.974447i	$0.427887\pi$
$744$	0	0
$745$	13.1412	0.481458
$746$	0	0
$747$	15.1580i	0.554602i
$748$	0	0
$749$	− 12.0843i	− 0.441550i
$750$	0	0
$751$	36.6667	1.33799	0.668994	−	0.743268i	$-0.266725\pi$
$751$	36.6667	1.33799	0.668994	+	0.743268i	$0.266725\pi$
$752$	0	0
$753$	13.6433	0.497189
$754$	0	0
$755$	4.68554i	0.170524i
$756$	0	0
$757$	10.2317i	0.371878i	0.982561	+	0.185939i	$0.0595327\pi$
$757$	10.2317i	0.371878i	−0.982561	+	0.185939i	$0.940467\pi$
$758$	0	0
$759$	14.4525	0.524593
$760$	0	0
$761$	37.2311	1.34963	0.674814	−	0.737988i	$-0.264224\pi$
$761$	37.2311	1.34963	0.674814	+	0.737988i	$0.264224\pi$
$762$	0	0
$763$	− 8.15571i	− 0.295257i
$764$	0	0
$765$	− 1.53073i	− 0.0553438i
$766$	0	0
$767$	−27.5422	−0.994493
$768$	0	0
$769$	−21.2244	−0.765373	−0.382687	−	0.923878i	$-0.625001\pi$
$769$	−21.2244	−0.765373	−0.382687	+	0.923878i	$0.625001\pi$
$770$	0	0
$771$	− 20.1344i	− 0.725124i
$772$	0	0
$773$	− 12.2678i	− 0.441241i	−0.975360	−	0.220621i	$-0.929192\pi$
$773$	− 12.2678i	− 0.441241i	0.975360	−	0.220621i	$-0.0708082\pi$
$774$	0	0
$775$	10.3086	0.370297
$776$	0	0
$777$	1.53073	0.0549148
$778$	0	0
$779$	16.1595i	0.578975i
$780$	0	0
$781$	35.2674i	1.26197i
$782$	0	0
$783$	4.52395	0.161673
$784$	0	0
$785$	−3.27677	−0.116953
$786$	0	0
$787$	20.3379i	0.724968i	0.931990	+	0.362484i	$0.118071\pi$
$787$	20.3379i	0.724968i	−0.931990	+	0.362484i	$0.881929\pi$
$788$	0	0
$789$	− 26.5786i	− 0.946224i
$790$	0	0
$791$	6.81657	0.242369
$792$	0	0
$793$	55.8201	1.98223
$794$	0	0
$795$	4.44834i	0.157766i
$796$	0	0
$797$	− 34.6675i	− 1.22798i	−0.789312	−	0.613992i	$-0.789563\pi$
$797$	− 34.6675i	− 1.22798i	0.789312	−	0.613992i	$-0.210437\pi$
$798$	0	0
$799$	0.718322	0.0254124
$800$	0	0
$801$	9.55582	0.337638
$802$	0	0
$803$	12.5626i	0.443325i
$804$	0	0
$805$	5.53073i	0.194933i
$806$	0	0
$807$	6.70549	0.236045
$808$	0	0
$809$	−26.4453	−0.929767	−0.464884	−	0.885372i	$-0.653904\pi$
$809$	−26.4453	−0.929767	−0.464884	+	0.885372i	$0.653904\pi$
$810$	0	0
$811$	− 36.8532i	− 1.29409i	−0.762452	−	0.647045i	$-0.776005\pi$
$811$	− 36.8532i	− 1.29409i	0.762452	−	0.647045i	$-0.223995\pi$
$812$	0	0
$813$	31.2200i	1.09493i
$814$	0	0
$815$	6.63405	0.232381
$816$	0	0
$817$	38.3835	1.34287
$818$	0	0
$819$	5.44155i	0.190143i
$820$	0	0
$821$	18.1872i	0.634737i	0.948302	+	0.317368i	$0.102799\pi$
$821$	18.1872i	0.634737i	−0.948302	+	0.317368i	$0.897201\pi$
$822$	0	0
$823$	−13.9219	−0.485288	−0.242644	−	0.970115i	$-0.578015\pi$
$823$	−13.9219	−0.485288	−0.242644	+	0.970115i	$0.578015\pi$
$824$	0	0
$825$	−2.61313	−0.0909774
$826$	0	0
$827$	− 23.9651i	− 0.833348i	−0.909056	−	0.416674i	$-0.863196\pi$
$827$	− 23.9651i	− 0.833348i	0.909056	−	0.416674i	$-0.136804\pi$
$828$	0	0
$829$	24.7762i	0.860513i	0.902707	+	0.430256i	$0.141577\pi$
$829$	24.7762i	0.860513i	−0.902707	+	0.430256i	$0.858423\pi$
$830$	0	0
$831$	11.6537	0.404261
$832$	0	0
$833$	−1.53073	−0.0530368
$834$	0	0
$835$	22.7981i	0.788960i
$836$	0	0
$837$	10.3086i	0.356319i
$838$	0	0
$839$	14.6309	0.505114	0.252557	−	0.967582i	$-0.418729\pi$
$839$	14.6309	0.505114	0.252557	+	0.967582i	$0.418729\pi$
$840$	0	0
$841$	8.53392	0.294273
$842$	0	0
$843$	30.8277i	1.06176i
$844$	0	0
$845$	16.6105i	0.571419i
$846$	0	0
$847$	−4.17157	−0.143337
$848$	0	0
$849$	30.4608	1.04541
$850$	0	0
$851$	8.46608i	0.290214i
$852$	0	0
$853$	− 30.6086i	− 1.04802i	−0.851712	−	0.524010i	$-0.824436\pi$
$853$	− 30.6086i	− 1.04802i	0.851712	−	0.524010i	$-0.175564\pi$
$854$	0	0
$855$	6.73925	0.230477
$856$	0	0
$857$	8.45914	0.288959	0.144479	−	0.989508i	$-0.453849\pi$
$857$	8.45914	0.288959	0.144479	+	0.989508i	$0.453849\pi$
$858$	0	0
$859$	− 7.38369i	− 0.251928i	−0.992035	−	0.125964i	$-0.959798\pi$
$859$	− 7.38369i	− 0.251928i	0.992035	−	0.125964i	$-0.0402024\pi$
$860$	0	0
$861$	2.39782i	0.0817176i
$862$	0	0
$863$	19.8641	0.676180	0.338090	−	0.941114i	$-0.390219\pi$
$863$	19.8641	0.676180	0.338090	+	0.941114i	$0.390219\pi$
$864$	0	0
$865$	−23.0060	−0.782229
$866$	0	0
$867$	− 14.6569i	− 0.497773i
$868$	0	0
$869$	37.5879i	1.27508i
$870$	0	0
$871$	86.6022	2.93440
$872$	0	0
$873$	10.8713	0.367936
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	− 49.6658i	− 1.67709i	−0.544830	−	0.838547i	$-0.683406\pi$
$877$	− 49.6658i	− 1.67709i	0.544830	−	0.838547i	$-0.316594\pi$
$878$	0	0
$879$	−24.8831	−0.839286
$880$	0	0
$881$	12.5143	0.421618	0.210809	−	0.977527i	$-0.432390\pi$
$881$	12.5143	0.421618	0.210809	+	0.977527i	$0.432390\pi$
$882$	0	0
$883$	− 43.8494i	− 1.47565i	−0.674992	−	0.737825i	$-0.735853\pi$
$883$	− 43.8494i	− 1.47565i	0.674992	−	0.737825i	$-0.264147\pi$
$884$	0	0
$885$	5.06147i	0.170139i
$886$	0	0
$887$	26.9538	0.905020	0.452510	−	0.891759i	$-0.350529\pi$
$887$	26.9538	0.905020	0.452510	+	0.891759i	$0.350529\pi$
$888$	0	0
$889$	8.15571	0.273534
$890$	0	0
$891$	− 2.61313i	− 0.0875430i
$892$	0	0
$893$	3.16250i	0.105829i
$894$	0	0
$895$	−3.47433	−0.116134
$896$	0	0
$897$	−30.0958	−1.00487
$898$	0	0
$899$	− 46.6357i	− 1.55539i
$900$	0	0
$901$	− 6.80923i	− 0.226848i
$902$	0	0
$903$	5.69552	0.189535
$904$	0	0
$905$	−12.7525	−0.423907
$906$	0	0
$907$	15.0036i	0.498186i	0.968480	+	0.249093i	$0.0801326\pi$
$907$	15.0036i	0.498186i	−0.968480	+	0.249093i	$0.919867\pi$
$908$	0	0
$909$	− 6.23304i	− 0.206737i
$910$	0	0
$911$	9.29416	0.307929	0.153965	−	0.988076i	$-0.450796\pi$
$911$	9.29416	0.307929	0.153965	+	0.988076i	$0.450796\pi$
$912$	0	0
$913$	−39.6098	−1.31089
$914$	0	0
$915$	− 10.2581i	− 0.339123i
$916$	0	0
$917$	− 3.22625i	− 0.106540i
$918$	0	0
$919$	−17.7571	−0.585754	−0.292877	−	0.956150i	$-0.594613\pi$
$919$	−17.7571	−0.585754	−0.292877	+	0.956150i	$0.594613\pi$
$920$	0	0
$921$	−3.82164	−0.125927
$922$	0	0
$923$	− 73.4405i	− 2.41732i
$924$	0	0
$925$	− 1.53073i	− 0.0503302i
$926$	0	0
$927$	−8.76017	−0.287722
$928$	0	0
$929$	22.5200	0.738858	0.369429	−	0.929259i	$-0.379553\pi$
$929$	22.5200	0.738858	0.369429	+	0.929259i	$0.379553\pi$
$930$	0	0
$931$	− 6.73925i	− 0.220870i
$932$	0	0
$933$	8.60896i	0.281845i
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	51.9595	1.69744	0.848721	−	0.528840i	$-0.177373\pi$
$937$	51.9595	1.69744	0.848721	+	0.528840i	$0.177373\pi$
$938$	0	0
$939$	− 26.2087i	− 0.855288i
$940$	0	0
$941$	− 37.5596i	− 1.22441i	−0.790700	−	0.612204i	$-0.790283\pi$
$941$	− 37.5596i	− 1.22441i	0.790700	−	0.612204i	$-0.209717\pi$
$942$	0	0
$943$	−13.2617	−0.431861
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	− 14.7151i	− 0.478178i	−0.970998	−	0.239089i	$-0.923151\pi$
$947$	− 14.7151i	− 0.478178i	0.970998	−	0.239089i	$-0.0768487\pi$
$948$	0	0
$949$	− 26.1603i	− 0.849199i
$950$	0	0
$951$	−9.32132	−0.302264
$952$	0	0
$953$	38.6465	1.25188	0.625942	−	0.779870i	$-0.284715\pi$
$953$	38.6465	1.25188	0.625942	+	0.779870i	$0.284715\pi$
$954$	0	0
$955$	12.9563i	0.419255i
$956$	0	0
$957$	11.8216i	0.382139i
$958$	0	0
$959$	−0.286743	−0.00925941
$960$	0	0
$961$	75.2681	2.42800
$962$	0	0
$963$	− 12.0843i	− 0.389410i
$964$	0	0
$965$	− 2.65914i	− 0.0856007i
$966$	0	0
$967$	14.2003	0.456650	0.228325	−	0.973585i	$-0.426675\pi$
$967$	14.2003	0.456650	0.228325	+	0.973585i	$0.426675\pi$
$968$	0	0
$969$	−10.3160	−0.331397
$970$	0	0
$971$	− 30.2843i	− 0.971869i	−0.873995	−	0.485934i	$-0.838479\pi$
$971$	− 30.2843i	− 0.971869i	0.873995	−	0.485934i	$-0.161521\pi$
$972$	0	0
$973$	4.61631i	0.147992i
$974$	0	0
$975$	5.44155	0.174269
$976$	0	0
$977$	−6.67369	−0.213510	−0.106755	−	0.994285i	$-0.534046\pi$
$977$	−6.67369	−0.213510	−0.106755	+	0.994285i	$0.534046\pi$
$978$	0	0
$979$	24.9706i	0.798063i
$980$	0	0
$981$	− 8.15571i	− 0.260392i
$982$	0	0
$983$	−7.84216	−0.250126	−0.125063	−	0.992149i	$-0.539913\pi$
$983$	−7.84216	−0.250126	−0.125063	+	0.992149i	$0.539913\pi$
$984$	0	0
$985$	−13.4962	−0.430026
$986$	0	0
$987$	0.469266i	0.0149369i
$988$	0	0
$989$	31.5004i	1.00165i
$990$	0	0
$991$	−15.9254	−0.505886	−0.252943	−	0.967481i	$-0.581399\pi$
$991$	−15.9254	−0.505886	−0.252943	+	0.967481i	$0.581399\pi$
$992$	0	0
$993$	−0.527131	−0.0167280
$994$	0	0
$995$	− 21.1144i	− 0.669372i
$996$	0	0
$997$	25.4317i	0.805431i	0.915325	+	0.402716i	$0.131934\pi$
$997$	25.4317i	0.805431i	−0.915325	+	0.402716i	$0.868066\pi$
$998$	0	0
$999$	1.53073	0.0484303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.g.a.1681.5		8
4.3	odd	2		840.2.g.a.421.2	yes	8
8.3	odd	2		840.2.g.a.421.1	✓	8
8.5	even	2	inner	3360.2.g.a.1681.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.g.a.421.1	✓	8	8.3	odd	2
840.2.g.a.421.2	yes	8	4.3	odd	2
3360.2.g.a.1681.4		8	8.5	even	2	inner
3360.2.g.a.1681.5		8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -2.61313i q^{11} +5.44155i q^{13} +1.00000 q^{15} -1.53073 q^{17} -6.73925i q^{19} -1.00000i q^{21} +5.53073 q^{23} -1.00000 q^{25} -1.00000i q^{27} +4.52395i q^{29} -10.3086 q^{31} +2.61313 q^{33} +1.00000i q^{35} +1.53073i q^{37} -5.44155 q^{39} -2.39782 q^{41} +5.69552i q^{43} +1.00000i q^{45} -0.469266 q^{47} +1.00000 q^{49} -1.53073i q^{51} +4.44834i q^{53} -2.61313 q^{55} +6.73925 q^{57} +5.06147i q^{59} -10.2581i q^{61} +1.00000 q^{63} +5.44155 q^{65} -15.9150i q^{67} +5.53073i q^{69} -13.4962 q^{71} -4.80750 q^{73} -1.00000i q^{75} +2.61313i q^{77} -14.3842 q^{79} +1.00000 q^{81} -15.1580i q^{83} +1.53073i q^{85} -4.52395 q^{87} -9.55582 q^{89} -5.44155i q^{91} -10.3086i q^{93} -6.73925 q^{95} -10.8713 q^{97} +2.61313i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} + 32 q^{23} - 8 q^{25} - 32 q^{31} - 16 q^{47} + 8 q^{49} + 8 q^{63} - 16 q^{79} + 8 q^{81} + 16 q^{87}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 + 32 * q^23 - 8 * q^25 - 32 * q^31 - 16 * q^47 + 8 * q^49 + 8 * q^63 - 16 * q^79 + 8 * q^81 + 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more