LMFDB - Embedded newform 3264.2.f.h.1633.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3264,2,Mod(1633,3264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3264, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3264.1633");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3264 = 2^{6} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3264.f (of order $2$, degree $1$, 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:	$26.0631712197$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1633.7
Root			$-0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	3264.1633
Dual form			3264.2.f.h.1633.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.792287i q^{5} -3.46410 q^{7} -1.00000 q^{9} +3.37228i q^{11} -5.84096i q^{13} -0.792287 q^{15} -1.00000 q^{17} +0.627719i q^{19} -3.46410i q^{21} -5.84096 q^{23} +4.37228 q^{25} -1.00000i q^{27} -5.04868i q^{29} +6.63325 q^{31} -3.37228 q^{33} -2.74456i q^{35} +3.16915i q^{37} +5.84096 q^{39} -8.11684 q^{41} -6.11684i q^{43} -0.792287i q^{45} +6.92820 q^{47} +5.00000 q^{49} -1.00000i q^{51} -1.87953i q^{53} -2.67181 q^{55} -0.627719 q^{57} +2.74456i q^{59} -5.34363i q^{61} +3.46410 q^{63} +4.62772 q^{65} +4.00000i q^{67} -5.84096i q^{69} +8.51278 q^{71} +16.7446 q^{73} +4.37228i q^{75} -11.6819i q^{77} +3.46410 q^{79} +1.00000 q^{81} -0.792287i q^{85} +5.04868 q^{87} -15.4891 q^{89} +20.2337i q^{91} +6.63325i q^{93} -0.497333 q^{95} +14.0000 q^{97} -3.37228i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{17} + 12 q^{25} - 4 q^{33} + 4 q^{41} + 40 q^{49} - 28 q^{57} + 60 q^{65} + 88 q^{73} + 8 q^{81} - 32 q^{89} + 112 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^17 + 12 * q^25 - 4 * q^33 + 4 * q^41 + 40 * q^49 - 28 * q^57 + 60 * q^65 + 88 * q^73 + 8 * q^81 - 32 * q^89 + 112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times$.

$n$	$511$	$2177$	$2245$	$2689$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	0.792287i	0.354322i	0.984182	+	0.177161i	$0.0566913\pi$
$5$	0.792287i	0.354322i	−0.984182	+	0.177161i	$0.943309\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.37228i	1.01678i	0.861127	+	0.508391i	$0.169759\pi$
$11$	3.37228i	1.01678i	−0.861127	+	0.508391i	$0.830241\pi$
$12$	0	0
$13$	− 5.84096i	− 1.61999i	−0.586436	−	0.809996i	$-0.699469\pi$
$13$	− 5.84096i	− 1.61999i	0.586436	−	0.809996i	$-0.300531\pi$
$14$	0	0
$15$	−0.792287	−0.204568
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.627719i	0.144009i	0.997404	+	0.0720043i	$0.0229395\pi$
$19$	0.627719i	0.144009i	−0.997404	+	0.0720043i	$0.977060\pi$
$20$	0	0
$21$	− 3.46410i	− 0.755929i
$22$	0	0
$23$	−5.84096	−1.21792	−0.608962	−	0.793199i	$-0.708414\pi$
$23$	−5.84096	−1.21792	−0.608962	+	0.793199i	$0.708414\pi$
$24$	0	0
$25$	4.37228	0.874456
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 5.04868i	− 0.937516i	−0.883327	−	0.468758i	$-0.844702\pi$
$29$	− 5.04868i	− 0.937516i	0.883327	−	0.468758i	$-0.155298\pi$
$30$	0	0
$31$	6.63325	1.19137	0.595683	−	0.803219i	$-0.296881\pi$
$31$	6.63325	1.19137	0.595683	+	0.803219i	$0.296881\pi$
$32$	0	0
$33$	−3.37228	−0.587039
$34$	0	0
$35$	− 2.74456i	− 0.463916i
$36$	0	0
$37$	3.16915i	0.521005i	0.965473	+	0.260502i	$0.0838882\pi$
$37$	3.16915i	0.521005i	−0.965473	+	0.260502i	$0.916112\pi$
$38$	0	0
$39$	5.84096	0.935303
$40$	0	0
$41$	−8.11684	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$41$	−8.11684	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$42$	0	0
$43$	− 6.11684i	− 0.932810i	−0.884571	−	0.466405i	$-0.845549\pi$
$43$	− 6.11684i	− 0.932810i	0.884571	−	0.466405i	$-0.154451\pi$
$44$	0	0
$45$	− 0.792287i	− 0.118107i
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	− 1.00000i	− 0.140028i
$52$	0	0
$53$	− 1.87953i	− 0.258173i	−0.991633	−	0.129086i	$-0.958796\pi$
$53$	− 1.87953i	− 0.258173i	0.991633	−	0.129086i	$-0.0412045\pi$
$54$	0	0
$55$	−2.67181	−0.360267
$56$	0	0
$57$	−0.627719	−0.0831434
$58$	0	0
$59$	2.74456i	0.357312i	0.983912	+	0.178656i	$0.0571749\pi$
$59$	2.74456i	0.357312i	−0.983912	+	0.178656i	$0.942825\pi$
$60$	0	0
$61$	− 5.34363i	− 0.684182i	−0.939667	−	0.342091i	$-0.888865\pi$
$61$	− 5.34363i	− 0.684182i	0.939667	−	0.342091i	$-0.111135\pi$
$62$	0	0
$63$	3.46410	0.436436
$64$	0	0
$65$	4.62772	0.573998
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	− 5.84096i	− 0.703169i
$70$	0	0
$71$	8.51278	1.01028	0.505140	−	0.863037i	$-0.331441\pi$
$71$	8.51278	1.01028	0.505140	+	0.863037i	$0.331441\pi$
$72$	0	0
$73$	16.7446	1.95980	0.979901	−	0.199482i	$-0.0639261\pi$
$73$	16.7446	1.95980	0.979901	+	0.199482i	$0.0639261\pi$
$74$	0	0
$75$	4.37228i	0.504868i
$76$	0	0
$77$	− 11.6819i	− 1.33128i
$78$	0	0
$79$	3.46410	0.389742	0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410	0.389742	0.194871	+	0.980829i	$0.437571\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 0.792287i	− 0.0859356i
$86$	0	0
$87$	5.04868	0.541275
$88$	0	0
$89$	−15.4891	−1.64184	−0.820922	−	0.571040i	$-0.806540\pi$
$89$	−15.4891	−1.64184	−0.820922	+	0.571040i	$0.806540\pi$
$90$	0	0
$91$	20.2337i	2.12107i
$92$	0	0
$93$	6.63325i	0.687836i
$94$	0	0
$95$	−0.497333	−0.0510253
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	− 3.37228i	− 0.338927i
$100$	0	0
$101$	− 1.87953i	− 0.187020i	−0.995618	−	0.0935100i	$-0.970191\pi$
$101$	− 1.87953i	− 0.187020i	0.995618	−	0.0935100i	$-0.0298087\pi$
$102$	0	0
$103$	−0.792287	−0.0780664	−0.0390332	−	0.999238i	$-0.512428\pi$
$103$	−0.792287	−0.0780664	−0.0390332	+	0.999238i	$0.512428\pi$
$104$	0	0
$105$	2.74456	0.267842
$106$	0	0
$107$	− 15.3723i	− 1.48609i	−0.669239	−	0.743047i	$-0.733380\pi$
$107$	− 15.3723i	− 1.48609i	0.669239	−	0.743047i	$-0.266620\pi$
$108$	0	0
$109$	− 6.92820i	− 0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$	− 6.92820i	− 0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$	0	0
$111$	−3.16915	−0.300802
$112$	0	0
$113$	13.3723	1.25796	0.628979	−	0.777422i	$-0.283473\pi$
$113$	13.3723	1.25796	0.628979	+	0.777422i	$0.283473\pi$
$114$	0	0
$115$	− 4.62772i	− 0.431537i
$116$	0	0
$117$	5.84096i	0.539997i
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	−0.372281	−0.0338438
$122$	0	0
$123$	− 8.11684i	− 0.731871i
$124$	0	0
$125$	7.42554i	0.664160i
$126$	0	0
$127$	6.13592	0.544475	0.272237	−	0.962230i	$-0.412236\pi$
$127$	6.13592	0.544475	0.272237	+	0.962230i	$0.412236\pi$
$128$	0	0
$129$	6.11684	0.538558
$130$	0	0
$131$	19.6060i	1.71298i	0.516162	+	0.856491i	$0.327360\pi$
$131$	19.6060i	1.71298i	−0.516162	+	0.856491i	$0.672640\pi$
$132$	0	0
$133$	− 2.17448i	− 0.188551i
$134$	0	0
$135$	0.792287	0.0681892
$136$	0	0
$137$	14.2337	1.21607	0.608033	−	0.793912i	$-0.291959\pi$
$137$	14.2337	1.21607	0.608033	+	0.793912i	$0.291959\pi$
$138$	0	0
$139$	− 9.48913i	− 0.804857i	−0.915451	−	0.402429i	$-0.868166\pi$
$139$	− 9.48913i	− 0.804857i	0.915451	−	0.402429i	$-0.131834\pi$
$140$	0	0
$141$	6.92820i	0.583460i
$142$	0	0
$143$	19.6974	1.64718
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	5.00000i	0.412393i
$148$	0	0
$149$	1.87953i	0.153977i	0.997032	+	0.0769885i	$0.0245305\pi$
$149$	1.87953i	0.153977i	−0.997032	+	0.0769885i	$0.975470\pi$
$150$	0	0
$151$	−19.8997	−1.61942	−0.809709	−	0.586831i	$-0.800375\pi$
$151$	−19.8997	−1.61942	−0.809709	+	0.586831i	$0.800375\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	5.25544i	0.422127i
$156$	0	0
$157$	− 21.2819i	− 1.69848i	−0.528004	−	0.849242i	$-0.677059\pi$
$157$	− 21.2819i	− 1.69848i	0.528004	−	0.849242i	$-0.322941\pi$
$158$	0	0
$159$	1.87953	0.149056
$160$	0	0
$161$	20.2337	1.59464
$162$	0	0
$163$	10.7446i	0.841579i	0.907158	+	0.420790i	$0.138247\pi$
$163$	10.7446i	0.841579i	−0.907158	+	0.420790i	$0.861753\pi$
$164$	0	0
$165$	− 2.67181i	− 0.208000i
$166$	0	0
$167$	15.9383	1.23334	0.616672	−	0.787220i	$-0.288481\pi$
$167$	15.9383	1.23334	0.616672	+	0.787220i	$0.288481\pi$
$168$	0	0
$169$	−21.1168	−1.62437
$170$	0	0
$171$	− 0.627719i	− 0.0480028i
$172$	0	0
$173$	1.38219i	0.105086i	0.998619	+	0.0525431i	$0.0167327\pi$
$173$	1.38219i	0.105086i	−0.998619	+	0.0525431i	$0.983267\pi$
$174$	0	0
$175$	−15.1460	−1.14493
$176$	0	0
$177$	−2.74456	−0.206294
$178$	0	0
$179$	5.25544i	0.392810i	0.980523	+	0.196405i	$0.0629267\pi$
$179$	5.25544i	0.392810i	−0.980523	+	0.196405i	$0.937073\pi$
$180$	0	0
$181$	− 13.2665i	− 0.986091i	−0.870003	−	0.493046i	$-0.835884\pi$
$181$	− 13.2665i	− 0.986091i	0.870003	−	0.493046i	$-0.164116\pi$
$182$	0	0
$183$	5.34363	0.395012
$184$	0	0
$185$	−2.51087	−0.184603
$186$	0	0
$187$	− 3.37228i	− 0.246606i
$188$	0	0
$189$	3.46410i	0.251976i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	7.48913	0.539079	0.269540	−	0.962989i	$-0.413129\pi$
$193$	7.48913	0.539079	0.269540	+	0.962989i	$0.413129\pi$
$194$	0	0
$195$	4.62772i	0.331398i
$196$	0	0
$197$	− 14.6487i	− 1.04368i	−0.853045	−	0.521838i	$-0.825247\pi$
$197$	− 14.6487i	− 1.04368i	0.853045	−	0.521838i	$-0.174753\pi$
$198$	0	0
$199$	3.46410	0.245564	0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410	0.245564	0.122782	+	0.992434i	$0.460818\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	17.4891i	1.22750i
$204$	0	0
$205$	− 6.43087i	− 0.449151i
$206$	0	0
$207$	5.84096	0.405975
$208$	0	0
$209$	−2.11684	−0.146425
$210$	0	0
$211$	− 5.25544i	− 0.361799i	−0.983502	−	0.180900i	$-0.942099\pi$
$211$	− 5.25544i	− 0.361799i	0.983502	−	0.180900i	$-0.0579009\pi$
$212$	0	0
$213$	8.51278i	0.583286i
$214$	0	0
$215$	4.84630	0.330515
$216$	0	0
$217$	−22.9783	−1.55987
$218$	0	0
$219$	16.7446i	1.13149i
$220$	0	0
$221$	5.84096i	0.392906i
$222$	0	0
$223$	20.9870	1.40539	0.702696	−	0.711490i	$-0.251979\pi$
$223$	20.9870	1.40539	0.702696	+	0.711490i	$0.251979\pi$
$224$	0	0
$225$	−4.37228	−0.291485
$226$	0	0
$227$	20.6277i	1.36911i	0.728961	+	0.684555i	$0.240003\pi$
$227$	20.6277i	1.36911i	−0.728961	+	0.684555i	$0.759997\pi$
$228$	0	0
$229$	− 17.0256i	− 1.12508i	−0.826770	−	0.562540i	$-0.809824\pi$
$229$	− 17.0256i	− 1.12508i	0.826770	−	0.562540i	$-0.190176\pi$
$230$	0	0
$231$	11.6819	0.768614
$232$	0	0
$233$	2.62772	0.172148	0.0860738	−	0.996289i	$-0.472568\pi$
$233$	2.62772	0.172148	0.0860738	+	0.996289i	$0.472568\pi$
$234$	0	0
$235$	5.48913i	0.358071i
$236$	0	0
$237$	3.46410i	0.225018i
$238$	0	0
$239$	−2.17448	−0.140656	−0.0703278	−	0.997524i	$-0.522405\pi$
$239$	−2.17448	−0.140656	−0.0703278	+	0.997524i	$0.522405\pi$
$240$	0	0
$241$	−11.4891	−0.740080	−0.370040	−	0.929016i	$-0.620656\pi$
$241$	−11.4891	−0.740080	−0.370040	+	0.929016i	$0.620656\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	3.96143i	0.253087i
$246$	0	0
$247$	3.66648	0.233293
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 26.9783i	− 1.70285i	−0.524475	−	0.851426i	$-0.675738\pi$
$251$	− 26.9783i	− 1.70285i	0.524475	−	0.851426i	$-0.324262\pi$
$252$	0	0
$253$	− 19.6974i	− 1.23836i
$254$	0	0
$255$	0.792287	0.0496149
$256$	0	0
$257$	−24.9783	−1.55810	−0.779050	−	0.626962i	$-0.784298\pi$
$257$	−24.9783	−1.55810	−0.779050	+	0.626962i	$0.784298\pi$
$258$	0	0
$259$	− 10.9783i	− 0.682155i
$260$	0	0
$261$	5.04868i	0.312505i
$262$	0	0
$263$	−9.50744	−0.586254	−0.293127	−	0.956073i	$-0.594696\pi$
$263$	−9.50744	−0.586254	−0.293127	+	0.956073i	$0.594696\pi$
$264$	0	0
$265$	1.48913	0.0914762
$266$	0	0
$267$	− 15.4891i	− 0.947919i
$268$	0	0
$269$	9.30506i	0.567340i	0.958922	+	0.283670i	$0.0915520\pi$
$269$	9.30506i	0.567340i	−0.958922	+	0.283670i	$0.908448\pi$
$270$	0	0
$271$	−21.5769	−1.31070	−0.655352	−	0.755324i	$-0.727480\pi$
$271$	−21.5769	−1.31070	−0.655352	+	0.755324i	$0.727480\pi$
$272$	0	0
$273$	−20.2337	−1.22460
$274$	0	0
$275$	14.7446i	0.889131i
$276$	0	0
$277$	23.9538i	1.43924i	0.694367	+	0.719621i	$0.255684\pi$
$277$	23.9538i	1.43924i	−0.694367	+	0.719621i	$0.744316\pi$
$278$	0	0
$279$	−6.63325	−0.397122
$280$	0	0
$281$	−7.48913	−0.446764	−0.223382	−	0.974731i	$-0.571710\pi$
$281$	−7.48913	−0.446764	−0.223382	+	0.974731i	$0.571710\pi$
$282$	0	0
$283$	− 24.2337i	− 1.44054i	−0.693692	−	0.720272i	$-0.744017\pi$
$283$	− 24.2337i	− 1.44054i	0.693692	−	0.720272i	$-0.255983\pi$
$284$	0	0
$285$	− 0.497333i	− 0.0294595i
$286$	0	0
$287$	28.1176	1.65973
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 28.4125i	− 1.65988i	−0.557855	−	0.829939i	$-0.688375\pi$
$293$	− 28.4125i	− 1.65988i	0.557855	−	0.829939i	$-0.311625\pi$
$294$	0	0
$295$	−2.17448	−0.126603
$296$	0	0
$297$	3.37228	0.195680
$298$	0	0
$299$	34.1168i	1.97303i
$300$	0	0
$301$	21.1894i	1.22133i
$302$	0	0
$303$	1.87953	0.107976
$304$	0	0
$305$	4.23369	0.242420
$306$	0	0
$307$	− 6.51087i	− 0.371595i	−0.982588	−	0.185798i	$-0.940513\pi$
$307$	− 6.51087i	− 0.371595i	0.982588	−	0.185798i	$-0.0594869\pi$
$308$	0	0
$309$	− 0.792287i	− 0.0450716i
$310$	0	0
$311$	12.2718	0.695872	0.347936	−	0.937518i	$-0.386883\pi$
$311$	12.2718	0.695872	0.347936	+	0.937518i	$0.386883\pi$
$312$	0	0
$313$	24.9783	1.41185	0.705927	−	0.708284i	$-0.250531\pi$
$313$	24.9783	1.41185	0.705927	+	0.708284i	$0.250531\pi$
$314$	0	0
$315$	2.74456i	0.154639i
$316$	0	0
$317$	32.1716i	1.80694i	0.428656	+	0.903468i	$0.358987\pi$
$317$	32.1716i	1.80694i	−0.428656	+	0.903468i	$0.641013\pi$
$318$	0	0
$319$	17.0256	0.953248
$320$	0	0
$321$	15.3723	0.857997
$322$	0	0
$323$	− 0.627719i	− 0.0349272i
$324$	0	0
$325$	− 25.5383i	− 1.41661i
$326$	0	0
$327$	6.92820	0.383131
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	4.86141i	0.267207i	0.991035	+	0.133604i	$0.0426549\pi$
$331$	4.86141i	0.267207i	−0.991035	+	0.133604i	$0.957345\pi$
$332$	0	0
$333$	− 3.16915i	− 0.173668i
$334$	0	0
$335$	−3.16915	−0.173149
$336$	0	0
$337$	7.25544	0.395229	0.197614	−	0.980280i	$-0.436681\pi$
$337$	7.25544	0.395229	0.197614	+	0.980280i	$0.436681\pi$
$338$	0	0
$339$	13.3723i	0.726283i
$340$	0	0
$341$	22.3692i	1.21136i
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	4.62772	0.249148
$346$	0	0
$347$	− 29.4891i	− 1.58306i	−0.611131	−	0.791530i	$-0.709285\pi$
$347$	− 29.4891i	− 1.58306i	0.611131	−	0.791530i	$-0.290715\pi$
$348$	0	0
$349$	− 32.9639i	− 1.76452i	−0.470767	−	0.882258i	$-0.656023\pi$
$349$	− 32.9639i	− 1.76452i	0.470767	−	0.882258i	$-0.343977\pi$
$350$	0	0
$351$	−5.84096	−0.311768
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	6.74456i	0.357964i
$356$	0	0
$357$	3.46410i	0.183340i
$358$	0	0
$359$	23.9538	1.26423	0.632115	−	0.774874i	$-0.282187\pi$
$359$	23.9538	1.26423	0.632115	+	0.774874i	$0.282187\pi$
$360$	0	0
$361$	18.6060	0.979262
$362$	0	0
$363$	− 0.372281i	− 0.0195397i
$364$	0	0
$365$	13.2665i	0.694400i
$366$	0	0
$367$	−28.4125	−1.48312	−0.741561	−	0.670886i	$-0.765914\pi$
$367$	−28.4125	−1.48312	−0.741561	+	0.670886i	$0.765914\pi$
$368$	0	0
$369$	8.11684	0.422546
$370$	0	0
$371$	6.51087i	0.338028i
$372$	0	0
$373$	− 10.0974i	− 0.522821i	−0.965228	−	0.261411i	$-0.915812\pi$
$373$	− 10.0974i	− 0.522821i	0.965228	−	0.261411i	$-0.0841876\pi$
$374$	0	0
$375$	−7.42554	−0.383453
$376$	0	0
$377$	−29.4891	−1.51877
$378$	0	0
$379$	− 13.2554i	− 0.680886i	−0.940265	−	0.340443i	$-0.889423\pi$
$379$	− 13.2554i	− 0.680886i	0.940265	−	0.340443i	$-0.110577\pi$
$380$	0	0
$381$	6.13592i	0.314353i
$382$	0	0
$383$	−28.7075	−1.46688	−0.733442	−	0.679752i	$-0.762087\pi$
$383$	−28.7075	−1.46688	−0.733442	+	0.679752i	$0.762087\pi$
$384$	0	0
$385$	9.25544	0.471701
$386$	0	0
$387$	6.11684i	0.310937i
$388$	0	0
$389$	5.04868i	0.255978i	0.991776	+	0.127989i	$0.0408522\pi$
$389$	5.04868i	0.255978i	−0.991776	+	0.127989i	$0.959148\pi$
$390$	0	0
$391$	5.84096	0.295390
$392$	0	0
$393$	−19.6060	−0.988990
$394$	0	0
$395$	2.74456i	0.138094i
$396$	0	0
$397$	5.34363i	0.268189i	0.990969	+	0.134095i	$0.0428126\pi$
$397$	5.34363i	0.268189i	−0.990969	+	0.134095i	$0.957187\pi$
$398$	0	0
$399$	2.17448	0.108860
$400$	0	0
$401$	9.37228	0.468029	0.234015	−	0.972233i	$-0.424814\pi$
$401$	9.37228	0.468029	0.234015	+	0.972233i	$0.424814\pi$
$402$	0	0
$403$	− 38.7446i	− 1.93000i
$404$	0	0
$405$	0.792287i	0.0393691i
$406$	0	0
$407$	−10.6873	−0.529748
$408$	0	0
$409$	−24.1168	−1.19250	−0.596251	−	0.802798i	$-0.703344\pi$
$409$	−24.1168	−1.19250	−0.596251	+	0.802798i	$0.703344\pi$
$410$	0	0
$411$	14.2337i	0.702096i
$412$	0	0
$413$	− 9.50744i	− 0.467831i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.48913	0.464684
$418$	0	0
$419$	2.51087i	0.122664i	0.998117	+	0.0613321i	$0.0195349\pi$
$419$	2.51087i	0.122664i	−0.998117	+	0.0613321i	$0.980465\pi$
$420$	0	0
$421$	19.6974i	0.959991i	0.877271	+	0.479996i	$0.159362\pi$
$421$	19.6974i	0.959991i	−0.877271	+	0.479996i	$0.840638\pi$
$422$	0	0
$423$	−6.92820	−0.336861
$424$	0	0
$425$	−4.37228	−0.212087
$426$	0	0
$427$	18.5109i	0.895804i
$428$	0	0
$429$	19.6974i	0.950998i
$430$	0	0
$431$	−2.17448	−0.104741	−0.0523705	−	0.998628i	$-0.516678\pi$
$431$	−2.17448	−0.104741	−0.0523705	+	0.998628i	$0.516678\pi$
$432$	0	0
$433$	25.3723	1.21931	0.609657	−	0.792665i	$-0.291307\pi$
$433$	25.3723	1.21931	0.609657	+	0.792665i	$0.291307\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	− 3.66648i	− 0.175392i
$438$	0	0
$439$	−5.04868	−0.240960	−0.120480	−	0.992716i	$-0.538443\pi$
$439$	−5.04868	−0.240960	−0.120480	+	0.992716i	$0.538443\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	− 5.48913i	− 0.260796i	−0.991462	−	0.130398i	$-0.958374\pi$
$443$	− 5.48913i	− 0.260796i	0.991462	−	0.130398i	$-0.0416255\pi$
$444$	0	0
$445$	− 12.2718i	− 0.581741i
$446$	0	0
$447$	−1.87953	−0.0888986
$448$	0	0
$449$	16.9783	0.801253	0.400627	−	0.916241i	$-0.368792\pi$
$449$	16.9783	0.801253	0.400627	+	0.916241i	$0.368792\pi$
$450$	0	0
$451$	− 27.3723i	− 1.28891i
$452$	0	0
$453$	− 19.8997i	− 0.934972i
$454$	0	0
$455$	−16.0309	−0.751540
$456$	0	0
$457$	−5.60597	−0.262236	−0.131118	−	0.991367i	$-0.541857\pi$
$457$	−5.60597	−0.262236	−0.131118	+	0.991367i	$0.541857\pi$
$458$	0	0
$459$	1.00000i	0.0466760i
$460$	0	0
$461$	− 19.8997i	− 0.926824i	−0.886143	−	0.463412i	$-0.846625\pi$
$461$	− 19.8997i	− 0.926824i	0.886143	−	0.463412i	$-0.153375\pi$
$462$	0	0
$463$	−7.22316	−0.335689	−0.167844	−	0.985814i	$-0.553681\pi$
$463$	−7.22316	−0.335689	−0.167844	+	0.985814i	$0.553681\pi$
$464$	0	0
$465$	−5.25544	−0.243715
$466$	0	0
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	− 13.8564i	− 0.639829i
$470$	0	0
$471$	21.2819	0.980620
$472$	0	0
$473$	20.6277	0.948464
$474$	0	0
$475$	2.74456i	0.125929i
$476$	0	0
$477$	1.87953i	0.0860577i
$478$	0	0
$479$	−22.8665	−1.04480	−0.522399	−	0.852701i	$-0.674963\pi$
$479$	−22.8665	−1.04480	−0.522399	+	0.852701i	$0.674963\pi$
$480$	0	0
$481$	18.5109	0.844023
$482$	0	0
$483$	20.2337i	0.920665i
$484$	0	0
$485$	11.0920i	0.503663i
$486$	0	0
$487$	−3.46410	−0.156973	−0.0784867	−	0.996915i	$-0.525009\pi$
$487$	−3.46410	−0.156973	−0.0784867	+	0.996915i	$0.525009\pi$
$488$	0	0
$489$	−10.7446	−0.485886
$490$	0	0
$491$	− 41.4891i	− 1.87238i	−0.351497	−	0.936189i	$-0.614327\pi$
$491$	− 41.4891i	− 1.87238i	0.351497	−	0.936189i	$-0.385673\pi$
$492$	0	0
$493$	5.04868i	0.227381i
$494$	0	0
$495$	2.67181	0.120089
$496$	0	0
$497$	−29.4891	−1.32277
$498$	0	0
$499$	− 33.4891i	− 1.49918i	−0.661903	−	0.749590i	$-0.730251\pi$
$499$	− 33.4891i	− 1.49918i	0.661903	−	0.749590i	$-0.269749\pi$
$500$	0	0
$501$	15.9383i	0.712071i
$502$	0	0
$503$	18.7027	0.833912	0.416956	−	0.908927i	$-0.363097\pi$
$503$	18.7027	0.833912	0.416956	+	0.908927i	$0.363097\pi$
$504$	0	0
$505$	1.48913	0.0662652
$506$	0	0
$507$	− 21.1168i	− 0.937832i
$508$	0	0
$509$	17.7253i	0.785659i	0.919611	+	0.392829i	$0.128504\pi$
$509$	17.7253i	0.785659i	−0.919611	+	0.392829i	$0.871496\pi$
$510$	0	0
$511$	−58.0049	−2.56598
$512$	0	0
$513$	0.627719	0.0277145
$514$	0	0
$515$	− 0.627719i	− 0.0276606i
$516$	0	0
$517$	23.3639i	1.02754i
$518$	0	0
$519$	−1.38219	−0.0606716
$520$	0	0
$521$	9.60597	0.420845	0.210423	−	0.977611i	$-0.432516\pi$
$521$	9.60597	0.420845	0.210423	+	0.977611i	$0.432516\pi$
$522$	0	0
$523$	− 30.9783i	− 1.35458i	−0.735714	−	0.677292i	$-0.763153\pi$
$523$	− 30.9783i	− 1.35458i	0.735714	−	0.677292i	$-0.236847\pi$
$524$	0	0
$525$	− 15.1460i	− 0.661027i
$526$	0	0
$527$	−6.63325	−0.288949
$528$	0	0
$529$	11.1168	0.483341
$530$	0	0
$531$	− 2.74456i	− 0.119104i
$532$	0	0
$533$	47.4102i	2.05356i
$534$	0	0
$535$	12.1793	0.526555
$536$	0	0
$537$	−5.25544	−0.226789
$538$	0	0
$539$	16.8614i	0.726272i
$540$	0	0
$541$	− 33.4612i	− 1.43861i	−0.694695	−	0.719305i	$-0.744461\pi$
$541$	− 33.4612i	− 1.43861i	0.694695	−	0.719305i	$-0.255539\pi$
$542$	0	0
$543$	13.2665	0.569320
$544$	0	0
$545$	5.48913	0.235128
$546$	0	0
$547$	− 10.7446i	− 0.459404i	−0.973261	−	0.229702i	$-0.926225\pi$
$547$	− 10.7446i	− 0.459404i	0.973261	−	0.229702i	$-0.0737752\pi$
$548$	0	0
$549$	5.34363i	0.228061i
$550$	0	0
$551$	3.16915	0.135010
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	− 2.51087i	− 0.106581i
$556$	0	0
$557$	34.3461i	1.45529i	0.685954	+	0.727645i	$0.259385\pi$
$557$	34.3461i	1.45529i	−0.685954	+	0.727645i	$0.740615\pi$
$558$	0	0
$559$	−35.7283	−1.51114
$560$	0	0
$561$	3.37228	0.142378
$562$	0	0
$563$	− 14.7446i	− 0.621409i	−0.950507	−	0.310705i	$-0.899435\pi$
$563$	− 14.7446i	− 0.621409i	0.950507	−	0.310705i	$-0.100565\pi$
$564$	0	0
$565$	10.5947i	0.445722i
$566$	0	0
$567$	−3.46410	−0.145479
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	24.2337i	1.01415i	0.861902	+	0.507074i	$0.169273\pi$
$571$	24.2337i	1.01415i	−0.861902	+	0.507074i	$0.830727\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−25.5383	−1.06502
$576$	0	0
$577$	−1.60597	−0.0668574	−0.0334287	−	0.999441i	$-0.510643\pi$
$577$	−1.60597	−0.0668574	−0.0334287	+	0.999441i	$0.510643\pi$
$578$	0	0
$579$	7.48913i	0.311237i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.33830	0.262505
$584$	0	0
$585$	−4.62772	−0.191333
$586$	0	0
$587$	1.48913i	0.0614628i	0.999528	+	0.0307314i	$0.00978365\pi$
$587$	1.48913i	0.0614628i	−0.999528	+	0.0307314i	$0.990216\pi$
$588$	0	0
$589$	4.16381i	0.171567i
$590$	0	0
$591$	14.6487	0.602567
$592$	0	0
$593$	−15.2554	−0.626466	−0.313233	−	0.949676i	$-0.601412\pi$
$593$	−15.2554	−0.626466	−0.313233	+	0.949676i	$0.601412\pi$
$594$	0	0
$595$	2.74456i	0.112516i
$596$	0	0
$597$	3.46410i	0.141776i
$598$	0	0
$599$	−27.1229	−1.10821	−0.554106	−	0.832446i	$-0.686940\pi$
$599$	−27.1229	−1.10821	−0.554106	+	0.832446i	$0.686940\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 0.294954i	− 0.0119916i
$606$	0	0
$607$	−21.4843	−0.872022	−0.436011	−	0.899941i	$-0.643609\pi$
$607$	−21.4843	−0.872022	−0.436011	+	0.899941i	$0.643609\pi$
$608$	0	0
$609$	−17.4891	−0.708695
$610$	0	0
$611$	− 40.4674i	− 1.63713i
$612$	0	0
$613$	− 5.25106i	− 0.212088i	−0.994361	−	0.106044i	$-0.966182\pi$
$613$	− 5.25106i	− 0.212088i	0.994361	−	0.106044i	$-0.0338185\pi$
$614$	0	0
$615$	6.43087	0.259318
$616$	0	0
$617$	−47.4891	−1.91184	−0.955920	−	0.293627i	$-0.905138\pi$
$617$	−47.4891	−1.91184	−0.955920	+	0.293627i	$0.905138\pi$
$618$	0	0
$619$	40.2337i	1.61713i	0.588408	+	0.808564i	$0.299755\pi$
$619$	40.2337i	1.61713i	−0.588408	+	0.808564i	$0.700245\pi$
$620$	0	0
$621$	5.84096i	0.234390i
$622$	0	0
$623$	53.6559	2.14968
$624$	0	0
$625$	15.9783	0.639130
$626$	0	0
$627$	− 2.11684i	− 0.0845386i
$628$	0	0
$629$	− 3.16915i	− 0.126362i
$630$	0	0
$631$	−16.2333	−0.646236	−0.323118	−	0.946359i	$-0.604731\pi$
$631$	−16.2333	−0.646236	−0.323118	+	0.946359i	$0.604731\pi$
$632$	0	0
$633$	5.25544	0.208885
$634$	0	0
$635$	4.86141i	0.192919i
$636$	0	0
$637$	− 29.2048i	− 1.15714i
$638$	0	0
$639$	−8.51278	−0.336760
$640$	0	0
$641$	11.8832	0.469356	0.234678	−	0.972073i	$-0.424596\pi$
$641$	11.8832	0.469356	0.234678	+	0.972073i	$0.424596\pi$
$642$	0	0
$643$	− 5.25544i	− 0.207254i	−0.994616	−	0.103627i	$-0.966955\pi$
$643$	− 5.25544i	− 0.207254i	0.994616	−	0.103627i	$-0.0330449\pi$
$644$	0	0
$645$	4.84630i	0.190823i
$646$	0	0
$647$	−33.4612	−1.31550	−0.657748	−	0.753238i	$-0.728491\pi$
$647$	−33.4612	−1.31550	−0.657748	+	0.753238i	$0.728491\pi$
$648$	0	0
$649$	−9.25544	−0.363308
$650$	0	0
$651$	− 22.9783i	− 0.900589i
$652$	0	0
$653$	− 42.3615i	− 1.65773i	−0.559446	−	0.828867i	$-0.688986\pi$
$653$	− 42.3615i	− 1.65773i	0.559446	−	0.828867i	$-0.311014\pi$
$654$	0	0
$655$	−15.5336	−0.606946
$656$	0	0
$657$	−16.7446	−0.653268
$658$	0	0
$659$	17.4891i	0.681280i	0.940194	+	0.340640i	$0.110644\pi$
$659$	17.4891i	0.681280i	−0.940194	+	0.340640i	$0.889356\pi$
$660$	0	0
$661$	1.08724i	0.0422888i	0.999776	+	0.0211444i	$0.00673097\pi$
$661$	1.08724i	0.0422888i	−0.999776	+	0.0211444i	$0.993269\pi$
$662$	0	0
$663$	−5.84096	−0.226844
$664$	0	0
$665$	1.72281	0.0668078
$666$	0	0
$667$	29.4891i	1.14182i
$668$	0	0
$669$	20.9870i	0.811404i
$670$	0	0
$671$	18.0202	0.695663
$672$	0	0
$673$	9.76631	0.376464	0.188232	−	0.982125i	$-0.439724\pi$
$673$	9.76631	0.376464	0.188232	+	0.982125i	$0.439724\pi$
$674$	0	0
$675$	− 4.37228i	− 0.168289i
$676$	0	0
$677$	37.0179i	1.42271i	0.702832	+	0.711356i	$0.251919\pi$
$677$	37.0179i	1.42271i	−0.702832	+	0.711356i	$0.748081\pi$
$678$	0	0
$679$	−48.4974	−1.86116
$680$	0	0
$681$	−20.6277	−0.790456
$682$	0	0
$683$	43.8397i	1.67748i	0.544534	+	0.838739i	$0.316707\pi$
$683$	43.8397i	1.67748i	−0.544534	+	0.838739i	$0.683293\pi$
$684$	0	0
$685$	11.2772i	0.430878i
$686$	0	0
$687$	17.0256	0.649565
$688$	0	0
$689$	−10.9783	−0.418238
$690$	0	0
$691$	6.97825i	0.265465i	0.991152	+	0.132733i	$0.0423751\pi$
$691$	6.97825i	0.265465i	−0.991152	+	0.132733i	$0.957625\pi$
$692$	0	0
$693$	11.6819i	0.443760i
$694$	0	0
$695$	7.51811	0.285178
$696$	0	0
$697$	8.11684	0.307447
$698$	0	0
$699$	2.62772i	0.0993894i
$700$	0	0
$701$	11.3870i	0.430080i	0.976605	+	0.215040i	$0.0689882\pi$
$701$	11.3870i	0.430080i	−0.976605	+	0.215040i	$0.931012\pi$
$702$	0	0
$703$	−1.98933	−0.0750291
$704$	0	0
$705$	−5.48913	−0.206732
$706$	0	0
$707$	6.51087i	0.244867i
$708$	0	0
$709$	− 40.9793i	− 1.53901i	−0.638641	−	0.769505i	$-0.720503\pi$
$709$	− 40.9793i	− 1.53901i	0.638641	−	0.769505i	$-0.279497\pi$
$710$	0	0
$711$	−3.46410	−0.129914
$712$	0	0
$713$	−38.7446	−1.45100
$714$	0	0
$715$	15.6060i	0.583630i
$716$	0	0
$717$	− 2.17448i	− 0.0812075i
$718$	0	0
$719$	14.3537	0.535304	0.267652	−	0.963516i	$-0.413752\pi$
$719$	14.3537	0.535304	0.267652	+	0.963516i	$0.413752\pi$
$720$	0	0
$721$	2.74456	0.102213
$722$	0	0
$723$	− 11.4891i	− 0.427285i
$724$	0	0
$725$	− 22.0742i	− 0.819816i
$726$	0	0
$727$	−44.4434	−1.64831	−0.824157	−	0.566361i	$-0.808351\pi$
$727$	−44.4434	−1.64831	−0.824157	+	0.566361i	$0.808351\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	6.11684i	0.226240i
$732$	0	0
$733$	37.2203i	1.37476i	0.726297	+	0.687381i	$0.241240\pi$
$733$	37.2203i	1.37476i	−0.726297	+	0.687381i	$0.758760\pi$
$734$	0	0
$735$	−3.96143	−0.146120
$736$	0	0
$737$	−13.4891	−0.496878
$738$	0	0
$739$	31.8397i	1.17124i	0.810585	+	0.585620i	$0.199149\pi$
$739$	31.8397i	1.17124i	−0.810585	+	0.585620i	$0.800851\pi$
$740$	0	0
$741$	3.66648i	0.134692i
$742$	0	0
$743$	−22.9591	−0.842287	−0.421144	−	0.906994i	$-0.638371\pi$
$743$	−22.9591	−0.842287	−0.421144	+	0.906994i	$0.638371\pi$
$744$	0	0
$745$	−1.48913	−0.0545573
$746$	0	0
$747$	0	0
$748$	0	0
$749$	53.2511i	1.94575i
$750$	0	0
$751$	−34.9360	−1.27483	−0.637416	−	0.770520i	$-0.719997\pi$
$751$	−34.9360	−1.27483	−0.637416	+	0.770520i	$0.719997\pi$
$752$	0	0
$753$	26.9783	0.983142
$754$	0	0
$755$	− 15.7663i	− 0.573795i
$756$	0	0
$757$	19.6974i	0.715913i	0.933738	+	0.357957i	$0.116526\pi$
$757$	19.6974i	0.715913i	−0.933738	+	0.357957i	$0.883474\pi$
$758$	0	0
$759$	19.6974	0.714969
$760$	0	0
$761$	42.4674	1.53944	0.769721	−	0.638381i	$-0.220396\pi$
$761$	42.4674	1.53944	0.769721	+	0.638381i	$0.220396\pi$
$762$	0	0
$763$	24.0000i	0.868858i
$764$	0	0
$765$	0.792287i	0.0286452i
$766$	0	0
$767$	16.0309	0.578842
$768$	0	0
$769$	17.6060	0.634887	0.317444	−	0.948277i	$-0.397176\pi$
$769$	17.6060	0.634887	0.317444	+	0.948277i	$0.397176\pi$
$770$	0	0
$771$	− 24.9783i	− 0.899570i
$772$	0	0
$773$	− 34.9360i	− 1.25656i	−0.777988	−	0.628280i	$-0.783759\pi$
$773$	− 34.9360i	− 1.25656i	0.777988	−	0.628280i	$-0.216241\pi$
$774$	0	0
$775$	29.0024	1.04180
$776$	0	0
$777$	10.9783	0.393843
$778$	0	0
$779$	− 5.09509i	− 0.182551i
$780$	0	0
$781$	28.7075i	1.02723i
$782$	0	0
$783$	−5.04868	−0.180425
$784$	0	0
$785$	16.8614	0.601809
$786$	0	0
$787$	− 18.7446i	− 0.668171i	−0.942543	−	0.334086i	$-0.891572\pi$
$787$	− 18.7446i	− 0.668171i	0.942543	−	0.334086i	$-0.108428\pi$
$788$	0	0
$789$	− 9.50744i	− 0.338474i
$790$	0	0
$791$	−46.3229	−1.64705
$792$	0	0
$793$	−31.2119	−1.10837
$794$	0	0
$795$	1.48913i	0.0528138i
$796$	0	0
$797$	− 8.21782i	− 0.291090i	−0.989352	−	0.145545i	$-0.953506\pi$
$797$	− 8.21782i	− 0.291090i	0.989352	−	0.145545i	$-0.0464936\pi$
$798$	0	0
$799$	−6.92820	−0.245102
$800$	0	0
$801$	15.4891	0.547281
$802$	0	0
$803$	56.4674i	1.99269i
$804$	0	0
$805$	16.0309i	0.565015i
$806$	0	0
$807$	−9.30506	−0.327554
$808$	0	0
$809$	−25.3723	−0.892042	−0.446021	−	0.895023i	$-0.647159\pi$
$809$	−25.3723	−0.892042	−0.446021	+	0.895023i	$0.647159\pi$
$810$	0	0
$811$	− 28.4674i	− 0.999625i	−0.866134	−	0.499812i	$-0.833402\pi$
$811$	− 28.4674i	− 0.999625i	0.866134	−	0.499812i	$-0.166598\pi$
$812$	0	0
$813$	− 21.5769i	− 0.756735i
$814$	0	0
$815$	−8.51278	−0.298190
$816$	0	0
$817$	3.83966	0.134333
$818$	0	0
$819$	− 20.2337i	− 0.707022i
$820$	0	0
$821$	32.2642i	1.12603i	0.826448	+	0.563013i	$0.190358\pi$
$821$	32.2642i	1.12603i	−0.826448	+	0.563013i	$0.809642\pi$
$822$	0	0
$823$	−41.6790	−1.45284	−0.726420	−	0.687251i	$-0.758817\pi$
$823$	−41.6790	−1.45284	−0.726420	+	0.687251i	$0.758817\pi$
$824$	0	0
$825$	−14.7446	−0.513340
$826$	0	0
$827$	− 6.11684i	− 0.212704i	−0.994329	−	0.106352i	$-0.966083\pi$
$827$	− 6.11684i	− 0.212704i	0.994329	−	0.106352i	$-0.0339170\pi$
$828$	0	0
$829$	20.1947i	0.701391i	0.936490	+	0.350696i	$0.114055\pi$
$829$	20.1947i	0.701391i	−0.936490	+	0.350696i	$0.885945\pi$
$830$	0	0
$831$	−23.9538	−0.830947
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	12.6277i	0.437000i
$836$	0	0
$837$	− 6.63325i	− 0.229279i
$838$	0	0
$839$	−42.0666	−1.45230	−0.726149	−	0.687537i	$-0.758692\pi$
$839$	−42.0666	−1.45230	−0.726149	+	0.687537i	$0.758692\pi$
$840$	0	0
$841$	3.51087	0.121065
$842$	0	0
$843$	− 7.48913i	− 0.257939i
$844$	0	0
$845$	− 16.7306i	− 0.575550i
$846$	0	0
$847$	1.28962	0.0443119
$848$	0	0
$849$	24.2337	0.831698
$850$	0	0
$851$	− 18.5109i	− 0.634545i
$852$	0	0
$853$	41.3841i	1.41696i	0.705729	+	0.708482i	$0.250620\pi$
$853$	41.3841i	1.41696i	−0.705729	+	0.708482i	$0.749380\pi$
$854$	0	0
$855$	0.497333	0.0170084
$856$	0	0
$857$	20.9783	0.716603	0.358302	−	0.933606i	$-0.383356\pi$
$857$	20.9783	0.716603	0.358302	+	0.933606i	$0.383356\pi$
$858$	0	0
$859$	− 30.9783i	− 1.05696i	−0.848944	−	0.528482i	$-0.822761\pi$
$859$	− 30.9783i	− 1.05696i	0.848944	−	0.528482i	$-0.177239\pi$
$860$	0	0
$861$	28.1176i	0.958244i
$862$	0	0
$863$	31.8766	1.08509	0.542547	−	0.840026i	$-0.317460\pi$
$863$	31.8766	1.08509	0.542547	+	0.840026i	$0.317460\pi$
$864$	0	0
$865$	−1.09509	−0.0372343
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	11.6819i	0.396282i
$870$	0	0
$871$	23.3639	0.791654
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	− 25.7228i	− 0.869590i
$876$	0	0
$877$	− 3.16915i	− 0.107015i	−0.998567	−	0.0535073i	$-0.982960\pi$
$877$	− 3.16915i	− 0.107015i	0.998567	−	0.0535073i	$-0.0170400\pi$
$878$	0	0
$879$	28.4125	0.958331
$880$	0	0
$881$	−0.978251	−0.0329581	−0.0164790	−	0.999864i	$-0.505246\pi$
$881$	−0.978251	−0.0329581	−0.0164790	+	0.999864i	$0.505246\pi$
$882$	0	0
$883$	− 34.3505i	− 1.15599i	−0.816041	−	0.577994i	$-0.803836\pi$
$883$	− 34.3505i	− 1.15599i	0.816041	−	0.577994i	$-0.196164\pi$
$884$	0	0
$885$	− 2.17448i	− 0.0730944i
$886$	0	0
$887$	55.3331	1.85790	0.928951	−	0.370203i	$-0.120712\pi$
$887$	55.3331	1.85790	0.928951	+	0.370203i	$0.120712\pi$
$888$	0	0
$889$	−21.2554	−0.712884
$890$	0	0
$891$	3.37228i	0.112976i
$892$	0	0
$893$	4.34896i	0.145532i
$894$	0	0
$895$	−4.16381	−0.139181
$896$	0	0
$897$	−34.1168	−1.13913
$898$	0	0
$899$	− 33.4891i	− 1.11692i
$900$	0	0
$901$	1.87953i	0.0626161i
$902$	0	0
$903$	−21.1894	−0.705138
$904$	0	0
$905$	10.5109	0.349393
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	1.87953i	0.0623400i
$910$	0	0
$911$	42.4713	1.40714	0.703569	−	0.710627i	$-0.251589\pi$
$911$	42.4713	1.40714	0.703569	+	0.710627i	$0.251589\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.23369i	0.139961i
$916$	0	0
$917$	− 67.9171i	− 2.24282i
$918$	0	0
$919$	8.12525	0.268027	0.134014	−	0.990979i	$-0.457213\pi$
$919$	8.12525	0.268027	0.134014	+	0.990979i	$0.457213\pi$
$920$	0	0
$921$	6.51087	0.214541
$922$	0	0
$923$	− 49.7228i	− 1.63665i
$924$	0	0
$925$	13.8564i	0.455596i
$926$	0	0
$927$	0.792287	0.0260221
$928$	0	0
$929$	5.13859	0.168592	0.0842959	−	0.996441i	$-0.473136\pi$
$929$	5.13859	0.168592	0.0842959	+	0.996441i	$0.473136\pi$
$930$	0	0
$931$	3.13859i	0.102863i
$932$	0	0
$933$	12.2718i	0.401762i
$934$	0	0
$935$	2.67181	0.0873777
$936$	0	0
$937$	8.51087	0.278038	0.139019	−	0.990290i	$-0.455605\pi$
$937$	8.51087	0.278038	0.139019	+	0.990290i	$0.455605\pi$
$938$	0	0
$939$	24.9783i	0.815134i
$940$	0	0
$941$	− 30.9918i	− 1.01030i	−0.863031	−	0.505151i	$-0.831437\pi$
$941$	− 30.9918i	− 1.01030i	0.863031	−	0.505151i	$-0.168563\pi$
$942$	0	0
$943$	47.4102	1.54389
$944$	0	0
$945$	−2.74456	−0.0892806
$946$	0	0
$947$	25.4891i	0.828285i	0.910212	+	0.414143i	$0.135919\pi$
$947$	25.4891i	0.828285i	−0.910212	+	0.414143i	$0.864081\pi$
$948$	0	0
$949$	− 97.8044i	− 3.17486i
$950$	0	0
$951$	−32.1716	−1.04323
$952$	0	0
$953$	−10.2337	−0.331502	−0.165751	−	0.986168i	$-0.553005\pi$
$953$	−10.2337	−0.331502	−0.165751	+	0.986168i	$0.553005\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	17.0256i	0.550358i
$958$	0	0
$959$	−49.3069	−1.59220
$960$	0	0
$961$	13.0000	0.419355
$962$	0	0
$963$	15.3723i	0.495365i
$964$	0	0
$965$	5.93354i	0.191007i
$966$	0	0
$967$	11.8843	0.382173	0.191087	−	0.981573i	$-0.438799\pi$
$967$	11.8843	0.382173	0.191087	+	0.981573i	$0.438799\pi$
$968$	0	0
$969$	0.627719	0.0201652
$970$	0	0
$971$	− 22.7446i	− 0.729908i	−0.931026	−	0.364954i	$-0.881085\pi$
$971$	− 22.7446i	− 0.729908i	0.931026	−	0.364954i	$-0.118915\pi$
$972$	0	0
$973$	32.8713i	1.05381i
$974$	0	0
$975$	25.5383	0.817881
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	− 52.2337i	− 1.66940i
$980$	0	0
$981$	6.92820i	0.221201i
$982$	0	0
$983$	31.9692	1.01966	0.509830	−	0.860275i	$-0.329708\pi$
$983$	31.9692	1.01966	0.509830	+	0.860275i	$0.329708\pi$
$984$	0	0
$985$	11.6060	0.369797
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	35.7283i	1.13609i
$990$	0	0
$991$	−17.9104	−0.568943	−0.284472	−	0.958684i	$-0.591818\pi$
$991$	−17.9104	−0.568943	−0.284472	+	0.958684i	$0.591818\pi$
$992$	0	0
$993$	−4.86141	−0.154272
$994$	0	0
$995$	2.74456i	0.0870085i
$996$	0	0
$997$	− 24.3585i	− 0.771442i	−0.922615	−	0.385721i	$-0.873953\pi$
$997$	− 24.3585i	− 0.771442i	0.922615	−	0.385721i	$-0.126047\pi$
$998$	0	0
$999$	3.16915	0.100267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.f.h.1633.7	yes	8
4.3	odd	2	inner	3264.2.f.h.1633.3	yes	8
8.3	odd	2	inner	3264.2.f.h.1633.6	yes	8
8.5	even	2	inner	3264.2.f.h.1633.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3264.2.f.h.1633.2	✓	8	8.5	even	2	inner
3264.2.f.h.1633.3	yes	8	4.3	odd	2	inner
3264.2.f.h.1633.6	yes	8	8.3	odd	2	inner
3264.2.f.h.1633.7	yes	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3264.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.792287i q^{5} -3.46410 q^{7} -1.00000 q^{9} +3.37228i q^{11} -5.84096i q^{13} -0.792287 q^{15} -1.00000 q^{17} +0.627719i q^{19} -3.46410i q^{21} -5.84096 q^{23} +4.37228 q^{25} -1.00000i q^{27} -5.04868i q^{29} +6.63325 q^{31} -3.37228 q^{33} -2.74456i q^{35} +3.16915i q^{37} +5.84096 q^{39} -8.11684 q^{41} -6.11684i q^{43} -0.792287i q^{45} +6.92820 q^{47} +5.00000 q^{49} -1.00000i q^{51} -1.87953i q^{53} -2.67181 q^{55} -0.627719 q^{57} +2.74456i q^{59} -5.34363i q^{61} +3.46410 q^{63} +4.62772 q^{65} +4.00000i q^{67} -5.84096i q^{69} +8.51278 q^{71} +16.7446 q^{73} +4.37228i q^{75} -11.6819i q^{77} +3.46410 q^{79} +1.00000 q^{81} -0.792287i q^{85} +5.04868 q^{87} -15.4891 q^{89} +20.2337i q^{91} +6.63325i q^{93} -0.497333 q^{95} +14.0000 q^{97} -3.37228i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{17} + 12 q^{25} - 4 q^{33} + 4 q^{41} + 40 q^{49} - 28 q^{57} + 60 q^{65} + 88 q^{73} + 8 q^{81} - 32 q^{89} + 112 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^17 + 12 * q^25 - 4 * q^33 + 4 * q^41 + 40 * q^49 - 28 * q^57 + 60 * q^65 + 88 * q^73 + 8 * q^81 - 32 * q^89 + 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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