LMFDB - Embedded newform 3264.2.f.h.1633.8

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			1633.8
Root			$-1.26217 - 1.18614i$ of defining polynomial
Character	$\chi$	$=$	3264.1633
Dual form			3264.2.f.h.1633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.52434i q^{5} +3.46410 q^{7} -1.00000 q^{9} -2.37228i q^{11} -4.10891i q^{13} -2.52434 q^{15} -1.00000 q^{17} +6.37228i q^{19} +3.46410i q^{21} -4.10891 q^{23} -1.37228 q^{25} -1.00000i q^{27} -1.58457i q^{29} +6.63325 q^{31} +2.37228 q^{33} +8.74456i q^{35} +10.0974i q^{37} +4.10891 q^{39} +9.11684 q^{41} +11.1168i q^{43} -2.52434i q^{45} -6.92820 q^{47} +5.00000 q^{49} -1.00000i q^{51} +8.51278i q^{53} +5.98844 q^{55} -6.37228 q^{57} -8.74456i q^{59} +11.9769i q^{61} -3.46410 q^{63} +10.3723 q^{65} +4.00000i q^{67} -4.10891i q^{69} -1.87953 q^{71} +5.25544 q^{73} -1.37228i q^{75} -8.21782i q^{77} -3.46410 q^{79} +1.00000 q^{81} -2.52434i q^{85} +1.58457 q^{87} +7.48913 q^{89} -14.2337i q^{91} +6.63325i q^{93} -16.0858 q^{95} +14.0000 q^{97} +2.37228i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.52434i	1.12892i	0.825461	+	0.564459i	$0.190915\pi$
$5$	2.52434i	1.12892i	−0.825461	+	0.564459i	$0.809085\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.37228i	− 0.715270i	−0.933862	−	0.357635i	$-0.883583\pi$
$11$	− 2.37228i	− 0.715270i	0.933862	−	0.357635i	$-0.116417\pi$
$12$	0	0
$13$	− 4.10891i	− 1.13961i	−0.821781	−	0.569804i	$-0.807019\pi$
$13$	− 4.10891i	− 1.13961i	0.821781	−	0.569804i	$-0.192981\pi$
$14$	0	0
$15$	−2.52434	−0.651781
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.37228i	1.46190i	0.682430	+	0.730951i	$0.260923\pi$
$19$	6.37228i	1.46190i	−0.682430	+	0.730951i	$0.739077\pi$
$20$	0	0
$21$	3.46410i	0.755929i
$22$	0	0
$23$	−4.10891	−0.856767	−0.428384	−	0.903597i	$-0.640917\pi$
$23$	−4.10891	−0.856767	−0.428384	+	0.903597i	$0.640917\pi$
$24$	0	0
$25$	−1.37228	−0.274456
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 1.58457i	− 0.294248i	−0.989118	−	0.147124i	$-0.952998\pi$
$29$	− 1.58457i	− 0.294248i	0.989118	−	0.147124i	$-0.0470016\pi$
$30$	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
$33$	2.37228	0.412961
$34$	0	0
$35$	8.74456i	1.47810i
$36$	0	0
$37$	10.0974i	1.65999i	0.557768	+	0.829997i	$0.311658\pi$
$37$	10.0974i	1.65999i	−0.557768	+	0.829997i	$0.688342\pi$
$38$	0	0
$39$	4.10891	0.657952
$40$	0	0
$41$	9.11684	1.42381	0.711906	−	0.702275i	$-0.247832\pi$
$41$	9.11684	1.42381	0.711906	+	0.702275i	$0.247832\pi$
$42$	0	0
$43$	11.1168i	1.69530i	0.530554	+	0.847651i	$0.321984\pi$
$43$	11.1168i	1.69530i	−0.530554	+	0.847651i	$0.678016\pi$
$44$	0	0
$45$	− 2.52434i	− 0.376306i
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	− 1.00000i	− 0.140028i
$52$	0	0
$53$	8.51278i	1.16932i	0.811278	+	0.584660i	$0.198772\pi$
$53$	8.51278i	1.16932i	−0.811278	+	0.584660i	$0.801228\pi$
$54$	0	0
$55$	5.98844	0.807481
$56$	0	0
$57$	−6.37228	−0.844029
$58$	0	0
$59$	− 8.74456i	− 1.13845i	−0.822183	−	0.569223i	$-0.807244\pi$
$59$	− 8.74456i	− 1.13845i	0.822183	−	0.569223i	$-0.192756\pi$
$60$	0	0
$61$	11.9769i	1.53348i	0.641956	+	0.766741i	$0.278123\pi$
$61$	11.9769i	1.53348i	−0.641956	+	0.766741i	$0.721877\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	10.3723	1.28652
$66$	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
$69$	− 4.10891i	− 0.494655i
$70$	0	0
$71$	−1.87953	−0.223059	−0.111529	−	0.993761i	$-0.535575\pi$
$71$	−1.87953	−0.223059	−0.111529	+	0.993761i	$0.535575\pi$
$72$	0	0
$73$	5.25544	0.615102	0.307551	−	0.951532i	$-0.400490\pi$
$73$	5.25544	0.615102	0.307551	+	0.951532i	$0.400490\pi$
$74$	0	0
$75$	− 1.37228i	− 0.158457i
$76$	0	0
$77$	− 8.21782i	− 0.936508i
$78$	0	0
$79$	−3.46410	−0.389742	−0.194871	−	0.980829i	$-0.562429\pi$
$79$	−3.46410	−0.389742	−0.194871	+	0.980829i	$0.562429\pi$
$80$	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
$85$	− 2.52434i	− 0.273803i
$86$	0	0
$87$	1.58457	0.169884
$88$	0	0
$89$	7.48913	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$89$	7.48913	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$90$	0	0
$91$	− 14.2337i	− 1.49210i
$92$	0	0
$93$	6.63325i	0.687836i
$94$	0	0
$95$	−16.0858	−1.65037
$96$	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
$99$	2.37228i	0.238423i
$100$	0	0
$101$	8.51278i	0.847053i	0.905884	+	0.423526i	$0.139208\pi$
$101$	8.51278i	0.847053i	−0.905884	+	0.423526i	$0.860792\pi$
$102$	0	0
$103$	−2.52434	−0.248730	−0.124365	−	0.992237i	$-0.539689\pi$
$103$	−2.52434	−0.248730	−0.124365	+	0.992237i	$0.539689\pi$
$104$	0	0
$105$	−8.74456	−0.853382
$106$	0	0
$107$	− 9.62772i	− 0.930747i	−0.885114	−	0.465373i	$-0.845920\pi$
$107$	− 9.62772i	− 0.930747i	0.885114	−	0.465373i	$-0.154080\pi$
$108$	0	0
$109$	6.92820i	0.663602i	0.943349	+	0.331801i	$0.107656\pi$
$109$	6.92820i	0.663602i	−0.943349	+	0.331801i	$0.892344\pi$
$110$	0	0
$111$	−10.0974	−0.958398
$112$	0	0
$113$	7.62772	0.717555	0.358778	−	0.933423i	$-0.383194\pi$
$113$	7.62772	0.717555	0.358778	+	0.933423i	$0.383194\pi$
$114$	0	0
$115$	− 10.3723i	− 0.967220i
$116$	0	0
$117$	4.10891i	0.379869i
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	5.37228	0.488389
$122$	0	0
$123$	9.11684i	0.822038i
$124$	0	0
$125$	9.15759i	0.819080i
$126$	0	0
$127$	−9.45254	−0.838777	−0.419389	−	0.907807i	$-0.637756\pi$
$127$	−9.45254	−0.838777	−0.419389	+	0.907807i	$0.637756\pi$
$128$	0	0
$129$	−11.1168	−0.978784
$130$	0	0
$131$	− 20.6060i	− 1.80035i	−0.435526	−	0.900176i	$-0.643438\pi$
$131$	− 20.6060i	− 1.80035i	0.435526	−	0.900176i	$-0.356562\pi$
$132$	0	0
$133$	22.0742i	1.91408i
$134$	0	0
$135$	2.52434	0.217260
$136$	0	0
$137$	−20.2337	−1.72868	−0.864340	−	0.502907i	$-0.832264\pi$
$137$	−20.2337	−1.72868	−0.864340	+	0.502907i	$0.832264\pi$
$138$	0	0
$139$	13.4891i	1.14413i	0.820207	+	0.572066i	$0.193858\pi$
$139$	13.4891i	1.14413i	−0.820207	+	0.572066i	$0.806142\pi$
$140$	0	0
$141$	− 6.92820i	− 0.583460i
$142$	0	0
$143$	−9.74749	−0.815126
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	5.00000i	0.412393i
$148$	0	0
$149$	− 8.51278i	− 0.697394i	−0.937236	−	0.348697i	$-0.886624\pi$
$149$	− 8.51278i	− 0.697394i	0.937236	−	0.348697i	$-0.113376\pi$
$150$	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
$155$	16.7446i	1.34496i
$156$	0	0
$157$	4.69882i	0.375007i	0.982264	+	0.187503i	$0.0600395\pi$
$157$	4.69882i	0.375007i	−0.982264	+	0.187503i	$0.939960\pi$
$158$	0	0
$159$	−8.51278	−0.675107
$160$	0	0
$161$	−14.2337	−1.12177
$162$	0	0
$163$	− 0.744563i	− 0.0583186i	−0.999575	−	0.0291593i	$-0.990717\pi$
$163$	− 0.744563i	− 0.0583186i	0.999575	−	0.0291593i	$-0.00928302\pi$
$164$	0	0
$165$	5.98844i	0.466199i
$166$	0	0
$167$	7.27806	0.563193	0.281597	−	0.959533i	$-0.409136\pi$
$167$	7.27806	0.563193	0.281597	+	0.959533i	$0.409136\pi$
$168$	0	0
$169$	−3.88316	−0.298704
$170$	0	0
$171$	− 6.37228i	− 0.487301i
$172$	0	0
$173$	− 24.5986i	− 1.87019i	−0.354391	−	0.935097i	$-0.615312\pi$
$173$	− 24.5986i	− 1.87019i	0.354391	−	0.935097i	$-0.384688\pi$
$174$	0	0
$175$	−4.75372	−0.359348
$176$	0	0
$177$	8.74456	0.657282
$178$	0	0
$179$	16.7446i	1.25155i	0.780005	+	0.625774i	$0.215217\pi$
$179$	16.7446i	1.25155i	−0.780005	+	0.625774i	$0.784783\pi$
$180$	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
$183$	−11.9769	−0.885356
$184$	0	0
$185$	−25.4891	−1.87400
$186$	0	0
$187$	2.37228i	0.173478i
$188$	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
$193$	−15.4891	−1.11493	−0.557466	−	0.830200i	$-0.688226\pi$
$193$	−15.4891	−1.11493	−0.557466	+	0.830200i	$0.688226\pi$
$194$	0	0
$195$	10.3723i	0.742774i
$196$	0	0
$197$	11.3321i	0.807376i	0.914897	+	0.403688i	$0.132272\pi$
$197$	11.3321i	0.807376i	−0.914897	+	0.403688i	$0.867728\pi$
$198$	0	0
$199$	−3.46410	−0.245564	−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410	−0.245564	−0.122782	+	0.992434i	$0.539182\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	− 5.48913i	− 0.385261i
$204$	0	0
$205$	23.0140i	1.60737i
$206$	0	0
$207$	4.10891	0.285589
$208$	0	0
$209$	15.1168	1.04565
$210$	0	0
$211$	− 16.7446i	− 1.15274i	−0.817188	−	0.576372i	$-0.804468\pi$
$211$	− 16.7446i	− 1.15274i	0.817188	−	0.576372i	$-0.195532\pi$
$212$	0	0
$213$	− 1.87953i	− 0.128783i
$214$	0	0
$215$	−28.0627	−1.91386
$216$	0	0
$217$	22.9783	1.55987
$218$	0	0
$219$	5.25544i	0.355130i
$220$	0	0
$221$	4.10891i	0.276395i
$222$	0	0
$223$	8.86263	0.593486	0.296743	−	0.954957i	$-0.404100\pi$
$223$	8.86263	0.593486	0.296743	+	0.954957i	$0.404100\pi$
$224$	0	0
$225$	1.37228	0.0914854
$226$	0	0
$227$	26.3723i	1.75039i	0.483770	+	0.875195i	$0.339267\pi$
$227$	26.3723i	1.75039i	−0.483770	+	0.875195i	$0.660733\pi$
$228$	0	0
$229$	3.75906i	0.248405i	0.992257	+	0.124203i	$0.0396373\pi$
$229$	3.75906i	0.248405i	−0.992257	+	0.124203i	$0.960363\pi$
$230$	0	0
$231$	8.21782	0.540693
$232$	0	0
$233$	8.37228	0.548486	0.274243	−	0.961660i	$-0.411573\pi$
$233$	8.37228	0.548486	0.274243	+	0.961660i	$0.411573\pi$
$234$	0	0
$235$	− 17.4891i	− 1.14086i
$236$	0	0
$237$	− 3.46410i	− 0.225018i
$238$	0	0
$239$	22.0742	1.42786	0.713932	−	0.700215i	$-0.246913\pi$
$239$	22.0742	1.42786	0.713932	+	0.700215i	$0.246913\pi$
$240$	0	0
$241$	11.4891	0.740080	0.370040	−	0.929016i	$-0.379344\pi$
$241$	11.4891	0.740080	0.370040	+	0.929016i	$0.379344\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	12.6217i	0.806370i
$246$	0	0
$247$	26.1831	1.66599
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.9783i	1.19790i	0.800788	+	0.598948i	$0.204415\pi$
$251$	18.9783i	1.19790i	−0.800788	+	0.598948i	$0.795585\pi$
$252$	0	0
$253$	9.74749i	0.612820i
$254$	0	0
$255$	2.52434	0.158080
$256$	0	0
$257$	20.9783	1.30859	0.654294	−	0.756241i	$-0.272966\pi$
$257$	20.9783	1.30859	0.654294	+	0.756241i	$0.272966\pi$
$258$	0	0
$259$	34.9783i	2.17344i
$260$	0	0
$261$	1.58457i	0.0980827i
$262$	0	0
$263$	−30.2921	−1.86789	−0.933944	−	0.357419i	$-0.883657\pi$
$263$	−30.2921	−1.86789	−0.933944	+	0.357419i	$0.883657\pi$
$264$	0	0
$265$	−21.4891	−1.32007
$266$	0	0
$267$	7.48913i	0.458327i
$268$	0	0
$269$	0.644810i	0.0393148i	0.999807	+	0.0196574i	$0.00625754\pi$
$269$	0.644810i	0.0393148i	−0.999807	+	0.0196574i	$0.993742\pi$
$270$	0	0
$271$	18.2603	1.10923	0.554616	−	0.832106i	$-0.312865\pi$
$271$	18.2603	1.10923	0.554616	+	0.832106i	$0.312865\pi$
$272$	0	0
$273$	14.2337	0.861462
$274$	0	0
$275$	3.25544i	0.196310i
$276$	0	0
$277$	− 10.6873i	− 0.642135i	−0.947056	−	0.321068i	$-0.895958\pi$
$277$	− 10.6873i	− 0.642135i	0.947056	−	0.321068i	$-0.104042\pi$
$278$	0	0
$279$	−6.63325	−0.397122
$280$	0	0
$281$	15.4891	0.924004	0.462002	−	0.886879i	$-0.347131\pi$
$281$	15.4891	0.924004	0.462002	+	0.886879i	$0.347131\pi$
$282$	0	0
$283$	10.2337i	0.608330i	0.952619	+	0.304165i	$0.0983774\pi$
$283$	10.2337i	0.608330i	−0.952619	+	0.304165i	$0.901623\pi$
$284$	0	0
$285$	− 16.0858i	− 0.952840i
$286$	0	0
$287$	31.5817	1.86421
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 18.0202i	− 1.05275i	−0.850252	−	0.526376i	$-0.823550\pi$
$293$	− 18.0202i	− 1.05275i	0.850252	−	0.526376i	$-0.176450\pi$
$294$	0	0
$295$	22.0742	1.28521
$296$	0	0
$297$	−2.37228	−0.137654
$298$	0	0
$299$	16.8832i	0.976378i
$300$	0	0
$301$	38.5099i	2.21967i
$302$	0	0
$303$	−8.51278	−0.489046
$304$	0	0
$305$	−30.2337	−1.73118
$306$	0	0
$307$	− 29.4891i	− 1.68303i	−0.540231	−	0.841517i	$-0.681663\pi$
$307$	− 29.4891i	− 1.68303i	0.540231	−	0.841517i	$-0.318337\pi$
$308$	0	0
$309$	− 2.52434i	− 0.143605i
$310$	0	0
$311$	−18.9051	−1.07201	−0.536004	−	0.844215i	$-0.680067\pi$
$311$	−18.9051	−1.07201	−0.536004	+	0.844215i	$0.680067\pi$
$312$	0	0
$313$	−20.9783	−1.18576	−0.592880	−	0.805291i	$-0.702009\pi$
$313$	−20.9783	−1.18576	−0.592880	+	0.805291i	$0.702009\pi$
$314$	0	0
$315$	− 8.74456i	− 0.492700i
$316$	0	0
$317$	0.994667i	0.0558660i	0.999610	+	0.0279330i	$0.00889251\pi$
$317$	0.994667i	0.0558660i	−0.999610	+	0.0279330i	$0.991107\pi$
$318$	0	0
$319$	−3.75906	−0.210467
$320$	0	0
$321$	9.62772	0.537367
$322$	0	0
$323$	− 6.37228i	− 0.354563i
$324$	0	0
$325$	5.63858i	0.312772i
$326$	0	0
$327$	−6.92820	−0.383131
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	− 23.8614i	− 1.31154i	−0.754960	−	0.655771i	$-0.772344\pi$
$331$	− 23.8614i	− 1.31154i	0.754960	−	0.655771i	$-0.227656\pi$
$332$	0	0
$333$	− 10.0974i	− 0.553331i
$334$	0	0
$335$	−10.0974	−0.551677
$336$	0	0
$337$	18.7446	1.02108	0.510541	−	0.859854i	$-0.329445\pi$
$337$	18.7446	1.02108	0.510541	+	0.859854i	$0.329445\pi$
$338$	0	0
$339$	7.62772i	0.414281i
$340$	0	0
$341$	− 15.7359i	− 0.852149i
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	10.3723	0.558425
$346$	0	0
$347$	− 6.51087i	− 0.349522i	−0.984611	−	0.174761i	$-0.944085\pi$
$347$	− 6.51087i	− 0.349522i	0.984611	−	0.174761i	$-0.0559153\pi$
$348$	0	0
$349$	− 3.51900i	− 0.188368i	−0.995555	−	0.0941840i	$-0.969976\pi$
$349$	− 3.51900i	− 0.188368i	0.995555	−	0.0941840i	$-0.0300242\pi$
$350$	0	0
$351$	−4.10891	−0.219317
$352$	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
$355$	− 4.74456i	− 0.251815i
$356$	0	0
$357$	− 3.46410i	− 0.183340i
$358$	0	0
$359$	−10.6873	−0.564052	−0.282026	−	0.959407i	$-0.591006\pi$
$359$	−10.6873	−0.564052	−0.282026	+	0.959407i	$0.591006\pi$
$360$	0	0
$361$	−21.6060	−1.13716
$362$	0	0
$363$	5.37228i	0.281972i
$364$	0	0
$365$	13.2665i	0.694400i
$366$	0	0
$367$	−18.0202	−0.940648	−0.470324	−	0.882494i	$-0.655863\pi$
$367$	−18.0202	−0.940648	−0.470324	+	0.882494i	$0.655863\pi$
$368$	0	0
$369$	−9.11684	−0.474604
$370$	0	0
$371$	29.4891i	1.53100i
$372$	0	0
$373$	− 3.16915i	− 0.164092i	−0.996629	−	0.0820461i	$-0.973855\pi$
$373$	− 3.16915i	− 0.164092i	0.996629	−	0.0820461i	$-0.0261455\pi$
$374$	0	0
$375$	−9.15759	−0.472896
$376$	0	0
$377$	−6.51087	−0.335327
$378$	0	0
$379$	− 24.7446i	− 1.27104i	−0.772083	−	0.635521i	$-0.780785\pi$
$379$	− 24.7446i	− 1.27104i	0.772083	−	0.635521i	$-0.219215\pi$
$380$	0	0
$381$	− 9.45254i	− 0.484268i
$382$	0	0
$383$	−4.45877	−0.227832	−0.113916	−	0.993490i	$-0.536340\pi$
$383$	−4.45877	−0.227832	−0.113916	+	0.993490i	$0.536340\pi$
$384$	0	0
$385$	20.7446	1.05724
$386$	0	0
$387$	− 11.1168i	− 0.565101i
$388$	0	0
$389$	1.58457i	0.0803411i	0.999193	+	0.0401705i	$0.0127901\pi$
$389$	1.58457i	0.0803411i	−0.999193	+	0.0401705i	$0.987210\pi$
$390$	0	0
$391$	4.10891	0.207797
$392$	0	0
$393$	20.6060	1.03943
$394$	0	0
$395$	− 8.74456i	− 0.439987i
$396$	0	0
$397$	− 11.9769i	− 0.601102i	−0.953766	−	0.300551i	$-0.902829\pi$
$397$	− 11.9769i	− 0.601102i	0.953766	−	0.300551i	$-0.0971706\pi$
$398$	0	0
$399$	−22.0742	−1.10509
$400$	0	0
$401$	3.62772	0.181160	0.0905798	−	0.995889i	$-0.471128\pi$
$401$	3.62772	0.181160	0.0905798	+	0.995889i	$0.471128\pi$
$402$	0	0
$403$	− 27.2554i	− 1.35769i
$404$	0	0
$405$	2.52434i	0.125435i
$406$	0	0
$407$	23.9538	1.18734
$408$	0	0
$409$	−6.88316	−0.340350	−0.170175	−	0.985414i	$-0.554433\pi$
$409$	−6.88316	−0.340350	−0.170175	+	0.985414i	$0.554433\pi$
$410$	0	0
$411$	− 20.2337i	− 0.998054i
$412$	0	0
$413$	− 30.2921i	− 1.49057i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.4891	−0.660565
$418$	0	0
$419$	25.4891i	1.24523i	0.782530	+	0.622613i	$0.213929\pi$
$419$	25.4891i	1.24523i	−0.782530	+	0.622613i	$0.786071\pi$
$420$	0	0
$421$	− 9.74749i	− 0.475064i	−0.971380	−	0.237532i	$-0.923662\pi$
$421$	− 9.74749i	− 0.475064i	0.971380	−	0.237532i	$-0.0763384\pi$
$422$	0	0
$423$	6.92820	0.336861
$424$	0	0
$425$	1.37228	0.0665654
$426$	0	0
$427$	41.4891i	2.00780i
$428$	0	0
$429$	− 9.74749i	− 0.470613i
$430$	0	0
$431$	22.0742	1.06328	0.531639	−	0.846971i	$-0.321576\pi$
$431$	22.0742	1.06328	0.531639	+	0.846971i	$0.321576\pi$
$432$	0	0
$433$	19.6277	0.943248	0.471624	−	0.881800i	$-0.343668\pi$
$433$	19.6277	0.943248	0.471624	+	0.881800i	$0.343668\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	− 26.1831i	− 1.25251i
$438$	0	0
$439$	−1.58457	−0.0756276	−0.0378138	−	0.999285i	$-0.512039\pi$
$439$	−1.58457	−0.0756276	−0.0378138	+	0.999285i	$0.512039\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	17.4891i	0.830933i	0.909608	+	0.415467i	$0.136382\pi$
$443$	17.4891i	0.830933i	−0.909608	+	0.415467i	$0.863618\pi$
$444$	0	0
$445$	18.9051i	0.896187i
$446$	0	0
$447$	8.51278	0.402641
$448$	0	0
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	0	0
$451$	− 21.6277i	− 1.01841i
$452$	0	0
$453$	− 19.8997i	− 0.934972i
$454$	0	0
$455$	35.9306	1.68445
$456$	0	0
$457$	34.6060	1.61880	0.809399	−	0.587258i	$-0.199793\pi$
$457$	34.6060	1.61880	0.809399	+	0.587258i	$0.199793\pi$
$458$	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
$463$	20.4897	0.952235	0.476118	−	0.879382i	$-0.342044\pi$
$463$	20.4897	0.952235	0.476118	+	0.879382i	$0.342044\pi$
$464$	0	0
$465$	−16.7446	−0.776510
$466$	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
$471$	−4.69882	−0.216510
$472$	0	0
$473$	26.3723	1.21260
$474$	0	0
$475$	− 8.74456i	− 0.401228i
$476$	0	0
$477$	− 8.51278i	− 0.389773i
$478$	0	0
$479$	−0.349857	−0.0159854	−0.00799268	−	0.999968i	$-0.502544\pi$
$479$	−0.349857	−0.0159854	−0.00799268	+	0.999968i	$0.502544\pi$
$480$	0	0
$481$	41.4891	1.89174
$482$	0	0
$483$	− 14.2337i	− 0.647655i
$484$	0	0
$485$	35.3407i	1.60474i
$486$	0	0
$487$	3.46410	0.156973	0.0784867	−	0.996915i	$-0.474991\pi$
$487$	3.46410	0.156973	0.0784867	+	0.996915i	$0.474991\pi$
$488$	0	0
$489$	0.744563	0.0336703
$490$	0	0
$491$	− 18.5109i	− 0.835384i	−0.908589	−	0.417692i	$-0.862839\pi$
$491$	− 18.5109i	− 0.835384i	0.908589	−	0.417692i	$-0.137161\pi$
$492$	0	0
$493$	1.58457i	0.0713656i
$494$	0	0
$495$	−5.98844	−0.269160
$496$	0	0
$497$	−6.51087	−0.292053
$498$	0	0
$499$	− 10.5109i	− 0.470531i	−0.971931	−	0.235266i	$-0.924404\pi$
$499$	− 10.5109i	− 0.470531i	0.971931	−	0.235266i	$-0.0755960\pi$
$500$	0	0
$501$	7.27806i	0.325160i
$502$	0	0
$503$	−41.9191	−1.86908	−0.934540	−	0.355859i	$-0.884188\pi$
$503$	−41.9191	−1.86908	−0.934540	+	0.355859i	$0.884188\pi$
$504$	0	0
$505$	−21.4891	−0.956254
$506$	0	0
$507$	− 3.88316i	− 0.172457i
$508$	0	0
$509$	41.9740i	1.86046i	0.366972	+	0.930232i	$0.380394\pi$
$509$	41.9740i	1.86046i	−0.366972	+	0.930232i	$0.619606\pi$
$510$	0	0
$511$	18.2054	0.805358
$512$	0	0
$513$	6.37228	0.281343
$514$	0	0
$515$	− 6.37228i	− 0.280796i
$516$	0	0
$517$	16.4356i	0.722839i
$518$	0	0
$519$	24.5986	1.07976
$520$	0	0
$521$	−30.6060	−1.34087	−0.670436	−	0.741967i	$-0.733893\pi$
$521$	−30.6060	−1.34087	−0.670436	+	0.741967i	$0.733893\pi$
$522$	0	0
$523$	14.9783i	0.654953i	0.944859	+	0.327477i	$0.106198\pi$
$523$	14.9783i	0.654953i	−0.944859	+	0.327477i	$0.893802\pi$
$524$	0	0
$525$	− 4.75372i	− 0.207469i
$526$	0	0
$527$	−6.63325	−0.288949
$528$	0	0
$529$	−6.11684	−0.265950
$530$	0	0
$531$	8.74456i	0.379482i
$532$	0	0
$533$	− 37.4603i	− 1.62259i
$534$	0	0
$535$	24.3036	1.05074
$536$	0	0
$537$	−16.7446	−0.722581
$538$	0	0
$539$	− 11.8614i	− 0.510907i
$540$	0	0
$541$	− 19.6048i	− 0.842876i	−0.906857	−	0.421438i	$-0.861526\pi$
$541$	− 19.6048i	− 0.842876i	0.906857	−	0.421438i	$-0.138474\pi$
$542$	0	0
$543$	13.2665	0.569320
$544$	0	0
$545$	−17.4891	−0.749152
$546$	0	0
$547$	0.744563i	0.0318352i	0.999873	+	0.0159176i	$0.00506694\pi$
$547$	0.744563i	0.0318352i	−0.999873	+	0.0159176i	$0.994933\pi$
$548$	0	0
$549$	− 11.9769i	− 0.511161i
$550$	0	0
$551$	10.0974	0.430162
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	− 25.4891i	− 1.08195i
$556$	0	0
$557$	− 21.0796i	− 0.893170i	−0.894741	−	0.446585i	$-0.852640\pi$
$557$	− 21.0796i	− 0.893170i	0.894741	−	0.446585i	$-0.147360\pi$
$558$	0	0
$559$	45.6781	1.93198
$560$	0	0
$561$	−2.37228	−0.100158
$562$	0	0
$563$	− 3.25544i	− 0.137200i	−0.997644	−	0.0686002i	$-0.978147\pi$
$563$	− 3.25544i	− 0.137200i	0.997644	−	0.0686002i	$-0.0218533\pi$
$564$	0	0
$565$	19.2549i	0.810061i
$566$	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
$571$	− 10.2337i	− 0.428267i	−0.976804	−	0.214133i	$-0.931307\pi$
$571$	− 10.2337i	− 0.428267i	0.976804	−	0.214133i	$-0.0686927\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.63858	0.235145
$576$	0	0
$577$	38.6060	1.60719	0.803594	−	0.595178i	$-0.202919\pi$
$577$	38.6060	1.60719	0.803594	+	0.595178i	$0.202919\pi$
$578$	0	0
$579$	− 15.4891i	− 0.643706i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20.1947	0.836379
$584$	0	0
$585$	−10.3723	−0.428841
$586$	0	0
$587$	− 21.4891i	− 0.886951i	−0.896286	−	0.443476i	$-0.853745\pi$
$587$	− 21.4891i	− 0.886951i	0.896286	−	0.443476i	$-0.146255\pi$
$588$	0	0
$589$	42.2689i	1.74166i
$590$	0	0
$591$	−11.3321	−0.466139
$592$	0	0
$593$	−26.7446	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$593$	−26.7446	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$594$	0	0
$595$	− 8.74456i	− 0.358492i
$596$	0	0
$597$	− 3.46410i	− 0.141776i
$598$	0	0
$599$	0.589907	0.0241030	0.0120515	−	0.999927i	$-0.496164\pi$
$599$	0.589907	0.0241030	0.0120515	+	0.999927i	$0.496164\pi$
$600$	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
$605$	13.5615i	0.551351i
$606$	0	0
$607$	−24.9484	−1.01263	−0.506313	−	0.862350i	$-0.668992\pi$
$607$	−24.9484	−1.01263	−0.506313	+	0.862350i	$0.668992\pi$
$608$	0	0
$609$	5.48913	0.222431
$610$	0	0
$611$	28.4674i	1.15167i
$612$	0	0
$613$	− 31.2318i	− 1.26144i	−0.776010	−	0.630721i	$-0.782759\pi$
$613$	− 31.2318i	− 1.26144i	0.776010	−	0.630721i	$-0.217241\pi$
$614$	0	0
$615$	−23.0140	−0.928014
$616$	0	0
$617$	−24.5109	−0.986771	−0.493385	−	0.869811i	$-0.664241\pi$
$617$	−24.5109	−0.986771	−0.493385	+	0.869811i	$0.664241\pi$
$618$	0	0
$619$	5.76631i	0.231768i	0.993263	+	0.115884i	$0.0369700\pi$
$619$	5.76631i	0.231768i	−0.993263	+	0.115884i	$0.963030\pi$
$620$	0	0
$621$	4.10891i	0.164885i
$622$	0	0
$623$	25.9431	1.03939
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	15.1168i	0.603709i
$628$	0	0
$629$	− 10.0974i	− 0.402608i
$630$	0	0
$631$	6.28339	0.250138	0.125069	−	0.992148i	$-0.460085\pi$
$631$	6.28339	0.250138	0.125069	+	0.992148i	$0.460085\pi$
$632$	0	0
$633$	16.7446	0.665537
$634$	0	0
$635$	− 23.8614i	− 0.946911i
$636$	0	0
$637$	− 20.5446i	− 0.814005i
$638$	0	0
$639$	1.87953	0.0743530
$640$	0	0
$641$	29.1168	1.15005	0.575023	−	0.818137i	$-0.304993\pi$
$641$	29.1168	1.15005	0.575023	+	0.818137i	$0.304993\pi$
$642$	0	0
$643$	− 16.7446i	− 0.660341i	−0.943921	−	0.330171i	$-0.892894\pi$
$643$	− 16.7446i	− 0.660341i	0.943921	−	0.330171i	$-0.107106\pi$
$644$	0	0
$645$	− 28.0627i	− 1.10497i
$646$	0	0
$647$	−19.6048	−0.770744	−0.385372	−	0.922761i	$-0.625927\pi$
$647$	−19.6048	−0.770744	−0.385372	+	0.922761i	$0.625927\pi$
$648$	0	0
$649$	−20.7446	−0.814295
$650$	0	0
$651$	22.9783i	0.900589i
$652$	0	0
$653$	39.0449i	1.52794i	0.645249	+	0.763972i	$0.276754\pi$
$653$	39.0449i	1.52794i	−0.645249	+	0.763972i	$0.723246\pi$
$654$	0	0
$655$	52.0164	2.03245
$656$	0	0
$657$	−5.25544	−0.205034
$658$	0	0
$659$	− 5.48913i	− 0.213826i	−0.994268	−	0.106913i	$-0.965903\pi$
$659$	− 5.48913i	− 0.213826i	0.994268	−	0.106913i	$-0.0340966\pi$
$660$	0	0
$661$	− 11.0371i	− 0.429294i	−0.976692	−	0.214647i	$-0.931140\pi$
$661$	− 11.0371i	− 0.429294i	0.976692	−	0.214647i	$-0.0688601\pi$
$662$	0	0
$663$	−4.10891	−0.159577
$664$	0	0
$665$	−55.7228	−2.16084
$666$	0	0
$667$	6.51087i	0.252102i
$668$	0	0
$669$	8.86263i	0.342649i
$670$	0	0
$671$	28.4125	1.09685
$672$	0	0
$673$	44.2337	1.70508	0.852542	−	0.522659i	$-0.175060\pi$
$673$	44.2337	1.70508	0.852542	+	0.522659i	$0.175060\pi$
$674$	0	0
$675$	1.37228i	0.0528191i
$676$	0	0
$677$	− 27.0680i	− 1.04031i	−0.854073	−	0.520154i	$-0.825875\pi$
$677$	− 27.0680i	− 1.04031i	0.854073	−	0.520154i	$-0.174125\pi$
$678$	0	0
$679$	48.4974	1.86116
$680$	0	0
$681$	−26.3723	−1.01059
$682$	0	0
$683$	− 30.8397i	− 1.18005i	−0.807386	−	0.590023i	$-0.799119\pi$
$683$	− 30.8397i	− 1.18005i	0.807386	−	0.590023i	$-0.200881\pi$
$684$	0	0
$685$	− 51.0767i	− 1.95154i
$686$	0	0
$687$	−3.75906	−0.143417
$688$	0	0
$689$	34.9783	1.33257
$690$	0	0
$691$	− 38.9783i	− 1.48280i	−0.671062	−	0.741401i	$-0.734162\pi$
$691$	− 38.9783i	− 1.48280i	0.671062	−	0.741401i	$-0.265838\pi$
$692$	0	0
$693$	8.21782i	0.312169i
$694$	0	0
$695$	−34.0511	−1.29163
$696$	0	0
$697$	−9.11684	−0.345325
$698$	0	0
$699$	8.37228i	0.316669i
$700$	0	0
$701$	21.7793i	0.822592i	0.911502	+	0.411296i	$0.134924\pi$
$701$	21.7793i	0.822592i	−0.911502	+	0.411296i	$0.865076\pi$
$702$	0	0
$703$	−64.3432	−2.42675
$704$	0	0
$705$	17.4891	0.658679
$706$	0	0
$707$	29.4891i	1.10905i
$708$	0	0
$709$	14.4463i	0.542543i	0.962503	+	0.271271i	$0.0874441\pi$
$709$	14.4463i	0.542543i	−0.962503	+	0.271271i	$0.912556\pi$
$710$	0	0
$711$	3.46410	0.129914
$712$	0	0
$713$	−27.2554	−1.02072
$714$	0	0
$715$	− 24.6060i	− 0.920211i
$716$	0	0
$717$	22.0742i	0.824377i
$718$	0	0
$719$	2.22938	0.0831420	0.0415710	−	0.999136i	$-0.486764\pi$
$719$	2.22938	0.0831420	0.0415710	+	0.999136i	$0.486764\pi$
$720$	0	0
$721$	−8.74456	−0.325665
$722$	0	0
$723$	11.4891i	0.427285i
$724$	0	0
$725$	2.17448i	0.0807582i
$726$	0	0
$727$	17.9104	0.664261	0.332130	−	0.943234i	$-0.392233\pi$
$727$	17.9104	0.664261	0.332130	+	0.943234i	$0.392233\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 11.1168i	− 0.411171i
$732$	0	0
$733$	2.57924i	0.0952664i	0.998865	+	0.0476332i	$0.0151679\pi$
$733$	2.57924i	0.0952664i	−0.998865	+	0.0476332i	$0.984832\pi$
$734$	0	0
$735$	−12.6217	−0.465558
$736$	0	0
$737$	9.48913	0.349536
$738$	0	0
$739$	− 42.8397i	− 1.57588i	−0.615751	−	0.787941i	$-0.711147\pi$
$739$	− 42.8397i	− 1.57588i	0.615751	−	0.787941i	$-0.288853\pi$
$740$	0	0
$741$	26.1831i	0.961862i
$742$	0	0
$743$	42.8588	1.57234	0.786169	−	0.618011i	$-0.212061\pi$
$743$	42.8588	1.57234	0.786169	+	0.618011i	$0.212061\pi$
$744$	0	0
$745$	21.4891	0.787301
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 33.3514i	− 1.21863i
$750$	0	0
$751$	48.2025	1.75893	0.879467	−	0.475961i	$-0.157900\pi$
$751$	48.2025	1.75893	0.879467	+	0.475961i	$0.157900\pi$
$752$	0	0
$753$	−18.9783	−0.691606
$754$	0	0
$755$	− 50.2337i	− 1.82819i
$756$	0	0
$757$	− 9.74749i	− 0.354279i	−0.984186	−	0.177139i	$-0.943316\pi$
$757$	− 9.74749i	− 0.354279i	0.984186	−	0.177139i	$-0.0566843\pi$
$758$	0	0
$759$	−9.74749	−0.353812
$760$	0	0
$761$	−26.4674	−0.959442	−0.479721	−	0.877421i	$-0.659262\pi$
$761$	−26.4674	−0.959442	−0.479721	+	0.877421i	$0.659262\pi$
$762$	0	0
$763$	24.0000i	0.868858i
$764$	0	0
$765$	2.52434i	0.0912676i
$766$	0	0
$767$	−35.9306	−1.29738
$768$	0	0
$769$	−22.6060	−0.815192	−0.407596	−	0.913162i	$-0.633633\pi$
$769$	−22.6060	−0.815192	−0.407596	+	0.913162i	$0.633633\pi$
$770$	0	0
$771$	20.9783i	0.755513i
$772$	0	0
$773$	48.2025i	1.73372i	0.498550	+	0.866861i	$0.333866\pi$
$773$	48.2025i	1.73372i	−0.498550	+	0.866861i	$0.666134\pi$
$774$	0	0
$775$	−9.10268	−0.326978
$776$	0	0
$777$	−34.9783	−1.25484
$778$	0	0
$779$	58.0951i	2.08147i
$780$	0	0
$781$	4.45877i	0.159547i
$782$	0	0
$783$	−1.58457	−0.0566281
$784$	0	0
$785$	−11.8614	−0.423352
$786$	0	0
$787$	− 7.25544i	− 0.258628i	−0.991604	−	0.129314i	$-0.958722\pi$
$787$	− 7.25544i	− 0.258628i	0.991604	−	0.129314i	$-0.0412776\pi$
$788$	0	0
$789$	− 30.2921i	− 1.07843i
$790$	0	0
$791$	26.4232	0.939501
$792$	0	0
$793$	49.2119	1.74757
$794$	0	0
$795$	− 21.4891i	− 0.762141i
$796$	0	0
$797$	− 11.6819i	− 0.413795i	−0.978363	−	0.206898i	$-0.933663\pi$
$797$	− 11.6819i	− 0.413795i	0.978363	−	0.206898i	$-0.0663367\pi$
$798$	0	0
$799$	6.92820	0.245102
$800$	0	0
$801$	−7.48913	−0.264615
$802$	0	0
$803$	− 12.4674i	− 0.439964i
$804$	0	0
$805$	− 35.9306i	− 1.26639i
$806$	0	0
$807$	−0.644810	−0.0226984
$808$	0	0
$809$	−19.6277	−0.690074	−0.345037	−	0.938589i	$-0.612134\pi$
$809$	−19.6277	−0.690074	−0.345037	+	0.938589i	$0.612134\pi$
$810$	0	0
$811$	40.4674i	1.42100i	0.703696	+	0.710501i	$0.251532\pi$
$811$	40.4674i	1.42100i	−0.703696	+	0.710501i	$0.748468\pi$
$812$	0	0
$813$	18.2603i	0.640416i
$814$	0	0
$815$	1.87953	0.0658370
$816$	0	0
$817$	−70.8397	−2.47837
$818$	0	0
$819$	14.2337i	0.497365i
$820$	0	0
$821$	− 42.2140i	− 1.47328i	−0.676285	−	0.736640i	$-0.736411\pi$
$821$	− 42.2140i	− 1.47328i	0.676285	−	0.736640i	$-0.263589\pi$
$822$	0	0
$823$	−31.2867	−1.09059	−0.545293	−	0.838245i	$-0.683582\pi$
$823$	−31.2867	−1.09059	−0.545293	+	0.838245i	$0.683582\pi$
$824$	0	0
$825$	−3.25544	−0.113340
$826$	0	0
$827$	11.1168i	0.386571i	0.981143	+	0.193285i	$0.0619143\pi$
$827$	11.1168i	0.386571i	−0.981143	+	0.193285i	$0.938086\pi$
$828$	0	0
$829$	6.33830i	0.220138i	0.993924	+	0.110069i	$0.0351072\pi$
$829$	6.33830i	0.220138i	−0.993924	+	0.110069i	$0.964893\pi$
$830$	0	0
$831$	10.6873	0.370737
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	18.3723i	0.635799i
$836$	0	0
$837$	− 6.63325i	− 0.229279i
$838$	0	0
$839$	25.4834	0.879786	0.439893	−	0.898050i	$-0.355016\pi$
$839$	25.4834	0.879786	0.439893	+	0.898050i	$0.355016\pi$
$840$	0	0
$841$	26.4891	0.913418
$842$	0	0
$843$	15.4891i	0.533474i
$844$	0	0
$845$	− 9.80240i	− 0.337213i
$846$	0	0
$847$	18.6101	0.639452
$848$	0	0
$849$	−10.2337	−0.351219
$850$	0	0
$851$	− 41.4891i	− 1.42223i
$852$	0	0
$853$	44.8482i	1.53557i	0.640706	+	0.767786i	$0.278642\pi$
$853$	44.8482i	1.53557i	−0.640706	+	0.767786i	$0.721358\pi$
$854$	0	0
$855$	16.0858	0.550122
$856$	0	0
$857$	−24.9783	−0.853241	−0.426620	−	0.904431i	$-0.640296\pi$
$857$	−24.9783	−0.853241	−0.426620	+	0.904431i	$0.640296\pi$
$858$	0	0
$859$	14.9783i	0.511051i	0.966802	+	0.255526i	$0.0822485\pi$
$859$	14.9783i	0.511051i	−0.966802	+	0.255526i	$0.917751\pi$
$860$	0	0
$861$	31.5817i	1.07630i
$862$	0	0
$863$	14.5561	0.495496	0.247748	−	0.968824i	$-0.420309\pi$
$863$	14.5561	0.495496	0.247748	+	0.968824i	$0.420309\pi$
$864$	0	0
$865$	62.0951	2.11130
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	8.21782i	0.278771i
$870$	0	0
$871$	16.4356	0.556901
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	31.7228i	1.07243i
$876$	0	0
$877$	− 10.0974i	− 0.340963i	−0.985361	−	0.170482i	$-0.945468\pi$
$877$	− 10.0974i	− 0.340963i	0.985361	−	0.170482i	$-0.0545324\pi$
$878$	0	0
$879$	18.0202	0.607807
$880$	0	0
$881$	44.9783	1.51536	0.757678	−	0.652629i	$-0.226334\pi$
$881$	44.9783	1.51536	0.757678	+	0.652629i	$0.226334\pi$
$882$	0	0
$883$	17.3505i	0.583892i	0.956435	+	0.291946i	$0.0943027\pi$
$883$	17.3505i	0.583892i	−0.956435	+	0.291946i	$0.905697\pi$
$884$	0	0
$885$	22.0742i	0.742017i
$886$	0	0
$887$	−12.2169	−0.410204	−0.205102	−	0.978741i	$-0.565753\pi$
$887$	−12.2169	−0.410204	−0.205102	+	0.978741i	$0.565753\pi$
$888$	0	0
$889$	−32.7446	−1.09822
$890$	0	0
$891$	− 2.37228i	− 0.0794744i
$892$	0	0
$893$	− 44.1485i	− 1.47737i
$894$	0	0
$895$	−42.2689	−1.41289
$896$	0	0
$897$	−16.8832	−0.563712
$898$	0	0
$899$	− 10.5109i	− 0.350557i
$900$	0	0
$901$	− 8.51278i	− 0.283602i
$902$	0	0
$903$	−38.5099	−1.28153
$904$	0	0
$905$	33.4891	1.11322
$906$	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
$909$	− 8.51278i	− 0.282351i
$910$	0	0
$911$	33.8111	1.12021	0.560105	−	0.828422i	$-0.310761\pi$
$911$	33.8111	1.12021	0.560105	+	0.828422i	$0.310761\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 30.2337i	− 0.999495i
$916$	0	0
$917$	− 71.3812i	− 2.35721i
$918$	0	0
$919$	54.8906	1.81067	0.905337	−	0.424693i	$-0.139618\pi$
$919$	54.8906	1.81067	0.905337	+	0.424693i	$0.139618\pi$
$920$	0	0
$921$	29.4891	0.971700
$922$	0	0
$923$	7.72281i	0.254199i
$924$	0	0
$925$	− 13.8564i	− 0.455596i
$926$	0	0
$927$	2.52434	0.0829101
$928$	0	0
$929$	33.8614	1.11096	0.555478	−	0.831531i	$-0.312535\pi$
$929$	33.8614	1.11096	0.555478	+	0.831531i	$0.312535\pi$
$930$	0	0
$931$	31.8614i	1.04422i
$932$	0	0
$933$	− 18.9051i	− 0.618925i
$934$	0	0
$935$	−5.98844	−0.195843
$936$	0	0
$937$	31.4891	1.02870	0.514352	−	0.857579i	$-0.328032\pi$
$937$	31.4891	1.02870	0.514352	+	0.857579i	$0.328032\pi$
$938$	0	0
$939$	− 20.9783i	− 0.684599i
$940$	0	0
$941$	− 55.2405i	− 1.80079i	−0.435075	−	0.900394i	$-0.643278\pi$
$941$	− 55.2405i	− 1.80079i	0.435075	−	0.900394i	$-0.356722\pi$
$942$	0	0
$943$	−37.4603	−1.21988
$944$	0	0
$945$	8.74456	0.284461
$946$	0	0
$947$	2.51087i	0.0815925i	0.999167	+	0.0407962i	$0.0129894\pi$
$947$	2.51087i	0.0815925i	−0.999167	+	0.0407962i	$0.987011\pi$
$948$	0	0
$949$	− 21.5941i	− 0.700975i
$950$	0	0
$951$	−0.994667	−0.0322543
$952$	0	0
$953$	24.2337	0.785006	0.392503	−	0.919751i	$-0.371609\pi$
$953$	24.2337	0.785006	0.392503	+	0.919751i	$0.371609\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.75906i	− 0.121513i
$958$	0	0
$959$	−70.0916	−2.26337
$960$	0	0
$961$	13.0000	0.419355
$962$	0	0
$963$	9.62772i	0.310249i
$964$	0	0
$965$	− 39.0998i	− 1.25867i
$966$	0	0
$967$	37.8651	1.21766	0.608829	−	0.793301i	$-0.291639\pi$
$967$	37.8651	1.21766	0.608829	+	0.793301i	$0.291639\pi$
$968$	0	0
$969$	6.37228	0.204707
$970$	0	0
$971$	− 11.2554i	− 0.361204i	−0.983556	−	0.180602i	$-0.942195\pi$
$971$	− 11.2554i	− 0.361204i	0.983556	−	0.180602i	$-0.0578046\pi$
$972$	0	0
$973$	46.7277i	1.49802i
$974$	0	0
$975$	−5.63858	−0.180579
$976$	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
$979$	− 17.7663i	− 0.567814i
$980$	0	0
$981$	− 6.92820i	− 0.221201i
$982$	0	0
$983$	−28.6526	−0.913875	−0.456938	−	0.889499i	$-0.651054\pi$
$983$	−28.6526	−0.913875	−0.456938	+	0.889499i	$0.651054\pi$
$984$	0	0
$985$	−28.6060	−0.911462
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	− 45.6781i	− 1.45248i
$990$	0	0
$991$	44.4434	1.41179	0.705896	−	0.708316i	$-0.250545\pi$
$991$	44.4434	1.41179	0.705896	+	0.708316i	$0.250545\pi$
$992$	0	0
$993$	23.8614	0.757219
$994$	0	0
$995$	− 8.74456i	− 0.277221i
$996$	0	0
$997$	− 48.6072i	− 1.53941i	−0.638402	−	0.769703i	$-0.720404\pi$
$997$	− 48.6072i	− 1.53941i	0.638402	−	0.769703i	$-0.279596\pi$
$998$	0	0
$999$	10.0974	0.319466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.f.h.1633.8	yes	8
4.3	odd	2	inner	3264.2.f.h.1633.4	yes	8
8.3	odd	2	inner	3264.2.f.h.1633.5	yes	8
8.5	even	2	inner	3264.2.f.h.1633.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3264.2.f.h.1633.1	✓	8	8.5	even	2	inner
3264.2.f.h.1633.4	yes	8	4.3	odd	2	inner
3264.2.f.h.1633.5	yes	8	8.3	odd	2	inner
3264.2.f.h.1633.8	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.52434i q^{5} +3.46410 q^{7} -1.00000 q^{9} -2.37228i q^{11} -4.10891i q^{13} -2.52434 q^{15} -1.00000 q^{17} +6.37228i q^{19} +3.46410i q^{21} -4.10891 q^{23} -1.37228 q^{25} -1.00000i q^{27} -1.58457i q^{29} +6.63325 q^{31} +2.37228 q^{33} +8.74456i q^{35} +10.0974i q^{37} +4.10891 q^{39} +9.11684 q^{41} +11.1168i q^{43} -2.52434i q^{45} -6.92820 q^{47} +5.00000 q^{49} -1.00000i q^{51} +8.51278i q^{53} +5.98844 q^{55} -6.37228 q^{57} -8.74456i q^{59} +11.9769i q^{61} -3.46410 q^{63} +10.3723 q^{65} +4.00000i q^{67} -4.10891i q^{69} -1.87953 q^{71} +5.25544 q^{73} -1.37228i q^{75} -8.21782i q^{77} -3.46410 q^{79} +1.00000 q^{81} -2.52434i q^{85} +1.58457 q^{87} +7.48913 q^{89} -14.2337i q^{91} +6.63325i q^{93} -16.0858 q^{95} +14.0000 q^{97} +2.37228i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{17} + 12 q^{25} - 4 q^{33} + 4 q^{41} + 40 q^{49} - 28 q^{57} + 60 q^{65} + 88 q^{73} + 8 q^{81} - 32 q^{89} + 112 q^{97}+O(q^{100}) \) 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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