LMFDB - Embedded newform 1792.2.e.i.895.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(895,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.5
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.i.895.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.874032i q^{3} -3.70246 q^{5} +(1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +O(q^{10})$	$q+0.874032i q^{3} -3.70246 q^{5} +(1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +3.23607 q^{11} -0.874032 q^{13} -3.23607i q^{15} -4.57649i q^{17} -1.95440i q^{19} +(-1.95440 + 1.23607i) q^{21} +1.23607i q^{23} +8.70820 q^{25} +4.57649i q^{27} -2.00000i q^{29} +10.2333 q^{31} +2.82843i q^{33} +(-5.23607 - 8.27895i) q^{35} +10.9443i q^{37} -0.763932i q^{39} +3.90879i q^{41} +1.70820 q^{43} -8.27895 q^{45} -8.07262 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.00000 q^{51} +0.472136i q^{53} -11.9814 q^{55} +1.70820 q^{57} -11.1074i q^{59} +8.94665 q^{61} +(3.16228 + 5.00000i) q^{63} +3.23607 q^{65} -9.70820 q^{67} -1.08036 q^{69} +10.0000i q^{71} +14.1421i q^{73} +7.61125i q^{75} +(4.57649 + 7.23607i) q^{77} -12.4721i q^{79} +2.70820 q^{81} +12.1877i q^{83} +16.9443i q^{85} +1.74806 q^{87} +9.82068i q^{89} +(-1.23607 - 1.95440i) q^{91} +8.94427i q^{93} +7.23607i q^{95} +4.57649i q^{97} +7.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} + 16 q^{25} - 24 q^{35} - 40 q^{43} - 24 q^{49} + 32 q^{51} - 40 q^{57} + 8 q^{65} - 24 q^{67} - 32 q^{81} + 8 q^{91} + 40 q^{99}+O(q^{100}) $ 8 * q + 8 * q^11 + 16 * q^25 - 24 * q^35 - 40 * q^43 - 24 * q^49 + 32 * q^51 - 40 * q^57 + 8 * q^65 - 24 * q^67 - 32 * q^81 + 8 * q^91 + 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			0.874032i			0.504623i	0.967646	+	0.252311i	$0.0811907\pi$
$3$			0.874032i			0.504623i	−0.967646	+	0.252311i	$0.918809\pi$
$4$		0			0
$5$	−3.70246			−1.65579			−0.827895	−	0.560883i	$-0.810462\pi$
$5$	−3.70246			−1.65579			−0.827895	+	0.560883i	$0.810462\pi$
$6$		0			0
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$8$		0			0
$9$	2.23607			0.745356
$10$		0			0
$11$	3.23607			0.975711			0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607			0.975711			0.487856	+	0.872924i	$0.337779\pi$
$12$		0			0
$13$	−0.874032			−0.242413			−0.121206	−	0.992627i	$-0.538676\pi$
$13$	−0.874032			−0.242413			−0.121206	+	0.992627i	$0.538676\pi$
$14$		0			0
$15$		−	3.23607i		−	0.835549i
$16$		0			0
$17$		−	4.57649i		−	1.10996i	−0.831863	−	0.554981i	$-0.812725\pi$
$17$		−	4.57649i		−	1.10996i	0.831863	−	0.554981i	$-0.187275\pi$
$18$		0			0
$19$		−	1.95440i		−	0.448369i	−0.974547	−	0.224184i	$-0.928028\pi$
$19$		−	1.95440i		−	0.448369i	0.974547	−	0.224184i	$-0.0719718\pi$
$20$		0			0
$21$	−1.95440	+	1.23607i	−0.426484	+	0.269732i
$22$		0			0
$23$			1.23607i			0.257738i	0.991662	+	0.128869i	$0.0411347\pi$
$23$			1.23607i			0.257738i	−0.991662	+	0.128869i	$0.958865\pi$
$24$		0			0
$25$	8.70820			1.74164
$26$		0			0
$27$			4.57649i			0.880746i
$28$		0			0
$29$		−	2.00000i		−	0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$		−	2.00000i		−	0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$		0			0
$31$	10.2333			1.83796			0.918982	−	0.394301i	$-0.129013\pi$
$31$	10.2333			1.83796			0.918982	+	0.394301i	$0.129013\pi$
$32$		0			0
$33$			2.82843i			0.492366i
$34$		0			0
$35$	−5.23607	−	8.27895i	−0.885057	−	1.39940i
$36$		0			0
$37$			10.9443i			1.79923i	0.436687	+	0.899614i	$0.356152\pi$
$37$			10.9443i			1.79923i	−0.436687	+	0.899614i	$0.643848\pi$
$38$		0			0
$39$		−	0.763932i		−	0.122327i
$40$		0			0
$41$			3.90879i			0.610450i	0.952280	+	0.305225i	$0.0987317\pi$
$41$			3.90879i			0.610450i	−0.952280	+	0.305225i	$0.901268\pi$
$42$		0			0
$43$	1.70820			0.260499			0.130249	−	0.991481i	$-0.458422\pi$
$43$	1.70820			0.260499			0.130249	+	0.991481i	$0.458422\pi$
$44$		0			0
$45$	−8.27895			−1.23415
$46$		0			0
$47$	−8.07262			−1.17751			−0.588756	−	0.808311i	$-0.700382\pi$
$47$	−8.07262			−1.17751			−0.588756	+	0.808311i	$0.700382\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$			0.472136i			0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$			0.472136i			0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$		0			0
$55$	−11.9814			−1.61557
$56$		0			0
$57$	1.70820			0.226257
$58$		0			0
$59$		−	11.1074i		−	1.44606i	−0.690818	−	0.723029i	$-0.742749\pi$
$59$		−	11.1074i		−	1.44606i	0.690818	−	0.723029i	$-0.257251\pi$
$60$		0			0
$61$	8.94665			1.14550			0.572751	−	0.819730i	$-0.305876\pi$
$61$	8.94665			1.14550			0.572751	+	0.819730i	$0.305876\pi$
$62$		0			0
$63$	3.16228	+	5.00000i	0.398410	+	0.629941i
$64$		0			0
$65$	3.23607			0.401385
$66$		0			0
$67$	−9.70820			−1.18605			−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820			−1.18605			−0.593023	+	0.805186i	$0.702066\pi$
$68$		0			0
$69$	−1.08036			−0.130060
$70$		0			0
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$			14.1421i			1.65521i	0.561310	+	0.827606i	$0.310298\pi$
$73$			14.1421i			1.65521i	−0.561310	+	0.827606i	$0.689702\pi$
$74$		0			0
$75$			7.61125i			0.878871i
$76$		0			0
$77$	4.57649	+	7.23607i	0.521540	+	0.824626i
$78$		0			0
$79$		−	12.4721i		−	1.40322i	−0.712559	−	0.701612i	$-0.752464\pi$
$79$		−	12.4721i		−	1.40322i	0.712559	−	0.701612i	$-0.247536\pi$
$80$		0			0
$81$	2.70820			0.300912
$82$		0			0
$83$			12.1877i			1.33778i	0.743362	+	0.668889i	$0.233230\pi$
$83$			12.1877i			1.33778i	−0.743362	+	0.668889i	$0.766770\pi$
$84$		0			0
$85$			16.9443i			1.83786i
$86$		0			0
$87$	1.74806			0.187412
$88$		0			0
$89$			9.82068i			1.04099i	0.853865	+	0.520495i	$0.174253\pi$
$89$			9.82068i			1.04099i	−0.853865	+	0.520495i	$0.825747\pi$
$90$		0			0
$91$	−1.23607	−	1.95440i	−0.129575	−	0.204876i
$92$		0			0
$93$			8.94427i			0.927478i
$94$		0			0
$95$			7.23607i			0.742405i
$96$		0			0
$97$			4.57649i			0.464672i	0.972636	+	0.232336i	$0.0746369\pi$
$97$			4.57649i			0.464672i	−0.972636	+	0.232336i	$0.925363\pi$
$98$		0			0
$99$	7.23607			0.727252
$100$		0			0
$101$	−4.37016			−0.434847			−0.217424	−	0.976077i	$-0.569765\pi$
$101$	−4.37016			−0.434847			−0.217424	+	0.976077i	$0.569765\pi$
$102$		0			0
$103$	5.24419			0.516726			0.258363	−	0.966048i	$-0.416817\pi$
$103$	5.24419			0.516726			0.258363	+	0.966048i	$0.416817\pi$
$104$		0			0
$105$	7.23607	−	4.57649i	0.706168	−	0.446620i
$106$		0			0
$107$	13.7082			1.32522			0.662611	−	0.748964i	$-0.269448\pi$
$107$	13.7082			1.32522			0.662611	+	0.748964i	$0.269448\pi$
$108$		0			0
$109$			12.4721i			1.19461i	0.802013	+	0.597307i	$0.203763\pi$
$109$			12.4721i			1.19461i	−0.802013	+	0.597307i	$0.796237\pi$
$110$		0			0
$111$	−9.56564			−0.907931
$112$		0			0
$113$	15.7082			1.47770			0.738852	−	0.673868i	$-0.235368\pi$
$113$	15.7082			1.47770			0.738852	+	0.673868i	$0.235368\pi$
$114$		0			0
$115$		−	4.57649i		−	0.426760i
$116$		0			0
$117$	−1.95440			−0.180684
$118$		0			0
$119$	10.2333	−	6.47214i	0.938089	−	0.593300i
$120$		0			0
$121$	−0.527864			−0.0479876
$122$		0			0
$123$	−3.41641			−0.308047
$124$		0			0
$125$	−13.7295			−1.22800
$126$		0			0
$127$		−	5.23607i		−	0.464626i	−0.972641	−	0.232313i	$-0.925371\pi$
$127$		−	5.23607i		−	0.464626i	0.972641	−	0.232313i	$-0.0746294\pi$
$128$		0			0
$129$			1.49302i			0.131454i
$130$		0			0
$131$			10.0270i			0.876064i	0.898959	+	0.438032i	$0.144324\pi$
$131$			10.0270i			0.876064i	−0.898959	+	0.438032i	$0.855676\pi$
$132$		0			0
$133$	4.37016	−	2.76393i	0.378941	−	0.239663i
$134$		0			0
$135$		−	16.9443i		−	1.45833i
$136$		0			0
$137$	3.52786			0.301406			0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786			0.301406			0.150703	+	0.988579i	$0.451846\pi$
$138$		0			0
$139$			18.5123i			1.57019i	0.619374	+	0.785096i	$0.287387\pi$
$139$			18.5123i			1.57019i	−0.619374	+	0.785096i	$0.712613\pi$
$140$		0			0
$141$		−	7.05573i		−	0.594199i
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$			7.40492i			0.614945i
$146$		0			0
$147$	−5.52786	−	2.62210i	−0.455931	−	0.216267i
$148$		0			0
$149$		−	13.4164i		−	1.09911i	−0.835456	−	0.549557i	$-0.814796\pi$
$149$		−	13.4164i		−	1.09911i	0.835456	−	0.549557i	$-0.185204\pi$
$150$		0			0
$151$		−	6.76393i		−	0.550441i	−0.961381	−	0.275220i	$-0.911249\pi$
$151$		−	6.76393i		−	0.550441i	0.961381	−	0.275220i	$-0.0887509\pi$
$152$		0			0
$153$		−	10.2333i		−	0.827317i
$154$		0			0
$155$	−37.8885			−3.04328
$156$		0			0
$157$	−4.37016			−0.348777			−0.174388	−	0.984677i	$-0.555795\pi$
$157$	−4.37016			−0.348777			−0.174388	+	0.984677i	$0.555795\pi$
$158$		0			0
$159$	−0.412662			−0.0327262
$160$		0			0
$161$	−2.76393	+	1.74806i	−0.217828	+	0.137767i
$162$		0			0
$163$	12.7639			0.999748			0.499874	−	0.866098i	$-0.333380\pi$
$163$	12.7639			0.999748			0.499874	+	0.866098i	$0.333380\pi$
$164$		0			0
$165$		−	10.4721i		−	0.815255i
$166$		0			0
$167$	−10.9010			−0.843548			−0.421774	−	0.906701i	$-0.638592\pi$
$167$	−10.9010			−0.843548			−0.421774	+	0.906701i	$0.638592\pi$
$168$		0			0
$169$	−12.2361			−0.941236
$170$		0			0
$171$		−	4.37016i		−	0.334195i
$172$		0			0
$173$	−8.69161			−0.660811			−0.330406	−	0.943839i	$-0.607186\pi$
$173$	−8.69161			−0.660811			−0.330406	+	0.943839i	$0.607186\pi$
$174$		0			0
$175$	12.3153	+	19.4721i	0.930946	+	1.47196i
$176$		0			0
$177$	9.70820			0.729713
$178$		0			0
$179$	10.6525			0.796203			0.398102	−	0.917341i	$-0.369669\pi$
$179$	10.6525			0.796203			0.398102	+	0.917341i	$0.369669\pi$
$180$		0			0
$181$	−3.70246			−0.275202			−0.137601	−	0.990488i	$-0.543939\pi$
$181$	−3.70246			−0.275202			−0.137601	+	0.990488i	$0.543939\pi$
$182$		0			0
$183$			7.81966i			0.578046i
$184$		0			0
$185$		−	40.5207i		−	2.97914i
$186$		0			0
$187$		−	14.8098i		−	1.08300i
$188$		0			0
$189$	−10.2333	+	6.47214i	−0.744366	+	0.470779i
$190$		0			0
$191$		−	5.05573i		−	0.365820i	−0.983130	−	0.182910i	$-0.941448\pi$
$191$		−	5.05573i		−	0.365820i	0.983130	−	0.182910i	$-0.0585516\pi$
$192$		0			0
$193$	−6.76393			−0.486878			−0.243439	−	0.969916i	$-0.578276\pi$
$193$	−6.76393			−0.486878			−0.243439	+	0.969916i	$0.578276\pi$
$194$		0			0
$195$			2.82843i			0.202548i
$196$		0			0
$197$			1.05573i			0.0752175i	0.999293	+	0.0376088i	$0.0119741\pi$
$197$			1.05573i			0.0752175i	−0.999293	+	0.0376088i	$0.988026\pi$
$198$		0			0
$199$	13.0618			0.925925			0.462962	−	0.886378i	$-0.346787\pi$
$199$	13.0618			0.925925			0.462962	+	0.886378i	$0.346787\pi$
$200$		0			0
$201$		−	8.48528i		−	0.598506i
$202$		0			0
$203$	4.47214	−	2.82843i	0.313882	−	0.198517i
$204$		0			0
$205$		−	14.4721i		−	1.01078i
$206$		0			0
$207$			2.76393i			0.192107i
$208$		0			0
$209$		−	6.32456i		−	0.437479i
$210$		0			0
$211$	3.23607			0.222780			0.111390	−	0.993777i	$-0.464470\pi$
$211$	3.23607			0.222780			0.111390	+	0.993777i	$0.464470\pi$
$212$		0			0
$213$	−8.74032			−0.598877
$214$		0			0
$215$	−6.32456			−0.431331
$216$		0			0
$217$	14.4721	+	22.8825i	0.982433	+	1.55336i
$218$		0			0
$219$	−12.3607			−0.835257
$220$		0			0
$221$			4.00000i			0.269069i
$222$		0			0
$223$	−21.8021			−1.45998			−0.729988	−	0.683460i	$-0.760474\pi$
$223$	−21.8021			−1.45998			−0.729988	+	0.683460i	$0.760474\pi$
$224$		0			0
$225$	19.4721			1.29814
$226$		0			0
$227$		−	20.2604i		−	1.34473i	−0.740221	−	0.672364i	$-0.765279\pi$
$227$		−	20.2604i		−	1.34473i	0.740221	−	0.672364i	$-0.234721\pi$
$228$		0			0
$229$	8.27895			0.547088			0.273544	−	0.961859i	$-0.411804\pi$
$229$	8.27895			0.547088			0.273544	+	0.961859i	$0.411804\pi$
$230$		0			0
$231$	−6.32456	+	4.00000i	−0.416125	+	0.263181i
$232$		0			0
$233$	−13.4164			−0.878938			−0.439469	−	0.898258i	$-0.644833\pi$
$233$	−13.4164			−0.878938			−0.439469	+	0.898258i	$0.644833\pi$
$234$		0			0
$235$	29.8885			1.94971
$236$		0			0
$237$	10.9010			0.708099
$238$		0			0
$239$			12.6525i			0.818421i	0.912440	+	0.409210i	$0.134196\pi$
$239$			12.6525i			0.818421i	−0.912440	+	0.409210i	$0.865804\pi$
$240$		0			0
$241$		−	4.57649i		−	0.294798i	−0.989077	−	0.147399i	$-0.952910\pi$
$241$		−	4.57649i		−	0.294798i	0.989077	−	0.147399i	$-0.0470901\pi$
$242$		0			0
$243$			16.0965i			1.03259i
$244$		0			0
$245$	11.1074	−	23.4164i	0.709624	−	1.49602i
$246$		0			0
$247$			1.70820i			0.108690i
$248$		0			0
$249$	−10.6525			−0.675073
$250$		0			0
$251$		−	16.0965i		−	1.01600i	−0.861356	−	0.508002i	$-0.830384\pi$
$251$		−	16.0965i		−	1.01600i	0.861356	−	0.508002i	$-0.169616\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$	−14.8098			−0.927428
$256$		0			0
$257$		−	16.9706i		−	1.05859i	−0.848436	−	0.529297i	$-0.822456\pi$
$257$		−	16.9706i		−	1.05859i	0.848436	−	0.529297i	$-0.177544\pi$
$258$		0			0
$259$	−24.4721	+	15.4775i	−1.52062	+	0.961727i
$260$		0			0
$261$		−	4.47214i		−	0.276818i
$262$		0			0
$263$		−	31.8885i		−	1.96633i	−0.182715	−	0.983166i	$-0.558489\pi$
$263$		−	31.8885i		−	1.96633i	0.182715	−	0.983166i	$-0.441511\pi$
$264$		0			0
$265$		−	1.74806i		−	0.107383i
$266$		0			0
$267$	−8.58359			−0.525307
$268$		0			0
$269$	16.0965			0.981423			0.490711	−	0.871322i	$-0.336737\pi$
$269$	16.0965			0.981423			0.490711	+	0.871322i	$0.336737\pi$
$270$		0			0
$271$	13.4744			0.818514			0.409257	−	0.912419i	$-0.365788\pi$
$271$	13.4744			0.818514			0.409257	+	0.912419i	$0.365788\pi$
$272$		0			0
$273$	1.70820	−	1.08036i	0.103385	−	0.0653865i
$274$		0			0
$275$	28.1803			1.69934
$276$		0			0
$277$		−	1.05573i		−	0.0634326i	−0.999497	−	0.0317163i	$-0.989903\pi$
$277$		−	1.05573i		−	0.0634326i	0.999497	−	0.0317163i	$-0.0100973\pi$
$278$		0			0
$279$	22.8825			1.36994
$280$		0			0
$281$	−31.8885			−1.90231			−0.951156	−	0.308712i	$-0.900102\pi$
$281$	−31.8885			−1.90231			−0.951156	+	0.308712i	$0.900102\pi$
$282$		0			0
$283$			7.86629i			0.467602i	0.972284	+	0.233801i	$0.0751165\pi$
$283$			7.86629i			0.467602i	−0.972284	+	0.233801i	$0.924884\pi$
$284$		0			0
$285$	−6.32456			−0.374634
$286$		0			0
$287$	−8.74032	+	5.52786i	−0.515925	+	0.326299i
$288$		0			0
$289$	−3.94427			−0.232016
$290$		0			0
$291$	−4.00000			−0.234484
$292$		0			0
$293$	12.6004			0.736123			0.368062	−	0.929801i	$-0.380022\pi$
$293$	12.6004			0.736123			0.368062	+	0.929801i	$0.380022\pi$
$294$		0			0
$295$			41.1246i			2.39437i
$296$		0			0
$297$			14.8098i			0.859354i
$298$		0			0
$299$		−	1.08036i		−	0.0624790i
$300$		0			0
$301$	2.41577	+	3.81966i	0.139242	+	0.220162i
$302$		0			0
$303$		−	3.81966i		−	0.219434i
$304$		0			0
$305$	−33.1246			−1.89671
$306$		0			0
$307$		−	10.4397i		−	0.595824i	−0.954593	−	0.297912i	$-0.903710\pi$
$307$		−	10.4397i		−	0.595824i	0.954593	−	0.297912i	$-0.0962902\pi$
$308$		0			0
$309$			4.58359i			0.260751i
$310$		0			0
$311$	14.1421			0.801927			0.400963	−	0.916094i	$-0.368675\pi$
$311$	14.1421			0.801927			0.400963	+	0.916094i	$0.368675\pi$
$312$		0			0
$313$			27.8716i			1.57540i	0.616061	+	0.787698i	$0.288727\pi$
$313$			27.8716i			1.57540i	−0.616061	+	0.787698i	$0.711273\pi$
$314$		0			0
$315$	−11.7082	−	18.5123i	−0.659683	−	1.04305i
$316$		0			0
$317$		−	15.8885i		−	0.892390i	−0.894936	−	0.446195i	$-0.852779\pi$
$317$		−	15.8885i		−	0.892390i	0.894936	−	0.446195i	$-0.147221\pi$
$318$		0			0
$319$		−	6.47214i		−	0.362370i
$320$		0			0
$321$			11.9814i			0.668737i
$322$		0			0
$323$	−8.94427			−0.497673
$324$		0			0
$325$	−7.61125			−0.422196
$326$		0			0
$327$	−10.9010			−0.602829
$328$		0			0
$329$	−11.4164	−	18.0509i	−0.629407	−	0.995180i
$330$		0			0
$331$	25.1246			1.38097			0.690487	−	0.723345i	$-0.257396\pi$
$331$	25.1246			1.38097			0.690487	+	0.723345i	$0.257396\pi$
$332$		0			0
$333$			24.4721i			1.34106i
$334$		0			0
$335$	35.9442			1.96384
$336$		0			0
$337$	−15.1246			−0.823890			−0.411945	−	0.911209i	$-0.635150\pi$
$337$	−15.1246			−0.823890			−0.411945	+	0.911209i	$0.635150\pi$
$338$		0			0
$339$			13.7295i			0.745683i
$340$		0			0
$341$	33.1158			1.79332
$342$		0			0
$343$	−18.3848	+	2.23607i	−0.992685	+	0.120736i
$344$		0			0
$345$	4.00000			0.215353
$346$		0			0
$347$	4.76393			0.255741			0.127871	−	0.991791i	$-0.459186\pi$
$347$	4.76393			0.255741			0.127871	+	0.991791i	$0.459186\pi$
$348$		0			0
$349$	22.4211			1.20017			0.600087	−	0.799935i	$-0.295133\pi$
$349$	22.4211			1.20017			0.600087	+	0.799935i	$0.295133\pi$
$350$		0			0
$351$		−	4.00000i		−	0.213504i
$352$		0			0
$353$			21.8021i			1.16041i	0.814471	+	0.580204i	$0.197027\pi$
$353$			21.8021i			1.16041i	−0.814471	+	0.580204i	$0.802973\pi$
$354$		0			0
$355$		−	37.0246i		−	1.96506i
$356$		0			0
$357$	5.65685	+	8.94427i	0.299392	+	0.473381i
$358$		0			0
$359$		−	22.7639i		−	1.20143i	−0.799462	−	0.600717i	$-0.794882\pi$
$359$		−	22.7639i		−	1.20143i	0.799462	−	0.600717i	$-0.205118\pi$
$360$		0			0
$361$	15.1803			0.798965
$362$		0			0
$363$		−	0.461370i		−	0.0242156i
$364$		0			0
$365$		−	52.3607i		−	2.74068i
$366$		0			0
$367$	5.65685			0.295285			0.147643	−	0.989041i	$-0.452831\pi$
$367$	5.65685			0.295285			0.147643	+	0.989041i	$0.452831\pi$
$368$		0			0
$369$			8.74032i			0.455003i
$370$		0			0
$371$	−1.05573	+	0.667701i	−0.0548107	+	0.0346653i
$372$		0			0
$373$			22.9443i			1.18801i	0.804462	+	0.594005i	$0.202454\pi$
$373$			22.9443i			1.18801i	−0.804462	+	0.594005i	$0.797546\pi$
$374$		0			0
$375$		−	12.0000i		−	0.619677i
$376$		0			0
$377$			1.74806i			0.0900299i
$378$		0			0
$379$	−21.7082			−1.11508			−0.557538	−	0.830152i	$-0.688254\pi$
$379$	−21.7082			−1.11508			−0.557538	+	0.830152i	$0.688254\pi$
$380$		0			0
$381$	4.57649			0.234461
$382$		0			0
$383$	15.0649			0.769779			0.384890	−	0.922963i	$-0.374240\pi$
$383$	15.0649			0.769779			0.384890	+	0.922963i	$0.374240\pi$
$384$		0			0
$385$	−16.9443	−	26.7912i	−0.863560	−	1.36541i
$386$		0			0
$387$	3.81966			0.194164
$388$		0			0
$389$		−	24.8328i		−	1.25907i	−0.776971	−	0.629537i	$-0.783245\pi$
$389$		−	24.8328i		−	1.25907i	0.776971	−	0.629537i	$-0.216755\pi$
$390$		0			0
$391$	5.65685			0.286079
$392$		0			0
$393$	−8.76393			−0.442082
$394$		0			0
$395$			46.1776i			2.32345i
$396$		0			0
$397$	15.4288			0.774351			0.387175	−	0.922006i	$-0.373451\pi$
$397$	15.4288			0.774351			0.387175	+	0.922006i	$0.373451\pi$
$398$		0			0
$399$	2.41577	+	3.81966i	0.120940	+	0.191222i
$400$		0			0
$401$	−16.6525			−0.831585			−0.415792	−	0.909460i	$-0.636496\pi$
$401$	−16.6525			−0.831585			−0.415792	+	0.909460i	$0.636496\pi$
$402$		0			0
$403$	−8.94427			−0.445546
$404$		0			0
$405$	−10.0270			−0.498246
$406$		0			0
$407$			35.4164i			1.75553i
$408$		0			0
$409$		−	25.7109i		−	1.27132i	−0.771969	−	0.635661i	$-0.780728\pi$
$409$		−	25.7109i		−	1.27132i	0.771969	−	0.635661i	$-0.219272\pi$
$410$		0			0
$411$			3.08347i			0.152096i
$412$		0			0
$413$	24.8369	−	15.7082i	1.22214	−	0.772950i
$414$		0			0
$415$		−	45.1246i		−	2.21508i
$416$		0			0
$417$	−16.1803			−0.792355
$418$		0			0
$419$		−	6.94355i		−	0.339215i	−0.985512	−	0.169607i	$-0.945750\pi$
$419$		−	6.94355i		−	0.339215i	0.985512	−	0.169607i	$-0.0542499\pi$
$420$		0			0
$421$			9.41641i			0.458928i	0.973317	+	0.229464i	$0.0736973\pi$
$421$			9.41641i			0.458928i	−0.973317	+	0.229464i	$0.926303\pi$
$422$		0			0
$423$	−18.0509			−0.877666
$424$		0			0
$425$		−	39.8530i		−	1.93316i
$426$		0			0
$427$	12.6525	+	20.0053i	0.612296	+	0.968125i
$428$		0			0
$429$		−	2.47214i		−	0.119356i
$430$		0			0
$431$			15.7082i			0.756638i	0.925675	+	0.378319i	$0.123498\pi$
$431$			15.7082i			0.756638i	−0.925675	+	0.378319i	$0.876502\pi$
$432$		0			0
$433$		−	17.2256i		−	0.827810i	−0.910320	−	0.413905i	$-0.864165\pi$
$433$		−	17.2256i		−	0.827810i	0.910320	−	0.413905i	$-0.135835\pi$
$434$		0			0
$435$	−6.47214			−0.310315
$436$		0			0
$437$	2.41577			0.115562
$438$		0			0
$439$	1.49302			0.0712582			0.0356291	−	0.999365i	$-0.488657\pi$
$439$	1.49302			0.0712582			0.0356291	+	0.999365i	$0.488657\pi$
$440$		0			0
$441$	−6.70820	+	14.1421i	−0.319438	+	0.673435i
$442$		0			0
$443$	18.2918			0.869069			0.434535	−	0.900655i	$-0.356913\pi$
$443$	18.2918			0.869069			0.434535	+	0.900655i	$0.356913\pi$
$444$		0			0
$445$		−	36.3607i		−	1.72366i
$446$		0			0
$447$	11.7264			0.554638
$448$		0			0
$449$	−14.3607			−0.677722			−0.338861	−	0.940836i	$-0.610042\pi$
$449$	−14.3607			−0.677722			−0.338861	+	0.940836i	$0.610042\pi$
$450$		0			0
$451$			12.6491i			0.595623i
$452$		0			0
$453$	5.91189			0.277765
$454$		0			0
$455$	4.57649	+	7.23607i	0.214549	+	0.339232i
$456$		0			0
$457$	0.652476			0.0305215			0.0152608	−	0.999884i	$-0.495142\pi$
$457$	0.652476			0.0305215			0.0152608	+	0.999884i	$0.495142\pi$
$458$		0			0
$459$	20.9443			0.977595
$460$		0			0
$461$	−10.6947			−0.498103			−0.249051	−	0.968490i	$-0.580119\pi$
$461$	−10.6947			−0.498103			−0.249051	+	0.968490i	$0.580119\pi$
$462$		0			0
$463$		−	3.88854i		−	0.180716i	−0.995909	−	0.0903580i	$-0.971199\pi$
$463$		−	3.88854i		−	0.180716i	0.995909	−	0.0903580i	$-0.0288011\pi$
$464$		0			0
$465$		−	33.1158i		−	1.53571i
$466$		0			0
$467$			3.70246i			0.171329i	0.996324	+	0.0856647i	$0.0273014\pi$
$467$			3.70246i			0.171329i	−0.996324	+	0.0856647i	$0.972699\pi$
$468$		0			0
$469$	−13.7295	−	21.7082i	−0.633968	−	1.00239i
$470$		0			0
$471$		−	3.81966i		−	0.176001i
$472$		0			0
$473$	5.52786			0.254171
$474$		0			0
$475$		−	17.0193i		−	0.780898i
$476$		0			0
$477$			1.05573i			0.0483385i
$478$		0			0
$479$	−8.89794			−0.406557			−0.203279	−	0.979121i	$-0.565160\pi$
$479$	−8.89794			−0.406557			−0.203279	+	0.979121i	$0.565160\pi$
$480$		0			0
$481$		−	9.56564i		−	0.436156i
$482$		0			0
$483$	−1.52786	−	2.41577i	−0.0695202	−	0.109921i
$484$		0			0
$485$		−	16.9443i		−	0.769400i
$486$		0			0
$487$			24.0689i			1.09067i	0.838220	+	0.545333i	$0.183597\pi$
$487$			24.0689i			1.09067i	−0.838220	+	0.545333i	$0.816403\pi$
$488$		0			0
$489$			11.1561i			0.504496i
$490$		0			0
$491$	−14.2918			−0.644980			−0.322490	−	0.946573i	$-0.604520\pi$
$491$	−14.2918			−0.644980			−0.322490	+	0.946573i	$0.604520\pi$
$492$		0			0
$493$	−9.15298			−0.412230
$494$		0			0
$495$	−26.7912			−1.20418
$496$		0			0
$497$	−22.3607	+	14.1421i	−1.00301	+	0.634361i
$498$		0			0
$499$	20.7639			0.929521			0.464761	−	0.885436i	$-0.346140\pi$
$499$	20.7639			0.929521			0.464761	+	0.885436i	$0.346140\pi$
$500$		0			0
$501$		−	9.52786i		−	0.425674i
$502$		0			0
$503$	22.4698			1.00188			0.500939	−	0.865482i	$-0.332988\pi$
$503$	22.4698			1.00188			0.500939	+	0.865482i	$0.332988\pi$
$504$		0			0
$505$	16.1803			0.720016
$506$		0			0
$507$		−	10.6947i		−	0.474969i
$508$		0			0
$509$	9.77198			0.433135			0.216568	−	0.976268i	$-0.430514\pi$
$509$	9.77198			0.433135			0.216568	+	0.976268i	$0.430514\pi$
$510$		0			0
$511$	−31.6228	+	20.0000i	−1.39891	+	0.884748i
$512$		0			0
$513$	8.94427			0.394899
$514$		0			0
$515$	−19.4164			−0.855589
$516$		0			0
$517$	−26.1235			−1.14891
$518$		0			0
$519$		−	7.59675i		−	0.333460i
$520$		0			0
$521$			29.2070i			1.27958i	0.768549	+	0.639791i	$0.220979\pi$
$521$			29.2070i			1.27958i	−0.768549	+	0.639791i	$0.779021\pi$
$522$		0			0
$523$			5.86319i			0.256379i	0.991750	+	0.128190i	$0.0409166\pi$
$523$			5.86319i			0.256379i	−0.991750	+	0.128190i	$0.959083\pi$
$524$		0			0
$525$	−17.0193	+	10.7639i	−0.742782	+	0.469777i
$526$		0			0
$527$		−	46.8328i		−	2.04007i
$528$		0			0
$529$	21.4721			0.933571
$530$		0			0
$531$		−	24.8369i		−	1.07783i
$532$		0			0
$533$		−	3.41641i		−	0.147981i
$534$		0			0
$535$	−50.7541			−2.19429
$536$		0			0
$537$			9.31061i			0.401782i
$538$		0			0
$539$	−9.70820	+	20.4667i	−0.418162	+	0.881563i
$540$		0			0
$541$			0.472136i			0.0202987i	0.999948	+	0.0101494i	$0.00323070\pi$
$541$			0.472136i			0.0202987i	−0.999948	+	0.0101494i	$0.996769\pi$
$542$		0			0
$543$		−	3.23607i		−	0.138873i
$544$		0			0
$545$		−	46.1776i		−	1.97803i
$546$		0			0
$547$	−37.1246			−1.58733			−0.793667	−	0.608353i	$-0.791831\pi$
$547$	−37.1246			−1.58733			−0.793667	+	0.608353i	$0.791831\pi$
$548$		0			0
$549$	20.0053			0.853806
$550$		0			0
$551$	−3.90879			−0.166520
$552$		0			0
$553$	27.8885	−	17.6383i	1.18594	−	0.750055i
$554$		0			0
$555$	35.4164			1.50334
$556$		0			0
$557$		−	2.36068i		−	0.100025i	−0.998749	−	0.0500126i	$-0.984074\pi$
$557$		−	2.36068i		−	0.100025i	0.998749	−	0.0500126i	$-0.0159262\pi$
$558$		0			0
$559$	−1.49302			−0.0631482
$560$		0			0
$561$	12.9443			0.546508
$562$		0			0
$563$			13.6808i			0.576576i	0.957544	+	0.288288i	$0.0930860\pi$
$563$			13.6808i			0.576576i	−0.957544	+	0.288288i	$0.906914\pi$
$564$		0			0
$565$	−58.1590			−2.44677
$566$		0			0
$567$	3.82998	+	6.05573i	0.160844	+	0.254317i
$568$		0			0
$569$	−24.0689			−1.00902			−0.504510	−	0.863406i	$-0.668327\pi$
$569$	−24.0689			−1.00902			−0.504510	+	0.863406i	$0.668327\pi$
$570$		0			0
$571$	−8.18034			−0.342337			−0.171168	−	0.985242i	$-0.554754\pi$
$571$	−8.18034			−0.342337			−0.171168	+	0.985242i	$0.554754\pi$
$572$		0			0
$573$	4.41887			0.184601
$574$		0			0
$575$			10.7639i			0.448887i
$576$		0			0
$577$		−	9.97831i		−	0.415402i	−0.978192	−	0.207701i	$-0.933402\pi$
$577$		−	9.97831i		−	0.415402i	0.978192	−	0.207701i	$-0.0665982\pi$
$578$		0			0
$579$		−	5.91189i		−	0.245690i
$580$		0			0
$581$	−27.2526	+	17.2361i	−1.13063	+	0.715073i
$582$		0			0
$583$			1.52786i			0.0632777i
$584$		0			0
$585$	7.23607			0.299175
$586$		0			0
$587$		−	38.5663i		−	1.59180i	−0.605426	−	0.795901i	$-0.706997\pi$
$587$		−	38.5663i		−	1.59180i	0.605426	−	0.795901i	$-0.293003\pi$
$588$		0			0
$589$		−	20.0000i		−	0.824086i
$590$		0			0
$591$	−0.922740			−0.0379565
$592$		0			0
$593$			11.3137i			0.464598i	0.972644	+	0.232299i	$0.0746248\pi$
$593$			11.3137i			0.464598i	−0.972644	+	0.232299i	$0.925375\pi$
$594$		0			0
$595$	−37.8885	+	23.9628i	−1.55328	+	0.982380i
$596$		0			0
$597$			11.4164i			0.467242i
$598$		0			0
$599$			12.4721i			0.509598i	0.966994	+	0.254799i	$0.0820093\pi$
$599$			12.4721i			0.509598i	−0.966994	+	0.254799i	$0.917991\pi$
$600$		0			0
$601$		−	13.3168i		−	0.543204i	−0.962410	−	0.271602i	$-0.912447\pi$
$601$		−	13.3168i		−	0.543204i	0.962410	−	0.271602i	$-0.0875534\pi$
$602$		0			0
$603$	−21.7082			−0.884026
$604$		0			0
$605$	1.95440			0.0794575
$606$		0			0
$607$	40.1081			1.62794			0.813968	−	0.580910i	$-0.197303\pi$
$607$	40.1081			1.62794			0.813968	+	0.580910i	$0.197303\pi$
$608$		0			0
$609$	2.47214	+	3.90879i	0.100176	+	0.158392i
$610$		0			0
$611$	7.05573			0.285444
$612$		0			0
$613$			17.4164i			0.703442i	0.936105	+	0.351721i	$0.114403\pi$
$613$			17.4164i			0.703442i	−0.936105	+	0.351721i	$0.885597\pi$
$614$		0			0
$615$	12.6491			0.510061
$616$		0			0
$617$	−2.76393			−0.111272			−0.0556359	−	0.998451i	$-0.517719\pi$
$617$	−2.76393			−0.111272			−0.0556359	+	0.998451i	$0.517719\pi$
$618$		0			0
$619$		−	17.4319i		−	0.700649i	−0.936628	−	0.350324i	$-0.886071\pi$
$619$		−	17.4319i		−	0.700649i	0.936628	−	0.350324i	$-0.113929\pi$
$620$		0			0
$621$	−5.65685			−0.227002
$622$		0			0
$623$	−21.9597	+	13.8885i	−0.879797	+	0.556433i
$624$		0			0
$625$	7.29180			0.291672
$626$		0			0
$627$	5.52786			0.220762
$628$		0			0
$629$	50.0864			1.99707
$630$		0			0
$631$		−	8.47214i		−	0.337270i	−0.985679	−	0.168635i	$-0.946064\pi$
$631$		−	8.47214i		−	0.337270i	0.985679	−	0.168635i	$-0.0539360\pi$
$632$		0			0
$633$			2.82843i			0.112420i
$634$		0			0
$635$			19.3863i			0.769323i
$636$		0			0
$637$	2.62210	−	5.52786i	0.103891	−	0.219022i
$638$		0			0
$639$			22.3607i			0.884575i
$640$		0			0
$641$	6.18034			0.244109			0.122054	−	0.992523i	$-0.461052\pi$
$641$	6.18034			0.244109			0.122054	+	0.992523i	$0.461052\pi$
$642$		0			0
$643$		−	18.0996i		−	0.713780i	−0.934146	−	0.356890i	$-0.883837\pi$
$643$		−	18.0996i		−	0.713780i	0.934146	−	0.356890i	$-0.116163\pi$
$644$		0			0
$645$		−	5.52786i		−	0.217659i
$646$		0			0
$647$	−16.5579			−0.650958			−0.325479	−	0.945549i	$-0.605526\pi$
$647$	−16.5579			−0.650958			−0.325479	+	0.945549i	$0.605526\pi$
$648$		0			0
$649$		−	35.9442i		−	1.41093i
$650$		0			0
$651$	−20.0000	+	12.6491i	−0.783862	+	0.495758i
$652$		0			0
$653$			13.4164i			0.525025i	0.964929	+	0.262512i	$0.0845510\pi$
$653$			13.4164i			0.525025i	−0.964929	+	0.262512i	$0.915449\pi$
$654$		0			0
$655$		−	37.1246i		−	1.45058i
$656$		0			0
$657$			31.6228i			1.23372i
$658$		0			0
$659$	35.0132			1.36392			0.681959	−	0.731390i	$-0.261128\pi$
$659$	35.0132			1.36392			0.681959	+	0.731390i	$0.261128\pi$
$660$		0			0
$661$	33.5772			1.30600			0.653000	−	0.757358i	$-0.273510\pi$
$661$	33.5772			1.30600			0.653000	+	0.757358i	$0.273510\pi$
$662$		0			0
$663$	−3.49613			−0.135778
$664$		0			0
$665$	−16.1803	+	10.2333i	−0.627447	+	0.396832i
$666$		0			0
$667$	2.47214			0.0957215
$668$		0			0
$669$		−	19.0557i		−	0.736737i
$670$		0			0
$671$	28.9520			1.11768
$672$		0			0
$673$	12.8328			0.494669			0.247334	−	0.968930i	$-0.420445\pi$
$673$	12.8328			0.494669			0.247334	+	0.968930i	$0.420445\pi$
$674$		0			0
$675$			39.8530i			1.53394i
$676$		0			0
$677$	−8.53399			−0.327988			−0.163994	−	0.986461i	$-0.552438\pi$
$677$	−8.53399			−0.327988			−0.163994	+	0.986461i	$0.552438\pi$
$678$		0			0
$679$	−10.2333	+	6.47214i	−0.392720	+	0.248378i
$680$		0			0
$681$	17.7082			0.678580
$682$		0			0
$683$	−12.1803			−0.466068			−0.233034	−	0.972469i	$-0.574865\pi$
$683$	−12.1803			−0.466068			−0.233034	+	0.972469i	$0.574865\pi$
$684$		0			0
$685$	−13.0618			−0.499065
$686$		0			0
$687$			7.23607i			0.276073i
$688$		0			0
$689$		−	0.412662i		−	0.0157212i
$690$		0			0
$691$		−	28.7456i		−	1.09354i	−0.837284	−	0.546768i	$-0.815858\pi$
$691$		−	28.7456i		−	1.09354i	0.837284	−	0.546768i	$-0.184142\pi$
$692$		0			0
$693$	10.2333	+	16.1803i	0.388733	+	0.614640i
$694$		0			0
$695$		−	68.5410i		−	2.59991i
$696$		0			0
$697$	17.8885			0.677577
$698$		0			0
$699$		−	11.7264i		−	0.443532i
$700$		0			0
$701$			40.4721i			1.52861i	0.644854	+	0.764306i	$0.276918\pi$
$701$			40.4721i			1.52861i	−0.644854	+	0.764306i	$0.723082\pi$
$702$		0			0
$703$	21.3894			0.806718
$704$		0			0
$705$			26.1235i			0.983870i
$706$		0			0
$707$	−6.18034	−	9.77198i	−0.232436	−	0.367513i
$708$		0			0
$709$			3.52786i			0.132492i	0.997803	+	0.0662459i	$0.0211022\pi$
$709$			3.52786i			0.132492i	−0.997803	+	0.0662459i	$0.978898\pi$
$710$		0			0
$711$		−	27.8885i		−	1.04590i
$712$		0			0
$713$			12.6491i			0.473713i
$714$		0			0
$715$	10.4721			0.391636
$716$		0			0
$717$	−11.0587			−0.412994
$718$		0			0
$719$	−36.8670			−1.37491			−0.687453	−	0.726229i	$-0.741271\pi$
$719$	−36.8670			−1.37491			−0.687453	+	0.726229i	$0.741271\pi$
$720$		0			0
$721$	7.41641	+	11.7264i	0.276201	+	0.436713i
$722$		0			0
$723$	4.00000			0.148762
$724$		0			0
$725$		−	17.4164i		−	0.646829i
$726$		0			0
$727$	−16.5579			−0.614099			−0.307049	−	0.951694i	$-0.599342\pi$
$727$	−16.5579			−0.614099			−0.307049	+	0.951694i	$0.599342\pi$
$728$		0			0
$729$	−5.94427			−0.220158
$730$		0			0
$731$		−	7.81758i		−	0.289144i
$732$		0			0
$733$	−48.1320			−1.77779			−0.888897	−	0.458106i	$-0.848528\pi$
$733$	−48.1320			−1.77779			−0.888897	+	0.458106i	$0.848528\pi$
$734$		0			0
$735$	20.4667	+	9.70820i	0.754925	+	0.358092i
$736$		0			0
$737$	−31.4164			−1.15724
$738$		0			0
$739$	51.5967			1.89802			0.949009	−	0.315250i	$-0.102089\pi$
$739$	51.5967			1.89802			0.949009	+	0.315250i	$0.102089\pi$
$740$		0			0
$741$	−1.49302			−0.0548476
$742$		0			0
$743$			4.29180i			0.157451i	0.996896	+	0.0787254i	$0.0250850\pi$
$743$			4.29180i			0.157451i	−0.996896	+	0.0787254i	$0.974915\pi$
$744$		0			0
$745$			49.6737i			1.81990i
$746$		0			0
$747$			27.2526i			0.997121i
$748$		0			0
$749$	19.3863	+	30.6525i	0.708361	+	1.12002i
$750$		0			0
$751$			14.5410i			0.530609i	0.964165	+	0.265305i	$0.0854725\pi$
$751$			14.5410i			0.530609i	−0.964165	+	0.265305i	$0.914527\pi$
$752$		0			0
$753$	14.0689			0.512699
$754$		0			0
$755$			25.0432i			0.911415i
$756$		0			0
$757$			22.3607i			0.812713i	0.913715	+	0.406356i	$0.133201\pi$
$757$			22.3607i			0.812713i	−0.913715	+	0.406356i	$0.866799\pi$
$758$		0			0
$759$	−3.49613			−0.126901
$760$		0			0
$761$			10.9010i			0.395163i	0.980287	+	0.197581i	$0.0633086\pi$
$761$			10.9010i			0.395163i	−0.980287	+	0.197581i	$0.936691\pi$
$762$		0			0
$763$	−27.8885	+	17.6383i	−1.00963	+	0.638548i
$764$		0			0
$765$			37.8885i			1.36986i
$766$		0			0
$767$			9.70820i			0.350543i
$768$		0			0
$769$			39.8530i			1.43714i	0.695456	+	0.718568i	$0.255202\pi$
$769$			39.8530i			1.43714i	−0.695456	+	0.718568i	$0.744798\pi$
$770$		0			0
$771$	14.8328			0.534191
$772$		0			0
$773$	22.9312			0.824777			0.412388	−	0.911008i	$-0.364695\pi$
$773$	22.9312			0.824777			0.412388	+	0.911008i	$0.364695\pi$
$774$		0			0
$775$	89.1141			3.20107
$776$		0			0
$777$	−13.5279	−	21.3894i	−0.485309	−	0.767342i
$778$		0			0
$779$	7.63932			0.273707
$780$		0			0
$781$			32.3607i			1.15796i
$782$		0			0
$783$	9.15298			0.327101
$784$		0			0
$785$	16.1803			0.577501
$786$		0			0
$787$		−	7.61125i		−	0.271312i	−0.990756	−	0.135656i	$-0.956686\pi$
$787$		−	7.61125i		−	0.271312i	0.990756	−	0.135656i	$-0.0433142\pi$
$788$		0			0
$789$	27.8716			0.992256
$790$		0			0
$791$	22.2148	+	35.1246i	0.789866	+	1.24889i
$792$		0			0
$793$	−7.81966			−0.277684
$794$		0			0
$795$	1.52786			0.0541878
$796$		0			0
$797$	10.4397			0.369792			0.184896	−	0.982758i	$-0.440805\pi$
$797$	10.4397			0.369792			0.184896	+	0.982758i	$0.440805\pi$
$798$		0			0
$799$			36.9443i			1.30699i
$800$		0			0
$801$			21.9597i			0.775908i
$802$		0			0
$803$			45.7649i			1.61501i
$804$		0			0
$805$	10.2333	−	6.47214i	0.360678	−	0.228113i
$806$		0			0
$807$			14.0689i			0.495248i
$808$		0			0
$809$	−56.0689			−1.97128			−0.985638	−	0.168869i	$-0.945988\pi$
$809$	−56.0689			−1.97128			−0.985638	+	0.168869i	$0.945988\pi$
$810$		0			0
$811$			29.6684i			1.04180i	0.853618	+	0.520899i	$0.174403\pi$
$811$			29.6684i			1.04180i	−0.853618	+	0.520899i	$0.825597\pi$
$812$		0			0
$813$			11.7771i			0.413040i
$814$		0			0
$815$	−47.2579			−1.65537
$816$		0			0
$817$		−	3.33851i		−	0.116800i
$818$		0			0
$819$	−2.76393	−	4.37016i	−0.0965796	−	0.152706i
$820$		0			0
$821$		−	23.8885i		−	0.833716i	−0.908972	−	0.416858i	$-0.863131\pi$
$821$		−	23.8885i		−	0.833716i	0.908972	−	0.416858i	$-0.136869\pi$
$822$		0			0
$823$		−	36.8328i		−	1.28391i	−0.766742	−	0.641956i	$-0.778123\pi$
$823$		−	36.8328i		−	1.28391i	0.766742	−	0.641956i	$-0.221877\pi$
$824$		0			0
$825$			24.6305i			0.857525i
$826$		0			0
$827$	34.0689			1.18469			0.592346	−	0.805684i	$-0.298202\pi$
$827$	34.0689			1.18469			0.592346	+	0.805684i	$0.298202\pi$
$828$		0			0
$829$	−13.6808			−0.475153			−0.237576	−	0.971369i	$-0.576353\pi$
$829$	−13.6808			−0.475153			−0.237576	+	0.971369i	$0.576353\pi$
$830$		0			0
$831$	0.922740			0.0320095
$832$		0			0
$833$	28.9443	+	13.7295i	1.00286	+	0.475698i
$834$		0			0
$835$	40.3607			1.39674
$836$		0			0
$837$			46.8328i			1.61878i
$838$		0			0
$839$	10.3910			0.358736			0.179368	−	0.983782i	$-0.442595\pi$
$839$	10.3910			0.358736			0.179368	+	0.983782i	$0.442595\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$		0			0
$843$		−	27.8716i		−	0.959949i
$844$		0			0
$845$	45.3035			1.55849
$846$		0			0
$847$	−0.746512	−	1.18034i	−0.0256505	−	0.0405570i
$848$		0			0
$849$	−6.87539			−0.235963
$850$		0			0
$851$	−13.5279			−0.463729
$852$		0			0
$853$	−45.3035			−1.55116			−0.775582	−	0.631247i	$-0.782543\pi$
$853$	−45.3035			−1.55116			−0.775582	+	0.631247i	$0.782543\pi$
$854$		0			0
$855$			16.1803i			0.553356i
$856$		0			0
$857$			0.922740i			0.0315202i	0.999876	+	0.0157601i	$0.00501680\pi$
$857$			0.922740i			0.0315202i	−0.999876	+	0.0157601i	$0.994983\pi$
$858$		0			0
$859$		−	54.0439i		−	1.84395i	−0.387246	−	0.921976i	$-0.626574\pi$
$859$		−	54.0439i		−	1.84395i	0.387246	−	0.921976i	$-0.373426\pi$
$860$		0			0
$861$	−4.83153	−	7.63932i	−0.164658	−	0.260347i
$862$		0			0
$863$		−	10.0000i		−	0.340404i	−0.985409	−	0.170202i	$-0.945558\pi$
$863$		−	10.0000i		−	0.340404i	0.985409	−	0.170202i	$-0.0544420\pi$
$864$		0			0
$865$	32.1803			1.09416
$866$		0			0
$867$		−	3.44742i		−	0.117081i
$868$		0			0
$869$		−	40.3607i		−	1.36914i
$870$		0			0
$871$	8.48528			0.287513
$872$		0			0
$873$			10.2333i			0.346346i
$874$		0			0
$875$	−19.4164	−	30.7000i	−0.656394	−	1.03785i
$876$		0			0
$877$		−	30.0000i		−	1.01303i	−0.862232	−	0.506514i	$-0.830934\pi$
$877$		−	30.0000i		−	1.01303i	0.862232	−	0.506514i	$-0.169066\pi$
$878$		0			0
$879$			11.0132i			0.371465i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−45.7082			−1.53820			−0.769102	−	0.639126i	$-0.779296\pi$
$883$	−45.7082			−1.53820			−0.769102	+	0.639126i	$0.779296\pi$
$884$		0			0
$885$	−35.9442			−1.20825
$886$		0			0
$887$	−16.5579			−0.555960			−0.277980	−	0.960587i	$-0.589665\pi$
$887$	−16.5579			−0.555960			−0.277980	+	0.960587i	$0.589665\pi$
$888$		0			0
$889$	11.7082	−	7.40492i	0.392681	−	0.248353i
$890$		0			0
$891$	8.76393			0.293603
$892$		0			0
$893$			15.7771i			0.527960i
$894$		0			0
$895$	−39.4404			−1.31835
$896$		0			0
$897$	0.944272			0.0315283
$898$		0			0
$899$		−	20.4667i		−	0.682602i
$900$		0			0
$901$	2.16073			0.0719842
$902$		0			0
$903$	−3.33851	+	2.11146i	−0.111099	+	0.0702649i
$904$		0			0
$905$	13.7082			0.455676
$906$		0			0
$907$	−13.7082			−0.455173			−0.227587	−	0.973758i	$-0.573084\pi$
$907$	−13.7082			−0.455173			−0.227587	+	0.973758i	$0.573084\pi$
$908$		0			0
$909$	−9.77198			−0.324116
$910$		0			0
$911$		−	15.1246i		−	0.501101i	−0.968104	−	0.250550i	$-0.919388\pi$
$911$		−	15.1246i		−	0.501101i	0.968104	−	0.250550i	$-0.0806116\pi$
$912$		0			0
$913$			39.4404i			1.30529i
$914$		0			0
$915$		−	28.9520i		−	0.957123i
$916$		0			0
$917$	−22.4211	+	14.1803i	−0.740409	+	0.468276i
$918$		0			0
$919$		−	10.9443i		−	0.361018i	−0.983573	−	0.180509i	$-0.942225\pi$
$919$		−	10.9443i		−	0.361018i	0.983573	−	0.180509i	$-0.0577745\pi$
$920$		0			0
$921$	9.12461			0.300666
$922$		0			0
$923$		−	8.74032i		−	0.287691i
$924$		0			0
$925$			95.3050i			3.13361i
$926$		0			0
$927$	11.7264			0.385145
$928$		0			0
$929$		−	36.8670i		−	1.20957i	−0.796390	−	0.604783i	$-0.793260\pi$
$929$		−	36.8670i		−	1.20957i	0.796390	−	0.604783i	$-0.206740\pi$
$930$		0			0
$931$	12.3607	+	5.86319i	0.405105	+	0.192158i
$932$		0			0
$933$			12.3607i			0.404670i
$934$		0			0
$935$			54.8328i			1.79322i
$936$		0			0
$937$		−	5.49923i		−	0.179652i	−0.995957	−	0.0898260i	$-0.971369\pi$
$937$		−	5.49923i		−	0.179652i	0.995957	−	0.0898260i	$-0.0286311\pi$
$938$		0			0
$939$	−24.3607			−0.794981
$940$		0			0
$941$	7.45363			0.242981			0.121491	−	0.992593i	$-0.461233\pi$
$941$	7.45363			0.242981			0.121491	+	0.992593i	$0.461233\pi$
$942$		0			0
$943$	−4.83153			−0.157336
$944$		0			0
$945$	37.8885	−	23.9628i	1.23251	−	0.779511i
$946$		0			0
$947$	25.1246			0.816440			0.408220	−	0.912884i	$-0.366150\pi$
$947$	25.1246			0.816440			0.408220	+	0.912884i	$0.366150\pi$
$948$		0			0
$949$		−	12.3607i		−	0.401245i
$950$		0			0
$951$	13.8871			0.450320
$952$		0			0
$953$	18.9443			0.613665			0.306833	−	0.951764i	$-0.400731\pi$
$953$	18.9443			0.613665			0.306833	+	0.951764i	$0.400731\pi$
$954$		0			0
$955$			18.7186i			0.605721i
$956$		0			0
$957$	5.65685			0.182860
$958$		0			0
$959$	4.98915	+	7.88854i	0.161108	+	0.254734i
$960$		0			0
$961$	73.7214			2.37811
$962$		0			0
$963$	30.6525			0.987762
$964$		0			0
$965$	25.0432			0.806169
$966$		0			0
$967$		−	48.6525i		−	1.56456i	−0.622928	−	0.782279i	$-0.714057\pi$
$967$		−	48.6525i		−	1.56456i	0.622928	−	0.782279i	$-0.285943\pi$
$968$		0			0
$969$		−	7.81758i		−	0.251137i
$970$		0			0
$971$		−	27.2526i		−	0.874578i	−0.899321	−	0.437289i	$-0.855939\pi$
$971$		−	27.2526i		−	0.874578i	0.899321	−	0.437289i	$-0.144061\pi$
$972$		0			0
$973$	−41.3948	+	26.1803i	−1.32705	+	0.839303i
$974$		0			0
$975$		−	6.65248i		−	0.213050i
$976$		0			0
$977$	6.58359			0.210628			0.105314	−	0.994439i	$-0.466415\pi$
$977$	6.58359			0.210628			0.105314	+	0.994439i	$0.466415\pi$
$978$		0			0
$979$			31.7804i			1.01571i
$980$		0			0
$981$			27.8885i			0.890413i
$982$		0			0
$983$	46.1776			1.47284			0.736418	−	0.676527i	$-0.236516\pi$
$983$	46.1776			1.47284			0.736418	+	0.676527i	$0.236516\pi$
$984$		0			0
$985$		−	3.90879i		−	0.124544i
$986$		0			0
$987$	15.7771	−	9.97831i	0.502190	−	0.317613i
$988$		0			0
$989$			2.11146i			0.0671404i
$990$		0			0
$991$			10.3607i			0.329118i	0.986367	+	0.164559i	$0.0526201\pi$
$991$			10.3607i			0.329118i	−0.986367	+	0.164559i	$0.947380\pi$
$992$		0			0
$993$			21.9597i			0.696871i
$994$		0			0
$995$	−48.3607			−1.53314
$996$		0			0
$997$	−28.4906			−0.902306			−0.451153	−	0.892447i	$-0.648987\pi$
$997$	−28.4906			−0.902306			−0.451153	+	0.892447i	$0.648987\pi$
$998$		0			0
$999$	−50.0864			−1.58466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.i.895.5		8
4.3	odd	2		1792.2.e.h.895.3		8
7.6	odd	2	inner	1792.2.e.i.895.4		8
8.3	odd	2	inner	1792.2.e.i.895.6		8
8.5	even	2		1792.2.e.h.895.4		8
16.3	odd	4		896.2.f.a.895.6	yes	8
16.5	even	4		896.2.f.b.895.5	yes	8
16.11	odd	4		896.2.f.b.895.3	yes	8
16.13	even	4		896.2.f.a.895.4	yes	8
28.27	even	2		1792.2.e.h.895.6		8
56.13	odd	2		1792.2.e.h.895.5		8
56.27	even	2	inner	1792.2.e.i.895.3		8
112.13	odd	4		896.2.f.a.895.5	yes	8
112.27	even	4		896.2.f.b.895.6	yes	8
112.69	odd	4		896.2.f.b.895.4	yes	8
112.83	even	4		896.2.f.a.895.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.f.a.895.3	✓	8	112.83	even	4
896.2.f.a.895.4	yes	8	16.13	even	4
896.2.f.a.895.5	yes	8	112.13	odd	4
896.2.f.a.895.6	yes	8	16.3	odd	4
896.2.f.b.895.3	yes	8	16.11	odd	4
896.2.f.b.895.4	yes	8	112.69	odd	4
896.2.f.b.895.5	yes	8	16.5	even	4
896.2.f.b.895.6	yes	8	112.27	even	4
1792.2.e.h.895.3		8	4.3	odd	2
1792.2.e.h.895.4		8	8.5	even	2
1792.2.e.h.895.5		8	56.13	odd	2
1792.2.e.h.895.6		8	28.27	even	2
1792.2.e.i.895.3		8	56.27	even	2	inner
1792.2.e.i.895.4		8	7.6	odd	2	inner
1792.2.e.i.895.5		8	1.1	even	1	trivial
1792.2.e.i.895.6		8	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.874032i q^{3} -3.70246 q^{5} +(1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +O(q^{10})\)	\(q+0.874032i q^{3} -3.70246 q^{5} +(1.41421 + 2.23607i) q^{7} +2.23607 q^{9} +3.23607 q^{11} -0.874032 q^{13} -3.23607i q^{15} -4.57649i q^{17} -1.95440i q^{19} +(-1.95440 + 1.23607i) q^{21} +1.23607i q^{23} +8.70820 q^{25} +4.57649i q^{27} -2.00000i q^{29} +10.2333 q^{31} +2.82843i q^{33} +(-5.23607 - 8.27895i) q^{35} +10.9443i q^{37} -0.763932i q^{39} +3.90879i q^{41} +1.70820 q^{43} -8.27895 q^{45} -8.07262 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.00000 q^{51} +0.472136i q^{53} -11.9814 q^{55} +1.70820 q^{57} -11.1074i q^{59} +8.94665 q^{61} +(3.16228 + 5.00000i) q^{63} +3.23607 q^{65} -9.70820 q^{67} -1.08036 q^{69} +10.0000i q^{71} +14.1421i q^{73} +7.61125i q^{75} +(4.57649 + 7.23607i) q^{77} -12.4721i q^{79} +2.70820 q^{81} +12.1877i q^{83} +16.9443i q^{85} +1.74806 q^{87} +9.82068i q^{89} +(-1.23607 - 1.95440i) q^{91} +8.94427i q^{93} +7.23607i q^{95} +4.57649i q^{97} +7.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} + 16 q^{25} - 24 q^{35} - 40 q^{43} - 24 q^{49} + 32 q^{51} - 40 q^{57} + 8 q^{65} - 24 q^{67} - 32 q^{81} + 8 q^{91} + 40 q^{99}+O(q^{100}) \) 8 * q + 8 * q^11 + 16 * q^25 - 24 * q^35 - 40 * q^43 - 24 * q^49 + 32 * q^51 - 40 * q^57 + 8 * q^65 - 24 * q^67 - 32 * q^81 + 8 * q^91 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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