LMFDB - Embedded newform 896.2.f.b.895.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(895,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("896.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.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:	$7.15459602111$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.6
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	896.895
Dual form			896.2.f.b.895.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.874032 q^{3} +3.70246i q^{5} +(-1.41421 + 2.23607i) q^{7} -2.23607 q^{9} +O(q^{10})$	$q+0.874032 q^{3} +3.70246i q^{5} +(-1.41421 + 2.23607i) q^{7} -2.23607 q^{9} -3.23607i q^{11} -0.874032i q^{13} +3.23607i q^{15} +4.57649i q^{17} -1.95440 q^{19} +(-1.23607 + 1.95440i) q^{21} +1.23607i q^{23} -8.70820 q^{25} -4.57649 q^{27} -2.00000 q^{29} +10.2333 q^{31} -2.82843i q^{33} +(-8.27895 - 5.23607i) q^{35} -10.9443 q^{37} -0.763932i q^{39} +3.90879i q^{41} -1.70820i q^{43} -8.27895i q^{45} -8.07262 q^{47} +(-3.00000 - 6.32456i) q^{49} +4.00000i q^{51} -0.472136 q^{53} +11.9814 q^{55} -1.70820 q^{57} +11.1074 q^{59} +8.94665i q^{61} +(3.16228 - 5.00000i) q^{63} +3.23607 q^{65} -9.70820i q^{67} +1.08036i q^{69} +10.0000i q^{71} +14.1421i q^{73} -7.61125 q^{75} +(7.23607 + 4.57649i) q^{77} +12.4721i q^{79} +2.70820 q^{81} +12.1877 q^{83} -16.9443 q^{85} -1.74806 q^{87} +9.82068i q^{89} +(1.95440 + 1.23607i) q^{91} +8.94427 q^{93} -7.23607i q^{95} -4.57649i q^{97} +7.23607i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{21} - 16 q^{25} - 16 q^{29} - 16 q^{37} - 24 q^{49} + 32 q^{53} + 40 q^{57} + 8 q^{65} + 40 q^{77} - 32 q^{81} - 64 q^{85}+O(q^{100}) $ 8 * q + 8 * q^21 - 16 * q^25 - 16 * q^29 - 16 * q^37 - 24 * q^49 + 32 * q^53 + 40 * q^57 + 8 * q^65 + 40 * q^77 - 32 * q^81 - 64 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0.874032			0.504623			0.252311	−	0.967646i	$-0.418809\pi$
$3$	0.874032			0.504623			0.252311	+	0.967646i	$0.418809\pi$
$4$		0			0
$5$			3.70246i			1.65579i	0.560883	+	0.827895i	$0.310462\pi$
$5$			3.70246i			1.65579i	−0.560883	+	0.827895i	$0.689538\pi$
$6$		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
$11$		−	3.23607i		−	0.975711i	−0.872924	−	0.487856i	$-0.837779\pi$
$11$		−	3.23607i		−	0.975711i	0.872924	−	0.487856i	$-0.162221\pi$
$12$		0			0
$13$		−	0.874032i		−	0.242413i	−0.992627	−	0.121206i	$-0.961324\pi$
$13$		−	0.874032i		−	0.242413i	0.992627	−	0.121206i	$-0.0386763\pi$
$14$		0			0
$15$			3.23607i			0.835549i
$16$		0			0
$17$			4.57649i			1.10996i	0.831863	+	0.554981i	$0.187275\pi$
$17$			4.57649i			1.10996i	−0.831863	+	0.554981i	$0.812725\pi$
$18$		0			0
$19$	−1.95440			−0.448369			−0.224184	−	0.974547i	$-0.571972\pi$
$19$	−1.95440			−0.448369			−0.224184	+	0.974547i	$0.571972\pi$
$20$		0			0
$21$	−1.23607	+	1.95440i	−0.269732	+	0.426484i
$22$		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
$27$	−4.57649			−0.880746
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		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
$35$	−8.27895	−	5.23607i	−1.39940	−	0.885057i
$36$		0			0
$37$	−10.9443			−1.79923			−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443			−1.79923			−0.899614	+	0.436687i	$0.856152\pi$
$38$		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
$43$		−	1.70820i		−	0.260499i	−0.991481	−	0.130249i	$-0.958422\pi$
$43$		−	1.70820i		−	0.260499i	0.991481	−	0.130249i	$-0.0415778\pi$
$44$		0			0
$45$		−	8.27895i		−	1.23415i
$46$		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
$51$			4.00000i			0.560112i
$52$		0			0
$53$	−0.472136			−0.0648529			−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136			−0.0648529			−0.0324264	+	0.999474i	$0.510323\pi$
$54$		0			0
$55$	11.9814			1.61557
$56$		0			0
$57$	−1.70820			−0.226257
$58$		0			0
$59$	11.1074			1.44606			0.723029	−	0.690818i	$-0.242749\pi$
$59$	11.1074			1.44606			0.723029	+	0.690818i	$0.242749\pi$
$60$		0			0
$61$			8.94665i			1.14550i	0.819730	+	0.572751i	$0.194124\pi$
$61$			8.94665i			1.14550i	−0.819730	+	0.572751i	$0.805876\pi$
$62$		0			0
$63$	3.16228	−	5.00000i	0.398410	−	0.629941i
$64$		0			0
$65$	3.23607			0.401385
$66$		0			0
$67$		−	9.70820i		−	1.18605i	−0.805186	−	0.593023i	$-0.797934\pi$
$67$		−	9.70820i		−	1.18605i	0.805186	−	0.593023i	$-0.202066\pi$
$68$		0			0
$69$			1.08036i			0.130060i
$70$		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
$75$	−7.61125			−0.878871
$76$		0			0
$77$	7.23607	+	4.57649i	0.824626	+	0.521540i
$78$		0			0
$79$			12.4721i			1.40322i	0.712559	+	0.701612i	$0.247536\pi$
$79$			12.4721i			1.40322i	−0.712559	+	0.701612i	$0.752464\pi$
$80$		0			0
$81$	2.70820			0.300912
$82$		0			0
$83$	12.1877			1.33778			0.668889	−	0.743362i	$-0.266770\pi$
$83$	12.1877			1.33778			0.668889	+	0.743362i	$0.266770\pi$
$84$		0			0
$85$	−16.9443			−1.83786
$86$		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
$91$	1.95440	+	1.23607i	0.204876	+	0.129575i
$92$		0			0
$93$	8.94427			0.927478
$94$		0			0
$95$		−	7.23607i		−	0.742405i
$96$		0			0
$97$		−	4.57649i		−	0.464672i	−0.972636	−	0.232336i	$-0.925363\pi$
$97$		−	4.57649i		−	0.464672i	0.972636	−	0.232336i	$-0.0746369\pi$
$98$		0			0
$99$			7.23607i			0.727252i
$100$		0			0
$101$			4.37016i			0.434847i	0.976077	+	0.217424i	$0.0697653\pi$
$101$			4.37016i			0.434847i	−0.976077	+	0.217424i	$0.930235\pi$
$102$		0			0
$103$	−5.24419			−0.516726			−0.258363	−	0.966048i	$-0.583183\pi$
$103$	−5.24419			−0.516726			−0.258363	+	0.966048i	$0.583183\pi$
$104$		0			0
$105$	−7.23607	−	4.57649i	−0.706168	−	0.446620i
$106$		0			0
$107$		−	13.7082i		−	1.32522i	−0.748964	−	0.662611i	$-0.769448\pi$
$107$		−	13.7082i		−	1.32522i	0.748964	−	0.662611i	$-0.230552\pi$
$108$		0			0
$109$	12.4721			1.19461			0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721			1.19461			0.597307	+	0.802013i	$0.296237\pi$
$110$		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
$115$	−4.57649			−0.426760
$116$		0			0
$117$			1.95440i			0.180684i
$118$		0			0
$119$	−10.2333	−	6.47214i	−0.938089	−	0.593300i
$120$		0			0
$121$	0.527864			0.0479876
$122$		0			0
$123$			3.41641i			0.308047i
$124$		0			0
$125$		−	13.7295i		−	1.22800i
$126$		0			0
$127$			5.23607i			0.464626i	0.972641	+	0.232313i	$0.0746294\pi$
$127$			5.23607i			0.464626i	−0.972641	+	0.232313i	$0.925371\pi$
$128$		0			0
$129$		−	1.49302i		−	0.131454i
$130$		0			0
$131$	10.0270			0.876064			0.438032	−	0.898959i	$-0.355676\pi$
$131$	10.0270			0.876064			0.438032	+	0.898959i	$0.355676\pi$
$132$		0			0
$133$	2.76393	−	4.37016i	0.239663	−	0.378941i
$134$		0			0
$135$		−	16.9443i		−	1.45833i
$136$		0			0
$137$	−3.52786			−0.301406			−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786			−0.301406			−0.150703	+	0.988579i	$0.548154\pi$
$138$		0			0
$139$	−18.5123			−1.57019			−0.785096	−	0.619374i	$-0.787387\pi$
$139$	−18.5123			−1.57019			−0.785096	+	0.619374i	$0.787387\pi$
$140$		0			0
$141$	−7.05573			−0.594199
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$		−	7.40492i		−	0.614945i
$146$		0			0
$147$	−2.62210	−	5.52786i	−0.216267	−	0.455931i
$148$		0			0
$149$	13.4164			1.09911			0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164			1.09911			0.549557	+	0.835456i	$0.314796\pi$
$150$		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
$155$			37.8885i			3.04328i
$156$		0			0
$157$		−	4.37016i		−	0.348777i	−0.984677	−	0.174388i	$-0.944205\pi$
$157$		−	4.37016i		−	0.348777i	0.984677	−	0.174388i	$-0.0557948\pi$
$158$		0			0
$159$	−0.412662			−0.0327262
$160$		0			0
$161$	−2.76393	−	1.74806i	−0.217828	−	0.137767i
$162$		0			0
$163$			12.7639i			0.999748i	0.866098	+	0.499874i	$0.166620\pi$
$163$			12.7639i			0.999748i	−0.866098	+	0.499874i	$0.833380\pi$
$164$		0			0
$165$	10.4721			0.815255
$166$		0			0
$167$	10.9010			0.843548			0.421774	−	0.906701i	$-0.361408\pi$
$167$	10.9010			0.843548			0.421774	+	0.906701i	$0.361408\pi$
$168$		0			0
$169$	12.2361			0.941236
$170$		0			0
$171$	4.37016			0.334195
$172$		0			0
$173$		−	8.69161i		−	0.660811i	−0.943839	−	0.330406i	$-0.892814\pi$
$173$		−	8.69161i		−	0.660811i	0.943839	−	0.330406i	$-0.107186\pi$
$174$		0			0
$175$	12.3153	−	19.4721i	0.930946	−	1.47196i
$176$		0			0
$177$	9.70820			0.729713
$178$		0			0
$179$			10.6525i			0.796203i	0.917341	+	0.398102i	$0.130331\pi$
$179$			10.6525i			0.796203i	−0.917341	+	0.398102i	$0.869669\pi$
$180$		0			0
$181$			3.70246i			0.275202i	0.990488	+	0.137601i	$0.0439391\pi$
$181$			3.70246i			0.275202i	−0.990488	+	0.137601i	$0.956061\pi$
$182$		0			0
$183$			7.81966i			0.578046i
$184$		0			0
$185$		−	40.5207i		−	2.97914i
$186$		0			0
$187$	14.8098			1.08300
$188$		0			0
$189$	6.47214	−	10.2333i	0.470779	−	0.744366i
$190$		0			0
$191$			5.05573i			0.365820i	0.983130	+	0.182910i	$0.0585516\pi$
$191$			5.05573i			0.365820i	−0.983130	+	0.182910i	$0.941448\pi$
$192$		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
$195$	2.82843			0.202548
$196$		0			0
$197$	−1.05573			−0.0752175			−0.0376088	−	0.999293i	$-0.511974\pi$
$197$	−1.05573			−0.0752175			−0.0376088	+	0.999293i	$0.511974\pi$
$198$		0			0
$199$	−13.0618			−0.925925			−0.462962	−	0.886378i	$-0.653213\pi$
$199$	−13.0618			−0.925925			−0.462962	+	0.886378i	$0.653213\pi$
$200$		0			0
$201$		−	8.48528i		−	0.598506i
$202$		0			0
$203$	2.82843	−	4.47214i	0.198517	−	0.313882i
$204$		0			0
$205$	−14.4721			−1.01078
$206$		0			0
$207$		−	2.76393i		−	0.192107i
$208$		0			0
$209$			6.32456i			0.437479i
$210$		0			0
$211$			3.23607i			0.222780i	0.993777	+	0.111390i	$0.0355303\pi$
$211$			3.23607i			0.222780i	−0.993777	+	0.111390i	$0.964470\pi$
$212$		0			0
$213$			8.74032i			0.598877i
$214$		0			0
$215$	6.32456			0.431331
$216$		0			0
$217$	−14.4721	+	22.8825i	−0.982433	+	1.55336i
$218$		0			0
$219$			12.3607i			0.835257i
$220$		0			0
$221$	4.00000			0.269069
$222$		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
$227$	−20.2604			−1.34473			−0.672364	−	0.740221i	$-0.734721\pi$
$227$	−20.2604			−1.34473			−0.672364	+	0.740221i	$0.734721\pi$
$228$		0			0
$229$		−	8.27895i		−	0.547088i	−0.961859	−	0.273544i	$-0.911804\pi$
$229$		−	8.27895i		−	0.547088i	0.961859	−	0.273544i	$-0.0881960\pi$
$230$		0			0
$231$	6.32456	+	4.00000i	0.416125	+	0.263181i
$232$		0			0
$233$	13.4164			0.878938			0.439469	−	0.898258i	$-0.355167\pi$
$233$	13.4164			0.878938			0.439469	+	0.898258i	$0.355167\pi$
$234$		0			0
$235$		−	29.8885i		−	1.94971i
$236$		0			0
$237$			10.9010i			0.708099i
$238$		0			0
$239$		−	12.6525i		−	0.818421i	−0.912440	−	0.409210i	$-0.865804\pi$
$239$		−	12.6525i		−	0.818421i	0.912440	−	0.409210i	$-0.134196\pi$
$240$		0			0
$241$			4.57649i			0.294798i	0.989077	+	0.147399i	$0.0470901\pi$
$241$			4.57649i			0.294798i	−0.989077	+	0.147399i	$0.952910\pi$
$242$		0			0
$243$	16.0965			1.03259
$244$		0			0
$245$	23.4164	−	11.1074i	1.49602	−	0.709624i
$246$		0			0
$247$			1.70820i			0.108690i
$248$		0			0
$249$	10.6525			0.675073
$250$		0			0
$251$	16.0965			1.01600			0.508002	−	0.861356i	$-0.330384\pi$
$251$	16.0965			1.01600			0.508002	+	0.861356i	$0.330384\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$	−14.8098			−0.927428
$256$		0			0
$257$			16.9706i			1.05859i	0.848436	+	0.529297i	$0.177544\pi$
$257$			16.9706i			1.05859i	−0.848436	+	0.529297i	$0.822456\pi$
$258$		0			0
$259$	15.4775	−	24.4721i	0.961727	−	1.52062i
$260$		0			0
$261$	4.47214			0.276818
$262$		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
$267$			8.58359i			0.525307i
$268$		0			0
$269$			16.0965i			0.981423i	0.871322	+	0.490711i	$0.163263\pi$
$269$			16.0965i			0.981423i	−0.871322	+	0.490711i	$0.836737\pi$
$270$		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
$275$			28.1803i			1.69934i
$276$		0			0
$277$	1.05573			0.0634326			0.0317163	−	0.999497i	$-0.489903\pi$
$277$	1.05573			0.0634326			0.0317163	+	0.999497i	$0.489903\pi$
$278$		0			0
$279$	−22.8825			−1.36994
$280$		0			0
$281$	31.8885			1.90231			0.951156	−	0.308712i	$-0.0998980\pi$
$281$	31.8885			1.90231			0.951156	+	0.308712i	$0.0998980\pi$
$282$		0			0
$283$	−7.86629			−0.467602			−0.233801	−	0.972284i	$-0.575116\pi$
$283$	−7.86629			−0.467602			−0.233801	+	0.972284i	$0.575116\pi$
$284$		0			0
$285$		−	6.32456i		−	0.374634i
$286$		0			0
$287$	−8.74032	−	5.52786i	−0.515925	−	0.326299i
$288$		0			0
$289$	−3.94427			−0.232016
$290$		0			0
$291$		−	4.00000i		−	0.234484i
$292$		0			0
$293$		−	12.6004i		−	0.736123i	−0.929801	−	0.368062i	$-0.880022\pi$
$293$		−	12.6004i		−	0.736123i	0.929801	−	0.368062i	$-0.119978\pi$
$294$		0			0
$295$			41.1246i			2.39437i
$296$		0			0
$297$			14.8098i			0.859354i
$298$		0			0
$299$	1.08036			0.0624790
$300$		0			0
$301$	3.81966	+	2.41577i	0.220162	+	0.139242i
$302$		0			0
$303$			3.81966i			0.219434i
$304$		0			0
$305$	−33.1246			−1.89671
$306$		0			0
$307$	−10.4397			−0.595824			−0.297912	−	0.954593i	$-0.596290\pi$
$307$	−10.4397			−0.595824			−0.297912	+	0.954593i	$0.596290\pi$
$308$		0			0
$309$	−4.58359			−0.260751
$310$		0			0
$311$	−14.1421			−0.801927			−0.400963	−	0.916094i	$-0.631325\pi$
$311$	−14.1421			−0.801927			−0.400963	+	0.916094i	$0.631325\pi$
$312$		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
$315$	18.5123	+	11.7082i	1.04305	+	0.659683i
$316$		0			0
$317$	−15.8885			−0.892390			−0.446195	−	0.894936i	$-0.647221\pi$
$317$	−15.8885			−0.892390			−0.446195	+	0.894936i	$0.647221\pi$
$318$		0			0
$319$			6.47214i			0.362370i
$320$		0			0
$321$		−	11.9814i		−	0.668737i
$322$		0			0
$323$		−	8.94427i		−	0.497673i
$324$		0			0
$325$			7.61125i			0.422196i
$326$		0			0
$327$	10.9010			0.602829
$328$		0			0
$329$	11.4164	−	18.0509i	0.629407	−	0.995180i
$330$		0			0
$331$		−	25.1246i		−	1.38097i	−0.723345	−	0.690487i	$-0.757396\pi$
$331$		−	25.1246i		−	1.38097i	0.723345	−	0.690487i	$-0.242604\pi$
$332$		0			0
$333$	24.4721			1.34106
$334$		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
$339$	13.7295			0.745683
$340$		0			0
$341$		−	33.1158i		−	1.79332i
$342$		0			0
$343$	18.3848	+	2.23607i	0.992685	+	0.120736i
$344$		0			0
$345$	−4.00000			−0.215353
$346$		0			0
$347$		−	4.76393i		−	0.255741i	−0.991791	−	0.127871i	$-0.959186\pi$
$347$		−	4.76393i		−	0.255741i	0.991791	−	0.127871i	$-0.0408142\pi$
$348$		0			0
$349$			22.4211i			1.20017i	0.799935	+	0.600087i	$0.204867\pi$
$349$			22.4211i			1.20017i	−0.799935	+	0.600087i	$0.795133\pi$
$350$		0			0
$351$			4.00000i			0.213504i
$352$		0			0
$353$		−	21.8021i		−	1.16041i	−0.814471	−	0.580204i	$-0.802973\pi$
$353$		−	21.8021i		−	1.16041i	0.814471	−	0.580204i	$-0.197027\pi$
$354$		0			0
$355$	−37.0246			−1.96506
$356$		0			0
$357$	−8.94427	−	5.65685i	−0.473381	−	0.299392i
$358$		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
$363$	0.461370			0.0242156
$364$		0			0
$365$	−52.3607			−2.74068
$366$		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
$371$	0.667701	−	1.05573i	0.0346653	−	0.0548107i
$372$		0			0
$373$	−22.9443			−1.18801			−0.594005	−	0.804462i	$-0.702454\pi$
$373$	−22.9443			−1.18801			−0.594005	+	0.804462i	$0.702454\pi$
$374$		0			0
$375$		−	12.0000i		−	0.619677i
$376$		0			0
$377$			1.74806i			0.0900299i
$378$		0			0
$379$			21.7082i			1.11508i	0.830152	+	0.557538i	$0.188254\pi$
$379$			21.7082i			1.11508i	−0.830152	+	0.557538i	$0.811746\pi$
$380$		0			0
$381$			4.57649i			0.234461i
$382$		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
$387$			3.81966i			0.194164i
$388$		0			0
$389$	24.8328			1.25907			0.629537	−	0.776971i	$-0.283245\pi$
$389$	24.8328			1.25907			0.629537	+	0.776971i	$0.283245\pi$
$390$		0			0
$391$	−5.65685			−0.286079
$392$		0			0
$393$	8.76393			0.442082
$394$		0			0
$395$	−46.1776			−2.32345
$396$		0			0
$397$			15.4288i			0.774351i	0.922006	+	0.387175i	$0.126549\pi$
$397$			15.4288i			0.774351i	−0.922006	+	0.387175i	$0.873451\pi$
$398$		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
$403$		−	8.94427i		−	0.445546i
$404$		0			0
$405$			10.0270i			0.498246i
$406$		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
$411$	−3.08347			−0.152096
$412$		0			0
$413$	−15.7082	+	24.8369i	−0.772950	+	1.22214i
$414$		0			0
$415$			45.1246i			2.21508i
$416$		0			0
$417$	−16.1803			−0.792355
$418$		0			0
$419$	−6.94355			−0.339215			−0.169607	−	0.985512i	$-0.554250\pi$
$419$	−6.94355			−0.339215			−0.169607	+	0.985512i	$0.554250\pi$
$420$		0			0
$421$	−9.41641			−0.458928			−0.229464	−	0.973317i	$-0.573697\pi$
$421$	−9.41641			−0.458928			−0.229464	+	0.973317i	$0.573697\pi$
$422$		0			0
$423$	18.0509			0.877666
$424$		0			0
$425$		−	39.8530i		−	1.93316i
$426$		0			0
$427$	−20.0053	−	12.6525i	−0.968125	−	0.612296i
$428$		0			0
$429$	−2.47214			−0.119356
$430$		0			0
$431$		−	15.7082i		−	0.756638i	−0.925675	−	0.378319i	$-0.876502\pi$
$431$		−	15.7082i		−	0.756638i	0.925675	−	0.378319i	$-0.123498\pi$
$432$		0			0
$433$			17.2256i			0.827810i	0.910320	+	0.413905i	$0.135835\pi$
$433$			17.2256i			0.827810i	−0.910320	+	0.413905i	$0.864165\pi$
$434$		0			0
$435$		−	6.47214i		−	0.310315i
$436$		0			0
$437$		−	2.41577i		−	0.115562i
$438$		0			0
$439$	−1.49302			−0.0712582			−0.0356291	−	0.999365i	$-0.511343\pi$
$439$	−1.49302			−0.0712582			−0.0356291	+	0.999365i	$0.511343\pi$
$440$		0			0
$441$	6.70820	+	14.1421i	0.319438	+	0.673435i
$442$		0			0
$443$		−	18.2918i		−	0.869069i	−0.900655	−	0.434535i	$-0.856913\pi$
$443$		−	18.2918i		−	0.869069i	0.900655	−	0.434535i	$-0.143087\pi$
$444$		0			0
$445$	−36.3607			−1.72366
$446$		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
$451$	12.6491			0.595623
$452$		0			0
$453$		−	5.91189i		−	0.277765i
$454$		0			0
$455$	−4.57649	+	7.23607i	−0.214549	+	0.339232i
$456$		0			0
$457$	−0.652476			−0.0305215			−0.0152608	−	0.999884i	$-0.504858\pi$
$457$	−0.652476			−0.0305215			−0.0152608	+	0.999884i	$0.504858\pi$
$458$		0			0
$459$		−	20.9443i		−	0.977595i
$460$		0			0
$461$		−	10.6947i		−	0.498103i	−0.968490	−	0.249051i	$-0.919881\pi$
$461$		−	10.6947i		−	0.498103i	0.968490	−	0.249051i	$-0.0801188\pi$
$462$		0			0
$463$			3.88854i			0.180716i	0.995909	+	0.0903580i	$0.0288011\pi$
$463$			3.88854i			0.180716i	−0.995909	+	0.0903580i	$0.971199\pi$
$464$		0			0
$465$			33.1158i			1.53571i
$466$		0			0
$467$	3.70246			0.171329			0.0856647	−	0.996324i	$-0.472699\pi$
$467$	3.70246			0.171329			0.0856647	+	0.996324i	$0.472699\pi$
$468$		0			0
$469$	21.7082	+	13.7295i	1.00239	+	0.633968i
$470$		0			0
$471$		−	3.81966i		−	0.176001i
$472$		0			0
$473$	−5.52786			−0.254171
$474$		0			0
$475$	17.0193			0.780898
$476$		0			0
$477$	1.05573			0.0483385
$478$		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
$483$	−2.41577	−	1.52786i	−0.109921	−	0.0695202i
$484$		0			0
$485$	16.9443			0.769400
$486$		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
$491$			14.2918i			0.644980i	0.946573	+	0.322490i	$0.104520\pi$
$491$			14.2918i			0.644980i	−0.946573	+	0.322490i	$0.895480\pi$
$492$		0			0
$493$		−	9.15298i		−	0.412230i
$494$		0			0
$495$	−26.7912			−1.20418
$496$		0			0
$497$	−22.3607	−	14.1421i	−1.00301	−	0.634361i
$498$		0			0
$499$			20.7639i			0.929521i	0.885436	+	0.464761i	$0.153860\pi$
$499$			20.7639i			0.929521i	−0.885436	+	0.464761i	$0.846140\pi$
$500$		0			0
$501$	9.52786			0.425674
$502$		0			0
$503$	−22.4698			−1.00188			−0.500939	−	0.865482i	$-0.667012\pi$
$503$	−22.4698			−1.00188			−0.500939	+	0.865482i	$0.667012\pi$
$504$		0			0
$505$	−16.1803			−0.720016
$506$		0			0
$507$	10.6947			0.474969
$508$		0			0
$509$			9.77198i			0.433135i	0.976268	+	0.216568i	$0.0694862\pi$
$509$			9.77198i			0.433135i	−0.976268	+	0.216568i	$0.930514\pi$
$510$		0			0
$511$	−31.6228	−	20.0000i	−1.39891	−	0.884748i
$512$		0			0
$513$	8.94427			0.394899
$514$		0			0
$515$		−	19.4164i		−	0.855589i
$516$		0			0
$517$			26.1235i			1.14891i
$518$		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
$523$	−5.86319			−0.256379			−0.128190	−	0.991750i	$-0.540917\pi$
$523$	−5.86319			−0.256379			−0.128190	+	0.991750i	$0.540917\pi$
$524$		0			0
$525$	10.7639	−	17.0193i	0.469777	−	0.742782i
$526$		0			0
$527$			46.8328i			2.04007i
$528$		0			0
$529$	21.4721			0.933571
$530$		0			0
$531$	−24.8369			−1.07783
$532$		0			0
$533$	3.41641			0.147981
$534$		0			0
$535$	50.7541			2.19429
$536$		0			0
$537$			9.31061i			0.401782i
$538$		0			0
$539$	−20.4667	+	9.70820i	−0.881563	+	0.418162i
$540$		0			0
$541$	0.472136			0.0202987			0.0101494	−	0.999948i	$-0.496769\pi$
$541$	0.472136			0.0202987			0.0101494	+	0.999948i	$0.496769\pi$
$542$		0			0
$543$			3.23607i			0.138873i
$544$		0			0
$545$			46.1776i			1.97803i
$546$		0			0
$547$		−	37.1246i		−	1.58733i	−0.608353	−	0.793667i	$-0.708169\pi$
$547$		−	37.1246i		−	1.58733i	0.608353	−	0.793667i	$-0.291831\pi$
$548$		0			0
$549$		−	20.0053i		−	0.853806i
$550$		0			0
$551$	3.90879			0.166520
$552$		0			0
$553$	−27.8885	−	17.6383i	−1.18594	−	0.750055i
$554$		0			0
$555$		−	35.4164i		−	1.50334i
$556$		0			0
$557$	−2.36068			−0.100025			−0.0500126	−	0.998749i	$-0.515926\pi$
$557$	−2.36068			−0.100025			−0.0500126	+	0.998749i	$0.515926\pi$
$558$		0			0
$559$	−1.49302			−0.0631482
$560$		0			0
$561$	12.9443			0.546508
$562$		0			0
$563$	13.6808			0.576576			0.288288	−	0.957544i	$-0.406914\pi$
$563$	13.6808			0.576576			0.288288	+	0.957544i	$0.406914\pi$
$564$		0			0
$565$			58.1590i			2.44677i
$566$		0			0
$567$	−3.82998	+	6.05573i	−0.160844	+	0.254317i
$568$		0			0
$569$	24.0689			1.00902			0.504510	−	0.863406i	$-0.331673\pi$
$569$	24.0689			1.00902			0.504510	+	0.863406i	$0.331673\pi$
$570$		0			0
$571$			8.18034i			0.342337i	0.985242	+	0.171168i	$0.0547542\pi$
$571$			8.18034i			0.342337i	−0.985242	+	0.171168i	$0.945246\pi$
$572$		0			0
$573$			4.41887i			0.184601i
$574$		0			0
$575$		−	10.7639i		−	0.448887i
$576$		0			0
$577$			9.97831i			0.415402i	0.978192	+	0.207701i	$0.0665982\pi$
$577$			9.97831i			0.415402i	−0.978192	+	0.207701i	$0.933402\pi$
$578$		0			0
$579$	−5.91189			−0.245690
$580$		0			0
$581$	−17.2361	+	27.2526i	−0.715073	+	1.13063i
$582$		0			0
$583$			1.52786i			0.0632777i
$584$		0			0
$585$	−7.23607			−0.299175
$586$		0			0
$587$	38.5663			1.59180			0.795901	−	0.605426i	$-0.206997\pi$
$587$	38.5663			1.59180			0.795901	+	0.605426i	$0.206997\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$	−0.922740			−0.0379565
$592$		0			0
$593$		−	11.3137i		−	0.464598i	−0.972644	−	0.232299i	$-0.925375\pi$
$593$		−	11.3137i		−	0.464598i	0.972644	−	0.232299i	$-0.0746248\pi$
$594$		0			0
$595$	23.9628	−	37.8885i	0.982380	−	1.55328i
$596$		0			0
$597$	−11.4164			−0.467242
$598$		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
$603$			21.7082i			0.884026i
$604$		0			0
$605$			1.95440i			0.0794575i
$606$		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
$611$			7.05573i			0.285444i
$612$		0			0
$613$	−17.4164			−0.703442			−0.351721	−	0.936105i	$-0.614403\pi$
$613$	−17.4164			−0.703442			−0.351721	+	0.936105i	$0.614403\pi$
$614$		0			0
$615$	−12.6491			−0.510061
$616$		0			0
$617$	2.76393			0.111272			0.0556359	−	0.998451i	$-0.482281\pi$
$617$	2.76393			0.111272			0.0556359	+	0.998451i	$0.482281\pi$
$618$		0			0
$619$	17.4319			0.700649			0.350324	−	0.936628i	$-0.386071\pi$
$619$	17.4319			0.700649			0.350324	+	0.936628i	$0.386071\pi$
$620$		0			0
$621$		−	5.65685i		−	0.227002i
$622$		0			0
$623$	−21.9597	−	13.8885i	−0.879797	−	0.556433i
$624$		0			0
$625$	7.29180			0.291672
$626$		0			0
$627$			5.52786i			0.220762i
$628$		0			0
$629$		−	50.0864i		−	1.99707i
$630$		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
$635$	−19.3863			−0.769323
$636$		0			0
$637$	−5.52786	+	2.62210i	−0.219022	+	0.103891i
$638$		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
$643$	−18.0996			−0.713780			−0.356890	−	0.934146i	$-0.616163\pi$
$643$	−18.0996			−0.713780			−0.356890	+	0.934146i	$0.616163\pi$
$644$		0			0
$645$	5.52786			0.217659
$646$		0			0
$647$	16.5579			0.650958			0.325479	−	0.945549i	$-0.394474\pi$
$647$	16.5579			0.650958			0.325479	+	0.945549i	$0.394474\pi$
$648$		0			0
$649$		−	35.9442i		−	1.41093i
$650$		0			0
$651$	−12.6491	+	20.0000i	−0.495758	+	0.783862i
$652$		0			0
$653$	13.4164			0.525025			0.262512	−	0.964929i	$-0.415449\pi$
$653$	13.4164			0.525025			0.262512	+	0.964929i	$0.415449\pi$
$654$		0			0
$655$			37.1246i			1.45058i
$656$		0			0
$657$		−	31.6228i		−	1.23372i
$658$		0			0
$659$			35.0132i			1.36392i	0.731390	+	0.681959i	$0.238872\pi$
$659$			35.0132i			1.36392i	−0.731390	+	0.681959i	$0.761128\pi$
$660$		0			0
$661$		−	33.5772i		−	1.30600i	−0.757358	−	0.653000i	$-0.773510\pi$
$661$		−	33.5772i		−	1.30600i	0.757358	−	0.653000i	$-0.226490\pi$
$662$		0			0
$663$	3.49613			0.135778
$664$		0			0
$665$	16.1803	+	10.2333i	0.627447	+	0.396832i
$666$		0			0
$667$		−	2.47214i		−	0.0957215i
$668$		0			0
$669$	−19.0557			−0.736737
$670$		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
$675$	39.8530			1.53394
$676$		0			0
$677$			8.53399i			0.327988i	0.986461	+	0.163994i	$0.0524377\pi$
$677$			8.53399i			0.327988i	−0.986461	+	0.163994i	$0.947562\pi$
$678$		0			0
$679$	10.2333	+	6.47214i	0.392720	+	0.248378i
$680$		0			0
$681$	−17.7082			−0.678580
$682$		0			0
$683$			12.1803i			0.466068i	0.972469	+	0.233034i	$0.0748653\pi$
$683$			12.1803i			0.466068i	−0.972469	+	0.233034i	$0.925135\pi$
$684$		0			0
$685$		−	13.0618i		−	0.499065i
$686$		0			0
$687$		−	7.23607i		−	0.276073i
$688$		0			0
$689$			0.412662i			0.0157212i
$690$		0			0
$691$	−28.7456			−1.09354			−0.546768	−	0.837284i	$-0.684142\pi$
$691$	−28.7456			−1.09354			−0.546768	+	0.837284i	$0.684142\pi$
$692$		0			0
$693$	−16.1803	−	10.2333i	−0.614640	−	0.388733i
$694$		0			0
$695$		−	68.5410i		−	2.59991i
$696$		0			0
$697$	−17.8885			−0.677577
$698$		0			0
$699$	11.7264			0.443532
$700$		0			0
$701$	40.4721			1.52861			0.764306	−	0.644854i	$-0.223082\pi$
$701$	40.4721			1.52861			0.764306	+	0.644854i	$0.223082\pi$
$702$		0			0
$703$	21.3894			0.806718
$704$		0			0
$705$		−	26.1235i		−	0.983870i
$706$		0			0
$707$	−9.77198	−	6.18034i	−0.367513	−	0.232436i
$708$		0			0
$709$	−3.52786			−0.132492			−0.0662459	−	0.997803i	$-0.521102\pi$
$709$	−3.52786			−0.132492			−0.0662459	+	0.997803i	$0.521102\pi$
$710$		0			0
$711$		−	27.8885i		−	1.04590i
$712$		0			0
$713$			12.6491i			0.473713i
$714$		0			0
$715$		−	10.4721i		−	0.391636i
$716$		0			0
$717$		−	11.0587i		−	0.412994i
$718$		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
$723$			4.00000i			0.148762i
$724$		0			0
$725$	17.4164			0.646829
$726$		0			0
$727$	16.5579			0.614099			0.307049	−	0.951694i	$-0.400658\pi$
$727$	16.5579			0.614099			0.307049	+	0.951694i	$0.400658\pi$
$728$		0			0
$729$	5.94427			0.220158
$730$		0			0
$731$	7.81758			0.289144
$732$		0			0
$733$		−	48.1320i		−	1.77779i	−0.458106	−	0.888897i	$-0.651472\pi$
$733$		−	48.1320i		−	1.77779i	0.458106	−	0.888897i	$-0.348528\pi$
$734$		0			0
$735$	20.4667	−	9.70820i	0.754925	−	0.358092i
$736$		0			0
$737$	−31.4164			−1.15724
$738$		0			0
$739$			51.5967i			1.89802i	0.315250	+	0.949009i	$0.397911\pi$
$739$			51.5967i			1.89802i	−0.315250	+	0.949009i	$0.602089\pi$
$740$		0			0
$741$			1.49302i			0.0548476i
$742$		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
$747$	−27.2526			−0.997121
$748$		0			0
$749$	30.6525	+	19.3863i	1.12002	+	0.708361i
$750$		0			0
$751$		−	14.5410i		−	0.530609i	−0.964165	−	0.265305i	$-0.914527\pi$
$751$		−	14.5410i		−	0.530609i	0.964165	−	0.265305i	$-0.0854725\pi$
$752$		0			0
$753$	14.0689			0.512699
$754$		0			0
$755$	25.0432			0.911415
$756$		0			0
$757$	−22.3607			−0.812713			−0.406356	−	0.913715i	$-0.633201\pi$
$757$	−22.3607			−0.812713			−0.406356	+	0.913715i	$0.633201\pi$
$758$		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
$763$	−17.6383	+	27.8885i	−0.638548	+	1.00963i
$764$		0			0
$765$	37.8885			1.36986
$766$		0			0
$767$		−	9.70820i		−	0.350543i
$768$		0			0
$769$		−	39.8530i		−	1.43714i	−0.695456	−	0.718568i	$-0.744798\pi$
$769$		−	39.8530i		−	1.43714i	0.695456	−	0.718568i	$-0.255202\pi$
$770$		0			0
$771$			14.8328i			0.534191i
$772$		0			0
$773$		−	22.9312i		−	0.824777i	−0.911008	−	0.412388i	$-0.864695\pi$
$773$		−	22.9312i		−	0.824777i	0.911008	−	0.412388i	$-0.135305\pi$
$774$		0			0
$775$	−89.1141			−3.20107
$776$		0			0
$777$	13.5279	−	21.3894i	0.485309	−	0.767342i
$778$		0			0
$779$		−	7.63932i		−	0.273707i
$780$		0			0
$781$	32.3607			1.15796
$782$		0			0
$783$	9.15298			0.327101
$784$		0			0
$785$	16.1803			0.577501
$786$		0			0
$787$	−7.61125			−0.271312			−0.135656	−	0.990756i	$-0.543314\pi$
$787$	−7.61125			−0.271312			−0.135656	+	0.990756i	$0.543314\pi$
$788$		0			0
$789$		−	27.8716i		−	0.992256i
$790$		0			0
$791$	−22.2148	+	35.1246i	−0.789866	+	1.24889i
$792$		0			0
$793$	7.81966			0.277684
$794$		0			0
$795$		−	1.52786i		−	0.0541878i
$796$		0			0
$797$			10.4397i			0.369792i	0.982758	+	0.184896i	$0.0591949\pi$
$797$			10.4397i			0.369792i	−0.982758	+	0.184896i	$0.940805\pi$
$798$		0			0
$799$		−	36.9443i		−	1.30699i
$800$		0			0
$801$		−	21.9597i		−	0.775908i
$802$		0			0
$803$	45.7649			1.61501
$804$		0			0
$805$	6.47214	−	10.2333i	0.228113	−	0.360678i
$806$		0			0
$807$			14.0689i			0.495248i
$808$		0			0
$809$	56.0689			1.97128			0.985638	−	0.168869i	$-0.0540115\pi$
$809$	56.0689			1.97128			0.985638	+	0.168869i	$0.0540115\pi$
$810$		0			0
$811$	−29.6684			−1.04180			−0.520899	−	0.853618i	$-0.674403\pi$
$811$	−29.6684			−1.04180			−0.520899	+	0.853618i	$0.674403\pi$
$812$		0			0
$813$	11.7771			0.413040
$814$		0			0
$815$	−47.2579			−1.65537
$816$		0			0
$817$			3.33851i			0.116800i
$818$		0			0
$819$	−4.37016	−	2.76393i	−0.152706	−	0.0965796i
$820$		0			0
$821$	23.8885			0.833716			0.416858	−	0.908972i	$-0.363131\pi$
$821$	23.8885			0.833716			0.416858	+	0.908972i	$0.363131\pi$
$822$		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
$827$		−	34.0689i		−	1.18469i	−0.805684	−	0.592346i	$-0.798202\pi$
$827$		−	34.0689i		−	1.18469i	0.805684	−	0.592346i	$-0.201798\pi$
$828$		0			0
$829$		−	13.6808i		−	0.475153i	−0.971369	−	0.237576i	$-0.923647\pi$
$829$		−	13.6808i		−	0.475153i	0.971369	−	0.237576i	$-0.0763530\pi$
$830$		0			0
$831$	0.922740			0.0320095
$832$		0			0
$833$	28.9443	−	13.7295i	1.00286	−	0.475698i
$834$		0			0
$835$			40.3607i			1.39674i
$836$		0			0
$837$	−46.8328			−1.61878
$838$		0			0
$839$	−10.3910			−0.358736			−0.179368	−	0.983782i	$-0.557405\pi$
$839$	−10.3910			−0.358736			−0.179368	+	0.983782i	$0.557405\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	27.8716			0.959949
$844$		0			0
$845$			45.3035i			1.55849i
$846$		0			0
$847$	−0.746512	+	1.18034i	−0.0256505	+	0.0405570i
$848$		0			0
$849$	−6.87539			−0.235963
$850$		0			0
$851$		−	13.5279i		−	0.463729i
$852$		0			0
$853$			45.3035i			1.55116i	0.631247	+	0.775582i	$0.282543\pi$
$853$			45.3035i			1.55116i	−0.631247	+	0.775582i	$0.717457\pi$
$854$		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
$859$	54.0439			1.84395			0.921976	−	0.387246i	$-0.126574\pi$
$859$	54.0439			1.84395			0.921976	+	0.387246i	$0.126574\pi$
$860$		0			0
$861$	−7.63932	−	4.83153i	−0.260347	−	0.164658i
$862$		0			0
$863$			10.0000i			0.340404i	0.985409	+	0.170202i	$0.0544420\pi$
$863$			10.0000i			0.340404i	−0.985409	+	0.170202i	$0.945558\pi$
$864$		0			0
$865$	32.1803			1.09416
$866$		0			0
$867$	−3.44742			−0.117081
$868$		0			0
$869$	40.3607			1.36914
$870$		0			0
$871$	−8.48528			−0.287513
$872$		0			0
$873$			10.2333i			0.346346i
$874$		0			0
$875$	30.7000	+	19.4164i	1.03785	+	0.656394i
$876$		0			0
$877$	−30.0000			−1.01303			−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000			−1.01303			−0.506514	+	0.862232i	$0.669066\pi$
$878$		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
$883$		−	45.7082i		−	1.53820i	−0.639126	−	0.769102i	$-0.720704\pi$
$883$		−	45.7082i		−	1.53820i	0.639126	−	0.769102i	$-0.279296\pi$
$884$		0			0
$885$			35.9442i			1.20825i
$886$		0			0
$887$	16.5579			0.555960			0.277980	−	0.960587i	$-0.410335\pi$
$887$	16.5579			0.555960			0.277980	+	0.960587i	$0.410335\pi$
$888$		0			0
$889$	−11.7082	−	7.40492i	−0.392681	−	0.248353i
$890$		0			0
$891$		−	8.76393i		−	0.293603i
$892$		0			0
$893$	15.7771			0.527960
$894$		0			0
$895$	−39.4404			−1.31835
$896$		0			0
$897$	0.944272			0.0315283
$898$		0			0
$899$	−20.4667			−0.682602
$900$		0			0
$901$		−	2.16073i		−	0.0719842i
$902$		0			0
$903$	3.33851	+	2.11146i	0.111099	+	0.0702649i
$904$		0			0
$905$	−13.7082			−0.455676
$906$		0			0
$907$			13.7082i			0.455173i	0.973758	+	0.227587i	$0.0730835\pi$
$907$			13.7082i			0.455173i	−0.973758	+	0.227587i	$0.926916\pi$
$908$		0			0
$909$		−	9.77198i		−	0.324116i
$910$		0			0
$911$			15.1246i			0.501101i	0.968104	+	0.250550i	$0.0806116\pi$
$911$			15.1246i			0.501101i	−0.968104	+	0.250550i	$0.919388\pi$
$912$		0			0
$913$		−	39.4404i		−	1.30529i
$914$		0			0
$915$	−28.9520			−0.957123
$916$		0			0
$917$	−14.1803	+	22.4211i	−0.468276	+	0.740409i
$918$		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
$923$	8.74032			0.287691
$924$		0			0
$925$	95.3050			3.13361
$926$		0			0
$927$	11.7264			0.385145
$928$		0			0
$929$			36.8670i			1.20957i	0.796390	+	0.604783i	$0.206740\pi$
$929$			36.8670i			1.20957i	−0.796390	+	0.604783i	$0.793260\pi$
$930$		0			0
$931$	5.86319	+	12.3607i	0.192158	+	0.405105i
$932$		0			0
$933$	−12.3607			−0.404670
$934$		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
$939$			24.3607i			0.794981i
$940$		0			0
$941$			7.45363i			0.242981i	0.992593	+	0.121491i	$0.0387674\pi$
$941$			7.45363i			0.242981i	−0.992593	+	0.121491i	$0.961233\pi$
$942$		0			0
$943$	−4.83153			−0.157336
$944$		0			0
$945$	37.8885	+	23.9628i	1.23251	+	0.779511i
$946$		0			0
$947$			25.1246i			0.816440i	0.912884	+	0.408220i	$0.133850\pi$
$947$			25.1246i			0.816440i	−0.912884	+	0.408220i	$0.866150\pi$
$948$		0			0
$949$	12.3607			0.401245
$950$		0			0
$951$	−13.8871			−0.450320
$952$		0			0
$953$	−18.9443			−0.613665			−0.306833	−	0.951764i	$-0.599269\pi$
$953$	−18.9443			−0.613665			−0.306833	+	0.951764i	$0.599269\pi$
$954$		0			0
$955$	−18.7186			−0.605721
$956$		0			0
$957$			5.65685i			0.182860i
$958$		0			0
$959$	4.98915	−	7.88854i	0.161108	−	0.254734i
$960$		0			0
$961$	73.7214			2.37811
$962$		0			0
$963$			30.6525i			0.987762i
$964$		0			0
$965$		−	25.0432i		−	0.806169i
$966$		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
$971$	27.2526			0.874578			0.437289	−	0.899321i	$-0.355939\pi$
$971$	27.2526			0.874578			0.437289	+	0.899321i	$0.355939\pi$
$972$		0			0
$973$	26.1803	−	41.3948i	0.839303	−	1.32705i
$974$		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
$979$	31.7804			1.01571
$980$		0			0
$981$	−27.8885			−0.890413
$982$		0			0
$983$	−46.1776			−1.47284			−0.736418	−	0.676527i	$-0.763484\pi$
$983$	−46.1776			−1.47284			−0.736418	+	0.676527i	$0.763484\pi$
$984$		0			0
$985$		−	3.90879i		−	0.124544i
$986$		0			0
$987$	9.97831	−	15.7771i	0.317613	−	0.502190i
$988$		0			0
$989$	2.11146			0.0671404
$990$		0			0
$991$		−	10.3607i		−	0.329118i	−0.986367	−	0.164559i	$-0.947380\pi$
$991$		−	10.3607i		−	0.329118i	0.986367	−	0.164559i	$-0.0526201\pi$
$992$		0			0
$993$		−	21.9597i		−	0.696871i
$994$		0			0
$995$		−	48.3607i		−	1.53314i
$996$		0			0
$997$			28.4906i			0.902306i	0.892447	+	0.451153i	$0.148987\pi$
$997$			28.4906i			0.902306i	−0.892447	+	0.451153i	$0.851013\pi$
$998$		0			0
$999$	50.0864			1.58466

Twists

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

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

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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.874032 q^{3} +3.70246i q^{5} +(-1.41421 + 2.23607i) q^{7} -2.23607 q^{9} +O(q^{10})\)	\(q+0.874032 q^{3} +3.70246i q^{5} +(-1.41421 + 2.23607i) q^{7} -2.23607 q^{9} -3.23607i q^{11} -0.874032i q^{13} +3.23607i q^{15} +4.57649i q^{17} -1.95440 q^{19} +(-1.23607 + 1.95440i) q^{21} +1.23607i q^{23} -8.70820 q^{25} -4.57649 q^{27} -2.00000 q^{29} +10.2333 q^{31} -2.82843i q^{33} +(-8.27895 - 5.23607i) q^{35} -10.9443 q^{37} -0.763932i q^{39} +3.90879i q^{41} -1.70820i q^{43} -8.27895i q^{45} -8.07262 q^{47} +(-3.00000 - 6.32456i) q^{49} +4.00000i q^{51} -0.472136 q^{53} +11.9814 q^{55} -1.70820 q^{57} +11.1074 q^{59} +8.94665i q^{61} +(3.16228 - 5.00000i) q^{63} +3.23607 q^{65} -9.70820i q^{67} +1.08036i q^{69} +10.0000i q^{71} +14.1421i q^{73} -7.61125 q^{75} +(7.23607 + 4.57649i) q^{77} +12.4721i q^{79} +2.70820 q^{81} +12.1877 q^{83} -16.9443 q^{85} -1.74806 q^{87} +9.82068i q^{89} +(1.95440 + 1.23607i) q^{91} +8.94427 q^{93} -7.23607i q^{95} -4.57649i q^{97} +7.23607i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{21} - 16 q^{25} - 16 q^{29} - 16 q^{37} - 24 q^{49} + 32 q^{53} + 40 q^{57} + 8 q^{65} + 40 q^{77} - 32 q^{81} - 64 q^{85}+O(q^{100}) \) 8 * q + 8 * q^21 - 16 * q^25 - 16 * q^29 - 16 * q^37 - 24 * q^49 + 32 * q^53 + 40 * q^57 + 8 * q^65 + 40 * q^77 - 32 * q^81 - 64 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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