LMFDB - Embedded newform 896.2.f.c.895.8

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:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*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			895.8
Root			$1.54779 + 1.54779i$ of defining polynomial
Character	$\chi$	$=$	896.895
Dual form			896.2.f.c.895.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09557 q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +6.58258 q^{9} +O(q^{10})$	$q+3.09557 q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +6.58258 q^{9} +5.58258i q^{11} +1.80341i q^{13} +9.58258i q^{15} -1.29217i q^{17} -4.38774 q^{19} +(-7.58258 - 3.09557i) q^{21} -3.58258i q^{23} -4.58258 q^{25} +11.0901 q^{27} +6.00000 q^{29} +1.29217 q^{31} +17.2813i q^{33} +(3.09557 - 7.58258i) q^{35} +2.00000 q^{37} +5.58258i q^{39} -6.19115i q^{41} -5.58258i q^{43} +20.3768i q^{45} +1.29217 q^{47} +(5.00000 + 4.89898i) q^{49} -4.00000i q^{51} +9.16515 q^{53} -17.2813 q^{55} -13.5826 q^{57} +1.80341 q^{59} +6.70239i q^{61} +(-16.1240 - 6.58258i) q^{63} -5.58258 q^{65} -13.5826i q^{67} -11.0901i q^{69} -9.16515i q^{71} +7.48331i q^{73} -14.1857 q^{75} +(5.58258 - 13.6745i) q^{77} -2.00000i q^{79} +14.5826 q^{81} +3.09557 q^{83} +4.00000 q^{85} +18.5734 q^{87} +7.48331i q^{89} +(1.80341 - 4.41742i) q^{91} +4.00000 q^{93} -13.5826i q^{95} -3.87650i q^{97} +36.7477i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} - 24 q^{21} + 48 q^{29} + 16 q^{37} + 40 q^{49} - 72 q^{57} - 8 q^{65} + 8 q^{77} + 80 q^{81} + 32 q^{85} + 32 q^{93}+O(q^{100}) $ 8 * q + 16 * q^9 - 24 * q^21 + 48 * q^29 + 16 * q^37 + 40 * q^49 - 72 * q^57 - 8 * q^65 + 8 * q^77 + 80 * q^81 + 32 * q^85 + 32 * q^93

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
$3$	3.09557			1.78723			0.893615	−	0.448834i	$-0.148161\pi$
$3$	3.09557			1.78723			0.893615	+	0.448834i	$0.148161\pi$
$4$		0			0
$5$			3.09557i			1.38438i	0.721714	+	0.692191i	$0.243355\pi$
$5$			3.09557i			1.38438i	−0.721714	+	0.692191i	$0.756645\pi$
$6$		0			0
$7$	−2.44949	−	1.00000i	−0.925820	−	0.377964i
$7$	−2.44949	−	1.00000i	−0.925820	−	0.377964i
$8$		0			0
$9$	6.58258			2.19419
$10$		0			0
$11$			5.58258i			1.68321i	0.540094	+	0.841605i	$0.318389\pi$
$11$			5.58258i			1.68321i	−0.540094	+	0.841605i	$0.681611\pi$
$12$		0			0
$13$			1.80341i			0.500175i	0.968223	+	0.250087i	$0.0804594\pi$
$13$			1.80341i			0.500175i	−0.968223	+	0.250087i	$0.919541\pi$
$14$		0			0
$15$			9.58258i			2.47421i
$16$		0			0
$17$		−	1.29217i		−	0.313397i	−0.987647	−	0.156698i	$-0.949915\pi$
$17$		−	1.29217i		−	0.313397i	0.987647	−	0.156698i	$-0.0500850\pi$
$18$		0			0
$19$	−4.38774			−1.00662			−0.503308	−	0.864107i	$-0.667884\pi$
$19$	−4.38774			−1.00662			−0.503308	+	0.864107i	$0.667884\pi$
$20$		0			0
$21$	−7.58258	−	3.09557i	−1.65465	−	0.675510i
$22$		0			0
$23$		−	3.58258i		−	0.747019i	−0.927626	−	0.373509i	$-0.878154\pi$
$23$		−	3.58258i		−	0.747019i	0.927626	−	0.373509i	$-0.121846\pi$
$24$		0			0
$25$	−4.58258			−0.916515
$26$		0			0
$27$	11.0901			2.13430
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	1.29217			0.232080			0.116040	−	0.993245i	$-0.462980\pi$
$31$	1.29217			0.232080			0.116040	+	0.993245i	$0.462980\pi$
$32$		0			0
$33$			17.2813i			3.00828i
$34$		0			0
$35$	3.09557	−	7.58258i	0.523247	−	1.28169i
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$			5.58258i			0.893928i
$40$		0			0
$41$		−	6.19115i		−	0.966895i	−0.875373	−	0.483447i	$-0.839384\pi$
$41$		−	6.19115i		−	0.966895i	0.875373	−	0.483447i	$-0.160616\pi$
$42$		0			0
$43$		−	5.58258i		−	0.851335i	−0.904880	−	0.425667i	$-0.860039\pi$
$43$		−	5.58258i		−	0.851335i	0.904880	−	0.425667i	$-0.139961\pi$
$44$		0			0
$45$			20.3768i			3.03760i
$46$		0			0
$47$	1.29217			0.188482			0.0942410	−	0.995549i	$-0.469958\pi$
$47$	1.29217			0.188482			0.0942410	+	0.995549i	$0.469958\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$		−	4.00000i		−	0.560112i
$52$		0			0
$53$	9.16515			1.25893			0.629465	−	0.777029i	$-0.283274\pi$
$53$	9.16515			1.25893			0.629465	+	0.777029i	$0.283274\pi$
$54$		0			0
$55$	−17.2813			−2.33021
$56$		0			0
$57$	−13.5826			−1.79906
$58$		0			0
$59$	1.80341			0.234783			0.117392	−	0.993086i	$-0.462547\pi$
$59$	1.80341			0.234783			0.117392	+	0.993086i	$0.462547\pi$
$60$		0			0
$61$			6.70239i			0.858153i	0.903268	+	0.429076i	$0.141161\pi$
$61$			6.70239i			0.858153i	−0.903268	+	0.429076i	$0.858839\pi$
$62$		0			0
$63$	−16.1240	−	6.58258i	−2.03143	−	0.829327i
$64$		0			0
$65$	−5.58258			−0.692433
$66$		0			0
$67$		−	13.5826i		−	1.65938i	−0.558228	−	0.829688i	$-0.688518\pi$
$67$		−	13.5826i		−	1.65938i	0.558228	−	0.829688i	$-0.311482\pi$
$68$		0			0
$69$		−	11.0901i		−	1.33509i
$70$		0			0
$71$		−	9.16515i		−	1.08770i	−0.839181	−	0.543852i	$-0.816965\pi$
$71$		−	9.16515i		−	1.08770i	0.839181	−	0.543852i	$-0.183035\pi$
$72$		0			0
$73$			7.48331i			0.875856i	0.899010	+	0.437928i	$0.144287\pi$
$73$			7.48331i			0.875856i	−0.899010	+	0.437928i	$0.855713\pi$
$74$		0			0
$75$	−14.1857			−1.63802
$76$		0			0
$77$	5.58258	−	13.6745i	0.636194	−	1.55835i
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$	14.5826			1.62029
$82$		0			0
$83$	3.09557			0.339783			0.169892	−	0.985463i	$-0.445658\pi$
$83$	3.09557			0.339783			0.169892	+	0.985463i	$0.445658\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	18.5734			1.99128
$88$		0			0
$89$			7.48331i			0.793230i	0.917985	+	0.396615i	$0.129815\pi$
$89$			7.48331i			0.793230i	−0.917985	+	0.396615i	$0.870185\pi$
$90$		0			0
$91$	1.80341	−	4.41742i	0.189048	−	0.463072i
$92$		0			0
$93$	4.00000			0.414781
$94$		0			0
$95$		−	13.5826i		−	1.39354i
$96$		0			0
$97$		−	3.87650i		−	0.393599i	−0.980444	−	0.196800i	$-0.936945\pi$
$97$		−	3.87650i		−	0.393599i	0.980444	−	0.196800i	$-0.0630548\pi$
$98$		0			0
$99$			36.7477i			3.69329i
$100$		0			0
$101$		−	14.1857i		−	1.41153i	−0.708446	−	0.705765i	$-0.750603\pi$
$101$		−	14.1857i		−	1.41153i	0.708446	−	0.705765i	$-0.249397\pi$
$102$		0			0
$103$	6.19115			0.610032			0.305016	−	0.952347i	$-0.401338\pi$
$103$	6.19115			0.610032			0.305016	+	0.952347i	$0.401338\pi$
$104$		0			0
$105$	9.58258	−	23.4724i	0.935164	−	2.29067i
$106$		0			0
$107$			6.41742i			0.620396i	0.950672	+	0.310198i	$0.100395\pi$
$107$			6.41742i			0.620396i	−0.950672	+	0.310198i	$0.899605\pi$
$108$		0			0
$109$	−13.1652			−1.26099			−0.630496	−	0.776192i	$-0.717149\pi$
$109$	−13.1652			−1.26099			−0.630496	+	0.776192i	$0.717149\pi$
$110$		0			0
$111$	6.19115			0.587638
$112$		0			0
$113$	−4.41742			−0.415556			−0.207778	−	0.978176i	$-0.566623\pi$
$113$	−4.41742			−0.415556			−0.207778	+	0.978176i	$0.566623\pi$
$114$		0			0
$115$	11.0901			1.03416
$116$		0			0
$117$			11.8711i			1.09748i
$118$		0			0
$119$	−1.29217	+	3.16515i	−0.118453	+	0.290149i
$120$		0			0
$121$	−20.1652			−1.83320
$122$		0			0
$123$		−	19.1652i		−	1.72806i
$124$		0			0
$125$			1.29217i			0.115575i
$126$		0			0
$127$			8.41742i			0.746926i	0.927645	+	0.373463i	$0.121830\pi$
$127$			8.41742i			0.746926i	−0.927645	+	0.373463i	$0.878170\pi$
$128$		0			0
$129$		−	17.2813i		−	1.52153i
$130$		0			0
$131$	−19.0847			−1.66744			−0.833718	−	0.552190i	$-0.813792\pi$
$131$	−19.0847			−1.66744			−0.833718	+	0.552190i	$0.813792\pi$
$132$		0			0
$133$	10.7477	+	4.38774i	0.931946	+	0.380465i
$134$		0			0
$135$			34.3303i			2.95468i
$136$		0			0
$137$	17.1652			1.46652			0.733259	−	0.679950i	$-0.237998\pi$
$137$	17.1652			1.46652			0.733259	+	0.679950i	$0.237998\pi$
$138$		0			0
$139$	6.97208			0.591364			0.295682	−	0.955286i	$-0.404453\pi$
$139$	6.97208			0.591364			0.295682	+	0.955286i	$0.404453\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$	−10.0677			−0.841899
$144$		0			0
$145$			18.5734i			1.54244i
$146$		0			0
$147$	15.4779	+	15.1652i	1.27659	+	1.25080i
$148$		0			0
$149$	−17.1652			−1.40622			−0.703112	−	0.711079i	$-0.748207\pi$
$149$	−17.1652			−1.40622			−0.703112	+	0.711079i	$0.748207\pi$
$150$		0			0
$151$		−	11.5826i		−	0.942577i	−0.881979	−	0.471288i	$-0.843789\pi$
$151$		−	11.5826i		−	0.942577i	0.881979	−	0.471288i	$-0.156211\pi$
$152$		0			0
$153$		−	8.50579i		−	0.687652i
$154$		0			0
$155$			4.00000i			0.321288i
$156$		0			0
$157$			14.1857i			1.13214i	0.824356	+	0.566071i	$0.191537\pi$
$157$			14.1857i			1.13214i	−0.824356	+	0.566071i	$0.808463\pi$
$158$		0			0
$159$	28.3714			2.25000
$160$		0			0
$161$	−3.58258	+	8.77548i	−0.282347	+	0.691605i
$162$		0			0
$163$			5.58258i			0.437261i	0.975808	+	0.218631i	$0.0701589\pi$
$163$			5.58258i			0.437261i	−0.975808	+	0.218631i	$0.929841\pi$
$164$		0			0
$165$	−53.4955			−4.16462
$166$		0			0
$167$	3.60681			0.279103			0.139552	−	0.990215i	$-0.455434\pi$
$167$	3.60681			0.279103			0.139552	+	0.990215i	$0.455434\pi$
$168$		0			0
$169$	9.74773			0.749825
$170$		0			0
$171$	−28.8826			−2.20871
$172$		0			0
$173$		−	5.41022i		−	0.411331i	−0.978622	−	0.205666i	$-0.934064\pi$
$173$		−	5.41022i		−	0.411331i	0.978622	−	0.205666i	$-0.0659359\pi$
$174$		0			0
$175$	11.2250	+	4.58258i	0.848528	+	0.346410i
$176$		0			0
$177$	5.58258			0.419612
$178$		0			0
$179$			1.58258i			0.118287i	0.998249	+	0.0591436i	$0.0188370\pi$
$179$			1.58258i			0.118287i	−0.998249	+	0.0591436i	$0.981163\pi$
$180$		0			0
$181$		−	21.6690i		−	1.61065i	−0.592837	−	0.805323i	$-0.701992\pi$
$181$		−	21.6690i		−	1.61065i	0.592837	−	0.805323i	$-0.298008\pi$
$182$		0			0
$183$			20.7477i			1.53372i
$184$		0			0
$185$			6.19115i			0.455182i
$186$		0			0
$187$	7.21362			0.527512
$188$		0			0
$189$	−27.1652	−	11.0901i	−1.97597	−	0.806688i
$190$		0			0
$191$		−	2.83485i		−	0.205122i	−0.994727	−	0.102561i	$-0.967296\pi$
$191$		−	2.83485i		−	0.205122i	0.994727	−	0.102561i	$-0.0327038\pi$
$192$		0			0
$193$	0.417424			0.0300469			0.0150234	−	0.999887i	$-0.495218\pi$
$193$	0.417424			0.0300469			0.0150234	+	0.999887i	$0.495218\pi$
$194$		0			0
$195$	−17.2813			−1.23754
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−6.19115			−0.438879			−0.219439	−	0.975626i	$-0.570423\pi$
$199$	−6.19115			−0.438879			−0.219439	+	0.975626i	$0.570423\pi$
$200$		0			0
$201$		−	42.0459i		−	2.96569i
$202$		0			0
$203$	−14.6969	−	6.00000i	−1.03152	−	0.421117i
$204$		0			0
$205$	19.1652			1.33855
$206$		0			0
$207$		−	23.5826i		−	1.63910i
$208$		0			0
$209$		−	24.4949i		−	1.69435i
$210$		0			0
$211$		−	19.9129i		−	1.37086i	−0.728139	−	0.685430i	$-0.759614\pi$
$211$		−	19.9129i		−	1.37086i	0.728139	−	0.685430i	$-0.240386\pi$
$212$		0			0
$213$		−	28.3714i		−	1.94398i
$214$		0			0
$215$	17.2813			1.17857
$216$		0			0
$217$	−3.16515	−	1.29217i	−0.214864	−	0.0877181i
$218$		0			0
$219$			23.1652i			1.56536i
$220$		0			0
$221$	2.33030			0.156753
$222$		0			0
$223$	−12.3823			−0.829180			−0.414590	−	0.910008i	$-0.636075\pi$
$223$	−12.3823			−0.829180			−0.414590	+	0.910008i	$0.636075\pi$
$224$		0			0
$225$	−30.1652			−2.01101
$226$		0			0
$227$	−23.9837			−1.59185			−0.795926	−	0.605394i	$-0.793015\pi$
$227$	−23.9837			−1.59185			−0.795926	+	0.605394i	$0.793015\pi$
$228$		0			0
$229$			20.3768i			1.34654i	0.739397	+	0.673270i	$0.235111\pi$
$229$			20.3768i			1.34654i	−0.739397	+	0.673270i	$0.764889\pi$
$230$		0			0
$231$	17.2813	−	42.3303i	1.13702	−	2.78513i
$232$		0			0
$233$	−17.1652			−1.12453			−0.562263	−	0.826958i	$-0.690069\pi$
$233$	−17.1652			−1.12453			−0.562263	+	0.826958i	$0.690069\pi$
$234$		0			0
$235$			4.00000i			0.260931i
$236$		0			0
$237$		−	6.19115i		−	0.402158i
$238$		0			0
$239$			16.4174i			1.06195i	0.847386	+	0.530977i	$0.178175\pi$
$239$			16.4174i			1.06195i	−0.847386	+	0.530977i	$0.821825\pi$
$240$		0			0
$241$		−	1.29217i		−	0.0832358i	−0.999134	−	0.0416179i	$-0.986749\pi$
$241$		−	1.29217i		−	0.0832358i	0.999134	−	0.0416179i	$-0.0132512\pi$
$242$		0			0
$243$	11.8711			0.761529
$244$		0			0
$245$	−15.1652	+	15.4779i	−0.968866	+	0.988845i
$246$		0			0
$247$		−	7.91288i		−	0.503484i
$248$		0			0
$249$	9.58258			0.607271
$250$		0			0
$251$	6.70239			0.423051			0.211525	−	0.977372i	$-0.432157\pi$
$251$	6.70239			0.423051			0.211525	+	0.977372i	$0.432157\pi$
$252$		0			0
$253$	20.0000			1.25739
$254$		0			0
$255$	12.3823			0.775409
$256$		0			0
$257$		−	9.79796i		−	0.611180i	−0.952163	−	0.305590i	$-0.901146\pi$
$257$		−	9.79796i		−	0.611180i	0.952163	−	0.305590i	$-0.0988537\pi$
$258$		0			0
$259$	−4.89898	−	2.00000i	−0.304408	−	0.124274i
$260$		0			0
$261$	39.4955			2.44471
$262$		0			0
$263$		−	25.1652i		−	1.55175i	−0.630887	−	0.775875i	$-0.717309\pi$
$263$		−	25.1652i		−	1.55175i	0.630887	−	0.775875i	$-0.282691\pi$
$264$		0			0
$265$			28.3714i			1.74284i
$266$		0			0
$267$			23.1652i			1.41768i
$268$		0			0
$269$		−	27.5905i		−	1.68222i	−0.540864	−	0.841110i	$-0.681902\pi$
$269$		−	27.5905i		−	1.68222i	0.540864	−	0.841110i	$-0.318098\pi$
$270$		0			0
$271$	−27.3489			−1.66133			−0.830664	−	0.556773i	$-0.812039\pi$
$271$	−27.3489			−1.66133			−0.830664	+	0.556773i	$0.812039\pi$
$272$		0			0
$273$	5.58258	−	13.6745i	0.337873	−	0.827616i
$274$		0			0
$275$		−	25.5826i		−	1.54269i
$276$		0			0
$277$	28.3303			1.70220			0.851101	−	0.525001i	$-0.175935\pi$
$277$	28.3303			1.70220			0.851101	+	0.525001i	$0.175935\pi$
$278$		0			0
$279$	8.50579			0.509228
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$	16.5003			0.980844			0.490422	−	0.871485i	$-0.336843\pi$
$283$	16.5003			0.980844			0.490422	+	0.871485i	$0.336843\pi$
$284$		0			0
$285$		−	42.0459i		−	2.49058i
$286$		0			0
$287$	−6.19115	+	15.1652i	−0.365452	+	0.895171i
$288$		0			0
$289$	15.3303			0.901783
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$			20.3768i			1.19043i	0.803567	+	0.595214i	$0.202933\pi$
$293$			20.3768i			1.19043i	−0.803567	+	0.595214i	$0.797067\pi$
$294$		0			0
$295$			5.58258i			0.325030i
$296$		0			0
$297$			61.9115i			3.59247i
$298$		0			0
$299$	6.46084			0.373640
$300$		0			0
$301$	−5.58258	+	13.6745i	−0.321774	+	0.788183i
$302$		0			0
$303$		−	43.9129i		−	2.52273i
$304$		0			0
$305$	−20.7477			−1.18801
$306$		0			0
$307$	−2.07310			−0.118318			−0.0591589	−	0.998249i	$-0.518842\pi$
$307$	−2.07310			−0.118318			−0.0591589	+	0.998249i	$0.518842\pi$
$308$		0			0
$309$	19.1652			1.09027
$310$		0			0
$311$	−29.6636			−1.68207			−0.841033	−	0.540983i	$-0.818052\pi$
$311$	−29.6636			−1.68207			−0.841033	+	0.540983i	$0.818052\pi$
$312$		0			0
$313$			3.60681i			0.203869i	0.994791	+	0.101935i	$0.0325032\pi$
$313$			3.60681i			0.203869i	−0.994791	+	0.101935i	$0.967497\pi$
$314$		0			0
$315$	20.3768	−	49.9129i	1.14811	−	2.81227i
$316$		0			0
$317$	−28.3303			−1.59119			−0.795594	−	0.605830i	$-0.792841\pi$
$317$	−28.3303			−1.59119			−0.795594	+	0.605830i	$0.792841\pi$
$318$		0			0
$319$			33.4955i			1.87539i
$320$		0			0
$321$			19.8656i			1.10879i
$322$		0			0
$323$			5.66970i			0.315470i
$324$		0			0
$325$		−	8.26424i		−	0.458418i
$326$		0			0
$327$	−40.7537			−2.25368
$328$		0			0
$329$	−3.16515	−	1.29217i	−0.174500	−	0.0712395i
$330$		0			0
$331$			33.5826i			1.84587i	0.384961	+	0.922933i	$0.374215\pi$
$331$			33.5826i			1.84587i	−0.384961	+	0.922933i	$0.625785\pi$
$332$		0			0
$333$	13.1652			0.721446
$334$		0			0
$335$	42.0459			2.29721
$336$		0			0
$337$	−18.7477			−1.02125			−0.510627	−	0.859802i	$-0.670587\pi$
$337$	−18.7477			−1.02125			−0.510627	+	0.859802i	$0.670587\pi$
$338$		0			0
$339$	−13.6745			−0.742695
$340$		0			0
$341$			7.21362i			0.390640i
$342$		0			0
$343$	−7.34847	−	17.0000i	−0.396780	−	0.917914i
$344$		0			0
$345$	34.3303			1.84828
$346$		0			0
$347$			0.747727i			0.0401401i	0.999799	+	0.0200700i	$0.00638892\pi$
$347$			0.747727i			0.0401401i	−0.999799	+	0.0200700i	$0.993611\pi$
$348$		0			0
$349$			9.28672i			0.497107i	0.968618	+	0.248553i	$0.0799551\pi$
$349$			9.28672i			0.497107i	−0.968618	+	0.248553i	$0.920045\pi$
$350$		0			0
$351$			20.0000i			1.06752i
$352$		0			0
$353$		−	31.9782i		−	1.70203i	−0.525143	−	0.851014i	$-0.675988\pi$
$353$		−	31.9782i		−	1.70203i	0.525143	−	0.851014i	$-0.324012\pi$
$354$		0			0
$355$	28.3714			1.50580
$356$		0			0
$357$	−4.00000	+	9.79796i	−0.211702	+	0.518563i
$358$		0			0
$359$			4.41742i			0.233143i	0.993182	+	0.116571i	$0.0371904\pi$
$359$			4.41742i			0.233143i	−0.993182	+	0.116571i	$0.962810\pi$
$360$		0			0
$361$	0.252273			0.0132775
$362$		0			0
$363$	−62.4227			−3.27634
$364$		0			0
$365$	−23.1652			−1.21252
$366$		0			0
$367$	−29.3939			−1.53435			−0.767174	−	0.641439i	$-0.778338\pi$
$367$	−29.3939			−1.53435			−0.767174	+	0.641439i	$0.778338\pi$
$368$		0			0
$369$		−	40.7537i		−	2.12155i
$370$		0			0
$371$	−22.4499	−	9.16515i	−1.16554	−	0.475831i
$372$		0			0
$373$	−8.33030			−0.431327			−0.215663	−	0.976468i	$-0.569191\pi$
$373$	−8.33030			−0.431327			−0.215663	+	0.976468i	$0.569191\pi$
$374$		0			0
$375$			4.00000i			0.206559i
$376$		0			0
$377$			10.8204i			0.557281i
$378$		0			0
$379$			15.9129i			0.817390i	0.912671	+	0.408695i	$0.134016\pi$
$379$			15.9129i			0.817390i	−0.912671	+	0.408695i	$0.865984\pi$
$380$		0			0
$381$			26.0568i			1.33493i
$382$		0			0
$383$	28.6411			1.46349			0.731746	−	0.681578i	$-0.238706\pi$
$383$	28.6411			1.46349			0.731746	+	0.681578i	$0.238706\pi$
$384$		0			0
$385$	42.3303	+	17.2813i	2.15735	+	0.880735i
$386$		0			0
$387$		−	36.7477i		−	1.86799i
$388$		0			0
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$		0			0
$391$	−4.62929			−0.234113
$392$		0			0
$393$	−59.0780			−2.98009
$394$		0			0
$395$	6.19115			0.311510
$396$		0			0
$397$			4.11805i			0.206679i	0.994646	+	0.103340i	$0.0329528\pi$
$397$			4.11805i			0.206679i	−0.994646	+	0.103340i	$0.967047\pi$
$398$		0			0
$399$	33.2704	+	13.5826i	1.66560	+	0.679979i
$400$		0			0
$401$	−15.5826			−0.778157			−0.389078	−	0.921205i	$-0.627206\pi$
$401$	−15.5826			−0.778157			−0.389078	+	0.921205i	$0.627206\pi$
$402$		0			0
$403$			2.33030i			0.116081i
$404$		0			0
$405$			45.1414i			2.24310i
$406$		0			0
$407$			11.1652i			0.553436i
$408$		0			0
$409$		−	36.1244i		−	1.78624i	−0.449822	−	0.893118i	$-0.648512\pi$
$409$		−	36.1244i		−	1.78624i	0.449822	−	0.893118i	$-0.351488\pi$
$410$		0			0
$411$	53.1360			2.62101
$412$		0			0
$413$	−4.41742	−	1.80341i	−0.217367	−	0.0887398i
$414$		0			0
$415$			9.58258i			0.470390i
$416$		0			0
$417$	21.5826			1.05690
$418$		0			0
$419$	20.6465			1.00865			0.504325	−	0.863514i	$-0.331741\pi$
$419$	20.6465			1.00865			0.504325	+	0.863514i	$0.331741\pi$
$420$		0			0
$421$	−23.4955			−1.14510			−0.572549	−	0.819870i	$-0.694045\pi$
$421$	−23.4955			−1.14510			−0.572549	+	0.819870i	$0.694045\pi$
$422$		0			0
$423$	8.50579			0.413566
$424$		0			0
$425$			5.92146i			0.287233i
$426$		0			0
$427$	6.70239	−	16.4174i	0.324351	−	0.794495i
$428$		0			0
$429$	−31.1652			−1.50467
$430$		0			0
$431$			22.7477i			1.09572i	0.836570	+	0.547860i	$0.184557\pi$
$431$			22.7477i			1.09572i	−0.836570	+	0.547860i	$0.815443\pi$
$432$		0			0
$433$			33.2704i			1.59887i	0.600751	+	0.799436i	$0.294868\pi$
$433$			33.2704i			1.59887i	−0.600751	+	0.799436i	$0.705132\pi$
$434$		0			0
$435$			57.4955i			2.75670i
$436$		0			0
$437$			15.7194i			0.751962i
$438$		0			0
$439$	10.0677			0.480503			0.240251	−	0.970711i	$-0.422770\pi$
$439$	10.0677			0.480503			0.240251	+	0.970711i	$0.422770\pi$
$440$		0			0
$441$	32.9129	+	32.2479i	1.56728	+	1.53561i
$442$		0			0
$443$		−	4.74773i		−	0.225571i	−0.993619	−	0.112786i	$-0.964023\pi$
$443$		−	4.74773i		−	0.225571i	0.993619	−	0.112786i	$-0.0359773\pi$
$444$		0			0
$445$	−23.1652			−1.09813
$446$		0			0
$447$	−53.1360			−2.51325
$448$		0			0
$449$	19.4955			0.920047			0.460024	−	0.887907i	$-0.347841\pi$
$449$	19.4955			0.920047			0.460024	+	0.887907i	$0.347841\pi$
$450$		0			0
$451$	34.5625			1.62749
$452$		0			0
$453$		−	35.8547i		−	1.68460i
$454$		0			0
$455$	13.6745	+	5.58258i	0.641069	+	0.261715i
$456$		0			0
$457$	22.7477			1.06409			0.532047	−	0.846715i	$-0.321423\pi$
$457$	22.7477			1.06409			0.532047	+	0.846715i	$0.321423\pi$
$458$		0			0
$459$		−	14.3303i		−	0.668881i
$460$		0			0
$461$			2.07310i			0.0965538i	0.998834	+	0.0482769i	$0.0153730\pi$
$461$			2.07310i			0.0965538i	−0.998834	+	0.0482769i	$0.984627\pi$
$462$		0			0
$463$		−	1.16515i		−	0.0541492i	−0.999633	−	0.0270746i	$-0.991381\pi$
$463$		−	1.16515i		−	0.0541492i	0.999633	−	0.0270746i	$-0.00861916\pi$
$464$		0			0
$465$			12.3823i			0.574215i
$466$		0			0
$467$	25.0061			1.15715			0.578573	−	0.815631i	$-0.303610\pi$
$467$	25.0061			1.15715			0.578573	+	0.815631i	$0.303610\pi$
$468$		0			0
$469$	−13.5826	+	33.2704i	−0.627185	+	1.53628i
$470$		0			0
$471$			43.9129i			2.02340i
$472$		0			0
$473$	31.1652			1.43298
$474$		0			0
$475$	20.1072			0.922580
$476$		0			0
$477$	60.3303			2.76233
$478$		0			0
$479$	8.50579			0.388640			0.194320	−	0.980938i	$-0.437750\pi$
$479$	8.50579			0.388640			0.194320	+	0.980938i	$0.437750\pi$
$480$		0			0
$481$			3.60681i			0.164456i
$482$		0			0
$483$	−11.0901	+	27.1652i	−0.504618	+	1.23606i
$484$		0			0
$485$	12.0000			0.544892
$486$		0			0
$487$			34.7477i			1.57457i	0.616589	+	0.787285i	$0.288514\pi$
$487$			34.7477i			1.57457i	−0.616589	+	0.787285i	$0.711486\pi$
$488$		0			0
$489$			17.2813i			0.781486i
$490$		0			0
$491$		−	5.58258i		−	0.251938i	−0.992034	−	0.125969i	$-0.959796\pi$
$491$		−	5.58258i		−	0.251938i	0.992034	−	0.125969i	$-0.0402040\pi$
$492$		0			0
$493$		−	7.75301i		−	0.349178i
$494$		0			0
$495$	−113.755			−5.11292
$496$		0			0
$497$	−9.16515	+	22.4499i	−0.411113	+	1.00702i
$498$		0			0
$499$		−	2.41742i		−	0.108219i	−0.998535	−	0.0541094i	$-0.982768\pi$
$499$		−	2.41742i		−	0.108219i	0.998535	−	0.0541094i	$-0.0172320\pi$
$500$		0			0
$501$	11.1652			0.498822
$502$		0			0
$503$	34.8322			1.55309			0.776546	−	0.630060i	$-0.216970\pi$
$503$	34.8322			1.55309			0.776546	+	0.630060i	$0.216970\pi$
$504$		0			0
$505$	43.9129			1.95410
$506$		0			0
$507$	30.1748			1.34011
$508$		0			0
$509$			4.65743i			0.206437i	0.994659	+	0.103219i	$0.0329141\pi$
$509$			4.65743i			0.206437i	−0.994659	+	0.103219i	$0.967086\pi$
$510$		0			0
$511$	7.48331	−	18.3303i	0.331042	−	0.810885i
$512$		0			0
$513$	−48.6606			−2.14842
$514$		0			0
$515$			19.1652i			0.844517i
$516$		0			0
$517$			7.21362i			0.317255i
$518$		0			0
$519$		−	16.7477i		−	0.735144i
$520$		0			0
$521$			13.4048i			0.587274i	0.955917	+	0.293637i	$0.0948656\pi$
$521$			13.4048i			0.587274i	−0.955917	+	0.293637i	$0.905134\pi$
$522$		0			0
$523$	−32.7591			−1.43246			−0.716229	−	0.697866i	$-0.754133\pi$
$523$	−32.7591			−1.43246			−0.716229	+	0.697866i	$0.754133\pi$
$524$		0			0
$525$	34.7477	+	14.1857i	1.51652	+	0.619115i
$526$		0			0
$527$		−	1.66970i		−	0.0727332i
$528$		0			0
$529$	10.1652			0.441963
$530$		0			0
$531$	11.8711			0.515160
$532$		0			0
$533$	11.1652			0.483616
$534$		0			0
$535$	−19.8656			−0.858865
$536$		0			0
$537$			4.89898i			0.211407i
$538$		0			0
$539$	−27.3489	+	27.9129i	−1.17800	+	1.20229i
$540$		0			0
$541$	37.1652			1.59785			0.798927	−	0.601428i	$-0.205401\pi$
$541$	37.1652			1.59785			0.798927	+	0.601428i	$0.205401\pi$
$542$		0			0
$543$		−	67.0780i		−	2.87859i
$544$		0			0
$545$		−	40.7537i		−	1.74570i
$546$		0			0
$547$		−	2.41742i		−	0.103362i	−0.998664	−	0.0516808i	$-0.983542\pi$
$547$		−	2.41742i		−	0.103362i	0.998664	−	0.0516808i	$-0.0164578\pi$
$548$		0			0
$549$			44.1190i			1.88295i
$550$		0			0
$551$	−26.3264			−1.12154
$552$		0			0
$553$	−2.00000	+	4.89898i	−0.0850487	+	0.208326i
$554$		0			0
$555$			19.1652i			0.813515i
$556$		0			0
$557$	−27.4955			−1.16502			−0.582510	−	0.812824i	$-0.697929\pi$
$557$	−27.4955			−1.16502			−0.582510	+	0.812824i	$0.697929\pi$
$558$		0			0
$559$	10.0677			0.425816
$560$		0			0
$561$	22.3303			0.942786
$562$		0			0
$563$	20.3768			0.858782			0.429391	−	0.903119i	$-0.358728\pi$
$563$	20.3768			0.858782			0.429391	+	0.903119i	$0.358728\pi$
$564$		0			0
$565$		−	13.6745i		−	0.575289i
$566$		0			0
$567$	−35.7199	−	14.5826i	−1.50009	−	0.612411i
$568$		0			0
$569$	8.41742			0.352877			0.176438	−	0.984312i	$-0.443542\pi$
$569$	8.41742			0.352877			0.176438	+	0.984312i	$0.443542\pi$
$570$		0			0
$571$		−	5.58258i		−	0.233624i	−0.993154	−	0.116812i	$-0.962733\pi$
$571$		−	5.58258i		−	0.233624i	0.993154	−	0.116812i	$-0.0372674\pi$
$572$		0			0
$573$		−	8.77548i		−	0.366601i
$574$		0			0
$575$			16.4174i			0.684654i
$576$		0			0
$577$			4.62929i			0.192720i	0.995347	+	0.0963599i	$0.0307200\pi$
$577$			4.62929i			0.192720i	−0.995347	+	0.0963599i	$0.969280\pi$
$578$		0			0
$579$	1.29217			0.0537007
$580$		0			0
$581$	−7.58258	−	3.09557i	−0.314578	−	0.128426i
$582$		0			0
$583$			51.1652i			2.11904i
$584$		0			0
$585$	−36.7477			−1.51933
$586$		0			0
$587$	4.38774			0.181101			0.0905507	−	0.995892i	$-0.471137\pi$
$587$	4.38774			0.181101			0.0905507	+	0.995892i	$0.471137\pi$
$588$		0			0
$589$	−5.66970			−0.233616
$590$		0			0
$591$	55.7203			2.29203
$592$		0			0
$593$		−	24.7646i		−	1.01696i	−0.861074	−	0.508480i	$-0.830208\pi$
$593$		−	24.7646i		−	1.01696i	0.861074	−	0.508480i	$-0.169792\pi$
$594$		0			0
$595$	−9.79796	−	4.00000i	−0.401677	−	0.163984i
$596$		0			0
$597$	−19.1652			−0.784377
$598$		0			0
$599$		−	10.0000i		−	0.408589i	−0.978909	−	0.204294i	$-0.934510\pi$
$599$		−	10.0000i		−	0.408589i	0.978909	−	0.204294i	$-0.0654900\pi$
$600$		0			0
$601$			10.0677i			0.410668i	0.978692	+	0.205334i	$0.0658281\pi$
$601$			10.0677i			0.410668i	−0.978692	+	0.205334i	$0.934172\pi$
$602$		0			0
$603$		−	89.4083i		−	3.64099i
$604$		0			0
$605$		−	62.4227i		−	2.53784i
$606$		0			0
$607$	17.5510			0.712372			0.356186	−	0.934415i	$-0.384077\pi$
$607$	17.5510			0.712372			0.356186	+	0.934415i	$0.384077\pi$
$608$		0			0
$609$	−45.4955	−	18.5734i	−1.84357	−	0.752634i
$610$		0			0
$611$			2.33030i			0.0942740i
$612$		0			0
$613$	−2.83485			−0.114498			−0.0572492	−	0.998360i	$-0.518233\pi$
$613$	−2.83485			−0.114498			−0.0572492	+	0.998360i	$0.518233\pi$
$614$		0			0
$615$	59.3271			2.39230
$616$		0			0
$617$	11.5826			0.466297			0.233148	−	0.972441i	$-0.425097\pi$
$617$	11.5826			0.466297			0.233148	+	0.972441i	$0.425097\pi$
$618$		0			0
$619$	26.8377			1.07870			0.539349	−	0.842082i	$-0.318670\pi$
$619$	26.8377			1.07870			0.539349	+	0.842082i	$0.318670\pi$
$620$		0			0
$621$		−	39.7312i		−	1.59436i
$622$		0			0
$623$	7.48331	−	18.3303i	0.299813	−	0.734388i
$624$		0			0
$625$	−26.9129			−1.07652
$626$		0			0
$627$		−	75.8258i		−	3.02819i
$628$		0			0
$629$		−	2.58434i		−	0.103044i
$630$		0			0
$631$		−	40.3303i		−	1.60552i	−0.596300	−	0.802762i	$-0.703363\pi$
$631$		−	40.3303i		−	1.60552i	0.596300	−	0.802762i	$-0.296637\pi$
$632$		0			0
$633$		−	61.6418i		−	2.45004i
$634$		0			0
$635$	−26.0568			−1.03403
$636$		0			0
$637$	−8.83485	+	9.01703i	−0.350049	+	0.357268i
$638$		0			0
$639$		−	60.3303i		−	2.38663i
$640$		0			0
$641$	6.74773			0.266519			0.133260	−	0.991081i	$-0.457456\pi$
$641$	6.74773			0.266519			0.133260	+	0.991081i	$0.457456\pi$
$642$		0			0
$643$	−31.7367			−1.25157			−0.625786	−	0.779995i	$-0.715222\pi$
$643$	−31.7367			−1.25157			−0.625786	+	0.779995i	$0.715222\pi$
$644$		0			0
$645$	53.4955			2.10638
$646$		0			0
$647$	−20.6184			−0.810593			−0.405296	−	0.914185i	$-0.632832\pi$
$647$	−20.6184			−0.810593			−0.405296	+	0.914185i	$0.632832\pi$
$648$		0			0
$649$			10.0677i			0.395190i
$650$		0			0
$651$	−9.79796	−	4.00000i	−0.384012	−	0.156772i
$652$		0			0
$653$	27.4955			1.07598			0.537990	−	0.842951i	$-0.319184\pi$
$653$	27.4955			1.07598			0.537990	+	0.842951i	$0.319184\pi$
$654$		0			0
$655$		−	59.0780i		−	2.30837i
$656$		0			0
$657$			49.2595i			1.92180i
$658$		0			0
$659$			44.7477i			1.74312i	0.490285	+	0.871562i	$0.336893\pi$
$659$			44.7477i			1.74312i	−0.490285	+	0.871562i	$0.663107\pi$
$660$		0			0
$661$			30.7142i			1.19464i	0.802002	+	0.597322i	$0.203768\pi$
$661$			30.7142i			1.19464i	−0.802002	+	0.597322i	$0.796232\pi$
$662$		0			0
$663$	7.21362			0.280154
$664$		0			0
$665$	−13.5826	+	33.2704i	−0.526710	+	1.29017i
$666$		0			0
$667$		−	21.4955i		−	0.832307i
$668$		0			0
$669$	−38.3303			−1.48194
$670$		0			0
$671$	−37.4166			−1.44445
$672$		0			0
$673$	−0.330303			−0.0127322			−0.00636612	−	0.999980i	$-0.502026\pi$
$673$	−0.330303			−0.0127322			−0.00636612	+	0.999980i	$0.502026\pi$
$674$		0			0
$675$	−50.8213			−1.95611
$676$		0			0
$677$			0.511238i			0.0196485i	0.999952	+	0.00982424i	$0.00312720\pi$
$677$			0.511238i			0.0196485i	−0.999952	+	0.00982424i	$0.996873\pi$
$678$		0			0
$679$	−3.87650	+	9.49545i	−0.148767	+	0.364402i
$680$		0			0
$681$	−74.2432			−2.84500
$682$		0			0
$683$		−	9.58258i		−	0.366667i	−0.983051	−	0.183334i	$-0.941311\pi$
$683$		−	9.58258i		−	0.366667i	0.983051	−	0.183334i	$-0.0586888\pi$
$684$		0			0
$685$			53.1360i			2.03022i
$686$		0			0
$687$			63.0780i			2.40658i
$688$		0			0
$689$			16.5285i			0.629685i
$690$		0			0
$691$	8.26424			0.314387			0.157193	−	0.987568i	$-0.449755\pi$
$691$	8.26424			0.314387			0.157193	+	0.987568i	$0.449755\pi$
$692$		0			0
$693$	36.7477	−	90.0132i	1.39593	−	3.41932i
$694$		0			0
$695$			21.5826i			0.818674i
$696$		0			0
$697$	−8.00000			−0.303022
$698$		0			0
$699$	−53.1360			−2.00979
$700$		0			0
$701$	−15.4955			−0.585255			−0.292628	−	0.956226i	$-0.594530\pi$
$701$	−15.4955			−0.585255			−0.292628	+	0.956226i	$0.594530\pi$
$702$		0			0
$703$	−8.77548			−0.330974
$704$		0			0
$705$			12.3823i			0.466344i
$706$		0			0
$707$	−14.1857	+	34.7477i	−0.533508	+	1.30682i
$708$		0			0
$709$	−13.1652			−0.494428			−0.247214	−	0.968961i	$-0.579515\pi$
$709$	−13.1652			−0.494428			−0.247214	+	0.968961i	$0.579515\pi$
$710$		0			0
$711$		−	13.1652i		−	0.493732i
$712$		0			0
$713$		−	4.62929i		−	0.173368i
$714$		0			0
$715$		−	31.1652i		−	1.16551i
$716$		0			0
$717$			50.8213i			1.89796i
$718$		0			0
$719$	3.33712			0.124454			0.0622268	−	0.998062i	$-0.480180\pi$
$719$	3.33712			0.124454			0.0622268	+	0.998062i	$0.480180\pi$
$720$		0			0
$721$	−15.1652	−	6.19115i	−0.564780	−	0.230570i
$722$		0			0
$723$		−	4.00000i		−	0.148762i
$724$		0			0
$725$	−27.4955			−1.02116
$726$		0			0
$727$	−36.1244			−1.33978			−0.669890	−	0.742460i	$-0.733659\pi$
$727$	−36.1244			−1.33978			−0.669890	+	0.742460i	$0.733659\pi$
$728$		0			0
$729$	−7.00000			−0.259259
$730$		0			0
$731$	−7.21362			−0.266806
$732$		0			0
$733$			34.0513i			1.25771i	0.777521	+	0.628857i	$0.216477\pi$
$733$			34.0513i			1.25771i	−0.777521	+	0.628857i	$0.783523\pi$
$734$		0			0
$735$	−46.9448	+	47.9129i	−1.73159	+	1.76729i
$736$		0			0
$737$	75.8258			2.79308
$738$		0			0
$739$		−	0.747727i		−	0.0275056i	−0.999905	−	0.0137528i	$-0.995622\pi$
$739$		−	0.747727i		−	0.0275056i	0.999905	−	0.0137528i	$-0.00437779\pi$
$740$		0			0
$741$		−	24.4949i		−	0.899843i
$742$		0			0
$743$		−	13.2523i		−	0.486179i	−0.970004	−	0.243089i	$-0.921839\pi$
$743$		−	13.2523i		−	0.486179i	0.970004	−	0.243089i	$-0.0781608\pi$
$744$		0			0
$745$		−	53.1360i		−	1.94675i
$746$		0			0
$747$	20.3768			0.745550
$748$		0			0
$749$	6.41742	−	15.7194i	0.234488	−	0.574375i
$750$		0			0
$751$			53.0780i			1.93684i	0.249315	+	0.968422i	$0.419795\pi$
$751$			53.0780i			1.93684i	−0.249315	+	0.968422i	$0.580205\pi$
$752$		0			0
$753$	20.7477			0.756089
$754$		0			0
$755$	35.8547			1.30489
$756$		0			0
$757$	−49.1652			−1.78694			−0.893469	−	0.449125i	$-0.851736\pi$
$757$	−49.1652			−1.78694			−0.893469	+	0.449125i	$0.851736\pi$
$758$		0			0
$759$	61.9115			2.24724
$760$		0			0
$761$			23.7421i			0.860651i	0.902674	+	0.430325i	$0.141601\pi$
$761$			23.7421i			0.860651i	−0.902674	+	0.430325i	$0.858399\pi$
$762$		0			0
$763$	32.2479	+	13.1652i	1.16745	+	0.476610i
$764$		0			0
$765$	26.3303			0.951974
$766$		0			0
$767$			3.25227i			0.117433i
$768$		0			0
$769$		−	28.1017i		−	1.01337i	−0.862130	−	0.506687i	$-0.830870\pi$
$769$		−	28.1017i		−	1.01337i	0.862130	−	0.506687i	$-0.169130\pi$
$770$		0			0
$771$		−	30.3303i		−	1.09232i
$772$		0			0
$773$		−	6.70239i		−	0.241068i	−0.992709	−	0.120534i	$-0.961539\pi$
$773$		−	6.70239i		−	0.241068i	0.992709	−	0.120534i	$-0.0384607\pi$
$774$		0			0
$775$	−5.92146			−0.212705
$776$		0			0
$777$	−15.1652	−	6.19115i	−0.544047	−	0.222106i
$778$		0			0
$779$			27.1652i			0.973293i
$780$		0			0
$781$	51.1652			1.83083
$782$		0			0
$783$	66.5408			2.37797
$784$		0			0
$785$	−43.9129			−1.56732
$786$		0			0
$787$	10.5789			0.377097			0.188548	−	0.982064i	$-0.439622\pi$
$787$	10.5789			0.377097			0.188548	+	0.982064i	$0.439622\pi$
$788$		0			0
$789$		−	77.9006i		−	2.77333i
$790$		0			0
$791$	10.8204	+	4.41742i	0.384730	+	0.157066i
$792$		0			0
$793$	−12.0871			−0.429226
$794$		0			0
$795$			87.8258i			3.11486i
$796$		0			0
$797$			1.80341i			0.0638799i	0.999490	+	0.0319400i	$0.0101685\pi$
$797$			1.80341i			0.0638799i	−0.999490	+	0.0319400i	$0.989831\pi$
$798$		0			0
$799$		−	1.66970i		−	0.0590696i
$800$		0			0
$801$			49.2595i			1.74050i
$802$		0			0
$803$	−41.7762			−1.47425
$804$		0			0
$805$	−27.1652	−	11.0901i	−0.957446	−	0.390876i
$806$		0			0
$807$		−	85.4083i		−	3.00652i
$808$		0			0
$809$	−7.58258			−0.266589			−0.133295	−	0.991076i	$-0.542556\pi$
$809$	−7.58258			−0.266589			−0.133295	+	0.991076i	$0.542556\pi$
$810$		0			0
$811$	9.28672			0.326101			0.163050	−	0.986618i	$-0.447867\pi$
$811$	9.28672			0.326101			0.163050	+	0.986618i	$0.447867\pi$
$812$		0			0
$813$	−84.6606			−2.96918
$814$		0			0
$815$	−17.2813			−0.605337
$816$		0			0
$817$			24.4949i			0.856968i
$818$		0			0
$819$	11.8711	−	29.0780i	0.414808	−	1.01607i
$820$		0			0
$821$	18.6606			0.651260			0.325630	−	0.945497i	$-0.394424\pi$
$821$	18.6606			0.651260			0.325630	+	0.945497i	$0.394424\pi$
$822$		0			0
$823$			37.1652i			1.29550i	0.761855	+	0.647748i	$0.224289\pi$
$823$			37.1652i			1.29550i	−0.761855	+	0.647748i	$0.775711\pi$
$824$		0			0
$825$		−	79.1927i		−	2.75714i
$826$		0			0
$827$			19.9129i			0.692439i	0.938154	+	0.346219i	$0.112535\pi$
$827$			19.9129i			0.692439i	−0.938154	+	0.346219i	$0.887465\pi$
$828$		0			0
$829$		−	37.6581i		−	1.30792i	−0.756529	−	0.653960i	$-0.773106\pi$
$829$		−	37.6581i		−	1.30792i	0.756529	−	0.653960i	$-0.226894\pi$
$830$		0			0
$831$	87.6985			3.04223
$832$		0			0
$833$	6.33030	−	6.46084i	0.219332	−	0.223855i
$834$		0			0
$835$			11.1652i			0.386386i
$836$		0			0
$837$	14.3303			0.495328
$838$		0			0
$839$	13.4048			0.462784			0.231392	−	0.972861i	$-0.425672\pi$
$839$	13.4048			0.462784			0.231392	+	0.972861i	$0.425672\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−30.9557			−1.06617
$844$		0			0
$845$			30.1748i			1.03804i
$846$		0			0
$847$	49.3943	+	20.1652i	1.69721	+	0.692883i
$848$		0			0
$849$	51.0780			1.75299
$850$		0			0
$851$		−	7.16515i		−	0.245618i
$852$		0			0
$853$		−	28.6129i		−	0.979689i	−0.871810	−	0.489844i	$-0.837054\pi$
$853$		−	28.6129i		−	0.979689i	0.871810	−	0.489844i	$-0.162946\pi$
$854$		0			0
$855$		−	89.4083i		−	3.05770i
$856$		0			0
$857$		−	40.7537i		−	1.39212i	−0.717984	−	0.696060i	$-0.754935\pi$
$857$		−	40.7537i		−	1.39212i	0.717984	−	0.696060i	$-0.245065\pi$
$858$		0			0
$859$	−47.4561			−1.61918			−0.809590	−	0.586995i	$-0.800311\pi$
$859$	−47.4561			−1.61918			−0.809590	+	0.586995i	$0.800311\pi$
$860$		0			0
$861$	−19.1652	+	46.9448i	−0.653147	+	1.59988i
$862$		0			0
$863$		−	33.1652i		−	1.12895i	−0.825448	−	0.564477i	$-0.809078\pi$
$863$		−	33.1652i		−	1.12895i	0.825448	−	0.564477i	$-0.190922\pi$
$864$		0			0
$865$	16.7477			0.569440
$866$		0			0
$867$	47.4561			1.61169
$868$		0			0
$869$	11.1652			0.378752
$870$		0			0
$871$	24.4949			0.829978
$872$		0			0
$873$		−	25.5174i		−	0.863632i
$874$		0			0
$875$	1.29217	−	3.16515i	0.0436832	−	0.107002i
$876$		0			0
$877$	−4.33030			−0.146224			−0.0731120	−	0.997324i	$-0.523293\pi$
$877$	−4.33030			−0.146224			−0.0731120	+	0.997324i	$0.523293\pi$
$878$		0			0
$879$			63.0780i			2.12757i
$880$		0			0
$881$		−	54.6978i		−	1.84282i	−0.388595	−	0.921409i	$-0.627039\pi$
$881$		−	54.6978i		−	1.84282i	0.388595	−	0.921409i	$-0.372961\pi$
$882$		0			0
$883$			51.0780i			1.71891i	0.511209	+	0.859456i	$0.329198\pi$
$883$			51.0780i			1.71891i	−0.511209	+	0.859456i	$0.670802\pi$
$884$		0			0
$885$			17.2813i			0.580904i
$886$		0			0
$887$	23.7421			0.797182			0.398591	−	0.917129i	$-0.369499\pi$
$887$	23.7421			0.797182			0.398591	+	0.917129i	$0.369499\pi$
$888$		0			0
$889$	8.41742	−	20.6184i	0.282311	−	0.691519i
$890$		0			0
$891$			81.4083i			2.72728i
$892$		0			0
$893$	−5.66970			−0.189729
$894$		0			0
$895$	−4.89898			−0.163755
$896$		0			0
$897$	20.0000			0.667781
$898$		0			0
$899$	7.75301			0.258577
$900$		0			0
$901$		−	11.8429i		−	0.394545i
$902$		0			0
$903$	−17.2813	+	42.3303i	−0.575085	+	1.40866i
$904$		0			0
$905$	67.0780			2.22975
$906$		0			0
$907$			39.9129i			1.32529i	0.748936	+	0.662643i	$0.230565\pi$
$907$			39.9129i			1.32529i	−0.748936	+	0.662643i	$0.769435\pi$
$908$		0			0
$909$		−	93.3784i		−	3.09717i
$910$		0			0
$911$		−	23.5826i		−	0.781326i	−0.920534	−	0.390663i	$-0.872246\pi$
$911$		−	23.5826i		−	0.781326i	0.920534	−	0.390663i	$-0.127754\pi$
$912$		0			0
$913$			17.2813i			0.571927i
$914$		0			0
$915$	−64.2261			−2.12325
$916$		0			0
$917$	46.7477	+	19.0847i	1.54375	+	0.630232i
$918$		0			0
$919$			5.16515i			0.170383i	0.996365	+	0.0851913i	$0.0271501\pi$
$919$			5.16515i			0.170383i	−0.996365	+	0.0851913i	$0.972850\pi$
$920$		0			0
$921$	−6.41742			−0.211461
$922$		0			0
$923$	16.5285			0.544042
$924$		0			0
$925$	−9.16515			−0.301348
$926$		0			0
$927$	40.7537			1.33853
$928$		0			0
$929$		−	40.4840i		−	1.32824i	−0.747627	−	0.664119i	$-0.768807\pi$
$929$		−	40.4840i		−	1.32824i	0.747627	−	0.664119i	$-0.231193\pi$
$930$		0			0
$931$	−21.9387	−	21.4955i	−0.719012	−	0.704485i
$932$		0			0
$933$	−91.8258			−3.00624
$934$		0			0
$935$			22.3303i			0.730279i
$936$		0			0
$937$		−	51.8438i		−	1.69366i	−0.531861	−	0.846832i	$-0.678507\pi$
$937$		−	51.8438i		−	1.69366i	0.531861	−	0.846832i	$-0.321493\pi$
$938$		0			0
$939$			11.1652i			0.364361i
$940$		0			0
$941$		−	37.9278i		−	1.23641i	−0.786016	−	0.618206i	$-0.787860\pi$
$941$		−	37.9278i		−	1.23641i	0.786016	−	0.618206i	$-0.212140\pi$
$942$		0			0
$943$	−22.1803			−0.722288
$944$		0			0
$945$	34.3303	−	84.0917i	1.11676	−	2.73550i
$946$		0			0
$947$		−	19.2523i		−	0.625615i	−0.949817	−	0.312807i	$-0.898731\pi$
$947$		−	19.2523i		−	0.625615i	0.949817	−	0.312807i	$-0.101269\pi$
$948$		0			0
$949$	−13.4955			−0.438081
$950$		0			0
$951$	−87.6985			−2.84382
$952$		0			0
$953$	−28.3303			−0.917709			−0.458854	−	0.888512i	$-0.651740\pi$
$953$	−28.3303			−0.917709			−0.458854	+	0.888512i	$0.651740\pi$
$954$		0			0
$955$	8.77548			0.283968
$956$		0			0
$957$			103.688i			3.35175i
$958$		0			0
$959$	−42.0459	−	17.1652i	−1.35773	−	0.554292i
$960$		0			0
$961$	−29.3303			−0.946139
$962$		0			0
$963$			42.2432i			1.36127i
$964$		0			0
$965$			1.29217i			0.0415963i
$966$		0			0
$967$			17.0780i			0.549192i	0.961560	+	0.274596i	$0.0885442\pi$
$967$			17.0780i			0.549192i	−0.961560	+	0.274596i	$0.911456\pi$
$968$		0			0
$969$			17.5510i			0.563818i
$970$		0			0
$971$	18.8150			0.603802			0.301901	−	0.953339i	$-0.402379\pi$
$971$	18.8150			0.603802			0.301901	+	0.953339i	$0.402379\pi$
$972$		0			0
$973$	−17.0780	−	6.97208i	−0.547497	−	0.223515i
$974$		0			0
$975$		−	25.5826i		−	0.819298i
$976$		0			0
$977$	−23.4955			−0.751686			−0.375843	−	0.926683i	$-0.622647\pi$
$977$	−23.4955			−0.751686			−0.375843	+	0.926683i	$0.622647\pi$
$978$		0			0
$979$	−41.7762			−1.33517
$980$		0			0
$981$	−86.6606			−2.76686
$982$		0			0
$983$	−4.14619			−0.132243			−0.0661215	−	0.997812i	$-0.521063\pi$
$983$	−4.14619			−0.132243			−0.0661215	+	0.997812i	$0.521063\pi$
$984$		0			0
$985$			55.7203i			1.77540i
$986$		0			0
$987$	−9.79796	−	4.00000i	−0.311872	−	0.127321i
$988$		0			0
$989$	−20.0000			−0.635963
$990$		0			0
$991$			20.3303i			0.645813i	0.946431	+	0.322907i	$0.104660\pi$
$991$			20.3303i			0.645813i	−0.946431	+	0.322907i	$0.895340\pi$
$992$		0			0
$993$			103.957i			3.29899i
$994$		0			0
$995$		−	19.1652i		−	0.607576i
$996$		0			0
$997$		−	19.0847i		−	0.604418i	−0.953242	−	0.302209i	$-0.902276\pi$
$997$		−	19.0847i		−	0.604418i	0.953242	−	0.302209i	$-0.0977240\pi$
$998$		0			0
$999$	22.1803			0.701752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.f.c.895.8	yes	8
4.3	odd	2	inner	896.2.f.c.895.2	yes	8
7.6	odd	2	inner	896.2.f.c.895.1	✓	8
8.3	odd	2		896.2.f.d.895.7	yes	8
8.5	even	2		896.2.f.d.895.1	yes	8
16.3	odd	4		1792.2.e.d.895.2		8
16.5	even	4		1792.2.e.d.895.1		8
16.11	odd	4		1792.2.e.e.895.7		8
16.13	even	4		1792.2.e.e.895.8		8
28.27	even	2	inner	896.2.f.c.895.7	yes	8
56.13	odd	2		896.2.f.d.895.8	yes	8
56.27	even	2		896.2.f.d.895.2	yes	8
112.13	odd	4		1792.2.e.e.895.1		8
112.27	even	4		1792.2.e.e.895.2		8
112.69	odd	4		1792.2.e.d.895.8		8
112.83	even	4		1792.2.e.d.895.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.f.c.895.1	✓	8	7.6	odd	2	inner
896.2.f.c.895.2	yes	8	4.3	odd	2	inner
896.2.f.c.895.7	yes	8	28.27	even	2	inner
896.2.f.c.895.8	yes	8	1.1	even	1	trivial
896.2.f.d.895.1	yes	8	8.5	even	2
896.2.f.d.895.2	yes	8	56.27	even	2
896.2.f.d.895.7	yes	8	8.3	odd	2
896.2.f.d.895.8	yes	8	56.13	odd	2
1792.2.e.d.895.1		8	16.5	even	4
1792.2.e.d.895.2		8	16.3	odd	4
1792.2.e.d.895.7		8	112.83	even	4
1792.2.e.d.895.8		8	112.69	odd	4
1792.2.e.e.895.1		8	112.13	odd	4
1792.2.e.e.895.2		8	112.27	even	4
1792.2.e.e.895.7		8	16.11	odd	4
1792.2.e.e.895.8		8	16.13	even	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)

\(f(q)\)	\(=\)	\(q+3.09557 q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +6.58258 q^{9} +O(q^{10})\)	\(q+3.09557 q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +6.58258 q^{9} +5.58258i q^{11} +1.80341i q^{13} +9.58258i q^{15} -1.29217i q^{17} -4.38774 q^{19} +(-7.58258 - 3.09557i) q^{21} -3.58258i q^{23} -4.58258 q^{25} +11.0901 q^{27} +6.00000 q^{29} +1.29217 q^{31} +17.2813i q^{33} +(3.09557 - 7.58258i) q^{35} +2.00000 q^{37} +5.58258i q^{39} -6.19115i q^{41} -5.58258i q^{43} +20.3768i q^{45} +1.29217 q^{47} +(5.00000 + 4.89898i) q^{49} -4.00000i q^{51} +9.16515 q^{53} -17.2813 q^{55} -13.5826 q^{57} +1.80341 q^{59} +6.70239i q^{61} +(-16.1240 - 6.58258i) q^{63} -5.58258 q^{65} -13.5826i q^{67} -11.0901i q^{69} -9.16515i q^{71} +7.48331i q^{73} -14.1857 q^{75} +(5.58258 - 13.6745i) q^{77} -2.00000i q^{79} +14.5826 q^{81} +3.09557 q^{83} +4.00000 q^{85} +18.5734 q^{87} +7.48331i q^{89} +(1.80341 - 4.41742i) q^{91} +4.00000 q^{93} -13.5826i q^{95} -3.87650i q^{97} +36.7477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} - 24 q^{21} + 48 q^{29} + 16 q^{37} + 40 q^{49} - 72 q^{57} - 8 q^{65} + 8 q^{77} + 80 q^{81} + 32 q^{85} + 32 q^{93}+O(q^{100}) \) 8 * q + 16 * q^9 - 24 * q^21 + 48 * q^29 + 16 * q^37 + 40 * q^49 - 72 * q^57 - 8 * q^65 + 8 * q^77 + 80 * q^81 + 32 * q^85 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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