LMFDB - Embedded newform 1792.2.e.e.895.7

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:	8.0.12745506816.1
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:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.7
Root			$-1.54779 + 1.54779i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.e.895.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09557i q^{3} -3.09557 q^{5} +(-2.44949 - 1.00000i) q^{7} -6.58258 q^{9} +O(q^{10})$	$q+3.09557i q^{3} -3.09557 q^{5} +(-2.44949 - 1.00000i) q^{7} -6.58258 q^{9} +5.58258 q^{11} +1.80341 q^{13} -9.58258i q^{15} -1.29217i q^{17} -4.38774i q^{19} +(3.09557 - 7.58258i) q^{21} -3.58258i q^{23} +4.58258 q^{25} -11.0901i q^{27} -6.00000i q^{29} -1.29217 q^{31} +17.2813i q^{33} +(7.58258 + 3.09557i) q^{35} +2.00000i q^{37} +5.58258i q^{39} +6.19115i q^{41} -5.58258 q^{43} +20.3768 q^{45} -1.29217 q^{47} +(5.00000 + 4.89898i) q^{49} +4.00000 q^{51} +9.16515i q^{53} -17.2813 q^{55} +13.5826 q^{57} -1.80341i q^{59} +6.70239 q^{61} +(16.1240 + 6.58258i) q^{63} -5.58258 q^{65} +13.5826 q^{67} +11.0901 q^{69} -9.16515i q^{71} -7.48331i q^{73} +14.1857i q^{75} +(-13.6745 - 5.58258i) q^{77} +2.00000i q^{79} +14.5826 q^{81} +3.09557i q^{83} +4.00000i q^{85} +18.5734 q^{87} -7.48331i q^{89} +(-4.41742 - 1.80341i) q^{91} -4.00000i q^{93} +13.5826i q^{95} -3.87650i q^{97} -36.7477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} + 8 q^{11} + 24 q^{35} - 8 q^{43} + 40 q^{49} + 32 q^{51} + 72 q^{57} - 8 q^{65} + 72 q^{67} + 80 q^{81} - 72 q^{91} - 184 q^{99}+O(q^{100}) $ 8 * q - 16 * q^9 + 8 * q^11 + 24 * q^35 - 8 * q^43 + 40 * q^49 + 32 * q^51 + 72 * q^57 - 8 * q^65 + 72 * q^67 + 80 * q^81 - 72 * q^91 - 184 * 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
$3$			3.09557i			1.78723i	0.448834	+	0.893615i	$0.351839\pi$
$3$			3.09557i			1.78723i	−0.448834	+	0.893615i	$0.648161\pi$
$4$		0			0
$5$	−3.09557			−1.38438			−0.692191	−	0.721714i	$-0.743355\pi$
$5$	−3.09557			−1.38438			−0.692191	+	0.721714i	$0.743355\pi$
$6$		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
$11$	5.58258			1.68321			0.841605	−	0.540094i	$-0.181611\pi$
$11$	5.58258			1.68321			0.841605	+	0.540094i	$0.181611\pi$
$12$		0			0
$13$	1.80341			0.500175			0.250087	−	0.968223i	$-0.419541\pi$
$13$	1.80341			0.500175			0.250087	+	0.968223i	$0.419541\pi$
$14$		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
$19$		−	4.38774i		−	1.00662i	−0.864107	−	0.503308i	$-0.832116\pi$
$19$		−	4.38774i		−	1.00662i	0.864107	−	0.503308i	$-0.167884\pi$
$20$		0			0
$21$	3.09557	−	7.58258i	0.675510	−	1.65465i
$22$		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
$27$		−	11.0901i		−	2.13430i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$	−1.29217			−0.232080			−0.116040	−	0.993245i	$-0.537020\pi$
$31$	−1.29217			−0.232080			−0.116040	+	0.993245i	$0.537020\pi$
$32$		0			0
$33$			17.2813i			3.00828i
$34$		0			0
$35$	7.58258	+	3.09557i	1.28169	+	0.523247i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$			5.58258i			0.893928i
$40$		0			0
$41$			6.19115i			0.966895i	0.875373	+	0.483447i	$0.160616\pi$
$41$			6.19115i			0.966895i	−0.875373	+	0.483447i	$0.839384\pi$
$42$		0			0
$43$	−5.58258			−0.851335			−0.425667	−	0.904880i	$-0.639961\pi$
$43$	−5.58258			−0.851335			−0.425667	+	0.904880i	$0.639961\pi$
$44$		0			0
$45$	20.3768			3.03760
$46$		0			0
$47$	−1.29217			−0.188482			−0.0942410	−	0.995549i	$-0.530042\pi$
$47$	−1.29217			−0.188482			−0.0942410	+	0.995549i	$0.530042\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$			9.16515i			1.25893i	0.777029	+	0.629465i	$0.216726\pi$
$53$			9.16515i			1.25893i	−0.777029	+	0.629465i	$0.783274\pi$
$54$		0			0
$55$	−17.2813			−2.33021
$56$		0			0
$57$	13.5826			1.79906
$58$		0			0
$59$		−	1.80341i		−	0.234783i	−0.993086	−	0.117392i	$-0.962547\pi$
$59$		−	1.80341i		−	0.234783i	0.993086	−	0.117392i	$-0.0374533\pi$
$60$		0			0
$61$	6.70239			0.858153			0.429076	−	0.903268i	$-0.358839\pi$
$61$	6.70239			0.858153			0.429076	+	0.903268i	$0.358839\pi$
$62$		0			0
$63$	16.1240	+	6.58258i	2.03143	+	0.829327i
$64$		0			0
$65$	−5.58258			−0.692433
$66$		0			0
$67$	13.5826			1.65938			0.829688	−	0.558228i	$-0.188518\pi$
$67$	13.5826			1.65938			0.829688	+	0.558228i	$0.188518\pi$
$68$		0			0
$69$	11.0901			1.33509
$70$		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	$-0.855713\pi$
$73$		−	7.48331i		−	0.875856i	0.899010	−	0.437928i	$-0.144287\pi$
$74$		0			0
$75$			14.1857i			1.63802i
$76$		0			0
$77$	−13.6745	−	5.58258i	−1.55835	−	0.636194i
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$	14.5826			1.62029
$82$		0			0
$83$			3.09557i			0.339783i	0.985463	+	0.169892i	$0.0543418\pi$
$83$			3.09557i			0.339783i	−0.985463	+	0.169892i	$0.945658\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$	18.5734			1.99128
$88$		0			0
$89$		−	7.48331i		−	0.793230i	−0.917985	−	0.396615i	$-0.870185\pi$
$89$		−	7.48331i		−	0.793230i	0.917985	−	0.396615i	$-0.129815\pi$
$90$		0			0
$91$	−4.41742	−	1.80341i	−0.463072	−	0.189048i
$92$		0			0
$93$		−	4.00000i		−	0.414781i
$94$		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
$99$	−36.7477			−3.69329
$100$		0			0
$101$	14.1857			1.41153			0.705765	−	0.708446i	$-0.250603\pi$
$101$	14.1857			1.41153			0.705765	+	0.708446i	$0.250603\pi$
$102$		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
$107$	6.41742			0.620396			0.310198	−	0.950672i	$-0.399605\pi$
$107$	6.41742			0.620396			0.310198	+	0.950672i	$0.399605\pi$
$108$		0			0
$109$			13.1652i			1.26099i	0.776192	+	0.630496i	$0.217149\pi$
$109$			13.1652i			1.26099i	−0.776192	+	0.630496i	$0.782851\pi$
$110$		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
$115$			11.0901i			1.03416i
$116$		0			0
$117$	−11.8711			−1.09748
$118$		0			0
$119$	−1.29217	+	3.16515i	−0.118453	+	0.290149i
$120$		0			0
$121$	20.1652			1.83320
$122$		0			0
$123$	−19.1652			−1.72806
$124$		0			0
$125$	1.29217			0.115575
$126$		0			0
$127$		−	8.41742i		−	0.746926i	−0.927645	−	0.373463i	$-0.878170\pi$
$127$		−	8.41742i		−	0.746926i	0.927645	−	0.373463i	$-0.121830\pi$
$128$		0			0
$129$		−	17.2813i		−	1.52153i
$130$		0			0
$131$		−	19.0847i		−	1.66744i	−0.552190	−	0.833718i	$-0.686208\pi$
$131$		−	19.0847i		−	1.66744i	0.552190	−	0.833718i	$-0.313792\pi$
$132$		0			0
$133$	−4.38774	+	10.7477i	−0.380465	+	0.931946i
$134$		0			0
$135$			34.3303i			2.95468i
$136$		0			0
$137$	−17.1652			−1.46652			−0.733259	−	0.679950i	$-0.762002\pi$
$137$	−17.1652			−1.46652			−0.733259	+	0.679950i	$0.762002\pi$
$138$		0			0
$139$		−	6.97208i		−	0.591364i	−0.955286	−	0.295682i	$-0.904453\pi$
$139$		−	6.97208i		−	0.591364i	0.955286	−	0.295682i	$-0.0955468\pi$
$140$		0			0
$141$		−	4.00000i		−	0.336861i
$142$		0			0
$143$	10.0677			0.841899
$144$		0			0
$145$			18.5734i			1.54244i
$146$		0			0
$147$	−15.1652	+	15.4779i	−1.25080	+	1.27659i
$148$		0			0
$149$		−	17.1652i		−	1.40622i	−0.711079	−	0.703112i	$-0.751793\pi$
$149$		−	17.1652i		−	1.40622i	0.711079	−	0.703112i	$-0.248207\pi$
$150$		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
$155$	4.00000			0.321288
$156$		0			0
$157$	14.1857			1.13214			0.566071	−	0.824356i	$-0.308463\pi$
$157$	14.1857			1.13214			0.566071	+	0.824356i	$0.308463\pi$
$158$		0			0
$159$	−28.3714			−2.25000
$160$		0			0
$161$	−3.58258	+	8.77548i	−0.282347	+	0.691605i
$162$		0			0
$163$	−5.58258			−0.437261			−0.218631	−	0.975808i	$-0.570159\pi$
$163$	−5.58258			−0.437261			−0.218631	+	0.975808i	$0.570159\pi$
$164$		0			0
$165$		−	53.4955i		−	4.16462i
$166$		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
$171$			28.8826i			2.20871i
$172$		0			0
$173$	−5.41022			−0.411331			−0.205666	−	0.978622i	$-0.565936\pi$
$173$	−5.41022			−0.411331			−0.205666	+	0.978622i	$0.565936\pi$
$174$		0			0
$175$	−11.2250	−	4.58258i	−0.848528	−	0.346410i
$176$		0			0
$177$	5.58258			0.419612
$178$		0			0
$179$	−1.58258			−0.118287			−0.0591436	−	0.998249i	$-0.518837\pi$
$179$	−1.58258			−0.118287			−0.0591436	+	0.998249i	$0.518837\pi$
$180$		0			0
$181$	21.6690			1.61065			0.805323	−	0.592837i	$-0.201992\pi$
$181$	21.6690			1.61065			0.805323	+	0.592837i	$0.201992\pi$
$182$		0			0
$183$			20.7477i			1.53372i
$184$		0			0
$185$		−	6.19115i		−	0.455182i
$186$		0			0
$187$		−	7.21362i		−	0.527512i
$188$		0			0
$189$	−11.0901	+	27.1652i	−0.806688	+	1.97597i
$190$		0			0
$191$			2.83485i			0.205122i	0.994727	+	0.102561i	$0.0327038\pi$
$191$			2.83485i			0.205122i	−0.994727	+	0.102561i	$0.967296\pi$
$192$		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
$195$		−	17.2813i		−	1.23754i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		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
$203$	−6.00000	+	14.6969i	−0.421117	+	1.03152i
$204$		0			0
$205$		−	19.1652i		−	1.33855i
$206$		0			0
$207$			23.5826i			1.63910i
$208$		0			0
$209$		−	24.4949i		−	1.69435i
$210$		0			0
$211$	19.9129			1.37086			0.685430	−	0.728139i	$-0.259614\pi$
$211$	19.9129			1.37086			0.685430	+	0.728139i	$0.259614\pi$
$212$		0			0
$213$	28.3714			1.94398
$214$		0			0
$215$	17.2813			1.17857
$216$		0			0
$217$	3.16515	+	1.29217i	0.214864	+	0.0877181i
$218$		0			0
$219$	23.1652			1.56536
$220$		0			0
$221$		−	2.33030i		−	0.156753i
$222$		0			0
$223$	12.3823			0.829180			0.414590	−	0.910008i	$-0.363925\pi$
$223$	12.3823			0.829180			0.414590	+	0.910008i	$0.363925\pi$
$224$		0			0
$225$	−30.1652			−2.01101
$226$		0			0
$227$		−	23.9837i		−	1.59185i	−0.605394	−	0.795926i	$-0.706985\pi$
$227$		−	23.9837i		−	1.59185i	0.605394	−	0.795926i	$-0.293015\pi$
$228$		0			0
$229$	−20.3768			−1.34654			−0.673270	−	0.739397i	$-0.735111\pi$
$229$	−20.3768			−1.34654			−0.673270	+	0.739397i	$0.735111\pi$
$230$		0			0
$231$	17.2813	−	42.3303i	1.13702	−	2.78513i
$232$		0			0
$233$	17.1652			1.12453			0.562263	−	0.826958i	$-0.309931\pi$
$233$	17.1652			1.12453			0.562263	+	0.826958i	$0.309931\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$		0			0
$237$	−6.19115			−0.402158
$238$		0			0
$239$		−	16.4174i		−	1.06195i	−0.847386	−	0.530977i	$-0.821825\pi$
$239$		−	16.4174i		−	1.06195i	0.847386	−	0.530977i	$-0.178175\pi$
$240$		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
$243$			11.8711i			0.761529i
$244$		0			0
$245$	−15.4779	−	15.1652i	−0.988845	−	0.968866i
$246$		0			0
$247$		−	7.91288i		−	0.503484i
$248$		0			0
$249$	−9.58258			−0.607271
$250$		0			0
$251$		−	6.70239i		−	0.423051i	−0.977372	−	0.211525i	$-0.932157\pi$
$251$		−	6.70239i		−	0.423051i	0.977372	−	0.211525i	$-0.0678431\pi$
$252$		0			0
$253$		−	20.0000i		−	1.25739i
$254$		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
$259$	2.00000	−	4.89898i	0.124274	−	0.304408i
$260$		0			0
$261$			39.4955i			2.44471i
$262$		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
$267$	23.1652			1.41768
$268$		0			0
$269$	−27.5905			−1.68222			−0.841110	−	0.540864i	$-0.818098\pi$
$269$	−27.5905			−1.68222			−0.841110	+	0.540864i	$0.818098\pi$
$270$		0			0
$271$	27.3489			1.66133			0.830664	−	0.556773i	$-0.187961\pi$
$271$	27.3489			1.66133			0.830664	+	0.556773i	$0.187961\pi$
$272$		0			0
$273$	5.58258	−	13.6745i	0.337873	−	0.827616i
$274$		0			0
$275$	25.5826			1.54269
$276$		0			0
$277$			28.3303i			1.70220i	0.525001	+	0.851101i	$0.324065\pi$
$277$			28.3303i			1.70220i	−0.525001	+	0.851101i	$0.675935\pi$
$278$		0			0
$279$	8.50579			0.509228
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		−	16.5003i		−	0.980844i	−0.871485	−	0.490422i	$-0.836843\pi$
$283$		−	16.5003i		−	0.980844i	0.871485	−	0.490422i	$-0.163157\pi$
$284$		0			0
$285$	−42.0459			−2.49058
$286$		0			0
$287$	6.19115	−	15.1652i	0.365452	−	0.895171i
$288$		0			0
$289$	15.3303			0.901783
$290$		0			0
$291$	12.0000			0.703452
$292$		0			0
$293$	−20.3768			−1.19043			−0.595214	−	0.803567i	$-0.702933\pi$
$293$	−20.3768			−1.19043			−0.595214	+	0.803567i	$0.702933\pi$
$294$		0			0
$295$			5.58258i			0.325030i
$296$		0			0
$297$		−	61.9115i		−	3.59247i
$298$		0			0
$299$		−	6.46084i		−	0.373640i
$300$		0			0
$301$	13.6745	+	5.58258i	0.788183	+	0.321774i
$302$		0			0
$303$			43.9129i			2.52273i
$304$		0			0
$305$	−20.7477			−1.18801
$306$		0			0
$307$		−	2.07310i		−	0.118318i	−0.998249	−	0.0591589i	$-0.981158\pi$
$307$		−	2.07310i		−	0.118318i	0.998249	−	0.0591589i	$-0.0188419\pi$
$308$		0			0
$309$			19.1652i			1.09027i
$310$		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	$-0.967497\pi$
$313$		−	3.60681i		−	0.203869i	0.994791	−	0.101935i	$-0.0325032\pi$
$314$		0			0
$315$	−49.9129	−	20.3768i	−2.81227	−	1.14811i
$316$		0			0
$317$			28.3303i			1.59119i	0.605830	+	0.795594i	$0.292841\pi$
$317$			28.3303i			1.59119i	−0.605830	+	0.795594i	$0.707159\pi$
$318$		0			0
$319$		−	33.4955i		−	1.87539i
$320$		0			0
$321$			19.8656i			1.10879i
$322$		0			0
$323$	−5.66970			−0.315470
$324$		0			0
$325$	8.26424			0.458418
$326$		0			0
$327$	−40.7537			−2.25368
$328$		0			0
$329$	3.16515	+	1.29217i	0.174500	+	0.0712395i
$330$		0			0
$331$	33.5826			1.84587			0.922933	−	0.384961i	$-0.125785\pi$
$331$	33.5826			1.84587			0.922933	+	0.384961i	$0.125785\pi$
$332$		0			0
$333$		−	13.1652i		−	0.721446i
$334$		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
$339$		−	13.6745i		−	0.742695i
$340$		0			0
$341$	−7.21362			−0.390640
$342$		0			0
$343$	−7.34847	−	17.0000i	−0.396780	−	0.917914i
$344$		0			0
$345$	−34.3303			−1.84828
$346$		0			0
$347$	0.747727			0.0401401			0.0200700	−	0.999799i	$-0.493611\pi$
$347$	0.747727			0.0401401			0.0200700	+	0.999799i	$0.493611\pi$
$348$		0			0
$349$	9.28672			0.497107			0.248553	−	0.968618i	$-0.420045\pi$
$349$	9.28672			0.497107			0.248553	+	0.968618i	$0.420045\pi$
$350$		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
$355$			28.3714i			1.50580i
$356$		0			0
$357$	−9.79796	−	4.00000i	−0.518563	−	0.211702i
$358$		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
$363$			62.4227i			3.27634i
$364$		0			0
$365$			23.1652i			1.21252i
$366$		0			0
$367$	29.3939			1.53435			0.767174	−	0.641439i	$-0.221662\pi$
$367$	29.3939			1.53435			0.767174	+	0.641439i	$0.221662\pi$
$368$		0			0
$369$		−	40.7537i		−	2.12155i
$370$		0			0
$371$	9.16515	−	22.4499i	0.475831	−	1.16554i
$372$		0			0
$373$		−	8.33030i		−	0.431327i	−0.976468	−	0.215663i	$-0.930809\pi$
$373$		−	8.33030i		−	0.431327i	0.976468	−	0.215663i	$-0.0691914\pi$
$374$		0			0
$375$			4.00000i			0.206559i
$376$		0			0
$377$		−	10.8204i		−	0.557281i
$378$		0			0
$379$	15.9129			0.817390			0.408695	−	0.912671i	$-0.365984\pi$
$379$	15.9129			0.817390			0.408695	+	0.912671i	$0.365984\pi$
$380$		0			0
$381$	26.0568			1.33493
$382$		0			0
$383$	−28.6411			−1.46349			−0.731746	−	0.681578i	$-0.761294\pi$
$383$	−28.6411			−1.46349			−0.731746	+	0.681578i	$0.761294\pi$
$384$		0			0
$385$	42.3303	+	17.2813i	2.15735	+	0.880735i
$386$		0			0
$387$	36.7477			1.86799
$388$		0			0
$389$			2.00000i			0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$			2.00000i			0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$		0			0
$391$	−4.62929			−0.234113
$392$		0			0
$393$	59.0780			2.98009
$394$		0			0
$395$		−	6.19115i		−	0.311510i
$396$		0			0
$397$	4.11805			0.206679			0.103340	−	0.994646i	$-0.467047\pi$
$397$	4.11805			0.206679			0.103340	+	0.994646i	$0.467047\pi$
$398$		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
$403$	−2.33030			−0.116081
$404$		0			0
$405$	−45.1414			−2.24310
$406$		0			0
$407$			11.1652i			0.553436i
$408$		0			0
$409$			36.1244i			1.78624i	0.449822	+	0.893118i	$0.351488\pi$
$409$			36.1244i			1.78624i	−0.449822	+	0.893118i	$0.648512\pi$
$410$		0			0
$411$		−	53.1360i		−	2.62101i
$412$		0			0
$413$	−1.80341	+	4.41742i	−0.0887398	+	0.217367i
$414$		0			0
$415$		−	9.58258i		−	0.470390i
$416$		0			0
$417$	21.5826			1.05690
$418$		0			0
$419$			20.6465i			1.00865i	0.863514	+	0.504325i	$0.168259\pi$
$419$			20.6465i			1.00865i	−0.863514	+	0.504325i	$0.831741\pi$
$420$		0			0
$421$		−	23.4955i		−	1.14510i	−0.819870	−	0.572549i	$-0.805955\pi$
$421$		−	23.4955i		−	1.14510i	0.819870	−	0.572549i	$-0.194045\pi$
$422$		0			0
$423$	8.50579			0.413566
$424$		0			0
$425$		−	5.92146i		−	0.287233i
$426$		0			0
$427$	−16.4174	−	6.70239i	−0.794495	−	0.324351i
$428$		0			0
$429$			31.1652i			1.50467i
$430$		0			0
$431$		−	22.7477i		−	1.09572i	−0.836570	−	0.547860i	$-0.815443\pi$
$431$		−	22.7477i		−	1.09572i	0.836570	−	0.547860i	$-0.184557\pi$
$432$		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
$435$	−57.4955			−2.75670
$436$		0			0
$437$	−15.7194			−0.751962
$438$		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
$443$	−4.74773			−0.225571			−0.112786	−	0.993619i	$-0.535977\pi$
$443$	−4.74773			−0.225571			−0.112786	+	0.993619i	$0.535977\pi$
$444$		0			0
$445$			23.1652i			1.09813i
$446$		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
$451$			34.5625i			1.62749i
$452$		0			0
$453$	35.8547			1.68460
$454$		0			0
$455$	13.6745	+	5.58258i	0.641069	+	0.261715i
$456$		0			0
$457$	−22.7477			−1.06409			−0.532047	−	0.846715i	$-0.678577\pi$
$457$	−22.7477			−1.06409			−0.532047	+	0.846715i	$0.678577\pi$
$458$		0			0
$459$	−14.3303			−0.668881
$460$		0			0
$461$	2.07310			0.0965538			0.0482769	−	0.998834i	$-0.484627\pi$
$461$	2.07310			0.0965538			0.0482769	+	0.998834i	$0.484627\pi$
$462$		0			0
$463$			1.16515i			0.0541492i	0.999633	+	0.0270746i	$0.00861916\pi$
$463$			1.16515i			0.0541492i	−0.999633	+	0.0270746i	$0.991381\pi$
$464$		0			0
$465$			12.3823i			0.574215i
$466$		0			0
$467$			25.0061i			1.15715i	0.815631	+	0.578573i	$0.196390\pi$
$467$			25.0061i			1.15715i	−0.815631	+	0.578573i	$0.803610\pi$
$468$		0			0
$469$	−33.2704	−	13.5826i	−1.53628	−	0.627185i
$470$		0			0
$471$			43.9129i			2.02340i
$472$		0			0
$473$	−31.1652			−1.43298
$474$		0			0
$475$		−	20.1072i		−	0.922580i
$476$		0			0
$477$		−	60.3303i		−	2.76233i
$478$		0			0
$479$	−8.50579			−0.388640			−0.194320	−	0.980938i	$-0.562250\pi$
$479$	−8.50579			−0.388640			−0.194320	+	0.980938i	$0.562250\pi$
$480$		0			0
$481$			3.60681i			0.164456i
$482$		0			0
$483$	−27.1652	−	11.0901i	−1.23606	−	0.504618i
$484$		0			0
$485$			12.0000i			0.544892i
$486$		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
$491$	−5.58258			−0.251938			−0.125969	−	0.992034i	$-0.540204\pi$
$491$	−5.58258			−0.251938			−0.125969	+	0.992034i	$0.540204\pi$
$492$		0			0
$493$	−7.75301			−0.349178
$494$		0			0
$495$	113.755			5.11292
$496$		0			0
$497$	−9.16515	+	22.4499i	−0.411113	+	1.00702i
$498$		0			0
$499$	2.41742			0.108219			0.0541094	−	0.998535i	$-0.482768\pi$
$499$	2.41742			0.108219			0.0541094	+	0.998535i	$0.482768\pi$
$500$		0			0
$501$			11.1652i			0.498822i
$502$		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
$507$		−	30.1748i		−	1.34011i
$508$		0			0
$509$	4.65743			0.206437			0.103219	−	0.994659i	$-0.467086\pi$
$509$	4.65743			0.206437			0.103219	+	0.994659i	$0.467086\pi$
$510$		0			0
$511$	−7.48331	+	18.3303i	−0.331042	+	0.810885i
$512$		0			0
$513$	−48.6606			−2.14842
$514$		0			0
$515$	−19.1652			−0.844517
$516$		0			0
$517$	−7.21362			−0.317255
$518$		0			0
$519$		−	16.7477i		−	0.735144i
$520$		0			0
$521$		−	13.4048i		−	0.587274i	−0.955917	−	0.293637i	$-0.905134\pi$
$521$		−	13.4048i		−	0.587274i	0.955917	−	0.293637i	$-0.0948656\pi$
$522$		0			0
$523$			32.7591i			1.43246i	0.697866	+	0.716229i	$0.254133\pi$
$523$			32.7591i			1.43246i	−0.697866	+	0.716229i	$0.745867\pi$
$524$		0			0
$525$	14.1857	−	34.7477i	0.619115	−	1.51652i
$526$		0			0
$527$			1.66970i			0.0727332i
$528$		0			0
$529$	10.1652			0.441963
$530$		0			0
$531$			11.8711i			0.515160i
$532$		0			0
$533$			11.1652i			0.483616i
$534$		0			0
$535$	−19.8656			−0.858865
$536$		0			0
$537$		−	4.89898i		−	0.211407i
$538$		0			0
$539$	27.9129	+	27.3489i	1.20229	+	1.17800i
$540$		0			0
$541$		−	37.1652i		−	1.59785i	−0.601428	−	0.798927i	$-0.705401\pi$
$541$		−	37.1652i		−	1.59785i	0.601428	−	0.798927i	$-0.294599\pi$
$542$		0			0
$543$			67.0780i			2.87859i
$544$		0			0
$545$		−	40.7537i		−	1.74570i
$546$		0			0
$547$	2.41742			0.103362			0.0516808	−	0.998664i	$-0.483542\pi$
$547$	2.41742			0.103362			0.0516808	+	0.998664i	$0.483542\pi$
$548$		0			0
$549$	−44.1190			−1.88295
$550$		0			0
$551$	−26.3264			−1.12154
$552$		0			0
$553$	2.00000	−	4.89898i	0.0850487	−	0.208326i
$554$		0			0
$555$	19.1652			0.813515
$556$		0			0
$557$			27.4955i			1.16502i	0.812824	+	0.582510i	$0.197929\pi$
$557$			27.4955i			1.16502i	−0.812824	+	0.582510i	$0.802071\pi$
$558$		0			0
$559$	−10.0677			−0.425816
$560$		0			0
$561$	22.3303			0.942786
$562$		0			0
$563$			20.3768i			0.858782i	0.903119	+	0.429391i	$0.141272\pi$
$563$			20.3768i			0.858782i	−0.903119	+	0.429391i	$0.858728\pi$
$564$		0			0
$565$	13.6745			0.575289
$566$		0			0
$567$	−35.7199	−	14.5826i	−1.50009	−	0.612411i
$568$		0			0
$569$	−8.41742			−0.352877			−0.176438	−	0.984312i	$-0.556458\pi$
$569$	−8.41742			−0.352877			−0.176438	+	0.984312i	$0.556458\pi$
$570$		0			0
$571$	−5.58258			−0.233624			−0.116812	−	0.993154i	$-0.537267\pi$
$571$	−5.58258			−0.233624			−0.116812	+	0.993154i	$0.537267\pi$
$572$		0			0
$573$	−8.77548			−0.366601
$574$		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
$579$			1.29217i			0.0537007i
$580$		0			0
$581$	3.09557	−	7.58258i	0.128426	−	0.314578i
$582$		0			0
$583$			51.1652i			2.11904i
$584$		0			0
$585$	36.7477			1.51933
$586$		0			0
$587$		−	4.38774i		−	0.181101i	−0.995892	−	0.0905507i	$-0.971137\pi$
$587$		−	4.38774i		−	0.181101i	0.995892	−	0.0905507i	$-0.0288627\pi$
$588$		0			0
$589$			5.66970i			0.233616i
$590$		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
$595$	4.00000	−	9.79796i	0.163984	−	0.401677i
$596$		0			0
$597$		−	19.1652i		−	0.784377i
$598$		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	$-0.934172\pi$
$601$		−	10.0677i		−	0.410668i	0.978692	−	0.205334i	$-0.0658281\pi$
$602$		0			0
$603$	−89.4083			−3.64099
$604$		0			0
$605$	−62.4227			−2.53784
$606$		0			0
$607$	−17.5510			−0.712372			−0.356186	−	0.934415i	$-0.615923\pi$
$607$	−17.5510			−0.712372			−0.356186	+	0.934415i	$0.615923\pi$
$608$		0			0
$609$	−45.4955	−	18.5734i	−1.84357	−	0.752634i
$610$		0			0
$611$	−2.33030			−0.0942740
$612$		0			0
$613$		−	2.83485i		−	0.114498i	−0.998360	−	0.0572492i	$-0.981767\pi$
$613$		−	2.83485i		−	0.114498i	0.998360	−	0.0572492i	$-0.0182330\pi$
$614$		0			0
$615$	59.3271			2.39230
$616$		0			0
$617$	−11.5826			−0.466297			−0.233148	−	0.972441i	$-0.574903\pi$
$617$	−11.5826			−0.466297			−0.233148	+	0.972441i	$0.574903\pi$
$618$		0			0
$619$		−	26.8377i		−	1.07870i	−0.842082	−	0.539349i	$-0.818670\pi$
$619$		−	26.8377i		−	1.07870i	0.842082	−	0.539349i	$-0.181330\pi$
$620$		0			0
$621$	−39.7312			−1.59436
$622$		0			0
$623$	−7.48331	+	18.3303i	−0.299813	+	0.734388i
$624$		0			0
$625$	−26.9129			−1.07652
$626$		0			0
$627$	75.8258			3.02819
$628$		0			0
$629$	2.58434			0.103044
$630$		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
$635$			26.0568i			1.03403i
$636$		0			0
$637$	9.01703	+	8.83485i	0.357268	+	0.350049i
$638$		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
$643$		−	31.7367i		−	1.25157i	−0.779995	−	0.625786i	$-0.784778\pi$
$643$		−	31.7367i		−	1.25157i	0.779995	−	0.625786i	$-0.215222\pi$
$644$		0			0
$645$			53.4955i			2.10638i
$646$		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
$651$	−4.00000	+	9.79796i	−0.156772	+	0.384012i
$652$		0			0
$653$		−	27.4955i		−	1.07598i	−0.842951	−	0.537990i	$-0.819184\pi$
$653$		−	27.4955i		−	1.07598i	0.842951	−	0.537990i	$-0.180816\pi$
$654$		0			0
$655$			59.0780i			2.30837i
$656$		0			0
$657$			49.2595i			1.92180i
$658$		0			0
$659$	−44.7477			−1.74312			−0.871562	−	0.490285i	$-0.836893\pi$
$659$	−44.7477			−1.74312			−0.871562	+	0.490285i	$0.836893\pi$
$660$		0			0
$661$	−30.7142			−1.19464			−0.597322	−	0.802002i	$-0.703768\pi$
$661$	−30.7142			−1.19464			−0.597322	+	0.802002i	$0.703768\pi$
$662$		0			0
$663$	7.21362			0.280154
$664$		0			0
$665$	13.5826	−	33.2704i	0.526710	−	1.29017i
$666$		0			0
$667$	−21.4955			−0.832307
$668$		0			0
$669$			38.3303i			1.48194i
$670$		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
$675$		−	50.8213i		−	1.95611i
$676$		0			0
$677$	−0.511238			−0.0196485			−0.00982424	−	0.999952i	$-0.503127\pi$
$677$	−0.511238			−0.0196485			−0.00982424	+	0.999952i	$0.503127\pi$
$678$		0			0
$679$	−3.87650	+	9.49545i	−0.148767	+	0.364402i
$680$		0			0
$681$	74.2432			2.84500
$682$		0			0
$683$	−9.58258			−0.366667			−0.183334	−	0.983051i	$-0.558689\pi$
$683$	−9.58258			−0.366667			−0.183334	+	0.983051i	$0.558689\pi$
$684$		0			0
$685$	53.1360			2.03022
$686$		0			0
$687$		−	63.0780i		−	2.40658i
$688$		0			0
$689$			16.5285i			0.629685i
$690$		0			0
$691$			8.26424i			0.314387i	0.987568	+	0.157193i	$0.0502446\pi$
$691$			8.26424i			0.314387i	−0.987568	+	0.157193i	$0.949755\pi$
$692$		0			0
$693$	90.0132	+	36.7477i	3.41932	+	1.39593i
$694$		0			0
$695$			21.5826i			0.818674i
$696$		0			0
$697$	8.00000			0.303022
$698$		0			0
$699$			53.1360i			2.00979i
$700$		0			0
$701$			15.4955i			0.585255i	0.956226	+	0.292628i	$0.0945296\pi$
$701$			15.4955i			0.585255i	−0.956226	+	0.292628i	$0.905470\pi$
$702$		0			0
$703$	8.77548			0.330974
$704$		0			0
$705$			12.3823i			0.466344i
$706$		0			0
$707$	−34.7477	−	14.1857i	−1.30682	−	0.533508i
$708$		0			0
$709$		−	13.1652i		−	0.494428i	−0.968961	−	0.247214i	$-0.920485\pi$
$709$		−	13.1652i		−	0.494428i	0.968961	−	0.247214i	$-0.0795150\pi$
$710$		0			0
$711$		−	13.1652i		−	0.493732i
$712$		0			0
$713$			4.62929i			0.173368i
$714$		0			0
$715$	−31.1652			−1.16551
$716$		0			0
$717$	50.8213			1.89796
$718$		0			0
$719$	−3.33712			−0.124454			−0.0622268	−	0.998062i	$-0.519820\pi$
$719$	−3.33712			−0.124454			−0.0622268	+	0.998062i	$0.519820\pi$
$720$		0			0
$721$	−15.1652	−	6.19115i	−0.564780	−	0.230570i
$722$		0			0
$723$	4.00000			0.148762
$724$		0			0
$725$		−	27.4955i		−	1.02116i
$726$		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
$731$			7.21362i			0.266806i
$732$		0			0
$733$	34.0513			1.25771			0.628857	−	0.777521i	$-0.283523\pi$
$733$	34.0513			1.25771			0.628857	+	0.777521i	$0.283523\pi$
$734$		0			0
$735$	46.9448	−	47.9129i	1.73159	−	1.76729i
$736$		0			0
$737$	75.8258			2.79308
$738$		0			0
$739$	0.747727			0.0275056			0.0137528	−	0.999905i	$-0.495622\pi$
$739$	0.747727			0.0275056			0.0137528	+	0.999905i	$0.495622\pi$
$740$		0			0
$741$	24.4949			0.899843
$742$		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
$747$		−	20.3768i		−	0.745550i
$748$		0			0
$749$	−15.7194	−	6.41742i	−0.574375	−	0.234488i
$750$		0			0
$751$		−	53.0780i		−	1.93684i	−0.249315	−	0.968422i	$-0.580205\pi$
$751$		−	53.0780i		−	1.93684i	0.249315	−	0.968422i	$-0.419795\pi$
$752$		0			0
$753$	20.7477			0.756089
$754$		0			0
$755$			35.8547i			1.30489i
$756$		0			0
$757$		−	49.1652i		−	1.78694i	−0.449125	−	0.893469i	$-0.648264\pi$
$757$		−	49.1652i		−	1.78694i	0.449125	−	0.893469i	$-0.351736\pi$
$758$		0			0
$759$	61.9115			2.24724
$760$		0			0
$761$		−	23.7421i		−	0.860651i	−0.902674	−	0.430325i	$-0.858399\pi$
$761$		−	23.7421i		−	0.860651i	0.902674	−	0.430325i	$-0.141601\pi$
$762$		0			0
$763$	13.1652	−	32.2479i	0.476610	−	1.16745i
$764$		0			0
$765$		−	26.3303i		−	0.951974i
$766$		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
$771$	30.3303			1.09232
$772$		0			0
$773$	6.70239			0.241068			0.120534	−	0.992709i	$-0.461539\pi$
$773$	6.70239			0.241068			0.120534	+	0.992709i	$0.461539\pi$
$774$		0			0
$775$	−5.92146			−0.212705
$776$		0			0
$777$	15.1652	+	6.19115i	0.544047	+	0.222106i
$778$		0			0
$779$	27.1652			0.973293
$780$		0			0
$781$		−	51.1652i		−	1.83083i
$782$		0			0
$783$	−66.5408			−2.37797
$784$		0			0
$785$	−43.9129			−1.56732
$786$		0			0
$787$			10.5789i			0.377097i	0.982064	+	0.188548i	$0.0603782\pi$
$787$			10.5789i			0.377097i	−0.982064	+	0.188548i	$0.939622\pi$
$788$		0			0
$789$	77.9006			2.77333
$790$		0			0
$791$	10.8204	+	4.41742i	0.384730	+	0.157066i
$792$		0			0
$793$	12.0871			0.429226
$794$		0			0
$795$	87.8258			3.11486
$796$		0			0
$797$	1.80341			0.0638799			0.0319400	−	0.999490i	$-0.489831\pi$
$797$	1.80341			0.0638799			0.0319400	+	0.999490i	$0.489831\pi$
$798$		0			0
$799$			1.66970i			0.0590696i
$800$		0			0
$801$			49.2595i			1.74050i
$802$		0			0
$803$		−	41.7762i		−	1.47425i
$804$		0			0
$805$	11.0901	−	27.1652i	0.390876	−	0.957446i
$806$		0			0
$807$		−	85.4083i		−	3.00652i
$808$		0			0
$809$	7.58258			0.266589			0.133295	−	0.991076i	$-0.457444\pi$
$809$	7.58258			0.266589			0.133295	+	0.991076i	$0.457444\pi$
$810$		0			0
$811$		−	9.28672i		−	0.326101i	−0.986618	−	0.163050i	$-0.947867\pi$
$811$		−	9.28672i		−	0.326101i	0.986618	−	0.163050i	$-0.0521333\pi$
$812$		0			0
$813$			84.6606i			2.96918i
$814$		0			0
$815$	17.2813			0.605337
$816$		0			0
$817$			24.4949i			0.856968i
$818$		0			0
$819$	29.0780	+	11.8711i	1.01607	+	0.414808i
$820$		0			0
$821$			18.6606i			0.651260i	0.945497	+	0.325630i	$0.105576\pi$
$821$			18.6606i			0.651260i	−0.945497	+	0.325630i	$0.894424\pi$
$822$		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
$827$	19.9129			0.692439			0.346219	−	0.938154i	$-0.387465\pi$
$827$	19.9129			0.692439			0.346219	+	0.938154i	$0.387465\pi$
$828$		0			0
$829$	−37.6581			−1.30792			−0.653960	−	0.756529i	$-0.726894\pi$
$829$	−37.6581			−1.30792			−0.653960	+	0.756529i	$0.726894\pi$
$830$		0			0
$831$	−87.6985			−3.04223
$832$		0			0
$833$	6.33030	−	6.46084i	0.219332	−	0.223855i
$834$		0			0
$835$	−11.1652			−0.386386
$836$		0			0
$837$			14.3303i			0.495328i
$838$		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
$843$			30.9557i			1.06617i
$844$		0			0
$845$	30.1748			1.03804
$846$		0			0
$847$	−49.3943	−	20.1652i	−1.69721	−	0.692883i
$848$		0			0
$849$	51.0780			1.75299
$850$		0			0
$851$	7.16515			0.245618
$852$		0			0
$853$	28.6129			0.979689			0.489844	−	0.871810i	$-0.337054\pi$
$853$	28.6129			0.979689			0.489844	+	0.871810i	$0.337054\pi$
$854$		0			0
$855$		−	89.4083i		−	3.05770i
$856$		0			0
$857$			40.7537i			1.39212i	0.717984	+	0.696060i	$0.245065\pi$
$857$			40.7537i			1.39212i	−0.717984	+	0.696060i	$0.754935\pi$
$858$		0			0
$859$			47.4561i			1.61918i	0.586995	+	0.809590i	$0.300311\pi$
$859$			47.4561i			1.61918i	−0.586995	+	0.809590i	$0.699689\pi$
$860$		0			0
$861$	46.9448	+	19.1652i	1.59988	+	0.653147i
$862$		0			0
$863$			33.1652i			1.12895i	0.825448	+	0.564477i	$0.190922\pi$
$863$			33.1652i			1.12895i	−0.825448	+	0.564477i	$0.809078\pi$
$864$		0			0
$865$	16.7477			0.569440
$866$		0			0
$867$			47.4561i			1.61169i
$868$		0			0
$869$			11.1652i			0.378752i
$870$		0			0
$871$	24.4949			0.829978
$872$		0			0
$873$			25.5174i			0.863632i
$874$		0			0
$875$	−3.16515	−	1.29217i	−0.107002	−	0.0436832i
$876$		0			0
$877$			4.33030i			0.146224i	0.997324	+	0.0731120i	$0.0232930\pi$
$877$			4.33030i			0.146224i	−0.997324	+	0.0731120i	$0.976707\pi$
$878$		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
$883$	−51.0780			−1.71891			−0.859456	−	0.511209i	$-0.829198\pi$
$883$	−51.0780			−1.71891			−0.859456	+	0.511209i	$0.829198\pi$
$884$		0			0
$885$	−17.2813			−0.580904
$886$		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
$891$	81.4083			2.72728
$892$		0			0
$893$			5.66970i			0.189729i
$894$		0			0
$895$	4.89898			0.163755
$896$		0			0
$897$	20.0000			0.667781
$898$		0			0
$899$			7.75301i			0.258577i
$900$		0			0
$901$	11.8429			0.394545
$902$		0			0
$903$	−17.2813	+	42.3303i	−0.575085	+	1.40866i
$904$		0			0
$905$	−67.0780			−2.22975
$906$		0			0
$907$	39.9129			1.32529			0.662643	−	0.748936i	$-0.269435\pi$
$907$	39.9129			1.32529			0.662643	+	0.748936i	$0.269435\pi$
$908$		0			0
$909$	−93.3784			−3.09717
$910$		0			0
$911$			23.5826i			0.781326i	0.920534	+	0.390663i	$0.127754\pi$
$911$			23.5826i			0.781326i	−0.920534	+	0.390663i	$0.872246\pi$
$912$		0			0
$913$			17.2813i			0.571927i
$914$		0			0
$915$		−	64.2261i		−	2.12325i
$916$		0			0
$917$	−19.0847	+	46.7477i	−0.630232	+	1.54375i
$918$		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
$923$		−	16.5285i		−	0.544042i
$924$		0			0
$925$			9.16515i			0.301348i
$926$		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
$931$	21.4955	−	21.9387i	0.704485	−	0.719012i
$932$		0			0
$933$		−	91.8258i		−	3.00624i
$934$		0			0
$935$			22.3303i			0.730279i
$936$		0			0
$937$			51.8438i			1.69366i	0.531861	+	0.846832i	$0.321493\pi$
$937$			51.8438i			1.69366i	−0.531861	+	0.846832i	$0.678507\pi$
$938$		0			0
$939$	11.1652			0.364361
$940$		0			0
$941$	−37.9278			−1.23641			−0.618206	−	0.786016i	$-0.712140\pi$
$941$	−37.9278			−1.23641			−0.618206	+	0.786016i	$0.712140\pi$
$942$		0			0
$943$	22.1803			0.722288
$944$		0			0
$945$	34.3303	−	84.0917i	1.11676	−	2.73550i
$946$		0			0
$947$	19.2523			0.625615			0.312807	−	0.949817i	$-0.398731\pi$
$947$	19.2523			0.625615			0.312807	+	0.949817i	$0.398731\pi$
$948$		0			0
$949$		−	13.4955i		−	0.438081i
$950$		0			0
$951$	−87.6985			−2.84382
$952$		0			0
$953$	28.3303			0.917709			0.458854	−	0.888512i	$-0.348260\pi$
$953$	28.3303			0.917709			0.458854	+	0.888512i	$0.348260\pi$
$954$		0			0
$955$		−	8.77548i		−	0.283968i
$956$		0			0
$957$	103.688			3.35175
$958$		0			0
$959$	42.0459	+	17.1652i	1.35773	+	0.554292i
$960$		0			0
$961$	−29.3303			−0.946139
$962$		0			0
$963$	−42.2432			−1.36127
$964$		0			0
$965$	−1.29217			−0.0415963
$966$		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
$971$		−	18.8150i		−	0.603802i	−0.953339	−	0.301901i	$-0.902379\pi$
$971$		−	18.8150i		−	0.603802i	0.953339	−	0.301901i	$-0.0976212\pi$
$972$		0			0
$973$	−6.97208	+	17.0780i	−0.223515	+	0.547497i
$974$		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
$979$		−	41.7762i		−	1.33517i
$980$		0			0
$981$		−	86.6606i		−	2.76686i
$982$		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
$987$	−4.00000	+	9.79796i	−0.127321	+	0.311872i
$988$		0			0
$989$			20.0000i			0.635963i
$990$		0			0
$991$		−	20.3303i		−	0.645813i	−0.946431	−	0.322907i	$-0.895340\pi$
$991$		−	20.3303i		−	0.645813i	0.946431	−	0.322907i	$-0.104660\pi$
$992$		0			0
$993$			103.957i			3.29899i
$994$		0			0
$995$	19.1652			0.607576
$996$		0			0
$997$	19.0847			0.604418			0.302209	−	0.953242i	$-0.402276\pi$
$997$	19.0847			0.604418			0.302209	+	0.953242i	$0.402276\pi$
$998$		0			0
$999$	22.1803			0.701752

Twists

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

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

\(f(q)\)	\(=\)	\(q+3.09557i q^{3} -3.09557 q^{5} +(-2.44949 - 1.00000i) q^{7} -6.58258 q^{9} +O(q^{10})\)	\(q+3.09557i q^{3} -3.09557 q^{5} +(-2.44949 - 1.00000i) q^{7} -6.58258 q^{9} +5.58258 q^{11} +1.80341 q^{13} -9.58258i q^{15} -1.29217i q^{17} -4.38774i q^{19} +(3.09557 - 7.58258i) q^{21} -3.58258i q^{23} +4.58258 q^{25} -11.0901i q^{27} -6.00000i q^{29} -1.29217 q^{31} +17.2813i q^{33} +(7.58258 + 3.09557i) q^{35} +2.00000i q^{37} +5.58258i q^{39} +6.19115i q^{41} -5.58258 q^{43} +20.3768 q^{45} -1.29217 q^{47} +(5.00000 + 4.89898i) q^{49} +4.00000 q^{51} +9.16515i q^{53} -17.2813 q^{55} +13.5826 q^{57} -1.80341i q^{59} +6.70239 q^{61} +(16.1240 + 6.58258i) q^{63} -5.58258 q^{65} +13.5826 q^{67} +11.0901 q^{69} -9.16515i q^{71} -7.48331i q^{73} +14.1857i q^{75} +(-13.6745 - 5.58258i) q^{77} +2.00000i q^{79} +14.5826 q^{81} +3.09557i q^{83} +4.00000i q^{85} +18.5734 q^{87} -7.48331i q^{89} +(-4.41742 - 1.80341i) q^{91} -4.00000i q^{93} +13.5826i q^{95} -3.87650i q^{97} -36.7477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} + 8 q^{11} + 24 q^{35} - 8 q^{43} + 40 q^{49} + 32 q^{51} + 72 q^{57} - 8 q^{65} + 72 q^{67} + 80 q^{81} - 72 q^{91} - 184 q^{99}+O(q^{100}) \) 8 * q - 16 * q^9 + 8 * q^11 + 24 * q^35 - 8 * q^43 + 40 * q^49 + 32 * q^51 + 72 * q^57 - 8 * q^65 + 72 * q^67 + 80 * q^81 - 72 * q^91 - 184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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