LMFDB - Embedded newform 320.2.f.b.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,2,Mod(289,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("320.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	320.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:	$2.55521286468$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	320.289
Dual form			320.2.f.b.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +2.00000i q^{11} +5.65685 q^{13} +(3.46410 + 4.24264i) q^{15} +4.89898i q^{17} +6.00000i q^{19} -3.46410i q^{21} -7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} +6.92820i q^{29} +6.92820 q^{31} -4.89898i q^{33} +(2.44949 - 2.00000i) q^{35} -2.82843 q^{37} -13.8564 q^{39} -4.00000 q^{41} -2.44949 q^{43} +(-4.24264 - 5.19615i) q^{45} +4.24264i q^{47} +5.00000 q^{49} -12.0000i q^{51} +(3.46410 - 2.82843i) q^{55} -14.6969i q^{57} -2.00000i q^{59} -3.46410i q^{61} +4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949 q^{67} +17.3205i q^{69} -6.92820 q^{71} +4.89898i q^{73} +(2.44949 - 12.0000i) q^{75} -2.82843 q^{77} -6.92820 q^{79} -9.00000 q^{81} +12.2474 q^{83} +(8.48528 - 6.92820i) q^{85} -16.9706i q^{87} -2.00000 q^{89} +8.00000i q^{91} -16.9706 q^{93} +(10.3923 - 8.48528i) q^{95} +14.6969i q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 8 q^{25} - 32 q^{41} + 40 q^{49} - 64 q^{65} - 72 q^{81} - 16 q^{89}+O(q^{100}) $ 8 * q + 24 * q^9 - 8 * q^25 - 32 * q^41 + 40 * q^49 - 64 * q^65 - 72 * q^81 - 16 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.44949			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−1.41421	−	1.73205i	−0.632456	−	0.774597i
$5$	−1.41421	−	1.73205i	−0.632456	−	0.774597i
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	5.65685			1.56893			0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685			1.56893			0.784465	+	0.620174i	$0.212938\pi$
$14$		0			0
$15$	3.46410	+	4.24264i	0.894427	+	1.09545i
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		−	3.46410i		−	0.755929i
$22$		0			0
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$	−1.00000	+	4.89898i	−0.200000	+	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$		0			0
$31$	6.92820			1.24434			0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820			1.24434			0.622171	+	0.782881i	$0.286251\pi$
$32$		0			0
$33$		−	4.89898i		−	0.852803i
$34$		0			0
$35$	2.44949	−	2.00000i	0.414039	−	0.338062i
$36$		0			0
$37$	−2.82843			−0.464991			−0.232495	−	0.972598i	$-0.574689\pi$
$37$	−2.82843			−0.464991			−0.232495	+	0.972598i	$0.574689\pi$
$38$		0			0
$39$	−13.8564			−2.21880
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	−2.44949			−0.373544			−0.186772	−	0.982403i	$-0.559803\pi$
$43$	−2.44949			−0.373544			−0.186772	+	0.982403i	$0.559803\pi$
$44$		0			0
$45$	−4.24264	−	5.19615i	−0.632456	−	0.774597i
$46$		0			0
$47$			4.24264i			0.618853i	0.950923	+	0.309426i	$0.100137\pi$
$47$			4.24264i			0.618853i	−0.950923	+	0.309426i	$0.899863\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		−	12.0000i		−	1.68034i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	3.46410	−	2.82843i	0.467099	−	0.381385i
$56$		0			0
$57$		−	14.6969i		−	1.94666i
$58$		0			0
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$		−	3.46410i		−	0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$		−	3.46410i		−	0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$		0			0
$63$			4.24264i			0.534522i
$64$		0			0
$65$	−8.00000	−	9.79796i	−0.992278	−	1.21529i
$66$		0			0
$67$	2.44949			0.299253			0.149626	−	0.988743i	$-0.452193\pi$
$67$	2.44949			0.299253			0.149626	+	0.988743i	$0.452193\pi$
$68$		0			0
$69$			17.3205i			2.08514i
$70$		0			0
$71$	−6.92820			−0.822226			−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820			−0.822226			−0.411113	+	0.911584i	$0.634860\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$	2.44949	−	12.0000i	0.282843	−	1.38564i
$76$		0			0
$77$	−2.82843			−0.322329
$78$		0			0
$79$	−6.92820			−0.779484			−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820			−0.779484			−0.389742	+	0.920924i	$0.627436\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	12.2474			1.34433			0.672166	−	0.740400i	$-0.265364\pi$
$83$	12.2474			1.34433			0.672166	+	0.740400i	$0.265364\pi$
$84$		0			0
$85$	8.48528	−	6.92820i	0.920358	−	0.751469i
$86$		0			0
$87$		−	16.9706i		−	1.81944i
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$		0			0
$93$	−16.9706			−1.75977
$94$		0			0
$95$	10.3923	−	8.48528i	1.06623	−	0.870572i
$96$		0			0
$97$			14.6969i			1.49225i	0.665807	+	0.746124i	$0.268087\pi$
$97$			14.6969i			1.49225i	−0.665807	+	0.746124i	$0.731913\pi$
$98$		0			0
$99$			6.00000i			0.603023i
$100$		0			0
$101$			6.92820i			0.689382i	0.938716	+	0.344691i	$0.112016\pi$
$101$			6.92820i			0.689382i	−0.938716	+	0.344691i	$0.887984\pi$
$102$		0			0
$103$			9.89949i			0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$			9.89949i			0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$		0			0
$105$	−6.00000	+	4.89898i	−0.585540	+	0.478091i
$106$		0			0
$107$	2.44949			0.236801			0.118401	−	0.992966i	$-0.462223\pi$
$107$	2.44949			0.236801			0.118401	+	0.992966i	$0.462223\pi$
$108$		0			0
$109$			3.46410i			0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$			3.46410i			0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$		0			0
$111$	6.92820			0.657596
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−12.2474	+	10.0000i	−1.14208	+	0.932505i
$116$		0			0
$117$	16.9706			1.56893
$118$		0			0
$119$	−6.92820			−0.635107
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	9.79796			0.883452
$124$		0			0
$125$	9.89949	−	5.19615i	0.885438	−	0.464758i
$126$		0			0
$127$			7.07107i			0.627456i	0.949513	+	0.313728i	$0.101578\pi$
$127$			7.07107i			0.627456i	−0.949513	+	0.313728i	$0.898422\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$		−	10.0000i		−	0.873704i	−0.899533	−	0.436852i	$-0.856093\pi$
$131$		−	10.0000i		−	0.873704i	0.899533	−	0.436852i	$-0.143907\pi$
$132$		0			0
$133$	−8.48528			−0.735767
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	9.79796i		−	0.837096i	−0.908195	−	0.418548i	$-0.862539\pi$
$137$		−	9.79796i		−	0.837096i	0.908195	−	0.418548i	$-0.137461\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$		0			0
$141$		−	10.3923i		−	0.875190i
$142$		0			0
$143$			11.3137i			0.946100i
$144$		0			0
$145$	12.0000	−	9.79796i	0.996546	−	0.813676i
$146$		0			0
$147$	−12.2474			−1.01015
$148$		0			0
$149$		−	10.3923i		−	0.851371i	−0.904871	−	0.425685i	$-0.860033\pi$
$149$		−	10.3923i		−	0.851371i	0.904871	−	0.425685i	$-0.139967\pi$
$150$		0			0
$151$	13.8564			1.12762			0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564			1.12762			0.563809	+	0.825905i	$0.309335\pi$
$152$		0			0
$153$			14.6969i			1.18818i
$154$		0			0
$155$	−9.79796	−	12.0000i	−0.786991	−	0.963863i
$156$		0			0
$157$	2.82843			0.225733			0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843			0.225733			0.112867	+	0.993610i	$0.463997\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.0000			0.788110
$162$		0			0
$163$	−22.0454			−1.72673			−0.863365	−	0.504580i	$-0.831647\pi$
$163$	−22.0454			−1.72673			−0.863365	+	0.504580i	$0.831647\pi$
$164$		0			0
$165$	−8.48528	+	6.92820i	−0.660578	+	0.539360i
$166$		0			0
$167$			9.89949i			0.766046i	0.923739	+	0.383023i	$0.125117\pi$
$167$			9.89949i			0.766046i	−0.923739	+	0.383023i	$0.874883\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$		0			0
$171$			18.0000i			1.37649i
$172$		0			0
$173$	8.48528			0.645124			0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528			0.645124			0.322562	+	0.946548i	$0.395456\pi$
$174$		0			0
$175$	−6.92820	−	1.41421i	−0.523723	−	0.106904i
$176$		0			0
$177$			4.89898i			0.368230i
$178$		0			0
$179$			2.00000i			0.149487i	0.997203	+	0.0747435i	$0.0238138\pi$
$179$			2.00000i			0.149487i	−0.997203	+	0.0747435i	$0.976186\pi$
$180$		0			0
$181$		−	20.7846i		−	1.54491i	−0.635071	−	0.772454i	$-0.719029\pi$
$181$		−	20.7846i		−	1.54491i	0.635071	−	0.772454i	$-0.280971\pi$
$182$		0			0
$183$			8.48528i			0.627250i
$184$		0			0
$185$	4.00000	+	4.89898i	0.294086	+	0.360180i
$186$		0			0
$187$	−9.79796			−0.716498
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		0			0
$193$			4.89898i			0.352636i	0.984333	+	0.176318i	$0.0564187\pi$
$193$			4.89898i			0.352636i	−0.984333	+	0.176318i	$0.943581\pi$
$194$		0			0
$195$	19.5959	+	24.0000i	1.40329	+	1.71868i
$196$		0			0
$197$	−22.6274			−1.61214			−0.806068	−	0.591822i	$-0.798409\pi$
$197$	−22.6274			−1.61214			−0.806068	+	0.591822i	$0.798409\pi$
$198$		0			0
$199$	20.7846			1.47338			0.736691	−	0.676230i	$-0.236387\pi$
$199$	20.7846			1.47338			0.736691	+	0.676230i	$0.236387\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$	−9.79796			−0.687682
$204$		0			0
$205$	5.65685	+	6.92820i	0.395092	+	0.483887i
$206$		0			0
$207$		−	21.2132i		−	1.47442i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$		0			0
$213$	16.9706			1.16280
$214$		0			0
$215$	3.46410	+	4.24264i	0.236250	+	0.289346i
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$			27.7128i			1.86417i
$222$		0			0
$223$		−	26.8701i		−	1.79935i	−0.436558	−	0.899676i	$-0.643803\pi$
$223$		−	26.8701i		−	1.79935i	0.436558	−	0.899676i	$-0.356197\pi$
$224$		0			0
$225$	−3.00000	+	14.6969i	−0.200000	+	0.979796i
$226$		0			0
$227$	17.1464			1.13805			0.569024	−	0.822321i	$-0.307321\pi$
$227$	17.1464			1.13805			0.569024	+	0.822321i	$0.307321\pi$
$228$		0			0
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$	6.92820			0.455842
$232$		0			0
$233$		−	14.6969i		−	0.962828i	−0.876493	−	0.481414i	$-0.840123\pi$
$233$		−	14.6969i		−	0.962828i	0.876493	−	0.481414i	$-0.159877\pi$
$234$		0			0
$235$	7.34847	−	6.00000i	0.479361	−	0.391397i
$236$		0			0
$237$	16.9706			1.10236
$238$		0			0
$239$	6.92820			0.448148			0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820			0.448148			0.224074	+	0.974572i	$0.428064\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	22.0454			1.41421
$244$		0			0
$245$	−7.07107	−	8.66025i	−0.451754	−	0.553283i
$246$		0			0
$247$			33.9411i			2.15962i
$248$		0			0
$249$	−30.0000			−1.90117
$250$		0			0
$251$			26.0000i			1.64111i	0.571571	+	0.820553i	$0.306334\pi$
$251$			26.0000i			1.64111i	−0.571571	+	0.820553i	$0.693666\pi$
$252$		0			0
$253$	14.1421			0.889108
$254$		0			0
$255$	−20.7846	+	16.9706i	−1.30158	+	1.06274i
$256$		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
$259$		−	4.00000i		−	0.248548i
$260$		0			0
$261$			20.7846i			1.28654i
$262$		0			0
$263$		−	26.8701i		−	1.65688i	−0.560079	−	0.828439i	$-0.689229\pi$
$263$		−	26.8701i		−	1.65688i	0.560079	−	0.828439i	$-0.310771\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.89898			0.299813
$268$		0			0
$269$		−	17.3205i		−	1.05605i	−0.849229	−	0.528025i	$-0.822933\pi$
$269$		−	17.3205i		−	1.05605i	0.849229	−	0.528025i	$-0.177067\pi$
$270$		0			0
$271$	−20.7846			−1.26258			−0.631288	−	0.775549i	$-0.717473\pi$
$271$	−20.7846			−1.26258			−0.631288	+	0.775549i	$0.717473\pi$
$272$		0			0
$273$		−	19.5959i		−	1.18600i
$274$		0			0
$275$	−9.79796	−	2.00000i	−0.590839	−	0.120605i
$276$		0			0
$277$	−14.1421			−0.849719			−0.424859	−	0.905259i	$-0.639676\pi$
$277$	−14.1421			−0.849719			−0.424859	+	0.905259i	$0.639676\pi$
$278$		0			0
$279$	20.7846			1.24434
$280$		0			0
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		0			0
$283$	2.44949			0.145607			0.0728035	−	0.997346i	$-0.476805\pi$
$283$	2.44949			0.145607			0.0728035	+	0.997346i	$0.476805\pi$
$284$		0			0
$285$	−25.4558	+	20.7846i	−1.50787	+	1.23117i
$286$		0			0
$287$		−	5.65685i		−	0.333914i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		−	36.0000i		−	2.11036i
$292$		0			0
$293$	−2.82843			−0.165238			−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843			−0.165238			−0.0826192	+	0.996581i	$0.526329\pi$
$294$		0			0
$295$	−3.46410	+	2.82843i	−0.201688	+	0.164677i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	40.0000i		−	2.31326i
$300$		0			0
$301$		−	3.46410i		−	0.199667i
$302$		0			0
$303$		−	16.9706i		−	0.974933i
$304$		0			0
$305$	−6.00000	+	4.89898i	−0.343559	+	0.280515i
$306$		0			0
$307$	2.44949			0.139800			0.0698999	−	0.997554i	$-0.477732\pi$
$307$	2.44949			0.139800			0.0698999	+	0.997554i	$0.477732\pi$
$308$		0			0
$309$		−	24.2487i		−	1.37946i
$310$		0			0
$311$	−27.7128			−1.57145			−0.785725	−	0.618576i	$-0.787710\pi$
$311$	−27.7128			−1.57145			−0.785725	+	0.618576i	$0.787710\pi$
$312$		0			0
$313$		−	29.3939i		−	1.66144i	−0.556690	−	0.830720i	$-0.687929\pi$
$313$		−	29.3939i		−	1.66144i	0.556690	−	0.830720i	$-0.312071\pi$
$314$		0			0
$315$	7.34847	−	6.00000i	0.414039	−	0.338062i
$316$		0			0
$317$	−22.6274			−1.27088			−0.635441	−	0.772149i	$-0.719182\pi$
$317$	−22.6274			−1.27088			−0.635441	+	0.772149i	$0.719182\pi$
$318$		0			0
$319$	−13.8564			−0.775810
$320$		0			0
$321$	−6.00000			−0.334887
$322$		0			0
$323$	−29.3939			−1.63552
$324$		0			0
$325$	−5.65685	+	27.7128i	−0.313786	+	1.53723i
$326$		0			0
$327$		−	8.48528i		−	0.469237i
$328$		0			0
$329$	−6.00000			−0.330791
$330$		0			0
$331$			6.00000i			0.329790i	0.986311	+	0.164895i	$0.0527285\pi$
$331$			6.00000i			0.329790i	−0.986311	+	0.164895i	$0.947272\pi$
$332$		0			0
$333$	−8.48528			−0.464991
$334$		0			0
$335$	−3.46410	−	4.24264i	−0.189264	−	0.231800i
$336$		0			0
$337$		−	19.5959i		−	1.06746i	−0.845656	−	0.533729i	$-0.820790\pi$
$337$		−	19.5959i		−	1.06746i	0.845656	−	0.533729i	$-0.179210\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.8564i			0.750366i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	30.0000	−	24.4949i	1.61515	−	1.31876i
$346$		0			0
$347$	17.1464			0.920468			0.460234	−	0.887798i	$-0.347765\pi$
$347$	17.1464			0.920468			0.460234	+	0.887798i	$0.347765\pi$
$348$		0			0
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.79796i			0.521493i	0.965407	+	0.260746i	$0.0839686\pi$
$353$			9.79796i			0.521493i	−0.965407	+	0.260746i	$0.916031\pi$
$354$		0			0
$355$	9.79796	+	12.0000i	0.520022	+	0.636894i
$356$		0			0
$357$	16.9706			0.898177
$358$		0			0
$359$	27.7128			1.46263			0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128			1.46263			0.731313	+	0.682042i	$0.238908\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−17.1464			−0.899954
$364$		0			0
$365$	8.48528	−	6.92820i	0.444140	−	0.362639i
$366$		0			0
$367$		−	9.89949i		−	0.516749i	−0.966045	−	0.258375i	$-0.916813\pi$
$367$		−	9.89949i		−	0.516749i	0.966045	−	0.258375i	$-0.0831869\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$		0			0
$372$		0			0
$373$	19.7990			1.02515			0.512576	−	0.858642i	$-0.328691\pi$
$373$	19.7990			1.02515			0.512576	+	0.858642i	$0.328691\pi$
$374$		0			0
$375$	−24.2487	+	12.7279i	−1.25220	+	0.657267i
$376$		0			0
$377$			39.1918i			2.01848i
$378$		0			0
$379$		−	14.0000i		−	0.719132i	−0.933120	−	0.359566i	$-0.882925\pi$
$379$		−	14.0000i		−	0.719132i	0.933120	−	0.359566i	$-0.117075\pi$
$380$		0			0
$381$		−	17.3205i		−	0.887357i
$382$		0			0
$383$			12.7279i			0.650366i	0.945651	+	0.325183i	$0.105426\pi$
$383$			12.7279i			0.650366i	−0.945651	+	0.325183i	$0.894574\pi$
$384$		0			0
$385$	4.00000	+	4.89898i	0.203859	+	0.249675i
$386$		0			0
$387$	−7.34847			−0.373544
$388$		0			0
$389$			3.46410i			0.175637i	0.996136	+	0.0878185i	$0.0279895\pi$
$389$			3.46410i			0.175637i	−0.996136	+	0.0878185i	$0.972010\pi$
$390$		0			0
$391$	34.6410			1.75187
$392$		0			0
$393$			24.4949i			1.23560i
$394$		0			0
$395$	9.79796	+	12.0000i	0.492989	+	0.603786i
$396$		0			0
$397$	−11.3137			−0.567819			−0.283909	−	0.958851i	$-0.591631\pi$
$397$	−11.3137			−0.567819			−0.283909	+	0.958851i	$0.591631\pi$
$398$		0			0
$399$	20.7846			1.04053
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$	39.1918			1.95228
$404$		0			0
$405$	12.7279	+	15.5885i	0.632456	+	0.774597i
$406$		0			0
$407$		−	5.65685i		−	0.280400i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$			24.0000i			1.18383i
$412$		0			0
$413$	2.82843			0.139178
$414$		0			0
$415$	−17.3205	−	21.2132i	−0.850230	−	1.04132i
$416$		0			0
$417$		−	24.4949i		−	1.19952i
$418$		0			0
$419$			26.0000i			1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$			26.0000i			1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$		0			0
$421$			3.46410i			0.168830i	0.996431	+	0.0844150i	$0.0269021\pi$
$421$			3.46410i			0.168830i	−0.996431	+	0.0844150i	$0.973098\pi$
$422$		0			0
$423$			12.7279i			0.618853i
$424$		0			0
$425$	−24.0000	−	4.89898i	−1.16417	−	0.237635i
$426$		0			0
$427$	4.89898			0.237078
$428$		0			0
$429$		−	27.7128i		−	1.33799i
$430$		0			0
$431$	−6.92820			−0.333720			−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820			−0.333720			−0.166860	+	0.985981i	$0.553363\pi$
$432$		0			0
$433$			34.2929i			1.64801i	0.566583	+	0.824005i	$0.308265\pi$
$433$			34.2929i			1.64801i	−0.566583	+	0.824005i	$0.691735\pi$
$434$		0			0
$435$	−29.3939	+	24.0000i	−1.40933	+	1.15071i
$436$		0			0
$437$	42.4264			2.02953
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$		0			0
$443$	22.0454			1.04741			0.523704	−	0.851900i	$-0.324550\pi$
$443$	22.0454			1.04741			0.523704	+	0.851900i	$0.324550\pi$
$444$		0			0
$445$	2.82843	+	3.46410i	0.134080	+	0.164214i
$446$		0			0
$447$			25.4558i			1.20402i
$448$		0			0
$449$	16.0000			0.755087			0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000			0.755087			0.377543	+	0.925992i	$0.376769\pi$
$450$		0			0
$451$		−	8.00000i		−	0.376705i
$452$		0			0
$453$	−33.9411			−1.59469
$454$		0			0
$455$	13.8564	−	11.3137i	0.649598	−	0.530395i
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			20.7846i			0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$			20.7846i			0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$		0			0
$463$		−	1.41421i		−	0.0657241i	−0.999460	−	0.0328620i	$-0.989538\pi$
$463$		−	1.41421i		−	0.0657241i	0.999460	−	0.0328620i	$-0.0104622\pi$
$464$		0			0
$465$	24.0000	+	29.3939i	1.11297	+	1.36311i
$466$		0			0
$467$	−31.8434			−1.47354			−0.736768	−	0.676146i	$-0.763649\pi$
$467$	−31.8434			−1.47354			−0.736768	+	0.676146i	$0.763649\pi$
$468$		0			0
$469$			3.46410i			0.159957i
$470$		0			0
$471$	−6.92820			−0.319235
$472$		0			0
$473$		−	4.89898i		−	0.225255i
$474$		0			0
$475$	−29.3939	−	6.00000i	−1.34868	−	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	41.5692			1.89935			0.949673	−	0.313243i	$-0.101415\pi$
$479$	41.5692			1.89935			0.949673	+	0.313243i	$0.101415\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$	−24.4949			−1.11456
$484$		0			0
$485$	25.4558	−	20.7846i	1.15589	−	0.943781i
$486$		0			0
$487$		−	26.8701i		−	1.21760i	−0.793324	−	0.608799i	$-0.791651\pi$
$487$		−	26.8701i		−	1.21760i	0.793324	−	0.608799i	$-0.208349\pi$
$488$		0			0
$489$	54.0000			2.44196
$490$		0			0
$491$			38.0000i			1.71492i	0.514554	+	0.857458i	$0.327958\pi$
$491$			38.0000i			1.71492i	−0.514554	+	0.857458i	$0.672042\pi$
$492$		0			0
$493$	−33.9411			−1.52863
$494$		0			0
$495$	10.3923	−	8.48528i	0.467099	−	0.381385i
$496$		0			0
$497$		−	9.79796i		−	0.439499i
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$		−	24.2487i		−	1.08335i
$502$		0			0
$503$			21.2132i			0.945850i	0.881103	+	0.472925i	$0.156802\pi$
$503$			21.2132i			0.945850i	−0.881103	+	0.472925i	$0.843198\pi$
$504$		0			0
$505$	12.0000	−	9.79796i	0.533993	−	0.436003i
$506$		0			0
$507$	−46.5403			−2.06693
$508$		0			0
$509$		−	20.7846i		−	0.921262i	−0.887592	−	0.460631i	$-0.847623\pi$
$509$		−	20.7846i		−	0.921262i	0.887592	−	0.460631i	$-0.152377\pi$
$510$		0			0
$511$	−6.92820			−0.306486
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.1464	−	14.0000i	0.755562	−	0.616914i
$516$		0			0
$517$	−8.48528			−0.373182
$518$		0			0
$519$	−20.7846			−0.912343
$520$		0			0
$521$	34.0000			1.48957			0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000			1.48957			0.744784	+	0.667306i	$0.232553\pi$
$522$		0			0
$523$	2.44949			0.107109			0.0535544	−	0.998565i	$-0.482945\pi$
$523$	2.44949			0.107109			0.0535544	+	0.998565i	$0.482945\pi$
$524$		0			0
$525$	16.9706	+	3.46410i	0.740656	+	0.151186i
$526$		0			0
$527$			33.9411i			1.47850i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$	−22.6274			−0.980102
$534$		0			0
$535$	−3.46410	−	4.24264i	−0.149766	−	0.183425i
$536$		0			0
$537$		−	4.89898i		−	0.211407i
$538$		0			0
$539$			10.0000i			0.430730i
$540$		0			0
$541$		−	6.92820i		−	0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$		−	6.92820i		−	0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$		0			0
$543$			50.9117i			2.18483i
$544$		0			0
$545$	6.00000	−	4.89898i	0.257012	−	0.209849i
$546$		0			0
$547$	26.9444			1.15206			0.576029	−	0.817429i	$-0.304601\pi$
$547$	26.9444			1.15206			0.576029	+	0.817429i	$0.304601\pi$
$548$		0			0
$549$		−	10.3923i		−	0.443533i
$550$		0			0
$551$	−41.5692			−1.77091
$552$		0			0
$553$		−	9.79796i		−	0.416652i
$554$		0			0
$555$	−9.79796	−	12.0000i	−0.415900	−	0.509372i
$556$		0			0
$557$	25.4558			1.07860			0.539299	−	0.842114i	$-0.318689\pi$
$557$	25.4558			1.07860			0.539299	+	0.842114i	$0.318689\pi$
$558$		0			0
$559$	−13.8564			−0.586064
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	7.34847			0.309701			0.154851	−	0.987938i	$-0.450510\pi$
$563$	7.34847			0.309701			0.154851	+	0.987938i	$0.450510\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	12.7279i		−	0.534522i
$568$		0			0
$569$	−28.0000			−1.17382			−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000			−1.17382			−0.586911	+	0.809652i	$0.699656\pi$
$570$		0			0
$571$		−	30.0000i		−	1.25546i	−0.778431	−	0.627730i	$-0.783984\pi$
$571$		−	30.0000i		−	1.25546i	0.778431	−	0.627730i	$-0.216016\pi$
$572$		0			0
$573$	33.9411			1.41791
$574$		0			0
$575$	34.6410	+	7.07107i	1.44463	+	0.294884i
$576$		0			0
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$		0			0
$579$		−	12.0000i		−	0.498703i
$580$		0			0
$581$			17.3205i			0.718576i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−24.0000	−	29.3939i	−0.992278	−	1.21529i
$586$		0			0
$587$	−12.2474			−0.505506			−0.252753	−	0.967531i	$-0.581336\pi$
$587$	−12.2474			−0.505506			−0.252753	+	0.967531i	$0.581336\pi$
$588$		0			0
$589$			41.5692i			1.71283i
$590$		0			0
$591$	55.4256			2.27991
$592$		0			0
$593$			29.3939i			1.20706i	0.797340	+	0.603531i	$0.206240\pi$
$593$			29.3939i			1.20706i	−0.797340	+	0.603531i	$0.793760\pi$
$594$		0			0
$595$	9.79796	+	12.0000i	0.401677	+	0.491952i
$596$		0			0
$597$	−50.9117			−2.08368
$598$		0			0
$599$	−20.7846			−0.849236			−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846			−0.849236			−0.424618	+	0.905373i	$0.639592\pi$
$600$		0			0
$601$	−24.0000			−0.978980			−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000			−0.978980			−0.489490	+	0.872009i	$0.662817\pi$
$602$		0			0
$603$	7.34847			0.299253
$604$		0			0
$605$	−9.89949	−	12.1244i	−0.402472	−	0.492925i
$606$		0			0
$607$		−	41.0122i		−	1.66463i	−0.554300	−	0.832317i	$-0.687014\pi$
$607$		−	41.0122i		−	1.66463i	0.554300	−	0.832317i	$-0.312986\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$			24.0000i			0.970936i
$612$		0			0
$613$	5.65685			0.228478			0.114239	−	0.993453i	$-0.463557\pi$
$613$	5.65685			0.228478			0.114239	+	0.993453i	$0.463557\pi$
$614$		0			0
$615$	−13.8564	−	16.9706i	−0.558744	−	0.684319i
$616$		0			0
$617$			24.4949i			0.986127i	0.869993	+	0.493064i	$0.164123\pi$
$617$			24.4949i			0.986127i	−0.869993	+	0.493064i	$0.835877\pi$
$618$		0			0
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.82843i		−	0.113319i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$	29.3939			1.17388
$628$		0			0
$629$		−	13.8564i		−	0.552491i
$630$		0			0
$631$	13.8564			0.551615			0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564			0.551615			0.275807	+	0.961213i	$0.411055\pi$
$632$		0			0
$633$			24.4949i			0.973585i
$634$		0			0
$635$	12.2474	−	10.0000i	0.486025	−	0.396838i
$636$		0			0
$637$	28.2843			1.12066
$638$		0			0
$639$	−20.7846			−0.822226
$640$		0			0
$641$	−40.0000			−1.57991			−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000			−1.57991			−0.789953	+	0.613168i	$0.789895\pi$
$642$		0			0
$643$	2.44949			0.0965984			0.0482992	−	0.998833i	$-0.484620\pi$
$643$	2.44949			0.0965984			0.0482992	+	0.998833i	$0.484620\pi$
$644$		0			0
$645$	−8.48528	−	10.3923i	−0.334108	−	0.409197i
$646$		0			0
$647$		−	4.24264i		−	0.166795i	−0.996516	−	0.0833977i	$-0.973423\pi$
$647$		−	4.24264i		−	0.166795i	0.996516	−	0.0833977i	$-0.0265772\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$		−	24.0000i		−	0.940634i
$652$		0			0
$653$	5.65685			0.221370			0.110685	−	0.993856i	$-0.464696\pi$
$653$	5.65685			0.221370			0.110685	+	0.993856i	$0.464696\pi$
$654$		0			0
$655$	−17.3205	+	14.1421i	−0.676768	+	0.552579i
$656$		0			0
$657$			14.6969i			0.573382i
$658$		0			0
$659$			2.00000i			0.0779089i	0.999241	+	0.0389545i	$0.0124027\pi$
$659$			2.00000i			0.0779089i	−0.999241	+	0.0389545i	$0.987597\pi$
$660$		0			0
$661$			10.3923i			0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$			10.3923i			0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$		0			0
$663$		−	67.8823i		−	2.63633i
$664$		0			0
$665$	12.0000	+	14.6969i	0.465340	+	0.569923i
$666$		0			0
$667$	48.9898			1.89689
$668$		0			0
$669$			65.8179i			2.54467i
$670$		0			0
$671$	6.92820			0.267460
$672$		0			0
$673$		−	24.4949i		−	0.944209i	−0.881543	−	0.472104i	$-0.843495\pi$
$673$		−	24.4949i		−	0.944209i	0.881543	−	0.472104i	$-0.156505\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.65685			0.217411			0.108705	−	0.994074i	$-0.465330\pi$
$677$	5.65685			0.217411			0.108705	+	0.994074i	$0.465330\pi$
$678$		0			0
$679$	−20.7846			−0.797640
$680$		0			0
$681$	−42.0000			−1.60944
$682$		0			0
$683$	−2.44949			−0.0937271			−0.0468636	−	0.998901i	$-0.514923\pi$
$683$	−2.44949			−0.0937271			−0.0468636	+	0.998901i	$0.514923\pi$
$684$		0			0
$685$	−16.9706	+	13.8564i	−0.648412	+	0.529426i
$686$		0			0
$687$			16.9706i			0.647467i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	42.0000i		−	1.59776i	−0.601494	−	0.798878i	$-0.705427\pi$
$691$		−	42.0000i		−	1.59776i	0.601494	−	0.798878i	$-0.294573\pi$
$692$		0			0
$693$	−8.48528			−0.322329
$694$		0			0
$695$	17.3205	−	14.1421i	0.657004	−	0.536442i
$696$		0			0
$697$		−	19.5959i		−	0.742248i
$698$		0			0
$699$			36.0000i			1.36165i
$700$		0			0
$701$			3.46410i			0.130837i	0.997858	+	0.0654187i	$0.0208383\pi$
$701$			3.46410i			0.130837i	−0.997858	+	0.0654187i	$0.979162\pi$
$702$		0			0
$703$		−	16.9706i		−	0.640057i
$704$		0			0
$705$	−18.0000	+	14.6969i	−0.677919	+	0.553519i
$706$		0			0
$707$	−9.79796			−0.368490
$708$		0			0
$709$			34.6410i			1.30097i	0.759519	+	0.650485i	$0.225434\pi$
$709$			34.6410i			1.30097i	−0.759519	+	0.650485i	$0.774566\pi$
$710$		0			0
$711$	−20.7846			−0.779484
$712$		0			0
$713$		−	48.9898i		−	1.83468i
$714$		0			0
$715$	19.5959	−	16.0000i	0.732846	−	0.598366i
$716$		0			0
$717$	−16.9706			−0.633777
$718$		0			0
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−33.9411	−	6.92820i	−1.26054	−	0.257307i
$726$		0			0
$727$			18.3848i			0.681854i	0.940090	+	0.340927i	$0.110741\pi$
$727$			18.3848i			0.681854i	−0.940090	+	0.340927i	$0.889259\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		−	12.0000i		−	0.443836i
$732$		0			0
$733$	−19.7990			−0.731292			−0.365646	−	0.930754i	$-0.619152\pi$
$733$	−19.7990			−0.731292			−0.365646	+	0.930754i	$0.619152\pi$
$734$		0			0
$735$	17.3205	+	21.2132i	0.638877	+	0.782461i
$736$		0			0
$737$			4.89898i			0.180456i
$738$		0			0
$739$		−	30.0000i		−	1.10357i	−0.833987	−	0.551784i	$-0.813947\pi$
$739$		−	30.0000i		−	1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$740$		0			0
$741$		−	83.1384i		−	3.05417i
$742$		0			0
$743$		−	4.24264i		−	0.155647i	−0.996967	−	0.0778237i	$-0.975203\pi$
$743$		−	4.24264i		−	0.155647i	0.996967	−	0.0778237i	$-0.0247971\pi$
$744$		0			0
$745$	−18.0000	+	14.6969i	−0.659469	+	0.538454i
$746$		0			0
$747$	36.7423			1.34433
$748$		0			0
$749$			3.46410i			0.126576i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		−	63.6867i		−	2.32087i
$754$		0			0
$755$	−19.5959	−	24.0000i	−0.713168	−	0.873449i
$756$		0			0
$757$	2.82843			0.102801			0.0514005	−	0.998678i	$-0.483632\pi$
$757$	2.82843			0.102801			0.0514005	+	0.998678i	$0.483632\pi$
$758$		0			0
$759$	−34.6410			−1.25739
$760$		0			0
$761$	38.0000			1.37750			0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$	−4.89898			−0.177355
$764$		0			0
$765$	25.4558	−	20.7846i	0.920358	−	0.751469i
$766$		0			0
$767$		−	11.3137i		−	0.408514i
$768$		0			0
$769$	42.0000			1.51456			0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000			1.51456			0.757279	+	0.653091i	$0.226528\pi$
$770$		0			0
$771$			24.0000i			0.864339i
$772$		0			0
$773$	45.2548			1.62770			0.813852	−	0.581073i	$-0.197367\pi$
$773$	45.2548			1.62770			0.813852	+	0.581073i	$0.197367\pi$
$774$		0			0
$775$	−6.92820	+	33.9411i	−0.248868	+	1.21920i
$776$		0			0
$777$			9.79796i			0.351500i
$778$		0			0
$779$		−	24.0000i		−	0.859889i
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.00000	−	4.89898i	−0.142766	−	0.174852i
$786$		0			0
$787$	−46.5403			−1.65898			−0.829491	−	0.558520i	$-0.811370\pi$
$787$	−46.5403			−1.65898			−0.829491	+	0.558520i	$0.811370\pi$
$788$		0			0
$789$			65.8179i			2.34318i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	19.5959i		−	0.695871i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.9411			−1.20226			−0.601128	−	0.799153i	$-0.705282\pi$
$797$	−33.9411			−1.20226			−0.601128	+	0.799153i	$0.705282\pi$
$798$		0			0
$799$	−20.7846			−0.735307
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$	−9.79796			−0.345762
$804$		0			0
$805$	−14.1421	−	17.3205i	−0.498445	−	0.610468i
$806$		0			0
$807$			42.4264i			1.49348i
$808$		0			0
$809$	50.0000			1.75791			0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$		0			0
$811$		−	22.0000i		−	0.772524i	−0.922389	−	0.386262i	$-0.873766\pi$
$811$		−	22.0000i		−	0.772524i	0.922389	−	0.386262i	$-0.126234\pi$
$812$		0			0
$813$	50.9117			1.78555
$814$		0			0
$815$	31.1769	+	38.1838i	1.09208	+	1.33752i
$816$		0			0
$817$		−	14.6969i		−	0.514181i
$818$		0			0
$819$			24.0000i			0.838628i
$820$		0			0
$821$			51.9615i			1.81347i	0.421701	+	0.906735i	$0.361433\pi$
$821$			51.9615i			1.81347i	−0.421701	+	0.906735i	$0.638567\pi$
$822$		0			0
$823$		−	35.3553i		−	1.23241i	−0.787586	−	0.616205i	$-0.788669\pi$
$823$		−	35.3553i		−	1.23241i	0.787586	−	0.616205i	$-0.211331\pi$
$824$		0			0
$825$	24.0000	+	4.89898i	0.835573	+	0.170561i
$826$		0			0
$827$	−31.8434			−1.10730			−0.553651	−	0.832749i	$-0.686766\pi$
$827$	−31.8434			−1.10730			−0.553651	+	0.832749i	$0.686766\pi$
$828$		0			0
$829$		−	45.0333i		−	1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$		−	45.0333i		−	1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$		0			0
$831$	34.6410			1.20168
$832$		0			0
$833$			24.4949i			0.848698i
$834$		0			0
$835$	17.1464	−	14.0000i	0.593377	−	0.484490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−41.5692			−1.43513			−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692			−1.43513			−0.717564	+	0.696492i	$0.754743\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$	−19.5959			−0.674919
$844$		0			0
$845$	−26.8701	−	32.9090i	−0.924358	−	1.13210i
$846$		0			0
$847$			9.89949i			0.340151i
$848$		0			0
$849$	−6.00000			−0.205919
$850$		0			0
$851$			20.0000i			0.685591i
$852$		0			0
$853$	5.65685			0.193687			0.0968435	−	0.995300i	$-0.469125\pi$
$853$	5.65685			0.193687			0.0968435	+	0.995300i	$0.469125\pi$
$854$		0			0
$855$	31.1769	−	25.4558i	1.06623	−	0.870572i
$856$		0			0
$857$		−	39.1918i		−	1.33877i	−0.742917	−	0.669384i	$-0.766558\pi$
$857$		−	39.1918i		−	1.33877i	0.742917	−	0.669384i	$-0.233442\pi$
$858$		0			0
$859$			22.0000i			0.750630i	0.926897	+	0.375315i	$0.122466\pi$
$859$			22.0000i			0.750630i	−0.926897	+	0.375315i	$0.877534\pi$
$860$		0			0
$861$			13.8564i			0.472225i
$862$		0			0
$863$		−	24.0416i		−	0.818387i	−0.912448	−	0.409193i	$-0.865810\pi$
$863$		−	24.0416i		−	0.818387i	0.912448	−	0.409193i	$-0.134190\pi$
$864$		0			0
$865$	−12.0000	−	14.6969i	−0.408012	−	0.499711i
$866$		0			0
$867$	17.1464			0.582323
$868$		0			0
$869$		−	13.8564i		−	0.470046i
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$			44.0908i			1.49225i
$874$		0			0
$875$	7.34847	+	14.0000i	0.248424	+	0.473286i
$876$		0			0
$877$	−36.7696			−1.24162			−0.620810	−	0.783961i	$-0.713196\pi$
$877$	−36.7696			−1.24162			−0.620810	+	0.783961i	$0.713196\pi$
$878$		0			0
$879$	6.92820			0.233682
$880$		0			0
$881$	52.0000			1.75192			0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000			1.75192			0.875962	+	0.482380i	$0.160227\pi$
$882$		0			0
$883$	−12.2474			−0.412159			−0.206080	−	0.978535i	$-0.566071\pi$
$883$	−12.2474			−0.412159			−0.206080	+	0.978535i	$0.566071\pi$
$884$		0			0
$885$	8.48528	−	6.92820i	0.285230	−	0.232889i
$886$		0			0
$887$		−	49.4975i		−	1.66196i	−0.556300	−	0.830981i	$-0.687780\pi$
$887$		−	49.4975i		−	1.66196i	0.556300	−	0.830981i	$-0.312220\pi$
$888$		0			0
$889$	−10.0000			−0.335389
$890$		0			0
$891$		−	18.0000i		−	0.603023i
$892$		0			0
$893$	−25.4558			−0.851847
$894$		0			0
$895$	3.46410	−	2.82843i	0.115792	−	0.0945439i
$896$		0			0
$897$			97.9796i			3.27144i
$898$		0			0
$899$			48.0000i			1.60089i
$900$		0			0
$901$		0			0
$902$		0			0
$903$			8.48528i			0.282372i
$904$		0			0
$905$	−36.0000	+	29.3939i	−1.19668	+	0.977086i
$906$		0			0
$907$	−26.9444			−0.894674			−0.447337	−	0.894366i	$-0.647627\pi$
$907$	−26.9444			−0.894674			−0.447337	+	0.894366i	$0.647627\pi$
$908$		0			0
$909$			20.7846i			0.689382i
$910$		0			0
$911$	−48.4974			−1.60679			−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974			−1.60679			−0.803396	+	0.595446i	$0.796976\pi$
$912$		0			0
$913$			24.4949i			0.810663i
$914$		0			0
$915$	14.6969	−	12.0000i	0.485866	−	0.396708i
$916$		0			0
$917$	14.1421			0.467014
$918$		0			0
$919$	6.92820			0.228540			0.114270	−	0.993450i	$-0.463547\pi$
$919$	6.92820			0.228540			0.114270	+	0.993450i	$0.463547\pi$
$920$		0			0
$921$	−6.00000			−0.197707
$922$		0			0
$923$	−39.1918			−1.29001
$924$		0			0
$925$	2.82843	−	13.8564i	0.0929981	−	0.455596i
$926$		0			0
$927$			29.6985i			0.975426i
$928$		0			0
$929$	−8.00000			−0.262471			−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000			−0.262471			−0.131236	+	0.991351i	$0.541894\pi$
$930$		0			0
$931$			30.0000i			0.983210i
$932$		0			0
$933$	67.8823			2.22237
$934$		0			0
$935$	13.8564	+	16.9706i	0.453153	+	0.554997i
$936$		0			0
$937$		−	24.4949i		−	0.800213i	−0.916469	−	0.400107i	$-0.868973\pi$
$937$		−	24.4949i		−	0.800213i	0.916469	−	0.400107i	$-0.131027\pi$
$938$		0			0
$939$			72.0000i			2.34963i
$940$		0			0
$941$		−	34.6410i		−	1.12926i	−0.825342	−	0.564632i	$-0.809018\pi$
$941$		−	34.6410i		−	1.12926i	0.825342	−	0.564632i	$-0.190982\pi$
$942$		0			0
$943$			28.2843i			0.921063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.6413			1.35316			0.676581	−	0.736369i	$-0.263461\pi$
$947$	41.6413			1.35316			0.676581	+	0.736369i	$0.263461\pi$
$948$		0			0
$949$			27.7128i			0.899596i
$950$		0			0
$951$	55.4256			1.79730
$952$		0			0
$953$			39.1918i			1.26955i	0.772698	+	0.634774i	$0.218907\pi$
$953$			39.1918i			1.26955i	−0.772698	+	0.634774i	$0.781093\pi$
$954$		0			0
$955$	19.5959	+	24.0000i	0.634109	+	0.776622i
$956$		0			0
$957$	33.9411			1.09716
$958$		0			0
$959$	13.8564			0.447447
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$	7.34847			0.236801
$964$		0			0
$965$	8.48528	−	6.92820i	0.273151	−	0.223027i
$966$		0			0
$967$			15.5563i			0.500258i	0.968212	+	0.250129i	$0.0804731\pi$
$967$			15.5563i			0.500258i	−0.968212	+	0.250129i	$0.919527\pi$
$968$		0			0
$969$	72.0000			2.31297
$970$		0			0
$971$			22.0000i			0.706014i	0.935621	+	0.353007i	$0.114841\pi$
$971$			22.0000i			0.706014i	−0.935621	+	0.353007i	$0.885159\pi$
$972$		0			0
$973$	−14.1421			−0.453376
$974$		0			0
$975$	13.8564	−	67.8823i	0.443760	−	2.17397i
$976$		0			0
$977$		−	4.89898i		−	0.156732i	−0.996925	−	0.0783661i	$-0.975030\pi$
$977$		−	4.89898i		−	0.156732i	0.996925	−	0.0783661i	$-0.0249703\pi$
$978$		0			0
$979$		−	4.00000i		−	0.127841i
$980$		0			0
$981$			10.3923i			0.331801i
$982$		0			0
$983$			46.6690i			1.48851i	0.667895	+	0.744256i	$0.267196\pi$
$983$			46.6690i			1.48851i	−0.667895	+	0.744256i	$0.732804\pi$
$984$		0			0
$985$	32.0000	+	39.1918i	1.01960	+	1.24876i
$986$		0			0
$987$	14.6969			0.467809
$988$		0			0
$989$			17.3205i			0.550760i
$990$		0			0
$991$	27.7128			0.880327			0.440163	−	0.897918i	$-0.354921\pi$
$991$	27.7128			0.880327			0.440163	+	0.897918i	$0.354921\pi$
$992$		0			0
$993$		−	14.6969i		−	0.466393i
$994$		0			0
$995$	−29.3939	−	36.0000i	−0.931849	−	1.14128i
$996$		0			0
$997$	56.5685			1.79154			0.895772	−	0.444514i	$-0.146624\pi$
$997$	56.5685			1.79154			0.895772	+	0.444514i	$0.146624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.f.b.289.1	✓	8
3.2	odd	2		2880.2.d.g.289.8		8
4.3	odd	2	inner	320.2.f.b.289.5	yes	8
5.2	odd	4		1600.2.d.i.801.6		8
5.3	odd	4		1600.2.d.i.801.4		8
5.4	even	2	inner	320.2.f.b.289.7	yes	8
8.3	odd	2	inner	320.2.f.b.289.4	yes	8
8.5	even	2	inner	320.2.f.b.289.8	yes	8
12.11	even	2		2880.2.d.g.289.7		8
15.14	odd	2		2880.2.d.g.289.3		8
16.3	odd	4		1280.2.c.g.769.3		4
16.5	even	4		1280.2.c.g.769.4		4
16.11	odd	4		1280.2.c.h.769.2		4
16.13	even	4		1280.2.c.h.769.1		4
20.3	even	4		1600.2.d.i.801.5		8
20.7	even	4		1600.2.d.i.801.3		8
20.19	odd	2	inner	320.2.f.b.289.3	yes	8
24.5	odd	2		2880.2.d.g.289.2		8
24.11	even	2		2880.2.d.g.289.1		8
40.3	even	4		1600.2.d.i.801.2		8
40.13	odd	4		1600.2.d.i.801.7		8
40.19	odd	2	inner	320.2.f.b.289.6	yes	8
40.27	even	4		1600.2.d.i.801.8		8
40.29	even	2	inner	320.2.f.b.289.2	yes	8
40.37	odd	4		1600.2.d.i.801.1		8
60.59	even	2		2880.2.d.g.289.4		8
80.3	even	4		6400.2.a.ct.1.2		4
80.13	odd	4		6400.2.a.cu.1.3		4
80.19	odd	4		1280.2.c.g.769.1		4
80.27	even	4		6400.2.a.cu.1.1		4
80.29	even	4		1280.2.c.h.769.3		4
80.37	odd	4		6400.2.a.ct.1.4		4
80.43	even	4		6400.2.a.cu.1.4		4
80.53	odd	4		6400.2.a.ct.1.1		4
80.59	odd	4		1280.2.c.h.769.4		4
80.67	even	4		6400.2.a.ct.1.3		4
80.69	even	4		1280.2.c.g.769.2		4
80.77	odd	4		6400.2.a.cu.1.2		4
120.29	odd	2		2880.2.d.g.289.5		8
120.59	even	2		2880.2.d.g.289.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.b.289.1	✓	8	1.1	even	1	trivial
320.2.f.b.289.2	yes	8	40.29	even	2	inner
320.2.f.b.289.3	yes	8	20.19	odd	2	inner
320.2.f.b.289.4	yes	8	8.3	odd	2	inner
320.2.f.b.289.5	yes	8	4.3	odd	2	inner
320.2.f.b.289.6	yes	8	40.19	odd	2	inner
320.2.f.b.289.7	yes	8	5.4	even	2	inner
320.2.f.b.289.8	yes	8	8.5	even	2	inner
1280.2.c.g.769.1		4	80.19	odd	4
1280.2.c.g.769.2		4	80.69	even	4
1280.2.c.g.769.3		4	16.3	odd	4
1280.2.c.g.769.4		4	16.5	even	4
1280.2.c.h.769.1		4	16.13	even	4
1280.2.c.h.769.2		4	16.11	odd	4
1280.2.c.h.769.3		4	80.29	even	4
1280.2.c.h.769.4		4	80.59	odd	4
1600.2.d.i.801.1		8	40.37	odd	4
1600.2.d.i.801.2		8	40.3	even	4
1600.2.d.i.801.3		8	20.7	even	4
1600.2.d.i.801.4		8	5.3	odd	4
1600.2.d.i.801.5		8	20.3	even	4
1600.2.d.i.801.6		8	5.2	odd	4
1600.2.d.i.801.7		8	40.13	odd	4
1600.2.d.i.801.8		8	40.27	even	4
2880.2.d.g.289.1		8	24.11	even	2
2880.2.d.g.289.2		8	24.5	odd	2
2880.2.d.g.289.3		8	15.14	odd	2
2880.2.d.g.289.4		8	60.59	even	2
2880.2.d.g.289.5		8	120.29	odd	2
2880.2.d.g.289.6		8	120.59	even	2
2880.2.d.g.289.7		8	12.11	even	2
2880.2.d.g.289.8		8	3.2	odd	2
6400.2.a.ct.1.1		4	80.53	odd	4
6400.2.a.ct.1.2		4	80.3	even	4
6400.2.a.ct.1.3		4	80.67	even	4
6400.2.a.ct.1.4		4	80.37	odd	4
6400.2.a.cu.1.1		4	80.27	even	4
6400.2.a.cu.1.2		4	80.77	odd	4
6400.2.a.cu.1.3		4	80.13	odd	4
6400.2.a.cu.1.4		4	80.43	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.44949 q^{3} +(-1.41421 - 1.73205i) q^{5} +1.41421i q^{7} +3.00000 q^{9} +2.00000i q^{11} +5.65685 q^{13} +(3.46410 + 4.24264i) q^{15} +4.89898i q^{17} +6.00000i q^{19} -3.46410i q^{21} -7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} +6.92820i q^{29} +6.92820 q^{31} -4.89898i q^{33} +(2.44949 - 2.00000i) q^{35} -2.82843 q^{37} -13.8564 q^{39} -4.00000 q^{41} -2.44949 q^{43} +(-4.24264 - 5.19615i) q^{45} +4.24264i q^{47} +5.00000 q^{49} -12.0000i q^{51} +(3.46410 - 2.82843i) q^{55} -14.6969i q^{57} -2.00000i q^{59} -3.46410i q^{61} +4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949 q^{67} +17.3205i q^{69} -6.92820 q^{71} +4.89898i q^{73} +(2.44949 - 12.0000i) q^{75} -2.82843 q^{77} -6.92820 q^{79} -9.00000 q^{81} +12.2474 q^{83} +(8.48528 - 6.92820i) q^{85} -16.9706i q^{87} -2.00000 q^{89} +8.00000i q^{91} -16.9706 q^{93} +(10.3923 - 8.48528i) q^{95} +14.6969i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 8 q^{25} - 32 q^{41} + 40 q^{49} - 64 q^{65} - 72 q^{81} - 16 q^{89}+O(q^{100}) \) 8 * q + 24 * q^9 - 8 * q^25 - 32 * q^41 + 40 * q^49 - 64 * q^65 - 72 * q^81 - 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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