LMFDB - Embedded newform 1280.2.c.h.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(769,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1280.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.c (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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	1280.769
Dual form			1280.2.c.h.769.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{3} +(1.73205 - 1.41421i) q^{5} +1.41421i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-2.44949i q^{3} +(1.73205 - 1.41421i) q^{5} +1.41421i q^{7} -3.00000 q^{9} +2.00000 q^{11} -5.65685i q^{13} +(-3.46410 - 4.24264i) q^{15} +4.89898i q^{17} -6.00000 q^{19} +3.46410 q^{21} -7.07107i q^{23} +(1.00000 - 4.89898i) q^{25} +6.92820 q^{29} -6.92820 q^{31} -4.89898i q^{33} +(2.00000 + 2.44949i) q^{35} -2.82843i q^{37} -13.8564 q^{39} +4.00000 q^{41} +2.44949i q^{43} +(-5.19615 + 4.24264i) q^{45} -4.24264i q^{47} +5.00000 q^{49} +12.0000 q^{51} +(3.46410 - 2.82843i) q^{55} +14.6969i q^{57} -2.00000 q^{59} -3.46410 q^{61} -4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949i q^{67} -17.3205 q^{69} -6.92820 q^{71} -4.89898i q^{73} +(-12.0000 - 2.44949i) q^{75} +2.82843i q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474i q^{83} +(6.92820 + 8.48528i) q^{85} -16.9706i q^{87} +2.00000 q^{89} +8.00000 q^{91} +16.9706i q^{93} +(-10.3923 + 8.48528i) q^{95} +14.6969i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 8 q^{11} - 24 q^{19} + 4 q^{25} + 8 q^{35} + 16 q^{41} + 20 q^{49} + 48 q^{51} - 8 q^{59} - 32 q^{65} - 48 q^{75} - 36 q^{81} + 8 q^{89} + 32 q^{91} - 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 8 * q^11 - 24 * q^19 + 4 * q^25 + 8 * q^35 + 16 * q^41 + 20 * q^49 + 48 * q^51 - 8 * q^59 - 32 * q^65 - 48 * q^75 - 36 * q^81 + 8 * q^89 + 32 * q^91 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	2.44949i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$		−	2.44949i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$		0			0
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$6$		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
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	5.65685i		−	1.56893i	−0.620174	−	0.784465i	$-0.712938\pi$
$13$		−	5.65685i		−	1.56893i	0.620174	−	0.784465i	$-0.287062\pi$
$14$		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
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	3.46410			0.755929
$22$		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
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$		0			0
$31$	−6.92820			−1.24434			−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820			−1.24434			−0.622171	+	0.782881i	$0.713749\pi$
$32$		0			0
$33$		−	4.89898i		−	0.852803i
$34$		0			0
$35$	2.00000	+	2.44949i	0.338062	+	0.414039i
$36$		0			0
$37$		−	2.82843i		−	0.464991i	−0.972598	−	0.232495i	$-0.925311\pi$
$37$		−	2.82843i		−	0.464991i	0.972598	−	0.232495i	$-0.0746890\pi$
$38$		0			0
$39$	−13.8564			−2.21880
$40$		0			0
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$			2.44949i			0.373544i	0.982403	+	0.186772i	$0.0598025\pi$
$43$			2.44949i			0.373544i	−0.982403	+	0.186772i	$0.940197\pi$
$44$		0			0
$45$	−5.19615	+	4.24264i	−0.774597	+	0.632456i
$46$		0			0
$47$		−	4.24264i		−	0.618853i	−0.950923	−	0.309426i	$-0.899863\pi$
$47$		−	4.24264i		−	0.618853i	0.950923	−	0.309426i	$-0.100137\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	12.0000			1.68034
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	3.46410	−	2.82843i	0.467099	−	0.381385i
$56$		0			0
$57$			14.6969i			1.94666i
$58$		0			0
$59$	−2.00000			−0.260378			−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000			−0.260378			−0.130189	+	0.991489i	$0.541558\pi$
$60$		0			0
$61$	−3.46410			−0.443533			−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410			−0.443533			−0.221766	+	0.975100i	$0.571182\pi$
$62$		0			0
$63$		−	4.24264i		−	0.534522i
$64$		0			0
$65$	−8.00000	−	9.79796i	−0.992278	−	1.21529i
$66$		0			0
$67$			2.44949i			0.299253i	0.988743	+	0.149626i	$0.0478071\pi$
$67$			2.44949i			0.299253i	−0.988743	+	0.149626i	$0.952193\pi$
$68$		0			0
$69$	−17.3205			−2.08514
$70$		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	$-0.907445\pi$
$73$		−	4.89898i		−	0.573382i	0.958023	−	0.286691i	$-0.0925553\pi$
$74$		0			0
$75$	−12.0000	−	2.44949i	−1.38564	−	0.282843i
$76$		0			0
$77$			2.82843i			0.322329i
$78$		0			0
$79$	6.92820			0.779484			0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820			0.779484			0.389742	+	0.920924i	$0.372564\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			12.2474i			1.34433i	0.740400	+	0.672166i	$0.234636\pi$
$83$			12.2474i			1.34433i	−0.740400	+	0.672166i	$0.765364\pi$
$84$		0			0
$85$	6.92820	+	8.48528i	0.751469	+	0.920358i
$86$		0			0
$87$		−	16.9706i		−	1.81944i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	8.00000			0.838628
$92$		0			0
$93$			16.9706i			1.75977i
$94$		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
$99$	−6.00000			−0.603023
$100$		0			0
$101$	−6.92820			−0.689382			−0.344691	−	0.938716i	$-0.612016\pi$
$101$	−6.92820			−0.689382			−0.344691	+	0.938716i	$0.612016\pi$
$102$		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
$107$		−	2.44949i		−	0.236801i	−0.992966	−	0.118401i	$-0.962223\pi$
$107$		−	2.44949i		−	0.236801i	0.992966	−	0.118401i	$-0.0377767\pi$
$108$		0			0
$109$	3.46410			0.331801			0.165900	−	0.986143i	$-0.446947\pi$
$109$	3.46410			0.331801			0.165900	+	0.986143i	$0.446947\pi$
$110$		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
$115$	−10.0000	−	12.2474i	−0.932505	−	1.14208i
$116$		0			0
$117$			16.9706i			1.56893i
$118$		0			0
$119$	−6.92820			−0.635107
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	9.79796i		−	0.883452i
$124$		0			0
$125$	−5.19615	−	9.89949i	−0.464758	−	0.885438i
$126$		0			0
$127$		−	7.07107i		−	0.627456i	−0.949513	−	0.313728i	$-0.898422\pi$
$127$		−	7.07107i		−	0.627456i	0.949513	−	0.313728i	$-0.101578\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$	10.0000			0.873704			0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000			0.873704			0.436852	+	0.899533i	$0.356093\pi$
$132$		0			0
$133$		−	8.48528i		−	0.735767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.79796i			0.837096i	0.908195	+	0.418548i	$0.137461\pi$
$137$			9.79796i			0.837096i	−0.908195	+	0.418548i	$0.862539\pi$
$138$		0			0
$139$	10.0000			0.848189			0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000			0.848189			0.424094	+	0.905618i	$0.360592\pi$
$140$		0			0
$141$	−10.3923			−0.875190
$142$		0			0
$143$		−	11.3137i		−	0.946100i
$144$		0			0
$145$	12.0000	−	9.79796i	0.996546	−	0.813676i
$146$		0			0
$147$		−	12.2474i		−	1.01015i
$148$		0			0
$149$	10.3923			0.851371			0.425685	−	0.904871i	$-0.360033\pi$
$149$	10.3923			0.851371			0.425685	+	0.904871i	$0.360033\pi$
$150$		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
$155$	−12.0000	+	9.79796i	−0.963863	+	0.786991i
$156$		0			0
$157$		−	2.82843i		−	0.225733i	−0.993610	−	0.112867i	$-0.963997\pi$
$157$		−	2.82843i		−	0.225733i	0.993610	−	0.112867i	$-0.0360032\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.0000			0.788110
$162$		0			0
$163$		−	22.0454i		−	1.72673i	−0.504580	−	0.863365i	$-0.668353\pi$
$163$		−	22.0454i		−	1.72673i	0.504580	−	0.863365i	$-0.331647\pi$
$164$		0			0
$165$	−6.92820	−	8.48528i	−0.539360	−	0.660578i
$166$		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
$171$	18.0000			1.37649
$172$		0			0
$173$		−	8.48528i		−	0.645124i	−0.946548	−	0.322562i	$-0.895456\pi$
$173$		−	8.48528i		−	0.645124i	0.946548	−	0.322562i	$-0.104544\pi$
$174$		0			0
$175$	6.92820	+	1.41421i	0.523723	+	0.106904i
$176$		0			0
$177$			4.89898i			0.368230i
$178$		0			0
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	20.7846			1.54491			0.772454	−	0.635071i	$-0.219029\pi$
$181$	20.7846			1.54491			0.772454	+	0.635071i	$0.219029\pi$
$182$		0			0
$183$			8.48528i			0.627250i
$184$		0			0
$185$	−4.00000	−	4.89898i	−0.294086	−	0.360180i
$186$		0			0
$187$			9.79796i			0.716498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.8564			1.00261			0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564			1.00261			0.501307	+	0.865269i	$0.332853\pi$
$192$		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
$195$	−24.0000	+	19.5959i	−1.71868	+	1.40329i
$196$		0			0
$197$		−	22.6274i		−	1.61214i	−0.591822	−	0.806068i	$-0.701591\pi$
$197$		−	22.6274i		−	1.61214i	0.591822	−	0.806068i	$-0.298409\pi$
$198$		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
$203$			9.79796i			0.687682i
$204$		0			0
$205$	6.92820	−	5.65685i	0.483887	−	0.395092i
$206$		0			0
$207$			21.2132i			1.47442i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	10.0000			0.688428			0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000			0.688428			0.344214	+	0.938891i	$0.388145\pi$
$212$		0			0
$213$			16.9706i			1.16280i
$214$		0			0
$215$	3.46410	+	4.24264i	0.236250	+	0.289346i
$216$		0			0
$217$		−	9.79796i		−	0.665129i
$218$		0			0
$219$	−12.0000			−0.810885
$220$		0			0
$221$	27.7128			1.86417
$222$		0			0
$223$			26.8701i			1.79935i	0.436558	+	0.899676i	$0.356197\pi$
$223$			26.8701i			1.79935i	−0.436558	+	0.899676i	$0.643803\pi$
$224$		0			0
$225$	−3.00000	+	14.6969i	−0.200000	+	0.979796i
$226$		0			0
$227$			17.1464i			1.13805i	0.822321	+	0.569024i	$0.192679\pi$
$227$			17.1464i			1.13805i	−0.822321	+	0.569024i	$0.807321\pi$
$228$		0			0
$229$	6.92820			0.457829			0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820			0.457829			0.228914	+	0.973447i	$0.426482\pi$
$230$		0			0
$231$	6.92820			0.455842
$232$		0			0
$233$			14.6969i			0.962828i	0.876493	+	0.481414i	$0.159877\pi$
$233$			14.6969i			0.962828i	−0.876493	+	0.481414i	$0.840123\pi$
$234$		0			0
$235$	−6.00000	−	7.34847i	−0.391397	−	0.479361i
$236$		0			0
$237$		−	16.9706i		−	1.10236i
$238$		0			0
$239$	−6.92820			−0.448148			−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820			−0.448148			−0.224074	+	0.974572i	$0.571936\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$			22.0454i			1.41421i
$244$		0			0
$245$	8.66025	−	7.07107i	0.553283	−	0.451754i
$246$		0			0
$247$			33.9411i			2.15962i
$248$		0			0
$249$	30.0000			1.90117
$250$		0			0
$251$	26.0000			1.64111			0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000			1.64111			0.820553	+	0.571571i	$0.193666\pi$
$252$		0			0
$253$		−	14.1421i		−	0.889108i
$254$		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
$259$	4.00000			0.248548
$260$		0			0
$261$	−20.7846			−1.28654
$262$		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
$267$		−	4.89898i		−	0.299813i
$268$		0			0
$269$	−17.3205			−1.05605			−0.528025	−	0.849229i	$-0.677067\pi$
$269$	−17.3205			−1.05605			−0.528025	+	0.849229i	$0.677067\pi$
$270$		0			0
$271$	20.7846			1.26258			0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846			1.26258			0.631288	+	0.775549i	$0.282527\pi$
$272$		0			0
$273$		−	19.5959i		−	1.18600i
$274$		0			0
$275$	2.00000	−	9.79796i	0.120605	−	0.590839i
$276$		0			0
$277$		−	14.1421i		−	0.849719i	−0.905259	−	0.424859i	$-0.860324\pi$
$277$		−	14.1421i		−	0.849719i	0.905259	−	0.424859i	$-0.139676\pi$
$278$		0			0
$279$	20.7846			1.24434
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$		−	2.44949i		−	0.145607i	−0.997346	−	0.0728035i	$-0.976805\pi$
$283$		−	2.44949i		−	0.145607i	0.997346	−	0.0728035i	$-0.0231946\pi$
$284$		0			0
$285$	20.7846	+	25.4558i	1.23117	+	1.50787i
$286$		0			0
$287$			5.65685i			0.333914i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	36.0000			2.11036
$292$		0			0
$293$		−	2.82843i		−	0.165238i	−0.996581	−	0.0826192i	$-0.973671\pi$
$293$		−	2.82843i		−	0.165238i	0.996581	−	0.0826192i	$-0.0263285\pi$
$294$		0			0
$295$	−3.46410	+	2.82843i	−0.201688	+	0.164677i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−40.0000			−2.31326
$300$		0			0
$301$	−3.46410			−0.199667
$302$		0			0
$303$			16.9706i			0.974933i
$304$		0			0
$305$	−6.00000	+	4.89898i	−0.343559	+	0.280515i
$306$		0			0
$307$			2.44949i			0.139800i	0.997554	+	0.0698999i	$0.0222680\pi$
$307$			2.44949i			0.139800i	−0.997554	+	0.0698999i	$0.977732\pi$
$308$		0			0
$309$	24.2487			1.37946
$310$		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	$0.312071\pi$
$313$			29.3939i			1.66144i	−0.556690	+	0.830720i	$0.687929\pi$
$314$		0			0
$315$	−6.00000	−	7.34847i	−0.338062	−	0.414039i
$316$		0			0
$317$			22.6274i			1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$			22.6274i			1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$		0			0
$319$	13.8564			0.775810
$320$		0			0
$321$	−6.00000			−0.334887
$322$		0			0
$323$		−	29.3939i		−	1.63552i
$324$		0			0
$325$	−27.7128	−	5.65685i	−1.53723	−	0.313786i
$326$		0			0
$327$		−	8.48528i		−	0.469237i
$328$		0			0
$329$	6.00000			0.330791
$330$		0			0
$331$	6.00000			0.329790			0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000			0.329790			0.164895	+	0.986311i	$0.447272\pi$
$332$		0			0
$333$			8.48528i			0.464991i
$334$		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
$341$	−13.8564			−0.750366
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	−30.0000	+	24.4949i	−1.61515	+	1.31876i
$346$		0			0
$347$		−	17.1464i		−	0.920468i	−0.887798	−	0.460234i	$-0.847765\pi$
$347$		−	17.1464i		−	0.920468i	0.887798	−	0.460234i	$-0.152235\pi$
$348$		0			0
$349$	−6.92820			−0.370858			−0.185429	−	0.982658i	$-0.559368\pi$
$349$	−6.92820			−0.370858			−0.185429	+	0.982658i	$0.559368\pi$
$350$		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
$355$	−12.0000	+	9.79796i	−0.636894	+	0.520022i
$356$		0			0
$357$			16.9706i			0.898177i
$358$		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
$363$			17.1464i			0.899954i
$364$		0			0
$365$	−6.92820	−	8.48528i	−0.362639	−	0.444140i
$366$		0			0
$367$			9.89949i			0.516749i	0.966045	+	0.258375i	$0.0831869\pi$
$367$			9.89949i			0.516749i	−0.966045	+	0.258375i	$0.916813\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$		0			0
$372$		0			0
$373$			19.7990i			1.02515i	0.858642	+	0.512576i	$0.171309\pi$
$373$			19.7990i			1.02515i	−0.858642	+	0.512576i	$0.828691\pi$
$374$		0			0
$375$	−24.2487	+	12.7279i	−1.25220	+	0.657267i
$376$		0			0
$377$		−	39.1918i		−	2.01848i
$378$		0			0
$379$	−14.0000			−0.719132			−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000			−0.719132			−0.359566	+	0.933120i	$0.617075\pi$
$380$		0			0
$381$	−17.3205			−0.887357
$382$		0			0
$383$		−	12.7279i		−	0.650366i	−0.945651	−	0.325183i	$-0.894574\pi$
$383$		−	12.7279i		−	0.650366i	0.945651	−	0.325183i	$-0.105426\pi$
$384$		0			0
$385$	4.00000	+	4.89898i	0.203859	+	0.249675i
$386$		0			0
$387$		−	7.34847i		−	0.373544i
$388$		0			0
$389$	−3.46410			−0.175637			−0.0878185	−	0.996136i	$-0.527990\pi$
$389$	−3.46410			−0.175637			−0.0878185	+	0.996136i	$0.527990\pi$
$390$		0			0
$391$	34.6410			1.75187
$392$		0			0
$393$		−	24.4949i		−	1.23560i
$394$		0			0
$395$	12.0000	−	9.79796i	0.603786	−	0.492989i
$396$		0			0
$397$			11.3137i			0.567819i	0.958851	+	0.283909i	$0.0916315\pi$
$397$			11.3137i			0.567819i	−0.958851	+	0.283909i	$0.908369\pi$
$398$		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
$403$			39.1918i			1.95228i
$404$		0			0
$405$	−15.5885	+	12.7279i	−0.774597	+	0.632456i
$406$		0			0
$407$		−	5.65685i		−	0.280400i
$408$		0			0
$409$	20.0000			0.988936			0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000			0.988936			0.494468	+	0.869196i	$0.335363\pi$
$410$		0			0
$411$	24.0000			1.18383
$412$		0			0
$413$		−	2.82843i		−	0.139178i
$414$		0			0
$415$	17.3205	+	21.2132i	0.850230	+	1.04132i
$416$		0			0
$417$		−	24.4949i		−	1.19952i
$418$		0			0
$419$	−26.0000			−1.27018			−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000			−1.27018			−0.635092	+	0.772437i	$0.719038\pi$
$420$		0			0
$421$	−3.46410			−0.168830			−0.0844150	−	0.996431i	$-0.526902\pi$
$421$	−3.46410			−0.168830			−0.0844150	+	0.996431i	$0.526902\pi$
$422$		0			0
$423$			12.7279i			0.618853i
$424$		0			0
$425$	24.0000	+	4.89898i	1.16417	+	0.237635i
$426$		0			0
$427$		−	4.89898i		−	0.237078i
$428$		0			0
$429$	−27.7128			−1.33799
$430$		0			0
$431$	6.92820			0.333720			0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820			0.333720			0.166860	+	0.985981i	$0.446637\pi$
$432$		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
$435$	−24.0000	−	29.3939i	−1.15071	−	1.40933i
$436$		0			0
$437$			42.4264i			2.02953i
$438$		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
$443$		−	22.0454i		−	1.04741i	−0.851900	−	0.523704i	$-0.824550\pi$
$443$		−	22.0454i		−	1.04741i	0.851900	−	0.523704i	$-0.175450\pi$
$444$		0			0
$445$	3.46410	−	2.82843i	0.164214	−	0.134080i
$446$		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
$451$	8.00000			0.376705
$452$		0			0
$453$		−	33.9411i		−	1.59469i
$454$		0			0
$455$	13.8564	−	11.3137i	0.649598	−	0.530395i
$456$		0			0
$457$		−	19.5959i		−	0.916658i	−0.888783	−	0.458329i	$-0.848448\pi$
$457$		−	19.5959i		−	0.916658i	0.888783	−	0.458329i	$-0.151552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.7846			0.968036			0.484018	−	0.875058i	$-0.339177\pi$
$461$	20.7846			0.968036			0.484018	+	0.875058i	$0.339177\pi$
$462$		0			0
$463$			1.41421i			0.0657241i	0.999460	+	0.0328620i	$0.0104622\pi$
$463$			1.41421i			0.0657241i	−0.999460	+	0.0328620i	$0.989538\pi$
$464$		0			0
$465$	24.0000	+	29.3939i	1.11297	+	1.36311i
$466$		0			0
$467$		−	31.8434i		−	1.47354i	−0.676146	−	0.736768i	$-0.736351\pi$
$467$		−	31.8434i		−	1.47354i	0.676146	−	0.736768i	$-0.263649\pi$
$468$		0			0
$469$	−3.46410			−0.159957
$470$		0			0
$471$	−6.92820			−0.319235
$472$		0			0
$473$			4.89898i			0.225255i
$474$		0			0
$475$	−6.00000	+	29.3939i	−0.275299	+	1.34868i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.5692			−1.89935			−0.949673	−	0.313243i	$-0.898585\pi$
$479$	−41.5692			−1.89935			−0.949673	+	0.313243i	$0.898585\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		−	24.4949i		−	1.11456i
$484$		0			0
$485$	20.7846	+	25.4558i	0.943781	+	1.15589i
$486$		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
$491$	38.0000			1.71492			0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000			1.71492			0.857458	+	0.514554i	$0.172042\pi$
$492$		0			0
$493$			33.9411i			1.52863i
$494$		0			0
$495$	−10.3923	+	8.48528i	−0.467099	+	0.381385i
$496$		0			0
$497$		−	9.79796i		−	0.439499i
$498$		0			0
$499$	6.00000			0.268597			0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000			0.268597			0.134298	+	0.990941i	$0.457122\pi$
$500$		0			0
$501$	24.2487			1.08335
$502$		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
$507$			46.5403i			2.06693i
$508$		0			0
$509$	−20.7846			−0.921262			−0.460631	−	0.887592i	$-0.652377\pi$
$509$	−20.7846			−0.921262			−0.460631	+	0.887592i	$0.652377\pi$
$510$		0			0
$511$	6.92820			0.306486
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14.0000	+	17.1464i	0.616914	+	0.755562i
$516$		0			0
$517$		−	8.48528i		−	0.373182i
$518$		0			0
$519$	−20.7846			−0.912343
$520$		0			0
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$		−	2.44949i		−	0.107109i	−0.998565	−	0.0535544i	$-0.982945\pi$
$523$		−	2.44949i		−	0.107109i	0.998565	−	0.0535544i	$-0.0170550\pi$
$524$		0			0
$525$	3.46410	−	16.9706i	0.151186	−	0.740656i
$526$		0			0
$527$		−	33.9411i		−	1.47850i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$		−	22.6274i		−	0.980102i
$534$		0			0
$535$	−3.46410	−	4.24264i	−0.149766	−	0.183425i
$536$		0			0
$537$			4.89898i			0.211407i
$538$		0			0
$539$	10.0000			0.430730
$540$		0			0
$541$	−6.92820			−0.297867			−0.148933	−	0.988847i	$-0.547584\pi$
$541$	−6.92820			−0.297867			−0.148933	+	0.988847i	$0.547584\pi$
$542$		0			0
$543$		−	50.9117i		−	2.18483i
$544$		0			0
$545$	6.00000	−	4.89898i	0.257012	−	0.209849i
$546$		0			0
$547$			26.9444i			1.15206i	0.817429	+	0.576029i	$0.195399\pi$
$547$			26.9444i			1.15206i	−0.817429	+	0.576029i	$0.804601\pi$
$548$		0			0
$549$	10.3923			0.443533
$550$		0			0
$551$	−41.5692			−1.77091
$552$		0			0
$553$			9.79796i			0.416652i
$554$		0			0
$555$	−12.0000	+	9.79796i	−0.509372	+	0.415900i
$556$		0			0
$557$		−	25.4558i		−	1.07860i	−0.842114	−	0.539299i	$-0.818689\pi$
$557$		−	25.4558i		−	1.07860i	0.842114	−	0.539299i	$-0.181311\pi$
$558$		0			0
$559$	13.8564			0.586064
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$			7.34847i			0.309701i	0.987938	+	0.154851i	$0.0494896\pi$
$563$			7.34847i			0.309701i	−0.987938	+	0.154851i	$0.950510\pi$
$564$		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	$-0.300344\pi$
$569$	28.0000			1.17382			0.586911	+	0.809652i	$0.300344\pi$
$570$		0			0
$571$	−30.0000			−1.25546			−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000			−1.25546			−0.627730	+	0.778431i	$0.716016\pi$
$572$		0			0
$573$		−	33.9411i		−	1.41791i
$574$		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
$579$	12.0000			0.498703
$580$		0			0
$581$	−17.3205			−0.718576
$582$		0			0
$583$		0			0
$584$		0			0
$585$	24.0000	+	29.3939i	0.992278	+	1.21529i
$586$		0			0
$587$			12.2474i			0.505506i	0.967531	+	0.252753i	$0.0813361\pi$
$587$			12.2474i			0.505506i	−0.967531	+	0.252753i	$0.918664\pi$
$588$		0			0
$589$	41.5692			1.71283
$590$		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
$595$	−12.0000	+	9.79796i	−0.491952	+	0.401677i
$596$		0			0
$597$		−	50.9117i		−	2.08368i
$598$		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	$-0.337183\pi$
$601$	24.0000			0.978980			0.489490	+	0.872009i	$0.337183\pi$
$602$		0			0
$603$		−	7.34847i		−	0.299253i
$604$		0			0
$605$	−12.1244	+	9.89949i	−0.492925	+	0.402472i
$606$		0			0
$607$			41.0122i			1.66463i	0.554300	+	0.832317i	$0.312986\pi$
$607$			41.0122i			1.66463i	−0.554300	+	0.832317i	$0.687014\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$			5.65685i			0.228478i	0.993453	+	0.114239i	$0.0364430\pi$
$613$			5.65685i			0.228478i	−0.993453	+	0.114239i	$0.963557\pi$
$614$		0			0
$615$	−13.8564	−	16.9706i	−0.558744	−	0.684319i
$616$		0			0
$617$		−	24.4949i		−	0.986127i	−0.869993	−	0.493064i	$-0.835877\pi$
$617$		−	24.4949i		−	0.986127i	0.869993	−	0.493064i	$-0.164123\pi$
$618$		0			0
$619$	14.0000			0.562708			0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000			0.562708			0.281354	+	0.959604i	$0.409217\pi$
$620$		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
$627$			29.3939i			1.17388i
$628$		0			0
$629$	13.8564			0.552491
$630$		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
$635$	−10.0000	−	12.2474i	−0.396838	−	0.486025i
$636$		0			0
$637$		−	28.2843i		−	1.12066i
$638$		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
$643$			2.44949i			0.0965984i	0.998833	+	0.0482992i	$0.0153801\pi$
$643$			2.44949i			0.0965984i	−0.998833	+	0.0482992i	$0.984620\pi$
$644$		0			0
$645$	10.3923	−	8.48528i	0.409197	−	0.334108i
$646$		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
$651$	−24.0000			−0.940634
$652$		0			0
$653$		−	5.65685i		−	0.221370i	−0.993856	−	0.110685i	$-0.964696\pi$
$653$		−	5.65685i		−	0.221370i	0.993856	−	0.110685i	$-0.0353044\pi$
$654$		0			0
$655$	17.3205	−	14.1421i	0.676768	−	0.552579i
$656$		0			0
$657$			14.6969i			0.573382i
$658$		0			0
$659$	−2.00000			−0.0779089			−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000			−0.0779089			−0.0389545	+	0.999241i	$0.512403\pi$
$660$		0			0
$661$	−10.3923			−0.404214			−0.202107	−	0.979363i	$-0.564779\pi$
$661$	−10.3923			−0.404214			−0.202107	+	0.979363i	$0.564779\pi$
$662$		0			0
$663$		−	67.8823i		−	2.63633i
$664$		0			0
$665$	−12.0000	−	14.6969i	−0.465340	−	0.569923i
$666$		0			0
$667$		−	48.9898i		−	1.89689i
$668$		0			0
$669$	65.8179			2.54467
$670$		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
$677$			5.65685i			0.217411i	0.994074	+	0.108705i	$0.0346705\pi$
$677$			5.65685i			0.217411i	−0.994074	+	0.108705i	$0.965330\pi$
$678$		0			0
$679$	−20.7846			−0.797640
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$			2.44949i			0.0937271i	0.998901	+	0.0468636i	$0.0149226\pi$
$683$			2.44949i			0.0937271i	−0.998901	+	0.0468636i	$0.985077\pi$
$684$		0			0
$685$	13.8564	+	16.9706i	0.529426	+	0.648412i
$686$		0			0
$687$		−	16.9706i		−	0.647467i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	42.0000			1.59776			0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000			1.59776			0.798878	+	0.601494i	$0.205427\pi$
$692$		0			0
$693$		−	8.48528i		−	0.322329i
$694$		0			0
$695$	17.3205	−	14.1421i	0.657004	−	0.536442i
$696$		0			0
$697$			19.5959i			0.742248i
$698$		0			0
$699$	36.0000			1.36165
$700$		0			0
$701$	3.46410			0.130837			0.0654187	−	0.997858i	$-0.479162\pi$
$701$	3.46410			0.130837			0.0654187	+	0.997858i	$0.479162\pi$
$702$		0			0
$703$			16.9706i			0.640057i
$704$		0			0
$705$	−18.0000	+	14.6969i	−0.677919	+	0.553519i
$706$		0			0
$707$		−	9.79796i		−	0.368490i
$708$		0			0
$709$	−34.6410			−1.30097			−0.650485	−	0.759519i	$-0.725434\pi$
$709$	−34.6410			−1.30097			−0.650485	+	0.759519i	$0.725434\pi$
$710$		0			0
$711$	−20.7846			−0.779484
$712$		0			0
$713$			48.9898i			1.83468i
$714$		0			0
$715$	−16.0000	−	19.5959i	−0.598366	−	0.732846i
$716$		0			0
$717$			16.9706i			0.633777i
$718$		0			0
$719$	6.92820			0.258378			0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820			0.258378			0.129189	+	0.991620i	$0.458763\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.92820	−	33.9411i	0.257307	−	1.26054i
$726$		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
$731$	−12.0000			−0.443836
$732$		0			0
$733$			19.7990i			0.731292i	0.930754	+	0.365646i	$0.119152\pi$
$733$			19.7990i			0.731292i	−0.930754	+	0.365646i	$0.880848\pi$
$734$		0			0
$735$	−17.3205	−	21.2132i	−0.638877	−	0.782461i
$736$		0			0
$737$			4.89898i			0.180456i
$738$		0			0
$739$	30.0000			1.10357			0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000			1.10357			0.551784	+	0.833987i	$0.313947\pi$
$740$		0			0
$741$	83.1384			3.05417
$742$		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
$747$		−	36.7423i		−	1.34433i
$748$		0			0
$749$	3.46410			0.126576
$750$		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
$755$	24.0000	−	19.5959i	0.873449	−	0.713168i
$756$		0			0
$757$			2.82843i			0.102801i	0.998678	+	0.0514005i	$0.0163685\pi$
$757$			2.82843i			0.102801i	−0.998678	+	0.0514005i	$0.983632\pi$
$758$		0			0
$759$	−34.6410			−1.25739
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$			4.89898i			0.177355i
$764$		0			0
$765$	−20.7846	−	25.4558i	−0.751469	−	0.920358i
$766$		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
$771$	−24.0000			−0.864339
$772$		0			0
$773$			45.2548i			1.62770i	0.581073	+	0.813852i	$0.302633\pi$
$773$			45.2548i			1.62770i	−0.581073	+	0.813852i	$0.697367\pi$
$774$		0			0
$775$	−6.92820	+	33.9411i	−0.248868	+	1.21920i
$776$		0			0
$777$		−	9.79796i		−	0.351500i
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	−13.8564			−0.495821
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.00000	−	4.89898i	−0.142766	−	0.174852i
$786$		0			0
$787$		−	46.5403i		−	1.65898i	−0.558520	−	0.829491i	$-0.688630\pi$
$787$		−	46.5403i		−	1.65898i	0.558520	−	0.829491i	$-0.311370\pi$
$788$		0			0
$789$	−65.8179			−2.34318
$790$		0			0
$791$		0			0
$792$		0			0
$793$			19.5959i			0.695871i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			33.9411i			1.20226i	0.799153	+	0.601128i	$0.205282\pi$
$797$			33.9411i			1.20226i	−0.799153	+	0.601128i	$0.794718\pi$
$798$		0			0
$799$	20.7846			0.735307
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$		−	9.79796i		−	0.345762i
$804$		0			0
$805$	17.3205	−	14.1421i	0.610468	−	0.498445i
$806$		0			0
$807$			42.4264i			1.49348i
$808$		0			0
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$		−	50.9117i		−	1.78555i
$814$		0			0
$815$	−31.1769	−	38.1838i	−1.09208	−	1.33752i
$816$		0			0
$817$		−	14.6969i		−	0.514181i
$818$		0			0
$819$	−24.0000			−0.838628
$820$		0			0
$821$	−51.9615			−1.81347			−0.906735	−	0.421701i	$-0.861433\pi$
$821$	−51.9615			−1.81347			−0.906735	+	0.421701i	$0.861433\pi$
$822$		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
$827$			31.8434i			1.10730i	0.832749	+	0.553651i	$0.186766\pi$
$827$			31.8434i			1.10730i	−0.832749	+	0.553651i	$0.813234\pi$
$828$		0			0
$829$	−45.0333			−1.56407			−0.782036	−	0.623233i	$-0.785819\pi$
$829$	−45.0333			−1.56407			−0.782036	+	0.623233i	$0.785819\pi$
$830$		0			0
$831$	−34.6410			−1.20168
$832$		0			0
$833$			24.4949i			0.848698i
$834$		0			0
$835$	14.0000	+	17.1464i	0.484490	+	0.593377i
$836$		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
$843$			19.5959i			0.674919i
$844$		0			0
$845$	−32.9090	+	26.8701i	−1.13210	+	0.924358i
$846$		0			0
$847$		−	9.89949i		−	0.340151i
$848$		0			0
$849$	−6.00000			−0.205919
$850$		0			0
$851$	−20.0000			−0.685591
$852$		0			0
$853$			5.65685i			0.193687i	0.995300	+	0.0968435i	$0.0308746\pi$
$853$			5.65685i			0.193687i	−0.995300	+	0.0968435i	$0.969125\pi$
$854$		0			0
$855$	31.1769	−	25.4558i	1.06623	−	0.870572i
$856$		0			0
$857$			39.1918i			1.33877i	0.742917	+	0.669384i	$0.233442\pi$
$857$			39.1918i			1.33877i	−0.742917	+	0.669384i	$0.766558\pi$
$858$		0			0
$859$	22.0000			0.750630			0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000			0.750630			0.375315	+	0.926897i	$0.377534\pi$
$860$		0			0
$861$	13.8564			0.472225
$862$		0			0
$863$			24.0416i			0.818387i	0.912448	+	0.409193i	$0.134190\pi$
$863$			24.0416i			0.818387i	−0.912448	+	0.409193i	$0.865810\pi$
$864$		0			0
$865$	−12.0000	−	14.6969i	−0.408012	−	0.499711i
$866$		0			0
$867$			17.1464i			0.582323i
$868$		0			0
$869$	13.8564			0.470046
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$		−	44.0908i		−	1.49225i
$874$		0			0
$875$	14.0000	−	7.34847i	0.473286	−	0.248424i
$876$		0			0
$877$			36.7696i			1.24162i	0.783961	+	0.620810i	$0.213196\pi$
$877$			36.7696i			1.24162i	−0.783961	+	0.620810i	$0.786804\pi$
$878$		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
$883$		−	12.2474i		−	0.412159i	−0.978535	−	0.206080i	$-0.933929\pi$
$883$		−	12.2474i		−	0.412159i	0.978535	−	0.206080i	$-0.0660706\pi$
$884$		0			0
$885$	6.92820	+	8.48528i	0.232889	+	0.285230i
$886$		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
$891$	−18.0000			−0.603023
$892$		0			0
$893$			25.4558i			0.851847i
$894$		0			0
$895$	−3.46410	+	2.82843i	−0.115792	+	0.0945439i
$896$		0			0
$897$			97.9796i			3.27144i
$898$		0			0
$899$	−48.0000			−1.60089
$900$		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
$907$			26.9444i			0.894674i	0.894366	+	0.447337i	$0.147627\pi$
$907$			26.9444i			0.894674i	−0.894366	+	0.447337i	$0.852373\pi$
$908$		0			0
$909$	20.7846			0.689382
$910$		0			0
$911$	48.4974			1.60679			0.803396	−	0.595446i	$-0.203024\pi$
$911$	48.4974			1.60679			0.803396	+	0.595446i	$0.203024\pi$
$912$		0			0
$913$			24.4949i			0.810663i
$914$		0			0
$915$	12.0000	+	14.6969i	0.396708	+	0.485866i
$916$		0			0
$917$			14.1421i			0.467014i
$918$		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
$923$			39.1918i			1.29001i
$924$		0			0
$925$	−13.8564	−	2.82843i	−0.455596	−	0.0929981i
$926$		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
$931$	−30.0000			−0.983210
$932$		0			0
$933$			67.8823i			2.22237i
$934$		0			0
$935$	13.8564	+	16.9706i	0.453153	+	0.554997i
$936$		0			0
$937$			24.4949i			0.800213i	0.916469	+	0.400107i	$0.131027\pi$
$937$			24.4949i			0.800213i	−0.916469	+	0.400107i	$0.868973\pi$
$938$		0			0
$939$	72.0000			2.34963
$940$		0			0
$941$	−34.6410			−1.12926			−0.564632	−	0.825342i	$-0.690982\pi$
$941$	−34.6410			−1.12926			−0.564632	+	0.825342i	$0.690982\pi$
$942$		0			0
$943$		−	28.2843i		−	0.921063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			41.6413i			1.35316i	0.736369	+	0.676581i	$0.236539\pi$
$947$			41.6413i			1.35316i	−0.736369	+	0.676581i	$0.763461\pi$
$948$		0			0
$949$	−27.7128			−0.899596
$950$		0			0
$951$	55.4256			1.79730
$952$		0			0
$953$		−	39.1918i		−	1.26955i	−0.772698	−	0.634774i	$-0.781093\pi$
$953$		−	39.1918i		−	1.26955i	0.772698	−	0.634774i	$-0.218907\pi$
$954$		0			0
$955$	24.0000	−	19.5959i	0.776622	−	0.634109i
$956$		0			0
$957$		−	33.9411i		−	1.09716i
$958$		0			0
$959$	−13.8564			−0.447447
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$			7.34847i			0.236801i
$964$		0			0
$965$	6.92820	+	8.48528i	0.223027	+	0.273151i
$966$		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
$971$	22.0000			0.706014			0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000			0.706014			0.353007	+	0.935621i	$0.385159\pi$
$972$		0			0
$973$			14.1421i			0.453376i
$974$		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
$979$	4.00000			0.127841
$980$		0			0
$981$	−10.3923			−0.331801
$982$		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
$987$		−	14.6969i		−	0.467809i
$988$		0			0
$989$	17.3205			0.550760
$990$		0			0
$991$	−27.7128			−0.880327			−0.440163	−	0.897918i	$-0.645079\pi$
$991$	−27.7128			−0.880327			−0.440163	+	0.897918i	$0.645079\pi$
$992$		0			0
$993$		−	14.6969i		−	0.466393i
$994$		0			0
$995$	36.0000	−	29.3939i	1.14128	−	0.931849i
$996$		0			0
$997$			56.5685i			1.79154i	0.444514	+	0.895772i	$0.353376\pi$
$997$			56.5685i			1.79154i	−0.444514	+	0.895772i	$0.646624\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.b.289.1	✓	8	16.3	odd	4
320.2.f.b.289.2	yes	8	80.59	odd	4
320.2.f.b.289.3	yes	8	80.29	even	4
320.2.f.b.289.4	yes	8	16.5	even	4
320.2.f.b.289.5	yes	8	16.13	even	4
320.2.f.b.289.6	yes	8	80.69	even	4
320.2.f.b.289.7	yes	8	80.19	odd	4
320.2.f.b.289.8	yes	8	16.11	odd	4
1280.2.c.g.769.1		4	40.29	even	2
1280.2.c.g.769.2		4	20.19	odd	2
1280.2.c.g.769.3		4	8.5	even	2
1280.2.c.g.769.4		4	4.3	odd	2
1280.2.c.h.769.1		4	8.3	odd	2	inner
1280.2.c.h.769.2		4	1.1	even	1	trivial
1280.2.c.h.769.3		4	40.19	odd	2	inner
1280.2.c.h.769.4		4	5.4	even	2	inner
1600.2.d.i.801.1		8	80.27	even	4
1600.2.d.i.801.2		8	80.53	odd	4
1600.2.d.i.801.3		8	80.77	odd	4
1600.2.d.i.801.4		8	80.3	even	4
1600.2.d.i.801.5		8	80.13	odd	4
1600.2.d.i.801.6		8	80.67	even	4
1600.2.d.i.801.7		8	80.43	even	4
1600.2.d.i.801.8		8	80.37	odd	4
2880.2.d.g.289.1		8	48.5	odd	4
2880.2.d.g.289.2		8	48.11	even	4
2880.2.d.g.289.3		8	240.179	even	4
2880.2.d.g.289.4		8	240.29	odd	4
2880.2.d.g.289.5		8	240.59	even	4
2880.2.d.g.289.6		8	240.149	odd	4
2880.2.d.g.289.7		8	48.29	odd	4
2880.2.d.g.289.8		8	48.35	even	4
6400.2.a.ct.1.1		4	20.3	even	4
6400.2.a.ct.1.2		4	40.13	odd	4
6400.2.a.ct.1.3		4	40.37	odd	4
6400.2.a.ct.1.4		4	20.7	even	4
6400.2.a.cu.1.1		4	5.2	odd	4
6400.2.a.cu.1.2		4	40.27	even	4
6400.2.a.cu.1.3		4	40.3	even	4
6400.2.a.cu.1.4		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.44949i q^{3} +(1.73205 - 1.41421i) q^{5} +1.41421i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-2.44949i q^{3} +(1.73205 - 1.41421i) q^{5} +1.41421i q^{7} -3.00000 q^{9} +2.00000 q^{11} -5.65685i q^{13} +(-3.46410 - 4.24264i) q^{15} +4.89898i q^{17} -6.00000 q^{19} +3.46410 q^{21} -7.07107i q^{23} +(1.00000 - 4.89898i) q^{25} +6.92820 q^{29} -6.92820 q^{31} -4.89898i q^{33} +(2.00000 + 2.44949i) q^{35} -2.82843i q^{37} -13.8564 q^{39} +4.00000 q^{41} +2.44949i q^{43} +(-5.19615 + 4.24264i) q^{45} -4.24264i q^{47} +5.00000 q^{49} +12.0000 q^{51} +(3.46410 - 2.82843i) q^{55} +14.6969i q^{57} -2.00000 q^{59} -3.46410 q^{61} -4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949i q^{67} -17.3205 q^{69} -6.92820 q^{71} -4.89898i q^{73} +(-12.0000 - 2.44949i) q^{75} +2.82843i q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474i q^{83} +(6.92820 + 8.48528i) q^{85} -16.9706i q^{87} +2.00000 q^{89} +8.00000 q^{91} +16.9706i q^{93} +(-10.3923 + 8.48528i) q^{95} +14.6969i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 8 q^{11} - 24 q^{19} + 4 q^{25} + 8 q^{35} + 16 q^{41} + 20 q^{49} + 48 q^{51} - 8 q^{59} - 32 q^{65} - 48 q^{75} - 36 q^{81} + 8 q^{89} + 32 q^{91} - 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 8 * q^11 - 24 * q^19 + 4 * q^25 + 8 * q^35 + 16 * q^41 + 20 * q^49 + 48 * q^51 - 8 * q^59 - 32 * q^65 - 48 * q^75 - 36 * q^81 + 8 * q^89 + 32 * q^91 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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