LMFDB - Embedded newform 6400.2.a.cu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.a (trivial)

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:	yes
Analytic conductor:	$51.1042572936$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
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^{2} $
Twist minimal:	no (minimal twist has level 320)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	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
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	−7.07107	−1.47442	−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107	−1.47442	−0.737210	+	0.675664i	$0.763857\pi$
$24$	0	0
$25$	0	0
$26$	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	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	4.89898	0.852803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.82843	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$38$	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
$43$	−2.44949	−0.373544	−0.186772	−	0.982403i	$-0.559803\pi$
$43$	−2.44949	−0.373544	−0.186772	+	0.982403i	$0.559803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.24264	0.618853	0.309426	−	0.950923i	$-0.399863\pi$
$47$	4.24264	0.618853	0.309426	+	0.950923i	$0.399863\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.6969	1.94666
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	3.46410	0.443533	0.221766	−	0.975100i	$-0.428818\pi$
$61$	3.46410	0.443533	0.221766	+	0.975100i	$0.428818\pi$
$62$	0	0
$63$	−4.24264	−0.534522
$64$	0	0
$65$	0	0
$66$	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
$69$	−17.3205	−2.08514
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	4.89898	0.573382	0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898	0.573382	0.286691	+	0.958023i	$0.407445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	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
$83$	−12.2474	−1.34433	−0.672166	−	0.740400i	$-0.734636\pi$
$83$	−12.2474	−1.34433	−0.672166	+	0.740400i	$0.734636\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.9706	1.81944
$88$	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
$91$	8.00000	0.838628
$92$	0	0
$93$	16.9706	1.75977
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6969	1.49225	0.746124	−	0.665807i	$-0.231913\pi$
$97$	14.6969	1.49225	0.746124	+	0.665807i	$0.231913\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	6.92820	0.689382	0.344691	−	0.938716i	$-0.387984\pi$
$101$	6.92820	0.689382	0.344691	+	0.938716i	$0.387984\pi$
$102$	0	0
$103$	9.89949	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$103$	9.89949	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.44949	−0.236801	−0.118401	−	0.992966i	$-0.537777\pi$
$107$	−2.44949	−0.236801	−0.118401	+	0.992966i	$0.537777\pi$
$108$	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		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	7.07107	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$127$	7.07107	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$128$	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
$133$	−8.48528	−0.735767
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.79796	0.837096	0.418548	−	0.908195i	$-0.362539\pi$
$137$	9.79796	0.837096	0.418548	+	0.908195i	$0.362539\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	10.3923	0.875190
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.2474	−1.01015
$148$	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	$-0.690665\pi$
$151$	−13.8564	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$152$	0	0
$153$	14.6969	1.18818
$154$	0	0
$155$	0	0
$156$	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	$-0.168353\pi$
$163$	22.0454	1.72673	0.863365	+	0.504580i	$0.168353\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.89949	−0.766046	−0.383023	−	0.923739i	$-0.625117\pi$
$167$	−9.89949	−0.766046	−0.383023	+	0.923739i	$0.625117\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	18.0000	1.37649
$172$	0	0
$173$	−8.48528	−0.645124	−0.322562	−	0.946548i	$-0.604544\pi$
$173$	−8.48528	−0.645124	−0.322562	+	0.946548i	$0.604544\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.89898	0.368230
$178$	0	0
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−20.7846	−1.54491	−0.772454	−	0.635071i	$-0.780971\pi$
$181$	−20.7846	−1.54491	−0.772454	+	0.635071i	$0.780971\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	0	0
$186$	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
$193$	−4.89898	−0.352636	−0.176318	−	0.984333i	$-0.556419\pi$
$193$	−4.89898	−0.352636	−0.176318	+	0.984333i	$0.556419\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.6274	1.61214	0.806068	−	0.591822i	$-0.201591\pi$
$197$	22.6274	1.61214	0.806068	+	0.591822i	$0.201591\pi$
$198$	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
$205$	0	0
$206$	0	0
$207$	−21.2132	−1.47442
$208$	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
$213$	16.9706	1.16280
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.79796	−0.665129
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	−27.7128	−1.86417
$222$	0	0
$223$	26.8701	1.79935	0.899676	−	0.436558i	$-0.143803\pi$
$223$	26.8701	1.79935	0.899676	+	0.436558i	$0.143803\pi$
$224$	0	0
$225$	0	0
$226$	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
$229$	6.92820	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
$233$	−14.6969	−0.962828	−0.481414	−	0.876493i	$-0.659877\pi$
$233$	−14.6969	−0.962828	−0.481414	+	0.876493i	$0.659877\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.9706	1.10236
$238$	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
$243$	−22.0454	−1.41421
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.9411	−2.15962
$248$	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
$253$	−14.1421	−0.889108
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.79796	−0.611180	−0.305590	−	0.952163i	$-0.598854\pi$
$257$	−9.79796	−0.611180	−0.305590	+	0.952163i	$0.598854\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	20.7846	1.28654
$262$	0	0
$263$	−26.8701	−1.65688	−0.828439	−	0.560079i	$-0.810771\pi$
$263$	−26.8701	−1.65688	−0.828439	+	0.560079i	$0.810771\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.89898	−0.299813
$268$	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	$-0.717473\pi$
$271$	−20.7846	−1.26258	−0.631288	+	0.775549i	$0.717473\pi$
$272$	0	0
$273$	19.5959	1.18600
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.1421	0.849719	0.424859	−	0.905259i	$-0.360324\pi$
$277$	14.1421	0.849719	0.424859	+	0.905259i	$0.360324\pi$
$278$	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
$283$	2.44949	0.145607	0.0728035	−	0.997346i	$-0.476805\pi$
$283$	2.44949	0.145607	0.0728035	+	0.997346i	$0.476805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	36.0000	2.11036
$292$	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
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	40.0000	2.31326
$300$	0	0
$301$	3.46410	0.199667
$302$	0	0
$303$	16.9706	0.974933
$304$	0	0
$305$	0	0
$306$	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
$309$	24.2487	1.37946
$310$	0	0
$311$	27.7128	1.57145	0.785725	−	0.618576i	$-0.212290\pi$
$311$	27.7128	1.57145	0.785725	+	0.618576i	$0.212290\pi$
$312$	0	0
$313$	−29.3939	−1.66144	−0.830720	−	0.556690i	$-0.812071\pi$
$313$	−29.3939	−1.66144	−0.830720	+	0.556690i	$0.812071\pi$
$314$	0	0
$315$	0	0
$316$	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
$325$	0	0
$326$	0	0
$327$	8.48528	0.469237
$328$	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
$333$	8.48528	0.464991
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.5959	−1.06746	−0.533729	−	0.845656i	$-0.679210\pi$
$337$	−19.5959	−1.06746	−0.533729	+	0.845656i	$0.679210\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.8564	0.750366
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1464	−0.920468	−0.460234	−	0.887798i	$-0.652235\pi$
$347$	−17.1464	−0.920468	−0.460234	+	0.887798i	$0.652235\pi$
$348$	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
$353$	−9.79796	−0.521493	−0.260746	−	0.965407i	$-0.583969\pi$
$353$	−9.79796	−0.521493	−0.260746	+	0.965407i	$0.583969\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	−9.89949	−0.516749	−0.258375	−	0.966045i	$-0.583187\pi$
$367$	−9.89949	−0.516749	−0.258375	+	0.966045i	$0.583187\pi$
$368$	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
$375$	0	0
$376$	0	0
$377$	−39.1918	−2.01848
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	17.3205	0.887357
$382$	0	0
$383$	−12.7279	−0.650366	−0.325183	−	0.945651i	$-0.605426\pi$
$383$	−12.7279	−0.650366	−0.325183	+	0.945651i	$0.605426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.34847	−0.373544
$388$	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
$393$	24.4949	1.23560
$394$	0	0
$395$	0	0
$396$	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
$405$	0	0
$406$	0	0
$407$	5.65685	0.280400
$408$	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
$411$	24.0000	1.18383
$412$	0	0
$413$	−2.82843	−0.139178
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−24.4949	−1.19952
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	3.46410	0.168830	0.0844150	−	0.996431i	$-0.473098\pi$
$421$	3.46410	0.168830	0.0844150	+	0.996431i	$0.473098\pi$
$422$	0	0
$423$	12.7279	0.618853
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.89898	−0.237078
$428$	0	0
$429$	−27.7128	−1.33799
$430$	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
$433$	−34.2929	−1.64801	−0.824005	−	0.566583i	$-0.808265\pi$
$433$	−34.2929	−1.64801	−0.824005	+	0.566583i	$0.808265\pi$
$434$	0	0
$435$	0	0
$436$	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
$445$	0	0
$446$	0	0
$447$	25.4558	1.20402
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	−33.9411	−1.59469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5959	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	−19.5959	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.7846	−0.968036	−0.484018	−	0.875058i	$-0.660823\pi$
$461$	−20.7846	−0.968036	−0.484018	+	0.875058i	$0.660823\pi$
$462$	0	0
$463$	1.41421	0.0657241	0.0328620	−	0.999460i	$-0.489538\pi$
$463$	1.41421	0.0657241	0.0328620	+	0.999460i	$0.489538\pi$
$464$	0	0
$465$	0	0
$466$	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
$469$	−3.46410	−0.159957
$470$	0	0
$471$	6.92820	0.319235
$472$	0	0
$473$	−4.89898	−0.225255
$474$	0	0
$475$	0	0
$476$	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
$483$	24.4949	1.11456
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8701	1.21760	0.608799	−	0.793324i	$-0.291651\pi$
$487$	26.8701	1.21760	0.608799	+	0.793324i	$0.291651\pi$
$488$	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
$493$	33.9411	1.52863
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.79796	−0.439499
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−24.2487	−1.08335
$502$	0	0
$503$	21.2132	0.945850	0.472925	−	0.881103i	$-0.343198\pi$
$503$	21.2132	0.945850	0.472925	+	0.881103i	$0.343198\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	46.5403	2.06693
$508$	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
$515$	0	0
$516$	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	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	33.9411	1.47850
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−22.6274	−0.980102
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.89898	0.211407
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	6.92820	0.297867	0.148933	−	0.988847i	$-0.452416\pi$
$541$	6.92820	0.297867	0.148933	+	0.988847i	$0.452416\pi$
$542$	0	0
$543$	−50.9117	−2.18483
$544$	0	0
$545$	0	0
$546$	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
$549$	10.3923	0.443533
$550$	0	0
$551$	41.5692	1.77091
$552$	0	0
$553$	−9.79796	−0.416652
$554$	0	0
$555$	0	0
$556$	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	$-0.549490\pi$
$563$	−7.34847	−0.309701	−0.154851	+	0.987938i	$0.549490\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.7279	0.534522
$568$	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
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	0	0
$573$	−33.9411	−1.41791
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.3939	−1.22368	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	−29.3939	−1.22368	−0.611842	+	0.790980i	$0.709571\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	17.3205	0.718576
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.2474	0.505506	0.252753	−	0.967531i	$-0.418664\pi$
$587$	12.2474	0.505506	0.252753	+	0.967531i	$0.418664\pi$
$588$	0	0
$589$	41.5692	1.71283
$590$	0	0
$591$	55.4256	2.27991
$592$	0	0
$593$	−29.3939	−1.20706	−0.603531	−	0.797340i	$-0.706240\pi$
$593$	−29.3939	−1.20706	−0.603531	+	0.797340i	$0.706240\pi$
$594$	0	0
$595$	0	0
$596$	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	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	7.34847	0.299253
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−41.0122	−1.66463	−0.832317	−	0.554300i	$-0.812986\pi$
$607$	−41.0122	−1.66463	−0.832317	+	0.554300i	$0.812986\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−24.0000	−0.970936
$612$	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
$615$	0	0
$616$	0	0
$617$	−24.4949	−0.986127	−0.493064	−	0.869993i	$-0.664123\pi$
$617$	−24.4949	−0.986127	−0.493064	+	0.869993i	$0.664123\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.82843	0.113319
$624$	0	0
$625$	0	0
$626$	0	0
$627$	29.3939	1.17388
$628$	0	0
$629$	13.8564	0.552491
$630$	0	0
$631$	−13.8564	−0.551615	−0.275807	−	0.961213i	$-0.588945\pi$
$631$	−13.8564	−0.551615	−0.275807	+	0.961213i	$0.588945\pi$
$632$	0	0
$633$	24.4949	0.973585
$634$	0	0
$635$	0	0
$636$	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	$-0.515380\pi$
$643$	−2.44949	−0.0965984	−0.0482992	+	0.998833i	$0.515380\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.24264	0.166795	0.0833977	−	0.996516i	$-0.473423\pi$
$647$	4.24264	0.166795	0.0833977	+	0.996516i	$0.473423\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	−5.65685	−0.221370	−0.110685	−	0.993856i	$-0.535304\pi$
$653$	−5.65685	−0.221370	−0.110685	+	0.993856i	$0.535304\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.6969	0.573382
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	10.3923	0.404214	0.202107	−	0.979363i	$-0.435221\pi$
$661$	10.3923	0.404214	0.202107	+	0.979363i	$0.435221\pi$
$662$	0	0
$663$	−67.8823	−2.63633
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−48.9898	−1.89689
$668$	0	0
$669$	65.8179	2.54467
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	24.4949	0.944209	0.472104	−	0.881543i	$-0.343495\pi$
$673$	24.4949	0.944209	0.472104	+	0.881543i	$0.343495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.65685	−0.217411	−0.108705	−	0.994074i	$-0.534670\pi$
$677$	−5.65685	−0.217411	−0.108705	+	0.994074i	$0.534670\pi$
$678$	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
$685$	0	0
$686$	0	0
$687$	16.9706	0.647467
$688$	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
$693$	−8.48528	−0.322329
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.5959	0.742248
$698$	0	0
$699$	−36.0000	−1.36165
$700$	0	0
$701$	−3.46410	−0.130837	−0.0654187	−	0.997858i	$-0.520838\pi$
$701$	−3.46410	−0.130837	−0.0654187	+	0.997858i	$0.520838\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.79796	−0.368490
$708$	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
$713$	−48.9898	−1.83468
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.9706	−0.633777
$718$	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
$725$	0	0
$726$	0	0
$727$	−18.3848	−0.681854	−0.340927	−	0.940090i	$-0.610741\pi$
$727$	−18.3848	−0.681854	−0.340927	+	0.940090i	$0.610741\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	19.7990	0.731292	0.365646	−	0.930754i	$-0.380848\pi$
$733$	19.7990	0.731292	0.365646	+	0.930754i	$0.380848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.89898	0.180456
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	−83.1384	−3.05417
$742$	0	0
$743$	−4.24264	−0.155647	−0.0778237	−	0.996967i	$-0.524797\pi$
$743$	−4.24264	−0.155647	−0.0778237	+	0.996967i	$0.524797\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−36.7423	−1.34433
$748$	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
$753$	63.6867	2.32087
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.82843	−0.102801	−0.0514005	−	0.998678i	$-0.516368\pi$
$757$	−2.82843	−0.102801	−0.0514005	+	0.998678i	$0.516368\pi$
$758$	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
$763$	−4.89898	−0.177355
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.3137	−0.408514
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	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
$775$	0	0
$776$	0	0
$777$	−9.79796	−0.351500
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	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
$789$	−65.8179	−2.34318
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−19.5959	−0.695871
$794$	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
$805$	0	0
$806$	0	0
$807$	−42.4264	−1.49348
$808$	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
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	−50.9117	−1.78555
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.6969	−0.514181
$818$	0	0
$819$	24.0000	0.838628
$820$	0	0
$821$	51.9615	1.81347	0.906735	−	0.421701i	$-0.138567\pi$
$821$	51.9615	1.81347	0.906735	+	0.421701i	$0.138567\pi$
$822$	0	0
$823$	−35.3553	−1.23241	−0.616205	−	0.787586i	$-0.711331\pi$
$823$	−35.3553	−1.23241	−0.616205	+	0.787586i	$0.711331\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.8434	1.10730	0.553651	−	0.832749i	$-0.313234\pi$
$827$	31.8434	1.10730	0.553651	+	0.832749i	$0.313234\pi$
$828$	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
$833$	−24.4949	−0.848698
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	9.89949	0.340151
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−20.0000	−0.685591
$852$	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
$855$	0	0
$856$	0	0
$857$	39.1918	1.33877	0.669384	−	0.742917i	$-0.266558\pi$
$857$	39.1918	1.33877	0.669384	+	0.742917i	$0.266558\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	−13.8564	−0.472225
$862$	0	0
$863$	24.0416	0.818387	0.409193	−	0.912448i	$-0.365810\pi$
$863$	24.0416	0.818387	0.409193	+	0.912448i	$0.365810\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.1464	0.582323
$868$	0	0
$869$	13.8564	0.470046
$870$	0	0
$871$	−13.8564	−0.469506
$872$	0	0
$873$	44.0908	1.49225
$874$	0	0
$875$	0	0
$876$	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	$-0.433929\pi$
$883$	12.2474	0.412159	0.206080	+	0.978535i	$0.433929\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.4975	1.66196	0.830981	−	0.556300i	$-0.187780\pi$
$887$	49.4975	1.66196	0.830981	+	0.556300i	$0.187780\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	−18.0000	−0.603023
$892$	0	0
$893$	25.4558	0.851847
$894$	0	0
$895$	0	0
$896$	0	0
$897$	97.9796	3.27144
$898$	0	0
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.48528	0.282372
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.9444	0.894674	0.447337	−	0.894366i	$-0.352373\pi$
$907$	26.9444	0.894674	0.447337	+	0.894366i	$0.352373\pi$
$908$	0	0
$909$	20.7846	0.689382
$910$	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
$913$	−24.4949	−0.810663
$914$	0	0
$915$	0	0
$916$	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
$925$	0	0
$926$	0	0
$927$	29.6985	0.975426
$928$	0	0
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	−30.0000	−0.983210
$932$	0	0
$933$	67.8823	2.22237
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.4949	0.800213	0.400107	−	0.916469i	$-0.368973\pi$
$937$	24.4949	0.800213	0.400107	+	0.916469i	$0.368973\pi$
$938$	0	0
$939$	−72.0000	−2.34963
$940$	0	0
$941$	34.6410	1.12926	0.564632	−	0.825342i	$-0.309018\pi$
$941$	34.6410	1.12926	0.564632	+	0.825342i	$0.309018\pi$
$942$	0	0
$943$	−28.2843	−0.921063
$944$	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
$949$	−27.7128	−0.899596
$950$	0	0
$951$	−55.4256	−1.79730
$952$	0	0
$953$	39.1918	1.26955	0.634774	−	0.772698i	$-0.281093\pi$
$953$	39.1918	1.26955	0.634774	+	0.772698i	$0.281093\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	−15.5563	−0.500258	−0.250129	−	0.968212i	$-0.580473\pi$
$967$	−15.5563	−0.500258	−0.250129	+	0.968212i	$0.580473\pi$
$968$	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
$973$	14.1421	0.453376
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.89898	−0.156732	−0.0783661	−	0.996925i	$-0.524970\pi$
$977$	−4.89898	−0.156732	−0.0783661	+	0.996925i	$0.524970\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	10.3923	0.331801
$982$	0	0
$983$	46.6690	1.48851	0.744256	−	0.667895i	$-0.232804\pi$
$983$	46.6690	1.48851	0.744256	+	0.667895i	$0.232804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−14.6969	−0.467809
$988$	0	0
$989$	17.3205	0.550760
$990$	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
$993$	14.6969	0.466393
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−56.5685	−1.79154	−0.895772	−	0.444514i	$-0.853376\pi$
$997$	−56.5685	−1.79154	−0.895772	+	0.444514i	$0.853376\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.b.289.1	✓	8	80.37	odd	4
320.2.f.b.289.2	yes	8	80.13	odd	4
320.2.f.b.289.3	yes	8	80.43	even	4
320.2.f.b.289.4	yes	8	80.67	even	4
320.2.f.b.289.5	yes	8	80.27	even	4
320.2.f.b.289.6	yes	8	80.3	even	4
320.2.f.b.289.7	yes	8	80.53	odd	4
320.2.f.b.289.8	yes	8	80.77	odd	4
1280.2.c.g.769.1		4	20.3	even	4
1280.2.c.g.769.2		4	40.13	odd	4
1280.2.c.g.769.3		4	20.7	even	4
1280.2.c.g.769.4		4	40.37	odd	4
1280.2.c.h.769.1		4	5.2	odd	4
1280.2.c.h.769.2		4	40.27	even	4
1280.2.c.h.769.3		4	5.3	odd	4
1280.2.c.h.769.4		4	40.3	even	4
1600.2.d.i.801.1		8	80.29	even	4
1600.2.d.i.801.2		8	16.3	odd	4
1600.2.d.i.801.3		8	80.59	odd	4
1600.2.d.i.801.4		8	16.5	even	4
1600.2.d.i.801.5		8	16.11	odd	4
1600.2.d.i.801.6		8	80.69	even	4
1600.2.d.i.801.7		8	16.13	even	4
1600.2.d.i.801.8		8	80.19	odd	4
2880.2.d.g.289.1		8	240.227	odd	4
2880.2.d.g.289.2		8	240.77	even	4
2880.2.d.g.289.3		8	240.53	even	4
2880.2.d.g.289.4		8	240.203	odd	4
2880.2.d.g.289.5		8	240.173	even	4
2880.2.d.g.289.6		8	240.83	odd	4
2880.2.d.g.289.7		8	240.107	odd	4
2880.2.d.g.289.8		8	240.197	even	4
6400.2.a.ct.1.1		4	8.5	even	2
6400.2.a.ct.1.2		4	4.3	odd	2
6400.2.a.ct.1.3		4	20.19	odd	2
6400.2.a.ct.1.4		4	40.29	even	2
6400.2.a.cu.1.1		4	40.19	odd	2	inner
6400.2.a.cu.1.2		4	5.4	even	2	inner
6400.2.a.cu.1.3		4	1.1	even	1	trivial
6400.2.a.cu.1.4		4	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

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

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