LMFDB - Embedded newform 1600.2.d.h.801.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(801,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1600.801");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.d (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			801.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.h.801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{3} +4.73205 q^{7} -4.46410 q^{9} +O(q^{10})$	$q+2.73205i q^{3} +4.73205 q^{7} -4.46410 q^{9} -3.46410i q^{11} +3.46410i q^{13} +3.46410 q^{17} +2.00000i q^{19} +12.9282i q^{21} -2.19615 q^{23} -4.00000i q^{27} +2.53590 q^{31} +9.46410 q^{33} +6.00000i q^{37} -9.46410 q^{39} +9.46410 q^{41} -0.196152i q^{43} +2.19615 q^{47} +15.3923 q^{49} +9.46410i q^{51} +10.3923i q^{53} -5.46410 q^{57} -6.00000i q^{59} +0.928203i q^{61} -21.1244 q^{63} -0.196152i q^{67} -6.00000i q^{69} -16.3923 q^{71} -6.39230 q^{73} -16.3923i q^{77} +12.0000 q^{79} -2.46410 q^{81} +1.26795i q^{83} -12.9282 q^{89} +16.3923i q^{91} +6.92820i q^{93} -14.3923 q^{97} +15.4641i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 12 * q^7 - 4 * q^9	$ 4 q + 12 q^{7} - 4 q^{9} + 12 q^{23} + 24 q^{31} + 24 q^{33} - 24 q^{39} + 24 q^{41} - 12 q^{47} + 20 q^{49} - 8 q^{57} - 36 q^{63} - 24 q^{71} + 16 q^{73} + 48 q^{79} + 4 q^{81} - 24 q^{89} - 16 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 - 4 * q^9 + 12 * q^23 + 24 * q^31 + 24 * q^33 - 24 * q^39 + 24 * q^41 - 12 * q^47 + 20 * q^49 - 8 * q^57 - 36 * q^63 - 24 * q^71 + 16 * q^73 + 48 * q^79 + 4 * q^81 - 24 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.73205i	1.57735i	0.614810	+	0.788675i	$0.289233\pi$
$3$	2.73205i	1.57735i	−0.614810	+	0.788675i	$0.710767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	−4.46410	−1.48803
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	12.9282i	2.82117i
$22$	0	0
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	2.53590	0.455461	0.227730	−	0.973724i	$-0.426870\pi$
$31$	2.53590	0.455461	0.227730	+	0.973724i	$0.426870\pi$
$32$	0	0
$33$	9.46410	1.64749
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−9.46410	−1.51547
$40$	0	0
$41$	9.46410	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$41$	9.46410	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$42$	0	0
$43$	− 0.196152i	− 0.0299130i	−0.999888	−	0.0149565i	$-0.995239\pi$
$43$	− 0.196152i	− 0.0299130i	0.999888	−	0.0149565i	$-0.00476097\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.19615	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$47$	2.19615	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	9.46410i	1.32524i
$52$	0	0
$53$	10.3923i	1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$	10.3923i	1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.46410	−0.723738
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	0.928203i	0.118844i	0.998233	+	0.0594221i	$0.0189258\pi$
$61$	0.928203i	0.118844i	−0.998233	+	0.0594221i	$0.981074\pi$
$62$	0	0
$63$	−21.1244	−2.66142
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.196152i	− 0.0239638i	−0.999928	−	0.0119819i	$-0.996186\pi$
$67$	− 0.196152i	− 0.0239638i	0.999928	−	0.0119819i	$-0.00381405\pi$
$68$	0	0
$69$	− 6.00000i	− 0.722315i
$70$	0	0
$71$	−16.3923	−1.94541	−0.972704	−	0.232048i	$-0.925457\pi$
$71$	−16.3923	−1.94541	−0.972704	+	0.232048i	$0.925457\pi$
$72$	0	0
$73$	−6.39230	−0.748163	−0.374081	−	0.927396i	$-0.622042\pi$
$73$	−6.39230	−0.748163	−0.374081	+	0.927396i	$0.622042\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.3923i	− 1.86808i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	1.26795i	0.139176i	0.997576	+	0.0695878i	$0.0221684\pi$
$83$	1.26795i	0.139176i	−0.997576	+	0.0695878i	$0.977832\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	16.3923i	1.71838i
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	15.4641i	1.55420i
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	−2.19615	−0.216393	−0.108197	−	0.994130i	$-0.534508\pi$
$103$	−2.19615	−0.216393	−0.108197	+	0.994130i	$0.534508\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.2679i	− 1.28266i	−0.767265	−	0.641331i	$-0.778383\pi$
$107$	− 13.2679i	− 1.28266i	0.767265	−	0.641331i	$-0.221617\pi$
$108$	0	0
$109$	12.9282i	1.23830i	0.785274	+	0.619149i	$0.212522\pi$
$109$	12.9282i	1.23830i	−0.785274	+	0.619149i	$0.787478\pi$
$110$	0	0
$111$	−16.3923	−1.55589
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 15.4641i	− 1.42966i
$118$	0	0
$119$	16.3923	1.50268
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	25.8564i	2.33139i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.1962	−1.25970	−0.629852	−	0.776715i	$-0.716885\pi$
$127$	−14.1962	−1.25970	−0.629852	+	0.776715i	$0.716885\pi$
$128$	0	0
$129$	0.535898	0.0471832
$130$	0	0
$131$	− 10.3923i	− 0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$	− 10.3923i	− 0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$	0	0
$133$	9.46410i	0.820642i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.928203	0.0793018	0.0396509	−	0.999214i	$-0.487375\pi$
$137$	0.928203	0.0793018	0.0396509	+	0.999214i	$0.487375\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	6.00000i	0.505291i
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	42.0526i	3.46844i
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	−9.46410	−0.770178	−0.385089	−	0.922880i	$-0.625829\pi$
$151$	−9.46410	−0.770178	−0.385089	+	0.922880i	$0.625829\pi$
$152$	0	0
$153$	−15.4641	−1.25020
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.9282i	1.03178i	0.856654	+	0.515891i	$0.172539\pi$
$157$	12.9282i	1.03178i	−0.856654	+	0.515891i	$0.827461\pi$
$158$	0	0
$159$	−28.3923	−2.25166
$160$	0	0
$161$	−10.3923	−0.819028
$162$	0	0
$163$	− 16.1962i	− 1.26858i	−0.773095	−	0.634290i	$-0.781292\pi$
$163$	− 16.1962i	− 1.26858i	0.773095	−	0.634290i	$-0.218708\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.19615	0.169943	0.0849717	−	0.996383i	$-0.472920\pi$
$167$	2.19615	0.169943	0.0849717	+	0.996383i	$0.472920\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	− 8.92820i	− 0.682757i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.3923	1.23212
$178$	0	0
$179$	− 7.85641i	− 0.587215i	−0.955926	−	0.293608i	$-0.905144\pi$
$179$	− 7.85641i	− 0.587215i	0.955926	−	0.293608i	$-0.0948559\pi$
$180$	0	0
$181$	− 6.92820i	− 0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	− 6.92820i	− 0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$	0	0
$183$	−2.53590	−0.187459
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	− 18.9282i	− 1.37682i
$190$	0	0
$191$	4.39230	0.317816	0.158908	−	0.987293i	$-0.449203\pi$
$191$	4.39230	0.317816	0.158908	+	0.987293i	$0.449203\pi$
$192$	0	0
$193$	14.3923	1.03598	0.517990	−	0.855386i	$-0.326680\pi$
$193$	14.3923	1.03598	0.517990	+	0.855386i	$0.326680\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.3923i	− 0.740421i	−0.928948	−	0.370211i	$-0.879286\pi$
$197$	− 10.3923i	− 0.740421i	0.928948	−	0.370211i	$-0.120714\pi$
$198$	0	0
$199$	−6.92820	−0.491127	−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820	−0.491127	−0.245564	+	0.969380i	$0.578973\pi$
$200$	0	0
$201$	0.535898	0.0377994
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.80385	0.681415
$208$	0	0
$209$	6.92820	0.479234
$210$	0	0
$211$	14.3923i	0.990807i	0.868663	+	0.495404i	$0.164980\pi$
$211$	14.3923i	0.990807i	−0.868663	+	0.495404i	$0.835020\pi$
$212$	0	0
$213$	− 44.7846i	− 3.06859i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	− 17.4641i	− 1.18011i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	9.12436	0.611012	0.305506	−	0.952190i	$-0.401174\pi$
$223$	9.12436	0.611012	0.305506	+	0.952190i	$0.401174\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.5885i	− 0.835525i	−0.908556	−	0.417763i	$-0.862814\pi$
$227$	− 12.5885i	− 0.835525i	0.908556	−	0.417763i	$-0.137186\pi$
$228$	0	0
$229$	− 5.07180i	− 0.335154i	−0.985859	−	0.167577i	$-0.946406\pi$
$229$	− 5.07180i	− 0.335154i	0.985859	−	0.167577i	$-0.0535942\pi$
$230$	0	0
$231$	44.7846	2.94661
$232$	0	0
$233$	22.3923	1.46697	0.733484	−	0.679706i	$-0.237893\pi$
$233$	22.3923	1.46697	0.733484	+	0.679706i	$0.237893\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.7846i	2.12959i
$238$	0	0
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	0.392305	0.0252706	0.0126353	−	0.999920i	$-0.495978\pi$
$241$	0.392305	0.0252706	0.0126353	+	0.999920i	$0.495978\pi$
$242$	0	0
$243$	− 18.7321i	− 1.20166i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.92820	−0.440831
$248$	0	0
$249$	−3.46410	−0.219529
$250$	0	0
$251$	− 8.53590i	− 0.538781i	−0.963031	−	0.269391i	$-0.913178\pi$
$251$	− 8.53590i	− 0.538781i	0.963031	−	0.269391i	$-0.0868223\pi$
$252$	0	0
$253$	7.60770i	0.478292i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	28.3923i	1.76421i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.80385	−0.604531	−0.302266	−	0.953224i	$-0.597743\pi$
$263$	−9.80385	−0.604531	−0.302266	+	0.953224i	$0.597743\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 35.3205i	− 2.16158i
$268$	0	0
$269$	− 26.7846i	− 1.63309i	−0.577284	−	0.816543i	$-0.695888\pi$
$269$	− 26.7846i	− 1.63309i	0.577284	−	0.816543i	$-0.304112\pi$
$270$	0	0
$271$	16.3923	0.995762	0.497881	−	0.867245i	$-0.334112\pi$
$271$	16.3923	0.995762	0.497881	+	0.867245i	$0.334112\pi$
$272$	0	0
$273$	−44.7846	−2.71049
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.85641i	0.472046i	0.971748	+	0.236023i	$0.0758440\pi$
$277$	7.85641i	0.472046i	−0.971748	+	0.236023i	$0.924156\pi$
$278$	0	0
$279$	−11.3205	−0.677741
$280$	0	0
$281$	−7.60770	−0.453837	−0.226919	−	0.973914i	$-0.572865\pi$
$281$	−7.60770	−0.453837	−0.226919	+	0.973914i	$0.572865\pi$
$282$	0	0
$283$	− 20.5885i	− 1.22386i	−0.790913	−	0.611928i	$-0.790394\pi$
$283$	− 20.5885i	− 1.22386i	0.790913	−	0.611928i	$-0.209606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.7846	2.64355
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 39.3205i	− 2.30501i
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−13.8564	−0.804030
$298$	0	0
$299$	− 7.60770i	− 0.439964i
$300$	0	0
$301$	− 0.928203i	− 0.0535007i
$302$	0	0
$303$	32.7846	1.88343
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.8038i	1.35856i	0.733880	+	0.679279i	$0.237707\pi$
$307$	23.8038i	1.35856i	−0.733880	+	0.679279i	$0.762293\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	0	0
$311$	28.3923	1.60998	0.804990	−	0.593288i	$-0.202171\pi$
$311$	28.3923	1.60998	0.804990	+	0.593288i	$0.202171\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.60770i	0.0902972i	0.998980	+	0.0451486i	$0.0143761\pi$
$317$	1.60770i	0.0902972i	−0.998980	+	0.0451486i	$0.985624\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	36.2487	2.02321
$322$	0	0
$323$	6.92820i	0.385496i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−35.3205	−1.95323
$328$	0	0
$329$	10.3923	0.572946
$330$	0	0
$331$	− 26.3923i	− 1.45065i	−0.688405	−	0.725326i	$-0.741689\pi$
$331$	− 26.3923i	− 1.45065i	0.688405	−	0.725326i	$-0.258311\pi$
$332$	0	0
$333$	− 26.7846i	− 1.46779i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	35.3205i	1.91835i
$340$	0	0
$341$	− 8.78461i	− 0.475713i
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.0526i	0.539650i	0.962909	+	0.269825i	$0.0869658\pi$
$347$	10.0526i	0.539650i	−0.962909	+	0.269825i	$0.913034\pi$
$348$	0	0
$349$	32.7846i	1.75492i	0.479650	+	0.877460i	$0.340764\pi$
$349$	32.7846i	1.75492i	−0.479650	+	0.877460i	$0.659236\pi$
$350$	0	0
$351$	13.8564	0.739600
$352$	0	0
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	44.7846i	2.37025i
$358$	0	0
$359$	32.7846	1.73031	0.865153	−	0.501508i	$-0.167221\pi$
$359$	32.7846	1.73031	0.865153	+	0.501508i	$0.167221\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 2.73205i	− 0.143395i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0526	−1.46433	−0.732166	−	0.681126i	$-0.761490\pi$
$367$	−28.0526	−1.46433	−0.732166	+	0.681126i	$0.761490\pi$
$368$	0	0
$369$	−42.2487	−2.19938
$370$	0	0
$371$	49.1769i	2.55314i
$372$	0	0
$373$	− 19.8564i	− 1.02813i	−0.857753	−	0.514063i	$-0.828140\pi$
$373$	− 19.8564i	− 1.02813i	0.857753	−	0.514063i	$-0.171860\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 2.00000i	− 0.102733i	−0.998680	−	0.0513665i	$-0.983642\pi$
$379$	− 2.00000i	− 0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$	0	0
$381$	− 38.7846i	− 1.98700i
$382$	0	0
$383$	14.1962	0.725390	0.362695	−	0.931908i	$-0.381857\pi$
$383$	14.1962	0.725390	0.362695	+	0.931908i	$0.381857\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.875644i	0.0445115i
$388$	0	0
$389$	− 14.7846i	− 0.749609i	−0.927104	−	0.374805i	$-0.877710\pi$
$389$	− 14.7846i	− 0.749609i	0.927104	−	0.374805i	$-0.122290\pi$
$390$	0	0
$391$	−7.60770	−0.384738
$392$	0	0
$393$	28.3923	1.43220
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.5359i	− 1.03067i	−0.856990	−	0.515334i	$-0.827668\pi$
$397$	− 20.5359i	− 1.03067i	0.856990	−	0.515334i	$-0.172332\pi$
$398$	0	0
$399$	−25.8564	−1.29444
$400$	0	0
$401$	−31.8564	−1.59083	−0.795417	−	0.606063i	$-0.792748\pi$
$401$	−31.8564	−1.59083	−0.795417	+	0.606063i	$0.792748\pi$
$402$	0	0
$403$	8.78461i	0.437593i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.7846	1.03025
$408$	0	0
$409$	24.3923	1.20612	0.603061	−	0.797695i	$-0.293948\pi$
$409$	24.3923	1.20612	0.603061	+	0.797695i	$0.293948\pi$
$410$	0	0
$411$	2.53590i	0.125087i
$412$	0	0
$413$	− 28.3923i	− 1.39709i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	27.3205	1.33789
$418$	0	0
$419$	− 12.9282i	− 0.631584i	−0.948828	−	0.315792i	$-0.897730\pi$
$419$	− 12.9282i	− 0.631584i	0.948828	−	0.315792i	$-0.102270\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	−9.80385	−0.476679
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.39230i	0.212559i
$428$	0	0
$429$	32.7846i	1.58286i
$430$	0	0
$431$	−7.60770	−0.366450	−0.183225	−	0.983071i	$-0.558654\pi$
$431$	−7.60770	−0.366450	−0.183225	+	0.983071i	$0.558654\pi$
$432$	0	0
$433$	−5.60770	−0.269489	−0.134744	−	0.990880i	$-0.543021\pi$
$433$	−5.60770	−0.269489	−0.134744	+	0.990880i	$0.543021\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.39230i	− 0.210112i
$438$	0	0
$439$	−5.07180	−0.242064	−0.121032	−	0.992649i	$-0.538620\pi$
$439$	−5.07180	−0.242064	−0.121032	+	0.992649i	$0.538620\pi$
$440$	0	0
$441$	−68.7128	−3.27204
$442$	0	0
$443$	− 17.6603i	− 0.839064i	−0.907741	−	0.419532i	$-0.862194\pi$
$443$	− 17.6603i	− 0.839064i	0.907741	−	0.419532i	$-0.137806\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−49.1769	−2.32599
$448$	0	0
$449$	−9.46410	−0.446639	−0.223319	−	0.974745i	$-0.571689\pi$
$449$	−9.46410	−0.446639	−0.223319	+	0.974745i	$0.571689\pi$
$450$	0	0
$451$	− 32.7846i	− 1.54377i
$452$	0	0
$453$	− 25.8564i	− 1.21484i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.7846	0.878707	0.439353	−	0.898314i	$-0.355208\pi$
$457$	18.7846	0.878707	0.439353	+	0.898314i	$0.355208\pi$
$458$	0	0
$459$	− 13.8564i	− 0.646762i
$460$	0	0
$461$	− 12.0000i	− 0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$	− 12.0000i	− 0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$	0	0
$463$	−26.1962	−1.21744	−0.608719	−	0.793386i	$-0.708316\pi$
$463$	−26.1962	−1.21744	−0.608719	+	0.793386i	$0.708316\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.9808i	1.34107i	0.741878	+	0.670535i	$0.233935\pi$
$467$	28.9808i	1.34107i	−0.741878	+	0.670535i	$0.766065\pi$
$468$	0	0
$469$	− 0.928203i	− 0.0428604i
$470$	0	0
$471$	−35.3205	−1.62748
$472$	0	0
$473$	−0.679492	−0.0312431
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 46.3923i	− 2.12416i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−20.7846	−0.947697
$482$	0	0
$483$	− 28.3923i	− 1.29189i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.8756	0.674080	0.337040	−	0.941490i	$-0.390574\pi$
$487$	14.8756	0.674080	0.337040	+	0.941490i	$0.390574\pi$
$488$	0	0
$489$	44.2487	2.00100
$490$	0	0
$491$	− 1.60770i	− 0.0725543i	−0.999342	−	0.0362771i	$-0.988450\pi$
$491$	− 1.60770i	− 0.0725543i	0.999342	−	0.0362771i	$-0.0115499\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−77.5692	−3.47946
$498$	0	0
$499$	− 43.5692i	− 1.95043i	−0.221268	−	0.975213i	$-0.571020\pi$
$499$	− 43.5692i	− 1.95043i	0.221268	−	0.975213i	$-0.428980\pi$
$500$	0	0
$501$	6.00000i	0.268060i
$502$	0	0
$503$	−2.19615	−0.0979216	−0.0489608	−	0.998801i	$-0.515591\pi$
$503$	−2.19615	−0.0979216	−0.0489608	+	0.998801i	$0.515591\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.73205i	0.121335i
$508$	0	0
$509$	− 32.7846i	− 1.45315i	−0.687086	−	0.726576i	$-0.741110\pi$
$509$	− 32.7846i	− 1.45315i	0.687086	−	0.726576i	$-0.258890\pi$
$510$	0	0
$511$	−30.2487	−1.33812
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.60770i	− 0.334586i
$518$	0	0
$519$	16.3923	0.719542
$520$	0	0
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	0	0
$523$	36.9808i	1.61706i	0.588458	+	0.808528i	$0.299735\pi$
$523$	36.9808i	1.61706i	−0.588458	+	0.808528i	$0.700265\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.78461	0.382664
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	26.7846i	1.16235i
$532$	0	0
$533$	32.7846i	1.42006i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.4641	0.926244
$538$	0	0
$539$	− 53.3205i	− 2.29668i
$540$	0	0
$541$	− 39.7128i	− 1.70739i	−0.520776	−	0.853694i	$-0.674357\pi$
$541$	− 39.7128i	− 1.70739i	0.520776	−	0.853694i	$-0.325643\pi$
$542$	0	0
$543$	18.9282	0.812287
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.1962i	0.521470i	0.965410	+	0.260735i	$0.0839649\pi$
$547$	12.1962i	0.521470i	−0.965410	+	0.260735i	$0.916035\pi$
$548$	0	0
$549$	− 4.14359i	− 0.176844i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	56.7846	2.41473
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26.7846i	− 1.13490i	−0.823408	−	0.567450i	$-0.807930\pi$
$557$	− 26.7846i	− 1.13490i	0.823408	−	0.567450i	$-0.192070\pi$
$558$	0	0
$559$	0.679492	0.0287394
$560$	0	0
$561$	32.7846	1.38417
$562$	0	0
$563$	15.8038i	0.666053i	0.942918	+	0.333026i	$0.108070\pi$
$563$	15.8038i	0.666053i	−0.942918	+	0.333026i	$0.891930\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.6603	−0.489685
$568$	0	0
$569$	6.24871	0.261960	0.130980	−	0.991385i	$-0.458188\pi$
$569$	6.24871	0.261960	0.130980	+	0.991385i	$0.458188\pi$
$570$	0	0
$571$	9.60770i	0.402070i	0.979584	+	0.201035i	$0.0644304\pi$
$571$	9.60770i	0.402070i	−0.979584	+	0.201035i	$0.935570\pi$
$572$	0	0
$573$	12.0000i	0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	39.3205i	1.63410i
$580$	0	0
$581$	6.00000i	0.248922i
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.5167i	− 1.30083i	−0.759578	−	0.650416i	$-0.774595\pi$
$587$	− 31.5167i	− 1.30083i	0.759578	−	0.650416i	$-0.225405\pi$
$588$	0	0
$589$	5.07180i	0.208980i
$590$	0	0
$591$	28.3923	1.16790
$592$	0	0
$593$	−12.9282	−0.530898	−0.265449	−	0.964125i	$-0.585520\pi$
$593$	−12.9282	−0.530898	−0.265449	+	0.964125i	$0.585520\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.9282i	− 0.774680i
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	0.392305	0.0160024	0.00800122	−	0.999968i	$-0.497453\pi$
$601$	0.392305	0.0160024	0.00800122	+	0.999968i	$0.497453\pi$
$602$	0	0
$603$	0.875644i	0.0356590i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.19615	0.0891391	0.0445695	−	0.999006i	$-0.485808\pi$
$607$	2.19615	0.0891391	0.0445695	+	0.999006i	$0.485808\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.60770i	0.307774i
$612$	0	0
$613$	− 34.3923i	− 1.38909i	−0.719448	−	0.694546i	$-0.755605\pi$
$613$	− 34.3923i	− 1.38909i	0.719448	−	0.694546i	$-0.244395\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.3923	−1.38458	−0.692291	−	0.721618i	$-0.743399\pi$
$617$	−34.3923	−1.38458	−0.692291	+	0.721618i	$0.743399\pi$
$618$	0	0
$619$	34.7846i	1.39811i	0.715067	+	0.699056i	$0.246396\pi$
$619$	34.7846i	1.39811i	−0.715067	+	0.699056i	$0.753604\pi$
$620$	0	0
$621$	8.78461i	0.352514i
$622$	0	0
$623$	−61.1769	−2.45100
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.9282i	0.755920i
$628$	0	0
$629$	20.7846i	0.828737i
$630$	0	0
$631$	−14.5359	−0.578665	−0.289332	−	0.957229i	$-0.593433\pi$
$631$	−14.5359	−0.578665	−0.289332	+	0.957229i	$0.593433\pi$
$632$	0	0
$633$	−39.3205	−1.56285
$634$	0	0
$635$	0	0
$636$	0	0
$637$	53.3205i	2.11264i
$638$	0	0
$639$	73.1769	2.89483
$640$	0	0
$641$	16.3923	0.647457	0.323729	−	0.946150i	$-0.395064\pi$
$641$	16.3923	0.647457	0.323729	+	0.946150i	$0.395064\pi$
$642$	0	0
$643$	− 20.5885i	− 0.811929i	−0.913889	−	0.405965i	$-0.866936\pi$
$643$	− 20.5885i	− 0.811929i	0.913889	−	0.405965i	$-0.133064\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.41154	−0.212750	−0.106375	−	0.994326i	$-0.533924\pi$
$647$	−5.41154	−0.212750	−0.106375	+	0.994326i	$0.533924\pi$
$648$	0	0
$649$	−20.7846	−0.815867
$650$	0	0
$651$	32.7846i	1.28493i
$652$	0	0
$653$	43.1769i	1.68964i	0.535048	+	0.844822i	$0.320293\pi$
$653$	43.1769i	1.68964i	−0.535048	+	0.844822i	$0.679707\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.5359	1.11329
$658$	0	0
$659$	28.6410i	1.11570i	0.829943	+	0.557848i	$0.188373\pi$
$659$	28.6410i	1.11570i	−0.829943	+	0.557848i	$0.811627\pi$
$660$	0	0
$661$	47.5692i	1.85023i	0.379689	+	0.925114i	$0.376031\pi$
$661$	47.5692i	1.85023i	−0.379689	+	0.925114i	$0.623969\pi$
$662$	0	0
$663$	−32.7846	−1.27325
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	24.9282i	0.963780i
$670$	0	0
$671$	3.21539	0.124129
$672$	0	0
$673$	−23.1769	−0.893404	−0.446702	−	0.894683i	$-0.647402\pi$
$673$	−23.1769	−0.893404	−0.446702	+	0.894683i	$0.647402\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19.1769i	− 0.737029i	−0.929622	−	0.368514i	$-0.879867\pi$
$677$	− 19.1769i	− 0.737029i	0.929622	−	0.368514i	$-0.120133\pi$
$678$	0	0
$679$	−68.1051	−2.61363
$680$	0	0
$681$	34.3923	1.31792
$682$	0	0
$683$	5.66025i	0.216584i	0.994119	+	0.108292i	$0.0345381\pi$
$683$	5.66025i	0.216584i	−0.994119	+	0.108292i	$0.965462\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.8564	0.528655
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	− 26.3923i	− 1.00401i	−0.864865	−	0.502005i	$-0.832596\pi$
$691$	− 26.3923i	− 1.00401i	0.864865	−	0.502005i	$-0.167404\pi$
$692$	0	0
$693$	73.1769i	2.77976i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	32.7846	1.24181
$698$	0	0
$699$	61.1769i	2.31392i
$700$	0	0
$701$	26.7846i	1.01164i	0.862639	+	0.505820i	$0.168810\pi$
$701$	26.7846i	1.01164i	−0.862639	+	0.505820i	$0.831190\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 56.7846i	− 2.13561i
$708$	0	0
$709$	37.8564i	1.42173i	0.703330	+	0.710864i	$0.251696\pi$
$709$	37.8564i	1.42173i	−0.703330	+	0.710864i	$0.748304\pi$
$710$	0	0
$711$	−53.5692	−2.00900
$712$	0	0
$713$	−5.56922	−0.208569
$714$	0	0
$715$	0	0
$716$	0	0
$717$	56.7846i	2.12066i
$718$	0	0
$719$	−3.21539	−0.119914	−0.0599569	−	0.998201i	$-0.519096\pi$
$719$	−3.21539	−0.119914	−0.0599569	+	0.998201i	$0.519096\pi$
$720$	0	0
$721$	−10.3923	−0.387030
$722$	0	0
$723$	1.07180i	0.0398606i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.05256	−0.150301	−0.0751505	−	0.997172i	$-0.523944\pi$
$727$	−4.05256	−0.150301	−0.0751505	+	0.997172i	$0.523944\pi$
$728$	0	0
$729$	43.7846	1.62165
$730$	0	0
$731$	− 0.679492i	− 0.0251319i
$732$	0	0
$733$	2.78461i	0.102852i	0.998677	+	0.0514260i	$0.0163766\pi$
$733$	2.78461i	0.102852i	−0.998677	+	0.0514260i	$0.983623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.679492	−0.0250294
$738$	0	0
$739$	− 2.00000i	− 0.0735712i	−0.999323	−	0.0367856i	$-0.988288\pi$
$739$	− 2.00000i	− 0.0735712i	0.999323	−	0.0367856i	$-0.0117119\pi$
$740$	0	0
$741$	− 18.9282i	− 0.695345i
$742$	0	0
$743$	5.41154	0.198530	0.0992651	−	0.995061i	$-0.468351\pi$
$743$	5.41154	0.198530	0.0992651	+	0.995061i	$0.468351\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.66025i	− 0.207098i
$748$	0	0
$749$	− 62.7846i	− 2.29410i
$750$	0	0
$751$	19.6077	0.715495	0.357747	−	0.933818i	$-0.383545\pi$
$751$	19.6077	0.715495	0.357747	+	0.933818i	$0.383545\pi$
$752$	0	0
$753$	23.3205	0.849847
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.9282i	1.34218i	0.741377	+	0.671089i	$0.234173\pi$
$757$	36.9282i	1.34218i	−0.741377	+	0.671089i	$0.765827\pi$
$758$	0	0
$759$	−20.7846	−0.754434
$760$	0	0
$761$	33.7128	1.22209	0.611044	−	0.791596i	$-0.290750\pi$
$761$	33.7128	1.22209	0.611044	+	0.791596i	$0.290750\pi$
$762$	0	0
$763$	61.1769i	2.21475i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.7846	0.750489
$768$	0	0
$769$	34.7846	1.25437	0.627183	−	0.778872i	$-0.284208\pi$
$769$	34.7846	1.25437	0.627183	+	0.778872i	$0.284208\pi$
$770$	0	0
$771$	− 16.3923i	− 0.590354i
$772$	0	0
$773$	− 25.6077i	− 0.921045i	−0.887648	−	0.460522i	$-0.847662\pi$
$773$	− 25.6077i	− 0.921045i	0.887648	−	0.460522i	$-0.152338\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−77.5692	−2.78278
$778$	0	0
$779$	18.9282i	0.678173i
$780$	0	0
$781$	56.7846i	2.03191i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16.1962i	− 0.577330i	−0.957430	−	0.288665i	$-0.906789\pi$
$787$	− 16.1962i	− 0.577330i	0.957430	−	0.288665i	$-0.0932115\pi$
$788$	0	0
$789$	− 26.7846i	− 0.953557i
$790$	0	0
$791$	61.1769	2.17520
$792$	0	0
$793$	−3.21539	−0.114182
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.3923i	− 0.793176i	−0.917997	−	0.396588i	$-0.870194\pi$
$797$	− 22.3923i	− 0.793176i	0.917997	−	0.396588i	$-0.129806\pi$
$798$	0	0
$799$	7.60770	0.269141
$800$	0	0
$801$	57.7128	2.03918
$802$	0	0
$803$	22.1436i	0.781430i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	73.1769	2.57595
$808$	0	0
$809$	−47.5692	−1.67244	−0.836222	−	0.548391i	$-0.815241\pi$
$809$	−47.5692	−1.67244	−0.836222	+	0.548391i	$0.815241\pi$
$810$	0	0
$811$	17.6077i	0.618290i	0.951015	+	0.309145i	$0.100043\pi$
$811$	17.6077i	0.618290i	−0.951015	+	0.309145i	$0.899957\pi$
$812$	0	0
$813$	44.7846i	1.57066i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.392305	0.0137250
$818$	0	0
$819$	− 73.1769i	− 2.55701i
$820$	0	0
$821$	− 9.21539i	− 0.321619i	−0.986985	−	0.160810i	$-0.948589\pi$
$821$	− 9.21539i	− 0.321619i	0.986985	−	0.160810i	$-0.0514105\pi$
$822$	0	0
$823$	−48.8372	−1.70236	−0.851178	−	0.524877i	$-0.824111\pi$
$823$	−48.8372	−1.70236	−0.851178	+	0.524877i	$0.824111\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.26795i	0.0440909i	0.999757	+	0.0220455i	$0.00701786\pi$
$827$	1.26795i	0.0440909i	−0.999757	+	0.0220455i	$0.992982\pi$
$828$	0	0
$829$	− 9.21539i	− 0.320064i	−0.987112	−	0.160032i	$-0.948840\pi$
$829$	− 9.21539i	− 0.320064i	0.987112	−	0.160032i	$-0.0511597\pi$
$830$	0	0
$831$	−21.4641	−0.744581
$832$	0	0
$833$	53.3205	1.84745
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10.1436i	− 0.350614i
$838$	0	0
$839$	32.7846	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$839$	32.7846	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	− 20.7846i	− 0.715860i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.73205	−0.162595
$848$	0	0
$849$	56.2487	1.93045
$850$	0	0
$851$	− 13.1769i	− 0.451699i
$852$	0	0
$853$	3.46410i	0.118609i	0.998240	+	0.0593043i	$0.0188882\pi$
$853$	3.46410i	0.118609i	−0.998240	+	0.0593043i	$0.981112\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.8564	1.49811	0.749053	−	0.662510i	$-0.230509\pi$
$857$	43.8564	1.49811	0.749053	+	0.662510i	$0.230509\pi$
$858$	0	0
$859$	− 31.5692i	− 1.07713i	−0.842585	−	0.538564i	$-0.818967\pi$
$859$	− 31.5692i	− 1.07713i	0.842585	−	0.538564i	$-0.181033\pi$
$860$	0	0
$861$	122.354i	4.16981i
$862$	0	0
$863$	−10.9808	−0.373789	−0.186895	−	0.982380i	$-0.559842\pi$
$863$	−10.9808	−0.373789	−0.186895	+	0.982380i	$0.559842\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.6603i	− 0.463927i
$868$	0	0
$869$	− 41.5692i	− 1.41014i
$870$	0	0
$871$	0.679492	0.0230237
$872$	0	0
$873$	64.2487	2.17449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.8564i	− 0.670503i	−0.942129	−	0.335252i	$-0.891179\pi$
$877$	− 19.8564i	− 0.670503i	0.942129	−	0.335252i	$-0.108821\pi$
$878$	0	0
$879$	−81.9615	−2.76449
$880$	0	0
$881$	−28.3923	−0.956561	−0.478281	−	0.878207i	$-0.658740\pi$
$881$	−28.3923	−0.956561	−0.478281	+	0.878207i	$0.658740\pi$
$882$	0	0
$883$	− 16.1962i	− 0.545044i	−0.962150	−	0.272522i	$-0.912142\pi$
$883$	− 16.1962i	− 0.545044i	0.962150	−	0.272522i	$-0.0878577\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	51.3731	1.72494	0.862469	−	0.506109i	$-0.168917\pi$
$887$	51.3731	1.72494	0.862469	+	0.506109i	$0.168917\pi$
$888$	0	0
$889$	−67.1769	−2.25304
$890$	0	0
$891$	8.53590i	0.285963i
$892$	0	0
$893$	4.39230i	0.146983i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.7846	0.693978
$898$	0	0
$899$	0	0
$900$	0	0
$901$	36.0000i	1.19933i
$902$	0	0
$903$	2.53590	0.0843894
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.5885i	1.08208i	0.840996	+	0.541041i	$0.181970\pi$
$907$	32.5885i	1.08208i	−0.840996	+	0.541041i	$0.818030\pi$
$908$	0	0
$909$	53.5692i	1.77678i
$910$	0	0
$911$	−25.1769	−0.834148	−0.417074	−	0.908872i	$-0.636944\pi$
$911$	−25.1769	−0.834148	−0.417074	+	0.908872i	$0.636944\pi$
$912$	0	0
$913$	4.39230	0.145364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 49.1769i	− 1.62396i
$918$	0	0
$919$	15.7128	0.518318	0.259159	−	0.965835i	$-0.416555\pi$
$919$	15.7128	0.518318	0.259159	+	0.965835i	$0.416555\pi$
$920$	0	0
$921$	−65.0333	−2.14292
$922$	0	0
$923$	− 56.7846i	− 1.86909i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.80385	0.322001
$928$	0	0
$929$	28.3923	0.931521	0.465761	−	0.884911i	$-0.345781\pi$
$929$	28.3923	0.931521	0.465761	+	0.884911i	$0.345781\pi$
$930$	0	0
$931$	30.7846i	1.00892i
$932$	0	0
$933$	77.5692i	2.53950i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.3923	0.992873	0.496437	−	0.868073i	$-0.334641\pi$
$937$	30.3923	0.992873	0.496437	+	0.868073i	$0.334641\pi$
$938$	0	0
$939$	− 60.1051i	− 1.96146i
$940$	0	0
$941$	− 20.7846i	− 0.677559i	−0.940866	−	0.338779i	$-0.889986\pi$
$941$	− 20.7846i	− 0.677559i	0.940866	−	0.338779i	$-0.110014\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 56.1962i	− 1.82613i	−0.407815	−	0.913065i	$-0.633709\pi$
$947$	− 56.1962i	− 1.82613i	0.407815	−	0.913065i	$-0.366291\pi$
$948$	0	0
$949$	− 22.1436i	− 0.718811i
$950$	0	0
$951$	−4.39230	−0.142430
$952$	0	0
$953$	−11.0718	−0.358651	−0.179325	−	0.983790i	$-0.557391\pi$
$953$	−11.0718	−0.358651	−0.179325	+	0.983790i	$0.557391\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.39230	0.141835
$960$	0	0
$961$	−24.5692	−0.792555
$962$	0	0
$963$	59.2295i	1.90864i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.19615	0.0706235	0.0353118	−	0.999376i	$-0.488758\pi$
$967$	2.19615	0.0706235	0.0353118	+	0.999376i	$0.488758\pi$
$968$	0	0
$969$	−18.9282	−0.608061
$970$	0	0
$971$	− 57.0333i	− 1.83029i	−0.403129	−	0.915143i	$-0.632077\pi$
$971$	− 57.0333i	− 1.83029i	0.403129	−	0.915143i	$-0.367923\pi$
$972$	0	0
$973$	− 47.3205i	− 1.51703i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.3205	0.938046	0.469023	−	0.883186i	$-0.344606\pi$
$977$	29.3205	0.938046	0.469023	+	0.883186i	$0.344606\pi$
$978$	0	0
$979$	44.7846i	1.43132i
$980$	0	0
$981$	− 57.7128i	− 1.84263i
$982$	0	0
$983$	−42.5885	−1.35836	−0.679180	−	0.733971i	$-0.737665\pi$
$983$	−42.5885	−1.35836	−0.679180	+	0.733971i	$0.737665\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28.3923i	0.903737i
$988$	0	0
$989$	0.430781i	0.0136980i
$990$	0	0
$991$	32.1051	1.01985	0.509926	−	0.860218i	$-0.329673\pi$
$991$	32.1051	1.01985	0.509926	+	0.860218i	$0.329673\pi$
$992$	0	0
$993$	72.1051	2.28819
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 46.3923i	− 1.46926i	−0.678468	−	0.734630i	$-0.737356\pi$
$997$	− 46.3923i	− 1.46926i	0.678468	−	0.734630i	$-0.262644\pi$
$998$	0	0
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.h.801.4		4
4.3	odd	2		1600.2.d.b.801.1		4
5.2	odd	4		1600.2.f.i.1249.4		4
5.3	odd	4		1600.2.f.e.1249.1		4
5.4	even	2		320.2.d.a.161.1	✓	4
8.3	odd	2		1600.2.d.b.801.4		4
8.5	even	2	inner	1600.2.d.h.801.1		4
15.14	odd	2		2880.2.k.e.1441.1		4
16.3	odd	4		6400.2.a.ck.1.2		2
16.5	even	4		6400.2.a.cd.1.2		2
16.11	odd	4		6400.2.a.bf.1.1		2
16.13	even	4		6400.2.a.y.1.1		2
20.3	even	4		1600.2.f.h.1249.4		4
20.7	even	4		1600.2.f.d.1249.1		4
20.19	odd	2		320.2.d.b.161.4	yes	4
40.3	even	4		1600.2.f.d.1249.2		4
40.13	odd	4		1600.2.f.i.1249.3		4
40.19	odd	2		320.2.d.b.161.1	yes	4
40.27	even	4		1600.2.f.h.1249.3		4
40.29	even	2		320.2.d.a.161.4	yes	4
40.37	odd	4		1600.2.f.e.1249.2		4
60.59	even	2		2880.2.k.l.1441.2		4
80.19	odd	4		1280.2.a.c.1.1		2
80.29	even	4		1280.2.a.p.1.2		2
80.59	odd	4		1280.2.a.m.1.2		2
80.69	even	4		1280.2.a.b.1.1		2
120.29	odd	2		2880.2.k.e.1441.3		4
120.59	even	2		2880.2.k.l.1441.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.d.a.161.1	✓	4	5.4	even	2
320.2.d.a.161.4	yes	4	40.29	even	2
320.2.d.b.161.1	yes	4	40.19	odd	2
320.2.d.b.161.4	yes	4	20.19	odd	2
1280.2.a.b.1.1		2	80.69	even	4
1280.2.a.c.1.1		2	80.19	odd	4
1280.2.a.m.1.2		2	80.59	odd	4
1280.2.a.p.1.2		2	80.29	even	4
1600.2.d.b.801.1		4	4.3	odd	2
1600.2.d.b.801.4		4	8.3	odd	2
1600.2.d.h.801.1		4	8.5	even	2	inner
1600.2.d.h.801.4		4	1.1	even	1	trivial
1600.2.f.d.1249.1		4	20.7	even	4
1600.2.f.d.1249.2		4	40.3	even	4
1600.2.f.e.1249.1		4	5.3	odd	4
1600.2.f.e.1249.2		4	40.37	odd	4
1600.2.f.h.1249.3		4	40.27	even	4
1600.2.f.h.1249.4		4	20.3	even	4
1600.2.f.i.1249.3		4	40.13	odd	4
1600.2.f.i.1249.4		4	5.2	odd	4
2880.2.k.e.1441.1		4	15.14	odd	2
2880.2.k.e.1441.3		4	120.29	odd	2
2880.2.k.l.1441.2		4	60.59	even	2
2880.2.k.l.1441.4		4	120.59	even	2
6400.2.a.y.1.1		2	16.13	even	4
6400.2.a.bf.1.1		2	16.11	odd	4
6400.2.a.cd.1.2		2	16.5	even	4
6400.2.a.ck.1.2		2	16.3	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.73205i q^{3} +4.73205 q^{7} -4.46410 q^{9} +O(q^{10})\)	\(q+2.73205i q^{3} +4.73205 q^{7} -4.46410 q^{9} -3.46410i q^{11} +3.46410i q^{13} +3.46410 q^{17} +2.00000i q^{19} +12.9282i q^{21} -2.19615 q^{23} -4.00000i q^{27} +2.53590 q^{31} +9.46410 q^{33} +6.00000i q^{37} -9.46410 q^{39} +9.46410 q^{41} -0.196152i q^{43} +2.19615 q^{47} +15.3923 q^{49} +9.46410i q^{51} +10.3923i q^{53} -5.46410 q^{57} -6.00000i q^{59} +0.928203i q^{61} -21.1244 q^{63} -0.196152i q^{67} -6.00000i q^{69} -16.3923 q^{71} -6.39230 q^{73} -16.3923i q^{77} +12.0000 q^{79} -2.46410 q^{81} +1.26795i q^{83} -12.9282 q^{89} +16.3923i q^{91} +6.92820i q^{93} -14.3923 q^{97} +15.4641i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 12 * q^7 - 4 * q^9	\( 4 q + 12 q^{7} - 4 q^{9} + 12 q^{23} + 24 q^{31} + 24 q^{33} - 24 q^{39} + 24 q^{41} - 12 q^{47} + 20 q^{49} - 8 q^{57} - 36 q^{63} - 24 q^{71} + 16 q^{73} + 48 q^{79} + 4 q^{81} - 24 q^{89} - 16 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 - 4 * q^9 + 12 * q^23 + 24 * q^31 + 24 * q^33 - 24 * q^39 + 24 * q^41 - 12 * q^47 + 20 * q^49 - 8 * q^57 - 36 * q^63 - 24 * q^71 + 16 * q^73 + 48 * q^79 + 4 * q^81 - 24 * q^89 - 16 * q^97

Introduction

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