LMFDB - Embedded newform 400.3.h.b.399.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,3,Mod(399,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("400.399");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	400.h (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:	$10.8992105744$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	400.399
Dual form			400.3.h.b.399.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +3.46410 q^{7} -6.00000 q^{9} +12.1244i q^{11} -16.0000i q^{13} -3.00000i q^{17} +22.5167i q^{19} -6.00000 q^{21} -45.0333 q^{23} +25.9808 q^{27} -12.0000 q^{29} +31.1769i q^{31} -21.0000i q^{33} +50.0000i q^{37} +27.7128i q^{39} -63.0000 q^{41} -62.3538 q^{43} -13.8564 q^{47} -37.0000 q^{49} +5.19615i q^{51} -18.0000i q^{53} -39.0000i q^{57} +55.4256i q^{59} +26.0000 q^{61} -20.7846 q^{63} +32.9090 q^{67} +78.0000 q^{69} +27.7128i q^{71} +17.0000i q^{73} +42.0000i q^{77} -86.6025i q^{79} +9.00000 q^{81} +98.7269 q^{83} +20.7846 q^{87} -99.0000 q^{89} -55.4256i q^{91} -54.0000i q^{93} -134.000i q^{97} -72.7461i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{9} - 24 q^{21} - 48 q^{29} - 252 q^{41} - 148 q^{49} + 104 q^{61} + 312 q^{69} + 36 q^{81} - 396 q^{89}+O(q^{100}) $ 4 * q - 24 * q^9 - 24 * q^21 - 48 * q^29 - 252 * q^41 - 148 * q^49 + 104 * q^61 + 312 * q^69 + 36 * q^81 - 396 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−1.73205	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.73205	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	0.494872	0.247436	−	0.968904i	$-0.420412\pi$
$7$	3.46410	0.494872	0.247436	+	0.968904i	$0.420412\pi$
$8$	0	0
$9$	−6.00000	−0.666667
$10$	0	0
$11$	12.1244i	1.10221i	0.834435	+	0.551107i	$0.185794\pi$
$11$	12.1244i	1.10221i	−0.834435	+	0.551107i	$0.814206\pi$
$12$	0	0
$13$	− 16.0000i	− 1.23077i	−0.788227	−	0.615385i	$-0.789001\pi$
$13$	− 16.0000i	− 1.23077i	0.788227	−	0.615385i	$-0.210999\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.00000i	− 0.176471i	−0.996100	−	0.0882353i	$-0.971877\pi$
$17$	− 3.00000i	− 0.176471i	0.996100	−	0.0882353i	$-0.0281227\pi$
$18$	0	0
$19$	22.5167i	1.18509i	0.805538	+	0.592544i	$0.201876\pi$
$19$	22.5167i	1.18509i	−0.805538	+	0.592544i	$0.798124\pi$
$20$	0	0
$21$	−6.00000	−0.285714
$22$	0	0
$23$	−45.0333	−1.95797	−0.978985	−	0.203931i	$-0.934628\pi$
$23$	−45.0333	−1.95797	−0.978985	+	0.203931i	$0.934628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	25.9808	0.962250
$28$	0	0
$29$	−12.0000	−0.413793	−0.206897	−	0.978363i	$-0.566336\pi$
$29$	−12.0000	−0.413793	−0.206897	+	0.978363i	$0.566336\pi$
$30$	0	0
$31$	31.1769i	1.00571i	0.864372	+	0.502853i	$0.167716\pi$
$31$	31.1769i	1.00571i	−0.864372	+	0.502853i	$0.832284\pi$
$32$	0	0
$33$	− 21.0000i	− 0.636364i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	50.0000i	1.35135i	0.737199	+	0.675676i	$0.236148\pi$
$37$	50.0000i	1.35135i	−0.737199	+	0.675676i	$0.763852\pi$
$38$	0	0
$39$	27.7128i	0.710585i
$40$	0	0
$41$	−63.0000	−1.53659	−0.768293	−	0.640099i	$-0.778893\pi$
$41$	−63.0000	−1.53659	−0.768293	+	0.640099i	$0.778893\pi$
$42$	0	0
$43$	−62.3538	−1.45009	−0.725045	−	0.688702i	$-0.758181\pi$
$43$	−62.3538	−1.45009	−0.725045	+	0.688702i	$0.758181\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.8564	−0.294817	−0.147409	−	0.989076i	$-0.547093\pi$
$47$	−13.8564	−0.294817	−0.147409	+	0.989076i	$0.547093\pi$
$48$	0	0
$49$	−37.0000	−0.755102
$50$	0	0
$51$	5.19615i	0.101885i
$52$	0	0
$53$	− 18.0000i	− 0.339623i	−0.985477	−	0.169811i	$-0.945684\pi$
$53$	− 18.0000i	− 0.339623i	0.985477	−	0.169811i	$-0.0543158\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 39.0000i	− 0.684211i
$58$	0	0
$59$	55.4256i	0.939417i	0.882821	+	0.469709i	$0.155641\pi$
$59$	55.4256i	0.939417i	−0.882821	+	0.469709i	$0.844359\pi$
$60$	0	0
$61$	26.0000	0.426230	0.213115	−	0.977027i	$-0.431639\pi$
$61$	26.0000	0.426230	0.213115	+	0.977027i	$0.431639\pi$
$62$	0	0
$63$	−20.7846	−0.329914
$64$	0	0
$65$	0	0
$66$	0	0
$67$	32.9090	0.491179	0.245589	−	0.969374i	$-0.421019\pi$
$67$	32.9090	0.491179	0.245589	+	0.969374i	$0.421019\pi$
$68$	0	0
$69$	78.0000	1.13043
$70$	0	0
$71$	27.7128i	0.390321i	0.980771	+	0.195161i	$0.0625228\pi$
$71$	27.7128i	0.390321i	−0.980771	+	0.195161i	$0.937477\pi$
$72$	0	0
$73$	17.0000i	0.232877i	0.993198	+	0.116438i	$0.0371477\pi$
$73$	17.0000i	0.232877i	−0.993198	+	0.116438i	$0.962852\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42.0000i	0.545455i
$78$	0	0
$79$	− 86.6025i	− 1.09623i	−0.836401	−	0.548117i	$-0.815345\pi$
$79$	− 86.6025i	− 1.09623i	0.836401	−	0.548117i	$-0.184655\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	98.7269	1.18948	0.594740	−	0.803918i	$-0.297255\pi$
$83$	98.7269	1.18948	0.594740	+	0.803918i	$0.297255\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	20.7846	0.238904
$88$	0	0
$89$	−99.0000	−1.11236	−0.556180	−	0.831062i	$-0.687733\pi$
$89$	−99.0000	−1.11236	−0.556180	+	0.831062i	$0.687733\pi$
$90$	0	0
$91$	− 55.4256i	− 0.609073i
$92$	0	0
$93$	− 54.0000i	− 0.580645i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 134.000i	− 1.38144i	−0.723121	−	0.690722i	$-0.757293\pi$
$97$	− 134.000i	− 1.38144i	0.723121	−	0.690722i	$-0.242707\pi$
$98$	0	0
$99$	− 72.7461i	− 0.734809i
$100$	0	0
$101$	150.000	1.48515	0.742574	−	0.669764i	$-0.233605\pi$
$101$	150.000	1.48515	0.742574	+	0.669764i	$0.233605\pi$
$102$	0	0
$103$	110.851	1.07623	0.538113	−	0.842873i	$-0.319137\pi$
$103$	110.851	1.07623	0.538113	+	0.842873i	$0.319137\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−164.545	−1.53780	−0.768901	−	0.639368i	$-0.779196\pi$
$107$	−164.545	−1.53780	−0.768901	+	0.639368i	$0.779196\pi$
$108$	0	0
$109$	74.0000	0.678899	0.339450	−	0.940624i	$-0.389759\pi$
$109$	74.0000	0.678899	0.339450	+	0.940624i	$0.389759\pi$
$110$	0	0
$111$	− 86.6025i	− 0.780203i
$112$	0	0
$113$	201.000i	1.77876i	0.457168	+	0.889381i	$0.348864\pi$
$113$	201.000i	1.77876i	−0.457168	+	0.889381i	$0.651136\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	96.0000i	0.820513i
$118$	0	0
$119$	− 10.3923i	− 0.0873303i
$120$	0	0
$121$	−26.0000	−0.214876
$122$	0	0
$123$	109.119	0.887148
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.3923	0.0818292	0.0409146	−	0.999163i	$-0.486973\pi$
$127$	10.3923	0.0818292	0.0409146	+	0.999163i	$0.486973\pi$
$128$	0	0
$129$	108.000	0.837209
$130$	0	0
$131$	6.92820i	0.0528870i	0.999650	+	0.0264435i	$0.00841822\pi$
$131$	6.92820i	0.0528870i	−0.999650	+	0.0264435i	$0.991582\pi$
$132$	0	0
$133$	78.0000i	0.586466i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 207.000i	− 1.51095i	−0.655178	−	0.755474i	$-0.727406\pi$
$137$	− 207.000i	− 1.51095i	0.655178	−	0.755474i	$-0.272594\pi$
$138$	0	0
$139$	− 178.401i	− 1.28346i	−0.766930	−	0.641731i	$-0.778217\pi$
$139$	− 178.401i	− 1.28346i	0.766930	−	0.641731i	$-0.221783\pi$
$140$	0	0
$141$	24.0000	0.170213
$142$	0	0
$143$	193.990	1.35657
$144$	0	0
$145$	0	0
$146$	0	0
$147$	64.0859	0.435958
$148$	0	0
$149$	108.000	0.724832	0.362416	−	0.932016i	$-0.381952\pi$
$149$	108.000	0.724832	0.362416	+	0.932016i	$0.381952\pi$
$150$	0	0
$151$	− 266.736i	− 1.76646i	−0.468938	−	0.883231i	$-0.655363\pi$
$151$	− 266.736i	− 1.76646i	0.468938	−	0.883231i	$-0.344637\pi$
$152$	0	0
$153$	18.0000i	0.117647i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	278.000i	1.77070i	0.464925	+	0.885350i	$0.346081\pi$
$157$	278.000i	1.77070i	−0.464925	+	0.885350i	$0.653919\pi$
$158$	0	0
$159$	31.1769i	0.196081i
$160$	0	0
$161$	−156.000	−0.968944
$162$	0	0
$163$	−19.0526	−0.116887	−0.0584434	−	0.998291i	$-0.518614\pi$
$163$	−19.0526	−0.116887	−0.0584434	+	0.998291i	$0.518614\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−263.272	−1.57648	−0.788239	−	0.615370i	$-0.789007\pi$
$167$	−263.272	−1.57648	−0.788239	+	0.615370i	$0.789007\pi$
$168$	0	0
$169$	−87.0000	−0.514793
$170$	0	0
$171$	− 135.100i	− 0.790058i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 96.0000i	− 0.542373i
$178$	0	0
$179$	− 12.1244i	− 0.0677338i	−0.999426	−	0.0338669i	$-0.989218\pi$
$179$	− 12.1244i	− 0.0677338i	0.999426	−	0.0338669i	$-0.0107822\pi$
$180$	0	0
$181$	166.000	0.917127	0.458564	−	0.888662i	$-0.348364\pi$
$181$	166.000	0.917127	0.458564	+	0.888662i	$0.348364\pi$
$182$	0	0
$183$	−45.0333	−0.246084
$184$	0	0
$185$	0	0
$186$	0	0
$187$	36.3731	0.194508
$188$	0	0
$189$	90.0000	0.476190
$190$	0	0
$191$	301.377i	1.57789i	0.614464	+	0.788945i	$0.289372\pi$
$191$	301.377i	1.57789i	−0.614464	+	0.788945i	$0.710628\pi$
$192$	0	0
$193$	139.000i	0.720207i	0.932912	+	0.360104i	$0.117259\pi$
$193$	139.000i	0.720207i	−0.932912	+	0.360104i	$0.882741\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 126.000i	− 0.639594i	−0.947486	−	0.319797i	$-0.896385\pi$
$197$	− 126.000i	− 0.639594i	0.947486	−	0.319797i	$-0.103615\pi$
$198$	0	0
$199$	76.2102i	0.382966i	0.981496	+	0.191483i	$0.0613297\pi$
$199$	76.2102i	0.382966i	−0.981496	+	0.191483i	$0.938670\pi$
$200$	0	0
$201$	−57.0000	−0.283582
$202$	0	0
$203$	−41.5692	−0.204774
$204$	0	0
$205$	0	0
$206$	0	0
$207$	270.200	1.30531
$208$	0	0
$209$	−273.000	−1.30622
$210$	0	0
$211$	188.794i	0.894756i	0.894345	+	0.447378i	$0.147642\pi$
$211$	188.794i	0.894756i	−0.894345	+	0.447378i	$0.852358\pi$
$212$	0	0
$213$	− 48.0000i	− 0.225352i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	108.000i	0.497696i
$218$	0	0
$219$	− 29.4449i	− 0.134451i
$220$	0	0
$221$	−48.0000	−0.217195
$222$	0	0
$223$	−145.492	−0.652432	−0.326216	−	0.945295i	$-0.605774\pi$
$223$	−145.492	−0.652432	−0.326216	+	0.945295i	$0.605774\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−173.205	−0.763018	−0.381509	−	0.924365i	$-0.624595\pi$
$227$	−173.205	−0.763018	−0.381509	+	0.924365i	$0.624595\pi$
$228$	0	0
$229$	140.000	0.611354	0.305677	−	0.952135i	$-0.401117\pi$
$229$	140.000	0.611354	0.305677	+	0.952135i	$0.401117\pi$
$230$	0	0
$231$	− 72.7461i	− 0.314918i
$232$	0	0
$233$	− 366.000i	− 1.57082i	−0.618979	−	0.785408i	$-0.712453\pi$
$233$	− 366.000i	− 1.57082i	0.618979	−	0.785408i	$-0.287547\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	150.000i	0.632911i
$238$	0	0
$239$	256.344i	1.07257i	0.844038	+	0.536284i	$0.180172\pi$
$239$	256.344i	1.07257i	−0.844038	+	0.536284i	$0.819828\pi$
$240$	0	0
$241$	73.0000	0.302905	0.151452	−	0.988465i	$-0.451605\pi$
$241$	73.0000	0.302905	0.151452	+	0.988465i	$0.451605\pi$
$242$	0	0
$243$	−249.415	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	360.267	1.45857
$248$	0	0
$249$	−171.000	−0.686747
$250$	0	0
$251$	168.009i	0.669358i	0.942332	+	0.334679i	$0.108628\pi$
$251$	168.009i	0.669358i	−0.942332	+	0.334679i	$0.891372\pi$
$252$	0	0
$253$	− 546.000i	− 2.15810i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 282.000i	− 1.09728i	−0.836060	−	0.548638i	$-0.815146\pi$
$257$	− 282.000i	− 1.09728i	0.836060	−	0.548638i	$-0.184854\pi$
$258$	0	0
$259$	173.205i	0.668745i
$260$	0	0
$261$	72.0000	0.275862
$262$	0	0
$263$	−169.741	−0.645403	−0.322701	−	0.946501i	$-0.604591\pi$
$263$	−169.741	−0.645403	−0.322701	+	0.946501i	$0.604591\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	171.473	0.642221
$268$	0	0
$269$	−288.000	−1.07063	−0.535316	−	0.844652i	$-0.679807\pi$
$269$	−288.000	−1.07063	−0.535316	+	0.844652i	$0.679807\pi$
$270$	0	0
$271$	169.741i	0.626350i	0.949695	+	0.313175i	$0.101393\pi$
$271$	169.741i	0.626350i	−0.949695	+	0.313175i	$0.898607\pi$
$272$	0	0
$273$	96.0000i	0.351648i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	76.0000i	0.274368i	0.990546	+	0.137184i	$0.0438052\pi$
$277$	76.0000i	0.274368i	−0.990546	+	0.137184i	$0.956195\pi$
$278$	0	0
$279$	− 187.061i	− 0.670471i
$280$	0	0
$281$	−354.000	−1.25979	−0.629893	−	0.776682i	$-0.716901\pi$
$281$	−354.000	−1.25979	−0.629893	+	0.776682i	$0.716901\pi$
$282$	0	0
$283$	−316.965	−1.12002	−0.560009	−	0.828486i	$-0.689202\pi$
$283$	−316.965	−1.12002	−0.560009	+	0.828486i	$0.689202\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−218.238	−0.760413
$288$	0	0
$289$	280.000	0.968858
$290$	0	0
$291$	232.095i	0.797577i
$292$	0	0
$293$	426.000i	1.45392i	0.686677	+	0.726962i	$0.259069\pi$
$293$	426.000i	1.45392i	−0.686677	+	0.726962i	$0.740931\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	315.000i	1.06061i
$298$	0	0
$299$	720.533i	2.40981i
$300$	0	0
$301$	−216.000	−0.717608
$302$	0	0
$303$	−259.808	−0.857451
$304$	0	0
$305$	0	0
$306$	0	0
$307$	223.435	0.727800	0.363900	−	0.931438i	$-0.381445\pi$
$307$	223.435	0.727800	0.363900	+	0.931438i	$0.381445\pi$
$308$	0	0
$309$	−192.000	−0.621359
$310$	0	0
$311$	433.013i	1.39232i	0.717885	+	0.696162i	$0.245110\pi$
$311$	433.013i	1.39232i	−0.717885	+	0.696162i	$0.754890\pi$
$312$	0	0
$313$	− 218.000i	− 0.696486i	−0.937404	−	0.348243i	$-0.886778\pi$
$313$	− 218.000i	− 0.696486i	0.937404	−	0.348243i	$-0.113222\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.0378549i	0.999821	+	0.0189274i	$0.00602515\pi$
$317$	12.0000i	0.0378549i	−0.999821	+	0.0189274i	$0.993975\pi$
$318$	0	0
$319$	− 145.492i	− 0.456089i
$320$	0	0
$321$	285.000	0.887850
$322$	0	0
$323$	67.5500	0.209133
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−128.172	−0.391963
$328$	0	0
$329$	−48.0000	−0.145897
$330$	0	0
$331$	− 465.922i	− 1.40762i	−0.710389	−	0.703809i	$-0.751481\pi$
$331$	− 465.922i	− 1.40762i	0.710389	−	0.703809i	$-0.248519\pi$
$332$	0	0
$333$	− 300.000i	− 0.900901i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	67.0000i	0.198813i	0.995047	+	0.0994065i	$0.0316944\pi$
$337$	67.0000i	0.198813i	−0.995047	+	0.0994065i	$0.968306\pi$
$338$	0	0
$339$	− 348.142i	− 1.02697i
$340$	0	0
$341$	−378.000	−1.10850
$342$	0	0
$343$	−297.913	−0.868550
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−192.258	−0.554057	−0.277028	−	0.960862i	$-0.589350\pi$
$347$	−192.258	−0.554057	−0.277028	+	0.960862i	$0.589350\pi$
$348$	0	0
$349$	−254.000	−0.727794	−0.363897	−	0.931439i	$-0.618554\pi$
$349$	−254.000	−0.727794	−0.363897	+	0.931439i	$0.618554\pi$
$350$	0	0
$351$	− 415.692i	− 1.18431i
$352$	0	0
$353$	114.000i	0.322946i	0.986877	+	0.161473i	$0.0516245\pi$
$353$	114.000i	0.322946i	−0.986877	+	0.161473i	$0.948376\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.0000i	0.0504202i
$358$	0	0
$359$	370.659i	1.03248i	0.856445	+	0.516238i	$0.172668\pi$
$359$	370.659i	1.03248i	−0.856445	+	0.516238i	$0.827332\pi$
$360$	0	0
$361$	−146.000	−0.404432
$362$	0	0
$363$	45.0333	0.124059
$364$	0	0
$365$	0	0
$366$	0	0
$367$	117.779	0.320925	0.160462	−	0.987042i	$-0.448701\pi$
$367$	117.779	0.320925	0.160462	+	0.987042i	$0.448701\pi$
$368$	0	0
$369$	378.000	1.02439
$370$	0	0
$371$	− 62.3538i	− 0.168070i
$372$	0	0
$373$	274.000i	0.734584i	0.930106	+	0.367292i	$0.119715\pi$
$373$	274.000i	0.734584i	−0.930106	+	0.367292i	$0.880285\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	192.000i	0.509284i
$378$	0	0
$379$	− 299.645i	− 0.790619i	−0.918548	−	0.395310i	$-0.870637\pi$
$379$	− 299.645i	− 0.790619i	0.918548	−	0.395310i	$-0.129363\pi$
$380$	0	0
$381$	−18.0000	−0.0472441
$382$	0	0
$383$	460.726	1.20294	0.601469	−	0.798896i	$-0.294582\pi$
$383$	460.726	1.20294	0.601469	+	0.798896i	$0.294582\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	374.123	0.966726
$388$	0	0
$389$	510.000	1.31105	0.655527	−	0.755172i	$-0.272447\pi$
$389$	510.000	1.31105	0.655527	+	0.755172i	$0.272447\pi$
$390$	0	0
$391$	135.100i	0.345524i
$392$	0	0
$393$	− 12.0000i	− 0.0305344i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 532.000i	− 1.34005i	−0.742338	−	0.670025i	$-0.766283\pi$
$397$	− 532.000i	− 1.34005i	0.742338	−	0.670025i	$-0.233717\pi$
$398$	0	0
$399$	− 135.100i	− 0.338596i
$400$	0	0
$401$	477.000	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$401$	477.000	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$402$	0	0
$403$	498.831	1.23779
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−606.218	−1.48948
$408$	0	0
$409$	−25.0000	−0.0611247	−0.0305623	−	0.999533i	$-0.509730\pi$
$409$	−25.0000	−0.0611247	−0.0305623	+	0.999533i	$0.509730\pi$
$410$	0	0
$411$	358.535i	0.872347i
$412$	0	0
$413$	192.000i	0.464891i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	309.000i	0.741007i
$418$	0	0
$419$	57.1577i	0.136415i	0.997671	+	0.0682073i	$0.0217279\pi$
$419$	57.1577i	0.136415i	−0.997671	+	0.0682073i	$0.978272\pi$
$420$	0	0
$421$	388.000	0.921615	0.460808	−	0.887500i	$-0.347560\pi$
$421$	388.000	0.921615	0.460808	+	0.887500i	$0.347560\pi$
$422$	0	0
$423$	83.1384	0.196545
$424$	0	0
$425$	0	0
$426$	0	0
$427$	90.0666	0.210929
$428$	0	0
$429$	−336.000	−0.783217
$430$	0	0
$431$	− 245.951i	− 0.570652i	−0.958430	−	0.285326i	$-0.907898\pi$
$431$	− 245.951i	− 0.570652i	0.958430	−	0.285326i	$-0.0921019\pi$
$432$	0	0
$433$	307.000i	0.709007i	0.935055	+	0.354503i	$0.115350\pi$
$433$	307.000i	0.709007i	−0.935055	+	0.354503i	$0.884650\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1014.00i	− 2.32037i
$438$	0	0
$439$	− 561.184i	− 1.27832i	−0.769072	−	0.639162i	$-0.779281\pi$
$439$	− 561.184i	− 1.27832i	0.769072	−	0.639162i	$-0.220719\pi$
$440$	0	0
$441$	222.000	0.503401
$442$	0	0
$443$	479.778	1.08302	0.541510	−	0.840694i	$-0.317853\pi$
$443$	479.778	1.08302	0.541510	+	0.840694i	$0.317853\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−187.061	−0.418482
$448$	0	0
$449$	39.0000	0.0868597	0.0434298	−	0.999056i	$-0.486172\pi$
$449$	39.0000	0.0868597	0.0434298	+	0.999056i	$0.486172\pi$
$450$	0	0
$451$	− 763.834i	− 1.69365i
$452$	0	0
$453$	462.000i	1.01987i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	349.000i	0.763676i	0.924229	+	0.381838i	$0.124709\pi$
$457$	349.000i	0.763676i	−0.924229	+	0.381838i	$0.875291\pi$
$458$	0	0
$459$	− 77.9423i	− 0.169809i
$460$	0	0
$461$	−372.000	−0.806941	−0.403471	−	0.914993i	$-0.632196\pi$
$461$	−372.000	−0.806941	−0.403471	+	0.914993i	$0.632196\pi$
$462$	0	0
$463$	422.620	0.912787	0.456394	−	0.889778i	$-0.349141\pi$
$463$	422.620	0.912787	0.456394	+	0.889778i	$0.349141\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−713.605	−1.52806	−0.764031	−	0.645180i	$-0.776783\pi$
$467$	−713.605	−1.52806	−0.764031	+	0.645180i	$0.776783\pi$
$468$	0	0
$469$	114.000	0.243070
$470$	0	0
$471$	− 481.510i	− 1.02231i
$472$	0	0
$473$	− 756.000i	− 1.59831i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	108.000i	0.226415i
$478$	0	0
$479$	− 758.638i	− 1.58380i	−0.610653	−	0.791898i	$-0.709093\pi$
$479$	− 758.638i	− 1.58380i	0.610653	−	0.791898i	$-0.290907\pi$
$480$	0	0
$481$	800.000	1.66320
$482$	0	0
$483$	270.200	0.559420
$484$	0	0
$485$	0	0
$486$	0	0
$487$	523.079	1.07408	0.537042	−	0.843555i	$-0.319541\pi$
$487$	523.079	1.07408	0.537042	+	0.843555i	$0.319541\pi$
$488$	0	0
$489$	33.0000	0.0674847
$490$	0	0
$491$	478.046i	0.973617i	0.873509	+	0.486809i	$0.161839\pi$
$491$	478.046i	0.973617i	−0.873509	+	0.486809i	$0.838161\pi$
$492$	0	0
$493$	36.0000i	0.0730223i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	96.0000i	0.193159i
$498$	0	0
$499$	62.3538i	0.124958i	0.998046	+	0.0624788i	$0.0199006\pi$
$499$	62.3538i	0.124958i	−0.998046	+	0.0624788i	$0.980099\pi$
$500$	0	0
$501$	456.000	0.910180
$502$	0	0
$503$	561.184	1.11567	0.557837	−	0.829950i	$-0.311631\pi$
$503$	561.184	1.11567	0.557837	+	0.829950i	$0.311631\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	150.688	0.297216
$508$	0	0
$509$	−390.000	−0.766208	−0.383104	−	0.923705i	$-0.625145\pi$
$509$	−390.000	−0.766208	−0.383104	+	0.923705i	$0.625145\pi$
$510$	0	0
$511$	58.8897i	0.115244i
$512$	0	0
$513$	585.000i	1.14035i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 168.000i	− 0.324952i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−183.000	−0.351248	−0.175624	−	0.984457i	$-0.556194\pi$
$521$	−183.000	−0.351248	−0.175624	+	0.984457i	$0.556194\pi$
$522$	0	0
$523$	185.329	0.354358	0.177179	−	0.984179i	$-0.443303\pi$
$523$	185.329	0.354358	0.177179	+	0.984179i	$0.443303\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	93.5307	0.177478
$528$	0	0
$529$	1499.00	2.83365
$530$	0	0
$531$	− 332.554i	− 0.626278i
$532$	0	0
$533$	1008.00i	1.89118i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.0000i	0.0391061i
$538$	0	0
$539$	− 448.601i	− 0.832284i
$540$	0	0
$541$	−968.000	−1.78928	−0.894640	−	0.446789i	$-0.852568\pi$
$541$	−968.000	−1.78928	−0.894640	+	0.446789i	$0.852568\pi$
$542$	0	0
$543$	−287.520	−0.529504
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−510.955	−0.934104	−0.467052	−	0.884230i	$-0.654684\pi$
$547$	−510.955	−0.934104	−0.467052	+	0.884230i	$0.654684\pi$
$548$	0	0
$549$	−156.000	−0.284153
$550$	0	0
$551$	− 270.200i	− 0.490381i
$552$	0	0
$553$	− 300.000i	− 0.542495i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 552.000i	− 0.991023i	−0.868601	−	0.495512i	$-0.834981\pi$
$557$	− 552.000i	− 0.991023i	0.868601	−	0.495512i	$-0.165019\pi$
$558$	0	0
$559$	997.661i	1.78472i
$560$	0	0
$561$	−63.0000	−0.112299
$562$	0	0
$563$	−401.836	−0.713740	−0.356870	−	0.934154i	$-0.616156\pi$
$563$	−401.836	−0.713740	−0.356870	+	0.934154i	$0.616156\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	31.1769	0.0549857
$568$	0	0
$569$	−117.000	−0.205624	−0.102812	−	0.994701i	$-0.532784\pi$
$569$	−117.000	−0.205624	−0.102812	+	0.994701i	$0.532784\pi$
$570$	0	0
$571$	− 297.913i	− 0.521739i	−0.965374	−	0.260869i	$-0.915991\pi$
$571$	− 297.913i	− 0.521739i	0.965374	−	0.260869i	$-0.0840092\pi$
$572$	0	0
$573$	− 522.000i	− 0.910995i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	491.000i	0.850953i	0.904969	+	0.425477i	$0.139894\pi$
$577$	491.000i	0.850953i	−0.904969	+	0.425477i	$0.860106\pi$
$578$	0	0
$579$	− 240.755i	− 0.415812i
$580$	0	0
$581$	342.000	0.588640
$582$	0	0
$583$	218.238	0.374337
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−244.219	−0.416046	−0.208023	−	0.978124i	$-0.566703\pi$
$587$	−244.219	−0.416046	−0.208023	+	0.978124i	$0.566703\pi$
$588$	0	0
$589$	−702.000	−1.19185
$590$	0	0
$591$	218.238i	0.369270i
$592$	0	0
$593$	57.0000i	0.0961214i	0.998844	+	0.0480607i	$0.0153041\pi$
$593$	57.0000i	0.0961214i	−0.998844	+	0.0480607i	$0.984696\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 132.000i	− 0.221106i
$598$	0	0
$599$	633.931i	1.05831i	0.848524	+	0.529157i	$0.177492\pi$
$599$	633.931i	1.05831i	−0.848524	+	0.529157i	$0.822508\pi$
$600$	0	0
$601$	263.000	0.437604	0.218802	−	0.975769i	$-0.429785\pi$
$601$	263.000	0.437604	0.218802	+	0.975769i	$0.429785\pi$
$602$	0	0
$603$	−197.454	−0.327452
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−484.974	−0.798969	−0.399485	−	0.916740i	$-0.630811\pi$
$607$	−484.974	−0.798969	−0.399485	+	0.916740i	$0.630811\pi$
$608$	0	0
$609$	72.0000	0.118227
$610$	0	0
$611$	221.703i	0.362852i
$612$	0	0
$613$	− 194.000i	− 0.316476i	−0.987401	−	0.158238i	$-0.949419\pi$
$613$	− 194.000i	− 0.316476i	0.987401	−	0.158238i	$-0.0505814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	726.000i	1.17666i	0.808620	+	0.588331i	$0.200215\pi$
$617$	726.000i	1.17666i	−0.808620	+	0.588331i	$0.799785\pi$
$618$	0	0
$619$	− 408.764i	− 0.660362i	−0.943918	−	0.330181i	$-0.892890\pi$
$619$	− 408.764i	− 0.660362i	0.943918	−	0.330181i	$-0.107110\pi$
$620$	0	0
$621$	−1170.00	−1.88406
$622$	0	0
$623$	−342.946	−0.550475
$624$	0	0
$625$	0	0
$626$	0	0
$627$	472.850	0.754147
$628$	0	0
$629$	150.000	0.238474
$630$	0	0
$631$	502.295i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$631$	502.295i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$632$	0	0
$633$	− 327.000i	− 0.516588i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	592.000i	0.929356i
$638$	0	0
$639$	− 166.277i	− 0.260214i
$640$	0	0
$641$	−174.000	−0.271451	−0.135725	−	0.990746i	$-0.543337\pi$
$641$	−174.000	−0.271451	−0.135725	+	0.990746i	$0.543337\pi$
$642$	0	0
$643$	−976.877	−1.51925	−0.759624	−	0.650362i	$-0.774617\pi$
$643$	−976.877	−1.51925	−0.759624	+	0.650362i	$0.774617\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	921.451	1.42419	0.712095	−	0.702083i	$-0.247746\pi$
$647$	921.451	1.42419	0.712095	+	0.702083i	$0.247746\pi$
$648$	0	0
$649$	−672.000	−1.03544
$650$	0	0
$651$	− 187.061i	− 0.287345i
$652$	0	0
$653$	− 222.000i	− 0.339969i	−0.985447	−	0.169985i	$-0.945628\pi$
$653$	− 222.000i	− 0.339969i	0.985447	−	0.169985i	$-0.0543718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 102.000i	− 0.155251i
$658$	0	0
$659$	857.365i	1.30101i	0.759502	+	0.650505i	$0.225443\pi$
$659$	857.365i	1.30101i	−0.759502	+	0.650505i	$0.774557\pi$
$660$	0	0
$661$	−268.000	−0.405446	−0.202723	−	0.979236i	$-0.564979\pi$
$661$	−268.000	−0.405446	−0.202723	+	0.979236i	$0.564979\pi$
$662$	0	0
$663$	83.1384	0.125397
$664$	0	0
$665$	0	0
$666$	0	0
$667$	540.400	0.810195
$668$	0	0
$669$	252.000	0.376682
$670$	0	0
$671$	315.233i	0.469796i
$672$	0	0
$673$	550.000i	0.817236i	0.912705	+	0.408618i	$0.133989\pi$
$673$	550.000i	0.817236i	−0.912705	+	0.408618i	$0.866011\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 600.000i	− 0.886263i	−0.896457	−	0.443131i	$-0.853867\pi$
$677$	− 600.000i	− 0.886263i	0.896457	−	0.443131i	$-0.146133\pi$
$678$	0	0
$679$	− 464.190i	− 0.683637i
$680$	0	0
$681$	300.000	0.440529
$682$	0	0
$683$	−729.193	−1.06763	−0.533817	−	0.845600i	$-0.679243\pi$
$683$	−729.193	−1.06763	−0.533817	+	0.845600i	$0.679243\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−242.487	−0.352965
$688$	0	0
$689$	−288.000	−0.417997
$690$	0	0
$691$	1231.49i	1.78218i	0.453824	+	0.891091i	$0.350059\pi$
$691$	1231.49i	1.78218i	−0.453824	+	0.891091i	$0.649941\pi$
$692$	0	0
$693$	− 252.000i	− 0.363636i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	189.000i	0.271162i
$698$	0	0
$699$	633.931i	0.906911i
$700$	0	0
$701$	1008.00	1.43795	0.718973	−	0.695038i	$-0.244613\pi$
$701$	1008.00	1.43795	0.718973	+	0.695038i	$0.244613\pi$
$702$	0	0
$703$	−1125.83	−1.60147
$704$	0	0
$705$	0	0
$706$	0	0
$707$	519.615	0.734958
$708$	0	0
$709$	−388.000	−0.547250	−0.273625	−	0.961837i	$-0.588223\pi$
$709$	−388.000	−0.547250	−0.273625	+	0.961837i	$0.588223\pi$
$710$	0	0
$711$	519.615i	0.730823i
$712$	0	0
$713$	− 1404.00i	− 1.96914i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 444.000i	− 0.619247i
$718$	0	0
$719$	973.413i	1.35384i	0.736056	+	0.676921i	$0.236686\pi$
$719$	973.413i	1.35384i	−0.736056	+	0.676921i	$0.763314\pi$
$720$	0	0
$721$	384.000	0.532594
$722$	0	0
$723$	−126.440	−0.174882
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−187.061	−0.257306	−0.128653	−	0.991690i	$-0.541065\pi$
$727$	−187.061	−0.257306	−0.128653	+	0.991690i	$0.541065\pi$
$728$	0	0
$729$	351.000	0.481481
$730$	0	0
$731$	187.061i	0.255898i
$732$	0	0
$733$	− 976.000i	− 1.33151i	−0.746168	−	0.665757i	$-0.768109\pi$
$733$	− 976.000i	− 1.33151i	0.746168	−	0.665757i	$-0.231891\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	399.000i	0.541384i
$738$	0	0
$739$	270.200i	0.365629i	0.983147	+	0.182815i	$0.0585208\pi$
$739$	270.200i	0.365629i	−0.983147	+	0.182815i	$0.941479\pi$
$740$	0	0
$741$	−624.000	−0.842105
$742$	0	0
$743$	38.1051	0.0512855	0.0256427	−	0.999671i	$-0.491837\pi$
$743$	38.1051	0.0512855	0.0256427	+	0.999671i	$0.491837\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−592.361	−0.792987
$748$	0	0
$749$	−570.000	−0.761015
$750$	0	0
$751$	− 921.451i	− 1.22697i	−0.789708	−	0.613483i	$-0.789768\pi$
$751$	− 921.451i	− 1.22697i	0.789708	−	0.613483i	$-0.210232\pi$
$752$	0	0
$753$	− 291.000i	− 0.386454i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 392.000i	− 0.517834i	−0.965900	−	0.258917i	$-0.916634\pi$
$757$	− 392.000i	− 0.517834i	0.965900	−	0.258917i	$-0.0833655\pi$
$758$	0	0
$759$	945.700i	1.24598i
$760$	0	0
$761$	−1335.00	−1.75427	−0.877135	−	0.480243i	$-0.840548\pi$
$761$	−1335.00	−1.75427	−0.877135	+	0.480243i	$0.840548\pi$
$762$	0	0
$763$	256.344	0.335968
$764$	0	0
$765$	0	0
$766$	0	0
$767$	886.810	1.15621
$768$	0	0
$769$	−509.000	−0.661899	−0.330949	−	0.943649i	$-0.607369\pi$
$769$	−509.000	−0.661899	−0.330949	+	0.943649i	$0.607369\pi$
$770$	0	0
$771$	488.438i	0.633513i
$772$	0	0
$773$	684.000i	0.884864i	0.896802	+	0.442432i	$0.145884\pi$
$773$	684.000i	0.884864i	−0.896802	+	0.442432i	$0.854116\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 300.000i	− 0.386100i
$778$	0	0
$779$	− 1418.55i	− 1.82099i
$780$	0	0
$781$	−336.000	−0.430218
$782$	0	0
$783$	−311.769	−0.398173
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.92820	0.00880331	0.00440165	−	0.999990i	$-0.498599\pi$
$787$	6.92820	0.00880331	0.00440165	+	0.999990i	$0.498599\pi$
$788$	0	0
$789$	294.000	0.372624
$790$	0	0
$791$	696.284i	0.880258i
$792$	0	0
$793$	− 416.000i	− 0.524590i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1374.00i	1.72396i	0.506938	+	0.861982i	$0.330777\pi$
$797$	1374.00i	1.72396i	−0.506938	+	0.861982i	$0.669223\pi$
$798$	0	0
$799$	41.5692i	0.0520266i
$800$	0	0
$801$	594.000	0.741573
$802$	0	0
$803$	−206.114	−0.256680
$804$	0	0
$805$	0	0
$806$	0	0
$807$	498.831	0.618130
$808$	0	0
$809$	−546.000	−0.674907	−0.337454	−	0.941342i	$-0.609566\pi$
$809$	−546.000	−0.674907	−0.337454	+	0.941342i	$0.609566\pi$
$810$	0	0
$811$	1309.43i	1.61459i	0.590150	+	0.807294i	$0.299069\pi$
$811$	1309.43i	1.61459i	−0.590150	+	0.807294i	$0.700931\pi$
$812$	0	0
$813$	− 294.000i	− 0.361624i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1404.00i	− 1.71848i
$818$	0	0
$819$	332.554i	0.406049i
$820$	0	0
$821$	1038.00	1.26431	0.632156	−	0.774841i	$-0.282170\pi$
$821$	1038.00	1.26431	0.632156	+	0.774841i	$0.282170\pi$
$822$	0	0
$823$	−381.051	−0.463003	−0.231501	−	0.972835i	$-0.574364\pi$
$823$	−381.051	−0.463003	−0.231501	+	0.972835i	$0.574364\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	102.191	0.123568	0.0617842	−	0.998090i	$-0.480321\pi$
$827$	102.191	0.123568	0.0617842	+	0.998090i	$0.480321\pi$
$828$	0	0
$829$	104.000	0.125452	0.0627262	−	0.998031i	$-0.480021\pi$
$829$	104.000	0.125452	0.0627262	+	0.998031i	$0.480021\pi$
$830$	0	0
$831$	− 131.636i	− 0.158407i
$832$	0	0
$833$	111.000i	0.133253i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	810.000i	0.967742i
$838$	0	0
$839$	− 1053.09i	− 1.25517i	−0.778548	−	0.627585i	$-0.784044\pi$
$839$	− 1053.09i	− 1.25517i	0.778548	−	0.627585i	$-0.215956\pi$
$840$	0	0
$841$	−697.000	−0.828775
$842$	0	0
$843$	613.146	0.727338
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−90.0666	−0.106336
$848$	0	0
$849$	549.000	0.646643
$850$	0	0
$851$	− 2251.67i	− 2.64591i
$852$	0	0
$853$	1234.00i	1.44666i	0.690503	+	0.723329i	$0.257389\pi$
$853$	1234.00i	1.44666i	−0.690503	+	0.723329i	$0.742611\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1185.00i	1.38273i	0.722505	+	0.691365i	$0.242990\pi$
$857$	1185.00i	1.38273i	−0.722505	+	0.691365i	$0.757010\pi$
$858$	0	0
$859$	1692.21i	1.96998i	0.172609	+	0.984990i	$0.444780\pi$
$859$	1692.21i	1.96998i	−0.172609	+	0.984990i	$0.555220\pi$
$860$	0	0
$861$	378.000	0.439024
$862$	0	0
$863$	−893.738	−1.03562	−0.517809	−	0.855496i	$-0.673252\pi$
$863$	−893.738	−1.03562	−0.517809	+	0.855496i	$0.673252\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−484.974	−0.559371
$868$	0	0
$869$	1050.00	1.20829
$870$	0	0
$871$	− 526.543i	− 0.604527i
$872$	0	0
$873$	804.000i	0.920962i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.0000i	0.0228050i	0.999935	+	0.0114025i	$0.00362961\pi$
$877$	20.0000i	0.0228050i	−0.999935	+	0.0114025i	$0.996370\pi$
$878$	0	0
$879$	− 737.854i	− 0.839424i
$880$	0	0
$881$	−330.000	−0.374574	−0.187287	−	0.982305i	$-0.559969\pi$
$881$	−330.000	−0.374574	−0.187287	+	0.982305i	$0.559969\pi$
$882$	0	0
$883$	95.2628	0.107885	0.0539427	−	0.998544i	$-0.482821\pi$
$883$	95.2628	0.107885	0.0539427	+	0.998544i	$0.482821\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1212.44	−1.36689	−0.683447	−	0.730000i	$-0.739520\pi$
$887$	−1212.44	−1.36689	−0.683447	+	0.730000i	$0.739520\pi$
$888$	0	0
$889$	36.0000	0.0404949
$890$	0	0
$891$	109.119i	0.122468i
$892$	0	0
$893$	− 312.000i	− 0.349384i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1248.00i	− 1.39130i
$898$	0	0
$899$	− 374.123i	− 0.416155i
$900$	0	0
$901$	−54.0000	−0.0599334
$902$	0	0
$903$	374.123	0.414311
$904$	0	0
$905$	0	0
$906$	0	0
$907$	297.913	0.328459	0.164230	−	0.986422i	$-0.447486\pi$
$907$	297.913	0.328459	0.164230	+	0.986422i	$0.447486\pi$
$908$	0	0
$909$	−900.000	−0.990099
$910$	0	0
$911$	− 48.4974i	− 0.0532354i	−0.999646	−	0.0266177i	$-0.991526\pi$
$911$	− 48.4974i	− 0.0532354i	0.999646	−	0.0266177i	$-0.00847367\pi$
$912$	0	0
$913$	1197.00i	1.31106i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.0000i	0.0261723i
$918$	0	0
$919$	− 758.638i	− 0.825504i	−0.910843	−	0.412752i	$-0.864568\pi$
$919$	− 758.638i	− 0.825504i	0.910843	−	0.412752i	$-0.135432\pi$
$920$	0	0
$921$	−387.000	−0.420195
$922$	0	0
$923$	443.405	0.480395
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−665.108	−0.717484
$928$	0	0
$929$	258.000	0.277718	0.138859	−	0.990312i	$-0.455657\pi$
$929$	258.000	0.277718	0.138859	+	0.990312i	$0.455657\pi$
$930$	0	0
$931$	− 833.116i	− 0.894862i
$932$	0	0
$933$	− 750.000i	− 0.803859i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 481.000i	− 0.513340i	−0.966499	−	0.256670i	$-0.917375\pi$
$937$	− 481.000i	− 0.513340i	0.966499	−	0.256670i	$-0.0826254\pi$
$938$	0	0
$939$	377.587i	0.402116i
$940$	0	0
$941$	−24.0000	−0.0255048	−0.0127524	−	0.999919i	$-0.504059\pi$
$941$	−24.0000	−0.0255048	−0.0127524	+	0.999919i	$0.504059\pi$
$942$	0	0
$943$	2837.10	3.00859
$944$	0	0
$945$	0	0
$946$	0	0
$947$	665.108	0.702331	0.351166	−	0.936313i	$-0.385785\pi$
$947$	665.108	0.702331	0.351166	+	0.936313i	$0.385785\pi$
$948$	0	0
$949$	272.000	0.286617
$950$	0	0
$951$	− 20.7846i	− 0.0218555i
$952$	0	0
$953$	291.000i	0.305352i	0.988276	+	0.152676i	$0.0487890\pi$
$953$	291.000i	0.305352i	−0.988276	+	0.152676i	$0.951211\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	252.000i	0.263323i
$958$	0	0
$959$	− 717.069i	− 0.747726i
$960$	0	0
$961$	−11.0000	−0.0114464
$962$	0	0
$963$	987.269	1.02520
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1094.66	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$967$	−1094.66	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$968$	0	0
$969$	−117.000	−0.120743
$970$	0	0
$971$	− 611.414i	− 0.629674i	−0.949146	−	0.314837i	$-0.898050\pi$
$971$	− 611.414i	− 0.629674i	0.949146	−	0.314837i	$-0.101950\pi$
$972$	0	0
$973$	− 618.000i	− 0.635149i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 231.000i	− 0.236438i	−0.992988	−	0.118219i	$-0.962282\pi$
$977$	− 231.000i	− 0.236438i	0.992988	−	0.118219i	$-0.0377185\pi$
$978$	0	0
$979$	− 1200.31i	− 1.22606i
$980$	0	0
$981$	−444.000	−0.452599
$982$	0	0
$983$	−162.813	−0.165628	−0.0828142	−	0.996565i	$-0.526391\pi$
$983$	−162.813	−0.165628	−0.0828142	+	0.996565i	$0.526391\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	83.1384	0.0842335
$988$	0	0
$989$	2808.00	2.83923
$990$	0	0
$991$	723.997i	0.730572i	0.930895	+	0.365286i	$0.119029\pi$
$991$	723.997i	0.730572i	−0.930895	+	0.365286i	$0.880971\pi$
$992$	0	0
$993$	807.000i	0.812689i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1100.00i	1.10331i	0.834072	+	0.551655i	$0.186003\pi$
$997$	1100.00i	1.10331i	−0.834072	+	0.551655i	$0.813997\pi$
$998$	0	0
$999$	1299.04i	1.30034i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.3.h.b.399.2		4
3.2	odd	2		3600.3.j.f.1999.3		4
4.3	odd	2	inner	400.3.h.b.399.3		4
5.2	odd	4		400.3.b.e.351.2	yes	2
5.3	odd	4		400.3.b.f.351.1	yes	2
5.4	even	2	inner	400.3.h.b.399.4		4
8.3	odd	2		1600.3.h.h.1599.2		4
8.5	even	2		1600.3.h.h.1599.3		4
12.11	even	2		3600.3.j.f.1999.2		4
15.2	even	4		3600.3.e.j.3151.2		2
15.8	even	4		3600.3.e.u.3151.1		2
15.14	odd	2		3600.3.j.f.1999.1		4
20.3	even	4		400.3.b.f.351.2	yes	2
20.7	even	4		400.3.b.e.351.1	✓	2
20.19	odd	2	inner	400.3.h.b.399.1		4
40.3	even	4		1600.3.b.i.1151.1		2
40.13	odd	4		1600.3.b.i.1151.2		2
40.19	odd	2		1600.3.h.h.1599.4		4
40.27	even	4		1600.3.b.j.1151.2		2
40.29	even	2		1600.3.h.h.1599.1		4
40.37	odd	4		1600.3.b.j.1151.1		2
60.23	odd	4		3600.3.e.u.3151.2		2
60.47	odd	4		3600.3.e.j.3151.1		2
60.59	even	2		3600.3.j.f.1999.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.3.b.e.351.1	✓	2	20.7	even	4
400.3.b.e.351.2	yes	2	5.2	odd	4
400.3.b.f.351.1	yes	2	5.3	odd	4
400.3.b.f.351.2	yes	2	20.3	even	4
400.3.h.b.399.1		4	20.19	odd	2	inner
400.3.h.b.399.2		4	1.1	even	1	trivial
400.3.h.b.399.3		4	4.3	odd	2	inner
400.3.h.b.399.4		4	5.4	even	2	inner
1600.3.b.i.1151.1		2	40.3	even	4
1600.3.b.i.1151.2		2	40.13	odd	4
1600.3.b.j.1151.1		2	40.37	odd	4
1600.3.b.j.1151.2		2	40.27	even	4
1600.3.h.h.1599.1		4	40.29	even	2
1600.3.h.h.1599.2		4	8.3	odd	2
1600.3.h.h.1599.3		4	8.5	even	2
1600.3.h.h.1599.4		4	40.19	odd	2
3600.3.e.j.3151.1		2	60.47	odd	4
3600.3.e.j.3151.2		2	15.2	even	4
3600.3.e.u.3151.1		2	15.8	even	4
3600.3.e.u.3151.2		2	60.23	odd	4
3600.3.j.f.1999.1		4	15.14	odd	2
3600.3.j.f.1999.2		4	12.11	even	2
3600.3.j.f.1999.3		4	3.2	odd	2
3600.3.j.f.1999.4		4	60.59	even	2

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +3.46410 q^{7} -6.00000 q^{9} +12.1244i q^{11} -16.0000i q^{13} -3.00000i q^{17} +22.5167i q^{19} -6.00000 q^{21} -45.0333 q^{23} +25.9808 q^{27} -12.0000 q^{29} +31.1769i q^{31} -21.0000i q^{33} +50.0000i q^{37} +27.7128i q^{39} -63.0000 q^{41} -62.3538 q^{43} -13.8564 q^{47} -37.0000 q^{49} +5.19615i q^{51} -18.0000i q^{53} -39.0000i q^{57} +55.4256i q^{59} +26.0000 q^{61} -20.7846 q^{63} +32.9090 q^{67} +78.0000 q^{69} +27.7128i q^{71} +17.0000i q^{73} +42.0000i q^{77} -86.6025i q^{79} +9.00000 q^{81} +98.7269 q^{83} +20.7846 q^{87} -99.0000 q^{89} -55.4256i q^{91} -54.0000i q^{93} -134.000i q^{97} -72.7461i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{9} - 24 q^{21} - 48 q^{29} - 252 q^{41} - 148 q^{49} + 104 q^{61} + 312 q^{69} + 36 q^{81} - 396 q^{89}+O(q^{100}) \) 4 * q - 24 * q^9 - 24 * q^21 - 48 * q^29 - 252 * q^41 - 148 * q^49 + 104 * q^61 + 312 * q^69 + 36 * q^81 - 396 * q^89

Introduction

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