LMFDB - Embedded newform 1080.4.f.d.649.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(649,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1080.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 50 x^{18} + 352 x^{17} + 21144 x^{16} - 183248 x^{15} + 837232 x^{14} + \cdots + 2209905230625 $ x^20 - 10x^19 + 50x^18 + 352x^17 + 21144x^16 - 183248x^15 + 837232x^14 + 4991426x^13 + 123172734x^12 - 846162224x^11 + 3147448836x^10 + 8830578242x^9 + 71410987721x^8 - 374031480668x^7 + 1119018321432x^6 + 1852681857730x^5 + 5531652507775x^4 - 15867459693850x^3 + 19223534162450x^2 - 9217612347750*x + 2209905230625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{12}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.18
Root			$-6.67988 - 6.67988i$ of defining polynomial
Character	$\chi$	$=$	1080.649
Dual form			1080.4.f.d.649.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.99584 + 7.81451i) q^{5} +22.1308i q^{7} +O(q^{10})$	$q+(7.99584 + 7.81451i) q^{5} +22.1308i q^{7} +49.2486 q^{11} +38.2445i q^{13} +15.2168i q^{17} -146.080 q^{19} +151.388i q^{23} +(2.86693 + 124.967i) q^{25} -72.6770 q^{29} +278.281 q^{31} +(-172.941 + 176.954i) q^{35} +95.6819i q^{37} +259.917 q^{41} -128.973i q^{43} -305.783i q^{47} -146.773 q^{49} -305.722i q^{53} +(393.784 + 384.853i) q^{55} -431.726 q^{59} -224.310 q^{61} +(-298.862 + 305.797i) q^{65} -756.489i q^{67} -774.302 q^{71} -137.234i q^{73} +1089.91i q^{77} -314.552 q^{79} +853.847i q^{83} +(-118.912 + 121.671i) q^{85} -1000.83 q^{89} -846.383 q^{91} +(-1168.04 - 1141.55i) q^{95} +989.972i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 100 q^{19} - 24 q^{25} + 228 q^{31} - 252 q^{49} - 64 q^{55} + 748 q^{61} - 668 q^{79} - 1500 q^{85} + 1744 q^{91}+O(q^{100}) $ 20 * q - 100 * q^19 - 24 * q^25 + 228 * q^31 - 252 * q^49 - 64 * q^55 + 748 * q^61 - 668 * q^79 - 1500 * q^85 + 1744 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.99584	+	7.81451i	0.715170	+	0.698951i
$5$	7.99584	+	7.81451i	0.715170	+	0.698951i
$6$		0			0
$7$			22.1308i			1.19495i	0.801887	+	0.597476i	$0.203830\pi$
$7$			22.1308i			1.19495i	−0.801887	+	0.597476i	$0.796170\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	49.2486			1.34991			0.674955	−	0.737859i	$-0.264163\pi$
$11$	49.2486			1.34991			0.674955	+	0.737859i	$0.264163\pi$
$12$		0			0
$13$			38.2445i			0.815933i	0.912997	+	0.407966i	$0.133762\pi$
$13$			38.2445i			0.815933i	−0.912997	+	0.407966i	$0.866238\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.2168i			0.217095i	0.994091	+	0.108547i	$0.0346199\pi$
$17$			15.2168i			0.217095i	−0.994091	+	0.108547i	$0.965380\pi$
$18$		0			0
$19$	−146.080			−1.76385			−0.881925	−	0.471389i	$-0.843753\pi$
$19$	−146.080			−1.76385			−0.881925	+	0.471389i	$0.843753\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			151.388i			1.37246i	0.727385	+	0.686229i	$0.240735\pi$
$23$			151.388i			1.37246i	−0.727385	+	0.686229i	$0.759265\pi$
$24$		0			0
$25$	2.86693	+	124.967i	0.0229355	+	0.999737i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−72.6770			−0.465372			−0.232686	−	0.972552i	$-0.574751\pi$
$29$	−72.6770			−0.465372			−0.232686	+	0.972552i	$0.574751\pi$
$30$		0			0
$31$	278.281			1.61228			0.806141	−	0.591723i	$-0.201552\pi$
$31$	278.281			1.61228			0.806141	+	0.591723i	$0.201552\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−172.941	+	176.954i	−0.835213	+	0.854593i
$36$		0			0
$37$			95.6819i			0.425135i	0.977146	+	0.212568i	$0.0681826\pi$
$37$			95.6819i			0.425135i	−0.977146	+	0.212568i	$0.931817\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	259.917			0.990055			0.495028	−	0.868877i	$-0.335158\pi$
$41$	259.917			0.990055			0.495028	+	0.868877i	$0.335158\pi$
$42$		0			0
$43$		−	128.973i		−	0.457400i	−0.973497	−	0.228700i	$-0.926553\pi$
$43$		−	128.973i		−	0.457400i	0.973497	−	0.228700i	$-0.0734474\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	305.783i		−	0.949000i	−0.880255	−	0.474500i	$-0.842629\pi$
$47$		−	305.783i		−	0.949000i	0.880255	−	0.474500i	$-0.157371\pi$
$48$		0			0
$49$	−146.773			−0.427910
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	305.722i		−	0.792342i	−0.918177	−	0.396171i	$-0.870339\pi$
$53$		−	305.722i		−	0.792342i	0.918177	−	0.396171i	$-0.129661\pi$
$54$		0			0
$55$	393.784	+	384.853i	0.965414	+	0.943520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−431.726			−0.952644			−0.476322	−	0.879271i	$-0.658030\pi$
$59$	−431.726			−0.952644			−0.476322	+	0.879271i	$0.658030\pi$
$60$		0			0
$61$	−224.310			−0.470820			−0.235410	−	0.971896i	$-0.575643\pi$
$61$	−224.310			−0.470820			−0.235410	+	0.971896i	$0.575643\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−298.862	+	305.797i	−0.570297	+	0.583530i
$66$		0			0
$67$		−	756.489i		−	1.37940i	−0.724095	−	0.689700i	$-0.757742\pi$
$67$		−	756.489i		−	1.37940i	0.724095	−	0.689700i	$-0.242258\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−774.302			−1.29426			−0.647132	−	0.762378i	$-0.724032\pi$
$71$	−774.302			−1.29426			−0.647132	+	0.762378i	$0.724032\pi$
$72$		0			0
$73$		−	137.234i		−	0.220028i	−0.993930	−	0.110014i	$-0.964910\pi$
$73$		−	137.234i		−	0.220028i	0.993930	−	0.110014i	$-0.0350896\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1089.91i			1.61308i
$78$		0			0
$79$	−314.552			−0.447973			−0.223986	−	0.974592i	$-0.571907\pi$
$79$	−314.552			−0.447973			−0.223986	+	0.974592i	$0.571907\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			853.847i			1.12918i	0.825372	+	0.564590i	$0.190966\pi$
$83$			853.847i			1.12918i	−0.825372	+	0.564590i	$0.809034\pi$
$84$		0			0
$85$	−118.912	+	121.671i	−0.151739	+	0.155260i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1000.83			−1.19200			−0.595999	−	0.802985i	$-0.703244\pi$
$89$	−1000.83			−1.19200			−0.595999	+	0.802985i	$0.703244\pi$
$90$		0			0
$91$	−846.383			−0.975000
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1168.04	−	1141.55i	−1.26145	−	1.23284i
$96$		0			0
$97$			989.972i			1.03625i	0.855304	+	0.518126i	$0.173370\pi$
$97$			989.972i			1.03625i	−0.855304	+	0.518126i	$0.826630\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1493.82			1.47169			0.735843	−	0.677152i	$-0.236786\pi$
$101$	1493.82			1.47169			0.735843	+	0.677152i	$0.236786\pi$
$102$		0			0
$103$		−	1916.03i		−	1.83293i	−0.400113	−	0.916466i	$-0.631029\pi$
$103$		−	1916.03i		−	1.83293i	0.400113	−	0.916466i	$-0.368971\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1231.04i			1.11223i	0.831105	+	0.556115i	$0.187709\pi$
$107$			1231.04i			1.11223i	−0.831105	+	0.556115i	$0.812291\pi$
$108$		0			0
$109$	1857.19			1.63199			0.815994	−	0.578060i	$-0.196190\pi$
$109$	1857.19			1.63199			0.815994	+	0.578060i	$0.196190\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			424.400i			0.353312i	0.984273	+	0.176656i	$0.0565280\pi$
$113$			424.400i			0.353312i	−0.984273	+	0.176656i	$0.943472\pi$
$114$		0			0
$115$	−1183.02	+	1210.47i	−0.959281	+	0.981541i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−336.760			−0.259418
$120$		0			0
$121$	1094.42			0.822255
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−953.633	+	1021.62i	−0.682364	+	0.731012i
$126$		0			0
$127$		−	878.419i		−	0.613756i	−0.951749	−	0.306878i	$-0.900716\pi$
$127$		−	878.419i		−	0.613756i	0.951749	−	0.306878i	$-0.0992844\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1118.47			−0.745963			−0.372982	−	0.927839i	$-0.621665\pi$
$131$	−1118.47			−0.745963			−0.372982	+	0.927839i	$0.621665\pi$
$132$		0			0
$133$		−	3232.88i		−	2.10772i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			632.442i			0.394403i	0.980363	+	0.197201i	$0.0631853\pi$
$137$			632.442i			0.394403i	−0.980363	+	0.197201i	$0.936815\pi$
$138$		0			0
$139$	−262.662			−0.160279			−0.0801393	−	0.996784i	$-0.525537\pi$
$139$	−262.662			−0.160279			−0.0801393	+	0.996784i	$0.525537\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1883.49i			1.10143i
$144$		0			0
$145$	−581.114	−	567.935i	−0.332820	−	0.325272i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1695.22			−0.932066			−0.466033	−	0.884767i	$-0.654317\pi$
$149$	−1695.22			−0.932066			−0.466033	+	0.884767i	$0.654317\pi$
$150$		0			0
$151$	1150.02			0.619785			0.309892	−	0.950772i	$-0.399707\pi$
$151$	1150.02			0.619785			0.309892	+	0.950772i	$0.399707\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2225.09	+	2174.63i	1.15306	+	1.12691i
$156$		0			0
$157$			2666.80i			1.35563i	0.735232	+	0.677815i	$0.237073\pi$
$157$			2666.80i			1.35563i	−0.735232	+	0.677815i	$0.762927\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3350.34			−1.64002
$162$		0			0
$163$			2824.91i			1.35745i	0.734394	+	0.678723i	$0.237466\pi$
$163$			2824.91i			1.35745i	−0.734394	+	0.678723i	$0.762534\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3331.39i			1.54366i	0.635831	+	0.771828i	$0.280658\pi$
$167$			3331.39i			1.54366i	−0.635831	+	0.771828i	$0.719342\pi$
$168$		0			0
$169$	734.356			0.334254
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2521.27i			1.10803i	0.832508	+	0.554014i	$0.186904\pi$
$173$			2521.27i			1.10803i	−0.832508	+	0.554014i	$0.813096\pi$
$174$		0			0
$175$	−2765.62	+	63.4476i	−1.19464	+	0.0274068i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2289.50			0.956005			0.478003	−	0.878358i	$-0.341361\pi$
$179$	2289.50			0.956005			0.478003	+	0.878358i	$0.341361\pi$
$180$		0			0
$181$	2786.11			1.14414			0.572071	−	0.820204i	$-0.306140\pi$
$181$	2786.11			1.14414			0.572071	+	0.820204i	$0.306140\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−747.707	+	765.057i	−0.297149	+	0.304044i
$186$		0			0
$187$			749.405i			0.293058i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4267.53			−1.61669			−0.808345	−	0.588710i	$-0.799636\pi$
$191$	−4267.53			−1.61669			−0.808345	+	0.588710i	$0.799636\pi$
$192$		0			0
$193$			2304.87i			0.859628i	0.902917	+	0.429814i	$0.141421\pi$
$193$			2304.87i			0.859628i	−0.902917	+	0.429814i	$0.858579\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2620.32i		−	0.947664i	−0.880615	−	0.473832i	$-0.842870\pi$
$197$		−	2620.32i		−	0.947664i	0.880615	−	0.473832i	$-0.157130\pi$
$198$		0			0
$199$	−1583.79			−0.564181			−0.282091	−	0.959388i	$-0.591028\pi$
$199$	−1583.79			−0.564181			−0.282091	+	0.959388i	$0.591028\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1608.40i		−	0.556097i
$204$		0			0
$205$	2078.26	+	2031.13i	0.708058	+	0.692000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7194.25			−2.38104
$210$		0			0
$211$	−4487.33			−1.46408			−0.732038	−	0.681263i	$-0.761431\pi$
$211$	−4487.33			−1.46408			−0.732038	+	0.681263i	$0.761431\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1007.86	−	1031.25i	0.319700	−	0.327118i
$216$		0			0
$217$			6158.59i			1.92660i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−581.959			−0.177135
$222$		0			0
$223$		−	4960.57i		−	1.48961i	−0.667280	−	0.744807i	$-0.732541\pi$
$223$		−	4960.57i		−	1.48961i	0.667280	−	0.744807i	$-0.267459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2702.59i		−	0.790207i	−0.918637	−	0.395104i	$-0.870709\pi$
$227$		−	2702.59i		−	0.790207i	0.918637	−	0.395104i	$-0.129291\pi$
$228$		0			0
$229$	−6166.71			−1.77951			−0.889755	−	0.456439i	$-0.849125\pi$
$229$	−6166.71			−1.77951			−0.889755	+	0.456439i	$0.849125\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3412.55i			0.959499i	0.877405	+	0.479750i	$0.159273\pi$
$233$			3412.55i			0.959499i	−0.877405	+	0.479750i	$0.840727\pi$
$234$		0			0
$235$	2389.54	−	2444.99i	0.663305	−	0.678696i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1282.13			0.347004			0.173502	−	0.984834i	$-0.444492\pi$
$239$	1282.13			0.347004			0.173502	+	0.984834i	$0.444492\pi$
$240$		0			0
$241$	2727.52			0.729024			0.364512	−	0.931199i	$-0.381236\pi$
$241$	2727.52			0.729024			0.364512	+	0.931199i	$0.381236\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1173.57	−	1146.96i	−0.306028	−	0.299088i
$246$		0			0
$247$		−	5586.78i		−	1.43918i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7035.14			−1.76914			−0.884569	−	0.466409i	$-0.845548\pi$
$251$	−7035.14			−1.76914			−0.884569	+	0.466409i	$0.845548\pi$
$252$		0			0
$253$			7455.64i			1.85269i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	36.9560i		−	0.00896984i	−0.999990	−	0.00448492i	$-0.998572\pi$
$257$		−	36.9560i		−	0.00896984i	0.999990	−	0.00448492i	$-0.00142760\pi$
$258$		0			0
$259$	−2117.52			−0.508016
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2080.87i		−	0.487879i	−0.969790	−	0.243939i	$-0.921560\pi$
$263$		−	2080.87i		−	0.487879i	0.969790	−	0.243939i	$-0.0784398\pi$
$264$		0			0
$265$	2389.06	−	2444.50i	0.553808	−	0.566659i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7280.62			1.65021			0.825106	−	0.564977i	$-0.191115\pi$
$269$	7280.62			1.65021			0.825106	+	0.564977i	$0.191115\pi$
$270$		0			0
$271$	7780.41			1.74401			0.872004	−	0.489499i	$-0.162820\pi$
$271$	7780.41			1.74401			0.872004	+	0.489499i	$0.162820\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	141.192	+	6154.45i	0.0309608	+	1.34955i
$276$		0			0
$277$		−	3486.25i		−	0.756203i	−0.925764	−	0.378101i	$-0.876577\pi$
$277$		−	3486.25i		−	0.756203i	0.925764	−	0.378101i	$-0.123423\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.7871			−0.00335152			−0.00167576	−	0.999999i	$-0.500533\pi$
$281$	−15.7871			−0.00335152			−0.00167576	+	0.999999i	$0.500533\pi$
$282$		0			0
$283$		−	4343.05i		−	0.912252i	−0.889915	−	0.456126i	$-0.849237\pi$
$283$		−	4343.05i		−	0.912252i	0.889915	−	0.456126i	$-0.150763\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5752.18i			1.18307i
$288$		0			0
$289$	4681.45			0.952870
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1317.55i			0.262703i	0.991336	+	0.131351i	$0.0419316\pi$
$293$			1317.55i			0.262703i	−0.991336	+	0.131351i	$0.958068\pi$
$294$		0			0
$295$	−3452.02	−	3373.73i	−0.681302	−	0.665851i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5789.76			−1.11983
$300$		0			0
$301$	2854.28			0.546570
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1793.55	−	1752.88i	−0.336716	−	0.329080i
$306$		0			0
$307$		−	4475.06i		−	0.831940i	−0.909378	−	0.415970i	$-0.863442\pi$
$307$		−	4475.06i		−	0.831940i	0.909378	−	0.415970i	$-0.136558\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3599.11			0.656228			0.328114	−	0.944638i	$-0.393587\pi$
$311$	3599.11			0.656228			0.328114	+	0.944638i	$0.393587\pi$
$312$		0			0
$313$			2137.83i			0.386061i	0.981193	+	0.193031i	$0.0618317\pi$
$313$			2137.83i			0.386061i	−0.981193	+	0.193031i	$0.938168\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			7563.30i			1.34005i	0.742337	+	0.670027i	$0.233717\pi$
$317$			7563.30i			1.34005i	−0.742337	+	0.670027i	$0.766283\pi$
$318$		0			0
$319$	−3579.24			−0.628210
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2222.87i		−	0.382923i
$324$		0			0
$325$	−4779.31	+	109.645i	−0.815718	+	0.0187138i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6767.22			1.13401
$330$		0			0
$331$	−3716.24			−0.617108			−0.308554	−	0.951207i	$-0.599845\pi$
$331$	−3716.24			−0.617108			−0.308554	+	0.951207i	$0.599845\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5911.59	−	6048.76i	0.964133	−	0.986505i
$336$		0			0
$337$		−	7090.32i		−	1.14610i	−0.819522	−	0.573048i	$-0.805761\pi$
$337$		−	7090.32i		−	1.14610i	0.819522	−	0.573048i	$-0.194239\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13704.9			2.17643
$342$		0			0
$343$			4342.66i			0.683620i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5471.00i		−	0.846394i	−0.906038	−	0.423197i	$-0.860908\pi$
$347$		−	5471.00i		−	0.846394i	0.906038	−	0.423197i	$-0.139092\pi$
$348$		0			0
$349$	−406.176			−0.0622982			−0.0311491	−	0.999515i	$-0.509917\pi$
$349$	−406.176			−0.0622982			−0.0311491	+	0.999515i	$0.509917\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	3123.98i		−	0.471027i	−0.971871	−	0.235514i	$-0.924323\pi$
$353$		−	3123.98i		−	0.471027i	0.971871	−	0.235514i	$-0.0756773\pi$
$354$		0			0
$355$	−6191.19	−	6050.79i	−0.925618	−	0.904627i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3682.80			0.541422			0.270711	−	0.962661i	$-0.412741\pi$
$359$	3682.80			0.541422			0.270711	+	0.962661i	$0.412741\pi$
$360$		0			0
$361$	14480.5			2.11117
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1072.42	−	1097.30i	0.153789	−	0.157357i
$366$		0			0
$367$		−	482.248i		−	0.0685917i	−0.999412	−	0.0342959i	$-0.989081\pi$
$367$		−	482.248i		−	0.0685917i	0.999412	−	0.0342959i	$-0.0109189\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6765.87			0.946810
$372$		0			0
$373$			5942.41i			0.824897i	0.910981	+	0.412449i	$0.135326\pi$
$373$			5942.41i			0.824897i	−0.910981	+	0.412449i	$0.864674\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2779.50i		−	0.379712i
$378$		0			0
$379$	−4498.82			−0.609733			−0.304867	−	0.952395i	$-0.598612\pi$
$379$	−4498.82			−0.609733			−0.304867	+	0.952395i	$0.598612\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	853.032i		−	0.113807i	−0.998380	−	0.0569033i	$-0.981877\pi$
$383$		−	853.032i		−	0.113807i	0.998380	−	0.0569033i	$-0.0181227\pi$
$384$		0			0
$385$	−8517.12	+	8714.76i	−1.12746	+	1.15362i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6450.62			−0.840770			−0.420385	−	0.907346i	$-0.638105\pi$
$389$	−6450.62			−0.840770			−0.420385	+	0.907346i	$0.638105\pi$
$390$		0			0
$391$	−2303.64			−0.297954
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2515.11	−	2458.07i	−0.320377	−	0.313111i
$396$		0			0
$397$			1920.34i			0.242769i	0.992606	+	0.121385i	$0.0387334\pi$
$397$			1920.34i			0.242769i	−0.992606	+	0.121385i	$0.961267\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4326.99			−0.538852			−0.269426	−	0.963021i	$-0.586834\pi$
$401$	−4326.99			−0.538852			−0.269426	+	0.963021i	$0.586834\pi$
$402$		0			0
$403$			10642.7i			1.31551i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4712.20i			0.573894i
$408$		0			0
$409$	10055.1			1.21563			0.607815	−	0.794079i	$-0.292046\pi$
$409$	10055.1			1.21563			0.607815	+	0.794079i	$0.292046\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	9554.46i		−	1.13836i
$414$		0			0
$415$	−6672.39	+	6827.22i	−0.789241	+	0.807555i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8156.01			−0.950948			−0.475474	−	0.879730i	$-0.657723\pi$
$419$	−8156.01			−0.950948			−0.475474	+	0.879730i	$0.657723\pi$
$420$		0			0
$421$	8614.86			0.997298			0.498649	−	0.866804i	$-0.333830\pi$
$421$	8614.86			0.997298			0.498649	+	0.866804i	$0.333830\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1901.60	+	43.6255i	−0.217038	+	0.00497917i
$426$		0			0
$427$		−	4964.17i		−	0.562607i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2046.83			−0.228753			−0.114376	−	0.993437i	$-0.536487\pi$
$431$	−2046.83			−0.228753			−0.114376	+	0.993437i	$0.536487\pi$
$432$		0			0
$433$			13467.6i			1.49471i	0.664422	+	0.747357i	$0.268678\pi$
$433$			13467.6i			1.49471i	−0.664422	+	0.747357i	$0.731322\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	22114.8i		−	2.42081i
$438$		0			0
$439$	−9260.44			−1.00678			−0.503390	−	0.864059i	$-0.667914\pi$
$439$	−9260.44			−1.00678			−0.503390	+	0.864059i	$0.667914\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	7.46000i		−	0.000800079i	−1.00000		0.000400040i	$-0.999873\pi$
$443$		−	7.46000i		−	0.000800079i	1.00000		0.000400040i	$-0.000127337\pi$
$444$		0			0
$445$	−8002.48	−	7821.00i	−0.852481	−	0.833148i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10460.6			1.09948			0.549740	−	0.835336i	$-0.314727\pi$
$449$	10460.6			1.09948			0.549740	+	0.835336i	$0.314727\pi$
$450$		0			0
$451$	12800.6			1.33648
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6767.54	−	6614.06i	−0.697291	−	0.681477i
$456$		0			0
$457$			4539.85i			0.464694i	0.972633	+	0.232347i	$0.0746405\pi$
$457$			4539.85i			0.464694i	−0.972633	+	0.232347i	$0.925359\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16139.2			1.63054			0.815269	−	0.579082i	$-0.196589\pi$
$461$	16139.2			1.63054			0.815269	+	0.579082i	$0.196589\pi$
$462$		0			0
$463$		−	16494.9i		−	1.65568i	−0.560963	−	0.827841i	$-0.689569\pi$
$463$		−	16494.9i		−	1.65568i	0.560963	−	0.827841i	$-0.310431\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	9259.72i		−	0.917534i	−0.888557	−	0.458767i	$-0.848291\pi$
$467$		−	9259.72i		−	0.917534i	0.888557	−	0.458767i	$-0.151709\pi$
$468$		0			0
$469$	16741.7			1.64832
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6351.73i		−	0.617448i
$474$		0			0
$475$	−418.803	−	18255.3i	−0.0404548	−	1.76339i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11778.3			1.12352			0.561758	−	0.827302i	$-0.310125\pi$
$479$	11778.3			1.12352			0.561758	+	0.827302i	$0.310125\pi$
$480$		0			0
$481$	−3659.31			−0.346882
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7736.14	+	7915.66i	−0.724289	+	0.741096i
$486$		0			0
$487$			20133.8i			1.87341i	0.350119	+	0.936705i	$0.386141\pi$
$487$			20133.8i			1.87341i	−0.350119	+	0.936705i	$0.613859\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18199.0			1.67272			0.836362	−	0.548177i	$-0.184678\pi$
$491$	18199.0			1.67272			0.836362	+	0.548177i	$0.184678\pi$
$492$		0			0
$493$		−	1105.91i		−	0.101030i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	17135.9i		−	1.54658i
$498$		0			0
$499$	−11949.1			−1.07197			−0.535987	−	0.844226i	$-0.680060\pi$
$499$	−11949.1			−1.07197			−0.535987	+	0.844226i	$0.680060\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	3940.53i		−	0.349303i	−0.984630	−	0.174652i	$-0.944120\pi$
$503$		−	3940.53i		−	0.349303i	0.984630	−	0.174652i	$-0.0558799\pi$
$504$		0			0
$505$	11944.3	+	11673.4i	1.05251	+	1.02864i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6175.68			0.537784			0.268892	−	0.963170i	$-0.413342\pi$
$509$	6175.68			0.537784			0.268892	+	0.963170i	$0.413342\pi$
$510$		0			0
$511$	3037.10			0.262923
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14972.8	−	15320.3i	1.28113	−	1.31086i
$516$		0			0
$517$		−	15059.4i		−	1.28106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6543.87			0.550273			0.275136	−	0.961405i	$-0.411277\pi$
$521$	6543.87			0.550273			0.275136	+	0.961405i	$0.411277\pi$
$522$		0			0
$523$			5888.49i			0.492324i	0.969229	+	0.246162i	$0.0791696\pi$
$523$			5888.49i			0.492324i	−0.969229	+	0.246162i	$0.920830\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4234.54i			0.350018i
$528$		0			0
$529$	−10751.3			−0.883643
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9940.42i			0.807818i
$534$		0			0
$535$	−9619.94	+	9843.16i	−0.777395	+	0.795434i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7228.36			−0.577640
$540$		0			0
$541$	3535.63			0.280977			0.140489	−	0.990082i	$-0.455133\pi$
$541$	3535.63			0.280977			0.140489	+	0.990082i	$0.455133\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14849.8	+	14513.0i	1.16715	+	1.14068i
$546$		0			0
$547$		−	3988.16i		−	0.311740i	−0.987778	−	0.155870i	$-0.950182\pi$
$547$		−	3988.16i		−	0.311740i	0.987778	−	0.155870i	$-0.0498180\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10616.7			0.820847
$552$		0			0
$553$		−	6961.29i		−	0.535306i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			518.167i			0.0394173i	0.999806	+	0.0197087i	$0.00627387\pi$
$557$			518.167i			0.0394173i	−0.999806	+	0.0197087i	$0.993726\pi$
$558$		0			0
$559$	4932.51			0.373207
$560$		0			0
$561$		0			0
$562$		0			0
$563$			2136.00i			0.159897i	0.996799	+	0.0799483i	$0.0254755\pi$
$563$			2136.00i			0.159897i	−0.996799	+	0.0799483i	$0.974524\pi$
$564$		0			0
$565$	−3316.48	+	3393.44i	−0.246948	+	0.252678i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	563.135			0.0414900			0.0207450	−	0.999785i	$-0.493396\pi$
$569$	563.135			0.0414900			0.0207450	+	0.999785i	$0.493396\pi$
$570$		0			0
$571$	21769.5			1.59549			0.797747	−	0.602993i	$-0.206025\pi$
$571$	21769.5			1.59549			0.797747	+	0.602993i	$0.206025\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−18918.5	+	434.019i	−1.37210	+	0.0314780i
$576$		0			0
$577$			16315.0i			1.17713i	0.808451	+	0.588564i	$0.200306\pi$
$577$			16315.0i			1.17713i	−0.808451	+	0.588564i	$0.799694\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18896.3			−1.34931
$582$		0			0
$583$		−	15056.4i		−	1.06959i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	5026.10i		−	0.353406i	−0.984264	−	0.176703i	$-0.943457\pi$
$587$		−	5026.10i		−	0.353406i	0.984264	−	0.176703i	$-0.0565432\pi$
$588$		0			0
$589$	−40651.4			−2.84382
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	9476.31i		−	0.656232i	−0.944637	−	0.328116i	$-0.893586\pi$
$593$		−	9476.31i		−	0.656232i	0.944637	−	0.328116i	$-0.106414\pi$
$594$		0			0
$595$	−2692.68	−	2631.61i	−0.185528	−	0.181320i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21096.9			1.43906			0.719528	−	0.694463i	$-0.244358\pi$
$599$	21096.9			1.43906			0.719528	+	0.694463i	$0.244358\pi$
$600$		0			0
$601$	27375.7			1.85803			0.929017	−	0.370037i	$-0.120655\pi$
$601$	27375.7			1.85803			0.929017	+	0.370037i	$0.120655\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8750.82	+	8552.36i	0.588052	+	0.574716i
$606$		0			0
$607$			19100.9i			1.27723i	0.769525	+	0.638617i	$0.220493\pi$
$607$			19100.9i			1.27723i	−0.769525	+	0.638617i	$0.779507\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11694.5			0.774320
$612$		0			0
$613$		−	26070.2i		−	1.71772i	−0.512207	−	0.858862i	$-0.671172\pi$
$613$		−	26070.2i		−	1.71772i	0.512207	−	0.858862i	$-0.328828\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			18552.1i			1.21050i	0.796036	+	0.605250i	$0.206927\pi$
$617$			18552.1i			1.21050i	−0.796036	+	0.605250i	$0.793073\pi$
$618$		0			0
$619$	3055.27			0.198387			0.0991937	−	0.995068i	$-0.468374\pi$
$619$	3055.27			0.198387			0.0991937	+	0.995068i	$0.468374\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	22149.2i		−	1.42438i
$624$		0			0
$625$	−15608.6	+	716.545i	−0.998948	+	0.0458589i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1455.97			−0.0922947
$630$		0			0
$631$	7082.20			0.446811			0.223406	−	0.974726i	$-0.428283\pi$
$631$	7082.20			0.446811			0.223406	+	0.974726i	$0.428283\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6864.41	−	7023.70i	0.428986	−	0.438940i
$636$		0			0
$637$		−	5613.27i		−	0.349146i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23751.8			−1.46356			−0.731779	−	0.681542i	$-0.761310\pi$
$641$	−23751.8			−1.46356			−0.731779	+	0.681542i	$0.761310\pi$
$642$		0			0
$643$		−	17726.8i		−	1.08721i	−0.839341	−	0.543606i	$-0.817059\pi$
$643$		−	17726.8i		−	1.08721i	0.839341	−	0.543606i	$-0.182941\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			10859.5i			0.659864i	0.944005	+	0.329932i	$0.107026\pi$
$647$			10859.5i			0.659864i	−0.944005	+	0.329932i	$0.892974\pi$
$648$		0			0
$649$	−21261.9			−1.28598
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	21819.8i		−	1.30762i	−0.756660	−	0.653809i	$-0.773170\pi$
$653$		−	21819.8i		−	1.30762i	0.756660	−	0.653809i	$-0.226830\pi$
$654$		0			0
$655$	−8943.11	−	8740.29i	−0.533490	−	0.521392i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17500.0			1.03445			0.517225	−	0.855850i	$-0.326965\pi$
$659$	17500.0			1.03445			0.517225	+	0.855850i	$0.326965\pi$
$660$		0			0
$661$	17559.9			1.03329			0.516643	−	0.856201i	$-0.327181\pi$
$661$	17559.9			1.03329			0.516643	+	0.856201i	$0.327181\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25263.4	−	25849.6i	1.47319	−	1.50738i
$666$		0			0
$667$		−	11002.4i		−	0.638704i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11047.0			−0.635564
$672$		0			0
$673$			5361.23i			0.307073i	0.988143	+	0.153536i	$0.0490663\pi$
$673$			5361.23i			0.307073i	−0.988143	+	0.153536i	$0.950934\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	5572.55i		−	0.316352i	−0.987411	−	0.158176i	$-0.949439\pi$
$677$		−	5572.55i		−	0.316352i	0.987411	−	0.158176i	$-0.0505614\pi$
$678$		0			0
$679$	−21908.9			−1.23827
$680$		0			0
$681$		0			0
$682$		0			0
$683$			26334.0i			1.47532i	0.675173	+	0.737659i	$0.264069\pi$
$683$			26334.0i			1.47532i	−0.675173	+	0.737659i	$0.735931\pi$
$684$		0			0
$685$	−4942.22	+	5056.91i	−0.275668	+	0.282065i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11692.2			0.646497
$690$		0			0
$691$	28230.8			1.55420			0.777099	−	0.629378i	$-0.216690\pi$
$691$	28230.8			1.55420			0.777099	+	0.629378i	$0.216690\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2100.20	−	2052.58i	−0.114626	−	0.112027i
$696$		0			0
$697$			3955.11i			0.214936i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24756.6			−1.33387			−0.666936	−	0.745115i	$-0.732394\pi$
$701$	−24756.6			−1.33387			−0.666936	+	0.745115i	$0.732394\pi$
$702$		0			0
$703$		−	13977.3i		−	0.749875i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			33059.4i			1.75859i
$708$		0			0
$709$	−5763.28			−0.305281			−0.152641	−	0.988282i	$-0.548778\pi$
$709$	−5763.28			−0.305281			−0.152641	+	0.988282i	$0.548778\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			42128.4i			2.21279i
$714$		0			0
$715$	−14718.5	+	15060.1i	−0.769849	+	0.787713i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12988.5			−0.673698			−0.336849	−	0.941559i	$-0.609361\pi$
$719$	−12988.5			−0.673698			−0.336849	+	0.941559i	$0.609361\pi$
$720$		0			0
$721$	42403.3			2.19027
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−208.360	−	9082.24i	−0.0106735	−	0.465250i
$726$		0			0
$727$		−	26135.5i		−	1.33330i	−0.745370	−	0.666651i	$-0.767727\pi$
$727$		−	26135.5i		−	1.33330i	0.745370	−	0.666651i	$-0.232273\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1962.55			0.0992991
$732$		0			0
$733$		−	15152.6i		−	0.763539i	−0.924257	−	0.381770i	$-0.875315\pi$
$733$		−	15152.6i		−	0.763539i	0.924257	−	0.381770i	$-0.124685\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	37256.0i		−	1.86207i
$738$		0			0
$739$	38081.7			1.89561			0.947807	−	0.318846i	$-0.103295\pi$
$739$	38081.7			1.89561			0.947807	+	0.318846i	$0.103295\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32872.6i		−	1.62312i	−0.584267	−	0.811561i	$-0.698618\pi$
$743$		−	32872.6i		−	1.62312i	0.584267	−	0.811561i	$-0.301382\pi$
$744$		0			0
$745$	−13554.7	−	13247.3i	−0.666585	−	0.651468i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27243.8			−1.32906
$750$		0			0
$751$	10531.5			0.511719			0.255860	−	0.966714i	$-0.417642\pi$
$751$	10531.5			0.511719			0.255860	+	0.966714i	$0.417642\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9195.40	+	8986.86i	0.443251	+	0.433199i
$756$		0			0
$757$			24131.8i			1.15863i	0.815103	+	0.579316i	$0.196680\pi$
$757$			24131.8i			1.15863i	−0.815103	+	0.579316i	$0.803320\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1915.59			−0.0912485			−0.0456242	−	0.998959i	$-0.514528\pi$
$761$	−1915.59			−0.0912485			−0.0456242	+	0.998959i	$0.514528\pi$
$762$		0			0
$763$			41101.2i			1.95015i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	16511.2i		−	0.777293i
$768$		0			0
$769$	32696.2			1.53323			0.766616	−	0.642105i	$-0.221939\pi$
$769$	32696.2			1.53323			0.766616	+	0.642105i	$0.221939\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			7482.57i			0.348162i	0.984731	+	0.174081i	$0.0556955\pi$
$773$			7482.57i			0.348162i	−0.984731	+	0.174081i	$0.944305\pi$
$774$		0			0
$775$	797.814	+	34776.0i	0.0369785	+	1.61186i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−37968.9			−1.74631
$780$		0			0
$781$	−38133.3			−1.74714
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20839.7	+	21323.3i	−0.947519	+	0.969506i
$786$		0			0
$787$			5828.55i			0.263997i	0.991250	+	0.131998i	$0.0421394\pi$
$787$			5828.55i			0.263997i	−0.991250	+	0.131998i	$0.957861\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9392.33			−0.422191
$792$		0			0
$793$		−	8578.65i		−	0.384157i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12604.7i			0.560201i	0.959971	+	0.280101i	$0.0903679\pi$
$797$			12604.7i			0.560201i	−0.959971	+	0.280101i	$0.909632\pi$
$798$		0			0
$799$	4653.03			0.206023
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	6758.58i		−	0.297018i
$804$		0			0
$805$	−26788.8	−	26181.2i	−1.17289	−	1.14629i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4937.28			−0.214568			−0.107284	−	0.994228i	$-0.534215\pi$
$809$	−4937.28			−0.214568			−0.107284	+	0.994228i	$0.534215\pi$
$810$		0			0
$811$	16096.2			0.696933			0.348467	−	0.937321i	$-0.386702\pi$
$811$	16096.2			0.696933			0.348467	+	0.937321i	$0.386702\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−22075.2	+	22587.5i	−0.948788	+	0.970804i
$816$		0			0
$817$			18840.4i			0.806785i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18828.0			−0.800370			−0.400185	−	0.916434i	$-0.631054\pi$
$821$	−18828.0			−0.800370			−0.400185	+	0.916434i	$0.631054\pi$
$822$		0			0
$823$			6689.04i			0.283311i	0.989916	+	0.141656i	$0.0452426\pi$
$823$			6689.04i			0.283311i	−0.989916	+	0.141656i	$0.954757\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			39525.0i			1.66193i	0.556321	+	0.830967i	$0.312212\pi$
$827$			39525.0i			1.66193i	−0.556321	+	0.830967i	$0.687788\pi$
$828$		0			0
$829$	−40885.3			−1.71291			−0.856457	−	0.516218i	$-0.827340\pi$
$829$	−40885.3			−1.71291			−0.856457	+	0.516218i	$0.827340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2233.41i		−	0.0928970i
$834$		0			0
$835$	−26033.2	+	26637.3i	−1.07894	+	1.10398i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28276.8			1.16355			0.581777	−	0.813348i	$-0.302358\pi$
$839$	28276.8			1.16355			0.581777	+	0.813348i	$0.302358\pi$
$840$		0			0
$841$	−19107.0			−0.783429
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5871.80	+	5738.63i	0.239048	+	0.233627i
$846$		0			0
$847$			24220.4i			0.982555i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14485.1			−0.583481
$852$		0			0
$853$		−	33085.8i		−	1.32806i	−0.747705	−	0.664032i	$-0.768844\pi$
$853$		−	33085.8i		−	1.32806i	0.747705	−	0.664032i	$-0.231156\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	2914.06i		−	0.116152i	−0.998312	−	0.0580761i	$-0.981503\pi$
$857$		−	2914.06i		−	0.116152i	0.998312	−	0.0580761i	$-0.0184966\pi$
$858$		0			0
$859$	−15497.3			−0.615554			−0.307777	−	0.951459i	$-0.599585\pi$
$859$	−15497.3			−0.615554			−0.307777	+	0.951459i	$0.599585\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17755.7i			0.700362i	0.936682	+	0.350181i	$0.113880\pi$
$863$			17755.7i			0.700362i	−0.936682	+	0.350181i	$0.886120\pi$
$864$		0			0
$865$	−19702.5	+	20159.7i	−0.774457	+	0.792428i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15491.2			−0.604723
$870$		0			0
$871$	28931.6			1.12550
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22609.3	−	21104.7i	−0.873525	−	0.815392i
$876$		0			0
$877$		−	35305.7i		−	1.35940i	−0.733492	−	0.679698i	$-0.762111\pi$
$877$		−	35305.7i		−	1.35940i	0.733492	−	0.679698i	$-0.237889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	29704.5			1.13595			0.567975	−	0.823046i	$-0.307727\pi$
$881$	29704.5			1.13595			0.567975	+	0.823046i	$0.307727\pi$
$882$		0			0
$883$			20091.2i			0.765711i	0.923808	+	0.382856i	$0.125059\pi$
$883$			20091.2i			0.765711i	−0.923808	+	0.382856i	$0.874941\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			11805.8i			0.446900i	0.974715	+	0.223450i	$0.0717320\pi$
$887$			11805.8i			0.446900i	−0.974715	+	0.223450i	$0.928268\pi$
$888$		0			0
$889$	19440.1			0.733409
$890$		0			0
$891$		0			0
$892$		0			0
$893$			44668.9i			1.67389i
$894$		0			0
$895$	18306.4	+	17891.3i	0.683706	+	0.668201i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20224.6			−0.750311
$900$		0			0
$901$	4652.10			0.172013
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22277.3	+	21772.0i	0.818255	+	0.799699i
$906$		0			0
$907$			11439.4i			0.418785i	0.977832	+	0.209393i	$0.0671486\pi$
$907$			11439.4i			0.418785i	−0.977832	+	0.209393i	$0.932851\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18860.2			−0.685912			−0.342956	−	0.939351i	$-0.611428\pi$
$911$	−18860.2			−0.685912			−0.342956	+	0.939351i	$0.611428\pi$
$912$		0			0
$913$			42050.7i			1.52429i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	24752.7i		−	0.891390i
$918$		0			0
$919$	14351.9			0.515153			0.257577	−	0.966258i	$-0.417076\pi$
$919$	14351.9			0.515153			0.257577	+	0.966258i	$0.417076\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	29612.8i		−	1.05603i
$924$		0			0
$925$	−11957.1	+	274.314i	−0.425024	+	0.00975068i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27122.4			0.957867			0.478934	−	0.877851i	$-0.341023\pi$
$929$	27122.4			0.957867			0.478934	+	0.877851i	$0.341023\pi$
$930$		0			0
$931$	21440.7			0.754769
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5856.23	+	5992.12i	−0.204833	+	0.209586i
$936$		0			0
$937$		−	41538.8i		−	1.44825i	−0.689668	−	0.724126i	$-0.742243\pi$
$937$		−	41538.8i		−	1.44825i	0.689668	−	0.724126i	$-0.257757\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17371.2			0.601792			0.300896	−	0.953657i	$-0.402714\pi$
$941$	17371.2			0.601792			0.300896	+	0.953657i	$0.402714\pi$
$942$		0			0
$943$			39348.3i			1.35881i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			27809.7i			0.954271i	0.878830	+	0.477136i	$0.158325\pi$
$947$			27809.7i			0.954271i	−0.878830	+	0.477136i	$0.841675\pi$
$948$		0			0
$949$	5248.45			0.179528
$950$		0			0
$951$		0			0
$952$		0			0
$953$			47023.6i			1.59837i	0.601088	+	0.799183i	$0.294734\pi$
$953$			47023.6i			1.59837i	−0.601088	+	0.799183i	$0.705266\pi$
$954$		0			0
$955$	−34122.5	−	33348.6i	−1.15621	−	1.12999i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13996.5			−0.471292
$960$		0			0
$961$	47649.3			1.59945
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18011.4	+	18429.4i	−0.600838	+	0.614780i
$966$		0			0
$967$		−	22050.7i		−	0.733302i	−0.930359	−	0.366651i	$-0.880504\pi$
$967$		−	22050.7i		−	0.733302i	0.930359	−	0.366651i	$-0.119496\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8198.43			−0.270958			−0.135479	−	0.990780i	$-0.543257\pi$
$971$	−8198.43			−0.270958			−0.135479	+	0.990780i	$0.543257\pi$
$972$		0			0
$973$		−	5812.93i		−	0.191525i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15262.5i		−	0.499787i	−0.968273	−	0.249893i	$-0.919604\pi$
$977$		−	15262.5i		−	0.499787i	0.968273	−	0.249893i	$-0.0803955\pi$
$978$		0			0
$979$	−49289.5			−1.60909
$980$		0			0
$981$		0			0
$982$		0			0
$983$			58803.2i			1.90797i	0.299860	+	0.953983i	$0.403060\pi$
$983$			58803.2i			1.90797i	−0.299860	+	0.953983i	$0.596940\pi$
$984$		0			0
$985$	20476.5	−	20951.6i	0.662370	−	0.677740i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19524.9			0.627762
$990$		0			0
$991$	7820.16			0.250672			0.125336	−	0.992114i	$-0.459999\pi$
$991$	7820.16			0.250672			0.125336	+	0.992114i	$0.459999\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−12663.8	−	12376.6i	−0.403485	−	0.394335i
$996$		0			0
$997$		−	28342.4i		−	0.900315i	−0.892949	−	0.450157i	$-0.851368\pi$
$997$		−	28342.4i		−	0.900315i	0.892949	−	0.450157i	$-0.148632\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.f.d.649.18	yes	20
3.2	odd	2	inner	1080.4.f.d.649.3	✓	20
5.4	even	2	inner	1080.4.f.d.649.17	yes	20
15.14	odd	2	inner	1080.4.f.d.649.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.f.d.649.3	✓	20	3.2	odd	2	inner
1080.4.f.d.649.4	yes	20	15.14	odd	2	inner
1080.4.f.d.649.17	yes	20	5.4	even	2	inner
1080.4.f.d.649.18	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(7.99584 + 7.81451i) q^{5} +22.1308i q^{7} +O(q^{10})\)	\(q+(7.99584 + 7.81451i) q^{5} +22.1308i q^{7} +49.2486 q^{11} +38.2445i q^{13} +15.2168i q^{17} -146.080 q^{19} +151.388i q^{23} +(2.86693 + 124.967i) q^{25} -72.6770 q^{29} +278.281 q^{31} +(-172.941 + 176.954i) q^{35} +95.6819i q^{37} +259.917 q^{41} -128.973i q^{43} -305.783i q^{47} -146.773 q^{49} -305.722i q^{53} +(393.784 + 384.853i) q^{55} -431.726 q^{59} -224.310 q^{61} +(-298.862 + 305.797i) q^{65} -756.489i q^{67} -774.302 q^{71} -137.234i q^{73} +1089.91i q^{77} -314.552 q^{79} +853.847i q^{83} +(-118.912 + 121.671i) q^{85} -1000.83 q^{89} -846.383 q^{91} +(-1168.04 - 1141.55i) q^{95} +989.972i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 100 q^{19} - 24 q^{25} + 228 q^{31} - 252 q^{49} - 64 q^{55} + 748 q^{61} - 668 q^{79} - 1500 q^{85} + 1744 q^{91}+O(q^{100}) \) 20 * q - 100 * q^19 - 24 * q^25 + 228 * q^31 - 252 * q^49 - 64 * q^55 + 748 * q^61 - 668 * q^79 - 1500 * q^85 + 1744 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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