LMFDB - Embedded newform 5040.2.t.t.1009.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1009,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5040.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.t (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:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.3
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.t.1009.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.224745 - 2.22474i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(0.224745 - 2.22474i) q^{5} +1.00000i q^{7} +4.89898 q^{11} +0.449490i q^{13} -2.00000i q^{17} +6.44949 q^{19} +6.89898i q^{23} +(-4.89898 - 1.00000i) q^{25} -2.89898 q^{29} +0.898979 q^{31} +(2.22474 + 0.224745i) q^{35} +2.00000i q^{37} +10.8990 q^{41} +8.89898i q^{43} -0.898979i q^{47} -1.00000 q^{49} -1.10102i q^{53} +(1.10102 - 10.8990i) q^{55} +6.44949 q^{59} +8.44949 q^{61} +(1.00000 + 0.101021i) q^{65} +8.00000i q^{67} -10.8990 q^{71} +6.89898i q^{73} +4.89898i q^{77} -2.89898 q^{79} -2.44949i q^{83} +(-4.44949 - 0.449490i) q^{85} -10.0000 q^{89} -0.449490 q^{91} +(1.44949 - 14.3485i) q^{95} -3.79796i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 16 q^{19} + 8 q^{29} - 16 q^{31} + 4 q^{35} + 24 q^{41} - 4 q^{49} + 24 q^{55} + 16 q^{59} + 24 q^{61} + 4 q^{65} - 24 q^{71} + 8 q^{79} - 8 q^{85} - 40 q^{89} + 8 q^{91} - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^5 + 16 * q^19 + 8 * q^29 - 16 * q^31 + 4 * q^35 + 24 * q^41 - 4 * q^49 + 24 * q^55 + 16 * q^59 + 24 * q^61 + 4 * q^65 - 24 * q^71 + 8 * q^79 - 8 * q^85 - 40 * q^89 + 8 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$5$	0.224745	−	2.22474i	0.100509	−	0.994936i
$5$	0.224745	−	2.22474i	0.100509	−	0.994936i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$			0.449490i			0.124666i	0.998055	+	0.0623330i	$0.0198541\pi$
$13$			0.449490i			0.124666i	−0.998055	+	0.0623330i	$0.980146\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	6.44949			1.47961			0.739807	−	0.672819i	$-0.234917\pi$
$19$	6.44949			1.47961			0.739807	+	0.672819i	$0.234917\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.89898i			1.43854i	0.694732	+	0.719268i	$0.255523\pi$
$23$			6.89898i			1.43854i	−0.694732	+	0.719268i	$0.744477\pi$
$24$		0			0
$25$	−4.89898	−	1.00000i	−0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.89898			−0.538327			−0.269163	−	0.963095i	$-0.586747\pi$
$29$	−2.89898			−0.538327			−0.269163	+	0.963095i	$0.586747\pi$
$30$		0			0
$31$	0.898979			0.161461			0.0807307	−	0.996736i	$-0.474275\pi$
$31$	0.898979			0.161461			0.0807307	+	0.996736i	$0.474275\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.22474	+	0.224745i	0.376051	+	0.0379888i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.8990			1.70213			0.851067	−	0.525057i	$-0.175956\pi$
$41$	10.8990			1.70213			0.851067	+	0.525057i	$0.175956\pi$
$42$		0			0
$43$			8.89898i			1.35708i	0.734563	+	0.678541i	$0.237387\pi$
$43$			8.89898i			1.35708i	−0.734563	+	0.678541i	$0.762613\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.898979i		−	0.131130i	−0.997848	−	0.0655648i	$-0.979115\pi$
$47$		−	0.898979i		−	0.131130i	0.997848	−	0.0655648i	$-0.0208849\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.10102i		−	0.151237i	−0.997137	−	0.0756184i	$-0.975907\pi$
$53$		−	1.10102i		−	0.151237i	0.997137	−	0.0756184i	$-0.0240931\pi$
$54$		0			0
$55$	1.10102	−	10.8990i	0.148462	−	1.46962i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.44949			0.839652			0.419826	−	0.907605i	$-0.362091\pi$
$59$	6.44949			0.839652			0.419826	+	0.907605i	$0.362091\pi$
$60$		0			0
$61$	8.44949			1.08185			0.540923	−	0.841072i	$-0.318075\pi$
$61$	8.44949			1.08185			0.540923	+	0.841072i	$0.318075\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	+	0.101021i	0.124035	+	0.0125301i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.8990			−1.29347			−0.646735	−	0.762714i	$-0.723866\pi$
$71$	−10.8990			−1.29347			−0.646735	+	0.762714i	$0.723866\pi$
$72$		0			0
$73$			6.89898i			0.807464i	0.914877	+	0.403732i	$0.132287\pi$
$73$			6.89898i			0.807464i	−0.914877	+	0.403732i	$0.867713\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.89898i			0.558291i
$78$		0			0
$79$	−2.89898			−0.326161			−0.163080	−	0.986613i	$-0.552143\pi$
$79$	−2.89898			−0.326161			−0.163080	+	0.986613i	$0.552143\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.44949i		−	0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$		−	2.44949i		−	0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$		0			0
$85$	−4.44949	−	0.449490i	−0.482615	−	0.0487540i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−0.449490			−0.0471193
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.44949	−	14.3485i	0.148715	−	1.47212i
$96$		0			0
$97$		−	3.79796i		−	0.385624i	−0.981236	−	0.192812i	$-0.938239\pi$
$97$		−	3.79796i		−	0.385624i	0.981236	−	0.192812i	$-0.0617608\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.44949			−0.840756			−0.420378	−	0.907349i	$-0.638102\pi$
$101$	−8.44949			−0.840756			−0.420378	+	0.907349i	$0.638102\pi$
$102$		0			0
$103$			3.10102i			0.305553i	0.988261	+	0.152776i	$0.0488214\pi$
$103$			3.10102i			0.305553i	−0.988261	+	0.152776i	$0.951179\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	2.89898			0.277672			0.138836	−	0.990315i	$-0.455664\pi$
$109$	2.89898			0.277672			0.138836	+	0.990315i	$0.455664\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			0.202041i			0.0190064i	0.999955	+	0.00950321i	$0.00302501\pi$
$113$			0.202041i			0.0190064i	−0.999955	+	0.00950321i	$0.996975\pi$
$114$		0			0
$115$	15.3485	+	1.55051i	1.43125	+	0.144586i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000			0.183340
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.32577	+	10.6742i	−0.297465	+	0.954733i
$126$		0			0
$127$			5.10102i			0.452642i	0.974053	+	0.226321i	$0.0726699\pi$
$127$			5.10102i			0.452642i	−0.974053	+	0.226321i	$0.927330\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.55051			−0.135469			−0.0677344	−	0.997703i	$-0.521577\pi$
$131$	−1.55051			−0.135469			−0.0677344	+	0.997703i	$0.521577\pi$
$132$		0			0
$133$			6.44949i			0.559242i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	17.7980i		−	1.52058i	−0.649582	−	0.760291i	$-0.725056\pi$
$137$		−	17.7980i		−	1.52058i	0.649582	−	0.760291i	$-0.274944\pi$
$138$		0			0
$139$	6.44949			0.547039			0.273519	−	0.961867i	$-0.411812\pi$
$139$	6.44949			0.547039			0.273519	+	0.961867i	$0.411812\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.20204i			0.184144i
$144$		0			0
$145$	−0.651531	+	6.44949i	−0.0541067	+	0.535601i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.7980			1.29422			0.647110	−	0.762397i	$-0.275978\pi$
$149$	15.7980			1.29422			0.647110	+	0.762397i	$0.275978\pi$
$150$		0			0
$151$	19.5959			1.59469			0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959			1.59469			0.797347	+	0.603522i	$0.206236\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.202041	−	2.00000i	0.0162283	−	0.160644i
$156$		0			0
$157$			8.44949i			0.674343i	0.941443	+	0.337171i	$0.109470\pi$
$157$			8.44949i			0.674343i	−0.941443	+	0.337171i	$0.890530\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.89898			−0.543716
$162$		0			0
$163$		−	16.8990i		−	1.32363i	−0.749667	−	0.661815i	$-0.769786\pi$
$163$		−	16.8990i		−	1.32363i	0.749667	−	0.661815i	$-0.230214\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.89898i			0.379094i	0.981872	+	0.189547i	$0.0607020\pi$
$167$			4.89898i			0.379094i	−0.981872	+	0.189547i	$0.939298\pi$
$168$		0			0
$169$	12.7980			0.984458
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.2474i			1.38733i	0.720299	+	0.693664i	$0.244005\pi$
$173$			18.2474i			1.38733i	−0.720299	+	0.693664i	$0.755995\pi$
$174$		0			0
$175$	1.00000	−	4.89898i	0.0755929	−	0.370328i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.79796			0.433360			0.216680	−	0.976243i	$-0.430477\pi$
$179$	5.79796			0.433360			0.216680	+	0.976243i	$0.430477\pi$
$180$		0			0
$181$	14.2474			1.05900			0.529502	−	0.848309i	$-0.322379\pi$
$181$	14.2474			1.05900			0.529502	+	0.848309i	$0.322379\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.44949	+	0.449490i	0.327133	+	0.0330471i
$186$		0			0
$187$		−	9.79796i		−	0.716498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.6969			−1.20815			−0.604074	−	0.796928i	$-0.706457\pi$
$191$	−16.6969			−1.20815			−0.604074	+	0.796928i	$0.706457\pi$
$192$		0			0
$193$		−	17.5959i		−	1.26658i	−0.773914	−	0.633291i	$-0.781704\pi$
$193$		−	17.5959i		−	1.26658i	0.773914	−	0.633291i	$-0.218296\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.10102i		−	0.648421i	−0.945985	−	0.324210i	$-0.894901\pi$
$197$		−	9.10102i		−	0.648421i	0.945985	−	0.324210i	$-0.105099\pi$
$198$		0			0
$199$	7.10102			0.503378			0.251689	−	0.967808i	$-0.419014\pi$
$199$	7.10102			0.503378			0.251689	+	0.967808i	$0.419014\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.89898i		−	0.203468i
$204$		0			0
$205$	2.44949	−	24.2474i	0.171080	−	1.69352i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	31.5959			2.18554
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.7980	+	2.00000i	1.35021	+	0.136399i
$216$		0			0
$217$			0.898979i			0.0610267i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.898979			0.0604719
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.34847i		−	0.487735i	−0.969809	−	0.243868i	$-0.921584\pi$
$227$		−	7.34847i		−	0.487735i	0.969809	−	0.243868i	$-0.0784162\pi$
$228$		0			0
$229$	−15.1464			−1.00090			−0.500452	−	0.865764i	$-0.666833\pi$
$229$	−15.1464			−1.00090			−0.500452	+	0.865764i	$0.666833\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			10.2020i			0.668358i	0.942510	+	0.334179i	$0.108459\pi$
$233$			10.2020i			0.668358i	−0.942510	+	0.334179i	$0.891541\pi$
$234$		0			0
$235$	−2.00000	−	0.202041i	−0.130466	−	0.0131797i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.7980			1.66873			0.834366	−	0.551211i	$-0.185834\pi$
$239$	25.7980			1.66873			0.834366	+	0.551211i	$0.185834\pi$
$240$		0			0
$241$	20.6969			1.33321			0.666604	−	0.745412i	$-0.267747\pi$
$241$	20.6969			1.33321			0.666604	+	0.745412i	$0.267747\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.224745	+	2.22474i	−0.0143584	+	0.142134i
$246$		0			0
$247$			2.89898i			0.184458i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.55051			−0.0978673			−0.0489337	−	0.998802i	$-0.515582\pi$
$251$	−1.55051			−0.0978673			−0.0489337	+	0.998802i	$0.515582\pi$
$252$		0			0
$253$			33.7980i			2.12486i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.6969i		−	1.29104i	−0.763744	−	0.645520i	$-0.776641\pi$
$257$		−	20.6969i		−	1.29104i	0.763744	−	0.645520i	$-0.223359\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$			9.79796i			0.604168i	0.953281	+	0.302084i	$0.0976823\pi$
$263$			9.79796i			0.604168i	−0.953281	+	0.302084i	$0.902318\pi$
$264$		0			0
$265$	−2.44949	−	0.247449i	−0.150471	−	0.0152007i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.1464			−0.923494			−0.461747	−	0.887012i	$-0.652777\pi$
$269$	−15.1464			−0.923494			−0.461747	+	0.887012i	$0.652777\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−24.0000	−	4.89898i	−1.44725	−	0.295420i
$276$		0			0
$277$		−	5.10102i		−	0.306491i	−0.988188	−	0.153245i	$-0.951028\pi$
$277$		−	5.10102i		−	0.306491i	0.988188	−	0.153245i	$-0.0489725\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$			28.2474i			1.67914i	0.543254	+	0.839568i	$0.317192\pi$
$283$			28.2474i			1.67914i	−0.543254	+	0.839568i	$0.682808\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.8990i			0.643346i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.24745i		−	0.364980i	−0.983208	−	0.182490i	$-0.941584\pi$
$293$		−	6.24745i		−	0.364980i	0.983208	−	0.182490i	$-0.0584157\pi$
$294$		0			0
$295$	1.44949	−	14.3485i	0.0843926	−	0.835400i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.10102			−0.179337
$300$		0			0
$301$	−8.89898			−0.512929
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.89898	−	18.7980i	0.108735	−	1.07637i
$306$		0			0
$307$		−	4.24745i		−	0.242415i	−0.992627	−	0.121207i	$-0.961323\pi$
$307$		−	4.24745i		−	0.242415i	0.992627	−	0.121207i	$-0.0386766\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	17.5959i		−	0.994580i	−0.867584	−	0.497290i	$-0.834328\pi$
$313$		−	17.5959i		−	0.994580i	0.867584	−	0.497290i	$-0.165672\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	26.4949i		−	1.48810i	−0.668123	−	0.744051i	$-0.732902\pi$
$317$		−	26.4949i		−	1.48810i	0.668123	−	0.744051i	$-0.267098\pi$
$318$		0			0
$319$	−14.2020			−0.795162
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.8990i		−	0.717718i
$324$		0			0
$325$	0.449490	−	2.20204i	0.0249332	−	0.122147i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.898979			0.0495623
$330$		0			0
$331$	−10.6969			−0.587957			−0.293978	−	0.955812i	$-0.594979\pi$
$331$	−10.6969			−0.587957			−0.293978	+	0.955812i	$0.594979\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.7980	+	1.79796i	0.972406	+	0.0982330i
$336$		0			0
$337$		−	29.5959i		−	1.61219i	−0.591785	−	0.806096i	$-0.701576\pi$
$337$		−	29.5959i		−	1.61219i	0.591785	−	0.806096i	$-0.298424\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.40408			0.238494
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			19.1010i			1.02540i	0.858569	+	0.512698i	$0.171354\pi$
$347$			19.1010i			1.02540i	−0.858569	+	0.512698i	$0.828646\pi$
$348$		0			0
$349$	3.55051			0.190054			0.0950272	−	0.995475i	$-0.469706\pi$
$349$	3.55051			0.190054			0.0950272	+	0.995475i	$0.469706\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.1010i			0.697297i	0.937254	+	0.348648i	$0.113359\pi$
$353$			13.1010i			0.697297i	−0.937254	+	0.348648i	$0.886641\pi$
$354$		0			0
$355$	−2.44949	+	24.2474i	−0.130005	+	1.28692i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.5959			−0.612009			−0.306005	−	0.952030i	$-0.598992\pi$
$359$	−11.5959			−0.612009			−0.306005	+	0.952030i	$0.598992\pi$
$360$		0			0
$361$	22.5959			1.18926
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.3485	+	1.55051i	0.803376	+	0.0811574i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.10102			0.0571621
$372$		0			0
$373$		−	24.6969i		−	1.27876i	−0.768891	−	0.639380i	$-0.779191\pi$
$373$		−	24.6969i		−	1.27876i	0.768891	−	0.639380i	$-0.220809\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.30306i		−	0.0671111i
$378$		0			0
$379$	1.30306			0.0669338			0.0334669	−	0.999440i	$-0.489345\pi$
$379$	1.30306			0.0669338			0.0334669	+	0.999440i	$0.489345\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.8990i			0.863498i	0.901994	+	0.431749i	$0.142103\pi$
$383$			16.8990i			0.863498i	−0.901994	+	0.431749i	$0.857897\pi$
$384$		0			0
$385$	10.8990	+	1.10102i	0.555463	+	0.0561132i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.8990			1.16102			0.580512	−	0.814252i	$-0.302852\pi$
$389$	22.8990			1.16102			0.580512	+	0.814252i	$0.302852\pi$
$390$		0			0
$391$	13.7980			0.697793
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.651531	+	6.44949i	−0.0327821	+	0.324509i
$396$		0			0
$397$		−	17.3485i		−	0.870695i	−0.900263	−	0.435347i	$-0.856626\pi$
$397$		−	17.3485i		−	0.870695i	0.900263	−	0.435347i	$-0.143374\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−29.3939			−1.46786			−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939			−1.46786			−0.733930	+	0.679225i	$0.762316\pi$
$402$		0			0
$403$			0.404082i			0.0201288i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9.79796i			0.485667i
$408$		0			0
$409$	14.4949			0.716727			0.358363	−	0.933582i	$-0.383335\pi$
$409$	14.4949			0.716727			0.358363	+	0.933582i	$0.383335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			6.44949i			0.317359i
$414$		0			0
$415$	−5.44949	−	0.550510i	−0.267505	−	0.0270235i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.44949			0.315078			0.157539	−	0.987513i	$-0.449644\pi$
$419$	6.44949			0.315078			0.157539	+	0.987513i	$0.449644\pi$
$420$		0			0
$421$	−23.7980			−1.15984			−0.579921	−	0.814673i	$-0.696917\pi$
$421$	−23.7980			−1.15984			−0.579921	+	0.814673i	$0.696917\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000	+	9.79796i	−0.0970143	+	0.475271i
$426$		0			0
$427$			8.44949i			0.408899i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.7980			0.857298			0.428649	−	0.903471i	$-0.358990\pi$
$431$	17.7980			0.857298			0.428649	+	0.903471i	$0.358990\pi$
$432$		0			0
$433$			19.7980i			0.951429i	0.879600	+	0.475715i	$0.157810\pi$
$433$			19.7980i			0.951429i	−0.879600	+	0.475715i	$0.842190\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			44.4949i			2.12848i
$438$		0			0
$439$	37.3939			1.78471			0.892356	−	0.451332i	$-0.149051\pi$
$439$	37.3939			1.78471			0.892356	+	0.451332i	$0.149051\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			9.79796i			0.465515i	0.972535	+	0.232758i	$0.0747749\pi$
$443$			9.79796i			0.465515i	−0.972535	+	0.232758i	$0.925225\pi$
$444$		0			0
$445$	−2.24745	+	22.2474i	−0.106539	+	1.05463i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$	53.3939			2.51422
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.101021	+	1.00000i	−0.00473591	+	0.0468807i
$456$		0			0
$457$		−	9.59592i		−	0.448878i	−0.974488	−	0.224439i	$-0.927945\pi$
$457$		−	9.59592i		−	0.448878i	0.974488	−	0.224439i	$-0.0720550\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.65153			−0.123494			−0.0617470	−	0.998092i	$-0.519667\pi$
$461$	−2.65153			−0.123494			−0.0617470	+	0.998092i	$0.519667\pi$
$462$		0			0
$463$		−	35.5959i		−	1.65428i	−0.561994	−	0.827141i	$-0.689966\pi$
$463$		−	35.5959i		−	1.65428i	0.561994	−	0.827141i	$-0.310034\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5.55051i			0.256847i	0.991719	+	0.128423i	$0.0409917\pi$
$467$			5.55051i			0.256847i	−0.991719	+	0.128423i	$0.959008\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$			43.5959i			2.00454i
$474$		0			0
$475$	−31.5959	−	6.44949i	−1.44972	−	0.295923i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.6969			−1.76811			−0.884054	−	0.467385i	$-0.845196\pi$
$479$	−38.6969			−1.76811			−0.884054	+	0.467385i	$0.845196\pi$
$480$		0			0
$481$	−0.898979			−0.0409899
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.44949	−	0.853572i	−0.383672	−	0.0387587i
$486$		0			0
$487$			36.6969i			1.66290i	0.555602	+	0.831449i	$0.312488\pi$
$487$			36.6969i			1.66290i	−0.555602	+	0.831449i	$0.687512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−19.5959			−0.884351			−0.442176	−	0.896928i	$-0.645793\pi$
$491$	−19.5959			−0.884351			−0.442176	+	0.896928i	$0.645793\pi$
$492$		0			0
$493$			5.79796i			0.261127i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10.8990i		−	0.488886i
$498$		0			0
$499$	25.7980			1.15488			0.577438	−	0.816435i	$-0.304053\pi$
$499$	25.7980			1.15488			0.577438	+	0.816435i	$0.304053\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			4.00000i			0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$			4.00000i			0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$		0			0
$505$	−1.89898	+	18.7980i	−0.0845035	+	0.836498i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−36.4495			−1.61560			−0.807798	−	0.589460i	$-0.799341\pi$
$509$	−36.4495			−1.61560			−0.807798	+	0.589460i	$0.799341\pi$
$510$		0			0
$511$	−6.89898			−0.305193
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.89898	+	0.696938i	0.304005	+	0.0307108i
$516$		0			0
$517$		−	4.40408i		−	0.193691i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.30306			−0.144710			−0.0723549	−	0.997379i	$-0.523051\pi$
$521$	−3.30306			−0.144710			−0.0723549	+	0.997379i	$0.523051\pi$
$522$		0			0
$523$			1.14643i			0.0501298i	0.999686	+	0.0250649i	$0.00797924\pi$
$523$			1.14643i			0.0501298i	−0.999686	+	0.0250649i	$0.992021\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.79796i		−	0.0783203i
$528$		0			0
$529$	−24.5959			−1.06939
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4.89898i			0.212198i
$534$		0			0
$535$	−17.7980	−	1.79796i	−0.769473	−	0.0777325i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.89898			−0.211014
$540$		0			0
$541$	−29.5959			−1.27243			−0.636214	−	0.771513i	$-0.719500\pi$
$541$	−29.5959			−1.27243			−0.636214	+	0.771513i	$0.719500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.651531	−	6.44949i	0.0279085	−	0.276266i
$546$		0			0
$547$		−	10.6969i		−	0.457368i	−0.973501	−	0.228684i	$-0.926558\pi$
$547$		−	10.6969i		−	0.457368i	0.973501	−	0.228684i	$-0.0734423\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.6969			−0.796516
$552$		0			0
$553$		−	2.89898i		−	0.123277i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			16.6969i			0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$557$			16.6969i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	14.0454i		−	0.591943i	−0.955197	−	0.295972i	$-0.904357\pi$
$563$		−	14.0454i		−	0.591943i	0.955197	−	0.295972i	$-0.0956434\pi$
$564$		0			0
$565$	0.449490	+	0.0454077i	0.0189102	+	0.00191032i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.2020			0.595381			0.297690	−	0.954663i	$-0.403784\pi$
$569$	14.2020			0.595381			0.297690	+	0.954663i	$0.403784\pi$
$570$		0			0
$571$	20.8990			0.874595			0.437298	−	0.899317i	$-0.355936\pi$
$571$	20.8990			0.874595			0.437298	+	0.899317i	$0.355936\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.89898	−	33.7980i	0.287707	−	1.40947i
$576$		0			0
$577$			46.4949i			1.93561i	0.251705	+	0.967804i	$0.419009\pi$
$577$			46.4949i			1.93561i	−0.251705	+	0.967804i	$0.580991\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.44949			0.101622
$582$		0			0
$583$		−	5.39388i		−	0.223392i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	33.1464i		−	1.36810i	−0.729435	−	0.684050i	$-0.760217\pi$
$587$		−	33.1464i		−	1.36810i	0.729435	−	0.684050i	$-0.239783\pi$
$588$		0			0
$589$	5.79796			0.238901
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.10102i		−	0.0452135i	−0.999744	−	0.0226067i	$-0.992803\pi$
$593$		−	1.10102i		−	0.0452135i	0.999744	−	0.0226067i	$-0.00719656\pi$
$594$		0			0
$595$	0.449490	−	4.44949i	0.0184273	−	0.182411i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.8990			0.935627			0.467813	−	0.883827i	$-0.345042\pi$
$599$	22.8990			0.935627			0.467813	+	0.883827i	$0.345042\pi$
$600$		0			0
$601$	19.3939			0.791093			0.395546	−	0.918446i	$-0.370555\pi$
$601$	19.3939			0.791093			0.395546	+	0.918446i	$0.370555\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.92168	−	28.9217i	0.118783	−	1.17583i
$606$		0			0
$607$			25.3939i			1.03071i	0.856978	+	0.515353i	$0.172339\pi$
$607$			25.3939i			1.03071i	−0.856978	+	0.515353i	$0.827661\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.404082			0.0163474
$612$		0			0
$613$			8.20204i			0.331277i	0.986187	+	0.165639i	$0.0529685\pi$
$613$			8.20204i			0.331277i	−0.986187	+	0.165639i	$0.947031\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			9.59592i			0.386317i	0.981168	+	0.193159i	$0.0618732\pi$
$617$			9.59592i			0.386317i	−0.981168	+	0.193159i	$0.938127\pi$
$618$		0			0
$619$	−46.4495			−1.86696			−0.933481	−	0.358626i	$-0.883245\pi$
$619$	−46.4495			−1.86696			−0.933481	+	0.358626i	$0.883245\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	10.0000i		−	0.400642i
$624$		0			0
$625$	23.0000	+	9.79796i	0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	−6.49490			−0.258558			−0.129279	−	0.991608i	$-0.541266\pi$
$631$	−6.49490			−0.258558			−0.129279	+	0.991608i	$0.541266\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.3485	+	1.14643i	0.450350	+	0.0454946i
$636$		0			0
$637$		−	0.449490i		−	0.0178094i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.20204			−0.244966			−0.122483	−	0.992471i	$-0.539086\pi$
$641$	−6.20204			−0.244966			−0.122483	+	0.992471i	$0.539086\pi$
$642$		0			0
$643$		−	9.14643i		−	0.360700i	−0.983603	−	0.180350i	$-0.942277\pi$
$643$		−	9.14643i		−	0.360700i	0.983603	−	0.180350i	$-0.0577230\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			22.2929i			0.876423i	0.898872	+	0.438211i	$0.144388\pi$
$647$			22.2929i			0.876423i	−0.898872	+	0.438211i	$0.855612\pi$
$648$		0			0
$649$	31.5959			1.24025
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	39.7980i		−	1.55741i	−0.627387	−	0.778707i	$-0.715876\pi$
$653$		−	39.7980i		−	1.55741i	0.627387	−	0.778707i	$-0.284124\pi$
$654$		0			0
$655$	−0.348469	+	3.44949i	−0.0136158	+	0.134783i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.10102			−0.276616			−0.138308	−	0.990389i	$-0.544166\pi$
$659$	−7.10102			−0.276616			−0.138308	+	0.990389i	$0.544166\pi$
$660$		0			0
$661$	12.9444			0.503478			0.251739	−	0.967795i	$-0.418997\pi$
$661$	12.9444			0.503478			0.251739	+	0.967795i	$0.418997\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.3485	+	1.44949i	0.556410	+	0.0562088i
$666$		0			0
$667$		−	20.0000i		−	0.774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	41.3939			1.59799
$672$		0			0
$673$		−	1.79796i		−	0.0693062i	−0.999399	−	0.0346531i	$-0.988967\pi$
$673$		−	1.79796i		−	0.0693062i	0.999399	−	0.0346531i	$-0.0110326\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			31.5505i			1.21258i	0.795242	+	0.606292i	$0.207344\pi$
$677$			31.5505i			1.21258i	−0.795242	+	0.606292i	$0.792656\pi$
$678$		0			0
$679$	3.79796			0.145752
$680$		0			0
$681$		0			0
$682$		0			0
$683$			35.5959i			1.36204i	0.732265	+	0.681020i	$0.238463\pi$
$683$			35.5959i			1.36204i	−0.732265	+	0.681020i	$0.761537\pi$
$684$		0			0
$685$	−39.5959	−	4.00000i	−1.51288	−	0.152832i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.494897			0.0188541
$690$		0			0
$691$	13.1464			0.500114			0.250057	−	0.968231i	$-0.419551\pi$
$691$	13.1464			0.500114			0.250057	+	0.968231i	$0.419551\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.44949	−	14.3485i	0.0549823	−	0.544268i
$696$		0			0
$697$		−	21.7980i		−	0.825657i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−40.6969			−1.53710			−0.768551	−	0.639788i	$-0.779022\pi$
$701$	−40.6969			−1.53710			−0.768551	+	0.639788i	$0.779022\pi$
$702$		0			0
$703$			12.8990i			0.486494i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.44949i		−	0.317776i
$708$		0			0
$709$	−40.2929			−1.51323			−0.756615	−	0.653861i	$-0.773148\pi$
$709$	−40.2929			−1.51323			−0.756615	+	0.653861i	$0.773148\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			6.20204i			0.232268i
$714$		0			0
$715$	4.89898	+	0.494897i	0.183211	+	0.0185081i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−44.4949			−1.65938			−0.829690	−	0.558225i	$-0.811483\pi$
$719$	−44.4949			−1.65938			−0.829690	+	0.558225i	$0.811483\pi$
$720$		0			0
$721$	−3.10102			−0.115488
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.2020	+	2.89898i	0.527451	+	0.107665i
$726$		0			0
$727$			6.69694i			0.248376i	0.992259	+	0.124188i	$0.0396325\pi$
$727$			6.69694i			0.248376i	−0.992259	+	0.124188i	$0.960367\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.7980			0.658281
$732$		0			0
$733$			43.6413i			1.61193i	0.591964	+	0.805965i	$0.298353\pi$
$733$			43.6413i			1.61193i	−0.591964	+	0.805965i	$0.701647\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			39.1918i			1.44365i
$738$		0			0
$739$	−44.4949			−1.63677			−0.818386	−	0.574669i	$-0.805131\pi$
$739$	−44.4949			−1.63677			−0.818386	+	0.574669i	$0.805131\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.3031i			0.561415i	0.959793	+	0.280707i	$0.0905691\pi$
$743$			15.3031i			0.561415i	−0.959793	+	0.280707i	$0.909431\pi$
$744$		0			0
$745$	3.55051	−	35.1464i	0.130081	−	1.28767i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	22.2020			0.810164			0.405082	−	0.914280i	$-0.367243\pi$
$751$	22.2020			0.810164			0.405082	+	0.914280i	$0.367243\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.40408	−	43.5959i	0.160281	−	1.58662i
$756$		0			0
$757$		−	32.2020i		−	1.17040i	−0.810888	−	0.585202i	$-0.801015\pi$
$757$		−	32.2020i		−	1.17040i	0.810888	−	0.585202i	$-0.198985\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	30.8990			1.12009			0.560044	−	0.828463i	$-0.310784\pi$
$761$	30.8990			1.12009			0.560044	+	0.828463i	$0.310784\pi$
$762$		0			0
$763$			2.89898i			0.104950i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			2.89898i			0.104676i
$768$		0			0
$769$	11.3031			0.407599			0.203799	−	0.979013i	$-0.434671\pi$
$769$	11.3031			0.407599			0.203799	+	0.979013i	$0.434671\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13.3485i		−	0.480111i	−0.970759	−	0.240056i	$-0.922834\pi$
$773$		−	13.3485i		−	0.480111i	0.970759	−	0.240056i	$-0.0771657\pi$
$774$		0			0
$775$	−4.40408	−	0.898979i	−0.158199	−	0.0322923i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	70.2929			2.51850
$780$		0			0
$781$	−53.3939			−1.91058
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.7980	+	1.89898i	0.670928	+	0.0677775i
$786$		0			0
$787$		−	45.5505i		−	1.62370i	−0.583866	−	0.811850i	$-0.698461\pi$
$787$		−	45.5505i		−	1.62370i	0.583866	−	0.811850i	$-0.301539\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.202041			−0.00718375
$792$		0			0
$793$			3.79796i			0.134869i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	52.9444i		−	1.87539i	−0.347464	−	0.937693i	$-0.612957\pi$
$797$		−	52.9444i		−	1.87539i	0.347464	−	0.937693i	$-0.387043\pi$
$798$		0			0
$799$	−1.79796			−0.0636072
$800$		0			0
$801$		0			0
$802$		0			0
$803$			33.7980i			1.19270i
$804$		0			0
$805$	−1.55051	+	15.3485i	−0.0546483	+	0.540962i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.40408			0.295472			0.147736	−	0.989027i	$-0.452801\pi$
$809$	8.40408			0.295472			0.147736	+	0.989027i	$0.452801\pi$
$810$		0			0
$811$	38.9444			1.36752			0.683761	−	0.729706i	$-0.260343\pi$
$811$	38.9444			1.36752			0.683761	+	0.729706i	$0.260343\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−37.5959	−	3.79796i	−1.31693	−	0.133037i
$816$		0			0
$817$			57.3939i			2.00796i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−27.7980			−0.970155			−0.485078	−	0.874471i	$-0.661209\pi$
$821$	−27.7980			−0.970155			−0.485078	+	0.874471i	$0.661209\pi$
$822$		0			0
$823$			39.1918i			1.36614i	0.730352	+	0.683071i	$0.239356\pi$
$823$			39.1918i			1.36614i	−0.730352	+	0.683071i	$0.760644\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			23.5959i			0.820510i	0.911971	+	0.410255i	$0.134560\pi$
$827$			23.5959i			0.820510i	−0.911971	+	0.410255i	$0.865440\pi$
$828$		0			0
$829$	−39.6413			−1.37680			−0.688400	−	0.725331i	$-0.741687\pi$
$829$	−39.6413			−1.37680			−0.688400	+	0.725331i	$0.741687\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	10.8990	+	1.10102i	0.377175	+	0.0381024i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.1010			−0.935631			−0.467816	−	0.883826i	$-0.654959\pi$
$839$	−27.1010			−0.935631			−0.467816	+	0.883826i	$0.654959\pi$
$840$		0			0
$841$	−20.5959			−0.710204
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.87628	−	28.4722i	0.0989469	−	0.979473i
$846$		0			0
$847$			13.0000i			0.446685i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.7980			−0.472988
$852$		0			0
$853$		−	29.8434i		−	1.02182i	−0.859635	−	0.510909i	$-0.829309\pi$
$853$		−	29.8434i		−	1.02182i	0.859635	−	0.510909i	$-0.170691\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	25.1918i		−	0.860537i	−0.902701	−	0.430268i	$-0.858419\pi$
$857$		−	25.1918i		−	0.860537i	0.902701	−	0.430268i	$-0.141581\pi$
$858$		0			0
$859$	−29.6413			−1.01135			−0.505674	−	0.862724i	$-0.668756\pi$
$859$	−29.6413			−1.01135			−0.505674	+	0.862724i	$0.668756\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	13.3939i		−	0.455933i	−0.973669	−	0.227966i	$-0.926792\pi$
$863$		−	13.3939i		−	0.455933i	0.973669	−	0.227966i	$-0.0732076\pi$
$864$		0			0
$865$	40.5959	+	4.10102i	1.38030	+	0.139439i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14.2020			−0.481771
$870$		0			0
$871$	−3.59592			−0.121843
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.6742	−	3.32577i	−0.360855	−	0.112431i
$876$		0			0
$877$			19.3939i			0.654885i	0.944871	+	0.327442i	$0.106187\pi$
$877$			19.3939i			0.654885i	−0.944871	+	0.327442i	$0.893813\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.7980			−0.936537			−0.468269	−	0.883586i	$-0.655122\pi$
$881$	−27.7980			−0.936537			−0.468269	+	0.883586i	$0.655122\pi$
$882$		0			0
$883$			41.7980i			1.40661i	0.710887	+	0.703307i	$0.248294\pi$
$883$			41.7980i			1.40661i	−0.710887	+	0.703307i	$0.751706\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	26.6969i		−	0.896395i	−0.893934	−	0.448198i	$-0.852066\pi$
$887$		−	26.6969i		−	0.896395i	0.893934	−	0.448198i	$-0.147934\pi$
$888$		0			0
$889$	−5.10102			−0.171083
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.79796i		−	0.194021i
$894$		0			0
$895$	1.30306	−	12.8990i	0.0435565	−	0.431165i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.60612			−0.0869191
$900$		0			0
$901$	−2.20204			−0.0733606
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.20204	−	31.6969i	0.106439	−	1.05364i
$906$		0			0
$907$			22.2020i			0.737207i	0.929587	+	0.368603i	$0.120164\pi$
$907$			22.2020i			0.737207i	−0.929587	+	0.368603i	$0.879836\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.59592			0.119138			0.0595690	−	0.998224i	$-0.481027\pi$
$911$	3.59592			0.119138			0.0595690	+	0.998224i	$0.481027\pi$
$912$		0			0
$913$		−	12.0000i		−	0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.55051i		−	0.0512024i
$918$		0			0
$919$	−17.1010			−0.564111			−0.282055	−	0.959398i	$-0.591016\pi$
$919$	−17.1010			−0.564111			−0.282055	+	0.959398i	$0.591016\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.89898i		−	0.161252i
$924$		0			0
$925$	2.00000	−	9.79796i	0.0657596	−	0.322155i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	40.2929			1.32197			0.660983	−	0.750401i	$-0.270140\pi$
$929$	40.2929			1.32197			0.660983	+	0.750401i	$0.270140\pi$
$930$		0			0
$931$	−6.44949			−0.211373
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.7980	−	2.20204i	−0.712869	−	0.0720144i
$936$		0			0
$937$		−	50.8990i		−	1.66280i	−0.555677	−	0.831399i	$-0.687541\pi$
$937$		−	50.8990i		−	1.66280i	0.555677	−	0.831399i	$-0.312459\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24.4495			0.797031			0.398515	−	0.917162i	$-0.369526\pi$
$941$	24.4495			0.797031			0.398515	+	0.917162i	$0.369526\pi$
$942$		0			0
$943$			75.1918i			2.44858i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	44.0908i		−	1.43276i	−0.697711	−	0.716379i	$-0.745798\pi$
$947$		−	44.0908i		−	1.43276i	0.697711	−	0.716379i	$-0.254202\pi$
$948$		0			0
$949$	−3.10102			−0.100663
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.7980i			0.706105i	0.935603	+	0.353053i	$0.114856\pi$
$953$			21.7980i			0.706105i	−0.935603	+	0.353053i	$0.885144\pi$
$954$		0			0
$955$	−3.75255	+	37.1464i	−0.121430	+	1.20203i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.7980			0.574726
$960$		0			0
$961$	−30.1918			−0.973930
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−39.1464	−	3.95459i	−1.26017	−	0.127303i
$966$		0			0
$967$		−	32.2929i		−	1.03847i	−0.854632	−	0.519234i	$-0.826217\pi$
$967$		−	32.2929i		−	1.03847i	0.854632	−	0.519234i	$-0.173783\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.4495			−0.463706			−0.231853	−	0.972751i	$-0.574479\pi$
$971$	−14.4495			−0.463706			−0.231853	+	0.972751i	$0.574479\pi$
$972$		0			0
$973$			6.44949i			0.206761i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	29.3939i		−	0.940393i	−0.882562	−	0.470197i	$-0.844183\pi$
$977$		−	29.3939i		−	0.940393i	0.882562	−	0.470197i	$-0.155817\pi$
$978$		0			0
$979$	−48.9898			−1.56572
$980$		0			0
$981$		0			0
$982$		0			0
$983$			42.6969i			1.36182i	0.732367	+	0.680910i	$0.238416\pi$
$983$			42.6969i			1.36182i	−0.732367	+	0.680910i	$0.761584\pi$
$984$		0			0
$985$	−20.2474	−	2.04541i	−0.645137	−	0.0651721i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−61.3939			−1.95221
$990$		0			0
$991$	−60.6969			−1.92810			−0.964051	−	0.265718i	$-0.914391\pi$
$991$	−60.6969			−1.92810			−0.964051	+	0.265718i	$0.914391\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.59592	−	15.7980i	0.0505940	−	0.500829i
$996$		0			0
$997$			42.6515i			1.35079i	0.737457	+	0.675394i	$0.236026\pi$
$997$			42.6515i			1.35079i	−0.737457	+	0.675394i	$0.763974\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.t.1009.3		4
3.2	odd	2		560.2.g.e.449.3		4
4.3	odd	2		630.2.g.g.379.4		4
5.4	even	2	inner	5040.2.t.t.1009.4		4
12.11	even	2		70.2.c.a.29.1	✓	4
15.2	even	4		2800.2.a.bl.1.2		2
15.8	even	4		2800.2.a.bm.1.1		2
15.14	odd	2		560.2.g.e.449.1		4
20.3	even	4		3150.2.a.bt.1.1		2
20.7	even	4		3150.2.a.bs.1.1		2
20.19	odd	2		630.2.g.g.379.2		4
24.5	odd	2		2240.2.g.i.449.2		4
24.11	even	2		2240.2.g.j.449.4		4
60.23	odd	4		350.2.a.g.1.2		2
60.47	odd	4		350.2.a.h.1.1		2
60.59	even	2		70.2.c.a.29.4	yes	4
84.11	even	6		490.2.i.c.79.3		8
84.23	even	6		490.2.i.c.459.2		8
84.47	odd	6		490.2.i.f.459.1		8
84.59	odd	6		490.2.i.f.79.4		8
84.83	odd	2		490.2.c.e.99.2		4
120.29	odd	2		2240.2.g.i.449.4		4
120.59	even	2		2240.2.g.j.449.2		4
420.59	odd	6		490.2.i.f.79.1		8
420.83	even	4		2450.2.a.bl.1.1		2
420.167	even	4		2450.2.a.bq.1.2		2
420.179	even	6		490.2.i.c.79.2		8
420.299	odd	6		490.2.i.f.459.4		8
420.359	even	6		490.2.i.c.459.3		8
420.419	odd	2		490.2.c.e.99.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.1	✓	4	12.11	even	2
70.2.c.a.29.4	yes	4	60.59	even	2
350.2.a.g.1.2		2	60.23	odd	4
350.2.a.h.1.1		2	60.47	odd	4
490.2.c.e.99.2		4	84.83	odd	2
490.2.c.e.99.3		4	420.419	odd	2
490.2.i.c.79.2		8	420.179	even	6
490.2.i.c.79.3		8	84.11	even	6
490.2.i.c.459.2		8	84.23	even	6
490.2.i.c.459.3		8	420.359	even	6
490.2.i.f.79.1		8	420.59	odd	6
490.2.i.f.79.4		8	84.59	odd	6
490.2.i.f.459.1		8	84.47	odd	6
490.2.i.f.459.4		8	420.299	odd	6
560.2.g.e.449.1		4	15.14	odd	2
560.2.g.e.449.3		4	3.2	odd	2
630.2.g.g.379.2		4	20.19	odd	2
630.2.g.g.379.4		4	4.3	odd	2
2240.2.g.i.449.2		4	24.5	odd	2
2240.2.g.i.449.4		4	120.29	odd	2
2240.2.g.j.449.2		4	120.59	even	2
2240.2.g.j.449.4		4	24.11	even	2
2450.2.a.bl.1.1		2	420.83	even	4
2450.2.a.bq.1.2		2	420.167	even	4
2800.2.a.bl.1.2		2	15.2	even	4
2800.2.a.bm.1.1		2	15.8	even	4
3150.2.a.bs.1.1		2	20.7	even	4
3150.2.a.bt.1.1		2	20.3	even	4
5040.2.t.t.1009.3		4	1.1	even	1	trivial
5040.2.t.t.1009.4		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.224745 - 2.22474i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(0.224745 - 2.22474i) q^{5} +1.00000i q^{7} +4.89898 q^{11} +0.449490i q^{13} -2.00000i q^{17} +6.44949 q^{19} +6.89898i q^{23} +(-4.89898 - 1.00000i) q^{25} -2.89898 q^{29} +0.898979 q^{31} +(2.22474 + 0.224745i) q^{35} +2.00000i q^{37} +10.8990 q^{41} +8.89898i q^{43} -0.898979i q^{47} -1.00000 q^{49} -1.10102i q^{53} +(1.10102 - 10.8990i) q^{55} +6.44949 q^{59} +8.44949 q^{61} +(1.00000 + 0.101021i) q^{65} +8.00000i q^{67} -10.8990 q^{71} +6.89898i q^{73} +4.89898i q^{77} -2.89898 q^{79} -2.44949i q^{83} +(-4.44949 - 0.449490i) q^{85} -10.0000 q^{89} -0.449490 q^{91} +(1.44949 - 14.3485i) q^{95} -3.79796i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 16 q^{19} + 8 q^{29} - 16 q^{31} + 4 q^{35} + 24 q^{41} - 4 q^{49} + 24 q^{55} + 16 q^{59} + 24 q^{61} + 4 q^{65} - 24 q^{71} + 8 q^{79} - 8 q^{85} - 40 q^{89} + 8 q^{91} - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^5 + 16 * q^19 + 8 * q^29 - 16 * q^31 + 4 * q^35 + 24 * q^41 - 4 * q^49 + 24 * q^55 + 16 * q^59 + 24 * q^61 + 4 * q^65 - 24 * q^71 + 8 * q^79 - 8 * q^85 - 40 * q^89 + 8 * q^91 - 4 * q^95

Introduction

L-functions

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