LMFDB - Embedded newform 5040.2.f.i.881.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(881,5040)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("5040.881");

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.f (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:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.5
Root			$2.73923i$ of defining polynomial
Character	$\chi$	$=$	5040.881
Dual form			5040.2.f.i.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +(1.80230 - 1.93693i) q^{7} +O(q^{10})$	$q+1.00000 q^{5} +(1.80230 - 1.93693i) q^{7} -3.87386i q^{11} -1.60461i q^{13} +8.11650 q^{17} -2.63803i q^{19} -5.47847i q^{23} +1.00000 q^{25} +5.47847i q^{29} +3.73074i q^{31} +(1.80230 - 1.93693i) q^{35} +4.51190 q^{37} -1.60461 q^{41} +10.1165 q^{43} -11.1097 q^{47} +(-0.503406 - 6.98188i) q^{49} -2.26926i q^{53} -3.87386i q^{55} -4.61142 q^{59} +11.8472i q^{61} -1.60461i q^{65} -6.90729 q^{67} -2.63803i q^{71} -13.7477i q^{73} +(-7.50341 - 6.98188i) q^{77} +8.01698 q^{79} -3.20921 q^{83} +8.11650 q^{85} -17.8376 q^{89} +(-3.10801 - 2.89199i) q^{91} -2.63803i q^{95} +8.68768i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 4 q^{7}+O(q^{10}) $ 8 * q + 8 * q^5 + 4 * q^7	$ 8 q + 8 q^{5} + 4 q^{7} + 8 q^{25} + 4 q^{35} - 8 q^{37} + 8 q^{41} + 16 q^{43} - 40 q^{47} + 4 q^{49} - 32 q^{67} - 52 q^{77} - 8 q^{79} + 16 q^{83} + 8 q^{89} + 4 q^{91}+O(q^{100}) $ 8 * q + 8 * q^5 + 4 * q^7 + 8 * q^25 + 4 * q^35 - 8 * q^37 + 8 * q^41 + 16 * q^43 - 40 * q^47 + 4 * q^49 - 32 * q^67 - 52 * q^77 - 8 * q^79 + 16 * q^83 + 8 * q^89 + 4 * q^91

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
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	1.80230	−	1.93693i	0.681207	−	0.732091i
$7$	1.80230	−	1.93693i	0.681207	−	0.732091i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.87386i		−	1.16801i	−0.811749	−	0.584007i	$-0.801484\pi$
$11$		−	3.87386i		−	1.16801i	0.811749	−	0.584007i	$-0.198516\pi$
$12$		0			0
$13$		−	1.60461i		−	0.445038i	−0.974928	−	0.222519i	$-0.928572\pi$
$13$		−	1.60461i		−	0.445038i	0.974928	−	0.222519i	$-0.0714279\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.11650			1.96854			0.984271	−	0.176667i	$-0.0565317\pi$
$17$	8.11650			1.96854			0.984271	+	0.176667i	$0.0565317\pi$
$18$		0			0
$19$		−	2.63803i		−	0.605207i	−0.953117	−	0.302603i	$-0.902144\pi$
$19$		−	2.63803i		−	0.605207i	0.953117	−	0.302603i	$-0.0978557\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.47847i		−	1.14234i	−0.820832	−	0.571170i	$-0.806490\pi$
$23$		−	5.47847i		−	1.14234i	0.820832	−	0.571170i	$-0.193510\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.47847i			1.01733i	0.860966	+	0.508663i	$0.169860\pi$
$29$			5.47847i			1.01733i	−0.860966	+	0.508663i	$0.830140\pi$
$30$		0			0
$31$			3.73074i			0.670061i	0.942207	+	0.335031i	$0.108747\pi$
$31$			3.73074i			0.670061i	−0.942207	+	0.335031i	$0.891253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.80230	−	1.93693i	0.304645	−	0.327401i
$36$		0			0
$37$	4.51190			0.741751			0.370876	−	0.928683i	$-0.379058\pi$
$37$	4.51190			0.741751			0.370876	+	0.928683i	$0.379058\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.60461			−0.250597			−0.125299	−	0.992119i	$-0.539989\pi$
$41$	−1.60461			−0.250597			−0.125299	+	0.992119i	$0.539989\pi$
$42$		0			0
$43$	10.1165			1.54275			0.771376	−	0.636379i	$-0.219569\pi$
$43$	10.1165			1.54275			0.771376	+	0.636379i	$0.219569\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.1097			−1.62052			−0.810258	−	0.586074i	$-0.800673\pi$
$47$	−11.1097			−1.62052			−0.810258	+	0.586074i	$0.800673\pi$
$48$		0			0
$49$	−0.503406	−	6.98188i	−0.0719152	−	0.997411i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2.26926i		−	0.311706i	−0.987780	−	0.155853i	$-0.950187\pi$
$53$		−	2.26926i		−	0.311706i	0.987780	−	0.155853i	$-0.0498127\pi$
$54$		0			0
$55$		−	3.87386i		−	0.522352i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.61142			−0.600356			−0.300178	−	0.953883i	$-0.597046\pi$
$59$	−4.61142			−0.600356			−0.300178	+	0.953883i	$0.597046\pi$
$60$		0			0
$61$			11.8472i			1.51688i	0.651740	+	0.758442i	$0.274039\pi$
$61$			11.8472i			1.51688i	−0.651740	+	0.758442i	$0.725961\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.60461i		−	0.199027i
$66$		0			0
$67$	−6.90729			−0.843860			−0.421930	−	0.906628i	$-0.638647\pi$
$67$	−6.90729			−0.843860			−0.421930	+	0.906628i	$0.638647\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.63803i		−	0.313077i	−0.987672	−	0.156539i	$-0.949966\pi$
$71$		−	2.63803i		−	0.313077i	0.987672	−	0.156539i	$-0.0500335\pi$
$72$		0			0
$73$		−	13.7477i		−	1.60905i	−0.593919	−	0.804525i	$-0.702420\pi$
$73$		−	13.7477i		−	1.60905i	0.593919	−	0.804525i	$-0.297580\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.50341	−	6.98188i	−0.855092	−	0.795659i
$78$		0			0
$79$	8.01698			0.901981			0.450990	−	0.892529i	$-0.351071\pi$
$79$	8.01698			0.901981			0.450990	+	0.892529i	$0.351071\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.20921			−0.352257			−0.176128	−	0.984367i	$-0.556357\pi$
$83$	−3.20921			−0.352257			−0.176128	+	0.984367i	$0.556357\pi$
$84$		0			0
$85$	8.11650			0.880358
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.8376			−1.89078			−0.945392	−	0.325937i	$-0.894320\pi$
$89$	−17.8376			−1.89078			−0.945392	+	0.325937i	$0.894320\pi$
$90$		0			0
$91$	−3.10801	−	2.89199i	−0.325808	−	0.303163i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	2.63803i		−	0.270657i
$96$		0			0
$97$			8.68768i			0.882100i	0.897482	+	0.441050i	$0.145394\pi$
$97$			8.68768i			0.882100i	−0.897482	+	0.441050i	$0.854606\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.21603			0.618518			0.309259	−	0.950978i	$-0.399919\pi$
$101$	6.21603			0.618518			0.309259	+	0.950978i	$0.399919\pi$
$102$		0			0
$103$		−	12.8807i		−	1.26917i	−0.772853	−	0.634585i	$-0.781171\pi$
$103$		−	12.8807i		−	1.26917i	0.772853	−	0.634585i	$-0.218829\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.9638i			1.92997i	0.262308	+	0.964984i	$0.415516\pi$
$107$			19.9638i			1.92997i	−0.262308	+	0.964984i	$0.584484\pi$
$108$		0			0
$109$	−15.0238			−1.43902			−0.719509	−	0.694483i	$-0.755633\pi$
$109$	−15.0238			−1.43902			−0.719509	+	0.694483i	$0.755633\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	10.2330i		−	0.962640i	−0.876545	−	0.481320i	$-0.840157\pi$
$113$		−	10.2330i		−	0.962640i	0.876545	−	0.481320i	$-0.159843\pi$
$114$		0			0
$115$		−	5.47847i		−	0.510870i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.6284	−	15.7211i	1.34098	−	1.44115i
$120$		0			0
$121$	−4.00681			−0.364256
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.81382			0.249686			0.124843	−	0.992177i	$-0.460157\pi$
$127$	2.81382			0.249686			0.124843	+	0.992177i	$0.460157\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.62840			0.753867			0.376933	−	0.926240i	$-0.376979\pi$
$131$	8.62840			0.753867			0.376933	+	0.926240i	$0.376979\pi$
$132$		0			0
$133$	−5.10969	−	4.75454i	−0.443066	−	0.412271i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	0.723932i		−	0.0618496i	−0.999522	−	0.0309248i	$-0.990155\pi$
$137$		−	0.723932i		−	0.0618496i	0.999522	−	0.0309248i	$-0.00984525\pi$
$138$		0			0
$139$			2.84043i			0.240923i	0.992718	+	0.120461i	$0.0384374\pi$
$139$			2.84043i			0.240923i	−0.992718	+	0.120461i	$0.961563\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.21603			−0.519810
$144$		0			0
$145$			5.47847i			0.454962i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.00681i			0.246328i	0.992386	+	0.123164i	$0.0393041\pi$
$149$			3.00681i			0.246328i	−0.992386	+	0.123164i	$0.960696\pi$
$150$		0			0
$151$	15.2262			1.23909			0.619545	−	0.784961i	$-0.287317\pi$
$151$	15.2262			1.23909			0.619545	+	0.784961i	$0.287317\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.73074i			0.299661i
$156$		0			0
$157$		−	2.64767i		−	0.211307i	−0.994403	−	0.105653i	$-0.966307\pi$
$157$		−	2.64767i		−	0.211307i	0.994403	−	0.105653i	$-0.0336934\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.6114	−	9.87386i	−0.836297	−	0.778169i
$162$		0			0
$163$	9.32572			0.730446			0.365223	−	0.930920i	$-0.380993\pi$
$163$	9.32572			0.730446			0.365223	+	0.930920i	$0.380993\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.3257			−1.34070			−0.670352	−	0.742043i	$-0.733857\pi$
$167$	−17.3257			−1.34070			−0.670352	+	0.742043i	$0.733857\pi$
$168$		0			0
$169$	10.4252			0.801941
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.99319			0.227568			0.113784	−	0.993506i	$-0.463703\pi$
$173$	2.99319			0.227568			0.113784	+	0.993506i	$0.463703\pi$
$174$		0			0
$175$	1.80230	−	1.93693i	0.136241	−	0.146418i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	7.38858i		−	0.552248i	−0.961122	−	0.276124i	$-0.910950\pi$
$179$		−	7.38858i		−	0.552248i	0.961122	−	0.276124i	$-0.0890501\pi$
$180$		0			0
$181$		−	11.1097i		−	0.825777i	−0.910782	−	0.412888i	$-0.864520\pi$
$181$		−	11.1097i		−	0.825777i	0.910782	−	0.412888i	$-0.135480\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.51190			0.331721
$186$		0			0
$187$		−	31.4422i		−	2.29928i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.36197i		−	0.677408i	−0.940893	−	0.338704i	$-0.890011\pi$
$191$		−	9.36197i		−	0.677408i	0.940893	−	0.338704i	$-0.109989\pi$
$192$		0			0
$193$	17.4422			1.25552			0.627759	−	0.778408i	$-0.283972\pi$
$193$	17.4422			1.25552			0.627759	+	0.778408i	$0.283972\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.27607i			0.375904i	0.982178	+	0.187952i	$0.0601850\pi$
$197$			5.27607i			0.375904i	−0.982178	+	0.187952i	$0.939815\pi$
$198$		0			0
$199$			12.2160i			0.865971i	0.901401	+	0.432986i	$0.142540\pi$
$199$			12.2160i			0.865971i	−0.901401	+	0.432986i	$0.857460\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.6114	+	9.87386i	0.744776	+	0.693009i
$204$		0			0
$205$	−1.60461			−0.112071
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.2194			−0.706889
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.1165			0.689940
$216$		0			0
$217$	7.22619	+	6.72393i	0.490546	+	0.456450i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	13.0238i		−	0.876075i
$222$		0			0
$223$			3.65784i			0.244947i	0.992472	+	0.122473i	$0.0390826\pi$
$223$			3.65784i			0.244947i	−0.992472	+	0.122473i	$0.960917\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.23301			−0.280955			−0.140477	−	0.990084i	$-0.544864\pi$
$227$	−4.23301			−0.280955			−0.140477	+	0.990084i	$0.544864\pi$
$228$		0			0
$229$		−	7.59497i		−	0.501890i	−0.968001	−	0.250945i	$-0.919259\pi$
$229$		−	7.59497i		−	0.501890i	0.968001	−	0.250945i	$-0.0807413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.94677i			0.651634i	0.945433	+	0.325817i	$0.105639\pi$
$233$			9.94677i			0.651634i	−0.945433	+	0.325817i	$0.894361\pi$
$234$		0			0
$235$	−11.1097			−0.724716
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.36197i			0.605575i	0.953058	+	0.302788i	$0.0979173\pi$
$239$			9.36197i			0.605575i	−0.953058	+	0.302788i	$0.902083\pi$
$240$		0			0
$241$		−	23.0408i		−	1.48419i	−0.670296	−	0.742093i	$-0.733833\pi$
$241$		−	23.0408i		−	1.48419i	0.670296	−	0.742093i	$-0.266167\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.503406	−	6.98188i	−0.0321614	−	0.446056i
$246$		0			0
$247$	−4.23301			−0.269340
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−23.8376			−1.50462			−0.752308	−	0.658811i	$-0.771060\pi$
$251$	−23.8376			−1.50462			−0.752308	+	0.658811i	$0.771060\pi$
$252$		0			0
$253$	−21.2228			−1.33427
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−29.7441			−1.85539			−0.927694	−	0.373341i	$-0.878212\pi$
$257$	−29.7441			−1.85539			−0.927694	+	0.373341i	$0.878212\pi$
$258$		0			0
$259$	8.13181	−	8.73923i	0.505286	−	0.543030i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.7545i			1.40310i	0.712618	+	0.701552i	$0.247509\pi$
$263$			22.7545i			1.40310i	−0.712618	+	0.701552i	$0.752491\pi$
$264$		0			0
$265$		−	2.26926i		−	0.139399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.202401			−0.0123406			−0.00617030	−	0.999981i	$-0.501964\pi$
$269$	−0.202401			−0.0123406			−0.00617030	+	0.999981i	$0.501964\pi$
$270$		0			0
$271$			11.4785i			0.697267i	0.937259	+	0.348634i	$0.113354\pi$
$271$			11.4785i			0.697267i	−0.937259	+	0.348634i	$0.886646\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.87386i		−	0.233603i
$276$		0			0
$277$	23.7211			1.42526			0.712632	−	0.701538i	$-0.247503\pi$
$277$	23.7211			1.42526			0.712632	+	0.701538i	$0.247503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5.57194i		−	0.332394i	−0.986093	−	0.166197i	$-0.946851\pi$
$281$		−	5.57194i		−	0.332394i	0.986093	−	0.166197i	$-0.0531488\pi$
$282$		0			0
$283$			32.1798i			1.91289i	0.291914	+	0.956445i	$0.405708\pi$
$283$			32.1798i			1.91289i	−0.291914	+	0.956445i	$0.594292\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.89199	+	3.10801i	−0.170709	+	0.183460i
$288$		0			0
$289$	48.8776			2.87515
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.8146			0.923898			0.461949	−	0.886907i	$-0.347150\pi$
$293$	15.8146			0.923898			0.461949	+	0.886907i	$0.347150\pi$
$294$		0			0
$295$	−4.61142			−0.268487
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.79079			−0.508384
$300$		0			0
$301$	18.2330	−	19.5950i	1.05093	−	1.12944i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			11.8472i			0.678371i
$306$		0			0
$307$			6.72393i			0.383755i	0.981419	+	0.191878i	$0.0614576\pi$
$307$			6.72393i			0.383755i	−0.981419	+	0.191878i	$0.938542\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.02379			−0.0580540			−0.0290270	−	0.999579i	$-0.509241\pi$
$311$	−1.02379			−0.0580540			−0.0290270	+	0.999579i	$0.509241\pi$
$312$		0			0
$313$		−	23.2568i		−	1.31455i	−0.753650	−	0.657276i	$-0.771709\pi$
$313$		−	23.2568i		−	1.31455i	0.753650	−	0.657276i	$-0.228291\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.46830i		−	0.250965i	−0.992096	−	0.125482i	$-0.959952\pi$
$317$		−	4.46830i		−	0.250965i	0.992096	−	0.125482i	$-0.0400478\pi$
$318$		0			0
$319$	21.2228			1.18825
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	21.4116i		−	1.19137i
$324$		0			0
$325$		−	1.60461i		−	0.0890076i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.0230	+	21.5187i	−1.10391	+	1.18636i
$330$		0			0
$331$	14.2160			0.781383			0.390692	−	0.920522i	$-0.372236\pi$
$331$	14.2160			0.781383			0.390692	+	0.920522i	$0.372236\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.90729			−0.377386
$336$		0			0
$337$	23.4286			1.27624			0.638118	−	0.769938i	$-0.279713\pi$
$337$	23.4286			1.27624			0.638118	+	0.769938i	$0.279713\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.4524			0.782641
$342$		0			0
$343$	−14.4307	−	11.6084i	−0.779185	−	0.626794i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.1798i		−	1.08331i	−0.840602	−	0.541654i	$-0.817798\pi$
$347$		−	20.1798i		−	1.08331i	0.840602	−	0.541654i	$-0.182202\pi$
$348$		0			0
$349$			27.0372i			1.44727i	0.690184	+	0.723634i	$0.257530\pi$
$349$			27.0372i			1.44727i	−0.690184	+	0.723634i	$0.742470\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.3495			0.657298			0.328649	−	0.944452i	$-0.393407\pi$
$353$	12.3495			0.657298			0.328649	+	0.944452i	$0.393407\pi$
$354$		0			0
$355$		−	2.63803i		−	0.140012i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	25.5757i		−	1.34983i	−0.737894	−	0.674917i	$-0.764179\pi$
$359$		−	25.5757i		−	1.34983i	0.737894	−	0.674917i	$-0.235821\pi$
$360$		0			0
$361$	12.0408			0.633725
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	13.7477i		−	0.719589i
$366$		0			0
$367$			20.8444i			1.08807i	0.839062	+	0.544035i	$0.183104\pi$
$367$			20.8444i			1.08807i	−0.839062	+	0.544035i	$0.816896\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.39539	−	4.08989i	−0.228197	−	0.212336i
$372$		0			0
$373$	7.48810			0.387719			0.193860	−	0.981029i	$-0.437899\pi$
$373$	7.48810			0.387719			0.193860	+	0.981029i	$0.437899\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.79079			0.452749
$378$		0			0
$379$	−32.6651			−1.67789			−0.838946	−	0.544215i	$-0.816827\pi$
$379$	−32.6651			−1.67789			−0.838946	+	0.544215i	$0.816827\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.890309			0.0454927			0.0227463	−	0.999741i	$-0.492759\pi$
$383$	0.890309			0.0454927			0.0227463	+	0.999741i	$0.492759\pi$
$384$		0			0
$385$	−7.50341	−	6.98188i	−0.382409	−	0.355829i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			13.5317i			0.686084i	0.939320	+	0.343042i	$0.111457\pi$
$389$			13.5317i			0.686084i	−0.939320	+	0.343042i	$0.888543\pi$
$390$		0			0
$391$		−	44.4660i		−	2.24874i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.01698			0.403378
$396$		0			0
$397$		−	25.2991i		−	1.26973i	−0.772625	−	0.634863i	$-0.781057\pi$
$397$		−	25.2991i		−	1.26973i	0.772625	−	0.634863i	$-0.218943\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	32.7619i		−	1.63605i	−0.575182	−	0.818025i	$-0.695069\pi$
$401$		−	32.7619i		−	1.63605i	0.575182	−	0.818025i	$-0.304931\pi$
$402$		0			0
$403$	5.98638			0.298203
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17.4785i		−	0.866376i
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.31117	+	8.93200i	−0.408966	+	0.439515i
$414$		0			0
$415$	−3.20921			−0.157534
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.17937			−0.204176			−0.102088	−	0.994775i	$-0.532552\pi$
$419$	−4.17937			−0.204176			−0.102088	+	0.994775i	$0.532552\pi$
$420$		0			0
$421$	−32.6514			−1.59133			−0.795667	−	0.605735i	$-0.792879\pi$
$421$	−32.6514			−1.59133			−0.795667	+	0.605735i	$0.792879\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.11650			0.393708
$426$		0			0
$427$	22.9473	+	21.3523i	1.11050	+	1.03331i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13.3087i			0.641059i	0.947239	+	0.320530i	$0.103861\pi$
$431$			13.3087i			0.641059i	−0.947239	+	0.320530i	$0.896139\pi$
$432$		0			0
$433$			6.95358i			0.334168i	0.985943	+	0.167084i	$0.0534351\pi$
$433$			6.95358i			0.334168i	−0.985943	+	0.167084i	$0.946565\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.4524			−0.691352
$438$		0			0
$439$		−	26.9739i		−	1.28739i	−0.765280	−	0.643697i	$-0.777400\pi$
$439$		−	26.9739i		−	1.28739i	0.765280	−	0.643697i	$-0.222600\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.80441i		−	0.133242i	−0.997778	−	0.0666208i	$-0.978778\pi$
$443$		−	2.80441i		−	0.133242i	0.997778	−	0.0666208i	$-0.0212218\pi$
$444$		0			0
$445$	−17.8376			−0.845584
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	30.1226i		−	1.42157i	−0.703409	−	0.710786i	$-0.748340\pi$
$449$		−	30.1226i		−	1.42157i	0.703409	−	0.710786i	$-0.251660\pi$
$450$		0			0
$451$			6.21603i			0.292701i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.10801	−	2.89199i	−0.145706	−	0.135578i
$456$		0			0
$457$	−22.0340			−1.03071			−0.515353	−	0.856978i	$-0.672339\pi$
$457$	−22.0340			−1.03071			−0.515353	+	0.856978i	$0.672339\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.4116			0.717790			0.358895	−	0.933378i	$-0.383154\pi$
$461$	15.4116			0.717790			0.358895	+	0.933378i	$0.383154\pi$
$462$		0			0
$463$	25.0605			1.16466			0.582329	−	0.812953i	$-0.302142\pi$
$463$	25.0605			1.16466			0.582329	+	0.812953i	$0.302142\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.62764			−0.445514			−0.222757	−	0.974874i	$-0.571506\pi$
$467$	−9.62764			−0.445514			−0.222757	+	0.974874i	$0.571506\pi$
$468$		0			0
$469$	−12.4490	+	13.3789i	−0.574843	+	0.617782i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	39.1899i		−	1.80196i
$474$		0			0
$475$		−	2.63803i		−	0.121041i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	37.0238			1.69166			0.845830	−	0.533452i	$-0.179106\pi$
$479$	37.0238			1.69166			0.845830	+	0.533452i	$0.179106\pi$
$480$		0			0
$481$		−	7.23982i		−	0.330107i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			8.68768i			0.394487i
$486$		0			0
$487$	11.6386			0.527394			0.263697	−	0.964606i	$-0.415058\pi$
$487$	11.6386			0.527394			0.263697	+	0.964606i	$0.415058\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	32.1069i		−	1.44896i	−0.689294	−	0.724481i	$-0.742079\pi$
$491$		−	32.1069i		−	1.44896i	0.689294	−	0.724481i	$-0.257921\pi$
$492$		0			0
$493$			44.4660i			2.00265i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.10969	−	4.75454i	−0.229201	−	0.213270i
$498$		0			0
$499$	−13.6122			−0.609365			−0.304682	−	0.952454i	$-0.598550\pi$
$499$	−13.6122			−0.609365			−0.304682	+	0.952454i	$0.598550\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.07573			−0.137140			−0.0685700	−	0.997646i	$-0.521844\pi$
$503$	−3.07573			−0.137140			−0.0685700	+	0.997646i	$0.521844\pi$
$504$		0			0
$505$	6.21603			0.276609
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.7908			−0.655590			−0.327795	−	0.944749i	$-0.606306\pi$
$509$	−14.7908			−0.655590			−0.327795	+	0.944749i	$0.606306\pi$
$510$		0			0
$511$	−26.6284	−	24.7776i	−1.17797	−	1.09610i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	12.8807i		−	0.567590i
$516$		0			0
$517$			43.0374i			1.89278i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.0332			1.18435			0.592173	−	0.805811i	$-0.298270\pi$
$521$	27.0332			1.18435			0.592173	+	0.805811i	$0.298270\pi$
$522$		0			0
$523$			3.51472i			0.153688i	0.997043	+	0.0768440i	$0.0244843\pi$
$523$			3.51472i			0.153688i	−0.997043	+	0.0768440i	$0.975516\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			30.2806i			1.31904i
$528$		0			0
$529$	−7.01362			−0.304940
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.57476i			0.111525i
$534$		0			0
$535$			19.9638i			0.863108i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.0468	+	1.95013i	−1.16499	+	0.0839979i
$540$		0			0
$541$	−39.6616			−1.70519			−0.852593	−	0.522576i	$-0.824971\pi$
$541$	−39.6616			−1.70519			−0.852593	+	0.522576i	$0.824971\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.0238			−0.643549
$546$		0			0
$547$	34.3495			1.46868			0.734339	−	0.678782i	$-0.237492\pi$
$547$	34.3495			1.46868			0.734339	+	0.678782i	$0.237492\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.4524			0.615692
$552$		0			0
$553$	14.4490	−	15.5283i	0.614435	−	0.660332i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			38.4660i			1.62986i	0.579561	+	0.814929i	$0.303224\pi$
$557$			38.4660i			1.62986i	−0.579561	+	0.814929i	$0.696776\pi$
$558$		0			0
$559$		−	16.2330i		−	0.686583i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.2194			−0.430696			−0.215348	−	0.976537i	$-0.569089\pi$
$563$	−10.2194			−0.430696			−0.215348	+	0.976537i	$0.569089\pi$
$564$		0			0
$565$		−	10.2330i		−	0.430506i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	2.48129i		−	0.104021i	−0.998647	−	0.0520106i	$-0.983437\pi$
$569$		−	2.48129i		−	0.104021i	0.998647	−	0.0520106i	$-0.0165629\pi$
$570$		0			0
$571$	−20.6345			−0.863525			−0.431762	−	0.901987i	$-0.642108\pi$
$571$	−20.6345			−0.863525			−0.431762	+	0.901987i	$0.642108\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	5.47847i		−	0.228468i
$576$		0			0
$577$			2.50455i			0.104266i	0.998640	+	0.0521329i	$0.0166019\pi$
$577$			2.50455i			0.104266i	−0.998640	+	0.0521329i	$0.983398\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.78397	+	6.21603i	−0.239960	+	0.257884i
$582$		0			0
$583$	−8.79079			−0.364077
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.4422			1.29776			0.648880	−	0.760891i	$-0.275238\pi$
$587$	31.4422			1.29776			0.648880	+	0.760891i	$0.275238\pi$
$588$		0			0
$589$	9.84183			0.405526
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.1267			−1.11396			−0.556979	−	0.830526i	$-0.688040\pi$
$593$	−27.1267			−1.11396			−0.556979	+	0.830526i	$0.688040\pi$
$594$		0			0
$595$	14.6284	−	15.7211i	0.599706	−	0.644503i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			33.7747i			1.38000i	0.723810	+	0.689999i	$0.242389\pi$
$599$			33.7747i			1.38000i	−0.723810	+	0.689999i	$0.757611\pi$
$600$		0			0
$601$		−	24.4886i		−	0.998912i	−0.866339	−	0.499456i	$-0.833533\pi$
$601$		−	24.4886i		−	0.998912i	0.866339	−	0.499456i	$-0.166467\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.00681			−0.162900
$606$		0			0
$607$			27.3331i			1.10941i	0.832045	+	0.554707i	$0.187170\pi$
$607$			27.3331i			1.10941i	−0.832045	+	0.554707i	$0.812830\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			17.8267i			0.721190i
$612$		0			0
$613$	11.3163			0.457061			0.228531	−	0.973537i	$-0.426608\pi$
$613$	11.3163			0.457061			0.228531	+	0.973537i	$0.426608\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10.6843i			0.430135i	0.976599	+	0.215067i	$0.0689971\pi$
$617$			10.6843i			0.430135i	−0.976599	+	0.215067i	$0.931003\pi$
$618$		0			0
$619$		−	9.56437i		−	0.384424i	−0.981353	−	0.192212i	$-0.938434\pi$
$619$		−	9.56437i		−	0.384424i	0.981353	−	0.192212i	$-0.0615662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−32.1488	+	34.5502i	−1.28801	+	1.38423i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	36.6208			1.46017
$630$		0			0
$631$	−17.1956			−0.684546			−0.342273	−	0.939601i	$-0.611197\pi$
$631$	−17.1956			−0.684546			−0.342273	+	0.939601i	$0.611197\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.81382			0.111663
$636$		0			0
$637$	−11.2032	+	0.807769i	−0.443885	+	0.0320050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	16.5289i		−	0.652851i	−0.945223	−	0.326426i	$-0.894156\pi$
$641$		−	16.5289i		−	0.652851i	0.945223	−	0.326426i	$-0.105844\pi$
$642$		0			0
$643$		−	37.4286i		−	1.47604i	−0.674779	−	0.738020i	$-0.735761\pi$
$643$		−	37.4286i		−	1.47604i	0.674779	−	0.738020i	$-0.264239\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.52812			−0.217333			−0.108666	−	0.994078i	$-0.534658\pi$
$647$	−5.52812			−0.217333			−0.108666	+	0.994078i	$0.534658\pi$
$648$		0			0
$649$			17.8640i			0.701223i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			18.5159i			0.724583i	0.932065	+	0.362291i	$0.118005\pi$
$653$			18.5159i			0.724583i	−0.932065	+	0.362291i	$0.881995\pi$
$654$		0			0
$655$	8.62840			0.337139
$656$		0			0
$657$		0			0
$658$		0			0
$659$			43.3693i			1.68943i	0.535217	+	0.844714i	$0.320230\pi$
$659$			43.3693i			1.68943i	−0.535217	+	0.844714i	$0.679770\pi$
$660$		0			0
$661$			36.1142i			1.40468i	0.711842	+	0.702340i	$0.247861\pi$
$661$			36.1142i			1.40468i	−0.711842	+	0.702340i	$0.752139\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.10969	−	4.75454i	−0.198145	−	0.184373i
$666$		0			0
$667$	30.0136			1.16213
$668$		0			0
$669$		0			0
$670$		0			0
$671$	45.8946			1.77174
$672$		0			0
$673$	−33.8743			−1.30576			−0.652879	−	0.757463i	$-0.726439\pi$
$673$	−33.8743			−1.30576			−0.652879	+	0.757463i	$0.726439\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.38446			0.168509			0.0842543	−	0.996444i	$-0.473149\pi$
$677$	4.38446			0.168509			0.0842543	+	0.996444i	$0.473149\pi$
$678$		0			0
$679$	16.8274	+	15.6578i	0.645778	+	0.600893i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.95013i		−	0.0746195i	−0.999304	−	0.0373098i	$-0.988121\pi$
$683$		−	1.95013i		−	0.0746195i	0.999304	−	0.0373098i	$-0.0118788\pi$
$684$		0			0
$685$		−	0.723932i		−	0.0276600i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.64126			−0.138721
$690$		0			0
$691$		−	12.1471i		−	0.462098i	−0.972942	−	0.231049i	$-0.925784\pi$
$691$		−	12.1471i		−	0.462098i	0.972942	−	0.231049i	$-0.0742157\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2.84043i			0.107744i
$696$		0			0
$697$	−13.0238			−0.493311
$698$		0			0
$699$		0			0
$700$		0			0
$701$			11.0600i			0.417732i	0.977944	+	0.208866i	$0.0669773\pi$
$701$			11.0600i			0.417732i	−0.977944	+	0.208866i	$0.933023\pi$
$702$		0			0
$703$		−	11.9025i		−	0.448913i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.2032	−	12.0400i	0.421338	−	0.452811i
$708$		0			0
$709$	14.2330			0.534532			0.267266	−	0.963623i	$-0.413880\pi$
$709$	14.2330			0.534532			0.267266	+	0.963623i	$0.413880\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	20.4388			0.765438
$714$		0			0
$715$	−6.21603			−0.232466
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.4660			−0.763254			−0.381627	−	0.924317i	$-0.624636\pi$
$719$	−20.4660			−0.763254			−0.381627	+	0.924317i	$0.624636\pi$
$720$		0			0
$721$	−24.9490	−	23.2149i	−0.929149	−	0.864567i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			5.47847i			0.203465i
$726$		0			0
$727$			25.5717i			0.948402i	0.880417	+	0.474201i	$0.157263\pi$
$727$			25.5717i			0.948402i	−0.880417	+	0.474201i	$0.842737\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	82.1106			3.03697
$732$		0			0
$733$			22.6624i			0.837053i	0.908204	+	0.418527i	$0.137453\pi$
$733$			22.6624i			0.837053i	−0.908204	+	0.418527i	$0.862547\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			26.7579i			0.985640i
$738$		0			0
$739$	44.8675			1.65048			0.825238	−	0.564785i	$-0.191041\pi$
$739$	44.8675			1.65048			0.825238	+	0.564785i	$0.191041\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5.37536i			0.197203i	0.995127	+	0.0986015i	$0.0314369\pi$
$743$			5.37536i			0.197203i	−0.995127	+	0.0986015i	$0.968563\pi$
$744$		0			0
$745$			3.00681i			0.110161i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	38.6684	+	35.9807i	1.41291	+	1.31471i
$750$		0			0
$751$	−13.6276			−0.497280			−0.248640	−	0.968596i	$-0.579984\pi$
$751$	−13.6276			−0.497280			−0.248640	+	0.968596i	$0.579984\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.2262			0.554138
$756$		0			0
$757$	−4.54586			−0.165222			−0.0826110	−	0.996582i	$-0.526326\pi$
$757$	−4.54586			−0.165222			−0.0826110	+	0.996582i	$0.526326\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.6522			0.567392			0.283696	−	0.958914i	$-0.408439\pi$
$761$	15.6522			0.567392			0.283696	+	0.958914i	$0.408439\pi$
$762$		0			0
$763$	−27.0774	+	29.1001i	−0.980269	+	1.05349i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7.39951i			0.267181i
$768$		0			0
$769$			17.1922i			0.619968i	0.950742	+	0.309984i	$0.100324\pi$
$769$			17.1922i			0.619968i	−0.950742	+	0.309984i	$0.899676\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.6990			−1.53578			−0.767889	−	0.640584i	$-0.778693\pi$
$773$	−42.6990			−1.53578			−0.767889	+	0.640584i	$0.778693\pi$
$774$		0			0
$775$			3.73074i			0.134012i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			4.23301i			0.151663i
$780$		0			0
$781$	−10.2194			−0.365678
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	2.64767i		−	0.0944993i
$786$		0			0
$787$			30.3992i			1.08361i	0.840503	+	0.541806i	$0.182259\pi$
$787$			30.3992i			1.08361i	−0.840503	+	0.541806i	$0.817741\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−19.8206	−	18.4430i	−0.704741	−	0.655757i
$792$		0			0
$793$	19.0102			0.675071
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.99319			−0.106024			−0.0530121	−	0.998594i	$-0.516882\pi$
$797$	−2.99319			−0.106024			−0.0530121	+	0.998594i	$0.516882\pi$
$798$		0			0
$799$	−90.1718			−3.19005
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−53.2568			−1.87939
$804$		0			0
$805$	−10.6114	−	9.87386i	−0.374003	−	0.348008i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			25.0334i			0.880128i	0.897966	+	0.440064i	$0.145044\pi$
$809$			25.0334i			0.880128i	−0.897966	+	0.440064i	$0.854956\pi$
$810$		0			0
$811$		−	19.4782i		−	0.683974i	−0.939705	−	0.341987i	$-0.888900\pi$
$811$		−	19.4782i		−	0.683974i	0.939705	−	0.341987i	$-0.111100\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.32572			0.326666
$816$		0			0
$817$		−	26.6877i		−	0.933684i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			45.3787i			1.58373i	0.610697	+	0.791864i	$0.290889\pi$
$821$			45.3787i			1.58373i	−0.610697	+	0.791864i	$0.709111\pi$
$822$		0			0
$823$	−25.8512			−0.901117			−0.450559	−	0.892747i	$-0.648775\pi$
$823$	−25.8512			−0.901117			−0.450559	+	0.892747i	$0.648775\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.3429i			0.672619i	0.941751	+	0.336310i	$0.109179\pi$
$827$			19.3429i			0.672619i	−0.941751	+	0.336310i	$0.890821\pi$
$828$		0			0
$829$			43.4764i			1.51000i	0.655726	+	0.754999i	$0.272363\pi$
$829$			43.4764i			1.51000i	−0.655726	+	0.754999i	$0.727637\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.08590	−	56.6684i	−0.141568	−	1.96344i
$834$		0			0
$835$	−17.3257			−0.599581
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.2670			−0.837789			−0.418894	−	0.908035i	$-0.637582\pi$
$839$	−24.2670			−0.837789			−0.418894	+	0.908035i	$0.637582\pi$
$840$		0			0
$841$	−1.01362			−0.0349526
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.4252			0.358639
$846$		0			0
$847$	−7.22149	+	7.76092i	−0.248133	+	0.266668i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	24.7183i		−	0.847332i
$852$		0			0
$853$			16.2439i			0.556182i	0.960555	+	0.278091i	$0.0897017\pi$
$853$			16.2439i			0.556182i	−0.960555	+	0.278091i	$0.910298\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−40.5825			−1.38627			−0.693136	−	0.720807i	$-0.743772\pi$
$857$	−40.5825			−1.38627			−0.693136	+	0.720807i	$0.743772\pi$
$858$		0			0
$859$		−	5.05310i		−	0.172410i	−0.996277	−	0.0862048i	$-0.972526\pi$
$859$		−	5.05310i		−	0.172410i	0.996277	−	0.0862048i	$-0.0274739\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	48.3289i		−	1.64514i	−0.568666	−	0.822568i	$-0.692540\pi$
$863$		−	48.3289i		−	1.64514i	0.568666	−	0.822568i	$-0.307460\pi$
$864$		0			0
$865$	2.99319			0.101771
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	31.0567i		−	1.05353i
$870$		0			0
$871$			11.0835i			0.375549i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.80230	−	1.93693i	0.0609290	−	0.0654802i
$876$		0			0
$877$	−55.3963			−1.87060			−0.935301	−	0.353854i	$-0.884871\pi$
$877$	−55.3963			−1.87060			−0.935301	+	0.353854i	$0.884871\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	49.8716			1.68022			0.840108	−	0.542419i	$-0.182491\pi$
$881$	49.8716			1.68022			0.840108	+	0.542419i	$0.182491\pi$
$882$		0			0
$883$	43.3393			1.45848			0.729242	−	0.684255i	$-0.239873\pi$
$883$	43.3393			1.45848			0.729242	+	0.684255i	$0.239873\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.1539			−1.11320			−0.556600	−	0.830781i	$-0.687894\pi$
$887$	−33.1539			−1.11320			−0.556600	+	0.830781i	$0.687894\pi$
$888$		0			0
$889$	5.07136	−	5.45018i	0.170088	−	0.182793i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			29.3077i			0.980746i
$894$		0			0
$895$		−	7.38858i		−	0.246973i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.4388			−0.681671
$900$		0			0
$901$		−	18.4184i		−	0.613607i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.1097i		−	0.369299i
$906$		0			0
$907$	−50.5553			−1.67866			−0.839330	−	0.543622i	$-0.817052\pi$
$907$	−50.5553			−1.67866			−0.839330	+	0.543622i	$0.817052\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	21.6755i		−	0.718140i	−0.933311	−	0.359070i	$-0.883094\pi$
$911$		−	21.6755i		−	0.718140i	0.933311	−	0.359070i	$-0.116906\pi$
$912$		0			0
$913$			12.4321i			0.411441i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.5510	−	16.7126i	0.513539	−	0.551899i
$918$		0			0
$919$	25.6616			0.846498			0.423249	−	0.906013i	$-0.360890\pi$
$919$	25.6616			0.846498			0.423249	+	0.906013i	$0.360890\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.23301			−0.139331
$924$		0			0
$925$	4.51190			0.148350
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31.6182			1.03736			0.518680	−	0.854968i	$-0.326424\pi$
$929$	31.6182			1.03736			0.518680	+	0.854968i	$0.326424\pi$
$930$		0			0
$931$	−18.4184	+	1.32800i	−0.603640	+	0.0435235i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	31.4422i		−	1.02827i
$936$		0			0
$937$		−	38.7013i		−	1.26432i	−0.774839	−	0.632158i	$-0.782169\pi$
$937$		−	38.7013i		−	1.26432i	0.774839	−	0.632158i	$-0.217831\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	32.9932			1.07555			0.537774	−	0.843089i	$-0.319266\pi$
$941$	32.9932			1.07555			0.537774	+	0.843089i	$0.319266\pi$
$942$		0			0
$943$			8.79079i			0.286267i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			31.8452i			1.03483i	0.855735	+	0.517415i	$0.173106\pi$
$947$			31.8452i			1.03483i	−0.855735	+	0.517415i	$0.826894\pi$
$948$		0			0
$949$	−22.0597			−0.716088
$950$		0			0
$951$		0			0
$952$		0			0
$953$			45.9003i			1.48686i	0.668817	+	0.743428i	$0.266801\pi$
$953$			45.9003i			1.48686i	−0.668817	+	0.743428i	$0.733199\pi$
$954$		0			0
$955$		−	9.36197i		−	0.302946i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.40221	−	1.30474i	−0.0452796	−	0.0421324i
$960$		0			0
$961$	17.0816			0.551018
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.4422			0.561485
$966$		0			0
$967$	41.2662			1.32703			0.663516	−	0.748162i	$-0.269064\pi$
$967$	41.2662			1.32703			0.663516	+	0.748162i	$0.269064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.0008			−0.353031			−0.176516	−	0.984298i	$-0.556483\pi$
$971$	−11.0008			−0.353031			−0.176516	+	0.984298i	$0.556483\pi$
$972$		0			0
$973$	5.50173	+	5.11933i	0.176377	+	0.164118i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.48528i		−	0.0795112i	−0.999209	−	0.0397556i	$-0.987342\pi$
$977$		−	2.48528i		−	0.0795112i	0.999209	−	0.0397556i	$-0.0126579\pi$
$978$		0			0
$979$			69.1005i			2.20846i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	31.7408			1.01237			0.506187	−	0.862424i	$-0.331055\pi$
$983$	31.7408			1.01237			0.506187	+	0.862424i	$0.331055\pi$
$984$		0			0
$985$			5.27607i			0.168110i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	55.4230i		−	1.76235i
$990$		0			0
$991$	−57.3368			−1.82136			−0.910682	−	0.413108i	$-0.864443\pi$
$991$	−57.3368			−1.82136			−0.910682	+	0.413108i	$0.864443\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			12.2160i			0.387274i
$996$		0			0
$997$			52.6760i			1.66827i	0.551564	+	0.834133i	$0.314031\pi$
$997$			52.6760i			1.66827i	−0.551564	+	0.834133i	$0.685969\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.f.i.881.5		8
3.2	odd	2		5040.2.f.f.881.5		8
4.3	odd	2		630.2.b.b.251.2	yes	8
7.6	odd	2		5040.2.f.f.881.6		8
12.11	even	2		630.2.b.a.251.6	yes	8
20.3	even	4		3150.2.d.a.3149.8		8
20.7	even	4		3150.2.d.f.3149.1		8
20.19	odd	2		3150.2.b.f.251.7		8
21.20	even	2	inner	5040.2.f.i.881.6		8
28.27	even	2		630.2.b.a.251.2	✓	8
60.23	odd	4		3150.2.d.d.3149.8		8
60.47	odd	4		3150.2.d.c.3149.1		8
60.59	even	2		3150.2.b.e.251.3		8
84.83	odd	2		630.2.b.b.251.6	yes	8
140.27	odd	4		3150.2.d.d.3149.7		8
140.83	odd	4		3150.2.d.c.3149.2		8
140.139	even	2		3150.2.b.e.251.7		8
420.83	even	4		3150.2.d.f.3149.2		8
420.167	even	4		3150.2.d.a.3149.7		8
420.419	odd	2		3150.2.b.f.251.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.2	✓	8	28.27	even	2
630.2.b.a.251.6	yes	8	12.11	even	2
630.2.b.b.251.2	yes	8	4.3	odd	2
630.2.b.b.251.6	yes	8	84.83	odd	2
3150.2.b.e.251.3		8	60.59	even	2
3150.2.b.e.251.7		8	140.139	even	2
3150.2.b.f.251.3		8	420.419	odd	2
3150.2.b.f.251.7		8	20.19	odd	2
3150.2.d.a.3149.7		8	420.167	even	4
3150.2.d.a.3149.8		8	20.3	even	4
3150.2.d.c.3149.1		8	60.47	odd	4
3150.2.d.c.3149.2		8	140.83	odd	4
3150.2.d.d.3149.7		8	140.27	odd	4
3150.2.d.d.3149.8		8	60.23	odd	4
3150.2.d.f.3149.1		8	20.7	even	4
3150.2.d.f.3149.2		8	420.83	even	4
5040.2.f.f.881.5		8	3.2	odd	2
5040.2.f.f.881.6		8	7.6	odd	2
5040.2.f.i.881.5		8	1.1	even	1	trivial
5040.2.f.i.881.6		8	21.20	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +(1.80230 - 1.93693i) q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +(1.80230 - 1.93693i) q^{7} -3.87386i q^{11} -1.60461i q^{13} +8.11650 q^{17} -2.63803i q^{19} -5.47847i q^{23} +1.00000 q^{25} +5.47847i q^{29} +3.73074i q^{31} +(1.80230 - 1.93693i) q^{35} +4.51190 q^{37} -1.60461 q^{41} +10.1165 q^{43} -11.1097 q^{47} +(-0.503406 - 6.98188i) q^{49} -2.26926i q^{53} -3.87386i q^{55} -4.61142 q^{59} +11.8472i q^{61} -1.60461i q^{65} -6.90729 q^{67} -2.63803i q^{71} -13.7477i q^{73} +(-7.50341 - 6.98188i) q^{77} +8.01698 q^{79} -3.20921 q^{83} +8.11650 q^{85} -17.8376 q^{89} +(-3.10801 - 2.89199i) q^{91} -2.63803i q^{95} +8.68768i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 4 q^{7}+O(q^{10}) \) 8 * q + 8 * q^5 + 4 * q^7	\( 8 q + 8 q^{5} + 4 q^{7} + 8 q^{25} + 4 q^{35} - 8 q^{37} + 8 q^{41} + 16 q^{43} - 40 q^{47} + 4 q^{49} - 32 q^{67} - 52 q^{77} - 8 q^{79} + 16 q^{83} + 8 q^{89} + 4 q^{91}+O(q^{100}) \) 8 * q + 8 * q^5 + 4 * q^7 + 8 * q^25 + 4 * q^35 - 8 * q^37 + 8 * q^41 + 16 * q^43 - 40 * q^47 + 4 * q^49 - 32 * q^67 - 52 * q^77 - 8 * q^79 + 16 * q^83 + 8 * q^89 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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