LMFDB - Embedded newform 5040.2.t.p.1009.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.p.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} +O(q^{10})$	$q+(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -3.00000 q^{11} +1.00000i q^{13} +7.00000i q^{17} +6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -5.00000 q^{29} -2.00000 q^{31} +(-1.00000 - 2.00000i) q^{35} -2.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +(-6.00000 + 3.00000i) q^{55} -10.0000 q^{59} -8.00000 q^{61} +(1.00000 + 2.00000i) q^{65} +2.00000i q^{67} -8.00000 q^{71} +6.00000i q^{73} +3.00000i q^{77} -5.00000 q^{79} -4.00000i q^{83} +(7.00000 + 14.0000i) q^{85} +1.00000 q^{91} -7.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} - 6 q^{11} + 6 q^{25} - 10 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{41} - 2 q^{49} - 12 q^{55} - 20 q^{59} - 16 q^{61} + 2 q^{65} - 16 q^{71} - 10 q^{79} + 14 q^{85} + 2 q^{91}+O(q^{100}) $ 2 * q + 4 * q^5 - 6 * q^11 + 6 * q^25 - 10 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^41 - 2 * q^49 - 12 * q^55 - 20 * q^59 - 16 * q^61 + 2 * q^65 - 16 * q^71 - 10 * q^79 + 14 * q^85 + 2 * 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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$			1.00000i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			1.00000i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.00000i			1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$			7.00000i			1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	−	2.00000i	−0.169031	−	0.338062i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.00000i			0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$			3.00000i			0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−6.00000	+	3.00000i	−0.809040	+	0.404520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.0000			−1.30189			−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	+	2.00000i	0.124035	+	0.248069i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.00000i			0.341882i
$78$		0			0
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	7.00000	+	14.0000i	0.759257	+	1.51851i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	1.00000			0.104828
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	7.00000i		−	0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$		−	7.00000i		−	0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$			19.0000i			1.87213i	0.351833	+	0.936063i	$0.385559\pi$
$103$			19.0000i			1.87213i	−0.351833	+	0.936063i	$0.614441\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	6.00000	+	12.0000i	0.559503	+	1.11901i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.00000			0.641689
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	22.0000			1.92215			0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000			1.92215			0.961074	+	0.276289i	$0.0891049\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3.00000i		−	0.250873i
$144$		0			0
$145$	−10.0000	+	5.00000i	−0.830455	+	0.415227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000	+	2.00000i	−0.321288	+	0.160644i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$			9.00000i			0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$			9.00000i			0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$		−	21.0000i		−	1.53567i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			5.00000i			0.350931i
$204$		0			0
$205$	−4.00000	+	2.00000i	−0.279372	+	0.139686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.00000			−0.470871
$222$		0			0
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	17.0000i		−	1.12833i	−0.825662	−	0.564165i	$-0.809198\pi$
$227$		−	17.0000i		−	1.12833i	0.825662	−	0.564165i	$-0.190802\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	3.00000	+	6.00000i	0.195698	+	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.0000			−0.970269			−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000			−0.970269			−0.485135	+	0.874439i	$0.661229\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.00000	+	1.00000i	−0.127775	+	0.0638877i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		−	18.0000i		−	1.13165i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.00000	+	12.0000i	−0.542720	+	0.723627i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.00000			−0.417585			−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000			−0.417585			−0.208792	+	0.977960i	$0.566953\pi$
$282$		0			0
$283$		−	11.0000i		−	0.653882i	−0.945045	−	0.326941i	$-0.893982\pi$
$283$		−	11.0000i		−	0.653882i	0.945045	−	0.326941i	$-0.106018\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.00000i			0.118056i
$288$		0			0
$289$	−32.0000			−1.88235
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	−20.0000	+	10.0000i	−1.16445	+	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.0000	+	8.00000i	−0.916157	+	0.458079i
$306$		0			0
$307$			7.00000i			0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$			7.00000i			0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$		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
$313$			21.0000i			1.18699i	0.804838	+	0.593495i	$0.202252\pi$
$313$			21.0000i			1.18699i	−0.804838	+	0.593495i	$0.797748\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$	15.0000			0.839839
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000	+	3.00000i	0.221880	+	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000			0.165395
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.00000	+	4.00000i	0.109272	+	0.218543i
$336$		0			0
$337$			18.0000i			0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$			18.0000i			0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	11.0000i		−	0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$		−	11.0000i		−	0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$		0			0
$355$	−16.0000	+	8.00000i	−0.849192	+	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000	+	12.0000i	0.314054	+	0.628109i
$366$		0			0
$367$		−	3.00000i		−	0.156599i	−0.996930	−	0.0782994i	$-0.975051\pi$
$367$		−	3.00000i		−	0.156599i	0.996930	−	0.0782994i	$-0.0249490\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000			−0.311504
$372$		0			0
$373$		−	24.0000i		−	1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$		−	24.0000i		−	1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.00000i		−	0.257513i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$	3.00000	+	6.00000i	0.152894	+	0.305788i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.00000			0.253510			0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000			0.253510			0.126755	+	0.991934i	$0.459544\pi$
$390$		0			0
$391$	−42.0000			−2.12403
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.0000	+	5.00000i	−0.503155	+	0.251577i
$396$		0			0
$397$		−	7.00000i		−	0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$		−	7.00000i		−	0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000			0.149813			0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000			0.149813			0.0749064	+	0.997191i	$0.476134\pi$
$402$		0			0
$403$		−	2.00000i		−	0.0996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.00000i			0.297409i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			10.0000i			0.492068i
$414$		0			0
$415$	−4.00000	−	8.00000i	−0.196352	−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	−3.00000			−0.146211			−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000			−0.146211			−0.0731055	+	0.997324i	$0.523291\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	28.0000	+	21.0000i	1.35820	+	1.01865i
$426$		0			0
$427$			8.00000i			0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.0000			−1.10787			−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000			−1.10787			−0.553936	+	0.832560i	$0.686875\pi$
$432$		0			0
$433$			26.0000i			1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$			26.0000i			1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	30.0000			1.43182			0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000			1.43182			0.715911	+	0.698192i	$0.246012\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.00000			−0.235965			−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000			−0.235965			−0.117982	+	0.993016i	$0.537643\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	−	1.00000i	0.0937614	−	0.0468807i
$456$		0			0
$457$			38.0000i			1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$			38.0000i			1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	27.0000i		−	1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$		−	27.0000i		−	1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$		0			0
$469$	2.00000			0.0923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.0000			1.37073			0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000			1.37073			0.685367	+	0.728197i	$0.259642\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00000	−	14.0000i	−0.317854	−	0.635707i
$486$		0			0
$487$			42.0000i			1.90320i	0.307337	+	0.951601i	$0.400562\pi$
$487$			42.0000i			1.90320i	−0.307337	+	0.951601i	$0.599438\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.00000			0.315906			0.157953	−	0.987447i	$-0.449511\pi$
$491$	7.00000			0.315906			0.157953	+	0.987447i	$0.449511\pi$
$492$		0			0
$493$		−	35.0000i		−	1.57632i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−35.0000			−1.56682			−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000			−1.56682			−0.783408	+	0.621508i	$0.786520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.0000	+	38.0000i	0.837240	+	1.67448i
$516$		0			0
$517$		−	9.00000i		−	0.395820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.0000			1.22670			0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000			1.22670			0.613351	+	0.789810i	$0.289821\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	14.0000i		−	0.609850i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.00000i		−	0.0866296i
$534$		0			0
$535$	8.00000	+	16.0000i	0.345870	+	0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000			0.129219
$540$		0			0
$541$	27.0000			1.16082			0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000			1.16082			0.580410	+	0.814324i	$0.302892\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.0000	+	5.00000i	−0.428353	+	0.214176i
$546$		0			0
$547$			32.0000i			1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$			32.0000i			1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			5.00000i			0.212622i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	28.0000i		−	1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$		−	28.0000i		−	1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−6.00000	−	12.0000i	−0.252422	−	0.504844i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		0			0
$577$			43.0000i			1.79011i	0.445952	+	0.895057i	$0.352865\pi$
$577$			43.0000i			1.79011i	−0.445952	+	0.895057i	$0.647135\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.00000			−0.165948
$582$		0			0
$583$			18.0000i			0.745484i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	41.0000i		−	1.68367i	−0.539736	−	0.841834i	$-0.681476\pi$
$593$		−	41.0000i		−	1.68367i	0.539736	−	0.841834i	$-0.318524\pi$
$594$		0			0
$595$	14.0000	−	7.00000i	0.573944	−	0.286972i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−25.0000			−1.02147			−0.510736	−	0.859738i	$-0.670627\pi$
$599$	−25.0000			−1.02147			−0.510736	+	0.859738i	$0.670627\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.00000	+	2.00000i	−0.162623	+	0.0813116i
$606$		0			0
$607$			27.0000i			1.09590i	0.836512	+	0.547948i	$0.184591\pi$
$607$			27.0000i			1.09590i	−0.836512	+	0.547948i	$0.815409\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00000			−0.121367
$612$		0			0
$613$		−	44.0000i		−	1.77714i	−0.458738	−	0.888572i	$-0.651698\pi$
$613$		−	44.0000i		−	1.77714i	0.458738	−	0.888572i	$-0.348302\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14.0000			0.558217
$630$		0			0
$631$	−37.0000			−1.47295			−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000			−1.47295			−0.736473	+	0.676467i	$0.763510\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00000	+	4.00000i	0.0793676	+	0.158735i
$636$		0			0
$637$		−	1.00000i		−	0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.0000			−0.868948			−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000			−0.868948			−0.434474	+	0.900684i	$0.643066\pi$
$642$		0			0
$643$		−	1.00000i		−	0.0394362i	−0.999806	−	0.0197181i	$-0.993723\pi$
$643$		−	1.00000i		−	0.0394362i	0.999806	−	0.0197181i	$-0.00627687\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8.00000i			0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$			8.00000i			0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$		0			0
$651$		0			0
$652$		0			0
$653$			4.00000i			0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$			4.00000i			0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$		0			0
$655$	44.0000	−	22.0000i	1.71922	−	0.859611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.0000			0.584317			0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000			0.584317			0.292159	+	0.956370i	$0.405627\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	30.0000i		−	1.16160i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	43.0000i		−	1.65262i	−0.563212	−	0.826312i	$-0.690435\pi$
$677$		−	43.0000i		−	1.65262i	0.563212	−	0.826312i	$-0.309565\pi$
$678$		0			0
$679$	−7.00000			−0.268635
$680$		0			0
$681$		0			0
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$	12.0000	+	24.0000i	0.458496	+	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000			0.228582
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.0000	+	10.0000i	−0.758643	+	0.379322i
$696$		0			0
$697$		−	14.0000i		−	0.530288i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.0000			0.491003			0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000			0.491003			0.245502	+	0.969396i	$0.421047\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			12.0000i			0.451306i
$708$		0			0
$709$	−35.0000			−1.31445			−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000			−1.31445			−0.657226	+	0.753693i	$0.728270\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	−3.00000	−	6.00000i	−0.112194	−	0.224387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	19.0000			0.707597
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0000	+	20.0000i	−0.557086	+	0.742781i
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−28.0000			−1.03562
$732$		0			0
$733$			1.00000i			0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$			1.00000i			0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.00000i		−	0.221013i
$738$		0			0
$739$	−35.0000			−1.28750			−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000			−1.28750			−0.643748	+	0.765238i	$0.722621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.0000i		−	0.513610i	−0.966463	−	0.256805i	$-0.917330\pi$
$743$		−	14.0000i		−	0.513610i	0.966463	−	0.256805i	$-0.0826698\pi$
$744$		0			0
$745$	20.0000	−	10.0000i	0.732743	−	0.366372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	33.0000			1.20419			0.602094	−	0.798426i	$-0.294333\pi$
$751$	33.0000			1.20419			0.602094	+	0.798426i	$0.294333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.0000	−	13.0000i	0.946237	−	0.473118i
$756$		0			0
$757$		−	32.0000i		−	1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$		−	32.0000i		−	1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.00000			−0.0724999			−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000			−0.0724999			−0.0362500	+	0.999343i	$0.511541\pi$
$762$		0			0
$763$			5.00000i			0.181012i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	10.0000i		−	0.361079i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$	−6.00000	+	8.00000i	−0.215526	+	0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000	+	36.0000i	0.642448	+	1.28490i
$786$		0			0
$787$			17.0000i			0.605985i	0.952993	+	0.302992i	$0.0979856\pi$
$787$			17.0000i			0.605985i	−0.952993	+	0.302992i	$0.902014\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$		−	8.00000i		−	0.284088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.0000i		−	0.460484i	−0.973133	−	0.230242i	$-0.926048\pi$
$797$		−	13.0000i		−	0.460484i	0.973133	−	0.230242i	$-0.0739517\pi$
$798$		0			0
$799$	−21.0000			−0.742927
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	18.0000i		−	0.635206i
$804$		0			0
$805$	12.0000	−	6.00000i	0.422944	−	0.211472i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.0000			1.58212			0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000			1.58212			0.791058	+	0.611741i	$0.209531\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.0000	+	28.0000i	0.490399	+	0.980797i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.0000			0.802706			0.401353	−	0.915924i	$-0.368540\pi$
$821$	23.0000			0.802706			0.401353	+	0.915924i	$0.368540\pi$
$822$		0			0
$823$		−	26.0000i		−	0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$		−	26.0000i		−	0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18.0000i			0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$			18.0000i			0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	7.00000i		−	0.242536i
$834$		0			0
$835$	3.00000	+	6.00000i	0.103819	+	0.207639i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.0000			−1.03572			−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000			−1.03572			−0.517858	+	0.855467i	$0.673270\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.0000	−	12.0000i	0.825625	−	0.412813i
$846$		0			0
$847$			2.00000i			0.0687208i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.0000			0.411355
$852$		0			0
$853$		−	54.0000i		−	1.84892i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$		−	54.0000i		−	1.84892i	0.381273	−	0.924462i	$-0.375486\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	54.0000i		−	1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$		−	54.0000i		−	1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$		0			0
$865$	9.00000	+	18.0000i	0.306009	+	0.612018i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.0000			0.508840
$870$		0			0
$871$	−2.00000			−0.0677674
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−11.0000	−	2.00000i	−0.371868	−	0.0676123i
$876$		0			0
$877$		−	32.0000i		−	1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$		−	32.0000i		−	1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.0000			−1.07811			−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28.0000i			0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$			28.0000i			0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$		0			0
$889$	2.00000			0.0670778
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	40.0000	−	20.0000i	1.33705	−	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.0000			0.333519
$900$		0			0
$901$	42.0000			1.39922
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.0000	+	18.0000i	−1.19668	+	0.598340i
$906$		0			0
$907$		−	38.0000i		−	1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$		−	38.0000i		−	1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$			12.0000i			0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	22.0000i		−	0.726504i
$918$		0			0
$919$	−5.00000			−0.164935			−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000			−0.164935			−0.0824674	+	0.996594i	$0.526280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	8.00000i		−	0.263323i
$924$		0			0
$925$	−8.00000	−	6.00000i	−0.263038	−	0.197279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.0000	−	42.0000i	−0.686773	−	1.37355i
$936$		0			0
$937$			13.0000i			0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$			13.0000i			0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	48.0000			1.56476			0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000			1.56476			0.782378	+	0.622804i	$0.214007\pi$
$942$		0			0
$943$		−	12.0000i		−	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	42.0000i		−	1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$		−	42.0000i		−	1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$	−6.00000	+	3.00000i	−0.194155	+	0.0970777i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.0000			0.387500
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.0000	+	32.0000i	0.515058	+	1.03012i
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.0000			0.706014			0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000			0.706014			0.353007	+	0.935621i	$0.385159\pi$
$972$		0			0
$973$			10.0000i			0.320585i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$			51.0000i			1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$			51.0000i			1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$		0			0
$985$	2.00000	+	4.00000i	0.0637253	+	0.127451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.0000			−0.763156
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.0000	+	10.0000i	−0.634043	+	0.317021i
$996$		0			0
$997$			13.0000i			0.411714i	0.978582	+	0.205857i	$0.0659982\pi$
$997$			13.0000i			0.411714i	−0.978582	+	0.205857i	$0.934002\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.p.1009.1		2
3.2	odd	2		560.2.g.b.449.1		2
4.3	odd	2		315.2.d.a.64.2		2
5.4	even	2	inner	5040.2.t.p.1009.2		2
12.11	even	2		35.2.b.a.29.1	✓	2
15.2	even	4		2800.2.a.l.1.1		1
15.8	even	4		2800.2.a.w.1.1		1
15.14	odd	2		560.2.g.b.449.2		2
20.3	even	4		1575.2.a.k.1.1		1
20.7	even	4		1575.2.a.a.1.1		1
20.19	odd	2		315.2.d.a.64.1		2
24.5	odd	2		2240.2.g.g.449.2		2
24.11	even	2		2240.2.g.h.449.1		2
28.27	even	2		2205.2.d.b.1324.2		2
60.23	odd	4		175.2.a.a.1.1		1
60.47	odd	4		175.2.a.c.1.1		1
60.59	even	2		35.2.b.a.29.2	yes	2
84.11	even	6		245.2.j.e.79.2		4
84.23	even	6		245.2.j.e.214.1		4
84.47	odd	6		245.2.j.d.214.1		4
84.59	odd	6		245.2.j.d.79.2		4
84.83	odd	2		245.2.b.a.99.1		2
120.29	odd	2		2240.2.g.g.449.1		2
120.59	even	2		2240.2.g.h.449.2		2
140.139	even	2		2205.2.d.b.1324.1		2
420.59	odd	6		245.2.j.d.79.1		4
420.83	even	4		1225.2.a.a.1.1		1
420.167	even	4		1225.2.a.i.1.1		1
420.179	even	6		245.2.j.e.79.1		4
420.299	odd	6		245.2.j.d.214.2		4
420.359	even	6		245.2.j.e.214.2		4
420.419	odd	2		245.2.b.a.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.b.a.29.1	✓	2	12.11	even	2
35.2.b.a.29.2	yes	2	60.59	even	2
175.2.a.a.1.1		1	60.23	odd	4
175.2.a.c.1.1		1	60.47	odd	4
245.2.b.a.99.1		2	84.83	odd	2
245.2.b.a.99.2		2	420.419	odd	2
245.2.j.d.79.1		4	420.59	odd	6
245.2.j.d.79.2		4	84.59	odd	6
245.2.j.d.214.1		4	84.47	odd	6
245.2.j.d.214.2		4	420.299	odd	6
245.2.j.e.79.1		4	420.179	even	6
245.2.j.e.79.2		4	84.11	even	6
245.2.j.e.214.1		4	84.23	even	6
245.2.j.e.214.2		4	420.359	even	6
315.2.d.a.64.1		2	20.19	odd	2
315.2.d.a.64.2		2	4.3	odd	2
560.2.g.b.449.1		2	3.2	odd	2
560.2.g.b.449.2		2	15.14	odd	2
1225.2.a.a.1.1		1	420.83	even	4
1225.2.a.i.1.1		1	420.167	even	4
1575.2.a.a.1.1		1	20.7	even	4
1575.2.a.k.1.1		1	20.3	even	4
2205.2.d.b.1324.1		2	140.139	even	2
2205.2.d.b.1324.2		2	28.27	even	2
2240.2.g.g.449.1		2	120.29	odd	2
2240.2.g.g.449.2		2	24.5	odd	2
2240.2.g.h.449.1		2	24.11	even	2
2240.2.g.h.449.2		2	120.59	even	2
2800.2.a.l.1.1		1	15.2	even	4
2800.2.a.w.1.1		1	15.8	even	4
5040.2.t.p.1009.1		2	1.1	even	1	trivial
5040.2.t.p.1009.2		2	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)

\(f(q)\)	\(=\)	\(q+(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} -3.00000 q^{11} +1.00000i q^{13} +7.00000i q^{17} +6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -5.00000 q^{29} -2.00000 q^{31} +(-1.00000 - 2.00000i) q^{35} -2.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +(-6.00000 + 3.00000i) q^{55} -10.0000 q^{59} -8.00000 q^{61} +(1.00000 + 2.00000i) q^{65} +2.00000i q^{67} -8.00000 q^{71} +6.00000i q^{73} +3.00000i q^{77} -5.00000 q^{79} -4.00000i q^{83} +(7.00000 + 14.0000i) q^{85} +1.00000 q^{91} -7.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} - 6 q^{11} + 6 q^{25} - 10 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{41} - 2 q^{49} - 12 q^{55} - 20 q^{59} - 16 q^{61} + 2 q^{65} - 16 q^{71} - 10 q^{79} + 14 q^{85} + 2 q^{91}+O(q^{100}) \) 2 * q + 4 * q^5 - 6 * q^11 + 6 * q^25 - 10 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^41 - 2 * q^49 - 12 * q^55 - 20 * q^59 - 16 * q^61 + 2 * q^65 - 16 * q^71 - 10 * q^79 + 14 * q^85 + 2 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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