LMFDB - Embedded newform 5040.2.t.h.1009.2

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{11} -6.00000i q^{13} -4.00000i q^{17} -6.00000 q^{19} +8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +2.00000 q^{31} +(-2.00000 - 1.00000i) q^{35} +4.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} +8.00000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} +8.00000 q^{59} -10.0000 q^{61} +(12.0000 + 6.00000i) q^{65} -8.00000i q^{67} -6.00000 q^{71} +14.0000i q^{73} +2.00000i q^{77} -12.0000 q^{79} +8.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -10.0000 q^{89} +6.00000 q^{91} +(6.00000 - 12.0000i) q^{95} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{11} - 12 q^{19} - 6 q^{25} + 12 q^{29} + 4 q^{31} - 4 q^{35} - 4 q^{41} - 2 q^{49} - 4 q^{55} + 16 q^{59} - 20 q^{61} + 24 q^{65} - 12 q^{71} - 24 q^{79} + 16 q^{85} - 20 q^{89} + 12 q^{91}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^11 - 12 * q^19 - 6 * q^25 + 12 * q^29 + 4 * q^31 - 4 * q^35 - 4 * q^41 - 2 * q^49 - 4 * q^55 + 16 * q^59 - 20 * q^61 + 24 * q^65 - 12 * q^71 - 24 * q^79 + 16 * q^85 - 20 * q^89 + 12 * q^91 + 12 * q^95

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	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		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
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		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	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	−2.00000	+	4.00000i	−0.269680	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.0000	+	6.00000i	1.48842	+	0.744208i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.00000i			0.227921i
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$	8.00000	+	4.00000i	0.867722	+	0.433861i
$86$		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
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	−	12.0000i	0.615587	−	1.23117i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	−16.0000	−	8.00000i	−1.49201	−	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		−	6.00000i		−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		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
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000	+	4.00000i	−0.160644	+	0.321288i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.00000			−0.630488
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$			8.00000i			0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$			8.00000i			0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.00000	−	4.00000i	−0.588172	−	0.294086i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	6.00000			0.425329			0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000			0.425329			0.212664	+	0.977125i	$0.431786\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.0000			−1.61441
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.0000i		−	0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$		−	10.0000i		−	0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$		0			0
$235$	−16.0000	−	8.00000i	−1.04372	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.0000			−1.68180			−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000			−1.68180			−0.840900	+	0.541190i	$0.817974\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	−	2.00000i	0.0638877	−	0.127775i
$246$		0			0
$247$			36.0000i			2.29063i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$	−12.0000	−	6.00000i	−0.737154	−	0.368577i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$			20.0000i			1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$			20.0000i			1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	2.00000i		−	0.118056i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	−8.00000	+	16.0000i	−0.465778	+	0.931556i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	48.0000			2.77591
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.0000	−	20.0000i	0.572598	−	1.14520i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$			2.00000i			0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$			2.00000i			0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		0			0
$322$		0			0
$323$			24.0000i			1.33540i
$324$		0			0
$325$	−24.0000	+	18.0000i	−1.33128	+	0.998460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.0000	+	8.00000i	0.874173	+	0.437087i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000			0.216612
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			36.0000i			1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$			36.0000i			1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.0000i			1.06449i	0.846590	+	0.532246i	$0.178652\pi$
$353$			20.0000i			1.06449i	−0.846590	+	0.532246i	$0.821348\pi$
$354$		0			0
$355$	6.00000	−	12.0000i	0.318447	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.0000	−	14.0000i	−1.46559	−	0.732793i
$366$		0			0
$367$			32.0000i			1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$			32.0000i			1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$		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	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	36.0000i		−	1.85409i
$378$		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
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$	−4.00000	−	2.00000i	−0.203859	−	0.101929i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$		0			0
$391$	32.0000			1.61831
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000	−	24.0000i	0.603786	−	1.20757i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0000			0.699127			0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000			0.699127			0.349563	+	0.936913i	$0.386330\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.00000i			0.396545i
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	−16.0000	−	8.00000i	−0.785409	−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.0000	+	12.0000i	−0.776114	+	0.582086i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.00000			0.0963366			0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000			0.0963366			0.0481683	+	0.998839i	$0.484662\pi$
$432$		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
$437$		−	48.0000i		−	2.29615i
$438$		0			0
$439$	−18.0000			−0.859093			−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000			−0.859093			−0.429547	+	0.903045i	$0.641327\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−4.00000			−0.188353
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000	+	12.0000i	−0.281284	+	0.562569i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$			36.0000i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$			36.0000i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.00000i			0.367840i
$474$		0			0
$475$	18.0000	+	24.0000i	0.825897	+	1.10120i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.0000	+	10.0000i	0.908153	+	0.454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.0000			−1.35388			−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000			−1.35388			−0.676941	+	0.736038i	$0.736695\pi$
$492$		0			0
$493$		−	24.0000i		−	1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.00000i		−	0.269137i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000	−	8.00000i	−0.705044	−	0.352522i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$		−	28.0000i		−	1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$		−	28.0000i		−	1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.0000i			0.519778i
$534$		0			0
$535$	−24.0000	−	12.0000i	−1.03761	−	0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.0000	+	28.0000i	−0.599694	+	1.19939i
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−36.0000			−1.53365
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			42.0000i			1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$			42.0000i			1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	−12.0000	−	6.00000i	−0.504844	−	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−46.0000			−1.92842			−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000			−1.92842			−0.964210	+	0.265139i	$0.914582\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.0000	−	24.0000i	1.33449	−	1.00087i
$576$		0			0
$577$		−	26.0000i		−	1.08239i	−0.840896	−	0.541197i	$-0.817971\pi$
$577$		−	26.0000i		−	1.08239i	0.840896	−	0.541197i	$-0.182029\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.00000			−0.331896
$582$		0			0
$583$			12.0000i			0.496989i
$584$		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
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$			12.0000i			0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$			12.0000i			0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.00000	−	14.0000i	0.284590	−	0.569181i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$		−	28.0000i		−	1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$		−	28.0000i		−	1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$	−22.0000			−0.884255			−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000			−0.884255			−0.442127	+	0.896952i	$0.645776\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	10.0000i		−	0.400642i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.0000			0.637962
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	4.00000i	0.317470	+	0.158735i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.0000			−0.552967			−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000			−0.552967			−0.276483	+	0.961019i	$0.589169\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.0000i			1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$			36.0000i			1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$	12.0000	−	24.0000i	0.468879	−	0.937758i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26.0000			1.01282			0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000			1.01282			0.506408	+	0.862294i	$0.330973\pi$
$660$		0			0
$661$	38.0000			1.47803			0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000			1.47803			0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	+	6.00000i	0.465340	+	0.232670i
$666$		0			0
$667$			48.0000i			1.85857i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20.0000			−0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	40.0000i		−	1.53732i	−0.639655	−	0.768662i	$-0.720923\pi$
$677$		−	40.0000i		−	1.53732i	0.639655	−	0.768662i	$-0.279077\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	−12.0000	−	6.00000i	−0.458496	−	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	36.0000			1.37149
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0000	−	28.0000i	0.531050	−	1.06210i
$696$		0			0
$697$			8.00000i			0.303022i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	10.0000i		−	0.376089i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$	24.0000	+	12.0000i	0.897549	+	0.448775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.00000			0.149175			0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000			0.149175			0.0745874	+	0.997214i	$0.476236\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	−	24.0000i	−0.668503	−	0.891338i
$726$		0			0
$727$		−	24.0000i		−	0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$		−	24.0000i		−	0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.0000			0.591781
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	16.0000i		−	0.589368i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			48.0000i			1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$			48.0000i			1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$		0			0
$745$	−10.0000	+	20.0000i	−0.366372	+	0.732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	+	16.0000i	−0.291150	+	0.582300i
$756$		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
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$			14.0000i			0.506834i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	48.0000i		−	1.73318i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−6.00000	−	8.00000i	−0.215526	−	0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	44.0000	+	22.0000i	1.57043	+	0.785214i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$			60.0000i			2.13066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$		0			0
$802$		0			0
$803$			28.0000i			0.988099i
$804$		0			0
$805$	8.00000	−	16.0000i	0.281963	−	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	14.0000			0.491606			0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000			0.491606			0.245803	+	0.969320i	$0.420948\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.00000	+	4.00000i	0.280228	+	0.140114i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.0000			−0.767805			−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000			−0.767805			−0.383903	+	0.923374i	$0.625420\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	20.0000i		−	0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$		−	20.0000i		−	0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			4.00000i			0.138592i
$834$		0			0
$835$	24.0000	+	12.0000i	0.830554	+	0.415277i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20.0000			0.690477			0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000			0.690477			0.345238	+	0.938515i	$0.387798\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.0000	−	46.0000i	0.791224	−	1.58245i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−32.0000			−1.09695
$852$		0			0
$853$		−	38.0000i		−	1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$		−	38.0000i		−	1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.0000i		−	0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$		−	24.0000i		−	0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$	−16.0000	−	8.00000i	−0.544016	−	0.272008i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	−48.0000			−1.62642
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$		−	28.0000i		−	0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$		−	28.0000i		−	0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$	10.0000	−	20.0000i	0.334263	−	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.00000	+	4.00000i	−0.0664822	+	0.132964i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$			16.0000i			0.529523i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	60.0000			1.97922			0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000			1.97922			0.989609	+	0.143787i	$0.0459280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			36.0000i			1.18495i
$924$		0			0
$925$	16.0000	−	12.0000i	0.526077	−	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16.0000	+	8.00000i	0.523256	+	0.261628i
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.0000			0.325991			0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000			0.325991			0.162995	+	0.986627i	$0.447884\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	84.0000			2.72676
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	10.0000i		−	0.323932i	−0.986796	−	0.161966i	$-0.948217\pi$
$953$		−	10.0000i		−	0.323932i	0.986796	−	0.161966i	$-0.0517835\pi$
$954$		0			0
$955$	18.0000	−	36.0000i	0.582466	−	1.16493i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.0000	−	8.00000i	−0.515058	−	0.257529i
$966$		0			0
$967$		−	56.0000i		−	1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$		−	56.0000i		−	1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$		−	14.0000i		−	0.448819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$	−20.0000			−0.639203
$980$		0			0
$981$		0			0
$982$		0			0
$983$			16.0000i			0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$			16.0000i			0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$		0			0
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	−4.00000			−0.127064			−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000			−0.127064			−0.0635321	+	0.997980i	$0.520237\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.00000	+	12.0000i	−0.190213	+	0.380426i
$996$		0			0
$997$		−	42.0000i		−	1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$		−	42.0000i		−	1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.h.1009.2		2
3.2	odd	2		1680.2.t.e.1009.2		2
4.3	odd	2		630.2.g.c.379.1		2
5.4	even	2	inner	5040.2.t.h.1009.1		2
12.11	even	2		210.2.g.b.169.2	yes	2
15.2	even	4		8400.2.a.bp.1.1		1
15.8	even	4		8400.2.a.w.1.1		1
15.14	odd	2		1680.2.t.e.1009.1		2
20.3	even	4		3150.2.a.d.1.1		1
20.7	even	4		3150.2.a.bk.1.1		1
20.19	odd	2		630.2.g.c.379.2		2
60.23	odd	4		1050.2.a.p.1.1		1
60.47	odd	4		1050.2.a.d.1.1		1
60.59	even	2		210.2.g.b.169.1	✓	2
84.11	even	6		1470.2.n.b.79.1		4
84.23	even	6		1470.2.n.b.949.2		4
84.47	odd	6		1470.2.n.f.949.2		4
84.59	odd	6		1470.2.n.f.79.1		4
84.83	odd	2		1470.2.g.b.589.2		2
420.59	odd	6		1470.2.n.f.79.2		4
420.83	even	4		7350.2.a.bz.1.1		1
420.167	even	4		7350.2.a.bk.1.1		1
420.179	even	6		1470.2.n.b.79.2		4
420.299	odd	6		1470.2.n.f.949.1		4
420.359	even	6		1470.2.n.b.949.1		4
420.419	odd	2		1470.2.g.b.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.b.169.1	✓	2	60.59	even	2
210.2.g.b.169.2	yes	2	12.11	even	2
630.2.g.c.379.1		2	4.3	odd	2
630.2.g.c.379.2		2	20.19	odd	2
1050.2.a.d.1.1		1	60.47	odd	4
1050.2.a.p.1.1		1	60.23	odd	4
1470.2.g.b.589.1		2	420.419	odd	2
1470.2.g.b.589.2		2	84.83	odd	2
1470.2.n.b.79.1		4	84.11	even	6
1470.2.n.b.79.2		4	420.179	even	6
1470.2.n.b.949.1		4	420.359	even	6
1470.2.n.b.949.2		4	84.23	even	6
1470.2.n.f.79.1		4	84.59	odd	6
1470.2.n.f.79.2		4	420.59	odd	6
1470.2.n.f.949.1		4	420.299	odd	6
1470.2.n.f.949.2		4	84.47	odd	6
1680.2.t.e.1009.1		2	15.14	odd	2
1680.2.t.e.1009.2		2	3.2	odd	2
3150.2.a.d.1.1		1	20.3	even	4
3150.2.a.bk.1.1		1	20.7	even	4
5040.2.t.h.1009.1		2	5.4	even	2	inner
5040.2.t.h.1009.2		2	1.1	even	1	trivial
7350.2.a.bk.1.1		1	420.167	even	4
7350.2.a.bz.1.1		1	420.83	even	4
8400.2.a.w.1.1		1	15.8	even	4
8400.2.a.bp.1.1		1	15.2	even	4

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{11} -6.00000i q^{13} -4.00000i q^{17} -6.00000 q^{19} +8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +2.00000 q^{31} +(-2.00000 - 1.00000i) q^{35} +4.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} +8.00000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} +8.00000 q^{59} -10.0000 q^{61} +(12.0000 + 6.00000i) q^{65} -8.00000i q^{67} -6.00000 q^{71} +14.0000i q^{73} +2.00000i q^{77} -12.0000 q^{79} +8.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -10.0000 q^{89} +6.00000 q^{91} +(6.00000 - 12.0000i) q^{95} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{11} - 12 q^{19} - 6 q^{25} + 12 q^{29} + 4 q^{31} - 4 q^{35} - 4 q^{41} - 2 q^{49} - 4 q^{55} + 16 q^{59} - 20 q^{61} + 24 q^{65} - 12 q^{71} - 24 q^{79} + 16 q^{85} - 20 q^{89} + 12 q^{91}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^11 - 12 * q^19 - 6 * q^25 + 12 * q^29 + 4 * q^31 - 4 * q^35 - 4 * q^41 - 2 * q^49 - 4 * q^55 + 16 * q^59 - 20 * q^61 + 24 * q^65 - 12 * q^71 - 24 * q^79 + 16 * q^85 - 20 * q^89 + 12 * q^91 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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