LMFDB - Embedded newform 2240.2.g.c.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,2,Mod(449,2240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2240.449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,6,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$17.8864900528$
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 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.c.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000 q^{9} +4.00000i q^{13} +4.00000i q^{17} -4.00000 q^{19} +8.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000 q^{29} +8.00000 q^{31} +(-2.00000 + 1.00000i) q^{35} +8.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} +(-3.00000 - 6.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} +4.00000 q^{59} +6.00000 q^{61} -3.00000i q^{63} +(8.00000 - 4.00000i) q^{65} +8.00000i q^{67} -12.0000 q^{71} +4.00000i q^{73} -4.00000 q^{79} +9.00000 q^{81} +(8.00000 - 4.00000i) q^{85} +10.0000 q^{89} +4.00000 q^{91} +(4.00000 + 8.00000i) q^{95} -12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 6 q^{9} - 8 q^{19} - 6 q^{25} + 4 q^{29} + 16 q^{31} - 4 q^{35} + 12 q^{41} - 6 q^{45} - 2 q^{49} + 8 q^{59} + 12 q^{61} + 16 q^{65} - 24 q^{71} - 8 q^{79} + 18 q^{81} + 16 q^{85} + 20 q^{89}+ \cdots + 8 q^{95}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^9 - 8 * q^19 - 6 * q^25 + 4 * q^29 + 16 * q^31 - 4 * q^35 + 12 * q^41 - 6 * q^45 - 2 * q^49 + 8 * q^59 + 12 * q^61 + 16 * q^65 - 24 * q^71 - 8 * q^79 + 18 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^91 + 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		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
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		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
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	+	1.00000i	−0.338062	+	0.169031i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$	−3.00000	−	6.00000i	−0.447214	−	0.894427i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		−	3.00000i		−	0.377964i
$64$		0			0
$65$	8.00000	−	4.00000i	0.992278	−	0.496139i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		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	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		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
$109$	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
$113$			16.0000i			1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$			16.0000i			1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	16.0000	−	8.00000i	1.49201	−	0.746004i
$116$		0			0
$117$			12.0000i			1.10940i
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$			4.00000i			0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−2.00000	−	4.00000i	−0.166091	−	0.332182i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$			12.0000i			0.970143i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$			24.0000i			1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$			24.0000i			1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.0000i		−	1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$		−	24.0000i		−	1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−12.0000			−0.917663
$172$		0			0
$173$			12.0000i			0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$			12.0000i			0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$		0			0
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.00000			0.597948			0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000			0.597948			0.298974	+	0.954261i	$0.403356\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.0000			1.44715			0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000			1.44715			0.723575	+	0.690246i	$0.242498\pi$
$192$		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
$197$		−	16.0000i		−	1.13995i	−0.821661	−	0.569976i	$-0.806952\pi$
$197$		−	16.0000i		−	1.13995i	0.821661	−	0.569976i	$-0.193048\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.00000i		−	0.140372i
$204$		0			0
$205$	−6.00000	−	12.0000i	−0.419058	−	0.838116i
$206$		0			0
$207$			24.0000i			1.66812i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−16.0000	+	8.00000i	−1.09119	+	0.545595i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.0000			−1.07628
$222$		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
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		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
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$		0			0
$235$	−16.0000	+	8.00000i	−1.04372	+	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	+	2.00000i	0.0638877	+	0.127775i
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	16.0000i		−	0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$		−	16.0000i		−	0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$		0			0
$279$	24.0000			1.43684
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	32.0000i		−	1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$		−	32.0000i		−	1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6.00000i		−	0.354169i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$			12.0000i			0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$			12.0000i			0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$		0			0
$295$	−4.00000	−	8.00000i	−0.232889	−	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−32.0000			−1.85061
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		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
$313$			4.00000i			0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$			4.00000i			0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$		0			0
$315$	−6.00000	+	3.00000i	−0.338062	+	0.169031i
$316$		0			0
$317$		−	16.0000i		−	0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$		−	16.0000i		−	0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	−16.0000			−0.879440			−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000			−0.879440			−0.439720	+	0.898135i	$0.644922\pi$
$332$		0			0
$333$			24.0000i			1.31519i
$334$		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
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.0000i			0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$			12.0000i			0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$		0			0
$355$	12.0000	+	24.0000i	0.636894	+	1.27379i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$	18.0000			0.937043
$370$		0			0
$371$		0			0
$372$		0			0
$373$			32.0000i			1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$			32.0000i			1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			8.00000i			0.412021i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	24.0000i		−	1.21999i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	−32.0000			−1.61831
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000	+	8.00000i	0.201262	+	0.402524i
$396$		0			0
$397$		−	28.0000i		−	1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$		−	28.0000i		−	1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$			32.0000i			1.59403i
$404$		0			0
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		−	24.0000i		−	1.16692i
$424$		0			0
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$		−	6.00000i		−	0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.0000			−0.963366			−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000			−0.963366			−0.481683	+	0.876346i	$0.659974\pi$
$432$		0			0
$433$		−	28.0000i		−	1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$		−	28.0000i		−	1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		0			0
$445$	−10.0000	−	20.0000i	−0.474045	−	0.948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.00000	−	8.00000i	−0.187523	−	0.375046i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.0000			1.21094			0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000			1.21094			0.605470	+	0.795868i	$0.292985\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			16.0000i			0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$467$			16.0000i			0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000	+	12.0000i	−1.08978	+	0.544892i
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			8.00000i			0.360302i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			12.0000i			0.538274i
$498$		0			0
$499$	8.00000			0.358129			0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000			0.358129			0.179065	+	0.983837i	$0.442693\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.00000i		−	0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$		−	8.00000i		−	0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$		0			0
$505$	−18.0000	−	36.0000i	−0.800989	−	1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−22.0000			−0.975133			−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000			−0.975133			−0.487566	+	0.873086i	$0.662115\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000	+	8.00000i	−0.705044	+	0.352522i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$			32.0000i			1.39926i	0.714504	+	0.699631i	$0.246652\pi$
$523$			32.0000i			1.39926i	−0.714504	+	0.699631i	$0.753348\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			32.0000i			1.39394i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$			24.0000i			1.03956i
$534$		0			0
$535$	16.0000	−	8.00000i	0.691740	−	0.345870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.0000	−	28.0000i	−0.599694	−	1.19939i
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$	18.0000			0.768221
$550$		0			0
$551$	−8.00000			−0.340811
$552$		0			0
$553$			4.00000i			0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			16.0000i			0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$			16.0000i			0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$		0			0
$559$	32.0000			1.35346
$560$		0			0
$561$		0			0
$562$		0			0
$563$			16.0000i			0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$			16.0000i			0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$		0			0
$565$	32.0000	−	16.0000i	1.34625	−	0.673125i
$566$		0			0
$567$		−	9.00000i		−	0.377964i
$568$		0			0
$569$	−42.0000			−1.76073			−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000			−1.76073			−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−32.0000	−	24.0000i	−1.33449	−	1.00087i
$576$		0			0
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	24.0000	−	12.0000i	0.992278	−	0.496139i
$586$		0			0
$587$		−	32.0000i		−	1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$		−	32.0000i		−	1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$			36.0000i			1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$			36.0000i			1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$		0			0
$595$	−4.00000	−	8.00000i	−0.163984	−	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−36.0000			−1.47092			−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000			−1.47092			−0.735460	+	0.677568i	$0.763034\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$		0			0
$605$	11.0000	+	22.0000i	0.447214	+	0.894427i
$606$		0			0
$607$			40.0000i			1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$			40.0000i			1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.0000			1.29458
$612$		0			0
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			8.00000i			0.322068i	0.986949	+	0.161034i	$0.0514829\pi$
$617$			8.00000i			0.322068i	−0.986949	+	0.161034i	$0.948517\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		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
$629$	−32.0000			−1.27592
$630$		0			0
$631$	36.0000			1.43314			0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000			1.43314			0.716569	+	0.697517i	$0.245712\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.0000	−	8.00000i	0.634941	−	0.317470i
$636$		0			0
$637$		−	4.00000i		−	0.158486i
$638$		0			0
$639$	−36.0000			−1.42414
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		−	16.0000i		−	0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$		−	16.0000i		−	0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$		0			0
$655$	4.00000	+	8.00000i	0.156293	+	0.312586i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	−34.0000			−1.32245			−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000			−1.32245			−0.661223	+	0.750189i	$0.729962\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000	−	4.00000i	0.310227	−	0.155113i
$666$		0			0
$667$			16.0000i			0.619522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	8.00000i		−	0.308377i	−0.988041	−	0.154189i	$-0.950724\pi$
$673$		−	8.00000i		−	0.308377i	0.988041	−	0.154189i	$-0.0492764\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	4.00000i		−	0.153732i	−0.997041	−	0.0768662i	$-0.975509\pi$
$677$		−	4.00000i		−	0.153732i	0.997041	−	0.0768662i	$-0.0244914\pi$
$678$		0			0
$679$	−12.0000			−0.460518
$680$		0			0
$681$		0			0
$682$		0			0
$683$			8.00000i			0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$			8.00000i			0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$		0			0
$685$	−16.0000	+	8.00000i	−0.611329	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.00000	+	8.00000i	0.151729	+	0.303457i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		−	32.0000i		−	1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	18.0000i		−	0.676960i
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		0			0
$713$			64.0000i			2.39682i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.0000			1.19340			0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000			1.19340			0.596699	+	0.802465i	$0.296479\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	+	8.00000i	−0.222834	+	0.297113i
$726$		0			0
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	32.0000			1.18356
$732$		0			0
$733$			20.0000i			0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$			20.0000i			0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.00000			0.294285			0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000			0.294285			0.147142	+	0.989115i	$0.452992\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$	−6.00000	−	12.0000i	−0.219823	−	0.439646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000	+	24.0000i	0.436725	+	0.873449i
$756$		0			0
$757$			24.0000i			0.872295i	0.899875	+	0.436147i	$0.143657\pi$
$757$			24.0000i			0.872295i	−0.899875	+	0.436147i	$0.856343\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$		−	14.0000i		−	0.506834i
$764$		0			0
$765$	24.0000	−	12.0000i	0.867722	−	0.433861i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			36.0000i			1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$			36.0000i			1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.0000	−	12.0000i	0.856597	−	0.428298i
$786$		0			0
$787$		−	48.0000i		−	1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$		−	48.0000i		−	1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.0000			0.568895
$792$		0			0
$793$			24.0000i			0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28.0000i		−	0.991811i	−0.868377	−	0.495905i	$-0.834836\pi$
$797$		−	28.0000i		−	0.991811i	0.868377	−	0.495905i	$-0.165164\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$	30.0000			1.06000
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−8.00000	−	16.0000i	−0.281963	−	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$		0			0
$811$	−36.0000			−1.26413			−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000			−1.26413			−0.632065	+	0.774915i	$0.717793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	48.0000	−	24.0000i	1.68137	−	0.840683i
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$	12.0000			0.419314
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.00000i			0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$			8.00000i			0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$		0			0
$829$	22.0000			0.764092			0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000			0.764092			0.382046	+	0.924143i	$0.375220\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.00000i		−	0.138592i
$834$		0			0
$835$	−48.0000	+	24.0000i	−1.66111	+	0.830554i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.0000			−0.552381			−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000			−0.552381			−0.276191	+	0.961103i	$0.589072\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000	+	6.00000i	0.103203	+	0.206406i
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−64.0000			−2.19389
$852$		0			0
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$		0			0
$855$	12.0000	+	24.0000i	0.410391	+	0.820783i
$856$		0			0
$857$		−	52.0000i		−	1.77629i	−0.459567	−	0.888143i	$-0.651995\pi$
$857$		−	52.0000i		−	1.77629i	0.459567	−	0.888143i	$-0.348005\pi$
$858$		0			0
$859$	−36.0000			−1.22830			−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000			−1.22830			−0.614152	+	0.789188i	$0.710502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$	24.0000	−	12.0000i	0.816024	−	0.408012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−32.0000			−1.08428
$872$		0			0
$873$		−	36.0000i		−	1.21842i
$874$		0			0
$875$	2.00000	−	11.0000i	0.0676123	−	0.371868i
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$			8.00000i			0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$			8.00000i			0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			24.0000i			0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$			24.0000i			0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$		0			0
$891$		0			0
$892$		0			0
$893$			32.0000i			1.07084i
$894$		0			0
$895$	−8.00000	−	16.0000i	−0.267411	−	0.534821i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.0000			0.533630
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	14.0000	+	28.0000i	0.465376	+	0.930751i
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$	54.0000			1.79107
$910$		0			0
$911$	28.0000			0.927681			0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000			0.927681			0.463841	+	0.885919i	$0.346471\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			4.00000i			0.132092i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$	−32.0000	−	24.0000i	−1.05215	−	0.789115i
$926$		0			0
$927$		−	24.0000i		−	0.788263i
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			4.00000i			0.130674i	0.997863	+	0.0653372i	$0.0208123\pi$
$937$			4.00000i			0.130674i	−0.997863	+	0.0653372i	$0.979188\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−38.0000			−1.23876			−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000			−1.23876			−0.619382	+	0.785090i	$0.712617\pi$
$942$		0			0
$943$			48.0000i			1.56310i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			24.0000i			0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$			24.0000i			0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$		0			0
$949$	−16.0000			−0.519382
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−20.0000	−	40.0000i	−0.647185	−	1.29437i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000			−0.258333
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			24.0000i			0.773389i
$964$		0			0
$965$	32.0000	−	16.0000i	1.03012	−	0.515058i
$966$		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
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$			4.00000i			0.128234i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	24.0000i		−	0.767828i	−0.923369	−	0.383914i	$-0.874576\pi$
$977$		−	24.0000i		−	0.767828i	0.923369	−	0.383914i	$-0.125424\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	42.0000			1.34096
$982$		0			0
$983$			40.0000i			1.27580i	0.770118	+	0.637901i	$0.220197\pi$
$983$			40.0000i			1.27580i	−0.770118	+	0.637901i	$0.779803\pi$
$984$		0			0
$985$	−32.0000	+	16.0000i	−1.01960	+	0.509802i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	64.0000			2.03508
$990$		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
$995$	24.0000	+	48.0000i	0.760851	+	1.52170i
$996$		0			0
$997$		−	28.0000i		−	0.886769i	−0.896332	−	0.443384i	$-0.853778\pi$
$997$		−	28.0000i		−	0.886769i	0.896332	−	0.443384i	$-0.146222\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.c.449.1		2
4.3	odd	2		2240.2.g.d.449.1		2
5.4	even	2	inner	2240.2.g.c.449.2		2
8.3	odd	2		140.2.e.b.29.2	yes	2
8.5	even	2		560.2.g.c.449.2		2
20.19	odd	2		2240.2.g.d.449.2		2
24.5	odd	2		5040.2.t.g.1009.1		2
24.11	even	2		1260.2.k.b.1009.1		2
40.3	even	4		700.2.a.h.1.1		1
40.13	odd	4		2800.2.a.o.1.1		1
40.19	odd	2		140.2.e.b.29.1	✓	2
40.27	even	4		700.2.a.f.1.1		1
40.29	even	2		560.2.g.c.449.1		2
40.37	odd	4		2800.2.a.s.1.1		1
56.3	even	6		980.2.q.e.569.2		4
56.11	odd	6		980.2.q.d.569.1		4
56.19	even	6		980.2.q.e.949.1		4
56.27	even	2		980.2.e.a.589.1		2
56.51	odd	6		980.2.q.d.949.2		4
120.29	odd	2		5040.2.t.g.1009.2		2
120.59	even	2		1260.2.k.b.1009.2		2
120.83	odd	4		6300.2.a.y.1.1		1
120.107	odd	4		6300.2.a.g.1.1		1
280.19	even	6		980.2.q.e.949.2		4
280.27	odd	4		4900.2.a.m.1.1		1
280.59	even	6		980.2.q.e.569.1		4
280.83	odd	4		4900.2.a.l.1.1		1
280.139	even	2		980.2.e.a.589.2		2
280.179	odd	6		980.2.q.d.569.2		4
280.219	odd	6		980.2.q.d.949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.e.b.29.1	✓	2	40.19	odd	2
140.2.e.b.29.2	yes	2	8.3	odd	2
560.2.g.c.449.1		2	40.29	even	2
560.2.g.c.449.2		2	8.5	even	2
700.2.a.f.1.1		1	40.27	even	4
700.2.a.h.1.1		1	40.3	even	4
980.2.e.a.589.1		2	56.27	even	2
980.2.e.a.589.2		2	280.139	even	2
980.2.q.d.569.1		4	56.11	odd	6
980.2.q.d.569.2		4	280.179	odd	6
980.2.q.d.949.1		4	280.219	odd	6
980.2.q.d.949.2		4	56.51	odd	6
980.2.q.e.569.1		4	280.59	even	6
980.2.q.e.569.2		4	56.3	even	6
980.2.q.e.949.1		4	56.19	even	6
980.2.q.e.949.2		4	280.19	even	6
1260.2.k.b.1009.1		2	24.11	even	2
1260.2.k.b.1009.2		2	120.59	even	2
2240.2.g.c.449.1		2	1.1	even	1	trivial
2240.2.g.c.449.2		2	5.4	even	2	inner
2240.2.g.d.449.1		2	4.3	odd	2
2240.2.g.d.449.2		2	20.19	odd	2
2800.2.a.o.1.1		1	40.13	odd	4
2800.2.a.s.1.1		1	40.37	odd	4
4900.2.a.l.1.1		1	280.83	odd	4
4900.2.a.m.1.1		1	280.27	odd	4
5040.2.t.g.1009.1		2	24.5	odd	2
5040.2.t.g.1009.2		2	120.29	odd	2
6300.2.a.g.1.1		1	120.107	odd	4
6300.2.a.y.1.1		1	120.83	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000 q^{9} +4.00000i q^{13} +4.00000i q^{17} -4.00000 q^{19} +8.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000 q^{29} +8.00000 q^{31} +(-2.00000 + 1.00000i) q^{35} +8.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} +(-3.00000 - 6.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} +4.00000 q^{59} +6.00000 q^{61} -3.00000i q^{63} +(8.00000 - 4.00000i) q^{65} +8.00000i q^{67} -12.0000 q^{71} +4.00000i q^{73} -4.00000 q^{79} +9.00000 q^{81} +(8.00000 - 4.00000i) q^{85} +10.0000 q^{89} +4.00000 q^{91} +(4.00000 + 8.00000i) q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 6 q^{9} - 8 q^{19} - 6 q^{25} + 4 q^{29} + 16 q^{31} - 4 q^{35} + 12 q^{41} - 6 q^{45} - 2 q^{49} + 8 q^{59} + 12 q^{61} + 16 q^{65} - 24 q^{71} - 8 q^{79} + 18 q^{81} + 16 q^{85} + 20 q^{89}+ \cdots + 8 q^{95}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^9 - 8 * q^19 - 6 * q^25 + 4 * q^29 + 16 * q^31 - 4 * q^35 + 12 * q^41 - 6 * q^45 - 2 * q^49 + 8 * q^59 + 12 * q^61 + 16 * q^65 - 24 * q^71 - 8 * q^79 + 18 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more