LMFDB - Embedded newform 3240.2.f.e.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(649,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("3240.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.f (of order $2$, degree $1$, 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:	$25.8715302549$
Analytic rank:	$1$
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 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3240.649
Dual form			3240.2.f.e.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +O(q^{10})$	$q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -2.00000 q^{11} +2.00000i q^{13} +6.00000i q^{17} -2.00000 q^{19} +1.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -7.00000 q^{29} -6.00000 q^{31} +(-2.00000 - 1.00000i) q^{35} -2.00000i q^{37} -5.00000 q^{41} -12.0000i q^{43} -9.00000i q^{47} +6.00000 q^{49} +8.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} -12.0000 q^{59} -7.00000 q^{61} +(4.00000 + 2.00000i) q^{65} +5.00000i q^{67} +10.0000 q^{71} +4.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +5.00000i q^{83} +(12.0000 + 6.00000i) q^{85} -15.0000 q^{89} +2.00000 q^{91} +(-2.00000 + 4.00000i) q^{95} +16.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 6 q^{25} - 14 q^{29} - 12 q^{31} - 4 q^{35} - 10 q^{41} + 12 q^{49} - 4 q^{55} - 24 q^{59} - 14 q^{61} + 8 q^{65} + 20 q^{71} - 8 q^{79} + 24 q^{85} - 30 q^{89} + 4 q^{91} - 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^11 - 4 * q^19 - 6 * q^25 - 14 * q^29 - 12 * q^31 - 4 * q^35 - 10 * q^41 + 12 * q^49 - 4 * q^55 - 24 * q^59 - 14 * q^61 + 8 * q^65 + 20 * q^71 - 8 * q^79 + 24 * q^85 - 30 * q^89 + 4 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		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	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.00000			−1.29987			−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000			−1.29987			−0.649934	+	0.759991i	$0.725203\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		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
$41$	−5.00000			−0.780869			−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000			−0.780869			−0.390434	+	0.920631i	$0.627675\pi$
$42$		0			0
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.00000i		−	1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$		−	9.00000i		−	1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.00000i			1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$			8.00000i			1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$		0			0
$55$	−2.00000	+	4.00000i	−0.269680	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$			5.00000i			0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$			5.00000i			0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		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
$77$			2.00000i			0.227921i
$78$		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
$81$		0			0
$82$		0			0
$83$			5.00000i			0.548821i	0.961613	+	0.274411i	$0.0884828\pi$
$83$			5.00000i			0.548821i	−0.961613	+	0.274411i	$0.911517\pi$
$84$		0			0
$85$	12.0000	+	6.00000i	1.30158	+	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.0000			−1.59000			−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000			−1.59000			−0.794998	+	0.606612i	$0.792528\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	+	4.00000i	−0.205196	+	0.410391i
$96$		0			0
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		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
$107$			13.0000i			1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$			13.0000i			1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$		0			0
$109$	9.00000			0.862044			0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000			0.862044			0.431022	+	0.902342i	$0.358153\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$	2.00000	+	1.00000i	0.186501	+	0.0932505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000			0.550019
$120$		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
$127$		−	3.00000i		−	0.266207i	−0.991102	−	0.133103i	$-0.957506\pi$
$127$		−	3.00000i		−	0.266207i	0.991102	−	0.133103i	$-0.0424943\pi$
$128$		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
$133$			2.00000i			0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.00000i			0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$			4.00000i			0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	−7.00000	+	14.0000i	−0.581318	+	1.16264i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.0000			1.22885			0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000			1.22885			0.614424	+	0.788976i	$0.289388\pi$
$150$		0			0
$151$	22.0000			1.79033			0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000			1.79033			0.895167	+	0.445730i	$0.147056\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.00000	+	12.0000i	−0.481932	+	0.963863i
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		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
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		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
$179$	−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
$181$	19.0000			1.41226			0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000			1.41226			0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	2.00000i	−0.294086	−	0.147043i
$186$		0			0
$187$		−	12.0000i		−	0.877527i
$188$		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
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.0000i		−	1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$		−	24.0000i		−	1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$		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
$203$			7.00000i			0.491304i
$204$		0			0
$205$	−5.00000	+	10.0000i	−0.349215	+	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−26.0000			−1.78991			−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000			−1.78991			−0.894957	+	0.446153i	$0.852794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−24.0000	−	12.0000i	−1.63679	−	0.818393i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$			9.00000i			0.602685i	0.953516	+	0.301342i	$0.0974347\pi$
$223$			9.00000i			0.602685i	−0.953516	+	0.301342i	$0.902565\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	−9.00000			−0.594737			−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000			−0.594737			−0.297368	+	0.954763i	$0.596109\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	−18.0000	−	9.00000i	−1.17419	−	0.587095i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.00000			−0.258738			−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000			−0.258738			−0.129369	+	0.991596i	$0.541295\pi$
$240$		0			0
$241$	17.0000			1.09507			0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000			1.09507			0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.00000	−	12.0000i	0.383326	−	0.766652i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			12.0000i			0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	16.0000	+	8.00000i	0.982872	+	0.491436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	+	8.00000i	0.361814	+	0.482418i
$276$		0			0
$277$			30.0000i			1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$			30.0000i			1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.00000			0.178965			0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000			0.178965			0.0894825	+	0.995988i	$0.471479\pi$
$282$		0			0
$283$		−	33.0000i		−	1.96165i	−0.194900	−	0.980823i	$-0.562438\pi$
$283$		−	33.0000i		−	1.96165i	0.194900	−	0.980823i	$-0.437562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.00000i			0.295141i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$	−12.0000	+	24.0000i	−0.698667	+	1.39733i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.00000			−0.115663
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.00000	+	14.0000i	−0.400819	+	0.801638i
$306$		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
$311$	6.00000			0.340229			0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000			0.340229			0.170114	+	0.985424i	$0.445586\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			30.0000i			1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$			30.0000i			1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$		0			0
$319$	14.0000			0.783850
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.00000			−0.496186
$330$		0			0
$331$	−32.0000			−1.75888			−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000			−1.75888			−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.0000	+	5.00000i	0.546358	+	0.273179i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.0000			0.649836
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	−5.00000			−0.267644			−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000			−0.267644			−0.133822	+	0.991005i	$0.542725\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$	10.0000	−	20.0000i	0.530745	−	1.06149i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.0000			−0.738892			−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000			−0.738892			−0.369446	+	0.929252i	$0.620452\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.00000	+	4.00000i	0.418739	+	0.209370i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.0000i		−	0.721037i
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	4.00000	+	2.00000i	0.203859	+	0.101929i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000			0.152106			0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000			0.152106			0.0760530	+	0.997104i	$0.475768\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$		0			0
$397$			12.0000i			0.602263i	0.953583	+	0.301131i	$0.0973643\pi$
$397$			12.0000i			0.602263i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4.00000i			0.198273i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			12.0000i			0.590481i
$414$		0			0
$415$	10.0000	+	5.00000i	0.490881	+	0.245440i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	24.0000	−	18.0000i	1.16417	−	0.873128i
$426$		0			0
$427$			7.00000i			0.338754i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$		−	30.0000i		−	1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.00000i		−	0.0956730i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9.00000i		−	0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$		−	9.00000i		−	0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$		0			0
$445$	−15.0000	+	30.0000i	−0.711068	+	1.42214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	−	4.00000i	0.0937614	−	0.187523i
$456$		0			0
$457$		−	6.00000i		−	0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$		−	6.00000i		−	0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.0000			−1.25752			−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000			−1.25752			−0.628758	+	0.777601i	$0.716436\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	5.00000			0.230879
$470$		0			0
$471$		0			0
$472$		0			0
$473$			24.0000i			1.10352i
$474$		0			0
$475$	6.00000	+	8.00000i	0.275299	+	0.367065i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.00000			−0.182765			−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$	32.0000	+	16.0000i	1.45305	+	0.726523i
$486$		0			0
$487$			20.0000i			0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$			20.0000i			0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	40.0000			1.80517			0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000			1.80517			0.902587	+	0.430507i	$0.141665\pi$
$492$		0			0
$493$		−	42.0000i		−	1.89158i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10.0000i		−	0.448561i
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			11.0000i			0.490466i	0.969464	+	0.245233i	$0.0788644\pi$
$503$			11.0000i			0.490466i	−0.969464	+	0.245233i	$0.921136\pi$
$504$		0			0
$505$	−18.0000	+	36.0000i	−0.800989	+	1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.00000			0.0443242			0.0221621	−	0.999754i	$-0.492945\pi$
$509$	1.00000			0.0443242			0.0221621	+	0.999754i	$0.492945\pi$
$510$		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
$517$			18.0000i			0.791639i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.00000			0.219054			0.109527	−	0.993984i	$-0.465066\pi$
$521$	5.00000			0.219054			0.109527	+	0.993984i	$0.465066\pi$
$522$		0			0
$523$		−	19.0000i		−	0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$		−	19.0000i		−	0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	36.0000i		−	1.56818i
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	10.0000i		−	0.433148i
$534$		0			0
$535$	26.0000	+	13.0000i	1.12408	+	0.562039i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	31.0000			1.33279			0.666397	−	0.745597i	$-0.267836\pi$
$541$	31.0000			1.33279			0.666397	+	0.745597i	$0.267836\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.00000	−	18.0000i	0.385518	−	0.771035i
$546$		0			0
$547$		−	13.0000i		−	0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$		−	13.0000i		−	0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.0000			0.596420
$552$		0			0
$553$			4.00000i			0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	21.0000i		−	0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$		−	21.0000i		−	0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$		0			0
$565$	8.00000	+	4.00000i	0.336563	+	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000	−	3.00000i	0.166812	−	0.125109i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.00000			0.207435
$582$		0			0
$583$		−	16.0000i		−	0.662652i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	35.0000i		−	1.44460i	−0.691577	−	0.722302i	$-0.743084\pi$
$587$		−	35.0000i		−	1.44460i	0.691577	−	0.722302i	$-0.256916\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	6.00000	−	12.0000i	0.245976	−	0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.0000			1.14405			0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000			1.14405			0.572024	+	0.820237i	$0.306158\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$		0			0
$607$			49.0000i			1.98885i	0.105453	+	0.994424i	$0.466371\pi$
$607$			49.0000i			1.98885i	−0.105453	+	0.994424i	$0.533629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.0000			0.728202
$612$		0			0
$613$			14.0000i			0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$			14.0000i			0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	16.0000i		−	0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$		−	16.0000i		−	0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15.0000i			0.600962i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.00000	−	3.00000i	−0.238103	−	0.119051i
$636$		0			0
$637$			12.0000i			0.475457i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.00000			0.118493			0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000			0.118493			0.0592464	+	0.998243i	$0.481130\pi$
$642$		0			0
$643$		−	15.0000i		−	0.591542i	−0.955259	−	0.295771i	$-0.904423\pi$
$643$		−	15.0000i		−	0.591542i	0.955259	−	0.295771i	$-0.0955766\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			31.0000i			1.21874i	0.792888	+	0.609368i	$0.208577\pi$
$647$			31.0000i			1.21874i	−0.792888	+	0.609368i	$0.791423\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	30.0000i		−	1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$		−	30.0000i		−	1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$		0			0
$655$	−12.0000	+	24.0000i	−0.468879	+	0.937758i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.00000	+	2.00000i	0.155113	+	0.0775567i
$666$		0			0
$667$		−	7.00000i		−	0.271041i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.0000			0.540464
$672$		0			0
$673$		−	2.00000i		−	0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$		−	2.00000i		−	0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	8.00000	+	4.00000i	0.305664	+	0.152832i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.0000			−0.609551
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	−	16.0000i	0.303457	−	0.606915i
$696$		0			0
$697$		−	30.0000i		−	1.13633i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$			4.00000i			0.150863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			18.0000i			0.676960i
$708$		0			0
$709$	−37.0000			−1.38956			−0.694782	−	0.719220i	$-0.744499\pi$
$709$	−37.0000			−1.38956			−0.694782	+	0.719220i	$0.744499\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.00000i		−	0.224702i
$714$		0			0
$715$	−8.00000	−	4.00000i	−0.299183	−	0.149592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	28.0000			1.04422			0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000			1.04422			0.522112	+	0.852877i	$0.325144\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	21.0000	+	28.0000i	0.779920	+	1.03989i
$726$		0			0
$727$			21.0000i			0.778847i	0.921059	+	0.389423i	$0.127326\pi$
$727$			21.0000i			0.778847i	−0.921059	+	0.389423i	$0.872674\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	72.0000			2.66302
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	10.0000i		−	0.368355i
$738$		0			0
$739$	−24.0000			−0.882854			−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000			−0.882854			−0.441427	+	0.897297i	$0.645528\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	15.0000i		−	0.550297i	−0.961402	−	0.275148i	$-0.911273\pi$
$743$		−	15.0000i		−	0.550297i	0.961402	−	0.275148i	$-0.0887270\pi$
$744$		0			0
$745$	15.0000	−	30.0000i	0.549557	−	1.09911i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.0000			0.475010
$750$		0			0
$751$	14.0000			0.510867			0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000			0.510867			0.255434	+	0.966827i	$0.417782\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.0000	−	44.0000i	0.800662	−	1.60132i
$756$		0			0
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.00000			−0.253750			−0.126875	−	0.991919i	$-0.540495\pi$
$761$	−7.00000			−0.253750			−0.126875	+	0.991919i	$0.540495\pi$
$762$		0			0
$763$		−	9.00000i		−	0.325822i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$		0			0
$769$	−1.00000			−0.0360609			−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000			−0.0360609			−0.0180305	+	0.999837i	$0.505740\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.00000i			0.143870i	0.997409	+	0.0719350i	$0.0229174\pi$
$773$			4.00000i			0.143870i	−0.997409	+	0.0719350i	$0.977083\pi$
$774$		0			0
$775$	18.0000	+	24.0000i	0.646579	+	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.0000			0.358287
$780$		0			0
$781$	−20.0000			−0.715656
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.00000	−	4.00000i	−0.285532	−	0.142766i
$786$		0			0
$787$		−	20.0000i		−	0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$		−	20.0000i		−	0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.00000			0.142224
$792$		0			0
$793$		−	14.0000i		−	0.497155i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$	54.0000			1.91038
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	8.00000i		−	0.282314i
$804$		0			0
$805$	1.00000	−	2.00000i	0.0352454	−	0.0704907i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−40.0000	−	20.0000i	−1.40114	−	0.700569i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0000			0.523504			0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000			0.523504			0.261752	+	0.965135i	$0.415700\pi$
$822$		0			0
$823$			17.0000i			0.592583i	0.955098	+	0.296291i	$0.0957499\pi$
$823$			17.0000i			0.592583i	−0.955098	+	0.296291i	$0.904250\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.00000i			0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$			3.00000i			0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$		0			0
$829$	−31.0000			−1.07667			−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000			−1.07667			−0.538337	+	0.842729i	$0.680947\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			36.0000i			1.24733i
$834$		0			0
$835$	6.00000	+	3.00000i	0.207639	+	0.103819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.0000			1.45000			0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000			1.45000			0.725001	+	0.688748i	$0.241839\pi$
$840$		0			0
$841$	20.0000			0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000			0.0685591
$852$		0			0
$853$		−	10.0000i		−	0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$		−	10.0000i		−	0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			38.0000i			1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$			38.0000i			1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$		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
$863$		−	45.0000i		−	1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$		−	45.0000i		−	1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$		0			0
$865$	24.0000	+	12.0000i	0.816024	+	0.408012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.00000			0.271381
$870$		0			0
$871$	−10.0000			−0.338837
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−49.0000			−1.65085			−0.825426	−	0.564510i	$-0.809065\pi$
$881$	−49.0000			−1.65085			−0.825426	+	0.564510i	$0.809065\pi$
$882$		0			0
$883$		−	35.0000i		−	1.17784i	−0.808190	−	0.588922i	$-0.799553\pi$
$883$		−	35.0000i		−	1.17784i	0.808190	−	0.588922i	$-0.200447\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−3.00000			−0.100617
$890$		0			0
$891$		0			0
$892$		0			0
$893$			18.0000i			0.602347i
$894$		0			0
$895$	−10.0000	+	20.0000i	−0.334263	+	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	42.0000			1.40078
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.0000	−	38.0000i	0.631581	−	1.26316i
$906$		0			0
$907$			1.00000i			0.0332045i	0.999862	+	0.0166022i	$0.00528490\pi$
$907$			1.00000i			0.0332045i	−0.999862	+	0.0166022i	$0.994715\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		−	10.0000i		−	0.330952i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	36.0000			1.18753			0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000			1.18753			0.593765	+	0.804638i	$0.297641\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			20.0000i			0.658308i
$924$		0			0
$925$	−8.00000	+	6.00000i	−0.263038	+	0.197279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−24.0000	−	12.0000i	−0.784884	−	0.392442i
$936$		0			0
$937$		−	28.0000i		−	0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$		−	28.0000i		−	0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.0000			0.814977			0.407488	−	0.913210i	$-0.366405\pi$
$941$	25.0000			0.814977			0.407488	+	0.913210i	$0.366405\pi$
$942$		0			0
$943$		−	5.00000i		−	0.162822i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			3.00000i			0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$			3.00000i			0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$		0			0
$949$	−8.00000			−0.259691
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		0			0
$955$	−18.0000	+	36.0000i	−0.582466	+	1.16493i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.00000			0.129167
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.00000	−	2.00000i	−0.128765	−	0.0643823i
$966$		0			0
$967$		−	1.00000i		−	0.0321578i	−0.999871	−	0.0160789i	$-0.994882\pi$
$967$		−	1.00000i		−	0.0321578i	0.999871	−	0.0160789i	$-0.00511830\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.00000i		−	0.0639857i	−0.999488	−	0.0319928i	$-0.989815\pi$
$977$		−	2.00000i		−	0.0639857i	0.999488	−	0.0319928i	$-0.0101854\pi$
$978$		0			0
$979$	30.0000			0.958804
$980$		0			0
$981$		0			0
$982$		0			0
$983$			13.0000i			0.414636i	0.978274	+	0.207318i	$0.0664734\pi$
$983$			13.0000i			0.414636i	−0.978274	+	0.207318i	$0.933527\pi$
$984$		0			0
$985$	−48.0000	−	24.0000i	−1.52941	−	0.764704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.0000	+	20.0000i	−0.317021	+	0.634043i
$996$		0			0
$997$			48.0000i			1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$			48.0000i			1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.f.e.649.1		2
3.2	odd	2		3240.2.f.b.649.2		2
5.4	even	2	inner	3240.2.f.e.649.2		2
9.2	odd	6		360.2.bi.a.49.1	✓	4
9.4	even	3		1080.2.bi.a.289.1		4
9.5	odd	6		360.2.bi.a.169.2	yes	4
9.7	even	3		1080.2.bi.a.1009.2		4
15.14	odd	2		3240.2.f.b.649.1		2
36.7	odd	6		2160.2.by.a.1009.2		4
36.11	even	6		720.2.by.b.49.2		4
36.23	even	6		720.2.by.b.529.1		4
36.31	odd	6		2160.2.by.a.289.1		4
45.4	even	6		1080.2.bi.a.289.2		4
45.14	odd	6		360.2.bi.a.169.1	yes	4
45.29	odd	6		360.2.bi.a.49.2	yes	4
45.34	even	6		1080.2.bi.a.1009.1		4
180.59	even	6		720.2.by.b.529.2		4
180.79	odd	6		2160.2.by.a.1009.1		4
180.119	even	6		720.2.by.b.49.1		4
180.139	odd	6		2160.2.by.a.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bi.a.49.1	✓	4	9.2	odd	6
360.2.bi.a.49.2	yes	4	45.29	odd	6
360.2.bi.a.169.1	yes	4	45.14	odd	6
360.2.bi.a.169.2	yes	4	9.5	odd	6
720.2.by.b.49.1		4	180.119	even	6
720.2.by.b.49.2		4	36.11	even	6
720.2.by.b.529.1		4	36.23	even	6
720.2.by.b.529.2		4	180.59	even	6
1080.2.bi.a.289.1		4	9.4	even	3
1080.2.bi.a.289.2		4	45.4	even	6
1080.2.bi.a.1009.1		4	45.34	even	6
1080.2.bi.a.1009.2		4	9.7	even	3
2160.2.by.a.289.1		4	36.31	odd	6
2160.2.by.a.289.2		4	180.139	odd	6
2160.2.by.a.1009.1		4	180.79	odd	6
2160.2.by.a.1009.2		4	36.7	odd	6
3240.2.f.b.649.1		2	15.14	odd	2
3240.2.f.b.649.2		2	3.2	odd	2
3240.2.f.e.649.1		2	1.1	even	1	trivial
3240.2.f.e.649.2		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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -2.00000 q^{11} +2.00000i q^{13} +6.00000i q^{17} -2.00000 q^{19} +1.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -7.00000 q^{29} -6.00000 q^{31} +(-2.00000 - 1.00000i) q^{35} -2.00000i q^{37} -5.00000 q^{41} -12.0000i q^{43} -9.00000i q^{47} +6.00000 q^{49} +8.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} -12.0000 q^{59} -7.00000 q^{61} +(4.00000 + 2.00000i) q^{65} +5.00000i q^{67} +10.0000 q^{71} +4.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +5.00000i q^{83} +(12.0000 + 6.00000i) q^{85} -15.0000 q^{89} +2.00000 q^{91} +(-2.00000 + 4.00000i) q^{95} +16.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 6 q^{25} - 14 q^{29} - 12 q^{31} - 4 q^{35} - 10 q^{41} + 12 q^{49} - 4 q^{55} - 24 q^{59} - 14 q^{61} + 8 q^{65} + 20 q^{71} - 8 q^{79} + 24 q^{85} - 30 q^{89} + 4 q^{91} - 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^11 - 4 * q^19 - 6 * q^25 - 14 * q^29 - 12 * q^31 - 4 * q^35 - 10 * q^41 + 12 * q^49 - 4 * q^55 - 24 * q^59 - 14 * q^61 + 8 * q^65 + 20 * q^71 - 8 * q^79 + 24 * q^85 - 30 * q^89 + 4 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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