LMFDB - Embedded newform 1620.2.d.e.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(649,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.d (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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.28356903014400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 20x^{4} - 75x^{2} + 625 $ x^8 - 3x^6 + 20x^4 - 75*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$2.07237 + 0.839805i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.2.d.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)

$f(q)$	$=$	$q+(-2.07237 - 0.839805i) q^{5} -4.93536i q^{7} +O(q^{10})$	$q+(-2.07237 - 0.839805i) q^{5} -4.93536i q^{7} -2.41269 q^{11} +2.90917i q^{13} -6.86869i q^{17} +4.17891 q^{19} +3.35922i q^{23} +(3.58945 + 3.48078i) q^{25} -5.19615 q^{29} -6.17891 q^{31} +(-4.14474 + 10.2279i) q^{35} -7.84453i q^{37} -5.87680 q^{41} +4.93536i q^{43} +11.9075i q^{47} -17.3578 q^{49} +8.54830i q^{53} +(5.00000 + 2.02619i) q^{55} +1.05141 q^{59} +9.17891 q^{61} +(2.44314 - 6.02889i) q^{65} -4.05239i q^{67} -14.1663 q^{71} +2.02619i q^{73} +11.9075i q^{77} -6.00000 q^{79} -5.18908i q^{83} +(-5.76836 + 14.2345i) q^{85} -3.09334 q^{89} +14.3578 q^{91} +(-8.66025 - 3.50947i) q^{95} -0.882978i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91}+O(q^{100}) $ 8 * q - 12 * q^19 + 6 * q^25 - 4 * q^31 - 48 * q^49 + 40 * q^55 + 28 * q^61 - 48 * q^79 + 22 * q^85 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.07237	−	0.839805i	−0.926793	−	0.375572i
$5$	−2.07237	−	0.839805i	−0.926793	−	0.375572i
$6$		0			0
$7$		−	4.93536i		−	1.86539i	−0.360663	−	0.932696i	$-0.617450\pi$
$7$		−	4.93536i		−	1.86539i	0.360663	−	0.932696i	$-0.382550\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.41269			−0.727455			−0.363727	−	0.931505i	$-0.618496\pi$
$11$	−2.41269			−0.727455			−0.363727	+	0.931505i	$0.618496\pi$
$12$		0			0
$13$			2.90917i			0.806859i	0.915011	+	0.403429i	$0.132182\pi$
$13$			2.90917i			0.806859i	−0.915011	+	0.403429i	$0.867818\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.86869i		−	1.66590i	−0.553347	−	0.832951i	$-0.686650\pi$
$17$		−	6.86869i		−	1.66590i	0.553347	−	0.832951i	$-0.313350\pi$
$18$		0			0
$19$	4.17891			0.958707			0.479354	−	0.877622i	$-0.340871\pi$
$19$	4.17891			0.958707			0.479354	+	0.877622i	$0.340871\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.35922i			0.700446i	0.936666	+	0.350223i	$0.113894\pi$
$23$			3.35922i			0.700446i	−0.936666	+	0.350223i	$0.886106\pi$
$24$		0			0
$25$	3.58945	+	3.48078i	0.717891	+	0.696156i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.19615			−0.964901			−0.482451	−	0.875923i	$-0.660253\pi$
$29$	−5.19615			−0.964901			−0.482451	+	0.875923i	$0.660253\pi$
$30$		0			0
$31$	−6.17891			−1.10976			−0.554882	−	0.831929i	$-0.687237\pi$
$31$	−6.17891			−1.10976			−0.554882	+	0.831929i	$0.687237\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.14474	+	10.2279i	−0.700590	+	1.72883i
$36$		0			0
$37$		−	7.84453i		−	1.28963i	−0.764337	−	0.644817i	$-0.776934\pi$
$37$		−	7.84453i		−	1.28963i	0.764337	−	0.644817i	$-0.223066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.87680			−0.917801			−0.458901	−	0.888488i	$-0.651757\pi$
$41$	−5.87680			−0.917801			−0.458901	+	0.888488i	$0.651757\pi$
$42$		0			0
$43$			4.93536i			0.752636i	0.926491	+	0.376318i	$0.122810\pi$
$43$			4.93536i			0.752636i	−0.926491	+	0.376318i	$0.877190\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			11.9075i			1.73689i	0.495785	+	0.868445i	$0.334880\pi$
$47$			11.9075i			1.73689i	−0.495785	+	0.868445i	$0.665120\pi$
$48$		0			0
$49$	−17.3578			−2.47969
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.54830i			1.17420i	0.809515	+	0.587100i	$0.199730\pi$
$53$			8.54830i			1.17420i	−0.809515	+	0.587100i	$0.800270\pi$
$54$		0			0
$55$	5.00000	+	2.02619i	0.674200	+	0.273212i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.05141			0.136882			0.0684408	−	0.997655i	$-0.478198\pi$
$59$	1.05141			0.136882			0.0684408	+	0.997655i	$0.478198\pi$
$60$		0			0
$61$	9.17891			1.17524			0.587619	−	0.809137i	$-0.300065\pi$
$61$	9.17891			1.17524			0.587619	+	0.809137i	$0.300065\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.44314	−	6.02889i	0.303034	−	0.747791i
$66$		0			0
$67$		−	4.05239i		−	0.495078i	−0.968878	−	0.247539i	$-0.920378\pi$
$67$		−	4.05239i		−	0.495078i	0.968878	−	0.247539i	$-0.0796218\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.1663			−1.68123			−0.840614	−	0.541634i	$-0.817806\pi$
$71$	−14.1663			−1.68123			−0.840614	+	0.541634i	$0.817806\pi$
$72$		0			0
$73$			2.02619i			0.237148i	0.992945	+	0.118574i	$0.0378323\pi$
$73$			2.02619i			0.237148i	−0.992945	+	0.118574i	$0.962168\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			11.9075i			1.35699i
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.18908i		−	0.569576i	−0.958591	−	0.284788i	$-0.908077\pi$
$83$		−	5.18908i		−	0.569576i	0.958591	−	0.284788i	$-0.0919231\pi$
$84$		0			0
$85$	−5.76836	+	14.2345i	−0.625667	+	1.54395i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.09334			−0.327893			−0.163947	−	0.986469i	$-0.552422\pi$
$89$	−3.09334			−0.327893			−0.163947	+	0.986469i	$0.552422\pi$
$90$		0			0
$91$	14.3578			1.50511
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.66025	−	3.50947i	−0.888523	−	0.360064i
$96$		0			0
$97$		−	0.882978i		−	0.0896528i	−0.998995	−	0.0448264i	$-0.985727\pi$
$97$		−	0.882978i		−	0.0896528i	0.998995	−	0.0448264i	$-0.0142735\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.87680			0.584763			0.292382	−	0.956302i	$-0.405552\pi$
$101$	5.87680			0.584763			0.292382	+	0.956302i	$0.405552\pi$
$102$		0			0
$103$			9.87073i			0.972592i	0.873794	+	0.486296i	$0.161652\pi$
$103$			9.87073i			0.972592i	−0.873794	+	0.486296i	$0.838348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	13.5871i		−	1.27817i	−0.769136	−	0.639085i	$-0.779313\pi$
$113$		−	13.5871i		−	1.27817i	0.769136	−	0.639085i	$-0.220687\pi$
$114$		0			0
$115$	2.82109	−	6.96156i	0.263068	−	0.649169i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−33.8995			−3.10756
$120$		0			0
$121$	−5.17891			−0.470810
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.51551	−	10.2279i	−0.403879	−	0.914812i
$126$		0			0
$127$		−	4.93536i		−	0.437943i	−0.975731	−	0.218971i	$-0.929730\pi$
$127$		−	4.93536i		−	0.437943i	0.975731	−	0.218971i	$-0.0702701\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.1663			−1.23771			−0.618857	−	0.785504i	$-0.712404\pi$
$131$	−14.1663			−1.23771			−0.618857	+	0.785504i	$0.712404\pi$
$132$		0			0
$133$		−	20.6244i		−	1.78837i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.03883i		−	0.430496i	−0.976559	−	0.215248i	$-0.930944\pi$
$137$		−	5.03883i		−	0.430496i	0.976559	−	0.215248i	$-0.0690561\pi$
$138$		0			0
$139$	−5.82109			−0.493739			−0.246869	−	0.969049i	$-0.579402\pi$
$139$	−5.82109			−0.493739			−0.246869	+	0.969049i	$0.579402\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	7.01894i		−	0.586953i
$144$		0			0
$145$	10.7684	+	4.36376i	0.894264	+	0.362390i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.1758			1.07940			0.539700	−	0.841857i	$-0.318538\pi$
$149$	13.1758			1.07940			0.539700	+	0.841857i	$0.318538\pi$
$150$		0			0
$151$	−6.17891			−0.502832			−0.251416	−	0.967879i	$-0.580896\pi$
$151$	−6.17891			−0.502832			−0.251416	+	0.967879i	$0.580896\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.8050	+	5.18908i	1.02852	+	0.416797i
$156$		0			0
$157$			7.84453i			0.626062i	0.949743	+	0.313031i	$0.101344\pi$
$157$			7.84453i			0.626062i	−0.949743	+	0.313031i	$0.898656\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.5790			1.30661
$162$		0			0
$163$		−	10.7537i		−	0.842295i	−0.906992	−	0.421148i	$-0.861627\pi$
$163$		−	10.7537i		−	0.842295i	0.906992	−	0.421148i	$-0.138373\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.54830i		−	0.661487i	−0.943721	−	0.330744i	$-0.892700\pi$
$167$		−	8.54830i		−	0.661487i	0.943721	−	0.330744i	$-0.107300\pi$
$168$		0			0
$169$	4.53673			0.348979
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.7762i			1.42753i	0.700386	+	0.713765i	$0.253011\pi$
$173$			18.7762i			1.42753i	−0.700386	+	0.713765i	$0.746989\pi$
$174$		0			0
$175$	17.1789	−	17.7153i	1.29860	−	1.33915i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.97961			0.596424			0.298212	−	0.954500i	$-0.403610\pi$
$179$	7.97961			0.596424			0.298212	+	0.954500i	$0.403610\pi$
$180$		0			0
$181$	3.82109			0.284020			0.142010	−	0.989865i	$-0.454644\pi$
$181$	3.82109			0.284020			0.142010	+	0.989865i	$0.454644\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.58788	+	16.2568i	−0.484351	+	1.19522i
$186$		0			0
$187$			16.5720i			1.21187i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.61832			0.478885			0.239443	−	0.970911i	$-0.423035\pi$
$191$	6.61832			0.478885			0.239443	+	0.970911i	$0.423035\pi$
$192$		0			0
$193$		−	21.7676i		−	1.56687i	−0.621474	−	0.783435i	$-0.713466\pi$
$193$		−	21.7676i		−	1.56687i	0.621474	−	0.783435i	$-0.286534\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.4170i			1.09842i	0.835686	+	0.549208i	$0.185070\pi$
$197$			15.4170i			1.09842i	−0.835686	+	0.549208i	$0.814930\pi$
$198$		0			0
$199$	−20.3578			−1.44313			−0.721564	−	0.692348i	$-0.756576\pi$
$199$	−20.3578			−1.44313			−0.721564	+	0.692348i	$0.756576\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			25.6449i			1.79992i
$204$		0			0
$205$	12.1789	+	4.93536i	0.850612	+	0.344701i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.0824			−0.697416
$210$		0			0
$211$	16.1789			1.11380			0.556901	−	0.830579i	$-0.311990\pi$
$211$	16.1789			1.11380			0.556901	+	0.830579i	$0.311990\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.14474	−	10.2279i	0.282669	−	0.697538i
$216$		0			0
$217$			30.4952i			2.07015i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	19.9822			1.34415
$222$		0			0
$223$			25.5598i			1.71161i	0.517298	+	0.855805i	$0.326938\pi$
$223$			25.5598i			1.71161i	−0.517298	+	0.855805i	$0.673062\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	18.9265i		−	1.25619i	−0.778135	−	0.628097i	$-0.783834\pi$
$227$		−	18.9265i		−	1.25619i	0.778135	−	0.628097i	$-0.216166\pi$
$228$		0			0
$229$	−6.82109			−0.450750			−0.225375	−	0.974272i	$-0.572361\pi$
$229$	−6.82109			−0.450750			−0.225375	+	0.974272i	$0.572361\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.150248i			0.00984309i	0.999988	+	0.00492154i	$0.00156658\pi$
$233$			0.150248i			0.00984309i	−0.999988	+	0.00492154i	$0.998433\pi$
$234$		0			0
$235$	10.0000	−	24.6768i	0.652328	−	1.60974i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.03102			0.584168			0.292084	−	0.956393i	$-0.405651\pi$
$239$	9.03102			0.584168			0.292084	+	0.956393i	$0.405651\pi$
$240$		0			0
$241$	19.0000			1.22390			0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000			1.22390			0.611949	+	0.790897i	$0.290386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	35.9719	+	14.5772i	2.29816	+	0.931302i
$246$		0			0
$247$			12.1572i			0.773541i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3205			−1.09326			−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205			−1.09326			−0.546630	+	0.837374i	$0.684090\pi$
$252$		0			0
$253$		−	8.10477i		−	0.509543i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.1354i		−	1.38077i	−0.723442	−	0.690385i	$-0.757441\pi$
$257$		−	22.1354i		−	1.38077i	0.723442	−	0.690385i	$-0.242559\pi$
$258$		0			0
$259$	−38.7156			−2.40567
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.71844i		−	0.414277i	−0.978312	−	0.207138i	$-0.933585\pi$
$263$		−	6.71844i		−	0.414277i	0.978312	−	0.207138i	$-0.0664151\pi$
$264$		0			0
$265$	7.17891	−	17.7153i	0.440997	−	1.08824i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	30.0646			1.83307			0.916536	−	0.399952i	$-0.130973\pi$
$269$	30.0646			1.83307			0.916536	+	0.399952i	$0.130973\pi$
$270$		0			0
$271$	−22.3578			−1.35814			−0.679070	−	0.734073i	$-0.737617\pi$
$271$	−22.3578			−1.35814			−0.679070	+	0.734073i	$0.737617\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.66025	−	8.39805i	−0.522233	−	0.506422i
$276$		0			0
$277$		−	10.7537i		−	0.646128i	−0.946377	−	0.323064i	$-0.895287\pi$
$277$		−	10.7537i		−	0.646128i	0.946377	−	0.323064i	$-0.104713\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.4342			0.741764			0.370882	−	0.928680i	$-0.379055\pi$
$281$	12.4342			0.741764			0.370882	+	0.928680i	$0.379055\pi$
$282$		0			0
$283$		−	1.76596i		−	0.104975i	−0.998622	−	0.0524876i	$-0.983285\pi$
$283$		−	1.76596i		−	0.104975i	0.998622	−	0.0524876i	$-0.0167150\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			29.0041i			1.71206i
$288$		0			0
$289$	−30.1789			−1.77523
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	20.6061i		−	1.20382i	−0.798564	−	0.601910i	$-0.794407\pi$
$293$		−	20.6061i		−	1.20382i	0.798564	−	0.601910i	$-0.205593\pi$
$294$		0			0
$295$	−2.17891	−	0.882978i	−0.126861	−	0.0514090i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.77255			−0.565161
$300$		0			0
$301$	24.3578			1.40396
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.0221	−	7.70850i	−1.08920	−	0.441387i
$306$		0			0
$307$			8.98775i			0.512958i	0.966550	+	0.256479i	$0.0825624\pi$
$307$			8.98775i			0.512958i	−0.966550	+	0.256479i	$0.917438\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.7022			−0.606865			−0.303433	−	0.952853i	$-0.598133\pi$
$311$	−10.7022			−0.606865			−0.303433	+	0.952853i	$0.598133\pi$
$312$		0			0
$313$		−	6.96156i		−	0.393490i	−0.980455	−	0.196745i	$-0.936963\pi$
$313$		−	6.96156i		−	0.393490i	0.980455	−	0.196745i	$-0.0630372\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	34.0430i		−	1.91204i	−0.293297	−	0.956021i	$-0.594752\pi$
$317$		−	34.0430i		−	1.91204i	0.293297	−	0.956021i	$-0.405248\pi$
$318$		0			0
$319$	12.5367			0.701922
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	28.7036i		−	1.59711i
$324$		0			0
$325$	−10.1262	+	10.4423i	−0.561699	+	0.579237i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	58.7680			3.23998
$330$		0			0
$331$	−32.5367			−1.78838			−0.894190	−	0.447688i	$-0.852248\pi$
$331$	−32.5367			−1.78838			−0.894190	+	0.447688i	$0.852248\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.40322	+	8.39805i	−0.185938	+	0.458835i
$336$		0			0
$337$			9.87073i			0.537693i	0.963183	+	0.268846i	$0.0866424\pi$
$337$			9.87073i			0.537693i	−0.963183	+	0.268846i	$0.913358\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.9078			0.807303
$342$		0			0
$343$			51.1196i			2.76020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.54830i		−	0.458897i	−0.973321	−	0.229448i	$-0.926308\pi$
$347$		−	8.54830i		−	0.458897i	0.973321	−	0.229448i	$-0.0736922\pi$
$348$		0			0
$349$	−20.1789			−1.08015			−0.540076	−	0.841616i	$-0.681605\pi$
$349$	−20.1789			−1.08015			−0.540076	+	0.841616i	$0.681605\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	21.9852i		−	1.17015i	−0.810978	−	0.585077i	$-0.801064\pi$
$353$		−	21.9852i		−	1.17015i	0.810978	−	0.585077i	$-0.198936\pi$
$354$		0			0
$355$	29.3578	+	11.8969i	1.55815	+	0.631423i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.309878			−0.0163548			−0.00817738	−	0.999967i	$-0.502603\pi$
$359$	−0.309878			−0.0163548			−0.00817738	+	0.999967i	$0.502603\pi$
$360$		0			0
$361$	−1.53673			−0.0808803
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.70161	−	4.19903i	0.0890662	−	0.219787i
$366$		0			0
$367$			9.87073i			0.515248i	0.966245	+	0.257624i	$0.0829396\pi$
$367$			9.87073i			0.515248i	−0.966245	+	0.257624i	$0.917060\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	42.1890			2.19034
$372$		0			0
$373$			8.98775i			0.465368i	0.972552	+	0.232684i	$0.0747508\pi$
$373$			8.98775i			0.465368i	−0.972552	+	0.232684i	$0.925249\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	15.1165i		−	0.778539i
$378$		0			0
$379$	0.357817			0.0183798			0.00918990	−	0.999958i	$-0.497075\pi$
$379$	0.357817			0.0183798			0.00918990	+	0.999958i	$0.497075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			13.7374i			0.701947i	0.936385	+	0.350974i	$0.114149\pi$
$383$			13.7374i			0.701947i	−0.936385	+	0.350974i	$0.885851\pi$
$384$		0			0
$385$	10.0000	−	24.6768i	0.509647	−	1.25765i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.6100			−1.29848			−0.649239	−	0.760584i	$-0.724913\pi$
$389$	−25.6100			−1.29848			−0.649239	+	0.760584i	$0.724913\pi$
$390$		0			0
$391$	23.0735			1.16687
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.4342	+	5.03883i	0.625634	+	0.253531i
$396$		0			0
$397$		−	11.0139i		−	0.552774i	−0.961046	−	0.276387i	$-0.910863\pi$
$397$		−	11.0139i		−	0.552774i	0.961046	−	0.276387i	$-0.0891371\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−20.7237			−1.03489			−0.517447	−	0.855715i	$-0.673117\pi$
$401$	−20.7237			−1.03489			−0.517447	+	0.855715i	$0.673117\pi$
$402$		0			0
$403$		−	17.9755i		−	0.895423i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			18.9265i			0.938150i
$408$		0			0
$409$	−6.82109			−0.337281			−0.168641	−	0.985678i	$-0.553938\pi$
$409$	−6.82109			−0.337281			−0.168641	+	0.985678i	$0.553938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	5.18908i		−	0.255338i
$414$		0			0
$415$	−4.35782	+	10.7537i	−0.213917	+	0.527879i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.5790			0.809936			0.404968	−	0.914331i	$-0.367283\pi$
$419$	16.5790			0.809936			0.404968	+	0.914331i	$0.367283\pi$
$420$		0			0
$421$	29.7156			1.44825			0.724126	−	0.689668i	$-0.242244\pi$
$421$	29.7156			1.44825			0.724126	+	0.689668i	$0.242244\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	23.9084	−	24.6548i	1.15973	−	1.19594i
$426$		0			0
$427$		−	45.3013i		−	2.19228i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.67116			0.0804972			0.0402486	−	0.999190i	$-0.487185\pi$
$431$	1.67116			0.0804972			0.0402486	+	0.999190i	$0.487185\pi$
$432$		0			0
$433$		−	36.5737i		−	1.75762i	−0.477170	−	0.878811i	$-0.658337\pi$
$433$		−	36.5737i		−	1.75762i	0.477170	−	0.878811i	$-0.341663\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			14.0379i			0.671523i
$438$		0			0
$439$	−18.5367			−0.884710			−0.442355	−	0.896840i	$-0.645857\pi$
$439$	−18.5367			−0.884710			−0.442355	+	0.896840i	$0.645857\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			18.6260i			0.884946i	0.896782	+	0.442473i	$0.145899\pi$
$443$			18.6260i			0.884946i	−0.896782	+	0.442473i	$0.854101\pi$
$444$		0			0
$445$	6.41055	+	2.59780i	0.303889	+	0.123148i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	25.1783			1.18824			0.594120	−	0.804377i	$-0.297500\pi$
$449$	25.1783			1.18824			0.594120	+	0.804377i	$0.297500\pi$
$450$		0			0
$451$	14.1789			0.667659
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−29.7547	−	12.0578i	−1.39492	−	0.565277i
$456$		0			0
$457$			27.5860i			1.29042i	0.764006	+	0.645209i	$0.223230\pi$
$457$			27.5860i			1.29042i	−0.764006	+	0.645209i	$0.776770\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.7022			0.498450			0.249225	−	0.968446i	$-0.419824\pi$
$461$	10.7022			0.498450			0.249225	+	0.968446i	$0.419824\pi$
$462$		0			0
$463$		−	29.6122i		−	1.37619i	−0.725618	−	0.688097i	$-0.758446\pi$
$463$		−	29.6122i		−	1.37619i	0.725618	−	0.688097i	$-0.241554\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	15.5672i		−	0.720366i	−0.932882	−	0.360183i	$-0.882714\pi$
$467$		−	15.5672i		−	0.720366i	0.932882	−	0.360183i	$-0.117286\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.9075i		−	0.547508i
$474$		0			0
$475$	15.0000	+	14.5459i	0.688247	+	0.667410i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.3001			−1.15599			−0.577996	−	0.816040i	$-0.696165\pi$
$479$	−25.3001			−1.15599			−0.577996	+	0.816040i	$0.696165\pi$
$480$		0			0
$481$	22.8211			1.04055
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.741529	+	1.82986i	−0.0336711	+	0.0830896i
$486$		0			0
$487$		−	17.9755i		−	0.814548i	−0.913306	−	0.407274i	$-0.866479\pi$
$487$		−	17.9755i		−	0.814548i	0.913306	−	0.407274i	$-0.133521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	35.5707			1.60528			0.802641	−	0.596463i	$-0.203428\pi$
$491$	35.5707			1.60528			0.802641	+	0.596463i	$0.203428\pi$
$492$		0			0
$493$			35.6908i			1.60743i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			69.9158i			3.13615i
$498$		0			0
$499$	−4.17891			−0.187074			−0.0935368	−	0.995616i	$-0.529817\pi$
$499$	−4.17891			−0.187074			−0.0935368	+	0.995616i	$0.529817\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			37.2519i			1.66098i	0.557032	+	0.830491i	$0.311940\pi$
$503$			37.2519i			1.66098i	−0.557032	+	0.830491i	$0.688060\pi$
$504$		0			0
$505$	−12.1789	−	4.93536i	−0.541954	−	0.219621i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.54796			−0.334557			−0.167279	−	0.985910i	$-0.553498\pi$
$509$	−7.54796			−0.334557			−0.167279	+	0.985910i	$0.553498\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.28949	−	20.4558i	0.365279	−	0.901391i
$516$		0			0
$517$		−	28.7292i		−	1.26351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.03102			0.395656			0.197828	−	0.980237i	$-0.436611\pi$
$521$	9.03102			0.395656			0.197828	+	0.980237i	$0.436611\pi$
$522$		0			0
$523$		−	7.58430i		−	0.331638i	−0.986156	−	0.165819i	$-0.946973\pi$
$523$		−	7.58430i		−	0.331638i	0.986156	−	0.165819i	$-0.0530268\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			42.4410i			1.84876i
$528$		0			0
$529$	11.7156			0.509375
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	17.0966i		−	0.740536i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	41.8791			1.80386
$540$		0			0
$541$	3.35782			0.144364			0.0721819	−	0.997391i	$-0.477004\pi$
$541$	3.35782			0.144364			0.0721819	+	0.997391i	$0.477004\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.5066	+	5.87864i	0.621395	+	0.251813i
$546$		0			0
$547$		−	19.7415i		−	0.844084i	−0.906576	−	0.422042i	$-0.861314\pi$
$547$		−	19.7415i		−	0.844084i	0.906576	−	0.422042i	$-0.138686\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.7142			−0.925058
$552$		0			0
$553$			29.6122i			1.25924i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	39.2320i		−	1.66231i	−0.556037	−	0.831157i	$-0.687679\pi$
$557$		−	39.2320i		−	1.66231i	0.556037	−	0.831157i	$-0.312321\pi$
$558$		0			0
$559$	−14.3578			−0.607271
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	11.9075i		−	0.501842i	−0.968008	−	0.250921i	$-0.919267\pi$
$563$		−	11.9075i		−	0.501842i	0.968008	−	0.250921i	$-0.0807335\pi$
$564$		0			0
$565$	−11.4105	+	28.1576i	−0.480045	+	1.18460i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.45462			0.186748			0.0933738	−	0.995631i	$-0.470235\pi$
$569$	4.45462			0.186748			0.0933738	+	0.995631i	$0.470235\pi$
$570$		0			0
$571$	−18.1789			−0.760764			−0.380382	−	0.924830i	$-0.624207\pi$
$571$	−18.1789			−0.760764			−0.380382	+	0.924830i	$0.624207\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.6927	+	12.0578i	−0.487619	+	0.502844i
$576$		0			0
$577$		−	7.84453i		−	0.326572i	−0.986579	−	0.163286i	$-0.947791\pi$
$577$		−	7.84453i		−	0.326572i	0.986579	−	0.163286i	$-0.0522094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−25.6100			−1.06248
$582$		0			0
$583$		−	20.6244i		−	0.854177i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11.6070i		−	0.479073i	−0.970887	−	0.239537i	$-0.923004\pi$
$587$		−	11.6070i		−	0.479073i	0.970887	−	0.239537i	$-0.0769955\pi$
$588$		0			0
$589$	−25.8211			−1.06394
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.03883i			0.206920i	0.994634	+	0.103460i	$0.0329914\pi$
$593$			5.03883i			0.206920i	−0.994634	+	0.103460i	$0.967009\pi$
$594$		0			0
$595$	70.2524	+	28.4690i	2.88007	+	1.16711i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	17.0106			0.695035			0.347518	−	0.937673i	$-0.387025\pi$
$599$	17.0106			0.695035			0.347518	+	0.937673i	$0.387025\pi$
$600$		0			0
$601$	−3.35782			−0.136968			−0.0684841	−	0.997652i	$-0.521816\pi$
$601$	−3.35782			−0.136968			−0.0684841	+	0.997652i	$0.521816\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.7326	+	4.34927i	0.436343	+	0.176823i
$606$		0			0
$607$			4.93536i			0.200320i	0.994971	+	0.100160i	$0.0319355\pi$
$607$			4.93536i			0.200320i	−0.994971	+	0.100160i	$0.968064\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−34.6410			−1.40143
$612$		0			0
$613$			37.7170i			1.52337i	0.647945	+	0.761687i	$0.275628\pi$
$613$			37.7170i			1.52337i	−0.647945	+	0.761687i	$0.724372\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	5.33933i		−	0.214953i	−0.994208	−	0.107477i	$-0.965723\pi$
$617$		−	5.33933i		−	0.214953i	0.994208	−	0.107477i	$-0.0342771\pi$
$618$		0			0
$619$	−11.6422			−0.467939			−0.233969	−	0.972244i	$-0.575172\pi$
$619$	−11.6422			−0.467939			−0.233969	+	0.972244i	$0.575172\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15.2667i			0.611649i
$624$		0			0
$625$	0.768363	+	24.9882i	0.0307345	+	0.999528i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−53.8817			−2.14840
$630$		0			0
$631$	−22.5367			−0.897173			−0.448586	−	0.893739i	$-0.648072\pi$
$631$	−22.5367			−0.897173			−0.448586	+	0.893739i	$0.648072\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.14474	+	10.2279i	−0.164479	+	0.405882i
$636$		0			0
$637$		−	50.4969i		−	2.00076i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.3110			−0.723242			−0.361621	−	0.932325i	$-0.617777\pi$
$641$	−18.3110			−0.723242			−0.361621	+	0.932325i	$0.617777\pi$
$642$		0			0
$643$			20.6244i			0.813348i	0.913573	+	0.406674i	$0.133312\pi$
$643$			20.6244i			0.813348i	−0.913573	+	0.406674i	$0.866688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	44.2709i		−	1.74047i	−0.492639	−	0.870234i	$-0.663968\pi$
$647$		−	44.2709i		−	1.74047i	0.492639	−	0.870234i	$-0.336032\pi$
$648$		0			0
$649$	−2.53673			−0.0995752
$650$		0			0
$651$		0			0
$652$		0			0
$653$			0.300496i			0.0117593i	0.999983	+	0.00587967i	$0.00187157\pi$
$653$			0.300496i			0.0117593i	−0.999983	+	0.00587967i	$0.998128\pi$
$654$		0			0
$655$	29.3578	+	11.8969i	1.14710	+	0.464851i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−41.4474			−1.61456			−0.807282	−	0.590166i	$-0.799062\pi$
$659$	−41.4474			−1.61456			−0.807282	+	0.590166i	$0.799062\pi$
$660$		0			0
$661$	−46.4313			−1.80597			−0.902983	−	0.429675i	$-0.858628\pi$
$661$	−46.4313			−1.80597			−0.902983	+	0.429675i	$0.858628\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−17.3205	+	42.7415i	−0.671660	+	1.65744i
$666$		0			0
$667$		−	17.4550i		−	0.675861i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.1459			−0.854933
$672$		0			0
$673$		−	14.5459i		−	0.560701i	−0.959898	−	0.280351i	$-0.909549\pi$
$673$		−	14.5459i		−	0.560701i	0.959898	−	0.280351i	$-0.0904508\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	34.4937i		−	1.32570i	−0.748752	−	0.662850i	$-0.769347\pi$
$677$		−	34.4937i		−	1.32570i	0.748752	−	0.662850i	$-0.230653\pi$
$678$		0			0
$679$	−4.35782			−0.167238
$680$		0			0
$681$		0			0
$682$		0			0
$683$			44.5714i			1.70548i	0.522339	+	0.852738i	$0.325060\pi$
$683$			44.5714i			1.70548i	−0.522339	+	0.852738i	$0.674940\pi$
$684$		0			0
$685$	−4.23164	+	10.4423i	−0.161683	+	0.398981i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.8685			−0.947413
$690$		0			0
$691$	−30.7156			−1.16848			−0.584239	−	0.811582i	$-0.698607\pi$
$691$	−30.7156			−1.16848			−0.584239	+	0.811582i	$0.698607\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.0635	+	4.88858i	0.457593	+	0.185435i
$696$		0			0
$697$			40.3659i			1.52897i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.3420			1.03269			0.516347	−	0.856379i	$-0.327291\pi$
$701$	27.3420			1.03269			0.516347	+	0.856379i	$0.327291\pi$
$702$		0			0
$703$		−	32.7816i		−	1.23638i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	29.0041i		−	1.09081i
$708$		0			0
$709$	−15.5367			−0.583494			−0.291747	−	0.956496i	$-0.594237\pi$
$709$	−15.5367			−0.583494			−0.291747	+	0.956496i	$0.594237\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	20.7563i		−	0.777330i
$714$		0			0
$715$	−5.89454	+	14.5459i	−0.220443	+	0.543984i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.3960			−1.50652			−0.753259	−	0.657724i	$-0.771519\pi$
$719$	−40.3960			−1.50652			−0.753259	+	0.657724i	$0.771519\pi$
$720$		0			0
$721$	48.7156			1.81426
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.6514	−	18.0867i	−0.692694	−	0.671722i
$726$		0			0
$727$		−	47.0672i		−	1.74563i	−0.488055	−	0.872813i	$-0.662293\pi$
$727$		−	47.0672i		−	1.74563i	0.488055	−	0.872813i	$-0.337707\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	33.8995			1.25382
$732$		0			0
$733$		−	19.7415i		−	0.729167i	−0.931171	−	0.364584i	$-0.881211\pi$
$733$		−	19.7415i		−	0.729167i	0.931171	−	0.364584i	$-0.118789\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9.77717i			0.360147i
$738$		0			0
$739$	22.8945			0.842189			0.421095	−	0.907017i	$-0.361646\pi$
$739$	22.8945			0.842189			0.421095	+	0.907017i	$0.361646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	49.7604i		−	1.82553i	−0.408481	−	0.912767i	$-0.633941\pi$
$743$		−	49.7604i		−	1.82553i	0.408481	−	0.912767i	$-0.366059\pi$
$744$		0			0
$745$	−27.3051	−	11.0651i	−1.00038	−	0.405393i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.35782			−0.232000			−0.116000	−	0.993249i	$-0.537007\pi$
$751$	−6.35782			−0.232000			−0.116000	+	0.993249i	$0.537007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.8050	+	5.18908i	0.466022	+	0.188850i
$756$		0			0
$757$		−	1.76596i		−	0.0641847i	−0.999485	−	0.0320924i	$-0.989783\pi$
$757$		−	1.76596i		−	0.0641847i	0.999485	−	0.0320924i	$-0.0102171\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.73205			0.0627868			0.0313934	−	0.999507i	$-0.490006\pi$
$761$	1.73205			0.0627868			0.0313934	+	0.999507i	$0.490006\pi$
$762$		0			0
$763$			34.5475i			1.25071i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.05872i			0.110444i
$768$		0			0
$769$	−25.7156			−0.927329			−0.463665	−	0.886011i	$-0.653466\pi$
$769$	−25.7156			−0.927329			−0.463665	+	0.886011i	$0.653466\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	30.6837i		−	1.10362i	−0.833971	−	0.551809i	$-0.813938\pi$
$773$		−	30.6837i		−	1.10362i	0.833971	−	0.551809i	$-0.186062\pi$
$774$		0			0
$775$	−22.1789	−	21.5074i	−0.796690	−	0.772569i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.5586			−0.879903
$780$		0			0
$781$	34.1789			1.22302
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.58788	−	16.2568i	0.235132	−	0.580230i
$786$		0			0
$787$			47.9502i			1.70924i	0.519254	+	0.854620i	$0.326210\pi$
$787$			47.9502i			1.70924i	−0.519254	+	0.854620i	$0.673790\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−67.0574			−2.38429
$792$		0			0
$793$			26.7030i			0.948252i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			11.7573i			0.416464i	0.978079	+	0.208232i	$0.0667709\pi$
$797$			11.7573i			0.416464i	−0.978079	+	0.208232i	$0.933229\pi$
$798$		0			0
$799$	81.7891			2.89349
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.88858i		−	0.172514i
$804$		0			0
$805$	−34.3578	−	13.9231i	−1.21095	−	0.490725i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.19615			−0.182687			−0.0913435	−	0.995819i	$-0.529116\pi$
$809$	−5.19615			−0.182687			−0.0913435	+	0.995819i	$0.529116\pi$
$810$		0			0
$811$	6.89454			0.242100			0.121050	−	0.992646i	$-0.461374\pi$
$811$	6.89454			0.242100			0.121050	+	0.992646i	$0.461374\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.03102	+	22.2857i	−0.316343	+	0.780633i
$816$		0			0
$817$			20.6244i			0.721558i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.2105			−1.82216			−0.911080	−	0.412230i	$-0.864750\pi$
$821$	−52.2105			−1.82216			−0.911080	+	0.412230i	$0.864750\pi$
$822$		0			0
$823$			11.6367i			0.405629i	0.979217	+	0.202815i	$0.0650089\pi$
$823$			11.6367i			0.405629i	−0.979217	+	0.202815i	$0.934991\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			13.7374i			0.477696i	0.971057	+	0.238848i	$0.0767697\pi$
$827$			13.7374i			0.477696i	−0.971057	+	0.238848i	$0.923230\pi$
$828$		0			0
$829$	4.17891			0.145139			0.0725697	−	0.997363i	$-0.476880\pi$
$829$	4.17891			0.145139			0.0725697	+	0.997363i	$0.476880\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			119.225i			4.13092i
$834$		0			0
$835$	−7.17891	+	17.7153i	−0.248436	+	0.613062i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23.8171			−0.822256			−0.411128	−	0.911578i	$-0.634865\pi$
$839$	−23.8171			−0.822256			−0.411128	+	0.911578i	$0.634865\pi$
$840$		0			0
$841$	−2.00000			−0.0689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.40178	−	3.80997i	−0.323431	−	0.131067i
$846$		0			0
$847$			25.5598i			0.878245i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	26.3515			0.903319
$852$		0			0
$853$			12.5197i			0.428665i	0.976761	+	0.214333i	$0.0687576\pi$
$853$			12.5197i			0.428665i	−0.976761	+	0.214333i	$0.931242\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			56.3286i			1.92415i	0.272786	+	0.962075i	$0.412055\pi$
$857$			56.3286i			1.92415i	−0.272786	+	0.962075i	$0.587945\pi$
$858$		0			0
$859$	−38.5367			−1.31486			−0.657428	−	0.753517i	$-0.728356\pi$
$859$	−38.5367			−1.31486			−0.657428	+	0.753517i	$0.728356\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.54830i		−	0.290988i	−0.989359	−	0.145494i	$-0.953523\pi$
$863$		−	8.54830i		−	0.290988i	0.989359	−	0.145494i	$-0.0464771\pi$
$864$		0			0
$865$	15.7684	−	38.9113i	0.536140	−	1.32302i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.4762			0.491070
$870$		0			0
$871$	11.7891			0.399458
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−50.4785	+	22.2857i	−1.70648	+	0.753394i
$876$		0			0
$877$		−	48.2104i		−	1.62795i	−0.580900	−	0.813975i	$-0.697299\pi$
$877$		−	48.2104i		−	1.62795i	0.580900	−	0.813975i	$-0.302701\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.9388			0.806520			0.403260	−	0.915085i	$-0.367877\pi$
$881$	23.9388			0.806520			0.403260	+	0.915085i	$0.367877\pi$
$882$		0			0
$883$		−	50.2366i		−	1.69060i	−0.534295	−	0.845298i	$-0.679423\pi$
$883$		−	50.2366i		−	1.69060i	0.534295	−	0.845298i	$-0.320577\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			25.6449i			0.861072i	0.902574	+	0.430536i	$0.141675\pi$
$887$			25.6449i			0.861072i	−0.902574	+	0.430536i	$0.858325\pi$
$888$		0			0
$889$	−24.3578			−0.816935
$890$		0			0
$891$		0			0
$892$		0			0
$893$			49.7604i			1.66517i
$894$		0			0
$895$	−16.5367	−	6.70132i	−0.552762	−	0.224000i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.1065			1.07081
$900$		0			0
$901$	58.7156			1.95610
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.91872	−	3.20897i	−0.263227	−	0.106670i
$906$		0			0
$907$			59.2244i			1.96651i	0.182227	+	0.983256i	$0.441669\pi$
$907$			59.2244i			1.96651i	−0.182227	+	0.983256i	$0.558331\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	35.5707			1.17851			0.589254	−	0.807948i	$-0.299422\pi$
$911$	35.5707			1.17851			0.589254	+	0.807948i	$0.299422\pi$
$912$		0			0
$913$			12.5197i			0.414340i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			69.9158i			2.30882i
$918$		0			0
$919$	19.8211			0.653837			0.326919	−	0.945052i	$-0.393990\pi$
$919$	19.8211			0.653837			0.326919	+	0.945052i	$0.393990\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	41.2121i		−	1.35651i
$924$		0			0
$925$	27.3051	−	28.1576i	0.897786	−	0.925816i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.7441			−0.418121			−0.209060	−	0.977903i	$-0.567041\pi$
$929$	−12.7441			−0.418121			−0.209060	+	0.977903i	$0.567041\pi$
$930$		0			0
$931$	−72.5367			−2.37730
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.9173	−	34.3435i	0.455144	−	1.12315i
$936$		0			0
$937$		−	12.7799i		−	0.417501i	−0.977969	−	0.208751i	$-0.933060\pi$
$937$		−	12.7799i		−	0.417501i	0.977969	−	0.208751i	$-0.0669397\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−53.8817			−1.75649			−0.878246	−	0.478209i	$-0.841286\pi$
$941$	−53.8817			−1.75649			−0.878246	+	0.478209i	$0.841286\pi$
$942$		0			0
$943$		−	19.7415i		−	0.642870i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			7.01894i			0.228085i	0.993476	+	0.114042i	$0.0363800\pi$
$947$			7.01894i			0.228085i	−0.993476	+	0.114042i	$0.963620\pi$
$948$		0			0
$949$	−5.89454			−0.191345
$950$		0			0
$951$		0			0
$952$		0			0
$953$			11.7573i			0.380855i	0.981701	+	0.190428i	$0.0609875\pi$
$953$			11.7573i			0.380855i	−0.981701	+	0.190428i	$0.939013\pi$
$954$		0			0
$955$	−13.7156	−	5.55810i	−0.443827	−	0.179856i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.8685			−0.803045
$960$		0			0
$961$	7.17891			0.231578
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.2806	+	45.1107i	−0.588473	+	1.45216i
$966$		0			0
$967$		−	13.0401i		−	0.419343i	−0.977772	−	0.209671i	$-0.932761\pi$
$967$		−	13.0401i		−	0.419343i	0.977772	−	0.209671i	$-0.0672394\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.7142			0.696843			0.348422	−	0.937338i	$-0.386718\pi$
$971$	21.7142			0.696843			0.348422	+	0.937338i	$0.386718\pi$
$972$		0			0
$973$			28.7292i			0.921016i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	27.1743i		−	0.869382i	−0.900580	−	0.434691i	$-0.856858\pi$
$977$		−	27.1743i		−	0.869382i	0.900580	−	0.434691i	$-0.143142\pi$
$978$		0			0
$979$	7.46327			0.238527
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	25.9454i		−	0.827530i	−0.910384	−	0.413765i	$-0.864214\pi$
$983$		−	25.9454i		−	0.827530i	0.910384	−	0.413765i	$-0.135786\pi$
$984$		0			0
$985$	12.9473	−	31.9497i	0.412534	−	1.01800i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.5790			−0.527181
$990$		0			0
$991$	−32.5367			−1.03356			−0.516782	−	0.856117i	$-0.672870\pi$
$991$	−32.5367			−1.03356			−0.516782	+	0.856117i	$0.672870\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	42.1890	+	17.0966i	1.33748	+	0.541999i
$996$		0			0
$997$			29.8724i			0.946069i	0.881044	+	0.473035i	$0.156841\pi$
$997$			29.8724i			0.946069i	−0.881044	+	0.473035i	$0.843159\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.d.e.649.1	✓	8
3.2	odd	2	inner	1620.2.d.e.649.8	yes	8
5.2	odd	4		8100.2.a.be.1.7		8
5.3	odd	4		8100.2.a.be.1.1		8
5.4	even	2	inner	1620.2.d.e.649.2	yes	8
9.2	odd	6		1620.2.r.h.109.2		16
9.4	even	3		1620.2.r.h.1189.5		16
9.5	odd	6		1620.2.r.h.1189.4		16
9.7	even	3		1620.2.r.h.109.7		16
15.2	even	4		8100.2.a.be.1.8		8
15.8	even	4		8100.2.a.be.1.2		8
15.14	odd	2	inner	1620.2.d.e.649.7	yes	8
45.4	even	6		1620.2.r.h.1189.7		16
45.14	odd	6		1620.2.r.h.1189.2		16
45.29	odd	6		1620.2.r.h.109.4		16
45.34	even	6		1620.2.r.h.109.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.d.e.649.1	✓	8	1.1	even	1	trivial
1620.2.d.e.649.2	yes	8	5.4	even	2	inner
1620.2.d.e.649.7	yes	8	15.14	odd	2	inner
1620.2.d.e.649.8	yes	8	3.2	odd	2	inner
1620.2.r.h.109.2		16	9.2	odd	6
1620.2.r.h.109.4		16	45.29	odd	6
1620.2.r.h.109.5		16	45.34	even	6
1620.2.r.h.109.7		16	9.7	even	3
1620.2.r.h.1189.2		16	45.14	odd	6
1620.2.r.h.1189.4		16	9.5	odd	6
1620.2.r.h.1189.5		16	9.4	even	3
1620.2.r.h.1189.7		16	45.4	even	6
8100.2.a.be.1.1		8	5.3	odd	4
8100.2.a.be.1.2		8	15.8	even	4
8100.2.a.be.1.7		8	5.2	odd	4
8100.2.a.be.1.8		8	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.07237 - 0.839805i) q^{5} -4.93536i q^{7} +O(q^{10})\)	\(q+(-2.07237 - 0.839805i) q^{5} -4.93536i q^{7} -2.41269 q^{11} +2.90917i q^{13} -6.86869i q^{17} +4.17891 q^{19} +3.35922i q^{23} +(3.58945 + 3.48078i) q^{25} -5.19615 q^{29} -6.17891 q^{31} +(-4.14474 + 10.2279i) q^{35} -7.84453i q^{37} -5.87680 q^{41} +4.93536i q^{43} +11.9075i q^{47} -17.3578 q^{49} +8.54830i q^{53} +(5.00000 + 2.02619i) q^{55} +1.05141 q^{59} +9.17891 q^{61} +(2.44314 - 6.02889i) q^{65} -4.05239i q^{67} -14.1663 q^{71} +2.02619i q^{73} +11.9075i q^{77} -6.00000 q^{79} -5.18908i q^{83} +(-5.76836 + 14.2345i) q^{85} -3.09334 q^{89} +14.3578 q^{91} +(-8.66025 - 3.50947i) q^{95} -0.882978i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91}+O(q^{100}) \) 8 * q - 12 * q^19 + 6 * q^25 - 4 * q^31 - 48 * q^49 + 40 * q^55 + 28 * q^61 - 48 * q^79 + 22 * q^85 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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