LMFDB - Embedded newform 1620.2.d.d.649.5

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:	$6$
Coefficient field:	6.0.301925376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 14x^{4} + 43x^{2} + 36 $ x^6 + 14x^4 + 43x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$1.56613i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.2.d.d.649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.20860 - 0.349411i) q^{5} +2.26496i q^{7} +O(q^{10})$	$q+(2.20860 - 0.349411i) q^{5} +2.26496i q^{7} +2.54722 q^{11} -5.02510i q^{13} +5.72392i q^{17} +1.86997 q^{19} -5.39722i q^{23} +(4.75582 - 1.54342i) q^{25} -3.00000 q^{29} +9.38162 q^{31} +(0.791400 + 5.00238i) q^{35} +2.59165i q^{37} -3.96442 q^{41} +10.2084i q^{43} +1.07095i q^{47} +1.86997 q^{49} -10.7036i q^{53} +(5.62580 - 0.890028i) q^{55} -1.58280 q^{59} +0.869975 q^{61} +(-1.75582 - 11.0984i) q^{65} +4.85661i q^{67} +1.86997 q^{71} +3.87652i q^{73} +5.76935i q^{77} +13.2516 q^{79} +14.7032i q^{83} +(2.00000 + 12.6418i) q^{85} -13.4172 q^{89} +11.3816 q^{91} +(4.13003 - 0.653390i) q^{95} -17.6669i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5}+O(q^{10}) $ 6 * q + q^5	$ 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) $ 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

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
$5$	2.20860	−	0.349411i	0.987716	−	0.156261i
$5$	2.20860	−	0.349411i	0.987716	−	0.156261i
$6$		0			0
$7$			2.26496i			0.856073i	0.903761	+	0.428036i	$0.140795\pi$
$7$			2.26496i			0.856073i	−0.903761	+	0.428036i	$0.859205\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.54722			0.768017			0.384009	−	0.923330i	$-0.374543\pi$
$11$	2.54722			0.768017			0.384009	+	0.923330i	$0.374543\pi$
$12$		0			0
$13$		−	5.02510i		−	1.39371i	−0.717212	−	0.696856i	$-0.754582\pi$
$13$		−	5.02510i		−	1.39371i	0.717212	−	0.696856i	$-0.245418\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.72392i			1.38825i	0.719852	+	0.694127i	$0.244209\pi$
$17$			5.72392i			1.38825i	−0.719852	+	0.694127i	$0.755791\pi$
$18$		0			0
$19$	1.86997			0.429002			0.214501	−	0.976724i	$-0.431188\pi$
$19$	1.86997			0.429002			0.214501	+	0.976724i	$0.431188\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.39722i		−	1.12540i	−0.826662	−	0.562699i	$-0.809763\pi$
$23$		−	5.39722i		−	1.12540i	0.826662	−	0.562699i	$-0.190237\pi$
$24$		0			0
$25$	4.75582	−	1.54342i	0.951165	−	0.308684i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	9.38162			1.68499			0.842495	−	0.538705i	$-0.181086\pi$
$31$	9.38162			1.68499			0.842495	+	0.538705i	$0.181086\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.791400	+	5.00238i	0.133771	+	0.845557i
$36$		0			0
$37$			2.59165i			0.426065i	0.977045	+	0.213032i	$0.0683340\pi$
$37$			2.59165i			0.426065i	−0.977045	+	0.213032i	$0.931666\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.96442			−0.619139			−0.309569	−	0.950877i	$-0.600185\pi$
$41$	−3.96442			−0.619139			−0.309569	+	0.950877i	$0.600185\pi$
$42$		0			0
$43$			10.2084i			1.55677i	0.627790	+	0.778383i	$0.283960\pi$
$43$			10.2084i			1.55677i	−0.627790	+	0.778383i	$0.716040\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.07095i			0.156214i	0.996945	+	0.0781070i	$0.0248876\pi$
$47$			1.07095i			0.156214i	−0.996945	+	0.0781070i	$0.975112\pi$
$48$		0			0
$49$	1.86997			0.267139
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.7036i		−	1.47025i	−0.677931	−	0.735125i	$-0.737123\pi$
$53$		−	10.7036i		−	1.47025i	0.677931	−	0.735125i	$-0.262877\pi$
$54$		0			0
$55$	5.62580	−	0.890028i	0.758583	−	0.120011i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.58280			−0.206063			−0.103032	−	0.994678i	$-0.532854\pi$
$59$	−1.58280			−0.206063			−0.103032	+	0.994678i	$0.532854\pi$
$60$		0			0
$61$	0.869975			0.111389			0.0556944	−	0.998448i	$-0.482263\pi$
$61$	0.869975			0.111389			0.0556944	+	0.998448i	$0.482263\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.75582	−	11.0984i	−0.217783	−	1.37659i
$66$		0			0
$67$			4.85661i			0.593329i	0.954982	+	0.296664i	$0.0958743\pi$
$67$			4.85661i			0.593329i	−0.954982	+	0.296664i	$0.904126\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.86997			0.221925			0.110963	−	0.993825i	$-0.464607\pi$
$71$	1.86997			0.221925			0.110963	+	0.993825i	$0.464607\pi$
$72$		0			0
$73$			3.87652i			0.453713i	0.973928	+	0.226856i	$0.0728448\pi$
$73$			3.87652i			0.453713i	−0.973928	+	0.226856i	$0.927155\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			5.76935i			0.657479i
$78$		0			0
$79$	13.2516			1.49092			0.745461	−	0.666550i	$-0.232230\pi$
$79$	13.2516			1.49092			0.745461	+	0.666550i	$0.232230\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			14.7032i			1.61388i	0.590632	+	0.806941i	$0.298878\pi$
$83$			14.7032i			1.61388i	−0.590632	+	0.806941i	$0.701122\pi$
$84$		0			0
$85$	2.00000	+	12.6418i	0.216930	+	1.37120i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.4172			−1.42222			−0.711110	−	0.703081i	$-0.751807\pi$
$89$	−13.4172			−1.42222			−0.711110	+	0.703081i	$0.751807\pi$
$90$		0			0
$91$	11.3816			1.19312
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.13003	−	0.653390i	0.423732	−	0.0670364i
$96$		0			0
$97$		−	17.6669i		−	1.79381i	−0.442227	−	0.896903i	$-0.645812\pi$
$97$		−	17.6669i		−	1.79381i	0.442227	−	0.896903i	$-0.354188\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.32275			0.529633			0.264817	−	0.964299i	$-0.414689\pi$
$101$	5.32275			0.529633			0.264817	+	0.964299i	$0.414689\pi$
$102$		0			0
$103$		−	6.01547i		−	0.592722i	−0.955076	−	0.296361i	$-0.904227\pi$
$103$		−	6.01547i		−	0.592722i	0.955076	−	0.296361i	$-0.0957731\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.69840i		−	0.454212i	−0.973870	−	0.227106i	$-0.927074\pi$
$107$		−	4.69840i		−	0.454212i	0.973870	−	0.227106i	$-0.0729263\pi$
$108$		0			0
$109$	6.64167			0.636157			0.318078	−	0.948064i	$-0.396962\pi$
$109$	6.64167			0.636157			0.318078	+	0.948064i	$0.396962\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.8000i			1.20413i	0.798448	+	0.602064i	$0.205655\pi$
$113$			12.8000i			1.20413i	−0.798448	+	0.602064i	$0.794345\pi$
$114$		0			0
$115$	−1.88585	−	11.9203i	−0.175856	−	1.11157i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.9644			−1.18845
$120$		0			0
$121$	−4.51165			−0.410150
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.96442	−	5.07053i	0.891245	−	0.453522i
$126$		0			0
$127$		−	12.8103i		−	1.13673i	−0.822775	−	0.568367i	$-0.807576\pi$
$127$		−	12.8103i		−	1.13673i	0.822775	−	0.568367i	$-0.192424\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.51165			0.831037			0.415518	−	0.909585i	$-0.363600\pi$
$131$	9.51165			0.831037			0.415518	+	0.909585i	$0.363600\pi$
$132$		0			0
$133$			4.23541i			0.367257i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11.4933i		−	0.981936i	−0.871178	−	0.490968i	$-0.836643\pi$
$137$		−	11.4933i		−	0.981936i	0.871178	−	0.490968i	$-0.163357\pi$
$138$		0			0
$139$	11.5116			0.976405			0.488203	−	0.872730i	$-0.337653\pi$
$139$	11.5116			0.976405			0.488203	+	0.872730i	$0.337653\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.8000i		−	1.07039i
$144$		0			0
$145$	−6.62580	+	1.04823i	−0.550243	+	0.0870510i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−19.7044			−1.61425			−0.807123	−	0.590384i	$-0.798976\pi$
$149$	−19.7044			−1.61425			−0.807123	+	0.590384i	$0.798976\pi$
$150$		0			0
$151$	15.3816			1.25174			0.625869	−	0.779928i	$-0.284744\pi$
$151$	15.3816			1.25174			0.625869	+	0.779928i	$0.284744\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.7202	−	3.27804i	1.66429	−	0.263299i
$156$		0			0
$157$			12.4836i			0.996303i	0.867090	+	0.498151i	$0.165988\pi$
$157$			12.4836i			0.996303i	−0.867090	+	0.498151i	$0.834012\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.2245			0.963424
$162$		0			0
$163$			1.93826i			0.151816i	0.997115	+	0.0759082i	$0.0241856\pi$
$163$			1.93826i			0.151816i	−0.997115	+	0.0759082i	$0.975814\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.326695i			0.0252804i	0.999920	+	0.0126402i	$0.00402361\pi$
$167$			0.326695i			0.0252804i	−0.999920	+	0.0126402i	$0.995976\pi$
$168$		0			0
$169$	−12.2516			−0.942431
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.7992i			1.58133i	0.612246	+	0.790667i	$0.290266\pi$
$173$			20.7992i			1.58133i	−0.612246	+	0.790667i	$0.709734\pi$
$174$		0			0
$175$	3.49577	+	10.7717i	0.264256	+	0.814266i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.79882			0.283937			0.141969	−	0.989871i	$-0.454657\pi$
$179$	3.79882			0.283937			0.141969	+	0.989871i	$0.454657\pi$
$180$		0			0
$181$	−16.5116			−1.22730			−0.613651	−	0.789578i	$-0.710300\pi$
$181$	−16.5116			−1.22730			−0.613651	+	0.789578i	$0.710300\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.905551	+	5.72392i	0.0665774	+	0.420831i
$186$		0			0
$187$			14.5801i			1.06620i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.3816			−0.823545			−0.411773	−	0.911287i	$-0.635090\pi$
$191$	−11.3816			−0.823545			−0.411773	+	0.911287i	$0.635090\pi$
$192$		0			0
$193$		−	14.0849i		−	1.01385i	−0.861989	−	0.506927i	$-0.830781\pi$
$193$		−	14.0849i		−	1.01385i	0.861989	−	0.506927i	$-0.169219\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.90830i			0.563443i	0.959496	+	0.281721i	$0.0909054\pi$
$197$			7.90830i			0.563443i	−0.959496	+	0.281721i	$0.909095\pi$
$198$		0			0
$199$	1.86997			0.132559			0.0662795	−	0.997801i	$-0.478887\pi$
$199$	1.86997			0.132559			0.0662795	+	0.997801i	$0.478887\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6.79487i		−	0.476906i
$204$		0			0
$205$	−8.75582	+	1.38521i	−0.611533	+	0.0967475i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.76325			0.329481
$210$		0			0
$211$	−11.3816			−0.783543			−0.391772	−	0.920063i	$-0.628138\pi$
$211$	−11.3816			−0.783543			−0.391772	+	0.920063i	$0.628138\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.56693	+	22.5463i	0.243262	+	1.53764i
$216$		0			0
$217$			21.2490i			1.44247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.7632			1.93483
$222$		0			0
$223$			4.85661i			0.325222i	0.986690	+	0.162611i	$0.0519916\pi$
$223$			4.85661i			0.325222i	−0.986690	+	0.162611i	$0.948008\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.07613i		−	0.469659i	−0.972037	−	0.234830i	$-0.924547\pi$
$227$		−	7.07613i		−	0.469659i	0.972037	−	0.234830i	$-0.0754532\pi$
$228$		0			0
$229$	−8.73995			−0.577552			−0.288776	−	0.957397i	$-0.593248\pi$
$229$	−8.73995			−0.577552			−0.288776	+	0.957397i	$0.593248\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.26346i			0.606869i	0.952852	+	0.303435i	$0.0981334\pi$
$233$			9.26346i			0.606869i	−0.952852	+	0.303435i	$0.901867\pi$
$234$		0			0
$235$	0.374201	+	2.36530i	0.0244102	+	0.154295i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.6772			−0.820023			−0.410012	−	0.912080i	$-0.634475\pi$
$239$	−12.6772			−0.820023			−0.410012	+	0.912080i	$0.634475\pi$
$240$		0			0
$241$	−10.6099			−0.683445			−0.341723	−	0.939801i	$-0.611010\pi$
$241$	−10.6099			−0.683445			−0.341723	+	0.939801i	$0.611010\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.13003	−	0.653390i	0.263858	−	0.0417435i
$246$		0			0
$247$		−	9.39680i		−	0.597904i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.6332			−1.17612			−0.588059	−	0.808818i	$-0.700108\pi$
$251$	−18.6332			−1.17612			−0.588059	+	0.808818i	$0.700108\pi$
$252$		0			0
$253$		−	13.7479i		−	0.864326i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	26.3898i		−	1.64615i	−0.567933	−	0.823075i	$-0.692257\pi$
$257$		−	26.3898i		−	1.64615i	0.567933	−	0.823075i	$-0.307743\pi$
$258$		0			0
$259$	−5.86997			−0.364742
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.6776i		−	0.843400i	−0.906735	−	0.421700i	$-0.861434\pi$
$263$		−	13.6776i		−	0.843400i	0.906735	−	0.421700i	$-0.138566\pi$
$264$		0			0
$265$	−3.73995	−	23.6399i	−0.229743	−	1.45219i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.6417			−1.38049			−0.690244	−	0.723577i	$-0.742497\pi$
$269$	−22.6417			−1.38049			−0.690244	+	0.723577i	$0.742497\pi$
$270$		0			0
$271$	5.86997			0.356576			0.178288	−	0.983978i	$-0.442944\pi$
$271$	5.86997			0.356576			0.178288	+	0.983978i	$0.442944\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.1142	−	3.93143i	0.730511	−	0.237074i
$276$		0			0
$277$		−	0.158205i		−	0.00950560i	−0.999989	−	0.00475280i	$-0.998487\pi$
$277$		−	0.158205i		−	0.00950560i	0.999989	−	0.00475280i	$-0.00151287\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.39008			0.202235			0.101117	−	0.994874i	$-0.467758\pi$
$281$	3.39008			0.202235			0.101117	+	0.994874i	$0.467758\pi$
$282$		0			0
$283$		−	11.9782i		−	0.712028i	−0.934481	−	0.356014i	$-0.884135\pi$
$283$		−	11.9782i		−	0.712028i	0.934481	−	0.356014i	$-0.115865\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.97924i		−	0.530028i
$288$		0			0
$289$	−15.7632			−0.927250
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.0753i		−	0.880708i	−0.897824	−	0.440354i	$-0.854853\pi$
$293$		−	15.0753i		−	0.880708i	0.897824	−	0.440354i	$-0.145147\pi$
$294$		0			0
$295$	−3.49577	+	0.553048i	−0.203532	+	0.0321997i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−27.1216			−1.56848
$300$		0			0
$301$	−23.1216			−1.33271
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.92143	−	0.303979i	0.110021	−	0.0174058i
$306$		0			0
$307$		−	6.00518i		−	0.342734i	−0.985207	−	0.171367i	$-0.945182\pi$
$307$		−	6.00518i		−	0.342734i	0.985207	−	0.171367i	$-0.0548183\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−30.4049			−1.72410			−0.862052	−	0.506819i	$-0.830821\pi$
$311$	−30.4049			−1.72410			−0.862052	+	0.506819i	$0.830821\pi$
$312$		0			0
$313$			18.2984i			1.03429i	0.855898	+	0.517144i	$0.173005\pi$
$313$			18.2984i			1.03429i	−0.855898	+	0.517144i	$0.826995\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			8.47377i			0.475935i	0.971273	+	0.237967i	$0.0764811\pi$
$317$			8.47377i			0.475935i	−0.971273	+	0.237967i	$0.923519\pi$
$318$		0			0
$319$	−7.64167			−0.427852
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10.7036i			0.595563i
$324$		0			0
$325$	−7.75582	−	23.8985i	−0.430216	−	1.32565i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.42565			−0.133731
$330$		0			0
$331$	−15.2516			−0.838304			−0.419152	−	0.907916i	$-0.637672\pi$
$331$	−15.2516			−0.838304			−0.419152	+	0.907916i	$0.637672\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.69695	+	10.7263i	0.0927144	+	0.586040i
$336$		0			0
$337$		−	17.9834i		−	0.979616i	−0.871830	−	0.489808i	$-0.837067\pi$
$337$		−	17.9834i		−	0.979616i	0.871830	−	0.489808i	$-0.162933\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.8971			1.29410
$342$		0			0
$343$			20.0901i			1.08476i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.3821i			0.879436i	0.898136	+	0.439718i	$0.144922\pi$
$347$			16.3821i			0.879436i	−0.898136	+	0.439718i	$0.855078\pi$
$348$		0			0
$349$	−17.7632			−0.950845			−0.475422	−	0.879758i	$-0.657705\pi$
$349$	−17.7632			−0.950845			−0.475422	+	0.879758i	$0.657705\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.7083i		−	1.10219i	−0.834441	−	0.551097i	$-0.814209\pi$
$353$		−	20.7083i		−	1.10219i	0.834441	−	0.551097i	$-0.185791\pi$
$354$		0			0
$355$	4.13003	−	0.653390i	0.219199	−	0.0346783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.3732			−1.81415			−0.907073	−	0.420973i	$-0.861689\pi$
$359$	−34.3732			−1.81415			−0.907073	+	0.420973i	$0.861689\pi$
$360$		0			0
$361$	−15.5032			−0.815958
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.35450	+	8.56169i	0.0708977	+	0.448139i
$366$		0			0
$367$			1.14857i			0.0599551i	0.999551	+	0.0299776i	$0.00954358\pi$
$367$			1.14857i			0.0599551i	−0.999551	+	0.0299776i	$0.990456\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.2431			1.25864
$372$		0			0
$373$			21.5435i			1.11548i	0.830016	+	0.557739i	$0.188331\pi$
$373$			21.5435i			1.11548i	−0.830016	+	0.557739i	$0.811669\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.0753i			0.776417i
$378$		0			0
$379$	−24.7632			−1.27200			−0.636001	−	0.771688i	$-0.719413\pi$
$379$	−24.7632			−1.27200			−0.636001	+	0.771688i	$0.719413\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.53659i		−	0.180711i	−0.995910	−	0.0903556i	$-0.971200\pi$
$383$		−	3.53659i		−	0.180711i	0.995910	−	0.0903556i	$-0.0288004\pi$
$384$		0			0
$385$	2.01587	+	12.7422i	0.102738	+	0.649402i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.7988			−1.56156			−0.780781	−	0.624805i	$-0.785179\pi$
$389$	−30.7988			−1.56156			−0.780781	+	0.624805i	$0.785179\pi$
$390$		0			0
$391$	30.8933			1.56234
$392$		0			0
$393$		0			0
$394$		0			0
$395$	29.2675	−	4.63025i	1.47261	−	0.232973i
$396$		0			0
$397$		−	18.1196i		−	0.909399i	−0.890645	−	0.454700i	$-0.849747\pi$
$397$		−	18.1196i		−	0.909399i	0.890645	−	0.454700i	$-0.150253\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.5828			0.678293			0.339146	−	0.940734i	$-0.389862\pi$
$401$	13.5828			0.678293			0.339146	+	0.940734i	$0.389862\pi$
$402$		0			0
$403$		−	47.1436i		−	2.34839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.60152i			0.327225i
$408$		0			0
$409$	−28.2749			−1.39810			−0.699052	−	0.715071i	$-0.746394\pi$
$409$	−28.2749			−1.39810			−0.699052	+	0.715071i	$0.746394\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	3.58497i		−	0.176405i
$414$		0			0
$415$	5.13745	+	32.4734i	0.252187	+	1.59406i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−22.0271			−1.07610			−0.538048	−	0.842914i	$-0.680838\pi$
$419$	−22.0271			−1.07610			−0.538048	+	0.842914i	$0.680838\pi$
$420$		0			0
$421$	−16.2749			−0.793190			−0.396595	−	0.917994i	$-0.629808\pi$
$421$	−16.2749			−0.793190			−0.396595	+	0.917994i	$0.629808\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.83440	+	27.2219i	0.428531	+	1.32046i
$426$		0			0
$427$			1.97045i			0.0953570i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.6921			0.900366			0.450183	−	0.892936i	$-0.351359\pi$
$431$	18.6921			0.900366			0.450183	+	0.892936i	$0.351359\pi$
$432$		0			0
$433$		−	11.9885i		−	0.576128i	−0.957611	−	0.288064i	$-0.906988\pi$
$433$		−	11.9885i		−	0.576128i	0.957611	−	0.288064i	$-0.0930116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	10.0927i		−	0.482798i
$438$		0			0
$439$	13.2516			0.632464			0.316232	−	0.948682i	$-0.397582\pi$
$439$	13.2516			0.632464			0.316232	+	0.948682i	$0.397582\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	23.3132i		−	1.10765i	−0.832635	−	0.553823i	$-0.813169\pi$
$443$		−	23.3132i		−	1.10765i	0.832635	−	0.553823i	$-0.186831\pi$
$444$		0			0
$445$	−29.6332	+	4.68812i	−1.40475	+	0.222238i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.7632			0.507949			0.253974	−	0.967211i	$-0.418262\pi$
$449$	10.7632			0.507949			0.253974	+	0.967211i	$0.418262\pi$
$450$		0			0
$451$	−10.0983			−0.475509
$452$		0			0
$453$		0			0
$454$		0			0
$455$	25.1374	−	3.97686i	1.17846	−	0.186438i
$456$		0			0
$457$		−	9.55501i		−	0.446964i	−0.974708	−	0.223482i	$-0.928258\pi$
$457$		−	9.55501i		−	0.446964i	0.974708	−	0.223482i	$-0.0717425\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.2601			0.803881			0.401940	−	0.915666i	$-0.368336\pi$
$461$	17.2601			0.803881			0.401940	+	0.915666i	$0.368336\pi$
$462$		0			0
$463$			35.8291i			1.66512i	0.553936	+	0.832559i	$0.313125\pi$
$463$			35.8291i			1.66512i	−0.553936	+	0.832559i	$0.686875\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.5287i		−	1.32015i	−0.751199	−	0.660076i	$-0.770524\pi$
$467$		−	28.5287i		−	1.32015i	0.751199	−	0.660076i	$-0.229476\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$			26.0031i			1.19562i
$474$		0			0
$475$	8.89327	−	2.88615i	0.408051	−	0.132426i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	25.8259			1.18002			0.590009	−	0.807397i	$-0.299124\pi$
$479$	25.8259			1.18002			0.590009	+	0.807397i	$0.299124\pi$
$480$		0			0
$481$	13.0233			0.593811
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.17302	−	39.0192i	−0.280303	−	1.77177i
$486$		0			0
$487$		−	4.88880i		−	0.221533i	−0.993846	−	0.110766i	$-0.964669\pi$
$487$		−	4.88880i		−	0.221533i	0.993846	−	0.110766i	$-0.0353305\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.51165			0.429255			0.214627	−	0.976696i	$-0.431146\pi$
$491$	9.51165			0.429255			0.214627	+	0.976696i	$0.431146\pi$
$492$		0			0
$493$		−	17.1718i		−	0.773377i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			4.23541i			0.189984i
$498$		0			0
$499$	−9.51165			−0.425800			−0.212900	−	0.977074i	$-0.568291\pi$
$499$	−9.51165			−0.425800			−0.212900	+	0.977074i	$0.568291\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			13.7801i			0.614426i	0.951641	+	0.307213i	$0.0993964\pi$
$503$			13.7801i			0.614426i	−0.951641	+	0.307213i	$0.900604\pi$
$504$		0			0
$505$	11.7558	−	1.85983i	0.523127	−	0.0827612i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.97670			−0.0876159			−0.0438079	−	0.999040i	$-0.513949\pi$
$509$	−1.97670			−0.0876159			−0.0438079	+	0.999040i	$0.513949\pi$
$510$		0			0
$511$	−8.78015			−0.388411
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.10187	−	13.2858i	−0.0926195	−	0.585440i
$516$		0			0
$517$			2.72795i			0.119975i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.1804			−0.796500			−0.398250	−	0.917277i	$-0.630382\pi$
$521$	−18.1804			−0.796500			−0.398250	+	0.917277i	$0.630382\pi$
$522$		0			0
$523$		−	15.0431i		−	0.657789i	−0.944367	−	0.328894i	$-0.893324\pi$
$523$		−	15.0431i		−	0.657789i	0.944367	−	0.328894i	$-0.106676\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			53.6996i			2.33919i
$528$		0			0
$529$	−6.13003			−0.266523
$530$		0			0
$531$		0			0
$532$		0			0
$533$			19.9216i			0.862901i
$534$		0			0
$535$	−1.64167	−	10.3769i	−0.0709757	−	0.448632i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.76325			0.205167
$540$		0			0
$541$	12.7717			0.549098			0.274549	−	0.961573i	$-0.411471\pi$
$541$	12.7717			0.549098			0.274549	+	0.961573i	$0.411471\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.6688	−	2.32067i	0.628342	−	0.0994067i
$546$		0			0
$547$			9.42900i			0.403155i	0.979473	+	0.201577i	$0.0646068\pi$
$547$			9.42900i			0.403155i	−0.979473	+	0.201577i	$0.935393\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.60992			−0.238991
$552$		0			0
$553$			30.0143i			1.27634i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	5.85726i		−	0.248180i	−0.992271	−	0.124090i	$-0.960399\pi$
$557$		−	5.85726i		−	0.248180i	0.992271	−	0.124090i	$-0.0396012\pi$
$558$		0			0
$559$	51.2982			2.16968
$560$		0			0
$561$		0			0
$562$		0			0
$563$			31.3124i			1.31966i	0.751415	+	0.659830i	$0.229372\pi$
$563$			31.3124i			1.31966i	−0.751415	+	0.659830i	$0.770628\pi$
$564$		0			0
$565$	4.47248	+	28.2702i	0.188159	+	1.18934i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−35.4405			−1.48574			−0.742871	−	0.669434i	$-0.766537\pi$
$569$	−35.4405			−1.48574			−0.742871	+	0.669434i	$0.766537\pi$
$570$		0			0
$571$	−38.7950			−1.62352			−0.811760	−	0.583991i	$-0.801490\pi$
$571$	−38.7950			−1.62352			−0.811760	+	0.583991i	$0.801490\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.33017	−	25.6682i	−0.347392	−	1.07044i
$576$		0			0
$577$			19.7634i			0.822761i	0.911464	+	0.411381i	$0.134953\pi$
$577$			19.7634i			0.822761i	−0.911464	+	0.411381i	$0.865047\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−33.3020			−1.38160
$582$		0			0
$583$		−	27.2644i		−	1.12918i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			21.0380i			0.868331i	0.900833	+	0.434165i	$0.142957\pi$
$587$			21.0380i			0.868331i	−0.900833	+	0.434165i	$0.857043\pi$
$588$		0			0
$589$	17.5434			0.722863
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.9390i			0.613471i	0.951795	+	0.306735i	$0.0992367\pi$
$593$			14.9390i			0.613471i	−0.951795	+	0.306735i	$0.900763\pi$
$594$		0			0
$595$	−28.6332	+	4.52991i	−1.17385	+	0.185708i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.5116			0.878942			0.439471	−	0.898257i	$-0.355166\pi$
$599$	21.5116			0.878942			0.439471	+	0.898257i	$0.355166\pi$
$600$		0			0
$601$	5.51165			0.224825			0.112412	−	0.993662i	$-0.464142\pi$
$601$	5.51165			0.224825			0.112412	+	0.993662i	$0.464142\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.96442	+	1.57642i	−0.405111	+	0.0640906i
$606$		0			0
$607$			9.70293i			0.393830i	0.980421	+	0.196915i	$0.0630923\pi$
$607$			9.70293i			0.393830i	−0.980421	+	0.196915i	$0.936908\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.38162			0.217717
$612$		0			0
$613$			2.25467i			0.0910653i	0.998963	+	0.0455326i	$0.0144985\pi$
$613$			2.25467i			0.0910653i	−0.998963	+	0.0455326i	$0.985501\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	40.1553i		−	1.61659i	−0.588776	−	0.808297i	$-0.700390\pi$
$617$		−	40.1553i		−	1.61659i	0.588776	−	0.808297i	$-0.299610\pi$
$618$		0			0
$619$	−5.12157			−0.205853			−0.102927	−	0.994689i	$-0.532821\pi$
$619$	−5.12157			−0.205853			−0.102927	+	0.994689i	$0.532821\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	30.3894i		−	1.21752i
$624$		0			0
$625$	20.2357	−	14.6804i	0.809429	−	0.587218i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.8344			−0.591486
$630$		0			0
$631$	−1.28335			−0.0510892			−0.0255446	−	0.999674i	$-0.508132\pi$
$631$	−1.28335			−0.0510892			−0.0255446	+	0.999674i	$0.508132\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.47607	−	28.2929i	−0.177627	−	1.12277i
$636$		0			0
$637$		−	9.39680i		−	0.372315i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.5388			1.12721			0.563607	−	0.826043i	$-0.309413\pi$
$641$	28.5388			1.12721			0.563607	+	0.826043i	$0.309413\pi$
$642$		0			0
$643$			23.3132i			0.919384i	0.888078	+	0.459692i	$0.152040\pi$
$643$			23.3132i			0.919384i	−0.888078	+	0.459692i	$0.847960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	8.93381i		−	0.351224i	−0.984459	−	0.175612i	$-0.943810\pi$
$647$		−	8.93381i		−	0.351224i	0.984459	−	0.175612i	$-0.0561905\pi$
$648$		0			0
$649$	−4.03175			−0.158260
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	14.3310i		−	0.560817i	−0.959881	−	0.280408i	$-0.909530\pi$
$653$		−	14.3310i		−	0.560817i	0.959881	−	0.280408i	$-0.0904699\pi$
$654$		0			0
$655$	21.0074	−	3.32347i	0.820828	−	0.129859i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.58280			−0.295384			−0.147692	−	0.989033i	$-0.547184\pi$
$659$	−7.58280			−0.295384			−0.147692	+	0.989033i	$0.547184\pi$
$660$		0			0
$661$	29.5116			1.14787			0.573935	−	0.818901i	$-0.305416\pi$
$661$	29.5116			1.14787			0.573935	+	0.818901i	$0.305416\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.47990	+	9.35433i	0.0573880	+	0.362745i
$666$		0			0
$667$			16.1917i			0.626944i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.21602			0.0855485
$672$		0			0
$673$		−	6.98527i		−	0.269262i	−0.990896	−	0.134631i	$-0.957015\pi$
$673$		−	6.98527i		−	0.269262i	0.990896	−	0.134631i	$-0.0429849\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			20.1033i			0.772634i	0.922366	+	0.386317i	$0.126253\pi$
$677$			20.1033i			0.772634i	−0.922366	+	0.386317i	$0.873747\pi$
$678$		0			0
$679$	40.0148			1.53563
$680$		0			0
$681$		0			0
$682$		0			0
$683$			44.5270i			1.70378i	0.523721	+	0.851890i	$0.324544\pi$
$683$			44.5270i			1.70378i	−0.523721	+	0.851890i	$0.675456\pi$
$684$		0			0
$685$	−4.01587	−	25.3840i	−0.153439	−	0.969874i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−53.7865			−2.04910
$690$		0			0
$691$	33.7717			1.28474			0.642368	−	0.766396i	$-0.277952\pi$
$691$	33.7717			1.28474			0.642368	+	0.766396i	$0.277952\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	25.4246	−	4.02230i	0.964411	−	0.152574i
$696$		0			0
$697$		−	22.6920i		−	0.859522i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.78398			0.256227			0.128114	−	0.991759i	$-0.459108\pi$
$701$	6.78398			0.256227			0.128114	+	0.991759i	$0.459108\pi$
$702$		0			0
$703$			4.84632i			0.182782i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			12.0558i			0.453405i
$708$		0			0
$709$	44.7865			1.68199			0.840997	−	0.541040i	$-0.181969\pi$
$709$	44.7865			1.68199			0.840997	+	0.541040i	$0.181969\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	50.6347i		−	1.89629i
$714$		0			0
$715$	−4.47248	−	28.2702i	−0.167261	−	1.05724i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.55105			0.207019			0.103510	−	0.994628i	$-0.466993\pi$
$719$	5.55105			0.207019			0.103510	+	0.994628i	$0.466993\pi$
$720$		0			0
$721$	13.6248			0.507413
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.2675	+	4.63025i	−0.529881	+	0.171963i
$726$		0			0
$727$			37.5564i			1.39289i	0.717611	+	0.696444i	$0.245236\pi$
$727$			37.5564i			1.39289i	−0.717611	+	0.696444i	$0.754764\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−58.4320			−2.16119
$732$		0			0
$733$		−	34.5442i		−	1.27592i	−0.770070	−	0.637959i	$-0.779779\pi$
$733$		−	34.5442i		−	1.27592i	0.770070	−	0.637959i	$-0.220221\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.3709i			0.455687i
$738$		0			0
$739$	0.520101			0.0191322			0.00956612	−	0.999954i	$-0.496955\pi$
$739$	0.520101			0.0191322			0.00956612	+	0.999954i	$0.496955\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	41.7918i		−	1.53319i	−0.642130	−	0.766596i	$-0.721949\pi$
$743$		−	41.7918i		−	1.53319i	0.642130	−	0.766596i	$-0.278051\pi$
$744$		0			0
$745$	−43.5191	+	6.88492i	−1.59442	+	0.252244i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.6417			0.388838
$750$		0			0
$751$	17.1216			0.624775			0.312388	−	0.949955i	$-0.398871\pi$
$751$	17.1216			0.624775			0.312388	+	0.949955i	$0.398871\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.9718	−	5.37451i	1.23636	−	0.195598i
$756$		0			0
$757$		−	6.80515i		−	0.247337i	−0.992324	−	0.123669i	$-0.960534\pi$
$757$		−	6.80515i		−	0.247337i	0.992324	−	0.123669i	$-0.0394660\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.2431			1.64006			0.820031	−	0.572319i	$-0.193956\pi$
$761$	45.2431			1.64006			0.820031	+	0.572319i	$0.193956\pi$
$762$		0			0
$763$			15.0431i			0.544597i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7.95373i			0.287192i
$768$		0			0
$769$	15.2431			0.549682			0.274841	−	0.961490i	$-0.411375\pi$
$769$	15.2431			0.549682			0.274841	+	0.961490i	$0.411375\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	44.6173	−	14.4798i	1.60270	−	0.520128i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.41337			−0.265612
$780$		0			0
$781$	4.76325			0.170442
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.36192	+	27.5714i	0.155684	+	0.984064i
$786$		0			0
$787$		−	20.2586i		−	0.722141i	−0.932539	−	0.361070i	$-0.882411\pi$
$787$		−	20.2586i		−	0.722141i	0.932539	−	0.361070i	$-0.117589\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−28.9915			−1.03082
$792$		0			0
$793$		−	4.37171i		−	0.155244i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	11.2691i		−	0.399171i	−0.979880	−	0.199585i	$-0.936040\pi$
$797$		−	11.2691i		−	0.399171i	0.979880	−	0.199585i	$-0.0639595\pi$
$798$		0			0
$799$	−6.13003			−0.216865
$800$		0			0
$801$		0			0
$802$		0			0
$803$			9.87437i			0.348459i
$804$		0			0
$805$	26.9990	−	4.27137i	0.951589	−	0.150546i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.5032			−0.931803			−0.465901	−	0.884837i	$-0.654270\pi$
$809$	−26.5032			−0.931803			−0.465901	+	0.884837i	$0.654270\pi$
$810$		0			0
$811$	−1.54340			−0.0541960			−0.0270980	−	0.999633i	$-0.508627\pi$
$811$	−1.54340			−0.0541960			−0.0270980	+	0.999633i	$0.508627\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.677250	+	4.28084i	0.0237230	+	0.149951i
$816$		0			0
$817$			19.0894i			0.667855i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.3189			−0.395033			−0.197517	−	0.980300i	$-0.563288\pi$
$821$	−11.3189			−0.395033			−0.197517	+	0.980300i	$0.563288\pi$
$822$		0			0
$823$		−	9.72350i		−	0.338940i	−0.985535	−	0.169470i	$-0.945794\pi$
$823$		−	9.72350i		−	0.338940i	0.985535	−	0.169470i	$-0.0542056\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			25.9722i			0.903143i	0.892235	+	0.451571i	$0.149136\pi$
$827$			25.9722i			0.903143i	−0.892235	+	0.451571i	$0.850864\pi$
$828$		0			0
$829$	9.61838			0.334060			0.167030	−	0.985952i	$-0.446582\pi$
$829$	9.61838			0.334060			0.167030	+	0.985952i	$0.446582\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			10.7036i			0.370857i
$834$		0			0
$835$	0.114151	+	0.721538i	0.00395035	+	0.0249699i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20.0906			0.693605			0.346803	−	0.937938i	$-0.387267\pi$
$839$	20.0906			0.693605			0.346803	+	0.937938i	$0.387267\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−27.0589	+	4.28084i	−0.930854	+	0.147265i
$846$		0			0
$847$		−	10.2187i		−	0.351118i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	13.9877			0.479493
$852$		0			0
$853$			18.9312i			0.648193i	0.946024	+	0.324097i	$0.105060\pi$
$853$			18.9312i			0.648193i	−0.946024	+	0.324097i	$0.894940\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	38.0618i		−	1.30017i	−0.759863	−	0.650084i	$-0.774734\pi$
$857$		−	38.0618i		−	1.30017i	0.759863	−	0.650084i	$-0.225266\pi$
$858$		0			0
$859$	−6.35833			−0.216943			−0.108472	−	0.994100i	$-0.534596\pi$
$859$	−6.35833			−0.216943			−0.108472	+	0.994100i	$0.534596\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			6.61609i			0.225214i	0.993640	+	0.112607i	$0.0359202\pi$
$863$			6.61609i			0.225214i	−0.993640	+	0.112607i	$0.964080\pi$
$864$		0			0
$865$	7.26747	+	45.9371i	0.247101	+	1.56191i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.7548			1.14505
$870$		0			0
$871$	24.4049			0.826929
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.4845	+	22.5690i	0.388248	+	0.762971i
$876$		0			0
$877$			21.5654i			0.728211i	0.931358	+	0.364105i	$0.118625\pi$
$877$			21.5654i			0.728211i	−0.931358	+	0.364105i	$0.881375\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43.1449			1.45359			0.726794	−	0.686856i	$-0.241010\pi$
$881$	43.1449			1.45359			0.726794	+	0.686856i	$0.241010\pi$
$882$		0			0
$883$			18.3306i			0.616874i	0.951245	+	0.308437i	$0.0998060\pi$
$883$			18.3306i			0.616874i	−0.951245	+	0.308437i	$0.900194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	38.0618i		−	1.27799i	−0.769210	−	0.638996i	$-0.779350\pi$
$887$		−	38.0618i		−	1.27799i	0.769210	−	0.638996i	$-0.220650\pi$
$888$		0			0
$889$	29.0148			0.973127
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.00265i			0.0670160i
$894$		0			0
$895$	8.39008	−	1.32735i	0.280449	−	0.0443684i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−28.1449			−0.938684
$900$		0			0
$901$	61.2664			2.04108
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.4676	+	5.76935i	−1.21222	+	0.191780i
$906$		0			0
$907$			29.1705i			0.968590i	0.874905	+	0.484295i	$0.160924\pi$
$907$			29.1705i			0.968590i	−0.874905	+	0.484295i	$0.839076\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.51165			0.116346			0.0581730	−	0.998307i	$-0.481472\pi$
$911$	3.51165			0.116346			0.0581730	+	0.998307i	$0.481472\pi$
$912$		0			0
$913$			37.4523i			1.23949i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			21.5435i			0.711428i
$918$		0			0
$919$	24.2431			0.799708			0.399854	−	0.916579i	$-0.369061\pi$
$919$	24.2431			0.799708			0.399854	+	0.916579i	$0.369061\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	9.39680i		−	0.309300i
$924$		0			0
$925$	4.00000	+	12.3254i	0.131519	+	0.405258i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.73150			−0.286471			−0.143236	−	0.989689i	$-0.545751\pi$
$929$	−8.73150			−0.286471			−0.143236	+	0.989689i	$0.545751\pi$
$930$		0			0
$931$	3.49681			0.114603
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.09445	+	32.2016i	0.166606	+	1.05311i
$936$		0			0
$937$			35.9873i			1.17565i	0.808987	+	0.587826i	$0.200016\pi$
$937$			35.9873i			1.17565i	−0.808987	+	0.587826i	$0.799984\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.6688			0.869378			0.434689	−	0.900581i	$-0.356858\pi$
$941$	26.6688			0.869378			0.434689	+	0.900581i	$0.356858\pi$
$942$		0			0
$943$			21.3969i			0.696778i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	8.05917i		−	0.261888i	−0.991390	−	0.130944i	$-0.958199\pi$
$947$		−	8.05917i		−	0.261888i	0.991390	−	0.130944i	$-0.0418008\pi$
$948$		0			0
$949$	19.4799			0.632344
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	23.7733i		−	0.770092i	−0.922897	−	0.385046i	$-0.874186\pi$
$953$		−	23.7733i		−	0.770092i	0.922897	−	0.385046i	$-0.125814\pi$
$954$		0			0
$955$	−25.1374	+	3.97686i	−0.813429	+	0.128688i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	26.0317			0.840609
$960$		0			0
$961$	57.0148			1.83919
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.92143	−	31.1079i	−0.158426	−	1.00140i
$966$		0			0
$967$		−	33.7223i		−	1.08444i	−0.840238	−	0.542218i	$-0.817585\pi$
$967$		−	33.7223i		−	1.08444i	0.840238	−	0.542218i	$-0.182415\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.1766			−1.16096			−0.580481	−	0.814273i	$-0.697136\pi$
$971$	−36.1766			−1.16096			−0.580481	+	0.814273i	$0.697136\pi$
$972$		0			0
$973$			26.0734i			0.835874i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			50.0722i			1.60195i	0.598697	+	0.800976i	$0.295685\pi$
$977$			50.0722i			1.60195i	−0.598697	+	0.800976i	$0.704315\pi$
$978$		0			0
$979$	−34.1766			−1.09229
$980$		0			0
$981$		0			0
$982$		0			0
$983$			40.3941i			1.28837i	0.764869	+	0.644186i	$0.222804\pi$
$983$			40.3941i			1.28837i	−0.764869	+	0.644186i	$0.777196\pi$
$984$		0			0
$985$	2.76325	+	17.4663i	0.0880443	+	0.556521i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	55.0970			1.75198
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.13003	−	0.653390i	0.130931	−	0.0207138i
$996$		0			0
$997$		−	42.9506i		−	1.36026i	−0.733091	−	0.680130i	$-0.761923\pi$
$997$		−	42.9506i		−	1.36026i	0.733091	−	0.680130i	$-0.238077\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.d.d.649.5		6
3.2	odd	2		1620.2.d.c.649.2		6
5.2	odd	4		8100.2.a.bd.1.2		6
5.3	odd	4		8100.2.a.bd.1.5		6
5.4	even	2	inner	1620.2.d.d.649.6		6
9.2	odd	6		180.2.r.a.49.2	✓	12
9.4	even	3		540.2.r.a.289.2		12
9.5	odd	6		180.2.r.a.169.5	yes	12
9.7	even	3		540.2.r.a.469.3		12
15.2	even	4		8100.2.a.bc.1.2		6
15.8	even	4		8100.2.a.bc.1.5		6
15.14	odd	2		1620.2.d.c.649.1		6
36.7	odd	6		2160.2.by.e.1009.3		12
36.11	even	6		720.2.by.e.49.5		12
36.23	even	6		720.2.by.e.529.2		12
36.31	odd	6		2160.2.by.e.289.2		12
45.2	even	12		900.2.i.f.301.5		12
45.4	even	6		540.2.r.a.289.3		12
45.7	odd	12		2700.2.i.f.901.5		12
45.13	odd	12		2700.2.i.f.1801.2		12
45.14	odd	6		180.2.r.a.169.2	yes	12
45.22	odd	12		2700.2.i.f.1801.5		12
45.23	even	12		900.2.i.f.601.2		12
45.29	odd	6		180.2.r.a.49.5	yes	12
45.32	even	12		900.2.i.f.601.5		12
45.34	even	6		540.2.r.a.469.2		12
45.38	even	12		900.2.i.f.301.2		12
45.43	odd	12		2700.2.i.f.901.2		12
180.59	even	6		720.2.by.e.529.5		12
180.79	odd	6		2160.2.by.e.1009.2		12
180.119	even	6		720.2.by.e.49.2		12
180.139	odd	6		2160.2.by.e.289.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.2	✓	12	9.2	odd	6
180.2.r.a.49.5	yes	12	45.29	odd	6
180.2.r.a.169.2	yes	12	45.14	odd	6
180.2.r.a.169.5	yes	12	9.5	odd	6
540.2.r.a.289.2		12	9.4	even	3
540.2.r.a.289.3		12	45.4	even	6
540.2.r.a.469.2		12	45.34	even	6
540.2.r.a.469.3		12	9.7	even	3
720.2.by.e.49.2		12	180.119	even	6
720.2.by.e.49.5		12	36.11	even	6
720.2.by.e.529.2		12	36.23	even	6
720.2.by.e.529.5		12	180.59	even	6
900.2.i.f.301.2		12	45.38	even	12
900.2.i.f.301.5		12	45.2	even	12
900.2.i.f.601.2		12	45.23	even	12
900.2.i.f.601.5		12	45.32	even	12
1620.2.d.c.649.1		6	15.14	odd	2
1620.2.d.c.649.2		6	3.2	odd	2
1620.2.d.d.649.5		6	1.1	even	1	trivial
1620.2.d.d.649.6		6	5.4	even	2	inner
2160.2.by.e.289.2		12	36.31	odd	6
2160.2.by.e.289.3		12	180.139	odd	6
2160.2.by.e.1009.2		12	180.79	odd	6
2160.2.by.e.1009.3		12	36.7	odd	6
2700.2.i.f.901.2		12	45.43	odd	12
2700.2.i.f.901.5		12	45.7	odd	12
2700.2.i.f.1801.2		12	45.13	odd	12
2700.2.i.f.1801.5		12	45.22	odd	12
8100.2.a.bc.1.2		6	15.2	even	4
8100.2.a.bc.1.5		6	15.8	even	4
8100.2.a.bd.1.2		6	5.2	odd	4
8100.2.a.bd.1.5		6	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.20860 - 0.349411i) q^{5} +2.26496i q^{7} +O(q^{10})\)	\(q+(2.20860 - 0.349411i) q^{5} +2.26496i q^{7} +2.54722 q^{11} -5.02510i q^{13} +5.72392i q^{17} +1.86997 q^{19} -5.39722i q^{23} +(4.75582 - 1.54342i) q^{25} -3.00000 q^{29} +9.38162 q^{31} +(0.791400 + 5.00238i) q^{35} +2.59165i q^{37} -3.96442 q^{41} +10.2084i q^{43} +1.07095i q^{47} +1.86997 q^{49} -10.7036i q^{53} +(5.62580 - 0.890028i) q^{55} -1.58280 q^{59} +0.869975 q^{61} +(-1.75582 - 11.0984i) q^{65} +4.85661i q^{67} +1.86997 q^{71} +3.87652i q^{73} +5.76935i q^{77} +13.2516 q^{79} +14.7032i q^{83} +(2.00000 + 12.6418i) q^{85} -13.4172 q^{89} +11.3816 q^{91} +(4.13003 - 0.653390i) q^{95} -17.6669i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5}+O(q^{10}) \) 6 * q + q^5	\( 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) \) 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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