LMFDB - Embedded newform 2052.3.m.a.881.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.19
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.497329i q^{5} +(5.73504 + 9.93338i) q^{7} +O(q^{10})$	$q-0.497329i q^{5} +(5.73504 + 9.93338i) q^{7} +(7.37787 - 4.25962i) q^{11} +(8.81568 + 15.2692i) q^{13} +(16.4117 - 9.47532i) q^{17} +(8.90449 - 16.7842i) q^{19} +(-0.354632 + 0.204747i) q^{23} +24.7527 q^{25} -38.4425i q^{29} +(9.63763 - 16.6929i) q^{31} +(4.94016 - 2.85220i) q^{35} -43.9804 q^{37} -29.4070i q^{41} +(-4.11810 + 7.13276i) q^{43} +41.4322i q^{47} +(-41.2813 + 71.5014i) q^{49} +(79.4128 + 45.8490i) q^{53} +(-2.11843 - 3.66923i) q^{55} +22.8606i q^{59} -42.6238 q^{61} +(7.59383 - 4.38430i) q^{65} +(-20.7667 - 35.9690i) q^{67} +(61.3593 - 35.4258i) q^{71} +(-6.81675 - 11.8070i) q^{73} +(84.6248 + 48.8581i) q^{77} +(-27.0355 + 46.8269i) q^{79} +(63.5614 - 36.6972i) q^{83} +(-4.71236 - 8.16204i) q^{85} +(-10.0596 - 5.80794i) q^{89} +(-101.117 + 175.139i) q^{91} +(-8.34729 - 4.42846i) q^{95} +(35.5215 - 61.5250i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$		−	0.497329i		−	0.0994659i	−0.998763	−	0.0497329i	$-0.984163\pi$
$5$		−	0.497329i		−	0.0994659i	0.998763	−	0.0497329i	$-0.0158370\pi$
$6$		0			0
$7$	5.73504	+	9.93338i	0.819291	+	1.41905i	0.906205	+	0.422838i	$0.138966\pi$
$7$	5.73504	+	9.93338i	0.819291	+	1.41905i	−0.0869140	+	0.996216i	$0.527701\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.37787	−	4.25962i	0.670716	−	0.387238i	−0.125632	−	0.992077i	$-0.540096\pi$
$11$	7.37787	−	4.25962i	0.670716	−	0.387238i	0.796348	+	0.604839i	$0.206763\pi$
$12$		0			0
$13$	8.81568	+	15.2692i	0.678130	+	1.17455i	0.975544	+	0.219806i	$0.0705425\pi$
$13$	8.81568	+	15.2692i	0.678130	+	1.17455i	−0.297414	+	0.954749i	$0.596124\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.4117	−	9.47532i	0.965397	−	0.557372i	0.0675668	−	0.997715i	$-0.478476\pi$
$17$	16.4117	−	9.47532i	0.965397	−	0.557372i	0.897830	+	0.440343i	$0.145143\pi$
$18$		0			0
$19$	8.90449	−	16.7842i	0.468657	−	0.883380i
$19$	8.90449	−	16.7842i	0.468657	−	0.883380i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.354632	+	0.204747i	−0.0154188	+	0.00890204i	−0.507690	−	0.861540i	$-0.669500\pi$
$23$	−0.354632	+	0.204747i	−0.0154188	+	0.00890204i	0.492271	+	0.870442i	$0.336167\pi$
$24$		0			0
$25$	24.7527			0.990107
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	38.4425i		−	1.32560i	−0.748796	−	0.662801i	$-0.769368\pi$
$29$		−	38.4425i		−	1.32560i	0.748796	−	0.662801i	$-0.230632\pi$
$30$		0			0
$31$	9.63763	−	16.6929i	0.310891	−	0.538479i	−0.667664	−	0.744462i	$-0.732706\pi$
$31$	9.63763	−	16.6929i	0.310891	−	0.538479i	0.978556	+	0.205983i	$0.0660392\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.94016	−	2.85220i	0.141147	−	0.0814915i
$36$		0			0
$37$	−43.9804			−1.18866			−0.594329	−	0.804222i	$-0.702582\pi$
$37$	−43.9804			−1.18866			−0.594329	+	0.804222i	$0.702582\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	29.4070i		−	0.717243i	−0.933483	−	0.358621i	$-0.883247\pi$
$41$		−	29.4070i		−	0.717243i	0.933483	−	0.358621i	$-0.116753\pi$
$42$		0			0
$43$	−4.11810	+	7.13276i	−0.0957698	+	0.165878i	−0.909930	−	0.414763i	$-0.863865\pi$
$43$	−4.11810	+	7.13276i	−0.0957698	+	0.165878i	0.814160	+	0.580641i	$0.197198\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			41.4322i			0.881535i	0.897621	+	0.440768i	$0.145294\pi$
$47$			41.4322i			0.881535i	−0.897621	+	0.440768i	$0.854706\pi$
$48$		0			0
$49$	−41.2813	+	71.5014i	−0.842476	+	1.45921i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	79.4128	+	45.8490i	1.49836	+	0.865076i	0.999998	−	0.00189596i	$-0.000603502\pi$
$53$	79.4128	+	45.8490i	1.49836	+	0.865076i	0.498357	+	0.866972i	$0.333937\pi$
$54$		0			0
$55$	−2.11843	−	3.66923i	−0.0385170	−	0.0667134i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			22.8606i			0.387467i	0.981054	+	0.193734i	$0.0620597\pi$
$59$			22.8606i			0.387467i	−0.981054	+	0.193734i	$0.937940\pi$
$60$		0			0
$61$	−42.6238			−0.698751			−0.349375	−	0.936983i	$-0.613606\pi$
$61$	−42.6238			−0.698751			−0.349375	+	0.936983i	$0.613606\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.59383	−	4.38430i	0.116828	−	0.0674508i
$66$		0			0
$67$	−20.7667	−	35.9690i	−0.309951	−	0.536851i	0.668400	−	0.743802i	$-0.266979\pi$
$67$	−20.7667	−	35.9690i	−0.309951	−	0.536851i	−0.978351	+	0.206950i	$0.933646\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	61.3593	−	35.4258i	0.864216	−	0.498955i	−0.00120604	−	0.999999i	$-0.500384\pi$
$71$	61.3593	−	35.4258i	0.864216	−	0.498955i	0.865422	+	0.501044i	$0.167051\pi$
$72$		0			0
$73$	−6.81675	−	11.8070i	−0.0933801	−	0.161739i	0.815551	−	0.578685i	$-0.196434\pi$
$73$	−6.81675	−	11.8070i	−0.0933801	−	0.161739i	−0.908931	+	0.416946i	$0.863101\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	84.6248	+	48.8581i	1.09902	+	0.634521i
$78$		0			0
$79$	−27.0355	+	46.8269i	−0.342222	+	0.592746i	−0.984845	−	0.173436i	$-0.944513\pi$
$79$	−27.0355	+	46.8269i	−0.342222	+	0.592746i	0.642623	+	0.766183i	$0.277846\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	63.5614	−	36.6972i	0.765801	−	0.442135i	−0.0655740	−	0.997848i	$-0.520888\pi$
$83$	63.5614	−	36.6972i	0.765801	−	0.442135i	0.831374	+	0.555713i	$0.187554\pi$
$84$		0			0
$85$	−4.71236	−	8.16204i	−0.0554395	−	0.0960240i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0596	−	5.80794i	−0.113030	−	0.0652577i	0.442419	−	0.896808i	$-0.354120\pi$
$89$	−10.0596	−	5.80794i	−0.113030	−	0.0652577i	−0.555449	+	0.831551i	$0.687454\pi$
$90$		0			0
$91$	−101.117	+	175.139i	−1.11117	+	1.92461i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.34729	−	4.42846i	−0.0878662	−	0.0466154i
$96$		0			0
$97$	35.5215	−	61.5250i	0.366201	−	0.634278i	−0.622767	−	0.782407i	$-0.713992\pi$
$97$	35.5215	−	61.5250i	0.366201	−	0.634278i	0.988968	+	0.148129i	$0.0473250\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			90.2859i			0.893920i	0.894554	+	0.446960i	$0.147493\pi$
$101$			90.2859i			0.893920i	−0.894554	+	0.446960i	$0.852507\pi$
$102$		0			0
$103$	36.6465	−	63.4737i	0.355792	−	0.616249i	−0.631461	−	0.775407i	$-0.717545\pi$
$103$	36.6465	−	63.4737i	0.355792	−	0.616249i	0.987253	+	0.159158i	$0.0508779\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			126.872i			1.18572i	0.805307	+	0.592858i	$0.202001\pi$
$107$			126.872i			1.18572i	−0.805307	+	0.592858i	$0.797999\pi$
$108$		0			0
$109$	48.3118	+	83.6785i	0.443227	+	0.767692i	0.997927	−	0.0643585i	$-0.0205001\pi$
$109$	48.3118	+	83.6785i	0.443227	+	0.767692i	−0.554700	+	0.832051i	$0.687167\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−161.886	−	93.4652i	−1.43262	−	0.827125i	−0.435303	−	0.900284i	$-0.643359\pi$
$113$	−161.886	−	93.4652i	−1.43262	−	0.827125i	−0.997320	+	0.0731588i	$0.976692\pi$
$114$		0			0
$115$	0.101827	+	0.176369i	0.000885450	+	0.00153364i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	188.244	+	108.683i	1.58188	+	0.913300i
$120$		0			0
$121$	−24.2113	+	41.9352i	−0.200093	+	0.346572i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	24.7435i		−	0.197948i
$126$		0			0
$127$	73.2518	−	126.876i	0.576786	−	0.999022i	−0.419060	−	0.907959i	$-0.637640\pi$
$127$	73.2518	−	126.876i	0.576786	−	0.999022i	0.995845	−	0.0910631i	$-0.0290265\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	89.7482i		−	0.685101i	−0.939499	−	0.342550i	$-0.888709\pi$
$131$		−	89.7482i		−	0.685101i	0.939499	−	0.342550i	$-0.111291\pi$
$132$		0			0
$133$	217.792	−	7.80653i	1.63753	−	0.0586957i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	74.0234i		−	0.540317i	−0.962816	−	0.270158i	$-0.912924\pi$
$137$		−	74.0234i		−	0.540317i	0.962816	−	0.270158i	$-0.0870760\pi$
$138$		0			0
$139$	123.835	+	214.489i	0.890902	+	1.54309i	0.838796	+	0.544446i	$0.183260\pi$
$139$	123.835	+	214.489i	0.890902	+	1.54309i	0.0521064	+	0.998642i	$0.483407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	130.082	+	75.1029i	0.909665	+	0.525195i
$144$		0			0
$145$	−19.1186			−0.131852
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	93.9704i		−	0.630674i	−0.948980	−	0.315337i	$-0.897882\pi$
$149$		−	93.9704i		−	0.630674i	0.948980	−	0.315337i	$-0.102118\pi$
$150$		0			0
$151$	−3.13122	−	5.42344i	−0.0207366	−	0.0359168i	0.855471	−	0.517851i	$-0.173268\pi$
$151$	−3.13122	−	5.42344i	−0.0207366	−	0.0359168i	−0.876207	+	0.481934i	$0.839934\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.30185	−	4.79308i	−0.0535603	−	0.0309231i
$156$		0			0
$157$	14.9879			0.0954644			0.0477322	−	0.998860i	$-0.484801\pi$
$157$	14.9879			0.0954644			0.0477322	+	0.998860i	$0.484801\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.06766	−	2.34846i	−0.0252650	−	0.0145867i
$162$		0			0
$163$	0.718713			0.00440928			0.00220464	−	0.999998i	$-0.499298\pi$
$163$	0.718713			0.00440928			0.00220464	+	0.999998i	$0.499298\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	277.850	−	160.417i	1.66377	−	0.960580i	0.692886	−	0.721047i	$-0.256339\pi$
$167$	277.850	−	160.417i	1.66377	−	0.960580i	0.970888	−	0.239533i	$-0.0769944\pi$
$168$		0			0
$169$	−70.9326	+	122.859i	−0.419720	+	0.726976i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−121.498	−	70.1467i	−0.702299	−	0.405473i	0.105904	−	0.994376i	$-0.466226\pi$
$173$	−121.498	−	70.1467i	−0.702299	−	0.405473i	−0.808203	+	0.588904i	$0.799560\pi$
$174$		0			0
$175$	141.957	+	245.878i	0.811186	+	1.40501i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	91.6384i		−	0.511946i	−0.966684	−	0.255973i	$-0.917604\pi$
$179$		−	91.6384i		−	0.511946i	0.966684	−	0.255973i	$-0.0823959\pi$
$180$		0			0
$181$	−122.574	+	212.304i	−0.677203	+	1.17295i	0.298617	+	0.954373i	$0.403475\pi$
$181$	−122.574	+	212.304i	−0.677203	+	1.17295i	−0.975820	+	0.218577i	$0.929859\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			21.8727i			0.118231i
$186$		0			0
$187$	80.7225	−	139.816i	0.431671	−	0.747676i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−302.171	+	174.459i	−1.58205	+	0.913396i	−0.587487	+	0.809233i	$0.699883\pi$
$191$	−302.171	+	174.459i	−1.58205	+	0.913396i	−0.994560	+	0.104162i	$0.966784\pi$
$192$		0			0
$193$	122.335			0.633858			0.316929	−	0.948449i	$-0.397348\pi$
$193$	122.335			0.633858			0.316929	+	0.948449i	$0.397348\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			55.4953i			0.281702i	0.990031	+	0.140851i	$0.0449839\pi$
$197$			55.4953i			0.281702i	−0.990031	+	0.140851i	$0.955016\pi$
$198$		0			0
$199$	−20.1790	+	34.9511i	−0.101402	+	0.175633i	−0.912263	−	0.409606i	$-0.865666\pi$
$199$	−20.1790	+	34.9511i	−0.101402	+	0.175633i	0.810861	+	0.585239i	$0.198999\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	381.863	−	220.469i	1.88110	−	1.08605i
$204$		0			0
$205$	−14.6249			−0.0713412
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.79819	−	161.762i	−0.0277425	−	0.773979i
$210$		0			0
$211$	284.667			1.34913			0.674565	−	0.738215i	$-0.264331\pi$
$211$	284.667			1.34913			0.674565	+	0.738215i	$0.264331\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.54733	+	2.04805i	0.0164992	+	0.00952583i
$216$		0			0
$217$	221.089			1.01884
$218$		0			0
$219$		0			0
$220$		0			0
$221$	289.361	+	167.063i	1.30933	+	0.755941i
$222$		0			0
$223$	99.7841	−	172.831i	0.447463	−	0.775028i	−0.550758	−	0.834665i	$-0.685661\pi$
$223$	99.7841	−	172.831i	0.447463	−	0.775028i	0.998220	+	0.0596374i	$0.0189944\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−373.638	+	215.720i	−1.64598	+	0.950307i	−0.667334	+	0.744759i	$0.732564\pi$
$227$	−373.638	+	215.720i	−1.64598	+	0.950307i	−0.978647	+	0.205548i	$0.934102\pi$
$228$		0			0
$229$	−120.604	+	208.892i	−0.526655	+	0.912193i	0.472862	+	0.881136i	$0.343221\pi$
$229$	−120.604	+	208.892i	−0.526655	+	0.912193i	−0.999518	+	0.0310572i	$0.990113\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	51.7582	−	29.8826i	0.222138	−	0.128252i	−0.384802	−	0.922999i	$-0.625730\pi$
$233$	51.7582	−	29.8826i	0.222138	−	0.128252i	0.606940	+	0.794748i	$0.292397\pi$
$234$		0			0
$235$	20.6054			0.0876827
$236$		0			0
$237$		0			0
$238$		0			0
$239$	321.259	+	185.479i	1.34418	+	0.776062i	0.987418	−	0.158134i	$-0.0505479\pi$
$239$	321.259	+	185.479i	1.34418	+	0.776062i	0.356760	+	0.934196i	$0.383881\pi$
$240$		0			0
$241$	233.378			0.968374			0.484187	−	0.874965i	$-0.339116\pi$
$241$	233.378			0.968374			0.484187	+	0.874965i	$0.339116\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	35.5597	+	20.5304i	0.145142	+	0.0837976i
$246$		0			0
$247$	334.781	−	11.9999i	1.35539	−	0.0485826i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−284.054	−	163.999i	−1.13169	−	0.653381i	−0.187330	−	0.982297i	$-0.559983\pi$
$251$	−284.054	−	163.999i	−1.13169	−	0.653381i	−0.944359	+	0.328916i	$0.893317\pi$
$252$		0			0
$253$	−1.74429	+	3.02120i	−0.00689442	+	0.0119415i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−271.570	+	156.791i	−1.05669	+	0.610083i	−0.924516	−	0.381143i	$-0.875530\pi$
$257$	−271.570	+	156.791i	−1.05669	+	0.610083i	−0.132178	+	0.991226i	$0.542197\pi$
$258$		0			0
$259$	−252.229	−	436.874i	−0.973858	−	1.68677i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−366.591	−	211.651i	−1.39388	−	0.804758i	−0.400140	−	0.916454i	$-0.631038\pi$
$263$	−366.591	−	211.651i	−1.39388	−	0.804758i	−0.993742	+	0.111696i	$0.964372\pi$
$264$		0			0
$265$	22.8021	−	39.4943i	0.0860455	−	0.149035i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	286.799	−	165.584i	1.06617	−	0.615553i	0.139037	−	0.990287i	$-0.455599\pi$
$269$	286.799	−	165.584i	1.06617	−	0.615553i	0.927132	+	0.374734i	$0.122266\pi$
$270$		0			0
$271$	136.655	+	236.694i	0.504263	+	0.873409i	0.999988	+	0.00492935i	$0.00156907\pi$
$271$	136.655	+	236.694i	0.504263	+	0.873409i	−0.495725	+	0.868480i	$0.665098\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	182.622	−	105.437i	0.664080	−	0.383407i
$276$		0			0
$277$	146.869	+	254.384i	0.530213	+	0.918355i	0.999379	+	0.0352453i	$0.0112212\pi$
$277$	146.869	+	254.384i	0.530213	+	0.918355i	−0.469166	+	0.883110i	$0.655445\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			220.492i			0.784670i	0.919822	+	0.392335i	$0.128333\pi$
$281$			220.492i			0.784670i	−0.919822	+	0.392335i	$0.871667\pi$
$282$		0			0
$283$	−172.057			−0.607974			−0.303987	−	0.952676i	$-0.598318\pi$
$283$	−172.057			−0.607974			−0.303987	+	0.952676i	$0.598318\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	292.110	−	168.650i	1.01781	−	0.587631i
$288$		0			0
$289$	35.0635	−	60.7318i	0.121327	−	0.210145i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	387.752	+	223.869i	1.32339	+	0.764057i	0.984267	−	0.176686i	$-0.0565377\pi$
$293$	387.752	+	223.869i	1.32339	+	0.764057i	0.339119	+	0.940743i	$0.389871\pi$
$294$		0			0
$295$	11.3692			0.0385398
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.25265	−	3.60997i	−0.0209119	−	0.0120735i
$300$		0			0
$301$	−94.4699			−0.313854
$302$		0			0
$303$		0			0
$304$		0			0
$305$			21.1981i			0.0695019i
$306$		0			0
$307$	−98.5500	−	170.694i	−0.321010	−	0.556005i	0.659687	−	0.751541i	$-0.270689\pi$
$307$	−98.5500	−	170.694i	−0.321010	−	0.556005i	−0.980697	+	0.195535i	$0.937356\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	143.204	+	82.6790i	0.460464	+	0.265849i	0.712239	−	0.701937i	$-0.247681\pi$
$311$	143.204	+	82.6790i	0.460464	+	0.265849i	−0.251776	+	0.967786i	$0.581014\pi$
$312$		0			0
$313$	−482.577			−1.54178			−0.770890	−	0.636968i	$-0.780188\pi$
$313$	−482.577			−1.54178			−0.770890	+	0.636968i	$0.780188\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			49.4053i			0.155853i	0.996959	+	0.0779264i	$0.0248299\pi$
$317$			49.4053i			0.155853i	−0.996959	+	0.0779264i	$0.975170\pi$
$318$		0			0
$319$	−163.750	−	283.624i	−0.513323	−	0.889102i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.8978	−	359.831i	−0.0399313	−	1.11403i
$324$		0			0
$325$	218.212	+	377.954i	0.671421	+	1.16293i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−411.561	+	237.615i	−1.25095	+	0.722234i
$330$		0			0
$331$	74.7326	+	129.441i	0.225778	+	0.391059i	0.956553	−	0.291560i	$-0.0941742\pi$
$331$	74.7326	+	129.441i	0.225778	+	0.391059i	−0.730774	+	0.682619i	$0.760841\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.8885	+	10.3279i	−0.0533984	+	0.0308296i
$336$		0			0
$337$	528.937			1.56955			0.784773	−	0.619784i	$-0.212780\pi$
$337$	528.937			1.56955			0.784773	+	0.619784i	$0.212780\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	164.210i		−	0.481556i
$342$		0			0
$343$	−384.966			−1.12235
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	410.266i		−	1.18232i	−0.806553	−	0.591161i	$-0.798670\pi$
$347$		−	410.266i		−	1.18232i	0.806553	−	0.591161i	$-0.201330\pi$
$348$		0			0
$349$	284.123	+	492.115i	0.814105	+	1.41007i	0.909969	+	0.414677i	$0.136105\pi$
$349$	284.123	+	492.115i	0.814105	+	1.41007i	−0.0958635	+	0.995394i	$0.530561\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−275.399	+	159.002i	−0.780167	+	0.450430i	−0.836490	−	0.547983i	$-0.815396\pi$
$353$	−275.399	+	159.002i	−0.780167	+	0.450430i	0.0563223	+	0.998413i	$0.482063\pi$
$354$		0			0
$355$	−17.6183	−	30.5158i	−0.0496290	−	0.0859600i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−400.232	+	231.074i	−1.11485	+	0.643661i	−0.940082	−	0.340948i	$-0.889252\pi$
$359$	−400.232	+	231.074i	−1.11485	+	0.643661i	−0.174771	+	0.984609i	$0.555919\pi$
$360$		0			0
$361$	−202.420	−	298.910i	−0.560721	−	0.828005i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.87195	+	3.39017i	−0.0160875	+	0.00928814i
$366$		0			0
$367$	−554.503			−1.51091			−0.755454	−	0.655201i	$-0.772584\pi$
$367$	−554.503			−1.51091			−0.755454	+	0.655201i	$0.772584\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1051.78i			2.83500i
$372$		0			0
$373$	−297.536	+	515.347i	−0.797682	+	1.38163i	0.123439	+	0.992352i	$0.460608\pi$
$373$	−297.536	+	515.347i	−0.797682	+	1.38163i	−0.921122	+	0.389274i	$0.872726\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	586.986	−	338.897i	1.55699	−	0.898930i
$378$		0			0
$379$	−459.057			−1.21123			−0.605616	−	0.795757i	$-0.707073\pi$
$379$	−459.057			−1.21123			−0.605616	+	0.795757i	$0.707073\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			306.676i			0.800720i	0.916358	+	0.400360i	$0.131115\pi$
$383$			306.676i			0.800720i	−0.916358	+	0.400360i	$0.868885\pi$
$384$		0			0
$385$	24.2986	−	42.0864i	0.0631132	−	0.109315i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			152.250i			0.391389i	0.980665	+	0.195694i	$0.0626961\pi$
$389$			152.250i			0.391389i	−0.980665	+	0.195694i	$0.937304\pi$
$390$		0			0
$391$	−3.88009	+	6.72051i	−0.00992350	+	0.0171880i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	23.2884	+	13.4456i	0.0589580	+	0.0340394i
$396$		0			0
$397$	106.852	+	185.072i	0.269148	+	0.466177i	0.968642	−	0.248461i	$-0.0799247\pi$
$397$	106.852	+	185.072i	0.269148	+	0.466177i	−0.699494	+	0.714638i	$0.746591\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			276.604i			0.689786i	0.938642	+	0.344893i	$0.112085\pi$
$401$			276.604i			0.689786i	−0.938642	+	0.344893i	$0.887915\pi$
$402$		0			0
$403$	339.849			0.843298
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−324.482	+	187.340i	−0.797252	+	0.460294i
$408$		0			0
$409$	−182.327	−	315.800i	−0.445787	−	0.772126i	0.552319	−	0.833633i	$-0.313743\pi$
$409$	−182.327	−	315.800i	−0.445787	−	0.772126i	−0.998107	+	0.0615063i	$0.980410\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−227.083	+	131.106i	−0.549837	+	0.317448i
$414$		0			0
$415$	−18.2506	−	31.6110i	−0.0439774	−	0.0761710i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	448.751	+	259.086i	1.07100	+	0.618344i	0.928456	−	0.371444i	$-0.121137\pi$
$419$	448.751	+	259.086i	1.07100	+	0.618344i	0.142548	+	0.989788i	$0.454470\pi$
$420$		0			0
$421$	−178.083	+	308.449i	−0.423001	+	0.732659i	−0.996231	−	0.0867352i	$-0.972357\pi$
$421$	−178.083	+	308.449i	−0.423001	+	0.732659i	0.573231	+	0.819394i	$0.305690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	406.234	−	234.539i	0.955845	−	0.551858i
$426$		0			0
$427$	−244.449	−	423.398i	−0.572480	−	0.991565i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−733.752	−	423.632i	−1.70244	−	0.982904i	−0.943279	−	0.332002i	$-0.892276\pi$
$431$	−733.752	−	423.632i	−1.70244	−	0.982904i	−0.759161	−	0.650902i	$-0.774391\pi$
$432$		0			0
$433$	22.9957	−	39.8296i	0.0531077	−	0.0919853i	−0.838249	−	0.545287i	$-0.816421\pi$
$433$	22.9957	−	39.8296i	0.0531077	−	0.0919853i	0.891357	+	0.453302i	$0.149754\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.278702	+	7.77539i	0.000637761	+	0.0177927i
$438$		0			0
$439$	178.960	−	309.968i	0.407653	−	0.706076i	−0.586973	−	0.809607i	$-0.699680\pi$
$439$	178.960	−	309.968i	0.407653	−	0.706076i	0.994626	+	0.103530i	$0.0330138\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	192.136i		−	0.433717i	−0.976203	−	0.216858i	$-0.930419\pi$
$443$		−	192.136i		−	0.433717i	0.976203	−	0.216858i	$-0.0695810\pi$
$444$		0			0
$445$	−2.88846	+	5.00296i	−0.00649092	+	0.0112426i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	332.574i		−	0.740700i	−0.928892	−	0.370350i	$-0.879238\pi$
$449$		−	332.574i		−	0.740700i	0.928892	−	0.370350i	$-0.120762\pi$
$450$		0			0
$451$	−125.262	−	216.961i	−0.277744	−	0.481066i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	87.1018	+	50.2883i	0.191433	+	0.110524i
$456$		0			0
$457$	391.702	+	678.447i	0.857115	+	1.48457i	0.874669	+	0.484721i	$0.161079\pi$
$457$	391.702	+	678.447i	0.857115	+	1.48457i	−0.0175537	+	0.999846i	$0.505588\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16.7213	−	9.65408i	−0.0362719	−	0.0209416i	0.481754	−	0.876306i	$-0.340000\pi$
$461$	−16.7213	−	9.65408i	−0.0362719	−	0.0209416i	−0.518026	+	0.855365i	$0.673333\pi$
$462$		0			0
$463$	−108.494	+	187.918i	−0.234329	+	0.405870i	−0.959078	−	0.283144i	$-0.908623\pi$
$463$	−108.494	+	187.918i	−0.234329	+	0.405870i	0.724748	+	0.689014i	$0.241956\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	472.704i		−	1.01221i	−0.862471	−	0.506107i	$-0.831084\pi$
$467$		−	472.704i		−	1.01221i	0.862471	−	0.506107i	$-0.168916\pi$
$468$		0			0
$469$	238.196	−	412.568i	0.507881	−	0.879675i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			70.1662i			0.148343i
$474$		0			0
$475$	220.410	−	415.454i	0.464021	−	0.874640i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	729.153i		−	1.52224i	−0.648611	−	0.761120i	$-0.724650\pi$
$479$		−	729.153i		−	1.52224i	0.648611	−	0.761120i	$-0.275350\pi$
$480$		0			0
$481$	−387.717	−	671.546i	−0.806065	−	1.39615i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.5982	−	17.6659i	−0.0630891	−	0.0364245i
$486$		0			0
$487$	346.901			0.712323			0.356162	−	0.934424i	$-0.384085\pi$
$487$	346.901			0.712323			0.356162	+	0.934424i	$0.384085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	956.995i		−	1.94907i	−0.224229	−	0.974536i	$-0.571986\pi$
$491$		−	956.995i		−	1.94907i	0.224229	−	0.974536i	$-0.428014\pi$
$492$		0			0
$493$	−364.255	−	630.908i	−0.738853	−	1.27973i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	703.796	+	406.337i	1.41609	+	0.817579i
$498$		0			0
$499$	−620.566			−1.24362			−0.621810	−	0.783168i	$-0.713602\pi$
$499$	−620.566			−1.24362			−0.621810	+	0.783168i	$0.713602\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.5529	+	15.9077i	0.0547771	+	0.0316256i	0.527138	−	0.849779i	$-0.323265\pi$
$503$	27.5529	+	15.9077i	0.0547771	+	0.0316256i	−0.472361	+	0.881405i	$0.656598\pi$
$504$		0			0
$505$	44.9019			0.0889146
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.37872	−	4.83746i	0.0164611	−	0.00950385i	−0.491747	−	0.870738i	$-0.663641\pi$
$509$	8.37872	−	4.83746i	0.0164611	−	0.00950385i	0.508208	+	0.861234i	$0.330308\pi$
$510$		0			0
$511$	78.1886	−	135.427i	0.153011	−	0.265023i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−31.5673	−	18.2254i	−0.0612958	−	0.0353891i
$516$		0			0
$517$	176.485	+	305.681i	0.341364	+	0.591260i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	758.567i		−	1.45598i	−0.685587	−	0.727991i	$-0.740454\pi$
$521$		−	758.567i		−	1.45598i	0.685587	−	0.727991i	$-0.259546\pi$
$522$		0			0
$523$	340.486	−	589.739i	0.651024	−	1.12761i	−0.331850	−	0.943332i	$-0.607673\pi$
$523$	340.486	−	589.739i	0.651024	−	1.12761i	0.982875	−	0.184275i	$-0.0589938\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	365.279i		−	0.693128i
$528$		0			0
$529$	−264.416	+	457.982i	−0.499842	+	0.865751i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	449.021	−	259.242i	0.842441	−	0.486384i
$534$		0			0
$535$	63.0970			0.117938
$536$		0			0
$537$		0			0
$538$		0			0
$539$			703.371i			1.30496i
$540$		0			0
$541$	−352.751	+	610.983i	−0.652035	+	1.12936i	0.330593	+	0.943774i	$0.392751\pi$
$541$	−352.751	+	610.983i	−0.652035	+	1.12936i	−0.982628	+	0.185585i	$0.940582\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	41.6158	−	24.0269i	0.0763592	−	0.0440860i
$546$		0			0
$547$	−184.307			−0.336942			−0.168471	−	0.985707i	$-0.553883\pi$
$547$	−184.307			−0.336942			−0.168471	+	0.985707i	$0.553883\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−645.227	−	342.310i	−1.17101	−	0.621253i
$552$		0			0
$553$	−620.200			−1.12152
$554$		0			0
$555$		0			0
$556$		0			0
$557$	406.427	+	234.651i	0.729672	+	0.421276i	0.818302	−	0.574788i	$-0.194916\pi$
$557$	406.427	+	234.651i	0.729672	+	0.421276i	−0.0886301	+	0.996065i	$0.528249\pi$
$558$		0			0
$559$	−145.216			−0.259777
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−143.084	−	82.6098i	−0.254146	−	0.146731i	0.367515	−	0.930018i	$-0.380209\pi$
$563$	−143.084	−	82.6098i	−0.254146	−	0.146731i	−0.621661	+	0.783286i	$0.713542\pi$
$564$		0			0
$565$	−46.4830	+	80.5109i	−0.0822708	+	0.142497i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−45.8090	+	26.4478i	−0.0805079	+	0.0464813i	−0.539714	−	0.841849i	$-0.681467\pi$
$569$	−45.8090	+	26.4478i	−0.0805079	+	0.0464813i	0.459206	+	0.888330i	$0.348134\pi$
$570$		0			0
$571$	−149.202	+	258.426i	−0.261300	+	0.452585i	−0.966588	−	0.256336i	$-0.917484\pi$
$571$	−149.202	+	258.426i	−0.261300	+	0.452585i	0.705288	+	0.708921i	$0.250818\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.77809	+	5.06803i	−0.0152662	+	0.00881397i
$576$		0			0
$577$	337.690			0.585251			0.292626	−	0.956227i	$-0.405471\pi$
$577$	337.690			0.585251			0.292626	+	0.956227i	$0.405471\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	729.055	+	420.920i	1.25483	+	0.724475i
$582$		0			0
$583$	781.197			1.33996
$584$		0			0
$585$		0			0
$586$		0			0
$587$	339.783	+	196.174i	0.578846	+	0.334197i	0.760675	−	0.649133i	$-0.224868\pi$
$587$	339.783	+	196.174i	0.578846	+	0.334197i	−0.181829	+	0.983330i	$0.558202\pi$
$588$		0			0
$589$	−194.359	−	310.401i	−0.329981	−	0.526997i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−86.2134	−	49.7753i	−0.145385	−	0.0839381i	0.425543	−	0.904938i	$-0.360083\pi$
$593$	−86.2134	−	49.7753i	−0.145385	−	0.0839381i	−0.570928	+	0.821000i	$0.693416\pi$
$594$		0			0
$595$	54.0511	−	93.6193i	0.0908422	−	0.157343i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−967.688	+	558.695i	−1.61551	+	0.932713i	−0.627443	+	0.778662i	$0.715898\pi$
$599$	−967.688	+	558.695i	−1.61551	+	0.932713i	−0.988063	+	0.154050i	$0.950768\pi$
$600$		0			0
$601$	105.661	+	183.010i	0.175808	+	0.304508i	0.940441	−	0.339958i	$-0.110413\pi$
$601$	105.661	+	183.010i	0.175808	+	0.304508i	−0.764633	+	0.644466i	$0.777080\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.8556	+	12.0410i	0.0344721	+	0.0199025i
$606$		0			0
$607$	176.208	−	305.202i	0.290294	−	0.502803i	−0.683585	−	0.729870i	$-0.739580\pi$
$607$	176.208	−	305.202i	0.290294	−	0.502803i	0.973879	+	0.227067i	$0.0729137\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−632.637	+	365.253i	−1.03541	+	0.597795i
$612$		0			0
$613$	−601.259	−	1041.41i	−0.980847	−	1.69888i	−0.659107	−	0.752049i	$-0.729066\pi$
$613$	−601.259	−	1041.41i	−0.980847	−	1.69888i	−0.321740	−	0.946828i	$-0.604268\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−699.402	+	403.800i	−1.13355	+	0.654457i	−0.944826	−	0.327572i	$-0.893769\pi$
$617$	−699.402	+	403.800i	−1.13355	+	0.654457i	−0.188727	+	0.982030i	$0.560436\pi$
$618$		0			0
$619$	−134.510	−	232.978i	−0.217302	−	0.376378i	0.736680	−	0.676241i	$-0.236392\pi$
$619$	−134.510	−	232.978i	−0.217302	−	0.376378i	−0.953982	+	0.299863i	$0.903059\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	133.235i		−	0.213860i
$624$		0			0
$625$	606.511			0.970417
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−721.795	+	416.728i	−1.14753	+	0.662525i
$630$		0			0
$631$	604.102	−	1046.34i	0.957373	−	1.65822i	0.228532	−	0.973537i	$-0.426608\pi$
$631$	604.102	−	1046.34i	0.957373	−	1.65822i	0.728842	−	0.684682i	$-0.240059\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−63.0991	−	36.4303i	−0.0993686	−	0.0573705i
$636$		0			0
$637$	−1455.69			−2.28523
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−631.243	−	364.448i	−0.984779	−	0.568562i	−0.0810691	−	0.996708i	$-0.525833\pi$
$641$	−631.243	−	364.448i	−0.984779	−	0.568562i	−0.903709	+	0.428146i	$0.859167\pi$
$642$		0			0
$643$	44.9271			0.0698711			0.0349356	−	0.999390i	$-0.488877\pi$
$643$	44.9271			0.0698711			0.0349356	+	0.999390i	$0.488877\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1018.65i		−	1.57442i	−0.616686	−	0.787209i	$-0.711525\pi$
$647$		−	1018.65i		−	1.57442i	0.616686	−	0.787209i	$-0.288475\pi$
$648$		0			0
$649$	97.3772	+	168.662i	0.150042	+	0.259880i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1025.90	+	592.301i	1.57105	+	0.907046i	0.996041	+	0.0888993i	$0.0283349\pi$
$653$	1025.90	+	592.301i	1.57105	+	0.907046i	0.575009	+	0.818147i	$0.304998\pi$
$654$		0			0
$655$	−44.6344			−0.0681441
$656$		0			0
$657$		0			0
$658$		0			0
$659$			985.529i			1.49549i	0.663985	+	0.747746i	$0.268864\pi$
$659$			985.529i			1.49549i	−0.663985	+	0.747746i	$0.731136\pi$
$660$		0			0
$661$	8.48723	+	14.7003i	0.0128400	+	0.0222395i	0.872374	−	0.488839i	$-0.162579\pi$
$661$	8.48723	+	14.7003i	0.0128400	+	0.0222395i	−0.859534	+	0.511079i	$0.829246\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.88242	−	108.314i	−0.00583822	−	0.162878i
$666$		0			0
$667$	7.87098	+	13.6329i	0.0118006	+	0.0204392i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−314.473	+	181.561i	−0.468663	+	0.270583i
$672$		0			0
$673$	−356.500	−	617.476i	−0.529718	−	0.917498i	−0.999399	−	0.0346619i	$-0.988965\pi$
$673$	−356.500	−	617.476i	−0.529718	−	0.917498i	0.469681	−	0.882836i	$-0.344369\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−245.650	+	141.826i	−0.362851	+	0.209492i	−0.670331	−	0.742063i	$-0.733848\pi$
$677$	−245.650	+	141.826i	−0.362851	+	0.209492i	0.307480	+	0.951555i	$0.400514\pi$
$678$		0			0
$679$	814.868			1.20010
$680$		0			0
$681$		0			0
$682$		0			0
$683$			168.166i			0.246216i	0.992393	+	0.123108i	$0.0392862\pi$
$683$			168.166i			0.246216i	−0.992393	+	0.123108i	$0.960714\pi$
$684$		0			0
$685$	−36.8140			−0.0537431
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1616.76i			2.34653i
$690$		0			0
$691$	−125.812	−	217.913i	−0.182073	−	0.315360i	0.760513	−	0.649322i	$-0.224947\pi$
$691$	−125.812	−	217.913i	−0.182073	−	0.315360i	−0.942586	+	0.333963i	$0.891614\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	106.672	−	61.5870i	0.153485	−	0.0886144i
$696$		0			0
$697$	−278.640	−	482.619i	−0.399771	−	0.692424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−748.072	+	431.899i	−1.06715	+	0.616119i	−0.927401	−	0.374068i	$-0.877963\pi$
$701$	−748.072	+	431.899i	−1.06715	+	0.616119i	−0.139748	+	0.990187i	$0.544629\pi$
$702$		0			0
$703$	−391.623	+	738.177i	−0.557074	+	1.05004i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−896.844	+	517.793i	−1.26852	+	0.732381i
$708$		0			0
$709$	−574.154			−0.809809			−0.404904	−	0.914359i	$-0.632695\pi$
$709$	−574.154			−0.809809			−0.404904	+	0.914359i	$0.632695\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			7.89310i			0.0110703i
$714$		0			0
$715$	37.3509	−	64.6936i	0.0522390	−	0.0904806i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−73.5548	+	42.4669i	−0.102302	+	0.0590638i	−0.550278	−	0.834982i	$-0.685478\pi$
$719$	−73.5548	+	42.4669i	−0.102302	+	0.0590638i	0.447976	+	0.894045i	$0.352145\pi$
$720$		0			0
$721$	840.677			1.16599
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	951.553i		−	1.31249i
$726$		0			0
$727$	−122.734	+	212.581i	−0.168822	+	0.292408i	−0.938006	−	0.346619i	$-0.887330\pi$
$727$	−122.734	+	212.581i	−0.168822	+	0.292408i	0.769184	+	0.639027i	$0.220663\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			156.081i			0.213518i
$732$		0			0
$733$	−588.181	+	1018.76i	−0.802430	+	1.38985i	0.115582	+	0.993298i	$0.463127\pi$
$733$	−588.181	+	1018.76i	−0.802430	+	1.38985i	−0.918012	+	0.396552i	$0.870207\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−306.429	−	176.917i	−0.415779	−	0.240050i
$738$		0			0
$739$	639.622	+	1107.86i	0.865524	+	1.49913i	0.866526	+	0.499131i	$0.166347\pi$
$739$	639.622	+	1107.86i	0.865524	+	1.49913i	−0.00100273	+	0.999999i	$0.500319\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1161.60i		−	1.56339i	−0.623658	−	0.781697i	$-0.714354\pi$
$743$		−	1161.60i		−	1.56339i	0.623658	−	0.781697i	$-0.285646\pi$
$744$		0			0
$745$	−46.7343			−0.0627305
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1260.26	+	727.614i	−1.68260	+	0.971447i
$750$		0			0
$751$	−390.247	−	675.928i	−0.519637	−	0.900037i	−0.999739	−	0.0228248i	$-0.992734\pi$
$751$	−390.247	−	675.928i	−0.519637	−	0.900037i	0.480103	−	0.877212i	$-0.340599\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.69723	+	1.55725i	−0.00357250	+	0.00206258i
$756$		0			0
$757$	−117.799	−	204.034i	−0.155613	−	0.269529i	0.777669	−	0.628674i	$-0.216402\pi$
$757$	−117.799	−	204.034i	−0.155613	−	0.269529i	−0.933282	+	0.359144i	$0.883069\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−314.907	−	181.812i	−0.413807	−	0.238911i	0.278617	−	0.960402i	$-0.410124\pi$
$761$	−314.907	−	181.812i	−0.413807	−	0.238911i	−0.692424	+	0.721491i	$0.743457\pi$
$762$		0			0
$763$	−554.140	+	959.798i	−0.726264	+	1.25793i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−349.063	+	201.531i	−0.455101	+	0.262753i
$768$		0			0
$769$	−546.768	−	947.030i	−0.711012	−	1.23151i	−0.964478	−	0.264164i	$-0.914904\pi$
$769$	−546.768	−	947.030i	−0.711012	−	1.23151i	0.253466	−	0.967344i	$-0.418429\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1301.99	−	751.704i	−1.68433	−	0.972451i	−0.958722	−	0.284347i	$-0.908223\pi$
$773$	−1301.99	−	751.704i	−1.68433	−	0.972451i	−0.725612	−	0.688104i	$-0.758443\pi$
$774$		0			0
$775$	238.557	−	413.193i	0.307815	−	0.533152i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−493.573	−	261.854i	−0.633598	−	0.336141i
$780$		0			0
$781$	301.801	−	522.734i	0.386429	−	0.669314i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7.45393i		−	0.00949546i
$786$		0			0
$787$	308.357	−	534.090i	0.391813	−	0.678640i	−0.600876	−	0.799342i	$-0.705181\pi$
$787$	308.357	−	534.090i	0.391813	−	0.678640i	0.992689	+	0.120703i	$0.0385147\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2144.11i		−	2.71063i
$792$		0			0
$793$	−375.758	−	650.832i	−0.473844	−	0.820721i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−175.903	−	101.558i	−0.220707	−	0.127425i	0.385571	−	0.922678i	$-0.374005\pi$
$797$	−175.903	−	101.558i	−0.220707	−	0.127425i	−0.606278	+	0.795253i	$0.707338\pi$
$798$		0			0
$799$	392.583	+	679.974i	0.491343	+	0.851031i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−100.586	−	58.0735i	−0.125263	−	0.0723207i
$804$		0			0
$805$	−1.16796	+	2.02297i	−0.00145088	+	0.00251300i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			589.547i			0.728736i	0.931255	+	0.364368i	$0.118715\pi$
$809$			589.547i			0.728736i	−0.931255	+	0.364368i	$0.881285\pi$
$810$		0			0
$811$	−332.344	+	575.637i	−0.409796	+	0.709787i	−0.994867	−	0.101195i	$-0.967733\pi$
$811$	−332.344	+	575.637i	−0.409796	+	0.709787i	0.585071	+	0.810982i	$0.301067\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	0.357437i		−	0.000438573i
$816$		0			0
$817$	83.0483	+	132.633i	0.101650	+	0.162341i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	966.861i		−	1.17766i	−0.808256	−	0.588831i	$-0.799588\pi$
$821$		−	966.861i		−	1.17766i	0.808256	−	0.588831i	$-0.200412\pi$
$822$		0			0
$823$	−358.318	−	620.625i	−0.435380	−	0.754101i	0.561946	−	0.827174i	$-0.310053\pi$
$823$	−358.318	−	620.625i	−0.435380	−	0.754101i	−0.997327	+	0.0730730i	$0.976719\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	89.6846	+	51.7794i	0.108446	+	0.0626112i	0.553242	−	0.833021i	$-0.313391\pi$
$827$	89.6846	+	51.7794i	0.108446	+	0.0626112i	−0.444796	+	0.895632i	$0.646724\pi$
$828$		0			0
$829$	33.8188			0.0407947			0.0203974	−	0.999792i	$-0.493507\pi$
$829$	33.8188			0.0407947			0.0203974	+	0.999792i	$0.493507\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1564.62i			1.87829i
$834$		0			0
$835$	−79.7801	−	138.183i	−0.0955450	−	0.165489i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	342.384	+	197.675i	0.408086	+	0.235608i	0.689967	−	0.723841i	$-0.257625\pi$
$839$	342.384	+	197.675i	0.408086	+	0.235608i	−0.281881	+	0.959449i	$0.590958\pi$
$840$		0			0
$841$	−636.822			−0.757220
$842$		0			0
$843$		0			0
$844$		0			0
$845$	61.1013	+	35.2769i	0.0723093	+	0.0417478i
$846$		0			0
$847$	−555.411			−0.655739
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.5969	−	9.00485i	0.0183277	−	0.0105815i
$852$		0			0
$853$	−350.270	+	606.686i	−0.410634	+	0.711238i	−0.994959	−	0.100281i	$-0.968026\pi$
$853$	−350.270	+	606.686i	−0.410634	+	0.711238i	0.584326	+	0.811519i	$0.301359\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1104.62	−	637.752i	−1.28894	−	0.744168i	−0.310472	−	0.950583i	$-0.600487\pi$
$857$	−1104.62	−	637.752i	−1.28894	−	0.744168i	−0.978465	+	0.206415i	$0.933820\pi$
$858$		0			0
$859$	−187.439	−	324.654i	−0.218206	−	0.377944i	0.736053	−	0.676924i	$-0.236687\pi$
$859$	−187.439	−	324.654i	−0.218206	−	0.377944i	−0.954260	+	0.298979i	$0.903354\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			282.004i			0.326772i	0.986562	+	0.163386i	$0.0522416\pi$
$863$			282.004i			0.326772i	−0.986562	+	0.163386i	$0.947758\pi$
$864$		0			0
$865$	−34.8860	+	60.4244i	−0.0403307	+	0.0698548i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			460.644i			0.530086i
$870$		0			0
$871$	366.146	−	634.184i	0.420374	−	0.728110i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	245.786	−	141.905i	0.280899	−	0.162177i
$876$		0			0
$877$	518.048			0.590705			0.295353	−	0.955388i	$-0.404563\pi$
$877$	518.048			0.590705			0.295353	+	0.955388i	$0.404563\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			489.787i			0.555945i	0.960589	+	0.277972i	$0.0896624\pi$
$881$			489.787i			0.555945i	−0.960589	+	0.277972i	$0.910338\pi$
$882$		0			0
$883$	292.251	−	506.194i	0.330975	−	0.573266i	−0.651728	−	0.758453i	$-0.725956\pi$
$883$	292.251	−	506.194i	0.330975	−	0.573266i	0.982703	+	0.185187i	$0.0592891\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	877.206	−	506.455i	0.988958	−	0.570975i	0.0839953	−	0.996466i	$-0.473232\pi$
$887$	877.206	−	506.455i	0.988958	−	0.570975i	0.904963	+	0.425491i	$0.139899\pi$
$888$		0			0
$889$	1680.41			1.89022
$890$		0			0
$891$		0			0
$892$		0			0
$893$	695.407	+	368.932i	0.778731	+	0.413138i
$894$		0			0
$895$	−45.5745			−0.0509212
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−641.715	−	370.494i	−0.713809	−	0.412118i
$900$		0			0
$901$	1737.74			1.92868
$902$		0			0
$903$		0			0
$904$		0			0
$905$	105.585	+	60.9595i	0.116668	+	0.0673586i
$906$		0			0
$907$	840.726	−	1456.18i	0.926930	−	1.60549i	0.138503	−	0.990362i	$-0.455771\pi$
$907$	840.726	−	1456.18i	0.926930	−	1.60549i	0.788427	−	0.615128i	$-0.210896\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−282.331	+	163.004i	−0.309913	+	0.178929i	−0.646888	−	0.762585i	$-0.723930\pi$
$911$	−282.331	+	163.004i	−0.309913	+	0.178929i	0.336974	+	0.941514i	$0.390596\pi$
$912$		0			0
$913$	312.632	−	541.495i	0.342423	−	0.593094i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	891.503	−	514.709i	0.972195	−	0.561297i
$918$		0			0
$919$	429.757			0.467636			0.233818	−	0.972280i	$-0.424878\pi$
$919$	429.757			0.467636			0.233818	+	0.972280i	$0.424878\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1081.85	+	624.606i	1.17210	+	0.676713i
$924$		0			0
$925$	−1088.63			−1.17690
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−253.818	−	146.542i	−0.273216	−	0.157742i	0.357132	−	0.934054i	$-0.383755\pi$
$929$	−253.818	−	146.542i	−0.273216	−	0.157742i	−0.630348	+	0.776312i	$0.717088\pi$
$930$		0			0
$931$	832.506	+	1329.56i	0.894206	+	1.42810i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−69.5344	−	40.1457i	−0.0743683	−	0.0429366i
$936$		0			0
$937$	42.8606	−	74.2368i	0.0457424	−	0.0792282i	−0.842248	−	0.539091i	$-0.818768\pi$
$937$	42.8606	−	74.2368i	0.0457424	−	0.0792282i	0.887990	+	0.459863i	$0.152101\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	629.814	−	363.623i	0.669303	−	0.386422i	−0.126510	−	0.991965i	$-0.540378\pi$
$941$	629.814	−	363.623i	0.669303	−	0.386422i	0.795812	+	0.605543i	$0.207044\pi$
$942$		0			0
$943$	6.02099	+	10.4287i	0.00638493	+	0.0110590i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1006.72	−	581.228i	−1.06306	−	0.613757i	−0.136782	−	0.990601i	$-0.543676\pi$
$947$	−1006.72	−	581.228i	−1.06306	−	0.613757i	−0.926277	+	0.376844i	$0.877009\pi$
$948$		0			0
$949$	120.189	−	208.173i	0.126648	−	0.219360i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−739.872	+	427.165i	−0.776361	+	0.448232i	−0.835139	−	0.550039i	$-0.814613\pi$
$953$	−739.872	+	427.165i	−0.776361	+	0.448232i	0.0587783	+	0.998271i	$0.481280\pi$
$954$		0			0
$955$	86.7634	+	150.279i	0.0908517	+	0.157360i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	735.302	−	424.527i	0.766738	−	0.442677i
$960$		0			0
$961$	294.732	+	510.491i	0.306693	+	0.531208i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	60.8406i		−	0.0630472i
$966$		0			0
$967$	1805.36			1.86697			0.933484	−	0.358620i	$-0.116753\pi$
$967$	1805.36			1.86697			0.933484	+	0.358620i	$0.116753\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1237.85	+	714.672i	−1.27482	+	0.736016i	−0.975891	−	0.218260i	$-0.929962\pi$
$971$	−1237.85	+	714.672i	−1.27482	+	0.736016i	−0.298927	+	0.954276i	$0.596629\pi$
$972$		0			0
$973$	−1420.40	+	2460.21i	−1.45982	+	2.52848i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1263.88	+	729.700i	1.29363	+	0.746878i	0.979296	−	0.202435i	$-0.0648854\pi$
$977$	1263.88	+	729.700i	1.29363	+	0.746878i	0.314334	+	0.949312i	$0.398219\pi$
$978$		0			0
$979$	−98.9584			−0.101081
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−788.050	−	454.981i	−0.801678	−	0.462849i	0.0423795	−	0.999102i	$-0.486506\pi$
$983$	−788.050	−	454.981i	−0.801678	−	0.462849i	−0.844058	+	0.536252i	$0.819839\pi$
$984$		0			0
$985$	27.5995			0.0280198
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	3.37268i		−	0.00341019i
$990$		0			0
$991$	−428.060	−	741.421i	−0.431947	−	0.748155i	0.565094	−	0.825027i	$-0.308840\pi$
$991$	−428.060	−	741.421i	−0.431947	−	0.748155i	−0.997041	+	0.0768721i	$0.975507\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.3822	+	10.0356i	0.0174695	+	0.0100860i
$996$		0			0
$997$	1048.19			1.05135			0.525674	−	0.850686i	$-0.323813\pi$
$997$	1048.19			1.05135			0.525674	+	0.850686i	$0.323813\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.19		80
3.2	odd	2		684.3.m.a.653.13	yes	80
9.2	odd	6		2052.3.be.a.197.19		80
9.7	even	3		684.3.be.a.425.39	yes	80
19.11	even	3		2052.3.be.a.125.19		80
57.11	odd	6		684.3.be.a.581.39	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.22		80
171.106	even	3		684.3.m.a.353.13	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.13	✓	80	171.106	even	3
684.3.m.a.653.13	yes	80	3.2	odd	2
684.3.be.a.425.39	yes	80	9.7	even	3
684.3.be.a.581.39	yes	80	57.11	odd	6
2052.3.m.a.881.19		80	1.1	even	1	trivial
2052.3.m.a.1493.22		80	171.11	odd	6	inner
2052.3.be.a.125.19		80	19.11	even	3
2052.3.be.a.197.19		80	9.2	odd	6

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-0.497329i q^{5} +(5.73504 + 9.93338i) q^{7} +O(q^{10})\)	\(q-0.497329i q^{5} +(5.73504 + 9.93338i) q^{7} +(7.37787 - 4.25962i) q^{11} +(8.81568 + 15.2692i) q^{13} +(16.4117 - 9.47532i) q^{17} +(8.90449 - 16.7842i) q^{19} +(-0.354632 + 0.204747i) q^{23} +24.7527 q^{25} -38.4425i q^{29} +(9.63763 - 16.6929i) q^{31} +(4.94016 - 2.85220i) q^{35} -43.9804 q^{37} -29.4070i q^{41} +(-4.11810 + 7.13276i) q^{43} +41.4322i q^{47} +(-41.2813 + 71.5014i) q^{49} +(79.4128 + 45.8490i) q^{53} +(-2.11843 - 3.66923i) q^{55} +22.8606i q^{59} -42.6238 q^{61} +(7.59383 - 4.38430i) q^{65} +(-20.7667 - 35.9690i) q^{67} +(61.3593 - 35.4258i) q^{71} +(-6.81675 - 11.8070i) q^{73} +(84.6248 + 48.8581i) q^{77} +(-27.0355 + 46.8269i) q^{79} +(63.5614 - 36.6972i) q^{83} +(-4.71236 - 8.16204i) q^{85} +(-10.0596 - 5.80794i) q^{89} +(-101.117 + 175.139i) q^{91} +(-8.34729 - 4.42846i) q^{95} +(35.5215 - 61.5250i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

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