LMFDB - Embedded newform 1053.2.b.i.649.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1053,2,Mod(649,1053)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1053, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1053.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.b (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:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 $ x^10 + 16x^8 + 91x^6 + 222x^4 + 228x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.9
Root			$2.28298i$ of defining polynomial
Character	$\chi$	$=$	1053.649
Dual form			1053.2.b.i.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.28298i q^{2} -3.21200 q^{4} +3.21586i q^{5} +2.42273i q^{7} -2.76698i q^{8} -7.34174 q^{10} +1.47490i q^{11} +(-1.31905 + 3.35561i) q^{13} -5.53105 q^{14} -0.107046 q^{16} +5.12974 q^{17} +1.13065i q^{19} -10.3293i q^{20} -3.36716 q^{22} +9.23470 q^{23} -5.34174 q^{25} +(-7.66079 - 3.01136i) q^{26} -7.78182i q^{28} -0.974585 q^{29} -3.65324i q^{31} -5.77834i q^{32} +11.7111i q^{34} -7.79116 q^{35} -4.22691i q^{37} -2.58125 q^{38} +8.89821 q^{40} -4.00898i q^{41} -8.66079 q^{43} -4.73737i q^{44} +21.0826i q^{46} +1.53964i q^{47} +1.13037 q^{49} -12.1951i q^{50} +(4.23679 - 10.7782i) q^{52} -0.739889 q^{53} -4.74305 q^{55} +6.70365 q^{56} -2.22496i q^{58} +7.76686i q^{59} +8.13562 q^{61} +8.34028 q^{62} +12.9777 q^{64} +(-10.7912 - 4.24187i) q^{65} -0.772969i q^{67} -16.4767 q^{68} -17.7871i q^{70} -3.01136i q^{71} -9.21010i q^{73} +9.64996 q^{74} -3.63165i q^{76} -3.57328 q^{77} +3.73717 q^{79} -0.344246i q^{80} +9.15243 q^{82} -14.2768i q^{83} +16.4965i q^{85} -19.7724i q^{86} +4.08100 q^{88} +8.21257i q^{89} +(-8.12974 - 3.19570i) q^{91} -29.6619 q^{92} -3.51497 q^{94} -3.63601 q^{95} -15.2281i q^{97} +2.58061i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} - 18 q^{14} - 4 q^{16} + 6 q^{17} + 10 q^{22} + 24 q^{23} + 12 q^{25} + 6 q^{26} + 12 q^{29} + 6 q^{35} + 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} - 54 q^{53} + 10 q^{55}+ \cdots + 24 q^{95}+O(q^{100}) $ 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 - 18 * q^14 - 4 * q^16 + 6 * q^17 + 10 * q^22 + 24 * q^23 + 12 * q^25 + 6 * q^26 + 12 * q^29 + 6 * q^35 + 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 - 54 * q^53 + 10 * q^55 + 36 * q^56 + 2 * q^61 + 36 * q^62 + 4 * q^64 - 24 * q^65 + 24 * q^68 - 42 * q^74 - 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 - 84 * q^92 - 20 * q^94 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$1$	$-1$

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 $

$2$			2.28298i			1.61431i	0.590339	+	0.807156i	$0.298994\pi$
$2$			2.28298i			1.61431i	−0.590339	+	0.807156i	$0.701006\pi$
$3$		0			0
$3$		0			0
$4$	−3.21200			−1.60600
$5$			3.21586i			1.43818i	0.694919	+	0.719088i	$0.255440\pi$
$5$			3.21586i			1.43818i	−0.694919	+	0.719088i	$0.744560\pi$
$6$		0			0
$7$			2.42273i			0.915707i	0.889028	+	0.457853i	$0.151382\pi$
$7$			2.42273i			0.915707i	−0.889028	+	0.457853i	$0.848618\pi$
$8$		−	2.76698i		−	0.978274i
$9$		0			0
$10$	−7.34174			−2.32166
$11$			1.47490i			0.444698i	0.974967	+	0.222349i	$0.0713724\pi$
$11$			1.47490i			0.444698i	−0.974967	+	0.222349i	$0.928628\pi$
$12$		0			0
$13$	−1.31905	+	3.35561i	−0.365838	+	0.930678i
$13$	−1.31905	+	3.35561i	−0.365838	+	0.930678i
$14$	−5.53105			−1.47824
$15$		0			0
$16$	−0.107046			−0.0267616
$17$	5.12974			1.24414			0.622072	−	0.782960i	$-0.286291\pi$
$17$	5.12974			1.24414			0.622072	+	0.782960i	$0.286291\pi$
$18$		0			0
$19$			1.13065i			0.259389i	0.991554	+	0.129694i	$0.0413996\pi$
$19$			1.13065i			0.259389i	−0.991554	+	0.129694i	$0.958600\pi$
$20$		−	10.3293i		−	2.30971i
$21$		0			0
$22$	−3.36716			−0.717880
$23$	9.23470			1.92557			0.962784	−	0.270273i	$-0.0871141\pi$
$23$	9.23470			1.92557			0.962784	+	0.270273i	$0.0871141\pi$
$24$		0			0
$25$	−5.34174			−1.06835
$26$	−7.66079	−	3.01136i	−1.50240	−	0.590577i
$27$		0			0
$28$		−	7.78182i		−	1.47063i
$29$	−0.974585			−0.180976			−0.0904880	−	0.995898i	$-0.528843\pi$
$29$	−0.974585			−0.180976			−0.0904880	+	0.995898i	$0.528843\pi$
$30$		0			0
$31$		−	3.65324i		−	0.656142i	−0.944653	−	0.328071i	$-0.893602\pi$
$31$		−	3.65324i		−	0.656142i	0.944653	−	0.328071i	$-0.106398\pi$
$32$		−	5.77834i		−	1.02148i
$33$		0			0
$34$			11.7111i			2.00844i
$35$	−7.79116			−1.31695
$36$		0			0
$37$		−	4.22691i		−	0.694900i	−0.937698	−	0.347450i	$-0.887048\pi$
$37$		−	4.22691i		−	0.694900i	0.937698	−	0.347450i	$-0.112952\pi$
$38$	−2.58125			−0.418734
$39$		0			0
$40$	8.89821			1.40693
$41$		−	4.00898i		−	0.626098i	−0.949737	−	0.313049i	$-0.898650\pi$
$41$		−	4.00898i		−	0.626098i	0.949737	−	0.313049i	$-0.101350\pi$
$42$		0			0
$43$	−8.66079			−1.32076			−0.660379	−	0.750932i	$-0.729604\pi$
$43$	−8.66079			−1.32076			−0.660379	+	0.750932i	$0.729604\pi$
$44$		−	4.73737i		−	0.714185i
$45$		0			0
$46$			21.0826i			3.10847i
$47$			1.53964i			0.224579i	0.993676	+	0.112290i	$0.0358185\pi$
$47$			1.53964i			0.224579i	−0.993676	+	0.112290i	$0.964182\pi$
$48$		0			0
$49$	1.13037			0.161481
$50$		−	12.1951i		−	1.72465i
$51$		0			0
$52$	4.23679	−	10.7782i	0.587537	−	1.49467i
$53$	−0.739889			−0.101632			−0.0508158	−	0.998708i	$-0.516182\pi$
$53$	−0.739889			−0.101632			−0.0508158	+	0.998708i	$0.516182\pi$
$54$		0			0
$55$	−4.74305			−0.639553
$56$	6.70365			0.895812
$57$		0			0
$58$		−	2.22496i		−	0.292152i
$59$			7.76686i			1.01116i	0.862780	+	0.505580i	$0.168721\pi$
$59$			7.76686i			1.01116i	−0.862780	+	0.505580i	$0.831279\pi$
$60$		0			0
$61$	8.13562			1.04166			0.520830	−	0.853660i	$-0.325623\pi$
$61$	8.13562			1.04166			0.520830	+	0.853660i	$0.325623\pi$
$62$	8.34028			1.05922
$63$		0			0
$64$	12.9777			1.62222
$65$	−10.7912	−	4.24187i	−1.33848	−	0.526140i
$66$		0			0
$67$		−	0.772969i		−	0.0944332i	−0.998885	−	0.0472166i	$-0.984965\pi$
$67$		−	0.772969i		−	0.0944332i	0.998885	−	0.0472166i	$-0.0150351\pi$
$68$	−16.4767			−1.99810
$69$		0			0
$70$		−	17.7871i		−	2.12596i
$71$		−	3.01136i		−	0.357383i	−0.983905	−	0.178692i	$-0.942814\pi$
$71$		−	3.01136i		−	0.357383i	0.983905	−	0.178692i	$-0.0571864\pi$
$72$		0			0
$73$		−	9.21010i		−	1.07796i	−0.842318	−	0.538980i	$-0.818810\pi$
$73$		−	9.21010i		−	1.07796i	0.842318	−	0.538980i	$-0.181190\pi$
$74$	9.64996			1.12179
$75$		0			0
$76$		−	3.63165i		−	0.416578i
$77$	−3.57328			−0.407212
$78$		0			0
$79$	3.73717			0.420464			0.210232	−	0.977651i	$-0.432578\pi$
$79$	3.73717			0.420464			0.210232	+	0.977651i	$0.432578\pi$
$80$		−	0.344246i		−	0.0384879i
$81$		0			0
$82$	9.15243			1.01072
$83$		−	14.2768i		−	1.56708i	−0.621343	−	0.783539i	$-0.713413\pi$
$83$		−	14.2768i		−	1.56708i	0.621343	−	0.783539i	$-0.286587\pi$
$84$		0			0
$85$			16.4965i			1.78930i
$86$		−	19.7724i		−	2.13212i
$87$		0			0
$88$	4.08100			0.435036
$89$			8.21257i			0.870531i	0.900302	+	0.435265i	$0.143345\pi$
$89$			8.21257i			0.870531i	−0.900302	+	0.435265i	$0.856655\pi$
$90$		0			0
$91$	−8.12974	−	3.19570i	−0.852228	−	0.335000i
$92$	−29.6619			−3.09246
$93$		0			0
$94$	−3.51497			−0.362541
$95$	−3.63601			−0.373046
$96$		0			0
$97$		−	15.2281i		−	1.54618i	−0.634294	−	0.773092i	$-0.718709\pi$
$97$		−	15.2281i		−	1.54618i	0.634294	−	0.773092i	$-0.281291\pi$
$98$			2.58061i			0.260681i
$99$		0			0
$100$	17.1577			1.71577
$101$	14.9063			1.48323			0.741617	−	0.670824i	$-0.234059\pi$
$101$	14.9063			1.48323			0.741617	+	0.670824i	$0.234059\pi$
$102$		0			0
$103$	−6.53105			−0.643524			−0.321762	−	0.946821i	$-0.604275\pi$
$103$	−6.53105			−0.643524			−0.321762	+	0.946821i	$0.604275\pi$
$104$	9.28490	+	3.64978i	0.910459	+	0.357890i
$105$		0			0
$106$		−	1.68915i		−	0.164065i
$107$	9.37527			0.906341			0.453171	−	0.891424i	$-0.350293\pi$
$107$	9.37527			0.906341			0.453171	+	0.891424i	$0.350293\pi$
$108$		0			0
$109$			14.5859i			1.39707i	0.715574	+	0.698537i	$0.246165\pi$
$109$			14.5859i			1.39707i	−0.715574	+	0.698537i	$0.753835\pi$
$110$		−	10.8283i		−	1.03244i
$111$		0			0
$112$		−	0.259345i		−	0.0245058i
$113$	−3.06764			−0.288579			−0.144290	−	0.989536i	$-0.546090\pi$
$113$	−3.06764			−0.288579			−0.144290	+	0.989536i	$0.546090\pi$
$114$		0			0
$115$			29.6975i			2.76930i
$116$	3.13037			0.290648
$117$		0			0
$118$	−17.7316			−1.63233
$119$			12.4280i			1.13927i
$120$		0			0
$121$	8.82468			0.802244
$122$			18.5735i			1.68156i
$123$		0			0
$124$			11.7342i			1.05376i
$125$		−	1.09900i		−	0.0982972i
$126$		0			0
$127$	0.163893			0.0145432			0.00727160	−	0.999974i	$-0.497685\pi$
$127$	0.163893			0.0145432			0.00727160	+	0.999974i	$0.497685\pi$
$128$			18.0713i			1.59729i
$129$		0			0
$130$	9.68412	−	24.6360i	0.849353	−	2.16072i
$131$	−6.53168			−0.570676			−0.285338	−	0.958427i	$-0.592106\pi$
$131$	−6.53168			−0.570676			−0.285338	+	0.958427i	$0.592106\pi$
$132$		0			0
$133$	−2.73926			−0.237524
$134$	1.76467			0.152445
$135$		0			0
$136$		−	14.1939i		−	1.21712i
$137$			13.9293i			1.19006i	0.803702	+	0.595032i	$0.202861\pi$
$137$			13.9293i			1.19006i	−0.803702	+	0.595032i	$0.797139\pi$
$138$		0			0
$139$	−14.1816			−1.20287			−0.601436	−	0.798921i	$-0.705404\pi$
$139$	−14.1816			−1.20287			−0.601436	+	0.798921i	$0.705404\pi$
$140$	25.0252			2.11502
$141$		0			0
$142$	6.87488			0.576927
$143$	−4.94917	−	1.94546i	−0.413870	−	0.162687i
$144$		0			0
$145$		−	3.13413i		−	0.260275i
$146$	21.0265			1.74016
$147$		0			0
$148$			13.5769i			1.11601i
$149$			18.5770i			1.52189i	0.648817	+	0.760945i	$0.275264\pi$
$149$			18.5770i			1.52189i	−0.648817	+	0.760945i	$0.724736\pi$
$150$		0			0
$151$		−	11.8922i		−	0.967772i	−0.875131	−	0.483886i	$-0.839225\pi$
$151$		−	11.8922i		−	0.967772i	0.875131	−	0.483886i	$-0.160775\pi$
$152$	3.12848			0.253753
$153$		0			0
$154$		−	8.15772i		−	0.657368i
$155$	11.7483			0.943647
$156$		0			0
$157$	3.98917			0.318371			0.159185	−	0.987249i	$-0.449113\pi$
$157$	3.98917			0.318371			0.159185	+	0.987249i	$0.449113\pi$
$158$			8.53188i			0.678760i
$159$		0			0
$160$	18.5823			1.46906
$161$			22.3732i			1.76325i
$162$		0			0
$163$			5.18096i			0.405804i	0.979199	+	0.202902i	$0.0650373\pi$
$163$			5.18096i			0.405804i	−0.979199	+	0.202902i	$0.934963\pi$
$164$			12.8769i			1.00551i
$165$		0			0
$166$	32.5936			2.52975
$167$		−	0.816030i		−	0.0631463i	−0.999501	−	0.0315732i	$-0.989948\pi$
$167$		−	0.816030i		−	0.0631463i	0.999501	−	0.0315732i	$-0.0100517\pi$
$168$		0			0
$169$	−9.52022	−	8.85242i	−0.732325	−	0.680956i
$170$	−37.6612			−2.88848
$171$		0			0
$172$	27.8185			2.12114
$173$	−0.523076			−0.0397687			−0.0198844	−	0.999802i	$-0.506330\pi$
$173$	−0.523076			−0.0397687			−0.0198844	+	0.999802i	$0.506330\pi$
$174$		0			0
$175$		−	12.9416i		−	0.978294i
$176$		−	0.157882i		−	0.0119008i
$177$		0			0
$178$	−18.7491			−1.40531
$179$	14.1750			1.05949			0.529746	−	0.848156i	$-0.322287\pi$
$179$	14.1750			1.05949			0.529746	+	0.848156i	$0.322287\pi$
$180$		0			0
$181$	−17.9757			−1.33612			−0.668060	−	0.744107i	$-0.732875\pi$
$181$	−17.9757			−1.33612			−0.668060	+	0.744107i	$0.732875\pi$
$182$	7.29572	−	18.5600i	0.540795	−	1.37576i
$183$		0			0
$184$		−	25.5522i		−	1.88373i
$185$	13.5932			0.999388
$186$		0			0
$187$			7.56583i			0.553268i
$188$		−	4.94532i		−	0.360675i
$189$		0			0
$190$		−	8.30093i		−	0.602213i
$191$	−9.29509			−0.672569			−0.336285	−	0.941760i	$-0.609170\pi$
$191$	−9.29509			−0.672569			−0.336285	+	0.941760i	$0.609170\pi$
$192$		0			0
$193$			11.4729i			0.825839i	0.910767	+	0.412920i	$0.135491\pi$
$193$			11.4729i			0.825839i	−0.910767	+	0.412920i	$0.864509\pi$
$194$	34.7656			2.49602
$195$		0			0
$196$	−3.63075			−0.259339
$197$			11.0296i			0.785828i	0.919575	+	0.392914i	$0.128533\pi$
$197$			11.0296i			0.785828i	−0.919575	+	0.392914i	$0.871467\pi$
$198$		0			0
$199$	−1.38397			−0.0981068			−0.0490534	−	0.998796i	$-0.515620\pi$
$199$	−1.38397			−0.0981068			−0.0490534	+	0.998796i	$0.515620\pi$
$200$			14.7805i			1.04514i
$201$		0			0
$202$			34.0308i			2.39440i
$203$		−	2.36116i		−	0.165721i
$204$		0			0
$205$	12.8923			0.900439
$206$		−	14.9103i		−	1.03885i
$207$		0			0
$208$	0.141199	−	0.359206i	0.00979042	−	0.0249064i
$209$	−1.66759			−0.115350
$210$		0			0
$211$	−9.18373			−0.632234			−0.316117	−	0.948720i	$-0.602379\pi$
$211$	−9.18373			−0.632234			−0.316117	+	0.948720i	$0.602379\pi$
$212$	2.37653			0.163220
$213$		0			0
$214$			21.4036i			1.46312i
$215$		−	27.8519i		−	1.89948i
$216$		0			0
$217$	8.85083			0.600833
$218$	−33.2993			−2.25531
$219$		0			0
$220$	15.2347			1.02712
$221$	−6.76638	+	17.2134i	−0.455156	+	1.15790i
$222$		0			0
$223$		−	6.36812i		−	0.426441i	−0.977004	−	0.213220i	$-0.931605\pi$
$223$		−	6.36812i		−	0.426441i	0.977004	−	0.213220i	$-0.0683952\pi$
$224$	13.9994			0.935372
$225$		0			0
$226$		−	7.00336i		−	0.465857i
$227$			10.1966i			0.676773i	0.941007	+	0.338386i	$0.109881\pi$
$227$			10.1966i			0.676773i	−0.941007	+	0.338386i	$0.890119\pi$
$228$		0			0
$229$		−	8.11732i		−	0.536407i	−0.963362	−	0.268204i	$-0.913570\pi$
$229$		−	8.11732i		−	0.536407i	0.963362	−	0.268204i	$-0.0864300\pi$
$230$	−67.7988			−4.47052
$231$		0			0
$232$			2.69666i			0.177044i
$233$	−8.37795			−0.548858			−0.274429	−	0.961607i	$-0.588489\pi$
$233$	−8.37795			−0.548858			−0.274429	+	0.961607i	$0.588489\pi$
$234$		0			0
$235$	−4.95126			−0.322985
$236$		−	24.9472i		−	1.62392i
$237$		0			0
$238$	−28.3729			−1.83914
$239$		−	4.61249i		−	0.298357i	−0.988810	−	0.149178i	$-0.952337\pi$
$239$		−	4.61249i		−	0.298357i	0.988810	−	0.149178i	$-0.0476629\pi$
$240$		0			0
$241$			8.03568i			0.517623i	0.965928	+	0.258812i	$0.0833309\pi$
$241$			8.03568i			0.517623i	−0.965928	+	0.258812i	$0.916669\pi$
$242$			20.1466i			1.29507i
$243$		0			0
$244$	−26.1316			−1.67291
$245$			3.63511i			0.232239i
$246$		0			0
$247$	−3.79402	−	1.49138i	−0.241407	−	0.0948943i
$248$	−10.1084			−0.641887
$249$		0			0
$250$	2.50899			0.158682
$251$	−17.1127			−1.08015			−0.540073	−	0.841618i	$-0.681604\pi$
$251$	−17.1127			−1.08015			−0.540073	+	0.841618i	$0.681604\pi$
$252$		0			0
$253$			13.6202i			0.856295i
$254$			0.374166i			0.0234772i
$255$		0			0
$256$	−15.3009			−0.956305
$257$	9.51424			0.593482			0.296741	−	0.954958i	$-0.404100\pi$
$257$	9.51424			0.593482			0.296741	+	0.954958i	$0.404100\pi$
$258$		0			0
$259$	10.2407			0.636325
$260$	34.6612	+	13.6249i	2.14960	+	0.844981i
$261$		0			0
$262$		−	14.9117i		−	0.921248i
$263$	27.9890			1.72587			0.862937	−	0.505311i	$-0.168622\pi$
$263$	27.9890			1.72587			0.862937	+	0.505311i	$0.168622\pi$
$264$		0			0
$265$		−	2.37938i		−	0.146164i
$266$		−	6.25368i		−	0.383438i
$267$		0			0
$268$			2.48278i			0.151660i
$269$	11.4199			0.696285			0.348143	−	0.937442i	$-0.386812\pi$
$269$	11.4199			0.696285			0.348143	+	0.937442i	$0.386812\pi$
$270$		0			0
$271$			13.2786i			0.806620i	0.915064	+	0.403310i	$0.132140\pi$
$271$			13.2786i			0.806620i	−0.915064	+	0.403310i	$0.867860\pi$
$272$	−0.549120			−0.0332953
$273$		0			0
$274$	−31.8004			−1.92113
$275$		−	7.87851i		−	0.475092i
$276$		0			0
$277$	1.80199			0.108271			0.0541356	−	0.998534i	$-0.482760\pi$
$277$	1.80199			0.108271			0.0541356	+	0.998534i	$0.482760\pi$
$278$		−	32.3764i		−	1.94181i
$279$		0			0
$280$			21.5580i			1.28834i
$281$		−	11.4781i		−	0.684727i	−0.939568	−	0.342364i	$-0.888773\pi$
$281$		−	11.4781i		−	0.684727i	0.939568	−	0.342364i	$-0.111227\pi$
$282$		0			0
$283$	−22.6554			−1.34672			−0.673360	−	0.739314i	$-0.735150\pi$
$283$	−22.6554			−1.34672			−0.673360	+	0.739314i	$0.735150\pi$
$284$			9.67250i			0.573958i
$285$		0			0
$286$	4.44144	−	11.2989i	0.262628	−	0.668116i
$287$	9.71269			0.573322
$288$		0			0
$289$	9.31424			0.547896
$290$	7.15516			0.420165
$291$		0			0
$292$			29.5829i			1.73121i
$293$		−	16.3683i		−	0.956244i	−0.878293	−	0.478122i	$-0.841318\pi$
$293$		−	16.3683i		−	0.956244i	0.878293	−	0.478122i	$-0.158682\pi$
$294$		0			0
$295$	−24.9771			−1.45422
$296$	−11.6958			−0.679803
$297$		0			0
$298$	−42.4110			−2.45680
$299$	−12.1810	+	30.9880i	−0.704446	+	1.79208i
$300$		0			0
$301$		−	20.9828i		−	1.20943i
$302$	27.1496			1.56228
$303$		0			0
$304$		−	0.121032i		−	0.00694166i
$305$			26.1630i			1.49809i
$306$		0			0
$307$			16.6786i			0.951898i	0.879473	+	0.475949i	$0.157895\pi$
$307$			16.6786i			0.951898i	−0.879473	+	0.475949i	$0.842105\pi$
$308$	11.4774			0.653984
$309$		0			0
$310$			26.8212i			1.52334i
$311$	−8.11230			−0.460006			−0.230003	−	0.973190i	$-0.573874\pi$
$311$	−8.11230			−0.460006			−0.230003	+	0.973190i	$0.573874\pi$
$312$		0			0
$313$	−22.0042			−1.24375			−0.621877	−	0.783115i	$-0.713629\pi$
$313$	−22.0042			−1.24375			−0.621877	+	0.783115i	$0.713629\pi$
$314$			9.10720i			0.513949i
$315$		0			0
$316$	−12.0038			−0.675266
$317$		−	6.72892i		−	0.377934i	−0.981983	−	0.188967i	$-0.939486\pi$
$317$		−	6.72892i		−	0.377934i	0.981983	−	0.188967i	$-0.0605139\pi$
$318$		0			0
$319$		−	1.43741i		−	0.0804796i
$320$			41.7346i			2.33303i
$321$		0			0
$322$	−51.0776			−2.84644
$323$			5.79994i			0.322717i
$324$		0			0
$325$	7.04602	−	17.9248i	0.390843	−	0.994289i
$326$	−11.8280			−0.655094
$327$		0			0
$328$	−11.0928			−0.612496
$329$	−3.73013			−0.205649
$330$		0			0
$331$			0.261295i			0.0143621i	0.999974	+	0.00718104i	$0.00228581\pi$
$331$			0.261295i			0.0143621i	−0.999974	+	0.00718104i	$0.997714\pi$
$332$			45.8570i			2.51673i
$333$		0			0
$334$	1.86298			0.101938
$335$	2.48576			0.135812
$336$		0			0
$337$	5.88805			0.320742			0.160371	−	0.987057i	$-0.448731\pi$
$337$	5.88805			0.320742			0.160371	+	0.987057i	$0.448731\pi$
$338$	20.2099	−	21.7345i	1.09927	−	1.18220i
$339$		0			0
$340$		−	52.9868i		−	2.87362i
$341$	5.38815			0.291785
$342$		0			0
$343$			19.6977i			1.06358i
$344$			23.9642i			1.29206i
$345$		0			0
$346$		−	1.19417i		−	0.0641991i
$347$	−3.19801			−0.171678			−0.0858391	−	0.996309i	$-0.527357\pi$
$347$	−3.19801			−0.171678			−0.0858391	+	0.996309i	$0.527357\pi$
$348$		0			0
$349$		−	13.9345i		−	0.745898i	−0.927852	−	0.372949i	$-0.878347\pi$
$349$		−	13.9345i		−	0.745898i	0.927852	−	0.372949i	$-0.121653\pi$
$350$	29.5454			1.57927
$351$		0			0
$352$	8.52245			0.454248
$353$			15.6885i			0.835016i	0.908673	+	0.417508i	$0.137096\pi$
$353$			15.6885i			0.835016i	−0.908673	+	0.417508i	$0.862904\pi$
$354$		0			0
$355$	9.68412			0.513979
$356$		−	26.3788i		−	1.39807i
$357$		0			0
$358$			32.3613i			1.71035i
$359$		−	5.86486i		−	0.309535i	−0.987951	−	0.154768i	$-0.950537\pi$
$359$		−	5.86486i		−	0.309535i	0.987951	−	0.154768i	$-0.0494629\pi$
$360$		0			0
$361$	17.7216			0.932718
$362$		−	41.0381i		−	2.15691i
$363$		0			0
$364$	26.1127	+	10.2646i	1.36868	+	0.538011i
$365$	29.6184			1.55030
$366$		0			0
$367$	3.33572			0.174123			0.0870617	−	0.996203i	$-0.472252\pi$
$367$	3.33572			0.174123			0.0870617	+	0.996203i	$0.472252\pi$
$368$	−0.988541			−0.0515313
$369$		0			0
$370$			31.0329i			1.61332i
$371$		−	1.79255i		−	0.0930647i
$372$		0			0
$373$	26.6413			1.37943			0.689716	−	0.724080i	$-0.257735\pi$
$373$	26.6413			1.37943			0.689716	+	0.724080i	$0.257735\pi$
$374$	−17.2726			−0.893147
$375$		0			0
$376$	4.26015			0.219700
$377$	1.28553	−	3.27033i	0.0662079	−	0.168430i
$378$		0			0
$379$		−	30.4926i		−	1.56630i	−0.621832	−	0.783150i	$-0.713611\pi$
$379$		−	30.4926i		−	1.56630i	0.621832	−	0.783150i	$-0.286389\pi$
$380$	11.6789			0.599113
$381$		0			0
$382$		−	21.2205i		−	1.08574i
$383$		−	25.0355i		−	1.27926i	−0.768685	−	0.639628i	$-0.779088\pi$
$383$		−	25.0355i		−	1.27926i	0.768685	−	0.639628i	$-0.220912\pi$
$384$		0			0
$385$		−	11.4911i		−	0.585643i
$386$	−26.1925			−1.33316
$387$		0			0
$388$			48.9128i			2.48317i
$389$	2.11230			0.107098			0.0535490	−	0.998565i	$-0.482947\pi$
$389$	2.11230			0.107098			0.0535490	+	0.998565i	$0.482947\pi$
$390$		0			0
$391$	47.3716			2.39568
$392$		−	3.12771i		−	0.157973i
$393$		0			0
$394$	−25.1804			−1.26857
$395$			12.0182i			0.604701i
$396$		0			0
$397$		−	28.7853i		−	1.44469i	−0.691533	−	0.722345i	$-0.743064\pi$
$397$		−	28.7853i		−	1.44469i	0.691533	−	0.722345i	$-0.256936\pi$
$398$		−	3.15957i		−	0.158375i
$399$		0			0
$400$	0.571814			0.0285907
$401$			11.3877i			0.568676i	0.958724	+	0.284338i	$0.0917738\pi$
$401$			11.3877i			0.568676i	−0.958724	+	0.284338i	$0.908226\pi$
$402$		0			0
$403$	12.2589	+	4.81880i	0.610657	+	0.240042i
$404$	−47.8791			−2.38208
$405$		0			0
$406$	5.39048			0.267525
$407$	6.23425			0.309020
$408$		0			0
$409$		−	3.35561i		−	0.165924i	−0.996553	−	0.0829621i	$-0.973562\pi$
$409$		−	3.35561i		−	0.165924i	0.996553	−	0.0829621i	$-0.0264380\pi$
$410$			29.4329i			1.45359i
$411$		0			0
$412$	20.9777			1.03350
$413$	−18.8170			−0.925925
$414$		0			0
$415$	45.9120			2.25373
$416$	19.3899	+	7.62191i	0.950666	+	0.373695i
$417$		0			0
$418$		−	3.80707i		−	0.186210i
$419$	−2.06559			−0.100910			−0.0504552	−	0.998726i	$-0.516067\pi$
$419$	−2.06559			−0.100910			−0.0504552	+	0.998726i	$0.516067\pi$
$420$		0			0
$421$			33.9026i			1.65231i	0.563441	+	0.826156i	$0.309477\pi$
$421$			33.9026i			1.65231i	−0.563441	+	0.826156i	$0.690523\pi$
$422$		−	20.9663i		−	1.02062i
$423$		0			0
$424$			2.04726i			0.0994236i
$425$	−27.4018			−1.32918
$426$		0			0
$427$			19.7104i			0.953855i
$428$	−30.1134			−1.45558
$429$		0			0
$430$	63.5853			3.06636
$431$		−	30.6212i		−	1.47497i	−0.675364	−	0.737485i	$-0.736013\pi$
$431$		−	30.6212i		−	1.47497i	0.675364	−	0.737485i	$-0.263987\pi$
$432$		0			0
$433$	8.63597			0.415018			0.207509	−	0.978233i	$-0.433464\pi$
$433$	8.63597			0.415018			0.207509	+	0.978233i	$0.433464\pi$
$434$			20.2063i			0.969932i
$435$		0			0
$436$		−	46.8499i		−	2.24370i
$437$			10.4412i			0.499470i
$438$		0			0
$439$	29.5653			1.41108			0.705538	−	0.708672i	$-0.250705\pi$
$439$	29.5653			1.41108			0.705538	+	0.708672i	$0.250705\pi$
$440$			13.1239i			0.625658i
$441$		0			0
$442$	−39.2979	−	15.4475i	−1.86921	−	0.734763i
$443$	−31.2662			−1.48550			−0.742751	−	0.669568i	$-0.766479\pi$
$443$	−31.2662			−1.48550			−0.742751	+	0.669568i	$0.766479\pi$
$444$		0			0
$445$	−26.4105			−1.25198
$446$	14.5383			0.688408
$447$		0			0
$448$			31.4416i			1.48548i
$449$			10.8346i			0.511316i	0.966767	+	0.255658i	$0.0822922\pi$
$449$			10.8346i			0.511316i	−0.966767	+	0.255658i	$0.917708\pi$
$450$		0			0
$451$	5.91283			0.278424
$452$	9.85326			0.463459
$453$		0			0
$454$	−23.2787			−1.09252
$455$	10.2769	−	26.1441i	0.481789	−	1.22565i
$456$		0			0
$457$		−	16.0429i		−	0.750457i	−0.926932	−	0.375229i	$-0.877564\pi$
$457$		−	16.0429i		−	0.750457i	0.926932	−	0.375229i	$-0.122436\pi$
$458$	18.5317			0.865929
$459$		0			0
$460$		−	95.3883i		−	4.44750i
$461$		−	25.6290i		−	1.19366i	−0.802368	−	0.596830i	$-0.796427\pi$
$461$		−	25.6290i		−	1.19366i	0.802368	−	0.596830i	$-0.203573\pi$
$462$		0			0
$463$		−	32.5540i		−	1.51291i	−0.654043	−	0.756457i	$-0.726929\pi$
$463$		−	32.5540i		−	1.51291i	0.654043	−	0.756457i	$-0.273071\pi$
$464$	0.104326			0.00484321
$465$		0			0
$466$		−	19.1267i		−	0.886027i
$467$	−16.2179			−0.750474			−0.375237	−	0.926929i	$-0.622439\pi$
$467$	−16.2179			−0.750474			−0.375237	+	0.926929i	$0.622439\pi$
$468$		0			0
$469$	1.87270			0.0864731
$470$		−	11.3036i		−	0.521398i
$471$		0			0
$472$	21.4907			0.989191
$473$		−	12.7738i		−	0.587338i
$474$		0			0
$475$		−	6.03964i		−	0.277117i
$476$		−	39.9187i		−	1.82967i
$477$		0			0
$478$	10.5302			0.481641
$479$			10.2360i			0.467695i	0.972273	+	0.233847i	$0.0751316\pi$
$479$			10.2360i			0.467695i	−0.972273	+	0.233847i	$0.924868\pi$
$480$		0			0
$481$	14.1839	+	5.57550i	0.646729	+	0.254221i
$482$	−18.3453			−0.835605
$483$		0			0
$484$	−28.3449			−1.28840
$485$	48.9715			2.22368
$486$		0			0
$487$		−	16.0863i		−	0.728940i	−0.931215	−	0.364470i	$-0.881250\pi$
$487$		−	16.0863i		−	0.728940i	0.931215	−	0.364470i	$-0.118750\pi$
$488$		−	22.5111i		−	1.01903i
$489$		0			0
$490$	−8.29889			−0.374906
$491$	−4.41885			−0.199420			−0.0997101	−	0.995017i	$-0.531792\pi$
$491$	−4.41885			−0.199420			−0.0997101	+	0.995017i	$0.531792\pi$
$492$		0			0
$493$	−4.99937			−0.225160
$494$	3.40479	−	8.66167i	0.153189	−	0.389707i
$495$		0			0
$496$			0.391066i			0.0175594i
$497$	7.29572			0.327258
$498$		0			0
$499$		−	7.38280i		−	0.330500i	−0.986252	−	0.165250i	$-0.947157\pi$
$499$		−	7.38280i		−	0.330500i	0.986252	−	0.165250i	$-0.0528430\pi$
$500$			3.52998i			0.157865i
$501$		0			0
$502$		−	39.0681i		−	1.74369i
$503$	−5.65418			−0.252107			−0.126054	−	0.992023i	$-0.540231\pi$
$503$	−5.65418			−0.252107			−0.126054	+	0.992023i	$0.540231\pi$
$504$		0			0
$505$			47.9366i			2.13315i
$506$	−31.0947			−1.38233
$507$		0			0
$508$	−0.526426			−0.0233564
$509$			26.8112i			1.18839i	0.804323	+	0.594193i	$0.202528\pi$
$509$			26.8112i			1.18839i	−0.804323	+	0.594193i	$0.797472\pi$
$510$		0			0
$511$	22.3136			0.987096
$512$			1.21094i			0.0535165i
$513$		0			0
$514$			21.7208i			0.958065i
$515$		−	21.0029i		−	0.925500i
$516$		0			0
$517$	−2.27081			−0.0998699
$518$			23.3793i			1.02723i
$519$		0			0
$520$	−11.7372	+	29.8589i	−0.514709	+	1.30940i
$521$	20.3407			0.891143			0.445572	−	0.895246i	$-0.353000\pi$
$521$	20.3407			0.891143			0.445572	+	0.895246i	$0.353000\pi$
$522$		0			0
$523$	5.74660			0.251281			0.125641	−	0.992076i	$-0.459901\pi$
$523$	5.74660			0.251281			0.125641	+	0.992076i	$0.459901\pi$
$524$	20.9798			0.916506
$525$		0			0
$526$			63.8983i			2.78610i
$527$		−	18.7402i		−	0.816335i
$528$		0			0
$529$	62.2796			2.70781
$530$	5.43208			0.235954
$531$		0			0
$532$	8.79851			0.381464
$533$	13.4526	+	5.28804i	0.582696	+	0.229051i
$534$		0			0
$535$			30.1495i			1.30348i
$536$	−2.13879			−0.0923816
$537$		0			0
$538$			26.0715i			1.12402i
$539$			1.66718i			0.0718104i
$540$		0			0
$541$		−	43.0286i		−	1.84994i	−0.380036	−	0.924972i	$-0.624088\pi$
$541$		−	43.0286i		−	1.84994i	0.380036	−	0.924972i	$-0.375912\pi$
$542$	−30.3149			−1.30214
$543$		0			0
$544$		−	29.6414i		−	1.27086i
$545$	−46.9061			−2.00924
$546$		0			0
$547$	−22.3163			−0.954176			−0.477088	−	0.878856i	$-0.658308\pi$
$547$	−22.3163			−0.954176			−0.477088	+	0.878856i	$0.658308\pi$
$548$		−	44.7411i		−	1.91124i
$549$		0			0
$550$	17.9865			0.766946
$551$		−	1.10191i		−	0.0469431i
$552$		0			0
$553$			9.05416i			0.385022i
$554$			4.11391i			0.174783i
$555$		0			0
$556$	45.5515			1.93181
$557$		−	28.7460i		−	1.21801i	−0.793168	−	0.609003i	$-0.791570\pi$
$557$		−	28.7460i		−	1.21801i	0.793168	−	0.609003i	$-0.208430\pi$
$558$		0			0
$559$	11.4240	−	29.0622i	0.483184	−	1.22920i
$560$	0.834016			0.0352436
$561$		0			0
$562$	26.2043			1.10536
$563$	11.8329			0.498699			0.249350	−	0.968414i	$-0.419783\pi$
$563$	11.8329			0.498699			0.249350	+	0.968414i	$0.419783\pi$
$564$		0			0
$565$		−	9.86509i		−	0.415028i
$566$		−	51.7217i		−	2.17403i
$567$		0			0
$568$	−8.33237			−0.349619
$569$	8.79322			0.368631			0.184315	−	0.982867i	$-0.440993\pi$
$569$	8.79322			0.368631			0.184315	+	0.982867i	$0.440993\pi$
$570$		0			0
$571$	31.2760			1.30886			0.654430	−	0.756123i	$-0.272909\pi$
$571$	31.2760			1.30886			0.654430	+	0.756123i	$0.272909\pi$
$572$	15.8967	+	6.24882i	0.664676	+	0.261276i
$573$		0			0
$574$			22.1739i			0.925520i
$575$	−49.3294			−2.05718
$576$		0			0
$577$			45.2450i			1.88357i	0.336210	+	0.941787i	$0.390855\pi$
$577$			45.2450i			1.88357i	−0.336210	+	0.941787i	$0.609145\pi$
$578$			21.2642i			0.884475i
$579$		0			0
$580$			10.0668i			0.418002i
$581$	34.5887			1.43498
$582$		0			0
$583$		−	1.09126i		−	0.0451953i
$584$	−25.4841			−1.05454
$585$		0			0
$586$	37.3684			1.54368
$587$		−	29.0150i		−	1.19758i	−0.800907	−	0.598788i	$-0.795649\pi$
$587$		−	29.0150i		−	1.19758i	0.800907	−	0.598788i	$-0.204351\pi$
$588$		0			0
$589$	4.13053			0.170196
$590$		−	57.0223i		−	2.34757i
$591$		0			0
$592$			0.452476i			0.0185966i
$593$			25.7497i			1.05741i	0.848805	+	0.528707i	$0.177323\pi$
$593$			25.7497i			1.05741i	−0.848805	+	0.528707i	$0.822677\pi$
$594$		0			0
$595$	−39.9666			−1.63847
$596$		−	59.6695i		−	2.44416i
$597$		0			0
$598$	−70.7451	−	27.8090i	−2.89298	−	1.13720i
$599$	−37.7340			−1.54177			−0.770885	−	0.636975i	$-0.780186\pi$
$599$	−37.7340			−1.54177			−0.770885	+	0.636975i	$0.780186\pi$
$600$		0			0
$601$	16.9290			0.690549			0.345274	−	0.938502i	$-0.387786\pi$
$601$	16.9290			0.690549			0.345274	+	0.938502i	$0.387786\pi$
$602$	47.9033			1.95239
$603$		0			0
$604$			38.1977i			1.55424i
$605$			28.3789i			1.15377i
$606$		0			0
$607$	−12.2789			−0.498386			−0.249193	−	0.968454i	$-0.580165\pi$
$607$	−12.2789			−0.498386			−0.249193	+	0.968454i	$0.580165\pi$
$608$	6.53327			0.264959
$609$		0			0
$610$	−59.7297			−2.41838
$611$	−5.16643	−	2.03086i	−0.209011	−	0.0821598i
$612$		0			0
$613$		−	3.84764i		−	0.155405i	−0.996977	−	0.0777023i	$-0.975242\pi$
$613$		−	3.84764i		−	0.155405i	0.996977	−	0.0777023i	$-0.0247584\pi$
$614$	−38.0769			−1.53666
$615$		0			0
$616$			9.88717i			0.398366i
$617$			34.3175i			1.38157i	0.723060	+	0.690785i	$0.242735\pi$
$617$			34.3175i			1.38157i	−0.723060	+	0.690785i	$0.757265\pi$
$618$		0			0
$619$		−	42.4792i		−	1.70738i	−0.520780	−	0.853691i	$-0.674359\pi$
$619$		−	42.4792i		−	1.70738i	0.520780	−	0.853691i	$-0.325641\pi$
$620$	−37.7356			−1.51550
$621$		0			0
$622$		−	18.5202i		−	0.742594i
$623$	−19.8969			−0.797151
$624$		0			0
$625$	−23.1745			−0.926980
$626$		−	50.2353i		−	2.00780i
$627$		0			0
$628$	−12.8132			−0.511303
$629$		−	21.6830i		−	0.864557i
$630$		0			0
$631$		−	17.9430i		−	0.714300i	−0.934047	−	0.357150i	$-0.883748\pi$
$631$		−	17.9430i		−	0.714300i	0.934047	−	0.357150i	$-0.116252\pi$
$632$		−	10.3407i		−	0.411329i
$633$		0			0
$634$	15.3620			0.610103
$635$			0.527058i			0.0209157i
$636$		0			0
$637$	−1.49101	+	3.79308i	−0.0590761	+	0.150287i
$638$	3.28158			0.129919
$639$		0			0
$640$	−58.1146			−2.29718
$641$	16.8623			0.666020			0.333010	−	0.942923i	$-0.391936\pi$
$641$	16.8623			0.666020			0.333010	+	0.942923i	$0.391936\pi$
$642$		0			0
$643$		−	49.9037i		−	1.96801i	−0.178137	−	0.984006i	$-0.557007\pi$
$643$		−	49.9037i		−	1.96801i	0.178137	−	0.984006i	$-0.442993\pi$
$644$		−	71.8627i		−	2.83179i
$645$		0			0
$646$	−13.2411			−0.520966
$647$	−28.5920			−1.12407			−0.562034	−	0.827114i	$-0.689981\pi$
$647$	−28.5920			−1.12407			−0.562034	+	0.827114i	$0.689981\pi$
$648$		0			0
$649$	−11.4553			−0.449660
$650$	40.9220	+	16.0859i	1.60509	+	0.630942i
$651$		0			0
$652$		−	16.6413i		−	0.651722i
$653$	−26.5468			−1.03886			−0.519429	−	0.854514i	$-0.673855\pi$
$653$	−26.5468			−1.03886			−0.519429	+	0.854514i	$0.673855\pi$
$654$		0			0
$655$		−	21.0050i		−	0.820732i
$656$			0.429147i			0.0167554i
$657$		0			0
$658$		−	8.51582i		−	0.331981i
$659$	43.4637			1.69310			0.846552	−	0.532306i	$-0.178674\pi$
$659$	43.4637			1.69310			0.846552	+	0.532306i	$0.178674\pi$
$660$		0			0
$661$			15.7866i			0.614029i	0.951705	+	0.307015i	$0.0993301\pi$
$661$			15.7866i			0.614029i	−0.951705	+	0.307015i	$0.900670\pi$
$662$	−0.596532			−0.0231849
$663$		0			0
$664$	−39.5035			−1.53303
$665$		−	8.80907i		−	0.341601i
$666$		0			0
$667$	−9.00000			−0.348481
$668$			2.62109i			0.101413i
$669$		0			0
$670$			5.67494i			0.219242i
$671$			11.9992i			0.463224i
$672$		0			0
$673$	−17.5271			−0.675620			−0.337810	−	0.941214i	$-0.609686\pi$
$673$	−17.5271			−0.675620			−0.337810	+	0.941214i	$0.609686\pi$
$674$			13.4423i			0.517778i
$675$		0			0
$676$	30.5790	+	28.4340i	1.17611	+	1.09362i
$677$	18.5007			0.711041			0.355520	−	0.934669i	$-0.384304\pi$
$677$	18.5007			0.711041			0.355520	+	0.934669i	$0.384304\pi$
$678$		0			0
$679$	36.8937			1.41585
$680$	45.6455			1.75042
$681$		0			0
$682$			12.3010i			0.471031i
$683$			4.68607i			0.179307i	0.995973	+	0.0896537i	$0.0285760\pi$
$683$			4.68607i			0.179307i	−0.995973	+	0.0896537i	$0.971424\pi$
$684$		0			0
$685$	−44.7948			−1.71152
$686$	−44.9695			−1.71694
$687$		0			0
$688$	0.927107			0.0353456
$689$	0.975950	−	2.48278i	0.0371807	−	0.0945863i
$690$		0			0
$691$			43.1163i			1.64022i	0.572206	+	0.820110i	$0.306088\pi$
$691$			43.1163i			1.64022i	−0.572206	+	0.820110i	$0.693912\pi$
$692$	1.68012			0.0638686
$693$		0			0
$694$		−	7.30099i		−	0.277142i
$695$		−	45.6061i		−	1.72994i
$696$		0			0
$697$		−	20.5650i		−	0.778957i
$698$	31.8123			1.20411
$699$		0			0
$700$			41.5685i			1.57114i
$701$	−35.9226			−1.35678			−0.678389	−	0.734703i	$-0.737322\pi$
$701$	−35.9226			−1.35678			−0.678389	+	0.734703i	$0.737322\pi$
$702$		0			0
$703$	4.77915			0.180249
$704$			19.1408i			0.721397i
$705$		0			0
$706$	−35.8166			−1.34798
$707$			36.1140i			1.35821i
$708$		0			0
$709$			24.1476i			0.906883i	0.891286	+	0.453442i	$0.149804\pi$
$709$			24.1476i			0.906883i	−0.891286	+	0.453442i	$0.850196\pi$
$710$			22.1087i			0.829723i
$711$		0			0
$712$	22.7240			0.851618
$713$		−	33.7366i		−	1.26344i
$714$		0			0
$715$	6.25632	−	15.9158i	0.233973	−	0.595218i
$716$	−45.5302			−1.70154
$717$		0			0
$718$	13.3894			0.499686
$719$	32.3717			1.20726			0.603630	−	0.797265i	$-0.293720\pi$
$719$	32.3717			1.20726			0.603630	+	0.797265i	$0.293720\pi$
$720$		0			0
$721$		−	15.8230i		−	0.589279i
$722$			40.4582i			1.50570i
$723$		0			0
$724$	57.7379			2.14581
$725$	5.20598			0.193345
$726$		0			0
$727$	19.5382			0.724632			0.362316	−	0.932055i	$-0.381986\pi$
$727$	19.5382			0.724632			0.362316	+	0.932055i	$0.381986\pi$
$728$	−8.84243	+	22.4948i	−0.327722	+	0.833713i
$729$		0			0
$730$			67.6182i			2.50266i
$731$	−44.4276			−1.64321
$732$		0			0
$733$			41.2487i			1.52355i	0.647839	+	0.761777i	$0.275673\pi$
$733$			41.2487i			1.52355i	−0.647839	+	0.761777i	$0.724327\pi$
$734$			7.61540i			0.281089i
$735$		0			0
$736$		−	53.3612i		−	1.96692i
$737$	1.14005			0.0419942
$738$		0			0
$739$			12.6677i			0.465988i	0.972478	+	0.232994i	$0.0748523\pi$
$739$			12.6677i			0.465988i	−0.972478	+	0.232994i	$0.925148\pi$
$740$	−43.6612			−1.60502
$741$		0			0
$742$	4.09236			0.150235
$743$			11.9538i			0.438542i	0.975664	+	0.219271i	$0.0703679\pi$
$743$			11.9538i			0.438542i	−0.975664	+	0.219271i	$0.929632\pi$
$744$		0			0
$745$	−59.7411			−2.18874
$746$			60.8215i			2.22683i
$747$		0			0
$748$		−	24.3015i		−	0.888549i
$749$			22.7138i			0.829943i
$750$		0			0
$751$	−13.1859			−0.481161			−0.240581	−	0.970629i	$-0.577338\pi$
$751$	−13.1859			−0.481161			−0.240581	+	0.970629i	$0.577338\pi$
$752$		−	0.164813i		−	0.00601011i
$753$		0			0
$754$	7.46610	+	2.93483i	0.271899	+	0.106880i
$755$	38.2436			1.39183
$756$		0			0
$757$	−26.1835			−0.951657			−0.475829	−	0.879538i	$-0.657852\pi$
$757$	−26.1835			−0.951657			−0.475829	+	0.879538i	$0.657852\pi$
$758$	69.6141			2.52850
$759$		0			0
$760$			10.0607i			0.364942i
$761$			6.93189i			0.251281i	0.992076	+	0.125640i	$0.0400986\pi$
$761$			6.93189i			0.251281i	−0.992076	+	0.125640i	$0.959901\pi$
$762$		0			0
$763$	−35.3377			−1.27931
$764$	29.8559			1.08015
$765$		0			0
$766$	57.1557			2.06512
$767$	−26.0625	−	10.2449i	−0.941064	−	0.369921i
$768$		0			0
$769$			38.5766i			1.39111i	0.718475	+	0.695553i	$0.244841\pi$
$769$			38.5766i			1.39111i	−0.718475	+	0.695553i	$0.755159\pi$
$770$	26.2341			0.945410
$771$		0			0
$772$		−	36.8511i		−	1.32630i
$773$			34.7210i			1.24883i	0.781093	+	0.624415i	$0.214662\pi$
$773$			34.7210i			1.24883i	−0.781093	+	0.624415i	$0.785338\pi$
$774$		0			0
$775$			19.5147i			0.700988i
$776$	−42.1359			−1.51259
$777$		0			0
$778$			4.82234i			0.172889i
$779$	4.53275			0.162403
$780$		0			0
$781$	4.44144			0.158927
$782$			108.148i			3.86738i
$783$		0			0
$784$	−0.121002			−0.00432150
$785$			12.8286i			0.457873i
$786$		0			0
$787$			35.8390i			1.27752i	0.769405	+	0.638761i	$0.220553\pi$
$787$			35.8390i			1.27752i	−0.769405	+	0.638761i	$0.779447\pi$
$788$		−	35.4272i		−	1.26204i
$789$		0			0
$790$	−27.4373			−0.976176
$791$		−	7.43207i		−	0.264254i
$792$		0			0
$793$	−10.7313	+	27.3000i	−0.381079	+	0.969451i
$794$	65.7162			2.33218
$795$		0			0
$796$	4.44531			0.157560
$797$	38.8542			1.37629			0.688144	−	0.725574i	$-0.258426\pi$
$797$	38.8542			1.37629			0.688144	+	0.725574i	$0.258426\pi$
$798$		0			0
$799$			7.89795i			0.279409i
$800$			30.8664i			1.09129i
$801$		0			0
$802$	−25.9980			−0.918020
$803$	13.5839			0.479367
$804$		0			0
$805$	−71.9490			−2.53587
$806$	−11.0012	+	27.9867i	−0.387502	+	0.985790i
$807$		0			0
$808$		−	41.2454i		−	1.45101i
$809$	12.2765			0.431619			0.215809	−	0.976435i	$-0.430761\pi$
$809$	12.2765			0.431619			0.215809	+	0.976435i	$0.430761\pi$
$810$		0			0
$811$		−	13.8855i		−	0.487585i	−0.969827	−	0.243792i	$-0.921608\pi$
$811$		−	13.8855i		−	0.487585i	0.969827	−	0.243792i	$-0.0783915\pi$
$812$			7.58405i			0.266148i
$813$		0			0
$814$			14.2327i			0.498855i
$815$	−16.6612			−0.583618
$816$		0			0
$817$		−	9.79231i		−	0.342590i
$818$	7.66079			0.267853
$819$		0			0
$820$	−41.4102			−1.44611
$821$			9.13784i			0.318913i	0.987205	+	0.159456i	$0.0509741\pi$
$821$			9.13784i			0.318913i	−0.987205	+	0.159456i	$0.949026\pi$
$822$		0			0
$823$	3.91482			0.136462			0.0682310	−	0.997670i	$-0.478265\pi$
$823$	3.91482			0.136462			0.0682310	+	0.997670i	$0.478265\pi$
$824$			18.0713i			0.629543i
$825$		0			0
$826$		−	42.9589i		−	1.49473i
$827$		−	18.1786i		−	0.632131i	−0.948737	−	0.316065i	$-0.897638\pi$
$827$		−	18.1786i		−	0.632131i	0.948737	−	0.316065i	$-0.102362\pi$
$828$		0			0
$829$	38.0468			1.32142			0.660711	−	0.750641i	$-0.270255\pi$
$829$	38.0468			1.32142			0.660711	+	0.750641i	$0.270255\pi$
$830$			104.816i			3.63823i
$831$		0			0
$832$	−17.1183	+	43.5483i	−0.593470	+	1.50976i
$833$	5.79851			0.200906
$834$		0			0
$835$	2.62424			0.0908155
$836$	5.35630			0.185251
$837$		0			0
$838$		−	4.71569i		−	0.162901i
$839$		−	28.6024i		−	0.987466i	−0.869614	−	0.493733i	$-0.835632\pi$
$839$		−	28.6024i		−	0.987466i	0.869614	−	0.493733i	$-0.164368\pi$
$840$		0			0
$841$	−28.0502			−0.967248
$842$	−77.3991			−2.66735
$843$		0			0
$844$	29.4982			1.01537
$845$	28.4681	−	30.6157i	0.979334	−	1.05321i
$846$		0			0
$847$			21.3798i			0.734620i
$848$	0.0792025			0.00271982
$849$		0			0
$850$		−	62.5577i		−	2.14571i
$851$		−	39.0343i		−	1.33808i
$852$		0			0
$853$			30.0536i			1.02902i	0.857485	+	0.514508i	$0.172026\pi$
$853$			30.0536i			1.02902i	−0.857485	+	0.514508i	$0.827974\pi$
$854$	−44.9986			−1.53982
$855$		0			0
$856$		−	25.9412i		−	0.886650i
$857$	1.33776			0.0456971			0.0228485	−	0.999739i	$-0.492726\pi$
$857$	1.33776			0.0456971			0.0228485	+	0.999739i	$0.492726\pi$
$858$		0			0
$859$	−41.2723			−1.40819			−0.704097	−	0.710104i	$-0.748648\pi$
$859$	−41.2723			−1.40819			−0.704097	+	0.710104i	$0.748648\pi$
$860$			89.4603i			3.05057i
$861$		0			0
$862$	69.9075			2.38106
$863$		−	37.3326i		−	1.27082i	−0.772176	−	0.635408i	$-0.780832\pi$
$863$		−	37.3326i		−	1.27082i	0.772176	−	0.635408i	$-0.219168\pi$
$864$		0			0
$865$		−	1.68214i		−	0.0571944i
$866$			19.7158i			0.669969i
$867$		0			0
$868$	−28.4289			−0.964939
$869$			5.51193i			0.186979i
$870$		0			0
$871$	2.59378	+	1.01958i	0.0878870	+	0.0345473i
$872$	40.3588			1.36672
$873$		0			0
$874$	−23.8371			−0.806301
$875$	2.66257			0.0900114
$876$		0			0
$877$		−	40.1885i		−	1.35707i	−0.734569	−	0.678534i	$-0.762616\pi$
$877$		−	40.1885i		−	1.35707i	0.734569	−	0.678534i	$-0.237384\pi$
$878$			67.4971i			2.27792i
$879$		0			0
$880$	0.507727			0.0171155
$881$	−13.8402			−0.466289			−0.233144	−	0.972442i	$-0.574901\pi$
$881$	−13.8402			−0.466289			−0.233144	+	0.972442i	$0.574901\pi$
$882$		0			0
$883$	−29.1280			−0.980235			−0.490118	−	0.871656i	$-0.663046\pi$
$883$	−29.1280			−0.980235			−0.490118	+	0.871656i	$0.663046\pi$
$884$	21.7336	−	55.2895i	0.730981	−	1.85959i
$885$		0			0
$886$		−	71.3801i		−	2.39806i
$887$	−18.1512			−0.609458			−0.304729	−	0.952439i	$-0.598566\pi$
$887$	−18.1512			−0.609458			−0.304729	+	0.952439i	$0.598566\pi$
$888$		0			0
$889$			0.397070i			0.0133173i
$890$		−	60.2946i		−	2.02108i
$891$		0			0
$892$			20.4544i			0.684864i
$893$	−1.74079			−0.0582534
$894$		0			0
$895$			45.5849i			1.52373i
$896$	−43.7818			−1.46265
$897$		0			0
$898$	−24.7352			−0.825424
$899$			3.56040i			0.118746i
$900$		0			0
$901$	−3.79544			−0.126444
$902$			13.4989i			0.449464i
$903$		0			0
$904$			8.48809i			0.282310i
$905$		−	57.8072i		−	1.92158i
$906$		0			0
$907$	5.11871			0.169964			0.0849820	−	0.996382i	$-0.472917\pi$
$907$	5.11871			0.169964			0.0849820	+	0.996382i	$0.472917\pi$
$908$		−	32.7515i		−	1.08690i
$909$		0			0
$910$	59.6865	+	23.4620i	1.97859	+	0.777758i
$911$	−42.2434			−1.39959			−0.699794	−	0.714345i	$-0.746725\pi$
$911$	−42.2434			−1.39959			−0.699794	+	0.714345i	$0.746725\pi$
$912$		0			0
$913$	21.0567			0.696876
$914$	36.6257			1.21147
$915$		0			0
$916$			26.0728i			0.861471i
$917$		−	15.8245i		−	0.522571i
$918$		0			0
$919$	37.2207			1.22780			0.613899	−	0.789384i	$-0.289600\pi$
$919$	37.2207			1.22780			0.613899	+	0.789384i	$0.289600\pi$
$920$	82.1722			2.70914
$921$		0			0
$922$	58.5104			1.92694
$923$	10.1050	+	3.97213i	0.332609	+	0.130744i
$924$		0			0
$925$			22.5791i			0.742396i
$926$	74.3202			2.44231
$927$		0			0
$928$			5.63149i			0.184863i
$929$			21.2444i			0.697005i	0.937308	+	0.348502i	$0.113310\pi$
$929$			21.2444i			0.697005i	−0.937308	+	0.348502i	$0.886690\pi$
$930$		0			0
$931$			1.27805i			0.0418865i
$932$	26.9100			0.881466
$933$		0			0
$934$		−	37.0251i		−	1.21150i
$935$	−24.3306			−0.795697
$936$		0			0
$937$	−55.6976			−1.81956			−0.909781	−	0.415089i	$-0.863750\pi$
$937$	−55.6976			−1.81956			−0.909781	+	0.415089i	$0.863750\pi$
$938$			4.27533i			0.139595i
$939$		0			0
$940$	15.9035			0.518714
$941$		−	19.9364i		−	0.649909i	−0.945730	−	0.324954i	$-0.894651\pi$
$941$		−	19.9364i		−	0.649909i	0.945730	−	0.324954i	$-0.105349\pi$
$942$		0			0
$943$		−	37.0218i		−	1.20559i
$944$		−	0.831414i		−	0.0270602i
$945$		0			0
$946$	29.1622			0.948146
$947$			32.6046i			1.05951i	0.848151	+	0.529754i	$0.177716\pi$
$947$			32.6046i			1.05951i	−0.848151	+	0.529754i	$0.822284\pi$
$948$		0			0
$949$	30.9055	+	12.1486i	1.00323	+	0.394359i
$950$	13.7884			0.447354
$951$		0			0
$952$	34.3880			1.11452
$953$	15.6146			0.505808			0.252904	−	0.967491i	$-0.418614\pi$
$953$	15.6146			0.505808			0.252904	+	0.967491i	$0.418614\pi$
$954$		0			0
$955$		−	29.8917i		−	0.967273i
$956$			14.8153i			0.479162i
$957$		0			0
$958$	−23.3686			−0.755005
$959$	−33.7470			−1.08975
$960$		0			0
$961$	17.6538			0.569478
$962$	−12.7288	+	32.3815i	−0.410392	+	1.04402i
$963$		0			0
$964$		−	25.8106i		−	0.831304i
$965$	−36.8953			−1.18770
$966$		0			0
$967$			4.01508i			0.129116i	0.997914	+	0.0645581i	$0.0205638\pi$
$967$			4.01508i			0.129116i	−0.997914	+	0.0645581i	$0.979436\pi$
$968$		−	24.4177i		−	0.784815i
$969$		0			0
$970$			111.801i			3.58972i
$971$	27.3969			0.879209			0.439604	−	0.898192i	$-0.355119\pi$
$971$	27.3969			0.879209			0.439604	+	0.898192i	$0.355119\pi$
$972$		0			0
$973$		−	34.3583i		−	1.10148i
$974$	36.7247			1.17674
$975$		0			0
$976$	−0.870889			−0.0278765
$977$		−	38.5516i		−	1.23337i	−0.787209	−	0.616687i	$-0.788475\pi$
$977$		−	38.5516i		−	1.23337i	0.787209	−	0.616687i	$-0.211525\pi$
$978$		0			0
$979$	−12.1127			−0.387123
$980$		−	11.6760i		−	0.372976i
$981$		0			0
$982$		−	10.0882i		−	0.321926i
$983$		−	46.5267i		−	1.48397i	−0.670416	−	0.741986i	$-0.733884\pi$
$983$		−	46.5267i		−	1.48397i	0.670416	−	0.741986i	$-0.266116\pi$
$984$		0			0
$985$	−35.4697			−1.13016
$986$		−	11.4135i		−	0.363479i
$987$		0			0
$988$	12.1864	+	4.79032i	0.387701	+	0.152400i
$989$	−79.9798			−2.54321
$990$		0			0
$991$	41.1031			1.30568			0.652841	−	0.757495i	$-0.273577\pi$
$991$	41.1031			1.30568			0.652841	+	0.757495i	$0.273577\pi$
$992$	−21.1097			−0.670233
$993$		0			0
$994$			16.6560i			0.528296i
$995$		−	4.45064i		−	0.141095i
$996$		0			0
$997$	−55.7772			−1.76648			−0.883241	−	0.468919i	$-0.844644\pi$
$997$	−55.7772			−1.76648			−0.883241	+	0.468919i	$0.844644\pi$
$998$	16.8548			0.533529
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.b.i.649.9		10
3.2	odd	2		1053.2.b.j.649.2		10
9.2	odd	6		117.2.t.c.103.2	yes	20
9.4	even	3		351.2.t.c.181.2		20
9.5	odd	6		117.2.t.c.25.9	yes	20
9.7	even	3		351.2.t.c.64.9		20
13.12	even	2	inner	1053.2.b.i.649.2		10
39.38	odd	2		1053.2.b.j.649.9		10
117.25	even	6		351.2.t.c.64.2		20
117.38	odd	6		117.2.t.c.103.9	yes	20
117.77	odd	6		117.2.t.c.25.2	✓	20
117.103	even	6		351.2.t.c.181.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.2	✓	20	117.77	odd	6
117.2.t.c.25.9	yes	20	9.5	odd	6
117.2.t.c.103.2	yes	20	9.2	odd	6
117.2.t.c.103.9	yes	20	117.38	odd	6
351.2.t.c.64.2		20	117.25	even	6
351.2.t.c.64.9		20	9.7	even	3
351.2.t.c.181.2		20	9.4	even	3
351.2.t.c.181.9		20	117.103	even	6
1053.2.b.i.649.2		10	13.12	even	2	inner
1053.2.b.i.649.9		10	1.1	even	1	trivial
1053.2.b.j.649.2		10	3.2	odd	2
1053.2.b.j.649.9		10	39.38	odd	2

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:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.28298i q^{2} -3.21200 q^{4} +3.21586i q^{5} +2.42273i q^{7} -2.76698i q^{8} -7.34174 q^{10} +1.47490i q^{11} +(-1.31905 + 3.35561i) q^{13} -5.53105 q^{14} -0.107046 q^{16} +5.12974 q^{17} +1.13065i q^{19} -10.3293i q^{20} -3.36716 q^{22} +9.23470 q^{23} -5.34174 q^{25} +(-7.66079 - 3.01136i) q^{26} -7.78182i q^{28} -0.974585 q^{29} -3.65324i q^{31} -5.77834i q^{32} +11.7111i q^{34} -7.79116 q^{35} -4.22691i q^{37} -2.58125 q^{38} +8.89821 q^{40} -4.00898i q^{41} -8.66079 q^{43} -4.73737i q^{44} +21.0826i q^{46} +1.53964i q^{47} +1.13037 q^{49} -12.1951i q^{50} +(4.23679 - 10.7782i) q^{52} -0.739889 q^{53} -4.74305 q^{55} +6.70365 q^{56} -2.22496i q^{58} +7.76686i q^{59} +8.13562 q^{61} +8.34028 q^{62} +12.9777 q^{64} +(-10.7912 - 4.24187i) q^{65} -0.772969i q^{67} -16.4767 q^{68} -17.7871i q^{70} -3.01136i q^{71} -9.21010i q^{73} +9.64996 q^{74} -3.63165i q^{76} -3.57328 q^{77} +3.73717 q^{79} -0.344246i q^{80} +9.15243 q^{82} -14.2768i q^{83} +16.4965i q^{85} -19.7724i q^{86} +4.08100 q^{88} +8.21257i q^{89} +(-8.12974 - 3.19570i) q^{91} -29.6619 q^{92} -3.51497 q^{94} -3.63601 q^{95} -15.2281i q^{97} +2.58061i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} - 18 q^{14} - 4 q^{16} + 6 q^{17} + 10 q^{22} + 24 q^{23} + 12 q^{25} + 6 q^{26} + 12 q^{29} + 6 q^{35} + 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} - 54 q^{53} + 10 q^{55}+ \cdots + 24 q^{95}+O(q^{100}) \) 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 - 18 * q^14 - 4 * q^16 + 6 * q^17 + 10 * q^22 + 24 * q^23 + 12 * q^25 + 6 * q^26 + 12 * q^29 + 6 * q^35 + 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 - 54 * q^53 + 10 * q^55 + 36 * q^56 + 2 * q^61 + 36 * q^62 + 4 * q^64 - 24 * q^65 + 24 * q^68 - 42 * q^74 - 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 - 84 * q^92 - 20 * q^94 + 24 * q^95

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