LMFDB - Embedded newform 531.3.c.c.235.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	531.c (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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.5
Root			$-2.72775i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72775i q^{2} -3.44061 q^{4} -5.71516 q^{5} +8.69147 q^{7} -1.52587i q^{8} +O(q^{10})$	\(q-2.72775i q^{2} -3.44061 q^{4} -5.71516 q^{5} +8.69147 q^{7} -1.52587i q^{8} +15.5895i q^{10} -6.01328i q^{11} -4.81519i q^{13} -23.7081i q^{14} -17.9246 q^{16} -10.8327 q^{17} -18.4655 q^{19} +19.6636 q^{20} -16.4027 q^{22} -15.4084i q^{23} +7.66303 q^{25} -13.1346 q^{26} -29.9040 q^{28} -15.5293 q^{29} -6.38687i q^{31} +42.7904i q^{32} +29.5490i q^{34} -49.6731 q^{35} -0.527041i q^{37} +50.3693i q^{38} +8.72058i q^{40} -9.31080 q^{41} +52.9611i q^{43} +20.6894i q^{44} -42.0302 q^{46} +78.0549i q^{47} +26.5417 q^{49} -20.9028i q^{50} +16.5672i q^{52} +10.2232 q^{53} +34.3668i q^{55} -13.2620i q^{56} +42.3600i q^{58} +(-29.4888 - 51.1020i) q^{59} -41.5604i q^{61} -17.4218 q^{62} +45.0230 q^{64} +27.5196i q^{65} -92.4125i q^{67} +37.2712 q^{68} +135.496i q^{70} -93.8038 q^{71} -88.0161i q^{73} -1.43764 q^{74} +63.5327 q^{76} -52.2642i q^{77} -81.2111 q^{79} +102.442 q^{80} +25.3975i q^{82} +2.92193i q^{83} +61.9108 q^{85} +144.465 q^{86} -9.17547 q^{88} +168.718i q^{89} -41.8511i q^{91} +53.0143i q^{92} +212.914 q^{94} +105.533 q^{95} -169.166i q^{97} -72.3990i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	2.72775i		−	1.36387i	−0.731411	−	0.681937i	$-0.761138\pi$
$2$		−	2.72775i		−	1.36387i	0.731411	−	0.681937i	$-0.238862\pi$
$3$		0			0
$3$		0			0
$4$	−3.44061			−0.860153
$5$	−5.71516			−1.14303			−0.571516	−	0.820591i	$-0.693644\pi$
$5$	−5.71516			−1.14303			−0.571516	+	0.820591i	$0.693644\pi$
$6$		0			0
$7$	8.69147			1.24164			0.620819	−	0.783954i	$-0.286800\pi$
$7$	8.69147			1.24164			0.620819	+	0.783954i	$0.286800\pi$
$8$		−	1.52587i		−	0.190733i
$9$		0			0
$10$			15.5895i			1.55895i
$11$		−	6.01328i		−	0.546662i	−0.961920	−	0.273331i	$-0.911875\pi$
$11$		−	6.01328i		−	0.546662i	0.961920	−	0.273331i	$-0.0881254\pi$
$12$		0			0
$13$		−	4.81519i		−	0.370399i	−0.982701	−	0.185200i	$-0.940707\pi$
$13$		−	4.81519i		−	0.370399i	0.982701	−	0.185200i	$-0.0592932\pi$
$14$		−	23.7081i		−	1.69344i
$15$		0			0
$16$	−17.9246			−1.12029
$17$	−10.8327			−0.637220			−0.318610	−	0.947886i	$-0.603216\pi$
$17$	−10.8327			−0.637220			−0.318610	+	0.947886i	$0.603216\pi$
$18$		0			0
$19$	−18.4655			−0.971869			−0.485935	−	0.873995i	$-0.661521\pi$
$19$	−18.4655			−0.971869			−0.485935	+	0.873995i	$0.661521\pi$
$20$	19.6636			0.983182
$21$		0			0
$22$	−16.4027			−0.745578
$23$		−	15.4084i		−	0.669930i	−0.942231	−	0.334965i	$-0.891276\pi$
$23$		−	15.4084i		−	0.669930i	0.942231	−	0.334965i	$-0.108724\pi$
$24$		0			0
$25$	7.66303			0.306521
$26$	−13.1346			−0.505178
$27$		0			0
$28$	−29.9040			−1.06800
$29$	−15.5293			−0.535492			−0.267746	−	0.963489i	$-0.586279\pi$
$29$	−15.5293			−0.535492			−0.267746	+	0.963489i	$0.586279\pi$
$30$		0			0
$31$		−	6.38687i		−	0.206028i	−0.994680	−	0.103014i	$-0.967151\pi$
$31$		−	6.38687i		−	0.206028i	0.994680	−	0.103014i	$-0.0328487\pi$
$32$			42.7904i			1.33720i
$33$		0			0
$34$			29.5490i			0.869087i
$35$	−49.6731			−1.41923
$36$		0			0
$37$		−	0.527041i		−	0.0142444i	−0.999975	−	0.00712218i	$-0.997733\pi$
$37$		−	0.527041i		−	0.0142444i	0.999975	−	0.00712218i	$-0.00226708\pi$
$38$			50.3693i			1.32551i
$39$		0			0
$40$			8.72058i			0.218014i
$41$	−9.31080			−0.227093			−0.113546	−	0.993533i	$-0.536221\pi$
$41$	−9.31080			−0.227093			−0.113546	+	0.993533i	$0.536221\pi$
$42$		0			0
$43$			52.9611i			1.23165i	0.787882	+	0.615827i	$0.211178\pi$
$43$			52.9611i			1.23165i	−0.787882	+	0.615827i	$0.788822\pi$
$44$			20.6894i			0.470213i
$45$		0			0
$46$	−42.0302			−0.913700
$47$			78.0549i			1.66074i	0.557211	+	0.830371i	$0.311872\pi$
$47$			78.0549i			1.66074i	−0.557211	+	0.830371i	$0.688128\pi$
$48$		0			0
$49$	26.5417			0.541666
$50$		−	20.9028i		−	0.418056i
$51$		0			0
$52$			16.5672i			0.318600i
$53$	10.2232			0.192891			0.0964457	−	0.995338i	$-0.469253\pi$
$53$	10.2232			0.192891			0.0964457	+	0.995338i	$0.469253\pi$
$54$		0			0
$55$			34.3668i			0.624851i
$56$		−	13.2620i		−	0.236822i
$57$		0			0
$58$			42.3600i			0.730344i
$59$	−29.4888	−	51.1020i	−0.499810	−	0.866135i
$59$	−29.4888	−	51.1020i	−0.499810	−	0.866135i
$60$		0			0
$61$		−	41.5604i		−	0.681318i	−0.940187	−	0.340659i	$-0.889350\pi$
$61$		−	41.5604i		−	0.681318i	0.940187	−	0.340659i	$-0.110650\pi$
$62$	−17.4218			−0.280997
$63$		0			0
$64$	45.0230			0.703484
$65$			27.5196i			0.423378i
$66$		0			0
$67$		−	92.4125i		−	1.37929i	−0.724147	−	0.689646i	$-0.757766\pi$
$67$		−	92.4125i		−	1.37929i	0.724147	−	0.689646i	$-0.242234\pi$
$68$	37.2712			0.548106
$69$		0			0
$70$			135.496i			1.93565i
$71$	−93.8038			−1.32118			−0.660590	−	0.750747i	$-0.729694\pi$
$71$	−93.8038			−1.32118			−0.660590	+	0.750747i	$0.729694\pi$
$72$		0			0
$73$		−	88.0161i		−	1.20570i	−0.797854	−	0.602850i	$-0.794032\pi$
$73$		−	88.0161i		−	1.20570i	0.797854	−	0.602850i	$-0.205968\pi$
$74$	−1.43764			−0.0194275
$75$		0			0
$76$	63.5327			0.835957
$77$		−	52.2642i		−	0.678756i
$78$		0			0
$79$	−81.2111			−1.02799			−0.513994	−	0.857794i	$-0.671835\pi$
$79$	−81.2111			−1.02799			−0.513994	+	0.857794i	$0.671835\pi$
$80$	102.442			1.28053
$81$		0			0
$82$			25.3975i			0.309726i
$83$			2.92193i			0.0352040i	0.999845	+	0.0176020i	$0.00560318\pi$
$83$			2.92193i			0.0352040i	−0.999845	+	0.0176020i	$0.994397\pi$
$84$		0			0
$85$	61.9108			0.728362
$86$	144.465			1.67982
$87$		0			0
$88$	−9.17547			−0.104267
$89$			168.718i			1.89571i	0.318699	+	0.947856i	$0.396754\pi$
$89$			168.718i			1.89571i	−0.318699	+	0.947856i	$0.603246\pi$
$90$		0			0
$91$		−	41.8511i		−	0.459902i
$92$			53.0143i			0.576242i
$93$		0			0
$94$	212.914			2.26504
$95$	105.533			1.11088
$96$		0			0
$97$		−	169.166i		−	1.74397i	−0.489528	−	0.871987i	$-0.662831\pi$
$97$		−	169.166i		−	1.74397i	0.489528	−	0.871987i	$-0.337169\pi$
$98$		−	72.3990i		−	0.738765i
$99$		0			0
$100$	−26.3655			−0.263655
$101$			6.43636i			0.0637263i	0.999492	+	0.0318631i	$0.0101441\pi$
$101$			6.43636i			0.0637263i	−0.999492	+	0.0318631i	$0.989856\pi$
$102$		0			0
$103$			176.969i			1.71814i	0.511854	+	0.859072i	$0.328959\pi$
$103$			176.969i			1.71814i	−0.511854	+	0.859072i	$0.671041\pi$
$104$	−7.34735			−0.0706476
$105$		0			0
$106$		−	27.8864i		−	0.263080i
$107$	183.573			1.71564			0.857818	−	0.513953i	$-0.171819\pi$
$107$	183.573			1.71564			0.857818	+	0.513953i	$0.171819\pi$
$108$		0			0
$109$		−	105.308i		−	0.966132i	−0.875584	−	0.483066i	$-0.839523\pi$
$109$		−	105.308i		−	0.966132i	0.875584	−	0.483066i	$-0.160477\pi$
$110$	93.7441			0.852219
$111$		0			0
$112$	−155.791			−1.39099
$113$		−	100.717i		−	0.891297i	−0.895208	−	0.445648i	$-0.852973\pi$
$113$		−	100.717i		−	0.891297i	0.895208	−	0.445648i	$-0.147027\pi$
$114$		0			0
$115$			88.0614i			0.765751i
$116$	53.4302			0.460606
$117$		0			0
$118$	−139.393	+	80.4379i	−1.18130	+	0.681677i
$119$	−94.1524			−0.791196
$120$		0			0
$121$	84.8405			0.701161
$122$	−113.366			−0.929233
$123$		0			0
$124$			21.9748i			0.177216i
$125$	99.0835			0.792668
$126$		0			0
$127$	−206.165			−1.62335			−0.811675	−	0.584110i	$-0.801444\pi$
$127$	−206.165			−1.62335			−0.811675	+	0.584110i	$0.801444\pi$
$128$			48.3503i			0.377737i
$129$		0			0
$130$	75.0665			0.577435
$131$		−	152.579i		−	1.16473i	−0.812929	−	0.582363i	$-0.802128\pi$
$131$		−	152.579i		−	1.16473i	0.812929	−	0.582363i	$-0.197872\pi$
$132$		0			0
$133$	−160.493			−1.20671
$134$	−252.078			−1.88118
$135$		0			0
$136$			16.5293i			0.121539i
$137$	−193.341			−1.41125			−0.705624	−	0.708586i	$-0.749333\pi$
$137$	−193.341			−1.41125			−0.705624	+	0.708586i	$0.749333\pi$
$138$		0			0
$139$	172.850			1.24353			0.621763	−	0.783206i	$-0.286417\pi$
$139$	172.850			1.24353			0.621763	+	0.783206i	$0.286417\pi$
$140$	170.906			1.22076
$141$		0			0
$142$			255.873i			1.80192i
$143$	−28.9551			−0.202483
$144$		0			0
$145$	88.7523			0.612085
$146$	−240.086			−1.64442
$147$		0			0
$148$			1.81335i			0.0122523i
$149$		−	181.616i		−	1.21890i	−0.792826	−	0.609448i	$-0.791391\pi$
$149$		−	181.616i		−	1.21890i	0.792826	−	0.609448i	$-0.208609\pi$
$150$		0			0
$151$		−	1.21185i		−	0.00802549i	−0.999992	−	0.00401275i	$-0.998723\pi$
$151$		−	1.21185i		−	0.00802549i	0.999992	−	0.00401275i	$-0.00127730\pi$
$152$			28.1759i			0.185368i
$153$		0			0
$154$	−142.564			−0.925738
$155$			36.5020i			0.235497i
$156$		0			0
$157$		−	184.394i		−	1.17449i	−0.809410	−	0.587244i	$-0.800213\pi$
$157$		−	184.394i		−	1.17449i	0.809410	−	0.587244i	$-0.199787\pi$
$158$			221.523i			1.40205i
$159$		0			0
$160$		−	244.554i		−	1.52846i
$161$		−	133.922i		−	0.831811i
$162$		0			0
$163$	8.96097			0.0549752			0.0274876	−	0.999622i	$-0.491249\pi$
$163$	8.96097			0.0549752			0.0274876	+	0.999622i	$0.491249\pi$
$164$	32.0348			0.195334
$165$		0			0
$166$	7.97030			0.0480139
$167$	143.336			0.858298			0.429149	−	0.903234i	$-0.358813\pi$
$167$	143.336			0.858298			0.429149	+	0.903234i	$0.358813\pi$
$168$		0			0
$169$	145.814			0.862804
$170$		−	168.877i		−	0.993394i
$171$		0			0
$172$		−	182.219i		−	1.05941i
$173$		−	57.4482i		−	0.332070i	−0.986120	−	0.166035i	$-0.946903\pi$
$173$		−	57.4482i		−	0.332070i	0.986120	−	0.166035i	$-0.0530965\pi$
$174$		0			0
$175$	66.6030			0.380588
$176$			107.786i			0.612419i
$177$		0			0
$178$	460.221			2.58551
$179$		−	90.1857i		−	0.503831i	−0.967749	−	0.251915i	$-0.918940\pi$
$179$		−	90.1857i		−	0.503831i	0.967749	−	0.251915i	$-0.0810604\pi$
$180$		0			0
$181$	123.939			0.684745			0.342373	−	0.939564i	$-0.388769\pi$
$181$	123.939			0.684745			0.342373	+	0.939564i	$0.388769\pi$
$182$	−114.159			−0.627249
$183$		0			0
$184$	−23.5112			−0.127778
$185$			3.01213i			0.0162818i
$186$		0			0
$187$			65.1402i			0.348343i
$188$		−	268.557i		−	1.42849i
$189$		0			0
$190$		−	287.868i		−	1.51510i
$191$		−	150.842i		−	0.789751i	−0.918735	−	0.394876i	$-0.870788\pi$
$191$		−	150.842i		−	0.789751i	0.918735	−	0.394876i	$-0.129212\pi$
$192$		0			0
$193$	−84.8208			−0.439486			−0.219743	−	0.975558i	$-0.570522\pi$
$193$	−84.8208			−0.439486			−0.219743	+	0.975558i	$0.570522\pi$
$194$	−461.441			−2.37856
$195$		0			0
$196$	−91.3195			−0.465916
$197$	190.721			0.968127			0.484063	−	0.875033i	$-0.339160\pi$
$197$	190.721			0.968127			0.484063	+	0.875033i	$0.339160\pi$
$198$		0			0
$199$	−200.054			−1.00529			−0.502647	−	0.864492i	$-0.667641\pi$
$199$	−200.054			−1.00529			−0.502647	+	0.864492i	$0.667641\pi$
$200$		−	11.6928i		−	0.0584638i
$201$		0			0
$202$	17.5568			0.0869147
$203$	−134.972			−0.664888
$204$		0			0
$205$	53.2127			0.259574
$206$	482.727			2.34333
$207$		0			0
$208$			86.3106i			0.414955i
$209$			111.038i			0.531284i
$210$		0			0
$211$		−	412.135i		−	1.95325i	−0.214955	−	0.976624i	$-0.568961\pi$
$211$		−	412.135i		−	1.95325i	0.214955	−	0.976624i	$-0.431039\pi$
$212$	−35.1742			−0.165916
$213$		0			0
$214$		−	500.741i		−	2.33991i
$215$		−	302.681i		−	1.40782i
$216$		0			0
$217$		−	55.5113i		−	0.255813i
$218$	−287.255			−1.31768
$219$		0			0
$220$		−	118.243i		−	0.537468i
$221$			52.1617i			0.236026i
$222$		0			0
$223$	−22.4832			−0.100821			−0.0504107	−	0.998729i	$-0.516053\pi$
$223$	−22.4832			−0.100821			−0.0504107	+	0.998729i	$0.516053\pi$
$224$			371.912i			1.66032i
$225$		0			0
$226$	−274.729			−1.21562
$227$			276.847i			1.21959i	0.792559	+	0.609795i	$0.208748\pi$
$227$			276.847i			1.21959i	−0.792559	+	0.609795i	$0.791252\pi$
$228$		0			0
$229$			160.926i			0.702732i	0.936238	+	0.351366i	$0.114283\pi$
$229$			160.926i			0.702732i	−0.936238	+	0.351366i	$0.885717\pi$
$230$	240.209			1.04439
$231$		0			0
$232$			23.6956i			0.102136i
$233$			10.6284i			0.0456153i	0.999740	+	0.0228076i	$0.00726052\pi$
$233$			10.6284i			0.0456153i	−0.999740	+	0.0228076i	$0.992739\pi$
$234$		0			0
$235$		−	446.096i		−	1.89828i
$236$	101.459	+	175.822i	0.429913	+	0.745009i
$237$		0			0
$238$			256.824i			1.07909i
$239$	−223.594			−0.935540			−0.467770	−	0.883850i	$-0.654942\pi$
$239$	−223.594			−0.935540			−0.467770	+	0.883850i	$0.654942\pi$
$240$		0			0
$241$	64.6207			0.268136			0.134068	−	0.990972i	$-0.457196\pi$
$241$	64.6207			0.268136			0.134068	+	0.990972i	$0.457196\pi$
$242$		−	231.424i		−	0.956296i
$243$		0			0
$244$			142.993i			0.586038i
$245$	−151.690			−0.619142
$246$		0			0
$247$			88.9150i			0.359980i
$248$	−9.74553			−0.0392965
$249$		0			0
$250$		−	270.275i		−	1.08110i
$251$	142.392			0.567299			0.283650	−	0.958928i	$-0.408455\pi$
$251$	142.392			0.567299			0.283650	+	0.958928i	$0.408455\pi$
$252$		0			0
$253$	−92.6549			−0.366225
$254$			562.367i			2.21405i
$255$		0			0
$256$	311.979			1.21867
$257$	−13.2725			−0.0516440			−0.0258220	−	0.999667i	$-0.508220\pi$
$257$	−13.2725			−0.0516440			−0.0258220	+	0.999667i	$0.508220\pi$
$258$		0			0
$259$		−	4.58077i		−	0.0176864i
$260$		−	94.6843i		−	0.364170i
$261$		0			0
$262$	−416.198			−1.58854
$263$	54.0179			0.205391			0.102696	−	0.994713i	$-0.467253\pi$
$263$	54.0179			0.205391			0.102696	+	0.994713i	$0.467253\pi$
$264$		0			0
$265$	−58.4274			−0.220481
$266$			437.783i			1.64580i
$267$		0			0
$268$			317.956i			1.18640i
$269$		−	372.720i		−	1.38558i	−0.721140	−	0.692789i	$-0.756382\pi$
$269$		−	372.720i		−	1.38558i	0.721140	−	0.692789i	$-0.243618\pi$
$270$		0			0
$271$	531.359			1.96074			0.980368	−	0.197177i	$-0.0631773\pi$
$271$	531.359			1.96074			0.980368	+	0.197177i	$0.0631773\pi$
$272$	194.173			0.713871
$273$		0			0
$274$			527.386i			1.92477i
$275$		−	46.0799i		−	0.167563i
$276$		0			0
$277$	136.910			0.494260			0.247130	−	0.968982i	$-0.420512\pi$
$277$	136.910			0.494260			0.247130	+	0.968982i	$0.420512\pi$
$278$		−	471.492i		−	1.69601i
$279$		0			0
$280$			75.7946i			0.270695i
$281$	186.333			0.663108			0.331554	−	0.943436i	$-0.392427\pi$
$281$	186.333			0.663108			0.331554	+	0.943436i	$0.392427\pi$
$282$		0			0
$283$			370.038i			1.30755i	0.756687	+	0.653777i	$0.226817\pi$
$283$			370.038i			1.30755i	−0.756687	+	0.653777i	$0.773183\pi$
$284$	322.743			1.13642
$285$		0			0
$286$			78.9822i			0.276162i
$287$	−80.9245			−0.281967
$288$		0			0
$289$	−171.652			−0.593951
$290$		−	242.094i		−	0.834807i
$291$		0			0
$292$			302.829i			1.03709i
$293$	−299.019			−1.02054			−0.510271	−	0.860013i	$-0.670455\pi$
$293$	−299.019			−1.02054			−0.510271	+	0.860013i	$0.670455\pi$
$294$		0			0
$295$	168.533	+	292.056i	0.571298	+	0.990020i
$296$	−0.804196			−0.00271688
$297$		0			0
$298$	−495.402			−1.66242
$299$	−74.1944			−0.248142
$300$		0			0
$301$			460.310i			1.52927i
$302$	−3.30562			−0.0109458
$303$		0			0
$304$	330.988			1.08878
$305$			237.524i			0.778768i
$306$		0			0
$307$	27.6515			0.0900700			0.0450350	−	0.998985i	$-0.485660\pi$
$307$	27.6515			0.0900700			0.0450350	+	0.998985i	$0.485660\pi$
$308$			179.821i			0.583834i
$309$		0			0
$310$	99.5682			0.321188
$311$	364.708			1.17269			0.586347	−	0.810060i	$-0.300566\pi$
$311$	364.708			1.17269			0.586347	+	0.810060i	$0.300566\pi$
$312$		0			0
$313$			174.644i			0.557969i	0.960296	+	0.278984i	$0.0899978\pi$
$313$			174.644i			0.557969i	−0.960296	+	0.278984i	$0.910002\pi$
$314$	−502.982			−1.60185
$315$		0			0
$316$	279.416			0.884228
$317$	505.102			1.59338			0.796691	−	0.604387i	$-0.206582\pi$
$317$	505.102			1.59338			0.796691	+	0.604387i	$0.206582\pi$
$318$		0			0
$319$			93.3819i			0.292733i
$320$	−257.313			−0.804105
$321$		0			0
$322$	−365.304			−1.13449
$323$	200.032			0.619294
$324$		0			0
$325$		−	36.8990i		−	0.113535i
$326$		−	24.4433i		−	0.0749793i
$327$		0			0
$328$			14.2070i			0.0433142i
$329$			678.412i			2.06204i
$330$		0			0
$331$	3.99311			0.0120638			0.00603188	−	0.999982i	$-0.498080\pi$
$331$	3.99311			0.0120638			0.00603188	+	0.999982i	$0.498080\pi$
$332$		−	10.0532i		−	0.0302809i
$333$		0			0
$334$		−	390.984i		−	1.17061i
$335$			528.152i			1.57657i
$336$		0			0
$337$		−	347.538i		−	1.03127i	−0.856808	−	0.515636i	$-0.827556\pi$
$337$		−	347.538i		−	1.03127i	0.856808	−	0.515636i	$-0.172444\pi$
$338$		−	397.744i		−	1.17676i
$339$		0			0
$340$	−213.011			−0.626503
$341$	−38.4060			−0.112628
$342$		0			0
$343$	−195.196			−0.569085
$344$	80.8116			0.234917
$345$		0			0
$346$	−156.704			−0.452902
$347$		−	587.431i		−	1.69289i	−0.532480	−	0.846443i	$-0.678740\pi$
$347$		−	587.431i		−	1.69289i	0.532480	−	0.846443i	$-0.321260\pi$
$348$		0			0
$349$			395.445i			1.13308i	0.824034	+	0.566540i	$0.191718\pi$
$349$			395.445i			1.13308i	−0.824034	+	0.566540i	$0.808282\pi$
$350$		−	181.676i		−	0.519075i
$351$		0			0
$352$	257.311			0.730996
$353$			201.884i			0.571909i	0.958243	+	0.285955i	$0.0923106\pi$
$353$			201.884i			0.571909i	−0.958243	+	0.285955i	$0.907689\pi$
$354$		0			0
$355$	536.104			1.51015
$356$		−	580.494i		−	1.63060i
$357$		0			0
$358$	−246.004			−0.687162
$359$	529.552			1.47507			0.737537	−	0.675307i	$-0.235989\pi$
$359$	529.552			1.47507			0.737537	+	0.675307i	$0.235989\pi$
$360$		0			0
$361$	−20.0246			−0.0554698
$362$		−	338.074i		−	0.933906i
$363$		0			0
$364$			143.993i			0.395586i
$365$			503.026i			1.37815i
$366$		0			0
$367$		−	359.579i		−	0.979779i	−0.871785	−	0.489889i	$-0.837037\pi$
$367$		−	359.579i		−	0.979779i	0.871785	−	0.489889i	$-0.162963\pi$
$368$			276.190i			0.750516i
$369$		0			0
$370$	8.21632			0.0222063
$371$	88.8550			0.239501
$372$		0			0
$373$	−439.679			−1.17876			−0.589382	−	0.807854i	$-0.700629\pi$
$373$	−439.679			−1.17876			−0.589382	+	0.807854i	$0.700629\pi$
$374$	177.686			0.475097
$375$		0			0
$376$	119.101			0.316759
$377$			74.7765i			0.198346i
$378$		0			0
$379$	−580.426			−1.53147			−0.765734	−	0.643158i	$-0.777624\pi$
$379$	−580.426			−1.53147			−0.765734	+	0.643158i	$0.777624\pi$
$380$	−363.099			−0.955525
$381$		0			0
$382$	−411.460			−1.07712
$383$	343.315			0.896384			0.448192	−	0.893937i	$-0.352068\pi$
$383$	343.315			0.896384			0.448192	+	0.893937i	$0.352068\pi$
$384$		0			0
$385$			298.698i			0.775840i
$386$			231.370i			0.599404i
$387$		0			0
$388$			582.033i			1.50009i
$389$	−303.425			−0.780013			−0.390007	−	0.920812i	$-0.627527\pi$
$389$	−303.425			−0.780013			−0.390007	+	0.920812i	$0.627527\pi$
$390$		0			0
$391$			166.915i			0.426893i
$392$		−	40.4991i		−	0.103314i
$393$		0			0
$394$		−	520.239i		−	1.32040i
$395$	464.134			1.17502
$396$		0			0
$397$			159.184i			0.400967i	0.979697	+	0.200484i	$0.0642513\pi$
$397$			159.184i			0.400967i	−0.979697	+	0.200484i	$0.935749\pi$
$398$			545.696i			1.37110i
$399$		0			0
$400$	−137.357			−0.343392
$401$			421.293i			1.05061i	0.850915	+	0.525303i	$0.176048\pi$
$401$			421.293i			1.05061i	−0.850915	+	0.525303i	$0.823952\pi$
$402$		0			0
$403$	−30.7540			−0.0763127
$404$		−	22.1450i		−	0.0548144i
$405$		0			0
$406$			368.170i			0.906824i
$407$	−3.16925			−0.00778685
$408$		0			0
$409$			662.202i			1.61908i	0.587068	+	0.809538i	$0.300282\pi$
$409$			662.202i			1.61908i	−0.587068	+	0.809538i	$0.699718\pi$
$410$		−	145.151i		−	0.354026i
$411$		0			0
$412$		−	608.881i		−	1.47787i
$413$	−256.301	−	444.151i	−0.620583	−	1.07543i
$414$		0			0
$415$		−	16.6993i		−	0.0402393i
$416$	206.044			0.495299
$417$		0			0
$418$	302.885			0.724604
$419$		−	338.057i		−	0.806818i	−0.915020	−	0.403409i	$-0.867825\pi$
$419$		−	338.057i		−	0.806818i	0.915020	−	0.403409i	$-0.132175\pi$
$420$		0			0
$421$			101.363i			0.240766i	0.992727	+	0.120383i	$0.0384123\pi$
$421$			101.363i			0.240766i	−0.992727	+	0.120383i	$0.961588\pi$
$422$	−1124.20			−2.66398
$423$		0			0
$424$		−	15.5993i		−	0.0367908i
$425$	−83.0115			−0.195321
$426$		0			0
$427$		−	361.221i		−	0.845951i
$428$	−631.604			−1.47571
$429$		0			0
$430$	−825.637			−1.92009
$431$			47.6635i			0.110588i	0.998470	+	0.0552941i	$0.0176096\pi$
$431$			47.6635i			0.110588i	−0.998470	+	0.0552941i	$0.982390\pi$
$432$		0			0
$433$	−172.895			−0.399294			−0.199647	−	0.979868i	$-0.563980\pi$
$433$	−172.895			−0.399294			−0.199647	+	0.979868i	$0.563980\pi$
$434$	−151.421			−0.348896
$435$		0			0
$436$			362.325i			0.831022i
$437$			284.524i			0.651084i
$438$		0			0
$439$	384.161			0.875081			0.437540	−	0.899199i	$-0.355850\pi$
$439$	384.161			0.875081			0.437540	+	0.899199i	$0.355850\pi$
$440$	52.4392			0.119180
$441$		0			0
$442$	142.284			0.321910
$443$			172.202i			0.388717i	0.980931	+	0.194359i	$0.0622625\pi$
$443$			172.202i			0.388717i	−0.980931	+	0.194359i	$0.937737\pi$
$444$		0			0
$445$		−	964.252i		−	2.16686i
$446$			61.3285i			0.137508i
$447$		0			0
$448$	391.316			0.873473
$449$	−613.309			−1.36595			−0.682973	−	0.730444i	$-0.739313\pi$
$449$	−613.309			−1.36595			−0.682973	+	0.730444i	$0.739313\pi$
$450$		0			0
$451$			55.9884i			0.124143i
$452$			346.527i			0.766652i
$453$		0			0
$454$	755.168			1.66337
$455$			239.186i			0.525683i
$456$		0			0
$457$			279.448i			0.611484i	0.952114	+	0.305742i	$0.0989046\pi$
$457$			279.448i			0.611484i	−0.952114	+	0.305742i	$0.901095\pi$
$458$	438.965			0.958438
$459$		0			0
$460$		−	302.985i		−	0.658663i
$461$	555.187			1.20431			0.602155	−	0.798379i	$-0.294309\pi$
$461$	555.187			1.20431			0.602155	+	0.798379i	$0.294309\pi$
$462$		0			0
$463$		−	672.785i		−	1.45310i	−0.687114	−	0.726550i	$-0.741123\pi$
$463$		−	672.785i		−	1.45310i	0.687114	−	0.726550i	$-0.258877\pi$
$464$	278.357			0.599907
$465$		0			0
$466$	28.9915			0.0622135
$467$		−	436.023i		−	0.933669i	−0.884345	−	0.466834i	$-0.845394\pi$
$467$		−	436.023i		−	0.933669i	0.884345	−	0.466834i	$-0.154606\pi$
$468$		0			0
$469$		−	803.201i		−	1.71258i
$470$	−1216.84			−2.58902
$471$		0			0
$472$	−77.9749	+	44.9960i	−0.165201	+	0.0953304i
$473$	318.470			0.673297
$474$		0			0
$475$	−141.502			−0.297899
$476$	323.942			0.680550
$477$		0			0
$478$			609.908i			1.27596i
$479$	−620.966			−1.29638			−0.648190	−	0.761479i	$-0.724474\pi$
$479$	−620.966			−1.29638			−0.648190	+	0.761479i	$0.724474\pi$
$480$		0			0
$481$	−2.53781			−0.00527611
$482$		−	176.269i		−	0.365704i
$483$		0			0
$484$	−291.903			−0.603106
$485$			966.808i			1.99342i
$486$		0			0
$487$	683.788			1.40408			0.702041	−	0.712136i	$-0.252272\pi$
$487$	683.788			1.40408			0.702041	+	0.712136i	$0.252272\pi$
$488$	−63.4157			−0.129950
$489$		0			0
$490$			413.771i			0.844432i
$491$	−650.610			−1.32507			−0.662536	−	0.749030i	$-0.730520\pi$
$491$	−650.610			−1.32507			−0.662536	+	0.749030i	$0.730520\pi$
$492$		0			0
$493$	168.225			0.341226
$494$	242.538			0.490967
$495$		0			0
$496$			114.482i			0.230811i
$497$	−815.293			−1.64043
$498$		0			0
$499$	−503.786			−1.00959			−0.504796	−	0.863239i	$-0.668432\pi$
$499$	−503.786			−1.00959			−0.504796	+	0.863239i	$0.668432\pi$
$500$	−340.908			−0.681816
$501$		0			0
$502$		−	388.410i		−	0.773725i
$503$		−	622.714i		−	1.23800i	−0.785391	−	0.619000i	$-0.787538\pi$
$503$		−	622.714i		−	1.23800i	0.785391	−	0.619000i	$-0.212462\pi$
$504$		0			0
$505$		−	36.7848i		−	0.0728412i
$506$			252.739i			0.499485i
$507$		0			0
$508$	709.335			1.39633
$509$		−	338.691i		−	0.665405i	−0.943032	−	0.332702i	$-0.892040\pi$
$509$		−	338.691i		−	0.665405i	0.943032	−	0.332702i	$-0.107960\pi$
$510$		0			0
$511$		−	764.990i		−	1.49704i
$512$		−	657.600i		−	1.28438i
$513$		0			0
$514$			36.2041i			0.0704360i
$515$		−	1011.41i		−	1.96389i
$516$		0			0
$517$	469.366			0.907864
$518$	−12.4952			−0.0241220
$519$		0			0
$520$	41.9913			0.0807524
$521$	−144.363			−0.277088			−0.138544	−	0.990356i	$-0.544242\pi$
$521$	−144.363			−0.277088			−0.138544	+	0.990356i	$0.544242\pi$
$522$		0			0
$523$	−81.3990			−0.155639			−0.0778193	−	0.996967i	$-0.524796\pi$
$523$	−81.3990			−0.155639			−0.0778193	+	0.996967i	$0.524796\pi$
$524$			524.966i			1.00184i
$525$		0			0
$526$		−	147.347i		−	0.280128i
$527$			69.1873i			0.131285i
$528$		0			0
$529$	291.582			0.551194
$530$			159.375i			0.300708i
$531$		0			0
$532$	552.193			1.03796
$533$			44.8333i			0.0841150i
$534$		0			0
$535$	−1049.15			−1.96103
$536$	−141.009			−0.263077
$537$		0			0
$538$	−1016.69			−1.88975
$539$		−	159.602i		−	0.296108i
$540$		0			0
$541$			52.0841i			0.0962738i	0.998841	+	0.0481369i	$0.0153284\pi$
$541$			52.0841i			0.0962738i	−0.998841	+	0.0481369i	$0.984672\pi$
$542$		−	1449.41i		−	2.67420i
$543$		0			0
$544$		−	463.537i		−	0.852091i
$545$			601.854i			1.10432i
$546$		0			0
$547$	477.454			0.872859			0.436429	−	0.899738i	$-0.356243\pi$
$547$	477.454			0.872859			0.436429	+	0.899738i	$0.356243\pi$
$548$	665.212			1.21389
$549$		0			0
$550$	−125.694			−0.228535
$551$	286.756			0.520429
$552$		0			0
$553$	−705.844			−1.27639
$554$		−	373.456i		−	0.674109i
$555$		0			0
$556$	−594.710			−1.06962
$557$	−313.535			−0.562899			−0.281449	−	0.959576i	$-0.590815\pi$
$557$	−313.535			−0.562899			−0.281449	+	0.959576i	$0.590815\pi$
$558$		0			0
$559$	255.018			0.456204
$560$	890.373			1.58995
$561$		0			0
$562$		−	508.270i		−	0.904395i
$563$		−	49.8387i		−	0.0885234i	−0.999020	−	0.0442617i	$-0.985906\pi$
$563$		−	49.8387i		−	0.0885234i	0.999020	−	0.0442617i	$-0.0140935\pi$
$564$		0			0
$565$			575.611i			1.01878i
$566$	1009.37			1.78334
$567$		0			0
$568$			143.132i			0.251993i
$569$		−	197.517i		−	0.347131i	−0.984822	−	0.173565i	$-0.944471\pi$
$569$		−	197.517i		−	0.347131i	0.984822	−	0.173565i	$-0.0555288\pi$
$570$		0			0
$571$		−	18.9985i		−	0.0332723i	−0.999862	−	0.0166362i	$-0.994704\pi$
$571$		−	18.9985i		−	0.0332723i	0.999862	−	0.0166362i	$-0.00529570\pi$
$572$	99.6233			0.174167
$573$		0			0
$574$			220.742i			0.384567i
$575$		−	118.075i		−	0.205348i
$576$		0			0
$577$	−248.509			−0.430691			−0.215346	−	0.976538i	$-0.569088\pi$
$577$	−248.509			−0.430691			−0.215346	+	0.976538i	$0.569088\pi$
$578$			468.223i			0.810075i
$579$		0			0
$580$	−305.362			−0.526487
$581$			25.3959i			0.0437107i
$582$		0			0
$583$		−	61.4752i		−	0.105446i
$584$	−134.301			−0.229967
$585$		0			0
$586$			815.649i			1.39189i
$587$		−	736.598i		−	1.25485i	−0.778676	−	0.627426i	$-0.784108\pi$
$587$		−	736.598i		−	1.25485i	0.778676	−	0.627426i	$-0.215892\pi$
$588$		0			0
$589$			117.937i			0.200232i
$590$	796.655	−	459.716i	1.35026	−	0.779179i
$591$		0			0
$592$			9.44703i			0.0159578i
$593$	83.9607			0.141586			0.0707932	−	0.997491i	$-0.477447\pi$
$593$	83.9607			0.141586			0.0707932	+	0.997491i	$0.477447\pi$
$594$		0			0
$595$	538.096			0.904362
$596$			624.869i			1.04844i
$597$		0			0
$598$			202.384i			0.338434i
$599$	93.6816			0.156397			0.0781983	−	0.996938i	$-0.475083\pi$
$599$	93.6816			0.156397			0.0781983	+	0.996938i	$0.475083\pi$
$600$		0			0
$601$			931.300i			1.54958i	0.632216	+	0.774792i	$0.282146\pi$
$601$			931.300i			1.54958i	−0.632216	+	0.774792i	$0.717854\pi$
$602$	1255.61			2.08573
$603$		0			0
$604$			4.16950i			0.00690315i
$605$	−484.877			−0.801449
$606$		0			0
$607$	−217.894			−0.358969			−0.179484	−	0.983761i	$-0.557443\pi$
$607$	−217.894			−0.358969			−0.179484	+	0.983761i	$0.557443\pi$
$608$		−	790.147i		−	1.29958i
$609$		0			0
$610$	647.907			1.06214
$611$	375.849			0.615138
$612$		0			0
$613$			46.8619i			0.0764468i	0.999269	+	0.0382234i	$0.0121698\pi$
$613$			46.8619i			0.0764468i	−0.999269	+	0.0382234i	$0.987830\pi$
$614$		−	75.4263i		−	0.122844i
$615$		0			0
$616$	−79.7483			−0.129462
$617$	−994.661			−1.61209			−0.806046	−	0.591853i	$-0.798397\pi$
$617$	−994.661			−1.61209			−0.806046	+	0.591853i	$0.798397\pi$
$618$		0			0
$619$	801.130			1.29423			0.647116	−	0.762391i	$-0.275975\pi$
$619$	801.130			1.29423			0.647116	+	0.762391i	$0.275975\pi$
$620$		−	125.589i		−	0.202563i
$621$		0			0
$622$		−	994.831i		−	1.59941i
$623$			1466.41i			2.35379i
$624$		0			0
$625$	−757.854			−1.21257
$626$	476.385			0.760999
$627$		0			0
$628$			634.430i			1.01024i
$629$			5.70930i			0.00907679i
$630$		0			0
$631$	891.594			1.41299			0.706493	−	0.707720i	$-0.250276\pi$
$631$	891.594			1.41299			0.706493	+	0.707720i	$0.250276\pi$
$632$			123.917i			0.196072i
$633$		0			0
$634$		−	1377.79i		−	2.17317i
$635$	1178.27			1.85554
$636$		0			0
$637$		−	127.803i		−	0.200633i
$638$	254.722			0.399251
$639$		0			0
$640$		−	276.330i		−	0.431765i
$641$	790.983			1.23398			0.616991	−	0.786970i	$-0.288351\pi$
$641$	790.983			1.23398			0.616991	+	0.786970i	$0.288351\pi$
$642$		0			0
$643$	−348.723			−0.542337			−0.271169	−	0.962532i	$-0.587410\pi$
$643$	−348.723			−0.542337			−0.271169	+	0.962532i	$0.587410\pi$
$644$			460.772i			0.715485i
$645$		0			0
$646$		−	545.637i		−	0.844639i
$647$	−746.195			−1.15332			−0.576658	−	0.816986i	$-0.695643\pi$
$647$	−746.195			−1.15332			−0.576658	+	0.816986i	$0.695643\pi$
$648$		0			0
$649$	−307.290	+	177.324i	−0.473483	+	0.273227i
$650$	−100.651			−0.154848
$651$		0			0
$652$	−30.8312			−0.0472871
$653$	945.003			1.44717			0.723586	−	0.690234i	$-0.242493\pi$
$653$	945.003			1.44717			0.723586	+	0.690234i	$0.242493\pi$
$654$		0			0
$655$			872.014i			1.33132i
$656$	166.893			0.254409
$657$		0			0
$658$	1850.54			2.81237
$659$			907.745i			1.37746i	0.725018	+	0.688729i	$0.241831\pi$
$659$			907.745i			1.37746i	−0.725018	+	0.688729i	$0.758169\pi$
$660$		0			0
$661$	−1024.26			−1.54957			−0.774783	−	0.632227i	$-0.782141\pi$
$661$	−1024.26			−1.54957			−0.774783	+	0.632227i	$0.782141\pi$
$662$		−	10.8922i		−	0.0164535i
$663$		0			0
$664$	4.45849			0.00671459
$665$	917.240			1.37931
$666$		0			0
$667$			239.281i			0.358742i
$668$	−493.163			−0.738268
$669$		0			0
$670$	1440.67			2.15025
$671$	−249.914			−0.372451
$672$		0			0
$673$			629.364i			0.935163i	0.883950	+	0.467581i	$0.154874\pi$
$673$			629.364i			0.935163i	−0.883950	+	0.467581i	$0.845126\pi$
$674$	−947.997			−1.40652
$675$		0			0
$676$	−501.689			−0.742144
$677$	−671.976			−0.992579			−0.496289	−	0.868157i	$-0.665305\pi$
$677$	−671.976			−0.992579			−0.496289	+	0.868157i	$0.665305\pi$
$678$		0			0
$679$		−	1470.30i		−	2.16539i
$680$		−	94.4677i		−	0.138923i
$681$		0			0
$682$			104.762i			0.153610i
$683$		−	683.994i		−	1.00145i	−0.865605	−	0.500727i	$-0.833066\pi$
$683$		−	683.994i		−	1.00145i	0.865605	−	0.500727i	$-0.166934\pi$
$684$		0			0
$685$	1104.97			1.61310
$686$			532.446i			0.776160i
$687$		0			0
$688$		−	949.308i		−	1.37981i
$689$		−	49.2269i		−	0.0714468i
$690$		0			0
$691$		−	69.2046i		−	0.100151i	−0.998745	−	0.0500757i	$-0.984054\pi$
$691$		−	69.2046i		−	0.100151i	0.998745	−	0.0500757i	$-0.0159463\pi$
$692$			197.657i			0.285631i
$693$		0			0
$694$	−1602.36			−2.30888
$695$	−987.865			−1.42139
$696$		0			0
$697$	100.861			0.144708
$698$	1078.67			1.54538
$699$		0			0
$700$	−229.155			−0.327364
$701$		−	1173.53i		−	1.67408i	−0.547143	−	0.837039i	$-0.684285\pi$
$701$		−	1173.53i		−	1.67408i	0.547143	−	0.837039i	$-0.315715\pi$
$702$		0			0
$703$			9.73209i			0.0138437i
$704$		−	270.736i		−	0.384568i
$705$		0			0
$706$	550.689			0.780012
$707$			55.9414i			0.0791250i
$708$		0			0
$709$	−291.385			−0.410980			−0.205490	−	0.978659i	$-0.565879\pi$
$709$	−291.385			−0.410980			−0.205490	+	0.978659i	$0.565879\pi$
$710$		−	1462.36i		−	2.05966i
$711$		0			0
$712$	257.442			0.361576
$713$	−98.4114			−0.138024
$714$		0			0
$715$	165.483			0.231445
$716$			310.294i			0.433371i
$717$		0			0
$718$		−	1444.48i		−	2.01182i
$719$			1045.70i			1.45438i	0.686434	+	0.727192i	$0.259175\pi$
$719$			1045.70i			1.45438i	−0.686434	+	0.727192i	$0.740825\pi$
$720$		0			0
$721$			1538.12i			2.13331i
$722$			54.6221i			0.0756538i
$723$		0			0
$724$	−426.426			−0.588986
$725$	−119.001			−0.164140
$726$		0			0
$727$	−96.8300			−0.133191			−0.0665956	−	0.997780i	$-0.521214\pi$
$727$	−96.8300			−0.133191			−0.0665956	+	0.997780i	$0.521214\pi$
$728$	−63.8593			−0.0877188
$729$		0			0
$730$	1372.13			1.87963
$731$		−	573.713i		−	0.784833i
$732$		0			0
$733$	63.9954			0.0873061			0.0436531	−	0.999047i	$-0.486100\pi$
$733$	63.9954			0.0873061			0.0436531	+	0.999047i	$0.486100\pi$
$734$	−980.840			−1.33629
$735$		0			0
$736$	659.332			0.895831
$737$	−555.702			−0.754006
$738$		0			0
$739$			234.644i			0.317515i	0.987318	+	0.158758i	$0.0507489\pi$
$739$			234.644i			0.317515i	−0.987318	+	0.158758i	$0.949251\pi$
$740$		−	10.3636i		−	0.0140048i
$741$		0			0
$742$		−	242.374i		−	0.326650i
$743$	−724.236			−0.974746			−0.487373	−	0.873194i	$-0.662045\pi$
$743$	−724.236			−0.974746			−0.487373	+	0.873194i	$0.662045\pi$
$744$		0			0
$745$			1037.96i			1.39324i
$746$			1199.33i			1.60769i
$747$		0			0
$748$		−	224.122i		−	0.299629i
$749$	1595.52			2.13020
$750$		0			0
$751$		−	539.064i		−	0.717795i	−0.933377	−	0.358897i	$-0.883153\pi$
$751$		−	539.064i		−	0.717795i	0.933377	−	0.358897i	$-0.116847\pi$
$752$		−	1399.11i		−	1.86051i
$753$		0			0
$754$	203.971			0.270519
$755$			6.92591i			0.00917339i
$756$		0			0
$757$	669.143			0.883941			0.441970	−	0.897030i	$-0.354280\pi$
$757$	669.143			0.883941			0.441970	+	0.897030i	$0.354280\pi$
$758$			1583.26i			2.08873i
$759$		0			0
$760$		−	161.030i		−	0.211882i
$761$	−896.609			−1.17820			−0.589099	−	0.808061i	$-0.700517\pi$
$761$	−896.609			−1.17820			−0.589099	+	0.808061i	$0.700517\pi$
$762$		0			0
$763$		−	915.285i		−	1.19959i
$764$			518.990i			0.679307i
$765$		0			0
$766$		−	936.477i		−	1.22255i
$767$	−246.066	+	141.994i	−0.320816	+	0.185129i
$768$		0			0
$769$			294.155i			0.382516i	0.981540	+	0.191258i	$0.0612567\pi$
$769$			294.155i			0.382516i	−0.981540	+	0.191258i	$0.938743\pi$
$770$	814.774			1.05815
$771$		0			0
$772$	291.836			0.378025
$773$			118.806i			0.153694i	0.997043	+	0.0768471i	$0.0244853\pi$
$773$			118.806i			0.153694i	−0.997043	+	0.0768471i	$0.975515\pi$
$774$		0			0
$775$		−	48.9428i		−	0.0631520i
$776$	−258.124			−0.332634
$777$		0			0
$778$			827.668i			1.06384i
$779$	171.929			0.220704
$780$		0			0
$781$			564.068i			0.722239i
$782$	455.302			0.582228
$783$		0			0
$784$	−475.749			−0.606823
$785$			1053.84i			1.34248i
$786$		0			0
$787$	67.0954			0.0852546			0.0426273	−	0.999091i	$-0.486427\pi$
$787$	67.0954			0.0852546			0.0426273	+	0.999091i	$0.486427\pi$
$788$	−656.197			−0.832737
$789$		0			0
$790$		−	1266.04i		−	1.60258i
$791$		−	875.375i		−	1.10667i
$792$		0			0
$793$	−200.121			−0.252360
$794$	434.214			0.546869
$795$		0			0
$796$	688.307			0.864707
$797$		−	390.810i		−	0.490352i	−0.969479	−	0.245176i	$-0.921154\pi$
$797$		−	390.810i		−	0.490352i	0.969479	−	0.245176i	$-0.0788457\pi$
$798$		0			0
$799$		−	845.548i		−	1.05826i
$800$			327.904i			0.409880i
$801$		0			0
$802$	1149.18			1.43290
$803$	−529.265			−0.659110
$804$		0			0
$805$			765.383i			0.950786i
$806$			83.8893i			0.104081i
$807$		0			0
$808$	9.82103			0.0121547
$809$			296.341i			0.366305i	0.983084	+	0.183153i	$0.0586302\pi$
$809$			296.341i			0.366305i	−0.983084	+	0.183153i	$0.941370\pi$
$810$		0			0
$811$			1396.97i			1.72253i	0.508158	+	0.861264i	$0.330327\pi$
$811$			1396.97i			1.72253i	−0.508158	+	0.861264i	$0.669673\pi$
$812$	464.387			0.571906
$813$		0			0
$814$			8.64491i			0.0106203i
$815$	−51.2133			−0.0628384
$816$		0			0
$817$		−	977.954i		−	1.19701i
$818$	1806.32			2.20821
$819$		0			0
$820$	−183.084			−0.223273
$821$		−	62.3621i		−	0.0759587i	−0.999279	−	0.0379793i	$-0.987908\pi$
$821$		−	62.3621i		−	0.0759587i	0.999279	−	0.0379793i	$-0.0120921\pi$
$822$		0			0
$823$		−	1509.54i		−	1.83419i	−0.398667	−	0.917096i	$-0.630527\pi$
$823$		−	1509.54i		−	1.83419i	0.398667	−	0.917096i	$-0.369473\pi$
$824$	270.031			0.327708
$825$		0			0
$826$	−1211.53	+	699.124i	−1.46675	+	0.846397i
$827$	−427.575			−0.517019			−0.258509	−	0.966009i	$-0.583231\pi$
$827$	−427.575			−0.517019			−0.258509	+	0.966009i	$0.583231\pi$
$828$		0			0
$829$	−1352.03			−1.63092			−0.815459	−	0.578814i	$-0.803516\pi$
$829$	−1352.03			−1.63092			−0.815459	+	0.578814i	$0.803516\pi$
$830$	−45.5515			−0.0548814
$831$		0			0
$832$		−	216.794i		−	0.260570i
$833$	−287.519			−0.345160
$834$		0			0
$835$	−819.187			−0.981062
$836$		−	382.040i		−	0.456985i
$837$		0			0
$838$	−922.134			−1.10040
$839$			849.147i			1.01209i	0.862506	+	0.506047i	$0.168894\pi$
$839$			849.147i			1.01209i	−0.862506	+	0.506047i	$0.831106\pi$
$840$		0			0
$841$	−599.841			−0.713248
$842$	276.492			0.328375
$843$		0			0
$844$			1418.00i			1.68009i
$845$	−833.350			−0.986212
$846$		0			0
$847$	737.389			0.870589
$848$	−183.248			−0.216094
$849$		0			0
$850$			226.435i			0.266394i
$851$	−8.12086			−0.00954273
$852$		0			0
$853$	−529.505			−0.620757			−0.310378	−	0.950613i	$-0.600456\pi$
$853$	−529.505			−0.620757			−0.310378	+	0.950613i	$0.600456\pi$
$854$	−985.321			−1.15377
$855$		0			0
$856$		−	280.108i		−	0.327229i
$857$		−	437.967i		−	0.511047i	−0.966803	−	0.255523i	$-0.917752\pi$
$857$		−	437.967i		−	0.511047i	0.966803	−	0.255523i	$-0.0822478\pi$
$858$		0			0
$859$			1046.52i			1.21831i	0.793053	+	0.609153i	$0.208490\pi$
$859$			1046.52i			1.21831i	−0.793053	+	0.609153i	$0.791510\pi$
$860$			1041.41i			1.21094i
$861$		0			0
$862$	130.014			0.150828
$863$		−	560.329i		−	0.649281i	−0.945838	−	0.324640i	$-0.894757\pi$
$863$		−	560.329i		−	0.649281i	0.945838	−	0.324640i	$-0.105243\pi$
$864$		0			0
$865$			328.325i			0.379567i
$866$			471.613i			0.544587i
$867$		0			0
$868$			190.993i			0.220038i
$869$			488.345i			0.561962i
$870$		0			0
$871$	−444.984			−0.510889
$872$	−160.687			−0.184274
$873$		0			0
$874$	776.110			0.887997
$875$	861.182			0.984207
$876$		0			0
$877$	−1008.15			−1.14954			−0.574771	−	0.818314i	$-0.694909\pi$
$877$	−1008.15			−1.14954			−0.574771	+	0.818314i	$0.694909\pi$
$878$		−	1047.89i		−	1.19350i
$879$		0			0
$880$		−	616.013i		−	0.700015i
$881$			498.377i			0.565695i	0.959165	+	0.282847i	$0.0912790\pi$
$881$			498.377i			0.565695i	−0.959165	+	0.282847i	$0.908721\pi$
$882$		0			0
$883$	−1428.37			−1.61763			−0.808817	−	0.588061i	$-0.799891\pi$
$883$	−1428.37			−1.61763			−0.808817	+	0.588061i	$0.799891\pi$
$884$		−	179.468i		−	0.203018i
$885$		0			0
$886$	469.723			0.530161
$887$			886.165i			0.999058i	0.866297	+	0.499529i	$0.166494\pi$
$887$			886.165i			0.999058i	−0.866297	+	0.499529i	$0.833506\pi$
$888$		0			0
$889$	−1791.88			−2.01561
$890$	−2630.24			−2.95532
$891$		0			0
$892$	77.3559			0.0867219
$893$		−	1441.32i		−	1.61402i
$894$		0			0
$895$			515.425i			0.575894i
$896$			420.235i			0.469013i
$897$		0			0
$898$			1672.95i			1.86298i
$899$			99.1836i			0.110327i
$900$		0			0
$901$	−110.746			−0.122914
$902$	152.722			0.169315
$903$		0			0
$904$	−153.680			−0.170000
$905$	−708.330			−0.782685
$906$		0			0
$907$	−1277.00			−1.40794			−0.703971	−	0.710229i	$-0.748591\pi$
$907$	−1277.00			−1.40794			−0.703971	+	0.710229i	$0.748591\pi$
$908$		−	952.522i		−	1.04903i
$909$		0			0
$910$	652.438			0.716965
$911$	−12.8533			−0.0141090			−0.00705452	−	0.999975i	$-0.502246\pi$
$911$	−12.8533			−0.0141090			−0.00705452	+	0.999975i	$0.502246\pi$
$912$		0			0
$913$	17.5704			0.0192447
$914$	762.265			0.833988
$915$		0			0
$916$		−	553.683i		−	0.604457i
$917$		−	1326.14i		−	1.44617i
$918$		0			0
$919$			728.935i			0.793182i	0.917995	+	0.396591i	$0.129807\pi$
$919$			728.935i			0.793182i	−0.917995	+	0.396591i	$0.870193\pi$
$920$	134.370			0.146054
$921$		0			0
$922$		−	1514.41i		−	1.64253i
$923$			451.684i			0.489365i
$924$		0			0
$925$		−	4.03873i		−	0.00436620i
$926$	−1835.19			−1.98184
$927$		0			0
$928$		−	664.505i		−	0.716061i
$929$		−	498.588i		−	0.536694i	−0.963322	−	0.268347i	$-0.913523\pi$
$929$		−	498.588i		−	0.536694i	0.963322	−	0.268347i	$-0.0864773\pi$
$930$		0			0
$931$	−490.105			−0.526429
$932$		−	36.5680i		−	0.0392361i
$933$		0			0
$934$	−1189.36			−1.27341
$935$		−	372.287i		−	0.398168i
$936$		0			0
$937$			12.3168i			0.0131449i	0.999978	+	0.00657247i	$0.00209210\pi$
$937$			12.3168i			0.0131449i	−0.999978	+	0.00657247i	$0.997908\pi$
$938$	−2190.93			−2.33575
$939$		0			0
$940$			1534.84i			1.63281i
$941$		−	974.832i		−	1.03595i	−0.855395	−	0.517976i	$-0.826686\pi$
$941$		−	974.832i		−	1.03595i	0.855395	−	0.517976i	$-0.173314\pi$
$942$		0			0
$943$			143.464i			0.152136i
$944$	528.575	+	915.984i	0.559932	+	0.970322i
$945$		0			0
$946$		−	868.705i		−	0.918293i
$947$	−247.178			−0.261012			−0.130506	−	0.991448i	$-0.541660\pi$
$947$	−247.178			−0.261012			−0.130506	+	0.991448i	$0.541660\pi$
$948$		0			0
$949$	−423.815			−0.446591
$950$			385.981i			0.406296i
$951$		0			0
$952$			143.664i			0.150908i
$953$	−19.6401			−0.0206087			−0.0103043	−	0.999947i	$-0.503280\pi$
$953$	−19.6401			−0.0206087			−0.0103043	+	0.999947i	$0.503280\pi$
$954$		0			0
$955$			862.088i			0.902710i
$956$	769.300			0.804707
$957$		0			0
$958$			1693.84i			1.76810i
$959$	−1680.42			−1.75226
$960$		0			0
$961$	920.208			0.957552
$962$			6.92250i			0.00719594i
$963$		0			0
$964$	−222.335			−0.230638
$965$	484.764			0.502347
$966$		0			0
$967$		−	1501.84i		−	1.55309i	−0.630062	−	0.776545i	$-0.716970\pi$
$967$		−	1501.84i		−	1.55309i	0.630062	−	0.776545i	$-0.283030\pi$
$968$		−	129.455i		−	0.133735i
$969$		0			0
$970$	2637.21			2.71877
$971$	1557.05			1.60355			0.801776	−	0.597624i	$-0.203889\pi$
$971$	1557.05			1.60355			0.801776	+	0.597624i	$0.203889\pi$
$972$		0			0
$973$	1502.32			1.54401
$974$		−	1865.20i		−	1.91499i
$975$		0			0
$976$			744.955i			0.763274i
$977$		−	220.988i		−	0.226191i	−0.993584	−	0.113095i	$-0.963923\pi$
$977$		−	220.988i		−	0.226191i	0.993584	−	0.113095i	$-0.0360766\pi$
$978$		0			0
$979$	1014.55			1.03631
$980$	521.906			0.532557
$981$		0			0
$982$			1774.70i			1.80723i
$983$		−	9.44684i		−	0.00961021i	−0.999988	−	0.00480510i	$-0.998470\pi$
$983$		−	9.44684i		−	0.00961021i	0.999988	−	0.00480510i	$-0.00152952\pi$
$984$		0			0
$985$	−1090.00			−1.10660
$986$		−	458.874i		−	0.465390i
$987$		0			0
$988$		−	305.922i		−	0.309638i
$989$	816.045			0.825121
$990$		0			0
$991$		−	86.9654i		−	0.0877552i	−0.999037	−	0.0438776i	$-0.986029\pi$
$991$		−	86.9654i		−	0.0877552i	0.999037	−	0.0438776i	$-0.0139712\pi$
$992$	273.297			0.275501
$993$		0			0
$994$			2223.91i			2.23734i
$995$	1143.34			1.14908
$996$		0			0
$997$	728.589			0.730781			0.365390	−	0.930854i	$-0.380935\pi$
$997$	728.589			0.730781			0.365390	+	0.930854i	$0.380935\pi$
$998$			1374.20i			1.37696i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.5		20
3.2	odd	2		177.3.c.a.58.16	yes	20
59.58	odd	2	inner	531.3.c.c.235.16		20
177.176	even	2		177.3.c.a.58.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.5	✓	20	177.176	even	2
177.3.c.a.58.16	yes	20	3.2	odd	2
531.3.c.c.235.5		20	1.1	even	1	trivial
531.3.c.c.235.16		20	59.58	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.72775i q^{2} -3.44061 q^{4} -5.71516 q^{5} +8.69147 q^{7} -1.52587i q^{8} +O(q^{10})\)	\(q-2.72775i q^{2} -3.44061 q^{4} -5.71516 q^{5} +8.69147 q^{7} -1.52587i q^{8} +15.5895i q^{10} -6.01328i q^{11} -4.81519i q^{13} -23.7081i q^{14} -17.9246 q^{16} -10.8327 q^{17} -18.4655 q^{19} +19.6636 q^{20} -16.4027 q^{22} -15.4084i q^{23} +7.66303 q^{25} -13.1346 q^{26} -29.9040 q^{28} -15.5293 q^{29} -6.38687i q^{31} +42.7904i q^{32} +29.5490i q^{34} -49.6731 q^{35} -0.527041i q^{37} +50.3693i q^{38} +8.72058i q^{40} -9.31080 q^{41} +52.9611i q^{43} +20.6894i q^{44} -42.0302 q^{46} +78.0549i q^{47} +26.5417 q^{49} -20.9028i q^{50} +16.5672i q^{52} +10.2232 q^{53} +34.3668i q^{55} -13.2620i q^{56} +42.3600i q^{58} +(-29.4888 - 51.1020i) q^{59} -41.5604i q^{61} -17.4218 q^{62} +45.0230 q^{64} +27.5196i q^{65} -92.4125i q^{67} +37.2712 q^{68} +135.496i q^{70} -93.8038 q^{71} -88.0161i q^{73} -1.43764 q^{74} +63.5327 q^{76} -52.2642i q^{77} -81.2111 q^{79} +102.442 q^{80} +25.3975i q^{82} +2.92193i q^{83} +61.9108 q^{85} +144.465 q^{86} -9.17547 q^{88} +168.718i q^{89} -41.8511i q^{91} +53.0143i q^{92} +212.914 q^{94} +105.533 q^{95} -169.166i q^{97} -72.3990i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more