LMFDB - Embedded newform 531.3.c.c.235.12

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			235.12
Root			$1.08628i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08628i q^{2} +2.82000 q^{4} -3.33671 q^{5} -1.22637 q^{7} +7.40842i q^{8} +O(q^{10})$	\(q+1.08628i q^{2} +2.82000 q^{4} -3.33671 q^{5} -1.22637 q^{7} +7.40842i q^{8} -3.62460i q^{10} -1.21764i q^{11} +12.6555i q^{13} -1.33218i q^{14} +3.23240 q^{16} +0.815320 q^{17} -7.74812 q^{19} -9.40954 q^{20} +1.32270 q^{22} +33.8014i q^{23} -13.8663 q^{25} -13.7474 q^{26} -3.45837 q^{28} -7.53415 q^{29} +7.71953i q^{31} +33.1450i q^{32} +0.885664i q^{34} +4.09206 q^{35} +16.7957i q^{37} -8.41661i q^{38} -24.7198i q^{40} +19.0497 q^{41} +48.9576i q^{43} -3.43375i q^{44} -36.7177 q^{46} +4.79310i q^{47} -47.4960 q^{49} -15.0627i q^{50} +35.6884i q^{52} +48.7328 q^{53} +4.06292i q^{55} -9.08548i q^{56} -8.18418i q^{58} +(-43.5656 - 39.7874i) q^{59} +87.6284i q^{61} -8.38556 q^{62} -23.0750 q^{64} -42.2277i q^{65} -42.6674i q^{67} +2.29920 q^{68} +4.44511i q^{70} +20.2245 q^{71} +45.6107i q^{73} -18.2448 q^{74} -21.8497 q^{76} +1.49328i q^{77} -62.5528 q^{79} -10.7856 q^{80} +20.6933i q^{82} -22.8390i q^{83} -2.72049 q^{85} -53.1815 q^{86} +9.02079 q^{88} -55.8982i q^{89} -15.5203i q^{91} +95.3199i q^{92} -5.20664 q^{94} +25.8533 q^{95} -123.320i q^{97} -51.5939i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			1.08628i			0.543139i	0.962419	+	0.271570i	$0.0875427\pi$
$2$			1.08628i			0.543139i	−0.962419	+	0.271570i	$0.912457\pi$
$3$		0			0
$3$		0			0
$4$	2.82000			0.705000
$5$	−3.33671			−0.667343			−0.333671	−	0.942689i	$-0.608288\pi$
$5$	−3.33671			−0.667343			−0.333671	+	0.942689i	$0.608288\pi$
$6$		0			0
$7$	−1.22637			−0.175196			−0.0875981	−	0.996156i	$-0.527919\pi$
$7$	−1.22637			−0.175196			−0.0875981	+	0.996156i	$0.527919\pi$
$8$			7.40842i			0.926052i
$9$		0			0
$10$		−	3.62460i		−	0.362460i
$11$		−	1.21764i		−	0.110695i	−0.998467	−	0.0553473i	$-0.982373\pi$
$11$		−	1.21764i		−	0.110695i	0.998467	−	0.0553473i	$-0.0176266\pi$
$12$		0			0
$13$			12.6555i			0.973497i	0.873542	+	0.486749i	$0.161817\pi$
$13$			12.6555i			0.973497i	−0.873542	+	0.486749i	$0.838183\pi$
$14$		−	1.33218i		−	0.0951559i
$15$		0			0
$16$	3.23240			0.202025
$17$	0.815320			0.0479600			0.0239800	−	0.999712i	$-0.492366\pi$
$17$	0.815320			0.0479600			0.0239800	+	0.999712i	$0.492366\pi$
$18$		0			0
$19$	−7.74812			−0.407796			−0.203898	−	0.978992i	$-0.565361\pi$
$19$	−7.74812			−0.407796			−0.203898	+	0.978992i	$0.565361\pi$
$20$	−9.40954			−0.470477
$21$		0			0
$22$	1.32270			0.0601226
$23$			33.8014i			1.46963i	0.678270	+	0.734813i	$0.262730\pi$
$23$			33.8014i			1.46963i	−0.678270	+	0.734813i	$0.737270\pi$
$24$		0			0
$25$	−13.8663			−0.554653
$26$	−13.7474			−0.528744
$27$		0			0
$28$	−3.45837			−0.123513
$29$	−7.53415			−0.259798			−0.129899	−	0.991527i	$-0.541465\pi$
$29$	−7.53415			−0.259798			−0.129899	+	0.991527i	$0.541465\pi$
$30$		0			0
$31$			7.71953i			0.249017i	0.992219	+	0.124509i	$0.0397354\pi$
$31$			7.71953i			0.249017i	−0.992219	+	0.124509i	$0.960265\pi$
$32$			33.1450i			1.03578i
$33$		0			0
$34$			0.885664i			0.0260489i
$35$	4.09206			0.116916
$36$		0			0
$37$			16.7957i			0.453937i	0.973902	+	0.226969i	$0.0728815\pi$
$37$			16.7957i			0.453937i	−0.973902	+	0.226969i	$0.927119\pi$
$38$		−	8.41661i		−	0.221490i
$39$		0			0
$40$		−	24.7198i		−	0.617994i
$41$	19.0497			0.464627			0.232314	−	0.972641i	$-0.425370\pi$
$41$	19.0497			0.464627			0.232314	+	0.972641i	$0.425370\pi$
$42$		0			0
$43$			48.9576i			1.13855i	0.822148	+	0.569274i	$0.192776\pi$
$43$			48.9576i			1.13855i	−0.822148	+	0.569274i	$0.807224\pi$
$44$		−	3.43375i		−	0.0780397i
$45$		0			0
$46$	−36.7177			−0.798211
$47$			4.79310i			0.101981i	0.998699	+	0.0509904i	$0.0162378\pi$
$47$			4.79310i			0.101981i	−0.998699	+	0.0509904i	$0.983762\pi$
$48$		0			0
$49$	−47.4960			−0.969306
$50$		−	15.0627i		−	0.301254i
$51$		0			0
$52$			35.6884i			0.686316i
$53$	48.7328			0.919488			0.459744	−	0.888052i	$-0.347941\pi$
$53$	48.7328			0.919488			0.459744	+	0.888052i	$0.347941\pi$
$54$		0			0
$55$			4.06292i			0.0738713i
$56$		−	9.08548i		−	0.162241i
$57$		0			0
$58$		−	8.18418i		−	0.141107i
$59$	−43.5656	−	39.7874i	−0.738400	−	0.674363i
$59$	−43.5656	−	39.7874i	−0.738400	−	0.674363i
$60$		0			0
$61$			87.6284i			1.43653i	0.695769	+	0.718265i	$0.255064\pi$
$61$			87.6284i			1.43653i	−0.695769	+	0.718265i	$0.744936\pi$
$62$	−8.38556			−0.135251
$63$		0			0
$64$	−23.0750			−0.360547
$65$		−	42.2277i		−	0.649657i
$66$		0			0
$67$		−	42.6674i		−	0.636826i	−0.947952	−	0.318413i	$-0.896850\pi$
$67$		−	42.6674i		−	0.636826i	0.947952	−	0.318413i	$-0.103150\pi$
$68$	2.29920			0.0338118
$69$		0			0
$70$			4.44511i			0.0635016i
$71$	20.2245			0.284852			0.142426	−	0.989805i	$-0.454510\pi$
$71$	20.2245			0.284852			0.142426	+	0.989805i	$0.454510\pi$
$72$		0			0
$73$			45.6107i			0.624804i	0.949950	+	0.312402i	$0.101134\pi$
$73$			45.6107i			0.624804i	−0.949950	+	0.312402i	$0.898866\pi$
$74$	−18.2448			−0.246551
$75$		0			0
$76$	−21.8497			−0.287496
$77$			1.49328i			0.0193933i
$78$		0			0
$79$	−62.5528			−0.791808			−0.395904	−	0.918292i	$-0.629569\pi$
$79$	−62.5528			−0.791808			−0.395904	+	0.918292i	$0.629569\pi$
$80$	−10.7856			−0.134820
$81$		0			0
$82$			20.6933i			0.252357i
$83$		−	22.8390i		−	0.275169i	−0.990490	−	0.137584i	$-0.956066\pi$
$83$		−	22.8390i		−	0.275169i	0.990490	−	0.137584i	$-0.0439338\pi$
$84$		0			0
$85$	−2.72049			−0.0320058
$86$	−53.1815			−0.618390
$87$		0			0
$88$	9.02079			0.102509
$89$		−	55.8982i		−	0.628070i	−0.949411	−	0.314035i	$-0.898319\pi$
$89$		−	55.8982i		−	0.628070i	0.949411	−	0.314035i	$-0.101681\pi$
$90$		0			0
$91$		−	15.5203i		−	0.170553i
$92$			95.3199i			1.03609i
$93$		0			0
$94$	−5.20664			−0.0553897
$95$	25.8533			0.272140
$96$		0			0
$97$		−	123.320i		−	1.27134i	−0.771962	−	0.635669i	$-0.780724\pi$
$97$		−	123.320i		−	1.27134i	0.771962	−	0.635669i	$-0.219276\pi$
$98$		−	51.5939i		−	0.526468i
$99$		0			0
$100$	−39.1031			−0.391031
$101$			39.0946i			0.387075i	0.981093	+	0.193538i	$0.0619962\pi$
$101$			39.0946i			0.387075i	−0.981093	+	0.193538i	$0.938004\pi$
$102$		0			0
$103$		−	56.8713i		−	0.552149i	−0.961136	−	0.276074i	$-0.910966\pi$
$103$		−	56.8713i		−	0.552149i	0.961136	−	0.276074i	$-0.0890336\pi$
$104$	−93.7570			−0.901509
$105$		0			0
$106$			52.9374i			0.499410i
$107$	−34.8029			−0.325261			−0.162630	−	0.986687i	$-0.551998\pi$
$107$	−34.8029			−0.325261			−0.162630	+	0.986687i	$0.551998\pi$
$108$		0			0
$109$			9.83542i			0.0902332i	0.998982	+	0.0451166i	$0.0143659\pi$
$109$			9.83542i			0.0902332i	−0.998982	+	0.0451166i	$0.985634\pi$
$110$	−4.41346			−0.0401224
$111$		0			0
$112$	−3.96413			−0.0353940
$113$			145.812i			1.29037i	0.764025	+	0.645186i	$0.223220\pi$
$113$			145.812i			1.29037i	−0.764025	+	0.645186i	$0.776780\pi$
$114$		0			0
$115$		−	112.786i		−	0.980744i
$116$	−21.2463			−0.183158
$117$		0			0
$118$	43.2202	−	47.3244i	0.366273	−	0.401054i
$119$	−0.999886			−0.00840241
$120$		0			0
$121$	119.517			0.987747
$122$	−95.1888			−0.780236
$123$		0			0
$124$			21.7691i			0.175557i
$125$	129.686			1.03749
$126$		0			0
$127$	145.827			1.14825			0.574123	−	0.818769i	$-0.305343\pi$
$127$	145.827			1.14825			0.574123	+	0.818769i	$0.305343\pi$
$128$			107.514i			0.839952i
$129$		0			0
$130$	45.8710			0.352854
$131$			30.5967i			0.233562i	0.993158	+	0.116781i	$0.0372576\pi$
$131$			30.5967i			0.233562i	−0.993158	+	0.116781i	$0.962742\pi$
$132$		0			0
$133$	9.50208			0.0714442
$134$	46.3486			0.345885
$135$		0			0
$136$			6.04023i			0.0444134i
$137$	39.4511			0.287964			0.143982	−	0.989580i	$-0.454009\pi$
$137$	39.4511			0.287964			0.143982	+	0.989580i	$0.454009\pi$
$138$		0			0
$139$	93.5556			0.673062			0.336531	−	0.941672i	$-0.390746\pi$
$139$	93.5556			0.673062			0.336531	+	0.941672i	$0.390746\pi$
$140$	11.5396			0.0824257
$141$		0			0
$142$			21.9694i			0.154714i
$143$	15.4098			0.107761
$144$		0			0
$145$	25.1393			0.173375
$146$	−49.5459			−0.339356
$147$		0			0
$148$			47.3638i			0.320026i
$149$		−	133.435i		−	0.895536i	−0.894150	−	0.447768i	$-0.852219\pi$
$149$		−	133.435i		−	0.895536i	0.894150	−	0.447768i	$-0.147781\pi$
$150$		0			0
$151$		−	95.5402i		−	0.632717i	−0.948640	−	0.316358i	$-0.897540\pi$
$151$		−	95.5402i		−	0.632717i	0.948640	−	0.316358i	$-0.102460\pi$
$152$		−	57.4013i		−	0.377640i
$153$		0			0
$154$	−1.62212			−0.0105332
$155$		−	25.7579i		−	0.166180i
$156$		0			0
$157$		−	245.077i		−	1.56100i	−0.625157	−	0.780499i	$-0.714965\pi$
$157$		−	245.077i		−	1.56100i	0.625157	−	0.780499i	$-0.285035\pi$
$158$		−	67.9498i		−	0.430062i
$159$		0			0
$160$		−	110.595i		−	0.691220i
$161$		−	41.4531i		−	0.257473i
$162$		0			0
$163$	204.601			1.25522			0.627611	−	0.778527i	$-0.284033\pi$
$163$	204.601			1.25522			0.627611	+	0.778527i	$0.284033\pi$
$164$	53.7202			0.327562
$165$		0			0
$166$	24.8095			0.149455
$167$	100.875			0.604043			0.302022	−	0.953301i	$-0.402339\pi$
$167$	100.875			0.604043			0.302022	+	0.953301i	$0.402339\pi$
$168$		0			0
$169$	8.83915			0.0523027
$170$		−	2.95521i		−	0.0173836i
$171$		0			0
$172$			138.060i			0.802677i
$173$		−	192.449i		−	1.11242i	−0.831041	−	0.556212i	$-0.812254\pi$
$173$		−	192.449i		−	1.11242i	0.831041	−	0.556212i	$-0.187746\pi$
$174$		0			0
$175$	17.0053			0.0971732
$176$		−	3.93590i		−	0.0223631i
$177$		0			0
$178$	60.7210			0.341129
$179$		−	66.4049i		−	0.370977i	−0.982646	−	0.185488i	$-0.940613\pi$
$179$		−	66.4049i		−	0.370977i	0.982646	−	0.185488i	$-0.0593867\pi$
$180$		0			0
$181$	100.811			0.556968			0.278484	−	0.960441i	$-0.410168\pi$
$181$	100.811			0.556968			0.278484	+	0.960441i	$0.410168\pi$
$182$	16.8594			0.0926340
$183$		0			0
$184$	−250.415			−1.36095
$185$		−	56.0424i		−	0.302932i
$186$		0			0
$187$		−	0.992766i		−	0.00530891i
$188$			13.5165i			0.0718965i
$189$		0			0
$190$			28.0838i			0.147810i
$191$			285.899i			1.49686i	0.663216	+	0.748428i	$0.269191\pi$
$191$			285.899i			1.49686i	−0.663216	+	0.748428i	$0.730809\pi$
$192$		0			0
$193$	11.7169			0.0607093			0.0303546	−	0.999539i	$-0.490336\pi$
$193$	11.7169			0.0607093			0.0303546	+	0.999539i	$0.490336\pi$
$194$	133.960			0.690513
$195$		0			0
$196$	−133.939			−0.683361
$197$	269.535			1.36820			0.684099	−	0.729389i	$-0.260195\pi$
$197$	269.535			1.36820			0.684099	+	0.729389i	$0.260195\pi$
$198$		0			0
$199$	164.357			0.825912			0.412956	−	0.910751i	$-0.364496\pi$
$199$	164.357			0.825912			0.412956	+	0.910751i	$0.364496\pi$
$200$		−	102.728i		−	0.513638i
$201$		0			0
$202$	−42.4676			−0.210236
$203$	9.23968			0.0455157
$204$		0			0
$205$	−63.5635			−0.310066
$206$	61.7781			0.299894
$207$		0			0
$208$			40.9075i			0.196671i
$209$			9.43442i			0.0451408i
$210$		0			0
$211$		−	155.167i		−	0.735391i	−0.929946	−	0.367695i	$-0.880147\pi$
$211$		−	155.167i		−	0.735391i	0.929946	−	0.367695i	$-0.119853\pi$
$212$	137.427			0.648239
$213$		0			0
$214$		−	37.8056i		−	0.176662i
$215$		−	163.357i		−	0.759802i
$216$		0			0
$217$		−	9.46703i		−	0.0436269i
$218$	−10.6840			−0.0490092
$219$		0			0
$220$			11.4574i			0.0520792i
$221$			10.3183i			0.0466889i
$222$		0			0
$223$	212.798			0.954251			0.477126	−	0.878835i	$-0.341679\pi$
$223$	212.798			0.954251			0.477126	+	0.878835i	$0.341679\pi$
$224$		−	40.6481i		−	0.181465i
$225$		0			0
$226$	−158.392			−0.700852
$227$			272.876i			1.20210i	0.799213	+	0.601048i	$0.205250\pi$
$227$			272.876i			1.20210i	−0.799213	+	0.601048i	$0.794750\pi$
$228$		0			0
$229$			350.552i			1.53080i	0.643557	+	0.765398i	$0.277458\pi$
$229$			350.552i			1.53080i	−0.643557	+	0.765398i	$0.722542\pi$
$230$	122.517			0.532681
$231$		0			0
$232$		−	55.8161i		−	0.240587i
$233$			1.64412i			0.00705630i	0.999994	+	0.00352815i	$0.00112305\pi$
$233$			1.64412i			0.00705630i	−0.999994	+	0.00352815i	$0.998877\pi$
$234$		0			0
$235$		−	15.9932i		−	0.0680562i
$236$	−122.855	−	112.200i	−0.520572	−	0.475426i
$237$		0			0
$238$		−	1.08615i		−	0.00456368i
$239$	74.5410			0.311887			0.155943	−	0.987766i	$-0.450158\pi$
$239$	74.5410			0.311887			0.155943	+	0.987766i	$0.450158\pi$
$240$		0			0
$241$	−56.4597			−0.234272			−0.117136	−	0.993116i	$-0.537371\pi$
$241$	−56.4597			−0.234272			−0.117136	+	0.993116i	$0.537371\pi$
$242$			129.829i			0.536484i
$243$		0			0
$244$			247.112i			1.01275i
$245$	158.481			0.646860
$246$		0			0
$247$		−	98.0560i		−	0.396988i
$248$	−57.1895			−0.230603
$249$		0			0
$250$			140.875i			0.563500i
$251$	21.6224			0.0861451			0.0430726	−	0.999072i	$-0.486285\pi$
$251$	21.6224			0.0861451			0.0430726	+	0.999072i	$0.486285\pi$
$252$		0			0
$253$	41.1580			0.162680
$254$			158.409i			0.623657i
$255$		0			0
$256$	−209.090			−0.816758
$257$	−70.0007			−0.272376			−0.136188	−	0.990683i	$-0.543485\pi$
$257$	−70.0007			−0.272376			−0.136188	+	0.990683i	$0.543485\pi$
$258$		0			0
$259$		−	20.5978i		−	0.0795281i
$260$		−	119.082i		−	0.458008i
$261$		0			0
$262$	−33.2365			−0.126857
$263$	−450.103			−1.71142			−0.855709	−	0.517457i	$-0.826879\pi$
$263$	−450.103			−1.71142			−0.855709	+	0.517457i	$0.826879\pi$
$264$		0			0
$265$	−162.608			−0.613614
$266$			10.3219i			0.0388042i
$267$		0			0
$268$		−	120.322i		−	0.448963i
$269$		−	167.392i		−	0.622275i	−0.950365	−	0.311137i	$-0.899290\pi$
$269$		−	167.392i		−	0.622275i	0.950365	−	0.311137i	$-0.100710\pi$
$270$		0			0
$271$	136.135			0.502342			0.251171	−	0.967943i	$-0.419184\pi$
$271$	136.135			0.502342			0.251171	+	0.967943i	$0.419184\pi$
$272$	2.63544			0.00968912
$273$		0			0
$274$			42.8549i			0.156405i
$275$			16.8842i			0.0613971i
$276$		0			0
$277$	−195.176			−0.704606			−0.352303	−	0.935886i	$-0.614601\pi$
$277$	−195.176			−0.704606			−0.352303	+	0.935886i	$0.614601\pi$
$278$			101.627i			0.365566i
$279$		0			0
$280$			30.3157i			0.108270i
$281$	−475.581			−1.69246			−0.846229	−	0.532820i	$-0.821132\pi$
$281$	−475.581			−1.69246			−0.846229	+	0.532820i	$0.821132\pi$
$282$		0			0
$283$		−	444.223i		−	1.56969i	−0.619691	−	0.784846i	$-0.712742\pi$
$283$		−	444.223i		−	1.56969i	0.619691	−	0.784846i	$-0.287258\pi$
$284$	57.0330			0.200821
$285$		0			0
$286$			16.7393i			0.0585292i
$287$	−23.3621			−0.0814009
$288$		0			0
$289$	−288.335			−0.997700
$290$			27.3083i			0.0941665i
$291$		0			0
$292$			128.622i			0.440487i
$293$	56.4950			0.192816			0.0964079	−	0.995342i	$-0.469265\pi$
$293$	56.4950			0.192816			0.0964079	+	0.995342i	$0.469265\pi$
$294$		0			0
$295$	145.366	+	132.759i	0.492766	+	0.450031i
$296$	−124.429			−0.420369
$297$		0			0
$298$	144.947			0.486401
$299$	−427.772			−1.43068
$300$		0			0
$301$		−	60.0403i		−	0.199469i
$302$	103.783			0.343653
$303$		0			0
$304$	−25.0450			−0.0823849
$305$		−	292.391i		−	0.958659i
$306$		0			0
$307$	279.533			0.910530			0.455265	−	0.890356i	$-0.349545\pi$
$307$	279.533			0.910530			0.455265	+	0.890356i	$0.349545\pi$
$308$			4.21106i			0.0136723i
$309$		0			0
$310$	27.9802			0.0902588
$311$	−210.784			−0.677762			−0.338881	−	0.940829i	$-0.610048\pi$
$311$	−210.784			−0.677762			−0.338881	+	0.940829i	$0.610048\pi$
$312$		0			0
$313$		−	372.662i		−	1.19061i	−0.803499	−	0.595306i	$-0.797031\pi$
$313$		−	372.662i		−	1.19061i	0.803499	−	0.595306i	$-0.202969\pi$
$314$	266.221			0.847839
$315$		0			0
$316$	−176.399			−0.558225
$317$	−299.104			−0.943544			−0.471772	−	0.881720i	$-0.656386\pi$
$317$	−299.104			−0.943544			−0.471772	+	0.881720i	$0.656386\pi$
$318$		0			0
$319$			9.17389i			0.0287583i
$320$	76.9948			0.240609
$321$		0			0
$322$	45.0296			0.139844
$323$	−6.31719			−0.0195579
$324$		0			0
$325$		−	175.485i		−	0.539954i
$326$			222.254i			0.681760i
$327$		0			0
$328$			141.128i			0.430269i
$329$		−	5.87813i		−	0.0178666i
$330$		0			0
$331$	268.182			0.810217			0.405109	−	0.914269i	$-0.367234\pi$
$331$	268.182			0.810217			0.405109	+	0.914269i	$0.367234\pi$
$332$		−	64.4060i		−	0.193994i
$333$		0			0
$334$			109.579i			0.328079i
$335$			142.369i			0.424982i
$336$		0			0
$337$			372.241i			1.10457i	0.833655	+	0.552286i	$0.186244\pi$
$337$			372.241i			1.10457i	−0.833655	+	0.552286i	$0.813756\pi$
$338$			9.60178i			0.0284076i
$339$		0			0
$340$	−7.67178			−0.0225641
$341$	9.39962			0.0275649
$342$		0			0
$343$	118.340			0.345015
$344$	−362.698			−1.05436
$345$		0			0
$346$	209.053			0.604201
$347$			162.680i			0.468820i	0.972138	+	0.234410i	$0.0753158\pi$
$347$			162.680i			0.468820i	−0.972138	+	0.234410i	$0.924684\pi$
$348$		0			0
$349$			474.478i			1.35954i	0.733427	+	0.679768i	$0.237920\pi$
$349$			474.478i			1.35954i	−0.733427	+	0.679768i	$0.762080\pi$
$350$			18.4725i			0.0527785i
$351$		0			0
$352$	40.3586			0.114655
$353$			553.231i			1.56723i	0.621248	+	0.783614i	$0.286626\pi$
$353$			553.231i			1.56723i	−0.621248	+	0.783614i	$0.713374\pi$
$354$		0			0
$355$	−67.4833			−0.190094
$356$		−	157.633i		−	0.442790i
$357$		0			0
$358$	72.1342			0.201492
$359$	347.072			0.966775			0.483387	−	0.875407i	$-0.339406\pi$
$359$	347.072			0.966775			0.483387	+	0.875407i	$0.339406\pi$
$360$		0			0
$361$	−300.967			−0.833703
$362$			109.509i			0.302511i
$363$		0			0
$364$		−	43.7673i		−	0.120240i
$365$		−	152.190i		−	0.416959i
$366$		0			0
$367$		−	617.839i		−	1.68349i	−0.539879	−	0.841743i	$-0.681530\pi$
$367$		−	617.839i		−	1.68349i	0.539879	−	0.841743i	$-0.318470\pi$
$368$			109.260i			0.296901i
$369$		0			0
$370$	60.8776			0.164534
$371$	−59.7647			−0.161091
$372$		0			0
$373$	−427.265			−1.14548			−0.572741	−	0.819736i	$-0.694120\pi$
$373$	−427.265			−1.14548			−0.572741	+	0.819736i	$0.694120\pi$
$374$	1.07842			0.00288348
$375$		0			0
$376$	−35.5093			−0.0944395
$377$		−	95.3482i		−	0.252913i
$378$		0			0
$379$	−400.731			−1.05734			−0.528668	−	0.848828i	$-0.677308\pi$
$379$	−400.731			−1.05734			−0.528668	+	0.848828i	$0.677308\pi$
$380$	72.9062			0.191858
$381$		0			0
$382$	−310.566			−0.813001
$383$	18.1585			0.0474111			0.0237056	−	0.999719i	$-0.492454\pi$
$383$	18.1585			0.0474111			0.0237056	+	0.999719i	$0.492454\pi$
$384$		0			0
$385$		−	4.98266i		−	0.0129420i
$386$			12.7278i			0.0329736i
$387$		0			0
$388$		−	347.762i		−	0.896293i
$389$	308.582			0.793269			0.396635	−	0.917977i	$-0.370178\pi$
$389$	308.582			0.793269			0.396635	+	0.917977i	$0.370178\pi$
$390$		0			0
$391$			27.5589i			0.0704832i
$392$		−	351.870i		−	0.897628i
$393$		0			0
$394$			292.790i			0.743122i
$395$	208.721			0.528407
$396$		0			0
$397$			193.629i			0.487730i	0.969809	+	0.243865i	$0.0784154\pi$
$397$			193.629i			0.487730i	−0.969809	+	0.243865i	$0.921585\pi$
$398$			178.537i			0.448585i
$399$		0			0
$400$	−44.8216			−0.112054
$401$			148.241i			0.369678i	0.982769	+	0.184839i	$0.0591763\pi$
$401$			148.241i			0.369678i	−0.982769	+	0.184839i	$0.940824\pi$
$402$		0			0
$403$	−97.6943			−0.242418
$404$			110.247i			0.272888i
$405$		0			0
$406$			10.0369i			0.0247213i
$407$	20.4511			0.0502484
$408$		0			0
$409$			486.132i			1.18859i	0.804249	+	0.594293i	$0.202568\pi$
$409$			486.132i			1.18859i	−0.804249	+	0.594293i	$0.797432\pi$
$410$		−	69.0476i		−	0.168409i
$411$		0			0
$412$		−	160.377i		−	0.389265i
$413$	53.4277	+	48.7942i	0.129365	+	0.118146i
$414$		0			0
$415$			76.2073i			0.183632i
$416$	−419.465			−1.00833
$417$		0			0
$418$	−10.2484			−0.0245177
$419$		−	393.978i		−	0.940281i	−0.882592	−	0.470140i	$-0.844203\pi$
$419$		−	393.978i		−	0.940281i	0.882592	−	0.470140i	$-0.155797\pi$
$420$		0			0
$421$			545.228i			1.29508i	0.762032	+	0.647539i	$0.224202\pi$
$421$			545.228i			1.29508i	−0.762032	+	0.647539i	$0.775798\pi$
$422$	168.555			0.399420
$423$		0			0
$424$			361.033i			0.851493i
$425$	−11.3055			−0.0266012
$426$		0			0
$427$		−	107.465i		−	0.251675i
$428$	−98.1441			−0.229309
$429$		0			0
$430$	177.452			0.412678
$431$		−	645.842i		−	1.49847i	−0.662302	−	0.749237i	$-0.730420\pi$
$431$		−	645.842i		−	1.49847i	0.662302	−	0.749237i	$-0.269580\pi$
$432$		0			0
$433$	−440.441			−1.01719			−0.508593	−	0.861007i	$-0.669834\pi$
$433$	−440.441			−1.01719			−0.508593	+	0.861007i	$0.669834\pi$
$434$	10.2838			0.0236954
$435$		0			0
$436$			27.7359i			0.0636144i
$437$		−	261.897i		−	0.599307i
$438$		0			0
$439$	−817.423			−1.86201			−0.931006	−	0.365003i	$-0.881068\pi$
$439$	−817.423			−1.86201			−0.931006	+	0.365003i	$0.881068\pi$
$440$	−30.0998			−0.0684086
$441$		0			0
$442$	−11.2085			−0.0253586
$443$			522.987i			1.18056i	0.807200	+	0.590278i	$0.200982\pi$
$443$			522.987i			1.18056i	−0.807200	+	0.590278i	$0.799018\pi$
$444$		0			0
$445$			186.517i			0.419138i
$446$			231.158i			0.518291i
$447$		0			0
$448$	28.2986			0.0631665
$449$	721.573			1.60707			0.803533	−	0.595260i	$-0.202951\pi$
$449$	721.573			1.60707			0.803533	+	0.595260i	$0.202951\pi$
$450$		0			0
$451$		−	23.1957i		−	0.0514317i
$452$			411.190i			0.909713i
$453$		0			0
$454$	−296.419			−0.652905
$455$			51.7869i			0.113817i
$456$		0			0
$457$			899.045i			1.96728i	0.180157	+	0.983638i	$0.442340\pi$
$457$			899.045i			1.96728i	−0.180157	+	0.983638i	$0.557660\pi$
$458$	−380.797			−0.831435
$459$		0			0
$460$		−	318.055i		−	0.691425i
$461$	−126.539			−0.274489			−0.137245	−	0.990537i	$-0.543825\pi$
$461$	−126.539			−0.274489			−0.137245	+	0.990537i	$0.543825\pi$
$462$		0			0
$463$			169.302i			0.365663i	0.983144	+	0.182831i	$0.0585263\pi$
$463$			169.302i			0.365663i	−0.983144	+	0.182831i	$0.941474\pi$
$464$	−24.3534			−0.0524858
$465$		0			0
$466$	−1.78597			−0.00383255
$467$		−	98.8969i		−	0.211771i	−0.994378	−	0.105885i	$-0.966232\pi$
$467$		−	98.8969i		−	0.211771i	0.994378	−	0.105885i	$-0.0337676\pi$
$468$		0			0
$469$			52.3261i			0.111570i
$470$	17.3731			0.0369640
$471$		0			0
$472$	294.762	−	322.752i	0.624495	−	0.683797i
$473$	59.6127			0.126031
$474$		0			0
$475$	107.438			0.226185
$476$	−2.81968			−0.00592370
$477$		0			0
$478$			80.9722i			0.169398i
$479$	91.0329			0.190048			0.0950240	−	0.995475i	$-0.469707\pi$
$479$	91.0329			0.190048			0.0950240	+	0.995475i	$0.469707\pi$
$480$		0			0
$481$	−212.557			−0.441907
$482$		−	61.3309i		−	0.127242i
$483$		0			0
$484$	337.039			0.696361
$485$			411.483i			0.848419i
$486$		0			0
$487$	−72.6279			−0.149133			−0.0745667	−	0.997216i	$-0.523757\pi$
$487$	−72.6279			−0.149133			−0.0745667	+	0.997216i	$0.523757\pi$
$488$	−649.187			−1.33030
$489$		0			0
$490$			172.154i			0.351335i
$491$	269.921			0.549736			0.274868	−	0.961482i	$-0.411366\pi$
$491$	269.921			0.549736			0.274868	+	0.961482i	$0.411366\pi$
$492$		0			0
$493$	−6.14274			−0.0124599
$494$	106.516			0.215620
$495$		0			0
$496$			24.9526i			0.0503077i
$497$	−24.8028			−0.0499050
$498$		0			0
$499$	219.786			0.440452			0.220226	−	0.975449i	$-0.429320\pi$
$499$	219.786			0.440452			0.220226	+	0.975449i	$0.429320\pi$
$500$	365.714			0.731428
$501$		0			0
$502$			23.4880i			0.0467888i
$503$		−	800.462i		−	1.59138i	−0.605707	−	0.795688i	$-0.707110\pi$
$503$		−	800.462i		−	1.59138i	0.605707	−	0.795688i	$-0.292890\pi$
$504$		0			0
$505$		−	130.448i		−	0.258312i
$506$			44.7090i			0.0883577i
$507$		0			0
$508$	411.233			0.809513
$509$		−	12.6193i		−	0.0247924i	−0.999923	−	0.0123962i	$-0.996054\pi$
$509$		−	12.6193i		−	0.0247924i	0.999923	−	0.0123962i	$-0.00394593\pi$
$510$		0			0
$511$		−	55.9358i		−	0.109463i
$512$			202.926i			0.396339i
$513$		0			0
$514$		−	76.0403i		−	0.147938i
$515$			189.763i			0.368473i
$516$		0			0
$517$	5.83627			0.0112887
$518$	22.3749			0.0431948
$519$		0			0
$520$	312.840			0.601616
$521$	−629.736			−1.20871			−0.604353	−	0.796717i	$-0.706568\pi$
$521$	−629.736			−1.20871			−0.604353	+	0.796717i	$0.706568\pi$
$522$		0			0
$523$	852.513			1.63004			0.815022	−	0.579430i	$-0.196725\pi$
$523$	852.513			1.63004			0.815022	+	0.579430i	$0.196725\pi$
$524$			86.2826i			0.164661i
$525$		0			0
$526$		−	488.937i		−	0.929538i
$527$			6.29389i			0.0119429i
$528$		0			0
$529$	−613.534			−1.15980
$530$		−	176.637i		−	0.333277i
$531$		0			0
$532$	26.7959			0.0503682
$533$			241.083i			0.452313i
$534$		0			0
$535$	116.127			0.217060
$536$	316.098			0.589734
$537$		0			0
$538$	181.834			0.337982
$539$			57.8331i			0.107297i
$540$		0			0
$541$		−	463.558i		−	0.856855i	−0.903576	−	0.428427i	$-0.859068\pi$
$541$		−	463.558i		−	0.856855i	0.903576	−	0.428427i	$-0.140932\pi$
$542$			147.880i			0.272842i
$543$		0			0
$544$			27.0237i			0.0496760i
$545$		−	32.8180i		−	0.0602165i
$546$		0			0
$547$	306.278			0.559922			0.279961	−	0.960011i	$-0.409678\pi$
$547$	306.278			0.559922			0.279961	+	0.960011i	$0.409678\pi$
$548$	111.252			0.203015
$549$		0			0
$550$	−18.3409			−0.0333472
$551$	58.3755			0.105945
$552$		0			0
$553$	76.7131			0.138722
$554$		−	212.015i		−	0.382699i
$555$		0			0
$556$	263.827			0.474508
$557$	994.672			1.78577			0.892883	−	0.450288i	$-0.148679\pi$
$557$	994.672			1.78577			0.892883	+	0.450288i	$0.148679\pi$
$558$		0			0
$559$	−619.581			−1.10837
$560$	13.2272			0.0236199
$561$		0			0
$562$		−	516.613i		−	0.919240i
$563$			206.256i			0.366352i	0.983080	+	0.183176i	$0.0586378\pi$
$563$			206.256i			0.366352i	−0.983080	+	0.183176i	$0.941362\pi$
$564$		0			0
$565$		−	486.533i		−	0.861121i
$566$	482.550			0.852561
$567$		0			0
$568$			149.831i			0.263788i
$569$			588.448i			1.03418i	0.855931	+	0.517089i	$0.172985\pi$
$569$			588.448i			1.03418i	−0.855931	+	0.517089i	$0.827015\pi$
$570$		0			0
$571$		−	139.611i		−	0.244503i	−0.992499	−	0.122251i	$-0.960989\pi$
$571$		−	139.611i		−	0.244503i	0.992499	−	0.122251i	$-0.0390114\pi$
$572$	43.4557			0.0759714
$573$		0			0
$574$		−	25.3777i		−	0.0442120i
$575$		−	468.701i		−	0.815133i
$576$		0			0
$577$	947.400			1.64194			0.820971	−	0.570970i	$-0.193433\pi$
$577$	947.400			1.64194			0.820971	+	0.570970i	$0.193433\pi$
$578$		−	313.212i		−	0.541890i
$579$		0			0
$580$	70.8929			0.122229
$581$			28.0092i			0.0482085i
$582$		0			0
$583$		−	59.3391i		−	0.101782i
$584$	−337.903			−0.578601
$585$		0			0
$586$			61.3693i			0.104726i
$587$		−	672.802i		−	1.14617i	−0.819496	−	0.573085i	$-0.805746\pi$
$587$		−	672.802i		−	1.14617i	0.819496	−	0.573085i	$-0.194254\pi$
$588$		0			0
$589$		−	59.8118i		−	0.101548i
$590$	−144.213	+	157.908i	−0.244429	+	0.267641i
$591$		0			0
$592$			54.2903i			0.0917067i
$593$	−235.268			−0.396742			−0.198371	−	0.980127i	$-0.563565\pi$
$593$	−235.268			−0.396742			−0.198371	+	0.980127i	$0.563565\pi$
$594$		0			0
$595$	3.33634			0.00560729
$596$		−	376.287i		−	0.631353i
$597$		0			0
$598$		−	464.680i		−	0.777056i
$599$	938.758			1.56721			0.783604	−	0.621260i	$-0.213379\pi$
$599$	938.758			1.56721			0.783604	+	0.621260i	$0.213379\pi$
$600$		0			0
$601$		−	396.943i		−	0.660470i	−0.943899	−	0.330235i	$-0.892872\pi$
$601$		−	396.943i		−	0.660470i	0.943899	−	0.330235i	$-0.107128\pi$
$602$	65.2204			0.108340
$603$		0			0
$604$		−	269.423i		−	0.446065i
$605$	−398.795			−0.659166
$606$		0			0
$607$	794.727			1.30927			0.654635	−	0.755945i	$-0.272822\pi$
$607$	794.727			1.30927			0.654635	+	0.755945i	$0.272822\pi$
$608$		−	256.811i		−	0.422386i
$609$		0			0
$610$	317.618			0.520685
$611$	−60.6589			−0.0992780
$612$		0			0
$613$			14.2456i			0.0232392i	0.999932	+	0.0116196i	$0.00369872\pi$
$613$			14.2456i			0.0232392i	−0.999932	+	0.0116196i	$0.996301\pi$
$614$			303.650i			0.494544i
$615$		0			0
$616$	−11.0629			−0.0179592
$617$	−739.753			−1.19895			−0.599476	−	0.800393i	$-0.704624\pi$
$617$	−739.753			−1.19895			−0.599476	+	0.800393i	$0.704624\pi$
$618$		0			0
$619$	202.230			0.326705			0.163352	−	0.986568i	$-0.447769\pi$
$619$	202.230			0.326705			0.163352	+	0.986568i	$0.447769\pi$
$620$		−	72.6372i		−	0.117157i
$621$		0			0
$622$		−	228.970i		−	0.368119i
$623$			68.5521i			0.110036i
$624$		0			0
$625$	−86.0664			−0.137706
$626$	404.814			0.646668
$627$		0			0
$628$		−	691.116i		−	1.10050i
$629$			13.6938i			0.0217708i
$630$		0			0
$631$	−504.848			−0.800076			−0.400038	−	0.916499i	$-0.631003\pi$
$631$	−504.848			−0.800076			−0.400038	+	0.916499i	$0.631003\pi$
$632$		−	463.417i		−	0.733255i
$633$		0			0
$634$		−	324.910i		−	0.512476i
$635$	−486.584			−0.766274
$636$		0			0
$637$		−	601.084i		−	0.943617i
$638$	−9.96539			−0.0156197
$639$		0			0
$640$		−	358.743i		−	0.560536i
$641$	−757.354			−1.18152			−0.590760	−	0.806847i	$-0.701172\pi$
$641$	−757.354			−1.18152			−0.590760	+	0.806847i	$0.701172\pi$
$642$		0			0
$643$	390.062			0.606628			0.303314	−	0.952891i	$-0.401907\pi$
$643$	390.062			0.606628			0.303314	+	0.952891i	$0.401907\pi$
$644$		−	116.898i		−	0.181518i
$645$		0			0
$646$		−	6.86223i		−	0.0106226i
$647$	992.824			1.53450			0.767252	−	0.641346i	$-0.221624\pi$
$647$	992.824			1.53450			0.767252	+	0.641346i	$0.221624\pi$
$648$		0			0
$649$	−48.4467	+	53.0473i	−0.0746483	+	0.0817369i
$650$	190.625			0.293270
$651$		0			0
$652$	576.976			0.884932
$653$	−1166.56			−1.78646			−0.893228	−	0.449604i	$-0.851565\pi$
$653$	−1166.56			−1.78646			−0.893228	+	0.449604i	$0.851565\pi$
$654$		0			0
$655$		−	102.092i		−	0.155866i
$656$	61.5763			0.0938663
$657$		0			0
$658$	6.38528			0.00970407
$659$			689.431i			1.04618i	0.852278	+	0.523089i	$0.175220\pi$
$659$			689.431i			1.04618i	−0.852278	+	0.523089i	$0.824780\pi$
$660$		0			0
$661$	1035.80			1.56703			0.783513	−	0.621376i	$-0.213426\pi$
$661$	1035.80			1.56703			0.783513	+	0.621376i	$0.213426\pi$
$662$			291.320i			0.440061i
$663$		0			0
$664$	169.201			0.254821
$665$	−31.7057			−0.0476778
$666$		0			0
$667$		−	254.665i		−	0.381806i
$668$	284.468			0.425851
$669$		0			0
$670$	−154.652			−0.230824
$671$	106.700			0.159016
$672$		0			0
$673$			504.708i			0.749937i	0.927037	+	0.374969i	$0.122347\pi$
$673$			504.708i			0.749937i	−0.927037	+	0.374969i	$0.877653\pi$
$674$	−404.357			−0.599936
$675$		0			0
$676$	24.9264			0.0368734
$677$	−184.647			−0.272743			−0.136372	−	0.990658i	$-0.543544\pi$
$677$	−184.647			−0.272743			−0.136372	+	0.990658i	$0.543544\pi$
$678$		0			0
$679$			151.236i			0.222734i
$680$		−	20.1545i		−	0.0296390i
$681$		0			0
$682$			10.2106i			0.0149715i
$683$			150.545i			0.220417i	0.993908	+	0.110209i	$0.0351519\pi$
$683$			150.545i			0.220417i	−0.993908	+	0.110209i	$0.964848\pi$
$684$		0			0
$685$	−131.637			−0.192171
$686$			128.550i			0.187391i
$687$		0			0
$688$			158.251i			0.230015i
$689$			616.737i			0.895119i
$690$		0			0
$691$		−	8.87982i		−	0.0128507i	−0.999979	−	0.00642534i	$-0.997955\pi$
$691$		−	8.87982i		−	0.0128507i	0.999979	−	0.00642534i	$-0.00204526\pi$
$692$		−	542.707i		−	0.784259i
$693$		0			0
$694$	−176.716			−0.254634
$695$	−312.168			−0.449163
$696$		0			0
$697$	15.5316			0.0222835
$698$	−515.415			−0.738417
$699$		0			0
$700$	47.9550			0.0685071
$701$		−	672.703i		−	0.959633i	−0.877369	−	0.479817i	$-0.840703\pi$
$701$		−	672.703i		−	0.959633i	0.877369	−	0.479817i	$-0.159297\pi$
$702$		0			0
$703$		−	130.135i		−	0.185114i
$704$			28.0971i			0.0399106i
$705$		0			0
$706$	−600.963			−0.851223
$707$		−	47.9446i		−	0.0678141i
$708$		0			0
$709$	−885.467			−1.24890			−0.624448	−	0.781067i	$-0.714676\pi$
$709$	−885.467			−1.24890			−0.624448	+	0.781067i	$0.714676\pi$
$710$		−	73.3057i		−	0.103247i
$711$		0			0
$712$	414.117			0.581626
$713$	−260.931			−0.365962
$714$		0			0
$715$	−51.4181			−0.0719135
$716$		−	187.262i		−	0.261539i
$717$		0			0
$718$			377.017i			0.525093i
$719$		−	900.012i		−	1.25175i	−0.779922	−	0.625877i	$-0.784741\pi$
$719$		−	900.012i		−	1.25175i	0.779922	−	0.625877i	$-0.215259\pi$
$720$		0			0
$721$			69.7455i			0.0967344i
$722$		−	326.934i		−	0.452816i
$723$		0			0
$724$	284.288			0.392662
$725$	104.471			0.144098
$726$		0			0
$727$	−702.800			−0.966713			−0.483357	−	0.875424i	$-0.660583\pi$
$727$	−702.800			−0.966713			−0.483357	+	0.875424i	$0.660583\pi$
$728$	114.981			0.157941
$729$		0			0
$730$	165.321			0.226467
$731$			39.9161i			0.0546048i
$732$		0			0
$733$	−389.940			−0.531978			−0.265989	−	0.963976i	$-0.585698\pi$
$733$	−389.940			−0.531978			−0.265989	+	0.963976i	$0.585698\pi$
$734$	671.145			0.914366
$735$		0			0
$736$	−1120.35			−1.52221
$737$	−51.9535			−0.0704933
$738$		0			0
$739$			1203.79i			1.62894i	0.580206	+	0.814470i	$0.302972\pi$
$739$			1203.79i			1.62894i	−0.580206	+	0.814470i	$0.697028\pi$
$740$		−	158.039i		−	0.213567i
$741$		0			0
$742$		−	64.9210i		−	0.0874947i
$743$	1379.37			1.85649			0.928247	−	0.371965i	$-0.121316\pi$
$743$	1379.37			1.85649			0.928247	+	0.371965i	$0.121316\pi$
$744$		0			0
$745$			445.234i			0.597630i
$746$		−	464.128i		−	0.622156i
$747$		0			0
$748$		−	2.79960i		−	0.00374278i
$749$	42.6813			0.0569844
$750$		0			0
$751$			855.872i			1.13964i	0.821769	+	0.569821i	$0.192988\pi$
$751$			855.872i			1.13964i	−0.821769	+	0.569821i	$0.807012\pi$
$752$			15.4932i			0.0206027i
$753$		0			0
$754$	103.575			0.137367
$755$			318.791i			0.422239i
$756$		0			0
$757$	−126.963			−0.167719			−0.0838593	−	0.996478i	$-0.526725\pi$
$757$	−126.963			−0.167719			−0.0838593	+	0.996478i	$0.526725\pi$
$758$		−	435.305i		−	0.574281i
$759$		0			0
$760$			191.532i			0.252015i
$761$	−615.423			−0.808703			−0.404351	−	0.914604i	$-0.632503\pi$
$761$	−615.423			−0.808703			−0.404351	+	0.914604i	$0.632503\pi$
$762$		0			0
$763$		−	12.0619i		−	0.0158085i
$764$			806.236i			1.05528i
$765$		0			0
$766$			19.7251i			0.0257508i
$767$	503.528	−	551.343i	0.656490	−	0.718831i
$768$		0			0
$769$			986.655i			1.28304i	0.767108	+	0.641518i	$0.221695\pi$
$769$			986.655i			1.28304i	−0.767108	+	0.641518i	$0.778305\pi$
$770$	5.41255			0.00702929
$771$		0			0
$772$	33.0416			0.0428000
$773$			6.18406i			0.00800008i	0.999992	+	0.00400004i	$0.00127326\pi$
$773$			6.18406i			0.00800008i	−0.999992	+	0.00400004i	$0.998727\pi$
$774$		0			0
$775$		−	107.042i		−	0.138118i
$776$	913.604			1.17733
$777$		0			0
$778$			335.206i			0.430856i
$779$	−147.599			−0.189473
$780$		0			0
$781$		−	24.6262i		−	0.0315316i
$782$	−29.9367			−0.0382822
$783$		0			0
$784$	−153.526			−0.195824
$785$			817.751i			1.04172i
$786$		0			0
$787$	545.010			0.692516			0.346258	−	0.938139i	$-0.387452\pi$
$787$	545.010			0.692516			0.346258	+	0.938139i	$0.387452\pi$
$788$	760.089			0.964580
$789$		0			0
$790$			226.729i			0.286999i
$791$		−	178.820i		−	0.226068i
$792$		0			0
$793$	−1108.98			−1.39846
$794$	−210.335			−0.264905
$795$		0			0
$796$	463.486			0.582268
$797$			722.233i			0.906190i	0.891462	+	0.453095i	$0.149680\pi$
$797$			722.233i			0.906190i	−0.891462	+	0.453095i	$0.850320\pi$
$798$		0			0
$799$			3.90791i			0.00489100i
$800$		−	459.599i		−	0.574499i
$801$		0			0
$802$	−161.031			−0.200786
$803$	55.5375			0.0691625
$804$		0			0
$805$			138.317i			0.171823i
$806$		−	106.123i		−	0.131666i
$807$		0			0
$808$	−289.629			−0.358452
$809$			1019.92i			1.26071i	0.776305	+	0.630357i	$0.217092\pi$
$809$			1019.92i			1.26071i	−0.776305	+	0.630357i	$0.782908\pi$
$810$		0			0
$811$		−	1310.48i		−	1.61588i	−0.589267	−	0.807939i	$-0.700583\pi$
$811$		−	1310.48i		−	1.61588i	0.589267	−	0.807939i	$-0.299417\pi$
$812$	26.0559			0.0320886
$813$		0			0
$814$			22.2156i			0.0272919i
$815$	−682.696			−0.837664
$816$		0			0
$817$		−	379.329i		−	0.464295i
$818$	−528.074			−0.645567
$819$		0			0
$820$	−179.249			−0.218596
$821$		−	504.163i		−	0.614084i	−0.951696	−	0.307042i	$-0.900661\pi$
$821$		−	504.163i		−	0.614084i	0.951696	−	0.307042i	$-0.0993392\pi$
$822$		0			0
$823$			307.365i			0.373469i	0.982410	+	0.186734i	$0.0597904\pi$
$823$			307.365i			0.373469i	−0.982410	+	0.186734i	$0.940210\pi$
$824$	421.327			0.511319
$825$		0			0
$826$	−53.0041	+	58.0374i	−0.0641696	+	0.0702631i
$827$	940.910			1.13774			0.568869	−	0.822428i	$-0.307381\pi$
$827$	940.910			1.13774			0.568869	+	0.822428i	$0.307381\pi$
$828$		0			0
$829$	398.747			0.480997			0.240499	−	0.970649i	$-0.422689\pi$
$829$	398.747			0.480997			0.240499	+	0.970649i	$0.422689\pi$
$830$	−82.7823			−0.0997377
$831$		0			0
$832$		−	292.025i		−	0.350992i
$833$	−38.7244			−0.0464879
$834$		0			0
$835$	−336.592			−0.403104
$836$			26.6051i			0.0318242i
$837$		0			0
$838$	427.969			0.510703
$839$		−	1542.84i		−	1.83890i	−0.393203	−	0.919452i	$-0.628633\pi$
$839$		−	1542.84i		−	1.83890i	0.393203	−	0.919452i	$-0.371367\pi$
$840$		0			0
$841$	−784.237			−0.932505
$842$	−592.269			−0.703407
$843$		0			0
$844$		−	437.572i		−	0.518451i
$845$	−29.4937			−0.0349038
$846$		0			0
$847$	−146.573			−0.173049
$848$	157.524			0.185760
$849$		0			0
$850$		−	12.2809i		−	0.0144481i
$851$	−567.717			−0.667118
$852$		0			0
$853$	−362.614			−0.425105			−0.212552	−	0.977150i	$-0.568178\pi$
$853$	−362.614			−0.425105			−0.212552	+	0.977150i	$0.568178\pi$
$854$	116.737			0.136694
$855$		0			0
$856$		−	257.834i		−	0.301208i
$857$			1243.72i			1.45124i	0.688093	+	0.725622i	$0.258448\pi$
$857$			1243.72i			1.45124i	−0.688093	+	0.725622i	$0.741552\pi$
$858$		0			0
$859$		−	1101.40i		−	1.28218i	−0.767464	−	0.641092i	$-0.778482\pi$
$859$		−	1101.40i		−	1.28218i	0.767464	−	0.641092i	$-0.221518\pi$
$860$		−	460.668i		−	0.535661i
$861$		0			0
$862$	701.564			0.813880
$863$			1561.83i			1.80977i	0.425654	+	0.904886i	$0.360044\pi$
$863$			1561.83i			1.80977i	−0.425654	+	0.904886i	$0.639956\pi$
$864$		0			0
$865$			642.148i			0.742368i
$866$		−	478.442i		−	0.552473i
$867$		0			0
$868$		−	26.6970i		−	0.0307569i
$869$			76.1669i			0.0876489i
$870$		0			0
$871$	539.976			0.619949
$872$	−72.8649			−0.0835606
$873$		0			0
$874$	284.493			0.325507
$875$	−159.043			−0.181764
$876$		0			0
$877$	1001.36			1.14181			0.570903	−	0.821017i	$-0.306593\pi$
$877$	1001.36			1.14181			0.570903	+	0.821017i	$0.306593\pi$
$878$		−	887.949i		−	1.01133i
$879$		0			0
$880$			13.1330i			0.0149238i
$881$		−	9.37327i		−	0.0106394i	−0.999986	−	0.00531968i	$-0.998307\pi$
$881$		−	9.37327i		−	0.0106394i	0.999986	−	0.00531968i	$-0.00169331\pi$
$882$		0			0
$883$	−1323.52			−1.49889			−0.749445	−	0.662067i	$-0.769679\pi$
$883$	−1323.52			−1.49889			−0.749445	+	0.662067i	$0.769679\pi$
$884$			29.0975i			0.0329157i
$885$		0			0
$886$	−568.109			−0.641207
$887$			351.395i			0.396161i	0.980186	+	0.198081i	$0.0634708\pi$
$887$			351.395i			0.396161i	−0.980186	+	0.198081i	$0.936529\pi$
$888$		0			0
$889$	−178.839			−0.201168
$890$	−202.609			−0.227650
$891$		0			0
$892$	600.090			0.672747
$893$		−	37.1375i		−	0.0415873i
$894$		0			0
$895$			221.574i			0.247569i
$896$		−	131.852i		−	0.147156i
$897$		0			0
$898$			783.829i			0.872860i
$899$		−	58.1601i		−	0.0646942i
$900$		0			0
$901$	39.7328			0.0440986
$902$	25.1970			0.0279346
$903$		0			0
$904$	−1080.24			−1.19495
$905$	−336.378			−0.371689
$906$		0			0
$907$	1042.34			1.14922			0.574610	−	0.818427i	$-0.305154\pi$
$907$	1042.34			1.14922			0.574610	+	0.818427i	$0.305154\pi$
$908$			769.510i			0.847478i
$909$		0			0
$910$	−56.2550			−0.0618187
$911$	1072.22			1.17697			0.588483	−	0.808510i	$-0.299725\pi$
$911$	1072.22			1.17697			0.588483	+	0.808510i	$0.299725\pi$
$912$		0			0
$913$	−27.8097			−0.0304597
$914$	−976.613			−1.06850
$915$		0			0
$916$			988.558i			1.07921i
$917$		−	37.5229i		−	0.0409192i
$918$		0			0
$919$			321.264i			0.349580i	0.984606	+	0.174790i	$0.0559247\pi$
$919$			321.264i			0.349580i	−0.984606	+	0.174790i	$0.944075\pi$
$920$	835.563			0.908220
$921$		0			0
$922$		−	137.457i		−	0.149086i
$923$			255.950i			0.277303i
$924$		0			0
$925$		−	232.894i		−	0.251778i
$926$	−183.909			−0.198606
$927$		0			0
$928$		−	249.719i		−	0.269094i
$929$			537.335i			0.578402i	0.957268	+	0.289201i	$0.0933896\pi$
$929$			537.335i			0.578402i	−0.957268	+	0.289201i	$0.906610\pi$
$930$		0			0
$931$	368.005			0.395279
$932$			4.63641i			0.00497469i
$933$		0			0
$934$	107.430			0.115021
$935$			3.31258i			0.00354286i
$936$		0			0
$937$		−	200.809i		−	0.214311i	−0.994242	−	0.107155i	$-0.965826\pi$
$937$		−	200.809i		−	0.214311i	0.994242	−	0.107155i	$-0.0341742\pi$
$938$	−56.8407			−0.0605978
$939$		0			0
$940$		−	45.1008i		−	0.0479796i
$941$			1718.91i			1.82668i	0.407197	+	0.913340i	$0.366506\pi$
$941$			1718.91i			1.82668i	−0.407197	+	0.913340i	$0.633494\pi$
$942$		0			0
$943$			643.907i			0.682828i
$944$	−140.822	−	128.609i	−0.149175	−	0.136238i
$945$		0			0
$946$			64.7560i			0.0684524i
$947$	−684.394			−0.722696			−0.361348	−	0.932431i	$-0.617683\pi$
$947$	−684.394			−0.722696			−0.361348	+	0.932431i	$0.617683\pi$
$948$		0			0
$949$	−577.225			−0.608245
$950$			116.708i			0.122850i
$951$		0			0
$952$		−	7.40758i		−	0.00778107i
$953$	1256.84			1.31882			0.659412	−	0.751781i	$-0.270805\pi$
$953$	1256.84			1.31882			0.659412	+	0.751781i	$0.270805\pi$
$954$		0			0
$955$		−	953.965i		−	0.998916i
$956$	210.206			0.219880
$957$		0			0
$958$			98.8871i			0.103222i
$959$	−48.3818			−0.0504503
$960$		0			0
$961$	901.409			0.937990
$962$		−	230.896i		−	0.240017i
$963$		0			0
$964$	−159.216			−0.165162
$965$	−39.0959			−0.0405139
$966$		0			0
$967$			1449.89i			1.49937i	0.661797	+	0.749683i	$0.269794\pi$
$967$			1449.89i			1.49937i	−0.661797	+	0.749683i	$0.730206\pi$
$968$			885.434i			0.914705i
$969$		0			0
$970$	−446.985			−0.460809
$971$	−947.360			−0.975654			−0.487827	−	0.872940i	$-0.662210\pi$
$971$	−947.360			−0.975654			−0.487827	+	0.872940i	$0.662210\pi$
$972$		0			0
$973$	−114.734			−0.117918
$974$		−	78.8941i		−	0.0810001i
$975$		0			0
$976$			283.250i			0.290215i
$977$		−	914.983i		−	0.936523i	−0.883590	−	0.468261i	$-0.844881\pi$
$977$		−	914.983i		−	0.936523i	0.883590	−	0.468261i	$-0.155119\pi$
$978$		0			0
$979$	−68.0640			−0.0695240
$980$	446.915			0.456036
$981$		0			0
$982$			293.209i			0.298583i
$983$			729.129i			0.741738i	0.928685	+	0.370869i	$0.120940\pi$
$983$			729.129i			0.741738i	−0.928685	+	0.370869i	$0.879060\pi$
$984$		0			0
$985$	−899.362			−0.913058
$986$		−	6.67273i		−	0.00676747i
$987$		0			0
$988$		−	276.518i		−	0.279877i
$989$	−1654.83			−1.67324
$990$		0			0
$991$			1025.34i			1.03465i	0.855788	+	0.517327i	$0.173073\pi$
$991$			1025.34i			1.03465i	−0.855788	+	0.517327i	$0.826927\pi$
$992$	−255.863			−0.257927
$993$		0			0
$994$		−	26.9427i		−	0.0271053i
$995$	−548.411			−0.551167
$996$		0			0
$997$	626.916			0.628802			0.314401	−	0.949290i	$-0.398196\pi$
$997$	626.916			0.628802			0.314401	+	0.949290i	$0.398196\pi$
$998$			238.748i			0.239227i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.12		20
3.2	odd	2		177.3.c.a.58.9	✓	20
59.58	odd	2	inner	531.3.c.c.235.9		20
177.176	even	2		177.3.c.a.58.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.9	✓	20	3.2	odd	2
177.3.c.a.58.12	yes	20	177.176	even	2
531.3.c.c.235.9		20	59.58	odd	2	inner
531.3.c.c.235.12		20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.08628i q^{2} +2.82000 q^{4} -3.33671 q^{5} -1.22637 q^{7} +7.40842i q^{8} +O(q^{10})\)	\(q+1.08628i q^{2} +2.82000 q^{4} -3.33671 q^{5} -1.22637 q^{7} +7.40842i q^{8} -3.62460i q^{10} -1.21764i q^{11} +12.6555i q^{13} -1.33218i q^{14} +3.23240 q^{16} +0.815320 q^{17} -7.74812 q^{19} -9.40954 q^{20} +1.32270 q^{22} +33.8014i q^{23} -13.8663 q^{25} -13.7474 q^{26} -3.45837 q^{28} -7.53415 q^{29} +7.71953i q^{31} +33.1450i q^{32} +0.885664i q^{34} +4.09206 q^{35} +16.7957i q^{37} -8.41661i q^{38} -24.7198i q^{40} +19.0497 q^{41} +48.9576i q^{43} -3.43375i q^{44} -36.7177 q^{46} +4.79310i q^{47} -47.4960 q^{49} -15.0627i q^{50} +35.6884i q^{52} +48.7328 q^{53} +4.06292i q^{55} -9.08548i q^{56} -8.18418i q^{58} +(-43.5656 - 39.7874i) q^{59} +87.6284i q^{61} -8.38556 q^{62} -23.0750 q^{64} -42.2277i q^{65} -42.6674i q^{67} +2.29920 q^{68} +4.44511i q^{70} +20.2245 q^{71} +45.6107i q^{73} -18.2448 q^{74} -21.8497 q^{76} +1.49328i q^{77} -62.5528 q^{79} -10.7856 q^{80} +20.6933i q^{82} -22.8390i q^{83} -2.72049 q^{85} -53.1815 q^{86} +9.02079 q^{88} -55.8982i q^{89} -15.5203i q^{91} +95.3199i q^{92} -5.20664 q^{94} +25.8533 q^{95} -123.320i q^{97} -51.5939i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 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

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