LMFDB - Embedded newform 531.2.d.a.530.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [531,2,Mod(530,531)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("531.530"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(531, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$4.24005634733$
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} - 10 x^{18} + 139 x^{16} - 476 x^{14} + 4681 x^{12} - 666 x^{10} + 82273 x^{8} + 168944 x^{6} + \cdots + 3374569 $ x^20 - 10x^18 + 139x^16 - 476x^14 + 4681x^12 - 666x^10 + 82273x^8 + 168944x^6 + 845867x^4 + 1613394*x^2 + 3374569
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			530.12
Root			$-0.535103 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	531.530
Dual form			531.2.d.a.530.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.535103 q^{2} -1.71366 q^{4} +0.0572139i q^{5} +1.50263 q^{7} -1.98719 q^{8} +0.0306154i q^{10} +4.66096 q^{11} +3.89497i q^{13} +0.804063 q^{14} +2.36397 q^{16} -1.25895i q^{17} +2.63275 q^{19} -0.0980455i q^{20} +2.49409 q^{22} +3.36966 q^{23} +4.99673 q^{25} +2.08421i q^{26} -2.57501 q^{28} -5.86470i q^{29} +7.43142i q^{31} +5.23936 q^{32} -0.673671i q^{34} +0.0859715i q^{35} +8.51607i q^{37} +1.40879 q^{38} -0.113695i q^{40} -9.39188i q^{41} +3.51460i q^{43} -7.98732 q^{44} +1.80311 q^{46} +1.49539 q^{47} -4.74210 q^{49} +2.67376 q^{50} -6.67467i q^{52} +5.49315i q^{53} +0.266672i q^{55} -2.98602 q^{56} -3.13822i q^{58} +(-6.04342 - 4.74100i) q^{59} -1.90926i q^{61} +3.97658i q^{62} -1.92435 q^{64} -0.222847 q^{65} -5.56817i q^{67} +2.15742i q^{68} +0.0460036i q^{70} +6.42027i q^{71} -12.1291i q^{73} +4.55698i q^{74} -4.51165 q^{76} +7.00371 q^{77} -4.35495 q^{79} +0.135252i q^{80} -5.02563i q^{82} -7.98732 q^{83} +0.0720297 q^{85} +1.88068i q^{86} -9.26223 q^{88} -12.3821 q^{89} +5.85271i q^{91} -5.77446 q^{92} +0.800187 q^{94} +0.150630i q^{95} +11.0266i q^{97} -2.53751 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) $ 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

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 $

$2$	0.535103			0.378375			0.189188	−	0.981941i	$-0.439415\pi$
$2$	0.535103			0.378375			0.189188	+	0.981941i	$0.439415\pi$
$3$		0			0
$3$		0			0
$4$	−1.71366			−0.856832
$5$			0.0572139i			0.0255868i	0.999918	+	0.0127934i	$0.00407238\pi$
$5$			0.0572139i			0.0255868i	−0.999918	+	0.0127934i	$0.995928\pi$
$6$		0			0
$7$	1.50263			0.567942			0.283971	−	0.958833i	$-0.408348\pi$
$7$	1.50263			0.567942			0.283971	+	0.958833i	$0.408348\pi$
$8$	−1.98719			−0.702579
$9$		0			0
$10$			0.0306154i			0.00968143i
$11$	4.66096			1.40533			0.702666	−	0.711520i	$-0.251993\pi$
$11$	4.66096			1.40533			0.702666	+	0.711520i	$0.251993\pi$
$12$		0			0
$13$			3.89497i			1.08027i	0.841578	+	0.540135i	$0.181627\pi$
$13$			3.89497i			1.08027i	−0.841578	+	0.540135i	$0.818373\pi$
$14$	0.804063			0.214895
$15$		0			0
$16$	2.36397			0.590994
$17$		−	1.25895i		−	0.305341i	−0.988277	−	0.152671i	$-0.951213\pi$
$17$		−	1.25895i		−	0.305341i	0.988277	−	0.152671i	$-0.0487874\pi$
$18$		0			0
$19$	2.63275			0.603995			0.301997	−	0.953309i	$-0.402347\pi$
$19$	2.63275			0.603995			0.301997	+	0.953309i	$0.402347\pi$
$20$		−	0.0980455i		−	0.0219236i
$21$		0			0
$22$	2.49409			0.531743
$23$	3.36966			0.702622			0.351311	−	0.936259i	$-0.385736\pi$
$23$	3.36966			0.702622			0.351311	+	0.936259i	$0.385736\pi$
$24$		0			0
$25$	4.99673			0.999345
$26$			2.08421i			0.408747i
$27$		0			0
$28$	−2.57501			−0.486631
$29$		−	5.86470i		−	1.08905i	−0.838746	−	0.544524i	$-0.816710\pi$
$29$		−	5.86470i		−	1.08905i	0.838746	−	0.544524i	$-0.183290\pi$
$30$		0			0
$31$			7.43142i			1.33472i	0.744734	+	0.667361i	$0.232576\pi$
$31$			7.43142i			1.33472i	−0.744734	+	0.667361i	$0.767424\pi$
$32$	5.23936			0.926197
$33$		0			0
$34$		−	0.673671i		−	0.115534i
$35$			0.0859715i			0.0145318i
$36$		0			0
$37$			8.51607i			1.40003i	0.714126	+	0.700017i	$0.246824\pi$
$37$			8.51607i			1.40003i	−0.714126	+	0.700017i	$0.753176\pi$
$38$	1.40879			0.228537
$39$		0			0
$40$		−	0.113695i		−	0.0179768i
$41$		−	9.39188i		−	1.46677i	−0.679816	−	0.733383i	$-0.737940\pi$
$41$		−	9.39188i		−	1.46677i	0.679816	−	0.733383i	$-0.262060\pi$
$42$		0			0
$43$			3.51460i			0.535972i	0.963423	+	0.267986i	$0.0863581\pi$
$43$			3.51460i			0.535972i	−0.963423	+	0.267986i	$0.913642\pi$
$44$	−7.98732			−1.20413
$45$		0			0
$46$	1.80311			0.265855
$47$	1.49539			0.218125			0.109062	−	0.994035i	$-0.465215\pi$
$47$	1.49539			0.218125			0.109062	+	0.994035i	$0.465215\pi$
$48$		0			0
$49$	−4.74210			−0.677442
$50$	2.67376			0.378127
$51$		0			0
$52$		−	6.67467i		−	0.925610i
$53$			5.49315i			0.754542i	0.926103	+	0.377271i	$0.123137\pi$
$53$			5.49315i			0.754542i	−0.926103	+	0.377271i	$0.876863\pi$
$54$		0			0
$55$			0.266672i			0.0359580i
$56$	−2.98602			−0.399024
$57$		0			0
$58$		−	3.13822i		−	0.412068i
$59$	−6.04342	−	4.74100i	−0.786787	−	0.617225i
$59$	−6.04342	−	4.74100i	−0.786787	−	0.617225i
$60$		0			0
$61$		−	1.90926i		−	0.244455i	−0.992502	−	0.122228i	$-0.960996\pi$
$61$		−	1.90926i		−	0.244455i	0.992502	−	0.122228i	$-0.0390038\pi$
$62$			3.97658i			0.505026i
$63$		0			0
$64$	−1.92435			−0.240544
$65$	−0.222847			−0.0276407
$66$		0			0
$67$		−	5.56817i		−	0.680260i	−0.940378	−	0.340130i	$-0.889529\pi$
$67$		−	5.56817i		−	0.680260i	0.940378	−	0.340130i	$-0.110471\pi$
$68$			2.15742i			0.261626i
$69$		0			0
$70$			0.0460036i			0.00549849i
$71$			6.42027i			0.761946i	0.924586	+	0.380973i	$0.124411\pi$
$71$			6.42027i			0.761946i	−0.924586	+	0.380973i	$0.875589\pi$
$72$		0			0
$73$		−	12.1291i		−	1.41961i	−0.704399	−	0.709805i	$-0.748783\pi$
$73$		−	12.1291i		−	1.41961i	0.704399	−	0.709805i	$-0.251217\pi$
$74$			4.55698i			0.529738i
$75$		0			0
$76$	−4.51165			−0.517522
$77$	7.00371			0.798146
$78$		0			0
$79$	−4.35495			−0.489971			−0.244985	−	0.969527i	$-0.578783\pi$
$79$	−4.35495			−0.489971			−0.244985	+	0.969527i	$0.578783\pi$
$80$			0.135252i			0.0151217i
$81$		0			0
$82$		−	5.02563i		−	0.554988i
$83$	−7.98732			−0.876722			−0.438361	−	0.898799i	$-0.644441\pi$
$83$	−7.98732			−0.876722			−0.438361	+	0.898799i	$0.644441\pi$
$84$		0			0
$85$	0.0720297			0.00781272
$86$			1.88068i			0.202798i
$87$		0			0
$88$	−9.26223			−0.987357
$89$	−12.3821			−1.31250			−0.656252	−	0.754542i	$-0.727859\pi$
$89$	−12.3821			−1.31250			−0.656252	+	0.754542i	$0.727859\pi$
$90$		0			0
$91$			5.85271i			0.613530i
$92$	−5.77446			−0.602029
$93$		0			0
$94$	0.800187			0.0825330
$95$			0.150630i			0.0154543i
$96$		0			0
$97$			11.0266i			1.11958i	0.828633	+	0.559792i	$0.189119\pi$
$97$			11.0266i			1.11958i	−0.828633	+	0.559792i	$0.810881\pi$
$98$	−2.53751			−0.256327
$99$		0			0
$100$	−8.56271			−0.856271
$101$	6.49193			0.645971			0.322986	−	0.946404i	$-0.395313\pi$
$101$	6.49193			0.645971			0.322986	+	0.946404i	$0.395313\pi$
$102$		0			0
$103$		−	14.1673i		−	1.39595i	−0.716123	−	0.697974i	$-0.754085\pi$
$103$		−	14.1673i		−	1.39595i	0.716123	−	0.697974i	$-0.245915\pi$
$104$		−	7.74006i		−	0.758975i
$105$		0			0
$106$			2.93940i			0.285500i
$107$		−	4.10739i		−	0.397076i	−0.980093	−	0.198538i	$-0.936381\pi$
$107$		−	4.10739i		−	0.397076i	0.980093	−	0.198538i	$-0.0636194\pi$
$108$		0			0
$109$		−	0.953525i		−	0.0913312i	−0.998957	−	0.0456656i	$-0.985459\pi$
$109$		−	0.953525i		−	0.0913312i	0.998957	−	0.0456656i	$-0.0145409\pi$
$110$			0.142697i			0.0136056i
$111$		0			0
$112$	3.55218			0.335650
$113$	6.15159			0.578693			0.289347	−	0.957224i	$-0.406562\pi$
$113$	6.15159			0.578693			0.289347	+	0.957224i	$0.406562\pi$
$114$		0			0
$115$			0.192791i			0.0179779i
$116$			10.0501i			0.933131i
$117$		0			0
$118$	−3.23386	−	2.53692i	−0.297700	−	0.233543i
$119$		−	1.89174i		−	0.173416i
$120$		0			0
$121$	10.7245			0.974958
$122$		−	1.02165i		−	0.0924958i
$123$		0			0
$124$		−	12.7350i		−	1.14363i
$125$			0.571952i			0.0511569i
$126$		0			0
$127$	3.42406			0.303836			0.151918	−	0.988393i	$-0.451455\pi$
$127$	3.42406			0.303836			0.151918	+	0.988393i	$0.451455\pi$
$128$	−11.5084			−1.01721
$129$		0			0
$130$	−0.119246			−0.0104586
$131$	4.30822			0.376411			0.188206	−	0.982130i	$-0.439733\pi$
$131$	4.30822			0.376411			0.188206	+	0.982130i	$0.439733\pi$
$132$		0			0
$133$	3.95606			0.343034
$134$		−	2.97955i		−	0.257394i
$135$		0			0
$136$			2.50179i			0.214526i
$137$			17.6475i			1.50773i	0.657031	+	0.753863i	$0.271812\pi$
$137$			17.6475i			1.50773i	−0.657031	+	0.753863i	$0.728188\pi$
$138$		0			0
$139$	−7.04266			−0.597351			−0.298675	−	0.954355i	$-0.596545\pi$
$139$	−7.04266			−0.597351			−0.298675	+	0.954355i	$0.596545\pi$
$140$		−	0.147326i		−	0.0124513i
$141$		0			0
$142$			3.43551i			0.288301i
$143$			18.1543i			1.51814i
$144$		0			0
$145$	0.335542			0.0278653
$146$		−	6.49035i		−	0.537145i
$147$		0			0
$148$		−	14.5937i		−	1.19959i
$149$	18.8497			1.54423			0.772113	−	0.635486i	$-0.219200\pi$
$149$	18.8497			1.54423			0.772113	+	0.635486i	$0.219200\pi$
$150$		0			0
$151$		−	20.0133i		−	1.62866i	−0.580402	−	0.814330i	$-0.697105\pi$
$151$		−	20.0133i		−	1.62866i	0.580402	−	0.814330i	$-0.302895\pi$
$152$	−5.23179			−0.424354
$153$		0			0
$154$	3.74771			0.301999
$155$	−0.425181			−0.0341513
$156$		0			0
$157$			23.8996i			1.90740i	0.300765	+	0.953698i	$0.402758\pi$
$157$			23.8996i			1.90740i	−0.300765	+	0.953698i	$0.597242\pi$
$158$	−2.33035			−0.185393
$159$		0			0
$160$			0.299764i			0.0236985i
$161$	5.06336			0.399048
$162$		0			0
$163$	−13.3857			−1.04845			−0.524225	−	0.851580i	$-0.675645\pi$
$163$	−13.3857			−1.04845			−0.524225	+	0.851580i	$0.675645\pi$
$164$			16.0945i			1.25677i
$165$		0			0
$166$	−4.27404			−0.331730
$167$		−	16.9603i		−	1.31243i	−0.754575	−	0.656214i	$-0.772157\pi$
$167$		−	16.9603i		−	1.31243i	0.754575	−	0.656214i	$-0.227843\pi$
$168$		0			0
$169$	−2.17079			−0.166984
$170$	0.0385433			0.00295614
$171$		0			0
$172$		−	6.02285i		−	0.459238i
$173$	−7.52360			−0.572008			−0.286004	−	0.958228i	$-0.592327\pi$
$173$	−7.52360			−0.572008			−0.286004	+	0.958228i	$0.592327\pi$
$174$		0			0
$175$	7.50824			0.567570
$176$	11.0184			0.830542
$177$		0			0
$178$	−6.62572			−0.496619
$179$	−19.5778			−1.46331			−0.731656	−	0.681674i	$-0.761252\pi$
$179$	−19.5778			−1.46331			−0.731656	+	0.681674i	$0.761252\pi$
$180$		0			0
$181$	12.2713			0.912115			0.456057	−	0.889950i	$-0.349261\pi$
$181$	12.2713			0.912115			0.456057	+	0.889950i	$0.349261\pi$
$182$			3.13180i			0.232145i
$183$		0			0
$184$	−6.69616			−0.493648
$185$	−0.487238			−0.0358225
$186$		0			0
$187$		−	5.86793i		−	0.429106i
$188$	−2.56259			−0.186896
$189$		0			0
$190$			0.0806027i			0.00584753i
$191$	−12.6097			−0.912408			−0.456204	−	0.889875i	$-0.650791\pi$
$191$	−12.6097			−0.912408			−0.456204	+	0.889875i	$0.650791\pi$
$192$		0			0
$193$	−9.92376			−0.714328			−0.357164	−	0.934042i	$-0.616256\pi$
$193$	−9.92376			−0.714328			−0.357164	+	0.934042i	$0.616256\pi$
$194$			5.90038i			0.423623i
$195$		0			0
$196$	8.12636			0.580454
$197$		−	18.9919i		−	1.35312i	−0.736388	−	0.676559i	$-0.763470\pi$
$197$		−	18.9919i		−	1.35312i	0.736388	−	0.676559i	$-0.236530\pi$
$198$		0			0
$199$	−15.5770			−1.10422			−0.552112	−	0.833770i	$-0.686178\pi$
$199$	−15.5770			−1.10422			−0.552112	+	0.833770i	$0.686178\pi$
$200$	−9.92947			−0.702119
$201$		0			0
$202$	3.47385			0.244420
$203$		−	8.81248i		−	0.618515i
$204$		0			0
$205$	0.537346			0.0375299
$206$		−	7.58098i		−	0.528192i
$207$		0			0
$208$			9.20761i			0.638433i
$209$	12.2711			0.848813
$210$		0			0
$211$			3.59521i			0.247504i	0.992313	+	0.123752i	$0.0394928\pi$
$211$			3.59521i			0.247504i	−0.992313	+	0.123752i	$0.960507\pi$
$212$		−	9.41341i		−	0.646516i
$213$		0			0
$214$		−	2.19788i		−	0.150244i
$215$	−0.201084			−0.0137138
$216$		0			0
$217$			11.1667i			0.758044i
$218$		−	0.510235i		−	0.0345574i
$219$		0			0
$220$		−	0.456986i		−	0.0308100i
$221$	4.90359			0.329851
$222$		0			0
$223$	18.0856			1.21110			0.605550	−	0.795807i	$-0.292953\pi$
$223$	18.0856			1.21110			0.605550	+	0.795807i	$0.292953\pi$
$224$	7.87283			0.526025
$225$		0			0
$226$	3.29174			0.218963
$227$	1.11175			0.0737895			0.0368948	−	0.999319i	$-0.488253\pi$
$227$	1.11175			0.0737895			0.0368948	+	0.999319i	$0.488253\pi$
$228$		0			0
$229$			1.43042i			0.0945249i	0.998883	+	0.0472624i	$0.0150497\pi$
$229$			1.43042i			0.0945249i	−0.998883	+	0.0472624i	$0.984950\pi$
$230$			0.103163i			0.00680239i
$231$		0			0
$232$			11.6543i			0.765142i
$233$	8.34181			0.546490			0.273245	−	0.961944i	$-0.411903\pi$
$233$	8.34181			0.546490			0.273245	+	0.961944i	$0.411903\pi$
$234$		0			0
$235$			0.0855570i			0.00558112i
$236$	10.3564	+	8.12448i	0.674144	+	0.528858i
$237$		0			0
$238$		−	1.01228i		−	0.0656163i
$239$			25.1068i			1.62402i	0.583641	+	0.812012i	$0.301627\pi$
$239$			25.1068i			1.62402i	−0.583641	+	0.812012i	$0.698373\pi$
$240$		0			0
$241$	19.9640			1.28599			0.642997	−	0.765868i	$-0.277691\pi$
$241$	19.9640			1.28599			0.642997	+	0.765868i	$0.277691\pi$
$242$	5.73874			0.368900
$243$		0			0
$244$			3.27182i			0.209457i
$245$		−	0.271314i		−	0.0173336i
$246$		0			0
$247$			10.2545i			0.652477i
$248$		−	14.7677i		−	0.937748i
$249$		0			0
$250$			0.306053i			0.0193565i
$251$			0.429389i			0.0271028i	0.999908	+	0.0135514i	$0.00431367\pi$
$251$			0.429389i			0.0271028i	−0.999908	+	0.0135514i	$0.995686\pi$
$252$		0			0
$253$	15.7058			0.987417
$254$	1.83222			0.114964
$255$		0			0
$256$	−2.30950			−0.144344
$257$		−	17.2872i		−	1.07835i	−0.842195	−	0.539173i	$-0.818737\pi$
$257$		−	17.2872i		−	1.07835i	0.842195	−	0.539173i	$-0.181263\pi$
$258$		0			0
$259$			12.7965i			0.795137i
$260$	0.381884			0.0236835
$261$		0			0
$262$	2.30534			0.142425
$263$			2.83668i			0.174917i	0.996168	+	0.0874586i	$0.0278746\pi$
$263$			2.83668i			0.174917i	−0.996168	+	0.0874586i	$0.972125\pi$
$264$		0			0
$265$	−0.314284			−0.0193063
$266$	2.11690			0.129795
$267$		0			0
$268$			9.54197i			0.582869i
$269$	−1.37965			−0.0841185			−0.0420593	−	0.999115i	$-0.513392\pi$
$269$	−1.37965			−0.0841185			−0.0420593	+	0.999115i	$0.513392\pi$
$270$		0			0
$271$	−15.7877			−0.959034			−0.479517	−	0.877533i	$-0.659188\pi$
$271$	−15.7877			−0.959034			−0.479517	+	0.877533i	$0.659188\pi$
$272$		−	2.97614i		−	0.180455i
$273$		0			0
$274$			9.44323i			0.570486i
$275$	23.2895			1.40441
$276$		0			0
$277$	−3.82862			−0.230039			−0.115020	−	0.993363i	$-0.536693\pi$
$277$	−3.82862			−0.230039			−0.115020	+	0.993363i	$0.536693\pi$
$278$	−3.76855			−0.226023
$279$		0			0
$280$		−	0.170842i		−	0.0102098i
$281$		−	12.1670i		−	0.725824i	−0.931824	−	0.362912i	$-0.881783\pi$
$281$		−	12.1670i		−	0.725824i	0.931824	−	0.362912i	$-0.118217\pi$
$282$		0			0
$283$		−	8.51457i		−	0.506139i	−0.967448	−	0.253069i	$-0.918560\pi$
$283$		−	8.51457i		−	0.506139i	0.967448	−	0.253069i	$-0.0814401\pi$
$284$		−	11.0022i		−	0.652860i
$285$		0			0
$286$			9.71442i			0.574426i
$287$		−	14.1125i		−	0.833037i
$288$		0			0
$289$	15.4150			0.906767
$290$	0.179550			0.0105435
$291$		0			0
$292$			20.7853i			1.21637i
$293$			13.5899i			0.793929i	0.917834	+	0.396964i	$0.129936\pi$
$293$			13.5899i			0.793929i	−0.917834	+	0.396964i	$0.870064\pi$
$294$		0			0
$295$	0.271251	−	0.345768i	0.0157928	−	0.0201314i
$296$		−	16.9231i		−	0.983635i
$297$		0			0
$298$	10.0865			0.584297
$299$			13.1247i			0.759022i
$300$		0			0
$301$			5.28115i			0.304401i
$302$		−	10.7092i		−	0.616244i
$303$		0			0
$304$	6.22376			0.356957
$305$	0.109236			0.00625484
$306$		0			0
$307$	7.72126			0.440676			0.220338	−	0.975424i	$-0.429284\pi$
$307$	7.72126			0.440676			0.220338	+	0.975424i	$0.429284\pi$
$308$	−12.0020			−0.683878
$309$		0			0
$310$	−0.227516			−0.0129220
$311$		−	2.34575i		−	0.133015i	−0.997786	−	0.0665076i	$-0.978814\pi$
$311$		−	2.34575i		−	0.133015i	0.997786	−	0.0665076i	$-0.0211857\pi$
$312$		0			0
$313$		−	6.94382i		−	0.392488i	−0.980555	−	0.196244i	$-0.937126\pi$
$313$		−	6.94382i		−	0.392488i	0.980555	−	0.196244i	$-0.0628745\pi$
$314$			12.7888i			0.721712i
$315$		0			0
$316$	7.46293			0.419823
$317$		−	32.1894i		−	1.80794i	−0.427596	−	0.903970i	$-0.640640\pi$
$317$		−	32.1894i		−	1.80794i	0.427596	−	0.903970i	$-0.359360\pi$
$318$		0			0
$319$		−	27.3351i		−	1.53047i
$320$		−	0.110100i		−	0.00615476i
$321$		0			0
$322$	2.70942			0.150990
$323$		−	3.31451i		−	0.184424i
$324$		0			0
$325$			19.4621i			1.07956i
$326$	−7.16274			−0.396708
$327$		0			0
$328$			18.6635i			1.03052i
$329$	2.24702			0.123882
$330$		0			0
$331$	−11.3990			−0.626548			−0.313274	−	0.949663i	$-0.601426\pi$
$331$	−11.3990			−0.626548			−0.313274	+	0.949663i	$0.601426\pi$
$332$	13.6876			0.751204
$333$		0			0
$334$		−	9.07551i		−	0.496590i
$335$	0.318577			0.0174057
$336$		0			0
$337$			16.0878i			0.876360i	0.898887	+	0.438180i	$0.144377\pi$
$337$			16.0878i			0.876360i	−0.898887	+	0.438180i	$0.855623\pi$
$338$	−1.16160			−0.0631825
$339$		0			0
$340$	−0.123435			−0.00669419
$341$			34.6376i			1.87573i
$342$		0			0
$343$	−17.6441			−0.952689
$344$		−	6.98420i		−	0.376563i
$345$		0			0
$346$	−4.02590			−0.216434
$347$	−8.52524			−0.457659			−0.228829	−	0.973467i	$-0.573490\pi$
$347$	−8.52524			−0.457659			−0.228829	+	0.973467i	$0.573490\pi$
$348$		0			0
$349$		−	0.170942i		−	0.00915033i	−0.999990	−	0.00457516i	$-0.998544\pi$
$349$		−	0.170942i		−	0.00915033i	0.999990	−	0.00457516i	$-0.00145633\pi$
$350$	4.01768			0.214754
$351$		0			0
$352$	24.4204			1.30161
$353$	−9.98798			−0.531606			−0.265803	−	0.964027i	$-0.585637\pi$
$353$	−9.98798			−0.531606			−0.265803	+	0.964027i	$0.585637\pi$
$354$		0			0
$355$	−0.367329			−0.0194958
$356$	21.2188			1.12460
$357$		0			0
$358$	−10.4761			−0.553681
$359$			31.6950i			1.67280i	0.548120	+	0.836399i	$0.315344\pi$
$359$			31.6950i			1.67280i	−0.548120	+	0.836399i	$0.684656\pi$
$360$		0			0
$361$	−12.0686			−0.635190
$362$	6.56639			0.345122
$363$		0			0
$364$		−	10.0296i		−	0.525693i
$365$	0.693956			0.0363233
$366$		0			0
$367$		−	14.0601i		−	0.733931i	−0.930235	−	0.366965i	$-0.880397\pi$
$367$		−	14.0601i		−	0.733931i	0.930235	−	0.366965i	$-0.119603\pi$
$368$	7.96579			0.415245
$369$		0			0
$370$	−0.260723			−0.0135543
$371$			8.25418i			0.428536i
$372$		0			0
$373$	28.8222			1.49235			0.746177	−	0.665747i	$-0.231887\pi$
$373$	28.8222			1.49235			0.746177	+	0.665747i	$0.231887\pi$
$374$		−	3.13995i		−	0.162363i
$375$		0			0
$376$	−2.97163			−0.153250
$377$	22.8428			1.17647
$378$		0			0
$379$	4.21089			0.216299			0.108150	−	0.994135i	$-0.465507\pi$
$379$	4.21089			0.216299			0.108150	+	0.994135i	$0.465507\pi$
$380$		−	0.258129i		−	0.0132418i
$381$		0			0
$382$	−6.74751			−0.345233
$383$		−	14.0507i		−	0.717958i	−0.933346	−	0.358979i	$-0.883125\pi$
$383$		−	14.0507i		−	0.717958i	0.933346	−	0.358979i	$-0.116875\pi$
$384$		0			0
$385$			0.400710i			0.0204221i
$386$	−5.31024			−0.270284
$387$		0			0
$388$		−	18.8959i		−	0.959296i
$389$		−	6.37231i		−	0.323089i	−0.986865	−	0.161545i	$-0.948352\pi$
$389$		−	6.37231i		−	0.323089i	0.986865	−	0.161545i	$-0.0516475\pi$
$390$		0			0
$391$		−	4.24224i		−	0.214540i
$392$	9.42347			0.475957
$393$		0			0
$394$		−	10.1626i		−	0.511987i
$395$		−	0.249164i		−	0.0125368i
$396$		0			0
$397$			7.29835i			0.366294i	0.983086	+	0.183147i	$0.0586284\pi$
$397$			7.29835i			0.366294i	−0.983086	+	0.183147i	$0.941372\pi$
$398$	−8.33530			−0.417811
$399$		0			0
$400$	11.8121			0.590607
$401$	−29.8174			−1.48901			−0.744504	−	0.667617i	$-0.767314\pi$
$401$	−29.8174			−1.48901			−0.744504	+	0.667617i	$0.767314\pi$
$402$		0			0
$403$	−28.9452			−1.44186
$404$	−11.1250			−0.553489
$405$		0			0
$406$		−	4.71559i		−	0.234031i
$407$			39.6931i			1.96751i
$408$		0			0
$409$		−	39.1449i		−	1.93559i	−0.251742	−	0.967795i	$-0.581003\pi$
$409$		−	39.1449i		−	1.93559i	0.251742	−	0.967795i	$-0.418997\pi$
$410$	0.287536			0.0142004
$411$		0			0
$412$			24.2780i			1.19609i
$413$	−9.08104	−	7.12397i	−0.446849	−	0.350548i
$414$		0			0
$415$		−	0.456986i		−	0.0224326i
$416$			20.4071i			1.00054i
$417$		0			0
$418$	6.56633			0.321170
$419$	−13.7418			−0.671329			−0.335665	−	0.941982i	$-0.608961\pi$
$419$	−13.7418			−0.671329			−0.335665	+	0.941982i	$0.608961\pi$
$420$		0			0
$421$		−	4.74347i		−	0.231182i	−0.993297	−	0.115591i	$-0.963124\pi$
$421$		−	4.74347i		−	0.231182i	0.993297	−	0.115591i	$-0.0368763\pi$
$422$			1.92381i			0.0936494i
$423$		0			0
$424$		−	10.9159i		−	0.530125i
$425$		−	6.29065i		−	0.305141i
$426$		0			0
$427$		−	2.86891i		−	0.138836i
$428$			7.03869i			0.340228i
$429$		0			0
$430$	−0.107601			−0.00518897
$431$	−38.9221			−1.87481			−0.937405	−	0.348240i	$-0.886779\pi$
$431$	−38.9221			−1.87481			−0.937405	+	0.348240i	$0.886779\pi$
$432$		0			0
$433$	−27.2141			−1.30783			−0.653913	−	0.756570i	$-0.726874\pi$
$433$	−27.2141			−1.30783			−0.653913	+	0.756570i	$0.726874\pi$
$434$			5.97533i			0.286825i
$435$		0			0
$436$			1.63402i			0.0782555i
$437$	8.87147			0.424380
$438$		0			0
$439$	−23.9897			−1.14497			−0.572483	−	0.819917i	$-0.694020\pi$
$439$	−23.9897			−1.14497			−0.572483	+	0.819917i	$0.694020\pi$
$440$		−	0.529929i		−	0.0252634i
$441$		0			0
$442$	2.62393			0.124807
$443$	29.7149			1.41180			0.705898	−	0.708313i	$-0.250543\pi$
$443$	29.7149			1.41180			0.705898	+	0.708313i	$0.250543\pi$
$444$		0			0
$445$		−	0.708431i		−	0.0335828i
$446$	9.67766			0.458250
$447$		0			0
$448$	−2.89159			−0.136615
$449$		−	7.04380i		−	0.332418i	−0.986091	−	0.166209i	$-0.946847\pi$
$449$		−	7.04380i		−	0.332418i	0.986091	−	0.166209i	$-0.0531526\pi$
$450$		0			0
$451$		−	43.7752i		−	2.06129i
$452$	−10.5418			−0.495843
$453$		0			0
$454$	0.594902			0.0279201
$455$	−0.334856			−0.0156983
$456$		0			0
$457$			26.6089i			1.24471i	0.782735	+	0.622356i	$0.213824\pi$
$457$			26.6089i			1.24471i	−0.782735	+	0.622356i	$0.786176\pi$
$458$			0.765423i			0.0357659i
$459$		0			0
$460$		−	0.330380i		−	0.0154040i
$461$		−	22.8883i		−	1.06602i	−0.846110	−	0.533008i	$-0.821062\pi$
$461$		−	22.8883i		−	1.06602i	0.846110	−	0.533008i	$-0.178938\pi$
$462$		0			0
$463$		−	28.5688i		−	1.32770i	−0.747864	−	0.663852i	$-0.768921\pi$
$463$		−	28.5688i		−	1.32770i	0.747864	−	0.663852i	$-0.231079\pi$
$464$		−	13.8640i		−	0.643620i
$465$		0			0
$466$	4.46373			0.206778
$467$	−32.7412			−1.51508			−0.757541	−	0.652787i	$-0.773599\pi$
$467$	−32.7412			−1.51508			−0.757541	+	0.652787i	$0.773599\pi$
$468$		0			0
$469$		−	8.36691i		−	0.386348i
$470$			0.0457818i			0.00211176i
$471$		0			0
$472$	12.0095	+	9.42128i	0.552780	+	0.433650i
$473$			16.3814i			0.753218i
$474$		0			0
$475$	13.1551			0.603599
$476$			3.24182i			0.148588i
$477$		0			0
$478$			13.4347i			0.614490i
$479$		−	21.2159i		−	0.969379i	−0.874686	−	0.484689i	$-0.838933\pi$
$479$		−	21.2159i		−	0.969379i	0.874686	−	0.484689i	$-0.161067\pi$
$480$		0			0
$481$	−33.1699			−1.51242
$482$	10.6828			0.486588
$483$		0			0
$484$	−18.3783			−0.835375
$485$	−0.630877			−0.0286466
$486$		0			0
$487$	4.69119			0.212578			0.106289	−	0.994335i	$-0.466103\pi$
$487$	4.69119			0.212578			0.106289	+	0.994335i	$0.466103\pi$
$488$			3.79406i			0.171749i
$489$		0			0
$490$		−	0.145181i		−	0.00655861i
$491$			41.4227i			1.86938i	0.355461	+	0.934691i	$0.384324\pi$
$491$			41.4227i			1.86938i	−0.355461	+	0.934691i	$0.615676\pi$
$492$		0			0
$493$	−7.38339			−0.332531
$494$			5.48721i			0.246881i
$495$		0			0
$496$			17.5677i			0.788813i
$497$			9.64730i			0.432741i
$498$		0			0
$499$	9.88636			0.442574			0.221287	−	0.975209i	$-0.428974\pi$
$499$	9.88636			0.442574			0.221287	+	0.975209i	$0.428974\pi$
$500$		−	0.980134i		−	0.0438329i
$501$		0			0
$502$			0.229767i			0.0102550i
$503$	−19.0690			−0.850245			−0.425123	−	0.905136i	$-0.639769\pi$
$503$	−19.0690			−0.850245			−0.425123	+	0.905136i	$0.639769\pi$
$504$		0			0
$505$			0.371429i			0.0165284i
$506$	8.40424			0.373614
$507$		0			0
$508$	−5.86768			−0.260336
$509$	−18.2367			−0.808327			−0.404163	−	0.914687i	$-0.632437\pi$
$509$	−18.2367			−0.808327			−0.404163	+	0.914687i	$0.632437\pi$
$510$		0			0
$511$		−	18.2256i		−	0.806255i
$512$	21.7811			0.962596
$513$		0			0
$514$		−	9.25044i		−	0.408019i
$515$	0.810568			0.0357179
$516$		0			0
$517$	6.96994			0.306538
$518$			6.84746i			0.300860i
$519$		0			0
$520$	0.442839			0.0194198
$521$			8.13363i			0.356341i	0.984000	+	0.178170i	$0.0570178\pi$
$521$			8.13363i			0.356341i	−0.984000	+	0.178170i	$0.942982\pi$
$522$		0			0
$523$	−18.5156			−0.809632			−0.404816	−	0.914398i	$-0.632664\pi$
$523$	−18.5156			−0.809632			−0.404816	+	0.914398i	$0.632664\pi$
$524$	−7.38285			−0.322521
$525$		0			0
$526$			1.51792i			0.0661843i
$527$	9.35582			0.407546
$528$		0			0
$529$	−11.6454			−0.506322
$530$	−0.168175			−0.00730504
$531$		0			0
$532$	−6.77935			−0.293922
$533$	36.5811			1.58450
$534$		0			0
$535$	0.235000			0.0101599
$536$			11.0650i			0.477937i
$537$		0			0
$538$	−0.738253			−0.0318284
$539$	−22.1027			−0.952031
$540$		0			0
$541$			3.51611i			0.151169i	0.997139	+	0.0755847i	$0.0240823\pi$
$541$			3.51611i			0.151169i	−0.997139	+	0.0755847i	$0.975918\pi$
$542$	−8.44804			−0.362874
$543$		0			0
$544$		−	6.59611i		−	0.282806i
$545$	0.0545549			0.00233688
$546$		0			0
$547$	20.4936			0.876245			0.438122	−	0.898915i	$-0.355644\pi$
$547$	20.4936			0.876245			0.438122	+	0.898915i	$0.355644\pi$
$548$		−	30.2419i		−	1.29187i
$549$		0			0
$550$	12.4623			0.531395
$551$		−	15.4403i		−	0.657779i
$552$		0			0
$553$	−6.54389			−0.278275
$554$	−2.04871			−0.0870412
$555$		0			0
$556$	12.0688			0.511829
$557$		−	22.4467i		−	0.951097i	−0.879689	−	0.475549i	$-0.842250\pi$
$557$		−	22.4467i		−	0.951097i	0.879689	−	0.475549i	$-0.157750\pi$
$558$		0			0
$559$	−13.6893			−0.578994
$560$			0.203234i			0.00858822i
$561$		0			0
$562$		−	6.51061i		−	0.274634i
$563$	−28.6890			−1.20910			−0.604549	−	0.796568i	$-0.706646\pi$
$563$	−28.6890			−1.20910			−0.604549	+	0.796568i	$0.706646\pi$
$564$		0			0
$565$			0.351957i			0.0148069i
$566$		−	4.55617i		−	0.191510i
$567$		0			0
$568$		−	12.7583i		−	0.535327i
$569$	−38.1279			−1.59840			−0.799202	−	0.601062i	$-0.794744\pi$
$569$	−38.1279			−1.59840			−0.799202	+	0.601062i	$0.794744\pi$
$570$		0			0
$571$		−	24.4240i		−	1.02211i	−0.859547	−	0.511057i	$-0.829254\pi$
$571$		−	24.4240i		−	1.02211i	0.859547	−	0.511057i	$-0.170746\pi$
$572$		−	31.1104i		−	1.30079i
$573$		0			0
$574$		−	7.55167i		−	0.315200i
$575$	16.8373			0.702162
$576$		0			0
$577$	19.2499			0.801382			0.400691	−	0.916213i	$-0.368770\pi$
$577$	19.2499			0.801382			0.400691	+	0.916213i	$0.368770\pi$
$578$	8.24864			0.343098
$579$		0			0
$580$	−0.575007			−0.0238759
$581$	−12.0020			−0.497927
$582$		0			0
$583$			25.6033i			1.06038i
$584$			24.1030i			0.997388i
$585$		0			0
$586$			7.27198i			0.300403i
$587$	27.0844			1.11789			0.558946	−	0.829204i	$-0.311206\pi$
$587$	27.0844			1.11789			0.558946	+	0.829204i	$0.311206\pi$
$588$		0			0
$589$			19.5651i			0.806165i
$590$	0.145147	−	0.185022i	0.00597562	−	0.00761722i
$591$		0			0
$592$			20.1318i			0.827411i
$593$			21.0005i			0.862389i	0.902259	+	0.431194i	$0.141908\pi$
$593$			21.0005i			0.862389i	−0.902259	+	0.431194i	$0.858092\pi$
$594$		0			0
$595$	0.108234			0.00443717
$596$	−32.3020			−1.32314
$597$		0			0
$598$			7.02308i			0.287195i
$599$		−	23.4727i		−	0.959068i	−0.877523	−	0.479534i	$-0.840806\pi$
$599$		−	23.4727i		−	0.959068i	0.877523	−	0.479534i	$-0.159194\pi$
$600$		0			0
$601$		−	36.8840i		−	1.50453i	−0.658861	−	0.752264i	$-0.728962\pi$
$601$		−	36.8840i		−	1.50453i	0.658861	−	0.752264i	$-0.271038\pi$
$602$			2.82596i			0.115178i
$603$		0			0
$604$			34.2961i			1.39549i
$605$			0.613593i			0.0249461i
$606$		0			0
$607$	−39.8036			−1.61558			−0.807789	−	0.589472i	$-0.799336\pi$
$607$	−39.8036			−1.61558			−0.807789	+	0.589472i	$0.799336\pi$
$608$	13.7939			0.559418
$609$		0			0
$610$	0.0584526			0.00236668
$611$			5.82449i			0.235634i
$612$		0			0
$613$			29.5183i			1.19223i	0.802898	+	0.596117i	$0.203290\pi$
$613$			29.5183i			1.19223i	−0.802898	+	0.596117i	$0.796710\pi$
$614$	4.13167			0.166741
$615$		0			0
$616$	−13.9177			−0.560761
$617$			31.1223i			1.25293i	0.779448	+	0.626467i	$0.215500\pi$
$617$			31.1223i			1.25293i	−0.779448	+	0.626467i	$0.784500\pi$
$618$		0			0
$619$	−42.1742			−1.69512			−0.847562	−	0.530697i	$-0.821930\pi$
$619$	−42.1742			−1.69512			−0.847562	+	0.530697i	$0.821930\pi$
$620$	0.728617			0.0292620
$621$		0			0
$622$		−	1.25522i		−	0.0503297i
$623$	−18.6058			−0.745425
$624$		0			0
$625$	24.9509			0.998036
$626$		−	3.71566i		−	0.148508i
$627$		0			0
$628$		−	40.9559i		−	1.63432i
$629$	10.7213			0.427488
$630$		0			0
$631$	−9.10110			−0.362309			−0.181154	−	0.983455i	$-0.557983\pi$
$631$	−9.10110			−0.362309			−0.181154	+	0.983455i	$0.557983\pi$
$632$	8.65414			0.344243
$633$		0			0
$634$		−	17.2247i		−	0.684079i
$635$			0.195904i			0.00777420i
$636$		0			0
$637$		−	18.4703i		−	0.731821i
$638$		−	14.6271i		−	0.579093i
$639$		0			0
$640$		−	0.658443i		−	0.0260273i
$641$		−	4.30662i		−	0.170101i	−0.996377	−	0.0850505i	$-0.972895\pi$
$641$		−	4.30662i		−	0.170101i	0.996377	−	0.0850505i	$-0.0271052\pi$
$642$		0			0
$643$	48.3224			1.90565			0.952825	−	0.303519i	$-0.0981617\pi$
$643$	48.3224			1.90565			0.952825	+	0.303519i	$0.0981617\pi$
$644$	−8.67689			−0.341917
$645$		0			0
$646$		−	1.77361i		−	0.0697816i
$647$			23.4111i			0.920387i	0.887819	+	0.460193i	$0.152220\pi$
$647$			23.4111i			0.920387i	−0.887819	+	0.460193i	$0.847780\pi$
$648$		0			0
$649$	−28.1681	−	22.0976i	−1.10570	−	0.867406i
$650$			10.4142i			0.408480i
$651$		0			0
$652$	22.9386			0.898346
$653$			11.8688i			0.464461i	0.972661	+	0.232231i	$0.0746024\pi$
$653$			11.8688i			0.464461i	−0.972661	+	0.232231i	$0.925398\pi$
$654$		0			0
$655$			0.246490i			0.00963118i
$656$		−	22.2022i		−	0.866849i
$657$		0			0
$658$	1.20239			0.0468739
$659$	36.5897			1.42533			0.712667	−	0.701503i	$-0.247487\pi$
$659$	36.5897			1.42533			0.712667	+	0.701503i	$0.247487\pi$
$660$		0			0
$661$	−26.2750			−1.02198			−0.510989	−	0.859587i	$-0.670721\pi$
$661$	−26.2750			−1.02198			−0.510989	+	0.859587i	$0.670721\pi$
$662$	−6.09966			−0.237070
$663$		0			0
$664$	15.8724			0.615967
$665$			0.226342i			0.00877715i
$666$		0			0
$667$		−	19.7620i		−	0.765189i
$668$			29.0643i			1.12453i
$669$		0			0
$670$	0.170472			0.00658589
$671$		−	8.89896i		−	0.343541i
$672$		0			0
$673$			5.26426i			0.202922i	0.994840	+	0.101461i	$0.0323518\pi$
$673$			5.26426i			0.202922i	−0.994840	+	0.101461i	$0.967648\pi$
$674$			8.60865i			0.331593i
$675$		0			0
$676$	3.72000			0.143077
$677$		−	24.6167i		−	0.946097i	−0.881036	−	0.473048i	$-0.843154\pi$
$677$		−	24.6167i		−	0.946097i	0.881036	−	0.473048i	$-0.156846\pi$
$678$		0			0
$679$			16.5690i			0.635858i
$680$	−0.143137			−0.00548905
$681$		0			0
$682$			18.5347i			0.709729i
$683$	−22.8844			−0.875646			−0.437823	−	0.899061i	$-0.644250\pi$
$683$	−22.8844			−0.875646			−0.437823	+	0.899061i	$0.644250\pi$
$684$		0			0
$685$	−1.00968			−0.0385780
$686$	−9.44139			−0.360474
$687$		0			0
$688$			8.30843i			0.316756i
$689$	−21.3956			−0.815109
$690$		0			0
$691$			19.8690i			0.755852i	0.925836	+	0.377926i	$0.123363\pi$
$691$			19.8690i			0.755852i	−0.925836	+	0.377926i	$0.876637\pi$
$692$	12.8929			0.490115
$693$		0			0
$694$	−4.56188			−0.173167
$695$		−	0.402938i		−	0.0152843i
$696$		0			0
$697$	−11.8239			−0.447864
$698$		−	0.0914718i		−	0.00346226i
$699$		0			0
$700$	−12.8666			−0.486312
$701$	38.7648			1.46413			0.732063	−	0.681237i	$-0.238558\pi$
$701$	38.7648			1.46413			0.732063	+	0.681237i	$0.238558\pi$
$702$		0			0
$703$			22.4207i			0.845613i
$704$	−8.96932			−0.338044
$705$		0			0
$706$	−5.34460			−0.201147
$707$	9.75499			0.366874
$708$		0			0
$709$	26.0280			0.977503			0.488752	−	0.872423i	$-0.337452\pi$
$709$	26.0280			0.977503			0.488752	+	0.872423i	$0.337452\pi$
$710$	−0.196559			−0.00737672
$711$		0			0
$712$	24.6057			0.922138
$713$			25.0413i			0.937806i
$714$		0			0
$715$	−1.03868			−0.0388444
$716$	33.5498			1.25381
$717$		0			0
$718$			16.9601i			0.632946i
$719$	−47.4001			−1.76773			−0.883864	−	0.467745i	$-0.845067\pi$
$719$	−47.4001			−1.76773			−0.883864	+	0.467745i	$0.845067\pi$
$720$		0			0
$721$		−	21.2883i		−	0.792817i
$722$	−6.45796			−0.240340
$723$		0			0
$724$	−21.0288			−0.781529
$725$		−	29.3043i		−	1.08833i
$726$		0			0
$727$	21.2881			0.789533			0.394767	−	0.918781i	$-0.370825\pi$
$727$	21.2881			0.789533			0.394767	+	0.918781i	$0.370825\pi$
$728$		−	11.6305i		−	0.431054i
$729$		0			0
$730$	0.371338			0.0137438
$731$	4.42472			0.163654
$732$		0			0
$733$	15.9687			0.589817			0.294909	−	0.955525i	$-0.404711\pi$
$733$	15.9687			0.589817			0.294909	+	0.955525i	$0.404711\pi$
$734$		−	7.52360i		−	0.277701i
$735$		0			0
$736$	17.6548			0.650766
$737$		−	25.9530i		−	0.955991i
$738$		0			0
$739$		−	23.5337i		−	0.865700i	−0.901466	−	0.432850i	$-0.857508\pi$
$739$		−	23.5337i		−	0.865700i	0.901466	−	0.432850i	$-0.142492\pi$
$740$	0.834963			0.0306938
$741$		0			0
$742$			4.41684i			0.162147i
$743$		−	24.9553i		−	0.915521i	−0.889076	−	0.457760i	$-0.848652\pi$
$743$		−	24.9553i		−	0.915521i	0.889076	−	0.457760i	$-0.151348\pi$
$744$		0			0
$745$			1.07846i			0.0395119i
$746$	15.4228			0.564670
$747$		0			0
$748$			10.0557i			0.367672i
$749$		−	6.17189i		−	0.225516i
$750$		0			0
$751$			27.8955i			1.01792i	0.860790	+	0.508961i	$0.169970\pi$
$751$			27.8955i			1.01792i	−0.860790	+	0.508961i	$0.830030\pi$
$752$	3.53506			0.128910
$753$		0			0
$754$	12.2233			0.445145
$755$	1.14504			0.0416723
$756$		0			0
$757$	36.7457			1.33554			0.667772	−	0.744366i	$-0.267248\pi$
$757$	36.7457			1.33554			0.667772	+	0.744366i	$0.267248\pi$
$758$	2.25326			0.0818422
$759$		0			0
$760$		−	0.299331i		−	0.0108579i
$761$			49.2771i			1.78629i	0.449767	+	0.893146i	$0.351507\pi$
$761$			49.2771i			1.78629i	−0.449767	+	0.893146i	$0.648493\pi$
$762$		0			0
$763$		−	1.43280i		−	0.0518708i
$764$	21.6089			0.781781
$765$		0			0
$766$		−	7.51859i		−	0.271658i
$767$	18.4660	−	23.5389i	0.666770	−	0.849942i
$768$		0			0
$769$			25.8526i			0.932267i	0.884714	+	0.466134i	$0.154353\pi$
$769$			25.8526i			0.932267i	−0.884714	+	0.466134i	$0.845647\pi$
$770$			0.214421i			0.00772720i
$771$		0			0
$772$	17.0060			0.612059
$773$	31.2038			1.12232			0.561161	−	0.827707i	$-0.310355\pi$
$773$	31.2038			1.12232			0.561161	+	0.827707i	$0.310355\pi$
$774$		0			0
$775$			37.1328i			1.33385i
$776$		−	21.9120i		−	0.786597i
$777$		0			0
$778$		−	3.40985i		−	0.122249i
$779$		−	24.7265i		−	0.885918i
$780$		0			0
$781$			29.9246i			1.07079i
$782$		−	2.27004i		−	0.0811764i
$783$		0			0
$784$	−11.2102			−0.400364
$785$	−1.36739			−0.0488043
$786$		0			0
$787$	28.7492			1.02480			0.512400	−	0.858747i	$-0.328757\pi$
$787$	28.7492			1.02480			0.512400	+	0.858747i	$0.328757\pi$
$788$			32.5458i			1.15940i
$789$		0			0
$790$		−	0.133329i		−	0.00474362i
$791$	9.24358			0.328664
$792$		0			0
$793$	7.43649			0.264078
$794$			3.90537i			0.138597i
$795$		0			0
$796$	26.6937			0.946135
$797$	24.7210			0.875662			0.437831	−	0.899057i	$-0.355747\pi$
$797$	24.7210			0.875662			0.437831	+	0.899057i	$0.355747\pi$
$798$		0			0
$799$		−	1.88262i		−	0.0666025i
$800$	26.1796			0.925590
$801$		0			0
$802$	−15.9554			−0.563404
$803$		−	56.5334i		−	1.99502i
$804$		0			0
$805$			0.289694i			0.0102104i
$806$	−15.4887			−0.545564
$807$		0			0
$808$	−12.9007			−0.453846
$809$	−4.02256			−0.141426			−0.0707128	−	0.997497i	$-0.522527\pi$
$809$	−4.02256			−0.141426			−0.0707128	+	0.997497i	$0.522527\pi$
$810$		0			0
$811$			31.8340i			1.11784i	0.829220	+	0.558922i	$0.188785\pi$
$811$			31.8340i			1.11784i	−0.829220	+	0.558922i	$0.811215\pi$
$812$			15.1016i			0.529964i
$813$		0			0
$814$			21.2399i			0.744458i
$815$		−	0.765850i		−	0.0268266i
$816$		0			0
$817$			9.25308i			0.323724i
$818$		−	20.9465i		−	0.732379i
$819$		0			0
$820$	−0.920832			−0.0321568
$821$	50.5547			1.76437			0.882186	−	0.470901i	$-0.156071\pi$
$821$	50.5547			1.76437			0.882186	+	0.470901i	$0.156071\pi$
$822$		0			0
$823$			26.8522i			0.936009i	0.883726	+	0.468004i	$0.155027\pi$
$823$			26.8522i			0.936009i	−0.883726	+	0.468004i	$0.844973\pi$
$824$			28.1532i			0.980764i
$825$		0			0
$826$	−4.85929	−	3.81206i	−0.169076	−	0.132639i
$827$			24.4845i			0.851408i	0.904862	+	0.425704i	$0.139974\pi$
$827$			24.4845i			0.851408i	−0.904862	+	0.425704i	$0.860026\pi$
$828$		0			0
$829$	23.8267			0.827536			0.413768	−	0.910382i	$-0.364213\pi$
$829$	23.8267			0.827536			0.413768	+	0.910382i	$0.364213\pi$
$830$		−	0.244535i		−	0.00848792i
$831$		0			0
$832$		−	7.49529i		−	0.259852i
$833$			5.97008i			0.206851i
$834$		0			0
$835$	0.970365			0.0335809
$836$	−21.0286			−0.727290
$837$		0			0
$838$	−7.35327			−0.254014
$839$	4.78134			0.165070			0.0825351	−	0.996588i	$-0.473698\pi$
$839$	4.78134			0.165070			0.0825351	+	0.996588i	$0.473698\pi$
$840$		0			0
$841$	−5.39469			−0.186024
$842$		−	2.53824i		−	0.0874737i
$843$		0			0
$844$		−	6.16098i		−	0.212070i
$845$		−	0.124199i		−	0.00427259i
$846$		0			0
$847$	16.1150			0.553719
$848$			12.9857i			0.445929i
$849$		0			0
$850$		−	3.36615i		−	0.115458i
$851$			28.6963i			0.983695i
$852$		0			0
$853$	41.6528			1.42616			0.713082	−	0.701080i	$-0.247299\pi$
$853$	41.6528			1.42616			0.713082	+	0.701080i	$0.247299\pi$
$854$		−	1.53516i		−	0.0525322i
$855$		0			0
$856$			8.16218i			0.278977i
$857$	20.2705			0.692427			0.346213	−	0.938156i	$-0.387467\pi$
$857$	20.2705			0.692427			0.346213	+	0.938156i	$0.387467\pi$
$858$		0			0
$859$		−	25.7983i		−	0.880225i	−0.897943	−	0.440113i	$-0.854939\pi$
$859$		−	25.7983i		−	0.880225i	0.897943	−	0.440113i	$-0.145061\pi$
$860$	0.344591			0.0117505
$861$		0			0
$862$	−20.8273			−0.709382
$863$	55.6395			1.89399			0.946995	−	0.321249i	$-0.104103\pi$
$863$	55.6395			1.89399			0.946995	+	0.321249i	$0.104103\pi$
$864$		0			0
$865$		−	0.430455i		−	0.0146359i
$866$	−14.5623			−0.494849
$867$		0			0
$868$		−	19.1360i		−	0.649517i
$869$	−20.2983			−0.688571
$870$		0			0
$871$	21.6879			0.734865
$872$			1.89484i			0.0641674i
$873$		0			0
$874$	4.74715			0.160575
$875$			0.859434i			0.0290542i
$876$		0			0
$877$	−5.28752			−0.178547			−0.0892734	−	0.996007i	$-0.528454\pi$
$877$	−5.28752			−0.178547			−0.0892734	+	0.996007i	$0.528454\pi$
$878$	−12.8370			−0.433227
$879$		0			0
$880$			0.630405i			0.0212510i
$881$	−6.52960			−0.219988			−0.109994	−	0.993932i	$-0.535083\pi$
$881$	−6.52960			−0.219988			−0.109994	+	0.993932i	$0.535083\pi$
$882$		0			0
$883$	53.3596			1.79569			0.897847	−	0.440308i	$-0.145131\pi$
$883$	53.3596			1.79569			0.897847	+	0.440308i	$0.145131\pi$
$884$	−8.40310			−0.282627
$885$		0			0
$886$	15.9005			0.534189
$887$	−22.7631			−0.764312			−0.382156	−	0.924098i	$-0.624818\pi$
$887$	−22.7631			−0.764312			−0.382156	+	0.924098i	$0.624818\pi$
$888$		0			0
$889$	5.14510			0.172561
$890$		−	0.379084i		−	0.0127069i
$891$		0			0
$892$	−30.9926			−1.03771
$893$	3.93698			0.131746
$894$		0			0
$895$		−	1.12012i		−	0.0374416i
$896$	−17.2930			−0.577717
$897$		0			0
$898$		−	3.76916i		−	0.125779i
$899$	43.5830			1.45358
$900$		0			0
$901$	6.91562			0.230393
$902$		−	23.4242i		−	0.779942i
$903$		0			0
$904$	−12.2244			−0.406578
$905$			0.702087i			0.0233381i
$906$		0			0
$907$	−27.1810			−0.902532			−0.451266	−	0.892390i	$-0.649027\pi$
$907$	−27.1810			−0.902532			−0.451266	+	0.892390i	$0.649027\pi$
$908$	−1.90517			−0.0632253
$909$		0			0
$910$	−0.179183			−0.00593985
$911$		−	55.0496i		−	1.82388i	−0.410328	−	0.911938i	$-0.634586\pi$
$911$		−	55.0496i		−	1.82388i	0.410328	−	0.911938i	$-0.365414\pi$
$912$		0			0
$913$	−37.2286			−1.23209
$914$			14.2385i			0.470968i
$915$		0			0
$916$		−	2.45126i		−	0.0809920i
$917$	6.47367			0.213780
$918$		0			0
$919$			0.0172712i			0.000569723i	1.00000		0.000284861i	$9.06742e-5\pi$
$919$			0.0172712i			0.000569723i	−1.00000		0.000284861i	$0.999909\pi$
$920$		−	0.383114i		−	0.0126309i
$921$		0			0
$922$		−	12.2476i		−	0.403354i
$923$	−25.0067			−0.823107
$924$		0			0
$925$			42.5525i			1.39912i
$926$		−	15.2872i		−	0.502370i
$927$		0			0
$928$		−	30.7273i		−	1.00867i
$929$	21.4405			0.703441			0.351721	−	0.936105i	$-0.385597\pi$
$929$	21.4405			0.703441			0.351721	+	0.936105i	$0.385597\pi$
$930$		0			0
$931$	−12.4848			−0.409172
$932$	−14.2951			−0.468250
$933$		0			0
$934$	−17.5199			−0.573269
$935$	0.335728			0.0109795
$936$		0			0
$937$			30.3318i			0.990896i	0.868638	+	0.495448i	$0.164996\pi$
$937$			30.3318i			0.990896i	−0.868638	+	0.495448i	$0.835004\pi$
$938$		−	4.47716i		−	0.146184i
$939$		0			0
$940$		−	0.146616i		−	0.00478209i
$941$	−53.0581			−1.72964			−0.864822	−	0.502078i	$-0.832569\pi$
$941$	−53.0581			−1.72964			−0.864822	+	0.502078i	$0.832569\pi$
$942$		0			0
$943$		−	31.6474i		−	1.03058i
$944$	−14.2865	−	11.2076i	−0.464986	−	0.364776i
$945$		0			0
$946$			8.76575i			0.284999i
$947$			30.9120i			1.00450i	0.864721	+	0.502252i	$0.167495\pi$
$947$			30.9120i			1.00450i	−0.864721	+	0.502252i	$0.832505\pi$
$948$		0			0
$949$	47.2427			1.53356
$950$	7.03936			0.228387
$951$		0			0
$952$			3.75926i			0.121838i
$953$		−	58.8468i		−	1.90624i	−0.302600	−	0.953118i	$-0.597855\pi$
$953$		−	58.8468i		−	1.90624i	0.302600	−	0.953118i	$-0.402145\pi$
$954$		0			0
$955$		−	0.721453i		−	0.0233457i
$956$		−	43.0246i		−	1.39152i
$957$		0			0
$958$		−	11.3527i		−	0.366789i
$959$			26.5177i			0.856301i
$960$		0			0
$961$	−24.2260			−0.781485
$962$	−17.7493			−0.572260
$963$		0			0
$964$	−34.2116			−1.10188
$965$		−	0.567777i		−	0.0182774i
$966$		0			0
$967$			26.2197i			0.843168i	0.906789	+	0.421584i	$0.138526\pi$
$967$			26.2197i			0.843168i	−0.906789	+	0.421584i	$0.861474\pi$
$968$	−21.3117			−0.684985
$969$		0			0
$970$	−0.337584			−0.0108392
$971$			46.1301i			1.48038i	0.672396	+	0.740192i	$0.265265\pi$
$971$			46.1301i			1.48038i	−0.672396	+	0.740192i	$0.734735\pi$
$972$		0			0
$973$	−10.5825			−0.339260
$974$	2.51027			0.0804342
$975$		0			0
$976$		−	4.51343i		−	0.144471i
$977$	−23.8245			−0.762212			−0.381106	−	0.924531i	$-0.624457\pi$
$977$	−23.8245			−0.762212			−0.381106	+	0.924531i	$0.624457\pi$
$978$		0			0
$979$	−57.7126			−1.84450
$980$			0.464941i			0.0148520i
$981$		0			0
$982$			22.1654i			0.707328i
$983$	7.06451			0.225323			0.112661	−	0.993633i	$-0.464062\pi$
$983$	7.06451			0.225323			0.112661	+	0.993633i	$0.464062\pi$
$984$		0			0
$985$	1.08660			0.0346220
$986$	−3.95087			−0.125821
$987$		0			0
$988$		−	17.5728i		−	0.559064i
$989$			11.8430i			0.376586i
$990$		0			0
$991$		−	42.1784i		−	1.33984i	−0.742432	−	0.669921i	$-0.766328\pi$
$991$		−	42.1784i		−	1.33984i	0.742432	−	0.669921i	$-0.233672\pi$
$992$			38.9359i			1.23622i
$993$		0			0
$994$			5.16230i			0.163738i
$995$		−	0.891221i		−	0.0282536i
$996$		0			0
$997$	15.8794			0.502904			0.251452	−	0.967870i	$-0.419092\pi$
$997$	15.8794			0.502904			0.251452	+	0.967870i	$0.419092\pi$
$998$	5.29023			0.167459
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.d.a.530.12	yes	20
3.2	odd	2	inner	531.2.d.a.530.9	✓	20
59.58	odd	2	inner	531.2.d.a.530.10	yes	20
177.176	even	2	inner	531.2.d.a.530.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.d.a.530.9	✓	20	3.2	odd	2	inner
531.2.d.a.530.10	yes	20	59.58	odd	2	inner
531.2.d.a.530.11	yes	20	177.176	even	2	inner
531.2.d.a.530.12	yes	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.535103 q^{2} -1.71366 q^{4} +0.0572139i q^{5} +1.50263 q^{7} -1.98719 q^{8} +0.0306154i q^{10} +4.66096 q^{11} +3.89497i q^{13} +0.804063 q^{14} +2.36397 q^{16} -1.25895i q^{17} +2.63275 q^{19} -0.0980455i q^{20} +2.49409 q^{22} +3.36966 q^{23} +4.99673 q^{25} +2.08421i q^{26} -2.57501 q^{28} -5.86470i q^{29} +7.43142i q^{31} +5.23936 q^{32} -0.673671i q^{34} +0.0859715i q^{35} +8.51607i q^{37} +1.40879 q^{38} -0.113695i q^{40} -9.39188i q^{41} +3.51460i q^{43} -7.98732 q^{44} +1.80311 q^{46} +1.49539 q^{47} -4.74210 q^{49} +2.67376 q^{50} -6.67467i q^{52} +5.49315i q^{53} +0.266672i q^{55} -2.98602 q^{56} -3.13822i q^{58} +(-6.04342 - 4.74100i) q^{59} -1.90926i q^{61} +3.97658i q^{62} -1.92435 q^{64} -0.222847 q^{65} -5.56817i q^{67} +2.15742i q^{68} +0.0460036i q^{70} +6.42027i q^{71} -12.1291i q^{73} +4.55698i q^{74} -4.51165 q^{76} +7.00371 q^{77} -4.35495 q^{79} +0.135252i q^{80} -5.02563i q^{82} -7.98732 q^{83} +0.0720297 q^{85} +1.88068i q^{86} -9.26223 q^{88} -12.3821 q^{89} +5.85271i q^{91} -5.77446 q^{92} +0.800187 q^{94} +0.150630i q^{95} +11.0266i q^{97} -2.53751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) \) 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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