LMFDB - Embedded newform 531.2.d.a.530.3

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			530.3
Root			$2.08812 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	531.530
Dual form			531.2.d.a.530.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.08812 q^{2} +2.36026 q^{4} -2.45675i q^{5} +4.05068 q^{7} -0.752258 q^{8} +5.12999i q^{10} +2.91471 q^{11} +1.21007i q^{13} -8.45832 q^{14} -3.14970 q^{16} +1.92759i q^{17} +2.11411 q^{19} -5.79855i q^{20} -6.08628 q^{22} -2.53929 q^{23} -1.03560 q^{25} -2.52677i q^{26} +9.56064 q^{28} +1.99371i q^{29} -5.80912i q^{31} +8.08148 q^{32} -4.02504i q^{34} -9.95149i q^{35} -5.66290i q^{37} -4.41451 q^{38} +1.84811i q^{40} +10.6010i q^{41} +10.2189i q^{43} +6.87947 q^{44} +5.30234 q^{46} +10.0953 q^{47} +9.40801 q^{49} +2.16245 q^{50} +2.85607i q^{52} -7.11916i q^{53} -7.16071i q^{55} -3.04716 q^{56} -4.16312i q^{58} +(0.376831 - 7.67190i) q^{59} -0.735026i q^{61} +12.1301i q^{62} -10.5757 q^{64} +2.97282 q^{65} -14.2358i q^{67} +4.54960i q^{68} +20.7799i q^{70} -1.02120i q^{71} -13.2278i q^{73} +11.8248i q^{74} +4.98983 q^{76} +11.8066 q^{77} -10.8908 q^{79} +7.73802i q^{80} -22.1362i q^{82} +6.87947 q^{83} +4.73559 q^{85} -21.3383i q^{86} -2.19262 q^{88} +8.24682 q^{89} +4.90159i q^{91} -5.99336 q^{92} -21.0802 q^{94} -5.19382i q^{95} +15.2551i q^{97} -19.6451 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.08812			−1.47653			−0.738263	−	0.674513i	$-0.764354\pi$
$2$	−2.08812			−1.47653			−0.738263	+	0.674513i	$0.764354\pi$
$3$		0			0
$3$		0			0
$4$	2.36026			1.18013
$5$		−	2.45675i		−	1.09869i	−0.835596	−	0.549345i	$-0.814877\pi$
$5$		−	2.45675i		−	1.09869i	0.835596	−	0.549345i	$-0.185123\pi$
$6$		0			0
$7$	4.05068			1.53101			0.765506	−	0.643428i	$-0.222488\pi$
$7$	4.05068			1.53101			0.765506	+	0.643428i	$0.222488\pi$
$8$	−0.752258			−0.265963
$9$		0			0
$10$			5.12999i			1.62224i
$11$	2.91471			0.878819			0.439409	−	0.898287i	$-0.355188\pi$
$11$	2.91471			0.878819			0.439409	+	0.898287i	$0.355188\pi$
$12$		0			0
$13$			1.21007i			0.335612i	0.985820	+	0.167806i	$0.0536682\pi$
$13$			1.21007i			0.335612i	−0.985820	+	0.167806i	$0.946332\pi$
$14$	−8.45832			−2.26058
$15$		0			0
$16$	−3.14970			−0.787426
$17$			1.92759i			0.467509i	0.972296	+	0.233754i	$0.0751012\pi$
$17$			1.92759i			0.467509i	−0.972296	+	0.233754i	$0.924899\pi$
$18$		0			0
$19$	2.11411			0.485009			0.242505	−	0.970150i	$-0.422031\pi$
$19$	2.11411			0.485009			0.242505	+	0.970150i	$0.422031\pi$
$20$		−	5.79855i		−	1.29659i
$21$		0			0
$22$	−6.08628			−1.29760
$23$	−2.53929			−0.529478			−0.264739	−	0.964320i	$-0.585286\pi$
$23$	−2.53929			−0.529478			−0.264739	+	0.964320i	$0.585286\pi$
$24$		0			0
$25$	−1.03560			−0.207120
$26$		−	2.52677i		−	0.495540i
$27$		0			0
$28$	9.56064			1.80679
$29$			1.99371i			0.370223i	0.982718	+	0.185112i	$0.0592647\pi$
$29$			1.99371i			0.370223i	−0.982718	+	0.185112i	$0.940735\pi$
$30$		0			0
$31$		−	5.80912i		−	1.04335i	−0.853145	−	0.521674i	$-0.825308\pi$
$31$		−	5.80912i		−	1.04335i	0.853145	−	0.521674i	$-0.174692\pi$
$32$	8.08148			1.42862
$33$		0			0
$34$		−	4.02504i		−	0.690289i
$35$		−	9.95149i		−	1.68211i
$36$		0			0
$37$		−	5.66290i		−	0.930976i	−0.885054	−	0.465488i	$-0.845879\pi$
$37$		−	5.66290i		−	0.930976i	0.885054	−	0.465488i	$-0.154121\pi$
$38$	−4.41451			−0.716129
$39$		0			0
$40$			1.84811i			0.292211i
$41$			10.6010i			1.65560i	0.561024	+	0.827800i	$0.310408\pi$
$41$			10.6010i			1.65560i	−0.561024	+	0.827800i	$0.689592\pi$
$42$		0			0
$43$			10.2189i			1.55836i	0.626797	+	0.779182i	$0.284365\pi$
$43$			10.2189i			1.55836i	−0.626797	+	0.779182i	$0.715635\pi$
$44$	6.87947			1.03712
$45$		0			0
$46$	5.30234			0.781787
$47$	10.0953			1.47255			0.736274	−	0.676684i	$-0.236584\pi$
$47$	10.0953			1.47255			0.736274	+	0.676684i	$0.236584\pi$
$48$		0			0
$49$	9.40801			1.34400
$50$	2.16245			0.305817
$51$		0			0
$52$			2.85607i			0.396065i
$53$		−	7.11916i		−	0.977893i	−0.872314	−	0.488946i	$-0.837381\pi$
$53$		−	7.11916i		−	0.977893i	0.872314	−	0.488946i	$-0.162619\pi$
$54$		0			0
$55$		−	7.16071i		−	0.965549i
$56$	−3.04716			−0.407193
$57$		0			0
$58$		−	4.16312i		−	0.546644i
$59$	0.376831	−	7.67190i	0.0490592	−	0.998796i
$59$	0.376831	−	7.67190i	0.0490592	−	0.998796i
$60$		0			0
$61$		−	0.735026i		−	0.0941104i	−0.998892	−	0.0470552i	$-0.985016\pi$
$61$		−	0.735026i		−	0.0941104i	0.998892	−	0.0470552i	$-0.0149837\pi$
$62$			12.1301i			1.54053i
$63$		0			0
$64$	−10.5757			−1.32197
$65$	2.97282			0.368733
$66$		0			0
$67$		−	14.2358i		−	1.73918i	−0.493776	−	0.869589i	$-0.664384\pi$
$67$		−	14.2358i		−	1.73918i	0.493776	−	0.869589i	$-0.335616\pi$
$68$			4.54960i			0.551720i
$69$		0			0
$70$			20.7799i			2.48368i
$71$		−	1.02120i		−	0.121194i	−0.998162	−	0.0605972i	$-0.980699\pi$
$71$		−	1.02120i		−	0.121194i	0.998162	−	0.0605972i	$-0.0193005\pi$
$72$		0			0
$73$		−	13.2278i		−	1.54820i	−0.633063	−	0.774100i	$-0.718203\pi$
$73$		−	13.2278i		−	1.54820i	0.633063	−	0.774100i	$-0.281797\pi$
$74$			11.8248i			1.37461i
$75$		0			0
$76$	4.98983			0.572373
$77$	11.8066			1.34548
$78$		0			0
$79$	−10.8908			−1.22531			−0.612656	−	0.790350i	$-0.709899\pi$
$79$	−10.8908			−1.22531			−0.612656	+	0.790350i	$0.709899\pi$
$80$			7.73802i			0.865137i
$81$		0			0
$82$		−	22.1362i		−	2.44453i
$83$	6.87947			0.755120			0.377560	−	0.925985i	$-0.376763\pi$
$83$	6.87947			0.755120			0.377560	+	0.925985i	$0.376763\pi$
$84$		0			0
$85$	4.73559			0.513647
$86$		−	21.3383i		−	2.30097i
$87$		0			0
$88$	−2.19262			−0.233734
$89$	8.24682			0.874162			0.437081	−	0.899422i	$-0.356012\pi$
$89$	8.24682			0.874162			0.437081	+	0.899422i	$0.356012\pi$
$90$		0			0
$91$			4.90159i			0.513826i
$92$	−5.99336			−0.624851
$93$		0			0
$94$	−21.0802			−2.17425
$95$		−	5.19382i		−	0.532875i
$96$		0			0
$97$			15.2551i			1.54892i	0.632622	+	0.774460i	$0.281979\pi$
$97$			15.2551i			1.54892i	−0.632622	+	0.774460i	$0.718021\pi$
$98$	−19.6451			−1.98445
$99$		0			0
$100$	−2.44428			−0.244428
$101$	−16.9747			−1.68905			−0.844525	−	0.535516i	$-0.820117\pi$
$101$	−16.9747			−1.68905			−0.844525	+	0.535516i	$0.820117\pi$
$102$		0			0
$103$		−	1.59479i		−	0.157139i	−0.996909	−	0.0785696i	$-0.974965\pi$
$103$		−	1.59479i		−	0.157139i	0.996909	−	0.0785696i	$-0.0250353\pi$
$104$		−	0.910282i		−	0.0892605i
$105$		0			0
$106$			14.8657i			1.44388i
$107$			3.49538i			0.337911i	0.985624	+	0.168956i	$0.0540395\pi$
$107$			3.49538i			0.337911i	−0.985624	+	0.168956i	$0.945961\pi$
$108$		0			0
$109$			11.7792i			1.12825i	0.825690	+	0.564124i	$0.190786\pi$
$109$			11.7792i			1.12825i	−0.825690	+	0.564124i	$0.809214\pi$
$110$			14.9524i			1.42566i
$111$		0			0
$112$	−12.7584			−1.20556
$113$	−4.13341			−0.388838			−0.194419	−	0.980919i	$-0.562282\pi$
$113$	−4.13341			−0.388838			−0.194419	+	0.980919i	$0.562282\pi$
$114$		0			0
$115$			6.23838i			0.581732i
$116$			4.70567i			0.436911i
$117$		0			0
$118$	−0.786868	+	16.0199i	−0.0724371	+	1.47475i
$119$			7.80804i			0.715762i
$120$		0			0
$121$	−2.50445			−0.227677
$122$			1.53482i			0.138956i
$123$		0			0
$124$		−	13.7110i		−	1.23128i
$125$		−	9.73953i		−	0.871130i
$126$		0			0
$127$	−10.7561			−0.954450			−0.477225	−	0.878781i	$-0.658357\pi$
$127$	−10.7561			−0.954450			−0.477225	+	0.878781i	$0.658357\pi$
$128$	5.92043			0.523297
$129$		0			0
$130$	−6.20762			−0.544444
$131$	−15.8771			−1.38719			−0.693596	−	0.720364i	$-0.743975\pi$
$131$	−15.8771			−1.38719			−0.693596	+	0.720364i	$0.743975\pi$
$132$		0			0
$133$	8.56357			0.742556
$134$			29.7261i			2.56794i
$135$		0			0
$136$		−	1.45004i		−	0.124340i
$137$		−	9.33648i		−	0.797669i	−0.917023	−	0.398835i	$-0.869415\pi$
$137$		−	9.33648i		−	0.797669i	0.917023	−	0.398835i	$-0.130585\pi$
$138$		0			0
$139$	4.53341			0.384518			0.192259	−	0.981344i	$-0.438419\pi$
$139$	4.53341			0.384518			0.192259	+	0.981344i	$0.438419\pi$
$140$		−	23.4881i		−	1.98510i
$141$		0			0
$142$			2.13240i			0.178947i
$143$			3.52700i			0.294942i
$144$		0			0
$145$	4.89805			0.406761
$146$			27.6213i			2.28596i
$147$		0			0
$148$		−	13.3659i		−	1.09867i
$149$	3.69706			0.302875			0.151438	−	0.988467i	$-0.451610\pi$
$149$	3.69706			0.302875			0.151438	+	0.988467i	$0.451610\pi$
$150$		0			0
$151$			17.7480i			1.44432i	0.691729	+	0.722158i	$0.256849\pi$
$151$			17.7480i			1.44432i	−0.691729	+	0.722158i	$0.743151\pi$
$152$	−1.59035			−0.128995
$153$		0			0
$154$	−24.6536			−1.98664
$155$	−14.2715			−1.14632
$156$		0			0
$157$		−	7.94350i		−	0.633960i	−0.948432	−	0.316980i	$-0.897331\pi$
$157$		−	7.94350i		−	0.633960i	0.948432	−	0.316980i	$-0.102669\pi$
$158$	22.7413			1.80920
$159$		0			0
$160$		−	19.8542i		−	1.56961i
$161$	−10.2858			−0.810637
$162$		0			0
$163$	10.4381			0.817575			0.408787	−	0.912630i	$-0.365952\pi$
$163$	10.4381			0.817575			0.408787	+	0.912630i	$0.365952\pi$
$164$			25.0211i			1.95382i
$165$		0			0
$166$	−14.3652			−1.11495
$167$			2.32049i			0.179565i	0.995961	+	0.0897825i	$0.0286172\pi$
$167$			2.32049i			0.179565i	−0.995961	+	0.0897825i	$0.971383\pi$
$168$		0			0
$169$	11.5357			0.887365
$170$	−9.88850			−0.758413
$171$		0			0
$172$			24.1192i			1.83907i
$173$	11.2625			0.856271			0.428136	−	0.903715i	$-0.359171\pi$
$173$	11.2625			0.856271			0.428136	+	0.903715i	$0.359171\pi$
$174$		0			0
$175$	−4.19487			−0.317103
$176$	−9.18048			−0.692005
$177$		0			0
$178$	−17.2204			−1.29072
$179$	−24.9652			−1.86599			−0.932994	−	0.359892i	$-0.882814\pi$
$179$	−24.9652			−1.86599			−0.932994	+	0.359892i	$0.882814\pi$
$180$		0			0
$181$	−6.84790			−0.509000			−0.254500	−	0.967073i	$-0.581911\pi$
$181$	−6.84790			−0.509000			−0.254500	+	0.967073i	$0.581911\pi$
$182$		−	10.2351i		−	0.758678i
$183$		0			0
$184$	1.91020			0.140822
$185$	−13.9123			−1.02285
$186$		0			0
$187$			5.61837i			0.410856i
$188$	23.8274			1.73779
$189$		0			0
$190$			10.8453i			0.786803i
$191$	−5.92374			−0.428627			−0.214314	−	0.976765i	$-0.568751\pi$
$191$	−5.92374			−0.428627			−0.214314	+	0.976765i	$0.568751\pi$
$192$		0			0
$193$	27.0879			1.94983			0.974916	−	0.222572i	$-0.0714452\pi$
$193$	27.0879			1.94983			0.974916	+	0.222572i	$0.0714452\pi$
$194$		−	31.8545i		−	2.28702i
$195$		0			0
$196$	22.2053			1.58609
$197$		−	6.14511i		−	0.437821i	−0.975745	−	0.218911i	$-0.929750\pi$
$197$		−	6.14511i		−	0.437821i	0.975745	−	0.218911i	$-0.0702503\pi$
$198$		0			0
$199$	−2.50512			−0.177583			−0.0887917	−	0.996050i	$-0.528301\pi$
$199$	−2.50512			−0.177583			−0.0887917	+	0.996050i	$0.528301\pi$
$200$	0.779037			0.0550862
$201$		0			0
$202$	35.4453			2.49393
$203$			8.07589i			0.566817i
$204$		0			0
$205$	26.0440			1.81899
$206$			3.33011i			0.232020i
$207$		0			0
$208$		−	3.81135i		−	0.264270i
$209$	6.16201			0.426235
$210$		0			0
$211$			21.0642i			1.45012i	0.688685	+	0.725060i	$0.258188\pi$
$211$			21.0642i			1.45012i	−0.688685	+	0.725060i	$0.741812\pi$
$212$		−	16.8030i		−	1.15404i
$213$		0			0
$214$		−	7.29878i		−	0.498935i
$215$	25.1052			1.71216
$216$		0			0
$217$		−	23.5309i		−	1.59738i
$218$		−	24.5965i		−	1.66589i
$219$		0			0
$220$		−	16.9011i		−	1.13947i
$221$	−2.33251			−0.156902
$222$		0			0
$223$	−26.0367			−1.74354			−0.871772	−	0.489912i	$-0.837029\pi$
$223$	−26.0367			−1.74354			−0.871772	+	0.489912i	$0.837029\pi$
$224$	32.7355			2.18723
$225$		0			0
$226$	8.63106			0.574129
$227$	15.6817			1.04083			0.520416	−	0.853913i	$-0.325777\pi$
$227$	15.6817			1.04083			0.520416	+	0.853913i	$0.325777\pi$
$228$		0			0
$229$			1.07199i			0.0708390i	0.999373	+	0.0354195i	$0.0112767\pi$
$229$			1.07199i			0.0708390i	−0.999373	+	0.0354195i	$0.988723\pi$
$230$		−	13.0265i		−	0.858942i
$231$		0			0
$232$		−	1.49979i		−	0.0984658i
$233$	−0.690675			−0.0452476			−0.0226238	−	0.999744i	$-0.507202\pi$
$233$	−0.690675			−0.0452476			−0.0226238	+	0.999744i	$0.507202\pi$
$234$		0			0
$235$		−	24.8015i		−	1.61787i
$236$	0.889416	−	18.1076i	0.0578961	−	1.17871i
$237$		0			0
$238$		−	16.3041i		−	1.05684i
$239$		−	20.3847i		−	1.31858i	−0.751889	−	0.659290i	$-0.770857\pi$
$239$		−	20.3847i		−	1.31858i	0.751889	−	0.659290i	$-0.229143\pi$
$240$		0			0
$241$	−9.96314			−0.641782			−0.320891	−	0.947116i	$-0.603982\pi$
$241$	−9.96314			−0.641782			−0.320891	+	0.947116i	$0.603982\pi$
$242$	5.22960			0.336171
$243$		0			0
$244$		−	1.73485i		−	0.111062i
$245$		−	23.1131i		−	1.47664i
$246$		0			0
$247$			2.55821i			0.162775i
$248$			4.36995i			0.277492i
$249$		0			0
$250$			20.3373i			1.28625i
$251$			25.1337i			1.58642i	0.608946	+	0.793211i	$0.291592\pi$
$251$			25.1337i			1.58642i	−0.608946	+	0.793211i	$0.708408\pi$
$252$		0			0
$253$	−7.40129			−0.465315
$254$	22.4601			1.40927
$255$		0			0
$256$	8.78885			0.549303
$257$			6.34893i			0.396035i	0.980198	+	0.198018i	$0.0634503\pi$
$257$			6.34893i			0.396035i	−0.980198	+	0.198018i	$0.936550\pi$
$258$		0			0
$259$		−	22.9386i		−	1.42534i
$260$	7.01663			0.435153
$261$		0			0
$262$	33.1534			2.04823
$263$			17.6799i			1.09019i	0.838375	+	0.545094i	$0.183506\pi$
$263$			17.6799i			1.09019i	−0.838375	+	0.545094i	$0.816494\pi$
$264$		0			0
$265$	−17.4900			−1.07440
$266$	−17.8818			−1.09640
$267$		0			0
$268$		−	33.6001i		−	2.05245i
$269$	−7.36072			−0.448791			−0.224396	−	0.974498i	$-0.572041\pi$
$269$	−7.36072			−0.448791			−0.224396	+	0.974498i	$0.572041\pi$
$270$		0			0
$271$	−15.4411			−0.937980			−0.468990	−	0.883203i	$-0.655382\pi$
$271$	−15.4411			−0.937980			−0.468990	+	0.883203i	$0.655382\pi$
$272$		−	6.07133i		−	0.368129i
$273$		0			0
$274$			19.4957i			1.17778i
$275$	−3.01847			−0.182021
$276$		0			0
$277$	28.1279			1.69004			0.845022	−	0.534732i	$-0.179588\pi$
$277$	28.1279			1.69004			0.845022	+	0.534732i	$0.179588\pi$
$278$	−9.46631			−0.567751
$279$		0			0
$280$			7.48609i			0.447379i
$281$			11.5328i			0.687987i	0.938972	+	0.343993i	$0.111780\pi$
$281$			11.5328i			0.687987i	−0.938972	+	0.343993i	$0.888220\pi$
$282$		0			0
$283$			8.42176i			0.500622i	0.968166	+	0.250311i	$0.0805328\pi$
$283$			8.42176i			0.500622i	−0.968166	+	0.250311i	$0.919467\pi$
$284$		−	2.41030i		−	0.143025i
$285$		0			0
$286$		−	7.36480i		−	0.435490i
$287$			42.9413i			2.53474i
$288$		0			0
$289$	13.2844			0.781435
$290$	−10.2277			−0.600592
$291$		0			0
$292$		−	31.2210i		−	1.82707i
$293$		−	15.5123i		−	0.906236i	−0.891451	−	0.453118i	$-0.850312\pi$
$293$		−	15.5123i		−	0.906236i	0.891451	−	0.453118i	$-0.149688\pi$
$294$		0			0
$295$	−18.8479	−	0.925777i	−1.09737	−	0.0539008i
$296$			4.25997i			0.247605i
$297$		0			0
$298$	−7.71992			−0.447203
$299$		−	3.07270i		−	0.177699i
$300$		0			0
$301$			41.3934i			2.38588i
$302$		−	37.0601i		−	2.13257i
$303$		0			0
$304$	−6.65881			−0.381909
$305$	−1.80577			−0.103398
$306$		0			0
$307$	−11.5400			−0.658625			−0.329313	−	0.944221i	$-0.606817\pi$
$307$	−11.5400			−0.658625			−0.329313	+	0.944221i	$0.606817\pi$
$308$	27.8665			1.58784
$309$		0			0
$310$	29.8007			1.69257
$311$			26.0654i			1.47803i	0.673687	+	0.739017i	$0.264710\pi$
$311$			26.0654i			1.47803i	−0.673687	+	0.739017i	$0.735290\pi$
$312$		0			0
$313$			27.8402i			1.57362i	0.617195	+	0.786811i	$0.288269\pi$
$313$			27.8402i			1.57362i	−0.617195	+	0.786811i	$0.711731\pi$
$314$			16.5870i			0.936058i
$315$		0			0
$316$	−25.7051			−1.44602
$317$		−	29.7970i		−	1.67357i	−0.547533	−	0.836784i	$-0.684433\pi$
$317$		−	29.7970i		−	1.67357i	0.547533	−	0.836784i	$-0.315567\pi$
$318$		0			0
$319$			5.81110i			0.325359i
$320$			25.9819i			1.45243i
$321$		0			0
$322$	21.4781			1.19693
$323$			4.07513i			0.226746i
$324$		0			0
$325$		−	1.25314i		−	0.0695118i
$326$	−21.7960			−1.20717
$327$		0			0
$328$		−	7.97469i		−	0.440329i
$329$	40.8927			2.25449
$330$		0			0
$331$	−9.49668			−0.521985			−0.260992	−	0.965341i	$-0.584050\pi$
$331$	−9.49668			−0.521985			−0.260992	+	0.965341i	$0.584050\pi$
$332$	16.2373			0.891138
$333$		0			0
$334$		−	4.84547i		−	0.265132i
$335$	−34.9737			−1.91082
$336$		0			0
$337$			6.87443i			0.374474i	0.982315	+	0.187237i	$0.0599533\pi$
$337$			6.87443i			0.374474i	−0.982315	+	0.187237i	$0.940047\pi$
$338$	−24.0880			−1.31022
$339$		0			0
$340$	11.1772			0.606169
$341$		−	16.9319i		−	0.916914i
$342$		0			0
$343$	9.75406			0.526670
$344$		−	7.68723i		−	0.414468i
$345$		0			0
$346$	−23.5175			−1.26431
$347$	19.6199			1.05325			0.526624	−	0.850098i	$-0.323457\pi$
$347$	19.6199			1.05325			0.526624	+	0.850098i	$0.323457\pi$
$348$		0			0
$349$		−	23.4764i		−	1.25666i	−0.777945	−	0.628332i	$-0.783738\pi$
$349$		−	23.4764i		−	1.25666i	0.777945	−	0.628332i	$-0.216262\pi$
$350$	8.75941			0.468210
$351$		0			0
$352$	23.5552			1.25550
$353$	−27.4568			−1.46138			−0.730689	−	0.682711i	$-0.760801\pi$
$353$	−27.4568			−1.46138			−0.730689	+	0.682711i	$0.760801\pi$
$354$		0			0
$355$	−2.50883			−0.133155
$356$	19.4646			1.03162
$357$		0			0
$358$	52.1304			2.75518
$359$			20.1509i			1.06352i	0.846895	+	0.531761i	$0.178469\pi$
$359$			20.1509i			1.06352i	−0.846895	+	0.531761i	$0.821531\pi$
$360$		0			0
$361$	−14.5306			−0.764766
$362$	14.2992			0.751552
$363$		0			0
$364$			11.5690i			0.606381i
$365$	−32.4974			−1.70099
$366$		0			0
$367$			29.4451i			1.53702i	0.639836	+	0.768511i	$0.279002\pi$
$367$			29.4451i			1.53702i	−0.639836	+	0.768511i	$0.720998\pi$
$368$	7.99800			0.416924
$369$		0			0
$370$	29.0506			1.51027
$371$		−	28.8375i		−	1.49717i
$372$		0			0
$373$	−13.5762			−0.702948			−0.351474	−	0.936198i	$-0.614319\pi$
$373$	−13.5762			−0.702948			−0.351474	+	0.936198i	$0.614319\pi$
$374$		−	11.7318i		−	0.606639i
$375$		0			0
$376$	−7.59425			−0.391644
$377$	−2.41252			−0.124251
$378$		0			0
$379$	1.41962			0.0729210			0.0364605	−	0.999335i	$-0.488392\pi$
$379$	1.41962			0.0729210			0.0364605	+	0.999335i	$0.488392\pi$
$380$		−	12.2587i		−	0.628861i
$381$		0			0
$382$	12.3695			0.632879
$383$			1.48973i			0.0761217i	0.999275	+	0.0380609i	$0.0121181\pi$
$383$			1.48973i			0.0761217i	−0.999275	+	0.0380609i	$0.987882\pi$
$384$		0			0
$385$		−	29.0057i		−	1.47827i
$386$	−56.5629			−2.87898
$387$		0			0
$388$			36.0059i			1.82792i
$389$		−	18.5068i		−	0.938332i	−0.883110	−	0.469166i	$-0.844555\pi$
$389$		−	18.5068i		−	0.938332i	0.883110	−	0.469166i	$-0.155445\pi$
$390$		0			0
$391$		−	4.89470i		−	0.247535i
$392$	−7.07725			−0.357455
$393$		0			0
$394$			12.8317i			0.646454i
$395$			26.7559i			1.34624i
$396$		0			0
$397$		−	4.54638i		−	0.228176i	−0.993471	−	0.114088i	$-0.963605\pi$
$397$		−	4.54638i		−	0.228176i	0.993471	−	0.114088i	$-0.0363946\pi$
$398$	5.23100			0.262206
$399$		0			0
$400$	3.26183			0.163091
$401$	−21.1693			−1.05714			−0.528572	−	0.848889i	$-0.677272\pi$
$401$	−21.1693			−1.05714			−0.528572	+	0.848889i	$0.677272\pi$
$402$		0			0
$403$	7.02942			0.350160
$404$	−40.0647			−1.99329
$405$		0			0
$406$		−	16.8635i		−	0.836919i
$407$		−	16.5057i		−	0.818159i
$408$		0			0
$409$			30.4469i			1.50550i	0.658305	+	0.752752i	$0.271274\pi$
$409$			30.4469i			1.50550i	−0.658305	+	0.752752i	$0.728726\pi$
$410$	−54.3830			−2.68579
$411$		0			0
$412$		−	3.76411i		−	0.185444i
$413$	1.52642	−	31.0764i	0.0751102	−	1.52917i
$414$		0			0
$415$		−	16.9011i		−	0.829642i
$416$			9.77913i			0.479461i
$417$		0			0
$418$	−12.8670			−0.629348
$419$	35.6215			1.74023			0.870113	−	0.492853i	$-0.164046\pi$
$419$	35.6215			1.74023			0.870113	+	0.492853i	$0.164046\pi$
$420$		0			0
$421$			17.3156i			0.843909i	0.906617	+	0.421954i	$0.138656\pi$
$421$			17.3156i			0.843909i	−0.906617	+	0.421954i	$0.861344\pi$
$422$		−	43.9847i		−	2.14114i
$423$		0			0
$424$			5.35545i			0.260084i
$425$		−	1.99621i		−	0.0968302i
$426$		0			0
$427$		−	2.97735i		−	0.144084i
$428$			8.24999i			0.398778i
$429$		0			0
$430$	−52.4227			−2.52805
$431$	−8.66700			−0.417475			−0.208737	−	0.977972i	$-0.566935\pi$
$431$	−8.66700			−0.417475			−0.208737	+	0.977972i	$0.566935\pi$
$432$		0			0
$433$	7.59616			0.365048			0.182524	−	0.983201i	$-0.441573\pi$
$433$	7.59616			0.365048			0.182524	+	0.983201i	$0.441573\pi$
$434$			49.1353i			2.35857i
$435$		0			0
$436$			27.8020i			1.33148i
$437$	−5.36832			−0.256802
$438$		0			0
$439$	−29.0707			−1.38747			−0.693734	−	0.720231i	$-0.744036\pi$
$439$	−29.0707			−1.38747			−0.693734	+	0.720231i	$0.744036\pi$
$440$			5.38670i			0.256801i
$441$		0			0
$442$	4.87057			0.231669
$443$	0.232426			0.0110429			0.00552144	−	0.999985i	$-0.498242\pi$
$443$	0.232426			0.0110429			0.00552144	+	0.999985i	$0.498242\pi$
$444$		0			0
$445$		−	20.2603i		−	0.960433i
$446$	54.3677			2.57439
$447$		0			0
$448$	−42.8389			−2.02395
$449$			10.5751i			0.499067i	0.968366	+	0.249534i	$0.0802773\pi$
$449$			10.5751i			0.499067i	−0.968366	+	0.249534i	$0.919723\pi$
$450$		0			0
$451$			30.8989i			1.45497i
$452$	−9.75590			−0.458879
$453$		0			0
$454$	−32.7453			−1.53682
$455$	12.0420			0.564536
$456$		0			0
$457$		−	28.6906i		−	1.34209i	−0.741417	−	0.671045i	$-0.765846\pi$
$457$		−	28.6906i		−	1.34209i	0.741417	−	0.671045i	$-0.234154\pi$
$458$		−	2.23845i		−	0.104596i
$459$		0			0
$460$			14.7242i			0.686518i
$461$			23.3762i			1.08874i	0.838846	+	0.544369i	$0.183231\pi$
$461$			23.3762i			1.08874i	−0.838846	+	0.544369i	$0.816769\pi$
$462$		0			0
$463$		−	4.08709i		−	0.189943i	−0.995480	−	0.0949715i	$-0.969724\pi$
$463$		−	4.08709i		−	0.189943i	0.995480	−	0.0949715i	$-0.0302760\pi$
$464$		−	6.27961i		−	0.291523i
$465$		0			0
$466$	1.44221			0.0668092
$467$	19.2120			0.889025			0.444512	−	0.895773i	$-0.353377\pi$
$467$	19.2120			0.889025			0.444512	+	0.895773i	$0.353377\pi$
$468$		0			0
$469$		−	57.6646i		−	2.66270i
$470$			51.7886i			2.38883i
$471$		0			0
$472$	−0.283474	+	5.77125i	−0.0130479	+	0.265643i
$473$			29.7851i			1.36952i
$474$		0			0
$475$	−2.18936			−0.100455
$476$			18.4290i			0.844691i
$477$		0			0
$478$			42.5658i			1.94692i
$479$			29.9336i			1.36770i	0.729621	+	0.683851i	$0.239696\pi$
$479$			29.9336i			1.36770i	−0.729621	+	0.683851i	$0.760304\pi$
$480$		0			0
$481$	6.85249			0.312447
$482$	20.8042			0.947607
$483$		0			0
$484$	−5.91114			−0.268688
$485$	37.4779			1.70178
$486$		0			0
$487$	−22.9050			−1.03792			−0.518962	−	0.854798i	$-0.673681\pi$
$487$	−22.9050			−1.03792			−0.518962	+	0.854798i	$0.673681\pi$
$488$			0.552929i			0.0250299i
$489$		0			0
$490$			48.2629i			2.18030i
$491$		−	12.4050i		−	0.559832i	−0.960025	−	0.279916i	$-0.909693\pi$
$491$		−	12.4050i		−	0.559832i	0.960025	−	0.279916i	$-0.0903065\pi$
$492$		0			0
$493$	−3.84306			−0.173083
$494$		−	5.34185i		−	0.240341i
$495$		0			0
$496$			18.2970i			0.821560i
$497$		−	4.13656i		−	0.185550i
$498$		0			0
$499$	−10.4532			−0.467948			−0.233974	−	0.972243i	$-0.575173\pi$
$499$	−10.4532			−0.467948			−0.233974	+	0.972243i	$0.575173\pi$
$500$		−	22.9878i		−	1.02804i
$501$		0			0
$502$		−	52.4822i		−	2.34239i
$503$	−25.9304			−1.15618			−0.578089	−	0.815974i	$-0.696201\pi$
$503$	−25.9304			−1.15618			−0.578089	+	0.815974i	$0.696201\pi$
$504$		0			0
$505$			41.7026i			1.85574i
$506$	15.4548			0.687049
$507$		0			0
$508$	−25.3872			−1.12637
$509$	−7.71600			−0.342006			−0.171003	−	0.985271i	$-0.554701\pi$
$509$	−7.71600			−0.342006			−0.171003	+	0.985271i	$0.554701\pi$
$510$		0			0
$511$		−	53.5817i		−	2.37031i
$512$	−30.1931			−1.33436
$513$		0			0
$514$		−	13.2573i		−	0.584756i
$515$	−3.91799			−0.172647
$516$		0			0
$517$	29.4248			1.29410
$518$			47.8986i			2.10454i
$519$		0			0
$520$	−2.23633			−0.0980696
$521$			6.07913i			0.266331i	0.991094	+	0.133166i	$0.0425142\pi$
$521$			6.07913i			0.266331i	−0.991094	+	0.133166i	$0.957486\pi$
$522$		0			0
$523$	−7.14169			−0.312284			−0.156142	−	0.987735i	$-0.549906\pi$
$523$	−7.14169			−0.312284			−0.156142	+	0.987735i	$0.549906\pi$
$524$	−37.4741			−1.63706
$525$		0			0
$526$		−	36.9177i		−	1.60969i
$527$	11.1976			0.487774
$528$		0			0
$529$	−16.5520			−0.719654
$530$	36.5212			1.58638
$531$		0			0
$532$	20.2122			0.876311
$533$	−12.8279			−0.555639
$534$		0			0
$535$	8.58726			0.371260
$536$			10.7090i			0.462558i
$537$		0			0
$538$	15.3701			0.662651
$539$	27.4216			1.18113
$540$		0			0
$541$			12.9777i			0.557956i	0.960297	+	0.278978i	$0.0899957\pi$
$541$			12.9777i			0.557956i	−0.960297	+	0.278978i	$0.910004\pi$
$542$	32.2429			1.38495
$543$		0			0
$544$			15.5778i			0.667892i
$545$	28.9386			1.23959
$546$		0			0
$547$	−37.5805			−1.60683			−0.803413	−	0.595422i	$-0.796985\pi$
$547$	−37.5805			−1.60683			−0.803413	+	0.595422i	$0.796985\pi$
$548$		−	22.0365i		−	0.941352i
$549$		0			0
$550$	6.30293			0.268758
$551$			4.21492i			0.179562i
$552$		0			0
$553$	−44.1152			−1.87597
$554$	−58.7345			−2.49539
$555$		0			0
$556$	10.7000			0.453781
$557$			1.90152i			0.0805700i	0.999188	+	0.0402850i	$0.0128266\pi$
$557$			1.90152i			0.0805700i	−0.999188	+	0.0402850i	$0.987173\pi$
$558$		0			0
$559$	−12.3655			−0.523006
$560$			31.3442i			1.32454i
$561$		0			0
$562$		−	24.0818i		−	1.01583i
$563$	−17.0033			−0.716602			−0.358301	−	0.933606i	$-0.616644\pi$
$563$	−17.0033			−0.716602			−0.358301	+	0.933606i	$0.616644\pi$
$564$		0			0
$565$			10.1547i			0.427213i
$566$		−	17.5857i		−	0.739181i
$567$		0			0
$568$			0.768208i			0.0322333i
$569$	47.3281			1.98410			0.992049	−	0.125853i	$-0.0401667\pi$
$569$	47.3281			1.98410			0.992049	+	0.125853i	$0.0401667\pi$
$570$		0			0
$571$		−	28.4465i		−	1.19045i	−0.803559	−	0.595225i	$-0.797063\pi$
$571$		−	28.4465i		−	1.19045i	0.803559	−	0.595225i	$-0.202937\pi$
$572$			8.32461i			0.348069i
$573$		0			0
$574$		−	89.6667i		−	3.74261i
$575$	2.62968			0.109665
$576$		0			0
$577$	16.5746			0.690009			0.345005	−	0.938601i	$-0.387877\pi$
$577$	16.5746			0.690009			0.345005	+	0.938601i	$0.387877\pi$
$578$	−27.7395			−1.15381
$579$		0			0
$580$	11.5606			0.480029
$581$	27.8665			1.15610
$582$		0			0
$583$		−	20.7503i		−	0.859390i
$584$			9.95074i			0.411764i
$585$		0			0
$586$			32.3915i			1.33808i
$587$	8.71377			0.359656			0.179828	−	0.983698i	$-0.442446\pi$
$587$	8.71377			0.359656			0.179828	+	0.983698i	$0.442446\pi$
$588$		0			0
$589$		−	12.2811i		−	0.506034i
$590$	39.3567	+	1.93314i	1.62029	+	0.0795859i
$591$		0			0
$592$			17.8365i			0.733075i
$593$			25.7454i			1.05724i	0.848860	+	0.528618i	$0.177290\pi$
$593$			25.7454i			1.05724i	−0.848860	+	0.528618i	$0.822710\pi$
$594$		0			0
$595$	19.1824			0.786401
$596$	8.72601			0.357431
$597$		0			0
$598$			6.41618i			0.262377i
$599$		−	17.6320i		−	0.720425i	−0.932870	−	0.360212i	$-0.882704\pi$
$599$		−	17.6320i		−	0.720425i	0.932870	−	0.360212i	$-0.117296\pi$
$600$		0			0
$601$		−	11.0031i		−	0.448827i	−0.974494	−	0.224413i	$-0.927953\pi$
$601$		−	11.0031i		−	0.448827i	0.974494	−	0.224413i	$-0.0720466\pi$
$602$		−	86.4345i		−	3.52281i
$603$		0			0
$604$			41.8899i			1.70448i
$605$			6.15280i			0.250147i
$606$		0			0
$607$	29.6123			1.20193			0.600963	−	0.799277i	$-0.294784\pi$
$607$	29.6123			1.20193			0.600963	+	0.799277i	$0.294784\pi$
$608$	17.0851			0.692893
$609$		0			0
$610$	3.77067			0.152670
$611$			12.2160i			0.494204i
$612$		0			0
$613$			7.37241i			0.297769i	0.988855	+	0.148884i	$0.0475683\pi$
$613$			7.37241i			0.297769i	−0.988855	+	0.148884i	$0.952432\pi$
$614$	24.0970			0.972477
$615$		0			0
$616$	−8.88159			−0.357849
$617$			22.2534i			0.895886i	0.894062	+	0.447943i	$0.147843\pi$
$617$			22.2534i			0.895886i	−0.894062	+	0.447943i	$0.852157\pi$
$618$		0			0
$619$	19.0679			0.766403			0.383201	−	0.923665i	$-0.374822\pi$
$619$	19.0679			0.766403			0.383201	+	0.923665i	$0.374822\pi$
$620$	−33.6844			−1.35280
$621$		0			0
$622$		−	54.4278i		−	2.18236i
$623$	33.4052			1.33835
$624$		0			0
$625$	−29.1055			−1.16422
$626$		−	58.1338i		−	2.32349i
$627$		0			0
$628$		−	18.7487i		−	0.748154i
$629$	10.9157			0.435239
$630$		0			0
$631$	29.5525			1.17647			0.588233	−	0.808692i	$-0.299824\pi$
$631$	29.5525			1.17647			0.588233	+	0.808692i	$0.299824\pi$
$632$	8.19270			0.325888
$633$		0			0
$634$			62.2199i			2.47107i
$635$			26.4250i			1.04864i
$636$		0			0
$637$			11.3843i			0.451063i
$638$		−	12.1343i		−	0.480401i
$639$		0			0
$640$		−	14.5450i		−	0.574941i
$641$		−	41.7782i		−	1.65014i	−0.565031	−	0.825070i	$-0.691136\pi$
$641$		−	41.7782i		−	1.65014i	0.565031	−	0.825070i	$-0.308864\pi$
$642$		0			0
$643$	−20.9815			−0.827431			−0.413715	−	0.910406i	$-0.635769\pi$
$643$	−20.9815			−0.827431			−0.413715	+	0.910406i	$0.635769\pi$
$644$	−24.2772			−0.956655
$645$		0			0
$646$		−	8.50937i		−	0.334797i
$647$		−	0.979600i		−	0.0385120i	−0.999815	−	0.0192560i	$-0.993870\pi$
$647$		−	0.979600i		−	0.0385120i	0.999815	−	0.0192560i	$-0.00612976\pi$
$648$		0			0
$649$	1.09835	−	22.3614i	0.0431141	−	0.877761i
$650$			2.61671i			0.102636i
$651$		0			0
$652$	24.6366			0.964842
$653$			30.2362i			1.18323i	0.806219	+	0.591617i	$0.201510\pi$
$653$			30.2362i			1.18323i	−0.806219	+	0.591617i	$0.798490\pi$
$654$		0			0
$655$			39.0061i			1.52409i
$656$		−	33.3900i		−	1.30366i
$657$		0			0
$658$	−85.3890			−3.32881
$659$	−4.38599			−0.170854			−0.0854269	−	0.996344i	$-0.527225\pi$
$659$	−4.38599			−0.170854			−0.0854269	+	0.996344i	$0.527225\pi$
$660$		0			0
$661$	−12.9447			−0.503492			−0.251746	−	0.967793i	$-0.581005\pi$
$661$	−12.9447			−0.503492			−0.251746	+	0.967793i	$0.581005\pi$
$662$	19.8302			0.770724
$663$		0			0
$664$	−5.17513			−0.200834
$665$		−	21.0385i		−	0.815838i
$666$		0			0
$667$		−	5.06261i		−	0.196025i
$668$			5.47695i			0.211910i
$669$		0			0
$670$	73.0294			2.82137
$671$		−	2.14239i		−	0.0827060i
$672$		0			0
$673$			22.9846i			0.885992i	0.896524	+	0.442996i	$0.146084\pi$
$673$			22.9846i			0.885992i	−0.896524	+	0.442996i	$0.853916\pi$
$674$		−	14.3547i		−	0.552921i
$675$		0			0
$676$	27.2273			1.04720
$677$		−	0.448150i		−	0.0172238i	−0.999963	−	0.00861191i	$-0.997259\pi$
$677$		−	0.448150i		−	0.0172238i	0.999963	−	0.00861191i	$-0.00274129\pi$
$678$		0			0
$679$			61.7935i			2.37142i
$680$	−3.56239			−0.136611
$681$		0			0
$682$			35.3559i			1.35385i
$683$	−17.4454			−0.667531			−0.333766	−	0.942656i	$-0.608319\pi$
$683$	−17.4454			−0.667531			−0.333766	+	0.942656i	$0.608319\pi$
$684$		0			0
$685$	−22.9373			−0.876391
$686$	−20.3677			−0.777641
$687$		0			0
$688$		−	32.1864i		−	1.22710i
$689$	8.61466			0.328192
$690$		0			0
$691$		−	21.0299i		−	0.800016i	−0.916512	−	0.400008i	$-0.869007\pi$
$691$		−	21.0299i		−	0.800016i	0.916512	−	0.400008i	$-0.130993\pi$
$692$	26.5824			1.01051
$693$		0			0
$694$	−40.9687			−1.55515
$695$		−	11.1374i		−	0.422467i
$696$		0			0
$697$	−20.4344			−0.774007
$698$			49.0217i			1.85550i
$699$		0			0
$700$	−9.90098			−0.374222
$701$	−13.9255			−0.525960			−0.262980	−	0.964801i	$-0.584705\pi$
$701$	−13.9255			−0.525960			−0.262980	+	0.964801i	$0.584705\pi$
$702$		0			0
$703$		−	11.9720i		−	0.451532i
$704$	−30.8252			−1.16177
$705$		0			0
$706$	57.3331			2.15776
$707$	−68.7592			−2.58596
$708$		0			0
$709$	40.7527			1.53050			0.765249	−	0.643734i	$-0.222616\pi$
$709$	40.7527			1.53050			0.765249	+	0.643734i	$0.222616\pi$
$710$	5.23875			0.196607
$711$		0			0
$712$	−6.20374			−0.232495
$713$			14.7510i			0.552429i
$714$		0			0
$715$	8.66493			0.324050
$716$	−58.9243			−2.20210
$717$		0			0
$718$		−	42.0775i		−	1.57032i
$719$	9.50629			0.354525			0.177262	−	0.984164i	$-0.443276\pi$
$719$	9.50629			0.354525			0.177262	+	0.984164i	$0.443276\pi$
$720$		0			0
$721$		−	6.45998i		−	0.240582i
$722$	30.3416			1.12920
$723$		0			0
$724$	−16.1628			−0.600685
$725$		−	2.06468i		−	0.0766805i
$726$		0			0
$727$	33.7101			1.25024			0.625119	−	0.780530i	$-0.285051\pi$
$727$	33.7101			1.25024			0.625119	+	0.780530i	$0.285051\pi$
$728$		−	3.68726i		−	0.136659i
$729$		0			0
$730$	67.8585			2.51156
$731$	−19.6978			−0.728549
$732$		0			0
$733$	−4.78826			−0.176858			−0.0884292	−	0.996082i	$-0.528185\pi$
$733$	−4.78826			−0.176858			−0.0884292	+	0.996082i	$0.528185\pi$
$734$		−	61.4850i		−	2.26945i
$735$		0			0
$736$	−20.5212			−0.756421
$737$		−	41.4932i		−	1.52842i
$738$		0			0
$739$		−	37.2975i		−	1.37201i	−0.727596	−	0.686006i	$-0.759362\pi$
$739$		−	37.2975i		−	1.37201i	0.727596	−	0.686006i	$-0.240638\pi$
$740$	−32.8366			−1.20710
$741$		0			0
$742$			60.2161i			2.21060i
$743$		−	32.4149i		−	1.18919i	−0.804026	−	0.594594i	$-0.797313\pi$
$743$		−	32.4149i		−	1.18919i	0.804026	−	0.594594i	$-0.202687\pi$
$744$		0			0
$745$		−	9.08274i		−	0.332766i
$746$	28.3487			1.03792
$747$		0			0
$748$			13.2608i			0.484862i
$749$			14.1587i			0.517346i
$750$		0			0
$751$		−	15.8850i		−	0.579652i	−0.957079	−	0.289826i	$-0.906403\pi$
$751$		−	15.8850i		−	0.579652i	0.957079	−	0.289826i	$-0.0935975\pi$
$752$	−31.7971			−1.15952
$753$		0			0
$754$	5.03765			0.183460
$755$	43.6024			1.58685
$756$		0			0
$757$	45.5239			1.65460			0.827298	−	0.561764i	$-0.189877\pi$
$757$	45.5239			1.65460			0.827298	+	0.561764i	$0.189877\pi$
$758$	−2.96434			−0.107670
$759$		0			0
$760$			3.90709i			0.141725i
$761$			45.9501i			1.66569i	0.553506	+	0.832845i	$0.313290\pi$
$761$			45.9501i			1.66569i	−0.553506	+	0.832845i	$0.686710\pi$
$762$		0			0
$763$			47.7140i			1.72736i
$764$	−13.9816			−0.505835
$765$		0			0
$766$		−	3.11074i		−	0.112396i
$767$	9.28350	+	0.455990i	0.335208	+	0.0164648i
$768$		0			0
$769$		−	41.6846i		−	1.50318i	−0.659628	−	0.751592i	$-0.729286\pi$
$769$		−	41.6846i		−	1.50318i	0.659628	−	0.751592i	$-0.270714\pi$
$770$			60.5675i			2.18270i
$771$		0			0
$772$	63.9345			2.30105
$773$	5.78737			0.208157			0.104079	−	0.994569i	$-0.466811\pi$
$773$	5.78737			0.208157			0.104079	+	0.994569i	$0.466811\pi$
$774$		0			0
$775$			6.01591i			0.216098i
$776$		−	11.4758i		−	0.411956i
$777$		0			0
$778$			38.6444i			1.38547i
$779$			22.4117i			0.802981i
$780$		0			0
$781$		−	2.97651i		−	0.106508i
$782$			10.2207i			0.365492i
$783$		0			0
$784$	−29.6324			−1.05830
$785$	−19.5151			−0.696526
$786$		0			0
$787$	−50.0820			−1.78523			−0.892616	−	0.450819i	$-0.851132\pi$
$787$	−50.0820			−1.78523			−0.892616	+	0.450819i	$0.851132\pi$
$788$		−	14.5040i		−	0.516685i
$789$		0			0
$790$		−	55.8697i		−	1.98775i
$791$	−16.7431			−0.595316
$792$		0			0
$793$	0.889430			0.0315846
$794$			9.49339i			0.336908i
$795$		0			0
$796$	−5.91273			−0.209571
$797$	−23.7485			−0.841217			−0.420608	−	0.907242i	$-0.638183\pi$
$797$	−23.7485			−0.841217			−0.420608	+	0.907242i	$0.638183\pi$
$798$		0			0
$799$			19.4595i			0.688429i
$800$	−8.36917			−0.295895
$801$		0			0
$802$	44.2040			1.56090
$803$		−	38.5553i		−	1.36059i
$804$		0			0
$805$			25.2697i			0.890639i
$806$	−14.6783			−0.517020
$807$		0			0
$808$	12.7694			0.449225
$809$	−30.5126			−1.07277			−0.536383	−	0.843975i	$-0.680210\pi$
$809$	−30.5126			−1.07277			−0.536383	+	0.843975i	$0.680210\pi$
$810$		0			0
$811$			23.9284i			0.840240i	0.907469	+	0.420120i	$0.138012\pi$
$811$			23.9284i			0.840240i	−0.907469	+	0.420120i	$0.861988\pi$
$812$			19.0612i			0.668916i
$813$		0			0
$814$			34.4660i			1.20803i
$815$		−	25.6437i		−	0.898261i
$816$		0			0
$817$			21.6038i			0.755821i
$818$		−	63.5769i		−	2.22291i
$819$		0			0
$820$	61.4705			2.14664
$821$	37.8171			1.31983			0.659913	−	0.751342i	$-0.270593\pi$
$821$	37.8171			1.31983			0.659913	+	0.751342i	$0.270593\pi$
$822$		0			0
$823$		−	21.3721i		−	0.744984i	−0.928035	−	0.372492i	$-0.878503\pi$
$823$		−	21.3721i		−	0.744984i	0.928035	−	0.372492i	$-0.121497\pi$
$824$			1.19969i			0.0417933i
$825$		0			0
$826$	−3.18735	+	64.8913i	−0.110902	+	2.25786i
$827$			24.0914i			0.837741i	0.908046	+	0.418871i	$0.137574\pi$
$827$			24.0914i			0.837741i	−0.908046	+	0.418871i	$0.862426\pi$
$828$		0			0
$829$	−5.57074			−0.193480			−0.0967399	−	0.995310i	$-0.530842\pi$
$829$	−5.57074			−0.193480			−0.0967399	+	0.995310i	$0.530842\pi$
$830$			35.2916i			1.22499i
$831$		0			0
$832$		−	12.7973i		−	0.443667i
$833$			18.1348i			0.628332i
$834$		0			0
$835$	5.70086			0.197286
$836$	14.5439			0.503012
$837$		0			0
$838$	−74.3821			−2.56949
$839$	−9.60739			−0.331684			−0.165842	−	0.986152i	$-0.553034\pi$
$839$	−9.60739			−0.331684			−0.165842	+	0.986152i	$0.553034\pi$
$840$		0			0
$841$	25.0251			0.862935
$842$		−	36.1570i		−	1.24605i
$843$		0			0
$844$			49.7169i			1.71133i
$845$		−	28.3404i		−	0.974939i
$846$		0			0
$847$	−10.1447			−0.348577
$848$			22.4233i			0.770018i
$849$		0			0
$850$			4.16832i			0.142972i
$851$			14.3797i			0.492931i
$852$		0			0
$853$	33.5799			1.14975			0.574876	−	0.818240i	$-0.305050\pi$
$853$	33.5799			1.14975			0.574876	+	0.818240i	$0.305050\pi$
$854$			6.21708i			0.212744i
$855$		0			0
$856$		−	2.62943i		−	0.0898720i
$857$	−49.8817			−1.70393			−0.851963	−	0.523601i	$-0.824588\pi$
$857$	−49.8817			−1.70393			−0.851963	+	0.523601i	$0.824588\pi$
$858$		0			0
$859$		−	14.5742i		−	0.497266i	−0.968598	−	0.248633i	$-0.920019\pi$
$859$		−	14.5742i		−	0.497266i	0.968598	−	0.248633i	$-0.0799812\pi$
$860$	59.2547			2.02057
$861$		0			0
$862$	18.0978			0.616412
$863$	−31.1338			−1.05981			−0.529904	−	0.848058i	$-0.677772\pi$
$863$	−31.1338			−1.05981			−0.529904	+	0.848058i	$0.677772\pi$
$864$		0			0
$865$		−	27.6691i		−	0.940776i
$866$	−15.8617			−0.539003
$867$		0			0
$868$		−	55.5389i		−	1.88511i
$869$	−31.7436			−1.07683
$870$		0			0
$871$	17.2262			0.583689
$872$		−	8.86104i		−	0.300073i
$873$		0			0
$874$	11.2097			0.379174
$875$		−	39.4517i		−	1.33371i
$876$		0			0
$877$	35.0300			1.18288			0.591440	−	0.806349i	$-0.298560\pi$
$877$	35.0300			1.18288			0.591440	+	0.806349i	$0.298560\pi$
$878$	60.7031			2.04863
$879$		0			0
$880$			22.5541i			0.760299i
$881$	−11.5847			−0.390298			−0.195149	−	0.980774i	$-0.562519\pi$
$881$	−11.5847			−0.390298			−0.195149	+	0.980774i	$0.562519\pi$
$882$		0			0
$883$	−24.1764			−0.813600			−0.406800	−	0.913517i	$-0.633355\pi$
$883$	−24.1764			−0.813600			−0.406800	+	0.913517i	$0.633355\pi$
$884$	−5.50532			−0.185164
$885$		0			0
$886$	−0.485333			−0.0163051
$887$	−15.5103			−0.520784			−0.260392	−	0.965503i	$-0.583852\pi$
$887$	−15.5103			−0.520784			−0.260392	+	0.965503i	$0.583852\pi$
$888$		0			0
$889$	−43.5696			−1.46128
$890$			42.3061i			1.41810i
$891$		0			0
$892$	−61.4532			−2.05760
$893$	21.3425			0.714199
$894$		0			0
$895$			61.3332i			2.05014i
$896$	23.9818			0.801175
$897$		0			0
$898$		−	22.0820i		−	0.736886i
$899$	11.5817			0.386272
$900$		0			0
$901$	13.7228			0.457173
$902$		−	64.5207i		−	2.14830i
$903$		0			0
$904$	3.10939			0.103417
$905$			16.8235i			0.559233i
$906$		0			0
$907$	−31.3281			−1.04023			−0.520116	−	0.854096i	$-0.674111\pi$
$907$	−31.3281			−1.04023			−0.520116	+	0.854096i	$0.674111\pi$
$908$	37.0129			1.22831
$909$		0			0
$910$	−25.1451			−0.833551
$911$		−	36.3805i		−	1.20534i	−0.797990	−	0.602671i	$-0.794103\pi$
$911$		−	36.3805i		−	1.20534i	0.797990	−	0.602671i	$-0.205897\pi$
$912$		0			0
$913$	20.0517			0.663613
$914$			59.9095i			1.98163i
$915$		0			0
$916$			2.53017i			0.0835991i
$917$	−64.3132			−2.12381
$918$		0			0
$919$		−	12.3963i		−	0.408917i	−0.978875	−	0.204459i	$-0.934457\pi$
$919$		−	12.3963i		−	0.408917i	0.978875	−	0.204459i	$-0.0655434\pi$
$920$		−	4.69287i		−	0.154719i
$921$		0			0
$922$		−	48.8123i		−	1.60755i
$923$	1.23572			0.0406743
$924$		0			0
$925$			5.86449i			0.192823i
$926$			8.53434i			0.280456i
$927$		0			0
$928$			16.1122i			0.528908i
$929$	6.54232			0.214646			0.107323	−	0.994224i	$-0.465772\pi$
$929$	6.54232			0.214646			0.107323	+	0.994224i	$0.465772\pi$
$930$		0			0
$931$	19.8895			0.651853
$932$	−1.63017			−0.0533980
$933$		0			0
$934$	−40.1170			−1.31267
$935$	13.8029			0.451403
$936$		0			0
$937$			34.3791i			1.12312i	0.827437	+	0.561559i	$0.189798\pi$
$937$			34.3791i			1.12312i	−0.827437	+	0.561559i	$0.810202\pi$
$938$			120.411i			3.93155i
$939$		0			0
$940$		−	58.5379i		−	1.90930i
$941$	−0.307719			−0.0100313			−0.00501567	−	0.999987i	$-0.501597\pi$
$941$	−0.307719			−0.0100313			−0.00501567	+	0.999987i	$0.501597\pi$
$942$		0			0
$943$		−	26.9190i		−	0.876603i
$944$	−1.18690	+	24.1642i	−0.0386305	+	0.786478i
$945$		0			0
$946$		−	62.1949i		−	2.02213i
$947$		−	45.9612i		−	1.49354i	−0.665084	−	0.746768i	$-0.731604\pi$
$947$		−	45.9612i		−	1.49354i	0.665084	−	0.746768i	$-0.268396\pi$
$948$		0			0
$949$	16.0065			0.519594
$950$	4.57166			0.148324
$951$		0			0
$952$		−	5.87366i		−	0.190367i
$953$		−	13.6688i		−	0.442777i	−0.975186	−	0.221388i	$-0.928941\pi$
$953$		−	13.6688i		−	0.442777i	0.975186	−	0.221388i	$-0.0710588\pi$
$954$		0			0
$955$			14.5531i			0.470928i
$956$		−	48.1132i		−	1.55609i
$957$		0			0
$958$		−	62.5051i		−	2.01945i
$959$		−	37.8191i		−	1.22124i
$960$		0			0
$961$	−2.74584			−0.0885754
$962$	−14.3088			−0.461335
$963$		0			0
$964$	−23.5155			−0.757385
$965$		−	66.5482i		−	2.14226i
$966$		0			0
$967$		−	37.2572i		−	1.19811i	−0.800708	−	0.599055i	$-0.795543\pi$
$967$		−	37.2572i		−	1.19811i	0.800708	−	0.599055i	$-0.204457\pi$
$968$	1.88399			0.0605538
$969$		0			0
$970$	−78.2584			−2.51273
$971$			16.1694i			0.518901i	0.965756	+	0.259451i	$0.0835415\pi$
$971$			16.1694i			0.518901i	−0.965756	+	0.259451i	$0.916458\pi$
$972$		0			0
$973$	18.3634			0.588703
$974$	47.8284			1.53252
$975$		0			0
$976$			2.31511i			0.0741050i
$977$	45.5053			1.45584			0.727922	−	0.685660i	$-0.240486\pi$
$977$	45.5053			1.45584			0.727922	+	0.685660i	$0.240486\pi$
$978$		0			0
$979$	24.0371			0.768230
$980$		−	54.5528i		−	1.74262i
$981$		0			0
$982$			25.9032i			0.826606i
$983$	−30.9022			−0.985626			−0.492813	−	0.870135i	$-0.664031\pi$
$983$	−30.9022			−0.985626			−0.492813	+	0.870135i	$0.664031\pi$
$984$		0			0
$985$	−15.0970			−0.481030
$986$	8.02478			0.255561
$987$		0			0
$988$			6.03803i			0.192095i
$989$		−	25.9486i		−	0.825119i
$990$		0			0
$991$			44.1846i			1.40357i	0.712389	+	0.701785i	$0.247613\pi$
$991$			44.1846i			1.40357i	−0.712389	+	0.701785i	$0.752387\pi$
$992$		−	46.9463i		−	1.49055i
$993$		0			0
$994$			8.63765i			0.273970i
$995$			6.15445i			0.195109i
$996$		0			0
$997$	56.3903			1.78590			0.892949	−	0.450158i	$-0.148632\pi$
$997$	56.3903			1.78590			0.892949	+	0.450158i	$0.148632\pi$
$998$	21.8275			0.690938
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.d.a.530.3	✓	20
3.2	odd	2	inner	531.2.d.a.530.18	yes	20
59.58	odd	2	inner	531.2.d.a.530.17	yes	20
177.176	even	2	inner	531.2.d.a.530.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.d.a.530.3	✓	20	1.1	even	1	trivial
531.2.d.a.530.4	yes	20	177.176	even	2	inner
531.2.d.a.530.17	yes	20	59.58	odd	2	inner
531.2.d.a.530.18	yes	20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.08812 q^{2} +2.36026 q^{4} -2.45675i q^{5} +4.05068 q^{7} -0.752258 q^{8} +5.12999i q^{10} +2.91471 q^{11} +1.21007i q^{13} -8.45832 q^{14} -3.14970 q^{16} +1.92759i q^{17} +2.11411 q^{19} -5.79855i q^{20} -6.08628 q^{22} -2.53929 q^{23} -1.03560 q^{25} -2.52677i q^{26} +9.56064 q^{28} +1.99371i q^{29} -5.80912i q^{31} +8.08148 q^{32} -4.02504i q^{34} -9.95149i q^{35} -5.66290i q^{37} -4.41451 q^{38} +1.84811i q^{40} +10.6010i q^{41} +10.2189i q^{43} +6.87947 q^{44} +5.30234 q^{46} +10.0953 q^{47} +9.40801 q^{49} +2.16245 q^{50} +2.85607i q^{52} -7.11916i q^{53} -7.16071i q^{55} -3.04716 q^{56} -4.16312i q^{58} +(0.376831 - 7.67190i) q^{59} -0.735026i q^{61} +12.1301i q^{62} -10.5757 q^{64} +2.97282 q^{65} -14.2358i q^{67} +4.54960i q^{68} +20.7799i q^{70} -1.02120i q^{71} -13.2278i q^{73} +11.8248i q^{74} +4.98983 q^{76} +11.8066 q^{77} -10.8908 q^{79} +7.73802i q^{80} -22.1362i q^{82} +6.87947 q^{83} +4.73559 q^{85} -21.3383i q^{86} -2.19262 q^{88} +8.24682 q^{89} +4.90159i q^{91} -5.99336 q^{92} -21.0802 q^{94} -5.19382i q^{95} +15.2551i q^{97} -19.6451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) \) 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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