LMFDB - Embedded newform 531.3.c.c.235.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

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

Embedding invariants

Embedding label			235.8
Root			$-1.39995i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39995i q^{2} +2.04015 q^{4} -0.273484 q^{5} -10.8938 q^{7} -8.45589i q^{8} +O(q^{10})$	\(q-1.39995i q^{2} +2.04015 q^{4} -0.273484 q^{5} -10.8938 q^{7} -8.45589i q^{8} +0.382863i q^{10} -15.0527i q^{11} +11.1132i q^{13} +15.2508i q^{14} -3.67721 q^{16} +3.40920 q^{17} -15.7121 q^{19} -0.557948 q^{20} -21.0729 q^{22} +15.2765i q^{23} -24.9252 q^{25} +15.5578 q^{26} -22.2251 q^{28} -22.5689 q^{29} +59.5188i q^{31} -28.6757i q^{32} -4.77271i q^{34} +2.97929 q^{35} -42.4500i q^{37} +21.9961i q^{38} +2.31255i q^{40} -78.4822 q^{41} -78.3986i q^{43} -30.7096i q^{44} +21.3862 q^{46} -35.3465i q^{47} +69.6759 q^{49} +34.8940i q^{50} +22.6725i q^{52} -37.2779 q^{53} +4.11666i q^{55} +92.1171i q^{56} +31.5952i q^{58} +(-52.6472 - 26.6321i) q^{59} +44.2464i q^{61} +83.3232 q^{62} -54.8532 q^{64} -3.03927i q^{65} -17.3224i q^{67} +6.95528 q^{68} -4.17086i q^{70} +59.0750 q^{71} +85.3122i q^{73} -59.4278 q^{74} -32.0550 q^{76} +163.981i q^{77} +62.2163 q^{79} +1.00566 q^{80} +109.871i q^{82} -98.9653i q^{83} -0.932363 q^{85} -109.754 q^{86} -127.284 q^{88} +35.9346i q^{89} -121.065i q^{91} +31.1663i q^{92} -49.4833 q^{94} +4.29702 q^{95} +11.0179i q^{97} -97.5426i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.39995i		−	0.699974i	−0.936755	−	0.349987i	$-0.886186\pi$
$2$		−	1.39995i		−	0.699974i	0.936755	−	0.349987i	$-0.113814\pi$
$3$		0			0
$3$		0			0
$4$	2.04015			0.510037
$5$	−0.273484			−0.0546968			−0.0273484	−	0.999626i	$-0.508706\pi$
$5$	−0.273484			−0.0546968			−0.0273484	+	0.999626i	$0.508706\pi$
$6$		0			0
$7$	−10.8938			−1.55626			−0.778132	−	0.628101i	$-0.783832\pi$
$7$	−10.8938			−1.55626			−0.778132	+	0.628101i	$0.783832\pi$
$8$		−	8.45589i		−	1.05699i
$9$		0			0
$10$			0.382863i			0.0382863i
$11$		−	15.0527i		−	1.36842i	−0.729284	−	0.684212i	$-0.760146\pi$
$11$		−	15.0527i		−	1.36842i	0.729284	−	0.684212i	$-0.239854\pi$
$12$		0			0
$13$			11.1132i			0.854858i	0.904049	+	0.427429i	$0.140581\pi$
$13$			11.1132i			0.854858i	−0.904049	+	0.427429i	$0.859419\pi$
$14$			15.2508i			1.08934i
$15$		0			0
$16$	−3.67721			−0.229825
$17$	3.40920			0.200541			0.100271	−	0.994960i	$-0.468029\pi$
$17$	3.40920			0.200541			0.100271	+	0.994960i	$0.468029\pi$
$18$		0			0
$19$	−15.7121			−0.826954			−0.413477	−	0.910515i	$-0.635686\pi$
$19$	−15.7121			−0.826954			−0.413477	+	0.910515i	$0.635686\pi$
$20$	−0.557948			−0.0278974
$21$		0			0
$22$	−21.0729			−0.957860
$23$			15.2765i			0.664194i	0.943245	+	0.332097i	$0.107756\pi$
$23$			15.2765i			0.664194i	−0.943245	+	0.332097i	$0.892244\pi$
$24$		0			0
$25$	−24.9252			−0.997008
$26$	15.5578			0.598378
$27$		0			0
$28$	−22.2251			−0.793752
$29$	−22.5689			−0.778237			−0.389118	−	0.921188i	$-0.627220\pi$
$29$	−22.5689			−0.778237			−0.389118	+	0.921188i	$0.627220\pi$
$30$		0			0
$31$			59.5188i			1.91996i	0.280067	+	0.959981i	$0.409643\pi$
$31$			59.5188i			1.91996i	−0.280067	+	0.959981i	$0.590357\pi$
$32$		−	28.6757i		−	0.896114i
$33$		0			0
$34$		−	4.77271i		−	0.140374i
$35$	2.97929			0.0851227
$36$		0			0
$37$		−	42.4500i		−	1.14730i	−0.819101	−	0.573649i	$-0.805527\pi$
$37$		−	42.4500i		−	1.14730i	0.819101	−	0.573649i	$-0.194473\pi$
$38$			21.9961i			0.578846i
$39$		0			0
$40$			2.31255i			0.0578138i
$41$	−78.4822			−1.91420			−0.957101	−	0.289756i	$-0.906426\pi$
$41$	−78.4822			−1.91420			−0.957101	+	0.289756i	$0.906426\pi$
$42$		0			0
$43$		−	78.3986i		−	1.82322i	−0.411052	−	0.911612i	$-0.634838\pi$
$43$		−	78.3986i		−	1.82322i	0.411052	−	0.911612i	$-0.365162\pi$
$44$		−	30.7096i		−	0.697946i
$45$		0			0
$46$	21.3862			0.464918
$47$		−	35.3465i		−	0.752054i	−0.926609	−	0.376027i	$-0.877290\pi$
$47$		−	35.3465i		−	0.752054i	0.926609	−	0.376027i	$-0.122710\pi$
$48$		0			0
$49$	69.6759			1.42196
$50$			34.8940i			0.697879i
$51$		0			0
$52$			22.6725i			0.436009i
$53$	−37.2779			−0.703357			−0.351678	−	0.936121i	$-0.614389\pi$
$53$	−37.2779			−0.703357			−0.351678	+	0.936121i	$0.614389\pi$
$54$		0			0
$55$			4.11666i			0.0748484i
$56$			92.1171i			1.64495i
$57$		0			0
$58$			31.5952i			0.544745i
$59$	−52.6472	−	26.6321i	−0.892326	−	0.451392i
$59$	−52.6472	−	26.6321i	−0.892326	−	0.451392i
$60$		0			0
$61$			44.2464i			0.725350i	0.931916	+	0.362675i	$0.118136\pi$
$61$			44.2464i			0.725350i	−0.931916	+	0.362675i	$0.881864\pi$
$62$	83.3232			1.34392
$63$		0			0
$64$	−54.8532			−0.857082
$65$		−	3.03927i		−	0.0467580i
$66$		0			0
$67$		−	17.3224i		−	0.258544i	−0.991609	−	0.129272i	$-0.958736\pi$
$67$		−	17.3224i		−	0.258544i	0.991609	−	0.129272i	$-0.0412640\pi$
$68$	6.95528			0.102284
$69$		0			0
$70$		−	4.17086i		−	0.0595836i
$71$	59.0750			0.832043			0.416021	−	0.909355i	$-0.363424\pi$
$71$	59.0750			0.832043			0.416021	+	0.909355i	$0.363424\pi$
$72$		0			0
$73$			85.3122i			1.16866i	0.811516	+	0.584330i	$0.198643\pi$
$73$			85.3122i			1.16866i	−0.811516	+	0.584330i	$0.801357\pi$
$74$	−59.4278			−0.803079
$75$		0			0
$76$	−32.0550			−0.421777
$77$			163.981i			2.12963i
$78$		0			0
$79$	62.2163			0.787548			0.393774	−	0.919207i	$-0.371169\pi$
$79$	62.2163			0.787548			0.393774	+	0.919207i	$0.371169\pi$
$80$	1.00566			0.0125707
$81$		0			0
$82$			109.871i			1.33989i
$83$		−	98.9653i		−	1.19235i	−0.802853	−	0.596177i	$-0.796686\pi$
$83$		−	98.9653i		−	1.19235i	0.802853	−	0.596177i	$-0.203314\pi$
$84$		0			0
$85$	−0.932363			−0.0109690
$86$	−109.754			−1.27621
$87$		0			0
$88$	−127.284			−1.44640
$89$			35.9346i			0.403759i	0.979410	+	0.201880i	$0.0647050\pi$
$89$			35.9346i			0.403759i	−0.979410	+	0.201880i	$0.935295\pi$
$90$		0			0
$91$		−	121.065i		−	1.33038i
$92$			31.1663i			0.338764i
$93$		0			0
$94$	−49.4833			−0.526418
$95$	4.29702			0.0452318
$96$		0			0
$97$			11.0179i			0.113587i	0.998386	+	0.0567935i	$0.0180877\pi$
$97$			11.0179i			0.113587i	−0.998386	+	0.0567935i	$0.981912\pi$
$98$		−	97.5426i		−	0.995332i
$99$		0			0
$100$	−50.8511			−0.508511
$101$			110.774i			1.09677i	0.836227	+	0.548384i	$0.184757\pi$
$101$			110.774i			1.09677i	−0.836227	+	0.548384i	$0.815243\pi$
$102$		0			0
$103$		−	43.0668i		−	0.418125i	−0.977902	−	0.209062i	$-0.932959\pi$
$103$		−	43.0668i		−	0.418125i	0.977902	−	0.209062i	$-0.0670412\pi$
$104$	93.9716			0.903573
$105$		0			0
$106$			52.1871i			0.492331i
$107$	146.405			1.36827			0.684137	−	0.729353i	$-0.260179\pi$
$107$	146.405			1.36827			0.684137	+	0.729353i	$0.260179\pi$
$108$		0			0
$109$		−	141.124i		−	1.29472i	−0.762185	−	0.647359i	$-0.775873\pi$
$109$		−	141.124i		−	1.29472i	0.762185	−	0.647359i	$-0.224127\pi$
$110$	5.76311			0.0523919
$111$		0			0
$112$	40.0589			0.357669
$113$		−	21.1887i		−	0.187511i	−0.995595	−	0.0937555i	$-0.970113\pi$
$113$		−	21.1887i		−	0.187511i	0.995595	−	0.0937555i	$-0.0298872\pi$
$114$		0			0
$115$		−	4.17787i		−	0.0363293i
$116$	−46.0438			−0.396929
$117$		0			0
$118$	−37.2836	+	73.7033i	−0.315963	+	0.624604i
$119$	−37.1393			−0.312095
$120$		0			0
$121$	−105.582			−0.872582
$122$	61.9426			0.507726
$123$		0			0
$124$			121.427i			0.979251i
$125$	13.6538			0.109230
$126$		0			0
$127$	215.084			1.69358			0.846789	−	0.531929i	$-0.178533\pi$
$127$	215.084			1.69358			0.846789	+	0.531929i	$0.178533\pi$
$128$		−	37.9110i		−	0.296180i
$129$		0			0
$130$	−4.25482			−0.0327294
$131$		−	79.8202i		−	0.609314i	−0.952462	−	0.304657i	$-0.901458\pi$
$131$		−	79.8202i		−	0.609314i	0.952462	−	0.304657i	$-0.0985419\pi$
$132$		0			0
$133$	171.165			1.28696
$134$	−24.2505			−0.180974
$135$		0			0
$136$		−	28.8278i		−	0.211969i
$137$	138.442			1.01053			0.505265	−	0.862965i	$-0.331395\pi$
$137$	138.442			1.01053			0.505265	+	0.862965i	$0.331395\pi$
$138$		0			0
$139$	59.8615			0.430658			0.215329	−	0.976542i	$-0.430918\pi$
$139$	59.8615			0.430658			0.215329	+	0.976542i	$0.430918\pi$
$140$	6.07820			0.0434157
$141$		0			0
$142$		−	82.7019i		−	0.582408i
$143$	167.282			1.16981
$144$		0			0
$145$	6.17223			0.0425671
$146$	119.433			0.818031
$147$		0			0
$148$		−	86.6044i		−	0.585165i
$149$		−	218.369i		−	1.46556i	−0.680464	−	0.732782i	$-0.738222\pi$
$149$		−	218.369i		−	1.46556i	0.680464	−	0.732782i	$-0.261778\pi$
$150$		0			0
$151$		−	26.4104i		−	0.174904i	−0.996169	−	0.0874518i	$-0.972128\pi$
$151$		−	26.4104i		−	0.174904i	0.996169	−	0.0874518i	$-0.0278724\pi$
$152$			132.860i			0.874078i
$153$		0			0
$154$	229.565			1.49068
$155$		−	16.2774i		−	0.105016i
$156$		0			0
$157$			18.4659i			0.117617i	0.998269	+	0.0588087i	$0.0187302\pi$
$157$			18.4659i			0.117617i	−0.998269	+	0.0588087i	$0.981270\pi$
$158$		−	87.0995i		−	0.551263i
$159$		0			0
$160$			7.84234i			0.0490146i
$161$		−	166.419i		−	1.03366i
$162$		0			0
$163$	−247.452			−1.51811			−0.759055	−	0.651026i	$-0.774339\pi$
$163$	−247.452			−1.51811			−0.759055	+	0.651026i	$0.774339\pi$
$164$	−160.115			−0.976313
$165$		0			0
$166$	−138.546			−0.834616
$167$	−63.0046			−0.377273			−0.188637	−	0.982047i	$-0.560407\pi$
$167$	−63.0046			−0.377273			−0.188637	+	0.982047i	$0.560407\pi$
$168$		0			0
$169$	45.4978			0.269218
$170$			1.30526i			0.00767800i
$171$		0			0
$172$		−	159.945i		−	0.929912i
$173$		−	163.258i		−	0.943690i	−0.881681	−	0.471845i	$-0.843588\pi$
$173$		−	163.258i		−	0.943690i	0.881681	−	0.471845i	$-0.156412\pi$
$174$		0			0
$175$	271.531			1.55161
$176$			55.3517i			0.314498i
$177$		0			0
$178$	50.3065			0.282621
$179$		−	80.5355i		−	0.449919i	−0.974368	−	0.224960i	$-0.927775\pi$
$179$		−	80.5355i		−	0.449919i	0.974368	−	0.224960i	$-0.0722250\pi$
$180$		0			0
$181$	−130.162			−0.719125			−0.359563	−	0.933121i	$-0.617074\pi$
$181$	−130.162			−0.719125			−0.359563	+	0.933121i	$0.617074\pi$
$182$	−169.485			−0.931234
$183$		0			0
$184$	129.176			0.702044
$185$			11.6094i			0.0627536i
$186$		0			0
$187$		−	51.3176i		−	0.274426i
$188$		−	72.1121i		−	0.383575i
$189$		0			0
$190$		−	6.01560i		−	0.0316610i
$191$			206.275i			1.07997i	0.841674	+	0.539987i	$0.181571\pi$
$191$			206.275i			1.07997i	−0.841674	+	0.539987i	$0.818429\pi$
$192$		0			0
$193$	174.658			0.904964			0.452482	−	0.891773i	$-0.350539\pi$
$193$	174.658			0.904964			0.452482	+	0.891773i	$0.350539\pi$
$194$	15.4245			0.0795079
$195$		0			0
$196$	142.149			0.725250
$197$	−262.732			−1.33367			−0.666833	−	0.745207i	$-0.732351\pi$
$197$	−262.732			−1.33367			−0.666833	+	0.745207i	$0.732351\pi$
$198$		0			0
$199$	−36.7705			−0.184776			−0.0923882	−	0.995723i	$-0.529450\pi$
$199$	−36.7705			−0.184776			−0.0923882	+	0.995723i	$0.529450\pi$
$200$			210.765i			1.05382i
$201$		0			0
$202$	155.077			0.767709
$203$	245.862			1.21114
$204$		0			0
$205$	21.4637			0.104701
$206$	−60.2913			−0.292676
$207$		0			0
$208$		−	40.8654i		−	0.196468i
$209$			236.509i			1.13162i
$210$		0			0
$211$			332.832i			1.57740i	0.614775	+	0.788702i	$0.289247\pi$
$211$			332.832i			1.57740i	−0.614775	+	0.788702i	$0.710753\pi$
$212$	−76.0525			−0.358738
$213$		0			0
$214$		−	204.960i		−	0.957756i
$215$			21.4408i			0.0997246i
$216$		0			0
$217$		−	648.389i		−	2.98797i
$218$	−197.567			−0.906269
$219$		0			0
$220$			8.39860i			0.0381755i
$221$			37.8870i			0.171434i
$222$		0			0
$223$	−68.0441			−0.305131			−0.152565	−	0.988293i	$-0.548753\pi$
$223$	−68.0441			−0.305131			−0.152565	+	0.988293i	$0.548753\pi$
$224$			312.388i			1.39459i
$225$		0			0
$226$	−29.6631			−0.131253
$227$		−	297.497i		−	1.31056i	−0.755386	−	0.655280i	$-0.772551\pi$
$227$		−	297.497i		−	1.31056i	0.755386	−	0.655280i	$-0.227449\pi$
$228$		0			0
$229$		−	27.5954i		−	0.120504i	−0.998183	−	0.0602519i	$-0.980810\pi$
$229$		−	27.5954i		−	0.120504i	0.998183	−	0.0602519i	$-0.0191904\pi$
$230$	−5.84880			−0.0254296
$231$		0			0
$232$			190.840i			0.822585i
$233$		−	359.245i		−	1.54182i	−0.636943	−	0.770911i	$-0.719801\pi$
$233$		−	359.245i		−	1.54182i	0.636943	−	0.770911i	$-0.280199\pi$
$234$		0			0
$235$			9.66672i			0.0411350i
$236$	−107.408	−	54.3335i	−0.455119	−	0.230227i
$237$		0			0
$238$			51.9931i			0.218458i
$239$	265.127			1.10932			0.554659	−	0.832078i	$-0.312849\pi$
$239$	265.127			1.10932			0.554659	+	0.832078i	$0.312849\pi$
$240$		0			0
$241$	−58.8858			−0.244340			−0.122170	−	0.992509i	$-0.538985\pi$
$241$	−58.8858			−0.244340			−0.122170	+	0.992509i	$0.538985\pi$
$242$			147.810i			0.610784i
$243$		0			0
$244$			90.2691i			0.369955i
$245$	−19.0553			−0.0777765
$246$		0			0
$247$		−	174.611i		−	0.706928i
$248$	503.284			2.02937
$249$		0			0
$250$		−	19.1145i		−	0.0764581i
$251$	−1.34813			−0.00537104			−0.00268552	−	0.999996i	$-0.500855\pi$
$251$	−1.34813			−0.00537104			−0.00268552	+	0.999996i	$0.500855\pi$
$252$		0			0
$253$	229.951			0.908899
$254$		−	301.107i		−	1.18546i
$255$		0			0
$256$	−272.486			−1.06440
$257$	−301.120			−1.17167			−0.585837	−	0.810429i	$-0.699234\pi$
$257$	−301.120			−1.17167			−0.585837	+	0.810429i	$0.699234\pi$
$258$		0			0
$259$			462.444i			1.78550i
$260$		−	6.20056i		−	0.0238483i
$261$		0			0
$262$	−111.744			−0.426504
$263$	−82.6739			−0.314350			−0.157175	−	0.987571i	$-0.550239\pi$
$263$	−82.6739			−0.314350			−0.157175	+	0.987571i	$0.550239\pi$
$264$		0			0
$265$	10.1949			0.0384714
$266$		−	239.623i		−	0.900837i
$267$		0			0
$268$		−	35.3403i		−	0.131867i
$269$			208.155i			0.773810i	0.922120	+	0.386905i	$0.126456\pi$
$269$			208.155i			0.773810i	−0.922120	+	0.386905i	$0.873544\pi$
$270$		0			0
$271$	38.7081			0.142834			0.0714171	−	0.997447i	$-0.477248\pi$
$271$	38.7081			0.142834			0.0714171	+	0.997447i	$0.477248\pi$
$272$	−12.5363			−0.0460895
$273$		0			0
$274$		−	193.812i		−	0.707344i
$275$			375.191i			1.36433i
$276$		0			0
$277$	209.375			0.755866			0.377933	−	0.925833i	$-0.376635\pi$
$277$	209.375			0.755866			0.377933	+	0.925833i	$0.376635\pi$
$278$		−	83.8030i		−	0.301450i
$279$		0			0
$280$		−	25.1926i		−	0.0899735i
$281$	−307.082			−1.09282			−0.546409	−	0.837519i	$-0.684005\pi$
$281$	−307.082			−1.09282			−0.546409	+	0.837519i	$0.684005\pi$
$282$		0			0
$283$		−	229.159i		−	0.809749i	−0.914372	−	0.404875i	$-0.867315\pi$
$283$		−	229.159i		−	0.809749i	0.914372	−	0.404875i	$-0.132685\pi$
$284$	120.522			0.424373
$285$		0			0
$286$		−	234.187i		−	0.818834i
$287$	854.974			2.97900
$288$		0			0
$289$	−277.377			−0.959783
$290$		−	8.64079i		−	0.0297958i
$291$		0			0
$292$			174.049i			0.596060i
$293$	−56.5578			−0.193030			−0.0965150	−	0.995332i	$-0.530770\pi$
$293$	−56.5578			−0.193030			−0.0965150	+	0.995332i	$0.530770\pi$
$294$		0			0
$295$	14.3982	+	7.28347i	0.0488074	+	0.0246897i
$296$	−358.953			−1.21268
$297$		0			0
$298$	−305.705			−1.02586
$299$	−169.770			−0.567792
$300$		0			0
$301$			854.063i			2.83742i
$302$	−36.9732			−0.122428
$303$		0			0
$304$	57.7767			0.190055
$305$		−	12.1007i		−	0.0396744i
$306$		0			0
$307$	−38.1615			−0.124305			−0.0621523	−	0.998067i	$-0.519796\pi$
$307$	−38.1615			−0.124305			−0.0621523	+	0.998067i	$0.519796\pi$
$308$			334.546i			1.08619i
$309$		0			0
$310$	−22.7876			−0.0735083
$311$	237.381			0.763282			0.381641	−	0.924311i	$-0.375359\pi$
$311$	237.381			0.763282			0.381641	+	0.924311i	$0.375359\pi$
$312$		0			0
$313$			468.363i			1.49637i	0.663491	+	0.748184i	$0.269074\pi$
$313$			468.363i			1.49637i	−0.663491	+	0.748184i	$0.730926\pi$
$314$	25.8513			0.0823291
$315$		0			0
$316$	126.930			0.401679
$317$	97.7779			0.308447			0.154224	−	0.988036i	$-0.450712\pi$
$317$	97.7779			0.308447			0.154224	+	0.988036i	$0.450712\pi$
$318$		0			0
$319$			339.721i			1.06496i
$320$	15.0015			0.0468797
$321$		0			0
$322$	−232.978			−0.723536
$323$	−53.5658			−0.165838
$324$		0			0
$325$		−	276.998i		−	0.852300i
$326$			346.420i			1.06264i
$327$		0			0
$328$			663.637i			2.02328i
$329$			385.060i			1.17039i
$330$		0			0
$331$	−248.132			−0.749644			−0.374822	−	0.927097i	$-0.622296\pi$
$331$	−248.132			−0.749644			−0.374822	+	0.927097i	$0.622296\pi$
$332$		−	201.904i		−	0.608144i
$333$		0			0
$334$			88.2032i			0.264081i
$335$			4.73741i			0.0141415i
$336$		0			0
$337$		−	84.1105i		−	0.249586i	−0.992183	−	0.124793i	$-0.960173\pi$
$337$		−	84.1105i		−	0.249586i	0.992183	−	0.124793i	$-0.0398267\pi$
$338$		−	63.6945i		−	0.188445i
$339$		0			0
$340$	−1.90216			−0.00559459
$341$	895.916			2.62732
$342$		0			0
$343$	−225.240			−0.656676
$344$	−662.930			−1.92712
$345$		0			0
$346$	−228.553			−0.660558
$347$		−	521.687i		−	1.50342i	−0.659493	−	0.751711i	$-0.729229\pi$
$347$		−	521.687i		−	1.50342i	0.659493	−	0.751711i	$-0.270771\pi$
$348$		0			0
$349$		−	299.873i		−	0.859234i	−0.903011	−	0.429617i	$-0.858649\pi$
$349$		−	299.873i		−	0.859234i	0.903011	−	0.429617i	$-0.141351\pi$
$350$		−	380.130i		−	1.08608i
$351$		0			0
$352$	−431.645			−1.22626
$353$			633.470i			1.79453i	0.441491	+	0.897266i	$0.354450\pi$
$353$			633.470i			1.79453i	−0.441491	+	0.897266i	$0.645550\pi$
$354$		0			0
$355$	−16.1561			−0.0455101
$356$			73.3119i			0.205932i
$357$		0			0
$358$	−112.745			−0.314932
$359$	−456.186			−1.27071			−0.635356	−	0.772219i	$-0.719147\pi$
$359$	−456.186			−1.27071			−0.635356	+	0.772219i	$0.719147\pi$
$360$		0			0
$361$	−114.129			−0.316148
$362$			182.220i			0.503369i
$363$		0			0
$364$		−	246.990i		−	0.678545i
$365$		−	23.3315i		−	0.0639220i
$366$		0			0
$367$		−	505.395i		−	1.37710i	−0.725190	−	0.688549i	$-0.758248\pi$
$367$		−	505.395i		−	1.37710i	0.725190	−	0.688549i	$-0.241752\pi$
$368$		−	56.1747i		−	0.152649i
$369$		0			0
$370$	16.2526			0.0439259
$371$	406.100			1.09461
$372$		0			0
$373$	−262.292			−0.703195			−0.351598	−	0.936151i	$-0.614361\pi$
$373$	−262.292			−0.703195			−0.351598	+	0.936151i	$0.614361\pi$
$374$	−71.8419			−0.192091
$375$		0			0
$376$	−298.886			−0.794910
$377$		−	250.811i		−	0.665282i
$378$		0			0
$379$	532.499			1.40501			0.702505	−	0.711679i	$-0.252065\pi$
$379$	532.499			1.40501			0.702505	+	0.711679i	$0.252065\pi$
$380$	8.76655			0.0230699
$381$		0			0
$382$	288.774			0.755953
$383$	−286.541			−0.748150			−0.374075	−	0.927398i	$-0.622040\pi$
$383$	−286.541			−0.748150			−0.374075	+	0.927398i	$0.622040\pi$
$384$		0			0
$385$		−	44.8463i		−	0.116484i
$386$		−	244.512i		−	0.633451i
$387$		0			0
$388$			22.4782i			0.0579336i
$389$	−620.394			−1.59484			−0.797421	−	0.603423i	$-0.793803\pi$
$389$	−620.394			−1.59484			−0.797421	+	0.603423i	$0.793803\pi$
$390$		0			0
$391$			52.0806i			0.133198i
$392$		−	589.171i		−	1.50299i
$393$		0			0
$394$			367.811i			0.933531i
$395$	−17.0152			−0.0430764
$396$		0			0
$397$			434.013i			1.09323i	0.837383	+	0.546616i	$0.184084\pi$
$397$			434.013i			1.09323i	−0.837383	+	0.546616i	$0.815916\pi$
$398$			51.4767i			0.129339i
$399$		0			0
$400$	91.6551			0.229138
$401$			107.207i			0.267350i	0.991025	+	0.133675i	$0.0426778\pi$
$401$			107.207i			0.267350i	−0.991025	+	0.133675i	$0.957322\pi$
$402$		0			0
$403$	−661.442			−1.64129
$404$			225.994i			0.559392i
$405$		0			0
$406$		−	344.193i		−	0.847767i
$407$	−638.986			−1.56999
$408$		0			0
$409$		−	550.503i		−	1.34597i	−0.739654	−	0.672987i	$-0.765011\pi$
$409$		−	550.503i		−	1.34597i	0.739654	−	0.672987i	$-0.234989\pi$
$410$		−	30.0480i		−	0.0732878i
$411$		0			0
$412$		−	87.8627i		−	0.213259i
$413$	573.531	+	290.126i	1.38869	+	0.702485i
$414$		0			0
$415$			27.0655i			0.0652180i
$416$	318.677			0.766050
$417$		0			0
$418$	331.100			0.792106
$419$			274.431i			0.654967i	0.944857	+	0.327483i	$0.106201\pi$
$419$			274.431i			0.654967i	−0.944857	+	0.327483i	$0.893799\pi$
$420$		0			0
$421$		−	46.0928i		−	0.109484i	−0.998501	−	0.0547420i	$-0.982566\pi$
$421$		−	46.0928i		−	0.109484i	0.998501	−	0.0547420i	$-0.0174336\pi$
$422$	465.948			1.10414
$423$		0			0
$424$			315.218i			0.743438i
$425$	−84.9751			−0.199941
$426$		0			0
$427$		−	482.013i		−	1.12884i
$428$	298.689			0.697871
$429$		0			0
$430$	30.0160			0.0698046
$431$			256.182i			0.594389i	0.954817	+	0.297195i	$0.0960510\pi$
$431$			256.182i			0.594389i	−0.954817	+	0.297195i	$0.903949\pi$
$432$		0			0
$433$	76.7056			0.177149			0.0885746	−	0.996070i	$-0.471769\pi$
$433$	76.7056			0.177149			0.0885746	+	0.996070i	$0.471769\pi$
$434$	−907.710			−2.09150
$435$		0			0
$436$		−	287.914i		−	0.660354i
$437$		−	240.026i		−	0.549258i
$438$		0			0
$439$	622.390			1.41775			0.708873	−	0.705336i	$-0.249204\pi$
$439$	622.390			1.41775			0.708873	+	0.705336i	$0.249204\pi$
$440$	34.8100			0.0791137
$441$		0			0
$442$	53.0398			0.120000
$443$			682.698i			1.54108i	0.637392	+	0.770540i	$0.280013\pi$
$443$			682.698i			1.54108i	−0.637392	+	0.770540i	$0.719987\pi$
$444$		0			0
$445$		−	9.82754i		−	0.0220844i
$446$			95.2582i			0.213583i
$447$		0			0
$448$	597.563			1.33385
$449$	144.491			0.321806			0.160903	−	0.986970i	$-0.448559\pi$
$449$	144.491			0.321806			0.160903	+	0.986970i	$0.448559\pi$
$450$		0			0
$451$			1181.37i			2.61944i
$452$		−	43.2282i		−	0.0956375i
$453$		0			0
$454$	−416.480			−0.917358
$455$			33.1094i			0.0727678i
$456$		0			0
$457$			293.853i			0.643004i	0.946909	+	0.321502i	$0.104188\pi$
$457$			293.853i			0.643004i	−0.946909	+	0.321502i	$0.895812\pi$
$458$	−38.6321			−0.0843495
$459$		0			0
$460$		−	8.52348i		−	0.0185293i
$461$	−457.634			−0.992698			−0.496349	−	0.868123i	$-0.665326\pi$
$461$	−457.634			−0.992698			−0.496349	+	0.868123i	$0.665326\pi$
$462$		0			0
$463$		−	671.285i		−	1.44986i	−0.688822	−	0.724930i	$-0.741872\pi$
$463$		−	671.285i		−	1.44986i	0.688822	−	0.724930i	$-0.258128\pi$
$464$	82.9904			0.178859
$465$		0			0
$466$	−502.923			−1.07923
$467$		−	358.075i		−	0.766756i	−0.923592	−	0.383378i	$-0.874761\pi$
$467$		−	358.075i		−	0.766756i	0.923592	−	0.383378i	$-0.125239\pi$
$468$		0			0
$469$			188.708i			0.402362i
$470$	13.5329			0.0287934
$471$		0			0
$472$	−225.198	+	445.179i	−0.477115	+	0.943176i
$473$	−1180.11			−2.49494
$474$		0			0
$475$	391.628			0.824480
$476$	−75.7697			−0.159180
$477$		0			0
$478$		−	371.164i		−	0.776493i
$479$	23.3301			0.0487059			0.0243530	−	0.999703i	$-0.492247\pi$
$479$	23.3301			0.0487059			0.0243530	+	0.999703i	$0.492247\pi$
$480$		0			0
$481$	471.754			0.980777
$482$			82.4371i			0.171031i
$483$		0			0
$484$	−215.404			−0.445049
$485$		−	3.01323i		−	0.00621285i
$486$		0			0
$487$	718.517			1.47539			0.737697	−	0.675132i	$-0.235913\pi$
$487$	718.517			1.47539			0.737697	+	0.675132i	$0.235913\pi$
$488$	374.142			0.766685
$489$		0			0
$490$			26.6763i			0.0544415i
$491$	−406.760			−0.828431			−0.414215	−	0.910179i	$-0.635944\pi$
$491$	−406.760			−0.828431			−0.414215	+	0.910179i	$0.635944\pi$
$492$		0			0
$493$	−76.9419			−0.156069
$494$	−244.446			−0.494831
$495$		0			0
$496$		−	218.863i		−	0.441256i
$497$	−643.554			−1.29488
$498$		0			0
$499$	−828.694			−1.66071			−0.830355	−	0.557235i	$-0.811862\pi$
$499$	−828.694			−1.66071			−0.830355	+	0.557235i	$0.811862\pi$
$500$	27.8557			0.0557114
$501$		0			0
$502$			1.88731i			0.00375959i
$503$		−	491.016i		−	0.976175i	−0.872795	−	0.488088i	$-0.837695\pi$
$503$		−	491.016i		−	0.976175i	0.872795	−	0.488088i	$-0.162305\pi$
$504$		0			0
$505$		−	30.2948i		−	0.0599897i
$506$		−	321.920i		−	0.636205i
$507$		0			0
$508$	438.804			0.863787
$509$		−	299.875i		−	0.589146i	−0.955629	−	0.294573i	$-0.904823\pi$
$509$		−	299.875i		−	0.589146i	0.955629	−	0.294573i	$-0.0951774\pi$
$510$		0			0
$511$		−	929.378i		−	1.81874i
$512$			229.822i			0.448872i
$513$		0			0
$514$			421.553i			0.820141i
$515$			11.7781i			0.0228701i
$516$		0			0
$517$	−532.059			−1.02913
$518$	647.398			1.24980
$519$		0			0
$520$	−25.6997			−0.0494226
$521$	793.658			1.52334			0.761668	−	0.647967i	$-0.224381\pi$
$521$	793.658			1.52334			0.761668	+	0.647967i	$0.224381\pi$
$522$		0			0
$523$	−991.881			−1.89652			−0.948261	−	0.317493i	$-0.897159\pi$
$523$	−991.881			−1.89652			−0.948261	+	0.317493i	$0.897159\pi$
$524$		−	162.845i		−	0.310773i
$525$		0			0
$526$			115.739i			0.220036i
$527$			202.912i			0.385032i
$528$		0			0
$529$	295.630			0.558846
$530$		−	14.2723i		−	0.0269290i
$531$		0			0
$532$	349.203			0.656396
$533$		−	872.185i		−	1.63637i
$534$		0			0
$535$	−40.0396			−0.0748403
$536$	−146.477			−0.273277
$537$		0			0
$538$	291.406			0.541646
$539$		−	1048.81i		−	1.94584i
$540$		0			0
$541$			200.875i			0.371303i	0.982616	+	0.185652i	$0.0594396\pi$
$541$			200.875i			0.371303i	−0.982616	+	0.185652i	$0.940560\pi$
$542$		−	54.1893i		−	0.0999802i
$543$		0			0
$544$		−	97.7612i		−	0.179708i
$545$			38.5953i			0.0708170i
$546$		0			0
$547$	−412.389			−0.753911			−0.376955	−	0.926231i	$-0.623029\pi$
$547$	−412.389			−0.753911			−0.376955	+	0.926231i	$0.623029\pi$
$548$	282.443			0.515407
$549$		0			0
$550$	525.247			0.954994
$551$	354.605			0.643566
$552$		0			0
$553$	−677.775			−1.22563
$554$		−	293.114i		−	0.529086i
$555$		0			0
$556$	122.126			0.219652
$557$	214.797			0.385632			0.192816	−	0.981235i	$-0.438238\pi$
$557$	214.797			0.385632			0.192816	+	0.981235i	$0.438238\pi$
$558$		0			0
$559$	871.256			1.55860
$560$	−10.9555			−0.0195634
$561$		0			0
$562$			429.898i			0.764943i
$563$		−	104.763i		−	0.186079i	−0.995662	−	0.0930396i	$-0.970342\pi$
$563$		−	104.763i		−	0.186079i	0.995662	−	0.0930396i	$-0.0296583\pi$
$564$		0			0
$565$			5.79479i			0.0102563i
$566$	−320.810			−0.566803
$567$		0			0
$568$		−	499.532i		−	0.879458i
$569$			765.436i			1.34523i	0.739992	+	0.672615i	$0.234829\pi$
$569$			765.436i			1.34523i	−0.739992	+	0.672615i	$0.765171\pi$
$570$		0			0
$571$		−	249.624i		−	0.437170i	−0.975818	−	0.218585i	$-0.929856\pi$
$571$		−	249.624i		−	0.437170i	0.975818	−	0.218585i	$-0.0701441\pi$
$572$	341.281			0.596645
$573$		0			0
$574$		−	1196.92i		−	2.08522i
$575$		−	380.769i		−	0.662207i
$576$		0			0
$577$	−268.842			−0.465930			−0.232965	−	0.972485i	$-0.574843\pi$
$577$	−268.842			−0.465930			−0.232965	+	0.972485i	$0.574843\pi$
$578$			388.314i			0.671823i
$579$		0			0
$580$	12.5923			0.0217108
$581$			1078.11i			1.85562i
$582$		0			0
$583$			561.132i			0.962490i
$584$	721.390			1.23526
$585$		0			0
$586$			79.1779i			0.135116i
$587$			793.085i			1.35108i	0.737322	+	0.675541i	$0.236090\pi$
$587$			793.085i			1.35108i	−0.737322	+	0.675541i	$0.763910\pi$
$588$		0			0
$589$		−	935.166i		−	1.58772i
$590$	10.1965	−	20.1567i	0.0172822	−	0.0341639i
$591$		0			0
$592$			156.098i			0.263678i
$593$	108.147			0.182373			0.0911866	−	0.995834i	$-0.470934\pi$
$593$	108.147			0.182373			0.0911866	+	0.995834i	$0.470934\pi$
$594$		0			0
$595$	10.1570			0.0170706
$596$		−	445.505i		−	0.747492i
$597$		0			0
$598$			237.669i			0.397439i
$599$	−21.0152			−0.0350837			−0.0175419	−	0.999846i	$-0.505584\pi$
$599$	−21.0152			−0.0350837			−0.0175419	+	0.999846i	$0.505584\pi$
$600$		0			0
$601$			792.854i			1.31922i	0.751606	+	0.659612i	$0.229279\pi$
$601$			792.854i			1.31922i	−0.751606	+	0.659612i	$0.770721\pi$
$602$	1195.64			1.98612
$603$		0			0
$604$		−	53.8812i		−	0.0892073i
$605$	28.8751			0.0477275
$606$		0			0
$607$	848.008			1.39705			0.698524	−	0.715587i	$-0.253841\pi$
$607$	848.008			1.39705			0.698524	+	0.715587i	$0.253841\pi$
$608$			450.555i			0.741045i
$609$		0			0
$610$	−16.9403			−0.0277710
$611$	392.811			0.642899
$612$		0			0
$613$		−	355.689i		−	0.580243i	−0.956990	−	0.290121i	$-0.906304\pi$
$613$		−	355.689i		−	0.580243i	0.956990	−	0.290121i	$-0.0936957\pi$
$614$			53.4241i			0.0870100i
$615$		0			0
$616$	1386.61			2.25099
$617$	−120.858			−0.195880			−0.0979402	−	0.995192i	$-0.531225\pi$
$617$	−120.858			−0.195880			−0.0979402	+	0.995192i	$0.531225\pi$
$618$		0			0
$619$	−112.377			−0.181546			−0.0907732	−	0.995872i	$-0.528934\pi$
$619$	−112.377			−0.181546			−0.0907732	+	0.995872i	$0.528934\pi$
$620$		−	33.2084i		−	0.0535619i
$621$		0			0
$622$		−	332.320i		−	0.534277i
$623$		−	391.466i		−	0.628356i
$624$		0			0
$625$	619.396			0.991034
$626$	655.684			1.04742
$627$		0			0
$628$			37.6732i			0.0599892i
$629$		−	144.721i		−	0.230081i
$630$		0			0
$631$	467.803			0.741368			0.370684	−	0.928759i	$-0.379123\pi$
$631$	467.803			0.741368			0.370684	+	0.928759i	$0.379123\pi$
$632$		−	526.094i		−	0.832427i
$633$		0			0
$634$		−	136.884i		−	0.215905i
$635$	−58.8222			−0.0926333
$636$		0			0
$637$			774.319i			1.21557i
$638$	475.592			0.745442
$639$		0			0
$640$			10.3681i			0.0162001i
$641$	1006.77			1.57062			0.785311	−	0.619102i	$-0.212503\pi$
$641$	1006.77			1.57062			0.785311	+	0.619102i	$0.212503\pi$
$642$		0			0
$643$	65.4231			0.101747			0.0508734	−	0.998705i	$-0.483800\pi$
$643$	65.4231			0.101747			0.0508734	+	0.998705i	$0.483800\pi$
$644$		−	339.520i		−	0.527205i
$645$		0			0
$646$			74.9893i			0.116083i
$647$	−1000.16			−1.54584			−0.772920	−	0.634503i	$-0.781205\pi$
$647$	−1000.16			−1.54584			−0.772920	+	0.634503i	$0.781205\pi$
$648$		0			0
$649$	−400.884	+	792.480i	−0.617696	+	1.22108i
$650$	−387.782			−0.596588
$651$		0			0
$652$	−504.839			−0.774293
$653$	635.037			0.972492			0.486246	−	0.873822i	$-0.338366\pi$
$653$	635.037			0.972492			0.486246	+	0.873822i	$0.338366\pi$
$654$		0			0
$655$			21.8296i			0.0333276i
$656$	288.595			0.439932
$657$		0			0
$658$	539.063			0.819245
$659$			445.774i			0.676440i	0.941067	+	0.338220i	$0.109825\pi$
$659$			445.774i			0.676440i	−0.941067	+	0.338220i	$0.890175\pi$
$660$		0			0
$661$	4.13501			0.00625568			0.00312784	−	0.999995i	$-0.499004\pi$
$661$	4.13501			0.00625568			0.00312784	+	0.999995i	$0.499004\pi$
$662$			347.372i			0.524731i
$663$		0			0
$664$	−836.840			−1.26030
$665$	−46.8110			−0.0703925
$666$		0			0
$667$		−	344.773i		−	0.516900i
$668$	−128.539			−0.192423
$669$		0			0
$670$	6.63213			0.00989870
$671$	666.025			0.992586
$672$		0			0
$673$			144.454i			0.214641i	0.994224	+	0.107321i	$0.0342271\pi$
$673$			144.454i			0.214641i	−0.994224	+	0.107321i	$0.965773\pi$
$674$	−117.750			−0.174704
$675$		0			0
$676$	92.8222			0.137311
$677$	−567.757			−0.838637			−0.419318	−	0.907839i	$-0.637731\pi$
$677$	−567.757			−0.838637			−0.419318	+	0.907839i	$0.637731\pi$
$678$		0			0
$679$		−	120.028i		−	0.176771i
$680$			7.88396i			0.0115941i
$681$		0			0
$682$		−	1254.24i		−	1.83905i
$683$			505.717i			0.740435i	0.928945	+	0.370218i	$0.120717\pi$
$683$			505.717i			0.740435i	−0.928945	+	0.370218i	$0.879283\pi$
$684$		0			0
$685$	−37.8618			−0.0552727
$686$			315.324i			0.459656i
$687$		0			0
$688$			288.288i			0.419023i
$689$		−	414.275i		−	0.601270i
$690$		0			0
$691$		−	843.080i		−	1.22009i	−0.792368	−	0.610044i	$-0.791152\pi$
$691$		−	843.080i		−	1.22009i	0.792368	−	0.610044i	$-0.208848\pi$
$692$		−	333.071i		−	0.481317i
$693$		0			0
$694$	−730.334			−1.05236
$695$	−16.3712			−0.0235557
$696$		0			0
$697$	−267.562			−0.383877
$698$	−419.806			−0.601441
$699$		0			0
$700$	553.964			0.791377
$701$			873.826i			1.24654i	0.782006	+	0.623271i	$0.214197\pi$
$701$			873.826i			1.24654i	−0.782006	+	0.623271i	$0.785803\pi$
$702$		0			0
$703$			666.980i			0.948763i
$704$			825.687i			1.17285i
$705$		0			0
$706$	886.824			1.25612
$707$		−	1206.75i		−	1.70686i
$708$		0			0
$709$	159.514			0.224985			0.112492	−	0.993653i	$-0.464117\pi$
$709$	159.514			0.224985			0.112492	+	0.993653i	$0.464117\pi$
$710$			22.6177i			0.0318559i
$711$		0			0
$712$	303.859			0.426768
$713$	−909.237			−1.27523
$714$		0			0
$715$	−45.7491			−0.0639848
$716$		−	164.304i		−	0.229475i
$717$		0			0
$718$			638.636i			0.889465i
$719$		−	35.0202i		−	0.0487068i	−0.999703	−	0.0243534i	$-0.992247\pi$
$719$		−	35.0202i		−	0.0487068i	0.999703	−	0.0243534i	$-0.00775270\pi$
$720$		0			0
$721$			469.164i			0.650712i
$722$			159.775i			0.221295i
$723$		0			0
$724$	−265.549			−0.366781
$725$	562.534			0.775908
$726$		0			0
$727$	1123.28			1.54509			0.772544	−	0.634961i	$-0.218984\pi$
$727$	1123.28			1.54509			0.772544	+	0.634961i	$0.218984\pi$
$728$	−1023.71			−1.40620
$729$		0			0
$730$	−32.6629			−0.0447437
$731$		−	267.277i		−	0.365632i
$732$		0			0
$733$	−1337.07			−1.82411			−0.912053	−	0.410073i	$-0.865503\pi$
$733$	−1337.07			−1.82411			−0.912053	+	0.410073i	$0.865503\pi$
$734$	−707.526			−0.963932
$735$		0			0
$736$	438.063			0.595194
$737$	−260.749			−0.353797
$738$		0			0
$739$		−	393.835i		−	0.532930i	−0.963845	−	0.266465i	$-0.914144\pi$
$739$		−	393.835i		−	0.532930i	0.963845	−	0.266465i	$-0.0858556\pi$
$740$			23.6849i			0.0320067i
$741$		0			0
$742$		−	568.518i		−	0.766197i
$743$	42.0432			0.0565857			0.0282929	−	0.999600i	$-0.490993\pi$
$743$	42.0432			0.0565857			0.0282929	+	0.999600i	$0.490993\pi$
$744$		0			0
$745$			59.7205i			0.0801617i
$746$			367.195i			0.492218i
$747$		0			0
$748$		−	104.695i		−	0.139967i
$749$	−1594.92			−2.12940
$750$		0			0
$751$			1078.71i			1.43636i	0.695858	+	0.718180i	$0.255025\pi$
$751$			1078.71i			1.43636i	−0.695858	+	0.718180i	$0.744975\pi$
$752$			129.976i			0.172841i
$753$		0			0
$754$	−351.123			−0.465680
$755$			7.22284i			0.00956667i
$756$		0			0
$757$	−247.277			−0.326654			−0.163327	−	0.986572i	$-0.552223\pi$
$757$	−247.277			−0.326654			−0.163327	+	0.986572i	$0.552223\pi$
$758$		−	745.470i		−	0.983470i
$759$		0			0
$760$		−	36.3351i		−	0.0478093i
$761$	−681.030			−0.894914			−0.447457	−	0.894305i	$-0.647670\pi$
$761$	−681.030			−0.894914			−0.447457	+	0.894305i	$0.647670\pi$
$762$		0			0
$763$			1537.39i			2.01492i
$764$			420.831i			0.550826i
$765$		0			0
$766$			401.143i			0.523685i
$767$	295.967	−	585.077i	0.385876	−	0.762812i
$768$		0			0
$769$		−	1018.02i		−	1.32382i	−0.749582	−	0.661912i	$-0.769745\pi$
$769$		−	1018.02i		−	1.32382i	0.749582	−	0.661912i	$-0.230255\pi$
$770$	−62.7824			−0.0815356
$771$		0			0
$772$	356.328			0.461565
$773$		−	1082.79i		−	1.40076i	−0.713770	−	0.700380i	$-0.753014\pi$
$773$		−	1082.79i		−	1.40076i	0.713770	−	0.700380i	$-0.246986\pi$
$774$		0			0
$775$		−	1483.52i		−	1.91422i
$776$	93.1664			0.120060
$777$		0			0
$778$			868.518i			1.11635i
$779$	1233.12			1.58296
$780$		0			0
$781$		−	889.236i		−	1.13859i
$782$	72.9101			0.0932354
$783$		0			0
$784$	−256.213			−0.326802
$785$		−	5.05014i		−	0.00643330i
$786$		0			0
$787$	302.623			0.384527			0.192263	−	0.981343i	$-0.438417\pi$
$787$	302.623			0.384527			0.192263	+	0.981343i	$0.438417\pi$
$788$	−536.013			−0.680219
$789$		0			0
$790$			23.8203i			0.0301523i
$791$			230.827i			0.291817i
$792$		0			0
$793$	−491.717			−0.620071
$794$	607.596			0.765234
$795$		0			0
$796$	−75.0172			−0.0942428
$797$			319.419i			0.400776i	0.979717	+	0.200388i	$0.0642204\pi$
$797$			319.419i			0.400776i	−0.979717	+	0.200388i	$0.935780\pi$
$798$		0			0
$799$		−	120.504i		−	0.150818i
$800$			714.747i			0.893433i
$801$		0			0
$802$	150.085			0.187138
$803$	1284.18			1.59922
$804$		0			0
$805$			45.5131i			0.0565380i
$806$			925.983i			1.14886i
$807$		0			0
$808$	936.689			1.15927
$809$			474.757i			0.586844i	0.955983	+	0.293422i	$0.0947941\pi$
$809$			474.757i			0.586844i	−0.955983	+	0.293422i	$0.905206\pi$
$810$		0			0
$811$		−	48.5354i		−	0.0598464i	−0.999552	−	0.0299232i	$-0.990474\pi$
$811$		−	48.5354i		−	0.0598464i	0.999552	−	0.0299232i	$-0.00952627\pi$
$812$	501.594			0.617727
$813$		0			0
$814$			894.547i			1.09895i
$815$	67.6742			0.0830359
$816$		0			0
$817$			1231.81i			1.50772i
$818$	−770.675			−0.942146
$819$		0			0
$820$	43.7890			0.0534012
$821$			60.2763i			0.0734182i	0.999326	+	0.0367091i	$0.0116875\pi$
$821$			60.2763i			0.0734182i	−0.999326	+	0.0367091i	$0.988313\pi$
$822$		0			0
$823$		−	1276.52i		−	1.55105i	−0.631316	−	0.775526i	$-0.717485\pi$
$823$		−	1276.52i		−	1.55105i	0.631316	−	0.775526i	$-0.282515\pi$
$824$	−364.168			−0.441952
$825$		0			0
$826$	406.162	−	802.913i	0.491721	−	0.972049i
$827$	−692.996			−0.837964			−0.418982	−	0.907995i	$-0.637613\pi$
$827$	−692.996			−0.837964			−0.418982	+	0.907995i	$0.637613\pi$
$828$		0			0
$829$	−1079.49			−1.30215			−0.651077	−	0.759012i	$-0.725682\pi$
$829$	−1079.49			−1.30215			−0.651077	+	0.759012i	$0.725682\pi$
$830$	37.8902			0.0456509
$831$		0			0
$832$		−	609.592i		−	0.732683i
$833$	237.539			0.285161
$834$		0			0
$835$	17.2308			0.0206357
$836$			482.514i			0.577169i
$837$		0			0
$838$	384.189			0.458459
$839$			974.349i			1.16132i	0.814145	+	0.580661i	$0.197206\pi$
$839$			974.349i			1.16132i	−0.814145	+	0.580661i	$0.802794\pi$
$840$		0			0
$841$	−331.646			−0.394348
$842$	−64.5275			−0.0766360
$843$		0			0
$844$			679.027i			0.804535i
$845$	−12.4429			−0.0147254
$846$		0			0
$847$	1150.20			1.35797
$848$	137.079			0.161649
$849$		0			0
$850$			118.961i			0.139954i
$851$	648.487			0.762029
$852$		0			0
$853$	−966.592			−1.13317			−0.566584	−	0.824004i	$-0.691735\pi$
$853$	−966.592			−1.13317			−0.566584	+	0.824004i	$0.691735\pi$
$854$	−674.793			−0.790155
$855$		0			0
$856$		−	1237.99i		−	1.44625i
$857$		−	1521.97i		−	1.77593i	−0.459912	−	0.887965i	$-0.652119\pi$
$857$		−	1521.97i		−	1.77593i	0.459912	−	0.887965i	$-0.347881\pi$
$858$		0			0
$859$		−	110.413i		−	0.128536i	−0.997933	−	0.0642681i	$-0.979529\pi$
$859$		−	110.413i		−	0.128536i	0.997933	−	0.0642681i	$-0.0204713\pi$
$860$			43.7424i			0.0508632i
$861$		0			0
$862$	358.641			0.416057
$863$			499.067i			0.578293i	0.957285	+	0.289146i	$0.0933714\pi$
$863$			499.067i			0.578293i	−0.957285	+	0.289146i	$0.906629\pi$
$864$		0			0
$865$			44.6486i			0.0516169i
$866$		−	107.384i		−	0.124000i
$867$		0			0
$868$		−	1322.81i		−	1.52397i
$869$		−	936.521i		−	1.07770i
$870$		0			0
$871$	192.507			0.221018
$872$	−1193.33			−1.36850
$873$		0			0
$874$	−336.023			−0.384466
$875$	−148.742			−0.169991
$876$		0			0
$877$	−91.5133			−0.104348			−0.0521741	−	0.998638i	$-0.516615\pi$
$877$	−91.5133			−0.104348			−0.0521741	+	0.998638i	$0.516615\pi$
$878$		−	871.314i		−	0.992385i
$879$		0			0
$880$		−	15.1378i		−	0.0172021i
$881$		−	711.614i		−	0.807734i	−0.914818	−	0.403867i	$-0.867666\pi$
$881$		−	711.614i		−	0.807734i	0.914818	−	0.403867i	$-0.132334\pi$
$882$		0			0
$883$	919.400			1.04122			0.520612	−	0.853794i	$-0.325704\pi$
$883$	919.400			1.04122			0.520612	+	0.853794i	$0.325704\pi$
$884$			77.2951i			0.0874379i
$885$		0			0
$886$	955.741			1.07871
$887$			1322.56i			1.49105i	0.666479	+	0.745523i	$0.267800\pi$
$887$			1322.56i			1.49105i	−0.666479	+	0.745523i	$0.732200\pi$
$888$		0			0
$889$	−2343.10			−2.63565
$890$	−13.7580			−0.0154585
$891$		0			0
$892$	−138.820			−0.155628
$893$			555.369i			0.621914i
$894$		0			0
$895$			22.0252i			0.0246092i
$896$			412.997i			0.460934i
$897$		0			0
$898$		−	202.280i		−	0.225256i
$899$		−	1343.27i		−	1.49418i
$900$		0			0
$901$	−127.088			−0.141052
$902$	1653.85			1.83354
$903$		0			0
$904$	−179.170			−0.198196
$905$	35.5972			0.0393339
$906$		0			0
$907$	735.997			0.811464			0.405732	−	0.913992i	$-0.367017\pi$
$907$	735.997			0.811464			0.405732	+	0.913992i	$0.367017\pi$
$908$		−	606.938i		−	0.668434i
$909$		0			0
$910$	46.3514			0.0509356
$911$	−1461.31			−1.60407			−0.802036	−	0.597276i	$-0.796250\pi$
$911$	−1461.31			−1.60407			−0.802036	+	0.597276i	$0.796250\pi$
$912$		0			0
$913$	−1489.69			−1.63164
$914$	411.378			0.450086
$915$		0			0
$916$		−	56.2986i		−	0.0614614i
$917$			869.549i			0.948254i
$918$		0			0
$919$			1461.36i			1.59016i	0.606506	+	0.795079i	$0.292571\pi$
$919$			1461.36i			1.59016i	−0.606506	+	0.795079i	$0.707429\pi$
$920$	−35.3276			−0.0383996
$921$		0			0
$922$			640.663i			0.694862i
$923$			656.510i			0.711278i
$924$		0			0
$925$			1058.08i			1.14387i
$926$	−939.764			−1.01486
$927$		0			0
$928$			647.177i			0.697389i
$929$			467.354i			0.503072i	0.967848	+	0.251536i	$0.0809357\pi$
$929$			467.354i			0.503072i	−0.967848	+	0.251536i	$0.919064\pi$
$930$		0			0
$931$	−1094.76			−1.17589
$932$		−	732.912i		−	0.786386i
$933$		0			0
$934$	−501.286			−0.536709
$935$			14.0345i			0.0150102i
$936$		0			0
$937$			939.907i			1.00310i	0.865128	+	0.501551i	$0.167237\pi$
$937$			939.907i			1.00310i	−0.865128	+	0.501551i	$0.832763\pi$
$938$	264.181			0.281643
$939$		0			0
$940$			19.7215i			0.0209804i
$941$		−	1495.70i		−	1.58948i	−0.606948	−	0.794741i	$-0.707606\pi$
$941$		−	1495.70i		−	1.58948i	0.606948	−	0.794741i	$-0.292394\pi$
$942$		0			0
$943$		−	1198.93i		−	1.27140i
$944$	193.595	+	97.9319i	0.205079	+	0.103741i
$945$		0			0
$946$			1652.09i			1.74639i
$947$	999.461			1.05540			0.527699	−	0.849432i	$-0.323055\pi$
$947$	999.461			1.05540			0.527699	+	0.849432i	$0.323055\pi$
$948$		0			0
$949$	−948.088			−0.999039
$950$		−	548.258i		−	0.577114i
$951$		0			0
$952$			314.046i			0.329880i
$953$	87.5531			0.0918710			0.0459355	−	0.998944i	$-0.485373\pi$
$953$	87.5531			0.0918710			0.0459355	+	0.998944i	$0.485373\pi$
$954$		0			0
$955$		−	56.4129i		−	0.0590711i
$956$	540.898			0.565793
$957$		0			0
$958$		−	32.6610i		−	0.0340929i
$959$	−1508.17			−1.57265
$960$		0			0
$961$	−2581.49			−2.68625
$962$		−	660.431i		−	0.686518i
$963$		0			0
$964$	−120.136			−0.124622
$965$	−47.7662			−0.0494987
$966$		0			0
$967$		−	1799.89i		−	1.86132i	−0.365887	−	0.930659i	$-0.619234\pi$
$967$		−	1799.89i		−	1.86132i	0.365887	−	0.930659i	$-0.380766\pi$
$968$			892.793i			0.922307i
$969$		0			0
$970$	−4.21836			−0.00434883
$971$	−1423.88			−1.46641			−0.733205	−	0.680008i	$-0.761976\pi$
$971$	−1423.88			−1.46641			−0.733205	+	0.680008i	$0.761976\pi$
$972$		0			0
$973$	−652.122			−0.670218
$974$		−	1005.89i		−	1.03274i
$975$		0			0
$976$		−	162.703i		−	0.166704i
$977$		−	535.083i		−	0.547679i	−0.961775	−	0.273840i	$-0.911706\pi$
$977$		−	535.083i		−	0.547679i	0.961775	−	0.273840i	$-0.0882938\pi$
$978$		0			0
$979$	540.911			0.552514
$980$	−38.8755			−0.0396689
$981$		0			0
$982$			569.442i			0.579880i
$983$		−	1248.96i		−	1.27056i	−0.772282	−	0.635280i	$-0.780885\pi$
$983$		−	1248.96i		−	1.27056i	0.772282	−	0.635280i	$-0.219115\pi$
$984$		0			0
$985$	71.8531			0.0729473
$986$			107.715i			0.109244i
$987$		0			0
$988$		−	356.233i		−	0.360559i
$989$	1197.65			1.21097
$990$		0			0
$991$		−	431.674i		−	0.435595i	−0.975994	−	0.217797i	$-0.930113\pi$
$991$		−	431.674i		−	0.435595i	0.975994	−	0.217797i	$-0.0698872\pi$
$992$	1706.74			1.72050
$993$		0			0
$994$			900.942i			0.906380i
$995$	10.0561			0.0101067
$996$		0			0
$997$	−133.711			−0.134113			−0.0670565	−	0.997749i	$-0.521361\pi$
$997$	−133.711			−0.134113			−0.0670565	+	0.997749i	$0.521361\pi$
$998$			1160.13i			1.16245i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.8		20
3.2	odd	2		177.3.c.a.58.13	yes	20
59.58	odd	2	inner	531.3.c.c.235.13		20
177.176	even	2		177.3.c.a.58.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.8	✓	20	177.176	even	2
177.3.c.a.58.13	yes	20	3.2	odd	2
531.3.c.c.235.8		20	1.1	even	1	trivial
531.3.c.c.235.13		20	59.58	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.39995i q^{2} +2.04015 q^{4} -0.273484 q^{5} -10.8938 q^{7} -8.45589i q^{8} +O(q^{10})\)	\(q-1.39995i q^{2} +2.04015 q^{4} -0.273484 q^{5} -10.8938 q^{7} -8.45589i q^{8} +0.382863i q^{10} -15.0527i q^{11} +11.1132i q^{13} +15.2508i q^{14} -3.67721 q^{16} +3.40920 q^{17} -15.7121 q^{19} -0.557948 q^{20} -21.0729 q^{22} +15.2765i q^{23} -24.9252 q^{25} +15.5578 q^{26} -22.2251 q^{28} -22.5689 q^{29} +59.5188i q^{31} -28.6757i q^{32} -4.77271i q^{34} +2.97929 q^{35} -42.4500i q^{37} +21.9961i q^{38} +2.31255i q^{40} -78.4822 q^{41} -78.3986i q^{43} -30.7096i q^{44} +21.3862 q^{46} -35.3465i q^{47} +69.6759 q^{49} +34.8940i q^{50} +22.6725i q^{52} -37.2779 q^{53} +4.11666i q^{55} +92.1171i q^{56} +31.5952i q^{58} +(-52.6472 - 26.6321i) q^{59} +44.2464i q^{61} +83.3232 q^{62} -54.8532 q^{64} -3.03927i q^{65} -17.3224i q^{67} +6.95528 q^{68} -4.17086i q^{70} +59.0750 q^{71} +85.3122i q^{73} -59.4278 q^{74} -32.0550 q^{76} +163.981i q^{77} +62.2163 q^{79} +1.00566 q^{80} +109.871i q^{82} -98.9653i q^{83} -0.932363 q^{85} -109.754 q^{86} -127.284 q^{88} +35.9346i q^{89} -121.065i q^{91} +31.1663i q^{92} -49.4833 q^{94} +4.29702 q^{95} +11.0179i q^{97} -97.5426i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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