LMFDB - Embedded newform 588.4.i.a.373.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.i (of order $3$, degree $2$, not 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:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			373.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.373
Dual form			588.4.i.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{3} +(-7.00000 + 12.1244i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(-1.50000 - 2.59808i) q^{3} +(-7.00000 + 12.1244i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-2.00000 - 3.46410i) q^{11} +54.0000 q^{13} +42.0000 q^{15} +(7.00000 + 12.1244i) q^{17} +(-46.0000 + 79.6743i) q^{19} +(76.0000 - 131.636i) q^{23} +(-35.5000 - 61.4878i) q^{25} +27.0000 q^{27} -106.000 q^{29} +(72.0000 + 124.708i) q^{31} +(-6.00000 + 10.3923i) q^{33} +(-79.0000 + 136.832i) q^{37} +(-81.0000 - 140.296i) q^{39} -390.000 q^{41} -508.000 q^{43} +(-63.0000 - 109.119i) q^{45} +(264.000 - 457.261i) q^{47} +(21.0000 - 36.3731i) q^{51} +(-303.000 - 524.811i) q^{53} +56.0000 q^{55} +276.000 q^{57} +(182.000 + 315.233i) q^{59} +(-339.000 + 587.165i) q^{61} +(-378.000 + 654.715i) q^{65} +(-422.000 - 730.925i) q^{67} -456.000 q^{69} -8.00000 q^{71} +(211.000 + 365.463i) q^{73} +(-106.500 + 184.463i) q^{75} +(-192.000 + 332.554i) q^{79} +(-40.5000 - 70.1481i) q^{81} -548.000 q^{83} -196.000 q^{85} +(159.000 + 275.396i) q^{87} +(-597.000 + 1034.03i) q^{89} +(216.000 - 374.123i) q^{93} +(-644.000 - 1115.44i) q^{95} -1502.00 q^{97} +36.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 14 q^{5} - 9 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 14 * q^5 - 9 * q^9	$ 2 q - 3 q^{3} - 14 q^{5} - 9 q^{9} - 4 q^{11} + 108 q^{13} + 84 q^{15} + 14 q^{17} - 92 q^{19} + 152 q^{23} - 71 q^{25} + 54 q^{27} - 212 q^{29} + 144 q^{31} - 12 q^{33} - 158 q^{37} - 162 q^{39} - 780 q^{41} - 1016 q^{43} - 126 q^{45} + 528 q^{47} + 42 q^{51} - 606 q^{53} + 112 q^{55} + 552 q^{57} + 364 q^{59} - 678 q^{61} - 756 q^{65} - 844 q^{67} - 912 q^{69} - 16 q^{71} + 422 q^{73} - 213 q^{75} - 384 q^{79} - 81 q^{81} - 1096 q^{83} - 392 q^{85} + 318 q^{87} - 1194 q^{89} + 432 q^{93} - 1288 q^{95} - 3004 q^{97} + 72 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 14 * q^5 - 9 * q^9 - 4 * q^11 + 108 * q^13 + 84 * q^15 + 14 * q^17 - 92 * q^19 + 152 * q^23 - 71 * q^25 + 54 * q^27 - 212 * q^29 + 144 * q^31 - 12 * q^33 - 158 * q^37 - 162 * q^39 - 780 * q^41 - 1016 * q^43 - 126 * q^45 + 528 * q^47 + 42 * q^51 - 606 * q^53 + 112 * q^55 + 552 * q^57 + 364 * q^59 - 678 * q^61 - 756 * q^65 - 844 * q^67 - 912 * q^69 - 16 * q^71 + 422 * q^73 - 213 * q^75 - 384 * q^79 - 81 * q^81 - 1096 * q^83 - 392 * q^85 + 318 * q^87 - 1194 * q^89 + 432 * q^93 - 1288 * q^95 - 3004 * q^97 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$4$		0			0
$5$	−7.00000	+	12.1244i	−0.626099	+	1.08444i	0.362228	+	0.932089i	$0.382016\pi$
$5$	−7.00000	+	12.1244i	−0.626099	+	1.08444i	−0.988327	+	0.152346i	$0.951317\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.0548202	−	0.0949514i	0.837313	−	0.546724i	$-0.184125\pi$
$11$	−2.00000	−	3.46410i	−0.0548202	−	0.0949514i	−0.892133	+	0.451772i	$0.850792\pi$
$12$		0			0
$13$	54.0000			1.15207			0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000			1.15207			0.576035	+	0.817425i	$0.304599\pi$
$14$		0			0
$15$	42.0000			0.722957
$16$		0			0
$17$	7.00000	+	12.1244i	0.0998676	+	0.172976i	0.911630	−	0.411012i	$-0.134825\pi$
$17$	7.00000	+	12.1244i	0.0998676	+	0.172976i	−0.811762	+	0.583988i	$0.801491\pi$
$18$		0			0
$19$	−46.0000	+	79.6743i	−0.555428	+	0.962029i	0.442443	+	0.896797i	$0.354112\pi$
$19$	−46.0000	+	79.6743i	−0.555428	+	0.962029i	−0.997870	+	0.0652319i	$0.979221\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	76.0000	−	131.636i	0.689004	−	1.19339i	−0.283156	−	0.959074i	$-0.591381\pi$
$23$	76.0000	−	131.636i	0.689004	−	1.19339i	0.972160	−	0.234316i	$-0.0752852\pi$
$24$		0			0
$25$	−35.5000	−	61.4878i	−0.284000	−	0.491902i
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−106.000			−0.678748			−0.339374	−	0.940651i	$-0.610215\pi$
$29$	−106.000			−0.678748			−0.339374	+	0.940651i	$0.610215\pi$
$30$		0			0
$31$	72.0000	+	124.708i	0.417148	+	0.722521i	0.995651	−	0.0931587i	$-0.0296964\pi$
$31$	72.0000	+	124.708i	0.417148	+	0.722521i	−0.578503	+	0.815680i	$0.696363\pi$
$32$		0			0
$33$	−6.00000	+	10.3923i	−0.0316505	+	0.0548202i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−79.0000	+	136.832i	−0.351014	+	0.607974i	−0.986427	−	0.164198i	$-0.947496\pi$
$37$	−79.0000	+	136.832i	−0.351014	+	0.607974i	0.635413	+	0.772172i	$0.280830\pi$
$38$		0			0
$39$	−81.0000	−	140.296i	−0.332574	−	0.576035i
$40$		0			0
$41$	−390.000			−1.48556			−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000			−1.48556			−0.742778	+	0.669538i	$0.766492\pi$
$42$		0			0
$43$	−508.000			−1.80161			−0.900806	−	0.434223i	$-0.857023\pi$
$43$	−508.000			−1.80161			−0.900806	+	0.434223i	$0.857023\pi$
$44$		0			0
$45$	−63.0000	−	109.119i	−0.208700	−	0.361478i
$46$		0			0
$47$	264.000	−	457.261i	0.819327	−	1.41912i	−0.0868522	−	0.996221i	$-0.527681\pi$
$47$	264.000	−	457.261i	0.819327	−	1.41912i	0.906179	−	0.422894i	$-0.138986\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	21.0000	−	36.3731i	0.0576586	−	0.0998676i
$52$		0			0
$53$	−303.000	−	524.811i	−0.785288	−	1.36016i	−0.928827	−	0.370514i	$-0.879182\pi$
$53$	−303.000	−	524.811i	−0.785288	−	1.36016i	0.143539	−	0.989645i	$-0.454152\pi$
$54$		0			0
$55$	56.0000			0.137292
$56$		0			0
$57$	276.000			0.641353
$58$		0			0
$59$	182.000	+	315.233i	0.401600	+	0.695591i	0.993919	−	0.110112i	$-0.0351210\pi$
$59$	182.000	+	315.233i	0.401600	+	0.695591i	−0.592319	+	0.805703i	$0.701788\pi$
$60$		0			0
$61$	−339.000	+	587.165i	−0.711549	+	1.23244i	0.252726	+	0.967538i	$0.418673\pi$
$61$	−339.000	+	587.165i	−0.711549	+	1.23244i	−0.964275	+	0.264902i	$0.914661\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−378.000	+	654.715i	−0.721310	+	1.24935i
$66$		0			0
$67$	−422.000	−	730.925i	−0.769485	−	1.33279i	−0.937842	−	0.347061i	$-0.887180\pi$
$67$	−422.000	−	730.925i	−0.769485	−	1.33279i	0.168357	−	0.985726i	$-0.446154\pi$
$68$		0			0
$69$	−456.000			−0.795593
$70$		0			0
$71$	−8.00000			−0.0133722			−0.00668609	−	0.999978i	$-0.502128\pi$
$71$	−8.00000			−0.0133722			−0.00668609	+	0.999978i	$0.502128\pi$
$72$		0			0
$73$	211.000	+	365.463i	0.338297	+	0.585948i	0.984113	−	0.177546i	$-0.0568158\pi$
$73$	211.000	+	365.463i	0.338297	+	0.585948i	−0.645816	+	0.763494i	$0.723482\pi$
$74$		0			0
$75$	−106.500	+	184.463i	−0.163967	+	0.284000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−192.000	+	332.554i	−0.273439	+	0.473610i	−0.969740	−	0.244139i	$-0.921495\pi$
$79$	−192.000	+	332.554i	−0.273439	+	0.473610i	0.696301	+	0.717750i	$0.254828\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−548.000			−0.724709			−0.362354	−	0.932040i	$-0.618027\pi$
$83$	−548.000			−0.724709			−0.362354	+	0.932040i	$0.618027\pi$
$84$		0			0
$85$	−196.000			−0.250108
$86$		0			0
$87$	159.000	+	275.396i	0.195938	+	0.339374i
$88$		0			0
$89$	−597.000	+	1034.03i	−0.711032	+	1.23154i	0.253438	+	0.967352i	$0.418439\pi$
$89$	−597.000	+	1034.03i	−0.711032	+	1.23154i	−0.964470	+	0.264192i	$0.914895\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	216.000	−	374.123i	0.240840	−	0.417148i
$94$		0			0
$95$	−644.000	−	1115.44i	−0.695505	−	1.20465i
$96$		0			0
$97$	−1502.00			−1.57222			−0.786108	−	0.618089i	$-0.787907\pi$
$97$	−1502.00			−1.57222			−0.786108	+	0.618089i	$0.787907\pi$
$98$		0			0
$99$	36.0000			0.0365468
$100$		0			0
$101$	−199.000	−	344.678i	−0.196052	−	0.339572i	0.751193	−	0.660083i	$-0.229479\pi$
$101$	−199.000	−	344.678i	−0.196052	−	0.339572i	−0.947245	+	0.320511i	$0.896145\pi$
$102$		0			0
$103$	−580.000	+	1004.59i	−0.554846	+	0.961021i	0.443070	+	0.896487i	$0.353889\pi$
$103$	−580.000	+	1004.59i	−0.554846	+	0.961021i	−0.997916	+	0.0645337i	$0.979444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−162.000	+	280.592i	−0.146366	+	0.253513i	−0.929882	−	0.367859i	$-0.880091\pi$
$107$	−162.000	+	280.592i	−0.146366	+	0.253513i	0.783516	+	0.621372i	$0.213424\pi$
$108$		0			0
$109$	469.000	+	812.332i	0.412129	+	0.713828i	0.995122	−	0.0986487i	$-0.0314520\pi$
$109$	469.000	+	812.332i	0.412129	+	0.713828i	−0.582993	+	0.812477i	$0.698119\pi$
$110$		0			0
$111$	474.000			0.405316
$112$		0			0
$113$	−622.000			−0.517813			−0.258906	−	0.965902i	$-0.583362\pi$
$113$	−622.000			−0.517813			−0.258906	+	0.965902i	$0.583362\pi$
$114$		0			0
$115$	1064.00	+	1842.90i	0.862770	+	1.49436i
$116$		0			0
$117$	−243.000	+	420.888i	−0.192012	+	0.332574i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	657.500	−	1138.82i	0.493989	−	0.855615i
$122$		0			0
$123$	585.000	+	1013.25i	0.428843	+	0.742778i
$124$		0			0
$125$	−756.000			−0.540950
$126$		0			0
$127$	1200.00			0.838447			0.419224	−	0.907883i	$-0.362302\pi$
$127$	1200.00			0.838447			0.419224	+	0.907883i	$0.362302\pi$
$128$		0			0
$129$	762.000	+	1319.82i	0.520080	+	0.900806i
$130$		0			0
$131$	698.000	−	1208.97i	0.465531	−	0.806323i	−0.533694	−	0.845677i	$-0.679197\pi$
$131$	698.000	−	1208.97i	0.465531	−	0.806323i	0.999225	+	0.0393543i	$0.0125301\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−189.000	+	327.358i	−0.120493	+	0.208700i
$136$		0			0
$137$	−1405.00	−	2433.53i	−0.876184	−	1.51760i	−0.855496	−	0.517810i	$-0.826747\pi$
$137$	−1405.00	−	2433.53i	−0.876184	−	1.51760i	−0.0206885	−	0.999786i	$-0.506586\pi$
$138$		0			0
$139$	4.00000			0.00244083			0.00122042	−	0.999999i	$-0.499612\pi$
$139$	4.00000			0.00244083			0.00122042	+	0.999999i	$0.499612\pi$
$140$		0			0
$141$	−1584.00			−0.946077
$142$		0			0
$143$	−108.000	−	187.061i	−0.0631567	−	0.109391i
$144$		0			0
$145$	742.000	−	1285.18i	0.424964	−	0.736059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−687.000	+	1189.92i	−0.377726	+	0.654241i	−0.990731	−	0.135838i	$-0.956627\pi$
$149$	−687.000	+	1189.92i	−0.377726	+	0.654241i	0.613005	+	0.790079i	$0.289961\pi$
$150$		0			0
$151$	−1052.00	−	1822.12i	−0.566957	−	0.981999i	−0.996865	−	0.0791258i	$-0.974787\pi$
$151$	−1052.00	−	1822.12i	−0.566957	−	0.981999i	0.429907	−	0.902873i	$-0.358546\pi$
$152$		0			0
$153$	−126.000			−0.0665784
$154$		0			0
$155$	−2016.00			−1.04470
$156$		0			0
$157$	−1603.00	−	2776.48i	−0.814862	−	1.41138i	−0.909427	−	0.415864i	$-0.863479\pi$
$157$	−1603.00	−	2776.48i	−0.814862	−	1.41138i	0.0945650	−	0.995519i	$-0.469854\pi$
$158$		0			0
$159$	−909.000	+	1574.43i	−0.453386	+	0.785288i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−166.000	+	287.520i	−0.0797676	+	0.138162i	−0.903150	−	0.429326i	$-0.858751\pi$
$163$	−166.000	+	287.520i	−0.0797676	+	0.138162i	0.823382	+	0.567488i	$0.192084\pi$
$164$		0			0
$165$	−84.0000	−	145.492i	−0.0396327	−	0.0686458i
$166$		0			0
$167$	1496.00			0.693197			0.346599	−	0.938014i	$-0.387337\pi$
$167$	1496.00			0.693197			0.346599	+	0.938014i	$0.387337\pi$
$168$		0			0
$169$	719.000			0.327264
$170$		0			0
$171$	−414.000	−	717.069i	−0.185143	−	0.320676i
$172$		0			0
$173$	1661.00	−	2876.94i	0.729962	−	1.26433i	−0.226936	−	0.973910i	$-0.572871\pi$
$173$	1661.00	−	2876.94i	0.729962	−	1.26433i	0.956899	−	0.290422i	$-0.0937958\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	546.000	−	945.700i	0.231864	−	0.401600i
$178$		0			0
$179$	450.000	+	779.423i	0.187903	+	0.325457i	0.944551	−	0.328365i	$-0.106498\pi$
$179$	450.000	+	779.423i	0.187903	+	0.325457i	−0.756648	+	0.653822i	$0.773164\pi$
$180$		0			0
$181$	1902.00			0.781075			0.390537	−	0.920587i	$-0.372289\pi$
$181$	1902.00			0.781075			0.390537	+	0.920587i	$0.372289\pi$
$182$		0			0
$183$	2034.00			0.821626
$184$		0			0
$185$	−1106.00	−	1915.65i	−0.439539	−	0.761304i
$186$		0			0
$187$	28.0000	−	48.4974i	0.0109495	−	0.0189651i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2064.00	+	3574.95i	−0.781915	+	1.35432i	0.148910	+	0.988851i	$0.452424\pi$
$191$	−2064.00	+	3574.95i	−0.781915	+	1.35432i	−0.930825	+	0.365466i	$0.880910\pi$
$192$		0			0
$193$	671.000	+	1162.21i	0.250257	+	0.433458i	0.963597	−	0.267361i	$-0.0861515\pi$
$193$	671.000	+	1162.21i	0.250257	+	0.433458i	−0.713339	+	0.700819i	$0.752818\pi$
$194$		0			0
$195$	2268.00			0.832897
$196$		0			0
$197$	−3506.00			−1.26798			−0.633990	−	0.773341i	$-0.718584\pi$
$197$	−3506.00			−1.26798			−0.633990	+	0.773341i	$0.718584\pi$
$198$		0			0
$199$	−340.000	−	588.897i	−0.121115	−	0.209778i	0.799092	−	0.601208i	$-0.205314\pi$
$199$	−340.000	−	588.897i	−0.121115	−	0.209778i	−0.920208	+	0.391430i	$0.871980\pi$
$200$		0			0
$201$	−1266.00	+	2192.78i	−0.444262	+	0.769485i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2730.00	−	4728.50i	0.930105	−	1.61099i
$206$		0			0
$207$	684.000	+	1184.72i	0.229668	+	0.397797i
$208$		0			0
$209$	368.000			0.121795
$210$		0			0
$211$	5372.00			1.75272			0.876360	−	0.481657i	$-0.159965\pi$
$211$	5372.00			1.75272			0.876360	+	0.481657i	$0.159965\pi$
$212$		0			0
$213$	12.0000	+	20.7846i	0.00386022	+	0.00668609i
$214$		0			0
$215$	3556.00	−	6159.17i	1.12799	−	1.95373i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	633.000	−	1096.39i	0.195316	−	0.338297i
$220$		0			0
$221$	378.000	+	654.715i	0.115054	+	0.199280i
$222$		0			0
$223$	−1072.00			−0.321912			−0.160956	−	0.986962i	$-0.551458\pi$
$223$	−1072.00			−0.321912			−0.160956	+	0.986962i	$0.551458\pi$
$224$		0			0
$225$	639.000			0.189333
$226$		0			0
$227$	1434.00	+	2483.76i	0.419286	+	0.726225i	0.995868	−	0.0908148i	$-0.0289471\pi$
$227$	1434.00	+	2483.76i	0.419286	+	0.726225i	−0.576582	+	0.817039i	$0.695614\pi$
$228$		0			0
$229$	−2399.00	+	4155.19i	−0.692272	+	1.19905i	0.278819	+	0.960344i	$0.410057\pi$
$229$	−2399.00	+	4155.19i	−0.692272	+	1.19905i	−0.971092	+	0.238707i	$0.923276\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2563.00	−	4439.25i	0.720634	−	1.24817i	−0.240112	−	0.970745i	$-0.577184\pi$
$233$	2563.00	−	4439.25i	0.720634	−	1.24817i	0.960746	−	0.277429i	$-0.0894825\pi$
$234$		0			0
$235$	3696.00	+	6401.66i	1.02596	+	1.77701i
$236$		0			0
$237$	1152.00			0.315740
$238$		0			0
$239$	−528.000			−0.142902			−0.0714508	−	0.997444i	$-0.522763\pi$
$239$	−528.000			−0.142902			−0.0714508	+	0.997444i	$0.522763\pi$
$240$		0			0
$241$	407.000	+	704.945i	0.108785	+	0.188421i	0.915278	−	0.402822i	$-0.131971\pi$
$241$	407.000	+	704.945i	0.108785	+	0.188421i	−0.806493	+	0.591243i	$0.798637\pi$
$242$		0			0
$243$	−121.500	+	210.444i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2484.00	+	4302.41i	−0.639891	+	1.10832i
$248$		0			0
$249$	822.000	+	1423.75i	0.209205	+	0.362354i
$250$		0			0
$251$	−1932.00			−0.485844			−0.242922	−	0.970046i	$-0.578106\pi$
$251$	−1932.00			−0.485844			−0.242922	+	0.970046i	$0.578106\pi$
$252$		0			0
$253$	−608.000			−0.151086
$254$		0			0
$255$	294.000	+	509.223i	0.0722000	+	0.125054i
$256$		0			0
$257$	1647.00	−	2852.69i	0.399755	−	0.692396i	−0.593940	−	0.804509i	$-0.702429\pi$
$257$	1647.00	−	2852.69i	0.399755	−	0.692396i	0.993695	+	0.112113i	$0.0357619\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	477.000	−	826.188i	0.113125	−	0.195938i
$262$		0			0
$263$	3540.00	+	6131.46i	0.829984	+	1.43757i	0.898050	+	0.439893i	$0.144984\pi$
$263$	3540.00	+	6131.46i	0.829984	+	1.43757i	−0.0680662	+	0.997681i	$0.521683\pi$
$264$		0			0
$265$	8484.00			1.96667
$266$		0			0
$267$	3582.00			0.821029
$268$		0			0
$269$	−3907.00	−	6767.12i	−0.885554	−	1.53382i	−0.845078	−	0.534644i	$-0.820446\pi$
$269$	−3907.00	−	6767.12i	−0.885554	−	1.53382i	−0.0404764	−	0.999180i	$-0.512888\pi$
$270$		0			0
$271$	−1584.00	+	2743.57i	−0.355060	+	0.614981i	−0.987128	−	0.159931i	$-0.948873\pi$
$271$	−1584.00	+	2743.57i	−0.355060	+	0.614981i	0.632069	+	0.774912i	$0.282206\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−142.000	+	245.951i	−0.0311379	+	0.0539324i
$276$		0			0
$277$	3929.00	+	6805.23i	0.852241	+	1.47612i	0.879181	+	0.476488i	$0.158090\pi$
$277$	3929.00	+	6805.23i	0.852241	+	1.47612i	−0.0269403	+	0.999637i	$0.508576\pi$
$278$		0			0
$279$	−1296.00			−0.278099
$280$		0			0
$281$	6730.00			1.42875			0.714374	−	0.699764i	$-0.246712\pi$
$281$	6730.00			1.42875			0.714374	+	0.699764i	$0.246712\pi$
$282$		0			0
$283$	1510.00	+	2615.40i	0.317174	+	0.549361i	0.979897	−	0.199503i	$-0.0639328\pi$
$283$	1510.00	+	2615.40i	0.317174	+	0.549361i	−0.662723	+	0.748864i	$0.730599\pi$
$284$		0			0
$285$	−1932.00	+	3346.32i	−0.401550	+	0.695505i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2358.50	−	4085.04i	0.480053	−	0.831476i
$290$		0			0
$291$	2253.00	+	3902.31i	0.453860	+	0.786108i
$292$		0			0
$293$	−6834.00			−1.36262			−0.681308	−	0.731997i	$-0.738589\pi$
$293$	−6834.00			−1.36262			−0.681308	+	0.731997i	$0.738589\pi$
$294$		0			0
$295$	−5096.00			−1.00576
$296$		0			0
$297$	−54.0000	−	93.5307i	−0.0105502	−	0.0182734i
$298$		0			0
$299$	4104.00	−	7108.34i	0.793781	−	1.37487i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−597.000	+	1034.03i	−0.113191	+	0.196052i
$304$		0			0
$305$	−4746.00	−	8220.31i	−0.891001	−	1.54326i
$306$		0			0
$307$	2332.00			0.433532			0.216766	−	0.976224i	$-0.430449\pi$
$307$	2332.00			0.433532			0.216766	+	0.976224i	$0.430449\pi$
$308$		0			0
$309$	3480.00			0.640681
$310$		0			0
$311$	−4420.00	−	7655.66i	−0.805901	−	1.39586i	−0.915681	−	0.401906i	$-0.868348\pi$
$311$	−4420.00	−	7655.66i	−0.805901	−	1.39586i	0.109780	−	0.993956i	$-0.464985\pi$
$312$		0			0
$313$	523.000	−	905.863i	0.0944464	−	0.163586i	−0.814931	−	0.579558i	$-0.803225\pi$
$313$	523.000	−	905.863i	0.0944464	−	0.163586i	0.909377	+	0.415972i	$0.136559\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3771.00	+	6531.56i	−0.668140	+	1.15725i	0.310283	+	0.950644i	$0.399576\pi$
$317$	−3771.00	+	6531.56i	−0.668140	+	1.15725i	−0.978424	+	0.206609i	$0.933757\pi$
$318$		0			0
$319$	212.000	+	367.195i	0.0372092	+	0.0644482i
$320$		0			0
$321$	972.000			0.169009
$322$		0			0
$323$	−1288.00			−0.221877
$324$		0			0
$325$	−1917.00	−	3320.34i	−0.327188	−	0.566706i
$326$		0			0
$327$	1407.00	−	2437.00i	0.237943	−	0.412129i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1378.00	+	2386.77i	−0.228827	+	0.396340i	−0.957461	−	0.288564i	$-0.906822\pi$
$331$	−1378.00	+	2386.77i	−0.228827	+	0.396340i	0.728634	+	0.684904i	$0.240156\pi$
$332$		0			0
$333$	−711.000	−	1231.49i	−0.117005	−	0.202658i
$334$		0			0
$335$	11816.0			1.92710
$336$		0			0
$337$	3954.00			0.639134			0.319567	−	0.947564i	$-0.396463\pi$
$337$	3954.00			0.639134			0.319567	+	0.947564i	$0.396463\pi$
$338$		0			0
$339$	933.000	+	1616.00i	0.149480	+	0.258906i
$340$		0			0
$341$	288.000	−	498.831i	0.0457363	−	0.0792176i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3192.00	−	5528.71i	0.498120	−	0.862770i
$346$		0			0
$347$	−3450.00	−	5975.58i	−0.533734	−	0.924454i	−0.999223	−	0.0394010i	$-0.987455\pi$
$347$	−3450.00	−	5975.58i	−0.533734	−	0.924454i	0.465489	−	0.885053i	$-0.345878\pi$
$348$		0			0
$349$	−2426.00			−0.372094			−0.186047	−	0.982541i	$-0.559568\pi$
$349$	−2426.00			−0.372094			−0.186047	+	0.982541i	$0.559568\pi$
$350$		0			0
$351$	1458.00			0.221716
$352$		0			0
$353$	735.000	+	1273.06i	0.110822	+	0.191949i	0.916102	−	0.400946i	$-0.131318\pi$
$353$	735.000	+	1273.06i	0.110822	+	0.191949i	−0.805280	+	0.592895i	$0.797985\pi$
$354$		0			0
$355$	56.0000	−	96.9948i	0.00837231	−	0.0145013i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3436.00	+	5951.33i	−0.505140	+	0.874928i	0.494843	+	0.868983i	$0.335226\pi$
$359$	−3436.00	+	5951.33i	−0.505140	+	0.874928i	−0.999982	+	0.00594499i	$0.998108\pi$
$360$		0			0
$361$	−802.500	−	1389.97i	−0.117000	−	0.202649i
$362$		0			0
$363$	−3945.00			−0.570410
$364$		0			0
$365$	−5908.00			−0.847230
$366$		0			0
$367$	3536.00	+	6124.53i	0.502937	+	0.871112i	0.999994	+	0.00339411i	$0.00108038\pi$
$367$	3536.00	+	6124.53i	0.502937	+	0.871112i	−0.497058	+	0.867717i	$0.665586\pi$
$368$		0			0
$369$	1755.00	−	3039.75i	0.247593	−	0.428843i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	409.000	−	708.409i	0.0567754	−	0.0983378i	−0.836241	−	0.548362i	$-0.815251\pi$
$373$	409.000	−	708.409i	0.0567754	−	0.0983378i	0.893016	+	0.450025i	$0.148585\pi$
$374$		0			0
$375$	1134.00	+	1964.15i	0.156159	+	0.270475i
$376$		0			0
$377$	−5724.00			−0.781966
$378$		0			0
$379$	−5132.00			−0.695549			−0.347775	−	0.937578i	$-0.613063\pi$
$379$	−5132.00			−0.695549			−0.347775	+	0.937578i	$0.613063\pi$
$380$		0			0
$381$	−1800.00	−	3117.69i	−0.242039	−	0.419224i
$382$		0			0
$383$	4288.00	−	7427.03i	0.572080	−	0.990871i	−0.424272	−	0.905535i	$-0.639470\pi$
$383$	4288.00	−	7427.03i	0.572080	−	0.990871i	0.996352	−	0.0853367i	$-0.0271966\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2286.00	−	3959.47i	0.300269	−	0.520080i
$388$		0			0
$389$	1865.00	+	3230.27i	0.243083	+	0.421032i	0.961591	−	0.274487i	$-0.0885080\pi$
$389$	1865.00	+	3230.27i	0.243083	+	0.421032i	−0.718508	+	0.695519i	$0.755175\pi$
$390$		0			0
$391$	2128.00			0.275237
$392$		0			0
$393$	−4188.00			−0.537549
$394$		0			0
$395$	−2688.00	−	4655.75i	−0.342400	−	0.593054i
$396$		0			0
$397$	−3339.00	+	5783.32i	−0.422115	+	0.731124i	−0.996146	−	0.0877090i	$-0.972045\pi$
$397$	−3339.00	+	5783.32i	−0.422115	+	0.731124i	0.574031	+	0.818833i	$0.305379\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1527.00	−	2644.84i	0.190161	−	0.329369i	−0.755142	−	0.655561i	$-0.772432\pi$
$401$	1527.00	−	2644.84i	0.190161	−	0.329369i	0.945304	+	0.326192i	$0.105766\pi$
$402$		0			0
$403$	3888.00	+	6734.21i	0.480583	+	0.832395i
$404$		0			0
$405$	1134.00			0.139133
$406$		0			0
$407$	632.000			0.0769707
$408$		0			0
$409$	−133.000	−	230.363i	−0.0160793	−	0.0278501i	0.857874	−	0.513860i	$-0.171785\pi$
$409$	−133.000	−	230.363i	−0.0160793	−	0.0278501i	−0.873953	+	0.486010i	$0.838452\pi$
$410$		0			0
$411$	−4215.00	+	7300.59i	−0.505865	+	0.876184i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3836.00	−	6644.15i	0.453739	−	0.785900i
$416$		0			0
$417$	−6.00000	−	10.3923i	−0.000704607	−	0.00122042i
$418$		0			0
$419$	8844.00			1.03116			0.515582	−	0.856840i	$-0.327576\pi$
$419$	8844.00			1.03116			0.515582	+	0.856840i	$0.327576\pi$
$420$		0			0
$421$	−4482.00			−0.518858			−0.259429	−	0.965762i	$-0.583534\pi$
$421$	−4482.00			−0.518858			−0.259429	+	0.965762i	$0.583534\pi$
$422$		0			0
$423$	2376.00	+	4115.35i	0.273109	+	0.473039i
$424$		0			0
$425$	497.000	−	860.829i	0.0567248	−	0.0982502i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−324.000	+	561.184i	−0.0364636	+	0.0631567i
$430$		0			0
$431$	−4968.00	−	8604.83i	−0.555221	−	0.961671i	−0.997886	−	0.0649838i	$-0.979300\pi$
$431$	−4968.00	−	8604.83i	−0.555221	−	0.961671i	0.442666	−	0.896687i	$-0.354033\pi$
$432$		0			0
$433$	−11758.0			−1.30497			−0.652487	−	0.757800i	$-0.726274\pi$
$433$	−11758.0			−1.30497			−0.652487	+	0.757800i	$0.726274\pi$
$434$		0			0
$435$	−4452.00			−0.490706
$436$		0			0
$437$	6992.00	+	12110.5i	0.765384	+	1.32568i
$438$		0			0
$439$	2052.00	−	3554.17i	0.223090	−	0.386404i	−0.732655	−	0.680601i	$-0.761719\pi$
$439$	2052.00	−	3554.17i	0.223090	−	0.386404i	0.955745	+	0.294197i	$0.0950522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4874.00	+	8442.02i	−0.522733	+	0.905400i	0.476917	+	0.878948i	$0.341754\pi$
$443$	−4874.00	+	8442.02i	−0.522733	+	0.905400i	−0.999650	+	0.0264519i	$0.991579\pi$
$444$		0			0
$445$	−8358.00	−	14476.5i	−0.890353	−	1.54214i
$446$		0			0
$447$	4122.00			0.436161
$448$		0			0
$449$	−478.000			−0.0502410			−0.0251205	−	0.999684i	$-0.507997\pi$
$449$	−478.000			−0.0502410			−0.0251205	+	0.999684i	$0.507997\pi$
$450$		0			0
$451$	780.000	+	1351.00i	0.0814385	+	0.141056i
$452$		0			0
$453$	−3156.00	+	5466.35i	−0.327333	+	0.566957i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5587.00	−	9676.97i	0.571879	−	0.990524i	−0.424494	−	0.905431i	$-0.639548\pi$
$457$	5587.00	−	9676.97i	0.571879	−	0.990524i	0.996373	−	0.0850931i	$-0.0271188\pi$
$458$		0			0
$459$	189.000	+	327.358i	0.0192195	+	0.0332892i
$460$		0			0
$461$	−11674.0			−1.17942			−0.589710	−	0.807615i	$-0.700758\pi$
$461$	−11674.0			−1.17942			−0.589710	+	0.807615i	$0.700758\pi$
$462$		0			0
$463$	10528.0			1.05676			0.528378	−	0.849009i	$-0.322801\pi$
$463$	10528.0			1.05676			0.528378	+	0.849009i	$0.322801\pi$
$464$		0			0
$465$	3024.00	+	5237.72i	0.301580	+	0.522352i
$466$		0			0
$467$	−8302.00	+	14379.5i	−0.822635	+	1.42485i	0.0810777	+	0.996708i	$0.474164\pi$
$467$	−8302.00	+	14379.5i	−0.822635	+	1.42485i	−0.903713	+	0.428139i	$0.859170\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4809.00	+	8329.43i	−0.470461	+	0.814862i
$472$		0			0
$473$	1016.00	+	1759.76i	0.0987648	+	0.171066i
$474$		0			0
$475$	6532.00			0.630966
$476$		0			0
$477$	5454.00			0.523525
$478$		0			0
$479$	4288.00	+	7427.03i	0.409027	+	0.708455i	0.994781	−	0.102034i	$-0.0325350\pi$
$479$	4288.00	+	7427.03i	0.409027	+	0.708455i	−0.585754	+	0.810489i	$0.699202\pi$
$480$		0			0
$481$	−4266.00	+	7388.93i	−0.404393	+	0.700429i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10514.0	−	18210.8i	0.984363	−	1.70497i
$486$		0			0
$487$	−4852.00	−	8403.91i	−0.451468	−	0.781966i	0.547009	−	0.837127i	$-0.315766\pi$
$487$	−4852.00	−	8403.91i	−0.451468	−	0.781966i	−0.998478	+	0.0551605i	$0.982433\pi$
$488$		0			0
$489$	996.000			0.0921077
$490$		0			0
$491$	−4092.00			−0.376109			−0.188054	−	0.982159i	$-0.560218\pi$
$491$	−4092.00			−0.376109			−0.188054	+	0.982159i	$0.560218\pi$
$492$		0			0
$493$	−742.000	−	1285.18i	−0.0677850	−	0.117407i
$494$		0			0
$495$	−252.000	+	436.477i	−0.0228819	+	0.0396327i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8942.00	+	15488.0i	−0.802202	+	1.38945i	0.115961	+	0.993254i	$0.463005\pi$
$499$	−8942.00	+	15488.0i	−0.802202	+	1.38945i	−0.918164	+	0.396201i	$0.870328\pi$
$500$		0			0
$501$	−2244.00	−	3886.72i	−0.200109	−	0.346599i
$502$		0			0
$503$	−7704.00			−0.682911			−0.341456	−	0.939898i	$-0.610920\pi$
$503$	−7704.00			−0.682911			−0.341456	+	0.939898i	$0.610920\pi$
$504$		0			0
$505$	5572.00			0.490992
$506$		0			0
$507$	−1078.50	−	1868.02i	−0.0944731	−	0.163632i
$508$		0			0
$509$	−7179.00	+	12434.4i	−0.625154	+	1.08280i	0.363357	+	0.931650i	$0.381631\pi$
$509$	−7179.00	+	12434.4i	−0.625154	+	1.08280i	−0.988511	+	0.151149i	$0.951703\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1242.00	+	2151.21i	−0.106892	+	0.185143i
$514$		0			0
$515$	−8120.00	−	14064.3i	−0.694777	−	1.20339i
$516$		0			0
$517$	−2112.00			−0.179663
$518$		0			0
$519$	−9966.00			−0.842888
$520$		0			0
$521$	−2541.00	−	4401.14i	−0.213672	−	0.370091i	0.739189	−	0.673498i	$-0.235209\pi$
$521$	−2541.00	−	4401.14i	−0.213672	−	0.370091i	−0.952861	+	0.303407i	$0.901876\pi$
$522$		0			0
$523$	878.000	−	1520.74i	0.0734078	−	0.127146i	−0.826985	−	0.562224i	$-0.809946\pi$
$523$	878.000	−	1520.74i	0.0734078	−	0.127146i	0.900393	+	0.435078i	$0.143279\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1008.00	+	1745.91i	−0.0833191	+	0.144313i
$528$		0			0
$529$	−5468.50	−	9471.72i	−0.449453	−	0.778476i
$530$		0			0
$531$	−3276.00			−0.267733
$532$		0			0
$533$	−21060.0			−1.71146
$534$		0			0
$535$	−2268.00	−	3928.29i	−0.183279	−	0.317448i
$536$		0			0
$537$	1350.00	−	2338.27i	0.108486	−	0.187903i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8115.00	+	14055.6i	−0.644900	+	1.11700i	0.339424	+	0.940633i	$0.389768\pi$
$541$	−8115.00	+	14055.6i	−0.644900	+	1.11700i	−0.984325	+	0.176367i	$0.943566\pi$
$542$		0			0
$543$	−2853.00	−	4941.54i	−0.225477	−	0.390537i
$544$		0			0
$545$	−13132.0			−1.03213
$546$		0			0
$547$	17676.0			1.38167			0.690833	−	0.723014i	$-0.257244\pi$
$547$	17676.0			1.38167			0.690833	+	0.723014i	$0.257244\pi$
$548$		0			0
$549$	−3051.00	−	5284.49i	−0.237183	−	0.410813i
$550$		0			0
$551$	4876.00	−	8445.48i	0.376996	−	0.652976i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−3318.00	+	5746.94i	−0.253768	+	0.439539i
$556$		0			0
$557$	6125.00	+	10608.8i	0.465933	+	0.807019i	0.999243	−	0.0389004i	$-0.0123855\pi$
$557$	6125.00	+	10608.8i	0.465933	+	0.807019i	−0.533310	+	0.845920i	$0.679052\pi$
$558$		0			0
$559$	−27432.0			−2.07558
$560$		0			0
$561$	−168.000			−0.0126434
$562$		0			0
$563$	5026.00	+	8705.29i	0.376236	+	0.651659i	0.990511	−	0.137432i	$-0.0438850\pi$
$563$	5026.00	+	8705.29i	0.376236	+	0.651659i	−0.614276	+	0.789092i	$0.710552\pi$
$564$		0			0
$565$	4354.00	−	7541.35i	0.324202	−	0.561534i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12837.0	+	22234.3i	−0.945791	+	1.63816i	−0.191631	+	0.981467i	$0.561378\pi$
$569$	−12837.0	+	22234.3i	−0.945791	+	1.63816i	−0.754160	+	0.656691i	$0.771956\pi$
$570$		0			0
$571$	−1866.00	−	3232.01i	−0.136759	−	0.236874i	0.789509	−	0.613739i	$-0.210335\pi$
$571$	−1866.00	−	3232.01i	−0.136759	−	0.236874i	−0.926268	+	0.376865i	$0.877002\pi$
$572$		0			0
$573$	12384.0			0.902878
$574$		0			0
$575$	−10792.0			−0.782709
$576$		0			0
$577$	607.000	+	1051.35i	0.0437950	+	0.0758552i	0.887092	−	0.461593i	$-0.152722\pi$
$577$	607.000	+	1051.35i	0.0437950	+	0.0758552i	−0.843297	+	0.537448i	$0.819388\pi$
$578$		0			0
$579$	2013.00	−	3486.62i	0.144486	−	0.250257i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1212.00	+	2099.25i	−0.0860993	+	0.149128i
$584$		0			0
$585$	−3402.00	−	5892.44i	−0.240437	−	0.416448i
$586$		0			0
$587$	7108.00			0.499793			0.249897	−	0.968273i	$-0.419603\pi$
$587$	7108.00			0.499793			0.249897	+	0.968273i	$0.419603\pi$
$588$		0			0
$589$	−13248.0			−0.926782
$590$		0			0
$591$	5259.00	+	9108.86i	0.366034	+	0.633990i
$592$		0			0
$593$	−3081.00	+	5336.45i	−0.213358	+	0.369548i	−0.952763	−	0.303713i	$-0.901774\pi$
$593$	−3081.00	+	5336.45i	−0.213358	+	0.369548i	0.739405	+	0.673261i	$0.235107\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1020.00	+	1766.69i	−0.0699260	+	0.121115i
$598$		0			0
$599$	−1236.00	−	2140.81i	−0.0843098	−	0.146029i	0.820787	−	0.571234i	$-0.193535\pi$
$599$	−1236.00	−	2140.81i	−0.0843098	−	0.146029i	−0.905097	+	0.425205i	$0.860202\pi$
$600$		0			0
$601$	−13750.0			−0.933235			−0.466617	−	0.884459i	$-0.654528\pi$
$601$	−13750.0			−0.933235			−0.466617	+	0.884459i	$0.654528\pi$
$602$		0			0
$603$	7596.00			0.512990
$604$		0			0
$605$	9205.00	+	15943.5i	0.618573	+	1.07140i
$606$		0			0
$607$	5688.00	−	9851.90i	0.380344	−	0.658775i	−0.610767	−	0.791810i	$-0.709139\pi$
$607$	5688.00	−	9851.90i	0.380344	−	0.658775i	0.991111	+	0.133035i	$0.0424723\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	14256.0	−	24692.1i	0.943921	−	1.63492i
$612$		0			0
$613$	−10191.0	−	17651.3i	−0.671469	−	1.16302i	−0.977488	−	0.210993i	$-0.932330\pi$
$613$	−10191.0	−	17651.3i	−0.671469	−	1.16302i	0.306018	−	0.952026i	$-0.401003\pi$
$614$		0			0
$615$	−16380.0			−1.07399
$616$		0			0
$617$	21178.0			1.38184			0.690919	−	0.722932i	$-0.257206\pi$
$617$	21178.0			1.38184			0.690919	+	0.722932i	$0.257206\pi$
$618$		0			0
$619$	2350.00	+	4070.32i	0.152592	+	0.264297i	0.932180	−	0.361996i	$-0.117905\pi$
$619$	2350.00	+	4070.32i	0.152592	+	0.264297i	−0.779588	+	0.626293i	$0.784571\pi$
$620$		0			0
$621$	2052.00	−	3554.17i	0.132599	−	0.229668i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9729.50	−	16852.0i	0.622688	−	1.07853i
$626$		0			0
$627$	−552.000	−	956.092i	−0.0351591	−	0.0608973i
$628$		0			0
$629$	−2212.00			−0.140220
$630$		0			0
$631$	−21736.0			−1.37131			−0.685655	−	0.727927i	$-0.740484\pi$
$631$	−21736.0			−1.37131			−0.685655	+	0.727927i	$0.740484\pi$
$632$		0			0
$633$	−8058.00	−	13956.9i	−0.505966	−	0.876360i
$634$		0			0
$635$	−8400.00	+	14549.2i	−0.524951	+	0.909242i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	36.0000	−	62.3538i	0.00222870	−	0.00386022i
$640$		0			0
$641$	6511.00	+	11277.4i	0.401200	+	0.694898i	0.993871	−	0.110546i	$-0.0352600\pi$
$641$	6511.00	+	11277.4i	0.401200	+	0.694898i	−0.592671	+	0.805444i	$0.701927\pi$
$642$		0			0
$643$	3308.00			0.202885			0.101442	−	0.994841i	$-0.467654\pi$
$643$	3308.00			0.202885			0.101442	+	0.994841i	$0.467654\pi$
$644$		0			0
$645$	−21336.0			−1.30249
$646$		0			0
$647$	6900.00	+	11951.2i	0.419269	+	0.726195i	0.995866	−	0.0908335i	$-0.0289531\pi$
$647$	6900.00	+	11951.2i	0.419269	+	0.726195i	−0.576597	+	0.817029i	$0.695620\pi$
$648$		0			0
$649$	728.000	−	1260.93i	0.0440316	−	0.0762649i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1341.00	−	2322.68i	0.0803635	−	0.139194i	−0.823043	−	0.567980i	$-0.807725\pi$
$653$	1341.00	−	2322.68i	0.0803635	−	0.139194i	0.903406	+	0.428786i	$0.141059\pi$
$654$		0			0
$655$	9772.00	+	16925.6i	0.582937	+	1.00968i
$656$		0			0
$657$	−3798.00			−0.225531
$658$		0			0
$659$	23836.0			1.40898			0.704491	−	0.709713i	$-0.251175\pi$
$659$	23836.0			1.40898			0.704491	+	0.709713i	$0.251175\pi$
$660$		0			0
$661$	5641.00	+	9770.50i	0.331936	+	0.574929i	0.982891	−	0.184186i	$-0.0589650\pi$
$661$	5641.00	+	9770.50i	0.331936	+	0.574929i	−0.650956	+	0.759116i	$0.725632\pi$
$662$		0			0
$663$	1134.00	−	1964.15i	0.0664267	−	0.115054i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8056.00	+	13953.4i	−0.467661	+	0.810012i
$668$		0			0
$669$	1608.00	+	2785.14i	0.0929281	+	0.160956i
$670$		0			0
$671$	2712.00			0.156029
$672$		0			0
$673$	−13726.0			−0.786179			−0.393089	−	0.919500i	$-0.628594\pi$
$673$	−13726.0			−0.786179			−0.393089	+	0.919500i	$0.628594\pi$
$674$		0			0
$675$	−958.500	−	1660.17i	−0.0546558	−	0.0946667i
$676$		0			0
$677$	−2487.00	+	4307.61i	−0.141186	+	0.244542i	−0.927944	−	0.372721i	$-0.878425\pi$
$677$	−2487.00	+	4307.61i	−0.141186	+	0.244542i	0.786757	+	0.617263i	$0.211758\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	4302.00	−	7451.28i	0.242075	−	0.419286i
$682$		0			0
$683$	4494.00	+	7783.84i	0.251769	+	0.436076i	0.964013	−	0.265855i	$-0.0856544\pi$
$683$	4494.00	+	7783.84i	0.251769	+	0.436076i	−0.712244	+	0.701932i	$0.752321\pi$
$684$		0			0
$685$	39340.0			2.19431
$686$		0			0
$687$	14394.0			0.799367
$688$		0			0
$689$	−16362.0	−	28339.8i	−0.904706	−	1.56700i
$690$		0			0
$691$	−5086.00	+	8809.21i	−0.280001	+	0.484976i	−0.971385	−	0.237512i	$-0.923668\pi$
$691$	−5086.00	+	8809.21i	−0.280001	+	0.484976i	0.691384	+	0.722488i	$0.257001\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−28.0000	+	48.4974i	−0.00152820	+	0.00264692i
$696$		0			0
$697$	−2730.00	−	4728.50i	−0.148359	−	0.256965i
$698$		0			0
$699$	−15378.0			−0.832116
$700$		0			0
$701$	27446.0			1.47877			0.739387	−	0.673280i	$-0.235115\pi$
$701$	27446.0			1.47877			0.739387	+	0.673280i	$0.235115\pi$
$702$		0			0
$703$	−7268.00	−	12588.5i	−0.389926	−	0.675371i
$704$		0			0
$705$	11088.0	−	19205.0i	0.592338	−	1.02596i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1999.00	+	3462.37i	−0.105887	+	0.183402i	−0.914100	−	0.405488i	$-0.867102\pi$
$709$	−1999.00	+	3462.37i	−0.105887	+	0.183402i	0.808213	+	0.588890i	$0.200435\pi$
$710$		0			0
$711$	−1728.00	−	2992.98i	−0.0911464	−	0.157870i
$712$		0			0
$713$	21888.0			1.14967
$714$		0			0
$715$	3024.00			0.158169
$716$		0			0
$717$	792.000	+	1371.78i	0.0412521	+	0.0714508i
$718$		0			0
$719$	−12936.0	+	22405.8i	−0.670976	+	1.16216i	0.306652	+	0.951822i	$0.400791\pi$
$719$	−12936.0	+	22405.8i	−0.670976	+	1.16216i	−0.977628	+	0.210342i	$0.932542\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	1221.00	−	2114.83i	0.0628070	−	0.108785i
$724$		0			0
$725$	3763.00	+	6517.71i	0.192765	+	0.333878i
$726$		0			0
$727$	12088.0			0.616670			0.308335	−	0.951278i	$-0.400228\pi$
$727$	12088.0			0.616670			0.308335	+	0.951278i	$0.400228\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−3556.00	−	6159.17i	−0.179923	−	0.311635i
$732$		0			0
$733$	−3987.00	+	6905.69i	−0.200905	+	0.347977i	−0.948820	−	0.315817i	$-0.897722\pi$
$733$	−3987.00	+	6905.69i	−0.200905	+	0.347977i	0.747915	+	0.663794i	$0.231055\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1688.00	+	2923.70i	−0.0843667	+	0.146127i
$738$		0			0
$739$	15882.0	+	27508.4i	0.790567	+	1.36930i	0.925616	+	0.378463i	$0.123547\pi$
$739$	15882.0	+	27508.4i	0.790567	+	1.36930i	−0.135050	+	0.990839i	$0.543119\pi$
$740$		0			0
$741$	14904.0			0.738883
$742$		0			0
$743$	888.000			0.0438460			0.0219230	−	0.999760i	$-0.493021\pi$
$743$	888.000			0.0438460			0.0219230	+	0.999760i	$0.493021\pi$
$744$		0			0
$745$	−9618.00	−	16658.9i	−0.472988	−	0.819240i
$746$		0			0
$747$	2466.00	−	4271.24i	0.120785	−	0.209205i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17328.0	−	30013.0i	0.841954	−	1.45831i	−0.0462858	−	0.998928i	$-0.514738\pi$
$751$	17328.0	−	30013.0i	0.841954	−	1.45831i	0.888240	−	0.459379i	$-0.151928\pi$
$752$		0			0
$753$	2898.00	+	5019.48i	0.140251	+	0.242922i
$754$		0			0
$755$	29456.0			1.41989
$756$		0			0
$757$	−22866.0			−1.09786			−0.548929	−	0.835869i	$-0.684964\pi$
$757$	−22866.0			−1.09786			−0.548929	+	0.835869i	$0.684964\pi$
$758$		0			0
$759$	912.000	+	1579.63i	0.0436146	+	0.0755428i
$760$		0			0
$761$	−11285.0	+	19546.2i	−0.537557	+	0.931076i	0.461478	+	0.887152i	$0.347319\pi$
$761$	−11285.0	+	19546.2i	−0.537557	+	0.931076i	−0.999035	+	0.0439244i	$0.986014\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	882.000	−	1527.67i	0.0416847	−	0.0722000i
$766$		0			0
$767$	9828.00	+	17022.6i	0.462671	+	0.801369i
$768$		0			0
$769$	−1790.00			−0.0839389			−0.0419695	−	0.999119i	$-0.513363\pi$
$769$	−1790.00			−0.0839389			−0.0419695	+	0.999119i	$0.513363\pi$
$770$		0			0
$771$	−9882.00			−0.461597
$772$		0			0
$773$	−1495.00	−	2589.42i	−0.0695620	−	0.120485i	0.829147	−	0.559031i	$-0.188827\pi$
$773$	−1495.00	−	2589.42i	−0.0695620	−	0.120485i	−0.898709	+	0.438546i	$0.855493\pi$
$774$		0			0
$775$	5112.00	−	8854.24i	0.236940	−	0.410392i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17940.0	−	31073.0i	0.825118	−	1.42915i
$780$		0			0
$781$	16.0000	+	27.7128i	0.000733067	+	0.00126971i
$782$		0			0
$783$	−2862.00			−0.130625
$784$		0			0
$785$	44884.0			2.04074
$786$		0			0
$787$	15378.0	+	26635.5i	0.696527	+	1.20642i	0.969663	+	0.244444i	$0.0786056\pi$
$787$	15378.0	+	26635.5i	0.696527	+	1.20642i	−0.273137	+	0.961975i	$0.588061\pi$
$788$		0			0

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-7.00000 + 12.1244i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-7.00000 + 12.1244i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-2.00000 - 3.46410i) q^{11} +54.0000 q^{13} +42.0000 q^{15} +(7.00000 + 12.1244i) q^{17} +(-46.0000 + 79.6743i) q^{19} +(76.0000 - 131.636i) q^{23} +(-35.5000 - 61.4878i) q^{25} +27.0000 q^{27} -106.000 q^{29} +(72.0000 + 124.708i) q^{31} +(-6.00000 + 10.3923i) q^{33} +(-79.0000 + 136.832i) q^{37} +(-81.0000 - 140.296i) q^{39} -390.000 q^{41} -508.000 q^{43} +(-63.0000 - 109.119i) q^{45} +(264.000 - 457.261i) q^{47} +(21.0000 - 36.3731i) q^{51} +(-303.000 - 524.811i) q^{53} +56.0000 q^{55} +276.000 q^{57} +(182.000 + 315.233i) q^{59} +(-339.000 + 587.165i) q^{61} +(-378.000 + 654.715i) q^{65} +(-422.000 - 730.925i) q^{67} -456.000 q^{69} -8.00000 q^{71} +(211.000 + 365.463i) q^{73} +(-106.500 + 184.463i) q^{75} +(-192.000 + 332.554i) q^{79} +(-40.5000 - 70.1481i) q^{81} -548.000 q^{83} -196.000 q^{85} +(159.000 + 275.396i) q^{87} +(-597.000 + 1034.03i) q^{89} +(216.000 - 374.123i) q^{93} +(-644.000 - 1115.44i) q^{95} -1502.00 q^{97} +36.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 14 q^{5} - 9 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 14 * q^5 - 9 * q^9	\( 2 q - 3 q^{3} - 14 q^{5} - 9 q^{9} - 4 q^{11} + 108 q^{13} + 84 q^{15} + 14 q^{17} - 92 q^{19} + 152 q^{23} - 71 q^{25} + 54 q^{27} - 212 q^{29} + 144 q^{31} - 12 q^{33} - 158 q^{37} - 162 q^{39} - 780 q^{41} - 1016 q^{43} - 126 q^{45} + 528 q^{47} + 42 q^{51} - 606 q^{53} + 112 q^{55} + 552 q^{57} + 364 q^{59} - 678 q^{61} - 756 q^{65} - 844 q^{67} - 912 q^{69} - 16 q^{71} + 422 q^{73} - 213 q^{75} - 384 q^{79} - 81 q^{81} - 1096 q^{83} - 392 q^{85} + 318 q^{87} - 1194 q^{89} + 432 q^{93} - 1288 q^{95} - 3004 q^{97} + 72 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 14 * q^5 - 9 * q^9 - 4 * q^11 + 108 * q^13 + 84 * q^15 + 14 * q^17 - 92 * q^19 + 152 * q^23 - 71 * q^25 + 54 * q^27 - 212 * q^29 + 144 * q^31 - 12 * q^33 - 158 * q^37 - 162 * q^39 - 780 * q^41 - 1016 * q^43 - 126 * q^45 + 528 * q^47 + 42 * q^51 - 606 * q^53 + 112 * q^55 + 552 * q^57 + 364 * q^59 - 678 * q^61 - 756 * q^65 - 844 * q^67 - 912 * q^69 - 16 * q^71 + 422 * q^73 - 213 * q^75 - 384 * q^79 - 81 * q^81 - 1096 * q^83 - 392 * q^85 + 318 * q^87 - 1194 * q^89 + 432 * q^93 - 1288 * q^95 - 3004 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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