LMFDB - Embedded newform 720.2.q.e.241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(241,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("720.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.q (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:	$5.74922894553$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			241.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	720.241
Dual form			720.2.q.e.481.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(1.50000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +(-2.50000 + 4.33013i) q^{11} +1.73205i q^{15} +3.00000 q^{17} -5.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.19615i q^{27} +(5.00000 - 8.66025i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-7.50000 + 4.33013i) q^{33} +4.00000 q^{37} +(1.50000 + 2.59808i) q^{41} +(1.50000 - 2.59808i) q^{43} +(-1.50000 + 2.59808i) q^{45} +(2.00000 - 3.46410i) q^{47} +(3.50000 + 6.06218i) q^{49} +(4.50000 + 2.59808i) q^{51} -6.00000 q^{53} -5.00000 q^{55} +(-7.50000 - 4.33013i) q^{57} +(-1.50000 - 2.59808i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-5.50000 - 9.52628i) q^{67} +10.3923i q^{69} +14.0000 q^{71} -15.0000 q^{73} +(-1.50000 + 0.866025i) q^{75} +(5.00000 - 8.66025i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-6.00000 + 10.3923i) q^{83} +(1.50000 + 2.59808i) q^{85} +(15.0000 - 8.66025i) q^{87} +14.0000 q^{89} -3.46410i q^{93} +(-2.50000 - 4.33013i) q^{95} +(6.50000 - 11.2583i) q^{97} -15.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + q^5 + 3 * q^9	$ 2 q + 3 q^{3} + q^{5} + 3 q^{9} - 5 q^{11} + 6 q^{17} - 10 q^{19} + 6 q^{23} - q^{25} + 10 q^{29} - 2 q^{31} - 15 q^{33} + 8 q^{37} + 3 q^{41} + 3 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} + 9 q^{51} - 12 q^{53} - 10 q^{55} - 15 q^{57} - 3 q^{59} - 2 q^{61} - 11 q^{67} + 28 q^{71} - 30 q^{73} - 3 q^{75} + 10 q^{79} - 9 q^{81} - 12 q^{83} + 3 q^{85} + 30 q^{87} + 28 q^{89} - 5 q^{95} + 13 q^{97} - 30 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + q^5 + 3 * q^9 - 5 * q^11 + 6 * q^17 - 10 * q^19 + 6 * q^23 - q^25 + 10 * q^29 - 2 * q^31 - 15 * q^33 + 8 * q^37 + 3 * q^41 + 3 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 + 9 * q^51 - 12 * q^53 - 10 * q^55 - 15 * q^57 - 3 * q^59 - 2 * q^61 - 11 * q^67 + 28 * q^71 - 30 * q^73 - 3 * q^75 + 10 * q^79 - 9 * q^81 - 12 * q^83 + 3 * q^85 + 30 * q^87 + 28 * q^89 - 5 * q^95 + 13 * q^97 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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	+	0.866025i	0.866025	+	0.500000i
$3$	1.50000	+	0.866025i	0.866025	+	0.500000i
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	$0.438437\pi$
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	−0.945979	+	0.324227i	$0.894896\pi$
$12$		0			0
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$			1.73205i			0.447214i
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$			5.19615i			1.00000i
$28$		0			0
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.142605	−	0.989780i	$-0.454452\pi$
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.785872	−	0.618389i	$-0.212214\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$		0			0
$33$	−7.50000	+	4.33013i	−1.30558	+	0.753778i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	0.959058	−	0.283211i	$-0.0913998\pi$
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	−0.724797	+	0.688963i	$0.758066\pi$
$42$		0			0
$43$	1.50000	−	2.59808i	0.228748	−	0.396203i	−0.728689	−	0.684844i	$-0.759870\pi$
$43$	1.50000	−	2.59808i	0.228748	−	0.396203i	0.957437	+	0.288641i	$0.0932035\pi$
$44$		0			0
$45$	−1.50000	+	2.59808i	−0.223607	+	0.387298i
$46$		0			0
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	−0.682489	−	0.730896i	$-0.739102\pi$
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	0.974219	+	0.225605i	$0.0724358\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$	4.50000	+	2.59808i	0.630126	+	0.363803i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$	−7.50000	−	4.33013i	−0.993399	−	0.573539i
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$			10.3923i			1.25109i
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	−15.0000			−1.75562			−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000			−1.75562			−0.877809	+	0.479012i	$0.840995\pi$
$74$		0			0
$75$	−1.50000	+	0.866025i	−0.173205	+	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	1.50000	+	2.59808i	0.162698	+	0.281801i
$86$		0			0
$87$	15.0000	−	8.66025i	1.60817	−	0.928477i
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	3.46410i		−	0.359211i
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	−0.320647	−	0.947199i	$-0.603900\pi$
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	0.980622	−	0.195911i	$-0.0627665\pi$
$98$		0			0
$99$	−15.0000			−1.50756
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.0000			1.64345			0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000			1.64345			0.821726	+	0.569883i	$0.193011\pi$
$108$		0			0
$109$	−12.0000			−1.14939			−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000			−1.14939			−0.574696	+	0.818367i	$0.694880\pi$
$110$		0			0
$111$	6.00000	+	3.46410i	0.569495	+	0.328798i
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	−3.00000	+	5.19615i	−0.279751	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$			5.19615i			0.468521i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−10.0000			−0.887357			−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000			−0.887357			−0.443678	+	0.896186i	$0.646327\pi$
$128$		0			0
$129$	4.50000	−	2.59808i	0.396203	−	0.228748i
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.50000	+	2.59808i	−0.387298	+	0.223607i
$136$		0			0
$137$	−3.50000	+	6.06218i	−0.299025	+	0.517927i	−0.975913	−	0.218159i	$-0.929995\pi$
$137$	−3.50000	+	6.06218i	−0.299025	+	0.517927i	0.676888	+	0.736086i	$0.263328\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$	6.00000	−	3.46410i	0.505291	−	0.291730i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$			12.1244i			1.00000i
$148$		0			0
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	0.772399	−	0.635138i	$-0.219057\pi$
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	−0.936245	+	0.351348i	$0.885723\pi$
$150$		0			0
$151$	11.0000	−	19.0526i	0.895167	−	1.55048i	0.0615699	−	0.998103i	$-0.480389\pi$
$151$	11.0000	−	19.0526i	0.895167	−	1.55048i	0.833597	−	0.552372i	$-0.186277\pi$
$152$		0			0
$153$	4.50000	+	7.79423i	0.363803	+	0.630126i
$154$		0			0
$155$	1.00000	−	1.73205i	0.0803219	−	0.139122i
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$	−9.00000	−	5.19615i	−0.713746	−	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$	−7.50000	−	4.33013i	−0.583874	−	0.337100i
$166$		0			0
$167$	−11.0000	−	19.0526i	−0.851206	−	1.47433i	−0.880121	−	0.474749i	$-0.842539\pi$
$167$	−11.0000	−	19.0526i	−0.851206	−	1.47433i	0.0289155	−	0.999582i	$-0.490795\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$	−7.50000	−	12.9904i	−0.573539	−	0.993399i
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	5.19615i		−	0.390567i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−4.00000			−0.297318			−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000			−0.297318			−0.148659	+	0.988889i	$0.547496\pi$
$182$		0			0
$183$	−3.00000	+	1.73205i	−0.221766	+	0.128037i
$184$		0			0
$185$	2.00000	+	3.46410i	0.147043	+	0.254686i
$186$		0			0
$187$	−7.50000	+	12.9904i	−0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		0			0
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	−0.973805	−	0.227387i	$-0.926982\pi$
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	0.289980	−	0.957033i	$-0.406351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$		−	19.0526i		−	1.34386i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.50000	+	2.59808i	−0.104765	+	0.181458i
$206$		0			0
$207$	−9.00000	+	15.5885i	−0.625543	+	1.08347i
$208$		0			0
$209$	12.5000	−	21.6506i	0.864643	−	1.49761i
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$	21.0000	+	12.1244i	1.43890	+	0.830747i
$214$		0			0
$215$	3.00000			0.204598
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−22.5000	−	12.9904i	−1.52041	−	0.877809i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	−0.830599	−	0.556871i	$-0.812002\pi$
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	0.897564	+	0.440884i	$0.145335\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	−0.565576	−	0.824696i	$-0.691346\pi$
$227$	6.50000	−	11.2583i	0.431420	−	0.747242i	0.996996	+	0.0774548i	$0.0246793\pi$
$228$		0			0
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	0.999442	+	0.0334101i	$0.0106368\pi$
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	−0.470787	+	0.882247i	$0.656030\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0000			−0.720634			−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000			−0.720634			−0.360317	+	0.932830i	$0.617331\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$		0			0
$237$	15.0000	−	8.66025i	0.974355	−	0.562544i
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	−0.211360	−	0.977408i	$-0.567789\pi$
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	0.952141	−	0.305661i	$-0.0988773\pi$
$242$		0			0
$243$	−13.5000	+	7.79423i	−0.866025	+	0.500000i
$244$		0			0
$245$	−3.50000	+	6.06218i	−0.223607	+	0.387298i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−18.0000	+	10.3923i	−1.14070	+	0.658586i
$250$		0			0
$251$	−11.0000			−0.694314			−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$			5.19615i			0.325396i
$256$		0			0
$257$	15.5000	+	26.8468i	0.966863	+	1.67466i	0.704523	+	0.709681i	$0.251161\pi$
$257$	15.5000	+	26.8468i	0.966863	+	1.67466i	0.262341	+	0.964975i	$0.415506\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	30.0000			1.85695
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−3.00000	−	5.19615i	−0.184289	−	0.319197i
$266$		0			0
$267$	21.0000	+	12.1244i	1.28518	+	0.741999i
$268$		0			0
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.50000	−	4.33013i	−0.150756	−	0.261116i
$276$		0			0
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	−0.575408	−	0.817867i	$-0.695157\pi$
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	0.995997	+	0.0893846i	$0.0284900\pi$
$278$		0			0
$279$	3.00000	−	5.19615i	0.179605	−	0.311086i
$280$		0			0
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	−0.975741	−	0.218926i	$-0.929745\pi$
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	0.677466	+	0.735554i	$0.263078\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		−	8.66025i		−	0.512989i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	19.5000	−	11.2583i	1.14311	−	0.659975i
$292$		0			0
$293$	5.00000	+	8.66025i	0.292103	+	0.505937i	0.974307	−	0.225225i	$-0.0723116\pi$
$293$	5.00000	+	8.66025i	0.292103	+	0.505937i	−0.682204	+	0.731162i	$0.738978\pi$
$294$		0			0
$295$	1.50000	−	2.59808i	0.0873334	−	0.151266i
$296$		0			0
$297$	−22.5000	−	12.9904i	−1.30558	−	0.753778i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	18.0000	−	10.3923i	1.03407	−	0.597022i
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	0.972250	−	0.233944i	$-0.0751631\pi$
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	−0.688726	+	0.725022i	$0.741830\pi$
$312$		0			0
$313$	−3.50000	+	6.06218i	−0.197832	+	0.342655i	−0.947825	−	0.318791i	$-0.896723\pi$
$313$	−3.50000	+	6.06218i	−0.197832	+	0.342655i	0.749993	+	0.661445i	$0.230057\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.0000	+	24.2487i	−0.786318	+	1.36194i	0.141890	+	0.989882i	$0.454682\pi$
$317$	−14.0000	+	24.2487i	−0.786318	+	1.36194i	−0.928208	+	0.372061i	$0.878651\pi$
$318$		0			0
$319$	25.0000	+	43.3013i	1.39973	+	2.42441i
$320$		0			0
$321$	25.5000	+	14.7224i	1.42327	+	0.821726i
$322$		0			0
$323$	−15.0000			−0.834622
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−18.0000	−	10.3923i	−0.995402	−	0.574696i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	−0.448649	−	0.893708i	$-0.648095\pi$
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	0.998298	−	0.0583130i	$-0.0185721\pi$
$332$		0			0
$333$	6.00000	+	10.3923i	0.328798	+	0.569495i
$334$		0			0
$335$	5.50000	−	9.52628i	0.300497	−	0.520476i
$336$		0			0
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	0.886194	+	0.463314i	$0.153340\pi$
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	−0.0418554	+	0.999124i	$0.513327\pi$
$338$		0			0
$339$			10.3923i			0.564433i
$340$		0			0
$341$	10.0000			0.541530
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−9.00000	+	5.19615i	−0.484544	+	0.279751i
$346$		0			0
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	0.822951	−	0.568112i	$-0.192326\pi$
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	−0.903475	+	0.428640i	$0.858993\pi$
$348$		0			0
$349$	−16.0000	+	27.7128i	−0.856460	+	1.48343i	0.0188232	+	0.999823i	$0.494008\pi$
$349$	−16.0000	+	27.7128i	−0.856460	+	1.48343i	−0.875284	+	0.483610i	$0.839325\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.5000	+	21.6506i	−0.665308	+	1.15235i	0.313894	+	0.949458i	$0.398366\pi$
$353$	−12.5000	+	21.6506i	−0.665308	+	1.15235i	−0.979202	+	0.202889i	$0.934967\pi$
$354$		0			0
$355$	7.00000	+	12.1244i	0.371521	+	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		−	24.2487i		−	1.27273i
$364$		0			0
$365$	−7.50000	−	12.9904i	−0.392568	−	0.679948i
$366$		0			0
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	−0.951336	−	0.308155i	$-0.900289\pi$
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	0.742538	+	0.669804i	$0.233622\pi$
$368$		0			0
$369$	−4.50000	+	7.79423i	−0.234261	+	0.405751i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	0.977016	+	0.213165i	$0.0683772\pi$
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	−0.303902	+	0.952703i	$0.598289\pi$
$374$		0			0
$375$	−1.50000	−	0.866025i	−0.0774597	−	0.0447214i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	35.0000			1.79783			0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000			1.79783			0.898915	+	0.438124i	$0.144357\pi$
$380$		0			0
$381$	−15.0000	−	8.66025i	−0.768473	−	0.443678i
$382$		0			0
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	0.977613	−	0.210411i	$-0.0674801\pi$
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	−0.671027	+	0.741433i	$0.734147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.00000			0.457496
$388$		0			0
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	−0.383076	−	0.923717i	$-0.625135\pi$
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	0.991500	−	0.130105i	$-0.0415314\pi$
$390$		0			0
$391$	9.00000	+	15.5885i	0.455150	+	0.788342i
$392$		0			0
$393$		−	6.92820i		−	0.349482i
$394$		0			0
$395$	10.0000			0.503155
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.50000	+	6.06218i	0.174782	+	0.302731i	0.940086	−	0.340938i	$-0.110745\pi$
$401$	3.50000	+	6.06218i	0.174782	+	0.302731i	−0.765304	+	0.643669i	$0.777411\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−9.00000			−0.447214
$406$		0			0
$407$	−10.0000	+	17.3205i	−0.495682	+	0.858546i
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$	−10.5000	+	6.06218i	−0.517927	+	0.299025i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$			12.1244i			0.593732i
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.136064	+	0.990700i	$0.543445\pi$
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.789940	+	0.613185i	$0.789888\pi$
$422$		0			0
$423$	12.0000			0.583460
$424$		0			0
$425$	−1.50000	+	2.59808i	−0.0727607	+	0.126025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.0000			−1.34871			−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000			−1.34871			−0.674356	+	0.738406i	$0.735579\pi$
$432$		0			0
$433$	27.0000			1.29754			0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000			1.29754			0.648769	+	0.760986i	$0.275284\pi$
$434$		0			0
$435$	15.0000	+	8.66025i	0.719195	+	0.415227i
$436$		0			0
$437$	−15.0000	−	25.9808i	−0.717547	−	1.24283i
$438$		0			0
$439$	−7.00000	+	12.1244i	−0.334092	+	0.578664i	−0.983310	−	0.181938i	$-0.941763\pi$
$439$	−7.00000	+	12.1244i	−0.334092	+	0.578664i	0.649218	+	0.760602i	$0.275096\pi$
$440$		0			0
$441$	−10.5000	+	18.1865i	−0.500000	+	0.866025i
$442$		0			0
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	0.283273	+	0.959039i	$0.408580\pi$
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	−0.972189	+	0.234198i	$0.924754\pi$
$444$		0			0
$445$	7.00000	+	12.1244i	0.331832	+	0.574750i
$446$		0			0
$447$		−	6.92820i		−	0.327693i
$448$		0			0
$449$	9.00000			0.424736			0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000			0.424736			0.212368	+	0.977190i	$0.431882\pi$
$450$		0			0
$451$	−15.0000			−0.706322
$452$		0			0
$453$	33.0000	−	19.0526i	1.55048	−	0.895167i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.5000	−	21.6506i	0.584725	−	1.01277i	−0.410184	−	0.912003i	$-0.634536\pi$
$457$	12.5000	−	21.6506i	0.584725	−	1.01277i	0.994910	−	0.100771i	$-0.0321310\pi$
$458$		0			0
$459$			15.5885i			0.727607i
$460$		0			0
$461$	−2.00000	+	3.46410i	−0.0931493	+	0.161339i	−0.908835	−	0.417156i	$-0.863027\pi$
$461$	−2.00000	+	3.46410i	−0.0931493	+	0.161339i	0.815685	+	0.578496i	$0.196360\pi$
$462$		0			0
$463$	−2.00000	−	3.46410i	−0.0929479	−	0.160990i	0.815802	−	0.578331i	$-0.196296\pi$
$463$	−2.00000	−	3.46410i	−0.0929479	−	0.160990i	−0.908750	+	0.417340i	$0.862962\pi$
$464$		0			0
$465$	3.00000	−	1.73205i	0.139122	−	0.0803219i
$466$		0			0
$467$	11.0000			0.509019			0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000			0.509019			0.254510	+	0.967070i	$0.418086\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.50000	+	12.9904i	0.344850	+	0.597298i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$	−9.00000	−	15.5885i	−0.412082	−	0.713746i
$478$		0			0
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	−0.399704	−	0.916644i	$-0.630887\pi$
$479$	13.0000	−	22.5167i	0.593985	−	1.02881i	0.993689	−	0.112168i	$-0.0357796\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.0000			0.590300
$486$		0			0
$487$	−34.0000			−1.54069			−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000			−1.54069			−0.770344	+	0.637629i	$0.779915\pi$
$488$		0			0
$489$	−24.0000	−	13.8564i	−1.08532	−	0.626608i
$490$		0			0
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	0.984145	−	0.177365i	$-0.0567572\pi$
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	−0.645675	+	0.763612i	$0.723424\pi$
$492$		0			0
$493$	15.0000	−	25.9808i	0.675566	−	1.17011i
$494$		0			0
$495$	−7.50000	−	12.9904i	−0.337100	−	0.583874i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−11.5000	−	19.9186i	−0.514811	−	0.891678i	−0.999852	−	0.0171872i	$-0.994529\pi$
$499$	−11.5000	−	19.9186i	−0.514811	−	0.891678i	0.485042	−	0.874491i	$-0.338804\pi$
$500$		0			0
$501$		−	38.1051i		−	1.70241i
$502$		0			0
$503$	−34.0000			−1.51599			−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000			−1.51599			−0.757993	+	0.652263i	$0.773820\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$	19.5000	−	11.2583i	0.866025	−	0.500000i
$508$		0			0
$509$	−22.0000	−	38.1051i	−0.975133	−	1.68898i	−0.679496	−	0.733679i	$-0.737801\pi$
$509$	−22.0000	−	38.1051i	−0.975133	−	1.68898i	−0.295637	−	0.955300i	$-0.595532\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	25.9808i		−	1.14708i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	10.0000	+	17.3205i	0.439799	+	0.761755i
$518$		0			0
$519$	3.00000	−	1.73205i	0.131685	−	0.0760286i
$520$		0			0
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	4.50000	−	7.79423i	0.195283	−	0.338241i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	8.50000	+	14.7224i	0.367487	+	0.636506i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−35.0000			−1.50756
$540$		0			0
$541$	24.0000			1.03184			0.515920	−	0.856637i	$-0.327450\pi$
$541$	24.0000			1.03184			0.515920	+	0.856637i	$0.327450\pi$
$542$		0			0
$543$	−6.00000	−	3.46410i	−0.257485	−	0.148659i
$544$		0			0
$545$	−6.00000	−	10.3923i	−0.257012	−	0.445157i
$546$		0			0
$547$	−4.50000	+	7.79423i	−0.192406	+	0.333257i	−0.946047	−	0.324029i	$-0.894962\pi$
$547$	−4.50000	+	7.79423i	−0.192406	+	0.333257i	0.753641	+	0.657286i	$0.228296\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$	−25.0000	+	43.3013i	−1.06504	+	1.84470i
$552$		0			0
$553$		0			0
$554$		0			0
$555$			6.92820i			0.294086i
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−22.5000	+	12.9904i	−0.949951	+	0.548454i
$562$		0			0
$563$	2.50000	+	4.33013i	0.105362	+	0.182493i	0.913886	−	0.405970i	$-0.133066\pi$
$563$	2.50000	+	4.33013i	0.105362	+	0.182493i	−0.808524	+	0.588463i	$0.799733\pi$
$564$		0			0
$565$	−3.00000	+	5.19615i	−0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.50000	−	12.9904i	0.314416	−	0.544585i	−0.664897	−	0.746935i	$-0.731525\pi$
$569$	7.50000	−	12.9904i	0.314416	−	0.544585i	0.979313	+	0.202350i	$0.0648579\pi$
$570$		0			0
$571$	−1.50000	−	2.59808i	−0.0627730	−	0.108726i	0.832931	−	0.553377i	$-0.186661\pi$
$571$	−1.50000	−	2.59808i	−0.0627730	−	0.108726i	−0.895704	+	0.444651i	$0.853328\pi$
$572$		0			0
$573$	18.0000	−	10.3923i	0.751961	−	0.434145i
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	−31.0000			−1.29055			−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000			−1.29055			−0.645273	+	0.763952i	$0.723257\pi$
$578$		0			0
$579$		−	32.9090i		−	1.36765i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	15.0000	−	25.9808i	0.621237	−	1.07601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.50000	+	14.7224i	−0.350833	+	0.607660i	−0.986396	−	0.164389i	$-0.947435\pi$
$587$	−8.50000	+	14.7224i	−0.350833	+	0.607660i	0.635563	+	0.772049i	$0.280768\pi$
$588$		0			0
$589$	5.00000	+	8.66025i	0.206021	+	0.356840i
$590$		0			0
$591$	−12.0000	−	6.92820i	−0.493614	−	0.284988i
$592$		0			0
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	36.0000	+	20.7846i	1.47338	+	0.850657i
$598$		0			0
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	0.981889	−	0.189456i	$-0.0606724\pi$
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	−0.655018	+	0.755613i	$0.727339\pi$
$600$		0			0
$601$	0.500000	−	0.866025i	0.0203954	−	0.0353259i	−0.855648	−	0.517559i	$-0.826841\pi$
$601$	0.500000	−	0.866025i	0.0203954	−	0.0353259i	0.876043	+	0.482233i	$0.160174\pi$
$602$		0			0
$603$	16.5000	−	28.5788i	0.671932	−	1.16382i
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$		0			0
$615$	−4.50000	+	2.59808i	−0.181458	+	0.104765i
$616$		0			0
$617$	−12.5000	−	21.6506i	−0.503231	−	0.871622i	−0.999993	−	0.00373492i	$-0.998811\pi$
$617$	−12.5000	−	21.6506i	−0.503231	−	0.871622i	0.496762	−	0.867887i	$-0.334522\pi$
$618$		0			0
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	−0.991325	−	0.131434i	$-0.958042\pi$
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	0.609488	+	0.792796i	$0.291375\pi$
$620$		0			0
$621$	−27.0000	+	15.5885i	−1.08347	+	0.625543i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	37.5000	−	21.6506i	1.49761	−	0.864643i
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$		0			0
$633$			20.7846i			0.826114i
$634$		0			0
$635$	−5.00000	−	8.66025i	−0.198419	−	0.343672i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	21.0000	+	36.3731i	0.830747	+	1.43890i
$640$		0			0
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	0.330997	+	0.943632i	$0.392615\pi$
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	−0.982708	+	0.185164i	$0.940718\pi$
$642$		0			0
$643$	−20.5000	−	35.5070i	−0.808441	−	1.40026i	−0.913943	−	0.405842i	$-0.866978\pi$
$643$	−20.5000	−	35.5070i	−0.808441	−	1.40026i	0.105502	−	0.994419i	$-0.466355\pi$
$644$		0			0
$645$	4.50000	+	2.59808i	0.177187	+	0.102299i
$646$		0			0
$647$	−30.0000			−1.17942			−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000			−1.17942			−0.589711	+	0.807614i	$0.700758\pi$
$648$		0			0
$649$	15.0000			0.588802
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	−0.979214	−	0.202828i	$-0.934987\pi$
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	0.313953	−	0.949439i	$-0.398347\pi$
$654$		0			0
$655$	2.00000	−	3.46410i	0.0781465	−	0.135354i
$656$		0			0
$657$	−22.5000	−	38.9711i	−0.877809	−	1.52041i
$658$		0			0
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	−0.0177803	−	0.999842i	$-0.505660\pi$
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	0.874779	−	0.484523i	$-0.161007\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	60.0000			2.32321
$668$		0			0
$669$	3.00000	−	1.73205i	0.115987	−	0.0669650i
$670$		0			0
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	−0.498886	−	0.866668i	$-0.666257\pi$
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	0.999999	−	0.00128586i	$-0.000409302\pi$
$674$		0			0
$675$	−4.50000	−	2.59808i	−0.173205	−	0.100000i
$676$		0			0
$677$	−14.0000	+	24.2487i	−0.538064	+	0.931954i	0.460945	+	0.887429i	$0.347511\pi$
$677$	−14.0000	+	24.2487i	−0.538064	+	0.931954i	−0.999008	+	0.0445248i	$0.985823\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	19.5000	−	11.2583i	0.747242	−	0.431420i
$682$		0			0
$683$	13.0000			0.497431			0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000			0.497431			0.248716	+	0.968577i	$0.419992\pi$
$684$		0			0
$685$	−7.00000			−0.267456
$686$		0			0
$687$			27.7128i			1.05731i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.50000	+	6.06218i	−0.132763	+	0.229952i
$696$		0			0
$697$	4.50000	+	7.79423i	0.170450	+	0.295227i
$698$		0			0
$699$	−16.5000	−	9.52628i	−0.624087	−	0.360317i
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$	6.00000	+	3.46410i	0.225973	+	0.130466i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$	30.0000			1.12509
$712$		0			0
$713$	6.00000	−	10.3923i	0.224702	−	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			10.3923i			0.388108i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	34.5000	−	19.9186i	1.28307	−	0.740780i
$724$		0			0
$725$	5.00000	+	8.66025i	0.185695	+	0.321634i
$726$		0			0
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	−0.678676	−	0.734438i	$-0.737446\pi$
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	0.975380	+	0.220532i	$0.0707793\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	4.50000	−	7.79423i	0.166439	−	0.288280i
$732$		0			0
$733$	−14.0000	−	24.2487i	−0.517102	−	0.895647i	−0.999803	−	0.0198613i	$-0.993678\pi$
$733$	−14.0000	−	24.2487i	−0.517102	−	0.895647i	0.482701	−	0.875785i	$-0.339656\pi$
$734$		0			0
$735$	−10.5000	+	6.06218i	−0.387298	+	0.223607i
$736$		0			0
$737$	55.0000			2.02595
$738$		0			0
$739$	−5.00000			−0.183928			−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000			−0.183928			−0.0919640	+	0.995762i	$0.529314\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	0.988797	+	0.149270i	$0.0476922\pi$
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	−0.365127	+	0.930958i	$0.618974\pi$
$744$		0			0
$745$	2.00000	−	3.46410i	0.0732743	−	0.126915i
$746$		0			0
$747$	−36.0000			−1.31717
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$		0			0
$753$	−16.5000	−	9.52628i	−0.601293	−	0.347157i
$754$		0			0
$755$	22.0000			0.800662
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$	−45.0000	−	25.9808i	−1.63340	−	0.943042i
$760$		0			0
$761$	−5.00000	−	8.66025i	−0.181250	−	0.313934i	0.761057	−	0.648686i	$-0.224681\pi$
$761$	−5.00000	−	8.66025i	−0.181250	−	0.313934i	−0.942306	+	0.334752i	$0.891348\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−4.50000	+	7.79423i	−0.162698	+	0.281801i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.684336	+	0.729167i	$0.260092\pi$
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.289309	+	0.957236i	$0.406575\pi$
$770$		0			0
$771$			53.6936i			1.93373i
$772$		0			0
$773$	−30.0000			−1.07903			−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000			−1.07903			−0.539513	+	0.841978i	$0.681391\pi$
$774$		0			0
$775$	2.00000			0.0718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.50000	−	12.9904i	−0.268715	−	0.465429i
$780$		0			0
$781$	−35.0000	+	60.6218i	−1.25240	+	2.16922i
$782$		0			0
$783$	45.0000	+	25.9808i	1.60817	+	0.928477i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	0.999999	−	0.00110111i	$-0.000350496\pi$
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	−0.500953	+	0.865474i	$0.667017\pi$
$788$		0			0
$789$	−18.0000	+	10.3923i	−0.640817	+	0.369976i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +(-2.50000 + 4.33013i) q^{11} +1.73205i q^{15} +3.00000 q^{17} -5.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.19615i q^{27} +(5.00000 - 8.66025i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-7.50000 + 4.33013i) q^{33} +4.00000 q^{37} +(1.50000 + 2.59808i) q^{41} +(1.50000 - 2.59808i) q^{43} +(-1.50000 + 2.59808i) q^{45} +(2.00000 - 3.46410i) q^{47} +(3.50000 + 6.06218i) q^{49} +(4.50000 + 2.59808i) q^{51} -6.00000 q^{53} -5.00000 q^{55} +(-7.50000 - 4.33013i) q^{57} +(-1.50000 - 2.59808i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-5.50000 - 9.52628i) q^{67} +10.3923i q^{69} +14.0000 q^{71} -15.0000 q^{73} +(-1.50000 + 0.866025i) q^{75} +(5.00000 - 8.66025i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-6.00000 + 10.3923i) q^{83} +(1.50000 + 2.59808i) q^{85} +(15.0000 - 8.66025i) q^{87} +14.0000 q^{89} -3.46410i q^{93} +(-2.50000 - 4.33013i) q^{95} +(6.50000 - 11.2583i) q^{97} -15.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + q^5 + 3 * q^9	\( 2 q + 3 q^{3} + q^{5} + 3 q^{9} - 5 q^{11} + 6 q^{17} - 10 q^{19} + 6 q^{23} - q^{25} + 10 q^{29} - 2 q^{31} - 15 q^{33} + 8 q^{37} + 3 q^{41} + 3 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} + 9 q^{51} - 12 q^{53} - 10 q^{55} - 15 q^{57} - 3 q^{59} - 2 q^{61} - 11 q^{67} + 28 q^{71} - 30 q^{73} - 3 q^{75} + 10 q^{79} - 9 q^{81} - 12 q^{83} + 3 q^{85} + 30 q^{87} + 28 q^{89} - 5 q^{95} + 13 q^{97} - 30 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + q^5 + 3 * q^9 - 5 * q^11 + 6 * q^17 - 10 * q^19 + 6 * q^23 - q^25 + 10 * q^29 - 2 * q^31 - 15 * q^33 + 8 * q^37 + 3 * q^41 + 3 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 + 9 * q^51 - 12 * q^53 - 10 * q^55 - 15 * q^57 - 3 * q^59 - 2 * q^61 - 11 * q^67 + 28 * q^71 - 30 * q^73 - 3 * q^75 + 10 * q^79 - 9 * q^81 - 12 * q^83 + 3 * q^85 + 30 * q^87 + 28 * q^89 - 5 * q^95 + 13 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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