LMFDB - Embedded newform 1638.2.j.b.235.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(235,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("1638.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.j (of order $3$, degree $2$, 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:	$13.0794958511$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1638.235
Dual form			1638.2.j.b.1171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +1.00000 q^{8} +(-1.00000 + 1.73205i) q^{10} +(1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{17} +(1.50000 + 2.59808i) q^{19} +2.00000 q^{20} -3.00000 q^{22} +(0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{26} +(-2.00000 - 1.73205i) q^{28} -3.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -1.00000 q^{34} +(5.00000 - 1.73205i) q^{35} +(1.00000 + 1.73205i) q^{37} +(1.50000 - 2.59808i) q^{38} +(-1.00000 - 1.73205i) q^{40} -2.00000 q^{41} +6.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{47} +(-6.50000 - 2.59808i) q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(0.500000 - 0.866025i) q^{53} -6.00000 q^{55} +(-0.500000 + 2.59808i) q^{56} +(1.50000 + 2.59808i) q^{58} +(5.50000 - 9.52628i) q^{59} +(-5.50000 - 9.52628i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(3.50000 - 6.06218i) q^{67} +(0.500000 + 0.866025i) q^{68} +(-4.00000 - 3.46410i) q^{70} -15.0000 q^{71} +(6.00000 - 10.3923i) q^{73} +(1.00000 - 1.73205i) q^{74} -3.00000 q^{76} +(6.00000 + 5.19615i) q^{77} +(-1.00000 - 1.73205i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(1.00000 + 1.73205i) q^{82} -2.00000 q^{85} +(-3.00000 - 5.19615i) q^{86} +(1.50000 - 2.59808i) q^{88} +(5.00000 + 8.66025i) q^{89} +(0.500000 - 2.59808i) q^{91} +(-4.50000 + 7.79423i) q^{94} +(3.00000 - 5.19615i) q^{95} -12.0000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} - q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 - q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 2 q^{5} - q^{7} + 2 q^{8} - 2 q^{10} + 3 q^{11} - 2 q^{13} + 5 q^{14} - q^{16} + q^{17} + 3 q^{19} + 4 q^{20} - 6 q^{22} + q^{25} + q^{26} - 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 2 q^{34} + 10 q^{35} + 2 q^{37} + 3 q^{38} - 2 q^{40} - 4 q^{41} + 12 q^{43} + 3 q^{44} - 9 q^{47} - 13 q^{49} - 2 q^{50} + q^{52} + q^{53} - 12 q^{55} - q^{56} + 3 q^{58} + 11 q^{59} - 11 q^{61} - 8 q^{62} + 2 q^{64} + 2 q^{65} + 7 q^{67} + q^{68} - 8 q^{70} - 30 q^{71} + 12 q^{73} + 2 q^{74} - 6 q^{76} + 12 q^{77} - 2 q^{79} - 2 q^{80} + 2 q^{82} - 4 q^{85} - 6 q^{86} + 3 q^{88} + 10 q^{89} + q^{91} - 9 q^{94} + 6 q^{95} - 24 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 - q^7 + 2 * q^8 - 2 * q^10 + 3 * q^11 - 2 * q^13 + 5 * q^14 - q^16 + q^17 + 3 * q^19 + 4 * q^20 - 6 * q^22 + q^25 + q^26 - 4 * q^28 - 6 * q^29 + 4 * q^31 - q^32 - 2 * q^34 + 10 * q^35 + 2 * q^37 + 3 * q^38 - 2 * q^40 - 4 * q^41 + 12 * q^43 + 3 * q^44 - 9 * q^47 - 13 * q^49 - 2 * q^50 + q^52 + q^53 - 12 * q^55 - q^56 + 3 * q^58 + 11 * q^59 - 11 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 7 * q^67 + q^68 - 8 * q^70 - 30 * q^71 + 12 * q^73 + 2 * q^74 - 6 * q^76 + 12 * q^77 - 2 * q^79 - 2 * q^80 + 2 * q^82 - 4 * q^85 - 6 * q^86 + 3 * q^88 + 10 * q^89 + q^91 - 9 * q^94 + 6 * q^95 - 24 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000	+	1.73205i	−0.316228	+	0.547723i
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$	2.50000	−	0.866025i	0.668153	−	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i	−0.799000	−	0.601331i	$-0.794637\pi$
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i	0.920268	+	0.391289i	$0.127971\pi$
$18$		0			0
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	0.985194	−	0.171442i	$-0.0548427\pi$
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	−0.641071	+	0.767482i	$0.721509\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	−3.00000			−0.639602
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$	0.500000	+	0.866025i	0.0980581	+	0.169842i
$27$		0			0
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−1.00000			−0.171499
$35$	5.00000	−	1.73205i	0.845154	−	0.292770i
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$	1.50000	−	2.59808i	0.243332	−	0.421464i
$39$		0			0
$40$	−1.00000	−	1.73205i	−0.158114	−	0.273861i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$	1.50000	+	2.59808i	0.226134	+	0.391675i
$45$		0			0
$46$		0			0
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	−0.981543	−	0.191243i	$-0.938748\pi$
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	0.325150	−	0.945662i	$-0.394585\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$	−1.00000			−0.141421
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	0.500000	−	0.866025i	0.0686803	−	0.118958i	−0.829640	−	0.558298i	$-0.811454\pi$
$53$	0.500000	−	0.866025i	0.0686803	−	0.118958i	0.898321	+	0.439340i	$0.144788\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	−0.500000	+	2.59808i	−0.0668153	+	0.347183i
$57$		0			0
$58$	1.50000	+	2.59808i	0.196960	+	0.341144i
$59$	5.50000	−	9.52628i	0.716039	−	1.24022i	−0.246518	−	0.969138i	$-0.579287\pi$
$59$	5.50000	−	9.52628i	0.716039	−	1.24022i	0.962557	−	0.271078i	$-0.0873801\pi$
$60$		0			0
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	−0.966978	−	0.254858i	$-0.917971\pi$
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	0.262776	−	0.964857i	$-0.415362\pi$
$62$	−4.00000			−0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	−0.569066	−	0.822292i	$-0.692695\pi$
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	0.996659	+	0.0816792i	$0.0260283\pi$
$68$	0.500000	+	0.866025i	0.0606339	+	0.105021i
$69$		0			0
$70$	−4.00000	−	3.46410i	−0.478091	−	0.414039i
$71$	−15.0000			−1.78017			−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000			−1.78017			−0.890086	+	0.455792i	$0.849356\pi$
$72$		0			0
$73$	6.00000	−	10.3923i	0.702247	−	1.21633i	−0.265429	−	0.964130i	$-0.585514\pi$
$73$	6.00000	−	10.3923i	0.702247	−	1.21633i	0.967676	−	0.252197i	$-0.0811531\pi$
$74$	1.00000	−	1.73205i	0.116248	−	0.201347i
$75$		0			0
$76$	−3.00000			−0.344124
$77$	6.00000	+	5.19615i	0.683763	+	0.592157i
$78$		0			0
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	0.804272	−	0.594261i	$-0.202555\pi$
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	−0.916781	+	0.399390i	$0.869222\pi$
$80$	−1.00000	+	1.73205i	−0.111803	+	0.193649i
$81$		0			0
$82$	1.00000	+	1.73205i	0.110432	+	0.191273i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$	−3.00000	−	5.19615i	−0.323498	−	0.560316i
$87$		0			0
$88$	1.50000	−	2.59808i	0.159901	−	0.276956i
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	0.999388	+	0.0349934i	$0.0111410\pi$
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	−0.469389	+	0.882992i	$0.655526\pi$
$90$		0			0
$91$	0.500000	−	2.59808i	0.0524142	−	0.272352i
$92$		0			0
$93$		0			0
$94$	−4.50000	+	7.79423i	−0.464140	+	0.803913i
$95$	3.00000	−	5.19615i	0.307794	−	0.533114i
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$	1.00000	+	6.92820i	0.101015	+	0.699854i
$99$		0			0
$100$	0.500000	+	0.866025i	0.0500000	+	0.0866025i
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	0.273138	+	0.961975i	$0.411939\pi$
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	−0.969664	+	0.244443i	$0.921395\pi$
$102$		0			0
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	0.282194	−	0.959357i	$-0.408938\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$	−1.00000			−0.0971286
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	0.753010	−	0.658009i	$-0.228601\pi$
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	−0.946357	+	0.323122i	$0.895268\pi$
$108$		0			0
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	−0.608377	−	0.793648i	$-0.708179\pi$
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	0.991508	+	0.130046i	$0.0415126\pi$
$110$	3.00000	+	5.19615i	0.286039	+	0.495434i
$111$		0			0
$112$	2.50000	−	0.866025i	0.236228	−	0.0818317i
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$		0			0
$115$		0			0
$116$	1.50000	−	2.59808i	0.139272	−	0.241225i
$117$		0			0
$118$	−11.0000			−1.01263
$119$	2.00000	+	1.73205i	0.183340	+	0.158777i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	−5.50000	+	9.52628i	−0.497947	+	0.862469i
$123$		0			0
$124$	2.00000	+	3.46410i	0.179605	+	0.311086i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	1.00000	−	1.73205i	0.0877058	−	0.151911i
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	−7.50000	+	2.59808i	−0.650332	+	0.225282i
$134$	−7.00000			−0.604708
$135$		0			0
$136$	0.500000	−	0.866025i	0.0428746	−	0.0742611i
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$	−1.00000	+	5.19615i	−0.0845154	+	0.439155i
$141$		0			0
$142$	7.50000	+	12.9904i	0.629386	+	1.09013i
$143$	−1.50000	+	2.59808i	−0.125436	+	0.217262i
$144$		0			0
$145$	3.00000	+	5.19615i	0.249136	+	0.431517i
$146$	−12.0000			−0.993127
$147$		0			0
$148$	−2.00000			−0.164399
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$	1.50000	+	2.59808i	0.121666	+	0.210732i
$153$		0			0
$154$	1.50000	−	7.79423i	0.120873	−	0.628077i
$155$	−8.00000			−0.642575
$156$		0			0
$157$	5.50000	−	9.52628i	0.438948	−	0.760280i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	5.50000	−	9.52628i	0.438948	−	0.760280i	0.997609	+	0.0691164i	$0.0220180\pi$
$158$	−1.00000	+	1.73205i	−0.0795557	+	0.137795i
$159$		0			0
$160$	2.00000			0.158114
$161$		0			0
$162$		0			0
$163$	12.5000	+	21.6506i	0.979076	+	1.69581i	0.665771	+	0.746156i	$0.268103\pi$
$163$	12.5000	+	21.6506i	0.979076	+	1.69581i	0.313304	+	0.949653i	$0.398564\pi$
$164$	1.00000	−	1.73205i	0.0780869	−	0.135250i
$165$		0			0
$166$		0			0
$167$	19.0000			1.47026			0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000			1.47026			0.735132	+	0.677924i	$0.237120\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$	1.00000	+	1.73205i	0.0766965	+	0.132842i
$171$		0			0
$172$	−3.00000	+	5.19615i	−0.228748	+	0.396203i
$173$	3.50000	+	6.06218i	0.266100	+	0.460899i	0.967851	−	0.251523i	$-0.0809315\pi$
$173$	3.50000	+	6.06218i	0.266100	+	0.460899i	−0.701751	+	0.712422i	$0.747598\pi$
$174$		0			0
$175$	2.00000	+	1.73205i	0.151186	+	0.130931i
$176$	−3.00000			−0.226134
$177$		0			0
$178$	5.00000	−	8.66025i	0.374766	−	0.649113i
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	−0.956088	−	0.293079i	$-0.905320\pi$
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	0.731858	+	0.681457i	$0.238654\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$	−2.50000	+	0.866025i	−0.185312	+	0.0641941i
$183$		0			0
$184$		0			0
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	−1.50000	−	2.59808i	−0.109691	−	0.189990i
$188$	9.00000			0.656392
$189$		0			0
$190$	−6.00000			−0.435286
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$		0			0
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	−0.496119	−	0.868255i	$-0.665242\pi$
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	0.999990	−	0.00447566i	$-0.00142465\pi$
$194$	6.00000	+	10.3923i	0.430775	+	0.746124i
$195$		0			0
$196$	5.50000	−	4.33013i	0.392857	−	0.309295i
$197$	−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
$199$	−6.00000	+	10.3923i	−0.425329	+	0.736691i	−0.996451	−	0.0841740i	$-0.973175\pi$
$199$	−6.00000	+	10.3923i	−0.425329	+	0.736691i	0.571122	+	0.820865i	$0.306508\pi$
$200$	0.500000	−	0.866025i	0.0353553	−	0.0612372i
$201$		0			0
$202$	14.0000			0.985037
$203$	1.50000	−	7.79423i	0.105279	−	0.547048i
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$	−7.00000	+	12.1244i	−0.487713	+	0.844744i
$207$		0			0
$208$	0.500000	+	0.866025i	0.0346688	+	0.0600481i
$209$	9.00000			0.622543
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$	0.500000	+	0.866025i	0.0343401	+	0.0594789i
$213$		0			0
$214$	−2.00000	+	3.46410i	−0.136717	+	0.236801i
$215$	−6.00000	−	10.3923i	−0.409197	−	0.708749i
$216$		0			0
$217$	8.00000	+	6.92820i	0.543075	+	0.470317i
$218$	−8.00000			−0.541828
$219$		0			0
$220$	3.00000	−	5.19615i	0.202260	−	0.350325i
$221$	−0.500000	+	0.866025i	−0.0336336	+	0.0582552i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−2.00000	−	1.73205i	−0.133631	−	0.115728i
$225$		0			0
$226$	1.50000	+	2.59808i	0.0997785	+	0.172821i
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	−0.702202	−	0.711977i	$-0.747800\pi$
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	0.967692	+	0.252136i	$0.0811332\pi$
$228$		0			0
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	0.897173	−	0.441679i	$-0.145617\pi$
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	−0.831092	+	0.556136i	$0.812283\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000			−0.196960
$233$	3.50000	+	6.06218i	0.229293	+	0.397146i	0.957599	−	0.288106i	$-0.0930254\pi$
$233$	3.50000	+	6.06218i	0.229293	+	0.397146i	−0.728306	+	0.685252i	$0.759692\pi$
$234$		0			0
$235$	−9.00000	+	15.5885i	−0.587095	+	1.01688i
$236$	5.50000	+	9.52628i	0.358020	+	0.620108i
$237$		0			0
$238$	0.500000	−	2.59808i	0.0324102	−	0.168408i
$239$	11.0000			0.711531			0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000			0.711531			0.355765	+	0.934575i	$0.384220\pi$
$240$		0			0
$241$	−6.00000	+	10.3923i	−0.386494	+	0.669427i	−0.991975	−	0.126432i	$-0.959647\pi$
$241$	−6.00000	+	10.3923i	−0.386494	+	0.669427i	0.605481	+	0.795860i	$0.292981\pi$
$242$	1.00000	−	1.73205i	0.0642824	−	0.111340i
$243$		0			0
$244$	11.0000			0.704203
$245$	2.00000	+	13.8564i	0.127775	+	0.885253i
$246$		0			0
$247$	−1.50000	−	2.59808i	−0.0954427	−	0.165312i
$248$	2.00000	−	3.46410i	0.127000	−	0.219971i
$249$		0			0
$250$	6.00000	+	10.3923i	0.379473	+	0.657267i
$251$	10.0000			0.631194			0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000			0.631194			0.315597	+	0.948893i	$0.397795\pi$
$252$		0			0
$253$		0			0
$254$	−4.00000	−	6.92820i	−0.250982	−	0.434714i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	−0.912222	−	0.409695i	$-0.865635\pi$
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	0.101305	−	0.994855i	$-0.467698\pi$
$258$		0			0
$259$	−5.00000	+	1.73205i	−0.310685	+	0.107624i
$260$	−2.00000			−0.124035
$261$		0			0
$262$	−6.00000	+	10.3923i	−0.370681	+	0.642039i
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	−0.669680	−	0.742650i	$-0.733569\pi$
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	0.977993	+	0.208635i	$0.0669022\pi$
$264$		0			0
$265$	−2.00000			−0.122859
$266$	6.00000	+	5.19615i	0.367884	+	0.318597i
$267$		0			0
$268$	3.50000	+	6.06218i	0.213797	+	0.370306i
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	0.345192	+	0.938532i	$0.387814\pi$
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	−0.985389	+	0.170321i	$0.945520\pi$
$270$		0			0
$271$	−7.50000	−	12.9904i	−0.455593	−	0.789109i	0.543130	−	0.839649i	$-0.317239\pi$
$271$	−7.50000	−	12.9904i	−0.455593	−	0.789109i	−0.998722	+	0.0505395i	$0.983906\pi$
$272$	−1.00000			−0.0606339
$273$		0			0
$274$	−18.0000			−1.08742
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	0.781094	+	0.624413i	$0.214662\pi$
$278$	7.00000	+	12.1244i	0.419832	+	0.727171i
$279$		0			0
$280$	5.00000	−	1.73205i	0.298807	−	0.103510i
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	7.50000	−	12.9904i	0.445043	−	0.770837i
$285$		0			0
$286$	3.00000			0.177394
$287$	1.00000	−	5.19615i	0.0590281	−	0.306719i
$288$		0			0
$289$	8.00000	+	13.8564i	0.470588	+	0.815083i
$290$	3.00000	−	5.19615i	0.176166	−	0.305129i
$291$		0			0
$292$	6.00000	+	10.3923i	0.351123	+	0.608164i
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	−22.0000			−1.28089
$296$	1.00000	+	1.73205i	0.0581238	+	0.100673i
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.00000	+	15.5885i	−0.172917	+	0.898504i
$302$	−17.0000			−0.978240
$303$		0			0
$304$	1.50000	−	2.59808i	0.0860309	−	0.149010i
$305$	−11.0000	+	19.0526i	−0.629858	+	1.09095i
$306$		0			0
$307$	−15.0000			−0.856095			−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000			−0.856095			−0.428048	+	0.903756i	$0.640798\pi$
$308$	−7.50000	+	2.59808i	−0.427352	+	0.148039i
$309$		0			0
$310$	4.00000	+	6.92820i	0.227185	+	0.393496i
$311$	7.00000	−	12.1244i	0.396934	−	0.687509i	−0.596412	−	0.802678i	$-0.703408\pi$
$311$	7.00000	−	12.1244i	0.396934	−	0.687509i	0.993346	+	0.115169i	$0.0367410\pi$
$312$		0			0
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	0.993186	−	0.116543i	$-0.0371814\pi$
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	−0.597522	+	0.801852i	$0.703848\pi$
$314$	−11.0000			−0.620766
$315$		0			0
$316$	2.00000			0.112509
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	−4.50000	+	7.79423i	−0.251952	+	0.436393i
$320$	−1.00000	−	1.73205i	−0.0559017	−	0.0968246i
$321$		0			0
$322$		0			0
$323$	3.00000			0.166924
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$	12.5000	−	21.6506i	0.692311	−	1.19912i
$327$		0			0
$328$	−2.00000			−0.110432
$329$	22.5000	−	7.79423i	1.24047	−	0.429710i
$330$		0			0
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	0.937829	+	0.347097i	$0.112833\pi$
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	−0.168320	+	0.985732i	$0.553834\pi$
$332$		0			0
$333$		0			0
$334$	−9.50000	−	16.4545i	−0.519817	−	0.900349i
$335$	−14.0000			−0.764902
$336$		0			0
$337$	7.00000			0.381314			0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000			0.381314			0.190657	+	0.981657i	$0.438938\pi$
$338$	−0.500000	−	0.866025i	−0.0271964	−	0.0471056i
$339$		0			0
$340$	1.00000	−	1.73205i	0.0542326	−	0.0939336i
$341$	−6.00000	−	10.3923i	−0.324918	−	0.562775i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$	6.00000			0.323498
$345$		0			0
$346$	3.50000	−	6.06218i	0.188161	−	0.325905i
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	−0.837935	−	0.545770i	$-0.816237\pi$
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	0.891618	+	0.452788i	$0.149571\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$	0.500000	−	2.59808i	0.0267261	−	0.138873i
$351$		0			0
$352$	1.50000	+	2.59808i	0.0799503	+	0.138478i
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	−0.999711	−	0.0240566i	$-0.992342\pi$
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	0.520689	+	0.853746i	$0.325675\pi$
$354$		0			0
$355$	15.0000	+	25.9808i	0.796117	+	1.37892i
$356$	−10.0000			−0.529999
$357$		0			0
$358$	6.00000			0.317110
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	0.996157	−	0.0875892i	$-0.0279163\pi$
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	−0.573933	+	0.818902i	$0.694583\pi$
$360$		0			0
$361$	5.00000	−	8.66025i	0.263158	−	0.455803i
$362$	2.50000	+	4.33013i	0.131397	+	0.227586i
$363$		0			0
$364$	2.00000	+	1.73205i	0.104828	+	0.0907841i
$365$	−24.0000			−1.25622
$366$		0			0
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	−0.999404	−	0.0345320i	$-0.989006\pi$
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	0.529607	+	0.848243i	$0.322339\pi$
$368$		0			0
$369$		0			0
$370$	−4.00000			−0.207950
$371$	2.00000	+	1.73205i	0.103835	+	0.0899236i
$372$		0			0
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$	−1.50000	+	2.59808i	−0.0775632	+	0.134343i
$375$		0			0
$376$	−4.50000	−	7.79423i	−0.232070	−	0.401957i
$377$	3.00000			0.154508
$378$		0			0
$379$	36.0000			1.84920			0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000			1.84920			0.924598	+	0.380945i	$0.124401\pi$
$380$	3.00000	+	5.19615i	0.153897	+	0.266557i
$381$		0			0
$382$	−6.00000	+	10.3923i	−0.306987	+	0.531717i
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	−0.907474	−	0.420109i	$-0.861992\pi$
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	0.0899119	−	0.995950i	$-0.471341\pi$
$384$		0			0
$385$	3.00000	−	15.5885i	0.152894	−	0.794461i
$386$	−14.0000			−0.712581
$387$		0			0
$388$	6.00000	−	10.3923i	0.304604	−	0.527589i
$389$	6.50000	−	11.2583i	0.329563	−	0.570820i	−0.652862	−	0.757477i	$-0.726432\pi$
$389$	6.50000	−	11.2583i	0.329563	−	0.570820i	0.982425	+	0.186657i	$0.0597652\pi$
$390$		0			0
$391$		0			0
$392$	−6.50000	−	2.59808i	−0.328300	−	0.131223i
$393$		0			0
$394$	4.00000	+	6.92820i	0.201517	+	0.349038i
$395$	−2.00000	+	3.46410i	−0.100631	+	0.174298i
$396$		0			0
$397$	10.0000	+	17.3205i	0.501886	+	0.869291i	0.999998	+	0.00217869i	$0.000693499\pi$
$397$	10.0000	+	17.3205i	0.501886	+	0.869291i	−0.498112	+	0.867113i	$0.665973\pi$
$398$	12.0000			0.601506
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	0.976050	−	0.217545i	$-0.0698049\pi$
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	−0.676425	+	0.736512i	$0.736472\pi$
$402$		0			0
$403$	−2.00000	+	3.46410i	−0.0996271	+	0.172559i
$404$	−7.00000	−	12.1244i	−0.348263	−	0.603209i
$405$		0			0
$406$	−7.50000	+	2.59808i	−0.372219	+	0.128940i
$407$	6.00000			0.297409
$408$		0			0
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.766426	−	0.642333i	$-0.222033\pi$
$410$	2.00000	−	3.46410i	0.0987730	−	0.171080i
$411$		0			0
$412$	14.0000			0.689730
$413$	22.0000	+	19.0526i	1.08255	+	0.937515i
$414$		0			0
$415$		0			0
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	−4.50000	−	7.79423i	−0.220102	−	0.381228i
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	8.00000	+	13.8564i	0.389434	+	0.674519i
$423$		0			0
$424$	0.500000	−	0.866025i	0.0242821	−	0.0420579i
$425$	−0.500000	−	0.866025i	−0.0242536	−	0.0420084i
$426$		0			0
$427$	27.5000	−	9.52628i	1.33082	−	0.461009i
$428$	4.00000			0.193347
$429$		0			0
$430$	−6.00000	+	10.3923i	−0.289346	+	0.501161i
$431$	−20.0000	+	34.6410i	−0.963366	+	1.66860i	−0.249424	+	0.968394i	$0.580241\pi$
$431$	−20.0000	+	34.6410i	−0.963366	+	1.66860i	−0.713942	+	0.700205i	$0.753092\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$	2.00000	−	10.3923i	0.0960031	−	0.498847i
$435$		0			0
$436$	4.00000	+	6.92820i	0.191565	+	0.331801i
$437$		0			0
$438$		0			0
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	0.609503	−	0.792784i	$-0.291369\pi$
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	−0.991322	+	0.131453i	$0.958036\pi$
$440$	−6.00000			−0.286039
$441$		0			0
$442$	1.00000			0.0475651
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	0.928450	−	0.371457i	$-0.121142\pi$
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	−0.785916	+	0.618333i	$0.787808\pi$
$444$		0			0
$445$	10.0000	−	17.3205i	0.474045	−	0.821071i
$446$	0.500000	+	0.866025i	0.0236757	+	0.0410075i
$447$		0			0
$448$	−0.500000	+	2.59808i	−0.0236228	+	0.122748i
$449$	40.0000			1.88772			0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000			1.88772			0.943858	+	0.330350i	$0.107167\pi$
$450$		0			0
$451$	−3.00000	+	5.19615i	−0.141264	+	0.244677i
$452$	1.50000	−	2.59808i	0.0705541	−	0.122203i
$453$		0			0
$454$	−8.00000			−0.375459
$455$	−5.00000	+	1.73205i	−0.234404	+	0.0811998i
$456$		0			0
$457$	16.0000	+	27.7128i	0.748448	+	1.29635i	0.948566	+	0.316579i	$0.102534\pi$
$457$	16.0000	+	27.7128i	0.748448	+	1.29635i	−0.200118	+	0.979772i	$0.564132\pi$
$458$	1.00000	−	1.73205i	0.0467269	−	0.0809334i
$459$		0			0
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	1.50000	+	2.59808i	0.0696358	+	0.120613i
$465$		0			0
$466$	3.50000	−	6.06218i	0.162134	−	0.280825i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	14.0000	+	12.1244i	0.646460	+	0.559851i
$470$	18.0000			0.830278
$471$		0			0
$472$	5.50000	−	9.52628i	0.253158	−	0.438483i
$473$	9.00000	−	15.5885i	0.413820	−	0.716758i
$474$		0			0
$475$	3.00000			0.137649
$476$	−2.50000	+	0.866025i	−0.114587	+	0.0396942i
$477$		0			0
$478$	−5.50000	−	9.52628i	−0.251564	−	0.435722i
$479$	7.50000	−	12.9904i	0.342684	−	0.593546i	−0.642246	−	0.766498i	$-0.721997\pi$
$479$	7.50000	−	12.9904i	0.342684	−	0.593546i	0.984930	+	0.172953i	$0.0553307\pi$
$480$		0			0
$481$	−1.00000	−	1.73205i	−0.0455961	−	0.0789747i
$482$	12.0000			0.546585
$483$		0			0
$484$	−2.00000			−0.0909091
$485$	12.0000	+	20.7846i	0.544892	+	0.943781i
$486$		0			0
$487$	16.5000	−	28.5788i	0.747686	−	1.29503i	−0.201243	−	0.979541i	$-0.564498\pi$
$487$	16.5000	−	28.5788i	0.747686	−	1.29503i	0.948929	−	0.315489i	$-0.102169\pi$
$488$	−5.50000	−	9.52628i	−0.248973	−	0.431234i
$489$		0			0
$490$	11.0000	−	8.66025i	0.496929	−	0.391230i
$491$	−22.0000			−0.992846			−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000			−0.992846			−0.496423	+	0.868081i	$0.665354\pi$
$492$		0			0
$493$	−1.50000	+	2.59808i	−0.0675566	+	0.117011i
$494$	−1.50000	+	2.59808i	−0.0674882	+	0.116893i
$495$		0			0
$496$	−4.00000			−0.179605
$497$	7.50000	−	38.9711i	0.336421	−	1.74809i
$498$		0			0
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$	6.00000	−	10.3923i	0.268328	−	0.464758i
$501$		0			0
$502$	−5.00000	−	8.66025i	−0.223161	−	0.386526i
$503$	20.0000			0.891756			0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000			0.891756			0.445878	+	0.895094i	$0.352892\pi$
$504$		0			0
$505$	28.0000			1.24598
$506$		0			0
$507$		0			0
$508$	−4.00000	+	6.92820i	−0.177471	+	0.307389i
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	0.967811	−	0.251679i	$-0.0809826\pi$
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	−0.701866	+	0.712309i	$0.747649\pi$
$510$		0			0
$511$	24.0000	+	20.7846i	1.06170	+	0.919457i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−13.0000	+	22.5167i	−0.573405	+	0.993167i
$515$	−14.0000	+	24.2487i	−0.616914	+	1.06853i
$516$		0			0
$517$	−27.0000			−1.18746
$518$	4.00000	+	3.46410i	0.175750	+	0.152204i
$519$		0			0
$520$	1.00000	+	1.73205i	0.0438529	+	0.0759555i
$521$	−19.0000	+	32.9090i	−0.832405	+	1.44177i	0.0637207	+	0.997968i	$0.479703\pi$
$521$	−19.0000	+	32.9090i	−0.832405	+	1.44177i	−0.896126	+	0.443800i	$0.853630\pi$
$522$		0			0
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	0.792951	−	0.609285i	$-0.208544\pi$
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	−0.924132	+	0.382073i	$0.875210\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$	−10.0000			−0.436021
$527$	−2.00000	−	3.46410i	−0.0871214	−	0.150899i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$	1.00000	+	1.73205i	0.0434372	+	0.0752355i
$531$		0			0
$532$	1.50000	−	7.79423i	0.0650332	−	0.337923i
$533$	2.00000			0.0866296
$534$		0			0
$535$	−4.00000	+	6.92820i	−0.172935	+	0.299532i
$536$	3.50000	−	6.06218i	0.151177	−	0.261846i
$537$		0			0
$538$	21.0000			0.905374
$539$	−16.5000	+	12.9904i	−0.710705	+	0.559535i
$540$		0			0
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	0.526593	−	0.850118i	$-0.323469\pi$
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	−0.999520	+	0.0309841i	$0.990136\pi$
$542$	−7.50000	+	12.9904i	−0.322153	+	0.557985i
$543$		0			0
$544$	0.500000	+	0.866025i	0.0214373	+	0.0371305i
$545$	−16.0000			−0.685365
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$	9.00000	+	15.5885i	0.384461	+	0.665906i
$549$		0			0
$550$	−1.50000	+	2.59808i	−0.0639602	+	0.110782i
$551$	−4.50000	−	7.79423i	−0.191706	−	0.332045i
$552$		0			0
$553$	5.00000	−	1.73205i	0.212622	−	0.0736543i
$554$	5.00000			0.212430
$555$		0			0
$556$	7.00000	−	12.1244i	0.296866	−	0.514187i
$557$	−11.0000	+	19.0526i	−0.466085	+	0.807283i	−0.999250	−	0.0387286i	$-0.987669\pi$
$557$	−11.0000	+	19.0526i	−0.466085	+	0.807283i	0.533165	+	0.846011i	$0.321003\pi$
$558$		0			0
$559$	−6.00000			−0.253773
$560$	−4.00000	−	3.46410i	−0.169031	−	0.146385i
$561$		0			0
$562$	−12.0000	−	20.7846i	−0.506189	−	0.876746i
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$	−4.00000			−0.168133
$567$		0			0
$568$	−15.0000			−0.629386
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	0.907532	+	0.419984i	$0.137964\pi$
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	−0.0900490	+	0.995937i	$0.528702\pi$
$570$		0			0
$571$	−22.0000	+	38.1051i	−0.920671	+	1.59465i	−0.122292	+	0.992494i	$0.539025\pi$
$571$	−22.0000	+	38.1051i	−0.920671	+	1.59465i	−0.798379	+	0.602155i	$0.794309\pi$
$572$	−1.50000	−	2.59808i	−0.0627182	−	0.108631i
$573$		0			0
$574$	−5.00000	+	1.73205i	−0.208696	+	0.0722944i
$575$		0			0
$576$		0			0
$577$	9.00000	−	15.5885i	0.374675	−	0.648956i	−0.615603	−	0.788056i	$-0.711088\pi$
$577$	9.00000	−	15.5885i	0.374675	−	0.648956i	0.990278	+	0.139100i	$0.0444210\pi$
$578$	8.00000	−	13.8564i	0.332756	−	0.576351i
$579$		0			0
$580$	−6.00000			−0.249136
$581$		0			0
$582$		0			0
$583$	−1.50000	−	2.59808i	−0.0621237	−	0.107601i
$584$	6.00000	−	10.3923i	0.248282	−	0.430037i
$585$		0			0
$586$	−6.00000	−	10.3923i	−0.247858	−	0.429302i
$587$	−39.0000			−1.60970			−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000			−1.60970			−0.804851	+	0.593477i	$0.797755\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$	11.0000	+	19.0526i	0.452863	+	0.784381i
$591$		0			0
$592$	1.00000	−	1.73205i	0.0410997	−	0.0711868i
$593$	−17.0000	−	29.4449i	−0.698106	−	1.20916i	−0.969122	−	0.246581i	$-0.920693\pi$
$593$	−17.0000	−	29.4449i	−0.698106	−	1.20916i	0.271016	−	0.962575i	$-0.412640\pi$
$594$		0			0
$595$	1.00000	−	5.19615i	0.0409960	−	0.213021i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	0.377869	+	0.925859i	$0.376657\pi$
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	−0.990752	+	0.135686i	$0.956676\pi$
$600$		0			0
$601$	−35.0000			−1.42768			−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000			−1.42768			−0.713840	+	0.700309i	$0.753046\pi$
$602$	15.0000	−	5.19615i	0.611354	−	0.211779i
$603$		0			0
$604$	8.50000	+	14.7224i	0.345860	+	0.599047i
$605$	2.00000	−	3.46410i	0.0813116	−	0.140836i
$606$		0			0
$607$	−14.0000	−	24.2487i	−0.568242	−	0.984225i	−0.996740	−	0.0806818i	$-0.974290\pi$
$607$	−14.0000	−	24.2487i	−0.568242	−	0.984225i	0.428497	−	0.903543i	$-0.359043\pi$
$608$	−3.00000			−0.121666
$609$		0			0
$610$	22.0000			0.890754
$611$	4.50000	+	7.79423i	0.182051	+	0.315321i
$612$		0			0
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$	7.50000	+	12.9904i	0.302675	+	0.524249i
$615$		0			0
$616$	6.00000	+	5.19615i	0.241747	+	0.209359i
$617$	14.0000			0.563619			0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000			0.563619			0.281809	+	0.959470i	$0.409065\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$	4.00000	−	6.92820i	0.160644	−	0.278243i
$621$		0			0
$622$	−14.0000			−0.561349
$623$	−25.0000	+	8.66025i	−1.00160	+	0.346966i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	7.00000	−	12.1244i	0.279776	−	0.484587i
$627$		0			0
$628$	5.50000	+	9.52628i	0.219474	+	0.380140i
$629$	2.00000			0.0797452
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$	−1.00000	−	1.73205i	−0.0397779	−	0.0688973i
$633$		0			0
$634$	9.00000	−	15.5885i	0.357436	−	0.619097i
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$	6.50000	+	2.59808i	0.257539	+	0.102940i
$638$	9.00000			0.356313
$639$		0			0
$640$	−1.00000	+	1.73205i	−0.0395285	+	0.0684653i
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$		0			0
$643$	1.00000			0.0394362			0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000			0.0394362			0.0197181	+	0.999806i	$0.493723\pi$
$644$		0			0
$645$		0			0
$646$	−1.50000	−	2.59808i	−0.0590167	−	0.102220i
$647$	−3.00000	+	5.19615i	−0.117942	+	0.204282i	−0.918952	−	0.394369i	$-0.870963\pi$
$647$	−3.00000	+	5.19615i	−0.117942	+	0.204282i	0.801010	+	0.598651i	$0.204296\pi$
$648$		0			0
$649$	−16.5000	−	28.5788i	−0.647682	−	1.12182i
$650$	1.00000			0.0392232
$651$		0			0
$652$	−25.0000			−0.979076
$653$	−15.0000	−	25.9808i	−0.586995	−	1.01671i	−0.994623	−	0.103558i	$-0.966977\pi$
$653$	−15.0000	−	25.9808i	−0.586995	−	1.01671i	0.407628	−	0.913148i	$-0.366356\pi$
$654$		0			0
$655$	−12.0000	+	20.7846i	−0.468879	+	0.812122i
$656$	1.00000	+	1.73205i	0.0390434	+	0.0676252i
$657$		0			0
$658$	−18.0000	−	15.5885i	−0.701713	−	0.607701i
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−19.0000	+	32.9090i	−0.739014	+	1.28001i	0.213925	+	0.976850i	$0.431375\pi$
$661$	−19.0000	+	32.9090i	−0.739014	+	1.28001i	−0.952940	+	0.303160i	$0.901958\pi$
$662$	14.0000	−	24.2487i	0.544125	−	0.942453i
$663$		0			0
$664$		0			0
$665$	12.0000	+	10.3923i	0.465340	+	0.402996i
$666$		0			0
$667$		0			0
$668$	−9.50000	+	16.4545i	−0.367566	+	0.636643i
$669$		0			0
$670$	7.00000	+	12.1244i	0.270434	+	0.468405i
$671$	−33.0000			−1.27395
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$	−3.50000	−	6.06218i	−0.134815	−	0.233506i
$675$		0			0
$676$	−0.500000	+	0.866025i	−0.0192308	+	0.0333087i
$677$	0.500000	+	0.866025i	0.0192166	+	0.0332841i	0.875474	−	0.483266i	$-0.160549\pi$
$677$	0.500000	+	0.866025i	0.0192166	+	0.0332841i	−0.856257	+	0.516550i	$0.827216\pi$
$678$		0			0
$679$	6.00000	−	31.1769i	0.230259	−	1.19646i
$680$	−2.00000			−0.0766965
$681$		0			0
$682$	−6.00000	+	10.3923i	−0.229752	+	0.397942i
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	0.0465592	+	0.998916i	$0.485174\pi$
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	−0.888366	+	0.459136i	$0.848159\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$	−18.5000	−	0.866025i	−0.706333	−	0.0330650i
$687$		0			0
$688$	−3.00000	−	5.19615i	−0.114374	−	0.198101i
$689$	−0.500000	+	0.866025i	−0.0190485	+	0.0329929i
$690$		0			0
$691$	−17.5000	−	30.3109i	−0.665731	−	1.15308i	−0.979086	−	0.203445i	$-0.934786\pi$
$691$	−17.5000	−	30.3109i	−0.665731	−	1.15308i	0.313355	−	0.949636i	$-0.398547\pi$
$692$	−7.00000			−0.266100
$693$		0			0
$694$	−2.00000			−0.0759190
$695$	14.0000	+	24.2487i	0.531050	+	0.919806i
$696$		0			0
$697$	−1.00000	+	1.73205i	−0.0378777	+	0.0656061i
$698$	−10.0000	−	17.3205i	−0.378506	−	0.655591i
$699$		0			0
$700$	−2.50000	+	0.866025i	−0.0944911	+	0.0327327i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	−3.00000	+	5.19615i	−0.113147	+	0.195977i
$704$	1.50000	−	2.59808i	0.0565334	−	0.0979187i
$705$		0			0
$706$	18.0000			0.677439
$707$	−28.0000	−	24.2487i	−1.05305	−	0.911967i
$708$		0			0
$709$	10.0000	+	17.3205i	0.375558	+	0.650485i	0.990410	−	0.138157i	$-0.0441178\pi$
$709$	10.0000	+	17.3205i	0.375558	+	0.650485i	−0.614852	+	0.788642i	$0.710784\pi$
$710$	15.0000	−	25.9808i	0.562940	−	0.975041i
$711$		0			0
$712$	5.00000	+	8.66025i	0.187383	+	0.324557i
$713$		0			0
$714$		0			0
$715$	6.00000			0.224387
$716$	−3.00000	−	5.19615i	−0.112115	−	0.194189i
$717$		0			0
$718$	8.00000	−	13.8564i	0.298557	−	0.517116i
$719$	19.0000	+	32.9090i	0.708580	+	1.22730i	0.965384	+	0.260834i	$0.0839974\pi$
$719$	19.0000	+	32.9090i	0.708580	+	1.22730i	−0.256803	+	0.966464i	$0.582669\pi$
$720$		0			0
$721$	35.0000	−	12.1244i	1.30347	−	0.451535i
$722$	−10.0000			−0.372161
$723$		0			0
$724$	2.50000	−	4.33013i	0.0929118	−	0.160928i
$725$	−1.50000	+	2.59808i	−0.0557086	+	0.0964901i
$726$		0			0
$727$	−12.0000			−0.445055			−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000			−0.445055			−0.222528	+	0.974926i	$0.571431\pi$
$728$	0.500000	−	2.59808i	0.0185312	−	0.0962911i
$729$		0			0
$730$	12.0000	+	20.7846i	0.444140	+	0.769273i
$731$	3.00000	−	5.19615i	0.110959	−	0.192187i
$732$		0			0
$733$	4.00000	+	6.92820i	0.147743	+	0.255899i	0.930393	−	0.366563i	$-0.119466\pi$
$733$	4.00000	+	6.92820i	0.147743	+	0.255899i	−0.782650	+	0.622462i	$0.786132\pi$
$734$	18.0000			0.664392
$735$		0			0
$736$		0			0
$737$	−10.5000	−	18.1865i	−0.386772	−	0.669910i
$738$		0			0
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$	2.00000	+	3.46410i	0.0735215	+	0.127343i
$741$		0			0
$742$	0.500000	−	2.59808i	0.0183556	−	0.0953784i
$743$	35.0000			1.28403			0.642013	−	0.766694i	$-0.278100\pi$
$743$	35.0000			1.28403			0.642013	+	0.766694i	$0.278100\pi$
$744$		0			0
$745$		0			0
$746$	5.50000	−	9.52628i	0.201369	−	0.348782i
$747$		0			0
$748$	3.00000			0.109691
$749$	10.0000	−	3.46410i	0.365392	−	0.126576i
$750$		0			0
$751$	−14.0000	−	24.2487i	−0.510867	−	0.884848i	−0.999921	−	0.0125942i	$-0.995991\pi$
$751$	−14.0000	−	24.2487i	−0.510867	−	0.884848i	0.489053	−	0.872254i	$-0.337342\pi$
$752$	−4.50000	+	7.79423i	−0.164098	+	0.284226i
$753$		0			0
$754$	−1.50000	−	2.59808i	−0.0546268	−	0.0946164i
$755$	−34.0000			−1.23739
$756$		0			0
$757$	19.0000			0.690567			0.345283	−	0.938498i	$-0.387783\pi$
$757$	19.0000			0.690567			0.345283	+	0.938498i	$0.387783\pi$
$758$	−18.0000	−	31.1769i	−0.653789	−	1.13240i
$759$		0			0
$760$	3.00000	−	5.19615i	0.108821	−	0.188484i
$761$	14.0000	+	24.2487i	0.507500	+	0.879015i	0.999962	+	0.00868155i	$0.00276346\pi$
$761$	14.0000	+	24.2487i	0.507500	+	0.879015i	−0.492463	+	0.870334i	$0.663903\pi$
$762$		0			0
$763$	16.0000	+	13.8564i	0.579239	+	0.501636i
$764$	12.0000			0.434145
$765$		0			0
$766$	−16.0000	+	27.7128i	−0.578103	+	1.00130i
$767$	−5.50000	+	9.52628i	−0.198593	+	0.343974i
$768$		0			0
$769$	−4.00000			−0.144244			−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000			−0.144244			−0.0721218	+	0.997396i	$0.522977\pi$
$770$	−15.0000	+	5.19615i	−0.540562	+	0.187256i
$771$		0			0
$772$	7.00000	+	12.1244i	0.251936	+	0.436365i
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	−0.189899	−	0.981804i	$-0.560816\pi$
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	0.945216	−	0.326445i	$-0.105851\pi$
$774$		0			0
$775$	−2.00000	−	3.46410i	−0.0718421	−	0.124434i
$776$	−12.0000			−0.430775
$777$		0			0
$778$	−13.0000			−0.466073
$779$	−3.00000	−	5.19615i	−0.107486	−	0.186171i
$780$		0			0
$781$	−22.5000	+	38.9711i	−0.805113	+	1.39450i
$782$		0			0
$783$		0			0
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$	−22.0000

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +1.00000 q^{8} +(-1.00000 + 1.73205i) q^{10} +(1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{17} +(1.50000 + 2.59808i) q^{19} +2.00000 q^{20} -3.00000 q^{22} +(0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{26} +(-2.00000 - 1.73205i) q^{28} -3.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -1.00000 q^{34} +(5.00000 - 1.73205i) q^{35} +(1.00000 + 1.73205i) q^{37} +(1.50000 - 2.59808i) q^{38} +(-1.00000 - 1.73205i) q^{40} -2.00000 q^{41} +6.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{47} +(-6.50000 - 2.59808i) q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(0.500000 - 0.866025i) q^{53} -6.00000 q^{55} +(-0.500000 + 2.59808i) q^{56} +(1.50000 + 2.59808i) q^{58} +(5.50000 - 9.52628i) q^{59} +(-5.50000 - 9.52628i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(3.50000 - 6.06218i) q^{67} +(0.500000 + 0.866025i) q^{68} +(-4.00000 - 3.46410i) q^{70} -15.0000 q^{71} +(6.00000 - 10.3923i) q^{73} +(1.00000 - 1.73205i) q^{74} -3.00000 q^{76} +(6.00000 + 5.19615i) q^{77} +(-1.00000 - 1.73205i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(1.00000 + 1.73205i) q^{82} -2.00000 q^{85} +(-3.00000 - 5.19615i) q^{86} +(1.50000 - 2.59808i) q^{88} +(5.00000 + 8.66025i) q^{89} +(0.500000 - 2.59808i) q^{91} +(-4.50000 + 7.79423i) q^{94} +(3.00000 - 5.19615i) q^{95} -12.0000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 - q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 2 q^{5} - q^{7} + 2 q^{8} - 2 q^{10} + 3 q^{11} - 2 q^{13} + 5 q^{14} - q^{16} + q^{17} + 3 q^{19} + 4 q^{20} - 6 q^{22} + q^{25} + q^{26} - 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 2 q^{34} + 10 q^{35} + 2 q^{37} + 3 q^{38} - 2 q^{40} - 4 q^{41} + 12 q^{43} + 3 q^{44} - 9 q^{47} - 13 q^{49} - 2 q^{50} + q^{52} + q^{53} - 12 q^{55} - q^{56} + 3 q^{58} + 11 q^{59} - 11 q^{61} - 8 q^{62} + 2 q^{64} + 2 q^{65} + 7 q^{67} + q^{68} - 8 q^{70} - 30 q^{71} + 12 q^{73} + 2 q^{74} - 6 q^{76} + 12 q^{77} - 2 q^{79} - 2 q^{80} + 2 q^{82} - 4 q^{85} - 6 q^{86} + 3 q^{88} + 10 q^{89} + q^{91} - 9 q^{94} + 6 q^{95} - 24 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 - q^7 + 2 * q^8 - 2 * q^10 + 3 * q^11 - 2 * q^13 + 5 * q^14 - q^16 + q^17 + 3 * q^19 + 4 * q^20 - 6 * q^22 + q^25 + q^26 - 4 * q^28 - 6 * q^29 + 4 * q^31 - q^32 - 2 * q^34 + 10 * q^35 + 2 * q^37 + 3 * q^38 - 2 * q^40 - 4 * q^41 + 12 * q^43 + 3 * q^44 - 9 * q^47 - 13 * q^49 - 2 * q^50 + q^52 + q^53 - 12 * q^55 - q^56 + 3 * q^58 + 11 * q^59 - 11 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 7 * q^67 + q^68 - 8 * q^70 - 30 * q^71 + 12 * q^73 + 2 * q^74 - 6 * q^76 + 12 * q^77 - 2 * q^79 - 2 * q^80 + 2 * q^82 - 4 * q^85 - 6 * q^86 + 3 * q^88 + 10 * q^89 + q^91 - 9 * q^94 + 6 * q^95 - 24 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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