LMFDB - Embedded newform 4830.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.a (trivial)

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:	yes
Analytic conductor:	$38.5677441763$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.74456 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.74456 q^{19} +1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -6.74456 q^{26} -1.00000 q^{27} -1.00000 q^{28} +4.74456 q^{29} +1.00000 q^{30} +4.74456 q^{31} -1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.74456 q^{38} -6.74456 q^{39} -1.00000 q^{40} -8.74456 q^{41} -1.00000 q^{42} -4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -12.7446 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.74456 q^{52} +12.7446 q^{53} +1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -4.74456 q^{57} -4.74456 q^{58} +4.00000 q^{59} -1.00000 q^{60} -11.4891 q^{61} -4.74456 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.74456 q^{65} -4.00000 q^{66} +8.00000 q^{67} +1.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} -0.744563 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.74456 q^{76} +4.00000 q^{77} +6.74456 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.74456 q^{82} +16.7446 q^{83} +1.00000 q^{84} -4.74456 q^{87} +4.00000 q^{88} -14.7446 q^{89} -1.00000 q^{90} -6.74456 q^{91} -1.00000 q^{92} -4.74456 q^{93} +12.7446 q^{94} +4.74456 q^{95} +1.00000 q^{96} +1.25544 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{19} + 2 q^{20} + 2 q^{21} + 8 q^{22} - 2 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{28} - 2 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} + 8 q^{33} - 2 q^{35} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 2 q^{40} - 6 q^{41} - 2 q^{42} - 8 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 2 q^{52} + 14 q^{53} + 2 q^{54} - 8 q^{55} + 2 q^{56} + 2 q^{57} + 2 q^{58} + 8 q^{59} - 2 q^{60} + 2 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{65} - 8 q^{66} + 16 q^{67} + 2 q^{69} + 2 q^{70} - 16 q^{71} - 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} + 8 q^{77} + 2 q^{78} - 16 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 22 q^{83} + 2 q^{84} + 2 q^{87} + 8 q^{88} - 18 q^{89} - 2 q^{90} - 2 q^{91} - 2 q^{92} + 2 q^{93} + 14 q^{94} - 2 q^{95} + 2 q^{96} + 14 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 8 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^18 - 2 * q^19 + 2 * q^20 + 2 * q^21 + 8 * q^22 - 2 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 - 2 * q^28 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 + 8 * q^33 - 2 * q^35 + 2 * q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 2 * q^40 - 6 * q^41 - 2 * q^42 - 8 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 2 * q^52 + 14 * q^53 + 2 * q^54 - 8 * q^55 + 2 * q^56 + 2 * q^57 + 2 * q^58 + 8 * q^59 - 2 * q^60 + 2 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^65 - 8 * q^66 + 16 * q^67 + 2 * q^69 + 2 * q^70 - 16 * q^71 - 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 + 8 * q^77 + 2 * q^78 - 16 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 22 * q^83 + 2 * q^84 + 2 * q^87 + 8 * q^88 - 18 * q^89 - 2 * q^90 - 2 * q^91 - 2 * q^92 + 2 * q^93 + 14 * q^94 - 2 * q^95 + 2 * q^96 + 14 * q^97 - 2 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	6.74456	1.87061	0.935303	−	0.353849i	$-0.115127\pi$
$13$	6.74456	1.87061	0.935303	+	0.353849i	$0.115127\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	4.74456	1.08848	0.544239	−	0.838930i	$-0.316819\pi$
$19$	4.74456	1.08848	0.544239	+	0.838930i	$0.316819\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	4.00000	0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−6.74456	−1.32272
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	4.74456	0.881043	0.440522	−	0.897742i	$-0.354794\pi$
$29$	4.74456	0.881043	0.440522	+	0.897742i	$0.354794\pi$
$30$	1.00000	0.182574
$31$	4.74456	0.852149	0.426074	−	0.904688i	$-0.359896\pi$
$31$	4.74456	0.852149	0.426074	+	0.904688i	$0.359896\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	0	0
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.74456	−0.769670
$39$	−6.74456	−1.07999
$40$	−1.00000	−0.158114
$41$	−8.74456	−1.36567	−0.682836	−	0.730572i	$-0.739253\pi$
$41$	−8.74456	−1.36567	−0.682836	+	0.730572i	$0.739253\pi$
$42$	−1.00000	−0.154303
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−12.7446	−1.85899	−0.929493	−	0.368840i	$-0.879755\pi$
$47$	−12.7446	−1.85899	−0.929493	+	0.368840i	$0.879755\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.74456	0.935303
$53$	12.7446	1.75060	0.875300	−	0.483580i	$-0.160664\pi$
$53$	12.7446	1.75060	0.875300	+	0.483580i	$0.160664\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	−4.74456	−0.628433
$58$	−4.74456	−0.622992
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−1.00000	−0.129099
$61$	−11.4891	−1.47103	−0.735516	−	0.677507i	$-0.763060\pi$
$61$	−11.4891	−1.47103	−0.735516	+	0.677507i	$0.763060\pi$
$62$	−4.74456	−0.602560
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	6.74456	0.836560
$66$	−4.00000	−0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	1.00000	0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	−0.744563	−0.0871445	−0.0435722	−	0.999050i	$-0.513874\pi$
$73$	−0.744563	−0.0871445	−0.0435722	+	0.999050i	$0.513874\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	4.74456	0.544239
$77$	4.00000	0.455842
$78$	6.74456	0.763671
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	8.74456	0.965675
$83$	16.7446	1.83795	0.918977	−	0.394311i	$-0.129017\pi$
$83$	16.7446	1.83795	0.918977	+	0.394311i	$0.129017\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	0	0
$87$	−4.74456	−0.508671
$88$	4.00000	0.426401
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	−1.00000	−0.105409
$91$	−6.74456	−0.707022
$92$	−1.00000	−0.104257
$93$	−4.74456	−0.491988
$94$	12.7446	1.31450
$95$	4.74456	0.486782
$96$	1.00000	0.102062
$97$	1.25544	0.127470	0.0637352	−	0.997967i	$-0.479699\pi$
$97$	1.25544	0.127470	0.0637352	+	0.997967i	$0.479699\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	18.7446	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$101$	18.7446	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−6.74456	−0.661359
$105$	1.00000	0.0975900
$106$	−12.7446	−1.23786
$107$	−6.74456	−0.652021	−0.326011	−	0.945366i	$-0.605705\pi$
$107$	−6.74456	−0.652021	−0.326011	+	0.945366i	$0.605705\pi$
$108$	−1.00000	−0.0962250
$109$	12.7446	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$109$	12.7446	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$110$	4.00000	0.381385
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	4.74456	0.444369
$115$	−1.00000	−0.0932505
$116$	4.74456	0.440522
$117$	6.74456	0.623535
$118$	−4.00000	−0.368230
$119$	0	0
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	11.4891	1.04018
$123$	8.74456	0.788471
$124$	4.74456	0.426074
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.74456	−0.591537
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	−4.74456	−0.411406
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.4891	0.981582	0.490791	−	0.871277i	$-0.336708\pi$
$137$	11.4891	0.981582	0.490791	+	0.871277i	$0.336708\pi$
$138$	−1.00000	−0.0851257
$139$	−13.4891	−1.14413	−0.572066	−	0.820207i	$-0.693858\pi$
$139$	−13.4891	−1.14413	−0.572066	+	0.820207i	$0.693858\pi$
$140$	−1.00000	−0.0845154
$141$	12.7446	1.07329
$142$	8.00000	0.671345
$143$	−26.9783	−2.25603
$144$	1.00000	0.0833333
$145$	4.74456	0.394014
$146$	0.744563	0.0616204
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	7.48913	0.613533	0.306767	−	0.951785i	$-0.400753\pi$
$149$	7.48913	0.613533	0.306767	+	0.951785i	$0.400753\pi$
$150$	1.00000	0.0816497
$151$	−9.48913	−0.772214	−0.386107	−	0.922454i	$-0.626180\pi$
$151$	−9.48913	−0.772214	−0.386107	+	0.922454i	$0.626180\pi$
$152$	−4.74456	−0.384835
$153$	0	0
$154$	−4.00000	−0.322329
$155$	4.74456	0.381092
$156$	−6.74456	−0.539997
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	8.00000	0.636446
$159$	−12.7446	−1.01071
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−10.7446	−0.841579	−0.420790	−	0.907158i	$-0.638247\pi$
$163$	−10.7446	−0.841579	−0.420790	+	0.907158i	$0.638247\pi$
$164$	−8.74456	−0.682836
$165$	4.00000	0.311400
$166$	−16.7446	−1.29963
$167$	−16.7446	−1.29573	−0.647867	−	0.761754i	$-0.724339\pi$
$167$	−16.7446	−1.29573	−0.647867	+	0.761754i	$0.724339\pi$
$168$	−1.00000	−0.0771517
$169$	32.4891	2.49916
$170$	0	0
$171$	4.74456	0.362826
$172$	0	0
$173$	6.74456	0.512780	0.256390	−	0.966573i	$-0.417467\pi$
$173$	6.74456	0.512780	0.256390	+	0.966573i	$0.417467\pi$
$174$	4.74456	0.359684
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	14.7446	1.10515
$179$	10.7446	0.803086	0.401543	−	0.915840i	$-0.368474\pi$
$179$	10.7446	0.803086	0.401543	+	0.915840i	$0.368474\pi$
$180$	1.00000	0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	6.74456	0.499940
$183$	11.4891	0.849301
$184$	1.00000	0.0737210
$185$	2.00000	0.147043
$186$	4.74456	0.347888
$187$	0	0
$188$	−12.7446	−0.929493
$189$	1.00000	0.0727393
$190$	−4.74456	−0.344207
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−1.25544	−0.0901351
$195$	−6.74456	−0.482988
$196$	1.00000	0.0714286
$197$	19.4891	1.38854	0.694271	−	0.719713i	$-0.255727\pi$
$197$	19.4891	1.38854	0.694271	+	0.719713i	$0.255727\pi$
$198$	4.00000	0.284268
$199$	24.2337	1.71788	0.858940	−	0.512076i	$-0.171123\pi$
$199$	24.2337	1.71788	0.858940	+	0.512076i	$0.171123\pi$
$200$	−1.00000	−0.0707107
$201$	−8.00000	−0.564276
$202$	−18.7446	−1.31886
$203$	−4.74456	−0.333003
$204$	0	0
$205$	−8.74456	−0.610747
$206$	−4.00000	−0.278693
$207$	−1.00000	−0.0695048
$208$	6.74456	0.467651
$209$	−18.9783	−1.31275
$210$	−1.00000	−0.0690066
$211$	9.48913	0.653258	0.326629	−	0.945153i	$-0.394087\pi$
$211$	9.48913	0.653258	0.326629	+	0.945153i	$0.394087\pi$
$212$	12.7446	0.875300
$213$	8.00000	0.548151
$214$	6.74456	0.461049
$215$	0	0
$216$	1.00000	0.0680414
$217$	−4.74456	−0.322082
$218$	−12.7446	−0.863171
$219$	0.744563	0.0503129
$220$	−4.00000	−0.269680
$221$	0	0
$222$	2.00000	0.134231
$223$	19.4891	1.30509	0.652544	−	0.757751i	$-0.273702\pi$
$223$	19.4891	1.30509	0.652544	+	0.757751i	$0.273702\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−2.00000	−0.133038
$227$	4.74456	0.314908	0.157454	−	0.987526i	$-0.449671\pi$
$227$	4.74456	0.314908	0.157454	+	0.987526i	$0.449671\pi$
$228$	−4.74456	−0.314216
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	1.00000	0.0659380
$231$	−4.00000	−0.263181
$232$	−4.74456	−0.311496
$233$	−23.4891	−1.53882	−0.769412	−	0.638753i	$-0.779451\pi$
$233$	−23.4891	−1.53882	−0.769412	+	0.638753i	$0.779451\pi$
$234$	−6.74456	−0.440906
$235$	−12.7446	−0.831364
$236$	4.00000	0.260378
$237$	8.00000	0.519656
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	−13.4891	−0.868911	−0.434455	−	0.900693i	$-0.643059\pi$
$241$	−13.4891	−0.868911	−0.434455	+	0.900693i	$0.643059\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−11.4891	−0.735516
$245$	1.00000	0.0638877
$246$	−8.74456	−0.557533
$247$	32.0000	2.03611
$248$	−4.74456	−0.301280
$249$	−16.7446	−1.06114
$250$	−1.00000	−0.0632456
$251$	18.2337	1.15090	0.575450	−	0.817837i	$-0.304827\pi$
$251$	18.2337	1.15090	0.575450	+	0.817837i	$0.304827\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	6.74456	0.418280
$261$	4.74456	0.293681
$262$	−4.00000	−0.247121
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−4.00000	−0.246183
$265$	12.7446	0.782892
$266$	4.74456	0.290908
$267$	14.7446	0.902353
$268$	8.00000	0.488678
$269$	−26.7446	−1.63064	−0.815322	−	0.579007i	$-0.803440\pi$
$269$	−26.7446	−1.63064	−0.815322	+	0.579007i	$0.803440\pi$
$270$	1.00000	0.0608581
$271$	12.7446	0.774177	0.387089	−	0.922043i	$-0.373481\pi$
$271$	12.7446	0.774177	0.387089	+	0.922043i	$0.373481\pi$
$272$	0	0
$273$	6.74456	0.408200
$274$	−11.4891	−0.694083
$275$	−4.00000	−0.241209
$276$	1.00000	0.0601929
$277$	32.7446	1.96743	0.983715	−	0.179735i	$-0.0575241\pi$
$277$	32.7446	1.96743	0.983715	+	0.179735i	$0.0575241\pi$
$278$	13.4891	0.809024
$279$	4.74456	0.284050
$280$	1.00000	0.0597614
$281$	−3.48913	−0.208144	−0.104072	−	0.994570i	$-0.533187\pi$
$281$	−3.48913	−0.208144	−0.104072	+	0.994570i	$0.533187\pi$
$282$	−12.7446	−0.758928
$283$	−7.25544	−0.431291	−0.215645	−	0.976472i	$-0.569186\pi$
$283$	−7.25544	−0.431291	−0.215645	+	0.976472i	$0.569186\pi$
$284$	−8.00000	−0.474713
$285$	−4.74456	−0.281044
$286$	26.9783	1.59526
$287$	8.74456	0.516175
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−4.74456	−0.278610
$291$	−1.25544	−0.0735950
$292$	−0.744563	−0.0435722
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	4.00000	0.232889
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	−7.48913	−0.433833
$299$	−6.74456	−0.390048
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	9.48913	0.546038
$303$	−18.7446	−1.07685
$304$	4.74456	0.272119
$305$	−11.4891	−0.657865
$306$	0	0
$307$	−18.9783	−1.08315	−0.541573	−	0.840654i	$-0.682171\pi$
$307$	−18.9783	−1.08315	−0.541573	+	0.840654i	$0.682171\pi$
$308$	4.00000	0.227921
$309$	−4.00000	−0.227552
$310$	−4.74456	−0.269473
$311$	3.48913	0.197850	0.0989251	−	0.995095i	$-0.468460\pi$
$311$	3.48913	0.197850	0.0989251	+	0.995095i	$0.468460\pi$
$312$	6.74456	0.381836
$313$	5.25544	0.297055	0.148527	−	0.988908i	$-0.452547\pi$
$313$	5.25544	0.297055	0.148527	+	0.988908i	$0.452547\pi$
$314$	2.00000	0.112867
$315$	−1.00000	−0.0563436
$316$	−8.00000	−0.450035
$317$	16.9783	0.953594	0.476797	−	0.879014i	$-0.341798\pi$
$317$	16.9783	0.953594	0.476797	+	0.879014i	$0.341798\pi$
$318$	12.7446	0.714680
$319$	−18.9783	−1.06258
$320$	1.00000	0.0559017
$321$	6.74456	0.376445
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	6.74456	0.374121
$326$	10.7446	0.595086
$327$	−12.7446	−0.704776
$328$	8.74456	0.482838
$329$	12.7446	0.702630
$330$	−4.00000	−0.220193
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	16.7446	0.918977
$333$	2.00000	0.109599
$334$	16.7446	0.916222
$335$	8.00000	0.437087
$336$	1.00000	0.0545545
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−32.4891	−1.76718
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−18.9783	−1.02773
$342$	−4.74456	−0.256557
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.00000	0.0538382
$346$	−6.74456	−0.362590
$347$	−22.9783	−1.23354	−0.616769	−	0.787145i	$-0.711559\pi$
$347$	−22.9783	−1.23354	−0.616769	+	0.787145i	$0.711559\pi$
$348$	−4.74456	−0.254335
$349$	9.25544	0.495432	0.247716	−	0.968833i	$-0.420320\pi$
$349$	9.25544	0.495432	0.247716	+	0.968833i	$0.420320\pi$
$350$	1.00000	0.0534522
$351$	−6.74456	−0.359998
$352$	4.00000	0.213201
$353$	19.4891	1.03730	0.518651	−	0.854986i	$-0.326435\pi$
$353$	19.4891	1.03730	0.518651	+	0.854986i	$0.326435\pi$
$354$	4.00000	0.212598
$355$	−8.00000	−0.424596
$356$	−14.7446	−0.781460
$357$	0	0
$358$	−10.7446	−0.567868
$359$	−26.9783	−1.42386	−0.711929	−	0.702252i	$-0.752178\pi$
$359$	−26.9783	−1.42386	−0.711929	+	0.702252i	$0.752178\pi$
$360$	−1.00000	−0.0527046
$361$	3.51087	0.184783
$362$	14.0000	0.735824
$363$	−5.00000	−0.262432
$364$	−6.74456	−0.353511
$365$	−0.744563	−0.0389722
$366$	−11.4891	−0.600546
$367$	6.51087	0.339865	0.169932	−	0.985456i	$-0.445645\pi$
$367$	6.51087	0.339865	0.169932	+	0.985456i	$0.445645\pi$
$368$	−1.00000	−0.0521286
$369$	−8.74456	−0.455224
$370$	−2.00000	−0.103975
$371$	−12.7446	−0.661665
$372$	−4.74456	−0.245994
$373$	23.4891	1.21622	0.608110	−	0.793852i	$-0.291928\pi$
$373$	23.4891	1.21622	0.608110	+	0.793852i	$0.291928\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	12.7446	0.657251
$377$	32.0000	1.64808
$378$	−1.00000	−0.0514344
$379$	−6.74456	−0.346445	−0.173222	−	0.984883i	$-0.555418\pi$
$379$	−6.74456	−0.346445	−0.173222	+	0.984883i	$0.555418\pi$
$380$	4.74456	0.243391
$381$	−8.00000	−0.409852
$382$	16.0000	0.818631
$383$	38.7446	1.97975	0.989877	−	0.141926i	$-0.0453294\pi$
$383$	38.7446	1.97975	0.989877	+	0.141926i	$0.0453294\pi$
$384$	1.00000	0.0510310
$385$	4.00000	0.203859
$386$	−14.0000	−0.712581
$387$	0	0
$388$	1.25544	0.0637352
$389$	3.48913	0.176906	0.0884528	−	0.996080i	$-0.471808\pi$
$389$	3.48913	0.176906	0.0884528	+	0.996080i	$0.471808\pi$
$390$	6.74456	0.341524
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−4.00000	−0.201773
$394$	−19.4891	−0.981848
$395$	−8.00000	−0.402524
$396$	−4.00000	−0.201008
$397$	33.7228	1.69250	0.846250	−	0.532786i	$-0.178855\pi$
$397$	33.7228	1.69250	0.846250	+	0.532786i	$0.178855\pi$
$398$	−24.2337	−1.21473
$399$	4.74456	0.237525
$400$	1.00000	0.0500000
$401$	36.9783	1.84661	0.923303	−	0.384073i	$-0.125479\pi$
$401$	36.9783	1.84661	0.923303	+	0.384073i	$0.125479\pi$
$402$	8.00000	0.399004
$403$	32.0000	1.59403
$404$	18.7446	0.932577
$405$	1.00000	0.0496904
$406$	4.74456	0.235469
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−8.97825	−0.443946	−0.221973	−	0.975053i	$-0.571250\pi$
$409$	−8.97825	−0.443946	−0.221973	+	0.975053i	$0.571250\pi$
$410$	8.74456	0.431863
$411$	−11.4891	−0.566717
$412$	4.00000	0.197066
$413$	−4.00000	−0.196827
$414$	1.00000	0.0491473
$415$	16.7446	0.821958
$416$	−6.74456	−0.330679
$417$	13.4891	0.660565
$418$	18.9783	0.928257
$419$	26.2337	1.28160	0.640800	−	0.767708i	$-0.278603\pi$
$419$	26.2337	1.28160	0.640800	+	0.767708i	$0.278603\pi$
$420$	1.00000	0.0487950
$421$	15.2554	0.743505	0.371752	−	0.928332i	$-0.378757\pi$
$421$	15.2554	0.743505	0.371752	+	0.928332i	$0.378757\pi$
$422$	−9.48913	−0.461923
$423$	−12.7446	−0.619662
$424$	−12.7446	−0.618931
$425$	0	0
$426$	−8.00000	−0.387601
$427$	11.4891	0.555998
$428$	−6.74456	−0.326011
$429$	26.9783	1.30252
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−1.00000	−0.0481125
$433$	4.23369	0.203458	0.101729	−	0.994812i	$-0.467563\pi$
$433$	4.23369	0.203458	0.101729	+	0.994812i	$0.467563\pi$
$434$	4.74456	0.227746
$435$	−4.74456	−0.227484
$436$	12.7446	0.610354
$437$	−4.74456	−0.226963
$438$	−0.744563	−0.0355766
$439$	19.2554	0.919012	0.459506	−	0.888175i	$-0.348026\pi$
$439$	19.2554	0.919012	0.459506	+	0.888175i	$0.348026\pi$
$440$	4.00000	0.190693
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−2.00000	−0.0949158
$445$	−14.7446	−0.698959
$446$	−19.4891	−0.922837
$447$	−7.48913	−0.354223
$448$	−1.00000	−0.0472456
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	−1.00000	−0.0471405
$451$	34.9783	1.64706
$452$	2.00000	0.0940721
$453$	9.48913	0.445838
$454$	−4.74456	−0.222673
$455$	−6.74456	−0.316190
$456$	4.74456	0.222185
$457$	27.4891	1.28589	0.642944	−	0.765914i	$-0.277713\pi$
$457$	27.4891	1.28589	0.642944	+	0.765914i	$0.277713\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	32.2337	1.50127	0.750636	−	0.660716i	$-0.229747\pi$
$461$	32.2337	1.50127	0.750636	+	0.660716i	$0.229747\pi$
$462$	4.00000	0.186097
$463$	36.4674	1.69478	0.847391	−	0.530969i	$-0.178172\pi$
$463$	36.4674	1.69478	0.847391	+	0.530969i	$0.178172\pi$
$464$	4.74456	0.220261
$465$	−4.74456	−0.220024
$466$	23.4891	1.08811
$467$	31.7228	1.46796	0.733978	−	0.679173i	$-0.237661\pi$
$467$	31.7228	1.46796	0.733978	+	0.679173i	$0.237661\pi$
$468$	6.74456	0.311768
$469$	−8.00000	−0.369406
$470$	12.7446	0.587863
$471$	2.00000	0.0921551
$472$	−4.00000	−0.184115
$473$	0	0
$474$	−8.00000	−0.367452
$475$	4.74456	0.217695
$476$	0	0
$477$	12.7446	0.583533
$478$	−8.00000	−0.365911
$479$	−6.51087	−0.297489	−0.148745	−	0.988876i	$-0.547523\pi$
$479$	−6.51087	−0.297489	−0.148745	+	0.988876i	$0.547523\pi$
$480$	1.00000	0.0456435
$481$	13.4891	0.615051
$482$	13.4891	0.614413
$483$	−1.00000	−0.0455016
$484$	5.00000	0.227273
$485$	1.25544	0.0570065
$486$	1.00000	0.0453609
$487$	−14.9783	−0.678729	−0.339365	−	0.940655i	$-0.610212\pi$
$487$	−14.9783	−0.678729	−0.339365	+	0.940655i	$0.610212\pi$
$488$	11.4891	0.520088
$489$	10.7446	0.485886
$490$	−1.00000	−0.0451754
$491$	−13.7228	−0.619302	−0.309651	−	0.950850i	$-0.600212\pi$
$491$	−13.7228	−0.619302	−0.309651	+	0.950850i	$0.600212\pi$
$492$	8.74456	0.394235
$493$	0	0
$494$	−32.0000	−1.43975
$495$	−4.00000	−0.179787
$496$	4.74456	0.213037
$497$	8.00000	0.358849
$498$	16.7446	0.750342
$499$	36.4674	1.63250	0.816252	−	0.577696i	$-0.196048\pi$
$499$	36.4674	1.63250	0.816252	+	0.577696i	$0.196048\pi$
$500$	1.00000	0.0447214
$501$	16.7446	0.748092
$502$	−18.2337	−0.813809
$503$	25.7228	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$503$	25.7228	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$504$	1.00000	0.0445435
$505$	18.7446	0.834122
$506$	−4.00000	−0.177822
$507$	−32.4891	−1.44289
$508$	8.00000	0.354943
$509$	−37.7228	−1.67203	−0.836017	−	0.548703i	$-0.815122\pi$
$509$	−37.7228	−1.67203	−0.836017	+	0.548703i	$0.815122\pi$
$510$	0	0
$511$	0.744563	0.0329375
$512$	−1.00000	−0.0441942
$513$	−4.74456	−0.209478
$514$	−18.0000	−0.793946
$515$	4.00000	0.176261
$516$	0	0
$517$	50.9783	2.24202
$518$	2.00000	0.0878750
$519$	−6.74456	−0.296053
$520$	−6.74456	−0.295769
$521$	−20.2337	−0.886454	−0.443227	−	0.896409i	$-0.646166\pi$
$521$	−20.2337	−0.886454	−0.443227	+	0.896409i	$0.646166\pi$
$522$	−4.74456	−0.207664
$523$	−10.2337	−0.447488	−0.223744	−	0.974648i	$-0.571828\pi$
$523$	−10.2337	−0.447488	−0.223744	+	0.974648i	$0.571828\pi$
$524$	4.00000	0.174741
$525$	1.00000	0.0436436
$526$	8.00000	0.348817
$527$	0	0
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−12.7446	−0.553588
$531$	4.00000	0.173585
$532$	−4.74456	−0.205703
$533$	−58.9783	−2.55463
$534$	−14.7446	−0.638060
$535$	−6.74456	−0.291593
$536$	−8.00000	−0.345547
$537$	−10.7446	−0.463662
$538$	26.7446	1.15304
$539$	−4.00000	−0.172292
$540$	−1.00000	−0.0430331
$541$	8.97825	0.386005	0.193003	−	0.981198i	$-0.438177\pi$
$541$	8.97825	0.386005	0.193003	+	0.981198i	$0.438177\pi$
$542$	−12.7446	−0.547426
$543$	14.0000	0.600798
$544$	0	0
$545$	12.7446	0.545917
$546$	−6.74456	−0.288641
$547$	−16.2337	−0.694102	−0.347051	−	0.937846i	$-0.612817\pi$
$547$	−16.2337	−0.694102	−0.347051	+	0.937846i	$0.612817\pi$
$548$	11.4891	0.490791
$549$	−11.4891	−0.490344
$550$	4.00000	0.170561
$551$	22.5109	0.958996
$552$	−1.00000	−0.0425628
$553$	8.00000	0.340195
$554$	−32.7446	−1.39118
$555$	−2.00000	−0.0848953
$556$	−13.4891	−0.572066
$557$	−34.2337	−1.45053	−0.725264	−	0.688471i	$-0.758282\pi$
$557$	−34.2337	−1.45053	−0.725264	+	0.688471i	$0.758282\pi$
$558$	−4.74456	−0.200853
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	3.48913	0.147180
$563$	−9.76631	−0.411601	−0.205800	−	0.978594i	$-0.565980\pi$
$563$	−9.76631	−0.411601	−0.205800	+	0.978594i	$0.565980\pi$
$564$	12.7446	0.536643
$565$	2.00000	0.0841406
$566$	7.25544	0.304969
$567$	−1.00000	−0.0419961
$568$	8.00000	0.335673
$569$	−24.9783	−1.04714	−0.523571	−	0.851982i	$-0.675401\pi$
$569$	−24.9783	−1.04714	−0.523571	+	0.851982i	$0.675401\pi$
$570$	4.74456	0.198728
$571$	13.7228	0.574282	0.287141	−	0.957888i	$-0.407295\pi$
$571$	13.7228	0.574282	0.287141	+	0.957888i	$0.407295\pi$
$572$	−26.9783	−1.12802
$573$	16.0000	0.668410
$574$	−8.74456	−0.364991
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	28.7446	1.19665	0.598326	−	0.801253i	$-0.295833\pi$
$577$	28.7446	1.19665	0.598326	+	0.801253i	$0.295833\pi$
$578$	17.0000	0.707107
$579$	−14.0000	−0.581820
$580$	4.74456	0.197007
$581$	−16.7446	−0.694682
$582$	1.25544	0.0520396
$583$	−50.9783	−2.11130
$584$	0.744563	0.0308102
$585$	6.74456	0.278853
$586$	−14.0000	−0.578335
$587$	29.4891	1.21715	0.608573	−	0.793498i	$-0.291742\pi$
$587$	29.4891	1.21715	0.608573	+	0.793498i	$0.291742\pi$
$588$	−1.00000	−0.0412393
$589$	22.5109	0.927544
$590$	−4.00000	−0.164677
$591$	−19.4891	−0.801675
$592$	2.00000	0.0821995
$593$	28.9783	1.18999	0.594997	−	0.803728i	$-0.297153\pi$
$593$	28.9783	1.18999	0.594997	+	0.803728i	$0.297153\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	7.48913	0.306767
$597$	−24.2337	−0.991819
$598$	6.74456	0.275806
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	1.00000	0.0408248
$601$	−42.4674	−1.73228	−0.866140	−	0.499801i	$-0.833406\pi$
$601$	−42.4674	−1.73228	−0.866140	+	0.499801i	$0.833406\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	−9.48913	−0.386107
$605$	5.00000	0.203279
$606$	18.7446	0.761446
$607$	47.4891	1.92752	0.963762	−	0.266763i	$-0.0859542\pi$
$607$	47.4891	1.92752	0.963762	+	0.266763i	$0.0859542\pi$
$608$	−4.74456	−0.192417
$609$	4.74456	0.192259
$610$	11.4891	0.465181
$611$	−85.9565	−3.47743
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	18.9783	0.765900
$615$	8.74456	0.352615
$616$	−4.00000	−0.161165
$617$	−28.9783	−1.16662	−0.583310	−	0.812249i	$-0.698243\pi$
$617$	−28.9783	−1.16662	−0.583310	+	0.812249i	$0.698243\pi$
$618$	4.00000	0.160904
$619$	6.23369	0.250553	0.125277	−	0.992122i	$-0.460018\pi$
$619$	6.23369	0.250553	0.125277	+	0.992122i	$0.460018\pi$
$620$	4.74456	0.190546
$621$	1.00000	0.0401286
$622$	−3.48913	−0.139901
$623$	14.7446	0.590728
$624$	−6.74456	−0.269999
$625$	1.00000	0.0400000
$626$	−5.25544	−0.210050
$627$	18.9783	0.757918
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	1.00000	0.0398410
$631$	−18.9783	−0.755512	−0.377756	−	0.925905i	$-0.623304\pi$
$631$	−18.9783	−0.755512	−0.377756	+	0.925905i	$0.623304\pi$
$632$	8.00000	0.318223
$633$	−9.48913	−0.377159
$634$	−16.9783	−0.674292
$635$	8.00000	0.317470
$636$	−12.7446	−0.505355
$637$	6.74456	0.267229
$638$	18.9783	0.751356
$639$	−8.00000	−0.316475
$640$	−1.00000	−0.0395285
$641$	12.5109	0.494150	0.247075	−	0.968996i	$-0.420531\pi$
$641$	12.5109	0.494150	0.247075	+	0.968996i	$0.420531\pi$
$642$	−6.74456	−0.266187
$643$	−27.7228	−1.09328	−0.546641	−	0.837367i	$-0.684094\pi$
$643$	−27.7228	−1.09328	−0.546641	+	0.837367i	$0.684094\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	0	0
$647$	−0.744563	−0.0292718	−0.0146359	−	0.999893i	$-0.504659\pi$
$647$	−0.744563	−0.0292718	−0.0146359	+	0.999893i	$0.504659\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	−6.74456	−0.264544
$651$	4.74456	0.185954
$652$	−10.7446	−0.420790
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	12.7446	0.498352
$655$	4.00000	0.156293
$656$	−8.74456	−0.341418
$657$	−0.744563	−0.0290482
$658$	−12.7446	−0.496835
$659$	34.9783	1.36256	0.681280	−	0.732023i	$-0.261424\pi$
$659$	34.9783	1.36256	0.681280	+	0.732023i	$0.261424\pi$
$660$	4.00000	0.155700
$661$	31.4891	1.22479	0.612393	−	0.790554i	$-0.290207\pi$
$661$	31.4891	1.22479	0.612393	+	0.790554i	$0.290207\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−16.7446	−0.649815
$665$	−4.74456	−0.183986
$666$	−2.00000	−0.0774984
$667$	−4.74456	−0.183710
$668$	−16.7446	−0.647867
$669$	−19.4891	−0.753493
$670$	−8.00000	−0.309067
$671$	45.9565	1.77413
$672$	−1.00000	−0.0385758
$673$	−4.51087	−0.173881	−0.0869407	−	0.996213i	$-0.527709\pi$
$673$	−4.51087	−0.173881	−0.0869407	+	0.996213i	$0.527709\pi$
$674$	−10.0000	−0.385186
$675$	−1.00000	−0.0384900
$676$	32.4891	1.24958
$677$	−7.48913	−0.287830	−0.143915	−	0.989590i	$-0.545969\pi$
$677$	−7.48913	−0.287830	−0.143915	+	0.989590i	$0.545969\pi$
$678$	2.00000	0.0768095
$679$	−1.25544	−0.0481793
$680$	0	0
$681$	−4.74456	−0.181812
$682$	18.9783	0.726715
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	4.74456	0.181413
$685$	11.4891	0.438977
$686$	1.00000	0.0381802
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	85.9565	3.27468
$690$	−1.00000	−0.0380693
$691$	−32.4674	−1.23512	−0.617559	−	0.786525i	$-0.711878\pi$
$691$	−32.4674	−1.23512	−0.617559	+	0.786525i	$0.711878\pi$
$692$	6.74456	0.256390
$693$	4.00000	0.151947
$694$	22.9783	0.872242
$695$	−13.4891	−0.511672
$696$	4.74456	0.179842
$697$	0	0
$698$	−9.25544	−0.350323
$699$	23.4891	0.888440
$700$	−1.00000	−0.0377964
$701$	−40.9783	−1.54773	−0.773864	−	0.633352i	$-0.781678\pi$
$701$	−40.9783	−1.54773	−0.773864	+	0.633352i	$0.781678\pi$
$702$	6.74456	0.254557
$703$	9.48913	0.357889
$704$	−4.00000	−0.150756
$705$	12.7446	0.479988
$706$	−19.4891	−0.733483
$707$	−18.7446	−0.704962
$708$	−4.00000	−0.150329
$709$	−17.7663	−0.667228	−0.333614	−	0.942710i	$-0.608268\pi$
$709$	−17.7663	−0.667228	−0.333614	+	0.942710i	$0.608268\pi$
$710$	8.00000	0.300235
$711$	−8.00000	−0.300023
$712$	14.7446	0.552576
$713$	−4.74456	−0.177685
$714$	0	0
$715$	−26.9783	−1.00893
$716$	10.7446	0.401543
$717$	−8.00000	−0.298765
$718$	26.9783	1.00682
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	1.00000	0.0372678
$721$	−4.00000	−0.148968
$722$	−3.51087	−0.130661
$723$	13.4891	0.501666
$724$	−14.0000	−0.520306
$725$	4.74456	0.176209
$726$	5.00000	0.185567
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	6.74456	0.249970
$729$	1.00000	0.0370370
$730$	0.744563	0.0275575
$731$	0	0
$732$	11.4891	0.424650
$733$	−15.4891	−0.572104	−0.286052	−	0.958214i	$-0.592343\pi$
$733$	−15.4891	−0.572104	−0.286052	+	0.958214i	$0.592343\pi$
$734$	−6.51087	−0.240321
$735$	−1.00000	−0.0368856
$736$	1.00000	0.0368605
$737$	−32.0000	−1.17874
$738$	8.74456	0.321892
$739$	36.4674	1.34147	0.670737	−	0.741695i	$-0.265978\pi$
$739$	36.4674	1.34147	0.670737	+	0.741695i	$0.265978\pi$
$740$	2.00000	0.0735215
$741$	−32.0000	−1.17555
$742$	12.7446	0.467868
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	4.74456	0.173944
$745$	7.48913	0.274380
$746$	−23.4891	−0.859998
$747$	16.7446	0.612652
$748$	0	0
$749$	6.74456	0.246441
$750$	1.00000	0.0365148
$751$	−10.5109	−0.383547	−0.191774	−	0.981439i	$-0.561424\pi$
$751$	−10.5109	−0.383547	−0.191774	+	0.981439i	$0.561424\pi$
$752$	−12.7446	−0.464746
$753$	−18.2337	−0.664473
$754$	−32.0000	−1.16537
$755$	−9.48913	−0.345345
$756$	1.00000	0.0363696
$757$	−7.48913	−0.272197	−0.136098	−	0.990695i	$-0.543456\pi$
$757$	−7.48913	−0.272197	−0.136098	+	0.990695i	$0.543456\pi$
$758$	6.74456	0.244974
$759$	−4.00000	−0.145191
$760$	−4.74456	−0.172103
$761$	19.7228	0.714951	0.357476	−	0.933922i	$-0.383637\pi$
$761$	19.7228	0.714951	0.357476	+	0.933922i	$0.383637\pi$
$762$	8.00000	0.289809
$763$	−12.7446	−0.461384
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−38.7446	−1.39990
$767$	26.9783	0.974128
$768$	−1.00000	−0.0360844
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	−4.00000	−0.144150
$771$	−18.0000	−0.648254
$772$	14.0000	0.503871
$773$	−35.4891	−1.27646	−0.638228	−	0.769848i	$-0.720332\pi$
$773$	−35.4891	−1.27646	−0.638228	+	0.769848i	$0.720332\pi$
$774$	0	0
$775$	4.74456	0.170430
$776$	−1.25544	−0.0450676
$777$	2.00000	0.0717496
$778$	−3.48913	−0.125091
$779$	−41.4891	−1.48650
$780$	−6.74456	−0.241494
$781$	32.0000	1.14505
$782$	0	0
$783$	−4.74456	−0.169557
$784$	1.00000	0.0357143
$785$	−2.00000	−0.0713831
$786$	4.00000	0.142675
$787$	−11.7228	−0.417873	−0.208937	−	0.977929i	$-0.567000\pi$
$787$	−11.7228	−0.417873	−0.208937	+	0.977929i	$0.567000\pi$
$788$	19.4891	0.694271
$789$	8.00000	0.284808
$790$	8.00000	0.284627
$791$	−2.00000	−0.0711118
$792$	4.00000	0.142134
$793$	−77.4891	−2.75172
$794$	−33.7228	−1.19678
$795$	−12.7446	−0.452003
$796$	24.2337	0.858940
$797$	52.9783	1.87659	0.938293	−	0.345841i	$-0.112407\pi$
$797$	52.9783	1.87659	0.938293	+	0.345841i	$0.112407\pi$
$798$	−4.74456	−0.167956
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−14.7446	−0.520974
$802$	−36.9783	−1.30575
$803$	2.97825	0.105100
$804$	−8.00000	−0.282138
$805$	1.00000	0.0352454
$806$	−32.0000	−1.12715
$807$	26.7446	0.941453
$808$	−18.7446	−0.659431
$809$	−52.9783	−1.86262	−0.931308	−	0.364233i	$-0.881331\pi$
$809$	−52.9783	−1.86262	−0.931308	+	0.364233i	$0.881331\pi$
$810$	−1.00000	−0.0351364
$811$	−46.9783	−1.64963	−0.824815	−	0.565403i	$-0.808721\pi$
$811$	−46.9783	−1.64963	−0.824815	+	0.565403i	$0.808721\pi$
$812$	−4.74456	−0.166502
$813$	−12.7446	−0.446971
$814$	8.00000	0.280400
$815$	−10.7446	−0.376366
$816$	0	0
$817$	0	0
$818$	8.97825	0.313917
$819$	−6.74456	−0.235674
$820$	−8.74456	−0.305373
$821$	49.2119	1.71751	0.858754	−	0.512388i	$-0.171239\pi$
$821$	49.2119	1.71751	0.858754	+	0.512388i	$0.171239\pi$
$822$	11.4891	0.400729
$823$	−45.9565	−1.60194	−0.800971	−	0.598703i	$-0.795683\pi$
$823$	−45.9565	−1.60194	−0.800971	+	0.598703i	$0.795683\pi$
$824$	−4.00000	−0.139347
$825$	4.00000	0.139262
$826$	4.00000	0.139178
$827$	25.7228	0.894470	0.447235	−	0.894417i	$-0.352409\pi$
$827$	25.7228	0.894470	0.447235	+	0.894417i	$0.352409\pi$
$828$	−1.00000	−0.0347524
$829$	−45.7228	−1.58802	−0.794009	−	0.607905i	$-0.792010\pi$
$829$	−45.7228	−1.58802	−0.794009	+	0.607905i	$0.792010\pi$
$830$	−16.7446	−0.581212
$831$	−32.7446	−1.13590
$832$	6.74456	0.233826
$833$	0	0
$834$	−13.4891	−0.467090
$835$	−16.7446	−0.579469
$836$	−18.9783	−0.656377
$837$	−4.74456	−0.163996
$838$	−26.2337	−0.906228
$839$	18.5109	0.639066	0.319533	−	0.947575i	$-0.396474\pi$
$839$	18.5109	0.639066	0.319533	+	0.947575i	$0.396474\pi$
$840$	−1.00000	−0.0345033
$841$	−6.48913	−0.223763
$842$	−15.2554	−0.525737
$843$	3.48913	0.120172
$844$	9.48913	0.326629
$845$	32.4891	1.11766
$846$	12.7446	0.438167
$847$	−5.00000	−0.171802
$848$	12.7446	0.437650
$849$	7.25544	0.249006
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	8.00000	0.274075
$853$	−36.2337	−1.24062	−0.620309	−	0.784358i	$-0.712993\pi$
$853$	−36.2337	−1.24062	−0.620309	+	0.784358i	$0.712993\pi$
$854$	−11.4891	−0.393150
$855$	4.74456	0.162261
$856$	6.74456	0.230524
$857$	−52.9783	−1.80970	−0.904851	−	0.425728i	$-0.860018\pi$
$857$	−52.9783	−1.80970	−0.904851	+	0.425728i	$0.860018\pi$
$858$	−26.9783	−0.921022
$859$	48.4674	1.65369	0.826843	−	0.562433i	$-0.190135\pi$
$859$	48.4674	1.65369	0.826843	+	0.562433i	$0.190135\pi$
$860$	0	0
$861$	−8.74456	−0.298014
$862$	−32.0000	−1.08992
$863$	−30.5109	−1.03860	−0.519301	−	0.854591i	$-0.673808\pi$
$863$	−30.5109	−1.03860	−0.519301	+	0.854591i	$0.673808\pi$
$864$	1.00000	0.0340207
$865$	6.74456	0.229322
$866$	−4.23369	−0.143867
$867$	17.0000	0.577350
$868$	−4.74456	−0.161041
$869$	32.0000	1.08553
$870$	4.74456	0.160856
$871$	53.9565	1.82825
$872$	−12.7446	−0.431585
$873$	1.25544	0.0424901
$874$	4.74456	0.160487
$875$	−1.00000	−0.0338062
$876$	0.744563	0.0251564
$877$	4.74456	0.160212	0.0801062	−	0.996786i	$-0.474474\pi$
$877$	4.74456	0.160212	0.0801062	+	0.996786i	$0.474474\pi$
$878$	−19.2554	−0.649840
$879$	−14.0000	−0.472208
$880$	−4.00000	−0.134840
$881$	26.7446	0.901047	0.450524	−	0.892765i	$-0.351237\pi$
$881$	26.7446	0.901047	0.450524	+	0.892765i	$0.351237\pi$
$882$	−1.00000	−0.0336718
$883$	−42.7446	−1.43847	−0.719235	−	0.694767i	$-0.755507\pi$
$883$	−42.7446	−1.43847	−0.719235	+	0.694767i	$0.755507\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	28.0000	0.940678
$887$	−14.2337	−0.477920	−0.238960	−	0.971029i	$-0.576807\pi$
$887$	−14.2337	−0.477920	−0.238960	+	0.971029i	$0.576807\pi$
$888$	2.00000	0.0671156
$889$	−8.00000	−0.268311
$890$	14.7446	0.494239
$891$	−4.00000	−0.134005
$892$	19.4891	0.652544
$893$	−60.4674	−2.02346
$894$	7.48913	0.250474
$895$	10.7446	0.359151
$896$	1.00000	0.0334077
$897$	6.74456	0.225194
$898$	28.9783	0.967017
$899$	22.5109	0.750780
$900$	1.00000	0.0333333
$901$	0	0
$902$	−34.9783	−1.16465
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	−14.0000	−0.465376
$906$	−9.48913	−0.315255
$907$	33.4891	1.11199	0.555994	−	0.831186i	$-0.312338\pi$
$907$	33.4891	1.11199	0.555994	+	0.831186i	$0.312338\pi$
$908$	4.74456	0.157454
$909$	18.7446	0.621718
$910$	6.74456	0.223580
$911$	10.9783	0.363726	0.181863	−	0.983324i	$-0.441787\pi$
$911$	10.9783	0.363726	0.181863	+	0.983324i	$0.441787\pi$
$912$	−4.74456	−0.157108
$913$	−66.9783	−2.21666
$914$	−27.4891	−0.909259
$915$	11.4891	0.379819
$916$	2.00000	0.0660819
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−21.4891	−0.708861	−0.354430	−	0.935082i	$-0.615325\pi$
$919$	−21.4891	−0.708861	−0.354430	+	0.935082i	$0.615325\pi$
$920$	1.00000	0.0329690
$921$	18.9783	0.625355
$922$	−32.2337	−1.06156
$923$	−53.9565	−1.77600
$924$	−4.00000	−0.131590
$925$	2.00000	0.0657596
$926$	−36.4674	−1.19839
$927$	4.00000	0.131377
$928$	−4.74456	−0.155748
$929$	−3.25544	−0.106807	−0.0534037	−	0.998573i	$-0.517007\pi$
$929$	−3.25544	−0.106807	−0.0534037	+	0.998573i	$0.517007\pi$
$930$	4.74456	0.155580
$931$	4.74456	0.155497
$932$	−23.4891	−0.769412
$933$	−3.48913	−0.114229
$934$	−31.7228	−1.03800
$935$	0	0
$936$	−6.74456	−0.220453
$937$	−2.74456	−0.0896610	−0.0448305	−	0.998995i	$-0.514275\pi$
$937$	−2.74456	−0.0896610	−0.0448305	+	0.998995i	$0.514275\pi$
$938$	8.00000	0.261209
$939$	−5.25544	−0.171505
$940$	−12.7446	−0.415682
$941$	−15.4891	−0.504931	−0.252466	−	0.967606i	$-0.581241\pi$
$941$	−15.4891	−0.504931	−0.252466	+	0.967606i	$0.581241\pi$
$942$	−2.00000	−0.0651635
$943$	8.74456	0.284762
$944$	4.00000	0.130189
$945$	1.00000	0.0325300
$946$	0	0
$947$	9.48913	0.308355	0.154178	−	0.988043i	$-0.450727\pi$
$947$	9.48913	0.308355	0.154178	+	0.988043i	$0.450727\pi$
$948$	8.00000	0.259828
$949$	−5.02175	−0.163013
$950$	−4.74456	−0.153934
$951$	−16.9783	−0.550557
$952$	0	0
$953$	−27.4891	−0.890460	−0.445230	−	0.895416i	$-0.646878\pi$
$953$	−27.4891	−0.890460	−0.445230	+	0.895416i	$0.646878\pi$
$954$	−12.7446	−0.412620
$955$	−16.0000	−0.517748
$956$	8.00000	0.258738
$957$	18.9783	0.613480
$958$	6.51087	0.210357
$959$	−11.4891	−0.371003
$960$	−1.00000	−0.0322749
$961$	−8.48913	−0.273843
$962$	−13.4891	−0.434907
$963$	−6.74456	−0.217340
$964$	−13.4891	−0.434455
$965$	14.0000	0.450676
$966$	1.00000	0.0321745
$967$	−9.02175	−0.290120	−0.145060	−	0.989423i	$-0.546338\pi$
$967$	−9.02175	−0.290120	−0.145060	+	0.989423i	$0.546338\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−1.25544	−0.0403097
$971$	−21.7663	−0.698514	−0.349257	−	0.937027i	$-0.613566\pi$
$971$	−21.7663	−0.698514	−0.349257	+	0.937027i	$0.613566\pi$
$972$	−1.00000	−0.0320750
$973$	13.4891	0.432442
$974$	14.9783	0.479934
$975$	−6.74456	−0.215999
$976$	−11.4891	−0.367758
$977$	−7.48913	−0.239598	−0.119799	−	0.992798i	$-0.538225\pi$
$977$	−7.48913	−0.239598	−0.119799	+	0.992798i	$0.538225\pi$
$978$	−10.7446	−0.343573
$979$	58.9783	1.88495
$980$	1.00000	0.0319438
$981$	12.7446	0.406903
$982$	13.7228	0.437913
$983$	−31.2119	−0.995506	−0.497753	−	0.867319i	$-0.665841\pi$
$983$	−31.2119	−0.995506	−0.497753	+	0.867319i	$0.665841\pi$
$984$	−8.74456	−0.278766
$985$	19.4891	0.620975
$986$	0	0
$987$	−12.7446	−0.405664
$988$	32.0000	1.01806
$989$	0	0
$990$	4.00000	0.127128
$991$	20.4674	0.650168	0.325084	−	0.945685i	$-0.394607\pi$
$991$	20.4674	0.650168	0.325084	+	0.945685i	$0.394607\pi$
$992$	−4.74456	−0.150640
$993$	12.0000	0.380808
$994$	−8.00000	−0.253745
$995$	24.2337	0.768260
$996$	−16.7446	−0.530572
$997$	−9.72281	−0.307925	−0.153962	−	0.988077i	$-0.549203\pi$
$997$	−9.72281	−0.307925	−0.153962	+	0.988077i	$0.549203\pi$
$998$	−36.4674	−1.15435
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.bm.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.bm.1.2	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.74456 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.74456 q^{19} +1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -6.74456 q^{26} -1.00000 q^{27} -1.00000 q^{28} +4.74456 q^{29} +1.00000 q^{30} +4.74456 q^{31} -1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.74456 q^{38} -6.74456 q^{39} -1.00000 q^{40} -8.74456 q^{41} -1.00000 q^{42} -4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -12.7446 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.74456 q^{52} +12.7446 q^{53} +1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -4.74456 q^{57} -4.74456 q^{58} +4.00000 q^{59} -1.00000 q^{60} -11.4891 q^{61} -4.74456 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.74456 q^{65} -4.00000 q^{66} +8.00000 q^{67} +1.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} -0.744563 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.74456 q^{76} +4.00000 q^{77} +6.74456 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.74456 q^{82} +16.7446 q^{83} +1.00000 q^{84} -4.74456 q^{87} +4.00000 q^{88} -14.7446 q^{89} -1.00000 q^{90} -6.74456 q^{91} -1.00000 q^{92} -4.74456 q^{93} +12.7446 q^{94} +4.74456 q^{95} +1.00000 q^{96} +1.25544 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{19} + 2 q^{20} + 2 q^{21} + 8 q^{22} - 2 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{28} - 2 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} + 8 q^{33} - 2 q^{35} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 2 q^{40} - 6 q^{41} - 2 q^{42} - 8 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 2 q^{52} + 14 q^{53} + 2 q^{54} - 8 q^{55} + 2 q^{56} + 2 q^{57} + 2 q^{58} + 8 q^{59} - 2 q^{60} + 2 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{65} - 8 q^{66} + 16 q^{67} + 2 q^{69} + 2 q^{70} - 16 q^{71} - 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} + 8 q^{77} + 2 q^{78} - 16 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 22 q^{83} + 2 q^{84} + 2 q^{87} + 8 q^{88} - 18 q^{89} - 2 q^{90} - 2 q^{91} - 2 q^{92} + 2 q^{93} + 14 q^{94} - 2 q^{95} + 2 q^{96} + 14 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 8 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^18 - 2 * q^19 + 2 * q^20 + 2 * q^21 + 8 * q^22 - 2 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 - 2 * q^28 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 + 8 * q^33 - 2 * q^35 + 2 * q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 2 * q^40 - 6 * q^41 - 2 * q^42 - 8 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 2 * q^52 + 14 * q^53 + 2 * q^54 - 8 * q^55 + 2 * q^56 + 2 * q^57 + 2 * q^58 + 8 * q^59 - 2 * q^60 + 2 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^65 - 8 * q^66 + 16 * q^67 + 2 * q^69 + 2 * q^70 - 16 * q^71 - 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 + 8 * q^77 + 2 * q^78 - 16 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 22 * q^83 + 2 * q^84 + 2 * q^87 + 8 * q^88 - 18 * q^89 - 2 * q^90 - 2 * q^91 - 2 * q^92 + 2 * q^93 + 14 * q^94 - 2 * q^95 + 2 * q^96 + 14 * q^97 - 2 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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