LMFDB - Embedded newform 4598.2.a.by.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 $ x^8 - 16x^6 - 4x^5 + 75x^4 + 32x^3 - 90x^2 - 28x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.55131$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} +0.103249 q^{6} +4.45623 q^{7} -1.00000 q^{8} -2.98934 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} +0.103249 q^{6} +4.45623 q^{7} -1.00000 q^{8} -2.98934 q^{9} +3.91798 q^{10} -0.103249 q^{12} +4.23103 q^{13} -4.45623 q^{14} +0.404529 q^{15} +1.00000 q^{16} +6.60160 q^{17} +2.98934 q^{18} +1.00000 q^{19} -3.91798 q^{20} -0.460102 q^{21} -1.59524 q^{23} +0.103249 q^{24} +10.3506 q^{25} -4.23103 q^{26} +0.618395 q^{27} +4.45623 q^{28} +6.26090 q^{29} -0.404529 q^{30} -6.39604 q^{31} -1.00000 q^{32} -6.60160 q^{34} -17.4594 q^{35} -2.98934 q^{36} +8.97366 q^{37} -1.00000 q^{38} -0.436851 q^{39} +3.91798 q^{40} -1.91418 q^{41} +0.460102 q^{42} -9.24310 q^{43} +11.7122 q^{45} +1.59524 q^{46} -4.59409 q^{47} -0.103249 q^{48} +12.8580 q^{49} -10.3506 q^{50} -0.681611 q^{51} +4.23103 q^{52} -4.30161 q^{53} -0.618395 q^{54} -4.45623 q^{56} -0.103249 q^{57} -6.26090 q^{58} -10.9528 q^{59} +0.404529 q^{60} +0.325652 q^{61} +6.39604 q^{62} -13.3212 q^{63} +1.00000 q^{64} -16.5771 q^{65} +8.88645 q^{67} +6.60160 q^{68} +0.164708 q^{69} +17.4594 q^{70} -2.66009 q^{71} +2.98934 q^{72} +6.64661 q^{73} -8.97366 q^{74} -1.06869 q^{75} +1.00000 q^{76} +0.436851 q^{78} +2.13991 q^{79} -3.91798 q^{80} +8.90417 q^{81} +1.91418 q^{82} -12.4605 q^{83} -0.460102 q^{84} -25.8650 q^{85} +9.24310 q^{86} -0.646434 q^{87} +8.88476 q^{89} -11.7122 q^{90} +18.8544 q^{91} -1.59524 q^{92} +0.660387 q^{93} +4.59409 q^{94} -3.91798 q^{95} +0.103249 q^{96} -11.0716 q^{97} -12.8580 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

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
$3$	−0.103249	−0.0596110	−0.0298055	−	0.999556i	$-0.509489\pi$
$3$	−0.103249	−0.0596110	−0.0298055	+	0.999556i	$0.509489\pi$
$4$	1.00000	0.500000
$5$	−3.91798	−1.75217	−0.876087	−	0.482153i	$-0.839855\pi$
$5$	−3.91798	−1.75217	−0.876087	+	0.482153i	$0.839855\pi$
$6$	0.103249	0.0421513
$7$	4.45623	1.68430	0.842148	−	0.539247i	$-0.181291\pi$
$7$	4.45623	1.68430	0.842148	+	0.539247i	$0.181291\pi$
$8$	−1.00000	−0.353553
$9$	−2.98934	−0.996447
$10$	3.91798	1.23897
$11$	0	0
$11$	0	0
$12$	−0.103249	−0.0298055
$13$	4.23103	1.17348	0.586739	−	0.809776i	$-0.300412\pi$
$13$	4.23103	1.17348	0.586739	+	0.809776i	$0.300412\pi$
$14$	−4.45623	−1.19098
$15$	0.404529	0.104449
$16$	1.00000	0.250000
$17$	6.60160	1.60112	0.800562	−	0.599250i	$-0.204534\pi$
$17$	6.60160	1.60112	0.800562	+	0.599250i	$0.204534\pi$
$18$	2.98934	0.704594
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.91798	−0.876087
$21$	−0.460102	−0.100403
$22$	0	0
$23$	−1.59524	−0.332631	−0.166315	−	0.986073i	$-0.553187\pi$
$23$	−1.59524	−0.332631	−0.166315	+	0.986073i	$0.553187\pi$
$24$	0.103249	0.0210757
$25$	10.3506	2.07011
$26$	−4.23103	−0.829774
$27$	0.618395	0.119010
$28$	4.45623	0.842148
$29$	6.26090	1.16262	0.581310	−	0.813682i	$-0.302540\pi$
$29$	6.26090	1.16262	0.581310	+	0.813682i	$0.302540\pi$
$30$	−0.404529	−0.0738565
$31$	−6.39604	−1.14876	−0.574382	−	0.818588i	$-0.694757\pi$
$31$	−6.39604	−1.14876	−0.574382	+	0.818588i	$0.694757\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.60160	−1.13217
$35$	−17.4594	−2.95118
$36$	−2.98934	−0.498223
$37$	8.97366	1.47526	0.737630	−	0.675205i	$-0.235945\pi$
$37$	8.97366	1.47526	0.737630	+	0.675205i	$0.235945\pi$
$38$	−1.00000	−0.162221
$39$	−0.436851	−0.0699521
$40$	3.91798	0.619487
$41$	−1.91418	−0.298945	−0.149472	−	0.988766i	$-0.547758\pi$
$41$	−1.91418	−0.298945	−0.149472	+	0.988766i	$0.547758\pi$
$42$	0.460102	0.0709953
$43$	−9.24310	−1.40956	−0.704780	−	0.709426i	$-0.748954\pi$
$43$	−9.24310	−1.40956	−0.704780	+	0.709426i	$0.748954\pi$
$44$	0	0
$45$	11.7122	1.74595
$46$	1.59524	0.235206
$47$	−4.59409	−0.670117	−0.335058	−	0.942197i	$-0.608756\pi$
$47$	−4.59409	−0.670117	−0.335058	+	0.942197i	$0.608756\pi$
$48$	−0.103249	−0.0149027
$49$	12.8580	1.83685
$50$	−10.3506	−1.46379
$51$	−0.681611	−0.0954446
$52$	4.23103	0.586739
$53$	−4.30161	−0.590872	−0.295436	−	0.955363i	$-0.595465\pi$
$53$	−4.30161	−0.590872	−0.295436	+	0.955363i	$0.595465\pi$
$54$	−0.618395	−0.0841529
$55$	0	0
$56$	−4.45623	−0.595489
$57$	−0.103249	−0.0136757
$58$	−6.26090	−0.822097
$59$	−10.9528	−1.42593	−0.712967	−	0.701198i	$-0.752649\pi$
$59$	−10.9528	−1.42593	−0.712967	+	0.701198i	$0.752649\pi$
$60$	0.404529	0.0522244
$61$	0.325652	0.0416955	0.0208477	−	0.999783i	$-0.493363\pi$
$61$	0.325652	0.0416955	0.0208477	+	0.999783i	$0.493363\pi$
$62$	6.39604	0.812298
$63$	−13.3212	−1.67831
$64$	1.00000	0.125000
$65$	−16.5771	−2.05614
$66$	0	0
$67$	8.88645	1.08565	0.542826	−	0.839845i	$-0.317354\pi$
$67$	8.88645	1.08565	0.542826	+	0.839845i	$0.317354\pi$
$68$	6.60160	0.800562
$69$	0.164708	0.0198285
$70$	17.4594	2.08680
$71$	−2.66009	−0.315695	−0.157848	−	0.987463i	$-0.550455\pi$
$71$	−2.66009	−0.315695	−0.157848	+	0.987463i	$0.550455\pi$
$72$	2.98934	0.352297
$73$	6.64661	0.777927	0.388964	−	0.921253i	$-0.372833\pi$
$73$	6.64661	0.777927	0.388964	+	0.921253i	$0.372833\pi$
$74$	−8.97366	−1.04317
$75$	−1.06869	−0.123402
$76$	1.00000	0.114708
$77$	0	0
$78$	0.436851	0.0494636
$79$	2.13991	0.240758	0.120379	−	0.992728i	$-0.461589\pi$
$79$	2.13991	0.240758	0.120379	+	0.992728i	$0.461589\pi$
$80$	−3.91798	−0.438044
$81$	8.90417	0.989352
$82$	1.91418	0.211386
$83$	−12.4605	−1.36772	−0.683858	−	0.729615i	$-0.739699\pi$
$83$	−12.4605	−1.36772	−0.683858	+	0.729615i	$0.739699\pi$
$84$	−0.460102	−0.0502013
$85$	−25.8650	−2.80545
$86$	9.24310	0.996709
$87$	−0.646434	−0.0693050
$88$	0	0
$89$	8.88476	0.941783	0.470891	−	0.882191i	$-0.343932\pi$
$89$	8.88476	0.941783	0.470891	+	0.882191i	$0.343932\pi$
$90$	−11.7122	−1.23457
$91$	18.8544	1.97648
$92$	−1.59524	−0.166315
$93$	0.660387	0.0684789
$94$	4.59409	0.473844
$95$	−3.91798	−0.401976
$96$	0.103249	0.0105378
$97$	−11.0716	−1.12415	−0.562073	−	0.827088i	$-0.689996\pi$
$97$	−11.0716	−1.12415	−0.562073	+	0.827088i	$0.689996\pi$
$98$	−12.8580	−1.29885
$99$	0	0
$100$	10.3506	1.03506
$101$	14.2396	1.41689	0.708445	−	0.705766i	$-0.249397\pi$
$101$	14.2396	1.41689	0.708445	+	0.705766i	$0.249397\pi$
$102$	0.681611	0.0674895
$103$	−16.9912	−1.67419	−0.837096	−	0.547056i	$-0.815749\pi$
$103$	−16.9912	−1.67419	−0.837096	+	0.547056i	$0.815749\pi$
$104$	−4.23103	−0.414887
$105$	1.80267	0.175923
$106$	4.30161	0.417810
$107$	13.8759	1.34144	0.670718	−	0.741712i	$-0.265986\pi$
$107$	13.8759	1.34144	0.670718	+	0.741712i	$0.265986\pi$
$108$	0.618395	0.0595051
$109$	6.11130	0.585357	0.292678	−	0.956211i	$-0.405453\pi$
$109$	6.11130	0.585357	0.292678	+	0.956211i	$0.405453\pi$
$110$	0	0
$111$	−0.926523	−0.0879417
$112$	4.45623	0.421074
$113$	−8.61809	−0.810722	−0.405361	−	0.914157i	$-0.632854\pi$
$113$	−8.61809	−0.810722	−0.405361	+	0.914157i	$0.632854\pi$
$114$	0.103249	0.00967018
$115$	6.25013	0.582827
$116$	6.26090	0.581310
$117$	−12.6480	−1.16931
$118$	10.9528	1.00829
$119$	29.4182	2.69677
$120$	−0.404529	−0.0369282
$121$	0	0
$122$	−0.325652	−0.0294832
$123$	0.197638	0.0178204
$124$	−6.39604	−0.574382
$125$	−20.9634	−1.87503
$126$	13.3212	1.18674
$127$	6.81088	0.604367	0.302184	−	0.953250i	$-0.402284\pi$
$127$	6.81088	0.604367	0.302184	+	0.953250i	$0.402284\pi$
$128$	−1.00000	−0.0883883
$129$	0.954343	0.0840252
$130$	16.5771	1.45391
$131$	18.3631	1.60439	0.802194	−	0.597064i	$-0.203666\pi$
$131$	18.3631	1.60439	0.802194	+	0.597064i	$0.203666\pi$
$132$	0	0
$133$	4.45623	0.386404
$134$	−8.88645	−0.767672
$135$	−2.42286	−0.208527
$136$	−6.60160	−0.566083
$137$	12.4294	1.06192	0.530958	−	0.847398i	$-0.321832\pi$
$137$	12.4294	1.06192	0.530958	+	0.847398i	$0.321832\pi$
$138$	−0.164708	−0.0140208
$139$	0.572791	0.0485835	0.0242917	−	0.999705i	$-0.492267\pi$
$139$	0.572791	0.0485835	0.0242917	+	0.999705i	$0.492267\pi$
$140$	−17.4594	−1.47559
$141$	0.474336	0.0399463
$142$	2.66009	0.223230
$143$	0	0
$144$	−2.98934	−0.249112
$145$	−24.5301	−2.03711
$146$	−6.64661	−0.550078
$147$	−1.32758	−0.109497
$148$	8.97366	0.737630
$149$	−12.9477	−1.06072	−0.530360	−	0.847773i	$-0.677943\pi$
$149$	−12.9477	−1.06072	−0.530360	+	0.847773i	$0.677943\pi$
$150$	1.06869	0.0872581
$151$	13.1700	1.07176	0.535879	−	0.844294i	$-0.319980\pi$
$151$	13.1700	1.07176	0.535879	+	0.844294i	$0.319980\pi$
$152$	−1.00000	−0.0811107
$153$	−19.7344	−1.59543
$154$	0	0
$155$	25.0596	2.01283
$156$	−0.436851	−0.0349761
$157$	16.3314	1.30339	0.651694	−	0.758482i	$-0.274058\pi$
$157$	16.3314	1.30339	0.651694	+	0.758482i	$0.274058\pi$
$158$	−2.13991	−0.170242
$159$	0.444138	0.0352225
$160$	3.91798	0.309744
$161$	−7.10876	−0.560249
$162$	−8.90417	−0.699578
$163$	3.96608	0.310648	0.155324	−	0.987864i	$-0.450358\pi$
$163$	3.96608	0.310648	0.155324	+	0.987864i	$0.450358\pi$
$164$	−1.91418	−0.149472
$165$	0	0
$166$	12.4605	0.967121
$167$	−13.6008	−1.05247	−0.526233	−	0.850341i	$-0.676396\pi$
$167$	−13.6008	−1.05247	−0.526233	+	0.850341i	$0.676396\pi$
$168$	0.460102	0.0354977
$169$	4.90163	0.377049
$170$	25.8650	1.98375
$171$	−2.98934	−0.228601
$172$	−9.24310	−0.704780
$173$	14.4150	1.09595	0.547976	−	0.836494i	$-0.315398\pi$
$173$	14.4150	1.09595	0.547976	+	0.836494i	$0.315398\pi$
$174$	0.646434	0.0490060
$175$	46.1245	3.48669
$176$	0	0
$177$	1.13087	0.0850013
$178$	−8.88476	−0.665941
$179$	−5.80484	−0.433874	−0.216937	−	0.976186i	$-0.569607\pi$
$179$	−5.80484	−0.433874	−0.216937	+	0.976186i	$0.569607\pi$
$180$	11.7122	0.872974
$181$	−0.148577	−0.0110436	−0.00552182	−	0.999985i	$-0.501758\pi$
$181$	−0.148577	−0.0110436	−0.00552182	+	0.999985i	$0.501758\pi$
$182$	−18.8544	−1.39758
$183$	−0.0336234	−0.00248551
$184$	1.59524	0.117603
$185$	−35.1586	−2.58491
$186$	−0.660387	−0.0484219
$187$	0	0
$188$	−4.59409	−0.335058
$189$	2.75571	0.200448
$190$	3.91798	0.284240
$191$	−6.74447	−0.488013	−0.244006	−	0.969774i	$-0.578462\pi$
$191$	−6.74447	−0.488013	−0.244006	+	0.969774i	$0.578462\pi$
$192$	−0.103249	−0.00745137
$193$	−8.15211	−0.586802	−0.293401	−	0.955989i	$-0.594787\pi$
$193$	−8.15211	−0.586802	−0.293401	+	0.955989i	$0.594787\pi$
$194$	11.0716	0.794891
$195$	1.71157	0.122568
$196$	12.8580	0.918426
$197$	−3.77994	−0.269309	−0.134655	−	0.990893i	$-0.542992\pi$
$197$	−3.77994	−0.269309	−0.134655	+	0.990893i	$0.542992\pi$
$198$	0	0
$199$	−0.977228	−0.0692739	−0.0346369	−	0.999400i	$-0.511027\pi$
$199$	−0.977228	−0.0692739	−0.0346369	+	0.999400i	$0.511027\pi$
$200$	−10.3506	−0.731896
$201$	−0.917519	−0.0647168
$202$	−14.2396	−1.00189
$203$	27.9000	1.95820
$204$	−0.681611	−0.0477223
$205$	7.49973	0.523804
$206$	16.9912	1.18383
$207$	4.76872	0.331449
$208$	4.23103	0.293369
$209$	0	0
$210$	−1.80267	−0.124396
$211$	11.5402	0.794463	0.397231	−	0.917718i	$-0.369971\pi$
$211$	11.5402	0.794463	0.397231	+	0.917718i	$0.369971\pi$
$212$	−4.30161	−0.295436
$213$	0.274653	0.0188189
$214$	−13.8759	−0.948539
$215$	36.2143	2.46979
$216$	−0.618395	−0.0420764
$217$	−28.5022	−1.93486
$218$	−6.11130	−0.413910
$219$	−0.686258	−0.0463730
$220$	0	0
$221$	27.9316	1.87888
$222$	0.926523	0.0621842
$223$	−16.0725	−1.07629	−0.538146	−	0.842852i	$-0.680875\pi$
$223$	−16.0725	−1.07629	−0.538146	+	0.842852i	$0.680875\pi$
$224$	−4.45623	−0.297744
$225$	−30.9414	−2.06276
$226$	8.61809	0.573267
$227$	−3.14962	−0.209047	−0.104524	−	0.994522i	$-0.533332\pi$
$227$	−3.14962	−0.209047	−0.104524	+	0.994522i	$0.533332\pi$
$228$	−0.103249	−0.00683785
$229$	22.0432	1.45665	0.728327	−	0.685229i	$-0.240298\pi$
$229$	22.0432	1.45665	0.728327	+	0.685229i	$0.240298\pi$
$230$	−6.25013	−0.412121
$231$	0	0
$232$	−6.26090	−0.411048
$233$	−10.0334	−0.657308	−0.328654	−	0.944450i	$-0.606595\pi$
$233$	−10.0334	−0.657308	−0.328654	+	0.944450i	$0.606595\pi$
$234$	12.6480	0.826825
$235$	17.9996	1.17416
$236$	−10.9528	−0.712967
$237$	−0.220944	−0.0143518
$238$	−29.4182	−1.90690
$239$	−11.4321	−0.739484	−0.369742	−	0.929134i	$-0.620554\pi$
$239$	−11.4321	−0.739484	−0.369742	+	0.929134i	$0.620554\pi$
$240$	0.404529	0.0261122
$241$	14.0982	0.908147	0.454073	−	0.890964i	$-0.349970\pi$
$241$	14.0982	0.908147	0.454073	+	0.890964i	$0.349970\pi$
$242$	0	0
$243$	−2.77453	−0.177986
$244$	0.325652	0.0208477
$245$	−50.3773	−3.21849
$246$	−0.197638	−0.0126009
$247$	4.23103	0.269214
$248$	6.39604	0.406149
$249$	1.28654	0.0815309
$250$	20.9634	1.32584
$251$	18.6997	1.18032	0.590158	−	0.807288i	$-0.299065\pi$
$251$	18.6997	1.18032	0.590158	+	0.807288i	$0.299065\pi$
$252$	−13.3212	−0.839155
$253$	0	0
$254$	−6.81088	−0.427352
$255$	2.67054	0.167236
$256$	1.00000	0.0625000
$257$	15.4448	0.963419	0.481709	−	0.876331i	$-0.340016\pi$
$257$	15.4448	0.963419	0.481709	+	0.876331i	$0.340016\pi$
$258$	−0.954343	−0.0594148
$259$	39.9887	2.48477
$260$	−16.5771	−1.02807
$261$	−18.7160	−1.15849
$262$	−18.3631	−1.13447
$263$	17.4417	1.07550	0.537751	−	0.843104i	$-0.319274\pi$
$263$	17.4417	1.07550	0.537751	+	0.843104i	$0.319274\pi$
$264$	0	0
$265$	16.8536	1.03531
$266$	−4.45623	−0.273229
$267$	−0.917345	−0.0561406
$268$	8.88645	0.542826
$269$	21.7774	1.32779	0.663894	−	0.747826i	$-0.268902\pi$
$269$	21.7774	1.32779	0.663894	+	0.747826i	$0.268902\pi$
$270$	2.42286	0.147451
$271$	−27.5183	−1.67162	−0.835808	−	0.549022i	$-0.815000\pi$
$271$	−27.5183	−1.67162	−0.835808	+	0.549022i	$0.815000\pi$
$272$	6.60160	0.400281
$273$	−1.94671	−0.117820
$274$	−12.4294	−0.750887
$275$	0	0
$276$	0.164708	0.00991423
$277$	5.10903	0.306972	0.153486	−	0.988151i	$-0.450950\pi$
$277$	5.10903	0.306972	0.153486	+	0.988151i	$0.450950\pi$
$278$	−0.572791	−0.0343537
$279$	19.1199	1.14468
$280$	17.4594	1.04340
$281$	27.4547	1.63781	0.818905	−	0.573929i	$-0.194581\pi$
$281$	27.4547	1.63781	0.818905	+	0.573929i	$0.194581\pi$
$282$	−0.474336	−0.0282463
$283$	6.79437	0.403883	0.201942	−	0.979398i	$-0.435275\pi$
$283$	6.79437	0.403883	0.201942	+	0.979398i	$0.435275\pi$
$284$	−2.66009	−0.157848
$285$	0.404529	0.0239622
$286$	0	0
$287$	−8.53003	−0.503512
$288$	2.98934	0.176149
$289$	26.5812	1.56360
$290$	24.5301	1.44046
$291$	1.14313	0.0670115
$292$	6.64661	0.388964
$293$	28.6595	1.67430	0.837152	−	0.546970i	$-0.184219\pi$
$293$	28.6595	1.67430	0.837152	+	0.546970i	$0.184219\pi$
$294$	1.32758	0.0774258
$295$	42.9129	2.49848
$296$	−8.97366	−0.521583
$297$	0	0
$298$	12.9477	0.750042
$299$	−6.74952	−0.390335
$300$	−1.06869	−0.0617008
$301$	−41.1894	−2.37412
$302$	−13.1700	−0.757848
$303$	−1.47022	−0.0844622
$304$	1.00000	0.0573539
$305$	−1.27590	−0.0730578
$306$	19.7344	1.12814
$307$	14.1122	0.805427	0.402713	−	0.915326i	$-0.368067\pi$
$307$	14.1122	0.805427	0.402713	+	0.915326i	$0.368067\pi$
$308$	0	0
$309$	1.75433	0.0998003
$310$	−25.0596	−1.42329
$311$	11.2126	0.635809	0.317905	−	0.948123i	$-0.397021\pi$
$311$	11.2126	0.635809	0.317905	+	0.948123i	$0.397021\pi$
$312$	0.436851	0.0247318
$313$	−11.6375	−0.657792	−0.328896	−	0.944366i	$-0.606677\pi$
$313$	−11.6375	−0.657792	−0.328896	+	0.944366i	$0.606677\pi$
$314$	−16.3314	−0.921635
$315$	52.1921	2.94069
$316$	2.13991	0.120379
$317$	−3.69201	−0.207364	−0.103682	−	0.994611i	$-0.533062\pi$
$317$	−3.69201	−0.207364	−0.103682	+	0.994611i	$0.533062\pi$
$318$	−0.444138	−0.0249060
$319$	0	0
$320$	−3.91798	−0.219022
$321$	−1.43268	−0.0799644
$322$	7.10876	0.396156
$323$	6.60160	0.367323
$324$	8.90417	0.494676
$325$	43.7936	2.42923
$326$	−3.96608	−0.219661
$327$	−0.630987	−0.0348937
$328$	1.91418	0.105693
$329$	−20.4723	−1.12868
$330$	0	0
$331$	13.9429	0.766373	0.383187	−	0.923671i	$-0.374827\pi$
$331$	13.9429	0.766373	0.383187	+	0.923671i	$0.374827\pi$
$332$	−12.4605	−0.683858
$333$	−26.8253	−1.47002
$334$	13.6008	0.744205
$335$	−34.8169	−1.90225
$336$	−0.460102	−0.0251006
$337$	18.9545	1.03252	0.516258	−	0.856433i	$-0.327324\pi$
$337$	18.9545	1.03252	0.516258	+	0.856433i	$0.327324\pi$
$338$	−4.90163	−0.266614
$339$	0.889812	0.0483279
$340$	−25.8650	−1.40272
$341$	0	0
$342$	2.98934	0.161645
$343$	26.1044	1.40951
$344$	9.24310	0.498354
$345$	−0.645321	−0.0347429
$346$	−14.4150	−0.774956
$347$	−36.6760	−1.96887	−0.984436	−	0.175745i	$-0.943767\pi$
$347$	−36.6760	−1.96887	−0.984436	+	0.175745i	$0.943767\pi$
$348$	−0.646434	−0.0346525
$349$	−6.99845	−0.374618	−0.187309	−	0.982301i	$-0.559977\pi$
$349$	−6.99845	−0.374618	−0.187309	+	0.982301i	$0.559977\pi$
$350$	−46.1245	−2.46546
$351$	2.61645	0.139656
$352$	0	0
$353$	−11.2690	−0.599786	−0.299893	−	0.953973i	$-0.596951\pi$
$353$	−11.2690	−0.599786	−0.299893	+	0.953973i	$0.596951\pi$
$354$	−1.13087	−0.0601050
$355$	10.4222	0.553153
$356$	8.88476	0.470891
$357$	−3.03741	−0.160757
$358$	5.80484	0.306795
$359$	5.70457	0.301076	0.150538	−	0.988604i	$-0.451899\pi$
$359$	5.70457	0.301076	0.150538	+	0.988604i	$0.451899\pi$
$360$	−11.7122	−0.617286
$361$	1.00000	0.0526316
$362$	0.148577	0.00780903
$363$	0	0
$364$	18.8544	0.988241
$365$	−26.0413	−1.36306
$366$	0.0336234	0.00175752
$367$	−18.7306	−0.977732	−0.488866	−	0.872359i	$-0.662589\pi$
$367$	−18.7306	−0.977732	−0.488866	+	0.872359i	$0.662589\pi$
$368$	−1.59524	−0.0831577
$369$	5.72214	0.297883
$370$	35.1586	1.82781
$371$	−19.1690	−0.995204
$372$	0.660387	0.0342395
$373$	11.8435	0.613233	0.306616	−	0.951833i	$-0.400803\pi$
$373$	11.8435	0.613233	0.306616	+	0.951833i	$0.400803\pi$
$374$	0	0
$375$	2.16446	0.111772
$376$	4.59409	0.236922
$377$	26.4901	1.36431
$378$	−2.75571	−0.141738
$379$	14.4703	0.743288	0.371644	−	0.928375i	$-0.378794\pi$
$379$	14.4703	0.743288	0.371644	+	0.928375i	$0.378794\pi$
$380$	−3.91798	−0.200988
$381$	−0.703218	−0.0360269
$382$	6.74447	0.345077
$383$	−23.9927	−1.22597	−0.612984	−	0.790095i	$-0.710031\pi$
$383$	−23.9927	−1.22597	−0.612984	+	0.790095i	$0.710031\pi$
$384$	0.103249	0.00526892
$385$	0	0
$386$	8.15211	0.414932
$387$	27.6308	1.40455
$388$	−11.0716	−0.562073
$389$	−27.2728	−1.38278	−0.691392	−	0.722480i	$-0.743002\pi$
$389$	−27.2728	−1.38278	−0.691392	+	0.722480i	$0.743002\pi$
$390$	−1.71157	−0.0866689
$391$	−10.5312	−0.532583
$392$	−12.8580	−0.649425
$393$	−1.89597	−0.0956391
$394$	3.77994	0.190430
$395$	−8.38411	−0.421850
$396$	0	0
$397$	20.4056	1.02413	0.512063	−	0.858948i	$-0.328881\pi$
$397$	20.4056	1.02413	0.512063	+	0.858948i	$0.328881\pi$
$398$	0.977228	0.0489840
$399$	−0.460102	−0.0230339
$400$	10.3506	0.517529
$401$	−13.8056	−0.689417	−0.344708	−	0.938710i	$-0.612022\pi$
$401$	−13.8056	−0.689417	−0.344708	+	0.938710i	$0.612022\pi$
$402$	0.917519	0.0457617
$403$	−27.0619	−1.34805
$404$	14.2396	0.708445
$405$	−34.8864	−1.73352
$406$	−27.9000	−1.38465
$407$	0	0
$408$	0.681611	0.0337448
$409$	21.2103	1.04878	0.524392	−	0.851477i	$-0.324293\pi$
$409$	21.2103	1.04878	0.524392	+	0.851477i	$0.324293\pi$
$410$	−7.49973	−0.370385
$411$	−1.28333	−0.0633018
$412$	−16.9912	−0.837096
$413$	−48.8082	−2.40169
$414$	−4.76872	−0.234370
$415$	48.8199	2.39648
$416$	−4.23103	−0.207443
$417$	−0.0591402	−0.00289611
$418$	0	0
$419$	−12.1697	−0.594527	−0.297264	−	0.954795i	$-0.596074\pi$
$419$	−12.1697	−0.594527	−0.297264	+	0.954795i	$0.596074\pi$
$420$	1.80267	0.0879614
$421$	−17.3500	−0.845585	−0.422793	−	0.906226i	$-0.638950\pi$
$421$	−17.3500	−0.845585	−0.422793	+	0.906226i	$0.638950\pi$
$422$	−11.5402	−0.561770
$423$	13.7333	0.667736
$424$	4.30161	0.208905
$425$	68.3304	3.31451
$426$	−0.274653	−0.0133070
$427$	1.45118	0.0702276
$428$	13.8759	0.670718
$429$	0	0
$430$	−36.2143	−1.74641
$431$	−16.6638	−0.802668	−0.401334	−	0.915932i	$-0.631453\pi$
$431$	−16.6638	−0.802668	−0.401334	+	0.915932i	$0.631453\pi$
$432$	0.618395	0.0297525
$433$	34.2652	1.64668	0.823341	−	0.567547i	$-0.192108\pi$
$433$	34.2652	1.64668	0.823341	+	0.567547i	$0.192108\pi$
$434$	28.5022	1.36815
$435$	2.53271	0.121434
$436$	6.11130	0.292678
$437$	−1.59524	−0.0763108
$438$	0.686258	0.0327907
$439$	−23.9778	−1.14440	−0.572198	−	0.820115i	$-0.693909\pi$
$439$	−23.9778	−1.14440	−0.572198	+	0.820115i	$0.693909\pi$
$440$	0	0
$441$	−38.4368	−1.83033
$442$	−27.9316	−1.32857
$443$	9.23108	0.438582	0.219291	−	0.975660i	$-0.429626\pi$
$443$	9.23108	0.438582	0.219291	+	0.975660i	$0.429626\pi$
$444$	−0.926523	−0.0439708
$445$	−34.8103	−1.65017
$446$	16.0725	0.761053
$447$	1.33684	0.0632305
$448$	4.45623	0.210537
$449$	4.61870	0.217970	0.108985	−	0.994043i	$-0.465240\pi$
$449$	4.61870	0.217970	0.108985	+	0.994043i	$0.465240\pi$
$450$	30.9414	1.45859
$451$	0	0
$452$	−8.61809	−0.405361
$453$	−1.35979	−0.0638886
$454$	3.14962	0.147819
$455$	−73.8713	−3.46314
$456$	0.103249	0.00483509
$457$	6.40059	0.299407	0.149703	−	0.988731i	$-0.452168\pi$
$457$	6.40059	0.299407	0.149703	+	0.988731i	$0.452168\pi$
$458$	−22.0432	−1.03001
$459$	4.08240	0.190550
$460$	6.25013	0.291414
$461$	27.9210	1.30041	0.650206	−	0.759758i	$-0.274682\pi$
$461$	27.9210	1.30041	0.650206	+	0.759758i	$0.274682\pi$
$462$	0	0
$463$	−8.30846	−0.386127	−0.193063	−	0.981186i	$-0.561842\pi$
$463$	−8.30846	−0.386127	−0.193063	+	0.981186i	$0.561842\pi$
$464$	6.26090	0.290655
$465$	−2.58738	−0.119987
$466$	10.0334	0.464787
$467$	14.7637	0.683182	0.341591	−	0.939849i	$-0.389034\pi$
$467$	14.7637	0.683182	0.341591	+	0.939849i	$0.389034\pi$
$468$	−12.6480	−0.584654
$469$	39.6000	1.82856
$470$	−17.9996	−0.830258
$471$	−1.68621	−0.0776963
$472$	10.9528	0.504144
$473$	0	0
$474$	0.220944	0.0101483
$475$	10.3506	0.474917
$476$	29.4182	1.34838
$477$	12.8590	0.588773
$478$	11.4321	0.522894
$479$	36.4821	1.66691	0.833456	−	0.552587i	$-0.186359\pi$
$479$	36.4821	1.66691	0.833456	+	0.552587i	$0.186359\pi$
$480$	−0.404529	−0.0184641
$481$	37.9678	1.73118
$482$	−14.0982	−0.642157
$483$	0.733974	0.0333970
$484$	0	0
$485$	43.3781	1.96970
$486$	2.77453	0.125855
$487$	24.8266	1.12500	0.562500	−	0.826797i	$-0.309840\pi$
$487$	24.8266	1.12500	0.562500	+	0.826797i	$0.309840\pi$
$488$	−0.325652	−0.0147416
$489$	−0.409495	−0.0185180
$490$	50.3773	2.27581
$491$	15.2207	0.686902	0.343451	−	0.939171i	$-0.388404\pi$
$491$	15.2207	0.686902	0.343451	+	0.939171i	$0.388404\pi$
$492$	0.197638	0.00891020
$493$	41.3320	1.86150
$494$	−4.23103	−0.190363
$495$	0	0
$496$	−6.39604	−0.287191
$497$	−11.8540	−0.531724
$498$	−1.28654	−0.0576510
$499$	−39.0356	−1.74747	−0.873737	−	0.486399i	$-0.838310\pi$
$499$	−39.0356	−1.74747	−0.873737	+	0.486399i	$0.838310\pi$
$500$	−20.9634	−0.937514
$501$	1.40428	0.0627385
$502$	−18.6997	−0.834610
$503$	23.9042	1.06584	0.532918	−	0.846167i	$-0.321095\pi$
$503$	23.9042	1.06584	0.532918	+	0.846167i	$0.321095\pi$
$504$	13.3212	0.593372
$505$	−55.7903	−2.48264
$506$	0	0
$507$	−0.506090	−0.0224762
$508$	6.81088	0.302184
$509$	39.8560	1.76659	0.883294	−	0.468819i	$-0.155320\pi$
$509$	39.8560	1.76659	0.883294	+	0.468819i	$0.155320\pi$
$510$	−2.67054	−0.118253
$511$	29.6188	1.31026
$512$	−1.00000	−0.0441942
$513$	0.618395	0.0273028
$514$	−15.4448	−0.681240
$515$	66.5712	2.93348
$516$	0.954343	0.0420126
$517$	0	0
$518$	−39.9887	−1.75700
$519$	−1.48834	−0.0653308
$520$	16.5771	0.726954
$521$	−34.7437	−1.52215	−0.761075	−	0.648664i	$-0.775328\pi$
$521$	−34.7437	−1.52215	−0.761075	+	0.648664i	$0.775328\pi$
$522$	18.7160	0.819176
$523$	−3.65461	−0.159805	−0.0799024	−	0.996803i	$-0.525461\pi$
$523$	−3.65461	−0.159805	−0.0799024	+	0.996803i	$0.525461\pi$
$524$	18.3631	0.802194
$525$	−4.76232	−0.207845
$526$	−17.4417	−0.760495
$527$	−42.2241	−1.83931
$528$	0	0
$529$	−20.4552	−0.889357
$530$	−16.8536	−0.732075
$531$	32.7416	1.42087
$532$	4.45623	0.193202
$533$	−8.09896	−0.350805
$534$	0.917345	0.0396974
$535$	−54.3656	−2.35043
$536$	−8.88645	−0.383836
$537$	0.599345	0.0258637
$538$	−21.7774	−0.938888
$539$	0	0
$540$	−2.42286	−0.104263
$541$	23.4833	1.00963	0.504814	−	0.863228i	$-0.331561\pi$
$541$	23.4833	1.00963	0.504814	+	0.863228i	$0.331561\pi$
$542$	27.5183	1.18201
$543$	0.0153405	0.000658322 0
$544$	−6.60160	−0.283041
$545$	−23.9440	−1.02565
$546$	1.94671	0.0833114
$547$	4.29735	0.183742	0.0918708	−	0.995771i	$-0.470715\pi$
$547$	4.29735	0.183742	0.0918708	+	0.995771i	$0.470715\pi$
$548$	12.4294	0.530958
$549$	−0.973485	−0.0415473
$550$	0	0
$551$	6.26090	0.266723
$552$	−0.164708	−0.00701042
$553$	9.53591	0.405508
$554$	−5.10903	−0.217062
$555$	3.63010	0.154089
$556$	0.572791	0.0242917
$557$	14.9520	0.633535	0.316767	−	0.948503i	$-0.397403\pi$
$557$	14.9520	0.633535	0.316767	+	0.948503i	$0.397403\pi$
$558$	−19.1199	−0.809412
$559$	−39.1078	−1.65409
$560$	−17.4594	−0.737795
$561$	0	0
$562$	−27.4547	−1.15811
$563$	−36.2456	−1.52757	−0.763786	−	0.645470i	$-0.776661\pi$
$563$	−36.2456	−1.52757	−0.763786	+	0.645470i	$0.776661\pi$
$564$	0.474336	0.0199732
$565$	33.7655	1.42053
$566$	−6.79437	−0.285589
$567$	39.6790	1.66636
$568$	2.66009	0.111615
$569$	−1.48123	−0.0620964	−0.0310482	−	0.999518i	$-0.509885\pi$
$569$	−1.48123	−0.0620964	−0.0310482	+	0.999518i	$0.509885\pi$
$570$	−0.404529	−0.0169438
$571$	36.9110	1.54468	0.772338	−	0.635211i	$-0.219087\pi$
$571$	36.9110	1.54468	0.772338	+	0.635211i	$0.219087\pi$
$572$	0	0
$573$	0.696362	0.0290909
$574$	8.53003	0.356037
$575$	−16.5117	−0.688584
$576$	−2.98934	−0.124556
$577$	37.8575	1.57603	0.788014	−	0.615657i	$-0.211109\pi$
$577$	37.8575	1.57603	0.788014	+	0.615657i	$0.211109\pi$
$578$	−26.5812	−1.10563
$579$	0.841700	0.0349798
$580$	−24.5301	−1.01856
$581$	−55.5267	−2.30364
$582$	−1.14313	−0.0473843
$583$	0	0
$584$	−6.64661	−0.275039
$585$	49.5546	2.04883
$586$	−28.6595	−1.18391
$587$	−2.40075	−0.0990896	−0.0495448	−	0.998772i	$-0.515777\pi$
$587$	−2.40075	−0.0990896	−0.0495448	+	0.998772i	$0.515777\pi$
$588$	−1.32758	−0.0547483
$589$	−6.39604	−0.263544
$590$	−42.9129	−1.76670
$591$	0.390276	0.0160538
$592$	8.97366	0.368815
$593$	6.20332	0.254740	0.127370	−	0.991855i	$-0.459346\pi$
$593$	6.20332	0.254740	0.127370	+	0.991855i	$0.459346\pi$
$594$	0	0
$595$	−115.260	−4.72520
$596$	−12.9477	−0.530360
$597$	0.100898	0.00412948
$598$	6.74952	0.276008
$599$	18.8744	0.771187	0.385593	−	0.922669i	$-0.373997\pi$
$599$	18.8744	0.771187	0.385593	+	0.922669i	$0.373997\pi$
$600$	1.06869	0.0436290
$601$	−40.5992	−1.65608	−0.828039	−	0.560671i	$-0.810543\pi$
$601$	−40.5992	−1.65608	−0.828039	+	0.560671i	$0.810543\pi$
$602$	41.1894	1.67875
$603$	−26.5646	−1.08179
$604$	13.1700	0.535879
$605$	0	0
$606$	1.47022	0.0597238
$607$	−2.96039	−0.120158	−0.0600792	−	0.998194i	$-0.519135\pi$
$607$	−2.96039	−0.120158	−0.0600792	+	0.998194i	$0.519135\pi$
$608$	−1.00000	−0.0405554
$609$	−2.88066	−0.116730
$610$	1.27590	0.0516596
$611$	−19.4377	−0.786367
$612$	−19.7344	−0.797717
$613$	0.763124	0.0308223	0.0154112	−	0.999881i	$-0.495094\pi$
$613$	0.763124	0.0308223	0.0154112	+	0.999881i	$0.495094\pi$
$614$	−14.1122	−0.569523
$615$	−0.774341	−0.0312245
$616$	0	0
$617$	1.06644	0.0429334	0.0214667	−	0.999770i	$-0.493166\pi$
$617$	1.06644	0.0429334	0.0214667	+	0.999770i	$0.493166\pi$
$618$	−1.75433	−0.0705694
$619$	12.5798	0.505626	0.252813	−	0.967515i	$-0.418644\pi$
$619$	12.5798	0.505626	0.252813	+	0.967515i	$0.418644\pi$
$620$	25.0596	1.00642
$621$	−0.986489	−0.0395865
$622$	−11.2126	−0.449585
$623$	39.5925	1.58624
$624$	−0.436851	−0.0174880
$625$	30.3815	1.21526
$626$	11.6375	0.465129
$627$	0	0
$628$	16.3314	0.651694
$629$	59.2405	2.36207
$630$	−52.1921	−2.07938
$631$	−22.9529	−0.913741	−0.456871	−	0.889533i	$-0.651030\pi$
$631$	−22.9529	−0.913741	−0.456871	+	0.889533i	$0.651030\pi$
$632$	−2.13991	−0.0851209
$633$	−1.19152	−0.0473587
$634$	3.69201	0.146628
$635$	−26.6849	−1.05896
$636$	0.444138	0.0176112
$637$	54.4025	2.15550
$638$	0	0
$639$	7.95192	0.314573
$640$	3.91798	0.154872
$641$	−2.91555	−0.115157	−0.0575787	−	0.998341i	$-0.518338\pi$
$641$	−2.91555	−0.115157	−0.0575787	+	0.998341i	$0.518338\pi$
$642$	1.43268	0.0565434
$643$	18.3286	0.722811	0.361406	−	0.932409i	$-0.382297\pi$
$643$	18.3286	0.722811	0.361406	+	0.932409i	$0.382297\pi$
$644$	−7.10876	−0.280124
$645$	−3.73910	−0.147227
$646$	−6.60160	−0.259737
$647$	5.12335	0.201420	0.100710	−	0.994916i	$-0.467889\pi$
$647$	5.12335	0.201420	0.100710	+	0.994916i	$0.467889\pi$
$648$	−8.90417	−0.349789
$649$	0	0
$650$	−43.7936	−1.71773
$651$	2.94283	0.115339
$652$	3.96608	0.155324
$653$	47.7078	1.86695	0.933475	−	0.358643i	$-0.116761\pi$
$653$	47.7078	1.86695	0.933475	+	0.358643i	$0.116761\pi$
$654$	0.630987	0.0246736
$655$	−71.9461	−2.81117
$656$	−1.91418	−0.0747362
$657$	−19.8690	−0.775163
$658$	20.4723	0.798094
$659$	−18.6754	−0.727491	−0.363746	−	0.931498i	$-0.618502\pi$
$659$	−18.6754	−0.727491	−0.363746	+	0.931498i	$0.618502\pi$
$660$	0	0
$661$	−37.8786	−1.47331	−0.736654	−	0.676270i	$-0.763595\pi$
$661$	−37.8786	−1.47331	−0.736654	+	0.676270i	$0.763595\pi$
$662$	−13.9429	−0.541908
$663$	−2.88392	−0.112002
$664$	12.4605	0.483560
$665$	−17.4594	−0.677047
$666$	26.8253	1.03946
$667$	−9.98766	−0.386724
$668$	−13.6008	−0.526233
$669$	1.65947	0.0641588
$670$	34.8169	1.34510
$671$	0	0
$672$	0.460102	0.0177488
$673$	−19.1798	−0.739328	−0.369664	−	0.929166i	$-0.620527\pi$
$673$	−19.1798	−0.739328	−0.369664	+	0.929166i	$0.620527\pi$
$674$	−18.9545	−0.730100
$675$	6.40074	0.246365
$676$	4.90163	0.188524
$677$	−0.144722	−0.00556210	−0.00278105	−	0.999996i	$-0.500885\pi$
$677$	−0.144722	−0.00556210	−0.00278105	+	0.999996i	$0.500885\pi$
$678$	−0.889812	−0.0341730
$679$	−49.3374	−1.89339
$680$	25.8650	0.991876
$681$	0.325196	0.0124615
$682$	0	0
$683$	−8.12408	−0.310859	−0.155430	−	0.987847i	$-0.549676\pi$
$683$	−8.12408	−0.310859	−0.155430	+	0.987847i	$0.549676\pi$
$684$	−2.98934	−0.114300
$685$	−48.6981	−1.86066
$686$	−26.1044	−0.996672
$687$	−2.27594	−0.0868326
$688$	−9.24310	−0.352390
$689$	−18.2003	−0.693375
$690$	0.645321	0.0245669
$691$	−38.0620	−1.44795	−0.723973	−	0.689828i	$-0.757686\pi$
$691$	−38.0620	−1.44795	−0.723973	+	0.689828i	$0.757686\pi$
$692$	14.4150	0.547976
$693$	0	0
$694$	36.6760	1.39220
$695$	−2.24418	−0.0851267
$696$	0.646434	0.0245030
$697$	−12.6367	−0.478648
$698$	6.99845	0.264895
$699$	1.03594	0.0391828
$700$	46.1245	1.74334
$701$	17.8921	0.675774	0.337887	−	0.941187i	$-0.390288\pi$
$701$	17.8921	0.675774	0.337887	+	0.941187i	$0.390288\pi$
$702$	−2.61645	−0.0987515
$703$	8.97366	0.338448
$704$	0	0
$705$	−1.85844	−0.0699929
$706$	11.2690	0.424113
$707$	63.4548	2.38646
$708$	1.13087	0.0425007
$709$	16.5211	0.620462	0.310231	−	0.950661i	$-0.399594\pi$
$709$	16.5211	0.620462	0.310231	+	0.950661i	$0.399594\pi$
$710$	−10.4222	−0.391138
$711$	−6.39691	−0.239903
$712$	−8.88476	−0.332971
$713$	10.2032	0.382114
$714$	3.03741	0.113672
$715$	0	0
$716$	−5.80484	−0.216937
$717$	1.18036	0.0440814
$718$	−5.70457	−0.212893
$719$	44.2488	1.65020	0.825101	−	0.564986i	$-0.191118\pi$
$719$	44.2488	1.65020	0.825101	+	0.564986i	$0.191118\pi$
$720$	11.7122	0.436487
$721$	−75.7166	−2.81984
$722$	−1.00000	−0.0372161
$723$	−1.45563	−0.0541355
$724$	−0.148577	−0.00552182
$725$	64.8039	2.40676
$726$	0	0
$727$	38.4604	1.42642	0.713209	−	0.700952i	$-0.247241\pi$
$727$	38.4604	1.42642	0.713209	+	0.700952i	$0.247241\pi$
$728$	−18.8544	−0.698792
$729$	−26.4260	−0.978742
$730$	26.0413	0.963832
$731$	−61.0193	−2.25688
$732$	−0.0336234	−0.00124275
$733$	−46.9096	−1.73265	−0.866323	−	0.499484i	$-0.833523\pi$
$733$	−46.9096	−1.73265	−0.866323	+	0.499484i	$0.833523\pi$
$734$	18.7306	0.691361
$735$	5.20142	0.191857
$736$	1.59524	0.0588014
$737$	0	0
$738$	−5.72214	−0.210635
$739$	10.9340	0.402215	0.201108	−	0.979569i	$-0.435546\pi$
$739$	10.9340	0.402215	0.201108	+	0.979569i	$0.435546\pi$
$740$	−35.1586	−1.29246
$741$	−0.436851	−0.0160481
$742$	19.1690	0.703715
$743$	31.3320	1.14946	0.574729	−	0.818344i	$-0.305107\pi$
$743$	31.3320	1.14946	0.574729	+	0.818344i	$0.305107\pi$
$744$	−0.660387	−0.0242110
$745$	50.7290	1.85857
$746$	−11.8435	−0.433621
$747$	37.2486	1.36286
$748$	0	0
$749$	61.8343	2.25938
$750$	−2.16446	−0.0790349
$751$	39.4660	1.44013	0.720067	−	0.693905i	$-0.244111\pi$
$751$	39.4660	1.44013	0.720067	+	0.693905i	$0.244111\pi$
$752$	−4.59409	−0.167529
$753$	−1.93073	−0.0703598
$754$	−26.4901	−0.964712
$755$	−51.5998	−1.87791
$756$	2.75571	0.100224
$757$	−0.203127	−0.00738276	−0.00369138	−	0.999993i	$-0.501175\pi$
$757$	−0.203127	−0.00738276	−0.00369138	+	0.999993i	$0.501175\pi$
$758$	−14.4703	−0.525584
$759$	0	0
$760$	3.91798	0.142120
$761$	−12.4379	−0.450873	−0.225437	−	0.974258i	$-0.572381\pi$
$761$	−12.4379	−0.450873	−0.225437	+	0.974258i	$0.572381\pi$
$762$	0.703218	0.0254749
$763$	27.2334	0.985914
$764$	−6.74447	−0.244006
$765$	77.3191	2.79548
$766$	23.9927	0.866890
$767$	−46.3417	−1.67330
$768$	−0.103249	−0.00372569
$769$	−24.4144	−0.880406	−0.440203	−	0.897898i	$-0.645094\pi$
$769$	−24.4144	−0.880406	−0.440203	+	0.897898i	$0.645094\pi$
$770$	0	0
$771$	−1.59466	−0.0574303
$772$	−8.15211	−0.293401
$773$	−27.8291	−1.00094	−0.500471	−	0.865753i	$-0.666840\pi$
$773$	−27.8291	−1.00094	−0.500471	+	0.865753i	$0.666840\pi$
$774$	−27.6308	−0.993167
$775$	−66.2027	−2.37807
$776$	11.0716	0.397446
$777$	−4.12880	−0.148120
$778$	27.2728	0.977776
$779$	−1.91418	−0.0685827
$780$	1.71157	0.0612842
$781$	0	0
$782$	10.5312	0.376593
$783$	3.87171	0.138364
$784$	12.8580	0.459213
$785$	−63.9862	−2.28376
$786$	1.89597	0.0676271
$787$	22.9480	0.818006	0.409003	−	0.912533i	$-0.365876\pi$
$787$	22.9480	0.818006	0.409003	+	0.912533i	$0.365876\pi$
$788$	−3.77994	−0.134655
$789$	−1.80084	−0.0641118
$790$	8.38411	0.298293
$791$	−38.4042	−1.36550
$792$	0	0
$793$	1.37785	0.0489287
$794$	−20.4056	−0.724166
$795$	−1.74013	−0.0617159
$796$	−0.977228	−0.0346369
$797$	−23.6863	−0.839013	−0.419507	−	0.907752i	$-0.637797\pi$
$797$	−23.6863	−0.839013	−0.419507	+	0.907752i	$0.637797\pi$
$798$	0.460102	0.0162874
$799$	−30.3284	−1.07294
$800$	−10.3506	−0.365948
$801$	−26.5596	−0.938436
$802$	13.8056	0.487491
$803$	0	0
$804$	−0.917519	−0.0323584
$805$	27.8520	0.981654
$806$	27.0619	0.953214
$807$	−2.24850	−0.0791508
$808$	−14.2396	−0.500946
$809$	20.9439	0.736349	0.368175	−	0.929757i	$-0.379983\pi$
$809$	20.9439	0.736349	0.368175	+	0.929757i	$0.379983\pi$
$810$	34.8864	1.22578
$811$	19.7226	0.692555	0.346277	−	0.938132i	$-0.387446\pi$
$811$	19.7226	0.692555	0.346277	+	0.938132i	$0.387446\pi$
$812$	27.9000	0.979099
$813$	2.84124	0.0996467
$814$	0	0
$815$	−15.5390	−0.544309
$816$	−0.681611	−0.0238611
$817$	−9.24310	−0.323375
$818$	−21.2103	−0.741602
$819$	−56.3623	−1.96946
$820$	7.49973	0.261902
$821$	35.4686	1.23786	0.618931	−	0.785445i	$-0.287566\pi$
$821$	35.4686	1.23786	0.618931	+	0.785445i	$0.287566\pi$
$822$	1.28333	0.0447611
$823$	18.8740	0.657906	0.328953	−	0.944346i	$-0.393304\pi$
$823$	18.8740	0.657906	0.328953	+	0.944346i	$0.393304\pi$
$824$	16.9912	0.591916
$825$	0	0
$826$	48.8082	1.69825
$827$	38.4943	1.33858	0.669290	−	0.743002i	$-0.266599\pi$
$827$	38.4943	1.33858	0.669290	+	0.743002i	$0.266599\pi$
$828$	4.76872	0.165724
$829$	20.0947	0.697917	0.348959	−	0.937138i	$-0.386535\pi$
$829$	20.0947	0.697917	0.348959	+	0.937138i	$0.386535\pi$
$830$	−48.8199	−1.69456
$831$	−0.527504	−0.0182989
$832$	4.23103	0.146685
$833$	84.8832	2.94103
$834$	0.0591402	0.00204786
$835$	53.2879	1.84410
$836$	0	0
$837$	−3.95528	−0.136714
$838$	12.1697	0.420394
$839$	18.8166	0.649621	0.324811	−	0.945779i	$-0.394699\pi$
$839$	18.8166	0.649621	0.324811	+	0.945779i	$0.394699\pi$
$840$	−1.80267	−0.0621981
$841$	10.1989	0.351687
$842$	17.3500	0.597919
$843$	−2.83468	−0.0976315
$844$	11.5402	0.397231
$845$	−19.2045	−0.660655
$846$	−13.7333	−0.472160
$847$	0	0
$848$	−4.30161	−0.147718
$849$	−0.701513	−0.0240759
$850$	−68.3304	−2.34371
$851$	−14.3152	−0.490717
$852$	0.274653	0.00940945
$853$	−50.7466	−1.73753	−0.868765	−	0.495225i	$-0.835086\pi$
$853$	−50.7466	−1.73753	−0.868765	+	0.495225i	$0.835086\pi$
$854$	−1.45118	−0.0496584
$855$	11.7122	0.400548
$856$	−13.8759	−0.474270
$857$	44.1640	1.50861	0.754307	−	0.656522i	$-0.227973\pi$
$857$	44.1640	1.50861	0.754307	+	0.656522i	$0.227973\pi$
$858$	0	0
$859$	23.1245	0.788999	0.394499	−	0.918896i	$-0.370918\pi$
$859$	23.1245	0.788999	0.394499	+	0.918896i	$0.370918\pi$
$860$	36.2143	1.23490
$861$	0.880719	0.0300148
$862$	16.6638	0.567572
$863$	47.1399	1.60466	0.802331	−	0.596879i	$-0.203593\pi$
$863$	47.1399	1.60466	0.802331	+	0.596879i	$0.203593\pi$
$864$	−0.618395	−0.0210382
$865$	−56.4777	−1.92030
$866$	−34.2652	−1.16438
$867$	−2.74448	−0.0932076
$868$	−28.5022	−0.967429
$869$	0	0
$870$	−2.53271	−0.0858671
$871$	37.5988	1.27399
$872$	−6.11130	−0.206955
$873$	33.0966	1.12015
$874$	1.59524	0.0539599
$875$	−93.4179	−3.15810
$876$	−0.686258	−0.0231865
$877$	−28.8821	−0.975280	−0.487640	−	0.873045i	$-0.662142\pi$
$877$	−28.8821	−0.975280	−0.487640	+	0.873045i	$0.662142\pi$
$878$	23.9778	0.809210
$879$	−2.95907	−0.0998069
$880$	0	0
$881$	−6.89494	−0.232296	−0.116148	−	0.993232i	$-0.537055\pi$
$881$	−6.89494	−0.232296	−0.116148	+	0.993232i	$0.537055\pi$
$882$	38.4368	1.29424
$883$	14.2757	0.480416	0.240208	−	0.970721i	$-0.422784\pi$
$883$	14.2757	0.480416	0.240208	+	0.970721i	$0.422784\pi$
$884$	27.9316	0.939441
$885$	−4.43072	−0.148937
$886$	−9.23108	−0.310124
$887$	−4.05611	−0.136191	−0.0680955	−	0.997679i	$-0.521692\pi$
$887$	−4.05611	−0.136191	−0.0680955	+	0.997679i	$0.521692\pi$
$888$	0.926523	0.0310921
$889$	30.3508	1.01793
$890$	34.8103	1.16684
$891$	0	0
$892$	−16.0725	−0.538146
$893$	−4.59409	−0.153735
$894$	−1.33684	−0.0447107
$895$	22.7432	0.760223
$896$	−4.45623	−0.148872
$897$	0.696883	0.0232682
$898$	−4.61870	−0.154128
$899$	−40.0450	−1.33558
$900$	−30.9414	−1.03138
$901$	−28.3975	−0.946059
$902$	0	0
$903$	4.25277	0.141523
$904$	8.61809	0.286634
$905$	0.582122	0.0193504
$906$	1.35979	0.0451761
$907$	0.368732	0.0122435	0.00612176	−	0.999981i	$-0.498051\pi$
$907$	0.368732	0.0122435	0.00612176	+	0.999981i	$0.498051\pi$
$908$	−3.14962	−0.104524
$909$	−42.5669	−1.41186
$910$	73.8713	2.44881
$911$	4.21077	0.139509	0.0697545	−	0.997564i	$-0.477778\pi$
$911$	4.21077	0.139509	0.0697545	+	0.997564i	$0.477778\pi$
$912$	−0.103249	−0.00341892
$913$	0	0
$914$	−6.40059	−0.211713
$915$	0.131736	0.00435505
$916$	22.0432	0.728327
$917$	81.8299	2.70226
$918$	−4.08240	−0.134739
$919$	−17.5496	−0.578907	−0.289453	−	0.957192i	$-0.593474\pi$
$919$	−17.5496	−0.578907	−0.289453	+	0.957192i	$0.593474\pi$
$920$	−6.25013	−0.206061
$921$	−1.45708	−0.0480123
$922$	−27.9210	−0.919531
$923$	−11.2549	−0.370461
$924$	0	0
$925$	92.8825	3.05396
$926$	8.30846	0.273033
$927$	50.7925	1.66824
$928$	−6.26090	−0.205524
$929$	−22.3125	−0.732048	−0.366024	−	0.930605i	$-0.619281\pi$
$929$	−22.3125	−0.732048	−0.366024	+	0.930605i	$0.619281\pi$
$930$	2.58738	0.0848436
$931$	12.8580	0.421403
$932$	−10.0334	−0.328654
$933$	−1.15769	−0.0379012
$934$	−14.7637	−0.483082
$935$	0	0
$936$	12.6480	0.413413
$937$	−46.2027	−1.50938	−0.754689	−	0.656083i	$-0.772212\pi$
$937$	−46.2027	−1.50938	−0.754689	+	0.656083i	$0.772212\pi$
$938$	−39.6000	−1.29299
$939$	1.20157	0.0392116
$940$	17.9996	0.587081
$941$	7.66776	0.249962	0.124981	−	0.992159i	$-0.460113\pi$
$941$	7.66776	0.249962	0.124981	+	0.992159i	$0.460113\pi$
$942$	1.68621	0.0549396
$943$	3.05358	0.0994383
$944$	−10.9528	−0.356483
$945$	−10.7968	−0.351220
$946$	0	0
$947$	−27.3140	−0.887585	−0.443793	−	0.896130i	$-0.646367\pi$
$947$	−27.3140	−0.887585	−0.443793	+	0.896130i	$0.646367\pi$
$948$	−0.220944	−0.00717592
$949$	28.1220	0.912880
$950$	−10.3506	−0.335817
$951$	0.381197	0.0123612
$952$	−29.4182	−0.953451
$953$	−3.09031	−0.100105	−0.0500525	−	0.998747i	$-0.515939\pi$
$953$	−3.09031	−0.100105	−0.0500525	+	0.998747i	$0.515939\pi$
$954$	−12.8590	−0.416325
$955$	26.4247	0.855084
$956$	−11.4321	−0.369742
$957$	0	0
$958$	−36.4821	−1.17868
$959$	55.3882	1.78858
$960$	0.404529	0.0130561
$961$	9.90937	0.319657
$962$	−37.9678	−1.22413
$963$	−41.4799	−1.33667
$964$	14.0982	0.454073
$965$	31.9398	1.02818
$966$	−0.733974	−0.0236152
$967$	−33.9900	−1.09305	−0.546523	−	0.837444i	$-0.684049\pi$
$967$	−33.9900	−1.09305	−0.546523	+	0.837444i	$0.684049\pi$
$968$	0	0
$969$	−0.681611	−0.0218965
$970$	−43.3781	−1.39279
$971$	−36.7362	−1.17892	−0.589461	−	0.807797i	$-0.700660\pi$
$971$	−36.7362	−1.17892	−0.589461	+	0.807797i	$0.700660\pi$
$972$	−2.77453	−0.0889932
$973$	2.55249	0.0818289
$974$	−24.8266	−0.795495
$975$	−4.52166	−0.144809
$976$	0.325652	0.0104239
$977$	−7.86284	−0.251554	−0.125777	−	0.992059i	$-0.540142\pi$
$977$	−7.86284	−0.251554	−0.125777	+	0.992059i	$0.540142\pi$
$978$	0.409495	0.0130942
$979$	0	0
$980$	−50.3773	−1.60924
$981$	−18.2688	−0.583277
$982$	−15.2207	−0.485713
$983$	3.75322	0.119709	0.0598546	−	0.998207i	$-0.480936\pi$
$983$	3.75322	0.119709	0.0598546	+	0.998207i	$0.480936\pi$
$984$	−0.197638	−0.00630046
$985$	14.8097	0.471877
$986$	−41.3320	−1.31628
$987$	2.11375	0.0672814
$988$	4.23103	0.134607
$989$	14.7450	0.468863
$990$	0	0
$991$	44.5118	1.41396	0.706982	−	0.707232i	$-0.250056\pi$
$991$	44.5118	1.41396	0.706982	+	0.707232i	$0.250056\pi$
$992$	6.39604	0.203075
$993$	−1.43960	−0.0456843
$994$	11.8540	0.375986
$995$	3.82876	0.121380
$996$	1.28654	0.0407654
$997$	−42.1484	−1.33485	−0.667426	−	0.744676i	$-0.732604\pi$
$997$	−42.1484	−1.33485	−0.667426	+	0.744676i	$0.732604\pi$
$998$	39.0356	1.23565
$999$	5.54926	0.175571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.by.1.3	✓	8
11.10	odd	2		4598.2.a.cb.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.by.1.3	✓	8	1.1	even	1	trivial
4598.2.a.cb.1.3	yes	8	11.10	odd	2

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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} +0.103249 q^{6} +4.45623 q^{7} -1.00000 q^{8} -2.98934 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.103249 q^{3} +1.00000 q^{4} -3.91798 q^{5} +0.103249 q^{6} +4.45623 q^{7} -1.00000 q^{8} -2.98934 q^{9} +3.91798 q^{10} -0.103249 q^{12} +4.23103 q^{13} -4.45623 q^{14} +0.404529 q^{15} +1.00000 q^{16} +6.60160 q^{17} +2.98934 q^{18} +1.00000 q^{19} -3.91798 q^{20} -0.460102 q^{21} -1.59524 q^{23} +0.103249 q^{24} +10.3506 q^{25} -4.23103 q^{26} +0.618395 q^{27} +4.45623 q^{28} +6.26090 q^{29} -0.404529 q^{30} -6.39604 q^{31} -1.00000 q^{32} -6.60160 q^{34} -17.4594 q^{35} -2.98934 q^{36} +8.97366 q^{37} -1.00000 q^{38} -0.436851 q^{39} +3.91798 q^{40} -1.91418 q^{41} +0.460102 q^{42} -9.24310 q^{43} +11.7122 q^{45} +1.59524 q^{46} -4.59409 q^{47} -0.103249 q^{48} +12.8580 q^{49} -10.3506 q^{50} -0.681611 q^{51} +4.23103 q^{52} -4.30161 q^{53} -0.618395 q^{54} -4.45623 q^{56} -0.103249 q^{57} -6.26090 q^{58} -10.9528 q^{59} +0.404529 q^{60} +0.325652 q^{61} +6.39604 q^{62} -13.3212 q^{63} +1.00000 q^{64} -16.5771 q^{65} +8.88645 q^{67} +6.60160 q^{68} +0.164708 q^{69} +17.4594 q^{70} -2.66009 q^{71} +2.98934 q^{72} +6.64661 q^{73} -8.97366 q^{74} -1.06869 q^{75} +1.00000 q^{76} +0.436851 q^{78} +2.13991 q^{79} -3.91798 q^{80} +8.90417 q^{81} +1.91418 q^{82} -12.4605 q^{83} -0.460102 q^{84} -25.8650 q^{85} +9.24310 q^{86} -0.646434 q^{87} +8.88476 q^{89} -11.7122 q^{90} +18.8544 q^{91} -1.59524 q^{92} +0.660387 q^{93} +4.59409 q^{94} -3.91798 q^{95} +0.103249 q^{96} -11.0716 q^{97} -12.8580 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more