LMFDB - Embedded newform 3549.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3549.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:	$28.3389076774$
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} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 $ x^8 - 2x^7 - 13x^6 + 22x^5 + 57x^4 - 72x^3 - 96x^2 + 64*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.14586$ of defining polynomial
Character	$\chi$	$=$	3549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14586 q^{2} -1.00000 q^{3} +2.60470 q^{4} +1.91954 q^{5} +2.14586 q^{6} -1.00000 q^{7} -1.29759 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.14586 q^{2} -1.00000 q^{3} +2.60470 q^{4} +1.91954 q^{5} +2.14586 q^{6} -1.00000 q^{7} -1.29759 q^{8} +1.00000 q^{9} -4.11906 q^{10} -1.42164 q^{11} -2.60470 q^{12} +2.14586 q^{14} -1.91954 q^{15} -2.42495 q^{16} -6.47107 q^{17} -2.14586 q^{18} -5.74461 q^{19} +4.99982 q^{20} +1.00000 q^{21} +3.05062 q^{22} -2.14271 q^{23} +1.29759 q^{24} -1.31537 q^{25} -1.00000 q^{27} -2.60470 q^{28} -7.63976 q^{29} +4.11906 q^{30} +2.50499 q^{31} +7.79877 q^{32} +1.42164 q^{33} +13.8860 q^{34} -1.91954 q^{35} +2.60470 q^{36} -7.82352 q^{37} +12.3271 q^{38} -2.49078 q^{40} +8.69450 q^{41} -2.14586 q^{42} +7.07445 q^{43} -3.70293 q^{44} +1.91954 q^{45} +4.59795 q^{46} +0.362849 q^{47} +2.42495 q^{48} +1.00000 q^{49} +2.82259 q^{50} +6.47107 q^{51} +0.524776 q^{53} +2.14586 q^{54} -2.72888 q^{55} +1.29759 q^{56} +5.74461 q^{57} +16.3938 q^{58} +9.00783 q^{59} -4.99982 q^{60} -7.90247 q^{61} -5.37534 q^{62} -1.00000 q^{63} -11.8851 q^{64} -3.05062 q^{66} +0.405361 q^{67} -16.8552 q^{68} +2.14271 q^{69} +4.11906 q^{70} +0.243311 q^{71} -1.29759 q^{72} -0.330581 q^{73} +16.7882 q^{74} +1.31537 q^{75} -14.9630 q^{76} +1.42164 q^{77} +8.64577 q^{79} -4.65478 q^{80} +1.00000 q^{81} -18.6571 q^{82} +11.4277 q^{83} +2.60470 q^{84} -12.4215 q^{85} -15.1807 q^{86} +7.63976 q^{87} +1.84470 q^{88} +12.6734 q^{89} -4.11906 q^{90} -5.58111 q^{92} -2.50499 q^{93} -0.778622 q^{94} -11.0270 q^{95} -7.79877 q^{96} +17.1484 q^{97} -2.14586 q^{98} -1.42164 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 8 * q^3 + 14 * q^4 + 6 * q^5 - 2 * q^6 - 8 * q^7 + 12 * q^8 + 8 * q^9	\( 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9} - 4 q^{10} + 16 q^{11} - 14 q^{12} - 2 q^{14} - 6 q^{15} + 10 q^{16} - 2 q^{17} + 2 q^{18} + 14 q^{19} - 10 q^{20} + 8 q^{21} + 18 q^{22} - 6 q^{23} - 12 q^{24} + 10 q^{25} - 8 q^{27} - 14 q^{28} + 12 q^{29} + 4 q^{30} + 16 q^{31} + 26 q^{32} - 16 q^{33} + 32 q^{34} - 6 q^{35} + 14 q^{36} - 8 q^{37} + 12 q^{38} - 14 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 68 q^{44} + 6 q^{45} - 28 q^{46} + 6 q^{47} - 10 q^{48} + 8 q^{49} + 32 q^{50} + 2 q^{51} + 14 q^{53} - 2 q^{54} - 2 q^{55} - 12 q^{56} - 14 q^{57} + 24 q^{58} + 50 q^{59} + 10 q^{60} - 2 q^{61} - 20 q^{62} - 8 q^{63} + 24 q^{64} - 18 q^{66} - 16 q^{67} - 36 q^{68} + 6 q^{69} + 4 q^{70} + 34 q^{71} + 12 q^{72} - 8 q^{73} - 6 q^{74} - 10 q^{75} - 4 q^{76} - 16 q^{77} + 46 q^{79} - 24 q^{80} + 8 q^{81} - 42 q^{82} + 42 q^{83} + 14 q^{84} + 20 q^{85} + 50 q^{86} - 12 q^{87} + 62 q^{88} + 28 q^{89} - 4 q^{90} - 58 q^{92} - 16 q^{93} + 24 q^{94} - 24 q^{95} - 26 q^{96} + 16 q^{97} + 2 q^{98} + 16 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 8 * q^3 + 14 * q^4 + 6 * q^5 - 2 * q^6 - 8 * q^7 + 12 * q^8 + 8 * q^9 - 4 * q^10 + 16 * q^11 - 14 * q^12 - 2 * q^14 - 6 * q^15 + 10 * q^16 - 2 * q^17 + 2 * q^18 + 14 * q^19 - 10 * q^20 + 8 * q^21 + 18 * q^22 - 6 * q^23 - 12 * q^24 + 10 * q^25 - 8 * q^27 - 14 * q^28 + 12 * q^29 + 4 * q^30 + 16 * q^31 + 26 * q^32 - 16 * q^33 + 32 * q^34 - 6 * q^35 + 14 * q^36 - 8 * q^37 + 12 * q^38 - 14 * q^40 - 4 * q^41 + 2 * q^42 + 14 * q^43 + 68 * q^44 + 6 * q^45 - 28 * q^46 + 6 * q^47 - 10 * q^48 + 8 * q^49 + 32 * q^50 + 2 * q^51 + 14 * q^53 - 2 * q^54 - 2 * q^55 - 12 * q^56 - 14 * q^57 + 24 * q^58 + 50 * q^59 + 10 * q^60 - 2 * q^61 - 20 * q^62 - 8 * q^63 + 24 * q^64 - 18 * q^66 - 16 * q^67 - 36 * q^68 + 6 * q^69 + 4 * q^70 + 34 * q^71 + 12 * q^72 - 8 * q^73 - 6 * q^74 - 10 * q^75 - 4 * q^76 - 16 * q^77 + 46 * q^79 - 24 * q^80 + 8 * q^81 - 42 * q^82 + 42 * q^83 + 14 * q^84 + 20 * q^85 + 50 * q^86 - 12 * q^87 + 62 * q^88 + 28 * q^89 - 4 * q^90 - 58 * q^92 - 16 * q^93 + 24 * q^94 - 24 * q^95 - 26 * q^96 + 16 * q^97 + 2 * q^98 + 16 * 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 $

$2$	−2.14586	−1.51735	−0.758675	−	0.651470i	$-0.774153\pi$
$2$	−2.14586	−1.51735	−0.758675	+	0.651470i	$0.774153\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.60470	1.30235
$5$	1.91954	0.858444	0.429222	−	0.903199i	$-0.358788\pi$
$5$	1.91954	0.858444	0.429222	+	0.903199i	$0.358788\pi$
$6$	2.14586	0.876042
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.29759	−0.458768
$9$	1.00000	0.333333
$10$	−4.11906	−1.30256
$11$	−1.42164	−0.428639	−0.214320	−	0.976764i	$-0.568753\pi$
$11$	−1.42164	−0.428639	−0.214320	+	0.976764i	$0.568753\pi$
$12$	−2.60470	−0.751911
$13$	0	0
$13$	0	0
$14$	2.14586	0.573504
$15$	−1.91954	−0.495623
$16$	−2.42495	−0.606237
$17$	−6.47107	−1.56946	−0.784732	−	0.619835i	$-0.787200\pi$
$17$	−6.47107	−1.56946	−0.784732	+	0.619835i	$0.787200\pi$
$18$	−2.14586	−0.505783
$19$	−5.74461	−1.31790	−0.658952	−	0.752185i	$-0.729000\pi$
$19$	−5.74461	−1.31790	−0.658952	+	0.752185i	$0.729000\pi$
$20$	4.99982	1.11799
$21$	1.00000	0.218218
$22$	3.05062	0.650395
$23$	−2.14271	−0.446786	−0.223393	−	0.974728i	$-0.571713\pi$
$23$	−2.14271	−0.446786	−0.223393	+	0.974728i	$0.571713\pi$
$24$	1.29759	0.264870
$25$	−1.31537	−0.263074
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.60470	−0.492242
$29$	−7.63976	−1.41867	−0.709334	−	0.704872i	$-0.751004\pi$
$29$	−7.63976	−1.41867	−0.709334	+	0.704872i	$0.751004\pi$
$30$	4.11906	0.752033
$31$	2.50499	0.449909	0.224954	−	0.974369i	$-0.427777\pi$
$31$	2.50499	0.449909	0.224954	+	0.974369i	$0.427777\pi$
$32$	7.79877	1.37864
$33$	1.42164	0.247475
$34$	13.8860	2.38143
$35$	−1.91954	−0.324461
$36$	2.60470	0.434116
$37$	−7.82352	−1.28618	−0.643090	−	0.765791i	$-0.722348\pi$
$37$	−7.82352	−1.28618	−0.643090	+	0.765791i	$0.722348\pi$
$38$	12.3271	1.99972
$39$	0	0
$40$	−2.49078	−0.393827
$41$	8.69450	1.35785	0.678926	−	0.734207i	$-0.262446\pi$
$41$	8.69450	1.35785	0.678926	+	0.734207i	$0.262446\pi$
$42$	−2.14586	−0.331113
$43$	7.07445	1.07884	0.539422	−	0.842036i	$-0.318643\pi$
$43$	7.07445	1.07884	0.539422	+	0.842036i	$0.318643\pi$
$44$	−3.70293	−0.558238
$45$	1.91954	0.286148
$46$	4.59795	0.677931
$47$	0.362849	0.0529270	0.0264635	−	0.999650i	$-0.491575\pi$
$47$	0.362849	0.0529270	0.0264635	+	0.999650i	$0.491575\pi$
$48$	2.42495	0.350011
$49$	1.00000	0.142857
$50$	2.82259	0.399174
$51$	6.47107	0.906131
$52$	0	0
$53$	0.524776	0.0720835	0.0360417	−	0.999350i	$-0.488525\pi$
$53$	0.524776	0.0720835	0.0360417	+	0.999350i	$0.488525\pi$
$54$	2.14586	0.292014
$55$	−2.72888	−0.367963
$56$	1.29759	0.173398
$57$	5.74461	0.760892
$58$	16.3938	2.15262
$59$	9.00783	1.17272	0.586360	−	0.810051i	$-0.300560\pi$
$59$	9.00783	1.17272	0.586360	+	0.810051i	$0.300560\pi$
$60$	−4.99982	−0.645474
$61$	−7.90247	−1.01181	−0.505904	−	0.862590i	$-0.668841\pi$
$61$	−7.90247	−1.01181	−0.505904	+	0.862590i	$0.668841\pi$
$62$	−5.37534	−0.682669
$63$	−1.00000	−0.125988
$64$	−11.8851	−1.48564
$65$	0	0
$66$	−3.05062	−0.375506
$67$	0.405361	0.0495227	0.0247614	−	0.999693i	$-0.492117\pi$
$67$	0.405361	0.0495227	0.0247614	+	0.999693i	$0.492117\pi$
$68$	−16.8552	−2.04399
$69$	2.14271	0.257952
$70$	4.11906	0.492321
$71$	0.243311	0.0288757	0.0144378	−	0.999896i	$-0.495404\pi$
$71$	0.243311	0.0288757	0.0144378	+	0.999896i	$0.495404\pi$
$72$	−1.29759	−0.152923
$73$	−0.330581	−0.0386916	−0.0193458	−	0.999813i	$-0.506158\pi$
$73$	−0.330581	−0.0386916	−0.0193458	+	0.999813i	$0.506158\pi$
$74$	16.7882	1.95158
$75$	1.31537	0.151886
$76$	−14.9630	−1.71637
$77$	1.42164	0.162010
$78$	0	0
$79$	8.64577	0.972725	0.486363	−	0.873757i	$-0.338323\pi$
$79$	8.64577	0.972725	0.486363	+	0.873757i	$0.338323\pi$
$80$	−4.65478	−0.520420
$81$	1.00000	0.111111
$82$	−18.6571	−2.06034
$83$	11.4277	1.25435	0.627177	−	0.778877i	$-0.284210\pi$
$83$	11.4277	1.25435	0.627177	+	0.778877i	$0.284210\pi$
$84$	2.60470	0.284196
$85$	−12.4215	−1.34730
$86$	−15.1807	−1.63698
$87$	7.63976	0.819068
$88$	1.84470	0.196646
$89$	12.6734	1.34338	0.671689	−	0.740833i	$-0.265569\pi$
$89$	12.6734	1.34338	0.671689	+	0.740833i	$0.265569\pi$
$90$	−4.11906	−0.434187
$91$	0	0
$92$	−5.58111	−0.581871
$93$	−2.50499	−0.259755
$94$	−0.778622	−0.0803087
$95$	−11.0270	−1.13135
$96$	−7.79877	−0.795959
$97$	17.1484	1.74115	0.870576	−	0.492035i	$-0.163747\pi$
$97$	17.1484	1.74115	0.870576	+	0.492035i	$0.163747\pi$
$98$	−2.14586	−0.216764
$99$	−1.42164	−0.142880
$100$	−3.42613	−0.342613
$101$	8.00571	0.796598	0.398299	−	0.917256i	$-0.369601\pi$
$101$	8.00571	0.796598	0.398299	+	0.917256i	$0.369601\pi$
$102$	−13.8860	−1.37492
$103$	−14.4716	−1.42593	−0.712965	−	0.701199i	$-0.752648\pi$
$103$	−14.4716	−1.42593	−0.712965	+	0.701199i	$0.752648\pi$
$104$	0	0
$105$	1.91954	0.187328
$106$	−1.12609	−0.109376
$107$	14.6485	1.41613	0.708063	−	0.706149i	$-0.249569\pi$
$107$	14.6485	1.41613	0.708063	+	0.706149i	$0.249569\pi$
$108$	−2.60470	−0.250637
$109$	13.6710	1.30944	0.654720	−	0.755871i	$-0.272786\pi$
$109$	13.6710	1.30944	0.654720	+	0.755871i	$0.272786\pi$
$110$	5.85579	0.558328
$111$	7.82352	0.742576
$112$	2.42495	0.229136
$113$	−1.46612	−0.137921	−0.0689607	−	0.997619i	$-0.521968\pi$
$113$	−1.46612	−0.137921	−0.0689607	+	0.997619i	$0.521968\pi$
$114$	−12.3271	−1.15454
$115$	−4.11302	−0.383541
$116$	−19.8993	−1.84760
$117$	0	0
$118$	−19.3295	−1.77943
$119$	6.47107	0.593202
$120$	2.49078	0.227376
$121$	−8.97895	−0.816268
$122$	16.9576	1.53527
$123$	−8.69450	−0.783956
$124$	6.52473	0.585938
$125$	−12.1226	−1.08428
$126$	2.14586	0.191168
$127$	−8.97273	−0.796201	−0.398101	−	0.917342i	$-0.630330\pi$
$127$	−8.97273	−0.796201	−0.398101	+	0.917342i	$0.630330\pi$
$128$	9.90627	0.875599
$129$	−7.07445	−0.622870
$130$	0	0
$131$	−1.49838	−0.130914	−0.0654569	−	0.997855i	$-0.520850\pi$
$131$	−1.49838	−0.130914	−0.0654569	+	0.997855i	$0.520850\pi$
$132$	3.70293	0.322299
$133$	5.74461	0.498121
$134$	−0.869846	−0.0751433
$135$	−1.91954	−0.165208
$136$	8.39682	0.720021
$137$	13.5700	1.15936	0.579681	−	0.814844i	$-0.303177\pi$
$137$	13.5700	1.15936	0.579681	+	0.814844i	$0.303177\pi$
$138$	−4.59795	−0.391403
$139$	6.33420	0.537260	0.268630	−	0.963243i	$-0.413429\pi$
$139$	6.33420	0.537260	0.268630	+	0.963243i	$0.413429\pi$
$140$	−4.99982	−0.422562
$141$	−0.362849	−0.0305574
$142$	−0.522110	−0.0438145
$143$	0	0
$144$	−2.42495	−0.202079
$145$	−14.6648	−1.21785
$146$	0.709380	0.0587087
$147$	−1.00000	−0.0824786
$148$	−20.3779	−1.67505
$149$	17.0188	1.39423	0.697115	−	0.716959i	$-0.254466\pi$
$149$	17.0188	1.39423	0.697115	+	0.716959i	$0.254466\pi$
$150$	−2.82259	−0.230463
$151$	−6.81585	−0.554666	−0.277333	−	0.960774i	$-0.589450\pi$
$151$	−6.81585	−0.554666	−0.277333	+	0.960774i	$0.589450\pi$
$152$	7.45416	0.604612
$153$	−6.47107	−0.523155
$154$	−3.05062	−0.245826
$155$	4.80842	0.386222
$156$	0	0
$157$	−11.7919	−0.941096	−0.470548	−	0.882374i	$-0.655944\pi$
$157$	−11.7919	−0.941096	−0.470548	+	0.882374i	$0.655944\pi$
$158$	−18.5526	−1.47596
$159$	−0.524776	−0.0416174
$160$	14.9701	1.18349
$161$	2.14271	0.168869
$162$	−2.14586	−0.168594
$163$	−8.90127	−0.697202	−0.348601	−	0.937271i	$-0.613343\pi$
$163$	−8.90127	−0.697202	−0.348601	+	0.937271i	$0.613343\pi$
$164$	22.6465	1.76840
$165$	2.72888	0.212443
$166$	−24.5222	−1.90329
$167$	−11.0481	−0.854930	−0.427465	−	0.904032i	$-0.640593\pi$
$167$	−11.0481	−0.854930	−0.427465	+	0.904032i	$0.640593\pi$
$168$	−1.29759	−0.100111
$169$	0	0
$170$	26.6547	2.04432
$171$	−5.74461	−0.439301
$172$	18.4268	1.40503
$173$	−12.7718	−0.971022	−0.485511	−	0.874231i	$-0.661366\pi$
$173$	−12.7718	−0.971022	−0.485511	+	0.874231i	$0.661366\pi$
$174$	−16.3938	−1.24281
$175$	1.31537	0.0994325
$176$	3.44739	0.259857
$177$	−9.00783	−0.677070
$178$	−27.1953	−2.03837
$179$	20.8586	1.55905	0.779524	−	0.626372i	$-0.215461\pi$
$179$	20.8586	1.55905	0.779524	+	0.626372i	$0.215461\pi$
$180$	4.99982	0.372665
$181$	0.377072	0.0280275	0.0140138	−	0.999902i	$-0.495539\pi$
$181$	0.377072	0.0280275	0.0140138	+	0.999902i	$0.495539\pi$
$182$	0	0
$183$	7.90247	0.584167
$184$	2.78037	0.204971
$185$	−15.0176	−1.10411
$186$	5.37534	0.394139
$187$	9.19950	0.672734
$188$	0.945112	0.0689294
$189$	1.00000	0.0727393
$190$	23.6623	1.71665
$191$	17.4101	1.25975	0.629876	−	0.776696i	$-0.283106\pi$
$191$	17.4101	1.25975	0.629876	+	0.776696i	$0.283106\pi$
$192$	11.8851	0.857737
$193$	−11.1529	−0.802800	−0.401400	−	0.915903i	$-0.631476\pi$
$193$	−11.1529	−0.802800	−0.401400	+	0.915903i	$0.631476\pi$
$194$	−36.7979	−2.64193
$195$	0	0
$196$	2.60470	0.186050
$197$	−8.30832	−0.591944	−0.295972	−	0.955197i	$-0.595643\pi$
$197$	−8.30832	−0.591944	−0.295972	+	0.955197i	$0.595643\pi$
$198$	3.05062	0.216798
$199$	−9.33095	−0.661454	−0.330727	−	0.943727i	$-0.607294\pi$
$199$	−9.33095	−0.661454	−0.330727	+	0.943727i	$0.607294\pi$
$200$	1.70681	0.120690
$201$	−0.405361	−0.0285920
$202$	−17.1791	−1.20872
$203$	7.63976	0.536206
$204$	16.8552	1.18010
$205$	16.6894	1.16564
$206$	31.0540	2.16363
$207$	−2.14271	−0.148929
$208$	0	0
$209$	8.16673	0.564905
$210$	−4.11906	−0.284242
$211$	27.5258	1.89496	0.947478	−	0.319820i	$-0.103622\pi$
$211$	27.5258	1.89496	0.947478	+	0.319820i	$0.103622\pi$
$212$	1.36688	0.0938778
$213$	−0.243311	−0.0166714
$214$	−31.4336	−2.14876
$215$	13.5797	0.926127
$216$	1.29759	0.0882900
$217$	−2.50499	−0.170050
$218$	−29.3359	−1.98688
$219$	0.330581	0.0223386
$220$	−7.10792	−0.479216
$221$	0	0
$222$	−16.7882	−1.12675
$223$	8.69099	0.581992	0.290996	−	0.956724i	$-0.406013\pi$
$223$	8.69099	0.581992	0.290996	+	0.956724i	$0.406013\pi$
$224$	−7.79877	−0.521077
$225$	−1.31537	−0.0876912
$226$	3.14609	0.209275
$227$	3.00725	0.199599	0.0997993	−	0.995008i	$-0.468180\pi$
$227$	3.00725	0.199599	0.0997993	+	0.995008i	$0.468180\pi$
$228$	14.9630	0.990946
$229$	−11.8292	−0.781696	−0.390848	−	0.920455i	$-0.627818\pi$
$229$	−11.8292	−0.781696	−0.390848	+	0.920455i	$0.627818\pi$
$230$	8.82595	0.581966
$231$	−1.42164	−0.0935367
$232$	9.91330	0.650840
$233$	−14.8027	−0.969755	−0.484877	−	0.874582i	$-0.661136\pi$
$233$	−14.8027	−0.969755	−0.484877	+	0.874582i	$0.661136\pi$
$234$	0	0
$235$	0.696503	0.0454348
$236$	23.4627	1.52729
$237$	−8.64577	−0.561603
$238$	−13.8860	−0.900094
$239$	8.41715	0.544460	0.272230	−	0.962232i	$-0.412239\pi$
$239$	8.41715	0.544460	0.272230	+	0.962232i	$0.412239\pi$
$240$	4.65478	0.300465
$241$	−17.9240	−1.15459	−0.577293	−	0.816537i	$-0.695891\pi$
$241$	−17.9240	−1.15459	−0.577293	+	0.816537i	$0.695891\pi$
$242$	19.2675	1.23856
$243$	−1.00000	−0.0641500
$244$	−20.5835	−1.31773
$245$	1.91954	0.122635
$246$	18.6571	1.18954
$247$	0	0
$248$	−3.25045	−0.206404
$249$	−11.4277	−0.724202
$250$	26.0133	1.64523
$251$	4.65312	0.293702	0.146851	−	0.989159i	$-0.453086\pi$
$251$	4.65312	0.293702	0.146851	+	0.989159i	$0.453086\pi$
$252$	−2.60470	−0.164081
$253$	3.04615	0.191510
$254$	19.2542	1.20811
$255$	12.4215	0.777863
$256$	2.51287	0.157054
$257$	27.0199	1.68546	0.842729	−	0.538339i	$-0.180948\pi$
$257$	27.0199	1.68546	0.842729	+	0.538339i	$0.180948\pi$
$258$	15.1807	0.945112
$259$	7.82352	0.486130
$260$	0	0
$261$	−7.63976	−0.472889
$262$	3.21530	0.198642
$263$	11.7397	0.723898	0.361949	−	0.932198i	$-0.382111\pi$
$263$	11.7397	0.723898	0.361949	+	0.932198i	$0.382111\pi$
$264$	−1.84470	−0.113534
$265$	1.00733	0.0618796
$266$	−12.3271	−0.755823
$267$	−12.6734	−0.775600
$268$	1.05584	0.0644959
$269$	3.52185	0.214731	0.107365	−	0.994220i	$-0.465759\pi$
$269$	3.52185	0.214731	0.107365	+	0.994220i	$0.465759\pi$
$270$	4.11906	0.250678
$271$	−21.4886	−1.30534	−0.652671	−	0.757641i	$-0.726352\pi$
$271$	−21.4886	−1.30534	−0.652671	+	0.757641i	$0.726352\pi$
$272$	15.6920	0.951467
$273$	0	0
$274$	−29.1192	−1.75916
$275$	1.86997	0.112764
$276$	5.58111	0.335944
$277$	−25.4165	−1.52713	−0.763565	−	0.645731i	$-0.776553\pi$
$277$	−25.4165	−1.52713	−0.763565	+	0.645731i	$0.776553\pi$
$278$	−13.5923	−0.815211
$279$	2.50499	0.149970
$280$	2.49078	0.148853
$281$	−2.42268	−0.144525	−0.0722624	−	0.997386i	$-0.523022\pi$
$281$	−2.42268	−0.144525	−0.0722624	+	0.997386i	$0.523022\pi$
$282$	0.778622	0.0463662
$283$	28.7879	1.71126	0.855632	−	0.517584i	$-0.173168\pi$
$283$	28.7879	1.71126	0.855632	+	0.517584i	$0.173168\pi$
$284$	0.633751	0.0376062
$285$	11.0270	0.653183
$286$	0	0
$287$	−8.69450	−0.513220
$288$	7.79877	0.459547
$289$	24.8747	1.46322
$290$	31.4686	1.84790
$291$	−17.1484	−1.00525
$292$	−0.861065	−0.0503900
$293$	13.1874	0.770415	0.385207	−	0.922830i	$-0.374130\pi$
$293$	13.1874	0.770415	0.385207	+	0.922830i	$0.374130\pi$
$294$	2.14586	0.125149
$295$	17.2909	1.00671
$296$	10.1518	0.590059
$297$	1.42164	0.0824916
$298$	−36.5198	−2.11554
$299$	0	0
$300$	3.42613	0.197808
$301$	−7.07445	−0.407764
$302$	14.6258	0.841622
$303$	−8.00571	−0.459916
$304$	13.9304	0.798961
$305$	−15.1691	−0.868580
$306$	13.8860	0.793809
$307$	6.41459	0.366100	0.183050	−	0.983104i	$-0.441403\pi$
$307$	6.41459	0.366100	0.183050	+	0.983104i	$0.441403\pi$
$308$	3.70293	0.210994
$309$	14.4716	0.823262
$310$	−10.3182	−0.586033
$311$	26.4924	1.50225	0.751124	−	0.660161i	$-0.229512\pi$
$311$	26.4924	1.50225	0.751124	+	0.660161i	$0.229512\pi$
$312$	0	0
$313$	5.44885	0.307987	0.153994	−	0.988072i	$-0.450786\pi$
$313$	5.44885	0.307987	0.153994	+	0.988072i	$0.450786\pi$
$314$	25.3037	1.42797
$315$	−1.91954	−0.108154
$316$	22.5196	1.26683
$317$	−26.2903	−1.47661	−0.738305	−	0.674467i	$-0.764374\pi$
$317$	−26.2903	−1.47661	−0.738305	+	0.674467i	$0.764374\pi$
$318$	1.12609	0.0631482
$319$	10.8610	0.608097
$320$	−22.8140	−1.27534
$321$	−14.6485	−0.817601
$322$	−4.59795	−0.256234
$323$	37.1737	2.06840
$324$	2.60470	0.144705
$325$	0	0
$326$	19.1009	1.05790
$327$	−13.6710	−0.756006
$328$	−11.2819	−0.622940
$329$	−0.362849	−0.0200045
$330$	−5.85579	−0.322351
$331$	36.0905	1.98372	0.991858	−	0.127351i	$-0.0406474\pi$
$331$	36.0905	1.98372	0.991858	+	0.127351i	$0.0406474\pi$
$332$	29.7657	1.63361
$333$	−7.82352	−0.428726
$334$	23.7077	1.29723
$335$	0.778107	0.0425125
$336$	−2.42495	−0.132292
$337$	13.7825	0.750781	0.375390	−	0.926867i	$-0.377509\pi$
$337$	13.7825	0.750781	0.375390	+	0.926867i	$0.377509\pi$
$338$	0	0
$339$	1.46612	0.0796290
$340$	−32.3542	−1.75465
$341$	−3.56118	−0.192848
$342$	12.3271	0.666573
$343$	−1.00000	−0.0539949
$344$	−9.17976	−0.494939
$345$	4.11302	0.221438
$346$	27.4064	1.47338
$347$	−17.8419	−0.957804	−0.478902	−	0.877868i	$-0.658965\pi$
$347$	−17.8419	−0.957804	−0.478902	+	0.877868i	$0.658965\pi$
$348$	19.8993	1.06671
$349$	22.8068	1.22082	0.610411	−	0.792085i	$-0.291004\pi$
$349$	22.8068	1.22082	0.610411	+	0.792085i	$0.291004\pi$
$350$	−2.82259	−0.150874
$351$	0	0
$352$	−11.0870	−0.590940
$353$	3.38095	0.179950	0.0899750	−	0.995944i	$-0.471321\pi$
$353$	3.38095	0.179950	0.0899750	+	0.995944i	$0.471321\pi$
$354$	19.3295	1.02735
$355$	0.467045	0.0247882
$356$	33.0104	1.74955
$357$	−6.47107	−0.342485
$358$	−44.7596	−2.36562
$359$	11.4402	0.603789	0.301894	−	0.953341i	$-0.402381\pi$
$359$	11.4402	0.603789	0.301894	+	0.953341i	$0.402381\pi$
$360$	−2.49078	−0.131276
$361$	14.0005	0.736868
$362$	−0.809142	−0.0425276
$363$	8.97895	0.471273
$364$	0	0
$365$	−0.634564	−0.0332146
$366$	−16.9576	−0.886386
$367$	−9.37584	−0.489415	−0.244707	−	0.969597i	$-0.578692\pi$
$367$	−9.37584	−0.489415	−0.244707	+	0.969597i	$0.578692\pi$
$368$	5.19596	0.270858
$369$	8.69450	0.452617
$370$	32.2255	1.67533
$371$	−0.524776	−0.0272450
$372$	−6.52473	−0.338291
$373$	−4.25300	−0.220212	−0.110106	−	0.993920i	$-0.535119\pi$
$373$	−4.25300	−0.220212	−0.110106	+	0.993920i	$0.535119\pi$
$374$	−19.7408	−1.02077
$375$	12.1226	0.626008
$376$	−0.470830	−0.0242812
$377$	0	0
$378$	−2.14586	−0.110371
$379$	22.0533	1.13280	0.566400	−	0.824131i	$-0.308336\pi$
$379$	22.0533	1.13280	0.566400	+	0.824131i	$0.308336\pi$
$380$	−28.7220	−1.47341
$381$	8.97273	0.459687
$382$	−37.3596	−1.91148
$383$	−14.0237	−0.716580	−0.358290	−	0.933610i	$-0.616640\pi$
$383$	−14.0237	−0.716580	−0.358290	+	0.933610i	$0.616640\pi$
$384$	−9.90627	−0.505527
$385$	2.72888	0.139077
$386$	23.9324	1.21813
$387$	7.07445	0.359614
$388$	44.6663	2.26759
$389$	9.60683	0.487086	0.243543	−	0.969890i	$-0.421690\pi$
$389$	9.60683	0.487086	0.243543	+	0.969890i	$0.421690\pi$
$390$	0	0
$391$	13.8656	0.701215
$392$	−1.29759	−0.0655384
$393$	1.49838	0.0755832
$394$	17.8285	0.898185
$395$	16.5959	0.835030
$396$	−3.70293	−0.186079
$397$	−39.2811	−1.97146	−0.985731	−	0.168327i	$-0.946163\pi$
$397$	−39.2811	−1.97146	−0.985731	+	0.168327i	$0.946163\pi$
$398$	20.0229	1.00366
$399$	−5.74461	−0.287590
$400$	3.18970	0.159485
$401$	2.07474	0.103608	0.0518039	−	0.998657i	$-0.483503\pi$
$401$	2.07474	0.103608	0.0518039	+	0.998657i	$0.483503\pi$
$402$	0.869846	0.0433840
$403$	0	0
$404$	20.8524	1.03745
$405$	1.91954	0.0953827
$406$	−16.3938	−0.813612
$407$	11.1222	0.551307
$408$	−8.39682	−0.415704
$409$	−37.3174	−1.84523	−0.922615	−	0.385723i	$-0.873952\pi$
$409$	−37.3174	−1.84523	−0.922615	+	0.385723i	$0.873952\pi$
$410$	−35.8131	−1.76868
$411$	−13.5700	−0.669358
$412$	−37.6942	−1.85706
$413$	−9.00783	−0.443246
$414$	4.59795	0.225977
$415$	21.9359	1.07679
$416$	0	0
$417$	−6.33420	−0.310187
$418$	−17.5246	−0.857158
$419$	14.0485	0.686316	0.343158	−	0.939278i	$-0.388503\pi$
$419$	14.0485	0.686316	0.343158	+	0.939278i	$0.388503\pi$
$420$	4.99982	0.243966
$421$	20.3637	0.992467	0.496233	−	0.868189i	$-0.334716\pi$
$421$	20.3637	0.992467	0.496233	+	0.868189i	$0.334716\pi$
$422$	−59.0665	−2.87531
$423$	0.362849	0.0176423
$424$	−0.680945	−0.0330696
$425$	8.51184	0.412885
$426$	0.522110	0.0252963
$427$	7.90247	0.382427
$428$	38.1550	1.84429
$429$	0	0
$430$	−29.1400	−1.40526
$431$	−37.7283	−1.81731	−0.908653	−	0.417552i	$-0.862888\pi$
$431$	−37.7283	−1.81731	−0.908653	+	0.417552i	$0.862888\pi$
$432$	2.42495	0.116670
$433$	29.5723	1.42115	0.710577	−	0.703620i	$-0.248434\pi$
$433$	29.5723	1.42115	0.710577	+	0.703620i	$0.248434\pi$
$434$	5.37534	0.258024
$435$	14.6648	0.703125
$436$	35.6087	1.70535
$437$	12.3090	0.588821
$438$	−0.709380	−0.0338955
$439$	−10.2284	−0.488175	−0.244087	−	0.969753i	$-0.578488\pi$
$439$	−10.2284	−0.488175	−0.244087	+	0.969753i	$0.578488\pi$
$440$	3.54098	0.168810
$441$	1.00000	0.0476190
$442$	0	0
$443$	−0.256887	−0.0122051	−0.00610253	−	0.999981i	$-0.501943\pi$
$443$	−0.256887	−0.0122051	−0.00610253	+	0.999981i	$0.501943\pi$
$444$	20.3779	0.967093
$445$	24.3271	1.15322
$446$	−18.6496	−0.883085
$447$	−17.0188	−0.804960
$448$	11.8851	0.561520
$449$	−8.88464	−0.419292	−0.209646	−	0.977777i	$-0.567231\pi$
$449$	−8.88464	−0.419292	−0.209646	+	0.977777i	$0.567231\pi$
$450$	2.82259	0.133058
$451$	−12.3604	−0.582029
$452$	−3.81881	−0.179622
$453$	6.81585	0.320236
$454$	−6.45314	−0.302861
$455$	0	0
$456$	−7.45416	−0.349073
$457$	27.1819	1.27152	0.635758	−	0.771889i	$-0.280688\pi$
$457$	27.1819	1.27152	0.635758	+	0.771889i	$0.280688\pi$
$458$	25.3838	1.18611
$459$	6.47107	0.302044
$460$	−10.7132	−0.499504
$461$	36.1557	1.68394	0.841969	−	0.539526i	$-0.181397\pi$
$461$	36.1557	1.68394	0.841969	+	0.539526i	$0.181397\pi$
$462$	3.05062	0.141928
$463$	13.0662	0.607240	0.303620	−	0.952793i	$-0.401805\pi$
$463$	13.0662	0.607240	0.303620	+	0.952793i	$0.401805\pi$
$464$	18.5260	0.860049
$465$	−4.80842	−0.222985
$466$	31.7644	1.47146
$467$	14.7172	0.681029	0.340514	−	0.940239i	$-0.389399\pi$
$467$	14.7172	0.681029	0.340514	+	0.940239i	$0.389399\pi$
$468$	0	0
$469$	−0.405361	−0.0187178
$470$	−1.49459	−0.0689405
$471$	11.7919	0.543342
$472$	−11.6885	−0.538007
$473$	−10.0573	−0.462434
$474$	18.5526	0.852148
$475$	7.55627	0.346705
$476$	16.8552	0.772556
$477$	0.524776	0.0240278
$478$	−18.0620	−0.826136
$479$	2.22535	0.101679	0.0508393	−	0.998707i	$-0.483810\pi$
$479$	2.22535	0.101679	0.0508393	+	0.998707i	$0.483810\pi$
$480$	−14.9701	−0.683286
$481$	0	0
$482$	38.4623	1.75191
$483$	−2.14271	−0.0974967
$484$	−23.3875	−1.06307
$485$	32.9169	1.49468
$486$	2.14586	0.0973380
$487$	14.0955	0.638730	0.319365	−	0.947632i	$-0.396530\pi$
$487$	14.0955	0.638730	0.319365	+	0.947632i	$0.396530\pi$
$488$	10.2542	0.464185
$489$	8.90127	0.402530
$490$	−4.11906	−0.186080
$491$	−32.1427	−1.45058	−0.725291	−	0.688443i	$-0.758295\pi$
$491$	−32.1427	−1.45058	−0.725291	+	0.688443i	$0.758295\pi$
$492$	−22.6465	−1.02098
$493$	49.4374	2.22655
$494$	0	0
$495$	−2.72888	−0.122654
$496$	−6.07446	−0.272751
$497$	−0.243311	−0.0109140
$498$	24.5222	1.09887
$499$	38.9893	1.74540	0.872701	−	0.488255i	$-0.162367\pi$
$499$	38.9893	1.74540	0.872701	+	0.488255i	$0.162367\pi$
$500$	−31.5757	−1.41211
$501$	11.0481	0.493594
$502$	−9.98492	−0.445649
$503$	25.1536	1.12154	0.560772	−	0.827970i	$-0.310504\pi$
$503$	25.1536	1.12154	0.560772	+	0.827970i	$0.310504\pi$
$504$	1.29759	0.0577994
$505$	15.3673	0.683835
$506$	−6.53661	−0.290588
$507$	0	0
$508$	−23.3712	−1.03693
$509$	10.6905	0.473850	0.236925	−	0.971528i	$-0.423861\pi$
$509$	10.6905	0.473850	0.236925	+	0.971528i	$0.423861\pi$
$510$	−26.6547	−1.18029
$511$	0.330581	0.0146241
$512$	−25.2048	−1.11391
$513$	5.74461	0.253631
$514$	−57.9809	−2.55743
$515$	−27.7788	−1.22408
$516$	−18.4268	−0.811194
$517$	−0.515839	−0.0226866
$518$	−16.7882	−0.737629
$519$	12.7718	0.560620
$520$	0	0
$521$	−25.5518	−1.11944	−0.559722	−	0.828680i	$-0.689092\pi$
$521$	−25.5518	−1.11944	−0.559722	+	0.828680i	$0.689092\pi$
$522$	16.3938	0.717538
$523$	−9.57078	−0.418501	−0.209251	−	0.977862i	$-0.567102\pi$
$523$	−9.57078	−0.418501	−0.209251	+	0.977862i	$0.567102\pi$
$524$	−3.90282	−0.170495
$525$	−1.31537	−0.0574074
$526$	−25.1916	−1.09841
$527$	−16.2099	−0.706116
$528$	−3.44739	−0.150028
$529$	−18.4088	−0.800382
$530$	−2.16158	−0.0938930
$531$	9.00783	0.390907
$532$	14.9630	0.648727
$533$	0	0
$534$	27.1953	1.17686
$535$	28.1184	1.21567
$536$	−0.525994	−0.0227195
$537$	−20.8586	−0.900117
$538$	−7.55738	−0.325822
$539$	−1.42164	−0.0612342
$540$	−4.99982	−0.215158
$541$	−36.9121	−1.58697	−0.793487	−	0.608587i	$-0.791737\pi$
$541$	−36.9121	−1.58697	−0.793487	+	0.608587i	$0.791737\pi$
$542$	46.1115	1.98066
$543$	−0.377072	−0.0161817
$544$	−50.4664	−2.16373
$545$	26.2419	1.12408
$546$	0	0
$547$	−11.8055	−0.504767	−0.252383	−	0.967627i	$-0.581214\pi$
$547$	−11.8055	−0.504767	−0.252383	+	0.967627i	$0.581214\pi$
$548$	35.3457	1.50989
$549$	−7.90247	−0.337269
$550$	−4.01269	−0.171102
$551$	43.8874	1.86967
$552$	−2.78037	−0.118340
$553$	−8.64577	−0.367656
$554$	54.5401	2.31719
$555$	15.0176	0.637460
$556$	16.4987	0.699700
$557$	−29.5755	−1.25316	−0.626578	−	0.779359i	$-0.715545\pi$
$557$	−29.5755	−1.25316	−0.626578	+	0.779359i	$0.715545\pi$
$558$	−5.37534	−0.227556
$559$	0	0
$560$	4.65478	0.196700
$561$	−9.19950	−0.388403
$562$	5.19872	0.219295
$563$	−12.7609	−0.537809	−0.268905	−	0.963167i	$-0.586662\pi$
$563$	−12.7609	−0.537809	−0.268905	+	0.963167i	$0.586662\pi$
$564$	−0.945112	−0.0397964
$565$	−2.81428	−0.118398
$566$	−61.7748	−2.59659
$567$	−1.00000	−0.0419961
$568$	−0.315719	−0.0132473
$569$	−34.5878	−1.45000	−0.724998	−	0.688751i	$-0.758159\pi$
$569$	−34.5878	−1.45000	−0.724998	+	0.688751i	$0.758159\pi$
$570$	−23.6623	−0.991107
$571$	−39.8897	−1.66933	−0.834666	−	0.550757i	$-0.814339\pi$
$571$	−39.8897	−1.66933	−0.834666	+	0.550757i	$0.814339\pi$
$572$	0	0
$573$	−17.4101	−0.727318
$574$	18.6571	0.778734
$575$	2.81845	0.117538
$576$	−11.8851	−0.495214
$577$	−21.8823	−0.910972	−0.455486	−	0.890243i	$-0.650535\pi$
$577$	−21.8823	−0.910972	−0.455486	+	0.890243i	$0.650535\pi$
$578$	−53.3776	−2.22022
$579$	11.1529	0.463497
$580$	−38.1974	−1.58606
$581$	−11.4277	−0.474101
$582$	36.7979	1.52532
$583$	−0.746039	−0.0308978
$584$	0.428960	0.0177505
$585$	0	0
$586$	−28.2982	−1.16899
$587$	−42.2588	−1.74421	−0.872103	−	0.489322i	$-0.837244\pi$
$587$	−42.2588	−1.74421	−0.872103	+	0.489322i	$0.837244\pi$
$588$	−2.60470	−0.107416
$589$	−14.3902	−0.592936
$590$	−37.1038	−1.52754
$591$	8.30832	0.341759
$592$	18.9716	0.779729
$593$	−8.62045	−0.353999	−0.177000	−	0.984211i	$-0.556639\pi$
$593$	−8.62045	−0.353999	−0.177000	+	0.984211i	$0.556639\pi$
$594$	−3.05062	−0.125169
$595$	12.4215	0.509231
$596$	44.3287	1.81577
$597$	9.33095	0.381890
$598$	0	0
$599$	45.0330	1.84000	0.919999	−	0.391921i	$-0.128189\pi$
$599$	45.0330	1.84000	0.919999	+	0.391921i	$0.128189\pi$
$600$	−1.70681	−0.0696803
$601$	−41.9152	−1.70976	−0.854878	−	0.518830i	$-0.826368\pi$
$601$	−41.9152	−1.70976	−0.854878	+	0.518830i	$0.826368\pi$
$602$	15.1807	0.618721
$603$	0.405361	0.0165076
$604$	−17.7532	−0.722368
$605$	−17.2355	−0.700721
$606$	17.1791	0.697853
$607$	8.11463	0.329363	0.164681	−	0.986347i	$-0.447340\pi$
$607$	8.11463	0.329363	0.164681	+	0.986347i	$0.447340\pi$
$608$	−44.8009	−1.81692
$609$	−7.63976	−0.309579
$610$	32.5507	1.31794
$611$	0	0
$612$	−16.8552	−0.681330
$613$	3.07336	0.124132	0.0620659	−	0.998072i	$-0.480231\pi$
$613$	3.07336	0.124132	0.0620659	+	0.998072i	$0.480231\pi$
$614$	−13.7648	−0.555501
$615$	−16.6894	−0.672983
$616$	−1.84470	−0.0743252
$617$	6.74860	0.271688	0.135844	−	0.990730i	$-0.456625\pi$
$617$	6.74860	0.271688	0.135844	+	0.990730i	$0.456625\pi$
$618$	−31.0540	−1.24918
$619$	1.73712	0.0698208	0.0349104	−	0.999390i	$-0.488885\pi$
$619$	1.73712	0.0698208	0.0349104	+	0.999390i	$0.488885\pi$
$620$	12.5245	0.502995
$621$	2.14271	0.0859840
$622$	−56.8490	−2.27944
$623$	−12.6734	−0.507749
$624$	0	0
$625$	−16.6930	−0.667719
$626$	−11.6924	−0.467324
$627$	−8.16673	−0.326148
$628$	−30.7143	−1.22564
$629$	50.6266	2.01861
$630$	4.11906	0.164107
$631$	42.0488	1.67394	0.836968	−	0.547251i	$-0.184326\pi$
$631$	42.0488	1.67394	0.836968	+	0.547251i	$0.184326\pi$
$632$	−11.2187	−0.446256
$633$	−27.5258	−1.09405
$634$	56.4152	2.24053
$635$	−17.2235	−0.683494
$636$	−1.36688	−0.0542004
$637$	0	0
$638$	−23.3060	−0.922695
$639$	0.243311	0.00962523
$640$	19.0155	0.751653
$641$	2.50873	0.0990887	0.0495444	−	0.998772i	$-0.484223\pi$
$641$	2.50873	0.0990887	0.0495444	+	0.998772i	$0.484223\pi$
$642$	31.4336	1.24059
$643$	−30.8814	−1.21785	−0.608923	−	0.793230i	$-0.708398\pi$
$643$	−30.8814	−1.21785	−0.608923	+	0.793230i	$0.708398\pi$
$644$	5.58111	0.219927
$645$	−13.5797	−0.534699
$646$	−79.7695	−3.13849
$647$	−10.0589	−0.395458	−0.197729	−	0.980257i	$-0.563357\pi$
$647$	−10.0589	−0.395458	−0.197729	+	0.980257i	$0.563357\pi$
$648$	−1.29759	−0.0509743
$649$	−12.8058	−0.502674
$650$	0	0
$651$	2.50499	0.0981781
$652$	−23.1851	−0.908000
$653$	−24.4478	−0.956716	−0.478358	−	0.878165i	$-0.658768\pi$
$653$	−24.4478	−0.956716	−0.478358	+	0.878165i	$0.658768\pi$
$654$	29.3359	1.14712
$655$	−2.87620	−0.112382
$656$	−21.0837	−0.823180
$657$	−0.330581	−0.0128972
$658$	0.778622	0.0303538
$659$	−47.5296	−1.85149	−0.925746	−	0.378146i	$-0.876562\pi$
$659$	−47.5296	−1.85149	−0.925746	+	0.378146i	$0.876562\pi$
$660$	7.10792	0.276675
$661$	29.6786	1.15436	0.577182	−	0.816616i	$-0.304152\pi$
$661$	29.6786	1.15436	0.577182	+	0.816616i	$0.304152\pi$
$662$	−77.4451	−3.00999
$663$	0	0
$664$	−14.8285	−0.575458
$665$	11.0270	0.427609
$666$	16.7882	0.650528
$667$	16.3698	0.633841
$668$	−28.7770	−1.11342
$669$	−8.69099	−0.336013
$670$	−1.66970	−0.0645063
$671$	11.2344	0.433700
$672$	7.79877	0.300844
$673$	1.22068	0.0470536	0.0235268	−	0.999723i	$-0.492511\pi$
$673$	1.22068	0.0470536	0.0235268	+	0.999723i	$0.492511\pi$
$674$	−29.5753	−1.13920
$675$	1.31537	0.0506285
$676$	0	0
$677$	6.05760	0.232812	0.116406	−	0.993202i	$-0.462863\pi$
$677$	6.05760	0.232812	0.116406	+	0.993202i	$0.462863\pi$
$678$	−3.14609	−0.120825
$679$	−17.1484	−0.658093
$680$	16.1180	0.618098
$681$	−3.00725	−0.115238
$682$	7.64177	0.292618
$683$	28.9040	1.10598	0.552991	−	0.833187i	$-0.313486\pi$
$683$	28.9040	1.10598	0.552991	+	0.833187i	$0.313486\pi$
$684$	−14.9630	−0.572123
$685$	26.0481	0.995247
$686$	2.14586	0.0819292
$687$	11.8292	0.451313
$688$	−17.1552	−0.654034
$689$	0	0
$690$	−8.82595	−0.335998
$691$	11.4433	0.435325	0.217663	−	0.976024i	$-0.430157\pi$
$691$	11.4433	0.435325	0.217663	+	0.976024i	$0.430157\pi$
$692$	−33.2667	−1.26461
$693$	1.42164	0.0540035
$694$	38.2862	1.45332
$695$	12.1588	0.461208
$696$	−9.91330	−0.375763
$697$	−56.2627	−2.13110
$698$	−48.9402	−1.85241
$699$	14.8027	0.559888
$700$	3.42613	0.129496
$701$	−10.1839	−0.384640	−0.192320	−	0.981332i	$-0.561601\pi$
$701$	−10.1839	−0.384640	−0.192320	+	0.981332i	$0.561601\pi$
$702$	0	0
$703$	44.9431	1.69506
$704$	16.8963	0.636805
$705$	−0.696503	−0.0262318
$706$	−7.25504	−0.273047
$707$	−8.00571	−0.301086
$708$	−23.4627	−0.881781
$709$	2.03360	0.0763736	0.0381868	−	0.999271i	$-0.487842\pi$
$709$	2.03360	0.0763736	0.0381868	+	0.999271i	$0.487842\pi$
$710$	−1.00221	−0.0376123
$711$	8.64577	0.324242
$712$	−16.4449	−0.616300
$713$	−5.36746	−0.201013
$714$	13.8860	0.519670
$715$	0	0
$716$	54.3304	2.03042
$717$	−8.41715	−0.314344
$718$	−24.5489	−0.916158
$719$	−7.74821	−0.288959	−0.144480	−	0.989508i	$-0.546151\pi$
$719$	−7.74821	−0.288959	−0.144480	+	0.989508i	$0.546151\pi$
$720$	−4.65478	−0.173473
$721$	14.4716	0.538951
$722$	−30.0431	−1.11809
$723$	17.9240	0.666601
$724$	0.982158	0.0365016
$725$	10.0491	0.373214
$726$	−19.2675	−0.715085
$727$	−13.8427	−0.513399	−0.256699	−	0.966491i	$-0.582635\pi$
$727$	−13.8427	−0.513399	−0.256699	+	0.966491i	$0.582635\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.36168	0.0503982
$731$	−45.7792	−1.69321
$732$	20.5835	0.760790
$733$	−35.7612	−1.32087	−0.660435	−	0.750883i	$-0.729628\pi$
$733$	−35.7612	−1.32087	−0.660435	+	0.750883i	$0.729628\pi$
$734$	20.1192	0.742613
$735$	−1.91954	−0.0708033
$736$	−16.7105	−0.615958
$737$	−0.576275	−0.0212274
$738$	−18.6571	−0.686779
$739$	11.4026	0.419452	0.209726	−	0.977760i	$-0.432743\pi$
$739$	11.4026	0.419452	0.209726	+	0.977760i	$0.432743\pi$
$740$	−39.1162	−1.43794
$741$	0	0
$742$	1.12609	0.0413402
$743$	34.4274	1.26302	0.631510	−	0.775368i	$-0.282436\pi$
$743$	34.4274	1.26302	0.631510	+	0.775368i	$0.282436\pi$
$744$	3.25045	0.119167
$745$	32.6682	1.19687
$746$	9.12632	0.334138
$747$	11.4277	0.418118
$748$	23.9619	0.876134
$749$	−14.6485	−0.535245
$750$	−26.0133	−0.949873
$751$	46.5980	1.70038	0.850192	−	0.526473i	$-0.176486\pi$
$751$	46.5980	1.70038	0.850192	+	0.526473i	$0.176486\pi$
$752$	−0.879889	−0.0320863
$753$	−4.65312	−0.169569
$754$	0	0
$755$	−13.0833	−0.476150
$756$	2.60470	0.0947319
$757$	52.0997	1.89360	0.946799	−	0.321826i	$-0.104297\pi$
$757$	52.0997	1.89360	0.946799	+	0.321826i	$0.104297\pi$
$758$	−47.3231	−1.71885
$759$	−3.04615	−0.110568
$760$	14.3086	0.519026
$761$	41.5101	1.50474	0.752370	−	0.658741i	$-0.228911\pi$
$761$	41.5101	1.50474	0.752370	+	0.658741i	$0.228911\pi$
$762$	−19.2542	−0.697506
$763$	−13.6710	−0.494922
$764$	45.3481	1.64064
$765$	−12.4215	−0.449099
$766$	30.0929	1.08730
$767$	0	0
$768$	−2.51287	−0.0906754
$769$	15.4047	0.555509	0.277754	−	0.960652i	$-0.410410\pi$
$769$	15.4047	0.555509	0.277754	+	0.960652i	$0.410410\pi$
$770$	−5.85579	−0.211028
$771$	−27.0199	−0.973099
$772$	−29.0498	−1.04553
$773$	5.19504	0.186853	0.0934264	−	0.995626i	$-0.470218\pi$
$773$	5.19504	0.186853	0.0934264	+	0.995626i	$0.470218\pi$
$774$	−15.1807	−0.545661
$775$	−3.29498	−0.118359
$776$	−22.2516	−0.798785
$777$	−7.82352	−0.280667
$778$	−20.6149	−0.739079
$779$	−49.9465	−1.78952
$780$	0	0
$781$	−0.345899	−0.0123773
$782$	−29.7536	−1.06399
$783$	7.63976	0.273023
$784$	−2.42495	−0.0866052
$785$	−22.6350	−0.807878
$786$	−3.21530	−0.114686
$787$	6.73938	0.240233	0.120116	−	0.992760i	$-0.461673\pi$
$787$	6.73938	0.240233	0.120116	+	0.992760i	$0.461673\pi$
$788$	−21.6407	−0.770917
$789$	−11.7397	−0.417943
$790$	−35.6124	−1.26703
$791$	1.46612	0.0521294
$792$	1.84470	0.0655487
$793$	0	0
$794$	84.2916	2.99140
$795$	−1.00733	−0.0357262
$796$	−24.3043	−0.861443
$797$	−37.0336	−1.31180	−0.655898	−	0.754849i	$-0.727710\pi$
$797$	−37.0336	−1.31180	−0.655898	+	0.754849i	$0.727710\pi$
$798$	12.3271	0.436374
$799$	−2.34802	−0.0830670
$800$	−10.2583	−0.362684
$801$	12.6734	0.447793
$802$	−4.45210	−0.157209
$803$	0.469966	0.0165847
$804$	−1.05584	−0.0372367
$805$	4.11302	0.144965
$806$	0	0
$807$	−3.52185	−0.123975
$808$	−10.3882	−0.365454
$809$	55.3200	1.94495	0.972474	−	0.233011i	$-0.0748578\pi$
$809$	55.3200	1.94495	0.972474	+	0.233011i	$0.0748578\pi$
$810$	−4.11906	−0.144729
$811$	−5.37007	−0.188569	−0.0942843	−	0.995545i	$-0.530056\pi$
$811$	−5.37007	−0.188569	−0.0942843	+	0.995545i	$0.530056\pi$
$812$	19.8993	0.698327
$813$	21.4886	0.753640
$814$	−23.8666	−0.836525
$815$	−17.0863	−0.598509
$816$	−15.6920	−0.549330
$817$	−40.6399	−1.42181
$818$	80.0779	2.79986
$819$	0	0
$820$	43.4709	1.51807
$821$	−36.9130	−1.28827	−0.644137	−	0.764910i	$-0.722783\pi$
$821$	−36.9130	−1.28827	−0.644137	+	0.764910i	$0.722783\pi$
$822$	29.1192	1.01565
$823$	−34.4893	−1.20222	−0.601110	−	0.799166i	$-0.705275\pi$
$823$	−34.4893	−1.20222	−0.601110	+	0.799166i	$0.705275\pi$
$824$	18.7783	0.654172
$825$	−1.86997	−0.0651041
$826$	19.3295	0.672559
$827$	35.2495	1.22575	0.612873	−	0.790181i	$-0.290014\pi$
$827$	35.2495	1.22575	0.612873	+	0.790181i	$0.290014\pi$
$828$	−5.58111	−0.193957
$829$	−5.42044	−0.188260	−0.0941299	−	0.995560i	$-0.530007\pi$
$829$	−5.42044	−0.188260	−0.0941299	+	0.995560i	$0.530007\pi$
$830$	−47.0714	−1.63387
$831$	25.4165	0.881689
$832$	0	0
$833$	−6.47107	−0.224209
$834$	13.5923	0.470662
$835$	−21.2073	−0.733910
$836$	21.2719	0.735703
$837$	−2.50499	−0.0865850
$838$	−30.1461	−1.04138
$839$	3.34238	0.115392	0.0576959	−	0.998334i	$-0.481625\pi$
$839$	3.34238	0.115392	0.0576959	+	0.998334i	$0.481625\pi$
$840$	−2.49078	−0.0859401
$841$	29.3660	1.01262
$842$	−43.6976	−1.50592
$843$	2.42268	0.0834414
$844$	71.6965	2.46789
$845$	0	0
$846$	−0.778622	−0.0267696
$847$	8.97895	0.308520
$848$	−1.27255	−0.0436996
$849$	−28.7879	−0.987999
$850$	−18.2652	−0.626490
$851$	16.7636	0.574647
$852$	−0.633751	−0.0217120
$853$	30.6155	1.04826	0.524128	−	0.851640i	$-0.324391\pi$
$853$	30.6155	1.04826	0.524128	+	0.851640i	$0.324391\pi$
$854$	−16.9576	−0.580276
$855$	−11.0270	−0.377115
$856$	−19.0078	−0.649674
$857$	14.5325	0.496422	0.248211	−	0.968706i	$-0.420157\pi$
$857$	14.5325	0.496422	0.248211	+	0.968706i	$0.420157\pi$
$858$	0	0
$859$	54.9295	1.87417	0.937085	−	0.349102i	$-0.113513\pi$
$859$	54.9295	1.87417	0.937085	+	0.349102i	$0.113513\pi$
$860$	35.3710	1.20614
$861$	8.69450	0.296308
$862$	80.9594	2.75749
$863$	31.6403	1.07705	0.538525	−	0.842610i	$-0.318982\pi$
$863$	31.6403	1.07705	0.538525	+	0.842610i	$0.318982\pi$
$864$	−7.79877	−0.265320
$865$	−24.5160	−0.833568
$866$	−63.4579	−2.15639
$867$	−24.8747	−0.844790
$868$	−6.52473	−0.221464
$869$	−12.2911	−0.416948
$870$	−31.4686	−1.06689
$871$	0	0
$872$	−17.7393	−0.600730
$873$	17.1484	0.580384
$874$	−26.4134	−0.893447
$875$	12.1226	0.409819
$876$	0.861065	0.0290927
$877$	4.68057	0.158052	0.0790258	−	0.996873i	$-0.474819\pi$
$877$	4.68057	0.158052	0.0790258	+	0.996873i	$0.474819\pi$
$878$	21.9487	0.740732
$879$	−13.1874	−0.444799
$880$	6.61740	0.223073
$881$	27.0026	0.909741	0.454870	−	0.890558i	$-0.349686\pi$
$881$	27.0026	0.909741	0.454870	+	0.890558i	$0.349686\pi$
$882$	−2.14586	−0.0722547
$883$	3.28945	0.110699	0.0553495	−	0.998467i	$-0.482373\pi$
$883$	3.28945	0.110699	0.0553495	+	0.998467i	$0.482373\pi$
$884$	0	0
$885$	−17.2909	−0.581227
$886$	0.551242	0.0185193
$887$	−38.8400	−1.30412	−0.652060	−	0.758168i	$-0.726095\pi$
$887$	−38.8400	−1.30412	−0.652060	+	0.758168i	$0.726095\pi$
$888$	−10.1518	−0.340670
$889$	8.97273	0.300936
$890$	−52.2025	−1.74983
$891$	−1.42164	−0.0476266
$892$	22.6374	0.757956
$893$	−2.08442	−0.0697526
$894$	36.5198	1.22140
$895$	40.0390	1.33836
$896$	−9.90627	−0.330945
$897$	0	0
$898$	19.0652	0.636213
$899$	−19.1375	−0.638271
$900$	−3.42613	−0.114204
$901$	−3.39586	−0.113132
$902$	26.5236	0.883141
$903$	7.07445	0.235423
$904$	1.90243	0.0632740
$905$	0.723805	0.0240601
$906$	−14.6258	−0.485910
$907$	42.9155	1.42499	0.712493	−	0.701679i	$-0.247566\pi$
$907$	42.9155	1.42499	0.712493	+	0.701679i	$0.247566\pi$
$908$	7.83299	0.259947
$909$	8.00571	0.265533
$910$	0	0
$911$	−10.9906	−0.364133	−0.182067	−	0.983286i	$-0.558279\pi$
$911$	−10.9906	−0.364133	−0.182067	+	0.983286i	$0.558279\pi$
$912$	−13.9304	−0.461280
$913$	−16.2460	−0.537665
$914$	−58.3284	−1.92933
$915$	15.1691	0.501475
$916$	−30.8115	−1.01804
$917$	1.49838	0.0494808
$918$	−13.8860	−0.458306
$919$	27.1924	0.896994	0.448497	−	0.893784i	$-0.351959\pi$
$919$	27.1924	0.896994	0.448497	+	0.893784i	$0.351959\pi$
$920$	5.33703	0.175957
$921$	−6.41459	−0.211368
$922$	−77.5849	−2.55512
$923$	0	0
$924$	−3.70293	−0.121817
$925$	10.2908	0.338360
$926$	−28.0383	−0.921395
$927$	−14.4716	−0.475310
$928$	−59.5808	−1.95583
$929$	17.5045	0.574304	0.287152	−	0.957885i	$-0.407292\pi$
$929$	17.5045	0.574304	0.287152	+	0.957885i	$0.407292\pi$
$930$	10.3182	0.338346
$931$	−5.74461	−0.188272
$932$	−38.5565	−1.26296
$933$	−26.4924	−0.867324
$934$	−31.5809	−1.03336
$935$	17.6588	0.577505
$936$	0	0
$937$	−10.8883	−0.355705	−0.177853	−	0.984057i	$-0.556915\pi$
$937$	−10.8883	−0.355705	−0.177853	+	0.984057i	$0.556915\pi$
$938$	0.869846	0.0284015
$939$	−5.44885	−0.177816
$940$	1.81418	0.0591720
$941$	−19.7105	−0.642542	−0.321271	−	0.946987i	$-0.604110\pi$
$941$	−19.7105	−0.642542	−0.321271	+	0.946987i	$0.604110\pi$
$942$	−25.3037	−0.824440
$943$	−18.6298	−0.606670
$944$	−21.8435	−0.710946
$945$	1.91954	0.0624426
$946$	21.5815	0.701674
$947$	9.17448	0.298131	0.149065	−	0.988827i	$-0.452374\pi$
$947$	9.17448	0.298131	0.149065	+	0.988827i	$0.452374\pi$
$948$	−22.5196	−0.731403
$949$	0	0
$950$	−16.2147	−0.526073
$951$	26.2903	0.852522
$952$	−8.39682	−0.272142
$953$	6.10028	0.197608	0.0988038	−	0.995107i	$-0.468498\pi$
$953$	6.10028	0.197608	0.0988038	+	0.995107i	$0.468498\pi$
$954$	−1.12609	−0.0364586
$955$	33.4194	1.08143
$956$	21.9241	0.709077
$957$	−10.8610	−0.351085
$958$	−4.77527	−0.154282
$959$	−13.5700	−0.438197
$960$	22.8140	0.736319
$961$	−24.7250	−0.797582
$962$	0	0
$963$	14.6485	0.472042
$964$	−46.6866	−1.50367
$965$	−21.4084	−0.689159
$966$	4.59795	0.147937
$967$	−27.5349	−0.885463	−0.442731	−	0.896654i	$-0.645990\pi$
$967$	−27.5349	−0.885463	−0.442731	+	0.896654i	$0.645990\pi$
$968$	11.6510	0.374478
$969$	−37.1737	−1.19419
$970$	−70.6350	−2.26795
$971$	−12.4000	−0.397935	−0.198967	−	0.980006i	$-0.563759\pi$
$971$	−12.4000	−0.397935	−0.198967	+	0.980006i	$0.563759\pi$
$972$	−2.60470	−0.0835457
$973$	−6.33420	−0.203065
$974$	−30.2470	−0.969177
$975$	0	0
$976$	19.1631	0.613395
$977$	−4.01737	−0.128527	−0.0642636	−	0.997933i	$-0.520470\pi$
$977$	−4.01737	−0.128527	−0.0642636	+	0.997933i	$0.520470\pi$
$978$	−19.1009	−0.610778
$979$	−18.0170	−0.575825
$980$	4.99982	0.159713
$981$	13.6710	0.436480
$982$	68.9737	2.20104
$983$	−34.4710	−1.09945	−0.549727	−	0.835345i	$-0.685268\pi$
$983$	−34.4710	−1.09945	−0.549727	+	0.835345i	$0.685268\pi$
$984$	11.2819	0.359654
$985$	−15.9482	−0.508151
$986$	−106.086	−3.37845
$987$	0.362849	0.0115496
$988$	0	0
$989$	−15.1585	−0.482012
$990$	5.85579	0.186109
$991$	16.3811	0.520363	0.260181	−	0.965560i	$-0.416218\pi$
$991$	16.3811	0.520363	0.260181	+	0.965560i	$0.416218\pi$
$992$	19.5358	0.620263
$993$	−36.0905	−1.14530
$994$	0.522110	0.0165603
$995$	−17.9111	−0.567821
$996$	−29.7657	−0.943163
$997$	−16.1378	−0.511089	−0.255544	−	0.966797i	$-0.582255\pi$
$997$	−16.1378	−0.511089	−0.255544	+	0.966797i	$0.582255\pi$
$998$	−83.6655	−2.64838
$999$	7.82352	0.247525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.bc.1.2		8
13.6	odd	12		273.2.bd.b.127.7	yes	16
13.11	odd	12		273.2.bd.b.43.7	✓	16
13.12	even	2		3549.2.a.ba.1.7		8
39.11	even	12		819.2.ct.c.316.2		16
39.32	even	12		819.2.ct.c.127.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bd.b.43.7	✓	16	13.11	odd	12
273.2.bd.b.127.7	yes	16	13.6	odd	12
819.2.ct.c.127.2		16	39.32	even	12
819.2.ct.c.316.2		16	39.11	even	12
3549.2.a.ba.1.7		8	13.12	even	2
3549.2.a.bc.1.2		8	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14586 q^{2} -1.00000 q^{3} +2.60470 q^{4} +1.91954 q^{5} +2.14586 q^{6} -1.00000 q^{7} -1.29759 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.14586 q^{2} -1.00000 q^{3} +2.60470 q^{4} +1.91954 q^{5} +2.14586 q^{6} -1.00000 q^{7} -1.29759 q^{8} +1.00000 q^{9} -4.11906 q^{10} -1.42164 q^{11} -2.60470 q^{12} +2.14586 q^{14} -1.91954 q^{15} -2.42495 q^{16} -6.47107 q^{17} -2.14586 q^{18} -5.74461 q^{19} +4.99982 q^{20} +1.00000 q^{21} +3.05062 q^{22} -2.14271 q^{23} +1.29759 q^{24} -1.31537 q^{25} -1.00000 q^{27} -2.60470 q^{28} -7.63976 q^{29} +4.11906 q^{30} +2.50499 q^{31} +7.79877 q^{32} +1.42164 q^{33} +13.8860 q^{34} -1.91954 q^{35} +2.60470 q^{36} -7.82352 q^{37} +12.3271 q^{38} -2.49078 q^{40} +8.69450 q^{41} -2.14586 q^{42} +7.07445 q^{43} -3.70293 q^{44} +1.91954 q^{45} +4.59795 q^{46} +0.362849 q^{47} +2.42495 q^{48} +1.00000 q^{49} +2.82259 q^{50} +6.47107 q^{51} +0.524776 q^{53} +2.14586 q^{54} -2.72888 q^{55} +1.29759 q^{56} +5.74461 q^{57} +16.3938 q^{58} +9.00783 q^{59} -4.99982 q^{60} -7.90247 q^{61} -5.37534 q^{62} -1.00000 q^{63} -11.8851 q^{64} -3.05062 q^{66} +0.405361 q^{67} -16.8552 q^{68} +2.14271 q^{69} +4.11906 q^{70} +0.243311 q^{71} -1.29759 q^{72} -0.330581 q^{73} +16.7882 q^{74} +1.31537 q^{75} -14.9630 q^{76} +1.42164 q^{77} +8.64577 q^{79} -4.65478 q^{80} +1.00000 q^{81} -18.6571 q^{82} +11.4277 q^{83} +2.60470 q^{84} -12.4215 q^{85} -15.1807 q^{86} +7.63976 q^{87} +1.84470 q^{88} +12.6734 q^{89} -4.11906 q^{90} -5.58111 q^{92} -2.50499 q^{93} -0.778622 q^{94} -11.0270 q^{95} -7.79877 q^{96} +17.1484 q^{97} -2.14586 q^{98} -1.42164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 8 * q^3 + 14 * q^4 + 6 * q^5 - 2 * q^6 - 8 * q^7 + 12 * q^8 + 8 * q^9	\( 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9} - 4 q^{10} + 16 q^{11} - 14 q^{12} - 2 q^{14} - 6 q^{15} + 10 q^{16} - 2 q^{17} + 2 q^{18} + 14 q^{19} - 10 q^{20} + 8 q^{21} + 18 q^{22} - 6 q^{23} - 12 q^{24} + 10 q^{25} - 8 q^{27} - 14 q^{28} + 12 q^{29} + 4 q^{30} + 16 q^{31} + 26 q^{32} - 16 q^{33} + 32 q^{34} - 6 q^{35} + 14 q^{36} - 8 q^{37} + 12 q^{38} - 14 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 68 q^{44} + 6 q^{45} - 28 q^{46} + 6 q^{47} - 10 q^{48} + 8 q^{49} + 32 q^{50} + 2 q^{51} + 14 q^{53} - 2 q^{54} - 2 q^{55} - 12 q^{56} - 14 q^{57} + 24 q^{58} + 50 q^{59} + 10 q^{60} - 2 q^{61} - 20 q^{62} - 8 q^{63} + 24 q^{64} - 18 q^{66} - 16 q^{67} - 36 q^{68} + 6 q^{69} + 4 q^{70} + 34 q^{71} + 12 q^{72} - 8 q^{73} - 6 q^{74} - 10 q^{75} - 4 q^{76} - 16 q^{77} + 46 q^{79} - 24 q^{80} + 8 q^{81} - 42 q^{82} + 42 q^{83} + 14 q^{84} + 20 q^{85} + 50 q^{86} - 12 q^{87} + 62 q^{88} + 28 q^{89} - 4 q^{90} - 58 q^{92} - 16 q^{93} + 24 q^{94} - 24 q^{95} - 26 q^{96} + 16 q^{97} + 2 q^{98} + 16 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 8 * q^3 + 14 * q^4 + 6 * q^5 - 2 * q^6 - 8 * q^7 + 12 * q^8 + 8 * q^9 - 4 * q^10 + 16 * q^11 - 14 * q^12 - 2 * q^14 - 6 * q^15 + 10 * q^16 - 2 * q^17 + 2 * q^18 + 14 * q^19 - 10 * q^20 + 8 * q^21 + 18 * q^22 - 6 * q^23 - 12 * q^24 + 10 * q^25 - 8 * q^27 - 14 * q^28 + 12 * q^29 + 4 * q^30 + 16 * q^31 + 26 * q^32 - 16 * q^33 + 32 * q^34 - 6 * q^35 + 14 * q^36 - 8 * q^37 + 12 * q^38 - 14 * q^40 - 4 * q^41 + 2 * q^42 + 14 * q^43 + 68 * q^44 + 6 * q^45 - 28 * q^46 + 6 * q^47 - 10 * q^48 + 8 * q^49 + 32 * q^50 + 2 * q^51 + 14 * q^53 - 2 * q^54 - 2 * q^55 - 12 * q^56 - 14 * q^57 + 24 * q^58 + 50 * q^59 + 10 * q^60 - 2 * q^61 - 20 * q^62 - 8 * q^63 + 24 * q^64 - 18 * q^66 - 16 * q^67 - 36 * q^68 + 6 * q^69 + 4 * q^70 + 34 * q^71 + 12 * q^72 - 8 * q^73 - 6 * q^74 - 10 * q^75 - 4 * q^76 - 16 * q^77 + 46 * q^79 - 24 * q^80 + 8 * q^81 - 42 * q^82 + 42 * q^83 + 14 * q^84 + 20 * q^85 + 50 * q^86 - 12 * q^87 + 62 * q^88 + 28 * q^89 - 4 * q^90 - 58 * q^92 - 16 * q^93 + 24 * q^94 - 24 * q^95 - 26 * q^96 + 16 * q^97 + 2 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more