LMFDB - Embedded newform 3549.2.a.bb.1.8

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:	$1$
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} - 9x^{6} + 14x^{5} + 25x^{4} - 24x^{3} - 16x^{2} + 8x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 25x^4 - 24x^3 - 16x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.8
Root			$-1.98765$ 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)

$f(q)$	$=$	$q+1.98765 q^{2} +1.00000 q^{3} +1.95074 q^{4} -2.85284 q^{5} +1.98765 q^{6} -1.00000 q^{7} -0.0979034 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.98765 q^{2} +1.00000 q^{3} +1.95074 q^{4} -2.85284 q^{5} +1.98765 q^{6} -1.00000 q^{7} -0.0979034 q^{8} +1.00000 q^{9} -5.67044 q^{10} +2.30669 q^{11} +1.95074 q^{12} -1.98765 q^{14} -2.85284 q^{15} -4.09609 q^{16} +2.52716 q^{17} +1.98765 q^{18} -4.60490 q^{19} -5.56516 q^{20} -1.00000 q^{21} +4.58489 q^{22} -4.10346 q^{23} -0.0979034 q^{24} +3.13870 q^{25} +1.00000 q^{27} -1.95074 q^{28} -2.68077 q^{29} -5.67044 q^{30} +0.917134 q^{31} -7.94577 q^{32} +2.30669 q^{33} +5.02309 q^{34} +2.85284 q^{35} +1.95074 q^{36} -5.93718 q^{37} -9.15293 q^{38} +0.279303 q^{40} -4.15216 q^{41} -1.98765 q^{42} -10.6622 q^{43} +4.49977 q^{44} -2.85284 q^{45} -8.15624 q^{46} -0.601438 q^{47} -4.09609 q^{48} +1.00000 q^{49} +6.23863 q^{50} +2.52716 q^{51} +1.41822 q^{53} +1.98765 q^{54} -6.58062 q^{55} +0.0979034 q^{56} -4.60490 q^{57} -5.32842 q^{58} -4.87711 q^{59} -5.56516 q^{60} +13.4491 q^{61} +1.82294 q^{62} -1.00000 q^{63} -7.60122 q^{64} +4.58489 q^{66} -14.2925 q^{67} +4.92983 q^{68} -4.10346 q^{69} +5.67044 q^{70} -12.3058 q^{71} -0.0979034 q^{72} +10.8999 q^{73} -11.8010 q^{74} +3.13870 q^{75} -8.98299 q^{76} -2.30669 q^{77} +6.67402 q^{79} +11.6855 q^{80} +1.00000 q^{81} -8.25302 q^{82} -17.4987 q^{83} -1.95074 q^{84} -7.20957 q^{85} -21.1926 q^{86} -2.68077 q^{87} -0.225833 q^{88} +10.8187 q^{89} -5.67044 q^{90} -8.00480 q^{92} +0.917134 q^{93} -1.19545 q^{94} +13.1371 q^{95} -7.94577 q^{96} -5.15334 q^{97} +1.98765 q^{98} +2.30669 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 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 + 6 * q^4 - 2 * q^5 - 2 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} - 4 q^{10} - 8 q^{11} + 6 q^{12} + 2 q^{14} - 2 q^{15} + 10 q^{16} + 10 q^{17} - 2 q^{18} - 18 q^{19} - 2 q^{20} - 8 q^{21} + 2 q^{22} - 2 q^{23} - 12 q^{24} - 6 q^{25} + 8 q^{27} - 6 q^{28} - 12 q^{29} - 4 q^{30} - 16 q^{31} - 26 q^{32} - 8 q^{33} - 24 q^{34} + 2 q^{35} + 6 q^{36} - 24 q^{37} + 16 q^{38} - 30 q^{40} + 4 q^{41} + 2 q^{42} - 10 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 10 q^{47} + 10 q^{48} + 8 q^{49} + 16 q^{50} + 10 q^{51} + 6 q^{53} - 2 q^{54} - 10 q^{55} + 12 q^{56} - 18 q^{57} - 16 q^{58} - 6 q^{59} - 2 q^{60} + 6 q^{61} + 16 q^{62} - 8 q^{63} - 8 q^{64} + 2 q^{66} - 24 q^{67} + 20 q^{68} - 2 q^{69} + 4 q^{70} - 42 q^{71} - 12 q^{72} - 32 q^{73} - 18 q^{74} - 6 q^{75} - 28 q^{76} + 8 q^{77} - 2 q^{79} + 40 q^{80} + 8 q^{81} - 18 q^{82} + 2 q^{83} - 6 q^{84} - 4 q^{85} - 26 q^{86} - 12 q^{87} - 2 q^{88} - 12 q^{89} - 4 q^{90} - 10 q^{92} - 16 q^{93} - 16 q^{94} - 4 q^{95} - 26 q^{96} - 64 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 - 4 * q^10 - 8 * q^11 + 6 * q^12 + 2 * q^14 - 2 * q^15 + 10 * q^16 + 10 * q^17 - 2 * q^18 - 18 * q^19 - 2 * q^20 - 8 * q^21 + 2 * q^22 - 2 * q^23 - 12 * q^24 - 6 * q^25 + 8 * q^27 - 6 * q^28 - 12 * q^29 - 4 * q^30 - 16 * q^31 - 26 * q^32 - 8 * q^33 - 24 * q^34 + 2 * q^35 + 6 * q^36 - 24 * q^37 + 16 * q^38 - 30 * q^40 + 4 * q^41 + 2 * q^42 - 10 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 10 * q^47 + 10 * q^48 + 8 * q^49 + 16 * q^50 + 10 * q^51 + 6 * q^53 - 2 * q^54 - 10 * q^55 + 12 * q^56 - 18 * q^57 - 16 * q^58 - 6 * q^59 - 2 * q^60 + 6 * q^61 + 16 * q^62 - 8 * q^63 - 8 * q^64 + 2 * q^66 - 24 * q^67 + 20 * q^68 - 2 * q^69 + 4 * q^70 - 42 * q^71 - 12 * q^72 - 32 * q^73 - 18 * q^74 - 6 * q^75 - 28 * q^76 + 8 * q^77 - 2 * q^79 + 40 * q^80 + 8 * q^81 - 18 * q^82 + 2 * q^83 - 6 * q^84 - 4 * q^85 - 26 * q^86 - 12 * q^87 - 2 * q^88 - 12 * q^89 - 4 * q^90 - 10 * q^92 - 16 * q^93 - 16 * q^94 - 4 * q^95 - 26 * q^96 - 64 * q^97 - 2 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.98765	1.40548	0.702740	−	0.711447i	$-0.251960\pi$
$2$	1.98765	1.40548	0.702740	+	0.711447i	$0.251960\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.95074	0.975372
$5$	−2.85284	−1.27583	−0.637915	−	0.770107i	$-0.720203\pi$
$5$	−2.85284	−1.27583	−0.637915	+	0.770107i	$0.720203\pi$
$6$	1.98765	0.811454
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.0979034	−0.0346141
$9$	1.00000	0.333333
$10$	−5.67044	−1.79315
$11$	2.30669	0.695494	0.347747	−	0.937588i	$-0.386947\pi$
$11$	2.30669	0.695494	0.347747	+	0.937588i	$0.386947\pi$
$12$	1.95074	0.563131
$13$	0	0
$13$	0	0
$14$	−1.98765	−0.531221
$15$	−2.85284	−0.736600
$16$	−4.09609	−1.02402
$17$	2.52716	0.612925	0.306463	−	0.951883i	$-0.400855\pi$
$17$	2.52716	0.612925	0.306463	+	0.951883i	$0.400855\pi$
$18$	1.98765	0.468493
$19$	−4.60490	−1.05644	−0.528219	−	0.849108i	$-0.677140\pi$
$19$	−4.60490	−1.05644	−0.528219	+	0.849108i	$0.677140\pi$
$20$	−5.56516	−1.24441
$21$	−1.00000	−0.218218
$22$	4.58489	0.977502
$23$	−4.10346	−0.855631	−0.427815	−	0.903866i	$-0.640717\pi$
$23$	−4.10346	−0.855631	−0.427815	+	0.903866i	$0.640717\pi$
$24$	−0.0979034	−0.0199844
$25$	3.13870	0.627740
$26$	0	0
$27$	1.00000	0.192450
$28$	−1.95074	−0.368656
$29$	−2.68077	−0.497806	−0.248903	−	0.968528i	$-0.580070\pi$
$29$	−2.68077	−0.497806	−0.248903	+	0.968528i	$0.580070\pi$
$30$	−5.67044	−1.03528
$31$	0.917134	0.164722	0.0823610	−	0.996603i	$-0.473754\pi$
$31$	0.917134	0.164722	0.0823610	+	0.996603i	$0.473754\pi$
$32$	−7.94577	−1.40463
$33$	2.30669	0.401544
$34$	5.02309	0.861454
$35$	2.85284	0.482218
$36$	1.95074	0.325124
$37$	−5.93718	−0.976067	−0.488034	−	0.872825i	$-0.662286\pi$
$37$	−5.93718	−0.976067	−0.488034	+	0.872825i	$0.662286\pi$
$38$	−9.15293	−1.48480
$39$	0	0
$40$	0.279303	0.0441616
$41$	−4.15216	−0.648458	−0.324229	−	0.945979i	$-0.605105\pi$
$41$	−4.15216	−0.648458	−0.324229	+	0.945979i	$0.605105\pi$
$42$	−1.98765	−0.306701
$43$	−10.6622	−1.62597	−0.812983	−	0.582287i	$-0.802158\pi$
$43$	−10.6622	−1.62597	−0.812983	+	0.582287i	$0.802158\pi$
$44$	4.49977	0.678365
$45$	−2.85284	−0.425276
$46$	−8.15624	−1.20257
$47$	−0.601438	−0.0877287	−0.0438643	−	0.999037i	$-0.513967\pi$
$47$	−0.601438	−0.0877287	−0.0438643	+	0.999037i	$0.513967\pi$
$48$	−4.09609	−0.591219
$49$	1.00000	0.142857
$50$	6.23863	0.882276
$51$	2.52716	0.353872
$52$	0	0
$53$	1.41822	0.194808	0.0974038	−	0.995245i	$-0.468946\pi$
$53$	1.41822	0.194808	0.0974038	+	0.995245i	$0.468946\pi$
$54$	1.98765	0.270485
$55$	−6.58062	−0.887331
$56$	0.0979034	0.0130829
$57$	−4.60490	−0.609934
$58$	−5.32842	−0.699656
$59$	−4.87711	−0.634945	−0.317473	−	0.948267i	$-0.602834\pi$
$59$	−4.87711	−0.634945	−0.317473	+	0.948267i	$0.602834\pi$
$60$	−5.56516	−0.718459
$61$	13.4491	1.72198	0.860989	−	0.508623i	$-0.169845\pi$
$61$	13.4491	1.72198	0.860989	+	0.508623i	$0.169845\pi$
$62$	1.82294	0.231513
$63$	−1.00000	−0.125988
$64$	−7.60122	−0.950152
$65$	0	0
$66$	4.58489	0.564361
$67$	−14.2925	−1.74610	−0.873052	−	0.487627i	$-0.837863\pi$
$67$	−14.2925	−1.74610	−0.873052	+	0.487627i	$0.837863\pi$
$68$	4.92983	0.597830
$69$	−4.10346	−0.493999
$70$	5.67044	0.677748
$71$	−12.3058	−1.46043	−0.730217	−	0.683215i	$-0.760581\pi$
$71$	−12.3058	−1.46043	−0.730217	+	0.683215i	$0.760581\pi$
$72$	−0.0979034	−0.0115380
$73$	10.8999	1.27573	0.637867	−	0.770146i	$-0.279817\pi$
$73$	10.8999	1.27573	0.637867	+	0.770146i	$0.279817\pi$
$74$	−11.8010	−1.37184
$75$	3.13870	0.362426
$76$	−8.98299	−1.03042
$77$	−2.30669	−0.262872
$78$	0	0
$79$	6.67402	0.750886	0.375443	−	0.926846i	$-0.377491\pi$
$79$	6.67402	0.750886	0.375443	+	0.926846i	$0.377491\pi$
$80$	11.6855	1.30648
$81$	1.00000	0.111111
$82$	−8.25302	−0.911394
$83$	−17.4987	−1.92073	−0.960366	−	0.278742i	$-0.910083\pi$
$83$	−17.4987	−1.92073	−0.960366	+	0.278742i	$0.910083\pi$
$84$	−1.95074	−0.212844
$85$	−7.20957	−0.781988
$86$	−21.1926	−2.28526
$87$	−2.68077	−0.287408
$88$	−0.225833	−0.0240739
$89$	10.8187	1.14678	0.573388	−	0.819284i	$-0.305629\pi$
$89$	10.8187	1.14678	0.573388	+	0.819284i	$0.305629\pi$
$90$	−5.67044	−0.597717
$91$	0	0
$92$	−8.00480	−0.834558
$93$	0.917134	0.0951023
$94$	−1.19545	−0.123301
$95$	13.1371	1.34783
$96$	−7.94577	−0.810962
$97$	−5.15334	−0.523243	−0.261621	−	0.965171i	$-0.584257\pi$
$97$	−5.15334	−0.523243	−0.261621	+	0.965171i	$0.584257\pi$
$98$	1.98765	0.200783
$99$	2.30669	0.231831
$100$	6.12280	0.612280
$101$	−11.8259	−1.17672	−0.588358	−	0.808600i	$-0.700225\pi$
$101$	−11.8259	−1.17672	−0.588358	+	0.808600i	$0.700225\pi$
$102$	5.02309	0.497360
$103$	18.1000	1.78345	0.891725	−	0.452578i	$-0.149496\pi$
$103$	18.1000	1.78345	0.891725	+	0.452578i	$0.149496\pi$
$104$	0	0
$105$	2.85284	0.278409
$106$	2.81892	0.273798
$107$	−3.56666	−0.344802	−0.172401	−	0.985027i	$-0.555152\pi$
$107$	−3.56666	−0.344802	−0.172401	+	0.985027i	$0.555152\pi$
$108$	1.95074	0.187710
$109$	−11.5102	−1.10248	−0.551238	−	0.834348i	$-0.685845\pi$
$109$	−11.5102	−1.10248	−0.551238	+	0.834348i	$0.685845\pi$
$110$	−13.0800	−1.24713
$111$	−5.93718	−0.563533
$112$	4.09609	0.387044
$113$	−2.05199	−0.193035	−0.0965176	−	0.995331i	$-0.530770\pi$
$113$	−2.05199	−0.193035	−0.0965176	+	0.995331i	$0.530770\pi$
$114$	−9.15293	−0.857250
$115$	11.7065	1.09164
$116$	−5.22949	−0.485546
$117$	0	0
$118$	−9.69397	−0.892403
$119$	−2.52716	−0.231664
$120$	0.279303	0.0254967
$121$	−5.67917	−0.516288
$122$	26.7320	2.42021
$123$	−4.15216	−0.374387
$124$	1.78909	0.160665
$125$	5.30999	0.474940
$126$	−1.98765	−0.177074
$127$	18.1578	1.61125	0.805623	−	0.592428i	$-0.201830\pi$
$127$	18.1578	1.61125	0.805623	+	0.592428i	$0.201830\pi$
$128$	0.782990	0.0692072
$129$	−10.6622	−0.938752
$130$	0	0
$131$	1.96602	0.171772	0.0858860	−	0.996305i	$-0.472628\pi$
$131$	1.96602	0.171772	0.0858860	+	0.996305i	$0.472628\pi$
$132$	4.49977	0.391654
$133$	4.60490	0.399296
$134$	−28.4084	−2.45411
$135$	−2.85284	−0.245533
$136$	−0.247417	−0.0212158
$137$	12.7013	1.08515	0.542574	−	0.840008i	$-0.317450\pi$
$137$	12.7013	1.08515	0.542574	+	0.840008i	$0.317450\pi$
$138$	−8.15624	−0.694305
$139$	−1.97930	−0.167882	−0.0839409	−	0.996471i	$-0.526751\pi$
$139$	−1.97930	−0.167882	−0.0839409	+	0.996471i	$0.526751\pi$
$140$	5.56516	0.470342
$141$	−0.601438	−0.0506502
$142$	−24.4597	−2.05261
$143$	0	0
$144$	−4.09609	−0.341340
$145$	7.64780	0.635116
$146$	21.6651	1.79302
$147$	1.00000	0.0824786
$148$	−11.5819	−0.952029
$149$	23.5264	1.92736	0.963678	−	0.267065i	$-0.0860539\pi$
$149$	23.5264	1.92736	0.963678	+	0.267065i	$0.0860539\pi$
$150$	6.23863	0.509382
$151$	−11.7828	−0.958872	−0.479436	−	0.877577i	$-0.659159\pi$
$151$	−11.7828	−0.958872	−0.479436	+	0.877577i	$0.659159\pi$
$152$	0.450836	0.0365676
$153$	2.52716	0.204308
$154$	−4.58489	−0.369461
$155$	−2.61644	−0.210157
$156$	0	0
$157$	−2.26601	−0.180848	−0.0904238	−	0.995903i	$-0.528822\pi$
$157$	−2.26601	−0.180848	−0.0904238	+	0.995903i	$0.528822\pi$
$158$	13.2656	1.05535
$159$	1.41822	0.112472
$160$	22.6680	1.79206
$161$	4.10346	0.323398
$162$	1.98765	0.156164
$163$	−17.3070	−1.35559	−0.677794	−	0.735252i	$-0.737064\pi$
$163$	−17.3070	−1.35559	−0.677794	+	0.735252i	$0.737064\pi$
$164$	−8.09979	−0.632488
$165$	−6.58062	−0.512301
$166$	−34.7813	−2.69955
$167$	14.8071	1.14581	0.572906	−	0.819621i	$-0.305816\pi$
$167$	14.8071	1.14581	0.572906	+	0.819621i	$0.305816\pi$
$168$	0.0979034	0.00755341
$169$	0	0
$170$	−14.3301	−1.09907
$171$	−4.60490	−0.352146
$172$	−20.7992	−1.58592
$173$	1.35021	0.102654	0.0513272	−	0.998682i	$-0.483655\pi$
$173$	1.35021	0.102654	0.0513272	+	0.998682i	$0.483655\pi$
$174$	−5.32842	−0.403947
$175$	−3.13870	−0.237263
$176$	−9.44841	−0.712201
$177$	−4.87711	−0.366586
$178$	21.5037	1.61177
$179$	−4.83204	−0.361164	−0.180582	−	0.983560i	$-0.557798\pi$
$179$	−4.83204	−0.361164	−0.180582	+	0.983560i	$0.557798\pi$
$180$	−5.56516	−0.414803
$181$	−17.8164	−1.32429	−0.662143	−	0.749378i	$-0.730353\pi$
$181$	−17.8164	−1.32429	−0.662143	+	0.749378i	$0.730353\pi$
$182$	0	0
$183$	13.4491	0.994185
$184$	0.401743	0.0296169
$185$	16.9378	1.24529
$186$	1.82294	0.133664
$187$	5.82937	0.426286
$188$	−1.17325	−0.0855681
$189$	−1.00000	−0.0727393
$190$	26.1118	1.89435
$191$	0.232424	0.0168176	0.00840879	−	0.999965i	$-0.497323\pi$
$191$	0.232424	0.0168176	0.00840879	+	0.999965i	$0.497323\pi$
$192$	−7.60122	−0.548571
$193$	−25.7953	−1.85678	−0.928392	−	0.371603i	$-0.878808\pi$
$193$	−25.7953	−1.85678	−0.928392	+	0.371603i	$0.878808\pi$
$194$	−10.2430	−0.735407
$195$	0	0
$196$	1.95074	0.139339
$197$	12.5949	0.897349	0.448674	−	0.893695i	$-0.351896\pi$
$197$	12.5949	0.897349	0.448674	+	0.893695i	$0.351896\pi$
$198$	4.58489	0.325834
$199$	−1.21688	−0.0862622	−0.0431311	−	0.999069i	$-0.513733\pi$
$199$	−1.21688	−0.0862622	−0.0431311	+	0.999069i	$0.513733\pi$
$200$	−0.307289	−0.0217286
$201$	−14.2925	−1.00811
$202$	−23.5056	−1.65385
$203$	2.68077	0.188153
$204$	4.92983	0.345157
$205$	11.8454	0.827321
$206$	35.9765	2.50660
$207$	−4.10346	−0.285210
$208$	0	0
$209$	−10.6221	−0.734746
$210$	5.67044	0.391298
$211$	1.99492	0.137336	0.0686681	−	0.997640i	$-0.478125\pi$
$211$	1.99492	0.137336	0.0686681	+	0.997640i	$0.478125\pi$
$212$	2.76659	0.190010
$213$	−12.3058	−0.843182
$214$	−7.08926	−0.484612
$215$	30.4175	2.07445
$216$	−0.0979034	−0.00666148
$217$	−0.917134	−0.0622591
$218$	−22.8782	−1.54951
$219$	10.8999	0.736546
$220$	−12.8371	−0.865478
$221$	0	0
$222$	−11.8010	−0.792033
$223$	28.6981	1.92177	0.960884	−	0.276950i	$-0.0893236\pi$
$223$	28.6981	1.92177	0.960884	+	0.276950i	$0.0893236\pi$
$224$	7.94577	0.530899
$225$	3.13870	0.209247
$226$	−4.07864	−0.271307
$227$	−8.32271	−0.552398	−0.276199	−	0.961101i	$-0.589075\pi$
$227$	−8.32271	−0.552398	−0.276199	+	0.961101i	$0.589075\pi$
$228$	−8.98299	−0.594913
$229$	1.24569	0.0823173	0.0411587	−	0.999153i	$-0.486895\pi$
$229$	1.24569	0.0823173	0.0411587	+	0.999153i	$0.486895\pi$
$230$	23.2684	1.53428
$231$	−2.30669	−0.151769
$232$	0.262456	0.0172311
$233$	9.67623	0.633911	0.316955	−	0.948440i	$-0.397339\pi$
$233$	9.67623	0.633911	0.316955	+	0.948440i	$0.397339\pi$
$234$	0	0
$235$	1.71581	0.111927
$236$	−9.51399	−0.619308
$237$	6.67402	0.433524
$238$	−5.02309	−0.325599
$239$	7.45095	0.481962	0.240981	−	0.970530i	$-0.422531\pi$
$239$	7.45095	0.481962	0.240981	+	0.970530i	$0.422531\pi$
$240$	11.6855	0.754294
$241$	−11.8394	−0.762643	−0.381321	−	0.924443i	$-0.624531\pi$
$241$	−11.8394	−0.762643	−0.381321	+	0.924443i	$0.624531\pi$
$242$	−11.2882	−0.725633
$243$	1.00000	0.0641500
$244$	26.2357	1.67957
$245$	−2.85284	−0.182261
$246$	−8.25302	−0.526194
$247$	0	0
$248$	−0.0897905	−0.00570170
$249$	−17.4987	−1.10894
$250$	10.5544	0.667519
$251$	−26.9359	−1.70018	−0.850091	−	0.526636i	$-0.823453\pi$
$251$	−26.9359	−1.70018	−0.850091	+	0.526636i	$0.823453\pi$
$252$	−1.95074	−0.122885
$253$	−9.46542	−0.595086
$254$	36.0914	2.26457
$255$	−7.20957	−0.451481
$256$	16.7587	1.04742
$257$	22.2392	1.38724	0.693621	−	0.720340i	$-0.256014\pi$
$257$	22.2392	1.38724	0.693621	+	0.720340i	$0.256014\pi$
$258$	−21.1926	−1.31940
$259$	5.93718	0.368919
$260$	0	0
$261$	−2.68077	−0.165935
$262$	3.90776	0.241422
$263$	4.90060	0.302184	0.151092	−	0.988520i	$-0.451721\pi$
$263$	4.90060	0.302184	0.151092	+	0.988520i	$0.451721\pi$
$264$	−0.225833	−0.0138991
$265$	−4.04596	−0.248541
$266$	9.15293	0.561202
$267$	10.8187	0.662091
$268$	−27.8810	−1.70310
$269$	27.4096	1.67119	0.835597	−	0.549343i	$-0.185122\pi$
$269$	27.4096	1.67119	0.835597	+	0.549343i	$0.185122\pi$
$270$	−5.67044	−0.345092
$271$	−20.4387	−1.24156	−0.620781	−	0.783984i	$-0.713184\pi$
$271$	−20.4387	−1.24156	−0.620781	+	0.783984i	$0.713184\pi$
$272$	−10.3514	−0.627648
$273$	0	0
$274$	25.2458	1.52515
$275$	7.24001	0.436589
$276$	−8.00480	−0.481833
$277$	−3.29054	−0.197710	−0.0988548	−	0.995102i	$-0.531518\pi$
$277$	−3.29054	−0.197710	−0.0988548	+	0.995102i	$0.531518\pi$
$278$	−3.93414	−0.235954
$279$	0.917134	0.0549074
$280$	−0.279303	−0.0166915
$281$	23.7022	1.41395	0.706977	−	0.707236i	$-0.250058\pi$
$281$	23.7022	1.41395	0.706977	+	0.707236i	$0.250058\pi$
$282$	−1.19545	−0.0711878
$283$	6.39122	0.379918	0.189959	−	0.981792i	$-0.439164\pi$
$283$	6.39122	0.379918	0.189959	+	0.981792i	$0.439164\pi$
$284$	−24.0056	−1.42447
$285$	13.1371	0.778172
$286$	0	0
$287$	4.15216	0.245094
$288$	−7.94577	−0.468209
$289$	−10.6135	−0.624323
$290$	15.2011	0.892642
$291$	−5.15334	−0.302094
$292$	21.2629	1.24432
$293$	22.0242	1.28667	0.643333	−	0.765586i	$-0.277551\pi$
$293$	22.0242	1.28667	0.643333	+	0.765586i	$0.277551\pi$
$294$	1.98765	0.115922
$295$	13.9136	0.810082
$296$	0.581270	0.0337857
$297$	2.30669	0.133848
$298$	46.7622	2.70886
$299$	0	0
$300$	6.12280	0.353500
$301$	10.6622	0.614557
$302$	−23.4201	−1.34768
$303$	−11.8259	−0.679377
$304$	18.8621	1.08181
$305$	−38.3681	−2.19695
$306$	5.02309	0.287151
$307$	−10.7195	−0.611797	−0.305898	−	0.952064i	$-0.598957\pi$
$307$	−10.7195	−0.611797	−0.305898	+	0.952064i	$0.598957\pi$
$308$	−4.49977	−0.256398
$309$	18.1000	1.02968
$310$	−5.20055	−0.295372
$311$	8.16438	0.462960	0.231480	−	0.972840i	$-0.425643\pi$
$311$	8.16438	0.462960	0.231480	+	0.972840i	$0.425643\pi$
$312$	0	0
$313$	−12.6727	−0.716304	−0.358152	−	0.933663i	$-0.616593\pi$
$313$	−12.6727	−0.716304	−0.358152	+	0.933663i	$0.616593\pi$
$314$	−4.50404	−0.254178
$315$	2.85284	0.160739
$316$	13.0193	0.732393
$317$	1.58156	0.0888291	0.0444146	−	0.999013i	$-0.485858\pi$
$317$	1.58156	0.0888291	0.0444146	+	0.999013i	$0.485858\pi$
$318$	2.81892	0.158077
$319$	−6.18371	−0.346221
$320$	21.6851	1.21223
$321$	−3.56666	−0.199071
$322$	8.15624	0.454529
$323$	−11.6373	−0.647517
$324$	1.95074	0.108375
$325$	0	0
$326$	−34.4002	−1.90525
$327$	−11.5102	−0.636515
$328$	0.406510	0.0224458
$329$	0.601438	0.0331583
$330$	−13.0800	−0.720028
$331$	24.4371	1.34318	0.671592	−	0.740921i	$-0.265611\pi$
$331$	24.4371	1.34318	0.671592	+	0.740921i	$0.265611\pi$
$332$	−34.1355	−1.87343
$333$	−5.93718	−0.325356
$334$	29.4314	1.61041
$335$	40.7742	2.22773
$336$	4.09609	0.223460
$337$	14.8194	0.807266	0.403633	−	0.914921i	$-0.367747\pi$
$337$	14.8194	0.807266	0.403633	+	0.914921i	$0.367747\pi$
$338$	0	0
$339$	−2.05199	−0.111449
$340$	−14.0640	−0.762729
$341$	2.11555	0.114563
$342$	−9.15293	−0.494934
$343$	−1.00000	−0.0539949
$344$	1.04386	0.0562813
$345$	11.7065	0.630258
$346$	2.68374	0.144279
$347$	−18.3751	−0.986428	−0.493214	−	0.869908i	$-0.664178\pi$
$347$	−18.3751	−0.986428	−0.493214	+	0.869908i	$0.664178\pi$
$348$	−5.22949	−0.280330
$349$	−11.1932	−0.599159	−0.299579	−	0.954071i	$-0.596846\pi$
$349$	−11.1932	−0.599159	−0.299579	+	0.954071i	$0.596846\pi$
$350$	−6.23863	−0.333469
$351$	0	0
$352$	−18.3284	−0.976909
$353$	20.3373	1.08244	0.541222	−	0.840880i	$-0.317962\pi$
$353$	20.3373	1.08244	0.541222	+	0.840880i	$0.317962\pi$
$354$	−9.69397	−0.515229
$355$	35.1066	1.86327
$356$	21.1044	1.11853
$357$	−2.52716	−0.133751
$358$	−9.60439	−0.507608
$359$	5.14966	0.271789	0.135894	−	0.990723i	$-0.456609\pi$
$359$	5.14966	0.271789	0.135894	+	0.990723i	$0.456609\pi$
$360$	0.279303	0.0147205
$361$	2.20514	0.116060
$362$	−35.4128	−1.86126
$363$	−5.67917	−0.298079
$364$	0	0
$365$	−31.0956	−1.62762
$366$	26.7320	1.39731
$367$	15.2906	0.798162	0.399081	−	0.916916i	$-0.369329\pi$
$367$	15.2906	0.798162	0.399081	+	0.916916i	$0.369329\pi$
$368$	16.8081	0.876184
$369$	−4.15216	−0.216153
$370$	33.6665	1.75024
$371$	−1.41822	−0.0736303
$372$	1.78909	0.0927602
$373$	−17.0855	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$373$	−17.0855	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$374$	11.5867	0.599136
$375$	5.30999	0.274207
$376$	0.0588828	0.00303665
$377$	0	0
$378$	−1.98765	−0.102234
$379$	−4.42300	−0.227194	−0.113597	−	0.993527i	$-0.536237\pi$
$379$	−4.42300	−0.227194	−0.113597	+	0.993527i	$0.536237\pi$
$380$	25.6270	1.31464
$381$	18.1578	0.930254
$382$	0.461976	0.0236368
$383$	−4.14725	−0.211915	−0.105957	−	0.994371i	$-0.533791\pi$
$383$	−4.14725	−0.211915	−0.105957	+	0.994371i	$0.533791\pi$
$384$	0.782990	0.0399568
$385$	6.58062	0.335380
$386$	−51.2719	−2.60967
$387$	−10.6622	−0.541989
$388$	−10.0529	−0.510356
$389$	34.7497	1.76188	0.880940	−	0.473228i	$-0.156911\pi$
$389$	34.7497	1.76188	0.880940	+	0.473228i	$0.156911\pi$
$390$	0	0
$391$	−10.3701	−0.524438
$392$	−0.0979034	−0.00494487
$393$	1.96602	0.0991726
$394$	25.0342	1.26121
$395$	−19.0399	−0.958002
$396$	4.49977	0.226122
$397$	8.30449	0.416790	0.208395	−	0.978045i	$-0.433176\pi$
$397$	8.30449	0.416790	0.208395	+	0.978045i	$0.433176\pi$
$398$	−2.41872	−0.121240
$399$	4.60490	0.230534
$400$	−12.8564	−0.642819
$401$	−35.5214	−1.77386	−0.886928	−	0.461908i	$-0.847165\pi$
$401$	−35.5214	−1.77386	−0.886928	+	0.461908i	$0.847165\pi$
$402$	−28.4084	−1.41688
$403$	0	0
$404$	−23.0692	−1.14774
$405$	−2.85284	−0.141759
$406$	5.32842	0.264445
$407$	−13.6953	−0.678849
$408$	−0.247417	−0.0122490
$409$	1.14005	0.0563716	0.0281858	−	0.999603i	$-0.491027\pi$
$409$	1.14005	0.0563716	0.0281858	+	0.999603i	$0.491027\pi$
$410$	23.5446	1.16278
$411$	12.7013	0.626511
$412$	35.3085	1.73953
$413$	4.87711	0.239987
$414$	−8.15624	−0.400857
$415$	49.9210	2.45053
$416$	0	0
$417$	−1.97930	−0.0969265
$418$	−21.1130	−1.03267
$419$	30.5236	1.49117	0.745587	−	0.666408i	$-0.232169\pi$
$419$	30.5236	1.49117	0.745587	+	0.666408i	$0.232169\pi$
$420$	5.56516	0.271552
$421$	15.4138	0.751223	0.375611	−	0.926777i	$-0.377433\pi$
$421$	15.4138	0.751223	0.375611	+	0.926777i	$0.377433\pi$
$422$	3.96520	0.193023
$423$	−0.601438	−0.0292429
$424$	−0.138849	−0.00674308
$425$	7.93198	0.384758
$426$	−24.4597	−1.18508
$427$	−13.4491	−0.650847
$428$	−6.95763	−0.336310
$429$	0	0
$430$	60.4592	2.91560
$431$	−13.6121	−0.655671	−0.327835	−	0.944735i	$-0.606319\pi$
$431$	−13.6121	−0.655671	−0.327835	+	0.944735i	$0.606319\pi$
$432$	−4.09609	−0.197073
$433$	−23.6374	−1.13594	−0.567969	−	0.823050i	$-0.692271\pi$
$433$	−23.6374	−1.13594	−0.567969	+	0.823050i	$0.692271\pi$
$434$	−1.82294	−0.0875039
$435$	7.64780	0.366684
$436$	−22.4534	−1.07533
$437$	18.8960	0.903920
$438$	21.6651	1.03520
$439$	−31.6718	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$439$	−31.6718	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$440$	0.644265	0.0307142
$441$	1.00000	0.0476190
$442$	0	0
$443$	10.5318	0.500379	0.250190	−	0.968197i	$-0.419507\pi$
$443$	10.5318	0.500379	0.250190	+	0.968197i	$0.419507\pi$
$444$	−11.5819	−0.549654
$445$	−30.8639	−1.46309
$446$	57.0418	2.70101
$447$	23.5264	1.11276
$448$	7.60122	0.359124
$449$	−14.9564	−0.705835	−0.352917	−	0.935654i	$-0.614810\pi$
$449$	−14.9564	−0.705835	−0.352917	+	0.935654i	$0.614810\pi$
$450$	6.23863	0.294092
$451$	−9.57774	−0.450998
$452$	−4.00291	−0.188281
$453$	−11.7828	−0.553605
$454$	−16.5426	−0.776384
$455$	0	0
$456$	0.450836	0.0211123
$457$	13.1863	0.616828	0.308414	−	0.951252i	$-0.400202\pi$
$457$	13.1863	0.616828	0.308414	+	0.951252i	$0.400202\pi$
$458$	2.47599	0.115695
$459$	2.52716	0.117957
$460$	22.8364	1.06475
$461$	4.78894	0.223043	0.111522	−	0.993762i	$-0.464428\pi$
$461$	4.78894	0.223043	0.111522	+	0.993762i	$0.464428\pi$
$462$	−4.58489	−0.213308
$463$	−19.4696	−0.904827	−0.452414	−	0.891808i	$-0.649437\pi$
$463$	−19.4696	−0.904827	−0.452414	+	0.891808i	$0.649437\pi$
$464$	10.9807	0.509764
$465$	−2.61644	−0.121334
$466$	19.2329	0.890949
$467$	7.28924	0.337306	0.168653	−	0.985675i	$-0.446058\pi$
$467$	7.28924	0.337306	0.168653	+	0.985675i	$0.446058\pi$
$468$	0	0
$469$	14.2925	0.659965
$470$	3.41042	0.157311
$471$	−2.26601	−0.104412
$472$	0.477485	0.0219780
$473$	−24.5943	−1.13085
$474$	13.2656	0.609309
$475$	−14.4534	−0.663168
$476$	−4.92983	−0.225959
$477$	1.41822	0.0649359
$478$	14.8099	0.677387
$479$	12.3287	0.563314	0.281657	−	0.959515i	$-0.409116\pi$
$479$	12.3287	0.563314	0.281657	+	0.959515i	$0.409116\pi$
$480$	22.6680	1.03465
$481$	0	0
$482$	−23.5326	−1.07188
$483$	4.10346	0.186714
$484$	−11.0786	−0.503573
$485$	14.7017	0.667568
$486$	1.98765	0.0901615
$487$	−14.1979	−0.643367	−0.321683	−	0.946847i	$-0.604249\pi$
$487$	−14.1979	−0.643367	−0.321683	+	0.946847i	$0.604249\pi$
$488$	−1.31671	−0.0596047
$489$	−17.3070	−0.782649
$490$	−5.67044	−0.256164
$491$	0.571501	0.0257915	0.0128957	−	0.999917i	$-0.495895\pi$
$491$	0.571501	0.0257915	0.0128957	+	0.999917i	$0.495895\pi$
$492$	−8.09979	−0.365167
$493$	−6.77472	−0.305118
$494$	0	0
$495$	−6.58062	−0.295777
$496$	−3.75666	−0.168679
$497$	12.3058	0.551993
$498$	−34.7813	−1.55859
$499$	−39.7298	−1.77855	−0.889274	−	0.457375i	$-0.848790\pi$
$499$	−39.7298	−1.77855	−0.889274	+	0.457375i	$0.848790\pi$
$500$	10.3584	0.463243
$501$	14.8071	0.661534
$502$	−53.5392	−2.38957
$503$	−32.8475	−1.46460	−0.732299	−	0.680984i	$-0.761552\pi$
$503$	−32.8475	−1.46460	−0.732299	+	0.680984i	$0.761552\pi$
$504$	0.0979034	0.00436096
$505$	33.7373	1.50129
$506$	−18.8139	−0.836381
$507$	0	0
$508$	35.4213	1.57157
$509$	−38.4199	−1.70293	−0.851467	−	0.524409i	$-0.824286\pi$
$509$	−38.4199	−1.70293	−0.851467	+	0.524409i	$0.824286\pi$
$510$	−14.3301	−0.634547
$511$	−10.8999	−0.482182
$512$	31.7445	1.40292
$513$	−4.60490	−0.203311
$514$	44.2036	1.94974
$515$	−51.6365	−2.27538
$516$	−20.7992	−0.915632
$517$	−1.38733	−0.0610148
$518$	11.8010	0.518508
$519$	1.35021	0.0592675
$520$	0	0
$521$	−9.38277	−0.411067	−0.205533	−	0.978650i	$-0.565893\pi$
$521$	−9.38277	−0.411067	−0.205533	+	0.978650i	$0.565893\pi$
$522$	−5.32842	−0.233219
$523$	−2.30465	−0.100776	−0.0503878	−	0.998730i	$-0.516046\pi$
$523$	−2.30465	−0.100776	−0.0503878	+	0.998730i	$0.516046\pi$
$524$	3.83520	0.167542
$525$	−3.13870	−0.136984
$526$	9.74066	0.424713
$527$	2.31774	0.100962
$528$	−9.44841	−0.411189
$529$	−6.16160	−0.267896
$530$	−8.04194	−0.349319
$531$	−4.87711	−0.211648
$532$	8.98299	0.389462
$533$	0	0
$534$	21.5037	0.930555
$535$	10.1751	0.439908
$536$	1.39928	0.0604398
$537$	−4.83204	−0.208518
$538$	54.4807	2.34883
$539$	2.30669	0.0993563
$540$	−5.56516	−0.239486
$541$	−6.63606	−0.285307	−0.142653	−	0.989773i	$-0.545563\pi$
$541$	−6.63606	−0.285307	−0.142653	+	0.989773i	$0.545563\pi$
$542$	−40.6249	−1.74499
$543$	−17.8164	−0.764577
$544$	−20.0802	−0.860931
$545$	32.8368	1.40657
$546$	0	0
$547$	4.37197	0.186932	0.0934660	−	0.995622i	$-0.470205\pi$
$547$	4.37197	0.186932	0.0934660	+	0.995622i	$0.470205\pi$
$548$	24.7771	1.05842
$549$	13.4491	0.573993
$550$	14.3906	0.613617
$551$	12.3447	0.525901
$552$	0.401743	0.0170993
$553$	−6.67402	−0.283808
$554$	−6.54044	−0.277877
$555$	16.9378	0.718971
$556$	−3.86110	−0.163747
$557$	25.9498	1.09953	0.549763	−	0.835320i	$-0.314718\pi$
$557$	25.9498	1.09953	0.549763	+	0.835320i	$0.314718\pi$
$558$	1.82294	0.0771712
$559$	0	0
$560$	−11.6855	−0.493802
$561$	5.82937	0.246116
$562$	47.1116	1.98728
$563$	−25.0126	−1.05416	−0.527078	−	0.849817i	$-0.676712\pi$
$563$	−25.0126	−1.05416	−0.527078	+	0.849817i	$0.676712\pi$
$564$	−1.17325	−0.0494028
$565$	5.85401	0.246280
$566$	12.7035	0.533968
$567$	−1.00000	−0.0419961
$568$	1.20478	0.0505516
$569$	17.4806	0.732825	0.366413	−	0.930452i	$-0.380586\pi$
$569$	17.4806	0.732825	0.366413	+	0.930452i	$0.380586\pi$
$570$	26.1118	1.09370
$571$	36.0126	1.50708	0.753539	−	0.657403i	$-0.228345\pi$
$571$	36.0126	1.50708	0.753539	+	0.657403i	$0.228345\pi$
$572$	0	0
$573$	0.232424	0.00970964
$574$	8.25302	0.344475
$575$	−12.8795	−0.537114
$576$	−7.60122	−0.316717
$577$	−25.7297	−1.07114	−0.535571	−	0.844491i	$-0.679903\pi$
$577$	−25.7297	−1.07114	−0.535571	+	0.844491i	$0.679903\pi$
$578$	−21.0959	−0.877473
$579$	−25.7953	−1.07201
$580$	14.9189	0.619474
$581$	17.4987	0.725969
$582$	−10.2430	−0.424587
$583$	3.27140	0.135487
$584$	−1.06714	−0.0441584
$585$	0	0
$586$	43.7763	1.80838
$587$	8.54575	0.352721	0.176361	−	0.984326i	$-0.443568\pi$
$587$	8.54575	0.352721	0.176361	+	0.984326i	$0.443568\pi$
$588$	1.95074	0.0804473
$589$	−4.22331	−0.174019
$590$	27.6554	1.13855
$591$	12.5949	0.518085
$592$	24.3192	0.999514
$593$	3.34043	0.137175	0.0685875	−	0.997645i	$-0.478151\pi$
$593$	3.34043	0.137175	0.0685875	+	0.997645i	$0.478151\pi$
$594$	4.58489	0.188120
$595$	7.20957	0.295564
$596$	45.8940	1.87989
$597$	−1.21688	−0.0498035
$598$	0	0
$599$	41.7075	1.70412	0.852062	−	0.523442i	$-0.175352\pi$
$599$	41.7075	1.70412	0.852062	+	0.523442i	$0.175352\pi$
$600$	−0.307289	−0.0125450
$601$	−0.602487	−0.0245760	−0.0122880	−	0.999924i	$-0.503911\pi$
$601$	−0.602487	−0.0245760	−0.0122880	+	0.999924i	$0.503911\pi$
$602$	21.1926	0.863748
$603$	−14.2925	−0.582035
$604$	−22.9853	−0.935257
$605$	16.2018	0.658696
$606$	−23.5056	−0.954851
$607$	−16.5620	−0.672232	−0.336116	−	0.941821i	$-0.609113\pi$
$607$	−16.5620	−0.672232	−0.336116	+	0.941821i	$0.609113\pi$
$608$	36.5895	1.48390
$609$	2.68077	0.108630
$610$	−76.2623	−3.08777
$611$	0	0
$612$	4.92983	0.199277
$613$	−1.90176	−0.0768115	−0.0384058	−	0.999262i	$-0.512228\pi$
$613$	−1.90176	−0.0768115	−0.0384058	+	0.999262i	$0.512228\pi$
$614$	−21.3067	−0.859868
$615$	11.8454	0.477654
$616$	0.225833	0.00909907
$617$	−18.1500	−0.730691	−0.365346	−	0.930872i	$-0.619049\pi$
$617$	−18.1500	−0.730691	−0.365346	+	0.930872i	$0.619049\pi$
$618$	35.9765	1.44719
$619$	10.0824	0.405247	0.202624	−	0.979257i	$-0.435053\pi$
$619$	10.0824	0.405247	0.202624	+	0.979257i	$0.435053\pi$
$620$	−5.10400	−0.204981
$621$	−4.10346	−0.164666
$622$	16.2279	0.650680
$623$	−10.8187	−0.433440
$624$	0	0
$625$	−30.8421	−1.23368
$626$	−25.1889	−1.00675
$627$	−10.6221	−0.424206
$628$	−4.42041	−0.176394
$629$	−15.0042	−0.598256
$630$	5.67044	0.225916
$631$	−31.5745	−1.25696	−0.628480	−	0.777826i	$-0.716322\pi$
$631$	−31.5745	−1.25696	−0.628480	+	0.777826i	$0.716322\pi$
$632$	−0.653409	−0.0259912
$633$	1.99492	0.0792910
$634$	3.14358	0.124848
$635$	−51.8014	−2.05568
$636$	2.76659	0.109702
$637$	0	0
$638$	−12.2910	−0.486607
$639$	−12.3058	−0.486812
$640$	−2.23374	−0.0882965
$641$	−21.7409	−0.858715	−0.429358	−	0.903135i	$-0.641260\pi$
$641$	−21.7409	−0.858715	−0.429358	+	0.903135i	$0.641260\pi$
$642$	−7.08926	−0.279791
$643$	34.3989	1.35656	0.678280	−	0.734803i	$-0.262725\pi$
$643$	34.3989	1.35656	0.678280	+	0.734803i	$0.262725\pi$
$644$	8.00480	0.315433
$645$	30.4175	1.19769
$646$	−23.1309	−0.910072
$647$	19.7962	0.778271	0.389135	−	0.921181i	$-0.372774\pi$
$647$	19.7962	0.778271	0.389135	+	0.921181i	$0.372774\pi$
$648$	−0.0979034	−0.00384601
$649$	−11.2500	−0.441601
$650$	0	0
$651$	−0.917134	−0.0359453
$652$	−33.7615	−1.32220
$653$	−3.67495	−0.143812	−0.0719059	−	0.997411i	$-0.522908\pi$
$653$	−3.67495	−0.143812	−0.0719059	+	0.997411i	$0.522908\pi$
$654$	−22.8782	−0.894609
$655$	−5.60874	−0.219152
$656$	17.0076	0.664035
$657$	10.8999	0.425245
$658$	1.19545	0.0466033
$659$	−42.6815	−1.66263	−0.831317	−	0.555798i	$-0.812413\pi$
$659$	−42.6815	−1.66263	−0.831317	+	0.555798i	$0.812413\pi$
$660$	−12.8371	−0.499684
$661$	0.825195	0.0320964	0.0160482	−	0.999871i	$-0.494891\pi$
$661$	0.825195	0.0320964	0.0160482	+	0.999871i	$0.494891\pi$
$662$	48.5723	1.88782
$663$	0	0
$664$	1.71318	0.0664844
$665$	−13.1371	−0.509433
$666$	−11.8010	−0.457281
$667$	11.0004	0.425938
$668$	28.8849	1.11759
$669$	28.6981	1.10953
$670$	81.0447	3.13103
$671$	31.0229	1.19763
$672$	7.94577	0.306515
$673$	−26.0367	−1.00364	−0.501820	−	0.864972i	$-0.667336\pi$
$673$	−26.0367	−1.00364	−0.501820	+	0.864972i	$0.667336\pi$
$674$	29.4558	1.13460
$675$	3.13870	0.120809
$676$	0	0
$677$	−50.7000	−1.94856	−0.974280	−	0.225342i	$-0.927650\pi$
$677$	−50.7000	−1.94856	−0.974280	+	0.225342i	$0.927650\pi$
$678$	−4.07864	−0.156639
$679$	5.15334	0.197767
$680$	0.705841	0.0270678
$681$	−8.32271	−0.318927
$682$	4.20496	0.161016
$683$	−10.5653	−0.404268	−0.202134	−	0.979358i	$-0.564788\pi$
$683$	−10.5653	−0.404268	−0.202134	+	0.979358i	$0.564788\pi$
$684$	−8.98299	−0.343473
$685$	−36.2349	−1.38446
$686$	−1.98765	−0.0758887
$687$	1.24569	0.0475259
$688$	43.6732	1.66502
$689$	0	0
$690$	23.2684	0.885815
$691$	51.7106	1.96716	0.983582	−	0.180459i	$-0.0577584\pi$
$691$	51.7106	1.96716	0.983582	+	0.180459i	$0.0577584\pi$
$692$	2.63391	0.100126
$693$	−2.30669	−0.0876240
$694$	−36.5233	−1.38640
$695$	5.64662	0.214188
$696$	0.262456	0.00994838
$697$	−10.4931	−0.397456
$698$	−22.2482	−0.842105
$699$	9.67623	0.365989
$700$	−6.12280	−0.231420
$701$	−47.4231	−1.79115	−0.895573	−	0.444915i	$-0.853234\pi$
$701$	−47.4231	−1.79115	−0.895573	+	0.444915i	$0.853234\pi$
$702$	0	0
$703$	27.3402	1.03115
$704$	−17.5337	−0.660825
$705$	1.71581	0.0646210
$706$	40.4233	1.52135
$707$	11.8259	0.444757
$708$	−9.51399	−0.357558
$709$	−19.3154	−0.725405	−0.362703	−	0.931905i	$-0.618146\pi$
$709$	−19.3154	−0.725405	−0.362703	+	0.931905i	$0.618146\pi$
$710$	69.7796	2.61878
$711$	6.67402	0.250295
$712$	−1.05918	−0.0396946
$713$	−3.76342	−0.140941
$714$	−5.02309	−0.187985
$715$	0	0
$716$	−9.42607	−0.352269
$717$	7.45095	0.278261
$718$	10.2357	0.381993
$719$	−14.5996	−0.544475	−0.272237	−	0.962230i	$-0.587764\pi$
$719$	−14.5996	−0.544475	−0.272237	+	0.962230i	$0.587764\pi$
$720$	11.6855	0.435492
$721$	−18.1000	−0.674081
$722$	4.38304	0.163120
$723$	−11.8394	−0.440312
$724$	−34.7553	−1.29167
$725$	−8.41413	−0.312493
$726$	−11.2882	−0.418944
$727$	−32.4357	−1.20297	−0.601487	−	0.798883i	$-0.705425\pi$
$727$	−32.4357	−1.20297	−0.601487	+	0.798883i	$0.705425\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−61.8071	−2.28758
$731$	−26.9450	−0.996595
$732$	26.2357	0.969700
$733$	33.4061	1.23388	0.616941	−	0.787009i	$-0.288372\pi$
$733$	33.4061	1.23388	0.616941	+	0.787009i	$0.288372\pi$
$734$	30.3923	1.12180
$735$	−2.85284	−0.105229
$736$	32.6052	1.20184
$737$	−32.9683	−1.21440
$738$	−8.25302	−0.303798
$739$	−41.6612	−1.53253	−0.766266	−	0.642523i	$-0.777888\pi$
$739$	−41.6612	−1.53253	−0.766266	+	0.642523i	$0.777888\pi$
$740$	33.0414	1.21463
$741$	0	0
$742$	−2.81892	−0.103486
$743$	2.62302	0.0962293	0.0481146	−	0.998842i	$-0.484679\pi$
$743$	2.62302	0.0962293	0.0481146	+	0.998842i	$0.484679\pi$
$744$	−0.0897905	−0.00329188
$745$	−67.1170	−2.45898
$746$	−33.9599	−1.24336
$747$	−17.4987	−0.640244
$748$	11.3716	0.415787
$749$	3.56666	0.130323
$750$	10.5544	0.385392
$751$	23.1087	0.843250	0.421625	−	0.906770i	$-0.361460\pi$
$751$	23.1087	0.843250	0.421625	+	0.906770i	$0.361460\pi$
$752$	2.46354	0.0898361
$753$	−26.9359	−0.981600
$754$	0	0
$755$	33.6145	1.22336
$756$	−1.95074	−0.0709479
$757$	−4.85108	−0.176316	−0.0881578	−	0.996107i	$-0.528098\pi$
$757$	−4.85108	−0.176316	−0.0881578	+	0.996107i	$0.528098\pi$
$758$	−8.79137	−0.319317
$759$	−9.46542	−0.343573
$760$	−1.28616	−0.0466540
$761$	−31.5172	−1.14250	−0.571248	−	0.820777i	$-0.693541\pi$
$761$	−31.5172	−1.14250	−0.571248	+	0.820777i	$0.693541\pi$
$762$	36.0914	1.30745
$763$	11.5102	0.416697
$764$	0.453399	0.0164034
$765$	−7.20957	−0.260663
$766$	−8.24328	−0.297842
$767$	0	0
$768$	16.7587	0.604729
$769$	−16.9004	−0.609442	−0.304721	−	0.952442i	$-0.598563\pi$
$769$	−16.9004	−0.609442	−0.304721	+	0.952442i	$0.598563\pi$
$770$	13.0800	0.471369
$771$	22.2392	0.800924
$772$	−50.3199	−1.81105
$773$	−26.5638	−0.955433	−0.477717	−	0.878514i	$-0.658535\pi$
$773$	−26.5638	−0.955433	−0.477717	+	0.878514i	$0.658535\pi$
$774$	−21.1926	−0.761754
$775$	2.87861	0.103403
$776$	0.504530	0.0181116
$777$	5.93718	0.212995
$778$	69.0702	2.47629
$779$	19.1203	0.685055
$780$	0	0
$781$	−28.3858	−1.01572
$782$	−20.6121	−0.737086
$783$	−2.68077	−0.0958028
$784$	−4.09609	−0.146289
$785$	6.46458	0.230731
$786$	3.90776	0.139385
$787$	−20.5315	−0.731868	−0.365934	−	0.930641i	$-0.619250\pi$
$787$	−20.5315	−0.731868	−0.365934	+	0.930641i	$0.619250\pi$
$788$	24.5694	0.875249
$789$	4.90060	0.174466
$790$	−37.8446	−1.34645
$791$	2.05199	0.0729605
$792$	−0.225833	−0.00802463
$793$	0	0
$794$	16.5064	0.585790
$795$	−4.04596	−0.143495
$796$	−2.37382	−0.0841377
$797$	−46.8622	−1.65994	−0.829971	−	0.557806i	$-0.811643\pi$
$797$	−46.8622	−1.65994	−0.829971	+	0.557806i	$0.811643\pi$
$798$	9.15293	0.324010
$799$	−1.51993	−0.0537711
$800$	−24.9394	−0.881740
$801$	10.8187	0.382258
$802$	−70.6041	−2.49312
$803$	25.1427	0.887265
$804$	−27.8810	−0.983286
$805$	−11.7065	−0.412601
$806$	0	0
$807$	27.4096	0.964864
$808$	1.15779	0.0407309
$809$	−40.5552	−1.42584	−0.712922	−	0.701243i	$-0.752629\pi$
$809$	−40.5552	−1.42584	−0.712922	+	0.701243i	$0.752629\pi$
$810$	−5.67044	−0.199239
$811$	−15.8829	−0.557724	−0.278862	−	0.960331i	$-0.589957\pi$
$811$	−15.8829	−0.557724	−0.278862	+	0.960331i	$0.589957\pi$
$812$	5.22949	0.183519
$813$	−20.4387	−0.716816
$814$	−27.2213	−0.954108
$815$	49.3741	1.72950
$816$	−10.3514	−0.362373
$817$	49.0983	1.71773
$818$	2.26601	0.0792292
$819$	0	0
$820$	23.1074	0.806946
$821$	−44.5658	−1.55536	−0.777678	−	0.628663i	$-0.783603\pi$
$821$	−44.5658	−1.55536	−0.777678	+	0.628663i	$0.783603\pi$
$822$	25.2458	0.880548
$823$	−28.1024	−0.979590	−0.489795	−	0.871838i	$-0.662928\pi$
$823$	−28.1024	−0.979590	−0.489795	+	0.871838i	$0.662928\pi$
$824$	−1.77205	−0.0617325
$825$	7.24001	0.252065
$826$	9.69397	0.337296
$827$	41.0346	1.42691	0.713457	−	0.700699i	$-0.247128\pi$
$827$	41.0346	1.42691	0.713457	+	0.700699i	$0.247128\pi$
$828$	−8.00480	−0.278186
$829$	10.2276	0.355220	0.177610	−	0.984101i	$-0.443163\pi$
$829$	10.2276	0.355220	0.177610	+	0.984101i	$0.443163\pi$
$830$	99.2254	3.44416
$831$	−3.29054	−0.114148
$832$	0	0
$833$	2.52716	0.0875607
$834$	−3.93414	−0.136228
$835$	−42.2424	−1.46186
$836$	−20.7210	−0.716650
$837$	0.917134	0.0317008
$838$	60.6701	2.09581
$839$	34.4244	1.18846	0.594230	−	0.804295i	$-0.297457\pi$
$839$	34.4244	1.18846	0.594230	+	0.804295i	$0.297457\pi$
$840$	−0.279303	−0.00963686
$841$	−21.8135	−0.752189
$842$	30.6372	1.05583
$843$	23.7022	0.816347
$844$	3.89158	0.133954
$845$	0	0
$846$	−1.19545	−0.0411003
$847$	5.67917	0.195139
$848$	−5.80915	−0.199487
$849$	6.39122	0.219346
$850$	15.7660	0.540769
$851$	24.3630	0.835153
$852$	−24.0056	−0.822417
$853$	19.9376	0.682650	0.341325	−	0.939945i	$-0.389124\pi$
$853$	19.9376	0.682650	0.341325	+	0.939945i	$0.389124\pi$
$854$	−26.7320	−0.914752
$855$	13.1371	0.449278
$856$	0.349188	0.0119350
$857$	5.68966	0.194355	0.0971776	−	0.995267i	$-0.469019\pi$
$857$	5.68966	0.194355	0.0971776	+	0.995267i	$0.469019\pi$
$858$	0	0
$859$	−17.3268	−0.591182	−0.295591	−	0.955315i	$-0.595517\pi$
$859$	−17.3268	−0.591182	−0.295591	+	0.955315i	$0.595517\pi$
$860$	59.3367	2.02337
$861$	4.15216	0.141505
$862$	−27.0560	−0.921532
$863$	53.7906	1.83105	0.915527	−	0.402257i	$-0.131774\pi$
$863$	53.7906	1.83105	0.915527	+	0.402257i	$0.131774\pi$
$864$	−7.94577	−0.270321
$865$	−3.85193	−0.130969
$866$	−46.9827	−1.59654
$867$	−10.6135	−0.360453
$868$	−1.78909	−0.0607258
$869$	15.3949	0.522236
$870$	15.2011	0.515367
$871$	0	0
$872$	1.12689	0.0381612
$873$	−5.15334	−0.174414
$874$	37.5587	1.27044
$875$	−5.30999	−0.179510
$876$	21.2629	0.718406
$877$	−47.8839	−1.61692	−0.808462	−	0.588548i	$-0.799700\pi$
$877$	−47.8839	−1.61692	−0.808462	+	0.588548i	$0.799700\pi$
$878$	−62.9524	−2.12454
$879$	22.0242	0.742857
$880$	26.9548	0.908646
$881$	−13.2800	−0.447416	−0.223708	−	0.974656i	$-0.571816\pi$
$881$	−13.2800	−0.447416	−0.223708	+	0.974656i	$0.571816\pi$
$882$	1.98765	0.0669276
$883$	−15.0727	−0.507236	−0.253618	−	0.967304i	$-0.581621\pi$
$883$	−15.0727	−0.507236	−0.253618	+	0.967304i	$0.581621\pi$
$884$	0	0
$885$	13.9136	0.467701
$886$	20.9334	0.703272
$887$	0.827796	0.0277947	0.0138973	−	0.999903i	$-0.495576\pi$
$887$	0.827796	0.0277947	0.0138973	+	0.999903i	$0.495576\pi$
$888$	0.581270	0.0195062
$889$	−18.1578	−0.608994
$890$	−61.3466	−2.05634
$891$	2.30669	0.0772771
$892$	55.9827	1.87444
$893$	2.76956	0.0926799
$894$	46.7622	1.56396
$895$	13.7850	0.460783
$896$	−0.782990	−0.0261578
$897$	0	0
$898$	−29.7280	−0.992036
$899$	−2.45862	−0.0819997
$900$	6.12280	0.204093
$901$	3.58406	0.119402
$902$	−19.0372	−0.633869
$903$	10.6622	0.354815
$904$	0.200897	0.00668174
$905$	50.8275	1.68956
$906$	−23.4201	−0.778081
$907$	−8.86006	−0.294193	−0.147097	−	0.989122i	$-0.546993\pi$
$907$	−8.86006	−0.294193	−0.147097	+	0.989122i	$0.546993\pi$
$908$	−16.2355	−0.538793
$909$	−11.8259	−0.392239
$910$	0	0
$911$	56.3640	1.86742	0.933711	−	0.358027i	$-0.116550\pi$
$911$	56.3640	1.86742	0.933711	+	0.358027i	$0.116550\pi$
$912$	18.8621	0.624586
$913$	−40.3641	−1.33586
$914$	26.2097	0.866939
$915$	−38.3681	−1.26841
$916$	2.43002	0.0802900
$917$	−1.96602	−0.0649237
$918$	5.02309	0.165787
$919$	−15.4469	−0.509546	−0.254773	−	0.967001i	$-0.582001\pi$
$919$	−15.4469	−0.509546	−0.254773	+	0.967001i	$0.582001\pi$
$920$	−1.14611	−0.0377861
$921$	−10.7195	−0.353221
$922$	9.51872	0.313482
$923$	0	0
$924$	−4.49977	−0.148031
$925$	−18.6350	−0.612716
$926$	−38.6986	−1.27172
$927$	18.1000	0.594483
$928$	21.3008	0.699232
$929$	7.24619	0.237740	0.118870	−	0.992910i	$-0.462073\pi$
$929$	7.24619	0.237740	0.118870	+	0.992910i	$0.462073\pi$
$930$	−5.20055	−0.170533
$931$	−4.60490	−0.150920
$932$	18.8758	0.618299
$933$	8.16438	0.267290
$934$	14.4885	0.474077
$935$	−16.6303	−0.543868
$936$	0	0
$937$	10.6692	0.348548	0.174274	−	0.984697i	$-0.444242\pi$
$937$	10.6692	0.348548	0.174274	+	0.984697i	$0.444242\pi$
$938$	28.4084	0.927567
$939$	−12.6727	−0.413559
$940$	3.34710	0.109170
$941$	34.4575	1.12328	0.561642	−	0.827381i	$-0.310170\pi$
$941$	34.4575	1.12328	0.561642	+	0.827381i	$0.310170\pi$
$942$	−4.50404	−0.146750
$943$	17.0382	0.554840
$944$	19.9771	0.650198
$945$	2.85284	0.0928029
$946$	−48.8849	−1.58939
$947$	−8.37245	−0.272068	−0.136034	−	0.990704i	$-0.543436\pi$
$947$	−8.37245	−0.272068	−0.136034	+	0.990704i	$0.543436\pi$
$948$	13.0193	0.422847
$949$	0	0
$950$	−28.7283	−0.932069
$951$	1.58156	0.0512855
$952$	0.247417	0.00801883
$953$	−18.4158	−0.596547	−0.298273	−	0.954481i	$-0.596411\pi$
$953$	−18.4158	−0.596547	−0.298273	+	0.954481i	$0.596411\pi$
$954$	2.81892	0.0912660
$955$	−0.663068	−0.0214564
$956$	14.5349	0.470092
$957$	−6.18371	−0.199891
$958$	24.5052	0.791726
$959$	−12.7013	−0.410148
$960$	21.6851	0.699883
$961$	−30.1589	−0.972867
$962$	0	0
$963$	−3.56666	−0.114934
$964$	−23.0956	−0.743860
$965$	73.5898	2.36894
$966$	8.15624	0.262423
$967$	24.6033	0.791189	0.395595	−	0.918425i	$-0.370539\pi$
$967$	24.6033	0.791189	0.395595	+	0.918425i	$0.370539\pi$
$968$	0.556010	0.0178708
$969$	−11.6373	−0.373844
$970$	29.2217	0.938254
$971$	−36.7326	−1.17881	−0.589403	−	0.807839i	$-0.700637\pi$
$971$	−36.7326	−1.17881	−0.589403	+	0.807839i	$0.700637\pi$
$972$	1.95074	0.0625701
$973$	1.97930	0.0634533
$974$	−28.2204	−0.904238
$975$	0	0
$976$	−55.0886	−1.76334
$977$	39.9740	1.27888	0.639442	−	0.768840i	$-0.279166\pi$
$977$	39.9740	1.27888	0.639442	+	0.768840i	$0.279166\pi$
$978$	−34.4002	−1.10000
$979$	24.9553	0.797575
$980$	−5.56516	−0.177773
$981$	−11.5102	−0.367492
$982$	1.13594	0.0362494
$983$	16.6132	0.529878	0.264939	−	0.964265i	$-0.414648\pi$
$983$	16.6132	0.529878	0.264939	+	0.964265i	$0.414648\pi$
$984$	0.406510	0.0129591
$985$	−35.9312	−1.14486
$986$	−13.4657	−0.428837
$987$	0.601438	0.0191440
$988$	0	0
$989$	43.7518	1.39123
$990$	−13.0800	−0.415709
$991$	−26.3358	−0.836584	−0.418292	−	0.908313i	$-0.637371\pi$
$991$	−26.3358	−0.836584	−0.418292	+	0.908313i	$0.637371\pi$
$992$	−7.28733	−0.231373
$993$	24.4371	0.775487
$994$	24.4597	0.775814
$995$	3.47156	0.110056
$996$	−34.1355	−1.08162
$997$	−25.6135	−0.811189	−0.405595	−	0.914053i	$-0.632935\pi$
$997$	−25.6135	−0.811189	−0.405595	+	0.914053i	$0.632935\pi$
$998$	−78.9688	−2.49971
$999$	−5.93718	−0.187844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.bb.1.8		8
13.2	odd	12		273.2.bd.a.43.7	✓	16
13.7	odd	12		273.2.bd.a.127.7	yes	16
13.12	even	2		3549.2.a.bd.1.1		8
39.2	even	12		819.2.ct.b.316.2		16
39.20	even	12		819.2.ct.b.127.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bd.a.43.7	✓	16	13.2	odd	12
273.2.bd.a.127.7	yes	16	13.7	odd	12
819.2.ct.b.127.2		16	39.20	even	12
819.2.ct.b.316.2		16	39.2	even	12
3549.2.a.bb.1.8		8	1.1	even	1	trivial
3549.2.a.bd.1.1		8	13.12	even	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98765 q^{2} +1.00000 q^{3} +1.95074 q^{4} -2.85284 q^{5} +1.98765 q^{6} -1.00000 q^{7} -0.0979034 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.98765 q^{2} +1.00000 q^{3} +1.95074 q^{4} -2.85284 q^{5} +1.98765 q^{6} -1.00000 q^{7} -0.0979034 q^{8} +1.00000 q^{9} -5.67044 q^{10} +2.30669 q^{11} +1.95074 q^{12} -1.98765 q^{14} -2.85284 q^{15} -4.09609 q^{16} +2.52716 q^{17} +1.98765 q^{18} -4.60490 q^{19} -5.56516 q^{20} -1.00000 q^{21} +4.58489 q^{22} -4.10346 q^{23} -0.0979034 q^{24} +3.13870 q^{25} +1.00000 q^{27} -1.95074 q^{28} -2.68077 q^{29} -5.67044 q^{30} +0.917134 q^{31} -7.94577 q^{32} +2.30669 q^{33} +5.02309 q^{34} +2.85284 q^{35} +1.95074 q^{36} -5.93718 q^{37} -9.15293 q^{38} +0.279303 q^{40} -4.15216 q^{41} -1.98765 q^{42} -10.6622 q^{43} +4.49977 q^{44} -2.85284 q^{45} -8.15624 q^{46} -0.601438 q^{47} -4.09609 q^{48} +1.00000 q^{49} +6.23863 q^{50} +2.52716 q^{51} +1.41822 q^{53} +1.98765 q^{54} -6.58062 q^{55} +0.0979034 q^{56} -4.60490 q^{57} -5.32842 q^{58} -4.87711 q^{59} -5.56516 q^{60} +13.4491 q^{61} +1.82294 q^{62} -1.00000 q^{63} -7.60122 q^{64} +4.58489 q^{66} -14.2925 q^{67} +4.92983 q^{68} -4.10346 q^{69} +5.67044 q^{70} -12.3058 q^{71} -0.0979034 q^{72} +10.8999 q^{73} -11.8010 q^{74} +3.13870 q^{75} -8.98299 q^{76} -2.30669 q^{77} +6.67402 q^{79} +11.6855 q^{80} +1.00000 q^{81} -8.25302 q^{82} -17.4987 q^{83} -1.95074 q^{84} -7.20957 q^{85} -21.1926 q^{86} -2.68077 q^{87} -0.225833 q^{88} +10.8187 q^{89} -5.67044 q^{90} -8.00480 q^{92} +0.917134 q^{93} -1.19545 q^{94} +13.1371 q^{95} -7.94577 q^{96} -5.15334 q^{97} +1.98765 q^{98} +2.30669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 8 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} + 8 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} - 4 q^{10} - 8 q^{11} + 6 q^{12} + 2 q^{14} - 2 q^{15} + 10 q^{16} + 10 q^{17} - 2 q^{18} - 18 q^{19} - 2 q^{20} - 8 q^{21} + 2 q^{22} - 2 q^{23} - 12 q^{24} - 6 q^{25} + 8 q^{27} - 6 q^{28} - 12 q^{29} - 4 q^{30} - 16 q^{31} - 26 q^{32} - 8 q^{33} - 24 q^{34} + 2 q^{35} + 6 q^{36} - 24 q^{37} + 16 q^{38} - 30 q^{40} + 4 q^{41} + 2 q^{42} - 10 q^{43} + 20 q^{44} - 2 q^{45} - 12 q^{46} - 10 q^{47} + 10 q^{48} + 8 q^{49} + 16 q^{50} + 10 q^{51} + 6 q^{53} - 2 q^{54} - 10 q^{55} + 12 q^{56} - 18 q^{57} - 16 q^{58} - 6 q^{59} - 2 q^{60} + 6 q^{61} + 16 q^{62} - 8 q^{63} - 8 q^{64} + 2 q^{66} - 24 q^{67} + 20 q^{68} - 2 q^{69} + 4 q^{70} - 42 q^{71} - 12 q^{72} - 32 q^{73} - 18 q^{74} - 6 q^{75} - 28 q^{76} + 8 q^{77} - 2 q^{79} + 40 q^{80} + 8 q^{81} - 18 q^{82} + 2 q^{83} - 6 q^{84} - 4 q^{85} - 26 q^{86} - 12 q^{87} - 2 q^{88} - 12 q^{89} - 4 q^{90} - 10 q^{92} - 16 q^{93} - 16 q^{94} - 4 q^{95} - 26 q^{96} - 64 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 - 4 * q^10 - 8 * q^11 + 6 * q^12 + 2 * q^14 - 2 * q^15 + 10 * q^16 + 10 * q^17 - 2 * q^18 - 18 * q^19 - 2 * q^20 - 8 * q^21 + 2 * q^22 - 2 * q^23 - 12 * q^24 - 6 * q^25 + 8 * q^27 - 6 * q^28 - 12 * q^29 - 4 * q^30 - 16 * q^31 - 26 * q^32 - 8 * q^33 - 24 * q^34 + 2 * q^35 + 6 * q^36 - 24 * q^37 + 16 * q^38 - 30 * q^40 + 4 * q^41 + 2 * q^42 - 10 * q^43 + 20 * q^44 - 2 * q^45 - 12 * q^46 - 10 * q^47 + 10 * q^48 + 8 * q^49 + 16 * q^50 + 10 * q^51 + 6 * q^53 - 2 * q^54 - 10 * q^55 + 12 * q^56 - 18 * q^57 - 16 * q^58 - 6 * q^59 - 2 * q^60 + 6 * q^61 + 16 * q^62 - 8 * q^63 - 8 * q^64 + 2 * q^66 - 24 * q^67 + 20 * q^68 - 2 * q^69 + 4 * q^70 - 42 * q^71 - 12 * q^72 - 32 * q^73 - 18 * q^74 - 6 * q^75 - 28 * q^76 + 8 * q^77 - 2 * q^79 + 40 * q^80 + 8 * q^81 - 18 * q^82 + 2 * q^83 - 6 * q^84 - 4 * q^85 - 26 * q^86 - 12 * q^87 - 2 * q^88 - 12 * q^89 - 4 * q^90 - 10 * q^92 - 16 * q^93 - 16 * q^94 - 4 * q^95 - 26 * q^96 - 64 * q^97 - 2 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more