LMFDB - Embedded newform 21.8.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [21,8,Mod(1,21)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.56008553517$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1065}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 266 $ x^2 - x - 266
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$16.8172$ of defining polynomial
Character	$\chi$	$=$	21.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} -3046.45 q^{10} -7491.69 q^{11} -8244.57 q^{12} +8553.34 q^{13} -7140.29 q^{14} -3951.27 q^{15} +37772.0 q^{16} -5219.65 q^{17} -15175.7 q^{18} -47136.9 q^{19} +44686.6 q^{20} -9261.00 q^{21} +155956. q^{22} +23833.8 q^{23} +99684.5 q^{24} -56708.6 q^{25} -178056. q^{26} -19683.0 q^{27} +104737. q^{28} -183928. q^{29} +82254.3 q^{30} -180272. q^{31} -313728. q^{32} +202276. q^{33} +108658. q^{34} +50195.8 q^{35} +222603. q^{36} -404121. q^{37} +981256. q^{38} -230940. q^{39} -540303. q^{40} +632653. q^{41} +192788. q^{42} +443131. q^{43} -2.28762e6 q^{44} +106684. q^{45} -496152. q^{46} -524989. q^{47} -1.01984e6 q^{48} +117649. q^{49} +1.18051e6 q^{50} +140931. q^{51} +2.61180e6 q^{52} +520667. q^{53} +409744. q^{54} -1.09636e6 q^{55} -1.26636e6 q^{56} +1.27270e6 q^{57} +3.82886e6 q^{58} -1.42159e6 q^{59} -1.20654e6 q^{60} -616389. q^{61} +3.75275e6 q^{62} +250047. q^{63} +1.69611e6 q^{64} +1.25173e6 q^{65} -4.21080e6 q^{66} -1.63770e6 q^{67} -1.59384e6 q^{68} -643513. q^{69} -1.04493e6 q^{70} -595589. q^{71} -2.69148e6 q^{72} -4.41186e6 q^{73} +8.41266e6 q^{74} +1.53113e6 q^{75} -1.43935e7 q^{76} -2.56965e6 q^{77} +4.80752e6 q^{78} +2.03908e6 q^{79} +5.52768e6 q^{80} +531441. q^{81} -1.31700e7 q^{82} -1.08171e6 q^{83} -2.82789e6 q^{84} -763862. q^{85} -9.22473e6 q^{86} +4.96605e6 q^{87} +2.76595e7 q^{88} +8.80625e6 q^{89} -2.22087e6 q^{90} +2.93380e6 q^{91} +7.27776e6 q^{92} +4.86734e6 q^{93} +1.09288e7 q^{94} -6.89817e6 q^{95} +8.47065e6 q^{96} +6.13951e6 q^{97} -2.44912e6 q^{98} -5.46144e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9} - 9030 q^{10} - 4932 q^{11} - 8559 q^{12} + 7708 q^{13} - 3087 q^{14} + 9720 q^{15} + 20033 q^{16} - 28584 q^{17}+ \cdots - 3595428 q^{99}+O(q^{100}) $ 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9 - 9030 * q^10 - 4932 * q^11 - 8559 * q^12 + 7708 * q^13 - 3087 * q^14 + 9720 * q^15 + 20033 * q^16 - 28584 * q^17 - 6561 * q^18 - 63728 * q^19 + 38790 * q^20 - 18522 * q^21 + 186204 * q^22 + 82260 * q^23 + 136809 * q^24 + 121550 * q^25 - 188046 * q^26 - 39366 * q^27 + 108731 * q^28 - 435996 * q^29 + 243810 * q^30 - 29240 * q^31 - 347355 * q^32 + 133164 * q^33 - 167442 * q^34 - 123480 * q^35 + 231093 * q^36 - 709556 * q^37 + 785196 * q^38 - 208116 * q^39 + 155910 * q^40 - 25056 * q^41 + 83349 * q^42 + 496216 * q^43 - 2257812 * q^44 - 262440 * q^45 + 194280 * q^46 - 1575000 * q^47 - 540891 * q^48 + 235298 * q^49 + 3287025 * q^50 + 771768 * q^51 + 2601958 * q^52 + 2057436 * q^53 + 177147 * q^54 - 2392440 * q^55 - 1737981 * q^56 + 1720656 * q^57 + 850122 * q^58 - 1101024 * q^59 - 1047330 * q^60 + 28996 * q^61 + 5537520 * q^62 + 500094 * q^63 + 3569321 * q^64 + 1679760 * q^65 - 5027508 * q^66 - 4480784 * q^67 - 1865934 * q^68 - 2221020 * q^69 - 3097290 * q^70 + 54540 * q^71 - 3693843 * q^72 + 666604 * q^73 + 4803282 * q^74 - 3281850 * q^75 - 14586668 * q^76 - 1691676 * q^77 + 5077242 * q^78 + 2322952 * q^79 + 14509710 * q^80 + 1062882 * q^81 - 20942298 * q^82 - 7384392 * q^83 - 2935737 * q^84 + 11066520 * q^85 - 8597412 * q^86 + 11771892 * q^87 + 24139932 * q^88 + 1784448 * q^89 - 6582870 * q^90 + 2643844 * q^91 + 7958160 * q^92 + 789480 * q^93 - 1479360 * q^94 + 1502640 * q^95 + 9378585 * q^96 + 16266412 * q^97 - 1058841 * q^98 - 3595428 * q^99

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−20.8172	−1.84000	−0.919998	−	0.391924i	$-0.871810\pi$
$2$	−20.8172	−1.84000	−0.919998	+	0.391924i	$0.871810\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	305.355	2.38558
$5$	146.343	0.523574	0.261787	−	0.965126i	$-0.415688\pi$
$5$	146.343	0.523574	0.261787	+	0.965126i	$0.415688\pi$
$6$	562.064	1.06232
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−3692.02	−2.54946
$9$	729.000	0.333333
$10$	−3046.45	−0.963374
$11$	−7491.69	−1.69709	−0.848546	−	0.529122i	$-0.822521\pi$
$11$	−7491.69	−1.69709	−0.848546	+	0.529122i	$0.822521\pi$
$12$	−8244.57	−1.37732
$13$	8553.34	1.07978	0.539889	−	0.841736i	$-0.318466\pi$
$13$	8553.34	1.07978	0.539889	+	0.841736i	$0.318466\pi$
$14$	−7140.29	−0.695453
$15$	−3951.27	−0.302286
$16$	37772.0	2.30542
$17$	−5219.65	−0.257674	−0.128837	−	0.991666i	$-0.541124\pi$
$17$	−5219.65	−0.257674	−0.128837	+	0.991666i	$0.541124\pi$
$18$	−15175.7	−0.613332
$19$	−47136.9	−1.57661	−0.788303	−	0.615287i	$-0.789040\pi$
$19$	−47136.9	−1.57661	−0.788303	+	0.615287i	$0.789040\pi$
$20$	44686.6	1.24903
$21$	−9261.00	−0.218218
$22$	155956.	3.12264
$23$	23833.8	0.408457	0.204228	−	0.978923i	$-0.434532\pi$
$23$	23833.8	0.408457	0.204228	+	0.978923i	$0.434532\pi$
$24$	99684.5	1.47193
$25$	−56708.6	−0.725870
$26$	−178056.	−1.98679
$27$	−19683.0	−0.192450
$28$	104737.	0.901665
$29$	−183928.	−1.40041	−0.700203	−	0.713943i	$-0.746907\pi$
$29$	−183928.	−1.40041	−0.700203	+	0.713943i	$0.746907\pi$
$30$	82254.3	0.556204
$31$	−180272.	−1.08683	−0.543416	−	0.839464i	$-0.682869\pi$
$31$	−180272.	−1.08683	−0.543416	+	0.839464i	$0.682869\pi$
$32$	−313728.	−1.69250
$33$	202276.	0.979816
$34$	108658.	0.474119
$35$	50195.8	0.197892
$36$	222603.	0.795194
$37$	−404121.	−1.31161	−0.655806	−	0.754929i	$-0.727671\pi$
$37$	−404121.	−1.31161	−0.655806	+	0.754929i	$0.727671\pi$
$38$	981256.	2.90095
$39$	−230940.	−0.623410
$40$	−540303.	−1.33483
$41$	632653.	1.43358	0.716790	−	0.697289i	$-0.245611\pi$
$41$	632653.	1.43358	0.716790	+	0.697289i	$0.245611\pi$
$42$	192788.	0.401520
$43$	443131.	0.849948	0.424974	−	0.905206i	$-0.360283\pi$
$43$	443131.	0.849948	0.424974	+	0.905206i	$0.360283\pi$
$44$	−2.28762e6	−4.04855
$45$	106684.	0.174525
$46$	−496152.	−0.751558
$47$	−524989.	−0.737578	−0.368789	−	0.929513i	$-0.620228\pi$
$47$	−524989.	−0.737578	−0.368789	+	0.929513i	$0.620228\pi$
$48$	−1.01984e6	−1.33103
$49$	117649.	0.142857
$50$	1.18051e6	1.33560
$51$	140931.	0.148768
$52$	2.61180e6	2.57590
$53$	520667.	0.480390	0.240195	−	0.970725i	$-0.422789\pi$
$53$	520667.	0.480390	0.240195	+	0.970725i	$0.422789\pi$
$54$	409744.	0.354107
$55$	−1.09636e6	−0.888553
$56$	−1.26636e6	−0.963607
$57$	1.27270e6	0.910254
$58$	3.82886e6	2.57674
$59$	−1.42159e6	−0.901139	−0.450569	−	0.892741i	$-0.648779\pi$
$59$	−1.42159e6	−0.901139	−0.450569	+	0.892741i	$0.648779\pi$
$60$	−1.20654e6	−0.721127
$61$	−616389.	−0.347697	−0.173848	−	0.984772i	$-0.555620\pi$
$61$	−616389.	−0.347697	−0.173848	+	0.984772i	$0.555620\pi$
$62$	3.75275e6	1.99976
$63$	250047.	0.125988
$64$	1.69611e6	0.808767
$65$	1.25173e6	0.565343
$66$	−4.21080e6	−1.80286
$67$	−1.63770e6	−0.665232	−0.332616	−	0.943062i	$-0.607931\pi$
$67$	−1.63770e6	−0.665232	−0.332616	+	0.943062i	$0.607931\pi$
$68$	−1.59384e6	−0.614702
$69$	−643513.	−0.235823
$70$	−1.04493e6	−0.364121
$71$	−595589.	−0.197489	−0.0987444	−	0.995113i	$-0.531483\pi$
$71$	−595589.	−0.197489	−0.0987444	+	0.995113i	$0.531483\pi$
$72$	−2.69148e6	−0.849822
$73$	−4.41186e6	−1.32737	−0.663685	−	0.748012i	$-0.731008\pi$
$73$	−4.41186e6	−1.32737	−0.663685	+	0.748012i	$0.731008\pi$
$74$	8.41266e6	2.41336
$75$	1.53113e6	0.419081
$76$	−1.43935e7	−3.76112
$77$	−2.56965e6	−0.641440
$78$	4.80752e6	1.14707
$79$	2.03908e6	0.465307	0.232653	−	0.972560i	$-0.425259\pi$
$79$	2.03908e6	0.465307	0.232653	+	0.972560i	$0.425259\pi$
$80$	5.52768e6	1.20706
$81$	531441.	0.111111
$82$	−1.31700e7	−2.63778
$83$	−1.08171e6	−0.207653	−0.103826	−	0.994595i	$-0.533109\pi$
$83$	−1.08171e6	−0.207653	−0.103826	+	0.994595i	$0.533109\pi$
$84$	−2.82789e6	−0.520577
$85$	−763862.	−0.134911
$86$	−9.22473e6	−1.56390
$87$	4.96605e6	0.808525
$88$	2.76595e7	4.32667
$89$	8.80625e6	1.32412	0.662058	−	0.749453i	$-0.269683\pi$
$89$	8.80625e6	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$90$	−2.22087e6	−0.321125
$91$	2.93380e6	0.408118
$92$	7.27776e6	0.974407
$93$	4.86734e6	0.627482
$94$	1.09288e7	1.35714
$95$	−6.89817e6	−0.825470
$96$	8.47065e6	0.977164
$97$	6.13951e6	0.683019	0.341509	−	0.939878i	$-0.389062\pi$
$97$	6.13951e6	0.683019	0.341509	+	0.939878i	$0.389062\pi$
$98$	−2.44912e6	−0.262856
$99$	−5.46144e6	−0.565697

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.8.a.b.1.1	✓	2
3.2	odd	2		63.8.a.f.1.2		2
4.3	odd	2		336.8.a.n.1.2		2
5.4	even	2		525.8.a.e.1.2		2
7.2	even	3		147.8.e.h.67.2		4
7.3	odd	6		147.8.e.g.79.2		4
7.4	even	3		147.8.e.h.79.2		4
7.5	odd	6		147.8.e.g.67.2		4
7.6	odd	2		147.8.a.c.1.1		2
21.20	even	2		441.8.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.b.1.1	✓	2	1.1	even	1	trivial
63.8.a.f.1.2		2	3.2	odd	2
147.8.a.c.1.1		2	7.6	odd	2
147.8.e.g.67.2		4	7.5	odd	6
147.8.e.g.79.2		4	7.3	odd	6
147.8.e.h.67.2		4	7.2	even	3
147.8.e.h.79.2		4	7.4	even	3
336.8.a.n.1.2		2	4.3	odd	2
441.8.a.m.1.2		2	21.20	even	2
525.8.a.e.1.2		2	5.4	even	2

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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

\(f(q)\)	\(=\)	\(q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} -3046.45 q^{10} -7491.69 q^{11} -8244.57 q^{12} +8553.34 q^{13} -7140.29 q^{14} -3951.27 q^{15} +37772.0 q^{16} -5219.65 q^{17} -15175.7 q^{18} -47136.9 q^{19} +44686.6 q^{20} -9261.00 q^{21} +155956. q^{22} +23833.8 q^{23} +99684.5 q^{24} -56708.6 q^{25} -178056. q^{26} -19683.0 q^{27} +104737. q^{28} -183928. q^{29} +82254.3 q^{30} -180272. q^{31} -313728. q^{32} +202276. q^{33} +108658. q^{34} +50195.8 q^{35} +222603. q^{36} -404121. q^{37} +981256. q^{38} -230940. q^{39} -540303. q^{40} +632653. q^{41} +192788. q^{42} +443131. q^{43} -2.28762e6 q^{44} +106684. q^{45} -496152. q^{46} -524989. q^{47} -1.01984e6 q^{48} +117649. q^{49} +1.18051e6 q^{50} +140931. q^{51} +2.61180e6 q^{52} +520667. q^{53} +409744. q^{54} -1.09636e6 q^{55} -1.26636e6 q^{56} +1.27270e6 q^{57} +3.82886e6 q^{58} -1.42159e6 q^{59} -1.20654e6 q^{60} -616389. q^{61} +3.75275e6 q^{62} +250047. q^{63} +1.69611e6 q^{64} +1.25173e6 q^{65} -4.21080e6 q^{66} -1.63770e6 q^{67} -1.59384e6 q^{68} -643513. q^{69} -1.04493e6 q^{70} -595589. q^{71} -2.69148e6 q^{72} -4.41186e6 q^{73} +8.41266e6 q^{74} +1.53113e6 q^{75} -1.43935e7 q^{76} -2.56965e6 q^{77} +4.80752e6 q^{78} +2.03908e6 q^{79} +5.52768e6 q^{80} +531441. q^{81} -1.31700e7 q^{82} -1.08171e6 q^{83} -2.82789e6 q^{84} -763862. q^{85} -9.22473e6 q^{86} +4.96605e6 q^{87} +2.76595e7 q^{88} +8.80625e6 q^{89} -2.22087e6 q^{90} +2.93380e6 q^{91} +7.27776e6 q^{92} +4.86734e6 q^{93} +1.09288e7 q^{94} -6.89817e6 q^{95} +8.47065e6 q^{96} +6.13951e6 q^{97} -2.44912e6 q^{98} -5.46144e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9} - 9030 q^{10} - 4932 q^{11} - 8559 q^{12} + 7708 q^{13} - 3087 q^{14} + 9720 q^{15} + 20033 q^{16} - 28584 q^{17}+ \cdots - 3595428 q^{99}+O(q^{100}) \) 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9 - 9030 * q^10 - 4932 * q^11 - 8559 * q^12 + 7708 * q^13 - 3087 * q^14 + 9720 * q^15 + 20033 * q^16 - 28584 * q^17 - 6561 * q^18 - 63728 * q^19 + 38790 * q^20 - 18522 * q^21 + 186204 * q^22 + 82260 * q^23 + 136809 * q^24 + 121550 * q^25 - 188046 * q^26 - 39366 * q^27 + 108731 * q^28 - 435996 * q^29 + 243810 * q^30 - 29240 * q^31 - 347355 * q^32 + 133164 * q^33 - 167442 * q^34 - 123480 * q^35 + 231093 * q^36 - 709556 * q^37 + 785196 * q^38 - 208116 * q^39 + 155910 * q^40 - 25056 * q^41 + 83349 * q^42 + 496216 * q^43 - 2257812 * q^44 - 262440 * q^45 + 194280 * q^46 - 1575000 * q^47 - 540891 * q^48 + 235298 * q^49 + 3287025 * q^50 + 771768 * q^51 + 2601958 * q^52 + 2057436 * q^53 + 177147 * q^54 - 2392440 * q^55 - 1737981 * q^56 + 1720656 * q^57 + 850122 * q^58 - 1101024 * q^59 - 1047330 * q^60 + 28996 * q^61 + 5537520 * q^62 + 500094 * q^63 + 3569321 * q^64 + 1679760 * q^65 - 5027508 * q^66 - 4480784 * q^67 - 1865934 * q^68 - 2221020 * q^69 - 3097290 * q^70 + 54540 * q^71 - 3693843 * q^72 + 666604 * q^73 + 4803282 * q^74 - 3281850 * q^75 - 14586668 * q^76 - 1691676 * q^77 + 5077242 * q^78 + 2322952 * q^79 + 14509710 * q^80 + 1062882 * q^81 - 20942298 * q^82 - 7384392 * q^83 - 2935737 * q^84 + 11066520 * q^85 - 8597412 * q^86 + 11771892 * q^87 + 24139932 * q^88 + 1784448 * q^89 - 6582870 * q^90 + 2643844 * q^91 + 7958160 * q^92 + 789480 * q^93 - 1479360 * q^94 + 1502640 * q^95 + 9378585 * q^96 + 16266412 * q^97 - 1058841 * q^98 - 3595428 * q^99

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