LMFDB - Embedded newform 21.8.a.b.1.2

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			1.2
Root			$-15.8172$ 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)

$f(q)$	$=$	\(q+11.8172 q^{2} -27.0000 q^{3} +11.6455 q^{4} -506.343 q^{5} -319.064 q^{6} +343.000 q^{7} -1374.98 q^{8} +729.000 q^{9} -5983.55 q^{10} +2559.69 q^{11} -314.428 q^{12} -845.345 q^{13} +4053.29 q^{14} +13671.3 q^{15} -17739.0 q^{16} -23364.3 q^{17} +8614.72 q^{18} -16591.1 q^{19} -5896.61 q^{20} -9261.00 q^{21} +30248.3 q^{22} +58426.2 q^{23} +37124.5 q^{24} +178259. q^{25} -9989.58 q^{26} -19683.0 q^{27} +3994.40 q^{28} -252068. q^{29} +161556. q^{30} +151032. q^{31} -33627.2 q^{32} -69111.6 q^{33} -276100. q^{34} -173676. q^{35} +8489.56 q^{36} -305435. q^{37} -196060. q^{38} +22824.3 q^{39} +696213. q^{40} -657709. q^{41} -109439. q^{42} +53085.2 q^{43} +29808.8 q^{44} -369124. q^{45} +690432. q^{46} -1.05001e6 q^{47} +478953. q^{48} +117649. q^{49} +2.10651e6 q^{50} +630837. q^{51} -9844.44 q^{52} +1.53677e6 q^{53} -232597. q^{54} -1.29608e6 q^{55} -471618. q^{56} +447961. q^{57} -2.97873e6 q^{58} +320564. q^{59} +159209. q^{60} +645385. q^{61} +1.78477e6 q^{62} +250047. q^{63} +1.87321e6 q^{64} +428035. q^{65} -816703. q^{66} -2.84308e6 q^{67} -272089. q^{68} -1.57751e6 q^{69} -2.05236e6 q^{70} +650129. q^{71} -1.00236e6 q^{72} +5.07847e6 q^{73} -3.60938e6 q^{74} -4.81298e6 q^{75} -193212. q^{76} +877973. q^{77} +269719. q^{78} +283873. q^{79} +8.98203e6 q^{80} +531441. q^{81} -7.77226e6 q^{82} -6.30268e6 q^{83} -107849. q^{84} +1.18304e7 q^{85} +627317. q^{86} +6.80584e6 q^{87} -3.51952e6 q^{88} -7.02180e6 q^{89} -4.36200e6 q^{90} -289953. q^{91} +680401. q^{92} -4.07786e6 q^{93} -1.24082e7 q^{94} +8.40081e6 q^{95} +907935. q^{96} +1.01269e7 q^{97} +1.39028e6 q^{98} +1.86601e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	11.8172	1.04450	0.522250	−	0.852792i	$-0.325093\pi$
$2$	11.8172	1.04450	0.522250	+	0.852792i	$0.325093\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	11.6455	0.0909803
$5$	−506.343	−1.81155	−0.905775	−	0.423760i	$-0.860710\pi$
$5$	−506.343	−1.81155	−0.905775	+	0.423760i	$0.860710\pi$
$6$	−319.064	−0.603042
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−1374.98	−0.949471
$9$	729.000	0.333333
$10$	−5983.55	−1.89216
$11$	2559.69	0.579846	0.289923	−	0.957050i	$-0.406370\pi$
$11$	2559.69	0.579846	0.289923	+	0.957050i	$0.406370\pi$
$12$	−314.428	−0.0525275
$13$	−845.345	−0.106717	−0.0533583	−	0.998575i	$-0.516993\pi$
$13$	−845.345	−0.106717	−0.0533583	+	0.998575i	$0.516993\pi$
$14$	4053.29	0.394784
$15$	13671.3	1.04590
$16$	−17739.0	−1.08270
$17$	−23364.3	−1.15341	−0.576703	−	0.816954i	$-0.695661\pi$
$17$	−23364.3	−1.15341	−0.576703	+	0.816954i	$0.695661\pi$
$18$	8614.72	0.348167
$19$	−16591.1	−0.554930	−0.277465	−	0.960736i	$-0.589494\pi$
$19$	−16591.1	−0.554930	−0.277465	+	0.960736i	$0.589494\pi$
$20$	−5896.61	−0.164815
$21$	−9261.00	−0.218218
$22$	30248.3	0.605649
$23$	58426.2	1.00129	0.500645	−	0.865652i	$-0.333096\pi$
$23$	58426.2	1.00129	0.500645	+	0.865652i	$0.333096\pi$
$24$	37124.5	0.548177
$25$	178259.	2.28171
$26$	−9989.58	−0.111466
$27$	−19683.0	−0.192450
$28$	3994.40	0.0343873
$29$	−252068.	−1.91922	−0.959611	−	0.281331i	$-0.909224\pi$
$29$	−252068.	−1.91922	−0.959611	+	0.281331i	$0.909224\pi$
$30$	161556.	1.09244
$31$	151032.	0.910548	0.455274	−	0.890351i	$-0.349541\pi$
$31$	151032.	0.910548	0.455274	+	0.890351i	$0.349541\pi$
$32$	−33627.2	−0.181412
$33$	−69111.6	−0.334774
$34$	−276100.	−1.20473
$35$	−173676.	−0.684701
$36$	8489.56	0.0303268
$37$	−305435.	−0.991317	−0.495659	−	0.868517i	$-0.665073\pi$
$37$	−305435.	−0.991317	−0.495659	+	0.868517i	$0.665073\pi$
$38$	−196060.	−0.579625
$39$	22824.3	0.0616129
$40$	696213.	1.72001
$41$	−657709.	−1.49036	−0.745178	−	0.666865i	$-0.767636\pi$
$41$	−657709.	−1.49036	−0.745178	+	0.666865i	$0.767636\pi$
$42$	−109439.	−0.227929
$43$	53085.2	0.101820	0.0509101	−	0.998703i	$-0.483788\pi$
$43$	53085.2	0.101820	0.0509101	+	0.998703i	$0.483788\pi$
$44$	29808.8	0.0527546
$45$	−369124.	−0.603850
$46$	690432.	1.04585
$47$	−1.05001e6	−1.47520	−0.737600	−	0.675237i	$-0.764041\pi$
$47$	−1.05001e6	−1.47520	−0.737600	+	0.675237i	$0.764041\pi$
$48$	478953.	0.625099
$49$	117649.	0.142857
$50$	2.10651e6	2.38325
$51$	630837.	0.665920
$52$	−9844.44	−0.00970911
$53$	1.53677e6	1.41789	0.708946	−	0.705263i	$-0.249171\pi$
$53$	1.53677e6	1.41789	0.708946	+	0.705263i	$0.249171\pi$
$54$	−232597.	−0.201014
$55$	−1.29608e6	−1.05042
$56$	−471618.	−0.358866
$57$	447961.	0.320389
$58$	−2.97873e6	−2.00463
$59$	320564.	0.203204	0.101602	−	0.994825i	$-0.467603\pi$
$59$	320564.	0.203204	0.101602	+	0.994825i	$0.467603\pi$
$60$	159209.	0.0951562
$61$	645385.	0.364053	0.182026	−	0.983294i	$-0.441734\pi$
$61$	645385.	0.364053	0.182026	+	0.983294i	$0.441734\pi$
$62$	1.78477e6	0.951067
$63$	250047.	0.125988
$64$	1.87321e6	0.893218
$65$	428035.	0.193322
$66$	−816703.	−0.349672
$67$	−2.84308e6	−1.15486	−0.577428	−	0.816442i	$-0.695944\pi$
$67$	−2.84308e6	−1.15486	−0.577428	+	0.816442i	$0.695944\pi$
$68$	−272089.	−0.104937
$69$	−1.57751e6	−0.578095
$70$	−2.05236e6	−0.715170
$71$	650129.	0.215573	0.107787	−	0.994174i	$-0.465624\pi$
$71$	650129.	0.215573	0.107787	+	0.994174i	$0.465624\pi$
$72$	−1.00236e6	−0.316490
$73$	5.07847e6	1.52793	0.763963	−	0.645260i	$-0.223251\pi$
$73$	5.07847e6	1.52793	0.763963	+	0.645260i	$0.223251\pi$
$74$	−3.60938e6	−1.03543
$75$	−4.81298e6	−1.31735
$76$	−193212.	−0.0504877
$77$	877973.	0.219161
$78$	269719.	0.0643546
$79$	283873.	0.0647783	0.0323892	−	0.999475i	$-0.489688\pi$
$79$	283873.	0.0647783	0.0323892	+	0.999475i	$0.489688\pi$
$80$	8.98203e6	1.96137
$81$	531441.	0.111111
$82$	−7.77226e6	−1.55668
$83$	−6.30268e6	−1.20991	−0.604954	−	0.796261i	$-0.706808\pi$
$83$	−6.30268e6	−1.20991	−0.604954	+	0.796261i	$0.706808\pi$
$84$	−107849.	−0.0198535
$85$	1.18304e7	2.08945
$86$	627317.	0.106351
$87$	6.80584e6	1.10806
$88$	−3.51952e6	−0.550547
$89$	−7.02180e6	−1.05580	−0.527902	−	0.849305i	$-0.677021\pi$
$89$	−7.02180e6	−1.05580	−0.527902	+	0.849305i	$0.677021\pi$
$90$	−4.36200e6	−0.630721
$91$	−289953.	−0.0403351
$92$	680401.	0.0910977
$93$	−4.07786e6	−0.525705
$94$	−1.24082e7	−1.54085
$95$	8.40081e6	1.00528
$96$	907935.	0.104738
$97$	1.01269e7	1.12662	0.563308	−	0.826247i	$-0.309529\pi$
$97$	1.01269e7	1.12662	0.563308	+	0.826247i	$0.309529\pi$
$98$	1.39028e6	0.149214
$99$	1.86601e6	0.193282

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.1.2	✓	2
3.2	odd	2		63.8.a.f.1.1		2
4.3	odd	2		336.8.a.n.1.1		2
5.4	even	2		525.8.a.e.1.1		2
7.2	even	3		147.8.e.h.67.1		4
7.3	odd	6		147.8.e.g.79.1		4
7.4	even	3		147.8.e.h.79.1		4
7.5	odd	6		147.8.e.g.67.1		4
7.6	odd	2		147.8.a.c.1.2		2
21.20	even	2		441.8.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.b.1.2	✓	2	1.1	even	1	trivial
63.8.a.f.1.1		2	3.2	odd	2
147.8.a.c.1.2		2	7.6	odd	2
147.8.e.g.67.1		4	7.5	odd	6
147.8.e.g.79.1		4	7.3	odd	6
147.8.e.h.67.1		4	7.2	even	3
147.8.e.h.79.1		4	7.4	even	3
336.8.a.n.1.1		2	4.3	odd	2
441.8.a.m.1.1		2	21.20	even	2
525.8.a.e.1.1		2	5.4	even	2

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Belyi maps

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

\(f(q)\)	\(=\)	\(q+11.8172 q^{2} -27.0000 q^{3} +11.6455 q^{4} -506.343 q^{5} -319.064 q^{6} +343.000 q^{7} -1374.98 q^{8} +729.000 q^{9} -5983.55 q^{10} +2559.69 q^{11} -314.428 q^{12} -845.345 q^{13} +4053.29 q^{14} +13671.3 q^{15} -17739.0 q^{16} -23364.3 q^{17} +8614.72 q^{18} -16591.1 q^{19} -5896.61 q^{20} -9261.00 q^{21} +30248.3 q^{22} +58426.2 q^{23} +37124.5 q^{24} +178259. q^{25} -9989.58 q^{26} -19683.0 q^{27} +3994.40 q^{28} -252068. q^{29} +161556. q^{30} +151032. q^{31} -33627.2 q^{32} -69111.6 q^{33} -276100. q^{34} -173676. q^{35} +8489.56 q^{36} -305435. q^{37} -196060. q^{38} +22824.3 q^{39} +696213. q^{40} -657709. q^{41} -109439. q^{42} +53085.2 q^{43} +29808.8 q^{44} -369124. q^{45} +690432. q^{46} -1.05001e6 q^{47} +478953. q^{48} +117649. q^{49} +2.10651e6 q^{50} +630837. q^{51} -9844.44 q^{52} +1.53677e6 q^{53} -232597. q^{54} -1.29608e6 q^{55} -471618. q^{56} +447961. q^{57} -2.97873e6 q^{58} +320564. q^{59} +159209. q^{60} +645385. q^{61} +1.78477e6 q^{62} +250047. q^{63} +1.87321e6 q^{64} +428035. q^{65} -816703. q^{66} -2.84308e6 q^{67} -272089. q^{68} -1.57751e6 q^{69} -2.05236e6 q^{70} +650129. q^{71} -1.00236e6 q^{72} +5.07847e6 q^{73} -3.60938e6 q^{74} -4.81298e6 q^{75} -193212. q^{76} +877973. q^{77} +269719. q^{78} +283873. q^{79} +8.98203e6 q^{80} +531441. q^{81} -7.77226e6 q^{82} -6.30268e6 q^{83} -107849. q^{84} +1.18304e7 q^{85} +627317. q^{86} +6.80584e6 q^{87} -3.51952e6 q^{88} -7.02180e6 q^{89} -4.36200e6 q^{90} -289953. q^{91} +680401. q^{92} -4.07786e6 q^{93} -1.24082e7 q^{94} +8.40081e6 q^{95} +907935. q^{96} +1.01269e7 q^{97} +1.39028e6 q^{98} +1.86601e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9} - 9030 q^{10} - 4932 q^{11} - 8559 q^{12} + 7708 q^{13} - 3087 q^{14} + 9720 q^{15} + 20033 q^{16} - 28584 q^{17}+ \cdots - 3595428 q^{99}+O(q^{100}) \) 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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