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Properties

Label 12.8.a.b.1.1
Level $12$
Weight $8$
Character 12.1
Self dual yes
Analytic conductor $3.749$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,8,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.74862030581\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 12.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +270.000 q^{5} +1112.00 q^{7} +729.000 q^{9} -5724.00 q^{11} -4570.00 q^{13} +7290.00 q^{15} -36558.0 q^{17} +51740.0 q^{19} +30024.0 q^{21} +22248.0 q^{23} -5225.00 q^{25} +19683.0 q^{27} -157194. q^{29} -103936. q^{31} -154548. q^{33} +300240. q^{35} -94834.0 q^{37} -123390. q^{39} +659610. q^{41} -75772.0 q^{43} +196830. q^{45} +405648. q^{47} +413001. q^{49} -987066. q^{51} -1.34627e6 q^{53} -1.54548e6 q^{55} +1.39698e6 q^{57} -1.30388e6 q^{59} +1.83378e6 q^{61} +810648. q^{63} -1.23390e6 q^{65} +1.36939e6 q^{67} +600696. q^{69} +2.71404e6 q^{71} +2.86879e6 q^{73} -141075. q^{75} -6.36509e6 q^{77} -1.12965e6 q^{79} +531441. q^{81} +5.91203e6 q^{83} -9.87066e6 q^{85} -4.24424e6 q^{87} -897750. q^{89} -5.08184e6 q^{91} -2.80627e6 q^{93} +1.39698e7 q^{95} +1.37191e7 q^{97} -4.17280e6 q^{99} +O(q^{100})\)

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 \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 270.000 0.965981 0.482991 − 0.875625i \(-0.339550\pi\)
0.482991 + 0.875625i \(0.339550\pi\)
\(6\) 0 0
\(7\) 1112.00 1.22535 0.612677 − 0.790333i \(-0.290093\pi\)
0.612677 + 0.790333i \(0.290093\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −5724.00 −1.29666 −0.648329 − 0.761361i \(-0.724532\pi\)
−0.648329 + 0.761361i \(0.724532\pi\)
\(12\) 0 0
\(13\) −4570.00 −0.576919 −0.288459 − 0.957492i \(-0.593143\pi\)
−0.288459 + 0.957492i \(0.593143\pi\)
\(14\) 0 0
\(15\) 7290.00 0.557710
\(16\) 0 0
\(17\) −36558.0 −1.80473 −0.902363 − 0.430977i \(-0.858169\pi\)
−0.902363 + 0.430977i \(0.858169\pi\)
\(18\) 0 0
\(19\) 51740.0 1.73057 0.865284 − 0.501281i \(-0.167138\pi\)
0.865284 + 0.501281i \(0.167138\pi\)
\(20\) 0 0
\(21\) 30024.0 0.707459
\(22\) 0 0
\(23\) 22248.0 0.381280 0.190640 − 0.981660i \(-0.438944\pi\)
0.190640 + 0.981660i \(0.438944\pi\)
\(24\) 0 0
\(25\) −5225.00 −0.0668800
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −157194. −1.19686 −0.598429 − 0.801175i \(-0.704208\pi\)
−0.598429 + 0.801175i \(0.704208\pi\)
\(30\) 0 0
\(31\) −103936. −0.626614 −0.313307 − 0.949652i \(-0.601437\pi\)
−0.313307 + 0.949652i \(0.601437\pi\)
\(32\) 0 0
\(33\) −154548. −0.748625
\(34\) 0 0
\(35\) 300240. 1.18367
\(36\) 0 0
\(37\) −94834.0 −0.307793 −0.153896 − 0.988087i \(-0.549182\pi\)
−0.153896 + 0.988087i \(0.549182\pi\)
\(38\) 0 0
\(39\) −123390. −0.333084
\(40\) 0 0
\(41\) 659610. 1.49466 0.747332 − 0.664451i \(-0.231334\pi\)
0.747332 + 0.664451i \(0.231334\pi\)
\(42\) 0 0
\(43\) −75772.0 −0.145335 −0.0726673 − 0.997356i \(-0.523151\pi\)
−0.0726673 + 0.997356i \(0.523151\pi\)
\(44\) 0 0
\(45\) 196830. 0.321994
\(46\) 0 0
\(47\) 405648. 0.569911 0.284955 − 0.958541i \(-0.408021\pi\)
0.284955 + 0.958541i \(0.408021\pi\)
\(48\) 0 0
\(49\) 413001. 0.501493
\(50\) 0 0
\(51\) −987066. −1.04196
\(52\) 0 0
\(53\) −1.34627e6 −1.24213 −0.621066 − 0.783758i \(-0.713300\pi\)
−0.621066 + 0.783758i \(0.713300\pi\)
\(54\) 0 0
\(55\) −1.54548e6 −1.25255
\(56\) 0 0
\(57\) 1.39698e6 0.999144
\(58\) 0 0
\(59\) −1.30388e6 −0.826527 −0.413263 − 0.910612i \(-0.635611\pi\)
−0.413263 + 0.910612i \(0.635611\pi\)
\(60\) 0 0
\(61\) 1.83378e6 1.03441 0.517206 − 0.855861i \(-0.326972\pi\)
0.517206 + 0.855861i \(0.326972\pi\)
\(62\) 0 0
\(63\) 810648. 0.408451
\(64\) 0 0
\(65\) −1.23390e6 −0.557293
\(66\) 0 0
\(67\) 1.36939e6 0.556243 0.278122 − 0.960546i \(-0.410288\pi\)
0.278122 + 0.960546i \(0.410288\pi\)
\(68\) 0 0
\(69\) 600696. 0.220132
\(70\) 0 0
\(71\) 2.71404e6 0.899937 0.449968 − 0.893044i \(-0.351435\pi\)
0.449968 + 0.893044i \(0.351435\pi\)
\(72\) 0 0
\(73\) 2.86879e6 0.863116 0.431558 − 0.902085i \(-0.357964\pi\)
0.431558 + 0.902085i \(0.357964\pi\)
\(74\) 0 0
\(75\) −141075. −0.0386132
\(76\) 0 0
\(77\) −6.36509e6 −1.58886
\(78\) 0 0
\(79\) −1.12965e6 −0.257779 −0.128890 − 0.991659i \(-0.541141\pi\)
−0.128890 + 0.991659i \(0.541141\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.91203e6 1.13491 0.567457 − 0.823403i \(-0.307927\pi\)
0.567457 + 0.823403i \(0.307927\pi\)
\(84\) 0 0
\(85\) −9.87066e6 −1.74333
\(86\) 0 0
\(87\) −4.24424e6 −0.691007
\(88\) 0 0
\(89\) −897750. −0.134987 −0.0674933 − 0.997720i \(-0.521500\pi\)
−0.0674933 + 0.997720i \(0.521500\pi\)
\(90\) 0 0
\(91\) −5.08184e6 −0.706930
\(92\) 0 0
\(93\) −2.80627e6 −0.361776
\(94\) 0 0
\(95\) 1.39698e7 1.67170
\(96\) 0 0
\(97\) 1.37191e7 1.52624 0.763122 − 0.646255i \(-0.223666\pi\)
0.763122 + 0.646255i \(0.223666\pi\)
\(98\) 0 0
\(99\) −4.17280e6 −0.432219
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 12.8.a.b.1.1 ✓ 1
3.2 odd 2 36.8.a.a.1.1 1
4.3 odd 2 48.8.a.d.1.1 1
5.2 odd 4 300.8.d.a.49.1 2
5.3 odd 4 300.8.d.a.49.2 2
5.4 even 2 300.8.a.a.1.1 1
7.2 even 3 588.8.i.b.361.1 2
7.3 odd 6 588.8.i.g.373.1 2
7.4 even 3 588.8.i.b.373.1 2
7.5 odd 6 588.8.i.g.361.1 2
7.6 odd 2 588.8.a.a.1.1 1
8.3 odd 2 192.8.a.j.1.1 1
8.5 even 2 192.8.a.b.1.1 1
9.2 odd 6 324.8.e.e.109.1 2
9.4 even 3 324.8.e.b.217.1 2
9.5 odd 6 324.8.e.e.217.1 2
9.7 even 3 324.8.e.b.109.1 2
12.11 even 2 144.8.a.c.1.1 1
24.5 odd 2 576.8.a.v.1.1 1
24.11 even 2 576.8.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.b.1.1 ✓ 1 1.1 even 1 trivial
36.8.a.a.1.1 1 3.2 odd 2
48.8.a.d.1.1 1 4.3 odd 2
144.8.a.c.1.1 1 12.11 even 2
192.8.a.b.1.1 1 8.5 even 2
192.8.a.j.1.1 1 8.3 odd 2
300.8.a.a.1.1 1 5.4 even 2
300.8.d.a.49.1 2 5.2 odd 4
300.8.d.a.49.2 2 5.3 odd 4
324.8.e.b.109.1 2 9.7 even 3
324.8.e.b.217.1 2 9.4 even 3
324.8.e.e.109.1 2 9.2 odd 6
324.8.e.e.217.1 2 9.5 odd 6
576.8.a.u.1.1 1 24.11 even 2
576.8.a.v.1.1 1 24.5 odd 2
588.8.a.a.1.1 1 7.6 odd 2
588.8.i.b.361.1 2 7.2 even 3
588.8.i.b.373.1 2 7.4 even 3
588.8.i.g.361.1 2 7.5 odd 6
588.8.i.g.373.1 2 7.3 odd 6