Properties

Label 12.16.a.b.1.2
Level $12$
Weight $16$
Character 12.1
Self dual yes
Analytic conductor $17.123$
Analytic rank $0$
Dimension $2$
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] = [12,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.1232206120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{8017}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2004 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-44.2689\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2187.00 q^{3} +292699. q^{5} -1.07507e6 q^{7} +4.78297e6 q^{9} +6.62625e7 q^{11} -3.32961e8 q^{13} +6.40132e8 q^{15} +1.81911e9 q^{17} +6.00869e9 q^{19} -2.35117e9 q^{21} +1.16112e10 q^{23} +5.51549e10 q^{25} +1.04604e10 q^{27} +5.31278e10 q^{29} -1.54744e11 q^{31} +1.44916e11 q^{33} -3.14670e11 q^{35} -5.64914e11 q^{37} -7.28185e11 q^{39} -1.71251e12 q^{41} +2.61772e12 q^{43} +1.39997e12 q^{45} -1.33544e12 q^{47} -3.59180e12 q^{49} +3.97840e12 q^{51} +6.50115e12 q^{53} +1.93949e13 q^{55} +1.31410e13 q^{57} -2.24202e13 q^{59} +6.03075e12 q^{61} -5.14200e12 q^{63} -9.74571e13 q^{65} +5.22895e13 q^{67} +2.53936e13 q^{69} -7.92879e13 q^{71} -1.39574e14 q^{73} +1.20624e14 q^{75} -7.12365e13 q^{77} +9.91857e13 q^{79} +2.28768e13 q^{81} +1.60896e13 q^{83} +5.32452e14 q^{85} +1.16191e14 q^{87} -5.06360e14 q^{89} +3.57954e14 q^{91} -3.38425e14 q^{93} +1.75874e15 q^{95} -7.99056e14 q^{97} +3.16931e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4374 q^{3} + 69660 q^{5} + 2491504 q^{7} + 9565938 q^{9} - 14975928 q^{11} + 11757580 q^{13} + 152346420 q^{15} + 4256077284 q^{17} + 9241689400 q^{19} + 5448919248 q^{21} + 24254822736 q^{23} + 74383511150 q^{25}+ \cdots - 71629399370232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2187.00 0.577350
\(4\) 0 0
\(5\) 292699. 1.67550 0.837752 − 0.546051i \(-0.183869\pi\)
0.837752 + 0.546051i \(0.183869\pi\)
\(6\) 0 0
\(7\) −1.07507e6 −0.493400 −0.246700 − 0.969092i \(-0.579346\pi\)
−0.246700 + 0.969092i \(0.579346\pi\)
\(8\) 0 0
\(9\) 4.78297e6 0.333333
\(10\) 0 0
\(11\) 6.62625e7 1.02523 0.512616 − 0.858618i \(-0.328676\pi\)
0.512616 + 0.858618i \(0.328676\pi\)
\(12\) 0 0
\(13\) −3.32961e8 −1.47169 −0.735847 − 0.677148i \(-0.763216\pi\)
−0.735847 + 0.677148i \(0.763216\pi\)
\(14\) 0 0
\(15\) 6.40132e8 0.967353
\(16\) 0 0
\(17\) 1.81911e9 1.07521 0.537604 − 0.843197i \(-0.319330\pi\)
0.537604 + 0.843197i \(0.319330\pi\)
\(18\) 0 0
\(19\) 6.00869e9 1.54216 0.771078 − 0.636741i \(-0.219718\pi\)
0.771078 + 0.636741i \(0.219718\pi\)
\(20\) 0 0
\(21\) −2.35117e9 −0.284865
\(22\) 0 0
\(23\) 1.16112e10 0.711077 0.355539 − 0.934662i \(-0.384297\pi\)
0.355539 + 0.934662i \(0.384297\pi\)
\(24\) 0 0
\(25\) 5.51549e10 1.80732
\(26\) 0 0
\(27\) 1.04604e10 0.192450
\(28\) 0 0
\(29\) 5.31278e10 0.571922 0.285961 − 0.958241i \(-0.407687\pi\)
0.285961 + 0.958241i \(0.407687\pi\)
\(30\) 0 0
\(31\) −1.54744e11 −1.01019 −0.505093 − 0.863065i \(-0.668542\pi\)
−0.505093 + 0.863065i \(0.668542\pi\)
\(32\) 0 0
\(33\) 1.44916e11 0.591918
\(34\) 0 0
\(35\) −3.14670e11 −0.826695
\(36\) 0 0
\(37\) −5.64914e11 −0.978294 −0.489147 − 0.872201i \(-0.662692\pi\)
−0.489147 + 0.872201i \(0.662692\pi\)
\(38\) 0 0
\(39\) −7.28185e11 −0.849683
\(40\) 0 0
\(41\) −1.71251e12 −1.37326 −0.686632 − 0.727006i \(-0.740911\pi\)
−0.686632 + 0.727006i \(0.740911\pi\)
\(42\) 0 0
\(43\) 2.61772e12 1.46862 0.734312 − 0.678812i \(-0.237505\pi\)
0.734312 + 0.678812i \(0.237505\pi\)
\(44\) 0 0
\(45\) 1.39997e12 0.558501
\(46\) 0 0
\(47\) −1.33544e12 −0.384494 −0.192247 − 0.981347i \(-0.561578\pi\)
−0.192247 + 0.981347i \(0.561578\pi\)
\(48\) 0 0
\(49\) −3.59180e12 −0.756556
\(50\) 0 0
\(51\) 3.97840e12 0.620772
\(52\) 0 0
\(53\) 6.50115e12 0.760188 0.380094 − 0.924948i \(-0.375892\pi\)
0.380094 + 0.924948i \(0.375892\pi\)
\(54\) 0 0
\(55\) 1.93949e13 1.71778
\(56\) 0 0
\(57\) 1.31410e13 0.890364
\(58\) 0 0
\(59\) −2.24202e13 −1.17287 −0.586434 − 0.809997i \(-0.699469\pi\)
−0.586434 + 0.809997i \(0.699469\pi\)
\(60\) 0 0
\(61\) 6.03075e12 0.245696 0.122848 − 0.992426i \(-0.460797\pi\)
0.122848 + 0.992426i \(0.460797\pi\)
\(62\) 0 0
\(63\) −5.14200e12 −0.164467
\(64\) 0 0
\(65\) −9.74571e13 −2.46583
\(66\) 0 0
\(67\) 5.22895e13 1.05403 0.527016 − 0.849856i \(-0.323311\pi\)
0.527016 + 0.849856i \(0.323311\pi\)
\(68\) 0 0
\(69\) 2.53936e13 0.410541
\(70\) 0 0
\(71\) −7.92879e13 −1.03459 −0.517296 − 0.855807i \(-0.673061\pi\)
−0.517296 + 0.855807i \(0.673061\pi\)
\(72\) 0 0
\(73\) −1.39574e14 −1.47871 −0.739355 − 0.673316i \(-0.764869\pi\)
−0.739355 + 0.673316i \(0.764869\pi\)
\(74\) 0 0
\(75\) 1.20624e14 1.04345
\(76\) 0 0
\(77\) −7.12365e13 −0.505850
\(78\) 0 0
\(79\) 9.91857e13 0.581093 0.290547 − 0.956861i \(-0.406163\pi\)
0.290547 + 0.956861i \(0.406163\pi\)
\(80\) 0 0
\(81\) 2.28768e13 0.111111
\(82\) 0 0
\(83\) 1.60896e13 0.0650819 0.0325410 − 0.999470i \(-0.489640\pi\)
0.0325410 + 0.999470i \(0.489640\pi\)
\(84\) 0 0
\(85\) 5.32452e14 1.80152
\(86\) 0 0
\(87\) 1.16191e14 0.330199
\(88\) 0 0
\(89\) −5.06360e14 −1.21348 −0.606742 − 0.794899i \(-0.707524\pi\)
−0.606742 + 0.794899i \(0.707524\pi\)
\(90\) 0 0
\(91\) 3.57954e14 0.726135
\(92\) 0 0
\(93\) −3.38425e14 −0.583231
\(94\) 0 0
\(95\) 1.75874e15 2.58389
\(96\) 0 0
\(97\) −7.99056e14 −1.00413 −0.502064 − 0.864830i \(-0.667426\pi\)
−0.502064 + 0.864830i \(0.667426\pi\)
\(98\) 0 0
\(99\) 3.16931e14 0.341744
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 12.16.a.b.1.2 ✓ 2
3.2 odd 2 36.16.a.c.1.1 2
4.3 odd 2 48.16.a.i.1.2 2
12.11 even 2 144.16.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.16.a.b.1.2 ✓ 2 1.1 even 1 trivial
36.16.a.c.1.1 2 3.2 odd 2
48.16.a.i.1.2 2 4.3 odd 2
144.16.a.r.1.1 2 12.11 even 2