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Properties

Label 12.16.a.b.1.1
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 1.1
Root \(45.2689\) 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)
 
\(f(q)\) \(=\) \(q+2187.00 q^{3} -223039. q^{5} +3.56657e6 q^{7} +4.78297e6 q^{9} -8.12384e7 q^{11} +3.44718e8 q^{13} -4.87785e8 q^{15} +2.43697e9 q^{17} +3.23300e9 q^{19} +7.80009e9 q^{21} +1.26437e10 q^{23} +1.92286e10 q^{25} +1.04604e10 q^{27} +9.41624e10 q^{29} +7.80665e10 q^{31} -1.77668e11 q^{33} -7.95483e11 q^{35} -4.06319e11 q^{37} +7.53898e11 q^{39} -2.98195e11 q^{41} -1.91876e12 q^{43} -1.06679e12 q^{45} +5.51593e11 q^{47} +7.97285e12 q^{49} +5.32964e12 q^{51} -1.20870e13 q^{53} +1.81193e13 q^{55} +7.07056e12 q^{57} +6.58404e12 q^{59} -4.21447e12 q^{61} +1.70588e13 q^{63} -7.68854e13 q^{65} -5.45476e13 q^{67} +2.76517e13 q^{69} -1.17707e14 q^{71} +1.31548e14 q^{73} +4.20530e13 q^{75} -2.89742e14 q^{77} +2.57200e14 q^{79} +2.28768e13 q^{81} +3.45415e14 q^{83} -5.43537e14 q^{85} +2.05933e14 q^{87} +3.21904e14 q^{89} +1.22946e15 q^{91} +1.70731e14 q^{93} -7.21083e14 q^{95} -1.01652e15 q^{97} -3.88561e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4374 q^{3} + 69660 q^{5} + 2491504 q^{7} + 9565938 q^{9} - 14975928 q^{11} + 11757580 q^{13} + 152346420 q^{15} + 4256077284 q^{17} + 9241689400 q^{19} + 5448919248 q^{21} + 24254822736 q^{23} + 74383511150 q^{25}+ \cdots - 71629399370232 q^{99}+O(q^{100}) \) 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
\(5\) −223039. −1.27675 −0.638374 − 0.769727i \(-0.720393\pi\)
−0.638374 + 0.769727i \(0.720393\pi\)
\(6\) 0 0
\(7\) 3.56657e6 1.63687 0.818437 − 0.574596i \(-0.194841\pi\)
0.818437 + 0.574596i \(0.194841\pi\)
\(8\) 0 0
\(9\) 4.78297e6 0.333333
\(10\) 0 0
\(11\) −8.12384e7 −1.25694 −0.628472 − 0.777832i \(-0.716319\pi\)
−0.628472 + 0.777832i \(0.716319\pi\)
\(12\) 0 0
\(13\) 3.44718e8 1.52366 0.761832 − 0.647775i \(-0.224300\pi\)
0.761832 + 0.647775i \(0.224300\pi\)
\(14\) 0 0
\(15\) −4.87785e8 −0.737130
\(16\) 0 0
\(17\) 2.43697e9 1.44040 0.720199 − 0.693768i \(-0.244051\pi\)
0.720199 + 0.693768i \(0.244051\pi\)
\(18\) 0 0
\(19\) 3.23300e9 0.829761 0.414881 − 0.909876i \(-0.363823\pi\)
0.414881 + 0.909876i \(0.363823\pi\)
\(20\) 0 0
\(21\) 7.80009e9 0.945050
\(22\) 0 0
\(23\) 1.26437e10 0.774309 0.387154 − 0.922015i \(-0.373458\pi\)
0.387154 + 0.922015i \(0.373458\pi\)
\(24\) 0 0
\(25\) 1.92286e10 0.630084
\(26\) 0 0
\(27\) 1.04604e10 0.192450
\(28\) 0 0
\(29\) 9.41624e10 1.01366 0.506830 − 0.862046i \(-0.330817\pi\)
0.506830 + 0.862046i \(0.330817\pi\)
\(30\) 0 0
\(31\) 7.80665e10 0.509626 0.254813 − 0.966990i \(-0.417986\pi\)
0.254813 + 0.966990i \(0.417986\pi\)
\(32\) 0 0
\(33\) −1.77668e11 −0.725697
\(34\) 0 0
\(35\) −7.95483e11 −2.08988
\(36\) 0 0
\(37\) −4.06319e11 −0.703646 −0.351823 − 0.936067i \(-0.614438\pi\)
−0.351823 + 0.936067i \(0.614438\pi\)
\(38\) 0 0
\(39\) 7.53898e11 0.879687
\(40\) 0 0
\(41\) −2.98195e11 −0.239123 −0.119562 − 0.992827i \(-0.538149\pi\)
−0.119562 + 0.992827i \(0.538149\pi\)
\(42\) 0 0
\(43\) −1.91876e12 −1.07648 −0.538241 − 0.842791i \(-0.680911\pi\)
−0.538241 + 0.842791i \(0.680911\pi\)
\(44\) 0 0
\(45\) −1.06679e12 −0.425582
\(46\) 0 0
\(47\) 5.51593e11 0.158813 0.0794063 − 0.996842i \(-0.474698\pi\)
0.0794063 + 0.996842i \(0.474698\pi\)
\(48\) 0 0
\(49\) 7.97285e12 1.67936
\(50\) 0 0
\(51\) 5.32964e12 0.831614
\(52\) 0 0
\(53\) −1.20870e13 −1.41335 −0.706676 − 0.707538i \(-0.749806\pi\)
−0.706676 + 0.707538i \(0.749806\pi\)
\(54\) 0 0
\(55\) 1.81193e13 1.60480
\(56\) 0 0
\(57\) 7.07056e12 0.479063
\(58\) 0 0
\(59\) 6.58404e12 0.344431 0.172216 − 0.985059i \(-0.444907\pi\)
0.172216 + 0.985059i \(0.444907\pi\)
\(60\) 0 0
\(61\) −4.21447e12 −0.171700 −0.0858498 − 0.996308i \(-0.527361\pi\)
−0.0858498 + 0.996308i \(0.527361\pi\)
\(62\) 0 0
\(63\) 1.70588e13 0.545625
\(64\) 0 0
\(65\) −7.68854e13 −1.94533
\(66\) 0 0
\(67\) −5.45476e13 −1.09955 −0.549774 − 0.835313i \(-0.685286\pi\)
−0.549774 + 0.835313i \(0.685286\pi\)
\(68\) 0 0
\(69\) 2.76517e13 0.447047
\(70\) 0 0
\(71\) −1.17707e14 −1.53591 −0.767956 − 0.640502i \(-0.778726\pi\)
−0.767956 + 0.640502i \(0.778726\pi\)
\(72\) 0 0
\(73\) 1.31548e14 1.39368 0.696840 − 0.717226i \(-0.254589\pi\)
0.696840 + 0.717226i \(0.254589\pi\)
\(74\) 0 0
\(75\) 4.20530e13 0.363779
\(76\) 0 0
\(77\) −2.89742e14 −2.05746
\(78\) 0 0
\(79\) 2.57200e14 1.50684 0.753422 − 0.657537i \(-0.228402\pi\)
0.753422 + 0.657537i \(0.228402\pi\)
\(80\) 0 0
\(81\) 2.28768e13 0.111111
\(82\) 0 0
\(83\) 3.45415e14 1.39719 0.698595 − 0.715518i \(-0.253809\pi\)
0.698595 + 0.715518i \(0.253809\pi\)
\(84\) 0 0
\(85\) −5.43537e14 −1.83902
\(86\) 0 0
\(87\) 2.05933e14 0.585237
\(88\) 0 0
\(89\) 3.21904e14 0.771438 0.385719 − 0.922616i \(-0.373953\pi\)
0.385719 + 0.922616i \(0.373953\pi\)
\(90\) 0 0
\(91\) 1.22946e15 2.49405
\(92\) 0 0
\(93\) 1.70731e14 0.294233
\(94\) 0 0
\(95\) −7.21083e14 −1.05940
\(96\) 0 0
\(97\) −1.01652e15 −1.27740 −0.638699 − 0.769457i \(-0.720527\pi\)
−0.638699 + 0.769457i \(0.720527\pi\)
\(98\) 0 0
\(99\) −3.88561e14 −0.418982
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.1.1 ✓ 2
3.2 odd 2 36.16.a.c.1.2 2
4.3 odd 2 48.16.a.i.1.1 2
12.11 even 2 144.16.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.16.a.b.1.1 ✓ 2 1.1 even 1 trivial
36.16.a.c.1.2 2 3.2 odd 2
48.16.a.i.1.1 2 4.3 odd 2
144.16.a.r.1.2 2 12.11 even 2