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Properties

Label 112.14.a.b.1.1
Level $112$
Weight $14$
Character 112.1
Self dual yes
Analytic conductor $120.099$
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] = [112,14,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1026] 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: \(120.098640426\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 112.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1026.00 q^{3} +4320.00 q^{5} -117649. q^{7} -541647. q^{9} +8.78731e6 q^{11} -2.04209e7 q^{13} +4.43232e6 q^{15} +1.71946e6 q^{17} +1.09703e8 q^{19} -1.20708e8 q^{21} +6.46760e8 q^{23} -1.20204e9 q^{25} -2.19151e9 q^{27} +7.28867e8 q^{29} -1.02805e9 q^{31} +9.01578e9 q^{33} -5.08244e8 q^{35} +1.42294e10 q^{37} -2.09519e10 q^{39} +4.45445e10 q^{41} +5.46898e10 q^{43} -2.33992e9 q^{45} -4.78683e10 q^{47} +1.38413e10 q^{49} +1.76417e9 q^{51} -1.69987e11 q^{53} +3.79612e10 q^{55} +1.12555e11 q^{57} +3.00766e11 q^{59} +3.69996e11 q^{61} +6.37242e10 q^{63} -8.82184e10 q^{65} +7.87011e11 q^{67} +6.63576e11 q^{69} -5.59441e11 q^{71} +1.21138e11 q^{73} -1.23329e12 q^{75} -1.03382e12 q^{77} -2.90427e11 q^{79} -1.38492e12 q^{81} +3.96511e12 q^{83} +7.42808e9 q^{85} +7.47818e11 q^{87} -6.02592e12 q^{89} +2.40250e12 q^{91} -1.05478e12 q^{93} +4.73917e11 q^{95} +1.13028e13 q^{97} -4.75962e12 q^{99} +O(q^{100})\)

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
\(3\) 1026.00 0.812567 0.406284 − 0.913747i \(-0.366825\pi\)
0.406284 + 0.913747i \(0.366825\pi\)
\(4\) 0 0
\(5\) 4320.00 0.123646 0.0618228 − 0.998087i \(-0.480309\pi\)
0.0618228 + 0.998087i \(0.480309\pi\)
\(6\) 0 0
\(7\) −117649. −0.377964
\(8\) 0 0
\(9\) −541647. −0.339735
\(10\) 0 0
\(11\) 8.78731e6 1.49556 0.747780 − 0.663947i \(-0.231120\pi\)
0.747780 + 0.663947i \(0.231120\pi\)
\(12\) 0 0
\(13\) −2.04209e7 −1.17339 −0.586697 − 0.809807i \(-0.699572\pi\)
−0.586697 + 0.809807i \(0.699572\pi\)
\(14\) 0 0
\(15\) 4.43232e6 0.100470
\(16\) 0 0
\(17\) 1.71946e6 0.0172772 0.00863862 − 0.999963i \(-0.497250\pi\)
0.00863862 + 0.999963i \(0.497250\pi\)
\(18\) 0 0
\(19\) 1.09703e8 0.534958 0.267479 − 0.963564i \(-0.413809\pi\)
0.267479 + 0.963564i \(0.413809\pi\)
\(20\) 0 0
\(21\) −1.20708e8 −0.307121
\(22\) 0 0
\(23\) 6.46760e8 0.910987 0.455494 − 0.890239i \(-0.349463\pi\)
0.455494 + 0.890239i \(0.349463\pi\)
\(24\) 0 0
\(25\) −1.20204e9 −0.984712
\(26\) 0 0
\(27\) −2.19151e9 −1.08862
\(28\) 0 0
\(29\) 7.28867e8 0.227542 0.113771 − 0.993507i \(-0.463707\pi\)
0.113771 + 0.993507i \(0.463707\pi\)
\(30\) 0 0
\(31\) −1.02805e9 −0.208048 −0.104024 − 0.994575i \(-0.533172\pi\)
−0.104024 + 0.994575i \(0.533172\pi\)
\(32\) 0 0
\(33\) 9.01578e9 1.21524
\(34\) 0 0
\(35\) −5.08244e8 −0.0467336
\(36\) 0 0
\(37\) 1.42294e10 0.911749 0.455874 − 0.890044i \(-0.349327\pi\)
0.455874 + 0.890044i \(0.349327\pi\)
\(38\) 0 0
\(39\) −2.09519e10 −0.953461
\(40\) 0 0
\(41\) 4.45445e10 1.46453 0.732266 − 0.681019i \(-0.238463\pi\)
0.732266 + 0.681019i \(0.238463\pi\)
\(42\) 0 0
\(43\) 5.46898e10 1.31935 0.659677 − 0.751549i \(-0.270693\pi\)
0.659677 + 0.751549i \(0.270693\pi\)
\(44\) 0 0
\(45\) −2.33992e9 −0.0420067
\(46\) 0 0
\(47\) −4.78683e10 −0.647757 −0.323879 − 0.946099i \(-0.604987\pi\)
−0.323879 + 0.946099i \(0.604987\pi\)
\(48\) 0 0
\(49\) 1.38413e10 0.142857
\(50\) 0 0
\(51\) 1.76417e9 0.0140389
\(52\) 0 0
\(53\) −1.69987e11 −1.05347 −0.526735 − 0.850029i \(-0.676584\pi\)
−0.526735 + 0.850029i \(0.676584\pi\)
\(54\) 0 0
\(55\) 3.79612e10 0.184919
\(56\) 0 0
\(57\) 1.12555e11 0.434689
\(58\) 0 0
\(59\) 3.00766e11 0.928304 0.464152 − 0.885756i \(-0.346359\pi\)
0.464152 + 0.885756i \(0.346359\pi\)
\(60\) 0 0
\(61\) 3.69996e11 0.919504 0.459752 − 0.888047i \(-0.347938\pi\)
0.459752 + 0.888047i \(0.347938\pi\)
\(62\) 0 0
\(63\) 6.37242e10 0.128408
\(64\) 0 0
\(65\) −8.82184e10 −0.145085
\(66\) 0 0
\(67\) 7.87011e11 1.06291 0.531453 − 0.847088i \(-0.321646\pi\)
0.531453 + 0.847088i \(0.321646\pi\)
\(68\) 0 0
\(69\) 6.63576e11 0.740238
\(70\) 0 0
\(71\) −5.59441e11 −0.518293 −0.259147 − 0.965838i \(-0.583441\pi\)
−0.259147 + 0.965838i \(0.583441\pi\)
\(72\) 0 0
\(73\) 1.21138e11 0.0936872 0.0468436 − 0.998902i \(-0.485084\pi\)
0.0468436 + 0.998902i \(0.485084\pi\)
\(74\) 0 0
\(75\) −1.23329e12 −0.800144
\(76\) 0 0
\(77\) −1.03382e12 −0.565268
\(78\) 0 0
\(79\) −2.90427e11 −0.134419 −0.0672095 − 0.997739i \(-0.521410\pi\)
−0.0672095 + 0.997739i \(0.521410\pi\)
\(80\) 0 0
\(81\) −1.38492e12 −0.544845
\(82\) 0 0
\(83\) 3.96511e12 1.33121 0.665606 − 0.746303i \(-0.268173\pi\)
0.665606 + 0.746303i \(0.268173\pi\)
\(84\) 0 0
\(85\) 7.42808e9 0.00213626
\(86\) 0 0
\(87\) 7.47818e11 0.184893
\(88\) 0 0
\(89\) −6.02592e12 −1.28525 −0.642626 − 0.766180i \(-0.722155\pi\)
−0.642626 + 0.766180i \(0.722155\pi\)
\(90\) 0 0
\(91\) 2.40250e12 0.443501
\(92\) 0 0
\(93\) −1.05478e12 −0.169053
\(94\) 0 0
\(95\) 4.73917e11 0.0661452
\(96\) 0 0
\(97\) 1.13028e13 1.37775 0.688875 − 0.724880i \(-0.258105\pi\)
0.688875 + 0.724880i \(0.258105\pi\)
\(98\) 0 0
\(99\) −4.75962e12 −0.508093
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 112.14.a.b.1.1 1
4.3 odd 2 14.14.a.b.1.1 ✓ 1
12.11 even 2 126.14.a.a.1.1 1
28.3 even 6 98.14.c.b.79.1 2
28.11 odd 6 98.14.c.c.79.1 2
28.19 even 6 98.14.c.b.67.1 2
28.23 odd 6 98.14.c.c.67.1 2
28.27 even 2 98.14.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.14.a.b.1.1 ✓ 1 4.3 odd 2
98.14.a.d.1.1 1 28.27 even 2
98.14.c.b.67.1 2 28.19 even 6
98.14.c.b.79.1 2 28.3 even 6
98.14.c.c.67.1 2 28.23 odd 6
98.14.c.c.79.1 2 28.11 odd 6
112.14.a.b.1.1 1 1.1 even 1 trivial
126.14.a.a.1.1 1 12.11 even 2