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Properties

Label 112.8.i.b.81.1
Level $112$
Weight $8$
Character 112.81
Analytic conductor $34.987$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [112,8,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,56] 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: no
Analytic conductor: \(34.9871228542\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{949})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(-7.45146 + 12.9063i\) of defining polynomial
Character \(\chi\) \(=\) 112.81
Dual form 112.8.i.b.65.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.40292 + 2.42993i) q^{3} +(219.141 + 379.563i) q^{5} +(893.282 - 159.971i) q^{7} +(1089.56 + 1887.18i) q^{9} +(-2740.43 + 4746.57i) q^{11} +4006.54 q^{13} -1229.75 q^{15} +(14011.5 - 24268.6i) q^{17} +(11920.6 + 20647.0i) q^{19} +(-864.487 + 2395.04i) q^{21} +(-36877.3 - 63873.3i) q^{23} +(-56983.0 + 98697.4i) q^{25} -12250.7 q^{27} -98721.3 q^{29} +(-23743.4 + 41124.8i) q^{31} +(-7689.23 - 13318.1i) q^{33} +(256474. + 304001. i) q^{35} +(50031.2 + 86656.7i) q^{37} +(-5620.87 + 9735.63i) q^{39} +489123. q^{41} -299600. q^{43} +(-477536. + 827116. i) q^{45} +(481369. + 833756. i) q^{47} +(772362. - 285798. i) q^{49} +(39314.1 + 68093.9i) q^{51} +(-918933. + 1.59164e6i) q^{53} -2.40216e6 q^{55} -66894.5 q^{57} +(-7255.29 + 12566.5i) q^{59} +(-1.01469e6 - 1.75749e6i) q^{61} +(1.27518e6 + 1.51148e6i) q^{63} +(877997. + 1.52074e6i) q^{65} +(-1.48449e6 + 2.57121e6i) q^{67} +206944. q^{69} +4.34296e6 q^{71} +(-750529. + 1.29995e6i) q^{73} +(-159885. - 276929. i) q^{75} +(-1.68867e6 + 4.67841e6i) q^{77} +(-886182. - 1.53491e6i) q^{79} +(-2.36569e6 + 4.09749e6i) q^{81} +1.57509e6 q^{83} +1.22820e7 q^{85} +(138498. - 239886. i) q^{87} +(-4.39727e6 - 7.61629e6i) q^{89} +(3.57897e6 - 640929. i) q^{91} +(-66620.4 - 115390. i) q^{93} +(-5.22457e6 + 9.04922e6i) q^{95} -1.03493e7 q^{97} -1.19435e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{3} + 14 q^{5} + 1848 q^{7} + 908 q^{9} + 2408 q^{11} + 21448 q^{13} - 52360 q^{15} + 35098 q^{17} - 2408 q^{19} - 92302 q^{21} - 61684 q^{23} - 215856 q^{25} + 83216 q^{27} - 191320 q^{29} - 166012 q^{31}+ \cdots - 43942528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −1.40292 + 2.42993i −0.0299992 + 0.0519601i −0.880635 − 0.473795i \(-0.842884\pi\)
0.850636 + 0.525755i \(0.176217\pi\)
\(4\) 0 0
\(5\) 219.141 + 379.563i 0.784022 + 1.35797i 0.929581 + 0.368617i \(0.120169\pi\)
−0.145559 + 0.989350i \(0.546498\pi\)
\(6\) 0 0
\(7\) 893.282 − 159.971i 0.984340 − 0.176278i
\(8\) 0 0
\(9\) 1089.56 + 1887.18i 0.498200 + 0.862908i
\(10\) 0 0
\(11\) −2740.43 + 4746.57i −0.620790 + 1.07524i 0.368549 + 0.929609i \(0.379855\pi\)
−0.989339 + 0.145632i \(0.953478\pi\)
\(12\) 0 0
\(13\) 4006.54 0.505787 0.252894 − 0.967494i \(-0.418618\pi\)
0.252894 + 0.967494i \(0.418618\pi\)
\(14\) 0 0
\(15\) −1229.75 −0.0940800
\(16\) 0 0
\(17\) 14011.5 − 24268.6i 0.691693 − 1.19805i −0.279590 − 0.960119i \(-0.590199\pi\)
0.971283 − 0.237927i \(-0.0764681\pi\)
\(18\) 0 0
\(19\) 11920.6 + 20647.0i 0.398712 + 0.690590i 0.993567 − 0.113243i \(-0.0361239\pi\)
−0.594855 + 0.803833i \(0.702791\pi\)
\(20\) 0 0
\(21\) −864.487 + 2395.04i −0.0203700 + 0.0564346i
\(22\) 0 0
\(23\) −36877.3 − 63873.3i −0.631992 − 1.09464i −0.987144 − 0.159833i \(-0.948904\pi\)
0.355152 − 0.934808i \(-0.384429\pi\)
\(24\) 0 0
\(25\) −56983.0 + 98697.4i −0.729382 + 1.26333i
\(26\) 0 0
\(27\) −12250.7 −0.119781
\(28\) 0 0
\(29\) −98721.3 −0.751653 −0.375827 − 0.926690i \(-0.622641\pi\)
−0.375827 + 0.926690i \(0.622641\pi\)
\(30\) 0 0
\(31\) −23743.4 + 41124.8i −0.143145 + 0.247935i −0.928680 − 0.370883i \(-0.879055\pi\)
0.785534 + 0.618818i \(0.212388\pi\)
\(32\) 0 0
\(33\) −7689.23 − 13318.1i −0.0372464 − 0.0645126i
\(34\) 0 0
\(35\) 256474. + 304001.i 1.01112 + 1.19850i
\(36\) 0 0
\(37\) 50031.2 + 86656.7i 0.162381 + 0.281252i 0.935722 − 0.352738i \(-0.114749\pi\)
−0.773341 + 0.633990i \(0.781416\pi\)
\(38\) 0 0
\(39\) −5620.87 + 9735.63i −0.0151732 + 0.0262808i
\(40\) 0 0
\(41\) 489123. 1.10834 0.554172 − 0.832402i \(-0.313035\pi\)
0.554172 + 0.832402i \(0.313035\pi\)
\(42\) 0 0
\(43\) −299600. −0.574649 −0.287324 − 0.957833i \(-0.592766\pi\)
−0.287324 + 0.957833i \(0.592766\pi\)
\(44\) 0 0
\(45\) −477536. + 827116.i −0.781200 + 1.35308i
\(46\) 0 0
\(47\) 481369. + 833756.i 0.676295 + 1.17138i 0.976089 + 0.217373i \(0.0697487\pi\)
−0.299794 + 0.954004i \(0.596918\pi\)
\(48\) 0 0
\(49\) 772362. − 285798.i 0.937852 − 0.347034i
\(50\) 0 0
\(51\) 39314.1 + 68093.9i 0.0415004 + 0.0718808i
\(52\) 0 0
\(53\) −918933. + 1.59164e6i −0.847849 + 1.46852i 0.0352758 + 0.999378i \(0.488769\pi\)
−0.883124 + 0.469139i \(0.844564\pi\)
\(54\) 0 0
\(55\) −2.40216e6 −1.94685
\(56\) 0 0
\(57\) −66894.5 −0.0478441
\(58\) 0 0
\(59\) −7255.29 + 12566.5i −0.00459910 + 0.00796587i −0.868316 − 0.496012i \(-0.834797\pi\)
0.863717 + 0.503978i \(0.168131\pi\)
\(60\) 0 0
\(61\) −1.01469e6 − 1.75749e6i −0.572370 − 0.991374i −0.996322 − 0.0856896i \(-0.972691\pi\)
0.423952 − 0.905685i \(-0.360643\pi\)
\(62\) 0 0
\(63\) 1.27518e6 + 1.51148e6i 0.642510 + 0.761574i
\(64\) 0 0
\(65\) 877997. + 1.52074e6i 0.396549 + 0.686842i
\(66\) 0 0
\(67\) −1.48449e6 + 2.57121e6i −0.602997 + 1.04442i 0.389368 + 0.921082i \(0.372694\pi\)
−0.992365 + 0.123339i \(0.960640\pi\)
\(68\) 0 0
\(69\) 206944. 0.0758369
\(70\) 0 0
\(71\) 4.34296e6 1.44006 0.720031 − 0.693942i \(-0.244128\pi\)
0.720031 + 0.693942i \(0.244128\pi\)
\(72\) 0 0
\(73\) −750529. + 1.29995e6i −0.225807 + 0.391109i −0.956561 − 0.291531i \(-0.905835\pi\)
0.730754 + 0.682641i \(0.239169\pi\)
\(74\) 0 0
\(75\) −159885. − 276929.i −0.0437617 − 0.0757975i
\(76\) 0 0
\(77\) −1.68867e6 + 4.67841e6i −0.421528 + 1.16783i
\(78\) 0 0
\(79\) −886182. − 1.53491e6i −0.202222 − 0.350259i 0.747022 − 0.664799i \(-0.231483\pi\)
−0.949244 + 0.314541i \(0.898150\pi\)
\(80\) 0 0
\(81\) −2.36569e6 + 4.09749e6i −0.494607 + 0.856684i
\(82\) 0 0
\(83\) 1.57509e6 0.302366 0.151183 − 0.988506i \(-0.451692\pi\)
0.151183 + 0.988506i \(0.451692\pi\)
\(84\) 0 0
\(85\) 1.22820e7 2.16921
\(86\) 0 0
\(87\) 138498. − 239886.i 0.0225490 − 0.0390560i
\(88\) 0 0
\(89\) −4.39727e6 − 7.61629e6i −0.661177 − 1.14519i −0.980307 − 0.197481i \(-0.936724\pi\)
0.319130 − 0.947711i \(-0.396609\pi\)
\(90\) 0 0
\(91\) 3.57897e6 − 640929.i 0.497867 − 0.0891590i
\(92\) 0 0
\(93\) −66620.4 − 115390.i −0.00858849 − 0.0148757i
\(94\) 0 0
\(95\) −5.22457e6 + 9.04922e6i −0.625199 + 1.08288i
\(96\) 0 0
\(97\) −1.03493e7 −1.15135 −0.575676 − 0.817678i \(-0.695261\pi\)
−0.575676 + 0.817678i \(0.695261\pi\)
\(98\) 0 0
\(99\) −1.19435e7 −1.23711
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 112.8.i.b.81.1 4
4.3 odd 2 14.8.c.b.11.2 yes 4
7.2 even 3 inner 112.8.i.b.65.1 4
12.11 even 2 126.8.g.d.109.1 4
28.3 even 6 98.8.a.d.1.2 2
28.11 odd 6 98.8.a.f.1.1 2
28.19 even 6 98.8.c.m.79.1 4
28.23 odd 6 14.8.c.b.9.2 ✓ 4
28.27 even 2 98.8.c.m.67.1 4
84.23 even 6 126.8.g.d.37.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.2 ✓ 4 28.23 odd 6
14.8.c.b.11.2 yes 4 4.3 odd 2
98.8.a.d.1.2 2 28.3 even 6
98.8.a.f.1.1 2 28.11 odd 6
98.8.c.m.67.1 4 28.27 even 2
98.8.c.m.79.1 4 28.19 even 6
112.8.i.b.65.1 4 7.2 even 3 inner
112.8.i.b.81.1 4 1.1 even 1 trivial
126.8.g.d.37.1 4 84.23 even 6
126.8.g.d.109.1 4 12.11 even 2