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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-266] 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: \(86.0544362227\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.2
Root \(-100.397 + 173.893i\) of defining polynomial
Character \(\chi\) \(=\) 112.81
Dual form 112.12.i.a.65.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-234.294 + 405.809i) q^{3} +(-3326.37 - 5761.45i) q^{5} +(-12255.3 + 42745.0i) q^{7} +(-21214.1 - 36743.8i) q^{9} +(60907.1 - 105494. i) q^{11} -1.81677e6 q^{13} +3.11740e6 q^{15} +(-2.70113e6 + 4.67849e6i) q^{17} +(-5.10442e6 - 8.84111e6i) q^{19} +(-1.44750e7 - 1.49882e7i) q^{21} +(1.88132e7 + 3.25854e7i) q^{23} +(2.28454e6 - 3.95694e6i) q^{25} -6.31277e7 q^{27} -1.87020e7 q^{29} +(-1.22797e8 + 2.12690e8i) q^{31} +(2.85404e7 + 4.94333e7i) q^{33} +(2.87039e8 - 7.15776e7i) q^{35} +(-3.25911e8 - 5.64495e8i) q^{37} +(4.25659e8 - 7.37263e8i) q^{39} -6.38428e8 q^{41} -8.54732e8 q^{43} +(-1.41132e8 + 2.44447e8i) q^{45} +(-4.23660e8 - 7.33801e8i) q^{47} +(-1.67694e9 - 1.04770e9i) q^{49} +(-1.26572e9 - 2.19228e9i) q^{51} +(2.11267e9 - 3.65925e9i) q^{53} -8.10399e8 q^{55} +4.78374e9 q^{57} +(-1.10225e9 + 1.90916e9i) q^{59} +(4.84994e9 + 8.40035e9i) q^{61} +(1.83060e9 - 4.56488e8i) q^{63} +(6.04326e9 + 1.04672e10i) q^{65} +(-4.86446e9 + 8.42549e9i) q^{67} -1.76313e10 q^{69} +2.83613e10 q^{71} +(1.83417e9 - 3.17688e9i) q^{73} +(1.07051e9 + 1.85417e9i) q^{75} +(3.76291e9 + 3.89633e9i) q^{77} +(6.34771e9 + 1.09946e10i) q^{79} +(1.85485e10 - 3.21269e10i) q^{81} +4.03358e10 q^{83} +3.59398e10 q^{85} +(4.38177e9 - 7.58944e9i) q^{87} +(1.64368e10 + 2.84694e10i) q^{89} +(2.22651e10 - 7.76579e10i) q^{91} +(-5.75412e10 - 9.96642e10i) q^{93} +(-3.39584e10 + 5.88177e10i) q^{95} +1.02279e11 q^{97} -5.16834e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 266 q^{3} + 7504 q^{5} + 42224 q^{7} - 123520 q^{9} - 213026 q^{11} - 2609712 q^{13} - 2275500 q^{15} + 8854244 q^{17} - 7232806 q^{19} + 49275128 q^{21} + 10649134 q^{23} - 119407256 q^{25} - 19012196 q^{27}+ \cdots - 393415805736 q^{99}+O(q^{100}) \) 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
\(3\) −234.294 + 405.809i −0.556666 + 0.964174i 0.441106 + 0.897455i \(0.354586\pi\)
−0.997772 + 0.0667186i \(0.978747\pi\)
\(4\) 0 0
\(5\) −3326.37 − 5761.45i −0.476032 − 0.824511i 0.523591 − 0.851970i \(-0.324592\pi\)
−0.999623 + 0.0274583i \(0.991259\pi\)
\(6\) 0 0
\(7\) −12255.3 + 42745.0i −0.275603 + 0.961271i
\(8\) 0 0
\(9\) −21214.1 − 36743.8i −0.119754 − 0.207420i
\(10\) 0 0
\(11\) 60907.1 − 105494.i 0.114027 − 0.197501i −0.803363 − 0.595489i \(-0.796958\pi\)
0.917390 + 0.397988i \(0.130292\pi\)
\(12\) 0 0
\(13\) −1.81677e6 −1.35710 −0.678550 − 0.734554i \(-0.737391\pi\)
−0.678550 + 0.734554i \(0.737391\pi\)
\(14\) 0 0
\(15\) 3.11740e6 1.05996
\(16\) 0 0
\(17\) −2.70113e6 + 4.67849e6i −0.461398 + 0.799165i −0.999031 − 0.0440143i \(-0.985985\pi\)
0.537633 + 0.843179i \(0.319319\pi\)
\(18\) 0 0
\(19\) −5.10442e6 − 8.84111e6i −0.472935 − 0.819148i 0.526585 − 0.850122i \(-0.323472\pi\)
−0.999520 + 0.0309748i \(0.990139\pi\)
\(20\) 0 0
\(21\) −1.44750e7 − 1.49882e7i −0.773414 − 0.800837i
\(22\) 0 0
\(23\) 1.88132e7 + 3.25854e7i 0.609481 + 1.05565i 0.991326 + 0.131425i \(0.0419554\pi\)
−0.381845 + 0.924226i \(0.624711\pi\)
\(24\) 0 0
\(25\) 2.28454e6 − 3.95694e6i 0.0467873 − 0.0810381i
\(26\) 0 0
\(27\) −6.31277e7 −0.846680
\(28\) 0 0
\(29\) −1.87020e7 −0.169316 −0.0846581 − 0.996410i \(-0.526980\pi\)
−0.0846581 + 0.996410i \(0.526980\pi\)
\(30\) 0 0
\(31\) −1.22797e8 + 2.12690e8i −0.770367 + 1.33431i 0.166995 + 0.985958i \(0.446594\pi\)
−0.937362 + 0.348357i \(0.886740\pi\)
\(32\) 0 0
\(33\) 2.85404e7 + 4.94333e7i 0.126950 + 0.219884i
\(34\) 0 0
\(35\) 2.87039e8 − 7.15776e7i 0.923775 − 0.230358i
\(36\) 0 0
\(37\) −3.25911e8 − 5.64495e8i −0.772663 − 1.33829i −0.936099 − 0.351737i \(-0.885591\pi\)
0.163436 − 0.986554i \(-0.447742\pi\)
\(38\) 0 0
\(39\) 4.25659e8 − 7.37263e8i 0.755452 − 1.30848i
\(40\) 0 0
\(41\) −6.38428e8 −0.860598 −0.430299 − 0.902686i \(-0.641592\pi\)
−0.430299 + 0.902686i \(0.641592\pi\)
\(42\) 0 0
\(43\) −8.54732e8 −0.886652 −0.443326 − 0.896360i \(-0.646202\pi\)
−0.443326 + 0.896360i \(0.646202\pi\)
\(44\) 0 0
\(45\) −1.41132e8 + 2.44447e8i −0.114013 + 0.197477i
\(46\) 0 0
\(47\) −4.23660e8 − 7.33801e8i −0.269451 − 0.466702i 0.699269 − 0.714858i \(-0.253509\pi\)
−0.968720 + 0.248156i \(0.920175\pi\)
\(48\) 0 0
\(49\) −1.67694e9 − 1.04770e9i −0.848086 − 0.529859i
\(50\) 0 0
\(51\) −1.26572e9 − 2.19228e9i −0.513689 − 0.889736i
\(52\) 0 0
\(53\) 2.11267e9 − 3.65925e9i 0.693927 − 1.20192i −0.276614 − 0.960981i \(-0.589212\pi\)
0.970541 − 0.240936i \(-0.0774544\pi\)
\(54\) 0 0
\(55\) −8.10399e8 −0.217122
\(56\) 0 0
\(57\) 4.78374e9 1.05307
\(58\) 0 0
\(59\) −1.10225e9 + 1.90916e9i −0.200722 + 0.347661i −0.948761 − 0.315994i \(-0.897662\pi\)
0.748039 + 0.663655i \(0.230995\pi\)
\(60\) 0 0
\(61\) 4.84994e9 + 8.40035e9i 0.735229 + 1.27345i 0.954623 + 0.297817i \(0.0962587\pi\)
−0.219394 + 0.975636i \(0.570408\pi\)
\(62\) 0 0
\(63\) 1.83060e9 − 4.56488e8i 0.232391 − 0.0579504i
\(64\) 0 0
\(65\) 6.04326e9 + 1.04672e10i 0.646023 + 1.11894i
\(66\) 0 0
\(67\) −4.86446e9 + 8.42549e9i −0.440173 + 0.762401i −0.997702 − 0.0677559i \(-0.978416\pi\)
0.557529 + 0.830157i \(0.311749\pi\)
\(68\) 0 0
\(69\) −1.76313e10 −1.35711
\(70\) 0 0
\(71\) 2.83613e10 1.86554 0.932770 − 0.360472i \(-0.117384\pi\)
0.932770 + 0.360472i \(0.117384\pi\)
\(72\) 0 0
\(73\) 1.83417e9 − 3.17688e9i 0.103553 − 0.179360i −0.809593 − 0.586992i \(-0.800312\pi\)
0.913146 + 0.407632i \(0.133645\pi\)
\(74\) 0 0
\(75\) 1.07051e9 + 1.85417e9i 0.0520898 + 0.0902223i
\(76\) 0 0
\(77\) 3.76291e9 + 3.89633e9i 0.158426 + 0.164043i
\(78\) 0 0
\(79\) 6.34771e9 + 1.09946e10i 0.232096 + 0.402002i 0.958425 − 0.285345i \(-0.0921083\pi\)
−0.726329 + 0.687348i \(0.758775\pi\)
\(80\) 0 0
\(81\) 1.85485e10 − 3.21269e10i 0.591072 − 1.02377i
\(82\) 0 0
\(83\) 4.03358e10 1.12399 0.561994 − 0.827141i \(-0.310034\pi\)
0.561994 + 0.827141i \(0.310034\pi\)
\(84\) 0 0
\(85\) 3.59398e10 0.878561
\(86\) 0 0
\(87\) 4.38177e9 − 7.58944e9i 0.0942526 − 0.163250i
\(88\) 0 0
\(89\) 1.64368e10 + 2.84694e10i 0.312013 + 0.540423i 0.978798 − 0.204827i \(-0.0656633\pi\)
−0.666785 + 0.745250i \(0.732330\pi\)
\(90\) 0 0
\(91\) 2.22651e10 − 7.76579e10i 0.374021 − 1.30454i
\(92\) 0 0
\(93\) −5.75412e10 − 9.96642e10i −0.857674 − 1.48554i
\(94\) 0 0
\(95\) −3.39584e10 + 5.88177e10i −0.450264 + 0.779881i
\(96\) 0 0
\(97\) 1.02279e11 1.20932 0.604661 − 0.796483i \(-0.293308\pi\)
0.604661 + 0.796483i \(0.293308\pi\)
\(98\) 0 0
\(99\) −5.16834e9 −0.0546208
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 112.12.i.a.81.2 8
4.3 odd 2 14.12.c.a.11.3 yes 8
7.2 even 3 inner 112.12.i.a.65.2 8
12.11 even 2 126.12.g.e.109.4 8
28.3 even 6 98.12.a.l.1.3 4
28.11 odd 6 98.12.a.j.1.2 4
28.19 even 6 98.12.c.l.79.2 8
28.23 odd 6 14.12.c.a.9.3 ✓ 8
28.27 even 2 98.12.c.l.67.2 8
84.23 even 6 126.12.g.e.37.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 28.23 odd 6
14.12.c.a.11.3 yes 8 4.3 odd 2
98.12.a.j.1.2 4 28.11 odd 6
98.12.a.l.1.3 4 28.3 even 6
98.12.c.l.67.2 8 28.27 even 2
98.12.c.l.79.2 8 28.19 even 6
112.12.i.a.65.2 8 7.2 even 3 inner
112.12.i.a.81.2 8 1.1 even 1 trivial
126.12.g.e.37.4 8 84.23 even 6
126.12.g.e.109.4 8 12.11 even 2