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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 65.2
Root \(-100.397 - 173.893i\) of defining polynomial
Character \(\chi\) \(=\) 112.65
Dual form 112.12.i.a.81.2

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

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

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 −0.997772 − 0.0667186i \(-0.978747\pi\)
0.441106 − 0.897455i \(-0.354586\pi\)
\(4\) 0 0
\(5\) −3326.37 + 5761.45i −0.476032 + 0.824511i −0.999623 − 0.0274583i \(-0.991259\pi\)
0.523591 + 0.851970i \(0.324592\pi\)
\(6\) 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 0.917390 − 0.397988i \(-0.130292\pi\)
−0.803363 + 0.595489i \(0.796958\pi\)
\(12\) 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 0.537633 − 0.843179i \(-0.319319\pi\)
−0.999031 + 0.0440143i \(0.985985\pi\)
\(18\) 0 0
\(19\) −5.10442e6 + 8.84111e6i −0.472935 + 0.819148i −0.999520 − 0.0309748i \(-0.990139\pi\)
0.526585 + 0.850122i \(0.323472\pi\)
\(20\) 0 0
\(21\) −1.44750e7 + 1.49882e7i −0.773414 + 0.800837i
\(22\) 0 0
\(23\) 1.88132e7 − 3.25854e7i 0.609481 − 1.05565i −0.381845 − 0.924226i \(-0.624711\pi\)
0.991326 − 0.131425i \(-0.0419554\pi\)
\(24\) 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 −0.937362 − 0.348357i \(-0.886740\pi\)
0.166995 − 0.985958i \(-0.446594\pi\)
\(32\) 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 0.163436 + 0.986554i \(0.447742\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(38\) 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 −0.968720 − 0.248156i \(-0.920175\pi\)
0.699269 + 0.714858i \(0.253509\pi\)
\(48\) 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 0.970541 + 0.240936i \(0.0774544\pi\)
−0.276614 + 0.960981i \(0.589212\pi\)
\(54\) 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 0.748039 − 0.663655i \(-0.230995\pi\)
−0.948761 + 0.315994i \(0.897662\pi\)
\(60\) 0 0
\(61\) 4.84994e9 − 8.40035e9i 0.735229 − 1.27345i −0.219394 − 0.975636i \(-0.570408\pi\)
0.954623 − 0.297817i \(-0.0962587\pi\)
\(62\) 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 0.557529 − 0.830157i \(-0.311749\pi\)
−0.997702 + 0.0677559i \(0.978416\pi\)
\(68\) 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 0.913146 − 0.407632i \(-0.133645\pi\)
−0.809593 + 0.586992i \(0.800312\pi\)
\(74\) 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 −0.726329 − 0.687348i \(-0.758775\pi\)
0.958425 + 0.285345i \(0.0921083\pi\)
\(80\) 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 −0.666785 − 0.745250i \(-0.732330\pi\)
0.978798 + 0.204827i \(0.0656633\pi\)
\(90\) 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.65.2 8
4.3 odd 2 14.12.c.a.9.3 ✓ 8
7.4 even 3 inner 112.12.i.a.81.2 8
12.11 even 2 126.12.g.e.37.4 8
28.3 even 6 98.12.c.l.67.2 8
28.11 odd 6 14.12.c.a.11.3 yes 8
28.19 even 6 98.12.a.l.1.3 4
28.23 odd 6 98.12.a.j.1.2 4
28.27 even 2 98.12.c.l.79.2 8
84.11 even 6 126.12.g.e.109.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 4.3 odd 2
14.12.c.a.11.3 yes 8 28.11 odd 6
98.12.a.j.1.2 4 28.23 odd 6
98.12.a.l.1.3 4 28.19 even 6
98.12.c.l.67.2 8 28.3 even 6
98.12.c.l.79.2 8 28.27 even 2
112.12.i.a.65.2 8 1.1 even 1 trivial
112.12.i.a.81.2 8 7.4 even 3 inner
126.12.g.e.37.4 8 12.11 even 2
126.12.g.e.109.4 8 84.11 even 6