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Properties

Label 792.2.f.h
Level $792$
Weight $2$
Character orbit 792.f
Analytic conductor $6.324$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [792,2,Mod(397,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.397"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 792.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.32415184009\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: 20.0.74831334220841134637329678336.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + \beta_{11} q^{4} + (\beta_{8} - \beta_{2}) q^{5} + (\beta_{17} - \beta_{3}) q^{7} + (\beta_{16} + \beta_{2}) q^{8} + ( - \beta_{9} + \beta_{6} + \cdots - \beta_{3}) q^{10} - \beta_{2} q^{11}+ \cdots + (2 \beta_{19} + 2 \beta_{18} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{4} + 8 q^{10} - 12 q^{16} - 12 q^{25} + 24 q^{28} + 40 q^{31} - 24 q^{34} - 32 q^{40} + 40 q^{46} + 36 q^{49} + 56 q^{52} - 16 q^{55} + 24 q^{58} + 4 q^{64} + 56 q^{70} - 56 q^{73} - 8 q^{76}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{17} + 5\nu^{13} - 14\nu^{11} + 32\nu^{9} - 16\nu^{7} + 80\nu^{5} - 96\nu^{3} + 384\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} + 2\nu^{12} - \nu^{10} + 8\nu^{6} + 32\nu^{4} - 16\nu^{2} - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{16} - 5\nu^{12} + 14\nu^{10} - 32\nu^{8} + 16\nu^{6} - 80\nu^{4} + 96\nu^{2} - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} - \nu^{10} - 4\nu^{8} - 16\nu^{4} + 80\nu^{2} - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{18} - 2\nu^{16} + 3\nu^{14} - 12\nu^{12} + 4\nu^{10} - 48\nu^{8} + 144\nu^{6} - 64\nu^{4} - 64\nu^{2} - 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{18} + 5\nu^{14} - 14\nu^{12} + 32\nu^{10} - 16\nu^{8} + 80\nu^{6} - 96\nu^{4} + 384\nu^{2} ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{19} - 4 \nu^{17} - 3 \nu^{15} + 14 \nu^{13} + 16 \nu^{11} + 16 \nu^{9} - 48 \nu^{7} + \cdots - 512 \nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{16} + 4\nu^{14} - 7\nu^{12} + 14\nu^{10} - 16\nu^{8} + 16\nu^{6} - 48\nu^{4} + 96\nu^{2} - 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{19} - 4\nu^{17} + 3\nu^{15} - 22\nu^{13} + 32\nu^{11} + 16\nu^{9} + 176\nu^{7} + 160\nu^{5} + 128\nu^{3} ) / 1024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{18} + 2 \nu^{16} - 5 \nu^{14} + 8 \nu^{12} - 28 \nu^{10} + 64 \nu^{8} - 112 \nu^{6} + \cdots + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{19} - 2 \nu^{17} + 5 \nu^{15} - 8 \nu^{13} + 28 \nu^{11} - 64 \nu^{9} + 112 \nu^{7} + \cdots - 512 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{18} - 2 \nu^{16} + 7 \nu^{14} - 4 \nu^{12} + 52 \nu^{10} - 80 \nu^{8} + 80 \nu^{6} + 320 \nu^{4} + \cdots - 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{19} - 6 \nu^{17} + 7 \nu^{15} - 24 \nu^{13} + 44 \nu^{11} - 80 \nu^{9} + 80 \nu^{7} + \cdots - 768 \nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3 \nu^{19} + 8 \nu^{17} - 7 \nu^{15} + 2 \nu^{13} - 72 \nu^{11} + 112 \nu^{9} - 176 \nu^{7} + \cdots + 512 \nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2\nu^{19} - \nu^{17} + 2\nu^{15} - \nu^{13} + 38\nu^{11} - 48\nu^{9} - 16\nu^{7} + 112\nu^{5} + 224\nu^{3} - 128\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3 \nu^{18} + 6 \nu^{16} - 7 \nu^{14} + 24 \nu^{12} - 44 \nu^{10} + 80 \nu^{8} - 80 \nu^{6} + \cdots + 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 3 \nu^{19} - 10 \nu^{17} + 7 \nu^{15} - 12 \nu^{13} + 36 \nu^{11} - 176 \nu^{9} + 144 \nu^{7} + \cdots - 768 \nu ) / 1024 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( \nu^{19} - 3\nu^{15} + 2\nu^{13} + 8\nu^{11} - 16\nu^{9} + 64\nu^{3} - 384\nu ) / 256 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - \beta_{11} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} + 2\beta_{12} + 2\beta_{8} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{17} + 2\beta_{13} + \beta_{11} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{18} + 2\beta_{16} + 2\beta_{15} - 2\beta_{14} + 2\beta_{10} + 2\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{17} - 2\beta_{13} - \beta_{11} - \beta_{9} + 2\beta_{7} + 3\beta_{6} - \beta_{5} - 3\beta_{4} + 4\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4 \beta_{19} - 4 \beta_{16} - 2 \beta_{15} - 4 \beta_{14} + 6 \beta_{12} + 4 \beta_{10} - 2 \beta_{8} + \cdots + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{17} - 6 \beta_{13} + 5 \beta_{11} + 5 \beta_{9} + 8 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + \cdots - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{18} - 2 \beta_{16} - 6 \beta_{15} + 10 \beta_{14} - 4 \beta_{12} + 6 \beta_{10} + \cdots - 5 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3 \beta_{17} - 2 \beta_{13} + 11 \beta_{11} + 3 \beta_{9} + 14 \beta_{7} - 9 \beta_{6} - 13 \beta_{5} + \cdots + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4 \beta_{19} - 20 \beta_{18} + 16 \beta_{16} - 26 \beta_{15} - 8 \beta_{14} - 14 \beta_{12} + \cdots - 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 17 \beta_{17} - 14 \beta_{13} - 39 \beta_{11} + 9 \beta_{9} - 12 \beta_{7} - 39 \beta_{6} - 15 \beta_{5} + \cdots + 47 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8 \beta_{19} - 14 \beta_{18} - 14 \beta_{16} - 30 \beta_{15} - 42 \beta_{14} + 40 \beta_{12} + \cdots + 7 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 39 \beta_{17} + 22 \beta_{13} - 49 \beta_{11} + 23 \beta_{9} - 22 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + \cdots - 95 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 108 \beta_{19} - 8 \beta_{18} + 52 \beta_{16} - 18 \beta_{15} + 20 \beta_{14} + 46 \beta_{12} + \cdots - 87 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 53 \beta_{17} + 42 \beta_{13} - 3 \beta_{11} - 163 \beta_{9} + 32 \beta_{7} + 37 \beta_{6} - 11 \beta_{5} + \cdots - 101 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 48 \beta_{19} - 50 \beta_{18} + 134 \beta_{16} - 22 \beta_{15} - 126 \beta_{14} + 20 \beta_{12} + \cdots - 109 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 309 \beta_{17} + 14 \beta_{13} - 13 \beta_{11} - 213 \beta_{9} - 26 \beta_{7} - 81 \beta_{6} + \cdots - 307 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 236 \beta_{19} + 4 \beta_{18} + 24 \beta_{16} - 10 \beta_{15} + 368 \beta_{14} - 22 \beta_{12} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/792\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
397.1
1.38682 − 0.277013i
1.38682 + 0.277013i
1.25295 − 0.655834i
1.25295 + 0.655834i
1.13849 − 0.838954i
1.13849 + 0.838954i
0.689696 − 1.23463i
0.689696 + 1.23463i
0.484785 − 1.32853i
0.484785 + 1.32853i
−0.484785 − 1.32853i
−0.484785 + 1.32853i
−0.689696 − 1.23463i
−0.689696 + 1.23463i
−1.13849 − 0.838954i
−1.13849 + 0.838954i
−1.25295 − 0.655834i
−1.25295 + 0.655834i
−1.38682 − 0.277013i
−1.38682 + 0.277013i
−1.38682 − 0.277013i 0 1.84653 + 0.768333i − 0.972802i 0 −0.372730 −2.34796 − 1.57705i 0 −0.269479 + 1.34910i
397.2 −1.38682 + 0.277013i 0 1.84653 − 0.768333i 0.972802i 0 −0.372730 −2.34796 + 1.57705i 0 −0.269479 − 1.34910i
397.3 −1.25295 − 0.655834i 0 1.13976 + 1.64345i 2.54164i 0 4.84329 −0.350232 − 2.80666i 0 1.66690 − 3.18455i
397.4 −1.25295 + 0.655834i 0 1.13976 − 1.64345i − 2.54164i 0 4.84329 −0.350232 + 2.80666i 0 1.66690 + 3.18455i
397.5 −1.13849 − 0.838954i 0 0.592314 + 1.91028i − 3.89064i 0 −1.77496 0.928292 − 2.67175i 0 −3.26407 + 4.42945i
397.6 −1.13849 + 0.838954i 0 0.592314 − 1.91028i 3.89064i 0 −1.77496 0.928292 + 2.67175i 0 −3.26407 − 4.42945i
397.7 −0.689696 − 1.23463i 0 −1.04864 + 1.70304i 0.752075i 0 −3.95740 2.82588 + 0.120102i 0 0.928537 − 0.518703i
397.8 −0.689696 + 1.23463i 0 −1.04864 − 1.70304i − 0.752075i 0 −3.95740 2.82588 − 0.120102i 0 0.928537 + 0.518703i
397.9 −0.484785 − 1.32853i 0 −1.52997 + 1.28810i 2.21156i 0 1.26179 2.45298 + 1.40815i 0 2.93811 − 1.07213i
397.10 −0.484785 + 1.32853i 0 −1.52997 − 1.28810i − 2.21156i 0 1.26179 2.45298 − 1.40815i 0 2.93811 + 1.07213i
397.11 0.484785 − 1.32853i 0 −1.52997 − 1.28810i 2.21156i 0 1.26179 −2.45298 + 1.40815i 0 2.93811 + 1.07213i
397.12 0.484785 + 1.32853i 0 −1.52997 + 1.28810i − 2.21156i 0 1.26179 −2.45298 − 1.40815i 0 2.93811 − 1.07213i
397.13 0.689696 − 1.23463i 0 −1.04864 − 1.70304i 0.752075i 0 −3.95740 −2.82588 + 0.120102i 0 0.928537 + 0.518703i
397.14 0.689696 + 1.23463i 0 −1.04864 + 1.70304i − 0.752075i 0 −3.95740 −2.82588 − 0.120102i 0 0.928537 − 0.518703i
397.15 1.13849 − 0.838954i 0 0.592314 − 1.91028i − 3.89064i 0 −1.77496 −0.928292 − 2.67175i 0 −3.26407 − 4.42945i
397.16 1.13849 + 0.838954i 0 0.592314 + 1.91028i 3.89064i 0 −1.77496 −0.928292 + 2.67175i 0 −3.26407 + 4.42945i
397.17 1.25295 − 0.655834i 0 1.13976 − 1.64345i 2.54164i 0 4.84329 0.350232 − 2.80666i 0 1.66690 + 3.18455i
397.18 1.25295 + 0.655834i 0 1.13976 + 1.64345i − 2.54164i 0 4.84329 0.350232 + 2.80666i 0 1.66690 − 3.18455i
397.19 1.38682 − 0.277013i 0 1.84653 − 0.768333i − 0.972802i 0 −0.372730 2.34796 − 1.57705i 0 −0.269479 − 1.34910i
397.20 1.38682 + 0.277013i 0 1.84653 + 0.768333i 0.972802i 0 −0.372730 2.34796 + 1.57705i 0 −0.269479 + 1.34910i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 397.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 792.2.f.h ✓ 20
3.b odd 2 1 inner 792.2.f.h ✓ 20
4.b odd 2 1 3168.2.f.h 20
8.b even 2 1 inner 792.2.f.h ✓ 20
8.d odd 2 1 3168.2.f.h 20
12.b even 2 1 3168.2.f.h 20
24.f even 2 1 3168.2.f.h 20
24.h odd 2 1 inner 792.2.f.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.f.h ✓ 20 1.a even 1 1 trivial
792.2.f.h ✓ 20 3.b odd 2 1 inner
792.2.f.h ✓ 20 8.b even 2 1 inner
792.2.f.h ✓ 20 24.h odd 2 1 inner
3168.2.f.h 20 4.b odd 2 1
3168.2.f.h 20 8.d odd 2 1
3168.2.f.h 20 12.b even 2 1
3168.2.f.h 20 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(792, [\chi])\):

\( T_{5}^{10} + 28T_{5}^{8} + 244T_{5}^{6} + 800T_{5}^{4} + 832T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{5} - 22T_{7}^{3} - 16T_{7}^{2} + 40T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 28 T^{8} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} - 22 T^{3} + \cdots + 16)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + 76 T^{8} + \cdots + 1792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 116 T^{8} + \cdots - 1835008)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 108 T^{8} + \cdots + 351232)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 140 T^{8} + \cdots - 2453248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 132 T^{8} + \cdots + 1478656)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 10 T^{4} + \cdots - 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 224 T^{8} + \cdots + 1835008)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 188 T^{8} + \cdots - 6028288)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 188 T^{8} + \cdots + 6028288)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 348 T^{8} + \cdots - 1507072)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 220 T^{8} + \cdots + 33362176)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 208 T^{8} + \cdots + 4194304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 428 T^{8} + \cdots + 143519488)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 408 T^{8} + \cdots + 1101463552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 156 T^{8} + \cdots - 646912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 14 T^{4} + \cdots + 224)^{4} \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots + 304)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 472 T^{8} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 416 T^{8} + \cdots - 7340032)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 2 T^{4} + \cdots - 4864)^{4} \) Copy content Toggle raw display
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