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Properties

Label 936.2.g.e
Level $936$
Weight $2$
Character orbit 936.g
Analytic conductor $7.474$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(469,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - \beta_{2} q^{4} - \beta_{14} q^{5} + ( - \beta_{8} - \beta_1) q^{7} - \beta_{13} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} - \beta_{2} q^{4} - \beta_{14} q^{5} + ( - \beta_{8} - \beta_1) q^{7} - \beta_{13} q^{8} + ( - \beta_{15} - \beta_{13} - \beta_{5} + \cdots - 1) q^{10}+ \cdots + (\beta_{15} + 4 \beta_{14} + 3 \beta_{13} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 4 q^{7} + 4 q^{8} - 8 q^{10} - 12 q^{14} - 12 q^{16} - 16 q^{17} - 24 q^{20} + 16 q^{22} + 8 q^{23} - 32 q^{25} + 2 q^{26} + 32 q^{28} - 4 q^{31} + 28 q^{32} - 32 q^{34} - 12 q^{38} - 12 q^{40} + 36 q^{41} - 16 q^{44} + 44 q^{46} - 24 q^{47} + 48 q^{49} + 34 q^{50} + 24 q^{55} + 64 q^{56} - 28 q^{58} - 52 q^{62} - 48 q^{64} - 4 q^{65} - 32 q^{68} + 84 q^{70} - 32 q^{73} + 60 q^{74} + 64 q^{76} + 68 q^{80} - 32 q^{82} - 56 q^{86} - 36 q^{88} + 60 q^{89} - 32 q^{92} + 24 q^{95} + 40 q^{97} + 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2 \nu^{15} + 3 \nu^{14} - 6 \nu^{13} + 12 \nu^{12} - 12 \nu^{11} + 7 \nu^{10} - 18 \nu^{9} + \cdots + 240 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{15} - 17 \nu^{14} + 6 \nu^{13} + 2 \nu^{12} + 82 \nu^{11} - 49 \nu^{10} - 66 \nu^{9} + \cdots - 2368 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{15} + 10 \nu^{14} - 2 \nu^{13} + 12 \nu^{12} - 51 \nu^{11} + 18 \nu^{10} + 30 \nu^{9} + \cdots + 1280 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} - 23 \nu^{14} + 11 \nu^{13} + 20 \nu^{12} + 85 \nu^{11} - 91 \nu^{10} - 93 \nu^{9} + \cdots - 2624 ) / 112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} + 8 \nu^{14} - 2 \nu^{13} + 8 \nu^{12} - 43 \nu^{11} + 16 \nu^{10} + 30 \nu^{9} + \cdots + 1024 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25 \nu^{15} + 13 \nu^{14} + 16 \nu^{13} + 94 \nu^{12} - 101 \nu^{11} - 91 \nu^{10} - 8 \nu^{9} + \cdots + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37 \nu^{15} - 66 \nu^{14} + 6 \nu^{13} - 124 \nu^{12} + 341 \nu^{11} - 42 \nu^{10} - 122 \nu^{9} + \cdots - 6400 ) / 448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19 \nu^{15} - 24 \nu^{14} + 20 \nu^{13} + 100 \nu^{12} + 61 \nu^{11} - 168 \nu^{10} - 164 \nu^{9} + \cdots - 3488 ) / 224 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 50 \nu^{15} - 75 \nu^{14} - 18 \nu^{13} - 202 \nu^{12} + 454 \nu^{11} + 21 \nu^{10} - 138 \nu^{9} + \cdots - 7680 ) / 448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13 \nu^{15} - 2 \nu^{14} - 10 \nu^{13} - 64 \nu^{12} + 29 \nu^{11} + 70 \nu^{10} + 54 \nu^{9} + \cdots + 512 ) / 112 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 117 \nu^{15} - 74 \nu^{14} - 34 \nu^{13} - 492 \nu^{12} + 485 \nu^{11} + 350 \nu^{10} + 94 \nu^{9} + \cdots - 2560 ) / 896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 33 \nu^{15} + 25 \nu^{14} + 20 \nu^{13} + 142 \nu^{12} - 177 \nu^{11} - 119 \nu^{10} + 4 \nu^{9} + \cdots + 1664 ) / 224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - 2\beta_{2} + 2\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} - \beta_{12} + \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 5 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} - \beta_{12} + \beta_{11} + 9 \beta_{10} + \beta_{9} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{14} + \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + 2 \beta_{8} + \cdots + 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7 \beta_{15} - 6 \beta_{14} - 19 \beta_{13} + 13 \beta_{12} - \beta_{11} + 11 \beta_{10} - \beta_{9} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14 \beta_{15} - 11 \beta_{14} - 21 \beta_{13} + \beta_{12} - 14 \beta_{11} + 18 \beta_{10} - 3 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( \beta_{15} + 16 \beta_{14} - 19 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 11 \beta_{9} - 17 \beta_{7} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4 \beta_{15} - 23 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} - 17 \beta_{11} - 19 \beta_{10} - 38 \beta_{9} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11 \beta_{15} + 2 \beta_{14} - 21 \beta_{13} - 5 \beta_{12} - 27 \beta_{11} + 57 \beta_{10} + \beta_{9} + \cdots - 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
−0.654116 + 1.25385i
−0.654116 − 1.25385i
0.791485 − 1.17199i
0.791485 + 1.17199i
−0.879244 + 1.10767i
−0.879244 − 1.10767i
1.27276 − 0.616518i
1.27276 + 0.616518i
1.40722 + 0.140463i
1.40722 − 0.140463i
−1.32561 − 0.492712i
−1.32561 + 0.492712i
0.802079 + 1.16476i
0.802079 − 1.16476i
−0.414573 − 1.35208i
−0.414573 + 1.35208i
−1.25385 − 0.654116i 0 1.14426 + 1.64032i − 1.87654i 0 0.584696 −0.361768 − 2.80520i 0 −1.22748 + 2.35290i
469.2 −1.25385 + 0.654116i 0 1.14426 − 1.64032i 1.87654i 0 0.584696 −0.361768 + 2.80520i 0 −1.22748 − 2.35290i
469.3 −1.17199 − 0.791485i 0 0.747102 + 1.85522i − 3.05343i 0 −0.397397 0.592787 − 2.76561i 0 −2.41675 + 3.57858i
469.4 −1.17199 + 0.791485i 0 0.747102 − 1.85522i 3.05343i 0 −0.397397 0.592787 + 2.76561i 0 −2.41675 − 3.57858i
469.5 −1.10767 − 0.879244i 0 0.453861 + 1.94782i 3.31390i 0 4.17825 1.20988 − 2.55660i 0 2.91372 − 3.67070i
469.6 −1.10767 + 0.879244i 0 0.453861 − 1.94782i − 3.31390i 0 4.17825 1.20988 + 2.55660i 0 2.91372 + 3.67070i
469.7 −0.616518 − 1.27276i 0 −1.23981 + 1.56935i 3.29521i 0 2.97802 2.76177 + 0.610443i 0 4.19399 − 2.03156i
469.8 −0.616518 + 1.27276i 0 −1.23981 − 1.56935i − 3.29521i 0 2.97802 2.76177 − 0.610443i 0 4.19399 + 2.03156i
469.9 0.140463 − 1.40722i 0 −1.96054 − 0.395325i − 0.218531i 0 −4.47783 −0.831693 + 2.70338i 0 −0.307521 − 0.0306955i
469.10 0.140463 + 1.40722i 0 −1.96054 + 0.395325i 0.218531i 0 −4.47783 −0.831693 − 2.70338i 0 −0.307521 + 0.0306955i
469.11 0.492712 − 1.32561i 0 −1.51447 − 1.30629i − 4.33571i 0 −2.30442 −2.47782 + 1.36397i 0 −5.74745 − 2.13626i
469.12 0.492712 + 1.32561i 0 −1.51447 + 1.30629i 4.33571i 0 −2.30442 −2.47782 − 1.36397i 0 −5.74745 + 2.13626i
469.13 1.16476 − 0.802079i 0 0.713337 − 1.86846i − 1.47174i 0 −2.93973 −0.667787 − 2.74847i 0 −1.18045 − 1.71422i
469.14 1.16476 + 0.802079i 0 0.713337 + 1.86846i 1.47174i 0 −2.93973 −0.667787 + 2.74847i 0 −1.18045 + 1.71422i
469.15 1.35208 − 0.414573i 0 1.65626 − 1.12108i − 0.550135i 0 4.37841 1.77463 − 2.20243i 0 −0.228071 − 0.743828i
469.16 1.35208 + 0.414573i 0 1.65626 + 1.12108i 0.550135i 0 4.37841 1.77463 + 2.20243i 0 −0.228071 + 0.743828i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.g.e 16
3.b odd 2 1 312.2.g.b ✓ 16
4.b odd 2 1 3744.2.g.e 16
8.b even 2 1 inner 936.2.g.e 16
8.d odd 2 1 3744.2.g.e 16
12.b even 2 1 1248.2.g.b 16
24.f even 2 1 1248.2.g.b 16
24.h odd 2 1 312.2.g.b ✓ 16
48.i odd 4 1 9984.2.a.bt 8
48.i odd 4 1 9984.2.a.bu 8
48.k even 4 1 9984.2.a.bs 8
48.k even 4 1 9984.2.a.bv 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.g.b ✓ 16 3.b odd 2 1
312.2.g.b ✓ 16 24.h odd 2 1
936.2.g.e 16 1.a even 1 1 trivial
936.2.g.e 16 8.b even 2 1 inner
1248.2.g.b 16 12.b even 2 1
1248.2.g.b 16 24.f even 2 1
3744.2.g.e 16 4.b odd 2 1
3744.2.g.e 16 8.d odd 2 1
9984.2.a.bs 8 48.k even 4 1
9984.2.a.bt 8 48.i odd 4 1
9984.2.a.bu 8 48.i odd 4 1
9984.2.a.bv 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 56 T_{5}^{14} + 1220 T_{5}^{12} + 13152 T_{5}^{10} + 73152 T_{5}^{8} + 197888 T_{5}^{6} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 56 T^{14} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots + 384)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 128 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} + 8 T^{7} + \cdots - 64256)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 224 T^{14} + \cdots + 22429696 \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots - 166912)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 4671995904 \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + \cdots + 110464)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$41$ \( (T^{8} - 18 T^{7} + \cdots - 139792)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 561468473344 \) Copy content Toggle raw display
$47$ \( (T^{8} + 12 T^{7} + \cdots + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 462422016 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 276642337024 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 329127100416 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 409763856384 \) Copy content Toggle raw display
$71$ \( (T^{8} - 408 T^{6} + \cdots + 21431088)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 3326208)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 336 T^{6} + \cdots + 2166016)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6293836597504 \) Copy content Toggle raw display
$89$ \( (T^{8} - 30 T^{7} + \cdots + 37581936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 20 T^{7} + \cdots - 637696)^{2} \) Copy content Toggle raw display
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