↑

Properties

Label 936.2.m.f
Level $936$
Weight $2$
Character orbit 936.m
Analytic conductor $7.474$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(181,936)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.m (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: \(8\)
Coefficient field: 8.0.4521217600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 104)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{6} q^{5} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{4} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{6} q^{5} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{4} - \beta_{3}) q^{8} + ( - \beta_{7} + \beta_{5} - 1) q^{10} + (\beta_{6} - \beta_{4} - \beta_1) q^{11} + (\beta_{7} - \beta_{4} - \beta_1) q^{13} + ( - \beta_{2} + 2) q^{14} + (\beta_{7} + \beta_{5} + 1) q^{16} + (\beta_{5} - \beta_{2} + 2) q^{17} + ( - \beta_{6} - 3 \beta_{4} - 3 \beta_1) q^{19} + (2 \beta_{6} + 3 \beta_{4} + \cdots + 2 \beta_1) q^{20}+ \cdots + ( - 3 \beta_{4} - \beta_{3} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 6 q^{10} + 18 q^{14} + 10 q^{16} + 20 q^{17} + 20 q^{22} - 32 q^{23} + 12 q^{25} + 14 q^{26} + 36 q^{38} - 26 q^{40} + 20 q^{49} - 4 q^{52} - 64 q^{55} - 14 q^{56} - 40 q^{62} - 38 q^{64} - 12 q^{65} - 46 q^{68} + 22 q^{74} + 16 q^{79} + 32 q^{88} + 8 q^{92} - 18 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{5} + 6\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 2\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{4} - 2\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{5} - 2\nu^{3} - 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 3\nu^{4} + 2\nu^{2} - 8 ) / 4 \) 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
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} - \beta_{4} + \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + 3\beta_{5} + 2\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{6} - 5\beta_{4} + \beta_{3} - 6\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.29437 + 0.569745i
1.29437 − 0.569745i
0.273147 + 1.38758i
0.273147 − 1.38758i
−0.273147 + 1.38758i
−0.273147 − 1.38758i
−1.29437 + 0.569745i
−1.29437 − 0.569745i
−1.29437 − 0.569745i 0 1.35078 + 1.47492i 1.81616 0 1.13949i −0.908080 − 2.67869i 0 −2.35078 − 1.03475i
181.2 −1.29437 + 0.569745i 0 1.35078 − 1.47492i 1.81616 0 − 1.13949i −0.908080 + 2.67869i 0 −2.35078 + 1.03475i
181.3 −0.273147 − 1.38758i 0 −1.85078 + 0.758030i −3.11473 0 2.77517i 1.55737 + 2.36106i 0 0.850781 + 4.32196i
181.4 −0.273147 + 1.38758i 0 −1.85078 − 0.758030i −3.11473 0 − 2.77517i 1.55737 − 2.36106i 0 0.850781 − 4.32196i
181.5 0.273147 − 1.38758i 0 −1.85078 − 0.758030i 3.11473 0 2.77517i −1.55737 + 2.36106i 0 0.850781 − 4.32196i
181.6 0.273147 + 1.38758i 0 −1.85078 + 0.758030i 3.11473 0 − 2.77517i −1.55737 − 2.36106i 0 0.850781 + 4.32196i
181.7 1.29437 − 0.569745i 0 1.35078 − 1.47492i −1.81616 0 1.13949i 0.908080 − 2.67869i 0 −2.35078 + 1.03475i
181.8 1.29437 + 0.569745i 0 1.35078 + 1.47492i −1.81616 0 − 1.13949i 0.908080 + 2.67869i 0 −2.35078 − 1.03475i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.b even 2 1 inner
104.e 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.m.f 8
3.b odd 2 1 104.2.e.c ✓ 8
4.b odd 2 1 3744.2.m.g 8
8.b even 2 1 inner 936.2.m.f 8
8.d odd 2 1 3744.2.m.g 8
12.b even 2 1 416.2.e.c 8
13.b even 2 1 inner 936.2.m.f 8
24.f even 2 1 416.2.e.c 8
24.h odd 2 1 104.2.e.c ✓ 8
39.d odd 2 1 104.2.e.c ✓ 8
52.b odd 2 1 3744.2.m.g 8
104.e even 2 1 inner 936.2.m.f 8
104.h odd 2 1 3744.2.m.g 8
156.h even 2 1 416.2.e.c 8
312.b odd 2 1 104.2.e.c ✓ 8
312.h even 2 1 416.2.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.e.c ✓ 8 3.b odd 2 1
104.2.e.c ✓ 8 24.h odd 2 1
104.2.e.c ✓ 8 39.d odd 2 1
104.2.e.c ✓ 8 312.b odd 2 1
416.2.e.c 8 12.b even 2 1
416.2.e.c 8 24.f even 2 1
416.2.e.c 8 156.h even 2 1
416.2.e.c 8 312.h even 2 1
936.2.m.f 8 1.a even 1 1 trivial
936.2.m.f 8 8.b even 2 1 inner
936.2.m.f 8 13.b even 2 1 inner
936.2.m.f 8 104.e even 2 1 inner
3744.2.m.g 8 4.b odd 2 1
3744.2.m.g 8 8.d odd 2 1
3744.2.m.g 8 52.b odd 2 1
3744.2.m.g 8 104.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 13T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 13 T^{2} + 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9 T^{2} + 10)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 26 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{2} - 5 T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 76 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 90 T^{2} + 1000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 29 T^{2} + 200)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 11 T^{2} + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 9 T^{2} + 10)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 176 T^{2} + 5120)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 202 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 185 T^{2} + 6250)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 296 T^{2} + 16000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 106 T^{2} + 800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 296 T^{2} + 16000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 184 T^{2} + 2560)^{2} \) Copy content Toggle raw display
show more
show less