Properties

Label 155.2.f.a
Level $155$
Weight $2$
Character orbit 155.f
Analytic conductor $1.238$
Analytic rank $0$
Dimension $12$
CM discriminant -31
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,2,Mod(92,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.92");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.f (of order \(4\), degree \(2\), 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: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.1931902335935778816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{7} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + \beta_{8} q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{7} + (\beta_{11} - \beta_{9} - 2 \beta_{3} + \beta_1) q^{8} - 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{7} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + \beta_{8} q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{7} + (\beta_{11} - \beta_{9} - 2 \beta_{3} + \beta_1) q^{8} - 3 \beta_1 q^{9} + (\beta_{9} + \beta_{6} - \beta_{5} - 2 \beta_1) q^{10} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{8} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3} - 1) q^{14} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{2} - 4) q^{16} + 3 \beta_{3} q^{18} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{19} + (\beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} + 3 \beta_1) q^{20} + ( - \beta_{10} - 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{25} + (\beta_{11} - \beta_{10} - \beta_{9} + 4 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots + 5) q^{28}+ \cdots + (\beta_{11} + 2 \beta_{10} - \beta_{9} + 5 \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \cdots + 10) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{8} - 48 q^{16} + 54 q^{28} - 12 q^{32} - 24 q^{35} + 72 q^{36} - 66 q^{38} + 36 q^{40} + 48 q^{47} - 72 q^{67} + 6 q^{70} + 18 q^{72} - 84 q^{80} - 108 q^{81} - 42 q^{82} - 54 q^{90} + 96 q^{95} + 114 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{6} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 7\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 15\nu^{4} - 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 7\nu^{2} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 16\nu^{7} + 15\nu^{5} - 112\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 4\nu^{8} + \nu^{4} - 28\nu^{2} + 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 4\nu^{10} - 15\nu^{5} + 28\nu^{4} ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 11\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} - 13\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 8\nu^{8} + 15\nu^{5} + 32\nu^{3} - 56\nu^{2} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} - 4\nu^{10} - 16\nu^{7} + 15\nu^{5} + 28\nu^{4} + 144\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} + 8\nu^{8} + 32\nu^{6} + 15\nu^{5} - 56\nu^{2} + 32\nu - 256 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} + 2\beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 10\beta_{7} + 2\beta_{6} + 3\beta_{5} + \beta_{4} + 7\beta_{3} - 3\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{10} + 2\beta_{9} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - 2\beta_{6} + 7\beta_{5} - \beta_{4} - 7\beta_{3} + 13\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{10} + 10\beta_{8} - 12\beta_{6} - 3\beta_{5} + 9\beta_{4} + 3\beta_{3} + 3\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{11} - \beta_{10} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{10} + 22\beta_{6} + 23\beta_{5} + 21\beta_{4} - 23\beta_{3} - 23\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11\beta_{10} + 70\beta_{7} + 22\beta_{6} + 33\beta_{5} + 11\beta_{4} + 77\beta_{3} - 33\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -7\beta_{10} + 14\beta_{9} - 7\beta_{5} - 7\beta_{4} - 7\beta_{3} + 7\beta_{2} + 16\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -31\beta_{10} - 62\beta_{6} + 57\beta_{5} - 31\beta_{4} - 57\beta_{3} + 83\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 39\beta_{10} + 150\beta_{8} - 52\beta_{6} - 13\beta_{5} + 39\beta_{4} + 13\beta_{3} + 13\beta_{2} ) / 10 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1
0.633359 1.26446i
1.41173 + 0.0837246i
−0.633359 1.26446i
0.778374 + 1.18073i
−1.41173 + 0.0837246i
−0.778374 + 1.18073i
0.633359 + 1.26446i
1.41173 0.0837246i
−0.633359 + 1.26446i
0.778374 1.18073i
−1.41173 0.0837246i
−0.778374 1.18073i
−1.89782 1.89782i 0 5.20343i 2.23507 + 0.0667460i 0 −3.43053 3.43053i 6.07952 6.07952i 3.00000i −4.11509 4.36843i
92.2 −1.32801 1.32801i 0 1.52721i 1.17534 + 1.90226i 0 3.71783 + 3.71783i −0.627865 + 0.627865i 3.00000i 0.965351 4.08707i
92.3 −0.631100 0.631100i 0 1.20343i −2.23507 + 0.0667460i 0 −1.49382 1.49382i −2.02168 + 2.02168i 3.00000i 1.45268 + 1.36843i
92.4 0.402360 + 0.402360i 0 1.67621i −1.05973 1.96900i 0 3.00895 + 3.00895i 1.47916 1.47916i 3.00000i 0.365854 1.21864i
92.5 1.49546 + 1.49546i 0 2.47279i −1.17534 + 1.90226i 0 0.421578 + 0.421578i −0.707033 + 0.707033i 3.00000i −4.60241 + 1.08707i
92.6 1.95911 + 1.95911i 0 5.67621i 1.05973 1.96900i 0 −2.22401 2.22401i −7.20210 + 7.20210i 3.00000i 5.93362 1.78136i
123.1 −1.89782 + 1.89782i 0 5.20343i 2.23507 0.0667460i 0 −3.43053 + 3.43053i 6.07952 + 6.07952i 3.00000i −4.11509 + 4.36843i
123.2 −1.32801 + 1.32801i 0 1.52721i 1.17534 1.90226i 0 3.71783 3.71783i −0.627865 0.627865i 3.00000i 0.965351 + 4.08707i
123.3 −0.631100 + 0.631100i 0 1.20343i −2.23507 0.0667460i 0 −1.49382 + 1.49382i −2.02168 2.02168i 3.00000i 1.45268 1.36843i
123.4 0.402360 0.402360i 0 1.67621i −1.05973 + 1.96900i 0 3.00895 3.00895i 1.47916 + 1.47916i 3.00000i 0.365854 + 1.21864i
123.5 1.49546 1.49546i 0 2.47279i −1.17534 1.90226i 0 0.421578 0.421578i −0.707033 0.707033i 3.00000i −4.60241 1.08707i
123.6 1.95911 1.95911i 0 5.67621i 1.05973 + 1.96900i 0 −2.22401 + 2.22401i −7.20210 7.20210i 3.00000i 5.93362 + 1.78136i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
5.c odd 4 1 inner
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.f.a 12
5.b even 2 1 775.2.f.e 12
5.c odd 4 1 inner 155.2.f.a 12
5.c odd 4 1 775.2.f.e 12
31.b odd 2 1 CM 155.2.f.a 12
155.c odd 2 1 775.2.f.e 12
155.f even 4 1 inner 155.2.f.a 12
155.f even 4 1 775.2.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.f.a 12 1.a even 1 1 trivial
155.2.f.a 12 5.c odd 4 1 inner
155.2.f.a 12 31.b odd 2 1 CM
155.2.f.a 12 155.f even 4 1 inner
775.2.f.e 12 5.b even 2 1
775.2.f.e 12 5.c odd 4 1
775.2.f.e 12 155.c odd 2 1
775.2.f.e 12 155.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 2T_{2}^{9} + 72T_{2}^{8} + 12T_{2}^{7} + 2T_{2}^{6} + 72T_{2}^{5} + 936T_{2}^{4} + 402T_{2}^{3} + 72T_{2}^{2} - 180T_{2} + 225 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{9} + 72 T^{8} + 12 T^{7} + \cdots + 225 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 246 T^{6} + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 32 T^{9} + 882 T^{8} + \cdots + 184900 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 114 T^{4} + 3249 T^{2} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{2} - 31)^{6} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( (T^{3} - 123 T + 278)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} + 128 T^{2} + 480 T + 900)^{3} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{6} + 354 T^{4} + 31329 T^{2} + \cdots + 273916)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( (T^{4} + 24 T^{3} + 288 T^{2} + 240 T + 100)^{3} \) Copy content Toggle raw display
$71$ \( (T^{6} - 426 T^{4} + 45369 T^{2} + \cdots - 431644)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 3812 T^{9} + \cdots + 3267020100100 \) Copy content Toggle raw display
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