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Properties

Label 1872.2.bi.e
Level $1872$
Weight $2$
Character orbit 1872.bi
Analytic conductor $14.948$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,Mod(161,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.bi (of order \(4\), degree \(2\), not 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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 27x^{8} + 107x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (\beta_{7} + \beta_{5} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + (\beta_{7} + \beta_{5} - \beta_1) q^{7} - \beta_{11} q^{11} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5}) q^{13} - \beta_{8} q^{17} + (2 \beta_{7} + \beta_{6} - \beta_{4} + \cdots - 1) q^{19}+ \cdots + ( - \beta_{7} - 6 \beta_{6} + 4 \beta_{4} + \cdots + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{13} - 16 q^{19} + 8 q^{31} + 28 q^{37} + 56 q^{61} - 8 q^{67} - 12 q^{73} + 64 q^{79} + 8 q^{85} + 24 q^{91} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 27x^{8} + 107x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 38\nu^{4} + 145 ) / 76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 38\nu^{5} + 373\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{9} - 190\nu^{5} - 711\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 2\nu^{8} + 76\nu^{6} - 38\nu^{4} + 245\nu^{2} - 24 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} - 2\nu^{8} - 76\nu^{6} - 38\nu^{4} - 245\nu^{2} - 24 ) / 76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{10} + 190\nu^{6} + 787\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{10} - 190\nu^{6} - 749\nu^{2} ) / 38 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\nu^{11} - 3\nu^{9} + 646\nu^{7} - 76\nu^{5} + 2530\nu^{3} - 245\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -24\nu^{11} - 3\nu^{9} - 646\nu^{7} - 76\nu^{5} - 2530\nu^{3} - 245\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -35\nu^{11} - 950\nu^{7} - 3859\nu^{3} ) / 76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -69\nu^{11} - 1862\nu^{7} - 7345\nu^{3} ) / 76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{11} - 3\beta_{10} - 5\beta_{9} + 5\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + 4\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25\beta_{9} + 25\beta_{8} - 23\beta_{3} - 11\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 28\beta_{6} + 7\beta_{5} - 7\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -107\beta_{11} + 45\beta_{10} + 121\beta_{9} - 121\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -38\beta_{5} - 38\beta_{4} - 76\beta _1 + 121 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -577\beta_{9} - 577\beta_{8} + 501\beta_{3} + 197\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 349\beta_{7} + 546\beta_{6} - 190\beta_{5} + 190\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2353\beta_{11} - 895\beta_{10} - 2733\beta_{9} + 2733\beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.04736 + 1.04736i
0.219986 + 0.219986i
1.53448 + 1.53448i
−1.53448 − 1.53448i
−0.219986 − 0.219986i
−1.04736 − 1.04736i
1.04736 − 1.04736i
0.219986 − 0.219986i
1.53448 − 1.53448i
−1.53448 + 1.53448i
−0.219986 + 0.219986i
−1.04736 + 1.04736i
0 0 0 −2.93897 − 2.93897i 0 1.67513 + 1.67513i 0 0 0
161.2 0 0 0 −1.07864 − 1.07864i 0 −2.21432 − 2.21432i 0 0 0
161.3 0 0 0 −0.446112 − 0.446112i 0 0.539189 + 0.539189i 0 0 0
161.4 0 0 0 0.446112 + 0.446112i 0 0.539189 + 0.539189i 0 0 0
161.5 0 0 0 1.07864 + 1.07864i 0 −2.21432 − 2.21432i 0 0 0
161.6 0 0 0 2.93897 + 2.93897i 0 1.67513 + 1.67513i 0 0 0
593.1 0 0 0 −2.93897 + 2.93897i 0 1.67513 − 1.67513i 0 0 0
593.2 0 0 0 −1.07864 + 1.07864i 0 −2.21432 + 2.21432i 0 0 0
593.3 0 0 0 −0.446112 + 0.446112i 0 0.539189 − 0.539189i 0 0 0
593.4 0 0 0 0.446112 − 0.446112i 0 0.539189 − 0.539189i 0 0 0
593.5 0 0 0 1.07864 − 1.07864i 0 −2.21432 + 2.21432i 0 0 0
593.6 0 0 0 2.93897 − 2.93897i 0 1.67513 − 1.67513i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.bi.e 12
3.b odd 2 1 inner 1872.2.bi.e 12
4.b odd 2 1 936.2.ba.b ✓ 12
12.b even 2 1 936.2.ba.b ✓ 12
13.d odd 4 1 inner 1872.2.bi.e 12
39.f even 4 1 inner 1872.2.bi.e 12
52.f even 4 1 936.2.ba.b ✓ 12
156.l odd 4 1 936.2.ba.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.ba.b ✓ 12 4.b odd 2 1
936.2.ba.b ✓ 12 12.b even 2 1
936.2.ba.b ✓ 12 52.f even 4 1
936.2.ba.b ✓ 12 156.l odd 4 1
1872.2.bi.e 12 1.a even 1 1 trivial
1872.2.bi.e 12 3.b odd 2 1 inner
1872.2.bi.e 12 13.d odd 4 1 inner
1872.2.bi.e 12 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{12} + 304T_{5}^{8} + 1664T_{5}^{4} + 256 \) Copy content Toggle raw display
\( T_{7}^{6} - 8T_{7}^{3} + 64T_{7}^{2} - 64T_{7} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 304 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{6} - 8 T^{3} + 64 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 80 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} + 8 T^{5} + \cdots + 11552)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 112 T^{4} + \cdots - 21632)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 22 T^{4} + 108 T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 4 T^{5} + 8 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 14 T^{5} + \cdots + 14792)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 181063936 \) Copy content Toggle raw display
$43$ \( (T^{6} + 112 T^{4} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 5158686976 \) Copy content Toggle raw display
$53$ \( (T^{6} + 70 T^{4} + 652 T^{2} + 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 40112 T^{8} + \cdots + 71639296 \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + \cdots + 472)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 4 T^{5} + \cdots + 800)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 3797883801856 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots + 5832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + \cdots - 32)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 16496 T^{8} + \cdots + 160000 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 41740124416 \) Copy content Toggle raw display
$97$ \( (T^{6} - 30 T^{5} + 450 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
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