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Properties

Label 1872.2.bf.n
Level $1872$
Weight $2$
Character orbit 1872.bf
Analytic conductor $14.948$
Analytic rank $0$
Dimension $8$
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] = [1872,2,Mod(1279,1872)]
 
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([2, 0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1279");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.bf (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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) 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 624)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{3} - \beta_{2}) q^{11} + (\beta_{7} + \beta_{5} + \beta_{2} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + 8 q^{7} + 4 q^{11} - 8 q^{19} - 8 q^{29} + 16 q^{31} - 24 q^{37} + 20 q^{41} - 32 q^{43} - 4 q^{47} - 8 q^{53} - 12 q^{59} + 16 q^{61} - 36 q^{65} + 32 q^{67} + 4 q^{71} + 16 q^{73} - 4 q^{83} - 8 q^{85} + 20 q^{89} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6\nu^{7} - 21\nu^{6} + 67\nu^{5} - 115\nu^{4} + 117\nu^{3} - 71\nu^{2} - 41\nu + 29 ) / 37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} - 9\nu^{6} - 3\nu^{5} - 192\nu^{4} + 251\nu^{3} - 522\nu^{2} + 363\nu - 199 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -16\nu^{7} + 56\nu^{6} - 228\nu^{5} + 393\nu^{4} - 682\nu^{3} + 473\nu^{2} - 310\nu - 65 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} + 228\nu^{5} - 467\nu^{4} + 830\nu^{3} - 991\nu^{2} + 754\nu - 379 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{7} - 86\nu^{6} + 292\nu^{5} - 737\nu^{4} + 1124\nu^{3} - 1325\nu^{2} + 780\nu - 290 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42\nu^{7} - 110\nu^{6} + 432\nu^{5} - 583\nu^{4} + 930\nu^{3} - 497\nu^{2} + 342\nu + 55 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -42\nu^{7} + 184\nu^{6} - 654\nu^{5} + 1397\nu^{4} - 2188\nu^{3} + 2347\nu^{2} - 1600\nu + 611 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{3} + 4\beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 4\beta_{5} - 2\beta_{4} + 2\beta_{3} - \beta_{2} - 7\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{7} - 6\beta_{6} + 4\beta_{5} - 11\beta_{4} + 7\beta_{3} - 20\beta_{2} - 16\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -21\beta_{7} - 29\beta_{6} - 36\beta_{5} + 8\beta_{4} - 18\beta_{3} - 14\beta_{2} + 52\beta _1 + 42 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33\beta_{7} - \beta_{6} - 50\beta_{5} + 69\beta_{4} - 55\beta_{3} + 78\beta_{2} + 128\beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 143\beta_{7} + 151\beta_{6} + 134\beta_{5} + 38\beta_{4} + 46\beta_{3} + 146\beta_{2} - 128\beta _1 - 180 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0.500000 − 1.19293i
0.500000 − 0.564882i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 + 1.19293i
0.500000 + 0.564882i
0.500000 − 2.19293i
0.500000 − 1.56488i
0 0 0 −2.05896 + 2.05896i 0 1.00000 − 1.00000i 0 0 0
1279.2 0 0 0 0.301143 − 0.301143i 0 1.00000 − 1.00000i 0 0 0
1279.3 0 0 0 1.32691 − 1.32691i 0 1.00000 − 1.00000i 0 0 0
1279.4 0 0 0 2.43091 − 2.43091i 0 1.00000 − 1.00000i 0 0 0
1711.1 0 0 0 −2.05896 − 2.05896i 0 1.00000 + 1.00000i 0 0 0
1711.2 0 0 0 0.301143 + 0.301143i 0 1.00000 + 1.00000i 0 0 0
1711.3 0 0 0 1.32691 + 1.32691i 0 1.00000 + 1.00000i 0 0 0
1711.4 0 0 0 2.43091 + 2.43091i 0 1.00000 + 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.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.bf.n 8
3.b odd 2 1 624.2.bc.f yes 8
4.b odd 2 1 1872.2.bf.m 8
12.b even 2 1 624.2.bc.e ✓ 8
13.d odd 4 1 1872.2.bf.m 8
39.f even 4 1 624.2.bc.e ✓ 8
52.f even 4 1 inner 1872.2.bf.n 8
156.l odd 4 1 624.2.bc.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.bc.e ✓ 8 12.b even 2 1
624.2.bc.e ✓ 8 39.f even 4 1
624.2.bc.f yes 8 3.b odd 2 1
624.2.bc.f yes 8 156.l odd 4 1
1872.2.bf.m 8 4.b odd 2 1
1872.2.bf.m 8 13.d odd 4 1
1872.2.bf.n 8 1.a even 1 1 trivial
1872.2.bf.n 8 52.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 \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 80T_{5}^{4} - 288T_{5}^{3} + 512T_{5}^{2} - 256T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{17}^{8} + 112T_{17}^{6} + 4128T_{17}^{4} + 50944T_{17}^{2} + 43264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} - 12 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 112 T^{6} + \cdots + 43264 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$23$ \( (T^{4} - 64 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 208)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + \cdots + 183184 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} + \cdots - 4544)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} - 96 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots - 704)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 32 T^{7} + \cdots + 26337424 \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} + \cdots + 937024 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{8} + 320 T^{6} + \cdots + 2768896 \) Copy content Toggle raw display
$83$ \( T^{8} + 4 T^{7} + \cdots + 4426816 \) Copy content Toggle raw display
$89$ \( T^{8} - 20 T^{7} + \cdots + 10705984 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots + 55696 \) Copy content Toggle raw display
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