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Properties

Label 1008.2.v.c
Level $1008$
Weight $2$
Character orbit 1008.v
Analytic conductor $8.049$
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] = [1008,2,Mod(323,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.v (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{7}) q^{5} - q^{7} + (\beta_{10} + \beta_{8} + \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{7}) q^{5} - q^{7} + (\beta_{10} + \beta_{8} + \beta_{3} + \beta_1) q^{8} + (\beta_{11} - \beta_{5}) q^{10} + (\beta_{10} + \beta_{9} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{11} - \beta_{6} - \beta_{4} + \cdots + 1) q^{13}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 12 q^{7} - 8 q^{10} + 16 q^{13} - 20 q^{16} - 16 q^{22} - 8 q^{28} - 12 q^{37} + 24 q^{40} - 20 q^{43} + 28 q^{46} + 12 q^{49} - 24 q^{52} + 32 q^{55} - 24 q^{58} - 32 q^{61} - 16 q^{64} + 44 q^{67} + 8 q^{70} + 48 q^{82} + 64 q^{85} + 24 q^{88} - 16 q^{91} + 16 q^{94} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 2\nu^{7} + \nu^{5} + 6\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 2\nu^{8} + 23\nu^{6} - 58\nu^{4} + 96\nu^{2} - 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + 2\nu^{6} - 5\nu^{4} + 6\nu^{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 34\nu^{4} - 64\nu^{2} + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 4\nu^{9} + 5\nu^{7} - 12\nu^{5} + 12\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{11} + 6\nu^{9} - 19\nu^{7} + 30\nu^{5} - 24\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 10\nu^{9} - 33\nu^{7} + 82\nu^{5} - 120\nu^{3} + 128\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{11} - 10\nu^{9} + 27\nu^{7} - 34\nu^{5} + 64\nu^{3} - 32\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 66\nu^{4} - 96\nu^{2} + 96 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 2\beta_{6} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{10} + \beta_{9} + 2\beta_{8} - \beta_{7} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} - 5\beta_{6} - \beta_{5} + 3\beta_{4} - 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{10} - \beta_{8} - 6\beta_{7} - 3\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -\beta_{11} - 6\beta_{5} + 6\beta_{4} - 3\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3\beta_{10} - \beta_{9} - 10\beta_{8} - 11\beta_{7} + 4\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -10\beta_{11} + \beta_{6} - 11\beta_{5} - 7\beta_{4} - 2\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -3\beta_{10} + 8\beta_{9} - 23\beta_{8} + 6\beta_{7} + 19\beta_{3} - 5\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
−1.35489 − 0.405301i
−1.16947 − 0.795191i
−0.892524 + 1.09700i
0.892524 − 1.09700i
1.16947 + 0.795191i
1.35489 + 0.405301i
−1.35489 + 0.405301i
−1.16947 + 0.795191i
−0.892524 − 1.09700i
0.892524 + 1.09700i
1.16947 − 0.795191i
1.35489 − 0.405301i
−1.35489 − 0.405301i 0 1.67146 + 1.09828i −1.41421 − 1.41421i 0 −1.00000 −1.81951 − 2.16549i 0 1.34292 + 2.48929i
323.2 −1.16947 − 0.795191i 0 0.735342 + 1.85991i 1.41421 + 1.41421i 0 −1.00000 0.619022 − 2.75986i 0 −0.529317 − 2.77846i
323.3 −0.892524 + 1.09700i 0 −0.406803 − 1.95819i 1.41421 + 1.41421i 0 −1.00000 2.51121 + 1.30147i 0 −2.81361 + 0.289169i
323.4 0.892524 − 1.09700i 0 −0.406803 − 1.95819i −1.41421 − 1.41421i 0 −1.00000 −2.51121 − 1.30147i 0 −2.81361 + 0.289169i
323.5 1.16947 + 0.795191i 0 0.735342 + 1.85991i −1.41421 − 1.41421i 0 −1.00000 −0.619022 + 2.75986i 0 −0.529317 − 2.77846i
323.6 1.35489 + 0.405301i 0 1.67146 + 1.09828i 1.41421 + 1.41421i 0 −1.00000 1.81951 + 2.16549i 0 1.34292 + 2.48929i
827.1 −1.35489 + 0.405301i 0 1.67146 − 1.09828i −1.41421 + 1.41421i 0 −1.00000 −1.81951 + 2.16549i 0 1.34292 − 2.48929i
827.2 −1.16947 + 0.795191i 0 0.735342 − 1.85991i 1.41421 − 1.41421i 0 −1.00000 0.619022 + 2.75986i 0 −0.529317 + 2.77846i
827.3 −0.892524 − 1.09700i 0 −0.406803 + 1.95819i 1.41421 − 1.41421i 0 −1.00000 2.51121 − 1.30147i 0 −2.81361 − 0.289169i
827.4 0.892524 + 1.09700i 0 −0.406803 + 1.95819i −1.41421 + 1.41421i 0 −1.00000 −2.51121 + 1.30147i 0 −2.81361 − 0.289169i
827.5 1.16947 − 0.795191i 0 0.735342 − 1.85991i −1.41421 + 1.41421i 0 −1.00000 −0.619022 − 2.75986i 0 −0.529317 + 2.77846i
827.6 1.35489 − 0.405301i 0 1.67146 − 1.09828i 1.41421 − 1.41421i 0 −1.00000 1.81951 − 2.16549i 0 1.34292 − 2.48929i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.v.c ✓ 12
3.b odd 2 1 inner 1008.2.v.c ✓ 12
4.b odd 2 1 4032.2.v.c 12
12.b even 2 1 4032.2.v.c 12
16.e even 4 1 4032.2.v.c 12
16.f odd 4 1 inner 1008.2.v.c ✓ 12
48.i odd 4 1 4032.2.v.c 12
48.k even 4 1 inner 1008.2.v.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.c ✓ 12 1.a even 1 1 trivial
1008.2.v.c ✓ 12 3.b odd 2 1 inner
1008.2.v.c ✓ 12 16.f odd 4 1 inner
1008.2.v.c ✓ 12 48.k even 4 1 inner
4032.2.v.c 12 4.b odd 2 1
4032.2.v.c 12 12.b even 2 1
4032.2.v.c 12 16.e even 4 1
4032.2.v.c 12 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{11}^{12} + 1056T_{11}^{8} + 53504T_{11}^{4} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 1056 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 104 T^{4} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 30 T^{4} + 188 T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$31$ \( (T^{6} + 80 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 2)^{6} \) Copy content Toggle raw display
$41$ \( (T^{6} - 128 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 10 T^{5} + \cdots + 412232)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 128 T^{4} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 18704 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 136651472896 \) Copy content Toggle raw display
$61$ \( (T^{6} + 16 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 22 T^{5} + \cdots + 2888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 238 T^{4} + \cdots + 412232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 156 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 320 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$89$ \( (T^{6} - 488 T^{4} + \cdots - 359552)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} + \cdots - 4664)^{4} \) Copy content Toggle raw display
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