Properties

Label 3312.2.i.b
Level $3312$
Weight $2$
Character orbit 3312.i
Analytic conductor $26.446$
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] = [3312,2,Mod(2575,3312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("3312.2575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.i (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: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_{4} q^{7} + (\beta_{5} - 3) q^{11} + 2 \beta_{5} q^{13} + \beta_{6} q^{17} - \beta_{2} q^{19} + ( - \beta_{7} - \beta_{5} + \beta_1 - 3) q^{23} + q^{25} + ( - 2 \beta_{4} - \beta_{2}) q^{29} + 3 \beta_{3} q^{31} - 2 \beta_1 q^{35} - 3 \beta_1 q^{37} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{41} + \beta_{2} q^{43} + ( - 2 \beta_{7} - 2 \beta_1) q^{47} + ( - 2 \beta_{5} + 1) q^{49} + (2 \beta_{6} + \beta_{3}) q^{53} + (2 \beta_{6} - 4 \beta_{3}) q^{55} + (4 \beta_{7} - 2 \beta_1) q^{59} + (4 \beta_{7} + \beta_1) q^{61} + (4 \beta_{6} - 2 \beta_{3}) q^{65} + ( - 2 \beta_{4} + \beta_{2}) q^{67} + (4 \beta_{7} + 2 \beta_1) q^{71} + (6 \beta_{5} - 2) q^{73} + ( - 4 \beta_{4} + \beta_{2}) q^{77} + (\beta_{4} - 2 \beta_{2}) q^{79} + ( - 3 \beta_{5} - 3) q^{83} + ( - 2 \beta_{5} - 2) q^{85} + (7 \beta_{6} - 4 \beta_{3}) q^{89} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{91} + 4 \beta_{7} q^{95} + ( - 4 \beta_{7} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{11} - 24 q^{23} + 8 q^{25} + 8 q^{49} - 16 q^{73} - 24 q^{83} - 16 q^{85}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 115\nu - 66 ) / 37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{4} - 4\nu^{3} + 14\nu^{2} - 12\nu + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\nu^{7} - 42\nu^{6} + 134\nu^{5} - 230\nu^{4} + 234\nu^{3} - 142\nu^{2} - 82\nu + 58 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 23\nu^{2} + 15\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} + 11\nu^{4} - 17\nu^{3} + 25\nu^{2} - 17\nu + 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{7} + 35\nu^{6} - 161\nu^{5} + 315\nu^{4} - 639\nu^{3} + 661\nu^{2} - 573\nu + 186 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\nu^{7} - 147\nu^{6} + 543\nu^{5} - 990\nu^{4} + 1559\nu^{3} - 1422\nu^{2} + 971\nu - 278 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} + 3\beta_{5} + 3\beta_{4} - 7\beta_{3} - 5\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} + 6\beta_{6} - 8\beta_{5} - 8\beta_{4} - 15\beta_{3} + 2\beta_{2} - 12\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} - 15\beta_{6} - 25\beta_{5} - 25\beta_{4} + 21\beta_{3} + 5\beta_{2} + 11\beta _1 + 67 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -34\beta_{7} - 60\beta_{6} + 18\beta_{5} + 14\beta_{4} + 101\beta_{3} - 7\beta_{2} + 64\beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 14\beta_{6} + 154\beta_{5} + 140\beta_{4} - \beta_{3} - 42\beta_{2} + 6\beta _1 - 320 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(2305\) \(2485\) \(2945\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1
0.500000 2.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 0.564882i
0.500000 1.19293i
0 0 0 2.00000i 0 −3.38587 0 0 0
2575.2 0 0 0 2.00000i 0 −2.12976 0 0 0
2575.3 0 0 0 2.00000i 0 2.12976 0 0 0
2575.4 0 0 0 2.00000i 0 3.38587 0 0 0
2575.5 0 0 0 2.00000i 0 −3.38587 0 0 0
2575.6 0 0 0 2.00000i 0 −2.12976 0 0 0
2575.7 0 0 0 2.00000i 0 2.12976 0 0 0
2575.8 0 0 0 2.00000i 0 3.38587 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
69.c even 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3312.2.i.b 8
3.b odd 2 1 3312.2.i.f yes 8
4.b odd 2 1 3312.2.i.f yes 8
12.b even 2 1 inner 3312.2.i.b 8
23.b odd 2 1 3312.2.i.f yes 8
69.c even 2 1 inner 3312.2.i.b 8
92.b even 2 1 inner 3312.2.i.b 8
276.h odd 2 1 3312.2.i.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3312.2.i.b 8 1.a even 1 1 trivial
3312.2.i.b 8 12.b even 2 1 inner
3312.2.i.b 8 69.c even 2 1 inner
3312.2.i.b 8 92.b even 2 1 inner
3312.2.i.f yes 8 3.b odd 2 1
3312.2.i.f yes 8 4.b odd 2 1
3312.2.i.f yes 8 23.b odd 2 1
3312.2.i.f yes 8 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3312, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 52 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 6 \) 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
$5$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 52)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{3} + 70 T^{2} + 276 T + 529)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 120 T^{2} + 1872)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 144 T^{2} + 4212)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 192 T^{2} + 7488)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 120 T^{2} + 1872)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 192 T^{2} + 7488)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 192 T^{2} + 468)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 256 T^{2} + 832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 104)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 296 T^{2} + 21316)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 192 T^{2} + 7488)^{2} \) Copy content Toggle raw display
show more
show less