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Properties

Label 3136.2.f.g
Level $3136$
Weight $2$
Character orbit 3136.f
Analytic conductor $25.041$
Analytic rank $0$
Dimension $8$
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] = [3136,2,Mod(3135,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3136.3135");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.f (of order \(2\), degree \(1\), 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: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1568)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + ( - \beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + ( - \beta_{6} - 1) q^{9} + 3 \beta_{3} q^{11} + ( - \beta_{4} - \beta_{2}) q^{13} + ( - 2 \beta_{3} + \beta_1) q^{15} + ( - 4 \beta_{4} + \beta_{2}) q^{17} + ( - 2 \beta_{7} - \beta_{5}) q^{19} + (4 \beta_{3} - \beta_1) q^{23} + (2 \beta_{6} + 1) q^{25} + ( - 3 \beta_{7} - \beta_{5}) q^{27} + 4 q^{29} + 4 \beta_{5} q^{31} + ( - 3 \beta_{4} + 3 \beta_{2}) q^{33} + ( - 2 \beta_{6} + 8) q^{37} - \beta_1 q^{39} + (3 \beta_{4} + 4 \beta_{2}) q^{41} + (\beta_{3} - 4 \beta_1) q^{43} + (\beta_{4} + \beta_{2}) q^{45} + ( - 2 \beta_{7} + 8 \beta_{5}) q^{47} + (5 \beta_{3} - 4 \beta_1) q^{51} + (2 \beta_{6} - 6) q^{53} + 6 \beta_{7} q^{55} + (\beta_{6} - 4) q^{57} + (6 \beta_{7} + 5 \beta_{5}) q^{59} + ( - 3 \beta_{4} - 7 \beta_{2}) q^{61} - 2 \beta_{6} q^{65} + (4 \beta_{3} + 2 \beta_1) q^{67} + ( - 6 \beta_{4} + 4 \beta_{2}) q^{69} + ( - 4 \beta_{3} + 2 \beta_1) q^{71} + (\beta_{4} + 2 \beta_{2}) q^{73} + ( - \beta_{7} + 2 \beta_{5}) q^{75} + (2 \beta_{3} + 4 \beta_1) q^{79} + (5 \beta_{6} - 3) q^{81} + (4 \beta_{7} - 3 \beta_{5}) q^{83} + ( - 8 \beta_{6} + 10) q^{85} + 4 \beta_{7} q^{87} + (3 \beta_{4} + 4 \beta_{2}) q^{89} + 4 \beta_{6} q^{93} + (4 \beta_{3} - \beta_1) q^{95} + (2 \beta_{4} - 3 \beta_{2}) q^{97} + ( - 3 \beta_{3} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 8 q^{25} + 32 q^{29} + 64 q^{37} - 48 q^{53} - 32 q^{57} - 24 q^{81} + 80 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( -\beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3135.1
0.382683 − 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
−0.923880 − 0.382683i
0.923880 + 0.382683i
0.923880 − 0.382683i
−0.382683 − 0.923880i
−0.382683 + 0.923880i
0 −1.84776 0 − 2.61313i 0 0 0 0.414214 0
3135.2 0 −1.84776 0 2.61313i 0 0 0 0.414214 0
3135.3 0 −0.765367 0 − 1.08239i 0 0 0 −2.41421 0
3135.4 0 −0.765367 0 1.08239i 0 0 0 −2.41421 0
3135.5 0 0.765367 0 − 1.08239i 0 0 0 −2.41421 0
3135.6 0 0.765367 0 1.08239i 0 0 0 −2.41421 0
3135.7 0 1.84776 0 − 2.61313i 0 0 0 0.414214 0
3135.8 0 1.84776 0 2.61313i 0 0 0 0.414214 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3135.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.f.g 8
4.b odd 2 1 inner 3136.2.f.g 8
7.b odd 2 1 inner 3136.2.f.g 8
8.b even 2 1 1568.2.f.a ✓ 8
8.d odd 2 1 1568.2.f.a ✓ 8
28.d even 2 1 inner 3136.2.f.g 8
56.e even 2 1 1568.2.f.a ✓ 8
56.h odd 2 1 1568.2.f.a ✓ 8
56.j odd 6 2 1568.2.p.c 16
56.k odd 6 2 1568.2.p.c 16
56.m even 6 2 1568.2.p.c 16
56.p even 6 2 1568.2.p.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.f.a ✓ 8 8.b even 2 1
1568.2.f.a ✓ 8 8.d odd 2 1
1568.2.f.a ✓ 8 56.e even 2 1
1568.2.f.a ✓ 8 56.h odd 2 1
1568.2.p.c 16 56.j odd 6 2
1568.2.p.c 16 56.k odd 6 2
1568.2.p.c 16 56.m even 6 2
1568.2.p.c 16 56.p even 6 2
3136.2.f.g 8 1.a even 1 1 trivial
3136.2.f.g 8 4.b odd 2 1 inner
3136.2.f.g 8 7.b odd 2 1 inner
3136.2.f.g 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 68 T^{2} + 98)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 20 T^{2} + 98)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 72 T^{2} + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T + 56)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 100 T^{2} + 578)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 132 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 272 T^{2} + 16928)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 28)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 244 T^{2} + 10082)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 232 T^{2} + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 144 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 100 T^{2} + 578)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 100 T^{2} + 578)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{2} + 578)^{2} \) Copy content Toggle raw display
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