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Properties

Label 756.2.bi.a
Level $756$
Weight $2$
Character orbit 756.bi
Analytic conductor $6.037$
Analytic rank $0$
Dimension $4$
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] = [756,2,Mod(307,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("756.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bi (of order \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{10} + \cdots + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \cdots - 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8} - 6 q^{13} + 2 q^{14} + 8 q^{16} + 2 q^{22} - 4 q^{25} - 20 q^{28} + 10 q^{29} + 8 q^{32} - 12 q^{34} - 24 q^{38} + 12 q^{40} + 30 q^{41} - 8 q^{44} - 20 q^{46} - 4 q^{49} - 4 q^{50} - 16 q^{53} + 16 q^{56} + 10 q^{58} + 18 q^{61} + 6 q^{65} - 12 q^{70} + 48 q^{76} + 2 q^{77} - 12 q^{85} + 2 q^{86} + 4 q^{88} + 20 q^{92} - 18 q^{94} + 18 q^{97} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−1.36603 − 0.366025i 0 1.73205 + 1.00000i −1.50000 − 0.866025i 0 −1.73205 + 2.00000i −2.00000 − 2.00000i 0 1.73205 + 1.73205i
307.2 0.366025 − 1.36603i 0 −1.73205 − 1.00000i −1.50000 − 0.866025i 0 1.73205 − 2.00000i −2.00000 + 2.00000i 0 −1.73205 + 1.73205i
559.1 −1.36603 + 0.366025i 0 1.73205 − 1.00000i −1.50000 + 0.866025i 0 −1.73205 − 2.00000i −2.00000 + 2.00000i 0 1.73205 − 1.73205i
559.2 0.366025 + 1.36603i 0 −1.73205 + 1.00000i −1.50000 + 0.866025i 0 1.73205 + 2.00000i −2.00000 − 2.00000i 0 −1.73205 − 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.l odd 6 1 inner
252.bi even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bi.a 4
3.b odd 2 1 252.2.bi.b yes 4
4.b odd 2 1 inner 756.2.bi.a 4
7.b odd 2 1 756.2.bi.b 4
9.c even 3 1 756.2.bi.b 4
9.d odd 6 1 252.2.bi.a ✓ 4
12.b even 2 1 252.2.bi.b yes 4
21.c even 2 1 252.2.bi.a ✓ 4
28.d even 2 1 756.2.bi.b 4
36.f odd 6 1 756.2.bi.b 4
36.h even 6 1 252.2.bi.a ✓ 4
63.l odd 6 1 inner 756.2.bi.a 4
63.o even 6 1 252.2.bi.b yes 4
84.h odd 2 1 252.2.bi.a ✓ 4
252.s odd 6 1 252.2.bi.b yes 4
252.bi even 6 1 inner 756.2.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bi.a ✓ 4 9.d odd 6 1
252.2.bi.a ✓ 4 21.c even 2 1
252.2.bi.a ✓ 4 36.h even 6 1
252.2.bi.a ✓ 4 84.h odd 2 1
252.2.bi.b yes 4 3.b odd 2 1
252.2.bi.b yes 4 12.b even 2 1
252.2.bi.b yes 4 63.o even 6 1
252.2.bi.b yes 4 252.s odd 6 1
756.2.bi.a 4 1.a even 1 1 trivial
756.2.bi.a 4 4.b odd 2 1 inner
756.2.bi.a 4 63.l odd 6 1 inner
756.2.bi.a 4 252.bi even 6 1 inner
756.2.bi.b 4 7.b odd 2 1
756.2.bi.b 4 9.c even 3 1
756.2.bi.b 4 28.d even 2 1
756.2.bi.b 4 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$53$ \( (T + 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$71$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$83$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
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