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Properties

Label 756.2.be.c
Level $756$
Weight $2$
Character orbit 756.be
Analytic conductor $6.037$
Analytic rank $0$
Dimension $28$
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(107,756)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.be (of order \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{4} - 2 q^{7} - 20 q^{10} + 8 q^{13} + 12 q^{16} + 42 q^{19} + 4 q^{22} + 6 q^{25} - 28 q^{28} - 30 q^{31} + 24 q^{34} + 12 q^{37} + 36 q^{40} - 12 q^{46} - 14 q^{49} + 84 q^{52} + 28 q^{58} + 6 q^{61} + 8 q^{64} - 24 q^{67} + 128 q^{70} - 22 q^{73} - 48 q^{79} - 36 q^{82} - 24 q^{85} - 16 q^{88} - 16 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.39533 + 0.230338i 0 1.89389 − 0.642794i −0.936239 − 0.540538i 0 −0.749282 − 2.53743i −2.49454 + 1.33314i 0 1.43087 + 0.538577i
107.2 −1.38410 − 0.290278i 0 1.83148 + 0.803549i 2.75822 + 1.59246i 0 −0.944233 + 2.47152i −2.30170 − 1.64383i 0 −3.35540 − 3.00478i
107.3 −1.04864 + 0.948869i 0 0.199295 − 1.99005i −0.479813 − 0.277020i 0 2.45883 − 0.976797i 1.67930 + 2.27595i 0 0.766008 − 0.164785i
107.4 −0.994183 + 1.00578i 0 −0.0232006 − 1.99987i 1.28692 + 0.743002i 0 −1.36953 + 2.26371i 2.03450 + 1.96490i 0 −2.02673 + 0.555680i
107.5 −0.687910 − 1.23563i 0 −1.05356 + 1.70000i 0.303357 + 0.175143i 0 −2.63538 − 0.234036i 2.82533 + 0.132362i 0 0.00772994 − 0.495319i
107.6 −0.253271 − 1.39135i 0 −1.87171 + 0.704776i 2.47695 + 1.43007i 0 2.52686 + 0.784194i 1.45464 + 2.42570i 0 1.36239 − 3.80850i
107.7 −0.109104 + 1.41000i 0 −1.97619 − 0.307674i 3.44992 + 1.99182i 0 0.212727 − 2.63719i 0.649430 − 2.75286i 0 −3.18486 + 4.64707i
107.8 0.109104 − 1.41000i 0 −1.97619 − 0.307674i −3.44992 − 1.99182i 0 0.212727 − 2.63719i −0.649430 + 2.75286i 0 −3.18486 + 4.64707i
107.9 0.253271 + 1.39135i 0 −1.87171 + 0.704776i −2.47695 − 1.43007i 0 2.52686 + 0.784194i −1.45464 − 2.42570i 0 1.36239 − 3.80850i
107.10 0.687910 + 1.23563i 0 −1.05356 + 1.70000i −0.303357 − 0.175143i 0 −2.63538 − 0.234036i −2.82533 − 0.132362i 0 0.00772994 − 0.495319i
107.11 0.994183 − 1.00578i 0 −0.0232006 − 1.99987i −1.28692 − 0.743002i 0 −1.36953 + 2.26371i −2.03450 − 1.96490i 0 −2.02673 + 0.555680i
107.12 1.04864 − 0.948869i 0 0.199295 − 1.99005i 0.479813 + 0.277020i 0 2.45883 − 0.976797i −1.67930 − 2.27595i 0 0.766008 − 0.164785i
107.13 1.38410 + 0.290278i 0 1.83148 + 0.803549i −2.75822 − 1.59246i 0 −0.944233 + 2.47152i 2.30170 + 1.64383i 0 −3.35540 − 3.00478i
107.14 1.39533 − 0.230338i 0 1.89389 − 0.642794i 0.936239 + 0.540538i 0 −0.749282 − 2.53743i 2.49454 − 1.33314i 0 1.43087 + 0.538577i
431.1 −1.39533 − 0.230338i 0 1.89389 + 0.642794i −0.936239 + 0.540538i 0 −0.749282 + 2.53743i −2.49454 − 1.33314i 0 1.43087 − 0.538577i
431.2 −1.38410 + 0.290278i 0 1.83148 − 0.803549i 2.75822 − 1.59246i 0 −0.944233 − 2.47152i −2.30170 + 1.64383i 0 −3.35540 + 3.00478i
431.3 −1.04864 − 0.948869i 0 0.199295 + 1.99005i −0.479813 + 0.277020i 0 2.45883 + 0.976797i 1.67930 − 2.27595i 0 0.766008 + 0.164785i
431.4 −0.994183 − 1.00578i 0 −0.0232006 + 1.99987i 1.28692 − 0.743002i 0 −1.36953 − 2.26371i 2.03450 − 1.96490i 0 −2.02673 − 0.555680i
431.5 −0.687910 + 1.23563i 0 −1.05356 − 1.70000i 0.303357 − 0.175143i 0 −2.63538 + 0.234036i 2.82533 − 0.132362i 0 0.00772994 + 0.495319i
431.6 −0.253271 + 1.39135i 0 −1.87171 − 0.704776i 2.47695 − 1.43007i 0 2.52686 − 0.784194i 1.45464 − 2.42570i 0 1.36239 + 3.80850i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n 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.be.c ✓ 28
3.b odd 2 1 inner 756.2.be.c ✓ 28
4.b odd 2 1 756.2.be.d yes 28
7.c even 3 1 756.2.be.d yes 28
12.b even 2 1 756.2.be.d yes 28
21.h odd 6 1 756.2.be.d yes 28
28.g odd 6 1 inner 756.2.be.c ✓ 28
84.n even 6 1 inner 756.2.be.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.c ✓ 28 1.a even 1 1 trivial
756.2.be.c ✓ 28 3.b odd 2 1 inner
756.2.be.c ✓ 28 28.g odd 6 1 inner
756.2.be.c ✓ 28 84.n even 6 1 inner
756.2.be.d yes 28 4.b odd 2 1
756.2.be.d yes 28 7.c even 3 1
756.2.be.d yes 28 12.b even 2 1
756.2.be.d yes 28 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{28} - 38 T_{5}^{26} + 936 T_{5}^{24} - 13544 T_{5}^{22} + 142048 T_{5}^{20} - 969088 T_{5}^{18} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{19}^{14} - 21 T_{19}^{13} + 152 T_{19}^{12} - 105 T_{19}^{11} - 3292 T_{19}^{10} + 6303 T_{19}^{9} + \cdots + 6083328 \) Copy content Toggle raw display