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Properties

Label 675.2.h.c
Level $675$
Weight $2$
Character orbit 675.h
Analytic conductor $5.390$
Analytic rank $0$
Dimension $40$
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] = [675,2,Mod(136,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.h (of order \(5\), degree \(4\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 12 q^{7} - 24 q^{10} + 8 q^{13} + 4 q^{16} - 14 q^{19} + 20 q^{22} - 12 q^{25} - 22 q^{28} - 10 q^{31} + 2 q^{34} + 26 q^{37} + 90 q^{40} - 64 q^{43} + 88 q^{46} + 116 q^{49} - 30 q^{52} + 26 q^{55} - 60 q^{58} + 16 q^{61} + 6 q^{64} + 38 q^{67} + 32 q^{70} + 10 q^{73} - 276 q^{76} + 38 q^{79} - 56 q^{82} + 4 q^{85} - 72 q^{88} + 4 q^{91} - 82 q^{94} - 18 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.14381 + 1.55757i 0 1.55186 − 4.77614i 2.10899 + 0.743063i 0 −3.78063 2.47454 + 7.61586i 0 −5.67865 + 1.69192i
136.2 −1.61759 + 1.17525i 0 0.617351 − 1.90001i 1.17012 − 1.90547i 0 4.57177 −0.00136471 − 0.00420014i 0 0.346625 + 4.45744i
136.3 −1.26030 + 0.915663i 0 0.131888 − 0.405909i −1.66307 + 1.49472i 0 −0.357750 −0.757327 − 2.33081i 0 0.727310 − 3.40661i
136.4 −1.18959 + 0.864290i 0 0.0501012 − 0.154196i −0.976745 − 2.01146i 0 −4.82878 −0.835099 − 2.57017i 0 2.90041 + 1.54863i
136.5 −0.0783628 + 0.0569339i 0 −0.615135 + 1.89319i 0.732172 + 2.11280i 0 1.77736 −0.119447 − 0.367620i 0 −0.177665 − 0.123880i
136.6 0.0783628 − 0.0569339i 0 −0.615135 + 1.89319i −0.732172 − 2.11280i 0 1.77736 0.119447 + 0.367620i 0 −0.177665 − 0.123880i
136.7 1.18959 − 0.864290i 0 0.0501012 − 0.154196i 0.976745 + 2.01146i 0 −4.82878 0.835099 + 2.57017i 0 2.90041 + 1.54863i
136.8 1.26030 − 0.915663i 0 0.131888 − 0.405909i 1.66307 − 1.49472i 0 −0.357750 0.757327 + 2.33081i 0 0.727310 − 3.40661i
136.9 1.61759 − 1.17525i 0 0.617351 − 1.90001i −1.17012 + 1.90547i 0 4.57177 0.00136471 + 0.00420014i 0 0.346625 + 4.45744i
136.10 2.14381 − 1.55757i 0 1.55186 − 4.77614i −2.10899 − 0.743063i 0 −3.78063 −2.47454 − 7.61586i 0 −5.67865 + 1.69192i
271.1 −0.839278 + 2.58303i 0 −4.34964 − 3.16020i 1.99911 + 1.00177i 0 0.261501 7.41894 − 5.39017i 0 −4.26542 + 4.32301i
271.2 −0.561545 + 1.72826i 0 −1.05350 − 0.765416i −1.24519 + 1.85728i 0 0.998076 −1.02586 + 0.745331i 0 −2.51063 − 3.19495i
271.3 −0.410672 + 1.26392i 0 0.189196 + 0.137459i 1.67523 − 1.48108i 0 −4.42748 −2.40174 + 1.74497i 0 1.18399 + 2.72559i
271.4 −0.282654 + 0.869919i 0 0.941168 + 0.683798i −1.13467 − 1.92679i 0 3.90830 −2.34087 + 1.70074i 0 1.99687 − 0.442454i
271.5 −0.0979733 + 0.301531i 0 1.53671 + 1.11649i 1.94582 + 1.10171i 0 −1.12236 −1.00021 + 0.726693i 0 −0.522839 + 0.478787i
271.6 0.0979733 − 0.301531i 0 1.53671 + 1.11649i −1.94582 − 1.10171i 0 −1.12236 1.00021 − 0.726693i 0 −0.522839 + 0.478787i
271.7 0.282654 − 0.869919i 0 0.941168 + 0.683798i 1.13467 + 1.92679i 0 3.90830 2.34087 − 1.70074i 0 1.99687 − 0.442454i
271.8 0.410672 − 1.26392i 0 0.189196 + 0.137459i −1.67523 + 1.48108i 0 −4.42748 2.40174 − 1.74497i 0 1.18399 + 2.72559i
271.9 0.561545 − 1.72826i 0 −1.05350 − 0.765416i 1.24519 − 1.85728i 0 0.998076 1.02586 − 0.745331i 0 −2.51063 − 3.19495i
271.10 0.839278 − 2.58303i 0 −4.34964 − 3.16020i −1.99911 − 1.00177i 0 0.261501 −7.41894 + 5.39017i 0 −4.26542 + 4.32301i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.h.c ✓ 40
3.b odd 2 1 inner 675.2.h.c ✓ 40
25.d even 5 1 inner 675.2.h.c ✓ 40
75.j odd 10 1 inner 675.2.h.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.h.c ✓ 40 1.a even 1 1 trivial
675.2.h.c ✓ 40 3.b odd 2 1 inner
675.2.h.c ✓ 40 25.d even 5 1 inner
675.2.h.c ✓ 40 75.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 12 T_{2}^{38} + 112 T_{2}^{36} + 911 T_{2}^{34} + 7086 T_{2}^{32} + 32932 T_{2}^{30} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display