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Properties

Label 950.2.u.e
Level $950$
Weight $2$
Character orbit 950.u
Analytic conductor $7.586$
Analytic rank $0$
Dimension $24$
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] = [950,2,Mod(99,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.u (of order \(18\), degree \(6\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 36 q^{9} + 12 q^{11} + 24 q^{21} + 12 q^{29} + 12 q^{31} + 36 q^{36} + 72 q^{39} - 12 q^{41} - 12 q^{44} - 24 q^{46} + 36 q^{49} + 24 q^{56} + 48 q^{59} - 60 q^{61} + 12 q^{64} + 48 q^{66} - 12 q^{69} - 84 q^{71} - 12 q^{74} + 36 q^{76} - 120 q^{79} + 36 q^{81} + 48 q^{84} - 72 q^{86} + 24 q^{89} + 48 q^{91} - 120 q^{94} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −0.342020 + 0.939693i −1.31143 + 0.231240i −0.766044 − 0.642788i 0 0.231240 − 1.31143i 3.78616 + 2.18594i 0.866025 − 0.500000i −1.15270 + 0.419550i 0
99.2 −0.342020 + 0.939693i 1.31143 − 0.231240i −0.766044 − 0.642788i 0 −0.231240 + 1.31143i −2.05411 − 1.18594i 0.866025 − 0.500000i −1.15270 + 0.419550i 0
99.3 0.342020 − 0.939693i −1.31143 + 0.231240i −0.766044 − 0.642788i 0 −0.231240 + 1.31143i 2.05411 + 1.18594i −0.866025 + 0.500000i −1.15270 + 0.419550i 0
99.4 0.342020 − 0.939693i 1.31143 − 0.231240i −0.766044 − 0.642788i 0 0.231240 − 1.31143i −3.78616 − 2.18594i −0.866025 + 0.500000i −1.15270 + 0.419550i 0
149.1 −0.642788 + 0.766044i −0.931116 − 2.55822i −0.173648 − 0.984808i 0 2.55822 + 0.931116i −2.31045 + 1.33394i 0.866025 + 0.500000i −3.37939 + 2.83564i 0
149.2 −0.642788 + 0.766044i 0.931116 + 2.55822i −0.173648 − 0.984808i 0 −2.55822 − 0.931116i 4.04250 − 2.33394i 0.866025 + 0.500000i −3.37939 + 2.83564i 0
149.3 0.642788 − 0.766044i −0.931116 − 2.55822i −0.173648 − 0.984808i 0 −2.55822 − 0.931116i −4.04250 + 2.33394i −0.866025 − 0.500000i −3.37939 + 2.83564i 0
149.4 0.642788 − 0.766044i 0.931116 + 2.55822i −0.173648 − 0.984808i 0 2.55822 + 0.931116i 2.31045 − 1.33394i −0.866025 − 0.500000i −3.37939 + 2.83564i 0
199.1 −0.984808 + 0.173648i −1.07851 − 1.28531i 0.939693 − 0.342020i 0 1.28531 + 1.07851i 0.411781 + 0.237742i −0.866025 + 0.500000i 0.0320889 − 0.181985i 0
199.2 −0.984808 + 0.173648i 1.07851 + 1.28531i 0.939693 − 0.342020i 0 −1.28531 − 1.07851i −2.14383 − 1.23774i −0.866025 + 0.500000i 0.0320889 − 0.181985i 0
199.3 0.984808 − 0.173648i −1.07851 − 1.28531i 0.939693 − 0.342020i 0 −1.28531 − 1.07851i 2.14383 + 1.23774i 0.866025 − 0.500000i 0.0320889 − 0.181985i 0
199.4 0.984808 − 0.173648i 1.07851 + 1.28531i 0.939693 − 0.342020i 0 1.28531 + 1.07851i −0.411781 − 0.237742i 0.866025 − 0.500000i 0.0320889 − 0.181985i 0
499.1 −0.342020 − 0.939693i −1.31143 − 0.231240i −0.766044 + 0.642788i 0 0.231240 + 1.31143i 3.78616 − 2.18594i 0.866025 + 0.500000i −1.15270 − 0.419550i 0
499.2 −0.342020 − 0.939693i 1.31143 + 0.231240i −0.766044 + 0.642788i 0 −0.231240 − 1.31143i −2.05411 + 1.18594i 0.866025 + 0.500000i −1.15270 − 0.419550i 0
499.3 0.342020 + 0.939693i −1.31143 − 0.231240i −0.766044 + 0.642788i 0 −0.231240 − 1.31143i 2.05411 − 1.18594i −0.866025 − 0.500000i −1.15270 − 0.419550i 0
499.4 0.342020 + 0.939693i 1.31143 + 0.231240i −0.766044 + 0.642788i 0 0.231240 + 1.31143i −3.78616 + 2.18594i −0.866025 − 0.500000i −1.15270 − 0.419550i 0
549.1 −0.984808 − 0.173648i −1.07851 + 1.28531i 0.939693 + 0.342020i 0 1.28531 − 1.07851i 0.411781 − 0.237742i −0.866025 − 0.500000i 0.0320889 + 0.181985i 0
549.2 −0.984808 − 0.173648i 1.07851 − 1.28531i 0.939693 + 0.342020i 0 −1.28531 + 1.07851i −2.14383 + 1.23774i −0.866025 − 0.500000i 0.0320889 + 0.181985i 0
549.3 0.984808 + 0.173648i −1.07851 + 1.28531i 0.939693 + 0.342020i 0 −1.28531 + 1.07851i 2.14383 − 1.23774i 0.866025 + 0.500000i 0.0320889 + 0.181985i 0
549.4 0.984808 + 0.173648i 1.07851 − 1.28531i 0.939693 + 0.342020i 0 1.28531 − 1.07851i −0.411781 + 0.237742i 0.866025 + 0.500000i 0.0320889 + 0.181985i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.e 24
5.b even 2 1 inner 950.2.u.e 24
5.c odd 4 1 190.2.k.b ✓ 12
5.c odd 4 1 950.2.l.h 12
19.e even 9 1 inner 950.2.u.e 24
95.p even 18 1 inner 950.2.u.e 24
95.q odd 36 1 190.2.k.b ✓ 12
95.q odd 36 1 950.2.l.h 12
95.q odd 36 1 3610.2.a.bc 6
95.r even 36 1 3610.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.b ✓ 12 5.c odd 4 1
190.2.k.b ✓ 12 95.q odd 36 1
950.2.l.h 12 5.c odd 4 1
950.2.l.h 12 95.q odd 36 1
950.2.u.e 24 1.a even 1 1 trivial
950.2.u.e 24 5.b even 2 1 inner
950.2.u.e 24 19.e even 9 1 inner
950.2.u.e 24 95.p even 18 1 inner
3610.2.a.bc 6 95.q odd 36 1
3610.2.a.be 6 95.r even 36 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{12} + 9T_{3}^{10} + 36T_{3}^{8} - 64T_{3}^{6} + 189T_{3}^{4} - 999T_{3}^{2} + 1369 \) Copy content Toggle raw display
\( T_{7}^{24} - 60 T_{7}^{22} + 2280 T_{7}^{20} - 52736 T_{7}^{18} + 886320 T_{7}^{16} - 10122624 T_{7}^{14} + \cdots + 533794816 \) Copy content Toggle raw display