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Properties

Label 350.3.i.c
Level $350$
Weight $3$
Character orbit 350.i
Analytic conductor $9.537$
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] = [350,3,Mod(199,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.i (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} - 72 q^{9} - 16 q^{11} - 56 q^{14} - 48 q^{16} + 96 q^{19} - 100 q^{21} + 48 q^{24} + 240 q^{26} + 48 q^{29} + 180 q^{31} - 288 q^{36} - 192 q^{39} + 32 q^{44} - 56 q^{46} - 224 q^{49} - 188 q^{51} + 504 q^{54} - 128 q^{56} + 444 q^{59} + 204 q^{61} - 192 q^{64} + 192 q^{66} + 488 q^{71} - 224 q^{74} - 532 q^{79} - 964 q^{81} - 208 q^{84} + 200 q^{86} - 96 q^{89} - 28 q^{91} + 24 q^{94} + 96 q^{96} + 408 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1 −1.22474 + 0.707107i −2.45842 + 4.25810i 1.00000 − 1.73205i 0 − 6.95345i 0.227632 + 6.99630i 2.82843i −7.58762 − 13.1421i 0
199.2 −1.22474 + 0.707107i −1.22377 + 2.11963i 1.00000 − 1.73205i 0 − 3.46134i −2.42173 + 6.56774i 2.82843i 1.50478 + 2.60635i 0
199.3 −1.22474 + 0.707107i 0.324566 − 0.562165i 1.00000 − 1.73205i 0 0.918011i 3.38492 − 6.12718i 2.82843i 4.28931 + 7.42931i 0
199.4 −1.22474 + 0.707107i 0.422371 − 0.731568i 1.00000 − 1.73205i 0 1.19465i 6.99576 + 0.243698i 2.82843i 4.14321 + 7.17624i 0
199.5 −1.22474 + 0.707107i 2.39476 − 4.14785i 1.00000 − 1.73205i 0 6.77342i −6.50595 − 2.58314i 2.82843i −6.96980 − 12.0720i 0
199.6 −1.22474 + 0.707107i 2.98997 − 5.17879i 1.00000 − 1.73205i 0 8.45692i 3.21835 + 6.21629i 2.82843i −13.3799 − 23.1746i 0
199.7 1.22474 − 0.707107i −2.98997 + 5.17879i 1.00000 − 1.73205i 0 8.45692i −3.21835 − 6.21629i − 2.82843i −13.3799 − 23.1746i 0
199.8 1.22474 − 0.707107i −2.39476 + 4.14785i 1.00000 − 1.73205i 0 6.77342i 6.50595 + 2.58314i − 2.82843i −6.96980 − 12.0720i 0
199.9 1.22474 − 0.707107i −0.422371 + 0.731568i 1.00000 − 1.73205i 0 1.19465i −6.99576 − 0.243698i − 2.82843i 4.14321 + 7.17624i 0
199.10 1.22474 − 0.707107i −0.324566 + 0.562165i 1.00000 − 1.73205i 0 0.918011i −3.38492 + 6.12718i − 2.82843i 4.28931 + 7.42931i 0
199.11 1.22474 − 0.707107i 1.22377 − 2.11963i 1.00000 − 1.73205i 0 − 3.46134i 2.42173 − 6.56774i − 2.82843i 1.50478 + 2.60635i 0
199.12 1.22474 − 0.707107i 2.45842 − 4.25810i 1.00000 − 1.73205i 0 − 6.95345i −0.227632 − 6.99630i − 2.82843i −7.58762 − 13.1421i 0
299.1 −1.22474 − 0.707107i −2.45842 − 4.25810i 1.00000 + 1.73205i 0 6.95345i 0.227632 − 6.99630i − 2.82843i −7.58762 + 13.1421i 0
299.2 −1.22474 − 0.707107i −1.22377 − 2.11963i 1.00000 + 1.73205i 0 3.46134i −2.42173 − 6.56774i − 2.82843i 1.50478 − 2.60635i 0
299.3 −1.22474 − 0.707107i 0.324566 + 0.562165i 1.00000 + 1.73205i 0 − 0.918011i 3.38492 + 6.12718i − 2.82843i 4.28931 − 7.42931i 0
299.4 −1.22474 − 0.707107i 0.422371 + 0.731568i 1.00000 + 1.73205i 0 − 1.19465i 6.99576 − 0.243698i − 2.82843i 4.14321 − 7.17624i 0
299.5 −1.22474 − 0.707107i 2.39476 + 4.14785i 1.00000 + 1.73205i 0 − 6.77342i −6.50595 + 2.58314i − 2.82843i −6.96980 + 12.0720i 0
299.6 −1.22474 − 0.707107i 2.98997 + 5.17879i 1.00000 + 1.73205i 0 − 8.45692i 3.21835 − 6.21629i − 2.82843i −13.3799 + 23.1746i 0
299.7 1.22474 + 0.707107i −2.98997 − 5.17879i 1.00000 + 1.73205i 0 − 8.45692i −3.21835 + 6.21629i 2.82843i −13.3799 + 23.1746i 0
299.8 1.22474 + 0.707107i −2.39476 − 4.14785i 1.00000 + 1.73205i 0 − 6.77342i 6.50595 − 2.58314i 2.82843i −6.96980 + 12.0720i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.i.c 24
5.b even 2 1 inner 350.3.i.c 24
5.c odd 4 1 350.3.k.c ✓ 12
5.c odd 4 1 350.3.k.d yes 12
7.d odd 6 1 inner 350.3.i.c 24
35.i odd 6 1 inner 350.3.i.c 24
35.k even 12 1 350.3.k.c ✓ 12
35.k even 12 1 350.3.k.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.i.c 24 1.a even 1 1 trivial
350.3.i.c 24 5.b even 2 1 inner
350.3.i.c 24 7.d odd 6 1 inner
350.3.i.c 24 35.i odd 6 1 inner
350.3.k.c ✓ 12 5.c odd 4 1
350.3.k.c ✓ 12 35.k even 12 1
350.3.k.d yes 12 5.c odd 4 1
350.3.k.d yes 12 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 90 T_{3}^{22} + 5263 T_{3}^{20} + 182574 T_{3}^{18} + 4617195 T_{3}^{16} + \cdots + 1275989841 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display