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Properties

Label 350.3.p.f
Level $350$
Weight $3$
Character orbit 350.p
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(93,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.93");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.p (of order \(12\), 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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 4 q^{3} + 16 q^{6} - 24 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 4 q^{3} + 16 q^{6} - 24 q^{7} - 48 q^{8} - 32 q^{11} - 8 q^{12} + 16 q^{13} + 48 q^{16} - 8 q^{17} - 24 q^{18} - 20 q^{21} + 64 q^{22} - 52 q^{23} - 16 q^{26} - 280 q^{27} + 24 q^{28} + 68 q^{31} + 48 q^{32} - 80 q^{33} + 96 q^{36} - 32 q^{37} - 112 q^{38} - 328 q^{41} - 44 q^{42} - 136 q^{43} - 104 q^{46} + 164 q^{47} - 32 q^{48} - 188 q^{51} - 16 q^{52} + 184 q^{53} + 96 q^{56} + 304 q^{57} - 136 q^{58} + 276 q^{61} - 136 q^{62} + 556 q^{63} - 160 q^{66} + 152 q^{67} + 16 q^{68} + 72 q^{71} - 48 q^{72} - 336 q^{73} + 448 q^{76} - 40 q^{77} - 352 q^{78} + 148 q^{81} + 164 q^{82} + 896 q^{83} + 136 q^{86} + 176 q^{87} + 64 q^{88} + 436 q^{91} + 208 q^{92} - 584 q^{93} + 32 q^{96} - 592 q^{97} + 424 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 0.366025 + 1.36603i −1.06318 + 3.96783i −1.73205 + 1.00000i 0 −5.80930 −6.83995 + 1.48830i −2.00000 − 2.00000i −6.81907 − 3.93699i 0
93.2 0.366025 + 1.36603i −0.384832 + 1.43621i −1.73205 + 1.00000i 0 −2.10276 −5.32696 + 4.54131i −2.00000 − 2.00000i 5.87961 + 3.39460i 0
93.3 0.366025 + 1.36603i −0.190466 + 0.710829i −1.73205 + 1.00000i 0 −1.04073 6.85529 − 1.41598i −2.00000 − 2.00000i 7.32523 + 4.22922i 0
93.4 0.366025 + 1.36603i −0.112937 + 0.421487i −1.73205 + 1.00000i 0 −0.617100 2.11986 − 6.67130i −2.00000 − 2.00000i 7.62933 + 4.40480i 0
93.5 0.366025 + 1.36603i 1.17138 − 4.37165i −1.73205 + 1.00000i 0 6.40054 −5.44416 − 4.40013i −2.00000 − 2.00000i −9.94495 − 5.74172i 0
93.6 0.366025 + 1.36603i 1.31208 − 4.89676i −1.73205 + 1.00000i 0 7.16935 −4.29228 + 5.52959i −2.00000 − 2.00000i −14.4625 − 8.34990i 0
107.1 −1.36603 + 0.366025i −4.89676 − 1.31208i 1.73205 − 1.00000i 0 7.16935 −5.52959 − 4.29228i −2.00000 + 2.00000i 14.4625 + 8.34990i 0
107.2 −1.36603 + 0.366025i −4.37165 − 1.17138i 1.73205 − 1.00000i 0 6.40054 4.40013 − 5.44416i −2.00000 + 2.00000i 9.94495 + 5.74172i 0
107.3 −1.36603 + 0.366025i 0.421487 + 0.112937i 1.73205 − 1.00000i 0 −0.617100 6.67130 + 2.11986i −2.00000 + 2.00000i −7.62933 − 4.40480i 0
107.4 −1.36603 + 0.366025i 0.710829 + 0.190466i 1.73205 − 1.00000i 0 −1.04073 1.41598 + 6.85529i −2.00000 + 2.00000i −7.32523 − 4.22922i 0
107.5 −1.36603 + 0.366025i 1.43621 + 0.384832i 1.73205 − 1.00000i 0 −2.10276 −4.54131 − 5.32696i −2.00000 + 2.00000i −5.87961 − 3.39460i 0
107.6 −1.36603 + 0.366025i 3.96783 + 1.06318i 1.73205 − 1.00000i 0 −5.80930 −1.48830 − 6.83995i −2.00000 + 2.00000i 6.81907 + 3.93699i 0
193.1 −1.36603 − 0.366025i −4.89676 + 1.31208i 1.73205 + 1.00000i 0 7.16935 −5.52959 + 4.29228i −2.00000 − 2.00000i 14.4625 − 8.34990i 0
193.2 −1.36603 − 0.366025i −4.37165 + 1.17138i 1.73205 + 1.00000i 0 6.40054 4.40013 + 5.44416i −2.00000 − 2.00000i 9.94495 − 5.74172i 0
193.3 −1.36603 − 0.366025i 0.421487 − 0.112937i 1.73205 + 1.00000i 0 −0.617100 6.67130 − 2.11986i −2.00000 − 2.00000i −7.62933 + 4.40480i 0
193.4 −1.36603 − 0.366025i 0.710829 − 0.190466i 1.73205 + 1.00000i 0 −1.04073 1.41598 − 6.85529i −2.00000 − 2.00000i −7.32523 + 4.22922i 0
193.5 −1.36603 − 0.366025i 1.43621 − 0.384832i 1.73205 + 1.00000i 0 −2.10276 −4.54131 + 5.32696i −2.00000 − 2.00000i −5.87961 + 3.39460i 0
193.6 −1.36603 − 0.366025i 3.96783 − 1.06318i 1.73205 + 1.00000i 0 −5.80930 −1.48830 + 6.83995i −2.00000 − 2.00000i 6.81907 − 3.93699i 0
207.1 0.366025 − 1.36603i −1.06318 − 3.96783i −1.73205 − 1.00000i 0 −5.80930 −6.83995 − 1.48830i −2.00000 + 2.00000i −6.81907 + 3.93699i 0
207.2 0.366025 − 1.36603i −0.384832 − 1.43621i −1.73205 − 1.00000i 0 −2.10276 −5.32696 − 4.54131i −2.00000 + 2.00000i 5.87961 − 3.39460i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.p.f ✓ 24
5.b even 2 1 350.3.p.g yes 24
5.c odd 4 1 inner 350.3.p.f ✓ 24
5.c odd 4 1 350.3.p.g yes 24
7.c even 3 1 inner 350.3.p.f ✓ 24
35.j even 6 1 350.3.p.g yes 24
35.l odd 12 1 inner 350.3.p.f ✓ 24
35.l odd 12 1 350.3.p.g yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.p.f ✓ 24 1.a even 1 1 trivial
350.3.p.f ✓ 24 5.c odd 4 1 inner
350.3.p.f ✓ 24 7.c even 3 1 inner
350.3.p.f ✓ 24 35.l odd 12 1 inner
350.3.p.g yes 24 5.b even 2 1
350.3.p.g yes 24 5.c odd 4 1
350.3.p.g yes 24 35.j even 6 1
350.3.p.g yes 24 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} + 152 T_{3}^{21} - 109 T_{3}^{20} - 2752 T_{3}^{19} + \cdots + 4100625 \) Copy content Toggle raw display
\( T_{11}^{12} + 16 T_{11}^{11} + 704 T_{11}^{10} + 6004 T_{11}^{9} + 272317 T_{11}^{8} + \cdots + 2099992248225 \) Copy content Toggle raw display