↑

Properties

Label 350.3.p.g
Level $350$
Weight $3$
Character orbit 350.p
Analytic conductor $9.537$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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 −1.31208 + 4.89676i −1.73205 + 1.00000i 0 7.16935 4.29228 − 5.52959i 2.00000 + 2.00000i −14.4625 − 8.34990i 0
93.2 −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
93.3 −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
93.4 −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
93.5 −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
93.6 −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
107.1 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
107.2 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
107.3 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
107.4 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
107.5 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
107.6 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
193.1 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
193.2 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
193.3 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
193.4 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
193.5 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
193.6 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
207.1 −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
207.2 −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
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.g yes 24
5.b even 2 1 350.3.p.f ✓ 24
5.c odd 4 1 350.3.p.f ✓ 24
5.c odd 4 1 inner 350.3.p.g yes 24
7.c even 3 1 inner 350.3.p.g yes 24
35.j even 6 1 350.3.p.f ✓ 24
35.l odd 12 1 350.3.p.f ✓ 24
35.l odd 12 1 inner 350.3.p.g yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.p.f ✓ 24 5.b even 2 1
350.3.p.f ✓ 24 5.c odd 4 1
350.3.p.f ✓ 24 35.j even 6 1
350.3.p.f ✓ 24 35.l odd 12 1
350.3.p.g yes 24 1.a even 1 1 trivial
350.3.p.g yes 24 5.c odd 4 1 inner
350.3.p.g yes 24 7.c even 3 1 inner
350.3.p.g yes 24 35.l odd 12 1 inner

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