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Properties

Label 350.2.m.a
Level $350$
Weight $2$
Character orbit 350.m
Analytic conductor $2.795$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(29,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.m (of order \(10\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{9} + 2 q^{11} + 10 q^{12} - 6 q^{14} + 20 q^{15} - 6 q^{16} - 22 q^{19} - 2 q^{21} - 10 q^{22} - 10 q^{23} + 8 q^{24} - 10 q^{25} - 4 q^{26} - 30 q^{27} - 12 q^{29} - 10 q^{30} + 20 q^{33} - 8 q^{36} + 10 q^{37} - 10 q^{38} - 48 q^{39} + 10 q^{40} + 42 q^{41} - 2 q^{44} - 40 q^{45} + 10 q^{46} + 30 q^{47} + 10 q^{48} - 24 q^{49} + 20 q^{50} - 52 q^{51} + 10 q^{53} + 4 q^{54} + 10 q^{55} + 6 q^{56} - 20 q^{58} - 10 q^{60} + 46 q^{61} - 20 q^{63} + 6 q^{64} + 10 q^{65} - 10 q^{66} + 10 q^{67} + 32 q^{71} + 30 q^{73} - 28 q^{74} - 10 q^{75} - 48 q^{76} + 20 q^{77} - 20 q^{78} - 44 q^{79} + 76 q^{81} + 50 q^{83} + 2 q^{84} - 50 q^{85} - 6 q^{86} - 20 q^{87} - 20 q^{88} - 4 q^{89} + 50 q^{90} - 6 q^{91} + 30 q^{92} - 6 q^{94} - 60 q^{95} + 2 q^{96} + 30 q^{97} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.951057 − 0.309017i −0.840443 + 1.15677i 0.809017 + 0.587785i −1.82116 + 1.29744i 1.15677 − 0.840443i 1.00000i −0.587785 − 0.809017i 0.295277 + 0.908768i 2.13296 − 0.671171i
29.2 −0.951057 − 0.309017i 0.310301 − 0.427092i 0.809017 + 0.587785i 1.73700 + 1.40813i −0.427092 + 0.310301i 1.00000i −0.587785 − 0.809017i 0.840930 + 2.58812i −1.21685 − 1.87597i
29.3 −0.951057 − 0.309017i 1.06039 − 1.45950i 0.809017 + 0.587785i 0.717612 − 2.11779i −1.45950 + 1.06039i 1.00000i −0.587785 − 0.809017i −0.0786699 − 0.242121i −1.33692 + 1.79238i
29.4 0.951057 + 0.309017i −0.974986 + 1.34195i 0.809017 + 0.587785i 2.09432 + 0.783465i −1.34195 + 0.974986i − 1.00000i 0.587785 + 0.809017i 0.0768108 + 0.236399i 1.74971 + 1.39230i
29.5 0.951057 + 0.309017i 1.26372 − 1.73935i 0.809017 + 0.587785i 0.915083 + 2.04025i 1.73935 − 1.26372i − 1.00000i 0.587785 + 0.809017i −0.501328 − 1.54293i 0.239823 + 2.22317i
29.6 0.951057 + 0.309017i 1.41709 − 1.95046i 0.809017 + 0.587785i −0.0248168 − 2.23593i 1.95046 − 1.41709i − 1.00000i 0.587785 + 0.809017i −0.869087 − 2.67478i 0.667338 − 2.13416i
169.1 −0.951057 + 0.309017i −0.840443 − 1.15677i 0.809017 − 0.587785i −1.82116 − 1.29744i 1.15677 + 0.840443i − 1.00000i −0.587785 + 0.809017i 0.295277 − 0.908768i 2.13296 + 0.671171i
169.2 −0.951057 + 0.309017i 0.310301 + 0.427092i 0.809017 − 0.587785i 1.73700 − 1.40813i −0.427092 − 0.310301i − 1.00000i −0.587785 + 0.809017i 0.840930 − 2.58812i −1.21685 + 1.87597i
169.3 −0.951057 + 0.309017i 1.06039 + 1.45950i 0.809017 − 0.587785i 0.717612 + 2.11779i −1.45950 − 1.06039i − 1.00000i −0.587785 + 0.809017i −0.0786699 + 0.242121i −1.33692 − 1.79238i
169.4 0.951057 − 0.309017i −0.974986 − 1.34195i 0.809017 − 0.587785i 2.09432 − 0.783465i −1.34195 − 0.974986i 1.00000i 0.587785 − 0.809017i 0.0768108 − 0.236399i 1.74971 − 1.39230i
169.5 0.951057 − 0.309017i 1.26372 + 1.73935i 0.809017 − 0.587785i 0.915083 − 2.04025i 1.73935 + 1.26372i 1.00000i 0.587785 − 0.809017i −0.501328 + 1.54293i 0.239823 − 2.22317i
169.6 0.951057 − 0.309017i 1.41709 + 1.95046i 0.809017 − 0.587785i −0.0248168 + 2.23593i 1.95046 + 1.41709i 1.00000i 0.587785 − 0.809017i −0.869087 + 2.67478i 0.667338 + 2.13416i
239.1 −0.587785 − 0.809017i −2.21568 − 0.719918i −0.309017 + 0.951057i 0.562587 + 2.16414i 0.719918 + 2.21568i − 1.00000i 0.951057 − 0.309017i 1.96390 + 1.42686i 1.42014 − 1.72719i
239.2 −0.587785 − 0.809017i 0.195447 + 0.0635047i −0.309017 + 0.951057i 0.0419655 − 2.23567i −0.0635047 − 0.195447i − 1.00000i 0.951057 − 0.309017i −2.39288 − 1.73853i −1.83337 + 1.28015i
239.3 −0.587785 − 0.809017i 1.85325 + 0.602159i −0.309017 + 0.951057i 1.98854 + 1.02259i −0.602159 − 1.85325i − 1.00000i 0.951057 − 0.309017i 0.644904 + 0.468550i −0.341542 − 2.20983i
239.4 0.587785 + 0.809017i −3.21572 − 1.04485i −0.309017 + 0.951057i −2.16379 − 0.563933i −1.04485 − 3.21572i 1.00000i −0.951057 + 0.309017i 6.82209 + 4.95654i −0.815611 − 2.08201i
239.5 0.587785 + 0.809017i −0.331395 − 0.107677i −0.309017 + 0.951057i −1.25842 + 1.84834i −0.107677 − 0.331395i 1.00000i −0.951057 + 0.309017i −2.32882 − 1.69199i −2.23502 + 0.0683444i
239.6 0.587785 + 0.809017i 1.47802 + 0.480239i −0.309017 + 0.951057i 2.21108 − 0.333354i 0.480239 + 1.47802i 1.00000i −0.951057 + 0.309017i −0.473123 − 0.343744i 1.56933 + 1.59286i
309.1 −0.587785 + 0.809017i −2.21568 + 0.719918i −0.309017 − 0.951057i 0.562587 − 2.16414i 0.719918 − 2.21568i 1.00000i 0.951057 + 0.309017i 1.96390 − 1.42686i 1.42014 + 1.72719i
309.2 −0.587785 + 0.809017i 0.195447 − 0.0635047i −0.309017 − 0.951057i 0.0419655 + 2.23567i −0.0635047 + 0.195447i 1.00000i 0.951057 + 0.309017i −2.39288 + 1.73853i −1.83337 − 1.28015i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.m.a ✓ 24
25.e even 10 1 inner 350.2.m.a ✓ 24
25.f odd 20 1 8750.2.a.z 12
25.f odd 20 1 8750.2.a.bb 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.m.a ✓ 24 1.a even 1 1 trivial
350.2.m.a ✓ 24 25.e even 10 1 inner
8750.2.a.z 12 25.f odd 20 1
8750.2.a.bb 12 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 13 T_{3}^{22} + 30 T_{3}^{21} + 88 T_{3}^{20} - 390 T_{3}^{19} + 119 T_{3}^{18} + \cdots + 400 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display