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Properties

Label 350.2.m.b
Level $350$
Weight $2$
Character orbit 350.m
Analytic conductor $2.795$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{4} + 6 q^{5} - 2 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{4} + 6 q^{5} - 2 q^{6} + 20 q^{9} - 4 q^{10} - 6 q^{11} + 10 q^{12} + 10 q^{14} - 12 q^{15} - 10 q^{16} - 2 q^{19} + 4 q^{20} - 2 q^{21} - 10 q^{22} - 10 q^{23} - 8 q^{24} - 10 q^{25} + 12 q^{26} - 30 q^{27} + 4 q^{29} - 22 q^{30} - 24 q^{31} - 60 q^{33} - 4 q^{35} - 20 q^{36} + 10 q^{37} + 10 q^{38} + 36 q^{39} - 6 q^{40} - 34 q^{41} + 6 q^{44} + 112 q^{45} - 6 q^{46} + 30 q^{47} + 10 q^{48} - 40 q^{49} - 16 q^{50} + 44 q^{51} + 10 q^{53} + 20 q^{54} + 34 q^{55} - 10 q^{56} + 20 q^{58} + 12 q^{59} + 2 q^{60} + 2 q^{61} + 10 q^{64} - 106 q^{65} + 10 q^{66} - 30 q^{67} + 84 q^{69} + 4 q^{70} + 16 q^{71} - 110 q^{73} - 60 q^{74} + 10 q^{75} + 32 q^{76} + 20 q^{77} - 20 q^{78} + 4 q^{79} - 4 q^{80} - 20 q^{81} + 10 q^{83} + 2 q^{84} - 42 q^{85} - 14 q^{86} - 20 q^{87} + 20 q^{88} - 38 q^{90} + 2 q^{91} - 30 q^{92} + 6 q^{94} + 64 q^{95} - 2 q^{96} + 30 q^{97} - 72 q^{99}+O(q^{100}) \) 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 −1.02238 + 1.40719i 0.809017 + 0.587785i −0.814517 − 2.08244i 1.40719 − 1.02238i − 1.00000i −0.587785 − 0.809017i −0.00786102 − 0.0241937i 0.131142 + 2.23222i
29.2 −0.951057 − 0.309017i −0.909805 + 1.25224i 0.809017 + 0.587785i 2.02853 − 0.940783i 1.25224 − 0.909805i − 1.00000i −0.587785 − 0.809017i 0.186694 + 0.574585i −2.21996 + 0.267888i
29.3 −0.951057 − 0.309017i 0.474883 − 0.653621i 0.809017 + 0.587785i −1.51703 + 1.64275i −0.653621 + 0.474883i − 1.00000i −0.587785 − 0.809017i 0.725345 + 2.23238i 1.95042 − 1.09356i
29.4 −0.951057 − 0.309017i 1.20334 − 1.65626i 0.809017 + 0.587785i −1.62675 − 1.53418i −1.65626 + 1.20334i − 1.00000i −0.587785 − 0.809017i −0.368112 − 1.13293i 1.07304 + 1.96178i
29.5 −0.951057 − 0.309017i 1.95978 − 2.69740i 0.809017 + 0.587785i 2.12074 + 0.708831i −2.69740 + 1.95978i − 1.00000i −0.587785 − 0.809017i −2.50820 − 7.71945i −1.79791 − 1.32948i
29.6 0.951057 + 0.309017i −1.58093 + 2.17596i 0.809017 + 0.587785i 0.479579 + 2.18403i −2.17596 + 1.58093i 1.00000i 0.587785 + 0.809017i −1.30843 − 4.02693i −0.218797 + 2.22534i
29.7 0.951057 + 0.309017i −0.832973 + 1.14649i 0.809017 + 0.587785i 1.21900 − 1.87457i −1.14649 + 0.832973i 1.00000i 0.587785 + 0.809017i 0.306458 + 0.943181i 1.73862 − 1.40613i
29.8 0.951057 + 0.309017i 0.454807 − 0.625988i 0.809017 + 0.587785i −1.27366 + 1.83787i 0.625988 − 0.454807i 1.00000i 0.587785 + 0.809017i 0.742040 + 2.28376i −1.77926 + 1.35434i
29.9 0.951057 + 0.309017i 0.578028 − 0.795587i 0.809017 + 0.587785i 1.99915 − 1.00169i 0.795587 − 0.578028i 1.00000i 0.587785 + 0.809017i 0.628209 + 1.93343i 2.21085 − 0.334894i
29.10 0.951057 + 0.309017i 1.91132 − 2.63070i 0.809017 + 0.587785i −2.23309 − 0.115392i 2.63070 − 1.91132i 1.00000i 0.587785 + 0.809017i −2.34041 − 7.20305i −2.08814 − 0.799807i
169.1 −0.951057 + 0.309017i −1.02238 − 1.40719i 0.809017 − 0.587785i −0.814517 + 2.08244i 1.40719 + 1.02238i 1.00000i −0.587785 + 0.809017i −0.00786102 + 0.0241937i 0.131142 − 2.23222i
169.2 −0.951057 + 0.309017i −0.909805 − 1.25224i 0.809017 − 0.587785i 2.02853 + 0.940783i 1.25224 + 0.909805i 1.00000i −0.587785 + 0.809017i 0.186694 − 0.574585i −2.21996 − 0.267888i
169.3 −0.951057 + 0.309017i 0.474883 + 0.653621i 0.809017 − 0.587785i −1.51703 − 1.64275i −0.653621 − 0.474883i 1.00000i −0.587785 + 0.809017i 0.725345 − 2.23238i 1.95042 + 1.09356i
169.4 −0.951057 + 0.309017i 1.20334 + 1.65626i 0.809017 − 0.587785i −1.62675 + 1.53418i −1.65626 − 1.20334i 1.00000i −0.587785 + 0.809017i −0.368112 + 1.13293i 1.07304 − 1.96178i
169.5 −0.951057 + 0.309017i 1.95978 + 2.69740i 0.809017 − 0.587785i 2.12074 − 0.708831i −2.69740 − 1.95978i 1.00000i −0.587785 + 0.809017i −2.50820 + 7.71945i −1.79791 + 1.32948i
169.6 0.951057 − 0.309017i −1.58093 − 2.17596i 0.809017 − 0.587785i 0.479579 − 2.18403i −2.17596 − 1.58093i − 1.00000i 0.587785 − 0.809017i −1.30843 + 4.02693i −0.218797 − 2.22534i
169.7 0.951057 − 0.309017i −0.832973 − 1.14649i 0.809017 − 0.587785i 1.21900 + 1.87457i −1.14649 − 0.832973i − 1.00000i 0.587785 − 0.809017i 0.306458 − 0.943181i 1.73862 + 1.40613i
169.8 0.951057 − 0.309017i 0.454807 + 0.625988i 0.809017 − 0.587785i −1.27366 − 1.83787i 0.625988 + 0.454807i − 1.00000i 0.587785 − 0.809017i 0.742040 − 2.28376i −1.77926 − 1.35434i
169.9 0.951057 − 0.309017i 0.578028 + 0.795587i 0.809017 − 0.587785i 1.99915 + 1.00169i 0.795587 + 0.578028i − 1.00000i 0.587785 − 0.809017i 0.628209 − 1.93343i 2.21085 + 0.334894i
169.10 0.951057 − 0.309017i 1.91132 + 2.63070i 0.809017 − 0.587785i −2.23309 + 0.115392i 2.63070 + 1.91132i − 1.00000i 0.587785 − 0.809017i −2.34041 + 7.20305i −2.08814 + 0.799807i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
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.b ✓ 40
25.e even 10 1 inner 350.2.m.b ✓ 40
25.f odd 20 1 8750.2.a.be 20
25.f odd 20 1 8750.2.a.bf 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.m.b ✓ 40 1.a even 1 1 trivial
350.2.m.b ✓ 40 25.e even 10 1 inner
8750.2.a.be 20 25.f odd 20 1
8750.2.a.bf 20 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 25 T_{3}^{38} + 30 T_{3}^{37} + 385 T_{3}^{36} - 750 T_{3}^{35} - 4430 T_{3}^{34} + \cdots + 2835856 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display