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Properties

Label 930.2.bg.h
Level $930$
Weight $2$
Character orbit 930.bg
Analytic conductor $7.426$
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] = [930,2,Mod(121,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bg (of order \(15\), degree \(8\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} - 3 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 11 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} - 3 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 11 q^{7} + 6 q^{8} + 3 q^{9} - 3 q^{10} - 5 q^{11} - 3 q^{12} - 13 q^{13} - 4 q^{14} + 6 q^{15} - 6 q^{16} + 31 q^{17} - 3 q^{18} - 3 q^{19} + 3 q^{20} - 4 q^{21} + 5 q^{22} - 9 q^{23} + 3 q^{24} - 12 q^{25} + 3 q^{26} + 6 q^{27} + 4 q^{28} + 11 q^{29} + 24 q^{30} + 17 q^{31} - 24 q^{32} + 10 q^{33} + 9 q^{34} + 7 q^{35} - 12 q^{36} + 8 q^{37} - 2 q^{38} - q^{39} - 3 q^{40} + 12 q^{41} + 4 q^{42} - 7 q^{43} - 10 q^{44} + 3 q^{45} - 6 q^{46} - 46 q^{47} - 3 q^{48} + 20 q^{49} - 3 q^{50} - 31 q^{51} + 17 q^{52} + 48 q^{53} - 6 q^{54} - 5 q^{55} + q^{56} - 2 q^{57} + 14 q^{58} + 12 q^{59} + 6 q^{60} - 4 q^{61} + 13 q^{62} + 2 q^{63} - 6 q^{64} + 17 q^{65} + 10 q^{66} - 33 q^{67} + q^{68} - 12 q^{69} - 7 q^{70} - 35 q^{71} - 3 q^{72} + 19 q^{73} + 7 q^{74} - 3 q^{75} + 2 q^{76} + 26 q^{77} - 4 q^{78} - 12 q^{79} + 3 q^{80} + 3 q^{81} - 12 q^{82} + 11 q^{84} - 2 q^{85} - 48 q^{86} + 3 q^{87} + 21 q^{89} - 3 q^{90} + 72 q^{91} + 6 q^{92} + 19 q^{93} - 14 q^{94} + 6 q^{95} + 3 q^{96} - 35 q^{97} - 5 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0.809017 − 0.587785i 0.104528 − 0.994522i 0.309017 − 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i −4.34565 + 0.923697i −0.309017 − 0.951057i −0.978148 − 0.207912i 0.104528 + 0.994522i
121.2 0.809017 − 0.587785i 0.104528 − 0.994522i 0.309017 − 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.105382 − 0.0223997i −0.309017 − 0.951057i −0.978148 − 0.207912i 0.104528 + 0.994522i
121.3 0.809017 − 0.587785i 0.104528 − 0.994522i 0.309017 − 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.744050 − 0.158153i −0.309017 − 0.951057i −0.978148 − 0.207912i 0.104528 + 0.994522i
361.1 −0.309017 − 0.951057i −0.669131 − 0.743145i −0.809017 + 0.587785i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.220951 − 2.10221i 0.809017 + 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
361.2 −0.309017 − 0.951057i −0.669131 − 0.743145i −0.809017 + 0.587785i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.0963276 + 0.916496i 0.809017 + 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
361.3 −0.309017 − 0.951057i −0.669131 − 0.743145i −0.809017 + 0.587785i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.167373 + 1.59245i 0.809017 + 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
391.1 0.809017 + 0.587785i −0.913545 − 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i −2.71613 + 3.01657i −0.309017 + 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
391.2 0.809017 + 0.587785i −0.913545 − 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.334019 − 0.370966i −0.309017 + 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
391.3 0.809017 + 0.587785i −0.913545 − 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i −0.500000 − 0.866025i 2.56932 − 2.85352i −0.309017 + 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
421.1 0.809017 − 0.587785i −0.913545 + 0.406737i 0.309017 − 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i −2.71613 − 3.01657i −0.309017 − 0.951057i 0.669131 − 0.743145i −0.913545 − 0.406737i
421.2 0.809017 − 0.587785i −0.913545 + 0.406737i 0.309017 − 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.334019 + 0.370966i −0.309017 − 0.951057i 0.669131 − 0.743145i −0.913545 − 0.406737i
421.3 0.809017 − 0.587785i −0.913545 + 0.406737i 0.309017 − 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i 2.56932 + 2.85352i −0.309017 − 0.951057i 0.669131 − 0.743145i −0.913545 − 0.406737i
541.1 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 − 0.587785i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.220951 + 2.10221i 0.809017 − 0.587785i −0.104528 − 0.994522i −0.669131 − 0.743145i
541.2 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 − 0.587785i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.0963276 − 0.916496i 0.809017 − 0.587785i −0.104528 − 0.994522i −0.669131 − 0.743145i
541.3 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 − 0.587785i −0.500000 + 0.866025i −0.500000 − 0.866025i 0.167373 − 1.59245i 0.809017 − 0.587785i −0.104528 − 0.994522i −0.669131 − 0.743145i
661.1 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i −4.34565 − 0.923697i −0.309017 + 0.951057i −0.978148 + 0.207912i 0.104528 − 0.994522i
661.2 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.105382 + 0.0223997i −0.309017 + 0.951057i −0.978148 + 0.207912i 0.104528 − 0.994522i
661.3 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −0.500000 − 0.866025i −0.500000 + 0.866025i 0.744050 + 0.158153i −0.309017 + 0.951057i −0.978148 + 0.207912i 0.104528 − 0.994522i
691.1 −0.309017 + 0.951057i 0.978148 + 0.207912i −0.809017 − 0.587785i −0.500000 − 0.866025i −0.500000 + 0.866025i −2.37522 + 1.05752i 0.809017 − 0.587785i 0.913545 + 0.406737i 0.978148 − 0.207912i
691.2 −0.309017 + 0.951057i 0.978148 + 0.207912i −0.809017 − 0.587785i −0.500000 − 0.866025i −0.500000 + 0.866025i −2.21753 + 0.987309i 0.809017 − 0.587785i 0.913545 + 0.406737i 0.978148 − 0.207912i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bg.h ✓ 24
31.g even 15 1 inner 930.2.bg.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.h ✓ 24 1.a even 1 1 trivial
930.2.bg.h ✓ 24 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 11 T_{7}^{23} + 40 T_{7}^{22} + 40 T_{7}^{21} + 163 T_{7}^{20} + 2264 T_{7}^{19} + \cdots + 20736 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display