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Properties

Label 650.2.l.c
Level $650$
Weight $2$
Character orbit 650.l
Analytic conductor $5.190$
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] = [650,2,Mod(131,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.l (of order \(5\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 3 q^{3} - 6 q^{4} - q^{5} + 3 q^{6} - 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 3 q^{3} - 6 q^{4} - q^{5} + 3 q^{6} - 6 q^{8} + 7 q^{9} - q^{10} + 2 q^{11} - 2 q^{12} - 6 q^{13} + 20 q^{15} - 6 q^{16} + 9 q^{17} + 22 q^{18} + 12 q^{19} + 4 q^{20} + 25 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} - 41 q^{25} + 24 q^{26} - 6 q^{27} - 2 q^{29} + 10 q^{30} + 13 q^{31} + 24 q^{32} - 34 q^{33} + 14 q^{34} + 7 q^{36} + 5 q^{37} + 22 q^{38} - 2 q^{39} - q^{40} - 10 q^{41} + 25 q^{42} - 18 q^{43} + 2 q^{44} + 3 q^{45} + 9 q^{46} - 5 q^{47} - 2 q^{48} + 32 q^{49} - 11 q^{50} - 56 q^{51} - 6 q^{52} + 34 q^{53} + 19 q^{54} + 20 q^{55} - 12 q^{57} - 2 q^{58} - 15 q^{60} - 2 q^{61} - 12 q^{62} + 10 q^{63} - 6 q^{64} - q^{65} + 26 q^{66} + 2 q^{67} - 46 q^{68} + 33 q^{69} - 20 q^{70} + 29 q^{71} - 18 q^{72} - 11 q^{73} - 30 q^{74} - 25 q^{75} - 68 q^{76} + 15 q^{77} + 3 q^{78} + 20 q^{79} - q^{80} - 9 q^{81} - 20 q^{82} - 69 q^{83} - 20 q^{84} - 27 q^{85} + 22 q^{86} - 18 q^{87} + 2 q^{88} + 19 q^{89} + 8 q^{90} + 9 q^{92} + 40 q^{93} - 5 q^{94} + 78 q^{95} + 3 q^{96} - 49 q^{97} + 2 q^{98} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 0.309017 + 0.951057i −2.36555 + 1.71867i −0.809017 + 0.587785i −0.618683 − 2.14877i −2.36555 − 1.71867i −4.94863 −0.809017 − 0.587785i 1.71494 − 5.27805i 1.85242 − 1.25241i
131.2 0.309017 + 0.951057i −0.970136 + 0.704845i −0.809017 + 0.587785i −1.50858 + 1.65051i −0.970136 − 0.704845i −1.35024 −0.809017 − 0.587785i −0.482694 + 1.48558i −2.03591 − 0.924706i
131.3 0.309017 + 0.951057i −0.892707 + 0.648590i −0.809017 + 0.587785i −1.63942 − 1.52062i −0.892707 − 0.648590i 4.94642 −0.809017 − 0.587785i −0.550793 + 1.69517i 0.939587 − 2.02908i
131.4 0.309017 + 0.951057i 0.859118 − 0.624186i −0.809017 + 0.587785i 1.70352 + 1.44845i 0.859118 + 0.624186i 0.905410 −0.809017 − 0.587785i −0.578575 + 1.78067i −0.851143 + 2.06774i
131.5 0.309017 + 0.951057i 2.22837 − 1.61900i −0.809017 + 0.587785i 0.656771 + 2.13744i 2.22837 + 1.61900i 1.40108 −0.809017 − 0.587785i 1.41739 − 4.36229i −1.82987 + 1.28513i
131.6 0.309017 + 0.951057i 2.44993 − 1.77998i −0.809017 + 0.587785i 0.597375 − 2.15480i 2.44993 + 1.77998i −0.954033 −0.809017 − 0.587785i 1.90678 − 5.86846i 2.23393 − 0.0977310i
261.1 −0.809017 + 0.587785i −0.854221 + 2.62902i 0.309017 − 0.951057i 0.249965 + 2.22205i −0.854221 − 2.62902i −2.02551 0.309017 + 0.951057i −3.75502 − 2.72818i −1.50832 − 1.65075i
261.2 −0.809017 + 0.587785i −0.217795 + 0.670303i 0.309017 − 0.951057i 0.980398 − 2.00968i −0.217795 − 0.670303i −4.30334 0.309017 + 0.951057i 2.02518 + 1.47138i 0.388103 + 2.20213i
261.3 −0.809017 + 0.587785i −0.102252 + 0.314699i 0.309017 − 0.951057i 2.08823 + 0.799566i −0.102252 − 0.314699i 4.37811 0.309017 + 0.951057i 2.33847 + 1.69900i −2.15938 + 0.580567i
261.4 −0.809017 + 0.587785i 0.0956093 − 0.294255i 0.309017 − 0.951057i −2.23331 − 0.110958i 0.0956093 + 0.294255i 1.22498 0.309017 + 0.951057i 2.34961 + 1.70709i 1.87201 − 1.22294i
261.5 −0.809017 + 0.587785i 0.402964 − 1.24019i 0.309017 − 0.951057i −0.529888 − 2.17238i 0.402964 + 1.24019i 1.35654 0.309017 + 0.951057i 1.05135 + 0.763850i 1.70558 + 1.44603i
261.6 −0.809017 + 0.587785i 0.866678 − 2.66736i 0.309017 − 0.951057i −0.246372 + 2.22245i 0.866678 + 2.66736i −0.630775 0.309017 + 0.951057i −3.93664 − 2.86013i −1.10701 − 1.94282i
391.1 −0.809017 − 0.587785i −0.854221 − 2.62902i 0.309017 + 0.951057i 0.249965 − 2.22205i −0.854221 + 2.62902i −2.02551 0.309017 − 0.951057i −3.75502 + 2.72818i −1.50832 + 1.65075i
391.2 −0.809017 − 0.587785i −0.217795 − 0.670303i 0.309017 + 0.951057i 0.980398 + 2.00968i −0.217795 + 0.670303i −4.30334 0.309017 − 0.951057i 2.02518 − 1.47138i 0.388103 − 2.20213i
391.3 −0.809017 − 0.587785i −0.102252 − 0.314699i 0.309017 + 0.951057i 2.08823 − 0.799566i −0.102252 + 0.314699i 4.37811 0.309017 − 0.951057i 2.33847 − 1.69900i −2.15938 − 0.580567i
391.4 −0.809017 − 0.587785i 0.0956093 + 0.294255i 0.309017 + 0.951057i −2.23331 + 0.110958i 0.0956093 − 0.294255i 1.22498 0.309017 − 0.951057i 2.34961 − 1.70709i 1.87201 + 1.22294i
391.5 −0.809017 − 0.587785i 0.402964 + 1.24019i 0.309017 + 0.951057i −0.529888 + 2.17238i 0.402964 − 1.24019i 1.35654 0.309017 − 0.951057i 1.05135 − 0.763850i 1.70558 − 1.44603i
391.6 −0.809017 − 0.587785i 0.866678 + 2.66736i 0.309017 + 0.951057i −0.246372 − 2.22245i 0.866678 − 2.66736i −0.630775 0.309017 − 0.951057i −3.93664 + 2.86013i −1.10701 + 1.94282i
521.1 0.309017 − 0.951057i −2.36555 − 1.71867i −0.809017 − 0.587785i −0.618683 + 2.14877i −2.36555 + 1.71867i −4.94863 −0.809017 + 0.587785i 1.71494 + 5.27805i 1.85242 + 1.25241i
521.2 0.309017 − 0.951057i −0.970136 − 0.704845i −0.809017 − 0.587785i −1.50858 − 1.65051i −0.970136 + 0.704845i −1.35024 −0.809017 + 0.587785i −0.482694 − 1.48558i −2.03591 + 0.924706i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.l.c ✓ 24
25.d even 5 1 inner 650.2.l.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.l.c ✓ 24 1.a even 1 1 trivial
650.2.l.c ✓ 24 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 3 T_{3}^{23} + 10 T_{3}^{22} - 15 T_{3}^{21} + 95 T_{3}^{20} - 149 T_{3}^{19} + 887 T_{3}^{18} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display