Properties

Label 650.2.a.g
Level $650$
Weight $2$
Character orbit 650.a
Self dual yes
Analytic conductor $5.190$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(1,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.a (trivial)

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: yes
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + 3 q^{3} + q^{4} - 3 q^{6} - q^{7} - q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 3 q^{3} + q^{4} - 3 q^{6} - q^{7} - q^{8} + 6 q^{9} - 2 q^{11} + 3 q^{12} + q^{13} + q^{14} + q^{16} + 3 q^{17} - 6 q^{18} + 6 q^{19} - 3 q^{21} + 2 q^{22} + 4 q^{23} - 3 q^{24} - q^{26} + 9 q^{27} - q^{28} + 2 q^{29} + 4 q^{31} - q^{32} - 6 q^{33} - 3 q^{34} + 6 q^{36} - 3 q^{37} - 6 q^{38} + 3 q^{39} + 3 q^{42} + 5 q^{43} - 2 q^{44} - 4 q^{46} - 13 q^{47} + 3 q^{48} - 6 q^{49} + 9 q^{51} + q^{52} - 12 q^{53} - 9 q^{54} + q^{56} + 18 q^{57} - 2 q^{58} - 10 q^{59} - 8 q^{61} - 4 q^{62} - 6 q^{63} + q^{64} + 6 q^{66} + 2 q^{67} + 3 q^{68} + 12 q^{69} - 5 q^{71} - 6 q^{72} + 10 q^{73} + 3 q^{74} + 6 q^{76} + 2 q^{77} - 3 q^{78} - 4 q^{79} + 9 q^{81} - 3 q^{84} - 5 q^{86} + 6 q^{87} + 2 q^{88} + 6 q^{89} - q^{91} + 4 q^{92} + 12 q^{93} + 13 q^{94} - 3 q^{96} - 14 q^{97} + 6 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 3.00000 1.00000 0 −3.00000 −1.00000 −1.00000 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.a.g 1
3.b odd 2 1 5850.2.a.bn 1
4.b odd 2 1 5200.2.a.c 1
5.b even 2 1 26.2.a.b 1
5.c odd 4 2 650.2.b.a 2
13.b even 2 1 8450.2.a.y 1
15.d odd 2 1 234.2.a.b 1
15.e even 4 2 5850.2.e.v 2
20.d odd 2 1 208.2.a.d 1
35.c odd 2 1 1274.2.a.o 1
35.i odd 6 2 1274.2.f.a 2
35.j even 6 2 1274.2.f.l 2
40.e odd 2 1 832.2.a.a 1
40.f even 2 1 832.2.a.j 1
45.h odd 6 2 2106.2.e.t 2
45.j even 6 2 2106.2.e.h 2
55.d odd 2 1 3146.2.a.a 1
60.h even 2 1 1872.2.a.m 1
65.d even 2 1 338.2.a.a 1
65.g odd 4 2 338.2.b.a 2
65.l even 6 2 338.2.c.g 2
65.n even 6 2 338.2.c.c 2
65.s odd 12 4 338.2.e.d 4
80.k odd 4 2 3328.2.b.k 2
80.q even 4 2 3328.2.b.g 2
85.c even 2 1 7514.2.a.i 1
95.d odd 2 1 9386.2.a.f 1
120.i odd 2 1 7488.2.a.w 1
120.m even 2 1 7488.2.a.v 1
195.e odd 2 1 3042.2.a.l 1
195.n even 4 2 3042.2.b.f 2
260.g odd 2 1 2704.2.a.n 1
260.u even 4 2 2704.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 5.b even 2 1
208.2.a.d 1 20.d odd 2 1
234.2.a.b 1 15.d odd 2 1
338.2.a.a 1 65.d even 2 1
338.2.b.a 2 65.g odd 4 2
338.2.c.c 2 65.n even 6 2
338.2.c.g 2 65.l even 6 2
338.2.e.d 4 65.s odd 12 4
650.2.a.g 1 1.a even 1 1 trivial
650.2.b.a 2 5.c odd 4 2
832.2.a.a 1 40.e odd 2 1
832.2.a.j 1 40.f even 2 1
1274.2.a.o 1 35.c odd 2 1
1274.2.f.a 2 35.i odd 6 2
1274.2.f.l 2 35.j even 6 2
1872.2.a.m 1 60.h even 2 1
2106.2.e.h 2 45.j even 6 2
2106.2.e.t 2 45.h odd 6 2
2704.2.a.n 1 260.g odd 2 1
2704.2.f.j 2 260.u even 4 2
3042.2.a.l 1 195.e odd 2 1
3042.2.b.f 2 195.n even 4 2
3146.2.a.a 1 55.d odd 2 1
3328.2.b.g 2 80.q even 4 2
3328.2.b.k 2 80.k odd 4 2
5200.2.a.c 1 4.b odd 2 1
5850.2.a.bn 1 3.b odd 2 1
5850.2.e.v 2 15.e even 4 2
7488.2.a.v 1 120.m even 2 1
7488.2.a.w 1 120.i odd 2 1
7514.2.a.i 1 85.c even 2 1
8450.2.a.y 1 13.b even 2 1
9386.2.a.f 1 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(650))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T + 13 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 5 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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