Properties

Label 130.2.g.a
Level $130$
Weight $2$
Character orbit 130.g
Analytic conductor $1.038$
Analytic rank $1$
Dimension $2$
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] = [130,2,Mod(57,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.g (of order \(4\), degree \(2\), 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: \(1.03805522628\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - 2 i - 2) q^{3} + q^{4} + ( - i - 2) q^{5} + (2 i + 2) q^{6} + 4 i q^{7} - q^{8} + 5 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - 2 i - 2) q^{3} + q^{4} + ( - i - 2) q^{5} + (2 i + 2) q^{6} + 4 i q^{7} - q^{8} + 5 i q^{9} + (i + 2) q^{10} + (2 i - 2) q^{11} + ( - 2 i - 2) q^{12} + ( - 3 i - 2) q^{13} - 4 i q^{14} + (6 i + 2) q^{15} + q^{16} + ( - 3 i - 3) q^{17} - 5 i q^{18} + (2 i - 2) q^{19} + ( - i - 2) q^{20} + ( - 8 i + 8) q^{21} + ( - 2 i + 2) q^{22} + (2 i - 2) q^{23} + (2 i + 2) q^{24} + (4 i + 3) q^{25} + (3 i + 2) q^{26} + ( - 4 i + 4) q^{27} + 4 i q^{28} - 6 i q^{29} + ( - 6 i - 2) q^{30} + ( - 6 i - 6) q^{31} - q^{32} + 8 q^{33} + (3 i + 3) q^{34} + ( - 8 i + 4) q^{35} + 5 i q^{36} + 4 i q^{37} + ( - 2 i + 2) q^{38} + (10 i - 2) q^{39} + (i + 2) q^{40} + ( - i - 1) q^{41} + (8 i - 8) q^{42} + ( - 2 i + 2) q^{43} + (2 i - 2) q^{44} + ( - 10 i + 5) q^{45} + ( - 2 i + 2) q^{46} + 4 i q^{47} + ( - 2 i - 2) q^{48} - 9 q^{49} + ( - 4 i - 3) q^{50} + 12 i q^{51} + ( - 3 i - 2) q^{52} + ( - i - 1) q^{53} + (4 i - 4) q^{54} + ( - 2 i + 6) q^{55} - 4 i q^{56} + 8 q^{57} + 6 i q^{58} + ( - 2 i - 2) q^{59} + (6 i + 2) q^{60} - 12 q^{61} + (6 i + 6) q^{62} - 20 q^{63} + q^{64} + (8 i + 1) q^{65} - 8 q^{66} + 4 q^{67} + ( - 3 i - 3) q^{68} + 8 q^{69} + (8 i - 4) q^{70} + ( - 2 i - 2) q^{71} - 5 i q^{72} + 10 q^{73} - 4 i q^{74} + ( - 14 i + 2) q^{75} + (2 i - 2) q^{76} + ( - 8 i - 8) q^{77} + ( - 10 i + 2) q^{78} - 4 i q^{79} + ( - i - 2) q^{80} - q^{81} + (i + 1) q^{82} + 8 i q^{83} + ( - 8 i + 8) q^{84} + (9 i + 3) q^{85} + (2 i - 2) q^{86} + (12 i - 12) q^{87} + ( - 2 i + 2) q^{88} + (5 i + 5) q^{89} + (10 i - 5) q^{90} + ( - 8 i + 12) q^{91} + (2 i - 2) q^{92} + 24 i q^{93} - 4 i q^{94} + ( - 2 i + 6) q^{95} + (2 i + 2) q^{96} + 9 q^{98} + ( - 10 i - 10) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{6} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{6} - 2 q^{8} + 4 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} + 4 q^{15} + 2 q^{16} - 6 q^{17} - 4 q^{19} - 4 q^{20} + 16 q^{21} + 4 q^{22} - 4 q^{23} + 4 q^{24} + 6 q^{25} + 4 q^{26} + 8 q^{27} - 4 q^{30} - 12 q^{31} - 2 q^{32} + 16 q^{33} + 6 q^{34} + 8 q^{35} + 4 q^{38} - 4 q^{39} + 4 q^{40} - 2 q^{41} - 16 q^{42} + 4 q^{43} - 4 q^{44} + 10 q^{45} + 4 q^{46} - 4 q^{48} - 18 q^{49} - 6 q^{50} - 4 q^{52} - 2 q^{53} - 8 q^{54} + 12 q^{55} + 16 q^{57} - 4 q^{59} + 4 q^{60} - 24 q^{61} + 12 q^{62} - 40 q^{63} + 2 q^{64} + 2 q^{65} - 16 q^{66} + 8 q^{67} - 6 q^{68} + 16 q^{69} - 8 q^{70} - 4 q^{71} + 20 q^{73} + 4 q^{75} - 4 q^{76} - 16 q^{77} + 4 q^{78} - 4 q^{80} - 2 q^{81} + 2 q^{82} + 16 q^{84} + 6 q^{85} - 4 q^{86} - 24 q^{87} + 4 q^{88} + 10 q^{89} - 10 q^{90} + 24 q^{91} - 4 q^{92} + 12 q^{95} + 4 q^{96} + 18 q^{98} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-i\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.00000i
1.00000i
−1.00000 −2.00000 + 2.00000i 1.00000 −2.00000 + 1.00000i 2.00000 2.00000i 4.00000i −1.00000 5.00000i 2.00000 1.00000i
73.1 −1.00000 −2.00000 2.00000i 1.00000 −2.00000 1.00000i 2.00000 + 2.00000i 4.00000i −1.00000 5.00000i 2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.g.a 2
3.b odd 2 1 1170.2.m.c 2
4.b odd 2 1 1040.2.bg.g 2
5.b even 2 1 650.2.g.e 2
5.c odd 4 1 130.2.j.a yes 2
5.c odd 4 1 650.2.j.e 2
13.d odd 4 1 130.2.j.a yes 2
15.e even 4 1 1170.2.w.b 2
20.e even 4 1 1040.2.cd.g 2
39.f even 4 1 1170.2.w.b 2
52.f even 4 1 1040.2.cd.g 2
65.f even 4 1 650.2.g.e 2
65.g odd 4 1 650.2.j.e 2
65.k even 4 1 inner 130.2.g.a 2
195.j odd 4 1 1170.2.m.c 2
260.s odd 4 1 1040.2.bg.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.a 2 1.a even 1 1 trivial
130.2.g.a 2 65.k even 4 1 inner
130.2.j.a yes 2 5.c odd 4 1
130.2.j.a yes 2 13.d odd 4 1
650.2.g.e 2 5.b even 2 1
650.2.g.e 2 65.f even 4 1
650.2.j.e 2 5.c odd 4 1
650.2.j.e 2 65.g odd 4 1
1040.2.bg.g 2 4.b odd 2 1
1040.2.bg.g 2 260.s odd 4 1
1040.2.cd.g 2 20.e even 4 1
1040.2.cd.g 2 52.f even 4 1
1170.2.m.c 2 3.b odd 2 1
1170.2.m.c 2 195.j odd 4 1
1170.2.w.b 2 15.e even 4 1
1170.2.w.b 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$61$ \( (T + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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