Properties

Label 130.2.j.a
Level $130$
Weight $2$
Character orbit 130.j
Analytic conductor $1.038$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.03805522628\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
1.00000i −2.00000 + 2.00000i −1.00000 −1.00000 2.00000i 2.00000 + 2.00000i −4.00000 1.00000i 5.00000i −2.00000 + 1.00000i
83.1 1.00000i −2.00000 2.00000i −1.00000 −1.00000 + 2.00000i 2.00000 2.00000i −4.00000 1.00000i 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.f 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.j.a yes 2
3.b odd 2 1 1170.2.w.b 2
4.b odd 2 1 1040.2.cd.g 2
5.b even 2 1 650.2.j.e 2
5.c odd 4 1 130.2.g.a 2
5.c odd 4 1 650.2.g.e 2
13.d odd 4 1 130.2.g.a 2
15.e even 4 1 1170.2.m.c 2
20.e even 4 1 1040.2.bg.g 2
39.f even 4 1 1170.2.m.c 2
52.f even 4 1 1040.2.bg.g 2
65.f even 4 1 inner 130.2.j.a yes 2
65.g odd 4 1 650.2.g.e 2
65.k even 4 1 650.2.j.e 2
195.u odd 4 1 1170.2.w.b 2
260.l odd 4 1 1040.2.cd.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.a 2 5.c odd 4 1
130.2.g.a 2 13.d odd 4 1
130.2.j.a yes 2 1.a even 1 1 trivial
130.2.j.a yes 2 65.f even 4 1 inner
650.2.g.e 2 5.c odd 4 1
650.2.g.e 2 65.g odd 4 1
650.2.j.e 2 5.b even 2 1
650.2.j.e 2 65.k even 4 1
1040.2.bg.g 2 20.e even 4 1
1040.2.bg.g 2 52.f even 4 1
1040.2.cd.g 2 4.b odd 2 1
1040.2.cd.g 2 260.l odd 4 1
1170.2.m.c 2 15.e even 4 1
1170.2.m.c 2 39.f even 4 1
1170.2.w.b 2 3.b odd 2 1
1170.2.w.b 2 195.u odd 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

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) 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
$37$ \( (T + 4)^{2} \) 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
$47$ \( (T + 4)^{2} \) 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
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) 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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