Properties

Label 3120.2.l.i
Level $3120$
Weight $2$
Character orbit 3120.l
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{3} + ( - i + 2) q^{5} + 4 i q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + ( - i + 2) q^{5} + 4 i q^{7} - q^{9} + 6 q^{11} - i q^{13} + (2 i + 1) q^{15} - 4 i q^{17} + 2 q^{19} - 4 q^{21} - 6 i q^{23} + ( - 4 i + 3) q^{25} - i q^{27} + 10 q^{29} - 4 q^{31} + 6 i q^{33} + (8 i + 4) q^{35} - 6 i q^{37} + q^{39} + 10 q^{41} + (i - 2) q^{45} - 8 i q^{47} - 9 q^{49} + 4 q^{51} + 6 i q^{53} + ( - 6 i + 12) q^{55} + 2 i q^{57} - 6 q^{59} - 6 q^{61} - 4 i q^{63} + ( - 2 i - 1) q^{65} + 12 i q^{67} + 6 q^{69} - 2 i q^{73} + (3 i + 4) q^{75} + 24 i q^{77} - 8 q^{79} + q^{81} - 4 i q^{83} + ( - 8 i - 4) q^{85} + 10 i q^{87} - 14 q^{89} + 4 q^{91} - 4 i q^{93} + ( - 2 i + 4) q^{95} + 14 i q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 20 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{39} + 20 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} + 24 q^{55} - 12 q^{59} - 12 q^{61} - 2 q^{65} + 12 q^{69} + 8 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 4.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 2.00000 1.00000i 0 4.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.2.l.i 2
4.b odd 2 1 390.2.e.d 2
5.b even 2 1 inner 3120.2.l.i 2
12.b even 2 1 1170.2.e.c 2
20.d odd 2 1 390.2.e.d 2
20.e even 4 1 1950.2.a.d 1
20.e even 4 1 1950.2.a.v 1
60.h even 2 1 1170.2.e.c 2
60.l odd 4 1 5850.2.a.e 1
60.l odd 4 1 5850.2.a.cc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.d 2 4.b odd 2 1
390.2.e.d 2 20.d odd 2 1
1170.2.e.c 2 12.b even 2 1
1170.2.e.c 2 60.h even 2 1
1950.2.a.d 1 20.e even 4 1
1950.2.a.v 1 20.e even 4 1
3120.2.l.i 2 1.a even 1 1 trivial
3120.2.l.i 2 5.b even 2 1 inner
5850.2.a.e 1 60.l odd 4 1
5850.2.a.cc 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3120, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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