Properties

Label 1520.2.d.b
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + ( -1 - 2 i ) q^{5} -2 i q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -1 - 2 i ) q^{5} -2 i q^{7} + 3 q^{9} + 4 q^{11} + 2 i q^{13} + 4 i q^{17} + q^{19} -6 i q^{23} + ( -3 + 4 i ) q^{25} + 6 q^{29} + 4 q^{31} + ( -4 + 2 i ) q^{35} -10 i q^{37} -10 q^{41} + 2 i q^{43} + ( -3 - 6 i ) q^{45} + 6 i q^{47} + 3 q^{49} -10 i q^{53} + ( -4 - 8 i ) q^{55} + 2 q^{61} -6 i q^{63} + ( 4 - 2 i ) q^{65} -8 i q^{67} -4 q^{71} -4 i q^{73} -8 i q^{77} + 4 q^{79} + 9 q^{81} -18 i q^{83} + ( 8 - 4 i ) q^{85} + 2 q^{89} + 4 q^{91} + ( -1 - 2 i ) q^{95} + 6 i q^{97} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{5} + 6q^{9} + 8q^{11} + 2q^{19} - 6q^{25} + 12q^{29} + 8q^{31} - 8q^{35} - 20q^{41} - 6q^{45} + 6q^{49} - 8q^{55} + 4q^{61} + 8q^{65} - 8q^{71} + 8q^{79} + 18q^{81} + 16q^{85} + 4q^{89} + 8q^{91} - 2q^{95} + 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.00000i
1.00000i
0 0 0 −1.00000 2.00000i 0 2.00000i 0 3.00000 0
609.2 0 0 0 −1.00000 + 2.00000i 0 2.00000i 0 3.00000 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 1520.2.d.b 2
4.b odd 2 1 95.2.b.a 2
5.b even 2 1 inner 1520.2.d.b 2
5.c odd 4 1 7600.2.a.i 1
5.c odd 4 1 7600.2.a.l 1
12.b even 2 1 855.2.c.b 2
20.d odd 2 1 95.2.b.a 2
20.e even 4 1 475.2.a.a 1
20.e even 4 1 475.2.a.c 1
60.h even 2 1 855.2.c.b 2
60.l odd 4 1 4275.2.a.e 1
60.l odd 4 1 4275.2.a.p 1
76.d even 2 1 1805.2.b.c 2
380.d even 2 1 1805.2.b.c 2
380.j odd 4 1 9025.2.a.c 1
380.j odd 4 1 9025.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 4.b odd 2 1
95.2.b.a 2 20.d odd 2 1
475.2.a.a 1 20.e even 4 1
475.2.a.c 1 20.e even 4 1
855.2.c.b 2 12.b even 2 1
855.2.c.b 2 60.h even 2 1
1520.2.d.b 2 1.a even 1 1 trivial
1520.2.d.b 2 5.b even 2 1 inner
1805.2.b.c 2 76.d even 2 1
1805.2.b.c 2 380.d even 2 1
4275.2.a.e 1 60.l odd 4 1
4275.2.a.p 1 60.l odd 4 1
7600.2.a.i 1 5.c odd 4 1
7600.2.a.l 1 5.c odd 4 1
9025.2.a.c 1 380.j odd 4 1
9025.2.a.h 1 380.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3} \)
\( T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 324 + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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