Properties

Label 1520.2.d.g
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 - 2 \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 - 2 \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + q^{9} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{13} + ( 3 \zeta_{8} - \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} - q^{19} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -3 - 4 \zeta_{8}^{2} ) q^{25} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{27} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} + ( -4 - 6 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{35} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{39} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 1 - 2 \zeta_{8}^{2} ) q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} -4 q^{51} + ( -7 \zeta_{8} - 2 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{53} + ( -2 + 2 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{55} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{57} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( -4 + 3 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{65} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{67} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{69} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( \zeta_{8} - 7 \zeta_{8}^{3} ) q^{75} + ( 8 \zeta_{8} + 12 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{77} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{79} -5 q^{81} + ( -4 \zeta_{8} - 10 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} + ( 6 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{85} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{87} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{91} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{93} + ( -1 + 2 \zeta_{8}^{2} ) q^{95} + ( -5 \zeta_{8} + 6 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{97} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 4q^{9} - 8q^{11} - 4q^{19} + 16q^{21} - 12q^{25} + 8q^{29} - 16q^{31} - 16q^{35} - 8q^{39} - 8q^{41} + 4q^{45} - 20q^{49} - 16q^{51} - 8q^{55} - 16q^{59} - 16q^{65} + 32q^{69} - 20q^{81} + 8q^{89} - 4q^{95} - 8q^{99} + O(q^{100}) \)

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
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0 1.41421i 0 1.00000 2.00000i 0 0.828427i 0 1.00000 0
609.2 0 1.41421i 0 1.00000 + 2.00000i 0 4.82843i 0 1.00000 0
609.3 0 1.41421i 0 1.00000 2.00000i 0 4.82843i 0 1.00000 0
609.4 0 1.41421i 0 1.00000 + 2.00000i 0 0.828427i 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 1520.2.d.g 4
4.b odd 2 1 760.2.d.d 4
5.b even 2 1 inner 1520.2.d.g 4
5.c odd 4 1 7600.2.a.z 2
5.c odd 4 1 7600.2.a.bb 2
20.d odd 2 1 760.2.d.d 4
20.e even 4 1 3800.2.a.m 2
20.e even 4 1 3800.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.d 4 4.b odd 2 1
760.2.d.d 4 20.d odd 2 1
1520.2.d.g 4 1.a even 1 1 trivial
1520.2.d.g 4 5.b even 2 1 inner
3800.2.a.m 2 20.e even 4 1
3800.2.a.o 2 20.e even 4 1
7600.2.a.z 2 5.c odd 4 1
7600.2.a.bb 2 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 2 \)
\( T_{7}^{4} + 24 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 2 + T^{2} )^{2} \)
$5$ \( ( 5 - 2 T + T^{2} )^{2} \)
$7$ \( 16 + 24 T^{2} + T^{4} \)
$11$ \( ( -4 + 4 T + T^{2} )^{2} \)
$13$ \( 4 + 12 T^{2} + T^{4} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 784 + 72 T^{2} + T^{4} \)
$29$ \( ( -28 - 4 T + T^{2} )^{2} \)
$31$ \( ( 8 + 8 T + T^{2} )^{2} \)
$37$ \( 4 + 12 T^{2} + T^{4} \)
$41$ \( ( -4 + 4 T + T^{2} )^{2} \)
$43$ \( 784 + 88 T^{2} + T^{4} \)
$47$ \( 16 + 24 T^{2} + T^{4} \)
$53$ \( 8836 + 204 T^{2} + T^{4} \)
$59$ \( ( 8 + 8 T + T^{2} )^{2} \)
$61$ \( ( -32 + T^{2} )^{2} \)
$67$ \( ( 98 + T^{2} )^{2} \)
$71$ \( ( -72 + T^{2} )^{2} \)
$73$ \( 64 + 48 T^{2} + T^{4} \)
$79$ \( ( -72 + T^{2} )^{2} \)
$83$ \( 4624 + 264 T^{2} + T^{4} \)
$89$ \( ( -124 - 4 T + T^{2} )^{2} \)
$97$ \( 196 + 172 T^{2} + T^{4} \)
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