Properties

Label 4680.2.l.c
Level $4680$
Weight $2$
Character orbit 4680.l
Analytic conductor $37.370$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 i ) q^{5} + 5 i q^{7} +O(q^{10})\) \( q + ( 1 - 2 i ) q^{5} + 5 i q^{7} -5 q^{11} + i q^{13} -3 i q^{17} + 4 q^{19} -5 i q^{23} + ( -3 - 4 i ) q^{25} -4 q^{29} + ( 10 + 5 i ) q^{35} + 7 i q^{37} -11 q^{41} -12 i q^{43} + 6 i q^{47} -18 q^{49} -i q^{53} + ( -5 + 10 i ) q^{55} + 12 q^{59} -7 q^{61} + ( 2 + i ) q^{65} -4 i q^{67} + 7 q^{71} -14 i q^{73} -25 i q^{77} + 5 q^{79} + 2 i q^{83} + ( -6 - 3 i ) q^{85} -3 q^{89} -5 q^{91} + ( 4 - 8 i ) q^{95} + i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} - 10 q^{11} + 8 q^{19} - 6 q^{25} - 8 q^{29} + 20 q^{35} - 22 q^{41} - 36 q^{49} - 10 q^{55} + 24 q^{59} - 14 q^{61} + 4 q^{65} + 14 q^{71} + 10 q^{79} - 12 q^{85} - 6 q^{89} - 10 q^{91} + 8 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2809.1
1.00000i
1.00000i
0 0 0 1.00000 2.00000i 0 5.00000i 0 0 0
2809.2 0 0 0 1.00000 + 2.00000i 0 5.00000i 0 0 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 4680.2.l.c 2
3.b odd 2 1 1560.2.l.a 2
5.b even 2 1 inner 4680.2.l.c 2
12.b even 2 1 3120.2.l.b 2
15.d odd 2 1 1560.2.l.a 2
15.e even 4 1 7800.2.a.l 1
15.e even 4 1 7800.2.a.m 1
60.h even 2 1 3120.2.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.a 2 3.b odd 2 1
1560.2.l.a 2 15.d odd 2 1
3120.2.l.b 2 12.b even 2 1
3120.2.l.b 2 60.h even 2 1
4680.2.l.c 2 1.a even 1 1 trivial
4680.2.l.c 2 5.b even 2 1 inner
7800.2.a.l 1 15.e even 4 1
7800.2.a.m 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{2} + 25 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 25 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( ( 11 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( 7 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -7 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 4 + T^{2} \)
$89$ \( ( 3 + T )^{2} \)
$97$ \( 1 + T^{2} \)
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