Properties

Label 1360.2.e.b
Level $1360$
Weight $2$
Character orbit 1360.e
Analytic conductor $10.860$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1360.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.8596546749\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
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.

\(f(q)\) \(=\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} + 2 q^{9} + 6 q^{11} -3 i q^{13} + ( 2 + i ) q^{15} + i q^{17} -7 q^{19} -8 i q^{23} + ( -3 - 4 i ) q^{25} + 5 i q^{27} + 5 q^{29} -5 q^{31} + 6 i q^{33} + 8 i q^{37} + 3 q^{39} -4 i q^{43} + ( 2 - 4 i ) q^{45} -3 i q^{47} + 7 q^{49} - q^{51} -9 i q^{53} + ( 6 - 12 i ) q^{55} -7 i q^{57} + 5 q^{59} -3 q^{61} + ( -6 - 3 i ) q^{65} + 2 i q^{67} + 8 q^{69} + 15 q^{71} + 11 i q^{73} + ( 4 - 3 i ) q^{75} + 8 q^{79} + q^{81} + 4 i q^{83} + ( 2 + i ) q^{85} + 5 i q^{87} + q^{89} -5 i q^{93} + ( -7 + 14 i ) q^{95} -9 i q^{97} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 4q^{9} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{9} + 12q^{11} + 4q^{15} - 14q^{19} - 6q^{25} + 10q^{29} - 10q^{31} + 6q^{39} + 4q^{45} + 14q^{49} - 2q^{51} + 12q^{55} + 10q^{59} - 6q^{61} - 12q^{65} + 16q^{69} + 30q^{71} + 8q^{75} + 16q^{79} + 2q^{81} + 4q^{85} + 2q^{89} - 14q^{95} + 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1089.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 0 0 2.00000 0
1089.2 0 1.00000i 0 1.00000 2.00000i 0 0 0 2.00000 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 1360.2.e.b 2
4.b odd 2 1 170.2.c.a 2
5.b even 2 1 inner 1360.2.e.b 2
5.c odd 4 1 6800.2.a.g 1
5.c odd 4 1 6800.2.a.r 1
12.b even 2 1 1530.2.d.b 2
20.d odd 2 1 170.2.c.a 2
20.e even 4 1 850.2.a.d 1
20.e even 4 1 850.2.a.h 1
60.h even 2 1 1530.2.d.b 2
60.l odd 4 1 7650.2.a.s 1
60.l odd 4 1 7650.2.a.cb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 4.b odd 2 1
170.2.c.a 2 20.d odd 2 1
850.2.a.d 1 20.e even 4 1
850.2.a.h 1 20.e even 4 1
1360.2.e.b 2 1.a even 1 1 trivial
1360.2.e.b 2 5.b even 2 1 inner
1530.2.d.b 2 12.b even 2 1
1530.2.d.b 2 60.h even 2 1
6800.2.a.g 1 5.c odd 4 1
6800.2.a.r 1 5.c odd 4 1
7650.2.a.s 1 60.l odd 4 1
7650.2.a.cb 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 \(S_{2}^{\mathrm{new}}(1360, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( ( -5 + T )^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( -15 + T )^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( -1 + T )^{2} \)
$97$ \( 81 + T^{2} \)
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