Properties

Label 1170.2.e.d
Level $1170$
Weight $2$
Character orbit 1170.e
Analytic conductor $9.342$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + ( 2 + i ) q^{5} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + ( 2 + i ) q^{5} + i q^{8} + ( 1 - 2 i ) q^{10} + 6 q^{11} -i q^{13} + q^{16} -6 q^{19} + ( -2 - i ) q^{20} -6 i q^{22} + 6 i q^{23} + ( 3 + 4 i ) q^{25} - q^{26} + 2 q^{29} + 4 q^{31} -i q^{32} + 10 i q^{37} + 6 i q^{38} + ( -1 + 2 i ) q^{40} + 6 q^{41} -8 i q^{43} -6 q^{44} + 6 q^{46} -8 i q^{47} + 7 q^{49} + ( 4 - 3 i ) q^{50} + i q^{52} -6 i q^{53} + ( 12 + 6 i ) q^{55} -2 i q^{58} + 10 q^{59} -6 q^{61} -4 i q^{62} - q^{64} + ( 1 - 2 i ) q^{65} + 4 i q^{67} + 8 q^{71} -6 i q^{73} + 10 q^{74} + 6 q^{76} -16 q^{79} + ( 2 + i ) q^{80} -6 i q^{82} + 4 i q^{83} -8 q^{86} + 6 i q^{88} -10 q^{89} -6 i q^{92} -8 q^{94} + ( -12 - 6 i ) q^{95} + 2 i q^{97} -7 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{5} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{5} + 2q^{10} + 12q^{11} + 2q^{16} - 12q^{19} - 4q^{20} + 6q^{25} - 2q^{26} + 4q^{29} + 8q^{31} - 2q^{40} + 12q^{41} - 12q^{44} + 12q^{46} + 14q^{49} + 8q^{50} + 24q^{55} + 20q^{59} - 12q^{61} - 2q^{64} + 2q^{65} + 16q^{71} + 20q^{74} + 12q^{76} - 32q^{79} + 4q^{80} - 16q^{86} - 20q^{89} - 16q^{94} - 24q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
1.00000i
1.00000i
1.00000i 0 −1.00000 2.00000 + 1.00000i 0 0 1.00000i 0 1.00000 2.00000i
469.2 1.00000i 0 −1.00000 2.00000 1.00000i 0 0 1.00000i 0 1.00000 + 2.00000i
\(n\): 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 1170.2.e.d 2
3.b odd 2 1 390.2.e.a 2
5.b even 2 1 inner 1170.2.e.d 2
5.c odd 4 1 5850.2.a.o 1
5.c odd 4 1 5850.2.a.bs 1
12.b even 2 1 3120.2.l.a 2
15.d odd 2 1 390.2.e.a 2
15.e even 4 1 1950.2.a.i 1
15.e even 4 1 1950.2.a.r 1
60.h even 2 1 3120.2.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.a 2 3.b odd 2 1
390.2.e.a 2 15.d odd 2 1
1170.2.e.d 2 1.a even 1 1 trivial
1170.2.e.d 2 5.b even 2 1 inner
1950.2.a.i 1 15.e even 4 1
1950.2.a.r 1 15.e even 4 1
3120.2.l.a 2 12.b even 2 1
3120.2.l.a 2 60.h even 2 1
5850.2.a.o 1 5.c odd 4 1
5850.2.a.bs 1 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7} \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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