Properties

Label 1170.2.e.b
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + ( -2 + i ) q^{5} + 4 i q^{7} + i q^{8} +O(q^{10})\) \( q -i q^{2} - q^{4} + ( -2 + i ) q^{5} + 4 i q^{7} + i q^{8} + ( 1 + 2 i ) q^{10} -2 q^{11} -i q^{13} + 4 q^{14} + q^{16} -4 i q^{17} -2 q^{19} + ( 2 - i ) q^{20} + 2 i q^{22} -6 i q^{23} + ( 3 - 4 i ) q^{25} - q^{26} -4 i q^{28} -2 q^{29} -4 q^{31} -i q^{32} -4 q^{34} + ( -4 - 8 i ) q^{35} -6 i q^{37} + 2 i q^{38} + ( -1 - 2 i ) q^{40} + 6 q^{41} -8 i q^{43} + 2 q^{44} -6 q^{46} -8 i q^{47} -9 q^{49} + ( -4 - 3 i ) q^{50} + i q^{52} + 10 i q^{53} + ( 4 - 2 i ) q^{55} -4 q^{56} + 2 i q^{58} -14 q^{59} + 10 q^{61} + 4 i q^{62} - q^{64} + ( 1 + 2 i ) q^{65} + 4 i q^{67} + 4 i q^{68} + ( -8 + 4 i ) q^{70} -8 q^{71} -10 i q^{73} -6 q^{74} + 2 q^{76} -8 i q^{77} + 8 q^{79} + ( -2 + i ) q^{80} -6 i q^{82} + 12 i q^{83} + ( 4 + 8 i ) q^{85} -8 q^{86} -2 i q^{88} -18 q^{89} + 4 q^{91} + 6 i q^{92} -8 q^{94} + ( 4 - 2 i ) q^{95} + 6 i q^{97} + 9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{5} + 2q^{10} - 4q^{11} + 8q^{14} + 2q^{16} - 4q^{19} + 4q^{20} + 6q^{25} - 2q^{26} - 4q^{29} - 8q^{31} - 8q^{34} - 8q^{35} - 2q^{40} + 12q^{41} + 4q^{44} - 12q^{46} - 18q^{49} - 8q^{50} + 8q^{55} - 8q^{56} - 28q^{59} + 20q^{61} - 2q^{64} + 2q^{65} - 16q^{70} - 16q^{71} - 12q^{74} + 4q^{76} + 16q^{79} - 4q^{80} + 8q^{85} - 16q^{86} - 36q^{89} + 8q^{91} - 16q^{94} + 8q^{95} + O(q^{100}) \)

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 4.00000i 1.00000i 0 1.00000 + 2.00000i
469.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 4.00000i 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.b 2
3.b odd 2 1 390.2.e.b 2
5.b even 2 1 inner 1170.2.e.b 2
5.c odd 4 1 5850.2.a.x 1
5.c odd 4 1 5850.2.a.bd 1
12.b even 2 1 3120.2.l.h 2
15.d odd 2 1 390.2.e.b 2
15.e even 4 1 1950.2.a.g 1
15.e even 4 1 1950.2.a.u 1
60.h even 2 1 3120.2.l.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.b 2 3.b odd 2 1
390.2.e.b 2 15.d odd 2 1
1170.2.e.b 2 1.a even 1 1 trivial
1170.2.e.b 2 5.b even 2 1 inner
1950.2.a.g 1 15.e even 4 1
1950.2.a.u 1 15.e even 4 1
3120.2.l.h 2 12.b even 2 1
3120.2.l.h 2 60.h even 2 1
5850.2.a.x 1 5.c odd 4 1
5850.2.a.bd 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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 14 + T )^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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