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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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