Properties

Label 1089.3.c.a
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 121)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 q^{4} + 7 q^{5} - 5 \beta q^{7} + 6 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 2 q^{4} + 7 q^{5} - 5 \beta q^{7} + 6 \beta q^{8} + 7 \beta q^{10} - 12 \beta q^{13} + 10 q^{14} - 4 q^{16} - 3 \beta q^{17} + 12 \beta q^{19} + 14 q^{20} + 9 q^{23} + 24 q^{25} + 24 q^{26} - 10 \beta q^{28} + 16 \beta q^{29} + 49 q^{31} + 20 \beta q^{32} + 6 q^{34} - 35 \beta q^{35} + 17 q^{37} - 24 q^{38} + 42 \beta q^{40} - 12 \beta q^{41} - 33 \beta q^{43} + 9 \beta q^{46} - 32 q^{47} - q^{49} + 24 \beta q^{50} - 24 \beta q^{52} - 16 q^{53} + 60 q^{56} - 32 q^{58} + 71 q^{59} - 8 \beta q^{61} + 49 \beta q^{62} - 56 q^{64} - 84 \beta q^{65} - 31 q^{67} - 6 \beta q^{68} + 70 q^{70} + 73 q^{71} + 28 \beta q^{73} + 17 \beta q^{74} + 24 \beta q^{76} - 111 \beta q^{79} - 28 q^{80} + 24 q^{82} - 25 \beta q^{83} - 21 \beta q^{85} + 66 q^{86} + 9 q^{89} - 120 q^{91} + 18 q^{92} - 32 \beta q^{94} + 84 \beta q^{95} - 17 q^{97} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 14 q^{5} + 20 q^{14} - 8 q^{16} + 28 q^{20} + 18 q^{23} + 48 q^{25} + 48 q^{26} + 98 q^{31} + 12 q^{34} + 34 q^{37} - 48 q^{38} - 64 q^{47} - 2 q^{49} - 32 q^{53} + 120 q^{56} - 64 q^{58} + 142 q^{59} - 112 q^{64} - 62 q^{67} + 140 q^{70} + 146 q^{71} - 56 q^{80} + 48 q^{82} + 132 q^{86} + 18 q^{89} - 240 q^{91} + 36 q^{92} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
1.41421i
1.41421i
1.41421i 0 2.00000 7.00000 0 7.07107i 8.48528i 0 9.89949i
604.2 1.41421i 0 2.00000 7.00000 0 7.07107i 8.48528i 0 9.89949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.a 2
3.b odd 2 1 121.3.b.a 2
11.b odd 2 1 inner 1089.3.c.a 2
33.d even 2 1 121.3.b.a 2
33.f even 10 4 121.3.d.e 8
33.h odd 10 4 121.3.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.3.b.a 2 3.b odd 2 1
121.3.b.a 2 33.d even 2 1
121.3.d.e 8 33.f even 10 4
121.3.d.e 8 33.h odd 10 4
1089.3.c.a 2 1.a even 1 1 trivial
1089.3.c.a 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 50 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 288 \) Copy content Toggle raw display
$17$ \( T^{2} + 18 \) Copy content Toggle raw display
$19$ \( T^{2} + 288 \) Copy content Toggle raw display
$23$ \( (T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 512 \) Copy content Toggle raw display
$31$ \( (T - 49)^{2} \) Copy content Toggle raw display
$37$ \( (T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 288 \) Copy content Toggle raw display
$43$ \( T^{2} + 2178 \) Copy content Toggle raw display
$47$ \( (T + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T - 71)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 128 \) Copy content Toggle raw display
$67$ \( (T + 31)^{2} \) Copy content Toggle raw display
$71$ \( (T - 73)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1568 \) Copy content Toggle raw display
$79$ \( T^{2} + 24642 \) Copy content Toggle raw display
$83$ \( T^{2} + 1250 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( (T + 17)^{2} \) Copy content Toggle raw display
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