Properties

Label 1183.2.e.b
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + (3 \zeta_{6} - 2) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + (3 \zeta_{6} - 2) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + (\zeta_{6} - 3) q^{14} + \zeta_{6} q^{16} + (7 \zeta_{6} - 7) q^{17} + (3 \zeta_{6} - 3) q^{18} - 7 \zeta_{6} q^{19} - 3 q^{22} + 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + (2 \zeta_{6} + 1) q^{28} - 5 q^{29} + ( - 5 \zeta_{6} + 5) q^{32} - 7 q^{34} + 3 q^{36} + 8 \zeta_{6} q^{37} + ( - 7 \zeta_{6} + 7) q^{38} + 2 q^{43} + 3 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{46} + 7 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + 5 q^{50} + ( - 3 \zeta_{6} + 3) q^{53} + (9 \zeta_{6} - 6) q^{56} - 5 \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} + 7 \zeta_{6} q^{61} + (3 \zeta_{6} - 9) q^{63} + 7 q^{64} + (3 \zeta_{6} - 3) q^{67} + 7 \zeta_{6} q^{68} + 5 q^{71} + 9 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + (8 \zeta_{6} - 8) q^{74} - 7 q^{76} + ( - 6 \zeta_{6} - 3) q^{77} + 6 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + 2 \zeta_{6} q^{86} + (9 \zeta_{6} - 9) q^{88} + 6 q^{92} + (7 \zeta_{6} - 7) q^{94} + 14 q^{97} + ( - 8 \zeta_{6} + 3) q^{98} - 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} - q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} - q^{7} + 6 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{14} + q^{16} - 7 q^{17} - 3 q^{18} - 7 q^{19} - 6 q^{22} + 6 q^{23} + 5 q^{25} + 4 q^{28} - 10 q^{29} + 5 q^{32} - 14 q^{34} + 6 q^{36} + 8 q^{37} + 7 q^{38} + 4 q^{43} + 3 q^{44} - 6 q^{46} + 7 q^{47} - 13 q^{49} + 10 q^{50} + 3 q^{53} - 3 q^{56} - 5 q^{58} - 7 q^{59} + 7 q^{61} - 15 q^{63} + 14 q^{64} - 3 q^{67} + 7 q^{68} + 10 q^{71} + 9 q^{72} + 14 q^{73} - 8 q^{74} - 14 q^{76} - 12 q^{77} + 6 q^{79} - 9 q^{81} + 2 q^{86} - 9 q^{88} + 12 q^{92} - 7 q^{94} + 28 q^{97} - 2 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 −0.500000 2.59808i 3.00000 1.50000 2.59808i 0
508.1 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 −0.500000 + 2.59808i 3.00000 1.50000 + 2.59808i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.b 2
7.c even 3 1 inner 1183.2.e.b 2
7.c even 3 1 8281.2.a.f 1
7.d odd 6 1 8281.2.a.e 1
13.b even 2 1 91.2.e.a 2
39.d odd 2 1 819.2.j.b 2
52.b odd 2 1 1456.2.r.g 2
91.b odd 2 1 637.2.e.a 2
91.r even 6 1 91.2.e.a 2
91.r even 6 1 637.2.a.c 1
91.s odd 6 1 637.2.a.d 1
91.s odd 6 1 637.2.e.a 2
273.w odd 6 1 819.2.j.b 2
273.w odd 6 1 5733.2.a.c 1
273.ba even 6 1 5733.2.a.d 1
364.bl odd 6 1 1456.2.r.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 13.b even 2 1
91.2.e.a 2 91.r even 6 1
637.2.a.c 1 91.r even 6 1
637.2.a.d 1 91.s odd 6 1
637.2.e.a 2 91.b odd 2 1
637.2.e.a 2 91.s odd 6 1
819.2.j.b 2 39.d odd 2 1
819.2.j.b 2 273.w odd 6 1
1183.2.e.b 2 1.a even 1 1 trivial
1183.2.e.b 2 7.c even 3 1 inner
1456.2.r.g 2 52.b odd 2 1
1456.2.r.g 2 364.bl odd 6 1
5733.2.a.c 1 273.w odd 6 1
5733.2.a.d 1 273.ba even 6 1
8281.2.a.e 1 7.d odd 6 1
8281.2.a.f 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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