Properties

Label 1183.2.a.f
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 3 + \beta ) q^{6} + q^{7} -\beta q^{8} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 3 + \beta ) q^{6} + q^{7} -\beta q^{8} + ( 1 + 2 \beta ) q^{9} + 3 q^{10} + ( 3 - \beta ) q^{11} + ( 1 + \beta ) q^{12} + \beta q^{14} + ( 3 + \beta ) q^{15} -5 q^{16} + ( -6 - \beta ) q^{17} + ( 6 + \beta ) q^{18} + 2 q^{19} + \beta q^{20} + ( 1 + \beta ) q^{21} + ( -3 + 3 \beta ) q^{22} + ( 3 + \beta ) q^{23} + ( -3 - \beta ) q^{24} -2 q^{25} + 4 q^{27} + q^{28} -3 q^{29} + ( 3 + 3 \beta ) q^{30} + ( -1 + 3 \beta ) q^{31} -3 \beta q^{32} + 2 \beta q^{33} + ( -3 - 6 \beta ) q^{34} + \beta q^{35} + ( 1 + 2 \beta ) q^{36} -7 q^{37} + 2 \beta q^{38} -3 q^{40} -3 \beta q^{41} + ( 3 + \beta ) q^{42} + ( 5 - 3 \beta ) q^{43} + ( 3 - \beta ) q^{44} + ( 6 + \beta ) q^{45} + ( 3 + 3 \beta ) q^{46} + ( 6 + 4 \beta ) q^{47} + ( -5 - 5 \beta ) q^{48} + q^{49} -2 \beta q^{50} + ( -9 - 7 \beta ) q^{51} + ( -3 - 4 \beta ) q^{53} + 4 \beta q^{54} + ( -3 + 3 \beta ) q^{55} -\beta q^{56} + ( 2 + 2 \beta ) q^{57} -3 \beta q^{58} + ( 9 - \beta ) q^{59} + ( 3 + \beta ) q^{60} + ( -10 + 3 \beta ) q^{61} + ( 9 - \beta ) q^{62} + ( 1 + 2 \beta ) q^{63} + q^{64} + 6 q^{66} + ( -1 - 3 \beta ) q^{67} + ( -6 - \beta ) q^{68} + ( 6 + 4 \beta ) q^{69} + 3 q^{70} + 6 q^{71} + ( -6 - \beta ) q^{72} + ( 2 - 3 \beta ) q^{73} -7 \beta q^{74} + ( -2 - 2 \beta ) q^{75} + 2 q^{76} + ( 3 - \beta ) q^{77} + ( 11 + 3 \beta ) q^{79} -5 \beta q^{80} + ( 1 - 2 \beta ) q^{81} -9 q^{82} + ( 3 - 3 \beta ) q^{83} + ( 1 + \beta ) q^{84} + ( -3 - 6 \beta ) q^{85} + ( -9 + 5 \beta ) q^{86} + ( -3 - 3 \beta ) q^{87} + ( 3 - 3 \beta ) q^{88} + ( 6 + 4 \beta ) q^{89} + ( 3 + 6 \beta ) q^{90} + ( 3 + \beta ) q^{92} + ( 8 + 2 \beta ) q^{93} + ( 12 + 6 \beta ) q^{94} + 2 \beta q^{95} + ( -9 - 3 \beta ) q^{96} + ( -4 + 6 \beta ) q^{97} + \beta q^{98} + ( -3 + 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} + 6q^{6} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} + 6q^{6} + 2q^{7} + 2q^{9} + 6q^{10} + 6q^{11} + 2q^{12} + 6q^{15} - 10q^{16} - 12q^{17} + 12q^{18} + 4q^{19} + 2q^{21} - 6q^{22} + 6q^{23} - 6q^{24} - 4q^{25} + 8q^{27} + 2q^{28} - 6q^{29} + 6q^{30} - 2q^{31} - 6q^{34} + 2q^{36} - 14q^{37} - 6q^{40} + 6q^{42} + 10q^{43} + 6q^{44} + 12q^{45} + 6q^{46} + 12q^{47} - 10q^{48} + 2q^{49} - 18q^{51} - 6q^{53} - 6q^{55} + 4q^{57} + 18q^{59} + 6q^{60} - 20q^{61} + 18q^{62} + 2q^{63} + 2q^{64} + 12q^{66} - 2q^{67} - 12q^{68} + 12q^{69} + 6q^{70} + 12q^{71} - 12q^{72} + 4q^{73} - 4q^{75} + 4q^{76} + 6q^{77} + 22q^{79} + 2q^{81} - 18q^{82} + 6q^{83} + 2q^{84} - 6q^{85} - 18q^{86} - 6q^{87} + 6q^{88} + 12q^{89} + 6q^{90} + 6q^{92} + 16q^{93} + 24q^{94} - 18q^{96} - 8q^{97} - 6q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 −0.732051 1.00000 −1.73205 1.26795 1.00000 1.73205 −2.46410 3.00000
1.2 1.73205 2.73205 1.00000 1.73205 4.73205 1.00000 −1.73205 4.46410 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.f 2
7.b odd 2 1 8281.2.a.r 2
13.b even 2 1 1183.2.a.e 2
13.c even 3 2 91.2.f.b 4
13.d odd 4 2 1183.2.c.e 4
39.i odd 6 2 819.2.o.b 4
52.j odd 6 2 1456.2.s.o 4
91.b odd 2 1 8281.2.a.t 2
91.g even 3 2 637.2.g.e 4
91.h even 3 2 637.2.h.d 4
91.m odd 6 2 637.2.g.d 4
91.n odd 6 2 637.2.f.d 4
91.v odd 6 2 637.2.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 13.c even 3 2
637.2.f.d 4 91.n odd 6 2
637.2.g.d 4 91.m odd 6 2
637.2.g.e 4 91.g even 3 2
637.2.h.d 4 91.h even 3 2
637.2.h.e 4 91.v odd 6 2
819.2.o.b 4 39.i odd 6 2
1183.2.a.e 2 13.b even 2 1
1183.2.a.f 2 1.a even 1 1 trivial
1183.2.c.e 4 13.d odd 4 2
1456.2.s.o 4 52.j odd 6 2
8281.2.a.r 2 7.b odd 2 1
8281.2.a.t 2 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{2} - 3 \)
\( T_{11}^{2} - 6 T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 6 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 33 + 12 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 6 - 6 T + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( -26 + 2 T + T^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( -27 + T^{2} \)
$43$ \( -2 - 10 T + T^{2} \)
$47$ \( -12 - 12 T + T^{2} \)
$53$ \( -39 + 6 T + T^{2} \)
$59$ \( 78 - 18 T + T^{2} \)
$61$ \( 73 + 20 T + T^{2} \)
$67$ \( -26 + 2 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( -23 - 4 T + T^{2} \)
$79$ \( 94 - 22 T + T^{2} \)
$83$ \( -18 - 6 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( -92 + 8 T + T^{2} \)
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