Properties

Label 322.2.a.f
Level $322$
Weight $2$
Character orbit 322.a
Self dual yes
Analytic conductor $2.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{6} + q^{7} + q^{8} + ( 1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{6} + q^{7} + q^{8} + ( 1 - 2 \beta ) q^{9} + ( 1 + \beta ) q^{10} -2 \beta q^{11} + ( -1 + \beta ) q^{12} + ( 2 - 2 \beta ) q^{13} + q^{14} + 2 q^{15} + q^{16} + ( 1 - \beta ) q^{17} + ( 1 - 2 \beta ) q^{18} -2 q^{19} + ( 1 + \beta ) q^{20} + ( -1 + \beta ) q^{21} -2 \beta q^{22} + q^{23} + ( -1 + \beta ) q^{24} + ( -1 + 2 \beta ) q^{25} + ( 2 - 2 \beta ) q^{26} -4 q^{27} + q^{28} + 8 q^{29} + 2 q^{30} + ( -5 + 3 \beta ) q^{31} + q^{32} + ( -6 + 2 \beta ) q^{33} + ( 1 - \beta ) q^{34} + ( 1 + \beta ) q^{35} + ( 1 - 2 \beta ) q^{36} + ( 4 - 2 \beta ) q^{37} -2 q^{38} + ( -8 + 4 \beta ) q^{39} + ( 1 + \beta ) q^{40} -2 q^{41} + ( -1 + \beta ) q^{42} -2 \beta q^{44} + ( -5 - \beta ) q^{45} + q^{46} + ( -3 - 3 \beta ) q^{47} + ( -1 + \beta ) q^{48} + q^{49} + ( -1 + 2 \beta ) q^{50} + ( -4 + 2 \beta ) q^{51} + ( 2 - 2 \beta ) q^{52} + 2 \beta q^{53} -4 q^{54} + ( -6 - 2 \beta ) q^{55} + q^{56} + ( 2 - 2 \beta ) q^{57} + 8 q^{58} + ( -3 - 5 \beta ) q^{59} + 2 q^{60} + ( 3 + 3 \beta ) q^{61} + ( -5 + 3 \beta ) q^{62} + ( 1 - 2 \beta ) q^{63} + q^{64} -4 q^{65} + ( -6 + 2 \beta ) q^{66} + ( -8 + 2 \beta ) q^{67} + ( 1 - \beta ) q^{68} + ( -1 + \beta ) q^{69} + ( 1 + \beta ) q^{70} + ( -2 + 6 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -4 - 2 \beta ) q^{73} + ( 4 - 2 \beta ) q^{74} + ( 7 - 3 \beta ) q^{75} -2 q^{76} -2 \beta q^{77} + ( -8 + 4 \beta ) q^{78} + ( -6 + 4 \beta ) q^{79} + ( 1 + \beta ) q^{80} + ( 1 + 2 \beta ) q^{81} -2 q^{82} + ( 4 - 6 \beta ) q^{83} + ( -1 + \beta ) q^{84} -2 q^{85} + ( -8 + 8 \beta ) q^{87} -2 \beta q^{88} + ( 9 + 3 \beta ) q^{89} + ( -5 - \beta ) q^{90} + ( 2 - 2 \beta ) q^{91} + q^{92} + ( 14 - 8 \beta ) q^{93} + ( -3 - 3 \beta ) q^{94} + ( -2 - 2 \beta ) q^{95} + ( -1 + \beta ) q^{96} + ( -7 - \beta ) q^{97} + q^{98} + ( 12 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} + 4q^{13} + 2q^{14} + 4q^{15} + 2q^{16} + 2q^{17} + 2q^{18} - 4q^{19} + 2q^{20} - 2q^{21} + 2q^{23} - 2q^{24} - 2q^{25} + 4q^{26} - 8q^{27} + 2q^{28} + 16q^{29} + 4q^{30} - 10q^{31} + 2q^{32} - 12q^{33} + 2q^{34} + 2q^{35} + 2q^{36} + 8q^{37} - 4q^{38} - 16q^{39} + 2q^{40} - 4q^{41} - 2q^{42} - 10q^{45} + 2q^{46} - 6q^{47} - 2q^{48} + 2q^{49} - 2q^{50} - 8q^{51} + 4q^{52} - 8q^{54} - 12q^{55} + 2q^{56} + 4q^{57} + 16q^{58} - 6q^{59} + 4q^{60} + 6q^{61} - 10q^{62} + 2q^{63} + 2q^{64} - 8q^{65} - 12q^{66} - 16q^{67} + 2q^{68} - 2q^{69} + 2q^{70} - 4q^{71} + 2q^{72} - 8q^{73} + 8q^{74} + 14q^{75} - 4q^{76} - 16q^{78} - 12q^{79} + 2q^{80} + 2q^{81} - 4q^{82} + 8q^{83} - 2q^{84} - 4q^{85} - 16q^{87} + 18q^{89} - 10q^{90} + 4q^{91} + 2q^{92} + 28q^{93} - 6q^{94} - 4q^{95} - 2q^{96} - 14q^{97} + 2q^{98} + 24q^{99} + O(q^{100}) \)

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
1.00000 −2.73205 1.00000 −0.732051 −2.73205 1.00000 1.00000 4.46410 −0.732051
1.2 1.00000 0.732051 1.00000 2.73205 0.732051 1.00000 1.00000 −2.46410 2.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 322.2.a.f 2
3.b odd 2 1 2898.2.a.w 2
4.b odd 2 1 2576.2.a.u 2
5.b even 2 1 8050.2.a.x 2
7.b odd 2 1 2254.2.a.o 2
23.b odd 2 1 7406.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.f 2 1.a even 1 1 trivial
2254.2.a.o 2 7.b odd 2 1
2576.2.a.u 2 4.b odd 2 1
2898.2.a.w 2 3.b odd 2 1
7406.2.a.s 2 23.b odd 2 1
8050.2.a.x 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\):

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{5}^{2} - 2 T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( -2 - 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -8 - 4 T + T^{2} \)
$17$ \( -2 - 2 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -2 + 10 T + T^{2} \)
$37$ \( 4 - 8 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -18 + 6 T + T^{2} \)
$53$ \( -12 + T^{2} \)
$59$ \( -66 + 6 T + T^{2} \)
$61$ \( -18 - 6 T + T^{2} \)
$67$ \( 52 + 16 T + T^{2} \)
$71$ \( -104 + 4 T + T^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( -12 + 12 T + T^{2} \)
$83$ \( -92 - 8 T + T^{2} \)
$89$ \( 54 - 18 T + T^{2} \)
$97$ \( 46 + 14 T + T^{2} \)
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