Properties

Label 322.2.c.a
Level $322$
Weight $2$
Character orbit 322.c
Analytic conductor $2.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} -\beta_{3} q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} -\beta_{3} q^{7} - q^{8} + q^{9} + \beta_{2} q^{11} + \beta_{1} q^{12} + 3 \beta_{1} q^{13} + \beta_{3} q^{14} + q^{16} - q^{18} -2 \beta_{3} q^{19} -\beta_{2} q^{21} -\beta_{2} q^{22} + ( -3 - \beta_{2} ) q^{23} -\beta_{1} q^{24} -5 q^{25} -3 \beta_{1} q^{26} + 4 \beta_{1} q^{27} -\beta_{3} q^{28} + 6 q^{29} + 6 \beta_{1} q^{31} - q^{32} -2 \beta_{3} q^{33} + q^{36} + 3 \beta_{2} q^{37} + 2 \beta_{3} q^{38} -6 q^{39} -4 \beta_{1} q^{41} + \beta_{2} q^{42} -3 \beta_{2} q^{43} + \beta_{2} q^{44} + ( 3 + \beta_{2} ) q^{46} + 2 \beta_{1} q^{47} + \beta_{1} q^{48} + 7 q^{49} + 5 q^{50} + 3 \beta_{1} q^{52} + \beta_{2} q^{53} -4 \beta_{1} q^{54} + \beta_{3} q^{56} -2 \beta_{2} q^{57} -6 q^{58} -7 \beta_{1} q^{59} + 4 \beta_{3} q^{61} -6 \beta_{1} q^{62} -\beta_{3} q^{63} + q^{64} + 2 \beta_{3} q^{66} + 3 \beta_{2} q^{67} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{69} + 6 q^{71} - q^{72} -6 \beta_{1} q^{73} -3 \beta_{2} q^{74} -5 \beta_{1} q^{75} -2 \beta_{3} q^{76} -7 \beta_{1} q^{77} + 6 q^{78} -5 q^{81} + 4 \beta_{1} q^{82} + 6 \beta_{3} q^{83} -\beta_{2} q^{84} + 3 \beta_{2} q^{86} + 6 \beta_{1} q^{87} -\beta_{2} q^{88} + 6 \beta_{3} q^{89} -3 \beta_{2} q^{91} + ( -3 - \beta_{2} ) q^{92} -12 q^{93} -2 \beta_{1} q^{94} -\beta_{1} q^{96} -2 \beta_{3} q^{97} -7 q^{98} + \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{9} + 4q^{16} - 4q^{18} - 12q^{23} - 20q^{25} + 24q^{29} - 4q^{32} + 4q^{36} - 24q^{39} + 12q^{46} + 28q^{49} + 20q^{50} - 24q^{58} + 4q^{64} + 24q^{71} - 4q^{72} + 24q^{78} - 20q^{81} - 12q^{92} - 48q^{93} - 28q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
1.16372i
2.57794i
1.16372i
2.57794i
−1.00000 1.41421i 1.00000 0 1.41421i −2.64575 −1.00000 1.00000 0
321.2 −1.00000 1.41421i 1.00000 0 1.41421i 2.64575 −1.00000 1.00000 0
321.3 −1.00000 1.41421i 1.00000 0 1.41421i −2.64575 −1.00000 1.00000 0
321.4 −1.00000 1.41421i 1.00000 0 1.41421i 2.64575 −1.00000 1.00000 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.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.c.a 4
3.b odd 2 1 2898.2.g.f 4
4.b odd 2 1 2576.2.f.d 4
7.b odd 2 1 inner 322.2.c.a 4
21.c even 2 1 2898.2.g.f 4
23.b odd 2 1 inner 322.2.c.a 4
28.d even 2 1 2576.2.f.d 4
69.c even 2 1 2898.2.g.f 4
92.b even 2 1 2576.2.f.d 4
161.c even 2 1 inner 322.2.c.a 4
483.c odd 2 1 2898.2.g.f 4
644.h odd 2 1 2576.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.a 4 1.a even 1 1 trivial
322.2.c.a 4 7.b odd 2 1 inner
322.2.c.a 4 23.b odd 2 1 inner
322.2.c.a 4 161.c even 2 1 inner
2576.2.f.d 4 4.b odd 2 1
2576.2.f.d 4 28.d even 2 1
2576.2.f.d 4 92.b even 2 1
2576.2.f.d 4 644.h odd 2 1
2898.2.g.f 4 3.b odd 2 1
2898.2.g.f 4 21.c even 2 1
2898.2.g.f 4 69.c even 2 1
2898.2.g.f 4 483.c odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 2 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( ( 14 + T^{2} )^{2} \)
$13$ \( ( 18 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -28 + T^{2} )^{2} \)
$23$ \( ( 23 + 6 T + T^{2} )^{2} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 72 + T^{2} )^{2} \)
$37$ \( ( 126 + T^{2} )^{2} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( ( 126 + T^{2} )^{2} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( ( 14 + T^{2} )^{2} \)
$59$ \( ( 98 + T^{2} )^{2} \)
$61$ \( ( -112 + T^{2} )^{2} \)
$67$ \( ( 126 + T^{2} )^{2} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( 72 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( -252 + T^{2} )^{2} \)
$89$ \( ( -252 + T^{2} )^{2} \)
$97$ \( ( -28 + T^{2} )^{2} \)
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