Properties

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

Related objects

Downloads

Learn more

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: 4.0.2312.1
Defining polynomial: \(x^{4} - x^{3} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{2} q^{3} + q^{4} -2 q^{5} -\beta_{2} q^{6} + ( -1 + \beta_{1} ) q^{7} + q^{8} + ( -2 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{2} q^{3} + q^{4} -2 q^{5} -\beta_{2} q^{6} + ( -1 + \beta_{1} ) q^{7} + q^{8} + ( -2 + \beta_{1} - \beta_{3} ) q^{9} -2 q^{10} -\beta_{2} q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} ) q^{14} + 2 \beta_{2} q^{15} + q^{16} + ( 3 - \beta_{1} + \beta_{3} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{18} + 4 q^{19} -2 q^{20} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{21} -\beta_{2} q^{22} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{23} -\beta_{2} q^{24} - q^{25} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{27} + ( -1 + \beta_{1} ) q^{28} -2 q^{29} + 2 \beta_{2} q^{30} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{31} + q^{32} + ( -5 + \beta_{1} - \beta_{3} ) q^{33} + ( 3 - \beta_{1} + \beta_{3} ) q^{34} + ( 2 - 2 \beta_{1} ) q^{35} + ( -2 + \beta_{1} - \beta_{3} ) q^{36} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + 4 q^{38} + ( 1 - \beta_{1} + \beta_{3} ) q^{39} -2 q^{40} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{42} -\beta_{2} q^{43} -\beta_{2} q^{44} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{45} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} -\beta_{2} q^{48} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{49} - q^{50} + ( 2 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{52} + ( -1 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{54} + 2 \beta_{2} q^{55} + ( -1 + \beta_{1} ) q^{56} -4 \beta_{2} q^{57} -2 q^{58} + 3 \beta_{2} q^{59} + 2 \beta_{2} q^{60} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{62} + ( 8 + 2 \beta_{2} + 3 \beta_{3} ) q^{63} + q^{64} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -5 + \beta_{1} - \beta_{3} ) q^{66} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{67} + ( 3 - \beta_{1} + \beta_{3} ) q^{68} + ( -3 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} ) q^{70} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{3} ) q^{72} -4 \beta_{2} q^{73} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{74} + \beta_{2} q^{75} + 4 q^{76} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{77} + ( 1 - \beta_{1} + \beta_{3} ) q^{78} + ( 3 - 3 \beta_{1} - 3 \beta_{3} ) q^{79} -2 q^{80} + ( 6 - \beta_{1} + \beta_{3} ) q^{81} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{82} + 4 q^{83} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{84} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{85} -\beta_{2} q^{86} + 2 \beta_{2} q^{87} -\beta_{2} q^{88} + 2 q^{89} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{90} + ( -4 - \beta_{2} + 2 \beta_{3} ) q^{91} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{92} + ( -8 + 4 \beta_{1} - 4 \beta_{3} ) q^{93} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{94} -8 q^{95} -\beta_{2} q^{96} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{98} + ( -2 + 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} - 8q^{5} - 2q^{7} + 4q^{8} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} - 8q^{5} - 2q^{7} + 4q^{8} - 8q^{9} - 8q^{10} - 2q^{14} + 4q^{16} + 12q^{17} - 8q^{18} + 16q^{19} - 8q^{20} - 8q^{21} + 2q^{23} - 4q^{25} - 2q^{28} - 8q^{29} + 4q^{32} - 20q^{33} + 12q^{34} + 4q^{35} - 8q^{36} + 16q^{38} + 4q^{39} - 8q^{40} - 8q^{42} + 16q^{45} + 2q^{46} + 8q^{49} - 4q^{50} - 2q^{56} - 8q^{58} + 16q^{61} + 38q^{63} + 4q^{64} - 20q^{66} + 12q^{68} - 16q^{69} + 4q^{70} + 12q^{71} - 8q^{72} + 16q^{76} - 8q^{77} + 4q^{78} - 8q^{80} + 24q^{81} + 16q^{83} - 8q^{84} - 24q^{85} + 8q^{89} + 16q^{90} - 12q^{91} + 2q^{92} - 32q^{93} - 32q^{95} - 16q^{97} + 8q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x + 4\):

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

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
−0.780776 + 1.17915i
1.28078 0.599676i
1.28078 + 0.599676i
−0.780776 1.17915i
1.00000 3.02045i 1.00000 −2.00000 3.02045i −2.56155 0.662153i 1.00000 −6.12311 −2.00000
321.2 1.00000 0.936426i 1.00000 −2.00000 0.936426i 1.56155 2.13578i 1.00000 2.12311 −2.00000
321.3 1.00000 0.936426i 1.00000 −2.00000 0.936426i 1.56155 + 2.13578i 1.00000 2.12311 −2.00000
321.4 1.00000 3.02045i 1.00000 −2.00000 3.02045i −2.56155 + 0.662153i 1.00000 −6.12311 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 4
3.b odd 2 1 2898.2.g.b 4
4.b odd 2 1 2576.2.f.b 4
7.b odd 2 1 322.2.c.d yes 4
21.c even 2 1 2898.2.g.a 4
23.b odd 2 1 322.2.c.d yes 4
28.d even 2 1 2576.2.f.e 4
69.c even 2 1 2898.2.g.a 4
92.b even 2 1 2576.2.f.e 4
161.c even 2 1 inner 322.2.c.c 4
483.c odd 2 1 2898.2.g.b 4
644.h odd 2 1 2576.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.c 4 1.a even 1 1 trivial
322.2.c.c 4 161.c even 2 1 inner
322.2.c.d yes 4 7.b odd 2 1
322.2.c.d yes 4 23.b odd 2 1
2576.2.f.b 4 4.b odd 2 1
2576.2.f.b 4 644.h odd 2 1
2576.2.f.e 4 28.d even 2 1
2576.2.f.e 4 92.b even 2 1
2898.2.g.a 4 21.c even 2 1
2898.2.g.a 4 69.c even 2 1
2898.2.g.b 4 3.b odd 2 1
2898.2.g.b 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 \(S_{2}^{\mathrm{new}}(322, [\chi])\):

\( T_{3}^{4} + 10 T_{3}^{2} + 8 \)
\( T_{5} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 8 + 10 T^{2} + T^{4} \)
$5$ \( ( 2 + T )^{4} \)
$7$ \( 49 + 14 T - 2 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 8 + 10 T^{2} + T^{4} \)
$13$ \( 32 + 14 T^{2} + T^{4} \)
$17$ \( ( -8 - 6 T + T^{2} )^{2} \)
$19$ \( ( -4 + T )^{4} \)
$23$ \( 529 - 46 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( 5408 + 148 T^{2} + T^{4} \)
$37$ \( 32 + 14 T^{2} + T^{4} \)
$41$ \( 512 + 56 T^{2} + T^{4} \)
$43$ \( 8 + 10 T^{2} + T^{4} \)
$47$ \( 2048 + 92 T^{2} + T^{4} \)
$53$ \( 32 + 190 T^{2} + T^{4} \)
$59$ \( 648 + 90 T^{2} + T^{4} \)
$61$ \( ( -52 - 8 T + T^{2} )^{2} \)
$67$ \( 8 + 58 T^{2} + T^{4} \)
$71$ \( ( -144 - 6 T + T^{2} )^{2} \)
$73$ \( 2048 + 160 T^{2} + T^{4} \)
$79$ \( 2592 + 180 T^{2} + T^{4} \)
$83$ \( ( -4 + T )^{4} \)
$89$ \( ( -2 + T )^{4} \)
$97$ \( ( -52 + 8 T + T^{2} )^{2} \)
show more
show less