Properties

Label 672.2.j.a
Level 672
Weight 2
Character orbit 672.j
Analytic conductor 5.366
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} -4 \zeta_{8}^{2} q^{13} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + 2 q^{19} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( 5 - \zeta_{8} - \zeta_{8}^{3} ) q^{27} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( 8 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{33} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + 8 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{39} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{41} + 10 q^{43} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} - q^{49} + ( 4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{53} -16 \zeta_{8}^{2} q^{55} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{57} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{59} + 12 \zeta_{8}^{2} q^{61} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} -6 q^{67} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{69} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + 10 q^{73} + ( -3 - 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{75} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{77} + 8 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} -8 \zeta_{8}^{2} q^{85} + ( -4 \zeta_{8} - 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} -4 q^{91} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} -2 q^{97} + ( -16 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{9} + 8q^{19} + 12q^{25} + 20q^{27} + 32q^{33} + 40q^{43} - 4q^{49} + 16q^{51} - 8q^{57} - 24q^{67} + 40q^{73} - 12q^{75} - 28q^{81} - 16q^{91} - 8q^{97} - 64q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 −1.00000 1.41421i 0 −2.82843 0 1.00000i 0 −1.00000 + 2.82843i 0
239.2 0 −1.00000 1.41421i 0 2.82843 0 1.00000i 0 −1.00000 + 2.82843i 0
239.3 0 −1.00000 + 1.41421i 0 −2.82843 0 1.00000i 0 −1.00000 2.82843i 0
239.4 0 −1.00000 + 1.41421i 0 2.82843 0 1.00000i 0 −1.00000 2.82843i 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.j.a 4
3.b odd 2 1 inner 672.2.j.a 4
4.b odd 2 1 168.2.j.b 4
8.b even 2 1 168.2.j.b 4
8.d odd 2 1 inner 672.2.j.a 4
12.b even 2 1 168.2.j.b 4
24.f even 2 1 inner 672.2.j.a 4
24.h odd 2 1 168.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.j.b 4 4.b odd 2 1
168.2.j.b 4 8.b even 2 1
168.2.j.b 4 12.b even 2 1
168.2.j.b 4 24.h odd 2 1
672.2.j.a 4 1.a even 1 1 trivial
672.2.j.a 4 3.b odd 2 1 inner
672.2.j.a 4 8.d odd 2 1 inner
672.2.j.a 4 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 26 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 74 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 10 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 74 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 10 T + 61 T^{2} )^{2}( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 6 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 70 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{4} \)
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