Properties

Label 672.2.k.c
Level 672
Weight 2
Character orbit 672.k
Analytic conductor 5.366
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{4} + \beta_{5} ) q^{3} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{4} + \beta_{5} ) q^{3} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 - \beta_{6} ) q^{9} + ( -2 \beta_{2} - \beta_{7} ) q^{13} + ( \beta_{1} - \beta_{3} ) q^{15} -2 \beta_{7} q^{17} -3 \beta_{4} q^{19} + ( -1 + \beta_{2} - \beta_{6} + 3 \beta_{7} ) q^{21} -6 \beta_{1} q^{23} -3 q^{25} + ( -4 \beta_{4} + \beta_{5} ) q^{27} + 2 \beta_{6} q^{29} -4 \beta_{4} q^{31} + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{35} -2 q^{37} + ( -5 \beta_{1} - \beta_{3} ) q^{39} -6 \beta_{7} q^{41} -4 \beta_{3} q^{43} + ( 2 \beta_{2} + 3 \beta_{7} ) q^{45} + ( -2 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 3 + 4 \beta_{2} + 2 \beta_{7} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{51} + 6 \beta_{6} q^{53} + ( -3 - 3 \beta_{6} ) q^{57} + ( \beta_{4} - 2 \beta_{5} ) q^{59} + ( -2 \beta_{2} - \beta_{7} ) q^{61} + ( 5 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{63} + 2 \beta_{6} q^{65} -6 \beta_{2} q^{69} + 8 \beta_{1} q^{71} + ( 8 \beta_{2} + 4 \beta_{7} ) q^{73} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{75} + 6 \beta_{3} q^{79} + ( -1 - 4 \beta_{6} ) q^{81} + ( -\beta_{4} + 2 \beta_{5} ) q^{83} -4 q^{85} + ( 4 \beta_{4} + 2 \beta_{5} ) q^{87} -8 \beta_{7} q^{89} + ( -2 \beta_{3} + 5 \beta_{4} ) q^{91} + ( -4 - 4 \beta_{6} ) q^{93} + 6 \beta_{1} q^{95} + ( -12 \beta_{2} - 6 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} - 8q^{21} - 24q^{25} - 16q^{37} + 24q^{49} - 24q^{57} - 8q^{81} - 32q^{85} - 32q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 8 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 8 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\( \nu^{6} + 6 \nu^{2} \)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + 3 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{3} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-8 \beta_{7} - 5 \beta_{5} + 8 \beta_{4} - 5 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{6} - 9 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(8 \beta_{7} + 13 \beta_{5} + 8 \beta_{4} - 13 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
−1.14412 1.14412i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
−0.437016 0.437016i
0.437016 + 0.437016i
1.14412 1.14412i
0.437016 0.437016i
1.14412 + 1.14412i
0 −1.58114 0.707107i 0 −1.41421 0 2.23607 1.41421i 0 2.00000 + 2.23607i 0
545.2 0 −1.58114 0.707107i 0 1.41421 0 −2.23607 1.41421i 0 2.00000 + 2.23607i 0
545.3 0 −1.58114 + 0.707107i 0 −1.41421 0 2.23607 + 1.41421i 0 2.00000 2.23607i 0
545.4 0 −1.58114 + 0.707107i 0 1.41421 0 −2.23607 + 1.41421i 0 2.00000 2.23607i 0
545.5 0 1.58114 0.707107i 0 −1.41421 0 −2.23607 1.41421i 0 2.00000 2.23607i 0
545.6 0 1.58114 0.707107i 0 1.41421 0 2.23607 1.41421i 0 2.00000 2.23607i 0
545.7 0 1.58114 + 0.707107i 0 −1.41421 0 −2.23607 + 1.41421i 0 2.00000 + 2.23607i 0
545.8 0 1.58114 + 0.707107i 0 1.41421 0 2.23607 + 1.41421i 0 2.00000 + 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.k.c 8
3.b odd 2 1 inner 672.2.k.c 8
4.b odd 2 1 inner 672.2.k.c 8
7.b odd 2 1 inner 672.2.k.c 8
8.b even 2 1 1344.2.k.h 8
8.d odd 2 1 1344.2.k.h 8
12.b even 2 1 inner 672.2.k.c 8
21.c even 2 1 inner 672.2.k.c 8
24.f even 2 1 1344.2.k.h 8
24.h odd 2 1 1344.2.k.h 8
28.d even 2 1 inner 672.2.k.c 8
56.e even 2 1 1344.2.k.h 8
56.h odd 2 1 1344.2.k.h 8
84.h odd 2 1 inner 672.2.k.c 8
168.e odd 2 1 1344.2.k.h 8
168.i even 2 1 1344.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.k.c 8 1.a even 1 1 trivial
672.2.k.c 8 3.b odd 2 1 inner
672.2.k.c 8 4.b odd 2 1 inner
672.2.k.c 8 7.b odd 2 1 inner
672.2.k.c 8 12.b even 2 1 inner
672.2.k.c 8 21.c even 2 1 inner
672.2.k.c 8 28.d even 2 1 inner
672.2.k.c 8 84.h odd 2 1 inner
1344.2.k.h 8 8.b even 2 1
1344.2.k.h 8 8.d odd 2 1
1344.2.k.h 8 24.f even 2 1
1344.2.k.h 8 24.h odd 2 1
1344.2.k.h 8 56.e even 2 1
1344.2.k.h 8 56.h odd 2 1
1344.2.k.h 8 168.e odd 2 1
1344.2.k.h 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{43}^{2} - 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 + 8 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{8} \)
$13$ \( ( 1 - 16 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 26 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 20 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 38 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 30 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{8} \)
$41$ \( ( 1 + 10 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 6 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 54 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 + 74 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 108 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 112 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 78 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 14 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 22 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 156 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 50 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 166 T^{2} + 9409 T^{4} )^{4} \)
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