Properties

Label 672.2.k.a
Level 672
Weight 2
Character orbit 672.k
Analytic conductor 5.366
Analytic rank 0
Dimension 8
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.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -\beta_{2} - \beta_{3} ) q^{5} + ( -1 - \beta_{4} ) q^{7} + ( -\beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -\beta_{2} - \beta_{3} ) q^{5} + ( -1 - \beta_{4} ) q^{7} + ( -\beta_{4} - \beta_{6} ) q^{9} + ( \beta_{1} + \beta_{7} ) q^{11} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{13} + ( -3 + \beta_{4} + \beta_{6} ) q^{15} + ( \beta_{1} - \beta_{7} ) q^{17} + ( -\beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{21} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( 1 - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{25} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{27} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{31} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{35} + ( -2 - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{37} + ( 5 + \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{39} + ( \beta_{1} - \beta_{7} ) q^{41} + 8 q^{43} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{45} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( 3 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{49} + ( -2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{51} + ( \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + ( -4 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{55} + ( 3 + \beta_{4} + \beta_{6} ) q^{57} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{59} + ( -5 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{63} + ( -2 \beta_{1} + 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{65} + ( -4 + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( -2 \beta_{2} - 3 \beta_{5} - 3 \beta_{6} ) q^{69} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{71} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{77} + ( -2 - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{79} + ( -1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{81} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{7} ) q^{83} + ( 4 + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{85} + ( 2 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{87} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{89} + ( -6 + 5 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -4 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{93} + ( \beta_{5} - \beta_{6} ) q^{95} + ( 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{97} + ( -4 + \beta_{1} - 4 \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{7} + 4q^{9} - 28q^{15} + 8q^{21} + 16q^{25} + 36q^{39} + 64q^{43} + 16q^{49} - 8q^{51} + 20q^{57} + 32q^{63} - 48q^{67} - 8q^{79} + 16q^{85} - 56q^{91} - 24q^{93} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + \nu^{3} \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + \nu^{5} - 3 \nu^{4} - 6 \nu^{3} - 10 \nu^{2} + 8 \nu - 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} + 6 \nu^{2} - 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} + 10 \nu^{3} - 12 \nu^{2} + 24 \nu \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 3 \nu^{5} + 2 \nu^{4} - 10 \nu^{3} - 12 \nu^{2} - 24 \nu \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - \nu^{5} \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{3} + 3 \beta_{2} + 7 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-6 \beta_{6} - 6 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-14 \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{3} - 3 \beta_{2} - 7 \beta_{1}\)\()/4\)

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
−0.599676 + 1.28078i
−0.599676 1.28078i
−1.17915 + 0.780776i
−1.17915 0.780776i
1.17915 0.780776i
1.17915 + 0.780776i
0.599676 1.28078i
0.599676 + 1.28078i
0 −1.66757 0.468213i 0 3.33513 0 1.56155 + 2.13578i 0 2.56155 + 1.56155i 0
545.2 0 −1.66757 + 0.468213i 0 3.33513 0 1.56155 2.13578i 0 2.56155 1.56155i 0
545.3 0 −0.848071 1.51022i 0 1.69614 0 −2.56155 + 0.662153i 0 −1.56155 + 2.56155i 0
545.4 0 −0.848071 + 1.51022i 0 1.69614 0 −2.56155 0.662153i 0 −1.56155 2.56155i 0
545.5 0 0.848071 1.51022i 0 −1.69614 0 −2.56155 + 0.662153i 0 −1.56155 2.56155i 0
545.6 0 0.848071 + 1.51022i 0 −1.69614 0 −2.56155 0.662153i 0 −1.56155 + 2.56155i 0
545.7 0 1.66757 0.468213i 0 −3.33513 0 1.56155 + 2.13578i 0 2.56155 1.56155i 0
545.8 0 1.66757 + 0.468213i 0 −3.33513 0 1.56155 2.13578i 0 2.56155 + 1.56155i 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
7.b odd 2 1 inner
21.c 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.k.a 8
3.b odd 2 1 inner 672.2.k.a 8
4.b odd 2 1 672.2.k.d yes 8
7.b odd 2 1 inner 672.2.k.a 8
8.b even 2 1 1344.2.k.e 8
8.d odd 2 1 1344.2.k.j 8
12.b even 2 1 672.2.k.d yes 8
21.c even 2 1 inner 672.2.k.a 8
24.f even 2 1 1344.2.k.j 8
24.h odd 2 1 1344.2.k.e 8
28.d even 2 1 672.2.k.d yes 8
56.e even 2 1 1344.2.k.j 8
56.h odd 2 1 1344.2.k.e 8
84.h odd 2 1 672.2.k.d yes 8
168.e odd 2 1 1344.2.k.j 8
168.i even 2 1 1344.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.k.a 8 1.a even 1 1 trivial
672.2.k.a 8 3.b odd 2 1 inner
672.2.k.a 8 7.b odd 2 1 inner
672.2.k.a 8 21.c even 2 1 inner
672.2.k.d yes 8 4.b odd 2 1
672.2.k.d yes 8 12.b even 2 1
672.2.k.d yes 8 28.d even 2 1
672.2.k.d yes 8 84.h odd 2 1
1344.2.k.e 8 8.b even 2 1
1344.2.k.e 8 24.h odd 2 1
1344.2.k.e 8 56.h odd 2 1
1344.2.k.e 8 168.i even 2 1
1344.2.k.j 8 8.d odd 2 1
1344.2.k.j 8 24.f even 2 1
1344.2.k.j 8 56.e even 2 1
1344.2.k.j 8 168.e odd 2 1

Hecke kernels

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

\( T_{5}^{4} - 14 T_{5}^{2} + 32 \)
\( T_{43} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \)
$5$ \( ( 1 + 6 T^{2} + 42 T^{4} + 150 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 6 T^{2} + 330 T^{4} - 1014 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 28 T^{2} + 502 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 40 T^{2} + 846 T^{4} - 21160 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 64 T^{2} + 2094 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 6 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 124 T^{2} + 6934 T^{4} + 208444 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 132 T^{2} + 8502 T^{4} + 291588 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 78 T^{2} + 7650 T^{4} + 271518 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 54 T^{2} + 7338 T^{4} - 200934 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 12 T + 102 T^{2} + 804 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 76 T^{2} + 1734 T^{4} - 383116 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 + 2 T + 142 T^{2} + 158 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 2 T^{2} - 2558 T^{4} - 13778 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 244 T^{2} + 29638 T^{4} + 1932724 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 140 T^{2} + 10390 T^{4} - 1317260 T^{6} + 88529281 T^{8} )^{2} \)
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