Properties

Label 672.2.i.e
Level 672
Weight 2
Character orbit 672.i
Analytic conductor 5.366
Analytic rank 0
Dimension 8
CM discriminant -56
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.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{13} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{15} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{5} + \beta_{6} ) q^{21} -2 \beta_{4} q^{23} + ( -5 - 2 \beta_{3} ) q^{25} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{27} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{35} + ( -5 - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{45} + 7 q^{49} + ( -1 - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{57} + ( -3 \beta_{1} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{59} + ( 4 \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{61} + ( 7 + \beta_{4} ) q^{63} + ( -2 \beta_{2} - 4 \beta_{4} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{69} -4 \beta_{2} q^{71} + ( -5 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} ) q^{75} + 2 \beta_{3} q^{79} + ( 5 + 2 \beta_{4} ) q^{81} + ( -3 \beta_{1} + \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{83} + ( -\beta_{1} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( 8 \beta_{2} - 2 \beta_{4} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{15} - 40q^{25} - 40q^{39} + 56q^{49} - 8q^{57} + 56q^{63} + 40q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 9 \nu^{5} + 37 \nu^{3} + 63 \nu \)\()/54\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} + 9 \nu^{5} + 20 \nu^{3} - 63 \nu \)\()/54\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{3} \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{5}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 5\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{3} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-19 \beta_{7} + \beta_{6} + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−1.68014 0.420861i
−1.68014 + 0.420861i
−0.420861 1.68014i
−0.420861 + 1.68014i
0.420861 1.68014i
0.420861 + 1.68014i
1.68014 0.420861i
1.68014 + 0.420861i
0 −1.68014 0.420861i 0 3.91044i 0 2.64575 0 2.64575 + 1.41421i 0
209.2 0 −1.68014 + 0.420861i 0 3.91044i 0 2.64575 0 2.64575 1.41421i 0
209.3 0 −0.420861 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 + 1.41421i 0
209.4 0 −0.420861 + 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 1.41421i 0
209.5 0 0.420861 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 1.41421i 0
209.6 0 0.420861 + 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 + 1.41421i 0
209.7 0 1.68014 0.420861i 0 3.91044i 0 2.64575 0 2.64575 1.41421i 0
209.8 0 1.68014 + 0.420861i 0 3.91044i 0 2.64575 0 2.64575 + 1.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
168.i 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.i.e 8
3.b odd 2 1 inner 672.2.i.e 8
4.b odd 2 1 168.2.i.d 8
7.b odd 2 1 inner 672.2.i.e 8
8.b even 2 1 inner 672.2.i.e 8
8.d odd 2 1 168.2.i.d 8
12.b even 2 1 168.2.i.d 8
21.c even 2 1 inner 672.2.i.e 8
24.f even 2 1 168.2.i.d 8
24.h odd 2 1 inner 672.2.i.e 8
28.d even 2 1 168.2.i.d 8
56.e even 2 1 168.2.i.d 8
56.h odd 2 1 CM 672.2.i.e 8
84.h odd 2 1 168.2.i.d 8
168.e odd 2 1 168.2.i.d 8
168.i even 2 1 inner 672.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.d 8 4.b odd 2 1
168.2.i.d 8 8.d odd 2 1
168.2.i.d 8 12.b even 2 1
168.2.i.d 8 24.f even 2 1
168.2.i.d 8 28.d even 2 1
168.2.i.d 8 56.e even 2 1
168.2.i.d 8 84.h odd 2 1
168.2.i.d 8 168.e odd 2 1
672.2.i.e 8 1.a even 1 1 trivial
672.2.i.e 8 3.b odd 2 1 inner
672.2.i.e 8 7.b odd 2 1 inner
672.2.i.e 8 8.b even 2 1 inner
672.2.i.e 8 21.c even 2 1 inner
672.2.i.e 8 24.h odd 2 1 inner
672.2.i.e 8 56.h odd 2 1 CM
672.2.i.e 8 168.i even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 20 T_{5}^{2} + 72 \)
\( T_{11} \)
\( T_{13}^{4} - 52 T_{13}^{2} + 648 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 10 T^{4} + 81 T^{8} \)
$5$ \( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{8} \)
$13$ \( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 + 41 T^{2} )^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 + 53 T^{2} )^{8} \)
$59$ \( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{8} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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