Properties

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

Related objects

Downloads

Learn more about

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: 8.0.3317760000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{3} ) q^{3} -\beta_{3} q^{5} + ( -1 - \beta_{6} ) q^{7} + ( 2 + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{3} ) q^{3} -\beta_{3} q^{5} + ( -1 - \beta_{6} ) q^{7} + ( 2 + \beta_{5} ) q^{9} + \beta_{4} q^{11} + ( -2 \beta_{2} - \beta_{3} ) q^{13} + ( -1 + \beta_{5} ) q^{15} + ( 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + 2 \beta_{5} q^{23} + 3 q^{25} + ( -\beta_{2} - 4 \beta_{3} ) q^{27} + 2 \beta_{4} q^{29} + 2 \beta_{6} q^{31} + ( \beta_{6} - \beta_{7} ) q^{33} + ( \beta_{3} + \beta_{4} ) q^{35} + ( 5 + \beta_{5} ) q^{39} + 2 \beta_{7} q^{41} + ( 2 \beta_{1} + \beta_{4} ) q^{43} + ( 2 \beta_{2} - \beta_{3} ) q^{45} + 2 \beta_{7} q^{47} + ( -5 + 2 \beta_{6} ) q^{49} + 2 \beta_{6} q^{55} + ( -5 - \beta_{5} ) q^{57} + 7 \beta_{3} q^{59} + ( -2 \beta_{2} - \beta_{3} ) q^{61} + ( -2 - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{63} + 2 \beta_{5} q^{65} + ( -2 \beta_{1} - \beta_{4} ) q^{67} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{69} -4 \beta_{5} q^{71} -6 \beta_{6} q^{73} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 6 \beta_{3} - \beta_{4} ) q^{77} + 10 q^{79} + ( -1 + 4 \beta_{5} ) q^{81} -5 \beta_{3} q^{83} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{93} -2 \beta_{5} q^{95} + 2 \beta_{6} q^{97} + ( -2 \beta_{1} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + 16q^{9} + O(q^{10}) \) \( 8q - 8q^{7} + 16q^{9} - 8q^{15} + 24q^{25} + 40q^{39} - 40q^{49} - 40q^{57} - 16q^{63} + 80q^{79} - 8q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 13 x^{4} + 12 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} - 10 \nu^{4} + 71 \nu^{2} - 6 \)\()/24\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 14 \nu^{5} - 97 \nu^{3} + 138 \nu \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 14 \nu^{5} - 25 \nu^{3} - 78 \nu \)\()/72\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 6 \nu^{4} - 7 \nu^{2} + 30 \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 6 \nu^{4} - 9 \nu^{2} + 2 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{5} + 5 \nu^{3} + 18 \nu \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 26 \nu^{5} - 59 \nu^{3} + 54 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} - 2 \beta_{3} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - 3 \beta_{4} + 8\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 3 \beta_{6} - 2 \beta_{3} - 14 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-8 \beta_{5} - 15 \beta_{4} - 6 \beta_{1} + 38\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(15 \beta_{7} + 21 \beta_{6} + 16 \beta_{3} - 50 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-62 \beta_{5} - 63 \beta_{4} - 36 \beta_{1} + 164\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(57 \beta_{7} + 141 \beta_{6} + 142 \beta_{3} - 194 \beta_{2}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−2.15988 + 0.258819i
0.578737 0.965926i
0.578737 + 0.965926i
−2.15988 0.258819i
2.15988 + 0.258819i
−0.578737 0.965926i
−0.578737 + 0.965926i
2.15988 0.258819i
0 −1.58114 0.707107i 0 1.41421i 0 −1.00000 2.44949i 0 2.00000 + 2.23607i 0
209.2 0 −1.58114 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 + 2.23607i 0
209.3 0 −1.58114 + 0.707107i 0 1.41421i 0 −1.00000 2.44949i 0 2.00000 2.23607i 0
209.4 0 −1.58114 + 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 2.23607i 0
209.5 0 1.58114 0.707107i 0 1.41421i 0 −1.00000 2.44949i 0 2.00000 2.23607i 0
209.6 0 1.58114 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 2.23607i 0
209.7 0 1.58114 + 0.707107i 0 1.41421i 0 −1.00000 2.44949i 0 2.00000 + 2.23607i 0
209.8 0 1.58114 + 0.707107i 0 1.41421i 0 −1.00000 + 2.44949i 0 2.00000 + 2.23607i 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
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
56.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.d 8
3.b odd 2 1 inner 672.2.i.d 8
4.b odd 2 1 168.2.i.e 8
7.b odd 2 1 inner 672.2.i.d 8
8.b even 2 1 inner 672.2.i.d 8
8.d odd 2 1 168.2.i.e 8
12.b even 2 1 168.2.i.e 8
21.c even 2 1 inner 672.2.i.d 8
24.f even 2 1 168.2.i.e 8
24.h odd 2 1 inner 672.2.i.d 8
28.d even 2 1 168.2.i.e 8
56.e even 2 1 168.2.i.e 8
56.h odd 2 1 inner 672.2.i.d 8
84.h odd 2 1 168.2.i.e 8
168.e odd 2 1 168.2.i.e 8
168.i even 2 1 inner 672.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.e 8 4.b odd 2 1
168.2.i.e 8 8.d odd 2 1
168.2.i.e 8 12.b even 2 1
168.2.i.e 8 24.f even 2 1
168.2.i.e 8 28.d even 2 1
168.2.i.e 8 56.e even 2 1
168.2.i.e 8 84.h odd 2 1
168.2.i.e 8 168.e odd 2 1
672.2.i.d 8 1.a even 1 1 trivial
672.2.i.d 8 3.b odd 2 1 inner
672.2.i.d 8 7.b odd 2 1 inner
672.2.i.d 8 8.b even 2 1 inner
672.2.i.d 8 21.c even 2 1 inner
672.2.i.d 8 24.h odd 2 1 inner
672.2.i.d 8 56.h odd 2 1 inner
672.2.i.d 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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 16 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 + 10 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 - 38 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 26 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 26 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{8} \)
$59$ \( ( 1 - 20 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 74 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 62 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 70 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{8} \)
$83$ \( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \)
show more
show less