Properties

Label 672.2.h.e
Level 672
Weight 2
Character orbit 672.h
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.h (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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} + \beta_{1} q^{7} + ( -2 + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} + \beta_{1} q^{7} + ( -2 + \beta_{6} ) q^{9} + 2 \beta_{3} q^{11} + ( -4 - \beta_{4} + 2 \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{15} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{19} + ( \beta_{4} - \beta_{5} ) q^{21} -4 \beta_{7} q^{23} + 3 q^{25} + ( -\beta_{2} + 3 \beta_{7} ) q^{27} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{29} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{31} + ( 4 \beta_{4} + 2 \beta_{5} ) q^{33} -\beta_{7} q^{35} + ( -2 + 2 \beta_{4} - 4 \beta_{5} ) q^{37} + ( 5 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{39} + ( -4 \beta_{4} + 2 \beta_{6} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{7} ) q^{43} + ( \beta_{4} + 2 \beta_{5} ) q^{45} + 4 \beta_{3} q^{47} - q^{49} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{7} ) q^{51} + ( -4 \beta_{4} - 2 \beta_{6} ) q^{53} + ( -4 \beta_{2} - 2 \beta_{7} ) q^{55} + ( -5 + 4 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{57} + \beta_{7} q^{59} + ( \beta_{4} - 2 \beta_{5} ) q^{61} + ( -2 \beta_{1} - \beta_{3} ) q^{63} + ( 4 \beta_{4} - 2 \beta_{6} ) q^{65} + 2 \beta_{1} q^{67} + ( 4 + 4 \beta_{6} ) q^{69} + ( -2 \beta_{3} - 4 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{4} - 4 \beta_{5} ) q^{73} + 3 \beta_{2} q^{75} + 2 \beta_{6} q^{77} + 4 \beta_{1} q^{79} + ( -1 - 4 \beta_{6} ) q^{81} + ( -4 \beta_{3} + \beta_{7} ) q^{83} + ( -4 - 2 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{7} ) q^{87} + ( -4 \beta_{4} - 2 \beta_{6} ) q^{89} + ( -4 \beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{91} + ( 10 - 2 \beta_{6} ) q^{93} + ( 2 \beta_{3} - 4 \beta_{7} ) q^{95} + ( 2 - 4 \beta_{4} + 8 \beta_{5} ) q^{97} + ( 10 \beta_{1} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{9} + O(q^{10}) \) \( 8q - 16q^{9} - 32q^{13} + 24q^{25} - 16q^{37} - 8q^{49} - 40q^{57} + 32q^{69} - 16q^{73} - 8q^{81} - 32q^{85} + 80q^{93} + 16q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
1.14412 1.14412i
−0.437016 0.437016i
−0.437016 + 0.437016i
1.14412 + 1.14412i
−1.14412 1.14412i
0.437016 0.437016i
0.437016 + 0.437016i
−1.14412 + 1.14412i
0 −0.707107 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 + 2.23607i 0
575.2 0 −0.707107 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 + 2.23607i 0
575.3 0 −0.707107 + 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 2.23607i 0
575.4 0 −0.707107 + 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 2.23607i 0
575.5 0 0.707107 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 2.23607i 0
575.6 0 0.707107 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 2.23607i 0
575.7 0 0.707107 + 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 + 2.23607i 0
575.8 0 0.707107 + 1.58114i 0 1.41421i 0 1.00000i 0 −2.00000 + 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.8
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
12.b 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.h.e 8
3.b odd 2 1 inner 672.2.h.e 8
4.b odd 2 1 inner 672.2.h.e 8
8.b even 2 1 1344.2.h.f 8
8.d odd 2 1 1344.2.h.f 8
12.b even 2 1 inner 672.2.h.e 8
24.f even 2 1 1344.2.h.f 8
24.h odd 2 1 1344.2.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.h.e 8 1.a even 1 1 trivial
672.2.h.e 8 3.b odd 2 1 inner
672.2.h.e 8 4.b odd 2 1 inner
672.2.h.e 8 12.b even 2 1 inner
1344.2.h.f 8 8.b even 2 1
1344.2.h.f 8 8.d odd 2 1
1344.2.h.f 8 24.f even 2 1
1344.2.h.f 8 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 + 2 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 12 T^{2} - 26 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 24 T^{2} + 226 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 14 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 60 T^{2} + 1942 T^{4} - 50460 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 22 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 4 T + 38 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 60 T^{2} + 1702 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 84 T^{2} + 4822 T^{4} - 155316 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 14 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 108 T^{2} + 5974 T^{4} - 303372 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 116 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 130 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 180 T^{2} + 15622 T^{4} + 907380 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 4 T + 110 T^{2} + 292 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 168 T^{2} + 20194 T^{4} + 1157352 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 252 T^{2} + 29158 T^{4} - 1996092 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 4 T + 38 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \)
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