Properties

Label 672.2.bc.a
Level 672
Weight 2
Character orbit 672.bc
Analytic conductor 5.366
Analytic rank 0
Dimension 4
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.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} + ( -1 - 4 \zeta_{12}^{2} ) q^{21} -5 \zeta_{12} q^{23} + 4 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 4 \zeta_{12}^{2} ) q^{29} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{31} + ( 10 - 5 \zeta_{12}^{2} ) q^{33} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{35} + ( -3 + 3 \zeta_{12}^{2} ) q^{37} + 6 \zeta_{12}^{3} q^{39} -8 q^{41} + ( -3 + 3 \zeta_{12}^{2} ) q^{45} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( -14 + 7 \zeta_{12}^{2} ) q^{53} -5 \zeta_{12}^{3} q^{55} + ( -1 - \zeta_{12}^{2} ) q^{57} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{59} + ( -7 - 7 \zeta_{12}^{2} ) q^{61} + ( 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{63} + ( 2 + 2 \zeta_{12}^{2} ) q^{65} + ( 7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} + ( 5 + 5 \zeta_{12}^{2} ) q^{69} -2 \zeta_{12}^{3} q^{71} + ( 10 - 5 \zeta_{12}^{2} ) q^{73} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( -10 - 5 \zeta_{12}^{2} ) q^{77} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} + q^{85} -6 \zeta_{12}^{3} q^{87} + ( 13 - 13 \zeta_{12}^{2} ) q^{89} + ( 8 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{91} + ( 18 - 9 \zeta_{12}^{2} ) q^{93} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{95} + ( -2 + 4 \zeta_{12}^{2} ) q^{97} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{5} + 12q^{9} - 2q^{17} - 12q^{21} + 8q^{25} + 30q^{33} - 6q^{37} - 32q^{41} - 6q^{45} - 4q^{49} - 42q^{53} - 6q^{57} - 42q^{61} + 12q^{65} + 30q^{69} + 30q^{73} - 50q^{77} + 36q^{81} + 4q^{85} + 26q^{89} + 54q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 −0.500000 + 0.866025i 0 1.73205 + 2.00000i 0 3.00000 0
257.2 0 1.73205 0 −0.500000 + 0.866025i 0 −1.73205 2.00000i 0 3.00000 0
353.1 0 −1.73205 0 −0.500000 0.866025i 0 1.73205 2.00000i 0 3.00000 0
353.2 0 1.73205 0 −0.500000 0.866025i 0 −1.73205 + 2.00000i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
21.g even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bc.a 4
3.b odd 2 1 672.2.bc.b yes 4
4.b odd 2 1 inner 672.2.bc.a 4
7.d odd 6 1 672.2.bc.b yes 4
12.b even 2 1 672.2.bc.b yes 4
21.g even 6 1 inner 672.2.bc.a 4
28.f even 6 1 672.2.bc.b yes 4
84.j odd 6 1 inner 672.2.bc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bc.a 4 1.a even 1 1 trivial
672.2.bc.a 4 4.b odd 2 1 inner
672.2.bc.a 4 21.g even 6 1 inner
672.2.bc.a 4 84.j odd 6 1 inner
672.2.bc.b yes 4 3.b odd 2 1
672.2.bc.b yes 4 7.d odd 6 1
672.2.bc.b yes 4 12.b even 2 1
672.2.bc.b yes 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( 1 - 3 T^{2} - 112 T^{4} - 363 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 11 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \)
$23$ \( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 19 T^{2} - 600 T^{4} - 18259 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 43 T^{2} )^{4} \)
$47$ \( 1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 21 T + 200 T^{2} + 1113 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \)
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