Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + q^{27} -8 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + 9 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{39} + 2 q^{41} - q^{43} + 8 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + 2 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -5 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} + ( -5 + 5 \zeta_{6} ) q^{67} -6 q^{69} -4 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + 5 \zeta_{6} q^{75} + ( -6 + 4 \zeta_{6} ) q^{77} -5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( 8 - 8 \zeta_{6} ) q^{87} -12 \zeta_{6} q^{89} + ( 2 + \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + 18 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 5q^{7} - q^{9} - 2q^{11} + 2q^{13} + 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} + 5q^{25} + 2q^{27} - 16q^{29} - 3q^{31} - 2q^{33} + 9q^{37} - q^{39} + 4q^{41} - 2q^{43} + 8q^{47} + 11q^{49} + 2q^{51} - 6q^{53} - 10q^{57} + 6q^{59} + 2q^{61} - q^{63} - 5q^{67} - 12q^{69} - 8q^{71} + 11q^{73} + 5q^{75} - 8q^{77} - 5q^{79} - q^{81} + 8q^{87} - 12q^{89} + 5q^{91} - 3q^{93} + 36q^{97} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 0 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.q.d 2
3.b odd 2 1 2016.2.s.h 2
4.b odd 2 1 672.2.q.h yes 2
7.c even 3 1 inner 672.2.q.d 2
7.c even 3 1 4704.2.a.ba 1
7.d odd 6 1 4704.2.a.j 1
8.b even 2 1 1344.2.q.q 2
8.d odd 2 1 1344.2.q.e 2
12.b even 2 1 2016.2.s.e 2
21.h odd 6 1 2016.2.s.h 2
28.f even 6 1 4704.2.a.y 1
28.g odd 6 1 672.2.q.h yes 2
28.g odd 6 1 4704.2.a.g 1
56.j odd 6 1 9408.2.a.ci 1
56.k odd 6 1 1344.2.q.e 2
56.k odd 6 1 9408.2.a.cn 1
56.m even 6 1 9408.2.a.x 1
56.p even 6 1 1344.2.q.q 2
56.p even 6 1 9408.2.a.t 1
84.n even 6 1 2016.2.s.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 1.a even 1 1 trivial
672.2.q.d 2 7.c even 3 1 inner
672.2.q.h yes 2 4.b odd 2 1
672.2.q.h yes 2 28.g odd 6 1
1344.2.q.e 2 8.d odd 2 1
1344.2.q.e 2 56.k odd 6 1
1344.2.q.q 2 8.b even 2 1
1344.2.q.q 2 56.p even 6 1
2016.2.s.e 2 12.b even 2 1
2016.2.s.e 2 84.n even 6 1
2016.2.s.h 2 3.b odd 2 1
2016.2.s.h 2 21.h odd 6 1
4704.2.a.g 1 28.g odd 6 1
4704.2.a.j 1 7.d odd 6 1
4704.2.a.y 1 28.f even 6 1
4704.2.a.ba 1 7.c even 3 1
9408.2.a.t 1 56.p even 6 1
9408.2.a.x 1 56.m even 6 1
9408.2.a.ci 1 56.j odd 6 1
9408.2.a.cn 1 56.k odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )^{2} \)
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