Properties

Label 672.2.h.b
Level $672$
Weight $2$
Character orbit 672.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} -2 q^{13} + ( 3 \zeta_{8} + 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 1 + \zeta_{8} + \zeta_{8}^{3} ) q^{21} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} -13 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{31} + ( 6 - 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{33} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{35} -4 q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -8 \zeta_{8}^{2} q^{43} + ( -12 - 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} - q^{49} + ( -5 \zeta_{8} - 10 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{51} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{53} + 18 \zeta_{8}^{2} q^{55} + ( -4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{57} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( -2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{63} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{65} -8 \zeta_{8}^{2} q^{67} + ( 2 - \zeta_{8} - \zeta_{8}^{3} ) q^{69} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{71} + 14 q^{73} + ( 13 \zeta_{8} - 13 \zeta_{8}^{2} - 13 \zeta_{8}^{3} ) q^{75} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{77} -4 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + 30 q^{85} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{87} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} + 2 \zeta_{8}^{2} q^{91} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{93} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{95} -10 q^{97} + ( -3 \zeta_{8} + 12 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 8q^{13} + 4q^{21} - 52q^{25} + 24q^{33} - 16q^{37} - 48q^{45} - 4q^{49} - 16q^{57} - 8q^{61} + 8q^{69} + 56q^{73} - 28q^{81} + 120q^{85} - 8q^{93} - 40q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 1.00000i 0 4.24264i 0 1.00000i 0 1.00000 + 2.82843i 0
575.2 0 −1.41421 + 1.00000i 0 4.24264i 0 1.00000i 0 1.00000 2.82843i 0
575.3 0 1.41421 1.00000i 0 4.24264i 0 1.00000i 0 1.00000 2.82843i 0
575.4 0 1.41421 + 1.00000i 0 4.24264i 0 1.00000i 0 1.00000 + 2.82843i 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
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.b 4
3.b odd 2 1 inner 672.2.h.b 4
4.b odd 2 1 inner 672.2.h.b 4
8.b even 2 1 1344.2.h.d 4
8.d odd 2 1 1344.2.h.d 4
12.b even 2 1 inner 672.2.h.b 4
24.f even 2 1 1344.2.h.d 4
24.h odd 2 1 1344.2.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.h.b 4 1.a even 1 1 trivial
672.2.h.b 4 3.b odd 2 1 inner
672.2.h.b 4 4.b odd 2 1 inner
672.2.h.b 4 12.b even 2 1 inner
1344.2.h.d 4 8.b even 2 1
1344.2.h.d 4 8.d odd 2 1
1344.2.h.d 4 24.f even 2 1
1344.2.h.d 4 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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 18 \)
\( T_{11}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 2 T^{2} + T^{4} \)
$5$ \( ( 18 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -18 + T^{2} )^{2} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( ( 50 + T^{2} )^{2} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -2 + T^{2} )^{2} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 4 + T^{2} )^{2} \)
$37$ \( ( 4 + T )^{4} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -162 + T^{2} )^{2} \)
$73$ \( ( -14 + T )^{4} \)
$79$ \( ( 16 + T^{2} )^{2} \)
$83$ \( ( -8 + T^{2} )^{2} \)
$89$ \( ( 98 + T^{2} )^{2} \)
$97$ \( ( 10 + T )^{4} \)
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