Properties

Label 672.2.h.a
Level 672
Weight 2
Character orbit 672.h
Analytic conductor 5.366
Analytic rank 0
Dimension 4
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.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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} -2 q^{11} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 3 + 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + ( -1 - \zeta_{8} + \zeta_{8}^{2} ) q^{21} + 4 q^{23} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{25} + ( -1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{29} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{31} + ( 2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{37} + ( -1 - 2 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} -10 \zeta_{8}^{2} q^{41} + ( 6 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{43} + ( -2 - 5 \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{45} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} - q^{49} + ( 4 + 2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} + ( 8 \zeta_{8} + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{53} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{55} + ( 5 + 2 \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{57} + ( 2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{59} + ( 10 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{61} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{65} + ( -8 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{67} + ( -4 - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{69} + ( 6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{73} + ( -3 + 8 \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{75} + 2 \zeta_{8}^{2} q^{77} -4 \zeta_{8}^{2} q^{79} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -10 - \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( -8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{85} + ( 4 + 2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{87} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{89} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{91} + ( -2 - 4 \zeta_{8} + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{93} + ( -10 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + ( -6 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{97} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + O(q^{10}) \) \( 4q - 4q^{3} - 8q^{11} + 8q^{13} + 12q^{15} - 4q^{21} + 16q^{23} - 4q^{25} - 4q^{27} + 8q^{33} + 8q^{35} + 8q^{37} - 4q^{39} - 8q^{45} + 16q^{47} - 4q^{49} + 16q^{51} + 20q^{57} + 8q^{59} + 40q^{61} + 4q^{63} - 16q^{69} + 24q^{71} - 24q^{73} - 12q^{75} + 28q^{81} - 40q^{83} - 32q^{85} + 16q^{87} - 8q^{93} - 40q^{95} - 24q^{97} + O(q^{100}) \)

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 −1.70711 0.292893i 0 0.585786i 0 1.00000i 0 2.82843 + 1.00000i 0
575.2 0 −1.70711 + 0.292893i 0 0.585786i 0 1.00000i 0 2.82843 1.00000i 0
575.3 0 −0.292893 1.70711i 0 3.41421i 0 1.00000i 0 −2.82843 + 1.00000i 0
575.4 0 −0.292893 + 1.70711i 0 3.41421i 0 1.00000i 0 −2.82843 1.00000i 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
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.a 4
3.b odd 2 1 672.2.h.d yes 4
4.b odd 2 1 672.2.h.d yes 4
8.b even 2 1 1344.2.h.e 4
8.d odd 2 1 1344.2.h.a 4
12.b even 2 1 inner 672.2.h.a 4
24.f even 2 1 1344.2.h.e 4
24.h odd 2 1 1344.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.h.a 4 1.a even 1 1 trivial
672.2.h.a 4 12.b even 2 1 inner
672.2.h.d yes 4 3.b odd 2 1
672.2.h.d yes 4 4.b odd 2 1
1344.2.h.a 4 8.d odd 2 1
1344.2.h.a 4 24.h odd 2 1
1344.2.h.e 4 8.b even 2 1
1344.2.h.e 4 24.f even 2 1

Hecke kernels

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

\( T_{5}^{4} + 12 T_{5}^{2} + 4 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \)
$5$ \( 1 - 8 T^{2} + 34 T^{4} - 200 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 - 4 T + 28 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 44 T^{2} + 934 T^{4} - 12716 T^{6} + 83521 T^{8} \)
$19$ \( 1 - 32 T^{2} + 690 T^{4} - 11552 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{4} \)
$29$ \( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} \)
$31$ \( 1 - 76 T^{2} + 2854 T^{4} - 73036 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 - 4 T + 70 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 20 T^{2} + 2646 T^{4} - 36980 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 52 T^{2} + 4246 T^{4} + 146068 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 4 T + 104 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 20 T + 204 T^{2} - 1220 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 4 T^{2} + 6934 T^{4} - 17956 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 12 T + 146 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 20 T + 264 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 284 T^{2} + 35494 T^{4} - 2249564 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2} \)
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