Properties

Label 672.4.a.c
Level $672$
Weight $4$
Character orbit 672.a
Self dual yes
Analytic conductor $39.649$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.6492835239\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 18q^{5} - 7q^{7} + 9q^{9} + 44q^{11} + 58q^{13} - 54q^{15} - 130q^{17} + 92q^{19} - 21q^{21} + 84q^{23} + 199q^{25} + 27q^{27} - 250q^{29} - 72q^{31} + 132q^{33} + 126q^{35} - 354q^{37} + 174q^{39} + 334q^{41} - 416q^{43} - 162q^{45} - 464q^{47} + 49q^{49} - 390q^{51} - 450q^{53} - 792q^{55} + 276q^{57} - 516q^{59} + 58q^{61} - 63q^{63} - 1044q^{65} - 656q^{67} + 252q^{69} - 940q^{71} + 178q^{73} + 597q^{75} - 308q^{77} + 1072q^{79} + 81q^{81} + 660q^{83} + 2340q^{85} - 750q^{87} + 1254q^{89} - 406q^{91} - 216q^{93} - 1656q^{95} + 210q^{97} + 396q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 0 −18.0000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.a.c yes 1
3.b odd 2 1 2016.4.a.e 1
4.b odd 2 1 672.4.a.a 1
8.b even 2 1 1344.4.a.m 1
8.d odd 2 1 1344.4.a.bb 1
12.b even 2 1 2016.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.a 1 4.b odd 2 1
672.4.a.c yes 1 1.a even 1 1 trivial
1344.4.a.m 1 8.b even 2 1
1344.4.a.bb 1 8.d odd 2 1
2016.4.a.e 1 3.b odd 2 1
2016.4.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(672))\):

\( T_{5} + 18 \)
\( T_{11} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 18 + T \)
$7$ \( 7 + T \)
$11$ \( -44 + T \)
$13$ \( -58 + T \)
$17$ \( 130 + T \)
$19$ \( -92 + T \)
$23$ \( -84 + T \)
$29$ \( 250 + T \)
$31$ \( 72 + T \)
$37$ \( 354 + T \)
$41$ \( -334 + T \)
$43$ \( 416 + T \)
$47$ \( 464 + T \)
$53$ \( 450 + T \)
$59$ \( 516 + T \)
$61$ \( -58 + T \)
$67$ \( 656 + T \)
$71$ \( 940 + T \)
$73$ \( -178 + T \)
$79$ \( -1072 + T \)
$83$ \( -660 + T \)
$89$ \( -1254 + T \)
$97$ \( -210 + T \)
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