Properties

Label 1680.4.a.s
Level 1680
Weight 4
Character orbit 1680.a
Self dual yes
Analytic conductor 99.123
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.1232088096\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 5q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 5q^{5} - 7q^{7} + 9q^{9} - 42q^{11} + 20q^{13} + 15q^{15} + 66q^{17} - 38q^{19} - 21q^{21} - 12q^{23} + 25q^{25} + 27q^{27} - 258q^{29} - 146q^{31} - 126q^{33} - 35q^{35} + 434q^{37} + 60q^{39} - 282q^{41} - 20q^{43} + 45q^{45} + 72q^{47} + 49q^{49} + 198q^{51} + 336q^{53} - 210q^{55} - 114q^{57} + 360q^{59} - 682q^{61} - 63q^{63} + 100q^{65} - 812q^{67} - 36q^{69} - 810q^{71} - 124q^{73} + 75q^{75} + 294q^{77} - 1136q^{79} + 81q^{81} - 156q^{83} + 330q^{85} - 774q^{87} - 1038q^{89} - 140q^{91} - 438q^{93} - 190q^{95} + 1208q^{97} - 378q^{99} + O(q^{100}) \)

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 5.00000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1680.4.a.s 1
4.b odd 2 1 105.4.a.a 1
12.b even 2 1 315.4.a.d 1
20.d odd 2 1 525.4.a.e 1
20.e even 4 2 525.4.d.f 2
28.d even 2 1 735.4.a.c 1
60.h even 2 1 1575.4.a.f 1
84.h odd 2 1 2205.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 4.b odd 2 1
315.4.a.d 1 12.b even 2 1
525.4.a.e 1 20.d odd 2 1
525.4.d.f 2 20.e even 4 2
735.4.a.c 1 28.d even 2 1
1575.4.a.f 1 60.h even 2 1
1680.4.a.s 1 1.a even 1 1 trivial
2205.4.a.o 1 84.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11} + 42 \)
\( T_{13} - 20 \)
\( T_{17} - 66 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T \)
$5$ \( 1 - 5 T \)
$7$ \( 1 + 7 T \)
$11$ \( 1 + 42 T + 1331 T^{2} \)
$13$ \( 1 - 20 T + 2197 T^{2} \)
$17$ \( 1 - 66 T + 4913 T^{2} \)
$19$ \( 1 + 38 T + 6859 T^{2} \)
$23$ \( 1 + 12 T + 12167 T^{2} \)
$29$ \( 1 + 258 T + 24389 T^{2} \)
$31$ \( 1 + 146 T + 29791 T^{2} \)
$37$ \( 1 - 434 T + 50653 T^{2} \)
$41$ \( 1 + 282 T + 68921 T^{2} \)
$43$ \( 1 + 20 T + 79507 T^{2} \)
$47$ \( 1 - 72 T + 103823 T^{2} \)
$53$ \( 1 - 336 T + 148877 T^{2} \)
$59$ \( 1 - 360 T + 205379 T^{2} \)
$61$ \( 1 + 682 T + 226981 T^{2} \)
$67$ \( 1 + 812 T + 300763 T^{2} \)
$71$ \( 1 + 810 T + 357911 T^{2} \)
$73$ \( 1 + 124 T + 389017 T^{2} \)
$79$ \( 1 + 1136 T + 493039 T^{2} \)
$83$ \( 1 + 156 T + 571787 T^{2} \)
$89$ \( 1 + 1038 T + 704969 T^{2} \)
$97$ \( 1 - 1208 T + 912673 T^{2} \)
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