Properties

Label 2178.4.a.e
Level 2178
Weight 4
Character orbit 2178.a
Self dual yes
Analytic conductor 128.506
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(128.506159993\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} - 6q^{5} + 16q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} - 6q^{5} + 16q^{7} - 8q^{8} + 12q^{10} - 38q^{13} - 32q^{14} + 16q^{16} - 126q^{17} - 20q^{19} - 24q^{20} - 168q^{23} - 89q^{25} + 76q^{26} + 64q^{28} + 30q^{29} - 88q^{31} - 32q^{32} + 252q^{34} - 96q^{35} + 254q^{37} + 40q^{38} + 48q^{40} + 42q^{41} + 52q^{43} + 336q^{46} + 96q^{47} - 87q^{49} + 178q^{50} - 152q^{52} - 198q^{53} - 128q^{56} - 60q^{58} + 660q^{59} + 538q^{61} + 176q^{62} + 64q^{64} + 228q^{65} + 884q^{67} - 504q^{68} + 192q^{70} - 792q^{71} - 218q^{73} - 508q^{74} - 80q^{76} + 520q^{79} - 96q^{80} - 84q^{82} - 492q^{83} + 756q^{85} - 104q^{86} - 810q^{89} - 608q^{91} - 672q^{92} - 192q^{94} + 120q^{95} + 1154q^{97} + 174q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 −6.00000 0 16.0000 −8.00000 0 12.0000
\(n\): 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 2178.4.a.e 1
3.b odd 2 1 726.4.a.f 1
11.b odd 2 1 18.4.a.a 1
33.d even 2 1 6.4.a.a 1
44.c even 2 1 144.4.a.c 1
55.d odd 2 1 450.4.a.h 1
55.e even 4 2 450.4.c.e 2
77.b even 2 1 882.4.a.n 1
77.h odd 6 2 882.4.g.i 2
77.i even 6 2 882.4.g.f 2
88.b odd 2 1 576.4.a.q 1
88.g even 2 1 576.4.a.r 1
99.g even 6 2 162.4.c.f 2
99.h odd 6 2 162.4.c.c 2
132.d odd 2 1 48.4.a.c 1
165.d even 2 1 150.4.a.i 1
165.l odd 4 2 150.4.c.d 2
231.h odd 2 1 294.4.a.e 1
231.k odd 6 2 294.4.e.g 2
231.l even 6 2 294.4.e.h 2
264.m even 2 1 192.4.a.i 1
264.p odd 2 1 192.4.a.c 1
429.e even 2 1 1014.4.a.g 1
429.l odd 4 2 1014.4.b.d 2
528.s odd 4 2 768.4.d.c 2
528.x even 4 2 768.4.d.n 2
561.h even 2 1 1734.4.a.d 1
627.b odd 2 1 2166.4.a.i 1
660.g odd 2 1 1200.4.a.b 1
660.q even 4 2 1200.4.f.j 2
924.n even 2 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 33.d even 2 1
18.4.a.a 1 11.b odd 2 1
48.4.a.c 1 132.d odd 2 1
144.4.a.c 1 44.c even 2 1
150.4.a.i 1 165.d even 2 1
150.4.c.d 2 165.l odd 4 2
162.4.c.c 2 99.h odd 6 2
162.4.c.f 2 99.g even 6 2
192.4.a.c 1 264.p odd 2 1
192.4.a.i 1 264.m even 2 1
294.4.a.e 1 231.h odd 2 1
294.4.e.g 2 231.k odd 6 2
294.4.e.h 2 231.l even 6 2
450.4.a.h 1 55.d odd 2 1
450.4.c.e 2 55.e even 4 2
576.4.a.q 1 88.b odd 2 1
576.4.a.r 1 88.g even 2 1
726.4.a.f 1 3.b odd 2 1
768.4.d.c 2 528.s odd 4 2
768.4.d.n 2 528.x even 4 2
882.4.a.n 1 77.b even 2 1
882.4.g.f 2 77.i even 6 2
882.4.g.i 2 77.h odd 6 2
1014.4.a.g 1 429.e even 2 1
1014.4.b.d 2 429.l odd 4 2
1200.4.a.b 1 660.g odd 2 1
1200.4.f.j 2 660.q even 4 2
1734.4.a.d 1 561.h even 2 1
2166.4.a.i 1 627.b odd 2 1
2178.4.a.e 1 1.a even 1 1 trivial
2352.4.a.e 1 924.n even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5} + 6 \)
\( T_{7} - 16 \)
\( T_{17} + 126 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ 1
$5$ \( 1 + 6 T + 125 T^{2} \)
$7$ \( 1 - 16 T + 343 T^{2} \)
$11$ 1
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 + 126 T + 4913 T^{2} \)
$19$ \( 1 + 20 T + 6859 T^{2} \)
$23$ \( 1 + 168 T + 12167 T^{2} \)
$29$ \( 1 - 30 T + 24389 T^{2} \)
$31$ \( 1 + 88 T + 29791 T^{2} \)
$37$ \( 1 - 254 T + 50653 T^{2} \)
$41$ \( 1 - 42 T + 68921 T^{2} \)
$43$ \( 1 - 52 T + 79507 T^{2} \)
$47$ \( 1 - 96 T + 103823 T^{2} \)
$53$ \( 1 + 198 T + 148877 T^{2} \)
$59$ \( 1 - 660 T + 205379 T^{2} \)
$61$ \( 1 - 538 T + 226981 T^{2} \)
$67$ \( 1 - 884 T + 300763 T^{2} \)
$71$ \( 1 + 792 T + 357911 T^{2} \)
$73$ \( 1 + 218 T + 389017 T^{2} \)
$79$ \( 1 - 520 T + 493039 T^{2} \)
$83$ \( 1 + 492 T + 571787 T^{2} \)
$89$ \( 1 + 810 T + 704969 T^{2} \)
$97$ \( 1 - 1154 T + 912673 T^{2} \)
show more
show less