Properties

Label 49.4.a.b
Level $49$
Weight $4$
Character orbit 49.a
Self dual yes
Analytic conductor $2.891$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89109359028\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 2q^{3} - 7q^{4} - 16q^{5} - 2q^{6} + 15q^{8} - 23q^{9} + O(q^{10}) \) \( q - q^{2} + 2q^{3} - 7q^{4} - 16q^{5} - 2q^{6} + 15q^{8} - 23q^{9} + 16q^{10} - 8q^{11} - 14q^{12} - 28q^{13} - 32q^{15} + 41q^{16} - 54q^{17} + 23q^{18} + 110q^{19} + 112q^{20} + 8q^{22} + 48q^{23} + 30q^{24} + 131q^{25} + 28q^{26} - 100q^{27} - 110q^{29} + 32q^{30} - 12q^{31} - 161q^{32} - 16q^{33} + 54q^{34} + 161q^{36} - 246q^{37} - 110q^{38} - 56q^{39} - 240q^{40} - 182q^{41} + 128q^{43} + 56q^{44} + 368q^{45} - 48q^{46} - 324q^{47} + 82q^{48} - 131q^{50} - 108q^{51} + 196q^{52} - 162q^{53} + 100q^{54} + 128q^{55} + 220q^{57} + 110q^{58} - 810q^{59} + 224q^{60} + 488q^{61} + 12q^{62} - 167q^{64} + 448q^{65} + 16q^{66} + 244q^{67} + 378q^{68} + 96q^{69} - 768q^{71} - 345q^{72} + 702q^{73} + 246q^{74} + 262q^{75} - 770q^{76} + 56q^{78} + 440q^{79} - 656q^{80} + 421q^{81} + 182q^{82} + 1302q^{83} + 864q^{85} - 128q^{86} - 220q^{87} - 120q^{88} - 730q^{89} - 368q^{90} - 336q^{92} - 24q^{93} + 324q^{94} - 1760q^{95} - 322q^{96} - 294q^{97} + 184q^{99} + O(q^{100}) \)

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
−1.00000 2.00000 −7.00000 −16.0000 −2.00000 0 15.0000 −23.0000 16.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.4.a.b 1
3.b odd 2 1 441.4.a.i 1
4.b odd 2 1 784.4.a.g 1
5.b even 2 1 1225.4.a.j 1
7.b odd 2 1 7.4.a.a 1
7.c even 3 2 49.4.c.b 2
7.d odd 6 2 49.4.c.c 2
21.c even 2 1 63.4.a.b 1
21.g even 6 2 441.4.e.h 2
21.h odd 6 2 441.4.e.e 2
28.d even 2 1 112.4.a.f 1
35.c odd 2 1 175.4.a.b 1
35.f even 4 2 175.4.b.b 2
56.e even 2 1 448.4.a.e 1
56.h odd 2 1 448.4.a.i 1
77.b even 2 1 847.4.a.b 1
84.h odd 2 1 1008.4.a.c 1
91.b odd 2 1 1183.4.a.b 1
105.g even 2 1 1575.4.a.e 1
119.d odd 2 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 7.b odd 2 1
49.4.a.b 1 1.a even 1 1 trivial
49.4.c.b 2 7.c even 3 2
49.4.c.c 2 7.d odd 6 2
63.4.a.b 1 21.c even 2 1
112.4.a.f 1 28.d even 2 1
175.4.a.b 1 35.c odd 2 1
175.4.b.b 2 35.f even 4 2
441.4.a.i 1 3.b odd 2 1
441.4.e.e 2 21.h odd 6 2
441.4.e.h 2 21.g even 6 2
448.4.a.e 1 56.e even 2 1
448.4.a.i 1 56.h odd 2 1
784.4.a.g 1 4.b odd 2 1
847.4.a.b 1 77.b even 2 1
1008.4.a.c 1 84.h odd 2 1
1183.4.a.b 1 91.b odd 2 1
1225.4.a.j 1 5.b even 2 1
1575.4.a.e 1 105.g even 2 1
2023.4.a.a 1 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} + 1 \)
\( T_{3} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -2 + T \)
$5$ \( 16 + T \)
$7$ \( T \)
$11$ \( 8 + T \)
$13$ \( 28 + T \)
$17$ \( 54 + T \)
$19$ \( -110 + T \)
$23$ \( -48 + T \)
$29$ \( 110 + T \)
$31$ \( 12 + T \)
$37$ \( 246 + T \)
$41$ \( 182 + T \)
$43$ \( -128 + T \)
$47$ \( 324 + T \)
$53$ \( 162 + T \)
$59$ \( 810 + T \)
$61$ \( -488 + T \)
$67$ \( -244 + T \)
$71$ \( 768 + T \)
$73$ \( -702 + T \)
$79$ \( -440 + T \)
$83$ \( -1302 + T \)
$89$ \( 730 + T \)
$97$ \( 294 + T \)
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