Properties

Label 49.4.a.d
Level $49$
Weight $4$
Character orbit 49.a
Self dual yes
Analytic conductor $2.891$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 7q^{3} - 4q^{4} + 7q^{5} + 14q^{6} - 24q^{8} + 22q^{9} + O(q^{10}) \) \( q + 2q^{2} + 7q^{3} - 4q^{4} + 7q^{5} + 14q^{6} - 24q^{8} + 22q^{9} + 14q^{10} - 5q^{11} - 28q^{12} - 14q^{13} + 49q^{15} - 16q^{16} - 21q^{17} + 44q^{18} + 49q^{19} - 28q^{20} - 10q^{22} - 159q^{23} - 168q^{24} - 76q^{25} - 28q^{26} - 35q^{27} + 58q^{29} + 98q^{30} + 147q^{31} + 160q^{32} - 35q^{33} - 42q^{34} - 88q^{36} + 219q^{37} + 98q^{38} - 98q^{39} - 168q^{40} + 350q^{41} - 124q^{43} + 20q^{44} + 154q^{45} - 318q^{46} + 525q^{47} - 112q^{48} - 152q^{50} - 147q^{51} + 56q^{52} + 303q^{53} - 70q^{54} - 35q^{55} + 343q^{57} + 116q^{58} - 105q^{59} - 196q^{60} - 413q^{61} + 294q^{62} + 448q^{64} - 98q^{65} - 70q^{66} + 415q^{67} + 84q^{68} - 1113q^{69} - 432q^{71} - 528q^{72} - 1113q^{73} + 438q^{74} - 532q^{75} - 196q^{76} - 196q^{78} - 103q^{79} - 112q^{80} - 839q^{81} + 700q^{82} + 1092q^{83} - 147q^{85} - 248q^{86} + 406q^{87} + 120q^{88} - 329q^{89} + 308q^{90} + 636q^{92} + 1029q^{93} + 1050q^{94} + 343q^{95} + 1120q^{96} - 882q^{97} - 110q^{99} + O(q^{100}) \)

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
2.00000 7.00000 −4.00000 7.00000 14.0000 0 −24.0000 22.0000 14.0000
\(n\): 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.d 1
3.b odd 2 1 441.4.a.d 1
4.b odd 2 1 784.4.a.b 1
5.b even 2 1 1225.4.a.c 1
7.b odd 2 1 49.4.a.c 1
7.c even 3 2 7.4.c.a 2
7.d odd 6 2 49.4.c.a 2
21.c even 2 1 441.4.a.e 1
21.g even 6 2 441.4.e.k 2
21.h odd 6 2 63.4.e.b 2
28.d even 2 1 784.4.a.r 1
28.g odd 6 2 112.4.i.c 2
35.c odd 2 1 1225.4.a.d 1
35.j even 6 2 175.4.e.a 2
35.l odd 12 4 175.4.k.a 4
56.k odd 6 2 448.4.i.a 2
56.p even 6 2 448.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 7.c even 3 2
49.4.a.c 1 7.b odd 2 1
49.4.a.d 1 1.a even 1 1 trivial
49.4.c.a 2 7.d odd 6 2
63.4.e.b 2 21.h odd 6 2
112.4.i.c 2 28.g odd 6 2
175.4.e.a 2 35.j even 6 2
175.4.k.a 4 35.l odd 12 4
441.4.a.d 1 3.b odd 2 1
441.4.a.e 1 21.c even 2 1
441.4.e.k 2 21.g even 6 2
448.4.i.a 2 56.k odd 6 2
448.4.i.f 2 56.p even 6 2
784.4.a.b 1 4.b odd 2 1
784.4.a.r 1 28.d even 2 1
1225.4.a.c 1 5.b even 2 1
1225.4.a.d 1 35.c odd 2 1

Hecke kernels

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

\( T_{2} - 2 \)
\( T_{3} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -7 + T \)
$5$ \( -7 + T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( 14 + T \)
$17$ \( 21 + T \)
$19$ \( -49 + T \)
$23$ \( 159 + T \)
$29$ \( -58 + T \)
$31$ \( -147 + T \)
$37$ \( -219 + T \)
$41$ \( -350 + T \)
$43$ \( 124 + T \)
$47$ \( -525 + T \)
$53$ \( -303 + T \)
$59$ \( 105 + T \)
$61$ \( 413 + T \)
$67$ \( -415 + T \)
$71$ \( 432 + T \)
$73$ \( 1113 + T \)
$79$ \( 103 + T \)
$83$ \( -1092 + T \)
$89$ \( 329 + T \)
$97$ \( 882 + T \)
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