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

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{3} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

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