Properties

Label 49.12.a.a
Level 49
Weight 12
Character orbit 49.a
Self dual yes
Analytic conductor 37.649
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 49 = 7^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.6488158474\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 24q^{2} - 252q^{3} - 1472q^{4} - 4830q^{5} + 6048q^{6} + 84480q^{8} - 113643q^{9} + O(q^{10}) \) \( q - 24q^{2} - 252q^{3} - 1472q^{4} - 4830q^{5} + 6048q^{6} + 84480q^{8} - 113643q^{9} + 115920q^{10} + 534612q^{11} + 370944q^{12} + 577738q^{13} + 1217160q^{15} + 987136q^{16} + 6905934q^{17} + 2727432q^{18} - 10661420q^{19} + 7109760q^{20} - 12830688q^{22} + 18643272q^{23} - 21288960q^{24} - 25499225q^{25} - 13865712q^{26} + 73279080q^{27} + 128406630q^{29} - 29211840q^{30} + 52843168q^{31} - 196706304q^{32} - 134722224q^{33} - 165742416q^{34} + 167282496q^{36} - 182213314q^{37} + 255874080q^{38} - 145589976q^{39} - 408038400q^{40} - 308120442q^{41} - 17125708q^{43} - 786948864q^{44} + 548895690q^{45} - 447438528q^{46} - 2687348496q^{47} - 248758272q^{48} + 611981400q^{50} - 1740295368q^{51} - 850430336q^{52} - 1596055698q^{53} - 1758697920q^{54} - 2582175960q^{55} + 2686677840q^{57} - 3081759120q^{58} + 5189203740q^{59} - 1791659520q^{60} - 6956478662q^{61} - 1268236032q^{62} + 2699296768q^{64} - 2790474540q^{65} + 3233333376q^{66} - 15481826884q^{67} - 10165534848q^{68} - 4698104544q^{69} + 9791485272q^{71} - 9600560640q^{72} - 1463791322q^{73} + 4373119536q^{74} + 6425804700q^{75} + 15693610240q^{76} + 3494159424q^{78} + 38116845680q^{79} - 4767866880q^{80} + 1665188361q^{81} + 7394890608q^{82} + 29335099668q^{83} - 33355661220q^{85} + 411016992q^{86} - 32358470760q^{87} + 45164021760q^{88} + 24992917110q^{89} - 13173496560q^{90} - 27442896384q^{92} - 13316478336q^{93} + 64496363904q^{94} + 51494658600q^{95} + 49569988608q^{96} - 75013568546q^{97} - 60754911516q^{99} + O(q^{100}) \)

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
−24.0000 −252.000 −1472.00 −4830.00 6048.00 0 84480.0 −113643. 115920.
\(n\): 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 49.12.a.a 1
7.b odd 2 1 1.12.a.a 1
7.c even 3 2 49.12.c.c 2
7.d odd 6 2 49.12.c.b 2
21.c even 2 1 9.12.a.b 1
28.d even 2 1 16.12.a.a 1
35.c odd 2 1 25.12.a.b 1
35.f even 4 2 25.12.b.b 2
56.e even 2 1 64.12.a.f 1
56.h odd 2 1 64.12.a.b 1
63.l odd 6 2 81.12.c.d 2
63.o even 6 2 81.12.c.b 2
77.b even 2 1 121.12.a.b 1
84.h odd 2 1 144.12.a.d 1
91.b odd 2 1 169.12.a.a 1
105.g even 2 1 225.12.a.b 1
105.k odd 4 2 225.12.b.d 2
112.j even 4 2 256.12.b.c 2
112.l odd 4 2 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 7.b odd 2 1
9.12.a.b 1 21.c even 2 1
16.12.a.a 1 28.d even 2 1
25.12.a.b 1 35.c odd 2 1
25.12.b.b 2 35.f even 4 2
49.12.a.a 1 1.a even 1 1 trivial
49.12.c.b 2 7.d odd 6 2
49.12.c.c 2 7.c even 3 2
64.12.a.b 1 56.h odd 2 1
64.12.a.f 1 56.e even 2 1
81.12.c.b 2 63.o even 6 2
81.12.c.d 2 63.l odd 6 2
121.12.a.b 1 77.b even 2 1
144.12.a.d 1 84.h odd 2 1
169.12.a.a 1 91.b odd 2 1
225.12.a.b 1 105.g even 2 1
225.12.b.d 2 105.k odd 4 2
256.12.b.c 2 112.j even 4 2
256.12.b.e 2 112.l odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} + 24 \)
\( T_{3} + 252 \)