Properties

Label 49.4.a.e
Level $49$
Weight $4$
Character orbit 49.a
Self dual yes
Analytic conductor $2.891$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 9 + \beta_{1} ) q^{4} -\beta_{3} q^{5} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 17 + \beta_{1} ) q^{8} + ( 7 - 6 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 9 + \beta_{1} ) q^{4} -\beta_{3} q^{5} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 17 + \beta_{1} ) q^{8} + ( 7 - 6 \beta_{1} ) q^{9} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{10} + ( 22 - 6 \beta_{1} ) q^{11} + ( 5 \beta_{2} + 2 \beta_{3} ) q^{12} + ( 6 \beta_{2} + \beta_{3} ) q^{13} + ( -20 - 8 \beta_{1} ) q^{15} + ( -39 + 9 \beta_{1} ) q^{16} + ( -\beta_{2} + 9 \beta_{3} ) q^{17} + ( -89 + 7 \beta_{1} ) q^{18} + ( -11 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -2 \beta_{2} - 12 \beta_{3} ) q^{20} + ( -74 + 22 \beta_{1} ) q^{22} + ( 92 + 8 \beta_{1} ) q^{23} + ( 13 \beta_{2} + 2 \beta_{3} ) q^{24} + ( -21 + 22 \beta_{1} ) q^{25} + ( -16 \beta_{2} + 16 \beta_{3} ) q^{26} + ( 4 \beta_{2} - 12 \beta_{3} ) q^{27} + ( 58 - 14 \beta_{1} ) q^{29} + ( -148 - 20 \beta_{1} ) q^{30} + ( -38 \beta_{2} + 4 \beta_{3} ) q^{31} + ( -31 - 47 \beta_{1} ) q^{32} + ( 46 \beta_{2} - 12 \beta_{3} ) q^{33} + ( 21 \beta_{2} + 34 \beta_{3} ) q^{34} + ( -33 - 41 \beta_{1} ) q^{36} + ( 50 - 6 \beta_{1} ) q^{37} + ( 25 \beta_{2} - 38 \beta_{3} ) q^{38} + ( 224 - 28 \beta_{1} ) q^{39} + ( -2 \beta_{2} - 20 \beta_{3} ) q^{40} + ( -31 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 170 + 70 \beta_{1} ) q^{43} + ( 102 - 26 \beta_{1} ) q^{44} + ( 12 \beta_{2} + 11 \beta_{3} ) q^{45} + ( 220 + 92 \beta_{1} ) q^{46} + ( -26 \beta_{2} + 32 \beta_{3} ) q^{47} + ( -75 \beta_{2} + 18 \beta_{3} ) q^{48} + ( 331 - 21 \beta_{1} ) q^{50} + ( 146 + 78 \beta_{1} ) q^{51} + ( 32 \beta_{2} + 24 \beta_{3} ) q^{52} + ( 22 + 36 \beta_{1} ) q^{53} + ( -36 \beta_{2} - 40 \beta_{3} ) q^{54} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -454 + 34 \beta_{1} ) q^{57} + ( -166 + 58 \beta_{1} ) q^{58} + ( 49 \beta_{2} - 20 \beta_{3} ) q^{59} + ( -308 - 84 \beta_{1} ) q^{60} + ( 68 \beta_{2} - 27 \beta_{3} ) q^{61} + ( 122 \beta_{2} - 60 \beta_{3} ) q^{62} + ( -471 - 103 \beta_{1} ) q^{64} + ( -224 - 70 \beta_{1} ) q^{65} + ( -162 \beta_{2} + 44 \beta_{3} ) q^{66} + ( -524 - 76 \beta_{1} ) q^{67} + ( 13 \beta_{2} + 106 \beta_{3} ) q^{68} + ( 60 \beta_{2} + 16 \beta_{3} ) q^{69} + ( 548 - 28 \beta_{1} ) q^{71} + ( 23 - 89 \beta_{1} ) q^{72} + ( -37 \beta_{2} - 25 \beta_{3} ) q^{73} + ( -46 + 50 \beta_{1} ) q^{74} + ( -109 \beta_{2} + 44 \beta_{3} ) q^{75} + ( -63 \beta_{2} - 70 \beta_{3} ) q^{76} + ( -224 + 224 \beta_{1} ) q^{78} + ( -356 - 188 \beta_{1} ) q^{79} + ( -18 \beta_{2} + 12 \beta_{3} ) q^{80} + ( -293 + 42 \beta_{1} ) q^{81} + ( 99 \beta_{2} - 50 \beta_{3} ) q^{82} + ( \beta_{2} - 8 \beta_{3} ) q^{83} + ( -916 - 190 \beta_{1} ) q^{85} + ( 1290 + 170 \beta_{1} ) q^{86} + ( 114 \beta_{2} - 28 \beta_{3} ) q^{87} + ( 278 - 74 \beta_{1} ) q^{88} + ( -157 \beta_{2} - 75 \beta_{3} ) q^{89} + ( -14 \beta_{2} + 68 \beta_{3} ) q^{90} + ( 956 + 156 \beta_{1} ) q^{92} + ( -1212 + 260 \beta_{1} ) q^{93} + ( 142 \beta_{2} + 76 \beta_{3} ) q^{94} + ( 636 + 176 \beta_{1} ) q^{95} + ( 157 \beta_{2} - 94 \beta_{3} ) q^{96} + ( -189 \beta_{2} + 91 \beta_{3} ) q^{97} + ( 730 - 210 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + 100q^{11} - 64q^{15} - 174q^{16} - 370q^{18} - 340q^{22} + 352q^{23} - 128q^{25} + 260q^{29} - 552q^{30} - 30q^{32} - 50q^{36} + 212q^{37} + 952q^{39} + 540q^{43} + 460q^{44} + 696q^{46} + 1366q^{50} + 428q^{51} + 16q^{53} - 1884q^{57} - 780q^{58} - 1064q^{60} - 1678q^{64} - 756q^{65} - 1944q^{67} + 2248q^{71} + 270q^{72} - 284q^{74} - 1344q^{78} - 1048q^{79} - 1256q^{81} - 3284q^{85} + 4820q^{86} + 1260q^{88} + 3512q^{92} - 5368q^{93} + 2192q^{95} + 3340q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 100 \nu - 79 \)\()/57\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 11 \nu^{2} + 12 \nu - 182 \)\()/19\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{3} + 12 \nu^{2} - 265 \nu - 392 \)\()/57\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 7 \beta_{1} + 7\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{3} + 10 \beta_{2} + 7 \beta_{1} + 133\)\()/7\)
\(\nu^{3}\)\(=\)\(8 \beta_{3} - 5 \beta_{2} + 23 \beta_{1} + 39\)

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
−2.11692
−4.94534
3.11692
5.94534
−3.53113 −7.82220 4.46887 −2.07730 27.6212 0 12.4689 34.1868 7.33521
1.2 −3.53113 7.82220 4.46887 2.07730 −27.6212 0 12.4689 34.1868 −7.33521
1.3 4.53113 −3.57956 12.5311 13.4791 −16.2194 0 20.5311 −14.1868 61.0753
1.4 4.53113 3.57956 12.5311 −13.4791 16.2194 0 20.5311 −14.1868 −61.0753
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.a.e 4
3.b odd 2 1 441.4.a.u 4
4.b odd 2 1 784.4.a.bf 4
5.b even 2 1 1225.4.a.bb 4
7.b odd 2 1 inner 49.4.a.e 4
7.c even 3 2 49.4.c.e 8
7.d odd 6 2 49.4.c.e 8
21.c even 2 1 441.4.a.u 4
21.g even 6 2 441.4.e.y 8
21.h odd 6 2 441.4.e.y 8
28.d even 2 1 784.4.a.bf 4
35.c odd 2 1 1225.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 1.a even 1 1 trivial
49.4.a.e 4 7.b odd 2 1 inner
49.4.c.e 8 7.c even 3 2
49.4.c.e 8 7.d odd 6 2
441.4.a.u 4 3.b odd 2 1
441.4.a.u 4 21.c even 2 1
441.4.e.y 8 21.g even 6 2
441.4.e.y 8 21.h odd 6 2
784.4.a.bf 4 4.b odd 2 1
784.4.a.bf 4 28.d even 2 1
1225.4.a.bb 4 5.b even 2 1
1225.4.a.bb 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} - T_{2} - 16 \)
\( T_{3}^{4} - 74 T_{3}^{2} + 784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -16 - T + T^{2} )^{2} \)
$3$ \( 784 - 74 T^{2} + T^{4} \)
$5$ \( 784 - 186 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 40 - 50 T + T^{2} )^{2} \)
$13$ \( 2458624 - 3234 T^{2} + T^{4} \)
$17$ \( 9746884 - 14564 T^{2} + T^{4} \)
$19$ \( 52591504 - 14746 T^{2} + T^{4} \)
$23$ \( ( 6704 - 176 T + T^{2} )^{2} \)
$29$ \( ( 1040 - 130 T + T^{2} )^{2} \)
$31$ \( 629407744 - 100104 T^{2} + T^{4} \)
$37$ \( ( 2224 - 106 T + T^{2} )^{2} \)
$41$ \( 307721764 - 66836 T^{2} + T^{4} \)
$43$ \( ( -61400 - 270 T + T^{2} )^{2} \)
$47$ \( 8332038400 - 187240 T^{2} + T^{4} \)
$53$ \( ( -21044 - 8 T + T^{2} )^{2} \)
$59$ \( 1600960144 - 189354 T^{2} + T^{4} \)
$61$ \( 5022273424 - 360266 T^{2} + T^{4} \)
$67$ \( ( 142336 + 972 T + T^{2} )^{2} \)
$71$ \( ( 303104 - 1124 T + T^{2} )^{2} \)
$73$ \( 12428236324 - 276756 T^{2} + T^{4} \)
$79$ \( ( -505696 + 524 T + T^{2} )^{2} \)
$83$ \( 6492304 - 11466 T^{2} + T^{4} \)
$89$ \( 2844693770884 - 3623876 T^{2} + T^{4} \)
$97$ \( 841222821124 - 3082884 T^{2} + T^{4} \)
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