Properties

Label 49.6.a.c
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{39}) \)
Defining polynomial: \( x^{2} - 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta q^{3} - 28 q^{4} + 3 \beta q^{5} - 2 \beta q^{6} + 120 q^{8} + 381 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta q^{3} - 28 q^{4} + 3 \beta q^{5} - 2 \beta q^{6} + 120 q^{8} + 381 q^{9} - 6 \beta q^{10} - 284 q^{11} - 28 \beta q^{12} + 21 \beta q^{13} + 1872 q^{15} + 656 q^{16} - 6 \beta q^{17} - 762 q^{18} + 87 \beta q^{19} - 84 \beta q^{20} + 568 q^{22} + 1496 q^{23} + 120 \beta q^{24} + 2491 q^{25} - 42 \beta q^{26} + 138 \beta q^{27} - 4366 q^{29} - 3744 q^{30} - 258 \beta q^{31} - 5152 q^{32} - 284 \beta q^{33} + 12 \beta q^{34} - 10668 q^{36} - 12630 q^{37} - 174 \beta q^{38} + 13104 q^{39} + 360 \beta q^{40} - 378 \beta q^{41} - 1356 q^{43} + 7952 q^{44} + 1143 \beta q^{45} - 2992 q^{46} + 402 \beta q^{47} + 656 \beta q^{48} - 4982 q^{50} - 3744 q^{51} - 588 \beta q^{52} + 14150 q^{53} - 276 \beta q^{54} - 852 \beta q^{55} + 54288 q^{57} + 8732 q^{58} - 1497 \beta q^{59} - 52416 q^{60} - 1425 \beta q^{61} + 516 \beta q^{62} - 10688 q^{64} + 39312 q^{65} + 568 \beta q^{66} - 3644 q^{67} + 168 \beta q^{68} + 1496 \beta q^{69} + 35632 q^{71} + 45720 q^{72} + 1632 \beta q^{73} + 25260 q^{74} + 2491 \beta q^{75} - 2436 \beta q^{76} - 26208 q^{78} - 54616 q^{79} + 1968 \beta q^{80} - 6471 q^{81} + 756 \beta q^{82} - 21 \beta q^{83} - 11232 q^{85} + 2712 q^{86} - 4366 \beta q^{87} - 34080 q^{88} - 816 \beta q^{89} - 2286 \beta q^{90} - 41888 q^{92} - 160992 q^{93} - 804 \beta q^{94} + 162864 q^{95} - 5152 \beta q^{96} + 7350 \beta q^{97} - 108204 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9} - 568 q^{11} + 3744 q^{15} + 1312 q^{16} - 1524 q^{18} + 1136 q^{22} + 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 7488 q^{30} - 10304 q^{32} - 21336 q^{36} - 25260 q^{37} + 26208 q^{39} - 2712 q^{43} + 15904 q^{44} - 5984 q^{46} - 9964 q^{50} - 7488 q^{51} + 28300 q^{53} + 108576 q^{57} + 17464 q^{58} - 104832 q^{60} - 21376 q^{64} + 78624 q^{65} - 7288 q^{67} + 71264 q^{71} + 91440 q^{72} + 50520 q^{74} - 52416 q^{78} - 109232 q^{79} - 12942 q^{81} - 22464 q^{85} + 5424 q^{86} - 68160 q^{88} - 83776 q^{92} - 321984 q^{93} + 325728 q^{95} - 216408 q^{99}+O(q^{100}) \) 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
−6.24500
6.24500
−2.00000 −24.9800 −28.0000 −74.9400 49.9600 0 120.000 381.000 149.880
1.2 −2.00000 24.9800 −28.0000 74.9400 −49.9600 0 120.000 381.000 −149.880
\(n\): 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 49.6.a.c 2
3.b odd 2 1 441.6.a.u 2
4.b odd 2 1 784.6.a.z 2
7.b odd 2 1 inner 49.6.a.c 2
7.c even 3 2 49.6.c.g 4
7.d odd 6 2 49.6.c.g 4
21.c even 2 1 441.6.a.u 2
28.d even 2 1 784.6.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 1.a even 1 1 trivial
49.6.a.c 2 7.b odd 2 1 inner
49.6.c.g 4 7.c even 3 2
49.6.c.g 4 7.d odd 6 2
441.6.a.u 2 3.b odd 2 1
441.6.a.u 2 21.c even 2 1
784.6.a.z 2 4.b odd 2 1
784.6.a.z 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{2} - 624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 624 \) Copy content Toggle raw display
$5$ \( T^{2} - 5616 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 284)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 275184 \) Copy content Toggle raw display
$17$ \( T^{2} - 22464 \) Copy content Toggle raw display
$19$ \( T^{2} - 4723056 \) Copy content Toggle raw display
$23$ \( (T - 1496)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4366)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 41535936 \) Copy content Toggle raw display
$37$ \( (T + 12630)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 89159616 \) Copy content Toggle raw display
$43$ \( (T + 1356)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 100840896 \) Copy content Toggle raw display
$53$ \( (T - 14150)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 1398389616 \) Copy content Toggle raw display
$61$ \( T^{2} - 1267110000 \) Copy content Toggle raw display
$67$ \( (T + 3644)^{2} \) Copy content Toggle raw display
$71$ \( (T - 35632)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1661976576 \) Copy content Toggle raw display
$79$ \( (T + 54616)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 275184 \) Copy content Toggle raw display
$89$ \( T^{2} - 415494144 \) Copy content Toggle raw display
$97$ \( T^{2} - 33710040000 \) Copy content Toggle raw display
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