Properties

Label 49.6.a.g
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + ( - \beta_{3} + 2 \beta_{2}) q^{3} - 5 \beta_1 q^{4} + (2 \beta_{3} - 8 \beta_{2}) q^{5} + (11 \beta_{3} - 11 \beta_{2}) q^{6} + ( - 17 \beta_1 - 76) q^{8} + ( - 48 \beta_1 + 31) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + ( - \beta_{3} + 2 \beta_{2}) q^{3} - 5 \beta_1 q^{4} + (2 \beta_{3} - 8 \beta_{2}) q^{5} + (11 \beta_{3} - 11 \beta_{2}) q^{6} + ( - 17 \beta_1 - 76) q^{8} + ( - 48 \beta_1 + 31) q^{9} + ( - 38 \beta_{3} + 30 \beta_{2}) q^{10} + ( - 12 \beta_1 - 494) q^{11} + ( - 45 \beta_{3} + 35 \beta_{2}) q^{12} + (64 \beta_{3} + 56 \beta_{2}) q^{13} + (136 \beta_1 - 956) q^{15} + (135 \beta_1 - 324) q^{16} + (7 \beta_{3} - 152 \beta_{2}) q^{17} + (175 \beta_1 - 1406) q^{18} + ( - 53 \beta_{3} - 78 \beta_{2}) q^{19} + (170 \beta_{3} - 70 \beta_{2}) q^{20} + ( - 458 \beta_1 + 652) q^{22} + (344 \beta_1 - 1612) q^{23} + ( - 77 \beta_{3} - 33 \beta_{2}) q^{24} + ( - 320 \beta_1 + 531) q^{25} + (32 \beta_{3} + 336 \beta_{2}) q^{26} + ( - 220 \beta_{3} - 88 \beta_{2}) q^{27} + (784 \beta_1 - 446) q^{29} + ( - 1364 \beta_1 + 5720) q^{30} + (246 \beta_{3} + 772 \beta_{2}) q^{31} + ( - 185 \beta_1 + 6860) q^{32} + (386 \beta_{3} - 904 \beta_{2}) q^{33} + ( - 629 \beta_{3} + 353 \beta_{2}) q^{34} + ( - 395 \beta_1 + 6720) q^{36} + ( - 48 \beta_1 - 2326) q^{37} + ( - 153 \beta_{3} - 215 \beta_{2}) q^{38} + (1232 \beta_1 + 1232) q^{39} + (426 \beta_{3} + 370 \beta_{2}) q^{40} + ( - 1103 \beta_{3} - 224 \beta_{2}) q^{41} + ( - 980 \beta_1 + 4622) q^{43} + (2410 \beta_1 + 1680) q^{44} + (1694 \beta_{3} - 920 \beta_{2}) q^{45} + ( - 2644 \beta_1 + 12856) q^{46} + ( - 590 \beta_{3} + 1644 \beta_{2}) q^{47} + (1539 \beta_{3} - 1593 \beta_{2}) q^{48} + (1491 \beta_1 - 10022) q^{50} + (1716 \beta_1 - 15994) q^{51} + ( - 800 \beta_{3} - 2240 \beta_{2}) q^{52} + (2592 \beta_1 - 24434) q^{53} + (308 \beta_{3} - 1364 \beta_{2}) q^{54} + ( - 580 \beta_{3} + 3784 \beta_{2}) q^{55} + ( - 704 \beta_1 - 4246) q^{57} + ( - 2798 \beta_1 + 22844) q^{58} + ( - 1425 \beta_{3} + 2914 \beta_{2}) q^{59} + (5460 \beta_1 - 19040) q^{60} + ( - 3326 \beta_{3} - 680 \beta_{2}) q^{61} + (2350 \beta_{3} + 178 \beta_{2}) q^{62} + (3095 \beta_1 - 8532) q^{64} + ( - 6496 \beta_1 - 19040) q^{65} + ( - 4774 \beta_{3} + 4510 \beta_{2}) q^{66} + ( - 11632 \beta_1 - 11540) q^{67} + (3075 \beta_{3} - 245 \beta_{2}) q^{68} + (4708 \beta_{3} - 5632 \beta_{2}) q^{69} + ( - 4312 \beta_1 - 40612) q^{71} + (2305 \beta_1 + 20492) q^{72} + (4407 \beta_{3} - 5080 \beta_{2}) q^{73} + ( - 2182 \beta_1 + 3308) q^{74} + ( - 3411 \beta_{3} + 3302 \beta_{2}) q^{75} + (1295 \beta_{3} + 1855 \beta_{2}) q^{76} + ( - 2464 \beta_1 + 32032) q^{78} + (4264 \beta_1 - 20540) q^{79} + ( - 5238 \beta_{3} + 4482 \beta_{2}) q^{80} + (6384 \beta_1 - 1109) q^{81} + (2413 \beta_{3} - 7273 \beta_{2}) q^{82} + ( - 4425 \beta_{3} - 6230 \beta_{2}) q^{83} + ( - 2608 \beta_1 + 66860) q^{85} + (7562 \beta_1 - 36684) q^{86} + (7502 \beta_{3} - 6380 \beta_{2}) q^{87} + (9106 \beta_1 + 43256) q^{88} + (6259 \beta_{3} - 1016 \beta_{2}) q^{89} + ( - 8762 \beta_{3} + 13698 \beta_{2}) q^{90} + (9780 \beta_1 - 48160) q^{92} + ( - 832 \beta_1 + 61524) q^{93} + (8346 \beta_{3} - 7418 \beta_{2}) q^{94} + (5000 \beta_1 + 29556) q^{95} + ( - 8525 \beta_{3} + 15015 \beta_{2}) q^{96} + ( - 5873 \beta_{3} + 7448 \beta_{2}) q^{97} + (22764 \beta_1 + 814) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36} - 9208 q^{37} + 2464 q^{39} + 20448 q^{43} + 1900 q^{44} + 56712 q^{46} - 43070 q^{50} - 67408 q^{51} - 102920 q^{53} - 15576 q^{57} + 96972 q^{58} - 87080 q^{60} - 40318 q^{64} - 63168 q^{65} - 22896 q^{67} - 153824 q^{71} + 77358 q^{72} + 17596 q^{74} + 133056 q^{78} - 90688 q^{79} - 17204 q^{81} + 272656 q^{85} - 161860 q^{86} + 154812 q^{88} - 212200 q^{92} + 247760 q^{93} + 108224 q^{95} - 42272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 172\nu - 139 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 7\beta _1 + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 86\beta_{3} + 21\beta_{2} + 245\beta _1 + 441 ) / 7 \) 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
−3.40086
−6.22929
4.40086
7.22929
−7.81507 −23.5186 29.0754 74.2753 183.799 0 22.8562 310.123 −580.467
1.2 −7.81507 23.5186 29.0754 −74.2753 −183.799 0 22.8562 310.123 580.467
1.3 2.81507 −6.54802 −24.0754 45.9910 −18.4331 0 −157.856 −200.123 129.468
1.4 2.81507 6.54802 −24.0754 −45.9910 18.4331 0 −157.856 −200.123 −129.468
\(n\): 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.g 4
3.b odd 2 1 441.6.a.z 4
4.b odd 2 1 784.6.a.bf 4
7.b odd 2 1 inner 49.6.a.g 4
7.c even 3 2 49.6.c.h 8
7.d odd 6 2 49.6.c.h 8
21.c even 2 1 441.6.a.z 4
28.d even 2 1 784.6.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 1.a even 1 1 trivial
49.6.a.g 4 7.b odd 2 1 inner
49.6.c.h 8 7.c even 3 2
49.6.c.h 8 7.d odd 6 2
441.6.a.z 4 3.b odd 2 1
441.6.a.z 4 21.c even 2 1
784.6.a.bf 4 4.b odd 2 1
784.6.a.bf 4 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} + 5T_{2} - 22 \) Copy content Toggle raw display
\( T_{3}^{4} - 596T_{3}^{2} + 23716 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5 T - 22)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 596 T^{2} + 23716 \) Copy content Toggle raw display
$5$ \( T^{4} - 7632 T^{2} + \cdots + 11669056 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 976 T + 234076)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 1260672 T^{2} + \cdots + 76158337024 \) Copy content Toggle raw display
$17$ \( T^{4} - 2613668 T^{2} + \cdots + 1700202150724 \) Copy content Toggle raw display
$19$ \( T^{4} - 1359892 T^{2} + \cdots + 56942116 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3568 T - 160336)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1676 T - 16661788)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 614334295349824 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4604 T + 5234116)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 251093444 T^{2} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10224 T - 998756)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 349180624 T^{2} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 51460 T + 472236292)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 1249753044 T^{2} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{4} - 2284246736 T^{2} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{2} + 11448 T - 3789557552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 76912 T + 953601968)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 6121721124 T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{2} + 45344 T + 386672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 9034370100 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} - 7617937028 T^{2} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{4} - 11859566628 T^{2} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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