Properties

Label 784.6.a.bf
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 49)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2}) q^{3} + (4 \beta_{3} - 2 \beta_{2}) q^{5} + (12 \beta_1 + 55) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2}) q^{3} + (4 \beta_{3} - 2 \beta_{2}) q^{5} + (12 \beta_1 + 55) q^{9} + ( - 3 \beta_1 + 488) q^{11} + ( - 28 \beta_{3} - 64 \beta_{2}) q^{13} + (34 \beta_1 + 1024) q^{15} + (76 \beta_{3} - 7 \beta_{2}) q^{17} + ( - 39 \beta_{3} - 53 \beta_{2}) q^{19} + (86 \beta_1 + 1784) q^{23} + (80 \beta_1 + 691) q^{25} + ( - 44 \beta_{3} - 220 \beta_{2}) q^{27} + ( - 196 \beta_1 - 838) q^{29} + (386 \beta_{3} + 246 \beta_{2}) q^{31} + (452 \beta_{3} - 386 \beta_{2}) q^{33} + (12 \beta_1 - 2302) q^{37} + (308 \beta_1 - 616) q^{39} + (112 \beta_{3} + 1103 \beta_{2}) q^{41} + ( - 245 \beta_1 - 5112) q^{43} + (460 \beta_{3} - 1694 \beta_{2}) q^{45} + (822 \beta_{3} - 590 \beta_{2}) q^{47} + (429 \beta_1 + 16852) q^{51} + ( - 648 \beta_1 - 25730) q^{53} + (1892 \beta_{3} - 580 \beta_{2}) q^{55} + (176 \beta_1 - 3894) q^{57} + (1457 \beta_{3} - 1425 \beta_{2}) q^{59} + (340 \beta_{3} + 3326 \beta_{2}) q^{61} + (1624 \beta_1 - 15792) q^{65} + ( - 2908 \beta_1 + 5724) q^{67} + (2816 \beta_{3} - 4708 \beta_{2}) q^{69} + ( - 1078 \beta_1 + 38456) q^{71} + (2540 \beta_{3} - 4407 \beta_{2}) q^{73} + (1651 \beta_{3} - 3411 \beta_{2}) q^{75} + (1066 \beta_1 + 22672) q^{79} + ( - 1596 \beta_1 - 4301) q^{81} + ( - 3115 \beta_{3} - 4425 \beta_{2}) q^{83} + (652 \beta_1 + 68164) q^{85} + ( - 3190 \beta_{3} + 7502 \beta_{2}) q^{87} + (508 \beta_{3} - 6259 \beta_{2}) q^{89} + (208 \beta_1 + 61940) q^{93} + (1250 \beta_1 - 27056) q^{95} + ( - 3724 \beta_{3} + 5873 \beta_{2}) q^{97} + (5691 \beta_1 + 10568) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} - 688\nu + 346 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 102\nu^{2} - 172\nu - 3116 ) / 105 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{2} - 7\beta _1 + 14 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} - \beta _1 + 122 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 42\beta_{3} + 344\beta_{2} - 245\beta _1 + 1274 ) / 28 \) 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
4.40086
7.22929
−6.22929
0 −23.5186 0 −74.2753 0 0 0 310.123 0
1.2 0 −6.54802 0 −45.9910 0 0 0 −200.123 0
1.3 0 6.54802 0 45.9910 0 0 0 −200.123 0
1.4 0 23.5186 0 74.2753 0 0 0 310.123 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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 784.6.a.bf 4
4.b odd 2 1 49.6.a.g 4
7.b odd 2 1 inner 784.6.a.bf 4
12.b even 2 1 441.6.a.z 4
28.d even 2 1 49.6.a.g 4
28.f even 6 2 49.6.c.h 8
28.g odd 6 2 49.6.c.h 8
84.h odd 2 1 441.6.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 4.b odd 2 1
49.6.a.g 4 28.d even 2 1
49.6.c.h 8 28.f even 6 2
49.6.c.h 8 28.g odd 6 2
441.6.a.z 4 12.b even 2 1
441.6.a.z 4 84.h odd 2 1
784.6.a.bf 4 1.a even 1 1 trivial
784.6.a.bf 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 596T_{3}^{2} + 23716 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) 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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