Properties

Label 448.6.a.bb
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.367637.1
Defining polynomial: \( x^{3} - x^{2} - 107x + 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{3} + (\beta_{2} + 5) q^{5} + 49 q^{7} + (3 \beta_{2} - \beta_1 + 51) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3) q^{3} + (\beta_{2} + 5) q^{5} + 49 q^{7} + (3 \beta_{2} - \beta_1 + 51) q^{9} + (\beta_{2} - 19 \beta_1 + 194) q^{11} + ( - 11 \beta_{2} - 6 \beta_1 + 319) q^{13} + (9 \beta_{2} + 39 \beta_1 - 24) q^{15} + (10 \beta_{2} + 4 \beta_1 + 244) q^{17} + (22 \beta_{2} - 55 \beta_1 - 363) q^{19} + (49 \beta_1 + 147) q^{21} + (15 \beta_{2} + 65 \beta_1 + 1280) q^{23} + ( - 25 \beta_{2} + 55 \beta_1 - 39) q^{25} + (24 \beta_{2} - 86 \beta_1 - 978) q^{27} + (22 \beta_{2} + 72 \beta_1 + 3076) q^{29} + (94 \beta_{2} + 148 \beta_1 - 402) q^{31} + ( - 48 \beta_{2} + 304 \beta_1 - 4872) q^{33} + (49 \beta_{2} + 245) q^{35} + (104 \beta_{2} + 502 \beta_1 + 1868) q^{37} + ( - 117 \beta_{2} - 31 \beta_1 - 324) q^{39} + (44 \beta_{2} + 390 \beta_1 - 7584) q^{41} + ( - 149 \beta_{2} - 25 \beta_1 + 9902) q^{43} + ( - 45 \beta_{2} + 126 \beta_1 + 9477) q^{45} + ( - 116 \beta_{2} - 542 \beta_1 + 3394) q^{47} + 2401 q^{49} + (102 \beta_{2} + 568 \beta_1 + 1482) q^{51} + ( - 294 \beta_{2} + 518 \beta_1 + 9410) q^{53} + (50 \beta_{2} - 686 \beta_1 + 4772) q^{55} + (33 \beta_{2} + 605 \beta_1 - 17622) q^{57} + ( - 592 \beta_{2} + 1311 \beta_1 - 5579) q^{59} + ( - 91 \beta_{2} - 820 \beta_1 + 30461) q^{61} + (147 \beta_{2} - 49 \beta_1 + 2499) q^{63} + (613 \beta_{2} - 839 \beta_1 - 31842) q^{65} + (879 \beta_{2} + \beta_1 + 18968) q^{67} + (330 \beta_{2} + 1530 \beta_1 + 21780) q^{69} + ( - 362 \beta_{2} - 850 \beta_1 + 25284) q^{71} + ( - 556 \beta_{2} + 1296 \beta_1 - 15202) q^{73} + ( - 60 \beta_{2} - 1109 \beta_1 + 16533) q^{75} + (49 \beta_{2} - 931 \beta_1 + 9506) q^{77} + (52 \beta_{2} + 1000 \beta_1 - 1108) q^{79} + ( - 771 \beta_{2} + 425 \beta_1 - 40773) q^{81} + ( - 874 \beta_{2} - 1239 \beta_1 + 35005) q^{83} + ( - 32 \beta_{2} + 706 \beta_1 + 31674) q^{85} + (414 \beta_{2} + 3536 \beta_1 + 28890) q^{87} + (390 \beta_{2} + 666 \beta_1 + 10290) q^{89} + ( - 539 \beta_{2} - 294 \beta_1 + 15631) q^{91} + (1290 \beta_{2} + 2202 \beta_1 + 37308) q^{93} + ( - 1353 \beta_{2} - 935 \beta_1 + 67672) q^{95} + ( - 172 \beta_{2} - 5478 \beta_1 - 54044) q^{97} + (237 \beta_{2} - 3103 \beta_1 + 26754) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{3} + 14 q^{5} + 147 q^{7} + 151 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{3} + 14 q^{5} + 147 q^{7} + 151 q^{9} + 600 q^{11} + 974 q^{13} - 120 q^{15} + 718 q^{17} - 1056 q^{19} + 392 q^{21} + 3760 q^{23} - 147 q^{25} - 2872 q^{27} + 9134 q^{29} - 1448 q^{31} - 14872 q^{33} + 686 q^{35} + 4998 q^{37} - 824 q^{39} - 23186 q^{41} + 29880 q^{43} + 28350 q^{45} + 10840 q^{47} + 7203 q^{49} + 3776 q^{51} + 28006 q^{53} + 14952 q^{55} - 53504 q^{57} - 17456 q^{59} + 92294 q^{61} + 7399 q^{63} - 95300 q^{65} + 56024 q^{67} + 63480 q^{69} + 77064 q^{71} - 46346 q^{73} + 50768 q^{75} + 29400 q^{77} - 4376 q^{79} - 121973 q^{81} + 107128 q^{83} + 94348 q^{85} + 82720 q^{87} + 29814 q^{89} + 47726 q^{91} + 108432 q^{93} + 205304 q^{95} - 156482 q^{97} + 83128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 107x + 282 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{2} + 10\nu - 291 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} - 5\beta _1 + 286 ) / 4 \) 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
−11.0251
2.76097
9.26413
0 −20.0502 0 33.3202 0 49.0000 0 159.011 0
1.2 0 7.52194 0 −72.6328 0 49.0000 0 −186.420 0
1.3 0 20.5283 0 53.3126 0 49.0000 0 178.410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 448.6.a.bb 3
4.b odd 2 1 448.6.a.ba 3
8.b even 2 1 224.6.a.e 3
8.d odd 2 1 224.6.a.f yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.e 3 8.b even 2 1
224.6.a.f yes 3 8.d odd 2 1
448.6.a.ba 3 4.b odd 2 1
448.6.a.bb 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{3} - 8T_{3}^{2} - 408T_{3} + 3096 \) Copy content Toggle raw display
\( T_{5}^{3} - 14T_{5}^{2} - 4516T_{5} + 129024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 8 T^{2} - 408 T + 3096 \) Copy content Toggle raw display
$5$ \( T^{3} - 14 T^{2} - 4516 T + 129024 \) Copy content Toggle raw display
$7$ \( (T - 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 600 T^{2} - 42560 T + 1824064 \) Copy content Toggle raw display
$13$ \( T^{3} - 974 T^{2} + \cdots + 53052784 \) Copy content Toggle raw display
$17$ \( T^{3} - 718 T^{2} + \cdots + 178338584 \) Copy content Toggle raw display
$19$ \( T^{3} + 1056 T^{2} + \cdots - 936225400 \) Copy content Toggle raw display
$23$ \( T^{3} - 3760 T^{2} + \cdots - 265872000 \) Copy content Toggle raw display
$29$ \( T^{3} - 9134 T^{2} + \cdots - 18655265000 \) Copy content Toggle raw display
$31$ \( T^{3} + 1448 T^{2} + \cdots - 54502545600 \) Copy content Toggle raw display
$37$ \( T^{3} - 4998 T^{2} + \cdots - 417364199688 \) Copy content Toggle raw display
$41$ \( T^{3} + 23186 T^{2} + \cdots - 195051110440 \) Copy content Toggle raw display
$43$ \( T^{3} - 29880 T^{2} + \cdots - 302765738048 \) Copy content Toggle raw display
$47$ \( T^{3} - 10840 T^{2} + \cdots + 1468178593088 \) Copy content Toggle raw display
$53$ \( T^{3} - 28006 T^{2} + \cdots + 1608716275704 \) Copy content Toggle raw display
$59$ \( T^{3} + 17456 T^{2} + \cdots - 27263200844760 \) Copy content Toggle raw display
$61$ \( T^{3} - 92294 T^{2} + \cdots - 18497589110240 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 132723178706432 \) Copy content Toggle raw display
$71$ \( T^{3} - 77064 T^{2} + \cdots + 12186764730368 \) Copy content Toggle raw display
$73$ \( T^{3} + 46346 T^{2} + \cdots - 39799282328008 \) Copy content Toggle raw display
$79$ \( T^{3} + 4376 T^{2} + \cdots - 267561011200 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 109229539940808 \) Copy content Toggle raw display
$89$ \( T^{3} - 29814 T^{2} + \cdots + 4144636813752 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 698292640604904 \) Copy content Toggle raw display
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