Properties

Label 448.6.a.bd
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 198x^{2} + 43x + 5999 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 4) q^{3} + (\beta_{2} + \beta_1 + 7) q^{5} - 49 q^{7} + (\beta_{3} + 3 \beta_{2} + 9 \beta_1 + 170) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 4) q^{3} + (\beta_{2} + \beta_1 + 7) q^{5} - 49 q^{7} + (\beta_{3} + 3 \beta_{2} + 9 \beta_1 + 170) q^{9} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 123) q^{11} + (2 \beta_{3} - 3 \beta_{2} - 19 \beta_1 - 161) q^{13} + (3 \beta_{3} + 23 \beta_{2} + 29 \beta_1 + 533) q^{15} + ( - 2 \beta_{3} - 14 \beta_{2} + 44 \beta_1 - 448) q^{17} + ( - 4 \beta_{3} + 2 \beta_{2} + 7 \beta_1 + 158) q^{19} + ( - 49 \beta_1 - 196) q^{21} + ( - \beta_{3} + 25 \beta_{2} + 45 \beta_1 - 57) q^{23} + (5 \beta_{3} + 15 \beta_{2} + 185 \beta_1 + 736) q^{25} + (4 \beta_{3} + 96 \beta_{2} + 110 \beta_1 + 3544) q^{27} + ( - 24 \beta_{3} - 2 \beta_{2} + 22 \beta_1 - 976) q^{29} + ( - 6 \beta_{3} - 46 \beta_{2} - 88 \beta_1 - 3146) q^{31} + ( - 26 \beta_{3} - 8 \beta_{2} + 270 \beta_1 - 532) q^{33} + ( - 49 \beta_{2} - 49 \beta_1 - 343) q^{35} + (26 \beta_{3} - 48 \beta_{2} + 164 \beta_1 - 274) q^{37} + ( - 47 \beta_{3} - 99 \beta_{2} - 133 \beta_1 - 8633) q^{39} + (30 \beta_{3} + 44 \beta_{2} + 140 \beta_1 + 5530) q^{41} + ( - 40 \beta_{3} + 69 \beta_{2} + 44 \beta_1 + 8189) q^{43} + (42 \beta_{3} + 331 \beta_{2} + 1087 \beta_1 + 14245) q^{45} + (96 \beta_{3} + 148 \beta_{2} + 22 \beta_1 + 692) q^{47} + 2401 q^{49} + (38 \beta_{3} - 166 \beta_{2} - 640 \beta_1 + 14286) q^{51} + ( - 54 \beta_{3} - 414 \beta_{2} + 438 \beta_1 - 5992) q^{53} + (10 \beta_{3} + 142 \beta_{2} + 386 \beta_1 - 6278) q^{55} + (55 \beta_{3} + 25 \beta_{2} - 121 \beta_1 + 3871) q^{57} + (72 \beta_{3} - 232 \beta_{2} + 47 \beta_1 + 27052) q^{59} + ( - 40 \beta_{3} - 347 \beta_{2} + 897 \beta_1 - 1533) q^{61} + ( - 49 \beta_{3} - 147 \beta_{2} - 441 \beta_1 - 8330) q^{63} + ( - 45 \beta_{3} - 443 \beta_{2} - 629 \beta_1 - 23465) q^{65} + ( - 116 \beta_{3} - 383 \beta_{2} + 278 \beta_1 + 27177) q^{67} + (106 \beta_{3} + 626 \beta_{2} + 506 \beta_1 + 20398) q^{69} + ( - 98 \beta_{3} + 98 \beta_{2} + 462 \beta_1 - 13930) q^{71} + ( - 118 \beta_{3} - 212 \beta_{2} - 906 \beta_1 - 2702) q^{73} + (160 \beta_{3} + 900 \beta_{2} + 2351 \beta_1 + 77704) q^{75} + ( - 98 \beta_{3} + 49 \beta_{2} + 98 \beta_1 - 6027) q^{77} + (14 \beta_{3} - 1036 \beta_{2} + 1258 \beta_1 - 20168) q^{79} + (15 \beta_{3} + 1557 \beta_{2} + 3887 \beta_1 + 26660) q^{81} + ( - 116 \beta_{3} - 726 \beta_{2} - 2705 \beta_1 + 30190) q^{83} + (86 \beta_{3} + 496 \beta_{2} - 1956 \beta_1 - 24392) q^{85} + (282 \beta_{3} - 190 \beta_{2} - 2988 \beta_1 + 6078) q^{87} + ( - 48 \beta_{3} + 14 \beta_{2} - 820 \beta_1 - 7504) q^{89} + ( - 98 \beta_{3} + 147 \beta_{2} + 931 \beta_1 + 7889) q^{91} + ( - 114 \beta_{3} - 1238 \beta_{2} - 4890 \beta_1 - 52122) q^{93} + ( - 11 \beta_{3} + 177 \beta_{2} - 293 \beta_1 + 16899) q^{95} + (468 \beta_{3} - 916 \beta_{2} - 418 \beta_1 + 10710) q^{97} + (54 \beta_{3} + 659 \beta_{2} - 1094 \beta_1 + 75895) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9} + 484 q^{11} - 686 q^{13} + 2184 q^{15} - 1700 q^{17} + 654 q^{19} - 882 q^{21} - 136 q^{23} + 3304 q^{25} + 14388 q^{27} - 3812 q^{29} - 12748 q^{31} - 1536 q^{33} - 1470 q^{35} - 820 q^{37} - 34704 q^{39} + 22340 q^{41} + 32924 q^{43} + 59070 q^{45} + 2620 q^{47} + 9604 q^{49} + 55788 q^{51} - 22984 q^{53} - 24360 q^{55} + 15132 q^{57} + 108158 q^{59} - 4258 q^{61} - 34104 q^{63} - 95028 q^{65} + 109496 q^{67} + 82392 q^{69} - 54600 q^{71} - 12384 q^{73} + 315198 q^{75} - 23716 q^{77} - 78184 q^{79} + 114384 q^{81} + 115582 q^{83} - 101652 q^{85} + 17772 q^{87} - 31560 q^{89} + 33614 q^{91} - 218040 q^{93} + 67032 q^{95} + 41068 q^{97} + 301284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 198x^{2} + 43x + 5999 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 8\nu^{2} - 272\nu - 959 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 20\nu^{2} + 258\nu - 1820 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + \beta _1 + 397 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 15\beta_{2} + 134\beta _1 + 165 ) / 2 \) 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
−12.2094
−6.08658
6.13256
13.1634
0 −20.4188 0 −21.4943 0 −49.0000 0 173.928 0
1.2 0 −8.17317 0 20.6340 0 −49.0000 0 −176.199 0
1.3 0 16.2651 0 −69.5406 0 −49.0000 0 21.5544 0
1.4 0 30.3268 0 100.401 0 −49.0000 0 676.717 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.bd 4
4.b odd 2 1 448.6.a.bc 4
8.b even 2 1 224.6.a.g 4
8.d odd 2 1 224.6.a.h yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.g 4 8.b even 2 1
224.6.a.h yes 4 8.d odd 2 1
448.6.a.bc 4 4.b odd 2 1
448.6.a.bd 4 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{4} - 18T_{3}^{3} - 672T_{3}^{2} + 6328T_{3} + 82320 \) Copy content Toggle raw display
\( T_{5}^{4} - 30T_{5}^{3} - 7452T_{5}^{2} + 7680T_{5} + 3096576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 18 T^{3} - 672 T^{2} + \cdots + 82320 \) Copy content Toggle raw display
$5$ \( T^{4} - 30 T^{3} - 7452 T^{2} + \cdots + 3096576 \) Copy content Toggle raw display
$7$ \( (T + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 484 T^{3} + \cdots - 9325648640 \) Copy content Toggle raw display
$13$ \( T^{4} + 686 T^{3} + \cdots - 40778482496 \) Copy content Toggle raw display
$17$ \( T^{4} + 1700 T^{3} + \cdots - 670305393104 \) Copy content Toggle raw display
$19$ \( T^{4} - 654 T^{3} + \cdots - 9671002992 \) Copy content Toggle raw display
$23$ \( T^{4} + 136 T^{3} + \cdots + 2509965337600 \) Copy content Toggle raw display
$29$ \( T^{4} + 3812 T^{3} + \cdots - 69185366122448 \) Copy content Toggle raw display
$31$ \( T^{4} + 12748 T^{3} + \cdots + 5102153647360 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 386774083587152 \) Copy content Toggle raw display
$41$ \( T^{4} - 22340 T^{3} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{4} - 32924 T^{3} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} - 2620 T^{3} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} + 22984 T^{3} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{4} - 108158 T^{3} + \cdots - 80\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{4} + 4258 T^{3} + \cdots - 43\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{4} - 109496 T^{3} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{4} + 54600 T^{3} + \cdots - 22\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{4} + 12384 T^{3} + \cdots + 60\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{4} + 78184 T^{3} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{4} - 115582 T^{3} + \cdots - 34\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{4} + 31560 T^{3} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{4} - 41068 T^{3} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
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