Properties

Label 448.6.a.bf
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + (\beta_{3} - 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 127) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + (\beta_{3} - 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 127) q^{9} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 9 \beta_1 - 23) q^{11} + ( - \beta_{4} + 6 \beta_{3} + \beta_{2} - 6 \beta_1 - 7) q^{13} + ( - 2 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} + 31 \beta_1 + 4) q^{15} + ( - 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 24 \beta_1 - 82) q^{17} + (5 \beta_{4} + 3 \beta_{3} + \beta_{2} - 57 \beta_1 + 718) q^{19} + ( - 49 \beta_1 + 98) q^{21} + (10 \beta_{4} + 8 \beta_{3} - \beta_{2} + 117 \beta_1 + 98) q^{23} + (16 \beta_{4} - 6 \beta_{3} - 7 \beta_{2} - 57 \beta_1 + 1925) q^{25} + ( - 7 \beta_{4} + 33 \beta_{3} - 3 \beta_{2} - 138 \beta_1 - 66) q^{27} + (10 \beta_{4} - 36 \beta_{3} - 10 \beta_{2} - 96 \beta_1 - 956) q^{29} + ( - 10 \beta_{4} + 30 \beta_{3} - 20 \beta_{2} + 48 \beta_1 - 2096) q^{31} + ( - 16 \beta_{4} + 62 \beta_{3} + 26 \beta_{2} - 116 \beta_1 + 3182) q^{33} + (49 \beta_{3} - 343) q^{35} + ( - 38 \beta_{3} + 6 \beta_{2} - 114 \beta_1 - 3936) q^{37} + ( - 24 \beta_{4} + 92 \beta_{3} + 35 \beta_{2} - 39 \beta_1 + 2188) q^{39} + ( - 2 \beta_{4} + 4 \beta_{3} - 28 \beta_{2} - 450 \beta_1 + 4680) q^{41} + ( - 3 \beta_{4} + 32 \beta_{3} - 39 \beta_{2} + 249 \beta_1 - 4403) q^{43} + ( - 47 \beta_{4} + 38 \beta_{3} + 11 \beta_{2} - 490 \beta_1 - 9897) q^{45} + (48 \beta_{4} + 28 \beta_{3} - 48 \beta_{2} - 354 \beta_1 + 3216) q^{47} + 2401 q^{49} + (26 \beta_{4} - 82 \beta_{3} + 44 \beta_{2} + 468 \beta_1 + 9124) q^{51} + (32 \beta_{4} + 124 \beta_{3} + 22 \beta_{2} - 546 \beta_1 - 10818) q^{53} + ( - 6 \beta_{4} - 290 \beta_{3} + 24 \beta_{2} - 798 \beta_1 + 10632) q^{55} + ( - 12 \beta_{4} - 154 \beta_{3} + 23 \beta_{2} - 851 \beta_1 + 21878) q^{57} + ( - 5 \beta_{4} - 41 \beta_{3} + 59 \beta_{2} + 537 \beta_1 + 14864) q^{59} + (38 \beta_{4} + 83 \beta_{3} + 22 \beta_{2} - 540 \beta_1 - 13639) q^{61} + ( - 98 \beta_{3} + 49 \beta_{2} - 49 \beta_1 + 6223) q^{63} + (64 \beta_{4} - 282 \beta_{3} - 13 \beta_{2} - 1119 \beta_1 + 30470) q^{65} + ( - 93 \beta_{4} - 402 \beta_{3} + 3 \beta_{2} + 615 \beta_1 + 5213) q^{67} + (5 \beta_{4} + 25 \beta_{3} - 185 \beta_{2} + 74 \beta_1 - 42938) q^{69} + ( - 50 \beta_{4} - 522 \beta_{3} + 44 \beta_{2} - 522 \beta_1 + 18500) q^{71} + ( - 58 \beta_{4} + 140 \beta_{3} + 4 \beta_{2} + 372 \beta_1 + 27236) q^{73} + (105 \beta_{4} - 815 \beta_{3} - 119 \beta_{2} - 739 \beta_1 + 24760) q^{75} + ( - 49 \beta_{4} + 98 \beta_{3} + 49 \beta_{2} - 441 \beta_1 - 1127) q^{77} + ( - 56 \beta_{4} + 406 \beta_{3} - 58 \beta_{2} + 204 \beta_1 + 19286) q^{79} + ( - 40 \beta_{4} + 414 \beta_{3} + 51 \beta_{2} + 1937 \beta_1 + 21035) q^{81} + (68 \beta_{4} + 678 \beta_{3} - 8 \beta_{2} - 945 \beta_1 + 29328) q^{83} + ( - 114 \beta_{4} - 664 \beta_{3} + 180 \beta_{2} + 930 \beta_1 - 23626) q^{85} + (192 \beta_{4} - 1088 \beta_{3} - 122 \beta_{2} + 2148 \beta_1 + 33596) q^{87} + (140 \beta_{4} + 46 \beta_{3} - 32 \beta_{2} + 2094 \beta_1 + 37748) q^{89} + ( - 49 \beta_{4} + 294 \beta_{3} + 49 \beta_{2} - 294 \beta_1 - 343) q^{91} + (150 \beta_{4} - 454 \beta_{3} + 92 \beta_{2} + 7018 \beta_1 - 17320) q^{93} + ( - 44 \beta_{4} + 1928 \beta_{3} + 59 \beta_{2} + 2601 \beta_1 + 15324) q^{95} + ( - 54 \beta_{4} + 910 \beta_{3} - 270 \beta_{2} + 822 \beta_1 - 17458) q^{97} + ( - 183 \beta_{4} + 1212 \beta_{3} + 255 \beta_{2} - 4597 \beta_1 + 52253) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9} - 116 q^{11} - 40 q^{13} + 16 q^{15} - 402 q^{17} + 3582 q^{19} + 490 q^{21} + 472 q^{23} + 9615 q^{25} - 356 q^{27} - 4754 q^{29} - 10500 q^{31} + 15864 q^{33} - 1764 q^{35} - 19642 q^{37} + 10872 q^{39} + 23398 q^{41} - 22044 q^{43} - 49476 q^{45} + 16004 q^{47} + 12005 q^{49} + 45676 q^{51} - 54246 q^{53} + 53456 q^{55} + 109556 q^{57} + 74366 q^{59} - 68316 q^{61} + 31213 q^{63} + 152568 q^{65} + 26560 q^{67} - 214720 q^{69} + 93072 q^{71} + 136098 q^{73} + 124510 q^{75} - 5684 q^{77} + 96080 q^{79} + 104801 q^{81} + 145894 q^{83} - 117352 q^{85} + 168876 q^{87} + 188554 q^{89} - 1960 q^{91} - 86296 q^{93} + 74736 q^{95} - 88146 q^{97} + 260236 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\nu^{4} - 64\nu^{3} - 8772\nu^{2} - 20218\nu - 3528 ) / 633 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\nu^{4} - 32\nu^{3} - 5652\nu^{2} - 8210\nu + 114075 ) / 633 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\nu^{4} + 200\nu^{3} - 8352\nu^{2} - 46908\nu + 206833 ) / 211 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} + \beta_{2} + 3\beta _1 + 366 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{4} - 45\beta_{3} + 9\beta_{2} + 630\beta _1 + 1298 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{4} - 339\beta_{3} + 207\beta_{2} + 1552\beta _1 + 58325 ) / 4 \) 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
14.6594
3.97668
2.24605
−8.98949
−11.8926
0 −27.3187 0 −52.2232 0 49.0000 0 503.312 0
1.2 0 −5.95336 0 −11.6830 0 49.0000 0 −207.557 0
1.3 0 −2.49210 0 99.5911 0 49.0000 0 −236.789 0
1.4 0 19.9790 0 −106.158 0 49.0000 0 156.159 0
1.5 0 25.7852 0 34.4729 0 49.0000 0 421.876 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bf 5
4.b odd 2 1 448.6.a.be 5
8.b even 2 1 224.6.a.i 5
8.d odd 2 1 224.6.a.j yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.i 5 8.b even 2 1
224.6.a.j yes 5 8.d odd 2 1
448.6.a.be 5 4.b odd 2 1
448.6.a.bf 5 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{5} - 10T_{3}^{4} - 876T_{3}^{3} + 7592T_{3}^{2} + 107952T_{3} + 208800 \) Copy content Toggle raw display
\( T_{5}^{5} + 36T_{5}^{4} - 11972T_{5}^{3} - 342672T_{5}^{2} + 16702736T_{5} + 222365696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 10 T^{4} - 876 T^{3} + \cdots + 208800 \) Copy content Toggle raw display
$5$ \( T^{5} + 36 T^{4} + \cdots + 222365696 \) Copy content Toggle raw display
$7$ \( (T - 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 116 T^{4} + \cdots - 183286596608 \) Copy content Toggle raw display
$13$ \( T^{5} + 40 T^{4} + \cdots - 247266196544 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 353511151373280 \) Copy content Toggle raw display
$19$ \( T^{5} - 3582 T^{4} + \cdots - 71\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{5} - 472 T^{4} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{5} + 4754 T^{4} + \cdots - 38\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{5} + 10500 T^{4} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{5} + 19642 T^{4} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} - 23398 T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + 22044 T^{4} + \cdots + 77\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{5} - 16004 T^{4} + \cdots + 62\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{5} + 54246 T^{4} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} - 74366 T^{4} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{5} + 68316 T^{4} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{5} - 26560 T^{4} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{5} - 93072 T^{4} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{5} - 136098 T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{5} - 96080 T^{4} + \cdots - 26\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{5} - 145894 T^{4} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{5} - 188554 T^{4} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{5} + 88146 T^{4} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
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