Properties

Label 256.8.a.r
Level $256$
Weight $8$
Character orbit 256.a
Self dual yes
Analytic conductor $79.971$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 163x^{4} + 4820x^{2} - 15296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{27} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 115) q^{7} + ( - \beta_{5} - \beta_{4} + 487) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 115) q^{7} + ( - \beta_{5} - \beta_{4} + 487) q^{9} + ( - \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{11} + ( - 2 \beta_{3} - 3 \beta_{2} - 21 \beta_1) q^{13} + ( - 8 \beta_{5} + \beta_{4} + 2981) q^{15} + (7 \beta_{5} - 25 \beta_{4} + 248) q^{17} + (5 \beta_{3} - 93 \beta_{2} - 105 \beta_1) q^{19} + (6 \beta_{3} + 6 \beta_{2} - 452 \beta_1) q^{21} + (8 \beta_{5} - 11 \beta_{4} + 217) q^{23} + ( - 6 \beta_{5} - 38 \beta_{4} + 6567) q^{25} + (3 \beta_{3} - 363 \beta_{2} - 310 \beta_1) q^{27} + (16 \beta_{3} + 79 \beta_{2} - 2337 \beta_1) q^{29} + (72 \beta_{5} + 12 \beta_{4} - 14908) q^{31} + ( - 63 \beta_{5} + 129 \beta_{4} + 8958) q^{33} + ( - 50 \beta_{3} - 478 \beta_{2} - 368 \beta_1) q^{35} + ( - 58 \beta_{3} + 33 \beta_{2} - 6633 \beta_1) q^{37} + ( - 64 \beta_{5} + 221 \beta_{4} + 54649) q^{39} + (142 \beta_{5} + 302 \beta_{4} - 87022) q^{41} + (38 \beta_{3} + 522 \beta_{2} - 579 \beta_1) q^{43} + ( - 30 \beta_{3} - 771 \beta_{2} - 13841 \beta_1) q^{45} + ( - 168 \beta_{5} - 210 \beta_{4} + 261198) q^{47} + (168 \beta_{5} + 168 \beta_{4} - 85287) q^{49} + (171 \beta_{3} + 2733 \beta_{2} + 3038 \beta_1) q^{51} + (170 \beta_{3} - 535 \beta_{2} - 17949 \beta_1) q^{53} + (96 \beta_{5} - 1237 \beta_{4} + 545423) q^{55} + ( - 701 \beta_{5} - 509 \beta_{4} + 315386) q^{57} + ( - 224 \beta_{3} + 3680 \beta_{2} + 10933 \beta_1) q^{59} + ( - 66 \beta_{3} + 3993 \beta_{2} - 7353 \beta_1) q^{61} + ( - 344 \beta_{5} + 1015 \beta_{4} + 962579) q^{63} + (50 \beta_{5} - 2030 \beta_{4} + 236740) q^{65} + ( - 123 \beta_{3} + 1443 \beta_{2} - 4017 \beta_1) q^{67} + (90 \beta_{3} + 3018 \beta_{2} + 9460 \beta_1) q^{69} + (504 \beta_{5} + 3279 \beta_{4} + 1264923) q^{71} + (1239 \beta_{5} - 809 \beta_{4} - 348404) q^{73} + (210 \beta_{3} - 1986 \beta_{2} - 29411 \beta_1) q^{75} + (218 \beta_{3} - 12598 \beta_{2} + 31860 \beta_1) q^{77} + (2016 \beta_{5} - 2054 \beta_{4} + 2669330) q^{79} + ( - 631 \beta_{5} + 2249 \beta_{4} - 121049) q^{81} + ( - 568 \beta_{3} - 9992 \beta_{2} + 13319 \beta_1) q^{83} + ( - 1390 \beta_{3} - 9876 \beta_{2} + 85134 \beta_1) q^{85} + ( - 1608 \beta_{5} - 4383 \beta_{4} + 6244293) q^{87} + (439 \beta_{5} + 1079 \beta_{4} - 362020) q^{89} + (1142 \beta_{3} - 19782 \beta_{2} + 56016 \beta_1) q^{91} + (144 \beta_{3} + 26496 \beta_{2} + 139408 \beta_1) q^{93} + ( - 1184 \beta_{5} + 1313 \beta_{4} + 8089613) q^{95} + ( - 849 \beta_{5} + 7183 \beta_{4} - 183496) q^{97} + (1224 \beta_{3} - 8712 \beta_{2} - 64323 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 163x^{4} + 4820x^{2} - 15296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 171\nu^{3} - 5468\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -37\nu^{5} + 5367\nu^{3} - 101516\nu ) / 600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -77\nu^{5} + 12207\nu^{3} - 13036\nu ) / 600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{4} - 728\nu^{2} - 5581 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -104\nu^{4} + 15864\nu^{2} - 275047 ) / 75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} - 8\beta_1 ) / 512 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 13\beta_{4} + 13904 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 105\beta_{3} - 425\beta_{2} + 1528\beta_1 ) / 512 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 273\beta_{5} + 1983\beta_{4} + 1443856 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12487\beta_{3} - 67207\beta_{2} + 243592\beta_1 ) / 512 \) 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
−1.89807
−5.81430
11.2068
−11.2068
5.81430
1.89807
0 −76.9497 0 −338.443 0 438.996 0 3734.25 0
1.2 0 −40.2163 0 324.492 0 956.960 0 −569.651 0
1.3 0 −21.9408 0 184.916 0 −1051.96 0 −1705.60 0
1.4 0 21.9408 0 −184.916 0 −1051.96 0 −1705.60 0
1.5 0 40.2163 0 −324.492 0 956.960 0 −569.651 0
1.6 0 76.9497 0 338.443 0 438.996 0 3734.25 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.8.a.r 6
4.b odd 2 1 256.8.a.q 6
8.b even 2 1 inner 256.8.a.r 6
8.d odd 2 1 256.8.a.q 6
16.e even 4 2 8.8.b.a 6
16.f odd 4 2 32.8.b.a 6
48.i odd 4 2 72.8.d.b 6
48.k even 4 2 288.8.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 16.e even 4 2
32.8.b.a 6 16.f odd 4 2
72.8.d.b 6 48.i odd 4 2
256.8.a.q 6 4.b odd 2 1
256.8.a.q 6 8.d odd 2 1
256.8.a.r 6 1.a even 1 1 trivial
256.8.a.r 6 8.b even 2 1 inner
288.8.d.b 6 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(256))\):

\( T_{3}^{6} - 8020T_{3}^{4} + 13205808T_{3}^{2} - 4610229696 \) Copy content Toggle raw display
\( T_{5}^{6} - 254032T_{5}^{4} + 19577926400T_{5}^{2} - 412405245440000 \) Copy content Toggle raw display
\( T_{7}^{3} - 344T_{7}^{2} - 1048384T_{7} + 441929216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 8020 T^{4} + \cdots - 4610229696 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 412405245440000 \) Copy content Toggle raw display
$7$ \( (T^{3} - 344 T^{2} - 1048384 T + 441929216)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 52294004 T^{4} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} - 171080144 T^{4} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{3} - 726 T^{2} + \cdots - 9112197964104)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 3360814100 T^{4} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{3} - 648 T^{2} + \cdots - 2134822184448)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 55662621776 T^{4} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + 44640 T^{2} + \cdots + 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 490654094672 T^{4} + \cdots - 61\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{3} + 260622 T^{2} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 124911737588 T^{4} + \cdots - 77\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{3} - 783216 T^{2} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 3916631783120 T^{4} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} - 6619585104052 T^{4} + \cdots - 55\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} - 4505952081744 T^{4} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} - 1291377394260 T^{4} + \cdots - 75\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} - 3798552 T^{2} + \cdots - 38\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 1044782 T^{2} + \cdots - 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 8007952 T^{2} + \cdots + 49\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 37884069033748 T^{4} + \cdots - 63\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1084542 T^{2} + \cdots - 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 544154 T^{2} + \cdots - 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
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