Properties

Label 288.8.d.b
Level $288$
Weight $8$
Character orbit 288.d
Analytic conductor $89.967$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7} + (\beta_{5} + 7 \beta_{3} + 3 \beta_{2}) q^{11} + ( - 2 \beta_{5} - 3 \beta_{3} - 21 \beta_{2}) q^{13} + (7 \beta_{4} - 25 \beta_1 - 248) q^{17} + ( - 5 \beta_{5} + 93 \beta_{3} + 105 \beta_{2}) q^{19} + (8 \beta_{4} - 11 \beta_1 - 217) q^{23} + ( - 6 \beta_{4} - 38 \beta_1 - 6567) q^{25} + ( - 16 \beta_{5} - 79 \beta_{3} + 2337 \beta_{2}) q^{29} + (72 \beta_{4} + 12 \beta_1 + 14908) q^{31} + ( - 50 \beta_{5} - 478 \beta_{3} - 368 \beta_{2}) q^{35} + (58 \beta_{5} - 33 \beta_{3} + 6633 \beta_{2}) q^{37} + ( - 142 \beta_{4} - 302 \beta_1 - 87022) q^{41} + (38 \beta_{5} + 522 \beta_{3} - 579 \beta_{2}) q^{43} + (168 \beta_{4} + 210 \beta_1 + 261198) q^{47} + ( - 168 \beta_{4} - 168 \beta_1 - 85287) q^{49} + (170 \beta_{5} - 535 \beta_{3} - 17949 \beta_{2}) q^{53} + ( - 96 \beta_{4} + 1237 \beta_1 + 545423) q^{55} + (224 \beta_{5} - 3680 \beta_{3} - 10933 \beta_{2}) q^{59} + ( - 66 \beta_{5} + 3993 \beta_{3} - 7353 \beta_{2}) q^{61} + (50 \beta_{4} - 2030 \beta_1 - 236740) q^{65} + (123 \beta_{5} - 1443 \beta_{3} + 4017 \beta_{2}) q^{67} + (504 \beta_{4} + 3279 \beta_1 - 1264923) q^{71} + (1239 \beta_{4} - 809 \beta_1 + 348404) q^{73} + ( - 218 \beta_{5} + 12598 \beta_{3} - 31860 \beta_{2}) q^{77} + (2016 \beta_{4} - 2054 \beta_1 - 2669330) q^{79} + ( - 568 \beta_{5} - 9992 \beta_{3} + 13319 \beta_{2}) q^{83} + (1390 \beta_{5} + 9876 \beta_{3} - 85134 \beta_{2}) q^{85} + ( - 439 \beta_{4} - 1079 \beta_1 - 362020) q^{89} + (1142 \beta_{5} - 19782 \beta_{3} + 56016 \beta_{2}) q^{91} + (1184 \beta_{4} - 1313 \beta_1 + 8089613) q^{95} + (849 \beta_{4} - 7183 \beta_1 - 183496) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 688 q^{7} - 1452 q^{17} - 1296 q^{23} - 39314 q^{25} + 89280 q^{31} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} - 1416480 q^{65} - 7597104 q^{71} + 2089564 q^{73} - 16015904 q^{79} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 3\nu^{3} + 42\nu^{2} - 72\nu + 106 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} - 7\nu^{4} - 50\nu^{3} + 104\nu^{2} + 1088\nu + 15872 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{5} - 47\nu^{4} + 222\nu^{3} - 1112\nu^{2} + 3136\nu + 60928 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 16\nu^{3} + 108\nu^{2} + 1200\nu + 2260 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 93\nu^{5} - 487\nu^{4} - 1458\nu^{3} - 3352\nu^{2} + 101952\nu - 205312 ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} - 8\beta_{2} - 2\beta _1 + 512 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 6\beta_{4} - 63\beta_{3} + 72\beta_{2} + 26\beta _1 + 4960 ) / 1024 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{5} + 38\beta_{4} + 329\beta_{3} - 2296\beta_{2} + 58\beta _1 + 32288 ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -141\beta_{5} + 222\beta_{4} - 1587\beta_{3} - 3288\beta_{2} - 638\beta _1 + 376864 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 807\beta_{5} - 218\beta_{4} - 4327\beta_{3} - 41848\beta_{2} + 698\beta _1 + 4357792 ) / 1024 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
5.57668 0.949035i
−4.85268 + 2.90715i
0.776001 5.60338i
0.776001 + 5.60338i
−4.85268 2.90715i
5.57668 + 0.949035i
0 0 0 338.443i 0 438.996 0 0 0
145.2 0 0 0 324.492i 0 956.960 0 0 0
145.3 0 0 0 184.916i 0 −1051.96 0 0 0
145.4 0 0 0 184.916i 0 −1051.96 0 0 0
145.5 0 0 0 324.492i 0 956.960 0 0 0
145.6 0 0 0 338.443i 0 438.996 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.6
Significant digits:
Format:

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 288.8.d.b 6
3.b odd 2 1 32.8.b.a 6
4.b odd 2 1 72.8.d.b 6
8.b even 2 1 inner 288.8.d.b 6
8.d odd 2 1 72.8.d.b 6
12.b even 2 1 8.8.b.a 6
24.f even 2 1 8.8.b.a 6
24.h odd 2 1 32.8.b.a 6
48.i odd 4 2 256.8.a.q 6
48.k even 4 2 256.8.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 12.b even 2 1
8.8.b.a 6 24.f even 2 1
32.8.b.a 6 3.b odd 2 1
32.8.b.a 6 24.h odd 2 1
72.8.d.b 6 4.b odd 2 1
72.8.d.b 6 8.d odd 2 1
256.8.a.q 6 48.i odd 4 2
256.8.a.r 6 48.k even 4 2
288.8.d.b 6 1.a even 1 1 trivial
288.8.d.b 6 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 254032T_{5}^{4} + 19577926400T_{5}^{2} + 412405245440000 \) acting on \(S_{8}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) 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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