Properties

Label 72.8.d.b
Level $72$
Weight $8$
Character orbit 72.d
Analytic conductor $22.492$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.4917218349\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 19) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + ( - 4 \beta_{5} + 20 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 - 250) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 19) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + ( - 4 \beta_{5} + 20 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 - 250) q^{8} + (4 \beta_{5} - 52 \beta_{4} + 8 \beta_{3} - 10 \beta_{2} - 12 \beta_1 - 280) q^{10} + (8 \beta_{5} + 11 \beta_{4} - 8 \beta_{3} - 16 \beta_{2} + 208 \beta_1) q^{11} + (5 \beta_{5} + 37 \beta_{4} - 5 \beta_{3} - 32 \beta_{2} + 438 \beta_1) q^{13} + (16 \beta_{5} - 80 \beta_{4} - 16 \beta_{3} + 40 \beta_{2} + 88 \beta_1 - 2008) q^{14} + ( - 40 \beta_{5} - 56 \beta_{4} - 12 \beta_{3} + 76 \beta_{2} + \cdots + 5908) q^{16}+ \cdots + (5376 \beta_{5} - 26880 \beta_{4} + 16128 \beta_{3} + \cdots + 2490039) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8} - 1656 q^{10} - 12048 q^{14} + 35344 q^{16} - 1452 q^{17} + 114768 q^{20} + 152860 q^{22} + 1296 q^{23} - 39314 q^{25} + 316968 q^{26} - 480800 q^{28} - 89280 q^{31} - 817056 q^{32} - 1009108 q^{34} - 974124 q^{38} + 954464 q^{40} - 521244 q^{41} + 1096344 q^{44} + 929840 q^{46} - 1566432 q^{47} - 511050 q^{49} + 148626 q^{50} + 823952 q^{52} - 3270256 q^{55} + 2468928 q^{56} + 3130744 q^{58} + 7055808 q^{62} - 4792768 q^{64} - 1416480 q^{65} - 6608040 q^{68} - 7406912 q^{70} + 7597104 q^{71} + 2089564 q^{73} - 7744200 q^{74} + 9241288 q^{76} + 16015904 q^{79} + 12600384 q^{80} + 10715932 q^{82} + 5639076 q^{86} + 1541200 q^{88} - 2169084 q^{89} - 669600 q^{92} + 15503712 q^{94} - 48537936 q^{95} - 1088308 q^{97} + 14983242 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 10\nu^{3} + 24\nu^{2} + 320\nu + 2560 ) / 512 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 10\nu^{3} - 24\nu^{2} + 7872\nu - 6656 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} - 49\nu^{4} + 242\nu^{3} + 760\nu^{2} + 3136\nu + 26624 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} + 7\nu^{4} + 50\nu^{3} - 104\nu^{2} - 1088\nu - 15872 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{5} + 51\nu^{4} - 182\nu^{3} + 3032\nu^{2} - 2496\nu - 90112 ) / 512 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 16 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} + 14\beta _1 + 152 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} + 68\beta_{4} + 28\beta_{3} - \beta_{2} + 206\beta _1 + 1000 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28\beta_{5} + 164\beta_{4} - 132\beta_{3} + 11\beta_{2} + 2086\beta _1 + 11816 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 140\beta_{5} + 1076\beta_{4} - 20\beta_{3} + 319\beta_{2} - 7090\beta _1 + 136136 ) / 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
5.57668 0.949035i
5.57668 + 0.949035i
0.776001 5.60338i
0.776001 + 5.60338i
−4.85268 2.90715i
−4.85268 + 2.90715i
−11.1534 1.89807i 0 120.795 + 42.3397i 338.443i 0 −438.996 −1266.90 701.506i 0 −642.387 + 3774.77i
37.2 −11.1534 + 1.89807i 0 120.795 42.3397i 338.443i 0 −438.996 −1266.90 + 701.506i 0 −642.387 3774.77i
37.3 −1.55200 11.2068i 0 −123.183 + 34.7858i 184.916i 0 1051.96 581.015 + 1326.49i 0 −2072.30 + 286.989i
37.4 −1.55200 + 11.2068i 0 −123.183 34.7858i 184.916i 0 1051.96 581.015 1326.49i 0 −2072.30 286.989i
37.5 9.70536 5.81430i 0 60.3879 112.860i 324.492i 0 −956.960 −70.1132 1446.46i 0 1886.69 + 3149.31i
37.6 9.70536 + 5.81430i 0 60.3879 + 112.860i 324.492i 0 −956.960 −70.1132 + 1446.46i 0 1886.69 3149.31i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
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 72.8.d.b 6
3.b odd 2 1 8.8.b.a 6
4.b odd 2 1 288.8.d.b 6
8.b even 2 1 inner 72.8.d.b 6
8.d odd 2 1 288.8.d.b 6
12.b even 2 1 32.8.b.a 6
24.f even 2 1 32.8.b.a 6
24.h odd 2 1 8.8.b.a 6
48.i odd 4 2 256.8.a.r 6
48.k even 4 2 256.8.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 3.b odd 2 1
8.8.b.a 6 24.h odd 2 1
32.8.b.a 6 12.b even 2 1
32.8.b.a 6 24.f even 2 1
72.8.d.b 6 1.a even 1 1 trivial
72.8.d.b 6 8.b even 2 1 inner
256.8.a.q 6 48.k even 4 2
256.8.a.r 6 48.i odd 4 2
288.8.d.b 6 4.b odd 2 1
288.8.d.b 6 8.d odd 2 1

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 \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{5} - 40 T^{4} + \cdots + 2097152 \) 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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