Properties

Label 72.8.d.c
Level $72$
Weight $8$
Character orbit 72.d
Analytic conductor $22.492$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{6}\cdot 5^{2}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 34) q^{4} + (\beta_{5} - 3 \beta_1) q^{5} + ( - \beta_{8} - \beta_{3} + 12) q^{7} + (\beta_{6} + 2 \beta_{5} + \beta_{2} + 34 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 34) q^{4} + (\beta_{5} - 3 \beta_1) q^{5} + ( - \beta_{8} - \beta_{3} + 12) q^{7} + (\beta_{6} + 2 \beta_{5} + \beta_{2} + 34 \beta_1) q^{8} + ( - \beta_{10} + \beta_{8} - 4 \beta_{3} + 391) q^{10} + (\beta_{7} + 2 \beta_{6} - 10 \beta_{5} - 2 \beta_{2} - 21 \beta_1) q^{11} + (2 \beta_{10} - \beta_{9} - \beta_{4} + 8 \beta_{3} - 3) q^{13} + (\beta_{11} - 2 \beta_{6} + 16 \beta_{5} + 5 \beta_{2} + 10 \beta_1) q^{14} + ( - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \beta_{4} + 40 \beta_{3} + \cdots - 1054) q^{16}+ \cdots + ( - 56 \beta_{11} + 4592 \beta_{6} + 8064 \beta_{5} + \cdots + 228397 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 206\nu^{9} + 24336\nu^{7} - 1510912\nu^{5} + 398721024\nu^{3} - 55297703936\nu ) / 34359738368 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{11} - 1854\nu^{9} + 219024\nu^{7} - 13598208\nu^{5} + 3588489216\nu^{3} + 601832292352\nu ) / 34359738368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} + 206\nu^{8} - 24336\nu^{6} + 1510912\nu^{4} - 398721024\nu^{2} + 46170898432 ) / 268435456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 206\nu^{8} - 24336\nu^{6} + 1510912\nu^{4} + 1748762624\nu^{2} - 27648851968 ) / 134217728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 213 \nu^{11} - 68454 \nu^{9} + 1857616 \nu^{7} - 265594368 \nu^{5} + 87565795328 \nu^{3} - 17613124009984 \nu ) / 171798691840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 389 \nu^{11} + 50938 \nu^{9} - 756912 \nu^{7} + 1293400576 \nu^{5} + 26052132864 \nu^{3} + 17324287459328 \nu ) / 171798691840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1057 \nu^{11} + 316046 \nu^{9} - 12419344 \nu^{7} + 1372114432 \nu^{5} - 88403607552 \nu^{3} + 65983045697536 \nu ) / 171798691840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -45\nu^{10} + 5174\nu^{8} - 251344\nu^{6} + 35419648\nu^{4} - 13230145536\nu^{2} + 1105417207808 ) / 268435456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{10} + 406\nu^{8} - 6864\nu^{6} + 10163712\nu^{4} + 355729408\nu^{2} + 137610919936 ) / 20971520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{10} - 43614\nu^{8} + 722576\nu^{6} - 96050688\nu^{4} + 23408672768\nu^{2} - 11655199064064 ) / 1342177280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4859 \nu^{11} + 378362 \nu^{9} - 23549104 \nu^{7} + 3397302784 \nu^{5} - 1392377593856 \nu^{3} + 74103755112448 \nu ) / 34359738368 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 9\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 2\beta_{3} + 550 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} + 32\beta_{5} + 23\beta_{2} + 121\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{10} + 16\beta_{9} + 16\beta_{8} + 3\beta_{4} - 134\beta_{3} - 8270 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32\beta_{11} + 120\beta_{7} + 1024\beta_{6} + 1888\beta_{5} + 269\beta_{2} + 18339\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -400\beta_{10} + 240\beta_{9} + 1648\beta_{8} - 719\beta_{4} - 62946\beta_{3} - 1156218 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1248\beta_{11} - 3544\beta_{7} + 15360\beta_{6} - 90336\beta_{5} + 21343\beta_{2} + 7776049\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -73008\beta_{10} - 7088\beta_{9} + 6864\beta_{8} + 63787\beta_{4} - 606422\beta_{3} - 498298078 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 66144 \beta_{11} + 415800 \beta_{7} - 453632 \beta_{6} - 6358944 \beta_{5} - 15820155 \beta_{2} + 215071403 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2064400\beta_{10} + 831600\beta_{9} - 6297616\beta_{8} - 13409015\beta_{4} + 76598894\beta_{3} - 13387313770 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8362016 \beta_{11} - 112307736 \beta_{7} + 53222400 \beta_{6} - 546233824 \beta_{5} - 553630585 \beta_{2} - 5901619319 \beta_1 \) 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
10.6405 3.84452i
10.6405 + 3.84452i
10.2927 4.69692i
10.2927 + 4.69692i
4.93370 10.1813i
4.93370 + 10.1813i
−4.93370 10.1813i
−4.93370 + 10.1813i
−10.2927 4.69692i
−10.2927 + 4.69692i
−10.6405 3.84452i
−10.6405 + 3.84452i
−10.6405 3.84452i 0 98.4393 + 81.8151i 209.469i 0 −1301.46 −732.900 1249.00i 0 805.310 2228.85i
37.2 −10.6405 + 3.84452i 0 98.4393 81.8151i 209.469i 0 −1301.46 −732.900 + 1249.00i 0 805.310 + 2228.85i
37.3 −10.2927 4.69692i 0 83.8779 + 96.6877i 316.260i 0 1193.16 −409.193 1389.14i 0 −1485.45 + 3255.16i
37.4 −10.2927 + 4.69692i 0 83.8779 96.6877i 316.260i 0 1193.16 −409.193 + 1389.14i 0 −1485.45 3255.16i
37.5 −4.93370 10.1813i 0 −79.3172 + 100.463i 181.720i 0 142.301 1414.17 + 311.897i 0 1850.14 896.551i
37.6 −4.93370 + 10.1813i 0 −79.3172 100.463i 181.720i 0 142.301 1414.17 311.897i 0 1850.14 + 896.551i
37.7 4.93370 10.1813i 0 −79.3172 100.463i 181.720i 0 142.301 −1414.17 + 311.897i 0 1850.14 + 896.551i
37.8 4.93370 + 10.1813i 0 −79.3172 + 100.463i 181.720i 0 142.301 −1414.17 311.897i 0 1850.14 896.551i
37.9 10.2927 4.69692i 0 83.8779 96.6877i 316.260i 0 1193.16 409.193 1389.14i 0 −1485.45 3255.16i
37.10 10.2927 + 4.69692i 0 83.8779 + 96.6877i 316.260i 0 1193.16 409.193 + 1389.14i 0 −1485.45 + 3255.16i
37.11 10.6405 3.84452i 0 98.4393 81.8151i 209.469i 0 −1301.46 732.900 1249.00i 0 805.310 + 2228.85i
37.12 10.6405 + 3.84452i 0 98.4393 + 81.8151i 209.469i 0 −1301.46 732.900 + 1249.00i 0 805.310 2228.85i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 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.c 12
3.b odd 2 1 inner 72.8.d.c 12
4.b odd 2 1 288.8.d.c 12
8.b even 2 1 inner 72.8.d.c 12
8.d odd 2 1 288.8.d.c 12
12.b even 2 1 288.8.d.c 12
24.f even 2 1 288.8.d.c 12
24.h odd 2 1 inner 72.8.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.d.c 12 1.a even 1 1 trivial
72.8.d.c 12 3.b odd 2 1 inner
72.8.d.c 12 8.b even 2 1 inner
72.8.d.c 12 24.h odd 2 1 inner
288.8.d.c 12 4.b odd 2 1
288.8.d.c 12 8.d odd 2 1
288.8.d.c 12 12.b even 2 1
288.8.d.c 12 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 176920T_{5}^{4} + 9140446400T_{5}^{2} + 144921907520000 \) acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 206 T^{10} + \cdots + 4398046511104 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 176920 T^{4} + \cdots + 144921907520000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - 34 T^{2} - 1568260 T + 220971400)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} + 60939872 T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 243382400 T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 1466921984 T^{4} + \cdots - 77\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 2939609600 T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 11881926656 T^{4} + \cdots - 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 47032252568 T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 9306 T^{2} + \cdots + 647461203489880)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 317465782400 T^{4} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 638791439360 T^{4} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 792374551040 T^{4} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 1408897142784 T^{4} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 1292368740440 T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 360667408768 T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 17526568763520 T^{4} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 38333221816320 T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 37178352599040 T^{4} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 3404762 T^{2} + \cdots + 34\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 5117638 T^{2} + \cdots + 54\!\cdots\!20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 57199026875488 T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 142801294807040 T^{4} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 6860614 T^{2} + \cdots - 16\!\cdots\!00)^{4} \) Copy content Toggle raw display
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