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Properties

Label 72.7.b.d
Level $72$
Weight $7$
Character orbit 72.b
Analytic conductor $16.564$
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: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.5638940206\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 13) q^{4} + ( - \beta_{8} - 2 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{10} - 13 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 13) q^{4} + ( - \beta_{8} - 2 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{10} - 13 \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - 2 \beta_{3} + 118) q^{10} + ( - \beta_{10} - \beta_{6} - \beta_{4} + 20 \beta_1) q^{11} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - 12 \beta_{3}) q^{13} + (5 \beta_{8} + 2 \beta_{6} + \beta_{2} + \beta_1) q^{14} + ( - 2 \beta_{11} - 4 \beta_{7} + \beta_{5} - 13 \beta_{3} - 122) q^{16} + ( - 6 \beta_{10} + 2 \beta_{8} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} + 65 \beta_1) q^{17} + (2 \beta_{11} - 6 \beta_{9} + 2 \beta_{7} + 22 \beta_{3} + 328) q^{19} + ( - 2 \beta_{10} + 26 \beta_{8} - 4 \beta_{6} - 6 \beta_{4} + 2 \beta_{2} + 118 \beta_1) q^{20} + (4 \beta_{11} - 2 \beta_{9} - 22 \beta_{7} - 6 \beta_{5} + 30 \beta_{3} + \cdots + 1324) q^{22}+ \cdots + (1376 \beta_{10} + 5040 \beta_{8} - 224 \beta_{6} - 3312 \beta_{4} + \cdots - 26315 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 156 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 156 q^{4} + 1416 q^{10} - 1464 q^{16} + 3936 q^{19} + 15888 q^{22} - 47796 q^{25} + 11256 q^{28} + 50016 q^{34} + 70896 q^{40} - 340704 q^{43} + 213696 q^{46} - 304644 q^{49} + 548016 q^{52} - 38616 q^{58} + 206544 q^{64} - 962112 q^{67} + 1074480 q^{70} - 1069560 q^{73} + 1064352 q^{76} - 694944 q^{82} - 3072672 q^{88} + 775008 q^{91} + 3752256 q^{94} - 86952 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 78\nu^{9} + 3408\nu^{7} + 73216\nu^{5} + 13959168\nu^{3} + 1308622848\nu ) / 1073741824 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 37 \nu^{11} - 17222 \nu^{9} - 720016 \nu^{7} + 22882816 \nu^{5} + 46324449280 \nu^{3} + 433556815872 \nu ) / 11811160064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 78\nu^{8} - 3408\nu^{6} - 73216\nu^{4} - 13959168\nu^{2} - 1090519040 ) / 16777216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{11} - 546\nu^{9} - 23856\nu^{7} - 512512\nu^{5} - 97714176\nu^{3} + 25199378432\nu ) / 1073741824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} - 78\nu^{8} - 3408\nu^{6} - 73216\nu^{4} + 254476288\nu^{2} + 2399141888 ) / 16777216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - 1102\nu^{9} + 47792\nu^{7} - 1728000\nu^{5} - 162332672\nu^{3} - 14998831104\nu ) / 134217728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{10} + 1390\nu^{8} - 48048\nu^{6} + 2776576\nu^{4} - 125763584\nu^{2} + 8908701696 ) / 92274688 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -43\nu^{11} + 230\nu^{9} + 1936\nu^{7} - 9546240\nu^{5} - 499318784\nu^{3} - 11408506880\nu ) / 2952790016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71\nu^{10} + 930\nu^{8} - 51920\nu^{6} + 21869056\nu^{4} + 872873984\nu^{2} + 40214986752 ) / 92274688 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -65\nu^{11} - 974\nu^{9} + 97968\nu^{7} + 9200128\nu^{5} - 607453184\nu^{3} - 27883732992\nu ) / 1073741824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -303\nu^{10} - 11346\nu^{8} + 712272\nu^{6} + 39091712\nu^{4} - 1438318592\nu^{2} - 202651992064 ) / 184549376 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} + 7\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - 208 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{8} - 7\beta_{4} + 4\beta_{2} + 27\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{11} + 24\beta_{9} + 24\beta_{7} - 7\beta_{5} + 31\beta_{3} - 976 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 320\beta_{10} - 1436\beta_{8} - 64\beta_{6} - 19\beta_{4} - 28\beta_{2} - 2393\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 328\beta_{11} - 296\beta_{9} - 1320\beta_{7} - 259\beta_{5} - 8997\beta_{3} + 68848 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5952\beta_{10} + 10612\beta_{8} + 4544\beta_{6} - 1423\beta_{4} + 756\beta_{2} + 581747\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3224\beta_{11} - 8904\beta_{9} + 63800\beta_{7} - 1279\beta_{5} - 198857\beta_{3} - 18562128 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 19520\beta_{10} + 772996\beta_{8} - 122688\beta_{6} - 736699\beta_{4} - 42236\beta_{2} + 7075183\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -226936\beta_{11} + 379800\beta_{9} - 1583208\beta_{7} - 537835\beta_{5} - 767613\beta_{3} - 234853008 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8761024 \beta_{10} - 29535788 \beta_{8} + 1499072 \beta_{6} - 4669783 \beta_{4} - 2230444 \beta_{2} + 14244635 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
7.38480 − 3.07648i
7.38480 + 3.07648i
4.24448 − 6.78118i
4.24448 + 6.78118i
1.98725 − 7.74925i
1.98725 + 7.74925i
−1.98725 − 7.74925i
−1.98725 + 7.74925i
−4.24448 − 6.78118i
−4.24448 + 6.78118i
−7.38480 − 3.07648i
−7.38480 + 3.07648i
−7.38480 − 3.07648i 0 45.0705 + 45.4384i − 70.4379i 0 87.7768i −193.046 − 474.212i 0 −216.701 + 520.170i
19.2 −7.38480 + 3.07648i 0 45.0705 − 45.4384i 70.4379i 0 − 87.7768i −193.046 + 474.212i 0 −216.701 − 520.170i
19.3 −4.24448 − 6.78118i 0 −27.9688 + 57.5651i 206.098i 0 210.403i 509.073 − 54.6722i 0 1397.59 − 874.779i
19.4 −4.24448 + 6.78118i 0 −27.9688 − 57.5651i − 206.098i 0 − 210.403i 509.073 + 54.6722i 0 1397.59 + 874.779i
19.5 −1.98725 − 7.74925i 0 −56.1017 + 30.7994i − 106.706i 0 − 614.112i 350.160 + 373.539i 0 −826.887 + 212.051i
19.6 −1.98725 + 7.74925i 0 −56.1017 − 30.7994i 106.706i 0 614.112i 350.160 − 373.539i 0 −826.887 − 212.051i
19.7 1.98725 − 7.74925i 0 −56.1017 − 30.7994i − 106.706i 0 614.112i −350.160 + 373.539i 0 −826.887 − 212.051i
19.8 1.98725 + 7.74925i 0 −56.1017 + 30.7994i 106.706i 0 − 614.112i −350.160 − 373.539i 0 −826.887 + 212.051i
19.9 4.24448 − 6.78118i 0 −27.9688 − 57.5651i 206.098i 0 − 210.403i −509.073 − 54.6722i 0 1397.59 + 874.779i
19.10 4.24448 + 6.78118i 0 −27.9688 + 57.5651i − 206.098i 0 210.403i −509.073 + 54.6722i 0 1397.59 − 874.779i
19.11 7.38480 − 3.07648i 0 45.0705 − 45.4384i − 70.4379i 0 − 87.7768i 193.046 − 474.212i 0 −216.701 − 520.170i
19.12 7.38480 + 3.07648i 0 45.0705 + 45.4384i 70.4379i 0 87.7768i 193.046 + 474.212i 0 −216.701 + 520.170i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.b.d ✓ 12
3.b odd 2 1 inner 72.7.b.d ✓ 12
4.b odd 2 1 288.7.b.c 12
8.b even 2 1 288.7.b.c 12
8.d odd 2 1 inner 72.7.b.d ✓ 12
12.b even 2 1 288.7.b.c 12
24.f even 2 1 inner 72.7.b.d ✓ 12
24.h odd 2 1 288.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.b.d ✓ 12 1.a even 1 1 trivial
72.7.b.d ✓ 12 3.b odd 2 1 inner
72.7.b.d ✓ 12 8.d odd 2 1 inner
72.7.b.d ✓ 12 24.f even 2 1 inner
288.7.b.c 12 4.b odd 2 1
288.7.b.c 12 8.b even 2 1
288.7.b.c 12 12.b even 2 1
288.7.b.c 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 58824T_{5}^{4} + 750878400T_{5}^{2} + 2399578560000 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 78 T^{10} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 58824 T^{4} + \cdots + 2399578560000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 429108 T^{4} + \cdots + 128634653983680)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 7021536 T^{4} + \cdots - 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 16498128 T^{4} + \cdots + 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 85317504 T^{4} + \cdots - 51\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 984 T^{2} + \cdots + 40086711808)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 405312000 T^{4} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 1501983624 T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 1586576052 T^{4} + \cdots + 28\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6158858832 T^{4} + \cdots + 91\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 7294871424 T^{4} + \cdots - 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 85176 T^{2} + \cdots - 657572380382720)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 9235517952 T^{4} + \cdots + 81\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 43574252040 T^{4} + \cdots + 44\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 149144563584 T^{4} + \cdots - 35\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 212406839760 T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 240528 T^{2} + \cdots - 18\!\cdots\!40)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 381073489920 T^{4} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 267390 T^{2} + \cdots - 19\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 692559891828 T^{4} + \cdots + 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 844970997984 T^{4} + \cdots - 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 3016772427264 T^{4} + \cdots - 62\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 21738 T^{2} + \cdots - 71\!\cdots\!00)^{4} \) Copy content Toggle raw display
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