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Properties

Label 72.7.b.c
Level $72$
Weight $7$
Character orbit 72.b
Analytic conductor $16.564$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} - 6345392 x^{5} + 7695616 x^{4} - 6850848 x^{3} - 19274256 x^{2} + \cdots + 767595744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{5} + \beta_1 + 2) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_1 + 1) q^{5} + (\beta_{9} + \beta_{5} + \beta_{4} + 9 \beta_1 + 2) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 67) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{5} + \beta_1 + 2) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_1 + 1) q^{5} + (\beta_{9} + \beta_{5} + \beta_{4} + 9 \beta_1 + 2) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 67) q^{8} + (\beta_{11} - \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + \cdots + 179) q^{10}+ \cdots + ( - 192 \beta_{11} + 512 \beta_{10} + 1200 \beta_{9} - 656 \beta_{8} + \cdots - 10485) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8} + 2172 q^{10} - 2720 q^{11} + 6444 q^{14} + 11640 q^{16} + 4888 q^{17} + 3936 q^{19} + 31608 q^{20} - 60432 q^{22} - 27204 q^{25} - 53952 q^{26} - 57072 q^{28} - 109480 q^{32} + 47388 q^{34} - 162336 q^{35} + 89080 q^{38} + 72120 q^{40} + 54280 q^{41} - 49824 q^{43} - 229184 q^{44} + 171864 q^{46} - 304644 q^{49} + 500078 q^{50} + 256848 q^{52} + 699816 q^{56} - 409524 q^{58} + 886144 q^{59} - 691356 q^{62} - 500640 q^{64} - 473376 q^{65} + 1565952 q^{67} - 669104 q^{68} + 473784 q^{70} + 555480 q^{73} + 753720 q^{74} - 293136 q^{76} + 251616 q^{80} + 2317716 q^{82} - 2497760 q^{83} - 476024 q^{86} + 971424 q^{88} - 367400 q^{89} - 4475808 q^{91} + 377376 q^{92} - 2642568 q^{94} - 1165656 q^{97} - 182674 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} - 6345392 x^{5} + 7695616 x^{4} - 6850848 x^{3} - 19274256 x^{2} + \cdots + 767595744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!35 \nu^{11} + \cdots - 14\!\cdots\!36 ) / 39\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 31\!\cdots\!93 \nu^{11} + \cdots - 43\!\cdots\!40 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!15 \nu^{11} + \cdots + 56\!\cdots\!96 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\!\cdots\!77 \nu^{11} + \cdots - 80\!\cdots\!88 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!33 \nu^{11} + \cdots + 12\!\cdots\!52 ) / 39\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!57 \nu^{11} + \cdots - 66\!\cdots\!08 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!36 \nu^{11} + \cdots + 40\!\cdots\!68 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!77 \nu^{11} + \cdots + 10\!\cdots\!88 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!99 \nu^{11} + \cdots - 37\!\cdots\!56 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!83 \nu^{11} + \cdots + 79\!\cdots\!16 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!41 \nu^{11} + \cdots - 25\!\cdots\!80 ) / 75\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} - 2\beta_{10} - 3\beta_{9} - 6\beta_{6} + 12\beta_{5} + 3\beta_{2} - 24\beta _1 + 42 ) / 288 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{11} + 46 \beta_{10} - 9 \beta_{9} - 12 \beta_{8} - 36 \beta_{7} + 54 \beta_{6} - 60 \beta_{5} + 48 \beta_{4} - 103 \beta_{2} + 944 \beta _1 - 1206 ) / 288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 33 \beta_{11} - 42 \beta_{10} + 57 \beta_{9} - 168 \beta_{8} + 72 \beta_{7} + 54 \beta_{6} + 741 \beta_{5} - 204 \beta_{4} - 99 \beta_{3} + 448 \beta_{2} - 1283 \beta _1 + 28965 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3177 \beta_{11} + 3910 \beta_{10} - 2373 \beta_{9} + 5220 \beta_{8} + 2412 \beta_{7} - 8346 \beta_{6} - 6414 \beta_{5} - 3720 \beta_{4} + 2502 \beta_{3} - 4749 \beta_{2} + 165894 \beta _1 - 427620 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16077 \beta_{11} - 20114 \beta_{10} - 1467 \beta_{9} + 4536 \beta_{8} - 55128 \beta_{7} + 84918 \beta_{6} - 231765 \beta_{5} + 73308 \beta_{4} - 261 \beta_{3} + 193322 \beta_{2} + \cdots + 14919327 ) / 288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 62829 \beta_{11} - 200466 \beta_{10} - 26409 \beta_{9} - 249468 \beta_{8} + 129228 \beta_{7} - 200178 \beta_{6} + 1779342 \beta_{5} - 283608 \beta_{4} - 9726 \beta_{3} - 197725 \beta_{2} + \cdots - 35834472 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 127233 \beta_{11} + 15815530 \beta_{10} - 1056489 \beta_{9} + 6833112 \beta_{8} - 2822136 \beta_{7} + 2670258 \beta_{6} - 41880759 \beta_{5} + 3208308 \beta_{4} + \cdots - 1128380043 ) / 288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 42965283 \beta_{11} - 124959842 \beta_{10} + 43704567 \beta_{9} - 61508076 \beta_{8} + 5987964 \beta_{7} + 91189950 \beta_{6} + 84916578 \beta_{5} + \cdots + 39833211204 ) / 288 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 108834915 \beta_{11} - 14175762 \beta_{10} - 80984555 \beta_{9} + 54900152 \beta_{8} + 86200488 \beta_{7} - 301366266 \beta_{6} + 469025271 \beta_{5} + \cdots - 43230321565 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7581327273 \beta_{11} + 16323434890 \beta_{10} + 1920596565 \beta_{9} + 4891236204 \beta_{8} - 17886519804 \beta_{7} + 31446709578 \beta_{6} + \cdots + 1360648800456 ) / 288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16616171415 \beta_{11} - 285454786022 \beta_{10} + 63370238031 \beta_{9} - 239403789192 \beta_{8} + 158088636648 \beta_{7} - 80548435134 \beta_{6} + \cdots + 23495738443101 ) / 288 \) 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
8.16014 − 0.886446i
8.16014 + 0.886446i
−1.87190 + 1.33932i
−1.87190 − 1.33932i
−0.0143493 − 10.5849i
−0.0143493 + 10.5849i
0.630233 + 2.92643i
0.630233 − 2.92643i
−9.37784 − 8.67520i
−9.37784 + 8.67520i
3.47372 + 1.02840i
3.47372 − 1.02840i
−7.97364 − 0.648923i 0 63.1578 + 10.3486i 232.265i 0 483.645i −496.882 − 123.500i 0 150.722 − 1852.00i
19.2 −7.97364 + 0.648923i 0 63.1578 − 10.3486i − 232.265i 0 − 483.645i −496.882 + 123.500i 0 150.722 + 1852.00i
19.3 −7.11414 − 3.65910i 0 37.2219 + 52.0627i − 87.0704i 0 355.505i −74.2991 − 506.580i 0 −318.599 + 619.431i
19.4 −7.11414 + 3.65910i 0 37.2219 − 52.0627i 87.0704i 0 − 355.505i −74.2991 + 506.580i 0 −318.599 − 619.431i
19.5 −1.98950 − 7.74867i 0 −56.0838 + 30.8319i − 82.3007i 0 351.467i 350.485 + 373.235i 0 −637.721 + 163.737i
19.6 −1.98950 + 7.74867i 0 −56.0838 − 30.8319i 82.3007i 0 − 351.467i 350.485 − 373.235i 0 −637.721 − 163.737i
19.7 −0.278171 − 7.99516i 0 −63.8452 + 4.44805i 111.403i 0 − 106.838i 53.3228 + 509.216i 0 890.684 − 30.9891i
19.8 −0.278171 + 7.99516i 0 −63.8452 − 4.44805i − 111.403i 0 106.838i 53.3228 − 509.216i 0 890.684 + 30.9891i
19.9 4.86506 − 6.35069i 0 −16.6624 − 61.7929i 100.822i 0 − 277.765i −473.491 − 194.808i 0 640.287 + 490.503i
19.10 4.86506 + 6.35069i 0 −16.6624 + 61.7929i − 100.822i 0 277.765i −473.491 + 194.808i 0 640.287 − 490.503i
19.11 7.49039 − 2.80965i 0 48.2118 − 42.0907i 128.353i 0 534.624i 242.865 − 450.734i 0 360.627 + 961.414i
19.12 7.49039 + 2.80965i 0 48.2118 + 42.0907i − 128.353i 0 − 534.624i 242.865 + 450.734i 0 360.627 − 961.414i
\(n\): 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
8.d odd 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.c 12
3.b odd 2 1 24.7.b.a ✓ 12
4.b odd 2 1 288.7.b.d 12
8.b even 2 1 288.7.b.d 12
8.d odd 2 1 inner 72.7.b.c 12
12.b even 2 1 96.7.b.a 12
24.f even 2 1 24.7.b.a ✓ 12
24.h odd 2 1 96.7.b.a 12
48.i odd 4 2 768.7.g.l 24
48.k even 4 2 768.7.g.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.b.a ✓ 12 3.b odd 2 1
24.7.b.a ✓ 12 24.f even 2 1
72.7.b.c 12 1.a even 1 1 trivial
72.7.b.c 12 8.d odd 2 1 inner
96.7.b.a 12 12.b even 2 1
96.7.b.a 12 24.h odd 2 1
288.7.b.d 12 4.b odd 2 1
288.7.b.d 12 8.b even 2 1
768.7.g.l 24 48.i odd 4 2
768.7.g.l 24 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 107352 T_{5}^{10} + 3991016688 T_{5}^{8} + 71113492512000 T_{5}^{6} + \cdots + 57\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 10 T^{11} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 107352 T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 858216 T^{10} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + 1360 T^{5} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 26982528 T^{10} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} - 2444 T^{5} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 1968 T^{5} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 879376800 T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + 4040599128 T^{10} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{12} + 7044987816 T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + 20697207840 T^{10} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} - 27140 T^{5} + \cdots - 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 24912 T^{5} + \cdots - 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 86670856608 T^{10} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + 172945812888 T^{10} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{6} - 443072 T^{5} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 348717987744 T^{10} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} - 782976 T^{5} + \cdots - 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 671412159648 T^{10} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} - 277740 T^{5} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 1736387849448 T^{10} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{6} + 1248880 T^{5} + \cdots + 53\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 183700 T^{5} + \cdots + 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 582828 T^{5} + \cdots - 50\!\cdots\!12)^{2} \) Copy content Toggle raw display
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