Properties

Label 72.4.d.d
Level $72$
Weight $4$
Character orbit 72.d
Analytic conductor $4.248$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
Defining polynomial: \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + 3) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 6) q^{7} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 4 \beta_1 + 14) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + 3) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 6) q^{7} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 4 \beta_1 + 14) q^{8} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 10) q^{10} + (4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{11} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2}) q^{13} + (2 \beta_{5} + 4 \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 16) q^{14} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 12 \beta_1 + 14) q^{16} + ( - 8 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 14) q^{17} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} - 12 \beta_1) q^{19} + (12 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 6) q^{20} + ( - 8 \beta_{5} - 16 \beta_{4} - 8 \beta_{2} + 32) q^{22} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 52) q^{23} + ( - 8 \beta_{5} - 8 \beta_{2} - 48 \beta_1 - 39) q^{25} + ( - 16 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} - 12) q^{26} + (6 \beta_{5} - 16 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 24 \beta_1 - 46) q^{28} + (4 \beta_{5} - 11 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - \beta_1) q^{29} + (9 \beta_{5} + \beta_{3} + 6 \beta_{2} + 45 \beta_1 - 86) q^{31} + (10 \beta_{5} - 20 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 16 \beta_1 + 48) q^{32} + ( - 8 \beta_{5} + 16 \beta_{4} - 24 \beta_{2} - 30 \beta_1 - 112) q^{34} + (4 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} - 20 \beta_1) q^{35} + ( - 22 \beta_{5} - 20 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} + 30 \beta_1) q^{37} + ( - 24 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 124) q^{38} + (6 \beta_{5} + 20 \beta_{4} + 24 \beta_{3} + 22 \beta_{2} + 48) q^{40} + (8 \beta_{5} + 12 \beta_{3} - 28 \beta_{2} - 60 \beta_1 - 66) q^{41} + (20 \beta_{5} - 18 \beta_{4} - 28 \beta_{3} - 8 \beta_{2} - 12 \beta_1) q^{43} + (16 \beta_{5} - 32 \beta_{3} + 32 \beta_1 - 176) q^{44} + (4 \beta_{5} + 8 \beta_{4} - 12 \beta_{2} - 60 \beta_1 + 32) q^{46} + (6 \beta_{5} - 2 \beta_{3} + 12 \beta_{2} + 54 \beta_1 + 92) q^{47} + ( - 24 \beta_{5} - 8 \beta_{3} - 72 \beta_1 + 77) q^{49} + ( - 32 \beta_{5} - 39 \beta_1 - 352) q^{50} + (8 \beta_{5} - 8 \beta_{4} - 32 \beta_{3} + 20 \beta_{2} - 52) q^{52} + ( - 20 \beta_{5} + 51 \beta_{4} - \beta_{3} - 21 \beta_{2} + 41 \beta_1) q^{53} + (32 \beta_{5} + 8 \beta_{3} + 8 \beta_{2} + 120 \beta_1 + 224) q^{55} + (10 \beta_{5} + 20 \beta_{4} - 32 \beta_{3} + 22 \beta_{2} - 40 \beta_1 + 308) q^{56} + ( - 2 \beta_{5} - 28 \beta_{4} - 22 \beta_{3} - 14 \beta_{2} + 18) q^{58} + ( - 16 \beta_{5} - 18 \beta_{4} - 16 \beta_{2} + 32 \beta_1) q^{59} + ( - 18 \beta_{5} + 80 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 30 \beta_1) q^{61} + (30 \beta_{5} - 4 \beta_{4} + 6 \beta_{2} - 82 \beta_1 + 336) q^{62} + (12 \beta_{5} - 16 \beta_{4} - 40 \beta_{3} + 40 \beta_{2} + 72 \beta_1 + 180) q^{64} + (8 \beta_{5} - 4 \beta_{3} + 20 \beta_{2} + 84 \beta_1 + 328) q^{65} + (58 \beta_{4} + 48 \beta_{3} + 48 \beta_{2} - 48 \beta_1) q^{67} + (2 \beta_{5} - 32 \beta_{4} + 32 \beta_{3} + 16 \beta_{2} - 96 \beta_1 - 522) q^{68} + ( - 40 \beta_{5} + 48 \beta_{4} + 4 \beta_{3} + 24 \beta_{2} + 220) q^{70} + (42 \beta_{5} + 18 \beta_{3} - 12 \beta_{2} + 90 \beta_1 + 324) q^{71} + (40 \beta_{5} + 24 \beta_{3} - 32 \beta_{2} + 24 \beta_1 + 170) q^{73} + (60 \beta_{5} + 56 \beta_{4} - 40 \beta_{3} + 28 \beta_{2} - 232) q^{74} + (16 \beta_{5} + 72 \beta_{4} + 32 \beta_{3} - 20 \beta_{2} + 96 \beta_1 - 260) q^{76} + ( - 40 \beta_{5} - 128 \beta_{4} + 40 \beta_{3} + 40 \beta_1) q^{77} + ( - 19 \beta_{5} - 11 \beta_{3} + 14 \beta_{2} - 15 \beta_1 - 14) q^{79} + ( - 28 \beta_{5} + 80 \beta_{4} + 40 \beta_{3} + 56 \beta_{2} + 56 \beta_1 + 380) q^{80} + ( - 40 \beta_{5} - 48 \beta_{4} + 72 \beta_{2} - 18 \beta_1 - 368) q^{82} + ( - 4 \beta_{5} + 56 \beta_{4} - 12 \beta_{3} - 16 \beta_{2} + 20 \beta_1) q^{83} + (56 \beta_{5} - 158 \beta_{4} - 70 \beta_{3} - 14 \beta_{2} - 42 \beta_1) q^{85} + ( - 24 \beta_{5} - 112 \beta_{4} - 36 \beta_{3} - 56 \beta_{2} + \cdots + 100) q^{86}+ \cdots + ( - 48 \beta_{5} + 32 \beta_{4} - 48 \beta_{2} + 45 \beta_1 - 576) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8} + 60 q^{10} + 100 q^{14} + 56 q^{16} - 52 q^{17} - 56 q^{20} + 224 q^{22} - 328 q^{23} - 106 q^{25} - 56 q^{26} - 352 q^{28} - 636 q^{31} + 248 q^{32} - 548 q^{34} + 776 q^{38} + 232 q^{40} - 236 q^{41} - 1152 q^{44} + 328 q^{46} + 408 q^{47} + 654 q^{49} - 1970 q^{50} - 368 q^{52} + 1024 q^{55} + 1864 q^{56} + 140 q^{58} + 2108 q^{62} + 832 q^{64} + 1744 q^{65} - 2976 q^{68} + 1352 q^{70} + 1704 q^{71} + 956 q^{73} - 1568 q^{74} - 1744 q^{76} - 44 q^{79} + 2112 q^{80} - 2236 q^{82} + 760 q^{86} + 1856 q^{88} + 220 q^{89} - 1728 q^{92} + 2088 q^{94} - 5104 q^{95} - 2444 q^{97} - 3354 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 3\nu^{3} - 18\nu^{2} - 8\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{5} - 6\nu^{4} + 15\nu^{3} - 18\nu^{2} + 96\nu - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 3\nu^{3} - 6\nu^{2} + 16\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 3\beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} - 3\beta_{2} - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{5} + 2\beta_{3} + 6\beta _1 + 37 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 6\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{5} - 24\beta_{4} + 2\beta_{3} - 12\beta_{2} + 30\beta _1 - 53 ) / 6 \) 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
1.88322 0.673417i
1.88322 + 0.673417i
−0.641412 + 1.89436i
−0.641412 1.89436i
−1.24181 + 1.56777i
−1.24181 1.56777i
−2.55664 1.20980i 0 5.07277 + 6.18604i 0.612661i 0 −22.7441 −5.48534 21.9525i 0 0.741198 1.56635i
37.2 −2.55664 + 1.20980i 0 5.07277 6.18604i 0.612661i 0 −22.7441 −5.48534 + 21.9525i 0 0.741198 + 1.56635i
37.3 −1.25295 2.53577i 0 −4.86025 + 6.35436i 9.15486i 0 27.4175 22.2028 + 4.36281i 0 23.2146 11.4705i
37.4 −1.25295 + 2.53577i 0 −4.86025 6.35436i 9.15486i 0 27.4175 22.2028 4.36281i 0 23.2146 + 11.4705i
37.5 2.80958 0.325969i 0 7.78749 1.83167i 18.5422i 0 9.32669 21.2825 7.68472i 0 6.04419 + 52.0958i
37.6 2.80958 + 0.325969i 0 7.78749 + 1.83167i 18.5422i 0 9.32669 21.2825 + 7.68472i 0 6.04419 52.0958i
\(n\): 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 72.4.d.d 6
3.b odd 2 1 24.4.d.a 6
4.b odd 2 1 288.4.d.d 6
8.b even 2 1 inner 72.4.d.d 6
8.d odd 2 1 288.4.d.d 6
12.b even 2 1 96.4.d.a 6
16.e even 4 1 2304.4.a.bt 3
16.e even 4 1 2304.4.a.bv 3
16.f odd 4 1 2304.4.a.bu 3
16.f odd 4 1 2304.4.a.bw 3
24.f even 2 1 96.4.d.a 6
24.h odd 2 1 24.4.d.a 6
48.i odd 4 1 768.4.a.r 3
48.i odd 4 1 768.4.a.s 3
48.k even 4 1 768.4.a.q 3
48.k even 4 1 768.4.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 3.b odd 2 1
24.4.d.a 6 24.h odd 2 1
72.4.d.d 6 1.a even 1 1 trivial
72.4.d.d 6 8.b even 2 1 inner
96.4.d.a 6 12.b even 2 1
96.4.d.a 6 24.f even 2 1
288.4.d.d 6 4.b odd 2 1
288.4.d.d 6 8.d odd 2 1
768.4.a.q 3 48.k even 4 1
768.4.a.r 3 48.i odd 4 1
768.4.a.s 3 48.i odd 4 1
768.4.a.t 3 48.k even 4 1
2304.4.a.bt 3 16.e even 4 1
2304.4.a.bu 3 16.f odd 4 1
2304.4.a.bv 3 16.e even 4 1
2304.4.a.bw 3 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 428T_{5}^{4} + 28976T_{5}^{2} + 10816 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} - 6 T^{4} - 40 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 428 T^{4} + 28976 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$7$ \( (T^{3} - 14 T^{2} - 580 T + 5816)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 5632 T^{4} + \cdots + 2415919104 \) Copy content Toggle raw display
$13$ \( T^{6} + 4912 T^{4} + \cdots + 3121680384 \) Copy content Toggle raw display
$17$ \( (T^{3} + 26 T^{2} - 11124 T - 477576)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 22960 T^{4} + \cdots + 75488661504 \) Copy content Toggle raw display
$23$ \( (T^{3} + 164 T^{2} + 6384 T + 45504)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 22348 T^{4} + \cdots + 3766031424 \) Copy content Toggle raw display
$31$ \( (T^{3} + 318 T^{2} + 4476 T - 3749624)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 179776 T^{4} + \cdots + 6879707136 \) Copy content Toggle raw display
$41$ \( (T^{3} + 118 T^{2} - 117300 T - 19985976)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 229552 T^{4} + \cdots + 73984219582464 \) Copy content Toggle raw display
$47$ \( (T^{3} - 204 T^{2} - 27792 T + 1964736)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 427051482970176 \) Copy content Toggle raw display
$59$ \( T^{6} + 138416 T^{4} + \cdots + 72651484205056 \) Copy content Toggle raw display
$61$ \( T^{6} + 902016 T^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{6} + 1054512 T^{4} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{3} - 852 T^{2} - 66960 T + 85084992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 478 T^{2} - 255956 T + 120833304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 22 T^{2} - 71524 T - 7902616)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 520448 T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} - 110 T^{2} - 41364 T - 1423656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 1222 T^{2} + 251660 T - 74802424)^{2} \) Copy content Toggle raw display
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