Properties

Label 72.7.b.b
Level $72$
Weight $7$
Character orbit 72.b
Analytic conductor $16.564$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.0.3803625.2
Defining polynomial: \( x^{4} - x^{3} + 6x^{2} - 16x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 - 11) q^{4} + ( - 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 + 4) q^{5} + (4 \beta_{3} - 20 \beta_{2} + 8 \beta_1 + 8) q^{7} + (2 \beta_{3} + 22 \beta_{2} + 30 \beta_1 - 74) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 - 11) q^{4} + ( - 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 + 4) q^{5} + (4 \beta_{3} - 20 \beta_{2} + 8 \beta_1 + 8) q^{7} + (2 \beta_{3} + 22 \beta_{2} + 30 \beta_1 - 74) q^{8} + ( - 4 \beta_{3} + 28 \beta_{2} + 68 \beta_1 - 492) q^{10} + ( - 114 \beta_{2} - 57 \beta_1 - 187) q^{11} + ( - 18 \beta_{3} + 202 \beta_{2} - 92 \beta_1 - 92) q^{13} + ( - 24 \beta_{3} - 24 \beta_{2} - 104 \beta_1 - 1416) q^{14} + (20 \beta_{3} + 60 \beta_{2} - 84 \beta_1 - 3684) q^{16} + (400 \beta_{2} + 200 \beta_1 - 1242) q^{17} + (130 \beta_{2} + 65 \beta_1 - 429) q^{19} + (32 \beta_{3} + 576 \beta_{2} + 96 \beta_1 - 8224) q^{20} + ( - 114 \beta_{3} + 244 \beta_{2} + 114 \beta_1 + 6042) q^{22} + ( - 172 \beta_{3} - 676 \beta_{2} + 424 \beta_1 + 424) q^{23} + (1760 \beta_{2} + 880 \beta_1 - 6855) q^{25} + (220 \beta_{3} - 4 \beta_{2} + 356 \beta_1 + 14772) q^{26} + (1600 \beta_{2} + 768 \beta_1 + 14080) q^{28} + ( - 102 \beta_{3} + 1742 \beta_{2} - 820 \beta_1 - 820) q^{29} + ( - 176 \beta_{3} + 1392 \beta_{2} - 608 \beta_1 - 608) q^{31} + (40 \beta_{3} + 3320 \beta_{2} - 680 \beta_1 + 10552) q^{32} + (400 \beta_{3} + 1042 \beta_{2} - 400 \beta_1 - 21200) q^{34} + (1600 \beta_{2} + 800 \beta_1 + 11680) q^{35} + ( - 526 \beta_{3} + 214 \beta_{2} + 156 \beta_1 + 156) q^{37} + (130 \beta_{3} + 364 \beta_{2} - 130 \beta_1 - 6890) q^{38} + (544 \beta_{3} + 7392 \beta_{2} - 1568 \beta_1 - 7328) q^{40} + ( - 2208 \beta_{2} - 1104 \beta_1 + 30590) q^{41} + (4242 \beta_{2} + 2121 \beta_1 + 47243) q^{43} + (358 \beta_{3} - 4918 \beta_{2} + 3290 \beta_1 - 7006) q^{44} + ( - 504 \beta_{3} + 2568 \beta_{2} + 6008 \beta_1 - 54312) q^{46} + ( - 1848 \beta_{3} + 3608 \beta_{2} - 880 \beta_1 - 880) q^{47} + ( - 11392 \beta_{2} - 5696 \beta_1 + 6225) q^{49} + (1760 \beta_{3} + 5975 \beta_{2} - 1760 \beta_1 - 93280) q^{50} + ( - 224 \beta_{3} - 16832 \beta_{2} - 6816 \beta_1 - 55072) q^{52} + ( - 2218 \beta_{3} - 4862 \beta_{2} + 3540 \beta_1 + 3540) q^{53} + ( - 652 \beta_{3} - 11076 \beta_{2} + 5864 \beta_1 + 5864) q^{55} + (1600 \beta_{3} - 14912 \beta_{2} - 1600 \beta_1 - 80704) q^{56} + (1844 \beta_{3} - 620 \beta_{2} + 1420 \beta_1 + 128508) q^{58} + (654 \beta_{2} + 327 \beta_1 - 135835) q^{59} + ( - 2346 \beta_{3} + 3458 \beta_{2} - 556 \beta_1 - 556) q^{61} + (1568 \beta_{3} + 544 \beta_{2} + 4064 \beta_1 + 100704) q^{62} + (3280 \beta_{3} - 14992 \beta_{2} - 4560 \beta_1 + 121840) q^{64} + ( - 25120 \beta_{2} - 12560 \beta_1 + 63920) q^{65} + ( - 11934 \beta_{2} - 5967 \beta_1 - 191581) q^{67} + (642 \beta_{3} + 15358 \beta_{2} - 13442 \beta_1 + 45462) q^{68} + (1600 \beta_{3} - 12480 \beta_{2} - 1600 \beta_1 - 84800) q^{70} + ( - 2468 \beta_{3} - 31180 \beta_{2} + 16824 \beta_1 + 16824) q^{71} + ( - 17072 \beta_{2} - 8536 \beta_1 + 119514) q^{73} + (740 \beta_{3} + 5572 \beta_{2} + 16092 \beta_1 + 5004) q^{74} + (234 \beta_{3} + 4966 \beta_{2} - 4394 \beta_1 + 15054) q^{76} + ( - 5992 \beta_{3} + 26312 \beta_{2} - 10160 \beta_1 - 10160) q^{77} + ( - 1144 \beta_{3} - 20904 \beta_{2} + 11024 \beta_1 + 11024) q^{79} + (6848 \beta_{3} - 7616 \beta_{2} - 24256 \beta_1 + 258624) q^{80} + ( - 2208 \beta_{3} - 29486 \beta_{2} + 2208 \beta_1 + 117024) q^{82} + ( - 6178 \beta_{2} - 3089 \beta_1 + 869341) q^{83} + (6084 \beta_{3} + 50252 \beta_{2} - 28168 \beta_1 - 28168) q^{85} + (4242 \beta_{3} - 49364 \beta_{2} - 4242 \beta_1 - 224826) q^{86} + ( - 5276 \beta_{3} + 11276 \beta_{2} - 6180 \beta_1 - 490612) q^{88} + (23856 \beta_{2} + 11928 \beta_1 - 202234) q^{89} + (79936 \beta_{2} + 39968 \beta_1 + 809632) q^{91} + (3072 \beta_{3} + 63296 \beta_{2} + 13056 \beta_1 - 719104) q^{92} + (5456 \beta_{3} + 16720 \beta_{2} + 53680 \beta_1 + 231792) q^{94} + (2028 \beta_{3} + 16484 \beta_{2} - 9256 \beta_1 - 9256) q^{95} + ( - 85392 \beta_{2} - 42696 \beta_1 - 188998) q^{97} + ( - 11392 \beta_{3} - 529 \beta_{2} + 11392 \beta_1 + 603776) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 44 q^{4} - 248 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 44 q^{4} - 248 q^{8} - 1920 q^{10} - 976 q^{11} - 5760 q^{14} - 14576 q^{16} - 4168 q^{17} - 1456 q^{19} - 31680 q^{20} + 24428 q^{22} - 23900 q^{25} + 59520 q^{26} + 59520 q^{28} + 48928 q^{32} - 81916 q^{34} + 49920 q^{35} - 26572 q^{38} - 13440 q^{40} + 117944 q^{41} + 197456 q^{43} - 37144 q^{44} - 213120 q^{46} + 2116 q^{49} - 357650 q^{50} - 254400 q^{52} - 349440 q^{56} + 516480 q^{58} - 542032 q^{59} + 407040 q^{62} + 463936 q^{64} + 205440 q^{65} - 790192 q^{67} + 213848 q^{68} - 360960 q^{70} + 443912 q^{73} + 32640 q^{74} + 70616 q^{76} + 1032960 q^{80} + 404708 q^{82} + 3465008 q^{83} - 989548 q^{86} - 1950448 q^{88} - 761224 q^{89} + 3398400 q^{91} - 2743680 q^{92} + 971520 q^{94} - 926776 q^{97} + 2391262 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 6x^{2} - 16x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} - 6\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 31\nu^{2} + 6\nu + 80 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 31\beta_{2} - 6\beta _1 + 46 ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
2.81174 2.84502i
2.81174 + 2.84502i
−2.31174 3.26433i
−2.31174 + 3.26433i
−5.62348 5.69004i 0 −0.753049 + 63.9956i 59.7107i 0 483.584i 368.372 355.593i 0 339.756 335.782i
19.2 −5.62348 + 5.69004i 0 −0.753049 63.9956i 59.7107i 0 483.584i 368.372 + 355.593i 0 339.756 + 335.782i
19.3 4.62348 6.52867i 0 −21.2470 60.3702i 199.084i 0 19.6656i −492.372 140.406i 0 −1299.76 920.462i
19.4 4.62348 + 6.52867i 0 −21.2470 + 60.3702i 199.084i 0 19.6656i −492.372 + 140.406i 0 −1299.76 + 920.462i
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 8.7.d.b 4
4.b odd 2 1 288.7.b.b 4
8.b even 2 1 288.7.b.b 4
8.d odd 2 1 inner 72.7.b.b 4
12.b even 2 1 32.7.d.b 4
24.f even 2 1 8.7.d.b 4
24.h odd 2 1 32.7.d.b 4
48.i odd 4 2 256.7.c.l 8
48.k even 4 2 256.7.c.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 3.b odd 2 1
8.7.d.b 4 24.f even 2 1
32.7.d.b 4 12.b even 2 1
32.7.d.b 4 24.h odd 2 1
72.7.b.b 4 1.a even 1 1 trivial
72.7.b.b 4 8.d odd 2 1 inner
256.7.c.l 8 48.i odd 4 2
256.7.c.l 8 48.k even 4 2
288.7.b.b 4 4.b odd 2 1
288.7.b.b 4 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 43200T_{5}^{2} + 141312000 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 24 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 43200 T^{2} + \cdots + 141312000 \) Copy content Toggle raw display
$7$ \( T^{4} + 234240 T^{2} + \cdots + 90439680 \) Copy content Toggle raw display
$11$ \( (T^{2} + 488 T - 1305044)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 14312640 T^{2} + \cdots + 28641121812480 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2084 T - 15714236)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 728 T - 1642004)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 380025600 T^{2} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + 969574080 T^{2} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{4} + 791654400 T^{2} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + 2141703360 T^{2} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( (T^{2} - 58972 T + 357521476)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 98728 T + 547375276)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 29538094080 T^{2} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{4} + 46462898880 T^{2} + \cdots + 29\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( (T^{2} + 271016 T + 18317507884)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 45212594880 T^{2} + \cdots + 33\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{2} + 395096 T + 24071074924)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 348036514560 T^{2} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( (T^{2} - 221956 T - 18286467836)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 144092482560 T^{2} + \cdots + 31\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1732504 T + 746384920684)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 380612 T - 23540043644)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 463388 T - 711956225084)^{2} \) Copy content Toggle raw display
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