# 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 $2$

# Related objects

## 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: $$x^{4} - x^{3} + 6x^{2} - 16x + 256$$ x^4 - x^3 + 6*x^2 - 16*x + 256 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1 - 11) q^{4} + ( - 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 + 4) q^{5} + (4 \beta_{3} - 20 \beta_{2} + 8 \beta_1 + 8) q^{7} + (2 \beta_{3} + 22 \beta_{2} + 30 \beta_1 - 74) q^{8}+O(q^{10})$$ q - b2 * q^2 + (-b3 + b2 + b1 - 11) * q^4 + (-2*b3 - 6*b2 + 4*b1 + 4) * q^5 + (4*b3 - 20*b2 + 8*b1 + 8) * q^7 + (2*b3 + 22*b2 + 30*b1 - 74) * q^8 $$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})$$ q - b2 * q^2 + (-b3 + b2 + b1 - 11) * q^4 + (-2*b3 - 6*b2 + 4*b1 + 4) * q^5 + (4*b3 - 20*b2 + 8*b1 + 8) * q^7 + (2*b3 + 22*b2 + 30*b1 - 74) * q^8 + (-4*b3 + 28*b2 + 68*b1 - 492) * q^10 + (-114*b2 - 57*b1 - 187) * q^11 + (-18*b3 + 202*b2 - 92*b1 - 92) * q^13 + (-24*b3 - 24*b2 - 104*b1 - 1416) * q^14 + (20*b3 + 60*b2 - 84*b1 - 3684) * q^16 + (400*b2 + 200*b1 - 1242) * q^17 + (130*b2 + 65*b1 - 429) * q^19 + (32*b3 + 576*b2 + 96*b1 - 8224) * q^20 + (-114*b3 + 244*b2 + 114*b1 + 6042) * q^22 + (-172*b3 - 676*b2 + 424*b1 + 424) * q^23 + (1760*b2 + 880*b1 - 6855) * q^25 + (220*b3 - 4*b2 + 356*b1 + 14772) * q^26 + (1600*b2 + 768*b1 + 14080) * q^28 + (-102*b3 + 1742*b2 - 820*b1 - 820) * q^29 + (-176*b3 + 1392*b2 - 608*b1 - 608) * q^31 + (40*b3 + 3320*b2 - 680*b1 + 10552) * q^32 + (400*b3 + 1042*b2 - 400*b1 - 21200) * q^34 + (1600*b2 + 800*b1 + 11680) * q^35 + (-526*b3 + 214*b2 + 156*b1 + 156) * q^37 + (130*b3 + 364*b2 - 130*b1 - 6890) * q^38 + (544*b3 + 7392*b2 - 1568*b1 - 7328) * q^40 + (-2208*b2 - 1104*b1 + 30590) * q^41 + (4242*b2 + 2121*b1 + 47243) * q^43 + (358*b3 - 4918*b2 + 3290*b1 - 7006) * q^44 + (-504*b3 + 2568*b2 + 6008*b1 - 54312) * q^46 + (-1848*b3 + 3608*b2 - 880*b1 - 880) * q^47 + (-11392*b2 - 5696*b1 + 6225) * q^49 + (1760*b3 + 5975*b2 - 1760*b1 - 93280) * q^50 + (-224*b3 - 16832*b2 - 6816*b1 - 55072) * q^52 + (-2218*b3 - 4862*b2 + 3540*b1 + 3540) * q^53 + (-652*b3 - 11076*b2 + 5864*b1 + 5864) * q^55 + (1600*b3 - 14912*b2 - 1600*b1 - 80704) * q^56 + (1844*b3 - 620*b2 + 1420*b1 + 128508) * q^58 + (654*b2 + 327*b1 - 135835) * q^59 + (-2346*b3 + 3458*b2 - 556*b1 - 556) * q^61 + (1568*b3 + 544*b2 + 4064*b1 + 100704) * q^62 + (3280*b3 - 14992*b2 - 4560*b1 + 121840) * q^64 + (-25120*b2 - 12560*b1 + 63920) * q^65 + (-11934*b2 - 5967*b1 - 191581) * q^67 + (642*b3 + 15358*b2 - 13442*b1 + 45462) * q^68 + (1600*b3 - 12480*b2 - 1600*b1 - 84800) * q^70 + (-2468*b3 - 31180*b2 + 16824*b1 + 16824) * q^71 + (-17072*b2 - 8536*b1 + 119514) * q^73 + (740*b3 + 5572*b2 + 16092*b1 + 5004) * q^74 + (234*b3 + 4966*b2 - 4394*b1 + 15054) * q^76 + (-5992*b3 + 26312*b2 - 10160*b1 - 10160) * q^77 + (-1144*b3 - 20904*b2 + 11024*b1 + 11024) * q^79 + (6848*b3 - 7616*b2 - 24256*b1 + 258624) * q^80 + (-2208*b3 - 29486*b2 + 2208*b1 + 117024) * q^82 + (-6178*b2 - 3089*b1 + 869341) * q^83 + (6084*b3 + 50252*b2 - 28168*b1 - 28168) * q^85 + (4242*b3 - 49364*b2 - 4242*b1 - 224826) * q^86 + (-5276*b3 + 11276*b2 - 6180*b1 - 490612) * q^88 + (23856*b2 + 11928*b1 - 202234) * q^89 + (79936*b2 + 39968*b1 + 809632) * q^91 + (3072*b3 + 63296*b2 + 13056*b1 - 719104) * q^92 + (5456*b3 + 16720*b2 + 53680*b1 + 231792) * q^94 + (2028*b3 + 16484*b2 - 9256*b1 - 9256) * q^95 + (-85392*b2 - 42696*b1 - 188998) * q^97 + (-11392*b3 - 529*b2 + 11392*b1 + 603776) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 44 q^{4} - 248 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 44 * q^4 - 248 * q^8 $$4 q - 2 q^{2} - 44 q^{4} - 248 q^{8} - 1920 q^{10} - 976 q^{11} - 5760 q^{14} - 14576 q^{16} - 4168 q^{17} - 1456 q^{19} - 31680 q^{20} + 24428 q^{22} - 23900 q^{25} + 59520 q^{26} + 59520 q^{28} + 48928 q^{32} - 81916 q^{34} + 49920 q^{35} - 26572 q^{38} - 13440 q^{40} + 117944 q^{41} + 197456 q^{43} - 37144 q^{44} - 213120 q^{46} + 2116 q^{49} - 357650 q^{50} - 254400 q^{52} - 349440 q^{56} + 516480 q^{58} - 542032 q^{59} + 407040 q^{62} + 463936 q^{64} + 205440 q^{65} - 790192 q^{67} + 213848 q^{68} - 360960 q^{70} + 443912 q^{73} + 32640 q^{74} + 70616 q^{76} + 1032960 q^{80} + 404708 q^{82} + 3465008 q^{83} - 989548 q^{86} - 1950448 q^{88} - 761224 q^{89} + 3398400 q^{91} - 2743680 q^{92} + 971520 q^{94} - 926776 q^{97} + 2391262 q^{98}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 6x^{2} - 16x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu - 1$$ 4*v - 1 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} - 6\nu + 16 ) / 8$$ (-v^3 + v^2 - 6*v + 16) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 31\nu^{2} + 6\nu + 80 ) / 8$$ (v^3 + 31*v^2 + 6*v + 80) / 8
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 4$$ (b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} - 12 ) / 4$$ (b3 + b2 - 12) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{3} - 31\beta_{2} - 6\beta _1 + 46 ) / 4$$ (b3 - 31*b2 - 6*b1 + 46) / 4

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{7}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

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