# Properties

 Label 72.3.h.a Level $72$ Weight $3$ Character orbit 72.h Analytic conductor $1.962$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 72.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.96185790339$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.33808912384.2 Defining polynomial: $$x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( 1 + \beta_{5} ) q^{4} + ( -\beta_{2} - \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} ) q^{7} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( 1 + \beta_{5} ) q^{4} + ( -\beta_{2} - \beta_{4} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} ) q^{7} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{8} + ( 3 + \beta_{1} + \beta_{5} + \beta_{7} ) q^{10} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{11} + ( -1 - \beta_{1} - \beta_{5} + 2 \beta_{7} ) q^{13} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{14} + ( -8 - 2 \beta_{7} ) q^{16} + ( 8 \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{17} + ( -4 \beta_{5} - 4 \beta_{7} ) q^{19} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{6} ) q^{20} + ( -14 + 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{22} + ( -4 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{23} + ( 1 + 4 \beta_{1} - 4 \beta_{5} ) q^{25} + ( 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 6 \beta_{6} ) q^{26} + ( 14 - 2 \beta_{5} - 4 \beta_{7} ) q^{28} + ( -7 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{29} + ( -13 - 3 \beta_{1} + 3 \beta_{5} ) q^{31} + ( 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{6} ) q^{32} + ( 23 - \beta_{1} - 7 \beta_{5} + \beta_{7} ) q^{34} + ( 10 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} ) q^{35} + ( 4 \beta_{5} + 4 \beta_{7} ) q^{37} + ( -4 \beta_{2} - 8 \beta_{4} - 8 \beta_{6} ) q^{38} + ( -26 - 4 \beta_{1} + 6 \beta_{5} + 2 \beta_{7} ) q^{40} + ( -8 \beta_{2} - 4 \beta_{3} - 3 \beta_{6} ) q^{41} + ( 4 + 4 \beta_{1} + 12 \beta_{5} ) q^{43} + ( 14 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 10 \beta_{6} ) q^{44} + ( -14 - 2 \beta_{1} + 6 \beta_{5} + 2 \beta_{7} ) q^{46} + ( -12 \beta_{2} - 6 \beta_{3} - 10 \beta_{6} ) q^{47} + 3 q^{49} + ( -5 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{6} ) q^{50} + ( 34 - 8 \beta_{1} + 2 \beta_{5} + 4 \beta_{7} ) q^{52} + ( -\beta_{2} + 4 \beta_{3} + 7 \beta_{4} ) q^{53} + ( 30 + 2 \beta_{1} - 2 \beta_{5} ) q^{55} + ( -18 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 10 \beta_{6} ) q^{56} + ( 37 - \beta_{1} + 7 \beta_{5} - \beta_{7} ) q^{58} + ( -16 \beta_{2} + 8 \beta_{3} ) q^{59} + ( 4 + 4 \beta_{1} + 8 \beta_{5} - 4 \beta_{7} ) q^{61} + ( 16 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{6} ) q^{62} + ( -12 + 8 \beta_{1} - 12 \beta_{5} - 4 \beta_{7} ) q^{64} + ( -8 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} ) q^{65} + ( -4 - 4 \beta_{1} - 8 \beta_{5} + 4 \beta_{7} ) q^{67} + ( -21 \beta_{2} + 11 \beta_{3} - 5 \beta_{4} + 9 \beta_{6} ) q^{68} + ( -38 - 6 \beta_{1} - 10 \beta_{5} - 6 \beta_{7} ) q^{70} + ( 12 \beta_{2} + 6 \beta_{3} + 18 \beta_{6} ) q^{71} + ( -12 - 8 \beta_{1} + 8 \beta_{5} ) q^{73} + ( 4 \beta_{2} + 8 \beta_{4} + 8 \beta_{6} ) q^{74} + ( 28 + 16 \beta_{1} - 4 \beta_{5} ) q^{76} + ( 28 \beta_{2} - 16 \beta_{3} - 4 \beta_{4} ) q^{77} + ( -59 + 11 \beta_{1} - 11 \beta_{5} ) q^{79} + ( 32 \beta_{2} + 8 \beta_{3} + 12 \beta_{4} - 4 \beta_{6} ) q^{80} + ( -21 + 3 \beta_{1} + 5 \beta_{5} - 3 \beta_{7} ) q^{82} + ( -10 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} ) q^{83} + ( -3 - 3 \beta_{1} - 23 \beta_{5} - 14 \beta_{7} ) q^{85} + ( -8 \beta_{2} - 24 \beta_{3} + 8 \beta_{4} - 8 \beta_{6} ) q^{86} + ( -36 - 8 \beta_{1} - 4 \beta_{5} + 12 \beta_{7} ) q^{88} + ( 16 \beta_{2} + 8 \beta_{3} + 11 \beta_{6} ) q^{89} + ( -8 - 8 \beta_{1} - 12 \beta_{5} + 12 \beta_{7} ) q^{91} + ( 18 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} - 2 \beta_{6} ) q^{92} + ( -26 + 10 \beta_{1} + 2 \beta_{5} - 10 \beta_{7} ) q^{94} + ( 56 \beta_{2} + 28 \beta_{3} - 20 \beta_{6} ) q^{95} + ( -20 - 4 \beta_{1} + 4 \beta_{5} ) q^{97} -3 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} + 28q^{10} - 72q^{16} - 88q^{22} + 40q^{25} + 104q^{28} - 128q^{31} + 212q^{34} - 240q^{40} - 136q^{46} + 24q^{49} + 248q^{52} + 256q^{55} + 260q^{58} - 32q^{64} - 312q^{70} - 160q^{73} + 304q^{76} - 384q^{79} - 188q^{82} - 256q^{88} - 216q^{94} - 192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-31 \nu^{7} - 2335 \nu^{6} + 5047 \nu^{5} - 30147 \nu^{4} + 48175 \nu^{3} - 134417 \nu^{2} + 129494 \nu + 56970$$$$)/19506$$ $$\beta_{2}$$ $$=$$ $$($$$$-679 \nu^{7} + 3284 \nu^{6} - 10895 \nu^{5} + 30888 \nu^{4} - 59171 \nu^{3} + 91534 \nu^{2} + 9854 \nu - 116964$$$$)/39012$$ $$\beta_{3}$$ $$=$$ $$($$$$-233 \nu^{7} + 1012 \nu^{6} - 3595 \nu^{5} + 10944 \nu^{4} - 21109 \nu^{3} + 45860 \nu^{2} - 12074 \nu + 2208$$$$)/9753$$ $$\beta_{4}$$ $$=$$ $$($$$$3271 \nu^{7} - 6254 \nu^{6} + 35651 \nu^{5} - 60462 \nu^{4} + 137447 \nu^{3} - 117568 \nu^{2} - 194498 \nu + 71472$$$$)/39012$$ $$\beta_{5}$$ $$=$$ $$($$$$1895 \nu^{7} - 5761 \nu^{6} + 23713 \nu^{5} - 57405 \nu^{4} + 120697 \nu^{3} - 154439 \nu^{2} - 32902 \nu + 100920$$$$)/19506$$ $$\beta_{6}$$ $$=$$ $$($$$$-3067 \nu^{7} + 6833 \nu^{6} - 36773 \nu^{5} + 65991 \nu^{4} - 176981 \nu^{3} + 149203 \nu^{2} - 2632 \nu - 110994$$$$)/19506$$ $$\beta_{7}$$ $$=$$ $$($$$$-5171 \nu^{7} + 11953 \nu^{6} - 64213 \nu^{5} + 118353 \nu^{4} - 317701 \nu^{3} + 288395 \nu^{2} + 30574 \nu - 209766$$$$)/19506$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} - 4 \beta_{6} - \beta_{5} + \beta_{1} + 1$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 4 \beta_{3} + \beta_{1} - 9$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} + 7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 9 \beta_{2} - 4 \beta_{1} + 5$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 8 \beta_{4} - 24 \beta_{3} - 24 \beta_{2} - 13 \beta_{1} + 19$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{7} - 16 \beta_{6} - 175 \beta_{5} + 120 \beta_{4} + 30 \beta_{3} - 240 \beta_{2} + 61 \beta_{1} - 353$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$-31 \beta_{7} + 58 \beta_{6} + 16 \beta_{5} + 34 \beta_{4} + 52 \beta_{3} + 126 \beta_{2} + 43 \beta_{1} + 92$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$338 \beta_{7} - 430 \beta_{6} + 941 \beta_{5} - 462 \beta_{4} - 630 \beta_{3} + 1974 \beta_{2} - 389 \beta_{1} + 4363$$$$)/12$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.651388 − 2.66948i −0.651388 + 2.66948i 1.15139 + 2.23537i 1.15139 − 2.23537i 1.15139 + 0.593052i 1.15139 − 0.593052i −0.651388 − 0.158947i −0.651388 + 0.158947i
−1.77521 0.921201i 0 2.30278 + 3.27066i 1.07498 0 7.21110 −1.07498 7.92745i 0 −1.90833 0.990277i
53.2 −1.77521 + 0.921201i 0 2.30278 3.27066i 1.07498 0 7.21110 −1.07498 + 7.92745i 0 −1.90833 + 0.990277i
53.3 −1.16130 1.62831i 0 −1.30278 + 3.78190i −7.67101 0 −7.21110 7.67101 2.27059i 0 8.90833 + 12.4908i
53.4 −1.16130 + 1.62831i 0 −1.30278 3.78190i −7.67101 0 −7.21110 7.67101 + 2.27059i 0 8.90833 12.4908i
53.5 1.16130 1.62831i 0 −1.30278 3.78190i 7.67101 0 −7.21110 −7.67101 2.27059i 0 8.90833 12.4908i
53.6 1.16130 + 1.62831i 0 −1.30278 + 3.78190i 7.67101 0 −7.21110 −7.67101 + 2.27059i 0 8.90833 + 12.4908i
53.7 1.77521 0.921201i 0 2.30278 3.27066i −1.07498 0 7.21110 1.07498 7.92745i 0 −1.90833 + 0.990277i
53.8 1.77521 + 0.921201i 0 2.30278 + 3.27066i −1.07498 0 7.21110 1.07498 + 7.92745i 0 −1.90833 0.990277i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.3.h.a 8
3.b odd 2 1 inner 72.3.h.a 8
4.b odd 2 1 288.3.h.a 8
8.b even 2 1 inner 72.3.h.a 8
8.d odd 2 1 288.3.h.a 8
12.b even 2 1 288.3.h.a 8
16.e even 4 2 2304.3.e.n 8
16.f odd 4 2 2304.3.e.o 8
24.f even 2 1 288.3.h.a 8
24.h odd 2 1 inner 72.3.h.a 8
48.i odd 4 2 2304.3.e.n 8
48.k even 4 2 2304.3.e.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a 8 1.a even 1 1 trivial
72.3.h.a 8 3.b odd 2 1 inner
72.3.h.a 8 8.b even 2 1 inner
72.3.h.a 8 24.h odd 2 1 inner
288.3.h.a 8 4.b odd 2 1
288.3.h.a 8 8.d odd 2 1
288.3.h.a 8 12.b even 2 1
288.3.h.a 8 24.f even 2 1
2304.3.e.n 8 16.e even 4 2
2304.3.e.n 8 48.i odd 4 2
2304.3.e.o 8 16.f odd 4 2
2304.3.e.o 8 48.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 32 T^{2} + 20 T^{4} - 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 68 - 60 T^{2} + T^{4} )^{2}$$
$7$ $$( -52 + T^{2} )^{4}$$
$11$ $$( 9792 - 304 T^{2} + T^{4} )^{2}$$
$13$ $$( 2448 + 472 T^{2} + T^{4} )^{2}$$
$17$ $$( 171396 + 836 T^{2} + T^{4} )^{2}$$
$19$ $$( 352512 + 1504 T^{2} + T^{4} )^{2}$$
$23$ $$( 576 + 464 T^{2} + T^{4} )^{2}$$
$29$ $$( 103428 - 988 T^{2} + T^{4} )^{2}$$
$31$ $$( -212 + 32 T + T^{2} )^{4}$$
$37$ $$( 352512 + 1504 T^{2} + T^{4} )^{2}$$
$41$ $$( 133956 + 932 T^{2} + T^{4} )^{2}$$
$43$ $$( 10027008 + 6400 T^{2} + T^{4} )^{2}$$
$47$ $$( 46656 + 4176 T^{2} + T^{4} )^{2}$$
$53$ $$( 1591812 - 2524 T^{2} + T^{4} )^{2}$$
$59$ $$( 4456448 - 4608 T^{2} + T^{4} )^{2}$$
$61$ $$( 3172608 + 5472 T^{2} + T^{4} )^{2}$$
$67$ $$( 3172608 + 5472 T^{2} + T^{4} )^{2}$$
$71$ $$( 13483584 + 11088 T^{2} + T^{4} )^{2}$$
$73$ $$( -2928 + 40 T + T^{2} )^{4}$$
$79$ $$( -3988 + 96 T + T^{2} )^{4}$$
$83$ $$( 183872 - 3120 T^{2} + T^{4} )^{2}$$
$89$ $$( 171396 + 5828 T^{2} + T^{4} )^{2}$$
$97$ $$( -256 + 48 T + T^{2} )^{4}$$