# Properties

 Label 162.7.d.a Level $162$ Weight $7$ Character orbit 162.d Analytic conductor $37.269$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.2687615464$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{3} + 2 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 15 \beta_1 q^{5} - 389 \beta_{2} q^{7} - 64 \beta_{3} q^{8}+O(q^{10})$$ q + (-2*b3 + 2*b1) * q^2 + (-32*b2 + 32) * q^4 + 15*b1 * q^5 - 389*b2 * q^7 - 64*b3 * q^8 $$q + ( - 2 \beta_{3} + 2 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 15 \beta_1 q^{5} - 389 \beta_{2} q^{7} - 64 \beta_{3} q^{8} + 240 q^{10} + ( - 735 \beta_{3} + 735 \beta_1) q^{11} + (1415 \beta_{2} - 1415) q^{13} - 778 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 837 \beta_{3} q^{17} - 3067 q^{19} + ( - 480 \beta_{3} + 480 \beta_1) q^{20} + ( - 11760 \beta_{2} + 11760) q^{22} - 7401 \beta_1 q^{23} - 13825 \beta_{2} q^{25} + 2830 \beta_{3} q^{26} - 12448 q^{28} + ( - 4578 \beta_{3} + 4578 \beta_1) q^{29} + ( - 11338 \beta_{2} + 11338) q^{31} - 2048 \beta_1 q^{32} + 13392 \beta_{2} q^{34} - 5835 \beta_{3} q^{35} + 47135 q^{37} + (6134 \beta_{3} - 6134 \beta_1) q^{38} + ( - 7680 \beta_{2} + 7680) q^{40} - 2190 \beta_1 q^{41} - 145118 \beta_{2} q^{43} - 23520 \beta_{3} q^{44} - 118416 q^{46} + (63705 \beta_{3} - 63705 \beta_1) q^{47} + (33672 \beta_{2} - 33672) q^{49} - 27650 \beta_1 q^{50} + 45280 \beta_{2} q^{52} - 93978 \beta_{3} q^{53} + 88200 q^{55} + (24896 \beta_{3} - 24896 \beta_1) q^{56} + ( - 73248 \beta_{2} + 73248) q^{58} - 127893 \beta_1 q^{59} + 350305 \beta_{2} q^{61} - 22676 \beta_{3} q^{62} - 32768 q^{64} + (21225 \beta_{3} - 21225 \beta_1) q^{65} + (120341 \beta_{2} - 120341) q^{67} + 26784 \beta_1 q^{68} - 93360 \beta_{2} q^{70} - 118476 \beta_{3} q^{71} + 175151 q^{73} + ( - 94270 \beta_{3} + 94270 \beta_1) q^{74} + (98144 \beta_{2} - 98144) q^{76} - 285915 \beta_1 q^{77} + 252259 \beta_{2} q^{79} - 15360 \beta_{3} q^{80} - 35040 q^{82} + ( - 119886 \beta_{3} + 119886 \beta_1) q^{83} + (100440 \beta_{2} - 100440) q^{85} - 290236 \beta_1 q^{86} - 376320 \beta_{2} q^{88} + 279297 \beta_{3} q^{89} + 550435 q^{91} + (236832 \beta_{3} - 236832 \beta_1) q^{92} + (1019280 \beta_{2} - 1019280) q^{94} - 46005 \beta_1 q^{95} + 1297105 \beta_{2} q^{97} + 67344 \beta_{3} q^{98}+O(q^{100})$$ q + (-2*b3 + 2*b1) * q^2 + (-32*b2 + 32) * q^4 + 15*b1 * q^5 - 389*b2 * q^7 - 64*b3 * q^8 + 240 * q^10 + (-735*b3 + 735*b1) * q^11 + (1415*b2 - 1415) * q^13 - 778*b1 * q^14 - 1024*b2 * q^16 + 837*b3 * q^17 - 3067 * q^19 + (-480*b3 + 480*b1) * q^20 + (-11760*b2 + 11760) * q^22 - 7401*b1 * q^23 - 13825*b2 * q^25 + 2830*b3 * q^26 - 12448 * q^28 + (-4578*b3 + 4578*b1) * q^29 + (-11338*b2 + 11338) * q^31 - 2048*b1 * q^32 + 13392*b2 * q^34 - 5835*b3 * q^35 + 47135 * q^37 + (6134*b3 - 6134*b1) * q^38 + (-7680*b2 + 7680) * q^40 - 2190*b1 * q^41 - 145118*b2 * q^43 - 23520*b3 * q^44 - 118416 * q^46 + (63705*b3 - 63705*b1) * q^47 + (33672*b2 - 33672) * q^49 - 27650*b1 * q^50 + 45280*b2 * q^52 - 93978*b3 * q^53 + 88200 * q^55 + (24896*b3 - 24896*b1) * q^56 + (-73248*b2 + 73248) * q^58 - 127893*b1 * q^59 + 350305*b2 * q^61 - 22676*b3 * q^62 - 32768 * q^64 + (21225*b3 - 21225*b1) * q^65 + (120341*b2 - 120341) * q^67 + 26784*b1 * q^68 - 93360*b2 * q^70 - 118476*b3 * q^71 + 175151 * q^73 + (-94270*b3 + 94270*b1) * q^74 + (98144*b2 - 98144) * q^76 - 285915*b1 * q^77 + 252259*b2 * q^79 - 15360*b3 * q^80 - 35040 * q^82 + (-119886*b3 + 119886*b1) * q^83 + (100440*b2 - 100440) * q^85 - 290236*b1 * q^86 - 376320*b2 * q^88 + 279297*b3 * q^89 + 550435 * q^91 + (236832*b3 - 236832*b1) * q^92 + (1019280*b2 - 1019280) * q^94 - 46005*b1 * q^95 + 1297105*b2 * q^97 + 67344*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 64 q^{4} - 778 q^{7}+O(q^{10})$$ 4 * q + 64 * q^4 - 778 * q^7 $$4 q + 64 q^{4} - 778 q^{7} + 960 q^{10} - 2830 q^{13} - 2048 q^{16} - 12268 q^{19} + 23520 q^{22} - 27650 q^{25} - 49792 q^{28} + 22676 q^{31} + 26784 q^{34} + 188540 q^{37} + 15360 q^{40} - 290236 q^{43} - 473664 q^{46} - 67344 q^{49} + 90560 q^{52} + 352800 q^{55} + 146496 q^{58} + 700610 q^{61} - 131072 q^{64} - 240682 q^{67} - 186720 q^{70} + 700604 q^{73} - 196288 q^{76} + 504518 q^{79} - 140160 q^{82} - 200880 q^{85} - 752640 q^{88} + 2201740 q^{91} - 2038560 q^{94} + 2594210 q^{97}+O(q^{100})$$ 4 * q + 64 * q^4 - 778 * q^7 + 960 * q^10 - 2830 * q^13 - 2048 * q^16 - 12268 * q^19 + 23520 * q^22 - 27650 * q^25 - 49792 * q^28 + 22676 * q^31 + 26784 * q^34 + 188540 * q^37 + 15360 * q^40 - 290236 * q^43 - 473664 * q^46 - 67344 * q^49 + 90560 * q^52 + 352800 * q^55 + 146496 * q^58 + 700610 * q^61 - 131072 * q^64 - 240682 * q^67 - 186720 * q^70 + 700604 * q^73 - 196288 * q^76 + 504518 * q^79 - 140160 * q^82 - 200880 * q^85 - 752640 * q^88 + 2201740 * q^91 - 2038560 * q^94 + 2594210 * q^97

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ v^3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$1 - \beta_{2}$$

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−4.89898 + 2.82843i 0 16.0000 27.7128i −36.7423 21.2132i 0 −194.500 336.884i 181.019i 0 240.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i 36.7423 + 21.2132i 0 −194.500 336.884i 181.019i 0 240.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −36.7423 + 21.2132i 0 −194.500 + 336.884i 181.019i 0 240.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i 36.7423 21.2132i 0 −194.500 + 336.884i 181.019i 0 240.000
 $$n$$: 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.7.d.a 4
3.b odd 2 1 inner 162.7.d.a 4
9.c even 3 1 54.7.b.b 2
9.c even 3 1 inner 162.7.d.a 4
9.d odd 6 1 54.7.b.b 2
9.d odd 6 1 inner 162.7.d.a 4
36.f odd 6 1 432.7.e.c 2
36.h even 6 1 432.7.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.b.b 2 9.c even 3 1
54.7.b.b 2 9.d odd 6 1
162.7.d.a 4 1.a even 1 1 trivial
162.7.d.a 4 3.b odd 2 1 inner
162.7.d.a 4 9.c even 3 1 inner
162.7.d.a 4 9.d odd 6 1 inner
432.7.e.c 2 36.f odd 6 1
432.7.e.c 2 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 1800T_{5}^{2} + 3240000$$ acting on $$S_{7}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 32T^{2} + 1024$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 1800 T^{2} + \cdots + 3240000$$
$7$ $$(T^{2} + 389 T + 151321)^{2}$$
$11$ $$T^{4} - 4321800 T^{2} + \cdots + 18677955240000$$
$13$ $$(T^{2} + 1415 T + 2002225)^{2}$$
$17$ $$(T^{2} + 5604552)^{2}$$
$19$ $$(T + 3067)^{4}$$
$23$ $$T^{4} - 438198408 T^{2} + \cdots + 19\!\cdots\!64$$
$29$ $$T^{4} - 167664672 T^{2} + \cdots + 28\!\cdots\!84$$
$31$ $$(T^{2} - 11338 T + 128550244)^{2}$$
$37$ $$(T - 47135)^{4}$$
$41$ $$T^{4} - 38368800 T^{2} + \cdots + 14\!\cdots\!00$$
$43$ $$(T^{2} + 145118 T + 21059233924)^{2}$$
$47$ $$T^{4} - 32466616200 T^{2} + \cdots + 10\!\cdots\!00$$
$53$ $$(T^{2} + 70654915872)^{2}$$
$59$ $$T^{4} - 130852955592 T^{2} + \cdots + 17\!\cdots\!64$$
$61$ $$(T^{2} - 350305 T + 122713593025)^{2}$$
$67$ $$(T^{2} + 120341 T + 14481956281)^{2}$$
$71$ $$(T^{2} + 112292500608)^{2}$$
$73$ $$(T - 175151)^{4}$$
$79$ $$(T^{2} - 252259 T + 63634603081)^{2}$$
$83$ $$T^{4} - 114981223968 T^{2} + \cdots + 13\!\cdots\!24$$
$89$ $$(T^{2} + 624054513672)^{2}$$
$97$ $$(T^{2} - 1297105 T + 1682481381025)^{2}$$