# Properties

 Label 162.7.d.c 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: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 6 \beta_1 q^{5} + 205 \beta_{2} q^{7} + 32 \beta_{3} q^{8}+O(q^{10})$$ q + (b3 - b1) * q^2 + (-32*b2 + 32) * q^4 + 6*b1 * q^5 + 205*b2 * q^7 + 32*b3 * q^8 $$q + (\beta_{3} - \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 6 \beta_1 q^{5} + 205 \beta_{2} q^{7} + 32 \beta_{3} q^{8} - 192 q^{10} + ( - 186 \beta_{3} + 186 \beta_1) q^{11} + ( - 2041 \beta_{2} + 2041) q^{13} - 205 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 1458 \beta_{3} q^{17} - 1501 q^{19} + ( - 192 \beta_{3} + 192 \beta_1) q^{20} + (5952 \beta_{2} - 5952) q^{22} - 1254 \beta_1 q^{23} - 14473 \beta_{2} q^{25} + 2041 \beta_{3} q^{26} + 6560 q^{28} + ( - 5028 \beta_{3} + 5028 \beta_1) q^{29} + ( - 34990 \beta_{2} + 34990) q^{31} + 1024 \beta_1 q^{32} - 46656 \beta_{2} q^{34} + 1230 \beta_{3} q^{35} - 57625 q^{37} + ( - 1501 \beta_{3} + 1501 \beta_1) q^{38} + (6144 \beta_{2} - 6144) q^{40} + 23748 \beta_1 q^{41} + 62566 \beta_{2} q^{43} - 5952 \beta_{3} q^{44} + 40128 q^{46} + (9174 \beta_{3} - 9174 \beta_1) q^{47} + ( - 75624 \beta_{2} + 75624) q^{49} + 14473 \beta_1 q^{50} - 65312 \beta_{2} q^{52} + 13644 \beta_{3} q^{53} + 35712 q^{55} + (6560 \beta_{3} - 6560 \beta_1) q^{56} + (160896 \beta_{2} - 160896) q^{58} + 65634 \beta_1 q^{59} + 61297 \beta_{2} q^{61} + 34990 \beta_{3} q^{62} - 32768 q^{64} + ( - 12246 \beta_{3} + 12246 \beta_1) q^{65} + (67691 \beta_{2} - 67691) q^{67} + 46656 \beta_1 q^{68} - 39360 \beta_{2} q^{70} + 90072 \beta_{3} q^{71} + 423983 q^{73} + ( - 57625 \beta_{3} + 57625 \beta_1) q^{74} + (48032 \beta_{2} - 48032) q^{76} + 38130 \beta_1 q^{77} + 707533 \beta_{2} q^{79} - 6144 \beta_{3} q^{80} - 759936 q^{82} + ( - 152196 \beta_{3} + 152196 \beta_1) q^{83} + (279936 \beta_{2} - 279936) q^{85} - 62566 \beta_1 q^{86} + 190464 \beta_{2} q^{88} + 118026 \beta_{3} q^{89} + 418405 q^{91} + (40128 \beta_{3} - 40128 \beta_1) q^{92} + ( - 293568 \beta_{2} + 293568) q^{94} - 9006 \beta_1 q^{95} - 526151 \beta_{2} q^{97} + 75624 \beta_{3} q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 + (-32*b2 + 32) * q^4 + 6*b1 * q^5 + 205*b2 * q^7 + 32*b3 * q^8 - 192 * q^10 + (-186*b3 + 186*b1) * q^11 + (-2041*b2 + 2041) * q^13 - 205*b1 * q^14 - 1024*b2 * q^16 + 1458*b3 * q^17 - 1501 * q^19 + (-192*b3 + 192*b1) * q^20 + (5952*b2 - 5952) * q^22 - 1254*b1 * q^23 - 14473*b2 * q^25 + 2041*b3 * q^26 + 6560 * q^28 + (-5028*b3 + 5028*b1) * q^29 + (-34990*b2 + 34990) * q^31 + 1024*b1 * q^32 - 46656*b2 * q^34 + 1230*b3 * q^35 - 57625 * q^37 + (-1501*b3 + 1501*b1) * q^38 + (6144*b2 - 6144) * q^40 + 23748*b1 * q^41 + 62566*b2 * q^43 - 5952*b3 * q^44 + 40128 * q^46 + (9174*b3 - 9174*b1) * q^47 + (-75624*b2 + 75624) * q^49 + 14473*b1 * q^50 - 65312*b2 * q^52 + 13644*b3 * q^53 + 35712 * q^55 + (6560*b3 - 6560*b1) * q^56 + (160896*b2 - 160896) * q^58 + 65634*b1 * q^59 + 61297*b2 * q^61 + 34990*b3 * q^62 - 32768 * q^64 + (-12246*b3 + 12246*b1) * q^65 + (67691*b2 - 67691) * q^67 + 46656*b1 * q^68 - 39360*b2 * q^70 + 90072*b3 * q^71 + 423983 * q^73 + (-57625*b3 + 57625*b1) * q^74 + (48032*b2 - 48032) * q^76 + 38130*b1 * q^77 + 707533*b2 * q^79 - 6144*b3 * q^80 - 759936 * q^82 + (-152196*b3 + 152196*b1) * q^83 + (279936*b2 - 279936) * q^85 - 62566*b1 * q^86 + 190464*b2 * q^88 + 118026*b3 * q^89 + 418405 * q^91 + (40128*b3 - 40128*b1) * q^92 + (-293568*b2 + 293568) * q^94 - 9006*b1 * q^95 - 526151*b2 * q^97 + 75624*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 64 q^{4} + 410 q^{7}+O(q^{10})$$ 4 * q + 64 * q^4 + 410 * q^7 $$4 q + 64 q^{4} + 410 q^{7} - 768 q^{10} + 4082 q^{13} - 2048 q^{16} - 6004 q^{19} - 11904 q^{22} - 28946 q^{25} + 26240 q^{28} + 69980 q^{31} - 93312 q^{34} - 230500 q^{37} - 12288 q^{40} + 125132 q^{43} + 160512 q^{46} + 151248 q^{49} - 130624 q^{52} + 142848 q^{55} - 321792 q^{58} + 122594 q^{61} - 131072 q^{64} - 135382 q^{67} - 78720 q^{70} + 1695932 q^{73} - 96064 q^{76} + 1415066 q^{79} - 3039744 q^{82} - 559872 q^{85} + 380928 q^{88} + 1673620 q^{91} + 587136 q^{94} - 1052302 q^{97}+O(q^{100})$$ 4 * q + 64 * q^4 + 410 * q^7 - 768 * q^10 + 4082 * q^13 - 2048 * q^16 - 6004 * q^19 - 11904 * q^22 - 28946 * q^25 + 26240 * q^28 + 69980 * q^31 - 93312 * q^34 - 230500 * q^37 - 12288 * q^40 + 125132 * q^43 + 160512 * q^46 + 151248 * q^49 - 130624 * q^52 + 142848 * q^55 - 321792 * q^58 + 122594 * q^61 - 131072 * q^64 - 135382 * q^67 - 78720 * q^70 + 1695932 * q^73 - 96064 * q^76 + 1415066 * q^79 - 3039744 * q^82 - 559872 * q^85 + 380928 * q^88 + 1673620 * q^91 + 587136 * q^94 - 1052302 * q^97

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

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$2\nu^{3}$$ 2*v^3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## 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 29.3939 + 16.9706i 0 102.500 + 177.535i 181.019i 0 −192.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i −29.3939 16.9706i 0 102.500 + 177.535i 181.019i 0 −192.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i 29.3939 16.9706i 0 102.500 177.535i 181.019i 0 −192.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i −29.3939 + 16.9706i 0 102.500 177.535i 181.019i 0 −192.000
 $$n$$: 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.c 4
3.b odd 2 1 inner 162.7.d.c 4
9.c even 3 1 54.7.b.a 2
9.c even 3 1 inner 162.7.d.c 4
9.d odd 6 1 54.7.b.a 2
9.d odd 6 1 inner 162.7.d.c 4
36.f odd 6 1 432.7.e.f 2
36.h even 6 1 432.7.e.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 1152T_{5}^{2} + 1327104$$ acting on $$S_{7}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 32T^{2} + 1024$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$7$ $$(T^{2} - 205 T + 42025)^{2}$$
$11$ $$T^{4} - 1107072 T^{2} + \cdots + 1225608413184$$
$13$ $$(T^{2} - 2041 T + 4165681)^{2}$$
$17$ $$(T^{2} + 68024448)^{2}$$
$19$ $$(T + 1501)^{4}$$
$23$ $$T^{4} - 50320512 T^{2} + \cdots + 25\!\cdots\!44$$
$29$ $$T^{4} - 808985088 T^{2} + \cdots + 65\!\cdots\!44$$
$31$ $$(T^{2} - 34990 T + 1224300100)^{2}$$
$37$ $$(T + 57625)^{4}$$
$41$ $$T^{4} - 18046960128 T^{2} + \cdots + 32\!\cdots\!84$$
$43$ $$(T^{2} - 62566 T + 3914504356)^{2}$$
$47$ $$T^{4} - 2693192832 T^{2} + \cdots + 72\!\cdots\!24$$
$53$ $$(T^{2} + 5957079552)^{2}$$
$59$ $$T^{4} - 137850302592 T^{2} + \cdots + 19\!\cdots\!64$$
$61$ $$(T^{2} - 61297 T + 3757322209)^{2}$$
$67$ $$(T^{2} + 67691 T + 4582071481)^{2}$$
$71$ $$(T^{2} + 259614885888)^{2}$$
$73$ $$(T - 423983)^{4}$$
$79$ $$(T^{2} - 707533 T + 500602946089)^{2}$$
$83$ $$T^{4} - 741235917312 T^{2} + \cdots + 54\!\cdots\!44$$
$89$ $$(T^{2} + 445764373632)^{2}$$
$97$ $$(T^{2} + 526151 T + 276834874801)^{2}$$