# Properties

 Label 162.7.d.d 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + (4 \beta_{3} - 4 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 123 \beta_1 q^{5} + 484 \beta_{2} q^{7} + 128 \beta_{3} q^{8}+O(q^{10})$$ q + (4*b3 - 4*b1) * q^2 + (-32*b2 + 32) * q^4 + 123*b1 * q^5 + 484*b2 * q^7 + 128*b3 * q^8 $$q + (4 \beta_{3} - 4 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 123 \beta_1 q^{5} + 484 \beta_{2} q^{7} + 128 \beta_{3} q^{8} - 984 q^{10} + (948 \beta_{3} - 948 \beta_1) q^{11} + (3368 \beta_{2} - 3368) q^{13} - 1936 \beta_1 q^{14} - 1024 \beta_{2} q^{16} - 9 \beta_{3} q^{17} + 5744 q^{19} + ( - 3936 \beta_{3} + 3936 \beta_1) q^{20} + ( - 7584 \beta_{2} + 7584) q^{22} - 2388 \beta_1 q^{23} + 14633 \beta_{2} q^{25} - 13472 \beta_{3} q^{26} + 15488 q^{28} + (20757 \beta_{3} - 20757 \beta_1) q^{29} + ( - 39796 \beta_{2} + 39796) q^{31} + 4096 \beta_1 q^{32} + 72 \beta_{2} q^{34} + 59532 \beta_{3} q^{35} + 52526 q^{37} + (22976 \beta_{3} - 22976 \beta_1) q^{38} + (31488 \beta_{2} - 31488) q^{40} - 26193 \beta_1 q^{41} - 3800 \beta_{2} q^{43} + 30336 \beta_{3} q^{44} + 19104 q^{46} + (54300 \beta_{3} - 54300 \beta_1) q^{47} + (116607 \beta_{2} - 116607) q^{49} - 58532 \beta_1 q^{50} + 107776 \beta_{2} q^{52} - 168813 \beta_{3} q^{53} - 233208 q^{55} + (61952 \beta_{3} - 61952 \beta_1) q^{56} + ( - 166056 \beta_{2} + 166056) q^{58} - 176664 \beta_1 q^{59} - 13250 \beta_{2} q^{61} + 159184 \beta_{3} q^{62} - 32768 q^{64} + (414264 \beta_{3} - 414264 \beta_1) q^{65} + (168968 \beta_{2} - 168968) q^{67} - 288 \beta_1 q^{68} - 476256 \beta_{2} q^{70} + 375804 \beta_{3} q^{71} + 236144 q^{73} + (210104 \beta_{3} - 210104 \beta_1) q^{74} + ( - 183808 \beta_{2} + 183808) q^{76} - 458832 \beta_1 q^{77} + 35116 \beta_{2} q^{79} - 125952 \beta_{3} q^{80} + 209544 q^{82} + ( - 7764 \beta_{3} + 7764 \beta_1) q^{83} + ( - 2214 \beta_{2} + 2214) q^{85} + 15200 \beta_1 q^{86} - 242688 \beta_{2} q^{88} + 91449 \beta_{3} q^{89} - 1630112 q^{91} + (76416 \beta_{3} - 76416 \beta_1) q^{92} + ( - 434400 \beta_{2} + 434400) q^{94} + 706512 \beta_1 q^{95} + 321424 \beta_{2} q^{97} - 466428 \beta_{3} q^{98}+O(q^{100})$$ q + (4*b3 - 4*b1) * q^2 + (-32*b2 + 32) * q^4 + 123*b1 * q^5 + 484*b2 * q^7 + 128*b3 * q^8 - 984 * q^10 + (948*b3 - 948*b1) * q^11 + (3368*b2 - 3368) * q^13 - 1936*b1 * q^14 - 1024*b2 * q^16 - 9*b3 * q^17 + 5744 * q^19 + (-3936*b3 + 3936*b1) * q^20 + (-7584*b2 + 7584) * q^22 - 2388*b1 * q^23 + 14633*b2 * q^25 - 13472*b3 * q^26 + 15488 * q^28 + (20757*b3 - 20757*b1) * q^29 + (-39796*b2 + 39796) * q^31 + 4096*b1 * q^32 + 72*b2 * q^34 + 59532*b3 * q^35 + 52526 * q^37 + (22976*b3 - 22976*b1) * q^38 + (31488*b2 - 31488) * q^40 - 26193*b1 * q^41 - 3800*b2 * q^43 + 30336*b3 * q^44 + 19104 * q^46 + (54300*b3 - 54300*b1) * q^47 + (116607*b2 - 116607) * q^49 - 58532*b1 * q^50 + 107776*b2 * q^52 - 168813*b3 * q^53 - 233208 * q^55 + (61952*b3 - 61952*b1) * q^56 + (-166056*b2 + 166056) * q^58 - 176664*b1 * q^59 - 13250*b2 * q^61 + 159184*b3 * q^62 - 32768 * q^64 + (414264*b3 - 414264*b1) * q^65 + (168968*b2 - 168968) * q^67 - 288*b1 * q^68 - 476256*b2 * q^70 + 375804*b3 * q^71 + 236144 * q^73 + (210104*b3 - 210104*b1) * q^74 + (-183808*b2 + 183808) * q^76 - 458832*b1 * q^77 + 35116*b2 * q^79 - 125952*b3 * q^80 + 209544 * q^82 + (-7764*b3 + 7764*b1) * q^83 + (-2214*b2 + 2214) * q^85 + 15200*b1 * q^86 - 242688*b2 * q^88 + 91449*b3 * q^89 - 1630112 * q^91 + (76416*b3 - 76416*b1) * q^92 + (-434400*b2 + 434400) * q^94 + 706512*b1 * q^95 + 321424*b2 * q^97 - 466428*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 64 q^{4} + 968 q^{7}+O(q^{10})$$ 4 * q + 64 * q^4 + 968 * q^7 $$4 q + 64 q^{4} + 968 q^{7} - 3936 q^{10} - 6736 q^{13} - 2048 q^{16} + 22976 q^{19} + 15168 q^{22} + 29266 q^{25} + 61952 q^{28} + 79592 q^{31} + 144 q^{34} + 210104 q^{37} - 62976 q^{40} - 7600 q^{43} + 76416 q^{46} - 233214 q^{49} + 215552 q^{52} - 932832 q^{55} + 332112 q^{58} - 26500 q^{61} - 131072 q^{64} - 337936 q^{67} - 952512 q^{70} + 944576 q^{73} + 367616 q^{76} + 70232 q^{79} + 838176 q^{82} + 4428 q^{85} - 485376 q^{88} - 6520448 q^{91} + 868800 q^{94} + 642848 q^{97}+O(q^{100})$$ 4 * q + 64 * q^4 + 968 * q^7 - 3936 * q^10 - 6736 * q^13 - 2048 * q^16 + 22976 * q^19 + 15168 * q^22 + 29266 * q^25 + 61952 * q^28 + 79592 * q^31 + 144 * q^34 + 210104 * q^37 - 62976 * q^40 - 7600 * q^43 + 76416 * q^46 - 233214 * q^49 + 215552 * q^52 - 932832 * q^55 + 332112 * q^58 - 26500 * q^61 - 131072 * q^64 - 337936 * q^67 - 952512 * q^70 + 944576 * q^73 + 367616 * q^76 + 70232 * q^79 + 838176 * q^82 + 4428 * q^85 - 485376 * q^88 - 6520448 * q^91 + 868800 * q^94 + 642848 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*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 150.644 + 86.9741i 0 242.000 + 419.156i 181.019i 0 −984.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i −150.644 86.9741i 0 242.000 + 419.156i 181.019i 0 −984.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i 150.644 86.9741i 0 242.000 419.156i 181.019i 0 −984.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i −150.644 + 86.9741i 0 242.000 419.156i 181.019i 0 −984.000
 $$n$$: 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.d 4
3.b odd 2 1 inner 162.7.d.d 4
9.c even 3 1 18.7.b.a 2
9.c even 3 1 inner 162.7.d.d 4
9.d odd 6 1 18.7.b.a 2
9.d odd 6 1 inner 162.7.d.d 4
36.f odd 6 1 144.7.e.d 2
36.h even 6 1 144.7.e.d 2
45.h odd 6 1 450.7.d.a 2
45.j even 6 1 450.7.d.a 2
45.k odd 12 2 450.7.b.a 4
45.l even 12 2 450.7.b.a 4
72.j odd 6 1 576.7.e.b 2
72.l even 6 1 576.7.e.k 2
72.n even 6 1 576.7.e.b 2
72.p odd 6 1 576.7.e.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 9.c even 3 1
18.7.b.a 2 9.d odd 6 1
144.7.e.d 2 36.f odd 6 1
144.7.e.d 2 36.h even 6 1
162.7.d.d 4 1.a even 1 1 trivial
162.7.d.d 4 3.b odd 2 1 inner
162.7.d.d 4 9.c even 3 1 inner
162.7.d.d 4 9.d odd 6 1 inner
450.7.b.a 4 45.k odd 12 2
450.7.b.a 4 45.l even 12 2
450.7.d.a 2 45.h odd 6 1
450.7.d.a 2 45.j even 6 1
576.7.e.b 2 72.j odd 6 1
576.7.e.b 2 72.n even 6 1
576.7.e.k 2 72.l even 6 1
576.7.e.k 2 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 30258T_{5}^{2} + 915546564$$ acting on $$S_{7}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 32T^{2} + 1024$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 30258 T^{2} + \cdots + 915546564$$
$7$ $$(T^{2} - 484 T + 234256)^{2}$$
$11$ $$T^{4} - 1797408 T^{2} + \cdots + 3230675518464$$
$13$ $$(T^{2} + 3368 T + 11343424)^{2}$$
$17$ $$(T^{2} + 162)^{2}$$
$19$ $$(T - 5744)^{4}$$
$23$ $$T^{4} + \cdots + 130076032287744$$
$29$ $$T^{4} - 861706098 T^{2} + \cdots + 74\!\cdots\!04$$
$31$ $$(T^{2} - 39796 T + 1583721616)^{2}$$
$37$ $$(T - 52526)^{4}$$
$41$ $$T^{4} - 1372146498 T^{2} + \cdots + 18\!\cdots\!04$$
$43$ $$(T^{2} + 3800 T + 14440000)^{2}$$
$47$ $$T^{4} - 5896980000 T^{2} + \cdots + 34\!\cdots\!00$$
$53$ $$(T^{2} + 56995657938)^{2}$$
$59$ $$T^{4} - 62420337792 T^{2} + \cdots + 38\!\cdots\!64$$
$61$ $$(T^{2} + 13250 T + 175562500)^{2}$$
$67$ $$(T^{2} + 168968 T + 28550185024)^{2}$$
$71$ $$(T^{2} + 282457292832)^{2}$$
$73$ $$(T - 236144)^{4}$$
$79$ $$(T^{2} - 35116 T + 1233133456)^{2}$$
$83$ $$T^{4} - 120559392 T^{2} + \cdots + 14\!\cdots\!64$$
$89$ $$(T^{2} + 16725839202)^{2}$$
$97$ $$(T^{2} - 321424 T + 103313387776)^{2}$$