# Properties

 Label 18.7.b.a Level $18$ Weight $7$ Character orbit 18.b Analytic conductor $4.141$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 18.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.14097350516$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{2} - 32 q^{4} + 123 \beta q^{5} - 484 q^{7} - 128 \beta q^{8} +O(q^{10})$$ q + 4*b * q^2 - 32 * q^4 + 123*b * q^5 - 484 * q^7 - 128*b * q^8 $$q + 4 \beta q^{2} - 32 q^{4} + 123 \beta q^{5} - 484 q^{7} - 128 \beta q^{8} - 984 q^{10} + 948 \beta q^{11} + 3368 q^{13} - 1936 \beta q^{14} + 1024 q^{16} + 9 \beta q^{17} + 5744 q^{19} - 3936 \beta q^{20} - 7584 q^{22} - 2388 \beta q^{23} - 14633 q^{25} + 13472 \beta q^{26} + 15488 q^{28} + 20757 \beta q^{29} - 39796 q^{31} + 4096 \beta q^{32} - 72 q^{34} - 59532 \beta q^{35} + 52526 q^{37} + 22976 \beta q^{38} + 31488 q^{40} - 26193 \beta q^{41} + 3800 q^{43} - 30336 \beta q^{44} + 19104 q^{46} + 54300 \beta q^{47} + 116607 q^{49} - 58532 \beta q^{50} - 107776 q^{52} + 168813 \beta q^{53} - 233208 q^{55} + 61952 \beta q^{56} - 166056 q^{58} - 176664 \beta q^{59} + 13250 q^{61} - 159184 \beta q^{62} - 32768 q^{64} + 414264 \beta q^{65} + 168968 q^{67} - 288 \beta q^{68} + 476256 q^{70} - 375804 \beta q^{71} + 236144 q^{73} + 210104 \beta q^{74} - 183808 q^{76} - 458832 \beta q^{77} - 35116 q^{79} + 125952 \beta q^{80} + 209544 q^{82} - 7764 \beta q^{83} - 2214 q^{85} + 15200 \beta q^{86} + 242688 q^{88} - 91449 \beta q^{89} - 1630112 q^{91} + 76416 \beta q^{92} - 434400 q^{94} + 706512 \beta q^{95} - 321424 q^{97} + 466428 \beta q^{98} +O(q^{100})$$ q + 4*b * q^2 - 32 * q^4 + 123*b * q^5 - 484 * q^7 - 128*b * q^8 - 984 * q^10 + 948*b * q^11 + 3368 * q^13 - 1936*b * q^14 + 1024 * q^16 + 9*b * q^17 + 5744 * q^19 - 3936*b * q^20 - 7584 * q^22 - 2388*b * q^23 - 14633 * q^25 + 13472*b * q^26 + 15488 * q^28 + 20757*b * q^29 - 39796 * q^31 + 4096*b * q^32 - 72 * q^34 - 59532*b * q^35 + 52526 * q^37 + 22976*b * q^38 + 31488 * q^40 - 26193*b * q^41 + 3800 * q^43 - 30336*b * q^44 + 19104 * q^46 + 54300*b * q^47 + 116607 * q^49 - 58532*b * q^50 - 107776 * q^52 + 168813*b * q^53 - 233208 * q^55 + 61952*b * q^56 - 166056 * q^58 - 176664*b * q^59 + 13250 * q^61 - 159184*b * q^62 - 32768 * q^64 + 414264*b * q^65 + 168968 * q^67 - 288*b * q^68 + 476256 * q^70 - 375804*b * q^71 + 236144 * q^73 + 210104*b * q^74 - 183808 * q^76 - 458832*b * q^77 - 35116 * q^79 + 125952*b * q^80 + 209544 * q^82 - 7764*b * q^83 - 2214 * q^85 + 15200*b * q^86 + 242688 * q^88 - 91449*b * q^89 - 1630112 * q^91 + 76416*b * q^92 - 434400 * q^94 + 706512*b * q^95 - 321424 * q^97 + 466428*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 64 q^{4} - 968 q^{7}+O(q^{10})$$ 2 * q - 64 * q^4 - 968 * q^7 $$2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43} + 38208 q^{46} + 233214 q^{49} - 215552 q^{52} - 466416 q^{55} - 332112 q^{58} + 26500 q^{61} - 65536 q^{64} + 337936 q^{67} + 952512 q^{70} + 472288 q^{73} - 367616 q^{76} - 70232 q^{79} + 419088 q^{82} - 4428 q^{85} + 485376 q^{88} - 3260224 q^{91} - 868800 q^{94} - 642848 q^{97}+O(q^{100})$$ 2 * q - 64 * q^4 - 968 * q^7 - 1968 * q^10 + 6736 * q^13 + 2048 * q^16 + 11488 * q^19 - 15168 * q^22 - 29266 * q^25 + 30976 * q^28 - 79592 * q^31 - 144 * q^34 + 105052 * q^37 + 62976 * q^40 + 7600 * q^43 + 38208 * q^46 + 233214 * q^49 - 215552 * q^52 - 466416 * q^55 - 332112 * q^58 + 26500 * q^61 - 65536 * q^64 + 337936 * q^67 + 952512 * q^70 + 472288 * q^73 - 367616 * q^76 - 70232 * q^79 + 419088 * q^82 - 4428 * q^85 + 485376 * q^88 - 3260224 * q^91 - 868800 * q^94 - 642848 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
5.65685i 0 −32.0000 173.948i 0 −484.000 181.019i 0 −984.000
17.2 5.65685i 0 −32.0000 173.948i 0 −484.000 181.019i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.7.b.a 2
3.b odd 2 1 inner 18.7.b.a 2
4.b odd 2 1 144.7.e.d 2
5.b even 2 1 450.7.d.a 2
5.c odd 4 2 450.7.b.a 4
8.b even 2 1 576.7.e.b 2
8.d odd 2 1 576.7.e.k 2
9.c even 3 2 162.7.d.d 4
9.d odd 6 2 162.7.d.d 4
12.b even 2 1 144.7.e.d 2
15.d odd 2 1 450.7.d.a 2
15.e even 4 2 450.7.b.a 4
24.f even 2 1 576.7.e.k 2
24.h odd 2 1 576.7.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 1.a even 1 1 trivial
18.7.b.a 2 3.b odd 2 1 inner
144.7.e.d 2 4.b odd 2 1
144.7.e.d 2 12.b even 2 1
162.7.d.d 4 9.c even 3 2
162.7.d.d 4 9.d odd 6 2
450.7.b.a 4 5.c odd 4 2
450.7.b.a 4 15.e even 4 2
450.7.d.a 2 5.b even 2 1
450.7.d.a 2 15.d odd 2 1
576.7.e.b 2 8.b even 2 1
576.7.e.b 2 24.h odd 2 1
576.7.e.k 2 8.d odd 2 1
576.7.e.k 2 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 32$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 30258$$
$7$ $$(T + 484)^{2}$$
$11$ $$T^{2} + 1797408$$
$13$ $$(T - 3368)^{2}$$
$17$ $$T^{2} + 162$$
$19$ $$(T - 5744)^{2}$$
$23$ $$T^{2} + 11405088$$
$29$ $$T^{2} + 861706098$$
$31$ $$(T + 39796)^{2}$$
$37$ $$(T - 52526)^{2}$$
$41$ $$T^{2} + 1372146498$$
$43$ $$(T - 3800)^{2}$$
$47$ $$T^{2} + 5896980000$$
$53$ $$T^{2} + 56995657938$$
$59$ $$T^{2} + 62420337792$$
$61$ $$(T - 13250)^{2}$$
$67$ $$(T - 168968)^{2}$$
$71$ $$T^{2} + 282457292832$$
$73$ $$(T - 236144)^{2}$$
$79$ $$(T + 35116)^{2}$$
$83$ $$T^{2} + 120559392$$
$89$ $$T^{2} + 16725839202$$
$97$ $$(T + 321424)^{2}$$