# Properties

 Label 144.7.e.a Level $144$ Weight $7$ Character orbit 144.e Analytic conductor $33.128$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 144.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1277880413$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 9\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} - 524 q^{7}+O(q^{10})$$ q + 5*b * q^5 - 524 * q^7 $$q + 5 \beta q^{5} - 524 q^{7} + 68 \beta q^{11} + 344 q^{13} - 561 \beta q^{17} + 2320 q^{19} - 452 \beta q^{23} + 11575 q^{25} + 1819 \beta q^{29} + 10564 q^{31} - 2620 \beta q^{35} - 24082 q^{37} - 8551 \beta q^{41} + 90952 q^{43} - 10132 \beta q^{47} + 156927 q^{49} - 15453 \beta q^{53} - 55080 q^{55} - 3128 \beta q^{59} + 251138 q^{61} + 1720 \beta q^{65} + 216088 q^{67} - 4236 \beta q^{71} - 308176 q^{73} - 35632 \beta q^{77} + 540124 q^{79} - 73252 \beta q^{83} + 454410 q^{85} + 17553 \beta q^{89} - 180256 q^{91} + 11600 \beta q^{95} - 37168 q^{97} +O(q^{100})$$ q + 5*b * q^5 - 524 * q^7 + 68*b * q^11 + 344 * q^13 - 561*b * q^17 + 2320 * q^19 - 452*b * q^23 + 11575 * q^25 + 1819*b * q^29 + 10564 * q^31 - 2620*b * q^35 - 24082 * q^37 - 8551*b * q^41 + 90952 * q^43 - 10132*b * q^47 + 156927 * q^49 - 15453*b * q^53 - 55080 * q^55 - 3128*b * q^59 + 251138 * q^61 + 1720*b * q^65 + 216088 * q^67 - 4236*b * q^71 - 308176 * q^73 - 35632*b * q^77 + 540124 * q^79 - 73252*b * q^83 + 454410 * q^85 + 17553*b * q^89 - 180256 * q^91 + 11600*b * q^95 - 37168 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1048 q^{7}+O(q^{10})$$ 2 * q - 1048 * q^7 $$2 q - 1048 q^{7} + 688 q^{13} + 4640 q^{19} + 23150 q^{25} + 21128 q^{31} - 48164 q^{37} + 181904 q^{43} + 313854 q^{49} - 110160 q^{55} + 502276 q^{61} + 432176 q^{67} - 616352 q^{73} + 1080248 q^{79} + 908820 q^{85} - 360512 q^{91} - 74336 q^{97}+O(q^{100})$$ 2 * q - 1048 * q^7 + 688 * q^13 + 4640 * q^19 + 23150 * q^25 + 21128 * q^31 - 48164 * q^37 + 181904 * q^43 + 313854 * q^49 - 110160 * q^55 + 502276 * q^61 + 432176 * q^67 - 616352 * q^73 + 1080248 * q^79 + 908820 * q^85 - 360512 * q^91 - 74336 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 0 0 63.6396i 0 −524.000 0 0 0
17.2 0 0 0 63.6396i 0 −524.000 0 0 0
 $$n$$: 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 144.7.e.a 2
3.b odd 2 1 inner 144.7.e.a 2
4.b odd 2 1 9.7.b.a 2
8.b even 2 1 576.7.e.a 2
8.d odd 2 1 576.7.e.l 2
12.b even 2 1 9.7.b.a 2
20.d odd 2 1 225.7.c.a 2
20.e even 4 2 225.7.d.a 4
24.f even 2 1 576.7.e.l 2
24.h odd 2 1 576.7.e.a 2
28.d even 2 1 441.7.b.a 2
36.f odd 6 2 81.7.d.d 4
36.h even 6 2 81.7.d.d 4
60.h even 2 1 225.7.c.a 2
60.l odd 4 2 225.7.d.a 4
84.h odd 2 1 441.7.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.b.a 2 4.b odd 2 1
9.7.b.a 2 12.b even 2 1
81.7.d.d 4 36.f odd 6 2
81.7.d.d 4 36.h even 6 2
144.7.e.a 2 1.a even 1 1 trivial
144.7.e.a 2 3.b odd 2 1 inner
225.7.c.a 2 20.d odd 2 1
225.7.c.a 2 60.h even 2 1
225.7.d.a 4 20.e even 4 2
225.7.d.a 4 60.l odd 4 2
441.7.b.a 2 28.d even 2 1
441.7.b.a 2 84.h odd 2 1
576.7.e.a 2 8.b even 2 1
576.7.e.a 2 24.h odd 2 1
576.7.e.l 2 8.d odd 2 1
576.7.e.l 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(144, [\chi])$$:

 $$T_{5}^{2} + 4050$$ T5^2 + 4050 $$T_{7} + 524$$ T7 + 524

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4050$$
$7$ $$(T + 524)^{2}$$
$11$ $$T^{2} + 749088$$
$13$ $$(T - 344)^{2}$$
$17$ $$T^{2} + 50984802$$
$19$ $$(T - 2320)^{2}$$
$23$ $$T^{2} + 33097248$$
$29$ $$T^{2} + 536019282$$
$31$ $$(T - 10564)^{2}$$
$37$ $$(T + 24082)^{2}$$
$41$ $$T^{2} + 11845375362$$
$43$ $$(T - 90952)^{2}$$
$47$ $$T^{2} + 16630502688$$
$53$ $$T^{2} + 38684823858$$
$59$ $$T^{2} + 1585070208$$
$61$ $$(T - 251138)^{2}$$
$67$ $$(T - 216088)^{2}$$
$71$ $$T^{2} + 2906878752$$
$73$ $$(T + 308176)^{2}$$
$79$ $$(T - 540124)^{2}$$
$83$ $$T^{2} + 869268591648$$
$89$ $$T^{2} + 49913465058$$
$97$ $$(T + 37168)^{2}$$