# Properties

 Label 144.7.e.c 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 36) 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.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} + 244 q^{7}+O(q^{10})$$ q + 5*b * q^5 + 244 * q^7 $$q + 5 \beta q^{5} + 244 q^{7} - 188 \beta q^{11} - 2728 q^{13} + 463 \beta q^{17} + 5392 q^{19} - 196 \beta q^{23} + 11575 q^{25} - 3301 \beta q^{29} - 10172 q^{31} + 1220 \beta q^{35} + 65006 q^{37} - 5479 \beta q^{41} + 49480 q^{43} - 12948 \beta q^{47} - 58113 q^{49} + 1955 \beta q^{53} + 152280 q^{55} - 28216 \beta q^{59} + 100610 q^{61} - 13640 \beta q^{65} + 435736 q^{67} + 4724 \beta q^{71} + 619568 q^{73} - 45872 \beta q^{77} - 514340 q^{79} + 19164 \beta q^{83} - 375030 q^{85} - 33647 \beta q^{89} - 665632 q^{91} + 26960 \beta q^{95} + 42704 q^{97} +O(q^{100})$$ q + 5*b * q^5 + 244 * q^7 - 188*b * q^11 - 2728 * q^13 + 463*b * q^17 + 5392 * q^19 - 196*b * q^23 + 11575 * q^25 - 3301*b * q^29 - 10172 * q^31 + 1220*b * q^35 + 65006 * q^37 - 5479*b * q^41 + 49480 * q^43 - 12948*b * q^47 - 58113 * q^49 + 1955*b * q^53 + 152280 * q^55 - 28216*b * q^59 + 100610 * q^61 - 13640*b * q^65 + 435736 * q^67 + 4724*b * q^71 + 619568 * q^73 - 45872*b * q^77 - 514340 * q^79 + 19164*b * q^83 - 375030 * q^85 - 33647*b * q^89 - 665632 * q^91 + 26960*b * q^95 + 42704 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 488 q^{7}+O(q^{10})$$ 2 * q + 488 * q^7 $$2 q + 488 q^{7} - 5456 q^{13} + 10784 q^{19} + 23150 q^{25} - 20344 q^{31} + 130012 q^{37} + 98960 q^{43} - 116226 q^{49} + 304560 q^{55} + 201220 q^{61} + 871472 q^{67} + 1239136 q^{73} - 1028680 q^{79} - 750060 q^{85} - 1331264 q^{91} + 85408 q^{97}+O(q^{100})$$ 2 * q + 488 * q^7 - 5456 * q^13 + 10784 * q^19 + 23150 * q^25 - 20344 * q^31 + 130012 * q^37 + 98960 * q^43 - 116226 * q^49 + 304560 * q^55 + 201220 * q^61 + 871472 * q^67 + 1239136 * q^73 - 1028680 * q^79 - 750060 * q^85 - 1331264 * q^91 + 85408 * 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 0 244.000 0 0 0
17.2 0 0 0 63.6396i 0 244.000 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.c 2
3.b odd 2 1 inner 144.7.e.c 2
4.b odd 2 1 36.7.c.a 2
8.b even 2 1 576.7.e.j 2
8.d odd 2 1 576.7.e.c 2
12.b even 2 1 36.7.c.a 2
24.f even 2 1 576.7.e.c 2
24.h odd 2 1 576.7.e.j 2
36.f odd 6 2 324.7.g.e 4
36.h even 6 2 324.7.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.7.c.a 2 4.b odd 2 1
36.7.c.a 2 12.b even 2 1
144.7.e.c 2 1.a even 1 1 trivial
144.7.e.c 2 3.b odd 2 1 inner
324.7.g.e 4 36.f odd 6 2
324.7.g.e 4 36.h even 6 2
576.7.e.c 2 8.d odd 2 1
576.7.e.c 2 24.f even 2 1
576.7.e.j 2 8.b even 2 1
576.7.e.j 2 24.h odd 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 $$T_{7} - 244$$ T7 - 244

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4050$$
$7$ $$(T - 244)^{2}$$
$11$ $$T^{2} + 5725728$$
$13$ $$(T + 2728)^{2}$$
$17$ $$T^{2} + 34727778$$
$19$ $$(T - 5392)^{2}$$
$23$ $$T^{2} + 6223392$$
$29$ $$T^{2} + 1765249362$$
$31$ $$(T + 10172)^{2}$$
$37$ $$(T - 65006)^{2}$$
$41$ $$T^{2} + 4863149442$$
$43$ $$(T - 49480)^{2}$$
$47$ $$T^{2} + 27159414048$$
$53$ $$T^{2} + 619168050$$
$59$ $$T^{2} + 128975110272$$
$61$ $$(T - 100610)^{2}$$
$67$ $$(T - 435736)^{2}$$
$71$ $$T^{2} + 3615220512$$
$73$ $$(T - 619568)^{2}$$
$79$ $$(T + 514340)^{2}$$
$83$ $$T^{2} + 59495941152$$
$89$ $$T^{2} + 183403538658$$
$97$ $$(T - 42704)^{2}$$