# Properties

 Label 441.2.e.h Level 441 Weight 2 Character orbit 441.e Analytic conductor 3.521 Analytic rank 0 Dimension 4 CM discriminant -7 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -11 - 11 \beta_{2} ) q^{16} -14 q^{22} + 2 \beta_{1} q^{23} -5 \beta_{2} q^{25} -4 \beta_{3} q^{29} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{32} + ( -6 - 6 \beta_{2} ) q^{37} + 12 q^{43} -10 \beta_{1} q^{44} + 14 \beta_{2} q^{46} -5 \beta_{3} q^{50} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{53} + ( 28 + 28 \beta_{2} ) q^{58} + 13 q^{64} + 4 \beta_{2} q^{67} -2 \beta_{3} q^{71} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{74} + ( -8 - 8 \beta_{2} ) q^{79} + 12 \beta_{1} q^{86} -42 \beta_{2} q^{88} + 10 \beta_{3} q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + O(q^{10})$$ $$4q - 10q^{4} - 22q^{16} - 56q^{22} + 10q^{25} - 12q^{37} + 48q^{43} - 28q^{46} + 56q^{58} + 52q^{64} - 8q^{67} - 16q^{79} + 84q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
−1.32288 + 2.29129i 0 −2.50000 4.33013i 0 0 0 7.93725 0 0
226.2 1.32288 2.29129i 0 −2.50000 4.33013i 0 0 0 −7.93725 0 0
361.1 −1.32288 2.29129i 0 −2.50000 + 4.33013i 0 0 0 7.93725 0 0
361.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i 0 0 0 −7.93725 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.h 4
3.b odd 2 1 inner 441.2.e.h 4
7.b odd 2 1 CM 441.2.e.h 4
7.c even 3 1 441.2.a.h 2
7.c even 3 1 inner 441.2.e.h 4
7.d odd 6 1 441.2.a.h 2
7.d odd 6 1 inner 441.2.e.h 4
21.c even 2 1 inner 441.2.e.h 4
21.g even 6 1 441.2.a.h 2
21.g even 6 1 inner 441.2.e.h 4
21.h odd 6 1 441.2.a.h 2
21.h odd 6 1 inner 441.2.e.h 4
28.f even 6 1 7056.2.a.co 2
28.g odd 6 1 7056.2.a.co 2
84.j odd 6 1 7056.2.a.co 2
84.n even 6 1 7056.2.a.co 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 7.c even 3 1
441.2.a.h 2 7.d odd 6 1
441.2.a.h 2 21.g even 6 1
441.2.a.h 2 21.h odd 6 1
441.2.e.h 4 1.a even 1 1 trivial
441.2.e.h 4 3.b odd 2 1 inner
441.2.e.h 4 7.b odd 2 1 CM
441.2.e.h 4 7.c even 3 1 inner
441.2.e.h 4 7.d odd 6 1 inner
441.2.e.h 4 21.c even 2 1 inner
441.2.e.h 4 21.g even 6 1 inner
441.2.e.h 4 21.h odd 6 1 inner
7056.2.a.co 2 28.f even 6 1
7056.2.a.co 2 28.g odd 6 1
7056.2.a.co 2 84.j odd 6 1
7056.2.a.co 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{4} + 7 T_{2}^{2} + 49$$ $$T_{5}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$
$3$ 1
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ 1
$11$ $$1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 12 T + 43 T^{2} )^{4}$$
$47$ $$( 1 - 47 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{4}$$