# Properties

 Label 441.4.e.t Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{4} -9 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{4} -9 \beta_{3} q^{8} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{11} + ( 55 + 55 \beta_{2} ) q^{16} + 70 q^{22} -82 \beta_{1} q^{23} -125 \beta_{2} q^{25} -100 \beta_{3} q^{29} + ( -17 \beta_{1} - 17 \beta_{3} ) q^{32} + ( 450 + 450 \beta_{2} ) q^{37} + 180 q^{43} -10 \beta_{1} q^{44} -574 \beta_{2} q^{46} -125 \beta_{3} q^{50} + ( -188 \beta_{1} - 188 \beta_{3} ) q^{53} + ( 700 + 700 \beta_{2} ) q^{58} + 559 q^{64} -740 \beta_{2} q^{67} + 370 \beta_{3} q^{71} + ( 450 \beta_{1} + 450 \beta_{3} ) q^{74} + ( 1384 + 1384 \beta_{2} ) q^{79} + 180 \beta_{1} q^{86} -630 \beta_{2} q^{88} + 82 \beta_{3} q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} + 110q^{16} + 280q^{22} + 250q^{25} + 900q^{37} + 720q^{43} + 1148q^{46} + 1400q^{58} + 2236q^{64} + 1480q^{67} + 2768q^{79} + 1260q^{88} + O(q^{100})$$

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 0.500000 + 0.866025i 0 0 0 −23.8118 0 0
226.2 1.32288 2.29129i 0 0.500000 + 0.866025i 0 0 0 23.8118 0 0
361.1 −1.32288 2.29129i 0 0.500000 0.866025i 0 0 0 −23.8118 0 0
361.2 1.32288 + 2.29129i 0 0.500000 0.866025i 0 0 0 23.8118 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 441.4.e.t 4
3.b odd 2 1 inner 441.4.e.t 4
7.b odd 2 1 CM 441.4.e.t 4
7.c even 3 1 441.4.a.p 2
7.c even 3 1 inner 441.4.e.t 4
7.d odd 6 1 441.4.a.p 2
7.d odd 6 1 inner 441.4.e.t 4
21.c even 2 1 inner 441.4.e.t 4
21.g even 6 1 441.4.a.p 2
21.g even 6 1 inner 441.4.e.t 4
21.h odd 6 1 441.4.a.p 2
21.h odd 6 1 inner 441.4.e.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.p 2 7.c even 3 1
441.4.a.p 2 7.d odd 6 1
441.4.a.p 2 21.g even 6 1
441.4.a.p 2 21.h odd 6 1
441.4.e.t 4 1.a even 1 1 trivial
441.4.e.t 4 3.b odd 2 1 inner
441.4.e.t 4 7.b odd 2 1 CM
441.4.e.t 4 7.c even 3 1 inner
441.4.e.t 4 7.d odd 6 1 inner
441.4.e.t 4 21.c even 2 1 inner
441.4.e.t 4 21.g even 6 1 inner
441.4.e.t 4 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 + 7 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$490000 + 700 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$2215396624 + 47068 T^{2} + T^{4}$$
$29$ $$( -70000 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 202500 - 450 T + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -180 + T )^{4}$$
$47$ $$T^{4}$$
$53$ $$61210718464 + 247408 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 547600 - 740 T + T^{2} )^{2}$$
$71$ $$( -958300 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 1915456 - 1384 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$