# Properties

 Label 441.2.e.f Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{8} + ( -\beta_{1} - \beta_{3} ) q^{10} + 2 \beta_{2} q^{11} + ( -4 - \beta_{3} ) q^{13} + ( -3 - 3 \beta_{2} ) q^{16} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{17} -2 \beta_{1} q^{19} + ( -2 + 3 \beta_{3} ) q^{20} + ( 2 + 2 \beta_{3} ) q^{22} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{25} + ( 6 - 5 \beta_{1} + 6 \beta_{2} ) q^{26} + ( 4 - 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( 8 + 5 \beta_{3} ) q^{34} + ( 4 + 4 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{40} + ( 2 - 3 \beta_{3} ) q^{41} -4 \beta_{3} q^{43} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{46} + 2 \beta_{1} q^{47} + ( -7 - 3 \beta_{3} ) q^{50} + ( 9 \beta_{1} - 8 \beta_{2} + 9 \beta_{3} ) q^{52} + 2 \beta_{2} q^{53} + ( 4 - 2 \beta_{3} ) q^{55} + 2 \beta_{1} q^{58} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 8 + 3 \beta_{1} + 8 \beta_{2} ) q^{61} + ( 8 + 6 \beta_{3} ) q^{62} + ( -7 - 2 \beta_{3} ) q^{64} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{65} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{67} + ( -14 + 7 \beta_{1} - 14 \beta_{2} ) q^{68} + ( 2 + 8 \beta_{3} ) q^{71} + ( -7 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{73} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -8 - 2 \beta_{3} ) q^{76} + ( -8 - 4 \beta_{1} - 8 \beta_{2} ) q^{79} + ( 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{80} + ( 4 - \beta_{1} + 4 \beta_{2} ) q^{82} + ( -4 - 8 \beta_{3} ) q^{83} + ( -2 + 4 \beta_{3} ) q^{85} + ( 8 - 4 \beta_{1} + 8 \beta_{2} ) q^{86} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{88} + ( 10 + 3 \beta_{1} + 10 \beta_{2} ) q^{89} -14 q^{92} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -4 - \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 4q^{5} + 12q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 4q^{5} + 12q^{8} - 4q^{11} - 16q^{13} - 6q^{16} - 4q^{17} - 8q^{20} + 8q^{22} - 4q^{23} - 2q^{25} + 12q^{26} + 16q^{29} - 8q^{31} + 6q^{32} + 32q^{34} + 8q^{37} + 8q^{38} - 8q^{40} + 8q^{41} - 4q^{44} + 12q^{46} - 28q^{50} + 16q^{52} - 4q^{53} + 16q^{55} + 8q^{59} + 16q^{61} + 32q^{62} - 28q^{64} + 12q^{65} - 28q^{68} + 8q^{71} + 8q^{73} + 8q^{74} - 32q^{76} - 16q^{79} - 12q^{80} + 8q^{82} - 16q^{83} - 8q^{85} + 16q^{86} - 12q^{88} + 20q^{89} - 56q^{92} - 8q^{94} - 8q^{95} - 16q^{97} + O(q^{100})$$

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

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

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−1.20711 + 2.09077i 0 −1.91421 3.31552i −0.292893 + 0.507306i 0 0 4.41421 0 −0.707107 1.22474i
226.2 0.207107 0.358719i 0 0.914214 + 1.58346i −1.70711 + 2.95680i 0 0 1.58579 0 0.707107 + 1.22474i
361.1 −1.20711 2.09077i 0 −1.91421 + 3.31552i −0.292893 0.507306i 0 0 4.41421 0 −0.707107 + 1.22474i
361.2 0.207107 + 0.358719i 0 0.914214 1.58346i −1.70711 2.95680i 0 0 1.58579 0 0.707107 1.22474i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.f 4
3.b odd 2 1 147.2.e.e 4
7.b odd 2 1 441.2.e.g 4
7.c even 3 1 441.2.a.j 2
7.c even 3 1 inner 441.2.e.f 4
7.d odd 6 1 441.2.a.i 2
7.d odd 6 1 441.2.e.g 4
12.b even 2 1 2352.2.q.bb 4
21.c even 2 1 147.2.e.d 4
21.g even 6 1 147.2.a.e yes 2
21.g even 6 1 147.2.e.d 4
21.h odd 6 1 147.2.a.d 2
21.h odd 6 1 147.2.e.e 4
28.f even 6 1 7056.2.a.cf 2
28.g odd 6 1 7056.2.a.cv 2
84.h odd 2 1 2352.2.q.bd 4
84.j odd 6 1 2352.2.a.bc 2
84.j odd 6 1 2352.2.q.bd 4
84.n even 6 1 2352.2.a.be 2
84.n even 6 1 2352.2.q.bb 4
105.o odd 6 1 3675.2.a.bf 2
105.p even 6 1 3675.2.a.bd 2
168.s odd 6 1 9408.2.a.ef 2
168.v even 6 1 9408.2.a.dq 2
168.ba even 6 1 9408.2.a.di 2
168.be odd 6 1 9408.2.a.dt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 21.h odd 6 1
147.2.a.e yes 2 21.g even 6 1
147.2.e.d 4 21.c even 2 1
147.2.e.d 4 21.g even 6 1
147.2.e.e 4 3.b odd 2 1
147.2.e.e 4 21.h odd 6 1
441.2.a.i 2 7.d odd 6 1
441.2.a.j 2 7.c even 3 1
441.2.e.f 4 1.a even 1 1 trivial
441.2.e.f 4 7.c even 3 1 inner
441.2.e.g 4 7.b odd 2 1
441.2.e.g 4 7.d odd 6 1
2352.2.a.bc 2 84.j odd 6 1
2352.2.a.be 2 84.n even 6 1
2352.2.q.bb 4 12.b even 2 1
2352.2.q.bb 4 84.n even 6 1
2352.2.q.bd 4 84.h odd 2 1
2352.2.q.bd 4 84.j odd 6 1
3675.2.a.bd 2 105.p even 6 1
3675.2.a.bf 2 105.o odd 6 1
7056.2.a.cf 2 28.f even 6 1
7056.2.a.cv 2 28.g odd 6 1
9408.2.a.di 2 168.ba even 6 1
9408.2.a.dq 2 168.v even 6 1
9408.2.a.dt 2 168.be odd 6 1
9408.2.a.ef 2 168.s odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{5}^{4} + 4 T_{5}^{3} + 14 T_{5}^{2} + 8 T_{5} + 4$$ $$T_{13}^{2} + 8 T_{13} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ 1
$5$ $$1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 40 T^{5} + 100 T^{6} + 500 T^{7} + 625 T^{8}$$
$7$ 1
$11$ $$( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8}$$
$23$ $$1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 2576 T^{5} - 1058 T^{6} + 48668 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 1984 T^{5} - 5766 T^{6} + 238328 T^{7} + 923521 T^{8}$$
$37$ $$( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 54 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 3776 T^{5} - 215822 T^{6} - 1643032 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 44896 T^{5} + 327448 T^{6} - 3631696 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 8 T + 656 T^{3} - 5905 T^{4} + 47888 T^{5} - 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 145960 T^{5} + 1108940 T^{6} - 14099380 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} )^{2}$$