# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{5} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{5} -2 \beta_{1} q^{8} -2 \beta_{2} q^{10} -\beta_{1} q^{11} + 3 \beta_{2} q^{13} -4 q^{16} + 2 \beta_{3} q^{17} -\beta_{2} q^{19} -2 q^{22} -4 \beta_{1} q^{23} + q^{25} + 3 \beta_{3} q^{26} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} -4 \beta_{2} q^{34} - q^{37} -\beta_{3} q^{38} -4 \beta_{2} q^{40} -3 \beta_{3} q^{41} - q^{43} -8 q^{46} -5 \beta_{3} q^{47} -\beta_{1} q^{50} + 2 \beta_{1} q^{53} -2 \beta_{2} q^{55} + 4 q^{58} + 2 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + \beta_{3} q^{62} -8 q^{64} + 9 \beta_{1} q^{65} + 11 q^{67} + 5 \beta_{1} q^{71} + \beta_{2} q^{73} + \beta_{1} q^{74} + 5 q^{79} -4 \beta_{3} q^{80} + 6 \beta_{2} q^{82} + 3 \beta_{3} q^{83} + 12 q^{85} + \beta_{1} q^{86} -4 q^{88} -2 \beta_{3} q^{89} + 10 \beta_{2} q^{94} -3 \beta_{1} q^{95} + 6 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{16} - 8q^{22} + 4q^{25} - 4q^{37} - 4q^{43} - 32q^{46} + 16q^{58} - 32q^{64} + 44q^{67} + 20q^{79} + 48q^{85} - 16q^{88} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i
1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.2 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
440.3 1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.4 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
 $$n$$: 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.c.a 4
3.b odd 2 1 inner 441.2.c.a 4
4.b odd 2 1 7056.2.k.b 4
7.b odd 2 1 inner 441.2.c.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.p.a 4
7.d odd 6 1 63.2.p.a 4
7.d odd 6 1 441.2.p.a 4
12.b even 2 1 7056.2.k.b 4
21.c even 2 1 inner 441.2.c.a 4
21.g even 6 1 63.2.p.a 4
21.g even 6 1 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.p.a 4
28.d even 2 1 7056.2.k.b 4
28.f even 6 1 1008.2.bt.b 4
28.g odd 6 1 1008.2.bt.b 4
35.i odd 6 1 1575.2.bk.c 4
35.j even 6 1 1575.2.bk.c 4
35.k even 12 2 1575.2.bc.a 8
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.s.d 4
63.h even 3 1 567.2.i.d 4
63.i even 6 1 567.2.i.d 4
63.j odd 6 1 567.2.i.d 4
63.k odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.s.d 4
63.s even 6 1 567.2.s.d 4
63.t odd 6 1 567.2.i.d 4
84.h odd 2 1 7056.2.k.b 4
84.j odd 6 1 1008.2.bt.b 4
84.n even 6 1 1008.2.bt.b 4
105.o odd 6 1 1575.2.bk.c 4
105.p even 6 1 1575.2.bk.c 4
105.w odd 12 2 1575.2.bc.a 8
105.x even 12 2 1575.2.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 7.c even 3 1
63.2.p.a 4 7.d odd 6 1
63.2.p.a 4 21.g even 6 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 1.a even 1 1 trivial
441.2.c.a 4 3.b odd 2 1 inner
441.2.c.a 4 7.b odd 2 1 inner
441.2.c.a 4 21.c even 2 1 inner
441.2.p.a 4 7.c even 3 1
441.2.p.a 4 7.d odd 6 1
441.2.p.a 4 21.g even 6 1
441.2.p.a 4 21.h odd 6 1
567.2.i.d 4 63.h even 3 1
567.2.i.d 4 63.i even 6 1
567.2.i.d 4 63.j odd 6 1
567.2.i.d 4 63.t odd 6 1
567.2.s.d 4 63.g even 3 1
567.2.s.d 4 63.k odd 6 1
567.2.s.d 4 63.n odd 6 1
567.2.s.d 4 63.s even 6 1
1008.2.bt.b 4 28.f even 6 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.j odd 6 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.k even 12 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.w odd 12 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.i odd 6 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.o odd 6 1
1575.2.bk.c 4 105.p even 6 1
7056.2.k.b 4 4.b odd 2 1
7056.2.k.b 4 12.b even 2 1
7056.2.k.b 4 28.d even 2 1
7056.2.k.b 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -6 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 2 + T^{2} )^{2}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 3 + T^{2} )^{2}$$
$23$ $$( 32 + T^{2} )^{2}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$( -54 + T^{2} )^{2}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$( -150 + T^{2} )^{2}$$
$53$ $$( 8 + T^{2} )^{2}$$
$59$ $$( -24 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( -11 + T )^{4}$$
$71$ $$( 50 + T^{2} )^{2}$$
$73$ $$( 3 + T^{2} )^{2}$$
$79$ $$( -5 + T )^{4}$$
$83$ $$( -54 + T^{2} )^{2}$$
$89$ $$( -24 + T^{2} )^{2}$$
$97$ $$( 108 + T^{2} )^{2}$$
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