# Properties

 Label 441.4.p.a Level $441$ Weight $4$ Character orbit 441.p Analytic conductor $26.020$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.9948826238976.7 Defining polynomial: $$x^{8} + 36 x^{6} + 935 x^{4} + 12996 x^{2} + 130321$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{4} q^{2} + ( \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{3} + 16 \beta_{4} ) q^{8} +O(q^{10})$$ $$q -2 \beta_{4} q^{2} + ( \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{3} + 16 \beta_{4} ) q^{8} -2 \beta_{7} q^{10} -11 \beta_{3} q^{11} -\beta_{6} q^{13} + 64 \beta_{1} q^{16} + 3 \beta_{2} q^{17} + ( -3 \beta_{6} - 3 \beta_{7} ) q^{19} -44 q^{22} + 55 \beta_{4} q^{23} + ( -319 + 319 \beta_{1} ) q^{25} + ( -4 \beta_{2} + 4 \beta_{5} ) q^{26} + ( 89 \beta_{3} + 89 \beta_{4} ) q^{29} + 8 \beta_{7} q^{31} + 6 \beta_{6} q^{34} + 184 \beta_{1} q^{37} -12 \beta_{2} q^{38} + ( 16 \beta_{6} + 16 \beta_{7} ) q^{40} -5 \beta_{5} q^{41} -190 q^{43} + ( -220 + 220 \beta_{1} ) q^{46} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{47} + ( 638 \beta_{3} + 638 \beta_{4} ) q^{50} + 253 \beta_{3} q^{53} -11 \beta_{6} q^{55} + 356 \beta_{1} q^{58} -4 \beta_{2} q^{59} + ( -22 \beta_{6} - 22 \beta_{7} ) q^{61} + 32 \beta_{5} q^{62} -512 q^{64} -444 \beta_{4} q^{65} + ( -296 + 296 \beta_{1} ) q^{67} + ( 233 \beta_{3} + 233 \beta_{4} ) q^{71} -27 \beta_{7} q^{73} + 368 \beta_{3} q^{74} -836 \beta_{1} q^{79} + 64 \beta_{2} q^{80} + ( -10 \beta_{6} - 10 \beta_{7} ) q^{82} -58 \beta_{5} q^{83} -1332 q^{85} + 380 \beta_{4} q^{86} + ( 352 - 352 \beta_{1} ) q^{88} + ( 33 \beta_{2} - 33 \beta_{5} ) q^{89} -4 \beta_{7} q^{94} + 1332 \beta_{3} q^{95} + 19 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 256q^{16} - 352q^{22} - 1276q^{25} + 736q^{37} - 1520q^{43} - 880q^{46} + 1424q^{58} - 4096q^{64} - 1184q^{67} - 3344q^{79} - 10656q^{85} + 1408q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 36 x^{6} + 935 x^{4} + 12996 x^{2} + 130321$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$36 \nu^{6} + 935 \nu^{4} + 33660 \nu^{2} + 467856$$$$)/337535$$ $$\beta_{2}$$ $$=$$ $$($$$$-148 \nu^{6} + 33660 \nu^{4} + 536690 \nu^{2} + 10227852$$$$)/337535$$ $$\beta_{3}$$ $$=$$ $$($$$$251 \nu^{7} + 15895 \nu^{5} + 234685 \nu^{3} + 3261996 \nu$$$$)/6413165$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 2899 \nu$$$$)/17765$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{6} + 72 \nu^{4} + 1148 \nu^{2} + 12996$$$$)/361$$ $$\beta_{6}$$ $$=$$ $$($$$$-148 \nu^{7} - 6050 \nu^{5} - 178090 \nu^{3} - 4107458 \nu$$$$)/377245$$ $$\beta_{7}$$ $$=$$ $$($$$$-4682 \nu^{7} - 102850 \nu^{5} - 3027530 \nu^{3} + 13410428 \nu$$$$)/6413165$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + 6 \beta_{4}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} + \beta_{2} + 108 \beta_{1} - 108$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-34 \beta_{7} - 17 \beta_{6} - 330 \beta_{4} - 330 \beta_{3}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{5} + 6 \beta_{2} - 287 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$251 \beta_{7} + 502 \beta_{6} + 9714 \beta_{3}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$935 \beta_{5} - 1870 \beta_{2} + 23004$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$2899 \beta_{7} - 2899 \beta_{6} + 230574 \beta_{4}$$$$)/12$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −1.53821 − 4.07847i 2.76295 + 3.37136i 1.53821 + 4.07847i −2.76295 − 3.37136i −1.53821 + 4.07847i 2.76295 − 3.37136i 1.53821 − 4.07847i −2.76295 + 3.37136i
−2.44949 + 1.41421i 0 0 −10.5357 18.2483i 0 0 22.6274i 0 51.6140 + 29.7993i
80.2 −2.44949 + 1.41421i 0 0 10.5357 + 18.2483i 0 0 22.6274i 0 −51.6140 29.7993i
80.3 2.44949 1.41421i 0 0 −10.5357 18.2483i 0 0 22.6274i 0 −51.6140 29.7993i
80.4 2.44949 1.41421i 0 0 10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 + 29.7993i
215.1 −2.44949 1.41421i 0 0 −10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 29.7993i
215.2 −2.44949 1.41421i 0 0 10.5357 18.2483i 0 0 22.6274i 0 −51.6140 + 29.7993i
215.3 2.44949 + 1.41421i 0 0 −10.5357 + 18.2483i 0 0 22.6274i 0 −51.6140 + 29.7993i
215.4 2.44949 + 1.41421i 0 0 10.5357 18.2483i 0 0 22.6274i 0 51.6140 29.7993i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.4 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
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.p.a 8
3.b odd 2 1 inner 441.4.p.a 8
7.b odd 2 1 inner 441.4.p.a 8
7.c even 3 1 63.4.c.b 4
7.c even 3 1 inner 441.4.p.a 8
7.d odd 6 1 63.4.c.b 4
7.d odd 6 1 inner 441.4.p.a 8
21.c even 2 1 inner 441.4.p.a 8
21.g even 6 1 63.4.c.b 4
21.g even 6 1 inner 441.4.p.a 8
21.h odd 6 1 63.4.c.b 4
21.h odd 6 1 inner 441.4.p.a 8
28.f even 6 1 1008.4.k.b 4
28.g odd 6 1 1008.4.k.b 4
84.j odd 6 1 1008.4.k.b 4
84.n even 6 1 1008.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b 4 7.c even 3 1
63.4.c.b 4 7.d odd 6 1
63.4.c.b 4 21.g even 6 1
63.4.c.b 4 21.h odd 6 1
441.4.p.a 8 1.a even 1 1 trivial
441.4.p.a 8 3.b odd 2 1 inner
441.4.p.a 8 7.b odd 2 1 inner
441.4.p.a 8 7.c even 3 1 inner
441.4.p.a 8 7.d odd 6 1 inner
441.4.p.a 8 21.c even 2 1 inner
441.4.p.a 8 21.g even 6 1 inner
441.4.p.a 8 21.h odd 6 1 inner
1008.4.k.b 4 28.f even 6 1
1008.4.k.b 4 28.g odd 6 1
1008.4.k.b 4 84.j odd 6 1
1008.4.k.b 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 8 T_{2}^{2} + 64$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 197136 + 444 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 58564 - 242 T^{2} + T^{4} )^{2}$$
$13$ $$( 888 + T^{2} )^{4}$$
$17$ $$( 15968016 + 3996 T^{2} + T^{4} )^{2}$$
$19$ $$( 63872064 - 7992 T^{2} + T^{4} )^{2}$$
$23$ $$( 36602500 - 6050 T^{2} + T^{4} )^{2}$$
$29$ $$( 15842 + T^{2} )^{4}$$
$31$ $$( 3229876224 - 56832 T^{2} + T^{4} )^{2}$$
$37$ $$( 33856 - 184 T + T^{2} )^{4}$$
$41$ $$( -11100 + T^{2} )^{4}$$
$43$ $$( 190 + T )^{8}$$
$47$ $$( 3154176 + 1776 T^{2} + T^{4} )^{2}$$
$53$ $$( 16388608324 - 128018 T^{2} + T^{4} )^{2}$$
$59$ $$( 50466816 + 7104 T^{2} + T^{4} )^{2}$$
$61$ $$( 184721163264 - 429792 T^{2} + T^{4} )^{2}$$
$67$ $$( 87616 + 296 T + T^{2} )^{4}$$
$71$ $$( 108578 + T^{2} )^{4}$$
$73$ $$( 419064611904 - 647352 T^{2} + T^{4} )^{2}$$
$79$ $$( 698896 + 836 T + T^{2} )^{4}$$
$83$ $$( -1493616 + T^{2} )^{4}$$
$89$ $$( 233787722256 + 483516 T^{2} + T^{4} )^{2}$$
$97$ $$( 320568 + T^{2} )^{4}$$