# Properties

 Label 441.4.e.u Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{4} + ( -10 - 7 \beta_{1} - 10 \beta_{2} ) q^{5} + ( 9 - 11 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{4} + ( -10 - 7 \beta_{1} - 10 \beta_{2} ) q^{5} + ( 9 - 11 \beta_{3} ) q^{8} + ( -17 \beta_{1} - 24 \beta_{2} - 17 \beta_{3} ) q^{10} + ( -24 \beta_{1} + 10 \beta_{2} - 24 \beta_{3} ) q^{11} + ( -52 + 25 \beta_{3} ) q^{13} + ( -9 + 36 \beta_{1} - 9 \beta_{2} ) q^{16} + ( -45 \beta_{1} + 58 \beta_{2} - 45 \beta_{3} ) q^{17} + ( 96 - 22 \beta_{1} + 96 \beta_{2} ) q^{19} + ( -22 + 15 \beta_{3} ) q^{20} + ( 38 - 14 \beta_{3} ) q^{22} + ( 14 - 28 \beta_{1} + 14 \beta_{2} ) q^{23} + ( 140 \beta_{1} + 73 \beta_{2} + 140 \beta_{3} ) q^{25} + ( -102 - 77 \beta_{1} - 102 \beta_{2} ) q^{26} + ( -148 - 62 \beta_{3} ) q^{29} + ( 50 \beta_{1} + 52 \beta_{2} + 50 \beta_{3} ) q^{31} + ( -61 \beta_{1} - 9 \beta_{2} - 61 \beta_{3} ) q^{32} + ( 32 + 13 \beta_{3} ) q^{34} + ( 124 - 48 \beta_{1} + 124 \beta_{2} ) q^{37} + ( 74 \beta_{1} + 52 \beta_{2} + 74 \beta_{3} ) q^{38} + ( -244 - 173 \beta_{1} - 244 \beta_{2} ) q^{40} + ( 10 + 219 \beta_{3} ) q^{41} + ( -360 + 100 \beta_{3} ) q^{43} + ( 146 - 140 \beta_{1} + 146 \beta_{2} ) q^{44} + ( -14 \beta_{1} - 42 \beta_{2} - 14 \beta_{3} ) q^{46} + ( 48 - 250 \beta_{1} + 48 \beta_{2} ) q^{47} + ( -353 + 213 \beta_{3} ) q^{50} + ( 21 \beta_{1} + 160 \beta_{2} + 21 \beta_{3} ) q^{52} + ( 360 \beta_{1} - 134 \beta_{2} + 360 \beta_{3} ) q^{53} + ( -236 + 170 \beta_{3} ) q^{55} + ( -24 - 86 \beta_{1} - 24 \beta_{2} ) q^{58} + ( -226 \beta_{1} - 308 \beta_{2} - 226 \beta_{3} ) q^{59} + ( 8 - 3 \beta_{1} + 8 \beta_{2} ) q^{61} + ( -152 + 102 \beta_{3} ) q^{62} + ( 59 - 358 \beta_{3} ) q^{64} + ( 870 + 614 \beta_{1} + 870 \beta_{2} ) q^{65} + ( -524 \beta_{1} - 72 \beta_{2} - 524 \beta_{3} ) q^{67} + ( 470 - 341 \beta_{1} + 470 \beta_{2} ) q^{68} + ( -494 - 232 \beta_{3} ) q^{71} + ( -401 \beta_{1} - 52 \beta_{2} - 401 \beta_{3} ) q^{73} + ( 76 \beta_{1} + 28 \beta_{2} + 76 \beta_{3} ) q^{74} + ( 568 + 302 \beta_{3} ) q^{76} + ( 472 - 236 \beta_{1} + 472 \beta_{2} ) q^{79} + ( -297 \beta_{1} - 414 \beta_{2} - 297 \beta_{3} ) q^{80} + ( -428 - 209 \beta_{1} - 428 \beta_{2} ) q^{82} + ( 508 - 80 \beta_{3} ) q^{83} + ( -50 + 44 \beta_{3} ) q^{85} + ( -560 - 460 \beta_{1} - 560 \beta_{2} ) q^{86} + ( -106 \beta_{1} - 438 \beta_{2} - 106 \beta_{3} ) q^{88} + ( 194 - 339 \beta_{1} + 194 \beta_{2} ) q^{89} + ( 182 + 168 \beta_{3} ) q^{92} + ( -202 \beta_{1} - 452 \beta_{2} - 202 \beta_{3} ) q^{94} + ( -452 \beta_{1} - 652 \beta_{2} - 452 \beta_{3} ) q^{95} + ( -244 - 599 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 10q^{4} - 20q^{5} + 36q^{8} + O(q^{10})$$ $$4q + 2q^{2} + 10q^{4} - 20q^{5} + 36q^{8} + 48q^{10} - 20q^{11} - 208q^{13} - 18q^{16} - 116q^{17} + 192q^{19} - 88q^{20} + 152q^{22} + 28q^{23} - 146q^{25} - 204q^{26} - 592q^{29} - 104q^{31} + 18q^{32} + 128q^{34} + 248q^{37} - 104q^{38} - 488q^{40} + 40q^{41} - 1440q^{43} + 292q^{44} + 84q^{46} + 96q^{47} - 1412q^{50} - 320q^{52} + 268q^{53} - 944q^{55} - 48q^{58} + 616q^{59} + 16q^{61} - 608q^{62} + 236q^{64} + 1740q^{65} + 144q^{67} + 940q^{68} - 1976q^{71} + 104q^{73} - 56q^{74} + 2272q^{76} + 944q^{79} + 828q^{80} - 856q^{82} + 2032q^{83} - 200q^{85} - 1120q^{86} + 876q^{88} + 388q^{89} + 728q^{92} + 904q^{94} + 1304q^{95} - 976q^{97} + O(q^{100})$$

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
−0.207107 + 0.358719i 0 3.91421 + 6.77962i −0.0502525 + 0.0870399i 0 0 −6.55635 0 −0.0208153 0.0360531i
226.2 1.20711 2.09077i 0 1.08579 + 1.88064i −9.94975 + 17.2335i 0 0 24.5563 0 24.0208 + 41.6053i
361.1 −0.207107 0.358719i 0 3.91421 6.77962i −0.0502525 0.0870399i 0 0 −6.55635 0 −0.0208153 + 0.0360531i
361.2 1.20711 + 2.09077i 0 1.08579 1.88064i −9.94975 17.2335i 0 0 24.5563 0 24.0208 41.6053i
 $$n$$: 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 441.4.e.u 4
3.b odd 2 1 147.4.e.j 4
7.b odd 2 1 441.4.e.v 4
7.c even 3 1 441.4.a.o 2
7.c even 3 1 inner 441.4.e.u 4
7.d odd 6 1 441.4.a.n 2
7.d odd 6 1 441.4.e.v 4
21.c even 2 1 147.4.e.k 4
21.g even 6 1 147.4.a.j 2
21.g even 6 1 147.4.e.k 4
21.h odd 6 1 147.4.a.k yes 2
21.h odd 6 1 147.4.e.j 4
84.j odd 6 1 2352.4.a.cf 2
84.n even 6 1 2352.4.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 21.g even 6 1
147.4.a.k yes 2 21.h odd 6 1
147.4.e.j 4 3.b odd 2 1
147.4.e.j 4 21.h odd 6 1
147.4.e.k 4 21.c even 2 1
147.4.e.k 4 21.g even 6 1
441.4.a.n 2 7.d odd 6 1
441.4.a.o 2 7.c even 3 1
441.4.e.u 4 1.a even 1 1 trivial
441.4.e.u 4 7.c even 3 1 inner
441.4.e.v 4 7.b odd 2 1
441.4.e.v 4 7.d odd 6 1
2352.4.a.bl 2 84.n even 6 1
2352.4.a.cf 2 84.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1$$ $$T_{5}^{4} + 20 T_{5}^{3} + 398 T_{5}^{2} + 40 T_{5} + 4$$ $$T_{13}^{2} + 104 T_{13} + 1454$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 + 40 T + 398 T^{2} + 20 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$1106704 - 21040 T + 1452 T^{2} + 20 T^{3} + T^{4}$$
$13$ $$( 1454 + 104 T + T^{2} )^{2}$$
$17$ $$470596 - 79576 T + 14142 T^{2} + 116 T^{3} + T^{4}$$
$19$ $$68029504 - 1583616 T + 28616 T^{2} - 192 T^{3} + T^{4}$$
$23$ $$1882384 + 38416 T + 2156 T^{2} - 28 T^{3} + T^{4}$$
$29$ $$( 14216 + 296 T + T^{2} )^{2}$$
$31$ $$5271616 - 238784 T + 13112 T^{2} + 104 T^{3} + T^{4}$$
$37$ $$115949824 - 2670464 T + 50736 T^{2} - 248 T^{3} + T^{4}$$
$41$ $$( -95822 - 20 T + T^{2} )^{2}$$
$43$ $$( 109600 + 720 T + T^{2} )^{2}$$
$47$ $$15054308416 + 11778816 T + 131912 T^{2} - 96 T^{3} + T^{4}$$
$53$ $$58198667536 + 64653392 T + 313068 T^{2} - 268 T^{3} + T^{4}$$
$59$ $$53114944 + 4489408 T + 386744 T^{2} - 616 T^{3} + T^{4}$$
$61$ $$2116 - 736 T + 210 T^{2} - 16 T^{3} + T^{4}$$
$67$ $$295901185024 + 78331392 T + 564704 T^{2} - 144 T^{3} + T^{4}$$
$71$ $$( 136388 + 988 T + T^{2} )^{2}$$
$73$ $$101695934404 + 33165392 T + 329714 T^{2} - 104 T^{3} + T^{4}$$
$79$ $$12408177664 - 105154048 T + 779744 T^{2} - 944 T^{3} + T^{4}$$
$83$ $$( 245264 - 1016 T + T^{2} )^{2}$$
$89$ $$36943146436 + 74575928 T + 342750 T^{2} - 388 T^{3} + T^{4}$$
$97$ $$( -658066 + 488 T + T^{2} )^{2}$$