# Properties

 Label 441.4.e.k Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 4 - 4 \zeta_{6} ) q^{4} -7 \zeta_{6} q^{5} + 24 q^{8} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 4 - 4 \zeta_{6} ) q^{4} -7 \zeta_{6} q^{5} + 24 q^{8} + ( 14 - 14 \zeta_{6} ) q^{10} + ( -5 + 5 \zeta_{6} ) q^{11} + 14 q^{13} + 16 \zeta_{6} q^{16} + ( 21 - 21 \zeta_{6} ) q^{17} + 49 \zeta_{6} q^{19} -28 q^{20} -10 q^{22} -159 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 28 \zeta_{6} q^{26} -58 q^{29} + ( 147 - 147 \zeta_{6} ) q^{31} + ( 160 - 160 \zeta_{6} ) q^{32} + 42 q^{34} -219 \zeta_{6} q^{37} + ( -98 + 98 \zeta_{6} ) q^{38} -168 \zeta_{6} q^{40} + 350 q^{41} -124 q^{43} + 20 \zeta_{6} q^{44} + ( 318 - 318 \zeta_{6} ) q^{46} -525 \zeta_{6} q^{47} + 152 q^{50} + ( 56 - 56 \zeta_{6} ) q^{52} + ( 303 - 303 \zeta_{6} ) q^{53} + 35 q^{55} -116 \zeta_{6} q^{58} + ( 105 - 105 \zeta_{6} ) q^{59} -413 \zeta_{6} q^{61} + 294 q^{62} + 448 q^{64} -98 \zeta_{6} q^{65} + ( -415 + 415 \zeta_{6} ) q^{67} -84 \zeta_{6} q^{68} + 432 q^{71} + ( -1113 + 1113 \zeta_{6} ) q^{73} + ( 438 - 438 \zeta_{6} ) q^{74} + 196 q^{76} + 103 \zeta_{6} q^{79} + ( 112 - 112 \zeta_{6} ) q^{80} + 700 \zeta_{6} q^{82} + 1092 q^{83} -147 q^{85} -248 \zeta_{6} q^{86} + ( -120 + 120 \zeta_{6} ) q^{88} + 329 \zeta_{6} q^{89} -636 q^{92} + ( 1050 - 1050 \zeta_{6} ) q^{94} + ( 343 - 343 \zeta_{6} ) q^{95} + 882 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 4q^{4} - 7q^{5} + 48q^{8} + 14q^{10} - 5q^{11} + 28q^{13} + 16q^{16} + 21q^{17} + 49q^{19} - 56q^{20} - 20q^{22} - 159q^{23} + 76q^{25} + 28q^{26} - 116q^{29} + 147q^{31} + 160q^{32} + 84q^{34} - 219q^{37} - 98q^{38} - 168q^{40} + 700q^{41} - 248q^{43} + 20q^{44} + 318q^{46} - 525q^{47} + 304q^{50} + 56q^{52} + 303q^{53} + 70q^{55} - 116q^{58} + 105q^{59} - 413q^{61} + 588q^{62} + 896q^{64} - 98q^{65} - 415q^{67} - 84q^{68} + 864q^{71} - 1113q^{73} + 438q^{74} + 392q^{76} + 103q^{79} + 112q^{80} + 700q^{82} + 2184q^{83} - 294q^{85} - 248q^{86} - 120q^{88} + 329q^{89} - 1272q^{92} + 1050q^{94} + 343q^{95} + 1764q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i 0 2.00000 + 3.46410i −3.50000 + 6.06218i 0 0 24.0000 0 7.00000 + 12.1244i
361.1 1.00000 + 1.73205i 0 2.00000 3.46410i −3.50000 6.06218i 0 0 24.0000 0 7.00000 12.1244i
 $$n$$: 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.k 2
3.b odd 2 1 49.4.c.a 2
7.b odd 2 1 63.4.e.b 2
7.c even 3 1 441.4.a.e 1
7.c even 3 1 inner 441.4.e.k 2
7.d odd 6 1 63.4.e.b 2
7.d odd 6 1 441.4.a.d 1
21.c even 2 1 7.4.c.a 2
21.g even 6 1 7.4.c.a 2
21.g even 6 1 49.4.a.d 1
21.h odd 6 1 49.4.a.c 1
21.h odd 6 1 49.4.c.a 2
84.h odd 2 1 112.4.i.c 2
84.j odd 6 1 112.4.i.c 2
84.j odd 6 1 784.4.a.b 1
84.n even 6 1 784.4.a.r 1
105.g even 2 1 175.4.e.a 2
105.k odd 4 2 175.4.k.a 4
105.o odd 6 1 1225.4.a.d 1
105.p even 6 1 175.4.e.a 2
105.p even 6 1 1225.4.a.c 1
105.w odd 12 2 175.4.k.a 4
168.e odd 2 1 448.4.i.a 2
168.i even 2 1 448.4.i.f 2
168.ba even 6 1 448.4.i.f 2
168.be odd 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 21.c even 2 1
7.4.c.a 2 21.g even 6 1
49.4.a.c 1 21.h odd 6 1
49.4.a.d 1 21.g even 6 1
49.4.c.a 2 3.b odd 2 1
49.4.c.a 2 21.h odd 6 1
63.4.e.b 2 7.b odd 2 1
63.4.e.b 2 7.d odd 6 1
112.4.i.c 2 84.h odd 2 1
112.4.i.c 2 84.j odd 6 1
175.4.e.a 2 105.g even 2 1
175.4.e.a 2 105.p even 6 1
175.4.k.a 4 105.k odd 4 2
175.4.k.a 4 105.w odd 12 2
441.4.a.d 1 7.d odd 6 1
441.4.a.e 1 7.c even 3 1
441.4.e.k 2 1.a even 1 1 trivial
441.4.e.k 2 7.c even 3 1 inner
448.4.i.a 2 168.e odd 2 1
448.4.i.a 2 168.be odd 6 1
448.4.i.f 2 168.i even 2 1
448.4.i.f 2 168.ba even 6 1
784.4.a.b 1 84.j odd 6 1
784.4.a.r 1 84.n even 6 1
1225.4.a.c 1 105.p even 6 1
1225.4.a.d 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2 T_{2} + 4$$ $$T_{5}^{2} + 7 T_{5} + 49$$ $$T_{13} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$49 + 7 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$( -14 + T )^{2}$$
$17$ $$441 - 21 T + T^{2}$$
$19$ $$2401 - 49 T + T^{2}$$
$23$ $$25281 + 159 T + T^{2}$$
$29$ $$( 58 + T )^{2}$$
$31$ $$21609 - 147 T + T^{2}$$
$37$ $$47961 + 219 T + T^{2}$$
$41$ $$( -350 + T )^{2}$$
$43$ $$( 124 + T )^{2}$$
$47$ $$275625 + 525 T + T^{2}$$
$53$ $$91809 - 303 T + T^{2}$$
$59$ $$11025 - 105 T + T^{2}$$
$61$ $$170569 + 413 T + T^{2}$$
$67$ $$172225 + 415 T + T^{2}$$
$71$ $$( -432 + T )^{2}$$
$73$ $$1238769 + 1113 T + T^{2}$$
$79$ $$10609 - 103 T + T^{2}$$
$83$ $$( -1092 + T )^{2}$$
$89$ $$108241 - 329 T + T^{2}$$
$97$ $$( -882 + T )^{2}$$