# Properties

 Label 441.2.u.a Level $441$ Weight $2$ Character orbit 441.u Analytic conductor $3.521$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.u (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{14} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{2} + ( -1 - \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} ) q^{4} + ( -1 + \zeta_{14} + \zeta_{14}^{4} ) q^{5} + ( 2 - \zeta_{14}^{3} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{7} + ( -1 + \zeta_{14} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{14} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{2} + ( -1 - \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} ) q^{4} + ( -1 + \zeta_{14} + \zeta_{14}^{4} ) q^{5} + ( 2 - \zeta_{14}^{3} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{7} + ( -1 + \zeta_{14} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{8} + ( \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{10} + ( -2 + \zeta_{14} - 3 \zeta_{14}^{2} + 3 \zeta_{14}^{3} - \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{11} + ( -2 - 5 \zeta_{14}^{2} + 5 \zeta_{14}^{3} + 2 \zeta_{14}^{5} ) q^{13} + ( 1 - 3 \zeta_{14} + 2 \zeta_{14}^{2} - \zeta_{14}^{3} + 3 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{14} + ( -\zeta_{14} - \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{16} + ( -2 + 3 \zeta_{14} + 3 \zeta_{14}^{3} - 2 \zeta_{14}^{4} ) q^{17} + ( -1 + 2 \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{19} + ( -1 + \zeta_{14} - 3 \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{20} + ( -2 + \zeta_{14} - \zeta_{14}^{2} + 2 \zeta_{14}^{3} ) q^{22} + ( 1 + \zeta_{14} - \zeta_{14}^{2} - \zeta_{14}^{3} + 3 \zeta_{14}^{5} ) q^{23} + ( 1 + 2 \zeta_{14} + \zeta_{14}^{2} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{25} + ( -5 + 5 \zeta_{14}^{3} - 3 \zeta_{14}^{5} ) q^{26} + ( -5 + 3 \zeta_{14} - \zeta_{14}^{2} + 6 \zeta_{14}^{3} - 4 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{28} + ( 2 + 3 \zeta_{14} + 3 \zeta_{14}^{3} + 2 \zeta_{14}^{4} ) q^{29} + ( -2 - \zeta_{14}^{2} + \zeta_{14}^{5} ) q^{31} + ( -3 - 2 \zeta_{14} + 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} - 3 \zeta_{14}^{4} ) q^{32} + ( -2 + 7 \zeta_{14} - 2 \zeta_{14}^{2} - 4 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{34} + ( 1 - 2 \zeta_{14} + 4 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{35} + ( 4 - 6 \zeta_{14} + 4 \zeta_{14}^{2} - 6 \zeta_{14}^{3} + 4 \zeta_{14}^{4} ) q^{37} + ( -4 + 7 \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} - 7 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{38} + ( -3 + 2 \zeta_{14} - 3 \zeta_{14}^{2} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{40} + ( 2 \zeta_{14}^{3} + 4 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{41} + ( 3 \zeta_{14} - 2 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + 3 \zeta_{14}^{5} ) q^{43} + ( 1 - \zeta_{14} - \zeta_{14}^{3} - 4 \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{44} + ( 6 - 6 \zeta_{14} - 2 \zeta_{14}^{3} + 7 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{46} + ( 5 - 4 \zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} + 4 \zeta_{14}^{4} - 5 \zeta_{14}^{5} ) q^{47} + ( 7 - 7 \zeta_{14} + 7 \zeta_{14}^{2} - 7 \zeta_{14}^{3} + 7 \zeta_{14}^{4} - 7 \zeta_{14}^{5} ) q^{49} + ( 5 - 2 \zeta_{14}^{2} - 2 \zeta_{14}^{3} + 2 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{50} + ( -3 + 3 \zeta_{14} + \zeta_{14}^{3} - 13 \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{52} + ( 8 - 4 \zeta_{14} + 4 \zeta_{14}^{2} - 8 \zeta_{14}^{3} - 6 \zeta_{14}^{5} ) q^{53} + ( -3 \zeta_{14} + 3 \zeta_{14}^{2} - 7 \zeta_{14}^{3} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{5} ) q^{55} + ( -4 + 8 \zeta_{14} - 4 \zeta_{14}^{2} + 5 \zeta_{14}^{3} - 3 \zeta_{14}^{4} + 4 \zeta_{14}^{5} ) q^{56} + ( 2 - \zeta_{14} + 2 \zeta_{14}^{2} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{58} + ( -\zeta_{14} + \zeta_{14}^{2} - 7 \zeta_{14}^{3} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{59} + ( -3 + 2 \zeta_{14} - 3 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 3 \zeta_{14}^{4} ) q^{61} + ( -1 + \zeta_{14}^{5} ) q^{62} + ( -1 + 6 \zeta_{14} - \zeta_{14}^{2} - 6 \zeta_{14}^{4} + 6 \zeta_{14}^{5} ) q^{64} + ( -5 \zeta_{14} + 6 \zeta_{14}^{2} - 13 \zeta_{14}^{3} + 6 \zeta_{14}^{4} - 5 \zeta_{14}^{5} ) q^{65} + ( -1 + 2 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{67} + ( 3 - 6 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} + 6 \zeta_{14}^{5} ) q^{68} + ( 2 - \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{70} + ( -1 + 3 \zeta_{14} - 3 \zeta_{14}^{2} + \zeta_{14}^{3} - 6 \zeta_{14}^{5} ) q^{71} + ( -3 + 9 \zeta_{14} - 3 \zeta_{14}^{2} - 5 \zeta_{14}^{4} + 5 \zeta_{14}^{5} ) q^{73} + ( 4 - 10 \zeta_{14} + 4 \zeta_{14}^{2} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{74} + ( 2 + 5 \zeta_{14} - 5 \zeta_{14}^{2} - 2 \zeta_{14}^{3} + 8 \zeta_{14}^{5} ) q^{76} + ( -6 + \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} + 4 \zeta_{14}^{4} + 2 \zeta_{14}^{5} ) q^{77} + ( 11 + 2 \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{79} + ( 2 + \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{80} + ( -4 \zeta_{14} + 4 \zeta_{14}^{2} + 2 \zeta_{14}^{3} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{5} ) q^{82} + ( -1 + \zeta_{14} - \zeta_{14}^{2} - 9 \zeta_{14}^{4} + 9 \zeta_{14}^{5} ) q^{83} + ( -1 - 3 \zeta_{14} + 3 \zeta_{14}^{2} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{85} + ( 4 - \zeta_{14} - 2 \zeta_{14}^{2} - \zeta_{14}^{3} + 4 \zeta_{14}^{4} ) q^{86} + ( \zeta_{14} - 6 \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 6 \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{88} + ( -8 + 9 \zeta_{14} - 8 \zeta_{14}^{2} - 9 \zeta_{14}^{4} + 9 \zeta_{14}^{5} ) q^{89} + ( -9 - 3 \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{3} + 11 \zeta_{14}^{4} - 2 \zeta_{14}^{5} ) q^{91} + ( -9 \zeta_{14} + 5 \zeta_{14}^{2} + 5 \zeta_{14}^{4} - 9 \zeta_{14}^{5} ) q^{92} + ( -3 - 4 \zeta_{14} + 4 \zeta_{14}^{2} + 3 \zeta_{14}^{3} - 8 \zeta_{14}^{5} ) q^{94} + ( \zeta_{14}^{3} + \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{95} + ( -1 - \zeta_{14}^{2} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{4} + \zeta_{14}^{5} ) q^{97} + ( -7 \zeta_{14} + 7 \zeta_{14}^{2} - 7 \zeta_{14}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} - 7q^{4} - 6q^{5} + 7q^{7} + q^{8} + O(q^{10})$$ $$6q + 3q^{2} - 7q^{4} - 6q^{5} + 7q^{7} + q^{8} - 3q^{10} - 2q^{11} - 7q^{14} - 3q^{16} - 4q^{17} - 8q^{19} - 8q^{22} + 10q^{23} + 11q^{25} - 28q^{26} - 14q^{28} + 16q^{29} - 10q^{31} - 21q^{32} + 5q^{34} + 4q^{37} - 4q^{38} - 15q^{40} + 12q^{43} + 7q^{44} + 19q^{46} + 15q^{47} + 7q^{49} + 30q^{50} + 26q^{53} - 19q^{55} + q^{58} - 11q^{59} - 8q^{61} - 5q^{62} + 13q^{64} - 35q^{65} - 12q^{67} + 28q^{68} + 14q^{70} - 5q^{71} + 4q^{73} + 2q^{74} + 28q^{76} - 35q^{77} + 60q^{79} + 10q^{80} - 14q^{82} + 14q^{83} - 17q^{85} + 20q^{86} + 16q^{88} - 13q^{89} - 70q^{91} - 28q^{92} - 31q^{94} + q^{95} - 21q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{14}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.222521 + 0.974928i 0.900969 − 0.433884i −0.623490 − 0.781831i −0.623490 + 0.781831i 0.900969 + 0.433884i 0.222521 − 0.974928i
0.500000 2.19064i 0 −2.74698 1.32288i −0.153989 + 0.193096i 0 2.06853 1.64960i −1.46950 + 1.84270i 0 0.346011 + 0.433884i
127.1 0.500000 + 0.240787i 0 −1.05496 1.32288i −0.321552 1.40881i 0 2.57942 + 0.588735i −0.455927 1.99755i 0 0.178448 0.781831i
190.1 0.500000 0.626980i 0 0.301938 + 1.32288i −2.52446 1.21572i 0 −1.14795 2.38374i 2.42543 + 1.16802i 0 −2.02446 + 0.974928i
253.1 0.500000 + 0.626980i 0 0.301938 1.32288i −2.52446 + 1.21572i 0 −1.14795 + 2.38374i 2.42543 1.16802i 0 −2.02446 0.974928i
316.1 0.500000 0.240787i 0 −1.05496 + 1.32288i −0.321552 + 1.40881i 0 2.57942 0.588735i −0.455927 + 1.99755i 0 0.178448 + 0.781831i
379.1 0.500000 + 2.19064i 0 −2.74698 + 1.32288i −0.153989 0.193096i 0 2.06853 + 1.64960i −1.46950 1.84270i 0 0.346011 0.433884i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.u.a 6
3.b odd 2 1 49.2.e.a 6
12.b even 2 1 784.2.u.a 6
21.c even 2 1 343.2.e.a 6
21.g even 6 2 343.2.g.e 12
21.h odd 6 2 343.2.g.f 12
49.e even 7 1 inner 441.2.u.a 6
147.k even 14 1 343.2.e.a 6
147.k even 14 1 2401.2.a.a 3
147.l odd 14 1 49.2.e.a 6
147.l odd 14 1 2401.2.a.b 3
147.n odd 42 2 343.2.g.f 12
147.o even 42 2 343.2.g.e 12
588.u even 14 1 784.2.u.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.a 6 3.b odd 2 1
49.2.e.a 6 147.l odd 14 1
343.2.e.a 6 21.c even 2 1
343.2.e.a 6 147.k even 14 1
343.2.g.e 12 21.g even 6 2
343.2.g.e 12 147.o even 42 2
343.2.g.f 12 21.h odd 6 2
343.2.g.f 12 147.n odd 42 2
441.2.u.a 6 1.a even 1 1 trivial
441.2.u.a 6 49.e even 7 1 inner
784.2.u.a 6 12.b even 2 1
784.2.u.a 6 588.u even 14 1
2401.2.a.a 3 147.k even 14 1
2401.2.a.b 3 147.l odd 14 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 13 T_{2}^{3} + 11 T_{2}^{2} - 5 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T + 11 T^{2} - 13 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$1 + 6 T + 22 T^{2} + 20 T^{3} + 15 T^{4} + 6 T^{5} + T^{6}$$
$7$ $$343 - 343 T + 147 T^{2} - 49 T^{3} + 21 T^{4} - 7 T^{5} + T^{6}$$
$11$ $$169 - 52 T + 142 T^{2} + 22 T^{3} - 3 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$41209 + 1421 T + 784 T^{2} + 259 T^{3} + 14 T^{4} + T^{6}$$
$17$ $$1681 - 369 T + 228 T^{2} + 29 T^{3} + 2 T^{4} + 4 T^{5} + T^{6}$$
$19$ $$( -1 - 11 T + 4 T^{2} + T^{3} )^{2}$$
$23$ $$169 + 65 T + 88 T^{2} + T^{3} + 30 T^{4} - 10 T^{5} + T^{6}$$
$29$ $$6889 - 7221 T + 3264 T^{2} - 827 T^{3} + 158 T^{4} - 16 T^{5} + T^{6}$$
$31$ $$( 1 + 6 T + 5 T^{2} + T^{3} )^{2}$$
$37$ $$10816 + 4576 T + 1600 T^{2} - 680 T^{3} + 72 T^{4} - 4 T^{5} + T^{6}$$
$41$ $$3136 + 3136 T + 784 T^{2} - 112 T^{3} + 56 T^{4} + T^{6}$$
$43$ $$841 - 899 T + 604 T^{2} - 265 T^{3} + 74 T^{4} - 12 T^{5} + T^{6}$$
$47$ $$9409 - 8730 T + 4180 T^{2} - 1079 T^{3} + 162 T^{4} - 15 T^{5} + T^{6}$$
$53$ $$53824 - 38048 T + 12784 T^{2} - 2680 T^{3} + 396 T^{4} - 26 T^{5} + T^{6}$$
$59$ $$57121 + 2151 T + 1257 T^{2} + 295 T^{3} + 65 T^{4} + 11 T^{5} + T^{6}$$
$61$ $$1849 + 1548 T + 680 T^{2} + 190 T^{3} + 43 T^{4} + 8 T^{5} + T^{6}$$
$67$ $$( -13 + 5 T + 6 T^{2} + T^{3} )^{2}$$
$71$ $$85849 + 23733 T + 1101 T^{2} + 97 T^{3} + 137 T^{4} + 5 T^{5} + T^{6}$$
$73$ $$5041 - 5112 T + 1019 T^{2} + 370 T^{3} + 184 T^{4} - 4 T^{5} + T^{6}$$
$79$ $$( -937 + 293 T - 30 T^{2} + T^{3} )^{2}$$
$83$ $$625681 + 155036 T + 7105 T^{2} - 280 T^{3} + 252 T^{4} - 14 T^{5} + T^{6}$$
$89$ $$187489 + 115611 T + 40419 T^{2} + 4955 T^{3} + 281 T^{4} + 13 T^{5} + T^{6}$$
$97$ $$( -7 - 7 T + T^{3} )^{2}$$