# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(226,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 19x^{6} + 319x^{4} + 798x^{2} + 1764$$ x^8 + 19*x^6 + 319*x^4 + 798*x^2 + 1764 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{8}+O(q^{10})$$ q + (b3 + b1) * q^2 + (-b6 - 2*b2 - 2) * q^4 + (-b7 - b3 - b1) * q^5 + (-b7 + b5 + b3) * q^8 $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{8} + (5 \beta_{6} + 8 \beta_{2} + 8) q^{10} + ( - 3 \beta_{5} + \beta_1) q^{11} + (\beta_{4} - 25) q^{13} + ( - 5 \beta_{6} + 5 \beta_{4} - 28 \beta_{2}) q^{16} + ( - 2 \beta_{5} + 26 \beta_1) q^{17} + ( - 11 \beta_{6} + 11 \beta_{4} - 61 \beta_{2}) q^{19} + ( - 3 \beta_{7} + 3 \beta_{5} + 25 \beta_{3}) q^{20} + (11 \beta_{4} - 16) q^{22} + (2 \beta_{7} + 46 \beta_{3} + 46 \beta_1) q^{23} + (17 \beta_{6} - 83 \beta_{2} - 83) q^{25} + (\beta_{7} - 20 \beta_{3} - 20 \beta_1) q^{26} + (\beta_{7} - \beta_{5} + 45 \beta_{3}) q^{29} + ( - 20 \beta_{6} + 45 \beta_{2} + 45) q^{31} + (13 \beta_{5} + 45 \beta_1) q^{32} + ( - 18 \beta_{4} - 264) q^{34} + (25 \beta_{6} - 25 \beta_{4} - 81 \beta_{2}) q^{37} + (11 \beta_{5} + 116 \beta_1) q^{38} + (27 \beta_{6} - 27 \beta_{4} - 192 \beta_{2}) q^{40} + (16 \beta_{7} - 16 \beta_{5} + 72 \beta_{3}) q^{41} + ( - 27 \beta_{4} - 223) q^{43} + ( - 13 \beta_{7} + 47 \beta_{3} + 47 \beta_1) q^{44} + ( - 54 \beta_{6} - 456 \beta_{2} - 456) q^{46} + (16 \beta_{7} + 44 \beta_{3} + 44 \beta_1) q^{47} + (17 \beta_{7} - 17 \beta_{5} + 2 \beta_{3}) q^{50} + (24 \beta_{6} + 2 \beta_{2} + 2) q^{52} + ( - 7 \beta_{5} + \beta_1) q^{53} + (71 \beta_{4} - 592) q^{55} + ( - 49 \beta_{6} + 49 \beta_{4} - 448 \beta_{2}) q^{58} + ( - 17 \beta_{5} - 73 \beta_1) q^{59} + (46 \beta_{6} - 46 \beta_{4} - 310 \beta_{2}) q^{61} + ( - 20 \beta_{7} + 20 \beta_{5} - 55 \beta_{3}) q^{62} + ( - 57 \beta_{4} - 200) q^{64} + (30 \beta_{7} + 2 \beta_{3} + 2 \beta_1) q^{65} + (19 \beta_{6} - 463 \beta_{2} - 463) q^{67} + ( - 34 \beta_{7} - 146 \beta_{3} - 146 \beta_1) q^{68} + ( - 36 \beta_{7} + 36 \beta_{5}) q^{71} + (7 \beta_{6} + 441 \beta_{2} + 441) q^{73} + ( - 25 \beta_{5} - 44 \beta_1) q^{74} + ( - 72 \beta_{4} - 650) q^{76} + (42 \beta_{6} - 42 \beta_{4} + 23 \beta_{2}) q^{79} + ( - 3 \beta_{5} - 143 \beta_1) q^{80} + ( - 136 \beta_{6} + 136 \beta_{4} - 688 \beta_{2}) q^{82} + ( - 21 \beta_{7} + 21 \beta_{5} - 189 \beta_{3}) q^{83} + (174 \beta_{4} - 192) q^{85} + ( - 27 \beta_{7} - 358 \beta_{3} - 358 \beta_1) q^{86} + (93 \beta_{6} - 624 \beta_{2} - 624) q^{88} + ( - 22 \beta_{7} + 314 \beta_{3} + 314 \beta_1) q^{89} + ( - 38 \beta_{7} + 38 \beta_{5} - 358 \beta_{3}) q^{92} + ( - 108 \beta_{6} - 408 \beta_{2} - 408) q^{94} + ( - 6 \beta_{5} - 314 \beta_1) q^{95} + ( - 91 \beta_{4} - 798) q^{97}+O(q^{100})$$ q + (b3 + b1) * q^2 + (-b6 - 2*b2 - 2) * q^4 + (-b7 - b3 - b1) * q^5 + (-b7 + b5 + b3) * q^8 + (5*b6 + 8*b2 + 8) * q^10 + (-3*b5 + b1) * q^11 + (b4 - 25) * q^13 + (-5*b6 + 5*b4 - 28*b2) * q^16 + (-2*b5 + 26*b1) * q^17 + (-11*b6 + 11*b4 - 61*b2) * q^19 + (-3*b7 + 3*b5 + 25*b3) * q^20 + (11*b4 - 16) * q^22 + (2*b7 + 46*b3 + 46*b1) * q^23 + (17*b6 - 83*b2 - 83) * q^25 + (b7 - 20*b3 - 20*b1) * q^26 + (b7 - b5 + 45*b3) * q^29 + (-20*b6 + 45*b2 + 45) * q^31 + (13*b5 + 45*b1) * q^32 + (-18*b4 - 264) * q^34 + (25*b6 - 25*b4 - 81*b2) * q^37 + (11*b5 + 116*b1) * q^38 + (27*b6 - 27*b4 - 192*b2) * q^40 + (16*b7 - 16*b5 + 72*b3) * q^41 + (-27*b4 - 223) * q^43 + (-13*b7 + 47*b3 + 47*b1) * q^44 + (-54*b6 - 456*b2 - 456) * q^46 + (16*b7 + 44*b3 + 44*b1) * q^47 + (17*b7 - 17*b5 + 2*b3) * q^50 + (24*b6 + 2*b2 + 2) * q^52 + (-7*b5 + b1) * q^53 + (71*b4 - 592) * q^55 + (-49*b6 + 49*b4 - 448*b2) * q^58 + (-17*b5 - 73*b1) * q^59 + (46*b6 - 46*b4 - 310*b2) * q^61 + (-20*b7 + 20*b5 - 55*b3) * q^62 + (-57*b4 - 200) * q^64 + (30*b7 + 2*b3 + 2*b1) * q^65 + (19*b6 - 463*b2 - 463) * q^67 + (-34*b7 - 146*b3 - 146*b1) * q^68 + (-36*b7 + 36*b5) * q^71 + (7*b6 + 441*b2 + 441) * q^73 + (-25*b5 - 44*b1) * q^74 + (-72*b4 - 650) * q^76 + (42*b6 - 42*b4 + 23*b2) * q^79 + (-3*b5 - 143*b1) * q^80 + (-136*b6 + 136*b4 - 688*b2) * q^82 + (-21*b7 + 21*b5 - 189*b3) * q^83 + (174*b4 - 192) * q^85 + (-27*b7 - 358*b3 - 358*b1) * q^86 + (93*b6 - 624*b2 - 624) * q^88 + (-22*b7 + 314*b3 + 314*b1) * q^89 + (-38*b7 + 38*b5 - 358*b3) * q^92 + (-108*b6 - 408*b2 - 408) * q^94 + (-6*b5 - 314*b1) * q^95 + (-91*b4 - 798) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{4}+O(q^{10})$$ 8 * q - 6 * q^4 $$8 q - 6 q^{4} + 22 q^{10} - 204 q^{13} + 102 q^{16} + 222 q^{19} - 172 q^{22} - 366 q^{25} + 220 q^{31} - 2040 q^{34} + 374 q^{37} + 822 q^{40} - 1676 q^{43} - 1716 q^{46} - 40 q^{52} - 5020 q^{55} + 1694 q^{58} + 1332 q^{61} - 1372 q^{64} - 1890 q^{67} + 1750 q^{73} - 4912 q^{76} - 8 q^{79} + 2480 q^{82} - 2232 q^{85} - 2682 q^{88} - 1416 q^{94} - 6020 q^{97}+O(q^{100})$$ 8 * q - 6 * q^4 + 22 * q^10 - 204 * q^13 + 102 * q^16 + 222 * q^19 - 172 * q^22 - 366 * q^25 + 220 * q^31 - 2040 * q^34 + 374 * q^37 + 822 * q^40 - 1676 * q^43 - 1716 * q^46 - 40 * q^52 - 5020 * q^55 + 1694 * q^58 + 1332 * q^61 - 1372 * q^64 - 1890 * q^67 + 1750 * q^73 - 4912 * q^76 - 8 * q^79 + 2480 * q^82 - 2232 * q^85 - 2682 * q^88 - 1416 * q^94 - 6020 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 19x^{6} + 319x^{4} + 798x^{2} + 1764$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 19\nu^{6} + 319\nu^{4} + 6061\nu^{2} + 1764 ) / 13398$$ (19*v^6 + 319*v^4 + 6061*v^2 + 1764) / 13398 $$\beta_{3}$$ $$=$$ $$( 19\nu^{7} + 319\nu^{5} + 6061\nu^{3} + 1764\nu ) / 13398$$ (19*v^7 + 319*v^5 + 6061*v^3 + 1764*v) / 13398 $$\beta_{4}$$ $$=$$ $$( \nu^{6} - 2392 ) / 319$$ (v^6 - 2392) / 319 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 3987\nu ) / 319$$ (v^7 - 3987*v) / 319 $$\beta_{6}$$ $$=$$ $$( -74\nu^{6} - 1595\nu^{4} - 23606\nu^{2} - 59052 ) / 6699$$ (-74*v^6 - 1595*v^4 - 23606*v^2 - 59052) / 6699 $$\beta_{7}$$ $$=$$ $$( -81\nu^{7} - 1595\nu^{5} - 25839\nu^{3} - 64638\nu ) / 4466$$ (-81*v^7 - 1595*v^5 - 25839*v^3 - 64638*v) / 4466
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{4} + 10\beta_{2}$$ b6 - b4 + 10*b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{5} + 15\beta_{3}$$ b7 - b5 + 15*b3 $$\nu^{4}$$ $$=$$ $$-19\beta_{6} - 148\beta_{2} - 148$$ -19*b6 - 148*b2 - 148 $$\nu^{5}$$ $$=$$ $$-19\beta_{7} - 243\beta_{3} - 243\beta_1$$ -19*b7 - 243*b3 - 243*b1 $$\nu^{6}$$ $$=$$ $$319\beta_{4} + 2392$$ 319*b4 + 2392 $$\nu^{7}$$ $$=$$ $$319\beta_{5} + 3987\beta_1$$ 319*b5 + 3987*b1

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 2.02770 + 3.51207i 0.799027 + 1.38396i −0.799027 − 1.38396i −2.02770 − 3.51207i 2.02770 − 3.51207i 0.799027 − 1.38396i −0.799027 + 1.38396i −2.02770 + 3.51207i
−2.02770 + 3.51207i 0 −4.22311 7.31464i 4.96020 8.59131i 0 0 1.80961 0 20.1156 + 34.8412i
226.2 −0.799027 + 1.38396i 0 2.72311 + 4.71657i −9.14584 + 15.8411i 0 0 −21.4878 0 −14.6156 25.3149i
226.3 0.799027 1.38396i 0 2.72311 + 4.71657i 9.14584 15.8411i 0 0 21.4878 0 −14.6156 25.3149i
226.4 2.02770 3.51207i 0 −4.22311 7.31464i −4.96020 + 8.59131i 0 0 −1.80961 0 20.1156 + 34.8412i
361.1 −2.02770 3.51207i 0 −4.22311 + 7.31464i 4.96020 + 8.59131i 0 0 1.80961 0 20.1156 34.8412i
361.2 −0.799027 1.38396i 0 2.72311 4.71657i −9.14584 15.8411i 0 0 −21.4878 0 −14.6156 + 25.3149i
361.3 0.799027 + 1.38396i 0 2.72311 4.71657i 9.14584 + 15.8411i 0 0 21.4878 0 −14.6156 + 25.3149i
361.4 2.02770 + 3.51207i 0 −4.22311 + 7.31464i −4.96020 8.59131i 0 0 −1.80961 0 20.1156 34.8412i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.4 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.c even 3 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.e.x 8
3.b odd 2 1 inner 441.4.e.x 8
7.b odd 2 1 63.4.e.d 8
7.c even 3 1 441.4.a.v 4
7.c even 3 1 inner 441.4.e.x 8
7.d odd 6 1 63.4.e.d 8
7.d odd 6 1 441.4.a.w 4
21.c even 2 1 63.4.e.d 8
21.g even 6 1 63.4.e.d 8
21.g even 6 1 441.4.a.w 4
21.h odd 6 1 441.4.a.v 4
21.h odd 6 1 inner 441.4.e.x 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.e.d 8 7.b odd 2 1
63.4.e.d 8 7.d odd 6 1
63.4.e.d 8 21.c even 2 1
63.4.e.d 8 21.g even 6 1
441.4.a.v 4 7.c even 3 1
441.4.a.v 4 21.h odd 6 1
441.4.a.w 4 7.d odd 6 1
441.4.a.w 4 21.g even 6 1
441.4.e.x 8 1.a even 1 1 trivial
441.4.e.x 8 3.b odd 2 1 inner
441.4.e.x 8 7.c even 3 1 inner
441.4.e.x 8 21.h odd 6 1 inner

## Hecke kernels

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

 $$T_{2}^{8} + 19T_{2}^{6} + 319T_{2}^{4} + 798T_{2}^{2} + 1764$$ T2^8 + 19*T2^6 + 319*T2^4 + 798*T2^2 + 1764 $$T_{5}^{8} + 433T_{5}^{6} + 154561T_{5}^{4} + 14257824T_{5}^{2} + 1084253184$$ T5^8 + 433*T5^6 + 154561*T5^4 + 14257824*T5^2 + 1084253184 $$T_{13}^{2} + 51T_{13} + 602$$ T13^2 + 51*T13 + 602

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 19 T^{6} + 319 T^{4} + \cdots + 1764$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 433 T^{6} + \cdots + 1084253184$$
$7$ $$T^{8}$$
$11$ $$T^{8} + 3937 T^{6} + \cdots + 473520144384$$
$13$ $$(T^{2} + 51 T + 602)^{4}$$
$17$ $$T^{8} + 15396 T^{6} + \cdots + 33\!\cdots\!64$$
$19$ $$(T^{4} - 111 T^{3} + 15079 T^{2} + \cdots + 7606564)^{2}$$
$23$ $$T^{8} + 40452 T^{6} + \cdots + 20\!\cdots\!64$$
$29$ $$(T^{4} - 38185 T^{2} + 96018048)^{2}$$
$31$ $$(T^{4} - 110 T^{3} + 28375 T^{2} + \cdots + 264875625)^{2}$$
$37$ $$(T^{4} - 187 T^{3} + 56383 T^{2} + \cdots + 458559396)^{2}$$
$41$ $$(T^{4} - 190144 T^{2} + \cdots + 6145155072)^{2}$$
$43$ $$(T^{2} + 419 T + 8716)^{4}$$
$47$ $$T^{8} + 135600 T^{6} + \cdots + 20\!\cdots\!04$$
$53$ $$T^{8} + \cdots + 737093829878784$$
$59$ $$T^{8} + 205665 T^{6} + \cdots + 59\!\cdots\!64$$
$61$ $$(T^{4} - 666 T^{3} + 434764 T^{2} + \cdots + 77299264)^{2}$$
$67$ $$(T^{4} + 945 T^{3} + \cdots + 42369282244)^{2}$$
$71$ $$(T^{4} - 557280 T^{2} + \cdots + 22856214528)^{2}$$
$73$ $$(T^{4} - 875 T^{3} + \cdots + 35736877764)^{2}$$
$79$ $$(T^{4} + 4 T^{3} + 85125 T^{2} + \cdots + 7243541881)^{2}$$
$83$ $$(T^{4} - 804825 T^{2} + \cdots + 10585989792)^{2}$$
$89$ $$T^{8} + 2191972 T^{6} + \cdots + 13\!\cdots\!84$$
$97$ $$(T^{2} + 1505 T + 166698)^{4}$$