# Properties

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

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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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 21) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{5} + (5 \beta_{2} + 41) q^{8}+O(q^{10})$$ q + (-b3 - b1 - 1) * q^2 + (3*b3 - 3*b2 + 7*b1) * q^4 + (-2*b3 - 2*b1 - 2) * q^5 + (5*b2 + 41) * q^8 $$q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{5} + (5 \beta_{2} + 41) q^{8} + (6 \beta_{3} - 6 \beta_{2} + 30 \beta_1) q^{10} + ( - 10 \beta_{3} + 10 \beta_{2} + 8 \beta_1) q^{11} + (12 \beta_{2} - 14) q^{13} + ( - 27 \beta_{3} - 55 \beta_1 - 55) q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{17} + ( - 24 \beta_{3} + 44 \beta_1 + 44) q^{19} + (26 \beta_{2} + 98) q^{20} + ( - 12 \beta_{2} - 132) q^{22} + ( - 34 \beta_{3} + 20 \beta_1 + 20) q^{23} + (12 \beta_{3} - 12 \beta_{2} - 65 \beta_1) q^{25} + ( - 10 \beta_{3} - 154 \beta_1 - 154) q^{26} + ( - 24 \beta_{2} + 138) q^{29} + ( - 72 \beta_{3} + 72 \beta_{2} + 16 \beta_1) q^{31} + (69 \beta_{3} - 69 \beta_{2} + 105 \beta_1) q^{32} + ( - 6 \beta_{2} - 30) q^{34} + (36 \beta_{3} + 106 \beta_1 + 106) q^{37} + (4 \beta_{3} - 4 \beta_{2} + 292 \beta_1) q^{38} + ( - 102 \beta_{3} - 222 \beta_1 - 222) q^{40} + ( - 30 \beta_{2} - 210) q^{41} + ( - 48 \beta_{2} + 212) q^{43} + (76 \beta_{3} + 364 \beta_1 + 364) q^{44} + (48 \beta_{3} - 48 \beta_{2} + 456 \beta_1) q^{46} + ( - 68 \beta_{3} + 40 \beta_1 + 40) q^{47} + ( - 41 \beta_{2} + 103) q^{50} + (78 \beta_{3} - 78 \beta_{2} + 406 \beta_1) q^{52} + ( - 4 \beta_{3} + 4 \beta_{2} + 554 \beta_1) q^{53} + ( - 24 \beta_{2} - 264) q^{55} + ( - 90 \beta_{3} + 198 \beta_1 + 198) q^{58} + ( - 116 \beta_{3} + 116 \beta_{2} + 460 \beta_1) q^{59} + (72 \beta_{3} - 250 \beta_1 - 250) q^{61} + ( - 128 \beta_{2} - 992) q^{62} + (27 \beta_{2} + 631) q^{64} + ( - 20 \beta_{3} - 308 \beta_1 - 308) q^{65} + (108 \beta_{3} - 108 \beta_{2} + 20 \beta_1) q^{67} + (26 \beta_{3} + 98 \beta_1 + 98) q^{68} + (30 \beta_{2} - 492) q^{71} + ( - 12 \beta_{3} + 12 \beta_{2} - 530 \beta_1) q^{73} + ( - 178 \beta_{3} + 178 \beta_{2} - 610 \beta_1) q^{74} + (108 \beta_{2} + 700) q^{76} + (108 \beta_{3} + 232 \beta_1 + 232) q^{79} + (218 \beta_{3} - 218 \beta_{2} + 866 \beta_1) q^{80} + (270 \beta_{3} + 630 \beta_1 + 630) q^{82} + (96 \beta_{2} + 924) q^{83} + ( - 12 \beta_{2} - 60) q^{85} + ( - 116 \beta_{3} + 460 \beta_1 + 460) q^{86} + ( - 420 \beta_{3} + 420 \beta_{2} - 372 \beta_1) q^{88} + (142 \beta_{3} - 254 \beta_1 - 254) q^{89} + (280 \beta_{2} + 1288) q^{92} + (96 \beta_{3} - 96 \beta_{2} + 912 \beta_1) q^{94} + (8 \beta_{3} - 8 \beta_{2} + 584 \beta_1) q^{95} + ( - 276 \beta_{2} - 266) q^{97}+O(q^{100})$$ q + (-b3 - b1 - 1) * q^2 + (3*b3 - 3*b2 + 7*b1) * q^4 + (-2*b3 - 2*b1 - 2) * q^5 + (5*b2 + 41) * q^8 + (6*b3 - 6*b2 + 30*b1) * q^10 + (-10*b3 + 10*b2 + 8*b1) * q^11 + (12*b2 - 14) * q^13 + (-27*b3 - 55*b1 - 55) * q^16 + (-2*b3 + 2*b2 - 2*b1) * q^17 + (-24*b3 + 44*b1 + 44) * q^19 + (26*b2 + 98) * q^20 + (-12*b2 - 132) * q^22 + (-34*b3 + 20*b1 + 20) * q^23 + (12*b3 - 12*b2 - 65*b1) * q^25 + (-10*b3 - 154*b1 - 154) * q^26 + (-24*b2 + 138) * q^29 + (-72*b3 + 72*b2 + 16*b1) * q^31 + (69*b3 - 69*b2 + 105*b1) * q^32 + (-6*b2 - 30) * q^34 + (36*b3 + 106*b1 + 106) * q^37 + (4*b3 - 4*b2 + 292*b1) * q^38 + (-102*b3 - 222*b1 - 222) * q^40 + (-30*b2 - 210) * q^41 + (-48*b2 + 212) * q^43 + (76*b3 + 364*b1 + 364) * q^44 + (48*b3 - 48*b2 + 456*b1) * q^46 + (-68*b3 + 40*b1 + 40) * q^47 + (-41*b2 + 103) * q^50 + (78*b3 - 78*b2 + 406*b1) * q^52 + (-4*b3 + 4*b2 + 554*b1) * q^53 + (-24*b2 - 264) * q^55 + (-90*b3 + 198*b1 + 198) * q^58 + (-116*b3 + 116*b2 + 460*b1) * q^59 + (72*b3 - 250*b1 - 250) * q^61 + (-128*b2 - 992) * q^62 + (27*b2 + 631) * q^64 + (-20*b3 - 308*b1 - 308) * q^65 + (108*b3 - 108*b2 + 20*b1) * q^67 + (26*b3 + 98*b1 + 98) * q^68 + (30*b2 - 492) * q^71 + (-12*b3 + 12*b2 - 530*b1) * q^73 + (-178*b3 + 178*b2 - 610*b1) * q^74 + (108*b2 + 700) * q^76 + (108*b3 + 232*b1 + 232) * q^79 + (218*b3 - 218*b2 + 866*b1) * q^80 + (270*b3 + 630*b1 + 630) * q^82 + (96*b2 + 924) * q^83 + (-12*b2 - 60) * q^85 + (-116*b3 + 460*b1 + 460) * q^86 + (-420*b3 + 420*b2 - 372*b1) * q^88 + (142*b3 - 254*b1 - 254) * q^89 + (280*b2 + 1288) * q^92 + (96*b3 - 96*b2 + 912*b1) * q^94 + (8*b3 - 8*b2 + 584*b1) * q^95 + (-276*b2 - 266) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 17 q^{4} - 6 q^{5} + 174 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 - 17 * q^4 - 6 * q^5 + 174 * q^8 $$4 q - 3 q^{2} - 17 q^{4} - 6 q^{5} + 174 q^{8} - 66 q^{10} - 6 q^{11} - 32 q^{13} - 137 q^{16} + 6 q^{17} + 64 q^{19} + 444 q^{20} - 552 q^{22} + 6 q^{23} + 118 q^{25} - 318 q^{26} + 504 q^{29} + 40 q^{31} - 279 q^{32} - 132 q^{34} + 248 q^{37} - 588 q^{38} - 546 q^{40} - 900 q^{41} + 752 q^{43} + 804 q^{44} - 960 q^{46} + 12 q^{47} + 330 q^{50} - 890 q^{52} - 1104 q^{53} - 1104 q^{55} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 4224 q^{62} + 2578 q^{64} - 636 q^{65} - 148 q^{67} + 222 q^{68} - 1908 q^{71} + 1072 q^{73} + 1398 q^{74} + 3016 q^{76} + 572 q^{79} - 1950 q^{80} + 1530 q^{82} + 3888 q^{83} - 264 q^{85} + 804 q^{86} + 1164 q^{88} - 366 q^{89} + 5712 q^{92} - 1920 q^{94} - 1176 q^{95} - 1616 q^{97}+O(q^{100})$$ 4 * q - 3 * q^2 - 17 * q^4 - 6 * q^5 + 174 * q^8 - 66 * q^10 - 6 * q^11 - 32 * q^13 - 137 * q^16 + 6 * q^17 + 64 * q^19 + 444 * q^20 - 552 * q^22 + 6 * q^23 + 118 * q^25 - 318 * q^26 + 504 * q^29 + 40 * q^31 - 279 * q^32 - 132 * q^34 + 248 * q^37 - 588 * q^38 - 546 * q^40 - 900 * q^41 + 752 * q^43 + 804 * q^44 - 960 * q^46 + 12 * q^47 + 330 * q^50 - 890 * q^52 - 1104 * q^53 - 1104 * q^55 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 4224 * q^62 + 2578 * q^64 - 636 * q^65 - 148 * q^67 + 222 * q^68 - 1908 * q^71 + 1072 * q^73 + 1398 * q^74 + 3016 * q^76 + 572 * q^79 - 1950 * q^80 + 1530 * q^82 + 3888 * q^83 - 264 * q^85 + 804 * q^86 + 1164 * q^88 - 366 * q^89 + 5712 * q^92 - 1920 * q^94 - 1176 * q^95 - 1616 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

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

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

## 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.

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
 2.13746 − 0.656712i −1.63746 + 1.52274i 2.13746 + 0.656712i −1.63746 − 1.52274i
−2.63746 + 4.56821i 0 −9.91238 17.1687i −5.27492 + 9.13642i 0 0 62.3746 0 −27.8248 48.1939i
226.2 1.13746 1.97014i 0 1.41238 + 2.44631i 2.27492 3.94027i 0 0 24.6254 0 −5.17525 8.96379i
361.1 −2.63746 4.56821i 0 −9.91238 + 17.1687i −5.27492 9.13642i 0 0 62.3746 0 −27.8248 + 48.1939i
361.2 1.13746 + 1.97014i 0 1.41238 2.44631i 2.27492 + 3.94027i 0 0 24.6254 0 −5.17525 + 8.96379i
 $$n$$: 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.p 4
3.b odd 2 1 147.4.e.m 4
7.b odd 2 1 441.4.e.q 4
7.c even 3 1 441.4.a.r 2
7.c even 3 1 inner 441.4.e.p 4
7.d odd 6 1 63.4.a.e 2
7.d odd 6 1 441.4.e.q 4
21.c even 2 1 147.4.e.l 4
21.g even 6 1 21.4.a.c 2
21.g even 6 1 147.4.e.l 4
21.h odd 6 1 147.4.a.i 2
21.h odd 6 1 147.4.e.m 4
28.f even 6 1 1008.4.a.ba 2
35.i odd 6 1 1575.4.a.p 2
84.j odd 6 1 336.4.a.m 2
84.n even 6 1 2352.4.a.bz 2
105.p even 6 1 525.4.a.n 2
105.w odd 12 2 525.4.d.g 4
168.ba even 6 1 1344.4.a.bg 2
168.be odd 6 1 1344.4.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 21.g even 6 1
63.4.a.e 2 7.d odd 6 1
147.4.a.i 2 21.h odd 6 1
147.4.e.l 4 21.c even 2 1
147.4.e.l 4 21.g even 6 1
147.4.e.m 4 3.b odd 2 1
147.4.e.m 4 21.h odd 6 1
336.4.a.m 2 84.j odd 6 1
441.4.a.r 2 7.c even 3 1
441.4.e.p 4 1.a even 1 1 trivial
441.4.e.p 4 7.c even 3 1 inner
441.4.e.q 4 7.b odd 2 1
441.4.e.q 4 7.d odd 6 1
525.4.a.n 2 105.p even 6 1
525.4.d.g 4 105.w odd 12 2
1008.4.a.ba 2 28.f even 6 1
1344.4.a.bg 2 168.ba even 6 1
1344.4.a.bo 2 168.be odd 6 1
1575.4.a.p 2 35.i odd 6 1
2352.4.a.bz 2 84.n even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{3} + 21T_{2}^{2} - 36T_{2} + 144$$ T2^4 + 3*T2^3 + 21*T2^2 - 36*T2 + 144 $$T_{5}^{4} + 6T_{5}^{3} + 84T_{5}^{2} - 288T_{5} + 2304$$ T5^4 + 6*T5^3 + 84*T5^2 - 288*T5 + 2304 $$T_{13}^{2} + 16T_{13} - 1988$$ T13^2 + 16*T13 - 1988

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + 21 T^{2} - 36 T + 144$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + 84 T^{2} + \cdots + 2304$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 6 T^{3} + 1452 T^{2} + \cdots + 2005056$$
$13$ $$(T^{2} + 16 T - 1988)^{2}$$
$17$ $$T^{4} - 6 T^{3} + 84 T^{2} + \cdots + 2304$$
$19$ $$T^{4} - 64 T^{3} + 11280 T^{2} + \cdots + 51609856$$
$23$ $$T^{4} - 6 T^{3} + 16500 T^{2} + \cdots + 271063296$$
$29$ $$(T^{2} - 252 T + 7668)^{2}$$
$31$ $$T^{4} - 40 T^{3} + \cdots + 5398134784$$
$37$ $$T^{4} - 248 T^{3} + 64596 T^{2} + \cdots + 9560464$$
$41$ $$(T^{2} + 450 T + 37800)^{2}$$
$43$ $$(T^{2} - 376 T + 2512)^{2}$$
$47$ $$T^{4} - 12 T^{3} + \cdots + 4337012736$$
$53$ $$T^{4} + 1104 T^{3} + \cdots + 92705634576$$
$59$ $$T^{4} + 804 T^{3} + \cdots + 908660736$$
$61$ $$T^{4} + 428 T^{3} + \cdots + 788261776$$
$67$ $$T^{4} + 148 T^{3} + \cdots + 25836061696$$
$71$ $$(T^{2} + 954 T + 214704)^{2}$$
$73$ $$T^{4} - 1072 T^{3} + \cdots + 81364139536$$
$79$ $$T^{4} - 572 T^{3} + \cdots + 7126061056$$
$83$ $$(T^{2} - 1944 T + 813456)^{2}$$
$89$ $$T^{4} + 366 T^{3} + \cdots + 64438807104$$
$97$ $$(T^{2} + 808 T - 922292)^{2}$$
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