# Properties

 Label 675.2.e.b Level $675$ Weight $2$ Character orbit 675.e Analytic conductor $5.390$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(226,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("675.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.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: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 45) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (-b5 + b4) * q^2 + (2*b3 - b2 - b1 - 2) * q^4 + (-b5 + b4 + 2*b3) * q^7 + (-b4 + b2 + 1) * q^8 $$q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8} + ( - \beta_{3} + \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} + ( - 2 \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{14} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{16} + ( - \beta_{2} - 1) q^{17} + ( - \beta_{2} + 1) q^{19} + (2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{22} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + (\beta_{2} + 1) q^{26} + ( - 3 \beta_{4} - \beta_{2} - 3) q^{28} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{29} + (4 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{31} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{32} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_1) q^{34} + ( - 2 \beta_{4} - \beta_{2} - 3) q^{37} + (\beta_{3} - \beta_1) q^{38} + (2 \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{41} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - \beta_1) q^{43} + ( - 2 \beta_{4} + \beta_{2} + 7) q^{44} + ( - 4 \beta_{4} + \beta_{2} - 2) q^{46} + (3 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} - \beta_1) q^{47} + ( - 4 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{49} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_1) q^{52} + 2 \beta_{4} q^{53} - 3 \beta_{3} q^{56} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{58} + 2 \beta_{5} q^{59} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{61} + (3 \beta_{2} + 15) q^{62} + ( - 2 \beta_{4} - \beta_{2} - 6) q^{64} + (\beta_{5} - 5 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 5) q^{67} + ( - 2 \beta_{5} - 7 \beta_{3} + \beta_{2} + \beta_1 + 7) q^{68} + (2 \beta_{4} - 3 \beta_{2} + 3) q^{71} + (4 \beta_{4} + 4) q^{73} + (4 \beta_{5} - 4 \beta_{4} - 7 \beta_{3} + \beta_1) q^{74} + ( - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{76} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{77} + (4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{79} + ( - 5 \beta_{4} + 3 \beta_{2} + 9) q^{82} + ( - 3 \beta_{5} + 3 \beta_{4} + 6 \beta_{3}) q^{83} + ( - 4 \beta_{5} - 9 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 9) q^{86} + ( - 4 \beta_{5} + 4 \beta_{4} - 7 \beta_{3} + \beta_1) q^{88} + 3 q^{89} + ( - \beta_{2} + 3) q^{91} + ( - \beta_{5} + \beta_{4} - 13 \beta_{3} + \beta_1) q^{92} + (4 \beta_{5} - 13 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 13) q^{94} + (2 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + 4 \beta_1) q^{97} + ( - 2 \beta_{4} - 3 \beta_{2} - 15) q^{98}+O(q^{100})$$ q + (-b5 + b4) * q^2 + (2*b3 - b2 - b1 - 2) * q^4 + (-b5 + b4 + 2*b3) * q^7 + (-b4 + b2 + 1) * q^8 + (-b3 + b1) * q^11 + (-b3 - b2 - b1 + 1) * q^13 + (-2*b5 + 4*b3 - b2 - b1 - 4) * q^14 + (-2*b5 + 2*b4 - b3) * q^16 + (-b2 - 1) * q^17 + (-b2 + 1) * q^19 + (2*b5 + b3 - b2 - b1 - 1) * q^22 + (-b5 + 2*b3 - 2*b2 - 2*b1 - 2) * q^23 + (b2 + 1) * q^26 + (-3*b4 - b2 - 3) * q^28 + (-2*b5 + 2*b4 - b3 - 2*b1) * q^29 + (4*b5 + b3 + b2 + b1 - 1) * q^31 + (-b5 + 6*b3 - 6) * q^32 + (2*b5 - 2*b4 + b3 - b1) * q^34 + (-2*b4 - b2 - 3) * q^37 + (b3 - b1) * q^38 + (2*b5 + 4*b3 - b2 - b1 - 4) * q^41 + (2*b5 - 2*b4 + 3*b3 - b1) * q^43 + (-2*b4 + b2 + 7) * q^44 + (-4*b4 + b2 - 2) * q^46 + (3*b5 - 3*b4 - 5*b3 - b1) * q^47 + (-4*b5 + b3 - b2 - b1 - 1) * q^49 + (-2*b5 + 2*b4 - 3*b3 - b1) * q^52 + 2*b4 * q^53 - 3*b3 * q^56 + (-b5 + 6*b3 - 6) * q^58 + 2*b5 * q^59 + (-2*b5 + 2*b4 + b1) * q^61 + (3*b2 + 15) * q^62 + (-2*b4 - b2 - 6) * q^64 + (b5 - 5*b3 + 3*b2 + 3*b1 + 5) * q^67 + (-2*b5 - 7*b3 + b2 + b1 + 7) * q^68 + (2*b4 - 3*b2 + 3) * q^71 + (4*b4 + 4) * q^73 + (4*b5 - 4*b4 - 7*b3 + b1) * q^74 + (-2*b5 - 3*b3 - b2 - b1 + 3) * q^76 + (2*b5 - b3 + b2 + b1 + 1) * q^77 + (4*b5 - 4*b4 - 2*b3) * q^79 + (-5*b4 + 3*b2 + 9) * q^82 + (-3*b5 + 3*b4 + 6*b3) * q^83 + (-4*b5 - 9*b3 + 3*b2 + 3*b1 + 9) * q^86 + (-4*b5 + 4*b4 - 7*b3 + b1) * q^88 + 3 * q^89 + (-b2 + 3) * q^91 + (-b5 + b4 - 13*b3 + b1) * q^92 + (4*b5 - 13*b3 + 4*b2 + 4*b1 + 13) * q^94 + (2*b5 - 2*b4 - 8*b3 + 4*b1) * q^97 + (-2*b4 - 3*b2 - 15) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8 $$6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 5 q^{16} - 4 q^{17} + 8 q^{19} - 4 q^{22} - 3 q^{23} + 4 q^{26} - 10 q^{28} - 7 q^{29} - 8 q^{31} - 17 q^{32} + 4 q^{34} - 12 q^{37} + 2 q^{38} - 13 q^{41} + 10 q^{43} + 44 q^{44} - 6 q^{46} - 13 q^{47} + 2 q^{49} - 12 q^{52} - 4 q^{53} - 9 q^{56} - 17 q^{58} - 2 q^{59} - q^{61} + 84 q^{62} - 30 q^{64} + 11 q^{67} + 22 q^{68} + 20 q^{71} + 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} + 58 q^{82} + 15 q^{83} + 28 q^{86} - 24 q^{88} + 18 q^{89} + 20 q^{91} - 39 q^{92} + 31 q^{94} - 18 q^{97} - 80 q^{98}+O(q^{100})$$ 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 5 * q^16 - 4 * q^17 + 8 * q^19 - 4 * q^22 - 3 * q^23 + 4 * q^26 - 10 * q^28 - 7 * q^29 - 8 * q^31 - 17 * q^32 + 4 * q^34 - 12 * q^37 + 2 * q^38 - 13 * q^41 + 10 * q^43 + 44 * q^44 - 6 * q^46 - 13 * q^47 + 2 * q^49 - 12 * q^52 - 4 * q^53 - 9 * q^56 - 17 * q^58 - 2 * q^59 - q^61 + 84 * q^62 - 30 * q^64 + 11 * q^67 + 22 * q^68 + 20 * q^71 + 16 * q^73 - 16 * q^74 + 12 * q^76 - 2 * q^79 + 58 * q^82 + 15 * q^83 + 28 * q^86 - 24 * q^88 + 18 * q^89 + 20 * q^91 - 39 * q^92 + 31 * q^94 - 18 * q^97 - 80 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27$$ (-v^5 + 4*v^4 - v^3 + 9*v^2 - 21*v - 9) / 27 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27$$ (-v^5 + 4*v^4 - v^3 - 18*v^2 + 33*v - 9) / 27 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27$$ (-2*v^5 - v^4 - 2*v^3 + 12*v + 36) / 27 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27$$ (-2*v^5 - v^4 + 7*v^3 + 9*v^2 + 12*v + 9) / 27 $$\beta_{5}$$ $$=$$ $$( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27$$ (4*v^5 + 2*v^4 - 5*v^3 + 18*v^2 + 3*v - 72) / 27
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b5 + b4 + b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3$$ (2*b5 + 2*b4 + 2*b3 - b2 + b1 + 2) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3$$ (-2*b5 + 7*b4 - 11*b3 + b2 - b1 + 7) / 3 $$\nu^{4}$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3$$ (2*b5 + 2*b4 - 7*b3 + 8*b2 + 10*b1 + 20) / 3 $$\nu^{5}$$ $$=$$ $$( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3$$ (7*b5 - 2*b4 - 20*b3 + b2 - 10*b1 + 43) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$1$$

## 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
 0.403374 + 1.68443i −1.62241 − 0.606458i 1.71903 − 0.211943i 0.403374 − 1.68443i −1.62241 + 0.606458i 1.71903 + 0.211943i
−1.25707 2.17731i 0 −2.16044 + 3.74200i 0 0 −0.257068 0.445256i 5.83502 0 0
226.2 −0.285997 0.495361i 0 0.836412 1.44871i 0 0 0.714003 + 1.23669i −2.10083 0 0
226.3 1.04307 + 1.80664i 0 −1.17597 + 2.03684i 0 0 2.04307 + 3.53869i −0.734191 0 0
451.1 −1.25707 + 2.17731i 0 −2.16044 3.74200i 0 0 −0.257068 + 0.445256i 5.83502 0 0
451.2 −0.285997 + 0.495361i 0 0.836412 + 1.44871i 0 0 0.714003 1.23669i −2.10083 0 0
451.3 1.04307 1.80664i 0 −1.17597 2.03684i 0 0 2.04307 3.53869i −0.734191 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.e.b 6
3.b odd 2 1 225.2.e.b 6
5.b even 2 1 135.2.e.b 6
5.c odd 4 2 675.2.k.b 12
9.c even 3 1 inner 675.2.e.b 6
9.c even 3 1 2025.2.a.o 3
9.d odd 6 1 225.2.e.b 6
9.d odd 6 1 2025.2.a.n 3
15.d odd 2 1 45.2.e.b 6
15.e even 4 2 225.2.k.b 12
20.d odd 2 1 2160.2.q.k 6
45.h odd 6 1 45.2.e.b 6
45.h odd 6 1 405.2.a.j 3
45.j even 6 1 135.2.e.b 6
45.j even 6 1 405.2.a.i 3
45.k odd 12 2 675.2.k.b 12
45.k odd 12 2 2025.2.b.m 6
45.l even 12 2 225.2.k.b 12
45.l even 12 2 2025.2.b.l 6
60.h even 2 1 720.2.q.i 6
180.n even 6 1 720.2.q.i 6
180.n even 6 1 6480.2.a.bv 3
180.p odd 6 1 2160.2.q.k 6
180.p odd 6 1 6480.2.a.bs 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 15.d odd 2 1
45.2.e.b 6 45.h odd 6 1
135.2.e.b 6 5.b even 2 1
135.2.e.b 6 45.j even 6 1
225.2.e.b 6 3.b odd 2 1
225.2.e.b 6 9.d odd 6 1
225.2.k.b 12 15.e even 4 2
225.2.k.b 12 45.l even 12 2
405.2.a.i 3 45.j even 6 1
405.2.a.j 3 45.h odd 6 1
675.2.e.b 6 1.a even 1 1 trivial
675.2.e.b 6 9.c even 3 1 inner
675.2.k.b 12 5.c odd 4 2
675.2.k.b 12 45.k odd 12 2
720.2.q.i 6 60.h even 2 1
720.2.q.i 6 180.n even 6 1
2025.2.a.n 3 9.d odd 6 1
2025.2.a.o 3 9.c even 3 1
2025.2.b.l 6 45.l even 12 2
2025.2.b.m 6 45.k odd 12 2
2160.2.q.k 6 20.d odd 2 1
2160.2.q.k 6 180.p odd 6 1
6480.2.a.bs 3 180.p odd 6 1
6480.2.a.bv 3 180.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 6T_{2}^{4} + T_{2}^{3} + 28T_{2}^{2} + 15T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 6 T^{4} + T^{3} + 28 T^{2} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 5 T^{5} + 22 T^{4} - 21 T^{3} + \cdots + 9$$
$11$ $$T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144$$
$13$ $$T^{6} - 4 T^{5} + 20 T^{4} + 8 T^{3} + \cdots + 16$$
$17$ $$(T^{3} + 2 T^{2} - 8 T - 12)^{2}$$
$19$ $$(T^{3} - 4 T^{2} - 4 T + 4)^{2}$$
$23$ $$T^{6} + 3 T^{5} + 42 T^{4} + \cdots + 13689$$
$29$ $$T^{6} + 7 T^{5} + 78 T^{4} + \cdots + 2601$$
$31$ $$T^{6} + 8 T^{5} + 124 T^{4} + \cdots + 219024$$
$37$ $$(T^{3} + 6 T^{2} - 12 T - 4)^{2}$$
$41$ $$T^{6} + 13 T^{5} + 150 T^{4} + 241 T^{3} + \cdots + 9$$
$43$ $$T^{6} - 10 T^{5} + 104 T^{4} + \cdots + 16$$
$47$ $$T^{6} + 13 T^{5} + 180 T^{4} + \cdots + 136161$$
$53$ $$(T^{3} + 2 T^{2} - 20 T - 24)^{2}$$
$59$ $$T^{6} + 2 T^{5} + 24 T^{4} + 8 T^{3} + \cdots + 576$$
$61$ $$T^{6} + T^{5} + 38 T^{4} - 179 T^{3} + \cdots + 5041$$
$67$ $$T^{6} - 11 T^{5} + 160 T^{4} + \cdots + 257049$$
$71$ $$(T^{3} - 10 T^{2} - 92 T + 708)^{2}$$
$73$ $$(T^{3} - 8 T^{2} - 64 T + 128)^{2}$$
$79$ $$T^{6} + 2 T^{5} + 88 T^{4} - 216 T^{3} + \cdots + 576$$
$83$ $$T^{6} - 15 T^{5} + 198 T^{4} + \cdots + 6561$$
$89$ $$(T - 3)^{6}$$
$97$ $$T^{6} + 18 T^{5} + 360 T^{4} + \cdots + 1700416$$