# Properties

 Label 975.2.bc.c Level $975$ Weight $2$ Character orbit 975.bc Analytic conductor $7.785$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(751,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("975.751");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.bc (of order $$6$$, degree $$2$$, 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: $$7.78541419707$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 - 2*z * q^4 + (-z + 2) * q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} - 2) q^{11} + 2 q^{12} + (\zeta_{6} - 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 2 \zeta_{6} + 4) q^{19} + (2 \zeta_{6} - 1) q^{21} + (6 \zeta_{6} - 6) q^{23} + q^{27} + ( - 2 \zeta_{6} - 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + (2 \zeta_{6} - 1) q^{31} + ( - 2 \zeta_{6} + 4) q^{33} + (2 \zeta_{6} - 2) q^{36} + ( - 4 \zeta_{6} + 3) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} - \zeta_{6} q^{43} + (8 \zeta_{6} - 4) q^{44} + ( - 4 \zeta_{6} + 2) q^{47} - 4 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{49} + (6 \zeta_{6} + 2) q^{52} - 12 q^{53} + (4 \zeta_{6} - 2) q^{57} + (2 \zeta_{6} - 4) q^{59} - \zeta_{6} q^{61} + ( - \zeta_{6} - 1) q^{63} + 8 q^{64} + ( - 5 \zeta_{6} - 5) q^{67} - 6 \zeta_{6} q^{69} + (6 \zeta_{6} - 12) q^{71} + ( - 2 \zeta_{6} + 1) q^{73} + ( - 4 \zeta_{6} - 4) q^{76} - 6 q^{77} - 11 q^{79} + (\zeta_{6} - 1) q^{81} + (16 \zeta_{6} - 8) q^{83} + ( - 2 \zeta_{6} + 4) q^{84} - 6 \zeta_{6} q^{87} + (4 \zeta_{6} + 4) q^{89} + (5 \zeta_{6} - 7) q^{91} + 12 q^{92} + ( - \zeta_{6} - 1) q^{93} + ( - 3 \zeta_{6} + 6) q^{97} + (4 \zeta_{6} - 2) q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - 2*z * q^4 + (-z + 2) * q^7 - z * q^9 + (-2*z - 2) * q^11 + 2 * q^12 + (z - 4) * q^13 + (4*z - 4) * q^16 + (-2*z + 4) * q^19 + (2*z - 1) * q^21 + (6*z - 6) * q^23 + q^27 + (-2*z - 2) * q^28 + (6*z - 6) * q^29 + (2*z - 1) * q^31 + (-2*z + 4) * q^33 + (2*z - 2) * q^36 + (-4*z + 3) * q^39 + (-4*z - 4) * q^41 - z * q^43 + (8*z - 4) * q^44 + (-4*z + 2) * q^47 - 4*z * q^48 + (4*z - 4) * q^49 + (6*z + 2) * q^52 - 12 * q^53 + (4*z - 2) * q^57 + (2*z - 4) * q^59 - z * q^61 + (-z - 1) * q^63 + 8 * q^64 + (-5*z - 5) * q^67 - 6*z * q^69 + (6*z - 12) * q^71 + (-2*z + 1) * q^73 + (-4*z - 4) * q^76 - 6 * q^77 - 11 * q^79 + (z - 1) * q^81 + (16*z - 8) * q^83 + (-2*z + 4) * q^84 - 6*z * q^87 + (4*z + 4) * q^89 + (5*z - 7) * q^91 + 12 * q^92 + (-z - 1) * q^93 + (-3*z + 6) * q^97 + (4*z - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9 $$2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9} - 6 q^{11} + 4 q^{12} - 7 q^{13} - 4 q^{16} + 6 q^{19} - 6 q^{23} + 2 q^{27} - 6 q^{28} - 6 q^{29} + 6 q^{33} - 2 q^{36} + 2 q^{39} - 12 q^{41} - q^{43} - 4 q^{48} - 4 q^{49} + 10 q^{52} - 24 q^{53} - 6 q^{59} - q^{61} - 3 q^{63} + 16 q^{64} - 15 q^{67} - 6 q^{69} - 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} - q^{81} + 6 q^{84} - 6 q^{87} + 12 q^{89} - 9 q^{91} + 24 q^{92} - 3 q^{93} + 9 q^{97}+O(q^{100})$$ 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9 - 6 * q^11 + 4 * q^12 - 7 * q^13 - 4 * q^16 + 6 * q^19 - 6 * q^23 + 2 * q^27 - 6 * q^28 - 6 * q^29 + 6 * q^33 - 2 * q^36 + 2 * q^39 - 12 * q^41 - q^43 - 4 * q^48 - 4 * q^49 + 10 * q^52 - 24 * q^53 - 6 * q^59 - q^61 - 3 * q^63 + 16 * q^64 - 15 * q^67 - 6 * q^69 - 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 - q^81 + 6 * q^84 - 6 * q^87 + 12 * q^89 - 9 * q^91 + 24 * q^92 - 3 * q^93 + 9 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
901.1 0 −0.500000 + 0.866025i −1.00000 1.73205i 0 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
 $$n$$: 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bc.c 2
5.b even 2 1 39.2.j.a 2
5.c odd 4 2 975.2.w.d 4
13.e even 6 1 inner 975.2.bc.c 2
15.d odd 2 1 117.2.q.a 2
20.d odd 2 1 624.2.bv.b 2
60.h even 2 1 1872.2.by.f 2
65.d even 2 1 507.2.j.b 2
65.g odd 4 2 507.2.e.f 4
65.l even 6 1 39.2.j.a 2
65.l even 6 1 507.2.b.c 2
65.n even 6 1 507.2.b.c 2
65.n even 6 1 507.2.j.b 2
65.r odd 12 2 975.2.w.d 4
65.s odd 12 2 507.2.a.e 2
65.s odd 12 2 507.2.e.f 4
195.x odd 6 1 1521.2.b.f 2
195.y odd 6 1 117.2.q.a 2
195.y odd 6 1 1521.2.b.f 2
195.bh even 12 2 1521.2.a.h 2
260.w odd 6 1 624.2.bv.b 2
260.bc even 12 2 8112.2.a.bu 2
780.cb even 6 1 1872.2.by.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 5.b even 2 1
39.2.j.a 2 65.l even 6 1
117.2.q.a 2 15.d odd 2 1
117.2.q.a 2 195.y odd 6 1
507.2.a.e 2 65.s odd 12 2
507.2.b.c 2 65.l even 6 1
507.2.b.c 2 65.n even 6 1
507.2.e.f 4 65.g odd 4 2
507.2.e.f 4 65.s odd 12 2
507.2.j.b 2 65.d even 2 1
507.2.j.b 2 65.n even 6 1
624.2.bv.b 2 20.d odd 2 1
624.2.bv.b 2 260.w odd 6 1
975.2.w.d 4 5.c odd 4 2
975.2.w.d 4 65.r odd 12 2
975.2.bc.c 2 1.a even 1 1 trivial
975.2.bc.c 2 13.e even 6 1 inner
1521.2.a.h 2 195.bh even 12 2
1521.2.b.f 2 195.x odd 6 1
1521.2.b.f 2 195.y odd 6 1
1872.2.by.f 2 60.h even 2 1
1872.2.by.f 2 780.cb even 6 1
8112.2.a.bu 2 260.bc even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} - 3T_{7} + 3$$ T7^2 - 3*T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3T + 3$$
$11$ $$T^{2} + 6T + 12$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 12T + 48$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 6T + 12$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 15T + 75$$
$71$ $$T^{2} + 18T + 108$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} - 9T + 27$$