# Properties

 Label 975.2.w.d Level $975$ Weight $2$ Character orbit 975.w Analytic conductor $7.785$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(49,975)]

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, 3, 5]))

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

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

chi := DirichletCharacter("975.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.w (of order $$6$$, 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: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{12} q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q - z * q^3 + 2*z^2 * q^4 + (-z^3 - z) * q^7 + z^2 * q^9 $$q - \zeta_{12} q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + ( - 2 \zeta_{12}^{2} - 2) q^{11} - 2 \zeta_{12}^{3} q^{12} + ( - 3 \zeta_{12}^{3} - \zeta_{12}) q^{13} + (4 \zeta_{12}^{2} - 4) q^{16} + (2 \zeta_{12}^{2} - 4) q^{19} + (2 \zeta_{12}^{2} - 1) q^{21} - 6 \zeta_{12} q^{23} - \zeta_{12}^{3} q^{27} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} + ( - 6 \zeta_{12}^{2} + 6) q^{29} + (2 \zeta_{12}^{2} - 1) q^{31} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{33} + (2 \zeta_{12}^{2} - 2) q^{36} + (4 \zeta_{12}^{2} - 3) q^{39} + ( - 4 \zeta_{12}^{2} - 4) q^{41} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{43} + ( - 8 \zeta_{12}^{2} + 4) q^{44} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{47} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 4 \zeta_{12}^{2} + 4) q^{49} + ( - 8 \zeta_{12}^{3} + 6 \zeta_{12}) q^{52} - 12 \zeta_{12}^{3} q^{53} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{57} + ( - 2 \zeta_{12}^{2} + 4) q^{59} - \zeta_{12}^{2} q^{61} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{63} - 8 q^{64} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{67} + 6 \zeta_{12}^{2} q^{69} + (6 \zeta_{12}^{2} - 12) q^{71} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{73} + ( - 4 \zeta_{12}^{2} - 4) q^{76} + 6 \zeta_{12}^{3} q^{77} + 11 q^{79} + (\zeta_{12}^{2} - 1) q^{81} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{83} + (2 \zeta_{12}^{2} - 4) q^{84} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{87} + ( - 4 \zeta_{12}^{2} - 4) q^{89} + (5 \zeta_{12}^{2} - 7) q^{91} - 12 \zeta_{12}^{3} q^{92} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{93} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{97} + ( - 4 \zeta_{12}^{2} + 2) q^{99} +O(q^{100})$$ q - z * q^3 + 2*z^2 * q^4 + (-z^3 - z) * q^7 + z^2 * q^9 + (-2*z^2 - 2) * q^11 - 2*z^3 * q^12 + (-3*z^3 - z) * q^13 + (4*z^2 - 4) * q^16 + (2*z^2 - 4) * q^19 + (2*z^2 - 1) * q^21 - 6*z * q^23 - z^3 * q^27 + (-4*z^3 + 2*z) * q^28 + (-6*z^2 + 6) * q^29 + (2*z^2 - 1) * q^31 + (2*z^3 + 2*z) * q^33 + (2*z^2 - 2) * q^36 + (4*z^2 - 3) * q^39 + (-4*z^2 - 4) * q^41 + (-z^3 + z) * q^43 + (-8*z^2 + 4) * q^44 + (2*z^3 - 4*z) * q^47 + (-4*z^3 + 4*z) * q^48 + (-4*z^2 + 4) * q^49 + (-8*z^3 + 6*z) * q^52 - 12*z^3 * q^53 + (-2*z^3 + 4*z) * q^57 + (-2*z^2 + 4) * q^59 - z^2 * q^61 + (-2*z^3 + z) * q^63 - 8 * q^64 + (10*z^3 - 5*z) * q^67 + 6*z^2 * q^69 + (6*z^2 - 12) * q^71 + (-z^3 + 2*z) * q^73 + (-4*z^2 - 4) * q^76 + 6*z^3 * q^77 + 11 * q^79 + (z^2 - 1) * q^81 + (8*z^3 - 16*z) * q^83 + (2*z^2 - 4) * q^84 + (6*z^3 - 6*z) * q^87 + (-4*z^2 - 4) * q^89 + (5*z^2 - 7) * q^91 - 12*z^3 * q^92 + (-2*z^3 + z) * q^93 + (-3*z^3 - 3*z) * q^97 + (-4*z^2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 2 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 + 2 * q^9 $$4 q + 4 q^{4} + 2 q^{9} - 12 q^{11} - 8 q^{16} - 12 q^{19} + 12 q^{29} - 4 q^{36} - 4 q^{39} - 24 q^{41} + 8 q^{49} + 12 q^{59} - 2 q^{61} - 32 q^{64} + 12 q^{69} - 36 q^{71} - 24 q^{76} + 44 q^{79} - 2 q^{81} - 12 q^{84} - 24 q^{89} - 18 q^{91}+O(q^{100})$$ 4 * q + 4 * q^4 + 2 * q^9 - 12 * q^11 - 8 * q^16 - 12 * q^19 + 12 * q^29 - 4 * q^36 - 4 * q^39 - 24 * q^41 + 8 * q^49 + 12 * q^59 - 2 * q^61 - 32 * q^64 + 12 * q^69 - 36 * q^71 - 24 * q^76 + 44 * q^79 - 2 * q^81 - 12 * q^84 - 24 * q^89 - 18 * q^91

## Character values

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

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 0.500000i 1.00000 1.73205i 0 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0
49.2 0 0.866025 0.500000i 1.00000 1.73205i 0 0 0.866025 1.50000i 0 0.500000 0.866025i 0
199.1 0 −0.866025 0.500000i 1.00000 + 1.73205i 0 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0
199.2 0 0.866025 + 0.500000i 1.00000 + 1.73205i 0 0 0.866025 + 1.50000i 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
5.b even 2 1 inner
13.e even 6 1 inner
65.l 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.w.d 4
5.b even 2 1 inner 975.2.w.d 4
5.c odd 4 1 39.2.j.a 2
5.c odd 4 1 975.2.bc.c 2
13.e even 6 1 inner 975.2.w.d 4
15.e even 4 1 117.2.q.a 2
20.e even 4 1 624.2.bv.b 2
60.l odd 4 1 1872.2.by.f 2
65.f even 4 1 507.2.e.f 4
65.h odd 4 1 507.2.j.b 2
65.k even 4 1 507.2.e.f 4
65.l even 6 1 inner 975.2.w.d 4
65.o even 12 1 507.2.a.e 2
65.o even 12 1 507.2.e.f 4
65.q odd 12 1 507.2.b.c 2
65.q odd 12 1 507.2.j.b 2
65.r odd 12 1 39.2.j.a 2
65.r odd 12 1 507.2.b.c 2
65.r odd 12 1 975.2.bc.c 2
65.t even 12 1 507.2.a.e 2
65.t even 12 1 507.2.e.f 4
195.bc odd 12 1 1521.2.a.h 2
195.bf even 12 1 117.2.q.a 2
195.bf even 12 1 1521.2.b.f 2
195.bl even 12 1 1521.2.b.f 2
195.bn odd 12 1 1521.2.a.h 2
260.be odd 12 1 8112.2.a.bu 2
260.bg even 12 1 624.2.bv.b 2
260.bl odd 12 1 8112.2.a.bu 2
780.cw odd 12 1 1872.2.by.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 5.c odd 4 1
39.2.j.a 2 65.r odd 12 1
117.2.q.a 2 15.e even 4 1
117.2.q.a 2 195.bf even 12 1
507.2.a.e 2 65.o even 12 1
507.2.a.e 2 65.t even 12 1
507.2.b.c 2 65.q odd 12 1
507.2.b.c 2 65.r odd 12 1
507.2.e.f 4 65.f even 4 1
507.2.e.f 4 65.k even 4 1
507.2.e.f 4 65.o even 12 1
507.2.e.f 4 65.t even 12 1
507.2.j.b 2 65.h odd 4 1
507.2.j.b 2 65.q odd 12 1
624.2.bv.b 2 20.e even 4 1
624.2.bv.b 2 260.bg even 12 1
975.2.w.d 4 1.a even 1 1 trivial
975.2.w.d 4 5.b even 2 1 inner
975.2.w.d 4 13.e even 6 1 inner
975.2.w.d 4 65.l even 6 1 inner
975.2.bc.c 2 5.c odd 4 1
975.2.bc.c 2 65.r odd 12 1
1521.2.a.h 2 195.bc odd 12 1
1521.2.a.h 2 195.bn odd 12 1
1521.2.b.f 2 195.bf even 12 1
1521.2.b.f 2 195.bl even 12 1
1872.2.by.f 2 60.l odd 4 1
1872.2.by.f 2 780.cw odd 12 1
8112.2.a.bu 2 260.be odd 12 1
8112.2.a.bu 2 260.bl odd 12 1

## 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 $$T_{7}^{4} + 3T_{7}^{2} + 9$$ T7^4 + 3*T7^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 3T^{2} + 9$$
$11$ $$(T^{2} + 6 T + 12)^{2}$$
$13$ $$T^{4} + 23T^{2} + 169$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 6 T + 12)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} - 6 T + 36)^{2}$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 12 T + 48)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$(T^{2} - 6 T + 12)^{2}$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4} + 75T^{2} + 5625$$
$71$ $$(T^{2} + 18 T + 108)^{2}$$
$73$ $$(T^{2} - 3)^{2}$$
$79$ $$(T - 11)^{4}$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} + 12 T + 48)^{2}$$
$97$ $$T^{4} + 27T^{2} + 729$$