Properties

 Label 1575.2.bc.a Level $1575$ Weight $2$ Character orbit 1575.bc Analytic conductor $12.576$ Analytic rank $0$ Dimension $8$ Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(899,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("1575.899");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.bc (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: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} - \beta_{4}) q^{2} + ( - 2 \beta_{3} + 3 \beta_1) q^{7} + 2 \beta_{7} q^{8}+O(q^{10})$$ q + (b7 - b4) * q^2 + (-2*b3 + 3*b1) * q^7 + 2*b7 * q^8 $$q + (\beta_{7} - \beta_{4}) q^{2} + ( - 2 \beta_{3} + 3 \beta_1) q^{7} + 2 \beta_{7} q^{8} + \beta_{5} q^{11} + ( - 3 \beta_{3} + 6 \beta_1) q^{13} + ( - 2 \beta_{6} + 3 \beta_{5}) q^{14} + ( - 4 \beta_{2} + 4) q^{16} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{2} - 1) q^{19} - 2 \beta_{3} q^{22} + ( - 4 \beta_{7} + 4 \beta_{4}) q^{23} + ( - 3 \beta_{6} + 6 \beta_{5}) q^{26} + (2 \beta_{6} - 2 \beta_{5}) q^{29} + ( - \beta_{2} + 2) q^{31} + (8 \beta_{2} - 4) q^{34} + \beta_1 q^{37} + ( - 2 \beta_{7} + \beta_{4}) q^{38} + (3 \beta_{6} + 3 \beta_{5}) q^{41} + \beta_{3} q^{43} + 8 \beta_{2} q^{46} + ( - 5 \beta_{7} - 5 \beta_{4}) q^{47} + ( - 3 \beta_{2} + 8) q^{49} + 2 \beta_{4} q^{53} + (2 \beta_{6} + 4 \beta_{5}) q^{56} + 4 \beta_1 q^{58} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{59} + ( - 2 \beta_{2} - 2) q^{61} + (\beta_{7} - 2 \beta_{4}) q^{62} + 8 q^{64} + ( - 11 \beta_{3} + 11 \beta_1) q^{67} + ( - 5 \beta_{6} + 5 \beta_{5}) q^{71} + ( - \beta_{3} - \beta_1) q^{73} + \beta_{5} q^{74} + (3 \beta_{7} - 2 \beta_{4}) q^{77} + ( - 5 \beta_{2} + 5) q^{79} + ( - 12 \beta_{3} + 6 \beta_1) q^{82} + ( - 3 \beta_{7} + 6 \beta_{4}) q^{83} + \beta_{6} q^{86} + ( - 4 \beta_{3} + 4 \beta_1) q^{88} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{89} + ( - 3 \beta_{2} + 15) q^{91} + (10 \beta_{2} - 20) q^{94} + (6 \beta_{3} - 12 \beta_1) q^{97} + (5 \beta_{7} - 8 \beta_{4}) q^{98}+O(q^{100})$$ q + (b7 - b4) * q^2 + (-2*b3 + 3*b1) * q^7 + 2*b7 * q^8 + b5 * q^11 + (-3*b3 + 6*b1) * q^13 + (-2*b6 + 3*b5) * q^14 + (-4*b2 + 4) * q^16 + (-4*b7 + 2*b4) * q^17 + (-b2 - 1) * q^19 - 2*b3 * q^22 + (-4*b7 + 4*b4) * q^23 + (-3*b6 + 6*b5) * q^26 + (2*b6 - 2*b5) * q^29 + (-b2 + 2) * q^31 + (8*b2 - 4) * q^34 + b1 * q^37 + (-2*b7 + b4) * q^38 + (3*b6 + 3*b5) * q^41 + b3 * q^43 + 8*b2 * q^46 + (-5*b7 - 5*b4) * q^47 + (-3*b2 + 8) * q^49 + 2*b4 * q^53 + (2*b6 + 4*b5) * q^56 + 4*b1 * q^58 + (-4*b6 + 2*b5) * q^59 + (-2*b2 - 2) * q^61 + (b7 - 2*b4) * q^62 + 8 * q^64 + (-11*b3 + 11*b1) * q^67 + (-5*b6 + 5*b5) * q^71 + (-b3 - b1) * q^73 + b5 * q^74 + (3*b7 - 2*b4) * q^77 + (-5*b2 + 5) * q^79 + (-12*b3 + 6*b1) * q^82 + (-3*b7 + 6*b4) * q^83 + b6 * q^86 + (-4*b3 + 4*b1) * q^88 + (-2*b6 + 4*b5) * q^89 + (-3*b2 + 15) * q^91 + (10*b2 - 20) * q^94 + (6*b3 - 12*b1) * q^97 + (5*b7 - 8*b4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 16 q^{16} - 12 q^{19} + 12 q^{31} + 32 q^{46} + 52 q^{49} - 24 q^{61} + 64 q^{64} + 20 q^{79} + 108 q^{91} - 120 q^{94}+O(q^{100})$$ 8 * q + 16 * q^16 - 12 * q^19 + 12 * q^31 + 32 * q^46 + 52 * q^49 - 24 * q^61 + 64 * q^64 + 20 * q^79 + 108 * q^91 - 120 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
899.1
 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i
−0.707107 1.22474i 0 0 0 0 −2.59808 0.500000i −2.82843 0 0
899.2 −0.707107 1.22474i 0 0 0 0 2.59808 + 0.500000i −2.82843 0 0
899.3 0.707107 + 1.22474i 0 0 0 0 −2.59808 0.500000i 2.82843 0 0
899.4 0.707107 + 1.22474i 0 0 0 0 2.59808 + 0.500000i 2.82843 0 0
1349.1 −0.707107 + 1.22474i 0 0 0 0 −2.59808 + 0.500000i −2.82843 0 0
1349.2 −0.707107 + 1.22474i 0 0 0 0 2.59808 0.500000i −2.82843 0 0
1349.3 0.707107 1.22474i 0 0 0 0 −2.59808 + 0.500000i 2.82843 0 0
1349.4 0.707107 1.22474i 0 0 0 0 2.59808 0.500000i 2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 899.4 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
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.bc.a 8
3.b odd 2 1 inner 1575.2.bc.a 8
5.b even 2 1 inner 1575.2.bc.a 8
5.c odd 4 1 63.2.p.a 4
5.c odd 4 1 1575.2.bk.c 4
7.d odd 6 1 inner 1575.2.bc.a 8
15.d odd 2 1 inner 1575.2.bc.a 8
15.e even 4 1 63.2.p.a 4
15.e even 4 1 1575.2.bk.c 4
20.e even 4 1 1008.2.bt.b 4
21.g even 6 1 inner 1575.2.bc.a 8
35.f even 4 1 441.2.p.a 4
35.i odd 6 1 inner 1575.2.bc.a 8
35.k even 12 1 63.2.p.a 4
35.k even 12 1 441.2.c.a 4
35.k even 12 1 1575.2.bk.c 4
35.l odd 12 1 441.2.c.a 4
35.l odd 12 1 441.2.p.a 4
45.k odd 12 1 567.2.i.d 4
45.k odd 12 1 567.2.s.d 4
45.l even 12 1 567.2.i.d 4
45.l even 12 1 567.2.s.d 4
60.l odd 4 1 1008.2.bt.b 4
105.k odd 4 1 441.2.p.a 4
105.p even 6 1 inner 1575.2.bc.a 8
105.w odd 12 1 63.2.p.a 4
105.w odd 12 1 441.2.c.a 4
105.w odd 12 1 1575.2.bk.c 4
105.x even 12 1 441.2.c.a 4
105.x even 12 1 441.2.p.a 4
140.w even 12 1 7056.2.k.b 4
140.x odd 12 1 1008.2.bt.b 4
140.x odd 12 1 7056.2.k.b 4
315.bs even 12 1 567.2.s.d 4
315.bu odd 12 1 567.2.s.d 4
315.bw odd 12 1 567.2.i.d 4
315.cg even 12 1 567.2.i.d 4
420.bp odd 12 1 7056.2.k.b 4
420.br even 12 1 1008.2.bt.b 4
420.br even 12 1 7056.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 5.c odd 4 1
63.2.p.a 4 15.e even 4 1
63.2.p.a 4 35.k even 12 1
63.2.p.a 4 105.w odd 12 1
441.2.c.a 4 35.k even 12 1
441.2.c.a 4 35.l odd 12 1
441.2.c.a 4 105.w odd 12 1
441.2.c.a 4 105.x even 12 1
441.2.p.a 4 35.f even 4 1
441.2.p.a 4 35.l odd 12 1
441.2.p.a 4 105.k odd 4 1
441.2.p.a 4 105.x even 12 1
567.2.i.d 4 45.k odd 12 1
567.2.i.d 4 45.l even 12 1
567.2.i.d 4 315.bw odd 12 1
567.2.i.d 4 315.cg even 12 1
567.2.s.d 4 45.k odd 12 1
567.2.s.d 4 45.l even 12 1
567.2.s.d 4 315.bs even 12 1
567.2.s.d 4 315.bu odd 12 1
1008.2.bt.b 4 20.e even 4 1
1008.2.bt.b 4 60.l odd 4 1
1008.2.bt.b 4 140.x odd 12 1
1008.2.bt.b 4 420.br even 12 1
1575.2.bc.a 8 1.a even 1 1 trivial
1575.2.bc.a 8 3.b odd 2 1 inner
1575.2.bc.a 8 5.b even 2 1 inner
1575.2.bc.a 8 7.d odd 6 1 inner
1575.2.bc.a 8 15.d odd 2 1 inner
1575.2.bc.a 8 21.g even 6 1 inner
1575.2.bc.a 8 35.i odd 6 1 inner
1575.2.bc.a 8 105.p even 6 1 inner
1575.2.bk.c 4 5.c odd 4 1
1575.2.bk.c 4 15.e even 4 1
1575.2.bk.c 4 35.k even 12 1
1575.2.bk.c 4 105.w odd 12 1
7056.2.k.b 4 140.w even 12 1
7056.2.k.b 4 140.x odd 12 1
7056.2.k.b 4 420.bp odd 12 1
7056.2.k.b 4 420.br even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1575, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{2} + 4$$ T2^4 + 2*T2^2 + 4 $$T_{11}^{4} - 2T_{11}^{2} + 4$$ T11^4 - 2*T11^2 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 13 T^{2} + 49)^{2}$$
$11$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$13$ $$(T^{2} - 27)^{4}$$
$17$ $$(T^{4} - 24 T^{2} + 576)^{2}$$
$19$ $$(T^{2} + 3 T + 3)^{4}$$
$23$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$29$ $$(T^{2} + 8)^{4}$$
$31$ $$(T^{2} - 3 T + 3)^{4}$$
$37$ $$(T^{4} - T^{2} + 1)^{2}$$
$41$ $$(T^{2} - 54)^{4}$$
$43$ $$(T^{2} + 1)^{4}$$
$47$ $$(T^{4} - 150 T^{2} + 22500)^{2}$$
$53$ $$(T^{4} + 8 T^{2} + 64)^{2}$$
$59$ $$(T^{4} + 24 T^{2} + 576)^{2}$$
$61$ $$(T^{2} + 6 T + 12)^{4}$$
$67$ $$(T^{4} - 121 T^{2} + 14641)^{2}$$
$71$ $$(T^{2} + 50)^{4}$$
$73$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$79$ $$(T^{2} - 5 T + 25)^{4}$$
$83$ $$(T^{2} + 54)^{4}$$
$89$ $$(T^{4} + 24 T^{2} + 576)^{2}$$
$97$ $$(T^{2} - 108)^{4}$$