# Properties

 Label 1260.2.c.a Level $1260$ Weight $2$ Character orbit 1260.c Analytic conductor $10.061$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(811,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("1260.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.c (of order $$2$$, degree $$1$$, 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: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{4} + \beta_{2} q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{2} + 2) q^{8}+O(q^{10})$$ q + (b3 + 1) * q^2 + (b3 - 2*b2 + b1) * q^4 + b2 * q^5 + (b3 - 3*b2 + b1) * q^7 + (-2*b2 + 2) * q^8 $$q + (\beta_{3} + 1) q^{2} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{4} + \beta_{2} q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{2} + 2) q^{8} + (\beta_1 - 1) q^{10} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{13} + ( - 2 \beta_{2} - \beta_1 + 3) q^{14} + (2 \beta_{3} - 2 \beta_1 + 4) q^{16} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{17} + 6 q^{19} + ( - \beta_{3} + \beta_1) q^{20} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{22} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{23} - q^{25} + ( - 2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 1) q^{26} + (3 \beta_{3} - 3 \beta_1 + 4) q^{28} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 1) q^{29} - 6 q^{31} + (4 \beta_{3} - 4 \beta_{2}) q^{32} + (2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 5) q^{34} + ( - \beta_{3} + \beta_1 + 1) q^{35} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + (6 \beta_{3} + 6) q^{38} + (2 \beta_{2} + 2) q^{40} + (2 \beta_{3} - 2 \beta_1 + 2) q^{41} - 2 \beta_{2} q^{43} + ( - \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{44} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{46} + (\beta_{3} - \beta_{2} + \beta_1) q^{47} + (4 \beta_{3} - 4 \beta_1 + 3) q^{49} + ( - \beta_{3} - 1) q^{50} + ( - \beta_{3} + 4 \beta_{2} + 5 \beta_1) q^{52} - 2 q^{53} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{55} + (4 \beta_{3} - 6 \beta_{2} - 2) q^{56} + ( - \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 3) q^{58} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{59} + (2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - 6 \beta_{3} - 6) q^{62} - 8 \beta_{2} q^{64} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{65} + (2 \beta_{3} - 2 \beta_1 + 2) q^{67} + ( - 5 \beta_{3} - 4 \beta_{2} + \beta_1) q^{68} + (\beta_{3} + 2 \beta_{2} + 3) q^{70} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{71} + (4 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{73} + ( - 6 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{74} + (6 \beta_{3} - 12 \beta_{2} + 6 \beta_1) q^{76} + (\beta_{2} + 4 \beta_1 + 2) q^{77} + ( - 5 \beta_{3} - 6 \beta_{2} + 5 \beta_1 - 5) q^{79} + (2 \beta_{3} + 2 \beta_1) q^{80} + (2 \beta_{3} - 4 \beta_{2} - 2) q^{82} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 12) q^{83} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{85} + ( - 2 \beta_1 + 2) q^{86} + (2 \beta_{2} + 4 \beta_1 + 2) q^{88} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 2) q^{89} + (\beta_{3} + 2 \beta_{2} + 7 \beta_1 + 3) q^{91} + ( - 4 \beta_{2} - 4 \beta_1) q^{92} + ( - 2 \beta_{2} + \beta_1 + 1) q^{94} + 6 \beta_{2} q^{95} + ( - 6 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 6) q^{97} + (3 \beta_{3} - 8 \beta_{2} - 5) q^{98}+O(q^{100})$$ q + (b3 + 1) * q^2 + (b3 - 2*b2 + b1) * q^4 + b2 * q^5 + (b3 - 3*b2 + b1) * q^7 + (-2*b2 + 2) * q^8 + (b1 - 1) * q^10 + (-b3 + 2*b2 + b1 - 1) * q^11 + (-2*b3 + 3*b2 + 2*b1 - 2) * q^13 + (-2*b2 - b1 + 3) * q^14 + (2*b3 - 2*b1 + 4) * q^16 + (2*b3 + 3*b2 - 2*b1 + 2) * q^17 + 6 * q^19 + (-b3 + b1) * q^20 + (-b3 + 2*b2 + 2*b1 - 1) * q^22 + (2*b3 - 2*b2 - 2*b1 + 2) * q^23 - q^25 + (-2*b3 + 4*b2 + 3*b1 - 1) * q^26 + (3*b3 - 3*b1 + 4) * q^28 + (4*b3 - 4*b2 + 4*b1 - 1) * q^29 - 6 * q^31 + (4*b3 - 4*b2) * q^32 + (2*b3 - 4*b2 + 3*b1 - 5) * q^34 + (-b3 + b1 + 1) * q^35 + (2*b3 - 2*b2 + 2*b1 - 6) * q^37 + (6*b3 + 6) * q^38 + (2*b2 + 2) * q^40 + (2*b3 - 2*b1 + 2) * q^41 - 2*b2 * q^43 + (-b3 + 2*b2 + 3*b1) * q^44 + (2*b3 - 4*b2 - 2*b1) * q^46 + (b3 - b2 + b1) * q^47 + (4*b3 - 4*b1 + 3) * q^49 + (-b3 - 1) * q^50 + (-b3 + 4*b2 + 5*b1) * q^52 - 2 * q^53 + (-b3 + b2 - b1 - 2) * q^55 + (4*b3 - 6*b2 - 2) * q^56 + (-b3 - 8*b2 + 4*b1 + 3) * q^58 + (-2*b3 + 2*b2 - 2*b1) * q^59 + (2*b3 + 6*b2 - 2*b1 + 2) * q^61 + (-6*b3 - 6) * q^62 - 8*b2 * q^64 + (-2*b3 + 2*b2 - 2*b1 - 3) * q^65 + (2*b3 - 2*b1 + 2) * q^67 + (-5*b3 - 4*b2 + b1) * q^68 + (b3 + 2*b2 + 3) * q^70 + (2*b3 + 4*b2 - 2*b1 + 2) * q^71 + (4*b3 + 6*b2 - 4*b1 + 4) * q^73 + (-6*b3 - 4*b2 + 2*b1 - 4) * q^74 + (6*b3 - 12*b2 + 6*b1) * q^76 + (b2 + 4*b1 + 2) * q^77 + (-5*b3 - 6*b2 + 5*b1 - 5) * q^79 + (2*b3 + 2*b1) * q^80 + (2*b3 - 4*b2 - 2) * q^82 + (2*b3 - 2*b2 + 2*b1 - 12) * q^83 + (2*b3 - 2*b2 + 2*b1 - 3) * q^85 + (-2*b1 + 2) * q^86 + (2*b2 + 4*b1 + 2) * q^88 + (2*b3 - 6*b2 - 2*b1 + 2) * q^89 + (b3 + 2*b2 + 7*b1 + 3) * q^91 + (-4*b2 - 4*b1) * q^92 + (-2*b2 + b1 + 1) * q^94 + 6*b2 * q^95 + (-6*b3 - 3*b2 + 6*b1 - 6) * q^97 + (3*b3 - 8*b2 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 8 * q^8 $$4 q + 2 q^{2} + 8 q^{8} - 2 q^{10} + 10 q^{14} + 8 q^{16} + 24 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{25} + 6 q^{26} + 4 q^{28} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 18 q^{34} + 8 q^{35} - 24 q^{37} + 12 q^{38} + 8 q^{40} + 8 q^{44} - 8 q^{46} - 4 q^{49} - 2 q^{50} + 12 q^{52} - 8 q^{53} - 8 q^{55} - 16 q^{56} + 22 q^{58} - 12 q^{62} - 12 q^{65} + 12 q^{68} + 10 q^{70} + 16 q^{77} - 12 q^{82} - 48 q^{83} - 12 q^{85} + 4 q^{86} + 16 q^{88} + 24 q^{91} - 8 q^{92} + 6 q^{94} - 26 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 8 * q^8 - 2 * q^10 + 10 * q^14 + 8 * q^16 + 24 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^25 + 6 * q^26 + 4 * q^28 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 18 * q^34 + 8 * q^35 - 24 * q^37 + 12 * q^38 + 8 * q^40 + 8 * q^44 - 8 * q^46 - 4 * q^49 - 2 * q^50 + 12 * q^52 - 8 * q^53 - 8 * q^55 - 16 * q^56 + 22 * q^58 - 12 * q^62 - 12 * q^65 + 12 * q^68 + 10 * q^70 + 16 * q^77 - 12 * q^82 - 48 * q^83 - 12 * q^85 + 4 * q^86 + 16 * q^88 + 24 * q^91 - 8 * q^92 + 6 * q^94 - 26 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2} + \zeta_{12}$$ v^2 + v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{2} + \zeta_{12}$$ -v^2 + v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( -\beta_{3} + \beta_1 ) / 2$$ (-b3 + b1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_{2}$$ b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i 1.00000i 0 −1.73205 + 2.00000i 2.00000 + 2.00000i 0 −1.36603 + 0.366025i
811.2 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.00000i 0 −1.73205 2.00000i 2.00000 2.00000i 0 −1.36603 0.366025i
811.3 1.36603 0.366025i 0 1.73205 1.00000i 1.00000i 0 1.73205 2.00000i 2.00000 2.00000i 0 0.366025 + 1.36603i
811.4 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.00000i 0 1.73205 + 2.00000i 2.00000 + 2.00000i 0 0.366025 1.36603i
 $$n$$: 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
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.c.a 4
3.b odd 2 1 140.2.g.a 4
4.b odd 2 1 1260.2.c.b 4
7.b odd 2 1 1260.2.c.b 4
12.b even 2 1 140.2.g.b yes 4
15.d odd 2 1 700.2.g.f 4
15.e even 4 1 700.2.c.b 4
15.e even 4 1 700.2.c.e 4
21.c even 2 1 140.2.g.b yes 4
21.g even 6 1 980.2.o.b 4
21.g even 6 1 980.2.o.d 4
21.h odd 6 1 980.2.o.a 4
21.h odd 6 1 980.2.o.c 4
24.f even 2 1 2240.2.k.b 4
24.h odd 2 1 2240.2.k.a 4
28.d even 2 1 inner 1260.2.c.a 4
60.h even 2 1 700.2.g.g 4
60.l odd 4 1 700.2.c.c 4
60.l odd 4 1 700.2.c.f 4
84.h odd 2 1 140.2.g.a 4
84.j odd 6 1 980.2.o.a 4
84.j odd 6 1 980.2.o.c 4
84.n even 6 1 980.2.o.b 4
84.n even 6 1 980.2.o.d 4
105.g even 2 1 700.2.g.g 4
105.k odd 4 1 700.2.c.c 4
105.k odd 4 1 700.2.c.f 4
168.e odd 2 1 2240.2.k.a 4
168.i even 2 1 2240.2.k.b 4
420.o odd 2 1 700.2.g.f 4
420.w even 4 1 700.2.c.b 4
420.w even 4 1 700.2.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 3.b odd 2 1
140.2.g.a 4 84.h odd 2 1
140.2.g.b yes 4 12.b even 2 1
140.2.g.b yes 4 21.c even 2 1
700.2.c.b 4 15.e even 4 1
700.2.c.b 4 420.w even 4 1
700.2.c.c 4 60.l odd 4 1
700.2.c.c 4 105.k odd 4 1
700.2.c.e 4 15.e even 4 1
700.2.c.e 4 420.w even 4 1
700.2.c.f 4 60.l odd 4 1
700.2.c.f 4 105.k odd 4 1
700.2.g.f 4 15.d odd 2 1
700.2.g.f 4 420.o odd 2 1
700.2.g.g 4 60.h even 2 1
700.2.g.g 4 105.g even 2 1
980.2.o.a 4 21.h odd 6 1
980.2.o.a 4 84.j odd 6 1
980.2.o.b 4 21.g even 6 1
980.2.o.b 4 84.n even 6 1
980.2.o.c 4 21.h odd 6 1
980.2.o.c 4 84.j odd 6 1
980.2.o.d 4 21.g even 6 1
980.2.o.d 4 84.n even 6 1
1260.2.c.a 4 1.a even 1 1 trivial
1260.2.c.a 4 28.d even 2 1 inner
1260.2.c.b 4 4.b odd 2 1
1260.2.c.b 4 7.b odd 2 1
2240.2.k.a 4 24.h odd 2 1
2240.2.k.a 4 168.e odd 2 1
2240.2.k.b 4 24.f even 2 1
2240.2.k.b 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{4} + 14T_{11}^{2} + 1$$ T11^4 + 14*T11^2 + 1 $$T_{19} - 6$$ T19 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} + 14T^{2} + 1$$
$13$ $$T^{4} + 42T^{2} + 9$$
$17$ $$T^{4} + 42T^{2} + 9$$
$19$ $$(T - 6)^{4}$$
$23$ $$T^{4} + 32T^{2} + 64$$
$29$ $$(T^{2} + 2 T - 47)^{2}$$
$31$ $$(T + 6)^{4}$$
$37$ $$(T^{2} + 12 T + 24)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 3)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$T^{4} + 96T^{2} + 576$$
$67$ $$(T^{2} + 12)^{2}$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$T^{4} + 168T^{2} + 144$$
$79$ $$T^{4} + 222T^{2} + 1521$$
$83$ $$(T^{2} + 24 T + 132)^{2}$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$T^{4} + 234T^{2} + 9801$$