# Properties

 Label 1035.2.b.a Level $1035$ Weight $2$ Character orbit 1035.b Analytic conductor $8.265$ 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] = [1035,2,Mod(829,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1035.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1035 = 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1035.b (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: $$8.26451660920$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times$$.

 $$n$$ $$461$$ $$622$$ $$856$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 −2.00000 1.00000i 0 1.00000i 0 0 −2.00000 + 4.00000i
829.2 2.00000i 0 −2.00000 −2.00000 + 1.00000i 0 1.00000i 0 0 −2.00000 4.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1035.2.b.a 2
3.b odd 2 1 115.2.b.a 2
5.b even 2 1 inner 1035.2.b.a 2
5.c odd 4 1 5175.2.a.a 1
5.c odd 4 1 5175.2.a.z 1
12.b even 2 1 1840.2.e.b 2
15.d odd 2 1 115.2.b.a 2
15.e even 4 1 575.2.a.a 1
15.e even 4 1 575.2.a.e 1
60.h even 2 1 1840.2.e.b 2
60.l odd 4 1 9200.2.a.g 1
60.l odd 4 1 9200.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 3.b odd 2 1
115.2.b.a 2 15.d odd 2 1
575.2.a.a 1 15.e even 4 1
575.2.a.e 1 15.e even 4 1
1035.2.b.a 2 1.a even 1 1 trivial
1035.2.b.a 2 5.b even 2 1 inner
1840.2.e.b 2 12.b even 2 1
1840.2.e.b 2 60.h even 2 1
5175.2.a.a 1 5.c odd 4 1
5175.2.a.z 1 5.c odd 4 1
9200.2.a.g 1 60.l odd 4 1
9200.2.a.bg 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1035, [\chi])$$:

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{7}^{2} + 1$$ T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 25$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T - 7)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 1$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T + 13)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} + 9$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 196$$