# Properties

 Label 1035.2.b.a Level $1035$ Weight $2$ Character orbit 1035.b Analytic conductor $8.265$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$8.26451660920$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.

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 $$S_{2}^{\mathrm{new}}(1035, [\chi])$$:

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

## Hecke characteristic polynomials

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