# Properties

 Label 35.3.c.a Level $35$ Weight $3$ Character orbit 35.c Self dual yes Analytic conductor $0.954$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 4q^{4} + 5q^{5} - 7q^{7} - 8q^{9} + O(q^{10})$$ $$q - q^{3} + 4q^{4} + 5q^{5} - 7q^{7} - 8q^{9} - 13q^{11} - 4q^{12} + 19q^{13} - 5q^{15} + 16q^{16} - 29q^{17} + 20q^{20} + 7q^{21} + 25q^{25} + 17q^{27} - 28q^{28} + 23q^{29} + 13q^{33} - 35q^{35} - 32q^{36} - 19q^{39} - 52q^{44} - 40q^{45} + 31q^{47} - 16q^{48} + 49q^{49} + 29q^{51} + 76q^{52} - 65q^{55} - 20q^{60} + 56q^{63} + 64q^{64} + 95q^{65} - 116q^{68} + 2q^{71} + 34q^{73} - 25q^{75} + 91q^{77} - 157q^{79} + 80q^{80} + 55q^{81} - 86q^{83} + 28q^{84} - 145q^{85} - 23q^{87} - 133q^{91} - 149q^{97} + 104q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 0
0 −1.00000 4.00000 5.00000 0 −7.00000 0 −8.00000 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.c.a 1
3.b odd 2 1 315.3.e.a 1
4.b odd 2 1 560.3.p.b 1
5.b even 2 1 35.3.c.b yes 1
5.c odd 4 2 175.3.d.e 2
7.b odd 2 1 35.3.c.b yes 1
7.c even 3 2 245.3.i.b 2
7.d odd 6 2 245.3.i.a 2
15.d odd 2 1 315.3.e.b 1
20.d odd 2 1 560.3.p.a 1
21.c even 2 1 315.3.e.b 1
28.d even 2 1 560.3.p.a 1
35.c odd 2 1 CM 35.3.c.a 1
35.f even 4 2 175.3.d.e 2
35.i odd 6 2 245.3.i.b 2
35.j even 6 2 245.3.i.a 2
105.g even 2 1 315.3.e.a 1
140.c even 2 1 560.3.p.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.c.a 1 1.a even 1 1 trivial
35.3.c.a 1 35.c odd 2 1 CM
35.3.c.b yes 1 5.b even 2 1
35.3.c.b yes 1 7.b odd 2 1
175.3.d.e 2 5.c odd 4 2
175.3.d.e 2 35.f even 4 2
245.3.i.a 2 7.d odd 6 2
245.3.i.a 2 35.j even 6 2
245.3.i.b 2 7.c even 3 2
245.3.i.b 2 35.i odd 6 2
315.3.e.a 1 3.b odd 2 1
315.3.e.a 1 105.g even 2 1
315.3.e.b 1 15.d odd 2 1
315.3.e.b 1 21.c even 2 1
560.3.p.a 1 20.d odd 2 1
560.3.p.a 1 28.d even 2 1
560.3.p.b 1 4.b odd 2 1
560.3.p.b 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )( 1 + 2 T )$$
$3$ $$1 + T + 9 T^{2}$$
$5$ $$1 - 5 T$$
$7$ $$1 + 7 T$$
$11$ $$1 + 13 T + 121 T^{2}$$
$13$ $$1 - 19 T + 169 T^{2}$$
$17$ $$1 + 29 T + 289 T^{2}$$
$19$ $$( 1 - 19 T )( 1 + 19 T )$$
$23$ $$( 1 - 23 T )( 1 + 23 T )$$
$29$ $$1 - 23 T + 841 T^{2}$$
$31$ $$( 1 - 31 T )( 1 + 31 T )$$
$37$ $$( 1 - 37 T )( 1 + 37 T )$$
$41$ $$( 1 - 41 T )( 1 + 41 T )$$
$43$ $$( 1 - 43 T )( 1 + 43 T )$$
$47$ $$1 - 31 T + 2209 T^{2}$$
$53$ $$( 1 - 53 T )( 1 + 53 T )$$
$59$ $$( 1 - 59 T )( 1 + 59 T )$$
$61$ $$( 1 - 61 T )( 1 + 61 T )$$
$67$ $$( 1 - 67 T )( 1 + 67 T )$$
$71$ $$1 - 2 T + 5041 T^{2}$$
$73$ $$1 - 34 T + 5329 T^{2}$$
$79$ $$1 + 157 T + 6241 T^{2}$$
$83$ $$1 + 86 T + 6889 T^{2}$$
$89$ $$( 1 - 89 T )( 1 + 89 T )$$
$97$ $$1 + 149 T + 9409 T^{2}$$