# Properties

 Label 35.5.c.a Level $35$ Weight $5$ Character orbit 35.c Self dual yes Analytic conductor $3.618$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$3.61794870793$$ 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

 $$f(q)$$ $$=$$ $$q - 17q^{3} + 16q^{4} + 25q^{5} + 49q^{7} + 208q^{9} + O(q^{10})$$ $$q - 17q^{3} + 16q^{4} + 25q^{5} + 49q^{7} + 208q^{9} - 73q^{11} - 272q^{12} + 23q^{13} - 425q^{15} + 256q^{16} + 263q^{17} + 400q^{20} - 833q^{21} + 625q^{25} - 2159q^{27} + 784q^{28} - 1153q^{29} + 1241q^{33} + 1225q^{35} + 3328q^{36} - 391q^{39} - 1168q^{44} + 5200q^{45} - 3457q^{47} - 4352q^{48} + 2401q^{49} - 4471q^{51} + 368q^{52} - 1825q^{55} - 6800q^{60} + 10192q^{63} + 4096q^{64} + 575q^{65} + 4208q^{68} - 10078q^{71} - 9502q^{73} - 10625q^{75} - 3577q^{77} + 12167q^{79} + 6400q^{80} + 19855q^{81} - 6382q^{83} - 13328q^{84} + 6575q^{85} + 19601q^{87} + 1127q^{91} + 3383q^{97} - 15184q^{99} + O(q^{100})$$

## 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 −17.0000 16.0000 25.0000 0 49.0000 0 208.000 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 35.5.c.a 1
3.b odd 2 1 315.5.e.a 1
4.b odd 2 1 560.5.p.b 1
5.b even 2 1 35.5.c.b yes 1
5.c odd 4 2 175.5.d.c 2
7.b odd 2 1 35.5.c.b yes 1
15.d odd 2 1 315.5.e.b 1
20.d odd 2 1 560.5.p.a 1
21.c even 2 1 315.5.e.b 1
28.d even 2 1 560.5.p.a 1
35.c odd 2 1 CM 35.5.c.a 1
35.f even 4 2 175.5.d.c 2
105.g even 2 1 315.5.e.a 1
140.c even 2 1 560.5.p.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 1.a even 1 1 trivial
35.5.c.a 1 35.c odd 2 1 CM
35.5.c.b yes 1 5.b even 2 1
35.5.c.b yes 1 7.b odd 2 1
175.5.d.c 2 5.c odd 4 2
175.5.d.c 2 35.f even 4 2
315.5.e.a 1 3.b odd 2 1
315.5.e.a 1 105.g even 2 1
315.5.e.b 1 15.d odd 2 1
315.5.e.b 1 21.c even 2 1
560.5.p.a 1 20.d odd 2 1
560.5.p.a 1 28.d even 2 1
560.5.p.b 1 4.b odd 2 1
560.5.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 $$S_{5}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} + 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$17 + T$$
$5$ $$-25 + T$$
$7$ $$-49 + T$$
$11$ $$73 + T$$
$13$ $$-23 + T$$
$17$ $$-263 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$1153 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$3457 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$10078 + T$$
$73$ $$9502 + T$$
$79$ $$-12167 + T$$
$83$ $$6382 + T$$
$89$ $$T$$
$97$ $$-3383 + T$$