# Properties

 Label 35.9.c.b Level $35$ Weight $9$ Character orbit 35.c Self dual yes Analytic conductor $14.258$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.2582513521$$ 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

 $$f(q)$$ $$=$$ $$q + 127q^{3} + 256q^{4} + 625q^{5} + 2401q^{7} + 9568q^{9} + O(q^{10})$$ $$q + 127q^{3} + 256q^{4} + 625q^{5} + 2401q^{7} + 9568q^{9} - 23953q^{11} + 32512q^{12} - 56593q^{13} + 79375q^{15} + 65536q^{16} - 97873q^{17} + 160000q^{20} + 304927q^{21} + 390625q^{25} + 381889q^{27} + 614656q^{28} - 85153q^{29} - 3042031q^{33} + 1500625q^{35} + 2449408q^{36} - 7187311q^{39} - 6131968q^{44} + 5980000q^{45} + 2191487q^{47} + 8323072q^{48} + 5764801q^{49} - 12429871q^{51} - 14487808q^{52} - 14970625q^{55} + 20320000q^{60} + 22972768q^{63} + 16777216q^{64} - 35370625q^{65} - 25055488q^{68} + 50742722q^{71} + 33491522q^{73} + 49609375q^{75} - 57511153q^{77} + 70135727q^{79} + 40960000q^{80} - 14275745q^{81} - 54186718q^{83} + 78061312q^{84} - 61170625q^{85} - 10814431q^{87} - 135879793q^{91} - 165613873q^{97} - 229182304q^{99} + O(q^{100})$$

## 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
0 127.000 256.000 625.000 0 2401.00 0 9568.00 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 35.9.c.b yes 1
5.b even 2 1 35.9.c.a 1
5.c odd 4 2 175.9.d.c 2
7.b odd 2 1 35.9.c.a 1
35.c odd 2 1 CM 35.9.c.b yes 1
35.f even 4 2 175.9.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.9.c.a 1 5.b even 2 1
35.9.c.a 1 7.b odd 2 1
35.9.c.b yes 1 1.a even 1 1 trivial
35.9.c.b yes 1 35.c odd 2 1 CM
175.9.d.c 2 5.c odd 4 2
175.9.d.c 2 35.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} - 127$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-127 + T$$
$5$ $$-625 + T$$
$7$ $$-2401 + T$$
$11$ $$23953 + T$$
$13$ $$56593 + T$$
$17$ $$97873 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$85153 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$-2191487 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$-50742722 + T$$
$73$ $$-33491522 + T$$
$79$ $$-70135727 + T$$
$83$ $$54186718 + T$$
$89$ $$T$$
$97$ $$165613873 + T$$