# Properties

 Label 35.17.c.a Level $35$ Weight $17$ Character orbit 35.c Self dual yes Analytic conductor $56.814$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$17$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.8135903498$$ 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

 $$f(q)$$ $$=$$ $$q - 3007q^{3} + 65536q^{4} - 390625q^{5} - 5764801q^{7} - 34004672q^{9} + O(q^{10})$$ $$q - 3007q^{3} + 65536q^{4} - 390625q^{5} - 5764801q^{7} - 34004672q^{9} + 145028447q^{11} - 197066752q^{12} - 1571306207q^{13} + 1174609375q^{15} + 4294967296q^{16} + 4372390753q^{17} - 25600000000q^{20} + 17334756607q^{21} + 152587890625q^{25} + 231693538751q^{27} - 377801998336q^{28} - 993241792513q^{29} - 436100540129q^{33} + 2251875390625q^{35} - 2228530184192q^{36} + 4724917764449q^{39} + 9504584302592q^{44} + 13283075000000q^{45} + 42819958052353q^{47} - 12914966659072q^{48} + 33232930569601q^{49} - 13147778994271q^{51} - 102977123581952q^{52} - 56651737109375q^{55} + 76979200000000q^{60} + 196030167150272q^{63} + 281474976710656q^{64} + 613791487109375q^{65} + 286549000388608q^{68} + 1283316773477762q^{71} + 491238137911678q^{73} - 458831787109375q^{75} - 836060136294047q^{77} + 1884802582005407q^{79} - 1677721600000000q^{80} + 767087157256255q^{81} + 1568384056666558q^{83} + 1136050608996352q^{84} - 1707965137890625q^{85} + 2986678070086591q^{87} + 9058267593419807q^{91} - 11753087741306207q^{97} - 4931644770904384q^{99} + O(q^{100})$$

## 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 −3007.00 65536.0 −390625. 0 −5.76480e6 0 −3.40047e7 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 35.17.c.a 1
5.b even 2 1 35.17.c.b yes 1
7.b odd 2 1 35.17.c.b yes 1
35.c odd 2 1 CM 35.17.c.a 1

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{17}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} + 3007$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3007 + T$$
$5$ $$390625 + T$$
$7$ $$5764801 + T$$
$11$ $$-145028447 + T$$
$13$ $$1571306207 + T$$
$17$ $$-4372390753 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$993241792513 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$-42819958052353 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$-1283316773477762 + T$$
$73$ $$-491238137911678 + T$$
$79$ $$-1884802582005407 + T$$
$83$ $$-1568384056666558 + T$$
$89$ $$T$$
$97$ $$11753087741306207 + T$$