Properties

 Label 35.15.c.a Level $35$ Weight $15$ Character orbit 35.c Self dual yes Analytic conductor $43.515$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: yes Analytic conductor: $$43.5151388532$$ 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

 $$f(q)$$ $$=$$ $$q - 4031q^{3} + 16384q^{4} - 78125q^{5} + 823543q^{7} + 11465992q^{9} + O(q^{10})$$ $$q - 4031q^{3} + 16384q^{4} - 78125q^{5} + 823543q^{7} + 11465992q^{9} - 37379173q^{11} - 66043904q^{12} - 65279611q^{13} + 314921875q^{15} + 268435456q^{16} - 626193259q^{17} - 1280000000q^{20} - 3319701833q^{21} + 6103515625q^{25} - 26939265713q^{27} + 13492928512q^{28} - 9775649497q^{29} + 150675446363q^{33} - 64339296875q^{35} + 187858812928q^{36} + 263142111941q^{39} - 612420370432q^{44} - 895780625000q^{45} + 719081600801q^{47} - 1082063323136q^{48} + 678223072849q^{49} + 2524185027029q^{51} - 1069541146624q^{52} + 2920247890625q^{55} + 5159680000000q^{60} + 9442737449656q^{63} + 4398046511104q^{64} + 5099969609375q^{65} - 10259550355456q^{68} - 1790558995678q^{71} + 22033597628414q^{73} - 24603271484375q^{75} - 30783356269939q^{77} - 27088287440917q^{79} - 20971520000000q^{80} + 53750695798855q^{81} + 33726754263974q^{83} - 54389994831872q^{84} + 48921348359375q^{85} + 39405643122407q^{87} - 53760566681773q^{91} + 24587561871581q^{97} - 428589298584616q^{99} + O(q^{100})$$

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 −4031.00 16384.0 −78125.0 0 823543. 0 1.14660e7 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 35.15.c.a 1
5.b even 2 1 35.15.c.b yes 1
7.b odd 2 1 35.15.c.b yes 1
35.c odd 2 1 CM 35.15.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.15.c.a 1 1.a even 1 1 trivial
35.15.c.a 1 35.c odd 2 1 CM
35.15.c.b yes 1 5.b even 2 1
35.15.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 $$S_{15}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} + 4031$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$4031 + T$$
$5$ $$78125 + T$$
$7$ $$-823543 + T$$
$11$ $$37379173 + T$$
$13$ $$65279611 + T$$
$17$ $$626193259 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$9775649497 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$-719081600801 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$1790558995678 + T$$
$73$ $$-22033597628414 + T$$
$79$ $$27088287440917 + T$$
$83$ $$-33726754263974 + T$$
$89$ $$T$$
$97$ $$-24587561871581 + T$$