Properties

 Label 35.7.c.b Level $35$ Weight $7$ Character orbit 35.c Self dual yes Analytic conductor $8.052$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: yes Analytic conductor: $$8.05189292669$$ 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

 $$f(q)$$ $$=$$ $$q + 26q^{3} + 64q^{4} + 125q^{5} - 343q^{7} - 53q^{9} + O(q^{10})$$ $$q + 26q^{3} + 64q^{4} + 125q^{5} - 343q^{7} - 53q^{9} + 2522q^{11} + 1664q^{12} - 2774q^{13} + 3250q^{15} + 4096q^{16} + 754q^{17} + 8000q^{20} - 8918q^{21} + 15625q^{25} - 20332q^{27} - 21952q^{28} - 45862q^{29} + 65572q^{33} - 42875q^{35} - 3392q^{36} - 72124q^{39} + 161408q^{44} - 6625q^{45} - 175646q^{47} + 106496q^{48} + 117649q^{49} + 19604q^{51} - 177536q^{52} + 315250q^{55} + 208000q^{60} + 18179q^{63} + 262144q^{64} - 346750q^{65} + 48256q^{68} - 30238q^{71} - 504254q^{73} + 406250q^{75} - 865046q^{77} - 930382q^{79} + 512000q^{80} - 489995q^{81} + 1141306q^{83} - 570752q^{84} + 94250q^{85} - 1192412q^{87} + 951482q^{91} + 897874q^{97} - 133666q^{99} + O(q^{100})$$

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 26.0000 64.0000 125.000 0 −343.000 0 −53.0000 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 35.7.c.b yes 1
5.b even 2 1 35.7.c.a 1
5.c odd 4 2 175.7.d.c 2
7.b odd 2 1 35.7.c.a 1
35.c odd 2 1 CM 35.7.c.b yes 1
35.f even 4 2 175.7.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.7.c.a 1 5.b even 2 1
35.7.c.a 1 7.b odd 2 1
35.7.c.b yes 1 1.a even 1 1 trivial
35.7.c.b yes 1 35.c odd 2 1 CM
175.7.d.c 2 5.c odd 4 2
175.7.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 $$S_{7}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}$$ $$T_{3} - 26$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-26 + T$$
$5$ $$-125 + T$$
$7$ $$343 + T$$
$11$ $$-2522 + T$$
$13$ $$2774 + T$$
$17$ $$-754 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$45862 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$175646 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$30238 + T$$
$73$ $$504254 + T$$
$79$ $$930382 + T$$
$83$ $$-1141306 + T$$
$89$ $$T$$
$97$ $$-897874 + T$$