# Properties

 Label 735.2.s.a Level $735$ Weight $2$ Character orbit 735.s Analytic conductor $5.869$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{6} ) q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 3 q^{6} + ( 1 - 2 \zeta_{6} ) q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{6} ) q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 3 q^{6} + ( 1 - 2 \zeta_{6} ) q^{8} + 3 \zeta_{6} q^{9} + ( 1 + \zeta_{6} ) q^{10} + ( -2 - 2 \zeta_{6} ) q^{11} + ( -2 + \zeta_{6} ) q^{12} + ( -1 + 2 \zeta_{6} ) q^{15} + 5 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -3 - 3 \zeta_{6} ) q^{18} + ( 4 - 2 \zeta_{6} ) q^{19} - q^{20} + 6 q^{22} + ( -4 + 2 \zeta_{6} ) q^{23} + ( -3 + 3 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -4 + 8 \zeta_{6} ) q^{29} -3 \zeta_{6} q^{30} + ( 2 + 2 \zeta_{6} ) q^{31} + ( -3 - 3 \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{33} + ( -6 + 12 \zeta_{6} ) q^{34} + 3 q^{36} + 2 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + ( -2 + \zeta_{6} ) q^{40} -6 q^{41} -8 q^{43} + ( -4 + 2 \zeta_{6} ) q^{44} + ( 3 - 3 \zeta_{6} ) q^{45} + ( 6 - 6 \zeta_{6} ) q^{46} -12 \zeta_{6} q^{47} + ( 5 - 10 \zeta_{6} ) q^{48} + ( 1 - 2 \zeta_{6} ) q^{50} + ( -12 + 6 \zeta_{6} ) q^{51} + 9 \zeta_{6} q^{54} + ( -2 + 4 \zeta_{6} ) q^{55} -6 q^{57} -12 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + ( 1 + \zeta_{6} ) q^{60} + ( -8 + 4 \zeta_{6} ) q^{61} -6 q^{62} - q^{64} + ( -6 - 6 \zeta_{6} ) q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + 6 q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( 6 - 3 \zeta_{6} ) q^{72} + ( -4 - 4 \zeta_{6} ) q^{73} + ( -2 - 2 \zeta_{6} ) q^{74} + ( 2 - \zeta_{6} ) q^{75} + ( 2 - 4 \zeta_{6} ) q^{76} -8 \zeta_{6} q^{79} + ( 5 - 5 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 6 \zeta_{6} ) q^{82} -6 q^{85} + ( 16 - 8 \zeta_{6} ) q^{86} + ( 12 - 12 \zeta_{6} ) q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + ( -3 + 6 \zeta_{6} ) q^{90} + ( -2 + 4 \zeta_{6} ) q^{92} -6 \zeta_{6} q^{93} + ( 12 + 12 \zeta_{6} ) q^{94} + ( -2 - 2 \zeta_{6} ) q^{95} + 9 \zeta_{6} q^{96} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 6 - 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 3q^{3} + q^{4} - q^{5} + 6q^{6} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 3q^{3} + q^{4} - q^{5} + 6q^{6} + 3q^{9} + 3q^{10} - 6q^{11} - 3q^{12} + 5q^{16} + 6q^{17} - 9q^{18} + 6q^{19} - 2q^{20} + 12q^{22} - 6q^{23} - 3q^{24} - q^{25} - 3q^{30} + 6q^{31} - 9q^{32} + 6q^{33} + 6q^{36} + 2q^{37} - 6q^{38} - 3q^{40} - 12q^{41} - 16q^{43} - 6q^{44} + 3q^{45} + 6q^{46} - 12q^{47} - 18q^{51} + 9q^{54} - 12q^{57} - 12q^{58} - 12q^{59} + 3q^{60} - 12q^{61} - 12q^{62} - 2q^{64} - 18q^{66} - 8q^{67} - 6q^{68} + 12q^{69} + 9q^{72} - 12q^{73} - 6q^{74} + 3q^{75} - 8q^{79} + 5q^{80} - 9q^{81} + 18q^{82} - 12q^{85} + 24q^{86} + 12q^{87} - 6q^{88} + 6q^{89} - 6q^{93} + 36q^{94} - 6q^{95} + 9q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
521.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 0.866025i −1.50000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 3.00000 0 1.73205i 1.50000 + 2.59808i 1.50000 + 0.866025i
656.1 −1.50000 0.866025i −1.50000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 3.00000 0 1.73205i 1.50000 2.59808i 1.50000 0.866025i
 $$n$$: 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
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.s.a 2
3.b odd 2 1 735.2.s.d 2
7.b odd 2 1 735.2.s.b 2
7.c even 3 1 105.2.b.b yes 2
7.c even 3 1 735.2.s.f 2
7.d odd 6 1 105.2.b.a 2
7.d odd 6 1 735.2.s.d 2
21.c even 2 1 735.2.s.f 2
21.g even 6 1 105.2.b.b yes 2
21.g even 6 1 inner 735.2.s.a 2
21.h odd 6 1 105.2.b.a 2
21.h odd 6 1 735.2.s.b 2
28.f even 6 1 1680.2.f.b 2
28.g odd 6 1 1680.2.f.c 2
35.i odd 6 1 525.2.b.a 2
35.j even 6 1 525.2.b.b 2
35.k even 12 2 525.2.g.b 4
35.l odd 12 2 525.2.g.c 4
84.j odd 6 1 1680.2.f.c 2
84.n even 6 1 1680.2.f.b 2
105.o odd 6 1 525.2.b.a 2
105.p even 6 1 525.2.b.b 2
105.w odd 12 2 525.2.g.c 4
105.x even 12 2 525.2.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 7.d odd 6 1
105.2.b.a 2 21.h odd 6 1
105.2.b.b yes 2 7.c even 3 1
105.2.b.b yes 2 21.g even 6 1
525.2.b.a 2 35.i odd 6 1
525.2.b.a 2 105.o odd 6 1
525.2.b.b 2 35.j even 6 1
525.2.b.b 2 105.p even 6 1
525.2.g.b 4 35.k even 12 2
525.2.g.b 4 105.x even 12 2
525.2.g.c 4 35.l odd 12 2
525.2.g.c 4 105.w odd 12 2
735.2.s.a 2 1.a even 1 1 trivial
735.2.s.a 2 21.g even 6 1 inner
735.2.s.b 2 7.b odd 2 1
735.2.s.b 2 21.h odd 6 1
735.2.s.d 2 3.b odd 2 1
735.2.s.d 2 7.d odd 6 1
735.2.s.f 2 7.c even 3 1
735.2.s.f 2 21.c even 2 1
1680.2.f.b 2 28.f even 6 1
1680.2.f.b 2 84.n even 6 1
1680.2.f.c 2 28.g odd 6 1
1680.2.f.c 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{2} + 3 T_{2} + 3$$ $$T_{13}$$ $$T_{17}^{2} - 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$12 + 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$12 - 6 T + T^{2}$$
$23$ $$12 + 6 T + T^{2}$$
$29$ $$48 + T^{2}$$
$31$ $$12 - 6 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$48 + 12 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$48 + 12 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$48 + T^{2}$$