# Properties

 Label 735.2.i.b Level $735$ Weight $2$ Character orbit 735.i Analytic conductor $5.869$ Analytic rank $0$ 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} -3 q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} -3 q^{8} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} -\zeta_{6} q^{12} + 6 q^{13} + q^{15} + \zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -8 \zeta_{6} q^{19} + q^{20} -8 \zeta_{6} q^{23} + ( -3 + 3 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -6 \zeta_{6} q^{26} - q^{27} -2 q^{29} -\zeta_{6} q^{30} + ( 4 - 4 \zeta_{6} ) q^{31} + ( -5 + 5 \zeta_{6} ) q^{32} -2 q^{34} - q^{36} + 2 \zeta_{6} q^{37} + ( -8 + 8 \zeta_{6} ) q^{38} + ( 6 - 6 \zeta_{6} ) q^{39} -3 \zeta_{6} q^{40} + 6 q^{41} + 4 q^{43} + ( 1 - \zeta_{6} ) q^{45} + ( -8 + 8 \zeta_{6} ) q^{46} + 8 \zeta_{6} q^{47} + q^{48} + q^{50} -2 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + ( -10 + 10 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} -8 q^{57} + 2 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} -2 \zeta_{6} q^{61} -4 q^{62} + 7 q^{64} + 6 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} -8 q^{69} -12 q^{71} + 3 \zeta_{6} q^{72} + ( -2 + 2 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} -8 q^{76} -6 q^{78} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 \zeta_{6} q^{82} + 4 q^{83} + 2 q^{85} -4 \zeta_{6} q^{86} + ( -2 + 2 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} - q^{90} -8 q^{92} -4 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{94} + ( 8 - 8 \zeta_{6} ) q^{95} + 5 \zeta_{6} q^{96} + 18 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} + q^{4} + q^{5} - 2q^{6} - 6q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} + q^{4} + q^{5} - 2q^{6} - 6q^{8} - q^{9} + q^{10} - q^{12} + 12q^{13} + 2q^{15} + q^{16} + 2q^{17} - q^{18} - 8q^{19} + 2q^{20} - 8q^{23} - 3q^{24} - q^{25} - 6q^{26} - 2q^{27} - 4q^{29} - q^{30} + 4q^{31} - 5q^{32} - 4q^{34} - 2q^{36} + 2q^{37} - 8q^{38} + 6q^{39} - 3q^{40} + 12q^{41} + 8q^{43} + q^{45} - 8q^{46} + 8q^{47} + 2q^{48} + 2q^{50} - 2q^{51} + 6q^{52} - 10q^{53} + q^{54} - 16q^{57} + 2q^{58} + 4q^{59} + q^{60} - 2q^{61} - 8q^{62} + 14q^{64} + 6q^{65} - 4q^{67} - 2q^{68} - 16q^{69} - 24q^{71} + 3q^{72} - 2q^{73} + 2q^{74} + q^{75} - 16q^{76} - 12q^{78} - 8q^{79} - q^{80} - q^{81} - 6q^{82} + 8q^{83} + 4q^{85} - 4q^{86} - 2q^{87} - 6q^{89} - 2q^{90} - 16q^{92} - 4q^{93} + 8q^{94} + 8q^{95} + 5q^{96} + 36q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 0 −3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
361.1 −0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 0 −3.00000 −0.500000 0.866025i 0.500000 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.i.b 2
7.b odd 2 1 735.2.i.a 2
7.c even 3 1 735.2.a.f 1
7.c even 3 1 inner 735.2.i.b 2
7.d odd 6 1 105.2.a.a 1
7.d odd 6 1 735.2.i.a 2
21.g even 6 1 315.2.a.a 1
21.h odd 6 1 2205.2.a.b 1
28.f even 6 1 1680.2.a.f 1
35.i odd 6 1 525.2.a.a 1
35.j even 6 1 3675.2.a.f 1
35.k even 12 2 525.2.d.b 2
56.j odd 6 1 6720.2.a.p 1
56.m even 6 1 6720.2.a.bk 1
84.j odd 6 1 5040.2.a.d 1
105.p even 6 1 1575.2.a.h 1
105.w odd 12 2 1575.2.d.b 2
140.s even 6 1 8400.2.a.co 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 7.d odd 6 1
315.2.a.a 1 21.g even 6 1
525.2.a.a 1 35.i odd 6 1
525.2.d.b 2 35.k even 12 2
735.2.a.f 1 7.c even 3 1
735.2.i.a 2 7.b odd 2 1
735.2.i.a 2 7.d odd 6 1
735.2.i.b 2 1.a even 1 1 trivial
735.2.i.b 2 7.c even 3 1 inner
1575.2.a.h 1 105.p even 6 1
1575.2.d.b 2 105.w odd 12 2
1680.2.a.f 1 28.f even 6 1
2205.2.a.b 1 21.h odd 6 1
3675.2.a.f 1 35.j even 6 1
5040.2.a.d 1 84.j odd 6 1
6720.2.a.p 1 56.j odd 6 1
6720.2.a.bk 1 56.m even 6 1
8400.2.a.co 1 140.s even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{13} - 6$$ $$T_{17}^{2} - 2 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -6 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$100 + 10 T + T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -18 + T )^{2}$$