# Properties

 Label 735.2.q.a Level $735$ Weight $2$ Character orbit 735.q Analytic conductor $5.869$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} - q^{6} -3 \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} - q^{6} -3 \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{10} + 6 \zeta_{12}^{2} q^{11} + \zeta_{12} q^{12} + 2 \zeta_{12}^{3} q^{13} + ( 2 - \zeta_{12}^{3} ) q^{15} + ( 1 - \zeta_{12}^{2} ) q^{16} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( -6 + 6 \zeta_{12}^{2} ) q^{19} + ( 1 + 2 \zeta_{12}^{3} ) q^{20} + 6 \zeta_{12}^{3} q^{22} + 3 \zeta_{12}^{2} q^{24} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + ( -2 + 2 \zeta_{12}^{2} ) q^{26} + \zeta_{12}^{3} q^{27} + 2 q^{29} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{30} + 10 \zeta_{12}^{2} q^{31} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{32} -6 \zeta_{12} q^{33} + 4 q^{34} - q^{36} + 4 \zeta_{12} q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{38} -2 \zeta_{12}^{2} q^{39} + ( -6 + 3 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{40} + 2 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( 6 - 6 \zeta_{12}^{2} ) q^{44} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{45} + \zeta_{12}^{3} q^{48} + ( 4 + 3 \zeta_{12}^{3} ) q^{50} + ( -4 + 4 \zeta_{12}^{2} ) q^{51} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{52} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( -1 + \zeta_{12}^{2} ) q^{54} + ( -6 - 12 \zeta_{12}^{3} ) q^{55} -6 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} -8 \zeta_{12}^{2} q^{59} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{60} + ( 2 - 2 \zeta_{12}^{2} ) q^{61} + 10 \zeta_{12}^{3} q^{62} -7 q^{64} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{65} -6 \zeta_{12}^{2} q^{66} + ( -16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{67} -4 \zeta_{12} q^{68} + 10 q^{71} -3 \zeta_{12} q^{72} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + 4 \zeta_{12}^{2} q^{74} + ( -4 - 3 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{75} + 6 q^{76} -2 \zeta_{12}^{3} q^{78} + ( 4 - 4 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + 2 \zeta_{12} q^{82} -8 \zeta_{12}^{3} q^{83} + ( -8 + 4 \zeta_{12}^{3} ) q^{85} + ( -4 + 4 \zeta_{12}^{2} ) q^{86} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 18 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{88} + ( 6 - 6 \zeta_{12}^{2} ) q^{89} + ( -2 + \zeta_{12}^{3} ) q^{90} -10 \zeta_{12} q^{93} + ( 12 \zeta_{12} - 6 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{95} + ( 5 - 5 \zeta_{12}^{2} ) q^{96} -2 \zeta_{12}^{3} q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{9} - 4 q^{10} + 12 q^{11} + 8 q^{15} + 2 q^{16} - 12 q^{19} + 4 q^{20} + 6 q^{24} + 6 q^{25} - 4 q^{26} + 8 q^{29} + 2 q^{30} + 20 q^{31} + 16 q^{34} - 4 q^{36} - 4 q^{39} - 12 q^{40} + 8 q^{41} + 12 q^{44} + 2 q^{45} + 16 q^{50} - 8 q^{51} - 2 q^{54} - 24 q^{55} - 16 q^{59} - 4 q^{60} + 4 q^{61} - 28 q^{64} + 8 q^{65} - 12 q^{66} + 40 q^{71} + 8 q^{74} - 8 q^{75} + 24 q^{76} + 8 q^{79} + 2 q^{80} - 2 q^{81} - 32 q^{85} - 8 q^{86} + 12 q^{89} - 8 q^{90} - 12 q^{95} + 10 q^{96} + 24 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 0.500000i −0.500000 0.866025i 1.23205 + 1.86603i −1.00000 0 3.00000i 0.500000 0.866025i −0.133975 2.23205i
79.2 0.866025 + 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i −2.23205 0.133975i −1.00000 0 3.00000i 0.500000 0.866025i −1.86603 1.23205i
214.1 −0.866025 + 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.23205 1.86603i −1.00000 0 3.00000i 0.500000 + 0.866025i −0.133975 + 2.23205i
214.2 0.866025 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i −2.23205 + 0.133975i −1.00000 0 3.00000i 0.500000 + 0.866025i −1.86603 + 1.23205i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.q.a 4
5.b even 2 1 inner 735.2.q.a 4
7.b odd 2 1 735.2.q.b 4
7.c even 3 1 105.2.d.a 2
7.c even 3 1 inner 735.2.q.a 4
7.d odd 6 1 735.2.d.a 2
7.d odd 6 1 735.2.q.b 4
21.g even 6 1 2205.2.d.f 2
21.h odd 6 1 315.2.d.c 2
28.g odd 6 1 1680.2.t.f 2
35.c odd 2 1 735.2.q.b 4
35.i odd 6 1 735.2.d.a 2
35.i odd 6 1 735.2.q.b 4
35.j even 6 1 105.2.d.a 2
35.j even 6 1 inner 735.2.q.a 4
35.k even 12 1 3675.2.a.d 1
35.k even 12 1 3675.2.a.l 1
35.l odd 12 1 525.2.a.b 1
35.l odd 12 1 525.2.a.c 1
84.n even 6 1 5040.2.t.e 2
105.o odd 6 1 315.2.d.c 2
105.p even 6 1 2205.2.d.f 2
105.x even 12 1 1575.2.a.e 1
105.x even 12 1 1575.2.a.i 1
140.p odd 6 1 1680.2.t.f 2
140.w even 12 1 8400.2.a.bj 1
140.w even 12 1 8400.2.a.ch 1
420.ba even 6 1 5040.2.t.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.c even 3 1
105.2.d.a 2 35.j even 6 1
315.2.d.c 2 21.h odd 6 1
315.2.d.c 2 105.o odd 6 1
525.2.a.b 1 35.l odd 12 1
525.2.a.c 1 35.l odd 12 1
735.2.d.a 2 7.d odd 6 1
735.2.d.a 2 35.i odd 6 1
735.2.q.a 4 1.a even 1 1 trivial
735.2.q.a 4 5.b even 2 1 inner
735.2.q.a 4 7.c even 3 1 inner
735.2.q.a 4 35.j even 6 1 inner
735.2.q.b 4 7.b odd 2 1
735.2.q.b 4 7.d odd 6 1
735.2.q.b 4 35.c odd 2 1
735.2.q.b 4 35.i odd 6 1
1575.2.a.e 1 105.x even 12 1
1575.2.a.i 1 105.x even 12 1
1680.2.t.f 2 28.g odd 6 1
1680.2.t.f 2 140.p odd 6 1
2205.2.d.f 2 21.g even 6 1
2205.2.d.f 2 105.p even 6 1
3675.2.a.d 1 35.k even 12 1
3675.2.a.l 1 35.k even 12 1
5040.2.t.e 2 84.n even 6 1
5040.2.t.e 2 420.ba even 6 1
8400.2.a.bj 1 140.w even 12 1
8400.2.a.ch 1 140.w even 12 1

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{2} + 1$$ $$T_{19}^{2} + 6 T_{19} + 36$$ $$T_{73}^{4} - 36 T_{73}^{2} + 1296$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 36 - 6 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 36 + 6 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$( 100 - 10 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 64 + 8 T + T^{2} )^{2}$$
$61$ $$( 4 - 2 T + T^{2} )^{2}$$
$67$ $$65536 - 256 T^{2} + T^{4}$$
$71$ $$( -10 + T )^{4}$$
$73$ $$1296 - 36 T^{2} + T^{4}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( 64 + T^{2} )^{2}$$
$89$ $$( 36 - 6 T + T^{2} )^{2}$$
$97$ $$( 4 + T^{2} )^{2}$$