# Properties

 Label 735.2.d.a Level $735$ Weight $2$ Character orbit 735.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} -i q^{3} + q^{4} + ( -1 + 2 i ) q^{5} + q^{6} + 3 i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} -i q^{3} + q^{4} + ( -1 + 2 i ) q^{5} + q^{6} + 3 i q^{8} - q^{9} + ( -2 - i ) q^{10} -6 q^{11} -i q^{12} + 2 i q^{13} + ( 2 + i ) q^{15} - q^{16} + 4 i q^{17} -i q^{18} -6 q^{19} + ( -1 + 2 i ) q^{20} -6 i q^{22} + 3 q^{24} + ( -3 - 4 i ) q^{25} -2 q^{26} + i q^{27} + 2 q^{29} + ( -1 + 2 i ) q^{30} + 10 q^{31} + 5 i q^{32} + 6 i q^{33} -4 q^{34} - q^{36} + 4 i q^{37} -6 i q^{38} + 2 q^{39} + ( -6 - 3 i ) q^{40} -2 q^{41} -4 i q^{43} -6 q^{44} + ( 1 - 2 i ) q^{45} + i q^{48} + ( 4 - 3 i ) q^{50} + 4 q^{51} + 2 i q^{52} + 6 i q^{53} - q^{54} + ( 6 - 12 i ) q^{55} + 6 i q^{57} + 2 i q^{58} -8 q^{59} + ( 2 + i ) q^{60} + 2 q^{61} + 10 i q^{62} -7 q^{64} + ( -4 - 2 i ) q^{65} -6 q^{66} + 16 i q^{67} + 4 i q^{68} + 10 q^{71} -3 i q^{72} + 6 i q^{73} -4 q^{74} + ( -4 + 3 i ) q^{75} -6 q^{76} + 2 i q^{78} -4 q^{79} + ( 1 - 2 i ) q^{80} + q^{81} -2 i q^{82} -8 i q^{83} + ( -8 - 4 i ) q^{85} + 4 q^{86} -2 i q^{87} -18 i q^{88} + 6 q^{89} + ( 2 + i ) q^{90} -10 i q^{93} + ( 6 - 12 i ) q^{95} + 5 q^{96} -2 i q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9} - 4 q^{10} - 12 q^{11} + 4 q^{15} - 2 q^{16} - 12 q^{19} - 2 q^{20} + 6 q^{24} - 6 q^{25} - 4 q^{26} + 4 q^{29} - 2 q^{30} + 20 q^{31} - 8 q^{34} - 2 q^{36} + 4 q^{39} - 12 q^{40} - 4 q^{41} - 12 q^{44} + 2 q^{45} + 8 q^{50} + 8 q^{51} - 2 q^{54} + 12 q^{55} - 16 q^{59} + 4 q^{60} + 4 q^{61} - 14 q^{64} - 8 q^{65} - 12 q^{66} + 20 q^{71} - 8 q^{74} - 8 q^{75} - 12 q^{76} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 16 q^{85} + 8 q^{86} + 12 q^{89} + 4 q^{90} + 12 q^{95} + 10 q^{96} + 12 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 −1.00000 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 + 1.00000i
589.2 1.00000i 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 0 3.00000i −1.00000 −2.00000 1.00000i
 $$n$$: 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

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.b odd 2 1
105.2.d.a 2 35.c odd 2 1
315.2.d.c 2 21.c even 2 1
315.2.d.c 2 105.g even 2 1
525.2.a.b 1 35.f even 4 1
525.2.a.c 1 35.f even 4 1
735.2.d.a 2 1.a even 1 1 trivial
735.2.d.a 2 5.b even 2 1 inner
735.2.q.a 4 7.d odd 6 2
735.2.q.a 4 35.i odd 6 2
735.2.q.b 4 7.c even 3 2
735.2.q.b 4 35.j even 6 2
1575.2.a.e 1 105.k odd 4 1
1575.2.a.i 1 105.k odd 4 1
1680.2.t.f 2 28.d even 2 1
1680.2.t.f 2 140.c even 2 1
2205.2.d.f 2 3.b odd 2 1
2205.2.d.f 2 15.d odd 2 1
3675.2.a.d 1 5.c odd 4 1
3675.2.a.l 1 5.c odd 4 1
5040.2.t.e 2 84.h odd 2 1
5040.2.t.e 2 420.o odd 2 1
8400.2.a.bj 1 140.j odd 4 1
8400.2.a.ch 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{19} + 6$$

## Hecke characteristic polynomials

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