# Properties

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

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## 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 15) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} + 3 q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} - q^{6} + 3 q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 2 q^{13} - q^{15} + \zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} + 4 \zeta_{6} q^{19} + q^{20} + 4 q^{22} + ( -3 + 3 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} + q^{27} -2 q^{29} -\zeta_{6} q^{30} + ( 5 - 5 \zeta_{6} ) q^{32} + 4 \zeta_{6} q^{33} + 2 q^{34} - q^{36} + 10 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + ( -2 + 2 \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{40} -10 q^{41} + 4 q^{43} -4 \zeta_{6} q^{44} + ( 1 - \zeta_{6} ) q^{45} + 8 \zeta_{6} q^{47} - q^{48} - q^{50} + 2 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 10 - 10 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + 4 q^{55} -4 q^{57} -2 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( -1 + \zeta_{6} ) q^{60} -2 \zeta_{6} q^{61} + 7 q^{64} + 2 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{66} + ( -12 + 12 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} -8 q^{71} -3 \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( -10 + 10 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} + 4 q^{76} -2 q^{78} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -10 \zeta_{6} q^{82} -12 q^{83} + 2 q^{85} + 4 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{87} + ( 12 - 12 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + q^{90} + ( -8 + 8 \zeta_{6} ) q^{94} + ( -4 + 4 \zeta_{6} ) q^{95} + 5 \zeta_{6} q^{96} -2 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} + q^{4} + q^{5} - 2q^{6} + 6q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} + q^{4} + q^{5} - 2q^{6} + 6q^{8} - q^{9} - q^{10} + 4q^{11} + q^{12} + 4q^{13} - 2q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + 2q^{20} + 8q^{22} - 3q^{24} - q^{25} + 2q^{26} + 2q^{27} - 4q^{29} - q^{30} + 5q^{32} + 4q^{33} + 4q^{34} - 2q^{36} + 10q^{37} - 4q^{38} - 2q^{39} + 3q^{40} - 20q^{41} + 8q^{43} - 4q^{44} + q^{45} + 8q^{47} - 2q^{48} - 2q^{50} + 2q^{51} + 2q^{52} + 10q^{53} + q^{54} + 8q^{55} - 8q^{57} - 2q^{58} - 4q^{59} - q^{60} - 2q^{61} + 14q^{64} + 2q^{65} - 4q^{66} - 12q^{67} - 2q^{68} - 16q^{71} - 3q^{72} + 10q^{73} - 10q^{74} - q^{75} + 8q^{76} - 4q^{78} - q^{80} - q^{81} - 10q^{82} - 24q^{83} + 4q^{85} + 4q^{86} + 2q^{87} + 12q^{88} - 6q^{89} + 2q^{90} - 8q^{94} - 4q^{95} + 5q^{96} - 4q^{97} - 8q^{99} + O(q^{100})$$

## Character values

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

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

## 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.d 2
7.b odd 2 1 735.2.i.e 2
7.c even 3 1 735.2.a.c 1
7.c even 3 1 inner 735.2.i.d 2
7.d odd 6 1 15.2.a.a 1
7.d odd 6 1 735.2.i.e 2
21.g even 6 1 45.2.a.a 1
21.h odd 6 1 2205.2.a.i 1
28.f even 6 1 240.2.a.d 1
35.i odd 6 1 75.2.a.b 1
35.j even 6 1 3675.2.a.j 1
35.k even 12 2 75.2.b.b 2
56.j odd 6 1 960.2.a.l 1
56.m even 6 1 960.2.a.a 1
63.i even 6 1 405.2.e.c 2
63.k odd 6 1 405.2.e.f 2
63.s even 6 1 405.2.e.c 2
63.t odd 6 1 405.2.e.f 2
77.i even 6 1 1815.2.a.d 1
84.j odd 6 1 720.2.a.c 1
91.s odd 6 1 2535.2.a.j 1
105.p even 6 1 225.2.a.b 1
105.w odd 12 2 225.2.b.b 2
112.v even 12 2 3840.2.k.r 2
112.x odd 12 2 3840.2.k.m 2
119.h odd 6 1 4335.2.a.c 1
133.o even 6 1 5415.2.a.j 1
140.s even 6 1 1200.2.a.e 1
140.x odd 12 2 1200.2.f.h 2
161.g even 6 1 7935.2.a.d 1
168.ba even 6 1 2880.2.a.y 1
168.be odd 6 1 2880.2.a.bc 1
231.k odd 6 1 5445.2.a.c 1
273.ba even 6 1 7605.2.a.g 1
280.ba even 6 1 4800.2.a.bz 1
280.bk odd 6 1 4800.2.a.t 1
280.bp odd 12 2 4800.2.f.c 2
280.bv even 12 2 4800.2.f.bf 2
385.o even 6 1 9075.2.a.g 1
420.be odd 6 1 3600.2.a.u 1
420.br even 12 2 3600.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 7.d odd 6 1
45.2.a.a 1 21.g even 6 1
75.2.a.b 1 35.i odd 6 1
75.2.b.b 2 35.k even 12 2
225.2.a.b 1 105.p even 6 1
225.2.b.b 2 105.w odd 12 2
240.2.a.d 1 28.f even 6 1
405.2.e.c 2 63.i even 6 1
405.2.e.c 2 63.s even 6 1
405.2.e.f 2 63.k odd 6 1
405.2.e.f 2 63.t odd 6 1
720.2.a.c 1 84.j odd 6 1
735.2.a.c 1 7.c even 3 1
735.2.i.d 2 1.a even 1 1 trivial
735.2.i.d 2 7.c even 3 1 inner
735.2.i.e 2 7.b odd 2 1
735.2.i.e 2 7.d odd 6 1
960.2.a.a 1 56.m even 6 1
960.2.a.l 1 56.j odd 6 1
1200.2.a.e 1 140.s even 6 1
1200.2.f.h 2 140.x odd 12 2
1815.2.a.d 1 77.i even 6 1
2205.2.a.i 1 21.h odd 6 1
2535.2.a.j 1 91.s odd 6 1
2880.2.a.y 1 168.ba even 6 1
2880.2.a.bc 1 168.be odd 6 1
3600.2.a.u 1 420.be odd 6 1
3600.2.f.e 2 420.br even 12 2
3675.2.a.j 1 35.j even 6 1
3840.2.k.m 2 112.x odd 12 2
3840.2.k.r 2 112.v even 12 2
4335.2.a.c 1 119.h odd 6 1
4800.2.a.t 1 280.bk odd 6 1
4800.2.a.bz 1 280.ba even 6 1
4800.2.f.c 2 280.bp odd 12 2
4800.2.f.bf 2 280.bv even 12 2
5415.2.a.j 1 133.o even 6 1
5445.2.a.c 1 231.k odd 6 1
7605.2.a.g 1 273.ba even 6 1
7935.2.a.d 1 161.g even 6 1
9075.2.a.g 1 385.o even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$100 - 10 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$
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