# Properties

 Label 735.2.i.k Level 735 Weight 2 Character orbit 735.i Analytic conductor 5.869 Analytic rank 0 Dimension 4 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} + 3 \beta_{1} q^{4} + ( 1 + \beta_{1} ) q^{5} + \beta_{3} q^{6} -\beta_{3} q^{8} + ( -1 - \beta_{1} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} + 3 \beta_{1} q^{4} + ( 1 + \beta_{1} ) q^{5} + \beta_{3} q^{6} -\beta_{3} q^{8} + ( -1 - \beta_{1} ) q^{9} + ( -\beta_{2} - \beta_{3} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 3 + 3 \beta_{1} ) q^{12} + 2 \beta_{3} q^{13} + q^{15} + ( 1 + \beta_{1} ) q^{16} -2 \beta_{1} q^{17} + ( \beta_{2} + \beta_{3} ) q^{18} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} -3 q^{20} + ( -10 - 2 \beta_{3} ) q^{22} + ( -4 - 4 \beta_{1} ) q^{23} + ( -\beta_{2} - \beta_{3} ) q^{24} + \beta_{1} q^{25} + ( 10 + 10 \beta_{1} ) q^{26} - q^{27} -2 q^{29} -\beta_{2} q^{30} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{32} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{33} + 2 \beta_{3} q^{34} + 3 q^{36} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{39} + \beta_{2} q^{40} -2 q^{41} + 4 \beta_{3} q^{43} + ( -6 - 6 \beta_{1} + 6 \beta_{2} ) q^{44} -\beta_{1} q^{45} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{46} + ( -4 - 4 \beta_{1} + 4 \beta_{2} ) q^{47} + q^{48} -\beta_{3} q^{50} + ( -2 - 2 \beta_{1} ) q^{51} + ( -6 \beta_{2} - 6 \beta_{3} ) q^{52} + ( -8 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + \beta_{2} q^{54} + ( -2 - 2 \beta_{3} ) q^{55} + ( -2 + 2 \beta_{3} ) q^{57} + 2 \beta_{2} q^{58} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{59} + 3 \beta_{1} q^{60} + ( 2 + 2 \beta_{1} ) q^{61} + ( 10 - 6 \beta_{3} ) q^{62} -13 q^{64} -2 \beta_{2} q^{65} + ( 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{66} -4 \beta_{1} q^{67} + ( 6 + 6 \beta_{1} ) q^{68} -4 q^{69} + ( 10 - 2 \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( -8 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( 20 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( 1 + \beta_{1} ) q^{75} + ( 6 - 6 \beta_{3} ) q^{76} + 10 q^{78} + ( -4 - 4 \beta_{1} + 4 \beta_{2} ) q^{79} + \beta_{1} q^{80} + \beta_{1} q^{81} + 2 \beta_{2} q^{82} + ( -8 - 4 \beta_{3} ) q^{83} + 2 q^{85} + ( 20 + 20 \beta_{1} ) q^{86} + 2 \beta_{1} q^{87} + ( -10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{88} + ( 2 + 2 \beta_{1} ) q^{89} + \beta_{3} q^{90} + 12 q^{92} + ( 6 + 6 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -20 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{95} -3 \beta_{2} q^{96} + ( 4 + 2 \beta_{3} ) q^{97} + ( 2 + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 6q^{4} + 2q^{5} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 6q^{4} + 2q^{5} - 2q^{9} - 4q^{11} + 6q^{12} + 4q^{15} + 2q^{16} + 4q^{17} - 4q^{19} - 12q^{20} - 40q^{22} - 8q^{23} - 2q^{25} + 20q^{26} - 4q^{27} - 8q^{29} - 12q^{31} + 4q^{33} + 12q^{36} - 4q^{37} - 20q^{38} - 8q^{41} - 12q^{44} + 2q^{45} - 8q^{47} + 4q^{48} - 4q^{51} + 16q^{53} - 8q^{55} - 8q^{57} - 6q^{60} + 4q^{61} + 40q^{62} - 52q^{64} - 20q^{66} + 8q^{67} + 12q^{68} - 16q^{69} + 40q^{71} + 16q^{73} - 40q^{74} + 2q^{75} + 24q^{76} + 40q^{78} - 8q^{79} - 2q^{80} - 2q^{81} - 32q^{83} + 8q^{85} + 40q^{86} - 4q^{87} + 20q^{88} + 4q^{89} + 48q^{92} + 12q^{93} + 40q^{94} + 4q^{95} + 16q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} + 6 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
−1.11803 + 1.93649i 0.500000 + 0.866025i −1.50000 2.59808i 0.500000 0.866025i −2.23607 0 2.23607 −0.500000 + 0.866025i 1.11803 + 1.93649i
226.2 1.11803 1.93649i 0.500000 + 0.866025i −1.50000 2.59808i 0.500000 0.866025i 2.23607 0 −2.23607 −0.500000 + 0.866025i −1.11803 1.93649i
361.1 −1.11803 1.93649i 0.500000 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i −2.23607 0 2.23607 −0.500000 0.866025i 1.11803 1.93649i
361.2 1.11803 + 1.93649i 0.500000 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i 2.23607 0 −2.23607 −0.500000 0.866025i −1.11803 + 1.93649i
 $$n$$: 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.k 4
7.b odd 2 1 735.2.i.i 4
7.c even 3 1 105.2.a.b 2
7.c even 3 1 inner 735.2.i.k 4
7.d odd 6 1 735.2.a.k 2
7.d odd 6 1 735.2.i.i 4
21.g even 6 1 2205.2.a.w 2
21.h odd 6 1 315.2.a.d 2
28.g odd 6 1 1680.2.a.v 2
35.i odd 6 1 3675.2.a.y 2
35.j even 6 1 525.2.a.g 2
35.l odd 12 2 525.2.d.c 4
56.k odd 6 1 6720.2.a.cs 2
56.p even 6 1 6720.2.a.cx 2
84.n even 6 1 5040.2.a.bw 2
105.o odd 6 1 1575.2.a.r 2
105.x even 12 2 1575.2.d.d 4
140.p odd 6 1 8400.2.a.cx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 7.c even 3 1
315.2.a.d 2 21.h odd 6 1
525.2.a.g 2 35.j even 6 1
525.2.d.c 4 35.l odd 12 2
735.2.a.k 2 7.d odd 6 1
735.2.i.i 4 7.b odd 2 1
735.2.i.i 4 7.d odd 6 1
735.2.i.k 4 1.a even 1 1 trivial
735.2.i.k 4 7.c even 3 1 inner
1575.2.a.r 2 105.o odd 6 1
1575.2.d.d 4 105.x even 12 2
1680.2.a.v 2 28.g odd 6 1
2205.2.a.w 2 21.g even 6 1
3675.2.a.y 2 35.i odd 6 1
5040.2.a.bw 2 84.n even 6 1
6720.2.a.cs 2 56.k odd 6 1
6720.2.a.cx 2 56.p even 6 1
8400.2.a.cx 2 140.p odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 5 T_{2}^{2} + 25$$ $$T_{13}^{2} - 20$$ $$T_{17}^{2} - 2 T_{17} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ 1
$11$ $$1 + 4 T + 10 T^{2} - 64 T^{3} - 261 T^{4} - 704 T^{5} + 1210 T^{6} + 5324 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 + 4 T - 6 T^{2} - 64 T^{3} - 181 T^{4} - 1216 T^{5} - 2166 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{4}$$
$31$ $$1 + 12 T + 66 T^{2} + 192 T^{3} + 659 T^{4} + 5952 T^{5} + 63426 T^{6} + 357492 T^{7} + 923521 T^{8}$$
$37$ $$1 + 4 T + 18 T^{2} - 304 T^{3} - 1957 T^{4} - 11248 T^{5} + 24642 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 6 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 8 T + 34 T^{2} - 512 T^{3} - 4317 T^{4} - 24064 T^{5} + 75106 T^{6} + 830584 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 106 T^{2} - 704 T^{3} + 6123 T^{4} - 37312 T^{5} + 297754 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 38 T^{2} - 2037 T^{4} - 132278 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 16 T + 66 T^{2} - 704 T^{3} + 12083 T^{4} - 51392 T^{5} + 351714 T^{6} - 6224272 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 8 T - 30 T^{2} - 512 T^{3} - 2461 T^{4} - 40448 T^{5} - 187230 T^{6} + 3944312 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 16 T + 150 T^{2} + 1328 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 8 T + 190 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$