# Properties

 Label 735.4.a.m Level 735 Weight 4 Character orbit 735.a Self dual yes Analytic conductor 43.366 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.3664038542$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 \beta ) q^{2} -3 q^{3} + ( -3 + 8 \beta ) q^{4} + 5 q^{5} + ( 3 + 6 \beta ) q^{6} + ( -5 - 2 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 - 2 \beta ) q^{2} -3 q^{3} + ( -3 + 8 \beta ) q^{4} + 5 q^{5} + ( 3 + 6 \beta ) q^{6} + ( -5 - 2 \beta ) q^{8} + 9 q^{9} + ( -5 - 10 \beta ) q^{10} + ( -48 + 4 \beta ) q^{11} + ( 9 - 24 \beta ) q^{12} + ( 34 - 76 \beta ) q^{13} -15 q^{15} + ( 33 - 48 \beta ) q^{16} + ( -22 + 88 \beta ) q^{17} + ( -9 - 18 \beta ) q^{18} + ( 28 + 52 \beta ) q^{19} + ( -15 + 40 \beta ) q^{20} + ( 40 + 84 \beta ) q^{22} + ( -180 + 40 \beta ) q^{23} + ( 15 + 6 \beta ) q^{24} + 25 q^{25} + ( 118 + 160 \beta ) q^{26} -27 q^{27} + ( -106 - 24 \beta ) q^{29} + ( 15 + 30 \beta ) q^{30} + ( -72 + 204 \beta ) q^{31} + ( 103 + 94 \beta ) q^{32} + ( 144 - 12 \beta ) q^{33} + ( -154 - 220 \beta ) q^{34} + ( -27 + 72 \beta ) q^{36} + ( 126 - 48 \beta ) q^{37} + ( -132 - 212 \beta ) q^{38} + ( -102 + 228 \beta ) q^{39} + ( -25 - 10 \beta ) q^{40} + ( 58 - 160 \beta ) q^{41} + ( 196 - 256 \beta ) q^{43} + ( 176 - 364 \beta ) q^{44} + 45 q^{45} + ( 100 + 240 \beta ) q^{46} + ( -32 - 336 \beta ) q^{47} + ( -99 + 144 \beta ) q^{48} + ( -25 - 50 \beta ) q^{50} + ( 66 - 264 \beta ) q^{51} + ( -710 - 108 \beta ) q^{52} + ( 94 - 172 \beta ) q^{53} + ( 27 + 54 \beta ) q^{54} + ( -240 + 20 \beta ) q^{55} + ( -84 - 156 \beta ) q^{57} + ( 154 + 284 \beta ) q^{58} + ( 268 - 72 \beta ) q^{59} + ( 45 - 120 \beta ) q^{60} + ( 258 + 168 \beta ) q^{61} + ( -336 - 468 \beta ) q^{62} + ( -555 - 104 \beta ) q^{64} + ( 170 - 380 \beta ) q^{65} + ( -120 - 252 \beta ) q^{66} + ( 532 - 328 \beta ) q^{67} + ( 770 + 264 \beta ) q^{68} + ( 540 - 120 \beta ) q^{69} + ( -508 + 276 \beta ) q^{71} + ( -45 - 18 \beta ) q^{72} + ( -90 - 244 \beta ) q^{73} + ( -30 - 108 \beta ) q^{74} -75 q^{75} + ( 332 + 484 \beta ) q^{76} + ( -354 - 480 \beta ) q^{78} + ( -688 + 968 \beta ) q^{79} + ( 165 - 240 \beta ) q^{80} + 81 q^{81} + ( 262 + 364 \beta ) q^{82} + ( -220 - 168 \beta ) q^{83} + ( -110 + 440 \beta ) q^{85} + ( 316 + 376 \beta ) q^{86} + ( 318 + 72 \beta ) q^{87} + ( 232 + 68 \beta ) q^{88} + ( 778 - 224 \beta ) q^{89} + ( -45 - 90 \beta ) q^{90} + ( 860 - 1240 \beta ) q^{92} + ( 216 - 612 \beta ) q^{93} + ( 704 + 1072 \beta ) q^{94} + ( 140 + 260 \beta ) q^{95} + ( -309 - 282 \beta ) q^{96} + ( 1310 - 172 \beta ) q^{97} + ( -432 + 36 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 12q^{6} - 12q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 12q^{6} - 12q^{8} + 18q^{9} - 20q^{10} - 92q^{11} - 6q^{12} - 8q^{13} - 30q^{15} + 18q^{16} + 44q^{17} - 36q^{18} + 108q^{19} + 10q^{20} + 164q^{22} - 320q^{23} + 36q^{24} + 50q^{25} + 396q^{26} - 54q^{27} - 236q^{29} + 60q^{30} + 60q^{31} + 300q^{32} + 276q^{33} - 528q^{34} + 18q^{36} + 204q^{37} - 476q^{38} + 24q^{39} - 60q^{40} - 44q^{41} + 136q^{43} - 12q^{44} + 90q^{45} + 440q^{46} - 400q^{47} - 54q^{48} - 100q^{50} - 132q^{51} - 1528q^{52} + 16q^{53} + 108q^{54} - 460q^{55} - 324q^{57} + 592q^{58} + 464q^{59} - 30q^{60} + 684q^{61} - 1140q^{62} - 1214q^{64} - 40q^{65} - 492q^{66} + 736q^{67} + 1804q^{68} + 960q^{69} - 740q^{71} - 108q^{72} - 424q^{73} - 168q^{74} - 150q^{75} + 1148q^{76} - 1188q^{78} - 408q^{79} + 90q^{80} + 162q^{81} + 888q^{82} - 608q^{83} + 220q^{85} + 1008q^{86} + 708q^{87} + 532q^{88} + 1332q^{89} - 180q^{90} + 480q^{92} - 180q^{93} + 2480q^{94} + 540q^{95} - 900q^{96} + 2448q^{97} - 828q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−4.23607 −3.00000 9.94427 5.00000 12.7082 0 −8.23607 9.00000 −21.1803
1.2 0.236068 −3.00000 −7.94427 5.00000 −0.708204 0 −3.76393 9.00000 1.18034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.4.a.m 2
3.b odd 2 1 2205.4.a.be 2
7.b odd 2 1 105.4.a.d 2
21.c even 2 1 315.4.a.l 2
28.d even 2 1 1680.4.a.bd 2
35.c odd 2 1 525.4.a.o 2
35.f even 4 2 525.4.d.k 4
105.g even 2 1 1575.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 7.b odd 2 1
315.4.a.l 2 21.c even 2 1
525.4.a.o 2 35.c odd 2 1
525.4.d.k 4 35.f even 4 2
735.4.a.m 2 1.a even 1 1 trivial
1575.4.a.n 2 105.g even 2 1
1680.4.a.bd 2 28.d even 2 1
2205.4.a.be 2 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2}^{2} + 4 T_{2} - 1$$ $$T_{11}^{2} + 92 T_{11} + 2096$$ $$T_{13}^{2} + 8 T_{13} - 7204$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 15 T^{2} + 32 T^{3} + 64 T^{4}$$
$3$ $$( 1 + 3 T )^{2}$$
$5$ $$( 1 - 5 T )^{2}$$
$7$ 1
$11$ $$1 + 92 T + 4758 T^{2} + 122452 T^{3} + 1771561 T^{4}$$
$13$ $$1 + 8 T - 2810 T^{2} + 17576 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 44 T + 630 T^{2} - 216172 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 108 T + 13254 T^{2} - 740772 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 320 T + 47934 T^{2} + 3893440 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 236 T + 61982 T^{2} + 5755804 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 60 T + 8462 T^{2} - 1787460 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 204 T + 108830 T^{2} - 10333212 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 44 T + 106326 T^{2} + 3032524 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 136 T + 81718 T^{2} - 10812952 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 400 T + 106526 T^{2} + 41529200 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 16 T + 260838 T^{2} - 2382032 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 684 T + 535646 T^{2} - 155255004 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 736 T + 602470 T^{2} - 221361568 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 740 T + 757502 T^{2} + 264854140 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 424 T + 748558 T^{2} + 164943208 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 408 T - 143586 T^{2} + 201159912 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 608 T + 1200710 T^{2} + 347646496 T^{3} + 326940373369 T^{4}$$
$89$ $$1 - 1332 T + 1790774 T^{2} - 939018708 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 2448 T + 3286542 T^{2} - 2234223504 T^{3} + 832972004929 T^{4}$$