# Properties

 Label 735.4.a.o 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: $$\Q(\sqrt{2})$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta ) q^{2} + 3 q^{3} + ( 1 - 4 \beta ) q^{4} -5 q^{5} + ( -3 + 6 \beta ) q^{6} + ( -9 - 10 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \beta ) q^{2} + 3 q^{3} + ( 1 - 4 \beta ) q^{4} -5 q^{5} + ( -3 + 6 \beta ) q^{6} + ( -9 - 10 \beta ) q^{8} + 9 q^{9} + ( 5 - 10 \beta ) q^{10} + ( -8 - 40 \beta ) q^{11} + ( 3 - 12 \beta ) q^{12} + ( 38 - 4 \beta ) q^{13} -15 q^{15} + ( -39 + 24 \beta ) q^{16} + ( 62 - 4 \beta ) q^{17} + ( -9 + 18 \beta ) q^{18} + ( 48 - 32 \beta ) q^{19} + ( -5 + 20 \beta ) q^{20} + ( -152 + 24 \beta ) q^{22} + ( -8 + 68 \beta ) q^{23} + ( -27 - 30 \beta ) q^{24} + 25 q^{25} + ( -54 + 80 \beta ) q^{26} + 27 q^{27} + ( 94 + 108 \beta ) q^{29} + ( 15 - 30 \beta ) q^{30} + ( 60 + 36 \beta ) q^{31} + ( 207 - 22 \beta ) q^{32} + ( -24 - 120 \beta ) q^{33} + ( -78 + 128 \beta ) q^{34} + ( 9 - 36 \beta ) q^{36} + ( -66 + 132 \beta ) q^{37} + ( -176 + 128 \beta ) q^{38} + ( 114 - 12 \beta ) q^{39} + ( 45 + 50 \beta ) q^{40} + ( -50 + 160 \beta ) q^{41} + ( -268 - 124 \beta ) q^{43} + ( 312 - 8 \beta ) q^{44} -45 q^{45} + ( 280 - 84 \beta ) q^{46} + ( 464 - 84 \beta ) q^{47} + ( -117 + 72 \beta ) q^{48} + ( -25 + 50 \beta ) q^{50} + ( 186 - 12 \beta ) q^{51} + ( 70 - 156 \beta ) q^{52} + ( 442 - 128 \beta ) q^{53} + ( -27 + 54 \beta ) q^{54} + ( 40 + 200 \beta ) q^{55} + ( 144 - 96 \beta ) q^{57} + ( 338 + 80 \beta ) q^{58} + ( -52 - 408 \beta ) q^{59} + ( -15 + 60 \beta ) q^{60} + ( 234 - 84 \beta ) q^{61} + ( 84 + 84 \beta ) q^{62} + ( 17 + 244 \beta ) q^{64} + ( -190 + 20 \beta ) q^{65} + ( -456 + 72 \beta ) q^{66} + ( -844 - 76 \beta ) q^{67} + ( 94 - 252 \beta ) q^{68} + ( -24 + 204 \beta ) q^{69} + ( -68 + 300 \beta ) q^{71} + ( -81 - 90 \beta ) q^{72} + ( -254 + 644 \beta ) q^{73} + ( 594 - 264 \beta ) q^{74} + 75 q^{75} + ( 304 - 224 \beta ) q^{76} + ( -162 + 240 \beta ) q^{78} + ( -216 + 464 \beta ) q^{79} + ( 195 - 120 \beta ) q^{80} + 81 q^{81} + ( 690 - 260 \beta ) q^{82} + ( 292 - 168 \beta ) q^{83} + ( -310 + 20 \beta ) q^{85} + ( -228 - 412 \beta ) q^{86} + ( 282 + 324 \beta ) q^{87} + ( 872 + 440 \beta ) q^{88} + ( 702 + 224 \beta ) q^{89} + ( 45 - 90 \beta ) q^{90} + ( -552 + 100 \beta ) q^{92} + ( 180 + 108 \beta ) q^{93} + ( -800 + 1012 \beta ) q^{94} + ( -240 + 160 \beta ) q^{95} + ( 621 - 66 \beta ) q^{96} + ( 594 + 92 \beta ) q^{97} + ( -72 - 360 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 6q^{3} + 2q^{4} - 10q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 10q^{10} - 16q^{11} + 6q^{12} + 76q^{13} - 30q^{15} - 78q^{16} + 124q^{17} - 18q^{18} + 96q^{19} - 10q^{20} - 304q^{22} - 16q^{23} - 54q^{24} + 50q^{25} - 108q^{26} + 54q^{27} + 188q^{29} + 30q^{30} + 120q^{31} + 414q^{32} - 48q^{33} - 156q^{34} + 18q^{36} - 132q^{37} - 352q^{38} + 228q^{39} + 90q^{40} - 100q^{41} - 536q^{43} + 624q^{44} - 90q^{45} + 560q^{46} + 928q^{47} - 234q^{48} - 50q^{50} + 372q^{51} + 140q^{52} + 884q^{53} - 54q^{54} + 80q^{55} + 288q^{57} + 676q^{58} - 104q^{59} - 30q^{60} + 468q^{61} + 168q^{62} + 34q^{64} - 380q^{65} - 912q^{66} - 1688q^{67} + 188q^{68} - 48q^{69} - 136q^{71} - 162q^{72} - 508q^{73} + 1188q^{74} + 150q^{75} + 608q^{76} - 324q^{78} - 432q^{79} + 390q^{80} + 162q^{81} + 1380q^{82} + 584q^{83} - 620q^{85} - 456q^{86} + 564q^{87} + 1744q^{88} + 1404q^{89} + 90q^{90} - 1104q^{92} + 360q^{93} - 1600q^{94} - 480q^{95} + 1242q^{96} + 1188q^{97} - 144q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−3.82843 3.00000 6.65685 −5.00000 −11.4853 0 5.14214 9.00000 19.1421
1.2 1.82843 3.00000 −4.65685 −5.00000 5.48528 0 −23.1421 9.00000 −9.14214
 $$n$$: 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.o 2
3.b odd 2 1 2205.4.a.bb 2
7.b odd 2 1 105.4.a.e 2
21.c even 2 1 315.4.a.k 2
28.d even 2 1 1680.4.a.bo 2
35.c odd 2 1 525.4.a.l 2
35.f even 4 2 525.4.d.l 4
105.g even 2 1 1575.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 7.b odd 2 1
315.4.a.k 2 21.c even 2 1
525.4.a.l 2 35.c odd 2 1
525.4.d.l 4 35.f even 4 2
735.4.a.o 2 1.a even 1 1 trivial
1575.4.a.q 2 105.g even 2 1
1680.4.a.bo 2 28.d even 2 1
2205.4.a.bb 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2}^{2} + 2 T_{2} - 7$$ $$T_{11}^{2} + 16 T_{11} - 3136$$ $$T_{13}^{2} - 76 T_{13} + 1412$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 9 T^{2} + 16 T^{3} + 64 T^{4}$$
$3$ $$( 1 - 3 T )^{2}$$
$5$ $$( 1 + 5 T )^{2}$$
$7$ 1
$11$ $$1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 76 T + 5806 T^{2} - 166972 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 124 T + 13638 T^{2} - 609212 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 96 T + 13974 T^{2} - 658464 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 16 T + 15150 T^{2} + 194672 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 188 T + 34286 T^{2} - 4585132 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 120 T + 60590 T^{2} - 3574920 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 132 T + 70814 T^{2} + 6686196 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 100 T + 89142 T^{2} + 6892100 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 536 T + 200086 T^{2} + 42615752 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 928 T + 408830 T^{2} - 96347744 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 884 T + 460350 T^{2} - 131607268 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 468 T + 494606 T^{2} - 106227108 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 1688 T + 1302310 T^{2} + 507687944 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 508 T + 13078 T^{2} + 197620636 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 584 T + 1172390 T^{2} - 333923608 T^{3} + 326940373369 T^{4}$$
$89$ $$1 - 1404 T + 1802390 T^{2} - 989776476 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 1188 T + 2161254 T^{2} - 1084255524 T^{3} + 832972004929 T^{4}$$