# 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 $1$

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## 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 1) q^{4} - 5 q^{5} + (3 \beta - 3) q^{6} + ( - 5 \beta - 9) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + 3 * q^3 + (-2*b + 1) * q^4 - 5 * q^5 + (3*b - 3) * q^6 + (-5*b - 9) * q^8 + 9 * q^9 $$q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 1) q^{4} - 5 q^{5} + (3 \beta - 3) q^{6} + ( - 5 \beta - 9) q^{8} + 9 q^{9} + ( - 5 \beta + 5) q^{10} + ( - 20 \beta - 8) q^{11} + ( - 6 \beta + 3) q^{12} + ( - 2 \beta + 38) q^{13} - 15 q^{15} + (12 \beta - 39) q^{16} + ( - 2 \beta + 62) q^{17} + (9 \beta - 9) q^{18} + ( - 16 \beta + 48) q^{19} + (10 \beta - 5) q^{20} + (12 \beta - 152) q^{22} + (34 \beta - 8) q^{23} + ( - 15 \beta - 27) q^{24} + 25 q^{25} + (40 \beta - 54) q^{26} + 27 q^{27} + (54 \beta + 94) q^{29} + ( - 15 \beta + 15) q^{30} + (18 \beta + 60) q^{31} + ( - 11 \beta + 207) q^{32} + ( - 60 \beta - 24) q^{33} + (64 \beta - 78) q^{34} + ( - 18 \beta + 9) q^{36} + (66 \beta - 66) q^{37} + (64 \beta - 176) q^{38} + ( - 6 \beta + 114) q^{39} + (25 \beta + 45) q^{40} + (80 \beta - 50) q^{41} + ( - 62 \beta - 268) q^{43} + ( - 4 \beta + 312) q^{44} - 45 q^{45} + ( - 42 \beta + 280) q^{46} + ( - 42 \beta + 464) q^{47} + (36 \beta - 117) q^{48} + (25 \beta - 25) q^{50} + ( - 6 \beta + 186) q^{51} + ( - 78 \beta + 70) q^{52} + ( - 64 \beta + 442) q^{53} + (27 \beta - 27) q^{54} + (100 \beta + 40) q^{55} + ( - 48 \beta + 144) q^{57} + (40 \beta + 338) q^{58} + ( - 204 \beta - 52) q^{59} + (30 \beta - 15) q^{60} + ( - 42 \beta + 234) q^{61} + (42 \beta + 84) q^{62} + (122 \beta + 17) q^{64} + (10 \beta - 190) q^{65} + (36 \beta - 456) q^{66} + ( - 38 \beta - 844) q^{67} + ( - 126 \beta + 94) q^{68} + (102 \beta - 24) q^{69} + (150 \beta - 68) q^{71} + ( - 45 \beta - 81) q^{72} + (322 \beta - 254) q^{73} + ( - 132 \beta + 594) q^{74} + 75 q^{75} + ( - 112 \beta + 304) q^{76} + (120 \beta - 162) q^{78} + (232 \beta - 216) q^{79} + ( - 60 \beta + 195) q^{80} + 81 q^{81} + ( - 130 \beta + 690) q^{82} + ( - 84 \beta + 292) q^{83} + (10 \beta - 310) q^{85} + ( - 206 \beta - 228) q^{86} + (162 \beta + 282) q^{87} + (220 \beta + 872) q^{88} + (112 \beta + 702) q^{89} + ( - 45 \beta + 45) q^{90} + (50 \beta - 552) q^{92} + (54 \beta + 180) q^{93} + (506 \beta - 800) q^{94} + (80 \beta - 240) q^{95} + ( - 33 \beta + 621) q^{96} + (46 \beta + 594) q^{97} + ( - 180 \beta - 72) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 + 3 * q^3 + (-2*b + 1) * q^4 - 5 * q^5 + (3*b - 3) * q^6 + (-5*b - 9) * q^8 + 9 * q^9 + (-5*b + 5) * q^10 + (-20*b - 8) * q^11 + (-6*b + 3) * q^12 + (-2*b + 38) * q^13 - 15 * q^15 + (12*b - 39) * q^16 + (-2*b + 62) * q^17 + (9*b - 9) * q^18 + (-16*b + 48) * q^19 + (10*b - 5) * q^20 + (12*b - 152) * q^22 + (34*b - 8) * q^23 + (-15*b - 27) * q^24 + 25 * q^25 + (40*b - 54) * q^26 + 27 * q^27 + (54*b + 94) * q^29 + (-15*b + 15) * q^30 + (18*b + 60) * q^31 + (-11*b + 207) * q^32 + (-60*b - 24) * q^33 + (64*b - 78) * q^34 + (-18*b + 9) * q^36 + (66*b - 66) * q^37 + (64*b - 176) * q^38 + (-6*b + 114) * q^39 + (25*b + 45) * q^40 + (80*b - 50) * q^41 + (-62*b - 268) * q^43 + (-4*b + 312) * q^44 - 45 * q^45 + (-42*b + 280) * q^46 + (-42*b + 464) * q^47 + (36*b - 117) * q^48 + (25*b - 25) * q^50 + (-6*b + 186) * q^51 + (-78*b + 70) * q^52 + (-64*b + 442) * q^53 + (27*b - 27) * q^54 + (100*b + 40) * q^55 + (-48*b + 144) * q^57 + (40*b + 338) * q^58 + (-204*b - 52) * q^59 + (30*b - 15) * q^60 + (-42*b + 234) * q^61 + (42*b + 84) * q^62 + (122*b + 17) * q^64 + (10*b - 190) * q^65 + (36*b - 456) * q^66 + (-38*b - 844) * q^67 + (-126*b + 94) * q^68 + (102*b - 24) * q^69 + (150*b - 68) * q^71 + (-45*b - 81) * q^72 + (322*b - 254) * q^73 + (-132*b + 594) * q^74 + 75 * q^75 + (-112*b + 304) * q^76 + (120*b - 162) * q^78 + (232*b - 216) * q^79 + (-60*b + 195) * q^80 + 81 * q^81 + (-130*b + 690) * q^82 + (-84*b + 292) * q^83 + (10*b - 310) * q^85 + (-206*b - 228) * q^86 + (162*b + 282) * q^87 + (220*b + 872) * q^88 + (112*b + 702) * q^89 + (-45*b + 45) * q^90 + (50*b - 552) * q^92 + (54*b + 180) * q^93 + (506*b - 800) * q^94 + (80*b - 240) * q^95 + (-33*b + 621) * q^96 + (46*b + 594) * q^97 + (-180*b - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 6 * q^6 - 18 * q^8 + 18 * q^9 $$2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9} + 10 q^{10} - 16 q^{11} + 6 q^{12} + 76 q^{13} - 30 q^{15} - 78 q^{16} + 124 q^{17} - 18 q^{18} + 96 q^{19} - 10 q^{20} - 304 q^{22} - 16 q^{23} - 54 q^{24} + 50 q^{25} - 108 q^{26} + 54 q^{27} + 188 q^{29} + 30 q^{30} + 120 q^{31} + 414 q^{32} - 48 q^{33} - 156 q^{34} + 18 q^{36} - 132 q^{37} - 352 q^{38} + 228 q^{39} + 90 q^{40} - 100 q^{41} - 536 q^{43} + 624 q^{44} - 90 q^{45} + 560 q^{46} + 928 q^{47} - 234 q^{48} - 50 q^{50} + 372 q^{51} + 140 q^{52} + 884 q^{53} - 54 q^{54} + 80 q^{55} + 288 q^{57} + 676 q^{58} - 104 q^{59} - 30 q^{60} + 468 q^{61} + 168 q^{62} + 34 q^{64} - 380 q^{65} - 912 q^{66} - 1688 q^{67} + 188 q^{68} - 48 q^{69} - 136 q^{71} - 162 q^{72} - 508 q^{73} + 1188 q^{74} + 150 q^{75} + 608 q^{76} - 324 q^{78} - 432 q^{79} + 390 q^{80} + 162 q^{81} + 1380 q^{82} + 584 q^{83} - 620 q^{85} - 456 q^{86} + 564 q^{87} + 1744 q^{88} + 1404 q^{89} + 90 q^{90} - 1104 q^{92} + 360 q^{93} - 1600 q^{94} - 480 q^{95} + 1242 q^{96} + 1188 q^{97} - 144 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 6 * q^6 - 18 * q^8 + 18 * q^9 + 10 * q^10 - 16 * q^11 + 6 * q^12 + 76 * q^13 - 30 * q^15 - 78 * q^16 + 124 * q^17 - 18 * q^18 + 96 * q^19 - 10 * q^20 - 304 * q^22 - 16 * q^23 - 54 * q^24 + 50 * q^25 - 108 * q^26 + 54 * q^27 + 188 * q^29 + 30 * q^30 + 120 * q^31 + 414 * q^32 - 48 * q^33 - 156 * q^34 + 18 * q^36 - 132 * q^37 - 352 * q^38 + 228 * q^39 + 90 * q^40 - 100 * q^41 - 536 * q^43 + 624 * q^44 - 90 * q^45 + 560 * q^46 + 928 * q^47 - 234 * q^48 - 50 * q^50 + 372 * q^51 + 140 * q^52 + 884 * q^53 - 54 * q^54 + 80 * q^55 + 288 * q^57 + 676 * q^58 - 104 * q^59 - 30 * q^60 + 468 * q^61 + 168 * q^62 + 34 * q^64 - 380 * q^65 - 912 * q^66 - 1688 * q^67 + 188 * q^68 - 48 * q^69 - 136 * q^71 - 162 * q^72 - 508 * q^73 + 1188 * q^74 + 150 * q^75 + 608 * q^76 - 324 * q^78 - 432 * q^79 + 390 * q^80 + 162 * q^81 + 1380 * q^82 + 584 * q^83 - 620 * q^85 - 456 * q^86 + 564 * q^87 + 1744 * q^88 + 1404 * q^89 + 90 * q^90 - 1104 * q^92 + 360 * q^93 - 1600 * q^94 - 480 * q^95 + 1242 * q^96 + 1188 * q^97 - 144 * q^99

## 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

## Atkin-Lehner signs

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

## 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

## Hecke kernels

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

 $$T_{2}^{2} + 2T_{2} - 7$$ T2^2 + 2*T2 - 7 $$T_{11}^{2} + 16T_{11} - 3136$$ T11^2 + 16*T11 - 3136 $$T_{13}^{2} - 76T_{13} + 1412$$ T13^2 - 76*T13 + 1412

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 7$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 16T - 3136$$
$13$ $$T^{2} - 76T + 1412$$
$17$ $$T^{2} - 124T + 3812$$
$19$ $$T^{2} - 96T + 256$$
$23$ $$T^{2} + 16T - 9184$$
$29$ $$T^{2} - 188T - 14492$$
$31$ $$T^{2} - 120T + 1008$$
$37$ $$T^{2} + 132T - 30492$$
$41$ $$T^{2} + 100T - 48700$$
$43$ $$T^{2} + 536T + 41072$$
$47$ $$T^{2} - 928T + 201184$$
$53$ $$T^{2} - 884T + 162596$$
$59$ $$T^{2} + 104T - 330224$$
$61$ $$T^{2} - 468T + 40644$$
$67$ $$T^{2} + 1688 T + 700784$$
$71$ $$T^{2} + 136T - 175376$$
$73$ $$T^{2} + 508T - 764956$$
$79$ $$T^{2} + 432T - 383936$$
$83$ $$T^{2} - 584T + 28816$$
$89$ $$T^{2} - 1404 T + 392452$$
$97$ $$T^{2} - 1188 T + 335908$$
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