Properties

 Label 3.10.a.a Level $3$ Weight $10$ Character orbit 3.a Self dual yes Analytic conductor $1.545$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$1.54510750849$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 36q^{2} - 81q^{3} + 784q^{4} - 1314q^{5} + 2916q^{6} - 4480q^{7} - 9792q^{8} + 6561q^{9} + O(q^{10})$$ $$q - 36q^{2} - 81q^{3} + 784q^{4} - 1314q^{5} + 2916q^{6} - 4480q^{7} - 9792q^{8} + 6561q^{9} + 47304q^{10} + 1476q^{11} - 63504q^{12} - 151522q^{13} + 161280q^{14} + 106434q^{15} - 48896q^{16} + 108162q^{17} - 236196q^{18} + 593084q^{19} - 1030176q^{20} + 362880q^{21} - 53136q^{22} - 969480q^{23} + 793152q^{24} - 226529q^{25} + 5454792q^{26} - 531441q^{27} - 3512320q^{28} - 6642522q^{29} - 3831624q^{30} + 7070600q^{31} + 6773760q^{32} - 119556q^{33} - 3893832q^{34} + 5886720q^{35} + 5143824q^{36} - 7472410q^{37} - 21351024q^{38} + 12273282q^{39} + 12866688q^{40} - 4350150q^{41} - 13063680q^{42} - 4358716q^{43} + 1157184q^{44} - 8621154q^{45} + 34901280q^{46} + 28309248q^{47} + 3960576q^{48} - 20283207q^{49} + 8155044q^{50} - 8761122q^{51} - 118793248q^{52} + 16111710q^{53} + 19131876q^{54} - 1939464q^{55} + 43868160q^{56} - 48039804q^{57} + 239130792q^{58} - 86075964q^{59} + 83444256q^{60} + 32213918q^{61} - 254541600q^{62} - 29393280q^{63} - 218820608q^{64} + 199099908q^{65} + 4304016q^{66} + 99531452q^{67} + 84799008q^{68} + 78527880q^{69} - 211921920q^{70} - 44170488q^{71} - 64245312q^{72} - 23560630q^{73} + 269006760q^{74} + 18348849q^{75} + 464977856q^{76} - 6612480q^{77} - 441838152q^{78} - 401754760q^{79} + 64249344q^{80} + 43046721q^{81} + 156605400q^{82} - 744528708q^{83} + 284497920q^{84} - 142124868q^{85} + 156913776q^{86} + 538044282q^{87} - 14452992q^{88} + 769871034q^{89} + 310361544q^{90} + 678818560q^{91} - 760072320q^{92} - 572718600q^{93} - 1019132928q^{94} - 779312376q^{95} - 548674560q^{96} + 907130882q^{97} + 730195452q^{98} + 9684036q^{99} + O(q^{100})$$

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
 0
−36.0000 −81.0000 784.000 −1314.00 2916.00 −4480.00 −9792.00 6561.00 47304.0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

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 3.10.a.a 1
3.b odd 2 1 9.10.a.c 1
4.b odd 2 1 48.10.a.e 1
5.b even 2 1 75.10.a.d 1
5.c odd 4 2 75.10.b.a 2
7.b odd 2 1 147.10.a.a 1
8.b even 2 1 192.10.a.m 1
8.d odd 2 1 192.10.a.f 1
9.c even 3 2 81.10.c.e 2
9.d odd 6 2 81.10.c.a 2
11.b odd 2 1 363.10.a.b 1
12.b even 2 1 144.10.a.l 1
15.d odd 2 1 225.10.a.a 1
15.e even 4 2 225.10.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.10.a.a 1 1.a even 1 1 trivial
9.10.a.c 1 3.b odd 2 1
48.10.a.e 1 4.b odd 2 1
75.10.a.d 1 5.b even 2 1
75.10.b.a 2 5.c odd 4 2
81.10.c.a 2 9.d odd 6 2
81.10.c.e 2 9.c even 3 2
144.10.a.l 1 12.b even 2 1
147.10.a.a 1 7.b odd 2 1
192.10.a.f 1 8.d odd 2 1
192.10.a.m 1 8.b even 2 1
225.10.a.a 1 15.d odd 2 1
225.10.b.a 2 15.e even 4 2
363.10.a.b 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 36$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(3))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 36 T + 512 T^{2}$$
$3$ $$1 + 81 T$$
$5$ $$1 + 1314 T + 1953125 T^{2}$$
$7$ $$1 + 4480 T + 40353607 T^{2}$$
$11$ $$1 - 1476 T + 2357947691 T^{2}$$
$13$ $$1 + 151522 T + 10604499373 T^{2}$$
$17$ $$1 - 108162 T + 118587876497 T^{2}$$
$19$ $$1 - 593084 T + 322687697779 T^{2}$$
$23$ $$1 + 969480 T + 1801152661463 T^{2}$$
$29$ $$1 + 6642522 T + 14507145975869 T^{2}$$
$31$ $$1 - 7070600 T + 26439622160671 T^{2}$$
$37$ $$1 + 7472410 T + 129961739795077 T^{2}$$
$41$ $$1 + 4350150 T + 327381934393961 T^{2}$$
$43$ $$1 + 4358716 T + 502592611936843 T^{2}$$
$47$ $$1 - 28309248 T + 1119130473102767 T^{2}$$
$53$ $$1 - 16111710 T + 3299763591802133 T^{2}$$
$59$ $$1 + 86075964 T + 8662995818654939 T^{2}$$
$61$ $$1 - 32213918 T + 11694146092834141 T^{2}$$
$67$ $$1 - 99531452 T + 27206534396294947 T^{2}$$
$71$ $$1 + 44170488 T + 45848500718449031 T^{2}$$
$73$ $$1 + 23560630 T + 58871586708267913 T^{2}$$
$79$ $$1 + 401754760 T + 119851595982618319 T^{2}$$
$83$ $$1 + 744528708 T + 186940255267540403 T^{2}$$
$89$ $$1 - 769871034 T + 350356403707485209 T^{2}$$
$97$ $$1 - 907130882 T + 760231058654565217 T^{2}$$