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 $1$

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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:

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

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