Properties

Label 3.22.a.c
Level $3$
Weight $22$
Character orbit 3.a
Self dual yes
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38432032861\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Defining polynomial: \( x^{2} - x - 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 333) q^{2} + 59049 q^{3} + ( - 666 \beta + 589618) q^{4} + ( - 13568 \beta + 498438) q^{5} + ( - 59049 \beta + 19663317) q^{6} + (182016 \beta + 339948056) q^{7} + (1285756 \beta + 1213527924) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 333) q^{2} + 59049 q^{3} + ( - 666 \beta + 589618) q^{4} + ( - 13568 \beta + 498438) q^{5} + ( - 59049 \beta + 19663317) q^{6} + (182016 \beta + 339948056) q^{7} + (1285756 \beta + 1213527924) q^{8} + 3486784401 q^{9} + ( - 5016582 \beta + 35115533262) q^{10} + ( - 31263232 \beta + 109934561484) q^{11} + ( - 39326634 \beta + 34816353282) q^{12} + (475236864 \beta - 24234454978) q^{13} + ( - 279336728 \beta - 355648853448) q^{14} + ( - 801176832 \beta + 29432265462) q^{15} + (611332056 \beta - 4144368220280) q^{16} + ( - 1677304320 \beta - 5666764520718) q^{17} + ( - 3486784401 \beta + 1161099205533) q^{18} + (22384332288 \beta + 5980292505812) q^{19} + ( - 8331896732 \beta + 23570290586412) q^{20} + (10747862784 \beta + 20073592758744) q^{21} + ( - 120345217740 \beta + 117138574281564) q^{22} + (73613782528 \beta - 73254195031752) q^{23} + (75922606044 \beta + 71657610384276) q^{24} + ( - 13525613568 \beta - 2393177123537) q^{25} + (182488330690 \beta - 12\!\cdots\!58) q^{26}+ \cdots + ( - 10\!\cdots\!32 \beta + 38\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 666 q^{2} + 118098 q^{3} + 1179236 q^{4} + 996876 q^{5} + 39326634 q^{6} + 679896112 q^{7} + 2427055848 q^{8} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 666 q^{2} + 118098 q^{3} + 1179236 q^{4} + 996876 q^{5} + 39326634 q^{6} + 679896112 q^{7} + 2427055848 q^{8} + 6973568802 q^{9} + 70231066524 q^{10} + 219869122968 q^{11} + 69632706564 q^{12} - 48468909956 q^{13} - 711297706896 q^{14} + 58864530924 q^{15} - 8288736440560 q^{16} - 11333529041436 q^{17} + 2322198411066 q^{18} + 11960585011624 q^{19} + 47140581172824 q^{20} + 40147185517488 q^{21} + 234277148563128 q^{22} - 146508390063504 q^{23} + 143315220768552 q^{24} - 4786354247074 q^{25} - 24\!\cdots\!16 q^{26}+ \cdots + 76\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
13.2377
−12.2377
−1271.96 59049.0 −479282. −2.12776e7 −7.51077e7 6.32076e8 3.27711e9 3.48678e9 2.70641e10
1.2 1937.96 59049.0 1.65852e6 2.22745e7 1.14434e8 4.78205e7 −8.50053e8 3.48678e9 4.31669e10
\(n\): 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 3.22.a.c 2
3.b odd 2 1 9.22.a.e 2
4.b odd 2 1 48.22.a.g 2
5.b even 2 1 75.22.a.d 2
5.c odd 4 2 75.22.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 1.a even 1 1 trivial
9.22.a.e 2 3.b odd 2 1
48.22.a.g 2 4.b odd 2 1
75.22.a.d 2 5.b even 2 1
75.22.b.d 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 666T_{2} - 2464992 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 666 T - 2464992 \) Copy content Toggle raw display
$3$ \( (T - 59049)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 473947100199900 \) Copy content Toggle raw display
$7$ \( T^{2} - 679896112 T + 30\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{2} - 219869122968 T + 95\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{2} + 48468909956 T - 58\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{2} + 11333529041436 T + 24\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 11960585011624 T - 12\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} + 146508390063504 T - 85\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 54\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 83\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 33\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
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