Properties

Label 3.36.a.a
Level $3$
Weight $36$
Character orbit 3.a
Self dual yes
Analytic conductor $23.279$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.2785391901\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2196841}) \)
Defining polynomial: \( x^{2} - x - 549210 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 168\sqrt{2196841}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 30456) q^{2} + 129140163 q^{3} + (60912 \beta + 28571469952) q^{4} + (1210960 \beta - 666889748370) q^{5} + ( - 129140163 \beta - 3933092804328) q^{6} + ( - 142131024 \beta - 600756391476472) q^{7} + (3933132544 \beta - 36\!\cdots\!12) q^{8}+ \cdots + 16\!\cdots\!69 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 30456) q^{2} + 129140163 q^{3} + (60912 \beta + 28571469952) q^{4} + (1210960 \beta - 666889748370) q^{5} + ( - 129140163 \beta - 3933092804328) q^{6} + ( - 142131024 \beta - 600756391476472) q^{7} + (3933132544 \beta - 36\!\cdots\!12) q^{8}+ \cdots + (59\!\cdots\!64 \beta - 12\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 60912 q^{2} + 258280326 q^{3} + 57142939904 q^{4} - 1333779496740 q^{5} - 7866185608656 q^{6} - 12\!\cdots\!44 q^{7}+ \cdots + 33\!\cdots\!38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 60912 q^{2} + 258280326 q^{3} + 57142939904 q^{4} - 1333779496740 q^{5} - 7866185608656 q^{6} - 12\!\cdots\!44 q^{7}+ \cdots - 24\!\cdots\!08 q^{99}+O(q^{100}) \) 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
741.587
−740.587
−279461. 1.29140e8 4.37389e10 −3.65354e11 −3.60897e13 −6.36148e14 −2.62111e15 1.66772e16 1.02102e17
1.2 218549. 1.29140e8 1.34041e10 −9.68425e11 2.82235e13 −5.65365e14 −4.57985e15 1.66772e16 −2.11649e17
\(n\): 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 3.36.a.a 2
3.b odd 2 1 9.36.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.36.a.a 2 1.a even 1 1 trivial
9.36.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 60912T_{2} - 61076072448 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 60912 T - 61076072448 \) Copy content Toggle raw display
$3$ \( (T - 129140163)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1333779496740 T + 35\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 24\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 13\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 57\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 15\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 13\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 40\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 61\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 51\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 24\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 37\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 67\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
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