Properties

Label 3.28.a.a
Level $3$
Weight $28$
Character orbit 3.a
Self dual yes
Analytic conductor $13.856$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.8556672451\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6469}) \)
Defining polynomial: \( x^{2} - x - 1617 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 144\sqrt{6469}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1584) q^{2} + 1594323 q^{3} + ( - 3168 \beta + 2432512) q^{4} + (196240 \beta - 2453032530) q^{5} + ( - 1594323 \beta + 2525407632) q^{6} + (22878576 \beta - 75828544792) q^{7} + (126767104 \beta + 216211488768) q^{8} + 2541865828329 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1584) q^{2} + 1594323 q^{3} + ( - 3168 \beta + 2432512) q^{4} + (196240 \beta - 2453032530) q^{5} + ( - 1594323 \beta + 2525407632) q^{6} + (22878576 \beta - 75828544792) q^{7} + (126767104 \beta + 216211488768) q^{8} + 2541865828329 q^{9} + (2763876690 \beta - 30209469475680) q^{10} + ( - 10511067424 \beta + 20427496063524) q^{11} + ( - 5050815264 \beta + 3878209829376) q^{12} + ( - 57087242784 \beta - 208698949319146) q^{13} + (112068209176 \beta - 31\!\cdots\!12) q^{14}+ \cdots + ( - 26\!\cdots\!96 \beta + 51\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3168 q^{2} + 3188646 q^{3} + 4865024 q^{4} - 4906065060 q^{5} + 5050815264 q^{6} - 151657089584 q^{7} + 432422977536 q^{8} + 5083731656658 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3168 q^{2} + 3188646 q^{3} + 4865024 q^{4} - 4906065060 q^{5} + 5050815264 q^{6} - 151657089584 q^{7} + 432422977536 q^{8} + 5083731656658 q^{9} - 60418938951360 q^{10} + 40854992127048 q^{11} + 7756419658752 q^{12} - 417397898638292 q^{13} - 63\!\cdots\!24 q^{14}+ \cdots + 10\!\cdots\!92 q^{99}+O(q^{100}) \) 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
40.7150
−39.7150
−9997.93 1.59432e6 −3.42591e7 −1.80194e8 −1.59399e10 1.89150e11 1.68442e12 2.54187e12 1.80157e12
1.2 13165.9 1.59432e6 3.91241e7 −4.72587e9 2.09908e10 −3.40807e11 −1.25200e12 2.54187e12 −6.22205e13
\(n\): 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 3.28.a.a 2
3.b odd 2 1 9.28.a.c 2
4.b odd 2 1 48.28.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.28.a.a 2 1.a even 1 1 trivial
9.28.a.c 2 3.b odd 2 1
48.28.a.d 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3168T_{2} - 131632128 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3168 T - 131632128 \) Copy content Toggle raw display
$3$ \( (T - 1594323)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4906065060 T + 85\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 151657089584 T - 64\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{2} - 40854992127048 T - 14\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{2} + 417397898638292 T - 39\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 90\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 42\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 31\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 34\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 43\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 28\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 68\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 61\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
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