Properties

Label 3.26.a.a
Level $3$
Weight $26$
Character orbit 3.a
Self dual yes
Analytic conductor $11.880$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8799033986\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1287001}) \)
Defining polynomial: \( x^{2} - x - 321750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{1287001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 162) q^{2} - 531441 q^{3} + (324 \beta + 12803848) q^{4} + (22912 \beta + 285430878) q^{5} + (531441 \beta + 86093442) q^{6} + (5152896 \beta - 14843692864) q^{7} + (20698096 \beta - 11649985056) q^{8} + 282429536481 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 162) q^{2} - 531441 q^{3} + (324 \beta + 12803848) q^{4} + (22912 \beta + 285430878) q^{5} + (531441 \beta + 86093442) q^{6} + (5152896 \beta - 14843692864) q^{7} + (20698096 \beta - 11649985056) q^{8} + 282429536481 q^{9} + ( - 289142622 \beta - 1107799411068) q^{10} + ( - 983057152 \beta - 9093892809276) q^{11} + ( - 172186884 \beta - 6804489784968) q^{12} + (8456327424 \beta - 52571818339618) q^{13} + (14008923712 \beta - 236339484732288) q^{14} + ( - 12176376192 \beta - 151689671235198) q^{15} + ( - 2574742464 \beta - 13\!\cdots\!20) q^{16}+ \cdots + ( - 27\!\cdots\!12 \beta - 25\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 324 q^{2} - 1062882 q^{3} + 25607696 q^{4} + 570861756 q^{5} + 172186884 q^{6} - 29687385728 q^{7} - 23299970112 q^{8} + 564859072962 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 324 q^{2} - 1062882 q^{3} + 25607696 q^{4} + 570861756 q^{5} + 172186884 q^{6} - 29687385728 q^{7} - 23299970112 q^{8} + 564859072962 q^{9} - 2215598822136 q^{10} - 18187785618552 q^{11} - 13608979569936 q^{12} - 105143636679236 q^{13} - 472678969464576 q^{14} - 303379342470396 q^{15} - 27\!\cdots\!40 q^{16}+ \cdots - 51\!\cdots\!12 q^{99}+O(q^{100}) \) 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
567.730
−566.730
−6968.76 −531441. 1.50092e7 4.41387e8 3.70349e9 2.02309e10 1.29237e11 2.82430e11 −3.07592e12
1.2 6644.76 −531441. 1.05985e7 1.29474e8 −3.53130e9 −4.99182e10 −1.52537e11 2.82430e11 8.60326e11
\(n\): 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 3.26.a.a 2
3.b odd 2 1 9.26.a.b 2
4.b odd 2 1 48.26.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.26.a.a 2 1.a even 1 1 trivial
9.26.a.b 2 3.b odd 2 1
48.26.a.f 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 324T_{2} - 46305792 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 324 T - 46305792 \) Copy content Toggle raw display
$3$ \( (T + 531441)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 570861756 T + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 29687385728 T - 10\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{2} + 18187785618552 T + 37\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{2} + 105143636679236 T - 54\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{2} + 456987364349436 T - 44\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 98\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 14\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 42\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 94\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 64\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 75\!\cdots\!64 \) Copy content Toggle raw display
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