Properties

Label 3.30.a.a
Level $3$
Weight $30$
Character orbit 3.a
Self dual yes
Analytic conductor $15.983$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9834127149\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{77089}) \)
Defining polynomial: \( x^{2} - x - 19272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 42\sqrt{77089}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 22518) q^{2} - 4782969 q^{3} + (45036 \beta + 106174408) q^{4} + ( - 1878560 \beta + 93806910) q^{5} + (4782969 \beta + 107702895942) q^{6} + (170898336 \beta + 1321988060624) q^{7} + ( - 583424144 \beta + 3574203597216) q^{8} + 22876792454961 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 22518) q^{2} - 4782969 q^{3} + (45036 \beta + 106174408) q^{4} + ( - 1878560 \beta + 93806910) q^{5} + (4782969 \beta + 107702895942) q^{6} + (170898336 \beta + 1321988060624) q^{7} + ( - 583424144 \beta + 3574203597216) q^{8} + 22876792454961 q^{9} + (42207607170 \beta + 253343630086380) q^{10} + ( - 57852695488 \beta + 591734923914564) q^{11} + ( - 215405791884 \beta - 507828902057352) q^{12} + (1127873657664 \beta - 10\!\cdots\!82) q^{13}+ \cdots + ( - 13\!\cdots\!68 \beta + 13\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 45036 q^{2} - 9565938 q^{3} + 212348816 q^{4} + 187613820 q^{5} + 215405791884 q^{6} + 2643976121248 q^{7} + 7148407194432 q^{8} + 45753584909922 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 45036 q^{2} - 9565938 q^{3} + 212348816 q^{4} + 187613820 q^{5} + 215405791884 q^{6} + 2643976121248 q^{7} + 7148407194432 q^{8} + 45753584909922 q^{9} + 506687260172760 q^{10} + 11\!\cdots\!28 q^{11}+ \cdots + 27\!\cdots\!08 q^{99}+O(q^{100}) \) 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
139.325
−138.325
−34179.3 −4.78297e6 6.31351e8 −2.18126e10 1.63478e11 3.31488e12 −3.22926e12 2.28768e13 7.45538e14
1.2 −10856.7 −4.78297e6 −4.19002e8 2.20002e10 5.19274e10 −6.70902e11 1.03777e13 2.28768e13 −2.38850e14
\(n\): 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 3.30.a.a 2
3.b odd 2 1 9.30.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.30.a.a 2 1.a even 1 1 trivial
9.30.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 45036T_{2} + 371075328 \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 45036 T + 371075328 \) Copy content Toggle raw display
$3$ \( (T + 4782969)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 187613820 T - 47\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} - 2643976121248 T - 22\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 56\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 23\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 20\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 77\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 16\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 59\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 26\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 52\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 73\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 25\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 51\!\cdots\!16 \) Copy content Toggle raw display
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