Properties

Label 3.28.a.b
Level $3$
Weight $28$
Character orbit 3.a
Self dual yes
Analytic conductor $13.856$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{30001}) \)
Defining polynomial: \( x^{2} - x - 7500 \) 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 \(\beta = 63\sqrt{30001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 10791) q^{2} - 1594323 q^{3} + ( - 21582 \beta + 101301922) q^{4} + ( - 17600 \beta - 885973050) q^{5} + (1594323 \beta - 17204339493) q^{6} + ( - 17982144 \beta + 184832599952) q^{7} + ( - 199975556 \beta + 2214659936412) q^{8} + 2541865828329 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 10791) q^{2} - 1594323 q^{3} + ( - 21582 \beta + 101301922) q^{4} + ( - 17600 \beta - 885973050) q^{5} + (1594323 \beta - 17204339493) q^{6} + ( - 17982144 \beta + 184832599952) q^{7} + ( - 199975556 \beta + 2214659936412) q^{8} + 2541865828329 q^{9} + (696051450 \beta - 7464833328150) q^{10} + ( - 8106238336 \beta + 37881167834124) q^{11} + (34408678986 \beta - 161507984188806) q^{12} + (115885051776 \beta - 51510539588794) q^{13} + ( - 378877915856 \beta + 41\!\cdots\!68) q^{14}+ \cdots + ( - 20\!\cdots\!44 \beta + 96\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9} - 14929666656300 q^{10} + 75762335668248 q^{11} - 323015968377612 q^{12} - 103021079177588 q^{13} + 82\!\cdots\!36 q^{14}+ \cdots + 19\!\cdots\!92 q^{99}+O(q^{100}) \) 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
87.1040
−86.1040
−121.102 −1.59432e6 −1.34203e8 −1.07803e9 1.93076e8 −1.13904e10 3.25063e10 2.54187e12 1.30551e11
1.2 21703.1 −1.59432e6 3.36807e8 −6.93920e8 −3.46018e10 3.81056e11 4.39681e12 2.54187e12 −1.50602e13
\(n\): 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.b 2
3.b odd 2 1 9.28.a.b 2
4.b odd 2 1 48.28.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.28.a.b 2 1.a even 1 1 trivial
9.28.a.b 2 3.b odd 2 1
48.28.a.e 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 21582T_{2} - 2628288 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 21582 T - 2628288 \) Copy content Toggle raw display
$3$ \( (T + 1594323)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1771946100 T + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} - 369665199904 T - 43\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{2} - 75762335668248 T - 63\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{2} + 103021079177588 T - 15\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 34\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 41\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 36\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 45\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
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