Properties

Label 3450.2.a.bb
Level $3450$
Weight $2$
Character orbit 3450.a
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + 5q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + 5q^{7} + q^{8} + q^{9} - 3q^{11} + q^{12} + 5q^{13} + 5q^{14} + q^{16} + 6q^{17} + q^{18} - q^{19} + 5q^{21} - 3q^{22} + q^{23} + q^{24} + 5q^{26} + q^{27} + 5q^{28} - 5q^{29} - 8q^{31} + q^{32} - 3q^{33} + 6q^{34} + q^{36} + 4q^{37} - q^{38} + 5q^{39} - 7q^{41} + 5q^{42} - 7q^{43} - 3q^{44} + q^{46} - 6q^{47} + q^{48} + 18q^{49} + 6q^{51} + 5q^{52} + 8q^{53} + q^{54} + 5q^{56} - q^{57} - 5q^{58} - 10q^{59} - 12q^{61} - 8q^{62} + 5q^{63} + q^{64} - 3q^{66} - 12q^{67} + 6q^{68} + q^{69} + 10q^{71} + q^{72} + 15q^{73} + 4q^{74} - q^{76} - 15q^{77} + 5q^{78} - 5q^{79} + q^{81} - 7q^{82} - 9q^{83} + 5q^{84} - 7q^{86} - 5q^{87} - 3q^{88} + 14q^{89} + 25q^{91} + q^{92} - 8q^{93} - 6q^{94} + q^{96} + 16q^{97} + 18q^{98} - 3q^{99} + O(q^{100}) \)

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
0
1.00000 1.00000 1.00000 0 1.00000 5.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 3450.2.a.bb yes 1
5.b even 2 1 3450.2.a.a 1
5.c odd 4 2 3450.2.d.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.a 1 5.b even 2 1
3450.2.a.bb yes 1 1.a even 1 1 trivial
3450.2.d.l 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\):

\( T_{7} - 5 \)
\( T_{11} + 3 \)
\( T_{13} - 5 \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -5 + T \)
$11$ \( 3 + T \)
$13$ \( -5 + T \)
$17$ \( -6 + T \)
$19$ \( 1 + T \)
$23$ \( -1 + T \)
$29$ \( 5 + T \)
$31$ \( 8 + T \)
$37$ \( -4 + T \)
$41$ \( 7 + T \)
$43$ \( 7 + T \)
$47$ \( 6 + T \)
$53$ \( -8 + T \)
$59$ \( 10 + T \)
$61$ \( 12 + T \)
$67$ \( 12 + T \)
$71$ \( -10 + T \)
$73$ \( -15 + T \)
$79$ \( 5 + T \)
$83$ \( 9 + T \)
$89$ \( -14 + T \)
$97$ \( -16 + T \)
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