Properties

Label 3450.2.a.bt
Level $3450$
Weight $2$
Character orbit 3450.a
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + ( 1 + \beta_{1} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + ( 1 + \beta_{1} ) q^{7} + q^{8} + q^{9} -\beta_{2} q^{11} - q^{12} + ( 1 - \beta_{1} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + q^{16} + ( 2 - \beta_{2} ) q^{17} + q^{18} + ( -1 + \beta_{1} ) q^{19} + ( -1 - \beta_{1} ) q^{21} -\beta_{2} q^{22} + q^{23} - q^{24} + ( 1 - \beta_{1} ) q^{26} - q^{27} + ( 1 + \beta_{1} ) q^{28} + ( 5 + \beta_{1} + 2 \beta_{2} ) q^{29} -4 q^{31} + q^{32} + \beta_{2} q^{33} + ( 2 - \beta_{2} ) q^{34} + q^{36} + ( 7 - \beta_{1} - \beta_{2} ) q^{37} + ( -1 + \beta_{1} ) q^{38} + ( -1 + \beta_{1} ) q^{39} + ( -2 + 2 \beta_{2} ) q^{41} + ( -1 - \beta_{1} ) q^{42} + ( 3 - \beta_{1} + \beta_{2} ) q^{43} -\beta_{2} q^{44} + q^{46} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{47} - q^{48} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -2 + \beta_{2} ) q^{51} + ( 1 - \beta_{1} ) q^{52} + ( 3 - \beta_{1} ) q^{53} - q^{54} + ( 1 + \beta_{1} ) q^{56} + ( 1 - \beta_{1} ) q^{57} + ( 5 + \beta_{1} + 2 \beta_{2} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{61} -4 q^{62} + ( 1 + \beta_{1} ) q^{63} + q^{64} + \beta_{2} q^{66} + ( 7 - \beta_{1} - \beta_{2} ) q^{67} + ( 2 - \beta_{2} ) q^{68} - q^{69} + ( 3 + 3 \beta_{1} + 2 \beta_{2} ) q^{71} + q^{72} + ( 2 - 2 \beta_{1} ) q^{73} + ( 7 - \beta_{1} - \beta_{2} ) q^{74} + ( -1 + \beta_{1} ) q^{76} + ( 2 - 2 \beta_{1} ) q^{77} + ( -1 + \beta_{1} ) q^{78} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -2 + 2 \beta_{2} ) q^{82} + ( 7 + \beta_{1} + \beta_{2} ) q^{83} + ( -1 - \beta_{1} ) q^{84} + ( 3 - \beta_{1} + \beta_{2} ) q^{86} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{87} -\beta_{2} q^{88} + ( 2 - 2 \beta_{1} ) q^{89} + ( -8 - 2 \beta_{2} ) q^{91} + q^{92} + 4 q^{93} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{94} - q^{96} + ( -1 + 3 \beta_{1} ) q^{97} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{98} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} - 3q^{12} + 4q^{13} + 2q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 4q^{19} - 2q^{21} + 3q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 2q^{28} + 14q^{29} - 12q^{31} + 3q^{32} + 6q^{34} + 3q^{36} + 22q^{37} - 4q^{38} - 4q^{39} - 6q^{41} - 2q^{42} + 10q^{43} + 3q^{46} - 8q^{47} - 3q^{48} + 7q^{49} - 6q^{51} + 4q^{52} + 10q^{53} - 3q^{54} + 2q^{56} + 4q^{57} + 14q^{58} + 2q^{59} - 14q^{61} - 12q^{62} + 2q^{63} + 3q^{64} + 22q^{67} + 6q^{68} - 3q^{69} + 6q^{71} + 3q^{72} + 8q^{73} + 22q^{74} - 4q^{76} + 8q^{77} - 4q^{78} - 4q^{79} + 3q^{81} - 6q^{82} + 20q^{83} - 2q^{84} + 10q^{86} - 14q^{87} + 8q^{89} - 24q^{91} + 3q^{92} + 12q^{93} - 8q^{94} - 3q^{96} - 6q^{97} + 7q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
−1.48119
0.311108
2.17009
1.00000 −1.00000 1.00000 0 −1.00000 −2.96239 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 0.622216 1.00000 1.00000 0
1.3 1.00000 −1.00000 1.00000 0 −1.00000 4.34017 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.bt 3
5.b even 2 1 3450.2.a.bo 3
5.c odd 4 2 690.2.d.c 6
15.e even 4 2 2070.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.c 6 5.c odd 4 2
2070.2.d.e 6 15.e even 4 2
3450.2.a.bo 3 5.b even 2 1
3450.2.a.bt 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{3} - 2 T_{7}^{2} - 12 T_{7} + 8 \)
\( T_{11}^{3} - 16 T_{11} - 16 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 8 T_{13} + 16 \)
\( T_{17}^{3} - 6 T_{17}^{2} - 4 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$11$ \( -16 - 16 T + T^{3} \)
$13$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$17$ \( 8 - 4 T - 6 T^{2} + T^{3} \)
$19$ \( -16 - 8 T + 4 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 344 + 4 T - 14 T^{2} + T^{3} \)
$31$ \( ( 4 + T )^{3} \)
$37$ \( -232 + 140 T - 22 T^{2} + T^{3} \)
$41$ \( 8 - 52 T + 6 T^{2} + T^{3} \)
$43$ \( 8 - 4 T - 10 T^{2} + T^{3} \)
$47$ \( -256 - 64 T + 8 T^{2} + T^{3} \)
$53$ \( 8 + 20 T - 10 T^{2} + T^{3} \)
$59$ \( 104 - 84 T - 2 T^{2} + T^{3} \)
$61$ \( -1112 - 68 T + 14 T^{2} + T^{3} \)
$67$ \( -232 + 140 T - 22 T^{2} + T^{3} \)
$71$ \( -232 - 124 T - 6 T^{2} + T^{3} \)
$73$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$79$ \( -592 - 144 T + 4 T^{2} + T^{3} \)
$83$ \( -160 + 112 T - 20 T^{2} + T^{3} \)
$89$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$97$ \( -248 - 108 T + 6 T^{2} + T^{3} \)
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