Properties

Label 3450.2.a.bs
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: 3.3.3368.1
Defining polynomial: \(x^{3} - x^{2} - 15 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} -\beta_{1} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{6} -\beta_{1} q^{7} + q^{8} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} - q^{12} + ( 1 - \beta_{1} + \beta_{2} ) q^{13} -\beta_{1} q^{14} + q^{16} + ( 1 + \beta_{2} ) q^{17} + q^{18} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( 1 - \beta_{1} - \beta_{2} ) q^{22} + q^{23} - q^{24} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} - q^{27} -\beta_{1} q^{28} - q^{29} + 6 q^{31} + q^{32} + ( -1 + \beta_{1} + \beta_{2} ) q^{33} + ( 1 + \beta_{2} ) q^{34} + q^{36} + ( -1 - \beta_{2} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{39} + ( 7 - \beta_{1} + \beta_{2} ) q^{41} + \beta_{1} q^{42} + ( -5 + \beta_{1} - \beta_{2} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} ) q^{44} + q^{46} + ( 2 + \beta_{1} - \beta_{2} ) q^{47} - q^{48} + ( 4 + 2 \beta_{2} ) q^{49} + ( -1 - \beta_{2} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} ) q^{52} + ( -4 - 2 \beta_{2} ) q^{53} - q^{54} -\beta_{1} q^{56} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} - q^{58} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + 6 q^{62} -\beta_{1} q^{63} + q^{64} + ( -1 + \beta_{1} + \beta_{2} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 1 + \beta_{2} ) q^{68} - q^{69} + ( 4 + \beta_{1} + \beta_{2} ) q^{71} + q^{72} + ( -1 + 2 \beta_{1} ) q^{73} + ( -1 - \beta_{2} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} ) q^{76} + ( 11 + \beta_{1} + 3 \beta_{2} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} ) q^{78} + ( -1 + \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 7 - \beta_{1} + \beta_{2} ) q^{82} + ( -4 - 3 \beta_{1} ) q^{83} + \beta_{1} q^{84} + ( -5 + \beta_{1} - \beta_{2} ) q^{86} + q^{87} + ( 1 - \beta_{1} - \beta_{2} ) q^{88} + ( 3 + \beta_{2} ) q^{89} + ( 11 - 3 \beta_{1} + \beta_{2} ) q^{91} + q^{92} -6 q^{93} + ( 2 + \beta_{1} - \beta_{2} ) q^{94} - q^{96} + ( -4 + 2 \beta_{1} ) q^{97} + ( 4 + 2 \beta_{2} ) q^{98} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + q^{13} - q^{14} + 3q^{16} + 2q^{17} + 3q^{18} - q^{19} + q^{21} + 3q^{22} + 3q^{23} - 3q^{24} + q^{26} - 3q^{27} - q^{28} - 3q^{29} + 18q^{31} + 3q^{32} - 3q^{33} + 2q^{34} + 3q^{36} - 2q^{37} - q^{38} - q^{39} + 19q^{41} + q^{42} - 13q^{43} + 3q^{44} + 3q^{46} + 8q^{47} - 3q^{48} + 10q^{49} - 2q^{51} + q^{52} - 10q^{53} - 3q^{54} - q^{56} + q^{57} - 3q^{58} + 20q^{61} + 18q^{62} - q^{63} + 3q^{64} - 3q^{66} + 4q^{67} + 2q^{68} - 3q^{69} + 12q^{71} + 3q^{72} - q^{73} - 2q^{74} - q^{76} + 31q^{77} - q^{78} - 3q^{79} + 3q^{81} + 19q^{82} - 15q^{83} + q^{84} - 13q^{86} + 3q^{87} + 3q^{88} + 8q^{89} + 29q^{91} + 3q^{92} - 18q^{93} + 8q^{94} - 3q^{96} - 10q^{97} + 10q^{98} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 15 x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 11 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + 11\)

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
4.03932
0.723686
−3.76300
1.00000 −1.00000 1.00000 0 −1.00000 −4.03932 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 −0.723686 1.00000 1.00000 0
1.3 1.00000 −1.00000 1.00000 0 −1.00000 3.76300 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.bs yes 3
5.b even 2 1 3450.2.a.bp 3
5.c odd 4 2 3450.2.d.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bp 3 5.b even 2 1
3450.2.a.bs yes 3 1.a even 1 1 trivial
3450.2.d.ba 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\):

\( T_{7}^{3} + T_{7}^{2} - 15 T_{7} - 11 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 32 T_{11} + 100 \)
\( T_{13}^{3} - T_{13}^{2} - 32 T_{13} - 12 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 17 T_{17} + 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -11 - 15 T + T^{2} + T^{3} \)
$11$ \( 100 - 32 T - 3 T^{2} + T^{3} \)
$13$ \( -12 - 32 T - T^{2} + T^{3} \)
$17$ \( 40 - 17 T - 2 T^{2} + T^{3} \)
$19$ \( 12 - 32 T + T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( ( 1 + T )^{3} \)
$31$ \( ( -6 + T )^{3} \)
$37$ \( -40 - 17 T + 2 T^{2} + T^{3} \)
$41$ \( -72 + 88 T - 19 T^{2} + T^{3} \)
$43$ \( -36 + 24 T + 13 T^{2} + T^{3} \)
$47$ \( 90 - 11 T - 8 T^{2} + T^{3} \)
$53$ \( -432 - 40 T + 10 T^{2} + T^{3} \)
$59$ \( -528 - 140 T + T^{3} \)
$61$ \( 160 + 72 T - 20 T^{2} + T^{3} \)
$67$ \( 352 - 124 T - 4 T^{2} + T^{3} \)
$71$ \( 10 + 13 T - 12 T^{2} + T^{3} \)
$73$ \( 27 - 61 T + T^{2} + T^{3} \)
$79$ \( -100 - 32 T + 3 T^{2} + T^{3} \)
$83$ \( -725 - 63 T + 15 T^{2} + T^{3} \)
$89$ \( 58 + 3 T - 8 T^{2} + T^{3} \)
$97$ \( -120 - 28 T + 10 T^{2} + T^{3} \)
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