Properties

Label 2070.2.a.t
Level $2070$
Weight $2$
Character orbit 2070.a
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} + q^{10} + ( -1 - \beta ) q^{11} + 2 q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( -3 - \beta ) q^{17} + 4 q^{19} - q^{20} + ( 1 + \beta ) q^{22} - q^{23} + q^{25} -2 q^{26} + ( -1 - \beta ) q^{28} -2 q^{29} - q^{32} + ( 3 + \beta ) q^{34} + ( 1 + \beta ) q^{35} + ( -3 - \beta ) q^{37} -4 q^{38} + q^{40} -2 q^{41} + ( -1 - \beta ) q^{44} + q^{46} + 8 q^{47} + ( 11 + 2 \beta ) q^{49} - q^{50} + 2 q^{52} + ( -4 + 2 \beta ) q^{53} + ( 1 + \beta ) q^{55} + ( 1 + \beta ) q^{56} + 2 q^{58} + ( 6 + 2 \beta ) q^{59} + ( 5 - \beta ) q^{61} + q^{64} -2 q^{65} -8 q^{67} + ( -3 - \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 2 - 2 \beta ) q^{71} + ( 4 + 2 \beta ) q^{73} + ( 3 + \beta ) q^{74} + 4 q^{76} + ( 18 + 2 \beta ) q^{77} + ( -1 - \beta ) q^{79} - q^{80} + 2 q^{82} + ( 1 - 3 \beta ) q^{83} + ( 3 + \beta ) q^{85} + ( 1 + \beta ) q^{88} + ( 1 - \beta ) q^{89} + ( -2 - 2 \beta ) q^{91} - q^{92} -8 q^{94} -4 q^{95} + ( -8 + 2 \beta ) q^{97} + ( -11 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + 2q^{10} - 2q^{11} + 4q^{13} + 2q^{14} + 2q^{16} - 6q^{17} + 8q^{19} - 2q^{20} + 2q^{22} - 2q^{23} + 2q^{25} - 4q^{26} - 2q^{28} - 4q^{29} - 2q^{32} + 6q^{34} + 2q^{35} - 6q^{37} - 8q^{38} + 2q^{40} - 4q^{41} - 2q^{44} + 2q^{46} + 16q^{47} + 22q^{49} - 2q^{50} + 4q^{52} - 8q^{53} + 2q^{55} + 2q^{56} + 4q^{58} + 12q^{59} + 10q^{61} + 2q^{64} - 4q^{65} - 16q^{67} - 6q^{68} - 2q^{70} + 4q^{71} + 8q^{73} + 6q^{74} + 8q^{76} + 36q^{77} - 2q^{79} - 2q^{80} + 4q^{82} + 2q^{83} + 6q^{85} + 2q^{88} + 2q^{89} - 4q^{91} - 2q^{92} - 16q^{94} - 8q^{95} - 16q^{97} - 22q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−1.00000 0 1.00000 −1.00000 0 −5.12311 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 3.12311 −1.00000 0 1.00000
\(n\): 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 2070.2.a.t 2
3.b odd 2 1 690.2.a.l 2
12.b even 2 1 5520.2.a.bs 2
15.d odd 2 1 3450.2.a.bi 2
15.e even 4 2 3450.2.d.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.l 2 3.b odd 2 1
2070.2.a.t 2 1.a even 1 1 trivial
3450.2.a.bi 2 15.d odd 2 1
3450.2.d.v 4 15.e even 4 2
5520.2.a.bs 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{2} + 2 T_{7} - 16 \)
\( T_{11}^{2} + 2 T_{11} - 16 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 6 T_{17} - 8 \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -16 + 2 T + T^{2} \)
$11$ \( -16 + 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -8 + 6 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( -8 + 6 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -52 + 8 T + T^{2} \)
$59$ \( -32 - 12 T + T^{2} \)
$61$ \( 8 - 10 T + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -64 - 4 T + T^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( -16 + 2 T + T^{2} \)
$83$ \( -152 - 2 T + T^{2} \)
$89$ \( -16 - 2 T + T^{2} \)
$97$ \( -4 + 16 T + T^{2} \)
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