Properties

Label 2070.2.a.w
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + ( 1 + \beta ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} + ( 1 + \beta ) q^{7} + q^{8} - q^{10} + ( 4 - \beta ) q^{11} + ( 1 + \beta ) q^{13} + ( 1 + \beta ) q^{14} + q^{16} -3 \beta q^{17} + ( -1 + 3 \beta ) q^{19} - q^{20} + ( 4 - \beta ) q^{22} + q^{23} + q^{25} + ( 1 + \beta ) q^{26} + ( 1 + \beta ) q^{28} + ( -2 + 2 \beta ) q^{29} + ( -1 - 3 \beta ) q^{31} + q^{32} -3 \beta q^{34} + ( -1 - \beta ) q^{35} + 8 q^{37} + ( -1 + 3 \beta ) q^{38} - q^{40} + ( 6 - 3 \beta ) q^{41} -4 \beta q^{43} + ( 4 - \beta ) q^{44} + q^{46} + ( -2 + 2 \beta ) q^{47} + ( -3 + 3 \beta ) q^{49} + q^{50} + ( 1 + \beta ) q^{52} + ( 2 + 4 \beta ) q^{53} + ( -4 + \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( -2 + 2 \beta ) q^{58} + ( 8 - 2 \beta ) q^{59} + 5 \beta q^{61} + ( -1 - 3 \beta ) q^{62} + q^{64} + ( -1 - \beta ) q^{65} -4 q^{67} -3 \beta q^{68} + ( -1 - \beta ) q^{70} + ( 14 + \beta ) q^{71} + ( 8 - 6 \beta ) q^{73} + 8 q^{74} + ( -1 + 3 \beta ) q^{76} + ( 1 + 2 \beta ) q^{77} + ( 4 - 8 \beta ) q^{79} - q^{80} + ( 6 - 3 \beta ) q^{82} + ( -2 - 4 \beta ) q^{83} + 3 \beta q^{85} -4 \beta q^{86} + ( 4 - \beta ) q^{88} + ( 4 + 3 \beta ) q^{91} + q^{92} + ( -2 + 2 \beta ) q^{94} + ( 1 - 3 \beta ) q^{95} + ( 4 + \beta ) q^{97} + ( -3 + 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 3q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 3q^{7} + 2q^{8} - 2q^{10} + 7q^{11} + 3q^{13} + 3q^{14} + 2q^{16} - 3q^{17} + q^{19} - 2q^{20} + 7q^{22} + 2q^{23} + 2q^{25} + 3q^{26} + 3q^{28} - 2q^{29} - 5q^{31} + 2q^{32} - 3q^{34} - 3q^{35} + 16q^{37} + q^{38} - 2q^{40} + 9q^{41} - 4q^{43} + 7q^{44} + 2q^{46} - 2q^{47} - 3q^{49} + 2q^{50} + 3q^{52} + 8q^{53} - 7q^{55} + 3q^{56} - 2q^{58} + 14q^{59} + 5q^{61} - 5q^{62} + 2q^{64} - 3q^{65} - 8q^{67} - 3q^{68} - 3q^{70} + 29q^{71} + 10q^{73} + 16q^{74} + q^{76} + 4q^{77} - 2q^{80} + 9q^{82} - 8q^{83} + 3q^{85} - 4q^{86} + 7q^{88} + 11q^{91} + 2q^{92} - 2q^{94} - q^{95} + 9q^{97} - 3q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
1.00000 0 1.00000 −1.00000 0 −0.302776 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 3.30278 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.w 2
3.b odd 2 1 230.2.a.b 2
12.b even 2 1 1840.2.a.j 2
15.d odd 2 1 1150.2.a.m 2
15.e even 4 2 1150.2.b.f 4
24.f even 2 1 7360.2.a.bu 2
24.h odd 2 1 7360.2.a.bc 2
60.h even 2 1 9200.2.a.ca 2
69.c even 2 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 3.b odd 2 1
1150.2.a.m 2 15.d odd 2 1
1150.2.b.f 4 15.e even 4 2
1840.2.a.j 2 12.b even 2 1
2070.2.a.w 2 1.a even 1 1 trivial
5290.2.a.j 2 69.c even 2 1
7360.2.a.bc 2 24.h odd 2 1
7360.2.a.bu 2 24.f even 2 1
9200.2.a.ca 2 60.h even 2 1

Hecke kernels

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

\( T_{7}^{2} - 3 T_{7} - 1 \)
\( T_{11}^{2} - 7 T_{11} + 9 \)
\( T_{13}^{2} - 3 T_{13} - 1 \)
\( T_{17}^{2} + 3 T_{17} - 27 \)
\( T_{29}^{2} + 2 T_{29} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -1 - 3 T + T^{2} \)
$11$ \( 9 - 7 T + T^{2} \)
$13$ \( -1 - 3 T + T^{2} \)
$17$ \( -27 + 3 T + T^{2} \)
$19$ \( -29 - T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -12 + 2 T + T^{2} \)
$31$ \( -23 + 5 T + T^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( -9 - 9 T + T^{2} \)
$43$ \( -48 + 4 T + T^{2} \)
$47$ \( -12 + 2 T + T^{2} \)
$53$ \( -36 - 8 T + T^{2} \)
$59$ \( 36 - 14 T + T^{2} \)
$61$ \( -75 - 5 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( 207 - 29 T + T^{2} \)
$73$ \( -92 - 10 T + T^{2} \)
$79$ \( -208 + T^{2} \)
$83$ \( -36 + 8 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 17 - 9 T + T^{2} \)
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