Properties

Label 2070.2.a.u
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8} + q^{10} + ( 1 - 3 \beta ) q^{11} + ( -4 + 5 \beta ) q^{13} -\beta q^{14} + q^{16} + ( -3 + 5 \beta ) q^{17} -3 \beta q^{19} - q^{20} + ( -1 + 3 \beta ) q^{22} - q^{23} + q^{25} + ( 4 - 5 \beta ) q^{26} + \beta q^{28} + ( 8 - 2 \beta ) q^{29} + ( 6 - 5 \beta ) q^{31} - q^{32} + ( 3 - 5 \beta ) q^{34} -\beta q^{35} + ( 4 - 4 \beta ) q^{37} + 3 \beta q^{38} + q^{40} + ( 1 + 7 \beta ) q^{41} + ( 1 - 3 \beta ) q^{44} + q^{46} -6 \beta q^{47} + ( -6 + \beta ) q^{49} - q^{50} + ( -4 + 5 \beta ) q^{52} + ( 2 + 4 \beta ) q^{53} + ( -1 + 3 \beta ) q^{55} -\beta q^{56} + ( -8 + 2 \beta ) q^{58} + ( 2 + 6 \beta ) q^{59} + ( -5 + 7 \beta ) q^{61} + ( -6 + 5 \beta ) q^{62} + q^{64} + ( 4 - 5 \beta ) q^{65} + ( 12 - 4 \beta ) q^{67} + ( -3 + 5 \beta ) q^{68} + \beta q^{70} + ( 1 - 5 \beta ) q^{71} + ( 2 - 2 \beta ) q^{73} + ( -4 + 4 \beta ) q^{74} -3 \beta q^{76} + ( -3 - 2 \beta ) q^{77} + ( 4 + 4 \beta ) q^{79} - q^{80} + ( -1 - 7 \beta ) q^{82} + ( 2 - 8 \beta ) q^{83} + ( 3 - 5 \beta ) q^{85} + ( -1 + 3 \beta ) q^{88} + ( 8 - 4 \beta ) q^{89} + ( 5 + \beta ) q^{91} - q^{92} + 6 \beta q^{94} + 3 \beta q^{95} + ( 13 + \beta ) q^{97} + ( 6 - \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−1.00000 0 1.00000 −1.00000 0 −0.618034 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.61803 −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.u 2
3.b odd 2 1 230.2.a.c 2
12.b even 2 1 1840.2.a.l 2
15.d odd 2 1 1150.2.a.j 2
15.e even 4 2 1150.2.b.i 4
24.f even 2 1 7360.2.a.bn 2
24.h odd 2 1 7360.2.a.bh 2
60.h even 2 1 9200.2.a.bu 2
69.c even 2 1 5290.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 3.b odd 2 1
1150.2.a.j 2 15.d odd 2 1
1150.2.b.i 4 15.e even 4 2
1840.2.a.l 2 12.b even 2 1
2070.2.a.u 2 1.a even 1 1 trivial
5290.2.a.o 2 69.c even 2 1
7360.2.a.bh 2 24.h odd 2 1
7360.2.a.bn 2 24.f even 2 1
9200.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{2} - T_{7} - 1 \)
\( T_{11}^{2} + T_{11} - 11 \)
\( T_{13}^{2} + 3 T_{13} - 29 \)
\( T_{17}^{2} + T_{17} - 31 \)
\( T_{29}^{2} - 14 T_{29} + 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -1 - T + T^{2} \)
$11$ \( -11 + T + T^{2} \)
$13$ \( -29 + 3 T + T^{2} \)
$17$ \( -31 + T + T^{2} \)
$19$ \( -9 + 3 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 44 - 14 T + T^{2} \)
$31$ \( -19 - 7 T + T^{2} \)
$37$ \( -16 - 4 T + T^{2} \)
$41$ \( -41 - 9 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -36 + 6 T + T^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( -20 - 10 T + T^{2} \)
$61$ \( -59 + 3 T + T^{2} \)
$67$ \( 80 - 20 T + T^{2} \)
$71$ \( -29 + 3 T + T^{2} \)
$73$ \( -4 - 2 T + T^{2} \)
$79$ \( 16 - 12 T + T^{2} \)
$83$ \( -76 + 4 T + T^{2} \)
$89$ \( 16 - 12 T + T^{2} \)
$97$ \( 181 - 27 T + T^{2} \)
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