Properties

Label 3870.2.a.bc
Level 3870
Weight 2
Character orbit 3870.a
Self dual yes
Analytic conductor 30.902
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3870.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9021055822\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 430)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + ( -2 - \beta ) q^{11} + ( 1 + 2 \beta ) q^{13} - q^{14} + q^{16} + \beta q^{17} - q^{19} - q^{20} + ( 2 + \beta ) q^{22} + ( -2 - 5 \beta ) q^{23} + q^{25} + ( -1 - 2 \beta ) q^{26} + q^{28} + ( 3 + 3 \beta ) q^{29} + ( 1 + \beta ) q^{31} - q^{32} -\beta q^{34} - q^{35} + ( -2 - 3 \beta ) q^{37} + q^{38} + q^{40} + ( -1 - 2 \beta ) q^{41} + q^{43} + ( -2 - \beta ) q^{44} + ( 2 + 5 \beta ) q^{46} + 5 \beta q^{47} -6 q^{49} - q^{50} + ( 1 + 2 \beta ) q^{52} -4 \beta q^{53} + ( 2 + \beta ) q^{55} - q^{56} + ( -3 - 3 \beta ) q^{58} + ( 4 + 2 \beta ) q^{59} + ( -7 - 5 \beta ) q^{61} + ( -1 - \beta ) q^{62} + q^{64} + ( -1 - 2 \beta ) q^{65} + ( 3 + 3 \beta ) q^{67} + \beta q^{68} + q^{70} + ( -2 + 7 \beta ) q^{71} + ( -5 + 3 \beta ) q^{73} + ( 2 + 3 \beta ) q^{74} - q^{76} + ( -2 - \beta ) q^{77} + ( -7 + \beta ) q^{79} - q^{80} + ( 1 + 2 \beta ) q^{82} + ( 2 + 4 \beta ) q^{83} -\beta q^{85} - q^{86} + ( 2 + \beta ) q^{88} + ( 2 - 5 \beta ) q^{89} + ( 1 + 2 \beta ) q^{91} + ( -2 - 5 \beta ) q^{92} -5 \beta q^{94} + q^{95} + ( -2 - 11 \beta ) q^{97} + 6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 2q^{10} - 4q^{11} + 2q^{13} - 2q^{14} + 2q^{16} - 2q^{19} - 2q^{20} + 4q^{22} - 4q^{23} + 2q^{25} - 2q^{26} + 2q^{28} + 6q^{29} + 2q^{31} - 2q^{32} - 2q^{35} - 4q^{37} + 2q^{38} + 2q^{40} - 2q^{41} + 2q^{43} - 4q^{44} + 4q^{46} - 12q^{49} - 2q^{50} + 2q^{52} + 4q^{55} - 2q^{56} - 6q^{58} + 8q^{59} - 14q^{61} - 2q^{62} + 2q^{64} - 2q^{65} + 6q^{67} + 2q^{70} - 4q^{71} - 10q^{73} + 4q^{74} - 2q^{76} - 4q^{77} - 14q^{79} - 2q^{80} + 2q^{82} + 4q^{83} - 2q^{86} + 4q^{88} + 4q^{89} + 2q^{91} - 4q^{92} + 2q^{95} - 4q^{97} + 12q^{98} + O(q^{100}) \)

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
1.41421
−1.41421
−1.00000 0 1.00000 −1.00000 0 1.00000 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.00000 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 3870.2.a.bc 2
3.b odd 2 1 430.2.a.g 2
12.b even 2 1 3440.2.a.j 2
15.d odd 2 1 2150.2.a.v 2
15.e even 4 2 2150.2.b.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.g 2 3.b odd 2 1
2150.2.a.v 2 15.d odd 2 1
2150.2.b.o 4 15.e even 4 2
3440.2.a.j 2 12.b even 2 1
3870.2.a.bc 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3870))\):

\( T_{7} - 1 \)
\( T_{11}^{2} + 4 T_{11} + 2 \)
\( T_{13}^{2} - 2 T_{13} - 7 \)
\( T_{17}^{2} - 2 \)
\( T_{19} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 + 4 T + 24 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 32 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 4 T + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 49 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 2 T + 75 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - T )^{2} \)
$47$ \( 1 + 44 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 74 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 125 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 48 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 14 T + 205 T^{2} + 1106 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T + 138 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 4 T + 132 T^{2} - 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 4 T - 44 T^{2} + 388 T^{3} + 9409 T^{4} \)
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