Properties

Label 3887.2.a.i
Level $3887$
Weight $2$
Character orbit 3887.a
Self dual yes
Analytic conductor $31.038$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3887 = 13^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3887.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.0378512657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 \beta q^{5} + ( 2 + \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + 2 q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 \beta q^{5} + ( 2 + \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + 2 q^{9} + ( 2 + 2 \beta ) q^{10} + ( 4 - 2 \beta ) q^{11} + ( 3 - \beta ) q^{12} + 2 q^{14} + ( 4 + 2 \beta ) q^{15} -3 \beta q^{16} + ( 2 + 2 \beta ) q^{17} + 2 \beta q^{18} + 2 q^{19} + 2 q^{20} + ( 6 - 2 \beta ) q^{21} + ( -2 + 2 \beta ) q^{22} + q^{23} -5 q^{24} + ( -1 + 4 \beta ) q^{25} + ( 1 - 2 \beta ) q^{27} + ( 4 - 2 \beta ) q^{28} -3 q^{29} + ( 2 + 6 \beta ) q^{30} + ( -3 + 6 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( -8 + 6 \beta ) q^{33} + ( 2 + 4 \beta ) q^{34} + 4 q^{35} + ( -2 + 2 \beta ) q^{36} -2 \beta q^{37} + 2 \beta q^{38} + ( -4 - 2 \beta ) q^{40} + ( 1 - 4 \beta ) q^{41} + ( -2 + 4 \beta ) q^{42} + ( -6 + 4 \beta ) q^{44} + 4 \beta q^{45} + \beta q^{46} + ( 1 - 2 \beta ) q^{47} + ( -6 - 3 \beta ) q^{48} + ( 1 - 4 \beta ) q^{49} + ( 4 + 3 \beta ) q^{50} + ( 2 + 6 \beta ) q^{51} + ( -2 - 4 \beta ) q^{53} + ( -2 - \beta ) q^{54} + ( -4 + 4 \beta ) q^{55} + ( -6 + 2 \beta ) q^{56} + ( -2 + 4 \beta ) q^{57} -3 \beta q^{58} + ( -4 + 4 \beta ) q^{59} + ( -2 + 4 \beta ) q^{60} + ( -2 + 8 \beta ) q^{61} + ( 6 + 3 \beta ) q^{62} + ( -4 + 4 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( 6 - 2 \beta ) q^{66} + ( 4 + 2 \beta ) q^{67} + 2 \beta q^{68} + ( -1 + 2 \beta ) q^{69} + 4 \beta q^{70} + ( -11 + 2 \beta ) q^{71} + ( 2 - 4 \beta ) q^{72} + ( -9 - 4 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} + ( 9 + 2 \beta ) q^{75} + ( -2 + 2 \beta ) q^{76} + ( -12 + 8 \beta ) q^{77} + ( -6 + 8 \beta ) q^{79} + ( -6 - 6 \beta ) q^{80} -11 q^{81} + ( -4 - 3 \beta ) q^{82} + ( 10 + 2 \beta ) q^{83} + ( -8 + 6 \beta ) q^{84} + ( 4 + 8 \beta ) q^{85} + ( 3 - 6 \beta ) q^{87} + ( 8 - 6 \beta ) q^{88} + ( 8 - 4 \beta ) q^{89} + ( 4 + 4 \beta ) q^{90} + ( -1 + \beta ) q^{92} + 15 q^{93} + ( -2 - \beta ) q^{94} + 4 \beta q^{95} + ( 7 - 9 \beta ) q^{96} + ( -14 + 6 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} + ( 8 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 2q^{5} + 5q^{6} - 2q^{7} + 4q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 2q^{5} + 5q^{6} - 2q^{7} + 4q^{9} + 6q^{10} + 6q^{11} + 5q^{12} + 4q^{14} + 10q^{15} - 3q^{16} + 6q^{17} + 2q^{18} + 4q^{19} + 4q^{20} + 10q^{21} - 2q^{22} + 2q^{23} - 10q^{24} + 2q^{25} + 6q^{28} - 6q^{29} + 10q^{30} - 9q^{32} - 10q^{33} + 8q^{34} + 8q^{35} - 2q^{36} - 2q^{37} + 2q^{38} - 10q^{40} - 2q^{41} - 8q^{44} + 4q^{45} + q^{46} - 15q^{48} - 2q^{49} + 11q^{50} + 10q^{51} - 8q^{53} - 5q^{54} - 4q^{55} - 10q^{56} - 3q^{58} - 4q^{59} + 4q^{61} + 15q^{62} - 4q^{63} + 4q^{64} + 10q^{66} + 10q^{67} + 2q^{68} + 4q^{70} - 20q^{71} - 22q^{73} - 6q^{74} + 20q^{75} - 2q^{76} - 16q^{77} - 4q^{79} - 18q^{80} - 22q^{81} - 11q^{82} + 22q^{83} - 10q^{84} + 16q^{85} + 10q^{88} + 12q^{89} + 12q^{90} - q^{92} + 30q^{93} - 5q^{94} + 4q^{95} + 5q^{96} - 22q^{97} - 11q^{98} + 12q^{99} + O(q^{100}) \)

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
−0.618034 −2.23607 −1.61803 −1.23607 1.38197 −3.23607 2.23607 2.00000 0.763932
1.2 1.61803 2.23607 0.618034 3.23607 3.61803 1.23607 −2.23607 2.00000 5.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(23\) \(-1\)

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 3887.2.a.i 2
13.b even 2 1 23.2.a.a 2
39.d odd 2 1 207.2.a.d 2
52.b odd 2 1 368.2.a.h 2
65.d even 2 1 575.2.a.f 2
65.h odd 4 2 575.2.b.d 4
91.b odd 2 1 1127.2.a.c 2
104.e even 2 1 1472.2.a.t 2
104.h odd 2 1 1472.2.a.s 2
143.d odd 2 1 2783.2.a.c 2
156.h even 2 1 3312.2.a.ba 2
195.e odd 2 1 5175.2.a.be 2
221.b even 2 1 6647.2.a.b 2
247.d odd 2 1 8303.2.a.e 2
260.g odd 2 1 9200.2.a.bt 2
299.c odd 2 1 529.2.a.a 2
299.o odd 22 10 529.2.c.n 20
299.p even 22 10 529.2.c.o 20
897.g even 2 1 4761.2.a.w 2
1196.d even 2 1 8464.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 13.b even 2 1
207.2.a.d 2 39.d odd 2 1
368.2.a.h 2 52.b odd 2 1
529.2.a.a 2 299.c odd 2 1
529.2.c.n 20 299.o odd 22 10
529.2.c.o 20 299.p even 22 10
575.2.a.f 2 65.d even 2 1
575.2.b.d 4 65.h odd 4 2
1127.2.a.c 2 91.b odd 2 1
1472.2.a.s 2 104.h odd 2 1
1472.2.a.t 2 104.e even 2 1
2783.2.a.c 2 143.d odd 2 1
3312.2.a.ba 2 156.h even 2 1
3887.2.a.i 2 1.a even 1 1 trivial
4761.2.a.w 2 897.g even 2 1
5175.2.a.be 2 195.e odd 2 1
6647.2.a.b 2 221.b even 2 1
8303.2.a.e 2 247.d odd 2 1
8464.2.a.bb 2 1196.d even 2 1
9200.2.a.bt 2 260.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3887))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{3}^{2} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( 4 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( -45 + T^{2} \)
$37$ \( -4 + 2 T + T^{2} \)
$41$ \( -19 + 2 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -5 + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -16 + 4 T + T^{2} \)
$61$ \( -76 - 4 T + T^{2} \)
$67$ \( 20 - 10 T + T^{2} \)
$71$ \( 95 + 20 T + T^{2} \)
$73$ \( 101 + 22 T + T^{2} \)
$79$ \( -76 + 4 T + T^{2} \)
$83$ \( 116 - 22 T + T^{2} \)
$89$ \( 16 - 12 T + T^{2} \)
$97$ \( 76 + 22 T + T^{2} \)
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