Properties

Label 31.2.a.a
Level 31
Weight 2
Character orbit 31.a
Self dual yes
Analytic conductor 0.248
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 \beta q^{3} + ( -1 + \beta ) q^{4} + q^{5} + ( -2 - 2 \beta ) q^{6} + ( -3 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( 1 + 4 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} -2 \beta q^{3} + ( -1 + \beta ) q^{4} + q^{5} + ( -2 - 2 \beta ) q^{6} + ( -3 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( 1 + 4 \beta ) q^{9} + \beta q^{10} + 2 q^{11} -2 q^{12} -2 \beta q^{13} + ( 2 - \beta ) q^{14} -2 \beta q^{15} -3 \beta q^{16} + ( 4 - 2 \beta ) q^{17} + ( 4 + 5 \beta ) q^{18} + ( 1 - 2 \beta ) q^{19} + ( -1 + \beta ) q^{20} + ( -4 + 2 \beta ) q^{21} + 2 \beta q^{22} + ( -4 + 6 \beta ) q^{23} + ( 4 + 2 \beta ) q^{24} -4 q^{25} + ( -2 - 2 \beta ) q^{26} + ( -8 - 4 \beta ) q^{27} + ( 5 - 3 \beta ) q^{28} + ( 6 - 2 \beta ) q^{29} + ( -2 - 2 \beta ) q^{30} + q^{31} + ( -5 + \beta ) q^{32} -4 \beta q^{33} + ( -2 + 2 \beta ) q^{34} + ( -3 + 2 \beta ) q^{35} + ( 3 + \beta ) q^{36} -2 q^{37} + ( -2 - \beta ) q^{38} + ( 4 + 4 \beta ) q^{39} + ( 1 - 2 \beta ) q^{40} + 7 q^{41} + ( 2 - 2 \beta ) q^{42} + ( -2 + 2 \beta ) q^{43} + ( -2 + 2 \beta ) q^{44} + ( 1 + 4 \beta ) q^{45} + ( 6 + 2 \beta ) q^{46} + ( -4 + 4 \beta ) q^{47} + ( 6 + 6 \beta ) q^{48} + ( 6 - 8 \beta ) q^{49} -4 \beta q^{50} + ( 4 - 4 \beta ) q^{51} -2 q^{52} + ( -4 - 4 \beta ) q^{53} + ( -4 - 12 \beta ) q^{54} + 2 q^{55} + ( -7 + 4 \beta ) q^{56} + ( 4 + 2 \beta ) q^{57} + ( -2 + 4 \beta ) q^{58} + ( -1 + 2 \beta ) q^{59} -2 q^{60} + ( -8 + 10 \beta ) q^{61} + \beta q^{62} + ( 5 - 2 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} -2 \beta q^{65} + ( -4 - 4 \beta ) q^{66} + 8 q^{67} + ( -6 + 4 \beta ) q^{68} + ( -12 - 4 \beta ) q^{69} + ( 2 - \beta ) q^{70} + ( 7 - 10 \beta ) q^{71} + ( -7 - 6 \beta ) q^{72} + ( 2 + 4 \beta ) q^{73} -2 \beta q^{74} + 8 \beta q^{75} + ( -3 + \beta ) q^{76} + ( -6 + 4 \beta ) q^{77} + ( 4 + 8 \beta ) q^{78} + ( -2 - 6 \beta ) q^{79} -3 \beta q^{80} + ( 5 + 12 \beta ) q^{81} + 7 \beta q^{82} + ( -2 - 8 \beta ) q^{83} + ( 6 - 4 \beta ) q^{84} + ( 4 - 2 \beta ) q^{85} + 2 q^{86} + ( 4 - 8 \beta ) q^{87} + ( 2 - 4 \beta ) q^{88} + ( 2 + 6 \beta ) q^{89} + ( 4 + 5 \beta ) q^{90} + ( -4 + 2 \beta ) q^{91} + ( 10 - 4 \beta ) q^{92} -2 \beta q^{93} + 4 q^{94} + ( 1 - 2 \beta ) q^{95} + ( -2 + 8 \beta ) q^{96} + ( -3 - 8 \beta ) q^{97} + ( -8 - 2 \beta ) q^{98} + ( 2 + 8 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 6q^{6} - 4q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} + 2q^{5} - 6q^{6} - 4q^{7} + 6q^{9} + q^{10} + 4q^{11} - 4q^{12} - 2q^{13} + 3q^{14} - 2q^{15} - 3q^{16} + 6q^{17} + 13q^{18} - q^{20} - 6q^{21} + 2q^{22} - 2q^{23} + 10q^{24} - 8q^{25} - 6q^{26} - 20q^{27} + 7q^{28} + 10q^{29} - 6q^{30} + 2q^{31} - 9q^{32} - 4q^{33} - 2q^{34} - 4q^{35} + 7q^{36} - 4q^{37} - 5q^{38} + 12q^{39} + 14q^{41} + 2q^{42} - 2q^{43} - 2q^{44} + 6q^{45} + 14q^{46} - 4q^{47} + 18q^{48} + 4q^{49} - 4q^{50} + 4q^{51} - 4q^{52} - 12q^{53} - 20q^{54} + 4q^{55} - 10q^{56} + 10q^{57} - 4q^{60} - 6q^{61} + q^{62} + 8q^{63} + 4q^{64} - 2q^{65} - 12q^{66} + 16q^{67} - 8q^{68} - 28q^{69} + 3q^{70} + 4q^{71} - 20q^{72} + 8q^{73} - 2q^{74} + 8q^{75} - 5q^{76} - 8q^{77} + 16q^{78} - 10q^{79} - 3q^{80} + 22q^{81} + 7q^{82} - 12q^{83} + 8q^{84} + 6q^{85} + 4q^{86} + 10q^{89} + 13q^{90} - 6q^{91} + 16q^{92} - 2q^{93} + 8q^{94} + 4q^{96} - 14q^{97} - 18q^{98} + 12q^{99} + O(q^{100}) \)

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 1.23607 −1.61803 1.00000 −0.763932 −4.23607 2.23607 −1.47214 −0.618034
1.2 1.61803 −3.23607 0.618034 1.00000 −5.23607 0.236068 −2.23607 7.47214 1.61803
\(n\): 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 31.2.a.a 2
3.b odd 2 1 279.2.a.a 2
4.b odd 2 1 496.2.a.i 2
5.b even 2 1 775.2.a.d 2
5.c odd 4 2 775.2.b.d 4
7.b odd 2 1 1519.2.a.a 2
8.b even 2 1 1984.2.a.r 2
8.d odd 2 1 1984.2.a.n 2
11.b odd 2 1 3751.2.a.b 2
12.b even 2 1 4464.2.a.bf 2
13.b even 2 1 5239.2.a.f 2
15.d odd 2 1 6975.2.a.y 2
17.b even 2 1 8959.2.a.b 2
31.b odd 2 1 961.2.a.f 2
31.c even 3 2 961.2.c.e 4
31.d even 5 2 961.2.d.c 4
31.d even 5 2 961.2.d.d 4
31.e odd 6 2 961.2.c.c 4
31.f odd 10 2 961.2.d.a 4
31.f odd 10 2 961.2.d.g 4
31.g even 15 4 961.2.g.a 8
31.g even 15 4 961.2.g.h 8
31.h odd 30 4 961.2.g.d 8
31.h odd 30 4 961.2.g.e 8
93.c even 2 1 8649.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.a.a 2 1.a even 1 1 trivial
279.2.a.a 2 3.b odd 2 1
496.2.a.i 2 4.b odd 2 1
775.2.a.d 2 5.b even 2 1
775.2.b.d 4 5.c odd 4 2
961.2.a.f 2 31.b odd 2 1
961.2.c.c 4 31.e odd 6 2
961.2.c.e 4 31.c even 3 2
961.2.d.a 4 31.f odd 10 2
961.2.d.c 4 31.d even 5 2
961.2.d.d 4 31.d even 5 2
961.2.d.g 4 31.f odd 10 2
961.2.g.a 8 31.g even 15 4
961.2.g.d 8 31.h odd 30 4
961.2.g.e 8 31.h odd 30 4
961.2.g.h 8 31.g even 15 4
1519.2.a.a 2 7.b odd 2 1
1984.2.a.n 2 8.d odd 2 1
1984.2.a.r 2 8.b even 2 1
3751.2.a.b 2 11.b odd 2 1
4464.2.a.bf 2 12.b even 2 1
5239.2.a.f 2 13.b even 2 1
6975.2.a.y 2 15.d odd 2 1
8649.2.a.c 2 93.c even 2 1
8959.2.a.b 2 17.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(31\) \(-1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ \( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ \( 1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 33 T^{2} + 361 T^{4} \)
$23$ \( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 10 T + 78 T^{2} - 290 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - T )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 7 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 2 T + 82 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 113 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 6 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 4 T + 21 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 122 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 10 T + 158 T^{2} - 890 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 14 T + 163 T^{2} + 1358 T^{3} + 9409 T^{4} \)
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