Properties

Label 3971.2.a.b
Level $3971$
Weight $2$
Character orbit 3971.a
Self dual yes
Analytic conductor $31.709$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.7085946427\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
2.00000 1.00000 2.00000 1.00000 2.00000 −2.00000 0 −2.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(19\) \(-1\)

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 3971.2.a.b 1
19.b odd 2 1 11.2.a.a 1
57.d even 2 1 99.2.a.d 1
76.d even 2 1 176.2.a.b 1
95.d odd 2 1 275.2.a.b 1
95.g even 4 2 275.2.b.a 2
133.c even 2 1 539.2.a.a 1
133.o even 6 2 539.2.e.g 2
133.r odd 6 2 539.2.e.h 2
152.b even 2 1 704.2.a.c 1
152.g odd 2 1 704.2.a.h 1
171.l even 6 2 891.2.e.b 2
171.o odd 6 2 891.2.e.k 2
209.d even 2 1 121.2.a.d 1
209.k even 10 4 121.2.c.a 4
209.m odd 10 4 121.2.c.e 4
228.b odd 2 1 1584.2.a.g 1
247.d odd 2 1 1859.2.a.b 1
285.b even 2 1 2475.2.a.a 1
285.j odd 4 2 2475.2.c.a 2
304.j odd 4 2 2816.2.c.j 2
304.m even 4 2 2816.2.c.f 2
323.c odd 2 1 3179.2.a.a 1
380.d even 2 1 4400.2.a.i 1
380.j odd 4 2 4400.2.b.h 2
399.h odd 2 1 4851.2.a.t 1
437.b even 2 1 5819.2.a.a 1
456.l odd 2 1 6336.2.a.bu 1
456.p even 2 1 6336.2.a.br 1
532.b odd 2 1 8624.2.a.j 1
551.d odd 2 1 9251.2.a.d 1
627.b odd 2 1 1089.2.a.b 1
836.h odd 2 1 1936.2.a.i 1
1045.e even 2 1 3025.2.a.a 1
1463.h odd 2 1 5929.2.a.h 1
1672.d odd 2 1 7744.2.a.k 1
1672.j even 2 1 7744.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 19.b odd 2 1
99.2.a.d 1 57.d even 2 1
121.2.a.d 1 209.d even 2 1
121.2.c.a 4 209.k even 10 4
121.2.c.e 4 209.m odd 10 4
176.2.a.b 1 76.d even 2 1
275.2.a.b 1 95.d odd 2 1
275.2.b.a 2 95.g even 4 2
539.2.a.a 1 133.c even 2 1
539.2.e.g 2 133.o even 6 2
539.2.e.h 2 133.r odd 6 2
704.2.a.c 1 152.b even 2 1
704.2.a.h 1 152.g odd 2 1
891.2.e.b 2 171.l even 6 2
891.2.e.k 2 171.o odd 6 2
1089.2.a.b 1 627.b odd 2 1
1584.2.a.g 1 228.b odd 2 1
1859.2.a.b 1 247.d odd 2 1
1936.2.a.i 1 836.h odd 2 1
2475.2.a.a 1 285.b even 2 1
2475.2.c.a 2 285.j odd 4 2
2816.2.c.f 2 304.m even 4 2
2816.2.c.j 2 304.j odd 4 2
3025.2.a.a 1 1045.e even 2 1
3179.2.a.a 1 323.c odd 2 1
3971.2.a.b 1 1.a even 1 1 trivial
4400.2.a.i 1 380.d even 2 1
4400.2.b.h 2 380.j odd 4 2
4851.2.a.t 1 399.h odd 2 1
5819.2.a.a 1 437.b even 2 1
5929.2.a.h 1 1463.h odd 2 1
6336.2.a.br 1 456.p even 2 1
6336.2.a.bu 1 456.l odd 2 1
7744.2.a.k 1 1672.d odd 2 1
7744.2.a.x 1 1672.j even 2 1
8624.2.a.j 1 532.b odd 2 1
9251.2.a.d 1 551.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3971))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 5 \) Copy content Toggle raw display
$61$ \( T - 12 \) Copy content Toggle raw display
$67$ \( T - 7 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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