Properties

Label 3969.2.a.l
Level 3969
Weight 2
Character orbit 3969.a
Self dual yes
Analytic conductor 31.693
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 3 + \beta - \beta^{2} ) q^{5} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 3 + \beta - \beta^{2} ) q^{5} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{8} + ( -4 - \beta + 2 \beta^{2} ) q^{10} + ( \beta - \beta^{2} ) q^{11} + ( -7 - 2 \beta + 4 \beta^{2} ) q^{13} + ( -5 - 3 \beta + 3 \beta^{2} ) q^{16} + ( 4 - \beta^{2} ) q^{17} + ( 3 + 2 \beta - \beta^{2} ) q^{19} + ( \beta - \beta^{2} ) q^{20} + ( -1 - 4 \beta + 2 \beta^{2} ) q^{22} + ( -\beta - 2 \beta^{2} ) q^{23} + ( 2 + \beta - 2 \beta^{2} ) q^{25} + ( 11 + 7 \beta - 6 \beta^{2} ) q^{26} + ( -11 - 5 \beta + 4 \beta^{2} ) q^{29} + ( 7 - 3 \beta - 3 \beta^{2} ) q^{31} + 3 \beta q^{32} + ( -5 + \beta + \beta^{2} ) q^{34} + ( -7 + 3 \beta + 3 \beta^{2} ) q^{37} + ( -4 - 2 \beta + 3 \beta^{2} ) q^{38} + ( 7 - 2 \beta - 2 \beta^{2} ) q^{40} + ( -2 + \beta + \beta^{2} ) q^{41} + ( 1 - \beta - \beta^{2} ) q^{43} + ( 3 + 7 \beta - 4 \beta^{2} ) q^{44} + ( -2 - 5 \beta + \beta^{2} ) q^{46} + ( -5 + 2 \beta + 3 \beta^{2} ) q^{47} + ( -4 - 5 \beta + 3 \beta^{2} ) q^{50} + ( -3 - 10 \beta + 5 \beta^{2} ) q^{52} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{53} + ( -2 - 2 \beta + \beta^{2} ) q^{55} + ( 15 + 6 \beta - 9 \beta^{2} ) q^{58} + ( -1 - 5 \beta ) q^{59} + ( -2 - 3 \beta ) q^{61} + ( -10 + \beta ) q^{62} + ( 10 + 3 \beta - 3 \beta^{2} ) q^{64} + ( -15 + \beta + 5 \beta^{2} ) q^{65} + ( -10 + 3 \beta^{2} ) q^{67} + ( -2 - 3 \beta + 2 \beta^{2} ) q^{68} + ( -15 - 3 \beta + 6 \beta^{2} ) q^{71} + ( 9 - 4 \beta - \beta^{2} ) q^{73} + ( 10 - \beta ) q^{74} + ( 1 + 3 \beta - 3 \beta^{2} ) q^{76} + ( -13 + 3 \beta^{2} ) q^{79} + ( -9 + \beta + 2 \beta^{2} ) q^{80} + 3 q^{82} + ( -4 - 4 \beta - \beta^{2} ) q^{83} + ( 11 + 2 \beta - 4 \beta^{2} ) q^{85} + ( -2 - \beta ) q^{86} + ( -5 - 8 \beta + 7 \beta^{2} ) q^{88} + ( -10 - 3 \beta + 7 \beta^{2} ) q^{89} + ( 3 + 8 \beta - 2 \beta^{2} ) q^{92} + ( 8 + 2 \beta - \beta^{2} ) q^{94} + ( 6 + \beta - \beta^{2} ) q^{95} + ( 17 + 7 \beta - 8 \beta^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} - 6q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} - 6q^{8} - 6q^{11} + 3q^{13} + 3q^{16} + 6q^{17} + 3q^{19} - 6q^{20} + 9q^{22} - 12q^{23} - 6q^{25} - 3q^{26} - 9q^{29} + 3q^{31} - 9q^{34} - 3q^{37} + 6q^{38} + 9q^{40} - 3q^{43} - 15q^{44} + 3q^{47} + 6q^{50} + 21q^{52} - 6q^{53} - 9q^{58} - 3q^{59} - 6q^{61} - 30q^{62} + 12q^{64} - 15q^{65} - 12q^{67} + 6q^{68} - 9q^{71} + 21q^{73} + 30q^{74} - 15q^{76} - 21q^{79} - 15q^{80} + 9q^{82} - 18q^{83} + 9q^{85} - 6q^{86} + 27q^{88} + 12q^{89} - 3q^{92} + 18q^{94} + 12q^{95} + 3q^{97} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
−2.53209 0 4.41147 −0.879385 0 0 −6.10607 0 2.22668
1.2 −1.34730 0 −0.184793 2.53209 0 0 2.94356 0 −3.41147
1.3 0.879385 0 −1.22668 1.34730 0 0 −2.83750 0 1.18479
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

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 3969.2.a.l 3
3.b odd 2 1 3969.2.a.q 3
7.b odd 2 1 567.2.a.c 3
9.c even 3 2 1323.2.f.d 6
9.d odd 6 2 441.2.f.c 6
21.c even 2 1 567.2.a.h 3
28.d even 2 1 9072.2.a.bs 3
63.g even 3 2 1323.2.g.e 6
63.h even 3 2 1323.2.h.b 6
63.i even 6 2 441.2.h.d 6
63.j odd 6 2 441.2.h.e 6
63.k odd 6 2 1323.2.g.d 6
63.l odd 6 2 189.2.f.b 6
63.n odd 6 2 441.2.g.b 6
63.o even 6 2 63.2.f.a 6
63.s even 6 2 441.2.g.c 6
63.t odd 6 2 1323.2.h.c 6
84.h odd 2 1 9072.2.a.ca 3
252.s odd 6 2 1008.2.r.h 6
252.bi even 6 2 3024.2.r.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 63.o even 6 2
189.2.f.b 6 63.l odd 6 2
441.2.f.c 6 9.d odd 6 2
441.2.g.b 6 63.n odd 6 2
441.2.g.c 6 63.s even 6 2
441.2.h.d 6 63.i even 6 2
441.2.h.e 6 63.j odd 6 2
567.2.a.c 3 7.b odd 2 1
567.2.a.h 3 21.c even 2 1
1008.2.r.h 6 252.s odd 6 2
1323.2.f.d 6 9.c even 3 2
1323.2.g.d 6 63.k odd 6 2
1323.2.g.e 6 63.g even 3 2
1323.2.h.b 6 63.h even 3 2
1323.2.h.c 6 63.t odd 6 2
3024.2.r.k 6 252.bi even 6 2
3969.2.a.l 3 1.a even 1 1 trivial
3969.2.a.q 3 3.b odd 2 1
9072.2.a.bs 3 28.d even 2 1
9072.2.a.ca 3 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{3} + 3 T_{2}^{2} - 3 \)
\( T_{5}^{3} - 3 T_{5}^{2} + 3 \)
\( T_{11}^{3} + 6 T_{11}^{2} + 9 T_{11} + 3 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 33 T_{13} + 107 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6} \)
$3$ 1
$5$ \( 1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6} \)
$7$ 1
$11$ \( 1 + 6 T + 42 T^{2} + 135 T^{3} + 462 T^{4} + 726 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 3 T + 6 T^{2} + 29 T^{3} + 78 T^{4} - 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 6 T + 60 T^{2} - 207 T^{3} + 1020 T^{4} - 1734 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 3 T + 51 T^{2} - 97 T^{3} + 969 T^{4} - 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 12 T + 96 T^{2} + 549 T^{3} + 2208 T^{4} + 6348 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 9 T + 51 T^{2} + 189 T^{3} + 1479 T^{4} + 7569 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 3 T + 15 T^{2} + 137 T^{3} + 465 T^{4} - 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 114 T^{2} - 9 T^{3} + 4674 T^{4} + 68921 T^{6} \)
$43$ \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 5289 T^{4} + 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 3 T + 87 T^{2} - 333 T^{3} + 4089 T^{4} - 6627 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 6 T + 150 T^{2} + 639 T^{3} + 7950 T^{4} + 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 3 T + 105 T^{2} + 405 T^{3} + 6195 T^{4} + 10443 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 168 T^{2} + 713 T^{3} + 10248 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 14874 T^{4} + 53868 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 11289 T^{4} + 45369 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 21 T + 303 T^{2} - 2797 T^{3} + 22119 T^{4} - 111909 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 28203 T^{4} + 131061 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 18 T + 294 T^{2} + 2997 T^{3} + 24402 T^{4} + 124002 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 18156 T^{4} - 95052 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 3 T + 123 T^{2} - 259 T^{3} + 11931 T^{4} - 28227 T^{5} + 912673 T^{6} \)
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