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Properties

Label 3969.2.a.bi
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3969,2,Mod(1,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3969.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{8} + 1) q^{4} - \beta_{10} q^{5} + ( - \beta_{9} + \beta_{8} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{8} + 1) q^{4} - \beta_{10} q^{5} + ( - \beta_{9} + \beta_{8} + 1) q^{8} + (\beta_{7} - \beta_{3} + \beta_1) q^{10} + (\beta_{11} + \beta_{8} - \beta_{2} + 2) q^{11} + ( - \beta_{10} + \beta_{6} + \cdots - \beta_1) q^{13}+ \cdots + (\beta_{10} + \beta_{7} + 3 \beta_{6} + \cdots + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29600 \nu^{11} + 377504 \nu^{10} - 178989 \nu^{9} - 9010433 \nu^{8} + 14881733 \nu^{7} + \cdots - 10235919 ) / 5915292 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 153653 \nu^{11} + 1106428 \nu^{10} + 2049118 \nu^{9} - 26580896 \nu^{8} + 9534853 \nu^{7} + \cdots - 14106727 ) / 8872938 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 153653 \nu^{11} + 1106428 \nu^{10} + 2049118 \nu^{9} - 26580896 \nu^{8} + 9534853 \nu^{7} + \cdots - 14106727 ) / 8872938 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 377200 \nu^{11} - 1719752 \nu^{10} - 7584581 \nu^{9} + 39941737 \nu^{8} + 30456613 \nu^{7} + \cdots + 4252571 ) / 17745876 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 352181 \nu^{11} + 1253056 \nu^{10} + 8078080 \nu^{9} - 28246487 \nu^{8} - 50779916 \nu^{7} + \cdots - 37815883 ) / 8872938 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 491816 \nu^{11} + 1331248 \nu^{10} + 12444820 \nu^{9} - 28741478 \nu^{8} - 96673697 \nu^{7} + \cdots - 16670305 ) / 8872938 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 371053 \nu^{11} - 814814 \nu^{10} - 9883901 \nu^{9} + 16809322 \nu^{8} + 83640367 \nu^{7} + \cdots - 16913407 ) / 5915292 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 491816 \nu^{11} - 1331248 \nu^{10} - 12444820 \nu^{9} + 28741478 \nu^{8} + 96673697 \nu^{7} + \cdots - 5512040 ) / 4436469 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 410574 \nu^{11} + 1474988 \nu^{10} + 9575126 \nu^{9} - 33756309 \nu^{8} - 63418903 \nu^{7} + \cdots - 13564244 ) / 2957646 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3199606 \nu^{11} + 10762397 \nu^{10} + 76422230 \nu^{9} - 243920686 \nu^{8} - 533002183 \nu^{7} + \cdots - 100038029 ) / 17745876 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 604072 \nu^{11} + 2002445 \nu^{10} + 14514229 \nu^{9} - 45298156 \nu^{8} - 102681496 \nu^{7} + \cdots - 18960010 ) / 2957646 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( -\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 2\beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + 3\beta_{6} - 3\beta_{4} - 8\beta_{3} + 10\beta_{2} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 19\beta_{8} - 8\beta_{7} + 28\beta_{6} + \beta_{5} - 4\beta_{4} + 4\beta_{3} + \beta_{2} - 4\beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + 10 \beta_{10} - 28 \beta_{9} + 30 \beta_{8} - 10 \beta_{7} + 60 \beta_{6} + \beta_{5} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} + 12 \beta_{10} - 41 \beta_{9} + 317 \beta_{8} - 176 \beta_{7} + 410 \beta_{6} + \cdots + 726 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 53 \beta_{11} + 266 \beta_{10} - 532 \beta_{9} + 644 \beta_{8} - 294 \beta_{7} + 1078 \beta_{6} + \cdots + 756 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 136 \beta_{11} + 432 \beta_{10} - 993 \beta_{9} + 5138 \beta_{8} - 3136 \beta_{7} + 6336 \beta_{6} + \cdots + 10379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1387 \beta_{11} + 5238 \beta_{10} - 9125 \beta_{9} + 12427 \beta_{8} - 6450 \beta_{7} + 18993 \beta_{6} + \cdots + 15729 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4052 \beta_{11} + 10676 \beta_{10} - 20438 \beta_{9} + 83197 \beta_{8} - 52708 \beta_{7} + \cdots + 155772 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 28918 \beta_{11} + 93324 \beta_{10} - 151655 \beta_{9} + 228997 \beta_{8} - 127292 \beta_{7} + \cdots + 306345 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.762097
−3.59052
−2.71409
0.114341
1.34584
−1.48259
2.51703
−0.311400
0.312397
3.14082
4.12935
1.30092
−2.17631 0 2.73633 −1.26829 0 0 −1.60248 0 2.76019
1.2 −2.17631 0 2.73633 1.26829 0 0 −1.60248 0 −2.76019
1.3 −1.29987 0 −0.310333 −3.52584 0 0 3.00314 0 4.58314
1.4 −1.29987 0 −0.310333 3.52584 0 0 3.00314 0 −4.58314
1.5 −0.0683740 0 −1.99532 −2.66379 0 0 0.273176 0 0.182134
1.6 −0.0683740 0 −1.99532 2.66379 0 0 0.273176 0 −0.182134
1.7 1.10281 0 −0.783802 −0.105466 0 0 −3.07001 0 −0.116309
1.8 1.10281 0 −0.783802 0.105466 0 0 −3.07001 0 0.116309
1.9 1.72661 0 0.981184 −3.51231 0 0 −1.75910 0 −6.06439
1.10 1.72661 0 0.981184 3.51231 0 0 −1.75910 0 6.06439
1.11 2.71513 0 5.37195 −1.58639 0 0 9.15528 0 −4.30727
1.12 2.71513 0 5.37195 1.58639 0 0 9.15528 0 4.30727
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.bi 12
3.b odd 2 1 3969.2.a.bh 12
7.b odd 2 1 inner 3969.2.a.bi 12
9.c even 3 2 1323.2.f.h 24
9.d odd 6 2 441.2.f.h ✓ 24
21.c even 2 1 3969.2.a.bh 12
63.g even 3 2 1323.2.g.h 24
63.h even 3 2 1323.2.h.h 24
63.i even 6 2 441.2.h.h 24
63.j odd 6 2 441.2.h.h 24
63.k odd 6 2 1323.2.g.h 24
63.l odd 6 2 1323.2.f.h 24
63.n odd 6 2 441.2.g.h 24
63.o even 6 2 441.2.f.h ✓ 24
63.s even 6 2 441.2.g.h 24
63.t odd 6 2 1323.2.h.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h ✓ 24 9.d odd 6 2
441.2.f.h ✓ 24 63.o even 6 2
441.2.g.h 24 63.n odd 6 2
441.2.g.h 24 63.s even 6 2
441.2.h.h 24 63.i even 6 2
441.2.h.h 24 63.j odd 6 2
1323.2.f.h 24 9.c even 3 2
1323.2.f.h 24 63.l odd 6 2
1323.2.g.h 24 63.g even 3 2
1323.2.g.h 24 63.k odd 6 2
1323.2.h.h 24 63.h even 3 2
1323.2.h.h 24 63.t odd 6 2
3969.2.a.bh 12 3.b odd 2 1
3969.2.a.bh 12 21.c even 2 1
3969.2.a.bi 12 1.a even 1 1 trivial
3969.2.a.bi 12 7.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 10T_{2}^{2} - 14T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 36T_{5}^{10} + 465T_{5}^{8} - 2580T_{5}^{6} + 5850T_{5}^{4} - 4470T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{6} - 10T_{11}^{5} + 8T_{11}^{4} + 134T_{11}^{3} - 211T_{11}^{2} - 160T_{11} + 283 \) Copy content Toggle raw display
\( T_{13}^{12} - 78T_{13}^{10} + 2283T_{13}^{8} - 31500T_{13}^{6} + 211779T_{13}^{4} - 621030T_{13}^{2} + 502681 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} - 7 T^{4} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 36 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 10 T^{5} + \cdots + 283)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 78 T^{10} + \cdots + 502681 \) Copy content Toggle raw display
$17$ \( T^{12} - 102 T^{10} + \cdots + 707281 \) Copy content Toggle raw display
$19$ \( T^{12} - 144 T^{10} + \cdots + 555025 \) Copy content Toggle raw display
$23$ \( (T^{6} - 16 T^{5} + \cdots + 47)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots + 18395)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 192 T^{10} + \cdots + 60025 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots + 4507)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 2344593241 \) Copy content Toggle raw display
$43$ \( (T^{6} - 171 T^{4} + \cdots + 65005)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 192 T^{10} + \cdots + 2393209 \) Copy content Toggle raw display
$53$ \( (T^{6} - 16 T^{5} + \cdots + 32827)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 354 T^{10} + \cdots + 40309801 \) Copy content Toggle raw display
$61$ \( T^{12} - 384 T^{10} + \cdots + 50481025 \) Copy content Toggle raw display
$67$ \( (T^{6} - 6 T^{5} + \cdots - 15893)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 28 T^{5} + \cdots - 4657)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 168 T^{10} + \cdots + 80089 \) Copy content Toggle raw display
$79$ \( (T^{6} + 6 T^{5} + \cdots + 8207)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 107723641 \) Copy content Toggle raw display
$89$ \( T^{12} - 246 T^{10} + \cdots + 6477025 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 64362167809 \) Copy content Toggle raw display
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