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Properties

Label 3971.2.a.t
Level $3971$
Weight $2$
Character orbit 3971.a
Self dual yes
Analytic conductor $31.709$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3971,2,Mod(1,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, 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("3971.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.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: \(31.7085946427\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 6 q^{2} + 15 q^{3} + 18 q^{4} + 6 q^{5} - 6 q^{6} + 6 q^{7} + 15 q^{8} + 18 q^{9} - 6 q^{10} + 21 q^{11} + 30 q^{12} + 36 q^{13} + 12 q^{14} + 12 q^{15} + 12 q^{16} + 21 q^{17} + 6 q^{18} + 15 q^{20}+ \cdots + 18 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.56175 2.84434 4.56256 2.10840 −7.28650 −3.15914 −6.56463 5.09029 −5.40120
1.2 −2.12376 0.453579 2.51036 −0.339819 −0.963294 3.66029 −1.08389 −2.79427 0.721696
1.3 −1.78817 2.91057 1.19754 0.686614 −5.20458 −2.35697 1.43493 5.47140 −1.22778
1.4 −1.68287 0.146409 0.832058 3.02083 −0.246387 −2.73938 1.96550 −2.97856 −5.08367
1.5 −1.68098 1.09464 0.825695 −0.508118 −1.84007 −0.0790567 1.97398 −1.80176 0.854136
1.6 −1.40278 2.92541 −0.0322206 1.23149 −4.10370 3.91477 2.85075 5.55805 −1.72751
1.7 −0.480744 −2.58126 −1.76889 1.32013 1.24092 −3.77419 1.81187 3.66289 −0.634645
1.8 −0.382798 0.191200 −1.85347 −2.09484 −0.0731910 0.573305 1.47510 −2.96344 0.801899
1.9 −0.264769 −2.12939 −1.92990 −3.70704 0.563796 3.17316 1.04051 1.53431 0.981509
1.10 −0.0299525 1.47813 −1.99910 −2.92924 −0.0442738 4.42920 0.119783 −0.815126 0.0877383
1.11 0.0844003 0.666890 −1.99288 2.36299 0.0562858 0.533060 −0.337000 −2.55526 0.199437
1.12 0.874945 −1.41672 −1.23447 1.02010 −1.23955 3.79602 −2.82998 −0.992901 0.892534
1.13 1.07764 1.92290 −0.838690 3.18383 2.07219 −3.17241 −3.05909 0.697526 3.43103
1.14 1.22492 1.35788 −0.499565 −2.26081 1.66330 −4.47757 −3.06177 −1.15617 −2.76932
1.15 1.23754 2.23273 −0.468490 3.48025 2.76309 1.94207 −3.05486 1.98507 4.30695
1.16 1.80744 −1.47121 1.26686 2.08596 −2.65913 −2.59855 −1.32512 −0.835544 3.77026
1.17 2.09359 2.89261 2.38311 −3.61022 6.05593 0.264846 0.802085 5.36718 −7.55832
1.18 2.38505 3.05691 3.68846 −1.11454 7.29087 0.620030 4.02705 6.34468 −2.65824
1.19 2.41158 −1.42647 3.81570 0.442094 −3.44003 3.70721 4.37870 −0.965195 1.06614
1.20 2.53856 1.67696 4.44428 2.98943 4.25705 1.55085 6.20496 −0.187815 7.58884
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.t 21
19.b odd 2 1 3971.2.a.s 21
19.e even 9 2 209.2.j.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.j.b ✓ 42 19.e even 9 2
3971.2.a.s 21 19.b odd 2 1
3971.2.a.t 21 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{21} - 6 T_{2}^{20} - 12 T_{2}^{19} + 131 T_{2}^{18} - 30 T_{2}^{17} - 1149 T_{2}^{16} + 1169 T_{2}^{15} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{21} - 15 T_{3}^{20} + 72 T_{3}^{19} + 7 T_{3}^{18} - 1185 T_{3}^{17} + 3141 T_{3}^{16} + \cdots - 856 \) Copy content Toggle raw display