Properties

Label 3971.2.a.k
Level $3971$
Weight $2$
Character orbit 3971.a
Self dual yes
Analytic conductor $31.709$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{6} + \beta_1) q^{5} + (\beta_{7} + \beta_{5} - \beta_1) q^{6} + (\beta_{4} + \beta_{3}) q^{7} + ( - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{6} + \beta_1) q^{5} + (\beta_{7} + \beta_{5} - \beta_1) q^{6} + (\beta_{4} + \beta_{3}) q^{7} + ( - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{8}+ \cdots + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} + 9 q^{4} + 3 q^{6} - 3 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{2} + 9 q^{4} + 3 q^{6} - 3 q^{7} + 11 q^{9} - 21 q^{10} - 9 q^{11} + 8 q^{12} - 11 q^{13} - q^{14} - 7 q^{15} + 13 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{20} - 24 q^{21} + q^{22} + q^{23} + 3 q^{24} + 7 q^{25} - 2 q^{26} + 3 q^{27} + 2 q^{28} - 13 q^{29} - 32 q^{30} - 16 q^{31} + 14 q^{32} - 8 q^{34} + 12 q^{35} - 10 q^{36} - 18 q^{37} + 10 q^{39} - 32 q^{40} - 4 q^{41} - 11 q^{42} - q^{43} - 9 q^{44} + 11 q^{45} - 7 q^{46} + 14 q^{47} - 37 q^{48} - 6 q^{49} + 15 q^{50} + 16 q^{51} - 57 q^{52} + 4 q^{53} + 33 q^{54} - 17 q^{56} + 4 q^{58} - 31 q^{59} - 9 q^{60} - 3 q^{61} - 5 q^{62} - 50 q^{63} + 32 q^{64} - 28 q^{65} - 3 q^{66} + 2 q^{67} - 48 q^{68} + 58 q^{70} + 12 q^{71} - 5 q^{72} + 26 q^{73} - 35 q^{74} + 25 q^{75} + 3 q^{77} - 3 q^{78} - 31 q^{79} - 31 q^{80} + 21 q^{81} - 25 q^{82} + 18 q^{83} - 51 q^{84} + 11 q^{85} - 41 q^{86} - 62 q^{87} + 11 q^{89} - 51 q^{90} + 20 q^{91} + 33 q^{92} - 20 q^{93} - 43 q^{94} + 30 q^{96} - 16 q^{97} - 4 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\nu^{8} - 24\nu^{7} - 111\nu^{6} + 290\nu^{5} + 323\nu^{4} - 949\nu^{3} - 335\nu^{2} + 842\nu + 230 ) / 73 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} + 22\nu^{7} - 26\nu^{6} - 278\nu^{5} + 2\nu^{4} + 1022\nu^{3} + 301\nu^{2} - 1076\nu - 369 ) / 73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{8} - 19\nu^{7} - 170\nu^{6} + 260\nu^{5} + 569\nu^{4} - 1022\nu^{3} - 542\nu^{2} + 1135\nu + 176 ) / 73 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -22\nu^{8} + 9\nu^{7} + 288\nu^{6} - 127\nu^{5} - 1134\nu^{4} + 511\nu^{3} + 1467\nu^{2} - 626\nu - 360 ) / 73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -28\nu^{8} + 38\nu^{7} + 340\nu^{6} - 447\nu^{5} - 1138\nu^{4} + 1387\nu^{3} + 1084\nu^{2} - 1102\nu - 206 ) / 73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\nu^{8} - 50\nu^{7} - 432\nu^{6} + 592\nu^{5} + 1701\nu^{4} - 1825\nu^{3} - 2237\nu^{2} + 1304\nu + 540 ) / 73 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{3} + 7\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{8} + 10\beta_{7} + 9\beta_{6} + 11\beta_{5} + 9\beta_{4} - 9\beta_{2} + 29\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{8} + 10\beta_{7} - 10\beta_{6} - 10\beta_{5} + \beta_{4} + 23\beta_{3} + 49\beta_{2} - 12\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68\beta_{8} + 81\beta_{7} + 67\beta_{6} + 92\beta_{5} + 70\beta_{4} - 71\beta_{2} + 184\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{8} + 78 \beta_{7} - 84 \beta_{6} - 82 \beta_{5} + 13 \beta_{4} + 198 \beta_{3} + 347 \beta_{2} + \cdots + 590 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.63637
2.01377
1.59566
1.05438
−0.138860
−0.396426
−1.23186
−1.83739
−2.69565
−2.63637 −1.26321 4.95045 1.51888 3.33030 1.94612 −7.77846 −1.40429 −4.00432
1.2 −2.01377 2.58057 2.05527 3.55226 −5.19667 −4.09817 −0.111292 3.65935 −7.15343
1.3 −1.59566 1.16701 0.546144 −0.413278 −1.86216 2.20560 2.31987 −1.63808 0.659453
1.4 −1.05438 −3.26124 −0.888282 1.30020 3.43859 −1.25919 3.04535 7.63569 −1.37091
1.5 0.138860 −1.49679 −1.98072 −3.51650 −0.207844 −1.46902 −0.552764 −0.759634 −0.488302
1.6 0.396426 1.26801 −1.84285 0.441921 0.502672 −0.667657 −1.52341 −1.39215 0.175189
1.7 1.23186 −2.02769 −0.482521 2.63096 −2.49783 3.75846 −3.05812 1.11154 3.24098
1.8 1.83739 3.12711 1.37600 −3.26983 5.74572 −3.63195 −1.14653 6.77879 −6.00795
1.9 2.69565 −0.0937671 5.26651 −2.24462 −0.252763 0.215810 8.80535 −2.99121 −6.05070
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.k 9
19.b odd 2 1 3971.2.a.l 9
19.c even 3 2 209.2.e.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.e.b 18 19.c even 3 2
3971.2.a.k 9 1.a even 1 1 trivial
3971.2.a.l 9 19.b odd 2 1

Hecke kernels

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

\( T_{2}^{9} + T_{2}^{8} - 13T_{2}^{7} - 12T_{2}^{6} + 51T_{2}^{5} + 38T_{2}^{4} - 70T_{2}^{3} - 30T_{2}^{2} + 27T_{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{9} - 19T_{3}^{7} - T_{3}^{6} + 110T_{3}^{5} + 18T_{3}^{4} - 224T_{3}^{3} - 35T_{3}^{2} + 148T_{3} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} - 13 T^{7} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{9} - 19 T^{7} + \cdots + 14 \) Copy content Toggle raw display
$5$ \( T^{9} - 26 T^{7} + \cdots - 87 \) Copy content Toggle raw display
$7$ \( T^{9} + 3 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} + 11 T^{8} + \cdots - 4798 \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots - 3684 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - T^{8} + \cdots + 270 \) Copy content Toggle raw display
$29$ \( T^{9} + 13 T^{8} + \cdots - 680694 \) Copy content Toggle raw display
$31$ \( T^{9} + 16 T^{8} + \cdots - 331264 \) Copy content Toggle raw display
$37$ \( T^{9} + 18 T^{8} + \cdots - 1086460 \) Copy content Toggle raw display
$41$ \( T^{9} + 4 T^{8} + \cdots + 11275008 \) Copy content Toggle raw display
$43$ \( T^{9} + T^{8} + \cdots + 6464 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} + \cdots - 318 \) Copy content Toggle raw display
$53$ \( T^{9} - 4 T^{8} + \cdots + 16365 \) Copy content Toggle raw display
$59$ \( T^{9} + 31 T^{8} + \cdots - 8751342 \) Copy content Toggle raw display
$61$ \( T^{9} + 3 T^{8} + \cdots + 4046560 \) Copy content Toggle raw display
$67$ \( T^{9} - 2 T^{8} + \cdots - 66714160 \) Copy content Toggle raw display
$71$ \( T^{9} - 12 T^{8} + \cdots + 3181524 \) Copy content Toggle raw display
$73$ \( T^{9} - 26 T^{8} + \cdots + 347318 \) Copy content Toggle raw display
$79$ \( T^{9} + 31 T^{8} + \cdots + 225935 \) Copy content Toggle raw display
$83$ \( T^{9} - 18 T^{8} + \cdots - 3773520 \) Copy content Toggle raw display
$89$ \( T^{9} - 11 T^{8} + \cdots - 54960 \) Copy content Toggle raw display
$97$ \( T^{9} + 16 T^{8} + \cdots - 391369283 \) Copy content Toggle raw display
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