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Properties

Label 4010.2.a.m
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $20$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, 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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + \cdots + 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{16} + 1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{16} + 1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9} - q^{10} + (\beta_{15} + 1) q^{11} + \beta_1 q^{12} + (\beta_{16} - \beta_{4}) q^{13} + ( - \beta_{16} - 1) q^{14} + \beta_1 q^{15} + q^{16} + (\beta_{19} - \beta_{18} + \beta_{15} + \cdots + 1) q^{17}+ \cdots + (2 \beta_{19} - 2 \beta_{18} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + \cdots + 104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!27 \nu^{19} + \cdots - 19\!\cdots\!52 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!11 \nu^{19} + \cdots + 84\!\cdots\!48 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!34 \nu^{19} + \cdots - 97\!\cdots\!84 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 99\!\cdots\!11 \nu^{19} + \cdots + 30\!\cdots\!68 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!06 \nu^{19} + \cdots + 93\!\cdots\!87 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46\!\cdots\!05 \nu^{19} + \cdots - 21\!\cdots\!41 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!55 \nu^{19} + \cdots + 76\!\cdots\!12 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 47\!\cdots\!62 \nu^{19} + \cdots - 22\!\cdots\!25 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!53 \nu^{19} + \cdots - 43\!\cdots\!52 ) / 63\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!01 \nu^{19} + \cdots - 60\!\cdots\!76 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 90\!\cdots\!82 \nu^{19} + \cdots + 14\!\cdots\!68 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36\!\cdots\!53 \nu^{19} + \cdots + 15\!\cdots\!16 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39\!\cdots\!35 \nu^{19} + \cdots - 61\!\cdots\!32 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!94 \nu^{19} + \cdots + 43\!\cdots\!53 ) / 31\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 54\!\cdots\!89 \nu^{19} + \cdots + 11\!\cdots\!60 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 27\!\cdots\!13 \nu^{19} + \cdots + 78\!\cdots\!26 ) / 63\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 11\!\cdots\!99 \nu^{19} + \cdots + 30\!\cdots\!84 ) / 12\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 2\beta_{18} - \beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} - 2 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{19} - 28 \beta_{18} - \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15 \beta_{19} - 16 \beta_{18} - 32 \beta_{16} - 45 \beta_{15} - 32 \beta_{14} - 15 \beta_{13} + \cdots + 190 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 197 \beta_{19} - 324 \beta_{18} - 13 \beta_{17} - 40 \beta_{16} + 34 \beta_{15} - 162 \beta_{14} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 179 \beta_{19} - 212 \beta_{18} + \beta_{17} - 394 \beta_{16} - 537 \beta_{15} - 400 \beta_{14} + \cdots + 1474 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2242 \beta_{19} - 3548 \beta_{18} - 130 \beta_{17} - 589 \beta_{16} + 458 \beta_{15} - 1775 \beta_{14} + \cdots + 792 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2022 \beta_{19} - 2637 \beta_{18} + \beta_{17} - 4475 \beta_{16} - 5936 \beta_{15} - 4580 \beta_{14} + \cdots + 11839 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 24694 \beta_{19} - 38052 \beta_{18} - 1220 \beta_{17} - 7703 \beta_{16} + 5632 \beta_{15} - 19042 \beta_{14} + \cdots + 10042 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 22669 \beta_{19} - 31860 \beta_{18} - 353 \beta_{17} - 49411 \beta_{16} - 63248 \beta_{15} + \cdots + 97185 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 267779 \beta_{19} - 404748 \beta_{18} - 11574 \beta_{17} - 94960 \beta_{16} + 65865 \beta_{15} + \cdots + 119688 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 255922 \beta_{19} - 378970 \beta_{18} - 9832 \beta_{17} - 540829 \beta_{16} - 659313 \beta_{15} + \cdots + 809827 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2881358 \beta_{19} - 4293131 \beta_{18} - 115183 \beta_{17} - 1132664 \beta_{16} + 746471 \beta_{15} + \cdots + 1369666 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2916094 \beta_{19} - 4463886 \beta_{18} - 185101 \beta_{17} - 5912648 \beta_{16} - 6771585 \beta_{15} + \cdots + 6824720 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 30888593 \beta_{19} - 45526694 \beta_{18} - 1217038 \beta_{17} - 13245715 \beta_{16} + 8285536 \beta_{15} + \cdots + 15231619 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 33465540 \beta_{19} - 52204864 \beta_{18} - 2926840 \beta_{17} - 64734513 \beta_{16} - 68796280 \beta_{15} + \cdots + 58044469 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 330626168 \beta_{19} - 483298046 \beta_{18} - 13558852 \beta_{17} - 152965455 \beta_{16} + \cdots + 165879535 \) 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
−3.16561
−2.99276
−2.78165
−2.20278
−1.97232
−1.72088
−0.330073
−0.323343
−0.241588
−0.0780369
0.622776
1.02602
1.03584
1.30183
1.96074
2.32194
2.35112
2.72853
3.13315
3.32710
−1.00000 −3.16561 1.00000 1.00000 3.16561 0.575814 −1.00000 7.02106 −1.00000
1.2 −1.00000 −2.99276 1.00000 1.00000 2.99276 0.839081 −1.00000 5.95662 −1.00000
1.3 −1.00000 −2.78165 1.00000 1.00000 2.78165 −3.35211 −1.00000 4.73759 −1.00000
1.4 −1.00000 −2.20278 1.00000 1.00000 2.20278 4.44205 −1.00000 1.85224 −1.00000
1.5 −1.00000 −1.97232 1.00000 1.00000 1.97232 4.46416 −1.00000 0.890065 −1.00000
1.6 −1.00000 −1.72088 1.00000 1.00000 1.72088 −0.251451 −1.00000 −0.0385865 −1.00000
1.7 −1.00000 −0.330073 1.00000 1.00000 0.330073 −4.66703 −1.00000 −2.89105 −1.00000
1.8 −1.00000 −0.323343 1.00000 1.00000 0.323343 4.27073 −1.00000 −2.89545 −1.00000
1.9 −1.00000 −0.241588 1.00000 1.00000 0.241588 0.516163 −1.00000 −2.94164 −1.00000
1.10 −1.00000 −0.0780369 1.00000 1.00000 0.0780369 −1.81697 −1.00000 −2.99391 −1.00000
1.11 −1.00000 0.622776 1.00000 1.00000 −0.622776 3.30687 −1.00000 −2.61215 −1.00000
1.12 −1.00000 1.02602 1.00000 1.00000 −1.02602 −3.81201 −1.00000 −1.94728 −1.00000
1.13 −1.00000 1.03584 1.00000 1.00000 −1.03584 −1.66553 −1.00000 −1.92705 −1.00000
1.14 −1.00000 1.30183 1.00000 1.00000 −1.30183 5.09963 −1.00000 −1.30525 −1.00000
1.15 −1.00000 1.96074 1.00000 1.00000 −1.96074 −2.08760 −1.00000 0.844520 −1.00000
1.16 −1.00000 2.32194 1.00000 1.00000 −2.32194 1.00226 −1.00000 2.39142 −1.00000
1.17 −1.00000 2.35112 1.00000 1.00000 −2.35112 2.99774 −1.00000 2.52776 −1.00000
1.18 −1.00000 2.72853 1.00000 1.00000 −2.72853 0.894798 −1.00000 4.44487 −1.00000
1.19 −1.00000 3.13315 1.00000 1.00000 −3.13315 3.12083 −1.00000 6.81662 −1.00000
1.20 −1.00000 3.32710 1.00000 1.00000 −3.32710 −2.87743 −1.00000 8.06959 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(1\)

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 4010.2.a.m ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.m ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{20} - 4 T_{3}^{19} - 35 T_{3}^{18} + 154 T_{3}^{17} + 460 T_{3}^{16} - 2392 T_{3}^{15} + \cdots + 104 \) Copy content Toggle raw display
\( T_{7}^{20} - 11 T_{7}^{19} - 31 T_{7}^{18} + 703 T_{7}^{17} - 594 T_{7}^{16} - 17399 T_{7}^{15} + \cdots + 814592 \) Copy content Toggle raw display
\( T_{11}^{20} - 10 T_{11}^{19} - 85 T_{11}^{18} + 1122 T_{11}^{17} + 1845 T_{11}^{16} - 49510 T_{11}^{15} + \cdots - 24378448 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 104 \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 11 T^{19} + \cdots + 814592 \) Copy content Toggle raw display
$11$ \( T^{20} - 10 T^{19} + \cdots - 24378448 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 520536064 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 7182899728 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6518996992 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 365433823232 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 291535028224 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 110810315648 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 2549161984 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 6441246038656 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 9417935028224 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 653474070528 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 23777614224384 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 585781273657344 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 139026391580128 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 4967019094592 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 695242894077952 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 666831390932992 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 10\!\cdots\!72 \) Copy content Toggle raw display
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