Properties

Label 2001.2.a.h
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $5$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.312617.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - q^{3} + ( - \beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - q^{3} + ( - \beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9} + ( - \beta_{4} - 2 \beta_1 + 3) q^{10} + (\beta_{4} - 2) q^{11} + (\beta_1 - 2) q^{12} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{13} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{14} + ( - \beta_{2} - \beta_1 + 1) q^{15} + (\beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{16} + \beta_1 q^{17} - \beta_{3} q^{18} + ( - 2 \beta_{4} - 1) q^{19} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{20} + (\beta_{2} + 1) q^{21} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{22} + q^{23} + (\beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{24} + \beta_{3} q^{25} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{26} - q^{27} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{28} + q^{29} + (\beta_{4} + 2 \beta_1 - 3) q^{30} + (2 \beta_{3} - \beta_{2} - 2) q^{31} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{32} + ( - \beta_{4} + 2) q^{33} + (\beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{34} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{35} + ( - \beta_1 + 2) q^{36} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{37} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 4) q^{38} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{39} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 4) q^{40} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{41} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{42} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{43} + (\beta_{4} + \beta_1 - 4) q^{44} + (\beta_{2} + \beta_1 - 1) q^{45} - \beta_{3} q^{46} + (3 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{47} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{48} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{49} + (\beta_1 - 4) q^{50} - \beta_1 q^{51} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{52} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{53} + \beta_{3} q^{54} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{55} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{56} + (2 \beta_{4} + 1) q^{57} - \beta_{3} q^{58} + ( - 4 \beta_{4} - 2 \beta_{3} + 4 \beta_1 + 2) q^{59} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{60} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{61} + (2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 10) q^{62} + ( - \beta_{2} - 1) q^{63} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{64} + ( - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 4) q^{65} + (\beta_{4} - \beta_{3} + \beta_{2} - 2) q^{66} + ( - 4 \beta_{4} - \beta_1) q^{67} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{68} - q^{69} + ( - \beta_{4} + 3 \beta_{3} + 5 \beta_1 - 6) q^{70} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{71} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{72} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 7) q^{73} + ( - 5 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 2) q^{74} - \beta_{3} q^{75} + ( - 2 \beta_{4} + 3 \beta_1 - 2) q^{76} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{78} + (2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{79} + ( - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{80} + q^{81} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - \beta_1 + 4) q^{82} + ( - 2 \beta_{3} - 7 \beta_1 + 5) q^{83} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{84} + (\beta_{4} + 2 \beta_{3} + \beta_{2}) q^{85} + (3 \beta_{4} + 4 \beta_{3} + \beta_{2} + 5 \beta_1 - 12) q^{86} - q^{87} + (2 \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{88} + (\beta_{4} + 2 \beta_{3} - 4 \beta_1 + 1) q^{89} + ( - \beta_{4} - 2 \beta_1 + 3) q^{90} + ( - 2 \beta_{3} + \beta_1 + 2) q^{91} + ( - \beta_1 + 2) q^{92} + ( - 2 \beta_{3} + \beta_{2} + 2) q^{93} + ( - 8 \beta_{4} + 6 \beta_1 - 7) q^{94} + (4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 1) q^{95} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{96} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 10) q^{98} + (\beta_{4} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - 5 q^{3} + 8 q^{4} - 3 q^{5} + 2 q^{6} - 5 q^{7} - 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - 5 q^{3} + 8 q^{4} - 3 q^{5} + 2 q^{6} - 5 q^{7} - 3 q^{8} + 5 q^{9} + 9 q^{10} - 8 q^{11} - 8 q^{12} + 5 q^{13} - 2 q^{14} + 3 q^{15} - 10 q^{16} + 2 q^{17} - 2 q^{18} - 9 q^{19} - 12 q^{20} + 5 q^{21} + 10 q^{22} + 5 q^{23} + 3 q^{24} + 2 q^{25} - 3 q^{26} - 5 q^{27} - 14 q^{28} + 5 q^{29} - 9 q^{30} - 6 q^{31} - 8 q^{32} + 8 q^{33} + 3 q^{34} - 15 q^{35} + 8 q^{36} + 10 q^{37} - 10 q^{38} - 5 q^{39} + 26 q^{40} - 11 q^{41} + 2 q^{42} - 9 q^{43} - 16 q^{44} - 3 q^{45} - 2 q^{46} - 13 q^{47} + 10 q^{48} - 6 q^{49} - 18 q^{50} - 2 q^{51} + 12 q^{52} + q^{53} + 2 q^{54} + 13 q^{55} - 4 q^{56} + 9 q^{57} - 2 q^{58} + 6 q^{59} + 12 q^{60} + 23 q^{61} - 36 q^{62} - 5 q^{63} - q^{64} - 20 q^{65} - 10 q^{66} - 10 q^{67} - 10 q^{68} - 5 q^{69} - 16 q^{70} - 11 q^{71} - 3 q^{72} + 31 q^{73} - 18 q^{74} - 2 q^{75} - 8 q^{76} + 3 q^{77} + 3 q^{78} + 8 q^{79} - 8 q^{80} + 5 q^{81} + 16 q^{82} + 7 q^{83} + 14 q^{84} + 6 q^{85} - 36 q^{86} - 5 q^{87} - 3 q^{88} + 3 q^{89} + 9 q^{90} + 8 q^{91} + 8 q^{92} + 6 q^{93} - 39 q^{94} - 11 q^{95} + 8 q^{96} + 3 q^{97} + 38 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + \nu^{3} + 6\nu^{2} - 5\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} + 6\beta_{3} + \beta_{2} + 8 \) 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
−2.28064
0.762877
2.26093
1.70431
−0.447481
−2.50612 −1.00000 4.28064 −2.73973 2.50612 −1.54091 −5.71555 1.00000 6.86609
1.2 −1.79920 −1.00000 1.23712 −2.60753 1.79920 1.37040 1.37257 1.00000 4.69147
1.3 −1.31874 −1.00000 −0.260930 2.51371 1.31874 −2.25278 2.98157 1.00000 −3.31493
1.4 1.51515 −1.00000 0.295689 −1.86677 −1.51515 1.57109 −2.58229 1.00000 −2.82845
1.5 2.10891 −1.00000 2.44748 1.70032 −2.10891 −4.14780 0.943696 1.00000 3.58582
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 2001.2.a.h 5
3.b odd 2 1 6003.2.a.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.h 5 1.a even 1 1 trivial
6003.2.a.h 5 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2}^{5} + 2T_{2}^{4} - 7T_{2}^{3} - 13T_{2}^{2} + 11T_{2} + 19 \) Copy content Toggle raw display
\( T_{5}^{5} + 3T_{5}^{4} - 9T_{5}^{3} - 28T_{5}^{2} + 17T_{5} + 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 7 T^{3} - 13 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 3 T^{4} - 9 T^{3} - 28 T^{2} + \cdots + 57 \) Copy content Toggle raw display
$7$ \( T^{5} + 5 T^{4} - 2 T^{3} - 25 T^{2} + \cdots + 31 \) Copy content Toggle raw display
$11$ \( T^{5} + 8 T^{4} + 16 T^{3} + 7 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 4 T^{3} + 23 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{5} - 2 T^{4} - 5 T^{3} + 11 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$19$ \( T^{5} + 9 T^{4} - 6 T^{3} - 246 T^{2} + \cdots - 563 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} - 40 T^{3} - 59 T^{2} + \cdots + 139 \) Copy content Toggle raw display
$37$ \( T^{5} - 10 T^{4} - 24 T^{3} + \cdots - 821 \) Copy content Toggle raw display
$41$ \( T^{5} + 11 T^{4} - 21 T^{3} + \cdots - 921 \) Copy content Toggle raw display
$43$ \( T^{5} + 9 T^{4} - 75 T^{3} + \cdots + 7447 \) Copy content Toggle raw display
$47$ \( T^{5} + 13 T^{4} - 145 T^{3} + \cdots + 49139 \) Copy content Toggle raw display
$53$ \( T^{5} - T^{4} - 49 T^{3} + 20 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$59$ \( T^{5} - 6 T^{4} - 204 T^{3} + \cdots + 1824 \) Copy content Toggle raw display
$61$ \( T^{5} - 23 T^{4} + 84 T^{3} + \cdots - 281 \) Copy content Toggle raw display
$67$ \( T^{5} + 10 T^{4} - 133 T^{3} + \cdots - 513 \) Copy content Toggle raw display
$71$ \( T^{5} + 11 T^{4} + 2 T^{3} - 119 T^{2} + \cdots + 57 \) Copy content Toggle raw display
$73$ \( T^{5} - 31 T^{4} + 283 T^{3} + \cdots + 1317 \) Copy content Toggle raw display
$79$ \( T^{5} - 8 T^{4} - 62 T^{3} + \cdots - 2921 \) Copy content Toggle raw display
$83$ \( T^{5} - 7 T^{4} - 285 T^{3} + \cdots + 40743 \) Copy content Toggle raw display
$89$ \( T^{5} - 3 T^{4} - 150 T^{3} + \cdots - 687 \) Copy content Toggle raw display
$97$ \( T^{5} - 3 T^{4} - 268 T^{3} + \cdots + 4337 \) Copy content Toggle raw display
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