Properties

Label 2001.2.a.i
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 \) 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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 13\nu^{2} - 5\nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_{2} + 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + \beta_{5} + 8\beta_{4} + 9\beta_{3} + 32\beta_{2} + 33\beta _1 + 31 \) 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
2.37349
2.22973
1.44008
0.743347
−0.586186
−1.21072
−1.98973
−2.37349 1.00000 3.63344 −1.31800 −2.37349 −0.859291 −3.87694 1.00000 3.12824
1.2 −2.22973 1.00000 2.97167 0.537118 −2.22973 −1.88241 −2.16657 1.00000 −1.19762
1.3 −1.44008 1.00000 0.0738306 0.734622 −1.44008 3.73618 2.77384 1.00000 −1.05791
1.4 −0.743347 1.00000 −1.44743 −3.36180 −0.743347 −2.98340 2.56264 1.00000 2.49898
1.5 0.586186 1.00000 −1.65639 −0.842866 0.586186 1.23837 −2.14332 1.00000 −0.494076
1.6 1.21072 1.00000 −0.534159 3.03795 1.21072 −3.40847 −3.06816 1.00000 3.67810
1.7 1.98973 1.00000 1.95904 −1.78703 1.98973 −0.840964 −0.0814996 1.00000 −3.55572
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.i 7
3.b odd 2 1 6003.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.i 7 1.a even 1 1 trivial
6003.2.a.j 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2}^{7} + 3T_{2}^{6} - 5T_{2}^{5} - 18T_{2}^{4} + 4T_{2}^{3} + 26T_{2}^{2} + T_{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{7} + 3T_{5}^{6} - 9T_{5}^{5} - 30T_{5}^{4} - 5T_{5}^{3} + 29T_{5}^{2} + 6T_{5} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 3 T^{6} - 5 T^{5} - 18 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 3 T^{6} - 9 T^{5} - 30 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{7} + 5 T^{6} - 8 T^{5} - 75 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{7} + 4 T^{6} - 34 T^{5} - 87 T^{4} + \cdots + 776 \) Copy content Toggle raw display
$13$ \( T^{7} + 18 T^{6} + 119 T^{5} + \cdots + 598 \) Copy content Toggle raw display
$17$ \( T^{7} + 3 T^{6} - 39 T^{5} + \cdots - 3088 \) Copy content Toggle raw display
$19$ \( T^{7} + 4 T^{6} - 37 T^{5} - 206 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( (T + 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} + 22 T^{6} + 168 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$37$ \( T^{7} + 25 T^{6} + 124 T^{5} + \cdots + 151376 \) Copy content Toggle raw display
$41$ \( T^{7} + 13 T^{6} - 21 T^{5} + \cdots + 5732 \) Copy content Toggle raw display
$43$ \( T^{7} + 2 T^{6} - 134 T^{5} + \cdots + 2258 \) Copy content Toggle raw display
$47$ \( T^{7} + 25 T^{6} + 103 T^{5} + \cdots - 105656 \) Copy content Toggle raw display
$53$ \( T^{7} + 5 T^{6} - 209 T^{5} + \cdots - 25268 \) Copy content Toggle raw display
$59$ \( T^{7} - 11 T^{6} - 210 T^{5} + \cdots - 7552 \) Copy content Toggle raw display
$61$ \( T^{7} + 33 T^{6} + 370 T^{5} + \cdots - 153712 \) Copy content Toggle raw display
$67$ \( T^{7} - 8 T^{6} - 287 T^{5} + \cdots - 1070848 \) Copy content Toggle raw display
$71$ \( T^{7} + 6 T^{6} - 195 T^{5} + \cdots + 78872 \) Copy content Toggle raw display
$73$ \( T^{7} - 15 T^{6} - 73 T^{5} + \cdots - 6928 \) Copy content Toggle raw display
$79$ \( T^{7} + 15 T^{6} + 12 T^{5} + \cdots + 142 \) Copy content Toggle raw display
$83$ \( T^{7} - 21 T^{6} - 173 T^{5} + \cdots - 1179352 \) Copy content Toggle raw display
$89$ \( T^{7} - 8 T^{6} - 191 T^{5} + \cdots + 12688 \) Copy content Toggle raw display
$97$ \( T^{7} - 13 T^{6} - 124 T^{5} + \cdots + 457432 \) Copy content Toggle raw display
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