Properties

Label 4002.2.a.bh
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
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: \( x^{7} - 3x^{6} - 16x^{5} + 51x^{4} + 45x^{3} - 152x^{2} - 54x + 70 \) 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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 16x^{5} + 51x^{4} + 45x^{3} - 152x^{2} - 54x + 70 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{6} + \nu^{5} + 100\nu^{4} - 18\nu^{3} - 499\nu^{2} - 18\nu + 383 ) / 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{6} - 3\nu^{5} - 257\nu^{4} + 54\nu^{3} + 981\nu^{2} + 269\nu - 547 ) / 43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{6} - 3\nu^{5} + 474\nu^{4} + 11\nu^{3} - 1771\nu^{2} - 720\nu + 743 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} - 8\nu^{5} - 671\nu^{4} + 187\nu^{3} + 2444\nu^{2} + 359\nu - 1215 ) / 43 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52\nu^{6} - 19\nu^{5} - 868\nu^{4} + 342\nu^{3} + 3031\nu^{2} + 428\nu - 1214 ) / 43 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{3} - \beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{6} + 12\beta_{5} + 12\beta_{4} + 38\beta_{3} + 18\beta_{2} - 5\beta _1 + 111 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{6} + 25\beta_{5} - 11\beta_{4} - 101\beta_{3} - 31\beta_{2} + 108\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{6} + 69\beta_{5} + 69\beta_{4} + 231\beta_{3} + 122\beta_{2} - 59\beta _1 + 574 \) 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
−0.924948
0.586166
−1.56994
3.08877
2.76200
−3.61135
2.66930
1.00000 1.00000 1.00000 −4.09187 1.00000 −2.85370 1.00000 1.00000 −4.09187
1.2 1.00000 1.00000 1.00000 −1.65866 1.00000 −1.05824 1.00000 1.00000 −1.65866
1.3 1.00000 1.00000 1.00000 −1.59086 1.00000 1.20696 1.00000 1.00000 −1.59086
1.4 1.00000 1.00000 1.00000 0.566335 1.00000 2.05767 1.00000 1.00000 0.566335
1.5 1.00000 1.00000 1.00000 0.888451 1.00000 4.32361 1.00000 1.00000 0.888451
1.6 1.00000 1.00000 1.00000 3.16777 1.00000 −1.07153 1.00000 1.00000 3.16777
1.7 1.00000 1.00000 1.00000 3.71883 1.00000 −4.60478 1.00000 1.00000 3.71883
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(23\) \( -1 \)
\(29\) \( +1 \)

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 4002.2.a.bh 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bh 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{7} - T_{5}^{6} - 23T_{5}^{5} + 21T_{5}^{4} + 120T_{5}^{3} - 44T_{5}^{2} - 128T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{7} + 2T_{7}^{6} - 26T_{7}^{5} - 44T_{7}^{4} + 136T_{7}^{3} + 168T_{7}^{2} - 144T_{7} - 160 \) Copy content Toggle raw display
\( T_{11}^{7} - 11T_{11}^{6} + 17T_{11}^{5} + 169T_{11}^{4} - 506T_{11}^{3} - 516T_{11}^{2} + 2432T_{11} - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{7} + 2 T^{6} + \cdots - 160 \) Copy content Toggle raw display
$11$ \( T^{7} - 11 T^{6} + \cdots - 1024 \) Copy content Toggle raw display
$13$ \( T^{7} + T^{6} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{7} - 18 T^{6} + \cdots + 512 \) Copy content Toggle raw display
$19$ \( T^{7} - 6 T^{6} + \cdots + 160 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( (T + 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} - 3 T^{6} + \cdots + 107776 \) Copy content Toggle raw display
$37$ \( T^{7} - 5 T^{6} + \cdots - 103184 \) Copy content Toggle raw display
$41$ \( T^{7} - 25 T^{6} + \cdots + 55808 \) Copy content Toggle raw display
$43$ \( T^{7} - 20 T^{6} + \cdots - 26176 \) Copy content Toggle raw display
$47$ \( T^{7} - 136 T^{5} + \cdots + 100352 \) Copy content Toggle raw display
$53$ \( T^{7} + 4 T^{6} + \cdots + 24448 \) Copy content Toggle raw display
$59$ \( T^{7} + 17 T^{6} + \cdots - 16000 \) Copy content Toggle raw display
$61$ \( T^{7} - 5 T^{6} + \cdots + 24176 \) Copy content Toggle raw display
$67$ \( T^{7} - 15 T^{6} + \cdots + 26816 \) Copy content Toggle raw display
$71$ \( T^{7} - 15 T^{6} + \cdots - 541760 \) Copy content Toggle raw display
$73$ \( T^{7} - 14 T^{6} + \cdots - 18880 \) Copy content Toggle raw display
$79$ \( T^{7} + 4 T^{6} + \cdots - 433600 \) Copy content Toggle raw display
$83$ \( T^{7} - 14 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$89$ \( T^{7} - 8 T^{6} + \cdots + 10519040 \) Copy content Toggle raw display
$97$ \( T^{7} + 18 T^{6} + \cdots + 105088 \) Copy content Toggle raw display
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