Properties

Label 4002.2.a.bi
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{7} - 79\nu^{6} + 173\nu^{5} + 1466\nu^{4} - 180\nu^{3} - 8105\nu^{2} - 2752\nu + 12248 ) / 1048 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} + 17\nu^{6} - 213\nu^{5} - 342\nu^{4} + 1362\nu^{3} + 1701\nu^{2} - 1726\nu - 732 ) / 524 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -21\nu^{7} + 135\nu^{6} + 235\nu^{5} - 2346\nu^{4} + 228\nu^{3} + 11489\nu^{2} - 3920\nu - 15584 ) / 1048 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 11368\nu + 4272 ) / 1048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 9272\nu + 4272 ) / 1048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -20\nu^{7} + 35\nu^{6} + 386\nu^{5} - 681\nu^{4} - 2197\nu^{3} + 3949\nu^{2} + 3690\nu - 6408 ) / 262 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 103\nu^{7} - 213\nu^{6} - 2001\nu^{5} + 3422\nu^{4} + 11308\nu^{3} - 14331\nu^{2} - 17104\nu + 15552 ) / 1048 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{4} + \beta_{3} - 2\beta_{2} - \beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 8\beta_{5} - 5\beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 4\beta_{6} + \beta_{5} - 16\beta_{4} + 13\beta_{3} - 26\beta_{2} - 13\beta _1 + 109 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -20\beta_{7} + 77\beta_{5} - 27\beta_{4} + 6\beta_{3} - 32\beta_{2} + 8\beta _1 + 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 99\beta_{7} - 72\beta_{6} + 23\beta_{5} - 188\beta_{4} + 147\beta_{3} - 286\beta_{2} - 153\beta _1 + 1017 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -140\beta_{7} - 8\beta_{6} + 399\beta_{5} - 69\beta_{4} + 9\beta_{3} - 204\beta_{2} + 11\beta _1 + 257 \) 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
−2.45069
−1.57534
−2.15995
1.98746
3.34107
2.85533
−3.18819
1.19031
−1.00000 1.00000 1.00000 −4.40973 −1.00000 −0.551242 −1.00000 1.00000 4.40973
1.2 −1.00000 1.00000 1.00000 −2.69359 −1.00000 −3.87906 −1.00000 1.00000 2.69359
1.3 −1.00000 1.00000 1.00000 −1.69873 −1.00000 3.41134 −1.00000 1.00000 1.69873
1.4 −1.00000 1.00000 1.00000 −0.881455 −1.00000 0.253486 −1.00000 1.00000 0.881455
1.5 −1.00000 1.00000 1.00000 −0.464023 −1.00000 −2.35800 −1.00000 1.00000 0.464023
1.6 −1.00000 1.00000 1.00000 1.27098 −1.00000 3.55617 −1.00000 1.00000 −1.27098
1.7 −1.00000 1.00000 1.00000 1.60812 −1.00000 −3.37648 −1.00000 1.00000 −1.60812
1.8 −1.00000 1.00000 1.00000 4.26843 −1.00000 −3.05622 −1.00000 1.00000 −4.26843
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bi 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{8} + 3T_{5}^{7} - 22T_{5}^{6} - 66T_{5}^{5} + 67T_{5}^{4} + 231T_{5}^{3} - 18T_{5}^{2} - 204T_{5} - 72 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 15T_{7}^{6} - 144T_{7}^{5} - 80T_{7}^{4} + 844T_{7}^{3} + 1400T_{7}^{2} + 224T_{7} - 160 \) Copy content Toggle raw display
\( T_{11}^{8} + 7T_{11}^{7} - 39T_{11}^{6} - 315T_{11}^{5} + 266T_{11}^{4} + 3846T_{11}^{3} + 2376T_{11}^{2} - 8424T_{11} - 3888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots - 72 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots - 160 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + \cdots - 3888 \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + \cdots + 1264 \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots - 89568 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 178784 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} + \cdots - 10368 \) Copy content Toggle raw display
$37$ \( T^{8} + 11 T^{7} + \cdots + 18680 \) Copy content Toggle raw display
$41$ \( T^{8} + 17 T^{7} + \cdots + 30024 \) Copy content Toggle raw display
$43$ \( T^{8} + 10 T^{7} + \cdots - 222912 \) Copy content Toggle raw display
$47$ \( T^{8} + 26 T^{7} + \cdots + 182016 \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + \cdots + 3939840 \) Copy content Toggle raw display
$59$ \( T^{8} + 25 T^{7} + \cdots - 42129216 \) Copy content Toggle raw display
$61$ \( T^{8} - 3 T^{7} + \cdots - 14816 \) Copy content Toggle raw display
$67$ \( T^{8} - T^{7} + \cdots + 7438336 \) Copy content Toggle raw display
$71$ \( T^{8} + 27 T^{7} + \cdots - 6184320 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 640384 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} + \cdots + 152000 \) Copy content Toggle raw display
$83$ \( T^{8} + 44 T^{7} + \cdots + 8418048 \) Copy content Toggle raw display
$89$ \( T^{8} + 52 T^{7} + \cdots - 27648 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots - 2388480 \) Copy content Toggle raw display
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