Properties

Label 7.20.a.b
Level $7$
Weight $20$
Character orbit 7.a
Self dual yes
Analytic conductor $16.017$
Analytic rank $0$
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] = [7,20,Mod(1,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(16.0171687589\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2204112x^{3} - 22116602x^{2} + 863582048991x - 99444286717974 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 23) q^{2} + (\beta_{2} + 4 \beta_1 - 6483) q^{3} + (\beta_{3} + 5 \beta_{2} + 62 \beta_1 + 357885) q^{4} + (\beta_{4} + \beta_{3} + 53 \beta_{2} - 812 \beta_1 + 1897109) q^{5} + ( - 5 \beta_{4} + 17 \beta_{3} + 510 \beta_{2} + 640 \beta_1 - 3151712) q^{6} - 40353607 q^{7} + ( - 5 \beta_{4} - 464 \beta_{3} + 11129 \beta_{2} + \cdots - 50000691) q^{8}+ \cdots + (205 \beta_{4} - 115 \beta_{3} - 22882 \beta_{2} - 697032 \beta_1 + 2918701) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 23) q^{2} + (\beta_{2} + 4 \beta_1 - 6483) q^{3} + (\beta_{3} + 5 \beta_{2} + 62 \beta_1 + 357885) q^{4} + (\beta_{4} + \beta_{3} + 53 \beta_{2} - 812 \beta_1 + 1897109) q^{5} + ( - 5 \beta_{4} + 17 \beta_{3} + 510 \beta_{2} + 640 \beta_1 - 3151712) q^{6} - 40353607 q^{7} + ( - 5 \beta_{4} - 464 \beta_{3} + 11129 \beta_{2} + \cdots - 50000691) q^{8}+ \cdots + (1449136201525 \beta_{4} + \cdots + 37\!\cdots\!80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 115 q^{2} - 32414 q^{3} + 1789429 q^{4} + 9485596 q^{5} - 15758062 q^{6} - 201768035 q^{7} - 249991857 q^{8} + 14570533 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 115 q^{2} - 32414 q^{3} + 1789429 q^{4} + 9485596 q^{5} - 15758062 q^{6} - 201768035 q^{7} - 249991857 q^{8} + 14570533 q^{9} + 3417623884 q^{10} + 8943902032 q^{11} + 15096861374 q^{12} + 902235824 q^{13} + 4640664805 q^{14} + 216213883888 q^{15} + 718680981361 q^{16} + 1562749140102 q^{17} + 3046240421405 q^{18} + 3477059903486 q^{19} + 5945119865048 q^{20} + 1308021817298 q^{21} + 21553377089920 q^{22} + 12272511787008 q^{23} + 58219119595794 q^{24} + 8645526747031 q^{25} + 570269803712 q^{26} - 100870197417764 q^{27} - 72209914620403 q^{28} - 101577258937886 q^{29} - 192135202588288 q^{30} + 172735355689956 q^{31} - 779637915241721 q^{32} - 655229182538384 q^{33} - 11\!\cdots\!86 q^{34}+ \cdots + 18\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2204112x^{3} - 22116602x^{2} + 863582048991x - 99444286717974 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} - 13001\nu^{3} + 6981503\nu^{2} + 18486426033\nu - 4730270045670 ) / 173803392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{4} + 65005\nu^{3} + 138895877\nu^{2} - 95212984437\nu - 129581367508098 ) / 173803392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4483\nu^{4} - 349019\nu^{3} + 8413964669\nu^{2} + 1122804345363\nu - 1799085118294194 ) / 289672320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} + 16\beta _1 + 881644 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{4} + 395\beta_{3} - 11474\beta_{2} + 1417429\beta _1 + 13272336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -65005\beta_{4} + 1846108\beta_{3} + 10277597\beta_{2} + 170135652\beta _1 + 1252376544926 \) 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
1332.90
613.497
119.891
−802.831
−1263.45
−1355.90 −10527.9 1.31417e6 97087.9 1.42747e7 −4.03536e7 −1.07100e9 −1.05143e9 −1.31641e8
1.2 −636.497 31039.9 −119160. 7.19630e6 −1.97568e7 −4.03536e7 4.09552e8 −1.98785e8 −4.58042e9
1.3 −142.891 −20020.3 −503870. −5.07605e6 2.86072e6 −4.03536e7 1.46915e8 −7.61450e8 7.25322e8
1.4 779.831 −60095.5 83849.0 3.49867e6 −4.68644e7 −4.03536e7 −3.43468e8 2.44921e9 2.72837e9
1.5 1240.45 27189.8 1.01444e6 3.76958e6 3.37277e7 −4.03536e7 6.08010e8 −4.22978e8 4.67599e9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

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 7.20.a.b 5
3.b odd 2 1 63.20.a.d 5
7.b odd 2 1 49.20.a.d 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.20.a.b 5 1.a even 1 1 trivial
49.20.a.d 5 7.b odd 2 1
63.20.a.d 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 115T_{2}^{4} - 2198822T_{2}^{3} - 129845456T_{2}^{2} + 861102886144T_{2} + 119291562532864 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + \cdots + 119291562532864 \) Copy content Toggle raw display
$3$ \( T^{5} + 32414 T^{4} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$5$ \( T^{5} - 9485596 T^{4} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 40353607)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 8943902032 T^{4} + \cdots + 54\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{5} - 902235824 T^{4} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{5} - 1562749140102 T^{4} + \cdots - 22\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{5} - 3477059903486 T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} - 12272511787008 T^{4} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{5} + 101577258937886 T^{4} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} - 172735355689956 T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 69\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 45\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{5} - 141597446651944 T^{4} + \cdots - 53\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 66\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 47\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 87\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
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