Properties

Label 5.20.a.a
Level $5$
Weight $20$
Character orbit 5.a
Self dual yes
Analytic conductor $11.441$
Analytic rank $1$
Dimension $3$
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] = [5,20,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, 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("5.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(11.4408348278\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 22777x - 646704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 335) q^{2} + ( - 13 \beta_{2} + 18 \beta_1 - 24478) q^{3} + (184 \beta_{2} - 398 \beta_1 + 318162) q^{4} + 1953125 q^{5} + (11216 \beta_{2} - 46152 \beta_1 + 17803448) q^{6} + ( - 30963 \beta_{2} - 44498 \beta_1 - 18318318) q^{7} + ( - 185104 \beta_{2} + 109668 \beta_1 - 171462364) q^{8} + (556788 \beta_{2} - 1678536 \beta_1 + 469399485) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 335) q^{2} + ( - 13 \beta_{2} + 18 \beta_1 - 24478) q^{3} + (184 \beta_{2} - 398 \beta_1 + 318162) q^{4} + 1953125 q^{5} + (11216 \beta_{2} - 46152 \beta_1 + 17803448) q^{6} + ( - 30963 \beta_{2} - 44498 \beta_1 - 18318318) q^{7} + ( - 185104 \beta_{2} + 109668 \beta_1 - 171462364) q^{8} + (556788 \beta_{2} - 1678536 \beta_1 + 469399485) q^{9} + (1953125 \beta_1 - 654296875) q^{10} + (1637230 \beta_{2} + 2097940 \beta_1 - 2854948528) q^{11} + ( - 8495552 \beta_{2} + 28995120 \beta_1 - 23777142992) q^{12} + (4569380 \beta_{2} - 16253608 \beta_1 - 28548921090) q^{13} + (10637872 \beta_{2} - 64436484 \beta_1 - 34790122052) q^{14} + ( - 25390625 \beta_{2} + 35156250 \beta_1 - 47808593750) q^{15} + (36253152 \beta_{2} - 262169144 \beta_1 - 79701977272) q^{16} + (205644916 \beta_{2} - 419322056 \beta_1 + 28946200346) q^{17} + ( - 647377728 \beta_{2} + 1454872293 \beta_1 - 1231308771243) q^{18} + (307290116 \beta_{2} + 823817048 \beta_1 + 427611297876) q^{19} + (359375000 \beta_{2} - 777343750 \beta_1 + 621410156250) q^{20} + (206921832 \beta_{2} + 227254896 \beta_1 + 2068504635984) q^{21} + ( - 609414880 \beta_{2} - 400295348 \beta_1 + 2934298505100) q^{22} + ( - 1481795237 \beta_{2} + 154917186 \beta_1 + 1362994724638) q^{23} + (4619983488 \beta_{2} - 14829867936 \beta_1 + 17490287734368) q^{24} + 3814697265625 q^{25} + ( - 5768846912 \beta_{2} - 20305323386 \beta_1 - 1060367722330) q^{26} + ( - 7437917106 \beta_{2} + \cdots - 35977406241996) q^{27}+ \cdots + ( - 24\!\cdots\!94 \beta_{2} + \cdots + 63\!\cdots\!80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 1006 q^{2} - 73452 q^{3} + 954884 q^{4} + 5859375 q^{5} + 53456496 q^{6} - 54910456 q^{7} - 514496760 q^{8} + 1409876991 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 1006 q^{2} - 73452 q^{3} + 954884 q^{4} + 5859375 q^{5} + 53456496 q^{6} - 54910456 q^{7} - 514496760 q^{8} + 1409876991 q^{9} - 1964843750 q^{10} - 8566943524 q^{11} - 71360424096 q^{12} - 85630509662 q^{13} - 104305929672 q^{14} - 143460937500 q^{15} - 238843762672 q^{16} + 87257923094 q^{17} - 3695381186022 q^{18} + 1282010076580 q^{19} + 1865007812500 q^{20} + 6205286653056 q^{21} + 8803295810648 q^{22} + 4088829256728 q^{23} + 52485693071040 q^{24} + 11444091796875 q^{25} - 3160797843604 q^{26} - 108009182325240 q^{27} - 68700385875088 q^{28} - 73280209082030 q^{29} + 104407218750000 q^{30} - 284526134418784 q^{31} - 194854359251936 q^{32} - 54624460283184 q^{33} - 780038743128412 q^{34} - 107246984375000 q^{35} + 31\!\cdots\!48 q^{36}+ \cdots + 19\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 22777x - 646704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} + 104\nu - 30381 ) / 35 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{2} + 738\nu + 91108 ) / 35 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -52\beta_{2} + 369\beta _1 + 455663 ) / 30 \) 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
−29.5226
−133.978
163.500
−1240.95 −64590.2 1.01566e6 1.95312e6 8.01531e7 −3.47039e7 −6.09772e8 3.00964e9 −2.42373e9
1.2 −575.417 14082.6 −193183. 1.95312e6 −8.10336e6 9.45293e7 4.12845e8 −9.63942e8 −1.12386e9
1.3 810.365 −22944.3 132403. 1.95312e6 −1.85933e7 −1.14736e8 −3.17570e8 −6.35819e8 1.58274e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 5.20.a.a 3
3.b odd 2 1 45.20.a.d 3
4.b odd 2 1 80.20.a.f 3
5.b even 2 1 25.20.a.b 3
5.c odd 4 2 25.20.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.a.a 3 1.a even 1 1 trivial
25.20.a.b 3 5.b even 2 1
25.20.b.b 6 5.c odd 4 2
45.20.a.d 3 3.b odd 2 1
80.20.a.f 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 1006T_{2}^{2} - 757856T_{2} - 578651136 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 1006 T^{2} + \cdots - 578651136 \) Copy content Toggle raw display
$3$ \( T^{3} + 73452 T^{2} + \cdots - 20870112837888 \) Copy content Toggle raw display
$5$ \( (T - 1953125)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 54910456 T^{2} + \cdots - 37\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{3} + 8566943524 T^{2} + \cdots - 30\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{3} + 85630509662 T^{2} + \cdots + 87\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{3} - 87257923094 T^{2} + \cdots - 64\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{3} - 1282010076580 T^{2} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 4088829256728 T^{2} + \cdots - 58\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{3} + 73280209082030 T^{2} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + 284526134418784 T^{2} + \cdots - 25\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{3} + 432614295273606 T^{2} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 93\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 66\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 57\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 21\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 76\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 55\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 65\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 44\!\cdots\!56 \) Copy content Toggle raw display
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