Properties

Label 20.5.d.c
Level $20$
Weight $5$
Character orbit 20.d
Analytic conductor $2.067$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,5,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1816805376000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 31x^{4} + 96x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 6) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1 - 5) q^{5} + (\beta_{7} + \beta_{5}) q^{6} + ( - 2 \beta_{4} + \beta_{2} + 6 \beta_1) q^{7} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 7 \beta_1) q^{8} + ( - 3 \beta_{7} - 3 \beta_{3} + 39) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 6) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1 - 5) q^{5} + (\beta_{7} + \beta_{5}) q^{6} + ( - 2 \beta_{4} + \beta_{2} + 6 \beta_1) q^{7} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 7 \beta_1) q^{8} + ( - 3 \beta_{7} - 3 \beta_{3} + 39) q^{9} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 20) q^{10} - 4 \beta_{5} q^{11} + (3 \beta_{7} + 6 \beta_{6} - 3 \beta_{4} + 3 \beta_{3} + 10 \beta_{2} + 3 \beta_1) q^{12} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_{4} - \beta_{3} - 18 \beta_1) q^{13} + ( - \beta_{7} - \beta_{5} + 8 \beta_{3} + 80) q^{14} + (4 \beta_{7} + 4 \beta_{5} + 6 \beta_{4} - 12 \beta_{3} - 3 \beta_{2} - 18 \beta_1) q^{15} + ( - 4 \beta_{7} + 4 \beta_{5} - 12 \beta_{3} - 104) q^{16} + (2 \beta_{7} + 4 \beta_{6} + 4 \beta_{4} + 2 \beta_{3} + 16 \beta_1) q^{17} + ( - 6 \beta_{7} - 12 \beta_{6} + 6 \beta_{4} - 6 \beta_{3} + 12 \beta_{2} + 33 \beta_1) q^{18} + ( - 8 \beta_{7} + 4 \beta_{5} + 24 \beta_{3}) q^{19} + (7 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - 11 \beta_{4} - 2 \beta_{3} - 22 \beta_{2} + \cdots - 50) q^{20}+ \cdots + ( - 216 \beta_{7} - 252 \beta_{5} + 648 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 48 q^{4} - 40 q^{5} + 312 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{4} - 40 q^{5} + 312 q^{9} - 160 q^{10} + 640 q^{14} - 832 q^{16} - 400 q^{20} - 960 q^{21} + 1920 q^{24} - 2040 q^{25} + 2496 q^{26} + 2704 q^{29} - 1920 q^{30} - 2176 q^{34} - 5712 q^{36} + 6400 q^{40} + 1456 q^{41} + 1920 q^{44} + 6120 q^{45} + 7040 q^{46} - 8008 q^{49} - 12480 q^{50} - 11520 q^{54} - 14720 q^{56} + 21120 q^{60} + 6576 q^{61} + 6912 q^{64} - 1920 q^{65} + 46080 q^{66} - 12480 q^{69} - 6400 q^{70} - 23616 q^{74} - 48000 q^{76} + 19520 q^{80} - 19512 q^{81} + 24960 q^{84} + 8960 q^{85} + 37120 q^{86} + 27664 q^{89} - 48480 q^{90} + 10880 q^{94} - 76800 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 31x^{4} + 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 10\nu^{5} + \nu^{3} + 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 6\nu^{5} + 31\nu^{3} + 80\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} + \nu^{2} + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 8\nu^{6} - 26\nu^{5} + 48\nu^{4} - 57\nu^{3} + 120\nu^{2} - 176\nu + 384 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} - 6\nu^{4} - 23\nu^{2} - 60 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 3\beta_{4} + \beta_{3} - 2\beta_{2} - 7\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + \beta_{5} - 3\beta_{3} - 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 10\beta_{2} - 13\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - 6\beta_{5} - 5\beta_{3} + 54 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{7} - 38\beta_{6} - 17\beta_{4} - 19\beta_{3} - 58\beta_{2} + 53\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.42849 1.39980i
−1.42849 + 1.39980i
−0.677813 1.88164i
−0.677813 + 1.88164i
0.677813 1.88164i
0.677813 + 1.88164i
1.42849 1.39980i
1.42849 + 1.39980i
−2.85697 2.79959i −6.64118 0.324555 + 15.9967i −17.6491 + 17.7062i 18.9737 + 18.5926i −39.0703 43.8570 46.6107i −36.8947 99.9931 1.17570i
19.2 −2.85697 + 2.79959i −6.64118 0.324555 15.9967i −17.6491 17.7062i 18.9737 18.5926i −39.0703 43.8570 + 46.6107i −36.8947 99.9931 + 1.17570i
19.3 −1.35563 3.76328i 13.9962 −12.3246 + 10.2032i 7.64911 23.8011i −18.9737 52.6718i −35.6863 55.1050 + 32.5490i 114.895 −99.9394 + 3.47961i
19.4 −1.35563 + 3.76328i 13.9962 −12.3246 10.2032i 7.64911 + 23.8011i −18.9737 + 52.6718i −35.6863 55.1050 32.5490i 114.895 −99.9394 3.47961i
19.5 1.35563 3.76328i −13.9962 −12.3246 10.2032i 7.64911 23.8011i −18.9737 + 52.6718i 35.6863 −55.1050 + 32.5490i 114.895 −79.2008 61.0511i
19.6 1.35563 + 3.76328i −13.9962 −12.3246 + 10.2032i 7.64911 + 23.8011i −18.9737 52.6718i 35.6863 −55.1050 32.5490i 114.895 −79.2008 + 61.0511i
19.7 2.85697 2.79959i 6.64118 0.324555 15.9967i −17.6491 + 17.7062i 18.9737 18.5926i 39.0703 −43.8570 46.6107i −36.8947 −0.852871 + 99.9964i
19.8 2.85697 + 2.79959i 6.64118 0.324555 + 15.9967i −17.6491 17.7062i 18.9737 + 18.5926i 39.0703 −43.8570 + 46.6107i −36.8947 −0.852871 99.9964i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.5.d.c 8
3.b odd 2 1 180.5.f.g 8
4.b odd 2 1 inner 20.5.d.c 8
5.b even 2 1 inner 20.5.d.c 8
5.c odd 4 2 100.5.b.e 8
8.b even 2 1 320.5.h.f 8
8.d odd 2 1 320.5.h.f 8
12.b even 2 1 180.5.f.g 8
15.d odd 2 1 180.5.f.g 8
20.d odd 2 1 inner 20.5.d.c 8
20.e even 4 2 100.5.b.e 8
40.e odd 2 1 320.5.h.f 8
40.f even 2 1 320.5.h.f 8
60.h even 2 1 180.5.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.c 8 1.a even 1 1 trivial
20.5.d.c 8 4.b odd 2 1 inner
20.5.d.c 8 5.b even 2 1 inner
20.5.d.c 8 20.d odd 2 1 inner
100.5.b.e 8 5.c odd 4 2
100.5.b.e 8 20.e even 4 2
180.5.f.g 8 3.b odd 2 1
180.5.f.g 8 12.b even 2 1
180.5.f.g 8 15.d odd 2 1
180.5.f.g 8 60.h even 2 1
320.5.h.f 8 8.b even 2 1
320.5.h.f 8 8.d odd 2 1
320.5.h.f 8 40.e odd 2 1
320.5.h.f 8 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 240T_{3}^{2} + 8640 \) acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 24 T^{6} + 496 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{4} - 240 T^{2} + 8640)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{3} + 710 T^{2} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2800 T^{2} + 1944000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 48000 T^{2} + \cdots + 552407040)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20928 T^{2} + 82861056)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 26368 T^{2} + 16367616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 447360 T^{2} + \cdots + 10372976640)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 350320 T^{2} + \cdots + 30678152640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 676 T + 104004)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1013760 T^{2} + \cdots + 1745879040)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3008448 T^{2} + \cdots + 126031666176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 364 T - 3961116)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 10034800 T^{2} + \cdots + 325194264000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 751600 T^{2} + \cdots + 49724936640)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 24538048 T^{2} + \cdots + 20248378776576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 30276480 T^{2} + \cdots + 225559394979840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 1644 T - 10307356)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 11037040 T^{2} + \cdots + 7632038946240)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 64765440 T^{2} + \cdots + 13443377725440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 68179968 T^{2} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 27279360 T^{2} + \cdots + 152966937968640)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 118219120 T^{2} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6916 T - 62026236)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 86769408 T^{2} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
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