↑

Properties

Label 1638.2.dm.e
Level $1638$
Weight $2$
Character orbit 1638.dm
Analytic conductor $13.079$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(415,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.415");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.dm (of order \(6\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 26 x^{18} + 431 x^{16} - 4370 x^{14} + 32381 x^{12} - 160412 x^{10} + 573820 x^{8} + \cdots + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{14} - \beta_{8}) q^{2} + ( - \beta_{3} + 1) q^{4} - \beta_1 q^{5} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{14}) q^{7}+ \cdots + \beta_{14} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} - \beta_{8}) q^{2} + ( - \beta_{3} + 1) q^{4} - \beta_1 q^{5} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{14}) q^{7}+ \cdots + (\beta_{18} - \beta_{17} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10 q^{4} + 4 q^{10} + 4 q^{13} + 2 q^{14} - 10 q^{16} + 6 q^{17} - 12 q^{22} + 16 q^{23} + 2 q^{25} + 4 q^{26} + 28 q^{29} - 16 q^{35} - 10 q^{38} - 4 q^{40} + 24 q^{43} + 2 q^{49} + 2 q^{52} + 22 q^{53} + 88 q^{55} + 10 q^{56} + 14 q^{61} - 40 q^{62} - 20 q^{64} - 20 q^{65} - 6 q^{68} - 24 q^{74} + 28 q^{77} + 4 q^{79} + 12 q^{82} - 6 q^{88} + 68 q^{91} + 32 q^{92} - 18 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 26 x^{18} + 431 x^{16} - 4370 x^{14} + 32381 x^{12} - 160412 x^{10} + 573820 x^{8} + \cdots + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 28019 \nu^{18} - 1353739 \nu^{16} + 26963794 \nu^{14} - 360314635 \nu^{12} + \cdots - 250163334900 ) / 50836113540 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 44475185183309 \nu^{18} + \cdots + 50\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 55035689845889 \nu^{18} + \cdots - 12\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 77692557753089 \nu^{18} + \cdots + 49\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33725347069901 \nu^{18} + \cdots - 10\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4648903677221 \nu^{18} - 113351516709967 \nu^{16} + \cdots + 34\!\cdots\!80 ) / 59\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 322032565057271 \nu^{19} + \cdots + 16\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44475185183309 \nu^{19} + \cdots - 50\!\cdots\!00 \nu ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 391520578290986 \nu^{19} + \cdots - 20\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4118545123555 \nu^{18} - 99803503973213 \nu^{16} + \cdots + 64\!\cdots\!88 ) / 14\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!01 \nu^{19} - 416452024229730 \nu^{18} + \cdots + 23\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!01 \nu^{19} - 416452024229730 \nu^{18} + \cdots + 23\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 92653087 \nu^{19} + 2400574562 \nu^{17} - 39527358797 \nu^{15} + \cdots + 47850783750000 \nu ) / 15250834062000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29\!\cdots\!49 \nu^{19} - 967658731840290 \nu^{18} + \cdots + 26\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!11 \nu^{19} + \cdots + 14\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!07 \nu^{19} + \cdots - 29\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 18\!\cdots\!42 \nu^{19} - 483829365920145 \nu^{18} + \cdots + 13\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 41250852113590 \nu^{19} - 993597754957391 \nu^{17} + \cdots + 22\!\cdots\!44 \nu ) / 44\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{7} + \beta_{4} - 4\beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - \beta_{17} + \beta_{13} - \beta_{12} + 2\beta_{10} + 6\beta_{9} + 2\beta_{8} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{13} + 10\beta_{12} - 10\beta_{7} + \beta_{6} - 10\beta_{5} - 26\beta_{3} - 2\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{18} - 9 \beta_{17} - 24 \beta_{16} - 11 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{11} + 36\beta_{7} - 88\beta_{5} - 88\beta_{4} - 378 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 73 \beta_{19} + 139 \beta_{18} - 242 \beta_{16} - 139 \beta_{15} - 114 \beta_{14} - 139 \beta_{13} + \cdots - 312 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6 \beta_{18} - 6 \beta_{15} - 760 \beta_{13} - 760 \beta_{12} - 169 \beta_{11} - 6 \beta_{9} + \cdots - 2430 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 603 \beta_{19} + 476 \beta_{18} + 603 \beta_{17} - 476 \beta_{15} - 1494 \beta_{14} - 923 \beta_{13} + \cdots - 2864 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 156 \beta_{18} + 156 \beta_{15} - 6712 \beta_{13} - 6712 \beta_{12} + 156 \beta_{9} + 6868 \beta_{7} + \cdots + 6868 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8119 \beta_{18} + 5149 \beta_{17} + 21794 \beta_{16} + 8119 \beta_{15} + 5556 \beta_{13} + \cdots - 5556 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5172 \beta_{18} + 5172 \beta_{15} - 2586 \beta_{13} - 2586 \beta_{12} + 16645 \beta_{11} + 5172 \beta_{9} + \cdots + 242914 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 45243 \beta_{19} - 131323 \beta_{18} + 202098 \beta_{16} + 131323 \beta_{15} + 177870 \beta_{14} + \cdots + 149076 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 35016 \beta_{18} + 35016 \beta_{15} + 492724 \beta_{13} + 492724 \beta_{12} + 156855 \beta_{11} + \cdots + 1593866 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 405901 \beta_{19} - 631644 \beta_{18} - 405901 \beta_{17} + 631644 \beta_{15} + 1792674 \beta_{14} + \cdots + 1842912 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 422982 \beta_{18} - 422982 \beta_{15} + 4724146 \beta_{13} + 4724146 \beta_{12} - 422982 \beta_{9} + \cdots - 5147128 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 5333051 \beta_{18} - 3692259 \beta_{17} - 17060442 \beta_{16} - 5333051 \beta_{15} + \cdots + 6394340 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 9507096 \beta_{18} - 9507096 \beta_{15} + 4753548 \beta_{13} + 4753548 \beta_{12} - 13368183 \beta_{11} + \cdots - 174041982 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 33873733 \beta_{19} + 109691851 \beta_{18} - 156041306 \beta_{16} - 109691851 \beta_{15} + \cdots - 83963220 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
2.38129 + 1.37484i
2.18313 + 1.26043i
0.935588 + 0.540162i
−1.14595 − 0.661615i
−2.62200 − 1.51381i
2.62200 + 1.51381i
1.14595 + 0.661615i
−0.935588 − 0.540162i
−2.18313 − 1.26043i
−2.38129 − 1.37484i
2.38129 − 1.37484i
2.18313 − 1.26043i
0.935588 − 0.540162i
−1.14595 + 0.661615i
−2.62200 + 1.51381i
2.62200 − 1.51381i
1.14595 − 0.661615i
−0.935588 + 0.540162i
−2.18313 + 1.26043i
−2.38129 + 1.37484i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.38129 − 1.37484i 0 0.588218 − 2.57953i − 1.00000i 0 1.37484 + 2.38129i
415.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.18313 − 1.26043i 0 −1.47350 + 2.19745i − 1.00000i 0 1.26043 + 2.18313i
415.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.935588 − 0.540162i 0 1.55401 + 2.14127i − 1.00000i 0 0.540162 + 0.935588i
415.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.14595 + 0.661615i 0 2.62831 − 0.303302i − 1.00000i 0 −0.661615 − 1.14595i
415.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.62200 + 1.51381i 0 −2.43101 + 1.04411i − 1.00000i 0 −1.51381 − 2.62200i
415.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.62200 − 1.51381i 0 2.43101 − 1.04411i 1.00000i 0 −1.51381 − 2.62200i
415.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.14595 − 0.661615i 0 −2.62831 + 0.303302i 1.00000i 0 −0.661615 − 1.14595i
415.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.935588 + 0.540162i 0 −1.55401 − 2.14127i 1.00000i 0 0.540162 + 0.935588i
415.9 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.18313 + 1.26043i 0 1.47350 − 2.19745i 1.00000i 0 1.26043 + 2.18313i
415.10 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.38129 + 1.37484i 0 −0.588218 + 2.57953i 1.00000i 0 1.37484 + 2.38129i
1117.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.38129 + 1.37484i 0 0.588218 + 2.57953i 1.00000i 0 1.37484 − 2.38129i
1117.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.18313 + 1.26043i 0 −1.47350 − 2.19745i 1.00000i 0 1.26043 − 2.18313i
1117.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.935588 + 0.540162i 0 1.55401 − 2.14127i 1.00000i 0 0.540162 − 0.935588i
1117.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.14595 − 0.661615i 0 2.62831 + 0.303302i 1.00000i 0 −0.661615 + 1.14595i
1117.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.62200 − 1.51381i 0 −2.43101 − 1.04411i 1.00000i 0 −1.51381 + 2.62200i
1117.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.62200 + 1.51381i 0 2.43101 + 1.04411i − 1.00000i 0 −1.51381 + 2.62200i
1117.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.14595 + 0.661615i 0 −2.62831 − 0.303302i − 1.00000i 0 −0.661615 + 1.14595i
1117.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.935588 − 0.540162i 0 −1.55401 + 2.14127i − 1.00000i 0 0.540162 − 0.935588i
1117.9 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.18313 − 1.26043i 0 1.47350 + 2.19745i − 1.00000i 0 1.26043 − 2.18313i
1117.10 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.38129 − 1.37484i 0 −0.588218 − 2.57953i − 1.00000i 0 1.37484 − 2.38129i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.dm.e 20
3.b odd 2 1 546.2.bk.c ✓ 20
7.c even 3 1 inner 1638.2.dm.e 20
13.b even 2 1 inner 1638.2.dm.e 20
21.g even 6 1 3822.2.c.n 10
21.h odd 6 1 546.2.bk.c ✓ 20
21.h odd 6 1 3822.2.c.m 10
39.d odd 2 1 546.2.bk.c ✓ 20
91.r even 6 1 inner 1638.2.dm.e 20
273.w odd 6 1 546.2.bk.c ✓ 20
273.w odd 6 1 3822.2.c.m 10
273.ba even 6 1 3822.2.c.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.c ✓ 20 3.b odd 2 1
546.2.bk.c ✓ 20 21.h odd 6 1
546.2.bk.c ✓ 20 39.d odd 2 1
546.2.bk.c ✓ 20 273.w odd 6 1
1638.2.dm.e 20 1.a even 1 1 trivial
1638.2.dm.e 20 7.c even 3 1 inner
1638.2.dm.e 20 13.b even 2 1 inner
1638.2.dm.e 20 91.r even 6 1 inner
3822.2.c.m 10 21.h odd 6 1
3822.2.c.m 10 273.w odd 6 1
3822.2.c.n 10 21.g even 6 1
3822.2.c.n 10 273.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 26 T_{5}^{18} + 431 T_{5}^{16} - 4370 T_{5}^{14} + 32381 T_{5}^{12} - 160412 T_{5}^{10} + \cdots + 810000 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 26 T^{18} + \cdots + 810000 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} - 85 T^{18} + \cdots + 31640625 \) Copy content Toggle raw display
$13$ \( (T^{10} - 2 T^{9} + \cdots + 371293)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 3 T^{9} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 207360000 \) Copy content Toggle raw display
$23$ \( (T^{10} - 8 T^{9} + \cdots + 944784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 7 T^{4} + \cdots - 3969)^{4} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1358954496 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 2821109907456 \) Copy content Toggle raw display
$41$ \( (T^{10} + 444 T^{8} + \cdots + 4671995904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 6 T^{4} + \cdots - 1712)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 3662186256 \) Copy content Toggle raw display
$53$ \( (T^{10} - 11 T^{9} + \cdots + 6395841)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{10} - 7 T^{9} + \cdots + 12376324)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 982045424460816 \) Copy content Toggle raw display
$71$ \( (T^{10} + 459 T^{8} + \cdots + 1913187600)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 167961600000000 \) Copy content Toggle raw display
$79$ \( (T^{10} - 2 T^{9} + \cdots + 3154176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 482 T^{8} + \cdots + 5817744)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + 894 T^{8} + \cdots + 15148194084)^{2} \) Copy content Toggle raw display
show more
show less