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Properties

Label 164.2.k.a
Level $164$
Weight $2$
Character orbit 164.k
Analytic conductor $1.310$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,2,Mod(25,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.k (of order \(10\), degree \(4\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 39x^{14} + 594x^{12} + 4428x^{10} + 16529x^{8} + 28236x^{6} + 17856x^{4} + 4032x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{12} q^{5} + (\beta_{14} - \beta_{9}) q^{7} + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{12} q^{5} + (\beta_{14} - \beta_{9}) q^{7} + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 2) q^{9}+ \cdots + ( - 2 \beta_{15} + 5 \beta_{13} + \cdots - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{5} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{5} - 30 q^{9} - 5 q^{11} - 10 q^{15} + 5 q^{17} + 15 q^{19} + 19 q^{21} + 12 q^{23} - 2 q^{25} + 20 q^{29} + 3 q^{31} - 25 q^{33} + 5 q^{35} + 2 q^{37} - 28 q^{39} - 4 q^{41} - 22 q^{43} - 48 q^{45} + 15 q^{47} - 28 q^{49} + 17 q^{51} + 25 q^{53} - 8 q^{57} + 8 q^{59} - 46 q^{61} - 40 q^{63} - 10 q^{65} - 45 q^{67} + 10 q^{69} + 15 q^{71} + 34 q^{73} + 135 q^{75} + 23 q^{77} + 108 q^{81} + 12 q^{83} + 14 q^{87} + 60 q^{93} - 30 q^{95} - 40 q^{97} + 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 39x^{14} + 594x^{12} + 4428x^{10} + 16529x^{8} + 28236x^{6} + 17856x^{4} + 4032x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{14} + 5 \nu^{13} - 31 \nu^{12} + 139 \nu^{11} - 354 \nu^{10} + 1330 \nu^{9} - 1780 \nu^{8} + \cdots + 2336 ) / 576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13 \nu^{15} - 2 \nu^{14} - 415 \nu^{13} - 214 \nu^{12} - 4854 \nu^{11} - 5116 \nu^{10} + \cdots - 21888 ) / 9216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -15\nu^{14} - 437\nu^{12} - 4594\nu^{10} - 21148\nu^{8} - 43311\nu^{6} - 47456\nu^{4} - 49408\nu^{2} - 10816 ) / 2304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} + 107 \nu^{13} + 194 \nu^{12} + 2558 \nu^{11} + 2356 \nu^{10} + 24708 \nu^{9} + \cdots + 2176 ) / 4608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 107 \nu^{13} + 194 \nu^{12} - 2558 \nu^{11} + 2356 \nu^{10} - 24708 \nu^{9} + \cdots + 2176 ) / 4608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17 \nu^{15} - 2 \nu^{14} - 651 \nu^{13} - 214 \nu^{12} - 9710 \nu^{11} - 5116 \nu^{10} + \cdots - 17280 ) / 9216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17 \nu^{15} - 2 \nu^{14} + 651 \nu^{13} - 214 \nu^{12} + 9710 \nu^{11} - 5116 \nu^{10} + 70564 \nu^{9} + \cdots - 17280 ) / 9216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} - 34 \nu^{14} - 107 \nu^{13} - 1046 \nu^{12} - 2558 \nu^{11} - 11996 \nu^{10} + \cdots - 6528 ) / 4608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{15} - 18 \nu^{14} - 731 \nu^{13} - 710 \nu^{12} - 11934 \nu^{11} - 10780 \nu^{10} + \cdots + 10880 ) / 9216 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19 \nu^{15} + 66 \nu^{14} + 753 \nu^{13} + 2326 \nu^{12} + 11674 \nu^{11} + 31868 \nu^{10} + \cdots + 22400 ) / 9216 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17 \nu^{15} - 46 \nu^{14} + 811 \nu^{13} - 1370 \nu^{12} + 14542 \nu^{11} - 14852 \nu^{10} + \cdots + 11136 ) / 9216 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51 \nu^{15} + 2 \nu^{14} + 1921 \nu^{13} + 214 \nu^{12} + 28202 \nu^{11} + 5116 \nu^{10} + \cdots + 21888 ) / 9216 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 103 \nu^{15} - 98 \nu^{14} - 4093 \nu^{13} - 3414 \nu^{12} - 63234 \nu^{11} - 46172 \nu^{10} + \cdots - 84352 ) / 9216 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 103 \nu^{15} + 98 \nu^{14} - 4093 \nu^{13} + 3414 \nu^{12} - 63234 \nu^{11} + 46172 \nu^{10} + \cdots + 84352 ) / 9216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} - 2\beta_{11} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{10} + \beta_{8} - 2\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{15} + 11 \beta_{14} - 9 \beta_{13} + 18 \beta_{11} - 4 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} + 25 \beta_{13} - 2 \beta_{12} + 14 \beta_{10} - 7 \beta_{8} + 25 \beta_{7} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 114 \beta_{15} - 114 \beta_{14} + 84 \beta_{13} - 168 \beta_{11} - 2 \beta_{10} + 66 \beta_{9} + \cdots - 432 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{15} - 54 \beta_{14} - 355 \beta_{13} + 24 \beta_{12} - 153 \beta_{10} + 35 \beta_{8} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1161 \beta_{15} + 1161 \beta_{14} - 810 \beta_{13} + 1620 \beta_{11} + 45 \beta_{10} - 816 \beta_{9} + \cdots + 4223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 714 \beta_{15} + 714 \beta_{14} + 4221 \beta_{13} - 204 \beta_{12} + 1557 \beta_{10} - 27 \beta_{8} + \cdots - 828 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11738 \beta_{15} - 11738 \beta_{14} + 7949 \beta_{13} - 15898 \beta_{11} - 675 \beta_{10} + 9108 \beta_{9} + \cdots - 41620 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8430 \beta_{15} - 8430 \beta_{14} - 46472 \beta_{13} + 1356 \beta_{12} - 15416 \beta_{10} + \cdots + 9960 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 118285 \beta_{15} + 118285 \beta_{14} - 78762 \beta_{13} + 157524 \beta_{11} + 8691 \beta_{10} + \cdots + 411981 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 94467 \beta_{15} + 94467 \beta_{14} + 492821 \beta_{13} - 5242 \beta_{12} + 150712 \beta_{10} + \cdots - 112509 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1190277 \beta_{15} - 1190277 \beta_{14} + 784716 \beta_{13} - 1569432 \beta_{11} - 104137 \beta_{10} + \cdots - 4088691 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1030527 \beta_{15} - 1030527 \beta_{14} - 5126360 \beta_{13} - 37938 \beta_{12} - 1463571 \beta_{10} + \cdots + 1230963 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
3.12433i
1.66505i
− 0.527140i
− 2.36013i
3.11423i
− 0.322385i
− 0.770134i
− 3.19728i
2.36013i
0.527140i
− 1.66505i
− 3.12433i
3.19728i
0.770134i
0.322385i
− 3.11423i
0 − 3.12433i 0 3.43428 + 2.49515i 0 1.24458 − 0.404388i 0 −6.76146 0
25.2 0 − 1.66505i 0 −2.75356 − 2.00058i 0 −0.676850 + 0.219922i 0 0.227602 0
25.3 0 0.527140i 0 −0.0209304 − 0.0152068i 0 1.98899 − 0.646262i 0 2.72212 0
25.4 0 2.36013i 0 1.45824 + 1.05947i 0 −2.55672 + 0.830728i 0 −2.57023 0
45.1 0 − 3.11423i 0 0.234613 − 0.722064i 0 −0.334979 + 0.461059i 0 −6.69841 0
45.2 0 0.322385i 0 −0.824490 + 2.53752i 0 −2.33895 + 3.21928i 0 2.89607 0
45.3 0 0.770134i 0 1.14437 − 3.52202i 0 0.0442579 − 0.0609158i 0 2.40689 0
45.4 0 3.19728i 0 −0.672530 + 2.06983i 0 2.62967 − 3.61943i 0 −7.22259 0
105.1 0 − 2.36013i 0 1.45824 − 1.05947i 0 −2.55672 − 0.830728i 0 −2.57023 0
105.2 0 − 0.527140i 0 −0.0209304 + 0.0152068i 0 1.98899 + 0.646262i 0 2.72212 0
105.3 0 1.66505i 0 −2.75356 + 2.00058i 0 −0.676850 − 0.219922i 0 0.227602 0
105.4 0 3.12433i 0 3.43428 − 2.49515i 0 1.24458 + 0.404388i 0 −6.76146 0
113.1 0 − 3.19728i 0 −0.672530 − 2.06983i 0 2.62967 + 3.61943i 0 −7.22259 0
113.2 0 − 0.770134i 0 1.14437 + 3.52202i 0 0.0442579 + 0.0609158i 0 2.40689 0
113.3 0 − 0.322385i 0 −0.824490 − 2.53752i 0 −2.33895 − 3.21928i 0 2.89607 0
113.4 0 3.11423i 0 0.234613 + 0.722064i 0 −0.334979 − 0.461059i 0 −6.69841 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.2.k.a ✓ 16
3.b odd 2 1 1476.2.bb.b 16
4.b odd 2 1 656.2.be.e 16
41.f even 10 1 inner 164.2.k.a ✓ 16
41.g even 20 2 6724.2.a.j 16
123.l odd 10 1 1476.2.bb.b 16
164.l odd 10 1 656.2.be.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.k.a ✓ 16 1.a even 1 1 trivial
164.2.k.a ✓ 16 41.f even 10 1 inner
656.2.be.e 16 4.b odd 2 1
656.2.be.e 16 164.l odd 10 1
1476.2.bb.b 16 3.b odd 2 1
1476.2.bb.b 16 123.l odd 10 1
6724.2.a.j 16 41.g even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(164, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 39 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{16} + 20 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{16} + 5 T^{15} + \cdots + 633616 \) Copy content Toggle raw display
$13$ \( T^{16} - 25 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{16} - 5 T^{15} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12204946576 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13577842576 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 203233536 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 324371255296 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3752665081 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$43$ \( T^{16} + 22 T^{15} + \cdots + 1948816 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 208925783056 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1151855721 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5388043718656 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 8895312215001 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 796242120976 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5235328391056 \) Copy content Toggle raw display
$73$ \( (T^{8} - 17 T^{7} + \cdots - 1437444)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 34930114816 \) Copy content Toggle raw display
$83$ \( (T^{8} - 6 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11764439296 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 733165775001 \) Copy content Toggle raw display
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