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Properties

Label 164.6.b.a
Level $164$
Weight $6$
Character orbit 164.b
Analytic conductor $26.303$
Analytic rank $0$
Dimension $18$
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] = [164,6,Mod(81,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.81");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 164.b (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: \(26.3029464493\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 3198 x^{16} + 4145894 x^{14} + 2811630520 x^{12} + 1078698643464 x^{10} + 238264942196960 x^{8} + \cdots + 89\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - 4) q^{5} + (\beta_{12} + \beta_1) q^{7} + (\beta_{2} - 112) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - 4) q^{5} + (\beta_{12} + \beta_1) q^{7} + (\beta_{2} - 112) q^{9} + ( - \beta_{13} + \beta_1) q^{11} + (\beta_{12} - \beta_{11} + 5 \beta_1) q^{13} + ( - \beta_{13} - 2 \beta_{12} + \cdots - 5 \beta_1) q^{15}+ \cdots + (9 \beta_{17} + 29 \beta_{16} + \cdots - 289 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 72 q^{5} - 2022 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 72 q^{5} - 2022 q^{9} - 5892 q^{21} + 2680 q^{23} + 11406 q^{25} - 2536 q^{31} - 3412 q^{33} + 2672 q^{37} - 29896 q^{39} - 5078 q^{41} + 12352 q^{43} + 13824 q^{45} + 2202 q^{49} + 42176 q^{51} - 82388 q^{57} - 2944 q^{59} + 7932 q^{61} - 167392 q^{73} + 91324 q^{77} + 271030 q^{81} + 37640 q^{83} - 377544 q^{87} - 288904 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 3198 x^{16} + 4145894 x^{14} + 2811630520 x^{12} + 1078698643464 x^{10} + 238264942196960 x^{8} + \cdots + 89\!\cdots\!32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65\!\cdots\!29 \nu^{16} + \cdots - 18\!\cdots\!24 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!15 \nu^{16} + \cdots - 46\!\cdots\!76 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!17 \nu^{16} + \cdots - 42\!\cdots\!40 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!73 \nu^{16} + \cdots + 42\!\cdots\!12 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!57 \nu^{16} + \cdots - 27\!\cdots\!24 ) / 29\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!81 \nu^{16} + \cdots + 21\!\cdots\!24 ) / 35\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34\!\cdots\!31 \nu^{16} + \cdots - 38\!\cdots\!60 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 75\!\cdots\!07 \nu^{17} + \cdots + 41\!\cdots\!84 \nu ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!75 \nu^{17} + \cdots - 65\!\cdots\!96 \nu ) / 13\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 91\!\cdots\!17 \nu^{17} + \cdots + 14\!\cdots\!76 \nu ) / 89\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!47 \nu^{17} + \cdots - 15\!\cdots\!36 \nu ) / 36\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!81 \nu^{17} + \cdots + 14\!\cdots\!04 \nu ) / 42\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!43 \nu^{17} + \cdots + 29\!\cdots\!80 \nu ) / 53\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!03 \nu^{17} + \cdots + 18\!\cdots\!68 \nu ) / 42\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 82\!\cdots\!61 \nu^{17} + \cdots + 13\!\cdots\!84 \nu ) / 42\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 355 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} - 7\beta_{12} + \beta_{11} - 606\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - \beta_{8} - 3\beta_{7} - 13\beta_{6} + \beta_{5} - 14\beta_{4} + 157\beta_{3} - 843\beta_{2} + 214765 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} - 336 \beta_{16} - 935 \beta_{15} + 1259 \beta_{14} - 284 \beta_{13} + \cdots + 428250 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1724 \beta_{9} + 251 \beta_{8} + 4794 \beta_{7} + 15918 \beta_{6} - 238 \beta_{5} + 18703 \beta_{4} + \cdots - 151533623 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10884 \beta_{17} + 426945 \beta_{16} + 766299 \beta_{15} - 1211775 \beta_{14} + 473702 \beta_{13} + \cdots - 320479304 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 972244 \beta_{9} + 443940 \beta_{8} - 6145839 \beta_{7} - 15415705 \beta_{6} - 781657 \beta_{5} + \cdots + 113221489877 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13808931 \beta_{17} - 419676654 \beta_{16} - 612491368 \beta_{15} + 1072373416 \beta_{14} + \cdots + 246724498664 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 313444618 \beta_{9} - 827720543 \beta_{8} + 6936209049 \beta_{7} + 13838928408 \beta_{6} + \cdots - 87045128603190 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13035210756 \beta_{17} + 379448567298 \beta_{16} + 488106831866 \beta_{15} + \cdots - 193153575071872 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 140217283896 \beta_{9} + 965743660780 \beta_{8} - 7193923867158 \beta_{7} - 11994521826586 \beta_{6} + \cdots + 68\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10627529705622 \beta_{17} - 330389009052576 \beta_{16} - 390298790642028 \beta_{15} + \cdots + 15\!\cdots\!40 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 410635898637660 \beta_{9} - 953972375423742 \beta_{8} + \cdots - 53\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 78\!\cdots\!40 \beta_{17} + \cdots - 12\!\cdots\!68 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 54\!\cdots\!36 \beta_{9} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 52\!\cdots\!40 \beta_{17} + \cdots + 98\!\cdots\!12 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
− 29.0201i
− 27.4898i
− 25.2642i
− 19.0249i
− 17.2784i
− 11.2815i
− 9.22322i
− 7.34383i
− 5.92502i
5.92502i
7.34383i
9.22322i
11.2815i
17.2784i
19.0249i
25.2642i
27.4898i
29.0201i
0 − 29.0201i 0 −95.4967 0 − 164.505i 0 −599.165 0
81.2 0 − 27.4898i 0 90.2281 0 1.96529i 0 −512.687 0
81.3 0 − 25.2642i 0 7.21142 0 − 8.54515i 0 −395.280 0
81.4 0 − 19.0249i 0 12.9614 0 141.096i 0 −118.948 0
81.5 0 − 17.2784i 0 −41.0764 0 141.083i 0 −55.5440 0
81.6 0 − 11.2815i 0 35.2361 0 − 248.823i 0 115.729 0
81.7 0 − 9.22322i 0 −36.1834 0 − 119.494i 0 157.932 0
81.8 0 − 7.34383i 0 −82.1247 0 73.7758i 0 189.068 0
81.9 0 − 5.92502i 0 73.2442 0 39.6968i 0 207.894 0
81.10 0 5.92502i 0 73.2442 0 − 39.6968i 0 207.894 0
81.11 0 7.34383i 0 −82.1247 0 − 73.7758i 0 189.068 0
81.12 0 9.22322i 0 −36.1834 0 119.494i 0 157.932 0
81.13 0 11.2815i 0 35.2361 0 248.823i 0 115.729 0
81.14 0 17.2784i 0 −41.0764 0 − 141.083i 0 −55.5440 0
81.15 0 19.0249i 0 12.9614 0 − 141.096i 0 −118.948 0
81.16 0 25.2642i 0 7.21142 0 8.54515i 0 −395.280 0
81.17 0 27.4898i 0 90.2281 0 − 1.96529i 0 −512.687 0
81.18 0 29.0201i 0 −95.4967 0 164.505i 0 −599.165 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.6.b.a ✓ 18
41.b even 2 1 inner 164.6.b.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.6.b.a ✓ 18 1.a even 1 1 trivial
164.6.b.a ✓ 18 41.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(164, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 89\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots - 253710214004736)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 30\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 36\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 75\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots + 83\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 19\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots + 85\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 70\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 68\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 26\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 15\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
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