Properties

Label 162.9.d.g
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,9,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not 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: \(65.9953348299\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11104 x^{14} + 77885403 x^{12} - 342246555016 x^{10} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{40} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_{8} q^{2} + ( - 128 \beta_1 + 128) q^{4} + (\beta_{13} - \beta_{12} + \beta_{11} + 151 \beta_{9} + \beta_{8}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{4} + 8 \beta_{2} - 186 \beta_1) q^{7} + (1024 \beta_{9} + 1024 \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{8} q^{2} + ( - 128 \beta_1 + 128) q^{4} + (\beta_{13} - \beta_{12} + \beta_{11} + 151 \beta_{9} + \beta_{8}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{4} + 8 \beta_{2} - 186 \beta_1) q^{7} + (1024 \beta_{9} + 1024 \beta_{8}) q^{8} + ( - 8 \beta_{5} + 8 \beta_{3} - 2400) q^{10} + ( - \beta_{14} + 14 \beta_{11} - 1640 \beta_{8}) q^{11} + (11 \beta_{6} - 16 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} + 5853 \beta_1 - 5853) q^{13} + (8 \beta_{15} + 128 \beta_{13} - 8 \beta_{12} + 128 \beta_{11} + 1560 \beta_{9} + 128 \beta_{8}) q^{14} - 16384 \beta_1 q^{16} + ( - 13 \beta_{15} + 13 \beta_{14} - 227 \beta_{13} - 155 \beta_{12} + \cdots + 24690 \beta_{8}) q^{17}+ \cdots + (11888 \beta_{15} - 11888 \beta_{14} + 525088 \beta_{13} + \cdots - 23685112 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1024 q^{4} - 1492 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 1024 q^{4} - 1492 q^{7} - 38400 q^{10} - 46780 q^{13} - 131072 q^{16} - 1280872 q^{19} - 210816 q^{22} + 156616 q^{25} - 381952 q^{28} - 107776 q^{31} - 3164928 q^{34} - 2918248 q^{37} - 2457600 q^{40} + 8465468 q^{43} - 10592640 q^{46} - 23911272 q^{49} + 5987840 q^{52} + 8196984 q^{55} + 626688 q^{58} + 28337396 q^{61} - 33554432 q^{64} - 93012796 q^{67} + 39994944 q^{70} - 278621824 q^{73} - 81975808 q^{76} + 40890356 q^{79} - 195950976 q^{82} - 479549772 q^{85} + 26984448 q^{88} - 1352438072 q^{91} - 406812480 q^{94} + 49442912 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11104 x^{14} + 77885403 x^{12} - 342246555016 x^{10} + \cdots + 28\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\!\cdots\!45 \nu^{14} + \cdots - 20\!\cdots\!48 ) / 21\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!27 \nu^{14} + \cdots + 49\!\cdots\!60 ) / 64\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{14} + \cdots + 12\!\cdots\!28 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!51 \nu^{14} + \cdots - 58\!\cdots\!76 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!63 \nu^{14} + \cdots - 46\!\cdots\!52 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!85 \nu^{14} + \cdots - 74\!\cdots\!76 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\!\cdots\!27 \nu^{14} + \cdots - 10\!\cdots\!00 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!91 \nu^{15} + \cdots - 19\!\cdots\!12 \nu ) / 42\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!61 \nu^{15} + \cdots - 42\!\cdots\!96 \nu ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 85\!\cdots\!67 \nu^{15} + \cdots + 18\!\cdots\!80 \nu ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 79\!\cdots\!73 \nu^{15} + \cdots - 24\!\cdots\!40 \nu ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!85 \nu^{15} + \cdots + 62\!\cdots\!88 \nu ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!79 \nu^{15} + \cdots + 24\!\cdots\!84 \nu ) / 51\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!89 \nu^{15} + \cdots - 76\!\cdots\!00 \nu ) / 42\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 95\!\cdots\!35 \nu^{15} + \cdots - 95\!\cdots\!76 \nu ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{15} + 4\beta_{13} - 13\beta_{12} + 4\beta_{11} + 49\beta_{9} + 4\beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 15\beta_{5} + 15\beta_{4} - 738\beta_{2} + 449711\beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1673 \beta_{15} + 1673 \beta_{14} + 18864 \beta_{13} - 41621 \beta_{12} - 41621 \beta_{10} + 501379 \beta_{9} + 501379 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11104\beta_{6} + 103125\beta_{4} + 4108806\beta_{3} - 4108806\beta_{2} + 1315110043\beta _1 - 1315110043 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1982509\beta_{14} - 81274620\beta_{11} - 136331191\beta_{10} + 2909594105\beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 48681374\beta_{7} - 48681374\beta_{6} - 530474235\beta_{5} + 17472818586\beta_{3} - 4023019402499 ) / 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1232327353 \beta_{15} - 334702500192 \beta_{13} + 453769993415 \beta_{12} - 334702500192 \beta_{11} - 14776449049837 \beta_{9} - 334702500192 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 198311421880 \beta_{7} - 2422307561835 \beta_{5} - 2422307561835 \beta_{4} + 67210123467450 \beta_{2} - 12\!\cdots\!17 \beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 21422959842329 \beta_{15} + 21422959842329 \beta_{14} + \cdots - 66\!\cdots\!59 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 784340321652002 \beta_{6} + \cdots + 41\!\cdots\!01 ) / 162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11\!\cdots\!87 \beta_{14} + \cdots - 27\!\cdots\!15 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 30\!\cdots\!44 \beta_{7} + \cdots + 13\!\cdots\!63 ) / 162 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 45\!\cdots\!51 \beta_{15} + \cdots + 20\!\cdots\!60 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11\!\cdots\!54 \beta_{7} + \cdots + 46\!\cdots\!59 \beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17\!\cdots\!53 \beta_{15} + \cdots + 45\!\cdots\!37 \beta_{8} ) / 162 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−49.7470 28.7215i
−40.2810 23.2563i
39.8327 + 22.9974i
51.4201 + 29.6874i
−51.4201 29.6874i
−39.8327 22.9974i
40.2810 + 23.2563i
49.7470 + 28.7215i
−49.7470 + 28.7215i
−40.2810 + 23.2563i
39.8327 22.9974i
51.4201 29.6874i
−51.4201 + 29.6874i
−39.8327 + 22.9974i
40.2810 23.2563i
49.7470 28.7215i
−9.79796 + 5.65685i 0 64.0000 110.851i −467.559 269.945i 0 108.294 + 187.571i 1448.15i 0 6108.16
53.2 −9.79796 + 5.65685i 0 64.0000 110.851i −35.4801 20.4844i 0 −1314.30 2276.44i 1448.15i 0 463.510
53.3 −9.79796 + 5.65685i 0 64.0000 110.851i 269.260 + 155.458i 0 2242.38 + 3883.91i 1448.15i 0 −3517.60
53.4 −9.79796 + 5.65685i 0 64.0000 110.851i 968.625 + 559.236i 0 −1409.37 2441.10i 1448.15i 0 −12654.1
53.5 9.79796 5.65685i 0 64.0000 110.851i −968.625 559.236i 0 −1409.37 2441.10i 1448.15i 0 −12654.1
53.6 9.79796 5.65685i 0 64.0000 110.851i −269.260 155.458i 0 2242.38 + 3883.91i 1448.15i 0 −3517.60
53.7 9.79796 5.65685i 0 64.0000 110.851i 35.4801 + 20.4844i 0 −1314.30 2276.44i 1448.15i 0 463.510
53.8 9.79796 5.65685i 0 64.0000 110.851i 467.559 + 269.945i 0 108.294 + 187.571i 1448.15i 0 6108.16
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i −467.559 + 269.945i 0 108.294 187.571i 1448.15i 0 6108.16
107.2 −9.79796 5.65685i 0 64.0000 + 110.851i −35.4801 + 20.4844i 0 −1314.30 + 2276.44i 1448.15i 0 463.510
107.3 −9.79796 5.65685i 0 64.0000 + 110.851i 269.260 155.458i 0 2242.38 3883.91i 1448.15i 0 −3517.60
107.4 −9.79796 5.65685i 0 64.0000 + 110.851i 968.625 559.236i 0 −1409.37 + 2441.10i 1448.15i 0 −12654.1
107.5 9.79796 + 5.65685i 0 64.0000 + 110.851i −968.625 + 559.236i 0 −1409.37 + 2441.10i 1448.15i 0 −12654.1
107.6 9.79796 + 5.65685i 0 64.0000 + 110.851i −269.260 + 155.458i 0 2242.38 3883.91i 1448.15i 0 −3517.60
107.7 9.79796 + 5.65685i 0 64.0000 + 110.851i 35.4801 20.4844i 0 −1314.30 + 2276.44i 1448.15i 0 463.510
107.8 9.79796 + 5.65685i 0 64.0000 + 110.851i 467.559 269.945i 0 108.294 187.571i 1448.15i 0 6108.16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.d.g 16
3.b odd 2 1 inner 162.9.d.g 16
9.c even 3 1 162.9.b.b 8
9.c even 3 1 inner 162.9.d.g 16
9.d odd 6 1 162.9.b.b 8
9.d odd 6 1 inner 162.9.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.9.b.b 8 9.c even 3 1
162.9.b.b 8 9.d odd 6 1
162.9.d.g 16 1.a even 1 1 trivial
162.9.d.g 16 3.b odd 2 1 inner
162.9.d.g 16 9.c even 3 1 inner
162.9.d.g 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 1640808 T_{5}^{14} + 2175755301198 T_{5}^{12} + \cdots + 35\!\cdots\!25 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 1640808 T^{14} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 746 T^{7} + \cdots + 51\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 86966028 T^{14} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{8} + 23390 T^{7} + \cdots + 45\!\cdots\!61)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 50461667568 T^{6} + \cdots + 42\!\cdots\!25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 320218 T^{3} + \cdots - 10\!\cdots\!04)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 222640275348 T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} - 1329947673696 T^{14} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( (T^{8} + 53888 T^{7} + \cdots + 22\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 729562 T^{3} + \cdots + 13\!\cdots\!53)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} - 37646502934896 T^{14} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} - 4232734 T^{7} + \cdots + 84\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 138220620047856 T^{14} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{8} + 18704070936720 T^{6} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 799197825064464 T^{14} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} - 14168698 T^{7} + \cdots + 15\!\cdots\!89)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 46506398 T^{7} + \cdots + 36\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 69655456 T^{3} + \cdots - 13\!\cdots\!11)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 20445178 T^{7} + \cdots + 41\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 23\!\cdots\!21)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 24721456 T^{7} + \cdots + 38\!\cdots\!64)^{2} \) Copy content Toggle raw display
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