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Properties

Label 162.11.d.f
Level $162$
Weight $11$
Character orbit 162.d
Analytic conductor $102.928$
Analytic rank $0$
Dimension $16$
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] = [162,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(102.927874933\)
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: \( x^{16} - 406 x^{14} + 104825 x^{12} - 16724454 x^{10} + 1962018884 x^{8} - 157478793654 x^{6} + \cdots + 78\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{72} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (512 \beta_1 + 512) q^{4} + (\beta_{10} + 13 \beta_{2}) q^{5} + (\beta_{3} + 1349 \beta_1) q^{7} + ( - 512 \beta_{4} - 512 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (512 \beta_1 + 512) q^{4} + (\beta_{10} + 13 \beta_{2}) q^{5} + (\beta_{3} + 1349 \beta_1) q^{7} + ( - 512 \beta_{4} - 512 \beta_{2}) q^{8} + ( - \beta_{8} - \beta_{6} + \cdots + 6720) q^{10}+ \cdots + ( - 480728 \beta_{14} + \cdots + 181864704 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4096 q^{4} - 10792 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4096 q^{4} - 10792 q^{7} + 107520 q^{10} + 487760 q^{13} - 2097152 q^{16} - 11345152 q^{19} - 10521600 q^{22} + 11839360 q^{25} - 11051008 q^{28} - 39988984 q^{31} + 54577152 q^{34} + 156663104 q^{37} + 27525120 q^{40} + 221580656 q^{43} - 599734272 q^{46} - 1454763840 q^{49} - 249733120 q^{52} - 6490829520 q^{55} - 2324447232 q^{58} + 1339406432 q^{61} - 2147483648 q^{64} - 7550950576 q^{67} + 3087022080 q^{70} - 8580588016 q^{73} - 2904358912 q^{76} + 9120386960 q^{79} - 14318008320 q^{82} - 38167504560 q^{85} + 5387059200 q^{88} - 71121563936 q^{91} - 16609560576 q^{94} + 2553001640 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 406 x^{14} + 104825 x^{12} - 16724454 x^{10} + 1962018884 x^{8} - 157478793654 x^{6} + \cdots + 78\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18\!\cdots\!00 \nu^{14} + \cdots - 45\!\cdots\!25 ) / 23\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!51 \nu^{15} + \cdots - 24\!\cdots\!79 \nu ) / 78\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!05 \nu^{14} + \cdots + 66\!\cdots\!85 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65844236536387 \nu^{15} + \cdots - 10\!\cdots\!21 \nu ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!81 \nu^{14} + \cdots - 55\!\cdots\!09 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48\!\cdots\!63 \nu^{14} + \cdots - 78\!\cdots\!45 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 72\!\cdots\!15 \nu^{14} + \cdots - 13\!\cdots\!77 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 51\!\cdots\!21 \nu^{14} + \cdots + 10\!\cdots\!01 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!41 \nu^{14} + \cdots - 44\!\cdots\!89 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!50 \nu^{15} + \cdots + 69\!\cdots\!15 \nu ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!37 \nu^{15} + \cdots - 14\!\cdots\!62 \nu ) / 61\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!25 \nu^{15} + \cdots - 59\!\cdots\!45 \nu ) / 24\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!83 \nu^{15} + \cdots + 98\!\cdots\!85 \nu ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!73 \nu^{15} + \cdots + 41\!\cdots\!03 \nu ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35\!\cdots\!43 \nu^{15} + \cdots + 48\!\cdots\!17 \nu ) / 13\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 27\beta_{15} - 115\beta_{14} - 115\beta_{12} - 1464\beta_{11} - 1464\beta_{10} - 216\beta_{4} ) / 629856 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 161\beta_{8} + 131\beta_{7} - 186\beta_{6} + 161\beta_{3} + 63930384\beta _1 + 63930384 ) / 629856 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6029\beta_{14} - 837\beta_{13} - 96312\beta_{11} + 576504\beta_{4} + 576504\beta_{2} ) / 314928 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2857\beta_{9} - 4542\beta_{5} + 2227\beta_{3} + 784065744\beta_1 ) / 69984 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -83835\beta_{15} - 83835\beta_{13} + 1322579\beta_{12} + 25560408\beta_{10} + 395905320\beta_{2} ) / 629856 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 242773\beta_{9} - 91723\beta_{8} - 242773\beta_{7} + 437358\beta_{6} - 437358\beta_{5} - 52061889294 ) / 39366 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 929151 \beta_{15} + 146546039 \beta_{14} + 146546039 \beta_{12} + 3427411800 \beta_{11} + \cdots - 87167728152 \beta_{4} ) / 629856 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 99421 \beta_{8} - 6597671 \beta_{7} + 13561346 \beta_{6} - 99421 \beta_{3} - 1274931369936 \beta _1 - 1274931369936 ) / 7776 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8013236849 \beta_{14} - 322662231 \beta_{13} + 232030908696 \beta_{11} + \cdots - 8027505385128 \beta_{2} ) / 314928 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -70512391805\beta_{9} + 165753517830\beta_{5} + 22465270945\beta_{3} - 13244948349840144\beta_1 ) / 629856 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 145456922097 \beta_{15} - 145456922097 \beta_{13} - 1698962107975 \beta_{12} + \cdots - 26\!\cdots\!76 \beta_{2} ) / 629856 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 31678112653 \beta_{9} - 20032751357 \beta_{8} + 31678112653 \beta_{7} - 85049714718 \beta_{6} + \cdots + 60\!\cdots\!07 ) / 2187 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24188818469229 \beta_{15} - 170932704755627 \beta_{14} - 170932704755627 \beta_{12} + \cdots + 42\!\cdots\!32 \beta_{4} ) / 629856 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10\!\cdots\!63 \beta_{8} + \cdots + 23\!\cdots\!04 ) / 629856 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 78\!\cdots\!93 \beta_{14} + \cdots + 33\!\cdots\!20 \beta_{2} ) / 314928 \) Copy content Toggle raw display

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

Character values

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

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

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
10.2895 − 5.94066i
6.96457 − 4.02100i
−7.83060 + 4.52100i
−9.42349 + 5.44066i
9.42349 − 5.44066i
7.83060 − 4.52100i
−6.96457 + 4.02100i
−10.2895 + 5.94066i
10.2895 + 5.94066i
6.96457 + 4.02100i
−7.83060 − 4.52100i
−9.42349 − 5.44066i
9.42349 + 5.44066i
7.83060 + 4.52100i
−6.96457 − 4.02100i
−10.2895 − 5.94066i
−19.5959 + 11.3137i 0 256.000 − 443.405i −5177.68 − 2989.34i 0 1371.22 + 2375.02i 11585.2i 0 135282.
53.2 −19.5959 + 11.3137i 0 256.000 − 443.405i 560.843 + 323.803i 0 14912.7 + 25829.5i 11585.2i 0 −14653.7
53.3 −19.5959 + 11.3137i 0 256.000 − 443.405i 1496.86 + 864.212i 0 −14963.4 − 25917.4i 11585.2i 0 −39109.8
53.4 −19.5959 + 11.3137i 0 256.000 − 443.405i 2091.20 + 1207.35i 0 −4018.47 − 6960.19i 11585.2i 0 −54638.5
53.5 19.5959 − 11.3137i 0 256.000 − 443.405i −2091.20 − 1207.35i 0 −4018.47 − 6960.19i − 11585.2i 0 −54638.5
53.6 19.5959 − 11.3137i 0 256.000 − 443.405i −1496.86 − 864.212i 0 −14963.4 − 25917.4i − 11585.2i 0 −39109.8
53.7 19.5959 − 11.3137i 0 256.000 − 443.405i −560.843 − 323.803i 0 14912.7 + 25829.5i − 11585.2i 0 −14653.7
53.8 19.5959 − 11.3137i 0 256.000 − 443.405i 5177.68 + 2989.34i 0 1371.22 + 2375.02i − 11585.2i 0 135282.
107.1 −19.5959 − 11.3137i 0 256.000 + 443.405i −5177.68 + 2989.34i 0 1371.22 − 2375.02i − 11585.2i 0 135282.
107.2 −19.5959 − 11.3137i 0 256.000 + 443.405i 560.843 − 323.803i 0 14912.7 − 25829.5i − 11585.2i 0 −14653.7
107.3 −19.5959 − 11.3137i 0 256.000 + 443.405i 1496.86 − 864.212i 0 −14963.4 + 25917.4i − 11585.2i 0 −39109.8
107.4 −19.5959 − 11.3137i 0 256.000 + 443.405i 2091.20 − 1207.35i 0 −4018.47 + 6960.19i − 11585.2i 0 −54638.5
107.5 19.5959 + 11.3137i 0 256.000 + 443.405i −2091.20 + 1207.35i 0 −4018.47 + 6960.19i 11585.2i 0 −54638.5
107.6 19.5959 + 11.3137i 0 256.000 + 443.405i −1496.86 + 864.212i 0 −14963.4 + 25917.4i 11585.2i 0 −39109.8
107.7 19.5959 + 11.3137i 0 256.000 + 443.405i −560.843 + 323.803i 0 14912.7 − 25829.5i 11585.2i 0 −14653.7
107.8 19.5959 + 11.3137i 0 256.000 + 443.405i 5177.68 − 2989.34i 0 1371.22 − 2375.02i 11585.2i 0 135282.
\(n\): 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 162.11.d.f 16
3.b odd 2 1 inner 162.11.d.f 16
9.c even 3 1 54.11.b.c ✓ 8
9.c even 3 1 inner 162.11.d.f 16
9.d odd 6 1 54.11.b.c ✓ 8
9.d odd 6 1 inner 162.11.d.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.11.b.c ✓ 8 9.c even 3 1
54.11.b.c ✓ 8 9.d odd 6 1
162.11.d.f 16 1.a even 1 1 trivial
162.11.d.f 16 3.b odd 2 1 inner
162.11.d.f 16 9.c even 3 1 inner
162.11.d.f 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 44982180 T_{5}^{14} + \cdots + 68\!\cdots\!25 \) acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 512 T^{2} + 262144)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 38\!\cdots\!61)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 57\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 15\!\cdots\!24)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 17\!\cdots\!21)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 66\!\cdots\!76)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 39\!\cdots\!21)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 38\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 20\!\cdots\!61)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 39\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 62\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 25\!\cdots\!01)^{2} \) Copy content Toggle raw display
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