Properties

Label 162.9.d.f
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $8$
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] = [162,9,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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.206763233181696.28
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - 128 \beta_1 q^{4} + ( - \beta_{4} + 21 \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - 41 \beta_1 - 41) q^{7} + (128 \beta_{3} - 128 \beta_{2}) q^{8} + ( - 2 \beta_{5} + 2688) q^{10} + ( - 29 \beta_{7} + 21 \beta_{3}) q^{11} + (\beta_{6} + 24911 \beta_1) q^{13} + ( - 64 \beta_{4} - 41 \beta_{2}) q^{14} + ( - 16384 \beta_1 - 16384) q^{16} + ( - 53 \beta_{7} + 53 \beta_{4} - 8379 \beta_{3} + 8379 \beta_{2}) q^{17} + ( - 22 \beta_{5} + 128009) q^{19} + ( - 128 \beta_{7} + 2688 \beta_{3}) q^{20} + ( - 58 \beta_{6} - 2688 \beta_1) q^{22} + (259 \beta_{4} - 17583 \beta_{2}) q^{23} + (84 \beta_{6} - 84 \beta_{5} + 126551 \beta_1 + 126551) q^{25} + (64 \beta_{7} - 64 \beta_{4} - 24911 \beta_{3} + 24911 \beta_{2}) q^{26} + ( - 128 \beta_{5} - 5248) q^{28} + (1306 \beta_{7} + 38838 \beta_{3}) q^{29} + (204 \beta_{6} + 148886 \beta_1) q^{31} - 16384 \beta_{2} q^{32} + ( - 106 \beta_{6} + 106 \beta_{5} + 1072512 \beta_1 + 1072512) q^{34} + (1303 \beta_{7} - 1303 \beta_{4} - 229503 \beta_{3} + 229503 \beta_{2}) q^{35} + (505 \beta_{5} - 252745) q^{37} + ( - 1408 \beta_{7} + 128009 \beta_{3}) q^{38} + ( - 256 \beta_{6} - 344064 \beta_1) q^{40} + (338 \beta_{4} - 204162 \beta_{2}) q^{41} + ( - 588 \beta_{6} + 588 \beta_{5} + 2023546 \beta_1 + 2023546) q^{43} + ( - 3712 \beta_{7} + 3712 \beta_{4} + 2688 \beta_{3} - 2688 \beta_{2}) q^{44} + (518 \beta_{5} - 2250624) q^{46} + ( - 2253 \beta_{7} + 296697 \beta_{3}) q^{47} + ( - 82 \beta_{6} + 8980176 \beta_1) q^{49} + ( - 5376 \beta_{4} + 126551 \beta_{2}) q^{50} + (128 \beta_{6} - 128 \beta_{5} + 3188608 \beta_1 + 3188608) q^{52} + ( - 1990 \beta_{7} + 1990 \beta_{4} + 229086 \beta_{3} - 229086 \beta_{2}) q^{53} + ( - 1260 \beta_{5} + 13417560) q^{55} + ( - 8192 \beta_{7} - 5248 \beta_{3}) q^{56} + (2612 \beta_{6} - 4971264 \beta_1) q^{58} + (5527 \beta_{4} + 1402905 \beta_{2}) q^{59} + (1761 \beta_{6} - 1761 \beta_{5} - 1196159 \beta_1 - 1196159) q^{61} + (13056 \beta_{7} - 13056 \beta_{4} - 148886 \beta_{3} + 148886 \beta_{2}) q^{62} - 2097152 q^{64} + (26255 \beta_{7} - 753495 \beta_{3}) q^{65} + ( - 5748 \beta_{6} + 14289425 \beta_1) q^{67} + (6784 \beta_{4} + 1072512 \beta_{2}) q^{68} + (2606 \beta_{6} - 2606 \beta_{5} + 29376384 \beta_1 + 29376384) q^{70} + (1300 \beta_{7} - 1300 \beta_{4} - 387324 \beta_{3} + 387324 \beta_{2}) q^{71} + ( - 6066 \beta_{5} + 3309023) q^{73} + (32320 \beta_{7} - 252745 \beta_{3}) q^{74} + ( - 2816 \beta_{6} - 16385152 \beta_1) q^{76} + ( - 155 \beta_{4} + 6679695 \beta_{2}) q^{77} + ( - 4537 \beta_{6} + 4537 \beta_{5} - 12395921 \beta_1 - 12395921) q^{79} + ( - 16384 \beta_{7} + 16384 \beta_{4} + 344064 \beta_{3} - 344064 \beta_{2}) q^{80} + (676 \beta_{5} - 26132736) q^{82} + ( - 101538 \beta_{7} - 7206 \beta_{3}) q^{83} + (14532 \beta_{6} - 1895832 \beta_1) q^{85} + (37632 \beta_{4} + 2023546 \beta_{2}) q^{86} + ( - 7424 \beta_{6} + 7424 \beta_{5} - 344064 \beta_1 - 344064) q^{88} + ( - 14585 \beta_{7} + 14585 \beta_{4} - 6311223 \beta_{3} + 6311223 \beta_{2}) q^{89} + (24870 \beta_{5} - 13721945) q^{91} + (33152 \beta_{7} - 2250624 \beta_{3}) q^{92} + ( - 4506 \beta_{6} - 37977216 \beta_1) q^{94} + ( - 157577 \beta_{4} + 7756197 \beta_{2}) q^{95} + (14114 \beta_{6} - 14114 \beta_{5} - 102086231 \beta_1 - 102086231) q^{97} + ( - 5248 \beta_{7} + 5248 \beta_{4} - 8980176 \beta_{3} + 8980176 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 512 q^{4} - 164 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 512 q^{4} - 164 q^{7} + 21504 q^{10} - 99644 q^{13} - 65536 q^{16} + 1024072 q^{19} + 10752 q^{22} + 506204 q^{25} - 41984 q^{28} - 595544 q^{31} + 4290048 q^{34} - 2021960 q^{37} + 1376256 q^{40} + 8094184 q^{43} - 18004992 q^{46} - 35920704 q^{49} + 12754432 q^{52} + 107340480 q^{55} + 19885056 q^{58} - 4784636 q^{61} - 16777216 q^{64} - 57157700 q^{67} + 117505536 q^{70} + 26472184 q^{73} + 65540608 q^{76} - 49583684 q^{79} - 209061888 q^{82} + 7583328 q^{85} - 1376256 q^{88} - 109775560 q^{91} + 151908864 q^{94} - 408344924 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 80x^{6} + 4879x^{4} - 121680x^{2} + 2313441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 80\nu^{6} - 4879\nu^{4} + 390320\nu^{2} - 9734400 ) / 7420959 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} - 626872\nu ) / 190281 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14072\nu^{7} - 1600312\nu^{5} + 68657288\nu^{3} - 1712280960\nu ) / 289417401 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -18\nu^{7} - 8260578\nu ) / 63427 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 432\nu^{6} + 31743360 ) / 4879 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -80592\nu^{6} + 9367680\nu^{4} - 393208368\nu^{2} + 9806434560 ) / 824551 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15998\nu^{7} - 1161202\nu^{5} + 78054242\nu^{3} - 1946636640\nu ) / 10719163 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -4\beta_{4} + 27\beta_{2} ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + 17280\beta _1 + 17280 ) / 432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 164\beta_{7} - 164\beta_{4} - 3213\beta_{3} + 3213\beta_{2} ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{6} + 45333\beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7036\beta_{7} - 215973\beta_{3} ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4879\beta_{5} - 31743360 ) / 432 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 313436\beta_{4} - 12390867\beta_{2} ) / 432 \) Copy content Toggle raw display

Character values

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

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

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
−6.05526 3.49600i
4.83051 + 2.78890i
−4.83051 2.78890i
6.05526 + 3.49600i
−6.05526 + 3.49600i
4.83051 2.78890i
−4.83051 + 2.78890i
6.05526 3.49600i
−9.79796 + 5.65685i 0 64.0000 110.851i −793.589 458.179i 0 1899.35 + 3289.77i 1448.15i 0 10367.4
53.2 −9.79796 + 5.65685i 0 64.0000 110.851i 382.074 + 220.591i 0 −1940.35 3360.78i 1448.15i 0 −4991.40
53.3 9.79796 5.65685i 0 64.0000 110.851i −382.074 220.591i 0 −1940.35 3360.78i 1448.15i 0 −4991.40
53.4 9.79796 5.65685i 0 64.0000 110.851i 793.589 + 458.179i 0 1899.35 + 3289.77i 1448.15i 0 10367.4
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i −793.589 + 458.179i 0 1899.35 3289.77i 1448.15i 0 10367.4
107.2 −9.79796 5.65685i 0 64.0000 + 110.851i 382.074 220.591i 0 −1940.35 + 3360.78i 1448.15i 0 −4991.40
107.3 9.79796 + 5.65685i 0 64.0000 + 110.851i −382.074 + 220.591i 0 −1940.35 + 3360.78i 1448.15i 0 −4991.40
107.4 9.79796 + 5.65685i 0 64.0000 + 110.851i 793.589 458.179i 0 1899.35 3289.77i 1448.15i 0 10367.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
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.f 8
3.b odd 2 1 inner 162.9.d.f 8
9.c even 3 1 54.9.b.b 4
9.c even 3 1 inner 162.9.d.f 8
9.d odd 6 1 54.9.b.b 4
9.d odd 6 1 inner 162.9.d.f 8
36.f odd 6 1 432.9.e.i 4
36.h even 6 1 432.9.e.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.b 4 9.c even 3 1
54.9.b.b 4 9.d odd 6 1
162.9.d.f 8 1.a even 1 1 trivial
162.9.d.f 8 3.b odd 2 1 inner
162.9.d.f 8 9.c even 3 1 inner
162.9.d.f 8 9.d odd 6 1 inner
432.9.e.i 4 36.f odd 6 1
432.9.e.i 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 1034352T_{5}^{6} + 906441741504T_{5}^{4} - 169056888921676800T_{5}^{2} + 26713391443966978560000 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 1034352 T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + 82 T^{3} + \cdots + 217315212808225)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 775057392 T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + 49822 T^{3} + \cdots + 36\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 20561526000 T^{2} + \cdots + 59\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 256018 T + 9250548817)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 140957633520 T^{6} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} - 1957816428480 T^{6} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{4} + 297772 T^{3} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 505490 T - 3696029027375)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 10775894113728 T^{6} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4047092 T^{3} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 27212771060208 T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + 17084039126976 T^{2} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 531992852563824 T^{6} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2392318 T^{3} + \cdots + 19\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 28578850 T^{3} + \cdots + 80\!\cdots\!81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 39962350169856 T^{2} + \cdots + 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6618046 T - 531549935014847)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24791842 T^{3} + \cdots + 22\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 204172462 T^{3} + \cdots + 56\!\cdots\!25)^{2} \) Copy content Toggle raw display
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