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Properties

Label 162.10.c.t
Level $162$
Weight $10$
Character orbit 162.c
Analytic conductor $83.436$
Analytic rank $0$
Dimension $8$
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] = [162,10,Mod(55,162)]
 
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([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), 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: \(83.4358054585\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1910x^{6} + 1303429x^{4} + 373918128x^{2} + 38299272804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 16 \beta_{2} q^{2} + (256 \beta_{2} - 256) q^{4} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 492 \beta_{2} + \beta_1 - 492) q^{5} + ( - 18 \beta_{7} + \beta_{6} - 1124 \beta_{2}) q^{7} - 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 \beta_{2} q^{2} + (256 \beta_{2} - 256) q^{4} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 492 \beta_{2} + \beta_1 - 492) q^{5} + ( - 18 \beta_{7} + \beta_{6} - 1124 \beta_{2}) q^{7} - 4096 q^{8} + ( - 16 \beta_{4} - 16 \beta_{3} + 16 \beta_1 - 7872) q^{10} + ( - 28 \beta_{7} + 15 \beta_{6} + 61 \beta_{5} + 61 \beta_{3} + 2196 \beta_{2}) q^{11} + ( - 166 \beta_{7} - 93 \beta_{6} - 52 \beta_{5} - 93 \beta_{4} + \cdots + 40639) q^{13}+ \cdots + (41472 \beta_{4} + 15488 \beta_{3} + 546048 \beta_1 + 509697072) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{2} - 1024 q^{4} - 1968 q^{5} - 4496 q^{7} - 32768 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{2} - 1024 q^{4} - 1968 q^{5} - 4496 q^{7} - 32768 q^{8} - 62976 q^{10} + 8784 q^{11} + 162556 q^{13} + 71936 q^{14} - 262144 q^{16} + 1076160 q^{17} - 16448 q^{19} - 503808 q^{20} - 140544 q^{22} - 2594736 q^{23} - 1873192 q^{25} + 5201792 q^{26} + 2301952 q^{28} - 3232656 q^{29} - 3482576 q^{31} + 4194304 q^{32} + 8609280 q^{34} + 15938592 q^{35} - 4975784 q^{37} - 131584 q^{38} + 8060928 q^{40} - 34657152 q^{41} + 41410000 q^{43} - 4497408 q^{44} - 83031552 q^{46} - 40558848 q^{47} + 127424268 q^{49} + 29971072 q^{50} + 41614336 q^{52} + 16842048 q^{53} - 458136000 q^{55} + 18415616 q^{56} + 51722496 q^{58} - 26843328 q^{59} + 111504484 q^{61} - 111442432 q^{62} + 134217728 q^{64} - 23507616 q^{65} + 208064512 q^{67} - 137748480 q^{68} + 127508736 q^{70} + 712016544 q^{71} - 74252216 q^{73} - 39806272 q^{74} + 2105344 q^{76} - 33010560 q^{77} + 645134848 q^{79} + 257949696 q^{80} - 1109028864 q^{82} - 1019036256 q^{83} - 80955540 q^{85} - 662560000 q^{86} - 35979264 q^{88} + 3096385536 q^{89} - 261553216 q^{91} - 664252416 q^{92} + 648941568 q^{94} - 3445933872 q^{95} - 1112014568 q^{97} + 4077576576 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1910x^{6} + 1303429x^{4} + 373918128x^{2} + 38299272804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -27\nu^{6} - 38691\nu^{4} - 15890526\nu^{2} - 1707648804 ) / 2152700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 1910\nu^{5} - 1107727\nu^{3} - 187022718\nu + 40314612 ) / 80629224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -307\nu^{6} - 408581\nu^{4} - 170116216\nu^{2} - 22532569914 ) / 538175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -617\nu^{6} - 1009561\nu^{4} - 521633546\nu^{2} - 82066424784 ) / 1076350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 81971 \nu^{7} + 20026838 \nu^{6} - 103529368 \nu^{5} + 26653372954 \nu^{4} - 27776247923 \nu^{3} + 11097361234544 \nu^{2} + \cdots + 14\!\cdots\!76 ) / 70214615900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 155513 \nu^{7} + 20124689 \nu^{6} + 213758629 \nu^{5} + 32928851137 \nu^{4} + 80304894194 \nu^{3} + 17014121369882 \nu^{2} + \cdots + 26\!\cdots\!28 ) / 70214615900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 661878 \nu^{7} + 880659 \nu^{6} + 992063349 \nu^{5} + 1261984347 \nu^{4} + 472757816439 \nu^{3} + 518301286542 \nu^{2} + \cdots + 55698381040068 ) / 140429231800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{7} - 18\beta_{6} - 18\beta_{5} - 9\beta_{4} - 9\beta_{3} + 2\beta_1 ) / 972 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} - 6\beta_{3} + 410\beta _1 - 154710 ) / 324 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 668 \beta_{7} + 6750 \beta_{6} + 12312 \beta_{5} + 3375 \beta_{4} + 6156 \beta_{3} + 300348 \beta_{2} + 334 \beta _1 - 150174 ) / 972 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1011\beta_{4} + 7584\beta_{3} - 391138\beta _1 + 84340602 ) / 324 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1927508 \beta_{7} - 2919906 \beta_{6} - 8242740 \beta_{5} - 1459953 \beta_{4} - 4121370 \beta_{3} - 478053900 \beta_{2} - 963754 \beta _1 + 239026950 ) / 972 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 105617\beta_{4} - 2445548\beta_{3} + 97789258\beta _1 - 16766384778 ) / 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2193487772 \beta_{7} + 1466272134 \beta_{6} + 5471707500 \beta_{5} + 733136067 \beta_{4} + 2735853750 \beta_{3} + 502007754276 \beta_{2} + \cdots - 251003877138 ) / 972 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
26.1057i
− 17.7872i
− 25.1057i
16.7872i
− 26.1057i
17.7872i
25.1057i
− 16.7872i
8.00000 − 13.8564i 0 −128.000 − 221.703i −992.635 − 1719.29i 0 −1748.17 + 3027.91i −4096.00 0 −31764.3
55.2 8.00000 − 13.8564i 0 −128.000 − 221.703i −957.062 − 1657.68i 0 24.9899 − 43.2838i −4096.00 0 −30626.0
55.3 8.00000 − 13.8564i 0 −128.000 − 221.703i 318.770 + 552.126i 0 −1890.77 + 3274.91i −4096.00 0 10200.6
55.4 8.00000 − 13.8564i 0 −128.000 − 221.703i 646.928 + 1120.51i 0 1365.95 − 2365.89i −4096.00 0 20701.7
109.1 8.00000 + 13.8564i 0 −128.000 + 221.703i −992.635 + 1719.29i 0 −1748.17 − 3027.91i −4096.00 0 −31764.3
109.2 8.00000 + 13.8564i 0 −128.000 + 221.703i −957.062 + 1657.68i 0 24.9899 + 43.2838i −4096.00 0 −30626.0
109.3 8.00000 + 13.8564i 0 −128.000 + 221.703i 318.770 − 552.126i 0 −1890.77 − 3274.91i −4096.00 0 10200.6
109.4 8.00000 + 13.8564i 0 −128.000 + 221.703i 646.928 − 1120.51i 0 1365.95 + 2365.89i −4096.00 0 20701.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.10.c.t 8
3.b odd 2 1 162.10.c.s 8
9.c even 3 1 162.10.a.f ✓ 4
9.c even 3 1 inner 162.10.c.t 8
9.d odd 6 1 162.10.a.g yes 4
9.d odd 6 1 162.10.c.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.10.a.f ✓ 4 9.c even 3 1
162.10.a.g yes 4 9.d odd 6 1
162.10.c.s 8 3.b odd 2 1
162.10.c.s 8 9.d odd 6 1
162.10.c.t 8 1.a even 1 1 trivial
162.10.c.t 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 1968 T_{5}^{7} + 6779358 T_{5}^{6} + 2526052608 T_{5}^{5} + 13425956722611 T_{5}^{4} - 355408098312960 T_{5}^{3} + \cdots + 98\!\cdots\!25 \) acting on \(S_{10}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 16 T + 256)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 1968 T^{7} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + 4496 T^{7} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{8} - 8784 T^{7} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{8} - 162556 T^{7} + \cdots + 16\!\cdots\!89 \) Copy content Toggle raw display
$17$ \( (T^{4} - 538080 T^{3} + \cdots - 66\!\cdots\!59)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8224 T^{3} + \cdots + 56\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2594736 T^{7} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + 3232656 T^{7} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$31$ \( T^{8} + 3482576 T^{7} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + 2487892 T^{3} + \cdots + 18\!\cdots\!21)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 34657152 T^{7} + \cdots + 78\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{8} - 41410000 T^{7} + \cdots + 90\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{8} + 40558848 T^{7} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8421024 T^{3} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 26843328 T^{7} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} - 111504484 T^{7} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{8} - 208064512 T^{7} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} - 356008272 T^{3} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 37126108 T^{3} + \cdots - 25\!\cdots\!99)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 645134848 T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + 1019036256 T^{7} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1548192768 T^{3} + \cdots + 13\!\cdots\!01)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 1112014568 T^{7} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
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