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Properties

Label 162.8.c.q
Level $162$
Weight $8$
Character orbit 162.c
Analytic conductor $50.606$
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] = [162,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(50.6063741284\)
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: \( x^{8} - 2x^{7} + 2x^{6} + 518x^{5} + 53377x^{4} + 11940x^{3} + 3528x^{2} + 1563408x + 346406544 \) 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.

\(f(q)\) \(=\) \( q + ( - 8 \beta_1 - 8) q^{2} + 64 \beta_1 q^{4} + ( - \beta_{4} + 132 \beta_1) q^{5} + ( - \beta_{7} + \beta_{3} - 140 \beta_1 - 140) q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \beta_1 - 8) q^{2} + 64 \beta_1 q^{4} + ( - \beta_{4} + 132 \beta_1) q^{5} + ( - \beta_{7} + \beta_{3} - 140 \beta_1 - 140) q^{7} + 512 q^{8} + (8 \beta_{5} + 1056) q^{10} + ( - 4 \beta_{7} - 5 \beta_{5} + \cdots - 540) q^{11}+ \cdots + (448 \beta_{7} + 448 \beta_{6} + \cdots + 2898792) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8} + 8448 q^{10} - 2160 q^{11} - 13460 q^{13} - 4480 q^{14} - 16384 q^{16} + 45120 q^{17} + 73408 q^{19} - 33792 q^{20} - 17280 q^{22} - 62640 q^{23} - 94696 q^{25} + 215360 q^{26} + 71680 q^{28} - 68400 q^{29} - 227504 q^{31} - 131072 q^{32} - 180480 q^{34} - 126048 q^{35} + 1047160 q^{37} - 293632 q^{38} - 270336 q^{40} - 67200 q^{41} - 562640 q^{43} + 276480 q^{44} + 1002240 q^{46} - 515328 q^{47} - 1449396 q^{49} - 757568 q^{50} - 861440 q^{52} + 4996032 q^{53} + 2645568 q^{55} - 286720 q^{56} - 547200 q^{58} - 4155840 q^{59} - 2130764 q^{61} + 3640064 q^{62} + 2097152 q^{64} - 8966496 q^{65} + 1205440 q^{67} - 1443840 q^{68} + 504192 q^{70} + 24972960 q^{71} - 9641720 q^{73} - 4188640 q^{74} - 2349056 q^{76} - 18686592 q^{77} + 11471680 q^{79} + 4325376 q^{80} + 1075200 q^{82} - 16811232 q^{83} + 21006396 q^{85} - 4501120 q^{86} - 1105920 q^{88} + 33715200 q^{89} - 46401088 q^{91} - 4008960 q^{92} - 4122624 q^{94} - 14090160 q^{95} + 27078040 q^{97} + 23190336 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 518x^{5} + 53377x^{4} + 11940x^{3} + 3528x^{2} + 1563408x + 346406544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 43855 \nu^{7} - 2805062 \nu^{6} + 9784562 \nu^{5} + 11661362 \nu^{4} + 241042495 \nu^{3} + \cdots - 3409479049680 ) / 6938096712624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 443726439501 \nu^{7} - 39084285383949 \nu^{6} + 196177707589548 \nu^{5} + \cdots - 18\!\cdots\!40 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1768347693753 \nu^{7} + 81480359331894 \nu^{6} + 96732571818426 \nu^{5} + \cdots - 37\!\cdots\!56 ) / 36\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26581205069137 \nu^{7} + 162467854891691 \nu^{6} + \cdots - 12\!\cdots\!20 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 161261082233 \nu^{7} - 262079734900 \nu^{6} - 11218568540246 \nu^{5} + \cdots - 37\!\cdots\!92 ) / 21\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52230739398496 \nu^{7} - 294121759941161 \nu^{6} + \cdots + 61\!\cdots\!12 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 117817034918123 \nu^{7} + 282711412090070 \nu^{6} + \cdots + 13\!\cdots\!28 ) / 10\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 54\beta _1 + 54 ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - 6\beta_{5} + 12\beta_{4} + 203\beta_{3} - 203\beta_{2} + 23652\beta _1 + 11826 ) / 324 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 111 \beta_{7} - 678 \beta_{6} + 924 \beta_{5} + 99 \beta_{4} - 545 \beta_{3} - 1111 \beta_{2} + \cdots - 17658 ) / 324 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -823\beta_{7} - 823\beta_{6} + 740\beta_{5} - 15217\beta_{3} - 15217\beta_{2} - 2938410 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 151677 \beta_{7} - 34410 \beta_{6} + 200124 \beta_{5} - 89193 \beta_{4} - 346379 \beta_{3} + \cdots - 39562830 ) / 324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 442161 \beta_{7} + 442161 \beta_{6} + 584694 \beta_{5} - 1169388 \beta_{4} - 7517765 \beta_{3} + \cdots - 903466710 ) / 324 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8586513 \beta_{7} + 34102302 \beta_{6} - 16441776 \beta_{5} - 26003289 \beta_{4} + 29297411 \beta_{3} + \cdots + 3024149094 ) / 324 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−6.80544 − 6.80544i
−9.64382 + 9.64382i
11.0098 − 11.0098i
6.43942 + 6.43942i
−6.80544 + 6.80544i
−9.64382 − 9.64382i
11.0098 + 11.0098i
6.43942 − 6.43942i
−4.00000 + 6.92820i 0 −32.0000 − 55.4256i −235.840 − 408.486i 0 −367.654 + 636.795i 512.000 0 3773.43
55.2 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −162.750 − 281.891i 0 725.227 − 1256.13i 512.000 0 2604.00
55.3 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −5.62316 − 9.73960i 0 −719.735 + 1246.62i 512.000 0 89.9705
55.4 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 140.213 + 242.855i 0 82.1617 − 142.308i 512.000 0 −2243.40
109.1 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −235.840 + 408.486i 0 −367.654 − 636.795i 512.000 0 3773.43
109.2 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −162.750 + 281.891i 0 725.227 + 1256.13i 512.000 0 2604.00
109.3 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −5.62316 + 9.73960i 0 −719.735 − 1246.62i 512.000 0 89.9705
109.4 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 140.213 − 242.855i 0 82.1617 + 142.308i 512.000 0 −2243.40
\(n\): 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 162.8.c.q 8
3.b odd 2 1 162.8.c.r 8
9.c even 3 1 162.8.a.j yes 4
9.c even 3 1 inner 162.8.c.q 8
9.d odd 6 1 162.8.a.g ✓ 4
9.d odd 6 1 162.8.c.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.g ✓ 4 9.d odd 6 1
162.8.a.j yes 4 9.c even 3 1
162.8.c.q 8 1.a even 1 1 trivial
162.8.c.q 8 9.c even 3 1 inner
162.8.c.r 8 3.b odd 2 1
162.8.c.r 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 528 T_{5}^{7} + 342990 T_{5}^{6} + 53782272 T_{5}^{5} + 27754932771 T_{5}^{4} + \cdots + 23\!\cdots\!25 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 16\!\cdots\!11)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 66\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 16\!\cdots\!39)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 32\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 48\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 13\!\cdots\!49)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 33\!\cdots\!81)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
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