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Properties

Label 162.11.b.a
Level $162$
Weight $11$
Character orbit 162.b
Analytic conductor $102.928$
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] = [162,11,Mod(161,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11512x^{6} + 49652044x^{4} + 95092179048x^{2} + 68231796144516 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 16 \beta_1 q^{2} - 512 q^{4} + (\beta_{7} + 6 \beta_{2} - 685 \beta_1) q^{5} + (2 \beta_{6} + 5 \beta_{3} + 5639) q^{7} + 8192 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 \beta_1 q^{2} - 512 q^{4} + (\beta_{7} + 6 \beta_{2} - 685 \beta_1) q^{5} + (2 \beta_{6} + 5 \beta_{3} + 5639) q^{7} + 8192 \beta_1 q^{8} + (16 \beta_{4} + 96 \beta_{3} - 21840) q^{10} + (7 \beta_{7} - 7 \beta_{5} - 323 \beta_{2} + 111 \beta_1) q^{11} + ( - 19 \beta_{6} + 112 \beta_{4} - 301 \beta_{3} - 90181) q^{13} + ( - 32 \beta_{7} - 32 \beta_{5} - 128 \beta_{2} - 90176 \beta_1) q^{14} + 262144 q^{16} + ( - 460 \beta_{7} - 58 \beta_{5} + 439 \beta_{2} + 274619 \beta_1) q^{17} + ( - 126 \beta_{6} - 697 \beta_{4} + 4736 \beta_{3} - 91399) q^{19} + ( - 512 \beta_{7} - 3072 \beta_{2} + 350720 \beta_1) q^{20} + ( - 224 \beta_{6} + 224 \beta_{4} - 5280 \beta_{3} - 1728) q^{22} + ( - 742 \beta_{7} + 490 \beta_{5} + 17858 \beta_{2} - 1063785 \beta_1) q^{23} + (350 \beta_{6} + 1190 \beta_{4} + 6765 \beta_{3} + 3448255) q^{25} + ( - 3280 \beta_{7} + 304 \beta_{5} + 9328 \beta_{2} + 1436592 \beta_1) q^{26} + ( - 1024 \beta_{6} - 2560 \beta_{3} - 2887168) q^{28} + (1624 \beta_{7} - 1281 \beta_{5} - 47871 \beta_{2} - 579922 \beta_1) q^{29} + ( - 1430 \beta_{6} + 1120 \beta_{4} - 172228 \beta_{3} + \cdots + 9062816) q^{31}+ \cdots + ( - 151808 \beta_{7} - 353152 \beta_{5} + \cdots - 385699600 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4096 q^{4} + 45112 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4096 q^{4} + 45112 q^{7} - 174720 q^{10} - 721448 q^{13} + 2097152 q^{16} - 731192 q^{19} - 13824 q^{22} + 27586040 q^{25} - 23097344 q^{28} + 72502528 q^{31} + 70417536 q^{34} - 166952240 q^{37} + 89456640 q^{40} - 12510440 q^{43} - 269948160 q^{46} + 188021592 q^{49} + 369381376 q^{52} - 181251000 q^{55} - 154795392 q^{58} - 1782273248 q^{61} - 1073741824 q^{64} + 108743656 q^{67} + 62250240 q^{70} + 5102234368 q^{73} + 374370304 q^{76} - 9246201752 q^{79} - 478119936 q^{82} + 20649676320 q^{85} + 7077888 q^{88} - 18226522120 q^{91} + 20130964224 q^{94} + 2247775936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11512x^{6} + 49652044x^{4} + 95092179048x^{2} + 68231796144516 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2875\nu^{7} + 24836746\nu^{5} + 71422333210\nu^{3} + 68368280214420\nu ) / 183724569468 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -114196\nu^{7} - 1417877527\nu^{5} - 5321515138840\nu^{3} - 6283667462882454\nu ) / 4593114236700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{6} + 77715\nu^{4} + 223182252\nu^{2} + 213158160144 ) / 185350 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} + 132143\nu^{4} + 636089948\nu^{2} + 853319477952 ) / 556050 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2875\nu^{7} + 24836746\nu^{5} + 71422333210\nu^{3} + 68735729353356\nu ) / 1701153421 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2689\nu^{6} + 23263999\nu^{4} + 66998000596\nu^{2} + 64227204227160 ) / 556050 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6013252\nu^{7} - 51931715275\nu^{5} - 149158877658136\nu^{3} - 142479928378757502\nu ) / 417555839700 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 108\beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{4} - 199\beta_{3} - 621648 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 675\beta_{7} - 2879\beta_{5} + 275\beta_{2} + 932564\beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5756\beta_{6} + 3553\beta_{4} + 572597\beta_{3} + 896995512 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1944000\beta_{7} + 4155635\beta_{5} - 1942000\beta_{2} - 2240935876\beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24905032\beta_{6} - 18281141\beta_{4} - 2474747109\beta_{3} - 2595631912776 ) / 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8409598875\beta_{7} - 12029266927\beta_{5} + 13360831275\beta_{2} + 9066551709828\beta_1 ) / 108 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 53.1246i
54.1327i
51.7104i
− 55.5469i
55.5469i
− 51.7104i
− 54.1327i
53.1246i
− 22.6274i 0 −512.000 − 3787.15i 0 −14083.2 11585.2i 0 −85693.4
161.2 − 22.6274i 0 −512.000 − 2601.86i 0 19795.7 11585.2i 0 −58873.3
161.3 − 22.6274i 0 −512.000 570.763i 0 24103.8 11585.2i 0 12914.9
161.4 − 22.6274i 0 −512.000 1957.44i 0 −7260.28 11585.2i 0 44291.8
161.5 22.6274i 0 −512.000 − 1957.44i 0 −7260.28 − 11585.2i 0 44291.8
161.6 22.6274i 0 −512.000 − 570.763i 0 24103.8 − 11585.2i 0 12914.9
161.7 22.6274i 0 −512.000 2601.86i 0 19795.7 − 11585.2i 0 −58873.3
161.8 22.6274i 0 −512.000 3787.15i 0 −14083.2 − 11585.2i 0 −85693.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.11.b.a ✓ 8
3.b odd 2 1 inner 162.11.b.a ✓ 8
9.c even 3 2 162.11.d.e 16
9.d odd 6 2 162.11.d.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.11.b.a ✓ 8 1.a even 1 1 trivial
162.11.b.a ✓ 8 3.b odd 2 1 inner
162.11.d.e 16 9.c even 3 2
162.11.d.e 16 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 25269480 T_{5}^{6} + 186112230469650 T_{5}^{4} + \cdots + 12\!\cdots\!25 \) acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 512)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 25269480 T^{6} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} - 22556 T^{3} + \cdots + 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 31767096888 T^{6} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{4} + 360724 T^{3} + \cdots + 53\!\cdots\!69)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 6373282768344 T^{6} + \cdots + 97\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( (T^{4} + 365596 T^{3} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 164231561749320 T^{6} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{4} - 36251264 T^{3} + \cdots + 47\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 83476120 T^{3} + \cdots - 45\!\cdots\!91)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6255220 T^{3} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} + 891136624 T^{3} + \cdots + 65\!\cdots\!81)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 54371828 T^{3} + \cdots - 90\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} - 2551117184 T^{3} + \cdots + 36\!\cdots\!61)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4623100876 T^{3} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{4} - 1123887968 T^{3} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
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