Properties

Label 162.9.b.a
Level $162$
Weight $9$
Character orbit 162.b
Analytic conductor $65.995$
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] = [162,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(65.9953348299\)
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: \( x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{16} \)
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.

\(f(q)\) \(=\) \( q + 8 \beta_{4} q^{2} - 128 q^{4} + (\beta_{7} - \beta_{6} - 66 \beta_{4}) q^{5} + ( - 7 \beta_{2} + 7 \beta_1 - 1113) q^{7} - 1024 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{4} q^{2} - 128 q^{4} + (\beta_{7} - \beta_{6} - 66 \beta_{4}) q^{5} + ( - 7 \beta_{2} + 7 \beta_1 - 1113) q^{7} - 1024 \beta_{4} q^{8} + (8 \beta_{3} - 8 \beta_1 + 1056) q^{10} + (62 \beta_{7} + \beta_{6} - 47 \beta_{5} + 1655 \beta_{4}) q^{11} + ( - 13 \beta_{3} - 67 \beta_{2} + 92 \beta_1 - 14706) q^{13} + (112 \beta_{7} + 56 \beta_{5} - 8792 \beta_{4}) q^{14} + 16384 q^{16} + (135 \beta_{7} - 120 \beta_{6} - 107 \beta_{5} + 5360 \beta_{4}) q^{17} + ( - 47 \beta_{3} - 639 \beta_{2} + 183 \beta_1 + 33458) q^{19} + ( - 128 \beta_{7} + 128 \beta_{6} + 8448 \beta_{4}) q^{20} + ( - 8 \beta_{3} - 752 \beta_{2} - 496 \beta_1 - 26728) q^{22} + (1828 \beta_{7} + 53 \beta_{6} + 686 \beta_{5} - 40094 \beta_{4}) q^{23} + ( - 167 \beta_{3} - 910 \beta_{2} + 1127 \beta_1 - 225664) q^{25} + (1472 \beta_{7} - 208 \beta_{6} + 536 \beta_{5} - 116480 \beta_{4}) q^{26} + (896 \beta_{2} - 896 \beta_1 + 142464) q^{28} + (445 \beta_{7} + 203 \beta_{6} - 1662 \beta_{5} + 257217 \beta_{4}) q^{29} + (293 \beta_{3} + 1786 \beta_{2} - 5938 \beta_1 - 48275) q^{31} + 131072 \beta_{4} q^{32} + (960 \beta_{3} - 1712 \beta_{2} - 1080 \beta_1 - 87352) q^{34} + ( - 14000 \beta_{7} + 1505 \beta_{6} - 2296 \beta_{5} + 138880 \beta_{4}) q^{35} + (175 \beta_{3} + 463 \beta_{2} - 5037 \beta_1 + 229105) q^{37} + (2928 \beta_{7} - 752 \beta_{6} + 5112 \beta_{5} + 273864 \beta_{4}) q^{38} + ( - 1024 \beta_{3} + 1024 \beta_1 - 135168) q^{40} + (4286 \beta_{7} + 268 \beta_{6} - 9023 \beta_{5} - 1132228 \beta_{4}) q^{41} + ( - 2190 \beta_{3} + 14361 \beta_{2} - 8969 \beta_1 + 1399085) q^{43} + ( - 7936 \beta_{7} - 128 \beta_{6} + 6016 \beta_{5} - 211840 \beta_{4}) q^{44} + ( - 424 \beta_{3} + 10976 \beta_{2} - 14624 \beta_1 + 667528) q^{46} + ( - 1750 \beta_{7} - 3620 \beta_{6} + 5183 \beta_{5} + 1641481 \beta_{4}) q^{47} + (2107 \beta_{3} + 13426 \beta_{2} - 11466 \beta_1 - 1691578) q^{49} + (18032 \beta_{7} - 2672 \beta_{6} + 7280 \beta_{5} - 1790352 \beta_{4}) q^{50} + (1664 \beta_{3} + 8576 \beta_{2} - 11776 \beta_1 + 1882368) q^{52} + ( - 78554 \beta_{7} - 1054 \beta_{6} - 1359 \beta_{5} - 2490006 \beta_{4}) q^{53} + (2284 \beta_{3} - 23365 \beta_{2} - 68299 \beta_1 - 209147) q^{55} + ( - 14336 \beta_{7} - 7168 \beta_{5} + 1125376 \beta_{4}) q^{56} + ( - 1624 \beta_{3} - 26592 \beta_{2} - 3560 \beta_1 - 4136880) q^{58} + ( - 43782 \beta_{7} - 720 \beta_{6} + 3337 \beta_{5} + 371255 \beta_{4}) q^{59} + (4582 \beta_{3} - 52831 \beta_{2} - 10087 \beta_1 + 1496610) q^{61} + ( - 95008 \beta_{7} + 4688 \beta_{6} - 14288 \beta_{5} - 445648 \beta_{4}) q^{62} - 2097152 q^{64} + ( - 168843 \beta_{7} + 21408 \beta_{6} - 38018 \beta_{5} + 9620513 \beta_{4}) q^{65} + ( - 3270 \beta_{3} - 31809 \beta_{2} - 78891 \beta_1 - 10060369) q^{67} + ( - 17280 \beta_{7} + 15360 \beta_{6} + 13696 \beta_{5} - 686080 \beta_{4}) q^{68} + ( - 12040 \beta_{3} - 36736 \beta_{2} + 112000 \beta_1 - 2358776) q^{70} + ( - 144708 \beta_{7} - 24399 \beta_{6} + 6184 \beta_{5} + 1363940 \beta_{4}) q^{71} + ( - 13573 \beta_{3} + 26806 \beta_{2} - 137885 \beta_1 + 24649123) q^{73} + ( - 80592 \beta_{7} + 2800 \beta_{6} - 3704 \beta_{5} + 1790240 \beta_{4}) q^{74} + (6016 \beta_{3} + 81792 \beta_{2} - 23424 \beta_1 - 4282624) q^{76} + ( - 73542 \beta_{7} - 126 \beta_{6} + 66199 \beta_{5} - 12330031 \beta_{4}) q^{77} + (3553 \beta_{3} + 39993 \beta_{2} + 350979 \beta_1 + 10576478) q^{79} + (16384 \beta_{7} - 16384 \beta_{6} - 1081344 \beta_{4}) q^{80} + ( - 2144 \beta_{3} - 144368 \beta_{2} - 34288 \beta_1 + 18007712) q^{82} + (38768 \beta_{7} - 31946 \beta_{6} + 123758 \beta_{5} - 19430234 \beta_{4}) q^{83} + ( - 3760 \beta_{3} - 119387 \beta_{2} - 44555 \beta_1 - 72889012) q^{85} + ( - 143504 \beta_{7} - 35040 \beta_{6} + \cdots + 10988520 \beta_{4}) q^{86}+ \cdots + ( - 183456 \beta_{7} + 33712 \beta_{6} + \cdots - 13714904 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1024 q^{4} - 8876 q^{7} + 8448 q^{10} - 117380 q^{13} + 131072 q^{16} + 270220 q^{19} - 210816 q^{22} - 1801672 q^{25} + 1136128 q^{28} - 393344 q^{31} - 691968 q^{34} + 1830988 q^{37} - 1081344 q^{40} + 11135236 q^{43} + 5296320 q^{46} - 13586328 q^{49} + 15024640 q^{52} - 1579716 q^{55} - 32988672 q^{58} + 12184204 q^{61} - 16777216 q^{64} - 80355716 q^{67} - 18723264 q^{70} + 197085760 q^{73} - 34588160 q^{76} + 84451852 q^{79} + 144639168 q^{82} - 582634548 q^{85} + 26984448 q^{88} + 373079588 q^{91} - 210121536 q^{94} + 341136928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 37\nu^{6} + 93556\nu^{4} + 75931333\nu^{2} + 19722241824 ) / 2006320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9859\nu^{6} - 25807324\nu^{4} - 22197698515\nu^{2} - 6262260164592 ) / 54170640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9829\nu^{6} - 27049132\nu^{4} - 24352443781\nu^{2} - 7141798486428 ) / 13542660 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1117\nu^{7} + 3077380\nu^{5} + 2773897261\nu^{3} + 814471345392\nu ) / 34160045760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 929183\nu^{7} + 6844468748\nu^{5} + 11352744767375\nu^{3} + 5136503071908144\nu ) / 27635477019840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18258415\nu^{7} + 47311753516\nu^{5} + 39826591020607\nu^{3} + 10500188456688432\nu ) / 13817738509920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -95130337\nu^{7} - 250569216868\nu^{5} - 216717212802097\nu^{3} - 61413480914212608\nu ) / 27635477019840 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} - 6\beta_{6} - 3\beta_{5} + 35\beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 30\beta_{2} - 178\beta _1 - 136257 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 250\beta_{7} + 1684\beta_{6} + 1679\beta_{5} - 43459\beta_{4} ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6711\beta_{3} + 57120\beta_{2} + 299596\beta _1 + 119083575 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 498074\beta_{7} - 4421202\beta_{6} - 6466773\beta_{5} + 237745069\beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3604147\beta_{3} - 27621430\beta_{2} - 127821642\beta _1 - 35944238537 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1776403958\beta_{7} + 4009684620\beta_{6} + 7495086327\beta_{5} - 351792378619\beta_{4} ) / 162 \) Copy content Toggle raw display

Character values

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

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

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
32.5932i
25.8318i
30.6613i
26.3494i
26.3494i
30.6613i
25.8318i
32.5932i
11.3137i 0 −128.000 724.855i 0 −1954.36 1448.15i 0 −8200.80
161.2 11.3137i 0 −128.000 618.080i 0 −549.440 1448.15i 0 −6992.78
161.3 11.3137i 0 −128.000 654.335i 0 1299.40 1448.15i 0 7402.96
161.4 11.3137i 0 −128.000 1061.95i 0 −3233.60 1448.15i 0 12014.6
161.5 11.3137i 0 −128.000 1061.95i 0 −3233.60 1448.15i 0 12014.6
161.6 11.3137i 0 −128.000 654.335i 0 1299.40 1448.15i 0 7402.96
161.7 11.3137i 0 −128.000 618.080i 0 −549.440 1448.15i 0 −6992.78
161.8 11.3137i 0 −128.000 724.855i 0 −1954.36 1448.15i 0 −8200.80
\(n\): 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 162.9.b.a 8
3.b odd 2 1 inner 162.9.b.a 8
9.c even 3 2 162.9.d.h 16
9.d odd 6 2 162.9.d.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.9.b.a 8 1.a even 1 1 trivial
162.9.b.a 8 3.b odd 2 1 inner
162.9.d.h 16 9.c even 3 2
162.9.d.h 16 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2463336T_{5}^{6} + 2095450167474T_{5}^{4} + 750455927372039400T_{5}^{2} + 96917652743250602030625 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2463336 T^{6} + \cdots + 96\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4438 T^{3} + \cdots - 4511860509296)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 1083686796 T^{6} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{4} + 58690 T^{3} + \cdots - 168608080827623)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 40156998768 T^{6} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{4} - 135110 T^{3} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 283534373268 T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + 1677572170848 T^{6} + \cdots + 31\!\cdots\!89 \) Copy content Toggle raw display
$31$ \( (T^{4} + 196672 T^{3} + \cdots + 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 915494 T^{3} + \cdots + 16\!\cdots\!21)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 42402792363120 T^{6} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} - 5567618 T^{3} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 62658471632112 T^{6} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + 288880676034192 T^{6} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{8} + 85968765926928 T^{6} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6092102 T^{3} + \cdots + 12\!\cdots\!01)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 40177858 T^{3} + \cdots - 44\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{4} - 98542880 T^{3} + \cdots - 56\!\cdots\!27)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 42225926 T^{3} + \cdots + 52\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 39\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{4} - 170568464 T^{3} + \cdots - 41\!\cdots\!32)^{2} \) Copy content Toggle raw display
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