Properties

Label 162.9.b.b
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: \( x^{8} + 11104x^{6} + 45413413x^{4} + 81011991468x^{2} + 53239758647364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{18} \)
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.

\(f(q)\) \(=\) \( q + 8 \beta_{4} q^{2} - 128 q^{4} + (\beta_{6} - \beta_{5} + 150 \beta_{4}) q^{5} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 + 186) q^{7} - 1024 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{4} q^{2} - 128 q^{4} + (\beta_{6} - \beta_{5} + 150 \beta_{4}) q^{5} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 + 186) q^{7} - 1024 \beta_{4} q^{8} + (8 \beta_{2} + 8 \beta_1 - 2400) q^{10} + ( - \beta_{7} - 14 \beta_{6} - 1654 \beta_{4}) q^{11} + (11 \beta_{3} + 16 \beta_{2} + 9 \beta_1 + 5853) q^{13} + ( - 8 \beta_{7} + 128 \beta_{6} - 8 \beta_{5} + 1552 \beta_{4}) q^{14} + 16384 q^{16} + ( - 13 \beta_{7} + 227 \beta_{6} + 155 \beta_{5} - 24535 \beta_{4}) q^{17} + ( - 9 \beta_{3} + 70 \beta_{2} + 318 \beta_1 - 80059) q^{19} + ( - 128 \beta_{6} + 128 \beta_{5} - 19200 \beta_{4}) q^{20} + (16 \beta_{3} + 8 \beta_{2} - 112 \beta_1 + 26360) q^{22} + (10 \beta_{7} + 590 \beta_{6} - 369 \beta_{5} + 41488 \beta_{4}) q^{23} + (14 \beta_{3} + 405 \beta_{2} + 2661 \beta_1 - 19570) q^{25} + (88 \beta_{7} - 144 \beta_{6} + 168 \beta_{5} + 46792 \beta_{4}) q^{26} + (128 \beta_{3} + 128 \beta_{2} + 1024 \beta_1 - 23808) q^{28} + (78 \beta_{7} + 2737 \beta_{6} - 727 \beta_{5} + 5901 \beta_{4}) q^{29} + ( - 218 \beta_{3} + 417 \beta_{2} + 350 \beta_1 + 13363) q^{31} + 131072 \beta_{4} q^{32} + (208 \beta_{3} - 1136 \beta_{2} + 1816 \beta_1 + 395720) q^{34} + ( - 176 \beta_{7} + 12350 \beta_{6} - 1551 \beta_{5} + 317860 \beta_{4}) q^{35} + ( - 263 \beta_{3} - 1726 \beta_{2} - 200 \beta_1 - 182522) q^{37} + ( - 72 \beta_{7} - 5088 \beta_{6} + 1192 \beta_{5} - 642384 \beta_{4}) q^{38} + ( - 1024 \beta_{2} - 1024 \beta_1 + 307200) q^{40} + (119 \beta_{7} - 25028 \beta_{6} + 2403 \beta_{5} + 754121 \beta_{4}) q^{41} + ( - 561 \beta_{3} + 525 \beta_{2} + 10052 \beta_1 - 1058464) q^{43} + (128 \beta_{7} + 1792 \beta_{6} + 211712 \beta_{4}) q^{44} + ( - 160 \beta_{3} + 2872 \beta_{2} + 4720 \beta_1 - 662120) q^{46} + ( - 743 \beta_{7} - 25332 \beta_{6} - 331 \beta_{5} - 3191054 \beta_{4}) q^{47} + (1486 \beta_{3} - 3805 \beta_{2} + 32818 \beta_1 + 2989652) q^{49} + (112 \beta_{7} - 42576 \beta_{6} + 6368 \beta_{5} - 174720 \beta_{4}) q^{50} + ( - 1408 \beta_{3} - 2048 \beta_{2} - 1152 \beta_1 - 749184) q^{52} + ( - 9 \beta_{7} + 5416 \beta_{6} - 1783 \beta_{5} + 1282803 \beta_{4}) q^{53} + (77 \beta_{3} - 2319 \beta_{2} + 3264 \beta_1 + 512350) q^{55} + (1024 \beta_{7} - 16384 \beta_{6} + 1024 \beta_{5} - 198656 \beta_{4}) q^{56} + ( - 1248 \beta_{3} + 5192 \beta_{2} + 21896 \beta_1 - 78960) q^{58} + ( - 1441 \beta_{7} - 85816 \beta_{6} - 13393 \beta_{5} + 3465350 \beta_{4}) q^{59} + (3623 \beta_{3} + 1401 \beta_{2} + 111138 \beta_1 - 3540363) q^{61} + ( - 1744 \beta_{7} - 5600 \beta_{6} + 8416 \beta_{5} + 109184 \beta_{4}) q^{62} - 2097152 q^{64} + (1250 \beta_{7} - 125593 \beta_{6} + 1058 \beta_{5} - 3497515 \beta_{4}) q^{65} + ( - 2919 \beta_{3} + 15807 \beta_{2} + 26400 \beta_1 + 11625140) q^{67} + (1664 \beta_{7} - 29056 \beta_{6} - 19840 \beta_{5} + 3140480 \beta_{4}) q^{68} + (2816 \beta_{3} + 13816 \beta_{2} + 98800 \beta_1 - 4997960) q^{70} + (5648 \beta_{7} - 120622 \beta_{6} + 12497 \beta_{5} + 733376 \beta_{4}) q^{71} + (5002 \beta_{3} + 13407 \beta_{2} - 73503 \beta_1 - 17411363) q^{73} + ( - 2104 \beta_{7} + 3200 \beta_{6} - 25512 \beta_{5} - 1470280 \beta_{4}) q^{74} + (1152 \beta_{3} - 8960 \beta_{2} - 40704 \beta_1 + 10247552) q^{76} + (1649 \beta_{7} - 112288 \beta_{6} + 2099 \beta_{5} + 5804810 \beta_{4}) q^{77} + ( - 6033 \beta_{3} - 1202 \beta_{2} + 252474 \beta_1 - 5114311) q^{79} + (16384 \beta_{6} - 16384 \beta_{5} + 2457600 \beta_{4}) q^{80} + ( - 1904 \beta_{3} - 20176 \beta_{2} - 200224 \beta_1 - 12247888) q^{82} + (3298 \beta_{7} - 369888 \beta_{6} + 33848 \beta_{5} + 6255592 \beta_{4}) q^{83} + (3907 \beta_{3} - 34917 \beta_{2} - 257874 \beta_1 + 59945675) q^{85} + ( - 4488 \beta_{7} - 160832 \beta_{6} + 12888 \beta_{5} - 8539440 \beta_{4}) q^{86} + ( - 2048 \beta_{3} - 1024 \beta_{2} + 14336 \beta_1 - 3374080) q^{88} + ( - 16426 \beta_{7} - 513937 \beta_{6} + \cdots - 22544404 \beta_{4}) q^{89}+ \cdots + (11888 \beta_{7} - 525088 \beta_{6} - 72768 \beta_{5} + 23612344 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1024 q^{4} + 1492 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1024 q^{4} + 1492 q^{7} - 19200 q^{10} + 46780 q^{13} + 131072 q^{16} - 640436 q^{19} + 210816 q^{22} - 156616 q^{25} - 190976 q^{28} + 107776 q^{31} + 3164928 q^{34} - 1459124 q^{37} + 2457600 q^{40} - 8465468 q^{43} - 5296320 q^{46} + 23911272 q^{49} - 5987840 q^{52} + 4098492 q^{55} - 626688 q^{58} - 28337396 q^{61} - 16777216 q^{64} + 93012796 q^{67} - 39994944 q^{70} - 139310912 q^{73} + 81975808 q^{76} - 40890356 q^{79} - 97975488 q^{82} + 479549772 q^{85} - 26984448 q^{88} - 676219036 q^{91} + 406812480 q^{94} - 49442912 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11104x^{6} + 45413413x^{4} + 81011991468x^{2} + 53239758647364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1323\nu^{6} - 11024262\nu^{4} - 29003973723\nu^{2} - 23874956935722 ) / 4784458360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 381\nu^{6} - 16353614\nu^{4} - 100069066675\nu^{2} - 135571528644342 ) / 2392229180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60309\nu^{6} + 539157111\nu^{4} + 1573880428884\nu^{2} + 1489905476964191 ) / 598057295 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3877\nu^{7} + 35753650\nu^{5} + 107261260261\nu^{3} + 103788422361222\nu ) / 5884382164680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1411527665 \nu^{7} - 12093379188626 \nu^{5} + \cdots - 30\!\cdots\!62 \nu ) / 52\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6873846745 \nu^{7} - 60374430342202 \nu^{5} + \cdots - 16\!\cdots\!34 \nu ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 649357487 \nu^{7} - 5324361232926 \nu^{5} + \cdots - 10\!\cdots\!34 \nu ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{6} - 13\beta_{5} + 36\beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 15\beta_{2} + 738\beta _1 - 449711 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1673\beta_{7} - 18864\beta_{6} + 41621\beta_{5} - 459758\beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11104\beta_{3} - 103125\beta_{2} - 4108806\beta _1 + 1315110043 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1982509\beta_{7} + 81274620\beta_{6} - 136331191\beta_{5} + 2854537534\beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 48681374\beta_{3} + 530474235\beta_{2} + 17472818586\beta _1 - 4023019402499 ) / 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1232327353\beta_{7} - 334702500192\beta_{6} + 453769993415\beta_{5} - 14322679056422\beta_{4} ) / 162 \) 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
59.3748i
45.9949i
46.5125i
57.4429i
57.4429i
46.5125i
45.9949i
59.3748i
11.3137i 0 −128.000 1118.47i 0 2818.74 1448.15i 0 −12654.1
161.2 11.3137i 0 −128.000 310.915i 0 −4484.76 1448.15i 0 −3517.60
161.3 11.3137i 0 −128.000 40.9689i 0 2628.61 1448.15i 0 463.510
161.4 11.3137i 0 −128.000 539.890i 0 −216.589 1448.15i 0 6108.16
161.5 11.3137i 0 −128.000 539.890i 0 −216.589 1448.15i 0 6108.16
161.6 11.3137i 0 −128.000 40.9689i 0 2628.61 1448.15i 0 463.510
161.7 11.3137i 0 −128.000 310.915i 0 −4484.76 1448.15i 0 −3517.60
161.8 11.3137i 0 −128.000 1118.47i 0 2818.74 1448.15i 0 −12654.1
\(n\): 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.b 8
3.b odd 2 1 inner 162.9.b.b 8
9.c even 3 2 162.9.d.g 16
9.d odd 6 2 162.9.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.9.b.b 8 1.a even 1 1 trivial
162.9.b.b 8 3.b odd 2 1 inner
162.9.d.g 16 9.c even 3 2
162.9.d.g 16 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 1640808T_{5}^{6} + 516495591666T_{5}^{4} + 36111140725818600T_{5}^{2} + 59163452262584150625 \) 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
$5$ \( T^{8} + 1640808 T^{6} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} - 746 T^{3} + \cdots + 7197064587664)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 86966028 T^{6} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} - 23390 T^{3} + \cdots + 67\!\cdots\!69)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 50461667568 T^{6} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{4} + 320218 T^{3} + \cdots - 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 222640275348 T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + 1329947673696 T^{6} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{4} - 53888 T^{3} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 729562 T^{3} + \cdots + 13\!\cdots\!53)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 37646502934896 T^{6} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4232734 T^{3} + \cdots - 29\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 138220620047856 T^{6} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 18704070936720 T^{6} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + 799197825064464 T^{6} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + 14168698 T^{3} + \cdots - 12\!\cdots\!83)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 46506398 T^{3} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{4} + 69655456 T^{3} + \cdots - 13\!\cdots\!11)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 20445178 T^{3} + \cdots + 64\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( (T^{4} + 24721456 T^{3} + \cdots - 19\!\cdots\!92)^{2} \) Copy content Toggle raw display
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