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Properties

Label 162.7.b.c
Level $162$
Weight $7$
Character orbit 162.b
Analytic conductor $37.269$
Analytic rank $0$
Dimension $12$
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] = [162,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{42} \)
Twist minimal: no (minimal twist has level 18)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 32 q^{4} + \beta_{8} q^{5} + ( - \beta_1 - 40) q^{7} - 32 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 32 q^{4} + \beta_{8} q^{5} + ( - \beta_1 - 40) q^{7} - 32 \beta_{2} q^{8} + \beta_{3} q^{10} + ( - \beta_{10} - 3 \beta_{8} - 19 \beta_{2}) q^{11} + (\beta_{4} - 280) q^{13} + (\beta_{9} + 2 \beta_{8} - 40 \beta_{2}) q^{14} + 1024 q^{16} + (\beta_{11} - 2 \beta_{10} + \cdots - 143 \beta_{2}) q^{17}+ \cdots + ( - 84 \beta_{11} + \cdots + 22525 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 384 q^{4} - 480 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49} + 107520 q^{52} + 580392 q^{55} - 318528 q^{58} - 271488 q^{61} - 393216 q^{64} + 579876 q^{67} - 311904 q^{70} - 977700 q^{73} + 90240 q^{76} + 1529592 q^{79} + 1073088 q^{82} - 3239136 q^{85} - 230400 q^{88} + 355584 q^{91} + 1473696 q^{94} + 77748 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13081 \nu^{10} - 5302927 \nu^{8} - 773457883 \nu^{6} - 48532556637 \nu^{4} + \cdots - 10233017042400 ) / 3086858160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26659 \nu^{11} + 7658854 \nu^{9} + 735955987 \nu^{7} + 28995600888 \nu^{5} + \cdots - 806248446336 \nu ) / 1753335434880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 210097 \nu^{10} + 67295716 \nu^{8} + 7678096429 \nu^{6} + 391082105466 \nu^{4} + \cdots + 66081730824000 ) / 3086858160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 225662 \nu^{10} + 73086479 \nu^{8} + 8515647236 \nu^{6} + 451736622819 \nu^{4} + \cdots + 98574178616640 ) / 3086858160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1009529 \nu^{10} + 306190910 \nu^{8} + 31261162145 \nu^{6} + 1265226360660 \nu^{4} + \cdots + 59476995947520 ) / 4115810880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3267469 \nu^{10} + 1216138966 \nu^{8} + 166433693269 \nu^{6} + 10303228617156 \nu^{4} + \cdots + 24\!\cdots\!00 ) / 12347432640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1385069 \nu^{11} + 3076022942 \nu^{9} + 982609562195 \nu^{7} + 97465949305360 \nu^{5} + \cdots + 37\!\cdots\!68 \nu ) / 2337780579840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9191227 \nu^{11} + 3200148886 \nu^{9} + 405374895019 \nu^{7} + 23127582217656 \nu^{5} + \cdots + 46\!\cdots\!80 \nu ) / 7013341739520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42893003 \nu^{11} - 14985851078 \nu^{9} - 1900173246107 \nu^{7} + \cdots - 21\!\cdots\!40 \nu ) / 3506670869760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 186692297 \nu^{11} + 65156440394 \nu^{9} + 8325664614761 \nu^{7} + \cdots + 10\!\cdots\!20 \nu ) / 14026683479040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 500693495 \nu^{11} - 172073646566 \nu^{9} - 21610924592375 \nu^{7} + \cdots - 28\!\cdots\!44 \nu ) / 14026683479040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} - 4\beta_{9} - 53\beta_{8} - 122\beta_{2} ) / 1458 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{6} + \beta_{5} + 79\beta_{4} - 75\beta_{3} - 221\beta _1 - 269730 ) / 4374 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 47\beta_{11} + 263\beta_{10} + 476\beta_{9} + 3679\beta_{8} - 36\beta_{7} - 55457\beta_{2} ) / 1458 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1531\beta_{6} + 275\beta_{5} - 10039\beta_{4} + 6819\beta_{3} + 47861\beta _1 + 24285906 ) / 4374 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17841 \beta_{11} - 153073 \beta_{10} - 176680 \beta_{9} - 910113 \beta_{8} + \cdots + 29162487 \beta_{2} ) / 4374 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -84433\beta_{6} - 30377\beta_{5} + 412409\beta_{4} - 164953\beta_{3} - 2565667\beta _1 - 884098638 ) / 1458 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2532885 \beta_{11} + 24684205 \beta_{10} + 23196280 \beta_{9} + 82540461 \beta_{8} + \cdots - 4223775003 \beta_{2} ) / 4374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 38381363 \beta_{6} + 17581675 \beta_{5} - 160101575 \beta_{4} + 29748507 \beta_{3} + \cdots + 322428746610 ) / 4374 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 117062251 \beta_{11} - 1228877051 \beta_{10} - 1049537384 \beta_{9} + \cdots + 196243285213 \beta_{2} ) / 1458 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5595014987 \beta_{6} - 2854696819 \beta_{5} + 21460960859 \beta_{4} - 946680195 \beta_{3} + \cdots - 41687065381770 ) / 4374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 48110080821 \beta_{11} + 531932678909 \beta_{10} + 433999027448 \beta_{9} + \cdots - 81411526600971 \beta_{2} ) / 4374 \) 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
4.28281i
8.15670i
3.87527i
− 11.8022i
7.20150i
− 8.88570i
8.88570i
− 7.20150i
11.8022i
− 3.87527i
− 8.15670i
− 4.28281i
− 5.65685i 0 −32.0000 − 181.232i 0 −208.612 181.019i 0 −1025.20
161.2 − 5.65685i 0 −32.0000 − 110.679i 0 326.338 181.019i 0 −626.092
161.3 − 5.65685i 0 −32.0000 1.84488i 0 −12.6882 181.019i 0 10.4362
161.4 − 5.65685i 0 −32.0000 10.8441i 0 −644.082 181.019i 0 61.3435
161.5 − 5.65685i 0 −32.0000 45.6802i 0 490.195 181.019i 0 258.406
161.6 − 5.65685i 0 −32.0000 233.541i 0 −191.150 181.019i 0 1321.11
161.7 5.65685i 0 −32.0000 − 233.541i 0 −191.150 − 181.019i 0 1321.11
161.8 5.65685i 0 −32.0000 − 45.6802i 0 490.195 − 181.019i 0 258.406
161.9 5.65685i 0 −32.0000 − 10.8441i 0 −644.082 − 181.019i 0 61.3435
161.10 5.65685i 0 −32.0000 − 1.84488i 0 −12.6882 − 181.019i 0 10.4362
161.11 5.65685i 0 −32.0000 110.679i 0 326.338 − 181.019i 0 −626.092
161.12 5.65685i 0 −32.0000 181.232i 0 −208.612 − 181.019i 0 −1025.20
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
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 162.7.b.c 12
3.b odd 2 1 inner 162.7.b.c 12
9.c even 3 1 18.7.d.a ✓ 12
9.c even 3 1 54.7.d.a 12
9.d odd 6 1 18.7.d.a ✓ 12
9.d odd 6 1 54.7.d.a 12
36.f odd 6 1 144.7.q.c 12
36.f odd 6 1 432.7.q.b 12
36.h even 6 1 144.7.q.c 12
36.h even 6 1 432.7.q.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a ✓ 12 9.c even 3 1
18.7.d.a ✓ 12 9.d odd 6 1
54.7.d.a 12 9.c even 3 1
54.7.d.a 12 9.d odd 6 1
144.7.q.c 12 36.f odd 6 1
144.7.q.c 12 36.h even 6 1
162.7.b.c 12 1.a even 1 1 trivial
162.7.b.c 12 3.b odd 2 1 inner
432.7.q.b 12 36.f odd 6 1
432.7.q.b 12 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 101844 T_{5}^{10} + 3082099734 T_{5}^{8} + 28287732473100 T_{5}^{6} + \cdots + 18\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 52130539391500)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 99\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 15\!\cdots\!40)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 13\!\cdots\!55)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 22\!\cdots\!45)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 49\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 12\!\cdots\!55)^{2} \) Copy content Toggle raw display
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