Properties

Label 162.7.b.a
Level $162$
Weight $7$
Character orbit 162.b
Analytic conductor $37.269$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_1 q^{2} - 32 q^{4} + ( - 25 \beta_{2} + 70 \beta_1) q^{5} + ( - \beta_{3} - 1) q^{7} + 128 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_1 q^{2} - 32 q^{4} + ( - 25 \beta_{2} + 70 \beta_1) q^{5} + ( - \beta_{3} - 1) q^{7} + 128 \beta_1 q^{8} + ( - 100 \beta_{3} + 660) q^{10} + (290 \beta_{2} - 701 \beta_1) q^{11} + ( - 362 \beta_{3} + 959) q^{13} + (8 \beta_{2} + 8 \beta_1) q^{14} + 1024 q^{16} + (193 \beta_{2} + 1988 \beta_1) q^{17} + (547 \beta_{3} + 2561) q^{19} + (800 \beta_{2} - 2240 \beta_1) q^{20} + (1160 \beta_{3} - 6768) q^{22} + ( - 2600 \beta_{2} + 3785 \beta_1) q^{23} + (4125 \beta_{3} - 6425) q^{25} + (2896 \beta_{2} - 2388 \beta_1) q^{26} + (32 \beta_{3} + 32) q^{28} + ( - 9805 \beta_{2} - 4133 \beta_1) q^{29} + (2826 \beta_{3} - 10024) q^{31} - 4096 \beta_1 q^{32} + (772 \beta_{3} + 15132) q^{34} + ( - 140 \beta_{2} + 185 \beta_1) q^{35} + (2303 \beta_{3} - 5050) q^{37} + ( - 4376 \beta_{2} - 12432 \beta_1) q^{38} + (3200 \beta_{3} - 21120) q^{40} + ( - 23146 \beta_{2} + 10483 \beta_1) q^{41} + (14799 \beta_{3} + 46235) q^{43} + ( - 9280 \beta_{2} + 22432 \beta_1) q^{44} + ( - 10400 \beta_{3} + 40680) q^{46} + ( - 2502 \beta_{2} - 62670 \beta_1) q^{47} + (2 \beta_{3} - 117621) q^{49} + ( - 33000 \beta_{2} + 9200 \beta_1) q^{50} + (11584 \beta_{3} - 30688) q^{52} + ( - 36874 \beta_{2} - 16193 \beta_1) q^{53} + ( - 45075 \beta_{3} + 237465) q^{55} + ( - 256 \beta_{2} - 256 \beta_1) q^{56} + ( - 39220 \beta_{3} + 6156) q^{58} + (16858 \beta_{2} - 55930 \beta_1) q^{59} + (32409 \beta_{3} + 152264) q^{61} + ( - 22608 \beta_{2} + 28792 \beta_1) q^{62} - 32768 q^{64} + ( - 83705 \beta_{2} + 159440 \beta_1) q^{65} + ( - 5307 \beta_{3} - 502243) q^{67} + ( - 6176 \beta_{2} - 63616 \beta_1) q^{68} + ( - 560 \beta_{3} + 2040) q^{70} + ( - 65072 \beta_{2} - 307903 \beta_1) q^{71} + ( - 38535 \beta_{3} + 131456) q^{73} + ( - 18424 \beta_{2} + 10988 \beta_1) q^{74} + ( - 17504 \beta_{3} - 81952) q^{76} + (1402 \beta_{2} - 2368 \beta_1) q^{77} + (131623 \beta_{3} - 212179) q^{79} + ( - 25600 \beta_{2} + 71680 \beta_1) q^{80} + ( - 92584 \beta_{3} + 176448) q^{82} + ( - 133176 \beta_{2} + 255690 \beta_1) q^{83} + (31365 \beta_{3} - 246960) q^{85} + ( - 118392 \beta_{2} - 244136 \beta_1) q^{86} + ( - 37120 \beta_{3} + 216576) q^{88} + ( - 245729 \beta_{2} - 202459 \beta_1) q^{89} + ( - 597 \beta_{3} + 8815) q^{91} + (83200 \beta_{2} - 121120 \beta_1) q^{92} + ( - 10008 \beta_{3} - 491352) q^{94} + (26230 \beta_{2} + 39785 \beta_1) q^{95} + ( - 93770 \beta_{3} + 905432) q^{97} + ( - 16 \beta_{2} + 470476 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 128 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 128 q^{4} - 4 q^{7} + 2640 q^{10} + 3836 q^{13} + 4096 q^{16} + 10244 q^{19} - 27072 q^{22} - 25700 q^{25} + 128 q^{28} - 40096 q^{31} + 60528 q^{34} - 20200 q^{37} - 84480 q^{40} + 184940 q^{43} + 162720 q^{46} - 470484 q^{49} - 122752 q^{52} + 949860 q^{55} + 24624 q^{58} + 609056 q^{61} - 131072 q^{64} - 2008972 q^{67} + 8160 q^{70} + 525824 q^{73} - 327808 q^{76} - 848716 q^{79} + 705792 q^{82} - 987840 q^{85} + 866304 q^{88} + 35260 q^{91} - 1965408 q^{94} + 3621728 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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
0.517638i
1.93185i
1.93185i
0.517638i
5.65685i 0 −32.0000 24.8168i 0 −6.19615 181.019i 0 140.385
161.2 5.65685i 0 −32.0000 208.528i 0 4.19615 181.019i 0 1179.62
161.3 5.65685i 0 −32.0000 208.528i 0 4.19615 181.019i 0 1179.62
161.4 5.65685i 0 −32.0000 24.8168i 0 −6.19615 181.019i 0 140.385
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 inner 162.7.b.a 4
9.c even 3 2 162.7.d.f 8
9.d odd 6 2 162.7.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.7.b.a 4 1.a even 1 1 trivial
162.7.b.a 4 3.b odd 2 1 inner
162.7.d.f 8 9.c even 3 2
162.7.d.f 8 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 44100T_{5}^{2} + 26780625 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 44100 T^{2} + 26780625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 87664550724 \) Copy content Toggle raw display
$13$ \( (T^{2} - 1918 T - 2618507)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 44258191098489 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5122 T - 1519922)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$31$ \( (T^{2} + 20048 T - 115148876)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10100 T - 117700343)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{2} - 92470 T - 3775605602)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{2} - 304528 T - 5174942891)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1004486 T + 251487596326)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 262912 T - 22812868139)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 424358 T - 422744653442)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 64\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{2} - 1810864 T + 582401158324)^{2} \) Copy content Toggle raw display
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