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Properties

Label 162.7.d.b
Level $162$
Weight $7$
Character orbit 162.d
Analytic conductor $37.269$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,7,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 30 \beta_1 q^{5} - 2 \beta_{2} q^{7} - 32 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 30 \beta_1 q^{5} - 2 \beta_{2} q^{7} - 32 \beta_{3} q^{8} + 960 q^{10} + (6 \beta_{3} - 6 \beta_1) q^{11} + ( - 2950 \beta_{2} + 2950) q^{13} - 2 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 792 \beta_{3} q^{17} + 5258 q^{19} + ( - 960 \beta_{3} + 960 \beta_1) q^{20} + (192 \beta_{2} - 192) q^{22} + 1812 \beta_1 q^{23} + 13175 \beta_{2} q^{25} - 2950 \beta_{3} q^{26} - 64 q^{28} + ( - 390 \beta_{3} + 390 \beta_1) q^{29} + (22898 \beta_{2} - 22898) q^{31} - 1024 \beta_1 q^{32} + 25344 \beta_{2} q^{34} - 60 \beta_{3} q^{35} + 34058 q^{37} + ( - 5258 \beta_{3} + 5258 \beta_1) q^{38} + ( - 30720 \beta_{2} + 30720) q^{40} + 2964 \beta_1 q^{41} + 6406 \beta_{2} q^{43} + 192 \beta_{3} q^{44} + 57984 q^{46} + ( - 31800 \beta_{3} + 31800 \beta_1) q^{47} + ( - 117645 \beta_{2} + 117645) q^{49} + 13175 \beta_1 q^{50} - 94400 \beta_{2} q^{52} - 34038 \beta_{3} q^{53} - 5760 q^{55} + (64 \beta_{3} - 64 \beta_1) q^{56} + ( - 12480 \beta_{2} + 12480) q^{58} + 57774 \beta_1 q^{59} + 62566 \beta_{2} q^{61} + 22898 \beta_{3} q^{62} - 32768 q^{64} + ( - 88500 \beta_{3} + 88500 \beta_1) q^{65} + (438698 \beta_{2} - 438698) q^{67} + 25344 \beta_1 q^{68} - 1920 \beta_{2} q^{70} - 12060 \beta_{3} q^{71} - 730510 q^{73} + ( - 34058 \beta_{3} + 34058 \beta_1) q^{74} + ( - 168256 \beta_{2} + 168256) q^{76} + 12 \beta_1 q^{77} - 340562 \beta_{2} q^{79} - 30720 \beta_{3} q^{80} + 94848 q^{82} + ( - 87726 \beta_{3} + 87726 \beta_1) q^{83} + (760320 \beta_{2} - 760320) q^{85} + 6406 \beta_1 q^{86} + 6144 \beta_{2} q^{88} - 68364 \beta_{3} q^{89} - 5900 q^{91} + ( - 57984 \beta_{3} + 57984 \beta_1) q^{92} + ( - 1017600 \beta_{2} + 1017600) q^{94} + 157740 \beta_1 q^{95} + 281086 \beta_{2} q^{97} - 117645 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 64 q^{4} - 4 q^{7} + 3840 q^{10} + 5900 q^{13} - 2048 q^{16} + 21032 q^{19} - 384 q^{22} + 26350 q^{25} - 256 q^{28} - 45796 q^{31} + 50688 q^{34} + 136232 q^{37} + 61440 q^{40} + 12812 q^{43} + 231936 q^{46} + 235290 q^{49} - 188800 q^{52} - 23040 q^{55} + 24960 q^{58} + 125132 q^{61} - 131072 q^{64} - 877396 q^{67} - 3840 q^{70} - 2922040 q^{73} + 336512 q^{76} - 681124 q^{79} + 379392 q^{82} - 1520640 q^{85} + 12288 q^{88} - 23600 q^{91} + 2035200 q^{94} + 562172 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−4.89898 + 2.82843i 0 16.0000 − 27.7128i −146.969 − 84.8528i 0 −1.00000 − 1.73205i 181.019i 0 960.000
53.2 4.89898 − 2.82843i 0 16.0000 − 27.7128i 146.969 + 84.8528i 0 −1.00000 − 1.73205i − 181.019i 0 960.000
107.1 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −146.969 + 84.8528i 0 −1.00000 + 1.73205i − 181.019i 0 960.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i 146.969 − 84.8528i 0 −1.00000 + 1.73205i 181.019i 0 960.000
\(n\): 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
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.7.d.b 4
3.b odd 2 1 inner 162.7.d.b 4
9.c even 3 1 6.7.b.a ✓ 2
9.c even 3 1 inner 162.7.d.b 4
9.d odd 6 1 6.7.b.a ✓ 2
9.d odd 6 1 inner 162.7.d.b 4
36.f odd 6 1 48.7.e.b 2
36.h even 6 1 48.7.e.b 2
45.h odd 6 1 150.7.d.a 2
45.j even 6 1 150.7.d.a 2
45.k odd 12 2 150.7.b.a 4
45.l even 12 2 150.7.b.a 4
63.l odd 6 1 294.7.b.a 2
63.o even 6 1 294.7.b.a 2
72.j odd 6 1 192.7.e.c 2
72.l even 6 1 192.7.e.f 2
72.n even 6 1 192.7.e.c 2
72.p odd 6 1 192.7.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a ✓ 2 9.c even 3 1
6.7.b.a ✓ 2 9.d odd 6 1
48.7.e.b 2 36.f odd 6 1
48.7.e.b 2 36.h even 6 1
150.7.b.a 4 45.k odd 12 2
150.7.b.a 4 45.l even 12 2
150.7.d.a 2 45.h odd 6 1
150.7.d.a 2 45.j even 6 1
162.7.d.b 4 1.a even 1 1 trivial
162.7.d.b 4 3.b odd 2 1 inner
162.7.d.b 4 9.c even 3 1 inner
162.7.d.b 4 9.d odd 6 1 inner
192.7.e.c 2 72.j odd 6 1
192.7.e.c 2 72.n even 6 1
192.7.e.f 2 72.l even 6 1
192.7.e.f 2 72.p odd 6 1
294.7.b.a 2 63.l odd 6 1
294.7.b.a 2 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 28800T_{5}^{2} + 829440000 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 28800 T^{2} + 829440000 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 1152 T^{2} + 1327104 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2950 T + 8702500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20072448)^{2} \) Copy content Toggle raw display
$19$ \( (T - 5258)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 23689635840000 \) Copy content Toggle raw display
$31$ \( (T^{2} + 22898 T + 524318404)^{2} \) Copy content Toggle raw display
$37$ \( (T - 34058)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6406 T + 41036836)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 37074734208)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{2} - 62566 T + 3914504356)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 438698 T + 192455935204)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4654195200)^{2} \) Copy content Toggle raw display
$73$ \( (T + 730510)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 340562 T + 115982475844)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} + 149556367872)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 281086 T + 79009339396)^{2} \) Copy content Toggle raw display
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