Properties

Label 165.4.a.h
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,4,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

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: yes
Analytic conductor: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{7} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 13) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{7} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 13) q^{8} + 9 q^{9} + ( - 5 \beta_1 + 5) q^{10} - 11 q^{11} + (3 \beta_{3} - 3 \beta_1 + 21) q^{12} + ( - 5 \beta_{3} + 5 \beta_{2} + \cdots - 2) q^{13}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9} + 20 q^{10} - 44 q^{11} + 78 q^{12} + 2 q^{13} - 52 q^{14} + 60 q^{15} + 66 q^{16} + 74 q^{17} + 36 q^{18} + 136 q^{19} + 130 q^{20} + 102 q^{21} - 44 q^{22} - 64 q^{23} + 144 q^{24} + 100 q^{25} - 320 q^{26} + 108 q^{27} - 20 q^{28} + 52 q^{29} + 60 q^{30} + 492 q^{31} + 208 q^{32} - 132 q^{33} + 244 q^{34} + 170 q^{35} + 234 q^{36} - 4 q^{37} - 404 q^{38} + 6 q^{39} + 240 q^{40} + 268 q^{41} - 156 q^{42} + 546 q^{43} - 286 q^{44} + 180 q^{45} + 368 q^{46} - 276 q^{47} + 198 q^{48} - 496 q^{49} + 100 q^{50} + 222 q^{51} - 1084 q^{52} - 184 q^{53} + 108 q^{54} - 220 q^{55} - 852 q^{56} + 408 q^{57} - 444 q^{58} - 1032 q^{59} + 390 q^{60} + 116 q^{61} - 1240 q^{62} + 306 q^{63} - 918 q^{64} + 10 q^{65} - 132 q^{66} - 552 q^{67} - 720 q^{68} - 192 q^{69} - 260 q^{70} - 920 q^{71} + 432 q^{72} + 926 q^{73} - 2856 q^{74} + 300 q^{75} + 1572 q^{76} - 374 q^{77} - 960 q^{78} + 1152 q^{79} + 330 q^{80} + 324 q^{81} - 1924 q^{82} - 134 q^{83} - 60 q^{84} + 370 q^{85} + 236 q^{86} + 156 q^{87} - 528 q^{88} - 1064 q^{89} + 180 q^{90} + 2780 q^{91} - 4896 q^{92} + 1476 q^{93} - 1432 q^{94} + 680 q^{95} + 624 q^{96} - 1648 q^{97} - 188 q^{98} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 27x^{2} - 18x + 92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 20\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 21\beta _1 + 14 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.20196
1.60719
−2.63835
−4.17080
−4.20196 3.00000 9.65650 5.00000 −12.6059 15.3793 −6.96057 9.00000 −21.0098
1.2 −0.607192 3.00000 −7.63132 5.00000 −1.82158 8.95080 9.49121 9.00000 −3.03596
1.3 3.63835 3.00000 5.23763 5.00000 10.9151 20.8444 −10.0505 9.00000 18.1918
1.4 5.17080 3.00000 18.7372 5.00000 15.5124 −11.1745 55.5199 9.00000 25.8540
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.a.h 4
3.b odd 2 1 495.4.a.m 4
5.b even 2 1 825.4.a.t 4
5.c odd 4 2 825.4.c.p 8
11.b odd 2 1 1815.4.a.t 4
15.d odd 2 1 2475.4.a.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.h 4 1.a even 1 1 trivial
495.4.a.m 4 3.b odd 2 1
825.4.a.t 4 5.b even 2 1
825.4.c.p 8 5.c odd 4 2
1815.4.a.t 4 11.b odd 2 1
2475.4.a.be 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{3} - 21T_{2}^{2} + 68T_{2} + 48 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 34 T^{3} + \cdots - 32064 \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 68144 \) Copy content Toggle raw display
$17$ \( T^{4} - 74 T^{3} + \cdots + 23770800 \) Copy content Toggle raw display
$19$ \( T^{4} - 136 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$23$ \( T^{4} + 64 T^{3} + \cdots + 247529472 \) Copy content Toggle raw display
$29$ \( T^{4} - 52 T^{3} + \cdots + 315474624 \) Copy content Toggle raw display
$31$ \( T^{4} - 492 T^{3} + \cdots - 903269376 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1260009136 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 6228069696 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 1200551616 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 5628791808 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 13030473936 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 2612441088 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 8546777488 \) Copy content Toggle raw display
$67$ \( T^{4} + 552 T^{3} + \cdots - 909580544 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 291456592896 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 133750796272 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 48148922944 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 17388663552 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 479041129296 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1607443600 \) Copy content Toggle raw display
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