Properties

Label 165.3.l.a
Level $165$
Weight $3$
Character orbit 165.l
Analytic conductor $4.496$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,3,Mod(32,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.32");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.l (of order \(4\), degree \(2\), 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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + \beta_{2} q^{4} + ( - 3 \beta_{2} - 4) q^{5} - 3 \beta_1 q^{6} - 2 \beta_{3} q^{7} - 3 \beta_{3} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + \beta_{2} q^{4} + ( - 3 \beta_{2} - 4) q^{5} - 3 \beta_1 q^{6} - 2 \beta_{3} q^{7} - 3 \beta_{3} q^{8} + 9 q^{9} + ( - 3 \beta_{3} - 4 \beta_1) q^{10} + ( - 2 \beta_{3} - 9 \beta_{2} + 2 \beta_1) q^{11} - 3 \beta_{2} q^{12} + 2 \beta_1 q^{13} + 10 q^{14} + (9 \beta_{2} + 12) q^{15} + 19 q^{16} - 14 \beta_1 q^{17} + 9 \beta_1 q^{18} + (4 \beta_{3} - 4 \beta_1) q^{19} + ( - 4 \beta_{2} + 3) q^{20} + 6 \beta_{3} q^{21} + ( - 9 \beta_{3} + 10 \beta_{2} + 10) q^{22} + ( - 7 \beta_{2} - 7) q^{23} + 9 \beta_{3} q^{24} + (24 \beta_{2} + 7) q^{25} + 10 \beta_{2} q^{26} - 27 q^{27} + 2 \beta_1 q^{28} + ( - 6 \beta_{3} - 6 \beta_1) q^{29} + (9 \beta_{3} + 12 \beta_1) q^{30} + 20 q^{31} + 7 \beta_1 q^{32} + (6 \beta_{3} + 27 \beta_{2} - 6 \beta_1) q^{33} - 70 \beta_{2} q^{34} + (8 \beta_{3} - 6 \beta_1) q^{35} + 9 \beta_{2} q^{36} + (7 \beta_{2} + 7) q^{37} + ( - 20 \beta_{2} - 20) q^{38} - 6 \beta_1 q^{39} + (12 \beta_{3} - 9 \beta_1) q^{40} + ( - 22 \beta_{3} + 22 \beta_1) q^{41} - 30 q^{42} + 14 \beta_1 q^{43} + (2 \beta_{3} + 2 \beta_1 + 9) q^{44} + ( - 27 \beta_{2} - 36) q^{45} + ( - 7 \beta_{3} - 7 \beta_1) q^{46} + (43 \beta_{2} - 43) q^{47} - 57 q^{48} + 29 \beta_{2} q^{49} + (24 \beta_{3} + 7 \beta_1) q^{50} + 42 \beta_1 q^{51} + 2 \beta_{3} q^{52} + (17 \beta_{2} + 17) q^{53} - 27 \beta_1 q^{54} + (2 \beta_{3} + 36 \beta_{2} - 14 \beta_1 - 27) q^{55} - 30 \beta_{2} q^{56} + ( - 12 \beta_{3} + 12 \beta_1) q^{57} + ( - 30 \beta_{2} + 30) q^{58} + 22 q^{59} + (12 \beta_{2} - 9) q^{60} + ( - 30 \beta_{3} - 30 \beta_1) q^{61} + 20 \beta_1 q^{62} - 18 \beta_{3} q^{63} - 41 \beta_{2} q^{64} + ( - 6 \beta_{3} - 8 \beta_1) q^{65} + (27 \beta_{3} - 30 \beta_{2} - 30) q^{66} + ( - 47 \beta_{2} - 47) q^{67} - 14 \beta_{3} q^{68} + (21 \beta_{2} + 21) q^{69} + ( - 30 \beta_{2} - 40) q^{70} + 120 \beta_{2} q^{71} - 27 \beta_{3} q^{72} + 14 \beta_1 q^{73} + (7 \beta_{3} + 7 \beta_1) q^{74} + ( - 72 \beta_{2} - 21) q^{75} + ( - 4 \beta_{3} - 4 \beta_1) q^{76} + ( - 20 \beta_{2} - 18 \beta_1 + 20) q^{77} - 30 \beta_{2} q^{78} + ( - 2 \beta_{3} + 2 \beta_1) q^{79} + ( - 57 \beta_{2} - 76) q^{80} + 81 q^{81} + (110 \beta_{2} + 110) q^{82} + 38 \beta_{3} q^{83} - 6 \beta_1 q^{84} + (42 \beta_{3} + 56 \beta_1) q^{85} + 70 \beta_{2} q^{86} + (18 \beta_{3} + 18 \beta_1) q^{87} + ( - 30 \beta_{2} - 27 \beta_1 + 30) q^{88} + 100 q^{89} + ( - 27 \beta_{3} - 36 \beta_1) q^{90} + 20 q^{91} + ( - 7 \beta_{2} + 7) q^{92} - 60 q^{93} + (43 \beta_{3} - 43 \beta_1) q^{94} + ( - 4 \beta_{3} + 28 \beta_1) q^{95} - 21 \beta_1 q^{96} + (43 \beta_{2} + 43) q^{97} + 29 \beta_{3} q^{98} + ( - 18 \beta_{3} - 81 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 16 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 16 q^{5} + 36 q^{9} + 40 q^{14} + 48 q^{15} + 76 q^{16} + 12 q^{20} + 40 q^{22} - 28 q^{23} + 28 q^{25} - 108 q^{27} + 80 q^{31} + 28 q^{37} - 80 q^{38} - 120 q^{42} + 36 q^{44} - 144 q^{45} - 172 q^{47} - 228 q^{48} + 68 q^{53} - 108 q^{55} + 120 q^{58} + 88 q^{59} - 36 q^{60} - 120 q^{66} - 188 q^{67} + 84 q^{69} - 160 q^{70} - 84 q^{75} + 80 q^{77} - 304 q^{80} + 324 q^{81} + 440 q^{82} + 120 q^{88} + 400 q^{89} + 80 q^{91} + 28 q^{92} - 240 q^{93} + 172 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.58114 1.58114i
1.58114 + 1.58114i
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i −3.00000 1.00000i −4.00000 3.00000i 4.74342 + 4.74342i −3.16228 + 3.16228i −4.74342 + 4.74342i 9.00000 1.58114 + 11.0680i
32.2 1.58114 + 1.58114i −3.00000 1.00000i −4.00000 3.00000i −4.74342 4.74342i 3.16228 3.16228i 4.74342 4.74342i 9.00000 −1.58114 11.0680i
98.1 −1.58114 + 1.58114i −3.00000 1.00000i −4.00000 + 3.00000i 4.74342 4.74342i −3.16228 3.16228i −4.74342 4.74342i 9.00000 1.58114 11.0680i
98.2 1.58114 1.58114i −3.00000 1.00000i −4.00000 + 3.00000i −4.74342 + 4.74342i 3.16228 + 3.16228i 4.74342 + 4.74342i 9.00000 −1.58114 + 11.0680i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
15.e even 4 1 inner
165.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.l.a 4
3.b odd 2 1 165.3.l.d yes 4
5.c odd 4 1 165.3.l.d yes 4
11.b odd 2 1 inner 165.3.l.a 4
15.e even 4 1 inner 165.3.l.a 4
33.d even 2 1 165.3.l.d yes 4
55.e even 4 1 165.3.l.d yes 4
165.l odd 4 1 inner 165.3.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.l.a 4 1.a even 1 1 trivial
165.3.l.a 4 11.b odd 2 1 inner
165.3.l.a 4 15.e even 4 1 inner
165.3.l.a 4 165.l odd 4 1 inner
165.3.l.d yes 4 3.b odd 2 1
165.3.l.d yes 4 5.c odd 4 1
165.3.l.d yes 4 33.d even 2 1
165.3.l.d yes 4 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(165, [\chi])\):

\( T_{2}^{4} + 25 \) Copy content Toggle raw display
\( T_{23}^{2} + 14T_{23} + 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 400 \) Copy content Toggle raw display
$11$ \( T^{4} + 82T^{2} + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 400 \) Copy content Toggle raw display
$17$ \( T^{4} + 960400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 160)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 360)^{2} \) Copy content Toggle raw display
$31$ \( (T - 20)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4840)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 960400 \) Copy content Toggle raw display
$47$ \( (T^{2} + 86 T + 3698)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 34 T + 578)^{2} \) Copy content Toggle raw display
$59$ \( (T - 22)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 94 T + 4418)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 14400)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 960400 \) Copy content Toggle raw display
$79$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 52128400 \) Copy content Toggle raw display
$89$ \( (T - 100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 86 T + 3698)^{2} \) Copy content Toggle raw display
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