Properties

Label 165.3.g.a
Level $165$
Weight $3$
Character orbit 165.g
Analytic conductor $4.496$
Analytic rank $0$
Dimension $2$
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] = [165,3,Mod(89,165)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.89");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.g (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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta - 2) q^{3} - 3 q^{4} - 5 q^{5} + (\beta + 2) q^{6} + ( - 6 \beta + 3) q^{7} + 7 q^{8} + (5 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta - 2) q^{3} - 3 q^{4} - 5 q^{5} + (\beta + 2) q^{6} + ( - 6 \beta + 3) q^{7} + 7 q^{8} + (5 \beta + 1) q^{9} + 5 q^{10} + (2 \beta - 1) q^{11} + (3 \beta + 6) q^{12} + (12 \beta - 6) q^{13} + (6 \beta - 3) q^{14} + (5 \beta + 10) q^{15} + 5 q^{16} - 19 q^{17} + ( - 5 \beta - 1) q^{18} + 11 q^{19} + 15 q^{20} + (15 \beta - 24) q^{21} + ( - 2 \beta + 1) q^{22} + 32 q^{23} + ( - 7 \beta - 14) q^{24} + 25 q^{25} + ( - 12 \beta + 6) q^{26} + ( - 16 \beta + 13) q^{27} + (18 \beta - 9) q^{28} + ( - 6 \beta + 3) q^{29} + ( - 5 \beta - 10) q^{30} - 31 q^{31} - 33 q^{32} + ( - 5 \beta + 8) q^{33} + 19 q^{34} + (30 \beta - 15) q^{35} + ( - 15 \beta - 3) q^{36} + (30 \beta - 15) q^{37} - 11 q^{38} + ( - 30 \beta + 48) q^{39} - 35 q^{40} + ( - 12 \beta + 6) q^{41} + ( - 15 \beta + 24) q^{42} + (36 \beta - 18) q^{43} + ( - 6 \beta + 3) q^{44} + ( - 25 \beta - 5) q^{45} - 32 q^{46} + 14 q^{47} + ( - 5 \beta - 10) q^{48} - 50 q^{49} - 25 q^{50} + (19 \beta + 38) q^{51} + ( - 36 \beta + 18) q^{52} - 13 q^{53} + (16 \beta - 13) q^{54} + ( - 10 \beta + 5) q^{55} + ( - 42 \beta + 21) q^{56} + ( - 11 \beta - 22) q^{57} + (6 \beta - 3) q^{58} + ( - 15 \beta - 30) q^{60} + 47 q^{61} + 31 q^{62} + ( - 21 \beta + 93) q^{63} + 13 q^{64} + ( - 60 \beta + 30) q^{65} + (5 \beta - 8) q^{66} + (24 \beta - 12) q^{67} + 57 q^{68} + ( - 32 \beta - 64) q^{69} + ( - 30 \beta + 15) q^{70} + (54 \beta - 27) q^{71} + (35 \beta + 7) q^{72} + (48 \beta - 24) q^{73} + ( - 30 \beta + 15) q^{74} + ( - 25 \beta - 50) q^{75} - 33 q^{76} + 33 q^{77} + (30 \beta - 48) q^{78} - 100 q^{79} - 25 q^{80} + (35 \beta - 74) q^{81} + (12 \beta - 6) q^{82} - 22 q^{83} + ( - 45 \beta + 72) q^{84} + 95 q^{85} + ( - 36 \beta + 18) q^{86} + (15 \beta - 24) q^{87} + (14 \beta - 7) q^{88} + (102 \beta - 51) q^{89} + (25 \beta + 5) q^{90} + 198 q^{91} - 96 q^{92} + (31 \beta + 62) q^{93} - 14 q^{94} - 55 q^{95} + (33 \beta + 66) q^{96} + ( - 96 \beta + 48) q^{97} + 50 q^{98} + (7 \beta - 31) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 5 q^{3} - 6 q^{4} - 10 q^{5} + 5 q^{6} + 14 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 5 q^{3} - 6 q^{4} - 10 q^{5} + 5 q^{6} + 14 q^{8} + 7 q^{9} + 10 q^{10} + 15 q^{12} + 25 q^{15} + 10 q^{16} - 38 q^{17} - 7 q^{18} + 22 q^{19} + 30 q^{20} - 33 q^{21} + 64 q^{23} - 35 q^{24} + 50 q^{25} + 10 q^{27} - 25 q^{30} - 62 q^{31} - 66 q^{32} + 11 q^{33} + 38 q^{34} - 21 q^{36} - 22 q^{38} + 66 q^{39} - 70 q^{40} + 33 q^{42} - 35 q^{45} - 64 q^{46} + 28 q^{47} - 25 q^{48} - 100 q^{49} - 50 q^{50} + 95 q^{51} - 26 q^{53} - 10 q^{54} - 55 q^{57} - 75 q^{60} + 94 q^{61} + 62 q^{62} + 165 q^{63} + 26 q^{64} - 11 q^{66} + 114 q^{68} - 160 q^{69} + 49 q^{72} - 125 q^{75} - 66 q^{76} + 66 q^{77} - 66 q^{78} - 200 q^{79} - 50 q^{80} - 113 q^{81} - 44 q^{83} + 99 q^{84} + 190 q^{85} - 33 q^{87} + 35 q^{90} + 396 q^{91} - 192 q^{92} + 155 q^{93} - 28 q^{94} - 110 q^{95} + 165 q^{96} + 100 q^{98} - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
0.500000 + 1.65831i
0.500000 1.65831i
−1.00000 −2.50000 1.65831i −3.00000 −5.00000 2.50000 + 1.65831i 9.94987i 7.00000 3.50000 + 8.29156i 5.00000
89.2 −1.00000 −2.50000 + 1.65831i −3.00000 −5.00000 2.50000 1.65831i 9.94987i 7.00000 3.50000 8.29156i 5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.g.a 2
3.b odd 2 1 165.3.g.b yes 2
5.b even 2 1 165.3.g.b yes 2
15.d odd 2 1 inner 165.3.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.g.a 2 1.a even 1 1 trivial
165.3.g.a 2 15.d odd 2 1 inner
165.3.g.b yes 2 3.b odd 2 1
165.3.g.b yes 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T + 9 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 99 \) Copy content Toggle raw display
$11$ \( T^{2} + 11 \) Copy content Toggle raw display
$13$ \( T^{2} + 396 \) Copy content Toggle raw display
$17$ \( (T + 19)^{2} \) Copy content Toggle raw display
$19$ \( (T - 11)^{2} \) Copy content Toggle raw display
$23$ \( (T - 32)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 99 \) Copy content Toggle raw display
$31$ \( (T + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2475 \) Copy content Toggle raw display
$41$ \( T^{2} + 396 \) Copy content Toggle raw display
$43$ \( T^{2} + 3564 \) Copy content Toggle raw display
$47$ \( (T - 14)^{2} \) Copy content Toggle raw display
$53$ \( (T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 47)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1584 \) Copy content Toggle raw display
$71$ \( T^{2} + 8019 \) Copy content Toggle raw display
$73$ \( T^{2} + 6336 \) Copy content Toggle raw display
$79$ \( (T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T + 22)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 28611 \) Copy content Toggle raw display
$97$ \( T^{2} + 25344 \) Copy content Toggle raw display
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