Properties

Label 45.3.c.a
Level $45$
Weight $3$
Character orbit 45.c
Analytic conductor $1.226$
Analytic rank $0$
Dimension $4$
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] = [45,3,Mod(26,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("45.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.c (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: \(1.22616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{3} + 4) q^{7} + (3 \beta_{2} + 9 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{3} + 4) q^{7} + (3 \beta_{2} + 9 \beta_1) q^{8} + ( - \beta_{3} - 5) q^{10} + (2 \beta_{2} - 7 \beta_1) q^{11} + (5 \beta_{3} - 6) q^{13} + ( - 6 \beta_{2} - 9 \beta_1) q^{14} + (4 \beta_{3} + 21) q^{16} + (8 \beta_{2} - 4 \beta_1) q^{17} + 2 \beta_{3} q^{19} + (3 \beta_{2} + 10 \beta_1) q^{20} + ( - 5 \beta_{3} - 4) q^{22} + ( - 6 \beta_{2} + 18 \beta_1) q^{23} - 5 q^{25} + ( - 4 \beta_{2} - 19 \beta_1) q^{26} + ( - 11 \beta_{3} - 32) q^{28} + (10 \beta_{2} + 16 \beta_1) q^{29} + ( - 14 \beta_{3} - 14) q^{31} + ( - 17 \beta_{2} - 5 \beta_1) q^{32} + (4 \beta_{3} + 32) q^{34} + ( - 4 \beta_{2} - 5 \beta_1) q^{35} + (9 \beta_{3} + 38) q^{37} + ( - 4 \beta_{2} - 10 \beta_1) q^{38} + (9 \beta_{3} + 15) q^{40} + ( - 8 \beta_{2} + \beta_1) q^{41} + ( - 10 \beta_{3} - 12) q^{43} + (22 \beta_{2} + \beta_1) q^{44} + (12 \beta_{3} + 6) q^{46} + ( - 4 \beta_{2} - 22 \beta_1) q^{47} + (8 \beta_{3} - 23) q^{49} + (5 \beta_{2} + 5 \beta_1) q^{50} + ( - 3 \beta_{3} - 82) q^{52} + (8 \beta_{2} - 22 \beta_1) q^{53} + ( - 7 \beta_{3} + 10) q^{55} + (30 \beta_{2} + 51 \beta_1) q^{56} + (26 \beta_{3} + 82) q^{58} + ( - 22 \beta_{2} + 17 \beta_1) q^{59} + (2 \beta_{3} - 42) q^{61} + (42 \beta_{2} + 84 \beta_1) q^{62} + ( - 6 \beta_{3} - 11) q^{64} + (6 \beta_{2} - 25 \beta_1) q^{65} + ( - 8 \beta_{3} + 52) q^{67} + ( - 8 \beta_{2} - 68 \beta_1) q^{68} + ( - 9 \beta_{3} - 30) q^{70} + ( - 44 \beta_{2} + 4 \beta_1) q^{71} + (2 \beta_{3} + 54) q^{73} + ( - 56 \beta_{2} - 83 \beta_1) q^{74} + ( - 6 \beta_{3} - 40) q^{76} + ( - 6 \beta_{2} - 18 \beta_1) q^{77} + ( - 26 \beta_{3} - 14) q^{79} + ( - 21 \beta_{2} - 20 \beta_1) q^{80} + ( - 7 \beta_{3} - 38) q^{82} + (24 \beta_{2} + 18 \beta_1) q^{83} + ( - 4 \beta_{3} + 40) q^{85} + (32 \beta_{2} + 62 \beta_1) q^{86} + (3 \beta_{3} + 96) q^{88} + (12 \beta_{2} + 57 \beta_1) q^{89} + (14 \beta_{3} + 26) q^{91} + ( - 54 \beta_{2} + 6 \beta_1) q^{92} + ( - 26 \beta_{3} - 64) q^{94} - 10 \beta_1 q^{95} + (18 \beta_{3} - 58) q^{97} + (7 \beta_{2} - 17 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
1.58114 + 0.707107i
−1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 0.707107i
3.65028i 0 −9.32456 2.23607i 0 7.16228 19.4361i 0 −8.16228
26.2 0.821854i 0 3.32456 2.23607i 0 0.837722 6.01972i 0 −1.83772
26.3 0.821854i 0 3.32456 2.23607i 0 0.837722 6.01972i 0 −1.83772
26.4 3.65028i 0 −9.32456 2.23607i 0 7.16228 19.4361i 0 −8.16228
\(n\): 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 45.3.c.a 4
3.b odd 2 1 inner 45.3.c.a 4
4.b odd 2 1 720.3.l.a 4
5.b even 2 1 225.3.c.c 4
5.c odd 4 2 225.3.d.b 8
8.b even 2 1 2880.3.l.g 4
8.d odd 2 1 2880.3.l.c 4
9.c even 3 2 405.3.i.d 8
9.d odd 6 2 405.3.i.d 8
12.b even 2 1 720.3.l.a 4
15.d odd 2 1 225.3.c.c 4
15.e even 4 2 225.3.d.b 8
20.d odd 2 1 3600.3.l.v 4
20.e even 4 2 3600.3.c.i 8
24.f even 2 1 2880.3.l.c 4
24.h odd 2 1 2880.3.l.g 4
60.h even 2 1 3600.3.l.v 4
60.l odd 4 2 3600.3.c.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.c.a 4 1.a even 1 1 trivial
45.3.c.a 4 3.b odd 2 1 inner
225.3.c.c 4 5.b even 2 1
225.3.c.c 4 15.d odd 2 1
225.3.d.b 8 5.c odd 4 2
225.3.d.b 8 15.e even 4 2
405.3.i.d 8 9.c even 3 2
405.3.i.d 8 9.d odd 6 2
720.3.l.a 4 4.b odd 2 1
720.3.l.a 4 12.b even 2 1
2880.3.l.c 4 8.d odd 2 1
2880.3.l.c 4 24.f even 2 1
2880.3.l.g 4 8.b even 2 1
2880.3.l.g 4 24.h odd 2 1
3600.3.c.i 8 20.e even 4 2
3600.3.c.i 8 60.l odd 4 2
3600.3.l.v 4 20.d odd 2 1
3600.3.l.v 4 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 14T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 236T^{2} + 6084 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12 T - 214)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 704 T^{2} + 82944 \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1656 T^{2} + 219024 \) Copy content Toggle raw display
$29$ \( T^{4} + 2024T^{2} + 144 \) Copy content Toggle raw display
$31$ \( (T^{2} + 28 T - 1764)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 76 T + 634)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 644 T^{2} + 101124 \) Copy content Toggle raw display
$43$ \( (T^{2} + 24 T - 856)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2096 T^{2} + 788544 \) Copy content Toggle raw display
$53$ \( T^{4} + 2576 T^{2} + 419904 \) Copy content Toggle raw display
$59$ \( T^{4} + 5996 T^{2} + \cdots + 3392964 \) Copy content Toggle raw display
$61$ \( (T^{2} + 84 T + 1724)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 104 T + 2064)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 19424 T^{2} + \cdots + 93083904 \) Copy content Toggle raw display
$73$ \( (T^{2} - 108 T + 2876)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 28 T - 6564)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 7056 T^{2} + \cdots + 4981824 \) Copy content Toggle raw display
$89$ \( T^{4} + 14436 T^{2} + \cdots + 33385284 \) Copy content Toggle raw display
$97$ \( (T^{2} + 116 T + 124)^{2} \) Copy content Toggle raw display
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