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Properties

Label 45.6.b.c
Level $45$
Weight $6$
Character orbit 45.b
Analytic conductor $7.217$
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] = [45,6,Mod(19,45)]
 
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([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.b (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: \(7.21727189158\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{89})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 45x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 11) q^{4} + (5 \beta_{2} - 5 \beta_1 - 30) q^{5} + (6 \beta_{2} - 18 \beta_1) q^{7} + ( - 2 \beta_{2} - 21 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 11) q^{4} + (5 \beta_{2} - 5 \beta_1 - 30) q^{5} + (6 \beta_{2} - 18 \beta_1) q^{7} + ( - 2 \beta_{2} - 21 \beta_1) q^{8} + ( - 5 \beta_{3} - 30 \beta_1 + 225) q^{10} + (12 \beta_{3} - 90) q^{11} + (114 \beta_{2} + 30 \beta_1) q^{13} + ( - 18 \beta_{3} + 786) q^{14} + (11 \beta_{3} + 547) q^{16} + (102 \beta_{2} - 38 \beta_1) q^{17} + ( - 8 \beta_{3} + 672) q^{19} + ( - 30 \beta_{3} + 170 \beta_{2} + \cdots + 330) q^{20}+ \cdots + ( - 576 \beta_{2} - 11165 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 42 q^{4} - 120 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 42 q^{4} - 120 q^{5} + 890 q^{10} - 336 q^{11} + 3108 q^{14} + 2210 q^{16} + 2672 q^{19} + 1260 q^{20} - 5300 q^{25} - 4188 q^{26} - 7104 q^{29} - 23296 q^{31} + 7276 q^{34} - 21360 q^{35} - 16910 q^{40} - 3624 q^{41} + 90036 q^{44} + 45304 q^{46} + 4300 q^{49} - 53400 q^{50} + 10080 q^{55} - 62940 q^{56} + 114672 q^{59} - 60280 q^{61} + 169822 q^{64} - 74760 q^{65} - 93240 q^{70} - 34848 q^{71} + 96948 q^{74} - 85728 q^{76} + 115040 q^{79} - 66300 q^{80} - 124600 q^{85} - 51576 q^{86} + 273528 q^{89} - 39456 q^{91} - 143792 q^{94} - 80160 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 45x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} - 35\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{3} - 137\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 9\nu^{2} + 203 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{2} + 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 203 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 70\beta_{2} - 137\beta_1 ) / 9 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 5.21699i
− 4.21699i
4.21699i
5.21699i
− 9.21699i 0 −52.9529 −30.0000 + 47.1699i 0 167.208i 193.123i 0 434.765 + 276.510i
19.2 − 0.216991i 0 31.9529 −30.0000 + 47.1699i 0 59.2078i − 13.8772i 0 10.2354 + 6.50972i
19.3 0.216991i 0 31.9529 −30.0000 − 47.1699i 0 − 59.2078i 13.8772i 0 10.2354 − 6.50972i
19.4 9.21699i 0 −52.9529 −30.0000 − 47.1699i 0 − 167.208i − 193.123i 0 434.765 − 276.510i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.6.b.c 4
3.b odd 2 1 15.6.b.a ✓ 4
4.b odd 2 1 720.6.f.h 4
5.b even 2 1 inner 45.6.b.c 4
5.c odd 4 1 225.6.a.i 2
5.c odd 4 1 225.6.a.u 2
12.b even 2 1 240.6.f.c 4
15.d odd 2 1 15.6.b.a ✓ 4
15.e even 4 1 75.6.a.f 2
15.e even 4 1 75.6.a.j 2
20.d odd 2 1 720.6.f.h 4
60.h even 2 1 240.6.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.b.a ✓ 4 3.b odd 2 1
15.6.b.a ✓ 4 15.d odd 2 1
45.6.b.c 4 1.a even 1 1 trivial
45.6.b.c 4 5.b even 2 1 inner
75.6.a.f 2 15.e even 4 1
75.6.a.j 2 15.e even 4 1
225.6.a.i 2 5.c odd 4 1
225.6.a.u 2 5.c odd 4 1
240.6.f.c 4 12.b even 2 1
240.6.f.c 4 60.h even 2 1
720.6.f.h 4 4.b odd 2 1
720.6.f.h 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 85T_{2}^{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 85T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 60 T + 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 31464 T^{2} + 98010000 \) Copy content Toggle raw display
$11$ \( (T^{2} + 168 T - 252468)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 69120616464 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 124719160336 \) Copy content Toggle raw display
$19$ \( (T^{2} - 1336 T + 330880)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 72965764000000 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3552 T - 13455360)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 11648 T + 33457600)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 1812 T - 202645980)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 670943443435776 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} - 57336 T + 572451660)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 30140 T + 3798916)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + 17424 T - 157672656)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 120354942009600 \) Copy content Toggle raw display
$79$ \( (T^{2} - 57520 T - 1122176000)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} - 136764 T + 4173659460)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
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