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Properties

Label 45.5.d.a
Level $45$
Weight $5$
Character orbit 45.d
Analytic conductor $4.652$
Analytic rank $0$
Dimension $8$
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] = [45,5,Mod(44,45)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.44");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.d (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: \(4.65164833877\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 74x^{6} + 1729x^{4} - 2880x^{2} + 32400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{2} + 5) q^{4} + (\beta_{5} + 2 \beta_{3}) q^{5} - \beta_{4} q^{7} + ( - \beta_{6} + \beta_{5} + 5 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{2} + 5) q^{4} + (\beta_{5} + 2 \beta_{3}) q^{5} - \beta_{4} q^{7} + ( - \beta_{6} + \beta_{5} + 5 \beta_{3}) q^{8} + (4 \beta_{2} + \beta_1 + 37) q^{10} + ( - \beta_{7} - 6 \beta_{6} - 6 \beta_{5}) q^{11} + ( - \beta_{4} + 2 \beta_1) q^{13} + (3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{14} + ( - 7 \beta_{2} + 15) q^{16} + ( - 3 \beta_{6} + 3 \beta_{5} - 74 \beta_{3}) q^{17} + (4 \beta_{2} - 188) q^{19} + ( - 5 \beta_{7} + 15 \beta_{6} + \cdots + 69 \beta_{3}) q^{20}+ \cdots + ( - 186 \beta_{6} + \cdots + 1951 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{4} + 280 q^{10} + 148 q^{16} - 1520 q^{19} + 1940 q^{25} + 2968 q^{31} - 12424 q^{34} + 7220 q^{40} + 10088 q^{46} - 8944 q^{49} + 19800 q^{55} + 544 q^{61} - 29188 q^{64} + 1800 q^{70} + 3600 q^{76} - 12632 q^{79} - 3260 q^{85} - 24552 q^{91} + 40928 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 74x^{6} + 1729x^{4} - 2880x^{2} + 32400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{6} + 652\nu^{4} - 21827\nu^{2} + 22320 ) / 8100 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 103\nu^{4} - 128\nu^{2} - 35595 ) / 2025 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\nu^{7} - 4154\nu^{5} + 86929\nu^{3} - 517140\nu ) / 729000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\nu^{6} - 3433\nu^{4} + 79958\nu^{2} - 100080 ) / 4050 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{7} + 2609\nu^{5} - 85009\nu^{3} + 808065\nu ) / 60750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 103\nu^{7} - 8342\nu^{5} + 215167\nu^{3} - 828720\nu ) / 81000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 367\nu^{7} - 25988\nu^{5} + 543913\nu^{3} + 1326420\nu ) / 121500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - 27\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} + 27\beta_{2} + 4\beta _1 + 513 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 37\beta_{7} - 91\beta_{6} - 145\beta_{5} - 837\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 44\beta_{4} + 783\beta_{2} + 304\beta _1 + 14013 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1939\beta_{7} - 6367\beta_{6} - 7501\beta_{5} - 18981\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2138\beta_{4} + 11259\beta_{2} + 15400\beta _1 + 208305 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 87793\beta_{7} - 312379\beta_{6} - 312649\beta_{5} - 6021\beta_{3} ) / 27 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
6.20986 + 1.41421i
6.20986 − 1.41421i
1.56128 − 1.41421i
1.56128 + 1.41421i
−1.56128 − 1.41421i
−1.56128 + 1.41421i
−6.20986 + 1.41421i
−6.20986 − 1.41421i
−6.20986 0 22.5624 −17.2708 − 18.0753i 0 − 12.6252i −40.7516 0 107.250 + 112.245i
44.2 −6.20986 0 22.5624 −17.2708 + 18.0753i 0 12.6252i −40.7516 0 107.250 − 112.245i
44.3 −1.56128 0 −13.5624 23.8583 − 7.46874i 0 82.9374i 46.1553 0 −37.2496 + 11.6608i
44.4 −1.56128 0 −13.5624 23.8583 + 7.46874i 0 − 82.9374i 46.1553 0 −37.2496 − 11.6608i
44.5 1.56128 0 −13.5624 −23.8583 − 7.46874i 0 − 82.9374i −46.1553 0 −37.2496 − 11.6608i
44.6 1.56128 0 −13.5624 −23.8583 + 7.46874i 0 82.9374i −46.1553 0 −37.2496 + 11.6608i
44.7 6.20986 0 22.5624 17.2708 − 18.0753i 0 12.6252i 40.7516 0 107.250 − 112.245i
44.8 6.20986 0 22.5624 17.2708 + 18.0753i 0 − 12.6252i 40.7516 0 107.250 + 112.245i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
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 45.5.d.a ✓ 8
3.b odd 2 1 inner 45.5.d.a ✓ 8
4.b odd 2 1 720.5.c.c 8
5.b even 2 1 inner 45.5.d.a ✓ 8
5.c odd 4 2 225.5.c.e 8
12.b even 2 1 720.5.c.c 8
15.d odd 2 1 inner 45.5.d.a ✓ 8
15.e even 4 2 225.5.c.e 8
20.d odd 2 1 720.5.c.c 8
60.h even 2 1 720.5.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.d.a ✓ 8 1.a even 1 1 trivial
45.5.d.a ✓ 8 3.b odd 2 1 inner
45.5.d.a ✓ 8 5.b even 2 1 inner
45.5.d.a ✓ 8 15.d odd 2 1 inner
225.5.c.e 8 5.c odd 4 2
225.5.c.e 8 15.e even 4 2
720.5.c.c 8 4.b odd 2 1
720.5.c.c 8 12.b even 2 1
720.5.c.c 8 20.d odd 2 1
720.5.c.c 8 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 41 T^{2} + 94)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7038 T^{2} + 1096416)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 66546 T^{2} + 940771584)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 56178 T^{2} + 506818296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 262226 T^{2} + 14258282464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 380 T + 30880)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 262844 T^{2} + 15652242304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 1032605533584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 742 T - 20264)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2298942 T^{2} + 818470158336)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 793764 T^{2} + 153399122244)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 25854296242176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 28238467170304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 34158390206464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 17863065566784)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 136 T - 4385396)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 83064055136256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 14696243610624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 88632165417696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 3158 T - 8313464)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1731344 T^{2} + 259048021504)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 27006709452804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 59408556493536)^{2} \) Copy content Toggle raw display
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