Properties

Label 45.3.g.a
Level $45$
Weight $3$
Character orbit 45.g
Analytic conductor $1.226$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.22616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
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

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{5} + ( -5 - 5 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{5} + ( -5 - 5 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} + ( -5 + 10 \beta_{2} ) q^{10} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{11} + ( 10 - 10 \beta_{2} ) q^{13} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{14} + 19 q^{16} -2 \beta_{1} q^{17} + 18 \beta_{2} q^{19} + ( -\beta_{1} + 2 \beta_{3} ) q^{20} + ( -25 - 25 \beta_{2} ) q^{22} + 2 \beta_{3} q^{23} + ( -20 + 15 \beta_{2} ) q^{25} + ( 10 \beta_{1} - 10 \beta_{3} ) q^{26} + ( 5 - 5 \beta_{2} ) q^{28} + ( 15 \beta_{1} + 15 \beta_{3} ) q^{29} + 8 q^{31} + 7 \beta_{1} q^{32} -10 \beta_{2} q^{34} + ( -5 \beta_{1} - 15 \beta_{3} ) q^{35} + ( 10 + 10 \beta_{2} ) q^{37} + 18 \beta_{3} q^{38} + ( 30 + 15 \beta_{2} ) q^{40} + ( 10 \beta_{1} - 10 \beta_{3} ) q^{41} + ( 10 - 10 \beta_{2} ) q^{43} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{44} -10 q^{46} -26 \beta_{1} q^{47} + \beta_{2} q^{49} + ( -20 \beta_{1} + 15 \beta_{3} ) q^{50} + ( 10 + 10 \beta_{2} ) q^{52} -16 \beta_{3} q^{53} + ( -25 - 75 \beta_{2} ) q^{55} + ( -15 \beta_{1} + 15 \beta_{3} ) q^{56} + ( -75 + 75 \beta_{2} ) q^{58} + ( -15 \beta_{1} - 15 \beta_{3} ) q^{59} -58 q^{61} + 8 \beta_{1} q^{62} -41 \beta_{2} q^{64} + ( 30 \beta_{1} - 10 \beta_{3} ) q^{65} + ( 70 + 70 \beta_{2} ) q^{67} -2 \beta_{3} q^{68} + ( 75 - 25 \beta_{2} ) q^{70} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{71} + ( 55 - 55 \beta_{2} ) q^{73} + ( 10 \beta_{1} + 10 \beta_{3} ) q^{74} -18 q^{76} + 50 \beta_{1} q^{77} + 12 \beta_{2} q^{79} + ( 38 \beta_{1} + 19 \beta_{3} ) q^{80} + ( 50 + 50 \beta_{2} ) q^{82} -34 \beta_{3} q^{83} + ( 10 - 20 \beta_{2} ) q^{85} + ( 10 \beta_{1} - 10 \beta_{3} ) q^{86} + ( -75 + 75 \beta_{2} ) q^{88} -100 q^{91} -2 \beta_{1} q^{92} -130 \beta_{2} q^{94} + ( -18 \beta_{1} + 36 \beta_{3} ) q^{95} + ( -5 - 5 \beta_{2} ) q^{97} + \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 20q^{7} + O(q^{10}) \) \( 4q - 20q^{7} - 20q^{10} + 40q^{13} + 76q^{16} - 100q^{22} - 80q^{25} + 20q^{28} + 32q^{31} + 40q^{37} + 120q^{40} + 40q^{43} - 40q^{46} + 40q^{52} - 100q^{55} - 300q^{58} - 232q^{61} + 280q^{67} + 300q^{70} + 220q^{73} - 72q^{76} + 200q^{82} + 40q^{85} - 300q^{88} - 400q^{91} - 20q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
−1.58114 + 1.58114i 0 1.00000i −1.58114 + 4.74342i 0 −5.00000 + 5.00000i −4.74342 4.74342i 0 −5.00000 10.0000i
28.2 1.58114 1.58114i 0 1.00000i 1.58114 4.74342i 0 −5.00000 + 5.00000i 4.74342 + 4.74342i 0 −5.00000 10.0000i
37.1 −1.58114 1.58114i 0 1.00000i −1.58114 4.74342i 0 −5.00000 5.00000i −4.74342 + 4.74342i 0 −5.00000 + 10.0000i
37.2 1.58114 + 1.58114i 0 1.00000i 1.58114 + 4.74342i 0 −5.00000 5.00000i 4.74342 4.74342i 0 −5.00000 + 10.0000i
\(n\): 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
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.3.g.a 4
3.b odd 2 1 inner 45.3.g.a 4
4.b odd 2 1 720.3.bh.j 4
5.b even 2 1 225.3.g.g 4
5.c odd 4 1 inner 45.3.g.a 4
5.c odd 4 1 225.3.g.g 4
9.c even 3 2 405.3.l.g 8
9.d odd 6 2 405.3.l.g 8
12.b even 2 1 720.3.bh.j 4
15.d odd 2 1 225.3.g.g 4
15.e even 4 1 inner 45.3.g.a 4
15.e even 4 1 225.3.g.g 4
20.e even 4 1 720.3.bh.j 4
45.k odd 12 2 405.3.l.g 8
45.l even 12 2 405.3.l.g 8
60.l odd 4 1 720.3.bh.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.g.a 4 1.a even 1 1 trivial
45.3.g.a 4 3.b odd 2 1 inner
45.3.g.a 4 5.c odd 4 1 inner
45.3.g.a 4 15.e even 4 1 inner
225.3.g.g 4 5.b even 2 1
225.3.g.g 4 5.c odd 4 1
225.3.g.g 4 15.d odd 2 1
225.3.g.g 4 15.e even 4 1
405.3.l.g 8 9.c even 3 2
405.3.l.g 8 9.d odd 6 2
405.3.l.g 8 45.k odd 12 2
405.3.l.g 8 45.l even 12 2
720.3.bh.j 4 4.b odd 2 1
720.3.bh.j 4 12.b even 2 1
720.3.bh.j 4 20.e even 4 1
720.3.bh.j 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25 \) acting on \(S_{3}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 625 + 40 T^{2} + T^{4} \)
$7$ \( ( 50 + 10 T + T^{2} )^{2} \)
$11$ \( ( -250 + T^{2} )^{2} \)
$13$ \( ( 200 - 20 T + T^{2} )^{2} \)
$17$ \( 400 + T^{4} \)
$19$ \( ( 324 + T^{2} )^{2} \)
$23$ \( 400 + T^{4} \)
$29$ \( ( 2250 + T^{2} )^{2} \)
$31$ \( ( -8 + T )^{4} \)
$37$ \( ( 200 - 20 T + T^{2} )^{2} \)
$41$ \( ( -1000 + T^{2} )^{2} \)
$43$ \( ( 200 - 20 T + T^{2} )^{2} \)
$47$ \( 11424400 + T^{4} \)
$53$ \( 1638400 + T^{4} \)
$59$ \( ( 2250 + T^{2} )^{2} \)
$61$ \( ( 58 + T )^{4} \)
$67$ \( ( 9800 - 140 T + T^{2} )^{2} \)
$71$ \( ( -4000 + T^{2} )^{2} \)
$73$ \( ( 6050 - 110 T + T^{2} )^{2} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( 33408400 + T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 50 + 10 T + T^{2} )^{2} \)
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