Properties

Label 45.4.b.b
Level $45$
Weight $4$
Character orbit 45.b
Analytic conductor $2.655$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8} + (\beta_{3} + 4 \beta_{2} + 10 \beta_1 + 3) q^{10} + ( - 2 \beta_{3} + 22) q^{11} + ( - 8 \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{3} - 58) q^{14} + ( - \beta_{3} + 13) q^{16} + ( - 6 \beta_{2} - 16 \beta_1) q^{17} + ( - 8 \beta_{3} - 24) q^{19} + ( - 6 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 102) q^{20} + ( - 8 \beta_{2} - 38 \beta_1) q^{22} + ( - 6 \beta_{2} + 2 \beta_1) q^{23} + ( - 6 \beta_{3} - 14 \beta_{2} + 10 \beta_1 + 67) q^{25} + (10 \beta_{3} - 2) q^{26} + (24 \beta_{2} + 42 \beta_1) q^{28} + ( - 14 \beta_{3} - 152) q^{29} + (4 \beta_{3} + 24) q^{31} + (28 \beta_{2} + 19 \beta_1) q^{32} + (22 \beta_{3} - 190) q^{34} + (10 \beta_{3} + 30 \beta_{2} + 30 \beta_1 + 70) q^{35} + (12 \beta_{2} + 66 \beta_1) q^{37} + ( - 32 \beta_{2} - 40 \beta_1) q^{38} + (7 \beta_{3} + 8 \beta_{2} - 70 \beta_1 + 101) q^{40} + ( - 4 \beta_{3} + 206) q^{41} + (14 \beta_{2} - 82 \beta_1) q^{43} + (30 \beta_{3} - 294) q^{44} + (4 \beta_{3} + 44) q^{46} + ( - 38 \beta_{2} + 54 \beta_1) q^{47} + ( - 12 \beta_{3} - 29) q^{49} + (4 \beta_{3} - 24 \beta_{2} - 115 \beta_1 + 172) q^{50} + ( - 24 \beta_{2} + 66 \beta_1) q^{52} + (54 \beta_{2} - 28 \beta_1) q^{53} + (24 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 228) q^{55} + ( - 50 \beta_{3} + 10) q^{56} + ( - 56 \beta_{2} + 40 \beta_1) q^{58} + (2 \beta_{3} - 94) q^{59} + ( - 32 \beta_{3} + 186) q^{61} + (16 \beta_{2} + 8 \beta_1) q^{62} + ( - 55 \beta_{3} + 267) q^{64} + ( - 22 \beta_{3} - 48 \beta_{2} + 60 \beta_1 - 226) q^{65} + (46 \beta_{2} + 106 \beta_1) q^{67} + (40 \beta_{2} + 238 \beta_1) q^{68} + ( - 60 \beta_{3} + 40 \beta_{2} + 10 \beta_1 + 300) q^{70} + (60 \beta_{3} - 12) q^{71} + ( - 84 \beta_{2} - 192 \beta_1) q^{73} + ( - 78 \beta_{3} + 822) q^{74} + (8 \beta_{3} - 616) q^{76} + ( - 24 \beta_{2} - 132 \beta_1) q^{77} + (100 \beta_{3} - 240) q^{79} + (14 \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 118) q^{80} + ( - 16 \beta_{2} - 238 \beta_1) q^{82} + ( - 30 \beta_{2} - 238 \beta_1) q^{83} + ( - 2 \beta_{3} + 22 \beta_{2} + 190 \beta_1 - 126) q^{85} + (68 \beta_{3} - 1108) q^{86} + (56 \beta_{2} + 230 \beta_1) q^{88} + (48 \beta_{3} + 534) q^{89} + (84 \beta_{3} + 444) q^{91} + ( - 32 \beta_{2} + 4 \beta_1) q^{92} + ( - 16 \beta_{3} + 816) q^{94} + ( - 16 \beta_{3} + 96 \beta_{2} - 40 \beta_1 - 688) q^{95} + ( - 4 \beta_{2} - 244 \beta_1) q^{97} + ( - 48 \beta_{2} - 67 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 65\beta_1 ) / 6 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
3.70156i
2.70156i
2.70156i
3.70156i
4.70156i 0 −14.1047 −11.1047 1.29844i 0 16.2094i 28.7016i 0 −6.10469 + 52.2094i
19.2 1.70156i 0 5.10469 8.10469 + 7.70156i 0 22.2094i 22.2984i 0 13.1047 13.7906i
19.3 1.70156i 0 5.10469 8.10469 7.70156i 0 22.2094i 22.2984i 0 13.1047 + 13.7906i
19.4 4.70156i 0 −14.1047 −11.1047 + 1.29844i 0 16.2094i 28.7016i 0 −6.10469 52.2094i
\(n\): 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 45.4.b.b 4
3.b odd 2 1 15.4.b.a 4
4.b odd 2 1 720.4.f.j 4
5.b even 2 1 inner 45.4.b.b 4
5.c odd 4 1 225.4.a.i 2
5.c odd 4 1 225.4.a.o 2
12.b even 2 1 240.4.f.f 4
15.d odd 2 1 15.4.b.a 4
15.e even 4 1 75.4.a.c 2
15.e even 4 1 75.4.a.f 2
20.d odd 2 1 720.4.f.j 4
24.f even 2 1 960.4.f.p 4
24.h odd 2 1 960.4.f.q 4
60.h even 2 1 240.4.f.f 4
60.l odd 4 1 1200.4.a.bn 2
60.l odd 4 1 1200.4.a.bt 2
120.i odd 2 1 960.4.f.q 4
120.m even 2 1 960.4.f.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 3.b odd 2 1
15.4.b.a 4 15.d odd 2 1
45.4.b.b 4 1.a even 1 1 trivial
45.4.b.b 4 5.b even 2 1 inner
75.4.a.c 2 15.e even 4 1
75.4.a.f 2 15.e even 4 1
225.4.a.i 2 5.c odd 4 1
225.4.a.o 2 5.c odd 4 1
240.4.f.f 4 12.b even 2 1
240.4.f.f 4 60.h even 2 1
720.4.f.j 4 4.b odd 2 1
720.4.f.j 4 20.d odd 2 1
960.4.f.p 4 24.f even 2 1
960.4.f.p 4 120.m even 2 1
960.4.f.q 4 24.h odd 2 1
960.4.f.q 4 120.i odd 2 1
1200.4.a.bn 2 60.l odd 4 1
1200.4.a.bt 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} - 110 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 756 T^{2} + 129600 \) Copy content Toggle raw display
$11$ \( (T^{2} - 42 T + 72)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3780 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$17$ \( T^{4} + 7252 T^{2} + \cdots + 2483776 \) Copy content Toggle raw display
$19$ \( (T^{2} + 56 T - 5120)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2464 T^{2} + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 318 T + 7200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 52 T - 800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 106596 T^{2} + \cdots + 41990400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 408 T + 40140)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 196128 T^{2} + \cdots + 8256266496 \) Copy content Toggle raw display
$47$ \( T^{4} + 189712 T^{2} + \cdots + 6186766336 \) Copy content Toggle raw display
$53$ \( T^{4} + 218644 T^{2} + \cdots + 813390400 \) Copy content Toggle raw display
$59$ \( (T^{2} + 186 T + 8280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 340 T - 65564)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 341712 T^{2} + \cdots + 9419867136 \) Copy content Toggle raw display
$71$ \( (T^{2} - 36 T - 331776)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1126224 T^{2} + \cdots + 104976000000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 380 T - 886400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1371040 T^{2} + \cdots + 40558737664 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1116 T + 98820)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1475712 T^{2} + \cdots + 196199387136 \) Copy content Toggle raw display
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