Properties

Label 15.4.b.a
Level $15$
Weight $4$
Character orbit 15.b
Analytic conductor $0.885$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 15.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.885028650086\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 5) q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{5} + ( - \beta_{3} + 5) q^{6} + ( - 4 \beta_{2} + 6 \beta_1) q^{7} + (8 \beta_{2} + \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 5) q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{5} + ( - \beta_{3} + 5) q^{6} + ( - 4 \beta_{2} + 6 \beta_1) q^{7} + (8 \beta_{2} + \beta_1) q^{8} - 9 q^{9} + (\beta_{3} - 8 \beta_{2} - 6 \beta_1 + 3) q^{10} + (2 \beta_{3} - 22) q^{11} - 9 \beta_1 q^{12} + (16 \beta_{2} - 6 \beta_1) q^{13} + ( - 2 \beta_{3} + 58) q^{14} + (\beta_{3} - 3 \beta_{2} + 9 \beta_1 + 13) q^{15} + ( - \beta_{3} + 13) q^{16} + ( - 12 \beta_{2} - 10 \beta_1) q^{17} + 9 \beta_1 q^{18} + ( - 8 \beta_{3} - 24) q^{19} + (6 \beta_{3} - 8 \beta_{2} + 9 \beta_1 - 102) q^{20} + (6 \beta_{3} + 6) q^{21} + (16 \beta_{2} + 30 \beta_1) q^{22} + ( - 12 \beta_{2} + 8 \beta_1) q^{23} + (\beta_{3} - 77) q^{24} + ( - 6 \beta_{3} + 28 \beta_{2} - 24 \beta_1 + 67) q^{25} + ( - 10 \beta_{3} + 2) q^{26} - 9 \beta_{2} q^{27} + ( - 48 \beta_{2} - 18 \beta_1) q^{28} + (14 \beta_{3} + 152) q^{29} + ( - 6 \beta_{3} + 8 \beta_{2} - 9 \beta_1 + 102) q^{30} + (4 \beta_{3} + 24) q^{31} + (56 \beta_{2} - 9 \beta_1) q^{32} + ( - 12 \beta_{2} - 18 \beta_1) q^{33} + (22 \beta_{3} - 190) q^{34} + ( - 10 \beta_{3} + 60 \beta_{2} - 70) q^{35} + ( - 9 \beta_{3} + 45) q^{36} + ( - 24 \beta_{2} - 54 \beta_1) q^{37} + ( - 64 \beta_{2} - 8 \beta_1) q^{38} + ( - 6 \beta_{3} - 114) q^{39} + (7 \beta_{3} - 16 \beta_{2} + 78 \beta_1 + 101) q^{40} + (4 \beta_{3} - 206) q^{41} + (48 \beta_{2} + 18 \beta_1) q^{42} + ( - 28 \beta_{2} + 96 \beta_1) q^{43} + ( - 30 \beta_{3} + 294) q^{44} + (9 \beta_{3} + 18 \beta_{2} - 9 \beta_1 - 18) q^{45} + (4 \beta_{3} + 44) q^{46} + ( - 76 \beta_{2} + 92 \beta_1) q^{47} + (8 \beta_{2} + 9 \beta_1) q^{48} + ( - 12 \beta_{3} - 29) q^{49} + ( - 4 \beta_{3} - 48 \beta_{2} - 91 \beta_1 - 172) q^{50} + ( - 10 \beta_{3} + 158) q^{51} + (48 \beta_{2} - 90 \beta_1) q^{52} + (108 \beta_{2} - 82 \beta_1) q^{53} + (9 \beta_{3} - 45) q^{54} + (24 \beta_{3} + 8 \beta_{2} + 6 \beta_1 - 228) q^{55} + (50 \beta_{3} - 10) q^{56} + ( - 64 \beta_{2} + 72 \beta_1) q^{57} + (112 \beta_{2} - 96 \beta_1) q^{58} + ( - 2 \beta_{3} + 94) q^{59} + (9 \beta_{3} - 72 \beta_{2} - 54 \beta_1 + 27) q^{60} + ( - 32 \beta_{3} + 186) q^{61} + (32 \beta_{2} - 8 \beta_1) q^{62} + (36 \beta_{2} - 54 \beta_1) q^{63} + ( - 55 \beta_{3} + 267) q^{64} + (22 \beta_{3} - 96 \beta_{2} + 108 \beta_1 + 226) q^{65} + (30 \beta_{3} - 294) q^{66} + ( - 92 \beta_{2} - 60 \beta_1) q^{67} + (80 \beta_{2} + 198 \beta_1) q^{68} + (8 \beta_{3} + 68) q^{69} + ( - 60 \beta_{3} - 80 \beta_{2} + 30 \beta_1 + 300) q^{70} + ( - 60 \beta_{3} + 12) q^{71} + ( - 72 \beta_{2} - 9 \beta_1) q^{72} + (168 \beta_{2} + 108 \beta_1) q^{73} + (78 \beta_{3} - 822) q^{74} + ( - 24 \beta_{3} + 37 \beta_{2} + 54 \beta_1 - 132) q^{75} + (8 \beta_{3} - 616) q^{76} + ( - 48 \beta_{2} - 108 \beta_1) q^{77} + ( - 48 \beta_{2} + 90 \beta_1) q^{78} + (100 \beta_{3} - 240) q^{79} + ( - 14 \beta_{3} - 8 \beta_{2} - \beta_1 + 118) q^{80} + 81 q^{81} + (32 \beta_{2} + 222 \beta_1) q^{82} + ( - 60 \beta_{2} - 208 \beta_1) q^{83} + ( - 18 \beta_{3} + 522) q^{84} + ( - 2 \beta_{3} - 44 \beta_{2} - 168 \beta_1 - 126) q^{85} + ( - 68 \beta_{3} + 1108) q^{86} + (222 \beta_{2} - 126 \beta_1) q^{87} + ( - 112 \beta_{2} - 174 \beta_1) q^{88} + ( - 48 \beta_{3} - 534) q^{89} + ( - 9 \beta_{3} + 72 \beta_{2} + 54 \beta_1 - 27) q^{90} + (84 \beta_{3} + 444) q^{91} + ( - 64 \beta_{2} + 36 \beta_1) q^{92} + (44 \beta_{2} - 36 \beta_1) q^{93} + ( - 16 \beta_{3} + 816) q^{94} + (16 \beta_{3} + 192 \beta_{2} - 136 \beta_1 + 688) q^{95} + ( - 9 \beta_{3} - 459) q^{96} + (8 \beta_{2} + 240 \beta_1) q^{97} + ( - 96 \beta_{2} - 19 \beta_1) q^{98} + ( - 18 \beta_{3} + 198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9} + 14 q^{10} - 84 q^{11} + 228 q^{14} + 54 q^{15} + 50 q^{16} - 112 q^{19} - 396 q^{20} + 36 q^{21} - 306 q^{24} + 256 q^{25} - 12 q^{26} + 636 q^{29} + 396 q^{30} + 104 q^{31} - 716 q^{34} - 300 q^{35} + 162 q^{36} - 468 q^{39} + 418 q^{40} - 816 q^{41} + 1116 q^{44} - 54 q^{45} + 184 q^{46} - 140 q^{49} - 696 q^{50} + 612 q^{51} - 162 q^{54} - 864 q^{55} + 60 q^{56} + 372 q^{59} + 126 q^{60} + 680 q^{61} + 958 q^{64} + 948 q^{65} - 1116 q^{66} + 288 q^{69} + 1080 q^{70} - 72 q^{71} - 3132 q^{74} - 576 q^{75} - 2448 q^{76} - 760 q^{79} + 444 q^{80} + 324 q^{81} + 2052 q^{84} - 508 q^{85} + 4296 q^{86} - 2232 q^{89} - 126 q^{90} + 1944 q^{91} + 3232 q^{94} + 2784 q^{95} - 1854 q^{96} + 756 q^{99}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 33\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 33\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
3.70156i
2.70156i
2.70156i
3.70156i
4.70156i 3.00000i −14.1047 11.1047 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 52.2094i
4.2 1.70156i 3.00000i 5.10469 −8.10469 + 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 + 13.7906i
4.3 1.70156i 3.00000i 5.10469 −8.10469 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 13.7906i
4.4 4.70156i 3.00000i −14.1047 11.1047 + 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −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 15.4.b.a 4
3.b odd 2 1 45.4.b.b 4
4.b odd 2 1 240.4.f.f 4
5.b even 2 1 inner 15.4.b.a 4
5.c odd 4 1 75.4.a.c 2
5.c odd 4 1 75.4.a.f 2
8.b even 2 1 960.4.f.q 4
8.d odd 2 1 960.4.f.p 4
12.b even 2 1 720.4.f.j 4
15.d odd 2 1 45.4.b.b 4
15.e even 4 1 225.4.a.i 2
15.e even 4 1 225.4.a.o 2
20.d odd 2 1 240.4.f.f 4
20.e even 4 1 1200.4.a.bn 2
20.e even 4 1 1200.4.a.bt 2
40.e odd 2 1 960.4.f.p 4
40.f even 2 1 960.4.f.q 4
60.h even 2 1 720.4.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 1.a even 1 1 trivial
15.4.b.a 4 5.b even 2 1 inner
45.4.b.b 4 3.b odd 2 1
45.4.b.b 4 15.d odd 2 1
75.4.a.c 2 5.c odd 4 1
75.4.a.f 2 5.c odd 4 1
225.4.a.i 2 15.e even 4 1
225.4.a.o 2 15.e even 4 1
240.4.f.f 4 4.b odd 2 1
240.4.f.f 4 20.d odd 2 1
720.4.f.j 4 12.b even 2 1
720.4.f.j 4 60.h even 2 1
960.4.f.p 4 8.d odd 2 1
960.4.f.p 4 40.e odd 2 1
960.4.f.q 4 8.b even 2 1
960.4.f.q 4 40.f even 2 1
1200.4.a.bn 2 20.e even 4 1
1200.4.a.bt 2 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) 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
show more
show less