Properties

Label 525.3.s.e
Level $525$
Weight $3$
Character orbit 525.s
Analytic conductor $14.305$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} - 15 \zeta_{12}^{2} q^{11} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{12} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{13} + ( - 24 \zeta_{12}^{2} + 15) q^{14} + ( - 11 \zeta_{12}^{2} + 11) q^{16} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{17} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{18} + (6 \zeta_{12}^{2} + 6) q^{19} + (11 \zeta_{12}^{2} - 13) q^{21} - 45 \zeta_{12}^{3} q^{22} + ( - 3 \zeta_{12}^{2} + 6) q^{24} + ( - 24 \zeta_{12}^{2} - 24) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 40 \zeta_{12}^{3} + 25 \zeta_{12}) q^{28} + 9 q^{29} + (7 \zeta_{12}^{2} - 14) q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} + (30 \zeta_{12}^{3} - 15 \zeta_{12}) q^{33} + ( - 36 \zeta_{12}^{2} + 18) q^{34} - 15 q^{36} + 10 \zeta_{12} q^{37} + (18 \zeta_{12}^{3} + 18 \zeta_{12}) q^{38} + 24 \zeta_{12}^{2} q^{39} + (12 \zeta_{12}^{2} - 6) q^{41} + (33 \zeta_{12}^{3} - 39 \zeta_{12}) q^{42} - 74 \zeta_{12}^{3} q^{43} + ( - 75 \zeta_{12}^{2} + 75) q^{44} + (11 \zeta_{12}^{3} - 22 \zeta_{12}) q^{48} + (39 \zeta_{12}^{2} - 55) q^{49} + (18 \zeta_{12}^{2} - 18) q^{51} + ( - 40 \zeta_{12}^{3} - 40 \zeta_{12}) q^{52} + ( - 33 \zeta_{12}^{3} + 33 \zeta_{12}) q^{53} + (9 \zeta_{12}^{2} + 9) q^{54} + ( - 9 \zeta_{12}^{2} + 24) q^{56} - 18 \zeta_{12}^{3} q^{57} + 27 \zeta_{12} q^{58} + (9 \zeta_{12}^{2} - 18) q^{59} + (52 \zeta_{12}^{2} + 52) q^{61} + (21 \zeta_{12}^{3} - 42 \zeta_{12}) q^{62} + ( - 9 \zeta_{12}^{3} + 24 \zeta_{12}) q^{63} + 91 q^{64} + (45 \zeta_{12}^{2} - 90) q^{66} + ( - 76 \zeta_{12}^{3} + 76 \zeta_{12}) q^{67} + ( - 60 \zeta_{12}^{3} + 30 \zeta_{12}) q^{68} + 84 q^{71} - 9 \zeta_{12} q^{72} + ( - 36 \zeta_{12}^{3} - 36 \zeta_{12}) q^{73} + 30 \zeta_{12}^{2} q^{74} + (60 \zeta_{12}^{2} - 30) q^{76} + (120 \zeta_{12}^{3} - 75 \zeta_{12}) q^{77} + 72 \zeta_{12}^{3} q^{78} + (43 \zeta_{12}^{2} - 43) q^{79} - 9 \zeta_{12}^{2} q^{81} + (36 \zeta_{12}^{3} - 18 \zeta_{12}) q^{82} + (69 \zeta_{12}^{3} - 138 \zeta_{12}) q^{83} + ( - 10 \zeta_{12}^{2} - 55) q^{84} + ( - 222 \zeta_{12}^{2} + 222) q^{86} + ( - 9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{87} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{88} + (42 \zeta_{12}^{2} + 42) q^{89} + (104 \zeta_{12}^{2} - 16) q^{91} + 21 \zeta_{12} q^{93} + ( - 45 \zeta_{12}^{2} - 45) q^{96} + (107 \zeta_{12}^{3} - 214 \zeta_{12}) q^{97} + (117 \zeta_{12}^{3} - 165 \zeta_{12}) q^{98} + 45 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} - 6 q^{9} - 30 q^{11} + 12 q^{14} + 22 q^{16} + 36 q^{19} - 30 q^{21} + 18 q^{24} - 144 q^{26} + 36 q^{29} - 42 q^{31} - 60 q^{36} + 48 q^{39} + 150 q^{44} - 142 q^{49} - 36 q^{51} + 54 q^{54} + 78 q^{56} - 54 q^{59} + 312 q^{61} + 364 q^{64} - 270 q^{66} + 336 q^{71} + 60 q^{74} - 86 q^{79} - 18 q^{81} - 240 q^{84} + 444 q^{86} + 252 q^{89} + 144 q^{91} - 270 q^{96} + 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−2.59808 1.50000i 0.866025 + 1.50000i 2.50000 + 4.33013i 0 5.19615i 2.59808 + 6.50000i 3.00000i −1.50000 + 2.59808i 0
124.2 2.59808 + 1.50000i −0.866025 1.50000i 2.50000 + 4.33013i 0 5.19615i −2.59808 6.50000i 3.00000i −1.50000 + 2.59808i 0
199.1 −2.59808 + 1.50000i 0.866025 1.50000i 2.50000 4.33013i 0 5.19615i 2.59808 6.50000i 3.00000i −1.50000 2.59808i 0
199.2 2.59808 1.50000i −0.866025 + 1.50000i 2.50000 4.33013i 0 5.19615i −2.59808 + 6.50000i 3.00000i −1.50000 2.59808i 0
\(n\): 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
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.e 4
5.b even 2 1 inner 525.3.s.e 4
5.c odd 4 1 21.3.f.a 2
5.c odd 4 1 525.3.o.h 2
7.d odd 6 1 inner 525.3.s.e 4
15.e even 4 1 63.3.m.d 2
20.e even 4 1 336.3.bh.d 2
35.f even 4 1 147.3.f.a 2
35.i odd 6 1 inner 525.3.s.e 4
35.k even 12 1 21.3.f.a 2
35.k even 12 1 147.3.d.c 2
35.k even 12 1 525.3.o.h 2
35.l odd 12 1 147.3.d.c 2
35.l odd 12 1 147.3.f.a 2
60.l odd 4 1 1008.3.cg.a 2
105.k odd 4 1 441.3.m.g 2
105.w odd 12 1 63.3.m.d 2
105.w odd 12 1 441.3.d.a 2
105.x even 12 1 441.3.d.a 2
105.x even 12 1 441.3.m.g 2
140.w even 12 1 2352.3.f.a 2
140.x odd 12 1 336.3.bh.d 2
140.x odd 12 1 2352.3.f.a 2
420.br even 12 1 1008.3.cg.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 5.c odd 4 1
21.3.f.a 2 35.k even 12 1
63.3.m.d 2 15.e even 4 1
63.3.m.d 2 105.w odd 12 1
147.3.d.c 2 35.k even 12 1
147.3.d.c 2 35.l odd 12 1
147.3.f.a 2 35.f even 4 1
147.3.f.a 2 35.l odd 12 1
336.3.bh.d 2 20.e even 4 1
336.3.bh.d 2 140.x odd 12 1
441.3.d.a 2 105.w odd 12 1
441.3.d.a 2 105.x even 12 1
441.3.m.g 2 105.k odd 4 1
441.3.m.g 2 105.x even 12 1
525.3.o.h 2 5.c odd 4 1
525.3.o.h 2 35.k even 12 1
525.3.s.e 4 1.a even 1 1 trivial
525.3.s.e 4 5.b even 2 1 inner
525.3.s.e 4 7.d odd 6 1 inner
525.3.s.e 4 35.i odd 6 1 inner
1008.3.cg.a 2 60.l odd 4 1
1008.3.cg.a 2 420.br even 12 1
2352.3.f.a 2 140.w even 12 1
2352.3.f.a 2 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - 9T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{11}^{2} + 15T_{11} + 225 \) Copy content Toggle raw display
\( T_{13}^{2} - 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 71T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$19$ \( (T^{2} - 18 T + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 5476)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 1089 T^{2} + \cdots + 1185921 \) Copy content Toggle raw display
$59$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 156 T + 8112)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 5776 T^{2} + \cdots + 33362176 \) Copy content Toggle raw display
$71$ \( (T - 84)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 3888 T^{2} + \cdots + 15116544 \) Copy content Toggle raw display
$79$ \( (T^{2} + 43 T + 1849)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14283)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 126 T + 5292)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 34347)^{2} \) Copy content Toggle raw display
show more
show less