Properties

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

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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: yes
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + 7 \zeta_{12} q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) 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} + 7 \zeta_{12} q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{12} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + 21 \zeta_{12}^{2} q^{14} + ( - 11 \zeta_{12}^{2} + 11) q^{16} + (9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{17} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{18} + (16 \zeta_{12}^{2} + 16) q^{19} + ( - 14 \zeta_{12}^{2} + 7) q^{21} + 15 \zeta_{12} q^{23} + ( - 3 \zeta_{12}^{2} + 6) q^{24} + (6 \zeta_{12}^{2} + 6) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + 35 \zeta_{12}^{3} q^{28} - 6 q^{29} + ( - 13 \zeta_{12}^{2} + 26) q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} + (54 \zeta_{12}^{2} - 27) q^{34} - 15 q^{36} - 70 \zeta_{12} q^{37} + (48 \zeta_{12}^{3} + 48 \zeta_{12}) q^{38} - 6 \zeta_{12}^{2} q^{39} + (42 \zeta_{12}^{2} - 21) q^{41} + ( - 42 \zeta_{12}^{3} + 21 \zeta_{12}) q^{42} - 34 \zeta_{12}^{3} q^{43} + 45 \zeta_{12}^{2} q^{46} + ( - 30 \zeta_{12}^{3} + 15 \zeta_{12}) q^{47} + (11 \zeta_{12}^{3} - 22 \zeta_{12}) q^{48} + 49 \zeta_{12}^{2} q^{49} + ( - 27 \zeta_{12}^{2} + 27) q^{51} + (10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{52} + (42 \zeta_{12}^{3} - 42 \zeta_{12}) q^{53} + (9 \zeta_{12}^{2} + 9) q^{54} + (21 \zeta_{12}^{2} - 21) q^{56} - 48 \zeta_{12}^{3} q^{57} - 18 \zeta_{12} q^{58} + (24 \zeta_{12}^{2} - 48) q^{59} + (42 \zeta_{12}^{2} + 42) q^{61} + ( - 39 \zeta_{12}^{3} + 78 \zeta_{12}) q^{62} + (21 \zeta_{12}^{3} - 21 \zeta_{12}) q^{63} + 91 q^{64} + (94 \zeta_{12}^{3} - 94 \zeta_{12}) q^{67} + (90 \zeta_{12}^{3} - 45 \zeta_{12}) q^{68} + ( - 30 \zeta_{12}^{2} + 15) q^{69} + 9 q^{71} - 9 \zeta_{12} q^{72} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{73} - 210 \zeta_{12}^{2} q^{74} + (160 \zeta_{12}^{2} - 80) q^{76} - 18 \zeta_{12}^{3} q^{78} + ( - 77 \zeta_{12}^{2} + 77) q^{79} - 9 \zeta_{12}^{2} q^{81} + (126 \zeta_{12}^{3} - 63 \zeta_{12}) q^{82} + (84 \zeta_{12}^{3} - 168 \zeta_{12}) q^{83} + ( - 35 \zeta_{12}^{2} + 70) q^{84} + ( - 102 \zeta_{12}^{2} + 102) q^{86} + (6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} + ( - 33 \zeta_{12}^{2} - 33) q^{89} + (14 \zeta_{12}^{2} + 14) q^{91} + 75 \zeta_{12}^{3} q^{92} - 39 \zeta_{12} q^{93} + ( - 45 \zeta_{12}^{2} + 90) q^{94} + ( - 45 \zeta_{12}^{2} - 45) q^{96} + (57 \zeta_{12}^{3} - 114 \zeta_{12}) q^{97} + 147 \zeta_{12}^{3} q^{98} +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 \( 4 q + 10 q^{4} - 6 q^{9} + 42 q^{14} + 22 q^{16} + 96 q^{19} + 18 q^{24} + 36 q^{26} - 24 q^{29} + 78 q^{31} - 60 q^{36} - 12 q^{39} + 90 q^{46} + 98 q^{49} + 54 q^{51} + 54 q^{54} - 42 q^{56} - 144 q^{59} + 252 q^{61} + 364 q^{64} + 36 q^{71} - 420 q^{74} + 154 q^{79} - 18 q^{81} + 210 q^{84} + 204 q^{86} - 198 q^{89} + 84 q^{91} + 270 q^{94} - 270 q^{96}+O(q^{100}) \) 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 −6.06218 3.50000i 3.00000i −1.50000 + 2.59808i 0
124.2 2.59808 + 1.50000i −0.866025 1.50000i 2.50000 + 4.33013i 0 5.19615i 6.06218 + 3.50000i 3.00000i −1.50000 + 2.59808i 0
199.1 −2.59808 + 1.50000i 0.866025 1.50000i 2.50000 4.33013i 0 5.19615i −6.06218 + 3.50000i 3.00000i −1.50000 2.59808i 0
199.2 2.59808 1.50000i −0.866025 + 1.50000i 2.50000 4.33013i 0 5.19615i 6.06218 3.50000i 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.f 4
5.b even 2 1 inner 525.3.s.f 4
5.c odd 4 1 525.3.o.a 2
5.c odd 4 1 525.3.o.i yes 2
7.d odd 6 1 inner 525.3.s.f 4
35.i odd 6 1 inner 525.3.s.f 4
35.k even 12 1 525.3.o.a 2
35.k even 12 1 525.3.o.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.a 2 5.c odd 4 1
525.3.o.a 2 35.k even 12 1
525.3.o.i yes 2 5.c odd 4 1
525.3.o.i yes 2 35.k even 12 1
525.3.s.f 4 1.a even 1 1 trivial
525.3.s.f 4 5.b even 2 1 inner
525.3.s.f 4 7.d odd 6 1 inner
525.3.s.f 4 35.i odd 6 1 inner

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
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 12 \) 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
$7$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 243 T^{2} + 59049 \) Copy content Toggle raw display
$19$ \( (T^{2} - 48 T + 768)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 39 T + 507)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4900 T^{2} + \cdots + 24010000 \) Copy content Toggle raw display
$41$ \( (T^{2} + 1323)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 675 T^{2} + 455625 \) Copy content Toggle raw display
$53$ \( T^{4} - 1764 T^{2} + \cdots + 3111696 \) Copy content Toggle raw display
$59$ \( (T^{2} + 72 T + 1728)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 126 T + 5292)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 8836 T^{2} + \cdots + 78074896 \) Copy content Toggle raw display
$71$ \( (T - 9)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$79$ \( (T^{2} - 77 T + 5929)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 21168)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 99 T + 3267)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 9747)^{2} \) Copy content Toggle raw display
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