Properties

Label 525.3.s.b
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} - 4 \zeta_{12}^{2} q^{11} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{12} + (10 \zeta_{12}^{3} - 20 \zeta_{12}) q^{13} + ( - 3 \zeta_{12}^{2} + 8) q^{14} + (5 \zeta_{12}^{2} - 5) q^{16} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + ( - 4 \zeta_{12}^{2} - 4) q^{19} + (2 \zeta_{12}^{2} + 11) q^{21} - 4 \zeta_{12}^{3} q^{22} - 31 \zeta_{12} q^{23} + ( - 7 \zeta_{12}^{2} + 14) q^{24} + ( - 10 \zeta_{12}^{2} - 10) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (9 \zeta_{12}^{3} - 24 \zeta_{12}) q^{28} - 10 q^{29} + ( - 17 \zeta_{12}^{2} + 34) q^{31} + (33 \zeta_{12}^{3} - 33 \zeta_{12}) q^{32} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{33} + ( - 10 \zeta_{12}^{2} + 5) q^{34} + 9 q^{36} + 50 \zeta_{12} q^{37} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{38} - 30 \zeta_{12}^{2} q^{39} + ( - 62 \zeta_{12}^{2} + 31) q^{41} + (2 \zeta_{12}^{3} + 11 \zeta_{12}) q^{42} - 34 \zeta_{12}^{3} q^{43} + (12 \zeta_{12}^{2} - 12) q^{44} - 31 \zeta_{12}^{2} q^{46} + ( - 50 \zeta_{12}^{3} + 25 \zeta_{12}) q^{47} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{48} + ( - 55 \zeta_{12}^{2} + 16) q^{49} + ( - 15 \zeta_{12}^{2} + 15) q^{51} + (30 \zeta_{12}^{3} + 30 \zeta_{12}) q^{52} + (50 \zeta_{12}^{3} - 50 \zeta_{12}) q^{53} + ( - 3 \zeta_{12}^{2} - 3) q^{54} + ( - 35 \zeta_{12}^{2} - 21) q^{56} - 12 \zeta_{12}^{3} q^{57} - 10 \zeta_{12} q^{58} + ( - 32 \zeta_{12}^{2} + 64) q^{59} + (14 \zeta_{12}^{2} + 14) q^{61} + ( - 17 \zeta_{12}^{3} + 34 \zeta_{12}) q^{62} + (15 \zeta_{12}^{3} + 9 \zeta_{12}) q^{63} - 13 q^{64} + ( - 4 \zeta_{12}^{2} + 8) q^{66} + (50 \zeta_{12}^{3} - 50 \zeta_{12}) q^{67} + (30 \zeta_{12}^{3} - 15 \zeta_{12}) q^{68} + ( - 62 \zeta_{12}^{2} + 31) q^{69} + 97 q^{71} + 21 \zeta_{12} q^{72} + (28 \zeta_{12}^{3} + 28 \zeta_{12}) q^{73} + 50 \zeta_{12}^{2} q^{74} + (24 \zeta_{12}^{2} - 12) q^{76} + (12 \zeta_{12}^{3} - 32 \zeta_{12}) q^{77} - 30 \zeta_{12}^{3} q^{78} + (7 \zeta_{12}^{2} - 7) q^{79} - 9 \zeta_{12}^{2} q^{81} + ( - 62 \zeta_{12}^{3} + 31 \zeta_{12}) q^{82} + (88 \zeta_{12}^{3} - 176 \zeta_{12}) q^{83} + ( - 39 \zeta_{12}^{2} + 6) q^{84} + ( - 34 \zeta_{12}^{2} + 34) q^{86} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{87} + (28 \zeta_{12}^{3} - 28 \zeta_{12}) q^{88} + (91 \zeta_{12}^{2} + 91) q^{89} + (110 \zeta_{12}^{2} - 130) q^{91} + 93 \zeta_{12}^{3} q^{92} + 51 \zeta_{12} q^{93} + ( - 25 \zeta_{12}^{2} + 50) q^{94} + ( - 33 \zeta_{12}^{2} - 33) q^{96} + ( - 65 \zeta_{12}^{3} + 130 \zeta_{12}) q^{97} + ( - 55 \zeta_{12}^{3} + 16 \zeta_{12}) q^{98} + 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 6 q^{9} - 8 q^{11} + 26 q^{14} - 10 q^{16} - 24 q^{19} + 48 q^{21} + 42 q^{24} - 60 q^{26} - 40 q^{29} + 102 q^{31} + 36 q^{36} - 60 q^{39} - 24 q^{44} - 62 q^{46} - 46 q^{49} + 30 q^{51} - 18 q^{54} - 154 q^{56} + 192 q^{59} + 84 q^{61} - 52 q^{64} + 24 q^{66} + 388 q^{71} + 100 q^{74} - 14 q^{79} - 18 q^{81} - 54 q^{84} + 68 q^{86} + 546 q^{89} - 300 q^{91} + 150 q^{94} - 198 q^{96} + 48 q^{99}+O(q^{100}) \) 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
−0.866025 0.500000i −0.866025 1.50000i −1.50000 2.59808i 0 1.73205i −4.33013 + 5.50000i 7.00000i −1.50000 + 2.59808i 0
124.2 0.866025 + 0.500000i 0.866025 + 1.50000i −1.50000 2.59808i 0 1.73205i 4.33013 5.50000i 7.00000i −1.50000 + 2.59808i 0
199.1 −0.866025 + 0.500000i −0.866025 + 1.50000i −1.50000 + 2.59808i 0 1.73205i −4.33013 5.50000i 7.00000i −1.50000 2.59808i 0
199.2 0.866025 0.500000i 0.866025 1.50000i −1.50000 + 2.59808i 0 1.73205i 4.33013 + 5.50000i 7.00000i −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.b 4
5.b even 2 1 inner 525.3.s.b 4
5.c odd 4 1 525.3.o.d 2
5.c odd 4 1 525.3.o.e yes 2
7.d odd 6 1 inner 525.3.s.b 4
35.i odd 6 1 inner 525.3.s.b 4
35.k even 12 1 525.3.o.d 2
35.k even 12 1 525.3.o.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.d 2 5.c odd 4 1
525.3.o.d 2 35.k even 12 1
525.3.o.e yes 2 5.c odd 4 1
525.3.o.e yes 2 35.k even 12 1
525.3.s.b 4 1.a even 1 1 trivial
525.3.s.b 4 5.b even 2 1 inner
525.3.s.b 4 7.d odd 6 1 inner
525.3.s.b 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 \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 300 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$19$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 961 T^{2} + 923521 \) Copy content Toggle raw display
$29$ \( (T + 10)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 51 T + 867)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 2500 T^{2} + \cdots + 6250000 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2883)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1875 T^{2} + \cdots + 3515625 \) Copy content Toggle raw display
$53$ \( T^{4} - 2500 T^{2} + \cdots + 6250000 \) Copy content Toggle raw display
$59$ \( (T^{2} - 96 T + 3072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 42 T + 588)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 2500 T^{2} + \cdots + 6250000 \) Copy content Toggle raw display
$71$ \( (T - 97)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2352 T^{2} + \cdots + 5531904 \) Copy content Toggle raw display
$79$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 23232)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 273 T + 24843)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12675)^{2} \) Copy content Toggle raw display
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