Properties

Label 525.3.s.g
Level $525$
Weight $3$
Character orbit 525.s
Analytic conductor $14.305$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{5} + 3 \beta_{2}) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 1) q^{6} + (5 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + 11 \beta_{3}) q^{8} + (3 \beta_{2} - 3) q^{9} + (2 \beta_{5} + 2 \beta_{2}) q^{11} + ( - 4 \beta_{7} + 2 \beta_{4} + 6 \beta_{3} - 3 \beta_1) q^{12} + ( - 2 \beta_{7} + 4 \beta_{4} + 5 \beta_{3} - 10 \beta_1) q^{13} + (5 \beta_{6} - 8 \beta_{5} + 8 \beta_{2} - 5) q^{14} + (4 \beta_{6} - 4 \beta_{5} + 5 \beta_{2} - 5) q^{16} + ( - 2 \beta_{7} - 2 \beta_{4} - 6 \beta_{3} - 6 \beta_1) q^{17} + ( - 3 \beta_{7} + 3 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{18} + ( - 6 \beta_{6} - 6 \beta_{5} - 5 \beta_{2} - 5) q^{19} + (11 \beta_{2} - 13) q^{21} - 10 \beta_{3} q^{22} + 20 \beta_1 q^{23} + (2 \beta_{6} - \beta_{5} + 11 \beta_{2} - 22) q^{24} + (7 \beta_{6} + 7 \beta_{5} - 17 \beta_{2} - 17) q^{26} + (3 \beta_{3} - 6 \beta_1) q^{27} + ( - 16 \beta_{7} + 10 \beta_{4} + 24 \beta_{3} - 15 \beta_1) q^{28} + ( - 12 \beta_{6} + 2) q^{29} + ( - 16 \beta_{6} + 8 \beta_{5}) q^{31} + ( - 5 \beta_{7} + 5 \beta_{4} - 15 \beta_{3} + 15 \beta_1) q^{32} + (4 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{33} + ( - 4 \beta_{6} + 8 \beta_{5} + 12 \beta_{2} - 6) q^{34} + (6 \beta_{6} - 9) q^{36} + (10 \beta_{4} - 25 \beta_1) q^{37} + ( - \beta_{7} - \beta_{4} + 31 \beta_{3} + 31 \beta_1) q^{38} + (6 \beta_{5} - 15 \beta_{2}) q^{39} + ( - 14 \beta_{6} + 28 \beta_{5} - 20 \beta_{2} + 10) q^{41} + ( - 11 \beta_{7} + 13 \beta_{4} + 11 \beta_{3} - 13 \beta_1) q^{42} + ( - 4 \beta_{7} + 50 \beta_{3}) q^{43} + ( - 2 \beta_{6} + 2 \beta_{5} - 18 \beta_{2} + 18) q^{44} + ( - 20 \beta_{5} + 20 \beta_{2}) q^{46} + ( - 12 \beta_{7} + 6 \beta_{4} + 48 \beta_{3} - 24 \beta_1) q^{47} + ( - 4 \beta_{7} + 8 \beta_{4} + 5 \beta_{3} - 10 \beta_1) q^{48} + (39 \beta_{2} - 55) q^{49} + (6 \beta_{6} - 6 \beta_{5} - 18 \beta_{2} + 18) q^{51} + (16 \beta_{7} + 16 \beta_{4} - 39 \beta_{3} - 39 \beta_1) q^{52} + (30 \beta_{7} - 30 \beta_{4} - 22 \beta_{3} + 22 \beta_1) q^{53} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} - 3) q^{54} + (8 \beta_{6} - 3 \beta_{5} + 33 \beta_{2} - 88) q^{56} + ( - 18 \beta_{7} - 15 \beta_{3}) q^{57} + ( - 14 \beta_{4} + 74 \beta_1) q^{58} + (36 \beta_{6} - 18 \beta_{5} - 4 \beta_{2} + 8) q^{59} + ( - 6 \beta_{6} - 6 \beta_{5} - 39 \beta_{2} - 39) q^{61} + (8 \beta_{7} - 16 \beta_{4} - 48 \beta_{3} + 96 \beta_1) q^{62} + (9 \beta_{3} - 24 \beta_1) q^{63} + ( - 26 \beta_{6} + 5) q^{64} + ( - 10 \beta_{2} + 20) q^{66} + (38 \beta_{7} - 38 \beta_{4} - \beta_{3} + \beta_1) q^{67} + (12 \beta_{7} - 6 \beta_{4} + 12 \beta_{3} - 6 \beta_1) q^{68} + (40 \beta_{2} - 20) q^{69} + ( - 20 \beta_{6} - 26) q^{71} + (3 \beta_{4} - 33 \beta_1) q^{72} + (4 \beta_{7} + 4 \beta_{4} - 15 \beta_{3} - 15 \beta_1) q^{73} + (35 \beta_{5} - 85 \beta_{2}) q^{74} + (8 \beta_{6} - 16 \beta_{5} + 114 \beta_{2} - 57) q^{76} + (16 \beta_{7} - 10 \beta_{4} + 16 \beta_{3} - 10 \beta_1) q^{77} + (21 \beta_{7} - 51 \beta_{3}) q^{78} + ( - 16 \beta_{6} + 16 \beta_{5} + 19 \beta_{2} - 19) q^{79} - 9 \beta_{2} q^{81} + (48 \beta_{7} - 24 \beta_{4} - 188 \beta_{3} + 94 \beta_1) q^{82} + ( - 10 \beta_{7} + 20 \beta_{4} - 24 \beta_{3} + 48 \beta_1) q^{83} + (22 \beta_{6} + 4 \beta_{5} - 6 \beta_{2} - 33) q^{84} + (54 \beta_{6} - 54 \beta_{5} + 74 \beta_{2} - 74) q^{86} + ( - 12 \beta_{7} - 12 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{87} + (20 \beta_{7} - 20 \beta_{4} + 10 \beta_{3} - 10 \beta_1) q^{88} + (10 \beta_{6} + 10 \beta_{5} + 12 \beta_{2} + 12) q^{89} + ( - 4 \beta_{6} + 26 \beta_{5} - 65 \beta_{2} + 10) q^{91} + ( - 40 \beta_{7} + 60 \beta_{3}) q^{92} - 24 \beta_{4} q^{93} + (60 \beta_{6} - 30 \beta_{5} + 60 \beta_{2} - 120) q^{94} + (5 \beta_{6} + 5 \beta_{5} + 15 \beta_{2} + 15) q^{96} + (24 \beta_{7} - 48 \beta_{4} + 17 \beta_{3} - 34 \beta_1) q^{97} + ( - 39 \beta_{7} + 55 \beta_{4} + 39 \beta_{3} - 55 \beta_1) q^{98} + ( - 6 \beta_{6} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} - 12 q^{9} + 8 q^{11} - 8 q^{14} - 20 q^{16} - 60 q^{19} - 60 q^{21} - 132 q^{24} - 204 q^{26} + 16 q^{29} - 72 q^{36} - 60 q^{39} + 72 q^{44} + 80 q^{46} - 284 q^{49} + 72 q^{51} - 36 q^{54} - 572 q^{56} + 48 q^{59} - 468 q^{61} + 40 q^{64} + 120 q^{66} - 208 q^{71} - 340 q^{74} - 76 q^{79} - 36 q^{81} - 288 q^{84} - 296 q^{86} + 144 q^{89} - 180 q^{91} - 720 q^{94} + 180 q^{96} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \beta_{2}\)

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
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−2.98735 1.72474i −0.866025 1.50000i 3.94949 + 6.84072i 0 5.97469i −2.59808 6.50000i 13.4495i −1.50000 + 2.59808i 0
124.2 −1.25529 0.724745i 0.866025 + 1.50000i −0.949490 1.64456i 0 2.51059i 2.59808 + 6.50000i 8.55051i −1.50000 + 2.59808i 0
124.3 1.25529 + 0.724745i −0.866025 1.50000i −0.949490 1.64456i 0 2.51059i −2.59808 6.50000i 8.55051i −1.50000 + 2.59808i 0
124.4 2.98735 + 1.72474i 0.866025 + 1.50000i 3.94949 + 6.84072i 0 5.97469i 2.59808 + 6.50000i 13.4495i −1.50000 + 2.59808i 0
199.1 −2.98735 + 1.72474i −0.866025 + 1.50000i 3.94949 6.84072i 0 5.97469i −2.59808 + 6.50000i 13.4495i −1.50000 2.59808i 0
199.2 −1.25529 + 0.724745i 0.866025 1.50000i −0.949490 + 1.64456i 0 2.51059i 2.59808 6.50000i 8.55051i −1.50000 2.59808i 0
199.3 1.25529 0.724745i −0.866025 + 1.50000i −0.949490 + 1.64456i 0 2.51059i −2.59808 + 6.50000i 8.55051i −1.50000 2.59808i 0
199.4 2.98735 1.72474i 0.866025 1.50000i 3.94949 6.84072i 0 5.97469i 2.59808 6.50000i 13.4495i −1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
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.g 8
5.b even 2 1 inner 525.3.s.g 8
5.c odd 4 1 525.3.o.j 4
5.c odd 4 1 525.3.o.k yes 4
7.d odd 6 1 inner 525.3.s.g 8
35.i odd 6 1 inner 525.3.s.g 8
35.k even 12 1 525.3.o.j 4
35.k even 12 1 525.3.o.k yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.j 4 5.c odd 4 1
525.3.o.j 4 35.k even 12 1
525.3.o.k yes 4 5.c odd 4 1
525.3.o.k yes 4 35.k even 12 1
525.3.s.g 8 1.a even 1 1 trivial
525.3.s.g 8 5.b even 2 1 inner
525.3.s.g 8 7.d odd 6 1 inner
525.3.s.g 8 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 \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{8} - 14T_{2}^{6} + 171T_{2}^{4} - 350T_{2}^{2} + 625 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 36T_{11}^{2} + 80T_{11} + 400 \) Copy content Toggle raw display
\( T_{13}^{4} - 294T_{13}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 71 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 294 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 360 T^{6} + 128304 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{4} + 30 T^{3} - 273 T^{2} + \cdots + 328329)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 400 T^{2} + 160000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 860)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 1152 T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 2450 T^{6} + 6001875 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7656 T^{2} + 10419984)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5192 T^{2} + 5779216)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 4752 T^{6} + \cdots + 1360488960000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 584046595707136 \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} - 5592 T^{2} + \cdots + 33454656)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 234 T^{3} + 22167 T^{2} + \cdots + 15327225)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 17330 T^{6} + \cdots + 56\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{2} + 52 T - 1724)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 1926 T^{6} + \cdots + 22430753361 \) Copy content Toggle raw display
$79$ \( (T^{4} + 38 T^{3} + 2619 T^{2} + \cdots + 1380625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 7056 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 72 T^{3} + 360 T^{2} + \cdots + 1871424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 22470 T^{2} + 90269001)^{2} \) Copy content Toggle raw display
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