Properties

Label 525.4.d.h
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{65})\)
Defining polynomial: \(x^{4} + 33 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -9 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -\beta_{1} + 16 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -9 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -\beta_{1} + 16 \beta_{2} ) q^{8} -9 q^{9} + ( -12 + 2 \beta_{3} ) q^{11} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{12} + ( -2 \beta_{1} - 12 \beta_{2} ) q^{13} + ( 7 - 7 \beta_{3} ) q^{14} + ( -39 - 9 \beta_{3} ) q^{16} + ( -16 \beta_{1} - 66 \beta_{2} ) q^{17} -9 \beta_{1} q^{18} + ( -44 - 14 \beta_{3} ) q^{19} -21 q^{21} + ( -12 \beta_{1} + 32 \beta_{2} ) q^{22} + ( 20 \beta_{1} + 140 \beta_{2} ) q^{23} + ( -51 + 3 \beta_{3} ) q^{24} + ( 22 + 10 \beta_{3} ) q^{26} -27 \beta_{2} q^{27} + ( 7 \beta_{1} - 56 \beta_{2} ) q^{28} + ( 122 - 48 \beta_{3} ) q^{29} + ( 96 - 42 \beta_{3} ) q^{31} + ( -47 \beta_{1} - 16 \beta_{2} ) q^{32} + ( 6 \beta_{1} - 30 \beta_{2} ) q^{33} + ( 206 + 50 \beta_{3} ) q^{34} + ( 81 - 9 \beta_{3} ) q^{36} + ( -36 \beta_{1} + 30 \beta_{2} ) q^{37} + ( -44 \beta_{1} - 224 \beta_{2} ) q^{38} + ( 30 + 6 \beta_{3} ) q^{39} + ( -38 - 100 \beta_{3} ) q^{41} -21 \beta_{1} q^{42} + ( 32 \beta_{1} - 156 \beta_{2} ) q^{43} + ( 140 - 28 \beta_{3} ) q^{44} + ( -200 - 120 \beta_{3} ) q^{46} + ( -48 \beta_{1} - 304 \beta_{2} ) q^{47} + ( -27 \beta_{1} - 144 \beta_{2} ) q^{48} -49 q^{49} + ( 150 + 48 \beta_{3} ) q^{51} + ( 6 \beta_{1} + 64 \beta_{2} ) q^{52} + ( -86 \beta_{1} + 120 \beta_{2} ) q^{53} + ( -27 + 27 \beta_{3} ) q^{54} + ( -119 + 7 \beta_{3} ) q^{56} + ( -42 \beta_{1} - 174 \beta_{2} ) q^{57} + ( 122 \beta_{1} - 768 \beta_{2} ) q^{58} + ( 380 + 84 \beta_{3} ) q^{59} + ( -90 - 24 \beta_{3} ) q^{61} + ( 96 \beta_{1} - 672 \beta_{2} ) q^{62} -63 \beta_{2} q^{63} + ( 471 - 103 \beta_{3} ) q^{64} + ( -132 + 36 \beta_{3} ) q^{66} + ( 64 \beta_{1} + 84 \beta_{2} ) q^{67} + ( 78 \beta_{1} + 272 \beta_{2} ) q^{68} + ( -360 - 60 \beta_{3} ) q^{69} + ( 856 - 42 \beta_{3} ) q^{71} + ( 9 \beta_{1} - 144 \beta_{2} ) q^{72} + ( 202 \beta_{1} - 92 \beta_{2} ) q^{73} + ( 642 - 66 \beta_{3} ) q^{74} + ( 172 + 68 \beta_{3} ) q^{76} + ( 14 \beta_{1} - 70 \beta_{2} ) q^{77} + ( 30 \beta_{1} + 96 \beta_{2} ) q^{78} + ( 496 - 104 \beta_{3} ) q^{79} + 81 q^{81} + ( -38 \beta_{1} - 1600 \beta_{2} ) q^{82} + ( -216 \beta_{1} + 356 \beta_{2} ) q^{83} + ( 189 - 21 \beta_{3} ) q^{84} + ( -700 + 188 \beta_{3} ) q^{86} + ( -144 \beta_{1} + 222 \beta_{2} ) q^{87} + ( 44 \beta_{1} - 192 \beta_{2} ) q^{88} + ( -298 + 8 \beta_{3} ) q^{89} + ( 70 + 14 \beta_{3} ) q^{91} + ( -40 \beta_{1} - 800 \beta_{2} ) q^{92} + ( -126 \beta_{1} + 162 \beta_{2} ) q^{93} + ( 512 + 256 \beta_{3} ) q^{94} + ( -93 + 141 \beta_{3} ) q^{96} + ( 314 \beta_{1} - 104 \beta_{2} ) q^{97} -49 \beta_{1} q^{98} + ( 108 - 18 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 34q^{4} + 6q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 34q^{4} + 6q^{6} - 36q^{9} - 44q^{11} + 14q^{14} - 174q^{16} - 204q^{19} - 84q^{21} - 198q^{24} + 108q^{26} + 392q^{29} + 300q^{31} + 924q^{34} + 306q^{36} + 132q^{39} - 352q^{41} + 504q^{44} - 1040q^{46} - 196q^{49} + 696q^{51} - 54q^{54} - 462q^{56} + 1688q^{59} - 408q^{61} + 1678q^{64} - 456q^{66} - 1560q^{69} + 3340q^{71} + 2436q^{74} + 824q^{76} + 1776q^{79} + 324q^{81} + 714q^{84} - 2424q^{86} - 1176q^{89} + 308q^{91} + 2560q^{94} - 90q^{96} + 396q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 33 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 17 \nu \)\()/16\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 17\)
\(\nu^{3}\)\(=\)\(16 \beta_{2} - 17 \beta_{1}\)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
4.53113i
3.53113i
3.53113i
4.53113i
4.53113i 3.00000i −12.5311 0 13.5934 7.00000i 20.5311i −9.00000 0
274.2 3.53113i 3.00000i −4.46887 0 −10.5934 7.00000i 12.4689i −9.00000 0
274.3 3.53113i 3.00000i −4.46887 0 −10.5934 7.00000i 12.4689i −9.00000 0
274.4 4.53113i 3.00000i −12.5311 0 13.5934 7.00000i 20.5311i −9.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.d.h 4
5.b even 2 1 inner 525.4.d.h 4
5.c odd 4 1 105.4.a.f 2
5.c odd 4 1 525.4.a.k 2
15.e even 4 1 315.4.a.i 2
15.e even 4 1 1575.4.a.w 2
20.e even 4 1 1680.4.a.bg 2
35.f even 4 1 735.4.a.p 2
105.k odd 4 1 2205.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 5.c odd 4 1
315.4.a.i 2 15.e even 4 1
525.4.a.k 2 5.c odd 4 1
525.4.d.h 4 1.a even 1 1 trivial
525.4.d.h 4 5.b even 2 1 inner
735.4.a.p 2 35.f even 4 1
1575.4.a.w 2 15.e even 4 1
1680.4.a.bg 2 20.e even 4 1
2205.4.a.z 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 33 T_{2}^{2} + 256 \)
\( T_{11}^{2} + 22 T_{11} + 56 \)