Properties

Label 525.4.d.b
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -7 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -7 i q^{7} -9 q^{9} + 62 q^{11} -24 i q^{12} + 62 i q^{13} + 28 q^{14} -64 q^{16} + 84 i q^{17} -36 i q^{18} -100 q^{19} + 21 q^{21} + 248 i q^{22} + 42 i q^{23} -248 q^{26} -27 i q^{27} + 56 i q^{28} + 10 q^{29} -48 q^{31} -256 i q^{32} + 186 i q^{33} -336 q^{34} + 72 q^{36} -246 i q^{37} -400 i q^{38} -186 q^{39} -248 q^{41} + 84 i q^{42} -68 i q^{43} -496 q^{44} -168 q^{46} + 324 i q^{47} -192 i q^{48} -49 q^{49} -252 q^{51} -496 i q^{52} -258 i q^{53} + 108 q^{54} -300 i q^{57} + 40 i q^{58} -120 q^{59} + 622 q^{61} -192 i q^{62} + 63 i q^{63} + 512 q^{64} -744 q^{66} + 904 i q^{67} -672 i q^{68} -126 q^{69} -678 q^{71} + 642 i q^{73} + 984 q^{74} + 800 q^{76} -434 i q^{77} -744 i q^{78} -740 q^{79} + 81 q^{81} -992 i q^{82} -468 i q^{83} -168 q^{84} + 272 q^{86} + 30 i q^{87} -200 q^{89} + 434 q^{91} -336 i q^{92} -144 i q^{93} -1296 q^{94} + 768 q^{96} -1266 i q^{97} -196 i q^{98} -558 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + 124q^{11} + 56q^{14} - 128q^{16} - 200q^{19} + 42q^{21} - 496q^{26} + 20q^{29} - 96q^{31} - 672q^{34} + 144q^{36} - 372q^{39} - 496q^{41} - 992q^{44} - 336q^{46} - 98q^{49} - 504q^{51} + 216q^{54} - 240q^{59} + 1244q^{61} + 1024q^{64} - 1488q^{66} - 252q^{69} - 1356q^{71} + 1968q^{74} + 1600q^{76} - 1480q^{79} + 162q^{81} - 336q^{84} + 544q^{86} - 400q^{89} + 868q^{91} - 2592q^{94} + 1536q^{96} - 1116q^{99} + O(q^{100}) \)

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
1.00000i
1.00000i
4.00000i 3.00000i −8.00000 0 −12.0000 7.00000i 0 −9.00000 0
274.2 4.00000i 3.00000i −8.00000 0 −12.0000 7.00000i 0 −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.b 2
5.b even 2 1 inner 525.4.d.b 2
5.c odd 4 1 21.4.a.b 1
5.c odd 4 1 525.4.a.b 1
15.e even 4 1 63.4.a.a 1
15.e even 4 1 1575.4.a.k 1
20.e even 4 1 336.4.a.h 1
35.f even 4 1 147.4.a.g 1
35.k even 12 2 147.4.e.b 2
35.l odd 12 2 147.4.e.c 2
40.i odd 4 1 1344.4.a.w 1
40.k even 4 1 1344.4.a.i 1
60.l odd 4 1 1008.4.a.m 1
105.k odd 4 1 441.4.a.b 1
105.w odd 12 2 441.4.e.n 2
105.x even 12 2 441.4.e.m 2
140.j odd 4 1 2352.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 5.c odd 4 1
63.4.a.a 1 15.e even 4 1
147.4.a.g 1 35.f even 4 1
147.4.e.b 2 35.k even 12 2
147.4.e.c 2 35.l odd 12 2
336.4.a.h 1 20.e even 4 1
441.4.a.b 1 105.k odd 4 1
441.4.e.m 2 105.x even 12 2
441.4.e.n 2 105.w odd 12 2
525.4.a.b 1 5.c odd 4 1
525.4.d.b 2 1.a even 1 1 trivial
525.4.d.b 2 5.b even 2 1 inner
1008.4.a.m 1 60.l odd 4 1
1344.4.a.i 1 40.k even 4 1
1344.4.a.w 1 40.i odd 4 1
1575.4.a.k 1 15.e even 4 1
2352.4.a.l 1 140.j odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 16 \)
\( T_{11} - 62 \)