Properties

Label 525.4.d.g
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: \(\Q(i, \sqrt{57})\)
Defining polynomial: \(x^{4} + 29 x^{2} + 196\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + ( -10 + 3 \beta_{3} ) q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -5 \beta_{1} + 41 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + ( -10 + 3 \beta_{3} ) q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -5 \beta_{1} + 41 \beta_{2} ) q^{8} -9 q^{9} + ( 2 - 10 \beta_{3} ) q^{11} + ( -9 \beta_{1} + 21 \beta_{2} ) q^{12} + ( -12 \beta_{1} - 14 \beta_{2} ) q^{13} + ( 14 - 7 \beta_{3} ) q^{14} + ( 82 - 27 \beta_{3} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + ( -20 - 24 \beta_{3} ) q^{19} + 21 q^{21} + ( 12 \beta_{1} - 132 \beta_{2} ) q^{22} + ( -34 \beta_{1} - 20 \beta_{2} ) q^{23} + ( 138 - 15 \beta_{3} ) q^{24} + ( 164 - 10 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( 21 \beta_{1} - 49 \beta_{2} ) q^{28} + ( 114 + 24 \beta_{3} ) q^{29} + ( 56 - 72 \beta_{3} ) q^{31} + ( 69 \beta_{1} - 105 \beta_{2} ) q^{32} + ( 30 \beta_{1} + 24 \beta_{2} ) q^{33} + ( -36 + 6 \beta_{3} ) q^{34} + ( 90 - 27 \beta_{3} ) q^{36} + ( 36 \beta_{1} - 106 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 292 \beta_{2} ) q^{38} + ( -6 - 36 \beta_{3} ) q^{39} + ( -240 + 30 \beta_{3} ) q^{41} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{42} + ( -48 \beta_{1} - 212 \beta_{2} ) q^{43} + ( -440 + 76 \beta_{3} ) q^{44} + ( 504 - 48 \beta_{3} ) q^{46} + ( -68 \beta_{1} - 40 \beta_{2} ) q^{47} + ( 81 \beta_{1} - 165 \beta_{2} ) q^{48} -49 q^{49} + ( -12 + 6 \beta_{3} ) q^{51} + ( 78 \beta_{1} - 406 \beta_{2} ) q^{52} + ( 4 \beta_{1} + 554 \beta_{2} ) q^{53} + ( 54 - 27 \beta_{3} ) q^{54} + ( -322 + 35 \beta_{3} ) q^{56} + ( 72 \beta_{1} + 132 \beta_{2} ) q^{57} + ( 90 \beta_{1} + 198 \beta_{2} ) q^{58} + ( -344 - 116 \beta_{3} ) q^{59} + ( -178 - 72 \beta_{3} ) q^{61} + ( 128 \beta_{1} - 992 \beta_{2} ) q^{62} -63 \beta_{2} q^{63} + ( -658 + 27 \beta_{3} ) q^{64} + ( -432 + 36 \beta_{3} ) q^{66} + ( -108 \beta_{1} + 20 \beta_{2} ) q^{67} + ( -26 \beta_{1} + 98 \beta_{2} ) q^{68} + ( 42 - 102 \beta_{3} ) q^{69} + ( 462 + 30 \beta_{3} ) q^{71} + ( 45 \beta_{1} - 369 \beta_{2} ) q^{72} + ( 12 \beta_{1} - 530 \beta_{2} ) q^{73} + ( -788 + 178 \beta_{3} ) q^{74} + ( -808 + 108 \beta_{3} ) q^{76} + ( -70 \beta_{1} - 56 \beta_{2} ) q^{77} + ( 30 \beta_{1} - 462 \beta_{2} ) q^{78} + ( 340 - 108 \beta_{3} ) q^{79} + 81 q^{81} + ( -270 \beta_{1} + 630 \beta_{2} ) q^{82} + ( 96 \beta_{1} - 924 \beta_{2} ) q^{83} + ( -210 + 63 \beta_{3} ) q^{84} + ( 344 + 116 \beta_{3} ) q^{86} + ( -72 \beta_{1} - 414 \beta_{2} ) q^{87} + ( -420 \beta_{1} + 372 \beta_{2} ) q^{88} + ( -112 - 142 \beta_{3} ) q^{89} + ( 14 + 84 \beta_{3} ) q^{91} + ( 280 \beta_{1} - 1288 \beta_{2} ) q^{92} + ( 216 \beta_{1} + 48 \beta_{2} ) q^{93} + ( 1008 - 96 \beta_{3} ) q^{94} + ( -522 + 207 \beta_{3} ) q^{96} + ( -276 \beta_{1} + 266 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( -18 + 90 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 34q^{4} - 18q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 34q^{4} - 18q^{6} - 36q^{9} - 12q^{11} + 42q^{14} + 274q^{16} - 128q^{19} + 84q^{21} + 522q^{24} + 636q^{26} + 504q^{29} + 80q^{31} - 132q^{34} + 306q^{36} - 96q^{39} - 900q^{41} - 1608q^{44} + 1920q^{46} - 196q^{49} - 36q^{51} + 162q^{54} - 1218q^{56} - 1608q^{59} - 856q^{61} - 2578q^{64} - 1656q^{66} - 36q^{69} + 1908q^{71} - 2796q^{74} - 3016q^{76} + 1144q^{79} + 324q^{81} - 714q^{84} + 1608q^{86} - 732q^{89} + 224q^{91} + 3840q^{94} - 1674q^{96} + 108q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 29 x^{2} + 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 15 \nu \)\()/14\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 15\)
\(\nu^{3}\)\(=\)\(14 \beta_{2} - 15 \beta_{1}\)

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
4.27492i
3.27492i
3.27492i
4.27492i
5.27492i 3.00000i −19.8248 0 −15.8248 7.00000i 62.3746i −9.00000 0
274.2 2.27492i 3.00000i 2.82475 0 6.82475 7.00000i 24.6254i −9.00000 0
274.3 2.27492i 3.00000i 2.82475 0 6.82475 7.00000i 24.6254i −9.00000 0
274.4 5.27492i 3.00000i −19.8248 0 −15.8248 7.00000i 62.3746i −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.g 4
5.b even 2 1 inner 525.4.d.g 4
5.c odd 4 1 21.4.a.c 2
5.c odd 4 1 525.4.a.n 2
15.e even 4 1 63.4.a.e 2
15.e even 4 1 1575.4.a.p 2
20.e even 4 1 336.4.a.m 2
35.f even 4 1 147.4.a.i 2
35.k even 12 2 147.4.e.m 4
35.l odd 12 2 147.4.e.l 4
40.i odd 4 1 1344.4.a.bg 2
40.k even 4 1 1344.4.a.bo 2
60.l odd 4 1 1008.4.a.ba 2
105.k odd 4 1 441.4.a.r 2
105.w odd 12 2 441.4.e.p 4
105.x even 12 2 441.4.e.q 4
140.j odd 4 1 2352.4.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 5.c odd 4 1
63.4.a.e 2 15.e even 4 1
147.4.a.i 2 35.f even 4 1
147.4.e.l 4 35.l odd 12 2
147.4.e.m 4 35.k even 12 2
336.4.a.m 2 20.e even 4 1
441.4.a.r 2 105.k odd 4 1
441.4.e.p 4 105.w odd 12 2
441.4.e.q 4 105.x even 12 2
525.4.a.n 2 5.c odd 4 1
525.4.d.g 4 1.a even 1 1 trivial
525.4.d.g 4 5.b even 2 1 inner
1008.4.a.ba 2 60.l odd 4 1
1344.4.a.bg 2 40.i odd 4 1
1344.4.a.bo 2 40.k even 4 1
1575.4.a.p 2 15.e even 4 1
2352.4.a.bz 2 140.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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 33 T_{2}^{2} + 144 \)
\( T_{11}^{2} + 6 T_{11} - 1416 \)