Properties

Label 525.4.d.c
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} -36 q^{11} + 3 i q^{12} -34 i q^{13} + 21 q^{14} -71 q^{16} -42 i q^{17} -27 i q^{18} + 124 q^{19} -21 q^{21} -108 i q^{22} + 63 q^{24} + 102 q^{26} + 27 i q^{27} + 7 i q^{28} -102 q^{29} -160 q^{31} -45 i q^{32} + 108 i q^{33} + 126 q^{34} + 9 q^{36} -398 i q^{37} + 372 i q^{38} -102 q^{39} -318 q^{41} -63 i q^{42} -268 i q^{43} + 36 q^{44} -240 i q^{47} + 213 i q^{48} -49 q^{49} -126 q^{51} + 34 i q^{52} -498 i q^{53} -81 q^{54} + 147 q^{56} -372 i q^{57} -306 i q^{58} + 132 q^{59} + 398 q^{61} -480 i q^{62} + 63 i q^{63} -433 q^{64} -324 q^{66} -92 i q^{67} + 42 i q^{68} -720 q^{71} -189 i q^{72} -502 i q^{73} + 1194 q^{74} -124 q^{76} + 252 i q^{77} -306 i q^{78} + 1024 q^{79} + 81 q^{81} -954 i q^{82} -204 i q^{83} + 21 q^{84} + 804 q^{86} + 306 i q^{87} -756 i q^{88} -354 q^{89} -238 q^{91} + 480 i q^{93} + 720 q^{94} -135 q^{96} + 286 i q^{97} -147 i q^{98} + 324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} - 72q^{11} + 42q^{14} - 142q^{16} + 248q^{19} - 42q^{21} + 126q^{24} + 204q^{26} - 204q^{29} - 320q^{31} + 252q^{34} + 18q^{36} - 204q^{39} - 636q^{41} + 72q^{44} - 98q^{49} - 252q^{51} - 162q^{54} + 294q^{56} + 264q^{59} + 796q^{61} - 866q^{64} - 648q^{66} - 1440q^{71} + 2388q^{74} - 248q^{76} + 2048q^{79} + 162q^{81} + 42q^{84} + 1608q^{86} - 708q^{89} - 476q^{91} + 1440q^{94} - 270q^{96} + 648q^{99} + O(q^{100}) \)

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
3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −9.00000 0
274.2 3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −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.c 2
5.b even 2 1 inner 525.4.d.c 2
5.c odd 4 1 21.4.a.a 1
5.c odd 4 1 525.4.a.g 1
15.e even 4 1 63.4.a.c 1
15.e even 4 1 1575.4.a.b 1
20.e even 4 1 336.4.a.f 1
35.f even 4 1 147.4.a.c 1
35.k even 12 2 147.4.e.g 2
35.l odd 12 2 147.4.e.i 2
40.i odd 4 1 1344.4.a.ba 1
40.k even 4 1 1344.4.a.n 1
60.l odd 4 1 1008.4.a.v 1
105.k odd 4 1 441.4.a.j 1
105.w odd 12 2 441.4.e.d 2
105.x even 12 2 441.4.e.b 2
140.j odd 4 1 2352.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 5.c odd 4 1
63.4.a.c 1 15.e even 4 1
147.4.a.c 1 35.f even 4 1
147.4.e.g 2 35.k even 12 2
147.4.e.i 2 35.l odd 12 2
336.4.a.f 1 20.e even 4 1
441.4.a.j 1 105.k odd 4 1
441.4.e.b 2 105.x even 12 2
441.4.e.d 2 105.w odd 12 2
525.4.a.g 1 5.c odd 4 1
525.4.d.c 2 1.a even 1 1 trivial
525.4.d.c 2 5.b even 2 1 inner
1008.4.a.v 1 60.l odd 4 1
1344.4.a.n 1 40.k even 4 1
1344.4.a.ba 1 40.i odd 4 1
1575.4.a.b 1 15.e even 4 1
2352.4.a.r 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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 9 \)
\( T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 64 T^{4} \)
$3$ \( 1 + 9 T^{2} \)
$5$ 1
$7$ \( 1 + 49 T^{2} \)
$11$ \( ( 1 + 36 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 3238 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 8062 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 124 T + 6859 T^{2} )^{2} \)
$23$ \( ( 1 - 12167 T^{2} )^{2} \)
$29$ \( ( 1 + 102 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 160 T + 29791 T^{2} )^{2} \)
$37$ \( 1 + 57098 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 318 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 87190 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 150046 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 49750 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 - 132 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 398 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 593062 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 720 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 526030 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 1024 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1101958 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 354 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1743550 T^{2} + 832972004929 T^{4} \)
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