Properties

Label 525.4.d.a
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 105)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} + 3 i q^{3} -17 q^{4} -15 q^{6} + 7 i q^{7} -45 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 5 i q^{2} + 3 i q^{3} -17 q^{4} -15 q^{6} + 7 i q^{7} -45 i q^{8} -9 q^{9} + 12 q^{11} -51 i q^{12} -30 i q^{13} -35 q^{14} + 89 q^{16} -134 i q^{17} -45 i q^{18} + 92 q^{19} -21 q^{21} + 60 i q^{22} -112 i q^{23} + 135 q^{24} + 150 q^{26} -27 i q^{27} -119 i q^{28} + 58 q^{29} -224 q^{31} + 85 i q^{32} + 36 i q^{33} + 670 q^{34} + 153 q^{36} -146 i q^{37} + 460 i q^{38} + 90 q^{39} + 18 q^{41} -105 i q^{42} -340 i q^{43} -204 q^{44} + 560 q^{46} + 208 i q^{47} + 267 i q^{48} -49 q^{49} + 402 q^{51} + 510 i q^{52} + 754 i q^{53} + 135 q^{54} + 315 q^{56} + 276 i q^{57} + 290 i q^{58} -380 q^{59} + 718 q^{61} -1120 i q^{62} -63 i q^{63} + 287 q^{64} -180 q^{66} + 412 i q^{67} + 2278 i q^{68} + 336 q^{69} -960 q^{71} + 405 i q^{72} -1066 i q^{73} + 730 q^{74} -1564 q^{76} + 84 i q^{77} + 450 i q^{78} -896 q^{79} + 81 q^{81} + 90 i q^{82} -436 i q^{83} + 357 q^{84} + 1700 q^{86} + 174 i q^{87} -540 i q^{88} + 1038 q^{89} + 210 q^{91} + 1904 i q^{92} -672 i q^{93} -1040 q^{94} -255 q^{96} -702 i q^{97} -245 i q^{98} -108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 34q^{4} - 30q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 34q^{4} - 30q^{6} - 18q^{9} + 24q^{11} - 70q^{14} + 178q^{16} + 184q^{19} - 42q^{21} + 270q^{24} + 300q^{26} + 116q^{29} - 448q^{31} + 1340q^{34} + 306q^{36} + 180q^{39} + 36q^{41} - 408q^{44} + 1120q^{46} - 98q^{49} + 804q^{51} + 270q^{54} + 630q^{56} - 760q^{59} + 1436q^{61} + 574q^{64} - 360q^{66} + 672q^{69} - 1920q^{71} + 1460q^{74} - 3128q^{76} - 1792q^{79} + 162q^{81} + 714q^{84} + 3400q^{86} + 2076q^{89} + 420q^{91} - 2080q^{94} - 510q^{96} - 216q^{99} + O(q^{100}) \)

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
5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −9.00000 0
274.2 5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −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.a 2
5.b even 2 1 inner 525.4.d.a 2
5.c odd 4 1 105.4.a.b 1
5.c odd 4 1 525.4.a.a 1
15.e even 4 1 315.4.a.a 1
15.e even 4 1 1575.4.a.l 1
20.e even 4 1 1680.4.a.u 1
35.f even 4 1 735.4.a.j 1
105.k odd 4 1 2205.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 5.c odd 4 1
315.4.a.a 1 15.e even 4 1
525.4.a.a 1 5.c odd 4 1
525.4.d.a 2 1.a even 1 1 trivial
525.4.d.a 2 5.b even 2 1 inner
735.4.a.j 1 35.f even 4 1
1575.4.a.l 1 15.e even 4 1
1680.4.a.u 1 20.e even 4 1
2205.4.a.b 1 105.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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 25 \)
\( T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 9 T^{2} + 64 T^{4} \)
$3$ \( 1 + 9 T^{2} \)
$5$ 1
$7$ \( 1 + 49 T^{2} \)
$11$ \( ( 1 - 12 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 3494 T^{2} + 4826809 T^{4} \)
$17$ \( 1 + 8130 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 92 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 11790 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 58 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 224 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 79990 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 18 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 43414 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 164382 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 + 270762 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 380 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 718 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 431782 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 960 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 358322 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 896 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 953478 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 1038 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1332542 T^{2} + 832972004929 T^{4} \)
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