Properties

Label 525.4.d.e
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: yes
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} -6 q^{6} + 7 i q^{7} + 24 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} -6 q^{6} + 7 i q^{7} + 24 i q^{8} -9 q^{9} -21 q^{11} + 12 i q^{12} + 24 i q^{13} -14 q^{14} -16 q^{16} + 22 i q^{17} -18 i q^{18} -16 q^{19} -21 q^{21} -42 i q^{22} -25 i q^{23} -72 q^{24} -48 q^{26} -27 i q^{27} + 28 i q^{28} -167 q^{29} + 10 q^{31} + 160 i q^{32} -63 i q^{33} -44 q^{34} -36 q^{36} + 133 i q^{37} -32 i q^{38} -72 q^{39} -168 q^{41} -42 i q^{42} -97 i q^{43} -84 q^{44} + 50 q^{46} + 400 i q^{47} -48 i q^{48} -49 q^{49} -66 q^{51} + 96 i q^{52} -182 i q^{53} + 54 q^{54} -168 q^{56} -48 i q^{57} -334 i q^{58} -488 q^{59} + 28 q^{61} + 20 i q^{62} -63 i q^{63} -448 q^{64} + 126 q^{66} + 967 i q^{67} + 88 i q^{68} + 75 q^{69} -285 q^{71} -216 i q^{72} -838 i q^{73} -266 q^{74} -64 q^{76} -147 i q^{77} -144 i q^{78} + 469 q^{79} + 81 q^{81} -336 i q^{82} -406 i q^{83} -84 q^{84} + 194 q^{86} -501 i q^{87} -504 i q^{88} -324 q^{89} -168 q^{91} -100 i q^{92} + 30 i q^{93} -800 q^{94} -480 q^{96} + 114 i q^{97} -98 i q^{98} + 189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} - 12q^{6} - 18q^{9} + O(q^{10}) \) \( 2q + 8q^{4} - 12q^{6} - 18q^{9} - 42q^{11} - 28q^{14} - 32q^{16} - 32q^{19} - 42q^{21} - 144q^{24} - 96q^{26} - 334q^{29} + 20q^{31} - 88q^{34} - 72q^{36} - 144q^{39} - 336q^{41} - 168q^{44} + 100q^{46} - 98q^{49} - 132q^{51} + 108q^{54} - 336q^{56} - 976q^{59} + 56q^{61} - 896q^{64} + 252q^{66} + 150q^{69} - 570q^{71} - 532q^{74} - 128q^{76} + 938q^{79} + 162q^{81} - 168q^{84} + 388q^{86} - 648q^{89} - 336q^{91} - 1600q^{94} - 960q^{96} + 378q^{99} + O(q^{100}) \)

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
2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −9.00000 0
274.2 2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −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.e 2
5.b even 2 1 inner 525.4.d.e 2
5.c odd 4 1 525.4.a.d 1
5.c odd 4 1 525.4.a.f yes 1
15.e even 4 1 1575.4.a.d 1
15.e even 4 1 1575.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 5.c odd 4 1
525.4.a.f yes 1 5.c odd 4 1
525.4.d.e 2 1.a even 1 1 trivial
525.4.d.e 2 5.b even 2 1 inner
1575.4.a.d 1 15.e even 4 1
1575.4.a.h 1 15.e even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \)
\( T_{11} + 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( 21 + T )^{2} \)
$13$ \( 576 + T^{2} \)
$17$ \( 484 + T^{2} \)
$19$ \( ( 16 + T )^{2} \)
$23$ \( 625 + T^{2} \)
$29$ \( ( 167 + T )^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( 17689 + T^{2} \)
$41$ \( ( 168 + T )^{2} \)
$43$ \( 9409 + T^{2} \)
$47$ \( 160000 + T^{2} \)
$53$ \( 33124 + T^{2} \)
$59$ \( ( 488 + T )^{2} \)
$61$ \( ( -28 + T )^{2} \)
$67$ \( 935089 + T^{2} \)
$71$ \( ( 285 + T )^{2} \)
$73$ \( 702244 + T^{2} \)
$79$ \( ( -469 + T )^{2} \)
$83$ \( 164836 + T^{2} \)
$89$ \( ( 324 + T )^{2} \)
$97$ \( 12996 + T^{2} \)
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