Properties

Label 525.6.d.d
Level $525$
Weight $6$
Character orbit 525.d
Analytic conductor $84.202$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(84.2015054018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 9 i q^{3} + 31 q^{4} -9 q^{6} -49 i q^{7} + 63 i q^{8} -81 q^{9} +O(q^{10})\) \( q + i q^{2} + 9 i q^{3} + 31 q^{4} -9 q^{6} -49 i q^{7} + 63 i q^{8} -81 q^{9} -340 q^{11} + 279 i q^{12} -454 i q^{13} + 49 q^{14} + 929 q^{16} -798 i q^{17} -81 i q^{18} -892 q^{19} + 441 q^{21} -340 i q^{22} + 3192 i q^{23} -567 q^{24} + 454 q^{26} -729 i q^{27} -1519 i q^{28} + 8242 q^{29} -2496 q^{31} + 2945 i q^{32} -3060 i q^{33} + 798 q^{34} -2511 q^{36} + 9798 i q^{37} -892 i q^{38} + 4086 q^{39} + 19834 q^{41} + 441 i q^{42} + 17236 i q^{43} -10540 q^{44} -3192 q^{46} + 8928 i q^{47} + 8361 i q^{48} -2401 q^{49} + 7182 q^{51} -14074 i q^{52} -150 i q^{53} + 729 q^{54} + 3087 q^{56} -8028 i q^{57} + 8242 i q^{58} + 42396 q^{59} + 14758 q^{61} -2496 i q^{62} + 3969 i q^{63} + 26783 q^{64} + 3060 q^{66} -1676 i q^{67} -24738 i q^{68} -28728 q^{69} + 14568 q^{71} -5103 i q^{72} -78378 i q^{73} -9798 q^{74} -27652 q^{76} + 16660 i q^{77} + 4086 i q^{78} + 2272 q^{79} + 6561 q^{81} + 19834 i q^{82} + 37764 i q^{83} + 13671 q^{84} -17236 q^{86} + 74178 i q^{87} -21420 i q^{88} + 117286 q^{89} -22246 q^{91} + 98952 i q^{92} -22464 i q^{93} -8928 q^{94} -26505 q^{96} + 10002 i q^{97} -2401 i q^{98} + 27540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 62q^{4} - 18q^{6} - 162q^{9} + O(q^{10}) \) \( 2q + 62q^{4} - 18q^{6} - 162q^{9} - 680q^{11} + 98q^{14} + 1858q^{16} - 1784q^{19} + 882q^{21} - 1134q^{24} + 908q^{26} + 16484q^{29} - 4992q^{31} + 1596q^{34} - 5022q^{36} + 8172q^{39} + 39668q^{41} - 21080q^{44} - 6384q^{46} - 4802q^{49} + 14364q^{51} + 1458q^{54} + 6174q^{56} + 84792q^{59} + 29516q^{61} + 53566q^{64} + 6120q^{66} - 57456q^{69} + 29136q^{71} - 19596q^{74} - 55304q^{76} + 4544q^{79} + 13122q^{81} + 27342q^{84} - 34472q^{86} + 234572q^{89} - 44492q^{91} - 17856q^{94} - 53010q^{96} + 55080q^{99} + O(q^{100}) \)

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
1.00000i 9.00000i 31.0000 0 −9.00000 49.0000i 63.0000i −81.0000 0
274.2 1.00000i 9.00000i 31.0000 0 −9.00000 49.0000i 63.0000i −81.0000 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 525.6.d.d 2
5.b even 2 1 inner 525.6.d.d 2
5.c odd 4 1 21.6.a.b 1
5.c odd 4 1 525.6.a.c 1
15.e even 4 1 63.6.a.c 1
20.e even 4 1 336.6.a.l 1
35.f even 4 1 147.6.a.e 1
35.k even 12 2 147.6.e.e 2
35.l odd 12 2 147.6.e.f 2
60.l odd 4 1 1008.6.a.t 1
105.k odd 4 1 441.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 5.c odd 4 1
63.6.a.c 1 15.e even 4 1
147.6.a.e 1 35.f even 4 1
147.6.e.e 2 35.k even 12 2
147.6.e.f 2 35.l odd 12 2
336.6.a.l 1 20.e even 4 1
441.6.a.d 1 105.k odd 4 1
525.6.a.c 1 5.c odd 4 1
525.6.d.d 2 1.a even 1 1 trivial
525.6.d.d 2 5.b even 2 1 inner
1008.6.a.t 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 81 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 2401 + T^{2} \)
$11$ \( ( 340 + T )^{2} \)
$13$ \( 206116 + T^{2} \)
$17$ \( 636804 + T^{2} \)
$19$ \( ( 892 + T )^{2} \)
$23$ \( 10188864 + T^{2} \)
$29$ \( ( -8242 + T )^{2} \)
$31$ \( ( 2496 + T )^{2} \)
$37$ \( 96000804 + T^{2} \)
$41$ \( ( -19834 + T )^{2} \)
$43$ \( 297079696 + T^{2} \)
$47$ \( 79709184 + T^{2} \)
$53$ \( 22500 + T^{2} \)
$59$ \( ( -42396 + T )^{2} \)
$61$ \( ( -14758 + T )^{2} \)
$67$ \( 2808976 + T^{2} \)
$71$ \( ( -14568 + T )^{2} \)
$73$ \( 6143110884 + T^{2} \)
$79$ \( ( -2272 + T )^{2} \)
$83$ \( 1426119696 + T^{2} \)
$89$ \( ( -117286 + T )^{2} \)
$97$ \( 100040004 + T^{2} \)
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