# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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