# Properties

 Label 525.3.f.a Level $525$ Weight $3$ Character orbit 525.f Analytic conductor $14.305$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 525.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3052138789$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4337012736.1 Defining polynomial: $$x^{8} - 4 x^{7} + 8 x^{6} + 4 x^{5} + 12 x^{4} - 40 x^{3} + 72 x^{2} + 24 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{3} + ( 3 + 2 \beta_{1} ) q^{4} + ( 3 + 2 \beta_{1} - \beta_{7} ) q^{6} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( -3 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{8} + ( 6 + 2 \beta_{1} - 3 \beta_{5} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{3} + ( 3 + 2 \beta_{1} ) q^{4} + ( 3 + 2 \beta_{1} - \beta_{7} ) q^{6} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( -3 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{8} + ( 6 + 2 \beta_{1} - 3 \beta_{5} + 2 \beta_{7} ) q^{9} -2 \beta_{5} q^{11} + ( -7 \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{12} + ( \beta_{2} - \beta_{3} + 8 \beta_{4} - \beta_{6} ) q^{13} + ( 1 + \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{14} + ( 9 + 4 \beta_{1} ) q^{16} + ( -2 \beta_{3} + 2 \beta_{6} ) q^{17} + ( -8 \beta_{2} + 2 \beta_{3} - 13 \beta_{4} ) q^{18} + ( -3 - 5 \beta_{1} ) q^{19} + ( 3 - \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{21} + ( -4 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} + 4 \beta_{6} ) q^{22} + ( 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{23} + ( 33 + 6 \beta_{1} - 6 \beta_{5} + 3 \beta_{7} ) q^{24} + ( 1 + \beta_{1} + 10 \beta_{5} + 2 \beta_{7} ) q^{26} + ( -14 \beta_{2} + 8 \beta_{3} + 11 \beta_{4} + 9 \beta_{6} ) q^{27} + ( 3 \beta_{2} - 3 \beta_{3} + 11 \beta_{4} - 3 \beta_{6} ) q^{28} + ( 2 + 2 \beta_{1} - 6 \beta_{5} + 4 \beta_{7} ) q^{29} + ( 34 - 2 \beta_{1} ) q^{31} + ( -5 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{32} + ( -4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 6 \beta_{6} ) q^{33} -6 \beta_{1} q^{34} + ( 24 + 8 \beta_{1} - 3 \beta_{5} - 10 \beta_{7} ) q^{36} + ( -14 \beta_{2} + 14 \beta_{3} + 18 \beta_{4} + 14 \beta_{6} ) q^{37} + ( 13 \beta_{2} - 5 \beta_{3} + 5 \beta_{6} ) q^{38} + ( 3 + 8 \beta_{1} + 3 \beta_{5} + 8 \beta_{7} ) q^{39} + ( 8 + 8 \beta_{1} + 6 \beta_{5} + 16 \beta_{7} ) q^{41} + ( \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{6} ) q^{42} + ( -6 \beta_{2} + 6 \beta_{3} + 46 \beta_{4} + 6 \beta_{6} ) q^{43} + ( -4 - 4 \beta_{1} - 10 \beta_{5} - 8 \beta_{7} ) q^{44} + ( -42 - 18 \beta_{1} ) q^{46} + ( -12 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{47} + ( -17 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} ) q^{48} -7 q^{49} + ( -24 + 6 \beta_{5} - 6 \beta_{7} ) q^{51} + ( 21 \beta_{2} - 21 \beta_{3} + 20 \beta_{4} - 21 \beta_{6} ) q^{52} + ( -6 \beta_{2} - 16 \beta_{3} + 16 \beta_{6} ) q^{53} + ( 30 + \beta_{1} - 6 \beta_{5} - 17 \beta_{7} ) q^{54} + ( -1 - \beta_{1} + 13 \beta_{5} - 2 \beta_{7} ) q^{56} + ( 13 \beta_{2} + 2 \beta_{3} + 17 \beta_{4} ) q^{57} + ( -2 \beta_{2} + 2 \beta_{3} - 26 \beta_{4} + 2 \beta_{6} ) q^{58} + ( 1 + \beta_{1} - 28 \beta_{5} + 2 \beta_{7} ) q^{59} + ( -39 - 7 \beta_{1} ) q^{61} + ( -30 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{62} + ( 4 \beta_{2} - \beta_{3} + 2 \beta_{4} - 9 \beta_{6} ) q^{63} + ( -1 - 18 \beta_{1} ) q^{64} + ( -12 - 10 \beta_{1} - 12 \beta_{5} - 10 \beta_{7} ) q^{66} + ( 8 \beta_{2} - 8 \beta_{3} - 14 \beta_{4} - 8 \beta_{6} ) q^{67} + ( 12 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{68} + ( -42 - 12 \beta_{1} + 6 \beta_{5} ) q^{69} + ( 6 + 6 \beta_{1} - 30 \beta_{5} + 12 \beta_{7} ) q^{71} + ( -54 \beta_{2} + 36 \beta_{3} + 27 \beta_{4} + 18 \beta_{6} ) q^{72} + ( 26 \beta_{2} - 26 \beta_{3} - 18 \beta_{4} - 26 \beta_{6} ) q^{73} + ( -14 - 14 \beta_{1} - 10 \beta_{5} - 28 \beta_{7} ) q^{74} + ( -79 - 21 \beta_{1} ) q^{76} + ( -4 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{77} + ( 19 \beta_{2} - 22 \beta_{3} + 23 \beta_{4} - 30 \beta_{6} ) q^{78} + ( -32 - 36 \beta_{1} ) q^{79} + ( -15 + 22 \beta_{1} - 24 \beta_{5} + 4 \beta_{7} ) q^{81} + ( 52 \beta_{2} - 52 \beta_{3} + 46 \beta_{4} - 52 \beta_{6} ) q^{82} + ( -27 \beta_{2} - 9 \beta_{3} + 9 \beta_{6} ) q^{83} + ( 9 + 11 \beta_{1} + 9 \beta_{5} + 11 \beta_{7} ) q^{84} + ( -6 - 6 \beta_{1} + 34 \beta_{5} - 12 \beta_{7} ) q^{86} + ( -14 \beta_{2} + 8 \beta_{3} + 20 \beta_{4} + 18 \beta_{6} ) q^{87} + ( -24 \beta_{2} + 24 \beta_{3} - 18 \beta_{4} + 24 \beta_{6} ) q^{88} + ( 16 + 16 \beta_{1} + 10 \beta_{5} + 32 \beta_{7} ) q^{89} + ( -7 - 9 \beta_{1} ) q^{91} + ( 54 \beta_{2} - 10 \beta_{3} + 10 \beta_{6} ) q^{92} + ( -30 \beta_{2} + 36 \beta_{3} + 42 \beta_{4} ) q^{93} + ( 84 + 12 \beta_{1} ) q^{94} + ( -33 + 10 \beta_{1} + 12 \beta_{5} - 17 \beta_{7} ) q^{96} + ( -8 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} + 8 \beta_{6} ) q^{97} + 7 \beta_{2} q^{98} + ( -30 - 4 \beta_{1} - 12 \beta_{5} - 4 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{4} + 28q^{6} + 40q^{9} + O(q^{10})$$ $$8q + 24q^{4} + 28q^{6} + 40q^{9} + 72q^{16} - 24q^{19} + 28q^{21} + 252q^{24} + 272q^{31} + 232q^{36} - 8q^{39} - 336q^{46} - 56q^{49} - 168q^{51} + 308q^{54} - 312q^{61} - 8q^{64} - 56q^{66} - 336q^{69} - 632q^{76} - 256q^{79} - 136q^{81} + 28q^{84} - 56q^{91} + 672q^{94} - 196q^{96} - 224q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 8 x^{6} + 4 x^{5} + 12 x^{4} - 40 x^{3} + 72 x^{2} + 24 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - 7 \nu^{6} + 29 \nu^{5} - 83 \nu^{4} + 110 \nu^{3} - 68 \nu^{2} - 26 \nu - 804$$$$)/302$$ $$\beta_{2}$$ $$=$$ $$($$$$40 \nu^{7} - 129 \nu^{6} + 254 \nu^{5} + 153 \nu^{4} + 1380 \nu^{3} - 1210 \nu^{2} - 436 \nu + 3476$$$$)/2718$$ $$\beta_{3}$$ $$=$$ $$($$$$-104 \nu^{7} + 879 \nu^{6} - 2563 \nu^{5} + 2592 \nu^{4} + 1848 \nu^{3} + 8582 \nu^{2} - 26590 \nu + 15968$$$$)/5436$$ $$\beta_{4}$$ $$=$$ $$($$$$-379 \nu^{7} + 1596 \nu^{6} - 3290 \nu^{5} - 1008 \nu^{4} - 4242 \nu^{3} + 17920 \nu^{2} - 29708 \nu - 4532$$$$)/5436$$ $$\beta_{5}$$ $$=$$ $$($$$$51 \nu^{7} - 206 \nu^{6} + 422 \nu^{5} + 146 \nu^{4} + 778 \nu^{3} - 2260 \nu^{2} + 4412 \nu + 672$$$$)/604$$ $$\beta_{6}$$ $$=$$ $$($$$$-494 \nu^{7} + 1797 \nu^{6} - 3001 \nu^{5} - 3996 \nu^{4} - 4812 \nu^{3} + 16982 \nu^{2} - 23698 \nu - 16564$$$$)/5436$$ $$\beta_{7}$$ $$=$$ $$($$$$-105 \nu^{7} + 433 \nu^{6} - 931 \nu^{5} - 194 \nu^{4} - 1282 \nu^{3} + 3818 \nu^{2} - 8142 \nu - 744$$$$)/604$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 3 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - \beta_{1} - 5$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} - 2 \beta_{3} + 12 \beta_{2} - 10 \beta_{1} - 30$$ $$\nu^{5}$$ $$=$$ $$-12 \beta_{7} + 26 \beta_{6} - 25 \beta_{5} - 37 \beta_{4} + 14 \beta_{3} + 11 \beta_{2} - 26 \beta_{1} - 63$$ $$\nu^{6}$$ $$=$$ $$-52 \beta_{7} + 100 \beta_{6} - 102 \beta_{5} - 174 \beta_{4} + 100 \beta_{3} - 100 \beta_{2} - 26 \beta_{1} - 26$$ $$\nu^{7}$$ $$=$$ $$-126 \beta_{7} + 154 \beta_{6} - 238 \beta_{5} - 368 \beta_{4} + 280 \beta_{3} - 518 \beta_{2} + 154 \beta_{1} + 522$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −1.25296 − 1.25296i −1.25296 + 1.25296i −0.153548 + 0.153548i −0.153548 − 0.153548i 1.15355 − 1.15355i 1.15355 + 1.15355i 2.25296 + 2.25296i 2.25296 − 2.25296i
−3.50592 −2.88494 0.822876i 8.29150 0 10.1144 + 2.88494i 2.64575i −15.0457 7.64575 + 4.74789i 0
449.2 −3.50592 −2.88494 + 0.822876i 8.29150 0 10.1144 2.88494i 2.64575i −15.0457 7.64575 4.74789i 0
449.3 −1.30710 2.38267 1.82288i −2.29150 0 −3.11438 + 2.38267i 2.64575i 8.22359 2.35425 8.68663i 0
449.4 −1.30710 2.38267 + 1.82288i −2.29150 0 −3.11438 2.38267i 2.64575i 8.22359 2.35425 + 8.68663i 0
449.5 1.30710 −2.38267 1.82288i −2.29150 0 −3.11438 2.38267i 2.64575i −8.22359 2.35425 + 8.68663i 0
449.6 1.30710 −2.38267 + 1.82288i −2.29150 0 −3.11438 + 2.38267i 2.64575i −8.22359 2.35425 8.68663i 0
449.7 3.50592 2.88494 0.822876i 8.29150 0 10.1144 2.88494i 2.64575i 15.0457 7.64575 4.74789i 0
449.8 3.50592 2.88494 + 0.822876i 8.29150 0 10.1144 + 2.88494i 2.64575i 15.0457 7.64575 + 4.74789i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.f.a 8
3.b odd 2 1 inner 525.3.f.a 8
5.b even 2 1 inner 525.3.f.a 8
5.c odd 4 1 21.3.b.a 4
5.c odd 4 1 525.3.c.a 4
15.d odd 2 1 inner 525.3.f.a 8
15.e even 4 1 21.3.b.a 4
15.e even 4 1 525.3.c.a 4
20.e even 4 1 336.3.d.c 4
35.f even 4 1 147.3.b.f 4
35.k even 12 2 147.3.h.c 8
35.l odd 12 2 147.3.h.e 8
40.i odd 4 1 1344.3.d.f 4
40.k even 4 1 1344.3.d.b 4
45.k odd 12 2 567.3.r.c 8
45.l even 12 2 567.3.r.c 8
60.l odd 4 1 336.3.d.c 4
105.k odd 4 1 147.3.b.f 4
105.w odd 12 2 147.3.h.c 8
105.x even 12 2 147.3.h.e 8
120.q odd 4 1 1344.3.d.b 4
120.w even 4 1 1344.3.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 5.c odd 4 1
21.3.b.a 4 15.e even 4 1
147.3.b.f 4 35.f even 4 1
147.3.b.f 4 105.k odd 4 1
147.3.h.c 8 35.k even 12 2
147.3.h.c 8 105.w odd 12 2
147.3.h.e 8 35.l odd 12 2
147.3.h.e 8 105.x even 12 2
336.3.d.c 4 20.e even 4 1
336.3.d.c 4 60.l odd 4 1
525.3.c.a 4 5.c odd 4 1
525.3.c.a 4 15.e even 4 1
525.3.f.a 8 1.a even 1 1 trivial
525.3.f.a 8 3.b odd 2 1 inner
525.3.f.a 8 5.b even 2 1 inner
525.3.f.a 8 15.d odd 2 1 inner
567.3.r.c 8 45.k odd 12 2
567.3.r.c 8 45.l even 12 2
1344.3.d.b 4 40.k even 4 1
1344.3.d.b 4 120.q odd 4 1
1344.3.d.f 4 40.i odd 4 1
1344.3.d.f 4 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 14 T_{2}^{2} + 21$$ acting on $$S_{3}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 5 T^{4} + 32 T^{6} + 256 T^{8} )^{2}$$
$3$ $$1 - 20 T^{2} + 234 T^{4} - 1620 T^{6} + 6561 T^{8}$$
$5$ 1
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 - 428 T^{2} + 74630 T^{4} - 6266348 T^{6} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 - 500 T^{2} + 117354 T^{4} - 14280500 T^{6} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 + 988 T^{2} + 407046 T^{4} + 82518748 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 + 6 T + 556 T^{2} + 2166 T^{3} + 130321 T^{4} )^{4}$$
$23$ $$( 1 + 1444 T^{2} + 980166 T^{4} + 404090404 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8} )^{2}$$
$31$ $$( 1 - 68 T + 3050 T^{2} - 65348 T^{3} + 923521 T^{4} )^{4}$$
$37$ $$( 1 - 2700 T^{2} + 5483014 T^{4} - 5060234700 T^{6} + 3512479453921 T^{8} )^{2}$$
$41$ $$( 1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 - 3692 T^{2} + 8632518 T^{4} - 12622213292 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 + 6148 T^{2} + 19144326 T^{4} + 30000278788 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$( 1 - 20 T^{2} - 13350138 T^{4} - 157809620 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 3676 T^{2} + 15964266 T^{4} - 44543419036 T^{6} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 + 78 T + 8620 T^{2} + 290238 T^{3} + 13845841 T^{4} )^{4}$$
$67$ $$( 1 - 16988 T^{2} + 112385766 T^{4} - 342327243548 T^{6} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 11724 T^{2} + 89948134 T^{4} - 332940977484 T^{6} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 + 64 T + 4434 T^{2} + 399424 T^{3} + 38950081 T^{4} )^{4}$$
$83$ $$( 1 + 13948 T^{2} + 141899946 T^{4} + 661948661308 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 36732 T^{2} + 514361350 T^{4} - 3251857549692 T^{6} + 7837433594376961 T^{8} )^{2}$$
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