# Properties

 Label 900.3.f.e Level $900$ Weight $3$ Character orbit 900.f Analytic conductor $24.523$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{20} q^{2} + 4 \zeta_{20}^{2} q^{4} + ( -8 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{7} -8 \zeta_{20}^{3} q^{8} +O(q^{10})$$ $$q -2 \zeta_{20} q^{2} + 4 \zeta_{20}^{2} q^{4} + ( -8 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{7} -8 \zeta_{20}^{3} q^{8} + ( -4 + 8 \zeta_{20}^{2} + 4 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{11} + ( 4 \zeta_{20}^{3} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{13} + ( 4 + 12 \zeta_{20}^{2} - 8 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{14} + 16 \zeta_{20}^{4} q^{16} + ( -16 \zeta_{20}^{3} + 14 \zeta_{20}^{5} - 16 \zeta_{20}^{7} ) q^{17} + ( -8 \zeta_{20}^{4} - 8 \zeta_{20}^{6} ) q^{19} + ( 8 \zeta_{20} - 16 \zeta_{20}^{3} - 8 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{22} + ( -8 \zeta_{20} + 10 \zeta_{20}^{3} - 4 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{23} + ( 8 - 8 \zeta_{20}^{2} - 12 \zeta_{20}^{6} ) q^{26} + ( -8 \zeta_{20} - 24 \zeta_{20}^{3} + 16 \zeta_{20}^{5} - 8 \zeta_{20}^{7} ) q^{28} + ( -6 - 8 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{29} + ( -20 + 40 \zeta_{20}^{2} - 28 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{31} -32 \zeta_{20}^{5} q^{32} + ( -32 + 32 \zeta_{20}^{2} + 4 \zeta_{20}^{6} ) q^{34} + ( -20 \zeta_{20}^{3} + 14 \zeta_{20}^{5} - 20 \zeta_{20}^{7} ) q^{37} + ( 16 \zeta_{20}^{5} + 16 \zeta_{20}^{7} ) q^{38} + ( 22 - 12 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{41} + ( -16 \zeta_{20} + 2 \zeta_{20}^{3} - 8 \zeta_{20}^{5} + 14 \zeta_{20}^{7} ) q^{43} + ( -48 + 32 \zeta_{20}^{2} - 16 \zeta_{20}^{4} + 64 \zeta_{20}^{6} ) q^{44} + ( -4 + 20 \zeta_{20}^{2} - 24 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{46} + ( -24 \zeta_{20} - 14 \zeta_{20}^{3} - 12 \zeta_{20}^{5} + 38 \zeta_{20}^{7} ) q^{47} + ( -9 - 20 \zeta_{20}^{4} + 20 \zeta_{20}^{6} ) q^{49} + ( -16 \zeta_{20} + 16 \zeta_{20}^{3} + 24 \zeta_{20}^{7} ) q^{52} + ( -20 \zeta_{20}^{3} - 34 \zeta_{20}^{5} - 20 \zeta_{20}^{7} ) q^{53} + ( -16 + 32 \zeta_{20}^{2} + 32 \zeta_{20}^{4} - 16 \zeta_{20}^{6} ) q^{56} + ( 12 \zeta_{20} + 16 \zeta_{20}^{5} - 16 \zeta_{20}^{7} ) q^{58} + ( -24 + 48 \zeta_{20}^{2} - 48 \zeta_{20}^{4} ) q^{59} + ( 58 + 52 \zeta_{20}^{4} - 52 \zeta_{20}^{6} ) q^{61} + ( 40 \zeta_{20} - 80 \zeta_{20}^{3} + 56 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{62} + 64 \zeta_{20}^{6} q^{64} + ( 80 \zeta_{20} - 62 \zeta_{20}^{3} + 40 \zeta_{20}^{5} - 18 \zeta_{20}^{7} ) q^{67} + ( 64 \zeta_{20} - 64 \zeta_{20}^{3} - 8 \zeta_{20}^{7} ) q^{68} + ( 4 - 8 \zeta_{20}^{2} - 36 \zeta_{20}^{4} - 44 \zeta_{20}^{6} ) q^{71} + ( 64 \zeta_{20}^{3} - 98 \zeta_{20}^{5} + 64 \zeta_{20}^{7} ) q^{73} + ( -40 + 40 \zeta_{20}^{2} + 12 \zeta_{20}^{6} ) q^{74} + ( 32 - 32 \zeta_{20}^{2} + 32 \zeta_{20}^{4} - 64 \zeta_{20}^{6} ) q^{76} + ( -40 \zeta_{20}^{3} - 40 \zeta_{20}^{5} - 40 \zeta_{20}^{7} ) q^{77} + ( -40 + 80 \zeta_{20}^{2} - 72 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{79} + ( -44 \zeta_{20} + 24 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{82} + ( -48 \zeta_{20} - 2 \zeta_{20}^{3} - 24 \zeta_{20}^{5} + 50 \zeta_{20}^{7} ) q^{83} + ( 28 + 4 \zeta_{20}^{2} + 24 \zeta_{20}^{4} - 12 \zeta_{20}^{6} ) q^{86} + ( 96 \zeta_{20} - 64 \zeta_{20}^{3} + 32 \zeta_{20}^{5} - 128 \zeta_{20}^{7} ) q^{88} + ( -62 - 80 \zeta_{20}^{4} + 80 \zeta_{20}^{6} ) q^{89} + ( 12 - 24 \zeta_{20}^{2} - 4 \zeta_{20}^{4} - 28 \zeta_{20}^{6} ) q^{91} + ( 8 \zeta_{20} - 40 \zeta_{20}^{3} + 48 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{92} + ( 76 - 28 \zeta_{20}^{2} + 104 \zeta_{20}^{4} - 52 \zeta_{20}^{6} ) q^{94} + ( -24 \zeta_{20}^{3} - 54 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{97} + ( 18 \zeta_{20} + 40 \zeta_{20}^{5} - 40 \zeta_{20}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + O(q^{10})$$ $$8q + 8q^{4} + 80q^{14} - 32q^{16} + 24q^{26} - 16q^{29} - 184q^{34} + 224q^{41} - 160q^{44} + 80q^{46} + 8q^{49} - 160q^{56} + 256q^{61} + 128q^{64} - 216q^{74} + 160q^{86} - 176q^{89} + 240q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.951057 + 0.309017i 0.951057 − 0.309017i 0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 + 0.809017i −0.587785 − 0.809017i −0.951057 + 0.309017i −0.951057 − 0.309017i
−1.90211 0.618034i 0 3.23607 + 2.35114i 0 0 −5.25731 −4.70228 6.47214i 0 0
199.2 −1.90211 + 0.618034i 0 3.23607 2.35114i 0 0 −5.25731 −4.70228 + 6.47214i 0 0
199.3 −1.17557 1.61803i 0 −1.23607 + 3.80423i 0 0 −8.50651 7.60845 2.47214i 0 0
199.4 −1.17557 + 1.61803i 0 −1.23607 3.80423i 0 0 −8.50651 7.60845 + 2.47214i 0 0
199.5 1.17557 1.61803i 0 −1.23607 3.80423i 0 0 8.50651 −7.60845 2.47214i 0 0
199.6 1.17557 + 1.61803i 0 −1.23607 + 3.80423i 0 0 8.50651 −7.60845 + 2.47214i 0 0
199.7 1.90211 0.618034i 0 3.23607 2.35114i 0 0 5.25731 4.70228 6.47214i 0 0
199.8 1.90211 + 0.618034i 0 3.23607 + 2.35114i 0 0 5.25731 4.70228 + 6.47214i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.e 8
3.b odd 2 1 100.3.d.b 8
4.b odd 2 1 inner 900.3.f.e 8
5.b even 2 1 inner 900.3.f.e 8
5.c odd 4 1 180.3.c.a 4
5.c odd 4 1 900.3.c.k 4
12.b even 2 1 100.3.d.b 8
15.d odd 2 1 100.3.d.b 8
15.e even 4 1 20.3.b.a 4
15.e even 4 1 100.3.b.f 4
20.d odd 2 1 inner 900.3.f.e 8
20.e even 4 1 180.3.c.a 4
20.e even 4 1 900.3.c.k 4
24.f even 2 1 1600.3.h.n 8
24.h odd 2 1 1600.3.h.n 8
40.i odd 4 1 2880.3.e.e 4
40.k even 4 1 2880.3.e.e 4
60.h even 2 1 100.3.d.b 8
60.l odd 4 1 20.3.b.a 4
60.l odd 4 1 100.3.b.f 4
120.i odd 2 1 1600.3.h.n 8
120.m even 2 1 1600.3.h.n 8
120.q odd 4 1 320.3.b.c 4
120.q odd 4 1 1600.3.b.s 4
120.w even 4 1 320.3.b.c 4
120.w even 4 1 1600.3.b.s 4
240.z odd 4 1 1280.3.g.e 8
240.bb even 4 1 1280.3.g.e 8
240.bd odd 4 1 1280.3.g.e 8
240.bf even 4 1 1280.3.g.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 15.e even 4 1
20.3.b.a 4 60.l odd 4 1
100.3.b.f 4 15.e even 4 1
100.3.b.f 4 60.l odd 4 1
100.3.d.b 8 3.b odd 2 1
100.3.d.b 8 12.b even 2 1
100.3.d.b 8 15.d odd 2 1
100.3.d.b 8 60.h even 2 1
180.3.c.a 4 5.c odd 4 1
180.3.c.a 4 20.e even 4 1
320.3.b.c 4 120.q odd 4 1
320.3.b.c 4 120.w even 4 1
900.3.c.k 4 5.c odd 4 1
900.3.c.k 4 20.e even 4 1
900.3.f.e 8 1.a even 1 1 trivial
900.3.f.e 8 4.b odd 2 1 inner
900.3.f.e 8 5.b even 2 1 inner
900.3.f.e 8 20.d odd 2 1 inner
1280.3.g.e 8 240.z odd 4 1
1280.3.g.e 8 240.bb even 4 1
1280.3.g.e 8 240.bd odd 4 1
1280.3.g.e 8 240.bf even 4 1
1600.3.b.s 4 120.q odd 4 1
1600.3.b.s 4 120.w even 4 1
1600.3.h.n 8 24.f even 2 1
1600.3.h.n 8 24.h odd 2 1
1600.3.h.n 8 120.i odd 2 1
1600.3.h.n 8 120.m even 2 1
2880.3.e.e 4 40.i odd 4 1
2880.3.e.e 4 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{4} - 100 T_{7}^{2} + 2000$$ $$T_{29}^{2} + 4 T_{29} - 76$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 64 T^{2} + 16 T^{4} - 4 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 2000 - 100 T^{2} + T^{4} )^{2}$$
$11$ $$( 1280 + 400 T^{2} + T^{4} )^{2}$$
$13$ $$( 16 + 72 T^{2} + T^{4} )^{2}$$
$17$ $$( 80656 + 712 T^{2} + T^{4} )^{2}$$
$19$ $$( 20480 + 320 T^{2} + T^{4} )^{2}$$
$23$ $$( 80 - 260 T^{2} + T^{4} )^{2}$$
$29$ $$( -76 + 4 T + T^{2} )^{4}$$
$31$ $$( 154880 + 2320 T^{2} + T^{4} )^{2}$$
$37$ $$( 234256 + 1032 T^{2} + T^{4} )^{2}$$
$41$ $$( 604 - 56 T + T^{2} )^{4}$$
$43$ $$( 2000 - 500 T^{2} + T^{4} )^{2}$$
$47$ $$( 3561680 - 4100 T^{2} + T^{4} )^{2}$$
$53$ $$( 2062096 + 4872 T^{2} + T^{4} )^{2}$$
$59$ $$( 1658880 + 5760 T^{2} + T^{4} )^{2}$$
$61$ $$( -2356 - 64 T + T^{2} )^{4}$$
$67$ $$( 19920080 - 10420 T^{2} + T^{4} )^{2}$$
$71$ $$( 10138880 + 8080 T^{2} + T^{4} )^{2}$$
$73$ $$( 583696 + 18952 T^{2} + T^{4} )^{2}$$
$79$ $$( 2478080 + 13120 T^{2} + T^{4} )^{2}$$
$83$ $$( 2620880 - 6260 T^{2} + T^{4} )^{2}$$
$89$ $$( -7516 + 44 T + T^{2} )^{4}$$
$97$ $$( 13220496 + 10152 T^{2} + T^{4} )^{2}$$
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