# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.77720518656.9 Defining polynomial: $$x^{8} - 121 x^{4} + 14641$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 180) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} -\beta_{7} q^{11} -\beta_{4} q^{13} + ( \beta_{6} + \beta_{7} ) q^{14} + ( -8 - 4 \beta_{5} ) q^{16} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{17} -6 \beta_{5} q^{19} + ( 3 \beta_{3} - \beta_{4} ) q^{22} -14 \beta_{2} q^{23} + ( -3 \beta_{6} + \beta_{7} ) q^{26} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{28} + \beta_{6} q^{29} -10 \beta_{5} q^{31} -16 \beta_{1} q^{32} + ( 18 - 3 \beta_{5} ) q^{34} + 3 \beta_{4} q^{37} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{38} + 2 \beta_{6} q^{41} + 4 \beta_{3} q^{43} + ( -6 \beta_{6} - 2 \beta_{7} ) q^{44} + ( -28 - 14 \beta_{5} ) q^{46} + 2 \beta_{2} q^{47} -39 q^{49} + ( -6 \beta_{3} - 2 \beta_{4} ) q^{52} + ( 18 \beta_{1} - 9 \beta_{2} ) q^{53} + 8 \beta_{6} q^{56} + ( \beta_{3} + \beta_{4} ) q^{58} -\beta_{7} q^{59} -58 q^{61} + ( -20 \beta_{1} - 20 \beta_{2} ) q^{62} -64 q^{64} + 2 \beta_{3} q^{67} + ( 12 \beta_{1} - 24 \beta_{2} ) q^{68} + 6 \beta_{7} q^{71} + ( 9 \beta_{6} - 3 \beta_{7} ) q^{74} + ( -72 - 12 \beta_{5} ) q^{76} + ( 88 \beta_{1} - 44 \beta_{2} ) q^{77} + 2 \beta_{5} q^{79} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{82} + 16 \beta_{2} q^{83} + ( -4 \beta_{6} - 4 \beta_{7} ) q^{86} -8 \beta_{4} q^{88} -8 \beta_{6} q^{89} -44 \beta_{5} q^{91} -56 \beta_{1} q^{92} + ( 4 + 2 \beta_{5} ) q^{94} -10 \beta_{4} q^{97} + ( -39 \beta_{1} + 39 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} + O(q^{10})$$ $$8q + 16q^{4} - 64q^{16} + 144q^{34} - 224q^{46} - 312q^{49} - 464q^{61} - 512q^{64} - 576q^{76} + 32q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 121 x^{4} + 14641$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/11$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{6}$$$$/1331$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - 22 \nu^{3} + 242 \nu$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} + 22 \nu^{5} + 242 \nu^{3} + 2662 \nu$$$$)/1331$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{4} - 242$$$$)/121$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{5} + 22 \nu^{3} + 242 \nu$$$$)/121$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} + 22 \nu^{5} - 242 \nu^{3} + 2662 \nu$$$$)/1331$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$11 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{6} - 11 \beta_{3}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$121 \beta_{5} + 242$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$121 \beta_{7} - 121 \beta_{6} + 121 \beta_{4} - 121 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$1331 \beta_{2}$$$$/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1331 \beta_{7} + 1331 \beta_{6} - 1331 \beta_{4} - 1331 \beta_{3}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.858406 + 3.20361i −0.858406 − 3.20361i −0.858406 + 3.20361i 0.858406 − 3.20361i −3.20361 − 0.858406i 3.20361 + 0.858406i 3.20361 − 0.858406i −3.20361 + 0.858406i
−1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.2 −1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.3 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.4 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.5 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.6 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.8 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
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.s 8
3.b odd 2 1 inner 900.3.c.s 8
4.b odd 2 1 inner 900.3.c.s 8
5.b even 2 1 inner 900.3.c.s 8
5.c odd 4 1 180.3.f.d 4
5.c odd 4 1 180.3.f.g yes 4
12.b even 2 1 inner 900.3.c.s 8
15.d odd 2 1 inner 900.3.c.s 8
15.e even 4 1 180.3.f.d 4
15.e even 4 1 180.3.f.g yes 4
20.d odd 2 1 inner 900.3.c.s 8
20.e even 4 1 180.3.f.d 4
20.e even 4 1 180.3.f.g yes 4
60.h even 2 1 inner 900.3.c.s 8
60.l odd 4 1 180.3.f.d 4
60.l odd 4 1 180.3.f.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.d 4 5.c odd 4 1
180.3.f.d 4 15.e even 4 1
180.3.f.d 4 20.e even 4 1
180.3.f.d 4 60.l odd 4 1
180.3.f.g yes 4 5.c odd 4 1
180.3.f.g yes 4 15.e even 4 1
180.3.f.g yes 4 20.e even 4 1
180.3.f.g yes 4 60.l odd 4 1
900.3.c.s 8 1.a even 1 1 trivial
900.3.c.s 8 3.b odd 2 1 inner
900.3.c.s 8 4.b odd 2 1 inner
900.3.c.s 8 5.b even 2 1 inner
900.3.c.s 8 12.b even 2 1 inner
900.3.c.s 8 15.d odd 2 1 inner
900.3.c.s 8 20.d odd 2 1 inner
900.3.c.s 8 60.h even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 88$$ $$T_{13}^{2} - 264$$ $$T_{17}^{2} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 88 + T^{2} )^{4}$$
$11$ $$( 264 + T^{2} )^{4}$$
$13$ $$( -264 + T^{2} )^{4}$$
$17$ $$( -108 + T^{2} )^{4}$$
$19$ $$( 432 + T^{2} )^{4}$$
$23$ $$( 784 + T^{2} )^{4}$$
$29$ $$( -88 + T^{2} )^{4}$$
$31$ $$( 1200 + T^{2} )^{4}$$
$37$ $$( -2376 + T^{2} )^{4}$$
$41$ $$( -352 + T^{2} )^{4}$$
$43$ $$( 1408 + T^{2} )^{4}$$
$47$ $$( 16 + T^{2} )^{4}$$
$53$ $$( -972 + T^{2} )^{4}$$
$59$ $$( 264 + T^{2} )^{4}$$
$61$ $$( 58 + T )^{8}$$
$67$ $$( 352 + T^{2} )^{4}$$
$71$ $$( 9504 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 48 + T^{2} )^{4}$$
$83$ $$( 1024 + T^{2} )^{4}$$
$89$ $$( -5632 + T^{2} )^{4}$$
$97$ $$( -26400 + T^{2} )^{4}$$
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