# Properties

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

# Related objects

## 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: 8.0.12239922073600.1 Defining polynomial: $$x^{8} - 2 x^{6} + 4 x^{4} - 32 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( 1 + \beta_{7} ) q^{4} -\beta_{2} q^{7} + ( -\beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} + ( 1 + \beta_{7} ) q^{4} -\beta_{2} q^{7} + ( -\beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{4} - \beta_{5} - \beta_{6} ) q^{11} + ( -2 - \beta_{3} - 2 \beta_{7} ) q^{13} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{14} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{16} + ( -\beta_{1} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{19} + ( 3 - 2 \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{22} + ( 2 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{5} - \beta_{6} ) q^{26} + ( -6 - 2 \beta_{2} - 3 \beta_{3} - \beta_{7} ) q^{28} + ( -\beta_{1} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{29} + ( -2 - \beta_{2} + 2 \beta_{3} - 4 \beta_{7} ) q^{31} + ( -4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{32} + ( -11 + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{7} ) q^{34} + ( -4 - 2 \beta_{3} - 4 \beta_{7} ) q^{37} + ( -\beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{38} + ( 3 \beta_{1} - \beta_{4} + \beta_{5} + 13 \beta_{6} ) q^{41} + ( 5 - \beta_{2} - 5 \beta_{3} + 10 \beta_{7} ) q^{43} + ( -4 \beta_{1} - 4 \beta_{4} + 4 \beta_{6} ) q^{44} + ( 15 + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{7} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + 14 \beta_{6} ) q^{47} + ( -14 - 4 \beta_{3} - 8 \beta_{7} ) q^{49} + ( -29 + 4 \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{52} + ( 5 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + 17 \beta_{6} ) q^{53} + ( -8 \beta_{1} - 4 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{56} + ( 7 + 6 \beta_{2} - 9 \beta_{3} + 2 \beta_{7} ) q^{58} + ( -6 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{59} + ( 12 + 7 \beta_{3} + 14 \beta_{7} ) q^{61} + ( 3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{62} + ( 18 - 4 \beta_{2} - 6 \beta_{3} - 4 \beta_{7} ) q^{64} + ( 5 + 7 \beta_{2} - 5 \beta_{3} + 10 \beta_{7} ) q^{67} + ( -4 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} ) q^{68} + ( 10 \beta_{1} + \beta_{4} + \beta_{5} - 39 \beta_{6} ) q^{71} + ( -40 - 10 \beta_{3} - 20 \beta_{7} ) q^{73} + ( -4 \beta_{1} + 4 \beta_{5} - 2 \beta_{6} ) q^{74} + ( -28 - 2 \beta_{2} + 5 \beta_{3} + \beta_{7} ) q^{76} + ( -7 \beta_{1} - \beta_{4} + \beta_{5} - 27 \beta_{6} ) q^{77} + ( 6 - 8 \beta_{2} - 6 \beta_{3} + 12 \beta_{7} ) q^{79} + ( 69 + 2 \beta_{2} - 3 \beta_{3} + 14 \beta_{7} ) q^{82} + ( 4 \beta_{1} - 10 \beta_{4} - 10 \beta_{5} - 26 \beta_{6} ) q^{83} + ( -11 \beta_{1} - 2 \beta_{4} - 11 \beta_{5} + 10 \beta_{6} ) q^{86} + ( -36 - 8 \beta_{3} + 4 \beta_{7} ) q^{88} + ( 8 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} + 28 \beta_{6} ) q^{89} + ( -5 + 7 \beta_{2} + 5 \beta_{3} - 10 \beta_{7} ) q^{91} + ( 12 \beta_{1} + 12 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} ) q^{92} + ( -42 - 4 \beta_{2} - 2 \beta_{3} + 12 \beta_{7} ) q^{94} -15 q^{97} + ( -8 \beta_{1} + 8 \beta_{5} - 10 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} - 8q^{13} - 8q^{16} + 32q^{22} - 44q^{28} - 80q^{34} - 16q^{37} + 128q^{46} - 80q^{49} - 236q^{52} + 48q^{58} + 40q^{61} + 160q^{64} - 240q^{73} - 228q^{76} + 496q^{82} - 304q^{88} - 384q^{94} - 120q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 4 x^{4} - 32 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 6 \nu^{4} + 4 \nu^{2} - 16$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{3} - 2 \nu$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} - 28 \nu^{3} + 32 \nu$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} - 4 \nu^{3} + 32 \nu$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} - 4 \nu^{2} + 16$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{6} + 4 \beta_{5} + 2 \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{7} - 2 \beta_{3} + 4 \beta_{2} + 14$$ $$\nu^{7}$$ $$=$$ $$-56 \beta_{6} + 8 \beta_{5} + 6 \beta_{1}$$

## 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
 −1.89639 + 0.635381i −1.89639 − 0.635381i −0.950636 + 1.75963i −0.950636 − 1.75963i 0.950636 + 1.75963i 0.950636 − 1.75963i 1.89639 + 0.635381i 1.89639 − 0.635381i
−1.89639 0.635381i 0 3.19258 + 2.40986i 0 0 10.1035i −4.52320 6.59853i 0 0
451.2 −1.89639 + 0.635381i 0 3.19258 2.40986i 0 0 10.1035i −4.52320 + 6.59853i 0 0
451.3 −0.950636 1.75963i 0 −2.19258 + 3.34553i 0 0 3.98982i 7.97124 + 0.677747i 0 0
451.4 −0.950636 + 1.75963i 0 −2.19258 3.34553i 0 0 3.98982i 7.97124 0.677747i 0 0
451.5 0.950636 1.75963i 0 −2.19258 3.34553i 0 0 3.98982i −7.97124 + 0.677747i 0 0
451.6 0.950636 + 1.75963i 0 −2.19258 + 3.34553i 0 0 3.98982i −7.97124 0.677747i 0 0
451.7 1.89639 0.635381i 0 3.19258 2.40986i 0 0 10.1035i 4.52320 6.59853i 0 0
451.8 1.89639 + 0.635381i 0 3.19258 + 2.40986i 0 0 10.1035i 4.52320 + 6.59853i 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
12.b 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.p 8
3.b odd 2 1 inner 900.3.c.p 8
4.b odd 2 1 inner 900.3.c.p 8
5.b even 2 1 900.3.c.q yes 8
5.c odd 4 2 900.3.f.g 16
12.b even 2 1 inner 900.3.c.p 8
15.d odd 2 1 900.3.c.q yes 8
15.e even 4 2 900.3.f.g 16
20.d odd 2 1 900.3.c.q yes 8
20.e even 4 2 900.3.f.g 16
60.h even 2 1 900.3.c.q yes 8
60.l odd 4 2 900.3.f.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.c.p 8 1.a even 1 1 trivial
900.3.c.p 8 3.b odd 2 1 inner
900.3.c.p 8 4.b odd 2 1 inner
900.3.c.p 8 12.b even 2 1 inner
900.3.c.q yes 8 5.b even 2 1
900.3.c.q yes 8 15.d odd 2 1
900.3.c.q yes 8 20.d odd 2 1
900.3.c.q yes 8 60.h even 2 1
900.3.f.g 16 5.c odd 4 2
900.3.f.g 16 15.e even 4 2
900.3.f.g 16 20.e even 4 2
900.3.f.g 16 60.l odd 4 2

## Hecke kernels

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

 $$T_{7}^{4} + 118 T_{7}^{2} + 1625$$ $$T_{13}^{2} + 2 T_{13} - 115$$ $$T_{17}^{4} - 424 T_{17}^{2} + 40768$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 32 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 1625 + 118 T^{2} + T^{4} )^{2}$$
$11$ $$( 8000 + 184 T^{2} + T^{4} )^{2}$$
$13$ $$( -115 + 2 T + T^{2} )^{4}$$
$17$ $$( 40768 - 424 T^{2} + T^{4} )^{2}$$
$19$ $$( 65 + 302 T^{2} + T^{4} )^{2}$$
$23$ $$( 15680 + 1784 T^{2} + T^{4} )^{2}$$
$29$ $$( 1019200 - 3048 T^{2} + T^{4} )^{2}$$
$31$ $$( 79625 + 1030 T^{2} + T^{4} )^{2}$$
$37$ $$( -460 + 4 T + T^{2} )^{4}$$
$41$ $$( 3515200 - 3752 T^{2} + T^{4} )^{2}$$
$43$ $$( 13456625 + 7358 T^{2} + T^{4} )^{2}$$
$47$ $$( 2708480 + 3296 T^{2} + T^{4} )^{2}$$
$53$ $$( 2896192 - 6888 T^{2} + T^{4} )^{2}$$
$59$ $$( 10368000 + 6624 T^{2} + T^{4} )^{2}$$
$61$ $$( -5659 - 10 T + T^{2} )^{4}$$
$67$ $$( 4564625 + 9502 T^{2} + T^{4} )^{2}$$
$71$ $$( 66248000 + 21304 T^{2} + T^{4} )^{2}$$
$73$ $$( -10700 + 60 T + T^{2} )^{4}$$
$79$ $$( 44994560 + 21568 T^{2} + T^{4} )^{2}$$
$83$ $$( 146232320 + 27104 T^{2} + T^{4} )^{2}$$
$89$ $$( 5324800 - 16512 T^{2} + T^{4} )^{2}$$
$97$ $$( 15 + T )^{8}$$