# Properties

 Label 900.3.c.o 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: 8.0.15012375625.1 Defining polynomial: $$x^{8} + 7 x^{6} + 28 x^{4} + 112 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{3} ) q^{4} + ( -\beta_{3} - \beta_{7} ) q^{7} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{3} ) q^{4} + ( -\beta_{3} - \beta_{7} ) q^{7} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} + ( -2 \beta_{5} - \beta_{6} ) q^{11} + ( -1 - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{13} + ( 2 \beta_{2} - 2 \beta_{6} ) q^{14} + ( -1 - 2 \beta_{3} + \beta_{4} ) q^{16} + ( -4 \beta_{1} - 2 \beta_{2} + 3 \beta_{6} ) q^{17} + ( 1 - 4 \beta_{3} - \beta_{4} + \beta_{7} ) q^{19} + ( -\beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{22} + ( 5 \beta_{1} + \beta_{2} + \beta_{6} ) q^{23} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} ) q^{26} + ( 12 + 2 \beta_{3} + 4 \beta_{7} ) q^{28} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{6} ) q^{29} + ( 1 - 2 \beta_{3} - \beta_{4} + 3 \beta_{7} ) q^{31} + ( -4 \beta_{1} + 6 \beta_{2} - 3 \beta_{5} - \beta_{6} ) q^{32} + ( -14 - 7 \beta_{3} - 5 \beta_{7} ) q^{34} + ( -35 + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{37} + ( 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} ) q^{38} + ( -8 \beta_{1} + 4 \beta_{6} ) q^{41} + ( -2 + 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{43} + ( 6 \beta_{1} - 10 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} ) q^{44} + ( -24 + 4 \beta_{3} ) q^{46} + ( 5 \beta_{1} + \beta_{2} - 8 \beta_{5} - 3 \beta_{6} ) q^{47} + ( -23 + 8 \beta_{3} - 4 \beta_{4} - 4 \beta_{7} ) q^{49} + ( -28 - 8 \beta_{3} - 4 \beta_{4} ) q^{52} + ( -15 \beta_{1} + 7 \beta_{2} + 4 \beta_{6} ) q^{53} + ( 12 \beta_{1} - 8 \beta_{2} - 2 \beta_{5} + 6 \beta_{6} ) q^{56} + ( -10 - 5 \beta_{3} - 3 \beta_{7} ) q^{58} + ( 30 \beta_{1} + 6 \beta_{2} + 6 \beta_{5} + 9 \beta_{6} ) q^{59} + ( 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{61} + ( 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{5} ) q^{62} + ( -21 - 6 \beta_{3} - 3 \beta_{4} + 4 \beta_{7} ) q^{64} + ( 4 - 26 \beta_{3} - 4 \beta_{4} - 6 \beta_{7} ) q^{67} + ( -14 \beta_{1} + 10 \beta_{2} - 2 \beta_{5} - 12 \beta_{6} ) q^{68} + ( -10 \beta_{1} - 2 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} ) q^{71} + ( 28 + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{73} + ( -32 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} ) q^{74} + ( 44 + 4 \beta_{3} - 4 \beta_{4} - 8 \beta_{7} ) q^{76} + ( -4 \beta_{1} + 16 \beta_{2} - 6 \beta_{6} ) q^{77} + ( 3 - 22 \beta_{3} - 3 \beta_{4} - 7 \beta_{7} ) q^{79} + ( -24 - 12 \beta_{3} - 4 \beta_{7} ) q^{82} + ( 40 \beta_{1} + 8 \beta_{2} - 4 \beta_{5} + 6 \beta_{6} ) q^{83} + ( -8 \beta_{1} + 16 \beta_{2} + 8 \beta_{5} + 8 \beta_{6} ) q^{86} + ( 78 + 12 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{88} + ( -34 \beta_{1} + 26 \beta_{2} + 4 \beta_{6} ) q^{89} + ( -2 - 4 \beta_{3} + 2 \beta_{4} - 14 \beta_{7} ) q^{91} + ( -24 \beta_{1} + 4 \beta_{5} + 4 \beta_{6} ) q^{92} + ( -24 - 8 \beta_{4} - 4 \beta_{7} ) q^{94} + ( -36 + 12 \beta_{3} - 6 \beta_{4} - 6 \beta_{7} ) q^{97} + ( -11 \beta_{1} - 16 \beta_{2} + 16 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 14q^{4} + O(q^{10})$$ $$8q - 14q^{4} - 16q^{13} - 14q^{16} + 4q^{22} + 92q^{28} - 116q^{34} - 272q^{37} - 184q^{46} - 152q^{49} - 232q^{52} - 84q^{58} + 16q^{61} - 182q^{64} + 240q^{73} + 384q^{76} - 208q^{82} + 652q^{88} - 168q^{94} - 240q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{6} + 28 x^{4} + 112 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 4 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 2 \nu^{2} + 5$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - 13 \nu^{5} - 28 \nu^{3} - 144 \nu$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + 13 \nu^{5} + 60 \nu^{3} + 208 \nu$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} + 12 \nu^{2} + 48$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 2 \beta_{3} - 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} - 3 \beta_{5} + 6 \beta_{2} - 4 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} - 21$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{6} - 7 \beta_{5} - 26 \beta_{2} - 12 \beta_{1}$$

## 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
 −1.46040 − 1.36646i −1.46040 + 1.36646i −0.342371 − 1.97048i −0.342371 + 1.97048i 0.342371 − 1.97048i 0.342371 + 1.97048i 1.46040 − 1.36646i 1.46040 + 1.36646i
−1.46040 1.36646i 0 0.265564 + 3.99117i 0 0 1.87135i 5.06596 6.19161i 0 0
451.2 −1.46040 + 1.36646i 0 0.265564 3.99117i 0 0 1.87135i 5.06596 + 6.19161i 0 0
451.3 −0.342371 1.97048i 0 −3.76556 + 1.34927i 0 0 11.5108i 3.94792 + 6.95801i 0 0
451.4 −0.342371 + 1.97048i 0 −3.76556 1.34927i 0 0 11.5108i 3.94792 6.95801i 0 0
451.5 0.342371 1.97048i 0 −3.76556 1.34927i 0 0 11.5108i −3.94792 + 6.95801i 0 0
451.6 0.342371 + 1.97048i 0 −3.76556 + 1.34927i 0 0 11.5108i −3.94792 6.95801i 0 0
451.7 1.46040 1.36646i 0 0.265564 3.99117i 0 0 1.87135i −5.06596 6.19161i 0 0
451.8 1.46040 + 1.36646i 0 0.265564 + 3.99117i 0 0 1.87135i −5.06596 + 6.19161i 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.o 8
3.b odd 2 1 inner 900.3.c.o 8
4.b odd 2 1 inner 900.3.c.o 8
5.b even 2 1 180.3.c.c 8
5.c odd 4 2 900.3.f.i 16
12.b even 2 1 inner 900.3.c.o 8
15.d odd 2 1 180.3.c.c 8
15.e even 4 2 900.3.f.i 16
20.d odd 2 1 180.3.c.c 8
20.e even 4 2 900.3.f.i 16
40.e odd 2 1 2880.3.e.i 8
40.f even 2 1 2880.3.e.i 8
60.h even 2 1 180.3.c.c 8
60.l odd 4 2 900.3.f.i 16
120.i odd 2 1 2880.3.e.i 8
120.m even 2 1 2880.3.e.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c 8 5.b even 2 1
180.3.c.c 8 15.d odd 2 1
180.3.c.c 8 20.d odd 2 1
180.3.c.c 8 60.h even 2 1
900.3.c.o 8 1.a even 1 1 trivial
900.3.c.o 8 3.b odd 2 1 inner
900.3.c.o 8 4.b odd 2 1 inner
900.3.c.o 8 12.b even 2 1 inner
900.3.f.i 16 5.c odd 4 2
900.3.f.i 16 15.e even 4 2
900.3.f.i 16 20.e even 4 2
900.3.f.i 16 60.l odd 4 2
2880.3.e.i 8 40.e odd 2 1
2880.3.e.i 8 40.f even 2 1
2880.3.e.i 8 120.i odd 2 1
2880.3.e.i 8 120.m even 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 136 T_{7}^{2} + 464$$ $$T_{13}^{2} + 4 T_{13} - 256$$ $$T_{17}^{4} - 1096 T_{17}^{2} + 150544$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 112 T^{2} + 28 T^{4} + 7 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 464 + 136 T^{2} + T^{4} )^{2}$$
$11$ $$( 22736 + 328 T^{2} + T^{4} )^{2}$$
$13$ $$( -256 + 4 T + T^{2} )^{4}$$
$17$ $$( 150544 - 1096 T^{2} + T^{4} )^{2}$$
$19$ $$( 118784 + 944 T^{2} + T^{4} )^{2}$$
$23$ $$( 29696 + 368 T^{2} + T^{4} )^{2}$$
$29$ $$( 35344 - 456 T^{2} + T^{4} )^{2}$$
$31$ $$( 742400 + 1840 T^{2} + T^{4} )^{2}$$
$37$ $$( 896 + 68 T + T^{2} )^{4}$$
$41$ $$( -832 + T^{2} )^{4}$$
$43$ $$( 1900544 + 3776 T^{2} + T^{4} )^{2}$$
$47$ $$( 7602176 + 5552 T^{2} + T^{4} )^{2}$$
$53$ $$( 21904 - 3624 T^{2} + T^{4} )^{2}$$
$59$ $$( 51452496 + 16488 T^{2} + T^{4} )^{2}$$
$61$ $$( -1036 - 4 T + T^{2} )^{4}$$
$67$ $$( 75732224 + 21664 T^{2} + T^{4} )^{2}$$
$71$ $$( 363776 + 2848 T^{2} + T^{4} )^{2}$$
$73$ $$( -140 - 60 T + T^{2} )^{4}$$
$79$ $$( 65598464 + 16816 T^{2} + T^{4} )^{2}$$
$83$ $$( 69852416 + 24608 T^{2} + T^{4} )^{2}$$
$89$ $$( 160985344 - 28704 T^{2} + T^{4} )^{2}$$
$97$ $$( -8460 + 60 T + T^{2} )^{4}$$