# Properties

 Label 900.2.k.h Level $900$ Weight $2$ Character orbit 900.k Analytic conductor $7.187$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{49}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 300) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{4} + \beta_{6} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{4} + \beta_{6} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( \beta_{5} + \beta_{7} ) q^{8} + ( -2 + 4 \beta_{2} ) q^{11} + ( -4 \beta_{5} + \beta_{7} ) q^{13} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{14} + ( -2 + 3 \beta_{2} ) q^{16} + ( 2 \beta_{4} - 4 \beta_{6} ) q^{19} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{22} + 2 \beta_{7} q^{23} + ( 8 - 2 \beta_{2} ) q^{26} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{28} + 8 \beta_{4} q^{29} + ( 2 - 4 \beta_{2} ) q^{31} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{32} + ( -3 \beta_{1} - \beta_{3} ) q^{37} + ( 2 \beta_{5} - 4 \beta_{7} ) q^{38} + 2 q^{41} + 4 \beta_{7} q^{43} + ( 10 \beta_{4} - 6 \beta_{6} ) q^{44} + 4 \beta_{2} q^{46} + 3 \beta_{4} q^{49} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{52} + ( -8 \beta_{5} + 2 \beta_{7} ) q^{53} + ( -4 - 2 \beta_{2} ) q^{56} + 8 \beta_{5} q^{58} + ( 2 \beta_{4} - 4 \beta_{6} ) q^{59} + 6 q^{61} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{62} + ( 7 \beta_{4} - 5 \beta_{6} ) q^{64} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{67} + ( -8 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -6 \beta_{4} - 2 \beta_{6} ) q^{74} + ( -4 - 6 \beta_{2} ) q^{76} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -2 \beta_{4} + 4 \beta_{6} ) q^{79} + 2 \beta_{1} q^{82} -6 \beta_{7} q^{83} + 8 \beta_{2} q^{86} + ( 10 \beta_{5} - 6 \beta_{7} ) q^{88} -6 \beta_{4} q^{89} + ( -4 + 8 \beta_{2} ) q^{91} + 4 \beta_{3} q^{92} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{97} + 3 \beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{16} + 56q^{26} + 16q^{41} + 16q^{46} - 40q^{56} + 48q^{61} - 56q^{76} + 32q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 2$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{3}$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 7 \nu^{2}$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{2} - 2$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{3} - 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{6} + 7 \beta_{4}$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{7} + 7 \beta_{5}$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −1.28897 − 0.581861i −0.581861 − 1.28897i 0.581861 + 1.28897i 1.28897 + 0.581861i −1.28897 + 0.581861i −0.581861 + 1.28897i 0.581861 − 1.28897i 1.28897 − 0.581861i
−1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 1.41421 1.41421i −0.832353 2.70318i 0 0
307.2 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −1.41421 + 1.41421i 2.70318 + 0.832353i 0 0
307.3 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 1.41421 1.41421i −2.70318 0.832353i 0 0
307.4 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 −1.41421 + 1.41421i 0.832353 + 2.70318i 0 0
343.1 −1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 1.41421 + 1.41421i −0.832353 + 2.70318i 0 0
343.2 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −1.41421 1.41421i 2.70318 0.832353i 0 0
343.3 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 1.41421 + 1.41421i −2.70318 + 0.832353i 0 0
343.4 1.28897 0.581861i 0 1.32288 1.50000i 0 0 −1.41421 1.41421i 0.832353 2.70318i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.4 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
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.h 8
3.b odd 2 1 300.2.j.b 8
4.b odd 2 1 inner 900.2.k.h 8
5.b even 2 1 inner 900.2.k.h 8
5.c odd 4 2 inner 900.2.k.h 8
12.b even 2 1 300.2.j.b 8
15.d odd 2 1 300.2.j.b 8
15.e even 4 2 300.2.j.b 8
20.d odd 2 1 inner 900.2.k.h 8
20.e even 4 2 inner 900.2.k.h 8
60.h even 2 1 300.2.j.b 8
60.l odd 4 2 300.2.j.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.b 8 3.b odd 2 1
300.2.j.b 8 12.b even 2 1
300.2.j.b 8 15.d odd 2 1
300.2.j.b 8 15.e even 4 2
300.2.j.b 8 60.h even 2 1
300.2.j.b 8 60.l odd 4 2
900.2.k.h 8 1.a even 1 1 trivial
900.2.k.h 8 4.b odd 2 1 inner
900.2.k.h 8 5.b even 2 1 inner
900.2.k.h 8 5.c odd 4 2 inner
900.2.k.h 8 20.d odd 2 1 inner
900.2.k.h 8 20.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{4} + 16$$ $$T_{11}^{2} + 28$$ $$T_{13}^{4} + 784$$ $$T_{17}$$ $$T_{19}^{2} - 28$$ $$T_{23}^{4} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 16 + T^{4} )^{2}$$
$11$ $$( 28 + T^{2} )^{4}$$
$13$ $$( 784 + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( -28 + T^{2} )^{4}$$
$23$ $$( 256 + T^{4} )^{2}$$
$29$ $$( 64 + T^{2} )^{4}$$
$31$ $$( 28 + T^{2} )^{4}$$
$37$ $$( 784 + T^{4} )^{2}$$
$41$ $$( -2 + T )^{8}$$
$43$ $$( 4096 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 12544 + T^{4} )^{2}$$
$59$ $$( -28 + T^{2} )^{4}$$
$61$ $$( -6 + T )^{8}$$
$67$ $$( 20736 + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 12544 + T^{4} )^{2}$$
$79$ $$( -28 + T^{2} )^{4}$$
$83$ $$( 20736 + T^{4} )^{2}$$
$89$ $$( 36 + T^{2} )^{4}$$
$97$ $$( 12544 + T^{4} )^{2}$$