# Properties

 Label 225.4.b.i Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -2 q^{4} + 3 \beta_{1} q^{7} -6 \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} -2 q^{4} + 3 \beta_{1} q^{7} -6 \beta_{2} q^{8} -4 \beta_{3} q^{11} -7 \beta_{1} q^{13} + 3 \beta_{3} q^{14} -76 q^{16} -28 \beta_{2} q^{17} -91 q^{19} + 40 \beta_{1} q^{22} -36 \beta_{2} q^{23} -7 \beta_{3} q^{26} -6 \beta_{1} q^{28} -4 \beta_{3} q^{29} -147 q^{31} + 28 \beta_{2} q^{32} -280 q^{34} + 74 \beta_{1} q^{37} + 91 \beta_{2} q^{38} + 28 \beta_{3} q^{41} -67 \beta_{1} q^{43} + 8 \beta_{3} q^{44} -360 q^{46} + 56 \beta_{2} q^{47} + 118 q^{49} + 14 \beta_{1} q^{52} + 28 \beta_{2} q^{53} + 18 \beta_{3} q^{56} + 40 \beta_{1} q^{58} -56 \beta_{3} q^{59} + 427 q^{61} + 147 \beta_{2} q^{62} -328 q^{64} + 3 \beta_{1} q^{67} + 56 \beta_{2} q^{68} + 4 \beta_{3} q^{71} + 14 \beta_{1} q^{73} + 74 \beta_{3} q^{74} + 182 q^{76} -300 \beta_{2} q^{77} + 876 q^{79} -280 \beta_{1} q^{82} -168 \beta_{2} q^{83} -67 \beta_{3} q^{86} + 240 \beta_{1} q^{88} + 525 q^{91} + 72 \beta_{2} q^{92} + 560 q^{94} -217 \beta_{1} q^{97} -118 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + O(q^{10})$$ $$4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 5 \beta_{2}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + 5 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
 −1.58114 + 1.58114i 1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i
3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.2 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.3 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.4 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.b.i 4
3.b odd 2 1 inner 225.4.b.i 4
5.b even 2 1 inner 225.4.b.i 4
5.c odd 4 1 225.4.a.l 2
5.c odd 4 1 225.4.a.m yes 2
15.d odd 2 1 inner 225.4.b.i 4
15.e even 4 1 225.4.a.l 2
15.e even 4 1 225.4.a.m yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.a.l 2 5.c odd 4 1
225.4.a.l 2 15.e even 4 1
225.4.a.m yes 2 5.c odd 4 1
225.4.a.m yes 2 15.e even 4 1
225.4.b.i 4 1.a even 1 1 trivial
225.4.b.i 4 3.b odd 2 1 inner
225.4.b.i 4 5.b even 2 1 inner
225.4.b.i 4 15.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 10$$ $$T_{7}^{2} + 225$$ $$T_{11}^{2} - 4000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 10 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 225 + T^{2} )^{2}$$
$11$ $$( -4000 + T^{2} )^{2}$$
$13$ $$( 1225 + T^{2} )^{2}$$
$17$ $$( 7840 + T^{2} )^{2}$$
$19$ $$( 91 + T )^{4}$$
$23$ $$( 12960 + T^{2} )^{2}$$
$29$ $$( -4000 + T^{2} )^{2}$$
$31$ $$( 147 + T )^{4}$$
$37$ $$( 136900 + T^{2} )^{2}$$
$41$ $$( -196000 + T^{2} )^{2}$$
$43$ $$( 112225 + T^{2} )^{2}$$
$47$ $$( 31360 + T^{2} )^{2}$$
$53$ $$( 7840 + T^{2} )^{2}$$
$59$ $$( -784000 + T^{2} )^{2}$$
$61$ $$( -427 + T )^{4}$$
$67$ $$( 225 + T^{2} )^{2}$$
$71$ $$( -4000 + T^{2} )^{2}$$
$73$ $$( 4900 + T^{2} )^{2}$$
$79$ $$( -876 + T )^{4}$$
$83$ $$( 282240 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 1177225 + T^{2} )^{2}$$