# Properties

 Label 225.4.f.a Level $225$ Weight $4$ Character orbit 225.f 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.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -9 + 9 \zeta_{8}^{2} ) q^{7} -21 \zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q + 3 \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -9 + 9 \zeta_{8}^{2} ) q^{7} -21 \zeta_{8}^{3} q^{8} + ( 27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{11} + ( 63 + 63 \zeta_{8}^{2} ) q^{13} + ( -27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{14} + 71 q^{16} + 42 \zeta_{8} q^{17} + 70 \zeta_{8}^{2} q^{19} + ( -81 + 81 \zeta_{8}^{2} ) q^{22} -102 \zeta_{8}^{3} q^{23} + ( 189 \zeta_{8} + 189 \zeta_{8}^{3} ) q^{26} + ( -9 - 9 \zeta_{8}^{2} ) q^{28} + ( -162 \zeta_{8} + 162 \zeta_{8}^{3} ) q^{29} + 196 q^{31} + 45 \zeta_{8} q^{32} + 126 \zeta_{8}^{2} q^{34} + ( 207 - 207 \zeta_{8}^{2} ) q^{37} + 210 \zeta_{8}^{3} q^{38} + ( 189 \zeta_{8} + 189 \zeta_{8}^{3} ) q^{41} + ( -144 - 144 \zeta_{8}^{2} ) q^{43} + ( -27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{44} + 306 q^{46} -504 \zeta_{8} q^{47} + 181 \zeta_{8}^{2} q^{49} + ( -63 + 63 \zeta_{8}^{2} ) q^{52} + 318 \zeta_{8}^{3} q^{53} + ( 189 \zeta_{8} + 189 \zeta_{8}^{3} ) q^{56} + ( -486 - 486 \zeta_{8}^{2} ) q^{58} + ( -189 \zeta_{8} + 189 \zeta_{8}^{3} ) q^{59} -322 q^{61} + 588 \zeta_{8} q^{62} -433 \zeta_{8}^{2} q^{64} + ( 378 - 378 \zeta_{8}^{2} ) q^{67} + 42 \zeta_{8}^{3} q^{68} + ( -594 \zeta_{8} - 594 \zeta_{8}^{3} ) q^{71} + ( 378 + 378 \zeta_{8}^{2} ) q^{73} + ( 621 \zeta_{8} - 621 \zeta_{8}^{3} ) q^{74} -70 q^{76} -486 \zeta_{8} q^{77} + 488 \zeta_{8}^{2} q^{79} + ( -567 + 567 \zeta_{8}^{2} ) q^{82} -1092 \zeta_{8}^{3} q^{83} + ( -432 \zeta_{8} - 432 \zeta_{8}^{3} ) q^{86} + ( 567 + 567 \zeta_{8}^{2} ) q^{88} + ( -189 \zeta_{8} + 189 \zeta_{8}^{3} ) q^{89} -1134 q^{91} + 102 \zeta_{8} q^{92} -1512 \zeta_{8}^{2} q^{94} + ( -252 + 252 \zeta_{8}^{2} ) q^{97} + 543 \zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{7} + O(q^{10})$$ $$4 q - 36 q^{7} + 252 q^{13} + 284 q^{16} - 324 q^{22} - 36 q^{28} + 784 q^{31} + 828 q^{37} - 576 q^{43} + 1224 q^{46} - 252 q^{52} - 1944 q^{58} - 1288 q^{61} + 1512 q^{67} + 1512 q^{73} - 280 q^{76} - 2268 q^{82} + 2268 q^{88} - 4536 q^{91} - 1008 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−2.12132 + 2.12132i 0 1.00000i 0 0 −9.00000 9.00000i −14.8492 14.8492i 0 0
107.2 2.12132 2.12132i 0 1.00000i 0 0 −9.00000 9.00000i 14.8492 + 14.8492i 0 0
143.1 −2.12132 2.12132i 0 1.00000i 0 0 −9.00000 + 9.00000i −14.8492 + 14.8492i 0 0
143.2 2.12132 + 2.12132i 0 1.00000i 0 0 −9.00000 + 9.00000i 14.8492 14.8492i 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 81$$ $$T_{7}^{2} + 18 T_{7} + 162$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 162 + 18 T + T^{2} )^{2}$$
$11$ $$( 1458 + T^{2} )^{2}$$
$13$ $$( 7938 - 126 T + T^{2} )^{2}$$
$17$ $$3111696 + T^{4}$$
$19$ $$( 4900 + T^{2} )^{2}$$
$23$ $$108243216 + T^{4}$$
$29$ $$( -52488 + T^{2} )^{2}$$
$31$ $$( -196 + T )^{4}$$
$37$ $$( 85698 - 414 T + T^{2} )^{2}$$
$41$ $$( 71442 + T^{2} )^{2}$$
$43$ $$( 41472 + 288 T + T^{2} )^{2}$$
$47$ $$64524128256 + T^{4}$$
$53$ $$10226063376 + T^{4}$$
$59$ $$( -71442 + T^{2} )^{2}$$
$61$ $$( 322 + T )^{4}$$
$67$ $$( 285768 - 756 T + T^{2} )^{2}$$
$71$ $$( 705672 + T^{2} )^{2}$$
$73$ $$( 285768 - 756 T + T^{2} )^{2}$$
$79$ $$( 238144 + T^{2} )^{2}$$
$83$ $$1421970391296 + T^{4}$$
$89$ $$( -71442 + T^{2} )^{2}$$
$97$ $$( 127008 + 504 T + T^{2} )^{2}$$