# Properties

 Label 225.4.b.e Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8} +O(q^{10})$$ $$q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8} -52 q^{11} + 22 i q^{13} -24 q^{14} + 41 q^{16} -14 i q^{17} + 20 q^{19} -52 i q^{22} + 168 i q^{23} -22 q^{26} + 168 i q^{28} + 230 q^{29} -288 q^{31} + 161 i q^{32} + 14 q^{34} + 34 i q^{37} + 20 i q^{38} -122 q^{41} -188 i q^{43} -364 q^{44} -168 q^{46} + 256 i q^{47} -233 q^{49} + 154 i q^{52} + 338 i q^{53} -360 q^{56} + 230 i q^{58} + 100 q^{59} + 742 q^{61} -288 i q^{62} + 167 q^{64} + 84 i q^{67} -98 i q^{68} + 328 q^{71} -38 i q^{73} -34 q^{74} + 140 q^{76} -1248 i q^{77} + 240 q^{79} -122 i q^{82} -1212 i q^{83} + 188 q^{86} -780 i q^{88} + 330 q^{89} -528 q^{91} + 1176 i q^{92} -256 q^{94} -866 i q^{97} -233 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{4} + O(q^{10})$$ $$2q + 14q^{4} - 104q^{11} - 48q^{14} + 82q^{16} + 40q^{19} - 44q^{26} + 460q^{29} - 576q^{31} + 28q^{34} - 244q^{41} - 728q^{44} - 336q^{46} - 466q^{49} - 720q^{56} + 200q^{59} + 1484q^{61} + 334q^{64} + 656q^{71} - 68q^{74} + 280q^{76} + 480q^{79} + 376q^{86} + 660q^{89} - 1056q^{91} - 512q^{94} + O(q^{100})$$

## Character values

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

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

## 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
 − 1.00000i 1.00000i
1.00000i 0 7.00000 0 0 24.0000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 24.0000i 15.0000i 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
5.b even 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.e 2
3.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 225.4.b.e 2
5.c odd 4 1 45.4.a.c 1
5.c odd 4 1 225.4.a.f 1
12.b even 2 1 1200.4.f.b 2
15.d odd 2 1 75.4.b.b 2
15.e even 4 1 15.4.a.a 1
15.e even 4 1 75.4.a.b 1
20.e even 4 1 720.4.a.n 1
35.f even 4 1 2205.4.a.l 1
45.k odd 12 2 405.4.e.i 2
45.l even 12 2 405.4.e.g 2
60.h even 2 1 1200.4.f.b 2
60.l odd 4 1 240.4.a.e 1
60.l odd 4 1 1200.4.a.t 1
105.k odd 4 1 735.4.a.e 1
120.q odd 4 1 960.4.a.ba 1
120.w even 4 1 960.4.a.b 1
165.l odd 4 1 1815.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 15.e even 4 1
45.4.a.c 1 5.c odd 4 1
75.4.a.b 1 15.e even 4 1
75.4.b.b 2 3.b odd 2 1
75.4.b.b 2 15.d odd 2 1
225.4.a.f 1 5.c odd 4 1
225.4.b.e 2 1.a even 1 1 trivial
225.4.b.e 2 5.b even 2 1 inner
240.4.a.e 1 60.l odd 4 1
405.4.e.g 2 45.l even 12 2
405.4.e.i 2 45.k odd 12 2
720.4.a.n 1 20.e even 4 1
735.4.a.e 1 105.k odd 4 1
960.4.a.b 1 120.w even 4 1
960.4.a.ba 1 120.q odd 4 1
1200.4.a.t 1 60.l odd 4 1
1200.4.f.b 2 12.b even 2 1
1200.4.f.b 2 60.h even 2 1
1815.4.a.e 1 165.l odd 4 1
2205.4.a.l 1 35.f even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{7}^{2} + 576$$ $$T_{11} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$576 + T^{2}$$
$11$ $$( 52 + T )^{2}$$
$13$ $$484 + T^{2}$$
$17$ $$196 + T^{2}$$
$19$ $$( -20 + T )^{2}$$
$23$ $$28224 + T^{2}$$
$29$ $$( -230 + T )^{2}$$
$31$ $$( 288 + T )^{2}$$
$37$ $$1156 + T^{2}$$
$41$ $$( 122 + T )^{2}$$
$43$ $$35344 + T^{2}$$
$47$ $$65536 + T^{2}$$
$53$ $$114244 + T^{2}$$
$59$ $$( -100 + T )^{2}$$
$61$ $$( -742 + T )^{2}$$
$67$ $$7056 + T^{2}$$
$71$ $$( -328 + T )^{2}$$
$73$ $$1444 + T^{2}$$
$79$ $$( -240 + T )^{2}$$
$83$ $$1468944 + T^{2}$$
$89$ $$( -330 + T )^{2}$$
$97$ $$749956 + T^{2}$$