# Properties

 Label 225.6.b.e Level $225$ Weight $6$ Character orbit 225.b Analytic conductor $36.086$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.0863594579$$ 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: $$2$$ Twist minimal: no (minimal twist has level 5) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 28 q^{4} -192 i q^{7} + 120 i q^{8} +O(q^{10})$$ $$q + 2 i q^{2} + 28 q^{4} -192 i q^{7} + 120 i q^{8} + 148 q^{11} + 286 i q^{13} + 384 q^{14} + 656 q^{16} -1678 i q^{17} -1060 q^{19} + 296 i q^{22} -2976 i q^{23} -572 q^{26} -5376 i q^{28} -3410 q^{29} -2448 q^{31} + 5152 i q^{32} + 3356 q^{34} -182 i q^{37} -2120 i q^{38} + 9398 q^{41} -1244 i q^{43} + 4144 q^{44} + 5952 q^{46} -12088 i q^{47} -20057 q^{49} + 8008 i q^{52} -23846 i q^{53} + 23040 q^{56} -6820 i q^{58} -20020 q^{59} + 32302 q^{61} -4896 i q^{62} + 10688 q^{64} -60972 i q^{67} -46984 i q^{68} + 32648 q^{71} -38774 i q^{73} + 364 q^{74} -29680 q^{76} -28416 i q^{77} + 33360 q^{79} + 18796 i q^{82} -16716 i q^{83} + 2488 q^{86} + 17760 i q^{88} + 101370 q^{89} + 54912 q^{91} -83328 i q^{92} + 24176 q^{94} + 119038 i q^{97} -40114 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 56q^{4} + O(q^{10})$$ $$2q + 56q^{4} + 296q^{11} + 768q^{14} + 1312q^{16} - 2120q^{19} - 1144q^{26} - 6820q^{29} - 4896q^{31} + 6712q^{34} + 18796q^{41} + 8288q^{44} + 11904q^{46} - 40114q^{49} + 46080q^{56} - 40040q^{59} + 64604q^{61} + 21376q^{64} + 65296q^{71} + 728q^{74} - 59360q^{76} + 66720q^{79} + 4976q^{86} + 202740q^{89} + 109824q^{91} + 48352q^{94} + O(q^{100})$$

## 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
2.00000i 0 28.0000 0 0 192.000i 120.000i 0 0
199.2 2.00000i 0 28.0000 0 0 192.000i 120.000i 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 225.6.b.e 2
3.b odd 2 1 25.6.b.a 2
5.b even 2 1 inner 225.6.b.e 2
5.c odd 4 1 45.6.a.b 1
5.c odd 4 1 225.6.a.f 1
12.b even 2 1 400.6.c.j 2
15.d odd 2 1 25.6.b.a 2
15.e even 4 1 5.6.a.a 1
15.e even 4 1 25.6.a.a 1
20.e even 4 1 720.6.a.a 1
60.h even 2 1 400.6.c.j 2
60.l odd 4 1 80.6.a.e 1
60.l odd 4 1 400.6.a.g 1
105.k odd 4 1 245.6.a.b 1
120.q odd 4 1 320.6.a.g 1
120.w even 4 1 320.6.a.j 1
165.l odd 4 1 605.6.a.a 1
195.s even 4 1 845.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 15.e even 4 1
25.6.a.a 1 15.e even 4 1
25.6.b.a 2 3.b odd 2 1
25.6.b.a 2 15.d odd 2 1
45.6.a.b 1 5.c odd 4 1
80.6.a.e 1 60.l odd 4 1
225.6.a.f 1 5.c odd 4 1
225.6.b.e 2 1.a even 1 1 trivial
225.6.b.e 2 5.b even 2 1 inner
245.6.a.b 1 105.k odd 4 1
320.6.a.g 1 120.q odd 4 1
320.6.a.j 1 120.w even 4 1
400.6.a.g 1 60.l odd 4 1
400.6.c.j 2 12.b even 2 1
400.6.c.j 2 60.h even 2 1
605.6.a.a 1 165.l odd 4 1
720.6.a.a 1 20.e even 4 1
845.6.a.b 1 195.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 4$$ $$T_{7}^{2} + 36864$$ $$T_{11} - 148$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$36864 + T^{2}$$
$11$ $$( -148 + T )^{2}$$
$13$ $$81796 + T^{2}$$
$17$ $$2815684 + T^{2}$$
$19$ $$( 1060 + T )^{2}$$
$23$ $$8856576 + T^{2}$$
$29$ $$( 3410 + T )^{2}$$
$31$ $$( 2448 + T )^{2}$$
$37$ $$33124 + T^{2}$$
$41$ $$( -9398 + T )^{2}$$
$43$ $$1547536 + T^{2}$$
$47$ $$146119744 + T^{2}$$
$53$ $$568631716 + T^{2}$$
$59$ $$( 20020 + T )^{2}$$
$61$ $$( -32302 + T )^{2}$$
$67$ $$3717584784 + T^{2}$$
$71$ $$( -32648 + T )^{2}$$
$73$ $$1503423076 + T^{2}$$
$79$ $$( -33360 + T )^{2}$$
$83$ $$279424656 + T^{2}$$
$89$ $$( -101370 + T )^{2}$$
$97$ $$14170045444 + T^{2}$$