# Properties

 Label 225.6.a.n Level 225 Weight 6 Character orbit 225.a Self dual yes Analytic conductor 36.086 Analytic rank 1 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.0863594579$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} -20 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} -20 \beta q^{8} -252 q^{11} -18 \beta q^{13} + 396 q^{14} -1264 q^{16} -104 \beta q^{17} + 220 q^{19} -252 \beta q^{22} -367 \beta q^{23} -792 q^{26} + 108 \beta q^{28} -6930 q^{29} + 6752 q^{31} -624 \beta q^{32} -4576 q^{34} -2106 \beta q^{37} + 220 \beta q^{38} + 198 q^{41} -63 \beta q^{43} -3024 q^{44} -16148 q^{46} -1589 \beta q^{47} -13243 q^{49} -216 \beta q^{52} + 878 \beta q^{53} -7920 q^{56} -6930 \beta q^{58} -24660 q^{59} -5698 q^{61} + 6752 \beta q^{62} + 12992 q^{64} + 6579 \beta q^{67} -1248 \beta q^{68} -53352 q^{71} + 10692 \beta q^{73} -92664 q^{74} + 2640 q^{76} -2268 \beta q^{77} -51920 q^{79} + 198 \beta q^{82} + 9323 \beta q^{83} -2772 q^{86} + 5040 \beta q^{88} -9990 q^{89} -7128 q^{91} -4404 \beta q^{92} -69916 q^{94} + 15264 \beta q^{97} -13243 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 24q^{4} + O(q^{10})$$ $$2q + 24q^{4} - 504q^{11} + 792q^{14} - 2528q^{16} + 440q^{19} - 1584q^{26} - 13860q^{29} + 13504q^{31} - 9152q^{34} + 396q^{41} - 6048q^{44} - 32296q^{46} - 26486q^{49} - 15840q^{56} - 49320q^{59} - 11396q^{61} + 25984q^{64} - 106704q^{71} - 185328q^{74} + 5280q^{76} - 103840q^{79} - 5544q^{86} - 19980q^{89} - 14256q^{91} - 139832q^{94} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.31662 3.31662
−6.63325 0 12.0000 0 0 −59.6992 132.665 0 0
1.2 6.63325 0 12.0000 0 0 59.6992 −132.665 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.a.n 2
3.b odd 2 1 25.6.a.c 2
5.b even 2 1 inner 225.6.a.n 2
5.c odd 4 2 45.6.b.b 2
12.b even 2 1 400.6.a.t 2
15.d odd 2 1 25.6.a.c 2
15.e even 4 2 5.6.b.a 2
20.e even 4 2 720.6.f.f 2
60.h even 2 1 400.6.a.t 2
60.l odd 4 2 80.6.c.a 2
105.k odd 4 2 245.6.b.a 2
120.q odd 4 2 320.6.c.g 2
120.w even 4 2 320.6.c.f 2

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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 44$$ $$T_{7}^{2} - 3564$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 20 T^{2} + 1024 T^{4}$$
$3$ 1
$5$ 1
$7$ $$1 + 30050 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 252 T + 161051 T^{2} )^{2}$$
$13$ $$1 + 728330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 2363810 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 + 6946370 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 6930 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 6752 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 56462470 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 198 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 293842250 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 347593490 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 802472090 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 24660 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 5698 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 795787610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 53352 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 883886830 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 51920 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 + 4053674810 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 9990 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 6923133890 T^{2} + 73742412689492826049 T^{4}$$