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

## 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

## Atkin-Lehner signs

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

## 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

## Hecke kernels

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

 $$T_{2}^{2} - 44$$ T2^2 - 44 $$T_{7}^{2} - 3564$$ T7^2 - 3564

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 44$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3564$$
$11$ $$(T + 252)^{2}$$
$13$ $$T^{2} - 14256$$
$17$ $$T^{2} - 475904$$
$19$ $$(T - 220)^{2}$$
$23$ $$T^{2} - 5926316$$
$29$ $$(T + 6930)^{2}$$
$31$ $$(T - 6752)^{2}$$
$37$ $$T^{2} - 195150384$$
$41$ $$(T - 198)^{2}$$
$43$ $$T^{2} - 174636$$
$47$ $$T^{2} - 111096524$$
$53$ $$T^{2} - 33918896$$
$59$ $$(T + 24660)^{2}$$
$61$ $$(T + 5698)^{2}$$
$67$ $$T^{2} - 1904462604$$
$71$ $$(T + 53352)^{2}$$
$73$ $$T^{2} - 5030030016$$
$79$ $$(T + 51920)^{2}$$
$83$ $$T^{2} - 3824406476$$
$89$ $$(T + 9990)^{2}$$
$97$ $$T^{2} - 10251546624$$