# Properties

 Label 225.6.a.t Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{31})$$ Defining polynomial: $$x^{2} - 31$$ x^2 - 31 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 3) q^{2} + (6 \beta + 8) q^{4} + ( - 24 \beta - 51) q^{7} + ( - 6 \beta + 114) q^{8}+O(q^{10})$$ q + (b + 3) * q^2 + (6*b + 8) * q^4 + (-24*b - 51) * q^7 + (-6*b + 114) * q^8 $$q + (\beta + 3) q^{2} + (6 \beta + 8) q^{4} + ( - 24 \beta - 51) q^{7} + ( - 6 \beta + 114) q^{8} + (88 \beta + 6) q^{11} + ( - 96 \beta - 527) q^{13} + ( - 123 \beta - 897) q^{14} + ( - 96 \beta - 100) q^{16} + (40 \beta - 858) q^{17} + ( - 168 \beta + 2107) q^{19} + (270 \beta + 2746) q^{22} + ( - 648 \beta - 222) q^{23} + ( - 815 \beta - 4557) q^{26} + ( - 498 \beta - 4872) q^{28} + (56 \beta - 2034) q^{29} + ( - 936 \beta - 1299) q^{31} + ( - 196 \beta - 6924) q^{32} + ( - 738 \beta - 1334) q^{34} + (1776 \beta - 2206) q^{37} + (1603 \beta + 1113) q^{38} + (296 \beta - 5616) q^{41} + (216 \beta + 4225) q^{43} + (740 \beta + 16416) q^{44} + ( - 2166 \beta - 20754) q^{46} + (3328 \beta + 1230) q^{47} + (2448 \beta + 3650) q^{49} + ( - 3930 \beta - 22072) q^{52} + ( - 248 \beta - 32532) q^{53} + ( - 2430 \beta - 1350) q^{56} + ( - 1866 \beta - 4366) q^{58} + ( - 3008 \beta + 31962) q^{59} + (1008 \beta + 3655) q^{61} + ( - 4107 \beta - 32913) q^{62} + ( - 4440 \beta - 23648) q^{64} + (5160 \beta - 30867) q^{67} + ( - 4828 \beta + 576) q^{68} + (6200 \beta - 49152) q^{71} + (3408 \beta + 13282) q^{73} + (3122 \beta + 48438) q^{74} + (11298 \beta - 14392) q^{76} + ( - 4632 \beta - 65778) q^{77} + (1920 \beta + 42000) q^{79} + ( - 4728 \beta - 7672) q^{82} + (768 \beta - 32886) q^{83} + (4873 \beta + 19371) q^{86} + (9996 \beta - 15684) q^{88} + (3168 \beta - 51552) q^{89} + (17544 \beta + 98301) q^{91} + ( - 6516 \beta - 122304) q^{92} + (11214 \beta + 106858) q^{94} + ( - 22080 \beta - 34687) q^{97} + (10994 \beta + 86838) q^{98}+O(q^{100})$$ q + (b + 3) * q^2 + (6*b + 8) * q^4 + (-24*b - 51) * q^7 + (-6*b + 114) * q^8 + (88*b + 6) * q^11 + (-96*b - 527) * q^13 + (-123*b - 897) * q^14 + (-96*b - 100) * q^16 + (40*b - 858) * q^17 + (-168*b + 2107) * q^19 + (270*b + 2746) * q^22 + (-648*b - 222) * q^23 + (-815*b - 4557) * q^26 + (-498*b - 4872) * q^28 + (56*b - 2034) * q^29 + (-936*b - 1299) * q^31 + (-196*b - 6924) * q^32 + (-738*b - 1334) * q^34 + (1776*b - 2206) * q^37 + (1603*b + 1113) * q^38 + (296*b - 5616) * q^41 + (216*b + 4225) * q^43 + (740*b + 16416) * q^44 + (-2166*b - 20754) * q^46 + (3328*b + 1230) * q^47 + (2448*b + 3650) * q^49 + (-3930*b - 22072) * q^52 + (-248*b - 32532) * q^53 + (-2430*b - 1350) * q^56 + (-1866*b - 4366) * q^58 + (-3008*b + 31962) * q^59 + (1008*b + 3655) * q^61 + (-4107*b - 32913) * q^62 + (-4440*b - 23648) * q^64 + (5160*b - 30867) * q^67 + (-4828*b + 576) * q^68 + (6200*b - 49152) * q^71 + (3408*b + 13282) * q^73 + (3122*b + 48438) * q^74 + (11298*b - 14392) * q^76 + (-4632*b - 65778) * q^77 + (1920*b + 42000) * q^79 + (-4728*b - 7672) * q^82 + (768*b - 32886) * q^83 + (4873*b + 19371) * q^86 + (9996*b - 15684) * q^88 + (3168*b - 51552) * q^89 + (17544*b + 98301) * q^91 + (-6516*b - 122304) * q^92 + (11214*b + 106858) * q^94 + (-22080*b - 34687) * q^97 + (10994*b + 86838) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8}+O(q^{10})$$ 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8 $$2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8} + 12 q^{11} - 1054 q^{13} - 1794 q^{14} - 200 q^{16} - 1716 q^{17} + 4214 q^{19} + 5492 q^{22} - 444 q^{23} - 9114 q^{26} - 9744 q^{28} - 4068 q^{29} - 2598 q^{31} - 13848 q^{32} - 2668 q^{34} - 4412 q^{37} + 2226 q^{38} - 11232 q^{41} + 8450 q^{43} + 32832 q^{44} - 41508 q^{46} + 2460 q^{47} + 7300 q^{49} - 44144 q^{52} - 65064 q^{53} - 2700 q^{56} - 8732 q^{58} + 63924 q^{59} + 7310 q^{61} - 65826 q^{62} - 47296 q^{64} - 61734 q^{67} + 1152 q^{68} - 98304 q^{71} + 26564 q^{73} + 96876 q^{74} - 28784 q^{76} - 131556 q^{77} + 84000 q^{79} - 15344 q^{82} - 65772 q^{83} + 38742 q^{86} - 31368 q^{88} - 103104 q^{89} + 196602 q^{91} - 244608 q^{92} + 213716 q^{94} - 69374 q^{97} + 173676 q^{98}+O(q^{100})$$ 2 * q + 6 * q^2 + 16 * q^4 - 102 * q^7 + 228 * q^8 + 12 * q^11 - 1054 * q^13 - 1794 * q^14 - 200 * q^16 - 1716 * q^17 + 4214 * q^19 + 5492 * q^22 - 444 * q^23 - 9114 * q^26 - 9744 * q^28 - 4068 * q^29 - 2598 * q^31 - 13848 * q^32 - 2668 * q^34 - 4412 * q^37 + 2226 * q^38 - 11232 * q^41 + 8450 * q^43 + 32832 * q^44 - 41508 * q^46 + 2460 * q^47 + 7300 * q^49 - 44144 * q^52 - 65064 * q^53 - 2700 * q^56 - 8732 * q^58 + 63924 * q^59 + 7310 * q^61 - 65826 * q^62 - 47296 * q^64 - 61734 * q^67 + 1152 * q^68 - 98304 * q^71 + 26564 * q^73 + 96876 * q^74 - 28784 * q^76 - 131556 * q^77 + 84000 * q^79 - 15344 * q^82 - 65772 * q^83 + 38742 * q^86 - 31368 * q^88 - 103104 * q^89 + 196602 * q^91 - 244608 * q^92 + 213716 * q^94 - 69374 * q^97 + 173676 * q^98

## 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
 −5.56776 5.56776
−2.56776 0 −25.4066 0 0 82.6263 147.407 0 0
1.2 8.56776 0 41.4066 0 0 −184.626 80.5934 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.a.t 2
3.b odd 2 1 75.6.a.g 2
5.b even 2 1 225.6.a.j 2
5.c odd 4 2 225.6.b.l 4
15.d odd 2 1 75.6.a.i yes 2
15.e even 4 2 75.6.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.a.g 2 3.b odd 2 1
75.6.a.i yes 2 15.d odd 2 1
75.6.b.f 4 15.e even 4 2
225.6.a.j 2 5.b even 2 1
225.6.a.t 2 1.a even 1 1 trivial
225.6.b.l 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} - 6T_{2} - 22$$ T2^2 - 6*T2 - 22 $$T_{7}^{2} + 102T_{7} - 15255$$ T7^2 + 102*T7 - 15255

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T - 22$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 102T - 15255$$
$11$ $$T^{2} - 12T - 240028$$
$13$ $$T^{2} + 1054T - 7967$$
$17$ $$T^{2} + 1716 T + 686564$$
$19$ $$T^{2} - 4214 T + 3564505$$
$23$ $$T^{2} + 444 T - 12967740$$
$29$ $$T^{2} + 4068 T + 4039940$$
$31$ $$T^{2} + 2598 T - 25471575$$
$37$ $$T^{2} + 4412 T - 92913020$$
$41$ $$T^{2} + 11232 T + 28823360$$
$43$ $$T^{2} - 8450 T + 16404289$$
$47$ $$T^{2} - 2460 T - 341830204$$
$53$ $$T^{2} + 65064 T + 1056424400$$
$59$ $$T^{2} - 63924 T + 741079460$$
$61$ $$T^{2} - 7310 T - 18138959$$
$67$ $$T^{2} + 61734 T + 127378089$$
$71$ $$T^{2} + 98304 T + 1224279104$$
$73$ $$T^{2} - 26564 T - 183636860$$
$79$ $$T^{2} - 84000 T + 1649721600$$
$83$ $$T^{2} + 65772 T + 1063204452$$
$89$ $$T^{2} + 103104 T + 2346485760$$
$97$ $$T^{2} + 69374 T - 13910130431$$