# Properties

 Label 225.6.a.s Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ x^2 - x - 60 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{241})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{2} + ( - 5 \beta + 37) q^{4} + ( - 4 \beta - 98) q^{7} + ( - 15 \beta + 315) q^{8}+O(q^{10})$$ q + (-b + 3) * q^2 + (-5*b + 37) * q^4 + (-4*b - 98) * q^7 + (-15*b + 315) * q^8 $$q + ( - \beta + 3) q^{2} + ( - 5 \beta + 37) q^{4} + ( - 4 \beta - 98) q^{7} + ( - 15 \beta + 315) q^{8} + ( - 50 \beta + 123) q^{11} + ( - 32 \beta + 196) q^{13} + (90 \beta - 54) q^{14} + ( - 185 \beta + 661) q^{16} + ( - 136 \beta + 813) q^{17} + ( - 70 \beta - 1555) q^{19} + ( - 223 \beta + 3369) q^{22} + (12 \beta + 774) q^{23} + ( - 260 \beta + 2508) q^{26} + (362 \beta - 2426) q^{28} + (80 \beta + 1920) q^{29} + ( - 1100 \beta + 2) q^{31} + ( - 551 \beta + 3003) q^{32} + ( - 1085 \beta + 10599) q^{34} + ( - 384 \beta - 818) q^{37} + (1415 \beta - 465) q^{38} + ( - 400 \beta - 13677) q^{41} + (2128 \beta + 436) q^{43} + ( - 2215 \beta + 19551) q^{44} + ( - 750 \beta + 1602) q^{46} + (1544 \beta + 12108) q^{47} + (800 \beta - 6243) q^{49} + ( - 2004 \beta + 16852) q^{52} + (752 \beta - 13866) q^{53} + (270 \beta - 27270) q^{56} + ( - 1760 \beta + 960) q^{58} + (1960 \beta - 6960) q^{59} + (2000 \beta - 13198) q^{61} + ( - 2202 \beta + 66006) q^{62} + (1815 \beta + 20917) q^{64} + (1586 \beta + 19237) q^{67} + ( - 8417 \beta + 70881) q^{68} + ( - 1000 \beta + 44148) q^{71} + ( - 1112 \beta + 35701) q^{73} + (50 \beta + 20586) q^{74} + (5535 \beta - 36535) q^{76} + (4608 \beta - 54) q^{77} + (5020 \beta + 30230) q^{79} + (12877 \beta - 17031) q^{82} + ( - 858 \beta + 46719) q^{83} + (3820 \beta - 126372) q^{86} + ( - 16845 \beta + 83745) q^{88} + (10440 \beta + 31185) q^{89} + (2480 \beta - 11528) q^{91} + ( - 3486 \beta + 25038) q^{92} + ( - 9020 \beta - 56316) q^{94} + ( - 10944 \beta + 68542) q^{97} + (7843 \beta - 66729) q^{98}+O(q^{100})$$ q + (-b + 3) * q^2 + (-5*b + 37) * q^4 + (-4*b - 98) * q^7 + (-15*b + 315) * q^8 + (-50*b + 123) * q^11 + (-32*b + 196) * q^13 + (90*b - 54) * q^14 + (-185*b + 661) * q^16 + (-136*b + 813) * q^17 + (-70*b - 1555) * q^19 + (-223*b + 3369) * q^22 + (12*b + 774) * q^23 + (-260*b + 2508) * q^26 + (362*b - 2426) * q^28 + (80*b + 1920) * q^29 + (-1100*b + 2) * q^31 + (-551*b + 3003) * q^32 + (-1085*b + 10599) * q^34 + (-384*b - 818) * q^37 + (1415*b - 465) * q^38 + (-400*b - 13677) * q^41 + (2128*b + 436) * q^43 + (-2215*b + 19551) * q^44 + (-750*b + 1602) * q^46 + (1544*b + 12108) * q^47 + (800*b - 6243) * q^49 + (-2004*b + 16852) * q^52 + (752*b - 13866) * q^53 + (270*b - 27270) * q^56 + (-1760*b + 960) * q^58 + (1960*b - 6960) * q^59 + (2000*b - 13198) * q^61 + (-2202*b + 66006) * q^62 + (1815*b + 20917) * q^64 + (1586*b + 19237) * q^67 + (-8417*b + 70881) * q^68 + (-1000*b + 44148) * q^71 + (-1112*b + 35701) * q^73 + (50*b + 20586) * q^74 + (5535*b - 36535) * q^76 + (4608*b - 54) * q^77 + (5020*b + 30230) * q^79 + (12877*b - 17031) * q^82 + (-858*b + 46719) * q^83 + (3820*b - 126372) * q^86 + (-16845*b + 83745) * q^88 + (10440*b + 31185) * q^89 + (2480*b - 11528) * q^91 + (-3486*b + 25038) * q^92 + (-9020*b - 56316) * q^94 + (-10944*b + 68542) * q^97 + (7843*b - 66729) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8}+O(q^{10})$$ 2 * q + 5 * q^2 + 69 * q^4 - 200 * q^7 + 615 * q^8 $$2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8} + 196 q^{11} + 360 q^{13} - 18 q^{14} + 1137 q^{16} + 1490 q^{17} - 3180 q^{19} + 6515 q^{22} + 1560 q^{23} + 4756 q^{26} - 4490 q^{28} + 3920 q^{29} - 1096 q^{31} + 5455 q^{32} + 20113 q^{34} - 2020 q^{37} + 485 q^{38} - 27754 q^{41} + 3000 q^{43} + 36887 q^{44} + 2454 q^{46} + 25760 q^{47} - 11686 q^{49} + 31700 q^{52} - 26980 q^{53} - 54270 q^{56} + 160 q^{58} - 11960 q^{59} - 24396 q^{61} + 129810 q^{62} + 43649 q^{64} + 40060 q^{67} + 133345 q^{68} + 87296 q^{71} + 70290 q^{73} + 41222 q^{74} - 67535 q^{76} + 4500 q^{77} + 65480 q^{79} - 21185 q^{82} + 92580 q^{83} - 248924 q^{86} + 150645 q^{88} + 72810 q^{89} - 20576 q^{91} + 46590 q^{92} - 121652 q^{94} + 126140 q^{97} - 125615 q^{98}+O(q^{100})$$ 2 * q + 5 * q^2 + 69 * q^4 - 200 * q^7 + 615 * q^8 + 196 * q^11 + 360 * q^13 - 18 * q^14 + 1137 * q^16 + 1490 * q^17 - 3180 * q^19 + 6515 * q^22 + 1560 * q^23 + 4756 * q^26 - 4490 * q^28 + 3920 * q^29 - 1096 * q^31 + 5455 * q^32 + 20113 * q^34 - 2020 * q^37 + 485 * q^38 - 27754 * q^41 + 3000 * q^43 + 36887 * q^44 + 2454 * q^46 + 25760 * q^47 - 11686 * q^49 + 31700 * q^52 - 26980 * q^53 - 54270 * q^56 + 160 * q^58 - 11960 * q^59 - 24396 * q^61 + 129810 * q^62 + 43649 * q^64 + 40060 * q^67 + 133345 * q^68 + 87296 * q^71 + 70290 * q^73 + 41222 * q^74 - 67535 * q^76 + 4500 * q^77 + 65480 * q^79 - 21185 * q^82 + 92580 * q^83 - 248924 * q^86 + 150645 * q^88 + 72810 * q^89 - 20576 * q^91 + 46590 * q^92 - 121652 * q^94 + 126140 * q^97 - 125615 * 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
 8.26209 −7.26209
−5.26209 0 −4.31044 0 0 −131.048 191.069 0 0
1.2 10.2621 0 73.3104 0 0 −68.9517 423.931 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.s 2
3.b odd 2 1 25.6.a.b 2
5.b even 2 1 225.6.a.l 2
5.c odd 4 2 225.6.b.i 4
12.b even 2 1 400.6.a.w 2
15.d odd 2 1 25.6.a.d yes 2
15.e even 4 2 25.6.b.b 4
60.h even 2 1 400.6.a.o 2
60.l odd 4 2 400.6.c.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 3.b odd 2 1
25.6.a.d yes 2 15.d odd 2 1
25.6.b.b 4 15.e even 4 2
225.6.a.l 2 5.b even 2 1
225.6.a.s 2 1.a even 1 1 trivial
225.6.b.i 4 5.c odd 4 2
400.6.a.o 2 60.h even 2 1
400.6.a.w 2 12.b even 2 1
400.6.c.n 4 60.l odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} - 5T_{2} - 54$$ T2^2 - 5*T2 - 54 $$T_{7}^{2} + 200T_{7} + 9036$$ T7^2 + 200*T7 + 9036

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5T - 54$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 200T + 9036$$
$11$ $$T^{2} - 196T - 141021$$
$13$ $$T^{2} - 360T - 29296$$
$17$ $$T^{2} - 1490 T - 559359$$
$19$ $$T^{2} + 3180 T + 2232875$$
$23$ $$T^{2} - 1560 T + 599724$$
$29$ $$T^{2} - 3920 T + 3456000$$
$31$ $$T^{2} + 1096 T - 72602196$$
$37$ $$T^{2} + 2020 T - 7864124$$
$41$ $$T^{2} + 27754 T + 182931129$$
$43$ $$T^{2} - 3000 T - 270585136$$
$47$ $$T^{2} - 25760 T + 22262256$$
$53$ $$T^{2} + 26980 T + 147908484$$
$59$ $$T^{2} + 11960 T - 195696000$$
$61$ $$T^{2} + 24396 T - 92208796$$
$67$ $$T^{2} - 40060 T + 249648291$$
$71$ $$T^{2} - 87296 T + 1844897904$$
$73$ $$T^{2} - 70290 T + 1160669249$$
$79$ $$T^{2} - 65480 T - 446416500$$
$83$ $$T^{2} - 92580 T + 2098410219$$
$89$ $$T^{2} - 72810 T - 5241540375$$
$97$ $$T^{2} - 126140 T - 3238386044$$