# Properties

 Label 225.6.a.l 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: $$\Q(\sqrt{241})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 2) q^{2} + (5 \beta + 32) q^{4} + ( - 4 \beta + 102) q^{7} + ( - 15 \beta - 300) q^{8}+O(q^{10})$$ q + (-b - 2) * q^2 + (5*b + 32) * q^4 + (-4*b + 102) * q^7 + (-15*b - 300) * q^8 $$q + ( - \beta - 2) q^{2} + (5 \beta + 32) q^{4} + ( - 4 \beta + 102) q^{7} + ( - 15 \beta - 300) q^{8} + (50 \beta + 73) q^{11} + ( - 32 \beta - 164) q^{13} + ( - 90 \beta + 36) q^{14} + (185 \beta + 476) q^{16} + ( - 136 \beta - 677) q^{17} + (70 \beta - 1625) q^{19} + ( - 223 \beta - 3146) q^{22} + (12 \beta - 786) q^{23} + (260 \beta + 2248) q^{26} + (362 \beta + 2064) q^{28} + ( - 80 \beta + 2000) q^{29} + (1100 \beta - 1098) q^{31} + ( - 551 \beta - 2452) q^{32} + (1085 \beta + 9514) q^{34} + ( - 384 \beta + 1202) q^{37} + (1415 \beta - 950) q^{38} + (400 \beta - 14077) q^{41} + (2128 \beta - 2564) q^{43} + (2215 \beta + 17336) q^{44} + (750 \beta + 852) q^{46} + (1544 \beta - 13652) q^{47} + ( - 800 \beta - 5443) q^{49} + ( - 2004 \beta - 14848) q^{52} + (752 \beta + 13114) q^{53} + ( - 270 \beta - 27000) q^{56} + ( - 1760 \beta + 800) q^{58} + ( - 1960 \beta - 5000) q^{59} + ( - 2000 \beta - 11198) q^{61} + ( - 2202 \beta - 63804) q^{62} + ( - 1815 \beta + 22732) q^{64} + (1586 \beta - 20823) q^{67} + ( - 8417 \beta - 62464) q^{68} + (1000 \beta + 43148) q^{71} + ( - 1112 \beta - 34589) q^{73} + ( - 50 \beta + 20636) q^{74} + ( - 5535 \beta - 31000) q^{76} + (4608 \beta - 4554) q^{77} + ( - 5020 \beta + 35250) q^{79} + (12877 \beta + 4154) q^{82} + ( - 858 \beta - 45861) q^{83} + ( - 3820 \beta - 122552) q^{86} + ( - 16845 \beta - 66900) q^{88} + ( - 10440 \beta + 41625) q^{89} + ( - 2480 \beta - 9048) q^{91} + ( - 3486 \beta - 21552) q^{92} + (9020 \beta - 65336) q^{94} + ( - 10944 \beta - 57598) q^{97} + (7843 \beta + 58886) q^{98}+O(q^{100})$$ q + (-b - 2) * q^2 + (5*b + 32) * q^4 + (-4*b + 102) * q^7 + (-15*b - 300) * q^8 + (50*b + 73) * q^11 + (-32*b - 164) * q^13 + (-90*b + 36) * q^14 + (185*b + 476) * q^16 + (-136*b - 677) * q^17 + (70*b - 1625) * q^19 + (-223*b - 3146) * q^22 + (12*b - 786) * q^23 + (260*b + 2248) * q^26 + (362*b + 2064) * q^28 + (-80*b + 2000) * q^29 + (1100*b - 1098) * q^31 + (-551*b - 2452) * q^32 + (1085*b + 9514) * q^34 + (-384*b + 1202) * q^37 + (1415*b - 950) * q^38 + (400*b - 14077) * q^41 + (2128*b - 2564) * q^43 + (2215*b + 17336) * q^44 + (750*b + 852) * q^46 + (1544*b - 13652) * q^47 + (-800*b - 5443) * q^49 + (-2004*b - 14848) * q^52 + (752*b + 13114) * q^53 + (-270*b - 27000) * q^56 + (-1760*b + 800) * q^58 + (-1960*b - 5000) * q^59 + (-2000*b - 11198) * q^61 + (-2202*b - 63804) * q^62 + (-1815*b + 22732) * q^64 + (1586*b - 20823) * q^67 + (-8417*b - 62464) * q^68 + (1000*b + 43148) * q^71 + (-1112*b - 34589) * q^73 + (-50*b + 20636) * q^74 + (-5535*b - 31000) * q^76 + (4608*b - 4554) * q^77 + (-5020*b + 35250) * q^79 + (12877*b + 4154) * q^82 + (-858*b - 45861) * q^83 + (-3820*b - 122552) * q^86 + (-16845*b - 66900) * q^88 + (-10440*b + 41625) * q^89 + (-2480*b - 9048) * q^91 + (-3486*b - 21552) * q^92 + (9020*b - 65336) * q^94 + (-10944*b - 57598) * q^97 + (7843*b + 58886) * 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
−10.2621 0 73.3104 0 0 68.9517 −423.931 0 0
1.2 5.26209 0 −4.31044 0 0 131.048 −191.069 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.l 2
3.b odd 2 1 25.6.a.d yes 2
5.b even 2 1 225.6.a.s 2
5.c odd 4 2 225.6.b.i 4
12.b even 2 1 400.6.a.o 2
15.d odd 2 1 25.6.a.b 2
15.e even 4 2 25.6.b.b 4
60.h even 2 1 400.6.a.w 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 15.d odd 2 1
25.6.a.d yes 2 3.b odd 2 1
25.6.b.b 4 15.e even 4 2
225.6.a.l 2 1.a even 1 1 trivial
225.6.a.s 2 5.b even 2 1
225.6.b.i 4 5.c odd 4 2
400.6.a.o 2 12.b even 2 1
400.6.a.w 2 60.h 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

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