# Properties

 Label 225.6.a.r 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: $$\Q(\sqrt{145})$$ Defining polynomial: $$x^{2} - x - 36$$ x^2 - x - 36 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{145})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{2} + ( - 5 \beta + 13) q^{4} + (20 \beta - 50) q^{7} + (9 \beta + 123) q^{8}+O(q^{10})$$ q + (-b + 3) * q^2 + (-5*b + 13) * q^4 + (20*b - 50) * q^7 + (9*b + 123) * q^8 $$q + ( - \beta + 3) q^{2} + ( - 5 \beta + 13) q^{4} + (20 \beta - 50) q^{7} + (9 \beta + 123) q^{8} + ( - 20 \beta - 390) q^{11} + ( - 80 \beta + 100) q^{13} + (90 \beta - 870) q^{14} + (55 \beta - 371) q^{16} + (32 \beta + 954) q^{17} + (320 \beta - 916) q^{19} + (350 \beta - 450) q^{22} + (504 \beta - 912) q^{23} + ( - 260 \beta + 3180) q^{26} + (410 \beta - 4250) q^{28} + (1280 \beta - 1290) q^{29} + ( - 440 \beta - 2692) q^{31} + (193 \beta - 7029) q^{32} + ( - 890 \beta + 1710) q^{34} + ( - 1440 \beta + 7000) q^{37} + (1556 \beta - 14268) q^{38} + ( - 1360 \beta + 480) q^{41} + ( - 1280 \beta - 12200) q^{43} + (1790 \beta - 1470) q^{44} + (1920 \beta - 20880) q^{46} + ( - 184 \beta + 9552) q^{47} + ( - 1600 \beta + 93) q^{49} + ( - 1140 \beta + 15700) q^{52} + (608 \beta + 24426) q^{53} + (2190 \beta + 330) q^{56} + (3850 \beta - 49950) q^{58} + ( - 980 \beta - 31110) q^{59} + ( - 5440 \beta - 21838) q^{61} + (1812 \beta + 7764) q^{62} + (5655 \beta - 16163) q^{64} + (8120 \beta - 7100) q^{67} + ( - 4514 \beta + 6642) q^{68} + ( - 1480 \beta - 31860) q^{71} + ( - 4640 \beta - 46550) q^{73} + ( - 9880 \beta + 72840) q^{74} + (7140 \beta - 69508) q^{76} + ( - 7200 \beta + 5100) q^{77} + (2680 \beta - 24484) q^{79} + ( - 3200 \beta + 50400) q^{82} + ( - 10728 \beta - 23316) q^{83} + (9640 \beta + 9480) q^{86} + ( - 6150 \beta - 54450) q^{88} + ( - 8400 \beta + 47700) q^{89} + (4400 \beta - 62600) q^{91} + (8592 \beta - 102576) q^{92} + ( - 9920 \beta + 35280) q^{94} + ( - 2880 \beta - 3650) q^{97} + ( - 3293 \beta + 57879) q^{98}+O(q^{100})$$ q + (-b + 3) * q^2 + (-5*b + 13) * q^4 + (20*b - 50) * q^7 + (9*b + 123) * q^8 + (-20*b - 390) * q^11 + (-80*b + 100) * q^13 + (90*b - 870) * q^14 + (55*b - 371) * q^16 + (32*b + 954) * q^17 + (320*b - 916) * q^19 + (350*b - 450) * q^22 + (504*b - 912) * q^23 + (-260*b + 3180) * q^26 + (410*b - 4250) * q^28 + (1280*b - 1290) * q^29 + (-440*b - 2692) * q^31 + (193*b - 7029) * q^32 + (-890*b + 1710) * q^34 + (-1440*b + 7000) * q^37 + (1556*b - 14268) * q^38 + (-1360*b + 480) * q^41 + (-1280*b - 12200) * q^43 + (1790*b - 1470) * q^44 + (1920*b - 20880) * q^46 + (-184*b + 9552) * q^47 + (-1600*b + 93) * q^49 + (-1140*b + 15700) * q^52 + (608*b + 24426) * q^53 + (2190*b + 330) * q^56 + (3850*b - 49950) * q^58 + (-980*b - 31110) * q^59 + (-5440*b - 21838) * q^61 + (1812*b + 7764) * q^62 + (5655*b - 16163) * q^64 + (8120*b - 7100) * q^67 + (-4514*b + 6642) * q^68 + (-1480*b - 31860) * q^71 + (-4640*b - 46550) * q^73 + (-9880*b + 72840) * q^74 + (7140*b - 69508) * q^76 + (-7200*b + 5100) * q^77 + (2680*b - 24484) * q^79 + (-3200*b + 50400) * q^82 + (-10728*b - 23316) * q^83 + (9640*b + 9480) * q^86 + (-6150*b - 54450) * q^88 + (-8400*b + 47700) * q^89 + (4400*b - 62600) * q^91 + (8592*b - 102576) * q^92 + (-9920*b + 35280) * q^94 + (-2880*b - 3650) * q^97 + (-3293*b + 57879) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 21 q^{4} - 80 q^{7} + 255 q^{8}+O(q^{10})$$ 2 * q + 5 * q^2 + 21 * q^4 - 80 * q^7 + 255 * q^8 $$2 q + 5 q^{2} + 21 q^{4} - 80 q^{7} + 255 q^{8} - 800 q^{11} + 120 q^{13} - 1650 q^{14} - 687 q^{16} + 1940 q^{17} - 1512 q^{19} - 550 q^{22} - 1320 q^{23} + 6100 q^{26} - 8090 q^{28} - 1300 q^{29} - 5824 q^{31} - 13865 q^{32} + 2530 q^{34} + 12560 q^{37} - 26980 q^{38} - 400 q^{41} - 25680 q^{43} - 1150 q^{44} - 39840 q^{46} + 18920 q^{47} - 1414 q^{49} + 30260 q^{52} + 49460 q^{53} + 2850 q^{56} - 96050 q^{58} - 63200 q^{59} - 49116 q^{61} + 17340 q^{62} - 26671 q^{64} - 6080 q^{67} + 8770 q^{68} - 65200 q^{71} - 97740 q^{73} + 135800 q^{74} - 131876 q^{76} + 3000 q^{77} - 46288 q^{79} + 97600 q^{82} - 57360 q^{83} + 28600 q^{86} - 115050 q^{88} + 87000 q^{89} - 120800 q^{91} - 196560 q^{92} + 60640 q^{94} - 10180 q^{97} + 112465 q^{98}+O(q^{100})$$ 2 * q + 5 * q^2 + 21 * q^4 - 80 * q^7 + 255 * q^8 - 800 * q^11 + 120 * q^13 - 1650 * q^14 - 687 * q^16 + 1940 * q^17 - 1512 * q^19 - 550 * q^22 - 1320 * q^23 + 6100 * q^26 - 8090 * q^28 - 1300 * q^29 - 5824 * q^31 - 13865 * q^32 + 2530 * q^34 + 12560 * q^37 - 26980 * q^38 - 400 * q^41 - 25680 * q^43 - 1150 * q^44 - 39840 * q^46 + 18920 * q^47 - 1414 * q^49 + 30260 * q^52 + 49460 * q^53 + 2850 * q^56 - 96050 * q^58 - 63200 * q^59 - 49116 * q^61 + 17340 * q^62 - 26671 * q^64 - 6080 * q^67 + 8770 * q^68 - 65200 * q^71 - 97740 * q^73 + 135800 * q^74 - 131876 * q^76 + 3000 * q^77 - 46288 * q^79 + 97600 * q^82 - 57360 * q^83 + 28600 * q^86 - 115050 * q^88 + 87000 * q^89 - 120800 * q^91 - 196560 * q^92 + 60640 * q^94 - 10180 * q^97 + 112465 * 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
 6.52080 −5.52080
−3.52080 0 −19.6040 0 0 80.4159 181.687 0 0
1.2 8.52080 0 40.6040 0 0 −160.416 73.3128 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.r 2
3.b odd 2 1 225.6.a.k 2
5.b even 2 1 45.6.a.d 2
5.c odd 4 2 225.6.b.j 4
15.d odd 2 1 45.6.a.f yes 2
15.e even 4 2 225.6.b.k 4
20.d odd 2 1 720.6.a.y 2
60.h even 2 1 720.6.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.a.d 2 5.b even 2 1
45.6.a.f yes 2 15.d odd 2 1
225.6.a.k 2 3.b odd 2 1
225.6.a.r 2 1.a even 1 1 trivial
225.6.b.j 4 5.c odd 4 2
225.6.b.k 4 15.e even 4 2
720.6.a.y 2 20.d odd 2 1
720.6.a.be 2 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 5T_{2} - 30$$ T2^2 - 5*T2 - 30 $$T_{7}^{2} + 80T_{7} - 12900$$ T7^2 + 80*T7 - 12900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5T - 30$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 80T - 12900$$
$11$ $$T^{2} + 800T + 145500$$
$13$ $$T^{2} - 120T - 228400$$
$17$ $$T^{2} - 1940 T + 903780$$
$19$ $$T^{2} + 1512 T - 3140464$$
$23$ $$T^{2} + 1320 T - 8772480$$
$29$ $$T^{2} + 1300 T - 58969500$$
$31$ $$T^{2} + 5824 T + 1461744$$
$37$ $$T^{2} - 12560 T - 35729600$$
$41$ $$T^{2} + 400 T - 67008000$$
$43$ $$T^{2} + 25680 T + 105473600$$
$47$ $$T^{2} - 18920 T + 88264320$$
$53$ $$T^{2} - 49460 T + 598172580$$
$59$ $$T^{2} + 63200 T + 963745500$$
$61$ $$T^{2} + 49116 T - 469672636$$
$67$ $$T^{2} + 6080 T - 2380880400$$
$71$ $$T^{2} + 65200 T + 983358000$$
$73$ $$T^{2} + 97740 T + 1607828900$$
$79$ $$T^{2} + 46288 T + 275282736$$
$83$ $$T^{2} + 57360 T - 3349469520$$
$89$ $$T^{2} - 87000 T - 665550000$$
$97$ $$T^{2} + 10180 T - 274763900$$