# Properties

 Label 4225.2.a.r Level $4225$ Weight $2$ Character orbit 4225.a Self dual yes Analytic conductor $33.737$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4225 = 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.7367948540$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 2) q^{6} + ( - 2 \beta + 2) q^{7} + (\beta - 3) q^{8} - q^{9} +O(q^{10})$$ q + (b - 1) * q^2 - b * q^3 + (-2*b + 1) * q^4 + (b - 2) * q^6 + (-2*b + 2) * q^7 + (b - 3) * q^8 - q^9 $$q + (\beta - 1) q^{2} - \beta q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 2) q^{6} + ( - 2 \beta + 2) q^{7} + (\beta - 3) q^{8} - q^{9} + (\beta - 2) q^{11} + ( - \beta + 4) q^{12} + (4 \beta - 6) q^{14} + 3 q^{16} + (2 \beta + 2) q^{17} + ( - \beta + 1) q^{18} + ( - \beta - 2) q^{19} + ( - 2 \beta + 4) q^{21} + ( - 3 \beta + 4) q^{22} + \beta q^{23} + (3 \beta - 2) q^{24} + 4 \beta q^{27} + ( - 6 \beta + 10) q^{28} + 4 \beta q^{29} + ( - 3 \beta - 6) q^{31} + (\beta + 3) q^{32} + (2 \beta - 2) q^{33} + 2 q^{34} + (2 \beta - 1) q^{36} + 6 \beta q^{37} - \beta q^{38} + (2 \beta + 6) q^{41} + (6 \beta - 8) q^{42} + ( - 5 \beta + 4) q^{43} + (5 \beta - 6) q^{44} + ( - \beta + 2) q^{46} + (2 \beta - 2) q^{47} - 3 \beta q^{48} + ( - 8 \beta + 5) q^{49} + ( - 2 \beta - 4) q^{51} + (6 \beta + 6) q^{53} + ( - 4 \beta + 8) q^{54} + (8 \beta - 10) q^{56} + (2 \beta + 2) q^{57} + ( - 4 \beta + 8) q^{58} + ( - 3 \beta - 6) q^{59} - 8 q^{61} - 3 \beta q^{62} + (2 \beta - 2) q^{63} + (2 \beta - 7) q^{64} + ( - 4 \beta + 6) q^{66} - 2 q^{67} + ( - 2 \beta - 6) q^{68} - 2 q^{69} + (7 \beta - 2) q^{71} + ( - \beta + 3) q^{72} - 6 \beta q^{73} + ( - 6 \beta + 12) q^{74} + (3 \beta + 2) q^{76} + (6 \beta - 8) q^{77} + 6 \beta q^{79} - 5 q^{81} + (4 \beta - 2) q^{82} + ( - 2 \beta - 6) q^{83} + ( - 10 \beta + 12) q^{84} + (9 \beta - 14) q^{86} - 8 q^{87} + ( - 5 \beta + 8) q^{88} - 6 q^{89} + (\beta - 4) q^{92} + (6 \beta + 6) q^{93} + ( - 4 \beta + 6) q^{94} + ( - 3 \beta - 2) q^{96} + (4 \beta - 2) q^{97} + (13 \beta - 21) q^{98} + ( - \beta + 2) q^{99} +O(q^{100})$$ q + (b - 1) * q^2 - b * q^3 + (-2*b + 1) * q^4 + (b - 2) * q^6 + (-2*b + 2) * q^7 + (b - 3) * q^8 - q^9 + (b - 2) * q^11 + (-b + 4) * q^12 + (4*b - 6) * q^14 + 3 * q^16 + (2*b + 2) * q^17 + (-b + 1) * q^18 + (-b - 2) * q^19 + (-2*b + 4) * q^21 + (-3*b + 4) * q^22 + b * q^23 + (3*b - 2) * q^24 + 4*b * q^27 + (-6*b + 10) * q^28 + 4*b * q^29 + (-3*b - 6) * q^31 + (b + 3) * q^32 + (2*b - 2) * q^33 + 2 * q^34 + (2*b - 1) * q^36 + 6*b * q^37 - b * q^38 + (2*b + 6) * q^41 + (6*b - 8) * q^42 + (-5*b + 4) * q^43 + (5*b - 6) * q^44 + (-b + 2) * q^46 + (2*b - 2) * q^47 - 3*b * q^48 + (-8*b + 5) * q^49 + (-2*b - 4) * q^51 + (6*b + 6) * q^53 + (-4*b + 8) * q^54 + (8*b - 10) * q^56 + (2*b + 2) * q^57 + (-4*b + 8) * q^58 + (-3*b - 6) * q^59 - 8 * q^61 - 3*b * q^62 + (2*b - 2) * q^63 + (2*b - 7) * q^64 + (-4*b + 6) * q^66 - 2 * q^67 + (-2*b - 6) * q^68 - 2 * q^69 + (7*b - 2) * q^71 + (-b + 3) * q^72 - 6*b * q^73 + (-6*b + 12) * q^74 + (3*b + 2) * q^76 + (6*b - 8) * q^77 + 6*b * q^79 - 5 * q^81 + (4*b - 2) * q^82 + (-2*b - 6) * q^83 + (-10*b + 12) * q^84 + (9*b - 14) * q^86 - 8 * q^87 + (-5*b + 8) * q^88 - 6 * q^89 + (b - 4) * q^92 + (6*b + 6) * q^93 + (-4*b + 6) * q^94 + (-3*b - 2) * q^96 + (4*b - 2) * q^97 + (13*b - 21) * q^98 + (-b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} - 4 q^{11} + 8 q^{12} - 12 q^{14} + 6 q^{16} + 4 q^{17} + 2 q^{18} - 4 q^{19} + 8 q^{21} + 8 q^{22} - 4 q^{24} + 20 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{33} + 4 q^{34} - 2 q^{36} + 12 q^{41} - 16 q^{42} + 8 q^{43} - 12 q^{44} + 4 q^{46} - 4 q^{47} + 10 q^{49} - 8 q^{51} + 12 q^{53} + 16 q^{54} - 20 q^{56} + 4 q^{57} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 4 q^{63} - 14 q^{64} + 12 q^{66} - 4 q^{67} - 12 q^{68} - 4 q^{69} - 4 q^{71} + 6 q^{72} + 24 q^{74} + 4 q^{76} - 16 q^{77} - 10 q^{81} - 4 q^{82} - 12 q^{83} + 24 q^{84} - 28 q^{86} - 16 q^{87} + 16 q^{88} - 12 q^{89} - 8 q^{92} + 12 q^{93} + 12 q^{94} - 4 q^{96} - 4 q^{97} - 42 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 - 4 * q^11 + 8 * q^12 - 12 * q^14 + 6 * q^16 + 4 * q^17 + 2 * q^18 - 4 * q^19 + 8 * q^21 + 8 * q^22 - 4 * q^24 + 20 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^33 + 4 * q^34 - 2 * q^36 + 12 * q^41 - 16 * q^42 + 8 * q^43 - 12 * q^44 + 4 * q^46 - 4 * q^47 + 10 * q^49 - 8 * q^51 + 12 * q^53 + 16 * q^54 - 20 * q^56 + 4 * q^57 + 16 * q^58 - 12 * q^59 - 16 * q^61 - 4 * q^63 - 14 * q^64 + 12 * q^66 - 4 * q^67 - 12 * q^68 - 4 * q^69 - 4 * q^71 + 6 * q^72 + 24 * q^74 + 4 * q^76 - 16 * q^77 - 10 * q^81 - 4 * q^82 - 12 * q^83 + 24 * q^84 - 28 * q^86 - 16 * q^87 + 16 * q^88 - 12 * q^89 - 8 * q^92 + 12 * q^93 + 12 * q^94 - 4 * q^96 - 4 * q^97 - 42 * q^98 + 4 * q^99

## 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
 −1.41421 1.41421
−2.41421 1.41421 3.82843 0 −3.41421 4.82843 −4.41421 −1.00000 0
1.2 0.414214 −1.41421 −1.82843 0 −0.585786 −0.828427 −1.58579 −1.00000 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
$$5$$ $$1$$
$$13$$ $$1$$

## 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 4225.2.a.r 2
5.b even 2 1 845.2.a.g 2
13.b even 2 1 325.2.a.i 2
15.d odd 2 1 7605.2.a.x 2
39.d odd 2 1 2925.2.a.u 2
52.b odd 2 1 5200.2.a.bu 2
65.d even 2 1 65.2.a.b 2
65.g odd 4 2 845.2.c.b 4
65.h odd 4 2 325.2.b.f 4
65.l even 6 2 845.2.e.h 4
65.n even 6 2 845.2.e.c 4
65.s odd 12 4 845.2.m.f 8
195.e odd 2 1 585.2.a.m 2
195.s even 4 2 2925.2.c.r 4
260.g odd 2 1 1040.2.a.j 2
455.h odd 2 1 3185.2.a.j 2
520.b odd 2 1 4160.2.a.z 2
520.p even 2 1 4160.2.a.bf 2
715.c odd 2 1 7865.2.a.j 2
780.d even 2 1 9360.2.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 65.d even 2 1
325.2.a.i 2 13.b even 2 1
325.2.b.f 4 65.h odd 4 2
585.2.a.m 2 195.e odd 2 1
845.2.a.g 2 5.b even 2 1
845.2.c.b 4 65.g odd 4 2
845.2.e.c 4 65.n even 6 2
845.2.e.h 4 65.l even 6 2
845.2.m.f 8 65.s odd 12 4
1040.2.a.j 2 260.g odd 2 1
2925.2.a.u 2 39.d odd 2 1
2925.2.c.r 4 195.s even 4 2
3185.2.a.j 2 455.h odd 2 1
4160.2.a.z 2 520.b odd 2 1
4160.2.a.bf 2 520.p even 2 1
4225.2.a.r 2 1.a even 1 1 trivial
5200.2.a.bu 2 52.b odd 2 1
7605.2.a.x 2 15.d odd 2 1
7865.2.a.j 2 715.c odd 2 1
9360.2.a.cd 2 780.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4225))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{7}^{2} - 4T_{7} - 4$$ T7^2 - 4*T7 - 4 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$T^{2} + 4T + 2$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} + 12T + 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 12T + 28$$
$43$ $$T^{2} - 8T - 34$$
$47$ $$T^{2} + 4T - 4$$
$53$ $$T^{2} - 12T - 36$$
$59$ $$T^{2} + 12T + 18$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$T^{2} + 4T - 94$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 72$$
$83$ $$T^{2} + 12T + 28$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4T - 28$$