# Properties

 Label 1225.4.a.q Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + 5 q^{3} + ( - 2 \beta + 4) q^{4} + (5 \beta - 5) q^{6} + ( - 2 \beta - 18) q^{8} - 2 q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + 5 * q^3 + (-2*b + 4) * q^4 + (5*b - 5) * q^6 + (-2*b - 18) * q^8 - 2 * q^9 $$q + (\beta - 1) q^{2} + 5 q^{3} + ( - 2 \beta + 4) q^{4} + (5 \beta - 5) q^{6} + ( - 2 \beta - 18) q^{8} - 2 q^{9} + (4 \beta + 33) q^{11} + ( - 10 \beta + 20) q^{12} + ( - 20 \beta + 5) q^{13} - 36 q^{16} + (20 \beta + 35) q^{17} + ( - 2 \beta + 2) q^{18} + (20 \beta - 70) q^{19} + (29 \beta + 11) q^{22} + ( - 28 \beta + 8) q^{23} + ( - 10 \beta - 90) q^{24} + (25 \beta - 225) q^{26} - 145 q^{27} + (48 \beta - 129) q^{29} + ( - 60 \beta + 10) q^{31} + ( - 20 \beta + 180) q^{32} + (20 \beta + 165) q^{33} + (15 \beta + 185) q^{34} + (4 \beta - 8) q^{36} + (44 \beta - 164) q^{37} + ( - 90 \beta + 290) q^{38} + ( - 100 \beta + 25) q^{39} + ( - 100 \beta - 150) q^{41} + (12 \beta + 58) q^{43} + ( - 50 \beta + 44) q^{44} + (36 \beta - 316) q^{46} + (40 \beta - 15) q^{47} - 180 q^{48} + (100 \beta + 175) q^{51} + ( - 90 \beta + 460) q^{52} + ( - 120 \beta - 270) q^{53} + ( - 145 \beta + 145) q^{54} + (100 \beta - 350) q^{57} + ( - 177 \beta + 657) q^{58} + (40 \beta - 190) q^{59} + ( - 60 \beta - 540) q^{61} + (70 \beta - 670) q^{62} + (200 \beta - 112) q^{64} + (145 \beta + 55) q^{66} + ( - 96 \beta - 234) q^{67} + (10 \beta - 300) q^{68} + ( - 140 \beta + 40) q^{69} + ( - 64 \beta - 528) q^{71} + (4 \beta + 36) q^{72} + (200 \beta - 430) q^{73} + ( - 208 \beta + 648) q^{74} + (220 \beta - 720) q^{76} + (125 \beta - 1125) q^{78} + ( - 348 \beta + 79) q^{79} - 671 q^{81} + ( - 50 \beta - 950) q^{82} + (200 \beta + 20) q^{83} + (46 \beta + 74) q^{86} + (240 \beta - 645) q^{87} + ( - 138 \beta - 682) q^{88} + (380 \beta + 120) q^{89} + ( - 128 \beta + 648) q^{92} + ( - 300 \beta + 50) q^{93} + ( - 55 \beta + 455) q^{94} + ( - 100 \beta + 900) q^{96} + ( - 180 \beta + 815) q^{97} + ( - 8 \beta - 66) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 + 5 * q^3 + (-2*b + 4) * q^4 + (5*b - 5) * q^6 + (-2*b - 18) * q^8 - 2 * q^9 + (4*b + 33) * q^11 + (-10*b + 20) * q^12 + (-20*b + 5) * q^13 - 36 * q^16 + (20*b + 35) * q^17 + (-2*b + 2) * q^18 + (20*b - 70) * q^19 + (29*b + 11) * q^22 + (-28*b + 8) * q^23 + (-10*b - 90) * q^24 + (25*b - 225) * q^26 - 145 * q^27 + (48*b - 129) * q^29 + (-60*b + 10) * q^31 + (-20*b + 180) * q^32 + (20*b + 165) * q^33 + (15*b + 185) * q^34 + (4*b - 8) * q^36 + (44*b - 164) * q^37 + (-90*b + 290) * q^38 + (-100*b + 25) * q^39 + (-100*b - 150) * q^41 + (12*b + 58) * q^43 + (-50*b + 44) * q^44 + (36*b - 316) * q^46 + (40*b - 15) * q^47 - 180 * q^48 + (100*b + 175) * q^51 + (-90*b + 460) * q^52 + (-120*b - 270) * q^53 + (-145*b + 145) * q^54 + (100*b - 350) * q^57 + (-177*b + 657) * q^58 + (40*b - 190) * q^59 + (-60*b - 540) * q^61 + (70*b - 670) * q^62 + (200*b - 112) * q^64 + (145*b + 55) * q^66 + (-96*b - 234) * q^67 + (10*b - 300) * q^68 + (-140*b + 40) * q^69 + (-64*b - 528) * q^71 + (4*b + 36) * q^72 + (200*b - 430) * q^73 + (-208*b + 648) * q^74 + (220*b - 720) * q^76 + (125*b - 1125) * q^78 + (-348*b + 79) * q^79 - 671 * q^81 + (-50*b - 950) * q^82 + (200*b + 20) * q^83 + (46*b + 74) * q^86 + (240*b - 645) * q^87 + (-138*b - 682) * q^88 + (380*b + 120) * q^89 + (-128*b + 648) * q^92 + (-300*b + 50) * q^93 + (-55*b + 455) * q^94 + (-100*b + 900) * q^96 + (-180*b + 815) * q^97 + (-8*b - 66) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 $$2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} + 40 q^{12} + 10 q^{13} - 72 q^{16} + 70 q^{17} + 4 q^{18} - 140 q^{19} + 22 q^{22} + 16 q^{23} - 180 q^{24} - 450 q^{26} - 290 q^{27} - 258 q^{29} + 20 q^{31} + 360 q^{32} + 330 q^{33} + 370 q^{34} - 16 q^{36} - 328 q^{37} + 580 q^{38} + 50 q^{39} - 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} - 30 q^{47} - 360 q^{48} + 350 q^{51} + 920 q^{52} - 540 q^{53} + 290 q^{54} - 700 q^{57} + 1314 q^{58} - 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} + 110 q^{66} - 468 q^{67} - 600 q^{68} + 80 q^{69} - 1056 q^{71} + 72 q^{72} - 860 q^{73} + 1296 q^{74} - 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} - 1900 q^{82} + 40 q^{83} + 148 q^{86} - 1290 q^{87} - 1364 q^{88} + 240 q^{89} + 1296 q^{92} + 100 q^{93} + 910 q^{94} + 1800 q^{96} + 1630 q^{97} - 132 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 + 40 * q^12 + 10 * q^13 - 72 * q^16 + 70 * q^17 + 4 * q^18 - 140 * q^19 + 22 * q^22 + 16 * q^23 - 180 * q^24 - 450 * q^26 - 290 * q^27 - 258 * q^29 + 20 * q^31 + 360 * q^32 + 330 * q^33 + 370 * q^34 - 16 * q^36 - 328 * q^37 + 580 * q^38 + 50 * q^39 - 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 - 30 * q^47 - 360 * q^48 + 350 * q^51 + 920 * q^52 - 540 * q^53 + 290 * q^54 - 700 * q^57 + 1314 * q^58 - 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 + 110 * q^66 - 468 * q^67 - 600 * q^68 + 80 * q^69 - 1056 * q^71 + 72 * q^72 - 860 * q^73 + 1296 * q^74 - 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 - 1900 * q^82 + 40 * q^83 + 148 * q^86 - 1290 * q^87 - 1364 * q^88 + 240 * q^89 + 1296 * q^92 + 100 * q^93 + 910 * q^94 + 1800 * q^96 + 1630 * q^97 - 132 * 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
 −3.31662 3.31662
−4.31662 5.00000 10.6332 0 −21.5831 0 −11.3668 −2.00000 0
1.2 2.31662 5.00000 −2.63325 0 11.5831 0 −24.6332 −2.00000 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1225.4.a.q 2
5.b even 2 1 245.4.a.i 2
7.b odd 2 1 1225.4.a.p 2
15.d odd 2 1 2205.4.a.x 2
35.c odd 2 1 245.4.a.j yes 2
35.i odd 6 2 245.4.e.j 4
35.j even 6 2 245.4.e.k 4
105.g even 2 1 2205.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 5.b even 2 1
245.4.a.j yes 2 35.c odd 2 1
245.4.e.j 4 35.i odd 6 2
245.4.e.k 4 35.j even 6 2
1225.4.a.p 2 7.b odd 2 1
1225.4.a.q 2 1.a even 1 1 trivial
2205.4.a.w 2 105.g even 2 1
2205.4.a.x 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{2} + 2T_{2} - 10$$ T2^2 + 2*T2 - 10 $$T_{3} - 5$$ T3 - 5 $$T_{19}^{2} + 140T_{19} + 500$$ T19^2 + 140*T19 + 500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 10$$
$3$ $$(T - 5)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 66T + 913$$
$13$ $$T^{2} - 10T - 4375$$
$17$ $$T^{2} - 70T - 3175$$
$19$ $$T^{2} + 140T + 500$$
$23$ $$T^{2} - 16T - 8560$$
$29$ $$T^{2} + 258T - 8703$$
$31$ $$T^{2} - 20T - 39500$$
$37$ $$T^{2} + 328T + 5600$$
$41$ $$T^{2} + 300T - 87500$$
$43$ $$T^{2} - 116T + 1780$$
$47$ $$T^{2} + 30T - 17375$$
$53$ $$T^{2} + 540T - 85500$$
$59$ $$T^{2} + 380T + 18500$$
$61$ $$T^{2} + 1080 T + 252000$$
$67$ $$T^{2} + 468T - 46620$$
$71$ $$T^{2} + 1056 T + 233728$$
$73$ $$T^{2} + 860T - 255100$$
$79$ $$T^{2} - 158 T - 1325903$$
$83$ $$T^{2} - 40T - 439600$$
$89$ $$T^{2} - 240 T - 1574000$$
$97$ $$T^{2} - 1630 T + 307825$$