# Properties

 Label 245.4.a.l Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4554679514$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.14360.1 Defining polynomial: $$x^{3} - 17x - 14$$ x^3 - 17*x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + ( - 3 \beta_{2} + 4 \beta_1 - 7) q^{6} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - b1 - 1) * q^3 + (b2 - b1 + 4) * q^4 - 5 * q^5 + (-3*b2 + 4*b1 - 7) * q^6 + (-3*b2 + b1 - 4) * q^8 + (-3*b2 - 9*b1 + 28) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + ( - 3 \beta_{2} + 4 \beta_1 - 7) q^{6} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9} + ( - 5 \beta_1 + 5) q^{10} + (\beta_{2} + 3 \beta_1 - 25) q^{11} + (2 \beta_{2} - 14 \beta_1 + 50) q^{12} + ( - 5 \beta_{2} + 13 \beta_1 - 13) q^{13} + ( - 5 \beta_{2} + 5 \beta_1 + 5) q^{15} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + ( - 11 \beta_{2} - 13 \beta_1 + 21) q^{17} + ( - 3 \beta_{2} + 13 \beta_1 - 136) q^{18} + ( - 6 \beta_{2} + 10 \beta_1 - 54) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 20) q^{20} + (\beta_{2} - 20 \beta_1 + 61) q^{22} + (2 \beta_{2} - 14 \beta_1 - 42) q^{23} + (6 \beta_{2} + 28 \beta_1 - 142) q^{24} + 25 q^{25} + (23 \beta_{2} - 38 \beta_1 + 141) q^{26} + (31 \beta_{2} - 7 \beta_1 - 67) q^{27} + (17 \beta_{2} + 19 \beta_1 + 105) q^{29} + (15 \beta_{2} - 20 \beta_1 + 35) q^{30} + (4 \beta_{2} - 24 \beta_1 - 108) q^{31} + (15 \beta_{2} - 39 \beta_1 - 66) q^{32} + ( - 35 \beta_{2} + 27 \beta_1 + 47) q^{33} + (9 \beta_{2} - 34 \beta_1 - 197) q^{34} + (43 \beta_{2} - 79 \beta_1 + 46) q^{36} + ( - 12 \beta_{2} + 16 \beta_1 - 14) q^{37} + (22 \beta_{2} - 84 \beta_1 + 146) q^{38} + ( - 19 \beta_{2} + 87 \beta_1 - 321) q^{39} + (15 \beta_{2} - 5 \beta_1 + 20) q^{40} + ( - 2 \beta_{2} - 10 \beta_1 - 120) q^{41} + ( - 34 \beta_{2} + 30 \beta_1 + 6) q^{43} + ( - 30 \beta_{2} + 42 \beta_1 - 78) q^{44} + (15 \beta_{2} + 45 \beta_1 - 140) q^{45} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + (13 \beta_{2} + 51 \beta_1 + 239) q^{47} + 68 q^{48} + (25 \beta_1 - 25) q^{50} + (91 \beta_{2} + 17 \beta_1 - 423) q^{51} + ( - 44 \beta_{2} + 152 \beta_1 - 386) q^{52} + ( - 22 \beta_{2} - 130 \beta_1 + 44) q^{53} + ( - 69 \beta_{2} + 88 \beta_1 + 83) q^{54} + ( - 5 \beta_{2} - 15 \beta_1 + 125) q^{55} + ( - 50 \beta_{2} + 126 \beta_1 - 302) q^{57} + ( - 15 \beta_{2} + 190 \beta_1 + 155) q^{58} + ( - 48 \beta_{2} + 176 \beta_1 + 76) q^{59} + ( - 10 \beta_{2} + 70 \beta_1 - 250) q^{60} + (26 \beta_{2} + 34 \beta_1 - 416) q^{61} + ( - 32 \beta_{2} - 88 \beta_1 - 144) q^{62} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + (25 \beta_{2} - 65 \beta_1 + 65) q^{65} + (97 \beta_{2} - 128 \beta_1 + 145) q^{66} + (108 \beta_{2} - 12 \beta_1 + 32) q^{67} + (36 \beta_{2} - 48 \beta_1 - 318) q^{68} + ( - 22 \beta_{2} - 14 \beta_1 + 246) q^{69} + ( - 40 \beta_{2} + 72 \beta_1 - 32) q^{71} + ( - 141 \beta_{2} + 157 \beta_1 + 302) q^{72} + ( - 76 \beta_{2} - 124 \beta_1 - 78) q^{73} + (40 \beta_{2} - 74 \beta_1 + 154) q^{74} + (25 \beta_{2} - 25 \beta_1 - 25) q^{75} + ( - 80 \beta_{2} + 176 \beta_1 - 572) q^{76} + (125 \beta_{2} - 416 \beta_1 + 1221) q^{78} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + (5 \beta_{2} + 55 \beta_1 + 130) q^{80} + ( - 96 \beta_{2} + 72 \beta_1 + 793) q^{81} + ( - 6 \beta_{2} - 130 \beta_1 + 4) q^{82} + ( - 8 \beta_{2} + 160 \beta_1 + 556) q^{83} + (55 \beta_{2} + 65 \beta_1 - 105) q^{85} + (98 \beta_{2} - 164 \beta_1 + 222) q^{86} + ( - \beta_{2} - 167 \beta_1 + 525) q^{87} + (94 \beta_{2} - 68 \beta_1 - 38) q^{88} + (82 \beta_{2} + 98 \beta_1 - 108) q^{89} + (15 \beta_{2} - 65 \beta_1 + 680) q^{90} + ( - 12 \beta_{2} - 84 \beta_1 + 36) q^{92} + ( - 76 \beta_{2} + 8 \beta_1 + 484) q^{93} + (25 \beta_{2} + 304 \beta_1 + 361) q^{94} + (30 \beta_{2} - 50 \beta_1 + 270) q^{95} + ( - 48 \beta_{2} - 156 \beta_1 + 1068) q^{96} + (65 \beta_{2} + 87 \beta_1 - 55) q^{97} + (106 \beta_{2} + 198 \beta_1 - 1198) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - b1 - 1) * q^3 + (b2 - b1 + 4) * q^4 - 5 * q^5 + (-3*b2 + 4*b1 - 7) * q^6 + (-3*b2 + b1 - 4) * q^8 + (-3*b2 - 9*b1 + 28) * q^9 + (-5*b1 + 5) * q^10 + (b2 + 3*b1 - 25) * q^11 + (2*b2 - 14*b1 + 50) * q^12 + (-5*b2 + 13*b1 - 13) * q^13 + (-5*b2 + 5*b1 + 5) * q^15 + (-b2 - 11*b1 - 26) * q^16 + (-11*b2 - 13*b1 + 21) * q^17 + (-3*b2 + 13*b1 - 136) * q^18 + (-6*b2 + 10*b1 - 54) * q^19 + (-5*b2 + 5*b1 - 20) * q^20 + (b2 - 20*b1 + 61) * q^22 + (2*b2 - 14*b1 - 42) * q^23 + (6*b2 + 28*b1 - 142) * q^24 + 25 * q^25 + (23*b2 - 38*b1 + 141) * q^26 + (31*b2 - 7*b1 - 67) * q^27 + (17*b2 + 19*b1 + 105) * q^29 + (15*b2 - 20*b1 + 35) * q^30 + (4*b2 - 24*b1 - 108) * q^31 + (15*b2 - 39*b1 - 66) * q^32 + (-35*b2 + 27*b1 + 47) * q^33 + (9*b2 - 34*b1 - 197) * q^34 + (43*b2 - 79*b1 + 46) * q^36 + (-12*b2 + 16*b1 - 14) * q^37 + (22*b2 - 84*b1 + 146) * q^38 + (-19*b2 + 87*b1 - 321) * q^39 + (15*b2 - 5*b1 + 20) * q^40 + (-2*b2 - 10*b1 - 120) * q^41 + (-34*b2 + 30*b1 + 6) * q^43 + (-30*b2 + 42*b1 - 78) * q^44 + (15*b2 + 45*b1 - 140) * q^45 + (-18*b2 - 32*b1 - 106) * q^46 + (13*b2 + 51*b1 + 239) * q^47 + 68 * q^48 + (25*b1 - 25) * q^50 + (91*b2 + 17*b1 - 423) * q^51 + (-44*b2 + 152*b1 - 386) * q^52 + (-22*b2 - 130*b1 + 44) * q^53 + (-69*b2 + 88*b1 + 83) * q^54 + (-5*b2 - 15*b1 + 125) * q^55 + (-50*b2 + 126*b1 - 302) * q^57 + (-15*b2 + 190*b1 + 155) * q^58 + (-48*b2 + 176*b1 + 76) * q^59 + (-10*b2 + 70*b1 - 250) * q^60 + (26*b2 + 34*b1 - 416) * q^61 + (-32*b2 - 88*b1 - 144) * q^62 + (-61*b2 + 97*b1 - 110) * q^64 + (25*b2 - 65*b1 + 65) * q^65 + (97*b2 - 128*b1 + 145) * q^66 + (108*b2 - 12*b1 + 32) * q^67 + (36*b2 - 48*b1 - 318) * q^68 + (-22*b2 - 14*b1 + 246) * q^69 + (-40*b2 + 72*b1 - 32) * q^71 + (-141*b2 + 157*b1 + 302) * q^72 + (-76*b2 - 124*b1 - 78) * q^73 + (40*b2 - 74*b1 + 154) * q^74 + (25*b2 - 25*b1 - 25) * q^75 + (-80*b2 + 176*b1 - 572) * q^76 + (125*b2 - 416*b1 + 1221) * q^78 + (-89*b2 - 83*b1 - 315) * q^79 + (5*b2 + 55*b1 + 130) * q^80 + (-96*b2 + 72*b1 + 793) * q^81 + (-6*b2 - 130*b1 + 4) * q^82 + (-8*b2 + 160*b1 + 556) * q^83 + (55*b2 + 65*b1 - 105) * q^85 + (98*b2 - 164*b1 + 222) * q^86 + (-b2 - 167*b1 + 525) * q^87 + (94*b2 - 68*b1 - 38) * q^88 + (82*b2 + 98*b1 - 108) * q^89 + (15*b2 - 65*b1 + 680) * q^90 + (-12*b2 - 84*b1 + 36) * q^92 + (-76*b2 + 8*b1 + 484) * q^93 + (25*b2 + 304*b1 + 361) * q^94 + (30*b2 - 50*b1 + 270) * q^95 + (-48*b2 - 156*b1 + 1068) * q^96 + (65*b2 + 87*b1 - 55) * q^97 + (106*b2 + 198*b1 - 1198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 2 * q^3 + 13 * q^4 - 15 * q^5 - 24 * q^6 - 15 * q^8 + 81 * q^9 $$3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20} + 184 q^{22} - 124 q^{23} - 420 q^{24} + 75 q^{25} + 446 q^{26} - 170 q^{27} + 332 q^{29} + 120 q^{30} - 320 q^{31} - 183 q^{32} + 106 q^{33} - 582 q^{34} + 181 q^{36} - 54 q^{37} + 460 q^{38} - 982 q^{39} + 75 q^{40} - 362 q^{41} - 16 q^{43} - 264 q^{44} - 405 q^{45} - 336 q^{46} + 730 q^{47} + 204 q^{48} - 75 q^{50} - 1178 q^{51} - 1202 q^{52} + 110 q^{53} + 180 q^{54} + 370 q^{55} - 956 q^{57} + 450 q^{58} + 180 q^{59} - 760 q^{60} - 1222 q^{61} - 464 q^{62} - 391 q^{64} + 220 q^{65} + 532 q^{66} + 204 q^{67} - 918 q^{68} + 716 q^{69} - 136 q^{71} + 765 q^{72} - 310 q^{73} + 502 q^{74} - 50 q^{75} - 1796 q^{76} + 3788 q^{78} - 1034 q^{79} + 395 q^{80} + 2283 q^{81} + 6 q^{82} + 1660 q^{83} - 260 q^{85} + 764 q^{86} + 1574 q^{87} - 20 q^{88} - 242 q^{89} + 2055 q^{90} + 96 q^{92} + 1376 q^{93} + 1108 q^{94} + 840 q^{95} + 3156 q^{96} - 100 q^{97} - 3488 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 2 * q^3 + 13 * q^4 - 15 * q^5 - 24 * q^6 - 15 * q^8 + 81 * q^9 + 15 * q^10 - 74 * q^11 + 152 * q^12 - 44 * q^13 + 10 * q^15 - 79 * q^16 + 52 * q^17 - 411 * q^18 - 168 * q^19 - 65 * q^20 + 184 * q^22 - 124 * q^23 - 420 * q^24 + 75 * q^25 + 446 * q^26 - 170 * q^27 + 332 * q^29 + 120 * q^30 - 320 * q^31 - 183 * q^32 + 106 * q^33 - 582 * q^34 + 181 * q^36 - 54 * q^37 + 460 * q^38 - 982 * q^39 + 75 * q^40 - 362 * q^41 - 16 * q^43 - 264 * q^44 - 405 * q^45 - 336 * q^46 + 730 * q^47 + 204 * q^48 - 75 * q^50 - 1178 * q^51 - 1202 * q^52 + 110 * q^53 + 180 * q^54 + 370 * q^55 - 956 * q^57 + 450 * q^58 + 180 * q^59 - 760 * q^60 - 1222 * q^61 - 464 * q^62 - 391 * q^64 + 220 * q^65 + 532 * q^66 + 204 * q^67 - 918 * q^68 + 716 * q^69 - 136 * q^71 + 765 * q^72 - 310 * q^73 + 502 * q^74 - 50 * q^75 - 1796 * q^76 + 3788 * q^78 - 1034 * q^79 + 395 * q^80 + 2283 * q^81 + 6 * q^82 + 1660 * q^83 - 260 * q^85 + 764 * q^86 + 1574 * q^87 - 20 * q^88 - 242 * q^89 + 2055 * q^90 + 96 * q^92 + 1376 * q^93 + 1108 * q^94 + 840 * q^95 + 3156 * q^96 - 100 * q^97 - 3488 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 17x - 14$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 11$$ v^2 - v - 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11

## 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
 −3.62456 −0.861086 4.48565
−4.62456 8.38660 13.3866 −5.00000 −38.7844 0 −24.9107 43.3350 23.1228
1.2 −1.86109 −9.53636 −4.53636 −5.00000 17.7480 0 23.3312 63.9421 9.30543
1.3 3.48565 −0.850238 4.14976 −5.00000 −2.96363 0 −13.4206 −26.2771 −17.4283
 $$n$$: 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 245.4.a.l 3
3.b odd 2 1 2205.4.a.bm 3
5.b even 2 1 1225.4.a.y 3
7.b odd 2 1 35.4.a.c 3
7.c even 3 2 245.4.e.n 6
7.d odd 6 2 245.4.e.m 6
21.c even 2 1 315.4.a.p 3
28.d even 2 1 560.4.a.u 3
35.c odd 2 1 175.4.a.f 3
35.f even 4 2 175.4.b.e 6
56.e even 2 1 2240.4.a.bv 3
56.h odd 2 1 2240.4.a.bt 3
105.g even 2 1 1575.4.a.ba 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 7.b odd 2 1
175.4.a.f 3 35.c odd 2 1
175.4.b.e 6 35.f even 4 2
245.4.a.l 3 1.a even 1 1 trivial
245.4.e.m 6 7.d odd 6 2
245.4.e.n 6 7.c even 3 2
315.4.a.p 3 21.c even 2 1
560.4.a.u 3 28.d even 2 1
1225.4.a.y 3 5.b even 2 1
1575.4.a.ba 3 105.g even 2 1
2205.4.a.bm 3 3.b odd 2 1
2240.4.a.bt 3 56.h odd 2 1
2240.4.a.bv 3 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - 14T_{2} - 30$$ T2^3 + 3*T2^2 - 14*T2 - 30 $$T_{3}^{3} + 2T_{3}^{2} - 79T_{3} - 68$$ T3^3 + 2*T3^2 - 79*T3 - 68

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3 T^{2} - 14 T - 30$$
$3$ $$T^{3} + 2 T^{2} - 79 T - 68$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 74 T^{2} + 1577 T + 7692$$
$13$ $$T^{3} + 44 T^{2} - 3491 T + 44870$$
$17$ $$T^{3} - 52 T^{2} - 11747 T + 56706$$
$19$ $$T^{3} + 168 T^{2} + 5620 T + 28720$$
$23$ $$T^{3} + 124 T^{2} + 1732 T - 94368$$
$29$ $$T^{3} - 332 T^{2} + 7405 T + 2565450$$
$31$ $$T^{3} + 320 T^{2} + 23968 T - 50176$$
$37$ $$T^{3} + 54 T^{2} - 12116 T + 25736$$
$41$ $$T^{3} + 362 T^{2} + 41536 T + 1536192$$
$43$ $$T^{3} + 16 T^{2} - 89516 T - 1524560$$
$47$ $$T^{3} - 730 T^{2} + 116057 T - 4968912$$
$53$ $$T^{3} - 110 T^{2} + \cdots + 90318336$$
$59$ $$T^{3} - 180 T^{2} + \cdots + 202459200$$
$61$ $$T^{3} + 1222 T^{2} + \cdots + 38393792$$
$67$ $$T^{3} - 204 T^{2} + \cdots + 324944128$$
$71$ $$T^{3} + 136 T^{2} + \cdots + 15575040$$
$73$ $$T^{3} + 310 T^{2} + \cdots - 48718616$$
$79$ $$T^{3} + 1034 T^{2} + \cdots - 343615600$$
$83$ $$T^{3} - 1660 T^{2} + \cdots + 42727104$$
$89$ $$T^{3} + 242 T^{2} - 687680 T + 6359520$$
$97$ $$T^{3} + 100 T^{2} - 471963 T + 1978018$$