# Properties

 Label 425.4.a.g Level $425$ Weight $4$ Character orbit 425.a Self dual yes Analytic conductor $25.076$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 425.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.0758117524$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{2} + \beta_1 - 2) q^{3} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + (2 \beta_{2} + 6 \beta_1 - 24) q^{6} + (4 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + ( - 2 \beta_{2} - 8 \beta_1 + 19) q^{9}+O(q^{10})$$ q + (b2 - b1) * q^2 + (-2*b2 + b1 - 2) * q^3 + (-b2 - 3*b1 + 8) * q^4 + (2*b2 + 6*b1 - 24) * q^6 + (4*b2 + b1 - 6) * q^7 + (9*b2 - 5*b1 + 16) * q^8 + (-2*b2 - 8*b1 + 19) * q^9 $$q + (\beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{2} + \beta_1 - 2) q^{3} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + (2 \beta_{2} + 6 \beta_1 - 24) q^{6} + (4 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + ( - 2 \beta_{2} - 8 \beta_1 + 19) q^{9} + ( - 2 \beta_{2} + 11 \beta_1 - 10) q^{11} + ( - 26 \beta_{2} + 26 \beta_1 - 16) q^{12} + ( - 6 \beta_{2} - 8 \beta_1 - 12) q^{13} + ( - 20 \beta_{2} + 4 \beta_1 + 24) q^{14} + (7 \beta_{2} - 11 \beta_1 + 48) q^{16} + 17 q^{17} + (41 \beta_{2} - 33 \beta_1 + 48) q^{18} + ( - 8 \beta_{2} + 22 \beta_1 + 24) q^{19} + (30 \beta_{2} - 12 \beta_1 - 54) q^{21} + ( - 26 \beta_{2} + 34 \beta_1 - 104) q^{22} + (4 \beta_{2} + 39 \beta_1 - 46) q^{23} + ( - 6 \beta_{2} + 46 \beta_1 - 224) q^{24} + (22 \beta_{2} + 2 \beta_1 + 16) q^{26} + ( - 8 \beta_{2} + 40 \beta_1 + 4) q^{27} + (44 \beta_{2} - 4 \beta_1 - 144) q^{28} + (30 \beta_{2} - 16 \beta_1 - 142) q^{29} + (16 \beta_{2} + 39 \beta_1 + 82) q^{31} + ( - 23 \beta_{2} - 37 \beta_1 + 16) q^{32} + (34 \beta_{2} - 76 \beta_1 + 122) q^{33} + (17 \beta_{2} - 17 \beta_1) q^{34} + (7 \beta_{2} - 91 \beta_1 + 440) q^{36} + (50 \beta_{2} + 28 \beta_1 - 102) q^{37} + (4 \beta_{2} + 28 \beta_1 - 240) q^{38} + ( - 16 \beta_{2} + 36 \beta_1 + 84) q^{39} + ( - 60 \beta_{2} - 52 \beta_1 - 118) q^{41} + ( - 120 \beta_{2} + 336) q^{42} + ( - 56 \beta_{2} - 2 \beta_1 - 204) q^{43} + ( - 78 \beta_{2} + 110 \beta_1 - 400) q^{44} + ( - 136 \beta_{2} + 120 \beta_1 - 280) q^{46} + ( - 44 \beta_{2} + 48 \beta_1 - 228) q^{47} + ( - 90 \beta_{2} + 114 \beta_1 - 288) q^{48} + ( - 94 \beta_{2} + 20 \beta_1 - 121) q^{49} + ( - 34 \beta_{2} + 17 \beta_1 - 34) q^{51} + ( - 6 \beta_{2} + 30 \beta_1 + 256) q^{52} + (8 \beta_{2} - 116 \beta_1 - 98) q^{53} + ( - 52 \beta_{2} + 84 \beta_1 - 384) q^{54} + ( - 108 \beta_{2} + 60 \beta_1 + 192) q^{56} + ( - 36 \beta_{2} - 108 \beta_1 + 228) q^{57} + ( - 200 \beta_{2} + 80 \beta_1 + 368) q^{58} + ( - 130 \beta_1 + 212) q^{59} + (78 \beta_{2} - 64 \beta_1 - 2) q^{61} + ( - 44 \beta_{2} - 20 \beta_1 - 184) q^{62} + (96 \beta_{2} - 9 \beta_1 - 342) q^{63} + (103 \beta_{2} + 21 \beta_1 - 272) q^{64} + (172 \beta_{2} - 308 \beta_1 + 880) q^{66} + (132 \beta_{2} - 24 \beta_1 - 292) q^{67} + ( - 17 \beta_{2} - 51 \beta_1 + 136) q^{68} + (186 \beta_{2} - 280 \beta_1 + 254) q^{69} + (72 \beta_{2} + 185 \beta_1 - 110) q^{71} + (273 \beta_{2} - 365 \beta_1 + 400) q^{72} + (16 \beta_{2} - 16 \beta_1 - 274) q^{73} + ( - 308 \beta_{2} + 108 \beta_1 + 176) q^{74} + ( - 244 \beta_{2} + 116 \beta_1 - 384) q^{76} + (18 \beta_{2} + 174) q^{77} + (60 \beta_{2} + 4 \beta_1 - 416) q^{78} + (180 \beta_{2} - 267 \beta_1 - 138) q^{79} + (94 \beta_{2} - 20 \beta_1 - 137) q^{81} + (166 \beta_{2} + 74 \beta_1 - 64) q^{82} + ( - 128 \beta_{2} - 82 \beta_1 + 756) q^{83} + (456 \beta_{2} - 120 \beta_1 - 528) q^{84} + ( - 32 \beta_{2} + 256 \beta_1 - 432) q^{86} + (372 \beta_{2} - 46 \beta_1 - 352) q^{87} + ( - 178 \beta_{2} + 426 \beta_1 - 672) q^{88} + (110 \beta_{2} - 276 \beta_1 - 20) q^{89} + (44 \beta_{2} - 64 \beta_1 - 324) q^{91} + ( - 144 \beta_{2} + 344 \beta_1 - 1680) q^{92} + ( - 22 \beta_{2} - 152 \beta_1 - 218) q^{93} + ( - 192 \beta_{2} + 368 \beta_1 - 736) q^{94} + ( - 198 \beta_{2} + 238 \beta_1 + 160) q^{96} + ( - 120 \beta_{2} - 140 \beta_1 + 50) q^{97} + (121 \beta_{2} + 255 \beta_1 - 912) q^{98} + ( - 206 \beta_{2} + 281 \beta_1 - 1042) q^{99}+O(q^{100})$$ q + (b2 - b1) * q^2 + (-2*b2 + b1 - 2) * q^3 + (-b2 - 3*b1 + 8) * q^4 + (2*b2 + 6*b1 - 24) * q^6 + (4*b2 + b1 - 6) * q^7 + (9*b2 - 5*b1 + 16) * q^8 + (-2*b2 - 8*b1 + 19) * q^9 + (-2*b2 + 11*b1 - 10) * q^11 + (-26*b2 + 26*b1 - 16) * q^12 + (-6*b2 - 8*b1 - 12) * q^13 + (-20*b2 + 4*b1 + 24) * q^14 + (7*b2 - 11*b1 + 48) * q^16 + 17 * q^17 + (41*b2 - 33*b1 + 48) * q^18 + (-8*b2 + 22*b1 + 24) * q^19 + (30*b2 - 12*b1 - 54) * q^21 + (-26*b2 + 34*b1 - 104) * q^22 + (4*b2 + 39*b1 - 46) * q^23 + (-6*b2 + 46*b1 - 224) * q^24 + (22*b2 + 2*b1 + 16) * q^26 + (-8*b2 + 40*b1 + 4) * q^27 + (44*b2 - 4*b1 - 144) * q^28 + (30*b2 - 16*b1 - 142) * q^29 + (16*b2 + 39*b1 + 82) * q^31 + (-23*b2 - 37*b1 + 16) * q^32 + (34*b2 - 76*b1 + 122) * q^33 + (17*b2 - 17*b1) * q^34 + (7*b2 - 91*b1 + 440) * q^36 + (50*b2 + 28*b1 - 102) * q^37 + (4*b2 + 28*b1 - 240) * q^38 + (-16*b2 + 36*b1 + 84) * q^39 + (-60*b2 - 52*b1 - 118) * q^41 + (-120*b2 + 336) * q^42 + (-56*b2 - 2*b1 - 204) * q^43 + (-78*b2 + 110*b1 - 400) * q^44 + (-136*b2 + 120*b1 - 280) * q^46 + (-44*b2 + 48*b1 - 228) * q^47 + (-90*b2 + 114*b1 - 288) * q^48 + (-94*b2 + 20*b1 - 121) * q^49 + (-34*b2 + 17*b1 - 34) * q^51 + (-6*b2 + 30*b1 + 256) * q^52 + (8*b2 - 116*b1 - 98) * q^53 + (-52*b2 + 84*b1 - 384) * q^54 + (-108*b2 + 60*b1 + 192) * q^56 + (-36*b2 - 108*b1 + 228) * q^57 + (-200*b2 + 80*b1 + 368) * q^58 + (-130*b1 + 212) * q^59 + (78*b2 - 64*b1 - 2) * q^61 + (-44*b2 - 20*b1 - 184) * q^62 + (96*b2 - 9*b1 - 342) * q^63 + (103*b2 + 21*b1 - 272) * q^64 + (172*b2 - 308*b1 + 880) * q^66 + (132*b2 - 24*b1 - 292) * q^67 + (-17*b2 - 51*b1 + 136) * q^68 + (186*b2 - 280*b1 + 254) * q^69 + (72*b2 + 185*b1 - 110) * q^71 + (273*b2 - 365*b1 + 400) * q^72 + (16*b2 - 16*b1 - 274) * q^73 + (-308*b2 + 108*b1 + 176) * q^74 + (-244*b2 + 116*b1 - 384) * q^76 + (18*b2 + 174) * q^77 + (60*b2 + 4*b1 - 416) * q^78 + (180*b2 - 267*b1 - 138) * q^79 + (94*b2 - 20*b1 - 137) * q^81 + (166*b2 + 74*b1 - 64) * q^82 + (-128*b2 - 82*b1 + 756) * q^83 + (456*b2 - 120*b1 - 528) * q^84 + (-32*b2 + 256*b1 - 432) * q^86 + (372*b2 - 46*b1 - 352) * q^87 + (-178*b2 + 426*b1 - 672) * q^88 + (110*b2 - 276*b1 - 20) * q^89 + (44*b2 - 64*b1 - 324) * q^91 + (-144*b2 + 344*b1 - 1680) * q^92 + (-22*b2 - 152*b1 - 218) * q^93 + (-192*b2 + 368*b1 - 736) * q^94 + (-198*b2 + 238*b1 + 160) * q^96 + (-120*b2 - 140*b1 + 50) * q^97 + (121*b2 + 255*b1 - 912) * q^98 + (-206*b2 + 281*b1 - 1042) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9}+O(q^{10})$$ 3 * q - q^2 - 4 * q^3 + 25 * q^4 - 74 * q^6 - 22 * q^7 + 39 * q^8 + 59 * q^9 $$3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9} - 28 q^{11} - 22 q^{12} - 30 q^{13} + 92 q^{14} + 137 q^{16} + 51 q^{17} + 103 q^{18} + 80 q^{19} - 192 q^{21} - 286 q^{22} - 142 q^{23} - 666 q^{24} + 26 q^{26} + 20 q^{27} - 476 q^{28} - 456 q^{29} + 230 q^{31} + 71 q^{32} + 332 q^{33} - 17 q^{34} + 1313 q^{36} - 356 q^{37} - 724 q^{38} + 268 q^{39} - 294 q^{41} + 1128 q^{42} - 556 q^{43} - 1122 q^{44} - 704 q^{46} - 640 q^{47} - 774 q^{48} - 269 q^{49} - 68 q^{51} + 774 q^{52} - 302 q^{53} - 1100 q^{54} + 684 q^{56} + 720 q^{57} + 1304 q^{58} + 636 q^{59} - 84 q^{61} - 508 q^{62} - 1122 q^{63} - 919 q^{64} + 2468 q^{66} - 1008 q^{67} + 425 q^{68} + 576 q^{69} - 402 q^{71} + 927 q^{72} - 838 q^{73} + 836 q^{74} - 908 q^{76} + 504 q^{77} - 1308 q^{78} - 594 q^{79} - 505 q^{81} - 358 q^{82} + 2396 q^{83} - 2040 q^{84} - 1264 q^{86} - 1428 q^{87} - 1838 q^{88} - 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 632 q^{93} - 2016 q^{94} + 678 q^{96} + 270 q^{97} - 2857 q^{98} - 2920 q^{99}+O(q^{100})$$ 3 * q - q^2 - 4 * q^3 + 25 * q^4 - 74 * q^6 - 22 * q^7 + 39 * q^8 + 59 * q^9 - 28 * q^11 - 22 * q^12 - 30 * q^13 + 92 * q^14 + 137 * q^16 + 51 * q^17 + 103 * q^18 + 80 * q^19 - 192 * q^21 - 286 * q^22 - 142 * q^23 - 666 * q^24 + 26 * q^26 + 20 * q^27 - 476 * q^28 - 456 * q^29 + 230 * q^31 + 71 * q^32 + 332 * q^33 - 17 * q^34 + 1313 * q^36 - 356 * q^37 - 724 * q^38 + 268 * q^39 - 294 * q^41 + 1128 * q^42 - 556 * q^43 - 1122 * q^44 - 704 * q^46 - 640 * q^47 - 774 * q^48 - 269 * q^49 - 68 * q^51 + 774 * q^52 - 302 * q^53 - 1100 * q^54 + 684 * q^56 + 720 * q^57 + 1304 * q^58 + 636 * q^59 - 84 * q^61 - 508 * q^62 - 1122 * q^63 - 919 * q^64 + 2468 * q^66 - 1008 * q^67 + 425 * q^68 + 576 * q^69 - 402 * q^71 + 927 * q^72 - 838 * q^73 + 836 * q^74 - 908 * q^76 + 504 * q^77 - 1308 * q^78 - 594 * q^79 - 505 * q^81 - 358 * q^82 + 2396 * q^83 - 2040 * q^84 - 1264 * q^86 - 1428 * q^87 - 1838 * q^88 - 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 632 * q^93 - 2016 * q^94 + 678 * q^96 + 270 * q^97 - 2857 * q^98 - 2920 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 14x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 10 ) / 2$$ (v^2 - 10) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + 10$$ 2*b2 + 10

## 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
 −0.287410 3.87707 −3.58966
−4.67129 7.62999 13.8209 0 −35.6419 −26.1222 −27.1912 31.2167 0
1.2 −1.36122 −3.15463 −6.14708 0 4.29415 7.94049 19.2573 −17.0483 0
1.3 5.03251 −8.47535 17.3261 0 −42.6523 −3.81828 46.9339 44.8316 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
$$5$$ $$1$$
$$17$$ $$-1$$

## 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 425.4.a.g 3
5.b even 2 1 17.4.a.b 3
5.c odd 4 2 425.4.b.f 6
15.d odd 2 1 153.4.a.g 3
20.d odd 2 1 272.4.a.h 3
35.c odd 2 1 833.4.a.d 3
40.e odd 2 1 1088.4.a.x 3
40.f even 2 1 1088.4.a.v 3
55.d odd 2 1 2057.4.a.e 3
60.h even 2 1 2448.4.a.bi 3
85.c even 2 1 289.4.a.b 3
85.j even 4 2 289.4.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 5.b even 2 1
153.4.a.g 3 15.d odd 2 1
272.4.a.h 3 20.d odd 2 1
289.4.a.b 3 85.c even 2 1
289.4.b.b 6 85.j even 4 2
425.4.a.g 3 1.a even 1 1 trivial
425.4.b.f 6 5.c odd 4 2
833.4.a.d 3 35.c odd 2 1
1088.4.a.v 3 40.f even 2 1
1088.4.a.x 3 40.e odd 2 1
2057.4.a.e 3 55.d odd 2 1
2448.4.a.bi 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(425))$$:

 $$T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32$$ T2^3 + T2^2 - 24*T2 - 32 $$T_{3}^{3} + 4T_{3}^{2} - 62T_{3} - 204$$ T3^3 + 4*T3^2 - 62*T3 - 204

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 24 T - 32$$
$3$ $$T^{3} + 4 T^{2} - 62 T - 204$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 22 T^{2} - 138 T - 792$$
$11$ $$T^{3} + 28 T^{2} - 1366 T - 4692$$
$13$ $$T^{3} + 30 T^{2} - 1472 T + 9392$$
$17$ $$(T - 17)^{3}$$
$19$ $$T^{3} - 80 T^{2} - 4632 T + 340128$$
$23$ $$T^{3} + 142 T^{2} - 15770 T - 1600544$$
$29$ $$T^{3} + 456 T^{2} + 53908 T + 1518624$$
$31$ $$T^{3} - 230 T^{2} - 11586 T - 81608$$
$37$ $$T^{3} + 356 T^{2} - 17964 T - 6176752$$
$41$ $$T^{3} + 294 T^{2} - 86564 T - 1638744$$
$43$ $$T^{3} + 556 T^{2} + 51096 T - 7270272$$
$47$ $$T^{3} + 640 T^{2} + 85328 T + 1671168$$
$53$ $$T^{3} + 302 T^{2} + \cdots - 18162072$$
$59$ $$T^{3} - 636 T^{2} + \cdots + 49419072$$
$61$ $$T^{3} + 84 T^{2} - 124412 T - 6792784$$
$67$ $$T^{3} + 1008 T^{2} + 65040 T + 765952$$
$71$ $$T^{3} + 402 T^{2} + \cdots - 274866016$$
$73$ $$T^{3} + 838 T^{2} + \cdots + 19957512$$
$79$ $$T^{3} + 594 T^{2} + \cdots - 742135824$$
$83$ $$T^{3} - 2396 T^{2} + \cdots - 142080704$$
$89$ $$T^{3} + 170 T^{2} + \cdots - 446571376$$
$97$ $$T^{3} - 270 T^{2} + \cdots + 206623000$$