# Properties

 Label 1450.4.a.h Level $1450$ Weight $4$ Character orbit 1450.a Self dual yes Analytic conductor $85.553$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$85.5527695083$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42x - 54$$ x^3 - x^2 - 42*x - 54 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 + 2) q^{6} + (4 \beta_{2} - 8) q^{7} - 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - 2 * q^2 + (b1 - 1) * q^3 + 4 * q^4 + (-2*b1 + 2) * q^6 + (4*b2 - 8) * q^7 - 8 * q^8 + (3*b2 + 2*b1 + 1) * q^9 $$q - 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 + 2) q^{6} + (4 \beta_{2} - 8) q^{7} - 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + ( - 2 \beta_{2} + \beta_1 + 3) q^{11} + (4 \beta_1 - 4) q^{12} + ( - 11 \beta_{2} - 8 \beta_1 + 4) q^{13} + ( - 8 \beta_{2} + 16) q^{14} + 16 q^{16} + (16 \beta_{2} + 12 \beta_1 + 18) q^{17} + ( - 6 \beta_{2} - 4 \beta_1 - 2) q^{18} + (10 \beta_{2} - 8 \beta_1 - 52) q^{19} + ( - 16 \beta_{2} - 4 \beta_1 - 28) q^{21} + (4 \beta_{2} - 2 \beta_1 - 6) q^{22} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + ( - 8 \beta_1 + 8) q^{24} + (22 \beta_{2} + 16 \beta_1 - 8) q^{26} + ( - 6 \beta_{2} - 17 \beta_1 + 53) q^{27} + (16 \beta_{2} - 32) q^{28} + 29 q^{29} + ( - 36 \beta_{2} - 11 \beta_1 - 25) q^{31} - 32 q^{32} + (11 \beta_{2} + 4 \beta_1 + 42) q^{33} + ( - 32 \beta_{2} - 24 \beta_1 - 36) q^{34} + (12 \beta_{2} + 8 \beta_1 + 4) q^{36} + ( - 38 \beta_{2} + 12 \beta_1 + 10) q^{37} + ( - 20 \beta_{2} + 16 \beta_1 + 104) q^{38} + (20 \beta_{2} - 31 \beta_1 - 121) q^{39} + ( - 6 \beta_{2} + 4 \beta_1 + 186) q^{41} + (32 \beta_{2} + 8 \beta_1 + 56) q^{42} + ( - 2 \beta_{2} + 15 \beta_1 - 11) q^{43} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{44} + (24 \beta_{2} - 12 \beta_1 - 132) q^{46} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + (16 \beta_1 - 16) q^{48} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + ( - 28 \beta_{2} + 70 \beta_1 + 162) q^{51} + ( - 44 \beta_{2} - 32 \beta_1 + 16) q^{52} + (27 \beta_{2} + 116 \beta_1 - 276) q^{53} + (12 \beta_{2} + 34 \beta_1 - 106) q^{54} + ( - 32 \beta_{2} + 64) q^{56} + ( - 64 \beta_{2} - 66 \beta_1 - 254) q^{57} - 58 q^{58} + (4 \beta_{2} - 86 \beta_1 + 90) q^{59} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + (72 \beta_{2} + 22 \beta_1 + 50) q^{62} + ( - 56 \beta_{2} - 56 \beta_1 + 280) q^{63} + 64 q^{64} + ( - 22 \beta_{2} - 8 \beta_1 - 84) q^{66} + (72 \beta_{2} - 36 \beta_1 + 88) q^{67} + (64 \beta_{2} + 48 \beta_1 + 72) q^{68} + (66 \beta_{2} + 72 \beta_1 + 204) q^{69} + ( - 4 \beta_{2} + 2 \beta_1 - 18) q^{71} + ( - 24 \beta_{2} - 16 \beta_1 - 8) q^{72} + ( - 30 \beta_{2} - 40 \beta_1 + 178) q^{73} + (76 \beta_{2} - 24 \beta_1 - 20) q^{74} + (40 \beta_{2} - 32 \beta_1 - 208) q^{76} + (24 \beta_{2} + 28 \beta_1 - 300) q^{77} + ( - 40 \beta_{2} + 62 \beta_1 + 242) q^{78} + ( - 38 \beta_{2} - 37 \beta_1 - 691) q^{79} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + (12 \beta_{2} - 8 \beta_1 - 372) q^{82} + ( - 12 \beta_{2} - 162 \beta_1 + 150) q^{83} + ( - 64 \beta_{2} - 16 \beta_1 - 112) q^{84} + (4 \beta_{2} - 30 \beta_1 + 22) q^{86} + (29 \beta_1 - 29) q^{87} + (16 \beta_{2} - 8 \beta_1 - 24) q^{88} + (154 \beta_{2} + 68 \beta_1 + 282) q^{89} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + ( - 48 \beta_{2} + 24 \beta_1 + 264) q^{92} + (111 \beta_{2} - 94 \beta_1 + 52) q^{93} + (4 \beta_{2} + 66 \beta_1 + 414) q^{94} + ( - 32 \beta_1 + 32) q^{96} + (34 \beta_{2} - 176 \beta_1 - 14) q^{97} + (160 \beta_{2} + 128 \beta_1 - 402) q^{98} + (22 \beta_{2} + 38 \beta_1 - 114) q^{99}+O(q^{100})$$ q - 2 * q^2 + (b1 - 1) * q^3 + 4 * q^4 + (-2*b1 + 2) * q^6 + (4*b2 - 8) * q^7 - 8 * q^8 + (3*b2 + 2*b1 + 1) * q^9 + (-2*b2 + b1 + 3) * q^11 + (4*b1 - 4) * q^12 + (-11*b2 - 8*b1 + 4) * q^13 + (-8*b2 + 16) * q^14 + 16 * q^16 + (16*b2 + 12*b1 + 18) * q^17 + (-6*b2 - 4*b1 - 2) * q^18 + (10*b2 - 8*b1 - 52) * q^19 + (-16*b2 - 4*b1 - 28) * q^21 + (4*b2 - 2*b1 - 6) * q^22 + (-12*b2 + 6*b1 + 66) * q^23 + (-8*b1 + 8) * q^24 + (22*b2 + 16*b1 - 8) * q^26 + (-6*b2 - 17*b1 + 53) * q^27 + (16*b2 - 32) * q^28 + 29 * q^29 + (-36*b2 - 11*b1 - 25) * q^31 - 32 * q^32 + (11*b2 + 4*b1 + 42) * q^33 + (-32*b2 - 24*b1 - 36) * q^34 + (12*b2 + 8*b1 + 4) * q^36 + (-38*b2 + 12*b1 + 10) * q^37 + (-20*b2 + 16*b1 + 104) * q^38 + (20*b2 - 31*b1 - 121) * q^39 + (-6*b2 + 4*b1 + 186) * q^41 + (32*b2 + 8*b1 + 56) * q^42 + (-2*b2 + 15*b1 - 11) * q^43 + (-8*b2 + 4*b1 + 12) * q^44 + (24*b2 - 12*b1 - 132) * q^46 + (-2*b2 - 33*b1 - 207) * q^47 + (16*b1 - 16) * q^48 + (-80*b2 - 64*b1 + 201) * q^49 + (-28*b2 + 70*b1 + 162) * q^51 + (-44*b2 - 32*b1 + 16) * q^52 + (27*b2 + 116*b1 - 276) * q^53 + (12*b2 + 34*b1 - 106) * q^54 + (-32*b2 + 64) * q^56 + (-64*b2 - 66*b1 - 254) * q^57 - 58 * q^58 + (4*b2 - 86*b1 + 90) * q^59 + (40*b2 - 80*b1 + 134) * q^61 + (72*b2 + 22*b1 + 50) * q^62 + (-56*b2 - 56*b1 + 280) * q^63 + 64 * q^64 + (-22*b2 - 8*b1 - 84) * q^66 + (72*b2 - 36*b1 + 88) * q^67 + (64*b2 + 48*b1 + 72) * q^68 + (66*b2 + 72*b1 + 204) * q^69 + (-4*b2 + 2*b1 - 18) * q^71 + (-24*b2 - 16*b1 - 8) * q^72 + (-30*b2 - 40*b1 + 178) * q^73 + (76*b2 - 24*b1 - 20) * q^74 + (40*b2 - 32*b1 - 208) * q^76 + (24*b2 + 28*b1 - 300) * q^77 + (-40*b2 + 62*b1 + 242) * q^78 + (-38*b2 - 37*b1 - 691) * q^79 + (-108*b2 - 58*b1 - 485) * q^81 + (12*b2 - 8*b1 - 372) * q^82 + (-12*b2 - 162*b1 + 150) * q^83 + (-64*b2 - 16*b1 - 112) * q^84 + (4*b2 - 30*b1 + 22) * q^86 + (29*b1 - 29) * q^87 + (16*b2 - 8*b1 - 24) * q^88 + (154*b2 + 68*b1 + 282) * q^89 + (244*b2 + 208*b1 - 1064) * q^91 + (-48*b2 + 24*b1 + 264) * q^92 + (111*b2 - 94*b1 + 52) * q^93 + (4*b2 + 66*b1 + 414) * q^94 + (-32*b1 + 32) * q^96 + (34*b2 - 176*b1 - 14) * q^97 + (160*b2 + 128*b1 - 402) * q^98 + (22*b2 + 38*b1 - 114) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 4 q^{6} - 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10})$$ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 4 * q^6 - 24 * q^7 - 24 * q^8 + 5 * q^9 $$3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 4 q^{6} - 24 q^{7} - 24 q^{8} + 5 q^{9} + 10 q^{11} - 8 q^{12} + 4 q^{13} + 48 q^{14} + 48 q^{16} + 66 q^{17} - 10 q^{18} - 164 q^{19} - 88 q^{21} - 20 q^{22} + 204 q^{23} + 16 q^{24} - 8 q^{26} + 142 q^{27} - 96 q^{28} + 87 q^{29} - 86 q^{31} - 96 q^{32} + 130 q^{33} - 132 q^{34} + 20 q^{36} + 42 q^{37} + 328 q^{38} - 394 q^{39} + 562 q^{41} + 176 q^{42} - 18 q^{43} + 40 q^{44} - 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} + 556 q^{51} + 16 q^{52} - 712 q^{53} - 284 q^{54} + 192 q^{56} - 828 q^{57} - 174 q^{58} + 184 q^{59} + 322 q^{61} + 172 q^{62} + 784 q^{63} + 192 q^{64} - 260 q^{66} + 228 q^{67} + 264 q^{68} + 684 q^{69} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} - 656 q^{76} - 872 q^{77} + 788 q^{78} - 2110 q^{79} - 1513 q^{81} - 1124 q^{82} + 288 q^{83} - 352 q^{84} + 36 q^{86} - 58 q^{87} - 80 q^{88} + 914 q^{89} - 2984 q^{91} + 816 q^{92} + 62 q^{93} + 1308 q^{94} + 64 q^{96} - 218 q^{97} - 1078 q^{98} - 304 q^{99}+O(q^{100})$$ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 4 * q^6 - 24 * q^7 - 24 * q^8 + 5 * q^9 + 10 * q^11 - 8 * q^12 + 4 * q^13 + 48 * q^14 + 48 * q^16 + 66 * q^17 - 10 * q^18 - 164 * q^19 - 88 * q^21 - 20 * q^22 + 204 * q^23 + 16 * q^24 - 8 * q^26 + 142 * q^27 - 96 * q^28 + 87 * q^29 - 86 * q^31 - 96 * q^32 + 130 * q^33 - 132 * q^34 + 20 * q^36 + 42 * q^37 + 328 * q^38 - 394 * q^39 + 562 * q^41 + 176 * q^42 - 18 * q^43 + 40 * q^44 - 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 + 556 * q^51 + 16 * q^52 - 712 * q^53 - 284 * q^54 + 192 * q^56 - 828 * q^57 - 174 * q^58 + 184 * q^59 + 322 * q^61 + 172 * q^62 + 784 * q^63 + 192 * q^64 - 260 * q^66 + 228 * q^67 + 264 * q^68 + 684 * q^69 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 - 656 * q^76 - 872 * q^77 + 788 * q^78 - 2110 * q^79 - 1513 * q^81 - 1124 * q^82 + 288 * q^83 - 352 * q^84 + 36 * q^86 - 58 * q^87 - 80 * q^88 + 914 * q^89 - 2984 * q^91 + 816 * q^92 + 62 * q^93 + 1308 * q^94 + 64 * q^96 - 218 * q^97 - 1078 * q^98 - 304 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 42x - 54$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 4\nu - 27 ) / 3$$ (v^2 - 4*v - 27) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 4\beta _1 + 27$$ 3*b2 + 4*b1 + 27

## 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
 −5.13291 −1.39712 7.53003
−2.00000 −6.13291 4.00000 0 12.2658 18.5045 −8.00000 10.6126 0
1.2 −2.00000 −2.39712 4.00000 0 4.79424 −33.9461 −8.00000 −21.2538 0
1.3 −2.00000 6.53003 4.00000 0 −13.0601 −8.55839 −8.00000 15.6413 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
$$2$$ $$1$$
$$5$$ $$1$$
$$29$$ $$-1$$

## 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 1450.4.a.h 3
5.b even 2 1 58.4.a.d 3
15.d odd 2 1 522.4.a.k 3
20.d odd 2 1 464.4.a.i 3
40.e odd 2 1 1856.4.a.s 3
40.f even 2 1 1856.4.a.r 3
145.d even 2 1 1682.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 5.b even 2 1
464.4.a.i 3 20.d odd 2 1
522.4.a.k 3 15.d odd 2 1
1450.4.a.h 3 1.a even 1 1 trivial
1682.4.a.d 3 145.d even 2 1
1856.4.a.r 3 40.f even 2 1
1856.4.a.s 3 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1450))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 41T_{3} - 96$$ T3^3 + 2*T3^2 - 41*T3 - 96 $$T_{7}^{3} + 24T_{7}^{2} - 496T_{7} - 5376$$ T7^3 + 24*T7^2 - 496*T7 - 5376

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{3}$$
$3$ $$T^{3} + 2 T^{2} - 41 T - 96$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 24 T^{2} - 496 T - 5376$$
$11$ $$T^{3} - 10 T^{2} - 233 T + 2424$$
$13$ $$T^{3} - 4 T^{2} - 5619 T - 131706$$
$17$ $$T^{3} - 66 T^{2} - 10660 T + 679368$$
$19$ $$T^{3} + 164 T^{2} - 124 T - 664448$$
$23$ $$T^{3} - 204 T^{2} + 4284 T + 677376$$
$29$ $$(T - 29)^{3}$$
$31$ $$T^{3} + 86 T^{2} - 48089 T - 4766172$$
$37$ $$T^{3} - 42 T^{2} - 79456 T + 7684896$$
$41$ $$T^{3} - 562 T^{2} + 102432 T - 5982048$$
$43$ $$T^{3} + 18 T^{2} - 10369 T - 196488$$
$47$ $$T^{3} + 654 T^{2} + 98015 T + 3425124$$
$53$ $$T^{3} + 712 T^{2} + \cdots - 252120546$$
$59$ $$T^{3} - 184 T^{2} + \cdots + 57362928$$
$61$ $$T^{3} - 322 T^{2} - 388372 T - 5254424$$
$67$ $$T^{3} - 228 T^{2} + \cdots - 47608192$$
$71$ $$T^{3} + 52 T^{2} - 164 T - 672$$
$73$ $$T^{3} - 494 T^{2} + 6112 T + 9410208$$
$79$ $$T^{3} + 2110 T^{2} + \cdots + 285187172$$
$83$ $$T^{3} - 288 T^{2} + \cdots + 437606064$$
$89$ $$T^{3} - 914 T^{2} + \cdots + 598011552$$
$97$ $$T^{3} + 218 T^{2} + \cdots + 17006112$$