# Properties

 Label 722.4.a.i Level $722$ Weight $4$ Character orbit 722.a Self dual yes Analytic conductor $42.599$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,4,Mod(1,722)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 722.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$42.5993790241$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 44$$ x^2 - x - 44 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - \beta q^{3} + 4 q^{4} + ( - 2 \beta + 6) q^{5} - 2 \beta q^{6} + (\beta + 28) q^{7} + 8 q^{8} + (\beta + 17) q^{9} +O(q^{10})$$ q + 2 * q^2 - b * q^3 + 4 * q^4 + (-2*b + 6) * q^5 - 2*b * q^6 + (b + 28) * q^7 + 8 * q^8 + (b + 17) * q^9 $$q + 2 q^{2} - \beta q^{3} + 4 q^{4} + ( - 2 \beta + 6) q^{5} - 2 \beta q^{6} + (\beta + 28) q^{7} + 8 q^{8} + (\beta + 17) q^{9} + ( - 4 \beta + 12) q^{10} + (2 \beta + 4) q^{11} - 4 \beta q^{12} + (7 \beta - 10) q^{13} + (2 \beta + 56) q^{14} + ( - 4 \beta + 88) q^{15} + 16 q^{16} + ( - 15 \beta - 18) q^{17} + (2 \beta + 34) q^{18} + ( - 8 \beta + 24) q^{20} + ( - 29 \beta - 44) q^{21} + (4 \beta + 8) q^{22} + (13 \beta - 84) q^{23} - 8 \beta q^{24} + ( - 20 \beta + 87) q^{25} + (14 \beta - 20) q^{26} + (9 \beta - 44) q^{27} + (4 \beta + 112) q^{28} + ( - 29 \beta + 54) q^{29} + ( - 8 \beta + 176) q^{30} + 16 \beta q^{31} + 32 q^{32} + ( - 6 \beta - 88) q^{33} + ( - 30 \beta - 36) q^{34} + ( - 52 \beta + 80) q^{35} + (4 \beta + 68) q^{36} + (16 \beta - 198) q^{37} + (3 \beta - 308) q^{39} + ( - 16 \beta + 48) q^{40} + ( - 6 \beta + 398) q^{41} + ( - 58 \beta - 88) q^{42} + (48 \beta + 124) q^{43} + (8 \beta + 16) q^{44} + ( - 30 \beta + 14) q^{45} + (26 \beta - 168) q^{46} + (40 \beta - 120) q^{47} - 16 \beta q^{48} + (57 \beta + 485) q^{49} + ( - 40 \beta + 174) q^{50} + (33 \beta + 660) q^{51} + (28 \beta - 40) q^{52} + ( - 9 \beta - 194) q^{53} + (18 \beta - 88) q^{54} - 152 q^{55} + (8 \beta + 224) q^{56} + ( - 58 \beta + 108) q^{58} + (71 \beta - 136) q^{59} + ( - 16 \beta + 352) q^{60} + (44 \beta - 362) q^{61} + 32 \beta q^{62} + (46 \beta + 520) q^{63} + 64 q^{64} + (48 \beta - 676) q^{65} + ( - 12 \beta - 176) q^{66} + (43 \beta + 448) q^{67} + ( - 60 \beta - 72) q^{68} + (71 \beta - 572) q^{69} + ( - 104 \beta + 160) q^{70} + ( - 22 \beta - 192) q^{71} + (8 \beta + 136) q^{72} + ( - \beta + 62) q^{73} + (32 \beta - 396) q^{74} + ( - 67 \beta + 880) q^{75} + (62 \beta + 200) q^{77} + (6 \beta - 616) q^{78} + ( - 58 \beta - 24) q^{79} + ( - 32 \beta + 96) q^{80} + (8 \beta - 855) q^{81} + ( - 12 \beta + 796) q^{82} + ( - 6 \beta + 1116) q^{83} + ( - 116 \beta - 176) q^{84} + ( - 24 \beta + 1212) q^{85} + (96 \beta + 248) q^{86} + ( - 25 \beta + 1276) q^{87} + (16 \beta + 32) q^{88} + (10 \beta + 430) q^{89} + ( - 60 \beta + 28) q^{90} + (193 \beta + 28) q^{91} + (52 \beta - 336) q^{92} + ( - 16 \beta - 704) q^{93} + (80 \beta - 240) q^{94} - 32 \beta q^{96} + (76 \beta + 894) q^{97} + (114 \beta + 970) q^{98} + (40 \beta + 156) q^{99} +O(q^{100})$$ q + 2 * q^2 - b * q^3 + 4 * q^4 + (-2*b + 6) * q^5 - 2*b * q^6 + (b + 28) * q^7 + 8 * q^8 + (b + 17) * q^9 + (-4*b + 12) * q^10 + (2*b + 4) * q^11 - 4*b * q^12 + (7*b - 10) * q^13 + (2*b + 56) * q^14 + (-4*b + 88) * q^15 + 16 * q^16 + (-15*b - 18) * q^17 + (2*b + 34) * q^18 + (-8*b + 24) * q^20 + (-29*b - 44) * q^21 + (4*b + 8) * q^22 + (13*b - 84) * q^23 - 8*b * q^24 + (-20*b + 87) * q^25 + (14*b - 20) * q^26 + (9*b - 44) * q^27 + (4*b + 112) * q^28 + (-29*b + 54) * q^29 + (-8*b + 176) * q^30 + 16*b * q^31 + 32 * q^32 + (-6*b - 88) * q^33 + (-30*b - 36) * q^34 + (-52*b + 80) * q^35 + (4*b + 68) * q^36 + (16*b - 198) * q^37 + (3*b - 308) * q^39 + (-16*b + 48) * q^40 + (-6*b + 398) * q^41 + (-58*b - 88) * q^42 + (48*b + 124) * q^43 + (8*b + 16) * q^44 + (-30*b + 14) * q^45 + (26*b - 168) * q^46 + (40*b - 120) * q^47 - 16*b * q^48 + (57*b + 485) * q^49 + (-40*b + 174) * q^50 + (33*b + 660) * q^51 + (28*b - 40) * q^52 + (-9*b - 194) * q^53 + (18*b - 88) * q^54 - 152 * q^55 + (8*b + 224) * q^56 + (-58*b + 108) * q^58 + (71*b - 136) * q^59 + (-16*b + 352) * q^60 + (44*b - 362) * q^61 + 32*b * q^62 + (46*b + 520) * q^63 + 64 * q^64 + (48*b - 676) * q^65 + (-12*b - 176) * q^66 + (43*b + 448) * q^67 + (-60*b - 72) * q^68 + (71*b - 572) * q^69 + (-104*b + 160) * q^70 + (-22*b - 192) * q^71 + (8*b + 136) * q^72 + (-b + 62) * q^73 + (32*b - 396) * q^74 + (-67*b + 880) * q^75 + (62*b + 200) * q^77 + (6*b - 616) * q^78 + (-58*b - 24) * q^79 + (-32*b + 96) * q^80 + (8*b - 855) * q^81 + (-12*b + 796) * q^82 + (-6*b + 1116) * q^83 + (-116*b - 176) * q^84 + (-24*b + 1212) * q^85 + (96*b + 248) * q^86 + (-25*b + 1276) * q^87 + (16*b + 32) * q^88 + (10*b + 430) * q^89 + (-60*b + 28) * q^90 + (193*b + 28) * q^91 + (52*b - 336) * q^92 + (-16*b - 704) * q^93 + (80*b - 240) * q^94 - 32*b * q^96 + (76*b + 894) * q^97 + (114*b + 970) * q^98 + (40*b + 156) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} + 16 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 + 16 * q^8 + 35 * q^9 $$2 q + 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} + 16 q^{8} + 35 q^{9} + 20 q^{10} + 10 q^{11} - 4 q^{12} - 13 q^{13} + 114 q^{14} + 172 q^{15} + 32 q^{16} - 51 q^{17} + 70 q^{18} + 40 q^{20} - 117 q^{21} + 20 q^{22} - 155 q^{23} - 8 q^{24} + 154 q^{25} - 26 q^{26} - 79 q^{27} + 228 q^{28} + 79 q^{29} + 344 q^{30} + 16 q^{31} + 64 q^{32} - 182 q^{33} - 102 q^{34} + 108 q^{35} + 140 q^{36} - 380 q^{37} - 613 q^{39} + 80 q^{40} + 790 q^{41} - 234 q^{42} + 296 q^{43} + 40 q^{44} - 2 q^{45} - 310 q^{46} - 200 q^{47} - 16 q^{48} + 1027 q^{49} + 308 q^{50} + 1353 q^{51} - 52 q^{52} - 397 q^{53} - 158 q^{54} - 304 q^{55} + 456 q^{56} + 158 q^{58} - 201 q^{59} + 688 q^{60} - 680 q^{61} + 32 q^{62} + 1086 q^{63} + 128 q^{64} - 1304 q^{65} - 364 q^{66} + 939 q^{67} - 204 q^{68} - 1073 q^{69} + 216 q^{70} - 406 q^{71} + 280 q^{72} + 123 q^{73} - 760 q^{74} + 1693 q^{75} + 462 q^{77} - 1226 q^{78} - 106 q^{79} + 160 q^{80} - 1702 q^{81} + 1580 q^{82} + 2226 q^{83} - 468 q^{84} + 2400 q^{85} + 592 q^{86} + 2527 q^{87} + 80 q^{88} + 870 q^{89} - 4 q^{90} + 249 q^{91} - 620 q^{92} - 1424 q^{93} - 400 q^{94} - 32 q^{96} + 1864 q^{97} + 2054 q^{98} + 352 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 + 16 * q^8 + 35 * q^9 + 20 * q^10 + 10 * q^11 - 4 * q^12 - 13 * q^13 + 114 * q^14 + 172 * q^15 + 32 * q^16 - 51 * q^17 + 70 * q^18 + 40 * q^20 - 117 * q^21 + 20 * q^22 - 155 * q^23 - 8 * q^24 + 154 * q^25 - 26 * q^26 - 79 * q^27 + 228 * q^28 + 79 * q^29 + 344 * q^30 + 16 * q^31 + 64 * q^32 - 182 * q^33 - 102 * q^34 + 108 * q^35 + 140 * q^36 - 380 * q^37 - 613 * q^39 + 80 * q^40 + 790 * q^41 - 234 * q^42 + 296 * q^43 + 40 * q^44 - 2 * q^45 - 310 * q^46 - 200 * q^47 - 16 * q^48 + 1027 * q^49 + 308 * q^50 + 1353 * q^51 - 52 * q^52 - 397 * q^53 - 158 * q^54 - 304 * q^55 + 456 * q^56 + 158 * q^58 - 201 * q^59 + 688 * q^60 - 680 * q^61 + 32 * q^62 + 1086 * q^63 + 128 * q^64 - 1304 * q^65 - 364 * q^66 + 939 * q^67 - 204 * q^68 - 1073 * q^69 + 216 * q^70 - 406 * q^71 + 280 * q^72 + 123 * q^73 - 760 * q^74 + 1693 * q^75 + 462 * q^77 - 1226 * q^78 - 106 * q^79 + 160 * q^80 - 1702 * q^81 + 1580 * q^82 + 2226 * q^83 - 468 * q^84 + 2400 * q^85 + 592 * q^86 + 2527 * q^87 + 80 * q^88 + 870 * q^89 - 4 * q^90 + 249 * q^91 - 620 * q^92 - 1424 * q^93 - 400 * q^94 - 32 * q^96 + 1864 * q^97 + 2054 * q^98 + 352 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 7.15207 −6.15207
2.00000 −7.15207 4.00000 −8.30413 −14.3041 35.1521 8.00000 24.1521 −16.6083
1.2 2.00000 6.15207 4.00000 18.3041 12.3041 21.8479 8.00000 10.8479 36.6083
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$19$$ $$-1$$

## 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 722.4.a.i 2
19.b odd 2 1 38.4.a.b 2
57.d even 2 1 342.4.a.k 2
76.d even 2 1 304.4.a.d 2
95.d odd 2 1 950.4.a.h 2
95.g even 4 2 950.4.b.g 4
133.c even 2 1 1862.4.a.b 2
152.b even 2 1 1216.4.a.l 2
152.g odd 2 1 1216.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 19.b odd 2 1
304.4.a.d 2 76.d even 2 1
342.4.a.k 2 57.d even 2 1
722.4.a.i 2 1.a even 1 1 trivial
950.4.a.h 2 95.d odd 2 1
950.4.b.g 4 95.g even 4 2
1216.4.a.j 2 152.g odd 2 1
1216.4.a.l 2 152.b even 2 1
1862.4.a.b 2 133.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(722))$$:

 $$T_{3}^{2} + T_{3} - 44$$ T3^2 + T3 - 44 $$T_{5}^{2} - 10T_{5} - 152$$ T5^2 - 10*T5 - 152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} + T - 44$$
$5$ $$T^{2} - 10T - 152$$
$7$ $$T^{2} - 57T + 768$$
$11$ $$T^{2} - 10T - 152$$
$13$ $$T^{2} + 13T - 2126$$
$17$ $$T^{2} + 51T - 9306$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 155T - 1472$$
$29$ $$T^{2} - 79T - 35654$$
$31$ $$T^{2} - 16T - 11264$$
$37$ $$T^{2} + 380T + 24772$$
$41$ $$T^{2} - 790T + 154432$$
$43$ $$T^{2} - 296T - 80048$$
$47$ $$T^{2} + 200T - 60800$$
$53$ $$T^{2} + 397T + 35818$$
$59$ $$T^{2} + 201T - 212964$$
$61$ $$T^{2} + 680T + 29932$$
$67$ $$T^{2} - 939T + 138612$$
$71$ $$T^{2} + 406T + 19792$$
$73$ $$T^{2} - 123T + 3738$$
$79$ $$T^{2} + 106T - 146048$$
$83$ $$T^{2} - 2226 T + 1237176$$
$89$ $$T^{2} - 870T + 184800$$
$97$ $$T^{2} - 1864 T + 613036$$