# Properties

 Label 58.4.a.c Level $58$ Weight $4$ Character orbit 58.a Self dual yes Analytic conductor $3.422$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 58.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.42211078033$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + (\beta - 1) q^{3} + 4 q^{4} + ( - 6 \beta - 5) q^{5} + ( - 2 \beta + 2) q^{6} + (8 \beta - 8) q^{7} - 8 q^{8} + ( - 2 \beta - 20) q^{9}+O(q^{10})$$ q - 2 * q^2 + (b - 1) * q^3 + 4 * q^4 + (-6*b - 5) * q^5 + (-2*b + 2) * q^6 + (8*b - 8) * q^7 - 8 * q^8 + (-2*b - 20) * q^9 $$q - 2 q^{2} + (\beta - 1) q^{3} + 4 q^{4} + ( - 6 \beta - 5) q^{5} + ( - 2 \beta + 2) q^{6} + (8 \beta - 8) q^{7} - 8 q^{8} + ( - 2 \beta - 20) q^{9} + (12 \beta + 10) q^{10} + (3 \beta - 45) q^{11} + (4 \beta - 4) q^{12} + ( - 8 \beta - 25) q^{13} + ( - 16 \beta + 16) q^{14} + (\beta - 31) q^{15} + 16 q^{16} + ( - 16 \beta - 22) q^{17} + (4 \beta + 40) q^{18} + (4 \beta + 54) q^{19} + ( - 24 \beta - 20) q^{20} + ( - 16 \beta + 56) q^{21} + ( - 6 \beta + 90) q^{22} + (78 \beta - 14) q^{23} + ( - 8 \beta + 8) q^{24} + (60 \beta + 116) q^{25} + (16 \beta + 50) q^{26} + ( - 45 \beta + 35) q^{27} + (32 \beta - 32) q^{28} + 29 q^{29} + ( - 2 \beta + 62) q^{30} + ( - 109 \beta + 33) q^{31} - 32 q^{32} + ( - 48 \beta + 63) q^{33} + (32 \beta + 44) q^{34} + (8 \beta - 248) q^{35} + ( - 8 \beta - 80) q^{36} + ( - 4 \beta + 20) q^{37} + ( - 8 \beta - 108) q^{38} + ( - 17 \beta - 23) q^{39} + (48 \beta + 40) q^{40} + (80 \beta + 152) q^{41} + (32 \beta - 112) q^{42} + ( - 53 \beta - 65) q^{43} + (12 \beta - 180) q^{44} + (130 \beta + 172) q^{45} + ( - 156 \beta + 28) q^{46} + (99 \beta - 257) q^{47} + (16 \beta - 16) q^{48} + ( - 128 \beta + 105) q^{49} + ( - 120 \beta - 232) q^{50} + ( - 6 \beta - 74) q^{51} + ( - 32 \beta - 100) q^{52} + ( - 52 \beta - 479) q^{53} + (90 \beta - 70) q^{54} + (255 \beta + 117) q^{55} + ( - 64 \beta + 64) q^{56} + (50 \beta - 30) q^{57} - 58 q^{58} + ( - 250 \beta - 90) q^{59} + (4 \beta - 124) q^{60} + (12 \beta + 514) q^{61} + (218 \beta - 66) q^{62} + ( - 144 \beta + 64) q^{63} + 64 q^{64} + (190 \beta + 413) q^{65} + (96 \beta - 126) q^{66} + (20 \beta - 456) q^{67} + ( - 64 \beta - 88) q^{68} + ( - 92 \beta + 482) q^{69} + ( - 16 \beta + 496) q^{70} + ( - 34 \beta + 398) q^{71} + (16 \beta + 160) q^{72} + ( - 176 \beta - 428) q^{73} + (8 \beta - 40) q^{74} + (56 \beta + 244) q^{75} + (16 \beta + 216) q^{76} + ( - 384 \beta + 504) q^{77} + (34 \beta + 46) q^{78} + (361 \beta - 159) q^{79} + ( - 96 \beta - 80) q^{80} + (134 \beta + 235) q^{81} + ( - 160 \beta - 304) q^{82} + (38 \beta - 914) q^{83} + ( - 64 \beta + 224) q^{84} + (212 \beta + 686) q^{85} + (106 \beta + 130) q^{86} + (29 \beta - 29) q^{87} + ( - 24 \beta + 360) q^{88} + ( - 72 \beta - 472) q^{89} + ( - 260 \beta - 344) q^{90} + ( - 136 \beta - 184) q^{91} + (312 \beta - 56) q^{92} + (142 \beta - 687) q^{93} + ( - 198 \beta + 514) q^{94} + ( - 344 \beta - 414) q^{95} + ( - 32 \beta + 32) q^{96} + ( - 100 \beta + 184) q^{97} + (256 \beta - 210) q^{98} + (30 \beta + 864) q^{99}+O(q^{100})$$ q - 2 * q^2 + (b - 1) * q^3 + 4 * q^4 + (-6*b - 5) * q^5 + (-2*b + 2) * q^6 + (8*b - 8) * q^7 - 8 * q^8 + (-2*b - 20) * q^9 + (12*b + 10) * q^10 + (3*b - 45) * q^11 + (4*b - 4) * q^12 + (-8*b - 25) * q^13 + (-16*b + 16) * q^14 + (b - 31) * q^15 + 16 * q^16 + (-16*b - 22) * q^17 + (4*b + 40) * q^18 + (4*b + 54) * q^19 + (-24*b - 20) * q^20 + (-16*b + 56) * q^21 + (-6*b + 90) * q^22 + (78*b - 14) * q^23 + (-8*b + 8) * q^24 + (60*b + 116) * q^25 + (16*b + 50) * q^26 + (-45*b + 35) * q^27 + (32*b - 32) * q^28 + 29 * q^29 + (-2*b + 62) * q^30 + (-109*b + 33) * q^31 - 32 * q^32 + (-48*b + 63) * q^33 + (32*b + 44) * q^34 + (8*b - 248) * q^35 + (-8*b - 80) * q^36 + (-4*b + 20) * q^37 + (-8*b - 108) * q^38 + (-17*b - 23) * q^39 + (48*b + 40) * q^40 + (80*b + 152) * q^41 + (32*b - 112) * q^42 + (-53*b - 65) * q^43 + (12*b - 180) * q^44 + (130*b + 172) * q^45 + (-156*b + 28) * q^46 + (99*b - 257) * q^47 + (16*b - 16) * q^48 + (-128*b + 105) * q^49 + (-120*b - 232) * q^50 + (-6*b - 74) * q^51 + (-32*b - 100) * q^52 + (-52*b - 479) * q^53 + (90*b - 70) * q^54 + (255*b + 117) * q^55 + (-64*b + 64) * q^56 + (50*b - 30) * q^57 - 58 * q^58 + (-250*b - 90) * q^59 + (4*b - 124) * q^60 + (12*b + 514) * q^61 + (218*b - 66) * q^62 + (-144*b + 64) * q^63 + 64 * q^64 + (190*b + 413) * q^65 + (96*b - 126) * q^66 + (20*b - 456) * q^67 + (-64*b - 88) * q^68 + (-92*b + 482) * q^69 + (-16*b + 496) * q^70 + (-34*b + 398) * q^71 + (16*b + 160) * q^72 + (-176*b - 428) * q^73 + (8*b - 40) * q^74 + (56*b + 244) * q^75 + (16*b + 216) * q^76 + (-384*b + 504) * q^77 + (34*b + 46) * q^78 + (361*b - 159) * q^79 + (-96*b - 80) * q^80 + (134*b + 235) * q^81 + (-160*b - 304) * q^82 + (38*b - 914) * q^83 + (-64*b + 224) * q^84 + (212*b + 686) * q^85 + (106*b + 130) * q^86 + (29*b - 29) * q^87 + (-24*b + 360) * q^88 + (-72*b - 472) * q^89 + (-260*b - 344) * q^90 + (-136*b - 184) * q^91 + (312*b - 56) * q^92 + (142*b - 687) * q^93 + (-198*b + 514) * q^94 + (-344*b - 414) * q^95 + (-32*b + 32) * q^96 + (-100*b + 184) * q^97 + (256*b - 210) * q^98 + (30*b + 864) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 $$2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9} + 20 q^{10} - 90 q^{11} - 8 q^{12} - 50 q^{13} + 32 q^{14} - 62 q^{15} + 32 q^{16} - 44 q^{17} + 80 q^{18} + 108 q^{19} - 40 q^{20} + 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} + 100 q^{26} + 70 q^{27} - 64 q^{28} + 58 q^{29} + 124 q^{30} + 66 q^{31} - 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} + 40 q^{37} - 216 q^{38} - 46 q^{39} + 80 q^{40} + 304 q^{41} - 224 q^{42} - 130 q^{43} - 360 q^{44} + 344 q^{45} + 56 q^{46} - 514 q^{47} - 32 q^{48} + 210 q^{49} - 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} + 234 q^{55} + 128 q^{56} - 60 q^{57} - 116 q^{58} - 180 q^{59} - 248 q^{60} + 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} - 252 q^{66} - 912 q^{67} - 176 q^{68} + 964 q^{69} + 992 q^{70} + 796 q^{71} + 320 q^{72} - 856 q^{73} - 80 q^{74} + 488 q^{75} + 432 q^{76} + 1008 q^{77} + 92 q^{78} - 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} + 448 q^{84} + 1372 q^{85} + 260 q^{86} - 58 q^{87} + 720 q^{88} - 944 q^{89} - 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} - 828 q^{95} + 64 q^{96} + 368 q^{97} - 420 q^{98} + 1728 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 + 20 * q^10 - 90 * q^11 - 8 * q^12 - 50 * q^13 + 32 * q^14 - 62 * q^15 + 32 * q^16 - 44 * q^17 + 80 * q^18 + 108 * q^19 - 40 * q^20 + 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 + 100 * q^26 + 70 * q^27 - 64 * q^28 + 58 * q^29 + 124 * q^30 + 66 * q^31 - 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 + 40 * q^37 - 216 * q^38 - 46 * q^39 + 80 * q^40 + 304 * q^41 - 224 * q^42 - 130 * q^43 - 360 * q^44 + 344 * q^45 + 56 * q^46 - 514 * q^47 - 32 * q^48 + 210 * q^49 - 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 + 234 * q^55 + 128 * q^56 - 60 * q^57 - 116 * q^58 - 180 * q^59 - 248 * q^60 + 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 - 252 * q^66 - 912 * q^67 - 176 * q^68 + 964 * q^69 + 992 * q^70 + 796 * q^71 + 320 * q^72 - 856 * q^73 - 80 * q^74 + 488 * q^75 + 432 * q^76 + 1008 * q^77 + 92 * q^78 - 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 + 448 * q^84 + 1372 * q^85 + 260 * q^86 - 58 * q^87 + 720 * q^88 - 944 * q^89 - 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 - 828 * q^95 + 64 * q^96 + 368 * q^97 - 420 * q^98 + 1728 * 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
 −2.44949 2.44949
−2.00000 −3.44949 4.00000 9.69694 6.89898 −27.5959 −8.00000 −15.1010 −19.3939
1.2 −2.00000 1.44949 4.00000 −19.6969 −2.89898 11.5959 −8.00000 −24.8990 39.3939
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$-1$$

## 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 58.4.a.c 2
3.b odd 2 1 522.4.a.j 2
4.b odd 2 1 464.4.a.e 2
5.b even 2 1 1450.4.a.g 2
8.b even 2 1 1856.4.a.l 2
8.d odd 2 1 1856.4.a.i 2
29.b even 2 1 1682.4.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 1.a even 1 1 trivial
464.4.a.e 2 4.b odd 2 1
522.4.a.j 2 3.b odd 2 1
1450.4.a.g 2 5.b even 2 1
1682.4.a.c 2 29.b even 2 1
1856.4.a.i 2 8.d odd 2 1
1856.4.a.l 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(58))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} + 2T - 5$$
$5$ $$T^{2} + 10T - 191$$
$7$ $$T^{2} + 16T - 320$$
$11$ $$T^{2} + 90T + 1971$$
$13$ $$T^{2} + 50T + 241$$
$17$ $$T^{2} + 44T - 1052$$
$19$ $$T^{2} - 108T + 2820$$
$23$ $$T^{2} + 28T - 36308$$
$29$ $$(T - 29)^{2}$$
$31$ $$T^{2} - 66T - 70197$$
$37$ $$T^{2} - 40T + 304$$
$41$ $$T^{2} - 304T - 15296$$
$43$ $$T^{2} + 130T - 12629$$
$47$ $$T^{2} + 514T + 7243$$
$53$ $$T^{2} + 958T + 213217$$
$59$ $$T^{2} + 180T - 366900$$
$61$ $$T^{2} - 1028 T + 263332$$
$67$ $$T^{2} + 912T + 205536$$
$71$ $$T^{2} - 796T + 151468$$
$73$ $$T^{2} + 856T - 2672$$
$79$ $$T^{2} + 318T - 756645$$
$83$ $$T^{2} + 1828 T + 826732$$
$89$ $$T^{2} + 944T + 191680$$
$97$ $$T^{2} - 368T - 26144$$