# Properties

 Label 46.4.a.d Level $46$ Weight $4$ Character orbit 46.a Self dual yes Analytic conductor $2.714$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$46 = 2 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 46.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.71408786026$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( - \beta + 2) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + ( - 2 \beta + 4) q^{6} + (4 \beta + 4) q^{7} + 8 q^{8} + ( - 3 \beta - 5) q^{9}+O(q^{10})$$ q + 2 * q^2 + (-b + 2) * q^3 + 4 * q^4 + (2*b + 4) * q^5 + (-2*b + 4) * q^6 + (4*b + 4) * q^7 + 8 * q^8 + (-3*b - 5) * q^9 $$q + 2 q^{2} + ( - \beta + 2) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + ( - 2 \beta + 4) q^{6} + (4 \beta + 4) q^{7} + 8 q^{8} + ( - 3 \beta - 5) q^{9} + (4 \beta + 8) q^{10} - 6 q^{11} + ( - 4 \beta + 8) q^{12} + ( - 11 \beta - 20) q^{13} + (8 \beta + 8) q^{14} + ( - 2 \beta - 28) q^{15} + 16 q^{16} + ( - 6 \beta - 30) q^{17} + ( - 6 \beta - 10) q^{18} + (20 \beta - 18) q^{19} + (8 \beta + 16) q^{20} - 64 q^{21} - 12 q^{22} + 23 q^{23} + ( - 8 \beta + 16) q^{24} + (20 \beta - 37) q^{25} + ( - 22 \beta - 40) q^{26} + (29 \beta - 10) q^{27} + (16 \beta + 16) q^{28} + ( - 33 \beta - 12) q^{29} + ( - 4 \beta - 56) q^{30} + ( - 69 \beta + 26) q^{31} + 32 q^{32} + (6 \beta - 12) q^{33} + ( - 12 \beta - 60) q^{34} + (32 \beta + 160) q^{35} + ( - 12 \beta - 20) q^{36} + (38 \beta + 84) q^{37} + (40 \beta - 36) q^{38} + (9 \beta + 158) q^{39} + (16 \beta + 32) q^{40} + (29 \beta + 172) q^{41} - 128 q^{42} + ( - 82 \beta + 198) q^{43} - 24 q^{44} + ( - 28 \beta - 128) q^{45} + 46 q^{46} + ( - 25 \beta + 442) q^{47} + ( - 16 \beta + 32) q^{48} + (48 \beta - 39) q^{49} + (40 \beta - 74) q^{50} + (24 \beta + 48) q^{51} + ( - 44 \beta - 80) q^{52} + ( - 38 \beta + 44) q^{53} + (58 \beta - 20) q^{54} + ( - 12 \beta - 24) q^{55} + (32 \beta + 32) q^{56} + (38 \beta - 396) q^{57} + ( - 66 \beta - 24) q^{58} + (60 \beta + 276) q^{59} + ( - 8 \beta - 112) q^{60} + (58 \beta - 560) q^{61} + ( - 138 \beta + 52) q^{62} + ( - 44 \beta - 236) q^{63} + 64 q^{64} + ( - 106 \beta - 476) q^{65} + (12 \beta - 24) q^{66} + (56 \beta - 450) q^{67} + ( - 24 \beta - 120) q^{68} + ( - 23 \beta + 46) q^{69} + (64 \beta + 320) q^{70} + ( - 21 \beta + 210) q^{71} + ( - 24 \beta - 40) q^{72} + (3 \beta - 640) q^{73} + (76 \beta + 168) q^{74} + (57 \beta - 434) q^{75} + (80 \beta - 72) q^{76} + ( - 24 \beta - 24) q^{77} + (18 \beta + 316) q^{78} + (26 \beta + 48) q^{79} + (32 \beta + 64) q^{80} + (120 \beta - 407) q^{81} + (58 \beta + 344) q^{82} + (22 \beta + 890) q^{83} - 256 q^{84} + ( - 96 \beta - 336) q^{85} + ( - 164 \beta + 396) q^{86} + ( - 21 \beta + 570) q^{87} - 48 q^{88} + (18 \beta + 1014) q^{89} + ( - 56 \beta - 256) q^{90} + ( - 168 \beta - 872) q^{91} + 92 q^{92} + ( - 95 \beta + 1294) q^{93} + ( - 50 \beta + 884) q^{94} + (84 \beta + 648) q^{95} + ( - 32 \beta + 64) q^{96} + ( - 274 \beta - 318) q^{97} + (96 \beta - 78) q^{98} + (18 \beta + 30) q^{99}+O(q^{100})$$ q + 2 * q^2 + (-b + 2) * q^3 + 4 * q^4 + (2*b + 4) * q^5 + (-2*b + 4) * q^6 + (4*b + 4) * q^7 + 8 * q^8 + (-3*b - 5) * q^9 + (4*b + 8) * q^10 - 6 * q^11 + (-4*b + 8) * q^12 + (-11*b - 20) * q^13 + (8*b + 8) * q^14 + (-2*b - 28) * q^15 + 16 * q^16 + (-6*b - 30) * q^17 + (-6*b - 10) * q^18 + (20*b - 18) * q^19 + (8*b + 16) * q^20 - 64 * q^21 - 12 * q^22 + 23 * q^23 + (-8*b + 16) * q^24 + (20*b - 37) * q^25 + (-22*b - 40) * q^26 + (29*b - 10) * q^27 + (16*b + 16) * q^28 + (-33*b - 12) * q^29 + (-4*b - 56) * q^30 + (-69*b + 26) * q^31 + 32 * q^32 + (6*b - 12) * q^33 + (-12*b - 60) * q^34 + (32*b + 160) * q^35 + (-12*b - 20) * q^36 + (38*b + 84) * q^37 + (40*b - 36) * q^38 + (9*b + 158) * q^39 + (16*b + 32) * q^40 + (29*b + 172) * q^41 - 128 * q^42 + (-82*b + 198) * q^43 - 24 * q^44 + (-28*b - 128) * q^45 + 46 * q^46 + (-25*b + 442) * q^47 + (-16*b + 32) * q^48 + (48*b - 39) * q^49 + (40*b - 74) * q^50 + (24*b + 48) * q^51 + (-44*b - 80) * q^52 + (-38*b + 44) * q^53 + (58*b - 20) * q^54 + (-12*b - 24) * q^55 + (32*b + 32) * q^56 + (38*b - 396) * q^57 + (-66*b - 24) * q^58 + (60*b + 276) * q^59 + (-8*b - 112) * q^60 + (58*b - 560) * q^61 + (-138*b + 52) * q^62 + (-44*b - 236) * q^63 + 64 * q^64 + (-106*b - 476) * q^65 + (12*b - 24) * q^66 + (56*b - 450) * q^67 + (-24*b - 120) * q^68 + (-23*b + 46) * q^69 + (64*b + 320) * q^70 + (-21*b + 210) * q^71 + (-24*b - 40) * q^72 + (3*b - 640) * q^73 + (76*b + 168) * q^74 + (57*b - 434) * q^75 + (80*b - 72) * q^76 + (-24*b - 24) * q^77 + (18*b + 316) * q^78 + (26*b + 48) * q^79 + (32*b + 64) * q^80 + (120*b - 407) * q^81 + (58*b + 344) * q^82 + (22*b + 890) * q^83 - 256 * q^84 + (-96*b - 336) * q^85 + (-164*b + 396) * q^86 + (-21*b + 570) * q^87 - 48 * q^88 + (18*b + 1014) * q^89 + (-56*b - 256) * q^90 + (-168*b - 872) * q^91 + 92 * q^92 + (-95*b + 1294) * q^93 + (-50*b + 884) * q^94 + (84*b + 648) * q^95 + (-32*b + 64) * q^96 + (-274*b - 318) * q^97 + (96*b - 78) * q^98 + (18*b + 30) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} + 12 q^{7} + 16 q^{8} - 13 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 3 * q^3 + 8 * q^4 + 10 * q^5 + 6 * q^6 + 12 * q^7 + 16 * q^8 - 13 * q^9 $$2 q + 4 q^{2} + 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} + 12 q^{7} + 16 q^{8} - 13 q^{9} + 20 q^{10} - 12 q^{11} + 12 q^{12} - 51 q^{13} + 24 q^{14} - 58 q^{15} + 32 q^{16} - 66 q^{17} - 26 q^{18} - 16 q^{19} + 40 q^{20} - 128 q^{21} - 24 q^{22} + 46 q^{23} + 24 q^{24} - 54 q^{25} - 102 q^{26} + 9 q^{27} + 48 q^{28} - 57 q^{29} - 116 q^{30} - 17 q^{31} + 64 q^{32} - 18 q^{33} - 132 q^{34} + 352 q^{35} - 52 q^{36} + 206 q^{37} - 32 q^{38} + 325 q^{39} + 80 q^{40} + 373 q^{41} - 256 q^{42} + 314 q^{43} - 48 q^{44} - 284 q^{45} + 92 q^{46} + 859 q^{47} + 48 q^{48} - 30 q^{49} - 108 q^{50} + 120 q^{51} - 204 q^{52} + 50 q^{53} + 18 q^{54} - 60 q^{55} + 96 q^{56} - 754 q^{57} - 114 q^{58} + 612 q^{59} - 232 q^{60} - 1062 q^{61} - 34 q^{62} - 516 q^{63} + 128 q^{64} - 1058 q^{65} - 36 q^{66} - 844 q^{67} - 264 q^{68} + 69 q^{69} + 704 q^{70} + 399 q^{71} - 104 q^{72} - 1277 q^{73} + 412 q^{74} - 811 q^{75} - 64 q^{76} - 72 q^{77} + 650 q^{78} + 122 q^{79} + 160 q^{80} - 694 q^{81} + 746 q^{82} + 1802 q^{83} - 512 q^{84} - 768 q^{85} + 628 q^{86} + 1119 q^{87} - 96 q^{88} + 2046 q^{89} - 568 q^{90} - 1912 q^{91} + 184 q^{92} + 2493 q^{93} + 1718 q^{94} + 1380 q^{95} + 96 q^{96} - 910 q^{97} - 60 q^{98} + 78 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 3 * q^3 + 8 * q^4 + 10 * q^5 + 6 * q^6 + 12 * q^7 + 16 * q^8 - 13 * q^9 + 20 * q^10 - 12 * q^11 + 12 * q^12 - 51 * q^13 + 24 * q^14 - 58 * q^15 + 32 * q^16 - 66 * q^17 - 26 * q^18 - 16 * q^19 + 40 * q^20 - 128 * q^21 - 24 * q^22 + 46 * q^23 + 24 * q^24 - 54 * q^25 - 102 * q^26 + 9 * q^27 + 48 * q^28 - 57 * q^29 - 116 * q^30 - 17 * q^31 + 64 * q^32 - 18 * q^33 - 132 * q^34 + 352 * q^35 - 52 * q^36 + 206 * q^37 - 32 * q^38 + 325 * q^39 + 80 * q^40 + 373 * q^41 - 256 * q^42 + 314 * q^43 - 48 * q^44 - 284 * q^45 + 92 * q^46 + 859 * q^47 + 48 * q^48 - 30 * q^49 - 108 * q^50 + 120 * q^51 - 204 * q^52 + 50 * q^53 + 18 * q^54 - 60 * q^55 + 96 * q^56 - 754 * q^57 - 114 * q^58 + 612 * q^59 - 232 * q^60 - 1062 * q^61 - 34 * q^62 - 516 * q^63 + 128 * q^64 - 1058 * q^65 - 36 * q^66 - 844 * q^67 - 264 * q^68 + 69 * q^69 + 704 * q^70 + 399 * q^71 - 104 * q^72 - 1277 * q^73 + 412 * q^74 - 811 * q^75 - 64 * q^76 - 72 * q^77 + 650 * q^78 + 122 * q^79 + 160 * q^80 - 694 * q^81 + 746 * q^82 + 1802 * q^83 - 512 * q^84 - 768 * q^85 + 628 * q^86 + 1119 * q^87 - 96 * q^88 + 2046 * q^89 - 568 * q^90 - 1912 * q^91 + 184 * q^92 + 2493 * q^93 + 1718 * q^94 + 1380 * q^95 + 96 * q^96 - 910 * q^97 - 60 * q^98 + 78 * 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
 4.77200 −3.77200
2.00000 −2.77200 4.00000 13.5440 −5.54400 23.0880 8.00000 −19.3160 27.0880
1.2 2.00000 5.77200 4.00000 −3.54400 11.5440 −11.0880 8.00000 6.31601 −7.08801
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$23$$ $$-1$$

## 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 46.4.a.d 2
3.b odd 2 1 414.4.a.f 2
4.b odd 2 1 368.4.a.f 2
5.b even 2 1 1150.4.a.j 2
5.c odd 4 2 1150.4.b.j 4
7.b odd 2 1 2254.4.a.f 2
8.b even 2 1 1472.4.a.k 2
8.d odd 2 1 1472.4.a.n 2
23.b odd 2 1 1058.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.d 2 1.a even 1 1 trivial
368.4.a.f 2 4.b odd 2 1
414.4.a.f 2 3.b odd 2 1
1058.4.a.j 2 23.b odd 2 1
1150.4.a.j 2 5.b even 2 1
1150.4.b.j 4 5.c odd 4 2
1472.4.a.k 2 8.b even 2 1
1472.4.a.n 2 8.d odd 2 1
2254.4.a.f 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3T_{3} - 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(46))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} - 3T - 16$$
$5$ $$T^{2} - 10T - 48$$
$7$ $$T^{2} - 12T - 256$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 51T - 1558$$
$17$ $$T^{2} + 66T + 432$$
$19$ $$T^{2} + 16T - 7236$$
$23$ $$(T - 23)^{2}$$
$29$ $$T^{2} + 57T - 19062$$
$31$ $$T^{2} + 17T - 86816$$
$37$ $$T^{2} - 206T - 15744$$
$41$ $$T^{2} - 373T + 19434$$
$43$ $$T^{2} - 314T - 98064$$
$47$ $$T^{2} - 859T + 173064$$
$53$ $$T^{2} - 50T - 25728$$
$59$ $$T^{2} - 612T + 27936$$
$61$ $$T^{2} + 1062 T + 220568$$
$67$ $$T^{2} + 844T + 120852$$
$71$ $$T^{2} - 399T + 31752$$
$73$ $$T^{2} + 1277 T + 407518$$
$79$ $$T^{2} - 122T - 8616$$
$83$ $$T^{2} - 1802 T + 802968$$
$89$ $$T^{2} - 2046 T + 1040616$$
$97$ $$T^{2} + 910 T - 1163112$$