# Properties

 Label 46.4.a.c 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: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + ( - 3 \beta + 1) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + (6 \beta - 2) q^{6} + (2 \beta + 2) q^{7} - 8 q^{8} + (3 \beta + 64) q^{9}+O(q^{10})$$ q - 2 * q^2 + (-3*b + 1) * q^3 + 4 * q^4 + (2*b + 4) * q^5 + (6*b - 2) * q^6 + (2*b + 2) * q^7 - 8 * q^8 + (3*b + 64) * q^9 $$q - 2 q^{2} + ( - 3 \beta + 1) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + (6 \beta - 2) q^{6} + (2 \beta + 2) q^{7} - 8 q^{8} + (3 \beta + 64) q^{9} + ( - 4 \beta - 8) q^{10} + ( - 8 \beta + 40) q^{11} + ( - 12 \beta + 4) q^{12} + (7 \beta - 59) q^{13} + ( - 4 \beta - 4) q^{14} + ( - 16 \beta - 56) q^{15} + 16 q^{16} + (24 \beta + 50) q^{17} + ( - 6 \beta - 128) q^{18} + (18 \beta + 2) q^{19} + (8 \beta + 16) q^{20} + ( - 10 \beta - 58) q^{21} + (16 \beta - 80) q^{22} - 23 q^{23} + (24 \beta - 8) q^{24} + (20 \beta - 69) q^{25} + ( - 14 \beta + 118) q^{26} + ( - 117 \beta - 53) q^{27} + (8 \beta + 8) q^{28} + (65 \beta - 25) q^{29} + (32 \beta + 112) q^{30} + (17 \beta + 25) q^{31} - 32 q^{32} + ( - 104 \beta + 280) q^{33} + ( - 48 \beta - 100) q^{34} + (16 \beta + 48) q^{35} + (12 \beta + 256) q^{36} + ( - 70 \beta + 44) q^{37} + ( - 36 \beta - 4) q^{38} + (163 \beta - 269) q^{39} + ( - 16 \beta - 32) q^{40} + ( - 21 \beta + 253) q^{41} + (20 \beta + 116) q^{42} + (56 \beta - 248) q^{43} + ( - 32 \beta + 160) q^{44} + (146 \beta + 316) q^{45} + 46 q^{46} + (93 \beta + 61) q^{47} + ( - 48 \beta + 16) q^{48} + (12 \beta - 299) q^{49} + ( - 40 \beta + 138) q^{50} + ( - 198 \beta - 670) q^{51} + (28 \beta - 236) q^{52} + ( - 124 \beta + 182) q^{53} + (234 \beta + 106) q^{54} + 32 \beta q^{55} + ( - 16 \beta - 16) q^{56} + ( - 42 \beta - 538) q^{57} + ( - 130 \beta + 50) q^{58} + ( - 32 \beta + 412) q^{59} + ( - 64 \beta - 224) q^{60} + ( - 212 \beta + 334) q^{61} + ( - 34 \beta - 50) q^{62} + (140 \beta + 188) q^{63} + 64 q^{64} + ( - 76 \beta - 96) q^{65} + (208 \beta - 560) q^{66} + (48 \beta + 96) q^{67} + (96 \beta + 200) q^{68} + (69 \beta - 23) q^{69} + ( - 32 \beta - 96) q^{70} + ( - 91 \beta - 307) q^{71} + ( - 24 \beta - 512) q^{72} + (29 \beta - 1) q^{73} + (140 \beta - 88) q^{74} + (167 \beta - 669) q^{75} + (72 \beta + 8) q^{76} + (48 \beta - 80) q^{77} + ( - 326 \beta + 538) q^{78} + ( - 74 \beta + 334) q^{79} + (32 \beta + 64) q^{80} + (312 \beta + 1729) q^{81} + (42 \beta - 506) q^{82} + ( - 178 \beta + 286) q^{83} + ( - 40 \beta - 232) q^{84} + (244 \beta + 680) q^{85} + ( - 112 \beta + 496) q^{86} + ( - 55 \beta - 1975) q^{87} + (64 \beta - 320) q^{88} + ( - 94 \beta - 196) q^{89} + ( - 292 \beta - 632) q^{90} + ( - 90 \beta + 22) q^{91} - 92 q^{92} + ( - 109 \beta - 485) q^{93} + ( - 186 \beta - 122) q^{94} + (112 \beta + 368) q^{95} + (96 \beta - 32) q^{96} + ( - 164 \beta + 1158) q^{97} + ( - 24 \beta + 598) q^{98} + ( - 416 \beta + 2320) q^{99}+O(q^{100})$$ q - 2 * q^2 + (-3*b + 1) * q^3 + 4 * q^4 + (2*b + 4) * q^5 + (6*b - 2) * q^6 + (2*b + 2) * q^7 - 8 * q^8 + (3*b + 64) * q^9 + (-4*b - 8) * q^10 + (-8*b + 40) * q^11 + (-12*b + 4) * q^12 + (7*b - 59) * q^13 + (-4*b - 4) * q^14 + (-16*b - 56) * q^15 + 16 * q^16 + (24*b + 50) * q^17 + (-6*b - 128) * q^18 + (18*b + 2) * q^19 + (8*b + 16) * q^20 + (-10*b - 58) * q^21 + (16*b - 80) * q^22 - 23 * q^23 + (24*b - 8) * q^24 + (20*b - 69) * q^25 + (-14*b + 118) * q^26 + (-117*b - 53) * q^27 + (8*b + 8) * q^28 + (65*b - 25) * q^29 + (32*b + 112) * q^30 + (17*b + 25) * q^31 - 32 * q^32 + (-104*b + 280) * q^33 + (-48*b - 100) * q^34 + (16*b + 48) * q^35 + (12*b + 256) * q^36 + (-70*b + 44) * q^37 + (-36*b - 4) * q^38 + (163*b - 269) * q^39 + (-16*b - 32) * q^40 + (-21*b + 253) * q^41 + (20*b + 116) * q^42 + (56*b - 248) * q^43 + (-32*b + 160) * q^44 + (146*b + 316) * q^45 + 46 * q^46 + (93*b + 61) * q^47 + (-48*b + 16) * q^48 + (12*b - 299) * q^49 + (-40*b + 138) * q^50 + (-198*b - 670) * q^51 + (28*b - 236) * q^52 + (-124*b + 182) * q^53 + (234*b + 106) * q^54 + 32*b * q^55 + (-16*b - 16) * q^56 + (-42*b - 538) * q^57 + (-130*b + 50) * q^58 + (-32*b + 412) * q^59 + (-64*b - 224) * q^60 + (-212*b + 334) * q^61 + (-34*b - 50) * q^62 + (140*b + 188) * q^63 + 64 * q^64 + (-76*b - 96) * q^65 + (208*b - 560) * q^66 + (48*b + 96) * q^67 + (96*b + 200) * q^68 + (69*b - 23) * q^69 + (-32*b - 96) * q^70 + (-91*b - 307) * q^71 + (-24*b - 512) * q^72 + (29*b - 1) * q^73 + (140*b - 88) * q^74 + (167*b - 669) * q^75 + (72*b + 8) * q^76 + (48*b - 80) * q^77 + (-326*b + 538) * q^78 + (-74*b + 334) * q^79 + (32*b + 64) * q^80 + (312*b + 1729) * q^81 + (42*b - 506) * q^82 + (-178*b + 286) * q^83 + (-40*b - 232) * q^84 + (244*b + 680) * q^85 + (-112*b + 496) * q^86 + (-55*b - 1975) * q^87 + (64*b - 320) * q^88 + (-94*b - 196) * q^89 + (-292*b - 632) * q^90 + (-90*b + 22) * q^91 - 92 * q^92 + (-109*b - 485) * q^93 + (-186*b - 122) * q^94 + (112*b + 368) * q^95 + (96*b - 32) * q^96 + (-164*b + 1158) * q^97 + (-24*b + 598) * q^98 + (-416*b + 2320) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9 $$2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9} - 20 q^{10} + 72 q^{11} - 4 q^{12} - 111 q^{13} - 12 q^{14} - 128 q^{15} + 32 q^{16} + 124 q^{17} - 262 q^{18} + 22 q^{19} + 40 q^{20} - 126 q^{21} - 144 q^{22} - 46 q^{23} + 8 q^{24} - 118 q^{25} + 222 q^{26} - 223 q^{27} + 24 q^{28} + 15 q^{29} + 256 q^{30} + 67 q^{31} - 64 q^{32} + 456 q^{33} - 248 q^{34} + 112 q^{35} + 524 q^{36} + 18 q^{37} - 44 q^{38} - 375 q^{39} - 80 q^{40} + 485 q^{41} + 252 q^{42} - 440 q^{43} + 288 q^{44} + 778 q^{45} + 92 q^{46} + 215 q^{47} - 16 q^{48} - 586 q^{49} + 236 q^{50} - 1538 q^{51} - 444 q^{52} + 240 q^{53} + 446 q^{54} + 32 q^{55} - 48 q^{56} - 1118 q^{57} - 30 q^{58} + 792 q^{59} - 512 q^{60} + 456 q^{61} - 134 q^{62} + 516 q^{63} + 128 q^{64} - 268 q^{65} - 912 q^{66} + 240 q^{67} + 496 q^{68} + 23 q^{69} - 224 q^{70} - 705 q^{71} - 1048 q^{72} + 27 q^{73} - 36 q^{74} - 1171 q^{75} + 88 q^{76} - 112 q^{77} + 750 q^{78} + 594 q^{79} + 160 q^{80} + 3770 q^{81} - 970 q^{82} + 394 q^{83} - 504 q^{84} + 1604 q^{85} + 880 q^{86} - 4005 q^{87} - 576 q^{88} - 486 q^{89} - 1556 q^{90} - 46 q^{91} - 184 q^{92} - 1079 q^{93} - 430 q^{94} + 848 q^{95} + 32 q^{96} + 2152 q^{97} + 1172 q^{98} + 4224 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9 - 20 * q^10 + 72 * q^11 - 4 * q^12 - 111 * q^13 - 12 * q^14 - 128 * q^15 + 32 * q^16 + 124 * q^17 - 262 * q^18 + 22 * q^19 + 40 * q^20 - 126 * q^21 - 144 * q^22 - 46 * q^23 + 8 * q^24 - 118 * q^25 + 222 * q^26 - 223 * q^27 + 24 * q^28 + 15 * q^29 + 256 * q^30 + 67 * q^31 - 64 * q^32 + 456 * q^33 - 248 * q^34 + 112 * q^35 + 524 * q^36 + 18 * q^37 - 44 * q^38 - 375 * q^39 - 80 * q^40 + 485 * q^41 + 252 * q^42 - 440 * q^43 + 288 * q^44 + 778 * q^45 + 92 * q^46 + 215 * q^47 - 16 * q^48 - 586 * q^49 + 236 * q^50 - 1538 * q^51 - 444 * q^52 + 240 * q^53 + 446 * q^54 + 32 * q^55 - 48 * q^56 - 1118 * q^57 - 30 * q^58 + 792 * q^59 - 512 * q^60 + 456 * q^61 - 134 * q^62 + 516 * q^63 + 128 * q^64 - 268 * q^65 - 912 * q^66 + 240 * q^67 + 496 * q^68 + 23 * q^69 - 224 * q^70 - 705 * q^71 - 1048 * q^72 + 27 * q^73 - 36 * q^74 - 1171 * q^75 + 88 * q^76 - 112 * q^77 + 750 * q^78 + 594 * q^79 + 160 * q^80 + 3770 * q^81 - 970 * q^82 + 394 * q^83 - 504 * q^84 + 1604 * q^85 + 880 * q^86 - 4005 * q^87 - 576 * q^88 - 486 * q^89 - 1556 * q^90 - 46 * q^91 - 184 * q^92 - 1079 * q^93 - 430 * q^94 + 848 * q^95 + 32 * q^96 + 2152 * q^97 + 1172 * q^98 + 4224 * 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
 3.70156 −2.70156
−2.00000 −10.1047 4.00000 11.4031 20.2094 9.40312 −8.00000 75.1047 −22.8062
1.2 −2.00000 9.10469 4.00000 −1.40312 −18.2094 −3.40312 −8.00000 55.8953 2.80625
 $$n$$: 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.c 2
3.b odd 2 1 414.4.a.j 2
4.b odd 2 1 368.4.a.g 2
5.b even 2 1 1150.4.a.k 2
5.c odd 4 2 1150.4.b.i 4
7.b odd 2 1 2254.4.a.d 2
8.b even 2 1 1472.4.a.m 2
8.d odd 2 1 1472.4.a.l 2
23.b odd 2 1 1058.4.a.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 92$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(46))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} + T - 92$$
$5$ $$T^{2} - 10T - 16$$
$7$ $$T^{2} - 6T - 32$$
$11$ $$T^{2} - 72T + 640$$
$13$ $$T^{2} + 111T + 2578$$
$17$ $$T^{2} - 124T - 2060$$
$19$ $$T^{2} - 22T - 3200$$
$23$ $$(T + 23)^{2}$$
$29$ $$T^{2} - 15T - 43250$$
$31$ $$T^{2} - 67T - 1840$$
$37$ $$T^{2} - 18T - 50144$$
$41$ $$T^{2} - 485T + 54286$$
$43$ $$T^{2} + 440T + 16256$$
$47$ $$T^{2} - 215T - 77096$$
$53$ $$T^{2} - 240T - 143204$$
$59$ $$T^{2} - 792T + 146320$$
$61$ $$T^{2} - 456T - 408692$$
$67$ $$T^{2} - 240T - 9216$$
$71$ $$T^{2} + 705T + 39376$$
$73$ $$T^{2} - 27T - 8438$$
$79$ $$T^{2} - 594T + 32080$$
$83$ $$T^{2} - 394T - 285952$$
$89$ $$T^{2} + 486T - 31520$$
$97$ $$T^{2} - 2152 T + 882092$$