# Properties

 Label 1815.4.a.i Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{3} + (\beta + 2) q^{4} + 5 q^{5} - 3 \beta q^{6} + (3 \beta - 2) q^{7} + (5 \beta - 10) q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 + 3 * q^3 + (b + 2) * q^4 + 5 * q^5 - 3*b * q^6 + (3*b - 2) * q^7 + (5*b - 10) * q^8 + 9 * q^9 $$q - \beta q^{2} + 3 q^{3} + (\beta + 2) q^{4} + 5 q^{5} - 3 \beta q^{6} + (3 \beta - 2) q^{7} + (5 \beta - 10) q^{8} + 9 q^{9} - 5 \beta q^{10} + (3 \beta + 6) q^{12} + (\beta - 24) q^{13} + ( - \beta - 30) q^{14} + 15 q^{15} + ( - 3 \beta - 66) q^{16} + (4 \beta + 47) q^{17} - 9 \beta q^{18} + ( - 9 \beta + 42) q^{19} + (5 \beta + 10) q^{20} + (9 \beta - 6) q^{21} + ( - 31 \beta - 13) q^{23} + (15 \beta - 30) q^{24} + 25 q^{25} + (23 \beta - 10) q^{26} + 27 q^{27} + (7 \beta + 26) q^{28} + (7 \beta + 60) q^{29} - 15 \beta q^{30} + ( - 21 \beta - 81) q^{31} + (29 \beta + 110) q^{32} + ( - 51 \beta - 40) q^{34} + (15 \beta - 10) q^{35} + (9 \beta + 18) q^{36} + (79 \beta - 154) q^{37} + ( - 33 \beta + 90) q^{38} + (3 \beta - 72) q^{39} + (25 \beta - 50) q^{40} + ( - 6 \beta - 6) q^{41} + ( - 3 \beta - 90) q^{42} + ( - 54 \beta - 66) q^{43} + 45 q^{45} + (44 \beta + 310) q^{46} + ( - 65 \beta - 275) q^{47} + ( - 9 \beta - 198) q^{48} + ( - 3 \beta - 249) q^{49} - 25 \beta q^{50} + (12 \beta + 141) q^{51} + ( - 21 \beta - 38) q^{52} + ( - \beta - 463) q^{53} - 27 \beta q^{54} + ( - 25 \beta + 170) q^{56} + ( - 27 \beta + 126) q^{57} + ( - 67 \beta - 70) q^{58} + (176 \beta - 278) q^{59} + (15 \beta + 30) q^{60} + (53 \beta - 281) q^{61} + (102 \beta + 210) q^{62} + (27 \beta - 18) q^{63} + ( - 115 \beta + 238) q^{64} + (5 \beta - 120) q^{65} + (150 \beta - 644) q^{67} + (59 \beta + 134) q^{68} + ( - 93 \beta - 39) q^{69} + ( - 5 \beta - 150) q^{70} + ( - 125 \beta - 74) q^{71} + (45 \beta - 90) q^{72} + ( - 32 \beta - 204) q^{73} + (75 \beta - 790) q^{74} + 75 q^{75} + (15 \beta - 6) q^{76} + (69 \beta - 30) q^{78} + (69 \beta - 521) q^{79} + ( - 15 \beta - 330) q^{80} + 81 q^{81} + (12 \beta + 60) q^{82} + (165 \beta - 90) q^{83} + (21 \beta + 78) q^{84} + (20 \beta + 235) q^{85} + (120 \beta + 540) q^{86} + (21 \beta + 180) q^{87} + (56 \beta - 672) q^{89} - 45 \beta q^{90} + ( - 71 \beta + 78) q^{91} + ( - 106 \beta - 336) q^{92} + ( - 63 \beta - 243) q^{93} + (340 \beta + 650) q^{94} + ( - 45 \beta + 210) q^{95} + (87 \beta + 330) q^{96} + ( - 52 \beta - 12) q^{97} + (252 \beta + 30) q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^3 + (b + 2) * q^4 + 5 * q^5 - 3*b * q^6 + (3*b - 2) * q^7 + (5*b - 10) * q^8 + 9 * q^9 - 5*b * q^10 + (3*b + 6) * q^12 + (b - 24) * q^13 + (-b - 30) * q^14 + 15 * q^15 + (-3*b - 66) * q^16 + (4*b + 47) * q^17 - 9*b * q^18 + (-9*b + 42) * q^19 + (5*b + 10) * q^20 + (9*b - 6) * q^21 + (-31*b - 13) * q^23 + (15*b - 30) * q^24 + 25 * q^25 + (23*b - 10) * q^26 + 27 * q^27 + (7*b + 26) * q^28 + (7*b + 60) * q^29 - 15*b * q^30 + (-21*b - 81) * q^31 + (29*b + 110) * q^32 + (-51*b - 40) * q^34 + (15*b - 10) * q^35 + (9*b + 18) * q^36 + (79*b - 154) * q^37 + (-33*b + 90) * q^38 + (3*b - 72) * q^39 + (25*b - 50) * q^40 + (-6*b - 6) * q^41 + (-3*b - 90) * q^42 + (-54*b - 66) * q^43 + 45 * q^45 + (44*b + 310) * q^46 + (-65*b - 275) * q^47 + (-9*b - 198) * q^48 + (-3*b - 249) * q^49 - 25*b * q^50 + (12*b + 141) * q^51 + (-21*b - 38) * q^52 + (-b - 463) * q^53 - 27*b * q^54 + (-25*b + 170) * q^56 + (-27*b + 126) * q^57 + (-67*b - 70) * q^58 + (176*b - 278) * q^59 + (15*b + 30) * q^60 + (53*b - 281) * q^61 + (102*b + 210) * q^62 + (27*b - 18) * q^63 + (-115*b + 238) * q^64 + (5*b - 120) * q^65 + (150*b - 644) * q^67 + (59*b + 134) * q^68 + (-93*b - 39) * q^69 + (-5*b - 150) * q^70 + (-125*b - 74) * q^71 + (45*b - 90) * q^72 + (-32*b - 204) * q^73 + (75*b - 790) * q^74 + 75 * q^75 + (15*b - 6) * q^76 + (69*b - 30) * q^78 + (69*b - 521) * q^79 + (-15*b - 330) * q^80 + 81 * q^81 + (12*b + 60) * q^82 + (165*b - 90) * q^83 + (21*b + 78) * q^84 + (20*b + 235) * q^85 + (120*b + 540) * q^86 + (21*b + 180) * q^87 + (56*b - 672) * q^89 - 45*b * q^90 + (-71*b + 78) * q^91 + (-106*b - 336) * q^92 + (-63*b - 243) * q^93 + (340*b + 650) * q^94 + (-45*b + 210) * q^95 + (87*b + 330) * q^96 + (-52*b - 12) * q^97 + (252*b + 30) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 6 q^{3} + 5 q^{4} + 10 q^{5} - 3 q^{6} - q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 + 6 * q^3 + 5 * q^4 + 10 * q^5 - 3 * q^6 - q^7 - 15 * q^8 + 18 * q^9 $$2 q - q^{2} + 6 q^{3} + 5 q^{4} + 10 q^{5} - 3 q^{6} - q^{7} - 15 q^{8} + 18 q^{9} - 5 q^{10} + 15 q^{12} - 47 q^{13} - 61 q^{14} + 30 q^{15} - 135 q^{16} + 98 q^{17} - 9 q^{18} + 75 q^{19} + 25 q^{20} - 3 q^{21} - 57 q^{23} - 45 q^{24} + 50 q^{25} + 3 q^{26} + 54 q^{27} + 59 q^{28} + 127 q^{29} - 15 q^{30} - 183 q^{31} + 249 q^{32} - 131 q^{34} - 5 q^{35} + 45 q^{36} - 229 q^{37} + 147 q^{38} - 141 q^{39} - 75 q^{40} - 18 q^{41} - 183 q^{42} - 186 q^{43} + 90 q^{45} + 664 q^{46} - 615 q^{47} - 405 q^{48} - 501 q^{49} - 25 q^{50} + 294 q^{51} - 97 q^{52} - 927 q^{53} - 27 q^{54} + 315 q^{56} + 225 q^{57} - 207 q^{58} - 380 q^{59} + 75 q^{60} - 509 q^{61} + 522 q^{62} - 9 q^{63} + 361 q^{64} - 235 q^{65} - 1138 q^{67} + 327 q^{68} - 171 q^{69} - 305 q^{70} - 273 q^{71} - 135 q^{72} - 440 q^{73} - 1505 q^{74} + 150 q^{75} + 3 q^{76} + 9 q^{78} - 973 q^{79} - 675 q^{80} + 162 q^{81} + 132 q^{82} - 15 q^{83} + 177 q^{84} + 490 q^{85} + 1200 q^{86} + 381 q^{87} - 1288 q^{89} - 45 q^{90} + 85 q^{91} - 778 q^{92} - 549 q^{93} + 1640 q^{94} + 375 q^{95} + 747 q^{96} - 76 q^{97} + 312 q^{98}+O(q^{100})$$ 2 * q - q^2 + 6 * q^3 + 5 * q^4 + 10 * q^5 - 3 * q^6 - q^7 - 15 * q^8 + 18 * q^9 - 5 * q^10 + 15 * q^12 - 47 * q^13 - 61 * q^14 + 30 * q^15 - 135 * q^16 + 98 * q^17 - 9 * q^18 + 75 * q^19 + 25 * q^20 - 3 * q^21 - 57 * q^23 - 45 * q^24 + 50 * q^25 + 3 * q^26 + 54 * q^27 + 59 * q^28 + 127 * q^29 - 15 * q^30 - 183 * q^31 + 249 * q^32 - 131 * q^34 - 5 * q^35 + 45 * q^36 - 229 * q^37 + 147 * q^38 - 141 * q^39 - 75 * q^40 - 18 * q^41 - 183 * q^42 - 186 * q^43 + 90 * q^45 + 664 * q^46 - 615 * q^47 - 405 * q^48 - 501 * q^49 - 25 * q^50 + 294 * q^51 - 97 * q^52 - 927 * q^53 - 27 * q^54 + 315 * q^56 + 225 * q^57 - 207 * q^58 - 380 * q^59 + 75 * q^60 - 509 * q^61 + 522 * q^62 - 9 * q^63 + 361 * q^64 - 235 * q^65 - 1138 * q^67 + 327 * q^68 - 171 * q^69 - 305 * q^70 - 273 * q^71 - 135 * q^72 - 440 * q^73 - 1505 * q^74 + 150 * q^75 + 3 * q^76 + 9 * q^78 - 973 * q^79 - 675 * q^80 + 162 * q^81 + 132 * q^82 - 15 * q^83 + 177 * q^84 + 490 * q^85 + 1200 * q^86 + 381 * q^87 - 1288 * q^89 - 45 * q^90 + 85 * q^91 - 778 * q^92 - 549 * q^93 + 1640 * q^94 + 375 * q^95 + 747 * q^96 - 76 * q^97 + 312 * q^98

## 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
−3.70156 3.00000 5.70156 5.00000 −11.1047 9.10469 8.50781 9.00000 −18.5078
1.2 2.70156 3.00000 −0.701562 5.00000 8.10469 −10.1047 −23.5078 9.00000 13.5078
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 1815.4.a.i 2
11.b odd 2 1 1815.4.a.o yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.i 2 1.a even 1 1 trivial
1815.4.a.o yes 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{2} + T_{2} - 10$$ T2^2 + T2 - 10 $$T_{7}^{2} + T_{7} - 92$$ T7^2 + T7 - 92

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 10$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} + T - 92$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 47T + 542$$
$17$ $$T^{2} - 98T + 2237$$
$19$ $$T^{2} - 75T + 576$$
$23$ $$T^{2} + 57T - 9038$$
$29$ $$T^{2} - 127T + 3530$$
$31$ $$T^{2} + 183T + 3852$$
$37$ $$T^{2} + 229T - 50860$$
$41$ $$T^{2} + 18T - 288$$
$43$ $$T^{2} + 186T - 21240$$
$47$ $$T^{2} + 615T + 51250$$
$53$ $$T^{2} + 927T + 214822$$
$59$ $$T^{2} + 380T - 281404$$
$61$ $$T^{2} + 509T + 35978$$
$67$ $$T^{2} + 1138T + 93136$$
$71$ $$T^{2} + 273T - 141524$$
$73$ $$T^{2} + 440T + 37904$$
$79$ $$T^{2} + 973T + 187882$$
$83$ $$T^{2} + 15T - 279000$$
$89$ $$T^{2} + 1288 T + 382592$$
$97$ $$T^{2} + 76T - 26272$$