# Properties

 Label 1815.4.a.n Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ 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] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.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: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} + (\beta - 4) q^{4} - 5 q^{5} + 3 \beta q^{6} + ( - 4 \beta + 4) q^{7} + ( - 11 \beta + 4) q^{8} + 9 q^{9}+O(q^{10})$$ q + b * q^2 + 3 * q^3 + (b - 4) * q^4 - 5 * q^5 + 3*b * q^6 + (-4*b + 4) * q^7 + (-11*b + 4) * q^8 + 9 * q^9 $$q + \beta q^{2} + 3 q^{3} + (\beta - 4) q^{4} - 5 q^{5} + 3 \beta q^{6} + ( - 4 \beta + 4) q^{7} + ( - 11 \beta + 4) q^{8} + 9 q^{9} - 5 \beta q^{10} + (3 \beta - 12) q^{12} + (2 \beta + 44) q^{13} - 16 q^{14} - 15 q^{15} + ( - 15 \beta - 12) q^{16} + ( - 44 \beta + 30) q^{17} + 9 \beta q^{18} + (22 \beta + 74) q^{19} + ( - 5 \beta + 20) q^{20} + ( - 12 \beta + 12) q^{21} + ( - 60 \beta - 32) q^{23} + ( - 33 \beta + 12) q^{24} + 25 q^{25} + (46 \beta + 8) q^{26} + 27 q^{27} + (16 \beta - 32) q^{28} + ( - 34 \beta + 96) q^{29} - 15 \beta q^{30} + ( - 12 \beta + 36) q^{31} + (61 \beta - 92) q^{32} + ( - 14 \beta - 176) q^{34} + (20 \beta - 20) q^{35} + (9 \beta - 36) q^{36} + ( - 112 \beta - 130) q^{37} + (96 \beta + 88) q^{38} + (6 \beta + 132) q^{39} + (55 \beta - 20) q^{40} + (154 \beta - 96) q^{41} - 48 q^{42} + (124 \beta + 196) q^{43} - 45 q^{45} + ( - 92 \beta - 240) q^{46} + (216 \beta + 4) q^{47} + ( - 45 \beta - 36) q^{48} + ( - 16 \beta - 263) q^{49} + 25 \beta q^{50} + ( - 132 \beta + 90) q^{51} + (38 \beta - 168) q^{52} + ( - 196 \beta + 334) q^{53} + 27 \beta q^{54} + ( - 16 \beta + 192) q^{56} + (66 \beta + 222) q^{57} + (62 \beta - 136) q^{58} + (240 \beta + 4) q^{59} + ( - 15 \beta + 60) q^{60} + ( - 364 \beta + 146) q^{61} + (24 \beta - 48) q^{62} + ( - 36 \beta + 36) q^{63} + (89 \beta + 340) q^{64} + ( - 10 \beta - 220) q^{65} + (16 \beta - 380) q^{67} + (162 \beta - 296) q^{68} + ( - 180 \beta - 96) q^{69} + 80 q^{70} + (44 \beta + 1008) q^{71} + ( - 99 \beta + 36) q^{72} + ( - 58 \beta + 272) q^{73} + ( - 242 \beta - 448) q^{74} + 75 q^{75} + (8 \beta - 208) q^{76} + (138 \beta + 24) q^{78} + (306 \beta - 474) q^{79} + (75 \beta + 60) q^{80} + 81 q^{81} + (58 \beta + 616) q^{82} + (426 \beta - 70) q^{83} + (48 \beta - 96) q^{84} + (220 \beta - 150) q^{85} + (320 \beta + 496) q^{86} + ( - 102 \beta + 288) q^{87} + ( - 128 \beta + 186) q^{89} - 45 \beta q^{90} + ( - 176 \beta + 144) q^{91} + (148 \beta - 112) q^{92} + ( - 36 \beta + 108) q^{93} + (220 \beta + 864) q^{94} + ( - 110 \beta - 370) q^{95} + (183 \beta - 276) q^{96} + (428 \beta - 298) q^{97} + ( - 279 \beta - 64) q^{98} +O(q^{100})$$ q + b * q^2 + 3 * q^3 + (b - 4) * q^4 - 5 * q^5 + 3*b * q^6 + (-4*b + 4) * q^7 + (-11*b + 4) * q^8 + 9 * q^9 - 5*b * q^10 + (3*b - 12) * q^12 + (2*b + 44) * q^13 - 16 * q^14 - 15 * q^15 + (-15*b - 12) * q^16 + (-44*b + 30) * q^17 + 9*b * q^18 + (22*b + 74) * q^19 + (-5*b + 20) * q^20 + (-12*b + 12) * q^21 + (-60*b - 32) * q^23 + (-33*b + 12) * q^24 + 25 * q^25 + (46*b + 8) * q^26 + 27 * q^27 + (16*b - 32) * q^28 + (-34*b + 96) * q^29 - 15*b * q^30 + (-12*b + 36) * q^31 + (61*b - 92) * q^32 + (-14*b - 176) * q^34 + (20*b - 20) * q^35 + (9*b - 36) * q^36 + (-112*b - 130) * q^37 + (96*b + 88) * q^38 + (6*b + 132) * q^39 + (55*b - 20) * q^40 + (154*b - 96) * q^41 - 48 * q^42 + (124*b + 196) * q^43 - 45 * q^45 + (-92*b - 240) * q^46 + (216*b + 4) * q^47 + (-45*b - 36) * q^48 + (-16*b - 263) * q^49 + 25*b * q^50 + (-132*b + 90) * q^51 + (38*b - 168) * q^52 + (-196*b + 334) * q^53 + 27*b * q^54 + (-16*b + 192) * q^56 + (66*b + 222) * q^57 + (62*b - 136) * q^58 + (240*b + 4) * q^59 + (-15*b + 60) * q^60 + (-364*b + 146) * q^61 + (24*b - 48) * q^62 + (-36*b + 36) * q^63 + (89*b + 340) * q^64 + (-10*b - 220) * q^65 + (16*b - 380) * q^67 + (162*b - 296) * q^68 + (-180*b - 96) * q^69 + 80 * q^70 + (44*b + 1008) * q^71 + (-99*b + 36) * q^72 + (-58*b + 272) * q^73 + (-242*b - 448) * q^74 + 75 * q^75 + (8*b - 208) * q^76 + (138*b + 24) * q^78 + (306*b - 474) * q^79 + (75*b + 60) * q^80 + 81 * q^81 + (58*b + 616) * q^82 + (426*b - 70) * q^83 + (48*b - 96) * q^84 + (220*b - 150) * q^85 + (320*b + 496) * q^86 + (-102*b + 288) * q^87 + (-128*b + 186) * q^89 - 45*b * q^90 + (-176*b + 144) * q^91 + (148*b - 112) * q^92 + (-36*b + 108) * q^93 + (220*b + 864) * q^94 + (-110*b - 370) * q^95 + (183*b - 276) * q^96 + (428*b - 298) * q^97 + (-279*b - 64) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 5 q^{10} - 21 q^{12} + 90 q^{13} - 32 q^{14} - 30 q^{15} - 39 q^{16} + 16 q^{17} + 9 q^{18} + 170 q^{19} + 35 q^{20} + 12 q^{21} - 124 q^{23} - 9 q^{24} + 50 q^{25} + 62 q^{26} + 54 q^{27} - 48 q^{28} + 158 q^{29} - 15 q^{30} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 63 q^{36} - 372 q^{37} + 272 q^{38} + 270 q^{39} + 15 q^{40} - 38 q^{41} - 96 q^{42} + 516 q^{43} - 90 q^{45} - 572 q^{46} + 224 q^{47} - 117 q^{48} - 542 q^{49} + 25 q^{50} + 48 q^{51} - 298 q^{52} + 472 q^{53} + 27 q^{54} + 368 q^{56} + 510 q^{57} - 210 q^{58} + 248 q^{59} + 105 q^{60} - 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} - 372 q^{69} + 160 q^{70} + 2060 q^{71} - 27 q^{72} + 486 q^{73} - 1138 q^{74} + 150 q^{75} - 408 q^{76} + 186 q^{78} - 642 q^{79} + 195 q^{80} + 162 q^{81} + 1290 q^{82} + 286 q^{83} - 144 q^{84} - 80 q^{85} + 1312 q^{86} + 474 q^{87} + 244 q^{89} - 45 q^{90} + 112 q^{91} - 76 q^{92} + 180 q^{93} + 1948 q^{94} - 850 q^{95} - 369 q^{96} - 168 q^{97} - 407 q^{98}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 - 5 * q^10 - 21 * q^12 + 90 * q^13 - 32 * q^14 - 30 * q^15 - 39 * q^16 + 16 * q^17 + 9 * q^18 + 170 * q^19 + 35 * q^20 + 12 * q^21 - 124 * q^23 - 9 * q^24 + 50 * q^25 + 62 * q^26 + 54 * q^27 - 48 * q^28 + 158 * q^29 - 15 * q^30 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 63 * q^36 - 372 * q^37 + 272 * q^38 + 270 * q^39 + 15 * q^40 - 38 * q^41 - 96 * q^42 + 516 * q^43 - 90 * q^45 - 572 * q^46 + 224 * q^47 - 117 * q^48 - 542 * q^49 + 25 * q^50 + 48 * q^51 - 298 * q^52 + 472 * q^53 + 27 * q^54 + 368 * q^56 + 510 * q^57 - 210 * q^58 + 248 * q^59 + 105 * q^60 - 72 * q^61 - 72 * q^62 + 36 * q^63 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 - 372 * q^69 + 160 * q^70 + 2060 * q^71 - 27 * q^72 + 486 * q^73 - 1138 * q^74 + 150 * q^75 - 408 * q^76 + 186 * q^78 - 642 * q^79 + 195 * q^80 + 162 * q^81 + 1290 * q^82 + 286 * q^83 - 144 * q^84 - 80 * q^85 + 1312 * q^86 + 474 * q^87 + 244 * q^89 - 45 * q^90 + 112 * q^91 - 76 * q^92 + 180 * q^93 + 1948 * q^94 - 850 * q^95 - 369 * q^96 - 168 * q^97 - 407 * 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.

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
 −1.56155 2.56155
−1.56155 3.00000 −5.56155 −5.00000 −4.68466 10.2462 21.1771 9.00000 7.80776
1.2 2.56155 3.00000 −1.43845 −5.00000 7.68466 −6.24621 −24.1771 9.00000 −12.8078
 $$n$$: 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.n 2
11.b odd 2 1 165.4.a.c 2
33.d even 2 1 495.4.a.d 2
55.d odd 2 1 825.4.a.m 2
55.e even 4 2 825.4.c.j 4
165.d even 2 1 2475.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 11.b odd 2 1
495.4.a.d 2 33.d even 2 1
825.4.a.m 2 55.d odd 2 1
825.4.c.j 4 55.e even 4 2
1815.4.a.n 2 1.a even 1 1 trivial
2475.4.a.n 2 165.d even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{7}^{2} - 4T_{7} - 64$$ T7^2 - 4*T7 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} - 4T - 64$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 90T + 2008$$
$17$ $$T^{2} - 16T - 8164$$
$19$ $$T^{2} - 170T + 5168$$
$23$ $$T^{2} + 124T - 11456$$
$29$ $$T^{2} - 158T + 1328$$
$31$ $$T^{2} - 60T + 288$$
$37$ $$T^{2} + 372T - 18716$$
$41$ $$T^{2} + 38T - 100432$$
$43$ $$T^{2} - 516T + 1216$$
$47$ $$T^{2} - 224T - 185744$$
$53$ $$T^{2} - 472T - 107572$$
$59$ $$T^{2} - 248T - 229424$$
$61$ $$T^{2} + 72T - 561812$$
$67$ $$T^{2} + 744T + 137296$$
$71$ $$T^{2} - 2060 T + 1052672$$
$73$ $$T^{2} - 486T + 44752$$
$79$ $$T^{2} + 642T - 294912$$
$83$ $$T^{2} - 286T - 750824$$
$89$ $$T^{2} - 244T - 54748$$
$97$ $$T^{2} + 168T - 771476$$