# Properties

 Label 1815.4.a.q Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1957.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 10$$ x^3 - x^2 - 9*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} + (2 \beta_{2} + \beta_1 + 6) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + ( - \beta_{2} + 3 \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 + (2*b2 + b1 + 6) * q^4 + 5 * q^5 - 3*b1 * q^6 + (-b2 + 3*b1 - 1) * q^7 + (-2*b2 + 3*b1 + 2) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} + (2 \beta_{2} + \beta_1 + 6) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + ( - \beta_{2} + 3 \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{8} + 9 q^{9} + 5 \beta_1 q^{10} + ( - 6 \beta_{2} - 3 \beta_1 - 18) q^{12} + (11 \beta_{2} + 13 \beta_1 + 11) q^{13} + (8 \beta_{2} + 48) q^{14} - 15 q^{15} + ( - 6 \beta_{2} - 7 \beta_1 + 6) q^{16} + ( - 7 \beta_{2} + 5 \beta_1 - 9) q^{17} + 9 \beta_1 q^{18} + (2 \beta_{2} + 8 \beta_1 - 36) q^{19} + (10 \beta_{2} + 5 \beta_1 + 30) q^{20} + (3 \beta_{2} - 9 \beta_1 + 3) q^{21} + ( - 16 \beta_{2} + 80) q^{23} + (6 \beta_{2} - 9 \beta_1 - 6) q^{24} + 25 q^{25} + (4 \beta_{2} + 46 \beta_1 + 116) q^{26} - 27 q^{27} + ( - 8 \beta_{2} + 40 \beta_1 - 40) q^{28} + ( - 26 \beta_1 - 88) q^{29} - 15 \beta_1 q^{30} + ( - 22 \beta_{2} + 34 \beta_1 + 42) q^{31} + (14 \beta_{2} - 37 \beta_1 - 78) q^{32} + (24 \beta_{2} - 18 \beta_1 + 112) q^{34} + ( - 5 \beta_{2} + 15 \beta_1 - 5) q^{35} + (18 \beta_{2} + 9 \beta_1 + 54) q^{36} + (42 \beta_{2} + 66 \beta_1 - 8) q^{37} + (12 \beta_{2} - 24 \beta_1 + 100) q^{38} + ( - 33 \beta_{2} - 39 \beta_1 - 33) q^{39} + ( - 10 \beta_{2} + 15 \beta_1 + 10) q^{40} + ( - 22 \beta_1 + 8) q^{41} + ( - 24 \beta_{2} - 144) q^{42} + (27 \beta_{2} + 19 \beta_1 + 51) q^{43} + 45 q^{45} + (32 \beta_{2} + 48 \beta_1 + 96) q^{46} + ( - 34 \beta_{2} + 58 \beta_1 - 54) q^{47} + (18 \beta_{2} + 21 \beta_1 - 18) q^{48} + (30 \beta_{2} - 14 \beta_1 - 157) q^{49} + 25 \beta_1 q^{50} + (21 \beta_{2} - 15 \beta_1 + 27) q^{51} + ( - 4 \beta_{2} + 66 \beta_1 + 532) q^{52} + ( - 56 \beta_{2} - 12 \beta_1 - 270) q^{53} - 27 \beta_1 q^{54} + (32 \beta_{2} - 16 \beta_1 + 224) q^{56} + ( - 6 \beta_{2} - 24 \beta_1 + 108) q^{57} + ( - 52 \beta_{2} - 114 \beta_1 - 364) q^{58} + ( - 8 \beta_{2} + 60 \beta_1 + 432) q^{59} + ( - 30 \beta_{2} - 15 \beta_1 - 90) q^{60} + ( - 22 \beta_{2} - 78 \beta_1 - 140) q^{61} + (112 \beta_{2} + 32 \beta_1 + 608) q^{62} + ( - 9 \beta_{2} + 27 \beta_1 - 9) q^{63} + ( - 54 \beta_{2} - 31 \beta_1 - 650) q^{64} + (55 \beta_{2} + 65 \beta_1 + 55) q^{65} + (104 \beta_{2} + 88 \beta_1 + 284) q^{67} + ( - 28 \beta_{2} + 102 \beta_1 - 324) q^{68} + (48 \beta_{2} - 240) q^{69} + (40 \beta_{2} + 240) q^{70} + (28 \beta_{2} - 16 \beta_1 + 600) q^{71} + ( - 18 \beta_{2} + 27 \beta_1 + 18) q^{72} + ( - 23 \beta_{2} + 43 \beta_1 - 19) q^{73} + (48 \beta_{2} + 142 \beta_1 + 672) q^{74} - 75 q^{75} + ( - 88 \beta_{2} + 36 \beta_1 - 120) q^{76} + ( - 12 \beta_{2} - 138 \beta_1 - 348) q^{78} + (154 \beta_{2} + 24 \beta_1 + 40) q^{79} + ( - 30 \beta_{2} - 35 \beta_1 + 30) q^{80} + 81 q^{81} + ( - 44 \beta_{2} - 14 \beta_1 - 308) q^{82} + ( - 195 \beta_{2} - 9 \beta_1 - 289) q^{83} + (24 \beta_{2} - 120 \beta_1 + 120) q^{84} + ( - 35 \beta_{2} + 25 \beta_1 - 45) q^{85} + ( - 16 \beta_{2} + 124 \beta_1 + 104) q^{86} + (78 \beta_1 + 264) q^{87} + ( - 292 \beta_1 + 182) q^{89} + 45 \beta_1 q^{90} + (38 \beta_{2} + 154 \beta_1 + 162) q^{91} + (160 \beta_{2} + 208 \beta_1 - 160) q^{92} + (66 \beta_{2} - 102 \beta_1 - 126) q^{93} + (184 \beta_{2} - 64 \beta_1 + 1016) q^{94} + (10 \beta_{2} + 40 \beta_1 - 180) q^{95} + ( - 42 \beta_{2} + 111 \beta_1 + 234) q^{96} + (148 \beta_{2} + 200 \beta_1 - 374) q^{97} + ( - 88 \beta_{2} - 111 \beta_1 - 376) q^{98}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 + (2*b2 + b1 + 6) * q^4 + 5 * q^5 - 3*b1 * q^6 + (-b2 + 3*b1 - 1) * q^7 + (-2*b2 + 3*b1 + 2) * q^8 + 9 * q^9 + 5*b1 * q^10 + (-6*b2 - 3*b1 - 18) * q^12 + (11*b2 + 13*b1 + 11) * q^13 + (8*b2 + 48) * q^14 - 15 * q^15 + (-6*b2 - 7*b1 + 6) * q^16 + (-7*b2 + 5*b1 - 9) * q^17 + 9*b1 * q^18 + (2*b2 + 8*b1 - 36) * q^19 + (10*b2 + 5*b1 + 30) * q^20 + (3*b2 - 9*b1 + 3) * q^21 + (-16*b2 + 80) * q^23 + (6*b2 - 9*b1 - 6) * q^24 + 25 * q^25 + (4*b2 + 46*b1 + 116) * q^26 - 27 * q^27 + (-8*b2 + 40*b1 - 40) * q^28 + (-26*b1 - 88) * q^29 - 15*b1 * q^30 + (-22*b2 + 34*b1 + 42) * q^31 + (14*b2 - 37*b1 - 78) * q^32 + (24*b2 - 18*b1 + 112) * q^34 + (-5*b2 + 15*b1 - 5) * q^35 + (18*b2 + 9*b1 + 54) * q^36 + (42*b2 + 66*b1 - 8) * q^37 + (12*b2 - 24*b1 + 100) * q^38 + (-33*b2 - 39*b1 - 33) * q^39 + (-10*b2 + 15*b1 + 10) * q^40 + (-22*b1 + 8) * q^41 + (-24*b2 - 144) * q^42 + (27*b2 + 19*b1 + 51) * q^43 + 45 * q^45 + (32*b2 + 48*b1 + 96) * q^46 + (-34*b2 + 58*b1 - 54) * q^47 + (18*b2 + 21*b1 - 18) * q^48 + (30*b2 - 14*b1 - 157) * q^49 + 25*b1 * q^50 + (21*b2 - 15*b1 + 27) * q^51 + (-4*b2 + 66*b1 + 532) * q^52 + (-56*b2 - 12*b1 - 270) * q^53 - 27*b1 * q^54 + (32*b2 - 16*b1 + 224) * q^56 + (-6*b2 - 24*b1 + 108) * q^57 + (-52*b2 - 114*b1 - 364) * q^58 + (-8*b2 + 60*b1 + 432) * q^59 + (-30*b2 - 15*b1 - 90) * q^60 + (-22*b2 - 78*b1 - 140) * q^61 + (112*b2 + 32*b1 + 608) * q^62 + (-9*b2 + 27*b1 - 9) * q^63 + (-54*b2 - 31*b1 - 650) * q^64 + (55*b2 + 65*b1 + 55) * q^65 + (104*b2 + 88*b1 + 284) * q^67 + (-28*b2 + 102*b1 - 324) * q^68 + (48*b2 - 240) * q^69 + (40*b2 + 240) * q^70 + (28*b2 - 16*b1 + 600) * q^71 + (-18*b2 + 27*b1 + 18) * q^72 + (-23*b2 + 43*b1 - 19) * q^73 + (48*b2 + 142*b1 + 672) * q^74 - 75 * q^75 + (-88*b2 + 36*b1 - 120) * q^76 + (-12*b2 - 138*b1 - 348) * q^78 + (154*b2 + 24*b1 + 40) * q^79 + (-30*b2 - 35*b1 + 30) * q^80 + 81 * q^81 + (-44*b2 - 14*b1 - 308) * q^82 + (-195*b2 - 9*b1 - 289) * q^83 + (24*b2 - 120*b1 + 120) * q^84 + (-35*b2 + 25*b1 - 45) * q^85 + (-16*b2 + 124*b1 + 104) * q^86 + (78*b1 + 264) * q^87 + (-292*b1 + 182) * q^89 + 45*b1 * q^90 + (38*b2 + 154*b1 + 162) * q^91 + (160*b2 + 208*b1 - 160) * q^92 + (66*b2 - 102*b1 - 126) * q^93 + (184*b2 - 64*b1 + 1016) * q^94 + (10*b2 + 40*b1 - 180) * q^95 + (-42*b2 + 111*b1 + 234) * q^96 + (148*b2 + 200*b1 - 374) * q^97 + (-88*b2 - 111*b1 - 376) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 + 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 $$3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} - 5 q^{10} - 51 q^{12} + 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} - 32 q^{17} - 9 q^{18} - 116 q^{19} + 85 q^{20} + 18 q^{21} + 240 q^{23} - 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} - 160 q^{28} - 238 q^{29} + 15 q^{30} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} - 60 q^{39} + 15 q^{40} + 46 q^{41} - 432 q^{42} + 134 q^{43} + 135 q^{45} + 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} - 25 q^{50} + 96 q^{51} + 1530 q^{52} - 798 q^{53} + 27 q^{54} + 688 q^{56} + 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} - 342 q^{61} + 1792 q^{62} - 54 q^{63} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} + 27 q^{72} - 100 q^{73} + 1874 q^{74} - 225 q^{75} - 396 q^{76} - 906 q^{78} + 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} - 858 q^{83} + 480 q^{84} - 160 q^{85} + 188 q^{86} + 714 q^{87} + 838 q^{89} - 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} + 3112 q^{94} - 580 q^{95} + 591 q^{96} - 1322 q^{97} - 1017 q^{98}+O(q^{100})$$ 3 * q - q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 + 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 - 5 * q^10 - 51 * q^12 + 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 - 32 * q^17 - 9 * q^18 - 116 * q^19 + 85 * q^20 + 18 * q^21 + 240 * q^23 - 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 - 160 * q^28 - 238 * q^29 + 15 * q^30 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 - 60 * q^39 + 15 * q^40 + 46 * q^41 - 432 * q^42 + 134 * q^43 + 135 * q^45 + 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 - 25 * q^50 + 96 * q^51 + 1530 * q^52 - 798 * q^53 + 27 * q^54 + 688 * q^56 + 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 - 342 * q^61 + 1792 * q^62 - 54 * q^63 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 + 27 * q^72 - 100 * q^73 + 1874 * q^74 - 225 * q^75 - 396 * q^76 - 906 * q^78 + 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 - 858 * q^83 + 480 * q^84 - 160 * q^85 + 188 * q^86 + 714 * q^87 + 838 * q^89 - 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 + 3112 * q^94 - 580 * q^95 + 591 * q^96 - 1322 * q^97 - 1017 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 10$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + \nu + 6$$ -v^2 + v + 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{2} + \beta _1 + 13 ) / 2$$ (-b2 + b1 + 13) / 2

## 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
 1.12946 −3.04096 2.91150
−4.59486 −3.00000 13.1127 5.00000 13.7846 −20.6383 −23.4921 9.00000 −22.9743
1.2 −0.793499 −3.00000 −7.37036 5.00000 2.38050 2.90793 12.1964 9.00000 −3.96749
1.3 4.38835 −3.00000 11.2577 5.00000 −13.1651 11.7304 14.2958 9.00000 21.9418
 $$n$$: 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.q 3
11.b odd 2 1 165.4.a.g 3
33.d even 2 1 495.4.a.i 3
55.d odd 2 1 825.4.a.p 3
55.e even 4 2 825.4.c.m 6
165.d even 2 1 2475.4.a.z 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 11.b odd 2 1
495.4.a.i 3 33.d even 2 1
825.4.a.p 3 55.d odd 2 1
825.4.c.m 6 55.e even 4 2
1815.4.a.q 3 1.a even 1 1 trivial
2475.4.a.z 3 165.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{3} + T_{2}^{2} - 20T_{2} - 16$$ T2^3 + T2^2 - 20*T2 - 16 $$T_{7}^{3} + 6T_{7}^{2} - 268T_{7} + 704$$ T7^3 + 6*T7^2 - 268*T7 + 704

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} + \cdots - 16$$
$3$ $$(T + 3)^{3}$$
$5$ $$(T - 5)^{3}$$
$7$ $$T^{3} + 6 T^{2} + \cdots + 704$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 20 T^{2} + \cdots + 78104$$
$17$ $$T^{3} + 32 T^{2} + \cdots + 22424$$
$19$ $$T^{3} + 116 T^{2} + \cdots + 80$$
$23$ $$T^{3} - 240 T^{2} + \cdots + 180224$$
$29$ $$T^{3} + 238 T^{2} + \cdots - 428416$$
$31$ $$T^{3} - 92 T^{2} + \cdots + 6769664$$
$37$ $$T^{3} + 90 T^{2} + \cdots - 6364168$$
$41$ $$T^{3} - 46 T^{2} + \cdots + 245888$$
$43$ $$T^{3} - 134 T^{2} + \cdots + 2381360$$
$47$ $$T^{3} + 220 T^{2} + \cdots + 10980224$$
$53$ $$T^{3} + 798 T^{2} + \cdots - 17262968$$
$59$ $$T^{3} - 1236 T^{2} + \cdots - 32923904$$
$61$ $$T^{3} + 342 T^{2} + \cdots + 2655176$$
$67$ $$T^{3} - 764 T^{2} + \cdots + 153685184$$
$71$ $$T^{3} - 1816 T^{2} + \cdots - 198158720$$
$73$ $$T^{3} + 100 T^{2} + \cdots + 5132984$$
$79$ $$T^{3} - 96 T^{2} + \cdots + 167159872$$
$83$ $$T^{3} + 858 T^{2} + \cdots - 542136176$$
$89$ $$T^{3} - 838 T^{2} + \cdots + 693013592$$
$97$ $$T^{3} + 1322 T^{2} + \cdots - 354601256$$