# Properties

 Label 1815.4.a.r 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: 3.3.47528.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 26x - 22$$ x^3 - x^2 - 26*x - 22 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} + 10) q^{4} - 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{7} + (2 \beta_{2} - 9 \beta_1 - 3) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 + 10) * q^4 - 5 * q^5 + (-3*b1 + 3) * q^6 + (-2*b2 + 2*b1 - 4) * q^7 + (2*b2 - 9*b1 - 3) * q^8 + 9 * q^9 $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} + 10) q^{4} - 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{7} + (2 \beta_{2} - 9 \beta_1 - 3) q^{8} + 9 q^{9} + (5 \beta_1 - 5) q^{10} + (3 \beta_{2} + 30) q^{12} + (2 \beta_{2} - 38) q^{13} + ( - 6 \beta_{2} + 16 \beta_1 - 28) q^{14} - 15 q^{15} + (5 \beta_{2} - 2 \beta_1 + 60) q^{16} + (2 \beta_{2} + 2 \beta_1 + 34) q^{17} + ( - 9 \beta_1 + 9) q^{18} + (8 \beta_{2} - 14 \beta_1 + 24) q^{19} + ( - 5 \beta_{2} - 50) q^{20} + ( - 6 \beta_{2} + 6 \beta_1 - 12) q^{21} + (4 \beta_{2} - 24 \beta_1 + 48) q^{23} + (6 \beta_{2} - 27 \beta_1 - 9) q^{24} + 25 q^{25} + (4 \beta_{2} + 24 \beta_1 - 48) q^{26} + 27 q^{27} + ( - 12 \beta_{2} + 38 \beta_1 - 238) q^{28} + (34 \beta_1 + 62) q^{29} + (15 \beta_1 - 15) q^{30} + ( - 8 \beta_{2} - 40 \beta_1 + 96) q^{31} + ( - 4 \beta_{2} - 21 \beta_1 + 93) q^{32} + (2 \beta_{2} - 50 \beta_1 - 10) q^{34} + (10 \beta_{2} - 10 \beta_1 + 20) q^{35} + (9 \beta_{2} + 90) q^{36} + ( - 20 \beta_1 + 286) q^{37} + (30 \beta_{2} - 66 \beta_1 + 222) q^{38} + (6 \beta_{2} - 114) q^{39} + ( - 10 \beta_{2} + 45 \beta_1 + 15) q^{40} + ( - 66 \beta_1 - 30) q^{41} + ( - 18 \beta_{2} + 48 \beta_1 - 84) q^{42} + (14 \beta_{2} + 22 \beta_1 - 48) q^{43} - 45 q^{45} + (32 \beta_{2} - 52 \beta_1 + 436) q^{46} + ( - 4 \beta_{2} + 168) q^{47} + (15 \beta_{2} - 6 \beta_1 + 180) q^{48} + ( - 72 \beta_1 + 117) q^{49} + ( - 25 \beta_1 + 25) q^{50} + (6 \beta_{2} + 6 \beta_1 + 102) q^{51} + ( - 32 \beta_{2} - 4 \beta_1 - 172) q^{52} + (16 \beta_{2} - 96 \beta_1 + 126) q^{53} + ( - 27 \beta_1 + 27) q^{54} + ( - 14 \beta_{2} + 156 \beta_1 - 600) q^{56} + (24 \beta_{2} - 42 \beta_1 + 72) q^{57} + ( - 34 \beta_{2} - 96 \beta_1 - 516) q^{58} + (8 \beta_{2} + 32 \beta_1 + 172) q^{59} + ( - 15 \beta_{2} - 150) q^{60} + ( - 68 \beta_{2} - 12 \beta_1 - 134) q^{61} + (24 \beta_{2} + 816) q^{62} + ( - 18 \beta_{2} + 18 \beta_1 - 36) q^{63} + ( - 27 \beta_{2} - 28 \beta_1 - 10) q^{64} + ( - 10 \beta_{2} + 190) q^{65} + ( - 16 \beta_{2} + 100 \beta_1 - 176) q^{67} + (38 \beta_{2} + 30 \beta_1 + 558) q^{68} + (12 \beta_{2} - 72 \beta_1 + 144) q^{69} + (30 \beta_{2} - 80 \beta_1 + 140) q^{70} + ( - 28 \beta_{2} + 60 \beta_1 - 324) q^{71} + (18 \beta_{2} - 81 \beta_1 - 27) q^{72} + (38 \beta_{2} - 36 \beta_1 - 194) q^{73} + (20 \beta_{2} - 266 \beta_1 + 626) q^{74} + 75 q^{75} + (62 \beta_{2} - 254 \beta_1 + 1002) q^{76} + (12 \beta_{2} + 72 \beta_1 - 144) q^{78} + ( - 8 \beta_{2} - 94 \beta_1 + 212) q^{79} + ( - 25 \beta_{2} + 10 \beta_1 - 300) q^{80} + 81 q^{81} + (66 \beta_{2} + 96 \beta_1 + 1092) q^{82} + ( - 54 \beta_{2} + 180 \beta_1 - 60) q^{83} + ( - 36 \beta_{2} + 114 \beta_1 - 714) q^{84} + ( - 10 \beta_{2} - 10 \beta_1 - 170) q^{85} + (6 \beta_{2} - 72 \beta_1 - 492) q^{86} + (102 \beta_1 + 186) q^{87} + (120 \beta_{2} + 28 \beta_1 + 254) q^{89} + (45 \beta_1 - 45) q^{90} + (92 \beta_{2} - 40 \beta_1 - 244) q^{91} + (84 \beta_{2} - 416 \beta_1 + 776) q^{92} + ( - 24 \beta_{2} - 120 \beta_1 + 288) q^{93} + ( - 8 \beta_{2} - 140 \beta_1 + 188) q^{94} + ( - 40 \beta_{2} + 70 \beta_1 - 120) q^{95} + ( - 12 \beta_{2} - 63 \beta_1 + 279) q^{96} + ( - 44 \beta_{2} + 100 \beta_1 + 658) q^{97} + (72 \beta_{2} - 45 \beta_1 + 1341) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 + 10) * q^4 - 5 * q^5 + (-3*b1 + 3) * q^6 + (-2*b2 + 2*b1 - 4) * q^7 + (2*b2 - 9*b1 - 3) * q^8 + 9 * q^9 + (5*b1 - 5) * q^10 + (3*b2 + 30) * q^12 + (2*b2 - 38) * q^13 + (-6*b2 + 16*b1 - 28) * q^14 - 15 * q^15 + (5*b2 - 2*b1 + 60) * q^16 + (2*b2 + 2*b1 + 34) * q^17 + (-9*b1 + 9) * q^18 + (8*b2 - 14*b1 + 24) * q^19 + (-5*b2 - 50) * q^20 + (-6*b2 + 6*b1 - 12) * q^21 + (4*b2 - 24*b1 + 48) * q^23 + (6*b2 - 27*b1 - 9) * q^24 + 25 * q^25 + (4*b2 + 24*b1 - 48) * q^26 + 27 * q^27 + (-12*b2 + 38*b1 - 238) * q^28 + (34*b1 + 62) * q^29 + (15*b1 - 15) * q^30 + (-8*b2 - 40*b1 + 96) * q^31 + (-4*b2 - 21*b1 + 93) * q^32 + (2*b2 - 50*b1 - 10) * q^34 + (10*b2 - 10*b1 + 20) * q^35 + (9*b2 + 90) * q^36 + (-20*b1 + 286) * q^37 + (30*b2 - 66*b1 + 222) * q^38 + (6*b2 - 114) * q^39 + (-10*b2 + 45*b1 + 15) * q^40 + (-66*b1 - 30) * q^41 + (-18*b2 + 48*b1 - 84) * q^42 + (14*b2 + 22*b1 - 48) * q^43 - 45 * q^45 + (32*b2 - 52*b1 + 436) * q^46 + (-4*b2 + 168) * q^47 + (15*b2 - 6*b1 + 180) * q^48 + (-72*b1 + 117) * q^49 + (-25*b1 + 25) * q^50 + (6*b2 + 6*b1 + 102) * q^51 + (-32*b2 - 4*b1 - 172) * q^52 + (16*b2 - 96*b1 + 126) * q^53 + (-27*b1 + 27) * q^54 + (-14*b2 + 156*b1 - 600) * q^56 + (24*b2 - 42*b1 + 72) * q^57 + (-34*b2 - 96*b1 - 516) * q^58 + (8*b2 + 32*b1 + 172) * q^59 + (-15*b2 - 150) * q^60 + (-68*b2 - 12*b1 - 134) * q^61 + (24*b2 + 816) * q^62 + (-18*b2 + 18*b1 - 36) * q^63 + (-27*b2 - 28*b1 - 10) * q^64 + (-10*b2 + 190) * q^65 + (-16*b2 + 100*b1 - 176) * q^67 + (38*b2 + 30*b1 + 558) * q^68 + (12*b2 - 72*b1 + 144) * q^69 + (30*b2 - 80*b1 + 140) * q^70 + (-28*b2 + 60*b1 - 324) * q^71 + (18*b2 - 81*b1 - 27) * q^72 + (38*b2 - 36*b1 - 194) * q^73 + (20*b2 - 266*b1 + 626) * q^74 + 75 * q^75 + (62*b2 - 254*b1 + 1002) * q^76 + (12*b2 + 72*b1 - 144) * q^78 + (-8*b2 - 94*b1 + 212) * q^79 + (-25*b2 + 10*b1 - 300) * q^80 + 81 * q^81 + (66*b2 + 96*b1 + 1092) * q^82 + (-54*b2 + 180*b1 - 60) * q^83 + (-36*b2 + 114*b1 - 714) * q^84 + (-10*b2 - 10*b1 - 170) * q^85 + (6*b2 - 72*b1 - 492) * q^86 + (102*b1 + 186) * q^87 + (120*b2 + 28*b1 + 254) * q^89 + (45*b1 - 45) * q^90 + (92*b2 - 40*b1 - 244) * q^91 + (84*b2 - 416*b1 + 776) * q^92 + (-24*b2 - 120*b1 + 288) * q^93 + (-8*b2 - 140*b1 + 188) * q^94 + (-40*b2 + 70*b1 - 120) * q^95 + (-12*b2 - 63*b1 + 279) * q^96 + (-44*b2 + 100*b1 + 658) * q^97 + (72*b2 - 45*b1 + 1341) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 + 6 * q^6 - 10 * q^7 - 18 * q^8 + 27 * q^9 $$3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} - 10 q^{10} + 90 q^{12} - 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} + 104 q^{17} + 18 q^{18} + 58 q^{19} - 150 q^{20} - 30 q^{21} + 120 q^{23} - 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} - 676 q^{28} + 220 q^{29} - 30 q^{30} + 248 q^{31} + 258 q^{32} - 80 q^{34} + 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} - 342 q^{39} + 90 q^{40} - 156 q^{41} - 204 q^{42} - 122 q^{43} - 135 q^{45} + 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} + 50 q^{50} + 312 q^{51} - 520 q^{52} + 282 q^{53} + 54 q^{54} - 1644 q^{56} + 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} - 414 q^{61} + 2448 q^{62} - 90 q^{63} - 58 q^{64} + 570 q^{65} - 428 q^{67} + 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} - 162 q^{72} - 618 q^{73} + 1612 q^{74} + 225 q^{75} + 2752 q^{76} - 360 q^{78} + 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} - 2028 q^{84} - 520 q^{85} - 1548 q^{86} + 660 q^{87} + 790 q^{89} - 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} + 424 q^{94} - 290 q^{95} + 774 q^{96} + 2074 q^{97} + 3978 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 9 * q^3 + 30 * q^4 - 15 * q^5 + 6 * q^6 - 10 * q^7 - 18 * q^8 + 27 * q^9 - 10 * q^10 + 90 * q^12 - 114 * q^13 - 68 * q^14 - 45 * q^15 + 178 * q^16 + 104 * q^17 + 18 * q^18 + 58 * q^19 - 150 * q^20 - 30 * q^21 + 120 * q^23 - 54 * q^24 + 75 * q^25 - 120 * q^26 + 81 * q^27 - 676 * q^28 + 220 * q^29 - 30 * q^30 + 248 * q^31 + 258 * q^32 - 80 * q^34 + 50 * q^35 + 270 * q^36 + 838 * q^37 + 600 * q^38 - 342 * q^39 + 90 * q^40 - 156 * q^41 - 204 * q^42 - 122 * q^43 - 135 * q^45 + 1256 * q^46 + 504 * q^47 + 534 * q^48 + 279 * q^49 + 50 * q^50 + 312 * q^51 - 520 * q^52 + 282 * q^53 + 54 * q^54 - 1644 * q^56 + 174 * q^57 - 1644 * q^58 + 548 * q^59 - 450 * q^60 - 414 * q^61 + 2448 * q^62 - 90 * q^63 - 58 * q^64 + 570 * q^65 - 428 * q^67 + 1704 * q^68 + 360 * q^69 + 340 * q^70 - 912 * q^71 - 162 * q^72 - 618 * q^73 + 1612 * q^74 + 225 * q^75 + 2752 * q^76 - 360 * q^78 + 542 * q^79 - 890 * q^80 + 243 * q^81 + 3372 * q^82 - 2028 * q^84 - 520 * q^85 - 1548 * q^86 + 660 * q^87 + 790 * q^89 - 90 * q^90 - 772 * q^91 + 1912 * q^92 + 744 * q^93 + 424 * q^94 - 290 * q^95 + 774 * q^96 + 2074 * q^97 + 3978 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 26x - 22$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 17$$ v^2 - 2*v - 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 17$$ b2 + 2*b1 + 17

## 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
 5.97123 −0.906392 −4.06484
−4.97123 3.00000 16.7131 −5.00000 −14.9137 −5.48376 −43.3148 9.00000 24.8561
1.2 1.90639 3.00000 −4.36567 −5.00000 5.71918 22.9186 −23.5738 9.00000 −9.53196
1.3 5.06484 3.00000 17.6526 −5.00000 15.1945 −27.4348 48.8887 9.00000 −25.3242
 $$n$$: 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.r 3
11.b odd 2 1 165.4.a.e 3
33.d even 2 1 495.4.a.k 3
55.d odd 2 1 825.4.a.r 3
55.e even 4 2 825.4.c.k 6
165.d even 2 1 2475.4.a.t 3

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

 $$T_{2}^{3} - 2T_{2}^{2} - 25T_{2} + 48$$ T2^3 - 2*T2^2 - 25*T2 + 48 $$T_{7}^{3} + 10T_{7}^{2} - 604T_{7} - 3448$$ T7^3 + 10*T7^2 - 604*T7 - 3448

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} + \cdots + 48$$
$3$ $$(T - 3)^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} + 10 T^{2} + \cdots - 3448$$
$11$ $$T^{3}$$
$13$ $$T^{3} + 114 T^{2} + \cdots + 37216$$
$17$ $$T^{3} - 104 T^{2} + \cdots - 8448$$
$19$ $$T^{3} - 58 T^{2} + \cdots - 65520$$
$23$ $$T^{3} - 120 T^{2} + \cdots + 148224$$
$29$ $$T^{3} - 220 T^{2} + \cdots + 629760$$
$31$ $$T^{3} - 248 T^{2} + \cdots + 9589248$$
$37$ $$T^{3} - 838 T^{2} + \cdots - 18607336$$
$41$ $$T^{3} + 156 T^{2} + \cdots + 3013632$$
$43$ $$T^{3} + 122 T^{2} + \cdots - 1445400$$
$47$ $$T^{3} - 504 T^{2} + \cdots - 4372224$$
$53$ $$T^{3} - 282 T^{2} + \cdots - 3654264$$
$59$ $$T^{3} - 548 T^{2} + \cdots - 1206720$$
$61$ $$T^{3} + 414 T^{2} + \cdots - 342344792$$
$67$ $$T^{3} + 428 T^{2} + \cdots - 8135552$$
$71$ $$T^{3} + 912 T^{2} + \cdots - 2867712$$
$73$ $$T^{3} + 618 T^{2} + \cdots - 26458592$$
$79$ $$T^{3} - 542 T^{2} + \cdots + 88503440$$
$83$ $$T^{3} - 1091340 T + 434328048$$
$89$ $$T^{3} + \cdots + 1941629400$$
$97$ $$T^{3} - 2074 T^{2} + \cdots + 98075336$$