# Properties

 Label 1815.4.a.t Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.1540841.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 27x^{2} - 18x + 92$$ x^4 - 27*x^2 - 18*x + 92 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + 5 q^{5} + (3 \beta_1 - 3) q^{6} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 8) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 13) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + 3 * q^3 + (b3 - b1 + 7) * q^4 + 5 * q^5 + (3*b1 - 3) * q^6 + (b3 - b2 - 2*b1 - 8) * q^7 + (-2*b3 + 2*b2 + 5*b1 - 13) * q^8 + 9 * q^9 $$q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + 5 q^{5} + (3 \beta_1 - 3) q^{6} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 8) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 13) q^{8}+ \cdots + (48 \beta_{3} - 40 \beta_{2} + \cdots + 71) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + 3 * q^3 + (b3 - b1 + 7) * q^4 + 5 * q^5 + (3*b1 - 3) * q^6 + (b3 - b2 - 2*b1 - 8) * q^7 + (-2*b3 + 2*b2 + 5*b1 - 13) * q^8 + 9 * q^9 + (5*b1 - 5) * q^10 + (3*b3 - 3*b1 + 21) * q^12 + (5*b3 - 5*b2 - 8*b1 + 2) * q^13 + (-6*b3 + 4*b2 - 4*b1 - 16) * q^14 + 15 * q^15 + (5*b3 - 8*b2 - 13*b1 + 19) * q^16 + (b3 + 7*b2 + 6*b1 - 18) * q^17 + (9*b1 - 9) * q^18 + (-4*b3 - 10*b1 - 36) * q^19 + (5*b3 - 5*b1 + 35) * q^20 + (3*b3 - 3*b2 - 6*b1 - 24) * q^21 + (-16*b3 - 8*b1 - 24) * q^23 + (-6*b3 + 6*b2 + 15*b1 - 39) * q^24 + 25 * q^25 + (-28*b3 + 20*b2 + 22*b1 - 94) * q^26 + 27 * q^27 + (6*b3 - 12*b2 - 28*b1 + 8) * q^28 + (-18*b3 - 2*b2 - 10*b1 - 22) * q^29 + (15*b1 - 15) * q^30 + (10*b3 - 10*b2 + 28*b1 + 128) * q^31 + (-26*b3 + 10*b2 - 7*b1 - 65) * q^32 + (26*b3 - 12*b2 + 2*b1 + 74) * q^34 + (5*b3 - 5*b2 - 10*b1 - 40) * q^35 + (9*b3 - 9*b1 + 63) * q^36 + (-18*b3 - 2*b2 + 52*b1 - 10) * q^37 + (-6*b3 - 8*b2 - 60*b1 - 104) * q^38 + (15*b3 - 15*b2 - 24*b1 + 6) * q^39 + (-10*b3 + 10*b2 + 25*b1 - 65) * q^40 + (-30*b3 + 26*b2 - 30*b1 - 82) * q^41 + (-18*b3 + 12*b2 - 12*b1 - 48) * q^42 + (-7*b3 + 19*b2 + 2*b1 - 140) * q^43 + 45 * q^45 + (8*b3 - 32*b2 - 120*b1 - 88) * q^46 + (10*b3 + 26*b2 + 32*b1 - 64) * q^47 + (15*b3 - 24*b2 - 39*b1 + 57) * q^48 + (-6*b3 + 14*b2 - 127) * q^49 + (25*b1 - 25) * q^50 + (3*b3 + 21*b2 + 18*b1 - 54) * q^51 + (70*b3 - 56*b2 - 158*b1 + 306) * q^52 + (8*b3 + 24*b2 - 68*b1 - 42) * q^53 + (27*b1 - 27) * q^54 + (-22*b3 + 4*b2 + 52*b1 - 224) * q^56 + (-12*b3 - 30*b1 - 108) * q^57 + (2*b3 - 32*b2 - 134*b1 - 110) * q^58 + (28*b3 - 48*b2 - 84*b1 - 244) * q^59 + (15*b3 - 15*b1 + 105) * q^60 + (46*b3 + 14*b2 - 52*b1 - 6) * q^61 + (-12*b3 + 40*b2 + 168*b1 + 304) * q^62 + (9*b3 - 9*b2 - 18*b1 - 72) * q^63 + (9*b3 - 8*b2 - 97*b1 - 225) * q^64 + (25*b3 - 25*b2 - 40*b1 + 10) * q^65 + (-4*b3 - 12*b2 + 32*b1 - 140) * q^67 + (-68*b3 + 20*b2 + 158*b1 + 146) * q^68 + (-48*b3 - 24*b1 - 72) * q^69 + (-30*b3 + 20*b2 - 20*b1 - 80) * q^70 + (-20*b3 + 72*b2 + 116*b1 - 240) * q^71 + (-18*b3 + 18*b2 + 45*b1 - 117) * q^72 + (-37*b3 - 47*b2 + 124*b1 - 250) * q^73 + (64*b3 - 32*b2 - 122*b1 + 746) * q^74 + 75 * q^75 + (-46*b3 + 4*b2 - 76*b1 - 416) * q^76 + (-84*b3 + 60*b2 + 66*b1 - 282) * q^78 + (-80*b3 + 24*b2 + 70*b1 - 328) * q^79 + (25*b3 - 40*b2 - 65*b1 + 95) * q^80 + 81 * q^81 + (78*b3 - 112*b2 - 210*b1 - 442) * q^82 + (-27*b3 + 35*b2 + 180*b1 + 20) * q^83 + (18*b3 - 36*b2 - 84*b1 + 24) * q^84 + (5*b3 + 35*b2 + 30*b1 - 90) * q^85 + (66*b3 - 52*b2 - 144*b1 + 92) * q^86 + (-54*b3 - 6*b2 - 30*b1 - 66) * q^87 + (24*b3 - 12*b2 + 332*b1 - 254) * q^89 + (45*b1 - 45) * q^90 + (2*b3 + 34*b2 - 96*b1 + 696) * q^91 + (-96*b3 + 80*b2 - 40*b1 - 1272) * q^92 + (30*b3 - 30*b2 + 84*b1 + 384) * q^93 + (100*b3 - 32*b2 + 48*b1 + 408) * q^94 + (-20*b3 - 50*b1 - 180) * q^95 + (-78*b3 + 30*b2 - 21*b1 - 195) * q^96 + (28*b3 + 24*b2 - 84*b1 - 398) * q^97 + (48*b3 - 40*b2 - 135*b1 + 71) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} - 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 12 * q^3 + 26 * q^4 + 20 * q^5 - 12 * q^6 - 34 * q^7 - 48 * q^8 + 36 * q^9 $$4 q - 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} - 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9} - 20 q^{10} + 78 q^{12} - 2 q^{13} - 52 q^{14} + 60 q^{15} + 66 q^{16} - 74 q^{17} - 36 q^{18} - 136 q^{19} + 130 q^{20} - 102 q^{21} - 64 q^{23} - 144 q^{24} + 100 q^{25} - 320 q^{26} + 108 q^{27} + 20 q^{28} - 52 q^{29} - 60 q^{30} + 492 q^{31} - 208 q^{32} + 244 q^{34} - 170 q^{35} + 234 q^{36} - 4 q^{37} - 404 q^{38} - 6 q^{39} - 240 q^{40} - 268 q^{41} - 156 q^{42} - 546 q^{43} + 180 q^{45} - 368 q^{46} - 276 q^{47} + 198 q^{48} - 496 q^{49} - 100 q^{50} - 222 q^{51} + 1084 q^{52} - 184 q^{53} - 108 q^{54} - 852 q^{56} - 408 q^{57} - 444 q^{58} - 1032 q^{59} + 390 q^{60} - 116 q^{61} + 1240 q^{62} - 306 q^{63} - 918 q^{64} - 10 q^{65} - 552 q^{67} + 720 q^{68} - 192 q^{69} - 260 q^{70} - 920 q^{71} - 432 q^{72} - 926 q^{73} + 2856 q^{74} + 300 q^{75} - 1572 q^{76} - 960 q^{78} - 1152 q^{79} + 330 q^{80} + 324 q^{81} - 1924 q^{82} + 134 q^{83} + 60 q^{84} - 370 q^{85} + 236 q^{86} - 156 q^{87} - 1064 q^{89} - 180 q^{90} + 2780 q^{91} - 4896 q^{92} + 1476 q^{93} + 1432 q^{94} - 680 q^{95} - 624 q^{96} - 1648 q^{97} + 188 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + 12 * q^3 + 26 * q^4 + 20 * q^5 - 12 * q^6 - 34 * q^7 - 48 * q^8 + 36 * q^9 - 20 * q^10 + 78 * q^12 - 2 * q^13 - 52 * q^14 + 60 * q^15 + 66 * q^16 - 74 * q^17 - 36 * q^18 - 136 * q^19 + 130 * q^20 - 102 * q^21 - 64 * q^23 - 144 * q^24 + 100 * q^25 - 320 * q^26 + 108 * q^27 + 20 * q^28 - 52 * q^29 - 60 * q^30 + 492 * q^31 - 208 * q^32 + 244 * q^34 - 170 * q^35 + 234 * q^36 - 4 * q^37 - 404 * q^38 - 6 * q^39 - 240 * q^40 - 268 * q^41 - 156 * q^42 - 546 * q^43 + 180 * q^45 - 368 * q^46 - 276 * q^47 + 198 * q^48 - 496 * q^49 - 100 * q^50 - 222 * q^51 + 1084 * q^52 - 184 * q^53 - 108 * q^54 - 852 * q^56 - 408 * q^57 - 444 * q^58 - 1032 * q^59 + 390 * q^60 - 116 * q^61 + 1240 * q^62 - 306 * q^63 - 918 * q^64 - 10 * q^65 - 552 * q^67 + 720 * q^68 - 192 * q^69 - 260 * q^70 - 920 * q^71 - 432 * q^72 - 926 * q^73 + 2856 * q^74 + 300 * q^75 - 1572 * q^76 - 960 * q^78 - 1152 * q^79 + 330 * q^80 + 324 * q^81 - 1924 * q^82 + 134 * q^83 + 60 * q^84 - 370 * q^85 + 236 * q^86 - 156 * q^87 - 1064 * q^89 - 180 * q^90 + 2780 * q^91 - 4896 * q^92 + 1476 * q^93 + 1432 * q^94 - 680 * q^95 - 624 * q^96 - 1648 * q^97 + 188 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 27x^{2} - 18x + 92$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 20\nu ) / 2$$ (v^3 - v^2 - 20*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 14$$ v^2 - v - 14
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 14$$ b3 + b1 + 14 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 21\beta _1 + 14$$ b3 + 2*b2 + 21*b1 + 14

## 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
 −4.17080 −2.63835 1.60719 5.20196
−5.17080 3.00000 18.7372 5.00000 −15.5124 11.1745 −55.5199 9.00000 −25.8540
1.2 −3.63835 3.00000 5.23763 5.00000 −10.9151 −20.8444 10.0505 9.00000 −18.1918
1.3 0.607192 3.00000 −7.63132 5.00000 1.82158 −8.95080 −9.49121 9.00000 3.03596
1.4 4.20196 3.00000 9.65650 5.00000 12.6059 −15.3793 6.96057 9.00000 21.0098
 $$n$$: 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.t 4
11.b odd 2 1 165.4.a.h 4
33.d even 2 1 495.4.a.m 4
55.d odd 2 1 825.4.a.t 4
55.e even 4 2 825.4.c.p 8
165.d even 2 1 2475.4.a.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.h 4 11.b odd 2 1
495.4.a.m 4 33.d even 2 1
825.4.a.t 4 55.d odd 2 1
825.4.c.p 8 55.e even 4 2
1815.4.a.t 4 1.a even 1 1 trivial
2475.4.a.be 4 165.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{4} + 4T_{2}^{3} - 21T_{2}^{2} - 68T_{2} + 48$$ T2^4 + 4*T2^3 - 21*T2^2 - 68*T2 + 48 $$T_{7}^{4} + 34T_{7}^{3} + 140T_{7}^{2} - 4336T_{7} - 32064$$ T7^4 + 34*T7^3 + 140*T7^2 - 4336*T7 - 32064

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + \cdots + 48$$
$3$ $$(T - 3)^{4}$$
$5$ $$(T - 5)^{4}$$
$7$ $$T^{4} + 34 T^{3} + \cdots - 32064$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 2 T^{3} + \cdots + 68144$$
$17$ $$T^{4} + 74 T^{3} + \cdots + 23770800$$
$19$ $$T^{4} + 136 T^{3} + \cdots + 576$$
$23$ $$T^{4} + 64 T^{3} + \cdots + 247529472$$
$29$ $$T^{4} + 52 T^{3} + \cdots + 315474624$$
$31$ $$T^{4} - 492 T^{3} + \cdots - 903269376$$
$37$ $$T^{4} + \cdots + 1260009136$$
$41$ $$T^{4} + \cdots - 6228069696$$
$43$ $$T^{4} + \cdots - 1200551616$$
$47$ $$T^{4} + \cdots + 5628791808$$
$53$ $$T^{4} + \cdots + 13030473936$$
$59$ $$T^{4} + \cdots - 2612441088$$
$61$ $$T^{4} + \cdots - 8546777488$$
$67$ $$T^{4} + 552 T^{3} + \cdots - 909580544$$
$71$ $$T^{4} + \cdots + 291456592896$$
$73$ $$T^{4} + \cdots - 133750796272$$
$79$ $$T^{4} + \cdots + 48148922944$$
$83$ $$T^{4} + \cdots - 17388663552$$
$89$ $$T^{4} + \cdots + 479041129296$$
$97$ $$T^{4} + \cdots + 1607443600$$