# Properties

 Label 1815.4.a.w 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: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 28x^{2} + 18x + 132$$ x^4 - x^3 - 28*x^2 + 18*x + 132 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 6) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} + 11) q^{7} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 6) * q^4 - 5 * q^5 - 3*b1 * q^6 + (-b3 - b2 + 11) * q^7 + (2*b3 + b2 + 2*b1 + 4) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 6) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} + 11) q^{7} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + ( - 3 \beta_{2} - 18) q^{12} + (2 \beta_{3} + \beta_{2} + 8 \beta_1 - 18) q^{13} + ( - 2 \beta_{3} - 6 \beta_{2} + \cdots - 8) q^{14}+ \cdots + ( - 54 \beta_{3} - 126 \beta_{2} + \cdots + 288) q^{98}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 6) * q^4 - 5 * q^5 - 3*b1 * q^6 + (-b3 - b2 + 11) * q^7 + (2*b3 + b2 + 2*b1 + 4) * q^8 + 9 * q^9 - 5*b1 * q^10 + (-3*b2 - 18) * q^12 + (2*b3 + b2 + 8*b1 - 18) * q^13 + (-2*b3 - 6*b2 + 11*b1 - 8) * q^14 + 15 * q^15 + (2*b3 + 5*b2 - 8) * q^16 + (-b3 - 8*b2 - 14) * q^17 + 9*b1 * q^18 + (-3*b3 + 7*b2 - 4*b1 + 13) * q^19 + (-5*b2 - 30) * q^20 + (3*b3 + 3*b2 - 33) * q^21 + (-b3 + 7*b2 - 28*b1 + 10) * q^23 + (-6*b3 - 3*b2 - 6*b1 - 12) * q^24 + 25 * q^25 + (2*b3 + 19*b2 - 22*b1 + 124) * q^26 - 27 * q^27 + (-4*b3 + 3*b2 - 24*b1 + 34) * q^28 + (-12*b3 - 3*b2 - 4*b1 - 12) * q^29 + 15*b1 * q^30 + (12*b3 + 7*b2 + 8*b1 - 41) * q^31 + (-6*b3 + 7*b2 - 12*b1 - 4) * q^32 + (-16*b3 - 13*b2 - 42*b1 - 36) * q^34 + (5*b3 + 5*b2 - 55) * q^35 + (9*b2 + 54) * q^36 + (b3 - 21*b2 + 8*b1 + 33) * q^37 + (14*b3 - 12*b2 + 53*b1 - 40) * q^38 + (-6*b3 - 3*b2 - 24*b1 + 54) * q^39 + (-10*b3 - 5*b2 - 10*b1 - 20) * q^40 + (4*b3 + 30*b2 - 20*b1 + 34) * q^41 + (6*b3 + 18*b2 - 33*b1 + 24) * q^42 + (6*b3 - 26*b2 + 32*b1 - 56) * q^43 - 45 * q^45 + (14*b3 - 26*b2 + 42*b1 - 368) * q^46 + (-17*b3 - 31*b2 + 28*b1 - 88) * q^47 + (-6*b3 - 15*b2 + 24) * q^48 + (-27*b3 - 27*b2 + 36*b1 - 54) * q^49 + 25*b1 * q^50 + (3*b3 + 24*b2 + 42) * q^51 + (22*b3 - b2 + 128*b1 - 80) * q^52 + (17*b3 + 25*b2 - 28*b1 - 110) * q^53 - 27*b1 * q^54 + (22*b3 + 7*b2 - 26*b1 - 276) * q^56 + (9*b3 - 21*b2 + 12*b1 - 39) * q^57 + (-6*b3 - 67*b2 + 24*b1 - 116) * q^58 + (2*b3 - 4*b2 + 52*b1 + 132) * q^59 + (15*b2 + 90) * q^60 + (24*b3 - 19*b2 - 48*b1 - 165) * q^61 + (14*b3 + 75*b2 - 61*b1 + 188) * q^62 + (-9*b3 - 9*b2 + 99) * q^63 + (-2*b3 - 75*b2 + 48*b1 - 100) * q^64 + (-10*b3 - 5*b2 - 40*b1 + 90) * q^65 + (-25*b3 - 58*b2 - 24*b1 - 175) * q^67 + (-18*b3 - 71*b2 - 24*b1 - 592) * q^68 + (3*b3 - 21*b2 + 84*b1 - 30) * q^69 + (10*b3 + 30*b2 - 55*b1 + 40) * q^70 + (-8*b3 - 31*b2 + 112*b1 + 434) * q^71 + (18*b3 + 9*b2 + 18*b1 + 36) * q^72 + (9*b3 - 8*b2 - 48*b1 + 175) * q^73 + (-42*b3 - 8*b2 - 55*b1 + 32) * q^74 - 75 * q^75 + (55*b2 - 112*b1 + 646) * q^76 + (-6*b3 - 57*b2 + 66*b1 - 372) * q^78 + (-8*b3 + 41*b2 - 136*b1 - 341) * q^79 + (-10*b3 - 25*b2 + 40) * q^80 + 81 * q^81 + (60*b3 + 30*b2 + 138*b1 - 144) * q^82 + (26*b3 + 39*b2 - 208*b1 + 404) * q^83 + (12*b3 - 9*b2 + 72*b1 - 102) * q^84 + (5*b3 + 40*b2 + 70) * q^85 + (-52*b3 + 36*b2 - 184*b1 + 368) * q^86 + (36*b3 + 9*b2 + 12*b1 + 36) * q^87 + (94*b3 - 40*b1 - 146) * q^89 - 45*b1 * q^90 + (32*b3 - 7*b2 + 40*b1 - 566) * q^91 + (-44*b3 + 30*b2 - 304*b1 + 460) * q^92 + (-36*b3 - 21*b2 - 24*b1 + 123) * q^93 + (-62*b3 - 88*b2 - 144*b1 + 200) * q^94 + (15*b3 - 35*b2 + 20*b1 - 65) * q^95 + (18*b3 - 21*b2 + 36*b1 + 12) * q^96 + (-67*b3 - 68*b2 - 48*b1 - 525) * q^97 + (-54*b3 - 126*b2 - 54*b1 + 288) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 12 q^{3} + 25 q^{4} - 20 q^{5} - 3 q^{6} + 45 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10})$$ 4 * q + q^2 - 12 * q^3 + 25 * q^4 - 20 * q^5 - 3 * q^6 + 45 * q^7 + 15 * q^8 + 36 * q^9 $$4 q + q^{2} - 12 q^{3} + 25 q^{4} - 20 q^{5} - 3 q^{6} + 45 q^{7} + 15 q^{8} + 36 q^{9} - 5 q^{10} - 75 q^{12} - 67 q^{13} - 23 q^{14} + 60 q^{15} - 31 q^{16} - 62 q^{17} + 9 q^{18} + 61 q^{19} - 125 q^{20} - 135 q^{21} + 21 q^{23} - 45 q^{24} + 100 q^{25} + 489 q^{26} - 108 q^{27} + 123 q^{28} - 31 q^{29} + 15 q^{30} - 173 q^{31} - 9 q^{32} - 167 q^{34} - 225 q^{35} + 225 q^{36} + 117 q^{37} - 147 q^{38} + 201 q^{39} - 75 q^{40} + 138 q^{41} + 69 q^{42} - 230 q^{43} - 180 q^{45} - 1484 q^{46} - 321 q^{47} + 93 q^{48} - 153 q^{49} + 25 q^{50} + 186 q^{51} - 237 q^{52} - 477 q^{53} - 27 q^{54} - 1167 q^{56} - 183 q^{57} - 495 q^{58} + 572 q^{59} + 375 q^{60} - 775 q^{61} + 738 q^{62} + 405 q^{63} - 423 q^{64} + 335 q^{65} - 732 q^{67} - 2427 q^{68} - 63 q^{69} + 115 q^{70} + 1833 q^{71} + 135 q^{72} + 626 q^{73} + 149 q^{74} - 300 q^{75} + 2527 q^{76} - 1467 q^{78} - 1443 q^{79} + 155 q^{80} + 324 q^{81} - 528 q^{82} + 1395 q^{83} - 369 q^{84} + 310 q^{85} + 1428 q^{86} + 93 q^{87} - 812 q^{89} - 45 q^{90} - 2295 q^{91} + 1654 q^{92} + 519 q^{93} + 692 q^{94} - 305 q^{95} + 27 q^{96} - 2082 q^{97} + 1080 q^{98}+O(q^{100})$$ 4 * q + q^2 - 12 * q^3 + 25 * q^4 - 20 * q^5 - 3 * q^6 + 45 * q^7 + 15 * q^8 + 36 * q^9 - 5 * q^10 - 75 * q^12 - 67 * q^13 - 23 * q^14 + 60 * q^15 - 31 * q^16 - 62 * q^17 + 9 * q^18 + 61 * q^19 - 125 * q^20 - 135 * q^21 + 21 * q^23 - 45 * q^24 + 100 * q^25 + 489 * q^26 - 108 * q^27 + 123 * q^28 - 31 * q^29 + 15 * q^30 - 173 * q^31 - 9 * q^32 - 167 * q^34 - 225 * q^35 + 225 * q^36 + 117 * q^37 - 147 * q^38 + 201 * q^39 - 75 * q^40 + 138 * q^41 + 69 * q^42 - 230 * q^43 - 180 * q^45 - 1484 * q^46 - 321 * q^47 + 93 * q^48 - 153 * q^49 + 25 * q^50 + 186 * q^51 - 237 * q^52 - 477 * q^53 - 27 * q^54 - 1167 * q^56 - 183 * q^57 - 495 * q^58 + 572 * q^59 + 375 * q^60 - 775 * q^61 + 738 * q^62 + 405 * q^63 - 423 * q^64 + 335 * q^65 - 732 * q^67 - 2427 * q^68 - 63 * q^69 + 115 * q^70 + 1833 * q^71 + 135 * q^72 + 626 * q^73 + 149 * q^74 - 300 * q^75 + 2527 * q^76 - 1467 * q^78 - 1443 * q^79 + 155 * q^80 + 324 * q^81 - 528 * q^82 + 1395 * q^83 - 369 * q^84 + 310 * q^85 + 1428 * q^86 + 93 * q^87 - 812 * q^89 - 45 * q^90 - 2295 * q^91 + 1654 * q^92 + 519 * q^93 + 692 * q^94 - 305 * q^95 + 27 * q^96 - 2082 * q^97 + 1080 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 28x^{2} + 18x + 132$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 14$$ v^2 - 14 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 18\nu + 10 ) / 2$$ (v^3 - v^2 - 18*v + 10) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 14$$ b2 + 14 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + \beta_{2} + 18\beta _1 + 4$$ 2*b3 + b2 + 18*b1 + 4

## 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
 −4.58977 −2.09331 2.83228 4.85080
−4.58977 −3.00000 13.0660 −5.00000 13.7693 16.5031 −23.2516 9.00000 22.9488
1.2 −2.09331 −3.00000 −3.61805 −5.00000 6.27993 3.55562 24.3202 9.00000 10.4666
1.3 2.83228 −3.00000 0.0217934 −5.00000 −8.49683 30.1196 −22.5965 9.00000 −14.1614
1.4 4.85080 −3.00000 15.5303 −5.00000 −14.5524 −5.17830 36.5279 9.00000 −24.2540
 $$n$$: 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.w yes 4
11.b odd 2 1 1815.4.a.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.u 4 11.b odd 2 1
1815.4.a.w yes 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{3} - 28T_{2}^{2} + 18T_{2} + 132$$ T2^4 - T2^3 - 28*T2^2 + 18*T2 + 132 $$T_{7}^{4} - 45T_{7}^{3} + 403T_{7}^{2} + 1665T_{7} - 9152$$ T7^4 - 45*T7^3 + 403*T7^2 + 1665*T7 - 9152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + \cdots + 132$$
$3$ $$(T + 3)^{4}$$
$5$ $$(T + 5)^{4}$$
$7$ $$T^{4} - 45 T^{3} + \cdots - 9152$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 67 T^{3} + \cdots - 897688$$
$17$ $$T^{4} + 62 T^{3} + \cdots + 12101364$$
$19$ $$T^{4} - 61 T^{3} + \cdots + 349632$$
$23$ $$T^{4} - 21 T^{3} + \cdots - 20390568$$
$29$ $$T^{4} + 31 T^{3} + \cdots + 519690048$$
$31$ $$T^{4} + 173 T^{3} + \cdots + 448608816$$
$37$ $$T^{4} - 117 T^{3} + \cdots + 792356168$$
$41$ $$T^{4} + \cdots + 2636989344$$
$43$ $$T^{4} + \cdots + 1281282048$$
$47$ $$T^{4} + \cdots - 4433877888$$
$53$ $$T^{4} + 477 T^{3} + \cdots - 1950072$$
$59$ $$T^{4} + \cdots - 1525313856$$
$61$ $$T^{4} + \cdots - 42346834486$$
$67$ $$T^{4} + 732 T^{3} + \cdots + 457300144$$
$71$ $$T^{4} + \cdots - 46964823936$$
$73$ $$T^{4} - 626 T^{3} + \cdots + 266171072$$
$79$ $$T^{4} + \cdots - 227131261706$$
$83$ $$T^{4} + \cdots + 54520777728$$
$89$ $$T^{4} + \cdots + 750833692224$$
$97$ $$T^{4} + \cdots + 60305152688$$