# Properties

 Label 1840.4.a.n Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$108.563514411$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ x^5 - x^4 - 27*x^3 + 7*x^2 + 168*x + 92 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 115) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - 1) q^{3} - 5 q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 16) q^{9}+O(q^{10})$$ q + (-b4 - 1) * q^3 - 5 * q^5 + (b4 - 2*b1) * q^7 + (3*b4 - 3*b3 - 3*b2 + 16) * q^9 $$q + ( - \beta_{4} - 1) q^{3} - 5 q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 16) q^{9} + (\beta_{3} - 2 \beta_{2} - 6 \beta_1 - 7) q^{11} + (4 \beta_{4} - 3 \beta_{3} + \beta_{2} + \cdots + 26) q^{13}+ \cdots + ( - 153 \beta_{4} + 208 \beta_{3} + \cdots - 436) q^{99}+O(q^{100})$$ q + (-b4 - 1) * q^3 - 5 * q^5 + (b4 - 2*b1) * q^7 + (3*b4 - 3*b3 - 3*b2 + 16) * q^9 + (b3 - 2*b2 - 6*b1 - 7) * q^11 + (4*b4 - 3*b3 + b2 - 3*b1 + 26) * q^13 + (5*b4 + 5) * q^15 + (-3*b4 - 5*b3 - 9*b2 - 10*b1) * q^17 + (-4*b4 + 8*b3 + 5*b2 + 4*b1 + 33) * q^19 + (8*b4 - 3*b3 + 9*b2 - 4*b1 - 12) * q^21 + 23 * q^23 + 25 * q^25 + (-13*b4 + 6*b3 - 9*b2 - 118) * q^27 + (19*b4 - 13*b3 - 5*b2 - 15*b1 + 78) * q^29 + (-5*b4 - 11*b3 - 12*b2 + b1 - 7) * q^31 + (37*b4 - 26*b3 + 27*b2 - 3*b1 + 44) * q^33 + (-5*b4 + 10*b1) * q^35 + (-21*b4 + 2*b3 - 53*b2 - 8*b1 - 15) * q^37 + (-29*b4 + 12*b3 - b2 - 18*b1 - 80) * q^39 + (-23*b4 + 38*b3 - 19*b2 - 5*b1 - 9) * q^41 + (30*b4 + 2*b3 - 4*b2 - 28*b1 + 108) * q^43 + (-15*b4 + 15*b3 + 15*b2 - 80) * q^45 + (-24*b4 + 31*b3 + 8*b2 - b1 - 189) * q^47 + (-7*b4 - 3*b3 + 5*b2 - 20*b1 - 157) * q^49 + (18*b4 - 56*b3 - 5*b2 - 8*b1 + 199) * q^51 + (-23*b4 - 22*b3 + 25*b2 + 36*b1 + 241) * q^53 + (-5*b3 + 10*b2 + 30*b1 + 35) * q^55 + (-3*b4 - b3 + 27*b2 + 17*b1 + 29) * q^57 + (60*b4 - 78*b3 + 9*b2 + 37*b1 - 238) * q^59 + (-65*b4 - 7*b3 - 60*b2 + 29*b1 - 336) * q^61 + (-5*b4 + 45*b3 + 6*b2 + 10*b1 - 51) * q^63 + (-20*b4 + 15*b3 - 5*b2 + 15*b1 - 130) * q^65 + (4*b4 - 16*b3 + 7*b2 + 7*b1 - 108) * q^67 + (-23*b4 - 23) * q^69 + (-106*b4 + 52*b3 - 77*b2 - 6*b1 - 2) * q^71 + (-79*b4 + 43*b3 + 84*b2 + 28*b1 + 230) * q^73 + (-25*b4 - 25) * q^75 + (-3*b4 - 3*b3 + 25*b2 - 45*b1 + 503) * q^77 + (-36*b4 + 8*b3 + 92*b2 - 24*b1 - 148) * q^79 + (69*b4 + 3*b3 + 93*b2 + 45*b1 - 32) * q^81 + (-2*b4 + 58*b3 + 9*b2 + 85*b1 + 50) * q^83 + (15*b4 + 25*b3 + 45*b2 + 50*b1) * q^85 + (-103*b4 + 23*b3 + 16*b2 - 54*b1 - 520) * q^87 + (-71*b4 - 56*b3 + 56*b2 - 7*b1 + 97) * q^89 + (-b4 + 3*b3 + 26*b2 - 79*b1 + 286) * q^91 + (-56*b4 - 26*b3 - 83*b2 + 5*b1 + 173) * q^93 + (20*b4 - 40*b3 - 25*b2 - 20*b1 - 165) * q^95 + (-12*b4 + 117*b3 + 26*b2 - 68*b1 + 97) * q^97 + (-153*b4 + 208*b3 - 35*b2 - 3*b1 - 436) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9}+O(q^{10})$$ 5 * q - 4 * q^3 - 25 * q^5 + 3 * q^7 + 77 * q^9 $$5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35} - 38 q^{37} - 335 q^{39} - 12 q^{41} + 566 q^{43} - 385 q^{45} - 919 q^{47} - 738 q^{49} + 993 q^{51} + 1156 q^{53} + 115 q^{55} + 114 q^{57} - 1324 q^{59} - 1673 q^{61} - 270 q^{63} - 660 q^{65} - 558 q^{67} - 92 q^{69} + 108 q^{71} + 1173 q^{73} - 100 q^{75} + 2608 q^{77} - 656 q^{79} - 319 q^{81} + 82 q^{83} - 115 q^{85} - 2389 q^{87} + 570 q^{89} + 1589 q^{91} + 911 q^{93} - 805 q^{95} + 633 q^{97} - 2021 q^{99}+O(q^{100})$$ 5 * q - 4 * q^3 - 25 * q^5 + 3 * q^7 + 77 * q^9 - 23 * q^11 + 132 * q^13 + 20 * q^15 + 23 * q^17 + 161 * q^19 - 60 * q^21 + 115 * q^23 + 125 * q^25 - 577 * q^27 + 401 * q^29 - 32 * q^31 + 189 * q^33 - 15 * q^35 - 38 * q^37 - 335 * q^39 - 12 * q^41 + 566 * q^43 - 385 * q^45 - 919 * q^47 - 738 * q^49 + 993 * q^51 + 1156 * q^53 + 115 * q^55 + 114 * q^57 - 1324 * q^59 - 1673 * q^61 - 270 * q^63 - 660 * q^65 - 558 * q^67 - 92 * q^69 + 108 * q^71 + 1173 * q^73 - 100 * q^75 + 2608 * q^77 - 656 * q^79 - 319 * q^81 + 82 * q^83 - 115 * q^85 - 2389 * q^87 + 570 * q^89 + 1589 * q^91 + 911 * q^93 - 805 * q^95 + 633 * q^97 - 2021 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 7\nu^{3} + 9\nu^{2} - 77\nu - 2 ) / 16$$ (-v^4 + 7*v^3 + 9*v^2 - 77*v - 2) / 16 $$\beta_{2}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 9\nu^{2} - 43\nu - 78 ) / 16$$ (v^4 + v^3 - 9*v^2 - 43*v - 78) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - \nu^{3} + 25\nu^{2} + 43\nu - 98 ) / 16$$ (-v^4 - v^3 + 25*v^2 + 43*v - 98) / 16 $$\beta_{4}$$ $$=$$ $$( -3\nu^{4} + 5\nu^{3} + 59\nu^{2} - 55\nu - 182 ) / 16$$ (-3*v^4 + 5*v^3 + 59*v^2 - 55*v - 182) / 16
 $$\nu$$ $$=$$ $$( -\beta_{4} + 2\beta_{3} + \beta _1 + 1 ) / 4$$ (-b4 + 2*b3 + b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 11$$ b3 + b2 + 11 $$\nu^{3}$$ $$=$$ $$( -15\beta_{4} + 30\beta_{3} + 8\beta_{2} + 23\beta _1 + 55 ) / 4$$ (-15*b4 + 30*b3 + 8*b2 + 23*b1 + 55) / 4 $$\nu^{4}$$ $$=$$ $$-7\beta_{4} + 23\beta_{3} + 23\beta_{2} + 5\beta _1 + 174$$ -7*b4 + 23*b3 + 23*b2 + 5*b1 + 174

## 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
 −2.49214 3.41740 4.60878 −3.93900 −0.595043
0 −9.02447 0 −5.00000 0 −4.33445 0 54.4411 0
1.2 0 −7.84147 0 −5.00000 0 8.97260 0 34.4886 0
1.3 0 1.89520 0 −5.00000 0 −11.4426 0 −23.4082 0
1.4 0 3.85751 0 −5.00000 0 23.5932 0 −12.1196 0
1.5 0 7.11323 0 −5.00000 0 −13.7888 0 23.5981 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 1840.4.a.n 5
4.b odd 2 1 115.4.a.e 5
12.b even 2 1 1035.4.a.k 5
20.d odd 2 1 575.4.a.j 5
20.e even 4 2 575.4.b.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 4.b odd 2 1
575.4.a.j 5 20.d odd 2 1
575.4.b.i 10 20.e even 4 2
1035.4.a.k 5 12.b even 2 1
1840.4.a.n 5 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{5} + 4T_{3}^{4} - 98T_{3}^{3} - 149T_{3}^{2} + 2536T_{3} - 3680$$ T3^5 + 4*T3^4 - 98*T3^3 - 149*T3^2 + 2536*T3 - 3680 $$T_{7}^{5} - 3T_{7}^{4} - 484T_{7}^{3} - 1757T_{7}^{2} + 34281T_{7} + 144774$$ T7^5 - 3*T7^4 - 484*T7^3 - 1757*T7^2 + 34281*T7 + 144774

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 4 T^{4} + \cdots - 3680$$
$5$ $$(T + 5)^{5}$$
$7$ $$T^{5} - 3 T^{4} + \cdots + 144774$$
$11$ $$T^{5} + 23 T^{4} + \cdots + 74136848$$
$13$ $$T^{5} - 132 T^{4} + \cdots + 1550116$$
$17$ $$T^{5} + \cdots - 1039045340$$
$19$ $$T^{5} - 161 T^{4} + \cdots + 801280$$
$23$ $$(T - 23)^{5}$$
$29$ $$T^{5} + \cdots - 6149898500$$
$31$ $$T^{5} + 32 T^{4} + \cdots + 438072447$$
$37$ $$T^{5} + \cdots + 1590700778176$$
$41$ $$T^{5} + \cdots + 114116030755$$
$43$ $$T^{5} + \cdots - 504784881664$$
$47$ $$T^{5} + \cdots - 117787714816$$
$53$ $$T^{5} + \cdots + 5720332226904$$
$59$ $$T^{5} + \cdots - 24279649927232$$
$61$ $$T^{5} + \cdots - 34095834816896$$
$67$ $$T^{5} + \cdots - 5644442112$$
$71$ $$T^{5} + \cdots - 15638892903635$$
$73$ $$T^{5} + \cdots - 100895881632176$$
$79$ $$T^{5} + \cdots + 90481602379776$$
$83$ $$T^{5} + \cdots - 18307318870176$$
$89$ $$T^{5} + \cdots - 115104799418880$$
$97$ $$T^{5} + \cdots - 480989167569272$$