# Properties

 Label 240.4.f.f Level $240$ Weight $4$ Character orbit 240.f Analytic conductor $14.160$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,4,Mod(49,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("240.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 240.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$14.1604584014$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} - 9 q^{9}+O(q^{10})$$ q - b2 * q^3 + (b3 - 2*b2 - b1 + 1) * q^5 + (b2 + 3*b1) * q^7 - 9 * q^9 $$q - \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} - 9 q^{9} + (2 \beta_{3} - \beta_{2} - \beta_1 + 20) q^{11} + (13 \beta_{2} + 3 \beta_1) q^{13} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 14) q^{15} + ( - 17 \beta_{2} + 5 \beta_1) q^{17} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 32) q^{19} + ( - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 12) q^{21} + (8 \beta_{2} + 4 \beta_1) q^{23} + (6 \beta_{3} + 13 \beta_{2} + 9 \beta_1 + 61) q^{25} + 9 \beta_{2} q^{27} + ( - 14 \beta_{3} + 7 \beta_{2} + 7 \beta_1 + 166) q^{29} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 28) q^{31} + ( - 21 \beta_{2} + 9 \beta_1) q^{33} + ( - 10 \beta_{3} - 55 \beta_{2} + 5 \beta_1 + 80) q^{35} + ( - 51 \beta_{2} + 27 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 120) q^{39} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 202) q^{41} + ( - 20 \beta_{2} + 48 \beta_1) q^{43} + ( - 9 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 9) q^{45} + (30 \beta_{2} + 46 \beta_1) q^{47} + (12 \beta_{3} - 6 \beta_{2} - 6 \beta_1 - 41) q^{49} + ( - 10 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 148) q^{51} + (67 \beta_{2} + 41 \beta_1) q^{53} + (24 \beta_{3} - 23 \beta_{2} - 9 \beta_1 + 204) q^{55} + ( - 28 \beta_{2} - 36 \beta_1) q^{57} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 92) q^{59} + (32 \beta_{3} - 16 \beta_{2} - 16 \beta_1 + 154) q^{61} + ( - 9 \beta_{2} - 27 \beta_1) q^{63} + ( - 22 \beta_{3} - 31 \beta_{2} - 43 \beta_1 + 248) q^{65} + (122 \beta_{2} - 30 \beta_1) q^{67} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 76) q^{69} + ( - 60 \beta_{3} + 30 \beta_{2} + 30 \beta_1 + 48) q^{71} + (222 \beta_{2} - 54 \beta_1) q^{73} + ( - 24 \beta_{3} - 52 \beta_{2} + 39 \beta_1 + 156) q^{75} + ( - 102 \beta_{2} + 54 \beta_1) q^{77} + (100 \beta_{3} - 50 \beta_{2} - 50 \beta_1 + 140) q^{79} + 81 q^{81} + (164 \beta_{2} - 104 \beta_1) q^{83} + (2 \beta_{3} - 129 \beta_{2} + 83 \beta_1 - 128) q^{85} + ( - 159 \beta_{2} - 63 \beta_1) q^{87} + (48 \beta_{3} - 24 \beta_{2} - 24 \beta_1 - 582) q^{89} + (84 \beta_{3} - 42 \beta_{2} - 42 \beta_1 - 528) q^{91} + (26 \beta_{2} + 18 \beta_1) q^{93} + (16 \beta_{3} - 132 \beta_{2} - 76 \beta_1 - 704) q^{95} + (128 \beta_{2} - 120 \beta_1) q^{97} + ( - 18 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 180) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b3 - 2*b2 - b1 + 1) * q^5 + (b2 + 3*b1) * q^7 - 9 * q^9 + (2*b3 - b2 - b1 + 20) * q^11 + (13*b2 + 3*b1) * q^13 + (b3 - 2*b2 + 4*b1 - 14) * q^15 + (-17*b2 + 5*b1) * q^17 + (-8*b3 + 4*b2 + 4*b1 + 32) * q^19 + (-6*b3 + 3*b2 + 3*b1 + 12) * q^21 + (8*b2 + 4*b1) * q^23 + (6*b3 + 13*b2 + 9*b1 + 61) * q^25 + 9*b2 * q^27 + (-14*b3 + 7*b2 + 7*b1 + 166) * q^29 + (4*b3 - 2*b2 - 2*b1 - 28) * q^31 + (-21*b2 + 9*b1) * q^33 + (-10*b3 - 55*b2 + 5*b1 + 80) * q^35 + (-51*b2 + 27*b1) * q^37 + (-6*b3 + 3*b2 + 3*b1 + 120) * q^39 + (-4*b3 + 2*b2 + 2*b1 - 202) * q^41 + (-20*b2 + 48*b1) * q^43 + (-9*b3 + 18*b2 + 9*b1 - 9) * q^45 + (30*b2 + 46*b1) * q^47 + (12*b3 - 6*b2 - 6*b1 - 41) * q^49 + (-10*b3 + 5*b2 + 5*b1 - 148) * q^51 + (67*b2 + 41*b1) * q^53 + (24*b3 - 23*b2 - 9*b1 + 204) * q^55 + (-28*b2 - 36*b1) * q^57 + (-2*b3 + b2 + b1 - 92) * q^59 + (32*b3 - 16*b2 - 16*b1 + 154) * q^61 + (-9*b2 - 27*b1) * q^63 + (-22*b3 - 31*b2 - 43*b1 + 248) * q^65 + (122*b2 - 30*b1) * q^67 + (-8*b3 + 4*b2 + 4*b1 + 76) * q^69 + (-60*b3 + 30*b2 + 30*b1 + 48) * q^71 + (222*b2 - 54*b1) * q^73 + (-24*b3 - 52*b2 + 39*b1 + 156) * q^75 + (-102*b2 + 54*b1) * q^77 + (100*b3 - 50*b2 - 50*b1 + 140) * q^79 + 81 * q^81 + (164*b2 - 104*b1) * q^83 + (2*b3 - 129*b2 + 83*b1 - 128) * q^85 + (-159*b2 - 63*b1) * q^87 + (48*b3 - 24*b2 - 24*b1 - 582) * q^89 + (84*b3 - 42*b2 - 42*b1 - 528) * q^91 + (26*b2 + 18*b1) * q^93 + (16*b3 - 132*b2 - 76*b1 - 704) * q^95 + (128*b2 - 120*b1) * q^97 + (-18*b3 + 9*b2 + 9*b1 - 180) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{5} - 36 q^{9}+O(q^{10})$$ 4 * q + 6 * q^5 - 36 * q^9 $$4 q + 6 q^{5} - 36 q^{9} + 84 q^{11} - 54 q^{15} + 112 q^{19} + 36 q^{21} + 256 q^{25} + 636 q^{29} - 104 q^{31} + 300 q^{35} + 468 q^{39} - 816 q^{41} - 54 q^{45} - 140 q^{49} - 612 q^{51} + 864 q^{55} - 372 q^{59} + 680 q^{61} + 948 q^{65} + 288 q^{69} + 72 q^{71} + 576 q^{75} + 760 q^{79} + 324 q^{81} - 508 q^{85} - 2232 q^{89} - 1944 q^{91} - 2784 q^{95} - 756 q^{99}+O(q^{100})$$ 4 * q + 6 * q^5 - 36 * q^9 + 84 * q^11 - 54 * q^15 + 112 * q^19 + 36 * q^21 + 256 * q^25 + 636 * q^29 - 104 * q^31 + 300 * q^35 + 468 * q^39 - 816 * q^41 - 54 * q^45 - 140 * q^49 - 612 * q^51 + 864 * q^55 - 372 * q^59 + 680 * q^61 + 948 * q^65 + 288 * q^69 + 72 * q^71 + 576 * q^75 + 760 * q^79 + 324 * q^81 - 508 * q^85 - 2232 * q^89 - 1944 * q^91 - 2784 * q^95 - 756 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 31\nu ) / 10$$ (v^3 + 31*v) / 10 $$\beta_{2}$$ $$=$$ $$( -3\nu^{3} - 33\nu ) / 10$$ (-3*v^3 - 33*v) / 10 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 30\nu^{2} - \nu + 320 ) / 10$$ (-v^3 + 30*v^2 - v + 320) / 10
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta_1 ) / 6$$ (b2 + 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} - \beta_{2} - \beta _1 - 64 ) / 6$$ (2*b3 - b2 - b1 - 64) / 6 $$\nu^{3}$$ $$=$$ $$( -31\beta_{2} - 33\beta_1 ) / 6$$ (-31*b2 - 33*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 3.70156i − 2.70156i − 3.70156i 2.70156i
0 3.00000i 0 −8.10469 7.70156i 0 22.2094i 0 −9.00000 0
49.2 0 3.00000i 0 11.1047 1.29844i 0 16.2094i 0 −9.00000 0
49.3 0 3.00000i 0 −8.10469 + 7.70156i 0 22.2094i 0 −9.00000 0
49.4 0 3.00000i 0 11.1047 + 1.29844i 0 16.2094i 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.f.f 4
3.b odd 2 1 720.4.f.j 4
4.b odd 2 1 15.4.b.a 4
5.b even 2 1 inner 240.4.f.f 4
5.c odd 4 1 1200.4.a.bn 2
5.c odd 4 1 1200.4.a.bt 2
8.b even 2 1 960.4.f.p 4
8.d odd 2 1 960.4.f.q 4
12.b even 2 1 45.4.b.b 4
15.d odd 2 1 720.4.f.j 4
20.d odd 2 1 15.4.b.a 4
20.e even 4 1 75.4.a.c 2
20.e even 4 1 75.4.a.f 2
40.e odd 2 1 960.4.f.q 4
40.f even 2 1 960.4.f.p 4
60.h even 2 1 45.4.b.b 4
60.l odd 4 1 225.4.a.i 2
60.l odd 4 1 225.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 4.b odd 2 1
15.4.b.a 4 20.d odd 2 1
45.4.b.b 4 12.b even 2 1
45.4.b.b 4 60.h even 2 1
75.4.a.c 2 20.e even 4 1
75.4.a.f 2 20.e even 4 1
225.4.a.i 2 60.l odd 4 1
225.4.a.o 2 60.l odd 4 1
240.4.f.f 4 1.a even 1 1 trivial
240.4.f.f 4 5.b even 2 1 inner
720.4.f.j 4 3.b odd 2 1
720.4.f.j 4 15.d odd 2 1
960.4.f.p 4 8.b even 2 1
960.4.f.p 4 40.f even 2 1
960.4.f.q 4 8.d odd 2 1
960.4.f.q 4 40.e odd 2 1
1200.4.a.bn 2 5.c odd 4 1
1200.4.a.bt 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{4} + 756T_{7}^{2} + 129600$$ T7^4 + 756*T7^2 + 129600 $$T_{11}^{2} - 42T_{11} + 72$$ T11^2 - 42*T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4} - 6 T^{3} - 110 T^{2} + \cdots + 15625$$
$7$ $$T^{4} + 756 T^{2} + 129600$$
$11$ $$(T^{2} - 42 T + 72)^{2}$$
$13$ $$T^{4} + 3780 T^{2} + \cdots + 1327104$$
$17$ $$T^{4} + 7252 T^{2} + \cdots + 2483776$$
$19$ $$(T^{2} - 56 T - 5120)^{2}$$
$23$ $$T^{4} + 2464 T^{2} + 6400$$
$29$ $$(T^{2} - 318 T + 7200)^{2}$$
$31$ $$(T^{2} + 52 T - 800)^{2}$$
$37$ $$T^{4} + 106596 T^{2} + \cdots + 41990400$$
$41$ $$(T^{2} + 408 T + 40140)^{2}$$
$43$ $$T^{4} + 196128 T^{2} + \cdots + 8256266496$$
$47$ $$T^{4} + 189712 T^{2} + \cdots + 6186766336$$
$53$ $$T^{4} + 218644 T^{2} + \cdots + 813390400$$
$59$ $$(T^{2} + 186 T + 8280)^{2}$$
$61$ $$(T^{2} - 340 T - 65564)^{2}$$
$67$ $$T^{4} + 341712 T^{2} + \cdots + 9419867136$$
$71$ $$(T^{2} - 36 T - 331776)^{2}$$
$73$ $$T^{4} + 1126224 T^{2} + \cdots + 104976000000$$
$79$ $$(T^{2} - 380 T - 886400)^{2}$$
$83$ $$T^{4} + 1371040 T^{2} + \cdots + 40558737664$$
$89$ $$(T^{2} + 1116 T + 98820)^{2}$$
$97$ $$T^{4} + 1475712 T^{2} + \cdots + 196199387136$$