# 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

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$14.1604584014$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 31 \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{3} - 33 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 30 \nu^{2} - \nu + 320$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - \beta_{2} - \beta_{1} - 64$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-31 \beta_{2} - 33 \beta_{1}$$$$)/6$$

## 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.

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 $$S_{4}^{\mathrm{new}}(240, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$15625 - 750 T - 110 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$129600 + 756 T^{2} + T^{4}$$
$11$ $$( 72 - 42 T + T^{2} )^{2}$$
$13$ $$1327104 + 3780 T^{2} + T^{4}$$
$17$ $$2483776 + 7252 T^{2} + T^{4}$$
$19$ $$( -5120 - 56 T + T^{2} )^{2}$$
$23$ $$6400 + 2464 T^{2} + T^{4}$$
$29$ $$( 7200 - 318 T + T^{2} )^{2}$$
$31$ $$( -800 + 52 T + T^{2} )^{2}$$
$37$ $$41990400 + 106596 T^{2} + T^{4}$$
$41$ $$( 40140 + 408 T + T^{2} )^{2}$$
$43$ $$8256266496 + 196128 T^{2} + T^{4}$$
$47$ $$6186766336 + 189712 T^{2} + T^{4}$$
$53$ $$813390400 + 218644 T^{2} + T^{4}$$
$59$ $$( 8280 + 186 T + T^{2} )^{2}$$
$61$ $$( -65564 - 340 T + T^{2} )^{2}$$
$67$ $$9419867136 + 341712 T^{2} + T^{4}$$
$71$ $$( -331776 - 36 T + T^{2} )^{2}$$
$73$ $$104976000000 + 1126224 T^{2} + T^{4}$$
$79$ $$( -886400 - 380 T + T^{2} )^{2}$$
$83$ $$40558737664 + 1371040 T^{2} + T^{4}$$
$89$ $$( 98820 + 1116 T + T^{2} )^{2}$$
$97$ $$196199387136 + 1475712 T^{2} + T^{4}$$