Properties

 Label 240.3.l.d Level $240$ Weight $3$ Character orbit 240.l Analytic conductor $6.540$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.681615360000.5 Defining polynomial: $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{5} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{5} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{9} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} -\beta_{1} q^{15} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 9 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{21} + ( 3 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{23} -5 q^{25} + ( -5 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{27} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} - 9 \beta_{6} + 2 \beta_{7} ) q^{29} + ( -16 + \beta_{1} + \beta_{2} - \beta_{3} + 6 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{31} + ( -16 + \beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} ) q^{33} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{37} + ( 2 + 3 \beta_{1} + \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{39} + ( -5 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} - 13 \beta_{6} - 4 \beta_{7} ) q^{41} + ( 45 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{43} + ( -9 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -1 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{47} + ( 5 + 3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{49} + ( -10 - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} ) q^{51} + ( -4 + 6 \beta_{1} + 10 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 12 - 3 \beta_{1} + 6 \beta_{2} - \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{55} + ( 16 + \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} + 17 \beta_{6} + 5 \beta_{7} ) q^{57} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 19 \beta_{6} - 5 \beta_{7} ) q^{59} + ( 8 - 7 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} + 2 \beta_{5} - 7 \beta_{6} - 8 \beta_{7} ) q^{61} + ( -1 - 5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 11 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{63} + ( -4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{65} + ( -31 + 3 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} - 9 \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{67} + ( 13 + 4 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} - \beta_{5} + 4 \beta_{6} - 15 \beta_{7} ) q^{69} + ( -6 + 4 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{73} + 5 \beta_{2} q^{75} + ( 12 + 12 \beta_{1} + 20 \beta_{2} - 12 \beta_{4} - 12 \beta_{5} - 20 \beta_{6} - 4 \beta_{7} ) q^{77} + ( -14 + \beta_{1} - 5 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{79} + ( 23 - 9 \beta_{1} - \beta_{2} + 5 \beta_{3} + 10 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{81} + ( 3 + 3 \beta_{1} + 13 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 40 \beta_{6} + 7 \beta_{7} ) q^{83} + ( 4 - 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{7} ) q^{85} + ( 20 + 9 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{87} + ( 8 + 2 \beta_{1} + 26 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} + 16 \beta_{7} ) q^{89} + ( -54 - 10 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{91} + ( -46 + 7 \beta_{1} + 20 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} ) q^{93} + ( -2 - 3 \beta_{1} - 11 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 6 \beta_{7} ) q^{95} + ( 6 + 8 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{97} + ( 10 - 5 \beta_{1} + 24 \beta_{2} - 7 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} + 35 \beta_{6} + 23 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + O(q^{10})$$ $$8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} - 8 q^{13} + 8 q^{19} + 28 q^{21} - 40 q^{25} - 20 q^{27} - 120 q^{31} - 112 q^{33} + 8 q^{37} + 72 q^{39} + 328 q^{43} - 60 q^{45} + 64 q^{49} - 64 q^{51} + 40 q^{55} + 72 q^{57} + 8 q^{61} - 88 q^{63} - 152 q^{67} + 100 q^{69} + 32 q^{73} - 20 q^{75} - 88 q^{79} + 224 q^{81} + 152 q^{87} - 560 q^{91} - 368 q^{93} + 144 q^{97} - 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 43 \nu^{6} + 27 \nu^{5} + 691 \nu^{4} - 866 \nu^{3} - 6528 \nu^{2} + 3264 \nu + 8424$$$$)/3720$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{7} + 7 \nu^{6} + 85 \nu^{5} - 230 \nu^{4} - 523 \nu^{3} + 243 \nu^{2} + 4668 \nu - 2124$$$$)/3100$$ $$\beta_{3}$$ $$=$$ $$($$$$17 \nu^{7} - 447 \nu^{6} + 1215 \nu^{5} + 1955 \nu^{4} - 3692 \nu^{3} - 18728 \nu^{2} + 11472 \nu - 33096$$$$)/18600$$ $$\beta_{4}$$ $$=$$ $$($$$$21 \nu^{7} - 151 \nu^{6} + 115 \nu^{5} + 2725 \nu^{4} - 4816 \nu^{3} - 16734 \nu^{2} + 2136 \nu + 56712$$$$)/18600$$ $$\beta_{5}$$ $$=$$ $$($$$$-69 \nu^{7} + 319 \nu^{6} + 375 \nu^{5} - 2045 \nu^{4} - 6186 \nu^{3} + 16366 \nu^{2} + 35496 \nu + 3912$$$$)/18600$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 28 \nu^{6} + 30 \nu^{5} - 145 \nu^{4} - 232 \nu^{3} + 507 \nu^{2} - 1788 \nu + 804$$$$)/1860$$ $$\beta_{7}$$ $$=$$ $$($$$$82 \nu^{7} - 287 \nu^{6} - 385 \nu^{5} + 1680 \nu^{4} + 5943 \nu^{3} - 8413 \nu^{2} + 13212 \nu - 5916$$$$)/9300$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 2 \beta_{5} + \beta_{3} - 3 \beta_{2} - \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{7} + 11 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{7} + 27 \beta_{6} + 18 \beta_{5} + 20 \beta_{4} + 5 \beta_{3} - 19 \beta_{2} - 9 \beta_{1} - 50$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$54 \beta_{7} + 123 \beta_{6} + 10 \beta_{5} + 72 \beta_{4} + 17 \beta_{3} + 31 \beta_{2} - 49 \beta_{1} - 78$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$188 \beta_{7} + 427 \beta_{6} + 10 \beta_{5} + 316 \beta_{4} - 75 \beta_{3} + 169 \beta_{2} - 137 \beta_{1} - 738$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$322 \beta_{7} + 619 \beta_{6} - 334 \beta_{5} + 1088 \beta_{4} - 255 \beta_{3} + 891 \beta_{2} - 609 \beta_{1} - 1774$$$$)/2$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.54294 + 1.41421i 1.54294 − 1.41421i −2.22255 + 1.41421i −2.22255 − 1.41421i −0.542939 + 1.41421i −0.542939 − 1.41421i 3.22255 − 1.41421i 3.22255 + 1.41421i
0 −2.98254 0.323191i 0 2.23607i 0 −4.72640 0 8.79110 + 1.92786i 0
161.2 0 −2.98254 + 0.323191i 0 2.23607i 0 −4.72640 0 8.79110 1.92786i 0
161.3 0 −0.291610 2.98579i 0 2.23607i 0 −4.46268 0 −8.82993 + 1.74137i 0
161.4 0 −0.291610 + 2.98579i 0 2.23607i 0 −4.46268 0 −8.82993 1.74137i 0
161.5 0 2.40140 1.79813i 0 2.23607i 0 10.2132 0 2.53346 8.63606i 0
161.6 0 2.40140 + 1.79813i 0 2.23607i 0 10.2132 0 2.53346 + 8.63606i 0
161.7 0 2.87275 0.864473i 0 2.23607i 0 −9.02416 0 7.50537 4.96683i 0
161.8 0 2.87275 + 0.864473i 0 2.23607i 0 −9.02416 0 7.50537 + 4.96683i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.l.d 8
3.b odd 2 1 inner 240.3.l.d 8
4.b odd 2 1 120.3.l.a 8
5.b even 2 1 1200.3.l.x 8
5.c odd 4 2 1200.3.c.m 16
8.b even 2 1 960.3.l.g 8
8.d odd 2 1 960.3.l.h 8
12.b even 2 1 120.3.l.a 8
15.d odd 2 1 1200.3.l.x 8
15.e even 4 2 1200.3.c.m 16
20.d odd 2 1 600.3.l.f 8
20.e even 4 2 600.3.c.d 16
24.f even 2 1 960.3.l.h 8
24.h odd 2 1 960.3.l.g 8
60.h even 2 1 600.3.l.f 8
60.l odd 4 2 600.3.c.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a 8 4.b odd 2 1
120.3.l.a 8 12.b even 2 1
240.3.l.d 8 1.a even 1 1 trivial
240.3.l.d 8 3.b odd 2 1 inner
600.3.c.d 16 20.e even 4 2
600.3.c.d 16 60.l odd 4 2
600.3.l.f 8 20.d odd 2 1
600.3.l.f 8 60.h even 2 1
960.3.l.g 8 8.b even 2 1
960.3.l.g 8 24.h odd 2 1
960.3.l.h 8 8.d odd 2 1
960.3.l.h 8 24.f even 2 1
1200.3.c.m 16 5.c odd 4 2
1200.3.c.m 16 15.e even 4 2
1200.3.l.x 8 5.b even 2 1
1200.3.l.x 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 8 T_{7}^{3} - 82 T_{7}^{2} - 872 T_{7} - 1944$$ acting on $$S_{3}^{\mathrm{new}}(240, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 - 2916 T - 162 T^{2} + 324 T^{3} - 102 T^{4} + 36 T^{5} - 2 T^{6} - 4 T^{7} + T^{8}$$
$5$ $$( 5 + T^{2} )^{4}$$
$7$ $$( -1944 - 872 T - 82 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$11$ $$232989696 + 14951168 T^{2} + 226128 T^{4} + 888 T^{6} + T^{8}$$
$13$ $$( -3456 - 3648 T - 380 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$17$ $$15872256 + 25369856 T^{2} + 312672 T^{4} + 1104 T^{6} + T^{8}$$
$19$ $$( 16736 + 3424 T - 708 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$93650688576 + 885144416 T^{2} + 2697012 T^{4} + 2964 T^{6} + T^{8}$$
$29$ $$3474395136 + 145293824 T^{2} + 1160592 T^{4} + 2376 T^{6} + T^{8}$$
$31$ $$( -151296 - 71360 T - 924 T^{2} + 60 T^{3} + T^{4} )^{2}$$
$37$ $$( -31104 + 39744 T - 3228 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$43961355472896 + 77260718592 T^{2} + 46587024 T^{4} + 11528 T^{6} + T^{8}$$
$43$ $$( 1582656 - 220576 T + 9482 T^{2} - 164 T^{3} + T^{4} )^{2}$$
$47$ $$13517317696 + 266492768 T^{2} + 1541364 T^{4} + 2612 T^{6} + T^{8}$$
$53$ $$6801580544256 + 88193961216 T^{2} + 82087776 T^{4} + 16976 T^{6} + T^{8}$$
$59$ $$15563214360576 + 64836527616 T^{2} + 49411216 T^{4} + 12616 T^{6} + T^{8}$$
$61$ $$( 30631296 + 12544 T - 12508 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$( -668224 - 236224 T - 4854 T^{2} + 76 T^{3} + T^{4} )^{2}$$
$71$ $$35499479924736 + 145459224576 T^{2} + 83806272 T^{4} + 16304 T^{6} + T^{8}$$
$73$ $$( 2938896 + 29376 T - 3528 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$79$ $$( -12384 + 8256 T - 1268 T^{2} + 44 T^{3} + T^{4} )^{2}$$
$83$ $$336130569170496 + 2057217800544 T^{2} + 532895092 T^{4} + 41716 T^{6} + T^{8}$$
$89$ $$1334603390386176 + 2220134105088 T^{2} + 649648384 T^{4} + 49312 T^{6} + T^{8}$$
$97$ $$( 10270096 + 535392 T - 10376 T^{2} - 72 T^{3} + T^{4} )^{2}$$