Properties

 Label 3840.2.f.l Level 3840 Weight 2 Character orbit 3840.f Analytic conductor 30.663 Analytic rank 0 Dimension 12 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.180227832610816.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + x^{10} - 8 x^{6} + 16 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{8} + \nu^{6} + 4 \nu^{4} - 4 \nu^{2}$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{9} + 6 \nu^{7} - 12 \nu^{5} + 24 \nu^{3}$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 8 \nu^{3}$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{10} + \nu^{6} + 4 \nu^{4} + 4 \nu^{2} - 8$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{9} - 2 \nu^{7} - 12 \nu^{5} + 24 \nu^{3} - 64 \nu$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{8} - \nu^{6} - 2 \nu^{4} - 8 \nu^{2} + 8$$$$)/8$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} - 3 \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 24 \nu^{3}$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} - \nu^{9} - 6 \nu^{7} - 4 \nu^{5} + 40 \nu^{3} + 32 \nu$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{10} - 3 \nu^{8} + 2 \nu^{6} + 4 \nu^{4} - 8 \nu^{2} - 32$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$3 \nu^{11} - 3 \nu^{9} - 10 \nu^{7} - 36 \nu^{5} - 8 \nu^{3} + 128 \nu$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{8} - 2 \beta_{6} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{10} - 2 \beta_{7} + \beta_{5} - \beta_{1} - 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} + 2 \beta_{9} + \beta_{8} - \beta_{3} + 2 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{10} - 2 \beta_{7} - \beta_{5} + 4 \beta_{4} + 5 \beta_{1} + 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{11} + 2 \beta_{9} - \beta_{8} - 4 \beta_{6} + 13 \beta_{3} - 6 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{10} - 6 \beta_{7} + \beta_{5} - 12 \beta_{4} + 3 \beta_{1} + 15$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{11} - 10 \beta_{9} + 9 \beta_{8} - 4 \beta_{6} + 19 \beta_{3} + 14 \beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-14 \beta_{10} + 6 \beta_{7} + 7 \beta_{5} - 4 \beta_{4} + 5 \beta_{1} - 23$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-11 \beta_{11} + 10 \beta_{9} - 17 \beta_{8} - 12 \beta_{6} - 27 \beta_{3} + 18 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-2 \beta_{10} - 22 \beta_{7} - 31 \beta_{5} + 4 \beta_{4} + 19 \beta_{1} - 17$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$3 \beta_{11} + 6 \beta_{9} - 39 \beta_{8} + 12 \beta_{6} + 147 \beta_{3} - 2 \beta_{2}$$$$)/4$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 0.450129 − 1.34067i 0.806504 + 1.16170i 1.37729 − 0.321037i −1.37729 − 0.321037i −0.806504 + 1.16170i −0.450129 − 1.34067i 0.450129 + 1.34067i 0.806504 − 1.16170i 1.37729 + 0.321037i −1.37729 + 0.321037i −0.806504 − 1.16170i −0.450129 + 1.34067i
0 1.00000i 0 −2.22158 0.254102i 0 2.64265i 0 −1.00000 0
769.2 0 1.00000i 0 −1.23992 1.86081i 0 0.746175i 0 −1.00000 0
769.3 0 1.00000i 0 −0.726062 + 2.11491i 0 4.05705i 0 −1.00000 0
769.4 0 1.00000i 0 0.726062 + 2.11491i 0 4.05705i 0 −1.00000 0
769.5 0 1.00000i 0 1.23992 1.86081i 0 0.746175i 0 −1.00000 0
769.6 0 1.00000i 0 2.22158 0.254102i 0 2.64265i 0 −1.00000 0
769.7 0 1.00000i 0 −2.22158 + 0.254102i 0 2.64265i 0 −1.00000 0
769.8 0 1.00000i 0 −1.23992 + 1.86081i 0 0.746175i 0 −1.00000 0
769.9 0 1.00000i 0 −0.726062 2.11491i 0 4.05705i 0 −1.00000 0
769.10 0 1.00000i 0 0.726062 2.11491i 0 4.05705i 0 −1.00000 0
769.11 0 1.00000i 0 1.23992 + 1.86081i 0 0.746175i 0 −1.00000 0
769.12 0 1.00000i 0 2.22158 + 0.254102i 0 2.64265i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.12 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
8.b even 2 1 inner
40.f even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.f.l 12
4.b odd 2 1 3840.2.f.m 12
5.b even 2 1 inner 3840.2.f.l 12
8.b even 2 1 inner 3840.2.f.l 12
8.d odd 2 1 3840.2.f.m 12
16.e even 4 1 120.2.d.a 6
16.e even 4 1 120.2.d.b yes 6
16.f odd 4 1 480.2.d.a 6
16.f odd 4 1 480.2.d.b 6
20.d odd 2 1 3840.2.f.m 12
40.e odd 2 1 3840.2.f.m 12
40.f even 2 1 inner 3840.2.f.l 12
48.i odd 4 1 360.2.d.e 6
48.i odd 4 1 360.2.d.f 6
48.k even 4 1 1440.2.d.e 6
48.k even 4 1 1440.2.d.f 6
80.i odd 4 2 600.2.k.f 12
80.j even 4 2 2400.2.k.f 12
80.k odd 4 1 480.2.d.a 6
80.k odd 4 1 480.2.d.b 6
80.q even 4 1 120.2.d.a 6
80.q even 4 1 120.2.d.b yes 6
80.s even 4 2 2400.2.k.f 12
80.t odd 4 2 600.2.k.f 12
240.t even 4 1 1440.2.d.e 6
240.t even 4 1 1440.2.d.f 6
240.z odd 4 2 7200.2.k.u 12
240.bb even 4 2 1800.2.k.u 12
240.bd odd 4 2 7200.2.k.u 12
240.bf even 4 2 1800.2.k.u 12
240.bm odd 4 1 360.2.d.e 6
240.bm odd 4 1 360.2.d.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 16.e even 4 1
120.2.d.a 6 80.q even 4 1
120.2.d.b yes 6 16.e even 4 1
120.2.d.b yes 6 80.q even 4 1
360.2.d.e 6 48.i odd 4 1
360.2.d.e 6 240.bm odd 4 1
360.2.d.f 6 48.i odd 4 1
360.2.d.f 6 240.bm odd 4 1
480.2.d.a 6 16.f odd 4 1
480.2.d.a 6 80.k odd 4 1
480.2.d.b 6 16.f odd 4 1
480.2.d.b 6 80.k odd 4 1
600.2.k.f 12 80.i odd 4 2
600.2.k.f 12 80.t odd 4 2
1440.2.d.e 6 48.k even 4 1
1440.2.d.e 6 240.t even 4 1
1440.2.d.f 6 48.k even 4 1
1440.2.d.f 6 240.t even 4 1
1800.2.k.u 12 240.bb even 4 2
1800.2.k.u 12 240.bf even 4 2
2400.2.k.f 12 80.j even 4 2
2400.2.k.f 12 80.s even 4 2
3840.2.f.l 12 1.a even 1 1 trivial
3840.2.f.l 12 5.b even 2 1 inner
3840.2.f.l 12 8.b even 2 1 inner
3840.2.f.l 12 40.f even 2 1 inner
3840.2.f.m 12 4.b odd 2 1
3840.2.f.m 12 8.d odd 2 1
3840.2.f.m 12 20.d odd 2 1
3840.2.f.m 12 40.e odd 2 1
7200.2.k.u 12 240.z odd 4 2
7200.2.k.u 12 240.bd odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3840, [\chi])$$:

 $$T_{7}^{6} + 24 T_{7}^{4} + 128 T_{7}^{2} + 64$$ $$T_{11}^{6} - 32 T_{11}^{4} + 96 T_{11}^{2} - 64$$ $$T_{29}^{6} - 108 T_{29}^{4} + 3120 T_{29}^{2} - 12544$$ $$T_{31}^{3} + 8 T_{31}^{2} - 4 T_{31} - 64$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{6}$$
$5$ $$1 + 2 T^{2} - 9 T^{4} - 196 T^{6} - 225 T^{8} + 1250 T^{10} + 15625 T^{12}$$
$7$ $$( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 9359 T^{8} - 43218 T^{10} + 117649 T^{12} )^{2}$$
$11$ $$( 1 + 34 T^{2} + 503 T^{4} + 5436 T^{6} + 60863 T^{8} + 497794 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 125567 T^{8} - 856830 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 651695 T^{8} - 5512386 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 + 54 T^{2} + 1367 T^{4} + 25652 T^{6} + 493487 T^{8} + 7037334 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$( 1 - 46 T^{2} + 1775 T^{4} - 40932 T^{6} + 938975 T^{8} - 12872686 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 + 66 T^{2} + 3207 T^{4} + 111228 T^{6} + 2697087 T^{8} + 46680546 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 + 8 T + 89 T^{2} + 432 T^{3} + 2759 T^{4} + 7688 T^{5} + 29791 T^{6} )^{4}$$
$37$ $$( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 15320479 T^{8} - 296117438 T^{10} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 - 2 T + 23 T^{2} - 220 T^{3} + 943 T^{4} - 3362 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 18147935 T^{8} - 444444130 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 49408703 T^{8} - 1083289182 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 74132319 T^{8} - 1877934478 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 + 178 T^{2} + 20567 T^{4} + 1418652 T^{6} + 71593727 T^{8} + 2156890258 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 + 190 T^{2} + 20039 T^{4} + 1419204 T^{6} + 74565119 T^{8} + 2630709790 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 166663103 T^{8} - 5521407154 T^{10} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 9443 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 12613743 T^{8} - 1533505014 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 18407 T^{4} - 49928 T^{5} + 493039 T^{6} )^{4}$$
$83$ $$( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 350340095 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 10 T + 103 T^{2} - 396 T^{3} + 9167 T^{4} - 79210 T^{5} + 704969 T^{6} )^{4}$$
$97$ $$( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 368672847 T^{8} - 21778203126 T^{10} + 832972004929 T^{12} )^{2}$$