# Properties

 Label 3840.2.f.m 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 Defining polynomial: $$x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64$$ x^12 + x^10 - 8*x^6 + 16*x^2 + 64 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{2} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10})$$ q - b3 * q^3 - b2 * q^5 - b1 * q^7 - q^9 $$q - \beta_{3} q^{3} - \beta_{2} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{11} - \beta_{3}) q^{13} + \beta_{7} q^{15} + ( - \beta_{10} - \beta_{7} - \beta_{4}) q^{17} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{19} - \beta_{8} q^{21} + (2 \beta_{10} + \beta_{7} + \beta_{4} + \beta_1) q^{23} + (\beta_{10} - \beta_{5}) q^{25} + \beta_{3} q^{27} + ( - 2 \beta_{9} + \beta_{6} + \beta_{2}) q^{29} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 3) q^{31} + (\beta_{10} + \beta_1) q^{33} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{3} + \beta_{2}) q^{35} + (\beta_{11} + 3 \beta_{3}) q^{37} + ( - \beta_{5} - 1) q^{39} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{41} + (2 \beta_{6} - 2 \beta_{2}) q^{43} + \beta_{2} q^{45} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{47} + (2 \beta_{7} - 2 \beta_{4} - 1) q^{49} + (\beta_{9} - \beta_{6} - \beta_{2}) q^{51} + ( - \beta_{6} - 4 \beta_{3} + \beta_{2}) q^{53} + ( - 2 \beta_{10} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{55} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{57} + ( - \beta_{9} + 3 \beta_{8}) q^{59} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{2}) q^{61} + \beta_1 q^{63} + ( - \beta_{10} + \beta_{7} - 3 \beta_{4} - 2 \beta_1 - 2) q^{65} + (2 \beta_{6} - 2 \beta_{2}) q^{67} + ( - 2 \beta_{9} + \beta_{8} + \beta_{6} + \beta_{2}) q^{69} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2) q^{71} + ( - 2 \beta_{10} - 4 \beta_{7} - 4 \beta_{4}) q^{73} + ( - \beta_{11} - \beta_{9}) q^{75} + 4 \beta_{3} q^{77} + (\beta_{7} + \beta_{5} - \beta_{4} - 3) q^{79} + q^{81} + ( - 2 \beta_{11} + 2 \beta_{3}) q^{83} + (\beta_{11} + 2 \beta_{8} - 2 \beta_{6} + 3 \beta_{3}) q^{85} + ( - 2 \beta_{10} - \beta_{7} - \beta_{4}) q^{87} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 4) q^{89} + ( - 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{6} + 4 \beta_{2}) q^{91} + ( - \beta_{11} + \beta_{6} - 3 \beta_{3} - \beta_{2}) q^{93} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1 - 6) q^{95} + (2 \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{97} + ( - \beta_{9} + \beta_{8}) q^{99}+O(q^{100})$$ q - b3 * q^3 - b2 * q^5 - b1 * q^7 - q^9 + (b9 - b8) * q^11 + (b11 - b3) * q^13 + b7 * q^15 + (-b10 - b7 - b4) * q^17 + (b8 + b6 + b2) * q^19 - b8 * q^21 + (2*b10 + b7 + b4 + b1) * q^23 + (b10 - b5) * q^25 + b3 * q^27 + (-2*b9 + b6 + b2) * q^29 + (-b7 - b5 + b4 + 3) * q^31 + (b10 + b1) * q^33 + (b11 - b9 - b8 + b6 + b3 + b2) * q^35 + (b11 + 3*b3) * q^37 + (-b5 - 1) * q^39 + (2*b7 + 2*b5 - 2*b4) * q^41 + (2*b6 - 2*b2) * q^43 + b2 * q^45 + (-b7 - b4 - b1) * q^47 + (2*b7 - 2*b4 - 1) * q^49 + (b9 - b6 - b2) * q^51 + (-b6 - 4*b3 + b2) * q^53 + (-2*b10 - b7 - b5 + b4 - b1 - 1) * q^55 + (-b7 - b4 - b1) * q^57 + (-b9 + 3*b8) * q^59 + (2*b8 - 2*b6 - 2*b2) * q^61 + b1 * q^63 + (-b10 + b7 - 3*b4 - 2*b1 - 2) * q^65 + (2*b6 - 2*b2) * q^67 + (-2*b9 + b8 + b6 + b2) * q^69 + (2*b7 + 2*b5 - 2*b4 + 2) * q^71 + (-2*b10 - 4*b7 - 4*b4) * q^73 + (-b11 - b9) * q^75 + 4*b3 * q^77 + (b7 + b5 - b4 - 3) * q^79 + q^81 + (-2*b11 + 2*b3) * q^83 + (b11 + 2*b8 - 2*b6 + 3*b3) * q^85 + (-2*b10 - b7 - b4) * q^87 + (2*b7 - 2*b5 - 2*b4 + 4) * q^89 + (-2*b9 - 2*b8 + 4*b6 + 4*b2) * q^91 + (-b11 + b6 - 3*b3 - b2) * q^93 + (b7 + 2*b5 + b4 - b1 - 6) * q^95 + (2*b10 - 2*b7 - 2*b4) * q^97 + (-b9 + b8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^9 $$12 q - 12 q^{9} - 4 q^{25} + 32 q^{31} - 16 q^{39} + 8 q^{41} - 12 q^{49} - 16 q^{55} - 24 q^{65} + 32 q^{71} - 32 q^{79} + 12 q^{81} + 40 q^{89} - 64 q^{95}+O(q^{100})$$ 12 * q - 12 * q^9 - 4 * q^25 + 32 * q^31 - 16 * q^39 + 8 * q^41 - 12 * q^49 - 16 * q^55 - 24 * q^65 + 32 * q^71 - 32 * q^79 + 12 * q^81 + 40 * q^89 - 64 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8$$ (v^8 + v^6 + 4*v^4 - 4*v^2) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{11} + 3\nu^{9} + 6\nu^{7} - 12\nu^{5} + 24\nu^{3} ) / 64$$ (v^11 + 3*v^9 + 6*v^7 - 12*v^5 + 24*v^3) / 64 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64$$ (v^11 - v^9 + 2*v^7 + 4*v^5 + 8*v^3) / 64 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + \nu^{4} + 2\nu^{2} + 4 ) / 4$$ (-v^6 + v^4 + 2*v^2 + 4) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8$$ (-v^10 + v^6 + 4*v^4 + 4*v^2 - 8) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{11} - 5\nu^{9} - 2\nu^{7} - 12\nu^{5} + 24\nu^{3} - 64\nu ) / 64$$ (v^11 - 5*v^9 - 2*v^7 - 12*v^5 + 24*v^3 - 64*v) / 64 $$\beta_{7}$$ $$=$$ $$( \nu^{8} - \nu^{6} - 2\nu^{4} - 8\nu^{2} + 8 ) / 8$$ (v^8 - v^6 - 2*v^4 - 8*v^2 + 8) / 8 $$\beta_{8}$$ $$=$$ $$( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32$$ (-v^11 - 3*v^9 + 2*v^7 + 4*v^5 + 24*v^3) / 32 $$\beta_{9}$$ $$=$$ $$( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32$$ (v^11 - v^9 - 6*v^7 - 4*v^5 + 40*v^3 + 32*v) / 32 $$\beta_{10}$$ $$=$$ $$( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} - 32 ) / 16$$ (-v^10 - 3*v^8 + 2*v^6 + 4*v^4 - 8*v^2 - 32) / 16 $$\beta_{11}$$ $$=$$ $$( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64$$ (3*v^11 - 3*v^9 - 10*v^7 - 36*v^5 - 8*v^3 + 128*v) / 64
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{8} - 2\beta_{6} + \beta_{3} ) / 4$$ (b11 + b8 - 2*b6 + b3) / 4 $$\nu^{2}$$ $$=$$ $$( -2\beta_{10} - 2\beta_{7} + \beta_{5} - \beta _1 - 1 ) / 4$$ (-2*b10 - 2*b7 + b5 - b1 - 1) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{11} + 2\beta_{9} + \beta_{8} - \beta_{3} + 2\beta_{2} ) / 4$$ (-b11 + 2*b9 + b8 - b3 + 2*b2) / 4 $$\nu^{4}$$ $$=$$ $$( 2\beta_{10} - 2\beta_{7} - \beta_{5} + 4\beta_{4} + 5\beta _1 + 1 ) / 4$$ (2*b10 - 2*b7 - b5 + 4*b4 + 5*b1 + 1) / 4 $$\nu^{5}$$ $$=$$ $$( -3\beta_{11} + 2\beta_{9} - \beta_{8} - 4\beta_{6} + 13\beta_{3} - 6\beta_{2} ) / 4$$ (-3*b11 + 2*b9 - b8 - 4*b6 + 13*b3 - 6*b2) / 4 $$\nu^{6}$$ $$=$$ $$( -2\beta_{10} - 6\beta_{7} + \beta_{5} - 12\beta_{4} + 3\beta _1 + 15 ) / 4$$ (-2*b10 - 6*b7 + b5 - 12*b4 + 3*b1 + 15) / 4 $$\nu^{7}$$ $$=$$ $$( 3\beta_{11} - 10\beta_{9} + 9\beta_{8} - 4\beta_{6} + 19\beta_{3} + 14\beta_{2} ) / 4$$ (3*b11 - 10*b9 + 9*b8 - 4*b6 + 19*b3 + 14*b2) / 4 $$\nu^{8}$$ $$=$$ $$( -14\beta_{10} + 6\beta_{7} + 7\beta_{5} - 4\beta_{4} + 5\beta _1 - 23 ) / 4$$ (-14*b10 + 6*b7 + 7*b5 - 4*b4 + 5*b1 - 23) / 4 $$\nu^{9}$$ $$=$$ $$( -11\beta_{11} + 10\beta_{9} - 17\beta_{8} - 12\beta_{6} - 27\beta_{3} + 18\beta_{2} ) / 4$$ (-11*b11 + 10*b9 - 17*b8 - 12*b6 - 27*b3 + 18*b2) / 4 $$\nu^{10}$$ $$=$$ $$( -2\beta_{10} - 22\beta_{7} - 31\beta_{5} + 4\beta_{4} + 19\beta _1 - 17 ) / 4$$ (-2*b10 - 22*b7 - 31*b5 + 4*b4 + 19*b1 - 17) / 4 $$\nu^{11}$$ $$=$$ $$( 3\beta_{11} + 6\beta_{9} - 39\beta_{8} + 12\beta_{6} + 147\beta_{3} - 2\beta_{2} ) / 4$$ (3*b11 + 6*b9 - 39*b8 + 12*b6 + 147*b3 - 2*b2) / 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.m 12
4.b odd 2 1 3840.2.f.l 12
5.b even 2 1 inner 3840.2.f.m 12
8.b even 2 1 inner 3840.2.f.m 12
8.d odd 2 1 3840.2.f.l 12
16.e even 4 1 480.2.d.a 6
16.e even 4 1 480.2.d.b 6
16.f odd 4 1 120.2.d.a 6
16.f odd 4 1 120.2.d.b yes 6
20.d odd 2 1 3840.2.f.l 12
40.e odd 2 1 3840.2.f.l 12
40.f even 2 1 inner 3840.2.f.m 12
48.i odd 4 1 1440.2.d.e 6
48.i odd 4 1 1440.2.d.f 6
48.k even 4 1 360.2.d.e 6
48.k even 4 1 360.2.d.f 6
80.i odd 4 2 2400.2.k.f 12
80.j even 4 2 600.2.k.f 12
80.k odd 4 1 120.2.d.a 6
80.k odd 4 1 120.2.d.b yes 6
80.q even 4 1 480.2.d.a 6
80.q even 4 1 480.2.d.b 6
80.s even 4 2 600.2.k.f 12
80.t odd 4 2 2400.2.k.f 12
240.t even 4 1 360.2.d.e 6
240.t even 4 1 360.2.d.f 6
240.z odd 4 2 1800.2.k.u 12
240.bb even 4 2 7200.2.k.u 12
240.bd odd 4 2 1800.2.k.u 12
240.bf even 4 2 7200.2.k.u 12
240.bm odd 4 1 1440.2.d.e 6
240.bm odd 4 1 1440.2.d.f 6

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

## Hecke kernels

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

 $$T_{7}^{6} + 24T_{7}^{4} + 128T_{7}^{2} + 64$$ T7^6 + 24*T7^4 + 128*T7^2 + 64 $$T_{11}^{6} - 32T_{11}^{4} + 96T_{11}^{2} - 64$$ T11^6 - 32*T11^4 + 96*T11^2 - 64 $$T_{29}^{6} - 108T_{29}^{4} + 3120T_{29}^{2} - 12544$$ T29^6 - 108*T29^4 + 3120*T29^2 - 12544 $$T_{31}^{3} - 8T_{31}^{2} - 4T_{31} + 64$$ T31^3 - 8*T31^2 - 4*T31 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{2} + 1)^{6}$$
$5$ $$T^{12} + 2 T^{10} - 9 T^{8} + \cdots + 15625$$
$7$ $$(T^{6} + 24 T^{4} + 128 T^{2} + 64)^{2}$$
$11$ $$(T^{6} - 32 T^{4} + 96 T^{2} - 64)^{2}$$
$13$ $$(T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2}$$
$17$ $$(T^{6} + 36 T^{4} + 368 T^{2} + 1024)^{2}$$
$19$ $$(T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2}$$
$23$ $$(T^{6} + 92 T^{4} + 2304 T^{2} + \cdots + 16384)^{2}$$
$29$ $$(T^{6} - 108 T^{4} + 3120 T^{2} + \cdots - 12544)^{2}$$
$31$ $$(T^{3} - 8 T^{2} - 4 T + 64)^{4}$$
$37$ $$(T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2}$$
$41$ $$(T^{3} - 2 T^{2} - 100 T - 56)^{4}$$
$43$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$47$ $$(T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2}$$
$53$ $$(T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2}$$
$59$ $$(T^{6} - 176 T^{4} + 9888 T^{2} + \cdots - 179776)^{2}$$
$61$ $$(T^{6} - 176 T^{4} + 7168 T^{2} + \cdots - 65536)^{2}$$
$67$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$71$ $$(T^{3} - 8 T^{2} - 80 T + 128)^{4}$$
$73$ $$(T^{6} + 384 T^{4} + 34560 T^{2} + \cdots + 16384)^{2}$$
$79$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{4}$$
$83$ $$(T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2}$$
$89$ $$(T^{3} - 10 T^{2} - 164 T + 1384)^{4}$$
$97$ $$(T^{6} + 336 T^{4} + 28416 T^{2} + \cdots + 262144)^{2}$$