# Properties

 Label 4032.2.v.c Level 4032 Weight 2 Character orbit 4032.v Analytic conductor 32.196 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.653473922154496.1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + ( 1 - \beta_{2} - \beta_{8} ) q^{13} + ( -\beta_{1} - 3 \beta_{3} - \beta_{7} - \beta_{10} ) q^{17} -\beta_{10} q^{23} + \beta_{2} q^{25} + ( \beta_{1} - \beta_{6} - \beta_{10} ) q^{29} + ( 2 \beta_{2} + \beta_{4} + \beta_{8} ) q^{31} + ( \beta_{3} - \beta_{5} ) q^{35} + ( -1 - \beta_{2} ) q^{37} + ( \beta_{1} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{8} + \beta_{11} ) q^{43} + ( 2 \beta_{5} - 2 \beta_{6} ) q^{47} + q^{49} + ( 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{53} + ( -4 + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{55} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{59} + ( -3 + 3 \beta_{2} - \beta_{8} ) q^{61} + ( 2 \beta_{3} + 2 \beta_{10} ) q^{65} + ( -4 - 4 \beta_{2} + \beta_{4} ) q^{67} + ( -\beta_{1} + 2 \beta_{3} - \beta_{7} - 3 \beta_{10} ) q^{71} + ( -4 \beta_{2} + \beta_{9} - \beta_{11} ) q^{73} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{77} + ( -4 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} ) q^{79} + ( -5 \beta_{3} + 5 \beta_{5} - \beta_{6} + \beta_{10} ) q^{83} + ( 6 + 6 \beta_{2} - 2 \beta_{9} ) q^{85} + ( \beta_{1} - 7 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{89} + ( 1 - \beta_{2} - \beta_{8} ) q^{91} + ( -6 + 3 \beta_{4} - 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{7} + O(q^{10})$$ $$12q + 12q^{7} + 16q^{13} - 12q^{37} + 20q^{43} + 12q^{49} - 32q^{55} - 32q^{61} - 44q^{67} + 64q^{85} + 16q^{91} - 56q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{11} + 6 \nu^{9} - 5 \nu^{7} + 46 \nu^{5} - 32 \nu^{3} + 32 \nu$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{10} + 18 \nu^{8} - 55 \nu^{6} + 138 \nu^{4} - 256 \nu^{2} + 352$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3}$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 18 \nu^{9} + 49 \nu^{7} - 90 \nu^{5} + 136 \nu^{3} - 192 \nu$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{11} + 10 \nu^{9} - 27 \nu^{7} + 98 \nu^{5} - 128 \nu^{3} + 224 \nu$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$-7 \nu^{10} + 18 \nu^{8} - 31 \nu^{6} + 74 \nu^{4} - 160 \nu^{2} + 96$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$-11 \nu^{10} + 10 \nu^{8} - 67 \nu^{6} + 98 \nu^{4} - 160 \nu^{2} + 96$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} - 12 \nu^{9} + 31 \nu^{7} - 68 \nu^{5} + 116 \nu^{3} - 80 \nu$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-13 \nu^{10} + 54 \nu^{8} - 117 \nu^{6} + 286 \nu^{4} - 320 \nu^{2} + 416$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - 2 \beta_{8} - \beta_{4} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{10} + \beta_{7} - \beta_{6} + 5 \beta_{5} - \beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{11} + \beta_{9} - \beta_{8} - 10 \beta_{2} - 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{10} + 3 \beta_{7} + 3 \beta_{6} + 7 \beta_{5} + 3 \beta_{3} + 5 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{11} - 2 \beta_{9} + 8 \beta_{8} - \beta_{4} - 16 \beta_{2} - 2$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{10} + 3 \beta_{7} + 9 \beta_{6} - 13 \beta_{5} - 15 \beta_{3} + 3 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$2 \beta_{11} - 13 \beta_{9} + 5 \beta_{8} - 4 \beta_{4} + 26 \beta_{2} + 8$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-\beta_{10} - 11 \beta_{7} - 3 \beta_{6} - 23 \beta_{5} - 51 \beta_{3} + 3 \beta_{1}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-\beta_{11} - 14 \beta_{9} - 24 \beta_{8} + 17 \beta_{4} + 32 \beta_{2} - 46$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$3 \beta_{10} - 19 \beta_{7} - 25 \beta_{6} - 19 \beta_{5} + 31 \beta_{3} + 29 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1583.1
 1.35489 + 0.405301i −1.16947 − 0.795191i −0.892524 + 1.09700i 0.892524 − 1.09700i 1.16947 + 0.795191i −1.35489 − 0.405301i 1.35489 − 0.405301i −1.16947 + 0.795191i −0.892524 − 1.09700i 0.892524 + 1.09700i 1.16947 − 0.795191i −1.35489 + 0.405301i
0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.2 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.3 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.4 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.5 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.6 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.2 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.3 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.4 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.5 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.6 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3599.6 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.c 12
3.b odd 2 1 inner 4032.2.v.c 12
4.b odd 2 1 1008.2.v.c 12
12.b even 2 1 1008.2.v.c 12
16.e even 4 1 1008.2.v.c 12
16.f odd 4 1 inner 4032.2.v.c 12
48.i odd 4 1 1008.2.v.c 12
48.k even 4 1 inner 4032.2.v.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.c 12 4.b odd 2 1
1008.2.v.c 12 12.b even 2 1
1008.2.v.c 12 16.e even 4 1
1008.2.v.c 12 48.i odd 4 1
4032.2.v.c 12 1.a even 1 1 trivial
4032.2.v.c 12 3.b odd 2 1 inner
4032.2.v.c 12 16.f odd 4 1 inner
4032.2.v.c 12 48.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} + 16$$ $$T_{11}^{12} + 1056 T_{11}^{8} + 53504 T_{11}^{4} + 65536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )^{3}( 1 + 8 T^{2} + 25 T^{4} )^{3}$$
$7$ $$( 1 - T )^{12}$$
$11$ $$1 + 22 T^{4} + 14047 T^{8} + 1317556 T^{12} + 205662127 T^{16} + 4715895382 T^{20} + 3138428376721 T^{24}$$
$13$ $$( 1 - 8 T + 32 T^{2} - 88 T^{3} + 315 T^{4} - 1728 T^{5} + 7616 T^{6} - 22464 T^{7} + 53235 T^{8} - 193336 T^{9} + 913952 T^{10} - 2970344 T^{11} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 + 2 T^{2} + 223 T^{4} - 3076 T^{6} + 64447 T^{8} + 167042 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 + 361 T^{4} )^{6}$$
$23$ $$( 1 - 108 T^{2} + 5363 T^{4} - 156760 T^{6} + 2837027 T^{8} - 30222828 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$1 - 3530 T^{4} + 5536831 T^{8} - 5509074188 T^{12} + 3916095366511 T^{16} - 1765869837752330 T^{20} + 353814783205469041 T^{24}$$
$31$ $$( 1 - 106 T^{2} + 5071 T^{4} - 169228 T^{6} + 4873231 T^{8} - 97893226 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{6}$$
$41$ $$( 1 + 118 T^{2} + 4879 T^{4} + 140692 T^{6} + 8201599 T^{8} + 333439798 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 - 10 T + 50 T^{2} - 398 T^{3} - 1785 T^{4} + 35220 T^{5} - 183748 T^{6} + 1514460 T^{7} - 3300465 T^{8} - 31643786 T^{9} + 170940050 T^{10} - 1470084430 T^{11} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 154 T^{2} + 13423 T^{4} + 756268 T^{6} + 29651407 T^{8} + 751470874 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$1 + 790 T^{4} - 2418209 T^{8} + 5899921588 T^{12} - 19080832168529 T^{16} + 49185155424975190 T^{20} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 - 6314 T^{4} + 48612511 T^{8} - 159936163532 T^{12} + 589055344903471 T^{16} - 927087383033682794 T^{20} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$( 1 + 16 T + 128 T^{2} + 1440 T^{3} + 18779 T^{4} + 143024 T^{5} + 921472 T^{6} + 8724464 T^{7} + 69876659 T^{8} + 326852640 T^{9} + 1772267648 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 + 22 T + 242 T^{2} + 2850 T^{3} + 35511 T^{4} + 315092 T^{5} + 2399612 T^{6} + 21111164 T^{7} + 159408879 T^{8} + 857174550 T^{9} + 4876571282 T^{10} + 29702752354 T^{11} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 188 T^{2} + 25939 T^{4} - 2091512 T^{6} + 130758499 T^{8} - 4777396028 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 282 T^{2} + 42047 T^{4} - 3793004 T^{6} + 224068463 T^{8} - 8008303962 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 - 154 T^{2} + 1711 T^{4} + 731348 T^{6} + 10678351 T^{8} - 5998312474 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$1 - 12810 T^{4} + 46034687 T^{8} + 36989567092 T^{12} + 2184728952780527 T^{16} - 28851863493701115210 T^{20} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 + 46 T^{2} + 3343 T^{4} + 916708 T^{6} + 26479903 T^{8} + 2886143086 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 + 14 T - 33 T^{2} - 1948 T^{3} - 3201 T^{4} + 131726 T^{5} + 912673 T^{6} )^{4}$$