# Properties

 Label 4032.2.k.h Level 4032 Weight 2 Character orbit 4032.k Analytic conductor 32.196 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.101415451701035401216.7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{26}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 2016) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + \beta_{3} q^{7} + \beta_{4} q^{11} + \beta_{9} q^{13} -\beta_{1} q^{17} + \beta_{10} q^{19} + \beta_{4} q^{23} + ( 1 - \beta_{8} ) q^{25} + \beta_{7} q^{29} + ( \beta_{2} - \beta_{3} + \beta_{10} ) q^{31} + ( -\beta_{4} + \beta_{11} + \beta_{13} ) q^{35} + ( -2 + \beta_{8} ) q^{37} + \beta_{12} q^{41} + ( \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{4} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{47} + ( 1 - \beta_{8} - \beta_{9} ) q^{49} -\beta_{14} q^{53} + \beta_{10} q^{55} + ( -\beta_{13} - \beta_{15} ) q^{59} -\beta_{6} q^{61} + ( -\beta_{7} + \beta_{14} ) q^{65} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{67} + \beta_{11} q^{71} + ( \beta_{6} - \beta_{9} ) q^{73} + ( -\beta_{12} - \beta_{14} ) q^{77} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{79} + ( -\beta_{4} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{83} + ( -6 + \beta_{8} ) q^{85} + ( 2 \beta_{1} + \beta_{12} ) q^{89} + ( 3 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} ) q^{91} + ( \beta_{4} + 3 \beta_{11} ) q^{95} + ( \beta_{6} + \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 16q^{25} - 32q^{37} + 16q^{49} - 96q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-49 \nu^{15} - 42 \nu^{13} + 910 \nu^{11} + 780 \nu^{9} - 7625 \nu^{7} - 6330 \nu^{5} + 7268 \nu^{3} + 4584 \nu$$$$)/1440$$ $$\beta_{2}$$ $$=$$ $$($$$$35 \nu^{15} + 51 \nu^{14} - 30 \nu^{13} - 2 \nu^{12} - 650 \nu^{11} - 930 \nu^{10} + 540 \nu^{9} + 20 \nu^{8} + 5395 \nu^{7} + 7635 \nu^{6} - 4470 \nu^{5} - 10 \nu^{4} - 5140 \nu^{3} - 6252 \nu^{2} + 3240 \nu - 1016$$$$)/1440$$ $$\beta_{3}$$ $$=$$ $$($$$$-35 \nu^{15} + 51 \nu^{14} + 30 \nu^{13} - 2 \nu^{12} + 650 \nu^{11} - 930 \nu^{10} - 540 \nu^{9} + 20 \nu^{8} - 5395 \nu^{7} + 7635 \nu^{6} + 4470 \nu^{5} - 10 \nu^{4} + 5140 \nu^{3} - 6252 \nu^{2} - 3240 \nu - 1016$$$$)/1440$$ $$\beta_{4}$$ $$=$$ $$($$$$-39 \nu^{14} + 64 \nu^{12} + 690 \nu^{10} - 1120 \nu^{8} - 5415 \nu^{6} + 8960 \nu^{4} + 948 \nu^{2} - 2288$$$$)/720$$ $$\beta_{5}$$ $$=$$ $$($$$$51 \nu^{14} + 10 \nu^{12} - 930 \nu^{10} - 100 \nu^{8} + 7635 \nu^{6} + 50 \nu^{4} - 6252 \nu^{2} + 5080$$$$)/720$$ $$\beta_{6}$$ $$=$$ $$($$$$143 \nu^{15} + 194 \nu^{13} - 2450 \nu^{11} - 3500 \nu^{9} + 18535 \nu^{7} + 27970 \nu^{5} + 6164 \nu^{3} - 10408 \nu$$$$)/720$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{14} - 36 \nu^{12} - 194 \nu^{10} + 648 \nu^{8} + 1531 \nu^{6} - 5076 \nu^{4} - 268 \nu^{2} + 1296$$$$)/144$$ $$\beta_{8}$$ $$=$$ $$($$$$7 \nu^{14} - 130 \nu^{10} + 1079 \nu^{6} - 884 \nu^{2}$$$$)/72$$ $$\beta_{9}$$ $$=$$ $$($$$$145 \nu^{15} + 34 \nu^{13} - 2590 \nu^{11} - 580 \nu^{9} + 20705 \nu^{7} + 4490 \nu^{5} - 7820 \nu^{3} + 1672 \nu$$$$)/720$$ $$\beta_{10}$$ $$=$$ $$($$$$17 \nu^{15} + 8 \nu^{13} - 302 \nu^{11} - 140 \nu^{9} + 2389 \nu^{7} + 1084 \nu^{5} - 592 \nu^{3} - 88 \nu$$$$)/72$$ $$\beta_{11}$$ $$=$$ $$($$$$117 \nu^{14} + 64 \nu^{12} - 2070 \nu^{10} - 1120 \nu^{8} + 16245 \nu^{6} + 8960 \nu^{4} - 2844 \nu^{2} - 2288$$$$)/720$$ $$\beta_{12}$$ $$=$$ $$($$$$-433 \nu^{15} + 262 \nu^{13} + 7630 \nu^{11} - 4660 \nu^{9} - 59945 \nu^{7} + 36950 \nu^{5} + 9476 \nu^{3} - 7064 \nu$$$$)/1440$$ $$\beta_{13}$$ $$=$$ $$($$$$275 \nu^{15} - 156 \nu^{14} + 10 \nu^{13} - 4970 \nu^{11} + 2760 \nu^{10} - 220 \nu^{9} + 40075 \nu^{7} - 21660 \nu^{6} + 2570 \nu^{5} - 21340 \nu^{3} + 3792 \nu^{2} - 8840 \nu$$$$)/1440$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{14} - 194 \nu^{10} + 1531 \nu^{6} - 268 \nu^{2}$$$$)/48$$ $$\beta_{15}$$ $$=$$ $$($$$$135 \nu^{15} + 52 \nu^{14} - 110 \nu^{13} - 2370 \nu^{11} - 920 \nu^{10} + 1940 \nu^{9} + 18495 \nu^{7} + 7220 \nu^{6} - 15310 \nu^{5} - 780 \nu^{3} - 1264 \nu^{2} + 4120 \nu$$$$)/480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} + 3 \beta_{12} - \beta_{11} - 3 \beta_{10} + 4 \beta_{9} + \beta_{6} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{14} - 3 \beta_{11} + 6 \beta_{8} - 2 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} - 10 \beta_{2}$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{15} - 3 \beta_{13} + 6 \beta_{12} - 5 \beta_{11} + 9 \beta_{10} - 2 \beta_{9} - 5 \beta_{6} + 5 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} - 12 \beta_{1}$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{14} + 9 \beta_{11} + 12 \beta_{7} - 12 \beta_{5} + 27 \beta_{4} + 12 \beta_{3} + 12 \beta_{2} + 108$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{15} + 27 \beta_{13} + 24 \beta_{12} + 7 \beta_{11} - 33 \beta_{10} + 42 \beta_{9} + 3 \beta_{6} - 7 \beta_{4} + 21 \beta_{3} - 21 \beta_{2} + 54 \beta_{1}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$100 \beta_{14} - 105 \beta_{11} + 42 \beta_{8} - 10 \beta_{5} + 105 \beta_{4} - 50 \beta_{3} - 50 \beta_{2}$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$35 \beta_{15} - 87 \beta_{13} + 6 \beta_{12} - 61 \beta_{11} + 9 \beta_{10} + 78 \beta_{9} - 45 \beta_{6} + 61 \beta_{4} + 183 \beta_{3} - 183 \beta_{2} - 252 \beta_{1}$$$$)/24$$ $$\nu^{8}$$ $$=$$ $$($$$$-24 \beta_{14} + 51 \beta_{11} + 72 \beta_{7} - 8 \beta_{5} + 153 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 68$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$-49 \beta_{15} + 333 \beta_{13} + 66 \beta_{12} + 191 \beta_{11} - 93 \beta_{10} + 358 \beta_{9} - 113 \beta_{6} - 191 \beta_{4} + 573 \beta_{3} - 573 \beta_{2} + 876 \beta_{1}$$$$)/24$$ $$\nu^{10}$$ $$=$$ $$($$$$1276 \beta_{14} - 1353 \beta_{11} - 246 \beta_{8} + 58 \beta_{5} + 1353 \beta_{4} + 290 \beta_{3} + 290 \beta_{2}$$$$)/24$$ $$\nu^{11}$$ $$=$$ $$($$$$-299 \beta_{15} - 1173 \beta_{13} - 732 \beta_{12} - 437 \beta_{11} - 1035 \beta_{10} + 1594 \beta_{9} - 65 \beta_{6} + 437 \beta_{4} + 1311 \beta_{3} - 1311 \beta_{2} - 2586 \beta_{1}$$$$)/24$$ $$\nu^{12}$$ $$=$$ $$($$$$-700 \beta_{14} + 1485 \beta_{11} + 2100 \beta_{7} + 1260 \beta_{5} + 4455 \beta_{4} - 1260 \beta_{3} - 1260 \beta_{2} - 10692$$$$)/24$$ $$\nu^{13}$$ $$=$$ $$($$$$-2651 \beta_{15} + 2115 \beta_{13} - 2064 \beta_{12} + 2383 \beta_{11} + 2919 \beta_{10} + 618 \beta_{9} - 2373 \beta_{6} - 2383 \beta_{4} + 7149 \beta_{3} - 7149 \beta_{2} + 8046 \beta_{1}$$$$)/24$$ $$\nu^{14}$$ $$=$$ $$($$$$8788 \beta_{14} - 9321 \beta_{11} - 10038 \beta_{8} + 2366 \beta_{5} + 9321 \beta_{4} + 11830 \beta_{3} + 11830 \beta_{2}$$$$)/24$$ $$\nu^{15}$$ $$=$$ $$($$$$-9961 \beta_{15} - 8691 \beta_{13} - 13638 \beta_{12} + 635 \beta_{11} - 19287 \beta_{10} + 17286 \beta_{9} + 4995 \beta_{6} - 635 \beta_{4} - 1905 \beta_{3} + 1905 \beta_{2} - 10944 \beta_{1}$$$$)/24$$

## Character values

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

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3905.1
 −0.752908 − 0.137538i −0.752908 + 0.137538i −0.332046 + 1.81768i −0.332046 − 1.81768i 1.81768 − 0.332046i 1.81768 + 0.332046i 0.137538 + 0.752908i 0.137538 − 0.752908i −1.81768 − 0.332046i −1.81768 + 0.332046i −0.137538 + 0.752908i −0.137538 − 0.752908i 0.752908 − 0.137538i 0.752908 + 0.137538i 0.332046 + 1.81768i 0.332046 − 1.81768i
0 0 0 −3.36028 0 −2.57794 0.595188i 0 0 0
3905.2 0 0 0 −3.36028 0 −2.57794 + 0.595188i 0 0 0
3905.3 0 0 0 −3.36028 0 2.57794 0.595188i 0 0 0
3905.4 0 0 0 −3.36028 0 2.57794 + 0.595188i 0 0 0
3905.5 0 0 0 −0.841723 0 −1.16372 2.37608i 0 0 0
3905.6 0 0 0 −0.841723 0 −1.16372 + 2.37608i 0 0 0
3905.7 0 0 0 −0.841723 0 1.16372 2.37608i 0 0 0
3905.8 0 0 0 −0.841723 0 1.16372 + 2.37608i 0 0 0
3905.9 0 0 0 0.841723 0 −1.16372 2.37608i 0 0 0
3905.10 0 0 0 0.841723 0 −1.16372 + 2.37608i 0 0 0
3905.11 0 0 0 0.841723 0 1.16372 2.37608i 0 0 0
3905.12 0 0 0 0.841723 0 1.16372 + 2.37608i 0 0 0
3905.13 0 0 0 3.36028 0 −2.57794 0.595188i 0 0 0
3905.14 0 0 0 3.36028 0 −2.57794 + 0.595188i 0 0 0
3905.15 0 0 0 3.36028 0 2.57794 0.595188i 0 0 0
3905.16 0 0 0 3.36028 0 2.57794 + 0.595188i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3905.16 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
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.k.h 16
3.b odd 2 1 inner 4032.2.k.h 16
4.b odd 2 1 inner 4032.2.k.h 16
7.b odd 2 1 inner 4032.2.k.h 16
8.b even 2 1 2016.2.k.b 16
8.d odd 2 1 2016.2.k.b 16
12.b even 2 1 inner 4032.2.k.h 16
21.c even 2 1 inner 4032.2.k.h 16
24.f even 2 1 2016.2.k.b 16
24.h odd 2 1 2016.2.k.b 16
28.d even 2 1 inner 4032.2.k.h 16
56.e even 2 1 2016.2.k.b 16
56.h odd 2 1 2016.2.k.b 16
84.h odd 2 1 inner 4032.2.k.h 16
168.e odd 2 1 2016.2.k.b 16
168.i even 2 1 2016.2.k.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.k.b 16 8.b even 2 1
2016.2.k.b 16 8.d odd 2 1
2016.2.k.b 16 24.f even 2 1
2016.2.k.b 16 24.h odd 2 1
2016.2.k.b 16 56.e even 2 1
2016.2.k.b 16 56.h odd 2 1
2016.2.k.b 16 168.e odd 2 1
2016.2.k.b 16 168.i even 2 1
4032.2.k.h 16 1.a even 1 1 trivial
4032.2.k.h 16 3.b odd 2 1 inner
4032.2.k.h 16 4.b odd 2 1 inner
4032.2.k.h 16 7.b odd 2 1 inner
4032.2.k.h 16 12.b even 2 1 inner
4032.2.k.h 16 21.c even 2 1 inner
4032.2.k.h 16 28.d even 2 1 inner
4032.2.k.h 16 84.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 12 T_{5}^{2} + 8$$ $$T_{43}^{4} - 32 T_{43}^{2} + 144$$ $$T_{67}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 8 T^{2} + 38 T^{4} + 200 T^{6} + 625 T^{8} )^{4}$$
$7$ $$( 1 - 4 T^{2} - 10 T^{4} - 196 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 28 T^{2} + 410 T^{4} - 3388 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 12 T^{2} + 262 T^{4} - 2028 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 + 56 T^{2} + 1334 T^{4} + 16184 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 36 T^{2} + 934 T^{4} - 12996 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 - 76 T^{2} + 2474 T^{4} - 40204 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 + 1234 T^{4} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 44 T^{2} + 1958 T^{4} - 42284 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 + 4 T + 50 T^{2} + 148 T^{3} + 1369 T^{4} )^{8}$$
$41$ $$( 1 + 88 T^{2} + 3926 T^{4} + 147928 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 140 T^{2} + 8486 T^{4} + 258860 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 - 88 T^{2} + 2809 T^{4} )^{8}$$
$59$ $$( 1 + 76 T^{2} + 6614 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 36 T^{2} + 7318 T^{4} - 133956 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 + 62 T^{2} + 4489 T^{4} )^{8}$$
$71$ $$( 1 - 252 T^{2} + 25706 T^{4} - 1270332 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 12 T^{2} + 5206 T^{4} - 63948 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 + 86 T^{2} + 6241 T^{4} )^{8}$$
$83$ $$( 1 + 124 T^{2} + 17174 T^{4} + 854236 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 + 248 T^{2} + 31190 T^{4} + 1964408 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 172 T^{2} + 17142 T^{4} - 1618348 T^{6} + 88529281 T^{8} )^{4}$$