# Properties

 Label 48.6.c.d Level 48 Weight 6 Character orbit 48.c Analytic conductor 7.698 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.69842335102$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-14})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} -\beta_{3} q^{5} + 11 \beta_{1} q^{7} + ( 93 - 5 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} -\beta_{3} q^{5} + 11 \beta_{1} q^{7} + ( 93 - 5 \beta_{3} ) q^{9} + ( 21 \beta_{1} + 42 \beta_{2} ) q^{11} + 814 q^{13} + ( 183 \beta_{1} + 30 \beta_{2} ) q^{15} -44 \beta_{3} q^{17} -11 \beta_{1} q^{19} + ( -330 - 11 \beta_{3} ) q^{21} + ( -54 \beta_{1} - 108 \beta_{2} ) q^{23} + 1109 q^{25} + ( 1101 \beta_{1} + 57 \beta_{2} ) q^{27} + 121 \beta_{3} q^{29} -3009 \beta_{1} q^{31} + ( -7056 + 105 \beta_{3} ) q^{33} + ( 66 \beta_{1} + 132 \beta_{2} ) q^{35} -1594 q^{37} + ( 1628 \beta_{1} - 814 \beta_{2} ) q^{39} + 198 \beta_{3} q^{41} -2607 \beta_{1} q^{43} + ( -10080 - 93 \beta_{3} ) q^{45} + ( -636 \beta_{1} - 1272 \beta_{2} ) q^{47} + 15355 q^{49} + ( 8052 \beta_{1} + 1320 \beta_{2} ) q^{51} -429 \beta_{3} q^{53} -7056 \beta_{1} q^{55} + ( 330 + 11 \beta_{3} ) q^{57} + ( 1761 \beta_{1} + 3522 \beta_{2} ) q^{59} + 23870 q^{61} + ( 1353 \beta_{1} + 660 \beta_{2} ) q^{63} -814 \beta_{3} q^{65} + 8773 \beta_{1} q^{67} + ( 18144 - 270 \beta_{3} ) q^{69} + ( -1266 \beta_{1} - 2532 \beta_{2} ) q^{71} + 17578 q^{73} + ( 2218 \beta_{1} - 1109 \beta_{2} ) q^{75} + 462 \beta_{3} q^{77} -2057 \beta_{1} q^{79} + ( -41751 - 930 \beta_{3} ) q^{81} + ( 627 \beta_{1} + 1254 \beta_{2} ) q^{83} -88704 q^{85} + ( -22143 \beta_{1} - 3630 \beta_{2} ) q^{87} + 1970 \beta_{3} q^{89} + 8954 \beta_{1} q^{91} + ( 90270 + 3009 \beta_{3} ) q^{93} + ( -66 \beta_{1} - 132 \beta_{2} ) q^{95} -49070 q^{97} + ( -33327 \beta_{1} + 3906 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 372q^{9} + O(q^{10})$$ $$4q + 372q^{9} + 3256q^{13} - 1320q^{21} + 4436q^{25} - 28224q^{33} - 6376q^{37} - 40320q^{45} + 61420q^{49} + 1320q^{57} + 95480q^{61} + 72576q^{69} + 70312q^{73} - 167004q^{81} - 354816q^{85} + 361080q^{93} - 196280q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 14 x^{2} + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{2} - 14$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - \nu^{2} + 28 \nu + 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 6 \beta_{2} + 3 \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{1} + 14$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$$$/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 3.24037 − 1.87083i 3.24037 + 1.87083i −3.24037 + 1.87083i −3.24037 − 1.87083i
0 −12.9615 8.66025i 0 44.8999i 0 38.1051i 0 93.0000 + 224.499i 0
47.2 0 −12.9615 + 8.66025i 0 44.8999i 0 38.1051i 0 93.0000 224.499i 0
47.3 0 12.9615 8.66025i 0 44.8999i 0 38.1051i 0 93.0000 224.499i 0
47.4 0 12.9615 + 8.66025i 0 44.8999i 0 38.1051i 0 93.0000 + 224.499i 0
 $$n$$: e.g. 2-40 or 990-1000 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.c.d 4
3.b odd 2 1 inner 48.6.c.d 4
4.b odd 2 1 inner 48.6.c.d 4
8.b even 2 1 192.6.c.d 4
8.d odd 2 1 192.6.c.d 4
12.b even 2 1 inner 48.6.c.d 4
24.f even 2 1 192.6.c.d 4
24.h odd 2 1 192.6.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.d 4 1.a even 1 1 trivial
48.6.c.d 4 3.b odd 2 1 inner
48.6.c.d 4 4.b odd 2 1 inner
48.6.c.d 4 12.b even 2 1 inner
192.6.c.d 4 8.b even 2 1
192.6.c.d 4 8.d odd 2 1
192.6.c.d 4 24.f even 2 1
192.6.c.d 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(48, [\chi])$$:

 $$T_{5}^{2} + 2016$$ $$T_{11}^{2} - 296352$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 186 T^{2} + 59049 T^{4}$$
$5$ $$( 1 - 4234 T^{2} + 9765625 T^{4} )^{2}$$
$7$ $$( 1 - 32162 T^{2} + 282475249 T^{4} )^{2}$$
$11$ $$( 1 + 25750 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 - 814 T + 371293 T^{2} )^{4}$$
$17$ $$( 1 + 1063262 T^{2} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 - 4950746 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$( 1 + 10913134 T^{2} + 41426511213649 T^{4} )^{2}$$
$29$ $$( 1 - 11506042 T^{2} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 51390670 T^{2} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 + 1594 T + 69343957 T^{2} )^{4}$$
$41$ $$( 1 - 152677138 T^{2} + 13422659310152401 T^{4} )^{2}$$
$43$ $$( 1 - 212459498 T^{2} + 21611482313284249 T^{4} )^{2}$$
$47$ $$( 1 + 186868702 T^{2} + 52599132235830049 T^{4} )^{2}$$
$53$ $$( 1 - 465364330 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 - 654104714 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 23870 T + 844596301 T^{2} )^{4}$$
$67$ $$( 1 - 1776663866 T^{2} + 1822837804551761449 T^{4} )^{2}$$
$71$ $$( 1 + 2531406670 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 17578 T + 2073071593 T^{2} )^{4}$$
$79$ $$( 1 - 6103337810 T^{2} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$( 1 + 7613898598 T^{2} + 15516041187205853449 T^{4} )^{2}$$
$89$ $$( 1 - 3344224498 T^{2} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 49070 T + 8587340257 T^{2} )^{4}$$