# Properties

 Label 882.5.c.a Level $882$ Weight $5$ Character orbit 882.c Analytic conductor $91.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 882.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$91.1723074400$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + 8 q^{4} + ( -11 \beta_{2} - \beta_{3} ) q^{5} -8 \beta_{1} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + 8 q^{4} + ( -11 \beta_{2} - \beta_{3} ) q^{5} -8 \beta_{1} q^{8} + ( -8 \beta_{2} - 11 \beta_{3} ) q^{10} + ( -81 + 6 \beta_{1} ) q^{11} + ( -36 \beta_{2} + 34 \beta_{3} ) q^{13} + 64 q^{16} + ( 34 \beta_{2} - 34 \beta_{3} ) q^{17} + ( 74 \beta_{2} - 34 \beta_{3} ) q^{19} + ( -88 \beta_{2} - 8 \beta_{3} ) q^{20} + ( -48 + 81 \beta_{1} ) q^{22} + ( 156 - 306 \beta_{1} ) q^{23} + ( 238 + 66 \beta_{1} ) q^{25} + ( 272 \beta_{2} - 36 \beta_{3} ) q^{26} + ( -681 - 21 \beta_{1} ) q^{29} + ( -631 \beta_{2} - 87 \beta_{3} ) q^{31} -64 \beta_{1} q^{32} + ( -272 \beta_{2} + 34 \beta_{3} ) q^{34} + ( -698 - 48 \beta_{1} ) q^{37} + ( -272 \beta_{2} + 74 \beta_{3} ) q^{38} + ( -64 \beta_{2} - 88 \beta_{3} ) q^{40} + ( 518 \beta_{2} - 392 \beta_{3} ) q^{41} + ( -158 + 1140 \beta_{1} ) q^{43} + ( -648 + 48 \beta_{1} ) q^{44} + ( 2448 - 156 \beta_{1} ) q^{46} + ( -1316 \beta_{2} - 340 \beta_{3} ) q^{47} + ( -528 - 238 \beta_{1} ) q^{50} + ( -288 \beta_{2} + 272 \beta_{3} ) q^{52} + ( -519 + 591 \beta_{1} ) q^{53} + ( 939 \beta_{2} + 147 \beta_{3} ) q^{55} + ( 168 + 681 \beta_{1} ) q^{58} + ( 161 \beta_{2} - 656 \beta_{3} ) q^{59} + ( -848 \beta_{2} + 164 \beta_{3} ) q^{61} + ( -696 \beta_{2} - 631 \beta_{3} ) q^{62} + 512 q^{64} + ( -372 - 1014 \beta_{1} ) q^{65} + ( -7300 - 486 \beta_{1} ) q^{67} + ( 272 \beta_{2} - 272 \beta_{3} ) q^{68} + ( 2424 - 732 \beta_{1} ) q^{71} + ( 3764 \beta_{2} + 500 \beta_{3} ) q^{73} + ( 384 + 698 \beta_{1} ) q^{74} + ( 592 \beta_{2} - 272 \beta_{3} ) q^{76} + ( -1987 + 2991 \beta_{1} ) q^{79} + ( -704 \beta_{2} - 64 \beta_{3} ) q^{80} + ( -3136 \beta_{2} + 518 \beta_{3} ) q^{82} + ( 2741 \beta_{2} - 710 \beta_{3} ) q^{83} + ( 306 + 1020 \beta_{1} ) q^{85} + ( -9120 + 158 \beta_{1} ) q^{86} + ( -384 + 648 \beta_{1} ) q^{88} + ( -5526 \beta_{2} + 1218 \beta_{3} ) q^{89} + ( 1248 - 2448 \beta_{1} ) q^{92} + ( -2720 \beta_{2} - 1316 \beta_{3} ) q^{94} + ( 1626 + 900 \beta_{1} ) q^{95} + ( 1013 \beta_{2} + 40 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{4} + O(q^{10})$$ $$4q + 32q^{4} - 324q^{11} + 256q^{16} - 192q^{22} + 624q^{23} + 952q^{25} - 2724q^{29} - 2792q^{37} - 632q^{43} - 2592q^{44} + 9792q^{46} - 2112q^{50} - 2076q^{53} + 672q^{58} + 2048q^{64} - 1488q^{65} - 29200q^{67} + 9696q^{71} + 1536q^{74} - 7948q^{79} + 1224q^{85} - 36480q^{86} - 1536q^{88} + 4992q^{92} + 6504q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i 0.707107 − 1.22474i
−2.82843 0 8.00000 14.1536i 0 0 −22.6274 0 40.0324i
685.2 −2.82843 0 8.00000 14.1536i 0 0 −22.6274 0 40.0324i
685.3 2.82843 0 8.00000 23.9515i 0 0 22.6274 0 67.7452i
685.4 2.82843 0 8.00000 23.9515i 0 0 22.6274 0 67.7452i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.5.c.a 4
3.b odd 2 1 294.5.c.a 4
7.b odd 2 1 inner 882.5.c.a 4
7.c even 3 1 126.5.n.b 4
7.d odd 6 1 126.5.n.b 4
21.c even 2 1 294.5.c.a 4
21.g even 6 1 42.5.g.a 4
21.g even 6 1 294.5.g.c 4
21.h odd 6 1 42.5.g.a 4
21.h odd 6 1 294.5.g.c 4
84.j odd 6 1 336.5.bh.d 4
84.n even 6 1 336.5.bh.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.g.a 4 21.g even 6 1
42.5.g.a 4 21.h odd 6 1
126.5.n.b 4 7.c even 3 1
126.5.n.b 4 7.d odd 6 1
294.5.c.a 4 3.b odd 2 1
294.5.c.a 4 21.c even 2 1
294.5.g.c 4 21.g even 6 1
294.5.g.c 4 21.h odd 6 1
336.5.bh.d 4 84.j odd 6 1
336.5.bh.d 4 84.n even 6 1
882.5.c.a 4 1.a even 1 1 trivial
882.5.c.a 4 7.b 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 $$S_{5}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 774 T_{5}^{2} + 114921$$ $$T_{11}^{2} + 162 T_{11} + 6273$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -8 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$114921 + 774 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 6273 + 162 T + T^{2} )^{2}$$
$13$ $$569108736 + 63264 T^{2} + T^{4}$$
$17$ $$589324176 + 62424 T^{2} + T^{4}$$
$19$ $$128051856 + 88344 T^{2} + T^{4}$$
$23$ $$( -724752 - 312 T + T^{2} )^{2}$$
$29$ $$( 460233 + 1362 T + T^{2} )^{2}$$
$31$ $$1025818531929 + 2752278 T^{2} + T^{4}$$
$37$ $$( 468772 + 1396 T + T^{2} )^{2}$$
$41$ $$8311481425296 + 8985816 T^{2} + T^{4}$$
$43$ $$( -10371836 + 316 T + T^{2} )^{2}$$
$47$ $$5862054484224 + 15939936 T^{2} + T^{4}$$
$53$ $$( -2524887 + 1038 T + T^{2} )^{2}$$
$59$ $$105068670590601 + 20811654 T^{2} + T^{4}$$
$61$ $$2285563428864 + 5605632 T^{2} + T^{4}$$
$67$ $$( 51400432 + 14600 T + T^{2} )^{2}$$
$71$ $$( 1589184 - 4848 T + T^{2} )^{2}$$
$73$ $$1332475433535744 + 97006176 T^{2} + T^{4}$$
$79$ $$( -67620479 + 3974 T + T^{2} )^{2}$$
$83$ $$109011202550649 + 69275286 T^{2} + T^{4}$$
$89$ $$3136610653724304 + 254429208 T^{2} + T^{4}$$
$97$ $$9242250571449 + 6233814 T^{2} + T^{4}$$