# Properties

 Label 1638.2.x.a Level $1638$ Weight $2$ Character orbit 1638.x Analytic conductor $13.079$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.836829184.2 Defining polynomial: $$x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{7} q^{4} + ( \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{7} q^{4} + ( \beta_{2} - \beta_{5} ) q^{5} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{2} q^{8} + ( \beta_{2} + \beta_{3} - \beta_{6} ) q^{10} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -2 + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{6} ) q^{14} - q^{16} + ( 2 - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{3} - \beta_{4} ) q^{20} + ( -2 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{22} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 5 \beta_{7} ) q^{23} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{25} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{26} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{28} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{31} + \beta_{3} q^{32} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{35} + ( 3 \beta_{2} - \beta_{5} ) q^{37} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{38} + ( \beta_{1} + \beta_{7} ) q^{40} + ( 1 + \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{41} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{44} + ( 3 - \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{46} -2 \beta_{3} q^{47} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{50} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{7} ) q^{52} + ( -6 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{53} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{7} ) q^{55} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{56} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{58} + ( -1 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} + 5 \beta_{7} ) q^{59} + ( 5 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{61} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{62} + \beta_{7} q^{64} + ( 2 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{65} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{68} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{70} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{73} + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} ) q^{74} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{76} + ( -3 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} - 5 \beta_{7} ) q^{77} + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{79} + ( -\beta_{2} + \beta_{5} ) q^{80} + ( -6 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{82} + ( 7 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + 6 \beta_{7} ) q^{83} + ( 2 - 3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{85} + ( 2 - 3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{86} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{88} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{89} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 4 - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{92} -2 \beta_{7} q^{94} + ( -\beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{95} + ( 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{97} + ( -2 - 2 \beta_{1} - \beta_{2} + 6 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{5} - 8q^{7} + O(q^{10})$$ $$8q - 4q^{5} - 8q^{7} - 4q^{10} - 16q^{13} + 4q^{14} - 8q^{16} + 4q^{17} + 8q^{19} - 4q^{20} - 12q^{22} - 16q^{26} - 12q^{29} + 8q^{31} + 8q^{34} + 24q^{35} - 4q^{37} - 4q^{38} + 12q^{41} + 24q^{46} - 32q^{49} + 8q^{50} + 4q^{52} - 40q^{53} - 4q^{56} + 4q^{58} - 8q^{59} + 8q^{62} + 12q^{65} + 32q^{67} + 12q^{71} - 20q^{73} - 20q^{74} + 8q^{76} + 8q^{77} + 24q^{79} + 4q^{80} - 40q^{82} + 44q^{83} + 20q^{85} + 20q^{86} + 16q^{89} - 12q^{91} + 28q^{92} + 8q^{97} - 16q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 11 \nu^{3} + 7 \nu^{2} + 26 \nu + 6$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 11 \nu^{3} + 7 \nu^{2} - 26 \nu + 6$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 11 \nu^{4} + 7 \nu^{3} + 26 \nu^{2} + 6 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + 11 \nu^{4} - 7 \nu^{3} + 26 \nu^{2} - 6 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 12$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 38 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{4} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$51 \beta_{6} - 47 \beta_{5} - 47 \beta_{4} - 44 \beta_{3} - 44 \beta_{2} - 87$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7} + 91 \beta_{5} - 91 \beta_{4} - 43 \beta_{3} + 43 \beta_{2} - 181 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\beta_{7}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 2.63640i − 0.222191i − 2.06644i 1.65222i − 2.63640i 0.222191i 2.06644i − 1.65222i
−0.707107 0.707107i 0 1.00000i −1.15711 + 1.15711i 0 −1.70711 2.02133i 0.707107 0.707107i 0 1.63640
307.2 −0.707107 0.707107i 0 1.00000i 0.864220 0.864220i 0 −1.70711 + 2.02133i 0.707107 0.707107i 0 −1.22219
307.3 0.707107 + 0.707107i 0 1.00000i −2.16830 + 2.16830i 0 −0.292893 2.62949i −0.707107 + 0.707107i 0 −3.06644
307.4 0.707107 + 0.707107i 0 1.00000i 0.461191 0.461191i 0 −0.292893 + 2.62949i −0.707107 + 0.707107i 0 0.652223
811.1 −0.707107 + 0.707107i 0 1.00000i −1.15711 1.15711i 0 −1.70711 + 2.02133i 0.707107 + 0.707107i 0 1.63640
811.2 −0.707107 + 0.707107i 0 1.00000i 0.864220 + 0.864220i 0 −1.70711 2.02133i 0.707107 + 0.707107i 0 −1.22219
811.3 0.707107 0.707107i 0 1.00000i −2.16830 2.16830i 0 −0.292893 + 2.62949i −0.707107 0.707107i 0 −3.06644
811.4 0.707107 0.707107i 0 1.00000i 0.461191 + 0.461191i 0 −0.292893 2.62949i −0.707107 0.707107i 0 0.652223
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.x.a 8
3.b odd 2 1 546.2.o.c yes 8
7.b odd 2 1 1638.2.x.c 8
13.d odd 4 1 1638.2.x.c 8
21.c even 2 1 546.2.o.b 8
39.f even 4 1 546.2.o.b 8
91.i even 4 1 inner 1638.2.x.a 8
273.o odd 4 1 546.2.o.c yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.o.b 8 21.c even 2 1
546.2.o.b 8 39.f even 4 1
546.2.o.c yes 8 3.b odd 2 1
546.2.o.c yes 8 273.o odd 4 1
1638.2.x.a 8 1.a even 1 1 trivial
1638.2.x.a 8 91.i even 4 1 inner
1638.2.x.c 8 7.b odd 2 1
1638.2.x.c 8 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} - 4 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$( 49 + 28 T + 16 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$256 + 768 T + 1152 T^{2} + 816 T^{3} + 321 T^{4} + 48 T^{5} + T^{8}$$
$13$ $$28561 + 35152 T + 19266 T^{2} + 6656 T^{3} + 1890 T^{4} + 512 T^{5} + 114 T^{6} + 16 T^{7} + T^{8}$$
$17$ $$( -8 - 32 T - 23 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$19$ $$4096 - 6144 T + 4608 T^{2} - 1376 T^{3} + 209 T^{4} - 24 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$23$ $$454276 + 103396 T^{2} + 6669 T^{4} + 146 T^{6} + T^{8}$$
$29$ $$( 98 - 224 T - 45 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$31$ $$4096 + 4096 T + 2048 T^{2} - 256 T^{3} + 16 T^{4} + 32 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$37$ $$64 - 192 T + 288 T^{2} + 328 T^{3} + 209 T^{4} - 36 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$41$ $$4096 + 22528 T + 61952 T^{2} - 11200 T^{3} + 1028 T^{4} + 56 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$43$ $$1024 + 6848 T^{2} + 1585 T^{4} + 86 T^{6} + T^{8}$$
$47$ $$( 16 + T^{4} )^{2}$$
$53$ $$( 128 - 288 T + 66 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$59$ $$1993744 - 1107008 T + 307328 T^{2} + 136096 T^{3} + 32520 T^{4} - 720 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$61$ $$134606404 + 6081228 T^{2} + 87229 T^{4} + 502 T^{6} + T^{8}$$
$67$ $$10837264 - 3371008 T + 524288 T^{2} - 27520 T^{3} + 12360 T^{4} - 3456 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$71$ $$817216 - 1417472 T + 1229312 T^{2} - 208416 T^{3} + 17684 T^{4} - 56 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$73$ $$23078416 - 7571104 T + 1241888 T^{2} + 97768 T^{3} + 5521 T^{4} - 884 T^{5} + 200 T^{6} + 20 T^{7} + T^{8}$$
$79$ $$( -1264 + 1008 T - 70 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$83$ $$1459264 - 2126080 T + 1548800 T^{2} - 513568 T^{3} + 101268 T^{4} - 12408 T^{5} + 968 T^{6} - 44 T^{7} + T^{8}$$
$89$ $$80656 - 99968 T + 61952 T^{2} - 14400 T^{3} + 1352 T^{4} + 96 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$97$ $$295936 + 1027072 T + 1782272 T^{2} - 380800 T^{3} + 40528 T^{4} - 256 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$