# Properties

 Label 1638.2.x.d 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: 8.0.7442857984.4 Defining polynomial: $$x^{8} + 26 x^{6} + 205 x^{4} + 540 x^{2} + 324$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} -\beta_{6} q^{4} + ( 1 + \beta_{1} + \beta_{3} - \beta_{7} ) q^{5} + \beta_{5} q^{7} -\beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} -\beta_{6} q^{4} + ( 1 + \beta_{1} + \beta_{3} - \beta_{7} ) q^{5} + \beta_{5} q^{7} -\beta_{2} q^{8} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{10} + ( 1 + \beta_{3} + \beta_{6} ) q^{11} + ( 2 - 3 \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{14} - q^{16} + ( 2 - 3 \beta_{2} + 3 \beta_{4} - \beta_{7} ) q^{17} + ( -\beta_{1} - 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{5} - \beta_{7} ) q^{20} + ( \beta_{2} - \beta_{4} + \beta_{7} ) q^{22} + ( -3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{23} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{6} ) q^{25} + ( -3 \beta_{2} - 2 \beta_{4} ) q^{26} + ( 1 + \beta_{3} - \beta_{7} ) q^{28} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{29} + ( 3 + \beta_{1} + 2 \beta_{6} - \beta_{7} ) q^{31} + \beta_{4} q^{32} + ( -3 - \beta_{1} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{35} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( \beta_{1} + \beta_{3} - \beta_{5} - 3 \beta_{7} ) q^{38} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{40} + ( -2 + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{41} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 5 \beta_{6} ) q^{43} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} ) q^{44} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{46} + ( 2 - 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{47} + ( -1 + \beta_{1} + 6 \beta_{2} - \beta_{6} + \beta_{7} ) q^{49} + ( -3 + 3 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{50} + ( -3 - 2 \beta_{6} ) q^{52} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{53} + ( -3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{55} + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{56} + ( -3 + \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{58} + ( -4 - \beta_{1} - 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{59} + ( -\beta_{1} + 3 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{61} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{62} + \beta_{6} q^{64} + ( 5 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} - 5 \beta_{7} ) q^{65} + ( 1 - 2 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{67} + ( -\beta_{1} + 3 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{68} + ( 3 + 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{70} + ( -6 + \beta_{1} + 6 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} - \beta_{7} ) q^{71} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{74} + ( -2 \beta_{1} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{76} + ( -4 - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( -4 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} ) q^{79} + ( -1 - \beta_{1} - \beta_{3} + \beta_{7} ) q^{80} + ( 6 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{82} + ( -4 + 2 \beta_{3} - 4 \beta_{6} ) q^{83} + ( 5 + 5 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} - 5 \beta_{7} ) q^{85} + ( 3 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{86} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{88} + ( -2 + 3 \beta_{1} - \beta_{6} + 3 \beta_{7} ) q^{89} + ( 3 + 3 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{91} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{94} + ( 2 \beta_{1} + 9 \beta_{2} + \beta_{3} + 9 \beta_{4} + \beta_{5} - 9 \beta_{6} ) q^{95} + ( -3 + 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{6} - 3 \beta_{7} ) q^{97} + ( 6 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} + O(q^{10})$$ $$8q + 4q^{5} + 4q^{10} + 8q^{11} + 16q^{13} - 8q^{16} + 12q^{17} + 4q^{19} + 4q^{20} + 4q^{22} + 4q^{28} + 12q^{29} + 20q^{31} - 24q^{34} - 32q^{35} - 8q^{37} - 12q^{38} - 16q^{41} + 8q^{44} - 20q^{46} + 16q^{47} - 4q^{49} - 24q^{50} - 24q^{52} + 24q^{53} + 4q^{56} - 16q^{58} - 28q^{59} + 20q^{65} + 8q^{67} + 24q^{70} - 52q^{71} - 8q^{73} + 4q^{74} + 4q^{76} - 32q^{77} - 48q^{79} - 4q^{80} + 40q^{82} - 32q^{83} + 20q^{85} + 20q^{86} - 4q^{89} + 12q^{91} - 20q^{92} - 36q^{97} + 48q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 26 x^{6} + 205 x^{4} + 540 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} + 3 \nu^{6} + 52 \nu^{5} + 60 \nu^{4} + 374 \nu^{3} + 219 \nu^{2} + 612 \nu - 162$$$$)/432$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 8 \nu^{5} + 24 \nu^{4} + 155 \nu^{3} - 249 \nu^{2} + 774 \nu - 810$$$$)/432$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 3 \nu^{6} + 52 \nu^{5} - 60 \nu^{4} + 374 \nu^{3} - 219 \nu^{2} + 612 \nu + 162$$$$)/432$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{6} - 8 \nu^{5} - 24 \nu^{4} + 155 \nu^{3} + 249 \nu^{2} + 1206 \nu + 810$$$$)/432$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} - 112 \nu^{5} - 665 \nu^{3} - 954 \nu$$$$)/432$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 20 \nu^{4} + 97 \nu^{2} + 114$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3 \beta_{4} - 3 \beta_{2} - 7$$ $$\nu^{3}$$ $$=$$ $$-6 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-13 \beta_{7} + 6 \beta_{5} - 45 \beta_{4} - 6 \beta_{3} + 45 \beta_{2} - 6 \beta_{1} + 73$$ $$\nu^{5}$$ $$=$$ $$114 \beta_{6} - 45 \beta_{5} + 120 \beta_{4} - 45 \beta_{3} + 120 \beta_{2} + 118 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$187 \beta_{7} - 120 \beta_{5} + 609 \beta_{4} + 120 \beta_{3} - 609 \beta_{2} + 120 \beta_{1} - 895$$ $$\nu^{7}$$ $$=$$ $$-1842 \beta_{6} + 609 \beta_{5} - 1890 \beta_{4} + 609 \beta_{3} - 1890 \beta_{2} - 1504 \beta_{1}$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$ $$1$$

## 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
 0.916813i − 1.91681i 2.73923i − 3.73923i − 0.916813i 1.91681i − 2.73923i 3.73923i
−0.707107 0.707107i 0 1.00000i −2.27220 + 2.27220i 0 1.35539 + 2.27220i 0.707107 0.707107i 0 3.21338
307.2 −0.707107 0.707107i 0 1.00000i 2.56510 2.56510i 0 −0.648285 2.56510i 0.707107 0.707107i 0 −3.62760
307.3 0.707107 + 0.707107i 0 1.00000i −0.0951965 + 0.0951965i 0 −2.64404 + 0.0951965i −0.707107 + 0.707107i 0 −0.134628
307.4 0.707107 + 0.707107i 0 1.00000i 1.80230 1.80230i 0 1.93693 1.80230i −0.707107 + 0.707107i 0 2.54884
811.1 −0.707107 + 0.707107i 0 1.00000i −2.27220 2.27220i 0 1.35539 2.27220i 0.707107 + 0.707107i 0 3.21338
811.2 −0.707107 + 0.707107i 0 1.00000i 2.56510 + 2.56510i 0 −0.648285 + 2.56510i 0.707107 + 0.707107i 0 −3.62760
811.3 0.707107 0.707107i 0 1.00000i −0.0951965 0.0951965i 0 −2.64404 0.0951965i −0.707107 0.707107i 0 −0.134628
811.4 0.707107 0.707107i 0 1.00000i 1.80230 + 1.80230i 0 1.93693 + 1.80230i −0.707107 0.707107i 0 2.54884
 $$n$$: 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.d 8
3.b odd 2 1 546.2.o.a 8
7.b odd 2 1 1638.2.x.b 8
13.d odd 4 1 1638.2.x.b 8
21.c even 2 1 546.2.o.d yes 8
39.f even 4 1 546.2.o.d yes 8
91.i even 4 1 inner 1638.2.x.d 8
273.o odd 4 1 546.2.o.a 8

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

## 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

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$16 + 160 T + 800 T^{2} - 424 T^{3} + 113 T^{4} + 4 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$7$ $$2401 + 98 T^{2} + 168 T^{3} + 2 T^{4} + 24 T^{5} + 2 T^{6} + T^{8}$$
$11$ $$64 + 192 T + 288 T^{2} + 40 T^{3} - 15 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$( 13 - 4 T + T^{2} )^{4}$$
$17$ $$( -92 + 204 T - 35 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$19$ $$777924 + 74088 T + 3528 T^{2} - 2100 T^{3} + 2725 T^{4} + 184 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$23$ $$135424 + 39264 T^{2} + 3265 T^{4} + 102 T^{6} + T^{8}$$
$29$ $$( 356 + 260 T - 67 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$31$ $$5184 - 1440 T^{3} + 2644 T^{4} - 1000 T^{5} + 200 T^{6} - 20 T^{7} + T^{8}$$
$37$ $$777924 - 222264 T + 31752 T^{2} + 6300 T^{3} + 1045 T^{4} - 172 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$41$ $$3936256 + 2285568 T + 663552 T^{2} + 91648 T^{3} + 6672 T^{4} + 320 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$43$ $$14961424 + 983096 T^{2} + 23873 T^{4} + 254 T^{6} + T^{8}$$
$47$ $$65536 - 98304 T + 73728 T^{2} - 5632 T^{3} + 528 T^{4} - 320 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$53$ $$( 128 + 128 T - 4 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$59$ $$73984 + 30464 T + 6272 T^{2} - 224 T^{3} + 5444 T^{4} + 2072 T^{5} + 392 T^{6} + 28 T^{7} + T^{8}$$
$61$ $$8464 + 35176 T^{2} + 3489 T^{4} + 106 T^{6} + T^{8}$$
$67$ $$16 + 384 T + 4608 T^{2} + 8416 T^{3} + 7752 T^{4} + 800 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$71$ $$1327104 - 2433024 T + 2230272 T^{2} + 968064 T^{3} + 187204 T^{4} + 20248 T^{5} + 1352 T^{6} + 52 T^{7} + T^{8}$$
$73$ $$777924 - 222264 T + 31752 T^{2} + 6300 T^{3} + 1045 T^{4} - 172 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$79$ $$( -20744 - 3480 T - 2 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$83$ $$541696 - 70656 T + 4608 T^{2} + 36992 T^{3} + 21072 T^{4} + 4384 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$
$89$ $$45050944 - 3006976 T + 100352 T^{2} + 72608 T^{3} + 35860 T^{4} - 440 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$97$ $$485409024 + 142767360 T + 20995200 T^{2} + 1143072 T^{3} + 46980 T^{4} + 4536 T^{5} + 648 T^{6} + 36 T^{7} + T^{8}$$