# Properties

 Label 138.3.b.a Level $138$ Weight $3$ Character orbit 138.b Analytic conductor $3.760$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 138.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76022764817$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1358954496.3 Defining polynomial: $$x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1$$ x^8 + 8*x^6 + 20*x^4 + 16*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + \beta_1 q^{3} + 2 q^{4} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{5} - \beta_{4} q^{6} + (2 \beta_{7} - \beta_{6} + 3 \beta_{3} + 2 \beta_{2}) q^{7} - 2 \beta_{5} q^{8} + 3 q^{9}+O(q^{10})$$ q - b5 * q^2 + b1 * q^3 + 2 * q^4 + (-b6 + 2*b3 - b2) * q^5 - b4 * q^6 + (2*b7 - b6 + 3*b3 + 2*b2) * q^7 - 2*b5 * q^8 + 3 * q^9 $$q - \beta_{5} q^{2} + \beta_1 q^{3} + 2 q^{4} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{5} - \beta_{4} q^{6} + (2 \beta_{7} - \beta_{6} + 3 \beta_{3} + 2 \beta_{2}) q^{7} - 2 \beta_{5} q^{8} + 3 q^{9} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{3} - \beta_{2}) q^{10} + (2 \beta_{7} - 2 \beta_{3} - 4 \beta_{2}) q^{11} + 2 \beta_1 q^{12} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_1 + 2) q^{13} + ( - 4 \beta_{7} - \beta_{6} + \beta_{3} - 5 \beta_{2}) q^{14} + ( - \beta_{7} + 3 \beta_{6} + \beta_{2}) q^{15} + 4 q^{16} + (6 \beta_{7} - 4 \beta_{6} - 3 \beta_{3}) q^{17} - 3 \beta_{5} q^{18} + (11 \beta_{7} + 7 \beta_{6} - 4 \beta_{3} + 3 \beta_{2}) q^{19} + ( - 2 \beta_{6} + 4 \beta_{3} - 2 \beta_{2}) q^{20} + (4 \beta_{7} + 4 \beta_{6} + \beta_{3} + 2 \beta_{2}) q^{21} + ( - 6 \beta_{6} + 2 \beta_{3} + 6 \beta_{2}) q^{22} + ( - 5 \beta_{7} + 8 \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} - 7 \beta_{2} + \cdots + 2) q^{23}+ \cdots + (6 \beta_{7} - 6 \beta_{3} - 12 \beta_{2}) q^{99}+O(q^{100})$$ q - b5 * q^2 + b1 * q^3 + 2 * q^4 + (-b6 + 2*b3 - b2) * q^5 - b4 * q^6 + (2*b7 - b6 + 3*b3 + 2*b2) * q^7 - 2*b5 * q^8 + 3 * q^9 + (-b7 - 2*b6 + 3*b3 - b2) * q^10 + (2*b7 - 2*b3 - 4*b2) * q^11 + 2*b1 * q^12 + (-2*b5 + 3*b4 + 2*b1 + 2) * q^13 + (-4*b7 - b6 + b3 - 5*b2) * q^14 + (-b7 + 3*b6 + b2) * q^15 + 4 * q^16 + (6*b7 - 4*b6 - 3*b3) * q^17 - 3*b5 * q^18 + (11*b7 + 7*b6 - 4*b3 + 3*b2) * q^19 + (-2*b6 + 4*b3 - 2*b2) * q^20 + (4*b7 + 4*b6 + b3 + 2*b2) * q^21 + (-6*b6 + 2*b3 + 6*b2) * q^22 + (-5*b7 + 8*b6 + 5*b5 + 4*b4 - 5*b3 - 7*b2 - 3*b1 + 2) * q^23 - 2*b4 * q^24 + (10*b5 - b4 + 2*b1 + 9) * q^25 + (-2*b5 - 2*b4 - 6*b1 + 4) * q^26 + 3*b1 * q^27 + (4*b7 - 2*b6 + 6*b3 + 4*b2) * q^28 + (-8*b5 + 6*b4 + 4*b1 - 18) * q^29 + (-2*b7 + 5*b6 - b3 - b2) * q^30 + (6*b5 - 4*b4 + 8*b1 - 16) * q^31 - 4*b5 * q^32 + (-2*b7 - 2*b6 - 2*b3 + 8*b2) * q^33 + (b7 - 10*b6 - 3*b3 + 3*b2) * q^34 + (9*b5 - 2*b4 - 2*b1 - 14) * q^35 + 6 * q^36 + (-14*b7 + 10*b6 - 4*b3 + 8*b2) * q^37 + (-14*b7 - b6 - 7*b3 + b2) * q^38 + (9*b5 - 2*b4 + 2*b1 + 6) * q^39 + (-2*b7 - 4*b6 + 6*b3 - 2*b2) * q^40 + (-4*b4 - 20*b1 - 2) * q^41 + (-9*b7 + 2*b6 - b3 - 3*b2) * q^42 + (-14*b7 + 9*b6 - 19*b3 - 6*b2) * q^43 + (4*b7 - 4*b3 - 8*b2) * q^44 + (-3*b6 + 6*b3 - 3*b2) * q^45 + (2*b7 + 6*b6 - 2*b5 + 3*b4 + 2*b3 + 12*b2 - 8*b1 - 10) * q^46 + (b5 - 2*b4 - 16*b1 - 14) * q^47 + 4*b1 * q^48 + (-14*b5 - 9*b4 - 14*b1 + 5) * q^49 + (-9*b5 - 2*b4 + 2*b1 - 20) * q^50 + (6*b7 + b6 - 11*b3 + 12*b2) * q^51 + (-4*b5 + 6*b4 + 4*b1 + 4) * q^52 + (-10*b7 - 36*b6 - 3*b3 - 4*b2) * q^53 - 3*b4 * q^54 + (-6*b5 - 4*b4 + 16*b1 - 8) * q^55 + (-8*b7 - 2*b6 + 2*b3 - 10*b2) * q^56 + (14*b7 - 11*b6 + 10*b3 + 19*b2) * q^57 + (18*b5 - 4*b4 - 12*b1 + 16) * q^58 + (-7*b5 + 22*b4 - 8*b1 + 10) * q^59 + (-2*b7 + 6*b6 + 2*b2) * q^60 + (-26*b7 + 14*b6 + 12*b3 - 36*b2) * q^61 + (16*b5 - 8*b4 + 8*b1 - 12) * q^62 + (6*b7 - 3*b6 + 9*b3 + 6*b2) * q^63 + 8 * q^64 + (2*b7 - 15*b6 + 13*b3 + b2) * q^65 + (6*b7 + 8*b6 - 10*b3 - 6*b2) * q^66 + (5*b7 + 29*b6 - 16*b3 + 13*b2) * q^67 + (12*b7 - 8*b6 - 6*b3) * q^68 + (-12*b7 - 13*b6 + 12*b5 + 5*b4 + 11*b3 - 3*b2 + 2*b1 - 9) * q^69 + (14*b5 + 2*b4 + 4*b1 - 18) * q^70 + (14*b5 + 16*b4 + 18*b1 + 4) * q^71 - 6*b5 * q^72 + (-36*b5 - 16*b4 + 8) * q^73 + (8*b7 + 32*b6 - 12*b3 - 4*b2) * q^74 + (-3*b5 + 10*b4 + 9*b1 + 6) * q^75 + (22*b7 + 14*b6 - 8*b3 + 6*b2) * q^76 + (16*b5 - 14*b4 + 28) * q^77 + (-6*b5 - 2*b4 + 4*b1 - 18) * q^78 + (19*b7 - 13*b6 - 4*b3 + 3*b2) * q^79 + (-4*b6 + 8*b3 - 4*b2) * q^80 + 9 * q^81 + (2*b5 + 20*b4 + 8*b1) * q^82 + (8*b7 + 20*b6 - 18*b2) * q^83 + (8*b7 + 8*b6 + 2*b3 + 4*b2) * q^84 + (-14*b5 - 21*b4 + 34*b1 + 6) * q^85 + (24*b7 + 17*b6 - 13*b3 + 25*b2) * q^86 + (18*b5 - 8*b4 - 18*b1 + 12) * q^87 + (-12*b6 + 4*b3 + 12*b2) * q^88 + (4*b7 - 29*b6 + 50*b3 + 27*b2) * q^89 + (-3*b7 - 6*b6 + 9*b3 - 3*b2) * q^90 + (31*b7 - 2*b6 + 13*b3 + 7*b2) * q^91 + (-10*b7 + 16*b6 + 10*b5 + 8*b4 - 10*b3 - 14*b2 - 6*b1 + 4) * q^92 + (-12*b5 + 6*b4 - 16*b1 + 24) * q^93 + (14*b5 + 16*b4 + 4*b1 - 2) * q^94 + (-37*b5 + 6*b4 - 4*b1 + 14) * q^95 - 4*b4 * q^96 + (-24*b7 + 2*b6 - 4*b3 - 50*b2) * q^97 + (-5*b5 + 14*b4 + 18*b1 + 28) * q^98 + (6*b7 - 6*b3 - 12*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{4} + 24 q^{9}+O(q^{10})$$ 8 * q + 16 * q^4 + 24 * q^9 $$8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100})$$ 8 * q + 16 * q^4 + 24 * q^9 + 16 * q^13 + 32 * q^16 + 16 * q^23 + 72 * q^25 + 32 * q^26 - 144 * q^29 - 128 * q^31 - 112 * q^35 + 48 * q^36 + 48 * q^39 - 16 * q^41 - 80 * q^46 - 112 * q^47 + 40 * q^49 - 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 + 80 * q^59 - 96 * q^62 + 64 * q^64 - 72 * q^69 - 144 * q^70 + 32 * q^71 + 64 * q^73 + 48 * q^75 + 224 * q^77 - 144 * q^78 + 72 * q^81 + 48 * q^85 + 96 * q^87 + 32 * q^92 + 192 * q^93 - 16 * q^94 + 112 * q^95 + 224 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} + 4\nu^{2} + 2$$ v^4 + 4*v^2 + 2 $$\beta_{2}$$ $$=$$ $$\nu^{5} + 5\nu^{3} + 4\nu$$ v^5 + 5*v^3 + 4*v $$\beta_{3}$$ $$=$$ $$\nu^{5} + 5\nu^{3} + 6\nu$$ v^5 + 5*v^3 + 6*v $$\beta_{4}$$ $$=$$ $$-\nu^{6} - 6\nu^{4} - 7\nu^{2} + 2$$ -v^6 - 6*v^4 - 7*v^2 + 2 $$\beta_{5}$$ $$=$$ $$\nu^{6} + 6\nu^{4} + 9\nu^{2} + 2$$ v^6 + 6*v^4 + 9*v^2 + 2 $$\beta_{6}$$ $$=$$ $$\nu^{7} + 7\nu^{5} + 13\nu^{3} + 5\nu$$ v^7 + 7*v^5 + 13*v^3 + 5*v $$\beta_{7}$$ $$=$$ $$\nu^{7} + 7\nu^{5} + 15\nu^{3} + 11\nu$$ v^7 + 7*v^5 + 15*v^3 + 11*v
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} ) / 2$$ (b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} - 4 ) / 2$$ (b5 + b4 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - \beta_{6} - 3\beta_{3} + 3\beta_{2} ) / 2$$ (b7 - b6 - 3*b3 + 3*b2) / 2 $$\nu^{4}$$ $$=$$ $$-2\beta_{5} - 2\beta_{4} + \beta _1 + 6$$ -2*b5 - 2*b4 + b1 + 6 $$\nu^{5}$$ $$=$$ $$( -5\beta_{7} + 5\beta_{6} + 11\beta_{3} - 9\beta_{2} ) / 2$$ (-5*b7 + 5*b6 + 11*b3 - 9*b2) / 2 $$\nu^{6}$$ $$=$$ $$( 17\beta_{5} + 15\beta_{4} - 12\beta _1 - 40 ) / 2$$ (17*b5 + 15*b4 - 12*b1 - 40) / 2 $$\nu^{7}$$ $$=$$ $$( 22\beta_{7} - 20\beta_{6} - 43\beta_{3} + 29\beta_{2} ) / 2$$ (22*b7 - 20*b6 - 43*b3 + 29*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 − 1.58671i 1.58671i − 0.261052i 0.261052i − 1.21752i 1.21752i − 1.98289i 1.98289i
−1.41421 −1.73205 2.00000 1.69484i 2.44949 4.18388i −2.82843 3.00000 2.39686i
91.2 −1.41421 −1.73205 2.00000 1.69484i 2.44949 4.18388i −2.82843 3.00000 2.39686i
91.3 −1.41421 1.73205 2.00000 0.918288i −2.44949 10.4925i −2.82843 3.00000 1.29866i
91.4 −1.41421 1.73205 2.00000 0.918288i −2.44949 10.4925i −2.82843 3.00000 1.29866i
91.5 1.41421 −1.73205 2.00000 6.00464i −2.44949 4.69017i 2.82843 3.00000 8.49185i
91.6 1.41421 −1.73205 2.00000 6.00464i −2.44949 4.69017i 2.82843 3.00000 8.49185i
91.7 1.41421 1.73205 2.00000 4.92225i 2.44949 5.13851i 2.82843 3.00000 6.96111i
91.8 1.41421 1.73205 2.00000 4.92225i 2.44949 5.13851i 2.82843 3.00000 6.96111i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.3.b.a 8
3.b odd 2 1 414.3.b.c 8
4.b odd 2 1 1104.3.c.c 8
23.b odd 2 1 inner 138.3.b.a 8
69.c even 2 1 414.3.b.c 8
92.b even 2 1 1104.3.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.b.a 8 1.a even 1 1 trivial
138.3.b.a 8 23.b odd 2 1 inner
414.3.b.c 8 3.b odd 2 1
414.3.b.c 8 69.c even 2 1
1104.3.c.c 8 4.b odd 2 1
1104.3.c.c 8 92.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(138, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{4}$$
$3$ $$(T^{2} - 3)^{4}$$
$5$ $$T^{8} + 64 T^{6} + 1100 T^{4} + \cdots + 2116$$
$7$ $$T^{8} + 176 T^{6} + 8684 T^{4} + \cdots + 1119364$$
$11$ $$T^{8} + 512 T^{6} + 43712 T^{4} + \cdots + 2262016$$
$13$ $$(T^{4} - 8 T^{3} - 124 T^{2} + 1136 T - 956)^{2}$$
$17$ $$T^{8} + 1168 T^{6} + \cdots + 1138792516$$
$19$ $$T^{8} + 2144 T^{6} + \cdots + 14491825924$$
$23$ $$T^{8} - 16 T^{7} + \cdots + 78310985281$$
$29$ $$(T^{4} + 72 T^{3} + 1160 T^{2} + \cdots - 6128)^{2}$$
$31$ $$(T^{4} + 64 T^{3} + 816 T^{2} + 2560 T + 1600)^{2}$$
$37$ $$T^{8} + 7168 T^{6} + \cdots + 2955978735616$$
$41$ $$(T^{4} + 8 T^{3} - 2568 T^{2} + \cdots + 1208464)^{2}$$
$43$ $$T^{8} + 6704 T^{6} + \cdots + 26854687876$$
$47$ $$(T^{4} + 56 T^{3} - 412 T^{2} + \cdots + 256036)^{2}$$
$53$ $$T^{8} + \cdots + 842259883497796$$
$59$ $$(T^{4} - 40 T^{3} - 5788 T^{2} + \cdots + 6720292)^{2}$$
$61$ $$T^{8} + 24704 T^{6} + \cdots + 138674176$$
$67$ $$T^{8} + 12192 T^{6} + \cdots + 15376496004$$
$71$ $$(T^{4} - 16 T^{3} - 5704 T^{2} + \cdots - 812912)^{2}$$
$73$ $$(T^{4} - 32 T^{3} - 7872 T^{2} + \cdots + 590848)^{2}$$
$79$ $$T^{8} + 8800 T^{6} + \cdots + 7772598291844$$
$83$ $$T^{8} + 14912 T^{6} + \cdots + 5440313672704$$
$89$ $$T^{8} + 40448 T^{6} + \cdots + 69\!\cdots\!16$$
$97$ $$T^{8} + \cdots + 380951699424256$$