# Properties

 Label 378.5.b.a Level $378$ Weight $5$ Character orbit 378.b Analytic conductor $39.074$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 378.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.0738460457$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.5443747577856.29 Defining polynomial: $$x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324$$ x^8 + 24*x^6 + 180*x^4 + 488*x^2 + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{4}\cdot 7^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 8 q^{4} + (\beta_{7} + \beta_{3} - 4 \beta_1) q^{5} + \beta_{4} q^{7} + 8 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 8 * q^4 + (b7 + b3 - 4*b1) * q^5 + b4 * q^7 + 8*b1 * q^8 $$q - \beta_1 q^{2} - 8 q^{4} + (\beta_{7} + \beta_{3} - 4 \beta_1) q^{5} + \beta_{4} q^{7} + 8 \beta_1 q^{8} + (\beta_{5} + \beta_{4} - \beta_{2} - 28) q^{10} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{3} - 15 \beta_1) q^{11} + ( - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 47) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{14} + 64 q^{16} + (16 \beta_{7} - 4 \beta_{6} - \beta_{3} - 38 \beta_1) q^{17} + (9 \beta_{5} + 4 \beta_{4} - 2 \beta_{2} + 140) q^{19} + ( - 8 \beta_{7} - 8 \beta_{3} + 32 \beta_1) q^{20} + (2 \beta_{5} + 12 \beta_{4} + 5 \beta_{2} - 140) q^{22} + ( - 17 \beta_{7} - 4 \beta_{6} - 28 \beta_{3} + 4 \beta_1) q^{23} + (9 \beta_{5} + 30 \beta_{4} - 10 \beta_{2} + 99) q^{25} + ( - 11 \beta_{7} + 5 \beta_{6} + 19 \beta_{3} + 55 \beta_1) q^{26} - 8 \beta_{4} q^{28} + (18 \beta_{7} + 6 \beta_{6} + 55 \beta_{3} + 112 \beta_1) q^{29} + ( - 12 \beta_{5} + 37 \beta_{4} + 6 \beta_{2} - 110) q^{31} - 64 \beta_1 q^{32} + ( - 5 \beta_{5} + 13 \beta_{4} - 20 \beta_{2} - 224) q^{34} + ( - 22 \beta_{7} - \beta_{6} - 13 \beta_{3} + 35 \beta_1) q^{35} + ( - 33 \beta_{5} - 13 \beta_{4} + 26 \beta_{2} + 197) q^{37} + ( - 25 \beta_{7} - 9 \beta_{6} - 63 \beta_{3} - 132 \beta_1) q^{38} + ( - 8 \beta_{5} - 8 \beta_{4} + 8 \beta_{2} + 224) q^{40} + (52 \beta_{7} - 13 \beta_{6} + 126 \beta_{3} - 85 \beta_1) q^{41} + (21 \beta_{5} + 76 \beta_{4} + 28 \beta_{2} - 721) q^{43} + (16 \beta_{7} - 24 \beta_{6} + 8 \beta_{3} + 120 \beta_1) q^{44} + ( - 32 \beta_{5} - 26 \beta_{4} + 13 \beta_{2} - 20) q^{46} + (22 \beta_{7} - 42 \beta_{6} - 27 \beta_{3} + 40 \beta_1) q^{47} + 343 q^{49} + ( - 99 \beta_{7} - 19 \beta_{6} - 53 \beta_{3} - 59 \beta_1) q^{50} + (24 \beta_{5} - 16 \beta_{4} + 16 \beta_{2} + 376) q^{52} + ( - 92 \beta_{7} - 50 \beta_{6} + 72 \beta_{3} + 78 \beta_1) q^{53} + (39 \beta_{5} + 7 \beta_{4} - 34 \beta_{2} + 61) q^{55} + (8 \beta_{7} + 8 \beta_{6} - 8 \beta_{3}) q^{56} + (61 \beta_{5} + 11 \beta_{4} - 12 \beta_{2} + 944) q^{58} + ( - 10 \beta_{7} - 76 \beta_{6} + 73 \beta_{3} - 488 \beta_1) q^{59} + ( - 30 \beta_{5} + 44 \beta_{4} - 92 \beta_{2} - 320) q^{61} + (11 \beta_{7} - 37 \beta_{6} + 133 \beta_{3} + 86 \beta_1) q^{62} - 512 q^{64} + ( - 124 \beta_{7} + 2 \beta_{6} - 27 \beta_{3} + 246 \beta_1) q^{65} + ( - 33 \beta_{5} + 64 \beta_{4} + 104 \beta_{2} + 2973) q^{67} + ( - 128 \beta_{7} + 32 \beta_{6} + 8 \beta_{3} + 304 \beta_1) q^{68} + ( - 14 \beta_{5} - 36 \beta_{4} + 21 \beta_{2} + 196) q^{70} + ( - 295 \beta_{7} - 92 \beta_{6} - 109 \beta_{3} - 756 \beta_1) q^{71} + ( - 3 \beta_{5} + 106 \beta_{4} + 44 \beta_{2} - 1647) q^{73} + (202 \beta_{7} - 6 \beta_{6} + 270 \beta_{3} - 301 \beta_1) q^{74} + ( - 72 \beta_{5} - 32 \beta_{4} + 16 \beta_{2} - 1120) q^{76} + (5 \beta_{7} - 2 \beta_{6} + 121 \beta_{3} + 560 \beta_1) q^{77} + ( - 15 \beta_{5} + 18 \beta_{4} - 94 \beta_{2} - 1199) q^{79} + (64 \beta_{7} + 64 \beta_{3} - 256 \beta_1) q^{80} + (113 \beta_{5} - 87 \beta_{4} - 65 \beta_{2} - 420) q^{82} + ( - 122 \beta_{7} + 8 \beta_{6} - 199 \beta_{3} - 416 \beta_1) q^{83} + (57 \beta_{5} + 364 \beta_{4} - 124 \beta_{2} - 4985) q^{85} + (71 \beta_{7} - 153 \beta_{6} - 15 \beta_{3} + 609 \beta_1) q^{86} + ( - 16 \beta_{5} - 96 \beta_{4} - 40 \beta_{2} + 1120) q^{88} + (16 \beta_{7} - 253 \beta_{6} - 12 \beta_{3} + 851 \beta_1) q^{89} + ( - 63 \beta_{5} - 48 \beta_{4} - 28 \beta_{2} + 637) q^{91} + (136 \beta_{7} + 32 \beta_{6} + 224 \beta_{3} - 32 \beta_1) q^{92} + ( - 69 \beta_{5} - 139 \beta_{4} - 64 \beta_{2} + 576) q^{94} + ( - 137 \beta_{7} - 96 \beta_{6} - 34 \beta_{3} + 1608 \beta_1) q^{95} + (132 \beta_{5} - 78 \beta_{4} + 172 \beta_{2} - 1752) q^{97} - 343 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 - 8 * q^4 + (b7 + b3 - 4*b1) * q^5 + b4 * q^7 + 8*b1 * q^8 + (b5 + b4 - b2 - 28) * q^10 + (-2*b7 + 3*b6 - b3 - 15*b1) * q^11 + (-3*b5 + 2*b4 - 2*b2 - 47) * q^13 + (-b7 - b6 + b3) * q^14 + 64 * q^16 + (16*b7 - 4*b6 - b3 - 38*b1) * q^17 + (9*b5 + 4*b4 - 2*b2 + 140) * q^19 + (-8*b7 - 8*b3 + 32*b1) * q^20 + (2*b5 + 12*b4 + 5*b2 - 140) * q^22 + (-17*b7 - 4*b6 - 28*b3 + 4*b1) * q^23 + (9*b5 + 30*b4 - 10*b2 + 99) * q^25 + (-11*b7 + 5*b6 + 19*b3 + 55*b1) * q^26 - 8*b4 * q^28 + (18*b7 + 6*b6 + 55*b3 + 112*b1) * q^29 + (-12*b5 + 37*b4 + 6*b2 - 110) * q^31 - 64*b1 * q^32 + (-5*b5 + 13*b4 - 20*b2 - 224) * q^34 + (-22*b7 - b6 - 13*b3 + 35*b1) * q^35 + (-33*b5 - 13*b4 + 26*b2 + 197) * q^37 + (-25*b7 - 9*b6 - 63*b3 - 132*b1) * q^38 + (-8*b5 - 8*b4 + 8*b2 + 224) * q^40 + (52*b7 - 13*b6 + 126*b3 - 85*b1) * q^41 + (21*b5 + 76*b4 + 28*b2 - 721) * q^43 + (16*b7 - 24*b6 + 8*b3 + 120*b1) * q^44 + (-32*b5 - 26*b4 + 13*b2 - 20) * q^46 + (22*b7 - 42*b6 - 27*b3 + 40*b1) * q^47 + 343 * q^49 + (-99*b7 - 19*b6 - 53*b3 - 59*b1) * q^50 + (24*b5 - 16*b4 + 16*b2 + 376) * q^52 + (-92*b7 - 50*b6 + 72*b3 + 78*b1) * q^53 + (39*b5 + 7*b4 - 34*b2 + 61) * q^55 + (8*b7 + 8*b6 - 8*b3) * q^56 + (61*b5 + 11*b4 - 12*b2 + 944) * q^58 + (-10*b7 - 76*b6 + 73*b3 - 488*b1) * q^59 + (-30*b5 + 44*b4 - 92*b2 - 320) * q^61 + (11*b7 - 37*b6 + 133*b3 + 86*b1) * q^62 - 512 * q^64 + (-124*b7 + 2*b6 - 27*b3 + 246*b1) * q^65 + (-33*b5 + 64*b4 + 104*b2 + 2973) * q^67 + (-128*b7 + 32*b6 + 8*b3 + 304*b1) * q^68 + (-14*b5 - 36*b4 + 21*b2 + 196) * q^70 + (-295*b7 - 92*b6 - 109*b3 - 756*b1) * q^71 + (-3*b5 + 106*b4 + 44*b2 - 1647) * q^73 + (202*b7 - 6*b6 + 270*b3 - 301*b1) * q^74 + (-72*b5 - 32*b4 + 16*b2 - 1120) * q^76 + (5*b7 - 2*b6 + 121*b3 + 560*b1) * q^77 + (-15*b5 + 18*b4 - 94*b2 - 1199) * q^79 + (64*b7 + 64*b3 - 256*b1) * q^80 + (113*b5 - 87*b4 - 65*b2 - 420) * q^82 + (-122*b7 + 8*b6 - 199*b3 - 416*b1) * q^83 + (57*b5 + 364*b4 - 124*b2 - 4985) * q^85 + (71*b7 - 153*b6 - 15*b3 + 609*b1) * q^86 + (-16*b5 - 96*b4 - 40*b2 + 1120) * q^88 + (16*b7 - 253*b6 - 12*b3 + 851*b1) * q^89 + (-63*b5 - 48*b4 - 28*b2 + 637) * q^91 + (136*b7 + 32*b6 + 224*b3 - 32*b1) * q^92 + (-69*b5 - 139*b4 - 64*b2 + 576) * q^94 + (-137*b7 - 96*b6 - 34*b3 + 1608*b1) * q^95 + (132*b5 - 78*b4 + 172*b2 - 1752) * q^97 - 343*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 64 q^{4}+O(q^{10})$$ 8 * q - 64 * q^4 $$8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100})$$ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} - 18\nu^{5} - 66\nu^{3} + 4\nu ) / 18$$ (-v^7 - 18*v^5 - 66*v^3 + 4*v) / 18 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} - 10\nu^{4} + 38\nu^{2} + 186 ) / 3$$ (-v^6 - 10*v^4 + 38*v^2 + 186) / 3 $$\beta_{3}$$ $$=$$ $$( 5\nu^{7} + 102\nu^{5} + 522\nu^{3} + 316\nu ) / 36$$ (5*v^7 + 102*v^5 + 522*v^3 + 316*v) / 36 $$\beta_{4}$$ $$=$$ $$( -7\nu^{6} - 133\nu^{4} - 616\nu^{2} - 588 ) / 6$$ (-7*v^6 - 133*v^4 - 616*v^2 - 588) / 6 $$\beta_{5}$$ $$=$$ $$( 13\nu^{6} + 247\nu^{4} + 1072\nu^{2} + 660 ) / 6$$ (13*v^6 + 247*v^4 + 1072*v^2 + 660) / 6 $$\beta_{6}$$ $$=$$ $$( 13\nu^{7} + 207\nu^{5} + 534\nu^{3} - 646\nu ) / 36$$ (13*v^7 + 207*v^5 + 534*v^3 - 646*v) / 36 $$\beta_{7}$$ $$=$$ $$( -22\nu^{7} - 441\nu^{5} - 2280\nu^{3} - 2846\nu ) / 36$$ (-22*v^7 - 441*v^5 - 2280*v^3 - 2846*v) / 36
 $$\nu$$ $$=$$ $$( -2\beta_{7} - 2\beta_{6} - 12\beta_{3} - 21\beta_1 ) / 84$$ (-2*b7 - 2*b6 - 12*b3 - 21*b1) / 84 $$\nu^{2}$$ $$=$$ $$( -7\beta_{5} - 13\beta_{4} - 504 ) / 84$$ (-7*b5 - 13*b4 - 504) / 84 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 31\beta_{6} + 81\beta_{3} + 371\beta_1 ) / 84$$ (3*b7 + 31*b6 + 81*b3 + 371*b1) / 84 $$\nu^{4}$$ $$=$$ $$( 49\beta_{5} + 87\beta_{4} + 14\beta_{2} + 2268 ) / 42$$ (49*b5 + 87*b4 + 14*b2 + 2268) / 42 $$\nu^{5}$$ $$=$$ $$( 4\beta_{7} - 220\beta_{6} - 354\beta_{3} - 2359\beta_1 ) / 42$$ (4*b7 - 220*b6 - 354*b3 - 2359*b1) / 42 $$\nu^{6}$$ $$=$$ $$( -623\beta_{5} - 1117\beta_{4} - 266\beta_{2} - 24444 ) / 42$$ (-623*b5 - 1117*b4 - 266*b2 - 24444) / 42 $$\nu^{7}$$ $$=$$ $$( -25\beta_{7} + 419\beta_{6} + 525\beta_{3} + 4203\beta_1 ) / 6$$ (-25*b7 + 419*b6 + 525*b3 + 4203*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1
 0.982345i 2.14697i − 3.56118i − 2.39656i 2.39656i 3.56118i − 2.14697i − 0.982345i
2.82843i 0 −8.00000 42.2558i 0 −18.5203 22.6274i 0 −119.518
323.2 2.82843i 0 −8.00000 9.19100i 0 18.5203 22.6274i 0 −25.9961
323.3 2.82843i 0 −8.00000 3.12468i 0 18.5203 22.6274i 0 −8.83793
323.4 2.82843i 0 −8.00000 14.9735i 0 −18.5203 22.6274i 0 42.3515
323.5 2.82843i 0 −8.00000 14.9735i 0 −18.5203 22.6274i 0 42.3515
323.6 2.82843i 0 −8.00000 3.12468i 0 18.5203 22.6274i 0 −8.83793
323.7 2.82843i 0 −8.00000 9.19100i 0 18.5203 22.6274i 0 −25.9961
323.8 2.82843i 0 −8.00000 42.2558i 0 −18.5203 22.6274i 0 −119.518
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.5.b.a 8
3.b odd 2 1 inner 378.5.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.5.b.a 8 1.a even 1 1 trivial
378.5.b.a 8 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2104T_{5}^{6} + 590554T_{5}^{4} + 39384216T_{5}^{2} + 330185241$$ acting on $$S_{5}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 8)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 2104 T^{6} + \cdots + 330185241$$
$7$ $$(T^{2} - 343)^{4}$$
$11$ $$T^{8} + 71752 T^{6} + \cdots + 16\!\cdots\!81$$
$13$ $$(T^{4} + 188 T^{3} - 40088 T^{2} + \cdots - 317096316)^{2}$$
$17$ $$T^{8} + 509464 T^{6} + \cdots + 12\!\cdots\!56$$
$19$ $$(T^{4} - 560 T^{3} + \cdots + 11482979377)^{2}$$
$23$ $$T^{8} + 957688 T^{6} + \cdots + 13\!\cdots\!01$$
$29$ $$T^{8} + 3390232 T^{6} + \cdots + 15\!\cdots\!36$$
$31$ $$(T^{4} + 440 T^{3} + \cdots + 88737781689)^{2}$$
$37$ $$(T^{4} - 788 T^{3} + \cdots + 1491560880553)^{2}$$
$41$ $$T^{8} + 15718392 T^{6} + \cdots + 54\!\cdots\!21$$
$43$ $$(T^{4} + 2884 T^{3} + \cdots - 16528157070236)^{2}$$
$47$ $$T^{8} + 13041720 T^{6} + \cdots + 18\!\cdots\!76$$
$53$ $$T^{8} + 44978400 T^{6} + \cdots + 96\!\cdots\!36$$
$59$ $$T^{8} + 52796776 T^{6} + \cdots + 20\!\cdots\!84$$
$61$ $$(T^{4} + 1280 T^{3} + \cdots + 394607747749392)^{2}$$
$67$ $$(T^{4} - 11892 T^{3} + \cdots - 650727964094204)^{2}$$
$71$ $$T^{8} + 216381064 T^{6} + \cdots + 22\!\cdots\!69$$
$73$ $$(T^{4} + 6588 T^{3} + \cdots - 20805309026012)^{2}$$
$79$ $$(T^{4} + 4796 T^{3} + \cdots + 345776930946468)^{2}$$
$83$ $$T^{8} + 49790920 T^{6} + \cdots + 31\!\cdots\!64$$
$89$ $$T^{8} + 452517768 T^{6} + \cdots + 21\!\cdots\!81$$
$97$ $$(T^{4} + 7008 T^{3} + \cdots + 44\!\cdots\!52)^{2}$$