# Properties

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

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## 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: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884$$ x^8 + 92*x^6 + 2949*x^4 + 37548*x^2 + 142884 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}\cdot 7^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 8 q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7} + 8 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 8 * q^4 + (-b2 - b1) * q^5 + b3 * q^7 + 8*b1 * q^8 $$q - \beta_1 q^{2} - 8 q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7} + 8 \beta_1 q^{8} + ( - \beta_{4} + \beta_{3} - 4) q^{10} + (\beta_{7} + \beta_{2} + 13 \beta_1) q^{11} + ( - \beta_{5} - 2 \beta_{3} + 37) q^{13} + \beta_{6} q^{14} + 64 q^{16} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} + 15 \beta_1) q^{17} + ( - \beta_{5} - 12 \beta_{3} - 10) q^{19} + (8 \beta_{2} + 8 \beta_1) q^{20} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} + 100) q^{22} + ( - \beta_{7} + \beta_{6} - 7 \beta_{2} - 72 \beta_1) q^{23} + (\beta_{5} - 14 \beta_{3} - 183) q^{25} + ( - 8 \beta_{7} + \beta_{6} - 37 \beta_1) q^{26} - 8 \beta_{3} q^{28} + (7 \beta_{7} - 9 \beta_{6} - 10 \beta_{2} + 41 \beta_1) q^{29} + ( - 6 \beta_{4} - 11 \beta_{3} - 128) q^{31} - 64 \beta_1 q^{32} + (\beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 112) q^{34} + (7 \beta_{7} - 2 \beta_{6} - 7 \beta_{2} + 21 \beta_1) q^{35} + (\beta_{5} - 51 \beta_{3} + 113) q^{37} + ( - 8 \beta_{7} - 9 \beta_{6} + 10 \beta_1) q^{38} + (8 \beta_{4} - 8 \beta_{3} + 32) q^{40} + (12 \beta_{7} - 17 \beta_{6} + 19 \beta_{2} - 46 \beta_1) q^{41} + (11 \beta_{5} + 6 \beta_{4} - 36 \beta_{3} - 403) q^{43} + ( - 8 \beta_{7} - 8 \beta_{2} - 104 \beta_1) q^{44} + (\beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 548) q^{46} + ( - \beta_{7} - 45 \beta_{6} - 18 \beta_{2} - 15 \beta_1) q^{47} + 343 q^{49} + (8 \beta_{7} - 17 \beta_{6} + 183 \beta_1) q^{50} + (8 \beta_{5} + 16 \beta_{3} - 296) q^{52} + (28 \beta_{7} - 34 \beta_{6} + 38 \beta_{2} + 364 \beta_1) q^{53} + ( - 5 \beta_{5} + 12 \beta_{4} - 97 \beta_{3} + 1039) q^{55} - 8 \beta_{6} q^{56} + ( - 7 \beta_{5} - 10 \beta_{4} + 61 \beta_{3} + 368) q^{58} + ( - 45 \beta_{7} - 35 \beta_{6} + 36 \beta_{2} + 31 \beta_1) q^{59} + (2 \beta_{5} + 48 \beta_{4} - 140 \beta_{3} + 16) q^{61} + ( - 17 \beta_{6} + 48 \beta_{2} + 152 \beta_1) q^{62} - 512 q^{64} + ( - 25 \beta_{7} - 89 \beta_{6} - 44 \beta_{2} - 199 \beta_1) q^{65} + ( - 11 \beta_{5} - 42 \beta_{4} + 56 \beta_{3} - 1269) q^{67} + (8 \beta_{7} - 8 \beta_{6} - 16 \beta_{2} - 120 \beta_1) q^{68} + ( - 7 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 196) q^{70} + ( - 10 \beta_{7} - 34 \beta_{6} - 87 \beta_{2} + 263 \beta_1) q^{71} + ( - 11 \beta_{5} - 6 \beta_{4} - 142 \beta_{3} + 1329) q^{73} + (8 \beta_{7} - 54 \beta_{6} - 113 \beta_1) q^{74} + (8 \beta_{5} + 96 \beta_{3} + 80) q^{76} + ( - 14 \beta_{7} - 4 \beta_{6} - 35 \beta_{2} - 189 \beta_1) q^{77} + (\beta_{5} - 60 \beta_{4} - 26 \beta_{3} - 509) q^{79} + ( - 64 \beta_{2} - 64 \beta_1) q^{80} + ( - 12 \beta_{5} + 19 \beta_{4} + 81 \beta_{3} - 444) q^{82} + ( - 37 \beta_{7} - 107 \beta_{6} + 92 \beta_{2} - 645 \beta_1) q^{83} + ( - 5 \beta_{5} + 6 \beta_{4} + 116 \beta_{3} + 1681) q^{85} + (88 \beta_{7} - 63 \beta_{6} - 48 \beta_{2} + 379 \beta_1) q^{86} + (8 \beta_{5} - 8 \beta_{4} + 32 \beta_{3} - 800) q^{88} + (66 \beta_{7} + \beta_{6} - 5 \beta_{2} - 1144 \beta_1) q^{89} + (7 \beta_{5} - 42 \beta_{4} + 52 \beta_{3} - 833) q^{91} + (8 \beta_{7} - 8 \beta_{6} + 56 \beta_{2} + 576 \beta_1) q^{92} + (\beta_{5} - 18 \beta_{4} + 381 \beta_{3} - 48) q^{94} + ( - 95 \beta_{7} - 69 \beta_{6} + 73 \beta_{2} - 362 \beta_1) q^{95} + ( - 28 \beta_{5} + 12 \beta_{4} - 94 \beta_{3} + 4452) q^{97} - 343 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 - 8 * q^4 + (-b2 - b1) * q^5 + b3 * q^7 + 8*b1 * q^8 + (-b4 + b3 - 4) * q^10 + (b7 + b2 + 13*b1) * q^11 + (-b5 - 2*b3 + 37) * q^13 + b6 * q^14 + 64 * q^16 + (-b7 + b6 + 2*b2 + 15*b1) * q^17 + (-b5 - 12*b3 - 10) * q^19 + (8*b2 + 8*b1) * q^20 + (-b5 + b4 - 4*b3 + 100) * q^22 + (-b7 + b6 - 7*b2 - 72*b1) * q^23 + (b5 - 14*b3 - 183) * q^25 + (-8*b7 + b6 - 37*b1) * q^26 - 8*b3 * q^28 + (7*b7 - 9*b6 - 10*b2 + 41*b1) * q^29 + (-6*b4 - 11*b3 - 128) * q^31 - 64*b1 * q^32 + (b5 + 2*b4 - 7*b3 + 112) * q^34 + (7*b7 - 2*b6 - 7*b2 + 21*b1) * q^35 + (b5 - 51*b3 + 113) * q^37 + (-8*b7 - 9*b6 + 10*b1) * q^38 + (8*b4 - 8*b3 + 32) * q^40 + (12*b7 - 17*b6 + 19*b2 - 46*b1) * q^41 + (11*b5 + 6*b4 - 36*b3 - 403) * q^43 + (-8*b7 - 8*b2 - 104*b1) * q^44 + (b5 - 7*b4 + 2*b3 - 548) * q^46 + (-b7 - 45*b6 - 18*b2 - 15*b1) * q^47 + 343 * q^49 + (8*b7 - 17*b6 + 183*b1) * q^50 + (8*b5 + 16*b3 - 296) * q^52 + (28*b7 - 34*b6 + 38*b2 + 364*b1) * q^53 + (-5*b5 + 12*b4 - 97*b3 + 1039) * q^55 - 8*b6 * q^56 + (-7*b5 - 10*b4 + 61*b3 + 368) * q^58 + (-45*b7 - 35*b6 + 36*b2 + 31*b1) * q^59 + (2*b5 + 48*b4 - 140*b3 + 16) * q^61 + (-17*b6 + 48*b2 + 152*b1) * q^62 - 512 * q^64 + (-25*b7 - 89*b6 - 44*b2 - 199*b1) * q^65 + (-11*b5 - 42*b4 + 56*b3 - 1269) * q^67 + (8*b7 - 8*b6 - 16*b2 - 120*b1) * q^68 + (-7*b5 - 7*b4 + 2*b3 + 196) * q^70 + (-10*b7 - 34*b6 - 87*b2 + 263*b1) * q^71 + (-11*b5 - 6*b4 - 142*b3 + 1329) * q^73 + (8*b7 - 54*b6 - 113*b1) * q^74 + (8*b5 + 96*b3 + 80) * q^76 + (-14*b7 - 4*b6 - 35*b2 - 189*b1) * q^77 + (b5 - 60*b4 - 26*b3 - 509) * q^79 + (-64*b2 - 64*b1) * q^80 + (-12*b5 + 19*b4 + 81*b3 - 444) * q^82 + (-37*b7 - 107*b6 + 92*b2 - 645*b1) * q^83 + (-5*b5 + 6*b4 + 116*b3 + 1681) * q^85 + (88*b7 - 63*b6 - 48*b2 + 379*b1) * q^86 + (8*b5 - 8*b4 + 32*b3 - 800) * q^88 + (66*b7 + b6 - 5*b2 - 1144*b1) * q^89 + (7*b5 - 42*b4 + 52*b3 - 833) * q^91 + (8*b7 - 8*b6 + 56*b2 + 576*b1) * q^92 + (b5 - 18*b4 + 381*b3 - 48) * q^94 + (-95*b7 - 69*b6 + 73*b2 - 362*b1) * q^95 + (-28*b5 + 12*b4 - 94*b3 + 4452) * q^97 - 343*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 64 q^{4}+O(q^{10})$$ 8 * q - 64 * q^4 $$8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100})$$ 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 34\nu^{5} - 3351\nu^{3} - 37674\nu ) / 15876$$ (v^7 - 34*v^5 - 3351*v^3 - 37674*v) / 15876 $$\beta_{2}$$ $$=$$ $$( -25\nu^{7} - 1502\nu^{5} - 23241\nu^{3} + 77490\nu ) / 21168$$ (-25*v^7 - 1502*v^5 - 23241*v^3 + 77490*v) / 21168 $$\beta_{3}$$ $$=$$ $$( -5\nu^{6} - 334\nu^{4} - 6429\nu^{2} - 30366 ) / 72$$ (-5*v^6 - 334*v^4 - 6429*v^2 - 30366) / 72 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 86\nu^{4} - 2073\nu^{2} - 11862 ) / 24$$ (-v^6 - 86*v^4 - 2073*v^2 - 11862) / 24 $$\beta_{5}$$ $$=$$ $$( -5\nu^{6} - 334\nu^{4} - 6237\nu^{2} - 25950 ) / 8$$ (-5*v^6 - 334*v^4 - 6237*v^2 - 25950) / 8 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} + 334\nu^{5} + 6177\nu^{3} + 24066\nu ) / 162$$ (5*v^7 + 334*v^5 + 6177*v^3 + 24066*v) / 162 $$\beta_{7}$$ $$=$$ $$( 29\nu^{7} + 1954\nu^{5} + 37914\nu^{3} + 182385\nu ) / 441$$ (29*v^7 + 1954*v^5 + 37914*v^3 + 182385*v) / 441
 $$\nu$$ $$=$$ $$( \beta_{6} + 28\beta_{2} + 35\beta_1 ) / 168$$ (b6 + 28*b2 + 35*b1) / 168 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 9\beta_{3} - 552 ) / 24$$ (b5 - 9*b3 - 552) / 24 $$\nu^{3}$$ $$=$$ $$( 8\beta_{7} - 21\beta_{6} - 108\beta_{2} - 87\beta_1 ) / 24$$ (8*b7 - 21*b6 - 108*b2 - 87*b1) / 24 $$\nu^{4}$$ $$=$$ $$( -41\beta_{5} - 30\beta_{4} + 387\beta_{3} + 15396 ) / 24$$ (-41*b5 - 30*b4 + 387*b3 + 15396) / 24 $$\nu^{5}$$ $$=$$ $$( -364\beta_{7} + 903\beta_{6} + 3228\beta_{2} - 1929\beta_1 ) / 24$$ (-364*b7 + 903*b6 + 3228*b2 - 1929*b1) / 24 $$\nu^{6}$$ $$=$$ $$( 1453\beta_{5} + 2004\beta_{4} - 14625\beta_{3} - 464448 ) / 24$$ (1453*b5 + 2004*b4 - 14625*b3 - 464448) / 24 $$\nu^{7}$$ $$=$$ $$( 14432\beta_{7} - 34287\beta_{6} - 101460\beta_{2} + 212271\beta_1 ) / 24$$ (14432*b7 - 34287*b6 - 101460*b2 + 212271*b1) / 24

## 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
 4.82885i 5.12618i − 2.54824i − 5.99257i 5.99257i 2.54824i − 5.12618i − 4.82885i
2.82843i 0 −8.00000 30.1368i 0 18.5203 22.6274i 0 −85.2398
323.2 2.82843i 0 −8.00000 28.1791i 0 −18.5203 22.6274i 0 −79.7026
323.3 2.82843i 0 −8.00000 17.8674i 0 −18.5203 22.6274i 0 50.5366
323.4 2.82843i 0 −8.00000 34.7917i 0 18.5203 22.6274i 0 98.4059
323.5 2.82843i 0 −8.00000 34.7917i 0 18.5203 22.6274i 0 98.4059
323.6 2.82843i 0 −8.00000 17.8674i 0 −18.5203 22.6274i 0 50.5366
323.7 2.82843i 0 −8.00000 28.1791i 0 −18.5203 22.6274i 0 −79.7026
323.8 2.82843i 0 −8.00000 30.1368i 0 18.5203 22.6274i 0 −85.2398
 $$n$$: 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.b 8
3.b odd 2 1 inner 378.5.b.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 3232T_{5}^{6} + 3711634T_{5}^{4} + 1761033888T_{5}^{2} + 278692135569$$ acting on $$S_{5}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 8)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 3232 T^{6} + \cdots + 278692135569$$
$7$ $$(T^{2} - 343)^{4}$$
$11$ $$T^{8} + 31504 T^{6} + \cdots + 7401278039841$$
$13$ $$(T^{4} - 148 T^{3} - 75032 T^{2} + \cdots + 1130516676)^{2}$$
$17$ $$T^{8} + 42760 T^{6} + \cdots + 39\!\cdots\!16$$
$19$ $$(T^{4} + 40 T^{3} - 184566 T^{2} + \cdots - 274761143)^{2}$$
$23$ $$T^{8} + 330784 T^{6} + \cdots + 44\!\cdots\!09$$
$29$ $$T^{8} + 1794088 T^{6} + \cdots + 40\!\cdots\!96$$
$31$ $$(T^{4} + 512 T^{3} + \cdots + 21940230321)^{2}$$
$37$ $$(T^{4} - 452 T^{3} + \cdots + 685159025593)^{2}$$
$41$ $$T^{8} + 5484096 T^{6} + \cdots + 59\!\cdots\!69$$
$43$ $$(T^{4} + 1612 T^{3} + \cdots + 19781954641348)^{2}$$
$47$ $$T^{8} + 25013736 T^{6} + \cdots + 10\!\cdots\!44$$
$53$ $$T^{8} + 28831680 T^{6} + \cdots + 16\!\cdots\!24$$
$59$ $$T^{8} + 72276952 T^{6} + \cdots + 65\!\cdots\!04$$
$61$ $$(T^{4} - 64 T^{3} + \cdots + 37203857115408)^{2}$$
$67$ $$(T^{4} + 5076 T^{3} + \cdots - 106304126923004)^{2}$$
$71$ $$T^{8} + 51389008 T^{6} + \cdots + 87\!\cdots\!41$$
$73$ $$(T^{4} - 5316 T^{3} + \cdots - 33552318651836)^{2}$$
$79$ $$(T^{4} + 2036 T^{3} + \cdots + 451461470055876)^{2}$$
$83$ $$T^{8} + 214925080 T^{6} + \cdots + 23\!\cdots\!64$$
$89$ $$T^{8} + 136948752 T^{6} + \cdots + 55\!\cdots\!61$$
$97$ $$(T^{4} - 17808 T^{3} + \cdots + 47506778460304)^{2}$$
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