Properties

 Label 378.3.b.c Level $378$ Weight $3$ Character orbit 378.b Analytic conductor $10.300$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$10.2997539928$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ x^8 + x^4 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} - 2 q^{4} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{7} + 2 \beta_{4} q^{8}+O(q^{10})$$ q - b4 * q^2 - 2 * q^4 + (-b6 + b4 - b1) * q^5 - b3 * q^7 + 2*b4 * q^8 $$q - \beta_{4} q^{2} - 2 q^{4} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{5} + 2 \beta_{3} + 2) q^{10} + (\beta_{6} - \beta_{4} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 5) q^{13} - \beta_{6} q^{14} + 4 q^{16} + (\beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{5} + 3 \beta_{3} + 5) q^{19} + (2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{20} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} - 2) q^{22} + ( - 2 \beta_{6} - 4 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{23} + ( - 2 \beta_{7} + 2 \beta_{5} - 4 \beta_{3}) q^{25} + ( - \beta_{6} + 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{26} + 2 \beta_{3} q^{28} + (5 \beta_{6} + 7 \beta_{4} + \beta_{2} + 3 \beta_1) q^{29} + ( - 3 \beta_{7} - 3 \beta_{5} - \beta_{3} - 5) q^{31} - 4 \beta_{4} q^{32} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{3} - 2) q^{34} + (\beta_{6} - 7 \beta_{4} + \beta_{2}) q^{35} + ( - 2 \beta_{7} + 2 \beta_{5} + 6 \beta_{3} - 1) q^{37} + (3 \beta_{6} - 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{5} - 4 \beta_{3} - 4) q^{40} + ( - 9 \beta_{6} + 9 \beta_{4} + \beta_1) q^{41} + ( - \beta_{7} - 11 \beta_{5} + 5) q^{43} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{44} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 8) q^{46} + (6 \beta_{6} + 18 \beta_{4} - 4 \beta_{2} - 13 \beta_1) q^{47} + 7 q^{49} + ( - 4 \beta_{6} - 4 \beta_{2} - 4 \beta_1) q^{50} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 10) q^{52} + ( - 6 \beta_{6} - 18 \beta_{4} + 4 \beta_{2} - 6 \beta_1) q^{53} + (\beta_{7} - 7 \beta_{5} + 13 \beta_{3} + 7) q^{55} + 2 \beta_{6} q^{56} + ( - \beta_{7} + 3 \beta_{5} - 10 \beta_{3} + 14) q^{58} + ( - 10 \beta_{6} - 2 \beta_{4} - 9 \beta_1) q^{59} + ( - \beta_{7} + 13 \beta_{5} - 13 \beta_{3} - 11) q^{61} + ( - \beta_{6} + 5 \beta_{4} - 6 \beta_{2} + 6 \beta_1) q^{62} - 8 q^{64} + (15 \beta_{6} - 21 \beta_{4} + 5 \beta_{2} + 21 \beta_1) q^{65} + (4 \beta_{7} + 8 \beta_{5} + 10 \beta_{3} + 30) q^{67} + ( - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{68} + ( - \beta_{7} - 2 \beta_{3} - 14) q^{70} + ( - 8 \beta_{6} - 4 \beta_{4} - 4 \beta_{2} + 8 \beta_1) q^{71} + ( - 2 \beta_{7} - 4 \beta_{5} - 14 \beta_{3} - 36) q^{73} + (6 \beta_{6} + \beta_{4} - 4 \beta_{2} - 4 \beta_1) q^{74} + (2 \beta_{7} - 2 \beta_{5} - 6 \beta_{3} - 10) q^{76} + ( - \beta_{6} + 7 \beta_{4} + \beta_{2} - 7 \beta_1) q^{77} + (\beta_{7} + 11 \beta_{5} - 10 \beta_{3} - 53) q^{79} + ( - 4 \beta_{6} + 4 \beta_{4} - 4 \beta_1) q^{80} + (\beta_{5} + 18 \beta_{3} + 18) q^{82} + (22 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} + 7 \beta_1) q^{83} + (4 \beta_{7} - 16 \beta_{5} + 22 \beta_{3} + 25) q^{85} + ( - 5 \beta_{4} - 2 \beta_{2} + 22 \beta_1) q^{86} + (2 \beta_{7} + 2 \beta_{5} + 4 \beta_{3} + 4) q^{88} + ( - 6 \beta_{6} + 30 \beta_{4} + 6 \beta_{2} - 14 \beta_1) q^{89} + (\beta_{7} - 7 \beta_{5} + 5 \beta_{3} + 7) q^{91} + (4 \beta_{6} + 8 \beta_{4} - 4 \beta_{2} - 8 \beta_1) q^{92} + (4 \beta_{7} - 13 \beta_{5} - 12 \beta_{3} + 36) q^{94} + (\beta_{6} + 17 \beta_{4} + \beta_{2} + 11 \beta_1) q^{95} + (5 \beta_{7} - 11 \beta_{5} - 23 \beta_{3} + 75) q^{97} - 7 \beta_{4} q^{98}+O(q^{100})$$ q - b4 * q^2 - 2 * q^4 + (-b6 + b4 - b1) * q^5 - b3 * q^7 + 2*b4 * q^8 + (-b5 + 2*b3 + 2) * q^10 + (b6 - b4 + b2 - b1) * q^11 + (-b7 + b5 - b3 - 5) * q^13 - b6 * q^14 + 4 * q^16 + (b6 - b4 + 2*b2 + b1) * q^17 + (-b7 + b5 + 3*b3 + 5) * q^19 + (2*b6 - 2*b4 + 2*b1) * q^20 + (-b7 - b5 - 2*b3 - 2) * q^22 + (-2*b6 - 4*b4 + 2*b2 + 4*b1) * q^23 + (-2*b7 + 2*b5 - 4*b3) * q^25 + (-b6 + 5*b4 - 2*b2 - 2*b1) * q^26 + 2*b3 * q^28 + (5*b6 + 7*b4 + b2 + 3*b1) * q^29 + (-3*b7 - 3*b5 - b3 - 5) * q^31 - 4*b4 * q^32 + (-2*b7 + b5 - 2*b3 - 2) * q^34 + (b6 - 7*b4 + b2) * q^35 + (-2*b7 + 2*b5 + 6*b3 - 1) * q^37 + (3*b6 - 5*b4 - 2*b2 - 2*b1) * q^38 + (2*b5 - 4*b3 - 4) * q^40 + (-9*b6 + 9*b4 + b1) * q^41 + (-b7 - 11*b5 + 5) * q^43 + (-2*b6 + 2*b4 - 2*b2 + 2*b1) * q^44 + (-2*b7 + 4*b5 + 4*b3 - 8) * q^46 + (6*b6 + 18*b4 - 4*b2 - 13*b1) * q^47 + 7 * q^49 + (-4*b6 - 4*b2 - 4*b1) * q^50 + (2*b7 - 2*b5 + 2*b3 + 10) * q^52 + (-6*b6 - 18*b4 + 4*b2 - 6*b1) * q^53 + (b7 - 7*b5 + 13*b3 + 7) * q^55 + 2*b6 * q^56 + (-b7 + 3*b5 - 10*b3 + 14) * q^58 + (-10*b6 - 2*b4 - 9*b1) * q^59 + (-b7 + 13*b5 - 13*b3 - 11) * q^61 + (-b6 + 5*b4 - 6*b2 + 6*b1) * q^62 - 8 * q^64 + (15*b6 - 21*b4 + 5*b2 + 21*b1) * q^65 + (4*b7 + 8*b5 + 10*b3 + 30) * q^67 + (-2*b6 + 2*b4 - 4*b2 - 2*b1) * q^68 + (-b7 - 2*b3 - 14) * q^70 + (-8*b6 - 4*b4 - 4*b2 + 8*b1) * q^71 + (-2*b7 - 4*b5 - 14*b3 - 36) * q^73 + (6*b6 + b4 - 4*b2 - 4*b1) * q^74 + (2*b7 - 2*b5 - 6*b3 - 10) * q^76 + (-b6 + 7*b4 + b2 - 7*b1) * q^77 + (b7 + 11*b5 - 10*b3 - 53) * q^79 + (-4*b6 + 4*b4 - 4*b1) * q^80 + (b5 + 18*b3 + 18) * q^82 + (22*b6 + 2*b4 + 2*b2 + 7*b1) * q^83 + (4*b7 - 16*b5 + 22*b3 + 25) * q^85 + (-5*b4 - 2*b2 + 22*b1) * q^86 + (2*b7 + 2*b5 + 4*b3 + 4) * q^88 + (-6*b6 + 30*b4 + 6*b2 - 14*b1) * q^89 + (b7 - 7*b5 + 5*b3 + 7) * q^91 + (4*b6 + 8*b4 - 4*b2 - 8*b1) * q^92 + (4*b7 - 13*b5 - 12*b3 + 36) * q^94 + (b6 + 17*b4 + b2 + 11*b1) * q^95 + (5*b7 - 11*b5 - 23*b3 + 75) * q^97 - 7*b4 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4}+O(q^{10})$$ 8 * q - 16 * q^4 $$8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100})$$ 8 * q - 16 * q^4 + 16 * q^10 - 40 * q^13 + 32 * q^16 + 40 * q^19 - 16 * q^22 - 40 * q^31 - 16 * q^34 - 8 * q^37 - 32 * q^40 + 40 * q^43 - 64 * q^46 + 56 * q^49 + 80 * q^52 + 56 * q^55 + 112 * q^58 - 88 * q^61 - 64 * q^64 + 240 * q^67 - 112 * q^70 - 288 * q^73 - 80 * q^76 - 424 * q^79 + 144 * q^82 + 200 * q^85 + 32 * q^88 + 56 * q^91 + 288 * q^94 + 600 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 5\nu^{2} ) / 4$$ (v^6 + 5*v^2) / 4 $$\beta_{2}$$ $$=$$ $$2\nu^{4} + 1$$ 2*v^4 + 1 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} + 3\nu^{2} ) / 4$$ (-v^6 + 3*v^2) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 4\nu^{5} - 7\nu^{3} + 4\nu ) / 24$$ (v^7 - 4*v^5 - 7*v^3 + 4*v) / 24 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 4\nu^{5} - 7\nu^{3} - 4\nu ) / 8$$ (v^7 + 4*v^5 - 7*v^3 - 4*v) / 8 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} + 4\nu^{5} + 13\nu^{3} + 44\nu ) / 24$$ (5*v^7 + 4*v^5 + 13*v^3 + 44*v) / 24 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 4\nu^{5} - 13\nu^{3} + 44\nu ) / 8$$ (-5*v^7 + 4*v^5 - 13*v^3 + 44*v) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} + 3\beta_{6} - \beta_{5} + 3\beta_{4} ) / 12$$ (b7 + 3*b6 - b5 + 3*b4) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} - 5\beta_{5} - 15\beta_{4} ) / 12$$ (-b7 + 3*b6 - 5*b5 - 15*b4) / 12 $$\nu^{4}$$ $$=$$ $$( \beta_{2} - 1 ) / 2$$ (b2 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} + 11\beta_{5} - 33\beta_{4} ) / 12$$ (b7 + 3*b6 + 11*b5 - 33*b4) / 12 $$\nu^{6}$$ $$=$$ $$( -5\beta_{3} + 3\beta_1 ) / 2$$ (-5*b3 + 3*b1) / 2 $$\nu^{7}$$ $$=$$ $$( -7\beta_{7} + 21\beta_{6} + 13\beta_{5} + 39\beta_{4} ) / 12$$ (-7*b7 + 21*b6 + 13*b5 + 39*b4) / 12

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
 0.581861 + 1.28897i −0.581861 + 1.28897i −1.28897 − 0.581861i 1.28897 − 0.581861i 1.28897 + 0.581861i −1.28897 + 0.581861i −0.581861 − 1.28897i 0.581861 − 1.28897i
1.41421i 0 −2.00000 5.32744i 0 2.64575 2.82843i 0 −7.53414
323.2 1.41421i 0 −2.00000 0.672556i 0 2.64575 2.82843i 0 0.951138
323.3 1.41421i 0 −2.00000 2.15587i 0 −2.64575 2.82843i 0 3.04886
323.4 1.41421i 0 −2.00000 8.15587i 0 −2.64575 2.82843i 0 11.5341
323.5 1.41421i 0 −2.00000 8.15587i 0 −2.64575 2.82843i 0 11.5341
323.6 1.41421i 0 −2.00000 2.15587i 0 −2.64575 2.82843i 0 3.04886
323.7 1.41421i 0 −2.00000 0.672556i 0 2.64575 2.82843i 0 0.951138
323.8 1.41421i 0 −2.00000 5.32744i 0 2.64575 2.82843i 0 −7.53414
 $$n$$: 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 378.3.b.c 8
3.b odd 2 1 inner 378.3.b.c 8
4.b odd 2 1 3024.3.d.h 8
9.c even 3 2 1134.3.q.f 16
9.d odd 6 2 1134.3.q.f 16
12.b even 2 1 3024.3.d.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.b.c 8 1.a even 1 1 trivial
378.3.b.c 8 3.b odd 2 1 inner
1134.3.q.f 16 9.c even 3 2
1134.3.q.f 16 9.d odd 6 2
3024.3.d.h 8 4.b odd 2 1
3024.3.d.h 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 100T_{5}^{6} + 2374T_{5}^{4} + 9828T_{5}^{2} + 3969$$ acting on $$S_{3}^{\mathrm{new}}(378, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 100 T^{6} + 2374 T^{4} + \cdots + 3969$$
$7$ $$(T^{2} - 7)^{4}$$
$11$ $$T^{8} + 352 T^{6} + 38536 T^{4} + \cdots + 63504$$
$13$ $$(T^{4} + 20 T^{3} - 152 T^{2} - 1512 T + 7812)^{2}$$
$17$ $$T^{8} + 1108 T^{6} + \cdots + 2793862449$$
$19$ $$(T^{4} - 20 T^{3} - 264 T^{2} + 616 T + 2884)^{2}$$
$23$ $$T^{8} + 1936 T^{6} + \cdots + 14876193024$$
$29$ $$T^{8} + 2368 T^{6} + \cdots + 29253313296$$
$31$ $$(T^{4} + 20 T^{3} - 2456 T^{2} + \cdots + 816804)^{2}$$
$37$ $$(T^{4} + 4 T^{3} - 1650 T^{2} + \cdots - 66023)^{2}$$
$41$ $$T^{8} + 5220 T^{6} + \cdots + 849230442369$$
$43$ $$(T^{4} - 20 T^{3} - 4458 T^{2} + \cdots + 4096129)^{2}$$
$47$ $$T^{8} + \cdots + 125526008337921$$
$53$ $$T^{8} + 9936 T^{6} + \cdots + 1049760000$$
$59$ $$T^{8} + 8548 T^{6} + \cdots + 173256570081$$
$61$ $$(T^{4} + 44 T^{3} - 7976 T^{2} + \cdots + 3242628)^{2}$$
$67$ $$(T^{4} - 120 T^{3} - 2336 T^{2} + \cdots - 19027904)^{2}$$
$71$ $$T^{8} + 10048 T^{6} + \cdots + 949829566464$$
$73$ $$(T^{4} + 144 T^{3} + 3448 T^{2} + \cdots - 8237936)^{2}$$
$79$ $$(T^{4} + 212 T^{3} + 10846 T^{2} + \cdots - 13387527)^{2}$$
$83$ $$T^{8} + 29908 T^{6} + \cdots + 12\!\cdots\!49$$
$89$ $$T^{8} + 25344 T^{6} + \cdots + 10\!\cdots\!56$$
$97$ $$(T^{4} - 300 T^{3} + 15688 T^{2} + \cdots - 190394300)^{2}$$