# Properties

 Label 378.2.f.b Level $378$ Weight $2$ Character orbit 378.f Analytic conductor $3.018$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 1.00000 0 −2.00000
253.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 1.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.f.b 2
3.b odd 2 1 126.2.f.b 2
4.b odd 2 1 3024.2.r.c 2
7.b odd 2 1 2646.2.f.b 2
7.c even 3 1 2646.2.e.i 2
7.c even 3 1 2646.2.h.b 2
7.d odd 6 1 2646.2.e.h 2
7.d odd 6 1 2646.2.h.c 2
9.c even 3 1 inner 378.2.f.b 2
9.c even 3 1 1134.2.a.f 1
9.d odd 6 1 126.2.f.b 2
9.d odd 6 1 1134.2.a.c 1
12.b even 2 1 1008.2.r.a 2
21.c even 2 1 882.2.f.f 2
21.g even 6 1 882.2.e.e 2
21.g even 6 1 882.2.h.g 2
21.h odd 6 1 882.2.e.a 2
21.h odd 6 1 882.2.h.h 2
36.f odd 6 1 3024.2.r.c 2
36.f odd 6 1 9072.2.a.f 1
36.h even 6 1 1008.2.r.a 2
36.h even 6 1 9072.2.a.t 1
63.g even 3 1 2646.2.e.i 2
63.h even 3 1 2646.2.h.b 2
63.i even 6 1 882.2.h.g 2
63.j odd 6 1 882.2.h.h 2
63.k odd 6 1 2646.2.e.h 2
63.l odd 6 1 2646.2.f.b 2
63.l odd 6 1 7938.2.a.bb 1
63.n odd 6 1 882.2.e.a 2
63.o even 6 1 882.2.f.f 2
63.o even 6 1 7938.2.a.e 1
63.s even 6 1 882.2.e.e 2
63.t odd 6 1 2646.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 3.b odd 2 1
126.2.f.b 2 9.d odd 6 1
378.2.f.b 2 1.a even 1 1 trivial
378.2.f.b 2 9.c even 3 1 inner
882.2.e.a 2 21.h odd 6 1
882.2.e.a 2 63.n odd 6 1
882.2.e.e 2 21.g even 6 1
882.2.e.e 2 63.s even 6 1
882.2.f.f 2 21.c even 2 1
882.2.f.f 2 63.o even 6 1
882.2.h.g 2 21.g even 6 1
882.2.h.g 2 63.i even 6 1
882.2.h.h 2 21.h odd 6 1
882.2.h.h 2 63.j odd 6 1
1008.2.r.a 2 12.b even 2 1
1008.2.r.a 2 36.h even 6 1
1134.2.a.c 1 9.d odd 6 1
1134.2.a.f 1 9.c even 3 1
2646.2.e.h 2 7.d odd 6 1
2646.2.e.h 2 63.k odd 6 1
2646.2.e.i 2 7.c even 3 1
2646.2.e.i 2 63.g even 3 1
2646.2.f.b 2 7.b odd 2 1
2646.2.f.b 2 63.l odd 6 1
2646.2.h.b 2 7.c even 3 1
2646.2.h.b 2 63.h even 3 1
2646.2.h.c 2 7.d odd 6 1
2646.2.h.c 2 63.t odd 6 1
3024.2.r.c 2 4.b odd 2 1
3024.2.r.c 2 36.f odd 6 1
7938.2.a.e 1 63.o even 6 1
7938.2.a.bb 1 63.l odd 6 1
9072.2.a.f 1 36.f odd 6 1
9072.2.a.t 1 36.h even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$(T - 5)^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$T^{2} - 6T + 36$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} - 12T + 144$$
$67$ $$T^{2} + 13T + 169$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 1)^{2}$$
$79$ $$T^{2} - 6T + 36$$
$83$ $$T^{2} - 16T + 256$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 5T + 25$$