Properties

 Label 304.2.u.b Level $304$ Weight $2$ Character orbit 304.u Analytic conductor $2.427$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.u (of order $$9$$, degree $$6$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.42745222145$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9}+O(q^{10})$$ q + (z^5 - z^3 - z + 1) * q^3 + (-z^5 - z^4 + z^2 - 1) * q^5 + (-z^5 - z^4 + z^2) * q^7 + (2*z^4 - 3*z^3 - 3*z + 2) * q^9 $$q + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{11} + ( - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 2) q^{13} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{15} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{17} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{19} + (\zeta_{18}^{4} - \zeta_{18}) q^{21} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}) q^{23} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{25} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2}) q^{27} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} - 1) q^{29} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{31} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{33} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{35} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{37} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 4) q^{39} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{41} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{43} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} - 5) q^{45} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{47} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{49} + (\zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 1) q^{51} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18} + 1) q^{53} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18} - 3) q^{55} + (5 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 4 \zeta_{18} + 5) q^{57} + (2 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18} - 2) q^{59} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4) q^{61} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{63} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{3} + 5 \zeta_{18}) q^{65} + (4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{67} + (4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 4) q^{69} + (10 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{71} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{73} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 5) q^{75} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{77} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{79} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{81} + ( - 6 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 9 \zeta_{18}) q^{83} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{85} + (6 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 6 \zeta_{18}) q^{87} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} - 1) q^{89} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{91} + (2 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 8 \zeta_{18} - 2) q^{93} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 6) q^{95} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 5 \zeta_{18} + 2) q^{97} + ( - 6 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18} - 4) q^{99}+O(q^{100})$$ q + (z^5 - z^3 - z + 1) * q^3 + (-z^5 - z^4 + z^2 - 1) * q^5 + (-z^5 - z^4 + z^2) * q^7 + (2*z^4 - 3*z^3 - 3*z + 2) * q^9 + (-2*z^5 + z^4 + z^2 - 2*z) * q^11 + (-z^4 + 3*z^3 - 2*z - 2) * q^13 + (-z^5 + z^4 + z^3 - 1) * q^15 + (-z^5 - 2*z^4 - z^3 + z^2 + 2*z + 1) * q^17 + (-z^5 - 2*z^4 - z^2 - 2*z + 2) * q^19 + (z^4 - z) * q^21 + (-2*z^5 - 2*z^4 - 2*z^3 + 2*z) * q^23 + (-2*z^5 + z^4 - 2*z^3 + 2*z^2 + z + 1) * q^25 + (3*z^4 - 2*z^3 + 3*z^2) * q^27 + (-z^4 + z^3 - 5*z^2 + z - 1) * q^29 + (z^5 + z^4 + 3*z^3 - 3*z^2 + 2*z - 3) * q^31 + (-2*z^5 + 3*z^4 - z^3 + z^2 - 1) * q^33 + (2*z^5 - 2*z^3 - 2*z^2 + z) * q^35 + (3*z^5 - z^4 - 2*z^2 - 2*z) * q^37 + (2*z^5 - z^4 - z^2 - z + 4) * q^39 + (z^5 - z^3 + 3*z^2 + 4*z + 4) * q^41 + (3*z^5 + 5*z^3 + 2*z^2 - 2) * q^43 + (-z^5 - z^4 + 5*z^3 + z^2 - 5) * q^45 + (2*z^4 - 3*z^3 + 2*z^2 - 3*z + 2) * q^47 + (z^5 - z^4 + 5*z^3 - z^2 + z) * q^49 + (z^5 + z^3 - z^2 - z - 1) * q^51 + (2*z^5 - 3*z^4 - 3*z^3 + 2*z + 1) * q^53 + (3*z^5 - 3*z^4 + 3*z - 3) * q^55 + (5*z^5 + z^4 - 2*z^3 - 4*z + 5) * q^57 + (2*z^5 - 7*z^4 + 7*z - 2) * q^59 + (-3*z^5 + 4*z^4 + 4*z^3 - 4) * q^61 + (-z^5 + z^4 + 2*z^3 + z^2 - 3*z - 3) * q^63 + (5*z^5 - 4*z^3 + 5*z) * q^65 + (4*z^4 + 2*z^3 - 6*z^2 + 2*z + 4) * q^67 + (4*z^3 - 2*z^2 + 2*z - 4) * q^69 + (10*z^4 - 2*z^3 - 2*z^2 + 2) * q^71 + (4*z^5 - 4*z^3 - 4*z^2) * q^73 + (-4*z^5 + 3*z^4 + z^2 + z - 5) * q^75 + (z^5 - 2*z^4 + z^2 + z - 3) * q^77 + (-z^5 + z^3 + 7*z^2 - 3*z + 6) * q^79 + (z^5 + 5*z^4 - z^2 + 1) * q^81 + (-6*z^5 - 6*z^4 - 3*z^2 + 9*z) * q^83 + (z^4 - 2*z^3 - 3*z^2 - 2*z + 1) * q^85 + (6*z^5 - 7*z^4 + 7*z^3 - 7*z^2 + 6*z) * q^87 + (-5*z^5 + 3*z^4 - 2*z^3 + 5*z^2 - z - 1) * q^89 + (5*z^5 - z^4 - z^3 + 3*z - 2) * q^91 + (2*z^5 - 8*z^4 + 7*z^3 - 7*z^2 + 8*z - 2) * q^93 + (3*z^5 - z^4 - z^3 + z^2 + z - 6) * q^95 + (-2*z^5 - 5*z^4 + 2*z^3 - 2*z^2 + 5*z + 2) * q^97 + (-6*z^5 + 5*z^4 + 5*z^3 - z - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 6 q^{5} + 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 6 * q^5 + 3 * q^9 $$6 q + 3 q^{3} - 6 q^{5} + 3 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{17} + 12 q^{19} - 6 q^{23} - 6 q^{27} - 3 q^{29} - 9 q^{31} - 9 q^{33} - 6 q^{35} + 24 q^{39} + 21 q^{41} + 3 q^{43} - 15 q^{45} + 3 q^{47} + 15 q^{49} - 3 q^{51} - 3 q^{53} - 18 q^{55} + 24 q^{57} - 12 q^{59} - 12 q^{61} - 12 q^{63} - 12 q^{65} + 30 q^{67} - 12 q^{69} + 6 q^{71} - 12 q^{73} - 30 q^{75} - 18 q^{77} + 39 q^{79} + 6 q^{81} + 21 q^{87} - 12 q^{89} - 15 q^{91} + 9 q^{93} - 39 q^{95} + 18 q^{97} - 9 q^{99}+O(q^{100})$$ 6 * q + 3 * q^3 - 6 * q^5 + 3 * q^9 - 3 * q^13 - 3 * q^15 + 3 * q^17 + 12 * q^19 - 6 * q^23 - 6 * q^27 - 3 * q^29 - 9 * q^31 - 9 * q^33 - 6 * q^35 + 24 * q^39 + 21 * q^41 + 3 * q^43 - 15 * q^45 + 3 * q^47 + 15 * q^49 - 3 * q^51 - 3 * q^53 - 18 * q^55 + 24 * q^57 - 12 * q^59 - 12 * q^61 - 12 * q^63 - 12 * q^65 + 30 * q^67 - 12 * q^69 + 6 * q^71 - 12 * q^73 - 30 * q^75 - 18 * q^77 + 39 * q^79 + 6 * q^81 + 21 * q^87 - 12 * q^89 - 15 * q^91 + 9 * q^93 - 39 * q^95 + 18 * q^97 - 9 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.766044 − 0.642788i −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.173648 − 0.984808i
0 2.20574 + 1.85083i 0 −0.826352 + 0.300767i 0 0.173648 + 0.300767i 0 0.918748 + 5.21048i 0
81.1 0 −0.0923963 + 0.524005i 0 −1.93969 1.62760i 0 −0.939693 1.62760i 0 2.55303 + 0.929228i 0
161.1 0 2.20574 1.85083i 0 −0.826352 0.300767i 0 0.173648 0.300767i 0 0.918748 5.21048i 0
177.1 0 −0.613341 0.223238i 0 −0.233956 1.32683i 0 0.766044 1.32683i 0 −1.97178 1.65452i 0
225.1 0 −0.613341 + 0.223238i 0 −0.233956 + 1.32683i 0 0.766044 + 1.32683i 0 −1.97178 + 1.65452i 0
289.1 0 −0.0923963 0.524005i 0 −1.93969 + 1.62760i 0 −0.939693 + 1.62760i 0 2.55303 0.929228i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.u.b 6
4.b odd 2 1 19.2.e.a 6
12.b even 2 1 171.2.u.c 6
19.e even 9 1 inner 304.2.u.b 6
19.e even 9 1 5776.2.a.br 3
19.f odd 18 1 5776.2.a.bi 3
20.d odd 2 1 475.2.l.a 6
20.e even 4 2 475.2.u.a 12
28.d even 2 1 931.2.w.a 6
28.f even 6 1 931.2.v.a 6
28.f even 6 1 931.2.x.b 6
28.g odd 6 1 931.2.v.b 6
28.g odd 6 1 931.2.x.a 6
76.d even 2 1 361.2.e.h 6
76.f even 6 1 361.2.e.a 6
76.f even 6 1 361.2.e.b 6
76.g odd 6 1 361.2.e.f 6
76.g odd 6 1 361.2.e.g 6
76.k even 18 1 361.2.a.h 3
76.k even 18 2 361.2.c.h 6
76.k even 18 1 361.2.e.a 6
76.k even 18 1 361.2.e.b 6
76.k even 18 1 361.2.e.h 6
76.l odd 18 1 19.2.e.a 6
76.l odd 18 1 361.2.a.g 3
76.l odd 18 2 361.2.c.i 6
76.l odd 18 1 361.2.e.f 6
76.l odd 18 1 361.2.e.g 6
228.u odd 18 1 3249.2.a.s 3
228.v even 18 1 171.2.u.c 6
228.v even 18 1 3249.2.a.z 3
380.ba odd 18 1 475.2.l.a 6
380.ba odd 18 1 9025.2.a.bd 3
380.bb even 18 1 9025.2.a.x 3
380.bj even 36 2 475.2.u.a 12
532.br even 18 1 931.2.v.a 6
532.bt odd 18 1 931.2.v.b 6
532.cc even 18 1 931.2.w.a 6
532.cd even 18 1 931.2.x.b 6
532.cf odd 18 1 931.2.x.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 4.b odd 2 1
19.2.e.a 6 76.l odd 18 1
171.2.u.c 6 12.b even 2 1
171.2.u.c 6 228.v even 18 1
304.2.u.b 6 1.a even 1 1 trivial
304.2.u.b 6 19.e even 9 1 inner
361.2.a.g 3 76.l odd 18 1
361.2.a.h 3 76.k even 18 1
361.2.c.h 6 76.k even 18 2
361.2.c.i 6 76.l odd 18 2
361.2.e.a 6 76.f even 6 1
361.2.e.a 6 76.k even 18 1
361.2.e.b 6 76.f even 6 1
361.2.e.b 6 76.k even 18 1
361.2.e.f 6 76.g odd 6 1
361.2.e.f 6 76.l odd 18 1
361.2.e.g 6 76.g odd 6 1
361.2.e.g 6 76.l odd 18 1
361.2.e.h 6 76.d even 2 1
361.2.e.h 6 76.k even 18 1
475.2.l.a 6 20.d odd 2 1
475.2.l.a 6 380.ba odd 18 1
475.2.u.a 12 20.e even 4 2
475.2.u.a 12 380.bj even 36 2
931.2.v.a 6 28.f even 6 1
931.2.v.a 6 532.br even 18 1
931.2.v.b 6 28.g odd 6 1
931.2.v.b 6 532.bt odd 18 1
931.2.w.a 6 28.d even 2 1
931.2.w.a 6 532.cc even 18 1
931.2.x.a 6 28.g odd 6 1
931.2.x.a 6 532.cf odd 18 1
931.2.x.b 6 28.f even 6 1
931.2.x.b 6 532.cd even 18 1
3249.2.a.s 3 228.u odd 18 1
3249.2.a.z 3 228.v even 18 1
5776.2.a.bi 3 19.f odd 18 1
5776.2.a.br 3 19.e even 9 1
9025.2.a.x 3 380.bb even 18 1
9025.2.a.bd 3 380.ba odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 3T_{3}^{5} + 3T_{3}^{4} + 8T_{3}^{3} + 6T_{3}^{2} + 3T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(304, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1$$
$5$ $$T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9$$
$7$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$11$ $$T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81$$
$13$ $$T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369$$
$17$ $$T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9$$
$19$ $$T^{6} - 12 T^{5} + 78 T^{4} + \cdots + 6859$$
$23$ $$T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576$$
$29$ $$T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321$$
$31$ $$T^{6} + 9 T^{5} + 75 T^{4} + \cdots + 2809$$
$37$ $$(T^{3} - 21 T - 17)^{2}$$
$41$ $$T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321$$
$43$ $$T^{6} - 3 T^{5} + 60 T^{4} + \cdots + 26569$$
$47$ $$T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9$$
$53$ $$T^{6} + 3 T^{5} - 84 T^{3} + \cdots + 2601$$
$59$ $$T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289$$
$61$ $$T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761$$
$67$ $$T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776$$
$71$ $$T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544$$
$73$ $$T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096$$
$79$ $$T^{6} - 39 T^{5} + 708 T^{4} + \cdots + 654481$$
$83$ $$T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681$$
$89$ $$T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249$$
$97$ $$T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129$$