Properties

 Label 390.2.bn.c Level $390$ Weight $2$ Character orbit 390.bn Analytic conductor $3.114$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.bn (of order $$12$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 16 q^{2} - 16 q^{4} - 12 q^{7} + 32 q^{8}+O(q^{10})$$ 32 * q - 16 * q^2 - 16 * q^4 - 12 * q^7 + 32 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 16 q^{2} - 16 q^{4} - 12 q^{7} + 32 q^{8} - 4 q^{11} + 8 q^{13} - 4 q^{15} - 16 q^{16} - 4 q^{17} + 20 q^{19} + 8 q^{21} + 8 q^{22} - 16 q^{23} - 8 q^{25} - 4 q^{26} + 12 q^{28} + 24 q^{29} + 8 q^{30} + 12 q^{31} - 16 q^{32} - 16 q^{34} + 12 q^{35} - 24 q^{37} - 4 q^{38} - 20 q^{39} - 28 q^{41} - 16 q^{42} + 4 q^{43} - 4 q^{44} - 4 q^{46} + 20 q^{49} - 8 q^{50} - 4 q^{52} - 4 q^{53} + 68 q^{55} - 12 q^{56} + 16 q^{57} - 24 q^{58} + 36 q^{59} - 4 q^{60} - 28 q^{61} - 48 q^{62} + 32 q^{64} - 28 q^{65} - 28 q^{67} + 20 q^{68} - 20 q^{69} - 24 q^{70} - 4 q^{71} - 48 q^{73} + 24 q^{74} + 24 q^{75} - 16 q^{76} + 20 q^{77} + 4 q^{78} + 16 q^{81} + 44 q^{82} + 8 q^{84} + 64 q^{85} - 8 q^{86} + 20 q^{87} - 4 q^{88} - 16 q^{89} - 40 q^{91} + 20 q^{92} - 24 q^{93} - 24 q^{94} + 68 q^{95} + 8 q^{97} + 20 q^{98} - 4 q^{99}+O(q^{100})$$ 32 * q - 16 * q^2 - 16 * q^4 - 12 * q^7 + 32 * q^8 - 4 * q^11 + 8 * q^13 - 4 * q^15 - 16 * q^16 - 4 * q^17 + 20 * q^19 + 8 * q^21 + 8 * q^22 - 16 * q^23 - 8 * q^25 - 4 * q^26 + 12 * q^28 + 24 * q^29 + 8 * q^30 + 12 * q^31 - 16 * q^32 - 16 * q^34 + 12 * q^35 - 24 * q^37 - 4 * q^38 - 20 * q^39 - 28 * q^41 - 16 * q^42 + 4 * q^43 - 4 * q^44 - 4 * q^46 + 20 * q^49 - 8 * q^50 - 4 * q^52 - 4 * q^53 + 68 * q^55 - 12 * q^56 + 16 * q^57 - 24 * q^58 + 36 * q^59 - 4 * q^60 - 28 * q^61 - 48 * q^62 + 32 * q^64 - 28 * q^65 - 28 * q^67 + 20 * q^68 - 20 * q^69 - 24 * q^70 - 4 * q^71 - 48 * q^73 + 24 * q^74 + 24 * q^75 - 16 * q^76 + 20 * q^77 + 4 * q^78 + 16 * q^81 + 44 * q^82 + 8 * q^84 + 64 * q^85 - 8 * q^86 + 20 * q^87 - 4 * q^88 - 16 * q^89 - 40 * q^91 + 20 * q^92 - 24 * q^93 - 24 * q^94 + 68 * q^95 + 8 * q^97 + 20 * q^98 - 4 * q^99

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −0.500000 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i −2.18531 + 0.473708i 0.258819 + 0.965926i 2.35599 + 1.36023i 1.00000 0.866025 + 0.500000i 1.50290 + 1.65568i
67.2 −0.500000 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 0.0142140 2.23602i 0.258819 + 0.965926i −3.50522 2.02374i 1.00000 0.866025 + 0.500000i −1.94356 + 1.10570i
67.3 −0.500000 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 0.933504 2.03189i 0.258819 + 0.965926i 3.73901 + 2.15872i 1.00000 0.866025 + 0.500000i −2.22642 + 0.207506i
67.4 −0.500000 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 1.23760 + 1.86235i 0.258819 + 0.965926i −3.19321 1.84360i 1.00000 0.866025 + 0.500000i 0.994046 2.00297i
67.5 −0.500000 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −2.23570 + 0.0407428i −0.258819 0.965926i 2.13987 + 1.23545i 1.00000 0.866025 + 0.500000i 1.15313 + 1.91580i
67.6 −0.500000 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −1.64894 + 1.51030i −0.258819 0.965926i −3.98101 2.29844i 1.00000 0.866025 + 0.500000i 2.13243 + 0.672876i
67.7 −0.500000 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i 1.83584 + 1.27660i −0.258819 0.965926i 0.838691 + 0.484219i 1.00000 0.866025 + 0.500000i 0.187644 2.22818i
67.8 −0.500000 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i 2.04880 0.895783i −0.258819 0.965926i −1.39412 0.804897i 1.00000 0.866025 + 0.500000i −1.80017 1.32642i
97.1 −0.500000 + 0.866025i −0.258819 0.965926i −0.500000 0.866025i −2.04652 + 0.900968i 0.965926 + 0.258819i −0.0517653 + 0.0298867i 1.00000 −0.866025 + 0.500000i 0.243000 2.22283i
97.2 −0.500000 + 0.866025i −0.258819 0.965926i −0.500000 0.866025i −0.849511 2.06841i 0.965926 + 0.258819i −1.91935 + 1.10814i 1.00000 −0.866025 + 0.500000i 2.21605 + 0.298508i
97.3 −0.500000 + 0.866025i −0.258819 0.965926i −0.500000 0.866025i 1.08182 + 1.95695i 0.965926 + 0.258819i −4.11737 + 2.37716i 1.00000 −0.866025 + 0.500000i −2.23568 0.0415914i
97.4 −0.500000 + 0.866025i −0.258819 0.965926i −0.500000 0.866025i 1.81421 1.30715i 0.965926 + 0.258819i 1.24242 0.717312i 1.00000 −0.866025 + 0.500000i 0.224915 + 2.22473i
97.5 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i −1.65293 1.50593i −0.965926 0.258819i −1.08688 + 0.627513i 1.00000 −0.866025 + 0.500000i 2.13064 0.678517i
97.6 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i −0.743195 + 2.10895i −0.965926 0.258819i −0.0594459 + 0.0343211i 1.00000 −0.866025 + 0.500000i −1.45481 1.69810i
97.7 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 1.13085 1.92904i −0.965926 0.258819i −0.245433 + 0.141701i 1.00000 −0.866025 + 0.500000i 1.10517 + 1.94386i
97.8 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 1.26528 + 1.84366i −0.965926 0.258819i 3.23783 1.86936i 1.00000 −0.866025 + 0.500000i −2.22929 + 0.173937i
163.1 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i −2.18531 0.473708i 0.258819 0.965926i 2.35599 1.36023i 1.00000 0.866025 0.500000i 1.50290 1.65568i
163.2 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 0.0142140 + 2.23602i 0.258819 0.965926i −3.50522 + 2.02374i 1.00000 0.866025 0.500000i −1.94356 1.10570i
163.3 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 0.933504 + 2.03189i 0.258819 0.965926i 3.73901 2.15872i 1.00000 0.866025 0.500000i −2.22642 0.207506i
163.4 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 1.23760 1.86235i 0.258819 0.965926i −3.19321 + 1.84360i 1.00000 0.866025 0.500000i 0.994046 + 2.00297i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bn.c yes 32
5.c odd 4 1 390.2.bd.c 32
13.f odd 12 1 390.2.bd.c 32
65.o even 12 1 inner 390.2.bn.c yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.c 32 5.c odd 4 1
390.2.bd.c 32 13.f odd 12 1
390.2.bn.c yes 32 1.a even 1 1 trivial
390.2.bn.c yes 32 65.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{32} + 12 T_{7}^{31} + 6 T_{7}^{30} - 504 T_{7}^{29} - 1189 T_{7}^{28} + 14124 T_{7}^{27} + 55822 T_{7}^{26} - 217728 T_{7}^{25} - 1213295 T_{7}^{24} + 2356404 T_{7}^{23} + 18547412 T_{7}^{22} - 12424368 T_{7}^{21} + \cdots + 65536$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.