Properties

 Label 154.2.m.c Level $154$ Weight $2$ Character orbit 154.m Analytic conductor $1.230$ Analytic rank $0$ Dimension $32$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(9,154)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([10, 18]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("154.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.m (of order $$15$$, degree $$8$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 7 q^{7} - 8 q^{8} + 4 q^{9}+O(q^{10})$$ 32 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 + 7 * q^7 - 8 * q^8 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 7 q^{7} - 8 q^{8} + 4 q^{9} - 12 q^{10} + 5 q^{11} - 8 q^{12} - 2 q^{13} - 8 q^{14} - 2 q^{15} + 4 q^{16} - q^{17} + 4 q^{18} + 3 q^{19} + 4 q^{20} - 14 q^{21} - 6 q^{23} + 2 q^{24} + 6 q^{25} + 6 q^{26} - 34 q^{27} + q^{28} - 34 q^{29} + q^{30} + 24 q^{31} - 16 q^{32} - q^{33} + 12 q^{34} - 29 q^{35} - 8 q^{36} + 3 q^{38} - 16 q^{39} + 8 q^{40} - 20 q^{41} - 10 q^{42} + 64 q^{43} + 5 q^{44} - 30 q^{45} - q^{46} - 26 q^{47} - 4 q^{48} - 47 q^{49} - 12 q^{50} - 40 q^{51} + 6 q^{52} - 27 q^{53} - 38 q^{54} + 22 q^{55} + 2 q^{56} + 36 q^{57} + 17 q^{58} + 11 q^{59} - 4 q^{60} + 12 q^{61} + 12 q^{62} + 46 q^{63} - 8 q^{64} + 32 q^{65} - 16 q^{66} + 50 q^{67} - q^{68} + 44 q^{69} + 5 q^{70} - 6 q^{71} + 4 q^{72} - 17 q^{73} + 51 q^{75} + 24 q^{76} - 19 q^{77} + 92 q^{78} + 5 q^{79} + 8 q^{80} + 28 q^{81} + 15 q^{82} + 28 q^{83} + 44 q^{84} - 86 q^{85} + 28 q^{86} + 58 q^{87} + 10 q^{88} - 10 q^{89} + 100 q^{90} - 63 q^{91} - 8 q^{92} + 18 q^{93} + 39 q^{94} - 33 q^{95} + 2 q^{96} - 54 q^{97} - 4 q^{98} - 12 q^{99}+O(q^{100})$$ 32 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 + 7 * q^7 - 8 * q^8 + 4 * q^9 - 12 * q^10 + 5 * q^11 - 8 * q^12 - 2 * q^13 - 8 * q^14 - 2 * q^15 + 4 * q^16 - q^17 + 4 * q^18 + 3 * q^19 + 4 * q^20 - 14 * q^21 - 6 * q^23 + 2 * q^24 + 6 * q^25 + 6 * q^26 - 34 * q^27 + q^28 - 34 * q^29 + q^30 + 24 * q^31 - 16 * q^32 - q^33 + 12 * q^34 - 29 * q^35 - 8 * q^36 + 3 * q^38 - 16 * q^39 + 8 * q^40 - 20 * q^41 - 10 * q^42 + 64 * q^43 + 5 * q^44 - 30 * q^45 - q^46 - 26 * q^47 - 4 * q^48 - 47 * q^49 - 12 * q^50 - 40 * q^51 + 6 * q^52 - 27 * q^53 - 38 * q^54 + 22 * q^55 + 2 * q^56 + 36 * q^57 + 17 * q^58 + 11 * q^59 - 4 * q^60 + 12 * q^61 + 12 * q^62 + 46 * q^63 - 8 * q^64 + 32 * q^65 - 16 * q^66 + 50 * q^67 - q^68 + 44 * q^69 + 5 * q^70 - 6 * q^71 + 4 * q^72 - 17 * q^73 + 51 * q^75 + 24 * q^76 - 19 * q^77 + 92 * q^78 + 5 * q^79 + 8 * q^80 + 28 * q^81 + 15 * q^82 + 28 * q^83 + 44 * q^84 - 86 * q^85 + 28 * q^86 + 58 * q^87 + 10 * q^88 - 10 * q^89 + 100 * q^90 - 63 * q^91 - 8 * q^92 + 18 * q^93 + 39 * q^94 - 33 * q^95 + 2 * q^96 - 54 * q^97 - 4 * q^98 - 12 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 0.669131 + 0.743145i −2.10800 0.938544i −0.104528 + 0.994522i −3.77835 0.803113i −0.713056 2.19456i −1.01275 + 2.44424i −0.809017 + 0.587785i 1.55542 + 1.72747i −1.93138 3.34525i
9.2 0.669131 + 0.743145i 0.128746 + 0.0573215i −0.104528 + 0.994522i 0.908627 + 0.193135i 0.0435498 + 0.134033i 2.64216 + 0.137793i −0.809017 + 0.587785i −1.99410 2.21467i 0.464463 + 0.804474i
9.3 0.669131 + 0.743145i 0.995253 + 0.443115i −0.104528 + 0.994522i 0.776533 + 0.165057i 0.336655 + 1.03612i −1.73148 + 2.00049i −0.809017 + 0.587785i −1.21321 1.34741i 0.396941 + 0.687521i
9.4 0.669131 + 0.743145i 2.81110 + 1.25158i −0.104528 + 0.994522i −3.02846 0.643719i 0.950885 + 2.92652i 0.432943 2.61009i −0.809017 + 0.587785i 4.32841 + 4.80719i −1.54806 2.68131i
25.1 0.913545 + 0.406737i −2.83199 0.601958i 0.669131 + 0.743145i 0.316143 3.00790i −2.34231 1.70179i 0.943723 2.47172i 0.309017 + 0.951057i 4.91717 + 2.18927i 1.51223 2.61927i
25.2 0.913545 + 0.406737i −1.84659 0.392505i 0.669131 + 0.743145i −0.328439 + 3.12489i −1.52730 1.10965i 1.60960 + 2.09981i 0.309017 + 0.951057i 0.515195 + 0.229380i −1.57105 + 2.72114i
25.3 0.913545 + 0.406737i 1.06712 + 0.226824i 0.669131 + 0.743145i −0.226107 + 2.15126i 0.882606 + 0.641251i −0.0253913 2.64563i 0.309017 + 0.951057i −1.65334 0.736114i −1.08156 + 1.87331i
25.4 0.913545 + 0.406737i 1.65516 + 0.351816i 0.669131 + 0.743145i 0.158550 1.50850i 1.36897 + 0.994615i −2.44148 + 1.01941i 0.309017 + 0.951057i −0.124848 0.0555860i 0.758407 1.31360i
37.1 0.913545 0.406737i −2.83199 + 0.601958i 0.669131 0.743145i 0.316143 + 3.00790i −2.34231 + 1.70179i 0.943723 + 2.47172i 0.309017 0.951057i 4.91717 2.18927i 1.51223 + 2.61927i
37.2 0.913545 0.406737i −1.84659 + 0.392505i 0.669131 0.743145i −0.328439 3.12489i −1.52730 + 1.10965i 1.60960 2.09981i 0.309017 0.951057i 0.515195 0.229380i −1.57105 2.72114i
37.3 0.913545 0.406737i 1.06712 0.226824i 0.669131 0.743145i −0.226107 2.15126i 0.882606 0.641251i −0.0253913 + 2.64563i 0.309017 0.951057i −1.65334 + 0.736114i −1.08156 1.87331i
37.4 0.913545 0.406737i 1.65516 0.351816i 0.669131 0.743145i 0.158550 + 1.50850i 1.36897 0.994615i −2.44148 1.01941i 0.309017 0.951057i −0.124848 + 0.0555860i 0.758407 + 1.31360i
53.1 −0.978148 + 0.207912i −0.321647 3.06027i 0.913545 0.406737i 2.07171 2.30086i 0.950885 + 2.92652i −2.34855 + 1.21831i −0.809017 + 0.587785i −6.32736 + 1.34492i −1.54806 + 2.68131i
53.2 −0.978148 + 0.207912i −0.113877 1.08347i 0.913545 0.406737i −0.531210 + 0.589969i 0.336655 + 1.03612i 1.36752 2.26492i −0.809017 + 0.587785i 1.77350 0.376969i 0.396941 0.687521i
53.3 −0.978148 + 0.207912i −0.0147312 0.140158i 0.913545 0.406737i −0.621573 + 0.690327i 0.0435498 + 0.134033i 0.947521 + 2.47026i −0.809017 + 0.587785i 2.91502 0.619606i 0.464463 0.804474i
53.4 −0.978148 + 0.207912i 0.241199 + 2.29486i 0.913545 0.406737i 2.58469 2.87059i −0.713056 2.19456i 2.01166 1.71850i −0.809017 + 0.587785i −2.27375 + 0.483300i −1.93138 + 3.34525i
81.1 −0.104528 + 0.994522i −1.13226 1.25750i −0.978148 0.207912i −1.38568 0.616944i 1.36897 0.994615i 1.37600 2.25978i 0.309017 0.951057i 0.0142852 0.135915i 0.758407 1.31360i
81.2 −0.104528 + 0.994522i −0.729995 0.810742i −0.978148 0.207912i 1.97610 + 0.879817i 0.882606 0.641251i 1.57560 + 2.12543i 0.309017 0.951057i 0.189176 1.79989i −1.08156 + 1.87331i
81.3 −0.104528 + 0.994522i 1.26321 + 1.40294i −0.978148 0.207912i 2.87045 + 1.27801i −1.52730 + 1.10965i −2.53643 0.752682i 0.309017 0.951057i −0.0589490 + 0.560862i −1.57105 + 2.72114i
81.4 −0.104528 + 0.994522i 1.93731 + 2.15160i −0.978148 0.207912i −2.76299 1.23016i −2.34231 + 1.70179i 0.689351 + 2.55437i 0.309017 0.951057i −0.562626 + 5.35303i 1.51223 2.61927i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.m.c 32
7.c even 3 1 inner 154.2.m.c 32
11.c even 5 1 inner 154.2.m.c 32
77.m even 15 1 inner 154.2.m.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.m.c 32 1.a even 1 1 trivial
154.2.m.c 32 7.c even 3 1 inner
154.2.m.c 32 11.c even 5 1 inner
154.2.m.c 32 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{32} - 2 T_{3}^{31} - 6 T_{3}^{30} + 34 T_{3}^{29} - 56 T_{3}^{28} + 68 T_{3}^{27} + 713 T_{3}^{26} + \cdots + 14641$$ acting on $$S_{2}^{\mathrm{new}}(154, [\chi])$$.