Properties

 Label 322.2.m.a Level $322$ Weight $2$ Character orbit 322.m Analytic conductor $2.571$ Analytic rank $0$ Dimension $160$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(9,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([22, 30]))

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

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

chi := DirichletCharacter("322.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.m (of order $$33$$, degree $$20$$, 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: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$8$$ over $$\Q(\zeta_{33})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10})$$ 160 * q - 8 * q^2 + 2 * q^3 + 8 * q^4 - 2 * q^5 + 4 * q^6 + 15 * q^7 + 16 * q^8 + 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9} + 2 q^{10} - 8 q^{11} - 9 q^{12} - 12 q^{13} + 7 q^{14} + 4 q^{15} + 8 q^{16} - 14 q^{17} - 28 q^{18} - 2 q^{19} - 18 q^{20} + 72 q^{21} - 16 q^{22} - 22 q^{23} - 2 q^{24} - 30 q^{25} - 6 q^{26} - 70 q^{27} + 14 q^{28} - 20 q^{30} + 4 q^{31} - 8 q^{32} + 40 q^{33} - 28 q^{34} - 79 q^{35} - 12 q^{36} - 14 q^{37} - 9 q^{38} + 16 q^{39} + 2 q^{40} + 12 q^{41} + 16 q^{42} + 106 q^{43} - 8 q^{44} + 40 q^{45} - 50 q^{47} + 18 q^{48} - 55 q^{49} + 28 q^{50} - 47 q^{51} - 16 q^{52} - 4 q^{53} - 24 q^{54} - 15 q^{56} + 8 q^{57} + 22 q^{58} - 42 q^{59} - 2 q^{60} - 80 q^{61} + 8 q^{62} + 5 q^{63} - 16 q^{64} + 71 q^{65} + 4 q^{66} + 12 q^{67} + 8 q^{68} - 72 q^{70} - 112 q^{71} + 16 q^{72} - 64 q^{73} + 14 q^{74} - 126 q^{75} + 4 q^{76} - 69 q^{77} + 54 q^{78} + 52 q^{79} - 2 q^{80} - 30 q^{81} - 49 q^{82} + 26 q^{83} - q^{84} - 22 q^{85} - 46 q^{86} + 4 q^{87} - 3 q^{88} - 72 q^{89} - 8 q^{90} + 44 q^{91} + 54 q^{93} + 6 q^{94} - 58 q^{95} - 2 q^{96} + 20 q^{97} - 19 q^{98} - 264 q^{99}+O(q^{100})$$ 160 * q - 8 * q^2 + 2 * q^3 + 8 * q^4 - 2 * q^5 + 4 * q^6 + 15 * q^7 + 16 * q^8 + 28 * q^9 + 2 * q^10 - 8 * q^11 - 9 * q^12 - 12 * q^13 + 7 * q^14 + 4 * q^15 + 8 * q^16 - 14 * q^17 - 28 * q^18 - 2 * q^19 - 18 * q^20 + 72 * q^21 - 16 * q^22 - 22 * q^23 - 2 * q^24 - 30 * q^25 - 6 * q^26 - 70 * q^27 + 14 * q^28 - 20 * q^30 + 4 * q^31 - 8 * q^32 + 40 * q^33 - 28 * q^34 - 79 * q^35 - 12 * q^36 - 14 * q^37 - 9 * q^38 + 16 * q^39 + 2 * q^40 + 12 * q^41 + 16 * q^42 + 106 * q^43 - 8 * q^44 + 40 * q^45 - 50 * q^47 + 18 * q^48 - 55 * q^49 + 28 * q^50 - 47 * q^51 - 16 * q^52 - 4 * q^53 - 24 * q^54 - 15 * q^56 + 8 * q^57 + 22 * q^58 - 42 * q^59 - 2 * q^60 - 80 * q^61 + 8 * q^62 + 5 * q^63 - 16 * q^64 + 71 * q^65 + 4 * q^66 + 12 * q^67 + 8 * q^68 - 72 * q^70 - 112 * q^71 + 16 * q^72 - 64 * q^73 + 14 * q^74 - 126 * q^75 + 4 * q^76 - 69 * q^77 + 54 * q^78 + 52 * q^79 - 2 * q^80 - 30 * q^81 - 49 * q^82 + 26 * q^83 - q^84 - 22 * q^85 - 46 * q^86 + 4 * q^87 - 3 * q^88 - 72 * q^89 - 8 * q^90 + 44 * q^91 + 54 * q^93 + 6 * q^94 - 58 * q^95 - 2 * q^96 + 20 * q^97 - 19 * q^98 - 264 * 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 −0.235759 + 0.971812i −0.899762 + 2.59969i −0.888835 0.458227i −0.684642 0.274089i −2.31428 1.48730i −2.64563 0.0257032i 0.654861 0.755750i −3.59067 2.82373i 0.427774 0.600724i
9.2 −0.235759 + 0.971812i −0.457151 + 1.32085i −0.888835 0.458227i −4.01090 1.60572i −1.17584 0.755667i 1.65184 2.06674i 0.654861 0.755750i 0.822498 + 0.646819i 2.50607 3.51928i
9.3 −0.235759 + 0.971812i −0.289551 + 0.836604i −0.888835 0.458227i 2.56790 + 1.02803i −0.744757 0.478626i 2.64344 0.110492i 0.654861 0.755750i 1.74209 + 1.37000i −1.60446 + 2.25314i
9.4 −0.235759 + 0.971812i −0.245677 + 0.709838i −0.888835 0.458227i 1.32131 + 0.528973i −0.631908 0.406103i −0.478358 2.60215i 0.654861 0.755750i 1.91465 + 1.50569i −0.825572 + 1.15935i
9.5 −0.235759 + 0.971812i 0.207322 0.599019i −0.888835 0.458227i −0.965092 0.386365i 0.533255 + 0.342702i −1.55248 + 2.14239i 0.654861 0.755750i 2.04232 + 1.60610i 0.603003 0.846799i
9.6 −0.235759 + 0.971812i 0.635615 1.83649i −0.888835 0.458227i 0.639202 + 0.255898i 1.63487 + 1.05067i −0.520133 2.59412i 0.654861 0.755750i −0.610530 0.480126i −0.399382 + 0.560854i
9.7 −0.235759 + 0.971812i 0.774374 2.23741i −0.888835 0.458227i 3.37590 + 1.35151i 1.99177 + 1.28003i −0.224436 + 2.63621i 0.654861 0.755750i −2.04818 1.61070i −2.10931 + 2.96211i
9.8 −0.235759 + 0.971812i 0.902469 2.60751i −0.888835 0.458227i −1.02777 0.411456i 2.32125 + 1.49177i 2.58489 + 0.564210i 0.654861 0.755750i −3.62651 2.85192i 0.642162 0.901791i
25.1 −0.0475819 + 0.998867i −3.01871 1.20851i −0.995472 0.0950560i −0.530176 2.18541i 1.35077 2.95778i −2.53173 0.768337i 0.142315 0.989821i 5.48089 + 5.22602i 2.20817 0.425589i
25.2 −0.0475819 + 0.998867i −1.62223 0.649442i −0.995472 0.0950560i 0.0716832 + 0.295482i 0.725895 1.58949i 2.51052 + 0.835042i 0.142315 0.989821i 0.0386478 + 0.0368506i −0.298558 + 0.0575424i
25.3 −0.0475819 + 0.998867i −0.846875 0.339038i −0.995472 0.0950560i 0.0344685 + 0.142081i 0.378950 0.829784i 0.462244 2.60506i 0.142315 0.989821i −1.56895 1.49599i −0.143560 + 0.0276690i
25.4 −0.0475819 + 0.998867i −0.473143 0.189418i −0.995472 0.0950560i 0.994090 + 4.09770i 0.211716 0.463594i −2.53885 + 0.744474i 0.142315 0.989821i −1.98322 1.89099i −4.14036 + 0.797988i
25.5 −0.0475819 + 0.998867i 0.421611 + 0.168788i −0.995472 0.0950560i −0.751910 3.09942i −0.188658 + 0.413103i −2.62384 + 0.339832i 0.142315 0.989821i −2.02194 1.92791i 3.13168 0.603582i
25.6 −0.0475819 + 0.998867i 2.32492 + 0.930757i −0.995472 0.0950560i 0.378372 + 1.55967i −1.04033 + 2.27800i −1.47479 2.19659i 0.142315 0.989821i 2.36774 + 2.25763i −1.57591 + 0.303731i
25.7 −0.0475819 + 0.998867i 2.33543 + 0.934966i −0.995472 0.0950560i −0.786581 3.24233i −1.04503 + 2.28830i 2.61565 0.397974i 0.142315 0.989821i 2.40888 + 2.29686i 3.27608 0.631414i
25.8 −0.0475819 + 0.998867i 2.44097 + 0.977219i −0.995472 0.0950560i 0.657157 + 2.70884i −1.09226 + 2.39171i 2.36993 + 1.17620i 0.142315 0.989821i 2.83220 + 2.70050i −2.73704 + 0.527521i
39.1 −0.981929 0.189251i −0.150806 3.16580i 0.928368 + 0.371662i −0.753896 + 1.05870i −0.451052 + 3.13713i −0.989718 + 2.45366i −0.841254 0.540641i −7.01315 + 0.669675i 0.940632 0.896891i
39.2 −0.981929 0.189251i −0.121146 2.54317i 0.928368 + 0.371662i 1.79650 2.52283i −0.362342 + 2.52014i −0.443953 2.60824i −0.841254 0.540641i −3.46663 + 0.331023i −2.24148 + 2.13725i
39.3 −0.981929 0.189251i −0.0679970 1.42743i 0.928368 + 0.371662i −1.39707 + 1.96191i −0.203375 + 1.41451i 2.62689 0.315333i −0.841254 0.540641i 0.953474 0.0910457i 1.74312 1.66206i
39.4 −0.981929 0.189251i −0.0525591 1.10335i 0.928368 + 0.371662i −1.42783 + 2.00510i −0.157201 + 1.09336i −2.61494 0.402583i −0.841254 0.540641i 1.77179 0.169186i 1.78149 1.69865i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 317.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.m.a 160
7.c even 3 1 inner 322.2.m.a 160
23.c even 11 1 inner 322.2.m.a 160
161.m even 33 1 inner 322.2.m.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.m.a 160 1.a even 1 1 trivial
322.2.m.a 160 7.c even 3 1 inner
322.2.m.a 160 23.c even 11 1 inner
322.2.m.a 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{160} - 2 T_{3}^{159} - 24 T_{3}^{158} + 82 T_{3}^{157} + 182 T_{3}^{156} - 1181 T_{3}^{155} + 1151 T_{3}^{154} + 5957 T_{3}^{153} - 33616 T_{3}^{152} + 49492 T_{3}^{151} + 274975 T_{3}^{150} - 1117439 T_{3}^{149} + \cdots + 35\!\cdots\!41$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.