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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9} + 2 q^{10} - 8 q^{11} - 9 q^{12} - 12 q^{13} + 7 q^{14} + 4 q^{15} + 8 q^{16} - 14 q^{17} - 28 q^{18} - 2 q^{19} - 18 q^{20} + 72 q^{21} - 16 q^{22} - 22 q^{23} - 2 q^{24} - 30 q^{25} - 6 q^{26} - 70 q^{27} + 14 q^{28} - 20 q^{30} + 4 q^{31} - 8 q^{32} + 40 q^{33} - 28 q^{34} - 79 q^{35} - 12 q^{36} - 14 q^{37} - 9 q^{38} + 16 q^{39} + 2 q^{40} + 12 q^{41} + 16 q^{42} + 106 q^{43} - 8 q^{44} + 40 q^{45} - 50 q^{47} + 18 q^{48} - 55 q^{49} + 28 q^{50} - 47 q^{51} - 16 q^{52} - 4 q^{53} - 24 q^{54} - 15 q^{56} + 8 q^{57} + 22 q^{58} - 42 q^{59} - 2 q^{60} - 80 q^{61} + 8 q^{62} + 5 q^{63} - 16 q^{64} + 71 q^{65} + 4 q^{66} + 12 q^{67} + 8 q^{68} - 72 q^{70} - 112 q^{71} + 16 q^{72} - 64 q^{73} + 14 q^{74} - 126 q^{75} + 4 q^{76} - 69 q^{77} + 54 q^{78} + 52 q^{79} - 2 q^{80} - 30 q^{81} - 49 q^{82} + 26 q^{83} - q^{84} - 22 q^{85} - 46 q^{86} + 4 q^{87} - 3 q^{88} - 72 q^{89} - 8 q^{90} + 44 q^{91} + 54 q^{93} + 6 q^{94} - 58 q^{95} - 2 q^{96} + 20 q^{97} - 19 q^{98} - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display