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Properties

Label 483.2.y.b
Level $483$
Weight $2$
Character orbit 483.y
Analytic conductor $3.857$
Analytic rank $0$
Dimension $320$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(4,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 44, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.y (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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(16\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9} + 2 q^{11} + 18 q^{12} + 18 q^{14} - 18 q^{15} - 8 q^{16} + 4 q^{17} + 2 q^{18} + 8 q^{19} - 162 q^{20} - 4 q^{21} + 144 q^{22} - 26 q^{23} + 6 q^{24} - 8 q^{25} - 14 q^{26} - 32 q^{27} + 86 q^{28} - 74 q^{29} - 56 q^{31} - 28 q^{32} + 13 q^{33} + 40 q^{34} - 32 q^{35} - 14 q^{36} + 2 q^{37} - 39 q^{38} - 52 q^{40} + 60 q^{41} - 61 q^{42} + 16 q^{43} - 75 q^{44} - 2 q^{45} - 4 q^{46} - 40 q^{47} - 28 q^{48} - 100 q^{49} + 146 q^{50} - 18 q^{51} - 18 q^{52} + 34 q^{53} + 2 q^{54} + 36 q^{55} - 102 q^{56} + 28 q^{57} - 17 q^{58} - 102 q^{59} - 18 q^{60} - 18 q^{61} - 88 q^{62} + 2 q^{63} - 252 q^{64} - 78 q^{65} + 16 q^{66} + 12 q^{67} + 34 q^{68} + 8 q^{69} + 264 q^{70} + 160 q^{71} + 6 q^{72} - 8 q^{73} + 70 q^{74} + 14 q^{75} - 40 q^{76} - 90 q^{77} - 16 q^{78} + 26 q^{79} - 103 q^{80} + 16 q^{81} - 30 q^{82} - 80 q^{83} - 52 q^{84} - 128 q^{85} + 90 q^{86} + 4 q^{87} - 293 q^{88} - 36 q^{89} + 16 q^{91} - 174 q^{92} + 32 q^{93} + 57 q^{94} - 85 q^{95} - 50 q^{96} - 8 q^{97} - 193 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.48158 + 2.08058i 0.235759 − 0.971812i −1.47962 − 4.27509i 0.193923 − 4.07094i 1.67264 + 1.93033i −2.46944 − 0.949661i 6.18541 + 1.81620i −0.888835 − 0.458227i 8.18263 + 6.43489i
4.2 −1.37000 + 1.92390i 0.235759 − 0.971812i −1.17034 − 3.38148i 0.00886507 − 0.186101i 1.54668 + 1.78496i 0.229059 + 2.63582i 3.57665 + 1.05020i −0.888835 − 0.458227i 0.345893 + 0.272014i
4.3 −1.32550 + 1.86141i 0.235759 − 0.971812i −1.05374 − 3.04459i −0.0837421 + 1.75796i 1.49644 + 1.72698i 0.00745284 − 2.64574i 2.67883 + 0.786576i −0.888835 − 0.458227i −3.16128 − 2.48606i
4.4 −1.09595 + 1.53904i 0.235759 − 0.971812i −0.513414 − 1.48341i −0.133721 + 2.80715i 1.23728 + 1.42790i 2.23077 + 1.42255i −0.779983 − 0.229024i −0.888835 − 0.458227i −4.17377 − 3.28229i
4.5 −0.848286 + 1.19125i 0.235759 − 0.971812i −0.0453539 − 0.131042i 0.176217 − 3.69925i 0.957681 + 1.10522i 2.34512 + 1.22490i −2.61178 − 0.766889i −0.888835 − 0.458227i 4.25725 + 3.34794i
4.6 −0.765883 + 1.07553i 0.235759 − 0.971812i 0.0839437 + 0.242539i −0.0355806 + 0.746929i 0.864651 + 0.997860i −2.58720 + 0.553515i −2.85890 − 0.839448i −0.888835 − 0.458227i −0.776095 − 0.610328i
4.7 −0.342635 + 0.481163i 0.235759 − 0.971812i 0.540016 + 1.56028i 0.0918733 − 1.92866i 0.386821 + 0.446415i −2.12937 + 1.57028i −2.06930 − 0.607603i −0.888835 − 0.458227i 0.896520 + 0.705031i
4.8 −0.111942 + 0.157200i 0.235759 − 0.971812i 0.641955 + 1.85481i −0.171106 + 3.59197i 0.126378 + 0.145848i −2.30883 − 1.29202i −0.733771 − 0.215455i −0.888835 − 0.458227i −0.545504 − 0.428989i
4.9 −0.0919807 + 0.129169i 0.235759 − 0.971812i 0.645912 + 1.86624i 0.0626824 − 1.31586i 0.103842 + 0.119841i 0.568266 − 2.58400i −0.604769 − 0.177576i −0.888835 − 0.458227i 0.164203 + 0.129131i
4.10 0.199911 − 0.280735i 0.235759 − 0.971812i 0.615288 + 1.77776i −0.0359296 + 0.754254i −0.225691 − 0.260461i 2.50738 − 0.844407i 1.28344 + 0.376852i −0.888835 − 0.458227i 0.204563 + 0.160870i
4.11 0.733120 − 1.02952i 0.235759 − 0.971812i 0.131683 + 0.380473i 0.00357914 − 0.0751355i −0.827663 − 0.955174i 1.84836 + 1.89303i 2.91361 + 0.855512i −0.888835 − 0.458227i −0.0747298 − 0.0587682i
4.12 0.746582 − 1.04843i 0.235759 − 0.971812i 0.112320 + 0.324527i 0.116881 − 2.45363i −0.842861 − 0.972713i −2.38903 − 1.13690i 2.89400 + 0.849754i −0.888835 − 0.458227i −2.48519 − 1.95438i
4.13 0.900476 − 1.26454i 0.235759 − 0.971812i −0.134072 − 0.387376i −0.164040 + 3.44363i −1.01660 − 1.17322i −1.20985 + 2.35293i 2.36844 + 0.695436i −0.888835 − 0.458227i 4.20690 + 3.30834i
4.14 1.20076 − 1.68624i 0.235759 − 0.971812i −0.747425 − 2.15954i −0.0741223 + 1.55602i −1.35561 − 1.56446i 2.38094 − 1.15374i −0.566529 − 0.166348i −0.888835 − 0.458227i 2.53481 + 1.99340i
4.15 1.38638 − 1.94690i 0.235759 − 0.971812i −1.21423 − 3.50828i 0.129724 − 2.72324i −1.56517 − 1.80630i −0.222669 + 2.63636i −3.92712 − 1.15311i −0.888835 − 0.458227i −5.12202 − 4.02800i
4.16 1.50681 − 2.11601i 0.235759 − 0.971812i −1.55291 − 4.48684i 0.00580718 − 0.121908i −1.70112 − 1.96320i −0.123513 − 2.64287i −6.84923 − 2.01112i −0.888835 − 0.458227i −0.249208 − 0.195979i
16.1 −0.860562 − 2.48643i −0.888835 − 0.458227i −3.86967 + 3.04314i 0.317869 + 0.0303528i −0.374451 + 2.60436i 1.99020 + 1.74330i 6.46974 + 4.15785i 0.580057 + 0.814576i −0.198076 − 0.816481i
16.2 −0.740363 − 2.13914i −0.888835 − 0.458227i −2.45566 + 1.93116i 2.48488 + 0.237277i −0.322149 + 2.24059i −2.63923 + 0.185589i 2.14051 + 1.37562i 0.580057 + 0.814576i −1.33214 − 5.49116i
16.3 −0.695372 − 2.00914i −0.888835 − 0.458227i −1.98101 + 1.55789i 1.53680 + 0.146746i −0.302572 + 2.10444i 0.278867 − 2.63101i 0.930423 + 0.597946i 0.580057 + 0.814576i −0.773810 − 3.18969i
16.4 −0.620909 − 1.79400i −0.888835 − 0.458227i −1.26080 + 0.991503i −3.16803 − 0.302510i −0.270172 + 1.87909i 1.40046 − 2.24471i −0.632491 − 0.406478i 0.580057 + 0.814576i 1.42436 + 5.87128i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
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 483.2.y.b ✓ 320
7.c even 3 1 inner 483.2.y.b ✓ 320
23.c even 11 1 inner 483.2.y.b ✓ 320
161.m even 33 1 inner 483.2.y.b ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.y.b ✓ 320 1.a even 1 1 trivial
483.2.y.b ✓ 320 7.c even 3 1 inner
483.2.y.b ✓ 320 23.c even 11 1 inner
483.2.y.b ✓ 320 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{320} - 2 T_{2}^{319} - 23 T_{2}^{318} + 58 T_{2}^{317} + 204 T_{2}^{316} - 720 T_{2}^{315} + \cdots + 13\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\). Copy content Toggle raw display