↑

Properties

Label 363.2.m.a
Level $363$
Weight $2$
Character orbit 363.m
Analytic conductor $2.899$
Analytic rank $0$
Dimension $440$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(4,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.m (of order \(55\), degree \(40\), 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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(11\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q + q^{2} + 110 q^{3} + 19 q^{4} + 3 q^{5} - q^{6} - q^{7} - 13 q^{8} - 110 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q + q^{2} + 110 q^{3} + 19 q^{4} + 3 q^{5} - q^{6} - q^{7} - 13 q^{8} - 110 q^{9} - 13 q^{10} + 46 q^{12} - 29 q^{13} - 20 q^{14} + 2 q^{15} + 5 q^{16} - 12 q^{17} - 4 q^{18} + 10 q^{19} - q^{20} - 4 q^{21} - 26 q^{22} - 14 q^{23} - 7 q^{24} + 8 q^{25} - 222 q^{26} + 110 q^{27} + 6 q^{28} - 6 q^{29} + 8 q^{30} + 3 q^{31} + 96 q^{32} + 20 q^{33} + 18 q^{34} + 19 q^{35} + 19 q^{36} - 40 q^{37} + 60 q^{38} - 26 q^{39} - 126 q^{40} + 25 q^{41} + 25 q^{42} + 22 q^{43} + 30 q^{44} - 2 q^{45} + 25 q^{46} - 4 q^{47} - 20 q^{48} - 14 q^{49} - 138 q^{50} - 3 q^{51} - 256 q^{52} - 290 q^{53} - 6 q^{54} - 46 q^{55} - 89 q^{56} + 5 q^{57} + 29 q^{58} + q^{60} + 21 q^{61} - 28 q^{62} - q^{63} - 27 q^{64} - 19 q^{65} - 84 q^{66} - 35 q^{67} + 27 q^{68} + 19 q^{69} + 87 q^{70} + 49 q^{71} + 7 q^{72} + 63 q^{73} + 103 q^{74} - 13 q^{75} - 93 q^{76} - q^{77} - 53 q^{78} - 407 q^{79} - 497 q^{80} - 110 q^{81} - 19 q^{82} - 13 q^{83} + 9 q^{84} + 31 q^{85} - 21 q^{86} - 24 q^{87} + 59 q^{88} - 44 q^{89} - 13 q^{90} - 36 q^{91} - 291 q^{92} - 58 q^{93} - 281 q^{94} - 5 q^{95} - 51 q^{96} - 18 q^{97} - 124 q^{98} - 15 q^{99}+O(q^{100}) \) 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 −2.38819 − 0.136561i −0.309017 + 0.951057i 3.69783 + 0.424286i −0.774218 − 1.46731i 0.867868 − 2.22910i 1.63760 + 0.692056i −4.05905 − 0.702446i −0.809017 − 0.587785i 1.64860 + 3.60993i
4.2 −1.78018 − 0.101795i −0.309017 + 0.951057i 1.17173 + 0.134444i 1.88355 + 3.56973i 0.646920 − 1.66160i 1.19508 + 0.505046i 1.44174 + 0.249503i −0.809017 − 0.587785i −2.98969 − 6.54651i
4.3 −1.29252 − 0.0739089i −0.309017 + 0.951057i −0.321820 − 0.0369254i −0.288837 − 0.547407i 0.469702 − 1.20642i −0.0650258 − 0.0274801i 2.96457 + 0.513039i −0.809017 − 0.587785i 0.332869 + 0.728882i
4.4 −0.652836 − 0.0373305i −0.309017 + 0.951057i −1.56216 − 0.179241i −0.949416 − 1.79934i 0.237241 − 0.609348i 1.02934 + 0.435003i 2.30179 + 0.398341i −0.809017 − 0.587785i 0.552643 + 1.21012i
4.5 −0.578766 − 0.0330950i −0.309017 + 0.951057i −1.65309 − 0.189674i 0.377821 + 0.716050i 0.210324 − 0.540212i −3.15749 − 1.33437i 2.09292 + 0.362193i −0.809017 − 0.587785i −0.194972 − 0.426929i
4.6 0.400915 + 0.0229252i −0.309017 + 0.951057i −1.82676 − 0.209601i 1.65322 + 3.13319i −0.145693 + 0.374209i 1.12941 + 0.477294i −1.51895 − 0.262864i −0.809017 − 0.587785i 0.590971 + 1.29404i
4.7 0.638211 + 0.0364942i −0.309017 + 0.951057i −1.58098 − 0.181401i −1.48633 − 2.81690i −0.231926 + 0.595697i 4.40207 + 1.86033i −2.26216 − 0.391483i −0.809017 − 0.587785i −0.845789 − 1.85202i
4.8 1.17580 + 0.0672344i −0.309017 + 0.951057i −0.608989 − 0.0698750i −1.13078 − 2.14307i −0.427284 + 1.09747i −3.25575 − 1.37589i −3.03228 − 0.524757i −0.809017 − 0.587785i −1.18548 − 2.59584i
4.9 2.23119 + 0.127584i −0.309017 + 0.951057i 2.97497 + 0.341346i 0.730566 + 1.38458i −0.810816 + 2.08256i 1.53419 + 0.648355i 2.18998 + 0.378990i −0.809017 − 0.587785i 1.45338 + 3.18246i
4.10 2.40271 + 0.137392i −0.309017 + 0.951057i 3.76719 + 0.432245i 1.24529 + 2.36009i −0.873147 + 2.24266i −3.98559 − 1.68432i 4.24931 + 0.735372i −0.809017 − 0.587785i 2.66782 + 5.84171i
4.11 2.68128 + 0.153321i −0.309017 + 0.951057i 5.17879 + 0.594210i −1.99669 − 3.78414i −0.974378 + 2.50267i 1.49490 + 0.631748i 8.50201 + 1.47133i −0.809017 − 0.587785i −4.77349 − 10.4525i
16.1 −2.51695 − 0.288793i 0.809017 + 0.587785i 4.30361 + 1.00076i 1.07741 − 1.57566i −1.86651 − 1.71306i 3.24756 + 3.34166i −5.77070 − 2.05898i 0.309017 + 0.951057i −3.16683 + 3.65471i
16.2 −2.35206 − 0.269873i 0.809017 + 0.587785i 3.51131 + 0.816520i −0.641522 + 0.938196i −1.74423 − 1.60084i −1.51712 − 1.56108i −3.57883 − 1.27692i 0.309017 + 0.951057i 1.76209 − 2.03356i
16.3 −1.60028 − 0.183615i 0.809017 + 0.587785i 0.579157 + 0.134677i 0.398130 − 0.582247i −1.18673 − 1.08917i 1.97766 + 2.03496i 2.13212 + 0.760740i 0.309017 + 0.951057i −0.744029 + 0.858655i
16.4 −1.34871 − 0.154750i 0.809017 + 0.587785i −0.152948 − 0.0355665i −0.899102 + 1.31489i −1.00017 − 0.917948i −1.86849 − 1.92263i 2.75800 + 0.984053i 0.309017 + 0.951057i 1.41611 − 1.63428i
16.5 −0.327735 − 0.0376041i 0.809017 + 0.587785i −1.84203 − 0.428345i 0.529023 − 0.773671i −0.243040 − 0.223060i 0.770675 + 0.793006i 1.20899 + 0.431367i 0.309017 + 0.951057i −0.202472 + 0.233665i
16.6 0.344542 + 0.0395325i 0.809017 + 0.587785i −1.83088 − 0.425752i 1.29351 − 1.89170i 0.255503 + 0.234499i −2.07865 − 2.13888i −1.26725 − 0.452154i 0.309017 + 0.951057i 0.520452 − 0.600634i
16.7 0.764200 + 0.0876838i 0.809017 + 0.587785i −1.37171 − 0.318978i −2.11073 + 3.08685i 0.566712 + 0.520123i −3.36427 − 3.46175i −2.46925 − 0.881027i 0.309017 + 0.951057i −1.88369 + 2.17389i
16.8 1.17646 + 0.134986i 0.809017 + 0.587785i −0.582192 − 0.135383i −1.45645 + 2.13000i 0.872432 + 0.800710i 2.04455 + 2.10379i −2.89727 − 1.03374i 0.309017 + 0.951057i −2.00098 + 2.30925i
16.9 1.70276 + 0.195374i 0.809017 + 0.587785i 0.913213 + 0.212359i 1.07786 − 1.57633i 1.26273 + 1.15892i 2.91685 + 3.00137i −1.71503 − 0.611920i 0.309017 + 0.951057i 2.14332 − 2.47353i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.m.a ✓ 440
121.g even 55 1 inner 363.2.m.a ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.m.a ✓ 440 1.a even 1 1 trivial
363.2.m.a ✓ 440 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{440} - T_{2}^{439} - 20 T_{2}^{438} + 30 T_{2}^{437} + 161 T_{2}^{436} - 486 T_{2}^{435} + \cdots + 15\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display