↑

Properties

Label 221.2.ba.a
Level $221$
Weight $2$
Character orbit 221.ba
Analytic conductor $1.765$
Analytic rank $0$
Dimension $152$
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] = [221,2,Mod(5,221)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(221, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([12, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("221.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.ba (of order \(16\), 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: \(1.76469388467\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(19\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9} - 8 q^{11} + 8 q^{15} + 16 q^{17} + 16 q^{18} - 8 q^{19} + 8 q^{20} - 16 q^{21} - 32 q^{22} + 24 q^{24} - 16 q^{27} - 88 q^{28} + 24 q^{29} - 40 q^{31} - 24 q^{32} - 48 q^{33} + 24 q^{34} - 32 q^{35} - 8 q^{37} - 80 q^{38} - 8 q^{39} - 16 q^{40} - 56 q^{41} + 32 q^{42} - 64 q^{43} + 24 q^{44} + 104 q^{45} + 24 q^{46} + 32 q^{48} + 16 q^{49} - 16 q^{52} - 40 q^{53} - 80 q^{54} - 48 q^{55} + 32 q^{57} - 40 q^{58} + 56 q^{59} + 48 q^{60} + 32 q^{61} + 96 q^{62} - 80 q^{63} - 48 q^{64} - 48 q^{65} - 224 q^{66} + 64 q^{67} - 16 q^{68} + 40 q^{70} + 56 q^{71} + 136 q^{72} + 32 q^{73} + 104 q^{74} - 112 q^{75} + 104 q^{76} - 72 q^{78} - 80 q^{79} + 64 q^{80} - 16 q^{81} - 8 q^{83} - 160 q^{84} - 112 q^{85} - 16 q^{86} + 80 q^{87} + 80 q^{89} + 8 q^{90} - 16 q^{91} - 16 q^{92} + 112 q^{93} - 16 q^{94} + 64 q^{95} + 16 q^{96} + 40 q^{97} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.51049 − 1.03988i 0.910248 + 1.36228i 3.80700 + 3.80700i −0.855423 − 1.28023i −0.868560 − 4.36655i 0.361166 − 0.540523i −3.51886 − 8.49528i 0.120788 − 0.291608i 0.816246 + 4.10355i
5.2 −2.29270 − 0.949668i −1.53353 − 2.29509i 2.94039 + 2.94039i −0.643946 − 0.963733i 1.33636 + 6.71831i −1.51680 + 2.27005i −2.04971 − 4.94844i −1.76768 + 4.26757i 0.561149 + 2.82109i
5.3 −1.86622 − 0.773015i −0.838707 − 1.25521i 1.47102 + 1.47102i 0.746921 + 1.11785i 0.594916 + 2.99084i 1.88767 − 2.82510i −0.0621058 − 0.149937i 0.275918 − 0.666124i −0.529810 − 2.66353i
5.4 −1.83809 − 0.761362i 1.39975 + 2.09487i 1.38469 + 1.38469i 1.11569 + 1.66974i −0.977906 − 4.91627i −2.08456 + 3.11977i 0.0317917 + 0.0767520i −1.28113 + 3.09293i −0.779453 − 3.91858i
5.5 −1.71341 − 0.709717i 0.119908 + 0.179454i 1.01786 + 1.01786i −0.267329 − 0.400087i −0.0780889 − 0.392579i 0.184011 − 0.275391i 0.397817 + 0.960416i 1.13022 − 2.72860i 0.174096 + 0.875241i
5.6 −0.933612 − 0.386715i 0.0429775 + 0.0643204i −0.692130 − 0.692130i 1.33309 + 1.99511i −0.0152507 − 0.0766704i −0.971255 + 1.45359i 1.15195 + 2.78106i 1.14576 − 2.76611i −0.473051 − 2.37819i
5.7 −0.804114 − 0.333075i 1.33191 + 1.99335i −0.878554 − 0.878554i −1.52559 − 2.28321i −0.407075 − 2.04651i 2.19920 − 3.29133i 1.07998 + 2.60731i −1.05140 + 2.53830i 0.466271 + 2.34410i
5.8 −0.695505 − 0.288088i −1.62039 − 2.42509i −1.01348 − 1.01348i −1.50875 − 2.25800i 0.428353 + 2.15348i 1.88062 − 2.81455i 0.989085 + 2.38786i −2.10734 + 5.08756i 0.398840 + 2.00510i
5.9 −0.273153 − 0.113144i 1.37505 + 2.05790i −1.35240 − 1.35240i 1.39301 + 2.08478i −0.142759 − 0.717699i 1.43172 − 2.14272i 0.442684 + 1.06873i −1.19616 + 2.88778i −0.144624 − 0.727074i
5.10 −0.251772 − 0.104287i −1.07447 − 1.60806i −1.36170 − 1.36170i −0.265841 − 0.397860i 0.102821 + 0.516918i −1.87287 + 2.80294i 0.409405 + 0.988390i −0.283318 + 0.683989i 0.0254397 + 0.127894i
5.11 0.278442 + 0.115335i 0.214819 + 0.321499i −1.34999 − 1.34999i −1.18264 − 1.76994i 0.0227347 + 0.114295i −0.276731 + 0.414157i −0.450862 − 1.08848i 1.09084 − 2.63351i −0.125161 − 0.629225i
5.12 0.409817 + 0.169752i −1.00181 − 1.49932i −1.27508 − 1.27508i 2.06727 + 3.09389i −0.156048 − 0.784506i 1.76379 − 2.63970i −0.645606 − 1.55863i −0.0962770 + 0.232433i 0.322009 + 1.61885i
5.13 1.28185 + 0.530961i 1.16795 + 1.74796i −0.0529848 − 0.0529848i 0.770336 + 1.15289i 0.569041 + 2.86076i 0.214771 − 0.321428i −1.10171 − 2.65976i −0.543206 + 1.31142i 0.375318 + 1.88685i
5.14 1.28671 + 0.532972i −0.475972 − 0.712342i −0.0426534 − 0.0426534i −1.73758 − 2.60047i −0.232779 − 1.17026i 0.669826 − 1.00247i −1.09809 − 2.65103i 0.867168 − 2.09353i −0.849779 − 4.27213i
5.15 1.59516 + 0.660736i 0.120163 + 0.179836i 0.693746 + 0.693746i 1.66060 + 2.48526i 0.0728543 + 0.366263i −2.79811 + 4.18767i −0.673221 − 1.62530i 1.13015 − 2.72842i 1.00682 + 5.06160i
5.16 1.74086 + 0.721088i −1.81248 − 2.71257i 1.09641 + 1.09641i −0.443834 − 0.664244i −1.19927 − 6.02916i 0.330817 − 0.495102i −0.324086 − 0.782412i −2.92489 + 7.06132i −0.293674 − 1.47640i
5.17 1.89272 + 0.783991i 1.61485 + 2.41679i 1.55354 + 1.55354i −2.04107 − 3.05468i 1.16172 + 5.84035i −0.303260 + 0.453861i 0.154476 + 0.372937i −2.08510 + 5.03388i −1.46834 − 7.38184i
5.18 2.09964 + 0.869700i −0.864818 − 1.29429i 2.23790 + 2.23790i 0.867285 + 1.29798i −0.690162 − 3.46968i 0.864770 − 1.29422i 1.01309 + 2.44582i 0.220771 − 0.532988i 0.692131 + 3.47958i
5.19 2.51774 + 1.04288i −0.310064 − 0.464044i 3.83722 + 3.83722i −1.13843 − 1.70379i −0.296719 − 1.49170i −1.65822 + 2.48170i 3.57359 + 8.62741i 1.02885 − 2.48387i −1.08943 − 5.47695i
31.1 −1.03706 + 2.50367i 0.364483 − 1.83238i −3.77868 − 3.77868i −0.218534 + 1.09865i 4.20969 + 2.81283i 0.684640 + 3.44192i 8.37194 − 3.46777i −0.453129 − 0.187692i −2.52402 − 1.68649i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
221.ba even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 221.2.ba.a yes 152
13.d odd 4 1 221.2.z.a ✓ 152
17.e odd 16 1 221.2.z.a ✓ 152
221.ba even 16 1 inner 221.2.ba.a yes 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.z.a ✓ 152 13.d odd 4 1
221.2.z.a ✓ 152 17.e odd 16 1
221.2.ba.a yes 152 1.a even 1 1 trivial
221.2.ba.a yes 152 221.ba even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(221, [\chi])\).