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Properties

Label 213.3.c.b
Level $213$
Weight $3$
Character orbit 213.c
Analytic conductor $5.804$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,3,Mod(143,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.143");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 213.c (of order \(2\), degree \(1\), 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: \(5.80382963087\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 8 q^{3} - 80 q^{4} - 14 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 8 q^{3} - 80 q^{4} - 14 q^{7} - 32 q^{9} + 34 q^{10} - 12 q^{12} + 8 q^{13} + 224 q^{16} - 9 q^{18} - 50 q^{19} - 20 q^{21} - 126 q^{22} - 19 q^{24} - 144 q^{25} + 104 q^{27} + 90 q^{28} - 37 q^{30} - 130 q^{31} + 102 q^{33} + 52 q^{34} + 100 q^{36} + 108 q^{37} - 56 q^{39} - 38 q^{40} + 110 q^{42} + 94 q^{43} + 104 q^{45} - 220 q^{46} - 361 q^{48} - 148 q^{49} - 172 q^{51} + 92 q^{52} - 100 q^{54} - 66 q^{55} + 138 q^{57} + 434 q^{58} - 87 q^{60} + 102 q^{61} + 12 q^{63} - 52 q^{64} + 424 q^{66} + 444 q^{67} - 96 q^{69} - 918 q^{70} - 16 q^{72} + 374 q^{73} + 356 q^{75} - 62 q^{76} - 66 q^{78} - 172 q^{79} - 592 q^{81} - 286 q^{82} + 150 q^{84} - 328 q^{85} + 18 q^{87} + 290 q^{88} + 521 q^{90} + 80 q^{91} + 130 q^{93} + 604 q^{94} - 308 q^{96} - 244 q^{97} - 262 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 − 3.86031i −2.01307 − 2.22431i −10.9020 − 3.56323i −8.58654 + 7.77106i 1.36638 26.6439i −0.895136 + 8.95537i −13.7552
143.2 − 3.65789i 1.12823 + 2.77976i −9.38017 2.20435i 10.1681 − 4.12695i −2.36269 19.6801i −6.45418 + 6.27244i 8.06326
143.3 − 3.62898i −2.62396 + 1.45424i −9.16946 6.03161i 5.27741 + 9.52229i −2.62324 18.7598i 4.77036 − 7.63176i 21.8886
143.4 − 3.50344i 2.59335 − 1.50816i −8.27407 2.45309i −5.28376 − 9.08562i −11.5773 14.9740i 4.45088 − 7.82238i 8.59426
143.5 − 3.31533i −1.18474 + 2.75615i −6.99143 − 7.31493i 9.13757 + 3.92782i 8.85269 9.91760i −6.19277 − 6.53067i −24.2514
143.6 − 3.21053i 2.95672 + 0.507735i −6.30749 3.49944i 1.63010 − 9.49264i 9.92104 7.40826i 8.48441 + 3.00246i 11.2351
143.7 − 3.21043i 1.65873 − 2.49972i −6.30689 − 5.86927i −8.02520 − 5.32524i 1.77887 7.40612i −3.49723 − 8.29273i −18.8429
143.8 − 3.04549i −1.24952 − 2.72740i −5.27498 8.71021i −8.30625 + 3.80539i −4.37569 3.88293i −5.87741 + 6.81587i 26.5268
143.9 − 2.59333i 1.64946 + 2.50585i −2.72535 − 5.52891i 6.49849 − 4.27759i −8.73495 − 3.30559i −3.55855 + 8.26660i −14.3383
143.10 − 2.21665i 0.000990803 − 3.00000i −0.913515 0.295476i −6.64993 − 0.00219626i 5.22401 − 6.84164i −9.00000 − 0.00594482i 0.654965
143.11 − 2.11843i −1.70335 + 2.46953i −0.487763 4.76805i 5.23155 + 3.60843i −7.49123 − 7.44044i −3.19720 − 8.41296i 10.1008
143.12 − 1.88662i 2.97994 + 0.346332i 0.440674 − 5.57066i 0.653396 − 5.62201i 2.66245 − 8.37785i 8.76011 + 2.06410i −10.5097
143.13 − 1.54156i 2.45797 − 1.71999i 1.62359 8.37764i −2.65147 − 3.78912i −0.930426 − 8.66911i 3.08327 − 8.45538i 12.9146
143.14 − 1.44252i 1.79437 + 2.40421i 1.91914 6.75396i 3.46812 − 2.58841i 0.838829 − 8.53847i −2.56048 + 8.62809i 9.74272
143.15 − 1.38668i −0.998629 + 2.82891i 2.07713 − 1.75520i 3.92278 + 1.38477i 3.86404 − 8.42701i −7.00548 − 5.65007i −2.43389
143.16 − 1.20313i −2.85700 − 0.915168i 2.55248 2.83051i −1.10106 + 3.43734i −1.42569 − 7.88348i 7.32494 + 5.22927i 3.40547
143.17 − 1.02828i −0.0784440 − 2.99897i 2.94264 − 3.16816i −3.08378 + 0.0806622i −13.6384 − 7.13897i −8.98769 + 0.470503i −3.25775
143.18 − 0.793238i −2.10846 − 2.13411i 3.37077 − 9.98824i −1.69285 + 1.67251i 8.68773 − 5.84677i −0.108815 + 8.99934i −7.92305
143.19 − 0.431539i 2.75734 − 1.18197i 3.81377 − 2.77352i −0.510067 − 1.18990i −6.08407 − 3.37195i 6.20589 − 6.51820i −1.19688
143.20 − 0.0827905i −2.51062 − 1.64219i 3.99315 5.07153i −0.135958 + 0.207856i −1.90929 − 0.661757i 3.60642 + 8.24583i 0.419875
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.42
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.3.c.b ✓ 42
3.b odd 2 1 inner 213.3.c.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.3.c.b ✓ 42 1.a even 1 1 trivial
213.3.c.b ✓ 42 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 124 T_{2}^{40} + 7056 T_{2}^{38} + 244318 T_{2}^{36} + 5755794 T_{2}^{34} + 97740520 T_{2}^{32} + \cdots + 465831 \) acting on \(S_{3}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display