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Properties

Label 215.2.q.a
Level $215$
Weight $2$
Character orbit 215.q
Analytic conductor $1.717$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [215,2,Mod(31,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.q (of order \(21\), degree \(12\), 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: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{3} + 6 q^{4} - 6 q^{5} + 14 q^{6} - 10 q^{7} - 20 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{3} + 6 q^{4} - 6 q^{5} + 14 q^{6} - 10 q^{7} - 20 q^{8} + 27 q^{9} - 14 q^{10} + 8 q^{11} + q^{12} - 6 q^{13} + q^{14} + 4 q^{15} - 6 q^{16} - 17 q^{17} - 14 q^{18} - 47 q^{19} - 18 q^{20} - 2 q^{21} + 23 q^{22} - 8 q^{23} - 21 q^{24} + 6 q^{25} + 44 q^{26} - 30 q^{27} - 101 q^{28} - q^{29} - 43 q^{31} + 41 q^{32} - 39 q^{33} + 29 q^{34} + q^{35} + 49 q^{36} + 7 q^{37} - 14 q^{38} + 27 q^{39} - 3 q^{40} + 4 q^{41} - 28 q^{42} + 9 q^{43} - 40 q^{44} + 12 q^{45} + 90 q^{46} + 30 q^{47} + 90 q^{48} + 10 q^{49} - 7 q^{50} - 77 q^{51} + 41 q^{52} - 95 q^{53} + 52 q^{54} + 4 q^{55} + 20 q^{56} - 19 q^{57} - 24 q^{58} - 23 q^{59} - q^{60} - 12 q^{61} - 101 q^{62} + 14 q^{63} - 18 q^{64} - 12 q^{65} + 32 q^{66} - 50 q^{67} - 13 q^{68} - 124 q^{69} + 9 q^{70} - 33 q^{71} + 2 q^{72} + 28 q^{73} - 3 q^{74} + 8 q^{75} - 52 q^{76} + 7 q^{77} + 42 q^{78} + 86 q^{79} + 11 q^{80} + 53 q^{81} - 14 q^{82} + 43 q^{83} + 44 q^{84} + 8 q^{85} + 68 q^{86} - 46 q^{87} + 40 q^{88} + 21 q^{89} + 98 q^{90} + 39 q^{91} + 18 q^{92} + 26 q^{93} - 92 q^{94} + 47 q^{95} - 132 q^{96} - 11 q^{97} + 24 q^{98} + 11 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.36959 + 1.14114i −0.297545 + 0.202863i 3.06581 − 3.84440i 0.733052 + 0.680173i 0.473566 − 0.820241i −1.95861 − 3.39240i −1.70725 + 7.47994i −1.04864 + 2.67190i −2.51321 − 0.775221i
31.2 −1.68748 + 0.812649i 0.984266 − 0.671061i 0.940219 − 1.17900i 0.733052 + 0.680173i −1.11559 + 1.93227i 0.588732 + 1.01971i 0.205055 − 0.898406i −0.577567 + 1.47162i −1.78975 − 0.552066i
31.3 −0.778151 + 0.374738i −1.66884 + 1.13779i −0.781889 + 0.980457i 0.733052 + 0.680173i 0.872233 − 1.51075i 2.01693 + 3.49342i 0.625388 − 2.74000i 0.394418 − 1.00496i −0.825312 − 0.254575i
31.4 −0.277024 + 0.133408i 2.54388 − 1.73439i −1.18804 + 1.48975i 0.733052 + 0.680173i −0.473335 + 0.819840i 1.08722 + 1.88311i 0.267209 − 1.17072i 2.36720 − 6.03153i −0.293813 − 0.0906293i
31.5 1.03504 − 0.498447i −1.17766 + 0.802916i −0.424130 + 0.531842i 0.733052 + 0.680173i −0.818711 + 1.41805i 0.451025 + 0.781199i −0.685160 + 3.00188i −0.353810 + 0.901494i 1.09777 + 0.338616i
31.6 1.45436 − 0.700385i 1.47244 − 1.00389i 0.377656 − 0.473566i 0.733052 + 0.680173i 1.43835 − 2.49130i −1.40348 − 2.43090i −0.500825 + 2.19426i 0.0642531 − 0.163714i 1.54251 + 0.475800i
56.1 −0.527991 + 2.31328i 2.09717 − 1.94589i −3.27055 − 1.57501i 0.988831 + 0.149042i 3.39409 + 5.87874i 0.271503 − 0.470256i 2.41147 − 3.02388i 0.387445 − 5.17009i −0.866870 + 2.20875i
56.2 −0.340014 + 1.48970i −1.86540 + 1.73084i −0.301657 − 0.145270i 0.988831 + 0.149042i −1.94417 − 3.36740i −1.44415 + 2.50134i −1.58642 + 1.98931i 0.259727 − 3.46581i −0.558245 + 1.42238i
56.3 −0.143409 + 0.628315i 1.49585 − 1.38794i 1.42772 + 0.687556i 0.988831 + 0.149042i 0.657548 + 1.13891i −1.11017 + 1.92286i −1.44039 + 1.80620i 0.0869821 − 1.16070i −0.235453 + 0.599923i
56.4 −0.0558980 + 0.244905i −0.899968 + 0.835048i 1.74508 + 0.840388i 0.988831 + 0.149042i −0.154201 − 0.267084i 2.14541 − 3.71596i −0.616607 + 0.773201i −0.111554 + 1.48858i −0.0917749 + 0.233839i
56.5 0.293323 − 1.28513i −1.15443 + 1.07115i 0.236416 + 0.113852i 0.988831 + 0.149042i 1.03795 + 1.79778i −1.46003 + 2.52884i 1.85940 − 2.33162i −0.0388545 + 0.518477i 0.481585 − 1.22706i
56.6 0.388876 − 1.70378i 0.914644 − 0.848665i −0.949696 − 0.457349i 0.988831 + 0.149042i −1.09025 − 1.88838i 0.229720 − 0.397887i 1.03068 − 1.29243i −0.107850 + 1.43916i 0.638468 − 1.62679i
66.1 −1.30151 − 1.63205i −0.0206681 − 0.0526614i −0.524595 + 2.29840i −0.0747301 + 0.997204i −0.0590461 + 0.102271i 2.09463 + 3.62801i 0.672381 − 0.323802i 2.19681 − 2.03834i 1.72474 − 1.17591i
66.2 −0.854239 − 1.07118i 0.156399 + 0.398499i 0.0273355 − 0.119765i −0.0747301 + 0.997204i 0.293262 − 0.507945i −1.32613 − 2.29692i −2.62046 + 1.26195i 2.06482 − 1.91587i 1.13202 − 0.771801i
66.3 0.214099 + 0.268471i 0.629300 + 1.60343i 0.418803 − 1.83490i −0.0747301 + 0.997204i −0.295743 + 0.512242i 0.656686 + 1.13741i 1.20105 − 0.578393i 0.0241829 − 0.0224385i −0.283720 + 0.193437i
66.4 0.668842 + 0.838702i −0.541001 − 1.37845i 0.188971 − 0.827938i −0.0747301 + 0.997204i 0.794263 − 1.37570i −1.93528 − 3.35200i 2.75379 − 1.32616i 0.591716 − 0.549033i −0.886339 + 0.604296i
66.5 1.29649 + 1.62575i 0.564847 + 1.43921i −0.517131 + 2.26570i −0.0747301 + 0.997204i −1.60747 + 2.78423i −0.521894 − 0.903947i −0.606948 + 0.292291i 0.446890 − 0.414653i −1.71809 + 1.17138i
66.6 1.63011 + 2.04409i −0.586129 − 1.49343i −1.07602 + 4.71434i −0.0747301 + 0.997204i 2.09726 − 3.63256i 1.22356 + 2.11927i −6.67941 + 3.21663i 0.312366 − 0.289833i −2.16020 + 1.47280i
81.1 −0.483918 + 2.12018i −2.66824 − 0.823043i −2.45906 − 1.18422i −0.365341 − 0.930874i 3.03621 − 5.25887i −0.269002 − 0.465926i 0.988925 − 1.24007i 3.96339 + 2.70219i 2.15042 − 0.324123i
81.2 −0.204279 + 0.895005i 2.14263 + 0.660912i 1.04263 + 0.502106i −0.365341 − 0.930874i −1.02921 + 1.78265i −0.103761 − 0.179720i −1.80713 + 2.26607i 1.67532 + 1.14221i 0.907768 − 0.136824i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.q.a ✓ 72
43.g even 21 1 inner 215.2.q.a ✓ 72
43.g even 21 1 9245.2.a.z 36
43.h odd 42 1 9245.2.a.y 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.q.a ✓ 72 1.a even 1 1 trivial
215.2.q.a ✓ 72 43.g even 21 1 inner
9245.2.a.y 36 43.h odd 42 1
9245.2.a.z 36 43.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 9 T_{2}^{70} + 16 T_{2}^{69} + 78 T_{2}^{68} + 123 T_{2}^{67} + 770 T_{2}^{66} + 441 T_{2}^{65} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(215, [\chi])\). Copy content Toggle raw display