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Properties

Label 633.2.bb.a
Level $633$
Weight $2$
Character orbit 633.bb
Analytic conductor $5.055$
Analytic rank $0$
Dimension $1632$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(8,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([35, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.bb (of order \(70\), degree \(24\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(1632\)
Relative dimension: \(68\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1632 q - 18 q^{3} + 18 q^{4} - 37 q^{6} - 26 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1632 q - 18 q^{3} + 18 q^{4} - 37 q^{6} - 26 q^{7} - 22 q^{9} - 50 q^{10} - 7 q^{12} - 62 q^{13} + 74 q^{16} - 8 q^{18} - 48 q^{19} - 18 q^{21} - 74 q^{22} - 30 q^{24} - 106 q^{25} + 12 q^{27} + 32 q^{28} - 62 q^{30} - 28 q^{31} - 28 q^{33} - 82 q^{34} - 97 q^{36} - 14 q^{37} - 109 q^{39} - 56 q^{40} - 25 q^{42} - 292 q^{43} - 40 q^{45} - 138 q^{46} - 13 q^{48} - 58 q^{49} - 80 q^{51} - 148 q^{52} - 275 q^{54} - 56 q^{55} + 22 q^{57} + 94 q^{58} + 68 q^{60} + 80 q^{61} + 7 q^{63} + 80 q^{64} + 17 q^{66} + 70 q^{67} - 47 q^{69} - 126 q^{72} - 22 q^{73} - 158 q^{75} - 16 q^{76} + 54 q^{78} - 26 q^{79} + 98 q^{81} + 218 q^{82} - 107 q^{84} - 296 q^{85} + 53 q^{87} - 70 q^{88} - 25 q^{90} - 240 q^{91} + 101 q^{93} - 14 q^{94} + 44 q^{96} - 4 q^{97} - 73 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.122198 + 2.72095i −1.57818 − 0.713686i −5.39668 − 0.485710i −1.21547 − 3.23862i 2.13475 − 4.20694i −4.06704 + 0.182651i 1.24984 − 9.22667i 1.98131 + 2.25265i 8.96064 − 2.91149i
8.2 −0.119895 + 2.66966i −1.52030 + 0.829875i −5.12077 − 0.460878i 1.01502 + 2.70450i −2.03321 − 4.15818i −0.623365 + 0.0279954i 1.12690 − 8.31914i 1.62261 − 2.52332i −7.34179 + 2.38549i
8.3 −0.119039 + 2.65062i 1.71553 + 0.238658i −5.01966 − 0.451778i −0.755945 − 2.01421i −0.836808 + 4.51881i 4.11853 − 0.184964i 1.08271 − 7.99288i 2.88608 + 0.818851i 5.42888 − 1.76395i
8.4 −0.118198 + 2.63187i 0.677577 − 1.59402i −4.92083 − 0.442883i −0.0271416 − 0.0723186i 4.11516 + 1.97170i −1.12485 + 0.0505171i 1.03996 − 7.67730i −2.08178 − 2.16014i 0.193541 − 0.0628854i
8.5 −0.114032 + 2.53913i 1.72882 + 0.105816i −4.44223 − 0.399808i 0.956068 + 2.54743i −0.465823 + 4.37762i −3.17520 + 0.142598i 0.839364 − 6.19643i 2.97761 + 0.365874i −6.57729 + 2.13709i
8.6 −0.112674 + 2.50889i −0.721869 − 1.57445i −4.28988 − 0.386096i −0.251300 − 0.669588i 4.03147 − 1.63369i 3.57651 − 0.160621i 0.777800 − 5.74195i −1.95781 + 2.27310i 1.70824 − 0.555040i
8.7 −0.102980 + 2.29302i 0.128439 + 1.72728i −3.25539 − 0.292990i 1.08025 + 2.87833i −3.97392 + 0.116639i 4.21043 − 0.189091i 0.390852 − 2.88539i −2.96701 + 0.443702i −6.71131 + 2.18064i
8.8 −0.100541 + 2.23873i −1.16699 + 1.27990i −3.00984 − 0.270891i −0.804571 − 2.14377i −2.74801 − 2.74125i −1.41334 + 0.0634732i 0.307435 − 2.26958i −0.276275 − 2.98725i 4.88021 − 1.58568i
8.9 −0.100425 + 2.23614i 0.940356 + 1.45456i −2.99828 − 0.269850i −0.760786 − 2.02711i −3.34702 + 1.95669i −0.681360 + 0.0305999i 0.303590 − 2.24119i −1.23146 + 2.73560i 4.60929 − 1.49765i
8.10 −0.0987041 + 2.19782i −1.72923 + 0.0987349i −2.82871 − 0.254589i −0.0287363 − 0.0765675i −0.0463187 − 3.81029i 2.95106 − 0.132532i 0.248110 − 1.83162i 2.98050 − 0.341471i 0.171118 − 0.0555995i
8.11 −0.0972916 + 2.16637i −0.0810395 − 1.73015i −2.69172 − 0.242260i 1.30759 + 3.48407i 3.75603 − 0.00723185i −0.166256 + 0.00746658i 0.204522 − 1.50984i −2.98687 + 0.280422i −7.67499 + 2.49376i
8.12 −0.0948077 + 2.11106i 1.52939 − 0.813003i −2.45563 − 0.221011i −1.20077 − 3.19943i 1.57130 + 3.30570i −2.50232 + 0.112379i 0.132059 − 0.974901i 1.67805 − 2.48679i 6.86803 − 2.23156i
8.13 −0.0914438 + 2.03615i −1.41631 − 0.997022i −2.14561 − 0.193109i 0.738958 + 1.96895i 2.15960 − 2.79266i −4.92918 + 0.221370i 0.0422115 − 0.311618i 1.01190 + 2.82419i −4.07665 + 1.32458i
8.14 −0.0887219 + 1.97555i −0.685837 + 1.59048i −1.90296 − 0.171270i −0.377331 − 1.00540i −3.08122 − 1.49601i −1.61903 + 0.0727107i −0.0237175 + 0.175089i −2.05926 − 2.18162i 2.01968 − 0.656234i
8.15 −0.0777799 + 1.73190i −0.687081 − 1.58994i −1.00149 − 0.0901359i −0.646118 − 1.72157i 2.80707 − 1.06629i 1.12347 − 0.0504551i −0.231424 + 1.70844i −2.05584 + 2.18484i 3.03186 − 0.985110i
8.16 −0.0776422 + 1.72884i 1.27833 − 1.16871i −0.990904 − 0.0891829i 0.677753 + 1.80587i 1.92125 + 2.30076i 1.93569 − 0.0869319i −0.233484 + 1.72365i 0.268245 − 2.98798i −3.17467 + 1.03151i
8.17 −0.0732737 + 1.63156i 1.59907 − 0.665566i −0.664686 − 0.0598229i 0.135900 + 0.362104i 0.968744 + 2.65775i 3.32592 − 0.149367i −0.292153 + 2.15676i 2.11404 − 2.12857i −0.600754 + 0.195197i
8.18 −0.0698666 + 1.55570i 1.67469 + 0.442058i −0.423373 − 0.0381042i −0.897992 − 2.39269i −0.804714 + 2.57443i −2.29139 + 0.102906i −0.329216 + 2.43037i 2.60917 + 1.48062i 3.78505 − 1.22984i
8.19 −0.0656524 + 1.46186i 0.831292 + 1.51952i −0.140789 − 0.0126712i 0.729734 + 1.94437i −2.27591 + 1.11548i −3.76997 + 0.169309i −0.365091 + 2.69521i −1.61791 + 2.52634i −2.89031 + 0.939119i
8.20 −0.0606966 + 1.35151i 0.433574 − 1.67691i 0.169042 + 0.0152141i −0.364916 − 0.972315i 2.24005 + 0.687764i −4.81530 + 0.216255i −0.394024 + 2.90880i −2.62403 − 1.45413i 1.33625 − 0.434173i
See next 80 embeddings (of 1632 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
211.n odd 70 1 inner
633.bb even 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.bb.a ✓ 1632
3.b odd 2 1 inner 633.2.bb.a ✓ 1632
211.n odd 70 1 inner 633.2.bb.a ✓ 1632
633.bb even 70 1 inner 633.2.bb.a ✓ 1632
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.bb.a ✓ 1632 1.a even 1 1 trivial
633.2.bb.a ✓ 1632 3.b odd 2 1 inner
633.2.bb.a ✓ 1632 211.n odd 70 1 inner
633.2.bb.a ✓ 1632 633.bb even 70 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(633, [\chi])\).