Properties

Label 667.2.h.b
Level $667$
Weight $2$
Character orbit 667.h
Analytic conductor $5.326$
Analytic rank $0$
Dimension $280$
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] = [667,2,Mod(59,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([14, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.h (of order \(11\), degree \(10\), 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: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(28\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 2 q^{3} - 32 q^{4} - 4 q^{6} + 6 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 2 q^{3} - 32 q^{4} - 4 q^{6} + 6 q^{7} - 26 q^{9} - 41 q^{10} - 20 q^{11} + 18 q^{12} - 3 q^{15} - 40 q^{16} + 9 q^{17} + 54 q^{18} + 10 q^{19} + 58 q^{20} - 4 q^{21} - 118 q^{22} - 4 q^{23} + 20 q^{24} - 44 q^{25} - 22 q^{26} + 4 q^{27} + 76 q^{28} + 28 q^{29} + 100 q^{30} - 5 q^{31} + 40 q^{32} - 81 q^{33} - 58 q^{34} + 28 q^{35} - 24 q^{36} + 40 q^{37} - 14 q^{38} + 4 q^{39} + 78 q^{40} + 12 q^{41} + 74 q^{42} + 40 q^{43} - 90 q^{44} - 246 q^{45} - 72 q^{46} + 50 q^{47} - 14 q^{48} - 34 q^{49} + 46 q^{50} + 48 q^{51} + 90 q^{52} + 4 q^{53} + 46 q^{54} - 18 q^{55} - 272 q^{56} - 142 q^{57} + 66 q^{59} + 42 q^{60} + 40 q^{61} - 26 q^{62} + 32 q^{63} + 44 q^{64} + 19 q^{65} + 78 q^{66} - 78 q^{67} - 130 q^{68} - 283 q^{69} + 120 q^{70} + 59 q^{71} + 82 q^{72} + 28 q^{73} + 62 q^{74} + 87 q^{75} + 203 q^{76} - 58 q^{77} - 36 q^{78} + 32 q^{79} - 182 q^{80} + 66 q^{81} + 60 q^{82} - 24 q^{83} + 90 q^{84} + 90 q^{85} + 49 q^{86} + 2 q^{87} + 26 q^{88} + 36 q^{89} - 113 q^{90} - 668 q^{91} - 175 q^{92} - 60 q^{93} + 49 q^{94} + 11 q^{95} + 313 q^{96} + 75 q^{97} - 48 q^{98} + 101 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.81528 2.09495i −1.45717 0.936468i −0.808926 + 5.62621i −0.296723 + 0.649732i 0.683329 + 4.75266i −0.244858 0.0718968i 8.59111 5.52117i 0.000135129 0 0.000295891i 1.89979 0.557829i
59.2 −1.64962 1.90376i 1.70338 + 1.09469i −0.618435 + 4.30131i 0.0924900 0.202525i −0.725886 5.04865i 4.03209 + 1.18393i 4.97055 3.19438i 0.456895 + 1.00046i −0.538132 + 0.158010i
59.3 −1.63657 1.88870i −2.49072 1.60069i −0.604204 + 4.20233i 0.541273 1.18522i 1.05301 + 7.32385i −2.12147 0.622919i 4.72101 3.03401i 2.39523 + 5.24482i −3.12436 + 0.917396i
59.4 −1.54727 1.78564i 2.09068 + 1.34360i −0.509849 + 3.54608i −1.74644 + 3.82416i −0.835654 5.81210i 0.842446 + 0.247365i 3.14556 2.02153i 1.31944 + 2.88917i 9.53078 2.79849i
59.5 −1.40842 1.62540i 0.188781 + 0.121322i −0.373655 + 2.59883i 0.368155 0.806146i −0.0686853 0.477717i 3.55482 + 1.04379i 1.13180 0.727366i −1.22533 2.68309i −1.82882 + 0.536991i
59.6 −1.23041 1.41997i 2.25145 + 1.44692i −0.217775 + 1.51466i −0.135129 + 0.295891i −0.715628 4.97730i −4.56831 1.34138i −0.742523 + 0.477191i 1.72921 + 3.78643i 0.586421 0.172189i
59.7 −1.11194 1.28325i 0.565284 + 0.363286i −0.125684 + 0.874153i 0.269428 0.589966i −0.162376 1.12935i −3.16259 0.928620i −1.59536 + 1.02527i −1.05868 2.31818i −1.05666 + 0.310264i
59.8 −0.829534 0.957333i −2.17115 1.39532i 0.0562693 0.391361i −0.0992720 + 0.217375i 0.465264 + 3.23598i 3.11975 + 0.916040i −2.55263 + 1.64047i 1.52076 + 3.33000i 0.290450 0.0852839i
59.9 −0.694016 0.800937i −2.29784 1.47674i 0.124788 0.867918i 1.68126 3.68144i 0.411969 + 2.86531i 1.17531 + 0.345102i −2.56486 + 1.64833i 1.85310 + 4.05772i −4.11542 + 1.20840i
59.10 −0.668480 0.771467i −2.53242 1.62749i 0.136334 0.948222i −0.338156 + 0.740459i 0.437320 + 3.04163i −0.901989 0.264848i −2.54016 + 1.63246i 2.51820 + 5.51409i 0.797291 0.234106i
59.11 −0.564521 0.651491i −0.188329 0.121032i 0.178872 1.24408i −1.11327 + 2.43772i 0.0274645 + 0.191020i 3.62972 + 1.06578i −2.36189 + 1.51789i −1.22543 2.68331i 2.21661 0.650857i
59.12 −0.515261 0.594643i 1.29259 + 0.830697i 0.196523 1.36685i 1.20112 2.63009i −0.172053 1.19666i 0.666888 + 0.195816i −2.23789 + 1.43820i −0.265515 0.581396i −2.18285 + 0.640944i
59.13 −0.455129 0.525247i 2.73097 + 1.75509i 0.215888 1.50153i −1.11348 + 2.43818i −0.321089 2.23322i 1.67462 + 0.491714i −2.05628 + 1.32149i 3.13162 + 6.85729i 1.78742 0.524834i
59.14 −0.152399 0.175878i −0.587160 0.377345i 0.276922 1.92604i 0.994539 2.17774i 0.0231161 + 0.160776i −2.61817 0.768764i −0.772505 + 0.496459i −1.04388 2.28577i −0.534584 + 0.156968i
59.15 0.0654127 + 0.0754903i 2.34114 + 1.50456i 0.283210 1.96977i 1.11825 2.44863i 0.0395606 + 0.275150i 3.56174 + 1.04582i 0.335286 0.215475i 1.97098 + 4.31584i 0.257996 0.0757543i
59.16 0.240264 + 0.277280i −0.500077 0.321380i 0.265473 1.84640i −1.01259 + 2.21725i −0.0310385 0.215877i 0.240790 + 0.0707024i 1.19306 0.766730i −1.09945 2.40747i −0.858088 + 0.251957i
59.17 0.339307 + 0.391581i −1.94329 1.24888i 0.246423 1.71391i −0.950274 + 2.08081i −0.170335 1.18471i −1.08596 0.318866i 1.62652 1.04530i 0.970441 + 2.12497i −1.13724 + 0.333924i
59.18 0.710735 + 0.820232i 2.06469 + 1.32690i 0.116993 0.813708i 0.256498 0.561653i 0.379086 + 2.63660i −1.04933 0.308110i 2.57664 1.65591i 1.25605 + 2.75037i 0.642989 0.188798i
59.19 0.766394 + 0.884466i 0.446241 + 0.286781i 0.0897096 0.623944i 0.351858 0.770461i 0.0883477 + 0.614472i −4.40986 1.29485i 2.58968 1.66428i −1.12936 2.47295i 0.951108 0.279271i
59.20 0.877124 + 1.01225i 0.298375 + 0.191754i 0.0293161 0.203898i −0.436441 + 0.955671i 0.0676079 + 0.470224i 2.79711 + 0.821304i 2.48567 1.59744i −1.19399 2.61447i −1.35020 + 0.396453i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.h.b 280
23.c even 11 1 inner 667.2.h.b 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.h.b 280 1.a even 1 1 trivial
667.2.h.b 280 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{280} + 44 T_{2}^{278} + 1082 T_{2}^{276} - 8 T_{2}^{275} + 19446 T_{2}^{274} - 540 T_{2}^{273} + 285429 T_{2}^{272} - 17853 T_{2}^{271} + 3635658 T_{2}^{270} - 401210 T_{2}^{269} + 41747513 T_{2}^{268} + \cdots + 39\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(667, [\chi])\). Copy content Toggle raw display