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Properties

Label 149.2.d.a
Level $149$
Weight $2$
Character orbit 149.d
Analytic conductor $1.190$
Analytic rank $0$
Dimension $396$
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] = [149,2,Mod(5,149)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(149, base_ring=CyclotomicField(74))
 
chi = DirichletCharacter(H, H._module([52]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("149.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 149.d (of order \(37\), degree \(36\), 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: \(1.18977099012\)
Analytic rank: \(0\)
Dimension: \(396\)
Relative dimension: \(11\) over \(\Q(\zeta_{37})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{37}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 396 q - 32 q^{2} - 35 q^{3} - 42 q^{4} - 27 q^{5} - 25 q^{6} - 27 q^{7} - 16 q^{8} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 396 q - 32 q^{2} - 35 q^{3} - 42 q^{4} - 27 q^{5} - 25 q^{6} - 27 q^{7} - 16 q^{8} - 34 q^{9} - 7 q^{10} - 25 q^{11} - 7 q^{12} - 27 q^{13} - 9 q^{14} - 21 q^{15} - 14 q^{16} - 19 q^{17} - 177 q^{18} - 29 q^{19} + 7 q^{20} - q^{21} - 5 q^{22} - 17 q^{23} + 27 q^{24} - 14 q^{25} + 19 q^{26} - 11 q^{27} - 195 q^{28} + 11 q^{29} + 61 q^{30} - 9 q^{31} + 60 q^{32} + 17 q^{33} + 5 q^{34} - 205 q^{35} + 78 q^{36} + 5 q^{37} - 76 q^{38} + 21 q^{39} + 53 q^{40} - 7 q^{41} + 73 q^{42} + 7 q^{43} + 14 q^{44} - 126 q^{45} + 39 q^{46} + 19 q^{47} + 77 q^{48} + 14 q^{49} + 68 q^{50} + 69 q^{51} - 168 q^{52} + 11 q^{53} + 87 q^{54} - 243 q^{55} + 83 q^{56} + 7 q^{57} + 55 q^{58} - 21 q^{59} + 103 q^{60} + 21 q^{61} - 195 q^{62} + 63 q^{63} + 68 q^{64} - 3 q^{65} + 73 q^{66} + 21 q^{67} + 95 q^{68} - 69 q^{69} + 107 q^{70} + 25 q^{71} - 305 q^{72} + 49 q^{73} + 17 q^{74} + 57 q^{75} + 45 q^{76} + 83 q^{77} + 151 q^{78} - 97 q^{79} + 93 q^{80} + 80 q^{81} - 41 q^{82} + 49 q^{83} + 159 q^{84} + 103 q^{85} - 239 q^{86} + 47 q^{87} + 113 q^{88} - 141 q^{89} + 247 q^{90} + 15 q^{91} + 82 q^{92} + 71 q^{93} + 87 q^{94} + 93 q^{95} - 243 q^{96} + 65 q^{97} + 100 q^{98} - 107 q^{99}+O(q^{100}) \) 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 −1.39453 + 1.88553i −1.56239 − 0.706244i −1.02486 − 3.34651i 3.77657 + 0.983341i 3.51044 − 1.96105i 2.16255 − 1.74613i 3.31667 + 1.17186i −0.0397462 − 0.0451604i −7.12065 + 5.74952i
5.2 −1.32476 + 1.79120i 2.81711 + 1.27341i −0.867749 − 2.83350i 0.938034 + 0.244245i −6.01294 + 3.35903i 0.772686 − 0.623900i 2.02371 + 0.715022i 4.33249 + 4.92266i −1.68017 + 1.35664i
5.3 −1.26915 + 1.71601i 0.244696 + 0.110609i −0.748292 − 2.44343i −1.36068 − 0.354294i −0.500363 + 0.279520i −3.02679 + 2.44396i 1.11779 + 0.394940i −1.93438 − 2.19789i 2.33489 − 1.88529i
5.4 −0.908228 + 1.22801i −1.70553 − 0.770950i −0.0974740 − 0.318286i −1.59294 − 0.414769i 2.49574 − 1.39421i 1.31679 − 1.06324i −2.40087 − 0.848282i 0.332456 + 0.377744i 1.95609 − 1.57943i
5.5 −0.418691 + 0.566108i 1.04210 + 0.471061i 0.440470 + 1.43828i 1.34433 + 0.350037i −0.702991 + 0.392714i −0.836535 + 0.675455i −2.32644 − 0.821982i −1.11794 − 1.27023i −0.761019 + 0.614480i
5.6 −0.0312766 + 0.0422888i −3.11429 − 1.40775i 0.584835 + 1.90969i 1.09130 + 0.284154i 0.156936 − 0.0876698i −1.68728 + 1.36238i −0.198237 − 0.0700417i 5.73501 + 6.51624i −0.0461488 + 0.0372626i
5.7 0.214479 − 0.289995i 2.27274 + 1.02735i 0.547550 + 1.78794i −2.84494 − 0.740764i 0.785382 − 0.438741i 0.796898 − 0.643450i 1.31611 + 0.465011i 2.12791 + 2.41777i −0.824998 + 0.666140i
5.8 0.554764 − 0.750090i −0.539288 − 0.243774i 0.330773 + 1.08009i −0.0731175 − 0.0190383i −0.482029 + 0.269278i 3.67843 − 2.97012i 2.75298 + 0.972690i −1.75062 − 1.98909i −0.0548434 + 0.0442829i
5.9 0.712561 − 0.963446i −0.434653 − 0.196475i 0.165161 + 0.539306i 3.48297 + 0.906896i −0.499010 + 0.278764i −2.22496 + 1.79653i 2.89702 + 1.02358i −1.83170 − 2.08122i 3.35558 − 2.70944i
5.10 1.24731 − 1.68647i −2.25079 − 1.01742i −0.702765 − 2.29477i −1.94771 − 0.507145i −4.52328 + 2.52685i 0.623794 − 0.503679i −0.791051 − 0.279496i 2.04888 + 2.32798i −3.28469 + 2.65220i
5.11 1.32000 − 1.78476i 1.56961 + 0.709510i −0.857325 − 2.79946i −1.30049 − 0.338620i 3.33820 − 1.86483i −1.56624 + 1.26465i −1.94193 − 0.686129i −0.0217449 − 0.0247070i −2.32100 + 1.87408i
6.1 −0.708990 + 2.31509i −0.830007 + 0.943071i −3.19997 − 2.16281i −1.59468 + 0.890842i −1.59483 − 2.59017i −0.563156 − 2.61305i 3.50825 − 2.83271i 0.180582 + 1.41019i −0.931771 − 4.32343i
6.2 −0.603658 + 1.97115i 0.565151 − 0.642136i −1.86401 − 1.25985i 0.732773 − 0.409352i 0.924589 + 1.50163i 0.664553 + 3.08354i 0.400718 − 0.323557i 0.288110 + 2.24989i 0.364549 + 1.69151i
6.3 −0.313929 + 1.02509i 1.72690 − 1.96214i 0.704770 + 0.476341i 0.144663 − 0.0808135i 1.46924 + 2.38619i −0.466383 − 2.16403i −2.37777 + 1.91991i −0.486754 − 3.80113i 0.0374269 + 0.173662i
6.4 −0.253719 + 0.828480i −0.873218 + 0.992168i 1.03501 + 0.699547i 3.76747 − 2.10464i −0.600439 − 0.975175i −0.331955 − 1.54028i −2.19044 + 1.76865i 0.159165 + 1.24294i 0.787769 + 3.65526i
6.5 −0.113839 + 0.371725i 0.238876 − 0.271416i 1.53180 + 1.03532i −3.42800 + 1.91500i 0.0736986 + 0.119694i 0.442749 + 2.05436i −1.16418 + 0.940008i 0.364449 + 2.84603i −0.321610 − 1.49227i
6.6 −0.0212460 + 0.0693756i −1.29192 + 1.46791i 1.65266 + 1.11700i −1.47894 + 0.826183i −0.0743886 − 0.120815i −0.224452 − 1.04146i −0.225507 + 0.182084i −0.104638 − 0.817130i −0.0258954 − 0.120155i
6.7 0.369036 − 1.20503i 0.220166 − 0.250157i 0.341115 + 0.230553i −0.634433 + 0.354416i −0.220197 − 0.357623i −1.03283 − 4.79234i 2.36478 − 1.90942i 0.366948 + 2.86555i 0.192952 + 0.895302i
6.8 0.398802 − 1.30223i −1.29256 + 1.46863i 0.120269 + 0.0812875i 1.25985 − 0.703794i 1.39702 + 2.26890i 0.573322 + 2.66022i 2.27307 − 1.83538i −0.105120 − 0.820900i −0.414068 − 1.92128i
6.9 0.493462 − 1.61132i 2.10195 − 2.38828i −0.695833 − 0.470301i −3.04917 + 1.70337i −2.81105 − 4.56544i 0.654563 + 3.03718i 1.52110 − 1.22820i −0.904627 − 7.06436i 1.24003 + 5.75374i
See next 80 embeddings (of 396 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
149.d even 37 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 149.2.d.a ✓ 396
149.d even 37 1 inner 149.2.d.a ✓ 396
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
149.2.d.a ✓ 396 1.a even 1 1 trivial
149.2.d.a ✓ 396 149.d even 37 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(149, [\chi])\).