Properties

Label 207.2.o.a
Level $207$
Weight $2$
Character orbit 207.o
Analytic conductor $1.653$
Analytic rank $0$
Dimension $440$
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] = [207,2,Mod(5,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([55, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.o (of order \(66\), degree \(20\), 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(22\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 27 q^{2} - 16 q^{3} - 29 q^{4} - 33 q^{5} - 25 q^{6} - 11 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q - 27 q^{2} - 16 q^{3} - 29 q^{4} - 33 q^{5} - 25 q^{6} - 11 q^{7} - 16 q^{9} - 44 q^{10} - 33 q^{11} - 22 q^{12} - 9 q^{13} - 33 q^{14} + 3 q^{16} - 39 q^{18} - 44 q^{19} - 33 q^{20} - 55 q^{21} - 27 q^{23} + 52 q^{24} + 11 q^{25} - 79 q^{27} - 44 q^{28} + 27 q^{29} - 66 q^{30} - 3 q^{31} - 33 q^{32} - 11 q^{34} + 23 q^{36} - 44 q^{37} - 33 q^{38} - 40 q^{39} - 77 q^{40} + 9 q^{41} - 22 q^{42} - 11 q^{43} - 36 q^{46} - 120 q^{47} - 56 q^{48} + 35 q^{49} - 3 q^{50} - 22 q^{51} - 38 q^{52} + 42 q^{54} - 44 q^{55} + 165 q^{56} + 11 q^{57} - 10 q^{58} - 9 q^{59} + 88 q^{60} - 11 q^{61} + 33 q^{63} - 22 q^{64} + 198 q^{65} + 33 q^{66} - 11 q^{67} + 3 q^{69} - 70 q^{70} + 14 q^{72} - 36 q^{73} + 231 q^{74} - 13 q^{75} - 11 q^{76} + 39 q^{77} + 3 q^{78} - 11 q^{79} + 172 q^{81} - 10 q^{82} + 66 q^{83} - 110 q^{84} + q^{85} - 33 q^{86} - 196 q^{87} - 99 q^{88} + 418 q^{90} + 63 q^{92} - 188 q^{93} - 42 q^{94} - 93 q^{95} - 82 q^{96} + 22 q^{97} + 242 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.62689 + 0.637277i 1.63894 0.560241i 4.71676 2.43166i 2.75970 + 2.17025i −3.94829 + 2.51615i 1.24506 + 0.886604i −6.75505 + 5.85328i 2.37226 1.83641i −8.63247 3.94232i
5.2 −2.45913 + 0.596579i 0.0507650 1.73131i 3.91376 2.01768i −2.75910 2.16978i 0.908024 + 4.28780i −2.46994 1.75883i −4.59596 + 3.98242i −2.99485 0.175779i 8.07946 + 3.68976i
5.3 −2.43968 + 0.591859i −0.572964 + 1.63454i 3.82406 1.97144i −0.573627 0.451105i 0.430433 4.32686i −0.847518 0.603515i −4.36812 + 3.78500i −2.34342 1.87306i 1.66646 + 0.761045i
5.4 −1.96150 + 0.475854i −1.72655 0.137906i 1.84336 0.950318i 2.46398 + 1.93770i 3.45225 0.551085i −3.19678 2.27642i −0.112732 + 0.0976829i 2.96196 + 0.476202i −5.75516 2.62829i
5.5 −1.90183 + 0.461379i 1.48247 + 0.895699i 1.62642 0.838476i −1.80924 1.42280i −3.23267 1.01949i 2.18638 + 1.55692i 0.251684 0.218086i 1.39545 + 2.65570i 4.09731 + 1.87118i
5.6 −1.87165 + 0.454057i −1.24149 1.20777i 1.51923 0.783220i −0.386973 0.304319i 2.87203 + 1.69682i 2.77945 + 1.97924i 0.423207 0.366711i 0.0825754 + 2.99886i 0.862456 + 0.393871i
5.7 −1.19377 + 0.289604i 0.337452 1.69886i −0.436467 + 0.225014i 1.95562 + 1.53792i 0.0891586 + 2.12577i 0.435801 + 0.310333i 2.31259 2.00387i −2.77225 1.14657i −2.77994 1.26955i
5.8 −1.07455 + 0.260682i −1.64011 + 0.556821i −0.690978 + 0.356224i −1.83293 1.44144i 1.61722 1.02588i 0.303689 + 0.216256i 2.32091 2.01108i 2.37990 1.82649i 2.34533 + 1.07108i
5.9 −0.942830 + 0.228728i 0.860190 + 1.50335i −0.941060 + 0.485150i 2.69473 + 2.11916i −1.15487 1.22066i 0.176132 + 0.125423i 2.24271 1.94332i −1.52015 + 2.58634i −3.02538 1.38165i
5.10 −0.565298 + 0.137140i 0.121015 + 1.72782i −1.47692 + 0.761403i −1.35334 1.06428i −0.305362 0.960136i −2.96537 2.11163i 1.60971 1.39482i −2.97071 + 0.418184i 0.910994 + 0.416037i
5.11 −0.158710 + 0.0385026i 1.72620 0.142203i −1.75396 + 0.904232i 0.845243 + 0.664707i −0.268490 + 0.0890323i 0.602097 + 0.428751i 0.490404 0.424938i 2.95956 0.490942i −0.159741 0.0729514i
5.12 −0.156007 + 0.0378470i 1.02630 1.39524i −1.75476 + 0.904644i −3.16235 2.48690i −0.107306 + 0.256511i 1.51977 + 1.08222i 0.482163 0.417797i −0.893396 2.86389i 0.587472 + 0.268290i
5.13 0.101474 0.0246174i −0.566247 1.63688i −1.76798 + 0.911457i 0.359165 + 0.282450i −0.0977550 0.152161i −3.37292 2.40185i −0.314794 + 0.272770i −2.35873 + 1.85375i 0.0433991 + 0.0198197i
5.14 0.554186 0.134444i −1.59629 0.672214i −1.48862 + 0.767439i 1.47451 + 1.15957i −0.975014 0.157920i 2.15103 + 1.53174i −1.58375 + 1.37232i 2.09626 + 2.14609i 0.973049 + 0.444377i
5.15 0.905370 0.219640i 0.0930743 + 1.72955i −1.00622 + 0.518741i −1.25721 0.988678i 0.464145 + 1.54544i 4.26264 + 3.03541i −2.20522 + 1.91084i −2.98267 + 0.321953i −1.35539 0.618986i
5.16 1.28415 0.311532i −1.70297 0.316062i −0.225678 + 0.116345i −2.59594 2.04147i −2.28533 + 0.124657i −2.08508 1.48478i −2.25085 + 1.95038i 2.80021 + 1.07649i −3.96956 1.81284i
5.17 1.29057 0.313088i −0.828584 + 1.52100i −0.210130 + 0.108329i 3.10558 + 2.44225i −0.593136 + 2.22238i −2.63088 1.87344i −2.24454 + 1.94491i −1.62690 2.52056i 4.77260 + 2.17957i
5.18 1.41889 0.344219i 1.58943 + 0.688262i 0.117094 0.0603664i 0.676831 + 0.532265i 2.49214 + 0.429455i −0.142829 0.101708i −2.06150 + 1.78630i 2.05259 + 2.18789i 1.14356 + 0.522249i
5.19 1.65908 0.402488i 1.26516 1.18295i 0.812874 0.419065i 0.0437969 + 0.0344423i 1.62288 2.47181i −1.22353 0.871275i −1.40048 + 1.21352i 0.201279 2.99324i 0.0865251 + 0.0395147i
5.20 2.17585 0.527855i −0.508290 1.65579i 2.67802 1.38061i −2.04245 1.60620i −1.97998 3.33445i 2.45761 + 1.75005i 1.71401 1.48520i −2.48328 + 1.68324i −5.29190 2.41673i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.d odd 22 1 inner
207.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.2.o.a 440
3.b odd 2 1 621.2.s.a 440
9.c even 3 1 621.2.s.a 440
9.d odd 6 1 inner 207.2.o.a 440
23.d odd 22 1 inner 207.2.o.a 440
69.g even 22 1 621.2.s.a 440
207.o even 66 1 inner 207.2.o.a 440
207.p odd 66 1 621.2.s.a 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.o.a 440 1.a even 1 1 trivial
207.2.o.a 440 9.d odd 6 1 inner
207.2.o.a 440 23.d odd 22 1 inner
207.2.o.a 440 207.o even 66 1 inner
621.2.s.a 440 3.b odd 2 1
621.2.s.a 440 9.c even 3 1
621.2.s.a 440 69.g even 22 1
621.2.s.a 440 207.p odd 66 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(207, [\chi])\).