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Properties

Label 669.2.m.b
Level $669$
Weight $2$
Character orbit 669.m
Analytic conductor $5.342$
Analytic rank $0$
Dimension $1368$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [669,2,Mod(19,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(222)) chi = DirichletCharacter(H, H._module([0, 172])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 669 = 3 \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 669.m (of order \(111\), degree \(72\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1368] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.34199189522\)
Analytic rank: \(0\)
Dimension: \(1368\)
Relative dimension: \(19\) over \(\Q(\zeta_{111})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{111}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1368 q - 2 q^{2} + 19 q^{3} - 38 q^{4} + q^{5} + q^{6} + 4 q^{7} + 19 q^{9} - 5 q^{10} - 4 q^{11} + 19 q^{12} - 12 q^{13} - 8 q^{14} - 2 q^{15} - 30 q^{16} + 8 q^{17} + q^{18} - 71 q^{19} + 10 q^{20}+ \cdots - 4 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.348890 − 2.72453i −0.759946 + 0.649986i −5.36589 + 1.39717i 1.62775 − 0.231898i 2.03605 + 1.84373i 1.59148 − 2.15182i 3.63044 + 9.03469i 0.155037 − 0.987909i −1.19972 − 4.35394i
19.2 −0.338150 − 2.64066i −0.759946 + 0.649986i −4.92328 + 1.28192i −1.33128 + 0.189661i 1.97337 + 1.78697i −1.37145 + 1.85432i 3.06468 + 7.62675i 0.155037 − 0.987909i 0.951004 + 3.45133i
19.3 −0.261508 − 2.04215i −0.759946 + 0.649986i −2.16654 + 0.564122i −3.76343 + 0.536159i 1.52610 + 1.38195i 0.345717 − 0.467440i 0.183294 + 0.456145i 0.155037 − 0.987909i 2.07909 + 7.54530i
19.4 −0.245630 − 1.91816i −0.759946 + 0.649986i −1.68353 + 0.438357i 1.54877 − 0.220646i 1.43344 + 1.29804i −1.56171 + 2.11157i −0.187712 − 0.467140i 0.155037 − 0.987909i −0.803657 − 2.91658i
19.5 −0.236396 − 1.84605i −0.759946 + 0.649986i −1.41656 + 0.368845i −2.01171 + 0.286599i 1.37956 + 1.24925i 0.583198 − 0.788536i −0.372090 − 0.925981i 0.155037 − 0.987909i 1.00464 + 3.64597i
19.6 −0.184517 − 1.44092i −0.759946 + 0.649986i −0.106730 + 0.0277902i 2.85330 − 0.406497i 1.07680 + 0.975086i 1.44616 − 1.95534i −1.02355 − 2.54720i 0.155037 − 0.987909i −1.11221 − 4.03637i
19.7 −0.149953 − 1.17100i −0.759946 + 0.649986i 0.586706 − 0.152766i −0.239221 + 0.0340808i 0.875091 + 0.792432i −2.71848 + 3.67563i −1.14723 − 2.85499i 0.155037 − 0.987909i 0.0757806 + 0.275018i
19.8 −0.107175 − 0.836946i −0.759946 + 0.649986i 1.24647 − 0.324557i −0.915816 + 0.130472i 0.625450 + 0.566371i 2.22100 − 3.00299i −1.03444 − 2.57432i 0.155037 − 0.987909i 0.207351 + 0.752505i
19.9 −0.0440153 − 0.343722i −0.759946 + 0.649986i 1.81926 − 0.473698i −3.74410 + 0.533405i 0.256864 + 0.232601i −1.52315 + 2.05943i −0.501306 − 1.24755i 0.155037 − 0.987909i 0.348140 + 1.26345i
19.10 −0.0219188 − 0.171167i −0.759946 + 0.649986i 1.90665 − 0.496453i 1.92777 − 0.274641i 0.127914 + 0.115831i 0.0710153 − 0.0960190i −0.255452 − 0.635717i 0.155037 − 0.987909i −0.0892641 − 0.323952i
19.11 0.0574183 + 0.448388i −0.759946 + 0.649986i 1.73771 − 0.452465i −0.981149 + 0.139780i −0.335081 − 0.303430i −1.27846 + 1.72860i 0.639755 + 1.59209i 0.155037 − 0.987909i −0.119012 − 0.431910i
19.12 0.131175 + 1.02436i −0.759946 + 0.649986i 0.903350 − 0.235214i −0.254696 + 0.0362853i −0.765508 − 0.693200i 1.60140 − 2.16524i 1.12956 + 2.81102i 0.155037 − 0.987909i −0.0705791 − 0.256141i
19.13 0.136992 + 1.06979i −0.759946 + 0.649986i 0.809785 − 0.210852i 3.61618 − 0.515180i −0.799454 − 0.723939i −0.951851 + 1.28699i 1.14077 + 2.83892i 0.155037 − 0.987909i 1.04652 + 3.79797i
19.14 0.164723 + 1.28634i −0.759946 + 0.649986i 0.307920 − 0.0801761i −2.05355 + 0.292560i −0.961286 − 0.870484i 1.65357 − 2.23577i 1.12093 + 2.78954i 0.155037 − 0.987909i −0.714600 − 2.59338i
19.15 0.194248 + 1.51691i −0.759946 + 0.649986i −0.327820 + 0.0853576i −1.81375 + 0.258396i −1.13359 − 1.02651i −1.18898 + 1.60761i 0.947259 + 2.35734i 0.155037 − 0.987909i −0.744281 − 2.70110i
19.16 0.211486 + 1.65153i −0.759946 + 0.649986i −0.747344 + 0.194593i 3.40291 − 0.484798i −1.23419 − 1.11761i −2.75199 + 3.72094i 0.762192 + 1.89679i 0.155037 − 0.987909i 1.52033 + 5.51747i
19.17 0.286670 + 2.23864i −0.759946 + 0.649986i −2.99388 + 0.779547i 1.02945 − 0.146661i −1.67294 − 1.51492i −0.336474 + 0.454943i −0.920366 − 2.29042i 0.155037 − 0.987909i 0.623435 + 2.26253i
19.18 0.313515 + 2.44829i −0.759946 + 0.649986i −3.96035 + 1.03119i 3.18451 − 0.453683i −1.82961 − 1.65679i 2.82138 − 3.81476i −1.92566 − 4.79219i 0.155037 − 0.987909i 2.10914 + 7.65436i
19.19 0.314908 + 2.45916i −0.759946 + 0.649986i −4.01285 + 1.04486i −3.67126 + 0.523028i −1.83773 − 1.66415i 1.49678 − 2.02379i −1.98437 − 4.93829i 0.155037 − 0.987909i −2.44232 − 8.86353i
25.1 −1.42823 − 2.31959i 0.812331 − 0.583196i −2.44026 + 4.83997i 1.07506 + 2.29135i −2.51297 − 1.05134i −0.332372 − 0.827139i 9.28357 − 0.790148i 0.319764 − 0.947497i 3.77956 − 5.76628i
See next 80 embeddings (of 1368 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.19
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.g even 111 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 669.2.m.b ✓ 1368
223.g even 111 1 inner 669.2.m.b ✓ 1368
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
669.2.m.b ✓ 1368 1.a even 1 1 trivial
669.2.m.b ✓ 1368 223.g even 111 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1368} + 2 T_{2}^{1367} + 59 T_{2}^{1366} + 118 T_{2}^{1365} + 1885 T_{2}^{1364} + \cdots + 43\!\cdots\!44 \) acting on \(S_{2}^{\mathrm{new}}(669, [\chi])\). Copy content Toggle raw display