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Properties

Label 600.2.be.b
Level $600$
Weight $2$
Character orbit 600.be
Analytic conductor $4.791$
Analytic rank $0$
Dimension $120$
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] = [600,2,Mod(109,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.be (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{3} - 10 q^{4} - 15 q^{8} - 30 q^{9} + q^{10} - 8 q^{13} + 6 q^{14} + 6 q^{16} - 9 q^{20} + 19 q^{22} - 2 q^{25} + 10 q^{26} + 30 q^{27} - 8 q^{28} + 19 q^{30} + 12 q^{31} + 10 q^{32} - 48 q^{34}+ \cdots - 109 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41286 − 0.0618720i −0.309017 − 0.951057i 1.99234 + 0.174833i −0.137607 + 2.23183i 0.377754 + 1.36283i − 5.06961i −2.80408 − 0.370285i −0.809017 + 0.587785i 0.332507 − 3.14475i
109.2 −1.40998 − 0.109285i −0.309017 − 0.951057i 1.97611 + 0.308181i −2.17022 − 0.538631i 0.331773 + 1.37475i 2.20134i −2.75261 − 0.650490i −0.809017 + 0.587785i 3.00112 + 0.996635i
109.3 −1.35254 − 0.413077i −0.309017 − 0.951057i 1.65873 + 1.11741i 2.21587 + 0.299831i 0.0250987 + 1.41399i 0.581515i −1.78193 − 2.19652i −0.809017 + 0.587785i −2.87321 − 1.32086i
109.4 −1.29421 + 0.570102i −0.309017 − 0.951057i 1.34997 − 1.47567i 0.665214 − 2.13483i 0.942133 + 1.05470i − 2.27192i −0.905862 + 2.67944i −0.809017 + 0.587785i 0.356143 + 3.14216i
109.5 −1.17403 + 0.788447i −0.309017 − 0.951057i 0.756702 − 1.85132i 2.15299 + 0.603864i 1.11265 + 0.872927i 0.170277i 0.571279 + 2.77013i −0.809017 + 0.587785i −3.00379 + 0.988560i
109.6 −1.13754 + 0.840246i −0.309017 − 0.951057i 0.587973 − 1.91162i −2.18469 + 0.476576i 1.15064 + 0.822210i − 1.14258i 0.937391 + 2.66858i −0.809017 + 0.587785i 2.08472 − 2.37780i
109.7 −1.13027 − 0.849991i −0.309017 − 0.951057i 0.555029 + 1.92144i 0.220430 + 2.22518i −0.459117 + 1.33761i 3.24236i 1.00588 − 2.64352i −0.809017 + 0.587785i 1.64224 − 2.70242i
109.8 −1.07149 − 0.922985i −0.309017 − 0.951057i 0.296198 + 1.97795i −0.169976 − 2.22960i −0.546701 + 1.30427i 0.993591i 1.50824 − 2.39274i −0.809017 + 0.587785i −1.87576 + 2.54589i
109.9 −0.842905 − 1.13557i −0.309017 − 0.951057i −0.579022 + 1.91435i −1.84303 + 1.26620i −0.819516 + 1.15256i − 1.03050i 2.66193 − 0.956096i −0.809017 + 0.587785i 2.99135 + 1.02559i
109.10 −0.815684 + 1.15527i −0.309017 − 0.951057i −0.669318 − 1.88468i 0.00633452 − 2.23606i 1.35079 + 0.418762i 4.12547i 2.72327 + 0.764056i −0.809017 + 0.587785i 2.57810 + 1.83124i
109.11 −0.444504 − 1.34254i −0.309017 − 0.951057i −1.60483 + 1.19353i 1.94711 − 1.09944i −1.13947 + 0.837616i 1.55008i 2.31572 + 1.62403i −0.809017 + 0.587785i −2.34154 − 2.12536i
109.12 −0.416185 + 1.35159i −0.309017 − 0.951057i −1.65358 − 1.12502i 2.09775 − 0.774239i 1.41404 − 0.0218486i − 2.82006i 2.20876 − 1.76674i −0.809017 + 0.587785i 0.173401 + 3.15752i
109.13 −0.240524 − 1.39361i −0.309017 − 0.951057i −1.88430 + 0.670392i −2.01169 − 0.976268i −1.25108 + 0.659401i − 3.89595i 1.38748 + 2.46473i −0.809017 + 0.587785i −0.876677 + 3.03833i
109.14 −0.0982749 + 1.41079i −0.309017 − 0.951057i −1.98068 − 0.277291i 1.06976 + 1.96357i 1.37211 − 0.342495i 3.44588i 0.585853 − 2.76709i −0.809017 + 0.587785i −2.87533 + 1.31624i
109.15 −0.0390780 + 1.41367i −0.309017 − 0.951057i −1.99695 − 0.110487i −1.59403 + 1.56814i 1.35656 − 0.399684i − 1.85305i 0.234229 − 2.81871i −0.809017 + 0.587785i −2.15454 − 2.31472i
109.16 0.00899160 − 1.41418i −0.309017 − 0.951057i −1.99984 − 0.0254316i 0.969475 + 2.01497i −1.34775 + 0.428456i 0.588151i −0.0539467 + 2.82791i −0.809017 + 0.587785i 2.85826 − 1.35290i
109.17 0.133418 − 1.40791i −0.309017 − 0.951057i −1.96440 − 0.375679i −1.74532 − 1.39780i −1.38023 + 0.308179i 3.69053i −0.791006 + 2.71557i −0.809017 + 0.587785i −2.20083 + 2.27076i
109.18 0.450849 + 1.34042i −0.309017 − 0.951057i −1.59347 + 1.20866i −0.432940 − 2.19376i 1.13550 − 0.842997i − 0.643164i −2.33853 − 1.59100i −0.809017 + 0.587785i 2.74537 − 1.56938i
109.19 0.565826 + 1.29609i −0.309017 − 0.951057i −1.35968 + 1.46672i −1.96621 − 1.06491i 1.05780 − 0.938645i 1.46554i −2.67034 − 0.932359i −0.809017 + 0.587785i 0.267679 − 3.15093i
109.20 0.573659 − 1.29264i −0.309017 − 0.951057i −1.34183 − 1.48307i 1.17919 − 1.89987i −1.40664 − 0.146134i − 4.86176i −2.68682 + 0.883729i −0.809017 + 0.587785i −1.77939 − 2.61415i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.30
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
200.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.be.b yes 120
8.b even 2 1 600.2.be.a ✓ 120
25.e even 10 1 600.2.be.a ✓ 120
200.o even 10 1 inner 600.2.be.b yes 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.be.a ✓ 120 8.b even 2 1
600.2.be.a ✓ 120 25.e even 10 1
600.2.be.b yes 120 1.a even 1 1 trivial
600.2.be.b yes 120 200.o even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{120} - 186 T_{11}^{118} + 18969 T_{11}^{116} - 1407040 T_{11}^{114} + 68880 T_{11}^{113} + \cdots + 20\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display