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Properties

Label 700.2.w.a
Level $700$
Weight $2$
Character orbit 700.w
Analytic conductor $5.590$
Analytic rank $0$
Dimension $464$
Inner twists $8$

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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] = [700,2,Mod(111,700)] 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("700.111"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 8, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(464\)
Relative dimension: \(116\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 6 q^{2} - 6 q^{4} + 12 q^{8} - 120 q^{9} + 9 q^{14} - 6 q^{16} + 4 q^{18} - 24 q^{21} - 14 q^{22} - 24 q^{25} + 31 q^{28} - 12 q^{29} - 90 q^{30} + 4 q^{32} + 52 q^{36} - 12 q^{37} - 31 q^{42} - 22 q^{44}+ \cdots + 11 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 −1.41418 + 0.0100953i −0.874876 + 2.69259i 1.99980 − 0.0285531i 0.871263 − 2.05934i 1.21005 − 3.81664i −1.67011 + 2.05201i −2.82778 + 0.0605676i −4.05759 − 2.94801i −1.21133 + 2.92107i
111.2 −1.41418 + 0.0100953i 0.874876 − 2.69259i 1.99980 − 0.0285531i −0.871263 + 2.05934i −1.21005 + 3.81664i 1.67011 + 2.05201i −2.82778 + 0.0605676i −4.05759 − 2.94801i 1.21133 − 2.92107i
111.3 −1.40612 + 0.151061i −0.626392 + 1.92784i 1.95436 − 0.424820i −1.94123 − 1.10978i 0.589564 − 2.80540i −2.38789 − 1.13929i −2.68390 + 0.892576i −0.897138 − 0.651809i 2.89726 + 1.26724i
111.4 −1.40612 + 0.151061i 0.626392 − 1.92784i 1.95436 − 0.424820i 1.94123 + 1.10978i −0.589564 + 2.80540i 2.38789 − 1.13929i −2.68390 + 0.892576i −0.897138 − 0.651809i −2.89726 − 1.26724i
111.5 −1.40391 + 0.170402i −0.240867 + 0.741312i 1.94193 − 0.478457i 1.64805 − 1.51127i 0.211835 − 1.08178i 2.60055 + 0.486980i −2.64476 + 1.00262i 1.93552 + 1.40624i −2.05619 + 2.40252i
111.6 −1.40391 + 0.170402i 0.240867 − 0.741312i 1.94193 − 0.478457i −1.64805 + 1.51127i −0.211835 + 1.08178i −2.60055 + 0.486980i −2.64476 + 1.00262i 1.93552 + 1.40624i 2.05619 − 2.40252i
111.7 −1.38705 + 0.275858i −0.601421 + 1.85098i 1.84780 − 0.765257i −0.221059 + 2.22511i 0.323591 − 2.73331i 0.280282 − 2.63086i −2.35189 + 1.57118i −0.637386 − 0.463088i −0.307195 − 3.14732i
111.8 −1.38705 + 0.275858i 0.601421 − 1.85098i 1.84780 − 0.765257i 0.221059 − 2.22511i −0.323591 + 2.73331i −0.280282 − 2.63086i −2.35189 + 1.57118i −0.637386 − 0.463088i 0.307195 + 3.14732i
111.9 −1.37918 − 0.312832i −0.315624 + 0.971392i 1.80427 + 0.862902i 2.23605 + 0.00930932i 0.739185 − 1.24099i −1.74983 − 1.98446i −2.21847 − 1.75453i 1.58307 + 1.15017i −3.08100 − 0.712346i
111.10 −1.37918 − 0.312832i 0.315624 − 0.971392i 1.80427 + 0.862902i −2.23605 − 0.00930932i −0.739185 + 1.24099i 1.74983 − 1.98446i −2.21847 − 1.75453i 1.58307 + 1.15017i 3.08100 + 0.712346i
111.11 −1.34430 − 0.439151i −0.0551538 + 0.169746i 1.61429 + 1.18070i −1.50775 − 1.65127i 0.148687 − 0.203969i 1.26171 + 2.32553i −1.65159 − 2.29614i 2.40128 + 1.74463i 1.30171 + 2.88194i
111.12 −1.34430 − 0.439151i 0.0551538 − 0.169746i 1.61429 + 1.18070i 1.50775 + 1.65127i −0.148687 + 0.203969i −1.26171 + 2.32553i −1.65159 − 2.29614i 2.40128 + 1.74463i −1.30171 − 2.88194i
111.13 −1.33076 − 0.478608i −0.723930 + 2.22803i 1.54187 + 1.27383i −1.71454 + 1.43539i 2.02973 − 2.61850i 2.60491 − 0.463059i −1.44220 − 2.43312i −2.01298 − 1.46251i 2.96864 − 1.08957i
111.14 −1.33076 − 0.478608i 0.723930 − 2.22803i 1.54187 + 1.27383i 1.71454 − 1.43539i −2.02973 + 2.61850i −2.60491 − 0.463059i −1.44220 − 2.43312i −2.01298 − 1.46251i −2.96864 + 1.08957i
111.15 −1.27775 + 0.606103i −0.285491 + 0.878652i 1.26528 − 1.54889i 0.283810 + 2.21798i −0.167768 − 1.29573i 1.45808 + 2.20771i −0.677918 + 2.74598i 1.73653 + 1.26166i −1.70696 − 2.66201i
111.16 −1.27775 + 0.606103i 0.285491 − 0.878652i 1.26528 − 1.54889i −0.283810 − 2.21798i 0.167768 + 1.29573i −1.45808 + 2.20771i −0.677918 + 2.74598i 1.73653 + 1.26166i 1.70696 + 2.66201i
111.17 −1.26122 − 0.639787i −0.697302 + 2.14608i 1.18135 + 1.61382i −1.61752 + 1.54391i 2.25248 − 2.26055i −1.66238 + 2.05828i −0.457432 − 2.79119i −1.69236 − 1.22957i 3.02781 − 0.912337i
111.18 −1.26122 − 0.639787i 0.697302 − 2.14608i 1.18135 + 1.61382i 1.61752 − 1.54391i −2.25248 + 2.26055i 1.66238 + 2.05828i −0.457432 − 2.79119i −1.69236 − 1.22957i −3.02781 + 0.912337i
111.19 −1.22851 + 0.700548i −0.408370 + 1.25683i 1.01846 − 1.72126i 2.15116 + 0.610344i −0.378787 − 1.83011i −1.94006 + 1.79893i −0.0453677 + 2.82806i 1.01418 + 0.736848i −3.07029 + 0.757177i
111.20 −1.22851 + 0.700548i 0.408370 − 1.25683i 1.01846 − 1.72126i −2.15116 − 0.610344i 0.378787 + 1.83011i 1.94006 + 1.79893i −0.0453677 + 2.82806i 1.01418 + 0.736848i 3.07029 − 0.757177i
See next 80 embeddings (of 464 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 111.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
25.d even 5 1 inner
28.d even 2 1 inner
100.j odd 10 1 inner
175.l odd 10 1 inner
700.w even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.w.a ✓ 464
4.b odd 2 1 inner 700.2.w.a ✓ 464
7.b odd 2 1 inner 700.2.w.a ✓ 464
25.d even 5 1 inner 700.2.w.a ✓ 464
28.d even 2 1 inner 700.2.w.a ✓ 464
100.j odd 10 1 inner 700.2.w.a ✓ 464
175.l odd 10 1 inner 700.2.w.a ✓ 464
700.w even 10 1 inner 700.2.w.a ✓ 464
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.w.a ✓ 464 1.a even 1 1 trivial
700.2.w.a ✓ 464 4.b odd 2 1 inner
700.2.w.a ✓ 464 7.b odd 2 1 inner
700.2.w.a ✓ 464 25.d even 5 1 inner
700.2.w.a ✓ 464 28.d even 2 1 inner
700.2.w.a ✓ 464 100.j odd 10 1 inner
700.2.w.a ✓ 464 175.l odd 10 1 inner
700.2.w.a ✓ 464 700.w even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).