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Properties

Label 790.2.k.a
Level $790$
Weight $2$
Character orbit 790.k
Analytic conductor $6.308$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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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] = [790,2,Mod(103,790)] 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("790.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.k (of order \(12\), 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: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 12 q^{7} - 8 q^{10} - 16 q^{11} + 80 q^{16} - 32 q^{18} + 48 q^{21} - 8 q^{22} - 12 q^{28} - 24 q^{31} - 36 q^{35} + 72 q^{36} - 36 q^{37} + 8 q^{38} + 20 q^{42} - 48 q^{43} - 4 q^{45} + 16 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −0.258819 − 0.965926i −0.774459 − 2.89032i −0.866025 + 0.500000i −2.21188 + 0.328031i −2.59139 + 1.49614i −0.448720 + 1.67465i 0.707107 + 0.707107i −5.15610 + 2.97687i 0.889330 + 2.05161i
103.2 −0.258819 − 0.965926i −0.682319 − 2.54645i −0.866025 + 0.500000i 1.91877 + 1.14818i −2.28309 + 1.31814i −0.648470 + 2.42012i 0.707107 + 0.707107i −3.42078 + 1.97499i 0.612438 − 2.15056i
103.3 −0.258819 − 0.965926i −0.602390 − 2.24815i −0.866025 + 0.500000i −0.404988 − 2.19909i −2.01563 + 1.16373i −0.954993 + 3.56408i 0.707107 + 0.707107i −2.09322 + 1.20852i −2.01934 + 0.960354i
103.4 −0.258819 − 0.965926i −0.596391 − 2.22576i −0.866025 + 0.500000i 2.00339 − 0.993188i −1.99556 + 1.15214i −0.105600 + 0.394104i 0.707107 + 0.707107i −2.00025 + 1.15485i −1.47786 − 1.67807i
103.5 −0.258819 − 0.965926i −0.485819 − 1.81310i −0.866025 + 0.500000i −1.27028 + 1.84022i −1.62558 + 0.938529i 0.662018 − 2.47069i 0.707107 + 0.707107i −0.453235 + 0.261675i 2.10628 + 0.750711i
103.6 −0.258819 − 0.965926i −0.346692 − 1.29387i −0.866025 + 0.500000i −0.721043 + 2.11662i −1.16005 + 0.669757i −0.774631 + 2.89096i 0.707107 + 0.707107i 1.04417 − 0.602852i 2.23112 + 0.148652i
103.7 −0.258819 − 0.965926i −0.344175 − 1.28448i −0.866025 + 0.500000i −2.23402 − 0.0957651i −1.15163 + 0.664896i 0.970810 − 3.62311i 0.707107 + 0.707107i 1.06664 − 0.615827i 0.485704 + 2.18268i
103.8 −0.258819 − 0.965926i −0.249711 − 0.931934i −0.866025 + 0.500000i −0.00291156 − 2.23607i −0.835550 + 0.482405i 0.678383 − 2.53176i 0.707107 + 0.707107i 1.79193 − 1.03457i −2.15912 + 0.581549i
103.9 −0.258819 − 0.965926i −0.245888 − 0.917666i −0.866025 + 0.500000i 2.23605 + 0.00764219i −0.822756 + 0.475019i 0.496377 − 1.85250i 0.707107 + 0.707107i 1.81643 − 1.04871i −0.571352 − 2.16184i
103.10 −0.258819 − 0.965926i −0.0226523 − 0.0845396i −0.866025 + 0.500000i 0.909451 + 2.04277i −0.0757961 + 0.0437609i −0.409923 + 1.52985i 0.707107 + 0.707107i 2.59144 − 1.49617i 1.73778 − 1.40717i
103.11 −0.258819 − 0.965926i 0.0373279 + 0.139310i −0.866025 + 0.500000i −1.20885 − 1.88114i 0.124902 − 0.0721120i −0.866269 + 3.23296i 0.707107 + 0.707107i 2.58006 − 1.48960i −1.50417 + 1.65453i
103.12 −0.258819 − 0.965926i 0.181193 + 0.676221i −0.866025 + 0.500000i 1.28855 − 1.82747i 0.606283 − 0.350038i 0.929635 − 3.46945i 0.707107 + 0.707107i 2.17363 − 1.25495i −2.09870 − 0.771656i
103.13 −0.258819 − 0.965926i 0.258477 + 0.964650i −0.866025 + 0.500000i 2.03068 − 0.936120i 0.864881 − 0.499339i −1.32344 + 4.93914i 0.707107 + 0.707107i 1.73434 − 1.00132i −1.42980 − 1.71920i
103.14 −0.258819 − 0.965926i 0.300996 + 1.12333i −0.866025 + 0.500000i −0.559961 + 2.16482i 1.00715 − 0.581480i 1.05325 − 3.93078i 0.707107 + 0.707107i 1.42680 − 0.823762i 2.23598 − 0.0194162i
103.15 −0.258819 − 0.965926i 0.327676 + 1.22290i −0.866025 + 0.500000i −2.16492 − 0.559557i 1.09643 − 0.633022i −0.174675 + 0.651897i 0.707107 + 0.707107i 1.20995 − 0.698566i 0.0198332 + 2.23598i
103.16 −0.258819 − 0.965926i 0.479313 + 1.78882i −0.866025 + 0.500000i 1.33016 + 1.79741i 1.60381 − 0.925961i 0.361030 − 1.34738i 0.707107 + 0.707107i −0.372059 + 0.214808i 1.39189 − 1.75004i
103.17 −0.258819 − 0.965926i 0.613790 + 2.29070i −0.866025 + 0.500000i 0.528417 − 2.17273i 2.05378 − 1.18575i 0.445979 − 1.66442i 0.707107 + 0.707107i −2.27248 + 1.31202i −2.23546 + 0.0519330i
103.18 −0.258819 − 0.965926i 0.658918 + 2.45912i −0.866025 + 0.500000i −1.66899 + 1.48811i 2.20478 − 1.27293i −0.862615 + 3.21932i 0.707107 + 0.707107i −3.01500 + 1.74071i 1.86937 + 1.22697i
103.19 −0.258819 − 0.965926i 0.674713 + 2.51806i −0.866025 + 0.500000i −1.73710 − 1.40801i 2.25763 − 1.30344i 0.640994 − 2.39222i 0.707107 + 0.707107i −3.28732 + 1.89794i −0.910442 + 2.04233i
103.20 −0.258819 − 0.965926i 0.818092 + 3.05316i −0.866025 + 0.500000i 2.19827 + 0.409419i 2.73739 − 1.58043i −0.303115 + 1.13124i 0.707107 + 0.707107i −6.05443 + 3.49553i −0.173485 − 2.22933i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 103.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
79.d odd 6 1 inner
395.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 790.2.k.a ✓ 160
5.c odd 4 1 inner 790.2.k.a ✓ 160
79.d odd 6 1 inner 790.2.k.a ✓ 160
395.l even 12 1 inner 790.2.k.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
790.2.k.a ✓ 160 1.a even 1 1 trivial
790.2.k.a ✓ 160 5.c odd 4 1 inner
790.2.k.a ✓ 160 79.d odd 6 1 inner
790.2.k.a ✓ 160 395.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(790, [\chi])\).