Properties

Label 171.2.bd.a
Level $171$
Weight $2$
Character orbit 171.bd
Analytic conductor $1.365$
Analytic rank $0$
Dimension $108$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(2,171)] 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("171.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.bd (of order \(18\), degree \(6\), 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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 9 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 9 q^{6} - 6 q^{7} + 3 q^{9} - 12 q^{10} - 9 q^{11} - 9 q^{12} - 6 q^{13} - 9 q^{14} - 9 q^{15} + 9 q^{16} - 27 q^{17} - 36 q^{18} - 15 q^{19} - 18 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.06424 − 1.73210i 1.65249 − 0.518905i 0.913607 + 5.18132i 0.404473 − 0.482032i −4.30994 − 1.79115i −1.01627 4.39400 − 7.61064i 2.46148 − 1.71497i −1.66986 + 0.294441i
2.2 −1.91073 − 1.60329i −1.54188 − 0.789062i 0.733045 + 4.15731i −1.20081 + 1.43107i 1.68101 + 3.97976i 2.44537 2.77045 − 4.79855i 1.75476 + 2.43327i 4.58886 − 0.809141i
2.3 −1.73314 − 1.45428i −1.29784 + 1.14700i 0.541559 + 3.07134i 1.93467 − 2.30565i 3.91740 − 0.100500i −4.02280 1.26552 − 2.19195i 0.368770 − 2.97725i −6.70611 + 1.18247i
2.4 −1.49018 − 1.25041i 0.939095 + 1.45537i 0.309818 + 1.75707i −1.38825 + 1.65445i 0.420388 − 3.34302i −0.126291 −0.209926 + 0.363602i −1.23620 + 2.73346i 4.13748 − 0.729550i
2.5 −1.20013 − 1.00703i 0.499246 − 1.65854i 0.0789109 + 0.447526i 1.56758 − 1.86817i −2.26936 + 1.48771i 4.63240 −1.21069 + 2.09698i −2.50151 − 1.65604i −3.76261 + 0.663450i
2.6 −1.09154 − 0.915910i 0.396744 − 1.68600i 0.00526991 + 0.0298872i −2.61333 + 3.11445i −1.97729 + 1.47695i −3.25194 −1.40328 + 2.43055i −2.68519 − 1.33782i 5.70511 − 1.00597i
2.7 −0.965067 − 0.809788i −1.15795 + 1.28808i −0.0716975 − 0.406617i −0.180362 + 0.214947i 2.16057 − 0.305396i 3.15185 −1.51989 + 2.63252i −0.318319 − 2.98306i 0.348122 − 0.0613834i
2.8 −0.586300 − 0.491964i −1.39355 − 1.02859i −0.245577 − 1.39274i 1.29387 − 1.54197i 0.311010 + 1.28864i −1.87387 −1.30656 + 2.26302i 0.883991 + 2.86680i −1.51719 + 0.267522i
2.9 −0.373949 − 0.313780i 1.65202 − 0.520407i −0.305917 − 1.73494i 1.23004 − 1.46591i −0.781065 − 0.323766i −3.29328 −0.918148 + 1.59028i 2.45835 − 1.71945i −0.919944 + 0.162211i
2.10 0.0307038 + 0.0257635i 1.68465 + 0.402423i −0.347017 − 1.96803i −1.41680 + 1.68848i 0.0413574 + 0.0557585i 2.99977 0.0801297 − 0.138789i 2.67611 + 1.35589i −0.0870023 + 0.0153409i
2.11 0.388217 + 0.325752i −1.51148 + 0.845833i −0.302699 − 1.71669i −2.48025 + 2.95585i −0.862313 − 0.164001i −2.21821 0.948484 − 1.64282i 1.56913 − 2.55692i −1.92575 + 0.339561i
2.12 0.400522 + 0.336078i 0.388639 + 1.68789i −0.299827 − 1.70040i 2.05581 − 2.45002i −0.411603 + 0.806648i 2.01117 0.974224 − 1.68741i −2.69792 + 1.31196i 1.64680 − 0.290374i
2.13 0.704973 + 0.591543i −1.72842 + 0.112107i −0.200232 − 1.13557i 0.899185 − 1.07161i −1.28480 − 0.943401i 1.63939 1.45086 − 2.51296i 2.97486 − 0.387535i 1.26780 − 0.223548i
2.14 0.742125 + 0.622716i −0.329326 − 1.70045i −0.184323 − 1.04535i 0.194712 − 0.232049i 0.814500 − 1.46703i −1.08691 1.48294 − 2.56853i −2.78309 + 1.12001i 0.289001 − 0.0509587i
2.15 1.37859 + 1.15678i 1.50503 + 0.857254i 0.215090 + 1.21983i 0.265173 − 0.316021i 1.08317 + 2.92279i −4.97178 0.685071 − 1.18658i 1.53023 + 2.58039i 0.731132 − 0.128918i
2.16 1.46425 + 1.22865i 1.16023 − 1.28602i 0.287148 + 1.62850i −1.02278 + 1.21890i 3.27895 − 0.457536i 0.784390 0.331042 − 0.573381i −0.307711 − 2.98418i −2.99522 + 0.528138i
2.17 1.78700 + 1.49947i −0.951155 + 1.44752i 0.597656 + 3.38948i 0.797436 − 0.950347i −3.87022 + 1.16048i −0.314886 −1.68164 + 2.91269i −1.19061 − 2.75363i 2.85003 − 0.502538i
2.18 1.90556 + 1.59895i −1.46656 − 0.921515i 0.727205 + 4.12418i −2.43276 + 2.89925i −1.32116 − 4.10097i 1.63251 −2.72112 + 4.71311i 1.30162 + 2.70292i −9.27152 + 1.63482i
32.1 −2.57424 − 0.936948i −1.03372 + 1.38976i 4.21676 + 3.53829i −1.12978 + 3.10405i 3.96317 − 2.60903i −0.922102 −4.80033 − 8.31442i −0.862851 − 2.87324i 5.81667 − 6.93203i
32.2 −2.26067 − 0.822816i 1.53353 + 0.805164i 2.90150 + 2.43465i 0.773557 − 2.12533i −2.80430 − 3.08202i −0.302529 −2.15031 − 3.72445i 1.70342 + 2.46949i −3.49751 + 4.16817i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.bd even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.bd.a yes 108
3.b odd 2 1 513.2.cd.a 108
9.c even 3 1 513.2.bo.a 108
9.d odd 6 1 171.2.x.a ✓ 108
19.f odd 18 1 171.2.x.a ✓ 108
57.j even 18 1 513.2.bo.a 108
171.bd even 18 1 inner 171.2.bd.a yes 108
171.be odd 18 1 513.2.cd.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.x.a ✓ 108 9.d odd 6 1
171.2.x.a ✓ 108 19.f odd 18 1
171.2.bd.a yes 108 1.a even 1 1 trivial
171.2.bd.a yes 108 171.bd even 18 1 inner
513.2.bo.a 108 9.c even 3 1
513.2.bo.a 108 57.j even 18 1
513.2.cd.a 108 3.b odd 2 1
513.2.cd.a 108 171.be odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(171, [\chi])\).