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Properties

Label 323.2.bb.a
Level $323$
Weight $2$
Character orbit 323.bb
Analytic conductor $2.579$
Analytic rank $0$
Dimension $672$
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] = [323,2,Mod(9,323)] 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("323.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(323, base_ring=CyclotomicField(72)) chi = DirichletCharacter(H, H._module([9, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 323 = 17 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 323.bb (of order \(72\), degree \(24\), 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: \(2.57916798529\)
Analytic rank: \(0\)
Dimension: \(672\)
Relative dimension: \(28\) over \(\Q(\zeta_{72})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 24 q^{2} - 24 q^{3} - 24 q^{5} - 48 q^{6} - 12 q^{7} - 12 q^{8} - 60 q^{9} - 24 q^{10} - 12 q^{12} - 24 q^{14} + 12 q^{15} - 48 q^{16} - 72 q^{17} + 48 q^{18} - 24 q^{19} - 48 q^{20} - 24 q^{23}+ \cdots - 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.19053 − 1.53383i −1.11035 + 0.350091i 1.76176 + 4.84039i −1.17473 + 1.28200i 2.96922 + 0.936193i 2.16426 + 0.284930i 2.18089 − 8.13920i −1.34715 + 0.943285i 4.53964 − 1.00641i
9.2 −2.12541 − 1.48823i 1.93797 − 0.611040i 1.61851 + 4.44682i 1.82578 − 1.99249i −5.02836 − 1.58544i 0.529382 + 0.0696945i 1.83480 − 6.84756i 0.924909 − 0.647629i −6.84581 + 1.51768i
9.3 −1.88515 − 1.32000i 1.53519 − 0.484043i 1.12736 + 3.09739i −1.46115 + 1.59456i −3.53300 − 1.11395i −2.23392 − 0.294101i 0.772044 − 2.88131i −0.334948 + 0.234533i 4.85929 − 1.07728i
9.4 −1.64621 − 1.15269i −2.33848 + 0.737321i 0.697279 + 1.91576i −2.05489 + 2.24252i 4.69954 + 1.48176i −4.99530 − 0.657644i 0.0201338 − 0.0751405i 2.46741 − 1.72770i 5.96772 − 1.32301i
9.5 −1.54128 − 1.07921i −1.26550 + 0.399012i 0.526792 + 1.44735i 1.90704 − 2.08117i 2.38111 + 0.750761i 2.25507 + 0.296886i −0.223896 + 0.835592i −1.01517 + 0.710827i −5.18530 + 1.14955i
9.6 −1.44243 − 1.01000i 0.00841580 − 0.00265349i 0.376467 + 1.03433i 1.46546 − 1.59927i −0.0148193 − 0.00467250i −2.18882 − 0.288163i −0.409849 + 1.52958i −2.45739 + 1.72068i −3.72909 + 0.826720i
9.7 −1.27910 − 0.895637i −0.249353 + 0.0786206i 0.149896 + 0.411835i −0.692391 + 0.755612i 0.389363 + 0.122766i 3.49035 + 0.459514i −0.631166 + 2.35554i −2.40146 + 1.68152i 1.56239 − 0.346374i
9.8 −0.996160 − 0.697518i 1.59930 − 0.504258i −0.178238 − 0.489706i −1.78614 + 1.94923i −1.94489 − 0.613221i 4.70388 + 0.619278i −0.793518 + 2.96145i −0.153968 + 0.107809i 3.13890 − 0.695878i
9.9 −0.937946 − 0.656757i −1.78597 + 0.563114i −0.235627 − 0.647381i −0.838166 + 0.914697i 2.04497 + 0.644777i −0.299445 − 0.0394227i −0.796872 + 2.97397i 0.415136 − 0.290681i 1.38689 − 0.307466i
9.10 −0.756034 − 0.529381i 3.09324 − 0.975295i −0.392697 − 1.07893i −0.0588444 + 0.0642174i −2.85490 − 0.900146i −2.70408 − 0.355999i −0.752023 + 2.80659i 6.15949 − 4.31292i 0.0784838 − 0.0173994i
9.11 −0.510179 − 0.357231i 2.21119 − 0.697187i −0.551372 − 1.51488i 2.43825 − 2.66089i −1.37716 − 0.434218i 1.48595 + 0.195629i −0.582257 + 2.17301i 1.94586 − 1.36250i −2.19450 + 0.486509i
9.12 −0.508263 − 0.355890i −3.02478 + 0.953710i −0.552366 − 1.51761i 1.83836 − 2.00622i 1.87680 + 0.591753i −0.389011 − 0.0512143i −0.580537 + 2.16659i 5.78229 − 4.04881i −1.64837 + 0.365434i
9.13 −0.252595 − 0.176869i 1.00909 − 0.318164i −0.651519 − 1.79003i −2.31792 + 2.52956i −0.311163 − 0.0981093i −2.86353 − 0.376990i −0.311650 + 1.16309i −1.54043 + 1.07862i 1.03289 − 0.228987i
9.14 −0.0361768 − 0.0253313i −1.65742 + 0.522584i −0.683373 − 1.87755i −0.496668 + 0.542018i 0.0731980 + 0.0230792i 0.0705367 + 0.00928633i −0.0456993 + 0.170552i 0.0165057 − 0.0115574i 0.0316978 − 0.00702724i
9.15 −0.00926327 − 0.00648621i 0.818709 − 0.258138i −0.683997 − 1.87927i 1.65303 − 1.80397i −0.00925826 − 0.00291912i 1.14584 + 0.150852i −0.0117069 + 0.0436907i −1.85381 + 1.29805i −0.0270134 + 0.00598873i
9.16 0.406248 + 0.284458i −2.53629 + 0.799691i −0.599919 − 1.64826i −2.53327 + 2.76458i −1.25784 − 0.396597i 3.41451 + 0.449529i 0.481862 − 1.79833i 3.33583 − 2.33577i −1.81554 + 0.402496i
9.17 0.643686 + 0.450714i −1.06646 + 0.336255i −0.472851 − 1.29915i 1.91383 − 2.08858i −0.838023 − 0.264228i −3.56783 − 0.469714i 0.687934 − 2.56741i −1.43318 + 1.00352i 2.17326 − 0.481799i
9.18 0.678793 + 0.475296i 2.58560 − 0.815236i −0.449187 − 1.23413i −1.52668 + 1.66607i 2.14256 + 0.675548i 2.59472 + 0.341601i 0.710615 − 2.65205i 3.56325 − 2.49501i −1.82818 + 0.405297i
9.19 0.834216 + 0.584124i 0.680336 − 0.214509i −0.329325 − 0.904813i 0.753017 − 0.821774i 0.692848 + 0.218454i 1.05324 + 0.138662i 0.780953 − 2.91455i −2.04061 + 1.42885i 1.10820 − 0.245682i
9.20 1.07612 + 0.753507i 2.36261 − 0.744929i −0.0937804 − 0.257659i 0.483471 − 0.527616i 3.10376 + 0.978613i −3.58518 − 0.471998i 0.773250 − 2.88581i 2.56957 − 1.79923i 0.917834 − 0.203479i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 9.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner
19.e even 9 1 inner
323.bb even 72 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 323.2.bb.a ✓ 672
17.d even 8 1 inner 323.2.bb.a ✓ 672
19.e even 9 1 inner 323.2.bb.a ✓ 672
323.bb even 72 1 inner 323.2.bb.a ✓ 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
323.2.bb.a ✓ 672 1.a even 1 1 trivial
323.2.bb.a ✓ 672 17.d even 8 1 inner
323.2.bb.a ✓ 672 19.e even 9 1 inner
323.2.bb.a ✓ 672 323.bb even 72 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(323, [\chi])\).