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Properties

Label 91.4.k.a
Level $91$
Weight $4$
Character orbit 91.k
Analytic conductor $5.369$
Analytic rank $0$
Dimension $52$
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] = [91,4,Mod(4,91)] 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("91.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 91.k (of order \(6\), degree \(2\), 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: \(5.36917381052\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 7 q^{3} - 194 q^{4} - 6 q^{6} - 39 q^{7} - 213 q^{9} - 23 q^{10} - 45 q^{11} - 112 q^{12} - 90 q^{13} - 116 q^{14} - 27 q^{15} + 598 q^{16} + 274 q^{17} + 321 q^{18} + 9 q^{19} - 138 q^{20} - 3 q^{21}+ \cdots + 2220 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 5.26157i 2.50015 + 4.33039i −19.6841 16.4599 − 9.50310i 22.7847 − 13.1547i 9.68237 − 15.7877i 61.4765i 0.998460 − 1.72938i −50.0012 − 86.6046i
4.2 − 5.10253i −0.690001 − 1.19512i −18.0358 −7.44591 + 4.29890i −6.09812 + 3.52075i −2.36742 + 18.3683i 51.2082i 12.5478 − 21.7334i 21.9353 + 37.9930i
4.3 − 4.70775i −3.69858 − 6.40612i −14.1629 11.3521 − 6.55415i −30.1584 + 17.4120i −18.2502 − 3.15133i 29.0133i −13.8590 + 24.0044i −30.8553 − 53.4429i
4.4 − 4.61101i 4.53158 + 7.84892i −13.2614 −17.0962 + 9.87051i 36.1915 − 20.8952i −14.8274 − 11.0973i 24.2605i −27.5704 + 47.7533i 45.5130 + 78.8309i
4.5 − 4.19790i −3.82753 − 6.62948i −9.62239 −8.84574 + 5.10709i −27.8299 + 16.0676i 16.2597 − 8.86685i 6.81063i −15.8000 + 27.3664i 21.4391 + 37.1335i
4.6 − 3.78836i 2.56525 + 4.44314i −6.35168 1.83377 − 1.05873i 16.8322 − 9.71808i 11.9318 + 14.1645i − 6.24442i 0.339011 − 0.587184i −4.01084 − 6.94698i
4.7 − 3.23981i 0.296826 + 0.514118i −2.49636 2.34750 − 1.35533i 1.66564 − 0.961660i −16.1084 − 9.13884i − 17.8307i 13.3238 − 23.0775i −4.39102 − 7.60546i
4.8 − 2.39875i −2.67138 − 4.62697i 2.24602 13.1062 − 7.56686i −11.0989 + 6.40797i 17.6553 + 5.59389i − 24.5776i −0.772575 + 1.33814i −18.1510 − 31.4384i
4.9 − 2.25442i 0.353738 + 0.612693i 2.91759 −8.95738 + 5.17154i 1.38127 − 0.797474i 7.58504 − 16.8958i − 24.6128i 13.2497 − 22.9492i 11.6588 + 20.1937i
4.10 − 1.73049i 4.84406 + 8.39017i 5.00540 14.0271 − 8.09857i 14.5191 − 8.38260i −17.8210 + 5.04096i − 22.5057i −33.4299 + 57.9023i −14.0145 − 24.2738i
4.11 − 1.52758i −3.14724 − 5.45118i 5.66650 −14.6324 + 8.44802i −8.32712 + 4.80766i −17.0793 + 7.16228i − 20.8767i −6.31025 + 10.9297i 12.9050 + 22.3522i
4.12 − 0.563215i 2.99433 + 5.18634i 7.68279 −12.8336 + 7.40950i 2.92103 − 1.68646i 5.70687 + 17.6191i − 8.83279i −4.43208 + 7.67659i 4.17315 + 7.22810i
4.13 − 0.280486i 3.19326 + 5.53088i 7.92133 3.50341 − 2.02269i 1.55133 − 0.895664i 12.2772 − 13.8661i − 4.46571i −6.89377 + 11.9404i −0.567337 − 0.982656i
4.14 − 0.267388i −1.33089 − 2.30517i 7.92850 8.34060 − 4.81545i −0.616375 + 0.355864i −6.91130 + 17.1824i − 4.25910i 9.95747 − 17.2468i −1.28760 − 2.23018i
4.15 0.488605i −4.96911 − 8.60676i 7.76127 5.68763 − 3.28375i 4.20530 − 2.42793i −4.38352 − 17.9940i 7.70103i −35.8842 + 62.1532i 1.60446 + 2.77900i
4.16 1.29936i 0.114368 + 0.198092i 6.31166 9.98475 − 5.76470i −0.257393 + 0.148606i −10.8866 − 14.9827i 18.5960i 13.4738 − 23.3374i 7.49042 + 12.9738i
4.17 1.34963i −2.25173 − 3.90011i 6.17849 −9.82721 + 5.67374i 5.26373 − 3.03901i 18.4673 + 1.39904i 19.1358i 3.35940 − 5.81866i −7.65747 − 13.2631i
4.18 2.42240i 2.28343 + 3.95502i 2.13196 −5.82815 + 3.36488i −9.58065 + 5.53139i −15.8397 + 9.59710i 24.5437i 3.07187 − 5.32064i −8.15110 − 14.1181i
4.19 3.14185i 1.72231 + 2.98312i −1.87124 12.8854 − 7.43937i −9.37253 + 5.41123i 14.4041 + 11.6414i 19.2557i 7.56732 − 13.1070i 23.3734 + 40.4839i
4.20 3.15774i 4.77658 + 8.27327i −1.97130 −6.28974 + 3.63139i −26.1248 + 15.0832i 6.90147 − 17.1863i 19.0371i −32.1314 + 55.6532i −11.4670 − 19.8614i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.26
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.4.k.a ✓ 52
7.c even 3 1 91.4.u.a yes 52
13.e even 6 1 91.4.u.a yes 52
91.k even 6 1 inner 91.4.k.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.4.k.a ✓ 52 1.a even 1 1 trivial
91.4.k.a ✓ 52 91.k even 6 1 inner
91.4.u.a yes 52 7.c even 3 1
91.4.u.a yes 52 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(91, [\chi])\).