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Properties

Label 615.2.bo.a
Level $615$
Weight $2$
Character orbit 615.bo
Analytic conductor $4.911$
Analytic rank $0$
Dimension $640$
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] = [615,2,Mod(23,615)] 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("615.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 15, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 615.bo (of order \(20\), degree \(8\), 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: \(4.91079972431\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(80\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 20 q^{6} - 20 q^{7} + 12 q^{10} - 40 q^{12} - 20 q^{13} - 10 q^{15} + 112 q^{16} - 18 q^{18} + 20 q^{21} - 60 q^{22} - 28 q^{25} + 20 q^{28} - 10 q^{30} - 24 q^{31} + 6 q^{33} + 60 q^{36} - 100 q^{37}+ \cdots + 140 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.26162 + 2.47607i 0.599153 + 1.62512i −3.36367 − 4.62970i 0.877195 − 2.05682i −4.77982 − 0.566741i −0.860770 + 0.438584i 10.2177 − 1.61832i −2.28203 + 1.94739i 3.98616 + 4.76693i
23.2 −1.24152 + 2.43661i −1.49476 − 0.875035i −3.22014 − 4.43214i 2.04725 + 0.899307i 3.98789 − 2.55578i −2.72413 + 1.38801i 9.39525 − 1.48806i 1.46863 + 2.61594i −4.73296 + 3.87186i
23.3 −1.23125 + 2.41646i 0.975068 − 1.43152i −3.14773 − 4.33248i −2.07946 − 0.822087i 2.25865 + 4.11876i −0.261766 + 0.133377i 8.98756 − 1.42349i −1.09849 − 2.79165i 4.54687 − 4.01275i
23.4 −1.19491 + 2.34514i 1.63566 − 0.569737i −2.89632 − 3.98644i 1.74827 + 1.39411i −0.618358 + 4.51666i 1.54409 − 0.786754i 7.61041 − 1.20537i 2.35080 − 1.86380i −5.35843 + 2.43410i
23.5 −1.13755 + 2.23257i 0.882871 + 1.49015i −2.51477 − 3.46128i −0.126497 + 2.23249i −4.33116 + 0.275951i 1.74312 − 0.888162i 5.63856 − 0.893061i −1.44108 + 2.63121i −4.84028 − 2.82198i
23.6 −1.08328 + 2.12605i −1.18712 − 1.26125i −2.17104 − 2.98818i −1.78353 + 1.34871i 3.96746 − 1.15761i 1.04689 − 0.533417i 3.99136 − 0.632170i −0.181480 + 2.99451i −0.935371 − 5.25291i
23.7 −1.07003 + 2.10006i −1.72816 + 0.116011i −2.08970 − 2.87622i −2.13592 + 0.661711i 1.60556 − 3.75337i 1.16382 − 0.592995i 3.62041 − 0.573417i 2.97308 − 0.400973i 0.895870 − 5.19360i
23.8 −1.06689 + 2.09389i −0.0736696 − 1.73048i −2.07056 − 2.84988i 1.69178 − 1.46215i 3.70205 + 1.69198i 3.67144 − 1.87069i 3.53423 − 0.559766i −2.98915 + 0.254968i 1.25663 + 5.10237i
23.9 −1.06411 + 2.08844i −1.26219 + 1.18612i −2.05366 − 2.82662i 1.85689 + 1.24577i −1.13404 − 3.89816i 2.80390 − 1.42866i 3.45845 − 0.547764i 0.186223 − 2.99421i −4.57766 + 2.55236i
23.10 −1.04549 + 2.05190i −1.49686 + 0.871441i −1.94166 − 2.67246i −1.16355 − 1.90949i −0.223150 − 3.98249i −4.24639 + 2.16364i 2.96451 − 0.469532i 1.48118 − 2.60885i 5.13456 − 0.391133i
23.11 −0.997129 + 1.95698i 1.67044 + 0.457859i −1.65992 − 2.28468i −1.73981 − 1.40465i −2.56166 + 2.81246i 3.00112 − 1.52915i 1.78757 − 0.283124i 2.58073 + 1.52965i 4.48369 − 2.00416i
23.12 −0.966983 + 1.89781i 0.412539 − 1.68220i −1.49106 − 2.05226i −0.189151 + 2.22805i 2.79359 + 2.40958i −2.87938 + 1.46712i 1.12916 − 0.178841i −2.65962 − 1.38795i −4.04552 − 2.51346i
23.13 −0.954632 + 1.87357i 0.0297901 + 1.73179i −1.42337 − 1.95910i −2.20663 − 0.361635i −3.27308 − 1.59741i 0.213222 − 0.108642i 0.875578 − 0.138678i −2.99823 + 0.103181i 2.78407 − 3.78905i
23.14 −0.953316 + 1.87099i −0.611053 − 1.62068i −1.41621 − 1.94925i −0.427900 − 2.19474i 3.61480 + 0.401749i −3.12469 + 1.59211i 0.849110 − 0.134486i −2.25323 + 1.98065i 4.51426 + 1.29169i
23.15 −0.934450 + 1.83396i 1.61045 − 0.637538i −1.31465 − 1.80946i 1.99584 − 1.00827i −0.335664 + 3.54925i −2.23530 + 1.13894i 0.481019 − 0.0761859i 2.18709 − 2.05344i −0.0158840 + 4.60248i
23.16 −0.898355 + 1.76312i −0.741781 + 1.56517i −1.12598 − 1.54979i 1.81462 − 1.30658i −2.09320 − 2.71393i −0.296419 + 0.151033i −0.164876 + 0.0261139i −1.89952 − 2.32203i 0.673480 + 4.37317i
23.17 −0.879450 + 1.72602i 1.21926 + 1.23021i −1.03013 − 1.41786i 2.04074 + 0.913987i −3.19563 + 1.02256i −3.75025 + 1.91085i −0.473415 + 0.0749816i −0.0268155 + 2.99988i −3.37229 + 2.71855i
23.18 −0.725889 + 1.42464i 1.72600 − 0.144609i −0.327108 − 0.450225i −0.968845 + 2.01528i −1.04687 + 2.56390i 1.59083 − 0.810568i −2.27960 + 0.361052i 2.95818 − 0.499192i −2.16776 − 2.84312i
23.19 −0.703332 + 1.38037i −1.70138 + 0.324500i −0.235166 − 0.323679i 1.00489 + 1.99755i 0.748708 − 2.57676i −2.25890 + 1.15097i −2.44810 + 0.387741i 2.78940 − 1.10420i −3.46412 − 0.0178164i
23.20 −0.688241 + 1.35075i 1.19571 − 1.25311i −0.175277 − 0.241248i −0.0206393 − 2.23597i 0.869700 + 2.47754i −0.0527459 + 0.0268754i −2.54814 + 0.403585i −0.140564 − 2.99671i 3.03444 + 1.51101i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 23.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
41.f even 10 1 inner
123.l odd 10 1 inner
205.v odd 20 1 inner
615.bo even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 615.2.bo.a ✓ 640
3.b odd 2 1 inner 615.2.bo.a ✓ 640
5.c odd 4 1 inner 615.2.bo.a ✓ 640
15.e even 4 1 inner 615.2.bo.a ✓ 640
41.f even 10 1 inner 615.2.bo.a ✓ 640
123.l odd 10 1 inner 615.2.bo.a ✓ 640
205.v odd 20 1 inner 615.2.bo.a ✓ 640
615.bo even 20 1 inner 615.2.bo.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
615.2.bo.a ✓ 640 1.a even 1 1 trivial
615.2.bo.a ✓ 640 3.b odd 2 1 inner
615.2.bo.a ✓ 640 5.c odd 4 1 inner
615.2.bo.a ✓ 640 15.e even 4 1 inner
615.2.bo.a ✓ 640 41.f even 10 1 inner
615.2.bo.a ✓ 640 123.l odd 10 1 inner
615.2.bo.a ✓ 640 205.v odd 20 1 inner
615.2.bo.a ✓ 640 615.bo even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(615, [\chi])\).