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Properties

Label 598.2.h.c
Level $598$
Weight $2$
Character orbit 598.h
Analytic conductor $4.775$
Analytic rank $0$
Dimension $28$
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] = [598,2,Mod(231,598)] 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("598.231"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(598, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 598 = 2 \cdot 13 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 598.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.77505404087\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{3} + 14 q^{4} - 6 q^{9} + 8 q^{10} - 8 q^{12} + 2 q^{13} + 4 q^{14} + 12 q^{15} - 14 q^{16} - 2 q^{17} - 12 q^{19} - 12 q^{22} + 14 q^{23} - 28 q^{25} - 10 q^{26} + 20 q^{27} - 14 q^{29} + 8 q^{30}+ \cdots + 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
231.1 −0.866025 + 0.500000i −1.46307 − 2.53412i 0.500000 − 0.866025i 1.30968i 2.53412 + 1.46307i −0.0644396 − 0.0372042i 1.00000i −2.78116 + 4.81712i −0.654838 − 1.13421i
231.2 −0.866025 + 0.500000i −1.29716 − 2.24675i 0.500000 − 0.866025i − 1.63382i 2.24675 + 1.29716i −2.94907 − 1.70264i 1.00000i −1.86526 + 3.23073i 0.816909 + 1.41493i
231.3 −0.866025 + 0.500000i −0.314251 − 0.544299i 0.500000 − 0.866025i − 4.29224i 0.544299 + 0.314251i 0.281733 + 0.162659i 1.00000i 1.30249 − 2.25598i 2.14612 + 3.71719i
231.4 −0.866025 + 0.500000i 0.0928940 + 0.160897i 0.500000 − 0.866025i − 0.931614i −0.160897 − 0.0928940i 3.98172 + 2.29885i 1.00000i 1.48274 − 2.56818i 0.465807 + 0.806802i
231.5 −0.866025 + 0.500000i 0.163491 + 0.283175i 0.500000 − 0.866025i 2.43918i −0.283175 − 0.163491i −2.12792 − 1.22855i 1.00000i 1.44654 − 2.50548i −1.21959 − 2.11239i
231.6 −0.866025 + 0.500000i 0.467505 + 0.809743i 0.500000 − 0.866025i 0.444674i −0.809743 − 0.467505i −3.03452 − 1.75198i 1.00000i 1.06288 − 1.84096i −0.222337 − 0.385099i
231.7 −0.866025 + 0.500000i 1.35060 + 2.33930i 0.500000 − 0.866025i − 1.33585i −2.33930 − 1.35060i 3.04647 + 1.75888i 1.00000i −2.14822 + 3.72083i 0.667927 + 1.15688i
231.8 0.866025 − 0.500000i −1.34433 − 2.32845i 0.500000 − 0.866025i 3.77055i −2.32845 − 1.34433i −3.89872 − 2.25093i − 1.00000i −2.11445 + 3.66234i 1.88528 + 3.26540i
231.9 0.866025 − 0.500000i −1.03572 − 1.79392i 0.500000 − 0.866025i 1.91535i −1.79392 − 1.03572i 1.82221 + 1.05206i − 1.00000i −0.645424 + 1.11791i 0.957675 + 1.65874i
231.10 0.866025 − 0.500000i −0.860950 − 1.49121i 0.500000 − 0.866025i − 3.67067i −1.49121 − 0.860950i 0.545394 + 0.314883i − 1.00000i 0.0175311 − 0.0303648i −1.83533 − 3.17889i
231.11 0.866025 − 0.500000i −0.0779423 − 0.135000i 0.500000 − 0.866025i 3.04588i −0.135000 − 0.0779423i 3.79453 + 2.19078i − 1.00000i 1.48785 − 2.57703i 1.52294 + 2.63781i
231.12 0.866025 − 0.500000i 0.219223 + 0.379705i 0.500000 − 0.866025i − 0.411787i 0.379705 + 0.219223i −3.89325 − 2.24777i − 1.00000i 1.40388 − 2.43160i −0.205894 − 0.356618i
231.13 0.866025 − 0.500000i 0.804621 + 1.39364i 0.500000 − 0.866025i − 2.70952i 1.39364 + 0.804621i 2.07244 + 1.19652i − 1.00000i 0.205169 − 0.355363i −1.35476 − 2.34652i
231.14 0.866025 − 0.500000i 1.29510 + 2.24317i 0.500000 − 0.866025i 2.06019i 2.24317 + 1.29510i 0.423414 + 0.244458i − 1.00000i −1.85455 + 3.21218i 1.03009 + 1.78418i
277.1 −0.866025 − 0.500000i −1.46307 + 2.53412i 0.500000 + 0.866025i − 1.30968i 2.53412 − 1.46307i −0.0644396 + 0.0372042i − 1.00000i −2.78116 − 4.81712i −0.654838 + 1.13421i
277.2 −0.866025 − 0.500000i −1.29716 + 2.24675i 0.500000 + 0.866025i 1.63382i 2.24675 − 1.29716i −2.94907 + 1.70264i − 1.00000i −1.86526 − 3.23073i 0.816909 − 1.41493i
277.3 −0.866025 − 0.500000i −0.314251 + 0.544299i 0.500000 + 0.866025i 4.29224i 0.544299 − 0.314251i 0.281733 − 0.162659i − 1.00000i 1.30249 + 2.25598i 2.14612 − 3.71719i
277.4 −0.866025 − 0.500000i 0.0928940 − 0.160897i 0.500000 + 0.866025i 0.931614i −0.160897 + 0.0928940i 3.98172 − 2.29885i − 1.00000i 1.48274 + 2.56818i 0.465807 − 0.806802i
277.5 −0.866025 − 0.500000i 0.163491 − 0.283175i 0.500000 + 0.866025i − 2.43918i −0.283175 + 0.163491i −2.12792 + 1.22855i − 1.00000i 1.44654 + 2.50548i −1.21959 + 2.11239i
277.6 −0.866025 − 0.500000i 0.467505 − 0.809743i 0.500000 + 0.866025i − 0.444674i −0.809743 + 0.467505i −3.03452 + 1.75198i − 1.00000i 1.06288 + 1.84096i −0.222337 + 0.385099i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 231.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 598.2.h.c ✓ 28
13.e even 6 1 inner 598.2.h.c ✓ 28
13.f odd 12 1 7774.2.a.bl 14
13.f odd 12 1 7774.2.a.bm 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
598.2.h.c ✓ 28 1.a even 1 1 trivial
598.2.h.c ✓ 28 13.e even 6 1 inner
7774.2.a.bl 14 13.f odd 12 1
7774.2.a.bm 14 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 4 T_{3}^{27} + 32 T_{3}^{26} + 84 T_{3}^{25} + 476 T_{3}^{24} + 1046 T_{3}^{23} + 4636 T_{3}^{22} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(598, [\chi])\). Copy content Toggle raw display