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Properties

Label 340.2.s.c
Level $340$
Weight $2$
Character orbit 340.s
Analytic conductor $2.715$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(183,340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("340.183"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{2} - 8 q^{5} - 8 q^{8} + 96 q^{9} - 10 q^{10} - 16 q^{12} - 8 q^{13} - 16 q^{14} - 8 q^{17} - 24 q^{18} + 18 q^{20} - 16 q^{21} - 12 q^{22} - 8 q^{24} - 52 q^{28} - 8 q^{29} - 24 q^{30} - 36 q^{32}+ \cdots - 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
183.1 −1.40654 − 0.147117i 1.04449 1.95671 + 0.413852i −1.92909 − 1.13075i −1.46912 − 0.153662i −0.199034 −2.69131 − 0.869965i −1.90904 2.54700 + 1.87425i
183.2 −1.39310 − 0.243459i −2.68200 1.88146 + 0.678324i 1.60549 + 1.55641i 3.73629 + 0.652955i 0.308840 −2.45591 − 1.40303i 4.19311 −1.85768 − 2.55910i
183.3 −1.38522 − 0.284916i 2.59680 1.83765 + 0.789341i −1.25353 + 1.85166i −3.59712 − 0.739869i −3.24046 −2.32064 − 1.61698i 3.74335 2.26398 − 2.20780i
183.4 −1.37910 + 0.313204i −0.891655 1.80381 − 0.863875i 1.45750 − 1.69578i 1.22968 − 0.279269i −4.80451 −2.21705 + 1.75633i −2.20495 −1.47891 + 2.79514i
183.5 −1.35957 − 0.389320i −0.791978 1.69686 + 1.05862i 0.707346 − 2.12124i 1.07675 + 0.308333i 3.51089 −1.89486 − 2.09988i −2.37277 −1.78753 + 2.60859i
183.6 −1.33826 + 0.457226i 1.47574 1.58189 − 1.22377i 1.94557 + 1.10216i −1.97493 + 0.674747i 2.84130 −1.55744 + 2.36101i −0.822184 −3.10762 − 0.585420i
183.7 −1.29346 + 0.571804i −3.35313 1.34608 − 1.47921i −1.17219 − 1.90420i 4.33714 − 1.91733i 2.29913 −0.895282 + 2.68300i 8.24346 2.60501 + 1.79274i
183.8 −1.28007 − 0.601190i 2.87659 1.27714 + 1.53913i 1.99672 − 1.00654i −3.68223 − 1.72938i 0.212186 −0.709520 − 2.73799i 5.27477 −3.16105 + 0.0880265i
183.9 −1.22566 + 0.705519i −0.136969 1.00449 − 1.72945i −2.15875 + 0.582914i 0.167877 − 0.0966342i 3.12506 −0.0109971 + 2.82841i −2.98124 2.23464 − 2.23750i
183.10 −1.17399 − 0.788511i −2.30159 0.756500 + 1.85141i −2.13907 − 0.651458i 2.70204 + 1.81483i −3.97452 0.571733 − 2.77004i 2.29731 1.99756 + 2.45148i
183.11 −1.05432 + 0.942559i −1.98485 0.223164 − 1.98751i −0.156098 + 2.23061i 2.09266 − 1.87084i −2.49962 1.63806 + 2.30581i 0.939636 −1.93791 − 2.49890i
183.12 −0.929802 + 1.06558i −0.834977 −0.270936 − 1.98156i 1.90939 − 1.16371i 0.776363 − 0.889737i 0.833757 2.36344 + 1.55376i −2.30281 −0.535328 + 3.11664i
183.13 −0.906765 + 1.08525i 3.22834 −0.355554 − 1.96814i −0.869778 + 2.05997i −2.92734 + 3.50357i 1.65546 2.45834 + 1.39878i 7.42216 −1.44691 − 2.81184i
183.14 −0.887474 + 1.10109i 1.49637 −0.424780 − 1.95437i −1.97804 − 1.04277i −1.32799 + 1.64763i −3.92725 2.52891 + 1.26673i −0.760875 2.90363 − 1.25256i
183.15 −0.788511 − 1.17399i 2.30159 −0.756500 + 1.85141i −2.13907 − 0.651458i −1.81483 − 2.70204i 3.97452 2.77004 − 0.571733i 2.29731 0.921873 + 3.02492i
183.16 −0.601190 − 1.28007i −2.87659 −1.27714 + 1.53913i 1.99672 − 1.00654i 1.72938 + 3.68223i −0.212186 2.73799 + 0.709520i 5.27477 −2.48884 − 1.95081i
183.17 −0.438467 + 1.34452i 1.86014 −1.61549 − 1.17906i 2.23007 − 0.163623i −0.815609 + 2.50100i −0.945309 2.29361 − 1.65509i 0.460113 −0.757818 + 3.07013i
183.18 −0.396151 + 1.35759i −1.53299 −1.68613 − 1.07563i −1.38805 − 1.75309i 0.607296 − 2.08118i 0.0810654 2.12823 − 1.86297i −0.649946 2.92986 − 1.18991i
183.19 −0.389320 − 1.35957i 0.791978 −1.69686 + 1.05862i 0.707346 − 2.12124i −0.308333 − 1.07675i −3.51089 2.09988 + 1.89486i −2.37277 −3.15936 − 0.135845i
183.20 −0.302726 + 1.38143i −2.00060 −1.81671 − 0.836391i 1.34314 + 1.78773i 0.605633 − 2.76369i 4.80889 1.70538 − 2.25647i 1.00240 −2.87623 + 1.31427i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 183.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
85.f odd 4 1 inner
340.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 340.2.s.c yes 96
4.b odd 2 1 inner 340.2.s.c yes 96
5.c odd 4 1 340.2.i.c ✓ 96
17.c even 4 1 340.2.i.c ✓ 96
20.e even 4 1 340.2.i.c ✓ 96
68.f odd 4 1 340.2.i.c ✓ 96
85.f odd 4 1 inner 340.2.s.c yes 96
340.s even 4 1 inner 340.2.s.c yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.i.c ✓ 96 5.c odd 4 1
340.2.i.c ✓ 96 17.c even 4 1
340.2.i.c ✓ 96 20.e even 4 1
340.2.i.c ✓ 96 68.f odd 4 1
340.2.s.c yes 96 1.a even 1 1 trivial
340.2.s.c yes 96 4.b odd 2 1 inner
340.2.s.c yes 96 85.f odd 4 1 inner
340.2.s.c yes 96 340.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(340, [\chi])\):

\( T_{3}^{48} - 96 T_{3}^{46} + 4286 T_{3}^{44} - 118276 T_{3}^{42} + 2261761 T_{3}^{40} - 31847908 T_{3}^{38} + \cdots + 186809600 \) Copy content Toggle raw display
\( T_{13}^{48} + 4 T_{13}^{47} + 8 T_{13}^{46} - 168 T_{13}^{45} + 3010 T_{13}^{44} + \cdots + 839905316454400 \) Copy content Toggle raw display