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Properties

Label 740.2.ca.c
Level $740$
Weight $2$
Character orbit 740.ca
Analytic conductor $5.909$
Analytic rank $0$
Dimension $1296$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [740,2,Mod(19,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([18, 18, 35])) 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("740.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.ca (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1296,0,0,-24,-24] 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(1296\)
Relative dimension: \(108\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q - 24 q^{4} - 24 q^{5} - 24 q^{6} - 48 q^{9} - 6 q^{10} - 24 q^{14} - 24 q^{16} - 48 q^{21} + 60 q^{24} - 12 q^{25} - 72 q^{26} - 48 q^{29} + 48 q^{30} - 60 q^{34} - 36 q^{40} - 96 q^{41} + 24 q^{44}+ \cdots - 108 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41346 + 0.0461680i −0.168670 − 0.201013i 1.99574 − 0.130513i −2.18823 − 0.460034i 0.247688 + 0.276336i 3.34899 + 1.21893i −2.81487 + 0.276614i 0.508988 − 2.88661i 3.11422 + 0.549213i
19.2 −1.41316 + 0.0545590i 1.13579 + 1.35359i 1.99405 − 0.154201i −2.18433 + 0.478216i −1.67891 − 1.85087i −3.30532 − 1.20304i −2.80950 + 0.326704i −0.0212246 + 0.120371i 3.06072 − 0.794971i
19.3 −1.40715 − 0.141126i −0.164873 − 0.196488i 1.96017 + 0.397172i 1.90521 + 1.17055i 0.204273 + 0.299757i −0.400434 − 0.145746i −2.70221 − 0.835512i 0.509520 − 2.88963i −2.51573 − 1.91601i
19.4 −1.40117 + 0.191663i 1.13579 + 1.35359i 1.92653 − 0.537102i 1.77040 − 1.36590i −1.85087 − 1.67891i −3.30532 − 1.20304i −2.59645 + 1.12181i −0.0212246 + 0.120371i −2.21883 + 2.25318i
19.5 −1.40000 + 0.199978i −0.168670 − 0.201013i 1.92002 − 0.559940i 1.05416 − 1.97199i 0.276336 + 0.247688i 3.34899 + 1.21893i −2.57605 + 1.16788i 0.508988 − 2.88661i −1.08148 + 2.97160i
19.6 −1.39227 − 0.248156i −1.68704 − 2.01053i 1.87684 + 0.691002i 1.82794 + 1.28787i 1.84989 + 3.21785i 2.43467 + 0.886149i −2.44159 − 1.42781i −0.675201 + 3.82926i −2.22540 − 2.24669i
19.7 −1.39011 − 0.259977i −1.82603 − 2.17618i 1.86482 + 0.722794i −1.96686 − 1.06371i 1.97263 + 3.49986i −0.634164 − 0.230817i −2.40440 − 1.48958i −0.880426 + 4.99314i 2.45761 + 1.99001i
19.8 −1.37153 − 0.344826i 1.41761 + 1.68944i 1.76219 + 0.945880i 0.00150107 + 2.23607i −1.36173 − 2.80595i −0.314336 − 0.114409i −2.09073 − 1.90495i −0.323650 + 1.83551i 0.768996 − 3.06735i
19.9 −1.36127 + 0.383332i −0.164873 − 0.196488i 1.70611 − 1.04364i −0.327953 + 2.21189i 0.299757 + 0.204273i −0.400434 − 0.145746i −1.92242 + 2.07468i 0.509520 − 2.88963i −0.401453 − 3.13669i
19.10 −1.35840 − 0.393370i 2.11811 + 2.52427i 1.69052 + 1.06871i 2.14020 − 0.647707i −1.88428 − 4.26218i 3.08263 + 1.12199i −1.87601 − 2.11674i −1.36458 + 7.73894i −3.16205 + 0.0379558i
19.11 −1.32803 + 0.486152i −1.68704 − 2.01053i 1.52731 − 1.29125i −0.188411 + 2.22812i 3.21785 + 1.84989i 2.43467 + 0.886149i −1.40057 + 2.45731i −0.675201 + 3.82926i −0.832987 − 3.05060i
19.12 −1.32385 + 0.497418i −1.82603 − 2.17618i 1.50515 − 1.31701i 0.449423 − 2.19044i 3.49986 + 1.97263i −0.634164 − 0.230817i −1.33749 + 2.49221i −0.880426 + 4.99314i 0.494595 + 3.12336i
19.13 −1.31553 − 0.519022i −0.177263 − 0.211254i 1.46123 + 1.36558i −0.256532 − 2.22130i 0.123549 + 0.369915i −3.01073 − 1.09582i −1.21353 − 2.55487i 0.507738 − 2.87953i −0.815431 + 3.05534i
19.14 −1.29081 + 0.577751i 1.41761 + 1.68944i 1.33241 − 1.49154i 1.71196 + 1.43847i −2.80595 − 1.36173i −0.314336 − 0.114409i −0.858151 + 2.69510i −0.323650 + 1.83551i −3.04090 − 0.867705i
19.15 −1.28499 − 0.590583i 0.817907 + 0.974744i 1.30242 + 1.51779i −1.70515 + 1.44654i −0.475339 − 1.73558i 2.80859 + 1.02224i −0.777222 − 2.71955i 0.239791 − 1.35992i 3.04541 − 0.851757i
19.16 −1.26946 + 0.623278i 2.11811 + 2.52427i 1.22305 − 1.58245i −1.87187 + 1.22315i −4.26218 − 1.88428i 3.08263 + 1.12199i −0.566305 + 2.77115i −1.36458 + 7.73894i 1.61390 − 2.71944i
19.17 −1.26182 − 0.638596i −1.35854 − 1.61904i 1.18439 + 1.61159i 1.66648 − 1.49093i 0.680319 + 2.91050i 1.40980 + 0.513124i −0.465337 − 2.78989i −0.254727 + 1.44463i −3.05489 + 0.817079i
19.18 −1.25279 − 0.656133i 1.18918 + 1.41720i 1.13898 + 1.64400i −0.867314 − 2.06101i −0.559915 − 2.55572i 1.79696 + 0.654041i −0.348221 − 2.80691i −0.0733852 + 0.416188i −0.265734 + 3.15109i
19.19 −1.20542 + 0.739576i −0.177263 − 0.211254i 0.906054 − 1.78299i −1.53672 − 1.62434i 0.369915 + 0.123549i −3.01073 − 1.09582i 0.226488 + 2.81934i 0.507738 − 2.87953i 3.05371 + 0.821483i
19.20 −1.20220 − 0.744796i −0.724395 − 0.863300i 0.890559 + 1.79078i −0.382793 + 2.20306i 0.227884 + 1.57738i −2.34929 − 0.855071i 0.263139 − 2.81616i 0.300405 − 1.70368i 2.10102 − 2.36341i
See next 80 embeddings (of 1296 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
37.i odd 36 1 inner
148.q even 36 1 inner
185.ba odd 36 1 inner
740.ca even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.ca.c ✓ 1296
4.b odd 2 1 inner 740.2.ca.c ✓ 1296
5.b even 2 1 inner 740.2.ca.c ✓ 1296
20.d odd 2 1 inner 740.2.ca.c ✓ 1296
37.i odd 36 1 inner 740.2.ca.c ✓ 1296
148.q even 36 1 inner 740.2.ca.c ✓ 1296
185.ba odd 36 1 inner 740.2.ca.c ✓ 1296
740.ca even 36 1 inner 740.2.ca.c ✓ 1296
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.ca.c ✓ 1296 1.a even 1 1 trivial
740.2.ca.c ✓ 1296 4.b odd 2 1 inner
740.2.ca.c ✓ 1296 5.b even 2 1 inner
740.2.ca.c ✓ 1296 20.d odd 2 1 inner
740.2.ca.c ✓ 1296 37.i odd 36 1 inner
740.2.ca.c ✓ 1296 148.q even 36 1 inner
740.2.ca.c ✓ 1296 185.ba odd 36 1 inner
740.2.ca.c ✓ 1296 740.ca even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\):

\( T_{3}^{648} + 12 T_{3}^{646} + 114 T_{3}^{644} - 11434 T_{3}^{642} - 140337 T_{3}^{640} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
\( T_{13}^{648} - 2265 T_{13}^{644} - 204624 T_{13}^{642} + 262755 T_{13}^{640} + 519053112 T_{13}^{638} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display