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Properties

Label 945.2.ca.a
Level $945$
Weight $2$
Character orbit 945.ca
Analytic conductor $7.546$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(368,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.368");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.ca (of order \(12\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 6 q^{2} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 6 q^{2} - 2 q^{7} - 4 q^{10} - 4 q^{13} + 68 q^{16} + 18 q^{17} + 12 q^{20} + 4 q^{22} - 4 q^{25} - 4 q^{28} + 4 q^{31} + 90 q^{32} - 4 q^{37} - 36 q^{40} + 36 q^{41} - 4 q^{43} + 4 q^{46} + 6 q^{47} - 36 q^{50} - 52 q^{52} + 4 q^{55} + 96 q^{56} - 12 q^{58} - 8 q^{61} + 78 q^{65} + 2 q^{67} - 42 q^{70} - 4 q^{73} - 24 q^{76} - 42 q^{77} - 36 q^{80} - 8 q^{82} + 12 q^{83} - 4 q^{85} + 12 q^{88} - 16 q^{91} - 72 q^{92} + 66 q^{95} - 4 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
368.1 −2.55373 + 0.684269i 0 4.32125 − 2.49487i −2.01493 − 0.969559i 0 −2.63527 − 0.235267i −5.58921 + 5.58921i 0 5.80903 + 1.09723i
368.2 −2.42876 + 0.650783i 0 3.74329 − 2.16119i 1.67326 + 1.48331i 0 2.32199 − 1.26821i −4.12912 + 4.12912i 0 −5.02925 − 2.51367i
368.3 −2.41263 + 0.646461i 0 3.67080 − 2.11934i 2.07655 − 0.829434i 0 −1.09925 − 2.40658i −3.95386 + 3.95386i 0 −4.47373 + 3.34352i
368.4 −2.40632 + 0.644772i 0 3.64261 − 2.10306i −0.935544 + 2.03095i 0 −1.30393 + 2.30212i −3.88619 + 3.88619i 0 0.941721 − 5.49033i
368.5 −2.31583 + 0.620525i 0 3.24597 − 1.87406i −0.708123 − 2.12098i 0 2.37173 − 1.17256i −2.96360 + 2.96360i 0 2.95602 + 4.47242i
368.6 −2.14761 + 0.575451i 0 2.54905 − 1.47169i 2.10613 + 0.751132i 0 1.83889 + 1.90223i −1.48316 + 1.48316i 0 −4.95540 − 0.401163i
368.7 −1.93179 + 0.517620i 0 1.73181 − 0.999863i 1.43910 − 1.71143i 0 0.522555 + 2.59363i 0.000387571 0 0.000387571i 0 −1.89416 + 4.05102i
368.8 −1.81917 + 0.487445i 0 1.33973 − 0.773492i −0.601260 + 2.15371i 0 −2.54492 + 0.723460i 0.603293 − 0.603293i 0 0.0439768 − 4.21106i
368.9 −1.77167 + 0.474718i 0 1.18141 − 0.682088i −1.69841 + 1.45445i 0 2.23175 − 1.42102i 0.824634 − 0.824634i 0 2.31857 − 3.38306i
368.10 −1.57017 + 0.420725i 0 0.556367 − 0.321219i −2.22760 − 0.194383i 0 −0.460381 − 2.60539i 1.56044 − 1.56044i 0 3.57949 − 0.631994i
368.11 −1.56147 + 0.418395i 0 0.531086 − 0.306623i 2.02767 + 0.942623i 0 −1.29986 − 2.30442i 1.58517 − 1.58517i 0 −3.56054 − 0.623511i
368.12 −1.47958 + 0.396452i 0 0.299929 − 0.173164i 1.06067 − 1.96850i 0 −2.39893 − 1.11585i 1.79114 − 1.79114i 0 −0.788924 + 3.33305i
368.13 −1.39960 + 0.375022i 0 0.0861894 − 0.0497615i −2.22170 − 0.253107i 0 1.21622 + 2.34964i 1.94719 − 1.94719i 0 3.20441 − 0.478936i
368.14 −1.23263 + 0.330283i 0 −0.321754 + 0.185765i 1.27451 + 1.83729i 0 −1.80378 + 1.93556i 2.13995 − 2.13995i 0 −2.17783 − 1.84375i
368.15 −0.971193 + 0.260230i 0 −0.856554 + 0.494532i 0.675134 − 2.13171i 0 2.27467 − 1.35126i 2.12511 − 2.12511i 0 −0.100949 + 2.24599i
368.16 −0.657323 + 0.176129i 0 −1.33100 + 0.768453i −0.742479 + 2.10920i 0 2.39381 + 1.12679i 1.70194 − 1.70194i 0 0.116557 − 1.51720i
368.17 −0.652018 + 0.174708i 0 −1.33745 + 0.772175i −1.39591 − 1.74683i 0 −2.59171 − 0.532031i 1.69176 − 1.69176i 0 1.21535 + 0.895092i
368.18 −0.576351 + 0.154433i 0 −1.42372 + 0.821985i 0.594062 − 2.15571i 0 −0.0615728 + 2.64503i 1.53746 − 1.53746i 0 −0.00947561 + 1.33419i
368.19 −0.534767 + 0.143291i 0 −1.46661 + 0.846746i 1.50743 + 1.65156i 0 1.59225 + 2.11299i 1.44592 − 1.44592i 0 −1.04278 − 0.667199i
368.20 −0.112117 + 0.0300417i 0 −1.72038 + 0.993264i −1.26170 + 1.84611i 0 −0.423666 − 2.61161i 0.327196 − 0.327196i 0 0.0859974 − 0.244884i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 368.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.n odd 6 1 inner
315.bx even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.ca.a 176
3.b odd 2 1 315.2.bx.a yes 176
5.c odd 4 1 inner 945.2.ca.a 176
7.c even 3 1 945.2.by.a 176
9.c even 3 1 315.2.bv.a ✓ 176
9.d odd 6 1 945.2.by.a 176
15.e even 4 1 315.2.bx.a yes 176
21.h odd 6 1 315.2.bv.a ✓ 176
35.l odd 12 1 945.2.by.a 176
45.k odd 12 1 315.2.bv.a ✓ 176
45.l even 12 1 945.2.by.a 176
63.g even 3 1 315.2.bx.a yes 176
63.n odd 6 1 inner 945.2.ca.a 176
105.x even 12 1 315.2.bv.a ✓ 176
315.bx even 12 1 inner 945.2.ca.a 176
315.ch odd 12 1 315.2.bx.a yes 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bv.a ✓ 176 9.c even 3 1
315.2.bv.a ✓ 176 21.h odd 6 1
315.2.bv.a ✓ 176 45.k odd 12 1
315.2.bv.a ✓ 176 105.x even 12 1
315.2.bx.a yes 176 3.b odd 2 1
315.2.bx.a yes 176 15.e even 4 1
315.2.bx.a yes 176 63.g even 3 1
315.2.bx.a yes 176 315.ch odd 12 1
945.2.by.a 176 7.c even 3 1
945.2.by.a 176 9.d odd 6 1
945.2.by.a 176 35.l odd 12 1
945.2.by.a 176 45.l even 12 1
945.2.ca.a 176 1.a even 1 1 trivial
945.2.ca.a 176 5.c odd 4 1 inner
945.2.ca.a 176 63.n odd 6 1 inner
945.2.ca.a 176 315.bx even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).