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Properties

Label 882.2.y.a
Level $882$
Weight $2$
Character orbit 882.y
Analytic conductor $7.043$
Analytic rank $0$
Dimension $336$
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] = [882,2,Mod(193,882)] 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("882.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([14, 22])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [336,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9} - 2 q^{10} + q^{12} - 23 q^{13} + q^{14} + q^{15} + 28 q^{16} + 7 q^{17} - q^{18} + 38 q^{19} + 2 q^{20} + 13 q^{21}+ \cdots - 157 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0.988831 − 0.149042i −1.68863 − 0.385381i 0.955573 − 0.294755i 1.02355 − 0.492917i −1.72721 − 0.129399i 2.30579 − 1.29744i 0.900969 − 0.433884i 2.70296 + 1.30153i 0.938655 − 0.639964i
193.2 0.988831 − 0.149042i −1.67319 − 0.447706i 0.955573 − 0.294755i −0.697157 + 0.335733i −1.72123 − 0.193329i −1.34035 + 2.28111i 0.900969 − 0.433884i 2.59912 + 1.49819i −0.639332 + 0.435889i
193.3 0.988831 − 0.149042i −1.64681 − 0.536661i 0.955573 − 0.294755i −3.76741 + 1.81429i −1.70841 − 0.285222i 2.57313 + 0.615639i 0.900969 − 0.433884i 2.42399 + 1.76756i −3.45492 + 2.35553i
193.4 0.988831 − 0.149042i −1.59771 + 0.668824i 0.955573 − 0.294755i −0.798057 + 0.384324i −1.48018 + 0.899480i −0.448308 + 2.60749i 0.900969 − 0.433884i 2.10535 − 2.13717i −0.731863 + 0.498976i
193.5 0.988831 − 0.149042i −1.57705 + 0.716170i 0.955573 − 0.294755i 3.20785 − 1.54482i −1.45270 + 0.943219i 2.59659 + 0.507667i 0.900969 − 0.433884i 1.97420 − 2.25888i 2.94177 − 2.00567i
193.6 0.988831 − 0.149042i −1.42463 − 0.985109i 0.955573 − 0.294755i 3.65590 − 1.76059i −1.55554 − 0.761776i −1.61191 + 2.09804i 0.900969 − 0.433884i 1.05912 + 2.80682i 3.35266 − 2.28581i
193.7 0.988831 − 0.149042i −1.39483 + 1.02686i 0.955573 − 0.294755i 2.49531 − 1.20168i −1.22620 + 1.22328i −2.56098 − 0.664373i 0.900969 − 0.433884i 0.891100 − 2.86460i 2.28834 − 1.56017i
193.8 0.988831 − 0.149042i −1.37307 + 1.05578i 0.955573 − 0.294755i −2.92067 + 1.40652i −1.20038 + 1.24863i −2.27392 − 1.35251i 0.900969 − 0.433884i 0.770659 − 2.89932i −2.67841 + 1.82611i
193.9 0.988831 − 0.149042i −0.959928 − 1.44171i 0.955573 − 0.294755i 0.760423 − 0.366201i −1.16408 − 1.28254i −2.64176 − 0.145192i 0.900969 − 0.433884i −1.15708 + 2.76788i 0.697351 − 0.475446i
193.10 0.988831 − 0.149042i −0.912991 − 1.47189i 0.955573 − 0.294755i −3.35691 + 1.61660i −1.12217 − 1.31937i −1.13854 − 2.38825i 0.900969 − 0.433884i −1.33289 + 2.68764i −3.07847 + 2.09887i
193.11 0.988831 − 0.149042i −0.694312 + 1.58680i 0.955573 − 0.294755i −0.275124 + 0.132493i −0.450057 + 1.67256i 2.62351 + 0.342365i 0.900969 − 0.433884i −2.03586 − 2.20347i −0.252304 + 0.172018i
193.12 0.988831 − 0.149042i −0.631098 + 1.61298i 0.955573 − 0.294755i −1.37316 + 0.661277i −0.383646 + 1.68903i 0.362482 − 2.62080i 0.900969 − 0.433884i −2.20343 − 2.03590i −1.25926 + 0.858550i
193.13 0.988831 − 0.149042i −0.578948 − 1.63243i 0.955573 − 0.294755i 0.355816 − 0.171352i −0.815782 − 1.52791i 1.84498 − 1.89633i 0.900969 − 0.433884i −2.32964 + 1.89018i 0.326303 − 0.222469i
193.14 0.988831 − 0.149042i −0.367749 − 1.69256i 0.955573 − 0.294755i −1.15518 + 0.556304i −0.615905 − 1.61885i 0.571215 + 2.58335i 0.900969 − 0.433884i −2.72952 + 1.24487i −1.05936 + 0.722261i
193.15 0.988831 − 0.149042i 0.0764713 − 1.73036i 0.955573 − 0.294755i 3.57321 − 1.72077i −0.182280 − 1.72243i 0.656435 − 2.56302i 0.900969 − 0.433884i −2.98830 − 0.264646i 3.27683 − 2.23411i
193.16 0.988831 − 0.149042i 0.312891 + 1.70355i 0.955573 − 0.294755i −0.662000 + 0.318803i 0.563298 + 1.63789i −2.61270 − 0.416886i 0.900969 − 0.433884i −2.80420 + 1.06606i −0.607091 + 0.413908i
193.17 0.988831 − 0.149042i 0.476395 + 1.66525i 0.955573 − 0.294755i 2.79471 − 1.34586i 0.719266 + 1.57564i −0.681045 − 2.55659i 0.900969 − 0.433884i −2.54610 + 1.58663i 2.56290 − 1.74736i
193.18 0.988831 − 0.149042i 0.549284 − 1.64265i 0.955573 − 0.294755i −2.77137 + 1.33462i 0.298325 − 1.70617i −2.03010 + 1.69667i 0.900969 − 0.433884i −2.39658 − 1.80456i −2.54150 + 1.73277i
193.19 0.988831 − 0.149042i 0.766846 + 1.55304i 0.955573 − 0.294755i 0.390697 − 0.188150i 0.989750 + 1.42141i 1.52140 + 2.16457i 0.900969 − 0.433884i −1.82390 + 2.38189i 0.358291 − 0.244278i
193.20 0.988831 − 0.149042i 0.896653 − 1.48190i 0.955573 − 0.294755i 1.94643 − 0.937350i 0.665773 − 1.59898i 1.39793 + 2.24628i 0.900969 − 0.433884i −1.39203 − 2.65749i 1.78498 − 1.21698i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.28
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.z even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.y.a ✓ 336
9.c even 3 1 882.2.bb.b yes 336
49.g even 21 1 882.2.bb.b yes 336
441.z even 21 1 inner 882.2.y.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.y.a ✓ 336 1.a even 1 1 trivial
882.2.y.a ✓ 336 441.z even 21 1 inner
882.2.bb.b yes 336 9.c even 3 1
882.2.bb.b yes 336 49.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{336} + 4 T_{5}^{335} + 172 T_{5}^{334} + 680 T_{5}^{333} + 15737 T_{5}^{332} + \cdots + 35\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display