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Properties

Label 882.2.bb.a
Level $882$
Weight $2$
Character orbit 882.bb
Analytic conductor $7.043$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(25,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([28, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.bb (of order \(21\), degree \(12\), 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: \(7.04280545828\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 56 q^{2} - 5 q^{3} - 56 q^{4} + 2 q^{5} - 5 q^{6} - 56 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 56 q^{2} - 5 q^{3} - 56 q^{4} + 2 q^{5} - 5 q^{6} - 56 q^{8} - 15 q^{9} + 2 q^{10} + 2 q^{12} - 21 q^{13} - 7 q^{15} - 56 q^{16} + 7 q^{17} + 6 q^{18} - 34 q^{19} + 2 q^{20} + 16 q^{21} - 36 q^{23} + 2 q^{24} + 32 q^{25} + 7 q^{26} + 49 q^{27} - 16 q^{29} - 7 q^{30} + 92 q^{31} - 56 q^{32} + 14 q^{33} + 7 q^{34} + 16 q^{35} - 15 q^{36} - 26 q^{37} + q^{38} + 44 q^{39} + 2 q^{40} + 3 q^{41} - 33 q^{42} - 3 q^{43} + 106 q^{45} - 36 q^{46} + 27 q^{47} + 2 q^{48} - 74 q^{49} - 164 q^{50} - 7 q^{51} + 21 q^{52} - 78 q^{53} - 28 q^{54} - 53 q^{55} - 18 q^{57} + 40 q^{58} - 16 q^{59} - 7 q^{60} - 45 q^{61} - 20 q^{62} - 50 q^{63} - 56 q^{64} - 14 q^{65} + 14 q^{66} + 86 q^{67} - 21 q^{68} - 85 q^{69} - 26 q^{70} + 7 q^{71} + 6 q^{72} + 36 q^{73} + 2 q^{74} - 35 q^{75} + 8 q^{76} - 37 q^{77} - 12 q^{78} - 2 q^{79} - 12 q^{80} + 41 q^{81} + 3 q^{82} + 191 q^{83} - 5 q^{84} + 159 q^{85} - 3 q^{86} + 72 q^{87} + 59 q^{89} + 22 q^{90} - 29 q^{91} + 27 q^{92} + 23 q^{93} + 6 q^{94} + 26 q^{95} + 2 q^{96} - 113 q^{97} - 18 q^{98} - 33 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.222521 + 0.974928i −1.73205 + 0.00402573i −0.900969 − 0.433884i −1.05576 − 2.69004i 0.381492 − 1.68952i −0.525426 + 2.59305i 0.623490 − 0.781831i 2.99997 − 0.0139455i 2.85752 − 0.430702i
25.2 −0.222521 + 0.974928i −1.72271 + 0.179669i −0.900969 − 0.433884i 1.05994 + 2.70069i 0.208174 − 1.71950i −2.64479 − 0.0712776i 0.623490 − 0.781831i 2.93544 − 0.619035i −2.86883 + 0.432407i
25.3 −0.222521 + 0.974928i −1.70549 − 0.302166i −0.900969 − 0.433884i −0.164649 − 0.419518i 0.674097 − 1.59549i −2.56050 − 0.666211i 0.623490 − 0.781831i 2.81739 + 1.03068i 0.445638 − 0.0671691i
25.4 −0.222521 + 0.974928i −1.61153 + 0.634811i −0.900969 − 0.433884i 1.28965 + 3.28597i −0.260296 − 1.71238i 1.77330 − 1.96352i 0.623490 − 0.781831i 2.19403 − 2.04603i −3.49056 + 0.526118i
25.5 −0.222521 + 0.974928i −1.36770 − 1.06273i −0.900969 − 0.433884i 0.398194 + 1.01458i 1.34043 − 1.09693i 2.60236 − 0.477177i 0.623490 − 0.781831i 0.741201 + 2.90700i −1.07775 + 0.162445i
25.6 −0.222521 + 0.974928i −1.34098 − 1.09625i −0.900969 − 0.433884i −1.24423 − 3.17025i 1.36716 − 1.06342i −0.633821 − 2.56871i 0.623490 − 0.781831i 0.596470 + 2.94011i 3.36764 − 0.507590i
25.7 −0.222521 + 0.974928i −1.28884 − 1.15711i −0.900969 − 0.433884i 1.18942 + 3.03058i 1.41489 − 0.999044i 0.0637114 + 2.64498i 0.623490 − 0.781831i 0.322207 + 2.98265i −3.21927 + 0.485227i
25.8 −0.222521 + 0.974928i −1.20543 + 1.24376i −0.900969 − 0.433884i 0.292541 + 0.745383i −0.944345 − 1.45197i −1.20289 + 2.35649i 0.623490 − 0.781831i −0.0938848 − 2.99853i −0.791791 + 0.119343i
25.9 −0.222521 + 0.974928i −1.00077 + 1.41367i −0.900969 − 0.433884i 0.197040 + 0.502050i −1.15553 − 1.29025i 2.15004 + 1.54186i 0.623490 − 0.781831i −0.996924 − 2.82951i −0.533309 + 0.0803833i
25.10 −0.222521 + 0.974928i −0.980031 + 1.42812i −0.900969 − 0.433884i −1.30088 − 3.31458i −1.17424 − 1.27325i −2.12929 − 1.57039i 0.623490 − 0.781831i −1.07908 − 2.79921i 3.52095 − 0.530697i
25.11 −0.222521 + 0.974928i −0.936103 + 1.45730i −0.900969 − 0.433884i 0.554482 + 1.41280i −1.21246 − 1.23691i 0.0992627 − 2.64389i 0.623490 − 0.781831i −1.24742 − 2.72836i −1.50076 + 0.226203i
25.12 −0.222521 + 0.974928i −0.821153 − 1.52503i −0.900969 − 0.433884i −0.121293 − 0.309050i 1.66952 − 0.461215i 2.60940 − 0.437094i 0.623490 − 0.781831i −1.65141 + 2.50456i 0.328292 − 0.0494820i
25.13 −0.222521 + 0.974928i −0.247644 − 1.71426i −0.900969 − 0.433884i −1.22665 − 3.12546i 1.72638 + 0.140022i 0.515923 + 2.59496i 0.623490 − 0.781831i −2.87734 + 0.849051i 3.32006 − 0.500418i
25.14 −0.222521 + 0.974928i −0.0636278 − 1.73088i −0.900969 − 0.433884i 0.114280 + 0.291182i 1.70164 + 0.323125i −1.56724 − 2.13161i 0.623490 − 0.781831i −2.99190 + 0.220264i −0.309311 + 0.0466211i
25.15 −0.222521 + 0.974928i 0.0522255 + 1.73126i −0.900969 − 0.433884i −0.703344 − 1.79209i −1.69948 − 0.334326i 2.20597 − 1.46072i 0.623490 − 0.781831i −2.99455 + 0.180832i 1.90367 − 0.286932i
25.16 −0.222521 + 0.974928i 0.234010 + 1.71617i −0.900969 − 0.433884i 1.42366 + 3.62742i −1.72521 − 0.153741i −2.41194 − 1.08745i 0.623490 − 0.781831i −2.89048 + 0.803201i −3.85326 + 0.580786i
25.17 −0.222521 + 0.974928i 0.272528 − 1.71048i −0.900969 − 0.433884i 0.357689 + 0.911378i 1.60695 + 0.646312i −2.25688 + 1.38077i 0.623490 − 0.781831i −2.85146 − 0.932306i −0.968121 + 0.145921i
25.18 −0.222521 + 0.974928i 0.631181 + 1.61295i −0.900969 − 0.433884i 1.46093 + 3.72239i −1.71296 + 0.256440i 1.56037 + 2.13665i 0.623490 − 0.781831i −2.20322 + 2.03613i −3.95415 + 0.595992i
25.19 −0.222521 + 0.974928i 0.739699 + 1.56616i −0.900969 − 0.433884i −1.31969 − 3.36251i −1.69149 + 0.372650i 1.48338 + 2.19079i 0.623490 − 0.781831i −1.90569 + 2.31697i 3.57186 − 0.538372i
25.20 −0.222521 + 0.974928i 0.841091 − 1.51412i −0.900969 − 0.433884i −1.15909 − 2.95331i 1.28900 + 1.15693i 2.26703 − 1.36402i 0.623490 − 0.781831i −1.58513 − 2.54703i 3.13718 − 0.472854i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.28
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{336} - 2 T_{5}^{335} - 84 T_{5}^{334} + 204 T_{5}^{333} + 2952 T_{5}^{332} + \cdots + 10\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display