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Properties

Label 441.3.bl.a
Level $441$
Weight $3$
Character orbit 441.bl
Analytic conductor $12.016$
Analytic rank $0$
Dimension $1320$
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] = [441,3,Mod(61,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([28, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.61");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.bl (of order \(42\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(1320\)
Relative dimension: \(110\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 8 q^{2} - 11 q^{3} + 208 q^{4} - 7 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 91 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 8 q^{2} - 11 q^{3} + 208 q^{4} - 7 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 91 q^{9} - 22 q^{10} - 5 q^{11} + 24 q^{12} - 22 q^{13} - 206 q^{14} - 22 q^{15} + 404 q^{16} + 5 q^{17} - 72 q^{18} - 36 q^{19} - 115 q^{20} - 26 q^{21} + 3 q^{22} - 50 q^{23} - 56 q^{24} + 1015 q^{25} - 82 q^{26} - 38 q^{27} - 12 q^{28} - 293 q^{29} - 85 q^{30} - 66 q^{31} - 130 q^{32} + 100 q^{33} - 19 q^{34} - 46 q^{35} - 266 q^{36} + 117 q^{37} - 63 q^{38} + 54 q^{39} - 7 q^{40} + 227 q^{41} - 187 q^{42} - 37 q^{43} - 31 q^{44} - 269 q^{45} - 174 q^{46} - 400 q^{47} - 147 q^{48} - 41 q^{49} - 27 q^{50} + 112 q^{51} + 273 q^{52} - 794 q^{53} - 392 q^{54} + 56 q^{55} - 2115 q^{56} + 71 q^{57} - 141 q^{58} - 49 q^{59} - 1104 q^{60} - 225 q^{61} + 208 q^{63} - 1516 q^{64} - 79 q^{65} + 1035 q^{66} + 41 q^{67} - 260 q^{69} - 271 q^{70} + 301 q^{71} + 417 q^{72} - 22 q^{73} + 627 q^{74} + 1369 q^{75} + 33 q^{76} + 25 q^{77} + 1041 q^{78} + 83 q^{79} + 567 q^{80} - 241 q^{81} - 10 q^{82} + 137 q^{83} + 3059 q^{84} + 20 q^{85} + 231 q^{86} - 577 q^{87} + 43 q^{88} - 574 q^{89} - 899 q^{90} - 67 q^{91} + 1151 q^{92} + 1225 q^{93} - 4 q^{94} - 171 q^{95} + 1303 q^{96} + 57 q^{97} + 169 q^{98} + 218 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −0.292550 − 3.90381i −0.936179 − 2.85019i −11.1988 + 1.68795i 5.73368 − 1.30867i −10.8527 + 4.48849i −4.43455 − 5.41615i 6.38118 + 27.9578i −7.24714 + 5.33657i −6.78620 − 22.0003i
61.2 −0.290836 − 3.88094i 2.89095 + 0.801488i −11.0218 + 1.66127i −3.62080 + 0.826423i 2.26973 − 11.4527i −5.36606 + 4.49505i 6.18876 + 27.1147i 7.71524 + 4.63413i 4.26036 + 13.8117i
61.3 −0.289850 − 3.86778i −1.61075 + 2.53091i −10.9204 + 1.64598i 2.09813 − 0.478884i 10.2559 + 5.49645i 1.57647 − 6.82017i 6.07927 + 26.6350i −3.81096 − 8.15332i −2.46036 − 7.97629i
61.4 −0.281014 − 3.74987i 2.66737 − 1.37300i −10.0272 + 1.51136i 8.73557 − 1.99384i −5.89813 − 9.61646i 4.24420 + 5.56658i 5.13812 + 22.5116i 5.22975 − 7.32460i −9.93144 − 32.1969i
61.5 −0.277151 − 3.69832i 0.677699 − 2.92245i −9.64544 + 1.45382i −5.51069 + 1.25778i −10.9960 − 1.69639i 5.42026 + 4.42954i 4.74888 + 20.8062i −8.08145 − 3.96108i 6.17896 + 20.0317i
61.6 −0.274190 − 3.65881i −2.83718 − 0.974876i −9.35641 + 1.41025i −2.85864 + 0.652467i −2.78896 + 10.6480i 6.78645 − 1.71584i 4.45951 + 19.5384i 7.09923 + 5.53181i 3.17107 + 10.2803i
61.7 −0.269165 − 3.59175i −2.99986 + 0.0287498i −8.87293 + 1.33738i −6.91031 + 1.57723i 0.910720 + 10.7670i −6.93461 + 0.954538i 3.98589 + 17.4633i 8.99835 − 0.172491i 7.52505 + 24.3956i
61.8 −0.267029 − 3.56325i 0.829875 + 2.88293i −8.67013 + 1.30681i 2.13909 − 0.488233i 10.0510 − 3.72688i 5.56629 + 4.24458i 3.79119 + 16.6103i −7.62261 + 4.78495i −2.31089 − 7.49173i
61.9 −0.256759 − 3.42620i 1.78610 + 2.41036i −7.71763 + 1.16325i 2.68445 − 0.612708i 7.79980 − 6.73842i −6.63750 − 2.22341i 2.90892 + 12.7448i −2.61970 + 8.61029i −2.78852 − 9.04016i
61.10 −0.255078 − 3.40378i 2.77030 − 1.15128i −7.56536 + 1.14029i −3.32469 + 0.758839i −4.62535 − 9.13584i 1.04611 − 6.92139i 2.77293 + 12.1490i 6.34912 − 6.37877i 3.43098 + 11.1230i
61.11 −0.253659 − 3.38485i −2.99997 − 0.0141595i −7.43752 + 1.12103i 6.40458 − 1.46180i 0.713041 + 10.1580i 4.32829 + 5.50144i 2.65985 + 11.6536i 8.99960 + 0.0849561i −6.57256 − 21.3077i
61.12 −0.250796 − 3.34664i −0.964612 + 2.84069i −7.18181 + 1.08248i −6.29098 + 1.43587i 9.74870 + 2.51578i 0.540008 + 6.97914i 2.43671 + 10.6759i −7.13905 − 5.48033i 6.38311 + 20.6935i
61.13 −0.242546 − 3.23655i −2.26171 + 1.97096i −6.46112 + 0.973857i 4.96704 − 1.13369i 6.92769 + 6.84208i −6.71964 + 1.96122i 1.83018 + 8.01853i 1.23063 − 8.91547i −4.87400 − 15.8011i
61.14 −0.240669 − 3.21151i −2.18359 − 2.05716i −6.30053 + 0.949653i 3.07645 − 0.702179i −6.08107 + 7.50772i −3.71151 + 5.93504i 1.69963 + 7.44658i 0.536165 + 8.98402i −2.99546 − 9.71103i
61.15 −0.237786 − 3.17303i 2.66139 + 1.38456i −6.05625 + 0.912834i 2.38077 − 0.543396i 3.76041 − 8.77390i 5.88889 − 3.78430i 1.50436 + 6.59102i 5.16598 + 7.36971i −2.29033 − 7.42506i
61.16 −0.230440 − 3.07501i −0.429622 − 2.96908i −5.44728 + 0.821045i −8.24133 + 1.88103i −9.03095 + 2.00529i −4.68723 − 5.19903i 1.03530 + 4.53596i −8.63085 + 2.55116i 7.68333 + 24.9087i
61.17 −0.215343 − 2.87355i 1.47745 − 2.61096i −4.25558 + 0.641425i 1.24640 − 0.284482i −7.82089 − 3.68327i −5.34889 + 4.51546i 0.194704 + 0.853053i −4.63428 − 7.71515i −1.08588 − 3.52032i
61.18 −0.214338 − 2.86015i 0.00720261 − 2.99999i −4.17918 + 0.629910i 4.58421 − 1.04632i −8.58196 + 0.622413i 5.50575 − 4.32281i 0.144487 + 0.633040i −8.99990 − 0.0432156i −3.97519 − 12.8873i
61.19 −0.210792 − 2.81282i 2.98015 − 0.344510i −3.91218 + 0.589667i 8.62878 − 1.96946i −1.59724 − 8.31001i −6.27921 − 3.09379i −0.0273824 − 0.119970i 8.76263 − 2.05339i −7.35861 − 23.8560i
61.20 −0.204594 − 2.73012i −2.38708 + 1.81710i −3.45635 + 0.520961i −4.72278 + 1.07794i 5.44928 + 6.14523i 5.32157 − 4.54763i −0.307413 − 1.34687i 2.39628 − 8.67513i 3.90916 + 12.6732i
See next 80 embeddings (of 1320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.110
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.bl odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.bl.a yes 1320
9.c even 3 1 441.3.bc.a ✓ 1320
49.h odd 42 1 441.3.bc.a ✓ 1320
441.bl odd 42 1 inner 441.3.bl.a yes 1320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.bc.a ✓ 1320 9.c even 3 1
441.3.bc.a ✓ 1320 49.h odd 42 1
441.3.bl.a yes 1320 1.a even 1 1 trivial
441.3.bl.a yes 1320 441.bl odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(441, [\chi])\).