Properties

Label 441.3.bm.a
Level $441$
Weight $3$
Character orbit 441.bm
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(11,441)]
 
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([7, 40]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.bm (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.

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 21 q^{2} - 13 q^{3} + 427 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 21 q^{2} - 13 q^{3} + 427 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} + 27 q^{9} - 34 q^{10} - 18 q^{11} - 46 q^{12} - 252 q^{14} - 16 q^{15} - 845 q^{16} - 27 q^{17} - 31 q^{18} + 2 q^{19} - 27 q^{20} + 3 q^{21} - 13 q^{22} - 45 q^{23} - 140 q^{24} - 518 q^{25} - 84 q^{26} + 131 q^{27} - 20 q^{28} + 93 q^{29} + 66 q^{30} + 20 q^{31} - 21 q^{32} + 17 q^{33} - q^{34} + 30 q^{35} - 286 q^{36} - 169 q^{37} + 132 q^{38} - 41 q^{39} + 41 q^{40} + 57 q^{41} + 46 q^{42} + 27 q^{43} + 183 q^{44} - 603 q^{45} + 122 q^{46} - 777 q^{47} - 197 q^{48} - 17 q^{49} - 255 q^{50} - 108 q^{51} + 536 q^{52} - 144 q^{53} - 447 q^{54} - 10 q^{55} - 573 q^{56} + 171 q^{57} + 296 q^{58} - 21 q^{59} - 227 q^{60} + 56 q^{61} - 84 q^{62} - 15 q^{63} + 1540 q^{64} - 21 q^{65} + 1956 q^{66} - 82 q^{67} + 519 q^{68} + 70 q^{69} + 14 q^{70} + 945 q^{71} - 358 q^{72} + 72 q^{73} + 174 q^{74} + 1116 q^{75} + 83 q^{76} + 405 q^{77} - 1803 q^{78} + 74 q^{79} + 3 q^{80} + 127 q^{81} - 22 q^{82} - 375 q^{83} - 1037 q^{84} + 20 q^{85} - 180 q^{86} - 5 q^{87} + 80 q^{88} + 1098 q^{89} - 324 q^{90} - 135 q^{91} + 639 q^{92} - 1043 q^{93} - 373 q^{94} - 21 q^{95} + 1356 q^{96} - 31 q^{97} + 123 q^{98} + 248 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.85226 + 0.879253i 2.99955 0.0522460i 10.4629 5.03868i 0.270126 + 1.79217i −11.5091 + 2.83862i 2.04933 6.69330i −23.5185 + 18.7554i 8.99454 0.313429i −2.61637 6.66639i
11.2 −3.76346 + 0.858985i 0.379079 + 2.97595i 9.82188 4.72997i 1.26468 + 8.39059i −3.98295 10.8742i 2.79668 + 6.41705i −20.8290 + 16.6106i −8.71260 + 2.25625i −11.9669 30.4913i
11.3 −3.69776 + 0.843989i −2.64374 1.41796i 9.35723 4.50620i −0.222314 1.47496i 10.9727 + 3.01199i −6.33007 + 2.98835i −18.9361 + 15.1010i 4.97876 + 7.49746i 2.06691 + 5.26641i
11.4 −3.69032 + 0.842292i 1.91071 2.31283i 9.30515 4.48112i −0.842237 5.58787i −5.10304 + 10.1445i 1.93246 + 6.72797i −18.7270 + 14.9343i −1.69840 8.83829i 7.81475 + 19.9117i
11.5 −3.67202 + 0.838116i −2.44499 1.73840i 9.17745 4.41963i 0.632965 + 4.19945i 10.4350 + 4.33425i 6.89263 + 1.22136i −18.2167 + 14.5273i 2.95595 + 8.50073i −5.84389 14.8900i
11.6 −3.64215 + 0.831296i −2.05762 + 2.18316i 8.97030 4.31987i −1.36579 9.06141i 5.67932 9.66188i 6.99915 0.109390i −17.3970 + 13.8736i −0.532367 8.98424i 12.5071 + 31.8676i
11.7 −3.49099 + 0.796796i −2.43344 + 1.75453i 7.94827 3.82768i 0.762633 + 5.05974i 7.09713 8.06399i −2.91713 6.36320i −13.4992 + 10.7653i 2.84328 8.53907i −6.69393 17.0558i
11.8 −3.48644 + 0.795757i −0.837676 + 2.88068i 7.91816 3.81319i −0.550933 3.65520i 0.628189 10.7099i −6.83606 + 1.50610i −13.3882 + 10.6767i −7.59660 4.82615i 4.82945 + 12.3052i
11.9 −3.37519 + 0.770366i 1.97862 2.25501i 7.19458 3.46473i 0.651352 + 4.32144i −4.94102 + 9.13536i −6.47385 + 2.66257i −10.7872 + 8.60252i −1.17016 8.92360i −5.52752 14.0839i
11.10 −3.37360 + 0.770003i −0.382520 2.97551i 7.18442 3.45983i −0.509050 3.37733i 3.58162 + 9.74366i 4.00587 5.74047i −10.7516 + 8.57412i −8.70736 + 2.27638i 4.31789 + 11.0018i
11.11 −3.33917 + 0.762143i 1.42379 + 2.64061i 6.96530 3.35431i −0.320505 2.12641i −6.76680 7.73230i −1.05395 6.92020i −9.99061 + 7.96725i −4.94563 + 7.51936i 2.69085 + 6.85617i
11.12 −3.09773 + 0.707036i 1.26663 2.71949i 5.49213 2.64487i 1.21443 + 8.05721i −2.00090 + 9.31980i 6.59924 2.33453i −5.20637 + 4.15194i −5.79129 6.88919i −9.45870 24.1004i
11.13 −3.07710 + 0.702327i −0.451137 2.96589i 5.37138 2.58672i −0.770464 5.11169i 3.47121 + 8.80947i −6.20114 3.24744i −4.84097 + 3.86055i −8.59295 + 2.67604i 5.96087 + 15.1881i
11.14 −3.06589 + 0.699769i 1.56870 + 2.55718i 5.30611 2.55529i −0.653283 4.33425i −6.59889 6.74230i 6.93916 + 0.920911i −4.64521 + 3.70443i −4.07837 + 8.02290i 5.03586 + 12.8312i
11.15 −3.04906 + 0.695927i 2.99824 + 0.102770i 5.20855 2.50831i −1.48219 9.83366i −9.21332 + 1.77320i −6.90823 1.12979i −4.35496 + 3.47296i 8.97888 + 0.616258i 11.3628 + 28.9519i
11.16 −3.03462 + 0.692632i 2.93053 + 0.641862i 5.12530 2.46821i 0.0346843 + 0.230115i −9.33762 + 0.0819742i 3.59973 + 6.00350i −4.10947 + 3.27719i 8.17603 + 3.76199i −0.264639 0.674289i
11.17 −2.98855 + 0.682116i −2.92784 + 0.654027i 4.86226 2.34154i −0.365541 2.42521i 8.30387 3.95172i −1.60483 + 6.81355i −3.34737 + 2.66944i 8.14450 3.82977i 2.74671 + 6.99851i
11.18 −2.98443 + 0.681177i 2.79051 + 1.10138i 4.83894 2.33031i 1.02417 + 6.79491i −9.07833 1.38614i −6.89710 + 1.19581i −3.28082 + 2.61637i 6.57395 + 6.14681i −7.68509 19.5813i
11.19 −2.97265 + 0.678489i −2.86246 + 0.897940i 4.77245 2.29829i 0.403378 + 2.67624i 7.89987 4.61141i 6.95934 0.753351i −3.09193 + 2.46573i 7.38741 5.14064i −3.01490 7.68183i
11.20 −2.91113 + 0.664446i −0.844404 2.87871i 4.42931 2.13304i 0.930053 + 6.17050i 4.37092 + 7.81924i −0.314731 + 6.99292i −2.13882 + 1.70566i −7.57396 + 4.86159i −6.80747 17.3451i
See next 80 embeddings (of 1320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.110
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.bm 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.bm.a yes 1320
9.d odd 6 1 441.3.bi.a 1320
49.g even 21 1 441.3.bi.a 1320
441.bm odd 42 1 inner 441.3.bm.a yes 1320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.bi.a 1320 9.d odd 6 1
441.3.bi.a 1320 49.g even 21 1
441.3.bm.a yes 1320 1.a even 1 1 trivial
441.3.bm.a yes 1320 441.bm odd 42 1 inner

Hecke kernels

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