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Properties

Label 441.3.bk.a
Level $441$
Weight $3$
Character orbit 441.bk
Analytic conductor $12.016$
Analytic rank $0$
Dimension $1320$
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] = [441,3,Mod(13,441)]
 
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([14, 33]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.bk (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.

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 5 q^{2} - 14 q^{3} + 211 q^{4} - 7 q^{5} - 70 q^{6} - 7 q^{7} - 36 q^{8} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 5 q^{2} - 14 q^{3} + 211 q^{4} - 7 q^{5} - 70 q^{6} - 7 q^{7} - 36 q^{8} - 50 q^{9} - 28 q^{10} - 5 q^{11} + 21 q^{12} - 7 q^{13} + 64 q^{14} + 14 q^{15} + 419 q^{16} - 28 q^{17} - 66 q^{18} - 7 q^{20} - 32 q^{21} - 21 q^{22} - 50 q^{23} - 14 q^{24} - 515 q^{25} - 28 q^{26} - 119 q^{27} - 48 q^{28} - 185 q^{29} - 154 q^{30} + 191 q^{32} - 14 q^{33} - 7 q^{34} + 236 q^{35} + 418 q^{36} + 90 q^{37} + 21 q^{38} + 132 q^{39} - 7 q^{40} - 7 q^{41} + 122 q^{42} + 59 q^{43} - 232 q^{44} + 7 q^{45} - 156 q^{46} + 245 q^{47} + 31 q^{49} + 18 q^{50} - 218 q^{51} + 273 q^{52} + 688 q^{53} - 14 q^{54} - 196 q^{55} + 921 q^{56} - 172 q^{57} - 141 q^{58} - 7 q^{59} + 156 q^{60} + 42 q^{61} - 84 q^{62} - 62 q^{63} - 1636 q^{64} + 137 q^{65} - 1470 q^{66} - 82 q^{67} + 322 q^{69} - 109 q^{70} - 662 q^{71} + 1056 q^{72} - 28 q^{73} + 651 q^{74} - 518 q^{75} + 105 q^{76} - 341 q^{77} + 834 q^{78} - 166 q^{79} - 550 q^{81} - 28 q^{82} + 875 q^{83} + 650 q^{84} - 55 q^{85} + 231 q^{86} - 574 q^{87} + 67 q^{88} + 1022 q^{89} - 1442 q^{90} + 356 q^{91} + 833 q^{92} - 302 q^{93} - 7 q^{94} + 696 q^{95} - 1925 q^{96} - 968 q^{98} - 508 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.90899 + 2.69914i 2.59177 + 1.51087i 0.877898 − 11.7147i −0.258873 − 1.71751i −11.6175 + 2.60047i 6.14763 + 3.34764i 19.1691 + 24.0373i 4.43455 + 7.83165i 5.38888 + 4.29748i
13.2 −2.86517 + 2.65849i 1.37610 − 2.66577i 0.842717 − 11.2453i 0.633542 + 4.20328i 3.14416 + 11.2963i 1.00876 − 6.92693i 17.7332 + 22.2367i −5.21268 − 7.33675i −12.9896 − 10.3589i
13.3 −2.80538 + 2.60301i −0.399054 + 2.97334i 0.795564 − 10.6161i −1.32057 − 8.76142i −6.62015 − 9.38010i −3.73843 − 5.91812i 15.8575 + 19.8847i −8.68151 − 2.37305i 26.5108 + 21.1417i
13.4 −2.77188 + 2.57193i −1.92475 + 2.30116i 0.769580 − 10.2693i 1.37677 + 9.13427i −0.583220 − 11.3289i 6.90645 + 1.14061i 14.8484 + 18.6194i −1.59064 − 8.85832i −27.3089 − 21.7781i
13.5 −2.71839 + 2.52229i −2.37643 − 1.83101i 0.728737 − 9.72433i 0.853333 + 5.66149i 11.0784 − 1.01665i −6.76324 + 1.80515i 13.2982 + 16.6755i 2.29480 + 8.70252i −16.5996 − 13.2378i
13.6 −2.68847 + 2.49453i 2.40958 − 1.78715i 0.706241 − 9.42413i −0.159880 − 1.06073i −2.01999 + 10.8155i −5.16924 + 4.72006i 12.4635 + 15.6287i 2.61219 − 8.61257i 3.07587 + 2.45292i
13.7 −2.65458 + 2.46309i −2.88072 + 0.837537i 0.681063 − 9.08815i −0.444701 − 2.95040i 5.58417 − 9.31879i −3.42610 + 6.10425i 11.5457 + 14.4779i 7.59706 − 4.82542i 8.44761 + 6.73674i
13.8 −2.62264 + 2.43345i −2.99739 − 0.125130i 0.657620 − 8.77533i −0.397029 − 2.63412i 8.16557 − 6.96584i 3.88353 − 5.82393i 10.7070 + 13.4262i 8.96868 + 0.750130i 7.45126 + 5.94218i
13.9 −2.59458 + 2.40742i 0.931947 + 2.85157i 0.637261 − 8.50365i 0.824446 + 5.46984i −9.28294 − 5.15505i −5.99151 − 3.61965i 9.99124 + 12.5286i −7.26295 + 5.31503i −15.3073 − 12.2071i
13.10 −2.57073 + 2.38529i −1.61441 − 2.52858i 0.620131 − 8.27507i −0.757170 − 5.02350i 10.1816 + 2.64947i 6.34589 + 2.95460i 9.39822 + 11.7850i −3.78739 + 8.16429i 13.9290 + 11.1080i
13.11 −2.40235 + 2.22906i −1.35792 + 2.67508i 0.503680 − 6.72114i −0.360858 − 2.39414i −2.70069 − 9.45337i −0.147903 + 6.99844i 5.59859 + 7.02041i −5.31208 − 7.26511i 6.20357 + 4.94718i
13.12 −2.37916 + 2.20754i 2.94969 + 0.547119i 0.488259 − 6.51536i −0.727396 − 4.82596i −8.22557 + 5.20986i −6.94816 − 0.850353i 5.12697 + 6.42902i 8.40132 + 3.22766i 12.3841 + 9.87598i
13.13 −2.34209 + 2.17314i −0.0690201 − 2.99921i 0.463916 − 6.19053i 0.632125 + 4.19388i 6.67934 + 6.87441i 1.90436 + 6.73598i 4.39820 + 5.51517i −8.99047 + 0.414011i −10.5944 − 8.44872i
13.14 −2.28203 + 2.11742i 2.08040 + 2.16147i 0.425298 − 5.67521i 0.795599 + 5.27846i −9.32426 − 0.527459i −1.48615 + 6.84042i 3.28241 + 4.11601i −0.343865 + 8.99343i −12.9923 − 10.3610i
13.15 −2.25760 + 2.09475i 0.268600 − 2.98795i 0.409875 − 5.46940i −0.614619 − 4.07773i 5.65262 + 7.30825i −2.47582 − 6.54754i 2.85096 + 3.57500i −8.85571 − 1.60513i 9.92939 + 7.91843i
13.16 −2.24511 + 2.08316i 2.60035 − 1.49605i 0.402055 − 5.36505i −1.14641 − 7.60595i −2.72158 + 8.77575i 6.36690 − 2.90905i 2.63537 + 3.30465i 4.52368 − 7.78051i 18.4182 + 14.6881i
13.17 −2.21889 + 2.05883i 2.86486 − 0.890275i 0.385777 − 5.14784i 0.839025 + 5.56656i −4.52389 + 7.87368i 6.09687 + 3.43922i 2.19351 + 2.75057i 7.41482 − 5.10102i −13.3223 − 10.6242i
13.18 −2.21733 + 2.05738i 2.99289 − 0.206403i 0.384811 − 5.13495i 1.22486 + 8.12638i −6.21157 + 6.61518i −1.84367 − 6.75284i 2.16759 + 2.71807i 8.91480 − 1.23549i −19.4350 − 15.4989i
13.19 −2.11338 + 1.96093i −0.944984 + 2.84728i 0.322206 − 4.29954i 0.244290 + 1.62076i −3.58620 − 7.87042i 5.35406 − 4.50933i 0.560094 + 0.702335i −7.21401 − 5.38127i −3.69446 − 2.94623i
13.20 −2.06622 + 1.91718i 1.48739 + 2.60532i 0.294799 − 3.93382i −0.162279 − 1.07665i −8.06813 − 2.53159i 5.59991 − 4.20012i −0.0969232 − 0.121538i −4.57537 + 7.75022i 2.39944 + 1.91349i
See next 80 embeddings (of 1320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.110
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
49.f odd 14 1 inner
441.bk 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.bk.a ✓ 1320
9.c even 3 1 inner 441.3.bk.a ✓ 1320
49.f odd 14 1 inner 441.3.bk.a ✓ 1320
441.bk odd 42 1 inner 441.3.bk.a ✓ 1320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.bk.a ✓ 1320 1.a even 1 1 trivial
441.3.bk.a ✓ 1320 9.c even 3 1 inner
441.3.bk.a ✓ 1320 49.f odd 14 1 inner
441.3.bk.a ✓ 1320 441.bk odd 42 1 inner

Hecke kernels

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