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Properties

Label 441.3.l.b
Level $441$
Weight $3$
Character orbit 441.l
Analytic conductor $12.016$
Analytic rank $0$
Dimension $28$
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(97,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.l (of order \(6\), degree \(2\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - 23 q^{4} + 3 q^{5} - 12 q^{6} - 16 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - 23 q^{4} + 3 q^{5} - 12 q^{6} - 16 q^{8} + 6 q^{9} + 7 q^{11} + 27 q^{12} - 15 q^{13} - 18 q^{15} - 27 q^{16} + 9 q^{18} - 108 q^{20} - 10 q^{22} + 34 q^{23} + 120 q^{24} + 31 q^{25} + 81 q^{27} + 70 q^{29} + 33 q^{30} + 45 q^{31} + 153 q^{32} - 111 q^{33} - 12 q^{34} - 174 q^{36} - 18 q^{37} - 87 q^{38} - 9 q^{39} + 102 q^{40} + 234 q^{41} + 30 q^{43} - 102 q^{44} + 3 q^{45} + 44 q^{46} - 111 q^{47} - 147 q^{48} + 241 q^{50} - 6 q^{51} + 219 q^{52} - 296 q^{53} + 207 q^{54} + 189 q^{57} + 17 q^{58} + 42 q^{59} - 489 q^{60} + 120 q^{61} - 48 q^{64} + 114 q^{65} - 705 q^{66} - 34 q^{67} + 18 q^{68} - 78 q^{69} - 350 q^{71} + 177 q^{72} + 359 q^{74} + 387 q^{75} - 72 q^{76} - 375 q^{78} - 82 q^{79} + 438 q^{81} - 738 q^{83} + 3 q^{85} + 17 q^{86} + 564 q^{87} + 25 q^{88} + 543 q^{90} + 288 q^{92} - 30 q^{93} - 3 q^{94} + 507 q^{95} - 813 q^{96} + 57 q^{97} - 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −1.67789 + 2.90618i 2.76141 − 1.17243i −3.63061 − 6.28839i 7.37564 − 4.25833i −1.22605 + 9.99238i 0 10.9439 6.25082 − 6.47512i 28.5800i
97.2 −1.67756 + 2.90562i −1.29550 + 2.70586i −3.62842 − 6.28461i 0.769575 − 0.444314i −5.68892 − 8.30348i 0 10.9271 −5.64335 − 7.01089i 2.98146i
97.3 −1.32841 + 2.30087i 2.68686 + 1.33447i −1.52933 − 2.64888i −7.97090 + 4.60200i −6.63967 + 4.40939i 0 −2.50096 5.43840 + 7.17104i − 24.4533i
97.4 −1.12025 + 1.94033i −1.02718 − 2.81867i −0.509909 − 0.883189i 1.67528 − 0.967222i 6.61983 + 1.16455i 0 −6.67708 −6.88982 + 5.79054i 4.33411i
97.5 −0.840995 + 1.45665i −2.99659 + 0.143035i 0.585454 + 1.01404i −2.03050 + 1.17231i 2.31176 − 4.48526i 0 −8.69742 8.95908 − 0.857237i − 3.94363i
97.6 −0.227576 + 0.394173i −0.216177 + 2.99220i 1.89642 + 3.28469i 3.78523 − 2.18540i −1.13025 − 0.766164i 0 −3.54692 −8.90653 − 1.29369i 1.98938i
97.7 −0.178911 + 0.309883i 1.55080 − 2.56808i 1.93598 + 3.35322i −3.97509 + 2.29502i 0.518348 + 0.940026i 0 −2.81677 −4.19003 − 7.96515i − 1.64242i
97.8 0.198068 − 0.343064i 2.98086 − 0.338326i 1.92154 + 3.32820i 2.57417 − 1.48620i 0.474346 − 1.08964i 0 3.10693 8.77107 − 2.01701i − 1.17747i
97.9 0.662399 − 1.14731i −2.99866 + 0.0895871i 1.12246 + 1.94415i 6.26581 − 3.61757i −1.88353 + 3.49973i 0 8.27324 8.98395 − 0.537283i − 9.58509i
97.10 0.826674 − 1.43184i 0.127743 + 2.99728i 0.633221 + 1.09677i −6.81496 + 3.93462i 4.39723 + 2.29486i 0 8.70726 −8.96736 + 0.765761i 13.0106i
97.11 0.902282 − 1.56280i −2.28529 − 1.94357i 0.371774 + 0.643931i −4.98393 + 2.87747i −5.09938 + 1.81779i 0 8.56004 1.44507 + 8.88323i 10.3852i
97.12 1.41697 − 2.45427i 2.70216 + 1.30320i −2.01561 − 3.49114i 2.07720 − 1.19927i 7.02729 − 4.78521i 0 −0.0884848 5.60332 + 7.04292i − 6.79734i
97.13 1.62718 − 2.81835i 0.248110 − 2.98972i −3.29541 − 5.70782i 2.94001 − 1.69741i −8.02237 − 5.56407i 0 −8.43145 −8.87688 − 1.48356i − 11.0480i
97.14 1.91801 − 3.32210i −2.23856 + 1.99721i −5.35756 − 9.27956i −0.187534 + 0.108273i 2.34136 + 11.2674i 0 −25.7594 1.02227 − 8.94175i 0.830676i
391.1 −1.67789 − 2.90618i 2.76141 + 1.17243i −3.63061 + 6.28839i 7.37564 + 4.25833i −1.22605 − 9.99238i 0 10.9439 6.25082 + 6.47512i − 28.5800i
391.2 −1.67756 − 2.90562i −1.29550 − 2.70586i −3.62842 + 6.28461i 0.769575 + 0.444314i −5.68892 + 8.30348i 0 10.9271 −5.64335 + 7.01089i − 2.98146i
391.3 −1.32841 − 2.30087i 2.68686 − 1.33447i −1.52933 + 2.64888i −7.97090 − 4.60200i −6.63967 − 4.40939i 0 −2.50096 5.43840 − 7.17104i 24.4533i
391.4 −1.12025 − 1.94033i −1.02718 + 2.81867i −0.509909 + 0.883189i 1.67528 + 0.967222i 6.61983 − 1.16455i 0 −6.67708 −6.88982 − 5.79054i − 4.33411i
391.5 −0.840995 − 1.45665i −2.99659 − 0.143035i 0.585454 − 1.01404i −2.03050 − 1.17231i 2.31176 + 4.48526i 0 −8.69742 8.95908 + 0.857237i 3.94363i
391.6 −0.227576 − 0.394173i −0.216177 − 2.99220i 1.89642 − 3.28469i 3.78523 + 2.18540i −1.13025 + 0.766164i 0 −3.54692 −8.90653 + 1.29369i − 1.98938i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.l.b 28
7.b odd 2 1 441.3.l.a 28
7.c even 3 1 63.3.t.a yes 28
7.c even 3 1 441.3.k.b 28
7.d odd 6 1 63.3.k.a ✓ 28
7.d odd 6 1 441.3.t.a 28
9.c even 3 1 441.3.l.a 28
21.g even 6 1 189.3.k.a 28
21.h odd 6 1 189.3.t.a 28
63.g even 3 1 441.3.t.a 28
63.h even 3 1 63.3.k.a ✓ 28
63.j odd 6 1 189.3.k.a 28
63.k odd 6 1 63.3.t.a yes 28
63.l odd 6 1 inner 441.3.l.b 28
63.s even 6 1 189.3.t.a 28
63.t odd 6 1 441.3.k.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a ✓ 28 7.d odd 6 1
63.3.k.a ✓ 28 63.h even 3 1
63.3.t.a yes 28 7.c even 3 1
63.3.t.a yes 28 63.k odd 6 1
189.3.k.a 28 21.g even 6 1
189.3.k.a 28 63.j odd 6 1
189.3.t.a 28 21.h odd 6 1
189.3.t.a 28 63.s even 6 1
441.3.k.b 28 7.c even 3 1
441.3.k.b 28 63.t odd 6 1
441.3.l.a 28 7.b odd 2 1
441.3.l.a 28 9.c even 3 1
441.3.l.b 28 1.a even 1 1 trivial
441.3.l.b 28 63.l odd 6 1 inner
441.3.t.a 28 7.d odd 6 1
441.3.t.a 28 63.g even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{28} - T_{2}^{27} + 40 T_{2}^{26} - 29 T_{2}^{25} + 977 T_{2}^{24} - 620 T_{2}^{23} + \cdots + 1034289 \) Copy content Toggle raw display
\( T_{5}^{28} - 3 T_{5}^{27} - 186 T_{5}^{26} + 567 T_{5}^{25} + 22593 T_{5}^{24} - 82521 T_{5}^{23} + \cdots + 118476672320601 \) Copy content Toggle raw display