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Properties

Label 441.3.v.a
Level $441$
Weight $3$
Character orbit 441.v
Analytic conductor $12.016$
Analytic rank $0$
Dimension $54$
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(55,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.v (of order \(14\), degree \(6\), 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: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{14})\)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 5 q^{2} - 25 q^{4} + 7 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 5 q^{2} - 25 q^{4} + 7 q^{5} + 3 q^{8} - 7 q^{10} + 38 q^{11} - 7 q^{13} - 77 q^{14} - 21 q^{16} - 42 q^{17} - 49 q^{20} + 95 q^{22} + 82 q^{23} - 2 q^{25} + 119 q^{26} + 238 q^{28} + 221 q^{29} - 55 q^{32} - 539 q^{34} - 133 q^{35} - 124 q^{37} - 161 q^{38} + 721 q^{40} - 91 q^{41} - 67 q^{43} - 521 q^{44} + 121 q^{46} - 112 q^{47} - 84 q^{49} - 164 q^{50} - 455 q^{52} - 242 q^{53} - 175 q^{55} + 994 q^{56} - 171 q^{58} + 357 q^{59} + 77 q^{61} + 175 q^{62} - 291 q^{64} + 7 q^{65} + 116 q^{67} - 1001 q^{70} - 682 q^{71} - 70 q^{73} - 149 q^{74} + 630 q^{76} - 56 q^{77} - 12 q^{79} + 1428 q^{82} + 161 q^{83} + 443 q^{85} - 235 q^{86} + 157 q^{88} + 161 q^{89} - 315 q^{91} - 950 q^{92} - 126 q^{94} - 230 q^{95} + 63 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −2.36993 + 2.97180i 0 −2.32493 − 10.1862i 2.94973 − 6.12518i 0 6.96434 − 0.705711i 22.0827 + 10.6345i 0 11.2122 + 23.2823i
55.2 −1.58614 + 1.98896i 0 −0.550031 − 2.40984i −3.04025 + 6.31315i 0 −6.63215 + 2.23934i −3.50266 − 1.68679i 0 −7.73433 − 16.0605i
55.3 −1.21271 + 1.52069i 0 0.0482519 + 0.211405i 2.79684 − 5.80770i 0 −6.81297 + 1.60731i −7.38966 − 3.55867i 0 5.43995 + 11.2962i
55.4 −0.712240 + 0.893121i 0 0.599705 + 2.62748i 0.756289 − 1.57045i 0 1.39870 − 6.85884i −6.89066 − 3.31837i 0 0.863943 + 1.79400i
55.5 0.303238 − 0.380248i 0 0.837448 + 3.66910i −1.33228 + 2.76651i 0 5.25745 − 4.62161i 3.40188 + 1.63826i 0 0.647963 + 1.34551i
55.6 0.523443 − 0.656377i 0 0.733246 + 3.21256i −0.0740213 + 0.153707i 0 −0.110195 + 6.99913i 5.51805 + 2.65735i 0 0.0621436 + 0.129043i
55.7 1.68012 − 2.10681i 0 −0.725737 − 3.17966i 3.73262 − 7.75087i 0 −4.15473 − 5.63367i 1.79312 + 0.863523i 0 −10.0583 − 20.8863i
55.8 1.96408 − 2.46288i 0 −1.31807 − 5.77485i −1.13327 + 2.35326i 0 6.18203 + 3.28367i −5.45883 − 2.62883i 0 3.56996 + 7.41309i
55.9 2.18762 − 2.74319i 0 −1.84933 − 8.10243i −3.43314 + 7.12898i 0 −5.30895 + 4.56235i −13.6273 − 6.56257i 0 12.0458 + 25.0133i
118.1 −3.21077 − 1.54623i 0 5.42428 + 6.80183i −4.36986 + 0.997392i 0 −1.50710 − 6.83584i −3.72698 − 16.3289i 0 15.5728 + 3.55439i
118.2 −2.40476 − 1.15807i 0 1.94776 + 2.44242i 7.38568 − 1.68573i 0 0.782673 − 6.95611i 0.520295 + 2.27956i 0 −19.7129 − 4.49935i
118.3 −1.92764 − 0.928303i 0 0.360091 + 0.451540i −2.57337 + 0.587354i 0 −1.17704 + 6.90033i 1.62939 + 7.13883i 0 5.50576 + 1.25665i
118.4 −0.158356 − 0.0762600i 0 −2.47470 − 3.10317i 6.56936 − 1.49941i 0 6.63963 + 2.21706i 0.311677 + 1.36554i 0 −1.15464 − 0.263539i
118.5 0.472237 + 0.227417i 0 −2.32267 − 2.91254i 2.27483 − 0.519215i 0 −4.38163 + 5.45906i −0.901021 − 3.94763i 0 1.19234 + 0.272143i
118.6 0.551246 + 0.265466i 0 −2.26056 − 2.83465i −6.43736 + 1.46929i 0 4.80439 − 5.09095i −1.03821 − 4.54868i 0 −3.93862 − 0.898963i
118.7 2.00938 + 0.967666i 0 0.607267 + 0.761488i 5.02682 − 1.14734i 0 −6.22696 − 3.19765i −1.50174 − 6.57955i 0 11.2110 + 2.55884i
118.8 2.79066 + 1.34391i 0 3.48771 + 4.37345i −6.99590 + 1.59677i 0 −0.883869 + 6.94397i 1.09854 + 4.81301i 0 −21.6691 − 4.94582i
118.9 3.50149 + 1.68623i 0 6.92313 + 8.68133i −0.503685 + 0.114963i 0 5.67726 − 4.09496i 6.14338 + 26.9159i 0 −1.95750 − 0.446788i
181.1 −0.836336 + 3.66423i 0 −9.12324 − 4.39352i 1.76769 + 1.40968i 0 6.23947 + 3.17318i 14.3555 − 18.0012i 0 −6.64379 + 5.29824i
181.2 −0.692224 + 3.03283i 0 −5.11501 − 2.46326i −5.95062 − 4.74546i 0 −4.61128 − 5.26651i 3.25311 − 4.07927i 0 18.5114 − 14.7623i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.v.a 54
3.b odd 2 1 49.3.f.a ✓ 54
21.c even 2 1 343.3.f.a 54
21.g even 6 2 343.3.h.d 108
21.h odd 6 2 343.3.h.e 108
49.f odd 14 1 inner 441.3.v.a 54
147.k even 14 1 49.3.f.a ✓ 54
147.l odd 14 1 343.3.f.a 54
147.n odd 42 2 343.3.h.d 108
147.o even 42 2 343.3.h.e 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.3.f.a ✓ 54 3.b odd 2 1
49.3.f.a ✓ 54 147.k even 14 1
343.3.f.a 54 21.c even 2 1
343.3.f.a 54 147.l odd 14 1
343.3.h.d 108 21.g even 6 2
343.3.h.d 108 147.n odd 42 2
343.3.h.e 108 21.h odd 6 2
343.3.h.e 108 147.o even 42 2
441.3.v.a 54 1.a even 1 1 trivial
441.3.v.a 54 49.f odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} - 5 T_{2}^{53} + 43 T_{2}^{52} - 169 T_{2}^{51} + 982 T_{2}^{50} - 3334 T_{2}^{49} + \cdots + 10909749546081 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display