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Properties

Label 445.2.q.a
Level $445$
Weight $2$
Character orbit 445.q
Analytic conductor $3.553$
Analytic rank $0$
Dimension $440$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [445,2,Mod(44,445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(445, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("445.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 445 = 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 445.q (of order \(22\), degree \(10\), 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: \(3.55334288995\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(44\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q + 26 q^{4} - 9 q^{5} - 22 q^{6} - 84 q^{9} - 39 q^{10} - 22 q^{11} - 22 q^{14} - 11 q^{15} - 46 q^{16} - 22 q^{19} - 33 q^{20} + 10 q^{21} - 66 q^{24} + 39 q^{25} - 22 q^{29} + 66 q^{30} - 22 q^{31}+ \cdots + 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −0.771427 + 2.62724i 1.16888 − 0.751194i −4.62478 − 2.97217i 2.22456 + 0.226566i 1.07186 + 3.65042i −0.483561 − 0.558060i 7.23757 − 6.27139i −0.444256 + 0.972785i −2.31133 + 5.66967i
44.2 −0.746742 + 2.54317i 0.829325 − 0.532975i −4.22759 − 2.71691i −2.12125 + 0.707320i 0.736154 + 2.50711i 2.23135 + 2.57511i 6.06019 − 5.25119i −0.842528 + 1.84488i −0.214808 − 5.92289i
44.3 −0.725159 + 2.46967i −1.46023 + 0.938430i −3.89089 − 2.50052i −0.451654 − 2.18998i −1.25871 − 4.28678i 0.00425364 + 0.00490897i 5.10648 − 4.42479i 0.00536306 − 0.0117435i 5.73604 + 0.472649i
44.4 −0.696237 + 2.37116i −2.06531 + 1.32729i −3.45517 − 2.22050i −1.62475 + 1.53629i −1.70928 − 5.82129i −1.27827 − 1.47520i 3.93547 − 3.41010i 1.25754 − 2.75363i −2.51157 − 4.92218i
44.5 −0.642049 + 2.18662i 2.61798 − 1.68247i −2.68656 − 1.72655i −2.15305 + 0.603651i 1.99805 + 6.80475i −2.52308 − 2.91179i 2.05561 − 1.78119i 2.77685 − 6.08046i 0.0624070 − 5.09546i
44.6 −0.632840 + 2.15526i −1.84334 + 1.18464i −2.56213 − 1.64658i 1.93305 + 1.12398i −1.38667 − 4.72256i 3.18264 + 3.67297i 1.77503 − 1.53808i 0.748280 − 1.63850i −3.64577 + 3.45492i
44.7 −0.611197 + 2.08155i 1.11110 − 0.714063i −2.27677 − 1.46319i 0.0399498 − 2.23571i 0.807252 + 2.74925i −3.18218 − 3.67243i 1.15817 − 1.00356i −0.521580 + 1.14210i 4.62932 + 1.44962i
44.8 −0.591955 + 2.01601i −0.536022 + 0.344480i −2.03139 − 1.30550i 1.17616 + 1.90175i −0.377176 − 1.28454i −2.16747 − 2.50139i 0.658547 − 0.570634i −1.07759 + 2.35960i −4.53018 + 1.24541i
44.9 −0.543405 + 1.85067i 2.62998 − 1.69018i −1.44717 − 0.930042i 1.35122 + 1.78163i 1.69882 + 5.78566i 1.94571 + 2.24547i −0.407776 + 0.353340i 2.81381 − 6.16139i −4.03146 + 1.53252i
44.10 −0.520500 + 1.77266i 0.773218 − 0.496917i −1.18889 − 0.764056i −1.54748 − 1.61410i 0.478405 + 1.62930i 1.73567 + 2.00307i −0.819257 + 0.709891i −0.895306 + 1.96045i 3.66671 − 1.90302i
44.11 −0.501558 + 1.70815i −2.74175 + 1.76201i −0.983713 − 0.632194i 2.07687 − 0.828623i −1.63464 − 5.56707i −1.38596 − 1.59948i −1.11760 + 0.968403i 3.16623 − 6.93309i 0.373743 + 3.96321i
44.12 −0.493536 + 1.68083i 1.73705 − 1.11633i −0.899101 − 0.577817i 1.80908 − 1.31424i 1.01907 + 3.47063i 1.26740 + 1.46266i −1.23288 + 1.06829i 0.524888 − 1.14934i 1.31617 + 3.68938i
44.13 −0.408437 + 1.39101i −0.158874 + 0.102102i −0.0855780 − 0.0549977i −0.427294 + 2.19486i −0.0771348 − 0.262697i 2.16670 + 2.50050i −2.07982 + 1.80217i −1.23143 + 2.69645i −2.87855 − 1.49083i
44.14 −0.354780 + 1.20827i 1.34406 − 0.863772i 0.348457 + 0.223940i −1.72647 + 1.42102i 0.566826 + 1.93043i −0.571711 − 0.659790i −2.29761 + 1.99089i −0.185862 + 0.406982i −1.10445 − 2.59020i
44.15 −0.326810 + 1.11301i −1.87919 + 1.20768i 0.550515 + 0.353795i −1.63684 − 1.52340i −0.730028 − 2.48625i 1.67144 + 1.92894i −2.32703 + 2.01638i 0.826616 − 1.81004i 2.23050 − 1.32396i
44.16 −0.292846 + 0.997342i −0.0142915 + 0.00918458i 0.773575 + 0.497147i 2.20178 + 0.390109i −0.00497496 − 0.0169432i −0.901739 − 1.04066i −2.29349 + 1.98732i −1.24613 + 2.72863i −1.03385 + 2.08168i
44.17 −0.247498 + 0.842901i −1.58587 + 1.01918i 1.03328 + 0.664049i 1.18264 − 1.89773i −0.466565 − 1.58898i −0.944195 − 1.08966i −2.14329 + 1.85717i 0.230017 − 0.503667i 1.30690 + 1.46653i
44.18 −0.148389 + 0.505365i 2.70891 − 1.74091i 1.44913 + 0.931301i −1.19341 − 1.89097i 0.477825 + 1.62732i 0.981314 + 1.13250i −1.48179 + 1.28398i 3.06119 − 6.70306i 1.13272 − 0.322509i
44.19 −0.143528 + 0.488812i −1.89175 + 1.21575i 1.46417 + 0.940965i −0.878884 + 2.05610i −0.322756 − 1.09921i 1.26025 + 1.45441i −1.44014 + 1.24788i 0.854416 − 1.87091i −0.878903 − 0.724718i
44.20 −0.113340 + 0.386000i 0.105745 − 0.0679584i 1.54636 + 0.993783i −0.248423 − 2.22223i 0.0142468 + 0.0485201i −2.20060 − 2.53962i −1.16693 + 1.01115i −1.23968 + 2.71452i 0.885935 + 0.155975i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
89.f even 22 1 inner
445.q even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 445.2.q.a ✓ 440
5.b even 2 1 inner 445.2.q.a ✓ 440
89.f even 22 1 inner 445.2.q.a ✓ 440
445.q even 22 1 inner 445.2.q.a ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.q.a ✓ 440 1.a even 1 1 trivial
445.2.q.a ✓ 440 5.b even 2 1 inner
445.2.q.a ✓ 440 89.f even 22 1 inner
445.2.q.a ✓ 440 445.q even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(445, [\chi])\).