Properties

Label 41.12.b.a
Level $41$
Weight $12$
Character orbit 41.b
Analytic conductor $31.502$
Analytic rank $0$
Dimension $38$
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] = [41,12,Mod(40,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.40");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 41.b (of order \(2\), degree \(1\), 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: \(31.5020704029\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 66 q^{2} + 38910 q^{4} - 16684 q^{5} - 355686 q^{8} - 2609286 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 66 q^{2} + 38910 q^{4} - 16684 q^{5} - 355686 q^{8} - 2609286 q^{9} - 33776 q^{10} + 35386182 q^{16} + 15486206 q^{18} - 84533616 q^{20} + 22359336 q^{21} + 48961136 q^{23} + 443253546 q^{25} + 27314544 q^{31} - 1549816934 q^{32} + 142959016 q^{33} - 3203096994 q^{36} - 72318812 q^{37} - 1004408304 q^{39} - 1855199808 q^{40} - 2101931682 q^{41} + 4801088572 q^{42} + 2000920056 q^{43} + 3757897980 q^{45} - 4447479312 q^{46} - 16146463854 q^{49} - 15661666326 q^{50} + 3239971344 q^{51} + 4021875464 q^{57} - 24696712216 q^{59} - 5981667852 q^{61} + 30498391104 q^{62} + 48415883318 q^{64} + 80781439420 q^{66} + 123025222122 q^{72} - 33322985444 q^{73} - 167742445248 q^{74} - 29127461384 q^{77} - 184636925680 q^{78} - 36307769616 q^{80} + 125136958990 q^{81} + 115144876422 q^{82} - 107567256312 q^{83} + 221501387324 q^{84} - 117435576984 q^{86} + 94834317696 q^{87} - 230367890688 q^{90} - 124086255328 q^{91} + 522338736912 q^{92} + 336160647078 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −86.6525 468.692i 5460.66 8950.34 40613.3i 66695.3i −295716. −42524.8 −775569.
40.2 −86.6525 468.692i 5460.66 8950.34 40613.3i 66695.3i −295716. −42524.8 −775569.
40.3 −84.4624 636.943i 5085.89 −13046.4 53797.7i 52159.6i −256588. −228549. 1.10193e6
40.4 −84.4624 636.943i 5085.89 −13046.4 53797.7i 52159.6i −256588. −228549. 1.10193e6
40.5 −71.4234 66.9659i 3053.30 −1937.46 4782.93i 230.167i −71802.1 172663. 138380.
40.6 −71.4234 66.9659i 3053.30 −1937.46 4782.93i 230.167i −71802.1 172663. 138380.
40.7 −62.7044 600.500i 1883.84 7906.03 37654.0i 61105.2i 10293.5 −183453. −495743.
40.8 −62.7044 600.500i 1883.84 7906.03 37654.0i 61105.2i 10293.5 −183453. −495743.
40.9 −55.5144 693.355i 1033.85 −5551.71 38491.2i 80006.7i 56299.8 −303594. 308200.
40.10 −55.5144 693.355i 1033.85 −5551.71 38491.2i 80006.7i 56299.8 −303594. 308200.
40.11 −40.3637 515.574i −418.774 −1318.86 20810.4i 11971.1i 99568.1 −88669.2 53234.1
40.12 −40.3637 515.574i −418.774 −1318.86 20810.4i 11971.1i 99568.1 −88669.2 53234.1
40.13 −34.9176 275.392i −828.758 10140.7 9616.05i 26269.1i 100450. 101306. −354088.
40.14 −34.9176 275.392i −828.758 10140.7 9616.05i 26269.1i 100450. 101306. −354088.
40.15 −32.1724 38.7160i −1012.94 −10804.4 1245.58i 61152.6i 98477.7 175648. 347603.
40.16 −32.1724 38.7160i −1012.94 −10804.4 1245.58i 61152.6i 98477.7 175648. 347603.
40.17 −13.4608 693.996i −1866.81 −4832.15 9341.76i 15652.4i 52696.5 −304484. 65044.7
40.18 −13.4608 693.996i −1866.81 −4832.15 9341.76i 15652.4i 52696.5 −304484. 65044.7
40.19 1.59232 160.353i −2045.46 4151.31 255.333i 65140.7i −6518.11 151434. 6610.22
40.20 1.59232 160.353i −2045.46 4151.31 255.333i 65140.7i −6518.11 151434. 6610.22
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.12.b.a 38
41.b even 2 1 inner 41.12.b.a 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.12.b.a 38 1.a even 1 1 trivial
41.12.b.a 38 41.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(41, [\chi])\).