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Properties

Label 1045.2.b.e
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,2,Mod(419,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 42 q^{4} + 12 q^{6} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 42 q^{4} + 12 q^{6} - 40 q^{9} + 10 q^{10} + 30 q^{11} + 4 q^{14} + 4 q^{15} + 66 q^{16} - 30 q^{19} + 10 q^{20} + 14 q^{21} - 22 q^{24} - 6 q^{25} - 30 q^{29} + 14 q^{30} + 26 q^{31} - 12 q^{34} + 6 q^{35} + 78 q^{36} - 64 q^{39} - 20 q^{40} + 22 q^{41} - 42 q^{44} + 6 q^{45} + 28 q^{46} - 60 q^{49} + 64 q^{51} - 62 q^{54} - 32 q^{56} + 14 q^{59} - 28 q^{60} + 78 q^{61} - 90 q^{64} + 40 q^{65} + 12 q^{66} + 28 q^{69} + 12 q^{70} + 20 q^{71} - 42 q^{74} + 50 q^{75} + 42 q^{76} - 102 q^{79} - 40 q^{80} + 42 q^{81} - 98 q^{84} - 2 q^{85} - 52 q^{86} + 8 q^{89} + 22 q^{90} + 56 q^{91} - 40 q^{94} - 74 q^{96} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1 − 2.79219i 1.83665i −5.79631 −2.23016 + 0.162474i 5.12828 0.971452i 10.6000i −0.373298 0.453657 + 6.22702i
419.2 − 2.77451i − 2.74433i −5.69792 0.316248 + 2.21359i −7.61418 3.20844i 10.2599i −4.53135 6.14164 − 0.877434i
419.3 − 2.44393i 0.661177i −3.97280 1.55248 + 1.60929i 1.61587 − 2.05193i 4.82139i 2.56284 3.93299 − 3.79415i
419.4 − 2.39831i − 0.881728i −3.75187 −0.259771 − 2.22093i −2.11465 − 2.42713i 4.20151i 2.22256 −5.32646 + 0.623010i
419.5 − 2.39110i 3.31399i −3.71736 1.04940 − 1.97453i 7.92409 0.907652i 4.10637i −7.98255 −4.72129 − 2.50922i
419.6 − 2.14697i 2.93729i −2.60947 −0.584477 + 2.15833i 6.30628 − 3.64055i 1.30851i −5.62770 4.63386 + 1.25485i
419.7 − 1.98136i − 2.35379i −1.92578 0.201454 − 2.22697i −4.66371 − 1.46477i − 0.147060i −2.54034 −4.41243 − 0.399153i
419.8 − 1.66433i − 1.25863i −0.769991 −1.73458 + 1.41111i −2.09478 4.13429i − 2.04714i 1.41584 2.34855 + 2.88691i
419.9 − 1.65738i 1.46266i −0.746921 2.23602 + 0.0141055i 2.42418 5.18504i − 2.07683i 0.860633 0.0233783 − 3.70595i
419.10 − 1.45151i − 0.0791235i −0.106883 −2.19310 + 0.436262i −0.114849 − 2.96150i − 2.74788i 2.99374 0.633239 + 3.18330i
419.11 − 0.961626i 0.510054i 1.07528 1.58322 + 1.57905i 0.490481 − 0.688271i − 2.95726i 2.73984 1.51846 − 1.52247i
419.12 − 0.661223i − 3.01131i 1.56278 −1.75827 − 1.38148i −1.99115 0.592449i − 2.35579i −6.06799 −0.913465 + 1.16261i
419.13 − 0.598278i − 1.31473i 1.64206 1.14673 + 1.91964i −0.786572 − 0.976473i − 2.17897i 1.27149 1.14848 − 0.686061i
419.14 − 0.379302i 2.60759i 1.85613 2.20907 − 0.346449i 0.989065 − 4.15654i − 1.46264i −3.79954 −0.131409 − 0.837903i
419.15 − 0.202376i 2.47875i 1.95904 −1.53426 − 1.62666i 0.501638 5.06314i − 0.801215i −3.14418 −0.329197 + 0.310498i
419.16 0.202376i − 2.47875i 1.95904 −1.53426 + 1.62666i 0.501638 − 5.06314i 0.801215i −3.14418 −0.329197 − 0.310498i
419.17 0.379302i − 2.60759i 1.85613 2.20907 + 0.346449i 0.989065 4.15654i 1.46264i −3.79954 −0.131409 + 0.837903i
419.18 0.598278i 1.31473i 1.64206 1.14673 − 1.91964i −0.786572 0.976473i 2.17897i 1.27149 1.14848 + 0.686061i
419.19 0.661223i 3.01131i 1.56278 −1.75827 + 1.38148i −1.99115 − 0.592449i 2.35579i −6.06799 −0.913465 − 1.16261i
419.20 0.961626i − 0.510054i 1.07528 1.58322 − 1.57905i 0.490481 0.688271i 2.95726i 2.73984 1.51846 + 1.52247i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.30
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.e ✓ 30
5.b even 2 1 inner 1045.2.b.e ✓ 30
5.c odd 4 2 5225.2.a.bc 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.e ✓ 30 1.a even 1 1 trivial
1045.2.b.e ✓ 30 5.b even 2 1 inner
5225.2.a.bc 30 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 51 T_{2}^{28} + 1161 T_{2}^{26} + 15571 T_{2}^{24} + 136754 T_{2}^{22} + 826847 T_{2}^{20} + \cdots + 2916 \) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display