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Properties

Label 1775.4.a.l
Level $1775$
Weight $4$
Character orbit 1775.a
Self dual yes
Analytic conductor $104.728$
Analytic rank $1$
Dimension $46$
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] = [1775,4,Mod(1,1775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1775.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1775 = 5^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1775.a (trivial)

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: yes
Analytic conductor: \(104.728390260\)
Analytic rank: \(1\)
Dimension: \(46\)
Twist minimal: no (minimal twist has level 355)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 154 q^{4} - 58 q^{6} + 320 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 154 q^{4} - 58 q^{6} + 320 q^{9} - 152 q^{11} - 64 q^{14} + 466 q^{16} - 610 q^{19} - 568 q^{21} - 2442 q^{24} - 598 q^{26} - 930 q^{29} - 1172 q^{31} - 2228 q^{34} + 924 q^{36} - 932 q^{39} - 2036 q^{41} - 2130 q^{44} - 1370 q^{46} - 1138 q^{49} - 952 q^{51} - 3316 q^{54} - 868 q^{56} - 1160 q^{59} - 4364 q^{61} + 1624 q^{64} - 974 q^{66} - 744 q^{69} - 3266 q^{71} + 138 q^{74} - 8448 q^{76} - 3326 q^{79} - 3214 q^{81} - 4210 q^{84} - 4052 q^{86} - 4238 q^{89} - 7708 q^{91} - 6926 q^{94} - 17210 q^{96} - 4944 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.54803 9.35334 22.7806 0 −51.8926 −15.0785 −82.0031 60.4851 0
1.2 −5.19777 3.38895 19.0168 0 −17.6150 2.10707 −57.2630 −15.5150 0
1.3 −4.87650 −2.82666 15.7802 0 13.7842 14.2142 −37.9403 −19.0100 0
1.4 −4.78355 −2.45768 14.8824 0 11.7564 4.11478 −32.9222 −20.9598 0
1.5 −4.65269 7.23949 13.6475 0 −33.6831 22.3978 −26.2762 25.4102 0
1.6 −4.53363 3.11710 12.5538 0 −14.1318 −21.9147 −20.6452 −17.2837 0
1.7 −4.52778 −6.05491 12.5008 0 27.4153 19.1460 −20.3787 9.66188 0
1.8 −3.89155 7.88665 7.14418 0 −30.6913 8.61949 3.33048 35.1992 0
1.9 −3.81699 −6.60916 6.56943 0 25.2271 −7.35015 5.46049 16.6810 0
1.10 −3.80253 −0.481665 6.45922 0 1.83154 −10.2602 5.85885 −26.7680 0
1.11 −2.89650 −3.72912 0.389699 0 10.8014 −29.7885 22.0432 −13.0937 0
1.12 −2.79318 −3.50495 −0.198131 0 9.78995 12.4505 22.8989 −14.7154 0
1.13 −2.76592 −9.71028 −0.349690 0 26.8578 19.1937 23.0946 67.2894 0
1.14 −2.41830 1.11611 −2.15181 0 −2.69909 −23.8240 24.5502 −25.7543 0
1.15 −2.29327 4.44417 −2.74092 0 −10.1917 12.7271 24.6318 −7.24932 0
1.16 −2.11726 −7.02811 −3.51723 0 14.8803 −18.9939 24.3849 22.3944 0
1.17 −1.90549 8.43697 −4.36910 0 −16.0766 −14.2166 23.5692 44.1825 0
1.18 −1.54734 4.20341 −5.60574 0 −6.50412 9.22141 21.0527 −9.33130 0
1.19 −1.39627 −8.03399 −6.05043 0 11.2176 18.0438 19.6182 37.5449 0
1.20 −1.00431 −7.78737 −6.99137 0 7.82091 0.895893 15.0559 33.6432 0
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(71\) \(1\)

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 1775.4.a.l 46
5.b even 2 1 inner 1775.4.a.l 46
5.c odd 4 2 355.4.b.a ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
355.4.b.a ✓ 46 5.c odd 4 2
1775.4.a.l 46 1.a even 1 1 trivial
1775.4.a.l 46 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} - 261 T_{2}^{44} + 31548 T_{2}^{42} - 2345289 T_{2}^{40} + 120106160 T_{2}^{38} + \cdots - 17\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\). Copy content Toggle raw display