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Properties

Label 755.2.b.d
Level $755$
Weight $2$
Character orbit 755.b
Analytic conductor $6.029$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [755,2,Mod(454,755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(755, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("755.454");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 755 = 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 755.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: \(6.02870535261\)
Analytic rank: \(0\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 50 q^{4} - q^{5} + 16 q^{6} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 50 q^{4} - q^{5} + 16 q^{6} - 70 q^{9} - q^{10} + 20 q^{11} - 30 q^{14} + 6 q^{15} + 58 q^{16} - 18 q^{19} - q^{20} + 30 q^{21} - 56 q^{24} + 7 q^{25} + 68 q^{26} - 30 q^{29} + 2 q^{30} + 12 q^{31} - 32 q^{34} + 3 q^{35} + 106 q^{36} - 10 q^{39} - 3 q^{40} + 102 q^{41} - 30 q^{44} - 19 q^{45} - 4 q^{46} - 84 q^{49} - 59 q^{50} + 48 q^{51} - 22 q^{54} - 23 q^{55} + 110 q^{56} - 78 q^{59} - 105 q^{60} + 28 q^{61} - 24 q^{64} - 31 q^{65} + 48 q^{66} + 18 q^{69} - 51 q^{70} + 78 q^{71} - 14 q^{74} - 55 q^{75} + 34 q^{76} - 14 q^{79} - 51 q^{80} + 160 q^{81} + 52 q^{84} - 26 q^{85} + 76 q^{86} - 182 q^{89} - 120 q^{90} + 30 q^{91} + 170 q^{94} - 30 q^{95} + 130 q^{96} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
454.1 − 2.78309i 1.70318i −5.74559 0.101464 − 2.23376i 4.74010 1.17703i 10.4243i 0.0991831 −6.21677 − 0.282384i
454.2 − 2.66530i − 0.611921i −5.10383 −2.21494 − 0.306687i −1.63095 − 1.02412i 8.27264i 2.62555 −0.817413 + 5.90347i
454.3 − 2.54808i − 0.456797i −4.49272 2.15557 + 0.594556i −1.16396 − 3.74960i 6.35164i 2.79134 1.51498 − 5.49258i
454.4 − 2.50245i 2.26285i −4.26228 −0.605296 + 2.15258i 5.66269 − 2.41817i 5.66125i −2.12051 5.38674 + 1.51473i
454.5 − 2.38097i 3.42268i −3.66904 1.99014 − 1.01948i 8.14931 − 2.42057i 3.97393i −8.71474 −2.42735 − 4.73848i
454.6 − 2.36549i − 3.15663i −3.59552 −1.54889 + 1.61274i −7.46697 − 4.87903i 3.77419i −6.96434 3.81491 + 3.66389i
454.7 − 2.20823i 2.61570i −2.87629 −2.16094 − 0.574742i 5.77608 − 0.554379i 1.93506i −3.84190 −1.26916 + 4.77186i
454.8 − 2.11165i − 0.873101i −2.45906 1.82237 − 1.29575i −1.84368 4.20624i 0.969368i 2.23769 −2.73617 − 3.84820i
454.9 − 1.95648i − 2.37441i −1.82783 1.29657 + 1.82179i −4.64550 − 0.105481i − 0.336847i −2.63783 3.56430 − 2.53672i
454.10 − 1.88281i 2.79975i −1.54496 0.266648 + 2.22011i 5.27139 3.81976i − 0.856747i −4.83860 4.18004 − 0.502046i
454.11 − 1.84174i − 2.98474i −1.39202 −2.09409 + 0.784079i −5.49712 5.05690i − 1.11975i −5.90867 1.44407 + 3.85678i
454.12 − 1.72651i 1.34061i −0.980836 1.22771 − 1.86889i 2.31458 − 2.82799i − 1.75960i 1.20276 −3.22665 − 2.11964i
454.13 − 1.27883i 0.440952i 0.364582 −2.12348 + 0.700586i 0.563905 2.70165i − 3.02391i 2.80556 0.895933 + 2.71558i
454.14 − 1.21263i − 3.11092i 0.529523 2.22909 − 0.176527i −3.77240 0.0525519i − 3.06738i −6.67782 −0.214062 − 2.70307i
454.15 − 1.15275i − 1.98556i 0.671165 2.18609 − 0.470128i −2.28886 − 4.70723i − 3.07919i −0.942447 −0.541940 − 2.52001i
454.16 − 0.980637i − 0.810289i 1.03835 1.03135 − 1.98402i −0.794600 2.23472i − 2.97952i 2.34343 −1.94560 − 1.01138i
454.17 − 0.844304i 0.365830i 1.28715 −1.77485 − 1.36011i 0.308872 2.69253i − 2.77536i 2.86617 −1.14835 + 1.49851i
454.18 − 0.783395i 2.94365i 1.38629 −1.21550 − 1.87685i 2.30604 − 2.34749i − 2.65280i −5.66505 −1.47031 + 0.952216i
454.19 − 0.368065i 1.94652i 1.86453 −1.83982 + 1.27086i 0.716445 − 4.27684i − 1.42240i −0.788922 0.467758 + 0.677173i
454.20 − 0.354420i 3.10701i 1.87439 1.18538 + 1.89602i 1.10119 − 1.68085i − 1.37316i −6.65351 0.671987 − 0.420122i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 454.44
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 755.2.b.d ✓ 44
5.b even 2 1 inner 755.2.b.d ✓ 44
5.c odd 4 2 3775.2.a.x 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.b.d ✓ 44 1.a even 1 1 trivial
755.2.b.d ✓ 44 5.b even 2 1 inner
3775.2.a.x 44 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} + 69 T_{2}^{42} + 2203 T_{2}^{40} + 43198 T_{2}^{38} + 582305 T_{2}^{36} + 5723215 T_{2}^{34} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\). Copy content Toggle raw display