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Properties

Label 775.2.l.a
Level $775$
Weight $2$
Character orbit 775.l
Analytic conductor $6.188$
Analytic rank $0$
Dimension $312$
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] = [775,2,Mod(66,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.66");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.l (of order \(5\), degree \(4\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(78\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - q^{2} - 12 q^{3} - 77 q^{4} - 6 q^{5} - 4 q^{6} - 8 q^{7} + 3 q^{8} + 284 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - q^{2} - 12 q^{3} - 77 q^{4} - 6 q^{5} - 4 q^{6} - 8 q^{7} + 3 q^{8} + 284 q^{9} - 2 q^{10} - 38 q^{11} - 5 q^{12} - q^{13} - 12 q^{14} - 28 q^{15} - 75 q^{16} - q^{17} - 11 q^{18} - 6 q^{19} + 30 q^{20} + 4 q^{21} - 44 q^{22} - 4 q^{23} + 27 q^{24} - 38 q^{25} - 36 q^{26} - 42 q^{27} + 18 q^{28} + 19 q^{29} - 61 q^{30} - 3 q^{31} + 80 q^{32} - 42 q^{33} + 5 q^{34} - 27 q^{35} - 37 q^{36} - q^{37} - 39 q^{38} - 19 q^{39} + 51 q^{40} + 11 q^{41} + 60 q^{42} + 8 q^{43} - 4 q^{44} - 17 q^{45} - 14 q^{46} + 9 q^{48} - 78 q^{49} - 33 q^{50} - 20 q^{51} - 16 q^{52} - 13 q^{53} + 3 q^{54} - 58 q^{55} + 34 q^{56} - 48 q^{57} - 83 q^{58} - 6 q^{59} + 42 q^{60} + 6 q^{61} + 81 q^{62} - 11 q^{63} - 65 q^{64} - 52 q^{65} - 52 q^{66} - 36 q^{67} - 100 q^{68} + 84 q^{69} + 13 q^{70} + 11 q^{71} + q^{72} - 6 q^{73} + 66 q^{74} + 42 q^{75} - 62 q^{76} - 41 q^{77} + 12 q^{78} + 25 q^{79} - 32 q^{80} + 176 q^{81} + 8 q^{82} + 20 q^{83} - 94 q^{84} + 58 q^{85} - 14 q^{86} - 37 q^{87} + 106 q^{88} - 70 q^{89} - 101 q^{90} + 4 q^{91} - 38 q^{92} - 132 q^{93} + 5 q^{94} - 4 q^{95} - 216 q^{96} + 25 q^{97} - 21 q^{98} - 142 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1 −0.850108 + 2.61636i 0.0335880 −4.50464 − 3.27282i −0.0938361 − 2.23410i −0.0285534 + 0.0878783i 1.14534 − 0.832138i 7.94109 − 5.76954i −2.99887 5.92499 + 1.65372i
66.2 −0.830916 + 2.55730i −0.737221 −4.23131 − 3.07423i 1.98662 + 1.02632i 0.612569 − 1.88529i 0.127512 − 0.0926431i 7.02684 − 5.10530i −2.45650 −4.27533 + 4.22759i
66.3 −0.821012 + 2.52681i 0.962596 −4.09270 − 2.97352i −2.04755 + 0.898626i −0.790303 + 2.43230i −3.31612 + 2.40931i 6.57481 − 4.77688i −2.07341 −0.589596 − 5.91157i
66.4 −0.814192 + 2.50583i 3.07079 −3.99822 − 2.90488i 0.834289 + 2.07460i −2.50022 + 7.69487i 0.00523390 − 0.00380265i 6.27126 − 4.55634i 6.42978 −5.87786 + 0.401460i
66.5 −0.801843 + 2.46782i −2.68018 −3.82914 − 2.78204i −2.23417 − 0.0920870i 2.14908 − 6.61419i 0.200846 − 0.145923i 5.73743 − 4.16849i 4.18334 2.01871 − 5.43969i
66.6 −0.753937 + 2.32038i 0.902305 −3.19771 − 2.32327i −1.19230 + 1.89167i −0.680281 + 2.09369i 3.80900 − 2.76740i 3.85408 − 2.80015i −2.18585 −3.49048 − 4.19279i
66.7 −0.748397 + 2.30333i −1.90709 −3.12720 − 2.27204i 0.299715 + 2.21589i 1.42726 − 4.39266i 2.32450 − 1.68885i 3.65499 − 2.65551i 0.636988 −5.32823 − 0.968025i
66.8 −0.739783 + 2.27682i 2.88542 −3.01859 − 2.19314i −1.84316 − 1.26601i −2.13458 + 6.56957i 1.95274 − 1.41875i 3.35292 − 2.43604i 5.32563 4.24600 − 3.25997i
66.9 −0.724742 + 2.23053i 1.62399 −2.83196 − 2.05754i 2.20835 − 0.350955i −1.17698 + 3.62236i −3.37271 + 2.45042i 2.84704 − 2.06849i −0.362641 −0.817674 + 5.18014i
66.10 −0.693523 + 2.13444i −1.80192 −2.45684 − 1.78500i 0.824241 − 2.07861i 1.24968 − 3.84611i −3.50399 + 2.54579i 1.88253 − 1.36773i 0.246932 3.86505 + 3.20086i
66.11 −0.669028 + 2.05906i −2.73182 −2.17408 − 1.57956i 1.45650 + 1.69665i 1.82767 − 5.62498i −3.44913 + 2.50594i 1.20385 − 0.874648i 4.46286 −4.46793 + 1.86391i
66.12 −0.668912 + 2.05870i −1.78335 −2.17277 − 1.57861i −1.88737 − 1.19911i 1.19291 − 3.67139i −0.687098 + 0.499206i 1.20081 − 0.872440i 0.180344 3.73108 − 3.08342i
66.13 −0.668658 + 2.05792i 1.35683 −2.16989 − 1.57651i 2.22325 − 0.239091i −0.907255 + 2.79224i 1.01048 − 0.734160i 1.19411 − 0.867572i −1.15901 −0.994564 + 4.73513i
66.14 −0.629617 + 1.93776i −1.46725 −1.74047 − 1.26452i 1.61427 − 1.54730i 0.923802 − 2.84317i 0.770345 − 0.559688i 0.249452 − 0.181238i −0.847190 1.98193 + 4.10227i
66.15 −0.609424 + 1.87561i 1.83988 −1.52850 − 1.11052i −1.97886 − 1.04120i −1.12127 + 3.45091i −0.955378 + 0.694123i −0.176576 + 0.128290i 0.385174 3.15886 − 3.07705i
66.16 −0.582185 + 1.79178i 0.872528 −1.25350 − 0.910723i 0.615812 − 2.14960i −0.507972 + 1.56338i 3.04627 − 2.21325i −0.686774 + 0.498970i −2.23870 3.49309 + 2.35486i
66.17 −0.548984 + 1.68960i −0.366909 −0.935325 − 0.679554i −0.872340 + 2.05889i 0.201427 − 0.619928i −1.83445 + 1.33281i −1.21287 + 0.881198i −2.86538 −2.99979 − 2.60420i
66.18 −0.522020 + 1.60661i 2.23044 −0.690661 − 0.501795i −0.629778 + 2.14555i −1.16433 + 3.58344i −1.17537 + 0.853954i −1.56660 + 1.13820i 1.97484 −3.11831 − 2.13183i
66.19 −0.488953 + 1.50484i 3.18226 −0.407443 − 0.296025i 2.16238 − 0.569310i −1.55598 + 4.78881i 2.71849 − 1.97510i −1.91550 + 1.39169i 7.12681 −0.200580 + 3.53241i
66.20 −0.458950 + 1.41250i −1.20931 −0.166492 − 0.120963i −2.23591 + 0.0269849i 0.555012 − 1.70815i 3.02036 − 2.19442i −2.15582 + 1.56629i −1.53757 0.988051 − 3.17060i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 66.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
775.l even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.l.a yes 312
25.d even 5 1 775.2.h.a ✓ 312
31.d even 5 1 775.2.h.a ✓ 312
775.l even 5 1 inner 775.2.l.a yes 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.h.a ✓ 312 25.d even 5 1
775.2.h.a ✓ 312 31.d even 5 1
775.2.l.a yes 312 1.a even 1 1 trivial
775.2.l.a yes 312 775.l even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).