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Properties

Label 77.4.f.b
Level $77$
Weight $4$
Character orbit 77.f
Analytic conductor $4.543$
Analytic rank $0$
Dimension $40$
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] = [77,4,Mod(15,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.15");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.f (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: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9} + 216 q^{10} - 42 q^{11} + 288 q^{12} + 49 q^{14} - 108 q^{15} - 98 q^{16} - 268 q^{17} - 173 q^{18} - 369 q^{19} - 549 q^{20} - 154 q^{21} + 14 q^{22} + 722 q^{23} + 588 q^{24} + 130 q^{25} - 221 q^{26} - 33 q^{27} + 413 q^{28} - 256 q^{29} - 368 q^{30} - 666 q^{31} + 892 q^{32} + 1275 q^{33} + 662 q^{34} + 168 q^{35} + 1008 q^{36} - 1883 q^{37} + 313 q^{38} - 10 q^{39} - 1034 q^{40} - 138 q^{41} - 210 q^{42} + 1252 q^{43} + 408 q^{44} + 1140 q^{45} - 1888 q^{46} - 738 q^{47} - 3636 q^{48} - 490 q^{49} - 193 q^{50} + 1857 q^{51} + 1769 q^{52} - 1847 q^{53} + 6808 q^{54} - 1544 q^{55} + 504 q^{56} - 2423 q^{57} + 2048 q^{58} - 2533 q^{59} + 1508 q^{60} + 558 q^{61} - 3811 q^{62} + 1197 q^{63} + 1794 q^{64} - 1908 q^{65} - 10372 q^{66} + 3880 q^{67} - 11248 q^{68} - 228 q^{69} - 882 q^{70} - 393 q^{71} + 7287 q^{72} + 1548 q^{73} + 3883 q^{74} + 4107 q^{75} + 10450 q^{76} - 931 q^{77} + 8274 q^{78} - 1951 q^{79} + 4549 q^{80} - 6879 q^{81} + 2862 q^{82} + 4759 q^{83} + 2044 q^{84} - 1050 q^{85} + 3715 q^{86} - 268 q^{87} - 18778 q^{88} + 7102 q^{89} - 16648 q^{90} + 70 q^{91} - 1259 q^{92} + 646 q^{93} + 10296 q^{94} + 1834 q^{95} - 6218 q^{96} - 4289 q^{97} - 98 q^{98} - 8829 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −4.23611 − 3.07771i −2.01219 + 6.19288i 6.00015 + 18.4666i −14.2597 + 10.3603i 27.5837 − 20.0407i −2.16312 − 6.65740i 18.4731 − 56.8544i −12.4593 − 9.05225i 92.2919
15.2 −3.13241 − 2.27583i 2.57135 − 7.91381i 2.16046 + 6.64921i −14.0082 + 10.1776i −26.0650 + 18.9373i −2.16312 − 6.65740i −1.20677 + 3.71406i −34.1731 − 24.8282i 67.0418
15.3 −2.08555 − 1.51524i −1.27325 + 3.91867i −0.418581 − 1.28826i 0.733295 − 0.532770i 8.59314 − 6.24328i −2.16312 − 6.65740i −7.45191 + 22.9346i 8.10867 + 5.89129i −2.33659
15.4 −1.91048 − 1.38804i 1.19654 − 3.68258i −0.748877 − 2.30480i 12.0553 − 8.75871i −7.39754 + 5.37463i −2.16312 − 6.65740i −7.60635 + 23.4100i 9.71380 + 7.05749i −35.1889
15.5 −0.281932 − 0.204836i −2.51790 + 7.74930i −2.43461 − 7.49295i −4.21014 + 3.05884i 2.29721 − 1.66902i −2.16312 − 6.65740i −1.70994 + 5.26265i −31.8684 − 23.1538i 1.81354
15.6 1.06715 + 0.775331i 2.12611 − 6.54350i −1.93446 − 5.95366i −8.06046 + 5.85627i 7.34226 − 5.33447i −2.16312 − 6.65740i 5.81262 − 17.8894i −16.4536 − 11.9543i −13.1423
15.7 2.07504 + 1.50761i 0.186723 − 0.574674i −0.439213 − 1.35176i 11.4299 − 8.30432i 1.25384 − 0.910969i −2.16312 − 6.65740i 7.46730 − 22.9820i 21.5481 + 15.6556i 36.2372
15.8 3.62073 + 2.63062i −0.489110 + 1.50533i 3.71743 + 11.4411i −16.0185 + 11.6381i −5.73088 + 4.16373i −2.16312 − 6.65740i −5.57330 + 17.1529i 19.8167 + 14.3977i −88.6140
15.9 3.84003 + 2.78994i −2.63098 + 8.09734i 4.48990 + 13.8185i 11.7384 − 8.52841i −32.6942 + 23.7537i −2.16312 − 6.65740i −9.57738 + 29.4761i −36.8013 − 26.7377i 68.8694
15.10 4.16155 + 3.02354i 2.81484 − 8.66318i 5.70454 + 17.5568i 1.18370 − 0.860011i 37.9076 − 27.5415i −2.16312 − 6.65740i −16.6274 + 51.1738i −45.2840 − 32.9007i 7.52632
36.1 −4.23611 + 3.07771i −2.01219 − 6.19288i 6.00015 − 18.4666i −14.2597 − 10.3603i 27.5837 + 20.0407i −2.16312 + 6.65740i 18.4731 + 56.8544i −12.4593 + 9.05225i 92.2919
36.2 −3.13241 + 2.27583i 2.57135 + 7.91381i 2.16046 − 6.64921i −14.0082 − 10.1776i −26.0650 − 18.9373i −2.16312 + 6.65740i −1.20677 − 3.71406i −34.1731 + 24.8282i 67.0418
36.3 −2.08555 + 1.51524i −1.27325 − 3.91867i −0.418581 + 1.28826i 0.733295 + 0.532770i 8.59314 + 6.24328i −2.16312 + 6.65740i −7.45191 − 22.9346i 8.10867 − 5.89129i −2.33659
36.4 −1.91048 + 1.38804i 1.19654 + 3.68258i −0.748877 + 2.30480i 12.0553 + 8.75871i −7.39754 − 5.37463i −2.16312 + 6.65740i −7.60635 − 23.4100i 9.71380 − 7.05749i −35.1889
36.5 −0.281932 + 0.204836i −2.51790 − 7.74930i −2.43461 + 7.49295i −4.21014 − 3.05884i 2.29721 + 1.66902i −2.16312 + 6.65740i −1.70994 − 5.26265i −31.8684 + 23.1538i 1.81354
36.6 1.06715 − 0.775331i 2.12611 + 6.54350i −1.93446 + 5.95366i −8.06046 − 5.85627i 7.34226 + 5.33447i −2.16312 + 6.65740i 5.81262 + 17.8894i −16.4536 + 11.9543i −13.1423
36.7 2.07504 − 1.50761i 0.186723 + 0.574674i −0.439213 + 1.35176i 11.4299 + 8.30432i 1.25384 + 0.910969i −2.16312 + 6.65740i 7.46730 + 22.9820i 21.5481 − 15.6556i 36.2372
36.8 3.62073 − 2.63062i −0.489110 − 1.50533i 3.71743 − 11.4411i −16.0185 − 11.6381i −5.73088 − 4.16373i −2.16312 + 6.65740i −5.57330 − 17.1529i 19.8167 − 14.3977i −88.6140
36.9 3.84003 − 2.78994i −2.63098 − 8.09734i 4.48990 − 13.8185i 11.7384 + 8.52841i −32.6942 − 23.7537i −2.16312 + 6.65740i −9.57738 − 29.4761i −36.8013 + 26.7377i 68.8694
36.10 4.16155 − 3.02354i 2.81484 + 8.66318i 5.70454 − 17.5568i 1.18370 + 0.860011i 37.9076 + 27.5415i −2.16312 + 6.65740i −16.6274 − 51.1738i −45.2840 + 32.9007i 7.52632
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.f.b ✓ 40
11.c even 5 1 inner 77.4.f.b ✓ 40
11.c even 5 1 847.4.a.q 20
11.d odd 10 1 847.4.a.r 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.f.b ✓ 40 1.a even 1 1 trivial
77.4.f.b ✓ 40 11.c even 5 1 inner
847.4.a.q 20 11.c even 5 1
847.4.a.r 20 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 8 T_{2}^{39} + 89 T_{2}^{38} - 480 T_{2}^{37} + 3827 T_{2}^{36} - 17852 T_{2}^{35} + \cdots + 11\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display