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Properties

Label 50.7.f.b
Level $50$
Weight $7$
Character orbit 50.f
Analytic conductor $11.503$
Analytic rank $0$
Dimension $64$
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] = [50,7,Mod(3,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 50.f (of order \(20\), degree \(8\), 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: \(11.5027041810\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 64 q^{2} + 32 q^{3} - 330 q^{5} + 256 q^{6} + 348 q^{7} + 2048 q^{8} - 760 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 64 q^{2} + 32 q^{3} - 330 q^{5} + 256 q^{6} + 348 q^{7} + 2048 q^{8} - 760 q^{9} + 1360 q^{10} + 1728 q^{11} + 1024 q^{12} + 272 q^{13} + 2880 q^{14} - 4470 q^{15} + 16384 q^{16} - 4792 q^{17} + 76544 q^{18} + 44270 q^{19} - 4480 q^{20} + 18388 q^{21} - 34368 q^{22} - 49728 q^{23} + 126840 q^{25} - 70784 q^{26} + 202010 q^{27} + 21504 q^{28} + 24800 q^{29} - 1200 q^{30} - 33252 q^{31} - 262144 q^{32} + 80004 q^{33} - 77600 q^{34} + 156530 q^{35} - 140928 q^{36} - 239982 q^{37} - 28880 q^{38} - 42080 q^{39} + 56320 q^{40} + 44888 q^{41} + 183712 q^{42} + 29472 q^{43} + 178560 q^{44} - 1357790 q^{45} + 298016 q^{46} - 428552 q^{47} - 32768 q^{48} + 85120 q^{50} - 1247832 q^{51} + 8704 q^{52} + 548582 q^{53} - 201280 q^{54} - 29500 q^{55} + 41472 q^{56} - 334520 q^{57} - 262000 q^{58} - 1434200 q^{59} + 139840 q^{60} + 161148 q^{61} + 1371152 q^{62} + 2623222 q^{63} + 2787320 q^{65} - 255088 q^{66} - 197702 q^{67} - 670976 q^{68} + 631680 q^{69} - 3122640 q^{70} + 13938 q^{71} - 456192 q^{72} - 691188 q^{73} + 50890 q^{75} - 231040 q^{76} + 2906396 q^{77} + 4477008 q^{78} + 3138640 q^{79} - 92160 q^{80} + 3878904 q^{81} + 468992 q^{82} - 1890068 q^{83} - 903680 q^{84} - 2110030 q^{85} - 452544 q^{86} + 876200 q^{87} - 1045504 q^{88} - 4091900 q^{89} + 303760 q^{90} + 1157838 q^{91} - 270976 q^{92} - 46716 q^{93} + 3952160 q^{94} - 3060800 q^{95} - 262144 q^{96} + 4387568 q^{97} + 2610256 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.56816 − 5.04029i −7.53135 − 47.5511i −18.8091 + 25.8885i 56.7542 + 111.373i −220.330 + 160.079i −158.728 + 158.728i 178.791 + 28.3177i −1511.06 + 490.974i 415.599 − 572.082i
3.2 −2.56816 − 5.04029i −5.66017 − 35.7369i −18.8091 + 25.8885i −83.9876 − 92.5801i −165.588 + 120.307i −288.370 + 288.370i 178.791 + 28.3177i −551.767 + 179.280i −250.937 + 661.083i
3.3 −2.56816 − 5.04029i −4.62916 − 29.2273i −18.8091 + 25.8885i 117.480 − 42.7023i −135.426 + 98.3928i 437.022 − 437.022i 178.791 + 28.3177i −139.489 + 45.3226i −516.939 − 482.467i
3.4 −2.56816 − 5.04029i −0.867997 − 5.48032i −18.8091 + 25.8885i −62.4621 + 108.275i −25.3933 + 18.4493i 128.992 − 128.992i 178.791 + 28.3177i 664.040 − 215.760i 706.151 + 36.7598i
3.5 −2.56816 − 5.04029i 0.756144 + 4.77411i −18.8091 + 25.8885i 86.6763 − 90.0679i 22.1210 − 16.0718i −203.996 + 203.996i 178.791 + 28.3177i 671.100 − 218.054i −676.567 − 205.566i
3.6 −2.56816 − 5.04029i 2.78783 + 17.6017i −18.8091 + 25.8885i −73.1723 − 101.345i 81.5581 − 59.2554i 55.6165 − 55.6165i 178.791 + 28.3177i 391.273 − 127.132i −322.891 + 629.080i
3.7 −2.56816 − 5.04029i 6.17165 + 38.9663i −18.8091 + 25.8885i 103.570 + 69.9869i 180.552 − 131.178i 178.415 − 178.415i 178.791 + 28.3177i −786.961 + 255.699i 86.7692 − 701.763i
3.8 −2.56816 − 5.04029i 6.70041 + 42.3047i −18.8091 + 25.8885i −114.140 + 50.9609i 196.020 − 142.417i −217.111 + 217.111i 178.791 + 28.3177i −1051.47 + 341.644i 549.988 + 444.425i
13.1 −5.58721 + 0.884927i −23.3017 + 45.7322i 30.4338 − 9.88854i −53.9211 − 112.772i 89.7220 − 276.136i −444.049 + 444.049i −161.289 + 82.1811i −1119.97 − 1541.51i 401.063 + 582.364i
13.2 −5.58721 + 0.884927i −20.0992 + 39.4470i 30.4338 − 9.88854i 120.893 + 31.7777i 77.3910 − 238.185i 452.271 − 452.271i −161.289 + 82.1811i −723.590 − 995.936i −703.577 − 70.5669i
13.3 −5.58721 + 0.884927i −8.42605 + 16.5371i 30.4338 − 9.88854i −50.0727 + 114.533i 32.4440 − 99.8524i −109.038 + 109.038i −161.289 + 82.1811i 226.020 + 311.089i 178.414 − 684.228i
13.4 −5.58721 + 0.884927i −7.43527 + 14.5925i 30.4338 − 9.88854i −119.974 − 35.0902i 28.6291 − 88.1112i 339.866 − 339.866i −161.289 + 82.1811i 270.837 + 372.775i 701.370 + 89.8882i
13.5 −5.58721 + 0.884927i 0.641223 − 1.25847i 30.4338 − 9.88854i 38.7204 − 118.852i −2.46899 + 7.59878i −71.7898 + 71.7898i −161.289 + 82.1811i 427.323 + 588.159i −111.164 + 698.314i
13.6 −5.58721 + 0.884927i 10.1653 − 19.9505i 30.4338 − 9.88854i 98.5109 + 76.9455i −39.1408 + 120.463i −104.115 + 104.115i −161.289 + 82.1811i 133.807 + 184.170i −618.492 − 342.736i
13.7 −5.58721 + 0.884927i 16.9188 − 33.2050i 30.4338 − 9.88854i 33.0662 − 120.547i −65.1448 + 200.495i 363.736 − 363.736i −161.289 + 82.1811i −387.830 − 533.803i −78.0721 + 702.784i
13.8 −5.58721 + 0.884927i 17.1882 − 33.7337i 30.4338 − 9.88854i −124.892 − 5.19064i −66.1821 + 203.688i −345.465 + 345.465i −161.289 + 82.1811i −414.034 − 569.869i 702.392 − 81.5193i
17.1 −2.56816 + 5.04029i −7.53135 + 47.5511i −18.8091 − 25.8885i 56.7542 − 111.373i −220.330 − 160.079i −158.728 − 158.728i 178.791 − 28.3177i −1511.06 − 490.974i 415.599 + 572.082i
17.2 −2.56816 + 5.04029i −5.66017 + 35.7369i −18.8091 − 25.8885i −83.9876 + 92.5801i −165.588 − 120.307i −288.370 − 288.370i 178.791 − 28.3177i −551.767 − 179.280i −250.937 − 661.083i
17.3 −2.56816 + 5.04029i −4.62916 + 29.2273i −18.8091 − 25.8885i 117.480 + 42.7023i −135.426 − 98.3928i 437.022 + 437.022i 178.791 − 28.3177i −139.489 − 45.3226i −516.939 + 482.467i
17.4 −2.56816 + 5.04029i −0.867997 + 5.48032i −18.8091 − 25.8885i −62.4621 − 108.275i −25.3933 − 18.4493i 128.992 + 128.992i 178.791 − 28.3177i 664.040 + 215.760i 706.151 − 36.7598i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.7.f.b ✓ 64
25.f odd 20 1 inner 50.7.f.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.7.f.b ✓ 64 1.a even 1 1 trivial
50.7.f.b ✓ 64 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 32 T_{3}^{63} + 892 T_{3}^{62} - 147166 T_{3}^{61} - 3588791 T_{3}^{60} + \cdots + 21\!\cdots\!36 \) acting on \(S_{7}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display