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Properties

Label 50.5.f.b
Level $50$
Weight $5$
Character orbit 50.f
Analytic conductor $5.168$
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] = [50,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(5.16849815419\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{2} - 10 q^{3} + 30 q^{5} + 40 q^{6} - 10 q^{7} - 160 q^{8} - 400 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{2} - 10 q^{3} + 30 q^{5} + 40 q^{6} - 10 q^{7} - 160 q^{8} - 400 q^{9} + 40 q^{10} - 80 q^{11} - 80 q^{12} - 90 q^{13} - 300 q^{14} - 90 q^{15} + 640 q^{16} - 300 q^{17} - 1840 q^{18} + 4300 q^{19} - 720 q^{20} + 1240 q^{21} + 960 q^{22} + 90 q^{23} + 1280 q^{25} - 1640 q^{26} - 4810 q^{27} - 1520 q^{28} - 3300 q^{29} - 3960 q^{30} + 1440 q^{31} + 5120 q^{32} + 360 q^{33} + 4900 q^{34} + 10360 q^{35} - 240 q^{36} - 150 q^{37} - 480 q^{38} - 11600 q^{39} - 160 q^{40} - 2560 q^{41} + 1120 q^{42} + 5670 q^{43} + 3200 q^{44} + 4720 q^{45} - 5460 q^{46} + 2790 q^{47} + 640 q^{48} + 1180 q^{50} - 9600 q^{51} - 720 q^{52} + 10010 q^{53} - 1600 q^{54} - 15300 q^{55} + 2880 q^{56} - 18560 q^{57} - 6680 q^{58} - 25250 q^{59} - 5600 q^{60} + 2760 q^{61} + 5820 q^{62} - 4940 q^{63} + 23360 q^{65} + 5060 q^{66} + 9750 q^{67} + 12000 q^{68} + 79350 q^{69} + 22160 q^{70} - 35900 q^{71} + 12480 q^{72} + 28310 q^{73} - 15170 q^{75} + 1920 q^{76} - 49980 q^{77} + 1380 q^{78} - 48200 q^{79} + 640 q^{80} + 87270 q^{81} - 29280 q^{82} - 107080 q^{83} - 42800 q^{84} - 2600 q^{85} + 3120 q^{86} + 36110 q^{87} + 2720 q^{88} + 22000 q^{89} + 4540 q^{90} - 33240 q^{91} + 15520 q^{92} + 36510 q^{93} + 32700 q^{94} + 31600 q^{95} - 2560 q^{96} + 3590 q^{97} - 8380 q^{98}+O(q^{100}) \) 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 1.28408 + 2.52015i −2.52774 − 15.9595i −4.70228 + 6.47214i −24.3640 + 5.60307i 36.9745 − 26.8636i −37.7016 + 37.7016i −22.3488 − 3.53971i −171.282 + 55.6527i −45.4059 − 54.2061i
3.2 1.28408 + 2.52015i −0.929779 − 5.87039i −4.70228 + 6.47214i 5.13137 − 24.4677i 13.6003 − 9.88123i 28.3982 − 28.3982i −22.3488 − 3.53971i 43.4386 − 14.1140i 68.2513 − 18.4867i
3.3 1.28408 + 2.52015i 0.0674480 + 0.425850i −4.70228 + 6.47214i 21.5386 + 12.6921i −0.986596 + 0.716804i −43.1930 + 43.1930i −22.3488 − 3.53971i 76.8588 − 24.9729i −4.32858 + 70.5781i
3.4 1.28408 + 2.52015i 1.33463 + 8.42650i −4.70228 + 6.47214i −25.0000 − 0.00814442i −19.5222 + 14.1837i −16.9570 + 16.9570i −22.3488 − 3.53971i 7.81094 − 2.53793i −32.0814 − 63.0141i
3.5 1.28408 + 2.52015i 2.76564 + 17.4616i −4.70228 + 6.47214i 24.1318 + 6.53103i −40.4545 + 29.3919i 46.1276 − 46.1276i −22.3488 − 3.53971i −220.223 + 71.5547i 14.5280 + 69.2021i
13.1 2.79360 − 0.442463i −4.63493 + 9.09656i 7.60845 − 2.47214i −23.8290 + 7.56181i −8.92326 + 27.4630i −36.5072 + 36.5072i 20.1612 − 10.2726i −13.6542 − 18.7934i −63.2228 + 31.6681i
13.2 2.79360 − 0.442463i −2.16532 + 4.24968i 7.60845 − 2.47214i 22.0031 + 11.8686i −4.16872 + 12.8300i −9.14924 + 9.14924i 20.1612 − 10.2726i 34.2394 + 47.1265i 66.7194 + 23.4205i
13.3 2.79360 − 0.442463i −0.343879 + 0.674901i 7.60845 − 2.47214i −8.93828 − 23.3475i −0.662043 + 2.03756i 60.6178 − 60.6178i 20.1612 − 10.2726i 47.2734 + 65.0662i −35.3005 − 61.2689i
13.4 2.79360 − 0.442463i 5.68645 − 11.1603i 7.60845 − 2.47214i −0.741874 − 24.9890i 10.9477 − 33.6935i −50.1479 + 50.1479i 20.1612 − 10.2726i −44.6057 − 61.3945i −13.1292 − 69.4811i
13.5 2.79360 − 0.442463i 5.94169 − 11.6612i 7.60845 − 2.47214i 6.69031 + 24.0882i 11.4391 − 35.2058i 25.6642 − 25.6642i 20.1612 − 10.2726i −53.0697 − 73.0442i 29.3482 + 64.3326i
17.1 1.28408 − 2.52015i −2.52774 + 15.9595i −4.70228 − 6.47214i −24.3640 − 5.60307i 36.9745 + 26.8636i −37.7016 − 37.7016i −22.3488 + 3.53971i −171.282 − 55.6527i −45.4059 + 54.2061i
17.2 1.28408 − 2.52015i −0.929779 + 5.87039i −4.70228 − 6.47214i 5.13137 + 24.4677i 13.6003 + 9.88123i 28.3982 + 28.3982i −22.3488 + 3.53971i 43.4386 + 14.1140i 68.2513 + 18.4867i
17.3 1.28408 − 2.52015i 0.0674480 − 0.425850i −4.70228 − 6.47214i 21.5386 − 12.6921i −0.986596 − 0.716804i −43.1930 − 43.1930i −22.3488 + 3.53971i 76.8588 + 24.9729i −4.32858 − 70.5781i
17.4 1.28408 − 2.52015i 1.33463 − 8.42650i −4.70228 − 6.47214i −25.0000 + 0.00814442i −19.5222 − 14.1837i −16.9570 − 16.9570i −22.3488 + 3.53971i 7.81094 + 2.53793i −32.0814 + 63.0141i
17.5 1.28408 − 2.52015i 2.76564 − 17.4616i −4.70228 − 6.47214i 24.1318 − 6.53103i −40.4545 − 29.3919i 46.1276 + 46.1276i −22.3488 + 3.53971i −220.223 − 71.5547i 14.5280 − 69.2021i
23.1 0.442463 − 2.79360i −12.4623 + 6.34987i −7.60845 − 2.47214i 23.9727 + 7.09291i 12.2249 + 37.6244i 47.1277 − 47.1277i −10.2726 + 20.1612i 67.3781 − 92.7380i 30.4218 − 63.8319i
23.2 0.442463 − 2.79360i −6.29238 + 3.20613i −7.60845 − 2.47214i 9.03489 − 23.3103i 6.17250 + 18.9970i −45.9477 + 45.9477i −10.2726 + 20.1612i −18.2958 + 25.1821i −61.1222 − 35.5539i
23.3 0.442463 − 2.79360i −2.26592 + 1.15455i −7.60845 − 2.47214i −15.2939 + 19.7762i 2.22276 + 6.84094i −13.4369 + 13.4369i −10.2726 + 20.1612i −43.8092 + 60.2981i 48.4797 + 51.4754i
23.4 0.442463 − 2.79360i 9.49853 − 4.83974i −7.60845 − 2.47214i 20.0138 + 14.9816i −9.31757 − 28.6765i 51.5120 − 51.5120i −10.2726 + 20.1612i 19.1883 − 26.4104i 50.7081 − 49.2817i
23.5 0.442463 − 2.79360i 10.1283 − 5.16061i −7.60845 − 2.47214i −13.1134 − 21.2847i −9.93531 − 30.5777i 6.89840 − 6.89840i −10.2726 + 20.1612i 28.3392 − 39.0056i −65.2633 + 27.2159i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
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 50.5.f.b ✓ 40
25.f odd 20 1 inner 50.5.f.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.5.f.b ✓ 40 1.a even 1 1 trivial
50.5.f.b ✓ 40 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 10 T_{3}^{39} + 250 T_{3}^{38} + 5930 T_{3}^{37} - 12695 T_{3}^{36} + 118388 T_{3}^{35} + \cdots + 27\!\cdots\!96 \) acting on \(S_{5}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display