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Properties

Label 50.7.f.a
Level $50$
Weight $7$
Character orbit 50.f
Analytic conductor $11.503$
Analytic rank $0$
Dimension $56$
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: \(56\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 56 q^{2} + 32 q^{3} + 150 q^{5} - 256 q^{6} + 348 q^{7} - 1792 q^{8} + 760 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 56 q^{2} + 32 q^{3} + 150 q^{5} - 256 q^{6} + 348 q^{7} - 1792 q^{8} + 760 q^{9} - 3640 q^{10} - 1728 q^{11} + 1024 q^{12} + 4342 q^{13} - 2880 q^{14} + 1150 q^{15} + 14336 q^{16} - 3802 q^{17} - 40096 q^{18} - 5870 q^{19} + 10880 q^{20} - 18388 q^{21} - 34368 q^{22} - 49728 q^{23} - 9930 q^{25} + 38224 q^{26} - 11890 q^{27} + 21504 q^{28} + 24800 q^{29} - 11280 q^{30} + 33252 q^{31} + 229376 q^{32} - 253196 q^{33} + 44600 q^{34} - 71050 q^{35} - 92352 q^{36} + 21708 q^{37} - 28880 q^{38} + 456880 q^{39} - 6400 q^{40} - 153448 q^{41} + 183712 q^{42} + 29472 q^{43} - 178560 q^{44} + 260180 q^{45} - 298016 q^{46} - 428552 q^{47} - 32768 q^{48} + 1480 q^{50} + 1247832 q^{51} + 138944 q^{52} + 453752 q^{53} + 201280 q^{54} + 936580 q^{55} - 41472 q^{56} - 334520 q^{57} - 211280 q^{58} - 2219000 q^{59} - 562240 q^{60} + 227292 q^{61} + 881552 q^{62} + 1947162 q^{63} - 178510 q^{65} + 255088 q^{66} - 682522 q^{67} - 702656 q^{68} - 6237280 q^{69} - 730320 q^{70} + 1503822 q^{71} + 67328 q^{72} - 1228818 q^{73} + 1342670 q^{75} + 231040 q^{76} + 3992156 q^{77} + 1901008 q^{78} + 1440560 q^{79} + 276480 q^{80} - 3845674 q^{81} + 903232 q^{82} + 3392812 q^{83} - 555520 q^{84} + 977770 q^{85} + 452544 q^{86} - 4839720 q^{87} + 388096 q^{88} - 4384250 q^{89} - 3630840 q^{90} - 1157838 q^{91} + 3012224 q^{92} + 996564 q^{93} - 3952160 q^{94} + 360040 q^{95} + 262144 q^{96} - 4583342 q^{97} - 1324184 q^{98}+O(q^{100}) \) 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 −5.41484 − 34.1880i −18.8091 + 25.8885i 119.511 + 36.6352i 158.411 − 115.093i −87.2994 + 87.2994i −178.791 − 28.3177i −446.177 + 144.972i 122.271 + 696.455i
3.2 2.56816 + 5.04029i −5.33209 − 33.6655i −18.8091 + 25.8885i −50.6969 + 114.258i 155.990 − 113.334i 125.471 − 125.471i −178.791 − 28.3177i −411.616 + 133.742i −706.090 + 37.9045i
3.3 2.56816 + 5.04029i −3.91496 − 24.7181i −18.8091 + 25.8885i 10.2190 − 124.582i 114.532 − 83.2124i −433.394 + 433.394i −178.791 − 28.3177i 97.6643 − 31.7330i 654.172 − 268.439i
3.4 2.56816 + 5.04029i −0.705848 − 4.45655i −18.8091 + 25.8885i −120.731 − 32.3891i 20.6496 − 15.0028i 155.694 − 155.694i −178.791 − 28.3177i 673.958 − 218.982i −146.805 − 691.699i
3.5 2.56816 + 5.04029i 1.66700 + 10.5250i −18.8091 + 25.8885i 108.564 − 61.9577i −48.7680 + 35.4321i 413.261 − 413.261i −178.791 − 28.3177i 585.323 − 190.183i 591.096 + 388.080i
3.6 2.56816 + 5.04029i 4.36021 + 27.5293i −18.8091 + 25.8885i 18.8739 + 123.567i −127.558 + 92.6762i −92.2825 + 92.2825i −178.791 − 28.3177i −45.5281 + 14.7930i −574.342 + 412.469i
3.7 2.56816 + 5.04029i 7.06791 + 44.6250i −18.8091 + 25.8885i −57.5851 − 110.946i −206.772 + 150.228i −149.611 + 149.611i −178.791 − 28.3177i −1248.12 + 405.537i 411.311 − 575.172i
13.1 5.58721 − 0.884927i −22.1944 + 43.5589i 30.4338 − 9.88854i 72.2199 + 102.026i −85.4582 + 263.013i −117.655 + 117.655i 161.289 − 82.1811i −976.294 − 1343.75i 493.793 + 506.131i
13.2 5.58721 − 0.884927i −15.2958 + 30.0198i 30.4338 − 9.88854i −112.996 − 53.4490i −58.8957 + 181.262i −7.35666 + 7.35666i 161.289 − 82.1811i −238.728 − 328.582i −678.633 − 198.637i
13.3 5.58721 − 0.884927i −11.2548 + 22.0888i 30.4338 − 9.88854i 78.3042 − 97.4344i −43.3361 + 133.375i 100.203 − 100.203i 161.289 − 82.1811i 67.2496 + 92.5611i 351.279 − 613.680i
13.4 5.58721 − 0.884927i 0.0529059 − 0.103834i 30.4338 − 9.88854i −80.4191 + 95.6963i 0.203711 − 0.626959i 359.821 − 359.821i 161.289 − 82.1811i 428.487 + 589.762i −364.634 + 605.840i
13.5 5.58721 − 0.884927i 2.71482 − 5.32814i 30.4338 − 9.88854i −12.8429 + 124.338i 10.4533 − 32.1718i −395.568 + 395.568i 161.289 − 82.1811i 407.477 + 560.844i 38.2745 + 706.070i
13.6 5.58721 − 0.884927i 13.4620 − 26.4208i 30.4338 − 9.88854i 123.280 − 20.6643i 51.8348 − 159.531i 33.9442 − 33.9442i 161.289 − 82.1811i −88.3344 − 121.582i 670.505 − 224.550i
13.7 5.58721 − 0.884927i 18.1664 − 35.6536i 30.4338 − 9.88854i −117.747 − 41.9610i 69.9488 − 215.280i 108.028 − 108.028i 161.289 − 82.1811i −512.667 − 705.625i −695.008 − 130.248i
17.1 2.56816 − 5.04029i −5.41484 + 34.1880i −18.8091 − 25.8885i 119.511 − 36.6352i 158.411 + 115.093i −87.2994 − 87.2994i −178.791 + 28.3177i −446.177 − 144.972i 122.271 − 696.455i
17.2 2.56816 − 5.04029i −5.33209 + 33.6655i −18.8091 − 25.8885i −50.6969 − 114.258i 155.990 + 113.334i 125.471 + 125.471i −178.791 + 28.3177i −411.616 − 133.742i −706.090 − 37.9045i
17.3 2.56816 − 5.04029i −3.91496 + 24.7181i −18.8091 − 25.8885i 10.2190 + 124.582i 114.532 + 83.2124i −433.394 − 433.394i −178.791 + 28.3177i 97.6643 + 31.7330i 654.172 + 268.439i
17.4 2.56816 − 5.04029i −0.705848 + 4.45655i −18.8091 − 25.8885i −120.731 + 32.3891i 20.6496 + 15.0028i 155.694 + 155.694i −178.791 + 28.3177i 673.958 + 218.982i −146.805 + 691.699i
17.5 2.56816 − 5.04029i 1.66700 − 10.5250i −18.8091 − 25.8885i 108.564 + 61.9577i −48.7680 − 35.4321i 413.261 + 413.261i −178.791 + 28.3177i 585.323 + 190.183i 591.096 − 388.080i
17.6 2.56816 − 5.04029i 4.36021 − 27.5293i −18.8091 − 25.8885i 18.8739 − 123.567i −127.558 − 92.6762i −92.2825 − 92.2825i −178.791 + 28.3177i −45.5281 − 14.7930i −574.342 − 412.469i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
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.a ✓ 56
25.f odd 20 1 inner 50.7.f.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.7.f.a ✓ 56 1.a even 1 1 trivial
50.7.f.a ✓ 56 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 32 T_{3}^{55} + 132 T_{3}^{54} - 51546 T_{3}^{53} - 2135899 T_{3}^{52} + \cdots + 36\!\cdots\!96 \) acting on \(S_{7}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display