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Properties

Label 209.4.f.a
Level $209$
Weight $4$
Character orbit 209.f
Analytic conductor $12.331$
Analytic rank $0$
Dimension $100$
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] = [209,4,Mod(20,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.20");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.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: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(25\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} + 8 q^{3} - 98 q^{4} + 26 q^{5} + 168 q^{6} + 119 q^{7} - 58 q^{8} - 217 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} + 8 q^{3} - 98 q^{4} + 26 q^{5} + 168 q^{6} + 119 q^{7} - 58 q^{8} - 217 q^{9} + 48 q^{10} - 73 q^{11} - 392 q^{12} - 24 q^{13} + 227 q^{14} + 268 q^{15} + 78 q^{16} + 121 q^{17} + 22 q^{18} - 475 q^{19} + 76 q^{20} + 48 q^{21} - 296 q^{22} - 1322 q^{23} + 1386 q^{24} - 55 q^{25} + 528 q^{26} + 248 q^{27} - 1177 q^{28} - 98 q^{29} + 312 q^{30} + 76 q^{31} + 568 q^{32} + 1036 q^{33} - 1044 q^{34} + 1711 q^{35} + 706 q^{36} + 566 q^{37} + 38 q^{38} - 2188 q^{39} - 1278 q^{40} + 156 q^{41} + 496 q^{42} - 602 q^{43} + 137 q^{44} - 6390 q^{45} - 51 q^{46} + 306 q^{47} + 704 q^{48} + 430 q^{49} - 1461 q^{50} - 2404 q^{51} + 3011 q^{52} + 2592 q^{53} - 908 q^{54} + 3525 q^{55} - 9732 q^{56} + 152 q^{57} + 739 q^{58} + 1456 q^{59} + 7778 q^{60} - 1857 q^{61} - 5775 q^{62} + 2519 q^{63} + 918 q^{64} - 616 q^{65} - 1815 q^{66} - 3532 q^{67} + 1817 q^{68} + 4326 q^{69} + 2594 q^{70} + 4600 q^{71} - 5255 q^{72} - 1122 q^{73} + 7152 q^{74} + 2750 q^{75} + 6498 q^{76} - 908 q^{77} - 10406 q^{78} + 62 q^{79} + 8263 q^{80} + 1459 q^{81} + 11646 q^{82} + 80 q^{83} - 9583 q^{84} - 157 q^{85} + 376 q^{86} + 4016 q^{87} - 7434 q^{88} - 13440 q^{89} - 5554 q^{90} + 4622 q^{91} - 2424 q^{92} + 6226 q^{93} - 8380 q^{94} + 969 q^{95} + 2630 q^{96} + 3102 q^{97} - 164 q^{98} - 3457 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −1.58346 + 4.87339i −4.36085 + 3.16834i −14.7705 − 10.7314i −2.30012 − 7.07905i −8.53534 − 26.2691i −13.9403 − 10.1282i 42.5223 − 30.8942i 0.635156 − 1.95481i 38.1412
20.2 −1.52391 + 4.69012i 3.63717 − 2.64256i −13.2028 − 9.59238i −5.95536 − 18.3287i 6.85119 + 21.0858i −0.194727 − 0.141477i 33.1920 − 24.1154i −2.09757 + 6.45565i 95.0393
20.3 −1.48626 + 4.57425i 6.76682 − 4.91638i −12.2426 − 8.89480i −1.39424 − 4.29104i 12.4315 + 38.2602i 21.7954 + 15.8353i 27.7541 − 20.1645i 13.2756 − 40.8580i 21.7005
20.4 −1.34173 + 4.12942i −1.02800 + 0.746889i −8.77974 − 6.37885i 2.42378 + 7.45962i −1.70491 − 5.24719i −18.2411 − 13.2530i 10.0194 − 7.27955i −7.84451 + 24.1429i −34.0560
20.5 −1.31901 + 4.05950i −1.33063 + 0.966762i −8.26761 − 6.00677i 1.54217 + 4.74631i −2.16945 − 6.67688i 18.3035 + 13.2983i 7.66382 − 5.56809i −7.50750 + 23.1057i −21.3018
20.6 −1.06853 + 3.28860i 5.09809 − 3.70398i −3.20099 − 2.32566i 6.14136 + 18.9012i 6.73344 + 20.7234i 0.197473 + 0.143473i −11.3111 + 8.21798i 3.92761 − 12.0880i −68.7206
20.7 −0.829401 + 2.55263i −7.29573 + 5.30066i 0.644099 + 0.467965i −0.167918 − 0.516799i −7.47956 − 23.0197i 2.16765 + 1.57489i −19.1000 + 13.8769i 16.7873 − 51.6659i 1.45847
20.8 −0.603412 + 1.85711i 4.60785 − 3.34780i 3.38738 + 2.46108i −3.24013 − 9.97209i 3.43680 + 10.5774i −20.9971 − 15.2553i −19.2525 + 13.9878i 1.68105 − 5.17375i 20.4744
20.9 −0.516182 + 1.58864i −4.40719 + 3.20201i 4.21479 + 3.06222i 2.75640 + 8.48334i −2.81195 − 8.65428i −5.48025 − 3.98164i −17.8514 + 12.9698i 0.826993 − 2.54522i −14.8998
20.10 −0.395400 + 1.21692i 2.40178 − 1.74500i 5.14759 + 3.73995i −0.676990 − 2.08356i 1.17385 + 3.61274i −16.8637 − 12.2522i −14.8679 + 10.8022i −5.61991 + 17.2963i 2.80320
20.11 −0.361404 + 1.11229i −5.94745 + 4.32107i 5.36557 + 3.89831i −6.33329 − 19.4919i −2.65684 − 8.17692i 28.2706 + 20.5398i −13.8445 + 10.0586i 8.35699 − 25.7202i 23.9694
20.12 −0.299910 + 0.923027i 2.43960 − 1.77247i 5.71010 + 4.14863i 0.742254 + 2.28442i 0.904380 + 2.78340i 25.2488 + 18.3443i −11.8232 + 8.59006i −5.53347 + 17.0303i −2.33119
20.13 −0.0948856 + 0.292028i 7.77796 − 5.65102i 6.39586 + 4.64686i 3.50031 + 10.7729i 0.912238 + 2.80758i −5.75051 − 4.17799i −3.95120 + 2.87071i 20.2192 − 62.2282i −3.47810
20.14 0.229892 − 0.707536i −3.65712 + 2.65706i 6.02438 + 4.37697i −3.23755 − 9.96416i 1.03922 + 3.19838i −1.39768 − 1.01548i 9.29675 − 6.75448i −2.02885 + 6.24416i −7.79429
20.15 0.532764 − 1.63968i −4.24013 + 3.08063i 4.06743 + 2.95516i 4.98901 + 15.3546i 2.79226 + 8.59369i 5.49775 + 3.99435i 18.1708 − 13.2019i 0.144919 − 0.446015i 27.8346
20.16 0.542636 − 1.67006i 4.37034 − 3.17524i 3.97749 + 2.88981i −3.69803 − 11.3814i −2.93134 − 9.02172i 18.8688 + 13.7090i 18.3496 − 13.3318i 0.674266 − 2.07518i −21.0142
20.17 0.571034 − 1.75746i 1.80147 − 1.30884i 3.70954 + 2.69514i 3.25160 + 10.0074i −1.27154 − 3.91341i 5.03665 + 3.65934i 18.8148 − 13.6697i −6.81124 + 20.9628i 19.4444
20.18 0.640851 − 1.97234i −0.639367 + 0.464527i 2.99271 + 2.17433i −6.32425 − 19.4640i 0.506465 + 1.55874i −19.7562 − 14.3537i 19.6286 − 14.2610i −8.15045 + 25.0845i −42.4425
20.19 1.02651 − 3.15927i 7.10579 − 5.16266i −2.45514 − 1.78376i −1.25742 − 3.86994i −9.01609 − 27.7487i −16.1371 − 11.7243i 13.3439 − 9.69489i 15.4958 − 47.6911i −13.5170
20.20 1.17440 − 3.61444i −7.19026 + 5.22403i −5.21282 − 3.78734i −4.75457 − 14.6331i 10.4377 + 32.1239i −3.35385 − 2.43671i 4.78595 − 3.47720i 16.0659 − 49.4456i −58.4741
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.25
Significant digits:
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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 209.4.f.a ✓ 100
11.c even 5 1 inner 209.4.f.a ✓ 100
11.c even 5 1 2299.4.a.v 50
11.d odd 10 1 2299.4.a.u 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.f.a ✓ 100 1.a even 1 1 trivial
209.4.f.a ✓ 100 11.c even 5 1 inner
2299.4.a.u 50 11.d odd 10 1
2299.4.a.v 50 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} - 2 T_{2}^{99} + 151 T_{2}^{98} - 264 T_{2}^{97} + 12547 T_{2}^{96} - 19628 T_{2}^{95} + \cdots + 10\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display