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Properties

Label 229.4.k.a
Level $229$
Weight $4$
Character orbit 229.k
Analytic conductor $13.511$
Analytic rank $0$
Dimension $2016$
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] = [229,4,Mod(5,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(114))
 
chi = DirichletCharacter(H, H._module([49]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 229.k (of order \(114\), degree \(36\), 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: \(13.5114373913\)
Analytic rank: \(0\)
Dimension: \(2016\)
Relative dimension: \(56\) over \(\Q(\zeta_{114})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{114}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2016 q - 38 q^{2} - 41 q^{3} + 374 q^{4} + 26 q^{5} - 59 q^{6} + 19 q^{7} - 38 q^{8} + 475 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2016 q - 38 q^{2} - 41 q^{3} + 374 q^{4} + 26 q^{5} - 59 q^{6} + 19 q^{7} - 38 q^{8} + 475 q^{9} + 25 q^{10} - 114 q^{11} + 22 q^{12} - 551 q^{13} - 197 q^{14} - 92 q^{15} - 2978 q^{16} - 180 q^{17} + 447 q^{18} - 151 q^{19} + 292 q^{20} - 38 q^{21} - 38 q^{22} + 839 q^{23} - 521 q^{24} + 2186 q^{25} - 1278 q^{26} - 620 q^{27} - 800 q^{28} - 35 q^{29} + 19 q^{30} - 3531 q^{31} - 38 q^{32} + 3310 q^{33} - 38 q^{34} + 927 q^{35} - 1827 q^{36} - 816 q^{37} + 922 q^{38} - 1478 q^{39} - 8225 q^{40} + 433 q^{41} - 2700 q^{42} + 3290 q^{43} + 2409 q^{44} + 1849 q^{45} - 287 q^{46} - 551 q^{47} + 7853 q^{48} - 3819 q^{49} + 616 q^{50} - 492 q^{51} - 38 q^{52} + 714 q^{53} - 6251 q^{54} - 490 q^{55} + 12732 q^{56} - 1178 q^{57} - 368 q^{58} + 3863 q^{59} + 4208 q^{60} + 1076 q^{61} + 498 q^{62} - 5220 q^{63} + 4806 q^{64} + 481 q^{65} - 1050 q^{66} - 2523 q^{67} + 1722 q^{68} - 2567 q^{69} - 657 q^{70} + 4005 q^{71} - 8856 q^{72} - 1292 q^{73} - 477 q^{74} - 15232 q^{75} + 9013 q^{76} + 3220 q^{77} - 20694 q^{78} - 6707 q^{79} + 25081 q^{80} + 14650 q^{81} - 12677 q^{82} - 3987 q^{83} + 21394 q^{84} + 163 q^{85} - 38 q^{86} - 1493 q^{87} - 38 q^{88} + 2508 q^{89} + 14058 q^{90} + 22986 q^{91} + 2299 q^{92} - 34352 q^{93} + 11099 q^{94} - 11700 q^{95} - 30358 q^{96} - 3498 q^{97} - 217 q^{98} - 1070 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −5.39377 − 0.446940i 1.83652 + 5.87168i 21.0021 + 3.50463i −8.96271 − 3.64034i −7.28147 − 32.4913i 3.07406 − 0.0847360i −69.7408 − 17.6608i −8.91586 + 6.18212i 46.7158 + 23.6409i
5.2 −5.23313 − 0.433629i −0.0707345 − 0.226151i 19.3067 + 3.22171i 11.6708 + 4.74026i 0.272097 + 1.21415i −8.28590 + 0.228399i −58.9142 − 14.9191i 22.1419 − 15.3528i −59.0192 − 29.8672i
5.3 −5.09262 − 0.421987i −0.639994 − 2.04618i 17.8659 + 2.98128i −12.4279 − 5.04779i 2.39579 + 10.6905i −32.4917 + 0.895627i −50.0963 − 12.6861i 18.4108 − 12.7657i 61.1606 + 30.9509i
5.4 −5.01440 − 0.415505i −2.27084 − 7.26027i 17.0807 + 2.85026i 9.31172 + 3.78209i 8.37022 + 37.3495i −3.24992 + 0.0895834i −45.4442 − 11.5080i −25.3669 + 17.5890i −45.1212 − 22.8340i
5.5 −4.99661 − 0.414031i −1.64200 − 5.24977i 16.9038 + 2.82075i −6.31420 − 2.56461i 6.03087 + 26.9109i 29.6752 − 0.817991i −44.4114 − 11.2465i −2.67587 + 1.85541i 30.4878 + 15.4286i
5.6 −4.85537 − 0.402327i 3.03655 + 9.70838i 15.5218 + 2.59013i 17.2227 + 6.99527i −10.8376 − 48.3594i 4.76893 − 0.131455i −36.5387 − 9.25285i −62.8441 + 43.5751i −80.8083 − 40.8938i
5.7 −4.34684 − 0.360190i 1.14804 + 3.67049i 10.8744 + 1.81462i 11.2548 + 4.57132i −3.66828 − 16.3686i −22.7515 + 0.627140i −12.7896 − 3.23877i 10.0335 − 6.95708i −47.2765 − 23.9247i
5.8 −4.34544 − 0.360073i −1.06655 − 3.40996i 10.8623 + 1.81260i −14.5004 − 5.88955i 3.40680 + 15.2018i 3.47156 − 0.0956929i −12.7335 − 3.22457i 11.6977 − 8.11103i 60.8899 + 30.8139i
5.9 −4.33446 − 0.359164i 1.32990 + 4.25194i 10.7677 + 1.79681i −1.79895 − 0.730670i −4.23727 − 18.9075i 30.7981 − 0.848945i −12.2969 − 3.11401i 5.87767 − 4.07549i 7.53505 + 3.81318i
5.10 −4.16242 − 0.344908i 1.49543 + 4.78117i 9.31591 + 1.55455i −3.51036 − 1.42578i −4.57556 − 20.4170i −1.09244 + 0.0301128i −5.84953 − 1.48130i 1.56478 − 1.08499i 14.1198 + 7.14547i
5.11 −3.59509 − 0.297898i −0.203129 − 0.649439i 4.94503 + 0.825179i 15.3479 + 6.23377i 0.536800 + 2.39530i 20.6055 − 0.567985i 10.4442 + 2.64483i 21.8075 − 15.1210i −53.3200 − 26.9831i
5.12 −3.48555 − 0.288821i 2.85728 + 9.13522i 4.17473 + 0.696638i −19.1383 − 7.77332i −7.32073 − 32.6665i −8.93891 + 0.246399i 12.7738 + 3.23476i −53.1002 + 36.8189i 64.4625 + 32.6218i
5.13 −3.43318 − 0.284481i −2.18431 − 6.98364i 3.81490 + 0.636594i −3.82791 − 1.55476i 5.51243 + 24.5975i −25.9177 + 0.714417i 13.8001 + 3.49466i −21.8120 + 15.1241i 12.6996 + 6.42676i
5.14 −3.42480 − 0.283787i −2.57515 − 8.23323i 3.75783 + 0.627071i 9.05824 + 3.67914i 6.48291 + 28.9280i 4.30078 − 0.118550i 13.9592 + 3.53494i −38.9666 + 27.0189i −29.9786 − 15.1709i
5.15 −3.12084 − 0.258600i 1.91157 + 6.11164i 1.78189 + 0.297345i 2.95972 + 1.20213i −4.38525 − 19.5678i −30.4639 + 0.839731i 18.8016 + 4.76121i −11.5100 + 7.98089i −8.92594 − 4.51705i
5.16 −2.77382 − 0.229845i −0.721880 − 2.30798i −0.249631 − 0.0416561i 7.59041 + 3.08296i 1.47189 + 6.56784i −16.6507 + 0.458972i 22.2681 + 5.63906i 17.3823 − 12.0527i −20.3459 − 10.2962i
5.17 −2.71994 − 0.225381i 0.394392 + 1.26094i −0.543604 − 0.0907114i −11.2463 − 4.56785i −0.788532 − 3.51858i −0.570455 + 0.0157245i 22.6241 + 5.72920i 20.7536 − 14.3902i 29.5598 + 14.9590i
5.18 −2.64262 − 0.218974i −2.91240 − 9.31146i −0.955402 − 0.159428i −17.8186 − 7.23727i 5.65740 + 25.2444i 13.3754 − 0.368691i 23.0541 + 5.83810i −56.0332 + 38.8526i 45.5029 + 23.0272i
5.19 −2.10930 − 0.174781i −0.810078 − 2.58996i −3.47231 − 0.579426i −6.46958 − 2.62772i 1.25602 + 5.60458i 14.0565 − 0.387464i 23.6369 + 5.98568i 16.1363 − 11.1887i 13.1870 + 6.67339i
5.20 −2.05332 − 0.170143i 2.69390 + 8.61288i −3.70370 − 0.618037i 3.51707 + 1.42851i −4.06603 − 18.1434i 13.3707 − 0.368562i 23.4783 + 5.94550i −44.7366 + 31.0197i −6.97863 − 3.53160i
See next 80 embeddings (of 2016 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
229.k even 114 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.4.k.a ✓ 2016
229.k even 114 1 inner 229.4.k.a ✓ 2016
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.4.k.a ✓ 2016 1.a even 1 1 trivial
229.4.k.a ✓ 2016 229.k even 114 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(229, [\chi])\).