↑

Properties

Label 201.4.p.a
Level $201$
Weight $4$
Character orbit 201.p
Analytic conductor $11.859$
Analytic rank $0$
Dimension $1320$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(2,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.2");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.p (of order \(66\), degree \(20\), 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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(1320\)
Relative dimension: \(66\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9} - 26 q^{10} - 4 q^{12} + 136 q^{13} + 166 q^{15} + 694 q^{16} - 181 q^{18} + 32 q^{19} + 1004 q^{21} + 544 q^{22} - 230 q^{24} - 2552 q^{25} - 22 q^{27} + 100 q^{28} + 810 q^{30} + 532 q^{31} + 800 q^{33} + 718 q^{34} - 243 q^{36} + 216 q^{37} - 1938 q^{39} + 820 q^{40} - 22 q^{42} + 1672 q^{43} + 4488 q^{45} - 3182 q^{46} + 2547 q^{48} - 2360 q^{49} + 287 q^{51} + 2156 q^{52} - 3793 q^{54} + 11272 q^{55} + 1091 q^{57} + 308 q^{58} - 56 q^{60} - 4544 q^{61} + 512 q^{63} - 22064 q^{64} - 1734 q^{67} + 350 q^{69} - 5588 q^{70} + 10648 q^{72} - 7992 q^{73} - 8459 q^{75} + 4540 q^{76} + 4664 q^{78} + 1178 q^{79} - 2448 q^{81} + 21556 q^{82} - 1183 q^{84} + 1864 q^{85} - 7051 q^{87} - 13694 q^{88} + 1138 q^{90} - 6308 q^{91} + 9792 q^{93} - 7172 q^{94} - 5417 q^{96} - 1140 q^{97} - 3678 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.263188 + 5.52500i 2.55934 + 4.52214i −22.4926 − 2.14778i 12.6391 − 14.5863i −25.6584 + 12.9502i −13.6688 − 26.5138i 11.4888 − 79.9067i −13.8995 + 23.1474i 77.2627 + 73.6698i
2.2 −0.257205 + 5.39940i 0.0986149 + 5.19522i −21.1236 − 2.01706i −9.22534 + 10.6466i −28.0764 − 0.803775i 10.8621 + 21.0696i 10.1697 − 70.7317i −26.9806 + 1.02465i −55.1125 − 52.5496i
2.3 −0.243816 + 5.11833i 0.0384771 − 5.19601i −18.1741 − 1.73542i 1.53871 − 1.77577i 26.5855 + 1.46381i −8.55363 − 16.5917i 7.47964 − 52.0221i −26.9970 − 0.399854i 8.71382 + 8.30861i
2.4 −0.241447 + 5.06860i −4.87773 + 1.79102i −17.6686 − 1.68715i −4.20480 + 4.85260i −7.90026 − 25.1557i −5.52731 − 10.7215i 7.04027 − 48.9662i 20.5845 − 17.4722i −23.5806 − 22.4841i
2.5 −0.240289 + 5.04428i 5.03306 − 1.29161i −17.4232 − 1.66372i −12.5554 + 14.4897i 5.30585 + 25.6985i −10.2423 − 19.8673i 6.82934 − 47.4991i 23.6635 − 13.0015i −70.0733 − 66.8147i
2.6 −0.239166 + 5.02071i 2.74669 − 4.41086i −17.1866 − 1.64112i 3.46556 − 3.99947i 21.4887 + 14.8453i 6.74721 + 13.0878i 6.62737 − 46.0944i −11.9113 − 24.2306i 19.2513 + 18.3561i
2.7 −0.235330 + 4.94018i 5.19611 − 0.0202274i −16.3863 − 1.56470i 2.61014 − 3.01227i −1.12287 + 25.6745i 8.63758 + 16.7546i 5.95520 − 41.4193i 26.9992 − 0.210208i 14.2669 + 13.6035i
2.8 −0.234299 + 4.91853i −4.94170 − 1.60611i −16.1733 − 1.54436i 9.16856 − 10.5811i 9.05754 − 23.9296i 6.12309 + 11.8771i 5.77918 − 40.1951i 21.8408 + 15.8738i 49.8952 + 47.5750i
2.9 −0.223576 + 4.69343i −2.83186 − 4.35667i −14.0145 − 1.33822i −11.6480 + 13.4425i 21.0808 − 12.3171i 15.7082 + 30.4697i 4.06453 − 28.2695i −10.9611 + 24.6750i −60.4871 − 57.6743i
2.10 −0.203341 + 4.26865i −2.80662 + 4.37297i −10.2162 − 0.975531i 1.38641 − 1.60001i −18.0960 − 12.8697i −2.67187 − 5.18271i 1.37612 − 9.57115i −11.2458 − 24.5465i 6.54795 + 6.24346i
2.11 −0.189440 + 3.97683i −0.298544 + 5.18757i −7.81552 − 0.746291i 7.60791 − 8.78000i −20.5735 − 2.16999i 14.2616 + 27.6636i −0.0843859 + 0.586917i −26.8217 − 3.09743i 33.4753 + 31.9187i
2.12 −0.181829 + 3.81706i 3.18386 + 4.10646i −6.57310 − 0.627655i −5.58017 + 6.43986i −16.2535 + 11.4063i −3.89368 − 7.55269i −0.759744 + 5.28414i −6.72605 + 26.1488i −23.5667 − 22.4708i
2.13 −0.178074 + 3.73823i 5.00732 + 1.38806i −5.97885 − 0.570910i 6.59041 − 7.60574i −6.08055 + 18.4713i 2.99850 + 5.81628i −1.06200 + 7.38637i 23.1466 + 13.9009i 27.2584 + 25.9908i
2.14 −0.176877 + 3.71311i −4.49634 − 2.60440i −5.79212 − 0.553080i −9.84104 + 11.3572i 10.4657 − 16.2348i −11.0761 − 21.4847i −1.15410 + 8.02692i 13.4342 + 23.4206i −40.4297 − 38.5497i
2.15 −0.168996 + 3.54767i −2.66059 − 4.46332i −4.59361 − 0.438637i 8.27468 − 9.54949i 16.2840 − 8.68460i −5.63397 − 10.9284i −1.71122 + 11.9018i −12.8425 + 23.7501i 32.4800 + 30.9696i
2.16 −0.160415 + 3.36752i 1.06419 − 5.08601i −3.35071 − 0.319954i −6.11753 + 7.06001i 16.9565 + 4.39956i −0.0381180 − 0.0739387i −2.22338 + 15.4640i −24.7350 − 10.8250i −22.7934 − 21.7335i
2.17 −0.147428 + 3.09489i −3.36537 + 3.95908i −1.59284 − 0.152098i 11.9344 − 13.7731i −11.7568 − 10.9991i −6.80969 − 13.2090i −2.82203 + 19.6277i −4.34859 − 26.6475i 40.8667 + 38.9663i
2.18 −0.146703 + 3.07967i 4.58334 − 2.44805i −1.49905 − 0.143142i 7.20079 − 8.31015i 6.86680 + 14.4743i −15.1390 − 29.3655i −2.84949 + 19.8186i 15.0141 − 22.4405i 24.5361 + 23.3951i
2.19 −0.134842 + 2.83067i −5.15314 + 0.667213i −0.0307503 − 0.00293630i −3.07901 + 3.55336i −1.19380 − 14.6768i 11.0994 + 21.5299i −3.21397 + 22.3537i 26.1097 − 6.87648i −9.64323 − 9.19480i
2.20 −0.119268 + 2.50375i 4.20022 − 3.05911i 1.70924 + 0.163213i −3.68755 + 4.25566i 7.15829 + 10.8812i 5.98781 + 11.6147i −3.46630 + 24.1086i 8.28370 − 25.6979i −10.2153 − 9.74027i
See next 80 embeddings (of 1320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.h odd 66 1 inner
201.p even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.p.a ✓ 1320
3.b odd 2 1 inner 201.4.p.a ✓ 1320
67.h odd 66 1 inner 201.4.p.a ✓ 1320
201.p even 66 1 inner 201.4.p.a ✓ 1320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.p.a ✓ 1320 1.a even 1 1 trivial
201.4.p.a ✓ 1320 3.b odd 2 1 inner
201.4.p.a ✓ 1320 67.h odd 66 1 inner
201.4.p.a ✓ 1320 201.p even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(201, [\chi])\).