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Properties

Label 235.4.g.b
Level $235$
Weight $4$
Character orbit 235.g
Analytic conductor $13.865$
Analytic rank $0$
Dimension $550$
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] = [235,4,Mod(6,235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(235, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([0, 38]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("235.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 235 = 5 \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 235.g (of order \(23\), degree \(22\), 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: \(13.8654488513\)
Analytic rank: \(0\)
Dimension: \(550\)
Relative dimension: \(25\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 550 q + 4 q^{2} - 2 q^{3} - 102 q^{4} + 125 q^{5} + 44 q^{6} - 2 q^{7} - 6 q^{8} - 175 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 550 q + 4 q^{2} - 2 q^{3} - 102 q^{4} + 125 q^{5} + 44 q^{6} - 2 q^{7} - 6 q^{8} - 175 q^{9} - 20 q^{10} + 74 q^{11} - 677 q^{12} + 132 q^{13} - 83 q^{14} + 10 q^{15} - 602 q^{16} - 62 q^{17} - 95 q^{18} + 590 q^{19} + 510 q^{20} - 148 q^{21} + 178 q^{22} - 2244 q^{23} + 625 q^{24} - 625 q^{25} - 2530 q^{26} + 472 q^{27} + 68 q^{28} - 40 q^{29} + 1505 q^{30} - 16 q^{31} + 301 q^{32} + 200 q^{33} + 104 q^{34} + 10 q^{35} - 778 q^{36} + 3435 q^{37} + 1724 q^{38} + 578 q^{39} + 30 q^{40} - 442 q^{41} - 3709 q^{42} - 484 q^{43} - 12170 q^{44} + 875 q^{45} + 614 q^{46} - 1743 q^{47} - 962 q^{48} - 2509 q^{49} + 100 q^{50} + 390 q^{51} - 7609 q^{52} + 758 q^{53} + 1373 q^{54} - 140 q^{55} + 106 q^{56} + 6740 q^{57} + 4800 q^{58} + 8566 q^{59} - 295 q^{60} + 1364 q^{61} - 7252 q^{62} - 3122 q^{63} - 4010 q^{64} - 660 q^{65} + 24646 q^{66} - 68 q^{67} - 905 q^{68} - 2624 q^{69} + 185 q^{70} - 318 q^{71} - 3352 q^{72} + 428 q^{73} + 506 q^{74} - 50 q^{75} + 2495 q^{76} + 18134 q^{77} + 16762 q^{78} + 4620 q^{79} - 785 q^{80} - 14403 q^{81} + 11899 q^{82} + 90 q^{83} + 15755 q^{84} + 310 q^{85} - 4050 q^{86} - 5440 q^{87} - 439 q^{88} - 2148 q^{89} + 475 q^{90} - 9933 q^{91} - 36335 q^{92} - 4472 q^{93} - 3725 q^{94} + 270 q^{95} - 1421 q^{96} - 7342 q^{97} - 39381 q^{98} - 11821 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −4.83947 + 2.10208i 0.348978 − 5.10188i 13.5413 − 14.4992i −4.27210 + 2.59792i 9.03569 + 25.4240i 4.15234 + 19.9822i −20.9189 + 58.8601i 0.841083 + 0.115604i 15.2136 − 21.5528i
6.2 −4.73955 + 2.05868i −0.618844 + 9.04717i 12.7647 − 13.6677i −4.27210 + 2.59792i −15.6922 − 44.1535i 5.19855 + 25.0168i −18.5182 + 52.1052i −54.7199 − 7.52107i 14.8995 − 21.1078i
6.3 −4.38199 + 1.90336i −0.347493 + 5.08017i 10.1186 − 10.8343i −4.27210 + 2.59792i −8.14671 − 22.9226i −5.60060 − 26.9516i −10.9186 + 30.7220i 1.06113 + 0.145849i 13.7755 − 19.5154i
6.4 −4.25988 + 1.85033i 0.368258 − 5.38374i 9.26246 − 9.91768i −4.27210 + 2.59792i 8.39295 + 23.6155i −2.49466 − 12.0049i −8.66357 + 24.3770i −2.10049 − 0.288705i 13.3916 − 18.9716i
6.5 −3.67791 + 1.59754i −0.200738 + 2.93469i 5.51447 − 5.90455i −4.27210 + 2.59792i −3.94999 − 11.1142i 1.65591 + 7.96866i −0.106316 + 0.299145i 18.1764 + 2.49829i 11.5621 − 16.3798i
6.6 −2.91026 + 1.26411i −0.0624765 + 0.913375i 1.41125 − 1.51107i −4.27210 + 2.59792i −0.972779 − 2.73714i 6.08400 + 29.2778i 6.30351 − 17.7364i 25.9182 + 3.56237i 9.14888 − 12.9610i
6.7 −2.38621 + 1.03648i −0.155733 + 2.27674i −0.840701 + 0.900170i −4.27210 + 2.59792i −1.98818 − 5.59420i −1.28967 − 6.20624i 8.04285 − 22.6304i 21.5892 + 2.96737i 7.50145 − 10.6271i
6.8 −2.23991 + 0.972929i 0.303787 − 4.44121i −1.38983 + 1.48815i −4.27210 + 2.59792i 3.64052 + 10.2435i 1.41632 + 6.81573i 8.20767 − 23.0942i 7.11650 + 0.978140i 7.04151 − 9.97555i
6.9 −1.89434 + 0.822828i 0.535965 − 7.83554i −2.54894 + 2.72925i −4.27210 + 2.59792i 5.43200 + 15.2842i −4.78679 − 23.0353i 8.11595 − 22.8361i −34.3598 − 4.72265i 5.95517 − 8.43655i
6.10 −1.65098 + 0.717122i −0.581264 + 8.49777i −3.24895 + 3.47878i −4.27210 + 2.59792i −5.13428 − 14.4465i −0.836956 − 4.02765i 7.69151 − 21.6418i −45.1257 − 6.20239i 5.19012 − 7.35273i
6.11 −0.261636 + 0.113644i −0.360867 + 5.27570i −5.40489 + 5.78722i −4.27210 + 2.59792i −0.505137 − 1.42132i −7.14630 − 34.3899i 1.52062 − 4.27863i −0.954221 − 0.131155i 0.822493 − 1.16521i
6.12 −0.162943 + 0.0707762i −0.578238 + 8.45354i −5.43888 + 5.82362i −4.27210 + 2.59792i −0.504090 − 1.41837i 5.47027 + 26.3244i 0.949988 − 2.67301i −44.3795 − 6.09983i 0.512238 − 0.725677i
6.13 −0.0106276 + 0.00461622i 0.179178 − 2.61949i −5.46033 + 5.84659i −4.27210 + 2.59792i 0.0101879 + 0.0286660i 0.261203 + 1.25698i 0.0620828 − 0.174684i 19.9189 + 2.73779i 0.0334096 − 0.0473306i
6.14 0.0794365 − 0.0345041i 0.567010 − 8.28939i −5.45531 + 5.84121i −4.27210 + 2.59792i −0.240977 − 0.678044i 4.63629 + 22.3111i −0.463827 + 1.30508i −41.6440 − 5.72383i −0.249721 + 0.353775i
6.15 1.12985 − 0.490765i −0.312030 + 4.56172i −4.42470 + 4.73770i −4.27210 + 2.59792i 1.88618 + 5.30721i 5.29616 + 25.4865i −5.97431 + 16.8101i 6.03657 + 0.829708i −3.55188 + 5.03187i
6.16 2.12058 − 0.921100i 0.0783936 − 1.14607i −1.81197 + 1.94015i −4.27210 + 2.59792i −0.889408 − 2.50255i −2.71783 − 13.0789i −8.24928 + 23.2112i 25.4412 + 3.49681i −6.66640 + 9.44414i
6.17 2.32609 − 1.01036i 0.445515 − 6.51320i −1.07057 + 1.14630i −4.27210 + 2.59792i −5.54439 − 15.6004i 4.14101 + 19.9277i −8.12622 + 22.8650i −15.4748 − 2.12696i −7.31244 + 10.3594i
6.18 2.47081 − 1.07322i −0.351491 + 5.13861i −0.507353 + 0.543242i −4.27210 + 2.59792i 4.64641 + 13.0737i −4.33565 − 20.8643i −7.88740 + 22.1930i 0.466704 + 0.0641470i −7.76737 + 11.0039i
6.19 2.74290 − 1.19141i 0.554042 − 8.09980i 0.643619 − 0.689147i −4.27210 + 2.59792i −8.13050 − 22.8770i −3.53213 − 16.9975i −7.06728 + 19.8854i −38.5513 − 5.29875i −8.62275 + 12.2157i
6.20 3.38952 − 1.47228i −0.625252 + 9.14087i 3.86083 − 4.13394i −4.27210 + 2.59792i 11.3386 + 31.9037i −0.499105 − 2.40183i −2.90022 + 8.16045i −56.4160 − 7.75420i −10.6555 + 15.0954i
See next 80 embeddings (of 550 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.c even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 235.4.g.b ✓ 550
47.c even 23 1 inner 235.4.g.b ✓ 550
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.4.g.b ✓ 550 1.a even 1 1 trivial
235.4.g.b ✓ 550 47.c even 23 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{550} - 4 T_{2}^{549} + 159 T_{2}^{548} - 634 T_{2}^{547} + 14259 T_{2}^{546} + \cdots + 77\!\cdots\!44 \) acting on \(S_{4}^{\mathrm{new}}(235, [\chi])\). Copy content Toggle raw display