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Properties

Label 74.3.i.b
Level $74$
Weight $3$
Character orbit 74.i
Analytic conductor $2.016$
Analytic rank $0$
Dimension $48$
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] = [74,3,Mod(5,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([23]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 74.i (of order \(36\), degree \(12\), 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: \(2.01635395627\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{8} - 30 q^{9} - 18 q^{10} - 36 q^{11} + 12 q^{12} - 24 q^{13} - 24 q^{14} + 114 q^{15} - 108 q^{17} + 60 q^{18} + 12 q^{19} - 174 q^{21} - 12 q^{22} + 216 q^{23} + 12 q^{24} + 54 q^{25} - 54 q^{26} - 216 q^{27} - 36 q^{28} + 96 q^{29} - 210 q^{30} - 300 q^{31} - 264 q^{33} - 54 q^{34} + 120 q^{37} - 204 q^{39} + 72 q^{40} - 48 q^{41} + 168 q^{42} + 252 q^{43} + 1224 q^{45} + 264 q^{46} + 30 q^{47} + 72 q^{48} + 378 q^{49} + 36 q^{50} - 108 q^{51} - 48 q^{52} + 6 q^{53} - 342 q^{54} + 114 q^{55} - 72 q^{56} + 168 q^{57} - 246 q^{58} + 78 q^{59} + 216 q^{60} - 672 q^{61} + 270 q^{62} - 864 q^{63} - 258 q^{65} + 270 q^{66} - 216 q^{67} - 12 q^{68} + 18 q^{69} + 6 q^{70} + 48 q^{71} - 120 q^{72} - 192 q^{74} + 276 q^{75} + 24 q^{76} + 516 q^{77} + 60 q^{78} - 600 q^{79} - 72 q^{80} + 804 q^{81} + 336 q^{82} + 96 q^{83} + 486 q^{85} + 12 q^{86} + 1038 q^{87} - 48 q^{88} + 612 q^{89} + 324 q^{90} - 756 q^{91} - 408 q^{92} - 1704 q^{93} - 66 q^{94} - 1842 q^{95} + 48 q^{96} - 330 q^{97} - 1044 q^{98} + 150 q^{99}+O(q^{100}) \) 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 −0.597672 + 1.28171i −1.99392 + 5.47824i −1.28558 − 1.53209i 0.872151 − 1.24556i −5.82982 − 5.82982i 0.722530 + 4.09767i 2.73205 − 0.732051i −19.1410 − 16.0612i 1.07519 + 1.86228i
5.2 −0.597672 + 1.28171i 0.243263 − 0.668361i −1.28558 − 1.53209i −3.95027 + 5.64157i 0.711255 + 0.711255i 1.80840 + 10.2559i 2.73205 − 0.732051i 6.50687 + 5.45991i −4.86990 − 8.43492i
5.3 −0.597672 + 1.28171i 0.318911 − 0.876201i −1.28558 − 1.53209i 4.81475 − 6.87617i 0.932433 + 0.932433i −0.0585572 − 0.332094i 2.73205 − 0.732051i 6.22838 + 5.22623i 5.93563 + 10.2808i
5.4 −0.597672 + 1.28171i 1.90536 − 5.23492i −1.28558 − 1.53209i −1.94180 + 2.77318i 5.57089 + 5.57089i −1.86179 − 10.5587i 2.73205 − 0.732051i −16.8796 − 14.1637i −2.39386 − 4.14629i
13.1 −0.811160 − 1.15846i −5.11108 + 0.901221i −0.684040 + 1.87939i 8.19128 + 0.716644i 5.18992 + 5.18992i 0.141127 − 0.118419i 2.73205 − 0.732051i 16.8537 − 6.13423i −5.81423 − 10.0705i
13.2 −0.811160 − 1.15846i −1.32062 + 0.232862i −0.684040 + 1.87939i −5.40245 − 0.472653i 1.34100 + 1.34100i −4.15470 + 3.48621i 2.73205 − 0.732051i −6.76741 + 2.46314i 3.83470 + 6.64190i
13.3 −0.811160 − 1.15846i 1.34344 − 0.236885i −0.684040 + 1.87939i 1.90770 + 0.166902i −1.36417 − 1.36417i 10.4001 − 8.72675i 2.73205 − 0.732051i −6.70851 + 2.44170i −1.35410 − 2.34537i
13.4 −0.811160 − 1.15846i 4.89133 − 0.862474i −0.684040 + 1.87939i 5.97319 + 0.522587i −4.96679 − 4.96679i −9.79144 + 8.21599i 2.73205 − 0.732051i 14.7240 − 5.35911i −4.23982 − 7.34358i
15.1 −0.597672 − 1.28171i −1.99392 − 5.47824i −1.28558 + 1.53209i 0.872151 + 1.24556i −5.82982 + 5.82982i 0.722530 − 4.09767i 2.73205 + 0.732051i −19.1410 + 16.0612i 1.07519 − 1.86228i
15.2 −0.597672 − 1.28171i 0.243263 + 0.668361i −1.28558 + 1.53209i −3.95027 − 5.64157i 0.711255 − 0.711255i 1.80840 − 10.2559i 2.73205 + 0.732051i 6.50687 − 5.45991i −4.86990 + 8.43492i
15.3 −0.597672 − 1.28171i 0.318911 + 0.876201i −1.28558 + 1.53209i 4.81475 + 6.87617i 0.932433 − 0.932433i −0.0585572 + 0.332094i 2.73205 + 0.732051i 6.22838 − 5.22623i 5.93563 − 10.2808i
15.4 −0.597672 − 1.28171i 1.90536 + 5.23492i −1.28558 + 1.53209i −1.94180 − 2.77318i 5.57089 − 5.57089i −1.86179 + 10.5587i 2.73205 + 0.732051i −16.8796 + 14.1637i −2.39386 + 4.14629i
17.1 1.15846 + 0.811160i −5.60626 − 0.988534i 0.684040 + 1.87939i −0.617239 − 7.05508i −5.69274 − 5.69274i −7.09684 − 5.95496i −0.732051 + 2.73205i 21.9957 + 8.00577i 5.00775 − 8.67368i
17.2 1.15846 + 0.811160i −2.80009 − 0.493731i 0.684040 + 1.87939i 0.640654 + 7.32270i −2.84329 − 2.84329i 6.19950 + 5.20200i −0.732051 + 2.73205i −0.860504 − 0.313198i −5.19771 + 9.00270i
17.3 1.15846 + 0.811160i 1.94430 + 0.342833i 0.684040 + 1.87939i −0.592752 − 6.77519i 1.97430 + 1.97430i 8.75112 + 7.34306i −0.732051 + 2.73205i −4.79446 − 1.74504i 4.80908 − 8.32957i
17.4 1.15846 + 0.811160i 3.24749 + 0.572621i 0.684040 + 1.87939i 0.134049 + 1.53219i 3.29759 + 3.29759i −7.51308 − 6.30422i −0.732051 + 2.73205i 1.76108 + 0.640982i −1.08756 + 1.88371i
19.1 1.40883 − 0.123257i −2.39987 − 2.86006i 1.96962 − 0.347296i −2.35689 − 5.05436i −3.73354 − 3.73354i −0.403710 − 0.146938i 2.73205 − 0.732051i −0.857704 + 4.86428i −3.94344 − 6.83024i
19.2 1.40883 − 0.123257i −0.0231618 − 0.0276032i 1.96962 − 0.347296i 0.534126 + 1.14544i −0.0360334 − 0.0360334i 8.81401 + 3.20804i 2.73205 − 0.732051i 1.56261 − 8.86199i 0.893677 + 1.54789i
19.3 1.40883 − 0.123257i 1.07390 + 1.27982i 1.96962 − 0.347296i 2.67929 + 5.74577i 1.67069 + 1.67069i −9.76513 − 3.55422i 2.73205 − 0.732051i 1.07814 − 6.11446i 4.48288 + 7.76458i
19.4 1.40883 − 0.123257i 3.67052 + 4.37436i 1.96962 − 0.347296i −3.52684 − 7.56334i 5.71032 + 5.71032i −3.64511 − 1.32671i 2.73205 − 0.732051i −4.09944 + 23.2491i −5.90096 − 10.2208i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.i.b ✓ 48
37.i odd 36 1 inner 74.3.i.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.i.b ✓ 48 1.a even 1 1 trivial
74.3.i.b ✓ 48 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 15 T_{3}^{46} + 180 T_{3}^{45} - 291 T_{3}^{44} + 5904 T_{3}^{43} - 28400 T_{3}^{42} + \cdots + 12\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display