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Properties

Label 65.4.k.b
Level $65$
Weight $4$
Character orbit 65.k
Analytic conductor $3.835$
Analytic rank $0$
Dimension $36$
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] = [65,4,Mod(8,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.k (of order \(4\), degree \(2\), 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: \(3.83512415037\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 6 q^{3} + 150 q^{4} - 26 q^{5} + 6 q^{6} + 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 6 q^{3} + 150 q^{4} - 26 q^{5} + 6 q^{6} + 42 q^{8} - 42 q^{10} - 2 q^{11} - 126 q^{12} - 26 q^{13} + 66 q^{15} + 326 q^{16} + 300 q^{17} - 166 q^{19} - 46 q^{20} - 280 q^{21} - 442 q^{22} + 186 q^{23} - 658 q^{24} + 604 q^{25} - 198 q^{26} - 132 q^{27} - 706 q^{30} - 518 q^{31} + 174 q^{32} - 340 q^{33} + 626 q^{34} + 504 q^{35} + 14 q^{38} - 482 q^{39} - 1022 q^{40} + 572 q^{41} + 108 q^{42} - 1242 q^{43} - 738 q^{44} + 1294 q^{45} + 598 q^{46} - 150 q^{48} + 676 q^{49} + 626 q^{50} - 1090 q^{52} + 536 q^{53} - 956 q^{54} - 994 q^{55} + 2348 q^{57} - 850 q^{59} - 86 q^{60} + 1348 q^{61} + 42 q^{62} + 7480 q^{63} + 910 q^{64} + 2932 q^{65} - 644 q^{66} - 28 q^{67} + 1150 q^{68} + 1116 q^{69} + 328 q^{70} + 762 q^{71} - 488 q^{73} - 2126 q^{75} - 4774 q^{76} + 1584 q^{77} - 2210 q^{78} - 3742 q^{80} - 3528 q^{81} + 4434 q^{82} + 24 q^{84} + 2028 q^{85} + 4950 q^{86} - 1520 q^{87} - 9242 q^{88} - 4732 q^{89} + 140 q^{90} + 420 q^{91} + 366 q^{92} - 2998 q^{95} - 1062 q^{96} - 7956 q^{97} - 3862 q^{98} + 3410 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −5.16850 −1.03806 − 1.03806i 18.7134 −10.3807 + 4.15234i 5.36523 + 5.36523i 23.9761i −55.3719 − 24.8448i 53.6524 − 21.4614i
8.2 −4.77932 −3.98774 − 3.98774i 14.8419 10.8554 − 2.67593i 19.0587 + 19.0587i − 14.1514i −32.6998 4.80418i −51.8814 + 12.7891i
8.3 −4.75622 5.41905 + 5.41905i 14.6216 6.70249 − 8.94855i −25.7742 − 25.7742i 1.86063i −31.4940 31.7322i −31.8785 + 42.5613i
8.4 −3.81387 3.94464 + 3.94464i 6.54563 −2.31695 + 10.9376i −15.0444 − 15.0444i − 7.55392i 5.54679 4.12041i 8.83656 − 41.7147i
8.5 −2.75261 −6.91673 − 6.91673i −0.423165 −11.0108 + 1.93945i 19.0390 + 19.0390i − 16.7708i 23.1856 68.6824i 30.3085 − 5.33853i
8.6 −2.47021 −3.25938 − 3.25938i −1.89808 −0.651659 − 11.1613i 8.05133 + 8.05133i 30.0815i 24.4503 − 5.75295i 1.60973 + 27.5708i
8.7 −2.30456 −1.40142 − 1.40142i −2.68902 6.43432 + 9.14328i 3.22965 + 3.22965i − 6.41824i 24.6334 − 23.0720i −14.8282 − 21.0712i
8.8 −2.16831 3.04119 + 3.04119i −3.29841 −7.13705 − 8.60596i −6.59426 − 6.59426i − 11.9829i 24.4985 − 8.50231i 15.4754 + 18.6604i
8.9 0.0453820 5.95117 + 5.95117i −7.99794 11.0966 + 1.36605i 0.270076 + 0.270076i 4.77624i −0.726019 43.8328i 0.503585 + 0.0619942i
8.10 0.410277 3.87142 + 3.87142i −7.83167 −10.2883 + 4.37616i 1.58835 + 1.58835i 30.8655i −6.49537 2.97581i −4.22105 + 1.79544i
8.11 0.545907 −0.683015 − 0.683015i −7.70199 7.92737 − 7.88396i −0.372862 − 0.372862i − 16.5537i −8.57182 − 26.0670i 4.32761 − 4.30391i
8.12 1.44022 −1.74614 − 1.74614i −5.92576 −10.5280 + 3.76320i −2.51483 − 2.51483i − 23.1100i −20.0562 − 20.9020i −15.1626 + 5.41984i
8.13 3.04409 −4.95613 − 4.95613i 1.26648 −5.77062 − 9.57601i −15.0869 − 15.0869i 6.50915i −20.4974 22.1265i −17.5663 − 29.1502i
8.14 3.76323 6.85128 + 6.85128i 6.16193 −6.91497 − 8.78540i 25.7830 + 25.7830i − 26.0054i −6.91708 66.8800i −26.0226 − 33.0615i
8.15 3.79801 2.69971 + 2.69971i 6.42486 1.19913 + 11.1158i 10.2535 + 10.2535i − 4.94611i −5.98239 − 12.4231i 4.55429 + 42.2181i
8.16 3.90194 1.35357 + 1.35357i 7.22514 8.64694 − 7.08734i 5.28156 + 5.28156i 17.9010i −3.02347 − 23.3357i 33.7399 − 27.6544i
8.17 4.75497 −5.71765 − 5.71765i 14.6097 10.1926 + 4.59458i −27.1872 − 27.1872i − 25.0062i 31.4290 38.3831i 48.4656 + 21.8471i
8.18 5.50957 −0.425760 − 0.425760i 22.3554 −11.0558 − 1.66406i −2.34575 − 2.34575i 12.5286i 79.0918 − 26.6375i −60.9127 − 9.16826i
57.1 −5.16850 −1.03806 + 1.03806i 18.7134 −10.3807 − 4.15234i 5.36523 − 5.36523i − 23.9761i −55.3719 24.8448i 53.6524 + 21.4614i
57.2 −4.77932 −3.98774 + 3.98774i 14.8419 10.8554 + 2.67593i 19.0587 − 19.0587i 14.1514i −32.6998 − 4.80418i −51.8814 − 12.7891i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.4.k.b yes 36
5.c odd 4 1 65.4.f.b ✓ 36
13.d odd 4 1 65.4.f.b ✓ 36
65.k even 4 1 inner 65.4.k.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.f.b ✓ 36 5.c odd 4 1
65.4.f.b ✓ 36 13.d odd 4 1
65.4.k.b yes 36 1.a even 1 1 trivial
65.4.k.b yes 36 65.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + T_{2}^{17} - 109 T_{2}^{16} - 111 T_{2}^{15} + 4872 T_{2}^{14} + 5004 T_{2}^{13} + \cdots + 991232 \) acting on \(S_{4}^{\mathrm{new}}(65, [\chi])\). Copy content Toggle raw display