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Properties

Label 3721.2.a.o
Level $3721$
Weight $2$
Character orbit 3721.a
Self dual yes
Analytic conductor $29.712$
Analytic rank $0$
Dimension $75$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3721,2,Mod(1,3721)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3721, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3721.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3721 = 61^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3721.a (trivial)

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: yes
Analytic conductor: \(29.7123345921\)
Analytic rank: \(0\)
Dimension: \(75\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 75 q + 14 q^{2} - 2 q^{3} + 74 q^{4} - q^{5} + 26 q^{6} + 28 q^{7} + 36 q^{8} + 69 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 75 q + 14 q^{2} - 2 q^{3} + 74 q^{4} - q^{5} + 26 q^{6} + 28 q^{7} + 36 q^{8} + 69 q^{9} + 25 q^{10} + 52 q^{11} - 10 q^{12} - 3 q^{13} - 10 q^{14} - 14 q^{15} + 64 q^{16} + 52 q^{17} + 25 q^{18} - 8 q^{19} - 23 q^{20} + 51 q^{21} - 26 q^{22} + 83 q^{23} + 68 q^{24} + 58 q^{25} + 72 q^{26} - 26 q^{27} + 32 q^{28} + 79 q^{29} + 29 q^{30} + 78 q^{31} + 81 q^{32} + 31 q^{33} - 29 q^{34} + 96 q^{35} + 38 q^{36} + 11 q^{37} + 54 q^{38} - 30 q^{39} + 48 q^{40} - 25 q^{41} - 48 q^{42} + 51 q^{43} + 106 q^{44} - 27 q^{45} + 86 q^{46} - 24 q^{47} + 56 q^{48} + 57 q^{49} + 70 q^{50} + 92 q^{51} + 83 q^{52} + 64 q^{53} + 56 q^{54} + 87 q^{55} - 64 q^{56} - 28 q^{57} + 71 q^{58} + 137 q^{59} - 92 q^{60} + 61 q^{62} + 112 q^{63} + 16 q^{64} + 61 q^{65} + 68 q^{66} + 27 q^{67} + 126 q^{68} + 63 q^{69} + 19 q^{70} + 178 q^{71} + 47 q^{72} - 31 q^{73} - 59 q^{74} - 64 q^{75} + 36 q^{76} + 55 q^{77} - 35 q^{78} + 107 q^{79} - 85 q^{80} + 23 q^{81} + 5 q^{82} - 40 q^{83} + 63 q^{84} + 23 q^{85} + 21 q^{86} + 118 q^{87} + 36 q^{88} + 145 q^{89} + 33 q^{90} + 71 q^{91} + 99 q^{92} - 16 q^{93} + 44 q^{94} - 68 q^{95} + 74 q^{96} + 46 q^{97} + 53 q^{98} + 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.66692 −1.05933 5.11247 −2.72215 2.82516 −2.42097 −8.30073 −1.87782 7.25977
1.2 −2.59431 −3.00318 4.73044 −0.890524 7.79118 −0.432047 −7.08359 6.01911 2.31029
1.3 −2.55275 0.342998 4.51655 −0.716142 −0.875589 2.30684 −6.42412 −2.88235 1.82813
1.4 −2.49665 0.894825 4.23329 −2.93477 −2.23407 0.332308 −5.57574 −2.19929 7.32710
1.5 −2.34283 −0.327888 3.48885 0.902209 0.768186 2.35196 −3.48813 −2.89249 −2.11372
1.6 −2.30567 −2.71116 3.31612 2.58866 6.25103 3.06593 −3.03453 4.35037 −5.96861
1.7 −2.28174 0.759526 3.20636 −1.94659 −1.73304 −1.76057 −2.75259 −2.42312 4.44162
1.8 −2.26535 1.07814 3.13182 0.267260 −2.44238 4.55239 −2.56398 −1.83760 −0.605438
1.9 −2.16879 −3.22253 2.70366 1.23741 6.98899 3.05509 −1.52609 7.38468 −2.68369
1.10 −2.13041 1.95700 2.53863 −3.52627 −4.16920 0.0411692 −1.14750 0.829842 7.51239
1.11 −2.04974 −0.101492 2.20143 0.352522 0.208032 1.13594 −0.412881 −2.98970 −0.722578
1.12 −1.97728 −1.57252 1.90963 −0.645666 3.10932 0.00698419 0.178681 −0.527174 1.27666
1.13 −1.92750 2.90194 1.71525 −1.16361 −5.59348 4.84668 0.548852 5.42123 2.24286
1.14 −1.75774 −1.90882 1.08965 1.08278 3.35521 −3.05118 1.60015 0.643594 −1.90324
1.15 −1.68257 −2.70847 0.831037 −3.79019 4.55719 −3.50528 1.96686 4.33581 6.37726
1.16 −1.66176 −1.76580 0.761457 3.83014 2.93435 1.72586 2.05817 0.118060 −6.36479
1.17 −1.65463 1.49680 0.737797 3.16720 −2.47665 4.49624 2.08848 −0.759593 −5.24053
1.18 −1.44193 2.82327 0.0791727 −3.00365 −4.07097 1.75646 2.76971 4.97084 4.33107
1.19 −1.43268 0.0200939 0.0525827 3.23768 −0.0287883 −2.81483 2.79003 −2.99960 −4.63857
1.20 −1.38611 1.00078 −0.0787125 −0.413618 −1.38718 0.936248 2.88131 −1.99844 0.573318
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(61\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3721.2.a.o yes 75
61.b even 2 1 3721.2.a.n ✓ 75
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3721.2.a.n ✓ 75 61.b even 2 1
3721.2.a.o yes 75 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{75} - 14 T_{2}^{74} - 14 T_{2}^{73} + 1080 T_{2}^{72} - 2951 T_{2}^{71} - 36891 T_{2}^{70} + \cdots + 114143 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\). Copy content Toggle raw display