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Properties

Label 3721.2.a.n
Level $3721$
Weight $2$
Character orbit 3721.a
Self dual yes
Analytic conductor $29.712$
Analytic rank $1$
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: \(1\)
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 −2.76020 0.777361 5.61869 0.628131 −2.14567 −2.30305 −9.98830 −2.39571 −1.73377
1.2 −2.72463 1.85140 5.42360 −2.62043 −5.04437 −1.38510 −9.32805 0.427667 7.13969
1.3 −2.68963 2.04146 5.23410 1.47003 −5.49076 3.18507 −8.69852 1.16756 −3.95384
1.4 −2.65031 −0.426401 5.02416 −3.76557 1.13010 1.73489 −8.01499 −2.81818 9.97995
1.5 −2.63397 −2.67803 4.93777 −0.239269 7.05385 3.36589 −7.73800 4.17187 0.630227
1.6 −2.61760 −2.16292 4.85181 2.86325 5.66166 −0.958009 −7.46490 1.67824 −7.49483
1.7 −2.58029 2.36485 4.65790 1.76009 −6.10201 −3.27872 −6.85816 2.59253 −4.54155
1.8 −2.46898 −0.972642 4.09585 −2.26783 2.40143 −5.00627 −5.17460 −2.05397 5.59921
1.9 −2.46542 2.10607 4.07828 −1.08921 −5.19235 2.79859 −5.12382 1.43555 2.68536
1.10 −2.33939 0.727632 3.47273 3.91568 −1.70221 −3.44376 −3.44529 −2.47055 −9.16029
1.11 −2.29420 0.708665 3.26334 1.27750 −1.62582 −0.00140676 −2.89835 −2.49779 −2.93084
1.12 −2.26399 3.31571 3.12567 −2.52823 −7.50675 0.155350 −2.54850 7.99394 5.72389
1.13 −2.24183 −0.302359 3.02582 3.40438 0.677840 3.08754 −2.29972 −2.90858 −7.63206
1.14 −2.11764 −1.59251 2.48440 −2.65577 3.37237 3.50453 −1.02579 −0.463898 5.62397
1.15 −2.08961 −2.58293 2.36645 4.15942 5.39730 −0.505717 −0.765739 3.67152 −8.69155
1.16 −2.02809 3.09306 2.11316 1.92824 −6.27300 −1.16188 −0.229489 6.56699 −3.91064
1.17 −2.02545 2.42966 2.10247 1.20652 −4.92117 −2.90805 −0.207540 2.90325 −2.44375
1.18 −1.85049 0.410201 1.42431 −3.23416 −0.759073 4.80487 1.06530 −2.83173 5.98479
1.19 −1.80267 −3.11674 1.24962 0.964493 5.61845 −2.17868 1.35269 6.71406 −1.73866
1.20 −1.76031 −1.92844 1.09868 0.693539 3.39464 −0.695360 1.58660 0.718877 −1.22084
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.n ✓ 75
61.b even 2 1 3721.2.a.o yes 75
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3721.2.a.n ✓ 75 1.a even 1 1 trivial
3721.2.a.o yes 75 61.b even 2 1

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