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Properties

Label 3997.2.a.c
Level $3997$
Weight $2$
Character orbit 3997.a
Self dual yes
Analytic conductor $31.916$
Analytic rank $1$
Dimension $63$
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] = [3997,2,Mod(1,3997)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3997, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3997.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3997 = 7 \cdot 571 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3997.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: \(31.9162056879\)
Analytic rank: \(1\)
Dimension: \(63\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 63 q - 21 q^{2} - 14 q^{3} + 57 q^{4} - 16 q^{5} - 3 q^{6} + 63 q^{7} - 54 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 63 q - 21 q^{2} - 14 q^{3} + 57 q^{4} - 16 q^{5} - 3 q^{6} + 63 q^{7} - 54 q^{8} + 39 q^{9} - 8 q^{10} - 54 q^{11} - 27 q^{12} - 10 q^{13} - 21 q^{14} - 33 q^{15} + 53 q^{16} - 25 q^{17} - 37 q^{18} - 23 q^{19} - 36 q^{20} - 14 q^{21} - 8 q^{22} - 62 q^{23} - 6 q^{24} + 37 q^{25} - 4 q^{26} - 56 q^{27} + 57 q^{28} - 96 q^{29} - 9 q^{30} - 9 q^{31} - 119 q^{32} - 23 q^{33} - 16 q^{35} + 16 q^{36} - 21 q^{37} - 21 q^{38} - 36 q^{39} + 3 q^{40} - 34 q^{41} - 3 q^{42} - 36 q^{43} - 79 q^{44} - 28 q^{45} - 15 q^{46} - 71 q^{47} - 16 q^{48} + 63 q^{49} - 69 q^{50} - 59 q^{51} + 4 q^{52} - 118 q^{53} - 4 q^{54} - 6 q^{55} - 54 q^{56} - 55 q^{57} + 16 q^{58} - 67 q^{59} - 32 q^{60} + q^{61} - 16 q^{62} + 39 q^{63} + 58 q^{64} - 100 q^{65} + 80 q^{66} - 82 q^{67} - 60 q^{68} + 7 q^{69} - 8 q^{70} - 115 q^{71} - 74 q^{72} - 9 q^{73} - 29 q^{74} - 38 q^{75} - 34 q^{76} - 54 q^{77} - 44 q^{78} - 21 q^{79} - 25 q^{80} + 3 q^{81} - q^{82} - 139 q^{83} - 27 q^{84} - 21 q^{85} - 58 q^{86} - q^{87} + 15 q^{88} - 29 q^{89} + 59 q^{90} - 10 q^{91} - 79 q^{92} - 45 q^{93} + 21 q^{94} - 83 q^{95} + 65 q^{96} + 22 q^{97} - 21 q^{98} - 73 q^{99}+O(q^{100}) \) 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 −2.77343 2.00613 5.69192 2.29701 −5.56386 1.00000 −10.2393 1.02455 −6.37059
1.2 −2.73580 −2.49872 5.48463 −0.283532 6.83602 1.00000 −9.53326 3.24361 0.775688
1.3 −2.72115 −1.64084 5.40464 −3.79956 4.46496 1.00000 −9.26451 −0.307651 10.3392
1.4 −2.71107 −2.75286 5.34989 3.30017 7.46321 1.00000 −9.08180 4.57827 −8.94700
1.5 −2.66758 0.671177 5.11600 −1.55965 −1.79042 1.00000 −8.31218 −2.54952 4.16048
1.6 −2.64489 1.29784 4.99543 −3.01727 −3.43265 1.00000 −7.92256 −1.31560 7.98035
1.7 −2.58317 −1.81136 4.67278 1.60435 4.67906 1.00000 −6.90426 0.281024 −4.14431
1.8 −2.53379 2.83648 4.42008 −4.00855 −7.18704 1.00000 −6.13198 5.04562 10.1568
1.9 −2.36500 0.166207 3.59324 0.549101 −0.393080 1.00000 −3.76801 −2.97238 −1.29862
1.10 −2.24680 2.42500 3.04812 −0.265457 −5.44850 1.00000 −2.35492 2.88064 0.596429
1.11 −2.23095 1.69776 2.97712 3.33002 −3.78762 1.00000 −2.17990 −0.117601 −7.42910
1.12 −2.17956 −2.80153 2.75047 −2.98927 6.10609 1.00000 −1.63568 4.84856 6.51528
1.13 −2.15959 1.31930 2.66383 1.06257 −2.84914 1.00000 −1.43359 −1.25945 −2.29471
1.14 −2.09676 −0.889167 2.39639 0.531264 1.86437 1.00000 −0.831142 −2.20938 −1.11393
1.15 −1.89400 −1.40315 1.58723 3.78261 2.65757 1.00000 0.781786 −1.03117 −7.16426
1.16 −1.85416 −0.558848 1.43789 −2.57345 1.03619 1.00000 1.04224 −2.68769 4.77158
1.17 −1.77165 −2.69105 1.13873 −1.03885 4.76758 1.00000 1.52587 4.24172 1.84047
1.18 −1.72027 −1.64604 0.959329 −2.39461 2.83163 1.00000 1.79024 −0.290557 4.11937
1.19 −1.64987 0.529692 0.722067 3.72123 −0.873922 1.00000 2.10842 −2.71943 −6.13954
1.20 −1.63355 2.61666 0.668475 −2.03536 −4.27443 1.00000 2.17511 3.84689 3.32486
See all 63 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.63
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(571\) \(-1\)

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 3997.2.a.c ✓ 63
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3997.2.a.c ✓ 63 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{63} + 21 T_{2}^{62} + 129 T_{2}^{61} - 332 T_{2}^{60} - 7145 T_{2}^{59} - 17879 T_{2}^{58} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3997))\). Copy content Toggle raw display