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Properties

Label 671.2.a.d
Level $671$
Weight $2$
Character orbit 671.a
Self dual yes
Analytic conductor $5.358$
Analytic rank $0$
Dimension $21$
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] = [671,2,Mod(1,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, 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("671.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 3 q^{3} + 32 q^{4} + 7 q^{5} + 5 q^{6} + 5 q^{7} - 6 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 3 q^{3} + 32 q^{4} + 7 q^{5} + 5 q^{6} + 5 q^{7} - 6 q^{8} + 40 q^{9} + q^{10} - 21 q^{11} + 6 q^{12} + 20 q^{13} + 17 q^{14} + 18 q^{15} + 50 q^{16} + q^{17} - 5 q^{18} + 15 q^{19} - 2 q^{20} + 16 q^{21} + 11 q^{23} - 16 q^{24} + 48 q^{25} - 5 q^{26} + 12 q^{27} - 16 q^{28} - 9 q^{29} + 16 q^{30} + 22 q^{31} + 3 q^{32} - 3 q^{33} + 33 q^{34} - 39 q^{35} + 57 q^{36} + 21 q^{37} + 11 q^{38} - 28 q^{39} - 16 q^{40} + 7 q^{41} - 55 q^{42} + 16 q^{43} - 32 q^{44} + 44 q^{45} - 3 q^{46} + 5 q^{47} - 71 q^{48} + 80 q^{49} - 33 q^{50} - 19 q^{51} + 60 q^{52} + 9 q^{53} + 13 q^{54} - 7 q^{55} + 44 q^{56} + 39 q^{57} - 27 q^{58} + 13 q^{59} + 70 q^{60} + 21 q^{61} - 23 q^{62} + 24 q^{63} + 66 q^{64} + 25 q^{65} - 5 q^{66} + 38 q^{67} - 74 q^{68} - 17 q^{69} - 33 q^{70} + 12 q^{71} - 75 q^{72} + 20 q^{73} - 12 q^{74} - 10 q^{75} + 59 q^{76} - 5 q^{77} - 14 q^{78} + q^{79} - 38 q^{80} + 89 q^{81} + 7 q^{82} - 19 q^{83} - 14 q^{84} + 38 q^{85} - 3 q^{86} + 4 q^{87} + 6 q^{88} + 37 q^{89} - 174 q^{90} + 24 q^{91} + 31 q^{92} - 15 q^{93} - 64 q^{94} - 43 q^{95} - 38 q^{96} + 68 q^{97} - 2 q^{98} - 40 q^{99}+O(q^{100}) \) 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 −2.73839 −3.22793 5.49879 0.170778 8.83935 −3.28872 −9.58106 7.41956 −0.467657
1.2 −2.60374 2.97894 4.77947 0.851432 −7.75638 3.04314 −7.23702 5.87407 −2.21691
1.3 −2.55546 1.15142 4.53040 4.10969 −2.94242 −4.64385 −6.46634 −1.67423 −10.5022
1.4 −2.40042 −0.672797 3.76203 −3.90458 1.61500 −3.05635 −4.22962 −2.54734 9.37264
1.5 −2.19302 1.13601 2.80936 −3.87665 −2.49129 3.34814 −1.77494 −1.70949 8.50160
1.6 −1.47389 −0.831223 0.172365 0.593969 1.22513 3.31761 2.69374 −2.30907 −0.875448
1.7 −1.45442 −2.91272 0.115325 3.82594 4.23631 3.26042 2.74110 5.48395 −5.56450
1.8 −1.29453 3.40741 −0.324186 2.94198 −4.41100 −0.962580 3.00873 8.61042 −3.80849
1.9 −0.634118 −1.50740 −1.59789 −1.64555 0.955868 −3.41198 2.28149 −0.727754 1.04347
1.10 −0.404310 1.66838 −1.83653 0.0975766 −0.674545 1.38056 1.55115 −0.216497 −0.0394512
1.11 0.472572 −1.64615 −1.77668 −3.20957 −0.777925 4.06194 −1.78475 −0.290187 −1.51675
1.12 0.543169 −1.77928 −1.70497 −0.382346 −0.966447 −4.84691 −2.01242 0.165824 −0.207678
1.13 0.731918 −2.97561 −1.46430 2.76447 −2.17790 −1.34613 −2.53558 5.85424 2.02337
1.14 0.942064 2.26946 −1.11252 4.16220 2.13797 −0.914635 −2.93219 2.15044 3.92106
1.15 1.29392 2.84985 −0.325780 −3.15298 3.68746 5.03078 −3.00937 5.12162 −4.07969
1.16 1.61525 2.06891 0.609041 2.50050 3.34180 4.06392 −2.24675 1.28037 4.03894
1.17 2.08860 3.36235 2.36225 −0.0546248 7.02260 −2.53368 0.756597 8.30537 −0.114089
1.18 2.16302 −0.957308 2.67865 3.42546 −2.07068 1.52570 1.46794 −2.08356 7.40934
1.19 2.54333 0.588033 4.46852 −0.268217 1.49556 2.27426 6.27827 −2.65422 −0.682163
1.20 2.67686 1.01551 5.16560 1.61394 2.71838 −4.48909 8.47390 −1.96874 4.32029
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(61\) \(-1\)

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 671.2.a.d ✓ 21
3.b odd 2 1 6039.2.a.l 21
11.b odd 2 1 7381.2.a.j 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.d ✓ 21 1.a even 1 1 trivial
6039.2.a.l 21 3.b odd 2 1
7381.2.a.j 21 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 37 T_{2}^{19} + 2 T_{2}^{18} + 582 T_{2}^{17} - 65 T_{2}^{16} - 5074 T_{2}^{15} + \cdots - 2082 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(671))\). Copy content Toggle raw display