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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Newspace parameters

Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.a (trivial)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 21q + 3q^{3} + 32q^{4} + 7q^{5} + 5q^{6} + 5q^{7} - 6q^{8} + 40q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 21q + 3q^{3} + 32q^{4} + 7q^{5} + 5q^{6} + 5q^{7} - 6q^{8} + 40q^{9} + q^{10} - 21q^{11} + 6q^{12} + 20q^{13} + 17q^{14} + 18q^{15} + 50q^{16} + q^{17} - 5q^{18} + 15q^{19} - 2q^{20} + 16q^{21} + 11q^{23} - 16q^{24} + 48q^{25} - 5q^{26} + 12q^{27} - 16q^{28} - 9q^{29} + 16q^{30} + 22q^{31} + 3q^{32} - 3q^{33} + 33q^{34} - 39q^{35} + 57q^{36} + 21q^{37} + 11q^{38} - 28q^{39} - 16q^{40} + 7q^{41} - 55q^{42} + 16q^{43} - 32q^{44} + 44q^{45} - 3q^{46} + 5q^{47} - 71q^{48} + 80q^{49} - 33q^{50} - 19q^{51} + 60q^{52} + 9q^{53} + 13q^{54} - 7q^{55} + 44q^{56} + 39q^{57} - 27q^{58} + 13q^{59} + 70q^{60} + 21q^{61} - 23q^{62} + 24q^{63} + 66q^{64} + 25q^{65} - 5q^{66} + 38q^{67} - 74q^{68} - 17q^{69} - 33q^{70} + 12q^{71} - 75q^{72} + 20q^{73} - 12q^{74} - 10q^{75} + 59q^{76} - 5q^{77} - 14q^{78} + q^{79} - 38q^{80} + 89q^{81} + 7q^{82} - 19q^{83} - 14q^{84} + 38q^{85} - 3q^{86} + 4q^{87} + 6q^{88} + 37q^{89} - 174q^{90} + 24q^{91} + 31q^{92} - 15q^{93} - 64q^{94} - 43q^{95} - 38q^{96} + 68q^{97} - 2q^{98} - 40q^{99} + O(q^{100}) \)

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.

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 \(T_{2}^{21} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(671))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database