Properties

Label 671.2.a.a
Level $671$
Weight $2$
Character orbit 671.a
Self dual yes
Analytic conductor $5.358$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
Defining polynomial: \(x^{5} - 5 x^{3} - x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{4} ) q^{3} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{6} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{8} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{4} ) q^{3} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{6} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{8} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + ( 1 - \beta_{3} ) q^{10} + q^{11} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{12} + ( -2 + \beta_{1} - \beta_{4} ) q^{13} + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{14} - q^{15} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{16} + ( -1 + \beta_{2} ) q^{17} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{18} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{20} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + \beta_{1} q^{22} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{24} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} + ( 1 - 4 \beta_{1} - \beta_{2} + \beta_{4} ) q^{26} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{27} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( -3 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{29} -\beta_{1} q^{30} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{31} + ( 1 + \beta_{3} + 2 \beta_{4} ) q^{32} + ( -\beta_{1} + \beta_{4} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} ) q^{34} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} + 3 \beta_{4} ) q^{36} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{37} + ( 1 - 7 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{38} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{40} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{42} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{43} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{44} + ( \beta_{1} + 3 \beta_{2} - \beta_{4} ) q^{45} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{46} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{47} + ( 1 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} ) q^{48} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( 1 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{50} + ( 1 + \beta_{1} - \beta_{4} ) q^{51} + ( -2 + 7 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{52} + ( 1 - 3 \beta_{1} - \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{53} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{54} -\beta_{2} q^{55} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{56} + ( -3 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{57} + ( -3 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{58} + ( -2 + 2 \beta_{1} - 3 \beta_{3} ) q^{59} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{60} + q^{61} + ( 5 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{62} + ( 2 + \beta_{1} - 4 \beta_{2} ) q^{63} + ( 2 + 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{64} + ( 1 + 2 \beta_{2} ) q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{66} + ( -1 + 4 \beta_{1} - 4 \beta_{4} ) q^{67} + ( \beta_{1} - \beta_{4} ) q^{68} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{69} + ( -3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{70} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 3 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{72} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{73} + ( 6 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{74} + ( 5 \beta_{1} + \beta_{2} - 5 \beta_{4} ) q^{75} + ( -3 + 11 \beta_{1} + 6 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{76} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{77} + ( 4 - 8 \beta_{1} - 4 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{78} + ( -5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{79} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{80} + ( 1 + 6 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{81} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{82} + ( -1 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( -3 + 7 \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{84} + ( -2 - \beta_{1} + \beta_{3} ) q^{85} + ( 1 + 8 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{86} + ( 2 + 4 \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{87} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{88} + ( -4 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{90} + ( 2 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( 3 - 3 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} ) q^{92} + ( -2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{93} + ( -4 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{94} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{95} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{96} + ( -1 + 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} ) q^{97} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{98} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} - 2q^{5} - 5q^{6} - q^{7} - 6q^{8} - 3q^{9} + O(q^{10}) \) \( 5q - 2q^{2} - 2q^{5} - 5q^{6} - q^{7} - 6q^{8} - 3q^{9} + 5q^{10} + 5q^{11} + 12q^{12} - 10q^{13} - 3q^{14} - 5q^{15} + 2q^{16} - 3q^{17} - 6q^{18} - 13q^{19} - 2q^{21} - 2q^{22} - 12q^{24} - 15q^{25} + 9q^{26} - 9q^{27} - 12q^{28} - 7q^{29} + 2q^{30} - 13q^{31} + q^{32} - 3q^{34} - 13q^{35} + 3q^{36} - 6q^{37} + 9q^{38} - 12q^{39} - 5q^{40} - 9q^{41} + 13q^{42} + 2q^{43} + 6q^{45} - 7q^{46} - 3q^{47} + q^{48} - 6q^{49} + 9q^{50} + 5q^{51} - 12q^{52} + 3q^{53} + 15q^{54} - 2q^{55} + 28q^{56} - 17q^{57} - 15q^{58} - 14q^{59} + 5q^{61} + 31q^{62} + 6q^{64} + 9q^{65} - 5q^{66} - 5q^{67} - 10q^{69} - 5q^{70} + 3q^{71} + 5q^{72} - 4q^{73} + 2q^{74} + 2q^{75} - 19q^{76} - q^{77} + 22q^{78} - 27q^{79} - 2q^{80} + q^{81} + 11q^{82} - 3q^{83} - 15q^{84} - 8q^{85} + 5q^{86} + 14q^{87} - 6q^{88} - 12q^{89} - 10q^{90} + 4q^{91} + 13q^{92} - 12q^{93} - 18q^{94} + 7q^{95} + 12q^{96} - 5q^{97} + 14q^{98} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 5 x^{3} - x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{4} + 5 \nu^{2} + \nu - 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 3 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - \nu^{3} - 9 \nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} + 5 \beta_{3} - 4 \beta_{1} + 8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.96003
0.878095
−0.722813
2.17442
−0.369680
−2.44983 2.04164 4.00166 −0.489803 −5.00166 −2.06203 −4.90374 1.16828 1.19993
1.2 −1.26073 0.467546 −0.410549 −2.13883 −0.589451 2.81011 3.03906 −2.78140 2.69649
1.3 −0.339328 −2.60767 −1.88486 0.383484 0.884856 0.840700 1.31824 3.79994 −0.130127
1.4 0.714533 0.684982 −1.48944 −1.45989 0.489443 1.23480 −2.49332 −2.53080 −1.04314
1.5 1.33536 −0.586497 −0.216816 1.70504 −0.783184 −3.82358 −2.96025 −2.65602 2.27684
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.a 5
3.b odd 2 1 6039.2.a.a 5
11.b odd 2 1 7381.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.a 5 1.a even 1 1 trivial
6039.2.a.a 5 3.b odd 2 1
7381.2.a.g 5 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 2 T_{2}^{4} - 3 T_{2}^{3} - 4 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(671))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 4 T^{2} - 3 T^{3} + 2 T^{4} + T^{5} \)
$3$ \( -1 + 2 T + 3 T^{2} - 6 T^{3} + T^{5} \)
$5$ \( 1 - 6 T^{2} - 3 T^{3} + 2 T^{4} + T^{5} \)
$7$ \( -23 + 37 T - T^{2} - 14 T^{3} + T^{4} + T^{5} \)
$11$ \( ( -1 + T )^{5} \)
$13$ \( -23 - 2 T + 41 T^{2} + 34 T^{3} + 10 T^{4} + T^{5} \)
$17$ \( 1 - 5 T^{2} - T^{3} + 3 T^{4} + T^{5} \)
$19$ \( -5 - 47 T + 36 T^{2} + 49 T^{3} + 13 T^{4} + T^{5} \)
$23$ \( -59 - 49 T + 101 T^{2} - 35 T^{3} + T^{5} \)
$29$ \( -2885 - 2681 T - 806 T^{2} - 67 T^{3} + 7 T^{4} + T^{5} \)
$31$ \( 313 + 39 T - 175 T^{2} + 8 T^{3} + 13 T^{4} + T^{5} \)
$37$ \( 97 + 308 T - 139 T^{2} - 32 T^{3} + 6 T^{4} + T^{5} \)
$41$ \( -43 - 199 T - 122 T^{2} + T^{3} + 9 T^{4} + T^{5} \)
$43$ \( -1037 + 228 T + 247 T^{2} - 66 T^{3} - 2 T^{4} + T^{5} \)
$47$ \( -17 - 129 T - 107 T^{2} - 22 T^{3} + 3 T^{4} + T^{5} \)
$53$ \( 2347 + 3077 T + 29 T^{2} - 130 T^{3} - 3 T^{4} + T^{5} \)
$59$ \( 655 - 989 T - 309 T^{2} + 27 T^{3} + 14 T^{4} + T^{5} \)
$61$ \( ( -1 + T )^{5} \)
$67$ \( 1249 - 155 T - 470 T^{2} - 86 T^{3} + 5 T^{4} + T^{5} \)
$71$ \( -149 - 340 T - 259 T^{2} - 73 T^{3} - 3 T^{4} + T^{5} \)
$73$ \( 431 + 818 T - 99 T^{2} - 118 T^{3} + 4 T^{4} + T^{5} \)
$79$ \( 1285 + 1959 T + 1076 T^{2} + 259 T^{3} + 27 T^{4} + T^{5} \)
$83$ \( 3457 + 1119 T - 718 T^{2} - 143 T^{3} + 3 T^{4} + T^{5} \)
$89$ \( 3995 - 253 T - 741 T^{2} - 75 T^{3} + 12 T^{4} + T^{5} \)
$97$ \( 62147 + 9859 T - 1362 T^{2} - 227 T^{3} + 5 T^{4} + T^{5} \)
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