Properties

Label 211.2.a.c
Level 211
Weight 2
Character orbit 211.a
Self dual yes
Analytic conductor 1.685
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.68484348265\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

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

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
2.11491
−0.254102
−1.86081
−2.11491 1.11491 2.47283 −1.35793 −2.35793 −3.11491 −1.00000 −1.75698 2.87189
1.2 0.254102 −1.25410 −1.93543 0.681331 −0.318669 −0.745898 −1.00000 −1.42723 0.173127
1.3 1.86081 −2.86081 1.46260 −4.32340 −5.32340 0.860806 −1.00000 5.18421 −8.04502
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(211\) \(1\)

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 211.2.a.c 3
3.b odd 2 1 1899.2.a.e 3
4.b odd 2 1 3376.2.a.m 3
5.b even 2 1 5275.2.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.a.c 3 1.a even 1 1 trivial
1899.2.a.e 3 3.b odd 2 1
3376.2.a.m 3 4.b odd 2 1
5275.2.a.h 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(211))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + T^{3} + 4 T^{4} + 8 T^{6} \)
$3$ \( 1 + 3 T + 8 T^{2} + 14 T^{3} + 24 T^{4} + 27 T^{5} + 27 T^{6} \)
$5$ \( 1 + 5 T + 17 T^{2} + 46 T^{3} + 85 T^{4} + 125 T^{5} + 125 T^{6} \)
$7$ \( 1 + 3 T + 20 T^{2} + 40 T^{3} + 140 T^{4} + 147 T^{5} + 343 T^{6} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{3} \)
$13$ \( 1 - T + 18 T^{2} + 11 T^{3} + 234 T^{4} - 169 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 17 T + 142 T^{2} + 726 T^{3} + 2414 T^{4} + 4913 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 2 T + 53 T^{2} - 69 T^{3} + 1007 T^{4} - 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 16 T + 142 T^{2} - 810 T^{3} + 3266 T^{4} - 8464 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 20 T + 208 T^{2} + 1386 T^{3} + 6032 T^{4} + 16820 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 3 T + 48 T^{2} - 240 T^{3} + 1488 T^{4} - 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 5 T + 107 T^{2} - 366 T^{3} + 3959 T^{4} - 6845 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 2 T + 34 T^{2} + 222 T^{3} + 1394 T^{4} + 3362 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 3 T + 68 T^{2} - 259 T^{3} + 2924 T^{4} - 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 4 T + 131 T^{2} + 335 T^{3} + 6157 T^{4} + 8836 T^{5} + 103823 T^{6} \)
$53$ \( 1 + T + 154 T^{2} + 108 T^{3} + 8162 T^{4} + 2809 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 12 T + 164 T^{2} + 1268 T^{3} + 9676 T^{4} + 41772 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 126 T^{2} + 52 T^{3} + 7686 T^{4} + 226981 T^{6} \)
$67$ \( ( 1 + 67 T^{2} )^{3} \)
$71$ \( 1 + 11 T + 95 T^{2} + 790 T^{3} + 6745 T^{4} + 55451 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 3 T + 218 T^{2} - 436 T^{3} + 15914 T^{4} - 15987 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 5 T + 11 T^{2} - 822 T^{3} + 869 T^{4} + 31205 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 28 T + 461 T^{2} - 5096 T^{3} + 38263 T^{4} - 192892 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 5 T + 78 T^{2} - 1626 T^{3} + 6942 T^{4} - 39605 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 7 T + 222 T^{2} - 1246 T^{3} + 21534 T^{4} - 65863 T^{5} + 912673 T^{6} \)
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