Properties

Label 406.2.e.a
Level $406$
Weight $2$
Character orbit 406.e
Analytic conductor $3.242$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 406 = 2 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 406.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.24192632206\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.3118758597603.1
Defining polynomial: \( x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - 1) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_1) q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 1) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_1) q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - 3) q^{9} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{10} + (\beta_{9} + \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{4} - 1) q^{12} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_1 + 2) q^{13} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{14} + ( - \beta_{9} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{15} + (\beta_{4} - 1) q^{16} + ( - \beta_{8} - 2 \beta_{4} - \beta_1) q^{17} + ( - \beta_{9} + \beta_{8} - 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{18} + ( - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{19} + ( - \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{20} + ( - 3 \beta_{9} + \beta_{8} - \beta_{4} - 2 \beta_{2} + 3) q^{21} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_1 - 1) q^{22} + ( - 2 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{2}) q^{23} + ( - \beta_{4} - \beta_{3}) q^{24} + (\beta_{9} + \beta_{8} - \beta_{4} + 4 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2) q^{26} + ( - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + \cdots + 5) q^{27}+ \cdots + (3 \beta_{9} - \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - 7 q^{5} + 6 q^{6} - 3 q^{7} + 10 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - 7 q^{5} + 6 q^{6} - 3 q^{7} + 10 q^{8} - 8 q^{9} - 7 q^{10} - 3 q^{12} + 20 q^{13} + 3 q^{14} - 20 q^{15} - 5 q^{16} - 8 q^{17} - 8 q^{18} - 2 q^{19} + 14 q^{20} + 19 q^{21} - q^{23} - 3 q^{24} - 12 q^{25} - 10 q^{26} + 30 q^{27} - 10 q^{29} + 10 q^{30} - 11 q^{31} - 5 q^{32} - 9 q^{33} + 16 q^{34} + 10 q^{35} + 16 q^{36} + 8 q^{37} - 2 q^{38} - 18 q^{39} - 7 q^{40} + 46 q^{41} - 8 q^{42} - 6 q^{43} - 4 q^{45} - q^{46} - 16 q^{47} + 6 q^{48} - 11 q^{49} + 24 q^{50} - 7 q^{51} - 10 q^{52} - 7 q^{53} - 15 q^{54} + 12 q^{55} - 3 q^{56} - 68 q^{57} + 5 q^{58} + 9 q^{59} + 10 q^{60} - 15 q^{61} + 22 q^{62} - 3 q^{63} + 10 q^{64} - 5 q^{65} - 9 q^{66} + 4 q^{67} - 8 q^{68} + 28 q^{69} + 4 q^{70} - 44 q^{71} - 8 q^{72} + 8 q^{74} + 34 q^{75} + 4 q^{76} + 39 q^{77} + 36 q^{78} + 13 q^{79} - 7 q^{80} - 17 q^{81} - 23 q^{82} + 56 q^{83} - 11 q^{84} - 14 q^{85} + 3 q^{86} + 3 q^{87} - 17 q^{89} + 8 q^{90} + 6 q^{91} + 2 q^{92} - 17 q^{93} - 16 q^{94} + 9 q^{95} - 3 q^{96} + 84 q^{97} - 20 q^{98} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 244169 \nu^{9} - 750258 \nu^{8} + 198422 \nu^{7} + 3041052 \nu^{6} - 8027063 \nu^{5} + 7504426 \nu^{4} + 14486388 \nu^{3} - 35106720 \nu^{2} + \cdots - 17035541 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 313192 \nu^{9} + 3118568 \nu^{8} - 2157819 \nu^{7} - 10818671 \nu^{6} + 20747939 \nu^{5} - 49339027 \nu^{4} - 63283058 \nu^{3} + \cdots + 139659043 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 893358 \nu^{9} + 475480 \nu^{8} + 5106016 \nu^{7} - 5222834 \nu^{6} + 14656748 \nu^{5} + 26556357 \nu^{4} - 84060972 \nu^{3} - 113795607 \nu^{2} + \cdots + 29238847 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5726 \nu^{9} + 8728 \nu^{8} + 15695 \nu^{7} - 30701 \nu^{6} + 113887 \nu^{5} + 47565 \nu^{4} - 438521 \nu^{3} - 328616 \nu^{2} - 105832 \nu + 163662 ) / 139597 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1616901 \nu^{9} - 715422 \nu^{8} - 6561850 \nu^{7} + 4318132 \nu^{6} - 26461030 \nu^{5} - 45525517 \nu^{4} + 100658881 \nu^{3} + 204814762 \nu^{2} + \cdots + 75110912 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1768918 \nu^{9} + 516157 \nu^{8} + 5805510 \nu^{7} + 1407113 \nu^{6} + 25732292 \nu^{5} + 50029236 \nu^{4} - 65805547 \nu^{3} + \cdots - 138763851 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1850159 \nu^{9} - 1056286 \nu^{8} + 10198817 \nu^{7} + 514287 \nu^{6} + 22231484 \nu^{5} + 89090025 \nu^{4} - 99565695 \nu^{3} + \cdots - 199826103 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2396972 \nu^{9} - 3681779 \nu^{8} - 7578548 \nu^{7} + 14487182 \nu^{6} - 48162975 \nu^{5} - 21260146 \nu^{4} + 196378044 \nu^{3} + 144963467 \nu^{2} + \cdots - 77176853 ) / 27779803 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2611328 \nu^{9} - 2640778 \nu^{8} - 9404480 \nu^{7} + 11479543 \nu^{6} - 50294265 \nu^{5} - 42008212 \nu^{4} + 188098184 \nu^{3} + 228513885 \nu^{2} + \cdots + 13015032 ) / 27779803 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 11 \beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{9} - 6\beta_{8} + 7\beta_{7} + 2\beta_{6} + 2\beta_{5} - 7\beta_{4} - \beta_{3} + 6\beta_{2} - \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} + 26 \beta_{6} + 31 \beta_{5} - 20 \beta_{4} + 25 \beta_{3} + 53 \beta_{2} - 2 \beta _1 + 61 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{9} - 2 \beta_{8} + 64 \beta_{7} + 16 \beta_{6} + 113 \beta_{5} + 29 \beta_{4} - 7 \beta_{3} + 31 \beta_{2} + 11 \beta _1 + 53 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{9} - 143 \beta_{8} + 121 \beta_{7} + 55 \beta_{6} + 302 \beta_{5} - 202 \beta_{4} + 17 \beta_{3} + 10 \beta_{2} - 127 \beta _1 + 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 124 \beta_{9} - 213 \beta_{8} + 134 \beta_{7} + 56 \beta_{6} + 131 \beta_{5} - 275 \beta_{4} - 47 \beta_{3} + 81 \beta_{2} - 33 \beta _1 + 184 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1048 \beta_{9} - 890 \beta_{8} + 1075 \beta_{7} + 544 \beta_{6} + 455 \beta_{5} - 1864 \beta_{4} + 113 \beta_{3} + 1321 \beta_{2} + 353 \beta _1 + 1898 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/406\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(379\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1
−0.359001 + 0.701254i
0.522109 + 2.12798i
−0.676693 0.583217i
2.31940 0.319028i
−1.80582 0.194943i
−0.359001 0.701254i
0.522109 2.12798i
−0.676693 + 0.583217i
2.31940 + 0.319028i
−1.80582 + 0.194943i
−0.500000 + 0.866025i −1.61748 2.80156i −0.500000 0.866025i −0.572197 + 0.991074i 3.23497 0.469216 + 2.60381i 1.00000 −3.73251 + 6.46489i −0.572197 0.991074i
233.2 −0.500000 + 0.866025i −1.20348 2.08450i −0.500000 0.866025i 1.10394 1.91209i 2.40697 −1.36646 2.26557i 1.00000 −1.39675 + 2.41924i 1.10394 + 1.91209i
233.3 −0.500000 + 0.866025i −0.257545 0.446080i −0.500000 0.866025i −1.84343 + 3.19291i 0.515089 −2.63485 + 0.239979i 1.00000 1.36734 2.36831i −1.84343 3.19291i
233.4 −0.500000 + 0.866025i 0.357289 + 0.618843i −0.500000 0.866025i −0.116584 + 0.201930i −0.714579 2.36799 + 1.18009i 1.00000 1.24469 2.15586i −0.116584 0.201930i
233.5 −0.500000 + 0.866025i 1.22122 + 2.11522i −0.500000 0.866025i −2.07173 + 3.58835i −2.44245 −0.335903 2.62434i 1.00000 −1.48277 + 2.56824i −2.07173 3.58835i
291.1 −0.500000 0.866025i −1.61748 + 2.80156i −0.500000 + 0.866025i −0.572197 0.991074i 3.23497 0.469216 2.60381i 1.00000 −3.73251 6.46489i −0.572197 + 0.991074i
291.2 −0.500000 0.866025i −1.20348 + 2.08450i −0.500000 + 0.866025i 1.10394 + 1.91209i 2.40697 −1.36646 + 2.26557i 1.00000 −1.39675 2.41924i 1.10394 1.91209i
291.3 −0.500000 0.866025i −0.257545 + 0.446080i −0.500000 + 0.866025i −1.84343 3.19291i 0.515089 −2.63485 0.239979i 1.00000 1.36734 + 2.36831i −1.84343 + 3.19291i
291.4 −0.500000 0.866025i 0.357289 0.618843i −0.500000 + 0.866025i −0.116584 0.201930i −0.714579 2.36799 1.18009i 1.00000 1.24469 + 2.15586i −0.116584 + 0.201930i
291.5 −0.500000 0.866025i 1.22122 2.11522i −0.500000 + 0.866025i −2.07173 3.58835i −2.44245 −0.335903 + 2.62434i 1.00000 −1.48277 2.56824i −2.07173 + 3.58835i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 291.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 406.2.e.a 10
7.c even 3 1 inner 406.2.e.a 10
7.c even 3 1 2842.2.a.z 5
7.d odd 6 1 2842.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.a 10 1.a even 1 1 trivial
406.2.e.a 10 7.c even 3 1 inner
2842.2.a.x 5 7.d odd 6 1
2842.2.a.z 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 3 T_{3}^{9} + 16 T_{3}^{8} + 17 T_{3}^{7} + 100 T_{3}^{6} + 104 T_{3}^{5} + 382 T_{3}^{4} - 16 T_{3}^{3} + 169 T_{3}^{2} + 42 T_{3} + 49 \) acting on \(S_{2}^{\mathrm{new}}(406, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + 16 T^{8} + 17 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{10} + 7 T^{9} + 43 T^{8} + 112 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + 10 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 33 T^{8} - 108 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{5} - 10 T^{4} + 28 T^{3} + 7 T^{2} + \cdots + 103)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 8 T^{9} + 55 T^{8} + 116 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + 67 T^{8} + \cdots + 2634129 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + 40 T^{8} - 233 T^{7} + \cdots + 1089 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} + 11 T^{9} + 145 T^{8} + \cdots + 149769 \) Copy content Toggle raw display
$37$ \( T^{10} - 8 T^{9} + 111 T^{8} + \cdots + 121801 \) Copy content Toggle raw display
$41$ \( (T^{5} - 23 T^{4} + 132 T^{3} + 295 T^{2} + \cdots + 7113)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 3 T^{4} - 163 T^{3} - 379 T^{2} + \cdots + 16843)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 16 T^{9} + 247 T^{8} + \cdots + 3969 \) Copy content Toggle raw display
$53$ \( T^{10} + 7 T^{9} + 106 T^{8} + \cdots + 33489 \) Copy content Toggle raw display
$59$ \( T^{10} - 9 T^{9} + 120 T^{8} + \cdots + 431649 \) Copy content Toggle raw display
$61$ \( T^{10} + 15 T^{9} + 235 T^{8} + \cdots + 28376929 \) Copy content Toggle raw display
$67$ \( T^{10} - 4 T^{9} + 190 T^{8} + \cdots + 131813361 \) Copy content Toggle raw display
$71$ \( (T^{5} + 22 T^{4} + 51 T^{3} - 1142 T^{2} + \cdots + 18189)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 247 T^{8} - 452 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$79$ \( T^{10} - 13 T^{9} + 202 T^{8} + \cdots + 2442969 \) Copy content Toggle raw display
$83$ \( (T^{5} - 28 T^{4} + 213 T^{3} + \cdots + 25851)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 17 T^{9} + \cdots + 26494398441 \) Copy content Toggle raw display
$97$ \( (T^{5} - 42 T^{4} + 539 T^{3} + \cdots + 15007)^{2} \) Copy content Toggle raw display
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