Properties

Label 414.2.e.c
Level $414$
Weight $2$
Character orbit 414.e
Analytic conductor $3.306$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.1481180578947.1
Defining polynomial: \( x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} + ( - \beta_{9} - \beta_{6} + \beta_{5} + 1) q^{3} + \beta_{4} q^{4} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{5} + \beta_{9} q^{6} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{7} + q^{8} + (\beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{2} + ( - \beta_{9} - \beta_{6} + \beta_{5} + 1) q^{3} + \beta_{4} q^{4} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{5} + \beta_{9} q^{6} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{7} + q^{8} + (\beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{9} + ( - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{10} + ( - 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{2} + 2 \beta_1 + 1) q^{11} + (\beta_{6} - \beta_{5} - 1) q^{12} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{13} + (\beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{14} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 4) q^{15}+ \cdots + ( - 5 \beta_{9} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} + q^{3} - 5 q^{4} + 5 q^{5} + q^{6} + 5 q^{7} + 10 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} + q^{3} - 5 q^{4} + 5 q^{5} + q^{6} + 5 q^{7} + 10 q^{8} + q^{9} - 10 q^{10} + 3 q^{11} - 2 q^{12} + 8 q^{13} + 5 q^{14} + 11 q^{15} - 5 q^{16} - 2 q^{17} - 8 q^{18} - 2 q^{19} + 5 q^{20} - 15 q^{21} + 3 q^{22} - 5 q^{23} + q^{24} - 16 q^{26} - 5 q^{27} - 10 q^{28} + 18 q^{29} + 5 q^{30} + 8 q^{31} - 5 q^{32} + 24 q^{33} + q^{34} + 2 q^{35} + 7 q^{36} - 12 q^{37} + q^{38} - 27 q^{39} + 5 q^{40} + 24 q^{41} - 3 q^{42} - 11 q^{43} - 6 q^{44} - 7 q^{45} + 10 q^{46} + 9 q^{47} + q^{48} - 4 q^{49} + 2 q^{51} + 8 q^{52} - 58 q^{53} - 20 q^{54} - 28 q^{55} + 5 q^{56} + 2 q^{57} + 18 q^{58} + 21 q^{59} - 16 q^{60} + 17 q^{61} - 16 q^{62} + 6 q^{63} + 10 q^{64} + 21 q^{65} + 21 q^{66} + 3 q^{67} + q^{68} - 2 q^{69} - q^{70} - 18 q^{71} + q^{72} - 14 q^{73} + 6 q^{74} + 13 q^{75} + q^{76} + 17 q^{77} + 15 q^{79} - 10 q^{80} + q^{81} - 48 q^{82} + 21 q^{83} + 18 q^{84} - 7 q^{85} - 11 q^{86} - 9 q^{87} + 3 q^{88} - 18 q^{89} - 7 q^{90} + 34 q^{91} - 5 q^{92} + 5 q^{93} + 9 q^{94} + 11 q^{95} - 2 q^{96} - 32 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 25\nu^{8} - 3\nu^{7} - 47\nu^{6} - 32\nu^{5} + 135\nu^{4} + 57\nu^{3} + 198\nu^{2} + 1188\nu + 648 ) / 1701 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{9} - 26\nu^{8} + 51\nu^{7} + \nu^{6} + 124\nu^{5} - 153\nu^{4} + 606\nu^{3} - 720\nu^{2} - 351\nu - 810 ) / 1701 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{9} - 37 \nu^{8} + 42 \nu^{7} - 121 \nu^{6} + 218 \nu^{5} - 339 \nu^{4} + 528 \nu^{3} - 810 \nu^{2} + 891 \nu - 2997 ) / 1701 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10 \nu^{9} + 23 \nu^{8} - 75 \nu^{7} + 92 \nu^{6} - 37 \nu^{5} + 372 \nu^{4} - 318 \nu^{3} + 729 \nu^{2} - 162 \nu - 810 ) / 1701 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{9} - 19 \nu^{8} + 51 \nu^{7} - 139 \nu^{6} + 425 \nu^{5} - 573 \nu^{4} + 1068 \nu^{3} - 1350 \nu^{2} + 3240 \nu - 2511 ) / 1701 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14 \nu^{9} + 25 \nu^{8} - 15 \nu^{7} + 64 \nu^{6} - 5 \nu^{5} + 357 \nu^{4} - 321 \nu^{3} - 135 \nu^{2} - 162 \nu - 891 ) / 1701 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{9} - 10\nu^{8} + 33\nu^{7} - 58\nu^{6} + 83\nu^{5} - 132\nu^{4} + 204\nu^{3} - 189\nu^{2} + 891\nu + 162 ) / 567 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37 \nu^{9} + 68 \nu^{8} - 111 \nu^{7} + 281 \nu^{6} - 451 \nu^{5} + 1011 \nu^{4} - 1425 \nu^{3} + 2079 \nu^{2} - 3564 \nu + 3321 ) / 1701 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - 2\beta_{7} + \beta_{5} + 3\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} + 4\beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{9} + 5\beta_{8} + 2\beta_{7} + 5\beta_{6} + 3\beta_{5} + 2\beta_{4} - 3\beta_{3} - 6\beta_{2} - 6\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} - 12\beta_{4} + 6\beta_{3} - 12\beta_{2} + 9\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{9} + 14\beta_{8} + 10\beta_{6} - 25\beta_{5} + 4\beta_{4} + 6\beta_{3} - 3\beta_{2} + 6\beta _1 - 35 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 34 \beta_{6} - 27 \beta_{5} - 35 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} - 36 \beta _1 - 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 63 \beta_{9} + 23 \beta_{8} + 22 \beta_{7} - 36 \beta_{6} + 52 \beta_{5} - 81 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 39 \beta _1 - 36 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−1.24278 1.20644i
0.785454 1.54372i
1.64906 0.529718i
−0.643944 + 1.60790i
0.452211 + 1.67198i
−1.24278 + 1.20644i
0.785454 + 1.54372i
1.64906 + 0.529718i
−0.643944 1.60790i
0.452211 1.67198i
−0.500000 0.866025i −1.66620 0.473061i −0.500000 + 0.866025i 0.274896 0.476134i 0.423416 + 1.67950i 0.708031 + 1.22635i 1.00000 2.55243 + 1.57642i −0.549792
139.2 −0.500000 0.866025i −0.944171 + 1.45208i −0.500000 + 0.866025i 1.85973 3.22115i 1.72963 + 0.0916356i 2.00591 + 3.47433i 1.00000 −1.21708 2.74203i −3.71947
139.3 −0.500000 0.866025i 0.365780 + 1.69299i −0.500000 + 0.866025i 0.231157 0.400376i 1.28328 1.16327i 0.165447 + 0.286563i 1.00000 −2.73241 + 1.23852i −0.462315
139.4 −0.500000 0.866025i 1.07051 1.36162i −0.500000 + 0.866025i −1.08471 + 1.87878i −1.71445 0.246277i 1.41120 + 2.44426i 1.00000 −0.708023 2.91525i 2.16943
139.5 −0.500000 0.866025i 1.67408 0.444362i −0.500000 + 0.866025i 1.21893 2.11124i −1.22187 1.22761i −1.79058 3.10138i 1.00000 2.60509 1.48779i −2.43785
277.1 −0.500000 + 0.866025i −1.66620 + 0.473061i −0.500000 0.866025i 0.274896 + 0.476134i 0.423416 1.67950i 0.708031 1.22635i 1.00000 2.55243 1.57642i −0.549792
277.2 −0.500000 + 0.866025i −0.944171 1.45208i −0.500000 0.866025i 1.85973 + 3.22115i 1.72963 0.0916356i 2.00591 3.47433i 1.00000 −1.21708 + 2.74203i −3.71947
277.3 −0.500000 + 0.866025i 0.365780 1.69299i −0.500000 0.866025i 0.231157 + 0.400376i 1.28328 + 1.16327i 0.165447 0.286563i 1.00000 −2.73241 1.23852i −0.462315
277.4 −0.500000 + 0.866025i 1.07051 + 1.36162i −0.500000 0.866025i −1.08471 1.87878i −1.71445 + 0.246277i 1.41120 2.44426i 1.00000 −0.708023 + 2.91525i 2.16943
277.5 −0.500000 + 0.866025i 1.67408 + 0.444362i −0.500000 0.866025i 1.21893 + 2.11124i −1.22187 + 1.22761i −1.79058 + 3.10138i 1.00000 2.60509 + 1.48779i −2.43785
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.e.c 10
3.b odd 2 1 1242.2.e.c 10
9.c even 3 1 inner 414.2.e.c 10
9.c even 3 1 3726.2.a.t 5
9.d odd 6 1 1242.2.e.c 10
9.d odd 6 1 3726.2.a.s 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.c 10 1.a even 1 1 trivial
414.2.e.c 10 9.c even 3 1 inner
1242.2.e.c 10 3.b odd 2 1
1242.2.e.c 10 9.d odd 6 1
3726.2.a.s 5 9.d odd 6 1
3726.2.a.t 5 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 5 T_{5}^{9} + 25 T_{5}^{8} - 46 T_{5}^{7} + 136 T_{5}^{6} - 205 T_{5}^{5} + 554 T_{5}^{4} - 483 T_{5}^{3} + 326 T_{5}^{2} - 105 T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + 2 T^{7} - 2 T^{6} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} + 25 T^{8} - 46 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + 32 T^{8} - 89 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{10} - 3 T^{9} + 46 T^{8} + \cdots + 710649 \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + 68 T^{8} - 122 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( (T^{5} + T^{4} - 45 T^{3} - 14 T^{2} + \cdots - 501)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + T^{4} - 93 T^{3} - 113 T^{2} + \cdots + 2043)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{10} - 18 T^{9} + 241 T^{8} + \cdots + 463761 \) Copy content Toggle raw display
$31$ \( T^{10} - 8 T^{9} + 190 T^{8} + \cdots + 112550881 \) Copy content Toggle raw display
$37$ \( (T^{5} + 6 T^{4} - 55 T^{3} - 255 T^{2} + \cdots + 2043)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 24 T^{9} + \cdots + 157527601 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + 179 T^{8} + \cdots + 1058841 \) Copy content Toggle raw display
$47$ \( T^{10} - 9 T^{9} + 91 T^{8} + \cdots + 305809 \) Copy content Toggle raw display
$53$ \( (T^{5} + 29 T^{4} + 323 T^{3} + 1717 T^{2} + \cdots + 3947)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 21 T^{9} + \cdots + 8393674689 \) Copy content Toggle raw display
$61$ \( T^{10} - 17 T^{9} + \cdots + 151560721 \) Copy content Toggle raw display
$67$ \( T^{10} - 3 T^{9} + 204 T^{8} + \cdots + 393824025 \) Copy content Toggle raw display
$71$ \( (T^{5} + 9 T^{4} - 202 T^{3} - 1875 T^{2} + \cdots + 17079)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 7 T^{4} - 97 T^{3} - 769 T^{2} + \cdots - 89)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 15 T^{9} + 145 T^{8} + \cdots + 1089 \) Copy content Toggle raw display
$83$ \( T^{10} - 21 T^{9} + 323 T^{8} + \cdots + 2253001 \) Copy content Toggle raw display
$89$ \( (T^{5} + 9 T^{4} - 115 T^{3} - 273 T^{2} + \cdots - 849)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 32 T^{9} + \cdots + 1093955625 \) Copy content Toggle raw display
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