Properties

Label 414.2.e.d
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: 10.0.288778218147.1
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + 1) q^{2} + \beta_1 q^{3} + \beta_{7} q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{9} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{2} + \beta_1 + 1) q^{7} - q^{8} + ( - \beta_{8} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + 1) q^{2} + \beta_1 q^{3} + \beta_{7} q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{9} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{2} + \beta_1 + 1) q^{7} - q^{8} + ( - \beta_{8} - \beta_{3} - \beta_1) q^{9} - \beta_{3} q^{10} + (\beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{11}+ \cdots + (3 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 3 q^{6} + 5 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 3 q^{6} + 5 q^{7} - 10 q^{8} - 3 q^{9} - 2 q^{10} - 11 q^{11} + 6 q^{13} - 5 q^{14} + 9 q^{15} - 5 q^{16} + 2 q^{17} - 6 q^{19} - q^{20} - 21 q^{21} + 11 q^{22} - 5 q^{23} + 3 q^{24} + 12 q^{26} + 27 q^{27} - 10 q^{28} - 8 q^{29} + 9 q^{30} + 4 q^{31} + 5 q^{32} - 24 q^{33} + q^{34} + 46 q^{35} + 3 q^{36} - 28 q^{37} - 3 q^{38} - 45 q^{39} + q^{40} - 24 q^{41} - 27 q^{42} + 27 q^{43} + 22 q^{44} + 27 q^{45} - 10 q^{46} - 9 q^{47} + 3 q^{48} - 12 q^{49} - 6 q^{51} + 6 q^{52} - 26 q^{53} + 18 q^{54} + 16 q^{55} - 5 q^{56} - 18 q^{57} + 8 q^{58} - 9 q^{59} + 3 q^{61} + 8 q^{62} + 42 q^{63} + 10 q^{64} + 5 q^{65} - 3 q^{66} + 5 q^{67} - q^{68} + 23 q^{70} + 54 q^{71} + 3 q^{72} + 34 q^{73} - 14 q^{74} - 45 q^{75} + 3 q^{76} - 13 q^{77} - 30 q^{78} - 11 q^{79} + 2 q^{80} + 33 q^{81} - 48 q^{82} - 23 q^{83} - 6 q^{84} + 23 q^{85} - 27 q^{86} + 63 q^{87} + 11 q^{88} + 78 q^{89} + 51 q^{90} - 30 q^{91} - 5 q^{92} - 27 q^{93} + 9 q^{94} - 37 q^{95} + 28 q^{97} - 24 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 658 \nu^{9} - 2394 \nu^{8} - 4352 \nu^{7} - 10326 \nu^{6} - 25351 \nu^{5} - 51907 \nu^{4} - 47450 \nu^{3} - 30472 \nu^{2} - 130790 \nu - 98232 ) / 72795 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 999 \nu^{9} - 537 \nu^{8} - 4321 \nu^{7} - 4688 \nu^{6} - 34543 \nu^{5} - 20431 \nu^{4} - 65255 \nu^{3} - 41986 \nu^{2} - 194145 \nu - 9081 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 339 \nu^{9} - 1348 \nu^{8} + 4381 \nu^{7} - 7882 \nu^{6} + 19883 \nu^{5} - 36059 \nu^{4} + 51145 \nu^{3} - 44484 \nu^{2} + 15165 \nu - 29709 ) / 24265 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1348 \nu^{9} + 3356 \nu^{8} - 10907 \nu^{7} + 16239 \nu^{6} - 49501 \nu^{5} + 89773 \nu^{4} - 119555 \nu^{3} + 110748 \nu^{2} - 110550 \nu + 245043 ) / 72795 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2533 \nu^{9} - 1626 \nu^{8} + 17417 \nu^{7} + 3021 \nu^{6} + 84646 \nu^{5} + 17167 \nu^{4} + 165845 \nu^{3} + 188992 \nu^{2} + 176015 \nu + 59532 ) / 72795 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2694 \nu^{9} - 6203 \nu^{8} + 26226 \nu^{7} - 34722 \nu^{6} + 133958 \nu^{5} - 159864 \nu^{4} + 369510 \nu^{3} - 180434 \nu^{2} + 342765 \nu - 139464 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 ) / 72795 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 ) / 14559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6042 \nu^{9} - 8994 \nu^{8} + 41363 \nu^{7} - 39341 \nu^{6} + 193324 \nu^{5} - 179927 \nu^{4} + 343585 \nu^{3} - 54152 \nu^{2} + 258905 \nu - 134607 ) / 72795 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{8} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} + 2\beta_{8} + 9\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{9} - 2 \beta_{8} - 36 \beta_{7} - 5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 7 \beta _1 - 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{9} - 8\beta_{8} - 11\beta_{6} - 8\beta_{5} + 11\beta_{4} + 29\beta_{3} - 29\beta_{2} + 14\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{9} - 8\beta_{8} - 7\beta_{6} - \beta_{5} - 15\beta_{4} + 8\beta_{3} - 7\beta_{2} + 8\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{9} + 65\beta_{8} - 43\beta_{6} + 86\beta_{5} - 89\beta_{4} + 43\beta_{3} + 47\beta_{2} - 110\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 198 \beta_{9} + 86 \beta_{8} + 666 \beta_{7} + 200 \beta_{6} - 226 \beta_{5} + 112 \beta_{4} - 227 \beta_{3} + 227 \beta_{2} - 2 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 47\beta_{9} - 58\beta_{8} + 126\beta_{6} - 105\beta_{5} + 68\beta_{4} - 268\beta_{3} + 126\beta_{2} + 58\beta _1 - 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(235\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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
0.187540 + 0.324828i
0.827154 + 1.43267i
−0.539982 0.935277i
1.07065 + 1.85442i
−1.04536 1.81062i
0.187540 0.324828i
0.827154 1.43267i
−0.539982 + 0.935277i
1.07065 1.85442i
−1.04536 + 1.81062i
0.500000 + 0.866025i −1.61720 0.620220i −0.500000 + 0.866025i 0.536235 0.928786i −0.271473 1.71064i −0.121951 0.211225i −1.00000 2.23065 + 2.00604i 1.07247
139.2 0.500000 + 0.866025i −0.958787 + 1.44247i −0.500000 + 0.866025i 0.217761 0.377174i −1.72861 0.109097i 2.31474 + 4.00925i −1.00000 −1.16146 2.76605i 0.435523
139.3 0.500000 + 0.866025i −0.376855 + 1.69056i −0.500000 + 0.866025i −0.990153 + 1.71499i −1.65249 + 0.518912i 0.245502 + 0.425221i −1.00000 −2.71596 1.27419i −1.98031
139.4 0.500000 + 0.866025i −0.278072 1.70958i −0.500000 + 0.866025i −1.69714 + 2.93953i 1.34151 1.09561i −1.74607 3.02428i −1.00000 −2.84535 + 0.950775i −3.39428
139.5 0.500000 + 0.866025i 1.73091 + 0.0627999i −0.500000 + 0.866025i 1.43330 2.48254i 0.811070 + 1.53041i 1.80778 + 3.13117i −1.00000 2.99211 + 0.217402i 2.86660
277.1 0.500000 0.866025i −1.61720 + 0.620220i −0.500000 0.866025i 0.536235 + 0.928786i −0.271473 + 1.71064i −0.121951 + 0.211225i −1.00000 2.23065 2.00604i 1.07247
277.2 0.500000 0.866025i −0.958787 1.44247i −0.500000 0.866025i 0.217761 + 0.377174i −1.72861 + 0.109097i 2.31474 4.00925i −1.00000 −1.16146 + 2.76605i 0.435523
277.3 0.500000 0.866025i −0.376855 1.69056i −0.500000 0.866025i −0.990153 1.71499i −1.65249 0.518912i 0.245502 0.425221i −1.00000 −2.71596 + 1.27419i −1.98031
277.4 0.500000 0.866025i −0.278072 + 1.70958i −0.500000 0.866025i −1.69714 2.93953i 1.34151 + 1.09561i −1.74607 + 3.02428i −1.00000 −2.84535 0.950775i −3.39428
277.5 0.500000 0.866025i 1.73091 0.0627999i −0.500000 0.866025i 1.43330 + 2.48254i 0.811070 1.53041i 1.80778 3.13117i −1.00000 2.99211 0.217402i 2.86660
\(n\): 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.d 10
3.b odd 2 1 1242.2.e.b 10
9.c even 3 1 inner 414.2.e.d 10
9.c even 3 1 3726.2.a.r 5
9.d odd 6 1 1242.2.e.b 10
9.d odd 6 1 3726.2.a.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.d 10 1.a even 1 1 trivial
414.2.e.d 10 9.c even 3 1 inner
1242.2.e.b 10 3.b odd 2 1
1242.2.e.b 10 9.d odd 6 1
3726.2.a.r 5 9.c even 3 1
3726.2.a.u 5 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + T_{5}^{9} + 13T_{5}^{8} - 2T_{5}^{7} + 124T_{5}^{6} + T_{5}^{5} + 334T_{5}^{4} - 341T_{5}^{3} + 580T_{5}^{2} - 225T_{5} + 81 \) 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
$3$ \( T^{10} + 3 T^{9} + 6 T^{8} - 24 T^{6} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + 13 T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + 36 T^{8} - 69 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + 106 T^{8} + \cdots + 84681 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + 70 T^{8} + \cdots + 1234321 \) Copy content Toggle raw display
$17$ \( (T^{5} - T^{4} - 75 T^{3} + 74 T^{2} + \cdots - 2037)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} - 49 T^{3} - 43 T^{2} + \cdots - 857)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} + 121 T^{8} + \cdots + 8590761 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + 100 T^{8} + \cdots + 3207681 \) Copy content Toggle raw display
$37$ \( (T^{5} + 14 T^{4} - 53 T^{3} - 1151 T^{2} + \cdots + 19189)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 24 T^{9} + 414 T^{8} + \cdots + 38900169 \) Copy content Toggle raw display
$43$ \( T^{10} - 27 T^{9} + 505 T^{8} + \cdots + 458329 \) Copy content Toggle raw display
$47$ \( T^{10} + 9 T^{9} + \cdots + 1378265625 \) Copy content Toggle raw display
$53$ \( (T^{5} + 13 T^{4} - 33 T^{3} - 617 T^{2} + \cdots + 213)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 9 T^{9} + 210 T^{8} + \cdots + 26594649 \) Copy content Toggle raw display
$61$ \( T^{10} - 3 T^{9} + 130 T^{8} + \cdots + 3374569 \) Copy content Toggle raw display
$67$ \( T^{10} - 5 T^{9} + 36 T^{8} - 69 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$71$ \( (T^{5} - 27 T^{4} + 150 T^{3} + \cdots + 21357)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 17 T^{4} + 15 T^{3} + 819 T^{2} + \cdots - 297)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 11 T^{9} + \cdots + 119968209 \) Copy content Toggle raw display
$83$ \( T^{10} + 23 T^{9} + \cdots + 107557641 \) Copy content Toggle raw display
$89$ \( (T^{5} - 39 T^{4} + 483 T^{3} + \cdots + 62667)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 28 T^{9} + 553 T^{8} + \cdots + 20007729 \) Copy content Toggle raw display
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