Properties

Label 414.2.i.a
Level $414$
Weight $2$
Character orbit 414.i
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.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + 2 \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - 1) q^{5} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}) q^{7} - \zeta_{22} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + 2 \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - 1) q^{5} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}) q^{7} - \zeta_{22} q^{8} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} - \zeta_{22}^{5} - \zeta_{22}^{3} + \zeta_{22} - 1) q^{10} + ( - 2 \zeta_{22}^{9} - 2 \zeta_{22}^{5} - \zeta_{22}^{3} + 1) q^{11} + ( - 2 \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 3 \zeta_{22}^{7} - 3 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \cdots + 3) q^{13} + \cdots + (5 \zeta_{22}^{9} - 5 \zeta_{22}^{8} + 2 \zeta_{22}^{5} + \zeta_{22}^{4} + 2 \zeta_{22}^{3} + \zeta_{22}^{2} + 2 \zeta_{22}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8} - 13 q^{10} + 5 q^{11} + 13 q^{13} - 9 q^{14} - q^{16} + 9 q^{20} - 6 q^{22} + 32 q^{23} + q^{25} + 13 q^{26} - 9 q^{28} - 27 q^{29} - 8 q^{31} - q^{32} - 11 q^{34} + 26 q^{35} - 11 q^{37} - 11 q^{38} + 9 q^{40} + 10 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 8 q^{47} + 25 q^{49} - 21 q^{50} + 2 q^{52} - 9 q^{53} - 23 q^{55} + 2 q^{56} - 5 q^{58} + 21 q^{59} - 4 q^{61} - 8 q^{62} - q^{64} - 29 q^{65} - 32 q^{67} - 22 q^{68} - 18 q^{70} - 22 q^{71} + 43 q^{73} - 11 q^{74} - 10 q^{77} - 16 q^{79} - 2 q^{80} + 32 q^{82} + 3 q^{83} + 33 q^{85} - 32 q^{86} - 6 q^{88} + 11 q^{89} - 70 q^{91} + 21 q^{92} + 3 q^{94} + 39 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(235\)
\(\chi(n)\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.654861 0.755750i
0.959493 + 0.281733i
0.142315 + 0.989821i
0.142315 0.989821i
0.654861 + 0.755750i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
−0.415415 0.909632i
−0.841254 0.540641i
0.959493 0.281733i
−0.959493 + 0.281733i 0 0.841254 0.540641i 0.0651865 + 0.453382i 0 −0.134858 + 0.295298i −0.654861 + 0.755750i 0 −0.190279 0.416652i
73.1 0.415415 + 0.909632i 0 −0.654861 + 0.755750i −2.47672 1.59169i 0 −0.151894 1.05645i −0.959493 0.281733i 0 0.418986 2.91411i
127.1 0.841254 0.540641i 0 0.415415 0.909632i −1.78074 + 0.522874i 0 −2.54487 2.93694i −0.142315 0.989821i 0 −1.21537 + 1.40261i
163.1 0.841254 + 0.540641i 0 0.415415 + 0.909632i −1.78074 0.522874i 0 −2.54487 + 2.93694i −0.142315 + 0.989821i 0 −1.21537 1.40261i
271.1 −0.959493 0.281733i 0 0.841254 + 0.540641i 0.0651865 0.453382i 0 −0.134858 0.295298i −0.654861 0.755750i 0 −0.190279 + 0.416652i
289.1 −0.654861 0.755750i 0 −0.142315 + 0.989821i 1.37787 3.01713i 0 3.66820 + 1.07708i 0.841254 0.540641i 0 −3.18251 + 0.934468i
307.1 −0.142315 + 0.989821i 0 −0.959493 0.281733i 1.81440 + 2.09393i 0 0.163423 + 0.105026i 0.415415 0.909632i 0 −2.33083 + 1.49793i
325.1 −0.142315 0.989821i 0 −0.959493 + 0.281733i 1.81440 2.09393i 0 0.163423 0.105026i 0.415415 + 0.909632i 0 −2.33083 1.49793i
361.1 −0.654861 + 0.755750i 0 −0.142315 0.989821i 1.37787 + 3.01713i 0 3.66820 1.07708i 0.841254 + 0.540641i 0 −3.18251 0.934468i
397.1 0.415415 0.909632i 0 −0.654861 0.755750i −2.47672 + 1.59169i 0 −0.151894 + 1.05645i −0.959493 + 0.281733i 0 0.418986 + 2.91411i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 397.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.i.a 10
3.b odd 2 1 138.2.e.d 10
23.c even 11 1 inner 414.2.i.a 10
23.c even 11 1 9522.2.a.bx 5
23.d odd 22 1 9522.2.a.by 5
69.g even 22 1 3174.2.a.w 5
69.h odd 22 1 138.2.e.d 10
69.h odd 22 1 3174.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.d 10 3.b odd 2 1
138.2.e.d 10 69.h odd 22 1
414.2.i.a 10 1.a even 1 1 trivial
414.2.i.a 10 23.c even 11 1 inner
3174.2.a.w 5 69.g even 22 1
3174.2.a.x 5 69.h odd 22 1
9522.2.a.bx 5 23.c even 11 1
9522.2.a.by 5 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 2 T_{5}^{9} + 4 T_{5}^{8} + 41 T_{5}^{7} + 137 T_{5}^{6} + 76 T_{5}^{5} + 460 T_{5}^{4} + 2185 T_{5}^{3} + 2324 T_{5}^{2} + 138 T_{5} + 529 \) acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + 4 T^{8} + 41 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} - 7 T^{8} - 41 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 5 T^{9} + 25 T^{8} - 37 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{10} - 13 T^{9} + 125 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{10} + 55 T^{8} + 121 T^{7} + \cdots + 7745089 \) Copy content Toggle raw display
$19$ \( T^{10} - 33 T^{7} - 165 T^{6} + \cdots + 64009 \) Copy content Toggle raw display
$23$ \( T^{10} - 32 T^{9} + 496 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 27 T^{9} + 322 T^{8} + \cdots + 7921 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + 108 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{9} + 66 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + 23 T^{8} + \cdots + 1515361 \) Copy content Toggle raw display
$43$ \( T^{10} - 34 T^{9} + 595 T^{8} + \cdots + 109561 \) Copy content Toggle raw display
$47$ \( (T^{5} + 4 T^{4} - 75 T^{3} + 28 T^{2} + \cdots - 857)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 9 T^{9} + 103 T^{8} + \cdots + 8427409 \) Copy content Toggle raw display
$59$ \( T^{10} - 21 T^{9} + 199 T^{8} + \cdots + 978121 \) Copy content Toggle raw display
$61$ \( T^{10} + 4 T^{9} + 5 T^{8} + \cdots + 9078169 \) Copy content Toggle raw display
$67$ \( T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 83375161 \) Copy content Toggle raw display
$71$ \( T^{10} + 22 T^{9} + 198 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$73$ \( T^{10} - 43 T^{9} + 1013 T^{8} + \cdots + 21132409 \) Copy content Toggle raw display
$79$ \( T^{10} + 16 T^{9} + 14 T^{8} + \cdots + 72471169 \) Copy content Toggle raw display
$83$ \( T^{10} - 3 T^{9} - 79 T^{8} + \cdots + 16752649 \) Copy content Toggle raw display
$89$ \( T^{10} - 11 T^{9} + \cdots + 517972081 \) Copy content Toggle raw display
$97$ \( T^{10} - 39 T^{9} + 773 T^{8} + \cdots + 5650129 \) Copy content Toggle raw display
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