Properties

Label 414.2.i.e
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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + ( - \zeta_{22}^{9} + \zeta_{22}^{6} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + ( - \zeta_{22}^{9} + \zeta_{22}^{6} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{3} - \zeta_{22} + 1) q^{10} + (2 \zeta_{22}^{7} + \zeta_{22}^{3} + 2 \zeta_{22}^{2} - 2 \zeta_{22} - 1) q^{11} + ( - 2 \zeta_{22}^{8} + \zeta_{22}^{7} + 2 \zeta_{22}^{6} + \zeta_{22}^{5} - 2 \zeta_{22}^{4} - \zeta_{22} + 1) q^{13} + (\zeta_{22}^{9} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} - \zeta_{22}^{3} + 2 \zeta_{22}^{2} - \zeta_{22} + 2) q^{14} - \zeta_{22}^{5} q^{16} + (3 \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{3} + \zeta_{22}^{2} + 3) q^{17} + (3 \zeta_{22}^{8} + \zeta_{22}^{5} - \zeta_{22}^{3} + \zeta_{22}^{2} - 1) q^{19} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2} + 1) q^{20} + ( - \zeta_{22}^{7} - 2 \zeta_{22}^{6} + 2 \zeta_{22}^{5} + \zeta_{22}^{4} + 2) q^{22} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + 5 \zeta_{22}^{3} + \zeta_{22}) q^{23} + ( - 3 \zeta_{22}^{8} - 3 \zeta_{22}^{7} + 2 \zeta_{22}^{6} - 2 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \cdots - 3) q^{25} + \cdots + (3 \zeta_{22}^{9} - 3 \zeta_{22}^{8} + 2 \zeta_{22}^{5} - 5 \zeta_{22}^{4} + 6 \zeta_{22}^{3} - 5 \zeta_{22}^{2} + 2 \zeta_{22}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{4} + 2 q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - q^{4} + 2 q^{5} + q^{8} + 9 q^{10} - 11 q^{11} + 13 q^{13} + 11 q^{14} - q^{16} + 24 q^{17} - 14 q^{19} + 13 q^{20} + 22 q^{22} + 10 q^{23} - 43 q^{25} + 9 q^{26} - 11 q^{28} - 13 q^{29} + 8 q^{31} + q^{32} + 9 q^{34} - 13 q^{37} + 3 q^{38} + 9 q^{40} + 10 q^{41} - 8 q^{43} - 11 q^{44} + q^{46} + 8 q^{47} + 29 q^{49} - 23 q^{50} + 2 q^{52} + 35 q^{53} - 11 q^{55} + 13 q^{58} - 37 q^{59} - 2 q^{61} - 8 q^{62} - q^{64} - 37 q^{65} + 14 q^{67} + 2 q^{68} - 22 q^{70} - 44 q^{71} - 49 q^{73} + 13 q^{74} + 8 q^{76} - 44 q^{77} - 8 q^{79} + 2 q^{80} + 12 q^{82} + 17 q^{83} - 37 q^{85} - 14 q^{86} - 59 q^{89} + 66 q^{91} - 45 q^{92} - 19 q^{94} + 28 q^{95} - 21 q^{97} + 26 q^{98}+O(q^{100}) \) 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 0.544078 + 3.78415i 0 1.20217 2.63238i 0.654861 0.755750i 0 1.58816 + 3.47758i
73.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −0.273100 0.175511i 0 0.346156 + 2.40757i 0.959493 + 0.281733i 0 −0.0462003 + 0.321330i
127.1 −0.841254 + 0.540641i 0 0.415415 0.909632i −0.186393 + 0.0547299i 0 −1.27819 1.47511i 0.142315 + 0.989821i 0 0.127214 0.146813i
163.1 −0.841254 0.540641i 0 0.415415 + 0.909632i −0.186393 0.0547299i 0 −1.27819 + 1.47511i 0.142315 0.989821i 0 0.127214 + 0.146813i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.544078 3.78415i 0 1.20217 + 2.63238i 0.654861 + 0.755750i 0 1.58816 3.47758i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 1.61435 3.53494i 0 −3.99283 1.17240i −0.841254 + 0.540641i 0 3.72871 1.09485i
307.1 0.142315 0.989821i 0 −0.959493 0.281733i −0.698939 0.806618i 0 3.72270 + 2.39243i −0.415415 + 0.909632i 0 −0.897877 + 0.577031i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −0.698939 + 0.806618i 0 3.72270 2.39243i −0.415415 0.909632i 0 −0.897877 0.577031i
361.1 0.654861 0.755750i 0 −0.142315 0.989821i 1.61435 + 3.53494i 0 −3.99283 + 1.17240i −0.841254 0.540641i 0 3.72871 + 1.09485i
397.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −0.273100 + 0.175511i 0 0.346156 2.40757i 0.959493 0.281733i 0 −0.0462003 0.321330i
\(n\): 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.e 10
3.b odd 2 1 138.2.e.b 10
23.c even 11 1 inner 414.2.i.e 10
23.c even 11 1 9522.2.a.bs 5
23.d odd 22 1 9522.2.a.br 5
69.g even 22 1 3174.2.a.bb 5
69.h odd 22 1 138.2.e.b 10
69.h odd 22 1 3174.2.a.ba 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.b 10 3.b odd 2 1
138.2.e.b 10 69.h odd 22 1
414.2.i.e 10 1.a even 1 1 trivial
414.2.i.e 10 23.c even 11 1 inner
3174.2.a.ba 5 69.h odd 22 1
3174.2.a.bb 5 69.g even 22 1
9522.2.a.br 5 23.d odd 22 1
9522.2.a.bs 5 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 2T_{5}^{9} + 26T_{5}^{8} + 3T_{5}^{7} + 159T_{5}^{6} + 386T_{5}^{5} + 526T_{5}^{4} + 323T_{5}^{3} + 102T_{5}^{2} + 16T_{5} + 1 \) 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
$5$ \( T^{10} - 2 T^{9} + 26 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 11 T^{8} + 55 T^{7} + \cdots + 64009 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + 55 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$13$ \( T^{10} - 13 T^{9} + 81 T^{8} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( T^{10} - 24 T^{9} + 279 T^{8} + \cdots + 279841 \) Copy content Toggle raw display
$19$ \( T^{10} + 14 T^{9} + 64 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + 34 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 13 T^{9} + 70 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( T^{10} - 8 T^{9} - 46 T^{8} + 137 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( T^{10} + 13 T^{9} + 70 T^{8} + \cdots + 139129 \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + 177 T^{8} + \cdots + 434281 \) Copy content Toggle raw display
$43$ \( T^{10} + 8 T^{9} + 31 T^{8} + \cdots + 50794129 \) Copy content Toggle raw display
$47$ \( (T^{5} - 4 T^{4} - 97 T^{3} + 82 T^{2} + \cdots + 4817)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 35 T^{9} + 587 T^{8} + \cdots + 279841 \) Copy content Toggle raw display
$59$ \( T^{10} + 37 T^{9} + \cdots + 127893481 \) Copy content Toggle raw display
$61$ \( T^{10} + 2 T^{9} + 147 T^{8} + \cdots + 11485321 \) Copy content Toggle raw display
$67$ \( T^{10} - 14 T^{9} + \cdots + 1051640041 \) Copy content Toggle raw display
$71$ \( T^{10} + 44 T^{9} + 968 T^{8} + \cdots + 25938649 \) Copy content Toggle raw display
$73$ \( T^{10} + 49 T^{9} + \cdots + 310499641 \) Copy content Toggle raw display
$79$ \( T^{10} + 8 T^{9} + 306 T^{8} + \cdots + 23203489 \) Copy content Toggle raw display
$83$ \( T^{10} - 17 T^{9} + 135 T^{8} - 590 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{10} + 59 T^{9} + \cdots + 3044501329 \) Copy content Toggle raw display
$97$ \( T^{10} + 21 T^{9} + 397 T^{8} + \cdots + 9078169 \) Copy content Toggle raw display
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