Properties

Label 414.2.i.f
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 46)
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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} - 2 \zeta_{22}^{2} + \zeta_{22}) q^{5} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{7} + \zeta_{22} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} - 2 \zeta_{22}^{2} + \zeta_{22}) q^{5} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{7} + \zeta_{22} q^{8} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{10} + (\zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{6} - \zeta_{22}^{3} - 2 \zeta_{22}^{2} + 2 \zeta_{22} + 1) q^{11} + (\zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} + 2 \zeta_{22} - 2) q^{13} + (2 \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}^{2} - 1) q^{14} - \zeta_{22}^{5} q^{16} + ( - 2 \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} + 3 \zeta_{22}^{3} - \zeta_{22}^{2} - \zeta_{22} - 2) q^{17} + (\zeta_{22}^{9} + 2 \zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{4} + 2 \zeta_{22}^{3} - 2 \zeta_{22}^{2} - \zeta_{22}) q^{19} + ( - \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - \zeta_{22}^{3} + \cdots + 1) q^{20} + \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + 4 \zeta_{22}^{6} + \zeta_{22}^{5} + 4 \zeta_{22}^{4} - 4 \zeta_{22}^{3} + 4 \zeta_{22}^{2} + \cdots + 4) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{4} + 6 q^{5} + 3 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - q^{4} + 6 q^{5} + 3 q^{7} + q^{8} - 6 q^{10} + 12 q^{11} - 14 q^{13} - 3 q^{14} - q^{16} - 15 q^{17} + 2 q^{19} - 5 q^{20} + 10 q^{22} + q^{23} + 13 q^{25} + 3 q^{26} + 3 q^{28} + 8 q^{29} - 21 q^{31} + q^{32} - 7 q^{34} - 7 q^{35} + 28 q^{37} + 9 q^{38} - 6 q^{40} + 31 q^{41} + 11 q^{43} + 12 q^{44} - 12 q^{46} - 18 q^{47} - 24 q^{49} - 2 q^{50} + 8 q^{52} + 21 q^{53} + 5 q^{55} - 3 q^{56} - 8 q^{58} + 5 q^{59} + 37 q^{61} - q^{62} - q^{64} - 37 q^{65} - 13 q^{67} - 26 q^{68} + 18 q^{70} - 49 q^{71} - 8 q^{73} - 28 q^{74} - 20 q^{76} + 8 q^{77} + 8 q^{79} - 5 q^{80} + 2 q^{82} + 7 q^{83} - 42 q^{85} - 22 q^{86} - q^{88} + 13 q^{89} - 24 q^{91} + 23 q^{92} - 37 q^{94} + 10 q^{95} - 32 q^{97} + 24 q^{98}+O(q^{100}) \) 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 −0.174863 1.21620i 0 0.260554 0.570534i 0.654861 0.755750i 0 −0.510424 1.11767i
73.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −0.767092 0.492980i 0 −0.601808 4.18567i 0.959493 + 0.281733i 0 −0.129769 + 0.902563i
127.1 −0.841254 + 0.540641i 0 0.415415 0.909632i 3.26024 0.957293i 0 −0.297176 0.342959i 0.142315 + 0.989821i 0 −2.22514 + 2.56794i
163.1 −0.841254 0.540641i 0 0.415415 + 0.909632i 3.26024 + 0.957293i 0 −0.297176 + 0.342959i 0.142315 0.989821i 0 −2.22514 2.56794i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i −0.174863 + 1.21620i 0 0.260554 + 0.570534i 0.654861 + 0.755750i 0 −0.510424 + 1.11767i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i −0.985691 + 2.15836i 0 0.381761 + 0.112095i −0.841254 + 0.540641i 0 −2.27667 + 0.668491i
307.1 0.142315 0.989821i 0 −0.959493 0.281733i 1.66741 + 1.92429i 0 1.75667 + 1.12894i −0.415415 + 0.909632i 0 2.14200 1.37658i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 1.66741 1.92429i 0 1.75667 1.12894i −0.415415 0.909632i 0 2.14200 + 1.37658i
361.1 0.654861 0.755750i 0 −0.142315 0.989821i −0.985691 2.15836i 0 0.381761 0.112095i −0.841254 0.540641i 0 −2.27667 0.668491i
397.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −0.767092 + 0.492980i 0 −0.601808 + 4.18567i 0.959493 0.281733i 0 −0.129769 0.902563i
\(n\): 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.f 10
3.b odd 2 1 46.2.c.a 10
12.b even 2 1 368.2.m.b 10
23.c even 11 1 inner 414.2.i.f 10
23.c even 11 1 9522.2.a.bp 5
23.d odd 22 1 9522.2.a.bu 5
69.g even 22 1 1058.2.a.l 5
69.h odd 22 1 46.2.c.a 10
69.h odd 22 1 1058.2.a.m 5
276.j odd 22 1 8464.2.a.bw 5
276.o even 22 1 368.2.m.b 10
276.o even 22 1 8464.2.a.bx 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.a 10 3.b odd 2 1
46.2.c.a 10 69.h odd 22 1
368.2.m.b 10 12.b even 2 1
368.2.m.b 10 276.o even 22 1
414.2.i.f 10 1.a even 1 1 trivial
414.2.i.f 10 23.c even 11 1 inner
1058.2.a.l 5 69.g even 22 1
1058.2.a.m 5 69.h odd 22 1
8464.2.a.bw 5 276.j odd 22 1
8464.2.a.bx 5 276.o even 22 1
9522.2.a.bp 5 23.c even 11 1
9522.2.a.bu 5 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 6 T_{5}^{9} + 14 T_{5}^{8} - 29 T_{5}^{7} + 86 T_{5}^{6} - 153 T_{5}^{5} + 126 T_{5}^{4} + 201 T_{5}^{3} + 587 T_{5}^{2} + 713 T_{5} + 529 \) 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
$5$ \( T^{10} - 6 T^{9} + 14 T^{8} - 29 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + 20 T^{8} - 71 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 12 T^{9} + 56 T^{8} + \cdots + 109561 \) Copy content Toggle raw display
$13$ \( T^{10} + 14 T^{9} + 86 T^{8} + 258 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} + 15 T^{9} + 137 T^{8} + \cdots + 214369 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$23$ \( T^{10} - T^{9} + 78 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} - 8 T^{9} - 2 T^{8} - 259 T^{7} + \cdots + 4489 \) Copy content Toggle raw display
$31$ \( T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 20529961 \) Copy content Toggle raw display
$37$ \( T^{10} - 28 T^{9} + 509 T^{8} + \cdots + 49857721 \) Copy content Toggle raw display
$41$ \( T^{10} - 31 T^{9} + \cdots + 172475689 \) Copy content Toggle raw display
$43$ \( T^{10} - 11 T^{9} + 66 T^{8} + \cdots + 7027801 \) Copy content Toggle raw display
$47$ \( (T^{5} + 9 T^{4} - 82 T^{3} - 922 T^{2} + \cdots + 529)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 21 T^{9} + 144 T^{8} + \cdots + 31236921 \) Copy content Toggle raw display
$59$ \( T^{10} - 5 T^{9} - 41 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$61$ \( T^{10} - 37 T^{9} + \cdots + 349727401 \) Copy content Toggle raw display
$67$ \( T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 94109401 \) Copy content Toggle raw display
$71$ \( T^{10} + 49 T^{9} + 1180 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$73$ \( T^{10} + 8 T^{9} + 42 T^{8} + \cdots + 7921 \) Copy content Toggle raw display
$79$ \( T^{10} - 8 T^{9} + 9 T^{8} + \cdots + 17161 \) Copy content Toggle raw display
$83$ \( T^{10} - 7 T^{9} + 49 T^{8} + \cdots + 667137241 \) Copy content Toggle raw display
$89$ \( T^{10} - 13 T^{9} + 147 T^{8} + \cdots + 26739241 \) Copy content Toggle raw display
$97$ \( T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 5031049 \) Copy content Toggle raw display
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