Properties

Label 414.2.i.c
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.

\(f(q)\) \(=\) \( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} + \zeta_{22}) q^{5} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2}) q^{7} - \zeta_{22} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} + \zeta_{22}) q^{5} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2}) q^{7} - \zeta_{22} q^{8} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{3} - \zeta_{22} + 1) q^{10} + (\zeta_{22}^{8} - \zeta_{22}^{7} + \zeta_{22}^{6} - \zeta_{22}^{3} + 2 \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{11} + (4 \zeta_{22}^{9} - 2 \zeta_{22}^{8} + 3 \zeta_{22}^{7} - 2 \zeta_{22}^{6} + 3 \zeta_{22}^{5} - 2 \zeta_{22}^{4} + \cdots - 2) q^{13} + \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} - 4 \zeta_{22}^{6} + 7 \zeta_{22}^{5} + 7 \zeta_{22} - 4) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{4} + 4 q^{5} - 7 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - q^{4} + 4 q^{5} - 7 q^{7} - q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{13} + 15 q^{14} - q^{16} + 9 q^{17} + 2 q^{19} - 7 q^{20} + 2 q^{22} - 21 q^{23} - 11 q^{25} - 9 q^{26} + 15 q^{28} + 2 q^{29} + 11 q^{31} - q^{32} - 13 q^{34} + 17 q^{35} - 18 q^{37} + 13 q^{38} + 4 q^{40} - 5 q^{41} - 21 q^{43} + 2 q^{44} - 10 q^{46} + 22 q^{47} + 24 q^{49} + 22 q^{50} - 20 q^{52} + 7 q^{53} + 3 q^{55} - 7 q^{56} + 24 q^{58} - 43 q^{59} - 3 q^{61} - 33 q^{62} - q^{64} - 41 q^{65} - q^{67} - 2 q^{68} + 6 q^{70} + 11 q^{71} - 28 q^{73} - 18 q^{74} + 2 q^{76} - 30 q^{77} + 34 q^{79} - 7 q^{80} + 6 q^{82} + 3 q^{83} + 8 q^{85} + 34 q^{86} - 9 q^{88} + 49 q^{89} - 52 q^{91} + q^{92} - 11 q^{94} + 36 q^{95} + 16 q^{97} - 20 q^{98}+O(q^{100}) \) 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 −0.459493 3.19584i 0 −0.497033 + 1.08835i −0.654861 + 0.755750i 0 1.34125 + 2.93694i
73.1 0.415415 + 0.909632i 0 −0.654861 + 0.755750i 0.915415 + 0.588302i 0 0.122916 + 0.854902i −0.959493 0.281733i 0 −0.154861 + 1.07708i
127.1 0.841254 0.540641i 0 0.415415 0.909632i 1.34125 0.393828i 0 2.81051 + 3.24350i −0.142315 0.989821i 0 0.915415 1.05645i
163.1 0.841254 + 0.540641i 0 0.415415 + 0.909632i 1.34125 + 0.393828i 0 2.81051 3.24350i −0.142315 + 0.989821i 0 0.915415 + 1.05645i
271.1 −0.959493 0.281733i 0 0.841254 + 0.540641i −0.459493 + 3.19584i 0 −0.497033 1.08835i −0.654861 0.755750i 0 1.34125 2.93694i
289.1 −0.654861 0.755750i 0 −0.142315 + 0.989821i −0.154861 + 0.339098i 0 −1.97611 0.580239i 0.841254 0.540641i 0 0.357685 0.105026i
307.1 −0.142315 + 0.989821i 0 −0.959493 0.281733i 0.357685 + 0.412791i 0 −3.96028 2.54512i 0.415415 0.909632i 0 −0.459493 + 0.295298i
325.1 −0.142315 0.989821i 0 −0.959493 + 0.281733i 0.357685 0.412791i 0 −3.96028 + 2.54512i 0.415415 + 0.909632i 0 −0.459493 0.295298i
361.1 −0.654861 + 0.755750i 0 −0.142315 0.989821i −0.154861 0.339098i 0 −1.97611 + 0.580239i 0.841254 + 0.540641i 0 0.357685 + 0.105026i
397.1 0.415415 0.909632i 0 −0.654861 0.755750i 0.915415 0.588302i 0 0.122916 0.854902i −0.959493 + 0.281733i 0 −0.154861 1.07708i
\(n\): 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.c 10
3.b odd 2 1 46.2.c.b 10
12.b even 2 1 368.2.m.a 10
23.c even 11 1 inner 414.2.i.c 10
23.c even 11 1 9522.2.a.bz 5
23.d odd 22 1 9522.2.a.bw 5
69.g even 22 1 1058.2.a.k 5
69.h odd 22 1 46.2.c.b 10
69.h odd 22 1 1058.2.a.j 5
276.j odd 22 1 8464.2.a.bv 5
276.o even 22 1 368.2.m.a 10
276.o even 22 1 8464.2.a.bu 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.b 10 3.b odd 2 1
46.2.c.b 10 69.h odd 22 1
368.2.m.a 10 12.b even 2 1
368.2.m.a 10 276.o even 22 1
414.2.i.c 10 1.a even 1 1 trivial
414.2.i.c 10 23.c even 11 1 inner
1058.2.a.j 5 69.h odd 22 1
1058.2.a.k 5 69.g even 22 1
8464.2.a.bu 5 276.o even 22 1
8464.2.a.bv 5 276.j odd 22 1
9522.2.a.bw 5 23.d odd 22 1
9522.2.a.bz 5 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 4T_{5}^{9} + 16T_{5}^{8} - 53T_{5}^{7} + 102T_{5}^{6} - 111T_{5}^{5} + 70T_{5}^{4} - 27T_{5}^{3} + 9T_{5}^{2} - 3T_{5} + 1 \) 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
$5$ \( T^{10} - 4 T^{9} + 16 T^{8} - 53 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + 7 T^{9} + 16 T^{8} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + 26 T^{8} - 74 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + 48 T^{8} + \cdots + 436921 \) Copy content Toggle raw display
$17$ \( T^{10} - 9 T^{9} + 81 T^{8} - 454 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + 15 T^{8} + \cdots + 139129 \) Copy content Toggle raw display
$23$ \( T^{10} + 21 T^{9} + 210 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + 48 T^{8} + 25 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) Copy content Toggle raw display
$37$ \( T^{10} + 18 T^{9} + 225 T^{8} + \cdots + 14645929 \) Copy content Toggle raw display
$41$ \( T^{10} + 5 T^{9} + 25 T^{8} + \cdots + 1985281 \) Copy content Toggle raw display
$43$ \( T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 53333809 \) Copy content Toggle raw display
$47$ \( (T^{5} - 11 T^{4} - 66 T^{3} + 726 T^{2} + \cdots - 3883)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 7 T^{9} + 16 T^{8} + \cdots + 1849 \) Copy content Toggle raw display
$59$ \( T^{10} + 43 T^{9} + 859 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$61$ \( T^{10} + 3 T^{9} + 53 T^{8} + 104 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} - 54 T^{8} + \cdots + 157609 \) Copy content Toggle raw display
$71$ \( T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) Copy content Toggle raw display
$73$ \( T^{10} + 28 T^{9} + 366 T^{8} + \cdots + 34774609 \) Copy content Toggle raw display
$79$ \( T^{10} - 34 T^{9} + \cdots + 118613881 \) Copy content Toggle raw display
$83$ \( T^{10} - 3 T^{9} - 277 T^{8} + \cdots + 923126689 \) Copy content Toggle raw display
$89$ \( T^{10} - 49 T^{9} + 1191 T^{8} + \cdots + 380689 \) Copy content Toggle raw display
$97$ \( T^{10} - 16 T^{9} + \cdots + 788093329 \) Copy content Toggle raw display
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