Properties

Label 966.2.q.a
Level $966$
Weight $2$
Character orbit 966.q
Analytic conductor $7.714$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(85,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.q (of order \(11\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \zeta_{22}^{4} q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3}+ \cdots - \zeta_{22}^{9} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22}^{4} q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3}+ \cdots + (\zeta_{22}^{9} - \zeta_{22}^{6} + \cdots + \zeta_{22}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{3} - q^{4} - 14 q^{5} - q^{6} + q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - q^{3} - q^{4} - 14 q^{5} - q^{6} + q^{7} - q^{8} - q^{9} + 8 q^{10} - 4 q^{11} - q^{12} + 14 q^{13} + q^{14} - 3 q^{15} - q^{16} + 15 q^{17} - q^{18} - 11 q^{19} - 3 q^{20} + q^{21} - 26 q^{22} - q^{23} + 10 q^{24} + 7 q^{25} - 8 q^{26} - q^{27} + q^{28} + 19 q^{29} + 8 q^{30} - 5 q^{31} - q^{32} + 7 q^{33} - 18 q^{34} - 8 q^{35} - q^{36} - 2 q^{37} + 11 q^{38} + 14 q^{39} - 3 q^{40} + 2 q^{41} + q^{42} + 25 q^{43} - 4 q^{44} + 8 q^{45} - 23 q^{46} - 4 q^{47} - q^{48} - q^{49} - 15 q^{50} + 15 q^{51} - 8 q^{52} + 15 q^{53} - q^{54} + 54 q^{55} + q^{56} + 8 q^{58} - 17 q^{59} - 14 q^{60} + q^{61} - 16 q^{62} + q^{63} - q^{64} - 13 q^{65} - 4 q^{66} - 4 q^{67} + 26 q^{68} + 21 q^{69} - 8 q^{70} + 6 q^{71} - q^{72} - 25 q^{73} - 2 q^{74} - 15 q^{75} - 22 q^{76} + 4 q^{77} + 14 q^{78} + 3 q^{79} + 8 q^{80} - q^{81} + 2 q^{82} + 7 q^{83} + q^{84} + 12 q^{85} - 41 q^{86} + 8 q^{87} + 7 q^{88} - 41 q^{89} + 8 q^{90} + 30 q^{91} - q^{92} - 16 q^{93} - 15 q^{94} - 44 q^{95} - q^{96} - 4 q^{97} - q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{22}^{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 + 0.281733i
−0.415415 0.909632i
0.654861 + 0.755750i
0.959493 0.281733i
0.142315 0.989821i
−0.841254 + 0.540641i
0.654861 0.755750i
−0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −1.27098 2.78305i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 2.93560 + 0.861971i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −3.15843 + 0.927399i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i −2.15565 + 2.48775i
169.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −2.42796 2.80202i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 3.11903 2.00448i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.114669 0.0736930i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.0193985 0.134919i
463.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −2.42796 + 2.80202i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 3.11903 + 2.00448i
547.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.0279611 + 0.194474i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.0816181 0.178719i
673.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.114669 + 0.0736930i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.0193985 + 0.134919i
715.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −3.15843 0.927399i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i −2.15565 2.48775i
841.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.27098 + 2.78305i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i 2.93560 0.861971i
883.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.0279611 0.194474i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.0816181 + 0.178719i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.1
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 966.2.q.a 10
23.c even 11 1 inner 966.2.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.a 10 1.a even 1 1 trivial
966.2.q.a 10 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 14 T_{5}^{9} + 97 T_{5}^{8} + 412 T_{5}^{7} + 1137 T_{5}^{6} + 2003 T_{5}^{5} + 1950 T_{5}^{4} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 14 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{10} - 14 T^{9} + \cdots + 69169 \) Copy content Toggle raw display
$17$ \( T^{10} - 15 T^{9} + \cdots + 124609 \) Copy content Toggle raw display
$19$ \( T^{10} + 11 T^{9} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} - 19 T^{9} + \cdots + 6285049 \) Copy content Toggle raw display
$31$ \( T^{10} + 5 T^{9} + \cdots + 109561 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots + 80982001 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 1400331241 \) Copy content Toggle raw display
$43$ \( T^{10} - 25 T^{9} + \cdots + 73599241 \) Copy content Toggle raw display
$47$ \( (T^{5} + 2 T^{4} + \cdots + 109)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 15 T^{9} + \cdots + 212521 \) Copy content Toggle raw display
$59$ \( T^{10} + 17 T^{9} + \cdots + 21557449 \) Copy content Toggle raw display
$61$ \( T^{10} - T^{9} + \cdots + 60668521 \) Copy content Toggle raw display
$67$ \( T^{10} + 4 T^{9} + \cdots + 29300569 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 460059601 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 127396369 \) Copy content Toggle raw display
$79$ \( T^{10} - 3 T^{9} + \cdots + 978121 \) Copy content Toggle raw display
$83$ \( T^{10} - 7 T^{9} + \cdots + 3659569 \) Copy content Toggle raw display
$89$ \( T^{10} + 41 T^{9} + \cdots + 23203489 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 18809299609 \) Copy content Toggle raw display
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