Properties

Label 23.2.c.a
Level $23$
Weight $2$
Character orbit 23.c
Analytic conductor $0.184$
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] = [23,2,Mod(2,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 23.c (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: \(0.183655924649\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{2}+ \cdots + ( - 4 \zeta_{22}^{8} + \zeta_{22}^{7} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 7 q^{2} - 7 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 5 q^{7} + 4 q^{8} - 2 q^{9} + q^{10} + 7 q^{11} + 12 q^{12} - 3 q^{13} + 9 q^{14} + 12 q^{15} + q^{16} - 10 q^{17} - 14 q^{18} + 2 q^{19} - 9 q^{20} - 2 q^{21} - 6 q^{22} - 12 q^{23} - 38 q^{24} - 4 q^{25} + 12 q^{26} - 4 q^{27} + 7 q^{28} + 14 q^{29} + 7 q^{30} + 10 q^{31} + 21 q^{32} + 16 q^{33} + 29 q^{34} + 7 q^{35} + 27 q^{36} - 19 q^{37} - 8 q^{38} + q^{39} + q^{40} + 7 q^{41} - 25 q^{42} - 11 q^{43} - 34 q^{44} - 6 q^{45} - 29 q^{46} - 18 q^{47} + 18 q^{48} - 18 q^{49} + 16 q^{50} + 7 q^{51} - 20 q^{52} + 29 q^{53} - 6 q^{54} - q^{55} - 2 q^{56} - 8 q^{57} - 23 q^{58} - 21 q^{59} + 25 q^{60} + 3 q^{61} + 4 q^{62} + 34 q^{63} + 24 q^{64} + 2 q^{65} + 2 q^{66} + 45 q^{67} - 30 q^{68} + 26 q^{69} + 38 q^{70} - 14 q^{71} + 19 q^{72} + 19 q^{73} + 10 q^{74} - 28 q^{75} - 16 q^{76} + 2 q^{77} - 4 q^{78} - 15 q^{79} - 52 q^{80} - 44 q^{81} + 16 q^{82} + 18 q^{83} - 17 q^{84} - 19 q^{85} - 11 q^{86} - 23 q^{87} + 27 q^{88} + 25 q^{89} - 20 q^{90} - 4 q^{91} + 52 q^{92} + 4 q^{93} + 17 q^{94} + 6 q^{95} - 51 q^{96} - 34 q^{97} + 17 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-\zeta_{22}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 0.909632i
−0.415415 + 0.909632i
0.142315 0.989821i
0.959493 0.281733i
−0.841254 + 0.540641i
0.654861 + 0.755750i
0.654861 0.755750i
0.959493 + 0.281733i
−2.11435 1.35881i −0.226900 1.57812i 1.79329 + 3.92676i 1.41899 + 0.416652i −1.66463 + 3.64502i −0.804632 + 0.928595i 0.828708 5.76379i 0.439490 0.129046i −2.43409 2.80909i
3.1 −0.313607 2.18119i −1.04408 + 2.28621i −2.74024 + 0.804606i 0.809721 0.934468i 5.31408 + 1.56036i −1.99611 + 1.28282i 0.783524 + 1.71568i −2.17208 2.50672i −2.29218 1.47310i
4.1 0.198939 + 0.435615i −2.11435 + 0.620830i 1.15954 1.33818i −2.18251 1.40261i −0.691070 0.797537i 0.483568 + 3.36329i 1.73259 + 0.508735i 1.56130 1.00339i 0.176814 1.22977i
6.1 0.198939 0.435615i −2.11435 0.620830i 1.15954 + 1.33818i −2.18251 + 1.40261i −0.691070 + 0.797537i 0.483568 3.36329i 1.73259 0.508735i 1.56130 + 1.00339i 0.176814 + 1.22977i
8.1 −0.313607 + 2.18119i −1.04408 2.28621i −2.74024 0.804606i 0.809721 + 0.934468i 5.31408 1.56036i −1.99611 1.28282i 0.783524 1.71568i −2.17208 + 2.50672i −2.29218 + 1.47310i
9.1 −0.226900 + 0.0666238i −0.313607 0.361922i −1.63546 + 1.05105i −0.215370 1.49793i 0.0952700 + 0.0612263i −1.05773 + 2.31611i 0.610783 0.704881i 0.394306 2.74246i 0.148666 + 0.325532i
12.1 −2.11435 + 1.35881i −0.226900 + 1.57812i 1.79329 3.92676i 1.41899 0.416652i −1.66463 3.64502i −0.804632 0.928595i 0.828708 + 5.76379i 0.439490 + 0.129046i −2.43409 + 2.80909i
13.1 −1.04408 1.20493i 0.198939 + 0.127850i −0.0771283 + 0.536439i −1.33083 + 2.91411i −0.0536570 0.373193i 0.874908 + 0.256896i −1.95561 + 1.25679i −1.22301 2.67803i 4.90079 1.43900i
16.1 −1.04408 + 1.20493i 0.198939 0.127850i −0.0771283 0.536439i −1.33083 2.91411i −0.0536570 + 0.373193i 0.874908 0.256896i −1.95561 1.25679i −1.22301 + 2.67803i 4.90079 + 1.43900i
18.1 −0.226900 0.0666238i −0.313607 + 0.361922i −1.63546 1.05105i −0.215370 + 1.49793i 0.0952700 0.0612263i −1.05773 2.31611i 0.610783 + 0.704881i 0.394306 + 2.74246i 0.148666 0.325532i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.1
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 23.2.c.a 10
3.b odd 2 1 207.2.i.c 10
4.b odd 2 1 368.2.m.c 10
5.b even 2 1 575.2.k.b 10
5.c odd 4 2 575.2.p.b 20
23.b odd 2 1 529.2.c.a 10
23.c even 11 1 inner 23.2.c.a 10
23.c even 11 1 529.2.a.i 5
23.c even 11 2 529.2.c.b 10
23.c even 11 2 529.2.c.d 10
23.c even 11 2 529.2.c.g 10
23.c even 11 2 529.2.c.i 10
23.d odd 22 1 529.2.a.j 5
23.d odd 22 1 529.2.c.a 10
23.d odd 22 2 529.2.c.c 10
23.d odd 22 2 529.2.c.e 10
23.d odd 22 2 529.2.c.f 10
23.d odd 22 2 529.2.c.h 10
69.g even 22 1 4761.2.a.bn 5
69.h odd 22 1 207.2.i.c 10
69.h odd 22 1 4761.2.a.bo 5
92.g odd 22 1 368.2.m.c 10
92.g odd 22 1 8464.2.a.bs 5
92.h even 22 1 8464.2.a.bt 5
115.j even 22 1 575.2.k.b 10
115.k odd 44 2 575.2.p.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.c.a 10 1.a even 1 1 trivial
23.2.c.a 10 23.c even 11 1 inner
207.2.i.c 10 3.b odd 2 1
207.2.i.c 10 69.h odd 22 1
368.2.m.c 10 4.b odd 2 1
368.2.m.c 10 92.g odd 22 1
529.2.a.i 5 23.c even 11 1
529.2.a.j 5 23.d odd 22 1
529.2.c.a 10 23.b odd 2 1
529.2.c.a 10 23.d odd 22 1
529.2.c.b 10 23.c even 11 2
529.2.c.c 10 23.d odd 22 2
529.2.c.d 10 23.c even 11 2
529.2.c.e 10 23.d odd 22 2
529.2.c.f 10 23.d odd 22 2
529.2.c.g 10 23.c even 11 2
529.2.c.h 10 23.d odd 22 2
529.2.c.i 10 23.c even 11 2
575.2.k.b 10 5.b even 2 1
575.2.k.b 10 115.j even 22 1
575.2.p.b 20 5.c odd 4 2
575.2.p.b 20 115.k odd 44 2
4761.2.a.bn 5 69.g even 22 1
4761.2.a.bo 5 69.h odd 22 1
8464.2.a.bs 5 92.g odd 22 1
8464.2.a.bt 5 92.h even 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(23, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 7 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} + 7 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{10} - 7 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{10} + 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} + 10 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 541696 \) Copy content Toggle raw display
$23$ \( T^{10} + 12 T^{9} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} - 14 T^{9} + \cdots + 4932841 \) Copy content Toggle raw display
$31$ \( T^{10} - 10 T^{9} + \cdots + 17161 \) Copy content Toggle raw display
$37$ \( T^{10} + 19 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{10} - 7 T^{9} + \cdots + 1849 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + \cdots + 64009 \) Copy content Toggle raw display
$47$ \( (T^{5} + 9 T^{4} - 5 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 517426009 \) Copy content Toggle raw display
$59$ \( T^{10} + 21 T^{9} + \cdots + 4489 \) Copy content Toggle raw display
$61$ \( T^{10} - 3 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1113757129 \) Copy content Toggle raw display
$71$ \( T^{10} + 14 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$73$ \( T^{10} - 19 T^{9} + \cdots + 982081 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 517426009 \) Copy content Toggle raw display
$83$ \( T^{10} - 18 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 78310985281 \) Copy content Toggle raw display
$97$ \( T^{10} + 34 T^{9} + \cdots + 2374681 \) Copy content Toggle raw display
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