Properties

Label 69.2.e.a
Level $69$
Weight $2$
Character orbit 69.e
Analytic conductor $0.551$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,2,Mod(4,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.e (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: \(0.550967773947\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} + \zeta_{22} - 1) q^{2} - \zeta_{22}^{8} q^{3} + (\zeta_{22}^{8} - \zeta_{22}^{7} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}^{2} + \zeta_{22} + 1) q^{4} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{6} + \zeta_{22}^{5} - \zeta_{22}^{4}) q^{5} + (\zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{2} - 1) q^{6} + ( - \zeta_{22}^{8} - \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - \zeta_{22} - 1) q^{7} + (\zeta_{22}^{9} - \zeta_{22}^{8} - 2 \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{3} - \zeta_{22}^{2} - 2) q^{8} - \zeta_{22}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} + \zeta_{22} - 1) q^{2} - \zeta_{22}^{8} q^{3} + (\zeta_{22}^{8} - \zeta_{22}^{7} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}^{2} + \zeta_{22} + 1) q^{4} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{6} + \zeta_{22}^{5} - \zeta_{22}^{4}) q^{5} + (\zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{2} - 1) q^{6} + ( - \zeta_{22}^{8} - \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - \zeta_{22} - 1) q^{7} + (\zeta_{22}^{9} - \zeta_{22}^{8} - 2 \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{3} - \zeta_{22}^{2} - 2) q^{8} - \zeta_{22}^{5} q^{9} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{10} + ( - 3 \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 2 \zeta_{22}^{7} + 3 \zeta_{22}^{6} - 3 \zeta_{22}^{5} + 3 \zeta_{22}^{4} + \cdots + 3) q^{11} + \cdots + (\zeta_{22}^{7} - \zeta_{22}^{6} - 3 \zeta_{22}^{4} - \zeta_{22}^{2} + \zeta_{22}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + q^{3} + 8 q^{4} + 5 q^{5} - 7 q^{6} - 8 q^{7} - 15 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} + q^{3} + 8 q^{4} + 5 q^{5} - 7 q^{6} - 8 q^{7} - 15 q^{8} - q^{9} - 2 q^{10} + 7 q^{11} + 14 q^{12} - 30 q^{13} + q^{14} - 5 q^{15} + 12 q^{16} - 2 q^{17} - 4 q^{18} + 10 q^{19} + 4 q^{20} - 3 q^{21} + 6 q^{22} - q^{23} - 18 q^{24} + 24 q^{25} + q^{26} + q^{27} + 9 q^{28} - 14 q^{29} + 2 q^{30} - 28 q^{31} + 23 q^{32} - 7 q^{33} - 8 q^{34} - 4 q^{35} - 3 q^{36} + 19 q^{37} - 15 q^{38} + 30 q^{39} - 13 q^{40} + 19 q^{41} + 21 q^{42} - 24 q^{43} + 54 q^{44} - 6 q^{45} + 18 q^{46} + 26 q^{47} + 10 q^{48} - 13 q^{49} - 36 q^{50} + 24 q^{51} - 57 q^{52} - q^{53} + 4 q^{54} - 24 q^{55} - 10 q^{56} + q^{57} + 10 q^{58} + 2 q^{59} + 7 q^{60} + 30 q^{61} - 24 q^{62} - 8 q^{63} + 13 q^{64} - 4 q^{65} - 28 q^{66} + 4 q^{67} - 50 q^{68} + q^{69} + 6 q^{70} - 14 q^{71} - 15 q^{72} - 26 q^{73} - 12 q^{74} - 13 q^{75} + 19 q^{76} - 43 q^{77} + 10 q^{78} + 20 q^{79} - 5 q^{80} - q^{81} + 10 q^{82} + 18 q^{83} - 42 q^{84} + 21 q^{85} + 14 q^{86} - 8 q^{87} - 38 q^{88} - 5 q^{89} + 9 q^{90} + 46 q^{91} + 52 q^{92} - 16 q^{93} - 6 q^{94} + 5 q^{95} - q^{96} + 15 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-\zeta_{22}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−0.415415 0.909632i
0.654861 + 0.755750i
0.654861 0.755750i
−0.841254 0.540641i
0.142315 0.989821i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 0.281733i
−0.841254 + 0.540641i
0.959493 + 0.281733i
−0.662317 1.45027i 0.959493 0.281733i −0.354905 + 0.409583i −0.438384 0.281733i −1.04408 1.20493i 0.188515 + 1.31115i −2.23047 0.654925i 0.841254 0.540641i −0.118239 + 0.822373i
13.1 1.44306 + 1.66538i −0.841254 0.540641i −0.406440 + 2.82685i 0.246902 0.540641i −0.313607 2.18119i −2.76921 0.813115i −1.58671 + 1.01971i 0.415415 + 0.909632i 1.25667 0.368991i
16.1 1.44306 1.66538i −0.841254 + 0.540641i −0.406440 2.82685i 0.246902 + 0.540641i −0.313607 + 2.18119i −2.76921 + 0.813115i −1.58671 1.01971i 0.415415 0.909632i 1.25667 + 0.368991i
25.1 −0.402869 0.258908i 0.142315 + 0.989821i −0.735560 1.61065i 3.37102 + 0.989821i 0.198939 0.435615i 0.527646 0.608936i −0.256983 + 1.78736i −0.959493 + 0.281733i −1.10181 1.27155i
31.1 0.0336545 0.234072i −0.415415 0.909632i 1.86533 + 0.547710i −0.788201 0.909632i −0.226900 + 0.0666238i 0.0566239 + 0.0363899i 0.387454 0.848406i −0.654861 + 0.755750i −0.239446 + 0.153882i
49.1 0.0336545 + 0.234072i −0.415415 + 0.909632i 1.86533 0.547710i −0.788201 + 0.909632i −0.226900 0.0666238i 0.0566239 0.0363899i 0.387454 + 0.848406i −0.654861 0.755750i −0.239446 0.153882i
52.1 −0.662317 + 1.45027i 0.959493 + 0.281733i −0.354905 0.409583i −0.438384 + 0.281733i −1.04408 + 1.20493i 0.188515 1.31115i −2.23047 + 0.654925i 0.841254 + 0.540641i −0.118239 0.822373i
55.1 −2.41153 + 0.708089i 0.654861 + 0.755750i 3.63158 2.33387i 0.108660 + 0.755750i −2.11435 1.35881i −2.00357 + 4.38721i −3.81329 + 4.40077i −0.142315 + 0.989821i −0.797176 1.74557i
58.1 −0.402869 + 0.258908i 0.142315 0.989821i −0.735560 + 1.61065i 3.37102 0.989821i 0.198939 + 0.435615i 0.527646 + 0.608936i −0.256983 1.78736i −0.959493 0.281733i −1.10181 + 1.27155i
64.1 −2.41153 0.708089i 0.654861 0.755750i 3.63158 + 2.33387i 0.108660 0.755750i −2.11435 + 1.35881i −2.00357 4.38721i −3.81329 4.40077i −0.142315 0.989821i −0.797176 + 1.74557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.1
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 69.2.e.a 10
3.b odd 2 1 207.2.i.b 10
23.c even 11 1 inner 69.2.e.a 10
23.c even 11 1 1587.2.a.o 5
23.d odd 22 1 1587.2.a.p 5
69.g even 22 1 4761.2.a.bq 5
69.h odd 22 1 207.2.i.b 10
69.h odd 22 1 4761.2.a.br 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.a 10 1.a even 1 1 trivial
69.2.e.a 10 23.c even 11 1 inner
207.2.i.b 10 3.b odd 2 1
207.2.i.b 10 69.h odd 22 1
1587.2.a.o 5 23.c even 11 1
1587.2.a.p 5 23.d odd 22 1
4761.2.a.bq 5 69.g even 22 1
4761.2.a.br 5 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 4T_{2}^{9} + 5T_{2}^{8} + 9T_{2}^{7} + 36T_{2}^{6} + 78T_{2}^{5} + 125T_{2}^{4} + 71T_{2}^{3} + 20T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + 5 T^{8} + 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} + 3 T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + 8 T^{9} + 42 T^{8} + 105 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 7 T^{9} + 16 T^{8} + 31 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{10} + 30 T^{9} + 438 T^{8} + \cdots + 2042041 \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} - 18 T^{8} + \cdots + 17161 \) Copy content Toggle raw display
$19$ \( T^{10} - 10 T^{9} + 56 T^{8} - 186 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + T^{8} - 21 T^{7} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 14 T^{9} + 108 T^{8} + \cdots + 734449 \) Copy content Toggle raw display
$31$ \( T^{10} + 28 T^{9} + 300 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$37$ \( T^{10} - 19 T^{9} + 152 T^{8} + \cdots + 14645929 \) Copy content Toggle raw display
$41$ \( T^{10} - 19 T^{9} + 196 T^{8} + \cdots + 38809 \) Copy content Toggle raw display
$43$ \( T^{10} + 24 T^{9} + 301 T^{8} + \cdots + 1739761 \) Copy content Toggle raw display
$47$ \( (T^{5} - 13 T^{4} - 27 T^{3} + 486 T^{2} + \cdots - 2507)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + T^{9} - 54 T^{8} - 615 T^{7} + \cdots + 94249 \) Copy content Toggle raw display
$59$ \( T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 1515361 \) Copy content Toggle raw display
$61$ \( T^{10} - 30 T^{9} + \cdots + 519520849 \) Copy content Toggle raw display
$67$ \( T^{10} - 4 T^{9} + 16 T^{8} + \cdots + 24930049 \) Copy content Toggle raw display
$71$ \( T^{10} + 14 T^{9} + \cdots + 4663114369 \) Copy content Toggle raw display
$73$ \( T^{10} + 26 T^{9} + 302 T^{8} + \cdots + 12243001 \) Copy content Toggle raw display
$79$ \( T^{10} - 20 T^{9} + 323 T^{8} + \cdots + 326041 \) Copy content Toggle raw display
$83$ \( T^{10} - 18 T^{9} + 192 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$89$ \( T^{10} + 5 T^{9} + \cdots + 3373402561 \) Copy content Toggle raw display
$97$ \( T^{10} - 15 T^{9} + 379 T^{8} + \cdots + 83740801 \) Copy content Toggle raw display
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