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Properties

Label 67.2.e.b
Level $67$
Weight $2$
Character orbit 67.e
Analytic conductor $0.535$
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] = [67,2,Mod(9,67)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(67, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("67.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 67.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: \(0.534997693543\)
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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{22}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.959493 + 0.281733i
−0.841254 − 0.540641i
0.959493 − 0.281733i
0.142315 + 0.989821i
−0.841254 + 0.540641i
0.654861 − 0.755750i
−0.415415 + 0.909632i
0.654861 + 0.755750i
−0.415415 − 0.909632i
0.142315 − 0.989821i
1.34125 + 0.861971i −0.297176 − 0.0872586i 0.225136 + 0.492980i −0.857685 + 0.989821i −0.323373 − 0.373193i 0.313607 + 0.201543i 0.330830 − 2.30097i −2.44306 − 1.57006i −2.00357 + 0.588302i
14.1 0.915415 + 2.00448i −0.601808 − 0.386758i −1.87023 + 2.15836i −0.0405070 − 0.281733i 0.224345 − 1.56036i 0.226900 + 0.496841i −1.80972 − 0.531382i −1.03365 − 2.26339i 0.527646 − 0.339098i
15.1 1.34125 − 0.861971i −0.297176 + 0.0872586i 0.225136 − 0.492980i −0.857685 − 0.989821i −0.323373 + 0.373193i 0.313607 − 0.201543i 0.330830 + 2.30097i −2.44306 + 1.57006i −2.00357 − 0.588302i
22.1 −0.459493 + 0.134919i 0.260554 + 1.81219i −1.48958 + 0.957293i −0.345139 + 0.755750i −0.364223 − 0.797537i 1.04408 − 0.306569i 1.18251 − 1.36469i −0.337683 + 0.0991526i 0.0566239 − 0.393828i
24.1 0.915415 − 2.00448i −0.601808 + 0.386758i −1.87023 − 2.15836i −0.0405070 + 0.281733i 0.224345 + 1.56036i 0.226900 − 0.496841i −1.80972 + 0.531382i −1.03365 + 2.26339i 0.527646 + 0.339098i
25.1 0.357685 + 2.48775i 1.75667 − 2.02730i −4.14200 + 1.21620i −1.41542 + 0.909632i 5.67177 + 3.64502i −0.198939 − 1.38365i −2.41899 − 5.29684i −0.597131 − 4.15314i −2.76921 − 3.19584i
40.1 −0.154861 − 0.178719i 0.381761 − 0.835939i 0.276671 − 1.92429i −1.84125 − 0.540641i −0.208518 + 0.0612263i 2.11435 + 2.44009i −0.784630 + 0.504251i 1.41153 + 1.62899i 0.188515 + 0.412791i
59.1 0.357685 − 2.48775i 1.75667 + 2.02730i −4.14200 − 1.21620i −1.41542 − 0.909632i 5.67177 − 3.64502i −0.198939 + 1.38365i −2.41899 + 5.29684i −0.597131 + 4.15314i −2.76921 + 3.19584i
62.1 −0.154861 + 0.178719i 0.381761 + 0.835939i 0.276671 + 1.92429i −1.84125 + 0.540641i −0.208518 − 0.0612263i 2.11435 − 2.44009i −0.784630 − 0.504251i 1.41153 − 1.62899i 0.188515 − 0.412791i
64.1 −0.459493 − 0.134919i 0.260554 − 1.81219i −1.48958 − 0.957293i −0.345139 − 0.755750i −0.364223 + 0.797537i 1.04408 + 0.306569i 1.18251 + 1.36469i −0.337683 − 0.0991526i 0.0566239 + 0.393828i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 67.2.e.b ✓ 10
3.b odd 2 1 603.2.u.a 10
67.e even 11 1 inner 67.2.e.b ✓ 10
67.e even 11 1 4489.2.a.i 5
67.f odd 22 1 4489.2.a.h 5
201.k odd 22 1 603.2.u.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.e.b ✓ 10 1.a even 1 1 trivial
67.2.e.b ✓ 10 67.e even 11 1 inner
603.2.u.a 10 3.b odd 2 1
603.2.u.a 10 201.k odd 22 1
4489.2.a.h 5 67.f odd 22 1
4489.2.a.i 5 67.e even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 4T_{2}^{9} + 16T_{2}^{8} - 31T_{2}^{7} + 47T_{2}^{6} - 23T_{2}^{5} - 18T_{2}^{4} + 39T_{2}^{3} + 31T_{2}^{2} + 8T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(67, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 9 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 7 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + \cdots + 17161 \) Copy content Toggle raw display
$13$ \( T^{10} - 15 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - T^{9} + \cdots + 39601 \) Copy content Toggle raw display
$19$ \( T^{10} + 13 T^{9} + \cdots + 6405961 \) Copy content Toggle raw display
$23$ \( T^{10} + 7 T^{9} + \cdots + 351649 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots - 131)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + T^{9} + \cdots + 1849 \) Copy content Toggle raw display
$37$ \( (T^{5} - T^{4} - 70 T^{3} + \cdots + 2507)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 7 T^{9} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{10} + 2 T^{9} + \cdots + 25816561 \) Copy content Toggle raw display
$47$ \( T^{10} - 33 T^{9} + \cdots + 64009 \) Copy content Toggle raw display
$53$ \( T^{10} - 21 T^{9} + \cdots + 35557369 \) Copy content Toggle raw display
$59$ \( T^{10} + 38 T^{9} + \cdots + 4363921 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 2766865201 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + 16 T^{9} + \cdots + 380689 \) Copy content Toggle raw display
$73$ \( T^{10} - 3 T^{9} + \cdots + 528529 \) Copy content Toggle raw display
$79$ \( T^{10} + 19 T^{9} + \cdots + 109561 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 2943171001 \) Copy content Toggle raw display
$89$ \( T^{10} - 7 T^{9} + \cdots + 71757841 \) Copy content Toggle raw display
$97$ \( (T^{5} - 27 T^{4} + \cdots + 1583)^{2} \) Copy content Toggle raw display
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