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Properties

Label 67.2.c.a
Level $67$
Weight $2$
Character orbit 67.c
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(29,67)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(67, base_ring=CyclotomicField(6))
 
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("67.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 67.c (of order \(3\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 6x^{7} + 35x^{6} - 21x^{5} + 84x^{4} - 2x^{3} + 76x^{2} - 30x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{4} + \beta_{3} - 1) q^{3} + ( - \beta_{8} - \beta_1) q^{4} + ( - \beta_{9} - 1) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{4} + \beta_{3} - 1) q^{3} + ( - \beta_{8} - \beta_1) q^{4} + ( - \beta_{9} - 1) q^{5} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + (8 \beta_{6} - 3 \beta_{3} - 2 \beta_{2} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 6 q^{5} - 6 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 6 q^{5} - 6 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9} + 3 q^{10} + 2 q^{11} + 7 q^{12} + 10 q^{14} - 2 q^{15} + 6 q^{16} - 6 q^{17} + 6 q^{18} - 7 q^{20} - 2 q^{21} - 16 q^{22} + 2 q^{23} + 42 q^{24} + 12 q^{25} + 5 q^{26} - 50 q^{27} + 11 q^{28} - 8 q^{29} - 35 q^{30} + 12 q^{31} - q^{32} - 14 q^{33} + 28 q^{34} - 21 q^{35} - 22 q^{36} - 7 q^{37} + 5 q^{38} + 17 q^{39} - 10 q^{40} + 16 q^{41} + 16 q^{42} + 20 q^{43} - 3 q^{44} + 18 q^{45} - 19 q^{46} + 17 q^{47} + 16 q^{48} + 5 q^{49} + 20 q^{50} - 32 q^{52} + 8 q^{53} - 18 q^{54} - 2 q^{55} - 8 q^{56} + 14 q^{57} - 32 q^{58} + 6 q^{59} + 42 q^{60} - 6 q^{61} + 8 q^{62} + 8 q^{63} - 12 q^{64} - 18 q^{65} + 6 q^{66} - 10 q^{67} + 94 q^{68} + 23 q^{69} - 44 q^{70} + 3 q^{71} - 106 q^{72} - 34 q^{73} - 20 q^{74} - 46 q^{75} - 32 q^{76} + 3 q^{77} - 2 q^{78} + 8 q^{79} - 31 q^{80} + 50 q^{81} + 42 q^{82} + 28 q^{83} + 3 q^{84} - 16 q^{85} + 9 q^{86} + 27 q^{87} - 19 q^{88} - 84 q^{89} + 77 q^{90} + 68 q^{91} + 18 q^{92} + q^{93} - 48 q^{94} + 31 q^{95} + 19 q^{96} - 20 q^{97} + 21 q^{98} + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 6x^{7} + 35x^{6} - 21x^{5} + 84x^{4} - 2x^{3} + 76x^{2} - 30x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34734 \nu^{9} + 357523 \nu^{8} - 423976 \nu^{7} + 1754799 \nu^{6} + 812430 \nu^{5} + \cdots + 3200950 ) / 6305615 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59348 \nu^{9} + 189706 \nu^{8} - 681217 \nu^{7} + 597638 \nu^{6} - 1075710 \nu^{5} + \cdots - 321850 ) / 6305615 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67007 \nu^{9} - 300674 \nu^{8} + 1079693 \nu^{7} - 2113577 \nu^{6} + 4964880 \nu^{5} + \cdots - 6343210 ) / 6305615 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13753 \nu^{9} - 43494 \nu^{8} + 156183 \nu^{7} - 198885 \nu^{6} + 535767 \nu^{5} - 687996 \nu^{4} + \cdots - 707190 ) / 1261123 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 141438 \nu^{9} - 214111 \nu^{8} + 1055472 \nu^{7} - 67713 \nu^{6} + 3955905 \nu^{5} + \cdots - 2266140 ) / 6305615 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29741 \nu^{9} + 75900 \nu^{8} - 272550 \nu^{7} + 253297 \nu^{6} - 934950 \nu^{5} + \cdots + 2885611 ) / 1261123 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 202936 \nu^{9} + 266192 \nu^{8} - 1529109 \nu^{7} - 136634 \nu^{6} - 5915895 \nu^{5} + \cdots - 6359825 ) / 6305615 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63073 \nu^{9} - 171226 \nu^{8} + 614857 \nu^{7} - 777224 \nu^{6} + 2109193 \nu^{5} + \cdots - 2550576 ) / 1261123 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{5} - \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} - 7\beta_{6} + \beta_{3} - \beta_{2} - 6\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - 7\beta_{8} - 7\beta_{7} - 8\beta_{6} - 18\beta_{5} + 7\beta_{4} - 2\beta_{2} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{9} - 25\beta_{7} - 33\beta_{5} + 11\beta_{4} - 11\beta_{3} + 31 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 44\beta_{8} + 50\beta_{6} - 43\beta_{3} + 20\beta_{2} + 89\beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 63\beta_{9} + 132\beta_{8} + 132\beta_{7} + 154\beta_{6} + 183\beta_{5} - 84\beta_{4} + 63\beta_{2} + 183\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 147\beta_{9} + 267\beta_{7} + 469\beta_{5} - 258\beta_{4} + 258\beta_{3} - 297 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.888396 − 1.53875i
−0.541705 − 0.938261i
0.288111 + 0.499023i
0.929156 + 1.60935i
1.21283 + 2.10069i
−0.888396 + 1.53875i
−0.541705 + 0.938261i
0.288111 − 0.499023i
0.929156 − 1.60935i
1.21283 − 2.10069i
−0.888396 + 1.53875i 1.43594 −0.578496 − 1.00199i 0.394369 −1.27568 + 2.20954i −0.118688 − 0.205573i −1.49785 −0.938087 −0.350356 + 0.606834i
29.2 −0.541705 + 0.938261i −2.80477 0.413111 + 0.715528i −3.21250 1.51936 − 2.63161i 0.693137 + 1.20055i −3.06196 4.86673 1.74023 − 3.01416i
29.3 0.288111 − 0.499023i −0.130627 0.833984 + 1.44450i 0.743238 −0.0376350 + 0.0651857i −1.70560 − 2.95419i 2.11356 −2.98294 0.214135 − 0.370893i
29.4 0.929156 − 1.60935i 0.610905 −0.726663 − 1.25862i −3.58858 0.567626 − 0.983157i 2.02095 + 3.50039i 1.01589 −2.62680 −3.33435 + 5.77526i
29.5 1.21283 − 2.10069i −3.11145 −1.94194 − 3.36353i 2.66347 −3.77367 + 6.53619i 0.110202 + 0.190876i −4.56965 6.68109 3.23034 − 5.59512i
37.1 −0.888396 − 1.53875i 1.43594 −0.578496 + 1.00199i 0.394369 −1.27568 − 2.20954i −0.118688 + 0.205573i −1.49785 −0.938087 −0.350356 − 0.606834i
37.2 −0.541705 − 0.938261i −2.80477 0.413111 − 0.715528i −3.21250 1.51936 + 2.63161i 0.693137 − 1.20055i −3.06196 4.86673 1.74023 + 3.01416i
37.3 0.288111 + 0.499023i −0.130627 0.833984 − 1.44450i 0.743238 −0.0376350 − 0.0651857i −1.70560 + 2.95419i 2.11356 −2.98294 0.214135 + 0.370893i
37.4 0.929156 + 1.60935i 0.610905 −0.726663 + 1.25862i −3.58858 0.567626 + 0.983157i 2.02095 − 3.50039i 1.01589 −2.62680 −3.33435 − 5.77526i
37.5 1.21283 + 2.10069i −3.11145 −1.94194 + 3.36353i 2.66347 −3.77367 − 6.53619i 0.110202 − 0.190876i −4.56965 6.68109 3.23034 + 5.59512i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 67.2.c.a ✓ 10
3.b odd 2 1 603.2.g.e 10
4.b odd 2 1 1072.2.i.e 10
67.c even 3 1 inner 67.2.c.a ✓ 10
67.c even 3 1 4489.2.a.g 5
67.d odd 6 1 4489.2.a.j 5
201.g odd 6 1 603.2.g.e 10
268.g odd 6 1 1072.2.i.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.c.a ✓ 10 1.a even 1 1 trivial
67.2.c.a ✓ 10 67.c even 3 1 inner
603.2.g.e 10 3.b odd 2 1
603.2.g.e 10 201.g odd 6 1
1072.2.i.e 10 4.b odd 2 1
1072.2.i.e 10 268.g odd 6 1
4489.2.a.g 5 67.c even 3 1
4489.2.a.j 5 67.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(67, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 2 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{5} + 4 T^{4} - 2 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} + 3 T^{4} - 11 T^{3} + \cdots - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} + 36 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{10} + 6 T^{9} + \cdots + 1447209 \) Copy content Toggle raw display
$19$ \( T^{10} + 33 T^{8} + \cdots + 6889 \) Copy content Toggle raw display
$23$ \( T^{10} - 2 T^{9} + \cdots + 91809 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} + \cdots + 10439361 \) Copy content Toggle raw display
$37$ \( T^{10} + 7 T^{9} + \cdots + 51529 \) Copy content Toggle raw display
$41$ \( T^{10} - 16 T^{9} + \cdots + 14630625 \) Copy content Toggle raw display
$43$ \( (T^{5} - 10 T^{4} + \cdots - 405)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 17 T^{9} + \cdots + 537289 \) Copy content Toggle raw display
$53$ \( (T^{5} - 4 T^{4} + \cdots - 8343)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 3 T^{4} + \cdots - 32805)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 6 T^{9} + \cdots + 77841 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 122744241 \) Copy content Toggle raw display
$73$ \( T^{10} + 34 T^{9} + \cdots + 2719201 \) Copy content Toggle raw display
$79$ \( T^{10} - 8 T^{9} + \cdots + 1585081 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 397085329 \) Copy content Toggle raw display
$89$ \( (T^{5} + 42 T^{4} + \cdots + 25367)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 602358849 \) Copy content Toggle raw display
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