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Properties

Label 266.2.u.c
Level $266$
Weight $2$
Character orbit 266.u
Analytic conductor $2.124$
Analytic rank $0$
Dimension $18$
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] = [266,2,Mod(43,266)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(266, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("266.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.u (of order \(9\), degree \(6\), 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: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 24 x^{16} - 35 x^{15} + 264 x^{14} - 276 x^{13} + 1919 x^{12} - 801 x^{11} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 24 x^{16} - 35 x^{15} + 264 x^{14} - 276 x^{13} + 1919 x^{12} - 801 x^{11} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52\!\cdots\!28 \nu^{17} + \cdots + 67\!\cdots\!31 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!94 \nu^{17} + \cdots + 37\!\cdots\!15 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!69 \nu^{17} + \cdots + 45\!\cdots\!28 ) / 95\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64\!\cdots\!02 \nu^{17} + \cdots - 38\!\cdots\!30 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!68 \nu^{17} + \cdots + 69\!\cdots\!49 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!55 \nu^{17} + \cdots - 68\!\cdots\!72 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!15 \nu^{17} + \cdots + 22\!\cdots\!04 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 38\!\cdots\!30 \nu^{17} + \cdots - 14\!\cdots\!89 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!06 \nu^{17} + \cdots - 23\!\cdots\!00 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!10 \nu^{17} + \cdots + 15\!\cdots\!87 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 68\!\cdots\!72 \nu^{17} + \cdots - 22\!\cdots\!60 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 70\!\cdots\!82 \nu^{17} + \cdots + 23\!\cdots\!87 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 45\!\cdots\!28 \nu^{17} + \cdots + 96\!\cdots\!35 ) / 95\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!97 \nu^{17} + \cdots + 31\!\cdots\!32 ) / 65\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!64 \nu^{17} + \cdots - 14\!\cdots\!92 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!03 \nu^{17} + \cdots + 15\!\cdots\!32 ) / 19\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{15} - 3\beta_{14} + \beta_{13} + \beta_{4} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - 2\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{4} + 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{17} - 6 \beta_{16} - 8 \beta_{15} + 17 \beta_{14} - 9 \beta_{13} - 4 \beta_{12} + 10 \beta_{11} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{17} - 4 \beta_{16} - 6 \beta_{15} + 22 \beta_{14} - 15 \beta_{13} - 8 \beta_{12} + \cdots - 46 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{17} + 30 \beta_{16} - 18 \beta_{13} - 30 \beta_{12} - 81 \beta_{11} + 99 \beta_{10} + \cdots + 125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 191 \beta_{17} + 150 \beta_{16} + 115 \beta_{15} - 228 \beta_{14} + 160 \beta_{13} + 82 \beta_{12} + \cdots + 228 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 756 \beta_{17} + 546 \beta_{16} + 749 \beta_{15} - 1077 \beta_{14} + 989 \beta_{13} + 896 \beta_{12} + \cdots + 981 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 525 \beta_{17} - 870 \beta_{16} + 525 \beta_{13} + 870 \beta_{12} + 1753 \beta_{11} - 2278 \beta_{10} + \cdots - 2420 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10076 \beta_{17} - 9672 \beta_{16} - 7897 \beta_{15} + 10181 \beta_{14} - 7337 \beta_{13} + \cdots - 10181 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 19051 \beta_{17} - 15214 \beta_{16} - 20271 \beta_{15} + 26164 \beta_{14} - 26229 \beta_{13} + \cdots - 37848 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 31044 \beta_{17} + 39654 \beta_{16} - 31044 \beta_{13} - 39654 \beta_{12} - 73630 \beta_{11} + \cdots + 101671 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 295916 \beta_{17} + 295458 \beta_{16} + 240001 \beta_{15} - 285290 \beta_{14} + 206981 \beta_{13} + \cdots + 285290 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 758358 \beta_{17} + 700342 \beta_{16} + 917786 \beta_{15} - 1049755 \beta_{14} + 1104554 \beta_{13} + \cdots + 1177454 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1046718 \beta_{17} - 1246806 \beta_{16} + 1046718 \beta_{13} + 1246806 \beta_{12} + 2251512 \beta_{11} + \cdots - 3120200 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 11789851 \beta_{17} - 12072954 \beta_{16} - 9978503 \beta_{15} + 11065569 \beta_{14} + \cdots - 11065569 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 24522133 \beta_{17} - 23867544 \beta_{16} - 31037666 \beta_{15} + 34149138 \beta_{14} + \cdots - 43575934 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.738263 + 1.27871i
0.840132 − 1.45515i
1.33782 − 2.31718i
−1.32423 − 2.29363i
−0.00239212 − 0.00414328i
1.65297 + 2.86303i
−0.738263 − 1.27871i
0.840132 + 1.45515i
1.33782 + 2.31718i
−1.32423 + 2.29363i
−0.00239212 + 0.00414328i
1.65297 − 2.86303i
1.09378 − 1.89448i
−0.453443 + 0.785387i
−0.906379 + 1.56989i
1.09378 + 1.89448i
−0.453443 − 0.785387i
−0.906379 − 1.56989i
−0.939693 + 0.342020i −0.256396 + 1.45409i 0.766044 − 0.642788i −0.945381 − 0.793269i −0.256396 − 1.45409i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.770427 + 0.280412i 1.15968 + 0.422090i
43.2 −0.939693 + 0.342020i 0.291775 − 1.65474i 0.766044 − 0.642788i 2.67348 + 2.24331i 0.291775 + 1.65474i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.166056 + 0.0604394i −3.27950 − 1.19364i
43.3 −0.939693 + 0.342020i 0.464621 − 2.63500i 0.766044 − 0.642788i −2.13570 − 1.79206i 0.464621 + 2.63500i 0.500000 + 0.866025i −0.500000 + 0.866025i −3.90826 − 1.42249i 2.61982 + 0.953537i
85.1 0.173648 + 0.984808i −2.02883 + 1.70239i −0.939693 + 0.342020i −2.17440 − 0.791417i −2.02883 − 1.70239i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.697076 − 3.95331i 0.401813 − 2.27880i
85.2 0.173648 + 0.984808i −0.00366495 + 0.00307525i −0.939693 + 0.342020i 0.977487 + 0.355776i −0.00366495 − 0.00307525i 0.500000 − 0.866025i −0.500000 − 0.866025i −0.520941 + 2.95440i −0.180632 + 1.02442i
85.3 0.173648 + 0.984808i 2.53250 − 2.12502i −0.939693 + 0.342020i −1.50882 − 0.549166i 2.53250 + 2.12502i 0.500000 − 0.866025i −0.500000 − 0.866025i 1.37690 − 7.80878i 0.278819 − 1.58126i
99.1 −0.939693 − 0.342020i −0.256396 − 1.45409i 0.766044 + 0.642788i −0.945381 + 0.793269i −0.256396 + 1.45409i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.770427 − 0.280412i 1.15968 − 0.422090i
99.2 −0.939693 − 0.342020i 0.291775 + 1.65474i 0.766044 + 0.642788i 2.67348 − 2.24331i 0.291775 − 1.65474i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.166056 − 0.0604394i −3.27950 + 1.19364i
99.3 −0.939693 − 0.342020i 0.464621 + 2.63500i 0.766044 + 0.642788i −2.13570 + 1.79206i 0.464621 − 2.63500i 0.500000 − 0.866025i −0.500000 − 0.866025i −3.90826 + 1.42249i 2.61982 − 0.953537i
169.1 0.173648 − 0.984808i −2.02883 − 1.70239i −0.939693 − 0.342020i −2.17440 + 0.791417i −2.02883 + 1.70239i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.697076 + 3.95331i 0.401813 + 2.27880i
169.2 0.173648 − 0.984808i −0.00366495 − 0.00307525i −0.939693 − 0.342020i 0.977487 − 0.355776i −0.00366495 + 0.00307525i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.520941 − 2.95440i −0.180632 − 1.02442i
169.3 0.173648 − 0.984808i 2.53250 + 2.12502i −0.939693 − 0.342020i −1.50882 + 0.549166i 2.53250 − 2.12502i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.37690 + 7.80878i 0.278819 + 1.58126i
225.1 0.766044 + 0.642788i −2.05563 + 0.748188i 0.173648 + 0.984808i −0.106991 + 0.606779i −2.05563 − 0.748188i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.36770 − 1.14763i −0.471990 + 0.396047i
225.2 0.766044 + 0.642788i 0.852195 − 0.310174i 0.173648 + 0.984808i −0.338055 + 1.91720i 0.852195 + 0.310174i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.66811 + 1.39971i −1.49132 + 1.25137i
225.3 0.766044 + 0.642788i 1.70344 − 0.620000i 0.173648 + 0.984808i 0.558387 − 3.16677i 1.70344 + 0.620000i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.219158 − 0.183895i 2.46331 − 2.06696i
253.1 0.766044 − 0.642788i −2.05563 − 0.748188i 0.173648 − 0.984808i −0.106991 − 0.606779i −2.05563 + 0.748188i 0.500000 − 0.866025i −0.500000 − 0.866025i 1.36770 + 1.14763i −0.471990 − 0.396047i
253.2 0.766044 − 0.642788i 0.852195 + 0.310174i 0.173648 − 0.984808i −0.338055 − 1.91720i 0.852195 − 0.310174i 0.500000 − 0.866025i −0.500000 − 0.866025i −1.66811 − 1.39971i −1.49132 − 1.25137i
253.3 0.766044 − 0.642788i 1.70344 + 0.620000i 0.173648 − 0.984808i 0.558387 + 3.16677i 1.70344 − 0.620000i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.219158 + 0.183895i 2.46331 + 2.06696i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 266.2.u.c ✓ 18
19.e even 9 1 inner 266.2.u.c ✓ 18
19.e even 9 1 5054.2.a.bk 9
19.f odd 18 1 5054.2.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.c ✓ 18 1.a even 1 1 trivial
266.2.u.c ✓ 18 19.e even 9 1 inner
5054.2.a.bj 9 19.f odd 18 1
5054.2.a.bk 9 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 3 T_{3}^{17} + 6 T_{3}^{16} + 5 T_{3}^{15} + 21 T_{3}^{14} - 30 T_{3}^{13} + 500 T_{3}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} + 6 T^{17} + \cdots + 32041 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} - 3 T^{17} + \cdots + 75759616 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 101989801 \) Copy content Toggle raw display
$17$ \( T^{18} + 21 T^{17} + \cdots + 1478656 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 2410417216 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 533794816 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 57170722816 \) Copy content Toggle raw display
$37$ \( (T^{9} - 9 T^{8} + \cdots + 2078144)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 161751186804736 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 299113981087744 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27065964630016 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 3193606144 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 69\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17296216312384 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 5436283949056 \) Copy content Toggle raw display
$71$ \( T^{18} + 27 T^{17} + \cdots + 494209 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 59136998761 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 96805507762441 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 139931108245504 \) Copy content Toggle raw display
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