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Properties

Label 266.2.u.b
Level $266$
Weight $2$
Character orbit 266.u
Analytic conductor $2.124$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 12 x^{10} - 16 x^{9} - 6 x^{8} + 27 x^{7} + 5 x^{6} - 27 x^{5} - 6 x^{4} + 16 x^{3} + \cdots + 1 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{8}) q^{2} + (\beta_{8} + \beta_{6} - \beta_1) q^{3} + \beta_{11} q^{4} + ( - \beta_{11} + \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + (\beta_{8} - \beta_{7} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{8}) q^{2} + (\beta_{8} + \beta_{6} - \beta_1) q^{3} + \beta_{11} q^{4} + ( - \beta_{11} + \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + (2 \beta_{11} - \beta_{10} - \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} + 6 q^{8} - 6 q^{9} - 9 q^{10} - 3 q^{11} - 3 q^{12} + 6 q^{13} - 6 q^{15} + 3 q^{17} + 6 q^{18} + 6 q^{19} - 6 q^{20} + 3 q^{22} + 18 q^{23} - 24 q^{25} + 6 q^{26} - 3 q^{27} - 9 q^{29} - 6 q^{30} - 3 q^{31} + 9 q^{33} + 24 q^{34} + 9 q^{35} - 6 q^{36} - 6 q^{37} - 3 q^{38} - 18 q^{39} - 9 q^{40} + 9 q^{41} + 6 q^{43} - 3 q^{44} + 12 q^{45} - 45 q^{47} - 6 q^{49} + 3 q^{50} - 3 q^{51} - 3 q^{52} + 15 q^{53} - 18 q^{54} + 3 q^{55} + 12 q^{56} - 9 q^{57} - 12 q^{58} + 6 q^{59} - 15 q^{60} - 18 q^{61} + 15 q^{62} - 12 q^{63} - 6 q^{64} - 3 q^{65} + 18 q^{66} + 18 q^{67} + 9 q^{68} - 24 q^{69} - 9 q^{70} + 15 q^{71} + 6 q^{72} + 39 q^{73} + 6 q^{74} + 66 q^{75} - 15 q^{76} - 6 q^{77} - 24 q^{79} + 12 q^{81} - 9 q^{82} + 33 q^{83} + 3 q^{84} - 78 q^{85} - 6 q^{86} - 15 q^{87} + 3 q^{88} + 30 q^{89} + 3 q^{90} + 3 q^{91} + 33 q^{93} - 18 q^{94} - 75 q^{95} - 6 q^{96} - 21 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 12 x^{10} - 16 x^{9} - 6 x^{8} + 27 x^{7} + 5 x^{6} - 27 x^{5} - 6 x^{4} + 16 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23 \nu^{11} - 177 \nu^{10} + 577 \nu^{9} - 1576 \nu^{8} + 1280 \nu^{7} + 1688 \nu^{6} - 2390 \nu^{5} + \cdots - 1265 ) / 1513 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75 \nu^{11} + 78 \nu^{10} - 473 \nu^{9} - 375 \nu^{8} + 2118 \nu^{7} + 1279 \nu^{6} - 8064 \nu^{5} + \cdots - 1304 ) / 1513 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72 \nu^{11} + 349 \nu^{10} - 1102 \nu^{9} + 2043 \nu^{8} + 1186 \nu^{7} - 8968 \nu^{6} + \cdots + 1379 ) / 1513 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34 \nu^{11} + 431 \nu^{10} - 1670 \nu^{9} + 7557 \nu^{8} - 15428 \nu^{7} + 9120 \nu^{6} + 11210 \nu^{5} + \cdots + 1601 ) / 1513 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 92 \nu^{11} - 263 \nu^{10} + 1062 \nu^{9} - 1320 \nu^{8} - 754 \nu^{7} + 2302 \nu^{6} + 942 \nu^{5} + \cdots + 102 ) / 1513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 108 \nu^{11} - 301 \nu^{10} + 1208 \nu^{9} - 1418 \nu^{8} - 1067 \nu^{7} + 2505 \nu^{6} + 3118 \nu^{5} + \cdots + 23 ) / 1513 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 102 \nu^{11} - 398 \nu^{10} + 1487 \nu^{9} - 2694 \nu^{8} + 708 \nu^{7} + 3508 \nu^{6} - 1792 \nu^{5} + \cdots - 1071 ) / 1513 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 231 \nu^{11} - 404 \nu^{10} + 1674 \nu^{9} + 710 \nu^{8} - 9553 \nu^{7} + 11358 \nu^{6} + 5695 \nu^{5} + \cdots + 1268 ) / 1513 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 232 \nu^{11} - 729 \nu^{10} + 3007 \nu^{9} - 4447 \nu^{8} + 401 \nu^{7} + 5213 \nu^{6} - 2624 \nu^{5} + \cdots + 3349 ) / 1513 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 765 \nu^{11} + 2629 \nu^{10} - 10307 \nu^{9} + 16734 \nu^{8} - 2551 \nu^{7} - 19546 \nu^{6} + \cdots - 1179 ) / 1513 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} + 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{9} - 3\beta_{7} - 4\beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{10} + 12 \beta_{9} - 13 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 11 \beta_{5} - 10 \beta_{4} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{11} + 2 \beta_{9} - 10 \beta_{8} + 33 \beta_{7} + 35 \beta_{6} - 12 \beta_{5} - 23 \beta_{4} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{11} + 114 \beta_{10} - 124 \beta_{9} + 126 \beta_{8} + 76 \beta_{7} + 101 \beta_{5} + \cdots - 114 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 101 \beta_{11} + 126 \beta_{10} - 162 \beta_{9} + 240 \beta_{8} - 238 \beta_{7} - 364 \beta_{6} + \cdots - 240 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 238 \beta_{11} - 1027 \beta_{10} + 1088 \beta_{9} - 1027 \beta_{8} - 1027 \beta_{7} - 402 \beta_{6} + \cdots + 901 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 764 \beta_{11} - 2431 \beta_{10} + 2879 \beta_{9} - 3597 \beta_{8} + 1292 \beta_{7} + 3281 \beta_{6} + \cdots + 3458 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3281 \beta_{11} + 7768 \beta_{10} - 7848 \beta_{9} + 6476 \beta_{8} + 11893 \beta_{7} + 7768 \beta_{6} + \cdots - 5337 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4125 \beta_{11} + 33404 \beta_{10} - 37971 \beta_{9} + 43866 \beta_{8} - 24786 \beta_{6} + \cdots - 41172 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\)
\(\chi(n)\) \(1\) \(\beta_{11}\)

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
1.05581 − 0.885927i
−0.555806 + 0.466377i
1.22238 − 0.444910i
−0.722380 + 0.262925i
1.05581 + 0.885927i
−0.555806 − 0.466377i
1.22238 + 0.444910i
−0.722380 − 0.262925i
−0.0543907 − 0.308465i
0.554391 + 3.14411i
−0.0543907 + 0.308465i
0.554391 − 3.14411i
0.939693 − 0.342020i −0.193030 + 1.09473i 0.766044 − 0.642788i 0.278530 + 0.233714i 0.193030 + 1.09473i 0.500000 + 0.866025i 0.500000 − 0.866025i 1.65792 + 0.603432i 0.341668 + 0.124357i
43.2 0.939693 − 0.342020i 0.366678 − 2.07953i 0.766044 − 0.642788i −2.75031 − 2.30779i −0.366678 − 2.07953i 0.500000 + 0.866025i 0.500000 − 0.866025i −1.37093 − 0.498976i −3.37376 − 1.22795i
85.1 −0.173648 − 0.984808i −1.10675 + 0.928673i −0.939693 + 0.342020i 0.688813 + 0.250707i 1.10675 + 0.928673i 0.500000 − 0.866025i 0.500000 + 0.866025i −0.158484 + 0.898805i 0.127287 − 0.721883i
85.2 −0.173648 − 0.984808i 1.87279 − 1.57146i −0.939693 + 0.342020i 1.36422 + 0.496536i −1.87279 − 1.57146i 0.500000 − 0.866025i 0.500000 + 0.866025i 0.516924 − 2.93162i 0.252098 − 1.42972i
99.1 0.939693 + 0.342020i −0.193030 − 1.09473i 0.766044 + 0.642788i 0.278530 − 0.233714i 0.193030 − 1.09473i 0.500000 − 0.866025i 0.500000 + 0.866025i 1.65792 − 0.603432i 0.341668 − 0.124357i
99.2 0.939693 + 0.342020i 0.366678 + 2.07953i 0.766044 + 0.642788i −2.75031 + 2.30779i −0.366678 + 2.07953i 0.500000 − 0.866025i 0.500000 + 0.866025i −1.37093 + 0.498976i −3.37376 + 1.22795i
169.1 −0.173648 + 0.984808i −1.10675 − 0.928673i −0.939693 − 0.342020i 0.688813 − 0.250707i 1.10675 − 0.928673i 0.500000 + 0.866025i 0.500000 − 0.866025i −0.158484 − 0.898805i 0.127287 + 0.721883i
169.2 −0.173648 + 0.984808i 1.87279 + 1.57146i −0.939693 − 0.342020i 1.36422 − 0.496536i −1.87279 + 1.57146i 0.500000 + 0.866025i 0.500000 − 0.866025i 0.516924 + 2.93162i 0.252098 + 1.42972i
225.1 −0.766044 − 0.642788i −1.04191 + 0.379226i 0.173648 + 0.984808i 0.675728 − 3.83224i 1.04191 + 0.379226i 0.500000 + 0.866025i 0.500000 − 0.866025i −1.35636 + 1.13812i −2.98095 + 2.50132i
225.2 −0.766044 − 0.642788i 0.102221 − 0.0372054i 0.173648 + 0.984808i −0.256980 + 1.45740i −0.102221 − 0.0372054i 0.500000 + 0.866025i 0.500000 − 0.866025i −2.28907 + 1.92076i 1.13366 − 0.951253i
253.1 −0.766044 + 0.642788i −1.04191 − 0.379226i 0.173648 − 0.984808i 0.675728 + 3.83224i 1.04191 − 0.379226i 0.500000 − 0.866025i 0.500000 + 0.866025i −1.35636 − 1.13812i −2.98095 − 2.50132i
253.2 −0.766044 + 0.642788i 0.102221 + 0.0372054i 0.173648 − 0.984808i −0.256980 − 1.45740i −0.102221 + 0.0372054i 0.500000 − 0.866025i 0.500000 + 0.866025i −2.28907 − 1.92076i 1.13366 + 0.951253i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.2
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.b ✓ 12
19.e even 9 1 inner 266.2.u.b ✓ 12
19.e even 9 1 5054.2.a.bb 6
19.f odd 18 1 5054.2.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.b ✓ 12 1.a even 1 1 trivial
266.2.u.b ✓ 12 19.e even 9 1 inner
5054.2.a.bb 6 19.e even 9 1
5054.2.a.bc 6 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 3 T_{3}^{10} + 7 T_{3}^{9} + 15 T_{3}^{8} + 39 T_{3}^{7} + 113 T_{3}^{6} + 183 T_{3}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + \cdots + 1478656 \) Copy content Toggle raw display
$29$ \( T^{12} + 9 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$31$ \( T^{12} + 3 T^{11} + \cdots + 5607424 \) Copy content Toggle raw display
$37$ \( (T^{6} + 3 T^{5} + \cdots - 296)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 9 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1928878561 \) Copy content Toggle raw display
$47$ \( T^{12} + 45 T^{11} + \cdots + 3474496 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2483228224 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1754520769 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1270779904 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2288952649 \) Copy content Toggle raw display
$71$ \( T^{12} - 15 T^{11} + \cdots + 95726656 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 5403867121 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 313431616 \) Copy content Toggle raw display
$83$ \( T^{12} - 33 T^{11} + \cdots + 87291649 \) Copy content Toggle raw display
$89$ \( T^{12} - 30 T^{11} + \cdots + 25130169 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 222367744 \) Copy content Toggle raw display
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