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Properties

Label 561.2.g.b
Level $561$
Weight $2$
Character orbit 561.g
Analytic conductor $4.480$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [561,2,Mod(67,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.g (of order \(2\), degree \(1\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 27x^{14} + 291x^{12} + 1585x^{10} + 4548x^{8} + 6536x^{6} + 4136x^{4} + 768x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{12} + \beta_1) q^{5} + \beta_1 q^{6} - \beta_{11} q^{7} + (\beta_{9} - \beta_{4}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{12} + \beta_1) q^{5} + \beta_1 q^{6} - \beta_{11} q^{7} + (\beta_{9} - \beta_{4}) q^{8} - q^{9} + ( - \beta_{15} - \beta_{14} + \cdots + 2 \beta_{8}) q^{10}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 22 q^{4} - 6 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 22 q^{4} - 6 q^{8} - 16 q^{9} + 12 q^{13} - 4 q^{15} + 34 q^{16} - 6 q^{17} + 2 q^{18} - 8 q^{19} + 2 q^{21} - 56 q^{25} - 8 q^{26} - 36 q^{30} - 34 q^{32} - 16 q^{33} + 30 q^{34} + 8 q^{35} - 22 q^{36} + 44 q^{38} - 12 q^{42} - 24 q^{43} + 18 q^{47} - 34 q^{49} + 30 q^{50} + 14 q^{51} + 28 q^{52} - 26 q^{53} - 4 q^{55} - 2 q^{59} - 12 q^{60} - 38 q^{64} + 2 q^{66} + 30 q^{67} + 42 q^{68} - 28 q^{69} + 88 q^{70} + 6 q^{72} - 84 q^{76} + 2 q^{77} + 16 q^{81} - 52 q^{83} + 24 q^{84} + 40 q^{85} - 52 q^{86} - 26 q^{87} + 78 q^{89} + 20 q^{93} - 16 q^{94} - 70 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 27x^{14} + 291x^{12} + 1585x^{10} + 4548x^{8} + 6536x^{6} + 4136x^{4} + 768x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{14} + 98\nu^{12} + 689\nu^{10} + 2024\nu^{8} + 2064\nu^{6} + 148\nu^{4} - 380\nu^{2} - 12 ) / 136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{14} - 98\nu^{12} - 689\nu^{10} - 2024\nu^{8} - 2064\nu^{6} - 12\nu^{4} + 1332\nu^{2} + 692 ) / 136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{14} - 23\nu^{12} - 199\nu^{10} - 789\nu^{8} - 1358\nu^{6} - 662\nu^{4} + 178\nu^{2} - 18 ) / 34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{14} - 98\nu^{12} - 689\nu^{10} - 1956\nu^{8} - 1180\nu^{6} + 3252\nu^{4} + 3780\nu^{2} + 284 ) / 136 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{15} + 86\nu^{13} + 971\nu^{11} + 5444\nu^{9} + 15668\nu^{7} + 21672\nu^{5} + 12556\nu^{3} + 1924\nu ) / 136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{14} - 69\nu^{12} - 614\nu^{10} - 2639\nu^{8} - 5553\nu^{6} - 5080\nu^{4} - 1438\nu^{2} - 20 ) / 34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9\nu^{14} + 190\nu^{12} + 1519\nu^{10} + 5724\nu^{8} + 10522\nu^{6} + 9596\nu^{4} + 4144\nu^{2} + 468 ) / 68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2\nu^{15} - 46\nu^{13} - 415\nu^{11} - 1867\nu^{9} - 4433\nu^{7} - 5421\nu^{5} - 2806\nu^{3} - 274\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2\nu^{15} + 46\nu^{13} + 415\nu^{11} + 1867\nu^{9} + 4450\nu^{7} + 5642\nu^{5} + 3622\nu^{3} + 920\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7\nu^{15} + 178\nu^{13} + 1801\nu^{11} + 9178\nu^{9} + 24534\nu^{7} + 32582\nu^{5} + 18644\nu^{3} + 2880\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5\nu^{15} + 166\nu^{13} + 2117\nu^{11} + 13312\nu^{9} + 43476\nu^{7} + 70800\nu^{5} + 50756\nu^{3} + 10596\nu ) / 136 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13 \nu^{15} - 350 \nu^{13} - 3777 \nu^{11} - 20712 \nu^{9} - 60256 \nu^{7} - 88336 \nu^{5} + \cdots - 9652 \nu ) / 136 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} - 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{11} - 2\beta_{8} - 7\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{6} - 9\beta_{5} - 11\beta_{4} + 44\beta_{2} - 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{13} + 2\beta_{12} - 11\beta_{11} + 26\beta_{8} + 43\beta_{3} - 175\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -13\beta_{10} - 13\beta_{9} + 2\beta_{7} + 13\beta_{6} + 67\beta_{5} + 95\beta_{4} - 272\beta_{2} + 568 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2\beta_{15} + 2\beta_{14} + 121\beta_{13} - 28\beta_{12} + 91\beta_{11} - 242\beta_{8} - 259\beta_{3} + 1071\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 121\beta_{10} + 119\beta_{9} - 32\beta_{7} - 115\beta_{6} - 471\beta_{5} - 745\beta_{4} + 1682\beta_{2} - 3472 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 42 \beta_{15} - 38 \beta_{14} - 989 \beta_{13} + 274 \beta_{12} - 675 \beta_{11} + 1966 \beta_{8} + \cdots - 6605 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 987 \beta_{10} - 951 \beta_{9} + 350 \beta_{7} + 869 \beta_{6} + 3223 \beta_{5} + 5539 \beta_{4} + \cdots + 21372 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 550 \beta_{15} + 468 \beta_{14} + 7559 \beta_{13} - 2324 \beta_{12} + 4729 \beta_{11} + \cdots + 40967 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7521 \beta_{10} + 7091 \beta_{9} - 3260 \beta_{7} - 6035 \beta_{6} - 21703 \beta_{5} - 39821 \beta_{4} + \cdots - 132272 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 5802 \beta_{15} - 4746 \beta_{14} - 55489 \beta_{13} + 18302 \beta_{12} - 31999 \beta_{11} + \cdots - 255321 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/561\mathbb{Z}\right)^\times\).

\(n\) \(188\) \(409\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
2.55224i
− 2.55224i
2.49230i
− 2.49230i
1.29673i
− 1.29673i
0.541497i
− 0.541497i
− 0.0732247i
0.0732247i
− 1.10818i
1.10818i
− 2.26409i
2.26409i
− 2.43727i
2.43727i
−2.55224 − 1.00000i 4.51391 3.57665i 2.55224i − 1.71829i −6.41610 −1.00000 − 9.12845i
67.2 −2.55224 1.00000i 4.51391 − 3.57665i − 2.55224i 1.71829i −6.41610 −1.00000 9.12845i
67.3 −2.49230 − 1.00000i 4.21156 − 2.17306i 2.49230i 2.46372i −5.51186 −1.00000 5.41591i
67.4 −2.49230 1.00000i 4.21156 2.17306i − 2.49230i − 2.46372i −5.51186 −1.00000 − 5.41591i
67.5 −1.29673 − 1.00000i −0.318494 3.74262i 1.29673i 4.49788i 3.00646 −1.00000 − 4.85316i
67.6 −1.29673 1.00000i −0.318494 − 3.74262i − 1.29673i − 4.49788i 3.00646 −1.00000 4.85316i
67.7 −0.541497 − 1.00000i −1.70678 − 3.34121i 0.541497i − 2.89088i 2.00721 −1.00000 1.80925i
67.8 −0.541497 1.00000i −1.70678 3.34121i − 0.541497i 2.89088i 2.00721 −1.00000 − 1.80925i
67.9 0.0732247 − 1.00000i −1.99464 1.86667i − 0.0732247i − 0.558036i −0.292506 −1.00000 0.136686i
67.10 0.0732247 1.00000i −1.99464 − 1.86667i 0.0732247i 0.558036i −0.292506 −1.00000 − 0.136686i
67.11 1.10818 − 1.00000i −0.771945 − 1.89953i − 1.10818i − 0.977937i −3.07180 −1.00000 − 2.10501i
67.12 1.10818 1.00000i −0.771945 1.89953i 1.10818i 0.977937i −3.07180 −1.00000 2.10501i
67.13 2.26409 − 1.00000i 3.12611 0.470551i − 2.26409i − 4.03769i 2.54961 −1.00000 1.06537i
67.14 2.26409 1.00000i 3.12611 − 0.470551i 2.26409i 4.03769i 2.54961 −1.00000 − 1.06537i
67.15 2.43727 − 1.00000i 3.94028 − 4.24270i − 2.43727i 4.22123i 4.72899 −1.00000 − 10.3406i
67.16 2.43727 1.00000i 3.94028 4.24270i 2.43727i − 4.22123i 4.72899 −1.00000 10.3406i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.g.b ✓ 16
3.b odd 2 1 1683.2.g.c 16
17.b even 2 1 inner 561.2.g.b ✓ 16
17.c even 4 1 9537.2.a.bm 8
17.c even 4 1 9537.2.a.bn 8
51.c odd 2 1 1683.2.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.g.b ✓ 16 1.a even 1 1 trivial
561.2.g.b ✓ 16 17.b even 2 1 inner
1683.2.g.c 16 3.b odd 2 1
1683.2.g.c 16 51.c odd 2 1
9537.2.a.bm 8 17.c even 4 1
9537.2.a.bn 8 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} - 13T_{2}^{6} - 11T_{2}^{5} + 50T_{2}^{4} + 34T_{2}^{3} - 48T_{2}^{2} - 24T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{7} - 13 T^{6} + \cdots + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 68 T^{14} + \cdots + 473344 \) Copy content Toggle raw display
$7$ \( T^{16} + 73 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} - 6 T^{7} + \cdots - 5776)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots - 3776)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 132 T^{14} + \cdots + 94633984 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 7676563456 \) Copy content Toggle raw display
$31$ \( T^{16} + 228 T^{14} + \cdots + 75759616 \) Copy content Toggle raw display
$37$ \( T^{16} + 200 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 231568838656 \) Copy content Toggle raw display
$43$ \( (T^{8} + 12 T^{7} + \cdots - 150256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 9 T^{7} + \cdots - 7307944)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 13 T^{7} + \cdots - 30616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + T^{7} + \cdots + 99416)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 405136384 \) Copy content Toggle raw display
$67$ \( (T^{8} - 15 T^{7} + \cdots + 70208)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 11955440336896 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 8947899620416 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 74193043456 \) Copy content Toggle raw display
$83$ \( (T^{8} + 26 T^{7} + \cdots - 25259392)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 39 T^{7} + \cdots + 275672)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 3833325420544 \) Copy content Toggle raw display
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