Properties

Label 570.2.u.j
Level $570$
Weight $2$
Character orbit 570.u
Analytic conductor $4.551$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,2,Mod(61,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.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: \(4.55147291521\)
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: \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{4} q^{3} + \beta_{3} q^{4} - \beta_{6} q^{5} + \beta_{9} q^{6} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{9} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{4} q^{3} + \beta_{3} q^{4} - \beta_{6} q^{5} + \beta_{9} q^{6} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{7} - 2 \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{7} - 6 q^{8} + 3 q^{11} + 6 q^{12} + 9 q^{13} + 9 q^{14} + 9 q^{17} + 12 q^{18} + 9 q^{19} + 12 q^{20} + 9 q^{21} - 6 q^{22} + 12 q^{23} + 9 q^{26} + 6 q^{27} - 9 q^{28} + 27 q^{29} + 6 q^{30} + 12 q^{31} - 3 q^{33} - 9 q^{34} - 42 q^{37} + 18 q^{38} + 18 q^{39} - 27 q^{41} - 9 q^{42} - 27 q^{43} - 6 q^{44} - 6 q^{45} + 9 q^{46} - 6 q^{47} + 3 q^{49} - 6 q^{50} + 9 q^{52} - 18 q^{53} + 3 q^{55} - 6 q^{56} + 6 q^{58} - 15 q^{59} - 9 q^{61} - 18 q^{62} - 6 q^{64} + 9 q^{65} - 3 q^{66} + 42 q^{67} + 6 q^{68} - 9 q^{69} + 24 q^{71} + 15 q^{73} + 18 q^{74} - 12 q^{75} + 3 q^{76} - 6 q^{77} + 18 q^{78} - 57 q^{79} - 27 q^{82} + 21 q^{83} - 3 q^{84} + 9 q^{85} + 9 q^{86} + 3 q^{87} + 3 q^{88} - 57 q^{89} + 21 q^{91} - 15 q^{92} - 9 q^{93} - 24 q^{94} - 18 q^{95} - 12 q^{96} - 6 q^{97} - 15 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + \cdots + 1060 ) / 218 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} + \cdots - 1706 ) / 218 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} + \cdots - 2263 ) / 218 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + \cdots + 524 ) / 218 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + \cdots + 555 ) / 218 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + \cdots - 83 ) / 218 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + \cdots + 4350 ) / 218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} + \cdots - 2201 ) / 218 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} + \cdots - 5194 ) / 218 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + \cdots - 5469 ) / 218 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} + \cdots - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} + \cdots + 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + \cdots + 77 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} + \cdots - 121 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} + \cdots - 535 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + \cdots + 257 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + \cdots + 3359 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} + \cdots + 1451 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} + \cdots - 18736 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 1.96356i
0.500000 0.677980i
0.500000 1.96356i
0.500000 + 0.677980i
0.500000 + 0.168222i
0.500000 + 1.80139i
0.500000 0.168222i
0.500000 1.80139i
0.500000 2.42499i
0.500000 + 1.74095i
0.500000 + 2.42499i
0.500000 1.74095i
−0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i −0.173648 + 0.984808i −0.897714 + 1.55489i −0.500000 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
61.2 −0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i −0.173648 + 0.984808i −0.308023 + 0.533512i −0.500000 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
271.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0.766044 + 0.642788i −0.173648 0.984808i −0.897714 1.55489i −0.500000 + 0.866025i −0.939693 0.342020i −0.939693 0.342020i
271.2 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0.766044 + 0.642788i −0.173648 0.984808i −0.308023 0.533512i −0.500000 + 0.866025i −0.939693 0.342020i −0.939693 0.342020i
301.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0.173648 0.984808i 0.939693 + 0.342020i −0.965167 1.67172i −0.500000 + 0.866025i 0.766044 0.642788i 0.766044 0.642788i
301.2 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0.173648 0.984808i 0.939693 + 0.342020i 2.05756 + 3.56380i −0.500000 + 0.866025i 0.766044 0.642788i 0.766044 0.642788i
481.1 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0.173648 + 0.984808i 0.939693 0.342020i −0.965167 + 1.67172i −0.500000 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
481.2 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0.173648 + 0.984808i 0.939693 0.342020i 2.05756 3.56380i −0.500000 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
511.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i −0.939693 + 0.342020i −0.766044 + 0.642788i −0.284816 0.493316i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
511.2 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i −0.939693 + 0.342020i −0.766044 + 0.642788i 1.89816 + 3.28770i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
541.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −0.939693 0.342020i −0.766044 0.642788i −0.284816 + 0.493316i −0.500000 0.866025i 0.173648 0.984808i 0.173648 0.984808i
541.2 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −0.939693 0.342020i −0.766044 0.642788i 1.89816 3.28770i −0.500000 0.866025i 0.173648 0.984808i 0.173648 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.2
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 570.2.u.j 12
19.e even 9 1 inner 570.2.u.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.j 12 1.a even 1 1 trivial
570.2.u.j 12 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{12} - 3 T_{7}^{11} + 24 T_{7}^{10} + 11 T_{7}^{9} + 189 T_{7}^{8} + 342 T_{7}^{7} + 1857 T_{7}^{6} + \cdots + 361 \) Copy content Toggle raw display
\( T_{11}^{12} - 3 T_{11}^{11} + 54 T_{11}^{10} - 15 T_{11}^{9} + 1755 T_{11}^{8} - 486 T_{11}^{7} + \cdots + 927369 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 927369 \) Copy content Toggle raw display
$13$ \( T^{12} - 9 T^{11} + \cdots + 47961 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + \cdots + 19855936 \) Copy content Toggle raw display
$19$ \( T^{12} - 9 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 178409449 \) Copy content Toggle raw display
$29$ \( T^{12} - 27 T^{11} + \cdots + 42094144 \) Copy content Toggle raw display
$31$ \( T^{12} - 12 T^{11} + \cdots + 36675136 \) Copy content Toggle raw display
$37$ \( (T^{6} + 21 T^{5} + \cdots + 85483)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 27 T^{11} + \cdots + 998001 \) Copy content Toggle raw display
$43$ \( T^{12} + 27 T^{11} + \cdots + 16192576 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots + 26286129 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 16922627569 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 289034001 \) Copy content Toggle raw display
$61$ \( T^{12} + 9 T^{11} + \cdots + 21827584 \) Copy content Toggle raw display
$67$ \( T^{12} - 42 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1455575104 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 504990784 \) Copy content Toggle raw display
$79$ \( T^{12} + 57 T^{11} + \cdots + 1478656 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17207142976 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 17461243881 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 399252114496 \) Copy content Toggle raw display
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