Properties

Label 570.2.u.i
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: \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} - \beta_{4} q^{3} - \beta_{2} q^{4} + \beta_1 q^{5} + \beta_{7} q^{6} - \beta_{5} q^{7} - \beta_{11} q^{8} + ( - \beta_{7} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{2} - \beta_{4} q^{3} - \beta_{2} q^{4} + \beta_1 q^{5} + \beta_{7} q^{6} - \beta_{5} q^{7} - \beta_{11} q^{8} + ( - \beta_{7} + \beta_{2}) q^{9} + ( - \beta_{7} + \beta_{2}) q^{10} + ( - 2 \beta_{11} + \beta_{7} + \cdots - \beta_1) q^{11}+ \cdots + ( - 2 \beta_{11} + \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) 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 \( 12 q - 3 q^{7} - 6 q^{8} - 9 q^{11} - 6 q^{12} - 3 q^{13} + 3 q^{14} + 9 q^{17} + 12 q^{18} - 3 q^{19} + 12 q^{20} + 3 q^{21} - 6 q^{22} + 12 q^{23} - 9 q^{26} - 6 q^{27} + 3 q^{28} - 3 q^{29} - 6 q^{30} - 12 q^{31} + 3 q^{33} - 9 q^{34} - 6 q^{35} + 42 q^{37} + 18 q^{39} + 21 q^{41} + 3 q^{42} + 9 q^{43} - 6 q^{44} - 6 q^{45} - 3 q^{46} + 3 q^{49} - 6 q^{50} - 3 q^{52} - 18 q^{53} + 3 q^{55} + 6 q^{56} + 6 q^{58} - 15 q^{59} + 9 q^{61} + 12 q^{62} - 6 q^{63} - 6 q^{64} - 9 q^{65} + 3 q^{66} + 18 q^{67} - 6 q^{68} - 3 q^{69} - 6 q^{70} + 12 q^{71} - 9 q^{73} + 12 q^{75} + 9 q^{76} - 54 q^{77} + 6 q^{78} - 9 q^{79} + 21 q^{82} + 15 q^{83} - 3 q^{84} + 9 q^{85} - 27 q^{86} - 3 q^{87} - 9 q^{88} + 3 q^{89} - 51 q^{91} + 3 q^{92} - 15 q^{93} + 12 q^{96} - 48 q^{97} - 15 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} + \cdots - 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + \cdots + 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{10} + 211\nu^{8} + 1387\nu^{6} + 3742\nu^{4} + 4204\nu^{2} + 2109 ) / 304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + \cdots + 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{11} + 11 \nu^{10} - 5 \nu^{9} + 211 \nu^{8} - 829 \nu^{7} + 1387 \nu^{6} - 6690 \nu^{5} + \cdots + 2109 ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} - 23 \nu^{10} - \nu^{9} - 455 \nu^{8} + 239 \nu^{7} - 2983 \nu^{6} + 2078 \nu^{5} + \cdots + 551 ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} + \cdots - 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 7 \nu^{10} + 53 \nu^{9} - 155 \nu^{8} + 793 \nu^{7} - 1311 \nu^{6} + 4686 \nu^{5} + \cdots - 3933 ) / 608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 11 \nu^{10} - 167 \nu^{9} - 211 \nu^{8} - 1479 \nu^{7} - 1387 \nu^{6} - 5822 \nu^{5} + \cdots - 1805 ) / 608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13 \nu^{11} + 49 \nu^{10} + 277 \nu^{9} + 1009 \nu^{8} + 2109 \nu^{7} + 7201 \nu^{6} + 6882 \nu^{5} + \cdots + 8303 ) / 608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 197\nu^{9} + 1545\nu^{7} + 5238\nu^{5} + 7304\nu^{3} + 2979\nu + 152 ) / 304 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} + \beta_{8} - 3\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{11} - 7 \beta_{9} - 4 \beta_{8} + 21 \beta_{7} - 4 \beta_{6} - 7 \beta_{5} + 13 \beta_{4} + \cdots + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 2 \beta_{10} + \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 8 \beta_{6} - \beta_{5} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 52 \beta_{11} + 64 \beta_{9} + 16 \beta_{8} - 141 \beta_{7} + 16 \beta_{6} + 46 \beta_{5} - 109 \beta_{4} + \cdots - 58 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{11} - 28 \beta_{10} - 14 \beta_{9} + 87 \beta_{8} + 6 \beta_{7} - 59 \beta_{6} + 14 \beta_{5} + \cdots - 167 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 424 \beta_{11} - 574 \beta_{9} - 52 \beta_{8} + 1020 \beta_{7} - 52 \beta_{6} - 310 \beta_{5} + \cdots + 499 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 149 \beta_{11} + 298 \beta_{10} + 149 \beta_{9} - 733 \beta_{8} + 99 \beta_{7} + 435 \beta_{6} + \cdots + 1253 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3610 \beta_{11} + 5026 \beta_{9} + 19 \beta_{8} - 7815 \beta_{7} + 19 \beta_{6} + 2128 \beta_{5} + \cdots - 4318 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1433 \beta_{11} - 2866 \beta_{10} - 1433 \beta_{9} + 6110 \beta_{8} - 1677 \beta_{7} - 3244 \beta_{6} + \cdots - 9805 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 31045 \beta_{11} - 43375 \beta_{9} + 2114 \beta_{8} + 61974 \beta_{7} + 2114 \beta_{6} - 14785 \beta_{5} + \cdots + 37210 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
0.918492i
2.88811i
0.918492i
2.88811i
2.57727i
1.89323i
2.57727i
1.89323i
0.728740i
2.01431i
0.728740i
2.01431i
−0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i 0.173648 0.984808i −0.814143 + 1.41014i −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 0.487791 0.844879i −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 −0.814143 1.41014i −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 0.487791 + 0.844879i −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 −2.15664 3.73541i −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 0.716948 + 1.24179i −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 −2.15664 + 3.73541i −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 0.716948 1.24179i −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 −1.21767 2.10906i −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 1.48371 + 2.56987i −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 −1.21767 + 2.10906i −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 1.48371 2.56987i −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.i 12
19.e even 9 1 inner 570.2.u.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.i 12 1.a even 1 1 trivial
570.2.u.i 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{12} + 3 T_{7}^{11} + 24 T_{7}^{10} + 17 T_{7}^{9} + 261 T_{7}^{8} + 180 T_{7}^{7} + 1503 T_{7}^{6} + \cdots + 5041 \) Copy content Toggle raw display
\( T_{11}^{12} + 9 T_{11}^{11} + 66 T_{11}^{10} + 217 T_{11}^{9} + 675 T_{11}^{8} + 882 T_{11}^{7} + \cdots + 1 \) 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
$7$ \( T^{12} + 3 T^{11} + \cdots + 5041 \) Copy content Toggle raw display
$11$ \( T^{12} + 9 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 1682209 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + \cdots + 23104 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 104366656 \) Copy content Toggle raw display
$37$ \( (T^{6} - 21 T^{5} + \cdots + 73)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots + 1923769 \) Copy content Toggle raw display
$43$ \( T^{12} - 9 T^{11} + \cdots + 46656 \) Copy content Toggle raw display
$47$ \( T^{12} + 24 T^{10} + \cdots + 4915089 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1285437609 \) Copy content Toggle raw display
$59$ \( T^{12} + 15 T^{11} + \cdots + 23409 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 427331584 \) Copy content Toggle raw display
$67$ \( T^{12} - 18 T^{11} + \cdots + 60341824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1254293056 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13809070144 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1430654976 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 60559303744 \) Copy content Toggle raw display
$89$ \( T^{12} - 3 T^{11} + \cdots + 86322681 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 51623475264 \) Copy content Toggle raw display
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