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Properties

Label 570.6.d.a
Level $570$
Weight $6$
Character orbit 570.d
Analytic conductor $91.419$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,6,Mod(229,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.229");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 570.d (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: \(91.4187772934\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 153194 x^{16} + 9054192701 x^{14} + 265982739083444 x^{12} + \cdots + 99\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta_{6} q^{2} - 9 \beta_{6} q^{3} - 16 q^{4} + ( - \beta_{14} - \beta_{6} - 3) q^{5} + 36 q^{6} + (\beta_{16} + \beta_{15} + \cdots - 2 \beta_{6}) q^{7}+ \cdots - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_{6} q^{2} - 9 \beta_{6} q^{3} - 16 q^{4} + ( - \beta_{14} - \beta_{6} - 3) q^{5} + 36 q^{6} + (\beta_{16} + \beta_{15} + \cdots - 2 \beta_{6}) q^{7}+ \cdots + (81 \beta_{14} + 81 \beta_{13} + \cdots + 12231) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 288 q^{4} - 60 q^{5} + 648 q^{6} - 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 288 q^{4} - 60 q^{5} + 648 q^{6} - 1458 q^{9} + 80 q^{10} - 2692 q^{11} - 180 q^{15} + 4608 q^{16} - 6498 q^{19} + 960 q^{20} - 10368 q^{24} + 6100 q^{25} - 6272 q^{26} + 7268 q^{29} - 2160 q^{30} - 17744 q^{31} + 24640 q^{34} + 6530 q^{35} + 23328 q^{36} + 14112 q^{39} - 1280 q^{40} - 39308 q^{41} + 43072 q^{44} + 4860 q^{45} - 38560 q^{46} - 3862 q^{49} - 12800 q^{50} - 55440 q^{51} - 52488 q^{54} + 68890 q^{55} + 57416 q^{59} + 2880 q^{60} - 208424 q^{61} - 73728 q^{64} + 118340 q^{65} - 96912 q^{66} + 86760 q^{69} + 75760 q^{70} - 83008 q^{71} + 12224 q^{74} + 28800 q^{75} + 103968 q^{76} + 302784 q^{79} - 15360 q^{80} + 118098 q^{81} + 62930 q^{85} + 90816 q^{86} + 91764 q^{89} - 6480 q^{90} - 319080 q^{91} + 5632 q^{94} + 21660 q^{95} + 165888 q^{96} + 218052 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 153194 x^{16} + 9054192701 x^{14} + 265982739083444 x^{12} + \cdots + 99\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29\!\cdots\!91 \nu^{16} + \cdots + 67\!\cdots\!04 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 58\!\cdots\!71 \nu^{16} + \cdots - 12\!\cdots\!76 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46\!\cdots\!37 \nu^{16} + \cdots - 28\!\cdots\!72 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 88\!\cdots\!23 \nu^{16} + \cdots - 39\!\cdots\!88 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28\!\cdots\!43 \nu^{16} + \cdots - 16\!\cdots\!08 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!53 \nu^{17} + \cdots + 76\!\cdots\!48 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!97 \nu^{17} + \cdots - 32\!\cdots\!20 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!97 \nu^{17} + \cdots - 32\!\cdots\!20 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!79 \nu^{17} + \cdots + 49\!\cdots\!60 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\!\cdots\!79 \nu^{17} + \cdots - 49\!\cdots\!60 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!93 \nu^{17} + \cdots - 61\!\cdots\!92 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 96\!\cdots\!39 \nu^{17} + \cdots + 50\!\cdots\!32 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!05 \nu^{17} + \cdots + 30\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23\!\cdots\!05 \nu^{17} + \cdots + 30\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 76\!\cdots\!29 \nu^{17} + \cdots - 56\!\cdots\!24 \nu ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!86 \nu^{17} + \cdots + 20\!\cdots\!16 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 36\!\cdots\!41 \nu^{17} + \cdots + 39\!\cdots\!96 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 5 \beta_{14} - 5 \beta_{13} + 56 \beta_{10} - 56 \beta_{9} + 158 \beta_{8} + 158 \beta_{7} + \cdots - 17079 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 8935 \beta_{17} - 13464 \beta_{16} - 22399 \beta_{15} + 26551 \beta_{14} + 19140 \beta_{13} + \cdots - 175931 \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 35305 \beta_{14} - 35305 \beta_{13} - 1644685 \beta_{10} + 1644685 \beta_{9} - 8964851 \beta_{8} + \cdots + 596321848 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 600078960 \beta_{17} + 156423516 \beta_{16} + 685341706 \beta_{15} - 734119692 \beta_{14} + \cdots + 15870741016 \beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2067416850 \beta_{14} - 2067416850 \beta_{13} + 47635900651 \beta_{10} - 47635900651 \beta_{9} + \cdots - 25740987947746 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 33309841936170 \beta_{17} + 3062589626456 \beta_{16} - 24363878462094 \beta_{15} + \cdots - 11\!\cdots\!94 \beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 622477564903240 \beta_{14} + 622477564903240 \beta_{13} + \cdots + 11\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17\!\cdots\!70 \beta_{17} + \cdots + 70\!\cdots\!14 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 62\!\cdots\!20 \beta_{14} + \cdots - 57\!\cdots\!58 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 90\!\cdots\!30 \beta_{17} + \cdots - 42\!\cdots\!62 \beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 46\!\cdots\!60 \beta_{14} + \cdots + 27\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 46\!\cdots\!30 \beta_{17} + \cdots + 24\!\cdots\!42 \beta_{6} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 30\!\cdots\!40 \beta_{14} + \cdots - 13\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 23\!\cdots\!90 \beta_{17} + \cdots - 13\!\cdots\!10 \beta_{6} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 18\!\cdots\!80 \beta_{14} + \cdots + 66\!\cdots\!34 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 12\!\cdots\!90 \beta_{17} + \cdots + 73\!\cdots\!50 \beta_{6} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
132.851i
− 43.6380i
− 129.128i
95.7754i
− 225.548i
25.3788i
35.6403i
− 102.355i
211.023i
− 132.851i
43.6380i
129.128i
− 95.7754i
225.548i
− 25.3788i
− 35.6403i
102.355i
− 211.023i
− 4.00000i 9.00000i −16.0000 −52.0945 + 20.2773i 36.0000 132.851i 64.0000i −81.0000 81.1090 + 208.378i
229.2 − 4.00000i 9.00000i −16.0000 −49.5668 + 25.8483i 36.0000 − 43.6380i 64.0000i −81.0000 103.393 + 198.267i
229.3 − 4.00000i 9.00000i −16.0000 −35.4185 − 43.2496i 36.0000 − 129.128i 64.0000i −81.0000 −172.998 + 141.674i
229.4 − 4.00000i 9.00000i −16.0000 −35.3071 − 43.3406i 36.0000 95.7754i 64.0000i −81.0000 −173.362 + 141.229i
229.5 − 4.00000i 9.00000i −16.0000 −23.1471 + 50.8843i 36.0000 − 225.548i 64.0000i −81.0000 203.537 + 92.5885i
229.6 − 4.00000i 9.00000i −16.0000 27.2119 + 48.8315i 36.0000 25.3788i 64.0000i −81.0000 195.326 − 108.848i
229.7 − 4.00000i 9.00000i −16.0000 32.9223 − 45.1788i 36.0000 35.6403i 64.0000i −81.0000 −180.715 − 131.689i
229.8 − 4.00000i 9.00000i −16.0000 51.9843 − 20.5581i 36.0000 − 102.355i 64.0000i −81.0000 −82.2325 − 207.937i
229.9 − 4.00000i 9.00000i −16.0000 53.4155 + 16.4858i 36.0000 211.023i 64.0000i −81.0000 65.9432 − 213.662i
229.10 4.00000i − 9.00000i −16.0000 −52.0945 − 20.2773i 36.0000 − 132.851i − 64.0000i −81.0000 81.1090 − 208.378i
229.11 4.00000i − 9.00000i −16.0000 −49.5668 − 25.8483i 36.0000 43.6380i − 64.0000i −81.0000 103.393 − 198.267i
229.12 4.00000i − 9.00000i −16.0000 −35.4185 + 43.2496i 36.0000 129.128i − 64.0000i −81.0000 −172.998 − 141.674i
229.13 4.00000i − 9.00000i −16.0000 −35.3071 + 43.3406i 36.0000 − 95.7754i − 64.0000i −81.0000 −173.362 − 141.229i
229.14 4.00000i − 9.00000i −16.0000 −23.1471 − 50.8843i 36.0000 225.548i − 64.0000i −81.0000 203.537 − 92.5885i
229.15 4.00000i − 9.00000i −16.0000 27.2119 − 48.8315i 36.0000 − 25.3788i − 64.0000i −81.0000 195.326 + 108.848i
229.16 4.00000i − 9.00000i −16.0000 32.9223 + 45.1788i 36.0000 − 35.6403i − 64.0000i −81.0000 −180.715 + 131.689i
229.17 4.00000i − 9.00000i −16.0000 51.9843 + 20.5581i 36.0000 102.355i − 64.0000i −81.0000 −82.2325 + 207.937i
229.18 4.00000i − 9.00000i −16.0000 53.4155 − 16.4858i 36.0000 − 211.023i − 64.0000i −81.0000 65.9432 + 213.662i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.6.d.a ✓ 18
5.b even 2 1 inner 570.6.d.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.d.a ✓ 18 1.a even 1 1 trivial
570.6.d.a ✓ 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 153194 T_{7}^{16} + 9054192701 T_{7}^{14} + 265982739083444 T_{7}^{12} + \cdots + 99\!\cdots\!24 \) acting on \(S_{6}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{9} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{9} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 99\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T + 361)^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 42\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
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