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Properties

Label 570.4.i.k
Level $570$
Weight $4$
Character orbit 570.i
Analytic conductor $33.631$
Analytic rank $0$
Dimension $8$
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] = [570,4,Mod(121,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.121");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 570.i (of order \(3\), degree \(2\), 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: \(33.6310887033\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 447x^{6} - 1570x^{5} + 192148x^{4} - 434016x^{3} + 4484160x^{2} + 7838208x + 60466176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + 4 \beta_{2} q^{4} + ( - 5 \beta_{2} - 5) q^{5} + 6 \beta_{2} q^{6} + ( - \beta_{7} - 12) q^{7} + 8 q^{8} + 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + 4 \beta_{2} q^{4} + ( - 5 \beta_{2} - 5) q^{5} + 6 \beta_{2} q^{6} + ( - \beta_{7} - 12) q^{7} + 8 q^{8} + 9 \beta_{2} q^{9} + 10 \beta_{2} q^{10} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{11} + 12 q^{12} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + (9 \beta_{7} - 9 \beta_{6} - 9 \beta_{4} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 12 q^{3} - 16 q^{4} - 20 q^{5} - 24 q^{6} - 100 q^{7} + 64 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 12 q^{3} - 16 q^{4} - 20 q^{5} - 24 q^{6} - 100 q^{7} + 64 q^{8} - 36 q^{9} - 40 q^{10} + 96 q^{12} + 59 q^{13} + 100 q^{14} - 60 q^{15} - 64 q^{16} - 81 q^{17} + 144 q^{18} + q^{19} + 160 q^{20} + 150 q^{21} - 185 q^{23} - 96 q^{24} - 100 q^{25} - 236 q^{26} + 216 q^{27} + 200 q^{28} + 403 q^{29} + 240 q^{30} - 534 q^{31} - 128 q^{32} - 162 q^{34} + 250 q^{35} - 144 q^{36} - 1060 q^{37} - 238 q^{38} - 354 q^{39} - 160 q^{40} - 263 q^{41} + 300 q^{42} + 859 q^{43} + 360 q^{45} + 740 q^{46} - 471 q^{47} - 192 q^{48} + 1476 q^{49} + 400 q^{50} - 243 q^{51} + 236 q^{52} + 328 q^{53} - 216 q^{54} - 800 q^{56} - 357 q^{57} - 1612 q^{58} + 97 q^{59} - 240 q^{60} - 406 q^{61} + 534 q^{62} + 450 q^{63} + 512 q^{64} - 590 q^{65} + 1561 q^{67} + 648 q^{68} + 1110 q^{69} + 500 q^{70} + 1021 q^{71} - 288 q^{72} + 183 q^{73} + 1060 q^{74} + 600 q^{75} + 472 q^{76} + 1314 q^{77} + 354 q^{78} + 340 q^{79} - 320 q^{80} - 324 q^{81} - 526 q^{82} + 1078 q^{83} - 1200 q^{84} - 405 q^{85} + 1718 q^{86} - 2418 q^{87} + 334 q^{89} - 360 q^{90} + 355 q^{91} - 740 q^{92} + 801 q^{93} + 1884 q^{94} - 595 q^{95} + 768 q^{96} + 1656 q^{97} - 1476 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 447x^{6} - 1570x^{5} + 192148x^{4} - 434016x^{3} + 4484160x^{2} + 7838208x + 60466176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 51911999 \nu^{7} + 434402045 \nu^{6} - 22667241135 \nu^{5} + 244941721808 \nu^{4} + \cdots - 348319137835008 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7083149 \nu^{7} - 9952267 \nu^{6} - 3026664507 \nu^{5} + 3841226642 \nu^{4} + \cdots - 58128149408256 ) / 31147862513472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 466404667 \nu^{7} - 4925430421 \nu^{6} + 257010574263 \nu^{5} - 2689997719636 \nu^{4} + \cdots - 15\!\cdots\!24 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 898932401 \nu^{7} + 4123237303 \nu^{6} + 384118249143 \nu^{5} - 487495475258 \nu^{4} + \cdots - 95\!\cdots\!08 ) / 22\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7078768775 \nu^{7} + 92291741963 \nu^{6} - 3018002984733 \nu^{5} + 47525590753034 \nu^{4} + \cdots - 42\!\cdots\!48 ) / 67\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1588059935 \nu^{7} - 2149603993 \nu^{6} - 678585843705 \nu^{5} + 861212736230 \nu^{4} + \cdots - 23\!\cdots\!96 ) / 11\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{7} - 4\beta_{6} + 2\beta_{4} + 3\beta_{3} + 224\beta_{2} - 3\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 70\beta_{7} - 4\beta_{5} - 443\beta_{3} + 498 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1788\beta_{6} - 1788\beta_{5} - 822\beta_{4} - 91560\beta_{2} + 1903\beta _1 - 91560 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32414\beta_{7} - 460\beta_{6} + 32414\beta_{4} + 190923\beta_{3} + 387560\beta_{2} - 190923\beta _1 + 32414 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 389206\beta_{7} + 761852\beta_{5} - 1081031\beta_{3} + 38889714 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1276716\beta_{6} - 1276716\beta_{5} - 14351694\beta_{4} - 221448552\beta_{2} + 82544083\beta _1 - 221448552 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
2.83562 + 4.91143i
9.94418 + 17.2238i
−10.6635 − 18.4697i
−1.61629 − 2.79950i
2.83562 − 4.91143i
9.94418 − 17.2238i
−10.6635 + 18.4697i
−1.61629 + 2.79950i
−1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i −3.00000 + 5.19615i −35.0435 8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
121.2 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i −3.00000 + 5.19615i −21.4303 8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
121.3 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i −3.00000 + 5.19615i −10.9360 8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
121.4 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i −3.00000 + 5.19615i 17.4098 8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
391.1 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i −3.00000 − 5.19615i −35.0435 8.00000 −4.50000 − 7.79423i −5.00000 − 8.66025i
391.2 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i −3.00000 − 5.19615i −21.4303 8.00000 −4.50000 − 7.79423i −5.00000 − 8.66025i
391.3 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i −3.00000 − 5.19615i −10.9360 8.00000 −4.50000 − 7.79423i −5.00000 − 8.66025i
391.4 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i −3.00000 − 5.19615i 17.4098 8.00000 −4.50000 − 7.79423i −5.00000 − 8.66025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.4.i.k ✓ 8
19.c even 3 1 inner 570.4.i.k ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.i.k ✓ 8 1.a even 1 1 trivial
570.4.i.k ✓ 8 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 50T_{7}^{3} + 195T_{7}^{2} - 15614T_{7} - 142984 \) acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 50 T^{3} + \cdots - 142984)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 5007 T^{2} + \cdots + 1498176)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 689643880704 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 128280088166400 \) Copy content Toggle raw display
$31$ \( (T^{4} + 267 T^{3} + \cdots - 1562912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 530 T^{3} + \cdots - 10075818400)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{4} - 539 T^{3} + \cdots + 100437334656)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
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