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Properties

Label 570.4.i.o
Level $570$
Weight $4$
Character orbit 570.i
Analytic conductor $33.631$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 1359 x^{10} - 1622 x^{9} + 1487219 x^{8} - 2474518 x^{7} + 474233266 x^{6} + \cdots + 12446784000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + 3 \beta_{4} q^{3} + ( - 4 \beta_{4} - 4) q^{4} + 5 \beta_{4} q^{5} + (6 \beta_{4} + 6) q^{6} + ( - \beta_{5} - 1) q^{7} - 8 q^{8} + ( - 9 \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{4} q^{2} + 3 \beta_{4} q^{3} + ( - 4 \beta_{4} - 4) q^{4} + 5 \beta_{4} q^{5} + (6 \beta_{4} + 6) q^{6} + ( - \beta_{5} - 1) q^{7} - 8 q^{8} + ( - 9 \beta_{4} - 9) q^{9} + (10 \beta_{4} + 10) q^{10} + (\beta_{7} + \beta_{5} - 3) q^{11} + 12 q^{12} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \cdots - 3) q^{13}+ \cdots + (9 \beta_{8} - 9 \beta_{7} + 27 \beta_{4} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 18 q^{3} - 24 q^{4} - 30 q^{5} + 36 q^{6} - 8 q^{7} - 96 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 18 q^{3} - 24 q^{4} - 30 q^{5} + 36 q^{6} - 8 q^{7} - 96 q^{8} - 54 q^{9} + 60 q^{10} - 38 q^{11} + 144 q^{12} - 20 q^{13} - 8 q^{14} - 90 q^{15} - 96 q^{16} + 15 q^{17} - 216 q^{18} + 34 q^{19} + 240 q^{20} + 12 q^{21} - 38 q^{22} + 108 q^{23} + 144 q^{24} - 150 q^{25} - 80 q^{26} + 324 q^{27} + 16 q^{28} - 161 q^{29} - 360 q^{30} + 384 q^{31} + 192 q^{32} + 57 q^{33} - 30 q^{34} + 20 q^{35} - 216 q^{36} + 528 q^{37} - 98 q^{38} + 120 q^{39} + 240 q^{40} + 36 q^{41} - 24 q^{42} - 260 q^{43} + 76 q^{44} + 540 q^{45} + 432 q^{46} + 357 q^{47} - 288 q^{48} + 1316 q^{49} - 600 q^{50} + 45 q^{51} - 80 q^{52} + 589 q^{53} + 324 q^{54} + 95 q^{55} + 64 q^{56} + 147 q^{57} - 644 q^{58} - 793 q^{59} - 360 q^{60} - 635 q^{61} + 384 q^{62} + 36 q^{63} + 768 q^{64} + 200 q^{65} - 114 q^{66} - 1740 q^{67} - 120 q^{68} - 648 q^{69} - 40 q^{70} + 1131 q^{71} + 432 q^{72} - 682 q^{73} + 528 q^{74} + 900 q^{75} - 332 q^{76} - 5396 q^{77} + 120 q^{78} + 275 q^{79} - 480 q^{80} - 486 q^{81} - 72 q^{82} + 3470 q^{83} - 96 q^{84} + 75 q^{85} + 520 q^{86} + 966 q^{87} + 304 q^{88} - 699 q^{89} + 540 q^{90} + 768 q^{91} + 432 q^{92} - 576 q^{93} + 1428 q^{94} + 245 q^{95} - 1152 q^{96} - 2440 q^{97} + 1316 q^{98} + 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 1359 x^{10} - 1622 x^{9} + 1487219 x^{8} - 2474518 x^{7} + 474233266 x^{6} + \cdots + 12446784000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29\!\cdots\!97 \nu^{11} + \cdots + 94\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\!\cdots\!11 \nu^{11} + \cdots - 40\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72\!\cdots\!67 \nu^{11} + \cdots - 22\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\!\cdots\!71 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!11 \nu^{11} + \cdots + 92\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!82 \nu^{11} + \cdots + 22\!\cdots\!00 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!53 \nu^{11} + \cdots + 52\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!21 \nu^{11} + \cdots - 30\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!71 \nu^{11} + \cdots - 51\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 30\!\cdots\!19 \nu^{11} + \cdots - 77\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -4\beta_{11} + 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - \beta_{5} + 451\beta_{4} + 5\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 24\beta_{11} + 9\beta_{10} + 48\beta_{9} + 97\beta_{7} - 123\beta_{6} + 849\beta_{5} + 15\beta_{3} + 24\beta _1 - 57 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1848 \beta_{11} - 10141 \beta_{10} - 1848 \beta_{9} - 273 \beta_{8} + 273 \beta_{7} + 113 \beta_{6} + \cdots - 375083 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 47132 \beta_{11} + 23566 \beta_{10} - 23566 \beta_{9} - 133253 \beta_{8} - 790769 \beta_{5} + \cdots - 767203 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1802616 \beta_{11} + 8747741 \beta_{10} + 3605232 \beta_{9} + 633095 \beta_{7} + 265355 \beta_{6} + \cdots + 353668375 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21949874 \beta_{11} - 37507163 \beta_{10} - 21949874 \beta_{9} + 147302341 \beta_{8} - 147302341 \beta_{7} + \cdots + 1179259021 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3562589008 \beta_{11} + 1781294504 \beta_{10} - 1781294504 \beta_{9} - 845330135 \beta_{8} + \cdots + 4903980909 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20087023690 \beta_{11} + 7766327545 \beta_{10} + 40174047380 \beta_{9} + 152820747861 \beta_{7} + \cdots - 1659863994861 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1768532836376 \beta_{11} - 10835826176757 \beta_{10} - 1768532836376 \beta_{9} + 807429150279 \beta_{8} + \cdots - 351205832681551 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36309197536020 \beta_{11} + 18154598768010 \beta_{10} - 18154598768010 \beta_{9} + \cdots - 769049495936967 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\)

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
−15.5777 + 26.9814i
−9.64201 + 16.7005i
−1.05369 + 1.82504i
2.50649 − 4.34137i
8.59041 − 14.8790i
16.1765 − 28.0185i
−15.5777 − 26.9814i
−9.64201 − 16.7005i
−1.05369 − 1.82504i
2.50649 + 4.34137i
8.59041 + 14.8790i
16.1765 + 28.0185i
1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i 3.00000 − 5.19615i −32.1554 −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 −20.2840 −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 −3.10738 −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 4.01299 −8.00000 −4.50000 + 7.79423i 5.00000 − 8.66025i
121.5 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i 3.00000 − 5.19615i 16.1808 −8.00000 −4.50000 + 7.79423i 5.00000 − 8.66025i
121.6 1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −2.50000 − 4.33013i 3.00000 − 5.19615i 31.3530 −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 −32.1554 −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 −20.2840 −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 −3.10738 −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 4.01299 −8.00000 −4.50000 − 7.79423i 5.00000 + 8.66025i
391.5 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i 3.00000 + 5.19615i 16.1808 −8.00000 −4.50000 − 7.79423i 5.00000 + 8.66025i
391.6 1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −2.50000 + 4.33013i 3.00000 + 5.19615i 31.3530 −8.00000 −4.50000 − 7.79423i 5.00000 + 8.66025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
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.o ✓ 12
19.c even 3 1 inner 570.4.i.o ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.i.o ✓ 12 1.a even 1 1 trivial
570.4.i.o ✓ 12 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 4T_{7}^{5} - 1350T_{7}^{4} - 3254T_{7}^{3} + 351501T_{7}^{2} - 244790T_{7} - 4126192 \) acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + \cdots - 4126192)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 19 T^{5} + \cdots - 1991692944)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{6} - 192 T^{5} + \cdots - 715057041240)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 10490803305408)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
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