Properties

Label 570.4.i.n
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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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: \(12\)
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: \( x^{12} - 2 x^{11} + 1297 x^{10} - 10878 x^{9} + 1304767 x^{8} - 10349528 x^{7} + 532040002 x^{6} - 5629645296 x^{5} + 166789197316 x^{4} + \cdots + 254354800896 \) 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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} - 3 \beta_{3} q^{3} + (4 \beta_{3} - 4) q^{4} + 5 \beta_{3} q^{5} + (6 \beta_{3} - 6) q^{6} + ( - \beta_{2} + 1) q^{7} + 8 q^{8} + (9 \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} - 3 \beta_{3} q^{3} + (4 \beta_{3} - 4) q^{4} + 5 \beta_{3} q^{5} + (6 \beta_{3} - 6) q^{6} + ( - \beta_{2} + 1) q^{7} + 8 q^{8} + (9 \beta_{3} - 9) q^{9} + ( - 10 \beta_{3} + 10) q^{10} + ( - \beta_{5} + \beta_{2} - 8) q^{11} + 12 q^{12} + (\beta_{11} - \beta_{6} + 5 \beta_{3} + \beta_1 - 5) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{14} + ( - 15 \beta_{3} + 15) q^{15} - 16 \beta_{3} q^{16} + ( - \beta_{7} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{17} + 18 q^{18} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_1 + 6) q^{19} - 20 q^{20} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{21} + (2 \beta_{10} + 2 \beta_{5} + 16 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{22} + ( - \beta_{10} + \beta_{7} + \beta_{4} - 13 \beta_{3} - \beta_1 + 13) q^{23} - 24 \beta_{3} q^{24} + (25 \beta_{3} - 25) q^{25} + (2 \beta_{6} - 2 \beta_{2} + 10) q^{26} + 27 q^{27} + (4 \beta_{3} + 4 \beta_1 - 4) q^{28} + ( - 2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - \beta_{8} + 3 \beta_{6} + 18 \beta_{3} + \cdots - 18) q^{29}+ \cdots + ( - 9 \beta_{10} - 72 \beta_{3} - 9 \beta_1 + 72) q^{99}+O(q^{100}) \) 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 \( 12 q - 12 q^{2} - 18 q^{3} - 24 q^{4} + 30 q^{5} - 36 q^{6} + 8 q^{7} + 96 q^{8} - 54 q^{9} + 60 q^{10} - 86 q^{11} + 144 q^{12} - 28 q^{13} - 8 q^{14} + 90 q^{15} - 96 q^{16} - 15 q^{17} + 216 q^{18} + 36 q^{19} - 240 q^{20} - 12 q^{21} + 86 q^{22} + 74 q^{23} - 144 q^{24} - 150 q^{25} + 112 q^{26} + 324 q^{27} - 16 q^{28} - 121 q^{29} - 360 q^{30} + 176 q^{31} - 192 q^{32} + 129 q^{33} - 30 q^{34} + 20 q^{35} - 216 q^{36} - 332 q^{37} - 162 q^{38} + 168 q^{39} + 240 q^{40} - 184 q^{41} - 24 q^{42} - 604 q^{43} + 172 q^{44} - 540 q^{45} - 296 q^{46} + 589 q^{47} - 288 q^{48} + 1068 q^{49} + 600 q^{50} - 45 q^{51} - 112 q^{52} - 767 q^{53} - 324 q^{54} - 215 q^{55} + 64 q^{56} - 243 q^{57} + 484 q^{58} + 527 q^{59} + 360 q^{60} + 107 q^{61} - 176 q^{62} - 36 q^{63} + 768 q^{64} - 280 q^{65} + 258 q^{66} + 266 q^{67} + 120 q^{68} - 444 q^{69} + 40 q^{70} - 545 q^{71} - 432 q^{72} - 1146 q^{73} + 332 q^{74} + 900 q^{75} + 180 q^{76} - 3264 q^{77} - 168 q^{78} + 199 q^{79} + 480 q^{80} - 486 q^{81} - 368 q^{82} + 3558 q^{83} + 96 q^{84} + 75 q^{85} - 1208 q^{86} + 726 q^{87} - 688 q^{88} - 1189 q^{89} + 540 q^{90} - 3132 q^{91} + 296 q^{92} - 264 q^{93} - 2356 q^{94} + 405 q^{95} + 1152 q^{96} + 1736 q^{97} - 1068 q^{98} + 387 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 1297 x^{10} - 10878 x^{9} + 1304767 x^{8} - 10349528 x^{7} + 532040002 x^{6} - 5629645296 x^{5} + 166789197316 x^{4} + \cdots + 254354800896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!09 \nu^{11} + \cdots + 18\!\cdots\!64 ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!33 \nu^{11} + \cdots + 25\!\cdots\!56 ) / 25\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\!\cdots\!35 \nu^{11} + \cdots + 12\!\cdots\!92 ) / 41\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!80 \nu^{11} + \cdots + 74\!\cdots\!12 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!73 \nu^{11} + \cdots - 39\!\cdots\!76 ) / 41\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\!\cdots\!61 \nu^{11} + \cdots - 86\!\cdots\!88 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\!\cdots\!13 \nu^{11} + \cdots + 31\!\cdots\!32 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!17 \nu^{11} + \cdots + 44\!\cdots\!76 ) / 73\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54\!\cdots\!39 \nu^{11} + \cdots + 16\!\cdots\!84 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!75 \nu^{11} + \cdots - 17\!\cdots\!52 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 9 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 435 \beta_{3} + 6 \beta_{2} - 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 12\beta_{9} + 12\beta_{8} + 75\beta_{6} + 77\beta_{5} - 21\beta_{4} - 706\beta_{2} + 2349 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7807 \beta_{11} + 4179 \beta_{10} + 2532 \beta_{9} - 1266 \beta_{8} - 1933 \beta_{7} - 6541 \beta_{6} - 1933 \beta_{4} + 302031 \beta_{3} + 6610 \beta _1 - 302031 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 84591 \beta_{11} - 115587 \beta_{10} + 7682 \beta_{9} - 15364 \beta_{8} - 19371 \beta_{7} - 7682 \beta_{6} - 115587 \beta_{5} - 2724207 \beta_{3} + 561526 \beta_{2} - 561526 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 810426 \beta_{9} - 810426 \beta_{8} + 5222835 \beta_{6} + 4548999 \beta_{5} + 2016193 \beta_{4} - 6725958 \beta_{2} + 234849291 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 100332543 \beta_{11} + 129685271 \beta_{10} - 9597772 \beta_{9} + 4798886 \beta_{8} + 11241903 \beta_{7} - 105131429 \beta_{6} + 11241903 \beta_{4} + 2814323547 \beta_{3} + \cdots - 2814323547 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6080698135 \beta_{11} - 4564924935 \beta_{10} - 560990290 \beta_{9} + 1121980580 \beta_{8} + 1829427985 \beta_{7} + 560990290 \beta_{6} - 4564924935 \beta_{5} + \cdots - 6784500046 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2951466718 \beta_{9} + 2951466718 \beta_{8} + 112600116875 \beta_{6} + 131916183183 \beta_{5} - 3361744839 \beta_{4} - 403054141534 \beta_{2} + \cdots + 2837901684195 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5448192276423 \beta_{11} + 4412354517831 \beta_{10} + 828490245668 \beta_{9} - 414245122834 \beta_{8} - 1589616848977 \beta_{7} + \cdots - 165669724138059 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 108098951444415 \beta_{11} - 128656731262415 \beta_{10} + 1604889301390 \beta_{9} - 3209778602780 \beta_{8} + 2691811079385 \beta_{7} + \cdots - 353224847642878 \beta_1 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
13.6473 23.6377i
9.09640 15.7554i
4.59215 7.95384i
0.0812068 0.140654i
−11.1488 + 19.3104i
−15.2682 + 26.4452i
13.6473 + 23.6377i
9.09640 + 15.7554i
4.59215 + 7.95384i
0.0812068 + 0.140654i
−11.1488 19.3104i
−15.2682 26.4452i
−1.00000 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i −3.00000 + 5.19615i −26.2945 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 −17.1928 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 −8.18431 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 0.837586 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 23.2977 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 31.5363 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 −26.2945 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 −17.1928 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 −8.18431 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 0.837586 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 23.2977 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 31.5363 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.n 12
19.c even 3 1 inner 570.4.i.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.i.n 12 1.a even 1 1 trivial
570.4.i.n 12 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 4T_{7}^{5} - 1288T_{7}^{4} - 1560T_{7}^{3} + 392979T_{7}^{2} + 2391124T_{7} - 2276916 \) 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
$7$ \( (T^{6} - 4 T^{5} - 1288 T^{4} + \cdots - 2276916)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 43 T^{5} - 5937 T^{4} + \cdots - 3783540528)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 28 T^{11} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + 15 T^{11} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{12} - 36 T^{11} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} - 74 T^{11} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + 121 T^{11} + \cdots + 68\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{6} - 88 T^{5} + \cdots - 73102333248616)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 166 T^{5} + \cdots - 7829761512672)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 184 T^{11} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{12} + 604 T^{11} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} - 589 T^{11} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + 767 T^{11} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} - 527 T^{11} + \cdots + 67\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} - 107 T^{11} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{12} - 266 T^{11} + \cdots + 72\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + 545 T^{11} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{12} + 1146 T^{11} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} - 199 T^{11} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{6} - 1779 T^{5} + \cdots - 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 1189 T^{11} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} - 1736 T^{11} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
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