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Properties

Label 570.4.d.e
Level $570$
Weight $4$
Character orbit 570.d
Analytic conductor $33.631$
Analytic rank $0$
Dimension $14$
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(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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.229");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(33.6310887033\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 399 x^{12} + 60535 x^{10} + 4347445 x^{8} + 149845100 x^{6} + 2267443016 x^{4} + \cdots + 15750250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} - 3 \beta_{3} q^{3} - 4 q^{4} + ( - \beta_{7} + \beta_{3} + 1) q^{5} - 6 q^{6} + ( - \beta_{7} - \beta_{5}) q^{7} + 8 \beta_{3} q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} - 3 \beta_{3} q^{3} - 4 q^{4} + ( - \beta_{7} + \beta_{3} + 1) q^{5} - 6 q^{6} + ( - \beta_{7} - \beta_{5}) q^{7} + 8 \beta_{3} q^{8} - 9 q^{9} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{10} + (\beta_{9} + \beta_{7} + \beta_1 - 8) q^{11} + 12 \beta_{3} q^{12} + ( - \beta_{13} - \beta_{11} - \beta_{6} + \cdots - 1) q^{13}+ \cdots + ( - 9 \beta_{9} - 9 \beta_{7} + \cdots + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 56 q^{4} + 16 q^{5} - 84 q^{6} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 56 q^{4} + 16 q^{5} - 84 q^{6} - 126 q^{9} + 36 q^{10} - 112 q^{11} + 54 q^{15} + 224 q^{16} + 266 q^{19} - 64 q^{20} + 336 q^{24} - 118 q^{25} - 24 q^{26} + 128 q^{29} - 96 q^{30} + 356 q^{31} + 216 q^{34} - 908 q^{35} + 504 q^{36} - 36 q^{39} - 144 q^{40} + 2220 q^{41} + 448 q^{44} - 144 q^{45} - 184 q^{46} + 666 q^{49} - 112 q^{50} + 324 q^{51} + 756 q^{54} - 1216 q^{55} - 464 q^{59} - 216 q^{60} + 2276 q^{61} - 896 q^{64} - 1196 q^{65} + 672 q^{66} - 276 q^{69} - 992 q^{70} + 3356 q^{71} - 1088 q^{74} - 168 q^{75} - 1064 q^{76} - 12 q^{79} + 256 q^{80} + 1134 q^{81} - 2548 q^{85} + 512 q^{86} - 2464 q^{89} - 324 q^{90} + 2824 q^{91} - 392 q^{94} + 304 q^{95} - 1344 q^{96} + 1008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 399 x^{12} + 60535 x^{10} + 4347445 x^{8} + 149845100 x^{6} + 2267443016 x^{4} + \cdots + 15750250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26276300489 \nu^{12} + 11048343064367 \nu^{10} + \cdots + 25\!\cdots\!50 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 63928429023 \nu^{12} + 20368110233594 \nu^{10} + \cdots - 33\!\cdots\!50 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11004803 \nu^{13} + 4008470709 \nu^{11} + 512843889701 \nu^{9} + 24277145375619 \nu^{7} + \cdots - 32\!\cdots\!60 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 125559362410 \nu^{12} + 46475487022118 \nu^{10} + \cdots + 29\!\cdots\!50 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15155621360377 \nu^{13} - 11770530939576 \nu^{12} + \cdots - 53\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19324986198767 \nu^{13} - 13190702845478 \nu^{12} + \cdots - 12\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42402605518047 \nu^{13} + 11770530939576 \nu^{12} + \cdots + 53\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 42402605518047 \nu^{13} - 197871220730696 \nu^{12} + \cdots - 56\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42402605518047 \nu^{13} + 265554661348600 \nu^{12} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21370650169989 \nu^{13} + 13016103054280 \nu^{12} + \cdots - 29\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14389556719606 \nu^{13} + 710085952951 \nu^{12} + \cdots + 58\!\cdots\!00 ) / 29\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15673451843357 \nu^{13} + 2922174950276 \nu^{12} + \cdots + 49\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 114726821148059 \nu^{13} + 11770530939576 \nu^{12} + \cdots + 53\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + 2\beta_{10} - \beta_{7} - \beta_{5} - 2\beta_{3} - 2\beta_{2} + 3\beta _1 - 1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{11} - 4 \beta_{10} - 5 \beta_{9} + 2 \beta_{7} - 3 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} + \cdots - 573 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35 \beta_{13} + 5 \beta_{12} + 101 \beta_{11} - 162 \beta_{10} + 96 \beta_{7} - 50 \beta_{6} + \cdots + 101 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 368 \beta_{11} + 439 \beta_{10} + 745 \beta_{9} - 250 \beta_{8} + 188 \beta_{7} + 368 \beta_{5} + \cdots + 54868 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5525 \beta_{13} - 2225 \beta_{12} - 9311 \beta_{11} + 15822 \beta_{10} - 11836 \beta_{7} + \cdots - 9311 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 25958 \beta_{11} - 44219 \beta_{10} - 96755 \beta_{9} + 46400 \beta_{8} - 72978 \beta_{7} + \cdots - 5909078 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 766735 \beta_{13} + 419255 \beta_{12} + 896061 \beta_{11} - 1664332 \beta_{10} + 1569556 \beta_{7} + \cdots + 896061 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 635598 \beta_{11} + 4042929 \beta_{10} + 12115695 \beta_{9} - 6938650 \beta_{8} + 14375818 \beta_{7} + \cdots + 668417798 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 103723925 \beta_{13} - 65514325 \beta_{12} - 90025721 \beta_{11} + 182465592 \beta_{10} + \cdots - 90025721 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 203416162 \beta_{11} - 334323959 \beta_{10} - 1503511755 \beta_{9} + 974663800 \beta_{8} + \cdots - 77923806758 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13895323435 \beta_{13} + 9538327255 \beta_{12} + 9358336121 \beta_{11} - 20611237952 \beta_{10} + \cdots + 9358336121 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 54830160122 \beta_{11} + 23712157069 \beta_{10} + 186819747395 \beta_{9} - 133405865550 \beta_{8} + \cdots + 9289976305278 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1850088230825 \beta_{13} - 1340652176925 \beta_{12} - 1000758642481 \beta_{11} + \cdots - 1000758642481 ) / 10 \) 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
11.3987i
6.75000i
4.58625i
− 1.34896i
− 2.72989i
− 9.72275i
− 9.93331i
− 11.3987i
− 6.75000i
− 4.58625i
1.34896i
2.72989i
9.72275i
9.93331i
− 2.00000i − 3.00000i −4.00000 −10.3987 + 4.10706i −6.00000 23.0708i 8.00000i −9.00000 8.21412 + 20.7973i
229.2 − 2.00000i − 3.00000i −4.00000 −5.75000 + 9.58840i −6.00000 12.2072i 8.00000i −9.00000 19.1768 + 11.5000i
229.3 − 2.00000i − 3.00000i −4.00000 −3.58625 − 10.5896i −6.00000 − 7.77374i 8.00000i −9.00000 −21.1791 + 7.17251i
229.4 − 2.00000i − 3.00000i −4.00000 2.34896 + 10.9308i −6.00000 − 16.5270i 8.00000i −9.00000 21.8616 − 4.69793i
229.5 − 2.00000i − 3.00000i −4.00000 3.72989 − 10.5398i −6.00000 − 27.3118i 8.00000i −9.00000 −21.0797 − 7.45978i
229.6 − 2.00000i − 3.00000i −4.00000 10.7227 + 3.16586i −6.00000 17.4895i 8.00000i −9.00000 6.33173 − 21.4455i
229.7 − 2.00000i − 3.00000i −4.00000 10.9333 + 2.33726i −6.00000 − 1.15487i 8.00000i −9.00000 4.67452 − 21.8666i
229.8 2.00000i 3.00000i −4.00000 −10.3987 − 4.10706i −6.00000 − 23.0708i − 8.00000i −9.00000 8.21412 − 20.7973i
229.9 2.00000i 3.00000i −4.00000 −5.75000 − 9.58840i −6.00000 − 12.2072i − 8.00000i −9.00000 19.1768 − 11.5000i
229.10 2.00000i 3.00000i −4.00000 −3.58625 + 10.5896i −6.00000 7.77374i − 8.00000i −9.00000 −21.1791 − 7.17251i
229.11 2.00000i 3.00000i −4.00000 2.34896 − 10.9308i −6.00000 16.5270i − 8.00000i −9.00000 21.8616 + 4.69793i
229.12 2.00000i 3.00000i −4.00000 3.72989 + 10.5398i −6.00000 27.3118i − 8.00000i −9.00000 −21.0797 + 7.45978i
229.13 2.00000i 3.00000i −4.00000 10.7227 − 3.16586i −6.00000 − 17.4895i − 8.00000i −9.00000 6.33173 + 21.4455i
229.14 2.00000i 3.00000i −4.00000 10.9333 − 2.33726i −6.00000 1.15487i − 8.00000i −9.00000 4.67452 + 21.8666i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.14
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 570.4.d.e ✓ 14
5.b even 2 1 inner 570.4.d.e ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.d.e ✓ 14 1.a even 1 1 trivial
570.4.d.e ✓ 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 2068 T_{7}^{12} + 1621436 T_{7}^{10} + 611234420 T_{7}^{8} + 115493323600 T_{7}^{6} + \cdots + 398401600000000 \) acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{7} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 476837158203125 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 398401600000000 \) Copy content Toggle raw display
$11$ \( (T^{7} + 56 T^{6} + \cdots + 13713693440)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T - 19)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 1448975603360)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 31629512681600)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 14481453960000)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 48\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
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