Properties

Label 462.6.a.n
Level $462$
Weight $6$
Character orbit 462.a
Self dual yes
Analytic conductor $74.097$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,6,Mod(1,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

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: yes
Analytic conductor: \(74.0973247536\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 1511x + 16911 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + (\beta_{2} - 19) q^{5} - 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + (\beta_{2} - 19) q^{5} - 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9} + (4 \beta_{2} - 76) q^{10} - 121 q^{11} - 144 q^{12} + (5 \beta_{2} + 2 \beta_1 - 185) q^{13} - 196 q^{14} + ( - 9 \beta_{2} + 171) q^{15} + 256 q^{16} + (6 \beta_{2} - 17 \beta_1 - 267) q^{17} + 324 q^{18} + (25 \beta_{2} - 7 \beta_1 - 22) q^{19} + (16 \beta_{2} - 304) q^{20} + 441 q^{21} - 484 q^{22} + ( - 10 \beta_{2} + 57 \beta_1 + 313) q^{23} - 576 q^{24} + ( - 45 \beta_{2} + 30 \beta_1 - 154) q^{25} + (20 \beta_{2} + 8 \beta_1 - 740) q^{26} - 729 q^{27} - 784 q^{28} + (81 \beta_{2} - 42 \beta_1 + 1543) q^{29} + ( - 36 \beta_{2} + 684) q^{30} + (80 \beta_{2} - 27 \beta_1 + 1003) q^{31} + 1024 q^{32} + 1089 q^{33} + (24 \beta_{2} - 68 \beta_1 - 1068) q^{34} + ( - 49 \beta_{2} + 931) q^{35} + 1296 q^{36} + ( - 115 \beta_{2} - 8 \beta_1 + 1625) q^{37} + (100 \beta_{2} - 28 \beta_1 - 88) q^{38} + ( - 45 \beta_{2} - 18 \beta_1 + 1665) q^{39} + (64 \beta_{2} - 1216) q^{40} + (26 \beta_{2} + 70 \beta_1 - 2374) q^{41} + 1764 q^{42} + ( - 30 \beta_{2} - 155 \beta_1 + 3461) q^{43} - 1936 q^{44} + (81 \beta_{2} - 1539) q^{45} + ( - 40 \beta_{2} + 228 \beta_1 + 1252) q^{46} + (35 \beta_{2} - 193 \beta_1 + 2518) q^{47} - 2304 q^{48} + 2401 q^{49} + ( - 180 \beta_{2} + 120 \beta_1 - 616) q^{50} + ( - 54 \beta_{2} + 153 \beta_1 + 2403) q^{51} + (80 \beta_{2} + 32 \beta_1 - 2960) q^{52} + ( - 4 \beta_{2} + 193 \beta_1 + 9313) q^{53} - 2916 q^{54} + ( - 121 \beta_{2} + 2299) q^{55} - 3136 q^{56} + ( - 225 \beta_{2} + 63 \beta_1 + 198) q^{57} + (324 \beta_{2} - 168 \beta_1 + 6172) q^{58} + ( - 201 \beta_{2} + 41 \beta_1 + 1364) q^{59} + ( - 144 \beta_{2} + 2736) q^{60} + ( - 56 \beta_{2} + 406 \beta_1 + 16856) q^{61} + (320 \beta_{2} - 108 \beta_1 + 4012) q^{62} - 3969 q^{63} + 4096 q^{64} + ( - 233 \beta_{2} + 128 \beta_1 + 18309) q^{65} + 4356 q^{66} + (501 \beta_{2} - 253 \beta_1 + 36644) q^{67} + (96 \beta_{2} - 272 \beta_1 - 4272) q^{68} + (90 \beta_{2} - 513 \beta_1 - 2817) q^{69} + ( - 196 \beta_{2} + 3724) q^{70} + ( - 240 \beta_{2} - 450 \beta_1 + 35170) q^{71} + 5184 q^{72} + ( - 165 \beta_{2} + 48 \beta_1 + 24919) q^{73} + ( - 460 \beta_{2} - 32 \beta_1 + 6500) q^{74} + (405 \beta_{2} - 270 \beta_1 + 1386) q^{75} + (400 \beta_{2} - 112 \beta_1 - 352) q^{76} + 5929 q^{77} + ( - 180 \beta_{2} - 72 \beta_1 + 6660) q^{78} + (516 \beta_{2} + 22 \beta_1 + 22486) q^{79} + (256 \beta_{2} - 4864) q^{80} + 6561 q^{81} + (104 \beta_{2} + 280 \beta_1 - 9496) q^{82} + ( - 1304 \beta_{2} + 1089 \beta_1 + 16203) q^{83} + 7056 q^{84} + ( - 1120 \beta_{2} + 367 \beta_1 + 5909) q^{85} + ( - 120 \beta_{2} - 620 \beta_1 + 13844) q^{86} + ( - 729 \beta_{2} + 378 \beta_1 - 13887) q^{87} - 7744 q^{88} + ( - 384 \beta_{2} - 1144 \beta_1 + 41330) q^{89} + (324 \beta_{2} - 6156) q^{90} + ( - 245 \beta_{2} - 98 \beta_1 + 9065) q^{91} + ( - 160 \beta_{2} + 912 \beta_1 + 5008) q^{92} + ( - 720 \beta_{2} + 243 \beta_1 - 9027) q^{93} + (140 \beta_{2} - 772 \beta_1 + 10072) q^{94} + ( - 959 \beta_{2} + 827 \beta_1 + 59564) q^{95} - 9216 q^{96} + (1316 \beta_{2} + 973 \beta_1 - 18151) q^{97} + 9604 q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} - 27 q^{3} + 48 q^{4} - 56 q^{5} - 108 q^{6} - 147 q^{7} + 192 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{2} - 27 q^{3} + 48 q^{4} - 56 q^{5} - 108 q^{6} - 147 q^{7} + 192 q^{8} + 243 q^{9} - 224 q^{10} - 363 q^{11} - 432 q^{12} - 552 q^{13} - 588 q^{14} + 504 q^{15} + 768 q^{16} - 778 q^{17} + 972 q^{18} - 34 q^{19} - 896 q^{20} + 1323 q^{21} - 1452 q^{22} + 872 q^{23} - 1728 q^{24} - 537 q^{25} - 2208 q^{26} - 2187 q^{27} - 2352 q^{28} + 4752 q^{29} + 2016 q^{30} + 3116 q^{31} + 3072 q^{32} + 3267 q^{33} - 3112 q^{34} + 2744 q^{35} + 3888 q^{36} + 4768 q^{37} - 136 q^{38} + 4968 q^{39} - 3584 q^{40} - 7166 q^{41} + 5292 q^{42} + 10508 q^{43} - 5808 q^{44} - 4536 q^{45} + 3488 q^{46} + 7782 q^{47} - 6912 q^{48} + 7203 q^{49} - 2148 q^{50} + 7002 q^{51} - 8832 q^{52} + 27742 q^{53} - 8748 q^{54} + 6776 q^{55} - 9408 q^{56} + 306 q^{57} + 19008 q^{58} + 3850 q^{59} + 8064 q^{60} + 50106 q^{61} + 12464 q^{62} - 11907 q^{63} + 12288 q^{64} + 54566 q^{65} + 13068 q^{66} + 110686 q^{67} - 12448 q^{68} - 7848 q^{69} + 10976 q^{70} + 105720 q^{71} + 15552 q^{72} + 74544 q^{73} + 19072 q^{74} + 4833 q^{75} - 544 q^{76} + 17787 q^{77} + 19872 q^{78} + 67952 q^{79} - 14336 q^{80} + 19683 q^{81} - 28664 q^{82} + 46216 q^{83} + 21168 q^{84} + 16240 q^{85} + 42032 q^{86} - 42768 q^{87} - 23232 q^{88} + 124750 q^{89} - 18144 q^{90} + 27048 q^{91} + 13952 q^{92} - 28044 q^{93} + 31128 q^{94} + 176906 q^{95} - 27648 q^{96} - 54110 q^{97} + 28812 q^{98} - 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 1511x + 16911 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 20\nu - 1011 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} - 10\beta _1 + 1001 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
12.3326
−43.1276
31.7949
4.00000 −9.00000 16.0000 −80.2253 −36.0000 −49.0000 64.0000 81.0000 −320.901
1.2 4.00000 −9.00000 16.0000 −20.3563 −36.0000 −49.0000 64.0000 81.0000 −81.4254
1.3 4.00000 −9.00000 16.0000 44.5816 −36.0000 −49.0000 64.0000 81.0000 178.327
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.6.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.6.a.n 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 56T_{5}^{2} - 2851T_{5} - 72806 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{3} \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 56 T^{2} - 2851 T - 72806 \) Copy content Toggle raw display
$7$ \( (T + 49)^{3} \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 552 T^{2} + \cdots - 41954486 \) Copy content Toggle raw display
$17$ \( T^{3} + 778 T^{2} + \cdots - 1189290768 \) Copy content Toggle raw display
$19$ \( T^{3} + 34 T^{2} + \cdots + 1076826276 \) Copy content Toggle raw display
$23$ \( T^{3} - 872 T^{2} + \cdots + 34291256000 \) Copy content Toggle raw display
$29$ \( T^{3} - 4752 T^{2} + \cdots + 91376488442 \) Copy content Toggle raw display
$31$ \( T^{3} - 3116 T^{2} + \cdots + 64837779776 \) Copy content Toggle raw display
$37$ \( T^{3} - 4768 T^{2} + \cdots + 130024130874 \) Copy content Toggle raw display
$41$ \( T^{3} + 7166 T^{2} + \cdots - 71994789696 \) Copy content Toggle raw display
$43$ \( T^{3} - 10508 T^{2} + \cdots + 226075983344 \) Copy content Toggle raw display
$47$ \( T^{3} - 7782 T^{2} + \cdots - 593934816200 \) Copy content Toggle raw display
$53$ \( T^{3} - 27742 T^{2} + \cdots + 2245924957512 \) Copy content Toggle raw display
$59$ \( T^{3} - 3850 T^{2} + \cdots - 251464184100 \) Copy content Toggle raw display
$61$ \( T^{3} - 50106 T^{2} + \cdots + 21398159861960 \) Copy content Toggle raw display
$67$ \( T^{3} - 110686 T^{2} + \cdots + 52537797876 \) Copy content Toggle raw display
$71$ \( T^{3} - 105720 T^{2} + \cdots + 24201305600000 \) Copy content Toggle raw display
$73$ \( T^{3} - 74544 T^{2} + \cdots - 13207717817050 \) Copy content Toggle raw display
$79$ \( T^{3} - 67952 T^{2} + \cdots + 9666171542016 \) Copy content Toggle raw display
$83$ \( T^{3} - 46216 T^{2} + \cdots + 13636328729760 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 292611013505880 \) Copy content Toggle raw display
$97$ \( T^{3} + 54110 T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
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