Properties

Label 462.4.a.j
Level $462$
Weight $4$
Character orbit 462.a
Self dual yes
Analytic conductor $27.259$
Analytic rank $0$
Dimension $2$
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] = [462,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(27.2588824227\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta + 1) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta + 1) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9} + ( - 2 \beta - 2) q^{10} + 11 q^{11} - 12 q^{12} + (3 \beta - 1) q^{13} - 14 q^{14} + ( - 3 \beta - 3) q^{15} + 16 q^{16} + (2 \beta + 56) q^{17} - 18 q^{18} + (13 \beta - 45) q^{19} + (4 \beta + 4) q^{20} - 21 q^{21} - 22 q^{22} + ( - 14 \beta + 22) q^{23} + 24 q^{24} + (3 \beta - 80) q^{25} + ( - 6 \beta + 2) q^{26} - 27 q^{27} + 28 q^{28} + (27 \beta + 39) q^{29} + (6 \beta + 6) q^{30} + ( - 32 \beta - 24) q^{31} - 32 q^{32} - 33 q^{33} + ( - 4 \beta - 112) q^{34} + (7 \beta + 7) q^{35} + 36 q^{36} + (25 \beta - 87) q^{37} + ( - 26 \beta + 90) q^{38} + ( - 9 \beta + 3) q^{39} + ( - 8 \beta - 8) q^{40} + ( - 70 \beta + 8) q^{41} + 42 q^{42} + (34 \beta - 102) q^{43} + 44 q^{44} + (9 \beta + 9) q^{45} + (28 \beta - 44) q^{46} + ( - 31 \beta + 131) q^{47} - 48 q^{48} + 49 q^{49} + ( - 6 \beta + 160) q^{50} + ( - 6 \beta - 168) q^{51} + (12 \beta - 4) q^{52} + (32 \beta + 326) q^{53} + 54 q^{54} + (11 \beta + 11) q^{55} - 56 q^{56} + ( - 39 \beta + 135) q^{57} + ( - 54 \beta - 78) q^{58} + (97 \beta + 175) q^{59} + ( - 12 \beta - 12) q^{60} + ( - 12 \beta + 626) q^{61} + (64 \beta + 48) q^{62} + 63 q^{63} + 64 q^{64} + (5 \beta + 131) q^{65} + 66 q^{66} + ( - 49 \beta + 145) q^{67} + (8 \beta + 224) q^{68} + (42 \beta - 66) q^{69} + ( - 14 \beta - 14) q^{70} + 528 q^{71} - 72 q^{72} + ( - 95 \beta + 141) q^{73} + ( - 50 \beta + 174) q^{74} + ( - 9 \beta + 240) q^{75} + (52 \beta - 180) q^{76} + 77 q^{77} + (18 \beta - 6) q^{78} + ( - 32 \beta + 48) q^{79} + (16 \beta + 16) q^{80} + 81 q^{81} + (140 \beta - 16) q^{82} + (36 \beta + 624) q^{83} - 84 q^{84} + (60 \beta + 144) q^{85} + ( - 68 \beta + 204) q^{86} + ( - 81 \beta - 117) q^{87} - 88 q^{88} + (56 \beta + 938) q^{89} + ( - 18 \beta - 18) q^{90} + (21 \beta - 7) q^{91} + ( - 56 \beta + 88) q^{92} + (96 \beta + 72) q^{93} + (62 \beta - 262) q^{94} + ( - 19 \beta + 527) q^{95} + 96 q^{96} + (132 \beta + 446) q^{97} - 98 q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 3 q^{5} + 12 q^{6} + 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 3 q^{5} + 12 q^{6} + 14 q^{7} - 16 q^{8} + 18 q^{9} - 6 q^{10} + 22 q^{11} - 24 q^{12} + q^{13} - 28 q^{14} - 9 q^{15} + 32 q^{16} + 114 q^{17} - 36 q^{18} - 77 q^{19} + 12 q^{20} - 42 q^{21} - 44 q^{22} + 30 q^{23} + 48 q^{24} - 157 q^{25} - 2 q^{26} - 54 q^{27} + 56 q^{28} + 105 q^{29} + 18 q^{30} - 80 q^{31} - 64 q^{32} - 66 q^{33} - 228 q^{34} + 21 q^{35} + 72 q^{36} - 149 q^{37} + 154 q^{38} - 3 q^{39} - 24 q^{40} - 54 q^{41} + 84 q^{42} - 170 q^{43} + 88 q^{44} + 27 q^{45} - 60 q^{46} + 231 q^{47} - 96 q^{48} + 98 q^{49} + 314 q^{50} - 342 q^{51} + 4 q^{52} + 684 q^{53} + 108 q^{54} + 33 q^{55} - 112 q^{56} + 231 q^{57} - 210 q^{58} + 447 q^{59} - 36 q^{60} + 1240 q^{61} + 160 q^{62} + 126 q^{63} + 128 q^{64} + 267 q^{65} + 132 q^{66} + 241 q^{67} + 456 q^{68} - 90 q^{69} - 42 q^{70} + 1056 q^{71} - 144 q^{72} + 187 q^{73} + 298 q^{74} + 471 q^{75} - 308 q^{76} + 154 q^{77} + 6 q^{78} + 64 q^{79} + 48 q^{80} + 162 q^{81} + 108 q^{82} + 1284 q^{83} - 168 q^{84} + 348 q^{85} + 340 q^{86} - 315 q^{87} - 176 q^{88} + 1932 q^{89} - 54 q^{90} + 7 q^{91} + 120 q^{92} + 240 q^{93} - 462 q^{94} + 1035 q^{95} + 192 q^{96} + 1024 q^{97} - 196 q^{98} + 198 q^{99}+O(q^{100}) \) 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
−6.15207
7.15207
−2.00000 −3.00000 4.00000 −5.15207 6.00000 7.00000 −8.00000 9.00000 10.3041
1.2 −2.00000 −3.00000 4.00000 8.15207 6.00000 7.00000 −8.00000 9.00000 −16.3041
\(n\): 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 462.4.a.j 2
3.b odd 2 1 1386.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.j 2 1.a even 1 1 trivial
1386.4.a.y 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} - 42 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 42 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 398 \) Copy content Toggle raw display
$17$ \( T^{2} - 114T + 3072 \) Copy content Toggle raw display
$19$ \( T^{2} + 77T - 5996 \) Copy content Toggle raw display
$23$ \( T^{2} - 30T - 8448 \) Copy content Toggle raw display
$29$ \( T^{2} - 105T - 29502 \) Copy content Toggle raw display
$31$ \( T^{2} + 80T - 43712 \) Copy content Toggle raw display
$37$ \( T^{2} + 149T - 22106 \) Copy content Toggle raw display
$41$ \( T^{2} + 54T - 216096 \) Copy content Toggle raw display
$43$ \( T^{2} + 170T - 43928 \) Copy content Toggle raw display
$47$ \( T^{2} - 231T - 29184 \) Copy content Toggle raw display
$53$ \( T^{2} - 684T + 71652 \) Copy content Toggle raw display
$59$ \( T^{2} - 447T - 366396 \) Copy content Toggle raw display
$61$ \( T^{2} - 1240 T + 378028 \) Copy content Toggle raw display
$67$ \( T^{2} - 241T - 91724 \) Copy content Toggle raw display
$71$ \( (T - 528)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 187T - 390614 \) Copy content Toggle raw display
$79$ \( T^{2} - 64T - 44288 \) Copy content Toggle raw display
$83$ \( T^{2} - 1284 T + 354816 \) Copy content Toggle raw display
$89$ \( T^{2} - 1932 T + 794388 \) Copy content Toggle raw display
$97$ \( T^{2} - 1024 T - 508868 \) Copy content Toggle raw display
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