Properties

Label 483.4.a.a
Level $483$
Weight $4$
Character orbit 483.a
Self dual yes
Analytic conductor $28.498$
Analytic rank $1$
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] = [483,4,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(28.4979225328\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3877.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 13x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 4 \beta_1 + 1) q^{4} + (\beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 3 \beta_1 - 3) q^{6} - 7 q^{7} + ( - 4 \beta_{2} - 8 \beta_1 - 27) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 4 \beta_1 + 1) q^{4} + (\beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 3 \beta_1 - 3) q^{6} - 7 q^{7} + ( - 4 \beta_{2} - 8 \beta_1 - 27) q^{8} + 9 q^{9} + (2 \beta_{2} + 2 \beta_1 + 13) q^{10} + ( - 9 \beta_{2} + 7 \beta_1 + 2) q^{11} + (3 \beta_{2} + 12 \beta_1 + 3) q^{12} + (4 \beta_{2} + \beta_1 + 9) q^{13} + (7 \beta_1 + 7) q^{14} + (3 \beta_{2} - 6 \beta_1 + 3) q^{15} + (31 \beta_1 + 91) q^{16} + ( - 7 \beta_{2} - \beta_1 - 26) q^{17} + ( - 9 \beta_1 - 9) q^{18} + ( - \beta_{2} + 15 \beta_1 + 6) q^{19} + ( - 10 \beta_{2} - 9 \beta_1 - 41) q^{20} - 21 q^{21} + ( - 7 \beta_{2} + 4 \beta_1 - 40) q^{22} - 23 q^{23} + ( - 12 \beta_{2} - 24 \beta_1 - 81) q^{24} + (8 \beta_{2} - 9 \beta_1 - 82) q^{25} + ( - \beta_{2} - 24 \beta_1 - 25) q^{26} + 27 q^{27} + ( - 7 \beta_{2} - 28 \beta_1 - 7) q^{28} + ( - 8 \beta_{2} + 6 \beta_1 - 113) q^{29} + (6 \beta_{2} + 6 \beta_1 + 39) q^{30} + (9 \beta_{2} + 7 \beta_1 - 156) q^{31} + (\beta_{2} - 120 \beta_1 - 123) q^{32} + ( - 27 \beta_{2} + 21 \beta_1 + 6) q^{33} + (\beta_{2} + 50 \beta_1 + 48) q^{34} + ( - 7 \beta_{2} + 14 \beta_1 - 7) q^{35} + (9 \beta_{2} + 36 \beta_1 + 9) q^{36} + (34 \beta_{2} + 38 \beta_1 - 85) q^{37} + ( - 15 \beta_{2} - 48 \beta_1 - 124) q^{38} + (12 \beta_{2} + 3 \beta_1 + 27) q^{39} + ( - 7 \beta_{2} + 82 \beta_1 + 29) q^{40} + ( - 40 \beta_{2} + 62 \beta_1 + 51) q^{41} + (21 \beta_1 + 21) q^{42} + (34 \beta_{2} + 33 \beta_1 - 129) q^{43} + (68 \beta_{2} - 7 \beta_1 + 6) q^{44} + (9 \beta_{2} - 18 \beta_1 + 9) q^{45} + (23 \beta_1 + 23) q^{46} + (51 \beta_{2} - 113 \beta_1 - 3) q^{47} + (93 \beta_1 + 273) q^{48} + 49 q^{49} + (9 \beta_{2} + 85 \beta_1 + 138) q^{50} + ( - 21 \beta_{2} - 3 \beta_1 - 78) q^{51} + ( - 8 \beta_{2} + 92 \beta_1 + 147) q^{52} + (94 \beta_{2} - 49 \beta_1 - 6) q^{53} + ( - 27 \beta_1 - 27) q^{54} + ( - 28 \beta_{2} + 59 \beta_1 - 222) q^{55} + (28 \beta_{2} + 56 \beta_1 + 189) q^{56} + ( - 3 \beta_{2} + 45 \beta_1 + 18) q^{57} + ( - 6 \beta_{2} + 119 \beta_1 + 81) q^{58} + (10 \beta_{2} + 97 \beta_1 - 564) q^{59} + ( - 30 \beta_{2} - 27 \beta_1 - 123) q^{60} + (9 \beta_{2} + 64 \beta_1 - 174) q^{61} + ( - 7 \beta_{2} + 108 \beta_1 + 82) q^{62} - 63 q^{63} + (120 \beta_{2} + 232 \beta_1 + 353) q^{64} + (10 \beta_{2} - 46 \beta_1 + 51) q^{65} + ( - 21 \beta_{2} + 12 \beta_1 - 120) q^{66} + ( - 110 \beta_{2} - 87 \beta_1 - 14) q^{67} + (6 \beta_{2} - 193 \beta_1 - 242) q^{68} - 69 q^{69} + ( - 14 \beta_{2} - 14 \beta_1 - 91) q^{70} + ( - 87 \beta_{2} + 52 \beta_1 + 430) q^{71} + ( - 36 \beta_{2} - 72 \beta_1 - 243) q^{72} + ( - 177 \beta_{2} + 141 \beta_1 - 102) q^{73} + ( - 38 \beta_{2} - 131 \beta_1 - 287) q^{74} + (24 \beta_{2} - 27 \beta_1 - 246) q^{75} + (56 \beta_{2} + 193 \beta_1 + 490) q^{76} + (63 \beta_{2} - 49 \beta_1 - 14) q^{77} + ( - 3 \beta_{2} - 72 \beta_1 - 75) q^{78} + (121 \beta_{2} - 9 \beta_1 + 294) q^{79} + ( - 2 \beta_{2} - 182 \beta_1 - 343) q^{80} + 81 q^{81} + ( - 62 \beta_{2} - 117 \beta_1 - 467) q^{82} + (99 \beta_{2} - 151 \beta_1 - 598) q^{83} + ( - 21 \beta_{2} - 84 \beta_1 - 21) q^{84} + ( - 30 \beta_{2} + 101 \beta_1 - 110) q^{85} + ( - 33 \beta_{2} - 72 \beta_1 - 203) q^{86} + ( - 24 \beta_{2} + 18 \beta_1 - 339) q^{87} + (63 \beta_{2} - 221 \beta_1 + 234) q^{88} + ( - 287 \beta_{2} + 54 \beta_1 - 226) q^{89} + (18 \beta_{2} + 18 \beta_1 + 117) q^{90} + ( - 28 \beta_{2} - 7 \beta_1 - 63) q^{91} + ( - 23 \beta_{2} - 92 \beta_1 - 23) q^{92} + (27 \beta_{2} + 21 \beta_1 - 468) q^{93} + (113 \beta_{2} + 189 \beta_1 + 805) q^{94} + ( - 40 \beta_{2} - 5 \beta_1 - 218) q^{95} + (3 \beta_{2} - 360 \beta_1 - 369) q^{96} + (20 \beta_{2} - 170 \beta_1 - 549) q^{97} + ( - 49 \beta_1 - 49) q^{98} + ( - 81 \beta_{2} + 63 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} + 9 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} - 21 q^{7} - 93 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} + 9 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} - 21 q^{7} - 93 q^{8} + 27 q^{9} + 43 q^{10} + 4 q^{11} + 24 q^{12} + 32 q^{13} + 28 q^{14} + 6 q^{15} + 304 q^{16} - 86 q^{17} - 36 q^{18} + 32 q^{19} - 142 q^{20} - 63 q^{21} - 123 q^{22} - 69 q^{23} - 279 q^{24} - 247 q^{25} - 100 q^{26} + 81 q^{27} - 56 q^{28} - 341 q^{29} + 129 q^{30} - 452 q^{31} - 488 q^{32} + 12 q^{33} + 195 q^{34} - 14 q^{35} + 72 q^{36} - 183 q^{37} - 435 q^{38} + 96 q^{39} + 162 q^{40} + 175 q^{41} + 84 q^{42} - 320 q^{43} + 79 q^{44} + 18 q^{45} + 92 q^{46} - 71 q^{47} + 912 q^{48} + 147 q^{49} + 508 q^{50} - 258 q^{51} + 525 q^{52} + 27 q^{53} - 108 q^{54} - 635 q^{55} + 651 q^{56} + 96 q^{57} + 356 q^{58} - 1585 q^{59} - 426 q^{60} - 449 q^{61} + 347 q^{62} - 189 q^{63} + 1411 q^{64} + 117 q^{65} - 369 q^{66} - 239 q^{67} - 913 q^{68} - 207 q^{69} - 301 q^{70} + 1255 q^{71} - 837 q^{72} - 342 q^{73} - 1030 q^{74} - 741 q^{75} + 1719 q^{76} - 28 q^{77} - 300 q^{78} + 994 q^{79} - 1213 q^{80} + 243 q^{81} - 1580 q^{82} - 1846 q^{83} - 168 q^{84} - 259 q^{85} - 714 q^{86} - 1023 q^{87} + 544 q^{88} - 911 q^{89} + 387 q^{90} - 224 q^{91} - 184 q^{92} - 1356 q^{93} + 2717 q^{94} - 699 q^{95} - 1464 q^{96} - 1797 q^{97} - 196 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 13x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 8 \) 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
4.43745
−0.881781
−2.55567
−5.43745 3.00000 21.5659 −5.05882 −16.3124 −7.00000 −73.7640 9.00000 27.5071
1.2 −0.118219 3.00000 −7.98602 −2.69534 −0.354656 −7.00000 1.88985 9.00000 0.318639
1.3 1.55567 3.00000 −5.57988 9.75415 4.66702 −7.00000 −21.1259 9.00000 15.1743
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)
\(23\) \( +1 \)

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 483.4.a.a 3
3.b odd 2 1 1449.4.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.a 3 1.a even 1 1 trivial
1449.4.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 4T_{2}^{2} - 8T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 4 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} + \cdots - 133 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} + \cdots + 16888 \) Copy content Toggle raw display
$13$ \( T^{3} - 32 T^{2} + \cdots + 7121 \) Copy content Toggle raw display
$17$ \( T^{3} + 86 T^{2} + \cdots - 32140 \) Copy content Toggle raw display
$19$ \( T^{3} - 32 T^{2} + \cdots - 4436 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 341 T^{2} + \cdots + 1279711 \) Copy content Toggle raw display
$31$ \( T^{3} + 452 T^{2} + \cdots + 2969438 \) Copy content Toggle raw display
$37$ \( T^{3} + 183 T^{2} + \cdots - 3177991 \) Copy content Toggle raw display
$41$ \( T^{3} - 175 T^{2} + \cdots + 11602699 \) Copy content Toggle raw display
$43$ \( T^{3} + 320 T^{2} + \cdots - 3480943 \) Copy content Toggle raw display
$47$ \( T^{3} + 71 T^{2} + \cdots - 30926812 \) Copy content Toggle raw display
$53$ \( T^{3} - 27 T^{2} + \cdots + 9069176 \) Copy content Toggle raw display
$59$ \( T^{3} + 1585 T^{2} + \cdots + 57554534 \) Copy content Toggle raw display
$61$ \( T^{3} + 449 T^{2} + \cdots - 11531794 \) Copy content Toggle raw display
$67$ \( T^{3} + 239 T^{2} + \cdots - 90556772 \) Copy content Toggle raw display
$71$ \( T^{3} - 1255 T^{2} + \cdots + 7078618 \) Copy content Toggle raw display
$73$ \( T^{3} + 342 T^{2} + \cdots + 20669796 \) Copy content Toggle raw display
$79$ \( T^{3} - 994 T^{2} + \cdots + 161629604 \) Copy content Toggle raw display
$83$ \( T^{3} + 1846 T^{2} + \cdots - 147725860 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 1448227670 \) Copy content Toggle raw display
$97$ \( T^{3} + 1797 T^{2} + \cdots + 26418199 \) Copy content Toggle raw display
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