Properties

Label 471.2.a.e
Level $471$
Weight $2$
Character orbit 471.a
Self dual yes
Analytic conductor $3.761$
Analytic rank $0$
Dimension $12$
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] = [471,2,Mod(1,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.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: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 20 x^{10} + 17 x^{9} + 149 x^{8} - 106 x^{7} - 500 x^{6} + 294 x^{5} + 711 x^{4} - 349 x^{3} - 290 x^{2} + 173 x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} - \beta_1 q^{6} + ( - \beta_{7} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} - \beta_1 q^{6} + ( - \beta_{7} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{11} + \beta_{10} + \beta_{3} - \beta_{2} + 1) q^{10} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{9} - \beta_{6} - \beta_{3} + \beta_{2} + 1) q^{13} + ( - \beta_{9} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{14} + \beta_{8} q^{15} + (\beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_1 + 1) q^{16} + ( - \beta_{11} - \beta_{6} + \beta_{5} + 1) q^{17} - \beta_1 q^{18} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{20}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 12 q^{3} + 17 q^{4} + 4 q^{5} - q^{6} + 8 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 12 q^{3} + 17 q^{4} + 4 q^{5} - q^{6} + 8 q^{7} - 6 q^{8} + 12 q^{9} + 9 q^{10} - q^{11} + 17 q^{12} + 15 q^{13} - 7 q^{14} + 4 q^{15} + 19 q^{16} + 10 q^{17} - q^{18} + 14 q^{19} - 11 q^{20} + 8 q^{21} - 9 q^{22} - 8 q^{23} - 6 q^{24} + 22 q^{25} - 17 q^{26} + 12 q^{27} + 7 q^{28} + 5 q^{29} + 9 q^{30} + 5 q^{31} - 33 q^{32} - q^{33} - 13 q^{34} - 14 q^{35} + 17 q^{36} + 4 q^{37} - 32 q^{38} + 15 q^{39} - 4 q^{40} + 18 q^{41} - 7 q^{42} + 28 q^{43} - 20 q^{44} + 4 q^{45} - 15 q^{46} - 3 q^{47} + 19 q^{48} + 48 q^{49} - 2 q^{50} + 10 q^{51} + 26 q^{52} - 18 q^{53} - q^{54} - 5 q^{55} - 10 q^{56} + 14 q^{57} - 6 q^{58} - 11 q^{60} + 20 q^{61} + q^{62} + 8 q^{63} - 14 q^{64} - 11 q^{65} - 9 q^{66} + 27 q^{67} - 21 q^{68} - 8 q^{69} - 29 q^{70} + 7 q^{71} - 6 q^{72} + 15 q^{73} - 14 q^{74} + 22 q^{75} + 30 q^{76} - 23 q^{77} - 17 q^{78} + 4 q^{79} - 31 q^{80} + 12 q^{81} - 31 q^{82} - 13 q^{83} + 7 q^{84} + 8 q^{85} - 26 q^{86} + 5 q^{87} - 11 q^{88} - 9 q^{89} + 9 q^{90} - 31 q^{91} - 21 q^{92} + 5 q^{93} - 4 q^{94} - 44 q^{95} - 33 q^{96} - 16 q^{97} - 54 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 20 x^{10} + 17 x^{9} + 149 x^{8} - 106 x^{7} - 500 x^{6} + 294 x^{5} + 711 x^{4} - 349 x^{3} - 290 x^{2} + 173 x - 15 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 10\nu^{4} - 2\nu^{3} + 25\nu^{2} + 7\nu - 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} + 16 \nu^{9} - 27 \nu^{8} - 94 \nu^{7} + 116 \nu^{6} + 250 \nu^{5} - 156 \nu^{4} - 293 \nu^{3} + 2 \nu^{2} + 74 \nu + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - \nu^{10} - 19 \nu^{9} + 14 \nu^{8} + 134 \nu^{7} - 64 \nu^{6} - 422 \nu^{5} + 100 \nu^{4} + 549 \nu^{3} - 21 \nu^{2} - 175 \nu + 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - \nu^{10} - 19 \nu^{9} + 14 \nu^{8} + 136 \nu^{7} - 64 \nu^{6} - 444 \nu^{5} + 98 \nu^{4} + 615 \nu^{3} - 19 \nu^{2} - 219 \nu + 34 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - \nu^{11} + 21 \nu^{9} + \nu^{8} - 162 \nu^{7} - 14 \nu^{6} + 553 \nu^{5} + 64 \nu^{4} - 776 \nu^{3} - 100 \nu^{2} + 285 \nu - 25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 54 \nu^{9} - 55 \nu^{8} - 364 \nu^{7} + 244 \nu^{6} + 1118 \nu^{5} - 356 \nu^{4} - 1475 \nu^{3} + 56 \nu^{2} + 502 \nu - 61 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\nu^{11} + 22\nu^{9} - 177\nu^{7} - 3\nu^{6} + 628\nu^{5} + 31\nu^{4} - 911\nu^{3} - 80\nu^{2} + 343\nu - 38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 2 \nu^{11} + \nu^{10} + 40 \nu^{9} - 13 \nu^{8} - 297 \nu^{7} + 50 \nu^{6} + 986 \nu^{5} - 34 \nu^{4} - 1359 \nu^{3} - 86 \nu^{2} + 486 \nu - 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{6} - \beta_{5} + 10\beta_{3} - \beta_{2} + 29\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{11} - 10\beta_{8} + 10\beta_{7} + 10\beta_{6} + \beta_{4} + 12\beta_{3} + 35\beta_{2} + 13\beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{11} + 11 \beta_{9} - 12 \beta_{8} + 24 \beta_{7} + 22 \beta_{6} - 11 \beta_{5} + 78 \beta_{3} - 6 \beta_{2} + 177 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 77 \beta_{11} + \beta_{10} - \beta_{9} - 77 \beta_{8} + 79 \beta_{7} + 77 \beta_{6} + 3 \beta_{5} + 15 \beta_{4} + 105 \beta_{3} + 211 \beta_{2} + 128 \beta _1 + 497 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105 \beta_{11} + 2 \beta_{10} + 89 \beta_{9} - 106 \beta_{8} + 212 \beta_{7} + 180 \beta_{6} - 87 \beta_{5} + 4 \beta_{4} + 561 \beta_{3} - 11 \beta_{2} + 1115 \beta _1 + 276 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 544 \beta_{11} + 19 \beta_{10} - 14 \beta_{9} - 547 \beta_{8} + 583 \beta_{7} + 543 \beta_{6} + 48 \beta_{5} + 155 \beta_{4} + 824 \beta_{3} + 1313 \beta_{2} + 1118 \beta _1 + 3053 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 815 \beta_{11} + 43 \beta_{10} + 639 \beta_{9} - 837 \beta_{8} + 1673 \beta_{7} + 1323 \beta_{6} - 595 \beta_{5} + 85 \beta_{4} + 3900 \beta_{3} + 193 \beta_{2} + 7193 \beta _1 + 2484 \) 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
2.67678
2.61941
1.94009
1.86194
0.576969
0.555677
0.108373
−0.971700
−1.40350
−2.27133
−2.29336
−2.39936
−2.67678 1.00000 5.16514 −1.96699 −2.67678 −4.23568 −8.47236 1.00000 5.26520
1.2 −2.61941 1.00000 4.86133 0.738753 −2.61941 4.13098 −7.49502 1.00000 −1.93510
1.3 −1.94009 1.00000 1.76394 −2.61988 −1.94009 5.01982 0.457975 1.00000 5.08280
1.4 −1.86194 1.00000 1.46683 2.94929 −1.86194 0.909688 0.992725 1.00000 −5.49141
1.5 −0.576969 1.00000 −1.66711 −4.25420 −0.576969 −0.240996 2.11581 1.00000 2.45454
1.6 −0.555677 1.00000 −1.69122 1.79523 −0.555677 1.92349 2.05113 1.00000 −0.997565
1.7 −0.108373 1.00000 −1.98826 3.74119 −0.108373 −4.41637 0.432219 1.00000 −0.405444
1.8 0.971700 1.00000 −1.05580 −0.360676 0.971700 3.09062 −2.96932 1.00000 −0.350469
1.9 1.40350 1.00000 −0.0301860 3.91565 1.40350 3.59994 −2.84937 1.00000 5.49562
1.10 2.27133 1.00000 3.15894 0.157525 2.27133 −1.90651 2.63234 1.00000 0.357791
1.11 2.29336 1.00000 3.25948 2.31994 2.29336 −3.22014 2.88843 1.00000 5.32046
1.12 2.39936 1.00000 3.75691 −2.41582 2.39936 3.34516 4.21544 1.00000 −5.79642
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(157\) \(1\)

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 471.2.a.e 12
3.b odd 2 1 1413.2.a.h 12
4.b odd 2 1 7536.2.a.bm 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.a.e 12 1.a even 1 1 trivial
1413.2.a.h 12 3.b odd 2 1
7536.2.a.bm 12 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} - 20 T_{2}^{10} - 17 T_{2}^{9} + 149 T_{2}^{8} + 106 T_{2}^{7} - 500 T_{2}^{6} - 294 T_{2}^{5} + 711 T_{2}^{4} + 349 T_{2}^{3} - 290 T_{2}^{2} - 173 T_{2} - 15 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} - 20 T^{10} - 17 T^{9} + \cdots - 15 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} - 33 T^{10} + 140 T^{9} + \cdots - 400 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} - 34 T^{10} + \cdots - 37376 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} - 87 T^{10} + \cdots + 393216 \) Copy content Toggle raw display
$13$ \( T^{12} - 15 T^{11} + 12 T^{10} + \cdots + 80384 \) Copy content Toggle raw display
$17$ \( T^{12} - 10 T^{11} - 68 T^{10} + \cdots + 1844992 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} - 20 T^{10} + \cdots - 34560 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} - 132 T^{10} + \cdots - 6124 \) Copy content Toggle raw display
$29$ \( T^{12} - 5 T^{11} - 195 T^{10} + \cdots - 32768 \) Copy content Toggle raw display
$31$ \( T^{12} - 5 T^{11} - 215 T^{10} + \cdots - 72516928 \) Copy content Toggle raw display
$37$ \( T^{12} - 4 T^{11} - 259 T^{10} + \cdots + 111040 \) Copy content Toggle raw display
$41$ \( T^{12} - 18 T^{11} + \cdots + 357250128 \) Copy content Toggle raw display
$43$ \( T^{12} - 28 T^{11} + 173 T^{10} + \cdots - 11950080 \) Copy content Toggle raw display
$47$ \( T^{12} + 3 T^{11} - 319 T^{10} + \cdots + 110166016 \) Copy content Toggle raw display
$53$ \( T^{12} + 18 T^{11} - 134 T^{10} + \cdots - 3297856 \) Copy content Toggle raw display
$59$ \( T^{12} - 337 T^{10} + 456 T^{9} + \cdots - 503760 \) Copy content Toggle raw display
$61$ \( T^{12} - 20 T^{11} - 77 T^{10} + \cdots - 131072 \) Copy content Toggle raw display
$67$ \( T^{12} - 27 T^{11} + \cdots - 2092751872 \) Copy content Toggle raw display
$71$ \( T^{12} - 7 T^{11} - 423 T^{10} + \cdots - 27557888 \) Copy content Toggle raw display
$73$ \( T^{12} - 15 T^{11} + \cdots - 21988393984 \) Copy content Toggle raw display
$79$ \( T^{12} - 4 T^{11} + \cdots - 5143724032 \) Copy content Toggle raw display
$83$ \( T^{12} + 13 T^{11} + \cdots + 4457318592 \) Copy content Toggle raw display
$89$ \( T^{12} + 9 T^{11} - 596 T^{10} + \cdots - 743226112 \) Copy content Toggle raw display
$97$ \( T^{12} + 16 T^{11} + \cdots - 127959040 \) Copy content Toggle raw display
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