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Properties

Label 460.6.a.b
Level $460$
Weight $6$
Character orbit 460.a
Self dual yes
Analytic conductor $73.777$
Analytic rank $1$
Dimension $9$
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] = [460,6,Mod(1,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, 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("460.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 460.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: \(73.7765571140\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 1380 x^{7} - 294 x^{6} + 623208 x^{5} + 755616 x^{4} - 104917195 x^{3} - 197773440 x^{2} + \cdots - 4082256000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + 25 q^{5} + (\beta_{4} + 2 \beta_1 - 11) q^{7} + (\beta_{2} + 2 \beta_1 + 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + 25 q^{5} + (\beta_{4} + 2 \beta_1 - 11) q^{7} + (\beta_{2} + 2 \beta_1 + 65) q^{9} + (\beta_{7} - \beta_{6} - \beta_{2} + \cdots - 69) q^{11}+ \cdots + (41 \beta_{8} - 33 \beta_{7} + \cdots - 48083) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} + 225 q^{5} - 95 q^{7} + 582 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} + 225 q^{5} - 95 q^{7} + 582 q^{9} - 620 q^{11} - 627 q^{13} - 225 q^{15} - 2445 q^{17} - 2720 q^{19} - 4684 q^{21} - 4761 q^{23} + 5625 q^{25} - 4797 q^{27} - 7712 q^{29} - 1336 q^{31} - 4210 q^{33} - 2375 q^{35} + 8689 q^{37} - 5945 q^{39} - 21154 q^{41} - 17816 q^{43} + 14550 q^{45} - 19651 q^{47} + 11992 q^{49} - 44700 q^{51} + 5005 q^{53} - 15500 q^{55} + 25942 q^{57} - 26341 q^{59} - 27362 q^{61} - 69505 q^{63} - 15675 q^{65} + 10853 q^{67} + 4761 q^{69} - 113802 q^{71} - 29261 q^{73} - 5625 q^{75} + 54734 q^{77} - 115842 q^{79} - 151095 q^{81} - 141093 q^{83} - 61125 q^{85} - 362239 q^{87} - 17598 q^{89} - 260198 q^{91} - 2593 q^{93} - 68000 q^{95} - 71284 q^{97} - 431686 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 1380 x^{7} - 294 x^{6} + 623208 x^{5} + 755616 x^{4} - 104917195 x^{3} - 197773440 x^{2} + \cdots - 4082256000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 307 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3459069547 \nu^{8} - 2921670629362 \nu^{7} + 3554907855986 \nu^{6} + \cdots - 11\!\cdots\!00 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70362868164 \nu^{8} + 1057863784345 \nu^{7} + 77230397585223 \nu^{6} + \cdots + 16\!\cdots\!56 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74700939409 \nu^{8} - 460223408078 \nu^{7} + 97215456677294 \nu^{6} + \cdots - 15\!\cdots\!48 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 85465511456 \nu^{8} - 1356354950489 \nu^{7} - 91413028843927 \nu^{6} + \cdots - 13\!\cdots\!64 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93435170661 \nu^{8} - 1639258353079 \nu^{7} - 94241310941913 \nu^{6} + \cdots - 10\!\cdots\!36 ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 243724393441 \nu^{8} - 2549809516487 \nu^{7} - 285508830912761 \nu^{6} + \cdots + 36\!\cdots\!12 ) / 26\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 307 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{8} + \beta_{7} + 13\beta_{6} - 12\beta_{5} + 2\beta_{4} + 3\beta_{3} - 6\beta_{2} + 478\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 51 \beta_{8} - 24 \beta_{7} - 204 \beta_{6} + 105 \beta_{5} - 213 \beta_{4} - 30 \beta_{3} + \cdots + 146584 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6992 \beta_{8} - 470 \beta_{7} + 12382 \beta_{6} - 9918 \beta_{5} + 608 \beta_{4} + 2400 \beta_{3} + \cdots - 199861 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48162 \beta_{8} - 5475 \beta_{7} - 206973 \beta_{6} + 109188 \beta_{5} - 206058 \beta_{4} + \cdots + 80925046 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5110757 \beta_{8} - 750236 \beta_{7} + 9524998 \beta_{6} - 7104327 \beta_{5} + 333665 \beta_{4} + \cdots - 368605672 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 41576940 \beta_{8} + 3582288 \beta_{7} - 168723048 \beta_{6} + 92185032 \beta_{5} - 156242844 \beta_{4} + \cdots + 48623771803 \) Copy content Toggle raw display

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

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
23.8191
21.4641
18.4612
5.38239
1.03619
−11.1768
−15.6036
−16.6088
−26.7738
0 −24.8191 0 25.0000 0 −197.204 0 372.986 0
1.2 0 −22.4641 0 25.0000 0 135.763 0 261.636 0
1.3 0 −19.4612 0 25.0000 0 180.613 0 135.740 0
1.4 0 −6.38239 0 25.0000 0 −83.1466 0 −202.265 0
1.5 0 −2.03619 0 25.0000 0 −29.9423 0 −238.854 0
1.6 0 10.1768 0 25.0000 0 62.6693 0 −139.434 0
1.7 0 14.6036 0 25.0000 0 −115.665 0 −29.7349 0
1.8 0 15.6088 0 25.0000 0 129.321 0 0.634528 0
1.9 0 25.7738 0 25.0000 0 −177.408 0 421.291 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 460.6.a.b ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.6.a.b ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + 9 T_{3}^{8} - 1344 T_{3}^{7} - 9282 T_{3}^{6} + 596118 T_{3}^{5} + 2316660 T_{3}^{4} + \cdots + 8430695568 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(460))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots + 8430695568 \) Copy content Toggle raw display
$5$ \( (T - 25)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 73\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 38\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T + 529)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 55\!\cdots\!38 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 79\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 13\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 34\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 14\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 90\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 41\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 90\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
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