Properties

Label 495.6.a.a
Level $495$
Weight $6$
Character orbit 495.a
Self dual yes
Analytic conductor $79.390$
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] = [495,6,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(79.3899908074\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.307532.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 76x + 168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + (\beta_{2} + 2 \beta_1 + 24) q^{4} + 25 q^{5} + (4 \beta_{2} - 6 \beta_1 + 34) q^{7} + ( - 7 \beta_{2} - 10 \beta_1 - 76) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + (\beta_{2} + 2 \beta_1 + 24) q^{4} + 25 q^{5} + (4 \beta_{2} - 6 \beta_1 + 34) q^{7} + ( - 7 \beta_{2} - 10 \beta_1 - 76) q^{8} + ( - 25 \beta_1 - 50) q^{10} - 121 q^{11} + ( - \beta_{2} + 113 \beta_1 - 68) q^{13} + ( - 14 \beta_{2} - 106 \beta_1 + 292) q^{14} + (13 \beta_{2} + 138 \beta_1 - 180) q^{16} + (12 \beta_{2} + 58 \beta_1 - 660) q^{17} + ( - 11 \beta_{2} + 57 \beta_1 + 672) q^{19} + (25 \beta_{2} + 50 \beta_1 + 600) q^{20} + (121 \beta_1 + 242) q^{22} + ( - 14 \beta_{2} - 286 \beta_1 - 316) q^{23} + 625 q^{25} + ( - 108 \beta_{2} + 86 \beta_1 - 5752) q^{26} + (48 \beta_{2} + 152 \beta_1 + 3672) q^{28} + ( - 145 \beta_{2} - 53 \beta_1 - 1498) q^{29} + ( - 34 \beta_{2} + 582 \beta_1 - 3768) q^{31} + (21 \beta_{2} + 266 \beta_1 - 4228) q^{32} + ( - 118 \beta_{2} + 444 \beta_1 - 1552) q^{34} + (100 \beta_{2} - 150 \beta_1 + 850) q^{35} + ( - 202 \beta_{2} - 274 \beta_1 - 2706) q^{37} + ( - 2 \beta_{2} - 474 \beta_1 - 4440) q^{38} + ( - 175 \beta_{2} - 250 \beta_1 - 1900) q^{40} + ( - 35 \beta_{2} - 879 \beta_1 - 1710) q^{41} + ( - 322 \beta_{2} - 1992 \beta_1 + 6846) q^{43} + ( - 121 \beta_{2} - 242 \beta_1 - 2904) q^{44} + (356 \beta_{2} + 568 \beta_1 + 15336) q^{46} + ( - 134 \beta_{2} + 442 \beta_1 - 18692) q^{47} + ( - 140 \beta_{2} - 672 \beta_1 + 813) q^{49} + ( - 625 \beta_1 - 1250) q^{50} + (486 \beta_{2} + 4080 \beta_1 + 7912) q^{52} + ( - 74 \beta_{2} + 1946 \beta_1 - 3742) q^{53} - 3025 q^{55} + (56 \beta_{2} - 1144 \beta_1 - 24016) q^{56} + (778 \beta_{2} + 4108 \beta_1 + 4012) q^{58} + (14 \beta_{2} + 566 \beta_1 + 19548) q^{59} + (544 \beta_{2} - 968 \beta_1 + 27138) q^{61} + ( - 412 \beta_{2} + 4380 \beta_1 - 23136) q^{62} + ( - 787 \beta_{2} - 566 \beta_1 + 636) q^{64} + ( - 25 \beta_{2} + 2825 \beta_1 - 1700) q^{65} + ( - 844 \beta_{2} + 2136 \beta_1 + 496) q^{67} + ( - 238 \beta_{2} + 1820 \beta_1 - 280) q^{68} + ( - 350 \beta_{2} - 2650 \beta_1 + 7300) q^{70} + (1064 \beta_{2} + 1228 \beta_1 + 25000) q^{71} + (563 \beta_{2} - 6451 \beta_1 - 18092) q^{73} + (1284 \beta_{2} + 6342 \beta_1 + 17236) q^{74} + (836 \beta_{2} + 2652 \beta_1 + 12000) q^{76} + ( - 484 \beta_{2} + 726 \beta_1 - 4114) q^{77} + ( - 1927 \beta_{2} + 2985 \beta_1 - 29492) q^{79} + (325 \beta_{2} + 3450 \beta_1 - 4500) q^{80} + (1054 \beta_{2} + 2340 \beta_1 + 48708) q^{82} + ( - 1715 \beta_{2} + 2811 \beta_1 - 26430) q^{83} + (300 \beta_{2} + 1450 \beta_1 - 16500) q^{85} + (3602 \beta_{2} - 1050 \beta_1 + 86028) q^{86} + (847 \beta_{2} + 1210 \beta_1 + 9196) q^{88} + (1410 \beta_{2} + 8246 \beta_1 - 14726) q^{89} + (466 \beta_{2} + 13682 \beta_1 - 46568) q^{91} + ( - 1900 \beta_{2} - 12592 \beta_1 - 45824) q^{92} + (228 \beta_{2} + 21104 \beta_1 + 12792) q^{94} + ( - 275 \beta_{2} + 1425 \beta_1 + 16800) q^{95} + ( - 2108 \beta_{2} + 88 \beta_1 + 10430) q^{97} + (1372 \beta_{2} + 1707 \beta_1 + 31638) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 76x + 168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 52 \) 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
7.91848
2.30119
−9.21967
−9.91848 0 66.3762 25.0000 0 92.6461 −340.959 0 −247.962
1.2 −4.30119 0 −13.4998 25.0000 0 −148.216 195.703 0 −107.530
1.3 7.21967 0 20.1236 25.0000 0 147.570 −85.7438 0 180.492
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 495.6.a.a 3
3.b odd 2 1 165.6.a.e 3
15.d odd 2 1 825.6.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.e 3 3.b odd 2 1
495.6.a.a 3 1.a even 1 1 trivial
825.6.a.f 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 7T_{2}^{2} - 60T_{2} - 308 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 7 T^{2} + \cdots - 308 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 92 T^{2} + \cdots + 2026368 \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 90 T^{2} + \cdots + 210673232 \) Copy content Toggle raw display
$17$ \( T^{3} + 1934 T^{2} + \cdots - 123909408 \) Copy content Toggle raw display
$19$ \( T^{3} - 2084 T^{2} + \cdots + 14437440 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 2404328128 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 80705567064 \) Copy content Toggle raw display
$31$ \( T^{3} + 10688 T^{2} + \cdots + 593236224 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 165140256344 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 127610311752 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 5672527691040 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 5576540180928 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 850686588776 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 7185357358784 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 2993338614376 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 6432661987328 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 31171869026560 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 152306824713328 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 283704612543488 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 200710881230832 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 340522035911288 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 93893760682568 \) Copy content Toggle raw display
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