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Properties

Label 495.4.a.p
Level $495$
Weight $4$
Character orbit 495.a
Self dual yes
Analytic conductor $29.206$
Analytic rank $0$
Dimension $7$
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] = [495,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{2} + 4) q^{7} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{2} + 4) q^{7} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 + 8) q^{8} + ( - 5 \beta_1 + 5) q^{10} + 11 q^{11} + ( - \beta_{6} - \beta_{5} + \beta_{2} + \cdots + 4) q^{13}+ \cdots + (7 \beta_{6} - 32 \beta_{5} + \cdots + 79) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8} + 25 q^{10} + 77 q^{11} + 38 q^{13} + 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} + 165 q^{20} + 55 q^{22} + 334 q^{23} + 175 q^{25} - 372 q^{26} + 812 q^{28} - 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} + 150 q^{35} + 466 q^{37} - 494 q^{38} + 225 q^{40} - 258 q^{41} + 308 q^{43} + 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} + 125 q^{50} + 512 q^{52} + 110 q^{53} + 385 q^{55} + 20 q^{56} + 1362 q^{58} - 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} + 190 q^{65} + 2268 q^{67} - 1186 q^{68} + 100 q^{70} - 166 q^{71} + 200 q^{73} - 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1545 q^{80} - 1006 q^{82} + 370 q^{83} + 60 q^{85} + 106 q^{86} + 495 q^{88} - 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} + 1130 q^{95} + 3698 q^{97} + 697 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 21\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{6} - 15\nu^{5} - 252\nu^{4} + 270\nu^{3} + 1832\nu^{2} - 1326\nu - 1302 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 13\nu^{5} + 108\nu^{4} - 372\nu^{3} - 1002\nu^{2} + 2336\nu + 1892 ) / 46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} + \nu^{5} + 95\nu^{4} - 18\nu^{3} - 1036\nu^{2} + 415\nu + 1752 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 23\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} + 33\beta_{2} + 56\beta _1 + 274 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} + 3\beta_{4} + 39\beta_{3} + 111\beta_{2} + 661\beta _1 + 700 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{6} + 51\beta_{5} + 49\beta_{4} + 153\beta_{3} + 1087\beta_{2} + 2473\beta _1 + 7908 \) 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
6.03128
3.28302
2.29255
−0.434062
−1.30247
−3.32942
−4.54090
−5.03128 0 17.3138 5.00000 0 18.5765 −46.8604 0 −25.1564
1.2 −2.28302 0 −2.78784 5.00000 0 13.7084 24.6288 0 −11.4151
1.3 −1.29255 0 −6.32931 5.00000 0 −23.9463 18.5214 0 −6.46276
1.4 1.43406 0 −5.94347 5.00000 0 −23.1010 −19.9958 0 7.17031
1.5 2.30247 0 −2.69863 5.00000 0 33.7164 −24.6333 0 11.5123
1.6 4.32942 0 10.7439 5.00000 0 −6.68577 11.8793 0 21.6471
1.7 5.54090 0 22.7016 5.00000 0 17.7319 81.4600 0 27.7045
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 495.4.a.p yes 7
3.b odd 2 1 495.4.a.o ✓ 7
5.b even 2 1 2475.4.a.bp 7
15.d odd 2 1 2475.4.a.bt 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.4.a.o ✓ 7 3.b odd 2 1
495.4.a.p yes 7 1.a even 1 1 trivial
2475.4.a.bp 7 5.b even 2 1
2475.4.a.bt 7 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(495))\):

\( T_{2}^{7} - 5T_{2}^{6} - 32T_{2}^{5} + 160T_{2}^{4} + 169T_{2}^{3} - 925T_{2}^{2} - 156T_{2} + 1176 \) Copy content Toggle raw display
\( T_{7}^{7} - 30T_{7}^{6} - 1118T_{7}^{5} + 33696T_{7}^{4} + 282492T_{7}^{3} - 10640472T_{7}^{2} + 12266712T_{7} + 563074784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 5 T^{6} + \cdots + 1176 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 5)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 30 T^{6} + \cdots + 563074784 \) Copy content Toggle raw display
$11$ \( (T - 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 118066003616 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 453649520592 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 570007071488 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 20611158859776 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 392596976050944 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 380921847353856 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 30\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 671867300186880 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 28\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 96\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 20\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 41\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 90\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 29\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 22\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 67\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 66\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
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