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Properties

Label 495.4.c.e
Level $495$
Weight $4$
Character orbit 495.c
Analytic conductor $29.206$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,4,Mod(199,495)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 103 x^{14} + 4248 x^{12} + 89496 x^{10} + 1015487 x^{8} + 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 5) q^{4} + \beta_{7} q^{5} + (\beta_{9} - \beta_1) q^{7} + (\beta_{3} - 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 5) q^{4} + \beta_{7} q^{5} + (\beta_{9} - \beta_1) q^{7} + (\beta_{3} - 5 \beta_1) q^{8} + ( - \beta_{8} + \beta_1 + 2) q^{10} - 11 q^{11} + ( - \beta_{13} - \beta_{7} + \cdots + 3 \beta_1) q^{13}+ \cdots + (10 \beta_{15} - 10 \beta_{14} + \cdots - 38 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 78 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 78 q^{4} + 26 q^{10} - 176 q^{11} + 124 q^{14} + 306 q^{16} - 84 q^{19} - 108 q^{20} + 180 q^{25} - 532 q^{26} + 652 q^{29} + 80 q^{31} - 360 q^{34} + 132 q^{35} + 174 q^{40} - 1204 q^{41} + 858 q^{44} - 672 q^{46} - 1016 q^{49} + 1396 q^{50} - 3332 q^{56} + 712 q^{59} + 880 q^{61} + 962 q^{64} + 1256 q^{65} - 136 q^{70} - 1968 q^{71} + 4152 q^{74} - 1048 q^{76} + 1636 q^{79} + 16 q^{80} - 864 q^{85} - 7284 q^{86} + 2776 q^{89} + 144 q^{91} - 1400 q^{94} + 1408 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 103 x^{14} + 4248 x^{12} + 89496 x^{10} + 1015487 x^{8} + 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 21\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5713 \nu^{14} - 534646 \nu^{12} - 19096082 \nu^{10} - 323388550 \nu^{8} - 2627255209 \nu^{6} + \cdots + 659977472 ) / 130706048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66059 \nu^{15} - 350639 \nu^{14} + 10540466 \nu^{13} - 30984006 \nu^{12} + 643543598 \nu^{11} + \cdots + 489170878976 ) / 37904753920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 145321 \nu^{15} - 2773241 \nu^{14} + 14208524 \nu^{13} - 263896694 \nu^{12} + \cdots - 3449127399936 ) / 37904753920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 145321 \nu^{15} + 350639 \nu^{14} + 14208524 \nu^{13} + 30984006 \nu^{12} + \cdots - 489170878976 ) / 37904753920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12091 \nu^{15} + 26191 \nu^{14} - 1068414 \nu^{13} + 2403874 \nu^{12} - 35440542 \nu^{11} + \cdots + 6929573376 ) / 1307060480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57002 \nu^{15} + 5705529 \nu^{13} + 226639762 \nu^{11} + 4547664614 \nu^{9} + \cdots + 243711549056 \nu ) / 3790475392 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 641281 \nu^{15} + 759539 \nu^{14} + 59401054 \nu^{13} + 69712346 \nu^{12} + \cdots + 200957627904 ) / 37904753920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 641281 \nu^{15} + 2416309 \nu^{14} + 59401054 \nu^{13} + 224759686 \nu^{12} + \cdots + 3155658736384 ) / 37904753920 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 145321 \nu^{15} - 350639 \nu^{14} + 14208524 \nu^{13} - 30984006 \nu^{12} + 547611262 \nu^{11} + \cdots + 489170878976 ) / 7580950784 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 44701 \nu^{15} + 4305155 \nu^{13} + 162772644 \nu^{11} + 3065070576 \nu^{9} + \cdots + 351980851136 \nu ) / 1895237696 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 584527 \nu^{15} - 1569103 \nu^{14} - 61801368 \nu^{13} - 141585192 \nu^{12} + \cdots - 681808296448 ) / 18952376960 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 262875 \nu^{15} - 697769 \nu^{14} + 27562252 \nu^{13} - 62830878 \nu^{12} + 1159302874 \nu^{11} + \cdots - 174889142784 ) / 7580950784 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{4} - 27\beta_{2} + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - 5 \beta_{9} - \beta_{8} + \cdots + 491 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{12} - 39 \beta_{11} + 44 \beta_{10} + 5 \beta_{8} + \cdots - 6184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 53 \beta_{15} + 53 \beta_{14} - 21 \beta_{13} + 61 \beta_{12} - 69 \beta_{10} + 223 \beta_{9} + \cdots - 12093 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 86 \beta_{15} + 86 \beta_{14} - 131 \beta_{12} + 1198 \beta_{11} - 1571 \beta_{10} - 373 \beta_{8} + \cdots + 151201 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2020 \beta_{15} - 2020 \beta_{14} + 108 \beta_{13} - 2620 \beta_{12} + 3020 \beta_{10} + \cdots + 307191 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2812 \beta_{15} - 2812 \beta_{14} + 4320 \beta_{12} - 34516 \beta_{11} + 51756 \beta_{10} + \cdots - 3826473 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 67988 \beta_{15} + 67988 \beta_{14} + 11212 \beta_{13} + 96180 \beta_{12} - 110116 \beta_{10} + \cdots - 7966949 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 87176 \beta_{15} + 87176 \beta_{14} - 132028 \beta_{12} + 978285 \beta_{11} - 1629313 \beta_{10} + \cdots + 99050664 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2153397 \beta_{15} - 2153397 \beta_{14} - 672475 \beta_{13} - 3237749 \beta_{12} + \cdots + 209801819 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2707338 \beta_{15} - 2707338 \beta_{14} + 3951567 \beta_{12} - 27693515 \beta_{11} + \cdots - 2605848300 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 65897505 \beta_{15} + 65897505 \beta_{14} + 27365039 \beta_{13} + 103403561 \beta_{12} + \cdots - 5591555581 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 5.31333i
− 4.85092i
− 4.59829i
− 3.54699i
− 3.37022i
− 2.28277i
− 0.919139i
− 0.312176i
0.312176i
0.919139i
2.28277i
3.37022i
3.54699i
4.59829i
4.85092i
5.31333i
− 5.31333i 0 −20.2314 7.55306 + 8.24326i 0 15.6271i 64.9896i 0 43.7991 − 40.1318i
199.2 − 4.85092i 0 −15.5314 −1.85171 − 11.0259i 0 24.5618i 36.5345i 0 −53.4859 + 8.98251i
199.3 − 4.59829i 0 −13.1443 −10.8778 − 2.58341i 0 − 8.59388i 23.6550i 0 −11.8793 + 50.0192i
199.4 − 3.54699i 0 −4.58114 10.9012 + 2.48289i 0 − 35.4720i − 12.1266i 0 8.80681 − 38.6663i
199.5 − 3.37022i 0 −3.35835 −6.14566 + 9.33975i 0 − 5.87458i − 15.6434i 0 31.4770 + 20.7122i
199.6 − 2.28277i 0 2.78895 8.77101 − 6.93321i 0 11.2428i − 24.6287i 0 −15.8269 − 20.0222i
199.7 − 0.919139i 0 7.15518 2.81781 + 10.8194i 0 26.1261i − 13.9297i 0 9.94456 − 2.58996i
199.8 − 0.312176i 0 7.90255 −11.1679 + 0.527603i 0 − 15.1298i − 4.96439i 0 0.164705 + 3.48634i
199.9 0.312176i 0 7.90255 −11.1679 − 0.527603i 0 15.1298i 4.96439i 0 0.164705 − 3.48634i
199.10 0.919139i 0 7.15518 2.81781 − 10.8194i 0 − 26.1261i 13.9297i 0 9.94456 + 2.58996i
199.11 2.28277i 0 2.78895 8.77101 + 6.93321i 0 − 11.2428i 24.6287i 0 −15.8269 + 20.0222i
199.12 3.37022i 0 −3.35835 −6.14566 − 9.33975i 0 5.87458i 15.6434i 0 31.4770 − 20.7122i
199.13 3.54699i 0 −4.58114 10.9012 − 2.48289i 0 35.4720i 12.1266i 0 8.80681 + 38.6663i
199.14 4.59829i 0 −13.1443 −10.8778 + 2.58341i 0 8.59388i − 23.6550i 0 −11.8793 − 50.0192i
199.15 4.85092i 0 −15.5314 −1.85171 + 11.0259i 0 − 24.5618i − 36.5345i 0 −53.4859 − 8.98251i
199.16 5.31333i 0 −20.2314 7.55306 − 8.24326i 0 − 15.6271i − 64.9896i 0 43.7991 + 40.1318i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.4.c.e ✓ 16
3.b odd 2 1 495.4.c.f yes 16
5.b even 2 1 inner 495.4.c.e ✓ 16
5.c odd 4 2 2475.4.a.bz 16
15.d odd 2 1 495.4.c.f yes 16
15.e even 4 2 2475.4.a.ca 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.4.c.e ✓ 16 1.a even 1 1 trivial
495.4.c.e ✓ 16 5.b even 2 1 inner
495.4.c.f yes 16 3.b odd 2 1
495.4.c.f yes 16 15.d odd 2 1
2475.4.a.bz 16 5.c odd 4 2
2475.4.a.ca 16 15.e even 4 2

Hecke kernels

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

\( T_{2}^{16} + 103 T_{2}^{14} + 4248 T_{2}^{12} + 89496 T_{2}^{10} + 1015487 T_{2}^{8} + 5956953 T_{2}^{6} + \cdots + 861184 \) Copy content Toggle raw display
\( T_{29}^{8} - 326 T_{29}^{7} - 92952 T_{29}^{6} + 36589120 T_{29}^{5} - 131527344 T_{29}^{4} + \cdots - 29\!\cdots\!36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 103 T^{14} + \cdots + 861184 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T + 11)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 271885102112768)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 35\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 83\!\cdots\!44 \) Copy content Toggle raw display
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