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Properties

Label 495.2.n.h
Level $495$
Weight $2$
Character orbit 495.n
Analytic conductor $3.953$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(91,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.n (of order \(5\), degree \(4\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} + 47 x^{12} + 32 x^{11} + 171 x^{10} + 26 x^{9} + 360 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{14} + \beta_{10} - \beta_{5} + \cdots - 1) q^{2}+ \cdots + ( - 2 \beta_{15} - \beta_{14} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} + \beta_{10} - \beta_{5} + \cdots - 1) q^{2}+ \cdots + (8 \beta_{14} - 3 \beta_{13} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 8 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} - 8 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} + 8 q^{10} - 4 q^{11} + 2 q^{13} + 22 q^{14} + 8 q^{16} + 4 q^{17} - 4 q^{19} - 2 q^{20} - 28 q^{22} - 8 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} + 26 q^{29} - 10 q^{31} - 56 q^{32} - 4 q^{34} + 4 q^{35} + 22 q^{37} + 30 q^{38} - 6 q^{40} + 6 q^{41} + 28 q^{43} - 68 q^{44} + 16 q^{46} + 20 q^{47} + 10 q^{49} + 2 q^{50} + 30 q^{52} - 14 q^{53} - 6 q^{55} - 68 q^{56} - 6 q^{58} + 16 q^{59} - 38 q^{61} + 20 q^{62} + 10 q^{64} - 12 q^{65} + 20 q^{67} + 48 q^{68} - 22 q^{70} + 54 q^{71} + 2 q^{73} - 28 q^{74} - 44 q^{76} - 34 q^{77} - 12 q^{79} + 22 q^{80} + 30 q^{82} + 28 q^{83} - 4 q^{85} - 74 q^{86} + 46 q^{88} - 76 q^{89} - 34 q^{91} + 8 q^{92} - 10 q^{94} + 4 q^{95} - 18 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} + 47 x^{12} + 32 x^{11} + 171 x^{10} + 26 x^{9} + 360 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 274909709246406 \nu^{15} - 784278699533661 \nu^{14} + 143020561508636 \nu^{13} + \cdots + 46\!\cdots\!04 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 553668747953762 \nu^{15} + \cdots - 30\!\cdots\!65 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!63 \nu^{15} + \cdots + 41\!\cdots\!35 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!73 \nu^{15} + \cdots + 274909709246406 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49\!\cdots\!38 \nu^{15} + \cdots - 55\!\cdots\!07 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 76\!\cdots\!39 \nu^{15} + \cdots - 37\!\cdots\!23 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79\!\cdots\!85 \nu^{15} + \cdots - 244926581437456 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 99\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!55 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54\!\cdots\!93 \nu^{15} + \cdots - 22\!\cdots\!40 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!53 \nu^{15} + \cdots + 18\!\cdots\!54 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!74 \nu^{15} + \cdots - 38\!\cdots\!23 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!06 \nu^{15} + \cdots - 11\!\cdots\!58 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22\!\cdots\!40 \nu^{15} + \cdots + 15\!\cdots\!45 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!22 \nu^{15} + \cdots - 49\!\cdots\!25 ) / 24\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 5 \beta_{10} + \beta_{9} + 5 \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - 10 \beta_{10} + 7 \beta_{8} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{15} + 6 \beta_{14} - \beta_{13} + 21 \beta_{12} - 14 \beta_{10} - 12 \beta_{9} + 5 \beta_{8} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 83 \beta_{12} - 15 \beta_{11} - 59 \beta_{9} + 25 \beta_{8} - 20 \beta_{7} + 79 \beta_{6} - 43 \beta_{5} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 47 \beta_{15} - 90 \beta_{14} + 19 \beta_{13} + 225 \beta_{12} - 114 \beta_{11} + 151 \beta_{10} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 365 \beta_{15} - 666 \beta_{14} + 178 \beta_{13} + 365 \beta_{12} - 365 \beta_{11} + 867 \beta_{10} + \cdots + 219 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1045 \beta_{15} - 1912 \beta_{14} + 543 \beta_{13} - 543 \beta_{12} - 532 \beta_{11} + 2677 \beta_{10} + \cdots + 1369 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1736 \beta_{15} - 3236 \beta_{14} + 768 \beta_{13} - 5576 \beta_{12} + 4618 \beta_{10} + 3220 \beta_{9} + \cdots + 4461 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22276 \beta_{12} + 5386 \beta_{11} + 14964 \beta_{9} - 7500 \beta_{8} + 4808 \beta_{7} - 19772 \beta_{6} + \cdots + 6934 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16902 \beta_{15} + 31300 \beta_{14} - 7890 \beta_{13} - 60515 \beta_{12} + 29227 \beta_{11} + \cdots - 1955 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 88227 \beta_{15} + 162068 \beta_{14} - 43613 \beta_{13} - 88227 \beta_{12} + 88227 \beta_{11} + \cdots - 70543 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 268732 \beta_{15} + 493851 \beta_{14} - 131840 \beta_{13} + 131840 \beta_{12} + 160781 \beta_{11} + \cdots - 362011 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 492987 \beta_{15} + 909118 \beta_{14} - 237637 \beta_{13} + 1454159 \beta_{12} - 1274254 \beta_{10} + \cdots - 1094186 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.659965 + 2.03116i
−0.458960 + 1.41253i
0.0698401 − 0.214946i
0.431051 − 1.32664i
−0.659965 − 2.03116i
−0.458960 − 1.41253i
0.0698401 + 0.214946i
0.431051 + 1.32664i
−1.41763 + 1.02997i
−0.166559 + 0.121012i
0.735494 − 0.534368i
2.46673 − 1.79218i
−1.41763 − 1.02997i
−0.166559 − 0.121012i
0.735494 + 0.534368i
2.46673 + 1.79218i
−0.918793 + 0.667542i 0 −0.219466 + 0.675446i 0.809017 + 0.587785i 0 1.26738 − 3.90059i −0.951141 − 2.92731i 0 −1.13569
91.2 −0.392557 + 0.285209i 0 −0.545277 + 1.67819i 0.809017 + 0.587785i 0 −0.500445 + 1.54021i −0.564470 − 1.73726i 0 −0.485227
91.3 0.991861 − 0.720629i 0 −0.153553 + 0.472586i 0.809017 + 0.587785i 0 −0.139581 + 0.429587i 0.945971 + 2.91140i 0 1.22601
91.4 1.93752 − 1.40769i 0 1.15436 − 3.55277i 0.809017 + 0.587785i 0 0.608715 − 1.87343i −1.28446 − 3.95317i 0 2.39491
136.1 −0.918793 − 0.667542i 0 −0.219466 − 0.675446i 0.809017 − 0.587785i 0 1.26738 + 3.90059i −0.951141 + 2.92731i 0 −1.13569
136.2 −0.392557 − 0.285209i 0 −0.545277 − 1.67819i 0.809017 − 0.587785i 0 −0.500445 − 1.54021i −0.564470 + 1.73726i 0 −0.485227
136.3 0.991861 + 0.720629i 0 −0.153553 − 0.472586i 0.809017 − 0.587785i 0 −0.139581 − 0.429587i 0.945971 − 2.91140i 0 1.22601
136.4 1.93752 + 1.40769i 0 1.15436 + 3.55277i 0.809017 − 0.587785i 0 0.608715 + 1.87343i −1.28446 + 3.95317i 0 2.39491
181.1 −0.850504 − 2.61758i 0 −4.51034 + 3.27695i −0.309017 + 0.951057i 0 −2.21013 + 1.60575i 7.96046 + 5.78361i 0 2.75229
181.2 −0.372637 − 1.14686i 0 0.441609 − 0.320848i −0.309017 + 0.951057i 0 −3.49122 + 2.53652i −2.48368 − 1.80450i 0 1.20588
181.3 −0.0280832 − 0.0864312i 0 1.61135 − 1.17072i −0.309017 + 0.951057i 0 1.98801 − 1.44438i −0.293484 − 0.213228i 0 0.0908791
181.4 0.633189 + 1.94876i 0 −1.77869 + 1.29229i −0.309017 + 0.951057i 0 0.477268 − 0.346756i −0.329192 − 0.239172i 0 −2.04904
361.1 −0.850504 + 2.61758i 0 −4.51034 − 3.27695i −0.309017 − 0.951057i 0 −2.21013 − 1.60575i 7.96046 − 5.78361i 0 2.75229
361.2 −0.372637 + 1.14686i 0 0.441609 + 0.320848i −0.309017 − 0.951057i 0 −3.49122 − 2.53652i −2.48368 + 1.80450i 0 1.20588
361.3 −0.0280832 + 0.0864312i 0 1.61135 + 1.17072i −0.309017 − 0.951057i 0 1.98801 + 1.44438i −0.293484 + 0.213228i 0 0.0908791
361.4 0.633189 − 1.94876i 0 −1.77869 − 1.29229i −0.309017 − 0.951057i 0 0.477268 + 0.346756i −0.329192 + 0.239172i 0 −2.04904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.h yes 16
3.b odd 2 1 495.2.n.g ✓ 16
11.c even 5 1 inner 495.2.n.h yes 16
11.c even 5 1 5445.2.a.ca 8
11.d odd 10 1 5445.2.a.cc 8
33.f even 10 1 5445.2.a.cb 8
33.h odd 10 1 495.2.n.g ✓ 16
33.h odd 10 1 5445.2.a.cd 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.n.g ✓ 16 3.b odd 2 1
495.2.n.g ✓ 16 33.h odd 10 1
495.2.n.h yes 16 1.a even 1 1 trivial
495.2.n.h yes 16 11.c even 5 1 inner
5445.2.a.ca 8 11.c even 5 1
5445.2.a.cb 8 33.f even 10 1
5445.2.a.cc 8 11.d odd 10 1
5445.2.a.cd 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 2 T_{2}^{15} + 10 T_{2}^{14} - 22 T_{2}^{13} + 61 T_{2}^{12} - 46 T_{2}^{11} + 113 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 10201 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + \cdots + 1771561 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + \cdots + 17901361 \) Copy content Toggle raw display
$19$ \( T^{16} + 4 T^{15} + \cdots + 35153041 \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} - 65 T^{6} + \cdots - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 26 T^{15} + \cdots + 3575881 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 532732561 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 15099740161 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7009547478025 \) Copy content Toggle raw display
$43$ \( (T^{8} - 14 T^{7} + \cdots - 2417279)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 20 T^{15} + \cdots + 346921 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 154209363025 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7826763961 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 213905325001 \) Copy content Toggle raw display
$67$ \( (T^{8} - 10 T^{7} + \cdots - 597971)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 81898970538025 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 9273553653001 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 32898311847025 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + 38 T^{7} + \cdots - 5696725)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 57744177883681 \) Copy content Toggle raw display
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