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Properties

Label 495.2.i.c
Level $495$
Weight $2$
Character orbit 495.i
Analytic conductor $3.953$
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] = [495,2,Mod(166,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.166");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.i (of order \(3\), degree \(2\), 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: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} + 6 x^{13} - 34 x^{12} + 82 x^{11} - 81 x^{10} - 91 x^{9} + 570 x^{8} + \cdots + 193 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{14} - \beta_{6} - \beta_{2} - 1) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{15} - \beta_{12} - \beta_{11} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{14} - \beta_{6} - \beta_{2} - 1) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 9 q^{4} - 8 q^{5} + 9 q^{6} - q^{7} + 18 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} - 9 q^{4} - 8 q^{5} + 9 q^{6} - q^{7} + 18 q^{8} + 12 q^{9} + 6 q^{10} + 8 q^{11} - 3 q^{12} - 13 q^{13} - 17 q^{14} - 3 q^{15} + q^{16} + 28 q^{17} + 39 q^{18} + 10 q^{19} - 9 q^{20} + 6 q^{21} + 3 q^{22} - 7 q^{23} - 12 q^{24} - 8 q^{25} + 4 q^{26} + 18 q^{27} + 32 q^{28} - 22 q^{29} - 9 q^{30} + 17 q^{31} + 4 q^{32} - 3 q^{33} - 17 q^{34} + 2 q^{35} + 3 q^{36} + 10 q^{37} - 14 q^{38} + 12 q^{39} - 9 q^{40} - 26 q^{41} + 15 q^{42} + 5 q^{43} - 18 q^{44} - 9 q^{45} - 12 q^{46} + 3 q^{47} - 18 q^{48} - 5 q^{49} - 3 q^{50} + 15 q^{51} - 11 q^{52} + 26 q^{53} + 18 q^{54} - 16 q^{55} - 36 q^{56} - 15 q^{57} - 23 q^{59} - 6 q^{60} - q^{61} - 4 q^{62} + 27 q^{63} - 18 q^{64} - 13 q^{65} + 3 q^{67} - 30 q^{68} - 9 q^{69} - 17 q^{70} + 34 q^{71} - 42 q^{72} - 22 q^{73} - 26 q^{74} + 3 q^{75} + 17 q^{76} + q^{77} - 9 q^{78} + 3 q^{79} - 2 q^{80} - 12 q^{81} + 58 q^{82} - 20 q^{83} - 69 q^{84} - 14 q^{85} + 11 q^{86} + 33 q^{87} + 9 q^{88} + 66 q^{89} - 9 q^{90} - 50 q^{91} + 32 q^{92} - 24 q^{93} - 22 q^{94} - 5 q^{95} + 36 q^{96} - 14 q^{97} + 46 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} + 6 x^{13} - 34 x^{12} + 82 x^{11} - 81 x^{10} - 91 x^{9} + 570 x^{8} + \cdots + 193 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!25 \nu^{15} + \cdots - 45\!\cdots\!76 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!33 \nu^{15} + \cdots - 25\!\cdots\!59 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!47 \nu^{15} + 355688587995454 \nu^{14} + \cdots - 62\!\cdots\!42 ) / 15\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!47 \nu^{15} + \cdots + 31\!\cdots\!11 ) / 15\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\!\cdots\!06 \nu^{15} + \cdots + 51\!\cdots\!05 ) / 15\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 318369684 \nu^{15} + 229555571 \nu^{14} + 390707337 \nu^{13} + 3002309077 \nu^{12} + \cdots - 236474229694 ) / 130952835387 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 66\!\cdots\!78 \nu^{15} + \cdots + 10\!\cdots\!30 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 82\!\cdots\!05 \nu^{15} + \cdots + 26\!\cdots\!99 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{15} + \cdots + 13\!\cdots\!53 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!52 \nu^{15} + \cdots - 11\!\cdots\!55 ) / 52\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!75 \nu^{15} + \cdots - 72\!\cdots\!84 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56\!\cdots\!64 \nu^{15} + \cdots + 55\!\cdots\!37 ) / 57\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!78 \nu^{15} + \cdots + 23\!\cdots\!64 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 20\!\cdots\!69 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 17\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!52 \nu^{15} + \cdots - 19\!\cdots\!50 ) / 17\!\cdots\!19 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} + \beta_{14} - 2\beta_{13} - 2\beta_{11} - \beta_{3} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} - 3 \beta_{9} + \beta_{8} - 4 \beta_{6} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{11} - 8 \beta_{10} - 3 \beta_{9} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{15} - 6 \beta_{14} - 4 \beta_{13} + 3 \beta_{12} + 15 \beta_{11} - 9 \beta_{10} + 3 \beta_{9} + \cdots + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29 \beta_{15} + 19 \beta_{14} - 7 \beta_{13} - 10 \beta_{12} - 41 \beta_{11} + 23 \beta_{10} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12 \beta_{15} - 27 \beta_{14} + 2 \beta_{13} + 7 \beta_{12} + 33 \beta_{11} + 46 \beta_{10} - 12 \beta_{9} + \cdots - 52 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{15} + 46 \beta_{14} + 89 \beta_{13} - 10 \beta_{12} - 80 \beta_{11} + 11 \beta_{10} + \cdots + 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 57 \beta_{15} - 52 \beta_{14} - 57 \beta_{13} + 76 \beta_{12} + 94 \beta_{11} - 44 \beta_{10} + 33 \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 419 \beta_{15} + 361 \beta_{14} + 133 \beta_{13} - 336 \beta_{12} - 329 \beta_{11} + 27 \beta_{10} + \cdots + 696 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21 \beta_{15} - 1116 \beta_{14} + 44 \beta_{13} + 98 \beta_{12} + 234 \beta_{11} - 127 \beta_{10} + \cdots - 1913 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 241 \beta_{15} + 2329 \beta_{14} + 940 \beta_{13} - 1154 \beta_{12} - 1199 \beta_{11} + 670 \beta_{10} + \cdots + 4994 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 48 \beta_{15} - 1896 \beta_{14} - 5482 \beta_{13} + 4422 \beta_{12} - 279 \beta_{11} - 723 \beta_{10} + \cdots - 12180 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1015 \beta_{15} + 6025 \beta_{14} + 5696 \beta_{13} - 7342 \beta_{12} + 9931 \beta_{11} - 277 \beta_{10} + \cdots + 22171 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5421 \beta_{15} - 26514 \beta_{14} + 7055 \beta_{13} + 5488 \beta_{12} - 15765 \beta_{11} + \cdots - 68638 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 33982 \beta_{15} + 34588 \beta_{14} + 6227 \beta_{13} - 14260 \beta_{12} + 42916 \beta_{11} + \cdots + 203107 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(-1 - \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1
−0.395968 − 0.576847i
−0.180193 + 0.753279i
0.892248 − 0.702363i
1.29880 − 0.0735422i
1.35669 + 1.17980i
0.545858 + 1.91602i
−0.253616 − 2.09142i
−2.26382 − 0.404936i
−0.395968 + 0.576847i
−0.180193 − 0.753279i
0.892248 + 0.702363i
1.29880 + 0.0735422i
1.35669 − 1.17980i
0.545858 − 1.91602i
−0.253616 + 2.09142i
−2.26382 + 0.404936i
−1.25523 − 2.17412i −1.14147 − 1.30271i −2.15120 + 3.72598i −0.500000 + 0.866025i −1.39944 + 4.11689i −1.32630 − 2.29721i 5.78006 −0.394092 + 2.97400i 2.51046
166.2 −1.20005 − 2.07855i 0.410968 + 1.68259i −1.88024 + 3.25668i −0.500000 + 0.866025i 3.00416 − 2.87341i −1.72643 − 2.99026i 4.22535 −2.66221 + 1.38298i 2.40010
166.3 −0.855290 − 1.48141i 1.63410 + 0.574207i −0.463042 + 0.802012i −0.500000 + 0.866025i −0.546997 − 2.91188i 1.71724 + 2.97434i −1.83702 2.34057 + 1.87662i 1.71058
166.4 −0.653858 − 1.13251i −1.59334 + 0.679170i 0.144940 − 0.251044i −0.500000 + 0.866025i 1.81099 + 1.36040i −1.01043 − 1.75012i −2.99451 2.07746 − 2.16429i 1.30772
166.5 0.116272 + 0.201389i −1.58947 − 0.688180i 0.972962 − 1.68522i −0.500000 + 0.866025i −0.0462187 − 0.400118i 1.29853 + 2.24911i 0.917601 2.05282 + 2.18768i −0.232544
166.6 0.484555 + 0.839274i 1.71377 + 0.251014i 0.530413 − 0.918702i −0.500000 + 0.866025i 0.619744 + 1.55995i −1.59232 − 2.75799i 2.96628 2.87398 + 0.860359i −0.969110
166.7 0.719179 + 1.24565i −0.878286 + 1.49285i −0.0344369 + 0.0596464i −0.500000 + 0.866025i −2.49123 − 0.0204111i 1.19158 + 2.06387i 2.77765 −1.45723 − 2.62231i −1.43836
166.8 1.14442 + 1.98219i 1.44373 − 0.956896i −1.61940 + 2.80488i −0.500000 + 0.866025i 3.54899 + 1.76666i 0.948144 + 1.64223i −2.83540 1.16870 − 2.76299i −2.28884
331.1 −1.25523 + 2.17412i −1.14147 + 1.30271i −2.15120 − 3.72598i −0.500000 − 0.866025i −1.39944 − 4.11689i −1.32630 + 2.29721i 5.78006 −0.394092 − 2.97400i 2.51046
331.2 −1.20005 + 2.07855i 0.410968 − 1.68259i −1.88024 − 3.25668i −0.500000 − 0.866025i 3.00416 + 2.87341i −1.72643 + 2.99026i 4.22535 −2.66221 − 1.38298i 2.40010
331.3 −0.855290 + 1.48141i 1.63410 − 0.574207i −0.463042 − 0.802012i −0.500000 − 0.866025i −0.546997 + 2.91188i 1.71724 − 2.97434i −1.83702 2.34057 − 1.87662i 1.71058
331.4 −0.653858 + 1.13251i −1.59334 − 0.679170i 0.144940 + 0.251044i −0.500000 − 0.866025i 1.81099 − 1.36040i −1.01043 + 1.75012i −2.99451 2.07746 + 2.16429i 1.30772
331.5 0.116272 − 0.201389i −1.58947 + 0.688180i 0.972962 + 1.68522i −0.500000 − 0.866025i −0.0462187 + 0.400118i 1.29853 − 2.24911i 0.917601 2.05282 − 2.18768i −0.232544
331.6 0.484555 − 0.839274i 1.71377 − 0.251014i 0.530413 + 0.918702i −0.500000 − 0.866025i 0.619744 − 1.55995i −1.59232 + 2.75799i 2.96628 2.87398 − 0.860359i −0.969110
331.7 0.719179 − 1.24565i −0.878286 − 1.49285i −0.0344369 − 0.0596464i −0.500000 − 0.866025i −2.49123 + 0.0204111i 1.19158 − 2.06387i 2.77765 −1.45723 + 2.62231i −1.43836
331.8 1.14442 − 1.98219i 1.44373 + 0.956896i −1.61940 − 2.80488i −0.500000 − 0.866025i 3.54899 − 1.76666i 0.948144 − 1.64223i −2.83540 1.16870 + 2.76299i −2.28884
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 166.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.i.c ✓ 16
3.b odd 2 1 1485.2.i.c 16
9.c even 3 1 inner 495.2.i.c ✓ 16
9.c even 3 1 4455.2.a.p 8
9.d odd 6 1 1485.2.i.c 16
9.d odd 6 1 4455.2.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.i.c ✓ 16 1.a even 1 1 trivial
495.2.i.c ✓ 16 9.c even 3 1 inner
1485.2.i.c 16 3.b odd 2 1
1485.2.i.c 16 9.d odd 6 1
4455.2.a.o 8 9.d odd 6 1
4455.2.a.p 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{16} + 3 T_{2}^{15} + 17 T_{2}^{14} + 28 T_{2}^{13} + 125 T_{2}^{12} + 168 T_{2}^{11} + 596 T_{2}^{10} + \cdots + 100 \) Copy content Toggle raw display
\( T_{13}^{16} + 13 T_{13}^{15} + 142 T_{13}^{14} + 835 T_{13}^{13} + 4956 T_{13}^{12} + 20533 T_{13}^{11} + \cdots + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} + \cdots + 5645376 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} + 13 T^{15} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T^{8} - 14 T^{7} + \cdots + 90)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} - 48 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 7 T^{15} + \cdots + 66049 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 31085216100 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 26346159225 \) Copy content Toggle raw display
$37$ \( (T^{8} - 5 T^{7} + \cdots - 188765)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 147185424 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1839895236 \) Copy content Toggle raw display
$47$ \( T^{16} - 3 T^{15} + \cdots + 51451929 \) Copy content Toggle raw display
$53$ \( (T^{8} - 13 T^{7} + \cdots - 828887)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 679406196121 \) Copy content Toggle raw display
$61$ \( T^{16} + T^{15} + \cdots + 18028516 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4466516224 \) Copy content Toggle raw display
$71$ \( (T^{8} - 17 T^{7} + \cdots + 38047)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 11 T^{7} + \cdots + 49246)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 311620732900 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 83651254963236 \) Copy content Toggle raw display
$89$ \( (T^{8} - 33 T^{7} + \cdots + 10090315)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9998200081 \) Copy content Toggle raw display
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