↑

Properties

Label 494.2.d.c
Level $494$
Weight $2$
Character orbit 494.d
Analytic conductor $3.945$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [494,2,Mod(77,494)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(494, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("494.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.d (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: \(3.94460985985\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 36x^{12} + 492x^{10} + 3171x^{8} + 9678x^{6} + 11765x^{4} + 1893x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{5} q^{3} - q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - \beta_{13} q^{7} + \beta_{6} q^{8} + (\beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{5} q^{3} - q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - \beta_{13} q^{7} + \beta_{6} q^{8} + (\beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{3} - 14 q^{4} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{3} - 14 q^{4} + 30 q^{9} - 4 q^{10} - 4 q^{12} - 2 q^{13} + 2 q^{14} + 14 q^{16} - 32 q^{17} + 10 q^{22} - 6 q^{23} - 14 q^{25} + 10 q^{26} + 10 q^{27} - 14 q^{29} - 26 q^{30} + 2 q^{35} - 30 q^{36} - 14 q^{38} + 36 q^{39} + 4 q^{40} - 6 q^{42} - 46 q^{43} + 4 q^{48} - 28 q^{49} + 32 q^{51} + 2 q^{52} + 42 q^{53} + 16 q^{55} - 2 q^{56} - 12 q^{61} - 22 q^{62} - 14 q^{64} + 2 q^{65} + 12 q^{66} + 32 q^{68} - 96 q^{69} - 6 q^{74} - 10 q^{75} + 52 q^{77} + 22 q^{78} - 8 q^{79} + 102 q^{81} - 30 q^{82} - 32 q^{87} - 10 q^{88} + 76 q^{90} + 22 q^{91} + 6 q^{92} - 2 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 36x^{12} + 492x^{10} + 3171x^{8} + 9678x^{6} + 11765x^{4} + 1893x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{12} + 297\nu^{10} + 9157\nu^{8} + 88140\nu^{6} + 339290\nu^{4} + 427475\nu^{2} - 28116 ) / 14920 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{12} + 707\nu^{10} + 8375\nu^{8} + 49372\nu^{6} + 152014\nu^{4} + 204025\nu^{2} + 42484 ) / 14920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\nu^{12} + 2623\nu^{10} + 35383\nu^{8} + 246712\nu^{6} + 885210\nu^{4} + 1296349\nu^{2} + 125684 ) / 29840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\nu^{12} + 1011\nu^{10} + 10539\nu^{8} + 47432\nu^{6} + 82450\nu^{4} + 26409\nu^{2} - 220 ) / 5968 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\nu^{13} + 2015\nu^{11} + 28071\nu^{9} + 184944\nu^{7} + 579722\nu^{5} + 729525\nu^{3} + 130524\nu ) / 5968 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 851 \nu^{13} + 183 \nu^{12} + 30635 \nu^{11} + 7247 \nu^{10} + 420283 \nu^{9} + 103871 \nu^{8} + \cdots + 97748 ) / 59680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 851 \nu^{13} - 183 \nu^{12} + 30635 \nu^{11} - 7247 \nu^{10} + 420283 \nu^{9} - 103871 \nu^{8} + \cdots - 97748 ) / 59680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 477 \nu^{13} - 16933 \nu^{11} - 227013 \nu^{9} - 1426448 \nu^{7} - 4233430 \nu^{5} + \cdots - 1265628 \nu ) / 29840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1693 \nu^{13} - 137 \nu^{12} - 60669 \nu^{11} - 5833 \nu^{10} - 826261 \nu^{9} - 87121 \nu^{8} + \cdots - 132140 ) / 59680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1693 \nu^{13} - 137 \nu^{12} + 60669 \nu^{11} - 5833 \nu^{10} + 826261 \nu^{9} - 87121 \nu^{8} + \cdots - 132140 ) / 59680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 487 \nu^{13} + 17435 \nu^{11} + 237271 \nu^{9} + 1525044 \nu^{7} + 4647678 \nu^{5} + \cdots + 920700 \nu ) / 14920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 807 \nu^{13} + 29023 \nu^{11} + 395439 \nu^{9} + 2533128 \nu^{7} + 7649098 \nu^{5} + \cdots + 1347532 \nu ) / 14920 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 2\beta_{12} - 2\beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{11} + 10\beta_{10} - 11\beta_{8} + 11\beta_{7} - \beta_{5} + 2\beta_{4} - 14\beta_{3} - 3\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 17 \beta_{13} - 25 \beta_{12} + 29 \beta_{11} - 29 \beta_{10} - 17 \beta_{9} - 18 \beta_{8} + \cdots + 79 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 97 \beta_{11} - 97 \beta_{10} + 111 \beta_{8} - 111 \beta_{7} + 11 \beta_{5} - 33 \beta_{4} + \cdots - 340 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 220 \beta_{13} + 261 \beta_{12} - 345 \beta_{11} + 345 \beta_{10} + 215 \beta_{9} + 246 \beta_{8} + \cdots - 838 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 957 \beta_{11} + 957 \beta_{10} - 1114 \beta_{8} + 1114 \beta_{7} - 97 \beta_{5} + 434 \beta_{4} + \cdots + 3612 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2616 \beta_{13} - 2639 \beta_{12} + 3887 \beta_{11} - 3887 \beta_{10} - 2496 \beta_{9} + \cdots + 9090 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9690 \beta_{11} - 9690 \beta_{10} + 11345 \beta_{8} - 11345 \beta_{7} + 804 \beta_{5} - 5255 \beta_{4} + \cdots - 39142 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 29990 \beta_{13} + 26804 \beta_{12} - 42990 \beta_{11} + 42990 \beta_{10} + 28045 \beta_{9} + \cdots - 99207 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 100388 \beta_{11} + 100388 \beta_{10} - 117536 \beta_{8} + 117536 \beta_{7} - 6397 \beta_{5} + \cdots + 426718 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 337419 \beta_{13} - 275765 \beta_{12} + 472103 \beta_{11} - 472103 \beta_{10} - 310594 \beta_{9} + \cdots + 1084135 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/494\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(457\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
3.29705i
1.95679i
0.431028i
0.0462763i
− 1.77366i
− 2.64643i
− 3.31105i
− 3.29705i
− 1.95679i
− 0.431028i
− 0.0462763i
1.77366i
2.64643i
3.31105i
− 1.00000i −3.29705 −1.00000 3.53833i 3.29705i − 0.150781i 1.00000i 7.87054 3.53833
77.2 − 1.00000i −1.95679 −1.00000 − 0.613661i 1.95679i 0.465830i 1.00000i 0.829012 −0.613661
77.3 − 1.00000i −0.431028 −1.00000 − 0.638027i 0.431028i 2.88689i 1.00000i −2.81422 −0.638027
77.4 − 1.00000i −0.0462763 −1.00000 − 2.81822i 0.0462763i − 4.11903i 1.00000i −2.99786 −2.81822
77.5 − 1.00000i 1.77366 −1.00000 0.491653i − 1.77366i 4.65991i 1.00000i 0.145874 0.491653
77.6 − 1.00000i 2.64643 −1.00000 − 4.02894i − 2.64643i 1.07639i 1.00000i 4.00358 −4.02894
77.7 − 1.00000i 3.31105 −1.00000 2.06886i − 3.31105i − 3.81920i 1.00000i 7.96307 2.06886
77.8 1.00000i −3.29705 −1.00000 − 3.53833i − 3.29705i 0.150781i − 1.00000i 7.87054 3.53833
77.9 1.00000i −1.95679 −1.00000 0.613661i − 1.95679i − 0.465830i − 1.00000i 0.829012 −0.613661
77.10 1.00000i −0.431028 −1.00000 0.638027i − 0.431028i − 2.88689i − 1.00000i −2.81422 −0.638027
77.11 1.00000i −0.0462763 −1.00000 2.81822i − 0.0462763i 4.11903i − 1.00000i −2.99786 −2.81822
77.12 1.00000i 1.77366 −1.00000 − 0.491653i 1.77366i − 4.65991i − 1.00000i 0.145874 0.491653
77.13 1.00000i 2.64643 −1.00000 4.02894i 2.64643i − 1.07639i − 1.00000i 4.00358 −4.02894
77.14 1.00000i 3.31105 −1.00000 − 2.06886i 3.31105i 3.81920i − 1.00000i 7.96307 2.06886
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.d.c ✓ 14
13.b even 2 1 inner 494.2.d.c ✓ 14
13.d odd 4 1 6422.2.a.be 7
13.d odd 4 1 6422.2.a.bf 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.d.c ✓ 14 1.a even 1 1 trivial
494.2.d.c ✓ 14 13.b even 2 1 inner
6422.2.a.be 7 13.d odd 4 1
6422.2.a.bf 7 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 2T_{3}^{6} - 16T_{3}^{5} + 29T_{3}^{4} + 60T_{3}^{3} - 79T_{3}^{2} - 47T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T^{7} - 2 T^{6} - 16 T^{5} + \cdots - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + 42 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{14} + 63 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{14} + 93 T^{12} + \cdots + 11075584 \) Copy content Toggle raw display
$13$ \( T^{14} + 2 T^{13} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} + 16 T^{6} + \cdots - 22622)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$23$ \( (T^{7} + 3 T^{6} + \cdots - 14944)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 7 T^{6} + \cdots - 1352)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 29948379136 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1321467904 \) Copy content Toggle raw display
$41$ \( T^{14} + 237 T^{12} + \cdots + 17305600 \) Copy content Toggle raw display
$43$ \( (T^{7} + 23 T^{6} + \cdots - 3328)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 11657089024 \) Copy content Toggle raw display
$53$ \( (T^{7} - 21 T^{6} + \cdots + 100256)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 5184109674496 \) Copy content Toggle raw display
$61$ \( (T^{7} + 6 T^{6} + \cdots - 58240)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 720298479616 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 6206918656 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 476519612416 \) Copy content Toggle raw display
$79$ \( (T^{7} + 4 T^{6} + \cdots - 2882560)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 39167570560000 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 182648521621504 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 7180189696 \) Copy content Toggle raw display
show more
show less