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Properties

Label 494.2.g.f
Level $494$
Weight $2$
Character orbit 494.g
Analytic conductor $3.945$
Analytic rank $0$
Dimension $12$
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] = [494,2,Mod(191,494)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(494, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("494.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.g (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: \(3.94460985985\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 16 x^{10} - 24 x^{9} + 179 x^{8} - 260 x^{7} + 694 x^{6} - 274 x^{5} + 805 x^{4} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 16 x^{10} - 24 x^{9} + 179 x^{8} - 260 x^{7} + 694 x^{6} - 274 x^{5} + 805 x^{4} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6148925 \nu^{11} + 39704558 \nu^{10} + 54890656 \nu^{9} + 517730002 \nu^{8} + \cdots - 9889208120 ) / 43894848351 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 473030975 \nu^{11} - 4207618851 \nu^{10} + 1097312037 \nu^{9} - 67239416991 \nu^{8} + \cdots - 1245412482180 ) / 263369090106 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1264189688 \nu^{11} + 4488085125 \nu^{10} - 23263010049 \nu^{9} + 59112823572 \nu^{8} + \cdots + 929568403371 ) / 263369090106 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1903241638 \nu^{11} - 791556792 \nu^{10} - 23647745724 \nu^{9} - 22105303473 \nu^{8} + \cdots - 180520004622 ) / 263369090106 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10944040468 \nu^{11} - 18875362001 \nu^{10} + 169612171173 \nu^{9} - 220415569002 \nu^{8} + \cdots - 144662197400 ) / 1316845450530 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1977841624 \nu^{11} - 3924938623 \nu^{10} + 31843988774 \nu^{9} - 47193745696 \nu^{8} + \cdots - 30663762825 ) / 219474241755 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9754759538 \nu^{11} - 14445679241 \nu^{10} + 148525612523 \nu^{9} - 157850808917 \nu^{8} + \cdots + 444515179225 ) / 438948483510 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42513126401 \nu^{11} + 92809420677 \nu^{10} - 683626416456 \nu^{9} + 1132613302929 \nu^{8} + \cdots + 6754292010450 ) / 1316845450530 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43274459454 \nu^{11} + 78022261103 \nu^{10} - 672815139909 \nu^{9} + 896549331711 \nu^{8} + \cdots - 907511630875 ) / 1316845450530 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51555102314 \nu^{11} + 93926626328 \nu^{10} - 810991334469 \nu^{9} + 1100616040221 \nu^{8} + \cdots + 2214603611525 ) / 1316845450530 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 4\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + 9\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{11} - 9 \beta_{10} + 11 \beta_{9} - 2 \beta_{8} - 45 \beta_{7} + 13 \beta_{6} + 2 \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 27 \beta_{8} + 17 \beta_{7} + 27 \beta_{6} - 27 \beta_{5} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{11} - 60 \beta_{10} - 89 \beta_{8} + 89 \beta_{7} - 149 \beta_{5} - 113 \beta_{4} + 36 \beta_{3} + \cdots + 425 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 312 \beta_{11} - 50 \beta_{10} + 50 \beta_{9} - 262 \beta_{8} - 279 \beta_{7} - 308 \beta_{6} + \cdots - 262 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 468 \beta_{11} + 1634 \beta_{10} - 1180 \beta_{9} + 1184 \beta_{8} + 3818 \beta_{7} - 1630 \beta_{6} + \cdots - 4268 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1162 \beta_{11} + 1162 \beta_{10} - 645 \beta_{8} + 645 \beta_{7} + 3331 \beta_{5} - 3463 \beta_{4} + \cdots + 1620 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12598 \beta_{11} - 9652 \beta_{10} + 12466 \beta_{9} - 2946 \beta_{8} - 50067 \beta_{7} + 17476 \beta_{6} + \cdots - 2946 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 50707 \beta_{11} - 5167 \beta_{10} - 7799 \beta_{9} + 37873 \beta_{8} + 32364 \beta_{7} + 35241 \beta_{6} + \cdots + 10676 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(287\) \(457\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.61923 − 2.80458i
1.09882 − 1.90322i
0.362179 − 0.627313i
0.130140 − 0.225410i
−0.566802 + 0.981729i
−1.64357 + 2.84674i
1.61923 + 2.80458i
1.09882 + 1.90322i
0.362179 + 0.627313i
0.130140 + 0.225410i
−0.566802 − 0.981729i
−1.64357 − 2.84674i
−0.500000 + 0.866025i −1.61923 + 2.80458i −0.500000 − 0.866025i −0.399620 −1.61923 − 2.80458i −0.558073 − 0.966610i 1.00000 −3.74379 − 6.48444i 0.199810 − 0.346081i
191.2 −0.500000 + 0.866025i −1.09882 + 1.90322i −0.500000 − 0.866025i 3.12703 −1.09882 − 1.90322i −1.08972 − 1.88745i 1.00000 −0.914826 − 1.58453i −1.56352 + 2.70809i
191.3 −0.500000 + 0.866025i −0.362179 + 0.627313i −0.500000 − 0.866025i −1.67578 −0.362179 − 0.627313i 1.23105 + 2.13225i 1.00000 1.23765 + 2.14368i 0.837892 − 1.45127i
191.4 −0.500000 + 0.866025i −0.130140 + 0.225410i −0.500000 − 0.866025i −4.36375 −0.130140 − 0.225410i −0.387567 − 0.671286i 1.00000 1.46613 + 2.53941i 2.18188 − 3.77912i
191.5 −0.500000 + 0.866025i 0.566802 − 0.981729i −0.500000 − 0.866025i 2.43014 0.566802 + 0.981729i 1.81634 + 3.14600i 1.00000 0.857472 + 1.48518i −1.21507 + 2.10456i
191.6 −0.500000 + 0.866025i 1.64357 − 2.84674i −0.500000 − 0.866025i 2.88198 1.64357 + 2.84674i −1.51204 − 2.61893i 1.00000 −3.90263 − 6.75956i −1.44099 + 2.49587i
419.1 −0.500000 − 0.866025i −1.61923 − 2.80458i −0.500000 + 0.866025i −0.399620 −1.61923 + 2.80458i −0.558073 + 0.966610i 1.00000 −3.74379 + 6.48444i 0.199810 + 0.346081i
419.2 −0.500000 − 0.866025i −1.09882 − 1.90322i −0.500000 + 0.866025i 3.12703 −1.09882 + 1.90322i −1.08972 + 1.88745i 1.00000 −0.914826 + 1.58453i −1.56352 − 2.70809i
419.3 −0.500000 − 0.866025i −0.362179 − 0.627313i −0.500000 + 0.866025i −1.67578 −0.362179 + 0.627313i 1.23105 − 2.13225i 1.00000 1.23765 − 2.14368i 0.837892 + 1.45127i
419.4 −0.500000 − 0.866025i −0.130140 − 0.225410i −0.500000 + 0.866025i −4.36375 −0.130140 + 0.225410i −0.387567 + 0.671286i 1.00000 1.46613 − 2.53941i 2.18188 + 3.77912i
419.5 −0.500000 − 0.866025i 0.566802 + 0.981729i −0.500000 + 0.866025i 2.43014 0.566802 − 0.981729i 1.81634 − 3.14600i 1.00000 0.857472 − 1.48518i −1.21507 − 2.10456i
419.6 −0.500000 − 0.866025i 1.64357 + 2.84674i −0.500000 + 0.866025i 2.88198 1.64357 − 2.84674i −1.51204 + 2.61893i 1.00000 −3.90263 + 6.75956i −1.44099 − 2.49587i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.g.f ✓ 12
13.c even 3 1 inner 494.2.g.f ✓ 12
13.c even 3 1 6422.2.a.bd 6
13.e even 6 1 6422.2.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.f ✓ 12 1.a even 1 1 trivial
494.2.g.f ✓ 12 13.c even 3 1 inner
6422.2.a.bc 6 13.e even 6 1
6422.2.a.bd 6 13.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 \(S_{2}^{\mathrm{new}}(494, [\chi])\):

\( T_{3}^{12} + 2 T_{3}^{11} + 16 T_{3}^{10} + 24 T_{3}^{9} + 179 T_{3}^{8} + 260 T_{3}^{7} + 694 T_{3}^{6} + \cdots + 25 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 21T_{5}^{4} + 51T_{5}^{3} + 64T_{5}^{2} - 144T_{5} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{6} - 2 T^{5} - 21 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 2601 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 24 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 2 T^{11} + \cdots + 14400 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} + \cdots + 18769 \) Copy content Toggle raw display
$29$ \( T^{12} + 20 T^{11} + \cdots + 1369 \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots - 16376)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1857265216 \) Copy content Toggle raw display
$41$ \( T^{12} + 3 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$43$ \( T^{12} + 13 T^{11} + \cdots + 40000 \) Copy content Toggle raw display
$47$ \( (T^{6} - 20 T^{5} + \cdots - 919)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 25 T^{5} + \cdots - 99821)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1807355169 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 33906066496 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 29752110144 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 13472709184 \) Copy content Toggle raw display
$73$ \( (T^{6} - 7 T^{5} + \cdots + 3033)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 18 T^{5} + \cdots - 87032)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 7 T^{5} + \cdots - 92760)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 9 T^{11} + \cdots + 419904 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1324598431744 \) Copy content Toggle raw display
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