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Properties

Label 494.2.m.a
Level $494$
Weight $2$
Character orbit 494.m
Analytic conductor $3.945$
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] = [494,2,Mod(153,494)]
 
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([5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("494.153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.m (of order \(6\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 198x^{12} + 718x^{10} + 1229x^{8} + 990x^{6} + 373x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 198x^{12} + 718x^{10} + 1229x^{8} + 990x^{6} + 373x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 61 \nu^{14} - 1427 \nu^{12} - 11191 \nu^{10} - 36549 \nu^{8} - 49739 \nu^{6} - 23005 \nu^{4} + \cdots - 526 ) / 1308 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 263 \nu^{15} + 6190 \nu^{13} + 49220 \nu^{11} + 166452 \nu^{9} + 250129 \nu^{7} + 160892 \nu^{5} + \cdots + 1308 ) / 2616 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4757 \nu^{15} + 5712 \nu^{14} - 112864 \nu^{13} + 135714 \nu^{12} - 910802 \nu^{11} + 1098426 \nu^{10} + \cdots + 99816 ) / 2616 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 661 \nu^{14} - 15690 \nu^{12} - 126745 \nu^{10} - 441168 \nu^{8} - 695726 \nu^{6} - 469721 \nu^{4} + \cdots - 9974 ) / 218 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11007 \nu^{15} - 3716 \nu^{14} + 260574 \nu^{13} - 87970 \nu^{12} + 2094270 \nu^{11} - 707036 \nu^{10} + \cdots - 43772 ) / 2616 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7291 \nu^{15} + 7932 \nu^{14} + 172604 \nu^{13} + 188280 \nu^{12} + 1387234 \nu^{11} + 1520940 \nu^{10} + \cdots + 123612 ) / 2616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14961 \nu^{15} - 13516 \nu^{14} + 355098 \nu^{13} - 320678 \nu^{12} + 2868138 \nu^{11} + \cdots - 203176 ) / 2616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5102 \nu^{15} + 3841 \nu^{14} + 121117 \nu^{13} + 91055 \nu^{12} + 978668 \nu^{11} + 733753 \nu^{10} + \cdots + 51154 ) / 1308 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8616 \nu^{15} + 125 \nu^{14} - 204324 \nu^{13} + 3085 \nu^{12} - 1647675 \nu^{11} + 26717 \nu^{10} + \cdots + 9998 ) / 1308 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4345 \nu^{15} - 20122 \nu^{14} - 104030 \nu^{13} - 477488 \nu^{12} - 854140 \nu^{11} + \cdots - 297964 ) / 2616 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14961 \nu^{15} - 13516 \nu^{14} - 355098 \nu^{13} - 320678 \nu^{12} - 2868138 \nu^{11} + \cdots - 203176 ) / 2616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5308 \nu^{15} - 13103 \nu^{14} - 125534 \nu^{13} - 311113 \nu^{12} - 1006999 \nu^{11} + \cdots - 206798 ) / 1308 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 20327 \nu^{15} + 7224 \nu^{14} - 482482 \nu^{13} + 171369 \nu^{12} - 3897362 \nu^{11} + \cdots + 102540 ) / 1308 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20327 \nu^{15} - 7224 \nu^{14} - 482482 \nu^{13} - 171369 \nu^{12} - 3897362 \nu^{11} + \cdots - 102540 ) / 1308 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{7} + 2\beta_{5} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} + \beta_{13} + 3 \beta_{12} - \beta_{11} - 2 \beta_{10} - 3 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} - 6 \beta_{9} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{15} + 12 \beta_{14} - 10 \beta_{13} - 37 \beta_{12} + 10 \beta_{11} + 34 \beta_{10} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 44 \beta_{15} + 44 \beta_{14} + 34 \beta_{13} - 36 \beta_{12} + 34 \beta_{11} + 108 \beta_{10} + \cdots - 232 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 121 \beta_{15} - 121 \beta_{14} + 108 \beta_{13} + 387 \beta_{12} - 108 \beta_{11} - 460 \beta_{10} + \cdots - 817 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 545 \beta_{15} - 545 \beta_{14} - 460 \beta_{13} + 496 \beta_{12} - 460 \beta_{11} - 1213 \beta_{10} + \cdots + 2704 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1225 \beta_{15} + 1225 \beta_{14} - 1213 \beta_{13} - 4050 \beta_{12} + 1213 \beta_{11} + 5702 \beta_{10} + \cdots + 9879 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6431 \beta_{15} + 6431 \beta_{14} + 5702 \beta_{13} - 6198 \beta_{12} + 5702 \beta_{11} + \cdots - 31230 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12833 \beta_{15} - 12833 \beta_{14} + 13756 \beta_{13} + 43472 \beta_{12} - 13756 \beta_{11} + \cdots - 115726 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 74373 \beta_{15} - 74373 \beta_{14} - 67738 \beta_{13} + 73936 \beta_{12} - 67738 \beta_{11} + \cdots + 357904 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 138562 \beta_{15} + 138562 \beta_{14} - 156193 \beta_{13} - 476789 \beta_{12} + 156193 \beta_{11} + \cdots + 1334500 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 851814 \beta_{15} + 851814 \beta_{14} + 787188 \beta_{13} - 861124 \beta_{12} + 787188 \beta_{11} + \cdots - 4081602 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1526902 \beta_{15} - 1526902 \beta_{14} + 1772675 \beta_{13} + 5303268 \beta_{12} - 1772675 \beta_{11} + \cdots - 15264161 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(287\) \(457\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
153.1
1.40354i
− 0.394661i
− 0.533995i
− 2.47489i
− 1.74849i
− 0.516521i
0.898616i
3.36639i
− 1.40354i
0.394661i
0.533995i
2.47489i
1.74849i
0.516521i
− 0.898616i
− 3.36639i
−0.866025 + 0.500000i −0.701771 − 1.21550i 0.500000 − 0.866025i 0.425845i 1.21550 + 0.701771i −2.53322 − 1.46256i 1.00000i 0.515036 − 0.892068i −0.212922 − 0.368793i
153.2 −0.866025 + 0.500000i 0.197330 + 0.341786i 0.500000 − 0.866025i 0.507765i −0.341786 − 0.197330i 1.08214 + 0.624774i 1.00000i 1.42212 − 2.46319i −0.253882 − 0.439737i
153.3 −0.866025 + 0.500000i 0.266997 + 0.462453i 0.500000 − 0.866025i − 2.79724i −0.462453 − 0.266997i 0.783658 + 0.452445i 1.00000i 1.35742 − 2.35113i 1.39862 + 2.42248i
153.4 −0.866025 + 0.500000i 1.23744 + 2.14331i 0.500000 − 0.866025i 2.86363i −2.14331 − 1.23744i 1.53345 + 0.885339i 1.00000i −1.56253 + 2.70638i −1.43182 − 2.47998i
153.5 0.866025 − 0.500000i −0.874243 − 1.51423i 0.500000 − 0.866025i − 2.50715i −1.51423 − 0.874243i 1.02707 + 0.592981i − 1.00000i −0.0286022 + 0.0495405i −1.25358 − 2.17126i
153.6 0.866025 − 0.500000i −0.258260 − 0.447320i 0.500000 − 0.866025i 1.53223i −0.447320 − 0.258260i −3.26735 − 1.88641i − 1.00000i 1.36660 − 2.36703i 0.766113 + 1.32695i
153.7 0.866025 − 0.500000i 0.449308 + 0.778224i 0.500000 − 0.866025i − 0.684128i 0.778224 + 0.449308i 2.72077 + 1.57084i − 1.00000i 1.09625 − 1.89875i −0.342064 − 0.592472i
153.8 0.866025 − 0.500000i 1.68320 + 2.91538i 0.500000 − 0.866025i 0.659053i 2.91538 + 1.68320i −1.34652 − 0.777413i − 1.00000i −4.16630 + 7.21624i 0.329526 + 0.570756i
381.1 −0.866025 − 0.500000i −0.701771 + 1.21550i 0.500000 + 0.866025i − 0.425845i 1.21550 − 0.701771i −2.53322 + 1.46256i − 1.00000i 0.515036 + 0.892068i −0.212922 + 0.368793i
381.2 −0.866025 − 0.500000i 0.197330 − 0.341786i 0.500000 + 0.866025i − 0.507765i −0.341786 + 0.197330i 1.08214 − 0.624774i − 1.00000i 1.42212 + 2.46319i −0.253882 + 0.439737i
381.3 −0.866025 − 0.500000i 0.266997 − 0.462453i 0.500000 + 0.866025i 2.79724i −0.462453 + 0.266997i 0.783658 − 0.452445i − 1.00000i 1.35742 + 2.35113i 1.39862 − 2.42248i
381.4 −0.866025 − 0.500000i 1.23744 − 2.14331i 0.500000 + 0.866025i − 2.86363i −2.14331 + 1.23744i 1.53345 − 0.885339i − 1.00000i −1.56253 − 2.70638i −1.43182 + 2.47998i
381.5 0.866025 + 0.500000i −0.874243 + 1.51423i 0.500000 + 0.866025i 2.50715i −1.51423 + 0.874243i 1.02707 − 0.592981i 1.00000i −0.0286022 − 0.0495405i −1.25358 + 2.17126i
381.6 0.866025 + 0.500000i −0.258260 + 0.447320i 0.500000 + 0.866025i − 1.53223i −0.447320 + 0.258260i −3.26735 + 1.88641i 1.00000i 1.36660 + 2.36703i 0.766113 − 1.32695i
381.7 0.866025 + 0.500000i 0.449308 − 0.778224i 0.500000 + 0.866025i 0.684128i 0.778224 − 0.449308i 2.72077 − 1.57084i 1.00000i 1.09625 + 1.89875i −0.342064 + 0.592472i
381.8 0.866025 + 0.500000i 1.68320 − 2.91538i 0.500000 + 0.866025i − 0.659053i 2.91538 − 1.68320i −1.34652 + 0.777413i 1.00000i −4.16630 − 7.21624i 0.329526 − 0.570756i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 153.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.m.a ✓ 16
13.e even 6 1 inner 494.2.m.a ✓ 16
13.f odd 12 1 6422.2.a.bg 8
13.f odd 12 1 6422.2.a.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.a ✓ 16 1.a even 1 1 trivial
494.2.m.a ✓ 16 13.e even 6 1 inner
6422.2.a.bg 8 13.f odd 12 1
6422.2.a.bh 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 4 T_{3}^{15} + 20 T_{3}^{14} - 28 T_{3}^{13} + 101 T_{3}^{12} - 92 T_{3}^{11} + 398 T_{3}^{10} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{16} + 26 T^{14} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{16} - 21 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + \cdots + 45796 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{15} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + 8 T^{15} + \cdots + 952576 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 101344489 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2117472256 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 36790308864 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 6746678968969 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 148840000 \) Copy content Toggle raw display
$47$ \( T^{16} + 226 T^{14} + \cdots + 248004 \) Copy content Toggle raw display
$53$ \( (T^{8} + 24 T^{7} + \cdots - 36195267)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 979826779044 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 239723289 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 38253101056 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 555868660356 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 52197170089 \) Copy content Toggle raw display
$79$ \( (T^{8} + 22 T^{7} + \cdots + 63006)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 4739093886916 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 298874944 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 13747093504 \) Copy content Toggle raw display
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