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Properties

Label 99.2.e.e
Level $99$
Weight $2$
Character orbit 99.e
Analytic conductor $0.791$
Analytic rank $0$
Dimension $8$
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] = [99,2,Mod(34,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.e (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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 79\nu^{6} - 177\nu^{5} + 459\nu^{4} - 1008\nu^{3} + 1011\nu^{2} - 752\nu - 478 ) / 933 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\nu^{7} + 164\nu^{6} - 395\nu^{5} + 260\nu^{4} - 2687\nu^{3} + 2894\nu^{2} + 1604\nu - 2193 ) / 1866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 217\nu^{7} + 146\nu^{6} + 39\nu^{5} - 876\nu^{4} - 4095\nu^{3} + 6498\nu^{2} + 1610\nu - 2525 ) / 5598 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 241\nu^{7} - 328\nu^{6} + 1101\nu^{5} - 3630\nu^{4} + 1953\nu^{3} - 5166\nu^{2} + 6122\nu + 343 ) / 5598 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -145\nu^{7} + 298\nu^{6} - 585\nu^{5} + 1944\nu^{4} - 2019\nu^{3} - 438\nu^{2} + 730\nu + 1799 ) / 1866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 701\nu^{7} - 1016\nu^{6} + 1863\nu^{5} - 6966\nu^{4} + 2181\nu^{3} + 10122\nu^{2} - 6296\nu - 6265 ) / 5598 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 5\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 6\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + 12\beta_{6} - \beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta_{2} - \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -29\beta_{7} - 35\beta_{6} - 7\beta_{5} + 32\beta_{4} - \beta_{3} + 2\beta_{2} - 18\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 77\beta_{6} + 20\beta_{5} - 13\beta_{4} - 10\beta_{3} + 92\beta_{2} + 52\beta _1 + 60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−0.734668 + 0.348716i
−0.577806 − 2.22188i
0.947217 + 0.807294i
1.86526 + 0.199842i
−0.734668 − 0.348716i
−0.577806 + 2.22188i
0.947217 − 0.807294i
1.86526 − 0.199842i
−1.23467 + 2.13851i 1.66933 − 0.461883i −2.04881 − 3.54864i 1.21814 + 2.10988i −1.07333 + 4.14015i −1.16933 + 2.02534i 5.17972 2.57333 − 1.54207i −6.01598
34.2 −1.07781 + 1.86682i −0.635299 − 1.61133i −1.32333 − 2.29208i −1.81197 − 3.13842i 3.69279 + 0.550720i 1.13530 − 1.96640i 1.39396 −2.19279 + 2.04736i 7.81179
34.3 0.447217 − 0.774602i 1.22553 + 1.22396i 0.599994 + 1.03922i −1.87447 − 3.24667i 1.49616 − 0.401921i −0.725528 + 1.25665i 2.86218 0.00384004 + 3.00000i −3.35317
34.4 1.36526 − 2.36469i 0.240440 + 1.71528i −2.72785 − 4.72478i 0.468293 + 0.811107i 4.38438 + 1.77323i 0.259560 − 0.449571i −9.43585 −2.88438 + 0.824844i 2.55736
67.1 −1.23467 − 2.13851i 1.66933 + 0.461883i −2.04881 + 3.54864i 1.21814 − 2.10988i −1.07333 − 4.14015i −1.16933 − 2.02534i 5.17972 2.57333 + 1.54207i −6.01598
67.2 −1.07781 − 1.86682i −0.635299 + 1.61133i −1.32333 + 2.29208i −1.81197 + 3.13842i 3.69279 − 0.550720i 1.13530 + 1.96640i 1.39396 −2.19279 − 2.04736i 7.81179
67.3 0.447217 + 0.774602i 1.22553 − 1.22396i 0.599994 − 1.03922i −1.87447 + 3.24667i 1.49616 + 0.401921i −0.725528 − 1.25665i 2.86218 0.00384004 − 3.00000i −3.35317
67.4 1.36526 + 2.36469i 0.240440 − 1.71528i −2.72785 + 4.72478i 0.468293 − 0.811107i 4.38438 − 1.77323i 0.259560 + 0.449571i −9.43585 −2.88438 − 0.824844i 2.55736
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.4
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 99.2.e.e ✓ 8
3.b odd 2 1 297.2.e.e 8
9.c even 3 1 inner 99.2.e.e ✓ 8
9.c even 3 1 891.2.a.q 4
9.d odd 6 1 297.2.e.e 8
9.d odd 6 1 891.2.a.p 4
11.b odd 2 1 1089.2.e.i 8
99.g even 6 1 9801.2.a.bl 4
99.h odd 6 1 1089.2.e.i 8
99.h odd 6 1 9801.2.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e ✓ 8 1.a even 1 1 trivial
99.2.e.e ✓ 8 9.c even 3 1 inner
297.2.e.e 8 3.b odd 2 1
297.2.e.e 8 9.d odd 6 1
891.2.a.p 4 9.d odd 6 1
891.2.a.q 4 9.c even 3 1
1089.2.e.i 8 11.b odd 2 1
1089.2.e.i 8 99.h odd 6 1
9801.2.a.bi 4 99.h odd 6 1
9801.2.a.bl 4 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + 10T_{2}^{6} + 7T_{2}^{5} + 76T_{2}^{4} + 46T_{2}^{3} + 181T_{2}^{2} - 104T_{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + \cdots + 24964 \) Copy content Toggle raw display
$17$ \( (T^{4} + 5 T^{3} + \cdots - 236)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} + 81 T - 54)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 14 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 22500 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$37$ \( (T^{4} - 3 T^{3} - 81 T^{2} + \cdots + 57)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 676 \) Copy content Toggle raw display
$43$ \( T^{8} - 21 T^{7} + \cdots + 248004 \) Copy content Toggle raw display
$47$ \( T^{8} - 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} + \cdots - 123)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( T^{8} + 15 T^{7} + \cdots + 14400 \) Copy content Toggle raw display
$67$ \( T^{8} + 14 T^{7} + \cdots + 4713241 \) Copy content Toggle raw display
$71$ \( (T^{4} + 3 T^{3} + \cdots - 195)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 22 T^{3} + \cdots - 1028)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 11 T^{7} + \cdots + 1600 \) Copy content Toggle raw display
$83$ \( T^{8} + 18 T^{7} + \cdots + 2396304 \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} + \cdots + 2064)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 26 T^{7} + \cdots + 1510441 \) Copy content Toggle raw display
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