Properties

Label 798.2.j.k
Level $798$
Weight $2$
Character orbit 798.j
Analytic conductor $6.372$
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] = [798,2,Mod(457,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.457");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.j (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 10x^{6} - 22x^{5} + 125x^{4} - 154x^{3} + 490x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 10x^{6} - 22x^{5} + 125x^{4} - 154x^{3} + 490x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40\nu^{7} + 53\nu^{6} - 307\nu^{5} - 2784\nu^{4} + 2809\nu^{3} - 8302\nu^{2} + 4361\nu - 116277 ) / 70658 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 22\nu^{6} + 5\nu^{5} - 368\nu^{4} + 276\nu^{3} - 567\nu^{2} - 273\nu - 10584 ) / 5047 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{7} + 101\nu^{6} + 272\nu^{5} + 313\nu^{4} + 306\nu^{3} + 2177\nu^{2} + 22785\nu + 13720 ) / 10094 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -82\nu^{7} - 361\nu^{6} + 377\nu^{5} - 2368\nu^{4} + 1055\nu^{3} - 34965\nu^{2} + 27146\nu - 67228 ) / 35329 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 22\nu^{4} + 125\nu^{3} - 154\nu^{2} + 490\nu - 686 ) / 343 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -339\nu^{7} + 398\nu^{6} - 3761\nu^{5} + 9607\nu^{4} - 22887\nu^{3} + 32543\nu^{2} - 107996\nu + 202027 ) / 70658 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{3} - 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + 5\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} - 4\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 5\beta_{6} + \beta_{5} - \beta_{4} - 9\beta_{3} - 9\beta_{2} - \beta _1 - 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{7} - 12\beta_{6} + 4\beta_{5} + 9\beta_{4} - \beta_{3} - 37\beta_{2} - 30\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - 13\beta_{6} + 3\beta_{5} + 50\beta_{4} - 69\beta_{3} + 190\beta_{2} - 109\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -135\beta_{7} - 78\beta_{6} - 41\beta_{5} - 137\beta_{4} - 268\beta_{3} + 119\beta_{2} + 224\beta _1 - 28 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
457.1
−1.55051 2.14381i
1.95143 + 1.78659i
1.64274 2.07398i
−1.04367 + 2.43121i
−1.55051 + 2.14381i
1.95143 1.78659i
1.64274 + 2.07398i
−1.04367 2.43121i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.85865 + 3.21928i 1.00000 1.55051 2.14381i 1.00000 −0.500000 + 0.866025i −1.85865 3.21928i
457.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.16476 + 2.01742i 1.00000 −1.95143 + 1.78659i 1.00000 −0.500000 + 0.866025i −1.16476 2.01742i
457.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.745555 1.29134i 1.00000 −1.64274 2.07398i 1.00000 −0.500000 + 0.866025i 0.745555 + 1.29134i
457.4 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.27785 2.21331i 1.00000 1.04367 + 2.43121i 1.00000 −0.500000 + 0.866025i 1.27785 + 2.21331i
571.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.85865 3.21928i 1.00000 1.55051 + 2.14381i 1.00000 −0.500000 0.866025i −1.85865 + 3.21928i
571.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.16476 2.01742i 1.00000 −1.95143 1.78659i 1.00000 −0.500000 0.866025i −1.16476 + 2.01742i
571.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.745555 + 1.29134i 1.00000 −1.64274 + 2.07398i 1.00000 −0.500000 0.866025i 0.745555 1.29134i
571.4 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.27785 + 2.21331i 1.00000 1.04367 2.43121i 1.00000 −0.500000 0.866025i 1.27785 2.21331i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 457.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.j.k 8
7.c even 3 1 inner 798.2.j.k 8
7.c even 3 1 5586.2.a.cb 4
7.d odd 6 1 5586.2.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.k 8 1.a even 1 1 trivial
798.2.j.k 8 7.c even 3 1 inner
5586.2.a.ca 4 7.d odd 6 1
5586.2.a.cb 4 7.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 \(S_{2}^{\mathrm{new}}(798, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 16T_{5}^{6} + 135T_{5}^{4} + 12T_{5}^{3} + 540T_{5}^{2} - 396T_{5} + 1089 \) Copy content Toggle raw display
\( T_{11}^{8} + 30T_{11}^{6} + 24T_{11}^{5} + 891T_{11}^{4} + 360T_{11}^{3} + 414T_{11}^{2} - 108T_{11} + 81 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + 16 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 30 T^{6} + 24 T^{5} + 891 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T - 2)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + 76 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 7 T^{7} + 103 T^{6} + \cdots + 799236 \) Copy content Toggle raw display
$29$ \( (T^{4} + 5 T^{3} - 21 T^{2} - 81 T + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + 78 T^{6} + \cdots + 28224 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + 16 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} - 54 T^{2} + 522 T + 1944)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 11 T^{3} + 12 T^{2} - 86 T + 14)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 9 T^{7} + 273 T^{6} + \cdots + 49787136 \) Copy content Toggle raw display
$53$ \( T^{8} + 11 T^{7} + 190 T^{6} + \cdots + 2903616 \) Copy content Toggle raw display
$59$ \( T^{8} + 21 T^{7} + \cdots + 351112644 \) Copy content Toggle raw display
$61$ \( T^{8} + 11 T^{7} + \cdots + 105349696 \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} + 412 T^{6} + \cdots + 82591744 \) Copy content Toggle raw display
$71$ \( (T^{4} + 18 T^{3} - 702 T + 972)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 15 T^{7} + 171 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$79$ \( T^{8} - 7 T^{7} + 106 T^{6} + \cdots + 126736 \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} + 18 T^{2} + 360 T + 453)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 2 T^{7} + 22 T^{6} + \cdots + 7056 \) Copy content Toggle raw display
$97$ \( (T^{4} - 19 T^{3} - 15 T^{2} + 679 T + 1556)^{2} \) Copy content Toggle raw display
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