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Properties

Label 798.2.ba.l
Level $798$
Weight $2$
Character orbit 798.ba
Analytic conductor $6.372$
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] = [798,2,Mod(407,798)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.407");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.ba (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.253930089113856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 14 x^{10} - 29 x^{9} + 52 x^{8} - 87 x^{7} + 131 x^{6} - 174 x^{5} + 208 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 14 x^{10} - 29 x^{9} + 52 x^{8} - 87 x^{7} + 131 x^{6} - 174 x^{5} + 208 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} - 76 \nu^{9} + 137 \nu^{8} - 246 \nu^{7} + 439 \nu^{6} - 781 \nu^{5} + \cdots + 1328 ) / 272 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 46 \nu^{10} - 117 \nu^{9} + 225 \nu^{8} - 389 \nu^{7} + 685 \nu^{6} - 982 \nu^{5} + \cdots + 416 ) / 272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{11} + 83 \nu^{10} - 142 \nu^{9} + 311 \nu^{8} - 584 \nu^{7} + 869 \nu^{6} - 1117 \nu^{5} + \cdots + 384 ) / 544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12 \nu^{11} + 73 \nu^{10} - 155 \nu^{9} + 268 \nu^{8} - 461 \nu^{7} + 794 \nu^{6} - 1143 \nu^{5} + \cdots + 400 ) / 272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15 \nu^{11} - 19 \nu^{10} + 62 \nu^{9} - 131 \nu^{8} + 232 \nu^{7} - 389 \nu^{6} + 481 \nu^{5} + \cdots - 432 ) / 272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25 \nu^{11} + 43 \nu^{10} - 24 \nu^{9} + 37 \nu^{8} - 58 \nu^{7} + 59 \nu^{6} + 139 \nu^{5} + \cdots - 640 ) / 272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55 \nu^{11} + 217 \nu^{10} - 488 \nu^{9} + 911 \nu^{8} - 1474 \nu^{7} + 2401 \nu^{6} - 3271 \nu^{5} + \cdots + 1312 ) / 544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31 \nu^{11} - 173 \nu^{10} + 484 \nu^{9} - 919 \nu^{8} + 1606 \nu^{7} - 2649 \nu^{6} + 4011 \nu^{5} + \cdots - 3232 ) / 272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21 \nu^{11} - 47 \nu^{10} + 97 \nu^{9} - 180 \nu^{8} + 318 \nu^{7} - 480 \nu^{6} + 619 \nu^{5} + \cdots - 224 ) / 136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 79 \nu^{11} - 329 \nu^{10} + 764 \nu^{9} - 1379 \nu^{8} + 2498 \nu^{7} - 3989 \nu^{6} + 5591 \nu^{5} + \cdots - 3200 ) / 544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 99 \nu^{11} - 343 \nu^{10} + 654 \nu^{9} - 1123 \nu^{8} + 1980 \nu^{7} - 3057 \nu^{6} + 3841 \nu^{5} + \cdots + 32 ) / 544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 2\beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + 2\beta_{4} + 3\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} - 2 \beta_{10} + 5 \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \cdots + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} + 2 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9 \beta_{11} - \beta_{10} + 4 \beta_{9} - \beta_{8} + 17 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9 \beta_{11} + 8 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} + 13 \beta_{6} + 8 \beta_{5} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6 \beta_{11} - 4 \beta_{10} + 19 \beta_{9} - 16 \beta_{8} - 5 \beta_{7} + 35 \beta_{6} + 15 \beta_{5} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9 \beta_{11} - 10 \beta_{10} + 29 \beta_{9} + 7 \beta_{8} + 14 \beta_{7} - 16 \beta_{6} - \beta_{5} + \cdots + 25 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
0.575305 − 1.29191i
0.763617 + 1.19033i
−0.456697 + 1.33844i
1.13636 + 0.841833i
1.40693 − 0.143364i
−0.925514 − 1.06931i
0.575305 + 1.29191i
0.763617 − 1.19033i
−0.456697 − 1.33844i
1.13636 − 0.841833i
1.40693 + 0.143364i
−0.925514 + 1.06931i
0.500000 − 0.866025i −0.913353 − 1.47166i −0.500000 − 0.866025i −0.945657 − 0.545975i −1.73117 + 0.0551566i 1.00000 −1.00000 −1.33157 + 2.68829i −0.945657 + 0.545975i
407.2 0.500000 − 0.866025i −0.597396 + 1.62577i −0.500000 − 0.866025i −3.61588 − 2.08763i 1.10926 + 1.33024i 1.00000 −1.00000 −2.28624 − 1.94245i −3.61588 + 2.08763i
407.3 0.500000 − 0.866025i −0.126158 + 1.72745i −0.500000 − 0.866025i 1.55455 + 0.897518i 1.43294 + 0.972981i 1.00000 −1.00000 −2.96817 − 0.435863i 1.55455 − 0.897518i
407.4 0.500000 − 0.866025i 0.446274 − 1.67357i −0.500000 − 0.866025i −2.69206 − 1.55426i −1.22622 − 1.22327i 1.00000 −1.00000 −2.60168 − 1.49374i −2.69206 + 1.55426i
407.5 0.500000 − 0.866025i 1.55197 − 0.769028i −0.500000 − 0.866025i 0.566916 + 0.327309i 0.109985 − 1.72856i 1.00000 −1.00000 1.81719 − 2.38701i 0.566916 − 0.327309i
407.6 0.500000 − 0.866025i 1.63867 + 0.561042i −0.500000 − 0.866025i 2.13213 + 1.23099i 1.30521 − 1.13861i 1.00000 −1.00000 2.37046 + 1.83872i 2.13213 − 1.23099i
449.1 0.500000 + 0.866025i −0.913353 + 1.47166i −0.500000 + 0.866025i −0.945657 + 0.545975i −1.73117 − 0.0551566i 1.00000 −1.00000 −1.33157 − 2.68829i −0.945657 − 0.545975i
449.2 0.500000 + 0.866025i −0.597396 − 1.62577i −0.500000 + 0.866025i −3.61588 + 2.08763i 1.10926 − 1.33024i 1.00000 −1.00000 −2.28624 + 1.94245i −3.61588 − 2.08763i
449.3 0.500000 + 0.866025i −0.126158 − 1.72745i −0.500000 + 0.866025i 1.55455 − 0.897518i 1.43294 − 0.972981i 1.00000 −1.00000 −2.96817 + 0.435863i 1.55455 + 0.897518i
449.4 0.500000 + 0.866025i 0.446274 + 1.67357i −0.500000 + 0.866025i −2.69206 + 1.55426i −1.22622 + 1.22327i 1.00000 −1.00000 −2.60168 + 1.49374i −2.69206 − 1.55426i
449.5 0.500000 + 0.866025i 1.55197 + 0.769028i −0.500000 + 0.866025i 0.566916 − 0.327309i 0.109985 + 1.72856i 1.00000 −1.00000 1.81719 + 2.38701i 0.566916 + 0.327309i
449.6 0.500000 + 0.866025i 1.63867 − 0.561042i −0.500000 + 0.866025i 2.13213 − 1.23099i 1.30521 + 1.13861i 1.00000 −1.00000 2.37046 − 1.83872i 2.13213 + 1.23099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.ba.l yes 12
3.b odd 2 1 798.2.ba.k ✓ 12
19.d odd 6 1 798.2.ba.k ✓ 12
57.f even 6 1 inner 798.2.ba.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.ba.k ✓ 12 3.b odd 2 1
798.2.ba.k ✓ 12 19.d odd 6 1
798.2.ba.l yes 12 1.a even 1 1 trivial
798.2.ba.l yes 12 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):

\( T_{5}^{12} + 6 T_{5}^{11} - T_{5}^{10} - 78 T_{5}^{9} + 4 T_{5}^{8} + 630 T_{5}^{7} + 169 T_{5}^{6} + \cdots + 1681 \) Copy content Toggle raw display
\( T_{11}^{12} + 132T_{11}^{10} + 6544T_{11}^{8} + 150544T_{11}^{6} + 1615168T_{11}^{4} + 7238144T_{11}^{2} + 8202496 \) Copy content Toggle raw display
\( T_{13}^{12} - 18 T_{13}^{11} + 113 T_{13}^{10} - 90 T_{13}^{9} - 1484 T_{13}^{8} + 714 T_{13}^{7} + \cdots + 32041 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 132 T^{10} + \cdots + 8202496 \) Copy content Toggle raw display
$13$ \( T^{12} - 18 T^{11} + \cdots + 32041 \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + \cdots + 1290496 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 1225 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{12} + 104 T^{10} + \cdots + 719104 \) Copy content Toggle raw display
$37$ \( T^{12} + 272 T^{10} + \cdots + 23658496 \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} + \cdots + 6801664 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 425291796736 \) Copy content Toggle raw display
$47$ \( T^{12} + 24 T^{11} + \cdots + 57274624 \) Copy content Toggle raw display
$53$ \( T^{12} + 6 T^{11} + \cdots + 7139584 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 6535590649 \) Copy content Toggle raw display
$61$ \( T^{12} + 16 T^{11} + \cdots + 12996025 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 134189056 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 57165462649 \) Copy content Toggle raw display
$73$ \( T^{12} - 12 T^{11} + \cdots + 80281600 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 3206448910336 \) Copy content Toggle raw display
$83$ \( T^{12} + 110 T^{10} + \cdots + 78961 \) Copy content Toggle raw display
$89$ \( T^{12} + 28 T^{11} + \cdots + 58003456 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1267360000 \) Copy content Toggle raw display
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