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Properties

Label 798.2.bo.g
Level $798$
Weight $2$
Character orbit 798.bo
Analytic conductor $6.372$
Analytic rank $0$
Dimension $18$
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(43,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bo (of order \(9\), degree \(6\), 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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 30x^{16} + 333x^{14} + 1704x^{12} + 4194x^{10} + 5049x^{8} + 3060x^{6} + 918x^{4} + 117x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 30x^{16} + 333x^{14} + 1704x^{12} + 4194x^{10} + 5049x^{8} + 3060x^{6} + 918x^{4} + 117x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7911 \nu^{16} + 234360 \nu^{14} + 2546364 \nu^{12} + 12523960 \nu^{10} + 28472514 \nu^{8} + \cdots + 62208 ) / 2738 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7911 \nu^{16} + 234360 \nu^{14} + 2546364 \nu^{12} + 12523960 \nu^{10} + 28472514 \nu^{8} + \cdots + 62208 ) / 2738 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1956883 \nu^{17} + 4061554 \nu^{16} - 58062977 \nu^{15} + 120064839 \nu^{14} - 632543815 \nu^{13} + \cdots + 27339506 ) / 982942 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1956883 \nu^{17} + 4061554 \nu^{16} + 58062977 \nu^{15} + 120064839 \nu^{14} + 632543815 \nu^{13} + \cdots + 27339506 ) / 982942 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2670703 \nu^{17} - 5357269 \nu^{16} + 78957993 \nu^{15} - 158566951 \nu^{14} + 854955549 \nu^{13} + \cdots - 41239219 ) / 982942 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20736 \nu^{17} - 614169 \nu^{15} - 6670728 \nu^{13} - 32787780 \nu^{11} - 74442824 \nu^{9} + \cdots - 1369 ) / 2738 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2670703 \nu^{17} - 5357269 \nu^{16} - 78957993 \nu^{15} - 158566951 \nu^{14} - 854955549 \nu^{13} + \cdots - 40256277 ) / 982942 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11461876 \nu^{17} - 1390851 \nu^{16} + 339142524 \nu^{15} - 41106846 \nu^{14} + 3677335981 \nu^{13} + \cdots - 8143297 ) / 982942 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11461876 \nu^{17} + 1390851 \nu^{16} + 339142524 \nu^{15} + 41106846 \nu^{14} + 3677335981 \nu^{13} + \cdots + 8143297 ) / 982942 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12852727 \nu^{17} - 6104607 \nu^{16} + 380249370 \nu^{15} - 180575573 \nu^{14} + 4122199082 \nu^{13} + \cdots - 43511867 ) / 982942 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16297015 \nu^{17} + 1346949 \nu^{16} + 482162330 \nu^{15} + 39697860 \nu^{14} + 5227232198 \nu^{13} + \cdots + 9714581 ) / 982942 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12852727 \nu^{17} + 6104607 \nu^{16} + 380249370 \nu^{15} + 180575573 \nu^{14} + 4122199082 \nu^{13} + \cdots + 42528925 ) / 982942 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5390787 \nu^{17} + 14809610 \nu^{16} - 159680557 \nu^{15} + 438312347 \nu^{14} + \cdots + 113182984 ) / 982942 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10906228 \nu^{17} - 13462661 \nu^{16} + 322481773 \nu^{15} - 398614487 \nu^{14} + 3492610861 \nu^{13} + \cdots - 103468403 ) / 982942 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10906228 \nu^{17} - 13462661 \nu^{16} - 322481773 \nu^{15} - 398614487 \nu^{14} + \cdots - 103468403 ) / 982942 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11461876 \nu^{16} + 339142524 \nu^{14} + 3677335981 \nu^{12} + 18018866734 \nu^{10} + \cdots + 82293731 ) / 491471 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22956129 \nu^{17} + 11461876 \nu^{16} - 679111564 \nu^{15} + 339142524 \nu^{14} + \cdots + 82293731 ) / 982942 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + \beta_{12} - \beta_{10} - \beta_{7} - \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{16} - 2 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} + 2 \beta_{11} + 10 \beta_{10} + 4 \beta_{9} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} - \beta_{16} + 11 \beta_{15} + 10 \beta_{14} + 21 \beta_{13} + 30 \beta_{12} - 21 \beta_{11} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 72 \beta_{16} + 3 \beta_{15} + 30 \beta_{14} - 27 \beta_{13} + 98 \beta_{12} - 27 \beta_{11} + \cdots - 195 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 48 \beta_{17} + 24 \beta_{16} - 113 \beta_{15} - 98 \beta_{14} - 211 \beta_{13} - 369 \beta_{12} + \cdots - 348 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 689 \beta_{16} - 71 \beta_{15} - 369 \beta_{14} + 298 \beta_{13} - 954 \beta_{12} + 298 \beta_{11} + \cdots + 1846 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 744 \beta_{17} - 372 \beta_{16} + 1179 \beta_{15} + 954 \beta_{14} + 2133 \beta_{13} + 4269 \beta_{12} + \cdots + 4555 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6801 \beta_{16} + 1164 \beta_{15} + 4269 \beta_{14} - 3105 \beta_{13} + 9225 \beta_{12} + \cdots - 18090 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9854 \beta_{17} + 4927 \beta_{16} - 12525 \beta_{15} - 9225 \beta_{14} - 21750 \beta_{13} + \cdots - 55666 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 68391 \beta_{16} - 16350 \beta_{15} - 48078 \beta_{14} + 31728 \beta_{13} - 88984 \beta_{12} + \cdots + 181115 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 121186 \beta_{17} - 60593 \beta_{16} + 134521 \beta_{15} + 88984 \beta_{14} + 223505 \beta_{13} + \cdots + 654799 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 696516 \beta_{16} + 211309 \beta_{15} + 533768 \beta_{14} - 322459 \beta_{13} + 859665 \beta_{12} + \cdots - 1839852 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1430784 \beta_{17} + 715392 \beta_{16} - 1452387 \beta_{15} - 859665 \beta_{14} - 2312052 \beta_{13} + \cdots - 7522454 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 7160472 \beta_{16} - 2595546 \beta_{15} - 5875212 \beta_{14} + 3279666 \beta_{13} - 8341720 \beta_{12} + \cdots + 18887384 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 16475692 \beta_{17} - 8237846 \beta_{16} + 15711787 \beta_{15} + 8341720 \beta_{14} + 24053507 \beta_{13} + \cdots + 85064338 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
3.27830i
0.614767i
− 0.637878i
2.36454i
− 0.183108i
− 1.57989i
− 2.36454i
0.183108i
1.57989i
− 0.995272i
0.662700i
2.98623i
− 3.27830i
− 0.614767i
0.637878i
0.995272i
− 0.662700i
− 2.98623i
−0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i −2.91005 − 2.44182i −0.173648 − 0.984808i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.939693 − 0.342020i 3.56970 + 1.29927i
43.2 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i 0.624002 + 0.523600i −0.173648 − 0.984808i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.939693 − 0.342020i −0.765452 − 0.278602i
43.3 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 − 0.642788i 2.28605 + 1.91822i −0.173648 − 0.984808i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.939693 − 0.342020i −2.80425 − 1.02066i
85.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −3.52215 − 1.28196i −0.766044 − 0.642788i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.173648 − 0.984808i 0.650868 − 3.69126i
85.2 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0.624378 + 0.227255i −0.766044 − 0.642788i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.173648 − 0.984808i −0.115380 + 0.654355i
85.3 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 2.89778 + 1.05470i −0.766044 − 0.642788i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.173648 − 0.984808i −0.535487 + 3.03690i
169.1 0.173648 − 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i −3.52215 + 1.28196i −0.766044 + 0.642788i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.650868 + 3.69126i
169.2 0.173648 − 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i 0.624378 − 0.227255i −0.766044 + 0.642788i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.115380 − 0.654355i
169.3 0.173648 − 0.984808i −0.766044 − 0.642788i −0.939693 − 0.342020i 2.89778 − 1.05470i −0.766044 + 0.642788i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.535487 − 3.03690i
253.1 0.766044 − 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.565390 − 3.20649i 0.939693 − 0.342020i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.766044 + 0.642788i −2.49420 − 2.09288i
253.2 0.766044 − 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i −0.0667260 − 0.378422i 0.939693 − 0.342020i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.766044 + 0.642788i −0.294360 − 0.246997i
253.3 0.766044 − 0.642788i 0.939693 + 0.342020i 0.173648 − 0.984808i 0.632116 + 3.58491i 0.939693 − 0.342020i 0.500000 − 0.866025i −0.500000 − 0.866025i 0.766044 + 0.642788i 2.78856 + 2.33988i
631.1 −0.939693 − 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i −2.91005 + 2.44182i −0.173648 + 0.984808i 0.500000 − 0.866025i −0.500000 − 0.866025i −0.939693 + 0.342020i 3.56970 − 1.29927i
631.2 −0.939693 − 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i 0.624002 − 0.523600i −0.173648 + 0.984808i 0.500000 − 0.866025i −0.500000 − 0.866025i −0.939693 + 0.342020i −0.765452 + 0.278602i
631.3 −0.939693 − 0.342020i −0.173648 − 0.984808i 0.766044 + 0.642788i 2.28605 − 1.91822i −0.173648 + 0.984808i 0.500000 − 0.866025i −0.500000 − 0.866025i −0.939693 + 0.342020i −2.80425 + 1.02066i
757.1 0.766044 + 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.565390 + 3.20649i 0.939693 + 0.342020i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.766044 − 0.642788i −2.49420 + 2.09288i
757.2 0.766044 + 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i −0.0667260 + 0.378422i 0.939693 + 0.342020i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.766044 − 0.642788i −0.294360 + 0.246997i
757.3 0.766044 + 0.642788i 0.939693 − 0.342020i 0.173648 + 0.984808i 0.632116 − 3.58491i 0.939693 + 0.342020i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.766044 − 0.642788i 2.78856 − 2.33988i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bo.g ✓ 18
19.e even 9 1 inner 798.2.bo.g ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bo.g ✓ 18 1.a even 1 1 trivial
798.2.bo.g ✓ 18 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 3 T_{5}^{15} - 324 T_{5}^{13} + 1815 T_{5}^{12} - 1296 T_{5}^{10} - 4772 T_{5}^{9} + \cdots + 104329 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + 3 T^{15} + \cdots + 104329 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} - 3 T^{17} + \cdots + 657721 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + \cdots + 788544 \) Copy content Toggle raw display
$17$ \( T^{18} - 6 T^{17} + \cdots + 5329 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 3376578977209 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 2661126639616 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 294643781721 \) Copy content Toggle raw display
$37$ \( (T^{9} - 24 T^{8} + \cdots - 3412457)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 2637541449 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 52583993344 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 74426233260096 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 20410372270656 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 23296527424 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 719317549449216 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 38583875174464 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 88171555380841 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 34948244183616 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 560902677691456 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 342852588005841 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 23104000000 \) Copy content Toggle raw display
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