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Properties

Label 399.2.bo.b
Level $399$
Weight $2$
Character orbit 399.bo
Analytic conductor $3.186$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(43,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, 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("399.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.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: \(3.18603104065\)
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: \( x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{13} q^{3} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_1) q^{4}+ \cdots - \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{13} q^{3} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{17} - 2 \beta_{16} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 3 q^{4} + 12 q^{5} + 3 q^{6} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} + 3 q^{4} + 12 q^{5} + 3 q^{6} + 9 q^{7} - 6 q^{8} + 6 q^{10} + 12 q^{11} + 9 q^{12} - 6 q^{14} - 12 q^{15} + 27 q^{16} - 3 q^{17} - 6 q^{18} - 3 q^{19} + 3 q^{22} - 18 q^{23} - 21 q^{24} - 6 q^{25} + 21 q^{26} + 9 q^{27} - 3 q^{28} - 15 q^{29} - 6 q^{30} - 27 q^{31} - 21 q^{32} - 3 q^{33} + 21 q^{34} + 6 q^{35} + 3 q^{36} - 60 q^{38} - 18 q^{39} + 24 q^{41} + 6 q^{42} + 3 q^{43} - 27 q^{44} + 3 q^{46} - 3 q^{47} - 9 q^{49} + 15 q^{50} + 12 q^{51} - 57 q^{52} + 45 q^{53} + 3 q^{54} - 12 q^{56} - 42 q^{58} + 45 q^{59} + 18 q^{60} - 30 q^{61} - 27 q^{62} + 24 q^{64} - 48 q^{65} - 12 q^{66} - 66 q^{67} + 18 q^{68} - 21 q^{69} - 6 q^{70} - 12 q^{71} + 21 q^{72} + 27 q^{73} + 12 q^{74} + 18 q^{75} - 27 q^{76} + 24 q^{77} - 3 q^{78} - 12 q^{79} + 36 q^{80} + 57 q^{82} - 30 q^{83} - 9 q^{84} + 18 q^{85} + 69 q^{86} - 15 q^{87} - 42 q^{88} - 21 q^{89} - 12 q^{90} + 48 q^{92} - 15 q^{93} + 114 q^{94} + 24 q^{95} + 12 q^{96} + 36 q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 50\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!19 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 964152485141125 \nu^{17} + \cdots + 23\!\cdots\!56 ) / 45\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!03 \nu^{17} + \cdots + 65\!\cdots\!83 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!07 \nu^{17} + \cdots - 98\!\cdots\!73 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!63 \nu^{17} + \cdots - 12\!\cdots\!12 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51\!\cdots\!07 \nu^{17} + \cdots - 20\!\cdots\!08 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\!\cdots\!83 \nu^{17} + \cdots + 31\!\cdots\!42 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69\!\cdots\!33 \nu^{17} + \cdots + 36\!\cdots\!53 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!93 \nu^{17} + \cdots + 28\!\cdots\!00 ) / 91\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!56 \nu^{17} + \cdots - 43\!\cdots\!34 ) / 45\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!56 \nu^{17} + \cdots - 18\!\cdots\!32 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!19 \nu^{17} + \cdots + 99\!\cdots\!80 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 85\!\cdots\!37 \nu^{17} + \cdots + 71\!\cdots\!05 ) / 91\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47\!\cdots\!18 \nu^{17} + \cdots + 80\!\cdots\!89 ) / 47\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 27\!\cdots\!72 \nu^{17} + \cdots - 16\!\cdots\!02 ) / 27\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 27\!\cdots\!71 \nu^{17} + \cdots - 33\!\cdots\!00 ) / 27\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{12} + 2\beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + 6 \beta_{14} - 5 \beta_{13} - 9 \beta_{11} - \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - 15 \beta_{16} - 6 \beta_{15} - 8 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 7 \beta_{9} + \cdots - 36 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} - 20 \beta_{16} - 34 \beta_{15} - 34 \beta_{14} + 25 \beta_{13} + 25 \beta_{12} + \cdots + 46 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 45 \beta_{17} - 36 \beta_{14} - 54 \beta_{13} - 61 \beta_{11} - 13 \beta_{10} + 45 \beta_{9} + 54 \beta_{8} + \cdots - 61 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24 \beta_{17} + 162 \beta_{16} + 199 \beta_{15} + 22 \beta_{13} - 152 \beta_{12} + 248 \beta_{11} + \cdots + 127 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 403 \beta_{17} + 691 \beta_{16} + 233 \beta_{15} + 233 \beta_{14} + 343 \beta_{13} + 343 \beta_{12} + \cdots + 378 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 504 \beta_{17} + 1209 \beta_{14} - 873 \beta_{13} + 180 \beta_{12} - 1381 \beta_{11} + 206 \beta_{10} + \cdots - 1381 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 882 \beta_{17} - 4596 \beta_{16} - 1587 \beta_{15} + 28 \beta_{13} - 2149 \beta_{12} + 2247 \beta_{11} + \cdots - 8711 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5163 \beta_{17} - 8758 \beta_{16} - 7559 \beta_{15} - 7559 \beta_{14} + 3744 \beta_{13} + 3744 \beta_{12} + \cdots + 7859 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12082 \beta_{17} - 11061 \beta_{14} - 13201 \beta_{13} + 246 \beta_{12} - 13048 \beta_{11} + \cdots - 13048 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11326 \beta_{17} + 61866 \beta_{16} + 48222 \beta_{15} + 9220 \beta_{13} - 29552 \beta_{12} + \cdots + 65974 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 123308 \beta_{17} + 202564 \beta_{16} + 77600 \beta_{15} + 77600 \beta_{14} + 78734 \beta_{13} + \cdots + 74626 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 175850 \beta_{17} + 311931 \beta_{14} - 172759 \beta_{13} + 62490 \beta_{12} - 265344 \beta_{11} + \cdots - 265344 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 300884 \beta_{17} - 1347894 \beta_{16} - 543853 \beta_{15} + 9498 \beta_{13} - 487563 \beta_{12} + \cdots - 2318046 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(134\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{8} + \beta_{12}\)

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
−1.15249 + 1.99616i
−0.0269184 + 0.0466240i
0.913360 − 1.58199i
1.30031 + 2.25221i
0.358173 + 0.620373i
−0.218795 − 0.378964i
1.30031 − 2.25221i
0.358173 − 0.620373i
−0.218795 + 0.378964i
−1.15249 − 1.99616i
−0.0269184 − 0.0466240i
0.913360 + 1.58199i
0.952616 + 1.64998i
0.585829 + 1.01469i
−1.21209 − 2.09941i
0.952616 − 1.64998i
0.585829 − 1.01469i
−1.21209 + 2.09941i
−2.16597 + 0.788347i −0.173648 + 0.984808i 2.53783 − 2.12949i −0.626322 − 0.525547i −0.400254 − 2.26995i 0.500000 + 0.866025i −1.51310 + 2.62076i −0.939693 − 0.342020i 1.77091 + 0.644557i
43.2 −0.0505900 + 0.0184132i −0.173648 + 0.984808i −1.52987 + 1.28371i 1.59327 + 1.33691i −0.00934865 − 0.0530188i 0.500000 + 0.866025i 0.107595 − 0.186361i −0.939693 − 0.342020i −0.105220 − 0.0382971i
43.3 1.71656 − 0.624775i −0.173648 + 0.984808i 1.02413 − 0.859347i 1.38035 + 1.15825i 0.317207 + 1.79897i 0.500000 + 0.866025i −0.605643 + 1.04900i −0.939693 − 0.342020i 3.09309 + 1.12579i
85.1 −0.451595 − 2.56112i −0.766044 + 0.642788i −4.47602 + 1.62914i 1.34902 + 0.491004i 1.99220 + 1.67165i 0.500000 − 0.866025i 3.59313 + 6.22348i 0.173648 − 0.984808i 0.648308 − 3.67674i
85.2 −0.124392 − 0.705462i −0.766044 + 0.642788i 1.39718 − 0.508532i −0.780211 − 0.283974i 0.548752 + 0.460458i 0.500000 − 0.866025i −1.24889 − 2.16315i 0.173648 − 0.984808i −0.103281 + 0.585733i
85.3 0.0759867 + 0.430942i −0.766044 + 0.642788i 1.69945 − 0.618549i 2.96328 + 1.07855i −0.335213 − 0.281277i 0.500000 − 0.866025i 0.833284 + 1.44329i 0.173648 − 0.984808i −0.239621 + 1.35896i
169.1 −0.451595 + 2.56112i −0.766044 − 0.642788i −4.47602 − 1.62914i 1.34902 − 0.491004i 1.99220 − 1.67165i 0.500000 + 0.866025i 3.59313 − 6.22348i 0.173648 + 0.984808i 0.648308 + 3.67674i
169.2 −0.124392 + 0.705462i −0.766044 − 0.642788i 1.39718 + 0.508532i −0.780211 + 0.283974i 0.548752 − 0.460458i 0.500000 + 0.866025i −1.24889 + 2.16315i 0.173648 + 0.984808i −0.103281 − 0.585733i
169.3 0.0759867 − 0.430942i −0.766044 − 0.642788i 1.69945 + 0.618549i 2.96328 − 1.07855i −0.335213 + 0.281277i 0.500000 + 0.866025i 0.833284 − 1.44329i 0.173648 + 0.984808i −0.239621 − 1.35896i
232.1 −2.16597 − 0.788347i −0.173648 − 0.984808i 2.53783 + 2.12949i −0.626322 + 0.525547i −0.400254 + 2.26995i 0.500000 − 0.866025i −1.51310 − 2.62076i −0.939693 + 0.342020i 1.77091 − 0.644557i
232.2 −0.0505900 − 0.0184132i −0.173648 − 0.984808i −1.52987 − 1.28371i 1.59327 − 1.33691i −0.00934865 + 0.0530188i 0.500000 − 0.866025i 0.107595 + 0.186361i −0.939693 + 0.342020i −0.105220 + 0.0382971i
232.3 1.71656 + 0.624775i −0.173648 − 0.984808i 1.02413 + 0.859347i 1.38035 − 1.15825i 0.317207 − 1.79897i 0.500000 − 0.866025i −0.605643 − 1.04900i −0.939693 + 0.342020i 3.09309 − 1.12579i
253.1 −1.45949 + 1.22466i 0.939693 + 0.342020i 0.283031 − 1.60515i 0.525672 + 2.98124i −1.79033 + 0.651628i 0.500000 − 0.866025i −0.352551 − 0.610637i 0.766044 + 0.642788i −4.41822 − 3.70732i
253.2 −0.897543 + 0.753128i 0.939693 + 0.342020i −0.108915 + 0.617687i −0.421473 − 2.39029i −1.10100 + 0.400731i 0.500000 − 0.866025i −1.53910 − 2.66580i 0.766044 + 0.642788i 2.17849 + 1.82797i
253.3 1.85704 − 1.55824i 0.939693 + 0.342020i 0.673180 − 3.81779i 0.0164156 + 0.0930973i 2.27799 − 0.829121i 0.500000 − 0.866025i −2.27472 − 3.93994i 0.766044 + 0.642788i 0.175552 + 0.147306i
358.1 −1.45949 − 1.22466i 0.939693 − 0.342020i 0.283031 + 1.60515i 0.525672 − 2.98124i −1.79033 − 0.651628i 0.500000 + 0.866025i −0.352551 + 0.610637i 0.766044 − 0.642788i −4.41822 + 3.70732i
358.2 −0.897543 − 0.753128i 0.939693 − 0.342020i −0.108915 − 0.617687i −0.421473 + 2.39029i −1.10100 − 0.400731i 0.500000 + 0.866025i −1.53910 + 2.66580i 0.766044 − 0.642788i 2.17849 − 1.82797i
358.3 1.85704 + 1.55824i 0.939693 − 0.342020i 0.673180 + 3.81779i 0.0164156 − 0.0930973i 2.27799 + 0.829121i 0.500000 + 0.866025i −2.27472 + 3.93994i 0.766044 − 0.642788i 0.175552 − 0.147306i
\(n\): 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 399.2.bo.b ✓ 18
19.e even 9 1 inner 399.2.bo.b ✓ 18
19.e even 9 1 7581.2.a.bg 9
19.f odd 18 1 7581.2.a.bh 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.bo.b ✓ 18 1.a even 1 1 trivial
399.2.bo.b ✓ 18 19.e even 9 1 inner
7581.2.a.bg 9 19.e even 9 1
7581.2.a.bh 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} + 3 T_{2}^{16} - 2 T_{2}^{15} - 6 T_{2}^{14} + 54 T_{2}^{13} + 298 T_{2}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 12 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} - 12 T^{17} + \cdots + 2809 \) Copy content Toggle raw display
$13$ \( T^{18} - 42 T^{16} + \cdots + 6713281 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 186622921 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 18 T^{17} + \cdots + 14295961 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 184065882841 \) Copy content Toggle raw display
$31$ \( T^{18} + 27 T^{17} + \cdots + 90649441 \) Copy content Toggle raw display
$37$ \( (T^{9} - 108 T^{7} + \cdots - 937)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 8135859601 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 3524356664929 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 39689803729 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1696029568489 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 14933084401 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 85\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 26484846273649 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 80577426579049 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 175843963220689 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 40179586755169 \) Copy content Toggle raw display
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