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Properties

Label 2002.2.g.c
Level $2002$
Weight $2$
Character orbit 2002.g
Analytic conductor $15.986$
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] = [2002,2,Mod(155,2002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2002.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2002.g (of order \(2\), degree \(1\), 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: \(15.9860504847\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 37 x^{16} + 542 x^{14} + 4033 x^{12} + 16250 x^{10} + 35009 x^{8} + 37245 x^{6} + 16064 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_{8} + \beta_{6}) q^{5} - \beta_1 q^{6} - \beta_{8} q^{7} + \beta_{8} q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_{8} + \beta_{6}) q^{5} - \beta_1 q^{6} - \beta_{8} q^{7} + \beta_{8} q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{7} + 1) q^{10} - \beta_{8} q^{11} + \beta_{3} q^{12} + \beta_{14} q^{13} - q^{14} + ( - \beta_{11} - \beta_{6} + 2 \beta_1) q^{15} + q^{16} + ( - \beta_{16} - \beta_{14} + \cdots + \beta_{3}) q^{17}+ \cdots + ( - \beta_{17} - \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{3} - 18 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{3} - 18 q^{4} + 20 q^{9} + 14 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 18 q^{16} + 10 q^{17} - 18 q^{22} - 24 q^{23} - 24 q^{25} + 2 q^{26} + 8 q^{27} + 16 q^{29} + 12 q^{30} + 14 q^{35} - 20 q^{36} + 10 q^{38} - 2 q^{39} - 14 q^{40} - 2 q^{42} + 2 q^{43} + 2 q^{48} - 18 q^{49} - 62 q^{51} + 4 q^{52} - 2 q^{53} + 14 q^{55} + 18 q^{56} - 14 q^{61} + 4 q^{62} - 18 q^{64} - 32 q^{65} - 2 q^{66} - 10 q^{68} + 48 q^{69} - 28 q^{74} + 52 q^{75} - 18 q^{77} - 14 q^{79} + 66 q^{81} + 8 q^{82} - 16 q^{87} + 18 q^{88} + 64 q^{90} + 2 q^{91} + 24 q^{92} - 16 q^{94} + 54 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 37 x^{16} + 542 x^{14} + 4033 x^{12} + 16250 x^{10} + 35009 x^{8} + 37245 x^{6} + 16064 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21329 \nu^{16} - 772467 \nu^{14} - 10961088 \nu^{12} - 77684477 \nu^{10} - 289889144 \nu^{8} + \cdots - 8444992 ) / 9527464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23550597 \nu^{17} - 34816521 \nu^{16} + 879024139 \nu^{15} - 1224235007 \nu^{14} + \cdots - 67664857136 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23550597 \nu^{17} + 34816521 \nu^{16} + 879024139 \nu^{15} + 1224235007 \nu^{14} + \cdots + 67664857136 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53638606 \nu^{17} - 1981243569 \nu^{15} - 28969901605 \nu^{13} - 215322256830 \nu^{11} + \cdots - 180788252362 \nu ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 85410799 \nu^{16} - 3171814333 \nu^{14} - 46698669332 \nu^{12} - 349851619187 \nu^{10} + \cdots - 86757784824 ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 131953 \nu^{17} + 4860932 \nu^{15} + 70746059 \nu^{13} + 521205361 \nu^{11} + 2066551773 \nu^{9} + \cdots + 114485366 \nu ) / 9527464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23550597 \nu^{17} + 190988911 \nu^{16} + 879024139 \nu^{15} + 7004619705 \nu^{14} + \cdots + 26762704848 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 107501185 \nu^{16} + 3951490739 \nu^{14} + 57313784676 \nu^{12} + 420074696189 \nu^{10} + \cdots + 97596913592 ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 139049405 \nu^{17} + 5153057902 \nu^{15} + 75668570937 \nu^{13} + 565173876017 \nu^{11} + \cdots + 275901623114 \nu ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 139049405 \nu^{17} - 5153057902 \nu^{15} - 75668570937 \nu^{13} - 565173876017 \nu^{11} + \cdots - 209056935690 \nu ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23550597 \nu^{17} - 314709315 \nu^{16} + 879024139 \nu^{15} - 11590200093 \nu^{14} + \cdots - 276974511360 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 489487306 \nu^{17} + 187837899 \nu^{16} - 18097721500 \nu^{15} + 6912581557 \nu^{14} + \cdots + 126021731840 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 332858381 \nu^{17} - 12306287615 \nu^{15} - 179986816496 \nu^{13} - 1334895305017 \nu^{11} + \cdots - 280653939600 \nu ) / 8355585928 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 489487306 \nu^{17} + 187837899 \nu^{16} + 18097721500 \nu^{15} + 6912581557 \nu^{14} + \cdots + 126021731840 ) / 16711171856 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 506483 \nu^{17} - 18671261 \nu^{15} - 272023148 \nu^{13} - 2007136967 \nu^{11} + \cdots - 449496472 \nu ) / 9527464 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{14} - \beta_{13} - \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{17} + 2 \beta_{16} - \beta_{15} - 2 \beta_{14} - 13 \beta_{12} - 15 \beta_{11} + \cdots + 74 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{16} - 20 \beta_{14} + 10 \beta_{13} + 22 \beta_{10} + 33 \beta_{9} - 9 \beta_{7} + \cdots - 283 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 103 \beta_{17} - 39 \beta_{16} + 15 \beta_{15} + 39 \beta_{14} + 151 \beta_{12} + 183 \beta_{11} + \cdots - 731 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 278 \beta_{16} + 278 \beta_{14} - 88 \beta_{13} - 333 \beta_{10} - 437 \beta_{9} + 50 \beta_{7} + \cdots + 2789 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1555 \beta_{17} + 547 \beta_{16} - 189 \beta_{15} - 547 \beta_{14} - 1725 \beta_{12} - 2099 \beta_{11} + \cdots + 7543 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3425 \beta_{16} - 3425 \beta_{14} + 820 \beta_{13} + 4448 \beta_{10} + 5379 \beta_{9} - 29 \beta_{7} + \cdots - 28789 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20835 \beta_{17} - 6847 \beta_{16} + 2285 \beta_{15} + 6847 \beta_{14} + 19726 \beta_{12} + \cdots - 80351 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 40299 \beta_{16} + 40299 \beta_{14} - 8317 \beta_{13} - 56214 \beta_{10} - 64183 \beta_{9} + \cdots + 306906 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 263166 \beta_{17} + 81773 \beta_{16} - 27082 \beta_{15} - 81773 \beta_{14} - 226506 \beta_{12} + \cdots + 876337 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 466035 \beta_{16} - 466035 \beta_{14} + 90042 \beta_{13} + 689930 \beta_{10} + 754712 \beta_{9} + \cdots - 3349690 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3216858 \beta_{17} - 956792 \beta_{16} + 316844 \beta_{15} + 956792 \beta_{14} + 2610445 \beta_{12} + \cdots - 9724702 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 5359695 \beta_{16} + 5359695 \beta_{14} - 1013705 \beta_{13} - 8319865 \beta_{10} - 8810286 \beta_{9} + \cdots + 37195721 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 38566297 \beta_{17} + 11096408 \beta_{16} - 3676845 \beta_{15} - 11096408 \beta_{14} + \cdots + 109283142 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2002\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(1431\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
− 2.86094i
− 2.65333i
− 1.38599i
− 0.348547i
0.224718i
0.796769i
1.63830i
2.18442i
3.40460i
2.86094i
2.65333i
1.38599i
0.348547i
− 0.224718i
− 0.796769i
− 1.63830i
− 2.18442i
− 3.40460i
− 1.00000i −2.86094 −1.00000 3.39727i 2.86094i − 1.00000i 1.00000i 5.18500 3.39727
155.2 − 1.00000i −2.65333 −1.00000 − 0.661288i 2.65333i − 1.00000i 1.00000i 4.04018 −0.661288
155.3 − 1.00000i −1.38599 −1.00000 − 2.87416i 1.38599i − 1.00000i 1.00000i −1.07904 −2.87416
155.4 − 1.00000i −0.348547 −1.00000 4.24228i 0.348547i − 1.00000i 1.00000i −2.87851 4.24228
155.5 − 1.00000i 0.224718 −1.00000 − 2.53436i − 0.224718i − 1.00000i 1.00000i −2.94950 −2.53436
155.6 − 1.00000i 0.796769 −1.00000 2.17621i − 0.796769i − 1.00000i 1.00000i −2.36516 2.17621
155.7 − 1.00000i 1.63830 −1.00000 0.269887i − 1.63830i − 1.00000i 1.00000i −0.315965 0.269887
155.8 − 1.00000i 2.18442 −1.00000 0.251278i − 2.18442i − 1.00000i 1.00000i 1.77171 0.251278
155.9 − 1.00000i 3.40460 −1.00000 2.73289i − 3.40460i − 1.00000i 1.00000i 8.59129 2.73289
155.10 1.00000i −2.86094 −1.00000 − 3.39727i − 2.86094i 1.00000i − 1.00000i 5.18500 3.39727
155.11 1.00000i −2.65333 −1.00000 0.661288i − 2.65333i 1.00000i − 1.00000i 4.04018 −0.661288
155.12 1.00000i −1.38599 −1.00000 2.87416i − 1.38599i 1.00000i − 1.00000i −1.07904 −2.87416
155.13 1.00000i −0.348547 −1.00000 − 4.24228i − 0.348547i 1.00000i − 1.00000i −2.87851 4.24228
155.14 1.00000i 0.224718 −1.00000 2.53436i 0.224718i 1.00000i − 1.00000i −2.94950 −2.53436
155.15 1.00000i 0.796769 −1.00000 − 2.17621i 0.796769i 1.00000i − 1.00000i −2.36516 2.17621
155.16 1.00000i 1.63830 −1.00000 − 0.269887i 1.63830i 1.00000i − 1.00000i −0.315965 0.269887
155.17 1.00000i 2.18442 −1.00000 − 0.251278i 2.18442i 1.00000i − 1.00000i 1.77171 0.251278
155.18 1.00000i 3.40460 −1.00000 − 2.73289i 3.40460i 1.00000i − 1.00000i 8.59129 2.73289
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2002.2.g.c ✓ 18
13.b even 2 1 inner 2002.2.g.c ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2002.2.g.c ✓ 18 1.a even 1 1 trivial
2002.2.g.c ✓ 18 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - T_{3}^{8} - 18T_{3}^{7} + 15T_{3}^{6} + 94T_{3}^{5} - 75T_{3}^{4} - 137T_{3}^{3} + 94T_{3}^{2} + 22T_{3} - 8 \) acting on \(S_{2}^{\mathrm{new}}(2002, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T^{9} - T^{8} - 18 T^{7} + \cdots - 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + 57 T^{16} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( (T^{9} - 5 T^{8} + \cdots + 59936)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + 137 T^{16} + \cdots + 2483776 \) Copy content Toggle raw display
$23$ \( (T^{9} + 12 T^{8} + \cdots + 25088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} - 8 T^{8} + \cdots - 14112)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + 292 T^{16} + \cdots + 2458624 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 14390429814784 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 200581731385344 \) Copy content Toggle raw display
$43$ \( (T^{9} - T^{8} - 176 T^{7} + \cdots - 2368)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + 212 T^{16} + \cdots + 6718464 \) Copy content Toggle raw display
$53$ \( (T^{9} + T^{8} + \cdots + 18648)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 168782932224 \) Copy content Toggle raw display
$61$ \( (T^{9} + 7 T^{8} + \cdots + 477104)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 2168480075776 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 17230237696 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 41402738774016 \) Copy content Toggle raw display
$79$ \( (T^{9} + 7 T^{8} + \cdots + 25747976)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 498098837424384 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
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