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Properties

Label 380.2.u.a
Level $380$
Weight $2$
Character orbit 380.u
Analytic conductor $3.034$
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] = [380,2,Mod(61,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.u (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: \(3.03431527681\)
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: \( x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{3}) q^{3} - \beta_{4} q^{5} + ( - \beta_{17} + \beta_{15} + \cdots + \beta_{4}) q^{7} + ( - \beta_{17} - \beta_{13} - \beta_{11} + \cdots - 1) q^{9} + ( - \beta_{17} + \beta_{14} - 2 \beta_{13} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{17} + \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{3} - 9 q^{9} + 9 q^{13} - 3 q^{15} + 12 q^{17} + 18 q^{19} + 15 q^{21} - 9 q^{23} - 33 q^{29} - 18 q^{31} + 9 q^{33} + 12 q^{35} + 60 q^{37} - 84 q^{39} - 18 q^{41} + 18 q^{43} + 6 q^{45} - 15 q^{47}+ \cdots - 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23\!\cdots\!01 \nu^{17} + \cdots - 93\!\cdots\!61 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 80\!\cdots\!23 \nu^{17} + \cdots + 42\!\cdots\!45 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!40 \nu^{17} + \cdots - 11\!\cdots\!00 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!41 \nu^{17} + \cdots - 15\!\cdots\!20 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!74 \nu^{17} + \cdots - 17\!\cdots\!96 ) / 16\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!66 \nu^{17} + \cdots + 13\!\cdots\!58 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47\!\cdots\!25 \nu^{17} + \cdots + 29\!\cdots\!10 ) / 12\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!15 \nu^{17} + \cdots + 13\!\cdots\!40 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!75 \nu^{17} + \cdots + 29\!\cdots\!70 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!47 \nu^{17} + \cdots + 90\!\cdots\!16 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!71 \nu^{17} + \cdots + 56\!\cdots\!50 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!44 \nu^{17} + \cdots + 82\!\cdots\!19 ) / 16\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 80\!\cdots\!05 \nu^{17} + \cdots - 19\!\cdots\!59 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 97\!\cdots\!98 \nu^{17} + \cdots + 64\!\cdots\!72 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!10 \nu^{17} + \cdots + 65\!\cdots\!98 ) / 36\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 60\!\cdots\!28 \nu^{17} + \cdots - 24\!\cdots\!27 ) / 12\!\cdots\!51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{12} - 3\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 4 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{17} - 8 \beta_{13} - 21 \beta_{12} - 8 \beta_{11} - 21 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{17} + 4 \beta_{16} + 7 \beta_{15} - 46 \beta_{13} - 48 \beta_{12} - 6 \beta_{11} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{17} + 55 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} - 189 \beta_{13} - 81 \beta_{12} + \cdots - 185 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 119 \beta_{16} + 59 \beta_{15} + 119 \beta_{14} - 550 \beta_{13} + 119 \beta_{12} + 199 \beta_{11} + \cdots - 411 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94 \beta_{17} + 94 \beta_{16} + 469 \beta_{15} + 469 \beta_{14} - 1234 \beta_{13} + 1328 \beta_{12} + \cdots - 634 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1138 \beta_{17} + 1138 \beta_{15} + 681 \beta_{14} + 5515 \beta_{12} - 727 \beta_{11} + 681 \beta_{10} + \cdots + 423 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4189 \beta_{17} - 1248 \beta_{16} + 1248 \beta_{15} + 906 \beta_{14} + 11563 \beta_{13} + 16578 \beta_{12} + \cdots + 5942 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7246 \beta_{17} - 10842 \beta_{16} + 2023 \beta_{14} + 54370 \beta_{13} + 31546 \beta_{12} + \cdots + 15578 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10177 \beta_{17} - 38476 \beta_{16} - 14438 \beta_{15} - 4261 \beta_{14} + 174807 \beta_{13} + \cdots + 4261 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13456 \beta_{17} - 74249 \beta_{16} - 103688 \beta_{15} - 29439 \beta_{14} + 407806 \beta_{13} + \cdots - 138370 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 48563 \beta_{17} - 107318 \beta_{16} - 359866 \beta_{15} + 490832 \beta_{13} - 673789 \beta_{12} + \cdots - 673789 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 251020 \beta_{17} - 88369 \beta_{16} - 745254 \beta_{15} + 251020 \beta_{14} - 789064 \beta_{13} + \cdots - 1534318 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 526665 \beta_{16} - 1092384 \beta_{15} + 526665 \beta_{14} - 5580452 \beta_{13} + 526665 \beta_{12} + \cdots + 1219807 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2221838 \beta_{17} + 2221838 \beta_{16} - 560842 \beta_{15} - 560842 \beta_{14} - 12855684 \beta_{13} + \cdots + 28353865 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\beta_{12}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−1.37827 − 1.15651i
−0.344606 − 0.289159i
2.39653 + 2.01093i
−1.37827 + 1.15651i
−0.344606 + 0.289159i
2.39653 − 2.01093i
0.128174 − 0.726910i
−0.124000 + 0.703239i
−0.443867 + 2.51729i
2.91971 − 1.06269i
−0.168248 + 0.0612373i
−1.48541 + 0.540646i
0.128174 + 0.726910i
−0.124000 − 0.703239i
−0.443867 − 2.51729i
2.91971 + 1.06269i
−0.168248 − 0.0612373i
−1.48541 − 0.540646i
0 −0.312429 − 1.77187i 0 −0.766044 + 0.642788i 0 −0.336777 + 0.583316i 0 −0.222848 + 0.0811099i 0
61.2 0 −0.0781159 − 0.443017i 0 −0.766044 + 0.642788i 0 −1.06981 + 1.85297i 0 2.62892 − 0.956847i 0
61.3 0 0.543249 + 3.08092i 0 −0.766044 + 0.642788i 0 0.0481505 − 0.0833990i 0 −6.37785 + 2.32135i 0
81.1 0 −0.312429 + 1.77187i 0 −0.766044 − 0.642788i 0 −0.336777 − 0.583316i 0 −0.222848 − 0.0811099i 0
81.2 0 −0.0781159 + 0.443017i 0 −0.766044 − 0.642788i 0 −1.06981 − 1.85297i 0 2.62892 + 0.956847i 0
81.3 0 0.543249 − 3.08092i 0 −0.766044 − 0.642788i 0 0.0481505 + 0.0833990i 0 −6.37785 − 2.32135i 0
101.1 0 −0.693610 − 0.252453i 0 −0.173648 − 0.984808i 0 −2.32856 + 4.03318i 0 −1.88077 − 1.57815i 0
101.2 0 0.671023 + 0.244232i 0 −0.173648 − 0.984808i 0 1.15961 − 2.00851i 0 −1.90751 − 1.60059i 0
101.3 0 2.40197 + 0.874246i 0 −0.173648 − 0.984808i 0 −0.118043 + 0.204456i 0 2.70703 + 2.27147i 0
161.1 0 −2.38017 + 1.99720i 0 0.939693 + 0.342020i 0 2.42255 − 4.19598i 0 1.15545 − 6.55290i 0
161.2 0 0.137157 − 0.115088i 0 0.939693 + 0.342020i 0 −1.55670 + 2.69628i 0 −0.515378 + 2.92285i 0
161.3 0 1.21092 − 1.01608i 0 0.939693 + 0.342020i 0 1.77958 − 3.08232i 0 −0.0870410 + 0.493634i 0
301.1 0 −0.693610 + 0.252453i 0 −0.173648 + 0.984808i 0 −2.32856 − 4.03318i 0 −1.88077 + 1.57815i 0
301.2 0 0.671023 − 0.244232i 0 −0.173648 + 0.984808i 0 1.15961 + 2.00851i 0 −1.90751 + 1.60059i 0
301.3 0 2.40197 − 0.874246i 0 −0.173648 + 0.984808i 0 −0.118043 − 0.204456i 0 2.70703 − 2.27147i 0
321.1 0 −2.38017 − 1.99720i 0 0.939693 − 0.342020i 0 2.42255 + 4.19598i 0 1.15545 + 6.55290i 0
321.2 0 0.137157 + 0.115088i 0 0.939693 − 0.342020i 0 −1.55670 − 2.69628i 0 −0.515378 − 2.92285i 0
321.3 0 1.21092 + 1.01608i 0 0.939693 − 0.342020i 0 1.77958 + 3.08232i 0 −0.0870410 − 0.493634i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.3
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 380.2.u.a ✓ 18
19.e even 9 1 inner 380.2.u.a ✓ 18
19.e even 9 1 7220.2.a.v 9
19.f odd 18 1 7220.2.a.x 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.a ✓ 18 1.a even 1 1 trivial
380.2.u.a ✓ 18 19.e even 9 1 inner
7220.2.a.v 9 19.e even 9 1
7220.2.a.x 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 3 T_{3}^{17} + 9 T_{3}^{16} - 24 T_{3}^{15} + 54 T_{3}^{14} - 297 T_{3}^{13} + 1135 T_{3}^{12} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 39 T^{16} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{18} + 60 T^{16} + \cdots + 531441 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 199402641 \) Copy content Toggle raw display
$17$ \( T^{18} - 12 T^{17} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 9 T^{17} + \cdots + 998001 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 387420489 \) Copy content Toggle raw display
$31$ \( T^{18} + 18 T^{17} + \cdots + 13082689 \) Copy content Toggle raw display
$37$ \( (T^{9} - 30 T^{8} + \cdots - 1864711)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 15485980059729 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 213364449 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 130045445640009 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1556065809 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 191850201 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 1917895384161 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 8000542361529 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 77975557962129 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 18281242781649 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 400527093527929 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 2004181007481 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 51059270662929 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 3020598812169 \) Copy content Toggle raw display
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