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Properties

Label 380.2.u.b
Level $380$
Weight $2$
Character orbit 380.u
Analytic conductor $3.034$
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] = [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: \( x^{18} - 3 x^{17} + 21 x^{16} - 30 x^{15} + 192 x^{14} - 207 x^{13} + 1178 x^{12} - 705 x^{11} + \cdots + 5329 \) 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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{3} + ( - \beta_{11} + \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{16} - \beta_{11} + \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{3} + ( - \beta_{11} + \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{17} + \beta_{16} - \beta_{15} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{3} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{3} + 15 q^{9} + 9 q^{13} + 3 q^{15} + 12 q^{17} - 18 q^{19} - 9 q^{21} + 21 q^{23} + 18 q^{27} - 9 q^{29} + 6 q^{31} - 21 q^{33} - 6 q^{35} - 36 q^{37} - 12 q^{39} + 6 q^{41} - 12 q^{43} - 6 q^{45} + 21 q^{47} - 3 q^{49} - 9 q^{51} + 36 q^{53} - 3 q^{55} - 24 q^{57} - 6 q^{61} + 36 q^{63} + 15 q^{65} + 60 q^{67} + 27 q^{69} - 36 q^{71} - 60 q^{73} - 6 q^{75} - 36 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{85} + 21 q^{87} + 6 q^{89} - 30 q^{91} - 48 q^{93} - 21 q^{95} - 57 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 21 x^{16} - 30 x^{15} + 192 x^{14} - 207 x^{13} + 1178 x^{12} - 705 x^{11} + \cdots + 5329 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72\!\cdots\!11 \nu^{17} + \cdots + 83\!\cdots\!10 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\!\cdots\!95 \nu^{17} + \cdots - 97\!\cdots\!37 ) / 14\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!69 \nu^{17} + \cdots - 14\!\cdots\!28 ) / 10\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44\!\cdots\!86 \nu^{17} + \cdots + 33\!\cdots\!19 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46\!\cdots\!86 \nu^{17} + \cdots - 55\!\cdots\!65 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 75\!\cdots\!09 \nu^{17} + \cdots - 14\!\cdots\!54 ) / 44\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!92 \nu^{17} + \cdots - 39\!\cdots\!07 ) / 17\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!53 \nu^{17} + \cdots - 60\!\cdots\!01 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!84 \nu^{17} + \cdots - 52\!\cdots\!03 ) / 44\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!82 \nu^{17} + \cdots + 65\!\cdots\!91 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!12 \nu^{17} + \cdots - 12\!\cdots\!98 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25\!\cdots\!17 \nu^{17} + \cdots - 10\!\cdots\!89 ) / 44\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!24 \nu^{17} + \cdots - 34\!\cdots\!78 ) / 44\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!40 \nu^{17} + \cdots + 11\!\cdots\!27 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 33\!\cdots\!30 \nu^{17} + \cdots + 16\!\cdots\!24 ) / 32\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 37\!\cdots\!50 \nu^{17} + \cdots - 50\!\cdots\!89 ) / 32\!\cdots\!31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{15} + \beta_{12} + \beta_{11} - \beta_{8} - 4\beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{17} + 9 \beta_{15} - \beta_{14} - 11 \beta_{12} - 10 \beta_{11} + \beta_{10} - 9 \beta_{9} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21 \beta_{17} - 12 \beta_{16} + 23 \beta_{15} - 8 \beta_{14} + 10 \beta_{13} - 16 \beta_{12} - 37 \beta_{11} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{17} - 31 \beta_{16} + 4 \beta_{15} + 14 \beta_{14} + 14 \beta_{13} + 28 \beta_{12} + \cdots + 181 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 228 \beta_{17} - 197 \beta_{15} + 81 \beta_{14} - 48 \beta_{13} + 381 \beta_{12} + 176 \beta_{11} + \cdots + 150 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 656 \beta_{17} + 373 \beta_{16} - 737 \beta_{15} - 8 \beta_{14} - 145 \beta_{13} + 719 \beta_{12} + \cdots - 1527 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 381 \beta_{17} + 1352 \beta_{16} - 381 \beta_{15} - 408 \beta_{14} - 408 \beta_{13} - 2032 \beta_{12} + \cdots - 3994 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6927 \beta_{17} + 5805 \beta_{15} - 1361 \beta_{14} - 216 \beta_{13} - 11470 \beta_{12} + \cdots - 2642 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16735 \beta_{17} - 13950 \beta_{16} + 21059 \beta_{15} - 571 \beta_{14} + 5114 \beta_{13} - 16638 \beta_{12} + \cdots + 38395 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13379 \beta_{17} - 43305 \beta_{16} + 13379 \beta_{15} + 15982 \beta_{14} + 15982 \beta_{13} + \cdots + 124772 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 201921 \beta_{17} - 154948 \beta_{15} + 42145 \beta_{14} + 5922 \beta_{13} + 362624 \beta_{12} + \cdots + 81467 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 480481 \beta_{17} + 445001 \beta_{16} - 628521 \beta_{15} - 50579 \beta_{14} - 110124 \beta_{13} + \cdots - 1163961 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 495580 \beta_{17} + 1432844 \beta_{16} - 495580 \beta_{15} - 494162 \beta_{14} - 494162 \beta_{13} + \cdots - 3499851 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6029028 \beta_{17} + 4458950 \beta_{15} - 984889 \beta_{14} - 621321 \beta_{13} - 11020913 \beta_{12} + \cdots - 2067007 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 13539874 \beta_{17} - 14325778 \beta_{16} + 18660336 \beta_{15} + 1895634 \beta_{14} + 3096251 \beta_{13} + \cdots + 33445352 \) 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
0.793225 − 1.37391i
−0.478554 + 0.828880i
−0.754364 + 1.30660i
0.793225 + 1.37391i
−0.478554 − 0.828880i
−0.754364 − 1.30660i
−0.838693 + 1.45266i
0.546970 − 0.947380i
1.55777 − 2.69813i
1.37427 − 2.38031i
0.443231 − 0.767698i
−1.14386 + 1.98122i
−0.838693 − 1.45266i
0.546970 + 0.947380i
1.55777 + 2.69813i
1.37427 + 2.38031i
0.443231 + 0.767698i
−1.14386 − 1.98122i
0 −0.275484 − 1.56235i 0 0.766044 − 0.642788i 0 0.778817 − 1.34895i 0 0.454036 − 0.165255i 0
61.2 0 0.166200 + 0.942568i 0 0.766044 − 0.642788i 0 −2.28066 + 3.95023i 0 1.95827 − 0.712751i 0
61.3 0 0.261988 + 1.48581i 0 0.766044 − 0.642788i 0 1.67550 − 2.90204i 0 0.680094 − 0.247534i 0
81.1 0 −0.275484 + 1.56235i 0 0.766044 + 0.642788i 0 0.778817 + 1.34895i 0 0.454036 + 0.165255i 0
81.2 0 0.166200 − 0.942568i 0 0.766044 + 0.642788i 0 −2.28066 − 3.95023i 0 1.95827 + 0.712751i 0
81.3 0 0.261988 − 1.48581i 0 0.766044 + 0.642788i 0 1.67550 + 2.90204i 0 0.680094 + 0.247534i 0
101.1 0 −1.57623 − 0.573700i 0 0.173648 + 0.984808i 0 0.230636 − 0.399473i 0 −0.142773 − 0.119801i 0
101.2 0 1.02797 + 0.374150i 0 0.173648 + 0.984808i 0 −1.81409 + 3.14209i 0 −1.38140 − 1.15914i 0
101.3 0 2.92764 + 1.06558i 0 0.173648 + 0.984808i 0 0.643760 − 1.11502i 0 5.13752 + 4.31089i 0
161.1 0 −2.10551 + 1.76673i 0 −0.939693 − 0.342020i 0 −1.23778 + 2.14391i 0 0.790885 − 4.48533i 0
161.2 0 −0.679069 + 0.569806i 0 −0.939693 − 0.342020i 0 0.460431 − 0.797489i 0 −0.384489 + 2.18055i 0
161.3 0 1.75249 − 1.47051i 0 −0.939693 − 0.342020i 0 1.54340 − 2.67324i 0 0.387867 − 2.19970i 0
301.1 0 −1.57623 + 0.573700i 0 0.173648 − 0.984808i 0 0.230636 + 0.399473i 0 −0.142773 + 0.119801i 0
301.2 0 1.02797 − 0.374150i 0 0.173648 − 0.984808i 0 −1.81409 − 3.14209i 0 −1.38140 + 1.15914i 0
301.3 0 2.92764 − 1.06558i 0 0.173648 − 0.984808i 0 0.643760 + 1.11502i 0 5.13752 − 4.31089i 0
321.1 0 −2.10551 − 1.76673i 0 −0.939693 + 0.342020i 0 −1.23778 − 2.14391i 0 0.790885 + 4.48533i 0
321.2 0 −0.679069 − 0.569806i 0 −0.939693 + 0.342020i 0 0.460431 + 0.797489i 0 −0.384489 − 2.18055i 0
321.3 0 1.75249 + 1.47051i 0 −0.939693 + 0.342020i 0 1.54340 + 2.67324i 0 0.387867 + 2.19970i 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.b ✓ 18
19.e even 9 1 inner 380.2.u.b ✓ 18
19.e even 9 1 7220.2.a.w 9
19.f odd 18 1 7220.2.a.y 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.b ✓ 18 1.a even 1 1 trivial
380.2.u.b ✓ 18 19.e even 9 1 inner
7220.2.a.w 9 19.e even 9 1
7220.2.a.y 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 3 T_{3}^{17} - 3 T_{3}^{16} + 6 T_{3}^{15} + 60 T_{3}^{14} - 111 T_{3}^{13} + 113 T_{3}^{12} + \cdots + 5329 \) 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
$3$ \( T^{18} - 3 T^{17} + \cdots + 5329 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + \cdots + 130321 \) Copy content Toggle raw display
$11$ \( T^{18} + 36 T^{16} + \cdots + 12321 \) Copy content Toggle raw display
$13$ \( T^{18} - 9 T^{17} + \cdots + 942841 \) Copy content Toggle raw display
$17$ \( T^{18} - 12 T^{17} + \cdots + 9162729 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} - 21 T^{17} + \cdots + 76055841 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 12637107826641 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1031244769 \) Copy content Toggle raw display
$37$ \( (T^{9} + 18 T^{8} + \cdots - 587421)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 4797190682001 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 16131286081 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 62131046121 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 55278122769 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 25345414942929 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 189464681156689 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 176384961 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 95\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 385979893469641 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 71913539889 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 156305980013121 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 45168117642729 \) Copy content Toggle raw display
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