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Properties

Label 1980.2.q.g
Level $1980$
Weight $2$
Character orbit 1980.q
Analytic conductor $15.810$
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] = [1980,2,Mod(661,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.661");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.q (of order \(3\), degree \(2\), 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: \(15.8103796002\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 7 x^{15} - 3 x^{14} + 3 x^{13} - 41 x^{12} + 45 x^{11} + 72 x^{10} - 216 x^{9} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + (\beta_{8} - \beta_{7} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{10} + \beta_{2}) q^{7} + (\beta_{8} - \beta_{7} - \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{3} - 9 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{3} - 9 q^{5} - 6 q^{7} + 3 q^{9} - 9 q^{11} - 5 q^{13} - 8 q^{17} + 6 q^{19} + 2 q^{21} - 16 q^{23} - 9 q^{25} - 6 q^{27} - 3 q^{29} + 3 q^{31} - 3 q^{33} + 12 q^{35} + 14 q^{37} - 20 q^{39} + 3 q^{41} + q^{43} + 9 q^{45} - 8 q^{47} - 23 q^{49} - 7 q^{51} + 54 q^{53} + 18 q^{55} + 5 q^{57} - 13 q^{59} + 2 q^{61} + 4 q^{63} - 5 q^{65} + 8 q^{67} - 48 q^{69} + 54 q^{71} + 18 q^{73} - 3 q^{75} - 6 q^{77} - 15 q^{79} - 21 q^{81} - 13 q^{83} + 4 q^{85} + 9 q^{87} - 44 q^{89} + 10 q^{91} - 50 q^{93} - 3 q^{95} - 36 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 7 x^{15} - 3 x^{14} + 3 x^{13} - 41 x^{12} + 45 x^{11} + 72 x^{10} - 216 x^{9} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 175 \nu^{17} - 3420 \nu^{16} - 2808 \nu^{15} + 8333 \nu^{14} - 1332 \nu^{13} - 9543 \nu^{12} + \cdots - 27352809 ) / 4605822 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 175 \nu^{17} + 3420 \nu^{16} + 2808 \nu^{15} - 8333 \nu^{14} + 1332 \nu^{13} + 9543 \nu^{12} + \cdots + 22746987 ) / 4605822 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{17} - 3 \nu^{16} + 7 \nu^{14} - 3 \nu^{13} + 3 \nu^{12} - 41 \nu^{11} + 45 \nu^{10} + \cdots - 19683 ) / 6561 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 832 \nu^{17} + 1800 \nu^{16} - 3024 \nu^{15} - 2276 \nu^{14} - 369 \nu^{13} + 16374 \nu^{12} + \cdots + 5327532 ) / 2302911 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3467 \nu^{17} - 10926 \nu^{16} - 10260 \nu^{15} + 15845 \nu^{14} + 14598 \nu^{13} + \cdots - 70497945 ) / 4605822 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1315 \nu^{17} - 936 \nu^{16} + 3186 \nu^{15} - 619 \nu^{14} - 3006 \nu^{13} - 14097 \nu^{12} + \cdots + 1148175 ) / 1535274 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10745 \nu^{17} - 21834 \nu^{16} - 32778 \nu^{15} + 44435 \nu^{14} + 15300 \nu^{13} + \cdots - 186037155 ) / 4605822 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6577 \nu^{17} - 5112 \nu^{16} - 12339 \nu^{15} + 14368 \nu^{14} + 23310 \nu^{13} + 55803 \nu^{12} + \cdots - 43584723 ) / 2302911 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6568 \nu^{17} - 12537 \nu^{16} - 17118 \nu^{15} + 24754 \nu^{14} + 13455 \nu^{13} + \cdots - 102102282 ) / 2302911 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1627 \nu^{17} - 1062 \nu^{16} - 2862 \nu^{15} + 2317 \nu^{14} + 3384 \nu^{13} + 14223 \nu^{12} + \cdots - 9139473 ) / 511758 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 847 \nu^{17} - 1386 \nu^{16} - 1728 \nu^{15} + 2527 \nu^{14} + 1899 \nu^{13} + 5889 \nu^{12} + \cdots - 9985842 ) / 255879 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1129 \nu^{17} - 1059 \nu^{16} - 2448 \nu^{15} + 2827 \nu^{14} + 2784 \nu^{13} + 8958 \nu^{12} + \cdots - 9467523 ) / 255879 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22937 \nu^{17} - 31176 \nu^{16} - 50598 \nu^{15} + 71945 \nu^{14} + 52722 \nu^{13} + \cdots - 255990537 ) / 4605822 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12035 \nu^{17} - 15786 \nu^{16} - 26784 \nu^{15} + 38642 \nu^{14} + 27639 \nu^{13} + \cdots - 138227148 ) / 2302911 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25720 \nu^{17} - 29277 \nu^{16} - 60534 \nu^{15} + 75955 \nu^{14} + 66780 \nu^{13} + \cdots - 268797609 ) / 2302911 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 26473 \nu^{17} - 35271 \nu^{16} - 60777 \nu^{15} + 81955 \nu^{14} + 62433 \nu^{13} + \cdots - 301195827 ) / 2302911 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{15} + \beta_{13} + \beta_{8} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} + \beta_{15} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 3 \beta_{11} - 4 \beta_{10} - \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{17} - 3 \beta_{16} + 2 \beta_{15} - 7 \beta_{14} + 3 \beta_{13} - \beta_{11} + 4 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{17} + 8 \beta_{16} + 8 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 2 \beta_{11} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{17} - 6 \beta_{16} - 6 \beta_{15} - 3 \beta_{14} + 12 \beta_{13} - 13 \beta_{11} - 9 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{17} + 6 \beta_{16} + 6 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 15 \beta_{12} + 4 \beta_{11} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22 \beta_{17} - 7 \beta_{16} - 18 \beta_{15} - 23 \beta_{14} + 19 \beta_{13} - 2 \beta_{12} - 30 \beta_{11} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 49 \beta_{17} + 9 \beta_{16} + 7 \beta_{15} + 14 \beta_{14} + 14 \beta_{13} + 42 \beta_{12} + \cdots - 109 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 32 \beta_{17} - 27 \beta_{16} - 35 \beta_{15} - 102 \beta_{14} + 97 \beta_{13} - 45 \beta_{11} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 141 \beta_{17} + 8 \beta_{16} + 70 \beta_{15} - 6 \beta_{14} + 66 \beta_{13} + 199 \beta_{12} + \cdots - 662 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 340 \beta_{17} - 9 \beta_{16} - 121 \beta_{15} - 77 \beta_{14} - 239 \beta_{13} + 72 \beta_{12} + \cdots + 386 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 622 \beta_{17} - 93 \beta_{16} - 130 \beta_{15} - 577 \beta_{14} - 72 \beta_{13} - 162 \beta_{12} + \cdots - 1449 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 376 \beta_{17} + 701 \beta_{16} - 232 \beta_{15} + 80 \beta_{14} + 17 \beta_{13} + 505 \beta_{12} + \cdots - 1076 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1356 \beta_{17} + 957 \beta_{16} - 3 \beta_{15} + 2361 \beta_{14} + 1098 \beta_{13} - 63 \beta_{12} + \cdots - 5959 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(991\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
661.1
−0.873678 + 1.49556i
−1.55784 + 0.757066i
−1.73149 + 0.0440122i
1.19065 + 1.25791i
1.70318 + 0.314950i
−0.639653 − 1.60961i
1.69677 − 0.347796i
1.35077 − 1.08416i
0.361281 − 1.69395i
−0.873678 − 1.49556i
−1.55784 − 0.757066i
−1.73149 − 0.0440122i
1.19065 − 1.25791i
1.70318 − 0.314950i
−0.639653 + 1.60961i
1.69677 + 0.347796i
1.35077 + 1.08416i
0.361281 + 1.69395i
0 −1.73203 + 0.00884936i 0 −0.500000 + 0.866025i 0 −2.30489 − 3.99219i 0 2.99984 − 0.0306547i 0
661.2 0 −1.43456 + 0.970592i 0 −0.500000 + 0.866025i 0 2.04206 + 3.53695i 0 1.11590 − 2.78474i 0
661.3 0 −0.903861 + 1.47751i 0 −0.500000 + 0.866025i 0 −0.607893 − 1.05290i 0 −1.36607 − 2.67093i 0
661.4 0 −0.494056 − 1.66009i 0 −0.500000 + 0.866025i 0 −0.275302 − 0.476836i 0 −2.51182 + 1.64036i 0
661.5 0 0.578833 − 1.63247i 0 −0.500000 + 0.866025i 0 1.91616 + 3.31888i 0 −2.32990 − 1.88985i 0
661.6 0 1.07414 + 1.35876i 0 −0.500000 + 0.866025i 0 −1.77346 − 3.07172i 0 −0.692463 + 2.91899i 0
661.7 0 1.14959 − 1.29555i 0 −0.500000 + 0.866025i 0 −0.541507 − 0.937917i 0 −0.356902 − 2.97869i 0
661.8 0 1.61430 − 0.627723i 0 −0.500000 + 0.866025i 0 −2.03207 − 3.51965i 0 2.21193 − 2.02667i 0
661.9 0 1.64765 + 0.534097i 0 −0.500000 + 0.866025i 0 0.576901 + 0.999221i 0 2.42948 + 1.76001i 0
1321.1 0 −1.73203 − 0.00884936i 0 −0.500000 − 0.866025i 0 −2.30489 + 3.99219i 0 2.99984 + 0.0306547i 0
1321.2 0 −1.43456 − 0.970592i 0 −0.500000 − 0.866025i 0 2.04206 − 3.53695i 0 1.11590 + 2.78474i 0
1321.3 0 −0.903861 − 1.47751i 0 −0.500000 − 0.866025i 0 −0.607893 + 1.05290i 0 −1.36607 + 2.67093i 0
1321.4 0 −0.494056 + 1.66009i 0 −0.500000 − 0.866025i 0 −0.275302 + 0.476836i 0 −2.51182 − 1.64036i 0
1321.5 0 0.578833 + 1.63247i 0 −0.500000 − 0.866025i 0 1.91616 − 3.31888i 0 −2.32990 + 1.88985i 0
1321.6 0 1.07414 − 1.35876i 0 −0.500000 − 0.866025i 0 −1.77346 + 3.07172i 0 −0.692463 − 2.91899i 0
1321.7 0 1.14959 + 1.29555i 0 −0.500000 − 0.866025i 0 −0.541507 + 0.937917i 0 −0.356902 + 2.97869i 0
1321.8 0 1.61430 + 0.627723i 0 −0.500000 − 0.866025i 0 −2.03207 + 3.51965i 0 2.21193 + 2.02667i 0
1321.9 0 1.64765 − 0.534097i 0 −0.500000 − 0.866025i 0 0.576901 − 0.999221i 0 2.42948 − 1.76001i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 661.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.q.g ✓ 18
3.b odd 2 1 5940.2.q.g 18
9.c even 3 1 inner 1980.2.q.g ✓ 18
9.d odd 6 1 5940.2.q.g 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.q.g ✓ 18 1.a even 1 1 trivial
1980.2.q.g ✓ 18 9.c even 3 1 inner
5940.2.q.g 18 3.b odd 2 1
5940.2.q.g 18 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 6 T_{7}^{17} + 61 T_{7}^{16} + 236 T_{7}^{15} + 1750 T_{7}^{14} + 6053 T_{7}^{13} + \cdots + 756900 \) acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} + 6 T^{17} + \cdots + 756900 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{18} + 5 T^{17} + \cdots + 9437184 \) Copy content Toggle raw display
$17$ \( (T^{9} + 4 T^{8} + \cdots + 45504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 3 T^{8} + \cdots - 3064)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 338081769 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 26524682496 \) Copy content Toggle raw display
$31$ \( T^{18} - 3 T^{17} + \cdots + 6260004 \) Copy content Toggle raw display
$37$ \( (T^{9} - 7 T^{8} + \cdots + 69222)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} - 3 T^{17} + \cdots + 50041476 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 479510141198400 \) Copy content Toggle raw display
$47$ \( T^{18} + 8 T^{17} + \cdots + 62583921 \) Copy content Toggle raw display
$53$ \( (T^{9} - 27 T^{8} + \cdots - 8128350)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 484474465764 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 54894615616 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{9} - 27 T^{8} + \cdots + 983736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 9 T^{8} + \cdots + 9389672)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 3198875331600 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{9} + 22 T^{8} + \cdots - 343361187)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
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