Properties

Label 1620.3.g.c
Level $1620$
Weight $3$
Character orbit 1620.g
Analytic conductor $44.142$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,3,Mod(161,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 148 x^{14} - 48 x^{13} + 8658 x^{12} + 5424 x^{11} - 251584 x^{10} - 237792 x^{9} + 3768217 x^{8} + 5306544 x^{7} - 26115036 x^{6} - 59547312 x^{5} + \cdots + 90699264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{18} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{9} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{9} + 1) q^{7} - \beta_{6} q^{11} + ( - \beta_{12} - 2) q^{13} + (\beta_{8} + \beta_{4}) q^{17} + ( - \beta_{11} + \beta_{9} + 3) q^{19} + ( - \beta_{14} - \beta_{10} + 3 \beta_1) q^{23} - 5 q^{25} + ( - \beta_{14} + \beta_{10} + \beta_{8} + 2 \beta_{6} + \beta_{4}) q^{29} + (\beta_{12} + \beta_{9} + 2 \beta_{7} - \beta_{3} - 5) q^{31} + (\beta_{10} + \beta_{2} - \beta_1) q^{35} + (\beta_{15} + \beta_{3} - 5) q^{37} + (\beta_{14} + \beta_{6} + 3 \beta_{4}) q^{41} + ( - \beta_{15} + 2 \beta_{12} + \beta_{9} - \beta_{5} + \beta_{3} + 6) q^{43} + (\beta_{14} + \beta_{13} + 2 \beta_{6} + \beta_{4}) q^{47} + (\beta_{12} - \beta_{11} + 2 \beta_{5} + \beta_{3} + 13) q^{49} + ( - 3 \beta_{14} + \beta_{13} - 2 \beta_{10} + 2 \beta_{8} + \beta_{6} - 2 \beta_{4} + \cdots + 3 \beta_1) q^{53}+ \cdots + ( - 3 \beta_{15} - 2 \beta_{12} - \beta_{9} - 4 \beta_{7} - 3 \beta_{5} + \beta_{3} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} - 40 q^{13} + 56 q^{19} - 80 q^{25} - 64 q^{31} - 88 q^{37} + 128 q^{43} + 216 q^{49} + 8 q^{61} - 40 q^{67} + 56 q^{73} - 136 q^{79} - 392 q^{91} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 148 x^{14} - 48 x^{13} + 8658 x^{12} + 5424 x^{11} - 251584 x^{10} - 237792 x^{9} + 3768217 x^{8} + 5306544 x^{7} - 26115036 x^{6} - 59547312 x^{5} + \cdots + 90699264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12658358851 \nu^{15} + 14668537302 \nu^{14} + 1844957174536 \nu^{13} - 1525122198024 \nu^{12} - 106126248301614 \nu^{11} + \cdots - 92\!\cdots\!40 ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 47\!\cdots\!43 \nu^{15} + \cdots + 20\!\cdots\!96 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36\!\cdots\!19 \nu^{15} + \cdots + 11\!\cdots\!36 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37\!\cdots\!19 \nu^{15} + \cdots - 16\!\cdots\!32 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!93 \nu^{15} + \cdots - 76\!\cdots\!04 ) / 40\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!25 \nu^{15} + \cdots - 10\!\cdots\!76 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!51 \nu^{15} + \cdots - 36\!\cdots\!20 ) / 40\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!75 \nu^{15} + \cdots + 22\!\cdots\!60 ) / 81\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50\!\cdots\!35 \nu^{15} + \cdots + 15\!\cdots\!64 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!99 \nu^{15} + \cdots + 17\!\cdots\!84 ) / 44\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!89 \nu^{15} + \cdots + 10\!\cdots\!84 ) / 57\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!02 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46\!\cdots\!91 \nu^{15} + \cdots + 35\!\cdots\!52 ) / 40\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!55 \nu^{15} + \cdots - 89\!\cdots\!96 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 88\!\cdots\!15 \nu^{15} + \cdots + 26\!\cdots\!92 ) / 60\!\cdots\!64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{11} + \beta_{8} - 3\beta_{7} + \beta_{5} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + 3 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + 171 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 23 \beta_{15} - 9 \beta_{14} + 27 \beta_{12} - 48 \beta_{11} - 21 \beta_{10} - 32 \beta_{9} + 55 \beta_{8} - 126 \beta_{7} + 9 \beta_{6} + 21 \beta_{5} - 9 \beta_{4} + 27 \beta_{3} - 18 \beta_{2} + 252 \beta _1 + 153 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{15} - 146 \beta_{14} - 40 \beta_{13} + 81 \beta_{12} + 58 \beta_{11} - 122 \beta_{10} - 478 \beta_{9} + 34 \beta_{8} + 123 \beta_{7} + 92 \beta_{6} + 190 \beta_{5} + 176 \beta_{4} + 27 \beta_{3} - 24 \beta_{2} + 12 \beta _1 + 5373 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1311 \beta_{15} - 855 \beta_{14} + 120 \beta_{13} + 1377 \beta_{12} - 914 \beta_{11} - 1715 \beta_{10} - 2418 \beta_{9} + 3369 \beta_{8} - 4926 \beta_{7} + 855 \beta_{6} + 727 \beta_{5} - 255 \beta_{4} + \cdots + 9315 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 771 \beta_{15} - 8894 \beta_{14} - 1177 \beta_{13} + 4617 \beta_{12} + 1740 \beta_{11} - 9543 \beta_{10} - 21882 \beta_{9} + 2762 \beta_{8} + 3366 \beta_{7} + 5465 \beta_{6} + 7023 \beta_{5} + 10799 \beta_{4} + \cdots + 184347 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 55465 \beta_{15} - 64239 \beta_{14} + 7896 \beta_{13} + 56673 \beta_{12} - 11728 \beta_{11} - 117859 \beta_{10} - 126952 \beta_{9} + 186899 \beta_{8} - 182184 \beta_{7} + 57435 \beta_{6} + \cdots + 565335 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 16996 \beta_{15} - 497228 \beta_{14} - 26944 \beta_{13} + 199341 \beta_{12} + 49982 \beta_{11} - 605572 \beta_{10} - 870422 \beta_{9} + 200676 \beta_{8} + 59349 \beta_{7} + 315788 \beta_{6} + \cdots + 6323679 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1992335 \beta_{15} - 4150359 \beta_{14} + 380808 \beta_{13} + 2133729 \beta_{12} + 59766 \beta_{11} - 7147107 \beta_{10} - 5603162 \beta_{9} + 9744253 \beta_{8} - 6169842 \beta_{7} + \cdots + 28546011 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8285 \beta_{15} - 26193980 \beta_{14} - 334345 \beta_{13} + 7360551 \beta_{12} + 1326004 \beta_{11} - 34157153 \beta_{10} - 30444790 \beta_{9} + 13161676 \beta_{8} - 424758 \beta_{7} + \cdots + 203597361 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 60064797 \beta_{15} - 242005995 \beta_{14} + 16733640 \beta_{13} + 71956269 \beta_{12} + 10990936 \beta_{11} - 398597903 \beta_{10} - 212098800 \beta_{9} + \cdots + 1188642411 ) / 18 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 24043794 \beta_{15} - 1317604610 \beta_{14} + 12001016 \beta_{13} + 224014167 \beta_{12} + 29440230 \beta_{11} - 1786323654 \beta_{10} - 877109010 \beta_{9} + \cdots + 5529557277 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1264219627 \beta_{15} - 13095389931 \beta_{14} + 721213896 \beta_{13} + 1903051557 \beta_{12} + 454356590 \beta_{11} - 20911998679 \beta_{10} + \cdots + 38292740775 ) / 18 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1476700361 \beta_{15} - 63705680546 \beta_{14} + 1319759255 \beta_{13} + 4151359593 \beta_{12} + 257066900 \beta_{11} - 88426356475 \beta_{10} + \cdots + 80782459551 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 7578431431 \beta_{15} - 667702351359 \beta_{14} + 31223290392 \beta_{13} + 13812495057 \beta_{12} + 9775746048 \beta_{11} - 1043457335283 \beta_{10} + \cdots + 592365390615 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
4.52064 + 0.707107i
−5.12934 + 0.707107i
−1.86964 0.707107i
−0.120607 0.707107i
3.60169 0.707107i
−1.61144 0.707107i
−6.25269 + 0.707107i
6.86139 + 0.707107i
4.52064 0.707107i
−5.12934 0.707107i
−1.86964 + 0.707107i
−0.120607 + 0.707107i
3.60169 + 0.707107i
−1.61144 + 0.707107i
−6.25269 0.707107i
6.86139 0.707107i
0 0 0 2.23607i 0 −11.5686 0 0 0
161.2 0 0 0 2.23607i 0 −7.14566 0 0 0
161.3 0 0 0 2.23607i 0 −6.97692 0 0 0
161.4 0 0 0 2.23607i 0 1.94751 0 0 0
161.5 0 0 0 2.23607i 0 2.49969 0 0 0
161.6 0 0 0 2.23607i 0 4.52972 0 0 0
161.7 0 0 0 2.23607i 0 7.09136 0 0 0
161.8 0 0 0 2.23607i 0 13.6229 0 0 0
161.9 0 0 0 2.23607i 0 −11.5686 0 0 0
161.10 0 0 0 2.23607i 0 −7.14566 0 0 0
161.11 0 0 0 2.23607i 0 −6.97692 0 0 0
161.12 0 0 0 2.23607i 0 1.94751 0 0 0
161.13 0 0 0 2.23607i 0 2.49969 0 0 0
161.14 0 0 0 2.23607i 0 4.52972 0 0 0
161.15 0 0 0 2.23607i 0 7.09136 0 0 0
161.16 0 0 0 2.23607i 0 13.6229 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.g.c 16
3.b odd 2 1 inner 1620.3.g.c 16
9.c even 3 2 1620.3.o.g 32
9.d odd 6 2 1620.3.o.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.3.g.c 16 1.a even 1 1 trivial
1620.3.g.c 16 3.b odd 2 1 inner
1620.3.o.g 32 9.c even 3 2
1620.3.o.g 32 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} - 242T_{7}^{6} + 740T_{7}^{5} + 15175T_{7}^{4} - 51028T_{7}^{3} - 253790T_{7}^{2} + 1202828T_{7} - 1228634 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} - 242 T^{6} + \cdots - 1228634)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 960 T^{14} + \cdots + 822424079376 \) Copy content Toggle raw display
$13$ \( (T^{8} + 20 T^{7} - 632 T^{6} + \cdots - 122845736)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 2148 T^{14} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} - 28 T^{7} - 752 T^{6} + \cdots + 14011609)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 3228 T^{14} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + 6072 T^{14} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} + 32 T^{7} - 4022 T^{6} + \cdots - 2745236384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 44 T^{7} - 3842 T^{6} + \cdots - 42707709224)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 12960 T^{14} + \cdots + 52\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{8} - 64 T^{7} + \cdots - 1225653312416)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 14064 T^{14} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + 28680 T^{14} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + 34512 T^{14} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots + 217364537682304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + \cdots + 2578090965376)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 37080 T^{14} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{8} - 28 T^{7} + \cdots + 276269820962176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 68 T^{7} - 23816 T^{6} + \cdots - 1341375104)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 73644 T^{14} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{16} + 77472 T^{14} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} - 64 T^{7} + \cdots + 1694320236544)^{2} \) Copy content Toggle raw display
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