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Properties

Label 9360.2.k.b
Level $9360$
Weight $2$
Character orbit 9360.k
Analytic conductor $74.740$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9360,2,Mod(4031,9360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 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("9360.4031");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.k (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: \(74.7399762919\)
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: \( x^{16} + 22x^{14} + 217x^{12} + 1296x^{10} + 6904x^{8} + 46656x^{6} + 281232x^{4} + 1026432x^{2} + 1679616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{5} + \beta_{3} q^{7} - \beta_{14} q^{11} + q^{13} + (\beta_{10} - \beta_{5} + 2 \beta_{2}) q^{17} + (\beta_{9} - \beta_{6} - \beta_{3}) q^{19} - \beta_{11} q^{23} - q^{25} + ( - \beta_{15} + \beta_{10}) q^{29} + (\beta_{12} + \beta_{9} - \beta_{6}) q^{31} - \beta_{13} q^{35} + ( - \beta_{8} + 2 \beta_{7} - 2) q^{37} + (\beta_{10} + \beta_{5} + \beta_{2}) q^{41} + ( - 3 \beta_{12} - 2 \beta_{9} + \beta_{3}) q^{43} + ( - \beta_{14} - \beta_{13} + 2 \beta_{11}) q^{47} + ( - \beta_{8} + 4 \beta_{7} - \beta_{4}) q^{49} + (3 \beta_{10} + \beta_{5}) q^{53} + \beta_{9} q^{55} + (\beta_{14} + 3 \beta_{13} + 2 \beta_1) q^{59} + (\beta_{8} + \beta_{4} + 3) q^{61} + \beta_{10} q^{65} + ( - 2 \beta_{12} + 2 \beta_{6} + 4 \beta_{3}) q^{67} + ( - \beta_{14} - 2 \beta_{11} + 2 \beta_1) q^{71} + (4 \beta_{7} + 2) q^{73} + (\beta_{15} + 2 \beta_{10} + \cdots + 2 \beta_{2}) q^{77}+ \cdots + (\beta_{8} + 2 \beta_{4} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{13} - 16 q^{25} - 24 q^{37} + 8 q^{49} + 40 q^{61} + 32 q^{73} - 8 q^{85} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 22x^{14} + 217x^{12} + 1296x^{10} + 6904x^{8} + 46656x^{6} + 281232x^{4} + 1026432x^{2} + 1679616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 184 \nu^{13} + 1945 \nu^{11} + 9666 \nu^{9} + 36172 \nu^{7} + 338904 \nu^{5} + \cdots + 7122816 \nu ) / 1492992 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{15} + 980 \nu^{13} + 16499 \nu^{11} + 91554 \nu^{9} + 298292 \nu^{7} + \cdots + 71570304 \nu ) / 20901888 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25 \nu^{14} + 1484 \nu^{12} + 18263 \nu^{10} + 91458 \nu^{8} + 342884 \nu^{6} + 3177720 \nu^{4} + \cdots + 67091328 ) / 5225472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{14} + 170 \nu^{12} + 3989 \nu^{10} + 25596 \nu^{8} + 78356 \nu^{6} + 673920 \nu^{4} + \cdots + 21275136 ) / 746496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 22\nu^{13} + 217\nu^{11} + 1296\nu^{9} + 6904\nu^{7} + 46656\nu^{5} + 281232\nu^{3} + 1306368\nu ) / 279936 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 967 \nu^{14} - 19348 \nu^{12} - 161959 \nu^{10} - 789930 \nu^{8} - 4454500 \nu^{6} + \cdots - 455642496 ) / 20901888 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41 \nu^{14} + 650 \nu^{12} + 4325 \nu^{10} + 19836 \nu^{8} + 132404 \nu^{6} + 989568 \nu^{4} + \cdots + 7464960 ) / 746496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59 \nu^{14} - 1118 \nu^{12} - 9167 \nu^{10} - 44532 \nu^{8} - 232700 \nu^{6} - 1906560 \nu^{4} + \cdots - 24261120 ) / 746496 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 421 \nu^{14} + 9604 \nu^{12} + 79765 \nu^{10} + 356742 \nu^{8} + 1977676 \nu^{6} + \cdots + 179998848 ) / 5225472 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1549 \nu^{15} - 26572 \nu^{13} - 198541 \nu^{11} - 932094 \nu^{9} - 5473036 \nu^{7} + \cdots - 433807488 \nu ) / 125411328 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 127 \nu^{15} - 1048 \nu^{13} - 5671 \nu^{11} - 26766 \nu^{9} - 240148 \nu^{7} + \cdots - 5692032 \nu ) / 8957952 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2255 \nu^{14} - 38612 \nu^{12} - 296303 \nu^{10} - 1361466 \nu^{8} - 8175812 \nu^{6} + \cdots - 638534016 ) / 20901888 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 305 \nu^{15} - 5864 \nu^{13} - 46601 \nu^{11} - 219474 \nu^{9} - 1264940 \nu^{7} + \cdots - 109268352 \nu ) / 8957952 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 413 \nu^{15} + 8168 \nu^{13} + 61973 \nu^{11} + 271242 \nu^{9} + 1630844 \nu^{7} + \cdots + 122332032 \nu ) / 8957952 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1091 \nu^{15} - 15540 \nu^{13} - 93891 \nu^{11} - 427426 \nu^{9} - 3078452 \nu^{7} + \cdots - 135851904 \nu ) / 13934592 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + 2\beta_{13} - \beta_{11} + 2\beta_{5} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{9} - 2\beta_{8} - 4\beta_{7} - 5\beta_{6} - 2\beta_{4} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} - 5\beta_{14} - 10\beta_{13} - 3\beta_{11} - 12\beta_{10} - 2\beta_{5} - 24\beta_{2} + 7\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33\beta_{12} + 27\beta_{9} - 2\beta_{8} + 16\beta_{7} - 5\beta_{6} + 10\beta_{4} - 44\beta_{3} + 24 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{15} + 7\beta_{14} + 42\beta_{13} + 53\beta_{11} - 200\beta_{10} + 22\beta_{5} + 24\beta_{2} - 73\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -91\beta_{12} - 45\beta_{9} + 158\beta_{8} + 20\beta_{7} - 121\beta_{6} + 6\beta_{4} + 72\beta_{3} - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28 \beta_{15} - 77 \beta_{14} - 634 \beta_{13} + 53 \beta_{11} + 1260 \beta_{10} + 110 \beta_{5} + \cdots - 249 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 433\beta_{12} - 317\beta_{9} - 450\beta_{8} + 1272\beta_{7} + 619\beta_{6} + 290\beta_{4} + 380\beta_{3} - 2848 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1104 \beta_{15} - 1977 \beta_{14} - 286 \beta_{13} + 2493 \beta_{11} - 3232 \beta_{10} + \cdots - 417 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8779 \beta_{12} + 603 \beta_{9} + 3534 \beta_{8} - 7588 \beta_{7} + 6263 \beta_{6} + 654 \beta_{4} + \cdots + 9596 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 964 \beta_{15} + 14371 \beta_{14} + 9870 \beta_{13} - 17987 \beta_{11} + 39508 \beta_{10} + \cdots - 1537 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 13905 \beta_{12} - 19429 \beta_{9} - 18130 \beta_{8} + 34464 \beta_{7} + 3467 \beta_{6} - 16758 \beta_{4} + \cdots + 23480 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 28680 \beta_{15} - 15833 \beta_{14} + 16330 \beta_{13} + 35733 \beta_{11} + 46568 \beta_{10} + \cdots + 117671 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 107995 \beta_{12} - 7037 \beta_{9} - 126562 \beta_{8} - 333148 \beta_{7} + 3975 \beta_{6} + \cdots - 5388 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 161068 \beta_{15} + 96819 \beta_{14} + 486774 \beta_{13} - 802491 \beta_{11} - 1402116 \beta_{10} + \cdots + 135191 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9360\mathbb{Z}\right)^\times\).

\(n\) \(2081\) \(2341\) \(5617\) \(5761\) \(8191\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4031.1
0.782537 − 2.32113i
0.286282 + 2.43270i
2.18150 + 1.11402i
1.02309 − 2.22560i
−1.02309 − 2.22560i
−2.18150 + 1.11402i
−0.286282 + 2.43270i
−0.782537 − 2.32113i
−0.782537 + 2.32113i
−0.286282 − 2.43270i
−2.18150 − 1.11402i
−1.02309 + 2.22560i
1.02309 + 2.22560i
2.18150 − 1.11402i
0.286282 − 2.43270i
0.782537 + 2.32113i
0 0 0 − 1.00000i 0 − 4.36301i 0 0 0
4031.2 0 0 0 − 1.00000i 0 − 2.04619i 0 0 0
4031.3 0 0 0 − 1.00000i 0 − 1.56507i 0 0 0
4031.4 0 0 0 − 1.00000i 0 − 0.572563i 0 0 0
4031.5 0 0 0 − 1.00000i 0 0.572563i 0 0 0
4031.6 0 0 0 − 1.00000i 0 1.56507i 0 0 0
4031.7 0 0 0 − 1.00000i 0 2.04619i 0 0 0
4031.8 0 0 0 − 1.00000i 0 4.36301i 0 0 0
4031.9 0 0 0 1.00000i 0 − 4.36301i 0 0 0
4031.10 0 0 0 1.00000i 0 − 2.04619i 0 0 0
4031.11 0 0 0 1.00000i 0 − 1.56507i 0 0 0
4031.12 0 0 0 1.00000i 0 − 0.572563i 0 0 0
4031.13 0 0 0 1.00000i 0 0.572563i 0 0 0
4031.14 0 0 0 1.00000i 0 1.56507i 0 0 0
4031.15 0 0 0 1.00000i 0 2.04619i 0 0 0
4031.16 0 0 0 1.00000i 0 4.36301i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4031.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9360.2.k.b ✓ 16
3.b odd 2 1 inner 9360.2.k.b ✓ 16
4.b odd 2 1 inner 9360.2.k.b ✓ 16
12.b even 2 1 inner 9360.2.k.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9360.2.k.b ✓ 16 1.a even 1 1 trivial
9360.2.k.b ✓ 16 3.b odd 2 1 inner
9360.2.k.b ✓ 16 4.b odd 2 1 inner
9360.2.k.b ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 26T_{7}^{6} + 145T_{7}^{4} + 240T_{7}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(9360, [\chi])\). 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
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 26 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 54 T^{6} + 541 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} + 86 T^{6} + \cdots + 8836)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 92 T^{6} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 30 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 140 T^{6} + \cdots + 906304)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{2} + 272)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6 T^{3} - 29 T^{2} + \cdots - 28)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 94 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 292 T^{6} + \cdots + 3625216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 116 T^{6} + \cdots + 341056)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 118 T^{6} + \cdots + 6724)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 308 T^{6} + \cdots + 906304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} + \cdots - 784)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 440 T^{6} + \cdots + 5456896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 278 T^{6} + \cdots + 276676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 28)^{8} \) Copy content Toggle raw display
$79$ \( (T^{8} + 574 T^{6} + \cdots + 58125376)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 284 T^{6} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 510 T^{6} + \cdots + 134838544)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 14 T^{3} + \cdots + 7004)^{4} \) Copy content Toggle raw display
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