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Properties

Label 6480.2.h.c
Level $6480$
Weight $2$
Character orbit 6480.h
Analytic conductor $51.743$
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] = [6480,2,Mod(2591,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6480.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.h (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: \(51.7430605098\)
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: \( x^{16} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 720)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + \beta_{12} q^{7} - \beta_{7} q^{11} + (\beta_{6} - 1) q^{13} + (2 \beta_{5} - \beta_1) q^{17} + ( - \beta_{11} + \beta_{10}) q^{19} + (\beta_{13} + \beta_{8} + \beta_{2}) q^{23} - q^{25} - \beta_{3} q^{29} + (2 \beta_{15} + \beta_{11} - \beta_{10}) q^{31} - \beta_{8} q^{35} + (\beta_{6} + 2 \beta_{4} + 1) q^{37} + (\beta_{5} + \beta_1) q^{41} + (\beta_{15} + \beta_{11} + \beta_{10}) q^{43} + ( - 2 \beta_{13} - \beta_{8} + \cdots - \beta_{2}) q^{47}+ \cdots + ( - 2 \beta_{9} + 2 \beta_{6} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{13} - 16 q^{25} + 8 q^{37} - 8 q^{49} + 56 q^{61} - 32 q^{73} + 24 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 99x^{12} - 432x^{10} + 1368x^{8} - 2214x^{6} + 2511x^{4} - 486x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1825 \nu^{14} - 50853 \nu^{12} + 541968 \nu^{10} - 4649490 \nu^{8} + 14820750 \nu^{6} + \cdots - 886221 ) / 14108742 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7562 \nu^{15} + 112407 \nu^{13} - 1024458 \nu^{11} + 5417334 \nu^{9} - 19376352 \nu^{7} + \cdots + 17730657 \nu ) / 7054371 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4378 \nu^{14} - 60494 \nu^{12} + 510600 \nu^{10} - 2531298 \nu^{8} + 8082612 \nu^{6} + \cdots - 1106811 ) / 2351457 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3845 \nu^{14} + 41273 \nu^{12} - 326056 \nu^{10} + 1213482 \nu^{8} - 3305898 \nu^{6} + \cdots - 1575081 ) / 1567638 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{14} - 285\nu^{12} + 2364\nu^{10} - 10530\nu^{8} + 33120\nu^{6} - 53730\nu^{4} + 55053\nu^{2} - 6237 ) / 7614 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5965 \nu^{14} - 63803 \nu^{12} + 505832 \nu^{10} - 1882554 \nu^{8} + 5287662 \nu^{6} + \cdots + 10322487 ) / 1567638 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44141 \nu^{15} - 541815 \nu^{13} + 4544394 \nu^{11} - 20410470 \nu^{9} + 66405600 \nu^{7} + \cdots - 48397581 \nu ) / 14108742 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46925 \nu^{15} + 610839 \nu^{13} - 5198514 \nu^{11} + 24737904 \nu^{9} - 82708632 \nu^{7} + \cdots + 67470489 \nu ) / 14108742 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7045 \nu^{14} + 78567 \nu^{12} - 597416 \nu^{10} + 2223402 \nu^{8} - 5557788 \nu^{6} + \cdots + 779355 ) / 1567638 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12193 \nu^{15} - 143136 \nu^{13} + 1173312 \nu^{11} - 4997712 \nu^{9} + 15676416 \nu^{7} + \cdots - 816480 \nu ) / 2351457 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12695 \nu^{15} - 152579 \nu^{13} + 1250718 \nu^{11} - 5399998 \nu^{9} + 16710624 \nu^{7} + \cdots - 870345 \nu ) / 1567638 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 24386 \nu^{15} + 286272 \nu^{13} - 2346624 \nu^{11} + 9995424 \nu^{9} - 31352832 \nu^{7} + \cdots + 1632960 \nu ) / 2351457 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 174161 \nu^{15} + 2125701 \nu^{13} - 17660274 \nu^{11} + 78688800 \nu^{9} + \cdots + 184212387 \nu ) / 14108742 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 145093 \nu^{14} - 1708332 \nu^{12} + 14010870 \nu^{10} - 60004602 \nu^{8} + 188610885 \nu^{6} + \cdots - 37167336 ) / 7054371 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47819 \nu^{15} - 566935 \nu^{13} + 4647270 \nu^{11} - 19933936 \nu^{9} + 62091360 \nu^{7} + \cdots - 3233925 \nu ) / 1567638 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 2\beta_{7} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{6} - 5\beta_{5} - \beta_{4} - 2\beta_{3} + 3\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + 2\beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{14} + 5\beta_{6} - 29\beta_{5} + 7\beta_{4} - 8\beta_{3} + 17\beta _1 - 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{15} + 12\beta_{13} + 14\beta_{12} - 2\beta_{11} + 19\beta_{10} - 18\beta_{8} + 36\beta_{7} + 21\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{9} + 13\beta_{6} + 22\beta_{4} - 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18 \beta_{15} + 66 \beta_{13} - 84 \beta_{12} + 24 \beta_{11} - 99 \beta_{10} - 120 \beta_{8} + \cdots + 123 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -183\beta_{14} - 24\beta_{9} + 141\beta_{6} + 1119\beta_{5} + 267\beta_{4} + 174\beta_{3} - 543\beta _1 - 648 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -63\beta_{15} - 243\beta_{12} + 99\beta_{11} - 270\beta_{10} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1107 \beta_{14} + 198 \beta_{9} - 783 \beta_{6} + 6849 \beta_{5} - 1593 \beta_{4} + 882 \beta_{3} + \cdots + 3456 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 810 \beta_{15} - 1980 \beta_{13} - 2790 \beta_{12} + 1404 \beta_{11} - 3015 \beta_{10} + \cdots - 4419 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 702\beta_{9} - 2205\beta_{6} - 4707\beta_{4} + 9441 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 5004 \beta_{15} - 11016 \beta_{13} + 16020 \beta_{12} - 9216 \beta_{11} + 17055 \beta_{10} + \cdots - 26271 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 39285 \beta_{14} + 9216 \beta_{9} - 25065 \beta_{6} - 246105 \beta_{5} - 55305 \beta_{4} - 24894 \beta_{3} + \cdots + 104922 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15120\beta_{15} + 46089\beta_{12} - 28944\beta_{11} + 48627\beta_{10} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6480\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(6401\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
2.08612 − 1.20442i
−1.20298 + 0.694538i
0.384853 + 0.222195i
−1.74724 − 1.00877i
1.74724 + 1.00877i
−0.384853 − 0.222195i
1.20298 − 0.694538i
−2.08612 + 1.20442i
−2.08612 − 1.20442i
1.20298 + 0.694538i
−0.384853 + 0.222195i
1.74724 − 1.00877i
−1.74724 + 1.00877i
0.384853 − 0.222195i
−1.20298 − 0.694538i
2.08612 + 1.20442i
0 0 0 − 1.00000i 0 − 4.34198i 0 0 0
2591.2 0 0 0 − 1.00000i 0 − 2.87604i 0 0 0
2591.3 0 0 0 − 1.00000i 0 − 1.68980i 0 0 0
2591.4 0 0 0 − 1.00000i 0 − 0.142168i 0 0 0
2591.5 0 0 0 − 1.00000i 0 0.142168i 0 0 0
2591.6 0 0 0 − 1.00000i 0 1.68980i 0 0 0
2591.7 0 0 0 − 1.00000i 0 2.87604i 0 0 0
2591.8 0 0 0 − 1.00000i 0 4.34198i 0 0 0
2591.9 0 0 0 1.00000i 0 − 4.34198i 0 0 0
2591.10 0 0 0 1.00000i 0 − 2.87604i 0 0 0
2591.11 0 0 0 1.00000i 0 − 1.68980i 0 0 0
2591.12 0 0 0 1.00000i 0 − 0.142168i 0 0 0
2591.13 0 0 0 1.00000i 0 0.142168i 0 0 0
2591.14 0 0 0 1.00000i 0 1.68980i 0 0 0
2591.15 0 0 0 1.00000i 0 2.87604i 0 0 0
2591.16 0 0 0 1.00000i 0 4.34198i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
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 6480.2.h.c 16
3.b odd 2 1 inner 6480.2.h.c 16
4.b odd 2 1 inner 6480.2.h.c 16
9.c even 3 1 720.2.bw.b ✓ 16
9.c even 3 1 2160.2.bw.b 16
9.d odd 6 1 720.2.bw.b ✓ 16
9.d odd 6 1 2160.2.bw.b 16
12.b even 2 1 inner 6480.2.h.c 16
36.f odd 6 1 720.2.bw.b ✓ 16
36.f odd 6 1 2160.2.bw.b 16
36.h even 6 1 720.2.bw.b ✓ 16
36.h even 6 1 2160.2.bw.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bw.b ✓ 16 9.c even 3 1
720.2.bw.b ✓ 16 9.d odd 6 1
720.2.bw.b ✓ 16 36.f odd 6 1
720.2.bw.b ✓ 16 36.h even 6 1
2160.2.bw.b 16 9.c even 3 1
2160.2.bw.b 16 9.d odd 6 1
2160.2.bw.b 16 36.f odd 6 1
2160.2.bw.b 16 36.h even 6 1
6480.2.h.c 16 1.a even 1 1 trivial
6480.2.h.c 16 3.b odd 2 1 inner
6480.2.h.c 16 4.b odd 2 1 inner
6480.2.h.c 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6480, [\chi])\):

\( T_{7}^{8} + 30T_{7}^{6} + 234T_{7}^{4} + 450T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{8} - 36T_{11}^{6} + 180T_{11}^{4} - 288T_{11}^{2} + 144 \) 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
$7$ \( (T^{8} + 30 T^{6} + 234 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 36 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 30 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 72 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 60 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 90 T^{6} + \cdots + 106929)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 54 T^{2} + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 156 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots - 284)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 72 T^{6} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 168 T^{6} + \cdots + 17424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 198 T^{6} + \cdots + 288369)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 216 T^{6} + \cdots + 278784)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + \cdots - 2591)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 306 T^{6} + \cdots + 2537649)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 468 T^{6} + \cdots + 17539344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots - 704)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 480 T^{6} + \cdots + 137170944)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 378 T^{6} + \cdots + 26594649)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 324 T^{6} + \cdots + 14220441)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 104)^{8} \) Copy content Toggle raw display
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