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Properties

Label 468.2.bv.a
Level $468$
Weight $2$
Character orbit 468.bv
Analytic conductor $3.737$
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] = [468,2,Mod(89,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.bv (of order \(12\), degree \(4\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} - \beta_{9}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} - \beta_{9}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - 1) q^{7} + (\beta_{15} + \beta_{14} + \cdots - \beta_{8}) q^{11}+ \cdots + ( - 3 \beta_{7} - 8 \beta_{6} + \cdots + 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 12 q^{13} - 4 q^{19} + 4 q^{31} + 16 q^{37} + 36 q^{43} + 36 q^{49} - 12 q^{61} + 32 q^{67} - 64 q^{73} - 96 q^{79} - 48 q^{85} - 28 q^{91} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 97553 \nu^{14} - 59673197 \nu^{12} + 497612320 \nu^{10} - 1386197292 \nu^{8} + \cdots + 33039693843 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4506447 \nu^{14} - 121241175 \nu^{12} + 1034556480 \nu^{10} - 3947923892 \nu^{8} + \cdots - 42326241815 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13282279 \nu^{14} + 200884791 \nu^{12} - 1177246560 \nu^{10} + 3219861076 \nu^{8} + \cdots - 296484914921 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7844936 \nu^{14} + 136368445 \nu^{12} - 773884672 \nu^{10} + 1974780696 \nu^{8} + \cdots - 181212319067 ) / 114021875864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1322533 \nu^{14} - 13200781 \nu^{12} + 41605136 \nu^{10} + 115043140 \nu^{8} + \cdots + 3326257155 ) / 17541827056 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30695 \nu^{14} - 419209 \nu^{12} + 2520928 \nu^{10} - 7255428 \nu^{8} + 1602740 \nu^{6} + \cdots + 379906423 ) / 289763344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2640659 \nu^{14} - 35680975 \nu^{12} + 174215264 \nu^{10} - 156875412 \nu^{8} + \cdots + 33548140145 ) / 17541827056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50666114 \nu^{15} + 732074351 \nu^{13} - 3701384776 \nu^{11} + 9907966024 \nu^{9} + \cdots - 1229225419617 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 69962947 \nu^{15} - 626126278 \nu^{13} + 1153913040 \nu^{11} + 9749257260 \nu^{9} + \cdots + 1139974663490 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 147179275 \nu^{15} - 1914617631 \nu^{13} + 11298132312 \nu^{11} - 33481415972 \nu^{9} + \cdots + 427559843945 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1582155 \nu^{15} + 21956880 \nu^{13} - 119050416 \nu^{11} + 229166428 \nu^{9} + \cdots - 29455777936 \nu ) / 14676083032 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1744076 \nu^{15} - 29470341 \nu^{13} + 189099312 \nu^{11} - 564140880 \nu^{9} + \cdots + 25716585843 \nu ) / 14676083032 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 271024417 \nu^{15} + 3673935498 \nu^{13} - 19294289752 \nu^{11} + \cdots - 3032420997190 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 348484208 \nu^{15} - 4736952005 \nu^{13} + 25606728568 \nu^{11} - 48182513944 \nu^{9} + \cdots + 2871441590667 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 389943419 \nu^{15} - 5627056128 \nu^{13} + 30944918552 \nu^{11} - 63467036404 \nu^{9} + \cdots + 4281857943584 \nu ) / 1482284386232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} - 6\beta_{11} - \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 3\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{15} - 7\beta_{14} + 5\beta_{13} - 18\beta_{12} - 12\beta_{11} + 6\beta_{10} - 18\beta_{9} - 7\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 2\beta_{5} - 2\beta_{4} + 12\beta_{3} - 7\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 91 \beta_{15} - 95 \beta_{14} + 5 \beta_{13} - 60 \beta_{12} - 48 \beta_{11} + 48 \beta_{10} + \cdots + 61 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{7} + 36\beta_{6} - 42\beta_{5} + 47\beta_{3} - 47\beta_{2} + 20\beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 167\beta_{15} - 485\beta_{14} - 437\beta_{13} - 30\beta_{12} - 48\beta_{11} + 96\beta_{10} + 613\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 233\beta_{7} + 44\beta_{6} - 260\beta_{5} + 130\beta_{4} + 167\beta_{3} - 74\beta_{2} - 44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1315 \beta_{15} - 469 \beta_{14} - 3491 \beta_{13} + 558 \beta_{12} - 1134 \beta_{10} + \cdots + 3203 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1320\beta_{7} + 99\beta_{6} - 1133\beta_{5} + 1133\beta_{4} + 1133\beta_{3} + 1035\beta_{2} - 660\beta _1 + 1122 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14519 \beta_{15} + 9143 \beta_{14} - 13495 \beta_{13} - 7326 \beta_{11} - 7326 \beta_{10} + \cdots + 7193 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4344 \beta_{7} + 4344 \beta_{6} - 2552 \beta_{5} + 5104 \beta_{4} + 12156 \beta_{3} + 7052 \beta_{2} + \cdots + 12335 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 56555 \beta_{15} + 62197 \beta_{14} - 4573 \beta_{13} - 57624 \beta_{12} - 100074 \beta_{11} + \cdots - 26867 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4948\beta_{7} + 64991\beta_{6} + 8353\beta_{4} + 103881\beta_{3} - 9896\beta _1 + 55095 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 75071 \beta_{15} + 56443 \beta_{14} + 247831 \beta_{13} - 623286 \beta_{12} - 720516 \beta_{11} + \cdots - 243941 \beta_{8} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−2.39101 + 0.123030i
−2.13219 + 1.08896i
2.13219 − 1.08896i
2.39101 − 0.123030i
−0.115299 + 1.50155i
0.850627 + 1.24273i
−0.850627 − 1.24273i
0.115299 − 1.50155i
−2.39101 − 0.123030i
−2.13219 − 1.08896i
2.13219 + 1.08896i
2.39101 + 0.123030i
−0.115299 − 1.50155i
0.850627 − 1.24273i
−0.850627 + 1.24273i
0.115299 + 1.50155i
0 0 0 −2.26798 + 2.26798i 0 −0.563685 − 2.10370i 0 0 0
89.2 0 0 0 −1.04323 + 1.04323i 0 0.0636852 + 0.237676i 0 0 0
89.3 0 0 0 1.04323 − 1.04323i 0 0.0636852 + 0.237676i 0 0 0
89.4 0 0 0 2.26798 − 2.26798i 0 −0.563685 − 2.10370i 0 0 0
197.1 0 0 0 −1.61685 − 1.61685i 0 2.40077 − 0.643285i 0 0 0
197.2 0 0 0 −0.392105 − 0.392105i 0 −2.90077 + 0.777260i 0 0 0
197.3 0 0 0 0.392105 + 0.392105i 0 −2.90077 + 0.777260i 0 0 0
197.4 0 0 0 1.61685 + 1.61685i 0 2.40077 − 0.643285i 0 0 0
305.1 0 0 0 −2.26798 − 2.26798i 0 −0.563685 + 2.10370i 0 0 0
305.2 0 0 0 −1.04323 − 1.04323i 0 0.0636852 − 0.237676i 0 0 0
305.3 0 0 0 1.04323 + 1.04323i 0 0.0636852 − 0.237676i 0 0 0
305.4 0 0 0 2.26798 + 2.26798i 0 −0.563685 + 2.10370i 0 0 0
449.1 0 0 0 −1.61685 + 1.61685i 0 2.40077 + 0.643285i 0 0 0
449.2 0 0 0 −0.392105 + 0.392105i 0 −2.90077 − 0.777260i 0 0 0
449.3 0 0 0 0.392105 − 0.392105i 0 −2.90077 − 0.777260i 0 0 0
449.4 0 0 0 1.61685 − 1.61685i 0 2.40077 + 0.643285i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.bv.a ✓ 16
3.b odd 2 1 inner 468.2.bv.a ✓ 16
13.c even 3 1 6084.2.p.d 16
13.e even 6 1 6084.2.p.e 16
13.f odd 12 1 inner 468.2.bv.a ✓ 16
13.f odd 12 1 6084.2.p.d 16
13.f odd 12 1 6084.2.p.e 16
39.h odd 6 1 6084.2.p.e 16
39.i odd 6 1 6084.2.p.d 16
39.k even 12 1 inner 468.2.bv.a ✓ 16
39.k even 12 1 6084.2.p.d 16
39.k even 12 1 6084.2.p.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.bv.a ✓ 16 1.a even 1 1 trivial
468.2.bv.a ✓ 16 3.b odd 2 1 inner
468.2.bv.a ✓ 16 13.f odd 12 1 inner
468.2.bv.a ✓ 16 39.k even 12 1 inner
6084.2.p.d 16 13.c even 3 1
6084.2.p.d 16 13.f odd 12 1
6084.2.p.d 16 39.i odd 6 1
6084.2.p.d 16 39.k even 12 1
6084.2.p.e 16 13.e even 6 1
6084.2.p.e 16 13.f odd 12 1
6084.2.p.e 16 39.h odd 6 1
6084.2.p.e 16 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 138 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} - 7 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 60 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 592240896 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2 T^{7} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1618961043456 \) Copy content Toggle raw display
$29$ \( T^{16} - 48 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 676)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 9475854336 \) Copy content Toggle raw display
$43$ \( (T^{8} - 18 T^{7} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 767544201216 \) Copy content Toggle raw display
$53$ \( (T^{8} + 252 T^{6} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 228 T^{14} + \cdots + 5308416 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 23242041)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots + 3139984)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 8430985875456 \) Copy content Toggle raw display
$73$ \( (T^{8} + 32 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + \cdots - 10056)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 151613669376 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 151613669376 \) Copy content Toggle raw display
$97$ \( (T^{8} + 28 T^{7} + \cdots + 65836996)^{2} \) Copy content Toggle raw display
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