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Properties

Label 5415.2.a.bm
Level $5415$
Weight $2$
Character orbit 5415.a
Self dual yes
Analytic conductor $43.239$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

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: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 19 x^{10} - 4 x^{9} + 116 x^{8} + 32 x^{7} - 293 x^{6} - 92 x^{5} + 309 x^{4} + 100 x^{3} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - q^{3} + ( - \beta_{9} + \beta_{7} + 1) q^{4} - q^{5} - \beta_{9} q^{6} + ( - \beta_{6} + \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{9} - \beta_{6} + \beta_{5} + \cdots - 1) q^{8}+ \cdots + (\beta_{8} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 12 q^{3} + 18 q^{4} - 12 q^{5} + 4 q^{6} - 10 q^{7} - 18 q^{8} + 12 q^{9} + 4 q^{10} + 22 q^{11} - 18 q^{12} - 16 q^{13} - 4 q^{14} + 12 q^{15} + 18 q^{16} + 8 q^{17} - 4 q^{18} - 18 q^{20}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 19 x^{10} - 4 x^{9} + 116 x^{8} + 32 x^{7} - 293 x^{6} - 92 x^{5} + 309 x^{4} + 100 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 991 \nu^{11} + 163110 \nu^{10} - 2627 \nu^{9} - 2906780 \nu^{8} - 407140 \nu^{7} + \cdots - 1277936 ) / 600841 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23056 \nu^{11} - 11520 \nu^{10} - 389361 \nu^{9} + 117332 \nu^{8} + 1915376 \nu^{7} - 556136 \nu^{6} + \cdots + 32792 ) / 600841 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32792 \nu^{11} - 23056 \nu^{10} - 611528 \nu^{9} + 258193 \nu^{8} + 3686540 \nu^{7} - 866032 \nu^{6} + \cdots - 862336 ) / 600841 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51734 \nu^{11} + 37107 \nu^{10} + 914787 \nu^{9} - 379815 \nu^{8} - 4870287 \nu^{7} + \cdots + 408844 ) / 600841 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53532 \nu^{11} - 149334 \nu^{10} - 951249 \nu^{9} + 2476064 \nu^{8} + 5607154 \nu^{7} + \cdots - 703997 ) / 600841 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 86856 \nu^{11} + 20465 \nu^{10} + 1625028 \nu^{9} - 15459 \nu^{8} - 9765692 \nu^{7} + \cdots + 2009223 ) / 600841 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 107336 \nu^{11} + 40288 \nu^{10} + 1997572 \nu^{9} - 363604 \nu^{8} - 11798968 \nu^{7} + \cdots - 2571453 ) / 600841 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 109046 \nu^{11} - 24047 \nu^{10} - 1897244 \nu^{9} - 49450 \nu^{8} + 9625224 \nu^{7} + \cdots + 1147878 ) / 600841 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 176834 \nu^{11} + 44238 \nu^{10} - 3378197 \nu^{9} - 1440703 \nu^{8} + 20498744 \nu^{7} + \cdots - 609101 ) / 600841 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 203398 \nu^{11} - 63268 \nu^{10} - 3703692 \nu^{9} + 359824 \nu^{8} + 20977243 \nu^{7} + \cdots - 492514 ) / 600841 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + 2\beta_{4} - \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{4} - 2\beta_{3} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{11} - \beta_{9} + 10 \beta_{8} + 13 \beta_{7} - 3 \beta_{6} + 10 \beta_{5} + 23 \beta_{4} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{11} - 10 \beta_{10} - 12 \beta_{9} + 7 \beta_{8} + 18 \beta_{7} - 15 \beta_{6} + 4 \beta_{5} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 136 \beta_{11} - 4 \beta_{10} - 21 \beta_{9} + 100 \beta_{8} + 145 \beta_{7} - 44 \beta_{6} + 96 \beta_{5} + \cdots + 275 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 377 \beta_{11} - 96 \beta_{10} - 130 \beta_{9} + 133 \beta_{8} + 252 \beta_{7} - 185 \beta_{6} + \cdots + 318 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1531 \beta_{11} - 88 \beta_{10} - 299 \beta_{9} + 1029 \beta_{8} + 1578 \beta_{7} - 541 \beta_{6} + \cdots + 2742 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4508 \beta_{11} - 961 \beta_{10} - 1411 \beta_{9} + 1889 \beta_{8} + 3249 \beta_{7} - 2148 \beta_{6} + \cdots + 4439 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17252 \beta_{11} - 1360 \beta_{10} - 3766 \beta_{9} + 10894 \beta_{8} + 17183 \beta_{7} - 6397 \beta_{6} + \cdots + 28576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 52870 \beta_{11} - 9966 \beta_{10} - 15554 \beta_{9} + 24253 \beta_{8} + 40230 \beta_{7} - 24469 \beta_{6} + \cdots + 57568 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.49550
1.04611
3.38350
−0.930473
−0.347787
1.74911
1.91379
−2.24742
−1.13360
0.921955
0.0257321
−2.88543
−2.73178 −1.00000 5.46264 −1.00000 2.73178 −5.10810 −9.45917 1.00000 2.73178
1.2 −2.57058 −1.00000 4.60786 −1.00000 2.57058 −0.896055 −6.70371 1.00000 2.57058
1.3 −2.52181 −1.00000 4.35951 −1.00000 2.52181 0.989250 −5.95024 1.00000 2.52181
1.4 −1.86435 −1.00000 1.47580 −1.00000 1.86435 2.59165 0.977288 1.00000 1.86435
1.5 −1.40468 −1.00000 −0.0268826 −1.00000 1.40468 1.48776 2.84712 1.00000 1.40468
1.6 −1.13623 −1.00000 −0.708973 −1.00000 1.13623 1.12789 3.07803 1.00000 1.13623
1.7 0.363549 −1.00000 −1.86783 −1.00000 −0.363549 −2.77102 −1.40615 1.00000 −0.363549
1.8 0.585827 −1.00000 −1.65681 −1.00000 −0.585827 −4.27434 −2.14226 1.00000 −0.585827
1.9 1.02437 −1.00000 −0.950664 −1.00000 −1.02437 −1.95686 −3.02257 1.00000 −1.02437
1.10 1.69245 −1.00000 0.864372 −1.00000 −1.69245 1.53774 −1.92199 1.00000 −1.69245
1.11 2.16032 −1.00000 2.66698 −1.00000 −2.16032 0.376786 1.44089 1.00000 −2.16032
1.12 2.40291 −1.00000 3.77400 −1.00000 −2.40291 −3.10470 4.26276 1.00000 −2.40291
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5415.2.a.bm ✓ 12
19.b odd 2 1 5415.2.a.br yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5415.2.a.bm ✓ 12 1.a even 1 1 trivial
5415.2.a.br yes 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 13 T_{2}^{10} - 62 T_{2}^{9} + 53 T_{2}^{8} + 348 T_{2}^{7} - 62 T_{2}^{6} + \cdots + 101 \) Copy content Toggle raw display
\( T_{7}^{12} + 10 T_{7}^{11} + 10 T_{7}^{10} - 152 T_{7}^{9} - 280 T_{7}^{8} + 930 T_{7}^{7} + 1519 T_{7}^{6} + \cdots + 821 \) Copy content Toggle raw display
\( T_{11}^{12} - 22 T_{11}^{11} + 178 T_{11}^{10} - 490 T_{11}^{9} - 1551 T_{11}^{8} + 14856 T_{11}^{7} + \cdots - 35839 \) Copy content Toggle raw display
\( T_{13}^{12} + 16 T_{13}^{11} + 38 T_{13}^{10} - 568 T_{13}^{9} - 2905 T_{13}^{8} + 4584 T_{13}^{7} + \cdots - 290159 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots + 101 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 10 T^{11} + \cdots + 821 \) Copy content Toggle raw display
$11$ \( T^{12} - 22 T^{11} + \cdots - 35839 \) Copy content Toggle raw display
$13$ \( T^{12} + 16 T^{11} + \cdots - 290159 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 21101 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + \cdots + 1307641 \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{11} + \cdots - 9738095 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 134107739 \) Copy content Toggle raw display
$37$ \( T^{12} + 10 T^{11} + \cdots + 832801 \) Copy content Toggle raw display
$41$ \( T^{12} + 12 T^{11} + \cdots + 42465601 \) Copy content Toggle raw display
$43$ \( T^{12} + 22 T^{11} + \cdots + 45271021 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 804824599 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 273139781 \) Copy content Toggle raw display
$59$ \( T^{12} + 10 T^{11} + \cdots - 2086219 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + \cdots + 8619805 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 345570139 \) Copy content Toggle raw display
$71$ \( T^{12} + 10 T^{11} + \cdots - 2708999 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 989107375 \) Copy content Toggle raw display
$79$ \( T^{12} - 4 T^{11} + \cdots + 95037221 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 12041145859 \) Copy content Toggle raw display
$89$ \( T^{12} + 34 T^{11} + \cdots - 2988295 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 23479621901 \) Copy content Toggle raw display
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