Properties

Label 1449.4.a.m
Level $1449$
Weight $4$
Character orbit 1449.a
Self dual yes
Analytic conductor $85.494$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1449,4,Mod(1,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1449.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1449.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: \(85.4937675983\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 57x^{7} - 13x^{6} + 1042x^{5} + 331x^{4} - 6570x^{3} - 1782x^{2} + 9424x + 5112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 483)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + ( - \beta_{7} - 3) q^{5} + 7 q^{7} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + ( - \beta_{7} - 3) q^{5} + 7 q^{7} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 5) q^{8} + (\beta_{8} - \beta_{6} + \beta_{4} - 5 \beta_1 - 7) q^{10} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{8} + \beta_{6} + 2 \beta_{2} + \beta_1 + 23) q^{13} + 7 \beta_1 q^{14} + ( - \beta_{8} + 3 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 9 \beta_1 + 20) q^{16} + (\beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 14) q^{17} + ( - \beta_{8} - 3 \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 31) q^{19} + (3 \beta_{8} - 7 \beta_{7} + \beta_{6} + 3 \beta_{5} - 8 \beta_{2} - 6 \beta_1 - 38) q^{20} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \cdots - 30) q^{22}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8} - 55 q^{10} - 12 q^{11} + 199 q^{13} + 170 q^{16} - 116 q^{17} + 260 q^{19} - 324 q^{20} - 265 q^{22} - 207 q^{23} + 438 q^{25} + 270 q^{26} + 294 q^{28} + 107 q^{29} + 440 q^{31} + 802 q^{32} + 295 q^{34} - 203 q^{35} + 563 q^{37} + 569 q^{38} - 640 q^{40} - 243 q^{41} + 435 q^{43} - 1025 q^{44} + 133 q^{47} + 441 q^{49} - 104 q^{50} + 2693 q^{52} - 958 q^{53} + 1846 q^{55} + 273 q^{56} + 2796 q^{58} - 538 q^{59} + 1374 q^{61} - 1263 q^{62} - 83 q^{64} - 745 q^{65} + 752 q^{67} - 5593 q^{68} - 385 q^{70} + 418 q^{71} + 2406 q^{73} - 352 q^{74} + 2765 q^{76} - 84 q^{77} - 486 q^{79} - 5709 q^{80} + 2726 q^{82} - 106 q^{83} + 4130 q^{85} + 2576 q^{86} + 1270 q^{88} - 234 q^{89} + 1393 q^{91} - 966 q^{92} + 4967 q^{94} + 3074 q^{95} + 2409 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 57x^{7} - 13x^{6} + 1042x^{5} + 331x^{4} - 6570x^{3} - 1782x^{2} + 9424x + 5112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93 \nu^{8} + 162 \nu^{7} + 7513 \nu^{6} - 12377 \nu^{5} - 193072 \nu^{4} + 262137 \nu^{3} + 1692304 \nu^{2} - 1540050 \nu - 2468812 ) / 30928 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75 \nu^{8} - 193 \nu^{7} - 4500 \nu^{6} + 10605 \nu^{5} + 86427 \nu^{4} - 184900 \nu^{3} - 553026 \nu^{2} + 1000912 \nu + 614432 ) / 15464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59 \nu^{8} + 209 \nu^{7} - 3540 \nu^{6} - 9441 \nu^{5} + 61649 \nu^{4} + 114856 \nu^{3} - 298500 \nu^{2} - 276436 \nu + 54588 ) / 7732 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 253 \nu^{8} - 316 \nu^{7} - 13247 \nu^{6} + 11805 \nu^{5} + 220490 \nu^{4} - 147567 \nu^{3} - 1237548 \nu^{2} + 743870 \nu + 1294020 ) / 30928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 645 \nu^{8} - 500 \nu^{7} - 36767 \nu^{6} + 17749 \nu^{5} + 670978 \nu^{4} - 227375 \nu^{3} - 4081020 \nu^{2} + 1586414 \nu + 4196996 ) / 30928 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + 3\beta_{7} - 2\beta_{5} - 2\beta_{4} + 26\beta_{2} + 9\beta _1 + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{8} + 8\beta_{7} + 2\beta_{6} + 3\beta_{5} - 3\beta_{4} + 33\beta_{3} + 42\beta_{2} + 452\beta _1 + 253 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -53\beta_{8} + 148\beta_{7} + 4\beta_{6} - 77\beta_{5} - 79\beta_{4} + 6\beta_{3} + 658\beta_{2} + 472\beta _1 + 6231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 264 \beta_{8} + 447 \beta_{7} + 120 \beta_{6} + 79 \beta_{5} - 183 \beta_{4} + 905 \beta_{3} + 1427 \beta_{2} + 10955 \beta _1 + 9840 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2000 \beta_{8} + 5442 \beta_{7} + 266 \beta_{6} - 2330 \beta_{5} - 2482 \beta_{4} + 488 \beta_{3} + 17091 \beta_{2} + 18188 \beta _1 + 154567 \) Copy content Toggle raw display

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
−4.81513
−4.30357
−3.40437
−1.05204
−0.644455
1.60936
2.78266
4.38536
5.44218
−4.81513 0 15.1855 −17.0256 0 7.00000 −34.5991 0 81.9804
1.2 −4.30357 0 10.5207 16.7192 0 7.00000 −10.8482 0 −71.9524
1.3 −3.40437 0 3.58971 −5.47889 0 7.00000 15.0142 0 18.6522
1.4 −1.05204 0 −6.89321 11.5074 0 7.00000 15.6682 0 −12.1063
1.5 −0.644455 0 −7.58468 −15.1553 0 7.00000 10.0436 0 9.76692
1.6 1.60936 0 −5.40995 −4.60745 0 7.00000 −21.5815 0 −7.41506
1.7 2.78266 0 −0.256781 −7.60787 0 7.00000 −22.9758 0 −21.1701
1.8 4.38536 0 11.2314 12.0618 0 7.00000 14.1708 0 52.8954
1.9 5.44218 0 21.6173 −19.4134 0 7.00000 74.1078 0 −105.651
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 1449.4.a.m 9
3.b odd 2 1 483.4.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.f 9 3.b odd 2 1
1449.4.a.m 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 57T_{2}^{7} - 13T_{2}^{6} + 1042T_{2}^{5} + 331T_{2}^{4} - 6570T_{2}^{3} - 1782T_{2}^{2} + 9424T_{2} + 5112 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 57 T^{7} - 13 T^{6} + \cdots + 5112 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 29 T^{8} + \cdots - 2232500544 \) Copy content Toggle raw display
$7$ \( (T - 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 12 T^{8} + \cdots + 592551340800 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 102208163239392 \) Copy content Toggle raw display
$17$ \( T^{9} + 116 T^{8} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{9} - 260 T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T + 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} - 107 T^{8} + \cdots - 91\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{9} - 440 T^{8} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{9} - 563 T^{8} + \cdots - 23\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{9} + 243 T^{8} + \cdots + 57\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{9} - 435 T^{8} + \cdots + 55\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{9} - 133 T^{8} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{9} + 958 T^{8} + \cdots - 59\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{9} + 538 T^{8} + \cdots - 23\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{9} - 1374 T^{8} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{9} - 752 T^{8} + \cdots - 53\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{9} - 418 T^{8} + \cdots + 57\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{9} - 2406 T^{8} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{9} + 486 T^{8} + \cdots - 26\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{9} + 106 T^{8} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{9} + 234 T^{8} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{9} - 2409 T^{8} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
show more
show less