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Properties

Label 1449.4.a.r
Level $1449$
Weight $4$
Character orbit 1449.a
Self dual yes
Analytic conductor $85.494$
Analytic rank $1$
Dimension $17$
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] = [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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 104 x^{15} + 4325 x^{13} - 18 x^{12} - 92339 x^{11} + 770 x^{10} + 1080612 x^{9} + \cdots + 4314624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 4) q^{4} + ( - \beta_{7} - 1) q^{5} - 7 q^{7} + ( - \beta_{3} - 5 \beta_1) q^{8} + ( - \beta_{10} + \beta_{7} + \beta_{3} + \cdots - 4) q^{10} + (\beta_{9} - \beta_{2} - 3) q^{11}+ \cdots - 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 72 q^{4} - 20 q^{5} - 119 q^{7} - 70 q^{10} - 54 q^{11} - 66 q^{13} + 428 q^{16} - 68 q^{17} - 26 q^{19} - 198 q^{20} - 4 q^{22} + 391 q^{23} + 589 q^{25} - 260 q^{26} - 504 q^{28} - 24 q^{29} + 4 q^{31}+ \cdots - 2204 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 104 x^{15} + 4325 x^{13} - 18 x^{12} - 92339 x^{11} + 770 x^{10} + 1080612 x^{9} + \cdots + 4314624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 104323083950013 \nu^{16} + 53950273237637 \nu^{15} + \cdots + 63\!\cdots\!76 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 301493772659815 \nu^{16} + 822271743740609 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2214723090399 \nu^{16} + 3300027988855 \nu^{15} - 213039900081225 \nu^{14} + \cdots - 75\!\cdots\!28 ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 629222184044787 \nu^{16} - 653034865749589 \nu^{15} + \cdots + 32\!\cdots\!80 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 317649814369545 \nu^{16} + 518035267871759 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 16\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 669695099281277 \nu^{16} + 335301836680517 \nu^{15} + \cdots - 38\!\cdots\!56 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 160282131224297 \nu^{16} - 250771061387791 \nu^{15} + \cdots + 72\!\cdots\!04 ) / 40\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 660164840477585 \nu^{16} + \cdots - 46\!\cdots\!12 ) / 16\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!75 \nu^{16} + \cdots + 97\!\cdots\!24 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{16} + 689051368373641 \nu^{15} + \cdots - 65\!\cdots\!68 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!91 \nu^{16} + \cdots + 14\!\cdots\!80 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!19 \nu^{16} + \cdots + 12\!\cdots\!64 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!39 \nu^{16} - 220085438322173 \nu^{15} + \cdots + 83\!\cdots\!48 ) / 16\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + 3\beta_{7} + \beta_{4} + 30\beta_{2} + 3\beta _1 + 248 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 3 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 44 \beta_{16} + \beta_{15} + 3 \beta_{14} - \beta_{13} + 38 \beta_{12} + 52 \beta_{11} + 43 \beta_{10} + \cdots + 5981 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{16} - 41 \beta_{15} + 159 \beta_{14} - 94 \beta_{13} - 107 \beta_{12} - 63 \beta_{11} + \cdots + 411 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1540 \beta_{16} + 14 \beta_{15} + 148 \beta_{14} - 54 \beta_{13} + 1196 \beta_{12} + 2000 \beta_{11} + \cdots + 155672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 220 \beta_{16} - 1310 \beta_{15} + 6266 \beta_{14} - 3340 \beta_{13} - 4204 \beta_{12} - 2804 \beta_{11} + \cdots + 13302 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 50521 \beta_{16} - 590 \beta_{15} + 5294 \beta_{14} - 2136 \beta_{13} + 36333 \beta_{12} + \cdots + 4246612 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 10702 \beta_{16} - 39875 \beta_{15} + 222637 \beta_{14} - 107744 \beta_{13} - 147744 \beta_{12} + \cdots + 194310 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1617010 \beta_{16} - 50059 \beta_{15} + 170439 \beta_{14} - 74949 \beta_{13} + 1101132 \beta_{12} + \cdots + 119573859 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 431175 \beta_{16} - 1213211 \beta_{15} + 7524957 \beta_{14} - 3337748 \beta_{13} - 4930211 \beta_{12} + \cdots - 5250863 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 51131762 \beta_{16} - 2387388 \beta_{15} + 5256834 \beta_{14} - 2455870 \beta_{13} + \cdots + 3442347430 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 15512102 \beta_{16} - 37241450 \beta_{15} + 247024234 \beta_{14} - 101599528 \beta_{13} + \cdots - 583164166 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1604843225 \beta_{16} - 94467596 \beta_{15} + 158549248 \beta_{14} - 76644096 \beta_{13} + \cdots + 100672619180 \) 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
5.46687
5.08708
4.32921
3.42888
2.82689
2.33706
2.11269
0.540052
−0.562483
−0.777480
−1.26182
−1.73807
−3.13070
−4.24387
−4.29964
−4.53705
−5.57763
−5.46687 0 21.8867 −17.3516 0 −7.00000 −75.9166 0 94.8589
1.2 −5.08708 0 17.8784 4.94565 0 −7.00000 −50.2523 0 −25.1589
1.3 −4.32921 0 10.7421 21.3803 0 −7.00000 −11.8710 0 −92.5598
1.4 −3.42888 0 3.75723 −17.0611 0 −7.00000 14.5480 0 58.5005
1.5 −2.82689 0 −0.00869765 2.03908 0 −7.00000 22.6397 0 −5.76426
1.6 −2.33706 0 −2.53813 12.1976 0 −7.00000 24.6283 0 −28.5067
1.7 −2.11269 0 −3.53654 7.48977 0 −7.00000 24.3731 0 −15.8236
1.8 −0.540052 0 −7.70834 −7.47775 0 −7.00000 8.48332 0 4.03838
1.9 0.562483 0 −7.68361 −15.7035 0 −7.00000 −8.82177 0 −8.83295
1.10 0.777480 0 −7.39553 8.19211 0 −7.00000 −11.9697 0 6.36920
1.11 1.26182 0 −6.40780 −21.6871 0 −7.00000 −18.1801 0 −27.3653
1.12 1.73807 0 −4.97911 13.6090 0 −7.00000 −22.5586 0 23.6534
1.13 3.13070 0 1.80126 −5.12959 0 −7.00000 −19.4064 0 −16.0592
1.14 4.24387 0 10.0104 12.8217 0 −7.00000 8.53197 0 54.4136
1.15 4.29964 0 10.4869 −9.72059 0 −7.00000 10.6926 0 −41.7950
1.16 4.53705 0 12.5849 2.22292 0 −7.00000 20.8018 0 10.0855
1.17 5.57763 0 23.1100 −10.7669 0 −7.00000 84.2777 0 −60.0538
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.r ✓ 17
3.b odd 2 1 1449.4.a.s yes 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1449.4.a.r ✓ 17 1.a even 1 1 trivial
1449.4.a.s yes 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 104 T_{2}^{15} + 4325 T_{2}^{13} + 18 T_{2}^{12} - 92339 T_{2}^{11} - 770 T_{2}^{10} + \cdots - 4314624 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 104 T^{15} + \cdots - 4314624 \) Copy content Toggle raw display
$3$ \( T^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 25\!\cdots\!60 \) Copy content Toggle raw display
$7$ \( (T + 7)^{17} \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 48\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots - 97\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T - 23)^{17} \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 26\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 33\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 25\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
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