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Properties

Label 2299.4.a.k
Level $2299$
Weight $4$
Character orbit 2299.a
Self dual yes
Analytic conductor $135.645$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2299,4,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2299.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2299.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,2,11,82,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(135.645391103\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{10} + \beta_{5} - \beta_1 + 1) q^{5} + (\beta_{12} - \beta_{10} - \beta_{7} + \cdots - 2) q^{6} + (\beta_{10} - \beta_{9} + \beta_{7} + \cdots - 2) q^{7}+ \cdots + ( - 59 \beta_{12} - 2 \beta_{11} + \cdots + 543) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} + 11 q^{3} + 82 q^{4} + 8 q^{5} - 13 q^{6} - 39 q^{7} - 6 q^{8} + 156 q^{9} - 124 q^{10} + 247 q^{12} + 23 q^{13} + 47 q^{14} + 278 q^{15} + 526 q^{16} - 73 q^{17} + 165 q^{18} + 247 q^{19}+ \cdots + 6273 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 179351 \nu^{12} - 6111956 \nu^{11} + 22627657 \nu^{10} + 624540444 \nu^{9} + \cdots + 665042042368 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 632365 \nu^{12} - 12143320 \nu^{11} + 16549089 \nu^{10} + 1022191756 \nu^{9} + \cdots + 4814040741376 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1490925 \nu^{12} - 3374106 \nu^{11} - 117105595 \nu^{10} + 266779852 \nu^{9} + \cdots - 863686138368 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1017601 \nu^{12} - 276553 \nu^{11} - 80542386 \nu^{10} - 32232160 \nu^{9} + \cdots - 637257679104 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1920205 \nu^{12} - 11132819 \nu^{11} - 167383848 \nu^{10} + 911265656 \nu^{9} + \cdots + 1253807937024 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 139235 \nu^{12} - 445304 \nu^{11} - 12795239 \nu^{10} + 39284284 \nu^{9} + 442283335 \nu^{8} + \cdots + 4926684672 ) / 3309692416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10489149 \nu^{12} - 29313602 \nu^{11} + 912896051 \nu^{10} + 2404111664 \nu^{9} + \cdots - 463328483328 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7125793 \nu^{12} + 3639059 \nu^{11} - 649932012 \nu^{10} - 291399086 \nu^{9} + \cdots + 456899993088 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16394151 \nu^{12} + 20435252 \nu^{11} + 1462858283 \nu^{10} - 1790208568 \nu^{9} + \cdots - 802773986816 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17020531 \nu^{12} + 12986530 \nu^{11} - 1507070505 \nu^{10} - 1046131624 \nu^{9} + \cdots + 548980252160 ) / 142316773888 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 23\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{12} - 4 \beta_{10} - 2 \beta_{9} - 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \cdots + 310 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{12} + 39\beta_{9} - 39\beta_{8} - 8\beta_{7} - 39\beta_{6} + 63\beta_{5} + 8\beta_{4} + 595\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 126 \beta_{12} + 8 \beta_{11} - 212 \beta_{10} - 78 \beta_{9} + 24 \beta_{8} - 32 \beta_{7} + \cdots + 7782 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1275 \beta_{12} + 72 \beta_{11} + 64 \beta_{10} + 1283 \beta_{9} - 1251 \beta_{8} - 416 \beta_{7} + \cdots - 1566 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5526 \beta_{12} + 480 \beta_{11} - 8460 \beta_{10} - 2454 \beta_{9} + 1392 \beta_{8} - 2016 \beta_{7} + \cdots + 208294 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 40243 \beta_{12} + 4864 \beta_{11} + 3776 \beta_{10} + 40307 \beta_{9} - 37843 \beta_{8} - 16536 \beta_{7} + \cdots - 32310 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 210454 \beta_{12} + 19096 \beta_{11} - 304948 \beta_{10} - 73094 \beta_{9} + 55848 \beta_{8} + \cdots + 5793542 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1263587 \beta_{12} + 225368 \beta_{11} + 150720 \beta_{10} + 1243259 \beta_{9} - 1122331 \beta_{8} + \cdots - 421022 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7470214 \beta_{12} + 645520 \beta_{11} - 10444476 \beta_{10} - 2139558 \beta_{9} + 1929504 \beta_{8} + \cdots + 165422118 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−5.49342
−4.99897
−4.33968
−3.02090
−2.57872
−0.264443
0.224790
1.33472
2.98386
3.21121
4.48413
4.82823
5.62918
−5.49342 −1.59182 22.1777 20.7070 8.74456 10.4694 −77.8840 −24.4661 −113.752
1.2 −4.99897 8.31216 16.9897 −8.69474 −41.5522 3.84543 −44.9392 42.0920 43.4648
1.3 −4.33968 2.40523 10.8328 6.68433 −10.4379 −24.5435 −12.2935 −21.2149 −29.0079
1.4 −3.02090 −8.02896 1.12583 −20.1089 24.2547 −33.1653 20.7662 37.4642 60.7468
1.5 −2.57872 8.66678 −1.35022 16.9434 −22.3492 6.89893 24.1116 48.1130 −43.6921
1.6 −0.264443 −4.63433 −7.93007 7.68704 1.22552 7.09559 4.21260 −5.52296 −2.03279
1.7 0.224790 −4.32964 −7.94947 −17.8280 −0.973257 19.8188 −3.58527 −8.25425 −4.00756
1.8 1.33472 8.27114 −6.21852 12.3033 11.0397 −31.4819 −18.9777 41.4117 16.4215
1.9 2.98386 3.69270 0.903438 −11.2324 11.0185 31.8562 −21.1752 −13.3640 −33.5161
1.10 3.21121 −2.40039 2.31189 14.8396 −7.70816 −20.3294 −18.2657 −21.2381 47.6532
1.11 4.48413 −7.61802 12.1074 0.924111 −34.1602 24.8241 18.4180 31.0342 4.14383
1.12 4.82823 −1.78306 15.3118 −12.0513 −8.60905 −32.7048 35.3032 −23.8207 −58.1866
1.13 5.62918 10.0382 23.6877 −2.17343 56.5070 −1.58350 88.3091 73.7660 −12.2346
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(19\) \( -1 \)

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 2299.4.a.k 13
11.b odd 2 1 209.4.a.c ✓ 13
33.d even 2 1 1881.4.a.k 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.a.c ✓ 13 11.b odd 2 1
1881.4.a.k 13 33.d even 2 1
2299.4.a.k 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):

\( T_{2}^{13} - 2 T_{2}^{12} - 91 T_{2}^{11} + 176 T_{2}^{10} + 3117 T_{2}^{9} - 5786 T_{2}^{8} + \cdots - 86016 \) Copy content Toggle raw display
\( T_{5}^{13} - 8 T_{5}^{12} - 1101 T_{5}^{11} + 8314 T_{5}^{10} + 448465 T_{5}^{9} - 3140328 T_{5}^{8} + \cdots - 2789379165656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 2 T^{12} + \cdots - 86016 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 444198064 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 2789379165656 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 833325521817216 \) Copy content Toggle raw display
$11$ \( T^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 49\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 90\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( (T - 19)^{13} \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 27\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 29\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 69\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 93\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 89\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 37\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 88\!\cdots\!40 \) Copy content Toggle raw display
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