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Properties

Label 2299.2.a.t
Level $2299$
Weight $2$
Character orbit 2299.a
Self dual yes
Analytic conductor $18.358$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2299,2,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, 2, 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, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2299.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-1,-2,7,-14,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.3576074247\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 17 x^{12} + 15 x^{11} + 107 x^{10} - 80 x^{9} - 310 x^{8} + 179 x^{7} + 415 x^{6} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{5} + \beta_{4}) q^{4} + ( - \beta_{7} - 1) q^{5} + (\beta_{6} - \beta_{3} + 1) q^{6} + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{7} + ( - \beta_{13} - \beta_{11} + \beta_{9} + \cdots - 1) q^{8}+ \cdots + (\beta_{13} - 3 \beta_{12} + 4 \beta_{11} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 2 q^{3} + 7 q^{4} - 14 q^{5} + 8 q^{6} + 4 q^{7} - 3 q^{8} - 2 q^{9} - 6 q^{10} - 8 q^{12} - 5 q^{13} - 6 q^{14} - 10 q^{15} + 5 q^{16} + 10 q^{17} + 6 q^{18} - 14 q^{19} - 16 q^{20} - 9 q^{21}+ \cdots + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 17 x^{12} + 15 x^{11} + 107 x^{10} - 80 x^{9} - 310 x^{8} + 179 x^{7} + 415 x^{6} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{13} + 16 \nu^{11} - 91 \nu^{9} + 5 \nu^{8} + 224 \nu^{7} - 40 \nu^{6} - 228 \nu^{5} + 96 \nu^{4} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 16 \nu^{11} + 92 \nu^{9} - 5 \nu^{8} - 237 \nu^{7} + 39 \nu^{6} + 281 \nu^{5} - 89 \nu^{4} + \cdots + 27 \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} - 2 \nu^{12} + 15 \nu^{11} + 31 \nu^{10} - 75 \nu^{9} - 163 \nu^{8} + 143 \nu^{7} + \cdots - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{13} - 2 \nu^{12} + 15 \nu^{11} + 31 \nu^{10} - 75 \nu^{9} - 163 \nu^{8} + 143 \nu^{7} + 347 \nu^{6} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2 \nu^{13} + \nu^{12} - 31 \nu^{11} - 16 \nu^{10} + 167 \nu^{9} + 81 \nu^{8} - 376 \nu^{7} - 148 \nu^{6} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{13} - 4 \nu^{12} + 31 \nu^{11} + 63 \nu^{10} - 165 \nu^{9} - 340 \nu^{8} + 360 \nu^{7} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{13} - 6 \nu^{12} + 28 \nu^{11} + 93 \nu^{10} - 119 \nu^{9} - 491 \nu^{8} + 126 \nu^{7} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4 \nu^{13} - 4 \nu^{12} + 60 \nu^{11} + 63 \nu^{10} - 302 \nu^{9} - 329 \nu^{8} + 579 \nu^{7} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4 \nu^{13} - \nu^{12} + 63 \nu^{11} + 17 \nu^{10} - 349 \nu^{9} - 88 \nu^{8} + 823 \nu^{7} + 167 \nu^{6} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{13} + 5 \nu^{12} - 77 \nu^{11} - 79 \nu^{10} + 408 \nu^{9} + 417 \nu^{8} - 883 \nu^{7} - 877 \nu^{6} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7 \nu^{13} + 2 \nu^{12} - 110 \nu^{11} - 32 \nu^{10} + 607 \nu^{9} + 151 \nu^{8} - 1424 \nu^{7} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8 \nu^{13} - 9 \nu^{12} + 121 \nu^{11} + 142 \nu^{10} - 619 \nu^{9} - 751 \nu^{8} + 1238 \nu^{7} + \cdots - 9 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{9} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} - 2\beta_{10} - \beta_{8} + \beta_{7} - 6\beta_{5} + 5\beta_{4} - \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{13} - \beta_{12} + 10 \beta_{11} - \beta_{10} - 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 21 \beta_{10} - 12 \beta_{8} + 9 \beta_{7} + \beta_{6} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 57 \beta_{13} - 12 \beta_{12} + 80 \beta_{11} - 12 \beta_{10} - 35 \beta_{9} - 11 \beta_{8} + 23 \beta_{7} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93 \beta_{13} - 67 \beta_{12} + 71 \beta_{11} - 169 \beta_{10} - 2 \beta_{9} - 102 \beta_{8} + \cdots + 213 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 396 \beta_{13} - 105 \beta_{12} + 587 \beta_{11} - 110 \beta_{10} - 212 \beta_{9} - 95 \beta_{8} + \cdots + 323 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 717 \beta_{13} - 472 \beta_{12} + 546 \beta_{11} - 1237 \beta_{10} - 34 \beta_{9} - 769 \beta_{8} + \cdots + 1265 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2731 \beta_{13} - 823 \beta_{12} + 4137 \beta_{11} - 921 \beta_{10} - 1328 \beta_{9} - 760 \beta_{8} + \cdots + 2256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5304 \beta_{13} - 3263 \beta_{12} + 4157 \beta_{11} - 8682 \beta_{10} - 388 \beta_{9} - 5499 \beta_{8} + \cdots + 7977 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18806 \beta_{13} - 6144 \beta_{12} + 28587 \beta_{11} - 7383 \beta_{10} - 8519 \beta_{9} - 5877 \beta_{8} + \cdots + 15868 \) 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
2.64334
1.95517
1.86740
1.86285
0.849796
0.626946
0.154596
−0.200027
−0.366340
−0.905029
−1.04769
−1.93293
−2.02502
−2.48307
−2.64334 −0.278680 4.98726 −3.23687 0.736647 1.52557 −7.89636 −2.92234 8.55617
1.2 −1.95517 −2.86067 1.82269 0.394240 5.59309 −0.353121 0.346674 5.18342 −0.770806
1.3 −1.86740 0.278036 1.48720 1.45924 −0.519205 3.87596 0.957603 −2.92270 −2.72498
1.4 −1.86285 0.912707 1.47022 −0.733992 −1.70024 0.296160 0.986904 −2.16697 1.36732
1.5 −0.849796 −1.98168 −1.27785 1.57133 1.68402 −0.302738 2.78550 0.927060 −1.33531
1.6 −0.626946 −1.68011 −1.60694 −4.03200 1.05334 −3.53548 2.26136 −0.177220 2.52785
1.7 −0.154596 1.75951 −1.97610 −0.465506 −0.272013 −2.01956 0.614689 0.0958795 0.0719653
1.8 0.200027 1.22622 −1.95999 0.0816230 0.245277 −0.895749 −0.792103 −1.49638 0.0163268
1.9 0.366340 −1.87701 −1.86579 −2.24160 −0.687624 4.43568 −1.41620 0.523161 −0.821188
1.10 0.905029 −0.561091 −1.18092 2.75301 −0.507804 0.551138 −2.87883 −2.68518 2.49156
1.11 1.04769 2.88767 −0.902343 −3.92698 3.02538 0.194286 −3.04076 5.33862 −4.11427
1.12 1.93293 2.02543 1.73621 −2.10790 3.91500 −3.28904 −0.509892 1.10235 −4.07443
1.13 2.02502 −0.0624840 2.10070 −3.35497 −0.126531 4.75396 0.203923 −2.99610 −6.79389
1.14 2.48307 −1.78784 4.16566 −0.159606 −4.43935 −1.23706 5.37748 0.196384 −0.396313
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 2299.2.a.t 14
11.b odd 2 1 2299.2.a.u 14
11.c even 5 2 209.2.f.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.f.b ✓ 28 11.c even 5 2
2299.2.a.t 14 1.a even 1 1 trivial
2299.2.a.u 14 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):

\( T_{2}^{14} + T_{2}^{13} - 17 T_{2}^{12} - 15 T_{2}^{11} + 107 T_{2}^{10} + 80 T_{2}^{9} - 310 T_{2}^{8} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{14} + 2 T_{3}^{13} - 18 T_{3}^{12} - 36 T_{3}^{11} + 113 T_{3}^{10} + 224 T_{3}^{9} - 315 T_{3}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{14} - 4 T_{7}^{13} - 37 T_{7}^{12} + 114 T_{7}^{11} + 529 T_{7}^{10} - 920 T_{7}^{9} - 3313 T_{7}^{8} + \cdots - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{14} + 14 T^{13} + \cdots - 9 \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots - 11 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 5 T^{13} + \cdots + 12419 \) Copy content Toggle raw display
$17$ \( T^{14} - 10 T^{13} + \cdots + 2737744 \) Copy content Toggle raw display
$19$ \( (T + 1)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + 21 T^{13} + \cdots - 2496119 \) Copy content Toggle raw display
$29$ \( T^{14} + 3 T^{13} + \cdots + 1961455 \) Copy content Toggle raw display
$31$ \( T^{14} + 13 T^{13} + \cdots - 98865679 \) Copy content Toggle raw display
$37$ \( T^{14} + 31 T^{13} + \cdots - 34143901 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 532426169 \) Copy content Toggle raw display
$43$ \( T^{14} + 6 T^{13} + \cdots + 11826491 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 30365724429 \) Copy content Toggle raw display
$53$ \( T^{14} + 42 T^{13} + \cdots - 1220029 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 179630369395 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 724582251 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 19444726741 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3465674591 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 137430711 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1933875295 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 1241650112679 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 9538214501275 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 254469301 \) Copy content Toggle raw display
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