Properties

Label 2299.2.a.x
Level $2299$
Weight $2$
Character orbit 2299.a
Self dual yes
Analytic conductor $18.358$
Analytic rank $0$
Dimension $20$
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] = [2299,2,Mod(1,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(18.3576074247\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 30 x^{18} + 28 x^{17} + 374 x^{16} - 321 x^{15} - 2521 x^{14} + 1965 x^{13} + \cdots + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} + 1) q^{5} + ( - \beta_{18} + \beta_{13} - \beta_{11} + \cdots + 1) q^{6} - \beta_{7} q^{7} + ( - \beta_{18} + \beta_{17} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - \beta_{19} + 3 \beta_{18} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + 4 q^{3} + 21 q^{4} + 18 q^{5} + 10 q^{6} + 2 q^{7} - 3 q^{8} + 32 q^{9} - 11 q^{10} + 10 q^{12} - q^{13} + 25 q^{14} + 28 q^{15} + 27 q^{16} - 2 q^{17} - 4 q^{18} + 20 q^{19} + 47 q^{20}+ \cdots - 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 30 x^{18} + 28 x^{17} + 374 x^{16} - 321 x^{15} - 2521 x^{14} + 1965 x^{13} + \cdots + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4636932 \nu^{19} - 13402747 \nu^{18} - 143832762 \nu^{17} + 385164970 \nu^{16} + \cdots + 1611018088 ) / 301131336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3096802 \nu^{19} - 27024535 \nu^{18} - 57736434 \nu^{17} + 743262844 \nu^{16} + \cdots + 1081989132 ) / 150565668 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 494148 \nu^{19} - 379215 \nu^{18} - 14366036 \nu^{17} + 10608318 \nu^{16} + 171253362 \nu^{15} + \cdots + 114980752 ) / 17713608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4881100 \nu^{19} + 18539595 \nu^{18} + 126025990 \nu^{17} - 509609896 \nu^{16} + \cdots - 69443936 ) / 150565668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10774594 \nu^{19} + 7115351 \nu^{18} + 325256132 \nu^{17} - 202155322 \nu^{16} + \cdots - 1976009944 ) / 301131336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4703 \nu^{19} - 12870 \nu^{18} - 126542 \nu^{17} + 353354 \nu^{16} + 1364456 \nu^{15} + \cdots + 538000 ) / 119544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 551098 \nu^{19} + 1442290 \nu^{18} - 15807527 \nu^{17} - 44343978 \nu^{16} + 182745084 \nu^{15} + \cdots - 277492608 ) / 13687788 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7640638 \nu^{19} + 8675578 \nu^{18} + 217741037 \nu^{17} - 229527116 \nu^{16} + \cdots - 605996384 ) / 150565668 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6725 \nu^{19} - 11428 \nu^{18} - 188880 \nu^{17} + 314842 \nu^{16} + 2161796 \nu^{15} + \cdots + 743312 ) / 119544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10870425 \nu^{19} - 25921136 \nu^{18} - 312578758 \nu^{17} + 740638530 \nu^{16} + \cdots + 4395011372 ) / 150565668 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1503682 \nu^{19} - 923117 \nu^{18} - 44373430 \nu^{17} + 24934650 \nu^{16} + 538918398 \nu^{15} + \cdots + 94802632 ) / 17713608 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13496715 \nu^{19} + 12845604 \nu^{18} + 376944662 \nu^{17} - 329618642 \nu^{16} + \cdots - 18467232 ) / 150565668 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30239133 \nu^{19} + 75826093 \nu^{18} + 777663530 \nu^{17} - 2030351764 \nu^{16} + \cdots - 1648026944 ) / 301131336 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 6004 \nu^{19} + 9868 \nu^{18} + 172485 \nu^{17} - 275030 \nu^{16} - 2034746 \nu^{15} + \cdots - 747532 ) / 59772 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3429191 \nu^{19} + 5389385 \nu^{18} + 94150694 \nu^{17} - 143955844 \nu^{16} + \cdots - 102497448 ) / 27375576 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 46769845 \nu^{19} - 95171694 \nu^{18} - 1283743412 \nu^{17} + 2583493782 \nu^{16} + \cdots + 4596345504 ) / 301131336 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 36265825 \nu^{19} - 44820186 \nu^{18} - 1022460594 \nu^{17} + 1202008150 \nu^{16} + \cdots + 2546435568 ) / 150565668 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{17} + \beta_{15} + \beta_{14} - \beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 9 \beta_{18} - 7 \beta_{17} + 9 \beta_{15} + 8 \beta_{14} - \beta_{12} - \beta_{11} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{19} + 8 \beta_{18} + \beta_{17} + 2 \beta_{16} - \beta_{15} + \beta_{13} + 11 \beta_{12} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{19} + 66 \beta_{18} - 45 \beta_{17} - 3 \beta_{16} + 67 \beta_{15} + 56 \beta_{14} - 13 \beta_{12} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 79 \beta_{19} + 47 \beta_{18} + 13 \beta_{17} + 35 \beta_{16} - 16 \beta_{15} + 19 \beta_{13} + \cdots + 556 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 127 \beta_{19} + 460 \beta_{18} - 288 \beta_{17} - 52 \beta_{16} + 479 \beta_{15} + 378 \beta_{14} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 587 \beta_{19} + 228 \beta_{18} + 124 \beta_{17} + 409 \beta_{16} - 182 \beta_{15} - 7 \beta_{14} + \cdots + 3651 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1122 \beta_{19} + 3170 \beta_{18} - 1856 \beta_{17} - 623 \beta_{16} + 3403 \beta_{15} + 2527 \beta_{14} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4297 \beta_{19} + 794 \beta_{18} + 1054 \beta_{17} + 4048 \beta_{16} - 1807 \beta_{15} - 162 \beta_{14} + \cdots + 24546 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9457 \beta_{19} + 21881 \beta_{18} - 12061 \beta_{17} - 6389 \beta_{16} + 24257 \beta_{15} + 16898 \beta_{14} + \cdots - 1722 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 31481 \beta_{19} - 332 \beta_{18} + 8471 \beta_{17} + 36731 \beta_{16} - 16690 \beta_{15} + \cdots + 167823 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 77615 \beta_{19} + 152018 \beta_{18} - 79041 \beta_{17} - 60218 \beta_{16} + 173960 \beta_{15} + \cdots - 31196 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 232120 \beta_{19} - 44614 \beta_{18} + 66063 \beta_{17} + 316557 \beta_{16} - 147419 \beta_{15} + \cdots + 1162855 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 626038 \beta_{19} + 1065216 \beta_{18} - 522501 \beta_{17} - 538846 \beta_{16} + 1255920 \beta_{15} + \cdots - 369366 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1725311 \beta_{19} - 614205 \beta_{18} + 506803 \beta_{17} + 2639374 \beta_{16} - 1263369 \beta_{15} + \cdots + 8149129 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 4987554 \beta_{19} + 7535021 \beta_{18} - 3485662 \beta_{17} - 4657996 \beta_{16} + 9127146 \beta_{15} + \cdots - 3751542 \) 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
2.61558
2.54411
2.49730
2.25904
1.47708
1.37878
1.28203
0.940270
0.703345
0.240189
−0.256781
−0.427654
−0.518694
−1.43854
−1.47058
−1.59318
−1.74171
−2.21256
−2.50173
−2.77630
−2.61558 −0.683274 4.84125 0.557842 1.78716 −4.84048 −7.43153 −2.53314 −1.45908
1.2 −2.54411 −3.21657 4.47252 2.95108 8.18333 0.392568 −6.29037 7.34634 −7.50789
1.3 −2.49730 2.95748 4.23649 1.79079 −7.38570 1.63600 −5.58519 5.74666 −4.47214
1.4 −2.25904 1.22882 3.10326 3.77789 −2.77595 −3.16066 −2.49230 −1.49001 −8.53441
1.5 −1.47708 −0.908640 0.181760 4.13119 1.34213 3.34266 2.68568 −2.17437 −6.10209
1.6 −1.37878 −2.73173 −0.0989593 −3.11821 3.76647 −1.02032 2.89401 4.46237 4.29934
1.7 −1.28203 0.500265 −0.356390 −1.89166 −0.641356 −3.41081 3.02097 −2.74973 2.42517
1.8 −0.940270 1.97032 −1.11589 2.74546 −1.85263 −4.99441 2.92978 0.882166 −2.58147
1.9 −0.703345 −0.621108 −1.50531 −1.36583 0.436853 2.39954 2.46544 −2.61422 0.960649
1.10 −0.240189 2.97370 −1.94231 3.63017 −0.714249 3.43180 0.946898 5.84289 −0.871926
1.11 0.256781 2.78089 −1.93406 −2.44163 0.714079 2.92963 −1.01019 4.73332 −0.626965
1.12 0.427654 0.244432 −1.81711 1.22029 0.104532 0.841281 −1.63240 −2.94025 0.521860
1.13 0.518694 −3.37771 −1.73096 −0.170573 −1.75200 3.86758 −1.93523 8.40893 −0.0884751
1.14 1.43854 −0.907108 0.0693936 −2.89790 −1.30491 −2.20559 −2.77725 −2.17715 −4.16874
1.15 1.47058 −2.06909 0.162607 −1.22321 −3.04277 −2.35709 −2.70203 1.28114 −1.79884
1.16 1.59318 3.02613 0.538218 3.60436 4.82117 0.228218 −2.32888 6.15749 5.74239
1.17 1.74171 −2.27462 1.03354 4.25996 −3.96172 4.07028 −1.68329 2.17391 7.41960
1.18 2.21256 2.84146 2.89542 2.20548 6.28689 −3.17511 1.98117 5.07388 4.87975
1.19 2.50173 1.10530 4.25867 −1.13710 2.76517 3.00049 5.65060 −1.77831 −2.84472
1.20 2.77630 1.16108 5.70785 1.37160 3.22350 1.02443 10.2941 −1.65190 3.80798
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.x 20
11.b odd 2 1 2299.2.a.y 20
11.c even 5 2 209.2.f.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.f.c 40 11.c even 5 2
2299.2.a.x 20 1.a even 1 1 trivial
2299.2.a.y 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):

\( T_{2}^{20} + T_{2}^{19} - 30 T_{2}^{18} - 28 T_{2}^{17} + 374 T_{2}^{16} + 321 T_{2}^{15} - 2521 T_{2}^{14} + \cdots + 80 \) Copy content Toggle raw display
\( T_{3}^{20} - 4 T_{3}^{19} - 38 T_{3}^{18} + 166 T_{3}^{17} + 549 T_{3}^{16} - 2760 T_{3}^{15} + \cdots - 3904 \) Copy content Toggle raw display
\( T_{7}^{20} - 2 T_{7}^{19} - 85 T_{7}^{18} + 203 T_{7}^{17} + 2920 T_{7}^{16} - 8058 T_{7}^{15} + \cdots + 2111951 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 80 \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots - 3904 \) Copy content Toggle raw display
$5$ \( T^{20} - 18 T^{19} + \cdots + 351521 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 2111951 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + T^{19} + \cdots + 23366656 \) Copy content Toggle raw display
$17$ \( T^{20} + 2 T^{19} + \cdots + 44530736 \) Copy content Toggle raw display
$19$ \( (T - 1)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 74508775105 \) Copy content Toggle raw display
$29$ \( T^{20} + 17 T^{19} + \cdots - 191680 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 405444393216 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 6918344238784 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 20500486272064 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 569751846491 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 2305142775379 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 21233292544 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 11924125120 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 218208966300391 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 654507853504 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 4382826261056 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 92849490844 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 286688586328000 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 42\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 9289360489280 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 562225937344 \) Copy content Toggle raw display
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