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Properties

Label 1849.2.a.r
Level $1849$
Weight $2$
Character orbit 1849.a
Self dual yes
Analytic conductor $14.764$
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] = [1849,2,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.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: \(14.7643393337\)
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: \( x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{17} + \beta_{12} - \beta_{7} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{6}+ \cdots + (\beta_{19} - 2 \beta_{18} - \beta_{17} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14230071983 \nu^{19} + 46615997016 \nu^{18} + 330972065901 \nu^{17} - 1171616127168 \nu^{16} + \cdots + 943797497903 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40192831361 \nu^{19} + 111749991106 \nu^{18} + 989932636217 \nu^{17} + \cdots + 1077447576383 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14228637027 \nu^{19} - 34498144612 \nu^{18} - 366459861464 \nu^{17} + 883733858774 \nu^{16} + \cdots + 327793978196 ) / 52625896924 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37344752748 \nu^{19} - 94788187709 \nu^{18} - 952275906137 \nu^{17} + 2426492911022 \nu^{16} + \cdots + 927358146403 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40076363964 \nu^{19} + 95400656607 \nu^{18} + 1041101282307 \nu^{17} - 2448128554906 \nu^{16} + \cdots - 775838268705 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22565527133 \nu^{19} + 52329290331 \nu^{18} + 589133963225 \nu^{17} - 1346077054352 \nu^{16} + \cdots - 1038060570543 ) / 52625896924 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12222313595 \nu^{19} - 35149906384 \nu^{18} - 301069644714 \nu^{17} + 897048356248 \nu^{16} + \cdots + 148383897244 ) / 26312948462 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 111811015775 \nu^{19} + 285669062450 \nu^{18} + 2847790709171 \nu^{17} + \cdots - 2574044282979 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71868444078 \nu^{19} - 192655709901 \nu^{18} - 1799642357535 \nu^{17} + 4914739593678 \nu^{16} + \cdots + 1778222415337 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 76368492269 \nu^{19} - 185555230662 \nu^{18} - 1971310193269 \nu^{17} + 4754809536988 \nu^{16} + \cdots + 1789038068597 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 155548732541 \nu^{19} - 402386853510 \nu^{18} - 3945463645905 \nu^{17} + 10289851728316 \nu^{16} + \cdots + 3615445347609 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 91935576031 \nu^{19} + 242725603925 \nu^{18} + 2316290687974 \nu^{17} + \cdots - 1605824368164 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 93833087507 \nu^{19} + 255780936564 \nu^{18} + 2344007607053 \nu^{17} + \cdots - 1234373979961 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 273386016425 \nu^{19} - 723381409446 \nu^{18} - 6885247962181 \nu^{17} + 18479122112404 \nu^{16} + \cdots + 4807873678325 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 375356809195 \nu^{19} - 990037727626 \nu^{18} - 9464837304391 \nu^{17} + 25295833347988 \nu^{16} + \cdots + 6688595499263 ) / 210503587696 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 305725917825 \nu^{19} - 806215480513 \nu^{18} - 7705358032622 \nu^{17} + 20593247638918 \nu^{16} + \cdots + 5366073004952 ) / 105251793848 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 633296538777 \nu^{19} + 1639968208882 \nu^{18} + 16054290614973 \nu^{17} + \cdots - 11881100268157 ) / 210503587696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{10} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{6} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - 2 \beta_{17} - \beta_{16} - \beta_{15} + 11 \beta_{13} - \beta_{11} + 8 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{19} + 11 \beta_{18} + 10 \beta_{17} - 11 \beta_{16} - \beta_{15} - 11 \beta_{13} + 11 \beta_{12} + \cdots + 95 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{18} - 26 \beta_{17} - 16 \beta_{16} - 13 \beta_{15} - \beta_{14} + 87 \beta_{13} + \beta_{12} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94 \beta_{19} + 95 \beta_{18} + 77 \beta_{17} - 96 \beta_{16} - 17 \beta_{15} - 93 \beta_{13} + \cdots + 594 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{19} + 155 \beta_{18} - 242 \beta_{17} - 170 \beta_{16} - 127 \beta_{15} - 16 \beta_{14} + \cdots - 113 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 734 \beta_{19} + 752 \beta_{18} + 547 \beta_{17} - 774 \beta_{16} - 190 \beta_{15} + 4 \beta_{14} + \cdots + 3820 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 52 \beta_{19} + 1378 \beta_{18} - 1981 \beta_{17} - 1536 \beta_{16} - 1101 \beta_{15} - 173 \beta_{14} + \cdots - 960 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5493 \beta_{19} + 5696 \beta_{18} + 3785 \beta_{17} - 5997 \beta_{16} - 1780 \beta_{15} + 88 \beta_{14} + \cdots + 24971 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 581 \beta_{19} + 11329 \beta_{18} - 15223 \beta_{17} - 12776 \beta_{16} - 8944 \beta_{15} - 1575 \beta_{14} + \cdots - 7659 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 40166 \beta_{19} + 42021 \beta_{18} + 26061 \beta_{17} - 45334 \beta_{16} - 15198 \beta_{15} + \cdots + 164944 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 5350 \beta_{19} + 88929 \beta_{18} - 112921 \beta_{17} - 101259 \beta_{16} - 69834 \beta_{15} + \cdots - 58534 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 289647 \beta_{19} + 304697 \beta_{18} + 180112 \beta_{17} - 336911 \beta_{16} - 122855 \beta_{15} + \cdots + 1097545 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 44295 \beta_{19} + 677703 \beta_{18} - 819862 \beta_{17} - 778323 \beta_{16} - 531369 \beta_{15} + \cdots - 433825 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2070183 \beta_{19} + 2183393 \beta_{18} + 1253150 \beta_{17} - 2472918 \beta_{16} - 959188 \beta_{15} + \cdots + 7344074 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 343673 \beta_{19} + 5062454 \beta_{18} - 5871055 \beta_{17} - 5860285 \beta_{16} - 3972692 \beta_{15} + \cdots - 3143881 \) 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
−2.64020
−2.55076
−1.97560
−1.90109
−1.59121
−1.22059
−0.528833
−0.369089
−0.277120
0.173392
0.415785
0.881156
1.14959
1.54977
1.62391
2.31191
2.37008
2.37157
2.54512
2.66221
−2.64020 −0.354022 4.97064 −1.23228 0.934688 −3.87263 −7.84307 −2.87467 3.25345
1.2 −2.55076 −2.76473 4.50637 0.117709 7.05217 −0.649562 −6.39315 4.64375 −0.300248
1.3 −1.97560 0.955865 1.90300 −3.57694 −1.88841 −2.56717 0.191632 −2.08632 7.06661
1.4 −1.90109 2.82058 1.61414 3.37266 −5.36217 1.68862 0.733548 4.95566 −6.41173
1.5 −1.59121 −2.69692 0.531955 −1.39893 4.29137 −3.40103 2.33597 4.27339 2.22600
1.6 −1.22059 1.36553 −0.510154 2.60720 −1.66675 5.05996 3.06388 −1.13534 −3.18232
1.7 −0.528833 0.931794 −1.72034 0.425701 −0.492764 2.15726 1.96744 −2.13176 −0.225125
1.8 −0.369089 0.00229289 −1.86377 −2.37597 −0.000846282 0 4.42543 1.42608 −2.99999 0.876946
1.9 −0.277120 −3.15485 −1.92320 0.330136 0.874272 −1.68241 1.08720 6.95305 −0.0914873
1.10 0.173392 0.896458 −1.96994 −3.74986 0.155439 −1.28742 −0.688355 −2.19636 −0.650196
1.11 0.415785 −2.94504 −1.82712 2.53117 −1.22450 3.71316 −1.59126 5.67325 1.05242
1.12 0.881156 2.54519 −1.22356 2.37402 2.24271 −3.19714 −2.84046 3.47799 2.09188
1.13 1.14959 1.67501 −0.678443 −3.60838 1.92557 −0.890099 −3.07911 −0.194345 −4.14816
1.14 1.54977 3.21447 0.401778 2.07911 4.98169 −0.871568 −2.47687 7.33285 3.22213
1.15 1.62391 −2.30521 0.637097 3.05171 −3.74346 −2.09950 −2.21324 2.31400 4.95571
1.16 2.31191 0.910509 3.34494 2.04187 2.10502 −2.45270 3.10939 −2.17097 4.72063
1.17 2.37008 −1.76176 3.61728 −2.72492 −4.17550 −3.47356 3.83309 0.103783 −6.45828
1.18 2.37157 −0.630839 3.62433 −1.61545 −1.49608 3.60874 3.85219 −2.60204 −3.83115
1.19 2.54512 2.55827 4.47761 −1.89319 6.51109 1.19543 6.30581 3.54474 −4.81839
1.20 2.66221 −2.26260 5.08738 0.244650 −6.02353 2.59618 8.21928 2.11936 0.651312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

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 1849.2.a.r yes 20
43.b odd 2 1 1849.2.a.p ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.2.a.p ✓ 20 43.b odd 2 1
1849.2.a.r yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 5 T_{2}^{19} - 19 T_{2}^{18} + 127 T_{2}^{17} + 95 T_{2}^{16} - 1293 T_{2}^{15} + 329 T_{2}^{14} + \cdots - 43 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 5 T^{19} + \cdots - 43 \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{20} + 3 T^{19} + \cdots - 4577 \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} + \cdots + 7159328 \) Copy content Toggle raw display
$11$ \( T^{20} - 21 T^{19} + \cdots + 273824 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 164019197 \) Copy content Toggle raw display
$17$ \( T^{20} - 15 T^{19} + \cdots + 15556529 \) Copy content Toggle raw display
$19$ \( T^{20} + 7 T^{19} + \cdots - 909536 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 52688021728 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 4975400431 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 1192113152 \) Copy content Toggle raw display
$37$ \( T^{20} + 19 T^{19} + \cdots + 650848 \) Copy content Toggle raw display
$41$ \( T^{20} - 27 T^{19} + \cdots + 19105109 \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 24461789605408 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 5611239811409 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 374695773728 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 273702209775941 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 481192411805984 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 55064594848 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 34047145350503 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 55\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 247888391228939 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 4313536817789 \) Copy content Toggle raw display
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