↑

Properties

Label 5819.2.a.o
Level $5819$
Weight $2$
Character orbit 5819.a
Self dual yes
Analytic conductor $46.465$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5819,2,Mod(1,5819)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5819, 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("5819.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5819 = 11 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5819.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,2,-2,18,8] 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: \(46.4649489362\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 25 x^{16} + 48 x^{15} + 251 x^{14} - 460 x^{13} - 1295 x^{12} + 2248 x^{11} + \cdots - 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} + (\beta_{14} - \beta_{13} - \beta_{8} + \cdots - 1) q^{6} + ( - \beta_{14} + 1) q^{7} + (\beta_{11} + \beta_{10} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + (\beta_{16} + \beta_{12} - \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} - 2 q^{3} + 18 q^{4} + 8 q^{5} - 6 q^{6} + 10 q^{7} + 6 q^{8} + 16 q^{9} + 12 q^{10} - 18 q^{11} + 10 q^{12} + 28 q^{14} + 8 q^{15} + 26 q^{16} - 20 q^{18} + 16 q^{19} + 40 q^{20} + 12 q^{21}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} - 25 x^{16} + 48 x^{15} + 251 x^{14} - 460 x^{13} - 1295 x^{12} + 2248 x^{11} + \cdots - 39 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 182307033 \nu^{17} - 2727044289 \nu^{16} + 10442949865 \nu^{15} + 67857115912 \nu^{14} + \cdots + 389519768808 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1285276592 \nu^{17} + 16249285962 \nu^{16} + 13234807060 \nu^{15} - 410176696018 \nu^{14} + \cdots + 1959112739557 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5785227488 \nu^{17} + 19385151640 \nu^{16} + 129615227360 \nu^{15} - 474403656716 \nu^{14} + \cdots + 1155792467423 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2739935 \nu^{17} - 7889782 \nu^{16} - 64666851 \nu^{15} + 192083322 \nu^{14} + 597021450 \nu^{13} + \cdots - 605858022 ) / 31175603 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6417801598 \nu^{17} + 20572894548 \nu^{16} + 147624309823 \nu^{15} - 501841686131 \nu^{14} + \cdots + 1607137807419 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7609798933 \nu^{17} - 14001576094 \nu^{16} - 191979711019 \nu^{15} + 329894754431 \nu^{14} + \cdots - 104070188529 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12537451366 \nu^{17} + 29963619426 \nu^{16} + 301651863459 \nu^{15} - 722157758355 \nu^{14} + \cdots + 1689865012733 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 629652705 \nu^{17} + 1109849840 \nu^{16} + 15684746913 \nu^{15} - 26236185531 \nu^{14} + \cdots + 39193054707 ) / 3040299023 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14932279292 \nu^{17} - 29441333517 \nu^{16} - 368315311597 \nu^{15} + 698727412540 \nu^{14} + \cdots - 823412074391 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18722473113 \nu^{17} + 33360289786 \nu^{16} + 464249480604 \nu^{15} - 780503608770 \nu^{14} + \cdots + 503441000748 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22728051727 \nu^{17} + 50432344927 \nu^{16} + 550219442283 \nu^{15} - 1198819323116 \nu^{14} + \cdots + 1524424730913 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22932949458 \nu^{17} + 51446228673 \nu^{16} + 554952468556 \nu^{15} - 1225663775071 \nu^{14} + \cdots + 1925685924532 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1204623975 \nu^{17} - 2983213387 \nu^{16} - 28829289821 \nu^{15} + 71637370804 \nu^{14} + \cdots - 132927404285 ) / 3040299023 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34093185046 \nu^{17} + 69733798969 \nu^{16} + 841193298815 \nu^{15} - 1662910171322 \nu^{14} + \cdots + 2526103100261 ) / 69926877529 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 54181039501 \nu^{17} - 117787526354 \nu^{16} - 1321916186635 \nu^{15} + 2816709146881 \nu^{14} + \cdots - 5046635759360 ) / 69926877529 \) 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
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{14} + 2\beta_{11} + \beta_{10} + \beta_{9} + 7\beta_{2} + 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} + 9 \beta_{10} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} + 14 \beta_{16} + 3 \beta_{15} + 12 \beta_{14} + \beta_{13} + \beta_{12} + 24 \beta_{11} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{17} + 4 \beta_{16} + 25 \beta_{15} + 15 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} + \cdots + 105 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93 \beta_{17} + 138 \beta_{16} + 46 \beta_{15} + 115 \beta_{14} + 15 \beta_{13} + 10 \beta_{12} + \cdots + 626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 121 \beta_{17} + 74 \beta_{16} + 239 \beta_{15} + 165 \beta_{14} + 221 \beta_{13} - 237 \beta_{12} + \cdots + 840 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 714 \beta_{17} + 1198 \beta_{16} + 499 \beta_{15} + 1010 \beta_{14} + 168 \beta_{13} + 53 \beta_{12} + \cdots + 4220 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 988 \beta_{17} + 916 \beta_{16} + 2082 \beta_{15} + 1598 \beta_{14} + 1862 \beta_{13} - 2035 \beta_{12} + \cdots + 6545 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5233 \beta_{17} + 9810 \beta_{16} + 4726 \beta_{15} + 8472 \beta_{14} + 1683 \beta_{13} + 24 \beta_{12} + \cdots + 29414 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7551 \beta_{17} + 9556 \beta_{16} + 17377 \beta_{15} + 14470 \beta_{14} + 15075 \beta_{13} + \cdots + 50713 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 37408 \beta_{17} + 77978 \beta_{16} + 41780 \beta_{15} + 69174 \beta_{14} + 15843 \beta_{13} + \cdots + 210380 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 55547 \beta_{17} + 90924 \beta_{16} + 141755 \beta_{15} + 125862 \beta_{14} + 119530 \beta_{13} + \cdots + 393405 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 263667 \beta_{17} + 610147 \beta_{16} + 354937 \beta_{15} + 555458 \beta_{14} + 143066 \beta_{13} + \cdots + 1534788 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 398781 \beta_{17} + 818318 \beta_{16} + 1140979 \beta_{15} + 1066575 \beta_{14} + 936574 \beta_{13} + \cdots + 3060675 \) 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
−2.60860
−2.40207
−2.05530
−2.00038
−1.50955
−0.889066
−0.517602
−0.378259
−0.303632
0.380461
0.797622
0.982168
1.11676
1.65948
2.20667
2.24352
2.48046
2.79732
−2.60860 −1.03474 4.80480 0.542141 2.69923 1.29798 −7.31659 −1.92931 −1.41423
1.2 −2.40207 2.84906 3.76995 1.08833 −6.84365 −0.398108 −4.25155 5.11714 −2.61424
1.3 −2.05530 0.964899 2.22425 2.82756 −1.98315 −1.89041 −0.460892 −2.06897 −5.81147
1.4 −2.00038 −3.10045 2.00153 −1.47500 6.20209 0.191472 −0.00306224 6.61281 2.95057
1.5 −1.50955 1.91247 0.278751 −2.20831 −2.88697 −2.16756 2.59832 0.657539 3.33356
1.6 −0.889066 −0.690733 −1.20956 3.36556 0.614107 −0.270300 2.85351 −2.52289 −2.99220
1.7 −0.517602 −1.74792 −1.73209 0.150041 0.904727 −3.21534 1.93174 0.0552240 −0.0776615
1.8 −0.378259 3.01307 −1.85692 0.184332 −1.13972 4.85283 1.45892 6.07860 −0.0697255
1.9 −0.303632 1.58132 −1.90781 −2.73916 −0.480139 1.58648 1.18654 −0.499424 0.831696
1.10 0.380461 −3.14561 −1.85525 −1.83833 −1.19678 −3.62513 −1.46677 6.89484 −0.699414
1.11 0.797622 0.704693 −1.36380 −2.69261 0.562078 3.03995 −2.68304 −2.50341 −2.14768
1.12 0.982168 −2.72901 −1.03535 2.65069 −2.68034 4.59744 −2.98122 4.44747 2.60342
1.13 1.11676 −0.0440645 −0.752841 −0.262759 −0.0492096 −1.32879 −3.07427 −2.99806 −0.293440
1.14 1.65948 −1.50381 0.753868 3.61083 −2.49554 −1.22717 −2.06793 −0.738551 5.99209
1.15 2.20667 −2.22371 2.86941 −2.44028 −4.90700 2.05536 1.91851 1.94487 −5.38490
1.16 2.24352 2.33974 3.03336 4.11350 5.24923 −1.26950 2.31836 2.47436 9.22871
1.17 2.48046 −0.125964 4.15270 0.544132 −0.312449 4.40992 5.33969 −2.98413 1.34970
1.18 2.79732 0.980758 5.82500 2.57933 2.74349 3.36086 10.6998 −2.03811 7.21522
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(23\) \( -1 \)

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 5819.2.a.o yes 18
23.b odd 2 1 5819.2.a.n ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5819.2.a.n ✓ 18 23.b odd 2 1
5819.2.a.o yes 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5819))\):

\( T_{2}^{18} - 2 T_{2}^{17} - 25 T_{2}^{16} + 48 T_{2}^{15} + 251 T_{2}^{14} - 460 T_{2}^{13} - 1295 T_{2}^{12} + \cdots - 39 \) Copy content Toggle raw display
\( T_{5}^{18} - 8 T_{5}^{17} - 16 T_{5}^{16} + 252 T_{5}^{15} - 72 T_{5}^{14} - 3228 T_{5}^{13} + \cdots - 243 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{17} + \cdots - 39 \) Copy content Toggle raw display
$3$ \( T^{18} + 2 T^{17} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{18} - 8 T^{17} + \cdots - 243 \) Copy content Toggle raw display
$7$ \( T^{18} - 10 T^{17} + \cdots - 8667 \) Copy content Toggle raw display
$11$ \( (T + 1)^{18} \) Copy content Toggle raw display
$13$ \( T^{18} - 146 T^{16} + \cdots - 311351 \) Copy content Toggle raw display
$17$ \( T^{18} - 179 T^{16} + \cdots + 45192249 \) Copy content Toggle raw display
$19$ \( T^{18} - 16 T^{17} + \cdots - 36715059 \) Copy content Toggle raw display
$23$ \( T^{18} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 2658438969 \) Copy content Toggle raw display
$31$ \( T^{18} - 234 T^{16} + \cdots - 5687471 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 84602266425 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 126115167 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 125075909943 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 5628431405091 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 474802013493 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 25792384473 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 1293783970167 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 155705952790341 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 796840076019 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 79133622023867 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 19\!\cdots\!43 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 32516046374733 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 162708037958079 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 14\!\cdots\!79 \) Copy content Toggle raw display
show more
show less