↑

Properties

Label 23.16.a.b
Level $23$
Weight $16$
Character orbit 23.a
Self dual yes
Analytic conductor $32.820$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,16,Mod(1,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(32.8195061730\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 376832 x^{13} - 2595165 x^{12} + 55941760876 x^{11} + 886520309896 x^{10} + \cdots + 26\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{27}\cdot 3^{5}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 467) q^{3} + (\beta_{3} - \beta_{2} + 10 \beta_1 + 17476) q^{4} + ( - \beta_{5} + 3 \beta_{2} + \cdots + 7201) q^{5}+ \cdots + (\beta_{14} + 3 \beta_{13} + \cdots + 6077246) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 467) q^{3} + (\beta_{3} - \beta_{2} + 10 \beta_1 + 17476) q^{4} + ( - \beta_{5} + 3 \beta_{2} + \cdots + 7201) q^{5}+ \cdots + ( - 48705860 \beta_{14} + \cdots + 162189327136247) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 7003 q^{3} + 262144 q^{4} + 108024 q^{5} + 681583 q^{6} - 135224 q^{7} - 7785495 q^{8} + 91157574 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 7003 q^{3} + 262144 q^{4} + 108024 q^{5} + 681583 q^{6} - 135224 q^{7} - 7785495 q^{8} + 91157574 q^{9} + 164331982 q^{10} + 53210612 q^{11} + 506583725 q^{12} + 683978393 q^{13} - 916105656 q^{14} - 576527390 q^{15} + 2255610448 q^{16} - 1728803440 q^{17} - 4955084537 q^{18} + 4149246772 q^{19} - 9717614716 q^{20} + 2921805612 q^{21} + 1933993764 q^{22} - 51072381705 q^{23} + 33226301404 q^{24} + 246891856341 q^{25} + 182052775663 q^{26} + 215969022205 q^{27} + 127563370986 q^{28} - 120612029353 q^{29} + 352813211090 q^{30} - 68213180161 q^{31} + 563355863720 q^{32} + 827251303670 q^{33} + 1644938617198 q^{34} + 515650672968 q^{35} + 3830793158071 q^{36} + 2848945111508 q^{37} + 1864473665804 q^{38} + 2993152625757 q^{39} + 10314230807046 q^{40} + 2887360381337 q^{41} + 15294532806472 q^{42} + 8285879754206 q^{43} + 14523816445630 q^{44} + 16130506291602 q^{45} + 13411036412483 q^{47} + 16133192633135 q^{48} + 21504566558799 q^{49} + 12540536392128 q^{50} + 21464277669572 q^{51} + 11377093677935 q^{52} + 5417079631228 q^{53} + 1186158166697 q^{54} + 21550376921836 q^{55} + 51362100318210 q^{56} + 37189380212906 q^{57} + 86677479318825 q^{58} - 55524580441888 q^{59} + 11127570927180 q^{60} + 15980601302412 q^{61} + 24107043971795 q^{62} - 130662066049788 q^{63} - 150294852150789 q^{64} - 62689564660198 q^{65} - 227299243744742 q^{66} - 105489546482644 q^{67} - 444475365007100 q^{68} - 23843992605341 q^{69} - 639801358673356 q^{70} - 514599529495945 q^{71} - 655430324373009 q^{72} + 67104292620675 q^{73} - 69299553411206 q^{74} - 508277257114275 q^{75} - 430714664215944 q^{76} - 256904998920632 q^{77} - 690862301613467 q^{78} + 52399072717482 q^{79} - 931252260872354 q^{80} + 170941949793751 q^{81} + 261821545121335 q^{82} + 433409172273282 q^{83} - 789088233312026 q^{84} + 538318389492780 q^{85} + 362127550949782 q^{86} - 10\!\cdots\!61 q^{87}+ \cdots + 24\!\cdots\!30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 376832 x^{13} - 2595165 x^{12} + 55941760876 x^{11} + 886520309896 x^{10} + \cdots + 26\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 62\!\cdots\!29 \nu^{14} + \cdots + 11\!\cdots\!52 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 62\!\cdots\!29 \nu^{14} + \cdots - 33\!\cdots\!48 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!39 \nu^{14} + \cdots - 88\!\cdots\!52 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!31 \nu^{14} + \cdots - 26\!\cdots\!28 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!73 \nu^{14} + \cdots + 70\!\cdots\!24 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 97\!\cdots\!97 \nu^{14} + \cdots - 74\!\cdots\!36 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 77\!\cdots\!99 \nu^{14} + \cdots + 44\!\cdots\!12 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43\!\cdots\!47 \nu^{14} + \cdots - 50\!\cdots\!36 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!21 \nu^{14} + \cdots - 78\!\cdots\!48 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!13 \nu^{14} + \cdots + 92\!\cdots\!44 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!57 \nu^{14} + \cdots + 31\!\cdots\!16 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!68 \nu^{14} + \cdots - 36\!\cdots\!84 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!01 \nu^{14} + \cdots + 30\!\cdots\!88 ) / 82\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 10\beta _1 + 50244 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 2 \beta_{11} - \beta_{8} + \beta_{6} + 13 \beta_{5} - \beta_{4} - 28 \beta_{3} + \cdots + 519037 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 61 \beta_{14} + 52 \beta_{13} - 3 \beta_{12} - 71 \beta_{11} - 75 \beta_{10} - 137 \beta_{9} + \cdots + 4015822454 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8976 \beta_{14} + 128335 \beta_{13} + 8649 \beta_{12} + 300265 \beta_{11} + 25425 \beta_{10} + \cdots + 30473757794 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6013932 \beta_{14} + 5393952 \beta_{13} - 1831368 \beta_{12} - 12694535 \beta_{11} + \cdots + 359422076226221 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1354841619 \beta_{14} + 13854927958 \beta_{13} + 1191153989 \beta_{12} + 35701393559 \beta_{11} + \cdots + 900665478176918 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 392071994652 \beta_{14} + 308194748779 \beta_{13} - 329528276343 \beta_{12} - 1799529374319 \beta_{11} + \cdots + 33\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 123169080154508 \beta_{14} + \cdots - 12\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 60\!\cdots\!65 \beta_{14} + \cdots + 33\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 64\!\cdots\!84 \beta_{14} + \cdots - 33\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 36\!\cdots\!64 \beta_{14} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 26\!\cdots\!49 \beta_{14} + \cdots - 55\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 85\!\cdots\!80 \beta_{14} + \cdots + 33\!\cdots\!94 \) 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
320.190
314.693
279.994
228.404
200.039
133.312
−11.8462
−27.8802
−48.6157
−95.1430
−170.486
−223.029
−256.822
−316.095
−326.716
−320.190 −4715.59 69753.9 −232950. 1.50989e6 2.81756e6 −1.18425e7 7.88788e6 7.45884e7
1.2 −314.693 4996.83 66263.8 −327437. −1.57247e6 −3.36082e6 −1.05409e7 1.06194e7 1.03042e8
1.3 −279.994 6752.70 45628.7 133649. −1.89072e6 1.73227e6 −3.60091e6 3.12500e7 −3.74210e7
1.4 −228.404 −5732.20 19400.2 179346. 1.30926e6 244191. 3.05324e6 1.85092e7 −4.09634e7
1.5 −200.039 1588.10 7247.79 −31005.4 −317682. −992491. 5.10505e6 −1.18269e7 6.20229e6
1.6 −133.312 −2503.30 −14996.0 215449. 333719. 3.44644e6 6.36750e6 −8.08242e6 −2.87218e7
1.7 11.8462 −2150.63 −32627.7 −91123.1 −25476.7 −2.18358e6 −774687. −9.72369e6 −1.07946e6
1.8 27.8802 4692.44 −31990.7 −68535.0 130826. −3.44375e6 −1.80549e6 7.67005e6 −1.91077e6
1.9 48.6157 3919.07 −30404.5 197314. 190528. 3.18607e6 −3.07117e6 1.01021e6 9.59255e6
1.10 95.1430 −354.805 −23715.8 −329717. −33757.2 2.35062e6 −5.37404e6 −1.42230e7 −3.13703e7
1.11 170.486 −6257.24 −3702.67 315716. −1.06677e6 −3.11221e6 −6.21772e6 2.48042e7 5.38250e7
1.12 223.029 −4472.03 16974.0 −217791. −997393. −2.16026e6 −3.52252e6 5.65014e6 −4.85738e7
1.13 256.822 7420.05 33189.6 192449. 1.90563e6 −1.14715e6 108266. 4.07082e7 4.94252e7
1.14 316.095 3947.86 67148.2 −121119. 1.24790e6 3.04583e6 1.08674e7 1.23665e6 −3.82851e7
1.15 326.716 −128.249 73975.1 293778. −41900.8 −557941. 1.34630e7 −1.43325e7 9.59819e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(1\)

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 23.16.a.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.16.a.b ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 376832 T_{2}^{13} + 2595165 T_{2}^{12} + 55941760876 T_{2}^{11} - 886520309896 T_{2}^{10} + \cdots - 26\!\cdots\!08 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 26\!\cdots\!08 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 82\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 3404825447)^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 74\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 18\!\cdots\!68 \) Copy content Toggle raw display
show more
show less