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Properties

Label 23.13.b.c
Level $23$
Weight $13$
Character orbit 23.b
Analytic conductor $21.022$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,13,Mod(22,23)]
 
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([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(21.0218577974\)
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: \( x^{20} - 10 x^{19} + 3347471347 x^{18} + 50192028136 x^{17} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: multiple of \( 2^{35}\cdot 3^{11}\cdot 67 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 4) q^{2} + ( - \beta_{3} + 2 \beta_1 + 15) q^{3} + (\beta_{4} - 2 \beta_{3} - 5 \beta_1 + 1805) q^{4} - \beta_{2} q^{5} + (\beta_{6} + 3 \beta_{4} + \cdots + 13142) q^{6}+ \cdots + ( - \beta_{11} + \beta_{6} + \cdots + 98474) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 4) q^{2} + ( - \beta_{3} + 2 \beta_1 + 15) q^{3} + (\beta_{4} - 2 \beta_{3} - 5 \beta_1 + 1805) q^{4} - \beta_{2} q^{5} + (\beta_{6} + 3 \beta_{4} + \cdots + 13142) q^{6}+ \cdots + (730 \beta_{19} + \cdots + 2788362 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9} + 19833480 q^{12} + 8049118 q^{13} + 81743632 q^{16} + 36238080 q^{18} - 367189708 q^{23} + 530348736 q^{24} - 1824317212 q^{25} - 2465696728 q^{26} - 1765163178 q^{27} - 417897362 q^{29} - 1574125298 q^{31} + 1240884960 q^{32} + 4723363152 q^{35} + 3607269840 q^{36} + 1926187110 q^{39} - 132511250 q^{41} + 62599086544 q^{46} - 45052376402 q^{47} + 77275023312 q^{48} - 65534183260 q^{49} - 57446928200 q^{50} - 20407259400 q^{52} - 15082009344 q^{54} - 12572534832 q^{55} + 97529544392 q^{58} - 110336269112 q^{59} + 553127354432 q^{62} + 13799515808 q^{64} + 7914588558 q^{69} + 590935322064 q^{70} - 40523102210 q^{71} - 539385055680 q^{72} - 449748966242 q^{73} + 2570431278 q^{75} - 345974135760 q^{77} - 48894418992 q^{78} - 2694979782144 q^{81} - 1210586085208 q^{82} + 1051198266576 q^{85} - 2237359632618 q^{87} + 1936236755640 q^{92} - 116415989178 q^{93} + 3420883097024 q^{94} + 3777482201184 q^{95} + 3567911213376 q^{96} + 3590387133208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 3347471347 x^{18} + 50192028136 x^{17} + \cdots + 27\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 61\!\cdots\!53 \nu^{19} + \cdots - 20\!\cdots\!48 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 50\!\cdots\!23 \nu^{19} + \cdots - 27\!\cdots\!20 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\!\cdots\!47 \nu^{19} + \cdots + 13\!\cdots\!44 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!37 \nu^{19} + \cdots - 15\!\cdots\!76 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!61 \nu^{19} + \cdots + 99\!\cdots\!80 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!13 \nu^{19} + \cdots + 34\!\cdots\!68 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!57 \nu^{19} + \cdots + 38\!\cdots\!00 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!43 \nu^{19} + \cdots - 66\!\cdots\!40 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!35 \nu^{19} + \cdots - 28\!\cdots\!64 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!79 \nu^{19} + \cdots + 39\!\cdots\!68 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!93 \nu^{19} + \cdots - 35\!\cdots\!12 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!27 \nu^{19} + \cdots + 26\!\cdots\!68 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 50\!\cdots\!13 \nu^{19} + \cdots + 84\!\cdots\!80 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!93 \nu^{19} + \cdots - 14\!\cdots\!60 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!79 \nu^{19} + \cdots - 13\!\cdots\!80 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34\!\cdots\!13 \nu^{19} + \cdots + 26\!\cdots\!60 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!20 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 17\!\cdots\!81 \nu^{19} + \cdots + 40\!\cdots\!00 ) / 98\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 39\!\cdots\!47 \nu^{19} + \cdots - 12\!\cdots\!80 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} - 139 \beta_{12} + 26 \beta_{11} + 146 \beta_{10} - 145 \beta_{9} + \cdots - 334574395 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 14221 \beta_{19} - 41709 \beta_{18} + 18551 \beta_{17} - 17522 \beta_{16} - 199770 \beta_{15} + \cdots - 13424786838 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 29866800 \beta_{19} - 27699220 \beta_{18} + 7012140 \beta_{17} + 264584820 \beta_{16} + \cdots + 19\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14356762574448 \beta_{19} + 41381024332035 \beta_{18} - 15546000000785 \beta_{17} + \cdots + 33\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\!\cdots\!50 \beta_{19} + \cdots - 13\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12\!\cdots\!33 \beta_{19} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 65\!\cdots\!56 \beta_{19} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11\!\cdots\!52 \beta_{19} + \cdots - 15\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 59\!\cdots\!10 \beta_{19} + \cdots - 64\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 94\!\cdots\!53 \beta_{19} + \cdots + 38\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 47\!\cdots\!36 \beta_{19} + \cdots + 46\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 77\!\cdots\!04 \beta_{19} + \cdots - 54\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 34\!\cdots\!02 \beta_{19} + \cdots - 33\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 63\!\cdots\!65 \beta_{19} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 22\!\cdots\!08 \beta_{19} + \cdots + 24\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 51\!\cdots\!76 \beta_{19} + \cdots - 66\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 12\!\cdots\!06 \beta_{19} + \cdots - 18\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 41\!\cdots\!93 \beta_{19} + \cdots + 65\!\cdots\!62 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−891.853 + 7568.36i
−891.853 − 7568.36i
726.027 + 22106.6i
726.027 − 22106.6i
−342.173 + 28280.5i
−342.173 − 28280.5i
1052.17 + 197.039i
1052.17 − 197.039i
273.019 + 7850.48i
273.019 − 7850.48i
−697.167 + 19533.2i
−697.167 − 19533.2i
1203.28 + 26094.4i
1203.28 − 26094.4i
−962.760 + 10393.2i
−962.760 − 10393.2i
345.826 + 9266.28i
345.826 − 9266.28i
−701.372 + 26263.6i
−701.372 − 26263.6i
−111.723 792.131 8385.93 − 7568.36i −88498.9 231962.i −479282. 96030.0 845556.i
22.2 −111.723 792.131 8385.93 7568.36i −88498.9 − 231962.i −479282. 96030.0 − 845556.i
22.3 −108.578 −822.605 7693.26 − 22106.6i 89317.1 − 37537.4i −390585. 145238. 2.40030e6i
22.4 −108.578 −822.605 7693.26 22106.6i 89317.1 37537.4i −390585. 145238. − 2.40030e6i
22.5 −52.0667 302.106 −1385.05 − 28280.5i −15729.7 − 52107.5i 285381. −440173. 1.47247e6i
22.6 −52.0667 302.106 −1385.05 28280.5i −15729.7 52107.5i 285381. −440173. − 1.47247e6i
22.7 −43.1697 −1083.34 −2232.38 − 197.039i 46767.4 − 138687.i 273194. 642183. 8506.13i
22.8 −43.1697 −1083.34 −2232.38 197.039i 46767.4 138687.i 273194. 642183. − 8506.13i
22.9 9.17467 −251.844 −4011.83 − 7850.48i −2310.59 186545.i −74386.7 −468016. − 72025.6i
22.10 9.17467 −251.844 −4011.83 7850.48i −2310.59 − 186545.i −74386.7 −468016. 72025.6i
22.11 20.5558 729.723 −3673.46 − 19533.2i 15000.0 − 5514.27i −159707. 1054.67 − 401520.i
22.12 20.5558 729.723 −3673.46 19533.2i 15000.0 5514.27i −159707. 1054.67 401520.i
22.13 52.2711 −1139.01 −1363.73 − 26094.4i −59537.5 − 15165.4i −285386. 765911. − 1.36399e6i
22.14 52.2711 −1139.01 −1363.73 26094.4i −59537.5 15165.4i −285386. 765911. 1.36399e6i
22.15 68.2708 1043.03 564.897 − 10393.2i 71208.5 − 148862.i −241071. 556472. − 709550.i
22.16 68.2708 1043.03 564.897 10393.2i 71208.5 148862.i −241071. 556472. 709550.i
22.17 91.3205 −242.505 4243.43 − 9266.28i −22145.7 − 93897.2i 13463.2 −472632. − 846201.i
22.18 91.3205 −242.505 4243.43 9266.28i −22145.7 93897.2i 13463.2 −472632. 846201.i
22.19 117.945 831.317 9814.93 − 26263.6i 98049.3 167269.i 674517. 159646. − 3.09765e6i
22.20 117.945 831.317 9814.93 26263.6i 98049.3 − 167269.i 674517. 159646. 3.09765e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.13.b.c ✓ 20
23.b odd 2 1 inner 23.13.b.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.13.b.c ✓ 20 1.a even 1 1 trivial
23.13.b.c ✓ 20 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 44 T_{2}^{9} - 28530 T_{2}^{8} + 1291520 T_{2}^{7} + 256814048 T_{2}^{6} + \cdots + 19\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 25\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 51\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 95\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
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