Properties

Label 23.4.a.b
Level $23$
Weight $4$
Character orbit 23.a
Self dual yes
Analytic conductor $1.357$
Analytic rank $0$
Dimension $4$
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] = [23,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(1.35704393013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.334189.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 16x^{2} - 5x + 4 \) 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

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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 6) q^{4}+ \cdots + (2 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 6) q^{4}+ \cdots + ( - 48 \beta_{3} - 46 \beta_{2} + \cdots - 340) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 16x^{2} - 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 20\nu - 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 5\nu^{2} + 28\nu - 1 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + 4\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_{2} + 24\beta _1 + 17 \) 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
−0.743529
5.22031
−2.83969
0.362907
−5.07751 1.55870 17.7811 10.0635 −7.91434 24.3381 −49.6639 −24.5704 −51.0976
1.2 −0.0323756 6.42170 −7.99895 14.1026 −0.207906 −14.0109 0.517976 14.2382 −0.456580
1.3 2.86845 3.43737 0.228032 −17.9704 9.85995 32.7301 −22.2935 −15.1845 −51.5473
1.4 4.24143 −4.41777 9.98977 7.80430 −18.7377 −27.0572 8.43948 −7.48328 33.1014
\(n\): e.g. 2-40 or 990-1000
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 23.4.a.b 4
3.b odd 2 1 207.4.a.e 4
4.b odd 2 1 368.4.a.l 4
5.b even 2 1 575.4.a.i 4
5.c odd 4 2 575.4.b.g 8
7.b odd 2 1 1127.4.a.c 4
8.b even 2 1 1472.4.a.y 4
8.d odd 2 1 1472.4.a.bf 4
23.b odd 2 1 529.4.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.b 4 1.a even 1 1 trivial
207.4.a.e 4 3.b odd 2 1
368.4.a.l 4 4.b odd 2 1
529.4.a.g 4 23.b odd 2 1
575.4.a.i 4 5.b even 2 1
575.4.b.g 8 5.c odd 4 2
1127.4.a.c 4 7.b odd 2 1
1472.4.a.y 4 8.b even 2 1
1472.4.a.bf 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 24T_{2}^{2} + 61T_{2} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( T^{4} - 7 T^{3} + \cdots - 152 \) Copy content Toggle raw display
$5$ \( T^{4} - 14 T^{3} + \cdots - 19904 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} + \cdots + 301984 \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} + \cdots - 81440 \) Copy content Toggle raw display
$13$ \( T^{4} - 111 T^{3} + \cdots + 1322658 \) Copy content Toggle raw display
$17$ \( T^{4} - 98 T^{3} + \cdots - 855280 \) Copy content Toggle raw display
$19$ \( T^{4} - 96 T^{3} + \cdots + 66996944 \) Copy content Toggle raw display
$23$ \( (T + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 21 T^{3} + \cdots + 325399050 \) Copy content Toggle raw display
$31$ \( T^{4} + 193 T^{3} + \cdots - 58104720 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 2389345472 \) Copy content Toggle raw display
$41$ \( T^{4} + 125 T^{3} + \cdots + 29467114 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots + 78004224 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 3169103456 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 7631805536 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1146071296 \) Copy content Toggle raw display
$61$ \( T^{4} - 754 T^{3} + \cdots - 621762112 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 1826338144 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 5581505296 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 14695752674 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 61908677856 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 7015211408 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 213195182848 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 60054540368 \) Copy content Toggle raw display
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