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Properties

Label 23.9.b.b
Level $23$
Weight $9$
Character orbit 23.b
Analytic conductor $9.370$
Analytic rank $0$
Dimension $12$
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] = [23,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(9.36970803141\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3434556 x^{10} + 4503520431468 x^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + (\beta_{3} - 6) q^{3} + (\beta_{4} - 2 \beta_{3} + 30) q^{4} - \beta_1 q^{5} + (\beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots - 73) q^{6}+ \cdots + (3 \beta_{6} - 3 \beta_{5} + \cdots + 477) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + (\beta_{3} - 6) q^{3} + (\beta_{4} - 2 \beta_{3} + 30) q^{4} - \beta_1 q^{5} + (\beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots - 73) q^{6}+ \cdots + (729 \beta_{11} - 771 \beta_{10} + \cdots + 2304 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{2} - 72 q^{3} + 360 q^{4} - 1104 q^{6} + 3056 q^{8} + 5892 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{2} - 72 q^{3} + 360 q^{4} - 1104 q^{6} + 3056 q^{8} + 5892 q^{9} - 176472 q^{12} + 33696 q^{13} - 136944 q^{16} + 82752 q^{18} - 634540 q^{23} - 436128 q^{24} - 2181612 q^{25} + 2907752 q^{26} + 2605608 q^{27} + 3183496 q^{29} - 2159088 q^{31} + 1401888 q^{32} - 5177928 q^{35} + 2446416 q^{36} - 5129184 q^{39} + 4482376 q^{41} + 3948336 q^{46} + 1780864 q^{47} + 7462416 q^{48} - 11810652 q^{49} - 31141640 q^{50} - 9867720 q^{52} + 12640032 q^{54} - 31910712 q^{55} + 63734184 q^{58} - 13809656 q^{59} + 44121440 q^{62} - 40692576 q^{64} + 28078488 q^{69} + 48672624 q^{70} + 51319096 q^{71} - 14464896 q^{72} - 967824 q^{73} + 61333488 q^{75} + 48071016 q^{77} - 44547696 q^{78} - 242317692 q^{81} + 159267336 q^{82} - 198448584 q^{85} + 1155768 q^{87} - 192278184 q^{92} - 60337752 q^{93} - 468343392 q^{94} + 141719424 q^{95} + 264031488 q^{96} + 459446680 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3434556 x^{10} + 4503520431468 x^{8} + \cdots + 22\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!83 \nu^{10} + \cdots + 28\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!97 \nu^{10} + \cdots - 60\!\cdots\!00 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59\!\cdots\!43 \nu^{10} + \cdots + 76\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!97 \nu^{10} + \cdots + 11\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!57 \nu^{10} + \cdots + 70\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!53 \nu^{11} + \cdots + 27\!\cdots\!00 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34\!\cdots\!53 \nu^{11} + \cdots + 26\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41\!\cdots\!81 \nu^{11} + \cdots + 29\!\cdots\!00 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!27 \nu^{11} + \cdots - 12\!\cdots\!00 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!39 \nu^{11} + \cdots - 31\!\cdots\!20 \nu ) / 50\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -184\beta_{6} + 354\beta_{5} - 661\beta_{4} + 404\beta_{3} - 9464\beta_{2} - 569210 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -323\beta_{11} - 579\beta_{10} + 16107\beta_{9} - 732\beta_{8} + 16439\beta_{7} - 819507\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 174803280 \beta_{6} - 342668736 \beta_{5} + 895695918 \beta_{4} - 603698892 \beta_{3} + \cdots + 458992783524 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 142595514 \beta_{11} + 540392430 \beta_{10} - 9997166586 \beta_{9} - 1344412716 \beta_{8} + \cdots + 752791649922 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 144279902866968 \beta_{6} + 328069763534664 \beta_{5} + \cdots - 41\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 677451524973228 \beta_{11} - 669674168086428 \beta_{10} + \cdots - 74\!\cdots\!56 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11\!\cdots\!28 \beta_{6} + \cdots + 41\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12\!\cdots\!96 \beta_{11} + \cdots + 75\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 84\!\cdots\!44 \beta_{6} + \cdots - 42\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19\!\cdots\!32 \beta_{11} + \cdots - 80\!\cdots\!92 \beta_1 \) 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
687.973i
− 687.973i
525.271i
− 525.271i
789.612i
− 789.612i
947.955i
− 947.955i
162.968i
− 162.968i
1066.15i
− 1066.15i
−24.5765 −90.1632 348.003 − 687.973i 2215.89 − 3769.56i −2261.10 1568.40 16907.9i
22.2 −24.5765 −90.1632 348.003 687.973i 2215.89 3769.56i −2261.10 1568.40 − 16907.9i
22.3 −15.1295 83.3518 −27.0996 − 525.271i −1261.07 2271.82i 4283.14 386.521 7947.06i
22.4 −15.1295 83.3518 −27.0996 525.271i −1261.07 − 2271.82i 4283.14 386.521 − 7947.06i
22.5 −7.72154 −44.2925 −196.378 − 789.612i 342.007 1308.38i 3493.05 −4599.17 6097.02i
22.6 −7.72154 −44.2925 −196.378 789.612i 342.007 − 1308.38i 3493.05 −4599.17 − 6097.02i
22.7 6.10528 125.487 −218.726 − 947.955i 766.132 − 2764.83i −2898.33 9185.93 − 5787.54i
22.8 6.10528 125.487 −218.726 947.955i 766.132 2764.83i −2898.33 9185.93 5787.54i
22.9 11.9630 −13.7539 −112.886 − 162.968i −164.539 3427.07i −4412.99 −6371.83 − 1949.59i
22.10 11.9630 −13.7539 −112.886 162.968i −164.539 − 3427.07i −4412.99 −6371.83 1949.59i
22.11 25.3591 −96.6289 387.086 − 1066.15i −2450.43 − 149.704i 3324.23 2776.15 − 27036.6i
22.12 25.3591 −96.6289 387.086 1066.15i −2450.43 149.704i 3324.23 2776.15 27036.6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.12
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 23.9.b.b ✓ 12
3.b odd 2 1 207.9.d.b 12
4.b odd 2 1 368.9.f.b 12
23.b odd 2 1 inner 23.9.b.b ✓ 12
69.c even 2 1 207.9.d.b 12
92.b even 2 1 368.9.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.9.b.b ✓ 12 1.a even 1 1 trivial
23.9.b.b ✓ 12 23.b odd 2 1 inner
207.9.d.b 12 3.b odd 2 1
207.9.d.b 12 69.c even 2 1
368.9.f.b 12 4.b odd 2 1
368.9.f.b 12 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 4T_{2}^{5} - 850T_{2}^{4} - 3248T_{2}^{3} + 147872T_{2}^{2} + 268672T_{2} - 5317760 \) acting on \(S_{9}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 4 T^{5} + \cdots - 5317760)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 36 T^{5} + \cdots + 55514443500)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 44\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 54\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 38\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 60\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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