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Properties

Label 21.9.b.a
Level $21$
Weight $9$
Character orbit 21.b
Analytic conductor $8.555$
Analytic rank $0$
Dimension $16$
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] = [21,9,Mod(8,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.8");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(8.55495081128\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2834 x^{14} + 3067809 x^{12} + 1610483612 x^{10} + 432397249972 x^{8} + 56638135387008 x^{6} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{20}\cdot 7^{13} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1 - 11) q^{3} + (\beta_{4} - \beta_{3} - 98) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} - 24 \beta_1 - 314) q^{6} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{12} - \beta_{8} + \cdots + 4) q^{8}+ \cdots + (\beta_{11} + \beta_{9} + 2 \beta_{8} + \cdots + 575) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1 - 11) q^{3} + (\beta_{4} - \beta_{3} - 98) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} - 24 \beta_1 - 314) q^{6} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{12} - \beta_{8} + \cdots + 4) q^{8}+ \cdots + ( - 3141 \beta_{15} - 7554 \beta_{14} + \cdots - 9755188) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 182 q^{3} - 1572 q^{4} - 5026 q^{6} + 9304 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 182 q^{3} - 1572 q^{4} - 5026 q^{6} + 9304 q^{9} - 7364 q^{10} + 87458 q^{12} - 47460 q^{13} + 115876 q^{15} + 487428 q^{16} - 525832 q^{18} - 377916 q^{19} - 81634 q^{21} + 275824 q^{22} - 281862 q^{24} - 1603704 q^{25} + 1070398 q^{27} - 595448 q^{28} + 1232108 q^{30} + 82936 q^{31} - 1601684 q^{33} + 10074288 q^{34} + 1486844 q^{36} - 10527200 q^{37} - 5636228 q^{39} - 412860 q^{40} + 5642350 q^{42} + 3062000 q^{43} + 4598860 q^{45} - 20247120 q^{46} - 17988082 q^{48} + 13176688 q^{49} + 14676 q^{51} + 36568868 q^{52} + 28779254 q^{54} + 39521384 q^{55} + 26828416 q^{57} - 18337424 q^{58} - 70421188 q^{60} - 55649580 q^{61} + 16240364 q^{63} - 127448396 q^{64} - 14627116 q^{66} + 48398760 q^{67} - 56397936 q^{69} - 32509540 q^{70} + 165877344 q^{72} - 82345928 q^{73} - 24802666 q^{75} + 333954740 q^{76} - 63454948 q^{78} - 1300864 q^{79} + 9523768 q^{81} + 108648344 q^{82} + 41225170 q^{84} - 64318944 q^{85} - 87414236 q^{87} + 172885848 q^{88} - 249384352 q^{90} - 240100 q^{91} + 137670888 q^{93} - 559369440 q^{94} + 504157654 q^{96} - 337523648 q^{97} - 156506576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2834 x^{14} + 3067809 x^{12} + 1610483612 x^{10} + 432397249972 x^{8} + 56638135387008 x^{6} + \cdots + 31\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 58\!\cdots\!43 \nu^{15} + \cdots + 26\!\cdots\!64 ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 58\!\cdots\!85 \nu^{15} + \cdots + 23\!\cdots\!20 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58\!\cdots\!85 \nu^{15} + \cdots + 51\!\cdots\!40 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58\!\cdots\!85 \nu^{15} + \cdots + 24\!\cdots\!76 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!81 \nu^{15} + \cdots + 11\!\cdots\!88 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!49 \nu^{15} + \cdots - 27\!\cdots\!80 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77\!\cdots\!55 \nu^{15} + \cdots + 23\!\cdots\!20 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92\!\cdots\!07 \nu^{15} + \cdots - 39\!\cdots\!60 ) / 68\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!67 \nu^{15} + \cdots + 52\!\cdots\!52 ) / 76\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\!\cdots\!39 \nu^{15} + \cdots - 15\!\cdots\!68 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!36 \nu^{15} + \cdots + 23\!\cdots\!08 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!71 \nu^{15} + \cdots - 33\!\cdots\!84 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!01 \nu^{15} + \cdots + 25\!\cdots\!84 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!67 \nu^{15} + \cdots - 62\!\cdots\!88 ) / 45\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 354 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{12} - \beta_{8} + \beta_{6} - 11\beta_{3} + \beta_{2} - 668\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} + 4 \beta_{14} - 7 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + 7 \beta_{9} + \cdots + 236750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 44 \beta_{15} + 1141 \beta_{14} - 64 \beta_{13} + 1153 \beta_{12} + 126 \beta_{11} + 126 \beta_{10} + \cdots - 6920 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5502 \beta_{15} - 3840 \beta_{14} + 9669 \beta_{13} - 3201 \beta_{12} - 5241 \beta_{11} + \cdots - 188523382 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 69774 \beta_{15} - 1109053 \beta_{14} + 124064 \beta_{13} - 1109089 \beta_{12} - 187674 \beta_{11} + \cdots + 8460084 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5493500 \beta_{15} + 2668060 \beta_{14} - 10660201 \beta_{13} + 2312717 \beta_{12} + \cdots + 162025732886 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 80247412 \beta_{15} + 1043347525 \beta_{14} - 172004992 \beta_{13} + 1024083641 \beta_{12} + \cdots - 9233873824 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4992364490 \beta_{15} - 1369871720 \beta_{14} + 10973384593 \beta_{13} - 1227105133 \beta_{12} + \cdots - 144411482218006 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 83014746634 \beta_{15} - 976656942581 \beta_{14} + 207805042784 \beta_{13} - 937609362185 \beta_{12} + \cdots + 9618396457756 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4409984343216 \beta_{15} + 216070832868 \beta_{14} - 10990780843209 \beta_{13} + \cdots + 13\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 82386014501976 \beta_{15} + 916220531882813 \beta_{14} - 233513543947456 \beta_{13} + \cdots - 97\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 38\!\cdots\!46 \beta_{15} + 727558784175952 \beta_{14} + \cdots - 12\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 80\!\cdots\!18 \beta_{15} + \cdots + 98\!\cdots\!84 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
− 31.0542i
− 28.0859i
− 20.5748i
− 18.3367i
− 13.8347i
− 11.0244i
− 2.52460i
− 1.39676i
1.39676i
2.52460i
11.0244i
13.8347i
18.3367i
20.5748i
28.0859i
31.0542i
− 31.0542i −68.9048 + 42.5808i −708.366 − 426.167i 1322.31 + 2139.79i 907.493 14047.9i 2934.75 − 5868.05i −13234.3
8.2 − 28.0859i 9.91526 − 80.3908i −532.817 850.805i −2257.85 − 278.479i −907.493 7774.65i −6364.38 − 1594.19i 23895.6
8.3 − 20.5748i 73.3512 − 34.3600i −167.321 − 494.107i −706.949 − 1509.18i 907.493 − 1824.55i 4199.78 − 5040.69i −10166.1
8.4 − 18.3367i −52.9258 − 61.3177i −80.2359 − 907.186i −1124.37 + 970.487i −907.493 − 3222.94i −958.710 + 6490.58i −16634.8
8.5 − 13.8347i −79.6358 − 14.8032i 64.6012 923.073i −204.797 + 1101.74i 907.493 − 4435.42i 6122.73 + 2357.72i 12770.4
8.6 − 11.0244i −56.8904 + 57.6584i 134.463 − 245.386i 635.648 + 627.181i −907.493 − 4304.61i −87.9733 − 6560.41i −2705.23
8.7 − 2.52460i 76.8898 − 25.4746i 249.626 1013.75i −64.3132 − 194.116i −907.493 − 1276.51i 5263.09 − 3917.47i 2559.31
8.8 − 1.39676i 7.20063 − 80.6793i 254.049 − 119.475i −112.690 − 10.0576i 907.493 − 712.416i −6457.30 − 1161.88i −166.878
8.9 1.39676i 7.20063 + 80.6793i 254.049 119.475i −112.690 + 10.0576i 907.493 712.416i −6457.30 + 1161.88i −166.878
8.10 2.52460i 76.8898 + 25.4746i 249.626 − 1013.75i −64.3132 + 194.116i −907.493 1276.51i 5263.09 + 3917.47i 2559.31
8.11 11.0244i −56.8904 − 57.6584i 134.463 245.386i 635.648 − 627.181i −907.493 4304.61i −87.9733 + 6560.41i −2705.23
8.12 13.8347i −79.6358 + 14.8032i 64.6012 − 923.073i −204.797 − 1101.74i 907.493 4435.42i 6122.73 − 2357.72i 12770.4
8.13 18.3367i −52.9258 + 61.3177i −80.2359 907.186i −1124.37 − 970.487i −907.493 3222.94i −958.710 − 6490.58i −16634.8
8.14 20.5748i 73.3512 + 34.3600i −167.321 494.107i −706.949 + 1509.18i 907.493 1824.55i 4199.78 + 5040.69i −10166.1
8.15 28.0859i 9.91526 + 80.3908i −532.817 − 850.805i −2257.85 + 278.479i −907.493 − 7774.65i −6364.38 + 1594.19i 23895.6
8.16 31.0542i −68.9048 − 42.5808i −708.366 426.167i 1322.31 − 2139.79i 907.493 − 14047.9i 2934.75 + 5868.05i −13234.3
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.9.b.a ✓ 16
3.b odd 2 1 inner 21.9.b.a ✓ 16
4.b odd 2 1 336.9.d.c 16
12.b even 2 1 336.9.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.b.a ✓ 16 1.a even 1 1 trivial
21.9.b.a ✓ 16 3.b odd 2 1 inner
336.9.d.c 16 4.b odd 2 1
336.9.d.c 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 823543)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 71\!\cdots\!08)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 53\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 29\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 37\!\cdots\!12)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 35\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 78\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 41\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 66\!\cdots\!68)^{2} \) Copy content Toggle raw display
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