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Properties

Label 42.9.b.a
Level $42$
Weight $9$
Character orbit 42.b
Analytic conductor $17.110$
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] = [42,9,Mod(29,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, 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("42.29");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 42.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: \(17.1099016226\)
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: \( x^{16} + 4360 x^{14} + 6995530 x^{12} + 4917350960 x^{10} + 1302964636433 x^{8} + 21295764676232 x^{6} + \cdots + 11049853553424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{33}\cdot 3^{21}\cdot 7^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 16) q^{3} - 128 q^{4} + ( - \beta_{6} - 7 \beta_1) q^{5} + (\beta_{5} + 16 \beta_1 + 112) q^{6} + (\beta_{3} - 2 \beta_{2}) q^{7} - 128 \beta_1 q^{8} + (\beta_{14} + \beta_{12} - \beta_{8} + \cdots - 562) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 16) q^{3} - 128 q^{4} + ( - \beta_{6} - 7 \beta_1) q^{5} + (\beta_{5} + 16 \beta_1 + 112) q^{6} + (\beta_{3} - 2 \beta_{2}) q^{7} - 128 \beta_1 q^{8} + (\beta_{14} + \beta_{12} - \beta_{8} + \cdots - 562) q^{9}+ \cdots + ( - 3051 \beta_{15} - 7553 \beta_{14} + \cdots - 18371370) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 252 q^{3} - 2048 q^{4} + 1792 q^{6} - 9088 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 252 q^{3} - 2048 q^{4} + 1792 q^{6} - 9088 q^{9} + 14336 q^{10} - 32256 q^{12} + 18760 q^{13} - 14056 q^{15} + 262144 q^{16} - 18432 q^{18} + 66584 q^{19} + 163268 q^{21} + 449536 q^{22} - 229376 q^{24} - 1538112 q^{25} - 180684 q^{27} - 2023936 q^{30} - 1995056 q^{31} + 4326952 q^{33} - 4831232 q^{34} + 1163264 q^{36} + 9791360 q^{37} - 1015384 q^{39} - 1835008 q^{40} + 8544224 q^{43} + 3099208 q^{45} - 9390080 q^{46} + 4128768 q^{48} + 13176688 q^{49} - 11224136 q^{51} - 2401280 q^{52} - 4417280 q^{54} - 12335120 q^{55} - 23273856 q^{57} + 6665216 q^{58} + 1799168 q^{60} + 13985944 q^{61} - 20571768 q^{63} - 33554432 q^{64} - 26733056 q^{66} - 31398480 q^{67} + 152693632 q^{69} + 20898304 q^{70} + 2359296 q^{72} + 3051888 q^{73} - 31931452 q^{75} - 8522752 q^{76} - 61194752 q^{78} + 87884032 q^{79} + 161678720 q^{81} + 44233728 q^{82} - 20898304 q^{84} - 151378784 q^{85} - 146242600 q^{87} - 57540608 q^{88} + 65820160 q^{90} - 12350744 q^{91} - 97629616 q^{93} - 84145152 q^{94} + 29360128 q^{96} + 249178272 q^{97} - 295557472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 4360 x^{14} + 6995530 x^{12} + 4917350960 x^{10} + 1302964636433 x^{8} + 21295764676232 x^{6} + \cdots + 11049853553424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!11 \nu^{15} + \cdots + 16\!\cdots\!36 \nu ) / 49\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67\!\cdots\!37 \nu^{15} + \cdots - 23\!\cdots\!36 ) / 55\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!74 \nu^{15} + \cdots - 67\!\cdots\!56 ) / 55\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67\!\cdots\!93 \nu^{15} + \cdots + 12\!\cdots\!60 ) / 16\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!89 \nu^{15} + \cdots - 52\!\cdots\!92 ) / 69\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67\!\cdots\!63 \nu^{15} + \cdots - 72\!\cdots\!20 \nu ) / 27\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 80\!\cdots\!60 \nu^{15} + \cdots - 34\!\cdots\!28 ) / 27\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!89 \nu^{15} + \cdots + 83\!\cdots\!44 ) / 83\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 97\!\cdots\!25 \nu^{15} + \cdots - 41\!\cdots\!80 ) / 16\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!01 \nu^{15} + \cdots - 79\!\cdots\!84 ) / 16\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 59\!\cdots\!63 \nu^{15} + \cdots + 41\!\cdots\!36 ) / 55\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!68 \nu^{15} + \cdots - 34\!\cdots\!48 ) / 16\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!13 \nu^{15} + \cdots + 20\!\cdots\!44 ) / 83\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!28 \nu^{15} + \cdots - 30\!\cdots\!08 ) / 16\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 44\!\cdots\!91 \nu^{15} + \cdots + 14\!\cdots\!28 ) / 16\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 40 \beta_{15} - 76 \beta_{14} - 9 \beta_{13} - 9 \beta_{12} - 220 \beta_{11} + 5 \beta_{9} + \cdots + 1937 ) / 74088 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 497 \beta_{15} + 3199 \beta_{14} + 4494 \beta_{13} - 1680 \beta_{12} - 2702 \beta_{11} + \cdots - 20178249 ) / 37044 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10874 \beta_{15} + 26374 \beta_{14} + 3875 \beta_{13} + 3875 \beta_{12} + 90096 \beta_{11} + \cdots - 606651 ) / 24696 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 442876 \beta_{15} - 2039702 \beta_{14} - 2355402 \beta_{13} + 1293432 \beta_{12} + \cdots + 11607775473 ) / 18522 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18610013 \beta_{15} - 92473361 \beta_{14} - 18465837 \beta_{13} - 18465837 \beta_{12} + \cdots + 1968001324 ) / 74088 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 433515033 \beta_{15} + 1717726759 \beta_{14} + 1694511280 \beta_{13} - 1313150762 \beta_{12} + \cdots - 9451972592013 ) / 12348 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 599314532 \beta_{15} + 15888883352 \beta_{14} + 3822392205 \beta_{13} + 3822392205 \beta_{12} + \cdots - 317494443103 ) / 10584 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 899929332617 \beta_{15} - 3286805468653 \beta_{14} - 2838496670688 \beta_{13} + \cdots + 18\!\cdots\!60 ) / 18522 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3740665040355 \beta_{15} - 45856509018433 \beta_{14} - 12399293514697 \beta_{13} + \cdots + 873167279666824 ) / 24696 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 24\!\cdots\!83 \beta_{15} + \cdots - 48\!\cdots\!15 ) / 37044 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 28\!\cdots\!60 \beta_{15} + \cdots - 32\!\cdots\!15 ) / 74088 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 55\!\cdots\!22 \beta_{15} + \cdots + 11\!\cdots\!65 ) / 6174 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 48\!\cdots\!03 \beta_{15} + \cdots + 41\!\cdots\!60 ) / 74088 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 65\!\cdots\!81 \beta_{15} + \cdots - 13\!\cdots\!09 ) / 5292 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 24\!\cdots\!20 \beta_{15} + \cdots - 18\!\cdots\!47 ) / 24696 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
− 38.7104i
3.43386i
− 1.35486i
31.6953i
− 0.349368i
1.85152i
33.2985i
− 27.0361i
38.7104i
− 3.43386i
1.35486i
− 31.6953i
0.349368i
− 1.85152i
− 33.2985i
27.0361i
− 11.3137i −80.9811 − 1.74792i −128.000 587.718i −19.7755 + 916.197i −907.493 1448.15i 6554.89 + 283.098i 6649.27
29.2 − 11.3137i −22.1845 + 77.9028i −128.000 − 387.711i 881.370 + 250.989i −907.493 1448.15i −5576.69 − 3456.47i −4386.45
29.3 − 11.3137i −14.5316 − 79.6858i −128.000 417.035i −901.542 + 164.406i 907.493 1448.15i −6138.67 + 2315.93i 4718.21
29.4 − 11.3137i −8.37043 + 80.5663i −128.000 1103.57i 911.504 + 94.7006i 907.493 1448.15i −6420.87 − 1348.75i 12485.4
29.5 − 11.3137i 45.9951 − 66.6742i −128.000 − 728.401i −754.332 − 520.376i −907.493 1448.15i −2329.89 − 6133.38i −8240.92
29.6 − 11.3137i 53.0624 + 61.1996i −128.000 − 1096.27i 692.394 − 600.332i 907.493 1448.15i −929.773 + 6494.79i −12402.9
29.7 − 11.3137i 75.1927 + 30.1173i −128.000 336.313i 340.738 − 850.709i −907.493 1448.15i 4746.90 + 4529.20i 3804.94
29.8 − 11.3137i 77.8175 − 22.4821i −128.000 401.320i −254.356 − 880.404i 907.493 1448.15i 5550.11 − 3499.00i 4540.42
29.9 11.3137i −80.9811 + 1.74792i −128.000 − 587.718i −19.7755 − 916.197i −907.493 − 1448.15i 6554.89 − 283.098i 6649.27
29.10 11.3137i −22.1845 − 77.9028i −128.000 387.711i 881.370 − 250.989i −907.493 − 1448.15i −5576.69 + 3456.47i −4386.45
29.11 11.3137i −14.5316 + 79.6858i −128.000 − 417.035i −901.542 − 164.406i 907.493 − 1448.15i −6138.67 − 2315.93i 4718.21
29.12 11.3137i −8.37043 − 80.5663i −128.000 − 1103.57i 911.504 − 94.7006i 907.493 − 1448.15i −6420.87 + 1348.75i 12485.4
29.13 11.3137i 45.9951 + 66.6742i −128.000 728.401i −754.332 + 520.376i −907.493 − 1448.15i −2329.89 + 6133.38i −8240.92
29.14 11.3137i 53.0624 − 61.1996i −128.000 1096.27i 692.394 + 600.332i 907.493 − 1448.15i −929.773 − 6494.79i −12402.9
29.15 11.3137i 75.1927 − 30.1173i −128.000 − 336.313i 340.738 + 850.709i −907.493 − 1448.15i 4746.90 − 4529.20i 3804.94
29.16 11.3137i 77.8175 + 22.4821i −128.000 − 401.320i −254.356 + 880.404i 907.493 − 1448.15i 5550.11 + 3499.00i 4540.42
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.16
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 42.9.b.a ✓ 16
3.b odd 2 1 inner 42.9.b.a ✓ 16
4.b odd 2 1 336.9.d.a 16
12.b even 2 1 336.9.d.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.b.a ✓ 16 1.a even 1 1 trivial
42.9.b.a ✓ 16 3.b odd 2 1 inner
336.9.d.a 16 4.b odd 2 1
336.9.d.a 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 823543)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 56\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 47\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 55\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 49\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 87\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 24\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 97\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 31\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 40\!\cdots\!72)^{2} \) Copy content Toggle raw display
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