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Properties

Label 19.9.b.b
Level $19$
Weight $9$
Character orbit 19.b
Analytic conductor $7.740$
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] = [19,9,Mod(18,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, 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("19.18");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 19.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: \(7.74019359116\)
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: \( x^{12} + 2323 x^{10} + 2010462 x^{8} + 803113072 x^{6} + 150633270400 x^{4} + 12173735396352 x^{2} + 333034797957120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 131) q^{4} + (\beta_{6} + 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 82) q^{6}+ \cdots + (\beta_{8} + \beta_{7} - 5 \beta_{5} + \cdots - 3391) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 131) q^{4} + (\beta_{6} + 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 82) q^{6}+ \cdots + ( - 22252 \beta_{8} - 22282 \beta_{7} + \cdots - 32330015) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 1574 q^{4} + 8 q^{5} + 986 q^{6} + 3686 q^{7} - 40706 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 1574 q^{4} + 8 q^{5} + 986 q^{6} + 3686 q^{7} - 40706 q^{9} - 8968 q^{11} - 30886 q^{16} - 163990 q^{17} - 297306 q^{19} + 286484 q^{20} + 1352102 q^{23} - 1485226 q^{24} + 1097344 q^{25} + 2423070 q^{26} - 2407770 q^{28} + 1776176 q^{30} - 3523616 q^{35} - 36708 q^{36} - 3002394 q^{38} - 1968898 q^{39} + 10898030 q^{42} - 13887524 q^{43} - 29515912 q^{44} - 819008 q^{45} + 27539792 q^{47} + 39928470 q^{49} - 25237238 q^{54} + 33455160 q^{55} - 8487050 q^{57} - 32278162 q^{58} + 47618848 q^{61} - 3995844 q^{62} - 96967460 q^{63} + 27278246 q^{64} + 91456852 q^{66} + 46476566 q^{68} - 104942458 q^{73} - 168372588 q^{74} - 109094088 q^{76} + 98262712 q^{77} + 191154560 q^{80} - 119895880 q^{81} - 17376524 q^{82} + 207713396 q^{83} - 262139212 q^{85} + 27034246 q^{87} - 19699030 q^{92} - 10067084 q^{93} + 251455412 q^{95} + 263607090 q^{96} - 387974044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2323 x^{10} + 2010462 x^{8} + 803113072 x^{6} + 150633270400 x^{4} + 12173735396352 x^{2} + 333034797957120 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 387 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4753 \nu^{11} - 10617499 \nu^{9} - 8709865350 \nu^{7} - 3171751273600 \nu^{5} + \cdots - 22\!\cdots\!92 \nu ) / 203073398341632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4753 \nu^{11} - 10617499 \nu^{9} - 8709865350 \nu^{7} - 3171751273600 \nu^{5} + \cdots - 12\!\cdots\!52 \nu ) / 16922783195136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 71905 \nu^{10} - 125266491 \nu^{8} - 73375143414 \nu^{6} - 16129766387488 \nu^{4} + \cdots + 20\!\cdots\!04 ) / 39486494121984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63047 \nu^{10} + 135576189 \nu^{8} + 104562087674 \nu^{6} + 34800796354272 \nu^{4} + \cdots + 20\!\cdots\!84 ) / 26324329414656 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 70037 \nu^{10} - 161713783 \nu^{8} - 139314142126 \nu^{6} - 54008239306528 \nu^{4} + \cdots - 40\!\cdots\!48 ) / 26324329414656 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 559673 \nu^{10} + 1282273923 \nu^{8} + 1042569320070 \nu^{6} + 352605624798752 \nu^{4} + \cdots + 13\!\cdots\!28 ) / 157945976487936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1768811 \nu^{11} + 3809202089 \nu^{9} + 2945114482098 \nu^{7} + 984026019395744 \nu^{5} + \cdots + 63\!\cdots\!16 \nu ) / 710756894195712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 668329 \nu^{11} - 1657746643 \nu^{9} - 1476312847782 \nu^{7} + \cdots - 45\!\cdots\!84 \nu ) / 236918964731904 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1702829 \nu^{11} - 3804944339 \nu^{9} - 3010498199994 \nu^{7} + \cdots - 31\!\cdots\!52 \nu ) / 355378447097856 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 387 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 12\beta_{3} - 590\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{8} + 4\beta_{7} + 38\beta_{6} + 13\beta_{5} - 853\beta_{2} + 229101 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{11} - 13\beta_{10} + 45\beta_{9} - 958\beta_{4} + 14572\beta_{3} + 399465\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17744\beta_{8} - 8972\beta_{7} - 49138\beta_{6} - 16991\beta_{5} + 655057\beta_{2} - 155586669 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -17744\beta_{11} + 16991\beta_{10} - 58863\beta_{9} + 775924\beta_{4} - 13985084\beta_{3} - 284876239\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14888208 \beta_{8} + 10810476 \beta_{7} + 47314818 \beta_{6} + 17464071 \beta_{5} + \cdots + 111232060257 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14888208 \beta_{11} - 17464071 \beta_{10} + 55549431 \beta_{9} - 600032776 \beta_{4} + \cdots + 208168052015 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11419251664 \beta_{8} - 10574892892 \beta_{7} - 40809186410 \beta_{6} - 16551273043 \beta_{5} + \cdots - 81433677024861 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 11419251664 \beta_{11} + 16551273043 \beta_{10} - 46252057923 \beta_{9} + \cdots - 154087490021739 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
− 27.7628i
− 26.9428i
− 20.7129i
− 15.4179i
− 10.1391i
− 7.53484i
7.53484i
10.1391i
15.4179i
20.7129i
26.9428i
27.7628i
− 27.7628i 123.321i −514.773 138.449 3423.74 3722.26 7184.27i −8647.06 − 3843.72i
18.2 − 26.9428i − 41.9384i −469.915 132.825 −1129.94 −3940.42 5763.46i 4802.17 − 3578.69i
18.3 − 20.7129i − 108.183i −173.026 −1019.95 −2240.79 4044.21 − 1718.63i −5142.56 21126.1i
18.4 − 15.4179i 21.8898i 18.2884 461.178 337.494 960.296 − 4228.95i 6081.84 − 7110.39i
18.5 − 10.1391i 110.688i 153.199 −737.472 1122.28 −2806.13 − 4148.90i −5690.92 7477.27i
18.6 − 7.53484i − 135.342i 199.226 1028.97 −1019.78 −137.220 − 3430.06i −11756.5 − 7753.10i
18.7 7.53484i 135.342i 199.226 1028.97 −1019.78 −137.220 3430.06i −11756.5 7753.10i
18.8 10.1391i − 110.688i 153.199 −737.472 1122.28 −2806.13 4148.90i −5690.92 − 7477.27i
18.9 15.4179i − 21.8898i 18.2884 461.178 337.494 960.296 4228.95i 6081.84 7110.39i
18.10 20.7129i 108.183i −173.026 −1019.95 −2240.79 4044.21 1718.63i −5142.56 − 21126.1i
18.11 26.9428i 41.9384i −469.915 132.825 −1129.94 −3940.42 − 5763.46i 4802.17 3578.69i
18.12 27.7628i − 123.321i −514.773 138.449 3423.74 3722.26 − 7184.27i −8647.06 3843.72i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.9.b.b ✓ 12
3.b odd 2 1 171.9.c.c 12
4.b odd 2 1 304.9.e.d 12
19.b odd 2 1 inner 19.9.b.b ✓ 12
57.d even 2 1 171.9.c.c 12
76.d even 2 1 304.9.e.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.9.b.b ✓ 12 1.a even 1 1 trivial
19.9.b.b ✓ 12 19.b odd 2 1 inner
171.9.c.c 12 3.b odd 2 1
171.9.c.c 12 57.d even 2 1
304.9.e.d 12 4.b odd 2 1
304.9.e.d 12 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 2323 T_{2}^{10} + 2010462 T_{2}^{8} + 803113072 T_{2}^{6} + 150633270400 T_{2}^{4} + \cdots + 333034797957120 \) acting on \(S_{9}^{\mathrm{new}}(19, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 333034797957120 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 33\!\cdots\!20 \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 21\!\cdots\!50)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 97\!\cdots\!60)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 29\!\cdots\!50)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 54\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 77\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 59\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 33\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
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