↑

Properties

Label 19.11.b.b
Level $19$
Weight $11$
Character orbit 19.b
Analytic conductor $12.072$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.18");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(12.0717878008\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 11363 x^{12} + 50841694 x^{10} + 114211512472 x^{8} + 135038975967104 x^{6} + \cdots + 76\!\cdots\!60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 599) q^{4} + ( - \beta_{5} - 203) q^{5} + ( - \beta_{7} + \beta_{2} + 393) q^{6} + (\beta_{6} + 3 \beta_{2} - 273) q^{7} + (\beta_{4} + 15 \beta_{3} - 370 \beta_1) q^{8} + (\beta_{11} - \beta_{7} + 3 \beta_{6} + \cdots - 21954) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 599) q^{4} + ( - \beta_{5} - 203) q^{5} + ( - \beta_{7} + \beta_{2} + 393) q^{6} + (\beta_{6} + 3 \beta_{2} - 273) q^{7} + (\beta_{4} + 15 \beta_{3} - 370 \beta_1) q^{8} + (\beta_{11} - \beta_{7} + 3 \beta_{6} + \cdots - 21954) q^{9}+ \cdots + (31694 \beta_{11} - 711 \beta_{8} + \cdots - 3667718625) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 8390 q^{4} - 2842 q^{5} + 5498 q^{6} - 3840 q^{7} - 307376 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 8390 q^{4} - 2842 q^{5} + 5498 q^{6} - 3840 q^{7} - 307376 q^{9} + 54802 q^{11} - 265446 q^{16} + 1219416 q^{17} + 1935436 q^{19} - 1920492 q^{20} - 11407694 q^{23} + 14430710 q^{24} - 20770608 q^{25} - 45069938 q^{26} + 56640854 q^{28} + 6192944 q^{30} - 7409770 q^{35} + 216003852 q^{36} + 238824566 q^{38} - 38751058 q^{39} - 619099570 q^{42} + 490977030 q^{43} - 15835416 q^{44} - 847425602 q^{45} + 219739034 q^{47} - 888067866 q^{49} + 947347354 q^{54} - 909598530 q^{55} + 1119469510 q^{57} + 2190695998 q^{58} - 2669740226 q^{61} + 1557378284 q^{62} + 8287575814 q^{63} - 333661674 q^{64} - 8676134108 q^{66} - 10238770650 q^{68} + 9355918916 q^{73} - 5556423948 q^{74} + 6698753656 q^{76} + 7785218410 q^{77} - 16166859488 q^{80} + 17654768906 q^{81} + 30215736644 q^{82} - 9301023920 q^{83} - 16692607370 q^{85} - 40691381306 q^{87} + 21755500026 q^{92} + 931962196 q^{93} + 24967225902 q^{95} + 1108342530 q^{96} - 51370432882 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11363 x^{12} + 50841694 x^{10} + 114211512472 x^{8} + 135038975967104 x^{6} + \cdots + 76\!\cdots\!60 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1623 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 803506617185 \nu^{13} + \cdots + 36\!\cdots\!68 \nu ) / 23\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4017533085925 \nu^{13} + \cdots + 59\!\cdots\!24 \nu ) / 77\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 820281933571955 \nu^{12} + \cdots - 11\!\cdots\!68 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 100255480970493 \nu^{12} + \cdots + 41\!\cdots\!16 ) / 32\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8743071679027 \nu^{12} + \cdots - 14\!\cdots\!24 ) / 22\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!97 \nu^{12} + \cdots + 86\!\cdots\!64 ) / 44\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!45 \nu^{13} + \cdots + 97\!\cdots\!00 \nu ) / 34\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23317805011399 \nu^{13} + \cdots + 32\!\cdots\!84 \nu ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!75 \nu^{12} + \cdots - 20\!\cdots\!84 ) / 33\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34\!\cdots\!75 \nu^{13} + \cdots + 18\!\cdots\!96 \nu ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!79 \nu^{13} + \cdots + 65\!\cdots\!88 \nu ) / 43\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1623 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 15\beta_{3} - 2418\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} + 9\beta_{8} - 9\beta_{7} + 38\beta_{6} + 156\beta_{5} - 3603\beta_{2} + 3918182 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{13} + 48\beta_{12} - 13\beta_{10} + 272\beta_{9} - 4780\beta_{4} - 60750\beta_{3} + 6760932\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28005 \beta_{11} - 46275 \beta_{8} + 79875 \beta_{7} - 214810 \beta_{6} - 1111848 \beta_{5} + \cdots - 10947579148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 85720 \beta_{13} - 297120 \beta_{12} + 106175 \beta_{10} - 1715128 \beta_{9} + 18849886 \beta_{4} + \cdots - 20622859464 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 152105973 \beta_{11} + 190146651 \beta_{8} - 478486491 \beta_{7} + 962066202 \beta_{6} + \cdots + 33398952572376 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 505691544 \beta_{13} + 1369010496 \beta_{12} - 581218839 \beta_{10} + 7785377352 \beta_{9} + \cdots + 66476277174480 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 686282572401 \beta_{11} - 728614893807 \beta_{8} + 2345777793519 \beta_{7} - 3920342139410 \beta_{6} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2378447708984 \beta_{13} - 5659589864832 \beta_{12} + 2712460374139 \beta_{10} + \cdots - 22\!\cdots\!76 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28\!\cdots\!33 \beta_{11} + \cdots + 36\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10\!\cdots\!56 \beta_{13} + \cdots + 76\!\cdots\!44 \beta_1 \) 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
− 59.9705i
− 50.5270i
− 44.5331i
− 40.7793i
− 30.9467i
− 24.6038i
− 2.08969i
2.08969i
24.6038i
30.9467i
40.7793i
44.5331i
50.5270i
59.9705i
− 59.9705i 62.9916i −2572.47 −2208.58 3777.64 2925.08 92862.4i 55081.1 132450.i
18.2 − 50.5270i − 430.788i −1528.98 770.370 −21766.4 −3278.66 25514.9i −126530. − 38924.5i
18.3 − 44.5331i 437.865i −959.198 2214.55 19499.5 −27438.5 − 2885.84i −132677. − 98620.8i
18.4 − 40.7793i − 20.8695i −638.953 4274.21 −851.043 12071.6 − 15701.9i 58613.5 − 174300.i
18.5 − 30.9467i 208.551i 66.3041 −4302.72 6453.95 10914.2 − 33741.3i 15555.6 133155.i
18.6 − 24.6038i − 204.268i 418.655 −3141.64 −5025.77 −14242.9 − 35494.7i 17323.4 77296.2i
18.7 − 2.08969i 316.392i 1019.63 972.806 661.160 17129.2 − 4270.55i −41054.9 − 2032.86i
18.8 2.08969i − 316.392i 1019.63 972.806 661.160 17129.2 4270.55i −41054.9 2032.86i
18.9 24.6038i 204.268i 418.655 −3141.64 −5025.77 −14242.9 35494.7i 17323.4 − 77296.2i
18.10 30.9467i − 208.551i 66.3041 −4302.72 6453.95 10914.2 33741.3i 15555.6 − 133155.i
18.11 40.7793i 20.8695i −638.953 4274.21 −851.043 12071.6 15701.9i 58613.5 174300.i
18.12 44.5331i − 437.865i −959.198 2214.55 19499.5 −27438.5 2885.84i −132677. 98620.8i
18.13 50.5270i 430.788i −1528.98 770.370 −21766.4 −3278.66 − 25514.9i −126530. 38924.5i
18.14 59.9705i − 62.9916i −2572.47 −2208.58 3777.64 2925.08 − 92862.4i 55081.1 − 132450.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.14
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 19.11.b.b ✓ 14
3.b odd 2 1 171.11.c.c 14
4.b odd 2 1 304.11.e.c 14
19.b odd 2 1 inner 19.11.b.b ✓ 14
57.d even 2 1 171.11.c.c 14
76.d even 2 1 304.11.e.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.11.b.b ✓ 14 1.a even 1 1 trivial
19.11.b.b ✓ 14 19.b odd 2 1 inner
171.11.c.c 14 3.b odd 2 1
171.11.c.c 14 57.d even 2 1
304.11.e.c 14 4.b odd 2 1
304.11.e.c 14 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 11363 T_{2}^{12} + 50841694 T_{2}^{10} + 114211512472 T_{2}^{8} + 135038975967104 T_{2}^{6} + \cdots + 76\!\cdots\!60 \) acting on \(S_{11}^{\mathrm{new}}(19, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 76\!\cdots\!60 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$5$ \( (T^{7} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 84\!\cdots\!50)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 27\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 38\!\cdots\!50)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 22\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 28\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 15\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 95\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
show more
show less