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Properties

Label 19.7.b.b
Level $19$
Weight $7$
Character orbit 19.b
Analytic conductor $4.371$
Analytic rank $0$
Dimension $8$
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] = [19,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.18");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(4.37102758878\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 29 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{3} - 57) q^{4} + ( - \beta_{6} + 14) q^{5} + (\beta_{6} + 7 \beta_{4} + 2 \beta_{3} - 48) q^{6} + ( - \beta_{6} + 2 \beta_{4} + \cdots - 16) q^{7}+ \cdots + (\beta_{6} + 8 \beta_{4} - 134) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{3} - 57) q^{4} + ( - \beta_{6} + 14) q^{5} + (\beta_{6} + 7 \beta_{4} + 2 \beta_{3} - 48) q^{6} + ( - \beta_{6} + 2 \beta_{4} + \cdots - 16) q^{7}+ \cdots + ( - 1342 \beta_{6} + 286 \beta_{4} + \cdots + 68231) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9} - 2024 q^{11} + 11546 q^{16} + 6008 q^{17} + 20552 q^{19} - 10732 q^{20} - 50252 q^{23} + 64310 q^{24} + 78492 q^{25} - 37522 q^{26} - 135818 q^{28} - 187696 q^{30} + 210800 q^{35} + 35052 q^{36} + 103318 q^{38} + 43724 q^{39} - 429970 q^{42} + 260800 q^{43} + 693512 q^{44} - 191012 q^{45} - 100248 q^{47} - 301872 q^{49} + 390202 q^{54} - 52480 q^{55} - 186860 q^{57} - 405186 q^{58} - 54548 q^{61} - 1461908 q^{62} - 137408 q^{63} - 858058 q^{64} + 1539556 q^{66} + 1243910 q^{68} + 479968 q^{73} + 2645844 q^{74} - 2569288 q^{76} - 1755300 q^{77} + 2344672 q^{80} - 4279648 q^{81} + 1847172 q^{82} + 483040 q^{83} + 2111780 q^{85} + 2802652 q^{87} + 3905498 q^{92} + 1507528 q^{93} - 2383888 q^{95} - 8462238 q^{96} + 528224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} - 5377\nu^{5} - 753578\nu^{3} - 25301016\nu ) / 1200192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{6} - 5377\nu^{4} - 603554\nu^{2} - 6097944 ) / 150024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{6} - 5377\nu^{4} - 753578\nu^{2} - 24250848 ) / 150024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} + 5377\nu^{5} + 925034\nu^{3} + 53934168\nu ) / 171456 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 107\nu^{6} + 38665\nu^{4} + 3811514\nu^{2} + 91462416 ) / 150024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1153\nu^{7} + 454499\nu^{5} + 50838718\nu^{3} + 1407625608\nu ) / 1200192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 121 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 7\beta_{2} - 167\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{6} + 151\beta_{4} - 258\beta_{3} + 20628 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{7} - 258\beta_{5} - 2959\beta_{2} + 31681\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5377\beta_{6} - 18943\beta_{4} + 57608\beta_{3} - 3998606 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5377\beta_{7} + 57608\beta_{5} + 857755\beta_{2} - 6345657\beta_1 \) 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
− 14.9269i
− 12.8592i
− 8.08057i
− 5.43437i
5.43437i
8.08057i
12.8592i
14.9269i
− 14.9269i − 33.0827i −158.814 159.622 −493.824 445.794 1415.28i −365.466 − 2382.67i
18.2 − 12.8592i 21.6433i −101.358 −216.848 278.315 −134.238 480.398i 260.567 2788.49i
18.3 − 8.08057i 27.2115i −1.29562 162.986 219.884 −68.0674 − 506.687i −11.4646 − 1317.02i
18.4 − 5.43437i − 33.7437i 34.4677 −51.7597 −183.376 −313.488 − 535.109i −409.636 281.281i
18.5 5.43437i 33.7437i 34.4677 −51.7597 −183.376 −313.488 535.109i −409.636 − 281.281i
18.6 8.08057i − 27.2115i −1.29562 162.986 219.884 −68.0674 506.687i −11.4646 1317.02i
18.7 12.8592i − 21.6433i −101.358 −216.848 278.315 −134.238 − 480.398i 260.567 − 2788.49i
18.8 14.9269i 33.0827i −158.814 159.622 −493.824 445.794 − 1415.28i −365.466 2382.67i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.8
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 19.7.b.b ✓ 8
3.b odd 2 1 171.7.c.d 8
4.b odd 2 1 304.7.e.d 8
19.b odd 2 1 inner 19.7.b.b ✓ 8
57.d even 2 1 171.7.c.d 8
76.d even 2 1 304.7.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.b.b ✓ 8 1.a even 1 1 trivial
19.7.b.b ✓ 8 19.b odd 2 1 inner
171.7.c.d 8 3.b odd 2 1
171.7.c.d 8 57.d even 2 1
304.7.e.d 8 4.b odd 2 1
304.7.e.d 8 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 483T_{2}^{6} + 75582T_{2}^{4} + 4242376T_{2}^{2} + 71047680 \) acting on \(S_{7}^{\mathrm{new}}(19, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 483 T^{6} + \cdots + 71047680 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 432254085120 \) Copy content Toggle raw display
$5$ \( (T^{4} - 54 T^{3} + \cdots + 292006000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 70 T^{3} + \cdots - 1276939885)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1012 T^{3} + \cdots + 401057409740)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 21923949558015)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 48\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 73\!\cdots\!80)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 17\!\cdots\!60)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 44\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 86\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 18\!\cdots\!15)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
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