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Properties

Label 7.19.b.b
Level $7$
Weight $19$
Character orbit 7.b
Analytic conductor $14.377$
Analytic rank $0$
Dimension $10$
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] = [7,19,Mod(6,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.6");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(14.3770296397\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 635494794 x^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{7}\cdot 5^{2}\cdot 7^{9} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 91) q^{2} + \beta_1 q^{3} + ( - \beta_{4} + 211 \beta_{2} + 65935) q^{4} + ( - \beta_{6} - 20 \beta_1) q^{5} + ( - \beta_{3} - 33 \beta_1) q^{6} + ( - \beta_{7} - 4 \beta_{6} + \cdots - 619419) q^{7}+ \cdots + ( - 4 \beta_{8} - 8 \beta_{7} + \cdots - 121014849) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 91) q^{2} + \beta_1 q^{3} + ( - \beta_{4} + 211 \beta_{2} + 65935) q^{4} + ( - \beta_{6} - 20 \beta_1) q^{5} + ( - \beta_{3} - 33 \beta_1) q^{6} + ( - \beta_{7} - 4 \beta_{6} + \cdots - 619419) q^{7}+ \cdots + ( - 413063347 \beta_{8} + \cdots + 27\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 912 q^{2} + 659776 q^{4} - 6200670 q^{7} + 496443648 q^{8} - 1209753462 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 912 q^{2} + 659776 q^{4} - 6200670 q^{7} + 496443648 q^{8} - 1209753462 q^{9} + 2811704772 q^{11} - 11011554288 q^{14} + 102088894800 q^{15} - 275266919424 q^{16} + 519186632208 q^{18} - 952214871216 q^{21} - 186167663968 q^{22} + 5931177144468 q^{23} - 14263974090710 q^{25} - 2339965496512 q^{28} + 43421110314756 q^{29} - 16668439687680 q^{30} - 19934498205696 q^{32} - 198891622589520 q^{35} + 211495869502272 q^{36} + 83444149554852 q^{37} - 15026878250832 q^{39} - 118016981967360 q^{42} - 800344062354012 q^{43} + 15\!\cdots\!52 q^{44}+ \cdots + 27\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 635494794 x^{8} + \cdots + 45\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!39 \nu^{8} + \cdots + 73\!\cdots\!50 ) / 75\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!39 \nu^{9} + \cdots - 82\!\cdots\!50 \nu ) / 37\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{8} + \cdots - 37\!\cdots\!00 ) / 75\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 71\!\cdots\!97 \nu^{8} + \cdots - 63\!\cdots\!25 ) / 37\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74\!\cdots\!44 \nu^{9} + \cdots + 70\!\cdots\!50 \nu ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39\!\cdots\!92 \nu^{9} + \cdots - 51\!\cdots\!50 ) / 33\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 71\!\cdots\!00 \nu^{9} + \cdots - 32\!\cdots\!00 ) / 27\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!04 \nu^{9} + \cdots - 23\!\cdots\!00 ) / 60\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{8} - 8\beta_{7} + 4\beta_{6} - 41\beta_{5} - 387\beta_{4} + 196765\beta_{2} + 4\beta _1 - 508435338 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3192 \beta_{9} - 12141 \beta_{8} + 158460 \beta_{7} - 1736871 \beta_{6} - 39615 \beta_{5} + \cdots + 51756 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1255545627 \beta_{8} + 2511091254 \beta_{7} - 1255545627 \beta_{6} + 2764556166 \beta_{5} + \cdots + 12\!\cdots\!26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1098863262108 \beta_{9} + 3429138230469 \beta_{8} - 45545111814060 \beta_{7} + 648467689485069 \beta_{6} + \cdots - 14815416183984 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 18\!\cdots\!99 \beta_{8} + \cdots - 17\!\cdots\!52 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 16\!\cdots\!86 \beta_{9} + \cdots + 20\!\cdots\!78 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 25\!\cdots\!58 \beta_{8} + \cdots + 23\!\cdots\!09 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24\!\cdots\!37 \beta_{9} + \cdots - 27\!\cdots\!01 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
− 7531.39i
7531.39i
− 17226.8i
17226.8i
− 4894.95i
4894.95i
− 15925.5i
15925.5i
− 2103.77i
2103.77i
−600.360 − 15062.8i 98288.5 − 2.47910e6i 9.04310e6i 3.69552e7 − 1.62088e7i 9.83723e7 1.60533e8 1.48835e9i
6.2 −600.360 15062.8i 98288.5 2.47910e6i − 9.04310e6i 3.69552e7 + 1.62088e7i 9.83723e7 1.60533e8 − 1.48835e9i
6.3 −466.518 − 34453.7i −44505.2 2.54521e6i 1.60732e7i −4.03019e7 − 2.04125e6i 1.43057e8 −7.99634e8 − 1.18739e9i
6.4 −466.518 34453.7i −44505.2 − 2.54521e6i − 1.60732e7i −4.03019e7 + 2.04125e6i 1.43057e8 −7.99634e8 1.18739e9i
6.5 98.2563 − 9789.91i −252490. − 1.22790e6i − 961920.i −9.46776e6 + 3.92272e7i −5.05660e7 2.91578e8 − 1.20649e8i
6.6 98.2563 9789.91i −252490. 1.22790e6i 961920.i −9.46776e6 − 3.92272e7i −5.05660e7 2.91578e8 1.20649e8i
6.7 574.273 − 31851.1i 67645.7 − 59718.3i − 1.82912e7i 3.42335e7 − 2.13655e7i −1.11695e8 −6.27071e8 − 3.42946e7i
6.8 574.273 31851.1i 67645.7 59718.3i 1.82912e7i 3.42335e7 + 2.13655e7i −1.11695e8 −6.27071e8 3.42946e7i
6.9 850.349 − 4207.54i 460949. 3.47421e6i − 3.57787e6i −2.45193e7 + 3.20502e7i 1.69053e8 3.69717e8 2.95429e9i
6.10 850.349 4207.54i 460949. − 3.47421e6i 3.57787e6i −2.45193e7 − 3.20502e7i 1.69053e8 3.69717e8 − 2.95429e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.19.b.b ✓ 10
3.b odd 2 1 63.19.d.d 10
7.b odd 2 1 inner 7.19.b.b ✓ 10
21.c even 2 1 63.19.d.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.19.b.b ✓ 10 1.a even 1 1 trivial
7.19.b.b ✓ 10 7.b odd 2 1 inner
63.19.d.d 10 3.b odd 2 1
63.19.d.d 10 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 456T_{2}^{4} - 716336T_{2}^{3} + 195823104T_{2}^{2} + 124785737728T_{2} - 13438656184320 \) acting on \(S_{19}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + \cdots - 13438656184320)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 53\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 76\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 61\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 60\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 44\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 14\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 18\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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