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Properties

Label 10.17.c.a
Level $10$
Weight $17$
Character orbit 10.c
Analytic conductor $16.232$
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] = [10,17,Mod(3,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), 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: \(16.2324543857\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7213550 x^{5} + 3043721913 x^{4} - 386278388950 x^{3} + 26017651801250 x^{2} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{8}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (128 \beta_1 - 128) q^{2} + ( - \beta_{3} - 673 \beta_1 - 673) q^{3} - 32768 \beta_1 q^{4} + ( - \beta_{6} + 4 \beta_{3} + \cdots + 23107) q^{5}+ \cdots + (66 \beta_{7} - 74 \beta_{6} + \cdots - 37) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (128 \beta_1 - 128) q^{2} + ( - \beta_{3} - 673 \beta_1 - 673) q^{3} - 32768 \beta_1 q^{4} + ( - \beta_{6} + 4 \beta_{3} + \cdots + 23107) q^{5}+ \cdots + (7638028233 \beta_{7} + \cdots - 2294793152) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1024 q^{2} - 5382 q^{3} + 184830 q^{5} + 1377792 q^{6} - 1586702 q^{7} + 33554432 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1024 q^{2} - 5382 q^{3} + 184830 q^{5} + 1377792 q^{6} - 1586702 q^{7} + 33554432 q^{8} - 179537920 q^{10} + 307957276 q^{11} - 176357376 q^{12} + 202095228 q^{13} - 4140218430 q^{15} - 8589934592 q^{16} - 10825054172 q^{17} - 19318270464 q^{18} + 39905198080 q^{20} - 279317583684 q^{21} - 39418531328 q^{22} - 58166716742 q^{23} + 48765928900 q^{25} - 51736378368 q^{26} - 433782808920 q^{27} + 51993051136 q^{28} + 881686264320 q^{30} - 1503757815484 q^{31} + 1099511627776 q^{32} + 3563163295596 q^{33} + 1361150225890 q^{35} + 4945477238784 q^{36} + 5719558248048 q^{37} + 10471905756160 q^{38} - 4332632145920 q^{40} - 21624661426724 q^{41} + 35752650711552 q^{42} + 694778360778 q^{43} - 125868018043710 q^{45} + 14890679485952 q^{46} + 17454156046938 q^{47} + 5778878496768 q^{48} - 38811694553600 q^{50} - 282689731949724 q^{51} + 6622256431104 q^{52} + 315933715243808 q^{53} - 170380752243540 q^{55} - 13310221090816 q^{56} + 218915538682320 q^{57} + 231112067379200 q^{58} - 90045006151680 q^{60} - 260671832048484 q^{61} + 192481000381952 q^{62} + 12\!\cdots\!78 q^{63}+ \cdots + 18\!\cdots\!04 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7213550 x^{5} + 3043721913 x^{4} - 386278388950 x^{3} + 26017651801250 x^{2} + \cdots + 77\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!75 \nu^{7} + \cdots - 38\!\cdots\!00 ) / 24\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!17 \nu^{7} + \cdots + 27\!\cdots\!16 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\!\cdots\!75 \nu^{7} + \cdots - 39\!\cdots\!84 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!29 \nu^{7} + \cdots - 18\!\cdots\!12 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!07 \nu^{7} + \cdots - 13\!\cdots\!04 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 66\!\cdots\!15 \nu^{7} + \cdots - 36\!\cdots\!52 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 80\!\cdots\!77 \nu^{7} + \cdots + 16\!\cdots\!96 ) / 30\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{7} + 2\beta_{6} + 6\beta_{5} - 5\beta_{4} + 17\beta_{3} - \beta_{2} + 8\beta _1 + 6 ) / 18000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{7} + 4\beta_{6} - 26\beta_{5} + 19\beta_{4} + 233\beta_{3} - 231\beta_{2} + 9638926\beta _1 + 2 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 312535 \beta_{7} - 361422 \beta_{6} + 165874 \beta_{5} - 244435 \beta_{4} - 48887 \beta_{3} + \cdots + 48691087904 ) / 18000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1738786 \beta_{7} + 1969358 \beta_{6} + 502738 \beta_{5} - 481941 \beta_{4} - 11071380 \beta_{3} + \cdots - 547874610637 ) / 360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5111067865 \beta_{7} - 4342021946 \beta_{6} - 8430876238 \beta_{5} + 7408662665 \beta_{4} + \cdots + 14\!\cdots\!62 ) / 6000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 31512905463 \beta_{7} - 41379522940 \beta_{6} + 145784856806 \beta_{5} - 134961712813 \beta_{4} + \cdots - 20689761470 ) / 360 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 539400346293155 \beta_{7} + 610685248912006 \beta_{6} - 325545638436602 \beta_{5} + \cdots - 11\!\cdots\!92 ) / 6000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
110.950 + 110.950i
15.9615 + 15.9615i
−191.774 − 191.774i
64.8621 + 64.8621i
110.950 − 110.950i
15.9615 − 15.9615i
−191.774 + 191.774i
64.8621 − 64.8621i
−128.000 + 128.000i −7760.96 − 7760.96i − 32768.0i 147024. + 361900.i 1.98681e6 7.60621e6 − 7.60621e6i 4.19430e6 + 4.19430e6i 7.74184e7i −6.51423e7 − 2.75042e7i
3.2 −128.000 + 128.000i −1354.68 − 1354.68i − 32768.0i −371534. − 120625.i 346799. 232276. − 232276.i 4.19430e6 + 4.19430e6i − 3.93764e7i 6.29964e7 − 3.21163e7i
3.3 −128.000 + 128.000i −1321.92 − 1321.92i − 32768.0i 390295. − 16042.9i 338411. −6.21755e6 + 6.21755e6i 4.19430e6 + 4.19430e6i − 3.95518e7i −4.79043e7 + 5.20113e7i
3.4 −128.000 + 128.000i 7746.56 + 7746.56i − 32768.0i −73370.4 + 383673.i −1.98312e6 −2.41428e6 + 2.41428e6i 4.19430e6 + 4.19430e6i 7.69718e7i −3.97187e7 − 5.85015e7i
7.1 −128.000 − 128.000i −7760.96 + 7760.96i 32768.0i 147024. − 361900.i 1.98681e6 7.60621e6 + 7.60621e6i 4.19430e6 − 4.19430e6i − 7.74184e7i −6.51423e7 + 2.75042e7i
7.2 −128.000 − 128.000i −1354.68 + 1354.68i 32768.0i −371534. + 120625.i 346799. 232276. + 232276.i 4.19430e6 − 4.19430e6i 3.93764e7i 6.29964e7 + 3.21163e7i
7.3 −128.000 − 128.000i −1321.92 + 1321.92i 32768.0i 390295. + 16042.9i 338411. −6.21755e6 − 6.21755e6i 4.19430e6 − 4.19430e6i 3.95518e7i −4.79043e7 − 5.20113e7i
7.4 −128.000 − 128.000i 7746.56 − 7746.56i 32768.0i −73370.4 − 383673.i −1.98312e6 −2.41428e6 − 2.41428e6i 4.19430e6 − 4.19430e6i − 7.69718e7i −3.97187e7 + 5.85015e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.17.c.a ✓ 8
3.b odd 2 1 90.17.g.b 8
4.b odd 2 1 80.17.p.a 8
5.b even 2 1 50.17.c.d 8
5.c odd 4 1 inner 10.17.c.a ✓ 8
5.c odd 4 1 50.17.c.d 8
15.e even 4 1 90.17.g.b 8
20.e even 4 1 80.17.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.17.c.a ✓ 8 1.a even 1 1 trivial
10.17.c.a ✓ 8 5.c odd 4 1 inner
50.17.c.d 8 5.b even 2 1
50.17.c.d 8 5.c odd 4 1
80.17.p.a 8 4.b odd 2 1
80.17.p.a 8 20.e even 4 1
90.17.g.b 8 3.b odd 2 1
90.17.g.b 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 5382 T_{3}^{7} + 14482962 T_{3}^{6} + 16125068520 T_{3}^{5} + \cdots + 18\!\cdots\!76 \) acting on \(S_{17}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 256 T + 32768)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 43\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 98\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 46\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
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