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Properties

Label 10.19.c.a
Level $10$
Weight $19$
Character orbit 10.c
Analytic conductor $20.539$
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] = [10,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(20.5386137710\)
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: \( x^{8} + 1012730x^{6} + 266454825649x^{4} + 6118030786524900x^{2} + 24596382356676000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4}\cdot 5^{12} \)
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.

\(f(q)\) \(=\) \( q + ( - 256 \beta_1 + 256) q^{2} + ( - \beta_{2} + 4628 \beta_1 + 4628) q^{3} - 131072 \beta_1 q^{4} + ( - \beta_{7} - 2 \beta_{4} + \cdots - 391556) q^{5}+ \cdots + (141 \beta_{7} - 235 \beta_{6} + \cdots - 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 256 \beta_1 + 256) q^{2} + ( - \beta_{2} + 4628 \beta_1 + 4628) q^{3} - 131072 \beta_1 q^{4} + ( - \beta_{7} - 2 \beta_{4} + \cdots - 391556) q^{5}+ \cdots + ( - 62290886181 \beta_{7} + \cdots + 20763628727) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2048 q^{2} + 37026 q^{3} - 3132450 q^{5} + 18957312 q^{6} - 43579766 q^{7} - 268435456 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2048 q^{2} + 37026 q^{3} - 3132450 q^{5} + 18957312 q^{6} - 43579766 q^{7} - 268435456 q^{8} - 361676800 q^{10} - 2678523284 q^{11} + 4853071872 q^{12} - 13618988004 q^{13} - 44901376950 q^{15} - 137438953472 q^{16} + 131385428404 q^{17} + 219304891392 q^{18} + 225397964800 q^{20} - 2519443347204 q^{21} - 685701960704 q^{22} + 2992780401346 q^{23} + 5025487502500 q^{25} - 6972921858048 q^{26} - 36827718166440 q^{27} + 5712087089152 q^{28} - 6097905484800 q^{30} - 41290623629644 q^{31} - 35184372088832 q^{32} + 3721928304252 q^{33} + 259024770537850 q^{35} + 112284104392704 q^{36} - 191725492483296 q^{37} + 93053262223360 q^{38} + 162809459507200 q^{40} + 538572935041756 q^{41} - 644977496884224 q^{42} + 278176126066386 q^{43} - 649523899006350 q^{45} + 15\!\cdots\!52 q^{46}+ \cdots + 14\!\cdots\!68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1012730x^{6} + 266454825649x^{4} + 6118030786524900x^{2} + 24596382356676000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4010593 \nu^{7} - 4062887717390 \nu^{5} + \cdots - 14\!\cdots\!00 \nu ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!39 \nu^{7} + \cdots + 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!39 \nu^{7} + \cdots + 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!31 \nu^{7} + \cdots + 70\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75\!\cdots\!71 \nu^{7} + \cdots + 62\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83\!\cdots\!49 \nu^{7} + \cdots - 90\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83\!\cdots\!49 \nu^{7} + \cdots - 25\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{7} - 10\beta_{6} + 5\beta_{5} - 17\beta_{4} + 163\beta_{3} - 159\beta_{2} + 14911\beta _1 - 2 ) / 30000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4669 \beta_{7} + 1885 \beta_{6} + 2320 \beta_{5} + 1392 \beta_{4} + 209387 \beta_{3} + \cdots - 7595372873 ) / 30000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3171954 \beta_{7} + 5286590 \beta_{6} - 1599845 \beta_{5} + 7943753 \beta_{4} - 66591967 \beta_{3} + \cdots + 1057318 ) / 30000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2910978331 \beta_{7} - 1116781915 \beta_{6} - 1495163680 \beta_{5} - 897098208 \beta_{4} + \cdots + 36\!\cdots\!27 ) / 30000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1620680781126 \beta_{7} - 2701134635210 \beta_{6} + 845460110855 \beta_{5} - 4086821673107 \beta_{4} + \cdots - 540226927042 ) / 30000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\!\cdots\!49 \beta_{7} + 607408334005585 \beta_{6} + 915160071472720 \beta_{5} + \cdots - 18\!\cdots\!33 ) / 30000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 83\!\cdots\!34 \beta_{7} + \cdots + 27\!\cdots\!78 ) / 30000 \) 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
− 71.7997i
140.480i
− 735.343i
668.663i
71.7997i
− 140.480i
735.343i
− 668.663i
256.000 − 256.000i −13138.9 − 13138.9i − 131072.i −1.63647e6 − 1.06614e6i −6.72711e6 −1.89897e7 + 1.89897e7i −3.35544e7 − 3.35544e7i − 4.21603e7i −6.91869e8 + 1.46007e8i
3.2 256.000 − 256.000i −5080.17 − 5080.17i − 131072.i 82450.2 + 1.95138e6i −2.60105e6 2.79429e7 − 2.79429e7i −3.35544e7 − 3.35544e7i − 3.35804e8i 5.20662e8 + 4.78447e8i
3.3 256.000 − 256.000i 10752.0 + 10752.0i − 131072.i 1.75470e6 − 857742.i 5.50503e6 −4.02524e6 + 4.02524e6i −3.35544e7 − 3.35544e7i − 1.56209e8i 2.29622e8 − 6.68785e8i
3.4 256.000 − 256.000i 25980.0 + 25980.0i − 131072.i −1.76690e6 + 832319.i 1.33018e7 −2.67178e7 + 2.67178e7i −3.35544e7 − 3.35544e7i 9.62503e8i −2.39253e8 + 6.65401e8i
7.1 256.000 + 256.000i −13138.9 + 13138.9i 131072.i −1.63647e6 + 1.06614e6i −6.72711e6 −1.89897e7 − 1.89897e7i −3.35544e7 + 3.35544e7i 4.21603e7i −6.91869e8 − 1.46007e8i
7.2 256.000 + 256.000i −5080.17 + 5080.17i 131072.i 82450.2 − 1.95138e6i −2.60105e6 2.79429e7 + 2.79429e7i −3.35544e7 + 3.35544e7i 3.35804e8i 5.20662e8 − 4.78447e8i
7.3 256.000 + 256.000i 10752.0 − 10752.0i 131072.i 1.75470e6 + 857742.i 5.50503e6 −4.02524e6 − 4.02524e6i −3.35544e7 + 3.35544e7i 1.56209e8i 2.29622e8 + 6.68785e8i
7.4 256.000 + 256.000i 25980.0 − 25980.0i 131072.i −1.76690e6 − 832319.i 1.33018e7 −2.67178e7 − 2.67178e7i −3.35544e7 + 3.35544e7i − 9.62503e8i −2.39253e8 − 6.65401e8i
\(n\): 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 10.19.c.a ✓ 8
3.b odd 2 1 90.19.g.a 8
5.b even 2 1 50.19.c.c 8
5.c odd 4 1 inner 10.19.c.a ✓ 8
5.c odd 4 1 50.19.c.c 8
15.e even 4 1 90.19.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.19.c.a ✓ 8 1.a even 1 1 trivial
10.19.c.a ✓ 8 5.c odd 4 1 inner
50.19.c.c 8 5.b even 2 1
50.19.c.c 8 5.c odd 4 1
90.19.g.a 8 3.b odd 2 1
90.19.g.a 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 37026 T_{3}^{7} + 685462338 T_{3}^{6} + 13379017230360 T_{3}^{5} + \cdots + 55\!\cdots\!76 \) acting on \(S_{19}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 512 T + 131072)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 63\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 56\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
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