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Properties

Label 10.19.c.b
Level $10$
Weight $19$
Character orbit 10.c
Analytic conductor $20.539$
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] = [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: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 15200484 x^{8} + 52963214026118 x^{6} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{17} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 256 \beta_1 - 256) q^{2} + ( - \beta_{3} - 323 \beta_1 + 323) q^{3} + 131072 \beta_1 q^{4} + (\beta_{5} - 7 \beta_{3} + \cdots - 143613) q^{5}+ \cdots + (11 \beta_{8} - 4 \beta_{7} + \cdots - 122787829 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 256 \beta_1 - 256) q^{2} + ( - \beta_{3} - 323 \beta_1 + 323) q^{3} + 131072 \beta_1 q^{4} + (\beta_{5} - 7 \beta_{3} + \cdots - 143613) q^{5}+ \cdots + ( - 24518990871 \beta_{9} + \cdots + 64\!\cdots\!62 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2560 q^{2} + 3230 q^{3} - 1436130 q^{5} - 1653760 q^{6} - 71828530 q^{7} + 335544320 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2560 q^{2} + 3230 q^{3} - 1436130 q^{5} - 1653760 q^{6} - 71828530 q^{7} + 335544320 q^{8} + 280000000 q^{10} - 4684617380 q^{11} + 423362560 q^{12} - 11858528670 q^{13} + 132580211870 q^{15} - 171798691840 q^{16} + 57425096770 q^{17} - 314336842240 q^{18} + 44876431360 q^{20} + 2599949895020 q^{21} + 1199262049280 q^{22} - 2409849379970 q^{23} - 9628701268750 q^{25} + 6071566679040 q^{26} + 10319468097200 q^{27} + 9414709084160 q^{28} - 42920067333120 q^{30} + 44076891672220 q^{31} + 43980465111040 q^{32} - 91028207725540 q^{33} - 92680242801650 q^{35} + 160940463226880 q^{36} + 285126548071770 q^{37} - 268015571763200 q^{38} - 59676892856320 q^{40} + 223001245455820 q^{41} - 665587173125120 q^{42} - 674098663802370 q^{43} + 13\!\cdots\!80 q^{45}+ \cdots + 14\!\cdots\!40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 15200484 x^{8} + 52963214026118 x^{6} + \cdots + 20\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 42\!\cdots\!37 \nu^{9} + \cdots + 18\!\cdots\!32 \nu ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!23 \nu^{9} + \cdots - 12\!\cdots\!68 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\!\cdots\!23 \nu^{9} + \cdots - 12\!\cdots\!68 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!79 \nu^{9} + \cdots - 28\!\cdots\!52 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!13 \nu^{9} + \cdots - 46\!\cdots\!16 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!20 \nu^{9} + \cdots - 20\!\cdots\!20 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!01 \nu^{9} + \cdots - 62\!\cdots\!12 ) / 50\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 79\!\cdots\!13 \nu^{9} + \cdots - 32\!\cdots\!28 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!97 \nu^{9} + \cdots + 76\!\cdots\!32 ) / 50\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{7} - 2\beta_{6} + 28\beta_{5} - 18\beta_{4} + 4087\beta_{3} - 4087\beta_{2} + 36000\beta_1 ) / 90000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6160 \beta_{9} - 1417 \beta_{8} - 9856 \beta_{7} + 2353 \beta_{6} + 17714 \beta_{5} + \cdots - 273608712000 ) / 90000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2413097 \beta_{9} - 4309000 \beta_{8} + 6891097 \beta_{7} + 22907194 \beta_{6} + \cdots + 444372046668000 \beta_1 ) / 90000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9531833704 \beta_{9} + 826991637 \beta_{8} + 18469220712 \beta_{7} - 5104015901 \beta_{6} + \cdots + 45\!\cdots\!00 ) / 18000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9082901147321 \beta_{9} + 83439080564240 \beta_{8} - 74138523758601 \beta_{7} + \cdots - 56\!\cdots\!00 \beta_1 ) / 90000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 41\!\cdots\!40 \beta_{9} + \cdots - 21\!\cdots\!00 ) / 90000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 40\!\cdots\!97 \beta_{9} + \cdots + 62\!\cdots\!00 \beta_1 ) / 90000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 78\!\cdots\!08 \beta_{9} + \cdots + 42\!\cdots\!00 ) / 18000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 20\!\cdots\!41 \beta_{9} + \cdots - 66\!\cdots\!00 \beta_1 ) / 90000 \) 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
3223.13i
− 369.775i
− 749.210i
− 79.4383i
− 2026.71i
− 3223.13i
369.775i
749.210i
79.4383i
2026.71i
−256.000 + 256.000i −22485.0 − 22485.0i − 131072.i −1.41140e6 + 1.35005e6i 1.15123e7 −5.59623e7 + 5.59623e7i 3.35544e7 + 3.35544e7i 6.23730e8i 1.57043e7 − 7.06932e8i
3.2 −256.000 + 256.000i −12815.7 − 12815.7i − 131072.i −120054. − 1.94943e6i 6.56166e6 2.37564e7 − 2.37564e7i 3.35544e7 + 3.35544e7i − 5.89343e7i 5.29788e8 + 4.68321e8i
3.3 −256.000 + 256.000i 4473.85 + 4473.85i − 131072.i 1.47317e6 + 1.28237e6i −2.29061e6 −6.84961e6 + 6.84961e6i 3.35544e7 + 3.35544e7i − 3.47390e8i −7.05418e8 + 4.88445e7i
3.4 −256.000 + 256.000i 10768.9 + 10768.9i − 131072.i −1.49934e6 + 1.25167e6i −5.51369e6 3.13009e7 − 3.13009e7i 3.35544e7 + 3.35544e7i − 1.55481e8i 6.34033e7 − 7.04258e8i
3.5 −256.000 + 256.000i 21673.0 + 21673.0i − 131072.i 839560. − 1.76347e6i −1.10966e7 −2.81596e7 + 2.81596e7i 3.35544e7 + 3.35544e7i 5.52014e8i 2.36522e8 + 6.66376e8i
7.1 −256.000 − 256.000i −22485.0 + 22485.0i 131072.i −1.41140e6 − 1.35005e6i 1.15123e7 −5.59623e7 − 5.59623e7i 3.35544e7 − 3.35544e7i − 6.23730e8i 1.57043e7 + 7.06932e8i
7.2 −256.000 − 256.000i −12815.7 + 12815.7i 131072.i −120054. + 1.94943e6i 6.56166e6 2.37564e7 + 2.37564e7i 3.35544e7 − 3.35544e7i 5.89343e7i 5.29788e8 − 4.68321e8i
7.3 −256.000 − 256.000i 4473.85 − 4473.85i 131072.i 1.47317e6 − 1.28237e6i −2.29061e6 −6.84961e6 − 6.84961e6i 3.35544e7 − 3.35544e7i 3.47390e8i −7.05418e8 − 4.88445e7i
7.4 −256.000 − 256.000i 10768.9 − 10768.9i 131072.i −1.49934e6 − 1.25167e6i −5.51369e6 3.13009e7 + 3.13009e7i 3.35544e7 − 3.35544e7i 1.55481e8i 6.34033e7 + 7.04258e8i
7.5 −256.000 − 256.000i 21673.0 − 21673.0i 131072.i 839560. + 1.76347e6i −1.10966e7 −2.81596e7 − 2.81596e7i 3.35544e7 − 3.35544e7i − 5.52014e8i 2.36522e8 − 6.66376e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
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.b ✓ 10
3.b odd 2 1 90.19.g.b 10
5.b even 2 1 50.19.c.d 10
5.c odd 4 1 inner 10.19.c.b ✓ 10
5.c odd 4 1 50.19.c.d 10
15.e even 4 1 90.19.g.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.19.c.b ✓ 10 1.a even 1 1 trivial
10.19.c.b ✓ 10 5.c odd 4 1 inner
50.19.c.d 10 5.b even 2 1
50.19.c.d 10 5.c odd 4 1
90.19.g.b 10 3.b odd 2 1
90.19.g.b 10 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 3230 T_{3}^{9} + 5216450 T_{3}^{8} - 2611193623920 T_{3}^{7} + \cdots + 28\!\cdots\!68 \) acting on \(S_{19}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 512 T + 131072)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 28\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 80\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 99\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 26\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 67\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 64\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 98\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 80\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 72\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 85\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 42\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 38\!\cdots\!68 \) Copy content Toggle raw display
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