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Properties

Label 100.7.f.b
Level $100$
Weight $7$
Character orbit 100.f
Analytic conductor $23.005$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,7,Mod(57,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.57");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 100.f (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: \(23.0054083620\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 450x^{3} + 23409x^{2} - 115668x + 285768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 5 \beta_1 - 5) q^{3} + (\beta_{4} - 6 \beta_{3} + 46 \beta_1 + 46) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - 295 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 5 \beta_1 - 5) q^{3} + (\beta_{4} - 6 \beta_{3} + 46 \beta_1 + 46) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - 295 \beta_1) q^{9}+ \cdots + ( - 1387 \beta_{5} + \cdots - 1063924 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 32 q^{3} + 264 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 32 q^{3} + 264 q^{7} + 2200 q^{11} - 858 q^{13} + 3278 q^{17} + 33176 q^{21} - 19984 q^{23} + 115528 q^{27} + 104976 q^{31} - 177320 q^{33} + 241554 q^{37} + 351736 q^{41} - 60720 q^{43} + 355248 q^{47} + 641872 q^{51} - 346526 q^{53} + 112816 q^{57} - 492888 q^{61} - 2288 q^{63} + 230304 q^{67} + 174128 q^{71} + 332442 q^{73} - 1618760 q^{77} - 3085166 q^{81} + 2190936 q^{83} - 2614304 q^{87} - 2186976 q^{91} - 242072 q^{93} - 3338406 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} - 450x^{3} + 23409x^{2} - 115668x + 285768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 38\nu^{5} + 17\nu^{4} + 25\nu^{3} - 2685\nu^{2} + 868617\nu - 2248722 ) / 2216970 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 325\nu^{5} - 1326\nu^{4} - 18050\nu^{3} - 251030\nu^{2} + 7760025\nu - 37081044 ) / 738990 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1375\nu^{5} - 5338\nu^{4} - 7850\nu^{3} + 843090\nu^{2} - 22977675\nu + 9970128 ) / 2216970 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 341\nu^{5} + 1377\nu^{4} + 2025\nu^{3} - 217485\nu^{2} + 8887694\nu - 2571912 ) / 82110 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10127\nu^{5} - 30073\nu^{4} + 197275\nu^{3} - 6079095\nu^{2} + 239272218\nu - 1163946672 ) / 2216970 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 7\beta_{3} + 11\beta _1 + 11 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} + 5\beta_{3} - 5\beta_{2} + 818\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 143\beta_{5} - 1121\beta_{2} - 9347\beta _1 + 9347 ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -50\beta_{5} + 50\beta_{4} - 115\beta_{3} - 115\beta_{2} - 29938 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -23659\beta_{4} - 157213\beta_{3} + 2377351\beta _1 + 2377351 ) / 40 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
7.73238 + 7.73238i
−9.34732 − 9.34732i
2.61494 + 2.61494i
7.73238 − 7.73238i
−9.34732 + 9.34732i
2.61494 − 2.61494i
0 −35.6344 + 35.6344i 0 0 0 −53.9521 − 53.9521i 0 − 1810.62i 0
57.2 0 2.90948 − 2.90948i 0 0 0 −236.070 − 236.070i 0 712.070i 0
57.3 0 16.7249 − 16.7249i 0 0 0 422.022 + 422.022i 0 169.553i 0
93.1 0 −35.6344 − 35.6344i 0 0 0 −53.9521 + 53.9521i 0 1810.62i 0
93.2 0 2.90948 + 2.90948i 0 0 0 −236.070 + 236.070i 0 − 712.070i 0
93.3 0 16.7249 + 16.7249i 0 0 0 422.022 − 422.022i 0 − 169.553i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.3
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 100.7.f.b 6
5.b even 2 1 20.7.f.a ✓ 6
5.c odd 4 1 20.7.f.a ✓ 6
5.c odd 4 1 inner 100.7.f.b 6
15.d odd 2 1 180.7.l.a 6
15.e even 4 1 180.7.l.a 6
20.d odd 2 1 80.7.p.d 6
20.e even 4 1 80.7.p.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.7.f.a ✓ 6 5.b even 2 1
20.7.f.a ✓ 6 5.c odd 4 1
80.7.p.d 6 20.d odd 2 1
80.7.p.d 6 20.e even 4 1
100.7.f.b 6 1.a even 1 1 trivial
100.7.f.b 6 5.c odd 4 1 inner
180.7.l.a 6 15.d odd 2 1
180.7.l.a 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 32T_{3}^{5} + 512T_{3}^{4} - 48600T_{3}^{3} + 1695204T_{3}^{2} - 9030672T_{3} + 24054048 \) acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 32 T^{5} + \cdots + 24054048 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 231130332030752 \) Copy content Toggle raw display
$11$ \( (T^{3} - 1100 T^{2} + \cdots + 1454420000)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 98\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 57\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 1745945230064)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 15081685496416)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 61\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 15\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 21\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 57\!\cdots\!52 \) Copy content Toggle raw display
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