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Properties

Label 100.11.d.a
Level $100$
Weight $11$
Character orbit 100.d
Analytic conductor $63.536$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,11,Mod(99,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.99");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 100.d (of order \(2\), degree \(1\), not 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: \(63.5357252674\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 114x^{6} + 3921x^{4} - 36284x^{2} + 112896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 4)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2 \beta_1) q^{2} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 4) q^{4} + (12 \beta_{6} - 16 \beta_{5} + \cdots + 1800) q^{6}+ \cdots + (72 \beta_{6} + 576 \beta_{4} + 7191) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2 \beta_1) q^{2} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 4) q^{4} + (12 \beta_{6} - 16 \beta_{5} + \cdots + 1800) q^{6}+ \cdots + ( - 5121792 \beta_{6} + \cdots + 42791481 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 14400 q^{6} + 57528 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 14400 q^{6} + 57528 q^{9} - 3803520 q^{14} - 7224832 q^{16} - 967680 q^{21} - 35804160 q^{24} - 64879344 q^{26} - 60093264 q^{29} + 18622256 q^{34} - 111928032 q^{36} - 680360304 q^{41} + 604151040 q^{44} + 482363520 q^{46} + 1609842808 q^{49} + 1263807360 q^{54} + 2784983040 q^{56} + 6824166736 q^{61} + 72736768 q^{64} + 1426429440 q^{66} - 8798376960 q^{69} - 3436515984 q^{74} - 14875948800 q^{76} - 9431561592 q^{81} + 7822725120 q^{84} - 17520498240 q^{86} - 10549443984 q^{89} + 14783343360 q^{94} + 10989158400 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 114x^{6} + 3921x^{4} - 36284x^{2} + 112896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{7} + 514\nu^{5} - 14845\nu^{3} + 69980\nu ) / 29904 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\nu^{7} - 2934\nu^{5} + 82527\nu^{3} + 1654604\nu ) / 29904 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{7} + 1202\nu^{5} - 34439\nu^{3} + 86316\nu ) / 7476 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\nu^{6} - 1384\nu^{4} + 53542\nu^{2} - 366198 ) / 1869 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -94\nu^{6} + 8524\nu^{4} - 220702\nu^{2} + 1168062 ) / 1869 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -136\nu^{6} + 15832\nu^{4} - 479968\nu^{2} + 1759632 ) / 1869 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 499\nu^{7} - 58702\nu^{5} + 1985627\nu^{3} - 13505924\nu ) / 9968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{3} + \beta_{2} - 13\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - 2\beta_{5} + 22\beta_{4} + 2736 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 404\beta_{3} + 85\beta_{2} - 2266\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 189\beta_{6} - 166\beta_{5} + 1010\beta_{4} + 123696 ) / 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 75\beta_{7} + 23980\beta_{3} + 4025\beta_{2} - 170054\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10095\beta_{6} - 12266\beta_{5} + 39934\beta_{4} + 5985936 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1197\beta_{7} + 1321652\beta_{3} + 189397\beta_{2} - 11691850\beta_1 ) / 96 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
2.50555 + 0.500000i
2.50555 − 0.500000i
−7.15697 − 0.500000i
−7.15697 + 0.500000i
7.15697 − 0.500000i
7.15697 + 0.500000i
−2.50555 + 0.500000i
−2.50555 − 0.500000i
−25.3936 − 19.4722i 120.267 265.666 + 988.937i 0 −3054.00 − 2341.85i 29129.9 12510.6 − 30285.8i −44585.0 0
99.2 −25.3936 + 19.4722i 120.267 265.666 − 988.937i 0 −3054.00 + 2341.85i 29129.9 12510.6 + 30285.8i −44585.0 0
99.3 −19.3692 − 25.4722i −343.535 −273.666 + 986.754i 0 6654.00 + 8750.58i 10902.2 30435.5 − 12141.8i 58967.0 0
99.4 −19.3692 + 25.4722i −343.535 −273.666 − 986.754i 0 6654.00 − 8750.58i 10902.2 30435.5 + 12141.8i 58967.0 0
99.5 19.3692 − 25.4722i 343.535 −273.666 − 986.754i 0 6654.00 − 8750.58i −10902.2 −30435.5 − 12141.8i 58967.0 0
99.6 19.3692 + 25.4722i 343.535 −273.666 + 986.754i 0 6654.00 + 8750.58i −10902.2 −30435.5 + 12141.8i 58967.0 0
99.7 25.3936 − 19.4722i −120.267 265.666 − 988.937i 0 −3054.00 + 2341.85i −29129.9 −12510.6 − 30285.8i −44585.0 0
99.8 25.3936 + 19.4722i −120.267 265.666 + 988.937i 0 −3054.00 − 2341.85i −29129.9 −12510.6 + 30285.8i −44585.0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.11.d.a 8
4.b odd 2 1 inner 100.11.d.a 8
5.b even 2 1 inner 100.11.d.a 8
5.c odd 4 1 4.11.b.a ✓ 4
5.c odd 4 1 100.11.b.d 4
15.e even 4 1 36.11.d.c 4
20.d odd 2 1 inner 100.11.d.a 8
20.e even 4 1 4.11.b.a ✓ 4
20.e even 4 1 100.11.b.d 4
40.i odd 4 1 64.11.c.d 4
40.k even 4 1 64.11.c.d 4
60.l odd 4 1 36.11.d.c 4
80.i odd 4 1 256.11.d.f 8
80.j even 4 1 256.11.d.f 8
80.s even 4 1 256.11.d.f 8
80.t odd 4 1 256.11.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.11.b.a ✓ 4 5.c odd 4 1
4.11.b.a ✓ 4 20.e even 4 1
36.11.d.c 4 15.e even 4 1
36.11.d.c 4 60.l odd 4 1
64.11.c.d 4 40.i odd 4 1
64.11.c.d 4 40.k even 4 1
100.11.b.d 4 5.c odd 4 1
100.11.b.d 4 20.e even 4 1
100.11.d.a 8 1.a even 1 1 trivial
100.11.d.a 8 4.b odd 2 1 inner
100.11.d.a 8 5.b even 2 1 inner
100.11.d.a 8 20.d odd 2 1 inner
256.11.d.f 8 80.i odd 4 1
256.11.d.f 8 80.j even 4 1
256.11.d.f 8 80.s even 4 1
256.11.d.f 8 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 132480T_{3}^{2} + 1706987520 \) acting on \(S_{11}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 1099511627776 \) Copy content Toggle raw display
$3$ \( (T^{4} - 132480 T^{2} + 1706987520)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 32\!\cdots\!20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 80\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 139887323593756)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 34\!\cdots\!20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 32\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 36\!\cdots\!24)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 75\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 27\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 51\!\cdots\!44)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 90\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 15\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 33\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 85\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 34\!\cdots\!96)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 70\!\cdots\!16)^{2} \) Copy content Toggle raw display
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