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Properties

Label 200.3.e.c
Level $200$
Weight $3$
Character orbit 200.e
Analytic conductor $5.450$
Analytic rank $0$
Dimension $12$
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] = [200,3,Mod(99,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.e (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: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{9} + \beta_{6} + 1) q^{4} + (\beta_{10} - \beta_{6}) q^{6} + ( - 2 \beta_{8} + \beta_{2} - \beta_1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{3} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{9} + \beta_{6} + 1) q^{4} + (\beta_{10} - \beta_{6}) q^{6} + ( - 2 \beta_{8} + \beta_{2} - \beta_1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{3} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{7} - 20 \beta_{6} - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 14 q^{4} + 2 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 14 q^{4} + 2 q^{6} + 16 q^{9} + 60 q^{11} - 8 q^{14} - 78 q^{16} + 4 q^{19} - 118 q^{24} - 72 q^{26} - 178 q^{34} - 16 q^{36} - 140 q^{41} + 154 q^{44} + 368 q^{46} + 276 q^{49} + 228 q^{51} + 138 q^{54} - 352 q^{56} + 88 q^{59} - 226 q^{64} - 202 q^{66} - 408 q^{74} + 310 q^{76} - 284 q^{81} + 800 q^{84} + 436 q^{86} - 196 q^{89} + 576 q^{91} + 632 q^{94} - 818 q^{96} - 608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 7\nu^{9} + 44\nu^{7} - 156\nu^{5} + 704\nu^{3} - 1280\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 11\nu^{9} + 40\nu^{7} - 236\nu^{5} + 560\nu^{3} - 2432\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 9\nu^{9} + 4\nu^{7} - 100\nu^{5} - 2048\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 3\nu^{9} + 16\nu^{7} - 44\nu^{5} + 272\nu^{3} - 256\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{10} + 13\nu^{8} - 76\nu^{6} + 244\nu^{4} - 992\nu^{2} + 1792 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 3\nu^{8} - 16\nu^{6} + 44\nu^{4} - 272\nu^{2} + 256 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} - 17\nu^{7} + 24\nu^{5} - 180\nu^{3} - 160\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{10} - 13\nu^{8} + 76\nu^{6} - 244\nu^{4} + 1248\nu^{2} - 2048 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 15\nu^{8} - 36\nu^{6} + 316\nu^{4} - 672\nu^{2} + 3840 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{10} - 31\nu^{8} + 84\nu^{6} - 444\nu^{4} + 992\nu^{2} - 6144 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{6} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{6} - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{8} + \beta_{5} + 4\beta_{4} - 7\beta_{3} + 3\beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 6\beta_{10} - \beta_{9} + 7\beta_{7} - 12\beta_{6} - 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -19\beta_{8} - 21\beta_{5} + 4\beta_{4} - 5\beta_{3} + 17\beta_{2} - 31\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -21\beta_{11} - 14\beta_{10} + 9\beta_{9} + 29\beta_{7} - 32\beta_{6} - 114 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{8} - 35\beta_{5} - 84\beta_{4} + 53\beta_{3} + 15\beta_{2} - 113\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -35\beta_{11} - 50\beta_{10} - 185\beta_{9} - 133\beta_{7} - 88\beta_{6} - 422 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -3\beta_{8} + 131\beta_{5} - 140\beta_{4} + 203\beta_{3} - 367\beta_{2} - 255\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
−1.83244 − 0.801352i
−1.83244 + 0.801352i
−1.76129 − 0.947553i
−1.76129 + 0.947553i
−1.13579 − 1.64620i
−1.13579 + 1.64620i
1.13579 − 1.64620i
1.13579 + 1.64620i
1.76129 − 0.947553i
1.76129 + 0.947553i
1.83244 − 0.801352i
1.83244 + 0.801352i
−1.83244 − 0.801352i − 4.03404i 2.71567 + 2.93686i 0 −3.23269 + 7.39214i 11.1194 −2.62284 − 7.55783i −7.27349 0
99.2 −1.83244 + 0.801352i 4.03404i 2.71567 − 2.93686i 0 −3.23269 − 7.39214i 11.1194 −2.62284 + 7.55783i −7.27349 0
99.3 −1.76129 − 0.947553i − 0.486535i 2.20429 + 3.33783i 0 −0.461018 + 0.856930i −5.23644 −0.719613 − 7.96757i 8.76328 0
99.4 −1.76129 + 0.947553i 0.486535i 2.20429 − 3.33783i 0 −0.461018 − 0.856930i −5.23644 −0.719613 + 7.96757i 8.76328 0
99.5 −1.13579 − 1.64620i 2.54751i −1.41995 + 3.73948i 0 4.19371 − 2.89344i −8.05846 7.76871 − 1.90974i 2.51021 0
99.6 −1.13579 + 1.64620i − 2.54751i −1.41995 − 3.73948i 0 4.19371 + 2.89344i −8.05846 7.76871 + 1.90974i 2.51021 0
99.7 1.13579 − 1.64620i 2.54751i −1.41995 − 3.73948i 0 4.19371 + 2.89344i 8.05846 −7.76871 − 1.90974i 2.51021 0
99.8 1.13579 + 1.64620i − 2.54751i −1.41995 + 3.73948i 0 4.19371 − 2.89344i 8.05846 −7.76871 + 1.90974i 2.51021 0
99.9 1.76129 − 0.947553i − 0.486535i 2.20429 − 3.33783i 0 −0.461018 − 0.856930i 5.23644 0.719613 − 7.96757i 8.76328 0
99.10 1.76129 + 0.947553i 0.486535i 2.20429 + 3.33783i 0 −0.461018 + 0.856930i 5.23644 0.719613 + 7.96757i 8.76328 0
99.11 1.83244 − 0.801352i − 4.03404i 2.71567 − 2.93686i 0 −3.23269 − 7.39214i −11.1194 2.62284 − 7.55783i −7.27349 0
99.12 1.83244 + 0.801352i 4.03404i 2.71567 + 2.93686i 0 −3.23269 + 7.39214i −11.1194 2.62284 + 7.55783i −7.27349 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.e.c 12
4.b odd 2 1 800.3.e.c 12
5.b even 2 1 inner 200.3.e.c 12
5.c odd 4 1 200.3.g.e ✓ 6
5.c odd 4 1 200.3.g.f yes 6
8.b even 2 1 800.3.e.c 12
8.d odd 2 1 inner 200.3.e.c 12
20.d odd 2 1 800.3.e.c 12
20.e even 4 1 800.3.g.e 6
20.e even 4 1 800.3.g.f 6
40.e odd 2 1 inner 200.3.e.c 12
40.f even 2 1 800.3.e.c 12
40.i odd 4 1 800.3.g.e 6
40.i odd 4 1 800.3.g.f 6
40.k even 4 1 200.3.g.e ✓ 6
40.k even 4 1 200.3.g.f yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.e.c 12 1.a even 1 1 trivial
200.3.e.c 12 5.b even 2 1 inner
200.3.e.c 12 8.d odd 2 1 inner
200.3.e.c 12 40.e odd 2 1 inner
200.3.g.e ✓ 6 5.c odd 4 1
200.3.g.e ✓ 6 40.k even 4 1
200.3.g.f yes 6 5.c odd 4 1
200.3.g.f yes 6 40.k even 4 1
800.3.e.c 12 4.b odd 2 1
800.3.e.c 12 8.b even 2 1
800.3.e.c 12 20.d odd 2 1
800.3.e.c 12 40.f even 2 1
800.3.g.e 6 20.e even 4 1
800.3.g.e 6 40.i odd 4 1
800.3.g.f 6 20.e even 4 1
800.3.g.f 6 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 23T_{3}^{4} + 111T_{3}^{2} + 25 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{6} + 23 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 216 T^{4} + \cdots - 220160)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 15 T^{2} + \cdots + 53)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 616 T^{4} + \cdots - 3522560)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 643 T^{4} + \cdots + 1113025)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 475 T - 373)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 3136 T^{4} + \cdots - 352256000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 2920 T^{4} + \cdots + 352256000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 5600 T^{4} + \cdots + 2201600000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 3336 T^{4} + \cdots - 220160)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 35 T^{2} + \cdots - 26447)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 5252 T^{4} + \cdots + 400000000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 5176 T^{4} + \cdots - 465858560)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 9536 T^{4} + \cdots - 4072079360)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 22 T^{2} + \cdots - 6752)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} + 17800 T^{4} + \cdots + 176353664000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 17303 T^{4} + \cdots + 329604025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 19960 T^{4} + \cdots + 110982656000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 20723 T^{4} + \cdots + 24628594225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8280 T^{4} + \cdots + 10176896000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 18663 T^{4} + \cdots + 17201634025)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 49 T^{2} + \cdots - 97393)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 25612 T^{4} + \cdots + 14046990400)^{2} \) Copy content Toggle raw display
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