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Properties

Label 201.2.d.a
Level $201$
Weight $2$
Character orbit 201.d
Analytic conductor $1.605$
Analytic rank $0$
Dimension $20$
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] = [201,2,Mod(200,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.200");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.d (of order \(2\), degree \(1\), 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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 7 x^{18} + 32 x^{16} + 128 x^{14} + 423 x^{12} + 1186 x^{10} + 3807 x^{8} + 10368 x^{6} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{5} q^{3} + (\beta_{16} + 1) q^{4} - \beta_{11} q^{5} + \beta_{8} q^{6} - \beta_{18} q^{7} + (\beta_{13} - \beta_{7}) q^{8} + (\beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{5} q^{3} + (\beta_{16} + 1) q^{4} - \beta_{11} q^{5} + \beta_{8} q^{6} - \beta_{18} q^{7} + (\beta_{13} - \beta_{7}) q^{8} + (\beta_{6} - 1) q^{9} + (\beta_{15} - 1) q^{10} + ( - \beta_{14} + \beta_{7}) q^{11} + (\beta_{19} + \beta_{14} + \cdots + \beta_{5}) q^{12}+ \cdots + ( - \beta_{19} + \beta_{18} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{4} - 4 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{4} - 4 q^{6} - 14 q^{9} - 12 q^{10} + 2 q^{15} + 12 q^{16} + 24 q^{19} + 12 q^{21} - 28 q^{22} + 2 q^{24} - 4 q^{25} + 10 q^{33} - 44 q^{36} + 24 q^{37} - 8 q^{39} - 32 q^{40} - 48 q^{49} - 26 q^{54} - 8 q^{55} - 38 q^{60} - 16 q^{64} + 32 q^{67} + 4 q^{73} + 116 q^{76} - 30 q^{81} - 32 q^{82} + 90 q^{84} - 40 q^{88} + 74 q^{90} + 20 q^{91} - 2 q^{93} + 30 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 7 x^{18} + 32 x^{16} + 128 x^{14} + 423 x^{12} + 1186 x^{10} + 3807 x^{8} + 10368 x^{6} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{18} + 60 \nu^{16} + 196 \nu^{14} + 780 \nu^{12} + 2203 \nu^{10} + 7153 \nu^{8} + 18308 \nu^{6} + \cdots + 229635 ) / 31104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{18} - 365 \nu^{16} - 1477 \nu^{14} - 5845 \nu^{12} - 16650 \nu^{10} - 47795 \nu^{8} + \cdots - 1482786 ) / 209952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{19} - 7 \nu^{17} - 32 \nu^{15} - 128 \nu^{13} - 423 \nu^{11} - 1186 \nu^{9} + \cdots - 45927 \nu ) / 19683 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{18} - 7 \nu^{16} - 32 \nu^{14} - 128 \nu^{12} - 423 \nu^{10} - 1186 \nu^{8} - 3807 \nu^{6} + \cdots - 39366 ) / 6561 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35 \nu^{19} - 236 \nu^{17} - 580 \nu^{15} - 2716 \nu^{13} - 7785 \nu^{11} - 21683 \nu^{9} + \cdots - 566433 \nu ) / 279936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35 \nu^{18} + 236 \nu^{16} + 580 \nu^{14} + 2716 \nu^{12} + 7785 \nu^{10} + 21683 \nu^{8} + \cdots + 566433 ) / 93312 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73 \nu^{18} + 814 \nu^{16} + 2486 \nu^{14} + 13262 \nu^{12} + 34293 \nu^{10} + 100969 \nu^{8} + \cdots + 4739229 ) / 139968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{19} + 76 \nu^{17} + 276 \nu^{15} + 1052 \nu^{13} + 3079 \nu^{11} + 9269 \nu^{9} + \cdots + 266895 \nu ) / 31104 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 511 \nu^{19} - 2524 \nu^{17} - 7604 \nu^{15} - 30092 \nu^{13} - 86877 \nu^{11} + \cdots - 4756725 \nu ) / 2519424 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 269 \nu^{18} + 2072 \nu^{16} + 6448 \nu^{14} + 25576 \nu^{12} + 73179 \nu^{10} + 203933 \nu^{8} + \cdots + 5058531 ) / 419904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23 \nu^{19} + 188 \nu^{17} + 736 \nu^{15} + 3700 \nu^{13} + 9081 \nu^{11} + 27143 \nu^{9} + \cdots + 1301265 \nu ) / 104976 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 619 \nu^{19} + 4108 \nu^{17} + 15236 \nu^{15} + 79484 \nu^{13} + 198369 \nu^{11} + \cdots + 28087641 \nu ) / 2519424 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 92 \nu^{18} - 401 \nu^{16} - 1729 \nu^{14} - 6673 \nu^{12} - 18261 \nu^{10} - 51764 \nu^{8} + \cdots - 1397493 ) / 104976 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 236 \nu^{18} + 1013 \nu^{16} + 4861 \nu^{14} + 18589 \nu^{12} + 49545 \nu^{10} + 153860 \nu^{8} + \cdots + 4258089 ) / 209952 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 553 \nu^{19} + 1162 \nu^{17} + 6914 \nu^{15} + 28970 \nu^{13} + 68769 \nu^{11} + \cdots + 6147657 \nu ) / 1259712 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 377 \nu^{19} - 1226 \nu^{17} - 5818 \nu^{15} - 21994 \nu^{13} - 55881 \nu^{11} + \cdots - 4730481 \nu ) / 629856 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 676 \nu^{19} - 2815 \nu^{17} - 13559 \nu^{15} - 51671 \nu^{13} - 135099 \nu^{11} + \cdots - 11934459 \nu ) / 629856 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + \beta_{10} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} - 2\beta_{15} - \beta_{12} + \beta_{4} + 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{18} - 2\beta_{17} - \beta_{14} + \beta_{13} + 3\beta_{11} - \beta_{10} - 3\beta_{7} - 2\beta_{5} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{16} + 3\beta_{15} + 5\beta_{12} + \beta_{9} - 6\beta_{8} - 2\beta_{6} + 4\beta_{4} - 2\beta_{3} - 2\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{19} - \beta_{18} - 5 \beta_{17} + 9 \beta_{14} - 6 \beta_{13} - 5 \beta_{11} - \beta_{10} + \cdots - 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{16} + 23 \beta_{15} - 6 \beta_{12} - 3 \beta_{9} + 19 \beta_{8} - 2 \beta_{6} - \beta_{4} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8 \beta_{18} + 20 \beta_{17} + 16 \beta_{14} - 25 \beta_{13} + 37 \beta_{11} + 14 \beta_{10} + \cdots + 30 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16 \beta_{16} - \beta_{15} + 6 \beta_{12} - 5 \beta_{9} + 15 \beta_{8} - 65 \beta_{6} - 39 \beta_{4} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 28 \beta_{19} + 82 \beta_{18} - 50 \beta_{14} + 35 \beta_{13} - 105 \beta_{11} - 52 \beta_{10} + \cdots - 61 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 24 \beta_{16} - 87 \beta_{15} - 94 \beta_{12} + 39 \beta_{9} + 37 \beta_{8} + 181 \beta_{6} + \cdots - 541 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 25 \beta_{19} + 61 \beta_{18} + 11 \beta_{17} - 108 \beta_{14} + 225 \beta_{13} - 25 \beta_{11} + \cdots - 494 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 119 \beta_{16} - 309 \beta_{15} + 143 \beta_{12} - 71 \beta_{9} - 471 \beta_{8} + 133 \beta_{6} + \cdots + 1371 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 612 \beta_{19} + 340 \beta_{18} - 742 \beta_{17} - 369 \beta_{14} + 156 \beta_{13} - 214 \beta_{11} + \cdots + 943 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 231 \beta_{16} + 860 \beta_{15} + 925 \beta_{12} - 28 \beta_{9} + \beta_{8} + 441 \beta_{6} + 187 \beta_{4} + \cdots + 423 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1673 \beta_{19} - 1231 \beta_{18} + 1745 \beta_{17} + 991 \beta_{14} - 1075 \beta_{13} + \cdots + 3908 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 384 \beta_{16} - 2536 \beta_{15} - 3108 \beta_{12} - 658 \beta_{9} + 4292 \beta_{8} + 409 \beta_{6} + \cdots + 5391 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3396 \beta_{19} - 6374 \beta_{18} + 2844 \beta_{17} - 3026 \beta_{14} + 1052 \beta_{13} + \cdots + 3942 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
200.1
−0.362922 + 1.69360i
−0.362922 − 1.69360i
1.16595 + 1.28084i
1.16595 − 1.28084i
−1.24906 + 1.19993i
−1.24906 − 1.19993i
1.58783 + 0.691953i
1.58783 − 0.691953i
−0.421281 + 1.68004i
−0.421281 − 1.68004i
0.421281 + 1.68004i
0.421281 − 1.68004i
−1.58783 + 0.691953i
−1.58783 − 0.691953i
1.24906 + 1.19993i
1.24906 − 1.19993i
−1.16595 + 1.28084i
−1.16595 − 1.28084i
0.362922 + 1.69360i
0.362922 − 1.69360i
−2.50456 −0.362922 − 1.69360i 4.27284 2.28042 0.908961 + 4.24173i 3.79080i −5.69246 −2.73658 + 1.22929i −5.71146
200.2 −2.50456 −0.362922 + 1.69360i 4.27284 2.28042 0.908961 − 4.24173i − 3.79080i −5.69246 −2.73658 − 1.22929i −5.71146
200.3 −2.09324 1.16595 − 1.28084i 2.38166 −1.47666 −2.44062 + 2.68111i − 2.11750i −0.798907 −0.281099 − 2.98680i 3.09102
200.4 −2.09324 1.16595 + 1.28084i 2.38166 −1.47666 −2.44062 − 2.68111i 2.11750i −0.798907 −0.281099 + 2.98680i 3.09102
200.5 −1.18817 −1.24906 − 1.19993i −0.588256 −2.08509 1.48410 + 1.42572i − 0.284155i 3.07528 0.120318 + 2.99759i 2.47744
200.6 −1.18817 −1.24906 + 1.19993i −0.588256 −2.08509 1.48410 − 1.42572i 0.284155i 3.07528 0.120318 − 2.99759i 2.47744
200.7 −0.758634 1.58783 − 0.691953i −1.42447 1.15744 −1.20458 + 0.524939i 4.81799i 2.59792 2.04240 − 2.19741i −0.878075
200.8 −0.758634 1.58783 + 0.691953i −1.42447 1.15744 −1.20458 − 0.524939i − 4.81799i 2.59792 2.04240 + 2.19741i −0.878075
200.9 −0.598526 −0.421281 − 1.68004i −1.64177 3.30634 0.252148 + 1.00555i − 2.20280i 2.17969 −2.64504 + 1.41554i −1.97893
200.10 −0.598526 −0.421281 + 1.68004i −1.64177 3.30634 0.252148 − 1.00555i 2.20280i 2.17969 −2.64504 − 1.41554i −1.97893
200.11 0.598526 0.421281 − 1.68004i −1.64177 −3.30634 0.252148 − 1.00555i − 2.20280i −2.17969 −2.64504 − 1.41554i −1.97893
200.12 0.598526 0.421281 + 1.68004i −1.64177 −3.30634 0.252148 + 1.00555i 2.20280i −2.17969 −2.64504 + 1.41554i −1.97893
200.13 0.758634 −1.58783 − 0.691953i −1.42447 −1.15744 −1.20458 − 0.524939i 4.81799i −2.59792 2.04240 + 2.19741i −0.878075
200.14 0.758634 −1.58783 + 0.691953i −1.42447 −1.15744 −1.20458 + 0.524939i − 4.81799i −2.59792 2.04240 − 2.19741i −0.878075
200.15 1.18817 1.24906 − 1.19993i −0.588256 2.08509 1.48410 − 1.42572i − 0.284155i −3.07528 0.120318 − 2.99759i 2.47744
200.16 1.18817 1.24906 + 1.19993i −0.588256 2.08509 1.48410 + 1.42572i 0.284155i −3.07528 0.120318 + 2.99759i 2.47744
200.17 2.09324 −1.16595 − 1.28084i 2.38166 1.47666 −2.44062 − 2.68111i − 2.11750i 0.798907 −0.281099 + 2.98680i 3.09102
200.18 2.09324 −1.16595 + 1.28084i 2.38166 1.47666 −2.44062 + 2.68111i 2.11750i 0.798907 −0.281099 − 2.98680i 3.09102
200.19 2.50456 0.362922 − 1.69360i 4.27284 −2.28042 0.908961 − 4.24173i 3.79080i 5.69246 −2.73658 − 1.22929i −5.71146
200.20 2.50456 0.362922 + 1.69360i 4.27284 −2.28042 0.908961 + 4.24173i − 3.79080i 5.69246 −2.73658 + 1.22929i −5.71146
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 200.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.b odd 2 1 inner
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.d.a ✓ 20
3.b odd 2 1 inner 201.2.d.a ✓ 20
67.b odd 2 1 inner 201.2.d.a ✓ 20
201.d even 2 1 inner 201.2.d.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.d.a ✓ 20 1.a even 1 1 trivial
201.2.d.a ✓ 20 3.b odd 2 1 inner
201.2.d.a ✓ 20 67.b odd 2 1 inner
201.2.d.a ✓ 20 201.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(201, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 13 T^{8} + \cdots - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 7 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( (T^{10} - 24 T^{8} + \cdots - 722)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 47 T^{8} + \cdots + 586)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 52 T^{8} + \cdots - 40328)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 73 T^{8} + \cdots + 2344)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 101 T^{8} + \cdots + 918848)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 6 T^{4} - 18 T^{3} + \cdots + 14)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + 95 T^{8} + \cdots + 1561397)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 135 T^{8} + \cdots + 229712)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 182 T^{8} + \cdots + 4641706)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 6 T^{4} + \cdots + 1117)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} - 137 T^{8} + \cdots - 2635808)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 237 T^{8} + \cdots + 1837696)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 302 T^{8} + \cdots + 84807092)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 222 T^{8} + \cdots - 3317888)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 465 T^{8} + \cdots + 3175318352)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 315 T^{8} + \cdots + 22511776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 16 T^{9} + \cdots + 1350125107)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 250 T^{8} + \cdots + 229712)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - T^{4} - 159 T^{3} + \cdots - 2524)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 379 T^{8} + \cdots + 600064)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 465 T^{8} + \cdots + 78618932)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 187 T^{8} + \cdots + 7314452)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 286 T^{8} + \cdots + 47264416)^{2} \) Copy content Toggle raw display
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