Properties

Label 201.4.a.b
Level $201$
Weight $4$
Character orbit 201.a
Self dual yes
Analytic conductor $11.859$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(1,201)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.a (trivial)

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: yes
Analytic conductor: \(11.8593839112\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 28x^{4} + 22x^{3} + 202x^{2} - 96x - 384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 3 \beta_{5} - \beta_{2} - \beta_1 - 10) q^{7} + ( - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} - 13) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 3 \beta_{5} - \beta_{2} - \beta_1 - 10) q^{7} + ( - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} - 13) q^{8} + 9 q^{9} + ( - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 9 \beta_{2} + 4 \beta_1 - 17) q^{10} + ( - 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_1 - 12) q^{11} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 6 \beta_1 + 6) q^{12} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 34) q^{13} + ( - \beta_{5} - 4 \beta_{3} + 3 \beta_{2} - 16 \beta_1 + 4) q^{14} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1 - 3) q^{15} + (2 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} - 11 \beta_1 + 2) q^{16} + (7 \beta_{5} + 15 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 24) q^{17} + (9 \beta_1 - 9) q^{18} + (11 \beta_{5} - 7 \beta_{4} + 5 \beta_{3} - 13 \beta_{2} + 15 \beta_1 - 44) q^{19} + (6 \beta_{5} - 13 \beta_{3} + 11 \beta_{2} - 30 \beta_1 + 50) q^{20} + ( - 9 \beta_{5} - 3 \beta_{2} - 3 \beta_1 - 30) q^{21} + (3 \beta_{5} - 3 \beta_{4} - 11 \beta_{3} + 17 \beta_{2} - 29 \beta_1 + 14) q^{22} + (11 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 11 \beta_{2} - 17 \beta_1 + 7) q^{23} + ( - 9 \beta_{5} - 3 \beta_{4} + 9 \beta_{3} - 3 \beta_{2} - 39) q^{24} + (15 \beta_{5} + 21 \beta_{4} + 24 \beta_{2} - 6 \beta_1 - 11) q^{25} + (13 \beta_{5} + 20 \beta_{4} + \beta_{3} - 4 \beta_{2} - 34 \beta_1 + 78) q^{26} + 27 q^{27} + ( - 4 \beta_{5} - 13 \beta_{4} + 10 \beta_{3} - 22 \beta_{2} + 37 \beta_1 - 57) q^{28} + ( - 6 \beta_{5} - 15 \beta_{4} + 4 \beta_{3} + 25 \beta_{2} + 7 \beta_1 - 42) q^{29} + ( - 12 \beta_{5} - 12 \beta_{4} + 6 \beta_{3} - 27 \beta_{2} + 12 \beta_1 - 51) q^{30} + ( - 14 \beta_{5} + \beta_{4} - 11 \beta_{3} - 48 \beta_{2} + 38 \beta_1 - 68) q^{31} + ( - 16 \beta_{5} - 12 \beta_{4} - 2 \beta_{3} - 52 \beta_{2} + 37 \beta_1 + 13) q^{32} + ( - 15 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} + 3 \beta_1 - 36) q^{33} + (3 \beta_{5} + 3 \beta_{4} + 11 \beta_{3} + 53 \beta_{2} - 3 \beta_1 - 38) q^{34} + ( - 8 \beta_{5} + 20 \beta_{4} + 22 \beta_{3} + 17 \beta_{2} + 31 \beta_1 - 65) q^{35} + (9 \beta_{5} + 9 \beta_{4} + 9 \beta_{2} - 18 \beta_1 + 18) q^{36} + ( - 5 \beta_{5} + \beta_{4} - 10 \beta_{3} - 30 \beta_{2} + 52 \beta_1 - 100) q^{37} + (37 \beta_{5} - 17 \beta_{4} - 19 \beta_{3} + 11 \beta_{2} - 67 \beta_1 + 132) q^{38} + (3 \beta_{5} - 6 \beta_{4} + 9 \beta_{3} + 24 \beta_{2} + 6 \beta_1 - 102) q^{39} + ( - 37 \beta_{5} + 5 \beta_{4} + 27 \beta_{3} - 14 \beta_{2} + 97 \beta_1 - 163) q^{40} + (8 \beta_{5} + 20 \beta_{4} + 81 \beta_{2} - \beta_1 - 11) q^{41} + ( - 3 \beta_{5} - 12 \beta_{3} + 9 \beta_{2} - 48 \beta_1 + 12) q^{42} + ( - 20 \beta_{5} + 16 \beta_{4} - 37 \beta_{3} + 43 \beta_{2} - 37 \beta_1 - 54) q^{43} + ( - 19 \beta_{5} - 17 \beta_{4} + 31 \beta_{3} - 51 \beta_{2} + 77 \beta_1 - 128) q^{44} + (9 \beta_{5} - 9 \beta_{4} - 9 \beta_{3} - 18 \beta_1 - 9) q^{45} + ( - 12 \beta_{5} - 3 \beta_{4} + 20 \beta_{3} - 18 \beta_{2} + 55 \beta_1 - 157) q^{46} + ( - 49 \beta_{5} - 60 \beta_{4} - 53 \beta_{3} - 58 \beta_{2} + 32 \beta_1 - 78) q^{47} + (6 \beta_{5} - 12 \beta_{4} - 33 \beta_{3} + 6 \beta_{2} - 33 \beta_1 + 6) q^{48} + (58 \beta_{5} - 20 \beta_{4} - 17 \beta_{3} - 13 \beta_{2} + 55 \beta_1 - 45) q^{49} + ( - 27 \beta_{5} + 27 \beta_{4} + 60 \beta_{3} + 18 \beta_{2} + 70 \beta_1 - 43) q^{50} + (21 \beta_{5} + 45 \beta_{4} + 21 \beta_{3} + 9 \beta_{2} - 9 \beta_1 - 72) q^{51} + ( - 59 \beta_{5} - 38 \beta_{4} + 3 \beta_{3} - 44 \beta_{2} + 136 \beta_1 - 190) q^{52} + (64 \beta_{5} - 12 \beta_{4} - 24 \beta_{3} - 19 \beta_{2} - 25 \beta_1 + 135) q^{53} + (27 \beta_1 - 27) q^{54} + ( - 19 \beta_{5} + 11 \beta_{4} + 33 \beta_{3} + 25 \beta_{2} + 73 \beta_1 - 196) q^{55} + (88 \beta_{5} + 7 \beta_{4} - 27 \beta_{3} + 30 \beta_{2} - 29 \beta_1 + 326) q^{56} + (33 \beta_{5} - 21 \beta_{4} + 15 \beta_{3} - 39 \beta_{2} + 45 \beta_1 - 132) q^{57} + (34 \beta_{5} + 67 \beta_{4} - 4 \beta_{3} - 45 \beta_{2} - 59 \beta_1 + 222) q^{58} + ( - 58 \beta_{5} + 22 \beta_{4} + 12 \beta_{3} - 103 \beta_{2} + 35 \beta_1 - 63) q^{59} + (18 \beta_{5} - 39 \beta_{3} + 33 \beta_{2} - 90 \beta_1 + 150) q^{60} + (39 \beta_{5} + 18 \beta_{4} - 3 \beta_{3} + 58 \beta_{2} - 44 \beta_1 - 240) q^{61} + (4 \beta_{5} - 55 \beta_{4} - 39 \beta_{3} + 70 \beta_{2} - 159 \beta_1 + 292) q^{62} + ( - 27 \beta_{5} - 9 \beta_{2} - 9 \beta_1 - 90) q^{63} + (27 \beta_{5} - 21 \beta_{4} + 12 \beta_{3} + 47 \beta_{2} - 28 \beta_1 + 204) q^{64} + ( - 77 \beta_{5} - \beta_{4} + 29 \beta_{3} - 105 \beta_{2} + 85 \beta_1 - 64) q^{65} + (9 \beta_{5} - 9 \beta_{4} - 33 \beta_{3} + 51 \beta_{2} - 87 \beta_1 + 42) q^{66} - 67 q^{67} + ( - 29 \beta_{5} - 9 \beta_{4} - 19 \beta_{3} - 41 \beta_{2} + 29 \beta_1 + 350) q^{68} + (33 \beta_{5} + 12 \beta_{4} + 9 \beta_{3} + 33 \beta_{2} - 51 \beta_1 + 21) q^{69} + (77 \beta_{5} + 95 \beta_{4} - 15 \beta_{3} + 148 \beta_{2} - 119 \beta_1 + 371) q^{70} + ( - 61 \beta_{5} - 98 \beta_{4} - 9 \beta_{3} - 16 \beta_{2} - 74 \beta_1 - 110) q^{71} + ( - 27 \beta_{5} - 9 \beta_{4} + 27 \beta_{3} - 9 \beta_{2} - 117) q^{72} + (58 \beta_{5} - 98 \beta_{4} + 15 \beta_{3} + 11 \beta_{2} - 109 \beta_1 + 12) q^{73} + (21 \beta_{5} - 13 \beta_{4} - 14 \beta_{3} + 60 \beta_{2} - 171 \beta_1 + 486) q^{74} + (45 \beta_{5} + 63 \beta_{4} + 72 \beta_{2} - 18 \beta_1 - 33) q^{75} + ( - 195 \beta_{5} - 45 \beta_{4} + 29 \beta_{3} - 119 \beta_{2} + \cdots - 390) q^{76}+ \cdots + ( - 45 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} - 45 \beta_{2} + 9 \beta_1 - 108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} + 18 q^{3} + 13 q^{4} - 12 q^{5} - 15 q^{6} - 62 q^{7} - 75 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} + 18 q^{3} + 13 q^{4} - 12 q^{5} - 15 q^{6} - 62 q^{7} - 75 q^{8} + 54 q^{9} - 111 q^{10} - 72 q^{11} + 39 q^{12} - 192 q^{13} + 3 q^{14} - 36 q^{15} - 27 q^{16} - 100 q^{17} - 45 q^{18} - 266 q^{19} + 255 q^{20} - 186 q^{21} + 44 q^{22} + 50 q^{23} - 225 q^{24} - 6 q^{25} + 472 q^{26} + 162 q^{27} - 333 q^{28} - 242 q^{29} - 333 q^{30} - 438 q^{31} + 35 q^{32} - 216 q^{33} - 150 q^{34} - 258 q^{35} + 117 q^{36} - 596 q^{37} + 664 q^{38} - 576 q^{39} - 831 q^{40} + 54 q^{41} + 9 q^{42} - 360 q^{43} - 714 q^{44} - 108 q^{45} - 871 q^{46} - 720 q^{47} - 81 q^{48} - 302 q^{49} + 4 q^{50} - 300 q^{51} - 1118 q^{52} + 694 q^{53} - 135 q^{54} - 990 q^{55} + 1917 q^{56} - 798 q^{57} + 1354 q^{58} - 378 q^{59} + 765 q^{60} - 1396 q^{61} + 1475 q^{62} - 558 q^{63} + 1225 q^{64} - 348 q^{65} + 132 q^{66} - 402 q^{67} + 2032 q^{68} + 150 q^{69} + 2415 q^{70} - 964 q^{71} - 675 q^{72} - 192 q^{73} + 2751 q^{74} - 18 q^{75} - 2306 q^{76} + 2724 q^{77} + 1416 q^{78} - 802 q^{79} + 4221 q^{80} + 486 q^{81} + 1735 q^{82} + 2126 q^{83} - 999 q^{84} - 1206 q^{85} - 609 q^{86} - 726 q^{87} + 3656 q^{88} + 432 q^{89} - 999 q^{90} + 1258 q^{91} + 3163 q^{92} - 1314 q^{93} + 1742 q^{94} + 936 q^{95} + 105 q^{96} + 1290 q^{97} + 2492 q^{98} - 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 28x^{4} + 22x^{3} + 202x^{2} - 96x - 384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 9\nu^{4} + 20\nu^{3} - 182\nu^{2} - 26\nu + 560 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} - 5\nu^{4} + 92\nu^{3} + 94\nu^{2} - 590\nu - 240 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{5} - 3\nu^{4} - 252\nu^{3} + 82\nu^{2} + 990\nu - 336 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{5} - 15\nu^{4} + 212\nu^{3} + 410\nu^{2} - 938\nu - 1936 ) / 128 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 13\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 20\beta_{5} + 22\beta_{4} + \beta_{3} + 28\beta_{2} - 3\beta _1 + 119 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{5} + 56\beta_{4} + 69\beta_{3} + 46\beta_{2} + 207\beta _1 - 27 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.17870
−2.57710
−1.43201
2.04345
2.81400
4.33036
−5.17870 3.00000 18.8190 17.1025 −15.5361 −19.8930 −56.0282 9.00000 −88.5686
1.2 −3.57710 3.00000 4.79568 −7.96859 −10.7313 0.508092 11.4622 9.00000 28.5045
1.3 −2.43201 3.00000 −2.08534 −2.10555 −7.29602 −2.84912 24.5276 9.00000 5.12071
1.4 1.04345 3.00000 −6.91121 −6.71512 3.13035 8.77104 −15.5591 9.00000 −7.00690
1.5 1.81400 3.00000 −4.70941 5.30362 5.44200 −31.2342 −23.0548 9.00000 9.62075
1.6 3.33036 3.00000 3.09132 −17.6168 9.99109 −17.3029 −16.3477 9.00000 −58.6704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.a.b 6
3.b odd 2 1 603.4.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.b 6 1.a even 1 1 trivial
603.4.a.b 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 5T_{2}^{5} - 18T_{2}^{4} - 80T_{2}^{3} + 105T_{2}^{2} + 263T_{2} - 284 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{5} - 18 T^{4} - 80 T^{3} + \cdots - 284 \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{5} - 300 T^{4} + \cdots + 180036 \) Copy content Toggle raw display
$7$ \( T^{6} + 62 T^{5} + 1044 T^{4} + \cdots + 136506 \) Copy content Toggle raw display
$11$ \( T^{6} + 72 T^{5} + \cdots + 291026784 \) Copy content Toggle raw display
$13$ \( T^{6} + 192 T^{5} + \cdots + 931980112 \) Copy content Toggle raw display
$17$ \( T^{6} + 100 T^{5} + \cdots - 37296518336 \) Copy content Toggle raw display
$19$ \( T^{6} + 266 T^{5} + \cdots - 10918171712 \) Copy content Toggle raw display
$23$ \( T^{6} - 50 T^{5} + \cdots + 326139706 \) Copy content Toggle raw display
$29$ \( T^{6} + 242 T^{5} + \cdots + 106089693184 \) Copy content Toggle raw display
$31$ \( T^{6} + 438 T^{5} + \cdots - 2597606002574 \) Copy content Toggle raw display
$37$ \( T^{6} + 596 T^{5} + \cdots + 1310694126564 \) Copy content Toggle raw display
$41$ \( T^{6} - 54 T^{5} + \cdots + 8708288608008 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 153851666161248 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 556675819402176 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 246603742693452 \) Copy content Toggle raw display
$59$ \( T^{6} + 378 T^{5} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 409773359945408 \) Copy content Toggle raw display
$67$ \( (T + 67)^{6} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 591280964816128 \) Copy content Toggle raw display
$73$ \( T^{6} + 192 T^{5} + \cdots - 21\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{6} + 802 T^{5} + \cdots - 26\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{6} - 2126 T^{5} + \cdots + 30\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{6} - 432 T^{5} + \cdots - 38\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{6} - 1290 T^{5} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
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