Properties

Label 2667.2.a.l
Level $2667$
Weight $2$
Character orbit 2667.a
Self dual yes
Analytic conductor $21.296$
Analytic rank $0$
Dimension $13$
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] = [2667,2,Mod(1,2667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2667, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2667.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2667 = 3 \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2667.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: \(21.2961022191\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} + 1) q^{5} - \beta_1 q^{6} + q^{7} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} + 1) q^{5} - \beta_1 q^{6} + q^{7} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{11} + \beta_1) q^{10} - \beta_{8} q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{12} - \beta_{7} + \beta_{4} + 2) q^{13} + \beta_1 q^{14} + (\beta_{10} - 1) q^{15} + ( - \beta_{12} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_1 + 1) q^{16} + ( - \beta_{9} - \beta_{5} + \beta_{4} + 1) q^{17} + \beta_1 q^{18} + (\beta_{12} - \beta_{8} - \beta_{4}) q^{19} + (\beta_{6} + \beta_{5} + \beta_{2} + \beta_1 + 3) q^{20} - q^{21} + (\beta_{12} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{3}) q^{22} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 3 \beta_1) q^{23} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{24} + ( - 3 \beta_{10} + \beta_{7} + \beta_{3} - \beta_{2} + \beta_1) q^{25} + ( - \beta_{12} - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{26} - q^{27} + (\beta_{2} + 1) q^{28} + (\beta_{11} - \beta_{10} - \beta_{6} + \beta_{4} - \beta_1 + 2) q^{29} + (\beta_{11} - \beta_1) q^{30} + (\beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{32}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} - 13 q^{3} + 10 q^{4} + 12 q^{5} - 4 q^{6} + 13 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 3 q^{11} - 10 q^{12} + 21 q^{13} + 4 q^{14} - 12 q^{15} + 8 q^{16} + 17 q^{17} + 4 q^{18} + 5 q^{19} + 29 q^{20} - 13 q^{21} + q^{22} + 4 q^{23} - 9 q^{24} + q^{25} + 22 q^{26} - 13 q^{27} + 10 q^{28} + 21 q^{29} - 6 q^{30} - 7 q^{31} + 12 q^{32} - 3 q^{33} + 2 q^{34} + 12 q^{35} + 10 q^{36} + 7 q^{37} - 9 q^{38} - 21 q^{39} + 29 q^{40} + 21 q^{41} - 4 q^{42} - 9 q^{43} - 2 q^{44} + 12 q^{45} - 28 q^{46} + 23 q^{47} - 8 q^{48} + 13 q^{49} + 15 q^{50} - 17 q^{51} + 15 q^{52} + 31 q^{53} - 4 q^{54} - 8 q^{55} + 9 q^{56} - 5 q^{57} - 25 q^{58} + 28 q^{59} - 29 q^{60} + 29 q^{61} - 3 q^{62} + 13 q^{63} + 9 q^{64} + 30 q^{65} - q^{66} - 18 q^{67} + 34 q^{68} - 4 q^{69} + 6 q^{70} + 10 q^{71} + 9 q^{72} + 24 q^{73} - 19 q^{74} - q^{75} + 3 q^{77} - 22 q^{78} - 28 q^{79} + 26 q^{80} + 13 q^{81} + 18 q^{82} + 26 q^{83} - 10 q^{84} + 20 q^{85} - 2 q^{86} - 21 q^{87} - 17 q^{88} + 44 q^{89} + 6 q^{90} + 21 q^{91} + 6 q^{92} + 7 q^{93} - 9 q^{94} - 2 q^{95} - 12 q^{96} + 17 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 18 \nu^{10} + 29 \nu^{9} + 125 \nu^{8} - 144 \nu^{7} - 411 \nu^{6} + 285 \nu^{5} + 615 \nu^{4} - 190 \nu^{3} - 334 \nu^{2} - 12 \nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 26 \nu^{9} + 84 \nu^{8} - 114 \nu^{7} - 224 \nu^{6} + 198 \nu^{5} + 287 \nu^{4} - 109 \nu^{3} - 150 \nu^{2} - 10 \nu + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} + 2 \nu^{11} - 22 \nu^{10} - 27 \nu^{9} + 169 \nu^{8} + 120 \nu^{7} - 571 \nu^{6} - 203 \nu^{5} + 871 \nu^{4} + 118 \nu^{3} - 502 \nu^{2} - 28 \nu + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 586 \nu^{3} - 154 \nu^{2} + 72 \nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 590 \nu^{3} - 154 \nu^{2} + 92 \nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{12} + 4 \nu^{11} + 8 \nu^{10} - 51 \nu^{9} + 9 \nu^{8} + 220 \nu^{7} - 193 \nu^{6} - 387 \nu^{5} + 455 \nu^{4} + 246 \nu^{3} - 302 \nu^{2} - 44 \nu + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2 \nu^{12} + 5 \nu^{11} + 27 \nu^{10} - 65 \nu^{9} - 128 \nu^{8} + 285 \nu^{7} + 269 \nu^{6} - 498 \nu^{5} - 265 \nu^{4} + 298 \nu^{3} + 138 \nu^{2} - 30 \nu - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 44 \nu^{10} + 109 \nu^{9} + 243 \nu^{8} - 510 \nu^{7} - 655 \nu^{6} + 975 \nu^{5} + 883 \nu^{4} - 670 \nu^{3} - 518 \nu^{2} + 60 \nu + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2 \nu^{12} - 7 \nu^{11} - 25 \nu^{10} + 93 \nu^{9} + 108 \nu^{8} - 419 \nu^{7} - 213 \nu^{6} + 758 \nu^{5} + 223 \nu^{4} - 478 \nu^{3} - 144 \nu^{2} + 42 \nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{12} - 7 \nu^{11} - 42 \nu^{10} + 91 \nu^{9} + 212 \nu^{8} - 399 \nu^{7} - 493 \nu^{6} + 696 \nu^{5} + 550 \nu^{4} - 405 \nu^{3} - 276 \nu^{2} + 12 \nu + 18 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 8 \beta_{7} + 9 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 30 \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} - 11 \beta_{9} - \beta_{8} - 10 \beta_{7} + 11 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} + 5 \beta_{3} + 35 \beta_{2} + 14 \beta _1 + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16 \beta_{12} + 14 \beta_{11} - 27 \beta_{10} - 15 \beta_{9} - 2 \beta_{8} - 58 \beta_{7} + 68 \beta_{6} + 14 \beta_{5} - 8 \beta_{4} + 29 \beta_{3} + 193 \beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 109 \beta_{12} + 31 \beta_{11} - 59 \beta_{10} - 95 \beta_{9} - 14 \beta_{8} - 84 \beta_{7} + 98 \beta_{6} + 31 \beta_{5} + 77 \beta_{4} + 72 \beta_{3} + 207 \beta_{2} + 141 \beta _1 + 571 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 174 \beta_{12} + 139 \beta_{11} - 262 \beta_{10} - 156 \beta_{9} - 31 \beta_{8} - 414 \beta_{7} + 495 \beta_{6} + 140 \beta_{5} - 44 \beta_{4} + 293 \beta_{3} + 4 \beta_{2} + 1284 \beta _1 + 694 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 894 \beta_{12} + 327 \beta_{11} - 607 \beta_{10} - 757 \beta_{9} - 140 \beta_{8} - 673 \beta_{7} + 812 \beta_{6} + 328 \beta_{5} + 546 \beta_{4} + 732 \beta_{3} + 1246 \beta_{2} + 1252 \beta _1 + 3811 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1613 \beta_{12} + 1210 \beta_{11} - 2248 \beta_{10} - 1404 \beta_{9} - 328 \beta_{8} - 2952 \beta_{7} + 3573 \beta_{6} + 1226 \beta_{5} - 175 \beta_{4} + 2571 \beta_{3} + 83 \beta_{2} + 8714 \beta _1 + 5431 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6983 \beta_{12} + 2953 \beta_{11} - 5411 \beta_{10} - 5816 \beta_{9} - 1226 \beta_{8} - 5269 \beta_{7} + 6480 \beta_{6} + 2974 \beta_{5} + 3757 \beta_{4} + 6486 \beta_{3} + 7632 \beta_{2} + 10446 \beta _1 + 26023 \) 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
−2.45160
−1.82297
−1.58856
−0.910949
−0.423652
−0.305711
0.307326
1.18273
1.27342
1.45503
2.28328
2.29288
2.70878
−2.45160 −1.00000 4.01034 1.23909 2.45160 1.00000 −4.92856 1.00000 −3.03775
1.2 −1.82297 −1.00000 1.32323 0.630622 1.82297 1.00000 1.23374 1.00000 −1.14961
1.3 −1.58856 −1.00000 0.523519 3.57671 1.58856 1.00000 2.34548 1.00000 −5.68181
1.4 −0.910949 −1.00000 −1.17017 −2.06751 0.910949 1.00000 2.88787 1.00000 1.88339
1.5 −0.423652 −1.00000 −1.82052 2.66196 0.423652 1.00000 1.61857 1.00000 −1.12774
1.6 −0.305711 −1.00000 −1.90654 −0.276458 0.305711 1.00000 1.19427 1.00000 0.0845162
1.7 0.307326 −1.00000 −1.90555 −0.988649 −0.307326 1.00000 −1.20028 1.00000 −0.303838
1.8 1.18273 −1.00000 −0.601156 4.39766 −1.18273 1.00000 −3.07646 1.00000 5.20123
1.9 1.27342 −1.00000 −0.378389 −2.08575 −1.27342 1.00000 −3.02870 1.00000 −2.65604
1.10 1.45503 −1.00000 0.117115 −0.420019 −1.45503 1.00000 −2.73966 1.00000 −0.611141
1.11 2.28328 −1.00000 3.21337 2.58645 −2.28328 1.00000 2.77047 1.00000 5.90560
1.12 2.29288 −1.00000 3.25728 −0.132682 −2.29288 1.00000 2.88278 1.00000 −0.304223
1.13 2.70878 −1.00000 5.33747 2.87857 −2.70878 1.00000 9.04047 1.00000 7.79741
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(127\) \(1\)

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 2667.2.a.l 13
3.b odd 2 1 8001.2.a.o 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.l 13 1.a even 1 1 trivial
8001.2.a.o 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2667))\):

\( T_{2}^{13} - 4 T_{2}^{12} - 10 T_{2}^{11} + 53 T_{2}^{10} + 19 T_{2}^{9} - 242 T_{2}^{8} + 61 T_{2}^{7} + 467 T_{2}^{6} - 211 T_{2}^{5} - 372 T_{2}^{4} + 146 T_{2}^{3} + 116 T_{2}^{2} - 12 T_{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{13} - 12 T_{5}^{12} + 39 T_{5}^{11} + 50 T_{5}^{10} - 457 T_{5}^{9} + 340 T_{5}^{8} + 1492 T_{5}^{7} - 1877 T_{5}^{6} - 1590 T_{5}^{5} + 1990 T_{5}^{4} + 784 T_{5}^{3} - 412 T_{5}^{2} - 184 T_{5} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 4 T^{12} - 10 T^{11} + 53 T^{10} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 12 T^{12} + 39 T^{11} + 50 T^{10} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} - 3 T^{12} - 60 T^{11} + \cdots + 284672 \) Copy content Toggle raw display
$13$ \( T^{13} - 21 T^{12} + 134 T^{11} + \cdots - 22336 \) Copy content Toggle raw display
$17$ \( T^{13} - 17 T^{12} + 37 T^{11} + \cdots + 27656 \) Copy content Toggle raw display
$19$ \( T^{13} - 5 T^{12} - 74 T^{11} + \cdots - 16768 \) Copy content Toggle raw display
$23$ \( T^{13} - 4 T^{12} - 115 T^{11} + \cdots + 975616 \) Copy content Toggle raw display
$29$ \( T^{13} - 21 T^{12} + 44 T^{11} + \cdots + 92930644 \) Copy content Toggle raw display
$31$ \( T^{13} + 7 T^{12} - 186 T^{11} + \cdots + 15003392 \) Copy content Toggle raw display
$37$ \( T^{13} - 7 T^{12} - 132 T^{11} + \cdots + 1844308 \) Copy content Toggle raw display
$41$ \( T^{13} - 21 T^{12} + 83 T^{11} + \cdots + 184216 \) Copy content Toggle raw display
$43$ \( T^{13} + 9 T^{12} - 196 T^{11} + \cdots - 28880896 \) Copy content Toggle raw display
$47$ \( T^{13} - 23 T^{12} - 4 T^{11} + \cdots - 9569152 \) Copy content Toggle raw display
$53$ \( T^{13} - 31 T^{12} + 247 T^{11} + \cdots + 405812 \) Copy content Toggle raw display
$59$ \( T^{13} - 28 T^{12} + \cdots + 43041792704 \) Copy content Toggle raw display
$61$ \( T^{13} - 29 T^{12} + 230 T^{11} + \cdots - 20477872 \) Copy content Toggle raw display
$67$ \( T^{13} + 18 T^{12} + \cdots + 49278934912 \) Copy content Toggle raw display
$71$ \( T^{13} - 10 T^{12} + \cdots - 15605244416 \) Copy content Toggle raw display
$73$ \( T^{13} - 24 T^{12} + \cdots + 169328048 \) Copy content Toggle raw display
$79$ \( T^{13} + 28 T^{12} - 2 T^{11} + \cdots - 18990976 \) Copy content Toggle raw display
$83$ \( T^{13} - 26 T^{12} + \cdots + 28884464384 \) Copy content Toggle raw display
$89$ \( T^{13} - 44 T^{12} + \cdots + 180656384 \) Copy content Toggle raw display
$97$ \( T^{13} - 17 T^{12} + \cdots - 416205928 \) Copy content Toggle raw display
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