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Properties

Label 2667.2.a.n
Level $2667$
Weight $2$
Character orbit 2667.a
Self dual yes
Analytic conductor $21.296$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) 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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9} + ( - \beta_{9} - \beta_{5}) q^{10} - \beta_{8} q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{11} + \beta_{9} + 1) q^{13} - \beta_1 q^{14} + \beta_{7} q^{15} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{16} + (\beta_{11} + \beta_{7} - \beta_{6}) q^{17} + \beta_1 q^{18} + ( - \beta_{15} + \beta_{12} - \beta_{6} + 1) q^{19} + (\beta_{15} + \beta_{12} + \cdots + \beta_1) q^{20}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 16 q^{3} + 20 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 16 q^{3} + 20 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{7} + 15 q^{8} + 16 q^{9} - 4 q^{10} + q^{11} + 20 q^{12} + 20 q^{13} - 4 q^{14} + 5 q^{15} + 32 q^{16} + 3 q^{17} + 4 q^{18} + 13 q^{19} + 17 q^{20} - 16 q^{21} + 13 q^{22} + 5 q^{23} + 15 q^{24} + 17 q^{25} - 2 q^{26} + 16 q^{27} - 20 q^{28} + 22 q^{29} - 4 q^{30} + 26 q^{31} + 54 q^{32} + q^{33} - 6 q^{34} - 5 q^{35} + 20 q^{36} + 30 q^{37} + 5 q^{38} + 20 q^{39} + 13 q^{40} + q^{41} - 4 q^{42} + 31 q^{43} + 22 q^{44} + 5 q^{45} - 2 q^{46} - q^{47} + 32 q^{48} + 16 q^{49} + 5 q^{50} + 3 q^{51} + 31 q^{52} + 24 q^{53} + 4 q^{54} + 8 q^{55} - 15 q^{56} + 13 q^{57} + 13 q^{58} - 17 q^{59} + 17 q^{60} + 32 q^{61} - 5 q^{62} - 16 q^{63} + 61 q^{64} - 3 q^{65} + 13 q^{66} + 16 q^{67} - 10 q^{68} + 5 q^{69} + 4 q^{70} - 10 q^{71} + 15 q^{72} + 23 q^{73} + q^{74} + 17 q^{75} + 18 q^{76} - q^{77} - 2 q^{78} + 48 q^{79} + 38 q^{80} + 16 q^{81} + 12 q^{82} + 9 q^{83} - 20 q^{84} + 22 q^{85} - 4 q^{86} + 22 q^{87} + 27 q^{88} + 17 q^{89} - 4 q^{90} - 20 q^{91} + 16 q^{92} + 26 q^{93} + 13 q^{94} + 22 q^{95} + 54 q^{96} + 17 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16172 \nu^{15} - 233047 \nu^{14} + 545740 \nu^{13} + 3569234 \nu^{12} - 13969268 \nu^{11} + \cdots - 1833530 ) / 739814 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55506 \nu^{15} + 171907 \nu^{14} + 1127781 \nu^{13} - 3471014 \nu^{12} - 8968081 \nu^{11} + \cdots - 3035278 ) / 739814 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 79649 \nu^{15} + 179031 \nu^{14} + 1937788 \nu^{13} - 4128253 \nu^{12} - 18731714 \nu^{11} + \cdots - 1805692 ) / 739814 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49011 \nu^{15} + 202993 \nu^{14} + 777233 \nu^{13} - 3856011 \nu^{12} - 3798508 \nu^{11} + \cdots + 482636 ) / 369907 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 123393 \nu^{15} - 271059 \nu^{14} - 3028354 \nu^{13} + 6241792 \nu^{12} + 29665237 \nu^{11} + \cdots + 3426510 ) / 739814 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 125049 \nu^{15} - 467479 \nu^{14} - 2288559 \nu^{13} + 9361224 \nu^{12} + 15315132 \nu^{11} + \cdots + 4758460 ) / 739814 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 129018 \nu^{15} - 383386 \nu^{14} - 2688667 \nu^{13} + 7911672 \nu^{12} + 21971299 \nu^{11} + \cdots + 725548 ) / 739814 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 137239 \nu^{15} + 515922 \nu^{14} + 2401726 \nu^{13} - 10047088 \nu^{12} - 14807929 \nu^{11} + \cdots - 4842758 ) / 739814 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 145145 \nu^{15} - 426142 \nu^{14} - 3134036 \nu^{13} + 9144531 \nu^{12} + 26680262 \nu^{11} + \cdots + 5228678 ) / 739814 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 158942 \nu^{15} - 398038 \nu^{14} - 3767459 \nu^{13} + 8967247 \nu^{12} + 35638598 \nu^{11} + \cdots + 7379728 ) / 739814 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 322032 \nu^{15} - 1082571 \nu^{14} - 6442221 \nu^{13} + 22367158 \nu^{12} + 49725923 \nu^{11} + \cdots + 7468534 ) / 739814 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{15} - 14 \beta_{14} - 13 \beta_{13} - 14 \beta_{12} - 17 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{15} - 19 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} - 14 \beta_{11} - 61 \beta_{10} + \cdots + 520 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 162 \beta_{15} - 146 \beta_{14} - 125 \beta_{13} - 139 \beta_{12} + 2 \beta_{11} - 196 \beta_{10} + \cdots + 203 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 378 \beta_{15} - 239 \beta_{14} - 59 \beta_{13} + 62 \beta_{12} - 129 \beta_{11} - 657 \beta_{10} + \cdots + 3232 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1538 \beta_{15} - 1365 \beta_{14} - 1076 \beta_{13} - 1217 \beta_{12} + 47 \beta_{11} - 1930 \beta_{10} + \cdots + 1641 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3733 \beta_{15} - 2530 \beta_{14} - 752 \beta_{13} + 223 \beta_{12} - 982 \beta_{11} - 6181 \beta_{10} + \cdots + 20471 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13671 \beta_{15} - 12084 \beta_{14} - 8781 \beta_{13} - 10051 \beta_{12} + 703 \beta_{11} - 17520 \beta_{10} + \cdots + 12767 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 34127 \beta_{15} - 24451 \beta_{14} - 7956 \beta_{13} - 756 \beta_{12} - 6564 \beta_{11} - 54356 \beta_{10} + \cdots + 131697 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 116930 \beta_{15} - 103619 \beta_{14} - 69592 \beta_{13} - 80533 \beta_{12} + 8580 \beta_{11} + \cdots + 97481 \) Copy content Toggle raw display

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

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
−2.57436
−2.35145
−1.84433
−1.40936
−1.35902
−1.14773
−0.0643619
0.284398
0.716430
0.825012
1.17466
1.50110
2.35655
2.45059
2.63870
2.80317
−2.57436 1.00000 4.62732 2.78465 −2.57436 −1.00000 −6.76367 1.00000 −7.16870
1.2 −2.35145 1.00000 3.52933 −2.01529 −2.35145 −1.00000 −3.59614 1.00000 4.73886
1.3 −1.84433 1.00000 1.40154 2.16633 −1.84433 −1.00000 1.10376 1.00000 −3.99542
1.4 −1.40936 1.00000 −0.0136985 2.06019 −1.40936 −1.00000 2.83803 1.00000 −2.90356
1.5 −1.35902 1.00000 −0.153057 −2.54404 −1.35902 −1.00000 2.92605 1.00000 3.45741
1.6 −1.14773 1.00000 −0.682718 1.07917 −1.14773 −1.00000 3.07903 1.00000 −1.23860
1.7 −0.0643619 1.00000 −1.99586 3.34910 −0.0643619 −1.00000 0.257181 1.00000 −0.215554
1.8 0.284398 1.00000 −1.91912 −3.66581 0.284398 −1.00000 −1.11459 1.00000 −1.04255
1.9 0.716430 1.00000 −1.48673 0.617976 0.716430 −1.00000 −2.49800 1.00000 0.442737
1.10 0.825012 1.00000 −1.31936 −1.66882 0.825012 −1.00000 −2.73851 1.00000 −1.37680
1.11 1.17466 1.00000 −0.620175 3.75278 1.17466 −1.00000 −3.07781 1.00000 4.40823
1.12 1.50110 1.00000 0.253312 −2.96917 1.50110 −1.00000 −2.62196 1.00000 −4.45703
1.13 2.35655 1.00000 3.55331 2.88159 2.35655 −1.00000 3.66045 1.00000 6.79060
1.14 2.45059 1.00000 4.00539 −2.70231 2.45059 −1.00000 4.91439 1.00000 −6.62224
1.15 2.63870 1.00000 4.96274 0.422965 2.63870 −1.00000 7.81779 1.00000 1.11608
1.16 2.80317 1.00000 5.85777 1.45069 2.80317 −1.00000 10.8140 1.00000 4.06653
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.n ✓ 16
3.b odd 2 1 8001.2.a.s 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.n ✓ 16 1.a even 1 1 trivial
8001.2.a.s 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2667))\):

\( T_{2}^{16} - 4 T_{2}^{15} - 18 T_{2}^{14} + 83 T_{2}^{13} + 112 T_{2}^{12} - 668 T_{2}^{11} - 235 T_{2}^{10} + \cdots - 20 \) Copy content Toggle raw display
\( T_{5}^{16} - 5 T_{5}^{15} - 36 T_{5}^{14} + 210 T_{5}^{13} + 435 T_{5}^{12} - 3467 T_{5}^{11} + \cdots + 46352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots - 20 \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 5 T^{15} + \cdots + 46352 \) Copy content Toggle raw display
$7$ \( (T + 1)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - T^{15} + \cdots - 4096 \) Copy content Toggle raw display
$13$ \( T^{16} - 20 T^{15} + \cdots - 143360 \) Copy content Toggle raw display
$17$ \( T^{16} - 3 T^{15} + \cdots - 27287872 \) Copy content Toggle raw display
$19$ \( T^{16} - 13 T^{15} + \cdots - 59975840 \) Copy content Toggle raw display
$23$ \( T^{16} - 5 T^{15} + \cdots - 92323840 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 265579040 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 2220064768 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2281961720 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 6198211456 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 270850329856 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 80836384768 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3330453274336 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2877741056 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 9236458496 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5716113605888 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4708249600 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 30744464128 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1732480000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6896650240 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 550072064 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 24693460838972 \) Copy content Toggle raw display
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