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Properties

Label 2667.2.a.p
Level $2667$
Weight $2$
Character orbit 2667.a
Self dual yes
Analytic conductor $21.296$
Analytic rank $1$
Dimension $18$
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: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9056232 \nu^{17} + 56952713 \nu^{16} + 63697503 \nu^{15} - 1083863790 \nu^{14} + \cdots - 228715354 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12516449 \nu^{17} - 35576866 \nu^{16} - 322120881 \nu^{15} + 878461675 \nu^{14} + \cdots - 1868332582 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25066381 \nu^{17} + 165457944 \nu^{16} + 197605069 \nu^{15} - 3294331535 \nu^{14} + \cdots - 2904833932 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32625911 \nu^{17} + 257402314 \nu^{16} + 38720489 \nu^{15} - 4899416105 \nu^{14} + \cdots - 523550232 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40185441 \nu^{17} + 349346684 \nu^{16} - 120164091 \nu^{15} - 6504500675 \nu^{14} + \cdots - 2527247372 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20878693 \nu^{17} + 77409897 \nu^{16} + 451307397 \nu^{15} - 1771179645 \nu^{14} + \cdots + 723097034 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8395741 \nu^{17} - 64482942 \nu^{16} - 11290113 \nu^{15} + 1199594231 \nu^{14} - 1945179675 \nu^{13} + \cdots - 76030588 ) / 146166028 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50383667 \nu^{17} - 161191148 \nu^{16} - 1204101853 \nu^{15} + 3787887405 \nu^{14} + \cdots - 2009629216 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80549203 \nu^{17} - 492648012 \nu^{16} - 821028837 \nu^{15} + 9925282045 \nu^{14} + \cdots + 3906261256 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 100575831 \nu^{17} - 597870654 \nu^{16} - 1070376159 \nu^{15} + 11961026425 \nu^{14} + \cdots + 4483050272 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 101911933 \nu^{17} + 591076942 \nu^{16} + 1177422127 \nu^{15} - 11988946255 \nu^{14} + \cdots + 1025152864 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10957014 \nu^{17} + 55541023 \nu^{16} + 166873740 \nu^{15} - 1164950524 \nu^{14} + \cdots + 267274566 ) / 73083014 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33634461 \nu^{17} - 212119814 \nu^{16} - 304035654 \nu^{15} + 4222090590 \nu^{14} + \cdots + 1153552087 ) / 182707535 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 100069624 \nu^{17} - 532974031 \nu^{16} - 1376337581 \nu^{15} + 10970946660 \nu^{14} + \cdots - 346733612 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 212880147 \nu^{17} - 1116211818 \nu^{16} - 2947927153 \nu^{15} + 22837760465 \nu^{14} + \cdots + 5373162964 ) / 730830140 \) 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
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{14} - \beta_{10} + \beta_{7} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{16} - \beta_{15} + 9 \beta_{14} - \beta_{13} - 9 \beta_{10} + 8 \beta_{7} + \beta_{6} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{16} + 11 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} - 2 \beta_{11} - 13 \beta_{10} + 12 \beta_{7} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70 \beta_{16} - 11 \beta_{15} + 69 \beta_{14} - 12 \beta_{13} - \beta_{12} - \beta_{11} - 72 \beta_{10} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{17} + 110 \beta_{16} + 92 \beta_{15} + 96 \beta_{14} + 77 \beta_{13} + 2 \beta_{12} - 29 \beta_{11} + \cdots + 575 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{17} + 525 \beta_{16} - 88 \beta_{15} + 510 \beta_{14} - 106 \beta_{13} - 14 \beta_{12} + \cdots + 698 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17 \beta_{17} + 926 \beta_{16} + 703 \beta_{15} + 786 \beta_{14} + 543 \beta_{13} + 32 \beta_{12} + \cdots + 3893 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 56 \beta_{17} + 3904 \beta_{16} - 606 \beta_{15} + 3746 \beta_{14} - 833 \beta_{13} - 137 \beta_{12} + \cdots + 5515 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 201 \beta_{17} + 7513 \beta_{16} + 5177 \beta_{15} + 6290 \beta_{14} + 3679 \beta_{13} + 330 \beta_{12} + \cdots + 27107 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 688 \beta_{17} + 29038 \beta_{16} - 3737 \beta_{15} + 27583 \beta_{14} - 6182 \beta_{13} - 1180 \beta_{12} + \cdots + 43005 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2053 \beta_{17} + 59783 \beta_{16} + 37549 \beta_{15} + 49823 \beta_{14} + 24374 \beta_{13} + \cdots + 192283 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 7080 \beta_{17} + 216596 \beta_{16} - 20422 \beta_{15} + 204059 \beta_{14} - 44489 \beta_{13} + \cdots + 333140 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 19439 \beta_{17} + 470153 \beta_{16} + 271056 \beta_{15} + 392329 \beta_{14} + 159039 \beta_{13} + \cdots + 1382442 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 66426 \beta_{17} + 1620828 \beta_{16} - 91207 \beta_{15} + 1516971 \beta_{14} - 314852 \beta_{13} + \cdots + 2572050 \) 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
2.77035
2.73355
2.59597
2.40246
1.77085
1.60620
1.52505
0.981962
0.803961
0.0366427
−0.375443
−0.432278
−0.665348
−1.33766
−1.70725
−1.90615
−2.24824
−2.55461
−2.77035 −1.00000 5.67483 −3.69380 2.77035 1.00000 −10.1806 1.00000 10.2331
1.2 −2.73355 −1.00000 5.47227 4.25416 2.73355 1.00000 −9.49162 1.00000 −11.6289
1.3 −2.59597 −1.00000 4.73905 −1.63234 2.59597 1.00000 −7.11048 1.00000 4.23750
1.4 −2.40246 −1.00000 3.77179 −0.666645 2.40246 1.00000 −4.25666 1.00000 1.60159
1.5 −1.77085 −1.00000 1.13590 1.38376 1.77085 1.00000 1.53019 1.00000 −2.45042
1.6 −1.60620 −1.00000 0.579871 −3.54808 1.60620 1.00000 2.28101 1.00000 5.69892
1.7 −1.52505 −1.00000 0.325779 −1.71268 1.52505 1.00000 2.55327 1.00000 2.61192
1.8 −0.981962 −1.00000 −1.03575 2.59441 0.981962 1.00000 2.98099 1.00000 −2.54761
1.9 −0.803961 −1.00000 −1.35365 0.343335 0.803961 1.00000 2.69620 1.00000 −0.276028
1.10 −0.0366427 −1.00000 −1.99866 1.02187 0.0366427 1.00000 0.146522 1.00000 −0.0374440
1.11 0.375443 −1.00000 −1.85904 2.49678 −0.375443 1.00000 −1.44885 1.00000 0.937397
1.12 0.432278 −1.00000 −1.81314 −4.10000 −0.432278 1.00000 −1.64833 1.00000 −1.77234
1.13 0.665348 −1.00000 −1.55731 −3.70415 −0.665348 1.00000 −2.36685 1.00000 −2.46455
1.14 1.33766 −1.00000 −0.210663 −0.660635 −1.33766 1.00000 −2.95712 1.00000 −0.883705
1.15 1.70725 −1.00000 0.914706 1.08728 −1.70725 1.00000 −1.85287 1.00000 1.85625
1.16 1.90615 −1.00000 1.63341 1.97979 −1.90615 1.00000 −0.698768 1.00000 3.77379
1.17 2.24824 −1.00000 3.05458 −3.31434 −2.24824 1.00000 2.37095 1.00000 −7.45143
1.18 2.55461 −1.00000 4.52602 −2.12871 −2.55461 1.00000 6.45298 1.00000 −5.43801
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.p ✓ 18
3.b odd 2 1 8001.2.a.u 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.p ✓ 18 1.a even 1 1 trivial
8001.2.a.u 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2667))\):

\( T_{2}^{18} + 6 T_{2}^{17} - 11 T_{2}^{16} - 123 T_{2}^{15} - 35 T_{2}^{14} + 982 T_{2}^{13} + 988 T_{2}^{12} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{18} + 10 T_{5}^{17} - 9 T_{5}^{16} - 374 T_{5}^{15} - 651 T_{5}^{14} + 4832 T_{5}^{13} + \cdots + 49792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 6 T^{17} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T + 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 10 T^{17} + \cdots + 49792 \) Copy content Toggle raw display
$7$ \( (T - 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + 9 T^{17} + \cdots + 1280000 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 623244800 \) Copy content Toggle raw display
$17$ \( T^{18} + 17 T^{17} + \cdots + 85910912 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 2183223712 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 48663006208 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 2717396224 \) Copy content Toggle raw display
$31$ \( T^{18} - 5 T^{17} + \cdots + 5492768 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 131904255436 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 1896176000 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 5710191104 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 31187414908224 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1454748563200 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 711880908800 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 167749132480000 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 13848455168 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 514861187238656 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 80005666705024 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 17570271507200 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 30\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 89133416579072 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 25023269459072 \) Copy content Toggle raw display
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