Properties

Label 2678.2.a.r
Level $2678$
Weight $2$
Character orbit 2678.a
Self dual yes
Analytic conductor $21.384$
Analytic rank $0$
Dimension $10$
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] = [2678,2,Mod(1,2678)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2678, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2678.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2678 = 2 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2678.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: \(21.3839376613\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 16x^{8} - 3x^{7} + 80x^{6} + 24x^{5} - 137x^{4} - 52x^{3} + 48x^{2} + x - 2 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} - 3x^{7} + 80x^{6} + 24x^{5} - 137x^{4} - 52x^{3} + 48x^{2} + x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 81 \nu^{9} - 61 \nu^{8} - 1265 \nu^{7} + 626 \nu^{6} + 6146 \nu^{5} - 1588 \nu^{4} - 9683 \nu^{3} - 275 \nu^{2} + 2183 \nu - 108 ) / 242 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51 \nu^{9} - 16 \nu^{8} - 792 \nu^{7} + 67 \nu^{6} + 3798 \nu^{5} + 367 \nu^{4} - 6025 \nu^{3} - 1903 \nu^{2} + 1612 \nu + 295 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 107 \nu^{9} + 21 \nu^{8} - 1683 \nu^{7} - 640 \nu^{6} + 8068 \nu^{5} + 4018 \nu^{4} - 12337 \nu^{3} - 7931 \nu^{2} + 1847 \nu + 906 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 107 \nu^{9} - 21 \nu^{8} + 1683 \nu^{7} + 640 \nu^{6} - 8068 \nu^{5} - 4018 \nu^{4} + 12337 \nu^{3} + 7689 \nu^{2} - 1605 \nu - 180 ) / 242 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 139 \nu^{9} + 27 \nu^{8} + 2123 \nu^{7} + 76 \nu^{6} - 9782 \nu^{5} - 2094 \nu^{4} + 13835 \nu^{3} + 6259 \nu^{2} - 1117 \nu - 460 ) / 242 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 137 \nu^{9} + 97 \nu^{8} - 2277 \nu^{7} - 1896 \nu^{6} + 11868 \nu^{5} + 10170 \nu^{4} - 21077 \nu^{3} - 16467 \nu^{2} + 5685 \nu + 382 ) / 242 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 211 \nu^{9} + 107 \nu^{8} - 3355 \nu^{7} - 2316 \nu^{6} + 16240 \nu^{5} + 13132 \nu^{4} - 24889 \nu^{3} - 23309 \nu^{2} + 2439 \nu + 2058 ) / 242 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 107 \nu^{9} - 21 \nu^{8} + 1683 \nu^{7} + 640 \nu^{6} - 8068 \nu^{5} - 4018 \nu^{4} + 12458 \nu^{3} + 7810 \nu^{2} - 2452 \nu - 664 ) / 121 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{5} + \beta_{4} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - 9\beta_{5} - 7\beta_{4} + 2\beta_{3} + 10\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{9} + \beta_{6} - 13\beta_{5} + 8\beta_{4} + 4\beta_{3} - 2\beta_{2} + 45\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{9} + 9 \beta_{8} - 11 \beta_{7} + 23 \beta_{6} - 79 \beta_{5} - 45 \beta_{4} + 22 \beta_{3} - \beta_{2} + 93 \beta _1 + 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 102 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 15 \beta_{6} - 140 \beta_{5} + 51 \beta_{4} + 57 \beta_{3} - 29 \beta_{2} + 371 \beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 155 \beta_{9} + 70 \beta_{8} - 99 \beta_{7} + 214 \beta_{6} - 692 \beta_{5} - 295 \beta_{4} + 206 \beta_{3} - 22 \beta_{2} + 865 \beta _1 + 896 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 906 \beta_{9} + 34 \beta_{8} - 56 \beta_{7} + 181 \beta_{6} - 1410 \beta_{5} + 294 \beta_{4} + 611 \beta_{3} - 307 \beta_{2} + 3202 \beta _1 + 1339 \) 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
−2.63388
−2.26430
−1.30337
−1.13321
−0.204243
0.257501
0.382373
1.80625
2.06311
3.02977
−1.00000 −2.63388 1.00000 2.52869 2.63388 −0.770546 −1.00000 3.93732 −2.52869
1.2 −1.00000 −2.26430 1.00000 −1.40930 2.26430 3.97629 −1.00000 2.12706 1.40930
1.3 −1.00000 −1.30337 1.00000 2.84126 1.30337 4.92791 −1.00000 −1.30123 −2.84126
1.4 −1.00000 −1.13321 1.00000 −0.171699 1.13321 −1.69110 −1.00000 −1.71583 0.171699
1.5 −1.00000 −0.204243 1.00000 1.52054 0.204243 −4.26071 −1.00000 −2.95828 −1.52054
1.6 −1.00000 0.257501 1.00000 −3.02407 −0.257501 2.86603 −1.00000 −2.93369 3.02407
1.7 −1.00000 0.382373 1.00000 −1.76419 −0.382373 −1.99315 −1.00000 −2.85379 1.76419
1.8 −1.00000 1.80625 1.00000 3.54615 −1.80625 −1.05176 −1.00000 0.262540 −3.54615
1.9 −1.00000 2.06311 1.00000 3.73268 −2.06311 4.82617 −1.00000 1.25642 −3.73268
1.10 −1.00000 3.02977 1.00000 1.19993 −3.02977 2.17086 −1.00000 6.17950 −1.19993
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(1\)
\(103\) \(-1\)

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 2678.2.a.r 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2678.2.a.r 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2678))\):

\( T_{3}^{10} - 16T_{3}^{8} - 3T_{3}^{7} + 80T_{3}^{6} + 24T_{3}^{5} - 137T_{3}^{4} - 52T_{3}^{3} + 48T_{3}^{2} + T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{10} - 9 T_{5}^{9} + 11 T_{5}^{8} + 109 T_{5}^{7} - 312 T_{5}^{6} - 204 T_{5}^{5} + 1348 T_{5}^{4} - 480 T_{5}^{3} - 1664 T_{5}^{2} + 1040 T_{5} + 224 \) Copy content Toggle raw display
\( T_{7}^{10} - 9 T_{7}^{9} - 11 T_{7}^{8} + 263 T_{7}^{7} - 232 T_{7}^{6} - 2252 T_{7}^{5} + 2012 T_{7}^{4} + 8864 T_{7}^{3} - 2416 T_{7}^{2} - 14416 T_{7} - 6848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 16 T^{8} - 3 T^{7} + 80 T^{6} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{10} - 9 T^{9} + 11 T^{8} + 109 T^{7} + \cdots + 224 \) Copy content Toggle raw display
$7$ \( T^{10} - 9 T^{9} - 11 T^{8} + \cdots - 6848 \) Copy content Toggle raw display
$11$ \( T^{10} - 9 T^{9} - 22 T^{8} + \cdots - 5888 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - T^{9} - 94 T^{8} + 111 T^{7} + \cdots - 24139 \) Copy content Toggle raw display
$19$ \( T^{10} - 8 T^{9} - 35 T^{8} + 319 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{10} - 9 T^{9} - 102 T^{8} + \cdots + 110336 \) Copy content Toggle raw display
$29$ \( T^{10} - 3 T^{9} - 82 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( T^{10} + 3 T^{9} - 152 T^{8} + \cdots + 267776 \) Copy content Toggle raw display
$37$ \( T^{10} - 11 T^{9} - 169 T^{8} + \cdots - 473728 \) Copy content Toggle raw display
$41$ \( T^{10} - 6 T^{9} - 186 T^{8} + \cdots - 191744 \) Copy content Toggle raw display
$43$ \( T^{10} - 23 T^{9} + 87 T^{8} + \cdots + 3790216 \) Copy content Toggle raw display
$47$ \( T^{10} - 28 T^{9} + 205 T^{8} + \cdots + 4910528 \) Copy content Toggle raw display
$53$ \( T^{10} - 19 T^{9} - 118 T^{8} + \cdots + 3731456 \) Copy content Toggle raw display
$59$ \( T^{10} + 20 T^{9} + 23 T^{8} + \cdots - 6376 \) Copy content Toggle raw display
$61$ \( T^{10} - 25 T^{9} + 186 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( T^{10} - 19 T^{9} - 30 T^{8} + \cdots - 1707008 \) Copy content Toggle raw display
$71$ \( T^{10} + 5 T^{9} - 375 T^{8} + \cdots + 21238436 \) Copy content Toggle raw display
$73$ \( T^{10} - 29 T^{9} + 175 T^{8} + \cdots + 4822772 \) Copy content Toggle raw display
$79$ \( T^{10} + 3 T^{9} - 344 T^{8} + \cdots - 144576064 \) Copy content Toggle raw display
$83$ \( T^{10} - 18 T^{9} + \cdots + 139212256 \) Copy content Toggle raw display
$89$ \( T^{10} + 10 T^{9} - 165 T^{8} + \cdots + 195944 \) Copy content Toggle raw display
$97$ \( T^{10} - 26 T^{9} - 336 T^{8} + \cdots + 24989696 \) Copy content Toggle raw display
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