Properties

Label 2678.2.a.s
Level $2678$
Weight $2$
Character orbit 2678.a
Self dual yes
Analytic conductor $21.384$
Analytic rank $1$
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: \(1\)
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: \( x^{10} - 4x^{9} - 8x^{8} + 37x^{7} + 20x^{6} - 106x^{5} - 17x^{4} + 90x^{3} + 2x^{2} - 17x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 8x^{8} + 37x^{7} + 20x^{6} - 106x^{5} - 17x^{4} + 90x^{3} + 2x^{2} - 17x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 5\nu^{8} + 24\nu^{6} - 14\nu^{5} - 16\nu^{4} - 5\nu^{3} - 15\nu^{2} + 16\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 15\nu^{3} + 16\nu^{2} - 10\nu - 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} + 5\nu^{8} + 2\nu^{7} - 32\nu^{6} + 6\nu^{5} + 60\nu^{4} + 7\nu^{3} - 35\nu^{2} - 10\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 5\nu^{8} - 2\nu^{7} + 34\nu^{6} - 14\nu^{5} - 66\nu^{4} + 29\nu^{3} + 33\nu^{2} - 8\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 3\nu^{8} - 12\nu^{7} + 32\nu^{6} + 44\nu^{5} - 94\nu^{4} - 53\nu^{3} + 57\nu^{2} + 10\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 4\nu^{8} - 9\nu^{7} + 40\nu^{6} + 26\nu^{5} - 118\nu^{4} - 27\nu^{3} + 86\nu^{2} - \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} + 6\nu^{8} - 3\nu^{7} - 36\nu^{6} + 46\nu^{5} + 60\nu^{4} - 89\nu^{3} - 38\nu^{2} + 41\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - 4\beta_{8} + 3\beta_{7} - \beta_{6} - 4\beta_{5} - 3\beta_{4} - 4\beta_{3} - 13\beta_{2} + 15\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{9} - 19 \beta_{8} + 15 \beta_{7} - 8 \beta_{6} - 20 \beta_{5} - 14 \beta_{4} - 19 \beta_{3} - 44 \beta_{2} + 65 \beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 32 \beta_{9} - 70 \beta_{8} + 51 \beta_{7} - 16 \beta_{6} - 73 \beta_{5} - 47 \beta_{4} - 70 \beta_{3} - 162 \beta_{2} + 189 \beta _1 + 236 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 127 \beta_{9} - 267 \beta_{8} + 197 \beta_{7} - 73 \beta_{6} - 282 \beta_{5} - 177 \beta_{4} - 266 \beta_{3} - 560 \beta_{2} + 701 \beta _1 + 741 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 417 \beta_{9} - 951 \beta_{8} + 685 \beta_{7} - 205 \beta_{6} - 1000 \beta_{5} - 609 \beta_{4} - 946 \beta_{3} - 1979 \beta_{2} + 2285 \beta _1 + 2678 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1508 \beta_{9} - 3410 \beta_{8} + 2464 \beta_{7} - 774 \beta_{6} - 3597 \beta_{5} - 2166 \beta_{4} - 3383 \beta_{3} - 6861 \beta_{2} + 8073 \beta _1 + 8979 \) 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
−1.94243
−1.93400
−1.06001
−0.401783
−0.126541
0.639130
0.894634
2.18512
2.28370
3.46218
1.00000 −2.94243 1.00000 −2.89430 −2.94243 −4.18233 1.00000 5.65791 −2.89430
1.2 1.00000 −2.93400 1.00000 1.44285 −2.93400 1.07406 1.00000 5.60834 1.44285
1.3 1.00000 −2.06001 1.00000 3.04261 −2.06001 −3.36532 1.00000 1.24362 3.04261
1.4 1.00000 −1.40178 1.00000 −0.640098 −1.40178 3.94750 1.00000 −1.03500 −0.640098
1.5 1.00000 −1.12654 1.00000 −1.35292 −1.12654 −1.19871 1.00000 −1.73091 −1.35292
1.6 1.00000 −0.360870 1.00000 4.09092 −0.360870 −3.96412 1.00000 −2.86977 4.09092
1.7 1.00000 −0.105366 1.00000 0.938819 −0.105366 1.20108 1.00000 −2.98890 0.938819
1.8 1.00000 1.18512 1.00000 −4.13063 1.18512 1.26664 1.00000 −1.59550 −4.13063
1.9 1.00000 1.28370 1.00000 −1.39563 1.28370 −1.74529 1.00000 −1.35211 −1.39563
1.10 1.00000 2.46218 1.00000 −2.10162 2.46218 −3.03351 1.00000 3.06232 −2.10162
\(n\): 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.s 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2678.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2678))\):

\( T_{3}^{10} + 6T_{3}^{9} + T_{3}^{8} - 51T_{3}^{7} - 71T_{3}^{6} + 91T_{3}^{5} + 194T_{3}^{4} - 7T_{3}^{3} - 136T_{3}^{2} - 52T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{10} + 3 T_{5}^{9} - 27 T_{5}^{8} - 87 T_{5}^{7} + 177 T_{5}^{6} + 707 T_{5}^{5} - 56 T_{5}^{4} - 1552 T_{5}^{3} - 672 T_{5}^{2} + 896 T_{5} + 512 \) Copy content Toggle raw display
\( T_{7}^{10} + 10 T_{7}^{9} + 11 T_{7}^{8} - 176 T_{7}^{7} - 556 T_{7}^{6} + 294 T_{7}^{5} + 2457 T_{7}^{4} + 358 T_{7}^{3} - 4009 T_{7}^{2} - 642 T_{7} + 2284 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 6 T^{9} + T^{8} - 51 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} - 27 T^{8} - 87 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( T^{10} + 10 T^{9} + 11 T^{8} + \cdots + 2284 \) Copy content Toggle raw display
$11$ \( T^{10} + 5 T^{9} - 53 T^{8} - 237 T^{7} + \cdots - 896 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 13 T^{9} - 22 T^{8} + \cdots + 13043 \) Copy content Toggle raw display
$19$ \( T^{10} + 21 T^{9} + 97 T^{8} + \cdots + 329728 \) Copy content Toggle raw display
$23$ \( T^{10} + 5 T^{9} - 124 T^{8} + \cdots - 794848 \) Copy content Toggle raw display
$29$ \( T^{10} - 6 T^{9} - 64 T^{8} + \cdots + 25088 \) Copy content Toggle raw display
$31$ \( T^{10} + 33 T^{9} + 351 T^{8} + \cdots - 16111616 \) Copy content Toggle raw display
$37$ \( T^{10} + 7 T^{9} - 117 T^{8} + \cdots - 84368 \) Copy content Toggle raw display
$41$ \( T^{10} + 20 T^{9} + 11 T^{8} + \cdots + 9812992 \) Copy content Toggle raw display
$43$ \( T^{10} + 22 T^{9} + 3 T^{8} + \cdots - 243808 \) Copy content Toggle raw display
$47$ \( T^{10} + 31 T^{9} + 296 T^{8} + \cdots - 16384 \) Copy content Toggle raw display
$53$ \( T^{10} - T^{9} - 356 T^{8} + \cdots + 9534896 \) Copy content Toggle raw display
$59$ \( T^{10} + 51 T^{9} + 937 T^{8} + \cdots - 286976 \) Copy content Toggle raw display
$61$ \( T^{10} + 6 T^{9} - 164 T^{8} + \cdots + 8192 \) Copy content Toggle raw display
$67$ \( T^{10} + 19 T^{9} - 98 T^{8} + \cdots + 10474088 \) Copy content Toggle raw display
$71$ \( T^{10} + 14 T^{9} + \cdots + 200385728 \) Copy content Toggle raw display
$73$ \( T^{10} + 16 T^{9} - 247 T^{8} + \cdots + 40136992 \) Copy content Toggle raw display
$79$ \( T^{10} - 11 T^{9} - 282 T^{8} + \cdots - 8075312 \) Copy content Toggle raw display
$83$ \( T^{10} - T^{9} - 543 T^{8} + \cdots + 4417073152 \) Copy content Toggle raw display
$89$ \( T^{10} + 47 T^{9} + 545 T^{8} + \cdots - 59988736 \) Copy content Toggle raw display
$97$ \( T^{10} + 44 T^{9} + \cdots + 1463781856 \) Copy content Toggle raw display
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