↑

Properties

Label 2678.2.a.w
Level $2678$
Weight $2$
Character orbit 2678.a
Self dual yes
Analytic conductor $21.384$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 6 x^{18} - 25 x^{17} + 203 x^{16} + 149 x^{15} - 2691 x^{14} + 997 x^{13} + 17945 x^{12} + \cdots - 992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 6 x^{18} - 25 x^{17} + 203 x^{16} + 149 x^{15} - 2691 x^{14} + 997 x^{13} + 17945 x^{12} + \cdots - 992 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!03 \nu^{18} + \cdots + 25\!\cdots\!12 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\!\cdots\!07 \nu^{18} + \cdots + 44\!\cdots\!16 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51\!\cdots\!61 \nu^{18} + \cdots + 21\!\cdots\!48 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 90\!\cdots\!50 \nu^{18} + \cdots - 14\!\cdots\!44 ) / 13\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43\!\cdots\!41 \nu^{18} + \cdots - 34\!\cdots\!68 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 45\!\cdots\!01 \nu^{18} + \cdots + 12\!\cdots\!12 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!37 \nu^{18} + \cdots - 10\!\cdots\!12 ) / 13\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!15 \nu^{18} + \cdots - 34\!\cdots\!20 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43\!\cdots\!85 \nu^{18} + \cdots - 82\!\cdots\!16 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57\!\cdots\!33 \nu^{18} + \cdots + 88\!\cdots\!96 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!97 \nu^{18} + \cdots - 17\!\cdots\!04 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 94\!\cdots\!87 \nu^{18} + \cdots - 22\!\cdots\!68 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 50\!\cdots\!26 \nu^{18} + \cdots - 16\!\cdots\!80 ) / 13\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{18} + \cdots - 35\!\cdots\!48 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{18} + \cdots - 43\!\cdots\!96 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 61\!\cdots\!84 \nu^{18} + \cdots - 16\!\cdots\!92 ) / 13\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{15} - \beta_{13} + \beta_{12} - \beta_{8} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} - 2 \beta_{10} + \beta_{9} + 2 \beta_{7} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{18} - 2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 11 \beta_{13} + 15 \beta_{12} + 2 \beta_{10} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{18} - 21 \beta_{17} + 20 \beta_{16} + 19 \beta_{15} + 17 \beta_{14} + \beta_{13} + \cdots + 384 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 97 \beta_{18} - 51 \beta_{17} + 18 \beta_{16} - 99 \beta_{15} + 3 \beta_{14} - 96 \beta_{13} + \cdots + 179 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 289 \beta_{18} - 318 \beta_{17} + 301 \beta_{16} + 257 \beta_{15} + 218 \beta_{14} + 19 \beta_{13} + \cdots + 3831 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 781 \beta_{18} - 868 \beta_{17} + 245 \beta_{16} - 836 \beta_{15} + 74 \beta_{14} - 754 \beta_{13} + \cdots + 1843 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3700 \beta_{18} - 4273 \beta_{17} + 4018 \beta_{16} + 3111 \beta_{15} + 2561 \beta_{14} + \cdots + 39536 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5872 \beta_{18} - 12562 \beta_{17} + 3061 \beta_{16} - 6846 \beta_{15} + 1202 \beta_{14} - 5344 \beta_{13} + \cdots + 18803 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 44682 \beta_{18} - 54257 \beta_{17} + 50383 \beta_{16} + 35916 \beta_{15} + 29043 \beta_{14} + \cdots + 416316 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 40406 \beta_{18} - 167525 \beta_{17} + 37189 \beta_{16} - 54851 \beta_{15} + 16483 \beta_{14} + \cdots + 195452 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 522682 \beta_{18} - 668144 \beta_{17} + 609611 \beta_{16} + 404872 \beta_{15} + 324118 \beta_{14} + \cdots + 4442982 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 233201 \beta_{18} - 2132884 \beta_{17} + 449246 \beta_{16} - 430031 \beta_{15} + 207950 \beta_{14} + \cdots + 2089995 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 6002651 \beta_{18} - 8080783 \beta_{17} + 7219185 \beta_{16} + 4502551 \beta_{15} + 3589909 \beta_{14} + \cdots + 47880403 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 713551 \beta_{18} - 26398584 \beta_{17} + 5442573 \beta_{16} - 3283093 \beta_{15} + 2510146 \beta_{14} + \cdots + 23002224 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 68188142 \beta_{18} - 96641169 \beta_{17} + 84355654 \beta_{16} + 49651657 \beta_{15} + \cdots + 519870514 \) 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
−3.30875
−2.94730
−2.15456
−1.97653
−1.89778
−1.09927
−0.934189
−0.348851
0.148626
0.419506
0.698461
1.35722
1.62671
1.93240
2.06516
2.77171
2.92335
3.31597
3.40812
1.00000 −3.30875 1.00000 0.511858 −3.30875 1.15812 1.00000 7.94780 0.511858
1.2 1.00000 −2.94730 1.00000 −2.85672 −2.94730 4.20904 1.00000 5.68655 −2.85672
1.3 1.00000 −2.15456 1.00000 −0.187336 −2.15456 −4.59782 1.00000 1.64213 −0.187336
1.4 1.00000 −1.97653 1.00000 1.61858 −1.97653 3.76870 1.00000 0.906656 1.61858
1.5 1.00000 −1.89778 1.00000 −1.03122 −1.89778 −0.726868 1.00000 0.601578 −1.03122
1.6 1.00000 −1.09927 1.00000 2.38513 −1.09927 2.83577 1.00000 −1.79160 2.38513
1.7 1.00000 −0.934189 1.00000 3.62623 −0.934189 −1.02197 1.00000 −2.12729 3.62623
1.8 1.00000 −0.348851 1.00000 −2.92888 −0.348851 −2.68252 1.00000 −2.87830 −2.92888
1.9 1.00000 0.148626 1.00000 1.94178 0.148626 5.04861 1.00000 −2.97791 1.94178
1.10 1.00000 0.419506 1.00000 −3.63755 0.419506 5.17252 1.00000 −2.82402 −3.63755
1.11 1.00000 0.698461 1.00000 −4.21650 0.698461 −4.34568 1.00000 −2.51215 −4.21650
1.12 1.00000 1.35722 1.00000 4.23335 1.35722 −2.31623 1.00000 −1.15796 4.23335
1.13 1.00000 1.62671 1.00000 −0.336687 1.62671 2.26230 1.00000 −0.353815 −0.336687
1.14 1.00000 1.93240 1.00000 3.29073 1.93240 0.677340 1.00000 0.734157 3.29073
1.15 1.00000 2.06516 1.00000 −0.610135 2.06516 −0.596117 1.00000 1.26487 −0.610135
1.16 1.00000 2.77171 1.00000 1.04536 2.77171 2.05871 1.00000 4.68236 1.04536
1.17 1.00000 2.92335 1.00000 2.10821 2.92335 3.13423 1.00000 5.54600 2.10821
1.18 1.00000 3.31597 1.00000 −4.43392 3.31597 2.19785 1.00000 7.99564 −4.43392
1.19 1.00000 3.40812 1.00000 0.477706 3.40812 −3.23600 1.00000 8.61531 0.477706
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
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.w ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2678.2.a.w ✓ 19 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2678))\):

\( T_{3}^{19} - 6 T_{3}^{18} - 25 T_{3}^{17} + 203 T_{3}^{16} + 149 T_{3}^{15} - 2691 T_{3}^{14} + \cdots - 992 \) Copy content Toggle raw display
\( T_{5}^{19} - T_{5}^{18} - 64 T_{5}^{17} + 86 T_{5}^{16} + 1600 T_{5}^{15} - 2718 T_{5}^{14} + \cdots + 4608 \) Copy content Toggle raw display
\( T_{7}^{19} - 13 T_{7}^{18} - 7 T_{7}^{17} + 745 T_{7}^{16} - 2114 T_{7}^{15} - 14470 T_{7}^{14} + \cdots - 5259264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{19} \) Copy content Toggle raw display
$3$ \( T^{19} - 6 T^{18} + \cdots - 992 \) Copy content Toggle raw display
$5$ \( T^{19} - T^{18} + \cdots + 4608 \) Copy content Toggle raw display
$7$ \( T^{19} - 13 T^{18} + \cdots - 5259264 \) Copy content Toggle raw display
$11$ \( T^{19} - 7 T^{18} + \cdots + 430080 \) Copy content Toggle raw display
$13$ \( (T - 1)^{19} \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 107633448 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 50474123264 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots + 2823290880 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 94529617920 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 284373200896 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots + 1672687360 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 797814695227392 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 15379144049408 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 105007036833792 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 341481750528 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 3043030204416 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 34770516967424 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 3427045660672 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 424927655424 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots + 80935700289792 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 2520625372160 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 380262238715904 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 883942621528064 \) Copy content Toggle raw display
show more
show less