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Properties

Label 2678.2.a.v
Level $2678$
Weight $2$
Character orbit 2678.a
Self dual yes
Analytic conductor $21.384$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 32 x^{13} - x^{12} + 395 x^{11} + 26 x^{10} - 2380 x^{9} - 205 x^{8} + 7309 x^{7} + 448 x^{6} + \cdots + 352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 32 x^{13} - x^{12} + 395 x^{11} + 26 x^{10} - 2380 x^{9} - 205 x^{8} + 7309 x^{7} + 448 x^{6} + \cdots + 352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1198420037 \nu^{14} + 13078769410 \nu^{13} - 48451496280 \nu^{12} - 407858202901 \nu^{11} + \cdots - 3806695849032 ) / 806414065132 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1199457150 \nu^{14} + 23942375558 \nu^{13} + 15864960283 \nu^{12} - 728255108710 \nu^{11} + \cdots - 12915279377698 ) / 403207032566 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5688031235 \nu^{14} - 17017366242 \nu^{13} + 189025392172 \nu^{12} + 542815057085 \nu^{11} + \cdots + 7725831982448 ) / 806414065132 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9990060380 \nu^{14} + 16032394458 \nu^{13} - 324160731091 \nu^{12} - 501109124534 \nu^{11} + \cdots - 9036279329552 ) / 403207032566 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36505598425 \nu^{14} - 10962797528 \nu^{13} + 1158035945528 \nu^{12} + \cdots + 25640022337168 ) / 806414065132 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3093159271 \nu^{14} - 747967404 \nu^{13} + 97101264868 \nu^{12} + 28187398693 \nu^{11} + \cdots + 1465814476424 ) / 62031851164 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55112743323 \nu^{14} - 19429384834 \nu^{13} + 1735918989742 \nu^{12} + \cdots + 33926165702592 ) / 806414065132 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29199734682 \nu^{14} - 10345419450 \nu^{13} + 922301814384 \nu^{12} + \cdots + 19033509022574 ) / 403207032566 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14873554190 \nu^{14} - 2549694691 \nu^{13} - 463098451181 \nu^{12} + 51790722212 \nu^{11} + \cdots - 6609485849740 ) / 201603516283 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 32010964397 \nu^{14} + 3892104624 \nu^{13} - 1003126605486 \nu^{12} - 166662265360 \nu^{11} + \cdots - 13467490606958 ) / 403207032566 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32021582015 \nu^{14} + 21644188528 \nu^{13} - 1026145351427 \nu^{12} - 709447338843 \nu^{11} + \cdots - 28629367865394 ) / 403207032566 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8376874065 \nu^{14} - 4237752802 \nu^{13} + 266624103300 \nu^{12} + 143310266201 \nu^{11} + \cdots + 6526722877152 ) / 62031851164 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - 2\beta_{12} - \beta_{8} - 3\beta_{7} + 2\beta_{6} - \beta_{4} + 9\beta_{2} - 2\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} - 3 \beta_{13} + 14 \beta_{11} - 3 \beta_{10} + 16 \beta_{9} - 12 \beta_{8} + 12 \beta_{7} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{14} - 15 \beta_{13} - 35 \beta_{12} + 2 \beta_{11} - \beta_{10} - 15 \beta_{8} - 41 \beta_{7} + \cdots + 260 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 18 \beta_{14} - 52 \beta_{13} + 8 \beta_{12} + 157 \beta_{11} - 51 \beta_{10} + 200 \beta_{9} + \cdots + 111 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 55 \beta_{14} - 184 \beta_{13} - 454 \beta_{12} + 39 \beta_{11} - 11 \beta_{10} + 5 \beta_{9} + \cdots + 2390 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 236 \beta_{14} - 675 \beta_{13} + 187 \beta_{12} + 1633 \beta_{11} - 682 \beta_{10} + 2291 \beta_{9} + \cdots + 952 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 954 \beta_{14} - 2118 \beta_{13} - 5317 \beta_{12} + 535 \beta_{11} - 42 \beta_{10} + 88 \beta_{9} + \cdots + 22688 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2749 \beta_{14} - 7893 \beta_{13} + 3042 \beta_{12} + 16503 \beta_{11} - 8347 \beta_{10} + \cdots + 7797 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13548 \beta_{14} - 23594 \beta_{13} - 59528 \beta_{12} + 6358 \beta_{11} + 747 \beta_{10} + \cdots + 219929 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 30214 \beta_{14} - 87849 \beta_{13} + 42619 \beta_{12} + 165154 \beta_{11} - 97417 \beta_{10} + \cdots + 60455 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 172851 \beta_{14} - 257200 \beta_{13} - 651354 \beta_{12} + 69818 \beta_{11} + 21973 \beta_{10} + \cdots + 2164427 \) 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
−3.26896
−2.86194
−2.23189
−1.93292
−1.45274
−1.38961
−0.494121
0.391431
0.510148
0.671527
1.31627
2.08097
2.58037
2.89543
3.18603
−1.00000 −3.26896 1.00000 3.33329 3.26896 0.974360 −1.00000 7.68611 −3.33329
1.2 −1.00000 −2.86194 1.00000 −0.869657 2.86194 −3.97445 −1.00000 5.19068 0.869657
1.3 −1.00000 −2.23189 1.00000 −2.75658 2.23189 1.87183 −1.00000 1.98135 2.75658
1.4 −1.00000 −1.93292 1.00000 4.37599 1.93292 2.99354 −1.00000 0.736181 −4.37599
1.5 −1.00000 −1.45274 1.00000 −2.96272 1.45274 2.52428 −1.00000 −0.889544 2.96272
1.6 −1.00000 −1.38961 1.00000 0.382188 1.38961 3.27832 −1.00000 −1.06899 −0.382188
1.7 −1.00000 −0.494121 1.00000 0.209166 0.494121 −3.13915 −1.00000 −2.75584 −0.209166
1.8 −1.00000 0.391431 1.00000 1.59794 −0.391431 2.48281 −1.00000 −2.84678 −1.59794
1.9 −1.00000 0.510148 1.00000 3.02894 −0.510148 −3.74168 −1.00000 −2.73975 −3.02894
1.10 −1.00000 0.671527 1.00000 −3.51428 −0.671527 −2.26638 −1.00000 −2.54905 3.51428
1.11 −1.00000 1.31627 1.00000 2.50381 −1.31627 1.82114 −1.00000 −1.26743 −2.50381
1.12 −1.00000 2.08097 1.00000 −3.85126 −2.08097 4.49186 −1.00000 1.33042 3.85126
1.13 −1.00000 2.58037 1.00000 0.926106 −2.58037 0.0700671 −1.00000 3.65832 −0.926106
1.14 −1.00000 2.89543 1.00000 −2.27073 −2.89543 −3.28770 −1.00000 5.38351 2.27073
1.15 −1.00000 3.18603 1.00000 2.86778 −3.18603 −1.09886 −1.00000 7.15082 −2.86778
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.v ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2678.2.a.v ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2678))\):

\( T_{3}^{15} - 32 T_{3}^{13} - T_{3}^{12} + 395 T_{3}^{11} + 26 T_{3}^{10} - 2380 T_{3}^{9} - 205 T_{3}^{8} + \cdots + 352 \) Copy content Toggle raw display
\( T_{5}^{15} - 3 T_{5}^{14} - 49 T_{5}^{13} + 145 T_{5}^{12} + 921 T_{5}^{11} - 2747 T_{5}^{10} + \cdots - 8192 \) Copy content Toggle raw display
\( T_{7}^{15} - 3 T_{7}^{14} - 54 T_{7}^{13} + 170 T_{7}^{12} + 1106 T_{7}^{11} - 3816 T_{7}^{10} + \cdots - 24576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 32 T^{13} + \cdots + 352 \) Copy content Toggle raw display
$5$ \( T^{15} - 3 T^{14} + \cdots - 8192 \) Copy content Toggle raw display
$7$ \( T^{15} - 3 T^{14} + \cdots - 24576 \) Copy content Toggle raw display
$11$ \( T^{15} + 5 T^{14} + \cdots - 685056 \) Copy content Toggle raw display
$13$ \( (T - 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 17 T^{14} + \cdots - 149174 \) Copy content Toggle raw display
$19$ \( T^{15} - 185 T^{13} + \cdots - 21188608 \) Copy content Toggle raw display
$23$ \( T^{15} - T^{14} + \cdots - 25600 \) Copy content Toggle raw display
$29$ \( T^{15} - 17 T^{14} + \cdots - 31840256 \) Copy content Toggle raw display
$31$ \( T^{15} - T^{14} + \cdots - 37117952 \) Copy content Toggle raw display
$37$ \( T^{15} - 19 T^{14} + \cdots + 27920128 \) Copy content Toggle raw display
$41$ \( T^{15} + 8 T^{14} + \cdots - 95744000 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 20682956800 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 31248093184 \) Copy content Toggle raw display
$53$ \( T^{15} - 9 T^{14} + \cdots + 10363200 \) Copy content Toggle raw display
$59$ \( T^{15} + 14 T^{14} + \cdots + 20949504 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 1315609542656 \) Copy content Toggle raw display
$67$ \( T^{15} + T^{14} + \cdots - 230144 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 1328384139008 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 87560482944 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 24891066828608 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 7001828352 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 9444523197952 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 76717665280 \) Copy content Toggle raw display
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