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Properties

Label 2668.2.a.e
Level $2668$
Weight $2$
Character orbit 2668.a
Self dual yes
Analytic conductor $21.304$
Analytic rank $0$
Dimension $14$
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] = [2668,2,Mod(1,2668)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2668, 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("2668.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2668 = 2^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2668.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: \(21.3040872593\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 23 x^{12} + 54 x^{11} + 213 x^{10} - 356 x^{9} - 970 x^{8} + 1082 x^{7} + 2188 x^{6} + \cdots - 134 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{8} q^{5} - \beta_{5} q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{8} q^{5} - \beta_{5} q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{12} + \beta_{8} + 1) q^{11} - \beta_{6} q^{13} + ( - \beta_{13} + \beta_{11} + \beta_{10} + \cdots + 1) q^{15}+ \cdots + ( - \beta_{13} - 4 \beta_{12} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 11 q^{3} - 2 q^{5} + 3 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 11 q^{3} - 2 q^{5} + 3 q^{7} + 21 q^{9} + 8 q^{11} + 3 q^{13} + 2 q^{15} + 14 q^{17} + 4 q^{19} - 15 q^{21} + 14 q^{23} + 20 q^{25} + 32 q^{27} + 14 q^{29} + 10 q^{31} + 23 q^{33} + 31 q^{35} - 7 q^{37} - 14 q^{39} + 35 q^{41} + 31 q^{43} - 14 q^{45} + 43 q^{47} + 37 q^{49} + 16 q^{51} - 21 q^{53} + 45 q^{55} + 12 q^{57} + 7 q^{59} - 14 q^{61} + 5 q^{63} + 9 q^{65} + 3 q^{67} + 11 q^{69} - 5 q^{71} + 8 q^{73} + 33 q^{75} - 11 q^{77} + 23 q^{79} + 70 q^{81} + 42 q^{83} - 12 q^{85} + 11 q^{87} + 22 q^{89} - 7 q^{91} + 42 q^{93} + 18 q^{95} - 4 q^{97} + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 23 x^{12} + 54 x^{11} + 213 x^{10} - 356 x^{9} - 970 x^{8} + 1082 x^{7} + 2188 x^{6} + \cdots - 134 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22535 \nu^{13} + 199472 \nu^{12} - 1448521 \nu^{11} - 3716085 \nu^{10} + 18439344 \nu^{9} + \cdots + 29008066 ) / 2694034 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30703 \nu^{13} - 135905 \nu^{12} + 1733615 \nu^{11} + 1298417 \nu^{10} - 20394920 \nu^{9} + \cdots - 6010401 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 123461 \nu^{13} - 691194 \nu^{12} - 1411057 \nu^{11} + 11417922 \nu^{10} + 4395706 \nu^{9} + \cdots - 20233070 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 139763 \nu^{13} - 429856 \nu^{12} - 2760431 \nu^{11} + 5783693 \nu^{10} + 23904853 \nu^{9} + \cdots + 15145821 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 140953 \nu^{13} + 556206 \nu^{12} + 2429261 \nu^{11} - 8523878 \nu^{10} - 18128845 \nu^{9} + \cdots - 6559922 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 166647 \nu^{13} - 873257 \nu^{12} - 1606356 \nu^{11} + 11781982 \nu^{10} + 3402378 \nu^{9} + \cdots + 372929 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 399047 \nu^{13} - 2173962 \nu^{12} - 3726999 \nu^{11} + 30541065 \nu^{10} + 5766964 \nu^{9} + \cdots + 22027324 ) / 2694034 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 526065 \nu^{13} - 2937200 \nu^{12} - 5234833 \nu^{11} + 44552129 \nu^{10} + 9772812 \nu^{9} + \cdots - 56357660 ) / 2694034 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 271664 \nu^{13} - 1598359 \nu^{12} - 2264145 \nu^{11} + 23901800 \nu^{10} - 1583652 \nu^{9} + \cdots - 13652284 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 289876 \nu^{13} + 1370026 \nu^{12} + 4134688 \nu^{11} - 21726167 \nu^{10} - 23268196 \nu^{9} + \cdots + 17511611 ) / 1347017 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86041 \nu^{13} - 362940 \nu^{12} - 1388271 \nu^{11} + 5676607 \nu^{10} + 9282248 \nu^{9} + \cdots - 2398756 ) / 384862 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{3} + 2\beta_{2} + 9\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{13} - 2 \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 4 \beta_{3} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{13} - 14 \beta_{12} + 3 \beta_{10} + 3 \beta_{9} - 6 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 70 \beta_{13} - 42 \beta_{12} + 13 \beta_{10} + 21 \beta_{9} - 32 \beta_{8} + 28 \beta_{7} + \cdots + 369 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 319 \beta_{13} - 202 \beta_{12} + 4 \beta_{11} + 83 \beta_{10} + 77 \beta_{9} - 156 \beta_{8} + \cdots + 1206 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1304 \beta_{13} - 727 \beta_{12} + 23 \beta_{11} + 356 \beta_{10} + 371 \beta_{9} - 696 \beta_{8} + \cdots + 5194 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5529 \beta_{13} - 3116 \beta_{12} + 153 \beta_{11} + 1691 \beta_{10} + 1453 \beta_{9} - 3075 \beta_{8} + \cdots + 19478 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 22702 \beta_{13} - 12087 \beta_{12} + 744 \beta_{11} + 7129 \beta_{10} + 6254 \beta_{9} + \cdots + 80333 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 93966 \beta_{13} - 49924 \beta_{12} + 3664 \beta_{11} + 30876 \beta_{10} + 25087 \beta_{9} + \cdots + 317168 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 384788 \beta_{13} - 199418 \beta_{12} + 16381 \beta_{11} + 128340 \beta_{10} + 103992 \beta_{9} + \cdots + 1293261 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1577818 \beta_{13} - 814424 \beta_{12} + 72712 \beta_{11} + 536643 \beta_{10} + 420884 \beta_{9} + \cdots + 5203321 \) 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
4.06992
2.85083
2.46874
1.43332
1.17499
0.921227
0.639339
−0.262595
−0.941383
−1.07050
−1.73186
−1.91152
−2.31905
−2.32145
0 −3.06992 0 −1.82406 0 1.92389 0 6.42439 0
1.2 0 −1.85083 0 2.69614 0 1.45194 0 0.425569 0
1.3 0 −1.46874 0 −2.55262 0 2.21862 0 −0.842811 0
1.4 0 −0.433317 0 −2.78011 0 −2.56928 0 −2.81224 0
1.5 0 −0.174986 0 3.83317 0 4.42495 0 −2.96938 0
1.6 0 0.0787726 0 0.484203 0 −1.43504 0 −2.99379 0
1.7 0 0.360661 0 −0.126353 0 −3.41789 0 −2.86992 0
1.8 0 1.26259 0 −0.946846 0 2.86599 0 −1.40585 0
1.9 0 1.94138 0 −4.00864 0 −4.97171 0 0.768968 0
1.10 0 2.07050 0 3.41586 0 0.786769 0 1.28698 0
1.11 0 2.73186 0 1.13054 0 2.31969 0 4.46306 0
1.12 0 2.91152 0 −0.797771 0 5.11710 0 5.47695 0
1.13 0 3.31905 0 2.86312 0 −4.06130 0 8.01609 0
1.14 0 3.32145 0 −3.38664 0 −1.65373 0 8.03200 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 2668.2.a.e ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2668.2.a.e ✓ 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 11 T_{3}^{13} + 29 T_{3}^{12} + 92 T_{3}^{11} - 568 T_{3}^{10} + 459 T_{3}^{9} + 2078 T_{3}^{8} + \cdots - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2668))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 11 T^{13} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 928 \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} + \cdots + 308864 \) Copy content Toggle raw display
$11$ \( T^{14} - 8 T^{13} + \cdots + 130948 \) Copy content Toggle raw display
$13$ \( T^{14} - 3 T^{13} + \cdots - 587861 \) Copy content Toggle raw display
$17$ \( T^{14} - 14 T^{13} + \cdots + 5252 \) Copy content Toggle raw display
$19$ \( T^{14} - 4 T^{13} + \cdots + 162284 \) Copy content Toggle raw display
$23$ \( (T - 1)^{14} \) Copy content Toggle raw display
$29$ \( (T - 1)^{14} \) Copy content Toggle raw display
$31$ \( T^{14} - 10 T^{13} + \cdots + 6672284 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 126287324284 \) Copy content Toggle raw display
$41$ \( T^{14} - 35 T^{13} + \cdots + 50482592 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 475423746661 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 5622474724 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 32982848212 \) Copy content Toggle raw display
$59$ \( T^{14} - 7 T^{13} + \cdots - 21134992 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 104748482912 \) Copy content Toggle raw display
$67$ \( T^{14} - 3 T^{13} + \cdots - 47536288 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 7502076556 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 5007939214784 \) Copy content Toggle raw display
$79$ \( T^{14} - 23 T^{13} + \cdots - 78287 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3965094159296 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 294338216 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 186758389727456 \) Copy content Toggle raw display
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