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Properties

Label 1340.4.a.b
Level $1340$
Weight $4$
Character orbit 1340.a
Self dual yes
Analytic conductor $79.063$
Analytic rank $1$
Dimension $16$
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] = [1340,4,Mod(1,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1340.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: \(79.0625594077\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 255 x^{14} + 1920 x^{13} + 25147 x^{12} - 173760 x^{11} - 1245648 x^{10} + \cdots - 19739450304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + 5 q^{5} + ( - \beta_{8} - 3) q^{7} + (\beta_{2} + \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + 5 q^{5} + ( - \beta_{8} - 3) q^{7} + (\beta_{2} + \beta_1 + 8) q^{9} + (\beta_{8} + \beta_{4} - \beta_{2} - 6) q^{11} + ( - \beta_{12} + 3 \beta_1 - 9) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{15} - \beta_{9} + \beta_{7} + \cdots - 3) q^{17}+ \cdots + ( - 11 \beta_{15} + 11 \beta_{14} + \cdots - 528) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 80 q^{5} - 43 q^{7} + 142 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 80 q^{5} - 43 q^{7} + 142 q^{9} - 115 q^{11} - 116 q^{13} - 40 q^{15} - 15 q^{17} - 397 q^{19} - 8 q^{21} - 159 q^{23} + 400 q^{25} - 440 q^{27} - 299 q^{29} - 368 q^{31} - 42 q^{33} - 215 q^{35} - 724 q^{37} - 1418 q^{39} - 562 q^{41} - 1066 q^{43} + 710 q^{45} - 585 q^{47} + 579 q^{49} - 1868 q^{51} - 188 q^{53} - 575 q^{55} - 488 q^{57} - 899 q^{59} - 1005 q^{61} - 1827 q^{63} - 580 q^{65} + 1072 q^{67} - 864 q^{69} - 777 q^{71} + 1905 q^{73} - 200 q^{75} - 1247 q^{77} - 2084 q^{79} + 2568 q^{81} - 1543 q^{83} - 75 q^{85} - 1318 q^{87} - 2412 q^{89} - 2230 q^{91} + 1434 q^{93} - 1985 q^{95} + 891 q^{97} - 8497 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 255 x^{14} + 1920 x^{13} + 25147 x^{12} - 173760 x^{11} - 1245648 x^{10} + \cdots - 19739450304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!49 \nu^{15} + \cdots + 23\!\cdots\!08 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!83 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 45\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59\!\cdots\!98 \nu^{15} + \cdots + 41\!\cdots\!54 ) / 13\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!14 \nu^{15} + \cdots - 55\!\cdots\!38 ) / 45\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31\!\cdots\!07 \nu^{15} + \cdots - 42\!\cdots\!44 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!94 \nu^{15} + \cdots - 55\!\cdots\!08 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84\!\cdots\!61 \nu^{15} + \cdots - 57\!\cdots\!92 ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87\!\cdots\!27 \nu^{15} + \cdots + 14\!\cdots\!12 ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!73 \nu^{15} + \cdots + 35\!\cdots\!36 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 64\!\cdots\!81 \nu^{15} + \cdots - 12\!\cdots\!92 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 74\!\cdots\!43 \nu^{15} + \cdots - 21\!\cdots\!76 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38\!\cdots\!07 \nu^{15} + \cdots + 42\!\cdots\!44 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 82\!\cdots\!37 \nu^{15} + \cdots + 13\!\cdots\!24 ) / 54\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{15} - 18 \beta_{14} - 15 \beta_{13} + 4 \beta_{12} + 16 \beta_{11} + 12 \beta_{10} + \cdots + 2239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 339 \beta_{15} - 299 \beta_{14} - 15 \beta_{13} - 204 \beta_{12} + 291 \beta_{11} + 138 \beta_{10} + \cdots + 2482 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1285 \beta_{15} - 2786 \beta_{14} - 2288 \beta_{13} + 317 \beta_{12} + 2326 \beta_{11} + \cdots + 176884 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42636 \beta_{15} - 34210 \beta_{14} - 3102 \beta_{13} - 27064 \beta_{12} + 33967 \beta_{11} + \cdots + 265229 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 175950 \beta_{15} - 329011 \beta_{14} - 271186 \beta_{13} + 11857 \beta_{12} + 265357 \beta_{11} + \cdots + 15254699 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4820223 \beta_{15} - 3600327 \beta_{14} - 466392 \beta_{13} - 3104004 \beta_{12} + 3640650 \beta_{11} + \cdots + 28364114 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 21677185 \beta_{15} - 35501146 \beta_{14} - 29429278 \beta_{13} - 1146687 \beta_{12} + \cdots + 1376767418 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 519147452 \beta_{15} - 367655021 \beta_{14} - 62271230 \beta_{13} - 334340060 \beta_{12} + \cdots + 3087200142 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2538567470 \beta_{15} - 3691871379 \beta_{14} - 3069863119 \beta_{13} - 358682699 \beta_{12} + \cdots + 127791457327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 54556666100 \beta_{15} - 37179555977 \beta_{14} - 7817840975 \beta_{13} - 34914771464 \beta_{12} + \cdots + 338706506762 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 288648053394 \beta_{15} - 378043340335 \beta_{14} - 314085710987 \beta_{13} - 60328276097 \beta_{12} + \cdots + 12090537314077 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 5658528028910 \beta_{15} - 3754818506900 \beta_{14} - 943181009701 \beta_{13} - 3586887474918 \beta_{12} + \cdots + 37168157910841 \) 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
10.1642
9.03513
7.23067
7.08874
3.87110
3.63995
3.19283
1.08897
1.03612
−2.62273
−3.58630
−4.52644
−4.95787
−5.16930
−7.89413
−9.59094
0 −10.1642 0 5.00000 0 20.5850 0 76.3110 0
1.2 0 −9.03513 0 5.00000 0 −35.7040 0 54.6335 0
1.3 0 −7.23067 0 5.00000 0 −3.36526 0 25.2826 0
1.4 0 −7.08874 0 5.00000 0 −4.26289 0 23.2502 0
1.5 0 −3.87110 0 5.00000 0 −26.2187 0 −12.0146 0
1.6 0 −3.63995 0 5.00000 0 −9.30334 0 −13.7508 0
1.7 0 −3.19283 0 5.00000 0 25.5537 0 −16.8058 0
1.8 0 −1.08897 0 5.00000 0 −12.2379 0 −25.8141 0
1.9 0 −1.03612 0 5.00000 0 28.6004 0 −25.9264 0
1.10 0 2.62273 0 5.00000 0 −1.42286 0 −20.1213 0
1.11 0 3.58630 0 5.00000 0 20.1076 0 −14.1384 0
1.12 0 4.52644 0 5.00000 0 1.99248 0 −6.51133 0
1.13 0 4.95787 0 5.00000 0 4.36756 0 −2.41957 0
1.14 0 5.16930 0 5.00000 0 −34.7170 0 −0.278338 0
1.15 0 7.89413 0 5.00000 0 −17.5854 0 35.3173 0
1.16 0 9.59094 0 5.00000 0 0.610689 0 64.9861 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(67\) \(-1\)

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 1340.4.a.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.4.a.b ✓ 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{15} - 255 T_{3}^{14} - 1920 T_{3}^{13} + 25147 T_{3}^{12} + 173760 T_{3}^{11} + \cdots - 19739450304 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1340))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots - 19739450304 \) Copy content Toggle raw display
$5$ \( (T - 5)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 21\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 74\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 75\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 87\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 88\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 48\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 80\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T - 67)^{16} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 83\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 73\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 12\!\cdots\!05 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
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