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Properties

Label 2340.2.r.e
Level $2340$
Weight $2$
Character orbit 2340.r
Analytic conductor $18.685$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,2,Mod(781,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 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("2340.781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.r (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 14 x^{18} - 29 x^{17} + 44 x^{16} - 56 x^{15} + 52 x^{14} + 3 x^{13} - 222 x^{12} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + ( - \beta_{7} - 1) q^{5} + \beta_{15} q^{7} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} + ( - \beta_{7} - 1) q^{5} + \beta_{15} q^{7} - \beta_{5} q^{9} + ( - \beta_{16} + \beta_{10}) q^{11} + (\beta_{7} + 1) q^{13} + (\beta_{9} - \beta_{8}) q^{15} + (\beta_{17} + \beta_{15} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{18} - 4 \beta_{17} + \beta_{16} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} - 10 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} - 10 q^{5} + 4 q^{7} - 6 q^{9} + q^{11} + 10 q^{13} + 5 q^{15} - 2 q^{19} - 17 q^{21} + q^{23} - 10 q^{25} + 2 q^{27} + 10 q^{29} + 6 q^{31} + 22 q^{33} - 8 q^{35} + 12 q^{37} - 5 q^{39} + 11 q^{41} - 5 q^{43} + 9 q^{45} - q^{47} + 12 q^{49} + 50 q^{51} + 4 q^{53} - 2 q^{55} + 13 q^{57} + 5 q^{59} + 7 q^{61} - 28 q^{63} + 10 q^{65} + 21 q^{67} + 29 q^{69} - 20 q^{71} - 52 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 18 q^{81} - 23 q^{83} + 56 q^{87} - 24 q^{89} + 8 q^{91} + 5 q^{93} + q^{95} + 23 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 14 x^{18} - 29 x^{17} + 44 x^{16} - 56 x^{15} + 52 x^{14} + 3 x^{13} - 222 x^{12} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{19} - 22 \nu^{18} + 55 \nu^{17} - 103 \nu^{16} + 133 \nu^{15} - 148 \nu^{14} + 92 \nu^{13} + \cdots - 216513 ) / 19683 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7996 \nu^{19} - 103841 \nu^{18} + 320342 \nu^{17} - 513137 \nu^{16} + 692756 \nu^{15} + \cdots - 865323729 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23911 \nu^{19} + 56182 \nu^{18} - 79657 \nu^{17} + 325054 \nu^{16} - 258007 \nu^{15} + \cdots + 933604056 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11997 \nu^{19} - 31678 \nu^{18} + 68141 \nu^{17} - 99158 \nu^{16} + 126047 \nu^{15} + \cdots - 119698884 ) / 6337926 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51586 \nu^{19} - 160205 \nu^{18} + 420698 \nu^{17} - 649583 \nu^{16} + 874148 \nu^{15} + \cdots - 1172024235 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58579 \nu^{19} + 138137 \nu^{18} - 339491 \nu^{17} + 436697 \nu^{16} - 628727 \nu^{15} + \cdots + 470640213 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 59545 \nu^{19} + 142967 \nu^{18} - 353015 \nu^{17} + 464711 \nu^{16} - 671231 \nu^{15} + \cdots + 565709103 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46837 \nu^{19} - 189605 \nu^{18} + 505295 \nu^{17} - 928436 \nu^{16} + 1160228 \nu^{15} + \cdots - 2235142431 ) / 9506889 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32575 \nu^{19} + 100502 \nu^{18} - 282137 \nu^{17} + 465212 \nu^{16} - 620693 \nu^{15} + \cdots + 1015367238 ) / 6337926 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 32849 \nu^{19} - 124444 \nu^{18} + 335761 \nu^{17} - 558910 \nu^{16} + 733783 \nu^{15} + \cdots - 1146022992 ) / 6337926 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6095 \nu^{19} + 16678 \nu^{18} - 42895 \nu^{17} + 65182 \nu^{16} - 88675 \nu^{15} + \cdots + 120774888 ) / 826686 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20791 \nu^{19} - 57971 \nu^{18} + 159821 \nu^{17} - 270869 \nu^{16} + 358535 \nu^{15} + \cdots - 641173725 ) / 2112642 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 201761 \nu^{19} - 670717 \nu^{18} + 1675051 \nu^{17} - 3074581 \nu^{16} + 3823879 \nu^{15} + \cdots - 7544159289 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 213013 \nu^{19} + 646052 \nu^{18} - 1594535 \nu^{17} + 2679428 \nu^{16} - 3458285 \nu^{15} + \cdots + 5729918130 ) / 19013778 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 82875 \nu^{19} - 283156 \nu^{18} + 709241 \nu^{17} - 1247624 \nu^{16} + 1574951 \nu^{15} + \cdots - 2813514264 ) / 6337926 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 133381 \nu^{19} - 480932 \nu^{18} + 1272269 \nu^{17} - 2263346 \nu^{16} + 2873714 \nu^{15} + \cdots - 5295691467 ) / 9506889 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 135741 \nu^{19} + 457042 \nu^{18} - 1164851 \nu^{17} + 2155412 \nu^{16} - 2681441 \nu^{15} + \cdots + 5383248012 ) / 6337926 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} + \beta_{17} + \beta_{16} + 4 \beta_{15} + \beta_{14} + 3 \beta_{13} - \beta_{9} - 3 \beta_{8} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - 3 \beta_{18} + \beta_{17} + \beta_{16} + 2 \beta_{15} - \beta_{14} + \beta_{13} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{19} - 2 \beta_{18} - 5 \beta_{17} - 7 \beta_{16} - 8 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{19} - 3 \beta_{18} - 10 \beta_{17} - 7 \beta_{16} - 6 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{19} - \beta_{18} - 14 \beta_{17} + 3 \beta_{16} + 10 \beta_{15} - 2 \beta_{14} - 11 \beta_{13} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 39 \beta_{19} + 27 \beta_{18} - 12 \beta_{17} - 12 \beta_{16} + 52 \beta_{15} + 17 \beta_{14} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 29 \beta_{19} - 12 \beta_{18} + 10 \beta_{17} + 7 \beta_{16} + 67 \beta_{15} + 10 \beta_{14} + 33 \beta_{13} + \cdots - 67 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 88 \beta_{19} - 12 \beta_{18} - 128 \beta_{17} - 38 \beta_{16} - 37 \beta_{15} - 22 \beta_{14} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 35 \beta_{19} - 2 \beta_{18} - 89 \beta_{17} - 64 \beta_{16} + 124 \beta_{15} - 77 \beta_{14} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 160 \beta_{19} - 156 \beta_{18} - 13 \beta_{17} + 161 \beta_{16} + 522 \beta_{15} + 263 \beta_{14} + \cdots + 267 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 226 \beta_{19} - 118 \beta_{18} - 503 \beta_{17} - 360 \beta_{16} - 308 \beta_{15} + 148 \beta_{14} + \cdots - 363 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 48 \beta_{19} + 9 \beta_{18} - 699 \beta_{17} - 1041 \beta_{16} - 743 \beta_{15} + 617 \beta_{14} + \cdots + 1421 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1030 \beta_{19} - 741 \beta_{18} - 956 \beta_{17} + 121 \beta_{16} - 98 \beta_{15} - 623 \beta_{14} + \cdots - 2503 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1589 \beta_{19} + 3120 \beta_{18} - 3488 \beta_{17} - 1670 \beta_{16} + 1418 \beta_{15} + 1451 \beta_{14} + \cdots + 2356 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 4556 \beta_{19} + 1483 \beta_{18} + 610 \beta_{17} + 446 \beta_{16} + 8302 \beta_{15} - 233 \beta_{14} + \cdots - 1922 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 863 \beta_{19} - 1200 \beta_{18} - 4822 \beta_{17} + 4352 \beta_{16} + 5883 \beta_{15} + 5570 \beta_{14} + \cdots + 17163 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
781.1
1.07876 − 1.35509i
0.371891 − 1.69166i
1.71680 − 0.229361i
−0.947423 − 1.44996i
1.71746 + 0.224314i
−1.20941 − 1.23989i
1.26177 + 1.18657i
−1.70993 + 0.275922i
0.384313 + 1.68888i
−0.164229 + 1.72425i
1.07876 + 1.35509i
0.371891 + 1.69166i
1.71680 + 0.229361i
−0.947423 + 1.44996i
1.71746 − 0.224314i
−1.20941 + 1.23989i
1.26177 − 1.18657i
−1.70993 − 0.275922i
0.384313 − 1.68888i
−0.164229 − 1.72425i
0 −1.71292 + 0.256693i 0 −0.500000 − 0.866025i 0 −0.158882 + 0.275192i 0 2.86822 − 0.879392i 0
781.2 0 −1.65096 − 0.523760i 0 −0.500000 − 0.866025i 0 0.668052 − 1.15710i 0 2.45135 + 1.72942i 0
781.3 0 −1.05703 + 1.37211i 0 −0.500000 − 0.866025i 0 2.03572 − 3.52596i 0 −0.765370 − 2.90073i 0
781.4 0 −0.781993 − 1.54547i 0 −0.500000 − 0.866025i 0 −0.568397 + 0.984493i 0 −1.77697 + 2.41710i 0
781.5 0 −0.664470 + 1.59952i 0 −0.500000 − 0.866025i 0 −1.60008 + 2.77141i 0 −2.11696 − 2.12567i 0
781.6 0 −0.469065 − 1.66733i 0 −0.500000 − 0.866025i 0 0.837169 − 1.45002i 0 −2.55996 + 1.56417i 0
781.7 0 0.396714 + 1.68601i 0 −0.500000 − 0.866025i 0 −0.671696 + 1.16341i 0 −2.68524 + 1.33772i 0
781.8 0 1.09392 − 1.34288i 0 −0.500000 − 0.866025i 0 1.90942 − 3.30722i 0 −0.606671 − 2.93802i 0
781.9 0 1.27045 + 1.17726i 0 −0.500000 − 0.866025i 0 0.799344 − 1.38450i 0 0.228104 + 2.99132i 0
781.10 0 1.57536 + 0.719897i 0 −0.500000 − 0.866025i 0 −1.25065 + 2.16619i 0 1.96350 + 2.26819i 0
1561.1 0 −1.71292 − 0.256693i 0 −0.500000 + 0.866025i 0 −0.158882 − 0.275192i 0 2.86822 + 0.879392i 0
1561.2 0 −1.65096 + 0.523760i 0 −0.500000 + 0.866025i 0 0.668052 + 1.15710i 0 2.45135 − 1.72942i 0
1561.3 0 −1.05703 − 1.37211i 0 −0.500000 + 0.866025i 0 2.03572 + 3.52596i 0 −0.765370 + 2.90073i 0
1561.4 0 −0.781993 + 1.54547i 0 −0.500000 + 0.866025i 0 −0.568397 − 0.984493i 0 −1.77697 − 2.41710i 0
1561.5 0 −0.664470 − 1.59952i 0 −0.500000 + 0.866025i 0 −1.60008 − 2.77141i 0 −2.11696 + 2.12567i 0
1561.6 0 −0.469065 + 1.66733i 0 −0.500000 + 0.866025i 0 0.837169 + 1.45002i 0 −2.55996 − 1.56417i 0
1561.7 0 0.396714 − 1.68601i 0 −0.500000 + 0.866025i 0 −0.671696 − 1.16341i 0 −2.68524 − 1.33772i 0
1561.8 0 1.09392 + 1.34288i 0 −0.500000 + 0.866025i 0 1.90942 + 3.30722i 0 −0.606671 + 2.93802i 0
1561.9 0 1.27045 − 1.17726i 0 −0.500000 + 0.866025i 0 0.799344 + 1.38450i 0 0.228104 − 2.99132i 0
1561.10 0 1.57536 − 0.719897i 0 −0.500000 + 0.866025i 0 −1.25065 − 2.16619i 0 1.96350 − 2.26819i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 781.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.r.e ✓ 20
3.b odd 2 1 7020.2.r.e 20
9.c even 3 1 inner 2340.2.r.e ✓ 20
9.d odd 6 1 7020.2.r.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.r.e ✓ 20 1.a even 1 1 trivial
2340.2.r.e ✓ 20 9.c even 3 1 inner
7020.2.r.e 20 3.b odd 2 1
7020.2.r.e 20 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} - 4 T_{7}^{19} + 37 T_{7}^{18} - 72 T_{7}^{17} + 608 T_{7}^{16} - 935 T_{7}^{15} + \cdots + 46656 \) acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} - 4 T^{19} + \cdots + 46656 \) Copy content Toggle raw display
$11$ \( T^{20} - T^{19} + \cdots + 1774224 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$17$ \( (T^{10} - 52 T^{8} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + T^{9} + \cdots - 4968)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 26665583616 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 550265756401 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 33418764864 \) Copy content Toggle raw display
$37$ \( (T^{10} - 6 T^{9} + \cdots + 3528)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 69551430062500 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 3689657039104 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 306900488196 \) Copy content Toggle raw display
$53$ \( (T^{10} - 2 T^{9} + \cdots - 80338824)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 2224834761744 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 826466446404 \) Copy content Toggle raw display
$71$ \( (T^{10} + 10 T^{9} + \cdots + 258001104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 26 T^{9} + \cdots + 1183794912)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 85853715369984 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} + 12 T^{9} + \cdots - 29222202)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 7461835236496 \) Copy content Toggle raw display
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