Properties

Label 740.2.i.b
Level $740$
Weight $2$
Character orbit 740.i
Analytic conductor $5.909$
Analytic rank $0$
Dimension $14$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(121,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.i (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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 13 x^{12} - 6 x^{11} + 130 x^{10} - 44 x^{9} + 466 x^{8} - 4 x^{7} + 1211 x^{6} - 162 x^{5} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} + 1) q^{5} + \beta_{11} q^{7} + (\beta_{8} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{5} + 1) q^{5} + \beta_{11} q^{7} + (\beta_{8} - \beta_{5}) q^{9} + ( - \beta_{13} - \beta_{12} + \beta_{3}) q^{11} + (\beta_{9} - \beta_{8} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{11} + 2 \beta_{9} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 7 q^{5} - 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 7 q^{5} - 2 q^{7} - 5 q^{9} - 10 q^{11} - 2 q^{13} - 5 q^{17} - 8 q^{19} + 9 q^{21} + 8 q^{23} - 7 q^{25} - 18 q^{27} - 4 q^{29} + 16 q^{31} - 7 q^{33} + 2 q^{35} + 4 q^{37} + 13 q^{39} + 9 q^{41} + 22 q^{43} - 10 q^{45} + 36 q^{47} - 7 q^{49} - 62 q^{51} + 2 q^{53} - 5 q^{55} - 38 q^{57} + 12 q^{59} - 20 q^{61} - 38 q^{63} + 2 q^{65} + 12 q^{67} + q^{69} - 13 q^{71} - 12 q^{73} - q^{77} - 21 q^{79} - 11 q^{81} + 7 q^{83} - 10 q^{85} + 41 q^{87} + 16 q^{89} - 6 q^{91} - 3 q^{93} + 8 q^{95} - 6 q^{97} + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 13 x^{12} - 6 x^{11} + 130 x^{10} - 44 x^{9} + 466 x^{8} - 4 x^{7} + 1211 x^{6} - 162 x^{5} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4964681246 \nu^{13} - 53146213338 \nu^{12} + 61333897174 \nu^{11} - 675325809833 \nu^{10} + \cdots - 153324396046686 ) / 37974187397487 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17625264898 \nu^{13} + 4964681246 \nu^{12} + 175982230336 \nu^{11} - 44417692214 \nu^{10} + \cdots + 13570254566379 ) / 37974187397487 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 100527566600 \nu^{13} - 943737361020 \nu^{12} + 1233649100200 \nu^{11} + \cdots - 254135293206537 ) / 37974187397487 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 502602020977 \nu^{13} + 52875794694 \nu^{12} - 6518932228963 \nu^{11} + \cdots + 112199806219077 ) / 113922562192461 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 759473947629 \nu^{13} + 6485308099790 \nu^{12} + 12379247830015 \nu^{11} + \cdots + 19\!\cdots\!83 ) / 113922562192461 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 544469451535 \nu^{13} - 1524651079067 \nu^{12} + 5819754702036 \nu^{11} + \cdots - 589952340165618 ) / 37974187397487 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2010408083908 \nu^{13} + 211503178776 \nu^{12} - 26075728915852 \nu^{11} + \cdots + 448799224876308 ) / 113922562192461 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3583643238985 \nu^{13} + 3721357453131 \nu^{12} + 46273743094051 \nu^{11} + \cdots + 124577625406023 ) / 113922562192461 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 192064798176 \nu^{13} + 257946709114 \nu^{12} - 2318682298207 \nu^{11} + \cdots + 105959772466956 ) / 5995924325919 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3380728818735 \nu^{13} - 1647768591811 \nu^{12} - 44215903918362 \nu^{11} + \cdots - 66709302888681 ) / 37974187397487 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1111116500645 \nu^{13} + 568647528233 \nu^{12} + 14344156023627 \nu^{11} + \cdots + 8131567198086 ) / 10356596562951 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12806917107547 \nu^{13} - 2169734251702 \nu^{12} - 164605593852051 \nu^{11} + \cdots + 16\!\cdots\!95 ) / 113922562192461 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{4} - 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - 9\beta_{8} + 2\beta_{6} + 27\beta_{5} - 9\beta_{2} + 4\beta _1 - 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 11 \beta_{12} - 12 \beta_{11} + 10 \beta_{10} + 11 \beta_{9} + 5 \beta_{8} + \cdots - 56 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{13} + 27 \beta_{12} - 15 \beta_{10} - 15 \beta_{7} - 15 \beta_{6} - 2 \beta_{4} - 69 \beta_{3} + \cdots + 219 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 107 \beta_{13} + 10 \beta_{12} + 120 \beta_{11} + 10 \beta_{10} - 104 \beta_{9} - 87 \beta_{8} + \cdots - 226 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 114 \beta_{13} - 184 \beta_{12} - 187 \beta_{11} + 298 \beta_{10} + 54 \beta_{9} + 729 \beta_{8} + \cdots - 883 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 967 \beta_{13} + 967 \beta_{12} - 1027 \beta_{10} - 1157 \beta_{7} - 1027 \beta_{6} - 954 \beta_{4} + \cdots + 3028 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2107 \beta_{13} - 1014 \beta_{12} + 2180 \beta_{11} - 1014 \beta_{10} - 890 \beta_{9} - 6818 \beta_{8} + \cdots - 17842 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 124 \beta_{13} - 9939 \beta_{12} - 11156 \beta_{11} + 9815 \beta_{10} + 8773 \beta_{9} + \cdots - 44014 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 32132 \beta_{13} + 32132 \beta_{12} - 23235 \beta_{10} - 24401 \beta_{7} - 23235 \beta_{6} + \cdots + 170213 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 97365 \beta_{13} - 3058 \beta_{12} + 108645 \beta_{11} - 3058 \beta_{10} - 81861 \beta_{9} + \cdots - 398649 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.32457 + 2.29422i
1.06888 + 1.85135i
0.297211 + 0.514784i
0.239489 + 0.414807i
−0.586370 1.01562i
−0.743604 1.28796i
−1.60017 2.77158i
1.32457 2.29422i
1.06888 1.85135i
0.297211 0.514784i
0.239489 0.414807i
−0.586370 + 1.01562i
−0.743604 + 1.28796i
−1.60017 + 2.77158i
0 −1.32457 2.29422i 0 0.500000 + 0.866025i 0 1.42866 + 2.47451i 0 −2.00895 + 3.47961i 0
121.2 0 −1.06888 1.85135i 0 0.500000 + 0.866025i 0 −0.0464432 0.0804420i 0 −0.785009 + 1.35968i 0
121.3 0 −0.297211 0.514784i 0 0.500000 + 0.866025i 0 0.855653 + 1.48203i 0 1.32333 2.29208i 0
121.4 0 −0.239489 0.414807i 0 0.500000 + 0.866025i 0 −2.25243 3.90131i 0 1.38529 2.39939i 0
121.5 0 0.586370 + 1.01562i 0 0.500000 + 0.866025i 0 −2.03636 3.52708i 0 0.812342 1.40702i 0
121.6 0 0.743604 + 1.28796i 0 0.500000 + 0.866025i 0 1.37759 + 2.38605i 0 0.394106 0.682612i 0
121.7 0 1.60017 + 2.77158i 0 0.500000 + 0.866025i 0 −0.326670 0.565809i 0 −3.62111 + 6.27194i 0
581.1 0 −1.32457 + 2.29422i 0 0.500000 0.866025i 0 1.42866 2.47451i 0 −2.00895 3.47961i 0
581.2 0 −1.06888 + 1.85135i 0 0.500000 0.866025i 0 −0.0464432 + 0.0804420i 0 −0.785009 1.35968i 0
581.3 0 −0.297211 + 0.514784i 0 0.500000 0.866025i 0 0.855653 1.48203i 0 1.32333 + 2.29208i 0
581.4 0 −0.239489 + 0.414807i 0 0.500000 0.866025i 0 −2.25243 + 3.90131i 0 1.38529 + 2.39939i 0
581.5 0 0.586370 1.01562i 0 0.500000 0.866025i 0 −2.03636 + 3.52708i 0 0.812342 + 1.40702i 0
581.6 0 0.743604 1.28796i 0 0.500000 0.866025i 0 1.37759 2.38605i 0 0.394106 + 0.682612i 0
581.7 0 1.60017 2.77158i 0 0.500000 0.866025i 0 −0.326670 + 0.565809i 0 −3.62111 6.27194i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.i.b 14
37.c even 3 1 inner 740.2.i.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.i.b 14 1.a even 1 1 trivial
740.2.i.b 14 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 13 T_{3}^{12} + 6 T_{3}^{11} + 130 T_{3}^{10} + 44 T_{3}^{9} + 466 T_{3}^{8} + 4 T_{3}^{7} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 13 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 2 T^{13} + \cdots + 225 \) Copy content Toggle raw display
$11$ \( (T^{7} + 5 T^{6} + \cdots - 2097)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 2 T^{13} + \cdots + 4431025 \) Copy content Toggle raw display
$17$ \( T^{14} + 5 T^{13} + \cdots + 21650409 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 178837129 \) Copy content Toggle raw display
$23$ \( (T^{7} - 4 T^{6} + \cdots - 2673)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 2 T^{6} + \cdots + 32643)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 8 T^{6} + \cdots + 39555)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 94931877133 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 257520726225 \) Copy content Toggle raw display
$43$ \( (T^{7} - 11 T^{6} + \cdots + 51341)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} - 18 T^{6} + \cdots - 123957)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} - 2 T^{13} + \cdots + 77633721 \) Copy content Toggle raw display
$59$ \( T^{14} - 12 T^{13} + \cdots + 342225 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 10193123521 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 474824355625 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 83598087953601 \) Copy content Toggle raw display
$73$ \( (T^{7} + 6 T^{6} + \cdots + 313603)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 21 T^{13} + \cdots + 1500625 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 15338574801 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 3961065969 \) Copy content Toggle raw display
$97$ \( (T^{7} + 3 T^{6} + \cdots + 38751)^{2} \) Copy content Toggle raw display
show more
show less