Properties

Label 740.2.bc.c
Level $740$
Weight $2$
Character orbit 740.bc
Analytic conductor $5.909$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(81,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bc (of order \(9\), degree \(6\), 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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 45 x^{16} - 46 x^{15} + 1434 x^{14} - 1842 x^{13} + 22726 x^{12} - 53385 x^{11} + 272919 x^{10} + \cdots + 651249 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{10} + \beta_{4}) q^{7} + (\beta_{11} + \beta_{8} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{10} + \beta_{4}) q^{7} + (\beta_{11} + \beta_{8} + \beta_{6}) q^{9} + (\beta_{6} - \beta_{2}) q^{11} + (\beta_{12} - \beta_{11} + \beta_{5}) q^{13} + (\beta_{8} - \beta_{6} - 1) q^{15} + (\beta_{17} - \beta_{13} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{10} + \beta_{7} - \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{3} + 3 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{3} + 3 q^{7} - 9 q^{9} - 9 q^{11} - 9 q^{13} - 9 q^{15} + 3 q^{17} + 6 q^{19} + 3 q^{21} - 3 q^{23} + 9 q^{27} + 12 q^{29} - 48 q^{31} + 15 q^{33} - 6 q^{35} + 18 q^{37} + 3 q^{39} - 21 q^{41} + 6 q^{43} + 9 q^{45} - 9 q^{47} - 15 q^{49} - 3 q^{51} - 18 q^{53} + 6 q^{55} + 24 q^{57} - 60 q^{59} - 48 q^{61} - 3 q^{63} - 9 q^{65} + 45 q^{67} - 3 q^{69} + 81 q^{71} + 6 q^{73} + 9 q^{77} + 24 q^{79} - 99 q^{81} - 27 q^{83} - 15 q^{87} + 12 q^{89} + 6 q^{91} + 33 q^{93} + 6 q^{95} + 6 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 45 x^{16} - 46 x^{15} + 1434 x^{14} - 1842 x^{13} + 22726 x^{12} - 53385 x^{11} + 272919 x^{10} + \cdots + 651249 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!89 \nu^{17} + \cdots + 21\!\cdots\!97 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 66\!\cdots\!70 \nu^{17} + \cdots - 14\!\cdots\!25 ) / 40\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 66\!\cdots\!70 \nu^{17} + \cdots + 14\!\cdots\!25 ) / 40\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!30 \nu^{17} + \cdots - 31\!\cdots\!03 ) / 55\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 55\!\cdots\!81 \nu^{17} + \cdots - 31\!\cdots\!07 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!25 \nu^{17} + \cdots - 16\!\cdots\!07 ) / 10\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!88 \nu^{17} + \cdots - 70\!\cdots\!24 ) / 55\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!43 \nu^{17} + \cdots + 17\!\cdots\!06 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 56\!\cdots\!26 \nu^{17} + \cdots - 13\!\cdots\!01 ) / 55\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!96 \nu^{17} + \cdots - 28\!\cdots\!69 ) / 55\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!63 \nu^{17} + \cdots - 28\!\cdots\!56 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!48 \nu^{17} + \cdots + 11\!\cdots\!59 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!94 \nu^{17} + \cdots + 11\!\cdots\!82 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 42\!\cdots\!76 \nu^{17} + \cdots + 18\!\cdots\!12 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43\!\cdots\!90 \nu^{17} + \cdots - 10\!\cdots\!32 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 60\!\cdots\!50 \nu^{17} + \cdots + 46\!\cdots\!51 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 62\!\cdots\!37 \nu^{17} + \cdots + 10\!\cdots\!32 ) / 14\!\cdots\!31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + 2 \beta_{16} + \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 22 \beta_{17} + 22 \beta_{15} + 19 \beta_{14} + 19 \beta_{13} + 3 \beta_{12} + 32 \beta_{11} + \cdots - 158 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 32 \beta_{17} - 70 \beta_{16} + 37 \beta_{15} - 37 \beta_{13} - 32 \beta_{12} + 465 \beta_{11} + \cdots - 69 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 98 \beta_{17} - 403 \beta_{16} - 414 \beta_{15} - 403 \beta_{14} - 512 \beta_{13} - 512 \beta_{12} + \cdots + 2876 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1950 \beta_{17} - 1950 \beta_{15} - 2026 \beta_{14} - 941 \beta_{13} - 1009 \beta_{12} - 9046 \beta_{11} + \cdots - 1776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9787 \beta_{17} + 9250 \beta_{16} - 2616 \beta_{15} + 2616 \beta_{13} + 9787 \beta_{12} + 4721 \beta_{11} + \cdots + 12403 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25904 \beta_{17} + 54610 \beta_{16} + 25791 \beta_{15} + 54610 \beta_{14} + 51695 \beta_{13} + \cdots + 75061 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 307482 \beta_{17} + 307482 \beta_{15} + 222478 \beta_{14} + 240083 \beta_{13} + 67399 \beta_{12} + \cdots - 1916268 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 679666 \beta_{17} - 1426354 \beta_{16} + 656718 \beta_{15} - 656718 \beta_{13} - 679666 \beta_{12} + \cdots - 1336384 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1723037 \beta_{17} - 5489263 \beta_{16} - 5991135 \beta_{15} - 5489263 \beta_{14} - 7714172 \beta_{13} + \cdots + 39783683 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 34177350 \beta_{17} - 34177350 \beta_{15} - 36703501 \beta_{14} - 17561711 \beta_{13} - 16615639 \beta_{12} + \cdots - 3745656 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 150739546 \beta_{17} + 137254669 \beta_{16} - 43918863 \beta_{15} + 43918863 \beta_{13} + \cdots + 194658409 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 420511025 \beta_{17} + 937619029 \beta_{16} + 449403564 \beta_{15} + 937619029 \beta_{14} + \cdots + 914791603 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4925343615 \beta_{17} + 4925343615 \beta_{15} + 3455427247 \beta_{14} + 3807963476 \beta_{13} + \cdots - 30004374174 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 11445753187 \beta_{17} - 23865894628 \beta_{16} + 10649028036 \beta_{15} - 10649028036 \beta_{13} + \cdots - 22094781223 \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(\beta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−1.74574 3.02371i
0.167614 + 0.290317i
2.51782 + 4.36099i
−1.87015 3.23920i
0.520967 + 0.902342i
1.17554 + 2.03609i
−1.74574 + 3.02371i
0.167614 0.290317i
2.51782 4.36099i
1.03934 + 1.80019i
0.713507 + 1.23583i
−2.51889 4.36284i
−1.87015 + 3.23920i
0.520967 0.902342i
1.17554 2.03609i
1.03934 1.80019i
0.713507 1.23583i
−2.51889 + 4.36284i
0 −1.43969 0.524005i 0 0.173648 + 0.984808i 0 −0.606289 3.43844i 0 −0.500000 0.419550i 0
81.2 0 −1.43969 0.524005i 0 0.173648 + 0.984808i 0 0.0582119 + 0.330136i 0 −0.500000 0.419550i 0
81.3 0 −1.43969 0.524005i 0 0.173648 + 0.984808i 0 0.874429 + 4.95913i 0 −0.500000 0.419550i 0
181.1 0 −0.326352 1.85083i 0 0.766044 0.642788i 0 −2.86524 + 2.40422i 0 −0.500000 + 0.181985i 0
181.2 0 −0.326352 1.85083i 0 0.766044 0.642788i 0 0.798168 0.669743i 0 −0.500000 + 0.181985i 0
181.3 0 −0.326352 1.85083i 0 0.766044 0.642788i 0 1.80103 1.51124i 0 −0.500000 + 0.181985i 0
201.1 0 −1.43969 + 0.524005i 0 0.173648 0.984808i 0 −0.606289 + 3.43844i 0 −0.500000 + 0.419550i 0
201.2 0 −1.43969 + 0.524005i 0 0.173648 0.984808i 0 0.0582119 0.330136i 0 −0.500000 + 0.419550i 0
201.3 0 −1.43969 + 0.524005i 0 0.173648 0.984808i 0 0.874429 4.95913i 0 −0.500000 + 0.419550i 0
441.1 0 0.266044 0.223238i 0 −0.939693 0.342020i 0 −1.95332 0.710949i 0 −0.500000 + 2.83564i 0
441.2 0 0.266044 0.223238i 0 −0.939693 0.342020i 0 −1.34095 0.488067i 0 −0.500000 + 2.83564i 0
441.3 0 0.266044 0.223238i 0 −0.939693 0.342020i 0 4.73396 + 1.72302i 0 −0.500000 + 2.83564i 0
601.1 0 −0.326352 + 1.85083i 0 0.766044 + 0.642788i 0 −2.86524 2.40422i 0 −0.500000 0.181985i 0
601.2 0 −0.326352 + 1.85083i 0 0.766044 + 0.642788i 0 0.798168 + 0.669743i 0 −0.500000 0.181985i 0
601.3 0 −0.326352 + 1.85083i 0 0.766044 + 0.642788i 0 1.80103 + 1.51124i 0 −0.500000 0.181985i 0
641.1 0 0.266044 + 0.223238i 0 −0.939693 + 0.342020i 0 −1.95332 + 0.710949i 0 −0.500000 2.83564i 0
641.2 0 0.266044 + 0.223238i 0 −0.939693 + 0.342020i 0 −1.34095 + 0.488067i 0 −0.500000 2.83564i 0
641.3 0 0.266044 + 0.223238i 0 −0.939693 + 0.342020i 0 4.73396 1.72302i 0 −0.500000 2.83564i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bc.c 18
37.f even 9 1 inner 740.2.bc.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bc.c 18 1.a even 1 1 trivial
740.2.bc.c 18 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} + 6T_{3}^{4} + 8T_{3}^{3} + 3T_{3}^{2} - 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} - 3 T^{17} + \cdots + 651249 \) Copy content Toggle raw display
$11$ \( T^{18} + 9 T^{17} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 2442237561 \) Copy content Toggle raw display
$17$ \( T^{18} - 3 T^{17} + \cdots + 2277081 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 255584169 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 165148201 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 71133690681 \) Copy content Toggle raw display
$31$ \( (T^{9} + 24 T^{8} + \cdots + 1219593)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 129961739795077 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 48254469561 \) Copy content Toggle raw display
$43$ \( (T^{9} - 3 T^{8} + \cdots - 39849)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 66095782281 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 8719813302489 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 9118159297641 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 21785994997209 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 334549377257481 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 86037738576409 \) Copy content Toggle raw display
$73$ \( (T^{9} - 3 T^{8} + \cdots - 141919047)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 129966452480089 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 955801477801 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
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