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Properties

Label 945.2.j.f
Level $945$
Weight $2$
Character orbit 945.j
Analytic conductor $7.546$
Analytic rank $0$
Dimension $10$
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] = [945,2,Mod(541,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.j (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 7x^{8} - 2x^{7} + 42x^{6} - 6x^{5} + 50x^{4} + 21x^{3} + 48x^{2} + 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{8} + \beta_{6} + \beta_{3} - 1) q^{4} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2}) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{8} + \beta_{6} + \beta_{3} - 1) q^{4} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2}) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{8} - \beta_1 q^{10} + (\beta_{9} - \beta_{8} + \cdots + \beta_{3}) q^{11}+ \cdots + ( - 2 \beta_{9} - 3 \beta_{8} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{4} - 5 q^{5} - q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{4} - 5 q^{5} - q^{7} - 6 q^{8} + 3 q^{11} - 20 q^{13} - 20 q^{14} - 6 q^{16} - q^{17} + 13 q^{19} + 8 q^{20} - 12 q^{22} - 4 q^{23} - 5 q^{25} - 5 q^{26} + 21 q^{28} - 24 q^{29} + 5 q^{31} + 16 q^{32} - 32 q^{34} - q^{35} - 8 q^{37} + 5 q^{38} + 3 q^{40} - 18 q^{41} + 16 q^{43} - 6 q^{44} + 26 q^{46} + 2 q^{47} - 11 q^{49} + 5 q^{52} - 6 q^{53} - 6 q^{55} + 24 q^{56} - 20 q^{58} - 14 q^{59} + 15 q^{61} + 40 q^{62} - 18 q^{64} + 10 q^{65} + 18 q^{67} + 30 q^{68} - 2 q^{70} - 26 q^{71} + 35 q^{73} - 9 q^{74} - 60 q^{76} - 46 q^{77} - q^{79} - 6 q^{80} + 39 q^{82} - 20 q^{83} + 2 q^{85} + 25 q^{86} + 46 q^{88} + 11 q^{89} + 26 q^{91} + 86 q^{92} - 29 q^{94} + 13 q^{95} - 44 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7x^{8} - 2x^{7} + 42x^{6} - 6x^{5} + 50x^{4} + 21x^{3} + 48x^{2} + 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1610 \nu^{9} + 2793 \nu^{8} - 3192 \nu^{7} + 13308 \nu^{6} - 25137 \nu^{5} + 120498 \nu^{4} + \cdots - 89964 ) / 634381 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2793 \nu^{9} + 14462 \nu^{8} - 16528 \nu^{7} + 92757 \nu^{6} - 130158 \nu^{5} + 623932 \nu^{4} + \cdots + 1904753 ) / 634381 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5257 \nu^{9} + 28427 \nu^{8} - 32488 \nu^{7} + 159297 \nu^{6} - 255843 \nu^{5} + 1226422 \nu^{4} + \cdots + 820552 ) / 634381 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10515 \nu^{9} + 78295 \nu^{8} - 89480 \nu^{7} + 582928 \nu^{6} - 704655 \nu^{5} + 3377870 \nu^{4} + \cdots + 1718413 ) / 634381 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 89964 \nu^{9} + 1610 \nu^{8} + 632541 \nu^{7} - 183120 \nu^{6} + 3791796 \nu^{5} - 564921 \nu^{4} + \cdots + 649299 ) / 634381 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 173516 \nu^{9} - 14162 \nu^{8} - 1161951 \nu^{7} + 286848 \nu^{6} - 6850733 \nu^{5} + \cdots - 845985 ) / 634381 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 269892 \nu^{9} + 4830 \nu^{8} + 1897623 \nu^{7} - 549360 \nu^{6} + 11375388 \nu^{5} + \cdots + 1947897 ) / 634381 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 276135 \nu^{9} + 3647 \nu^{8} - 1907311 \nu^{7} + 522974 \nu^{6} - 11451681 \nu^{5} + \cdots - 1753507 ) / 634381 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 6\beta_{8} + 15\beta_{6} - \beta_{5} + 6\beta_{3} + 2\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{8} + 7\beta_{7} - 11\beta_{6} + 28\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} + \beta_{4} - 36\beta_{3} - 20\beta_{2} + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{9} - 56\beta_{8} - 42\beta_{7} + 88\beta_{6} - \beta_{5} - 42\beta_{4} + 56\beta_{3} + 163\beta _1 - 88 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 42\beta_{9} + 219\beta_{8} + 15\beta_{7} - 502\beta_{6} + 159\beta_{2} - 159\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15\beta_{5} + 246\beta_{4} - 378\beta_{3} - 967\beta_{2} + 639 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
1.12879 + 1.95513i
0.655317 + 1.13504i
−0.0746830 − 0.129355i
−0.448675 − 0.777128i
−1.26075 − 2.18369i
1.12879 − 1.95513i
0.655317 − 1.13504i
−0.0746830 + 0.129355i
−0.448675 + 0.777128i
−1.26075 + 2.18369i
−1.12879 + 1.95513i 0 −1.54835 − 2.68182i −0.500000 + 0.866025i 0 0.801918 + 2.52129i 2.47590 0 −1.12879 − 1.95513i
541.2 −0.655317 + 1.13504i 0 0.141120 + 0.244426i −0.500000 + 0.866025i 0 2.17793 − 1.50220i −2.99118 0 −0.655317 − 1.13504i
541.3 0.0746830 − 0.129355i 0 0.988845 + 1.71273i −0.500000 + 0.866025i 0 −1.43332 + 2.22387i 0.594132 0 0.0746830 + 0.129355i
541.4 0.448675 − 0.777128i 0 0.597381 + 1.03469i −0.500000 + 0.866025i 0 0.591510 − 2.57878i 2.86682 0 0.448675 + 0.777128i
541.5 1.26075 − 2.18369i 0 −2.17899 − 3.77413i −0.500000 + 0.866025i 0 −2.63804 + 0.201846i −5.94568 0 1.26075 + 2.18369i
676.1 −1.12879 − 1.95513i 0 −1.54835 + 2.68182i −0.500000 − 0.866025i 0 0.801918 − 2.52129i 2.47590 0 −1.12879 + 1.95513i
676.2 −0.655317 − 1.13504i 0 0.141120 − 0.244426i −0.500000 − 0.866025i 0 2.17793 + 1.50220i −2.99118 0 −0.655317 + 1.13504i
676.3 0.0746830 + 0.129355i 0 0.988845 − 1.71273i −0.500000 − 0.866025i 0 −1.43332 − 2.22387i 0.594132 0 0.0746830 − 0.129355i
676.4 0.448675 + 0.777128i 0 0.597381 − 1.03469i −0.500000 − 0.866025i 0 0.591510 + 2.57878i 2.86682 0 0.448675 − 0.777128i
676.5 1.26075 + 2.18369i 0 −2.17899 + 3.77413i −0.500000 − 0.866025i 0 −2.63804 − 0.201846i −5.94568 0 1.26075 − 2.18369i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.j.f ✓ 10
3.b odd 2 1 945.2.j.h yes 10
7.c even 3 1 inner 945.2.j.f ✓ 10
7.c even 3 1 6615.2.a.bp 5
7.d odd 6 1 6615.2.a.bl 5
21.g even 6 1 6615.2.a.bq 5
21.h odd 6 1 945.2.j.h yes 10
21.h odd 6 1 6615.2.a.bm 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.j.f ✓ 10 1.a even 1 1 trivial
945.2.j.f ✓ 10 7.c even 3 1 inner
945.2.j.h yes 10 3.b odd 2 1
945.2.j.h yes 10 21.h odd 6 1
6615.2.a.bl 5 7.d odd 6 1
6615.2.a.bm 5 21.h odd 6 1
6615.2.a.bp 5 7.c even 3 1
6615.2.a.bq 5 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 7T_{2}^{8} + 2T_{2}^{7} + 42T_{2}^{6} + 6T_{2}^{5} + 50T_{2}^{4} - 21T_{2}^{3} + 48T_{2}^{2} - 7T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 7 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} - 3 T^{9} + \cdots + 123201 \) Copy content Toggle raw display
$13$ \( (T^{5} + 10 T^{4} + \cdots + 141)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + T^{9} + \cdots + 13225 \) Copy content Toggle raw display
$19$ \( T^{10} - 13 T^{9} + \cdots + 351649 \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + \cdots + 1530169 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots + 431)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 5 T^{9} + \cdots + 1946025 \) Copy content Toggle raw display
$37$ \( T^{10} + 8 T^{9} + \cdots + 149769 \) Copy content Toggle raw display
$41$ \( (T^{5} + 9 T^{4} + \cdots - 459)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 8 T^{4} + \cdots - 245)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 2 T^{9} + \cdots + 5583769 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 1455804025 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 2171280409 \) Copy content Toggle raw display
$61$ \( T^{10} - 15 T^{9} + \cdots + 86434209 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 15351457801 \) Copy content Toggle raw display
$71$ \( (T^{5} + 13 T^{4} + \cdots - 3963)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1547399569 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + \cdots + 3003289 \) Copy content Toggle raw display
$83$ \( (T^{5} + 10 T^{4} + \cdots - 6151)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 11 T^{9} + \cdots + 622521 \) Copy content Toggle raw display
$97$ \( (T^{5} + 22 T^{4} + \cdots - 122775)^{2} \) Copy content Toggle raw display
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