Properties

Label 975.2.i.l
Level $975$
Weight $2$
Character orbit 975.i
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
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] = [975,2,Mod(451,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, 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("975.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1714608.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{4} + \beta_{5} q^{6} + (\beta_{5} - \beta_{4}) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{4} + \beta_{5} q^{6} + (\beta_{5} - \beta_{4}) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + \beta_{4} q^{9} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{11} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{12} + (3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{14} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4) q^{16} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{17} - \beta_{3} q^{18} + ( - \beta_{5} + 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{3} - 1) q^{21} + ( - 2 \beta_{5} + 6 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2) q^{24} + ( - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 7) q^{26} + q^{27} + ( - 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{28} + ( - 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{29} + (3 \beta_{4} + \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 4) q^{31} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{32} + 2 \beta_{5} q^{33} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{34} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{36} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - 6 \beta_{3} - 2) q^{38} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{39} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{42} + ( - 2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{43} + ( - 8 \beta_{3} - 4) q^{44} + ( - 4 \beta_{5} + 4 \beta_{4}) q^{46} + ( - 3 \beta_{3} + 4) q^{47} + ( - 2 \beta_{5} + 4 \beta_{4}) q^{48} + (3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + (2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{51} + (7 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 5) q^{52} + 2 \beta_{3} q^{53} + ( - \beta_{5} + \beta_{3}) q^{54} + ( - 6 \beta_{5} + 8 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{56} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{57} + (\beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{58} + (2 \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{59} + (\beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1) q^{61} + ( - 5 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{62} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{63} + (4 \beta_{3} - 4) q^{64} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 10) q^{66} + (3 \beta_{4} + 3 \beta_{2} - 3 \beta_1 + 6) q^{67} + ( - 14 \beta_{4} - 14) q^{68} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{69} + (2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 6 \beta_1) q^{71} + (2 \beta_{5} - 2 \beta_{4}) q^{72} + (3 \beta_{3} - 7) q^{73} + (2 \beta_{5} - 4 \beta_{4}) q^{74} + (6 \beta_{5} - 16 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 20) q^{76} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 10) q^{77} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{78} + ( - 3 \beta_{4} - \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 4) q^{79} + ( - \beta_{4} - 1) q^{81} + ( - 3 \beta_{5} + 5 \beta_{4} - \beta_{2} - 2 \beta_1) q^{82} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{83} + (5 \beta_{5} - 3 \beta_{4} + \beta_{2} + 2 \beta_1) q^{84} + ( - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{86} + (3 \beta_{4} + \beta_{2} + 2 \beta_1) q^{87} + (8 \beta_{5} - 16 \beta_{4} - 8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 20) q^{88} + ( - 10 \beta_{4} + \beta_{2} - \beta_1 - 9) q^{89} + ( - 3 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{91} + ( - 4 \beta_{3} - 16) q^{92} + (\beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 4) q^{93} + ( - \beta_{5} - 12 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 15) q^{94} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 6) q^{96} + ( - 4 \beta_{5} - 6 \beta_{4} - \beta_{2} - 2 \beta_1) q^{97} + (2 \beta_{5} - 8 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{4} + 3 q^{7} - 12 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} - 6 q^{4} + 3 q^{7} - 12 q^{8} - 3 q^{9} + 12 q^{12} - 3 q^{13} + 24 q^{14} - 12 q^{16} - 12 q^{19} - 6 q^{21} - 24 q^{22} + 6 q^{24} - 18 q^{26} + 6 q^{27} + 12 q^{28} - 6 q^{29} + 6 q^{31} + 12 q^{32} - 6 q^{36} + 6 q^{37} - 12 q^{38} + 12 q^{39} - 12 q^{42} + 9 q^{43} - 24 q^{44} - 12 q^{46} + 24 q^{47} - 12 q^{48} + 6 q^{49} - 12 q^{52} - 30 q^{56} + 24 q^{57} + 6 q^{58} + 6 q^{59} + 3 q^{61} - 6 q^{62} + 3 q^{63} - 24 q^{64} + 48 q^{66} + 9 q^{67} - 42 q^{68} + 12 q^{71} + 6 q^{72} - 42 q^{73} + 12 q^{74} - 48 q^{76} + 48 q^{77} - 12 q^{78} - 6 q^{79} - 3 q^{81} - 18 q^{82} + 36 q^{83} + 12 q^{84} - 12 q^{86} - 6 q^{87} - 48 q^{88} - 30 q^{89} - 3 q^{91} - 96 q^{92} - 3 q^{93} - 36 q^{94} - 24 q^{96} + 15 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 2.23871i
0.500000 1.75780i
0.500000 + 0.385124i
0.500000 2.23871i
0.500000 + 1.75780i
0.500000 0.385124i
−1.13090 + 1.95878i −0.500000 + 0.866025i −1.55787 2.69832i 0 −1.13090 1.95878i −0.630901 1.09275i 2.52360 −0.500000 0.866025i 0
451.2 −0.169938 + 0.294342i −0.500000 + 0.866025i 0.942242 + 1.63201i 0 −0.169938 0.294342i 0.330062 + 0.571683i −1.32025 −0.500000 0.866025i 0
451.3 1.30084 2.25312i −0.500000 + 0.866025i −2.38437 4.12985i 0 1.30084 + 2.25312i 1.80084 + 3.11915i −7.20336 −0.500000 0.866025i 0
601.1 −1.13090 1.95878i −0.500000 0.866025i −1.55787 + 2.69832i 0 −1.13090 + 1.95878i −0.630901 + 1.09275i 2.52360 −0.500000 + 0.866025i 0
601.2 −0.169938 0.294342i −0.500000 0.866025i 0.942242 1.63201i 0 −0.169938 + 0.294342i 0.330062 0.571683i −1.32025 −0.500000 + 0.866025i 0
601.3 1.30084 + 2.25312i −0.500000 0.866025i −2.38437 + 4.12985i 0 1.30084 2.25312i 1.80084 3.11915i −7.20336 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.i.l 6
5.b even 2 1 195.2.i.d 6
5.c odd 4 2 975.2.bb.k 12
13.c even 3 1 inner 975.2.i.l 6
15.d odd 2 1 585.2.j.f 6
65.l even 6 1 2535.2.a.ba 3
65.n even 6 1 195.2.i.d 6
65.n even 6 1 2535.2.a.bb 3
65.q odd 12 2 975.2.bb.k 12
195.x odd 6 1 585.2.j.f 6
195.x odd 6 1 7605.2.a.bv 3
195.y odd 6 1 7605.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 5.b even 2 1
195.2.i.d 6 65.n even 6 1
585.2.j.f 6 15.d odd 2 1
585.2.j.f 6 195.x odd 6 1
975.2.i.l 6 1.a even 1 1 trivial
975.2.i.l 6 13.c even 3 1 inner
975.2.bb.k 12 5.c odd 4 2
975.2.bb.k 12 65.q odd 12 2
2535.2.a.ba 3 65.l even 6 1
2535.2.a.bb 3 65.n even 6 1
7605.2.a.bv 3 195.x odd 6 1
7605.2.a.bw 3 195.y odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{6} + 6T_{2}^{4} + 4T_{2}^{3} + 36T_{2}^{2} + 12T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} + 12T_{7}^{4} + 3T_{7}^{3} + 18T_{7}^{2} - 9T_{7} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{4} + 4 T^{3} + 36 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 12 T^{4} + 3 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} + 32 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} - 6 T^{4} - 83 T^{3} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 42 T^{4} + 196 T^{3} + \cdots + 9604 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 108 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} - 192 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + 36 T^{4} + 28 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 93 T + 363)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + 72 T^{4} + 232 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} + 24 T^{4} + 52 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + 66 T^{4} - 157 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( (T^{3} - 12 T^{2} - 6 T + 206)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 24 T - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + 48 T^{4} + 44 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + 30 T^{4} + \cdots + 4489 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 123201 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + 192 T^{4} + \cdots + 465124 \) Copy content Toggle raw display
$73$ \( (T^{3} + 21 T^{2} + 93 T - 89)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 3 T^{2} - 93 T - 363)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 18 T^{2} + 60 T + 168)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 30 T^{5} + 612 T^{4} + \cdots + 777924 \) Copy content Toggle raw display
$97$ \( T^{6} - 15 T^{5} + 234 T^{4} + \cdots + 346921 \) Copy content Toggle raw display
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