Properties

Label 975.2.i.m
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: 6.0.591408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 16\nu^{3} + 10\nu^{2} - 3\nu + 12 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{5} + 16\nu^{4} - 27\nu^{3} + 40\nu^{2} - 12\nu + 85 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{5} + 26\nu^{4} - 104\nu^{3} - 9\nu^{2} - 260\nu + 78 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} - 7\beta_{4} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 17\beta_{2} - 17\beta _1 + 8 \) 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
−0.740597 1.28275i
1.08504 + 1.87935i
0.155554 + 0.269427i
−0.740597 + 1.28275i
1.08504 1.87935i
0.155554 0.269427i
−0.837565 + 1.45071i −0.500000 + 0.866025i −0.403032 0.698071i 0 −0.837565 1.45071i 1.81876 + 3.15018i −2.00000 −0.500000 0.866025i 0
451.2 −0.269594 + 0.466951i −0.500000 + 0.866025i 0.854638 + 1.48028i 0 −0.269594 0.466951i −2.40049 4.15777i −2.00000 −0.500000 0.866025i 0
451.3 1.10716 1.91766i −0.500000 + 0.866025i −1.45161 2.51426i 0 1.10716 + 1.91766i −1.91827 3.32254i −2.00000 −0.500000 0.866025i 0
601.1 −0.837565 1.45071i −0.500000 0.866025i −0.403032 + 0.698071i 0 −0.837565 + 1.45071i 1.81876 3.15018i −2.00000 −0.500000 + 0.866025i 0
601.2 −0.269594 0.466951i −0.500000 0.866025i 0.854638 1.48028i 0 −0.269594 + 0.466951i −2.40049 + 4.15777i −2.00000 −0.500000 + 0.866025i 0
601.3 1.10716 + 1.91766i −0.500000 0.866025i −1.45161 + 2.51426i 0 1.10716 1.91766i −1.91827 + 3.32254i −2.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.3
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.m 6
5.b even 2 1 195.2.i.e 6
5.c odd 4 2 975.2.bb.j 12
13.c even 3 1 inner 975.2.i.m 6
15.d odd 2 1 585.2.j.g 6
65.l even 6 1 2535.2.a.z 3
65.n even 6 1 195.2.i.e 6
65.n even 6 1 2535.2.a.y 3
65.q odd 12 2 975.2.bb.j 12
195.x odd 6 1 585.2.j.g 6
195.x odd 6 1 7605.2.a.bu 3
195.y odd 6 1 7605.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e 6 5.b even 2 1
195.2.i.e 6 65.n even 6 1
585.2.j.g 6 15.d odd 2 1
585.2.j.g 6 195.x odd 6 1
975.2.i.m 6 1.a even 1 1 trivial
975.2.i.m 6 13.c even 3 1 inner
975.2.bb.j 12 5.c odd 4 2
975.2.bb.j 12 65.q odd 12 2
2535.2.a.y 3 65.n even 6 1
2535.2.a.z 3 65.l even 6 1
7605.2.a.bt 3 195.y odd 6 1
7605.2.a.bu 3 195.x odd 6 1

Hecke kernels

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

\( T_{2}^{6} + 4T_{2}^{4} + 4T_{2}^{3} + 16T_{2}^{2} + 8T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 5T_{7}^{5} + 38T_{7}^{4} + 69T_{7}^{3} + 504T_{7}^{2} + 871T_{7} + 4489 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 4 T^{4} + 4 T^{3} + 16 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + 38 T^{4} + \cdots + 4489 \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + 32 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 12 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + 24 T^{4} + \cdots + 1156 \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + 152 T^{3} + \cdots + 5776 \) Copy content Toggle raw display
$23$ \( T^{6} - 8 T^{5} + 104 T^{4} + \cdots + 92416 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + 54 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} - T - 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 14 T^{5} + 184 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$41$ \( T^{6} - 20 T^{5} + 282 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$43$ \( T^{6} + 15 T^{5} + 184 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 16 T^{2} + 32 T - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 10 T^{5} + 202 T^{4} + \cdots + 940900 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + 82 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$67$ \( T^{6} + 11 T^{5} + 164 T^{4} + \cdots + 61009 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} + 34 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$73$ \( (T^{3} - 15 T^{2} + 71 T - 103)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 17 T^{2} + 63 T - 67)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 2 T^{2} - 12 T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 22 T^{5} + 398 T^{4} + \cdots + 17956 \) Copy content Toggle raw display
$97$ \( T^{6} + T^{5} + 152 T^{4} + \cdots + 299209 \) Copy content Toggle raw display
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