↑

Properties

Label 975.2.w.g
Level $975$
Weight $2$
Character orbit 975.w
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(49,975)]
 
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, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.w (of order \(6\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{2} - \beta_{4} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots - 1) q^{4}+ \cdots + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{2} - \beta_{4} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 4 q^{4} - 6 q^{6} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 4 q^{4} - 6 q^{6} + 6 q^{7} + 4 q^{9} + 12 q^{11} - 6 q^{13} + 4 q^{14} - 6 q^{17} - 4 q^{18} - 12 q^{19} + 24 q^{22} + 12 q^{24} - 4 q^{26} + 4 q^{28} + 6 q^{29} - 12 q^{32} - 8 q^{33} + 4 q^{36} - 4 q^{37} - 30 q^{41} - 18 q^{42} - 30 q^{43} + 36 q^{46} - 76 q^{47} - 24 q^{48} + 8 q^{49} + 4 q^{51} + 20 q^{52} - 6 q^{54} + 4 q^{56} - 28 q^{58} + 30 q^{59} - 18 q^{62} - 6 q^{63} - 32 q^{64} - 14 q^{67} + 36 q^{68} - 4 q^{69} + 18 q^{71} + 44 q^{73} - 12 q^{74} - 26 q^{78} - 56 q^{79} - 4 q^{81} + 48 q^{82} - 40 q^{83} - 12 q^{84} - 6 q^{87} - 48 q^{88} + 18 q^{89} - 16 q^{91} + 16 q^{94} + 10 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 + 0.564882i
0.500000 − 1.19293i
0.500000 − 1.56488i
0.500000 + 2.19293i
0.500000 − 0.564882i
0.500000 + 1.19293i
0.500000 + 1.56488i
0.500000 − 2.19293i
−1.21545 − 2.10523i 0.866025 − 0.500000i −1.95466 + 3.38556i 0 −2.10523 − 1.21545i 0.650571 − 1.12682i 4.64136 0.500000 − 0.866025i 0
49.2 −0.663454 − 1.14914i −0.866025 + 0.500000i 0.119657 − 0.207252i 0 1.14914 + 0.663454i −0.529480 + 0.917086i −2.97136 0.500000 − 0.866025i 0
49.3 −0.150571 − 0.260797i 0.866025 − 0.500000i 0.954656 − 1.65351i 0 −0.260797 − 0.150571i 1.71545 − 2.97125i −1.17726 0.500000 − 0.866025i 0
49.4 1.02948 + 1.78311i −0.866025 + 0.500000i −1.11966 + 1.93930i 0 −1.78311 − 1.02948i 1.16345 − 2.01516i −0.492737 0.500000 − 0.866025i 0
199.1 −1.21545 + 2.10523i 0.866025 + 0.500000i −1.95466 − 3.38556i 0 −2.10523 + 1.21545i 0.650571 + 1.12682i 4.64136 0.500000 + 0.866025i 0
199.2 −0.663454 + 1.14914i −0.866025 − 0.500000i 0.119657 + 0.207252i 0 1.14914 − 0.663454i −0.529480 − 0.917086i −2.97136 0.500000 + 0.866025i 0
199.3 −0.150571 + 0.260797i 0.866025 + 0.500000i 0.954656 + 1.65351i 0 −0.260797 + 0.150571i 1.71545 + 2.97125i −1.17726 0.500000 + 0.866025i 0
199.4 1.02948 − 1.78311i −0.866025 − 0.500000i −1.11966 − 1.93930i 0 −1.78311 + 1.02948i 1.16345 + 2.01516i −0.492737 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.w.g 8
5.b even 2 1 975.2.w.j 8
5.c odd 4 1 195.2.bb.c ✓ 8
5.c odd 4 1 975.2.bc.i 8
13.e even 6 1 975.2.w.j 8
15.e even 4 1 585.2.bu.b 8
65.l even 6 1 inner 975.2.w.g 8
65.o even 12 1 2535.2.a.bl 4
65.r odd 12 1 195.2.bb.c ✓ 8
65.r odd 12 1 975.2.bc.i 8
65.t even 12 1 2535.2.a.bi 4
195.bc odd 12 1 7605.2.a.ck 4
195.bf even 12 1 585.2.bu.b 8
195.bn odd 12 1 7605.2.a.cg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.c ✓ 8 5.c odd 4 1
195.2.bb.c ✓ 8 65.r odd 12 1
585.2.bu.b 8 15.e even 4 1
585.2.bu.b 8 195.bf even 12 1
975.2.w.g 8 1.a even 1 1 trivial
975.2.w.g 8 65.l even 6 1 inner
975.2.w.j 8 5.b even 2 1
975.2.w.j 8 13.e even 6 1
975.2.bc.i 8 5.c odd 4 1
975.2.bc.i 8 65.r odd 12 1
2535.2.a.bi 4 65.t even 12 1
2535.2.a.bl 4 65.o even 12 1
7605.2.a.cg 4 195.bn odd 12 1
7605.2.a.ck 4 195.bc odd 12 1

Hecke kernels

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

\( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 8T_{2}^{5} + 34T_{2}^{4} + 40T_{2}^{3} + 56T_{2}^{2} + 16T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{7} + 28T_{7}^{6} - 60T_{7}^{5} + 111T_{7}^{4} - 84T_{7}^{3} + 124T_{7}^{2} - 66T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( T^{8} - 24 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$31$ \( T^{8} + 108 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 30 T^{7} + \cdots + 15507844 \) Copy content Toggle raw display
$43$ \( T^{8} + 30 T^{7} + \cdots + 316969 \) Copy content Toggle raw display
$47$ \( (T^{4} + 38 T^{3} + \cdots + 6862)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 192 T^{6} + \cdots + 2768896 \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + \cdots + 209764 \) Copy content Toggle raw display
$61$ \( T^{8} + 178 T^{6} + \cdots + 2474329 \) Copy content Toggle raw display
$67$ \( T^{8} + 14 T^{7} + \cdots + 5470921 \) Copy content Toggle raw display
$71$ \( T^{8} - 18 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$73$ \( (T^{4} - 22 T^{3} + \cdots + 421)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 28 T^{3} + \cdots - 407)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20 T^{3} + \cdots - 25832)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 18 T^{7} + \cdots + 94828644 \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots + 39400729 \) Copy content Toggle raw display
show more
show less