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Properties

Label 675.4.b.p
Level $675$
Weight $4$
Character orbit 675.b
Analytic conductor $39.826$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,4,Mod(649,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), 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: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 25x^{6} + 186x^{4} + 441x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} - 5) q^{4} + ( - \beta_{6} + \beta_{3} + \cdots + \beta_1) q^{7} + (4 \beta_{3} - 3 \beta_{2} + \beta_1) q^{8} + ( - \beta_{7} - 2 \beta_{4} - 14) q^{11} + ( - \beta_{6} - 2 \beta_{3} + 9 \beta_{2}) q^{13}+ \cdots + ( - 54 \beta_{6} - 1026 \beta_{3} + \cdots + 9 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 38 q^{4} - 104 q^{11} + 276 q^{14} - 10 q^{16} + 92 q^{19} - 938 q^{26} + 940 q^{29} - 524 q^{31} + 84 q^{34} - 1396 q^{41} + 838 q^{44} + 1074 q^{46} - 1560 q^{49} - 4068 q^{56} + 200 q^{59} + 148 q^{61}+ \cdots - 5694 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 25x^{6} + 186x^{4} + 441x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} + 44\nu^{3} + 258\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 22\nu^{5} - 120\nu^{3} - 108\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 23\nu^{5} - 142\nu^{3} - 183\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 25\nu^{4} + 168\nu^{2} + 216 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 25\nu^{4} + 186\nu^{2} + 333 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{7} + 589\nu^{5} + 3750\nu^{3} + 4977\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\nu^{6} - 46\nu^{4} - 284\nu^{2} - 365 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{3} - 6\beta_{2} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} - 17\beta_{3} + 13\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 26\beta_{5} + 44\beta_{4} + 271 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 44\beta_{6} + 576\beta_{3} - 314\beta_{2} + 19\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25\beta_{7} + 314\beta_{5} - 728\beta_{4} - 3271 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -364\beta_{6} - 4368\beta_{3} + 1948\beta_{2} - 107\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
3.67875i
2.82516i
− 1.39127i
− 1.24486i
1.24486i
1.39127i
− 2.82516i
− 3.67875i
− 4.72651i 0 −14.3399 0 0 17.6256i 29.9655i 0 0
649.2 − 4.52654i 0 −12.4896 0 0 30.8119i 20.3223i 0 0
649.3 − 2.53008i 0 1.59867 0 0 − 27.8841i − 24.2855i 0 0
649.4 − 1.33012i 0 6.23078 0 0 − 10.6979i − 18.9286i 0 0
649.5 1.33012i 0 6.23078 0 0 10.6979i 18.9286i 0 0
649.6 2.53008i 0 1.59867 0 0 27.8841i 24.2855i 0 0
649.7 4.52654i 0 −12.4896 0 0 − 30.8119i − 20.3223i 0 0
649.8 4.72651i 0 −14.3399 0 0 − 17.6256i − 29.9655i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.b.p 8
3.b odd 2 1 675.4.b.q 8
5.b even 2 1 inner 675.4.b.p 8
5.c odd 4 1 675.4.a.u ✓ 4
5.c odd 4 1 675.4.a.z yes 4
15.d odd 2 1 675.4.b.q 8
15.e even 4 1 675.4.a.v yes 4
15.e even 4 1 675.4.a.y yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.4.a.u ✓ 4 5.c odd 4 1
675.4.a.v yes 4 15.e even 4 1
675.4.a.y yes 4 15.e even 4 1
675.4.a.z yes 4 5.c odd 4 1
675.4.b.p 8 1.a even 1 1 trivial
675.4.b.p 8 5.b even 2 1 inner
675.4.b.q 8 3.b odd 2 1
675.4.b.q 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{8} + 51T_{2}^{6} + 819T_{2}^{4} + 4225T_{2}^{2} + 5184 \) Copy content Toggle raw display
\( T_{7}^{8} + 2152T_{7}^{6} + 1507824T_{7}^{4} + 375192000T_{7}^{2} + 26244000000 \) Copy content Toggle raw display
\( T_{11}^{4} + 52T_{11}^{3} + 38T_{11}^{2} - 16780T_{11} + 93825 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 51 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 26244000000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 52 T^{3} + \cdots + 93825)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 15945375625 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 541169774459136 \) Copy content Toggle raw display
$19$ \( (T^{4} - 46 T^{3} + \cdots - 23014672)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 68492647521 \) Copy content Toggle raw display
$29$ \( (T^{4} - 470 T^{3} + \cdots - 776494800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 262 T^{3} + \cdots + 254274336)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{4} + 698 T^{3} + \cdots + 1081900800)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{4} - 100 T^{3} + \cdots + 39525145425)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 74 T^{3} + \cdots + 13956747407)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1622 T^{3} + \cdots - 83536909425)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} - 508 T^{3} + \cdots + 3926930544)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 95\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} - 756 T^{3} + \cdots + 671702058000)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
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