Properties

Label 675.4.a.n
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $2$
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(1,675)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

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: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 10 q^{4} - 11 q^{7} + 2 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 10 q^{4} - 11 q^{7} + 2 \beta q^{8} - 4 \beta q^{11} - 29 q^{13} - 11 \beta q^{14} - 44 q^{16} - 12 \beta q^{17} + 29 q^{19} - 72 q^{22} + 20 \beta q^{23} - 29 \beta q^{26} - 110 q^{28} - 64 \beta q^{29} - 268 q^{31} - 60 \beta q^{32} - 216 q^{34} - 83 q^{37} + 29 \beta q^{38} + 64 \beta q^{41} + 232 q^{43} - 40 \beta q^{44} + 360 q^{46} - 92 \beta q^{47} - 222 q^{49} - 290 q^{52} + 72 \beta q^{53} - 22 \beta q^{56} - 1152 q^{58} - 68 \beta q^{59} + 767 q^{61} - 268 \beta q^{62} - 728 q^{64} + 511 q^{67} - 120 \beta q^{68} - 168 \beta q^{71} - 137 q^{73} - 83 \beta q^{74} + 290 q^{76} + 44 \beta q^{77} - 475 q^{79} + 1152 q^{82} + 136 \beta q^{83} + 232 \beta q^{86} - 144 q^{88} + 60 \beta q^{89} + 319 q^{91} + 200 \beta q^{92} - 1656 q^{94} - 821 q^{97} - 222 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{4} - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{4} - 22 q^{7} - 58 q^{13} - 88 q^{16} + 58 q^{19} - 144 q^{22} - 220 q^{28} - 536 q^{31} - 432 q^{34} - 166 q^{37} + 464 q^{43} + 720 q^{46} - 444 q^{49} - 580 q^{52} - 2304 q^{58} + 1534 q^{61} - 1456 q^{64} + 1022 q^{67} - 274 q^{73} + 580 q^{76} - 950 q^{79} + 2304 q^{82} - 288 q^{88} + 638 q^{91} - 3312 q^{94} - 1642 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−4.24264 0 10.0000 0 0 −11.0000 −8.48528 0 0
1.2 4.24264 0 10.0000 0 0 −11.0000 8.48528 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.a.n 2
3.b odd 2 1 inner 675.4.a.n 2
5.b even 2 1 27.4.a.c 2
5.c odd 4 2 675.4.b.i 4
15.d odd 2 1 27.4.a.c 2
15.e even 4 2 675.4.b.i 4
20.d odd 2 1 432.4.a.q 2
35.c odd 2 1 1323.4.a.t 2
40.e odd 2 1 1728.4.a.bk 2
40.f even 2 1 1728.4.a.bp 2
45.h odd 6 2 81.4.c.e 4
45.j even 6 2 81.4.c.e 4
60.h even 2 1 432.4.a.q 2
105.g even 2 1 1323.4.a.t 2
120.i odd 2 1 1728.4.a.bp 2
120.m even 2 1 1728.4.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.c 2 5.b even 2 1
27.4.a.c 2 15.d odd 2 1
81.4.c.e 4 45.h odd 6 2
81.4.c.e 4 45.j even 6 2
432.4.a.q 2 20.d odd 2 1
432.4.a.q 2 60.h even 2 1
675.4.a.n 2 1.a even 1 1 trivial
675.4.a.n 2 3.b odd 2 1 inner
675.4.b.i 4 5.c odd 4 2
675.4.b.i 4 15.e even 4 2
1323.4.a.t 2 35.c odd 2 1
1323.4.a.t 2 105.g even 2 1
1728.4.a.bk 2 40.e odd 2 1
1728.4.a.bk 2 120.m even 2 1
1728.4.a.bp 2 40.f even 2 1
1728.4.a.bp 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2}^{2} - 18 \) Copy content Toggle raw display
\( T_{7} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 18 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 11)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 288 \) Copy content Toggle raw display
$13$ \( (T + 29)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2592 \) Copy content Toggle raw display
$19$ \( (T - 29)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 7200 \) Copy content Toggle raw display
$29$ \( T^{2} - 73728 \) Copy content Toggle raw display
$31$ \( (T + 268)^{2} \) Copy content Toggle raw display
$37$ \( (T + 83)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 73728 \) Copy content Toggle raw display
$43$ \( (T - 232)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 152352 \) Copy content Toggle raw display
$53$ \( T^{2} - 93312 \) Copy content Toggle raw display
$59$ \( T^{2} - 83232 \) Copy content Toggle raw display
$61$ \( (T - 767)^{2} \) Copy content Toggle raw display
$67$ \( (T - 511)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 508032 \) Copy content Toggle raw display
$73$ \( (T + 137)^{2} \) Copy content Toggle raw display
$79$ \( (T + 475)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 332928 \) Copy content Toggle raw display
$89$ \( T^{2} - 64800 \) Copy content Toggle raw display
$97$ \( (T + 821)^{2} \) Copy content Toggle raw display
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