Properties

Label 675.4.b.b
Level $675$
Weight $4$
Character orbit 675.b
Analytic conductor $39.826$
Analytic rank $0$
Dimension $2$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{2} - q^{4} + 25 i q^{7} + 21 i q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} - q^{4} + 25 i q^{7} + 21 i q^{8} + 15 q^{11} + 20 i q^{13} - 75 q^{14} - 71 q^{16} + 72 i q^{17} - 2 q^{19} + 45 i q^{22} - 114 i q^{23} - 60 q^{26} - 25 i q^{28} + 30 q^{29} + 101 q^{31} - 45 i q^{32} - 216 q^{34} + 430 i q^{37} - 6 i q^{38} + 30 q^{41} + 110 i q^{43} - 15 q^{44} + 342 q^{46} - 330 i q^{47} - 282 q^{49} - 20 i q^{52} - 621 i q^{53} - 525 q^{56} + 90 i q^{58} - 660 q^{59} - 376 q^{61} + 303 i q^{62} - 433 q^{64} + 250 i q^{67} - 72 i q^{68} + 360 q^{71} + 785 i q^{73} - 1290 q^{74} + 2 q^{76} + 375 i q^{77} - 488 q^{79} + 90 i q^{82} - 489 i q^{83} - 330 q^{86} + 315 i q^{88} - 450 q^{89} - 500 q^{91} + 114 i q^{92} + 990 q^{94} + 1105 i q^{97} - 846 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 30 q^{11} - 150 q^{14} - 142 q^{16} - 4 q^{19} - 120 q^{26} + 60 q^{29} + 202 q^{31} - 432 q^{34} + 60 q^{41} - 30 q^{44} + 684 q^{46} - 564 q^{49} - 1050 q^{56} - 1320 q^{59} - 752 q^{61} - 866 q^{64} + 720 q^{71} - 2580 q^{74} + 4 q^{76} - 976 q^{79} - 660 q^{86} - 900 q^{89} - 1000 q^{91} + 1980 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
3.00000i 0 −1.00000 0 0 25.0000i 21.0000i 0 0
649.2 3.00000i 0 −1.00000 0 0 25.0000i 21.0000i 0 0
\(n\): e.g. 2-40 or 990-1000
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.b 2
3.b odd 2 1 675.4.b.a 2
5.b even 2 1 inner 675.4.b.b 2
5.c odd 4 1 27.4.a.a 1
5.c odd 4 1 675.4.a.j 1
15.d odd 2 1 675.4.b.a 2
15.e even 4 1 27.4.a.b yes 1
15.e even 4 1 675.4.a.a 1
20.e even 4 1 432.4.a.a 1
35.f even 4 1 1323.4.a.d 1
40.i odd 4 1 1728.4.a.bc 1
40.k even 4 1 1728.4.a.bd 1
45.k odd 12 2 81.4.c.c 2
45.l even 12 2 81.4.c.a 2
60.l odd 4 1 432.4.a.n 1
105.k odd 4 1 1323.4.a.k 1
120.q odd 4 1 1728.4.a.d 1
120.w even 4 1 1728.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 5.c odd 4 1
27.4.a.b yes 1 15.e even 4 1
81.4.c.a 2 45.l even 12 2
81.4.c.c 2 45.k odd 12 2
432.4.a.a 1 20.e even 4 1
432.4.a.n 1 60.l odd 4 1
675.4.a.a 1 15.e even 4 1
675.4.a.j 1 5.c odd 4 1
675.4.b.a 2 3.b odd 2 1
675.4.b.a 2 15.d odd 2 1
675.4.b.b 2 1.a even 1 1 trivial
675.4.b.b 2 5.b even 2 1 inner
1323.4.a.d 1 35.f even 4 1
1323.4.a.k 1 105.k odd 4 1
1728.4.a.c 1 120.w even 4 1
1728.4.a.d 1 120.q odd 4 1
1728.4.a.bc 1 40.i odd 4 1
1728.4.a.bd 1 40.k even 4 1

Hecke kernels

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

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 625 \) Copy content Toggle raw display
\( T_{11} - 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 625 \) Copy content Toggle raw display
$11$ \( (T - 15)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( T^{2} + 5184 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12996 \) Copy content Toggle raw display
$29$ \( (T - 30)^{2} \) Copy content Toggle raw display
$31$ \( (T - 101)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 184900 \) Copy content Toggle raw display
$41$ \( (T - 30)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 12100 \) Copy content Toggle raw display
$47$ \( T^{2} + 108900 \) Copy content Toggle raw display
$53$ \( T^{2} + 385641 \) Copy content Toggle raw display
$59$ \( (T + 660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 376)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 62500 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 616225 \) Copy content Toggle raw display
$79$ \( (T + 488)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 239121 \) Copy content Toggle raw display
$89$ \( (T + 450)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1221025 \) Copy content Toggle raw display
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