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Properties

Label 675.5.d.d
Level $675$
Weight $5$
Character orbit 675.d
Analytic conductor $69.775$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,5,Mod(674,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.674"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 675.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,6,0,-14,0,0,0,-138,0,0,0,0,0,0,0,-190,-828,0,608] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.7747250816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + 3 q^{2} - 7 q^{4} - 19 i q^{7} - 69 q^{8} + 123 i q^{11} - 302 i q^{13} - 57 i q^{14} - 95 q^{16} - 414 q^{17} + 304 q^{19} + 369 i q^{22} + 300 q^{23} - 906 i q^{26} + 133 i q^{28} + 678 i q^{29} + \cdots + 6120 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 14 q^{4} - 138 q^{8} - 190 q^{16} - 828 q^{17} + 608 q^{19} + 600 q^{23} + 478 q^{31} + 1638 q^{32} - 2484 q^{34} + 1824 q^{38} + 1800 q^{46} - 4332 q^{47} + 4080 q^{49} - 3186 q^{53} - 632 q^{61}+ \cdots + 12240 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
674.1
1.00000i
− 1.00000i
3.00000 0 −7.00000 0 0 − 19.0000i −69.0000 0 0
674.2 3.00000 0 −7.00000 0 0 19.0000i −69.0000 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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.5.d.d 2
3.b odd 2 1 675.5.d.a 2
5.b even 2 1 675.5.d.a 2
5.c odd 4 1 27.5.b.c ✓ 2
5.c odd 4 1 675.5.c.h 2
15.d odd 2 1 inner 675.5.d.d 2
15.e even 4 1 27.5.b.c ✓ 2
15.e even 4 1 675.5.c.h 2
20.e even 4 1 432.5.e.e 2
45.k odd 12 2 81.5.d.b 4
45.l even 12 2 81.5.d.b 4
60.l odd 4 1 432.5.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.5.b.c ✓ 2 5.c odd 4 1
27.5.b.c ✓ 2 15.e even 4 1
81.5.d.b 4 45.k odd 12 2
81.5.d.b 4 45.l even 12 2
432.5.e.e 2 20.e even 4 1
432.5.e.e 2 60.l odd 4 1
675.5.c.h 2 5.c odd 4 1
675.5.c.h 2 15.e even 4 1
675.5.d.a 2 3.b odd 2 1
675.5.d.a 2 5.b even 2 1
675.5.d.d 2 1.a even 1 1 trivial
675.5.d.d 2 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(675, [\chi])\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 361 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 361 \) Copy content Toggle raw display
$11$ \( T^{2} + 15129 \) Copy content Toggle raw display
$13$ \( T^{2} + 91204 \) Copy content Toggle raw display
$17$ \( (T + 414)^{2} \) Copy content Toggle raw display
$19$ \( (T - 304)^{2} \) Copy content Toggle raw display
$23$ \( (T - 300)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 459684 \) Copy content Toggle raw display
$31$ \( (T - 239)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 547600 \) Copy content Toggle raw display
$41$ \( T^{2} + 51984 \) Copy content Toggle raw display
$43$ \( T^{2} + 964324 \) Copy content Toggle raw display
$47$ \( (T + 2166)^{2} \) Copy content Toggle raw display
$53$ \( (T + 1593)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8538084 \) Copy content Toggle raw display
$61$ \( (T + 316)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 21362884 \) Copy content Toggle raw display
$71$ \( T^{2} + 3305124 \) Copy content Toggle raw display
$73$ \( T^{2} + 9186961 \) Copy content Toggle raw display
$79$ \( (T - 10450)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12633)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 49028004 \) Copy content Toggle raw display
$97$ \( T^{2} + 42471289 \) Copy content Toggle raw display
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