Properties

Label 675.2.b.f
Level $675$
Weight $2$
Character orbit 675.b
Analytic conductor $5.390$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + i q^{7} +O(q^{10})\) \( q + 2 q^{4} + i q^{7} + 5 i q^{13} + 4 q^{16} + 7 q^{19} + 2 i q^{28} -4 q^{31} -11 i q^{37} + 8 i q^{43} + 6 q^{49} + 10 i q^{52} - q^{61} + 8 q^{64} -5 i q^{67} -7 i q^{73} + 14 q^{76} -17 q^{79} -5 q^{91} + 19 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + O(q^{10}) \) \( 2 q + 4 q^{4} + 8 q^{16} + 14 q^{19} - 8 q^{31} + 12 q^{49} - 2 q^{61} + 16 q^{64} + 28 q^{76} - 34 q^{79} - 10 q^{91} + O(q^{100}) \)

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.

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
0 0 2.00000 0 0 1.00000i 0 0 0
649.2 0 0 2.00000 0 0 1.00000i 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
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 675.2.b.f 2
3.b odd 2 1 CM 675.2.b.f 2
5.b even 2 1 inner 675.2.b.f 2
5.c odd 4 1 27.2.a.a 1
5.c odd 4 1 675.2.a.e 1
15.d odd 2 1 inner 675.2.b.f 2
15.e even 4 1 27.2.a.a 1
15.e even 4 1 675.2.a.e 1
20.e even 4 1 432.2.a.e 1
35.f even 4 1 1323.2.a.i 1
40.i odd 4 1 1728.2.a.n 1
40.k even 4 1 1728.2.a.o 1
45.k odd 12 2 81.2.c.a 2
45.l even 12 2 81.2.c.a 2
55.e even 4 1 3267.2.a.f 1
60.l odd 4 1 432.2.a.e 1
65.h odd 4 1 4563.2.a.e 1
85.g odd 4 1 7803.2.a.k 1
95.g even 4 1 9747.2.a.f 1
105.k odd 4 1 1323.2.a.i 1
120.q odd 4 1 1728.2.a.o 1
120.w even 4 1 1728.2.a.n 1
135.q even 36 6 729.2.e.f 6
135.r odd 36 6 729.2.e.f 6
165.l odd 4 1 3267.2.a.f 1
180.v odd 12 2 1296.2.i.i 2
180.x even 12 2 1296.2.i.i 2
195.s even 4 1 4563.2.a.e 1
255.o even 4 1 7803.2.a.k 1
285.j odd 4 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 5.c odd 4 1
27.2.a.a 1 15.e even 4 1
81.2.c.a 2 45.k odd 12 2
81.2.c.a 2 45.l even 12 2
432.2.a.e 1 20.e even 4 1
432.2.a.e 1 60.l odd 4 1
675.2.a.e 1 5.c odd 4 1
675.2.a.e 1 15.e even 4 1
675.2.b.f 2 1.a even 1 1 trivial
675.2.b.f 2 3.b odd 2 1 CM
675.2.b.f 2 5.b even 2 1 inner
675.2.b.f 2 15.d odd 2 1 inner
729.2.e.f 6 135.q even 36 6
729.2.e.f 6 135.r odd 36 6
1296.2.i.i 2 180.v odd 12 2
1296.2.i.i 2 180.x even 12 2
1323.2.a.i 1 35.f even 4 1
1323.2.a.i 1 105.k odd 4 1
1728.2.a.n 1 40.i odd 4 1
1728.2.a.n 1 120.w even 4 1
1728.2.a.o 1 40.k even 4 1
1728.2.a.o 1 120.q odd 4 1
3267.2.a.f 1 55.e even 4 1
3267.2.a.f 1 165.l odd 4 1
4563.2.a.e 1 65.h odd 4 1
4563.2.a.e 1 195.s even 4 1
7803.2.a.k 1 85.g odd 4 1
7803.2.a.k 1 255.o even 4 1
9747.2.a.f 1 95.g even 4 1
9747.2.a.f 1 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\):

\( T_{2} \)
\( T_{7}^{2} + 1 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 121 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( 25 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 49 + T^{2} \)
$79$ \( ( 17 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 361 + T^{2} \)
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