Properties

Label 675.2.k.a
Level $675$
Weight $2$
Character orbit 675.k
Analytic conductor $5.390$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} - \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{7} - 3 \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} - \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{7} - 3 \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 2) q^{11} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{13} + 3 \zeta_{12}^{2} q^{14} + ( - \zeta_{12}^{2} + 1) q^{16} - 4 \zeta_{12}^{3} q^{17} + 8 q^{19} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{22} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{23} + 2 q^{26} - 3 \zeta_{12}^{3} q^{28} + ( - \zeta_{12}^{2} + 1) q^{29} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{32} + ( - 4 \zeta_{12}^{2} + 4) q^{34} - 4 \zeta_{12}^{3} q^{37} + 8 \zeta_{12} q^{38} + 5 \zeta_{12}^{2} q^{41} - 8 \zeta_{12} q^{43} + 2 q^{44} + 3 q^{46} + 7 \zeta_{12} q^{47} + 2 \zeta_{12}^{2} q^{49} - 2 \zeta_{12} q^{52} - 2 \zeta_{12}^{3} q^{53} + ( - 9 \zeta_{12}^{2} + 9) q^{56} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{58} + 14 \zeta_{12}^{2} q^{59} + (7 \zeta_{12}^{2} - 7) q^{61} - 7 q^{64} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{67} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{68} - 2 q^{71} - 4 \zeta_{12}^{3} q^{73} + ( - 4 \zeta_{12}^{2} + 4) q^{74} - 8 \zeta_{12}^{2} q^{76} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{77} + (6 \zeta_{12}^{2} - 6) q^{79} + 5 \zeta_{12}^{3} q^{82} - 9 \zeta_{12} q^{83} - 8 \zeta_{12}^{2} q^{86} + 6 \zeta_{12} q^{88} - 15 q^{89} + 6 q^{91} - 3 \zeta_{12} q^{92} + 7 \zeta_{12}^{2} q^{94} - 2 \zeta_{12} q^{97} + 2 \zeta_{12}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} - 4 q^{11} + 6 q^{14} + 2 q^{16} + 32 q^{19} + 8 q^{26} + 2 q^{29} + 8 q^{34} + 10 q^{41} + 8 q^{44} + 12 q^{46} + 4 q^{49} + 18 q^{56} + 28 q^{59} - 14 q^{61} - 28 q^{64} - 8 q^{71} + 8 q^{74} - 16 q^{76} - 12 q^{79} - 16 q^{86} - 60 q^{89} + 24 q^{91} + 14 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0 −2.59808 + 1.50000i 3.00000i 0 0
199.2 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0 2.59808 1.50000i 3.00000i 0 0
424.1 −0.866025 0.500000i 0 −0.500000 0.866025i 0 0 −2.59808 1.50000i 3.00000i 0 0
424.2 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 0 2.59808 + 1.50000i 3.00000i 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.k.a 4
3.b odd 2 1 225.2.k.a 4
5.b even 2 1 inner 675.2.k.a 4
5.c odd 4 1 135.2.e.a 2
5.c odd 4 1 675.2.e.a 2
9.c even 3 1 inner 675.2.k.a 4
9.c even 3 1 2025.2.b.d 2
9.d odd 6 1 225.2.k.a 4
9.d odd 6 1 2025.2.b.c 2
15.d odd 2 1 225.2.k.a 4
15.e even 4 1 45.2.e.a 2
15.e even 4 1 225.2.e.a 2
20.e even 4 1 2160.2.q.a 2
45.h odd 6 1 225.2.k.a 4
45.h odd 6 1 2025.2.b.c 2
45.j even 6 1 inner 675.2.k.a 4
45.j even 6 1 2025.2.b.d 2
45.k odd 12 1 135.2.e.a 2
45.k odd 12 1 405.2.a.b 1
45.k odd 12 1 675.2.e.a 2
45.k odd 12 1 2025.2.a.e 1
45.l even 12 1 45.2.e.a 2
45.l even 12 1 225.2.e.a 2
45.l even 12 1 405.2.a.e 1
45.l even 12 1 2025.2.a.b 1
60.l odd 4 1 720.2.q.d 2
180.v odd 12 1 720.2.q.d 2
180.v odd 12 1 6480.2.a.k 1
180.x even 12 1 2160.2.q.a 2
180.x even 12 1 6480.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 15.e even 4 1
45.2.e.a 2 45.l even 12 1
135.2.e.a 2 5.c odd 4 1
135.2.e.a 2 45.k odd 12 1
225.2.e.a 2 15.e even 4 1
225.2.e.a 2 45.l even 12 1
225.2.k.a 4 3.b odd 2 1
225.2.k.a 4 9.d odd 6 1
225.2.k.a 4 15.d odd 2 1
225.2.k.a 4 45.h odd 6 1
405.2.a.b 1 45.k odd 12 1
405.2.a.e 1 45.l even 12 1
675.2.e.a 2 5.c odd 4 1
675.2.e.a 2 45.k odd 12 1
675.2.k.a 4 1.a even 1 1 trivial
675.2.k.a 4 5.b even 2 1 inner
675.2.k.a 4 9.c even 3 1 inner
675.2.k.a 4 45.j even 6 1 inner
720.2.q.d 2 60.l odd 4 1
720.2.q.d 2 180.v odd 12 1
2025.2.a.b 1 45.l even 12 1
2025.2.a.e 1 45.k odd 12 1
2025.2.b.c 2 9.d odd 6 1
2025.2.b.c 2 45.h odd 6 1
2025.2.b.d 2 9.c even 3 1
2025.2.b.d 2 45.j even 6 1
2160.2.q.a 2 20.e even 4 1
2160.2.q.a 2 180.x even 12 1
6480.2.a.k 1 180.v odd 12 1
6480.2.a.x 1 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T - 8)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$47$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$89$ \( (T + 15)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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