Properties

Label 675.3.i.a
Level $675$
Weight $3$
Character orbit 675.i
Analytic conductor $18.392$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3924178443\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} + ( 2 - \zeta_{12}^{2} ) q^{11} + 4 \zeta_{12} q^{13} + ( 2 + 2 \zeta_{12}^{2} ) q^{14} + 11 \zeta_{12}^{2} q^{16} + ( -18 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{17} -11 q^{19} -3 \zeta_{12} q^{22} + ( -16 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{23} + ( 4 - 8 \zeta_{12}^{2} ) q^{26} + 2 \zeta_{12}^{3} q^{28} + ( 52 - 26 \zeta_{12}^{2} ) q^{29} + ( -32 + 32 \zeta_{12}^{2} ) q^{31} + ( -9 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{32} + 27 \zeta_{12}^{2} q^{34} + 34 \zeta_{12}^{3} q^{37} + ( 11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{38} + ( 7 + 7 \zeta_{12}^{2} ) q^{41} + ( -61 \zeta_{12} + 61 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} ) q^{44} + 48 q^{46} + ( -28 \zeta_{12} - 28 \zeta_{12}^{3} ) q^{47} + ( -45 + 45 \zeta_{12}^{2} ) q^{49} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} + ( 20 - 10 \zeta_{12}^{2} ) q^{56} -78 \zeta_{12} q^{58} + ( 29 + 29 \zeta_{12}^{2} ) q^{59} -56 \zeta_{12}^{2} q^{61} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{62} + 71 q^{64} -31 \zeta_{12} q^{67} + ( -9 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{68} + ( -18 + 36 \zeta_{12}^{2} ) q^{71} + 65 \zeta_{12}^{3} q^{73} + ( 68 - 34 \zeta_{12}^{2} ) q^{74} + ( -11 + 11 \zeta_{12}^{2} ) q^{76} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{77} + 38 \zeta_{12}^{2} q^{79} -21 \zeta_{12}^{3} q^{82} + ( 28 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{83} + ( 61 + 61 \zeta_{12}^{2} ) q^{86} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{88} + ( -72 + 144 \zeta_{12}^{2} ) q^{89} -8 q^{91} + ( 16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{92} + ( -84 + 84 \zeta_{12}^{2} ) q^{94} + ( 115 \zeta_{12} - 115 \zeta_{12}^{3} ) q^{97} + ( 90 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + O(q^{10}) \) \( 4 q + 2 q^{4} + 6 q^{11} + 12 q^{14} + 22 q^{16} - 44 q^{19} + 156 q^{29} - 64 q^{31} + 54 q^{34} + 42 q^{41} + 192 q^{46} - 90 q^{49} + 60 q^{56} + 174 q^{59} - 112 q^{61} + 284 q^{64} + 204 q^{74} - 22 q^{76} + 76 q^{79} + 366 q^{86} - 32 q^{91} - 168 q^{94} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0 −1.73205 1.00000i −8.66025 0 0
224.2 0.866025 1.50000i 0 0.500000 + 0.866025i 0 0 1.73205 + 1.00000i 8.66025 0 0
449.1 −0.866025 1.50000i 0 0.500000 0.866025i 0 0 −1.73205 + 1.00000i −8.66025 0 0
449.2 0.866025 + 1.50000i 0 0.500000 0.866025i 0 0 1.73205 1.00000i 8.66025 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.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.3.i.a 4
3.b odd 2 1 225.3.i.a 4
5.b even 2 1 inner 675.3.i.a 4
5.c odd 4 1 27.3.d.a 2
5.c odd 4 1 675.3.j.a 2
9.c even 3 1 225.3.i.a 4
9.d odd 6 1 inner 675.3.i.a 4
15.d odd 2 1 225.3.i.a 4
15.e even 4 1 9.3.d.a 2
15.e even 4 1 225.3.j.a 2
20.e even 4 1 432.3.q.a 2
40.i odd 4 1 1728.3.q.a 2
40.k even 4 1 1728.3.q.b 2
45.h odd 6 1 inner 675.3.i.a 4
45.j even 6 1 225.3.i.a 4
45.k odd 12 1 9.3.d.a 2
45.k odd 12 1 81.3.b.a 2
45.k odd 12 1 225.3.j.a 2
45.l even 12 1 27.3.d.a 2
45.l even 12 1 81.3.b.a 2
45.l even 12 1 675.3.j.a 2
60.l odd 4 1 144.3.q.a 2
105.k odd 4 1 441.3.r.a 2
105.w odd 12 1 441.3.j.b 2
105.w odd 12 1 441.3.n.a 2
105.x even 12 1 441.3.j.a 2
105.x even 12 1 441.3.n.b 2
120.q odd 4 1 576.3.q.a 2
120.w even 4 1 576.3.q.b 2
180.v odd 12 1 432.3.q.a 2
180.v odd 12 1 1296.3.e.a 2
180.x even 12 1 144.3.q.a 2
180.x even 12 1 1296.3.e.a 2
315.bs even 12 1 441.3.n.a 2
315.bt odd 12 1 441.3.n.b 2
315.cb even 12 1 441.3.r.a 2
315.cg even 12 1 441.3.j.b 2
315.ch odd 12 1 441.3.j.a 2
360.bo even 12 1 576.3.q.a 2
360.br even 12 1 1728.3.q.a 2
360.bt odd 12 1 1728.3.q.b 2
360.bu odd 12 1 576.3.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 15.e even 4 1
9.3.d.a 2 45.k odd 12 1
27.3.d.a 2 5.c odd 4 1
27.3.d.a 2 45.l even 12 1
81.3.b.a 2 45.k odd 12 1
81.3.b.a 2 45.l even 12 1
144.3.q.a 2 60.l odd 4 1
144.3.q.a 2 180.x even 12 1
225.3.i.a 4 3.b odd 2 1
225.3.i.a 4 9.c even 3 1
225.3.i.a 4 15.d odd 2 1
225.3.i.a 4 45.j even 6 1
225.3.j.a 2 15.e even 4 1
225.3.j.a 2 45.k odd 12 1
432.3.q.a 2 20.e even 4 1
432.3.q.a 2 180.v odd 12 1
441.3.j.a 2 105.x even 12 1
441.3.j.a 2 315.ch odd 12 1
441.3.j.b 2 105.w odd 12 1
441.3.j.b 2 315.cg even 12 1
441.3.n.a 2 105.w odd 12 1
441.3.n.a 2 315.bs even 12 1
441.3.n.b 2 105.x even 12 1
441.3.n.b 2 315.bt odd 12 1
441.3.r.a 2 105.k odd 4 1
441.3.r.a 2 315.cb even 12 1
576.3.q.a 2 120.q odd 4 1
576.3.q.a 2 360.bo even 12 1
576.3.q.b 2 120.w even 4 1
576.3.q.b 2 360.bu odd 12 1
675.3.i.a 4 1.a even 1 1 trivial
675.3.i.a 4 5.b even 2 1 inner
675.3.i.a 4 9.d odd 6 1 inner
675.3.i.a 4 45.h odd 6 1 inner
675.3.j.a 2 5.c odd 4 1
675.3.j.a 2 45.l even 12 1
1296.3.e.a 2 180.v odd 12 1
1296.3.e.a 2 180.x even 12 1
1728.3.q.a 2 40.i odd 4 1
1728.3.q.a 2 360.br even 12 1
1728.3.q.b 2 40.k even 4 1
1728.3.q.b 2 360.bt odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(675, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( ( 3 - 3 T + T^{2} )^{2} \)
$13$ \( 256 - 16 T^{2} + T^{4} \)
$17$ \( ( -243 + T^{2} )^{2} \)
$19$ \( ( 11 + T )^{4} \)
$23$ \( 589824 + 768 T^{2} + T^{4} \)
$29$ \( ( 2028 - 78 T + T^{2} )^{2} \)
$31$ \( ( 1024 + 32 T + T^{2} )^{2} \)
$37$ \( ( 1156 + T^{2} )^{2} \)
$41$ \( ( 147 - 21 T + T^{2} )^{2} \)
$43$ \( 13845841 - 3721 T^{2} + T^{4} \)
$47$ \( 5531904 + 2352 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 2523 - 87 T + T^{2} )^{2} \)
$61$ \( ( 3136 + 56 T + T^{2} )^{2} \)
$67$ \( 923521 - 961 T^{2} + T^{4} \)
$71$ \( ( 972 + T^{2} )^{2} \)
$73$ \( ( 4225 + T^{2} )^{2} \)
$79$ \( ( 1444 - 38 T + T^{2} )^{2} \)
$83$ \( 5531904 + 2352 T^{2} + T^{4} \)
$89$ \( ( 15552 + T^{2} )^{2} \)
$97$ \( 174900625 - 13225 T^{2} + T^{4} \)
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