Properties

Label 225.3.i.a
Level $225$
Weight $3$
Character orbit 225.i
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13080594811\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 3 + 3 \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} + ( 9 - 9 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 3 + 3 \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} + ( 9 - 9 \zeta_{12}^{2} ) q^{9} + ( -2 + \zeta_{12}^{2} ) q^{11} + 3 \zeta_{12}^{3} q^{12} -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} + ( -18 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{18} -11 q^{19} + ( -6 + 6 \zeta_{12}^{2} ) q^{21} + 3 \zeta_{12} q^{22} + ( -16 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{23} + ( 30 - 15 \zeta_{12}^{2} ) q^{24} + ( -4 + 8 \zeta_{12}^{2} ) q^{26} + 27 \zeta_{12}^{3} q^{27} -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} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{33} + 27 \zeta_{12}^{2} q^{34} -9 \zeta_{12}^{2} q^{36} -34 \zeta_{12}^{3} q^{37} + ( 11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{38} + 12 q^{39} + ( -7 - 7 \zeta_{12}^{2} ) q^{41} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{42} + ( 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} -33 \zeta_{12} q^{48} + ( -45 + 45 \zeta_{12}^{2} ) q^{49} + ( 54 - 27 \zeta_{12}^{2} ) q^{51} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{52} + ( 54 - 27 \zeta_{12}^{2} ) q^{54} + ( -20 + 10 \zeta_{12}^{2} ) q^{56} + ( 33 \zeta_{12} - 33 \zeta_{12}^{3} ) q^{57} + 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} -18 \zeta_{12}^{3} q^{63} + 71 q^{64} -9 q^{66} + 31 \zeta_{12} q^{67} + ( -9 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{68} + ( 48 - 96 \zeta_{12}^{2} ) q^{69} + ( 18 - 36 \zeta_{12}^{2} ) q^{71} + ( -45 \zeta_{12} + 90 \zeta_{12}^{3} ) q^{72} -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} + ( -12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{78} + 38 \zeta_{12}^{2} q^{79} -81 \zeta_{12}^{2} q^{81} + 21 \zeta_{12}^{3} q^{82} + ( 28 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{83} + 6 \zeta_{12}^{2} q^{84} + ( -61 - 61 \zeta_{12}^{2} ) q^{86} + ( 78 \zeta_{12} - 156 \zeta_{12}^{3} ) q^{87} + ( 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} -96 \zeta_{12}^{3} q^{93} + ( -84 + 84 \zeta_{12}^{2} ) q^{94} + ( 27 - 54 \zeta_{12}^{2} ) q^{96} + ( -115 \zeta_{12} + 115 \zeta_{12}^{3} ) q^{97} + ( 90 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{98} + ( -9 + 18 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 18 q^{6} + 18 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} + 18 q^{6} + 18 q^{9} - 6 q^{11} - 12 q^{14} + 22 q^{16} - 44 q^{19} - 12 q^{21} + 90 q^{24} - 156 q^{29} - 64 q^{31} + 54 q^{34} - 18 q^{36} + 48 q^{39} - 42 q^{41} + 192 q^{46} - 90 q^{49} + 162 q^{51} + 162 q^{54} - 60 q^{56} - 174 q^{59} - 112 q^{61} + 284 q^{64} - 36 q^{66} - 204 q^{74} - 22 q^{76} + 76 q^{79} - 162 q^{81} + 12 q^{84} - 366 q^{86} - 32 q^{91} - 168 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −2.59808 1.50000i 0.500000 + 0.866025i 0 4.50000 2.59808i 1.73205 + 1.00000i −8.66025 4.50000 + 7.79423i 0
74.2 0.866025 1.50000i 2.59808 + 1.50000i 0.500000 + 0.866025i 0 4.50000 2.59808i −1.73205 1.00000i 8.66025 4.50000 + 7.79423i 0
149.1 −0.866025 1.50000i −2.59808 + 1.50000i 0.500000 0.866025i 0 4.50000 + 2.59808i 1.73205 1.00000i −8.66025 4.50000 7.79423i 0
149.2 0.866025 + 1.50000i 2.59808 1.50000i 0.500000 0.866025i 0 4.50000 + 2.59808i −1.73205 + 1.00000i 8.66025 4.50000 7.79423i 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 225.3.i.a 4
3.b odd 2 1 675.3.i.a 4
5.b even 2 1 inner 225.3.i.a 4
5.c odd 4 1 9.3.d.a 2
5.c odd 4 1 225.3.j.a 2
9.c even 3 1 675.3.i.a 4
9.d odd 6 1 inner 225.3.i.a 4
15.d odd 2 1 675.3.i.a 4
15.e even 4 1 27.3.d.a 2
15.e even 4 1 675.3.j.a 2
20.e even 4 1 144.3.q.a 2
35.f even 4 1 441.3.r.a 2
35.k even 12 1 441.3.j.b 2
35.k even 12 1 441.3.n.a 2
35.l odd 12 1 441.3.j.a 2
35.l odd 12 1 441.3.n.b 2
40.i odd 4 1 576.3.q.b 2
40.k even 4 1 576.3.q.a 2
45.h odd 6 1 inner 225.3.i.a 4
45.j even 6 1 675.3.i.a 4
45.k odd 12 1 27.3.d.a 2
45.k odd 12 1 81.3.b.a 2
45.k odd 12 1 675.3.j.a 2
45.l even 12 1 9.3.d.a 2
45.l even 12 1 81.3.b.a 2
45.l even 12 1 225.3.j.a 2
60.l odd 4 1 432.3.q.a 2
120.q odd 4 1 1728.3.q.b 2
120.w even 4 1 1728.3.q.a 2
180.v odd 12 1 144.3.q.a 2
180.v odd 12 1 1296.3.e.a 2
180.x even 12 1 432.3.q.a 2
180.x even 12 1 1296.3.e.a 2
315.bu odd 12 1 441.3.n.a 2
315.bv even 12 1 441.3.n.b 2
315.bw odd 12 1 441.3.j.b 2
315.bx even 12 1 441.3.j.a 2
315.cf odd 12 1 441.3.r.a 2
360.bo even 12 1 1728.3.q.b 2
360.br even 12 1 576.3.q.b 2
360.bt odd 12 1 576.3.q.a 2
360.bu odd 12 1 1728.3.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 5.c odd 4 1
9.3.d.a 2 45.l even 12 1
27.3.d.a 2 15.e even 4 1
27.3.d.a 2 45.k odd 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 20.e even 4 1
144.3.q.a 2 180.v odd 12 1
225.3.i.a 4 1.a even 1 1 trivial
225.3.i.a 4 5.b even 2 1 inner
225.3.i.a 4 9.d odd 6 1 inner
225.3.i.a 4 45.h odd 6 1 inner
225.3.j.a 2 5.c odd 4 1
225.3.j.a 2 45.l even 12 1
432.3.q.a 2 60.l odd 4 1
432.3.q.a 2 180.x even 12 1
441.3.j.a 2 35.l odd 12 1
441.3.j.a 2 315.bx even 12 1
441.3.j.b 2 35.k even 12 1
441.3.j.b 2 315.bw odd 12 1
441.3.n.a 2 35.k even 12 1
441.3.n.a 2 315.bu odd 12 1
441.3.n.b 2 35.l odd 12 1
441.3.n.b 2 315.bv even 12 1
441.3.r.a 2 35.f even 4 1
441.3.r.a 2 315.cf odd 12 1
576.3.q.a 2 40.k even 4 1
576.3.q.a 2 360.bt odd 12 1
576.3.q.b 2 40.i odd 4 1
576.3.q.b 2 360.br even 12 1
675.3.i.a 4 3.b odd 2 1
675.3.i.a 4 9.c even 3 1
675.3.i.a 4 15.d odd 2 1
675.3.i.a 4 45.j even 6 1
675.3.j.a 2 15.e even 4 1
675.3.j.a 2 45.k odd 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 2 120.w even 4 1
1728.3.q.a 2 360.bu odd 12 1
1728.3.q.b 2 120.q odd 4 1
1728.3.q.b 2 360.bo even 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 \(S_{3}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( 81 - 9 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( ( 3 + 3 T + T^{2} )^{2} \)
$13$ \( 256 - 16 T^{2} + T^{4} \)
$17$ \( ( -243 + T^{2} )^{2} \)
$19$ \( ( 11 + T )^{4} \)
$23$ \( 589824 + 768 T^{2} + T^{4} \)
$29$ \( ( 2028 + 78 T + T^{2} )^{2} \)
$31$ \( ( 1024 + 32 T + T^{2} )^{2} \)
$37$ \( ( 1156 + T^{2} )^{2} \)
$41$ \( ( 147 + 21 T + T^{2} )^{2} \)
$43$ \( 13845841 - 3721 T^{2} + T^{4} \)
$47$ \( 5531904 + 2352 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 2523 + 87 T + T^{2} )^{2} \)
$61$ \( ( 3136 + 56 T + T^{2} )^{2} \)
$67$ \( 923521 - 961 T^{2} + T^{4} \)
$71$ \( ( 972 + T^{2} )^{2} \)
$73$ \( ( 4225 + T^{2} )^{2} \)
$79$ \( ( 1444 - 38 T + T^{2} )^{2} \)
$83$ \( 5531904 + 2352 T^{2} + T^{4} \)
$89$ \( ( 15552 + T^{2} )^{2} \)
$97$ \( 174900625 - 13225 T^{2} + T^{4} \)
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