Properties

Label 450.2.p.d
Level 450
Weight 2
Character orbit 450.p
Analytic conductor 3.593
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{5} ) q^{3} -\zeta_{24}^{2} q^{4} + ( 1 - 2 \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + \zeta_{24}^{7} q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{5} ) q^{3} -\zeta_{24}^{2} q^{4} + ( 1 - 2 \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} + ( 4 - 2 \zeta_{24}^{4} ) q^{11} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{12} + ( -4 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{13} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{14} + \zeta_{24}^{4} q^{16} + 6 \zeta_{24}^{3} q^{17} + 3 \zeta_{24}^{5} q^{18} -2 \zeta_{24}^{6} q^{19} -3 q^{21} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{22} + 3 \zeta_{24}^{5} q^{23} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{24} + ( 2 - 4 \zeta_{24}^{4} ) q^{26} + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{27} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{28} + ( 5 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{29} + ( -10 + 10 \zeta_{24}^{4} ) q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} ) q^{32} + ( -6 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{33} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{34} -3 q^{36} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{37} + 2 \zeta_{24} q^{38} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{39} + ( 5 + 5 \zeta_{24}^{4} ) q^{41} -3 \zeta_{24}^{7} q^{42} + ( 6 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{43} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{44} -3 q^{46} + 3 \zeta_{24}^{7} q^{47} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{48} -4 \zeta_{24}^{2} q^{49} + ( -6 - 6 \zeta_{24}^{4} ) q^{51} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{54} + ( 2 - \zeta_{24}^{4} ) q^{56} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{57} + ( -10 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{58} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{59} + 13 \zeta_{24}^{4} q^{61} -10 \zeta_{24}^{3} q^{62} + ( 6 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{63} -\zeta_{24}^{6} q^{64} -6 \zeta_{24}^{4} q^{66} + ( -7 \zeta_{24}^{3} - 7 \zeta_{24}^{7} ) q^{67} -6 \zeta_{24}^{5} q^{68} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{69} + ( 2 - 4 \zeta_{24}^{4} ) q^{71} -3 \zeta_{24}^{7} q^{72} + ( 8 \zeta_{24} + 8 \zeta_{24}^{5} ) q^{73} + ( 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{74} + ( -2 + 2 \zeta_{24}^{4} ) q^{76} + ( 6 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{77} + 6 \zeta_{24}^{5} q^{78} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{79} + ( 9 - 9 \zeta_{24}^{4} ) q^{81} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{82} + 3 \zeta_{24} q^{83} + 3 \zeta_{24}^{2} q^{84} + ( 6 + 6 \zeta_{24}^{4} ) q^{86} -15 \zeta_{24}^{3} q^{87} + ( 2 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{88} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{89} -6 q^{91} -3 \zeta_{24}^{7} q^{92} + ( 10 \zeta_{24} - 20 \zeta_{24}^{5} ) q^{93} -3 \zeta_{24}^{2} q^{94} + ( 2 - \zeta_{24}^{4} ) q^{96} + ( -8 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{98} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 24q^{11} + 4q^{16} - 24q^{21} - 40q^{31} - 24q^{36} + 60q^{41} - 24q^{46} - 72q^{51} + 12q^{56} + 52q^{61} - 24q^{66} - 8q^{76} + 36q^{81} + 72q^{86} - 48q^{91} + 12q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i 0.448288 + 1.67303i 0.866025 + 0.500000i 0 1.73205i −0.448288 + 1.67303i −0.707107 0.707107i −2.59808 + 1.50000i 0
257.2 0.965926 + 0.258819i −0.448288 1.67303i 0.866025 + 0.500000i 0 1.73205i 0.448288 1.67303i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
293.1 −0.258819 + 0.965926i −1.67303 + 0.448288i −0.866025 0.500000i 0 1.73205i 1.67303 + 0.448288i 0.707107 0.707107i 2.59808 1.50000i 0
293.2 0.258819 0.965926i 1.67303 0.448288i −0.866025 0.500000i 0 1.73205i −1.67303 0.448288i −0.707107 + 0.707107i 2.59808 1.50000i 0
407.1 −0.258819 0.965926i −1.67303 0.448288i −0.866025 + 0.500000i 0 1.73205i 1.67303 0.448288i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
407.2 0.258819 + 0.965926i 1.67303 + 0.448288i −0.866025 + 0.500000i 0 1.73205i −1.67303 + 0.448288i −0.707107 0.707107i 2.59808 + 1.50000i 0
443.1 −0.965926 + 0.258819i 0.448288 1.67303i 0.866025 0.500000i 0 1.73205i −0.448288 1.67303i −0.707107 + 0.707107i −2.59808 1.50000i 0
443.2 0.965926 0.258819i −0.448288 + 1.67303i 0.866025 0.500000i 0 1.73205i 0.448288 + 1.67303i 0.707107 0.707107i −2.59808 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.p.d 8
3.b odd 2 1 1350.2.q.b 8
5.b even 2 1 inner 450.2.p.d 8
5.c odd 4 2 inner 450.2.p.d 8
9.c even 3 1 1350.2.q.b 8
9.d odd 6 1 inner 450.2.p.d 8
15.d odd 2 1 1350.2.q.b 8
15.e even 4 2 1350.2.q.b 8
45.h odd 6 1 inner 450.2.p.d 8
45.j even 6 1 1350.2.q.b 8
45.k odd 12 2 1350.2.q.b 8
45.l even 12 2 inner 450.2.p.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.d 8 1.a even 1 1 trivial
450.2.p.d 8 5.b even 2 1 inner
450.2.p.d 8 5.c odd 4 2 inner
450.2.p.d 8 9.d odd 6 1 inner
450.2.p.d 8 45.h odd 6 1 inner
450.2.p.d 8 45.l even 12 2 inner
1350.2.q.b 8 3.b odd 2 1
1350.2.q.b 8 9.c even 3 1
1350.2.q.b 8 15.d odd 2 1
1350.2.q.b 8 15.e even 4 2
1350.2.q.b 8 45.j even 6 1
1350.2.q.b 8 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{8} - 9 T_{7}^{4} + 81 \)
\( T_{11}^{2} - 6 T_{11} + 12 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( 1 - 9 T^{4} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 - 94 T^{4} + 2401 T^{8} )( 1 + 71 T^{4} + 2401 T^{8} ) \)
$11$ \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )^{4} \)
$13$ \( 1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16} \)
$17$ \( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 34 T^{2} + 361 T^{4} )^{4} \)
$23$ \( 1 - 311 T^{4} - 183120 T^{8} - 87030551 T^{12} + 78310985281 T^{16} \)
$29$ \( ( 1 + 17 T^{2} - 552 T^{4} + 14297 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 1106 T^{4} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 15 T + 116 T^{2} - 615 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 23 T^{4} + 3418801 T^{8} )( 1 + 3191 T^{4} + 3418801 T^{8} ) \)
$47$ \( 1 - 2807 T^{4} + 2999568 T^{8} - 13697264567 T^{12} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 2903 T^{4} + 20151121 T^{8} )( 1 + 5906 T^{4} + 20151121 T^{8} ) \)
$71$ \( ( 1 - 130 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 8542 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 11 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 10871 T^{4} + 70720320 T^{8} - 515919407591 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 175 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( 1 - 2498 T^{4} - 82289277 T^{8} - 221146143938 T^{12} + 7837433594376961 T^{16} \)
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