Properties

Label 450.3.i.b
Level $450$
Weight $3$
Character orbit 450.i
Analytic conductor $12.262$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 6) q^{11} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{12} + ( - 12 \beta_{3} - 5 \beta_{2} + 6 \beta_1 + 5) q^{13} + (6 \beta_{2} - \beta_1 + 6) q^{14} - 4 \beta_{2} q^{16} + (6 \beta_{3} - 12 \beta_{2} + 6) q^{17} + ( - 3 \beta_{3} + 12 \beta_{2} + 3 \beta_1) q^{18} + ( - 6 \beta_{3} + 12 \beta_1 - 10) q^{19} + ( - 8 \beta_{3} - 17 \beta_{2} + 10 \beta_1 - 2) q^{21} + ( - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 6) q^{22} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{23} + (2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 4) q^{24} + ( - 5 \beta_{3} - 24 \beta_{2} + 12) q^{26} + ( - 3 \beta_{3} - 30 \beta_{2} + 6 \beta_1 + 15) q^{27} + ( - 6 \beta_{3} + 12 \beta_1 - 2) q^{28} + (6 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 6) q^{29} + ( - 18 \beta_{3} - 19 \beta_{2} + 9 \beta_1 + 19) q^{31} - 4 \beta_1 q^{32} + (6 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 12) q^{33} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{34} + ( - 6 \beta_{2} + 12 \beta_1 + 6) q^{36} + ( - 6 \beta_{3} + 12 \beta_1 - 32) q^{37} + (10 \beta_{3} - 12 \beta_{2} - 10 \beta_1 + 24) q^{38} + (10 \beta_{3} + 31 \beta_{2} - 23 \beta_1 - 41) q^{39} + ( - 21 \beta_{2} - 18 \beta_1 - 21) q^{41} + (2 \beta_{3} - 16 \beta_{2} - 19 \beta_1 + 20) q^{42} + ( - 9 \beta_{3} + 23 \beta_{2} - 9 \beta_1) q^{43} + ( - 6 \beta_{3} - 12 \beta_{2} + 6) q^{44} + (3 \beta_{3} - 6 \beta_1 - 6) q^{46} + ( - 21 \beta_{3} + 9 \beta_{2} + 21 \beta_1 - 18) q^{47} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 8) q^{48} + ( - 12 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 6) q^{49} + (12 \beta_{3} - 24 \beta_{2} + 12 \beta_1 - 6) q^{51} + ( - 12 \beta_{3} - 10 \beta_{2} - 12 \beta_1) q^{52} + (30 \beta_{3} - 60 \beta_{2} + 30) q^{53} + ( - 15 \beta_{3} - 6 \beta_{2} - 15 \beta_1 + 12) q^{54} + (2 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 24) q^{56} + ( - 28 \beta_{3} + 20 \beta_{2} + 20 \beta_1 - 46) q^{57} + ( - 6 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 12) q^{58} + (21 \beta_{2} + 39 \beta_1 + 21) q^{59} + (18 \beta_{3} + 31 \beta_{2} + 18 \beta_1) q^{61} + ( - 19 \beta_{3} - 36 \beta_{2} + 18) q^{62} + (3 \beta_{3} + 33 \beta_{2} + 15 \beta_1 - 72) q^{63} - 8 q^{64} + (12 \beta_{3} + 12 \beta_{2} - 15 \beta_1 - 24) q^{66} + (18 \beta_{3} - 53 \beta_{2} - 9 \beta_1 + 53) q^{67} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{68} + (6 \beta_{3} + 15 \beta_{2} + 6 \beta_1 - 3) q^{69} + ( - 24 \beta_{3} - 60 \beta_{2} + 30) q^{71} + ( - 6 \beta_{3} + 24) q^{72} + ( - 18 \beta_{3} + 36 \beta_1 + 52) q^{73} + (32 \beta_{3} - 12 \beta_{2} - 32 \beta_1 + 24) q^{74} + ( - 24 \beta_{3} + 20 \beta_{2} + 12 \beta_1 - 20) q^{76} + (15 \beta_{2} + 24 \beta_1 + 15) q^{77} + (41 \beta_{3} + 20 \beta_{2} - 10 \beta_1 - 46) q^{78} + ( - 15 \beta_{3} + 7 \beta_{2} - 15 \beta_1) q^{79} + (36 \beta_{3} - 63) q^{81} + (21 \beta_{3} - 42 \beta_1 - 36) q^{82} + (15 \beta_{3} - 63 \beta_{2} - 15 \beta_1 + 126) q^{83} + ( - 20 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 38) q^{84} + ( - 18 \beta_{2} + 23 \beta_1 - 18) q^{86} + (15 \beta_{3} - 21 \beta_{2} - 3 \beta_1 + 24) q^{87} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{88} + (66 \beta_{3} - 60 \beta_{2} + 30) q^{89} + ( - 9 \beta_{3} + 18 \beta_1 + 103) q^{91} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 - 12) q^{92} + (38 \beta_{3} + 35 \beta_{2} - 46 \beta_1 - 73) q^{93} + (18 \beta_{3} - 42 \beta_{2} - 9 \beta_1 + 42) q^{94} + (8 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 8) q^{96} + (42 \beta_{3} - 7 \beta_{2} + 42 \beta_1) q^{97} + (6 \beta_{3} - 24 \beta_{2} + 12) q^{98} + (9 \beta_{3} + 27 \beta_{2} + 27 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9} + 18 q^{11} - 12 q^{12} + 10 q^{13} + 36 q^{14} - 8 q^{16} + 24 q^{18} - 40 q^{19} - 42 q^{21} + 12 q^{22} - 18 q^{23} - 8 q^{28} + 18 q^{29} + 38 q^{31} - 54 q^{33} + 24 q^{34} + 12 q^{36} - 128 q^{37} + 72 q^{38} - 102 q^{39} - 126 q^{41} + 48 q^{42} + 46 q^{43} - 24 q^{46} - 54 q^{47} - 24 q^{48} - 12 q^{49} - 72 q^{51} - 20 q^{52} + 36 q^{54} + 72 q^{56} - 144 q^{57} - 24 q^{58} + 126 q^{59} + 62 q^{61} - 222 q^{63} - 32 q^{64} - 72 q^{66} + 106 q^{67} - 72 q^{68} + 18 q^{69} + 96 q^{72} + 208 q^{73} + 72 q^{74} - 40 q^{76} + 90 q^{77} - 144 q^{78} + 14 q^{79} - 252 q^{81} - 144 q^{82} + 378 q^{83} - 144 q^{84} - 108 q^{86} + 54 q^{87} - 24 q^{88} + 412 q^{91} - 36 q^{92} - 222 q^{93} + 84 q^{94} + 48 q^{96} - 14 q^{97} + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i 2.44949 + 1.73205i 1.00000 + 1.73205i 0 −1.77526 3.85337i −4.17423 + 7.22999i 2.82843i 3.00000 + 8.48528i 0
101.2 1.22474 + 0.707107i −2.44949 + 1.73205i 1.00000 + 1.73205i 0 −4.22474 + 0.389270i 3.17423 5.49794i 2.82843i 3.00000 8.48528i 0
401.1 −1.22474 + 0.707107i 2.44949 1.73205i 1.00000 1.73205i 0 −1.77526 + 3.85337i −4.17423 7.22999i 2.82843i 3.00000 8.48528i 0
401.2 1.22474 0.707107i −2.44949 1.73205i 1.00000 1.73205i 0 −4.22474 0.389270i 3.17423 + 5.49794i 2.82843i 3.00000 + 8.48528i 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
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.i.b 4
3.b odd 2 1 1350.3.i.b 4
5.b even 2 1 18.3.d.a 4
5.c odd 4 2 450.3.k.a 8
9.c even 3 1 1350.3.i.b 4
9.d odd 6 1 inner 450.3.i.b 4
15.d odd 2 1 54.3.d.a 4
15.e even 4 2 1350.3.k.a 8
20.d odd 2 1 144.3.q.c 4
40.e odd 2 1 576.3.q.e 4
40.f even 2 1 576.3.q.f 4
45.h odd 6 1 18.3.d.a 4
45.h odd 6 1 162.3.b.a 4
45.j even 6 1 54.3.d.a 4
45.j even 6 1 162.3.b.a 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
60.h even 2 1 432.3.q.d 4
120.i odd 2 1 1728.3.q.d 4
120.m even 2 1 1728.3.q.c 4
180.n even 6 1 144.3.q.c 4
180.n even 6 1 1296.3.e.g 4
180.p odd 6 1 432.3.q.d 4
180.p odd 6 1 1296.3.e.g 4
360.z odd 6 1 1728.3.q.c 4
360.bd even 6 1 576.3.q.e 4
360.bh odd 6 1 576.3.q.f 4
360.bk even 6 1 1728.3.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 5.b even 2 1
18.3.d.a 4 45.h odd 6 1
54.3.d.a 4 15.d odd 2 1
54.3.d.a 4 45.j even 6 1
144.3.q.c 4 20.d odd 2 1
144.3.q.c 4 180.n even 6 1
162.3.b.a 4 45.h odd 6 1
162.3.b.a 4 45.j even 6 1
432.3.q.d 4 60.h even 2 1
432.3.q.d 4 180.p odd 6 1
450.3.i.b 4 1.a even 1 1 trivial
450.3.i.b 4 9.d odd 6 1 inner
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 40.e odd 2 1
576.3.q.e 4 360.bd even 6 1
576.3.q.f 4 40.f even 2 1
576.3.q.f 4 360.bh odd 6 1
1296.3.e.g 4 180.n even 6 1
1296.3.e.g 4 180.p odd 6 1
1350.3.i.b 4 3.b odd 2 1
1350.3.i.b 4 9.c even 3 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 120.m even 2 1
1728.3.q.c 4 360.z odd 6 1
1728.3.q.d 4 120.i odd 2 1
1728.3.q.d 4 360.bk even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809 \) acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + 63 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} + 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} - 46 T^{3} + 2073 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} + 54 T^{3} + 333 T^{2} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} - 106 T^{3} + 8913 T^{2} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + \cdots + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} - 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} - 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + \cdots + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
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