Properties

Label 450.3.k.a
Level $450$
Weight $3$
Character orbit 450.k
Analytic conductor $12.262$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4}) q^{2} + ( - \beta_{7} + 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - 2) q^{6} + (3 \beta_{7} + 3 \beta_{4} + \beta_1) q^{7} + 2 \beta_{7} q^{8} + ( - 6 \beta_{6} + 6 \beta_{5} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4}) q^{2} + ( - \beta_{7} + 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - 2) q^{6} + (3 \beta_{7} + 3 \beta_{4} + \beta_1) q^{7} + 2 \beta_{7} q^{8} + ( - 6 \beta_{6} + 6 \beta_{5} - 3) q^{9} + ( - 3 \beta_{6} + 3 \beta_{2} + 3) q^{11} + (4 \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{12} + (12 \beta_{7} - 6 \beta_{4} + 5 \beta_{3} - 5 \beta_1) q^{13} + ( - \beta_{5} + 6 \beta_{2} - 12) q^{14} + (4 \beta_{2} - 4) q^{16} + (6 \beta_{7} - 6 \beta_{3} + 12 \beta_1) q^{17} + (3 \beta_{7} - 3 \beta_{4} + 12 \beta_1) q^{18} + (6 \beta_{6} + 6 \beta_{5} + 10) q^{19} + ( - 8 \beta_{6} - 2 \beta_{5} + 17 \beta_{2} - 19) q^{21} + ( - 6 \beta_{7} + 3 \beta_{4} - 6 \beta_{3} + 6 \beta_1) q^{22} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_1) q^{23} + ( - 2 \beta_{6} - 2 \beta_{5} + 8 \beta_{2} - 4) q^{24} + ( - 5 \beta_{6} + 5 \beta_{5} + 24 \beta_{2} - 12) q^{26} + ( - 3 \beta_{7} + 6 \beta_{4} - 15 \beta_{3} + 30 \beta_1) q^{27} + (6 \beta_{7} - 12 \beta_{4} - 2 \beta_{3}) q^{28} + ( - 6 \beta_{6} - 3 \beta_{2} - 3) q^{29} + ( - 18 \beta_{6} + 9 \beta_{5} + 19 \beta_{2}) q^{31} - 4 \beta_{4} q^{32} + ( - 6 \beta_{7} + 12 \beta_{4} - 12 \beta_{3} - 3 \beta_1) q^{33} + (6 \beta_{6} - 12 \beta_{5} + 12 \beta_{2} - 12) q^{34} + ( - 12 \beta_{5} + 6 \beta_{2}) q^{36} + ( - 6 \beta_{7} + 12 \beta_{4} + 32 \beta_{3}) q^{37} + ( - 10 \beta_{7} + 10 \beta_{4} + 24 \beta_{3} - 12 \beta_1) q^{38} + ( - 10 \beta_{6} - 13 \beta_{5} + 31 \beta_{2} + 10) q^{39} + (18 \beta_{5} + 21 \beta_{2} - 42) q^{41} + (2 \beta_{7} - 19 \beta_{4} - 20 \beta_{3} + 16 \beta_1) q^{42} + (9 \beta_{7} + 9 \beta_{4} + 23 \beta_1) q^{43} + (6 \beta_{6} - 6 \beta_{5} - 12 \beta_{2} + 6) q^{44} + (3 \beta_{6} + 3 \beta_{5} - 6) q^{46} + ( - 21 \beta_{7} + 21 \beta_{4} + 18 \beta_{3} - 9 \beta_1) q^{47} + ( - 4 \beta_{7} - 4 \beta_{4} - 8 \beta_{3} + 4 \beta_1) q^{48} + (12 \beta_{6} - 6 \beta_{5} + 6 \beta_{2}) q^{49} + (12 \beta_{6} - 24 \beta_{5} + 24 \beta_{2} - 30) q^{51} + ( - 12 \beta_{7} - 12 \beta_{4} + 10 \beta_1) q^{52} + ( - 30 \beta_{7} + 30 \beta_{3} - 60 \beta_1) q^{53} + (15 \beta_{6} - 30 \beta_{5} - 6 \beta_{2} - 6) q^{54} + (2 \beta_{6} + 12 \beta_{2} + 12) q^{56} + ( - 28 \beta_{7} + 20 \beta_{4} + 46 \beta_{3} - 20 \beta_1) q^{57} + (6 \beta_{7} - 3 \beta_{4} - 12 \beta_{3} + 12 \beta_1) q^{58} + (39 \beta_{5} + 21 \beta_{2} - 42) q^{59} + (18 \beta_{6} - 36 \beta_{5} - 31 \beta_{2} + 31) q^{61} + ( - 19 \beta_{7} - 18 \beta_{3} + 36 \beta_1) q^{62} + ( - 3 \beta_{7} - 15 \beta_{4} - 72 \beta_{3} + 33 \beta_1) q^{63} + 8 q^{64} + (12 \beta_{6} + 3 \beta_{5} - 12 \beta_{2} - 12) q^{66} + (18 \beta_{7} - 9 \beta_{4} - 53 \beta_{3} + 53 \beta_1) q^{67} + ( - 12 \beta_{4} - 12 \beta_{3} - 12 \beta_1) q^{68} + ( - 6 \beta_{6} + 12 \beta_{5} + 15 \beta_{2} - 12) q^{69} + ( - 24 \beta_{6} + 24 \beta_{5} + 60 \beta_{2} - 30) q^{71} + ( - 6 \beta_{7} - 24 \beta_{3}) q^{72} + (18 \beta_{7} - 36 \beta_{4} + 52 \beta_{3}) q^{73} + ( - 32 \beta_{6} - 12 \beta_{2} - 12) q^{74} + ( - 24 \beta_{6} + 12 \beta_{5} - 20 \beta_{2}) q^{76} + (24 \beta_{4} - 15 \beta_{3} - 15 \beta_1) q^{77} + ( - 41 \beta_{7} + 10 \beta_{4} - 46 \beta_{3} + 20 \beta_1) q^{78} + (15 \beta_{6} - 30 \beta_{5} + 7 \beta_{2} - 7) q^{79} + (36 \beta_{6} - 36 \beta_{5} - 63) q^{81} + (21 \beta_{7} - 42 \beta_{4} + 36 \beta_{3}) q^{82} + ( - 15 \beta_{7} + 15 \beta_{4} + 126 \beta_{3} - 63 \beta_1) q^{83} + (20 \beta_{6} - 16 \beta_{5} + 4 \beta_{2} + 34) q^{84} + ( - 23 \beta_{5} + 18 \beta_{2} - 36) q^{86} + (15 \beta_{7} - 3 \beta_{4} - 24 \beta_{3} + 21 \beta_1) q^{87} + (6 \beta_{7} + 6 \beta_{4} - 12 \beta_1) q^{88} + ( - 66 \beta_{6} + 66 \beta_{5} - 60 \beta_{2} + 30) q^{89} + ( - 9 \beta_{6} - 9 \beta_{5} + 103) q^{91} + (6 \beta_{7} - 6 \beta_{4} + 12 \beta_{3} - 6 \beta_1) q^{92} + ( - 38 \beta_{7} + 46 \beta_{4} - 73 \beta_{3} + 35 \beta_1) q^{93} + ( - 18 \beta_{6} + 9 \beta_{5} - 42 \beta_{2}) q^{94} + (8 \beta_{6} - 4 \beta_{5} - 8 \beta_{2} + 16) q^{96} + (42 \beta_{7} + 42 \beta_{4} + 7 \beta_1) q^{97} + ( - 6 \beta_{7} + 12 \beta_{3} - 24 \beta_1) q^{98} + ( - 9 \beta_{6} + 36 \beta_{5} + 27 \beta_{2} - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 24 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 24 q^{6} - 24 q^{9} + 36 q^{11} - 72 q^{14} - 16 q^{16} + 80 q^{19} - 84 q^{21} - 36 q^{29} + 76 q^{31} - 48 q^{34} + 24 q^{36} + 204 q^{39} - 252 q^{41} - 48 q^{46} + 24 q^{49} - 144 q^{51} - 72 q^{54} + 144 q^{56} - 252 q^{59} + 124 q^{61} + 64 q^{64} - 144 q^{66} - 36 q^{69} - 144 q^{74} - 80 q^{76} - 28 q^{79} - 504 q^{81} + 288 q^{84} - 216 q^{86} + 824 q^{91} - 168 q^{94} + 96 q^{96} - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
−0.707107 1.22474i −1.73205 2.44949i −1.00000 + 1.73205i 0 −1.77526 + 3.85337i 7.22999 4.17423i 2.82843 −3.00000 + 8.48528i 0
149.2 −0.707107 1.22474i 1.73205 2.44949i −1.00000 + 1.73205i 0 −4.22474 0.389270i 5.49794 3.17423i 2.82843 −3.00000 8.48528i 0
149.3 0.707107 + 1.22474i −1.73205 + 2.44949i −1.00000 + 1.73205i 0 −4.22474 0.389270i −5.49794 + 3.17423i −2.82843 −3.00000 8.48528i 0
149.4 0.707107 + 1.22474i 1.73205 + 2.44949i −1.00000 + 1.73205i 0 −1.77526 + 3.85337i −7.22999 + 4.17423i −2.82843 −3.00000 + 8.48528i 0
299.1 −0.707107 + 1.22474i −1.73205 + 2.44949i −1.00000 1.73205i 0 −1.77526 3.85337i 7.22999 + 4.17423i 2.82843 −3.00000 8.48528i 0
299.2 −0.707107 + 1.22474i 1.73205 + 2.44949i −1.00000 1.73205i 0 −4.22474 + 0.389270i 5.49794 + 3.17423i 2.82843 −3.00000 + 8.48528i 0
299.3 0.707107 1.22474i −1.73205 2.44949i −1.00000 1.73205i 0 −4.22474 + 0.389270i −5.49794 3.17423i −2.82843 −3.00000 + 8.48528i 0
299.4 0.707107 1.22474i 1.73205 2.44949i −1.00000 1.73205i 0 −1.77526 3.85337i −7.22999 4.17423i −2.82843 −3.00000 8.48528i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.4
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 450.3.k.a 8
3.b odd 2 1 1350.3.k.a 8
5.b even 2 1 inner 450.3.k.a 8
5.c odd 4 1 18.3.d.a 4
5.c odd 4 1 450.3.i.b 4
9.c even 3 1 1350.3.k.a 8
9.d odd 6 1 inner 450.3.k.a 8
15.d odd 2 1 1350.3.k.a 8
15.e even 4 1 54.3.d.a 4
15.e even 4 1 1350.3.i.b 4
20.e even 4 1 144.3.q.c 4
40.i odd 4 1 576.3.q.f 4
40.k even 4 1 576.3.q.e 4
45.h odd 6 1 inner 450.3.k.a 8
45.j even 6 1 1350.3.k.a 8
45.k odd 12 1 54.3.d.a 4
45.k odd 12 1 162.3.b.a 4
45.k odd 12 1 1350.3.i.b 4
45.l even 12 1 18.3.d.a 4
45.l even 12 1 162.3.b.a 4
45.l even 12 1 450.3.i.b 4
60.l odd 4 1 432.3.q.d 4
120.q odd 4 1 1728.3.q.c 4
120.w even 4 1 1728.3.q.d 4
180.v odd 12 1 144.3.q.c 4
180.v odd 12 1 1296.3.e.g 4
180.x even 12 1 432.3.q.d 4
180.x even 12 1 1296.3.e.g 4
360.bo even 12 1 1728.3.q.c 4
360.br even 12 1 576.3.q.f 4
360.bt odd 12 1 576.3.q.e 4
360.bu odd 12 1 1728.3.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 5.c odd 4 1
18.3.d.a 4 45.l even 12 1
54.3.d.a 4 15.e even 4 1
54.3.d.a 4 45.k odd 12 1
144.3.q.c 4 20.e even 4 1
144.3.q.c 4 180.v odd 12 1
162.3.b.a 4 45.k odd 12 1
162.3.b.a 4 45.l even 12 1
432.3.q.d 4 60.l odd 4 1
432.3.q.d 4 180.x even 12 1
450.3.i.b 4 5.c odd 4 1
450.3.i.b 4 45.l even 12 1
450.3.k.a 8 1.a even 1 1 trivial
450.3.k.a 8 5.b even 2 1 inner
450.3.k.a 8 9.d odd 6 1 inner
450.3.k.a 8 45.h odd 6 1 inner
576.3.q.e 4 40.k even 4 1
576.3.q.e 4 360.bt odd 12 1
576.3.q.f 4 40.i odd 4 1
576.3.q.f 4 360.br even 12 1
1296.3.e.g 4 180.v odd 12 1
1296.3.e.g 4 180.x even 12 1
1350.3.i.b 4 15.e even 4 1
1350.3.i.b 4 45.k odd 12 1
1350.3.k.a 8 3.b odd 2 1
1350.3.k.a 8 9.c even 3 1
1350.3.k.a 8 15.d odd 2 1
1350.3.k.a 8 45.j even 6 1
1728.3.q.c 4 120.q odd 4 1
1728.3.q.c 4 360.bo even 12 1
1728.3.q.d 4 120.w even 4 1
1728.3.q.d 4 360.bu odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 110T_{7}^{6} + 9291T_{7}^{4} - 308990T_{7}^{2} + 7890481 \) acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 6 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 110 T^{6} + 9291 T^{4} + \cdots + 7890481 \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{3} + 117 T^{2} - 162 T + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 482 T^{6} + \cdots + 1330863361 \) Copy content Toggle raw display
$17$ \( (T^{4} - 360 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 20 T - 116)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 90 T^{6} + 8019 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$29$ \( (T^{4} + 18 T^{3} + 63 T^{2} - 810 T + 2025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2480 T^{2} + 652864)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 2030 T^{6} + \cdots + 3418801 \) Copy content Toggle raw display
$47$ \( T^{8} + 2250 T^{6} + \cdots + 166726039041 \) Copy content Toggle raw display
$53$ \( (T^{4} - 9000 T^{2} + 810000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 126 T^{3} + 3573 T^{2} + \cdots + 2954961)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 6590 T^{6} + \cdots + 29120366676241 \) Copy content Toggle raw display
$71$ \( (T^{4} + 7704 T^{2} + 2396304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9296 T^{2} + 577600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 14 T^{3} + 1497 T^{2} + \cdots + 1692601)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 24714 T^{6} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + 22824 T^{2} + 36144144)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 21266 T^{6} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
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