Properties

Label 450.6.c.n
Level $450$
Weight $6$
Character orbit 450.c
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + 233 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + 233 i q^{7} -64 i q^{8} + 498 q^{11} -809 i q^{13} -932 q^{14} + 256 q^{16} + 1002 i q^{17} + 1705 q^{19} + 1992 i q^{22} + 1554 i q^{23} + 3236 q^{26} -3728 i q^{28} + 7830 q^{29} + 977 q^{31} + 1024 i q^{32} -4008 q^{34} -4822 i q^{37} + 6820 i q^{38} + 8148 q^{41} -19469 i q^{43} -7968 q^{44} -6216 q^{46} -8418 i q^{47} -37482 q^{49} + 12944 i q^{52} + 17664 i q^{53} + 14912 q^{56} + 31320 i q^{58} + 35910 q^{59} + 3527 q^{61} + 3908 i q^{62} -4096 q^{64} + 57473 i q^{67} -16032 i q^{68} + 7548 q^{71} + 646 i q^{73} + 19288 q^{74} -27280 q^{76} + 116034 i q^{77} + 22720 q^{79} + 32592 i q^{82} + 11574 i q^{83} + 77876 q^{86} -31872 i q^{88} -78960 q^{89} + 188497 q^{91} -24864 i q^{92} + 33672 q^{94} + 54593 i q^{97} -149928 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} + 996q^{11} - 1864q^{14} + 512q^{16} + 3410q^{19} + 6472q^{26} + 15660q^{29} + 1954q^{31} - 8016q^{34} + 16296q^{41} - 15936q^{44} - 12432q^{46} - 74964q^{49} + 29824q^{56} + 71820q^{59} + 7054q^{61} - 8192q^{64} + 15096q^{71} + 38576q^{74} - 54560q^{76} + 45440q^{79} + 155752q^{86} - 157920q^{89} + 376994q^{91} + 67344q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 0 −16.0000 0 0 233.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 233.000i 64.0000i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.c.n 2
3.b odd 2 1 150.6.c.e 2
5.b even 2 1 inner 450.6.c.n 2
5.c odd 4 1 450.6.a.a 1
5.c odd 4 1 450.6.a.x 1
15.d odd 2 1 150.6.c.e 2
15.e even 4 1 150.6.a.c 1
15.e even 4 1 150.6.a.l yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 15.e even 4 1
150.6.a.l yes 1 15.e even 4 1
150.6.c.e 2 3.b odd 2 1
150.6.c.e 2 15.d odd 2 1
450.6.a.a 1 5.c odd 4 1
450.6.a.x 1 5.c odd 4 1
450.6.c.n 2 1.a even 1 1 trivial
450.6.c.n 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 54289 \)
\( T_{11} - 498 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 54289 + T^{2} \)
$11$ \( ( -498 + T )^{2} \)
$13$ \( 654481 + T^{2} \)
$17$ \( 1004004 + T^{2} \)
$19$ \( ( -1705 + T )^{2} \)
$23$ \( 2414916 + T^{2} \)
$29$ \( ( -7830 + T )^{2} \)
$31$ \( ( -977 + T )^{2} \)
$37$ \( 23251684 + T^{2} \)
$41$ \( ( -8148 + T )^{2} \)
$43$ \( 379041961 + T^{2} \)
$47$ \( 70862724 + T^{2} \)
$53$ \( 312016896 + T^{2} \)
$59$ \( ( -35910 + T )^{2} \)
$61$ \( ( -3527 + T )^{2} \)
$67$ \( 3303145729 + T^{2} \)
$71$ \( ( -7548 + T )^{2} \)
$73$ \( 417316 + T^{2} \)
$79$ \( ( -22720 + T )^{2} \)
$83$ \( 133957476 + T^{2} \)
$89$ \( ( 78960 + T )^{2} \)
$97$ \( 2980395649 + T^{2} \)
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