Properties

Label 450.6.c.j
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q -4 i q^{2} -16 q^{4} + 176 i q^{7} + 64 i q^{8} +O(q^{10})\) \( q -4 i q^{2} -16 q^{4} + 176 i q^{7} + 64 i q^{8} + 60 q^{11} + 658 i q^{13} + 704 q^{14} + 256 q^{16} + 414 i q^{17} -956 q^{19} -240 i q^{22} + 600 i q^{23} + 2632 q^{26} -2816 i q^{28} + 5574 q^{29} -3592 q^{31} -1024 i q^{32} + 1656 q^{34} -8458 i q^{37} + 3824 i q^{38} -19194 q^{41} -13316 i q^{43} -960 q^{44} + 2400 q^{46} + 19680 i q^{47} -14169 q^{49} -10528 i q^{52} -31266 i q^{53} -11264 q^{56} -22296 i q^{58} + 26340 q^{59} -31090 q^{61} + 14368 i q^{62} -4096 q^{64} -16804 i q^{67} -6624 i q^{68} -6120 q^{71} + 25558 i q^{73} -33832 q^{74} + 15296 q^{76} + 10560 i q^{77} -74408 q^{79} + 76776 i q^{82} -6468 i q^{83} -53264 q^{86} + 3840 i q^{88} -32742 q^{89} -115808 q^{91} -9600 i q^{92} + 78720 q^{94} + 166082 i q^{97} + 56676 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} + 120q^{11} + 1408q^{14} + 512q^{16} - 1912q^{19} + 5264q^{26} + 11148q^{29} - 7184q^{31} + 3312q^{34} - 38388q^{41} - 1920q^{44} + 4800q^{46} - 28338q^{49} - 22528q^{56} + 52680q^{59} - 62180q^{61} - 8192q^{64} - 12240q^{71} - 67664q^{74} + 30592q^{76} - 148816q^{79} - 106528q^{86} - 65484q^{89} - 231616q^{91} + 157440q^{94} + O(q^{100}) \)

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 0 176.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 176.000i 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.j 2
3.b odd 2 1 150.6.c.b 2
5.b even 2 1 inner 450.6.c.j 2
5.c odd 4 1 18.6.a.b 1
5.c odd 4 1 450.6.a.m 1
15.d odd 2 1 150.6.c.b 2
15.e even 4 1 6.6.a.a 1
15.e even 4 1 150.6.a.d 1
20.e even 4 1 144.6.a.j 1
35.f even 4 1 882.6.a.a 1
40.i odd 4 1 576.6.a.j 1
40.k even 4 1 576.6.a.i 1
45.k odd 12 2 162.6.c.h 2
45.l even 12 2 162.6.c.e 2
60.l odd 4 1 48.6.a.c 1
105.k odd 4 1 294.6.a.m 1
105.w odd 12 2 294.6.e.a 2
105.x even 12 2 294.6.e.g 2
120.q odd 4 1 192.6.a.g 1
120.w even 4 1 192.6.a.o 1
165.l odd 4 1 726.6.a.a 1
195.s even 4 1 1014.6.a.c 1
240.z odd 4 1 768.6.d.p 2
240.bb even 4 1 768.6.d.c 2
240.bd odd 4 1 768.6.d.p 2
240.bf even 4 1 768.6.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 15.e even 4 1
18.6.a.b 1 5.c odd 4 1
48.6.a.c 1 60.l odd 4 1
144.6.a.j 1 20.e even 4 1
150.6.a.d 1 15.e even 4 1
150.6.c.b 2 3.b odd 2 1
150.6.c.b 2 15.d odd 2 1
162.6.c.e 2 45.l even 12 2
162.6.c.h 2 45.k odd 12 2
192.6.a.g 1 120.q odd 4 1
192.6.a.o 1 120.w even 4 1
294.6.a.m 1 105.k odd 4 1
294.6.e.a 2 105.w odd 12 2
294.6.e.g 2 105.x even 12 2
450.6.a.m 1 5.c odd 4 1
450.6.c.j 2 1.a even 1 1 trivial
450.6.c.j 2 5.b even 2 1 inner
576.6.a.i 1 40.k even 4 1
576.6.a.j 1 40.i odd 4 1
726.6.a.a 1 165.l odd 4 1
768.6.d.c 2 240.bb even 4 1
768.6.d.c 2 240.bf even 4 1
768.6.d.p 2 240.z odd 4 1
768.6.d.p 2 240.bd odd 4 1
882.6.a.a 1 35.f even 4 1
1014.6.a.c 1 195.s even 4 1

Hecke kernels

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

\( T_{7}^{2} + 30976 \)
\( T_{11} - 60 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - 2638 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 60 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 309622 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 2668318 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 956 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 12512686 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 - 5574 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 3592 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 67150150 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 19194 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 116701030 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 71387614 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 141171770 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 - 26340 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 31090 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 2417875798 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 6120 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3492931822 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 + 74408 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7836246262 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 32742 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 10408550210 T^{2} + 73742412689492826049 T^{4} \)
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