Properties

Label 90.6.c.b
Level $90$
Weight $6$
Character orbit 90.c
Analytic conductor $14.435$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4345437832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + ( 55 + 10 i ) q^{5} + 4 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + ( 55 + 10 i ) q^{5} + 4 i q^{7} -64 i q^{8} + ( -40 + 220 i ) q^{10} + 500 q^{11} -288 i q^{13} -16 q^{14} + 256 q^{16} + 1516 i q^{17} + 1344 q^{19} + ( -880 - 160 i ) q^{20} + 2000 i q^{22} + 4100 i q^{23} + ( 2925 + 1100 i ) q^{25} + 1152 q^{26} -64 i q^{28} -2646 q^{29} -5612 q^{31} + 1024 i q^{32} -6064 q^{34} + ( -40 + 220 i ) q^{35} + 7288 i q^{37} + 5376 i q^{38} + ( 640 - 3520 i ) q^{40} + 18986 q^{41} -2404 i q^{43} -8000 q^{44} -16400 q^{46} + 8900 i q^{47} + 16791 q^{49} + ( -4400 + 11700 i ) q^{50} + 4608 i q^{52} -39804 i q^{53} + ( 27500 + 5000 i ) q^{55} + 256 q^{56} -10584 i q^{58} -28300 q^{59} + 18290 q^{61} -22448 i q^{62} -4096 q^{64} + ( 2880 - 15840 i ) q^{65} -65956 i q^{67} -24256 i q^{68} + ( -880 - 160 i ) q^{70} + 28800 q^{71} -30808 i q^{73} -29152 q^{74} -21504 q^{76} + 2000 i q^{77} -60228 q^{79} + ( 14080 + 2560 i ) q^{80} + 75944 i q^{82} + 2468 i q^{83} + ( -15160 + 83380 i ) q^{85} + 9616 q^{86} -32000 i q^{88} + 22678 q^{89} + 1152 q^{91} -65600 i q^{92} -35600 q^{94} + ( 73920 + 13440 i ) q^{95} + 36968 i q^{97} + 67164 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 110q^{5} + O(q^{10}) \) \( 2q - 32q^{4} + 110q^{5} - 80q^{10} + 1000q^{11} - 32q^{14} + 512q^{16} + 2688q^{19} - 1760q^{20} + 5850q^{25} + 2304q^{26} - 5292q^{29} - 11224q^{31} - 12128q^{34} - 80q^{35} + 1280q^{40} + 37972q^{41} - 16000q^{44} - 32800q^{46} + 33582q^{49} - 8800q^{50} + 55000q^{55} + 512q^{56} - 56600q^{59} + 36580q^{61} - 8192q^{64} + 5760q^{65} - 1760q^{70} + 57600q^{71} - 58304q^{74} - 43008q^{76} - 120456q^{79} + 28160q^{80} - 30320q^{85} + 19232q^{86} + 45356q^{89} + 2304q^{91} - 71200q^{94} + 147840q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.00000i
1.00000i
4.00000i 0 −16.0000 55.0000 10.0000i 0 4.00000i 64.0000i 0 −40.0000 220.000i
19.2 4.00000i 0 −16.0000 55.0000 + 10.0000i 0 4.00000i 64.0000i 0 −40.0000 + 220.000i
\(n\): 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 90.6.c.b 2
3.b odd 2 1 30.6.c.a 2
4.b odd 2 1 720.6.f.g 2
5.b even 2 1 inner 90.6.c.b 2
5.c odd 4 1 450.6.a.f 1
5.c odd 4 1 450.6.a.s 1
12.b even 2 1 240.6.f.a 2
15.d odd 2 1 30.6.c.a 2
15.e even 4 1 150.6.a.f 1
15.e even 4 1 150.6.a.j 1
20.d odd 2 1 720.6.f.g 2
60.h even 2 1 240.6.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 3.b odd 2 1
30.6.c.a 2 15.d odd 2 1
90.6.c.b 2 1.a even 1 1 trivial
90.6.c.b 2 5.b even 2 1 inner
150.6.a.f 1 15.e even 4 1
150.6.a.j 1 15.e even 4 1
240.6.f.a 2 12.b even 2 1
240.6.f.a 2 60.h even 2 1
450.6.a.f 1 5.c odd 4 1
450.6.a.s 1 5.c odd 4 1
720.6.f.g 2 4.b odd 2 1
720.6.f.g 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16 \) acting on \(S_{6}^{\mathrm{new}}(90, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 3125 - 110 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( -500 + T )^{2} \)
$13$ \( 82944 + T^{2} \)
$17$ \( 2298256 + T^{2} \)
$19$ \( ( -1344 + T )^{2} \)
$23$ \( 16810000 + T^{2} \)
$29$ \( ( 2646 + T )^{2} \)
$31$ \( ( 5612 + T )^{2} \)
$37$ \( 53114944 + T^{2} \)
$41$ \( ( -18986 + T )^{2} \)
$43$ \( 5779216 + T^{2} \)
$47$ \( 79210000 + T^{2} \)
$53$ \( 1584358416 + T^{2} \)
$59$ \( ( 28300 + T )^{2} \)
$61$ \( ( -18290 + T )^{2} \)
$67$ \( 4350193936 + T^{2} \)
$71$ \( ( -28800 + T )^{2} \)
$73$ \( 949132864 + T^{2} \)
$79$ \( ( 60228 + T )^{2} \)
$83$ \( 6091024 + T^{2} \)
$89$ \( ( -22678 + T )^{2} \)
$97$ \( 1366633024 + T^{2} \)
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