Properties

Label 90.4.c.b
Level $90$
Weight $4$
Character orbit 90.c
Analytic conductor $5.310$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.31017190052\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -4 q^{4} + ( 5 - 10 i ) q^{5} -26 i q^{7} -8 i q^{8} +O(q^{10})\) \( q + 2 i q^{2} -4 q^{4} + ( 5 - 10 i ) q^{5} -26 i q^{7} -8 i q^{8} + ( 20 + 10 i ) q^{10} + 28 q^{11} + 12 i q^{13} + 52 q^{14} + 16 q^{16} -64 i q^{17} + 60 q^{19} + ( -20 + 40 i ) q^{20} + 56 i q^{22} + 58 i q^{23} + ( -75 - 100 i ) q^{25} -24 q^{26} + 104 i q^{28} + 90 q^{29} -128 q^{31} + 32 i q^{32} + 128 q^{34} + ( -260 - 130 i ) q^{35} -236 i q^{37} + 120 i q^{38} + ( -80 - 40 i ) q^{40} -242 q^{41} + 362 i q^{43} -112 q^{44} -116 q^{46} + 226 i q^{47} -333 q^{49} + ( 200 - 150 i ) q^{50} -48 i q^{52} + 108 i q^{53} + ( 140 - 280 i ) q^{55} -208 q^{56} + 180 i q^{58} -20 q^{59} + 542 q^{61} -256 i q^{62} -64 q^{64} + ( 120 + 60 i ) q^{65} + 434 i q^{67} + 256 i q^{68} + ( 260 - 520 i ) q^{70} + 1128 q^{71} + 632 i q^{73} + 472 q^{74} -240 q^{76} -728 i q^{77} + 720 q^{79} + ( 80 - 160 i ) q^{80} -484 i q^{82} + 478 i q^{83} + ( -640 - 320 i ) q^{85} -724 q^{86} -224 i q^{88} -490 q^{89} + 312 q^{91} -232 i q^{92} -452 q^{94} + ( 300 - 600 i ) q^{95} -1456 i q^{97} -666 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + 10q^{5} + O(q^{10}) \) \( 2q - 8q^{4} + 10q^{5} + 40q^{10} + 56q^{11} + 104q^{14} + 32q^{16} + 120q^{19} - 40q^{20} - 150q^{25} - 48q^{26} + 180q^{29} - 256q^{31} + 256q^{34} - 520q^{35} - 160q^{40} - 484q^{41} - 224q^{44} - 232q^{46} - 666q^{49} + 400q^{50} + 280q^{55} - 416q^{56} - 40q^{59} + 1084q^{61} - 128q^{64} + 240q^{65} + 520q^{70} + 2256q^{71} + 944q^{74} - 480q^{76} + 1440q^{79} + 160q^{80} - 1280q^{85} - 1448q^{86} - 980q^{89} + 624q^{91} - 904q^{94} + 600q^{95} + O(q^{100}) \)

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
2.00000i 0 −4.00000 5.00000 + 10.0000i 0 26.0000i 8.00000i 0 20.0000 10.0000i
19.2 2.00000i 0 −4.00000 5.00000 10.0000i 0 26.0000i 8.00000i 0 20.0000 + 10.0000i
\(n\): 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 90.4.c.b 2
3.b odd 2 1 10.4.b.a 2
4.b odd 2 1 720.4.f.f 2
5.b even 2 1 inner 90.4.c.b 2
5.c odd 4 1 450.4.a.j 1
5.c odd 4 1 450.4.a.k 1
12.b even 2 1 80.4.c.a 2
15.d odd 2 1 10.4.b.a 2
15.e even 4 1 50.4.a.b 1
15.e even 4 1 50.4.a.d 1
20.d odd 2 1 720.4.f.f 2
21.c even 2 1 490.4.c.b 2
24.f even 2 1 320.4.c.c 2
24.h odd 2 1 320.4.c.d 2
60.h even 2 1 80.4.c.a 2
60.l odd 4 1 400.4.a.h 1
60.l odd 4 1 400.4.a.n 1
105.g even 2 1 490.4.c.b 2
105.k odd 4 1 2450.4.a.o 1
105.k odd 4 1 2450.4.a.bb 1
120.i odd 2 1 320.4.c.d 2
120.m even 2 1 320.4.c.c 2
120.q odd 4 1 1600.4.a.t 1
120.q odd 4 1 1600.4.a.bg 1
120.w even 4 1 1600.4.a.u 1
120.w even 4 1 1600.4.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 3.b odd 2 1
10.4.b.a 2 15.d odd 2 1
50.4.a.b 1 15.e even 4 1
50.4.a.d 1 15.e even 4 1
80.4.c.a 2 12.b even 2 1
80.4.c.a 2 60.h even 2 1
90.4.c.b 2 1.a even 1 1 trivial
90.4.c.b 2 5.b even 2 1 inner
320.4.c.c 2 24.f even 2 1
320.4.c.c 2 120.m even 2 1
320.4.c.d 2 24.h odd 2 1
320.4.c.d 2 120.i odd 2 1
400.4.a.h 1 60.l odd 4 1
400.4.a.n 1 60.l odd 4 1
450.4.a.j 1 5.c odd 4 1
450.4.a.k 1 5.c odd 4 1
490.4.c.b 2 21.c even 2 1
490.4.c.b 2 105.g even 2 1
720.4.f.f 2 4.b odd 2 1
720.4.f.f 2 20.d odd 2 1
1600.4.a.t 1 120.q odd 4 1
1600.4.a.u 1 120.w even 4 1
1600.4.a.bg 1 120.q odd 4 1
1600.4.a.bh 1 120.w even 4 1
2450.4.a.o 1 105.k odd 4 1
2450.4.a.bb 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 676 \) acting on \(S_{4}^{\mathrm{new}}(90, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 125 - 10 T + T^{2} \)
$7$ \( 676 + T^{2} \)
$11$ \( ( -28 + T )^{2} \)
$13$ \( 144 + T^{2} \)
$17$ \( 4096 + T^{2} \)
$19$ \( ( -60 + T )^{2} \)
$23$ \( 3364 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( ( 128 + T )^{2} \)
$37$ \( 55696 + T^{2} \)
$41$ \( ( 242 + T )^{2} \)
$43$ \( 131044 + T^{2} \)
$47$ \( 51076 + T^{2} \)
$53$ \( 11664 + T^{2} \)
$59$ \( ( 20 + T )^{2} \)
$61$ \( ( -542 + T )^{2} \)
$67$ \( 188356 + T^{2} \)
$71$ \( ( -1128 + T )^{2} \)
$73$ \( 399424 + T^{2} \)
$79$ \( ( -720 + T )^{2} \)
$83$ \( 228484 + T^{2} \)
$89$ \( ( 490 + T )^{2} \)
$97$ \( 2119936 + T^{2} \)
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