Properties

Label 90.6.c.d
Level $90$
Weight $6$
Character orbit 90.c
Analytic conductor $14.435$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \( x^{4} - 9x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} - 16 q^{4} + ( - 15 \beta_{2} + 5 \beta_1) q^{5} - \beta_{3} q^{7} - 32 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} - 16 q^{4} + ( - 15 \beta_{2} + 5 \beta_1) q^{5} - \beta_{3} q^{7} - 32 \beta_{2} q^{8} + ( - 10 \beta_{3} + 140) q^{10} + ( - 37 \beta_{2} - 74 \beta_1) q^{11} + 63 \beta_{3} q^{13} + ( - 4 \beta_{2} - 8 \beta_1) q^{14} + 256 q^{16} - 583 \beta_{2} q^{17} + 2244 q^{19} + (240 \beta_{2} - 80 \beta_1) q^{20} + 148 \beta_{3} q^{22} + 250 \beta_{2} q^{23} + (175 \beta_{3} + 675) q^{25} + (252 \beta_{2} + 504 \beta_1) q^{26} + 16 \beta_{3} q^{28} + ( - 27 \beta_{2} - 54 \beta_1) q^{29} + 3856 q^{31} + 512 \beta_{2} q^{32} + 4664 q^{34} + (415 \beta_{2} + 70 \beta_1) q^{35} + 401 \beta_{3} q^{37} + 4488 \beta_{2} q^{38} + (160 \beta_{3} - 2240) q^{40} + ( - 550 \beta_{2} - 1100 \beta_1) q^{41} + 304 \beta_{3} q^{43} + (592 \beta_{2} + 1184 \beta_1) q^{44} - 2000 q^{46} - 9950 \beta_{2} q^{47} + 16503 q^{49} + (2050 \beta_{2} + 1400 \beta_1) q^{50} - 1008 \beta_{3} q^{52} + 573 \beta_{2} q^{53} + ( - 1295 \beta_{3} - 28120) q^{55} + (64 \beta_{2} + 128 \beta_1) q^{56} + 108 \beta_{3} q^{58} + ( - 2143 \beta_{2} - 4286 \beta_1) q^{59} - 38158 q^{61} + 7712 \beta_{2} q^{62} - 4096 q^{64} + ( - 26145 \beta_{2} - 4410 \beta_1) q^{65} - 2078 \beta_{3} q^{67} + 9328 \beta_{2} q^{68} + ( - 140 \beta_{3} - 3040) q^{70} + (954 \beta_{2} + 1908 \beta_1) q^{71} - 3980 \beta_{3} q^{73} + (1604 \beta_{2} + 3208 \beta_1) q^{74} - 35904 q^{76} - 5624 \beta_{2} q^{77} + 20664 q^{79} + ( - 3840 \beta_{2} + 1280 \beta_1) q^{80} + 2200 \beta_{3} q^{82} + 48484 \beta_{2} q^{83} + (2915 \beta_{3} - 40810) q^{85} + (1216 \beta_{2} + 2432 \beta_1) q^{86} - 2368 \beta_{3} q^{88} + (3424 \beta_{2} + 6848 \beta_1) q^{89} + 19152 q^{91} - 4000 \beta_{2} q^{92} + 79600 q^{94} + ( - 33660 \beta_{2} + 11220 \beta_1) q^{95} + 334 \beta_{3} q^{97} + 33006 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{4} + 560 q^{10} + 1024 q^{16} + 8976 q^{19} + 2700 q^{25} + 15424 q^{31} + 18656 q^{34} - 8960 q^{40} - 8000 q^{46} + 66012 q^{49} - 112480 q^{55} - 152632 q^{61} - 16384 q^{64} - 12160 q^{70} - 143616 q^{76} + 82656 q^{79} - 163240 q^{85} + 76608 q^{91} + 318400 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 8\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 36 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{2} + \beta_1 \) Copy content Toggle raw display

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
−2.17945 + 0.500000i
2.17945 + 0.500000i
−2.17945 0.500000i
2.17945 0.500000i
4.00000i 0 −16.0000 −43.5890 + 35.0000i 0 17.4356i 64.0000i 0 140.000 + 174.356i
19.2 4.00000i 0 −16.0000 43.5890 + 35.0000i 0 17.4356i 64.0000i 0 140.000 174.356i
19.3 4.00000i 0 −16.0000 −43.5890 35.0000i 0 17.4356i 64.0000i 0 140.000 174.356i
19.4 4.00000i 0 −16.0000 43.5890 35.0000i 0 17.4356i 64.0000i 0 140.000 + 174.356i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 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.d 4
3.b odd 2 1 inner 90.6.c.d 4
4.b odd 2 1 720.6.f.j 4
5.b even 2 1 inner 90.6.c.d 4
5.c odd 4 1 450.6.a.z 2
5.c odd 4 1 450.6.a.be 2
12.b even 2 1 720.6.f.j 4
15.d odd 2 1 inner 90.6.c.d 4
15.e even 4 1 450.6.a.z 2
15.e even 4 1 450.6.a.be 2
20.d odd 2 1 720.6.f.j 4
60.h even 2 1 720.6.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.c.d 4 1.a even 1 1 trivial
90.6.c.d 4 3.b odd 2 1 inner
90.6.c.d 4 5.b even 2 1 inner
90.6.c.d 4 15.d odd 2 1 inner
450.6.a.z 2 5.c odd 4 1
450.6.a.z 2 15.e even 4 1
450.6.a.be 2 5.c odd 4 1
450.6.a.be 2 15.e even 4 1
720.6.f.j 4 4.b odd 2 1
720.6.f.j 4 12.b even 2 1
720.6.f.j 4 20.d odd 2 1
720.6.f.j 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 304 \) acting on \(S_{6}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 1350 T^{2} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 416176)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1206576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1359556)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2244)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 250000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 221616)^{2} \) Copy content Toggle raw display
$31$ \( (T - 3856)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48883504)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 91960000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 28094464)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 396010000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1313316)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 1396104496)^{2} \) Copy content Toggle raw display
$61$ \( (T + 38158)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1312697536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 276675264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4815481600)^{2} \) Copy content Toggle raw display
$79$ \( (T - 20664)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9402793024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3564027904)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 33913024)^{2} \) Copy content Toggle raw display
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