Properties

Label 900.3.f.e
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \zeta_{20} q^{2} + 4 \zeta_{20}^{2} q^{4} + ( -8 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{7} -8 \zeta_{20}^{3} q^{8} +O(q^{10})\) \( q -2 \zeta_{20} q^{2} + 4 \zeta_{20}^{2} q^{4} + ( -8 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{7} -8 \zeta_{20}^{3} q^{8} + ( -4 + 8 \zeta_{20}^{2} + 4 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{11} + ( 4 \zeta_{20}^{3} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{13} + ( 4 + 12 \zeta_{20}^{2} - 8 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{14} + 16 \zeta_{20}^{4} q^{16} + ( -16 \zeta_{20}^{3} + 14 \zeta_{20}^{5} - 16 \zeta_{20}^{7} ) q^{17} + ( -8 \zeta_{20}^{4} - 8 \zeta_{20}^{6} ) q^{19} + ( 8 \zeta_{20} - 16 \zeta_{20}^{3} - 8 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{22} + ( -8 \zeta_{20} + 10 \zeta_{20}^{3} - 4 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{23} + ( 8 - 8 \zeta_{20}^{2} - 12 \zeta_{20}^{6} ) q^{26} + ( -8 \zeta_{20} - 24 \zeta_{20}^{3} + 16 \zeta_{20}^{5} - 8 \zeta_{20}^{7} ) q^{28} + ( -6 - 8 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{29} + ( -20 + 40 \zeta_{20}^{2} - 28 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{31} -32 \zeta_{20}^{5} q^{32} + ( -32 + 32 \zeta_{20}^{2} + 4 \zeta_{20}^{6} ) q^{34} + ( -20 \zeta_{20}^{3} + 14 \zeta_{20}^{5} - 20 \zeta_{20}^{7} ) q^{37} + ( 16 \zeta_{20}^{5} + 16 \zeta_{20}^{7} ) q^{38} + ( 22 - 12 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{41} + ( -16 \zeta_{20} + 2 \zeta_{20}^{3} - 8 \zeta_{20}^{5} + 14 \zeta_{20}^{7} ) q^{43} + ( -48 + 32 \zeta_{20}^{2} - 16 \zeta_{20}^{4} + 64 \zeta_{20}^{6} ) q^{44} + ( -4 + 20 \zeta_{20}^{2} - 24 \zeta_{20}^{4} + 12 \zeta_{20}^{6} ) q^{46} + ( -24 \zeta_{20} - 14 \zeta_{20}^{3} - 12 \zeta_{20}^{5} + 38 \zeta_{20}^{7} ) q^{47} + ( -9 - 20 \zeta_{20}^{4} + 20 \zeta_{20}^{6} ) q^{49} + ( -16 \zeta_{20} + 16 \zeta_{20}^{3} + 24 \zeta_{20}^{7} ) q^{52} + ( -20 \zeta_{20}^{3} - 34 \zeta_{20}^{5} - 20 \zeta_{20}^{7} ) q^{53} + ( -16 + 32 \zeta_{20}^{2} + 32 \zeta_{20}^{4} - 16 \zeta_{20}^{6} ) q^{56} + ( 12 \zeta_{20} + 16 \zeta_{20}^{5} - 16 \zeta_{20}^{7} ) q^{58} + ( -24 + 48 \zeta_{20}^{2} - 48 \zeta_{20}^{4} ) q^{59} + ( 58 + 52 \zeta_{20}^{4} - 52 \zeta_{20}^{6} ) q^{61} + ( 40 \zeta_{20} - 80 \zeta_{20}^{3} + 56 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{62} + 64 \zeta_{20}^{6} q^{64} + ( 80 \zeta_{20} - 62 \zeta_{20}^{3} + 40 \zeta_{20}^{5} - 18 \zeta_{20}^{7} ) q^{67} + ( 64 \zeta_{20} - 64 \zeta_{20}^{3} - 8 \zeta_{20}^{7} ) q^{68} + ( 4 - 8 \zeta_{20}^{2} - 36 \zeta_{20}^{4} - 44 \zeta_{20}^{6} ) q^{71} + ( 64 \zeta_{20}^{3} - 98 \zeta_{20}^{5} + 64 \zeta_{20}^{7} ) q^{73} + ( -40 + 40 \zeta_{20}^{2} + 12 \zeta_{20}^{6} ) q^{74} + ( 32 - 32 \zeta_{20}^{2} + 32 \zeta_{20}^{4} - 64 \zeta_{20}^{6} ) q^{76} + ( -40 \zeta_{20}^{3} - 40 \zeta_{20}^{5} - 40 \zeta_{20}^{7} ) q^{77} + ( -40 + 80 \zeta_{20}^{2} - 72 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{79} + ( -44 \zeta_{20} + 24 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{82} + ( -48 \zeta_{20} - 2 \zeta_{20}^{3} - 24 \zeta_{20}^{5} + 50 \zeta_{20}^{7} ) q^{83} + ( 28 + 4 \zeta_{20}^{2} + 24 \zeta_{20}^{4} - 12 \zeta_{20}^{6} ) q^{86} + ( 96 \zeta_{20} - 64 \zeta_{20}^{3} + 32 \zeta_{20}^{5} - 128 \zeta_{20}^{7} ) q^{88} + ( -62 - 80 \zeta_{20}^{4} + 80 \zeta_{20}^{6} ) q^{89} + ( 12 - 24 \zeta_{20}^{2} - 4 \zeta_{20}^{4} - 28 \zeta_{20}^{6} ) q^{91} + ( 8 \zeta_{20} - 40 \zeta_{20}^{3} + 48 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{92} + ( 76 - 28 \zeta_{20}^{2} + 104 \zeta_{20}^{4} - 52 \zeta_{20}^{6} ) q^{94} + ( -24 \zeta_{20}^{3} - 54 \zeta_{20}^{5} - 24 \zeta_{20}^{7} ) q^{97} + ( 18 \zeta_{20} + 40 \zeta_{20}^{5} - 40 \zeta_{20}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} + 80q^{14} - 32q^{16} + 24q^{26} - 16q^{29} - 184q^{34} + 224q^{41} - 160q^{44} + 80q^{46} + 8q^{49} - 160q^{56} + 256q^{61} + 128q^{64} - 216q^{74} + 160q^{86} - 176q^{89} + 240q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
0.951057 + 0.309017i
0.951057 0.309017i
0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 + 0.309017i
−0.951057 0.309017i
−1.90211 0.618034i 0 3.23607 + 2.35114i 0 0 −5.25731 −4.70228 6.47214i 0 0
199.2 −1.90211 + 0.618034i 0 3.23607 2.35114i 0 0 −5.25731 −4.70228 + 6.47214i 0 0
199.3 −1.17557 1.61803i 0 −1.23607 + 3.80423i 0 0 −8.50651 7.60845 2.47214i 0 0
199.4 −1.17557 + 1.61803i 0 −1.23607 3.80423i 0 0 −8.50651 7.60845 + 2.47214i 0 0
199.5 1.17557 1.61803i 0 −1.23607 3.80423i 0 0 8.50651 −7.60845 2.47214i 0 0
199.6 1.17557 + 1.61803i 0 −1.23607 + 3.80423i 0 0 8.50651 −7.60845 + 2.47214i 0 0
199.7 1.90211 0.618034i 0 3.23607 2.35114i 0 0 5.25731 4.70228 6.47214i 0 0
199.8 1.90211 + 0.618034i 0 3.23607 + 2.35114i 0 0 5.25731 4.70228 + 6.47214i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.e 8
3.b odd 2 1 100.3.d.b 8
4.b odd 2 1 inner 900.3.f.e 8
5.b even 2 1 inner 900.3.f.e 8
5.c odd 4 1 180.3.c.a 4
5.c odd 4 1 900.3.c.k 4
12.b even 2 1 100.3.d.b 8
15.d odd 2 1 100.3.d.b 8
15.e even 4 1 20.3.b.a 4
15.e even 4 1 100.3.b.f 4
20.d odd 2 1 inner 900.3.f.e 8
20.e even 4 1 180.3.c.a 4
20.e even 4 1 900.3.c.k 4
24.f even 2 1 1600.3.h.n 8
24.h odd 2 1 1600.3.h.n 8
40.i odd 4 1 2880.3.e.e 4
40.k even 4 1 2880.3.e.e 4
60.h even 2 1 100.3.d.b 8
60.l odd 4 1 20.3.b.a 4
60.l odd 4 1 100.3.b.f 4
120.i odd 2 1 1600.3.h.n 8
120.m even 2 1 1600.3.h.n 8
120.q odd 4 1 320.3.b.c 4
120.q odd 4 1 1600.3.b.s 4
120.w even 4 1 320.3.b.c 4
120.w even 4 1 1600.3.b.s 4
240.z odd 4 1 1280.3.g.e 8
240.bb even 4 1 1280.3.g.e 8
240.bd odd 4 1 1280.3.g.e 8
240.bf even 4 1 1280.3.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 15.e even 4 1
20.3.b.a 4 60.l odd 4 1
100.3.b.f 4 15.e even 4 1
100.3.b.f 4 60.l odd 4 1
100.3.d.b 8 3.b odd 2 1
100.3.d.b 8 12.b even 2 1
100.3.d.b 8 15.d odd 2 1
100.3.d.b 8 60.h even 2 1
180.3.c.a 4 5.c odd 4 1
180.3.c.a 4 20.e even 4 1
320.3.b.c 4 120.q odd 4 1
320.3.b.c 4 120.w even 4 1
900.3.c.k 4 5.c odd 4 1
900.3.c.k 4 20.e even 4 1
900.3.f.e 8 1.a even 1 1 trivial
900.3.f.e 8 4.b odd 2 1 inner
900.3.f.e 8 5.b even 2 1 inner
900.3.f.e 8 20.d odd 2 1 inner
1280.3.g.e 8 240.z odd 4 1
1280.3.g.e 8 240.bb even 4 1
1280.3.g.e 8 240.bd odd 4 1
1280.3.g.e 8 240.bf even 4 1
1600.3.b.s 4 120.q odd 4 1
1600.3.b.s 4 120.w even 4 1
1600.3.h.n 8 24.f even 2 1
1600.3.h.n 8 24.h odd 2 1
1600.3.h.n 8 120.i odd 2 1
1600.3.h.n 8 120.m even 2 1
2880.3.e.e 4 40.i odd 4 1
2880.3.e.e 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} - 100 T_{7}^{2} + 2000 \)
\( T_{29}^{2} + 4 T_{29} - 76 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 64 T^{2} + 16 T^{4} - 4 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 2000 - 100 T^{2} + T^{4} )^{2} \)
$11$ \( ( 1280 + 400 T^{2} + T^{4} )^{2} \)
$13$ \( ( 16 + 72 T^{2} + T^{4} )^{2} \)
$17$ \( ( 80656 + 712 T^{2} + T^{4} )^{2} \)
$19$ \( ( 20480 + 320 T^{2} + T^{4} )^{2} \)
$23$ \( ( 80 - 260 T^{2} + T^{4} )^{2} \)
$29$ \( ( -76 + 4 T + T^{2} )^{4} \)
$31$ \( ( 154880 + 2320 T^{2} + T^{4} )^{2} \)
$37$ \( ( 234256 + 1032 T^{2} + T^{4} )^{2} \)
$41$ \( ( 604 - 56 T + T^{2} )^{4} \)
$43$ \( ( 2000 - 500 T^{2} + T^{4} )^{2} \)
$47$ \( ( 3561680 - 4100 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2062096 + 4872 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1658880 + 5760 T^{2} + T^{4} )^{2} \)
$61$ \( ( -2356 - 64 T + T^{2} )^{4} \)
$67$ \( ( 19920080 - 10420 T^{2} + T^{4} )^{2} \)
$71$ \( ( 10138880 + 8080 T^{2} + T^{4} )^{2} \)
$73$ \( ( 583696 + 18952 T^{2} + T^{4} )^{2} \)
$79$ \( ( 2478080 + 13120 T^{2} + T^{4} )^{2} \)
$83$ \( ( 2620880 - 6260 T^{2} + T^{4} )^{2} \)
$89$ \( ( -7516 + 44 T + T^{2} )^{4} \)
$97$ \( ( 13220496 + 10152 T^{2} + T^{4} )^{2} \)
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