Properties

Label 900.3.f.g
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{12} + 25 x^{8} - 16 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{7} + ( \beta_{1} - \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{7} + ( \beta_{1} - \beta_{4} ) q^{8} + \beta_{6} q^{11} -\beta_{13} q^{13} + ( -\beta_{6} + \beta_{9} ) q^{14} + ( -1 - 2 \beta_{3} - \beta_{5} ) q^{16} + ( 4 \beta_{1} - \beta_{4} + \beta_{10} + \beta_{12} ) q^{17} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{7} - 2 \beta_{11} + \beta_{13} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{4} + 2 \beta_{10} + 3 \beta_{12} ) q^{23} + ( -2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{26} + ( \beta_{7} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{28} + ( \beta_{6} + 3 \beta_{9} + \beta_{15} ) q^{29} + ( -2 + 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( 2 \beta_{1} + 2 \beta_{4} + 4 \beta_{12} ) q^{32} + ( 11 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{34} + 2 \beta_{13} q^{37} + ( \beta_{1} - \beta_{4} - \beta_{10} + 2 \beta_{12} ) q^{38} + ( \beta_{6} - 4 \beta_{8} + \beta_{9} - \beta_{15} ) q^{41} + ( \beta_{7} + \beta_{11} - \beta_{13} + 4 \beta_{14} ) q^{43} + 4 \beta_{9} q^{44} + ( 15 + 2 \beta_{2} + 6 \beta_{3} - 3 \beta_{5} ) q^{46} + ( -10 \beta_{1} + 2 \beta_{4} + 4 \beta_{10} + 2 \beta_{12} ) q^{47} + ( 14 - 8 \beta_{2} - 4 \beta_{5} ) q^{49} + ( -6 \beta_{7} + 4 \beta_{11} + \beta_{13} - \beta_{14} ) q^{52} + ( 22 \beta_{1} - 3 \beta_{4} + 5 \beta_{10} + 3 \beta_{12} ) q^{53} + ( -\beta_{6} + 3 \beta_{8} - 6 \beta_{9} + \beta_{15} ) q^{56} + ( -\beta_{7} - 6 \beta_{11} - 5 \beta_{13} + 4 \beta_{14} ) q^{58} -6 \beta_{6} q^{59} + ( 12 - 14 \beta_{2} - 7 \beta_{5} ) q^{61} + ( \beta_{1} - 3 \beta_{4} - 3 \beta_{10} + 2 \beta_{12} ) q^{62} + ( -18 - 4 \beta_{2} + 4 \beta_{3} - 6 \beta_{5} ) q^{64} + ( -\beta_{7} + 7 \beta_{11} + \beta_{13} - 4 \beta_{14} ) q^{67} + ( -12 \beta_{1} + 4 \beta_{4} - 4 \beta_{10} + 4 \beta_{12} ) q^{68} + ( 3 \beta_{6} + 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{15} ) q^{71} + ( 4 \beta_{7} - 10 \beta_{13} ) q^{73} + ( 4 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} ) q^{74} + ( -28 - \beta_{2} - 2 \beta_{3} - 5 \beta_{5} ) q^{76} + ( 34 \beta_{1} - \beta_{4} + 7 \beta_{10} + \beta_{12} ) q^{77} + ( -6 + 12 \beta_{2} + 8 \beta_{3} - 6 \beta_{5} ) q^{79} + ( 11 \beta_{7} + 2 \beta_{11} - \beta_{13} - 4 \beta_{14} ) q^{82} + ( -22 \beta_{1} - 10 \beta_{4} + 4 \beta_{10} - 10 \beta_{12} ) q^{83} + ( 3 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + 4 \beta_{15} ) q^{86} + ( 8 \beta_{7} - 4 \beta_{13} + 4 \beta_{14} ) q^{88} + ( -4 \beta_{6} + 8 \beta_{8} - 8 \beta_{9} ) q^{89} + ( -5 + 10 \beta_{2} + 7 \beta_{3} - 5 \beta_{5} ) q^{91} + ( -24 \beta_{1} - 8 \beta_{4} - 12 \beta_{10} - 12 \beta_{12} ) q^{92} + ( 42 + 12 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{94} -3 \beta_{7} q^{97} + ( -18 \beta_{1} + 8 \beta_{4} - 8 \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{4} + O(q^{10}) \) \( 16q - 8q^{4} - 16q^{16} + 160q^{34} + 256q^{46} + 160q^{49} + 80q^{61} - 320q^{64} - 456q^{76} + 768q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{12} + 25 x^{8} - 16 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{15} + 19 \nu^{11} - 27 \nu^{7} + 192 \nu^{3} - 448 \nu \)\()/448\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{12} - 3 \nu^{8} - 9 \nu^{4} - 34 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{12} - 19 \nu^{8} + 139 \nu^{4} - 248 \)\()/56\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{15} + 32 \nu^{13} + 19 \nu^{11} + 96 \nu^{9} - 27 \nu^{7} + 288 \nu^{5} + 1088 \nu^{3} + 1088 \nu \)\()/448\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{12} + 19 \nu^{8} - 27 \nu^{4} + 220 \)\()/28\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{15} - 16 \nu^{13} + 67 \nu^{11} - 48 \nu^{9} + 117 \nu^{7} + 304 \nu^{5} + 1408 \nu^{3} - 320 \nu \)\()/448\)
\(\beta_{7}\)\(=\)\((\)\( -15 \nu^{14} + 95 \nu^{10} - 135 \nu^{6} + 960 \nu^{2} \)\()/448\)
\(\beta_{8}\)\(=\)\((\)\( -15 \nu^{15} + 95 \nu^{11} - 135 \nu^{7} + 960 \nu^{3} + 2240 \nu \)\()/448\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{15} - 5 \nu^{13} - 3 \nu^{11} + 13 \nu^{9} + 19 \nu^{7} - 101 \nu^{5} - 20 \nu^{3} + 208 \nu \)\()/56\)
\(\beta_{10}\)\(=\)\((\)\( -11 \nu^{15} - 24 \nu^{13} - 5 \nu^{11} + 152 \nu^{9} - 323 \nu^{7} - 216 \nu^{5} + 32 \nu^{3} + 1984 \nu \)\()/448\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{14} + 13 \nu^{10} - 5 \nu^{6} + 224 \nu^{2} \)\()/64\)
\(\beta_{12}\)\(=\)\((\)\( 19 \nu^{15} - 16 \nu^{13} + 29 \nu^{11} - 48 \nu^{9} + 171 \nu^{7} - 592 \nu^{5} + 128 \nu^{3} - 320 \nu \)\()/448\)
\(\beta_{13}\)\(=\)\((\)\( 25 \nu^{14} - 9 \nu^{10} + 673 \nu^{6} + 1536 \nu^{2} \)\()/448\)
\(\beta_{14}\)\(=\)\((\)\( -25 \nu^{14} + 9 \nu^{10} - 673 \nu^{6} + 2944 \nu^{2} \)\()/448\)
\(\beta_{15}\)\(=\)\((\)\( -19 \nu^{15} - 16 \nu^{13} - 29 \nu^{11} - 272 \nu^{9} - 395 \nu^{7} - 368 \nu^{5} + 768 \nu^{3} - 3904 \nu \)\()/224\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} - 5 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} + \beta_{13}\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + 2 \beta_{9} - \beta_{8} + 3 \beta_{6} + 5 \beta_{4} - 5 \beta_{1}\)\()/20\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3} + 1\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{15} - 10 \beta_{12} - 2 \beta_{9} - \beta_{8} + 7 \beta_{6} - 5 \beta_{4} - 5 \beta_{1}\)\()/20\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{14} + 11 \beta_{13} - 10 \beta_{11} + 11 \beta_{7}\)\()/20\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{15} - 5 \beta_{12} - 10 \beta_{10} + 8 \beta_{9} + 2 \beta_{6} + 5 \beta_{1}\)\()/10\)
\(\nu^{8}\)\(=\)\((\)\(2 \beta_{5} - 3 \beta_{2} - 23\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-7 \beta_{15} + 20 \beta_{10} + 6 \beta_{9} - 29 \beta_{8} - \beta_{6} + 15 \beta_{4} + 125 \beta_{1}\)\()/20\)
\(\nu^{10}\)\(=\)\((\)\(-30 \beta_{14} - 15 \beta_{13} + 30 \beta_{11} + 67 \beta_{7}\)\()/20\)
\(\nu^{11}\)\(=\)\((\)\(-15 \beta_{15} + 30 \beta_{12} - 30 \beta_{9} + 41 \beta_{8} - 15 \beta_{6} - 45 \beta_{4} + 235 \beta_{1}\)\()/20\)
\(\nu^{12}\)\(=\)\((\)\(-21 \beta_{5} - 18 \beta_{3} - 38 \beta_{2} - 7\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(\beta_{15} + 45 \beta_{12} - 30 \beta_{10} - 28 \beta_{9} + 14 \beta_{8} - 72 \beta_{6} + 70 \beta_{4} + 5 \beta_{1}\)\()/10\)
\(\nu^{14}\)\(=\)\((\)\(-13 \beta_{14} - 33 \beta_{13} + 140 \beta_{11} - 136 \beta_{7}\)\()/10\)
\(\nu^{15}\)\(=\)\((\)\(-13 \beta_{15} + 280 \beta_{12} + 180 \beta_{10} - 206 \beta_{9} - 103 \beta_{8} + 61 \beta_{6} + 35 \beta_{4} - 415 \beta_{1}\)\()/20\)

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
−0.404496 1.35513i
−1.35513 + 0.404496i
−0.404496 + 1.35513i
−1.35513 0.404496i
−1.26588 0.630504i
0.630504 1.26588i
−1.26588 + 0.630504i
0.630504 + 1.26588i
1.26588 0.630504i
−0.630504 1.26588i
1.26588 + 0.630504i
−0.630504 + 1.26588i
0.404496 1.35513i
1.35513 + 0.404496i
0.404496 + 1.35513i
1.35513 0.404496i
−1.75963 0.950636i 0 2.19258 + 3.34553i 0 0 −3.98982 −0.677747 7.97124i 0 0
199.2 −1.75963 0.950636i 0 2.19258 + 3.34553i 0 0 3.98982 −0.677747 7.97124i 0 0
199.3 −1.75963 + 0.950636i 0 2.19258 3.34553i 0 0 −3.98982 −0.677747 + 7.97124i 0 0
199.4 −1.75963 + 0.950636i 0 2.19258 3.34553i 0 0 3.98982 −0.677747 + 7.97124i 0 0
199.5 −0.635381 1.89639i 0 −3.19258 + 2.40986i 0 0 −10.1035 6.59853 + 4.52320i 0 0
199.6 −0.635381 1.89639i 0 −3.19258 + 2.40986i 0 0 10.1035 6.59853 + 4.52320i 0 0
199.7 −0.635381 + 1.89639i 0 −3.19258 2.40986i 0 0 −10.1035 6.59853 4.52320i 0 0
199.8 −0.635381 + 1.89639i 0 −3.19258 2.40986i 0 0 10.1035 6.59853 4.52320i 0 0
199.9 0.635381 1.89639i 0 −3.19258 2.40986i 0 0 −10.1035 −6.59853 + 4.52320i 0 0
199.10 0.635381 1.89639i 0 −3.19258 2.40986i 0 0 10.1035 −6.59853 + 4.52320i 0 0
199.11 0.635381 + 1.89639i 0 −3.19258 + 2.40986i 0 0 −10.1035 −6.59853 4.52320i 0 0
199.12 0.635381 + 1.89639i 0 −3.19258 + 2.40986i 0 0 10.1035 −6.59853 4.52320i 0 0
199.13 1.75963 0.950636i 0 2.19258 3.34553i 0 0 −3.98982 0.677747 7.97124i 0 0
199.14 1.75963 0.950636i 0 2.19258 3.34553i 0 0 3.98982 0.677747 7.97124i 0 0
199.15 1.75963 + 0.950636i 0 2.19258 + 3.34553i 0 0 −3.98982 0.677747 + 7.97124i 0 0
199.16 1.75963 + 0.950636i 0 2.19258 + 3.34553i 0 0 3.98982 0.677747 + 7.97124i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 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.g 16
3.b odd 2 1 inner 900.3.f.g 16
4.b odd 2 1 inner 900.3.f.g 16
5.b even 2 1 inner 900.3.f.g 16
5.c odd 4 1 900.3.c.p 8
5.c odd 4 1 900.3.c.q yes 8
12.b even 2 1 inner 900.3.f.g 16
15.d odd 2 1 inner 900.3.f.g 16
15.e even 4 1 900.3.c.p 8
15.e even 4 1 900.3.c.q yes 8
20.d odd 2 1 inner 900.3.f.g 16
20.e even 4 1 900.3.c.p 8
20.e even 4 1 900.3.c.q yes 8
60.h even 2 1 inner 900.3.f.g 16
60.l odd 4 1 900.3.c.p 8
60.l odd 4 1 900.3.c.q yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.c.p 8 5.c odd 4 1
900.3.c.p 8 15.e even 4 1
900.3.c.p 8 20.e even 4 1
900.3.c.p 8 60.l odd 4 1
900.3.c.q yes 8 5.c odd 4 1
900.3.c.q yes 8 15.e even 4 1
900.3.c.q yes 8 20.e even 4 1
900.3.c.q yes 8 60.l odd 4 1
900.3.f.g 16 1.a even 1 1 trivial
900.3.f.g 16 3.b odd 2 1 inner
900.3.f.g 16 4.b odd 2 1 inner
900.3.f.g 16 5.b even 2 1 inner
900.3.f.g 16 12.b even 2 1 inner
900.3.f.g 16 15.d odd 2 1 inner
900.3.f.g 16 20.d odd 2 1 inner
900.3.f.g 16 60.h even 2 1 inner

Hecke kernels

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

\( T_{7}^{4} - 118 T_{7}^{2} + 1625 \)
\( T_{29}^{4} - 3048 T_{29}^{2} + 1019200 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 + 32 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 1625 - 118 T^{2} + T^{4} )^{4} \)
$11$ \( ( 8000 + 184 T^{2} + T^{4} )^{4} \)
$13$ \( ( 13225 + 234 T^{2} + T^{4} )^{4} \)
$17$ \( ( 40768 + 424 T^{2} + T^{4} )^{4} \)
$19$ \( ( 65 + 302 T^{2} + T^{4} )^{4} \)
$23$ \( ( 15680 - 1784 T^{2} + T^{4} )^{4} \)
$29$ \( ( 1019200 - 3048 T^{2} + T^{4} )^{4} \)
$31$ \( ( 79625 + 1030 T^{2} + T^{4} )^{4} \)
$37$ \( ( 211600 + 936 T^{2} + T^{4} )^{4} \)
$41$ \( ( 3515200 - 3752 T^{2} + T^{4} )^{4} \)
$43$ \( ( 13456625 - 7358 T^{2} + T^{4} )^{4} \)
$47$ \( ( 2708480 - 3296 T^{2} + T^{4} )^{4} \)
$53$ \( ( 2896192 + 6888 T^{2} + T^{4} )^{4} \)
$59$ \( ( 10368000 + 6624 T^{2} + T^{4} )^{4} \)
$61$ \( ( -5659 - 10 T + T^{2} )^{8} \)
$67$ \( ( 4564625 - 9502 T^{2} + T^{4} )^{4} \)
$71$ \( ( 66248000 + 21304 T^{2} + T^{4} )^{4} \)
$73$ \( ( 114490000 + 25000 T^{2} + T^{4} )^{4} \)
$79$ \( ( 44994560 + 21568 T^{2} + T^{4} )^{4} \)
$83$ \( ( 146232320 - 27104 T^{2} + T^{4} )^{4} \)
$89$ \( ( 5324800 - 16512 T^{2} + T^{4} )^{4} \)
$97$ \( ( 225 + T^{2} )^{8} \)
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