Properties

Label 900.3.c.r
Level $900$
Weight $3$
Character orbit 900.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6080256576.2
Defining polynomial: \(x^{8} - 3 x^{7} + 7 x^{6} - 12 x^{5} + 12 x^{4} - 48 x^{3} + 112 x^{2} - 192 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( 1 + \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{4} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( 1 + \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{4} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( 1 - \beta_{5} - 3 \beta_{7} ) q^{11} + ( 3 \beta_{2} + \beta_{3} ) q^{13} + ( 3 - \beta_{5} + \beta_{6} - \beta_{7} ) q^{14} + ( -6 - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{16} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{17} + 2 \beta_{6} q^{19} + ( -2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{22} + ( 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{23} + ( -10 - 2 \beta_{7} ) q^{26} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{28} + ( -23 - 3 \beta_{5} + 3 \beta_{7} ) q^{29} + ( 2 - 2 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{31} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} ) q^{32} + ( -6 \beta_{5} - 2 \beta_{6} ) q^{34} + ( 9 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{38} + ( 32 - 6 \beta_{5} + 6 \beta_{7} ) q^{41} + ( 8 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} ) q^{43} + ( 44 + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{44} + ( 15 + \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{46} + ( -8 \beta_{2} + 9 \beta_{3} + \beta_{4} ) q^{47} + ( 3 + 6 \beta_{5} - 6 \beta_{7} ) q^{49} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( 9 \beta_{2} + 3 \beta_{3} ) q^{53} + ( 32 - 6 \beta_{5} + 2 \beta_{6} ) q^{56} + ( 6 \beta_{1} + 20 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} ) q^{58} + ( 1 - \beta_{5} - 8 \beta_{6} - 3 \beta_{7} ) q^{59} + 38 q^{61} + ( -8 \beta_{3} + 8 \beta_{4} ) q^{62} + ( 2 + 3 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{64} + ( -14 \beta_{2} + 15 \beta_{3} + \beta_{4} ) q^{67} + ( 2 \beta_{1} - 10 \beta_{2} - 22 \beta_{3} - 10 \beta_{4} ) q^{68} + ( 6 - 6 \beta_{5} - 4 \beta_{6} - 18 \beta_{7} ) q^{71} + ( 8 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{73} + ( -30 - 6 \beta_{7} ) q^{74} + ( -12 \beta_{5} + 4 \beta_{6} ) q^{76} + ( -8 \beta_{1} + 22 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} ) q^{77} + ( 2 - 2 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{79} + ( 12 \beta_{1} - 38 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} ) q^{82} + ( -4 \beta_{2} + 9 \beta_{3} + 5 \beta_{4} ) q^{83} + ( 39 - 3 \beta_{5} + 3 \beta_{6} - 13 \beta_{7} ) q^{86} + ( -2 \beta_{1} - 38 \beta_{2} + 6 \beta_{3} + 10 \beta_{4} ) q^{88} + ( 70 - 4 \beta_{5} + 4 \beta_{7} ) q^{89} + ( -2 + 2 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{91} + ( -8 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{92} + ( -51 - \beta_{5} + \beta_{6} + 17 \beta_{7} ) q^{94} + ( 8 \beta_{1} + 68 \beta_{2} + 24 \beta_{3} - 4 \beta_{4} ) q^{97} + ( -12 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 12 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 10q^{4} + O(q^{10}) \) \( 8q + 10q^{4} + 20q^{14} - 46q^{16} - 84q^{26} - 184q^{29} - 12q^{34} + 256q^{41} + 348q^{44} + 112q^{46} + 24q^{49} + 244q^{56} + 304q^{61} + 10q^{64} - 252q^{74} - 24q^{76} + 280q^{86} + 560q^{89} - 376q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 7 x^{6} - 12 x^{5} + 12 x^{4} - 48 x^{3} + 112 x^{2} - 192 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} + 17 \nu^{5} + 10 \nu^{4} - 64 \nu^{3} - 88 \nu^{2} + 176 \nu - 544 \)\()/112\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} + 15 \nu^{6} - 23 \nu^{5} - 2 \nu^{4} - 60 \nu^{3} + 152 \nu^{2} - 528 \nu + 736 \)\()/224\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} + 25 \nu^{6} - \nu^{5} + 34 \nu^{4} + 12 \nu^{3} + 104 \nu^{2} - 880 \nu + 32 \)\()/224\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{6} + 11 \nu^{5} + 2 \nu^{4} - 4 \nu^{3} + 40 \nu^{2} + 80 \nu - 352 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 36 \nu^{6} - 58 \nu^{5} + 61 \nu^{4} - 116 \nu^{3} + 292 \nu^{2} - 976 \nu + 1744 \)\()/112\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 8 \nu^{6} + 30 \nu^{5} - 33 \nu^{4} + 60 \nu^{3} - 180 \nu^{2} + 528 \nu - 848 \)\()/56\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{7} - 20 \nu^{6} + 26 \nu^{5} - 37 \nu^{4} + 52 \nu^{3} - 324 \nu^{2} + 592 \nu - 384 \)\()/112\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 3\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} - 3\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 3 \beta_{2} - 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} - 25 \beta_{2} + 2 \beta_{1} + 11\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 2 \beta_{6} + \beta_{5} + 8 \beta_{4} + 11 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 105\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{7} + 10 \beta_{6} + 9 \beta_{5} + 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(71 \beta_{7} - 2 \beta_{6} + 43 \beta_{5} + 40 \beta_{4} - \beta_{3} + 11 \beta_{2} - 38 \beta_{1} - 237\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.51328 1.30766i
−1.51328 + 1.30766i
0.670410 + 1.88429i
0.670410 1.88429i
1.96705 0.361553i
1.96705 + 0.361553i
0.375825 + 1.96437i
0.375825 1.96437i
−1.88911 0.656712i 0 3.13746 + 2.48120i 0 0 9.55505i −4.29756 6.74766i 0 0
451.2 −1.88911 + 0.656712i 0 3.13746 2.48120i 0 0 9.55505i −4.29756 + 6.74766i 0 0
451.3 −1.29664 1.52274i 0 −0.637459 + 3.94888i 0 0 0.837253i 6.83966 4.14959i 0 0
451.4 −1.29664 + 1.52274i 0 −0.637459 3.94888i 0 0 0.837253i 6.83966 + 4.14959i 0 0
451.5 1.29664 1.52274i 0 −0.637459 3.94888i 0 0 0.837253i −6.83966 4.14959i 0 0
451.6 1.29664 + 1.52274i 0 −0.637459 + 3.94888i 0 0 0.837253i −6.83966 + 4.14959i 0 0
451.7 1.88911 0.656712i 0 3.13746 2.48120i 0 0 9.55505i 4.29756 6.74766i 0 0
451.8 1.88911 + 0.656712i 0 3.13746 + 2.48120i 0 0 9.55505i 4.29756 + 6.74766i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.8
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.c.r 8
3.b odd 2 1 300.3.c.f 8
4.b odd 2 1 inner 900.3.c.r 8
5.b even 2 1 inner 900.3.c.r 8
5.c odd 4 2 180.3.f.h 8
12.b even 2 1 300.3.c.f 8
15.d odd 2 1 300.3.c.f 8
15.e even 4 2 60.3.f.b 8
20.d odd 2 1 inner 900.3.c.r 8
20.e even 4 2 180.3.f.h 8
60.h even 2 1 300.3.c.f 8
60.l odd 4 2 60.3.f.b 8
120.q odd 4 2 960.3.j.e 8
120.w even 4 2 960.3.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 15.e even 4 2
60.3.f.b 8 60.l odd 4 2
180.3.f.h 8 5.c odd 4 2
180.3.f.h 8 20.e even 4 2
300.3.c.f 8 3.b odd 2 1
300.3.c.f 8 12.b even 2 1
300.3.c.f 8 15.d odd 2 1
300.3.c.f 8 60.h even 2 1
900.3.c.r 8 1.a even 1 1 trivial
900.3.c.r 8 4.b odd 2 1 inner
900.3.c.r 8 5.b even 2 1 inner
900.3.c.r 8 20.d odd 2 1 inner
960.3.j.e 8 120.q odd 4 2
960.3.j.e 8 120.w even 4 2

Hecke kernels

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

\( T_{7}^{4} + 92 T_{7}^{2} + 64 \)
\( T_{13}^{4} - 84 T_{13}^{2} + 1536 \)
\( T_{17}^{4} - 1044 T_{17}^{2} + 221184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 80 T^{2} + 24 T^{4} - 5 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 64 + 92 T^{2} + T^{4} )^{2} \)
$11$ \( ( 24576 + 348 T^{2} + T^{4} )^{2} \)
$13$ \( ( 1536 - 84 T^{2} + T^{4} )^{2} \)
$17$ \( ( 221184 - 1044 T^{2} + T^{4} )^{2} \)
$19$ \( ( 221184 + 1008 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1024 + 368 T^{2} + T^{4} )^{2} \)
$29$ \( ( 16 + 46 T + T^{2} )^{4} \)
$31$ \( ( 393216 + 2304 T^{2} + T^{4} )^{2} \)
$37$ \( ( 124416 - 756 T^{2} + T^{4} )^{2} \)
$41$ \( ( -1028 - 64 T + T^{2} )^{4} \)
$43$ \( ( 1364224 + 2528 T^{2} + T^{4} )^{2} \)
$47$ \( ( 614656 + 3296 T^{2} + T^{4} )^{2} \)
$53$ \( ( 124416 - 756 T^{2} + T^{4} )^{2} \)
$59$ \( ( 69033984 + 16668 T^{2} + T^{4} )^{2} \)
$61$ \( ( -38 + T )^{8} \)
$67$ \( ( 7573504 + 9392 T^{2} + T^{4} )^{2} \)
$71$ \( ( 884736 + 17136 T^{2} + T^{4} )^{2} \)
$73$ \( ( 3538944 - 4176 T^{2} + T^{4} )^{2} \)
$79$ \( ( 393216 + 2304 T^{2} + T^{4} )^{2} \)
$83$ \( ( 2027776 + 4064 T^{2} + T^{4} )^{2} \)
$89$ \( ( 3988 - 140 T + T^{2} )^{4} \)
$97$ \( ( 495550464 - 45888 T^{2} + T^{4} )^{2} \)
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