Properties

Label 3150.3.c.b
Level $3150$
Weight $3$
Character orbit 3150.c
Analytic conductor $85.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + 2 q^{4} + \beta_{1} q^{7} -2 \beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + 2 q^{4} + \beta_{1} q^{7} -2 \beta_{3} q^{8} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{11} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{13} -\beta_{6} q^{14} + 4 q^{16} + ( -13 \beta_{3} + 2 \beta_{7} ) q^{17} -20 q^{19} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -2 \beta_{3} + 4 \beta_{7} ) q^{23} + ( -8 \beta_{5} + 4 \beta_{6} ) q^{26} + 2 \beta_{1} q^{28} + ( 19 \beta_{5} - 4 \beta_{6} ) q^{29} + ( 4 + 4 \beta_{4} ) q^{31} -4 \beta_{3} q^{32} + ( 26 - 2 \beta_{4} ) q^{34} -19 \beta_{2} q^{37} + 20 \beta_{3} q^{38} + ( 27 \beta_{5} - 6 \beta_{6} ) q^{41} + ( 24 \beta_{1} + 10 \beta_{2} ) q^{43} + ( -4 \beta_{5} - 8 \beta_{6} ) q^{44} + ( 4 - 4 \beta_{4} ) q^{46} -12 \beta_{3} q^{47} -7 q^{49} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{52} + ( -3 \beta_{3} - 24 \beta_{7} ) q^{53} -2 \beta_{6} q^{56} + ( 8 \beta_{1} + 19 \beta_{2} ) q^{58} + ( -20 \beta_{5} + 8 \beta_{6} ) q^{59} + ( 58 - 8 \beta_{4} ) q^{61} + ( -4 \beta_{3} - 8 \beta_{7} ) q^{62} + 8 q^{64} + ( 32 \beta_{1} + 24 \beta_{2} ) q^{67} + ( -26 \beta_{3} + 4 \beta_{7} ) q^{68} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -20 \beta_{1} - 12 \beta_{2} ) q^{73} -38 \beta_{5} q^{74} -40 q^{76} + ( 28 \beta_{3} - 2 \beta_{7} ) q^{77} + ( -76 + 8 \beta_{4} ) q^{79} + ( 12 \beta_{1} + 27 \beta_{2} ) q^{82} + ( 64 \beta_{3} - 8 \beta_{7} ) q^{83} + ( 20 \beta_{5} - 24 \beta_{6} ) q^{86} + ( 16 \beta_{1} - 4 \beta_{2} ) q^{88} + ( -51 \beta_{5} - 18 \beta_{6} ) q^{89} + ( 28 + 4 \beta_{4} ) q^{91} + ( -4 \beta_{3} + 8 \beta_{7} ) q^{92} + 24 q^{94} + ( 44 \beta_{1} + 36 \beta_{2} ) q^{97} + 7 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{4} + O(q^{10}) \) \( 8q + 16q^{4} + 32q^{16} - 160q^{19} + 32q^{31} + 208q^{34} + 32q^{46} - 56q^{49} + 464q^{61} + 64q^{64} - 320q^{76} - 608q^{79} + 224q^{91} + 192q^{94} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 1 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{2} \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} - 7 \nu^{3} + 4 \nu \)\()/24\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 3 \beta_{2}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - 5 \beta_{5} + 5 \beta_{3}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - 11 \beta_{5} - 11 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{4} + 9 \beta_{2}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} + 7 \beta_{6} + 13 \beta_{5} - 13 \beta_{3}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.28897 0.581861i
−0.581861 1.28897i
−0.581861 + 1.28897i
1.28897 + 0.581861i
0.581861 + 1.28897i
−1.28897 + 0.581861i
−1.28897 0.581861i
0.581861 1.28897i
−1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.5 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.6 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.7 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.8 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
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 3150.3.c.b 8
3.b odd 2 1 inner 3150.3.c.b 8
5.b even 2 1 inner 3150.3.c.b 8
5.c odd 4 1 126.3.b.a 4
5.c odd 4 1 3150.3.e.e 4
15.d odd 2 1 inner 3150.3.c.b 8
15.e even 4 1 126.3.b.a 4
15.e even 4 1 3150.3.e.e 4
20.e even 4 1 1008.3.d.a 4
35.f even 4 1 882.3.b.f 4
35.k even 12 2 882.3.s.i 8
35.l odd 12 2 882.3.s.e 8
40.i odd 4 1 4032.3.d.i 4
40.k even 4 1 4032.3.d.j 4
45.k odd 12 2 1134.3.q.c 8
45.l even 12 2 1134.3.q.c 8
60.l odd 4 1 1008.3.d.a 4
105.k odd 4 1 882.3.b.f 4
105.w odd 12 2 882.3.s.i 8
105.x even 12 2 882.3.s.e 8
120.q odd 4 1 4032.3.d.j 4
120.w even 4 1 4032.3.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 5.c odd 4 1
126.3.b.a 4 15.e even 4 1
882.3.b.f 4 35.f even 4 1
882.3.b.f 4 105.k odd 4 1
882.3.s.e 8 35.l odd 12 2
882.3.s.e 8 105.x even 12 2
882.3.s.i 8 35.k even 12 2
882.3.s.i 8 105.w odd 12 2
1008.3.d.a 4 20.e even 4 1
1008.3.d.a 4 60.l odd 4 1
1134.3.q.c 8 45.k odd 12 2
1134.3.q.c 8 45.l even 12 2
3150.3.c.b 8 1.a even 1 1 trivial
3150.3.c.b 8 3.b odd 2 1 inner
3150.3.c.b 8 5.b even 2 1 inner
3150.3.c.b 8 15.d odd 2 1 inner
3150.3.e.e 4 5.c odd 4 1
3150.3.e.e 4 15.e even 4 1
4032.3.d.i 4 40.i odd 4 1
4032.3.d.i 4 120.w even 4 1
4032.3.d.j 4 40.k even 4 1
4032.3.d.j 4 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 464 T_{11}^{2} + 46656 \) acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( ( 46656 + 464 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2304 + 352 T^{2} + T^{4} )^{2} \)
$17$ \( ( 79524 - 788 T^{2} + T^{4} )^{2} \)
$19$ \( ( 20 + T )^{8} \)
$23$ \( ( 46656 - 464 T^{2} + T^{4} )^{2} \)
$29$ \( ( 248004 + 1892 T^{2} + T^{4} )^{2} \)
$31$ \( ( -432 - 8 T + T^{2} )^{4} \)
$37$ \( ( 1444 + T^{2} )^{4} \)
$41$ \( ( 910116 + 3924 T^{2} + T^{4} )^{2} \)
$43$ \( ( 13191424 + 8864 T^{2} + T^{4} )^{2} \)
$47$ \( ( -288 + T^{2} )^{4} \)
$53$ \( ( 64738116 - 16164 T^{2} + T^{4} )^{2} \)
$59$ \( ( 9216 + 3392 T^{2} + T^{4} )^{2} \)
$61$ \( ( 1572 - 116 T + T^{2} )^{4} \)
$67$ \( ( 23658496 + 18944 T^{2} + T^{4} )^{2} \)
$71$ \( ( 46656 + 464 T^{2} + T^{4} )^{2} \)
$73$ \( ( 4946176 + 6752 T^{2} + T^{4} )^{2} \)
$79$ \( ( 3984 + 152 T + T^{2} )^{4} \)
$83$ \( ( 53231616 - 18176 T^{2} + T^{4} )^{2} \)
$89$ \( ( 443556 + 19476 T^{2} + T^{4} )^{2} \)
$97$ \( ( 70023424 + 37472 T^{2} + T^{4} )^{2} \)
show more
show less