Properties

Label 3150.3.e.e
Level $3150$
Weight $3$
Character orbit 3150.e
Analytic conductor $85.831$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -2 q^{4} + \beta_{3} q^{7} -2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -2 q^{4} + \beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{11} + ( 8 + 4 \beta_{3} ) q^{13} + \beta_{2} q^{14} + 4 q^{16} + ( 13 \beta_{1} - 2 \beta_{2} ) q^{17} + 20 q^{19} + ( -4 + 8 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{26} -2 \beta_{3} q^{28} + ( 19 \beta_{1} + 4 \beta_{2} ) q^{29} + ( 4 + 8 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -26 + 4 \beta_{3} ) q^{34} -38 q^{37} + 20 \beta_{1} q^{38} + ( -27 \beta_{1} - 6 \beta_{2} ) q^{41} + ( -20 - 24 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 4 - 8 \beta_{3} ) q^{46} + 12 \beta_{1} q^{47} + 7 q^{49} + ( -16 - 8 \beta_{3} ) q^{52} + ( -3 \beta_{1} - 24 \beta_{2} ) q^{53} -2 \beta_{2} q^{56} + ( -38 - 8 \beta_{3} ) q^{58} + ( -20 \beta_{1} - 8 \beta_{2} ) q^{59} + ( 58 - 16 \beta_{3} ) q^{61} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{62} -8 q^{64} + ( 48 + 32 \beta_{3} ) q^{67} + ( -26 \beta_{1} + 4 \beta_{2} ) q^{68} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{71} + ( 24 + 20 \beta_{3} ) q^{73} -38 \beta_{1} q^{74} -40 q^{76} + ( -28 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 76 - 16 \beta_{3} ) q^{79} + ( 54 + 12 \beta_{3} ) q^{82} + ( 64 \beta_{1} - 8 \beta_{2} ) q^{83} + ( -20 \beta_{1} - 24 \beta_{2} ) q^{86} + ( 8 - 16 \beta_{3} ) q^{88} + ( -51 \beta_{1} + 18 \beta_{2} ) q^{89} + ( 28 + 8 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{92} -24 q^{94} + ( 72 + 44 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 32q^{13} + 16q^{16} + 80q^{19} - 16q^{22} + 16q^{31} - 104q^{34} - 152q^{37} - 80q^{43} + 16q^{46} + 28q^{49} - 64q^{52} - 152q^{58} + 232q^{61} - 32q^{64} + 192q^{67} + 96q^{73} - 160q^{76} + 304q^{79} + 216q^{82} + 32q^{88} + 112q^{91} - 96q^{94} + 288q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
2.57794i
1.16372i
2.57794i
1.16372i
1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
701.2 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
701.3 1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
701.4 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.e.e 4
3.b odd 2 1 inner 3150.3.e.e 4
5.b even 2 1 126.3.b.a 4
5.c odd 4 2 3150.3.c.b 8
15.d odd 2 1 126.3.b.a 4
15.e even 4 2 3150.3.c.b 8
20.d odd 2 1 1008.3.d.a 4
35.c odd 2 1 882.3.b.f 4
35.i odd 6 2 882.3.s.i 8
35.j even 6 2 882.3.s.e 8
40.e odd 2 1 4032.3.d.j 4
40.f even 2 1 4032.3.d.i 4
45.h odd 6 2 1134.3.q.c 8
45.j even 6 2 1134.3.q.c 8
60.h even 2 1 1008.3.d.a 4
105.g even 2 1 882.3.b.f 4
105.o odd 6 2 882.3.s.e 8
105.p even 6 2 882.3.s.i 8
120.i odd 2 1 4032.3.d.i 4
120.m even 2 1 4032.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 5.b even 2 1
126.3.b.a 4 15.d odd 2 1
882.3.b.f 4 35.c odd 2 1
882.3.b.f 4 105.g even 2 1
882.3.s.e 8 35.j even 6 2
882.3.s.e 8 105.o odd 6 2
882.3.s.i 8 35.i odd 6 2
882.3.s.i 8 105.p even 6 2
1008.3.d.a 4 20.d odd 2 1
1008.3.d.a 4 60.h even 2 1
1134.3.q.c 8 45.h odd 6 2
1134.3.q.c 8 45.j even 6 2
3150.3.c.b 8 5.c odd 4 2
3150.3.c.b 8 15.e even 4 2
3150.3.e.e 4 1.a even 1 1 trivial
3150.3.e.e 4 3.b odd 2 1 inner
4032.3.d.i 4 40.f even 2 1
4032.3.d.i 4 120.i odd 2 1
4032.3.d.j 4 40.e odd 2 1
4032.3.d.j 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{4} + 464 T_{11}^{2} + 46656 \)
\( T_{13}^{2} - 16 T_{13} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 46656 + 464 T^{2} + T^{4} \)
$13$ \( ( -48 - 16 T + T^{2} )^{2} \)
$17$ \( 79524 + 788 T^{2} + T^{4} \)
$19$ \( ( -20 + T )^{4} \)
$23$ \( 46656 + 464 T^{2} + T^{4} \)
$29$ \( 248004 + 1892 T^{2} + T^{4} \)
$31$ \( ( -432 - 8 T + T^{2} )^{2} \)
$37$ \( ( 38 + T )^{4} \)
$41$ \( 910116 + 3924 T^{2} + T^{4} \)
$43$ \( ( -3632 + 40 T + T^{2} )^{2} \)
$47$ \( ( 288 + T^{2} )^{2} \)
$53$ \( 64738116 + 16164 T^{2} + T^{4} \)
$59$ \( 9216 + 3392 T^{2} + T^{4} \)
$61$ \( ( 1572 - 116 T + T^{2} )^{2} \)
$67$ \( ( -4864 - 96 T + T^{2} )^{2} \)
$71$ \( 46656 + 464 T^{2} + T^{4} \)
$73$ \( ( -2224 - 48 T + T^{2} )^{2} \)
$79$ \( ( 3984 - 152 T + T^{2} )^{2} \)
$83$ \( 53231616 + 18176 T^{2} + T^{4} \)
$89$ \( 443556 + 19476 T^{2} + T^{4} \)
$97$ \( ( -8368 - 144 T + T^{2} )^{2} \)
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