Properties

Label 975.2.a.o
Level $975$
Weight $2$
Character orbit 975.a
Self dual yes
Analytic conductor $7.785$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + \beta_1 q^{6} + \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + \beta_1 q^{6} + \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8} + q^{9} - \beta_{2} q^{11} + (\beta_{2} + 3) q^{12} - q^{13} + (2 \beta_1 - 2) q^{14} + (\beta_{2} - 2 \beta_1 + 9) q^{16} + ( - \beta_{2} - 2 \beta_1) q^{17} + \beta_1 q^{18} + (2 \beta_1 + 2) q^{19} + \beta_{2} q^{21} + ( - 2 \beta_1 + 2) q^{22} + ( - \beta_{2} - 2 \beta_1 + 2) q^{23} + (3 \beta_1 - 2) q^{24} - \beta_1 q^{26} + q^{27} + ( - 2 \beta_1 + 10) q^{28} + 6 q^{29} + ( - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{32} - \beta_{2} q^{33} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{34} + (\beta_{2} + 3) q^{36} + (\beta_{2} - 2 \beta_1 - 4) q^{37} + (2 \beta_{2} + 2 \beta_1 + 10) q^{38} - q^{39} + ( - \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_1 - 2) q^{42} + 4 \beta_1 q^{43} + (2 \beta_1 - 10) q^{44} + ( - 2 \beta_{2} - 8) q^{46} + ( - 2 \beta_1 + 6) q^{47} + (\beta_{2} - 2 \beta_1 + 9) q^{48} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} - 2 \beta_1) q^{51} + ( - \beta_{2} - 3) q^{52} + ( - \beta_{2} - 2 \beta_1 - 4) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{56} + (2 \beta_1 + 2) q^{57} + 6 \beta_1 q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (3 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{62} + \beta_{2} q^{63} + (3 \beta_{2} - 8 \beta_1 + 11) q^{64} + ( - 2 \beta_1 + 2) q^{66} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - 8 \beta_1 - 6) q^{68} + ( - \beta_{2} - 2 \beta_1 + 2) q^{69} + ( - \beta_{2} - 4) q^{71} + (3 \beta_1 - 2) q^{72} + (4 \beta_1 + 2) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 12) q^{74} + (2 \beta_{2} + 10 \beta_1 + 2) q^{76} + (3 \beta_{2} + 2 \beta_1 - 10) q^{77} - \beta_1 q^{78} + (\beta_{2} - 2 \beta_1 + 2) q^{79} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{82} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + ( - 2 \beta_1 + 10) q^{84} + (4 \beta_{2} + 20) q^{86} + 6 q^{87} + (2 \beta_{2} - 6 \beta_1 + 6) q^{88} + (\beta_{2} + 2 \beta_1 + 4) q^{89} - \beta_{2} q^{91} + (2 \beta_{2} - 8 \beta_1) q^{92} + ( - 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + 6 \beta_1 - 10) q^{94} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{96} + ( - \beta_{2} - 2 \beta_1 + 8) q^{97} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{98} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9} + q^{11} + 8 q^{12} - 3 q^{13} - 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} - q^{21} + 6 q^{22} + 7 q^{23} - 6 q^{24} + 3 q^{27} + 30 q^{28} + 18 q^{29} + 6 q^{31} - 22 q^{32} + q^{33} - 22 q^{34} + 8 q^{36} - 13 q^{37} + 28 q^{38} - 3 q^{39} + q^{41} - 6 q^{42} - 30 q^{44} - 22 q^{46} + 18 q^{47} + 26 q^{48} + 12 q^{49} + q^{51} - 8 q^{52} - 11 q^{53} - 16 q^{56} + 6 q^{57} - 8 q^{59} + 9 q^{61} - 28 q^{62} - q^{63} + 30 q^{64} + 6 q^{66} - 4 q^{67} - 18 q^{68} + 7 q^{69} - 11 q^{71} - 6 q^{72} + 6 q^{73} - 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 3 q^{81} - 22 q^{82} - 8 q^{83} + 30 q^{84} + 56 q^{86} + 18 q^{87} + 16 q^{88} + 11 q^{89} + q^{91} - 2 q^{92} + 6 q^{93} - 28 q^{94} - 22 q^{96} + 25 q^{97} - 10 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.470683
−1.81361
2.34292
−2.77846 1.00000 5.71982 0 −2.77846 2.71982 −10.3354 1.00000 0
1.2 0.289169 1.00000 −1.91638 0 0.289169 −4.91638 −1.13249 1.00000 0
1.3 2.48929 1.00000 4.19656 0 2.48929 1.19656 5.46787 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.a.o 3
3.b odd 2 1 2925.2.a.bh 3
5.b even 2 1 195.2.a.e 3
5.c odd 4 2 975.2.c.i 6
15.d odd 2 1 585.2.a.n 3
15.e even 4 2 2925.2.c.w 6
20.d odd 2 1 3120.2.a.bj 3
35.c odd 2 1 9555.2.a.bq 3
60.h even 2 1 9360.2.a.dd 3
65.d even 2 1 2535.2.a.bc 3
195.e odd 2 1 7605.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 5.b even 2 1
585.2.a.n 3 15.d odd 2 1
975.2.a.o 3 1.a even 1 1 trivial
975.2.c.i 6 5.c odd 4 2
2535.2.a.bc 3 65.d even 2 1
2925.2.a.bh 3 3.b odd 2 1
2925.2.c.w 6 15.e even 4 2
3120.2.a.bj 3 20.d odd 2 1
7605.2.a.bx 3 195.e odd 2 1
9360.2.a.dd 3 60.h even 2 1
9555.2.a.bq 3 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(975))\):

\( T_{2}^{3} - 7T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 7T + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 16 T - 16 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 32 T + 76 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 16 T + 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 7 T^{2} - 16 T + 128 \) Copy content Toggle raw display
$29$ \( (T - 6)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} + 13T^{2} - 316 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 32 T + 76 \) Copy content Toggle raw display
$43$ \( T^{3} - 112T + 128 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + 80 T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 11 T^{2} + 8 T - 4 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 48 T - 128 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 112 T + 844 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$71$ \( T^{3} + 11 T^{2} + 24 T - 32 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 100 T + 344 \) Copy content Toggle raw display
$79$ \( T^{3} - 5 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 48 T - 128 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$97$ \( T^{3} - 25 T^{2} + 176 T - 244 \) Copy content Toggle raw display
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