Properties

Label 975.2.c.j
Level $975$
Weight $2$
Character orbit 975.c
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} - \beta_1 q^{6} + (\beta_{5} - 2 \beta_{4}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} - \beta_1 q^{6} + (\beta_{5} - 2 \beta_{4}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8} - q^{9} + (\beta_{3} - \beta_1 - 1) q^{11} + ( - 2 \beta_{4} - \beta_{2}) q^{12} + \beta_{4} q^{13} + ( - \beta_{3} + 2 \beta_1 + 4) q^{14} + ( - 2 \beta_1 + 1) q^{16} + (2 \beta_{5} - 3 \beta_{4}) q^{17} + \beta_{5} q^{18} + (\beta_{3} - 2 \beta_1 - 3) q^{19} + (\beta_1 + 2) q^{21} + (2 \beta_{5} - 3 \beta_{4}) q^{22} + (2 \beta_{5} + \beta_{4} + \beta_{2}) q^{23} + (\beta_{3} + \beta_1 - 1) q^{24} - \beta_1 q^{26} - \beta_{4} q^{27} + ( - 3 \beta_{5} + 3 \beta_{4} + \beta_{2}) q^{28} + ( - 2 \beta_{3} + 5) q^{29} + (\beta_{3} - 3 \beta_1 + 1) q^{31} + (\beta_{5} - 6 \beta_{4}) q^{32} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{33} + ( - 2 \beta_{3} + 3 \beta_1 + 8) q^{34} + ( - \beta_{3} + 2) q^{36} + (2 \beta_{5} - \beta_{4} + \beta_{2}) q^{37} + (4 \beta_{5} - 7 \beta_{4} - \beta_{2}) q^{38} - q^{39} + ( - \beta_{3} + 2 \beta_1 + 1) q^{41} + ( - 2 \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{42} + ( - \beta_{4} - 3 \beta_{2}) q^{43} + (\beta_1 + 6) q^{44} + ( - 3 \beta_{3} - 2 \beta_1 + 9) q^{46} + ( - \beta_{5} + 4 \beta_{4} + 4 \beta_{2}) q^{47} + (2 \beta_{5} + \beta_{4}) q^{48} + (\beta_{3} - 4 \beta_1 - 1) q^{49} + (2 \beta_1 + 3) q^{51} + ( - 2 \beta_{4} - \beta_{2}) q^{52} + (2 \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{53} + \beta_1 q^{54} - 3 q^{56} + (2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{57} + ( - 7 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{58} + (2 \beta_{3} - \beta_1 - 2) q^{59} + (2 \beta_{3} - 2 \beta_1 - 3) q^{61} + ( - 11 \beta_{4} - 2 \beta_{2}) q^{62} + ( - \beta_{5} + 2 \beta_{4}) q^{63} + ( - \beta_{3} + 2 \beta_1 + 6) q^{64} + (2 \beta_1 + 3) q^{66} + ( - 3 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{67} + ( - 6 \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{68} + (\beta_{3} + 2 \beta_1 - 1) q^{69} + ( - \beta_{3} + 2 \beta_1 - 5) q^{71} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{72} + (9 \beta_{4} + \beta_{2}) q^{73} + ( - 3 \beta_{3} + 9) q^{74} + ( - \beta_{3} + 4 \beta_1 + 9) q^{76} + ( - 4 \beta_{5} + 5 \beta_{4} + 2 \beta_{2}) q^{77} + \beta_{5} q^{78} + (\beta_{3} - 4 \beta_1 - 3) q^{79} + q^{81} + ( - 2 \beta_{5} + 7 \beta_{4} + \beta_{2}) q^{82} + ( - 5 \beta_{5} + 6 \beta_{4}) q^{83} + (\beta_{3} - 3 \beta_1 - 3) q^{84} + (3 \beta_{3} + 4 \beta_1 - 3) q^{86} + (5 \beta_{4} + 2 \beta_{2}) q^{87} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{88} + (2 \beta_1 + 6) q^{89} + (\beta_1 + 2) q^{91} + ( - 8 \beta_{5} - 9 \beta_{4} - 3 \beta_{2}) q^{92} + (3 \beta_{5} + \beta_{4} - \beta_{2}) q^{93} + ( - 3 \beta_{3} - 8 \beta_1) q^{94} + (\beta_1 + 6) q^{96} + (2 \beta_{5} - 10 \beta_{4} - 2 \beta_{2}) q^{97} + (2 \beta_{5} - 15 \beta_{4} - 3 \beta_{2}) q^{98} + ( - \beta_{3} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 2 q^{6} - 6 q^{9} - 2 q^{11} + 18 q^{14} + 10 q^{16} - 12 q^{19} + 10 q^{21} - 6 q^{24} + 2 q^{26} + 26 q^{29} + 14 q^{31} + 38 q^{34} + 10 q^{36} - 6 q^{39} + 34 q^{44} + 52 q^{46} + 4 q^{49} + 14 q^{51} - 2 q^{54} - 18 q^{56} - 6 q^{59} - 10 q^{61} + 30 q^{64} + 14 q^{66} - 8 q^{69} - 36 q^{71} + 48 q^{74} + 44 q^{76} - 8 q^{79} + 6 q^{81} - 10 q^{84} - 20 q^{86} + 32 q^{89} + 10 q^{91} + 10 q^{94} + 34 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 6\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
1.66044 + 1.66044i
0.675970 0.675970i
−1.33641 1.33641i
−1.33641 + 1.33641i
0.675970 + 0.675970i
1.66044 1.66044i
2.51414i 1.00000i −4.32088 0 2.51414 0.514137i 5.83502i −1.00000 0
274.2 2.08613i 1.00000i −2.35194 0 −2.08613 4.08613i 0.734191i −1.00000 0
274.3 0.571993i 1.00000i 1.67282 0 0.571993 1.42801i 2.10083i −1.00000 0
274.4 0.571993i 1.00000i 1.67282 0 0.571993 1.42801i 2.10083i −1.00000 0
274.5 2.08613i 1.00000i −2.35194 0 −2.08613 4.08613i 0.734191i −1.00000 0
274.6 2.51414i 1.00000i −4.32088 0 2.51414 0.514137i 5.83502i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.c.j 6
3.b odd 2 1 2925.2.c.x 6
5.b even 2 1 inner 975.2.c.j 6
5.c odd 4 1 975.2.a.n 3
5.c odd 4 1 975.2.a.p yes 3
15.d odd 2 1 2925.2.c.x 6
15.e even 4 1 2925.2.a.bg 3
15.e even 4 1 2925.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.a.n 3 5.c odd 4 1
975.2.a.p yes 3 5.c odd 4 1
975.2.c.j 6 1.a even 1 1 trivial
975.2.c.j 6 5.b even 2 1 inner
2925.2.a.bg 3 15.e even 4 1
2925.2.a.bi 3 15.e even 4 1
2925.2.c.x 6 3.b odd 2 1
2925.2.c.x 6 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 19T_{7}^{4} + 39T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 11T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 31 T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + 39 T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 11 T - 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 59 T^{4} + 403 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 12 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 80 T^{4} + 736 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{3} - 13 T^{2} + 19 T + 129)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 7 T^{2} - 31 T + 223)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 76 T^{4} + 1728 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$41$ \( (T^{3} - 24 T - 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 168 T^{4} + 7056 T^{2} + \cdots + 26896 \) Copy content Toggle raw display
$47$ \( T^{6} + 299 T^{4} + 28351 T^{2} + \cdots + 826281 \) Copy content Toggle raw display
$53$ \( T^{6} + 75 T^{4} + 1467 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( (T^{3} + 3 T^{2} - 33 T - 117)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 5 T^{2} - 37 T - 113)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 327 T^{4} + 30459 T^{2} + \cdots + 744769 \) Copy content Toggle raw display
$71$ \( (T^{3} + 18 T^{2} + 84 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 244 T^{4} + 17328 T^{2} + \cdots + 318096 \) Copy content Toggle raw display
$79$ \( (T^{3} + 4 T^{2} - 76 T + 164)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 323 T^{4} + 14743 T^{2} + \cdots + 114921 \) Copy content Toggle raw display
$89$ \( (T^{3} - 16 T^{2} + 64 T - 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 316 T^{4} + 21168 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
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