Properties

Label 975.2.c.i
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: 6.0.399424.1
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 195)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{4} - 3) q^{4} + \beta_1 q^{6} + \beta_{5} q^{7} + (3 \beta_{3} + 2 \beta_{2}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{4} - 3) q^{4} + \beta_1 q^{6} + \beta_{5} q^{7} + (3 \beta_{3} + 2 \beta_{2}) q^{8} - q^{9} + \beta_{4} q^{11} + (\beta_{5} + 3 \beta_{2}) q^{12} + \beta_{2} q^{13} + ( - 2 \beta_1 + 2) q^{14} + ( - \beta_{4} - 2 \beta_1 + 9) q^{16} + ( - \beta_{5} + 2 \beta_{3}) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_1 - 2) q^{19} - \beta_{4} q^{21} + (2 \beta_{3} + 2 \beta_{2}) q^{22} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{23} + ( - 3 \beta_1 + 2) q^{24} - \beta_1 q^{26} + \beta_{2} q^{27} + ( - 2 \beta_{3} - 10 \beta_{2}) q^{28} - 6 q^{29} + ( - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{5} - 5 \beta_{3} - 8 \beta_{2}) q^{32} + \beta_{5} q^{33} + ( - 2 \beta_{4} + 2 \beta_1 + 8) q^{34} + ( - \beta_{4} + 3) q^{36} + (\beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{37} + ( - 2 \beta_{5} + 2 \beta_{3} - 10 \beta_{2}) q^{38} + q^{39} + (\beta_{4} - 2 \beta_1) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{42} + 4 \beta_{3} q^{43} + ( - 2 \beta_1 + 10) q^{44} + (2 \beta_{4} - 8) q^{46} + (2 \beta_{3} + 6 \beta_{2}) q^{47} + ( - \beta_{5} - 2 \beta_{3} - 9 \beta_{2}) q^{48} + ( - 3 \beta_{4} + 2 \beta_1 - 3) q^{49} + (\beta_{4} - 2 \beta_1) q^{51} + ( - \beta_{5} - 3 \beta_{2}) q^{52} + (\beta_{5} - 2 \beta_{3} + 4 \beta_{2}) q^{53} - \beta_1 q^{54} + (2 \beta_{4} + 6 \beta_1 - 6) q^{56} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{57} + 6 \beta_{3} q^{58} + (2 \beta_{4} - 2 \beta_1 + 2) q^{59} + ( - 3 \beta_{4} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{5} - 2 \beta_{3} - 10 \beta_{2}) q^{62} - \beta_{5} q^{63} + (3 \beta_{4} + 8 \beta_1 - 11) q^{64} + ( - 2 \beta_1 + 2) q^{66} + ( - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{67} + ( - 8 \beta_{3} + 6 \beta_{2}) q^{68} + ( - \beta_{4} + 2 \beta_1 - 2) q^{69} + (\beta_{4} - 4) q^{71} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{72} + (4 \beta_{3} - 2 \beta_{2}) q^{73} + ( - 2 \beta_{4} + 2 \beta_1 + 12) q^{74} + ( - 2 \beta_{4} + 10 \beta_1 + 2) q^{76} + (3 \beta_{5} - 2 \beta_{3} - 10 \beta_{2}) q^{77} - \beta_{3} q^{78} + (\beta_{4} + 2 \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{5} + 2 \beta_{3} - 8 \beta_{2}) q^{82} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{83} + (2 \beta_1 - 10) q^{84} + ( - 4 \beta_{4} + 20) q^{86} + 6 \beta_{2} q^{87} + ( - 2 \beta_{5} - 6 \beta_{3} - 6 \beta_{2}) q^{88} + (\beta_{4} - 2 \beta_1 - 4) q^{89} + \beta_{4} q^{91} + (2 \beta_{5} + 8 \beta_{3}) q^{92} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{93} + ( - 2 \beta_{4} - 6 \beta_1 + 10) q^{94} + (2 \beta_{4} + 5 \beta_1 - 8) q^{96} + ( - \beta_{5} + 2 \beta_{3} + 8 \beta_{2}) q^{97} + (2 \beta_{5} - 3 \beta_{3} + 4 \beta_{2}) q^{98} - \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} - 3\nu^{4} + 8\nu^{3} - 11\nu^{2} + 8\nu - 20 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - 11\beta_{2} - 4\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + 3\beta_{4} - 2\beta_{3} + 8\beta_{2} - 7\beta _1 + 7 ) / 4 \) 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
0.264658 + 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
1.40680 0.144584i
−0.671462 1.24464i
0.264658 1.38923i
2.77846i 1.00000i −5.71982 0 −2.77846 2.71982i 10.3354i −1.00000 0
274.2 2.48929i 1.00000i −4.19656 0 2.48929 1.19656i 5.46787i −1.00000 0
274.3 0.289169i 1.00000i 1.91638 0 0.289169 4.91638i 1.13249i −1.00000 0
274.4 0.289169i 1.00000i 1.91638 0 0.289169 4.91638i 1.13249i −1.00000 0
274.5 2.48929i 1.00000i −4.19656 0 2.48929 1.19656i 5.46787i −1.00000 0
274.6 2.77846i 1.00000i −5.71982 0 −2.77846 2.71982i 10.3354i −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.i 6
3.b odd 2 1 2925.2.c.w 6
5.b even 2 1 inner 975.2.c.i 6
5.c odd 4 1 195.2.a.e 3
5.c odd 4 1 975.2.a.o 3
15.d odd 2 1 2925.2.c.w 6
15.e even 4 1 585.2.a.n 3
15.e even 4 1 2925.2.a.bh 3
20.e even 4 1 3120.2.a.bj 3
35.f even 4 1 9555.2.a.bq 3
60.l odd 4 1 9360.2.a.dd 3
65.h odd 4 1 2535.2.a.bc 3
195.s even 4 1 7605.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 5.c odd 4 1
585.2.a.n 3 15.e even 4 1
975.2.a.o 3 5.c odd 4 1
975.2.c.i 6 1.a even 1 1 trivial
975.2.c.i 6 5.b even 2 1 inner
2535.2.a.bc 3 65.h odd 4 1
2925.2.a.bh 3 15.e even 4 1
2925.2.c.w 6 3.b odd 2 1
2925.2.c.w 6 15.d odd 2 1
3120.2.a.bj 3 20.e even 4 1
7605.2.a.bx 3 195.s even 4 1
9360.2.a.dd 3 60.l odd 4 1
9555.2.a.bq 3 35.f even 4 1

Hecke kernels

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

\( T_{2}^{6} + 14T_{2}^{4} + 49T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 33T_{7}^{4} + 224T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 16T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 14 T^{4} + 49 T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 33 T^{4} + 224 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 65 T^{4} + 1176 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 81 T^{4} + 2048 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T + 6)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 169 T^{4} + 8216 T^{2} + \cdots + 99856 \) Copy content Toggle raw display
$41$ \( (T^{3} - T^{2} - 32 T + 76)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 224 T^{4} + 12544 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{6} + 164 T^{4} + 4096 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{6} + 105 T^{4} + 152 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 48 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 9 T^{2} - 112 T + 844)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( (T^{3} + 11 T^{2} + 24 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 236 T^{4} + 14128 T^{2} + \cdots + 118336 \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 48 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 160 T^{4} + 4352 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$89$ \( (T^{3} + 11 T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 273 T^{4} + 18776 T^{2} + \cdots + 59536 \) Copy content Toggle raw display
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