Properties

Label 975.2.c.h
Level $975$
Weight $2$
Character orbit 975.c
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 −2.41421 2.82843i 4.41421i −1.00000 0
274.2 0.414214i 1.00000i 1.82843 0 0.414214 2.82843i 1.58579i −1.00000 0
274.3 0.414214i 1.00000i 1.82843 0 0.414214 2.82843i 1.58579i −1.00000 0
274.4 2.41421i 1.00000i −3.82843 0 −2.41421 2.82843i 4.41421i −1.00000 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
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.h 4
3.b odd 2 1 2925.2.c.u 4
5.b even 2 1 inner 975.2.c.h 4
5.c odd 4 1 39.2.a.b 2
5.c odd 4 1 975.2.a.l 2
15.d odd 2 1 2925.2.c.u 4
15.e even 4 1 117.2.a.c 2
15.e even 4 1 2925.2.a.v 2
20.e even 4 1 624.2.a.k 2
35.f even 4 1 1911.2.a.h 2
40.i odd 4 1 2496.2.a.bf 2
40.k even 4 1 2496.2.a.bi 2
45.k odd 12 2 1053.2.e.m 4
45.l even 12 2 1053.2.e.e 4
55.e even 4 1 4719.2.a.p 2
60.l odd 4 1 1872.2.a.w 2
65.f even 4 1 507.2.b.e 4
65.h odd 4 1 507.2.a.h 2
65.k even 4 1 507.2.b.e 4
65.o even 12 2 507.2.j.f 8
65.q odd 12 2 507.2.e.h 4
65.r odd 12 2 507.2.e.d 4
65.t even 12 2 507.2.j.f 8
105.k odd 4 1 5733.2.a.u 2
120.q odd 4 1 7488.2.a.co 2
120.w even 4 1 7488.2.a.cl 2
195.j odd 4 1 1521.2.b.j 4
195.s even 4 1 1521.2.a.f 2
195.u odd 4 1 1521.2.b.j 4
260.p even 4 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 5.c odd 4 1
117.2.a.c 2 15.e even 4 1
507.2.a.h 2 65.h odd 4 1
507.2.b.e 4 65.f even 4 1
507.2.b.e 4 65.k even 4 1
507.2.e.d 4 65.r odd 12 2
507.2.e.h 4 65.q odd 12 2
507.2.j.f 8 65.o even 12 2
507.2.j.f 8 65.t even 12 2
624.2.a.k 2 20.e even 4 1
975.2.a.l 2 5.c odd 4 1
975.2.c.h 4 1.a even 1 1 trivial
975.2.c.h 4 5.b even 2 1 inner
1053.2.e.e 4 45.l even 12 2
1053.2.e.m 4 45.k odd 12 2
1521.2.a.f 2 195.s even 4 1
1521.2.b.j 4 195.j odd 4 1
1521.2.b.j 4 195.u odd 4 1
1872.2.a.w 2 60.l odd 4 1
1911.2.a.h 2 35.f even 4 1
2496.2.a.bf 2 40.i odd 4 1
2496.2.a.bi 2 40.k even 4 1
2925.2.a.v 2 15.e even 4 1
2925.2.c.u 4 3.b odd 2 1
2925.2.c.u 4 15.d odd 2 1
4719.2.a.p 2 55.e even 4 1
5733.2.a.u 2 105.k odd 4 1
7488.2.a.cl 2 120.w even 4 1
7488.2.a.co 2 120.q odd 4 1
8112.2.a.bm 2 260.p even 4 1

Hecke kernels

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

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T^{2} + 24 T + 136)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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