Properties

Label 975.2.h.f
Level $975$
Weight $2$
Character orbit 975.h
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.73205 1.00000i 1.00000 0 1.73205i 3.46410 1.73205 −1.00000 0
649.2 −1.73205 1.00000i 1.00000 0 1.73205i 3.46410 1.73205 −1.00000 0
649.3 1.73205 1.00000i 1.00000 0 1.73205i −3.46410 −1.73205 −1.00000 0
649.4 1.73205 1.00000i 1.00000 0 1.73205i −3.46410 −1.73205 −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
13.b even 2 1 inner
65.d 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.h.f 4
5.b even 2 1 inner 975.2.h.f 4
5.c odd 4 1 39.2.b.a 2
5.c odd 4 1 975.2.b.d 2
13.b even 2 1 inner 975.2.h.f 4
15.e even 4 1 117.2.b.a 2
20.e even 4 1 624.2.c.e 2
35.f even 4 1 1911.2.c.d 2
40.i odd 4 1 2496.2.c.k 2
40.k even 4 1 2496.2.c.d 2
60.l odd 4 1 1872.2.c.e 2
65.d even 2 1 inner 975.2.h.f 4
65.f even 4 1 507.2.a.f 2
65.h odd 4 1 39.2.b.a 2
65.h odd 4 1 975.2.b.d 2
65.k even 4 1 507.2.a.f 2
65.o even 12 2 507.2.e.e 4
65.q odd 12 1 507.2.j.a 2
65.q odd 12 1 507.2.j.c 2
65.r odd 12 1 507.2.j.a 2
65.r odd 12 1 507.2.j.c 2
65.t even 12 2 507.2.e.e 4
195.j odd 4 1 1521.2.a.l 2
195.s even 4 1 117.2.b.a 2
195.u odd 4 1 1521.2.a.l 2
260.l odd 4 1 8112.2.a.bv 2
260.p even 4 1 624.2.c.e 2
260.s odd 4 1 8112.2.a.bv 2
455.s even 4 1 1911.2.c.d 2
520.bc even 4 1 2496.2.c.d 2
520.bg odd 4 1 2496.2.c.k 2
780.w odd 4 1 1872.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 5.c odd 4 1
39.2.b.a 2 65.h odd 4 1
117.2.b.a 2 15.e even 4 1
117.2.b.a 2 195.s even 4 1
507.2.a.f 2 65.f even 4 1
507.2.a.f 2 65.k even 4 1
507.2.e.e 4 65.o even 12 2
507.2.e.e 4 65.t even 12 2
507.2.j.a 2 65.q odd 12 1
507.2.j.a 2 65.r odd 12 1
507.2.j.c 2 65.q odd 12 1
507.2.j.c 2 65.r odd 12 1
624.2.c.e 2 20.e even 4 1
624.2.c.e 2 260.p even 4 1
975.2.b.d 2 5.c odd 4 1
975.2.b.d 2 65.h odd 4 1
975.2.h.f 4 1.a even 1 1 trivial
975.2.h.f 4 5.b even 2 1 inner
975.2.h.f 4 13.b even 2 1 inner
975.2.h.f 4 65.d even 2 1 inner
1521.2.a.l 2 195.j odd 4 1
1521.2.a.l 2 195.u odd 4 1
1872.2.c.e 2 60.l odd 4 1
1872.2.c.e 2 780.w odd 4 1
1911.2.c.d 2 35.f even 4 1
1911.2.c.d 2 455.s even 4 1
2496.2.c.d 2 40.k even 4 1
2496.2.c.d 2 520.bc even 4 1
2496.2.c.k 2 40.i odd 4 1
2496.2.c.k 2 520.bg odd 4 1
8112.2.a.bv 2 260.l odd 4 1
8112.2.a.bv 2 260.s odd 4 1

Hecke kernels

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

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
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