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: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{2} - 3 \)
\( T_{7}^{2} - 12 \)

Hecke characteristic polynomials

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