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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{4} + 6 T_{2}^{2} + 1 \)
\( T_{7}^{2} + 8 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

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