Properties

Label 975.2.c.e
Level $975$
Weight $2$
Character orbit 975.c
Analytic conductor $7.785$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 0 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 −1.00000 0 3.00000i −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.e 2
3.b odd 2 1 2925.2.c.f 2
5.b even 2 1 inner 975.2.c.e 2
5.c odd 4 1 195.2.a.a 1
5.c odd 4 1 975.2.a.i 1
15.d odd 2 1 2925.2.c.f 2
15.e even 4 1 585.2.a.g 1
15.e even 4 1 2925.2.a.d 1
20.e even 4 1 3120.2.a.k 1
35.f even 4 1 9555.2.a.b 1
60.l odd 4 1 9360.2.a.o 1
65.h odd 4 1 2535.2.a.k 1
195.s even 4 1 7605.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 5.c odd 4 1
585.2.a.g 1 15.e even 4 1
975.2.a.i 1 5.c odd 4 1
975.2.c.e 2 1.a even 1 1 trivial
975.2.c.e 2 5.b even 2 1 inner
2535.2.a.k 1 65.h odd 4 1
2925.2.a.d 1 15.e even 4 1
2925.2.c.f 2 3.b odd 2 1
2925.2.c.f 2 15.d odd 2 1
3120.2.a.k 1 20.e even 4 1
7605.2.a.h 1 195.s even 4 1
9360.2.a.o 1 60.l odd 4 1
9555.2.a.b 1 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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 324 + T^{2} \)
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