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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + 3 i q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + 3 i q^{8} - q^{9} + 4 q^{11} + i q^{12} + i q^{13} - q^{16} - 2 i q^{17} - i q^{18} + 4 q^{19} + 4 i q^{22} + 8 i q^{23} - 3 q^{24} - q^{26} - i q^{27} + 2 q^{29} - 8 q^{31} + 5 i q^{32} + 4 i q^{33} + 2 q^{34} - q^{36} - 6 i q^{37} + 4 i q^{38} - q^{39} - 6 q^{41} - 4 i q^{43} + 4 q^{44} - 8 q^{46} + 8 i q^{47} - i q^{48} + 7 q^{49} + 2 q^{51} + i q^{52} + 6 i q^{53} + q^{54} + 4 i q^{57} + 2 i q^{58} + 12 q^{59} - 2 q^{61} - 8 i q^{62} - 7 q^{64} - 4 q^{66} + 4 i q^{67} - 2 i q^{68} - 8 q^{69} - 3 i q^{72} - 6 i q^{73} + 6 q^{74} + 4 q^{76} - i q^{78} - 16 q^{79} + q^{81} - 6 i q^{82} - 4 i q^{83} + 4 q^{86} + 2 i q^{87} + 12 i q^{88} - 10 q^{89} + 8 i q^{92} - 8 i q^{93} - 8 q^{94} - 5 q^{96} - 18 i q^{97} + 7 i q^{98} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{16} + 8 q^{19} - 6 q^{24} - 2 q^{26} + 4 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 2 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{69} + 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 16 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 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
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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

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

Hecke characteristic polynomials

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