Properties

Label 1575.2.d.a
Level $1575$
Weight $2$
Character orbit 1575.d
Analytic conductor $12.576$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
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 21)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} - 4 q^{11} + 2 i q^{13} + q^{14} - q^{16} + 6 i q^{17} - 4 q^{19} - 4 i q^{22} - 2 q^{26} - i q^{28} - 2 q^{29} + 5 i q^{32} - 6 q^{34} + 6 i q^{37} - 4 i q^{38} - 2 q^{41} + 4 i q^{43} - 4 q^{44} - q^{49} + 2 i q^{52} + 6 i q^{53} + 3 q^{56} - 2 i q^{58} + 12 q^{59} - 2 q^{61} - 7 q^{64} + 4 i q^{67} + 6 i q^{68} + 6 i q^{73} - 6 q^{74} - 4 q^{76} + 4 i q^{77} + 16 q^{79} - 2 i q^{82} - 12 i q^{83} - 4 q^{86} - 12 i q^{88} - 14 q^{89} + 2 q^{91} + 18 i q^{97} - i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 8 q^{11} + 2 q^{14} - 2 q^{16} - 8 q^{19} - 4 q^{26} - 4 q^{29} - 12 q^{34} - 4 q^{41} - 8 q^{44} - 2 q^{49} + 6 q^{56} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 12 q^{74} - 8 q^{76} + 32 q^{79} - 8 q^{86} - 28 q^{89} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 1.00000i 3.00000i 0 0
1324.2 1.00000i 0 1.00000 0 0 1.00000i 3.00000i 0 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 1575.2.d.a 2
3.b odd 2 1 525.2.d.a 2
5.b even 2 1 inner 1575.2.d.a 2
5.c odd 4 1 63.2.a.a 1
5.c odd 4 1 1575.2.a.c 1
15.d odd 2 1 525.2.d.a 2
15.e even 4 1 21.2.a.a 1
15.e even 4 1 525.2.a.d 1
20.e even 4 1 1008.2.a.l 1
35.f even 4 1 441.2.a.f 1
35.k even 12 2 441.2.e.b 2
35.l odd 12 2 441.2.e.a 2
40.i odd 4 1 4032.2.a.h 1
40.k even 4 1 4032.2.a.k 1
45.k odd 12 2 567.2.f.b 2
45.l even 12 2 567.2.f.g 2
55.e even 4 1 7623.2.a.g 1
60.l odd 4 1 336.2.a.a 1
60.l odd 4 1 8400.2.a.bn 1
105.k odd 4 1 147.2.a.a 1
105.k odd 4 1 3675.2.a.n 1
105.w odd 12 2 147.2.e.c 2
105.x even 12 2 147.2.e.b 2
120.q odd 4 1 1344.2.a.s 1
120.w even 4 1 1344.2.a.g 1
140.j odd 4 1 7056.2.a.p 1
165.l odd 4 1 2541.2.a.j 1
195.s even 4 1 3549.2.a.c 1
240.z odd 4 1 5376.2.c.l 2
240.bb even 4 1 5376.2.c.r 2
240.bd odd 4 1 5376.2.c.l 2
240.bf even 4 1 5376.2.c.r 2
255.o even 4 1 6069.2.a.b 1
285.j odd 4 1 7581.2.a.d 1
420.w even 4 1 2352.2.a.v 1
420.bp odd 12 2 2352.2.q.x 2
420.br even 12 2 2352.2.q.e 2
840.bm even 4 1 9408.2.a.m 1
840.bp odd 4 1 9408.2.a.bv 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 15.e even 4 1
63.2.a.a 1 5.c odd 4 1
147.2.a.a 1 105.k odd 4 1
147.2.e.b 2 105.x even 12 2
147.2.e.c 2 105.w odd 12 2
336.2.a.a 1 60.l odd 4 1
441.2.a.f 1 35.f even 4 1
441.2.e.a 2 35.l odd 12 2
441.2.e.b 2 35.k even 12 2
525.2.a.d 1 15.e even 4 1
525.2.d.a 2 3.b odd 2 1
525.2.d.a 2 15.d odd 2 1
567.2.f.b 2 45.k odd 12 2
567.2.f.g 2 45.l even 12 2
1008.2.a.l 1 20.e even 4 1
1344.2.a.g 1 120.w even 4 1
1344.2.a.s 1 120.q odd 4 1
1575.2.a.c 1 5.c odd 4 1
1575.2.d.a 2 1.a even 1 1 trivial
1575.2.d.a 2 5.b even 2 1 inner
2352.2.a.v 1 420.w even 4 1
2352.2.q.e 2 420.br even 12 2
2352.2.q.x 2 420.bp odd 12 2
2541.2.a.j 1 165.l odd 4 1
3549.2.a.c 1 195.s even 4 1
3675.2.a.n 1 105.k odd 4 1
4032.2.a.h 1 40.i odd 4 1
4032.2.a.k 1 40.k even 4 1
5376.2.c.l 2 240.z odd 4 1
5376.2.c.l 2 240.bd odd 4 1
5376.2.c.r 2 240.bb even 4 1
5376.2.c.r 2 240.bf even 4 1
6069.2.a.b 1 255.o even 4 1
7056.2.a.p 1 140.j odd 4 1
7581.2.a.d 1 285.j odd 4 1
7623.2.a.g 1 55.e even 4 1
8400.2.a.bn 1 60.l odd 4 1
9408.2.a.m 1 840.bm even 4 1
9408.2.a.bv 1 840.bp odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{2} + 1 \) 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
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) 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
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 324 \) Copy content Toggle raw display
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