Properties

Label 525.2.d.b
Level $525$
Weight $2$
Character orbit 525.d
Analytic conductor $4.192$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
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 105)
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.

\(f(q)\) \(=\) \( q + i q^{2} -i q^{3} + q^{4} + q^{6} + i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -i q^{3} + q^{4} + q^{6} + i q^{7} + 3 i q^{8} - q^{9} -i q^{12} + 6 i q^{13} - q^{14} - q^{16} + 2 i q^{17} -i q^{18} + 8 q^{19} + q^{21} -8 i q^{23} + 3 q^{24} -6 q^{26} + i q^{27} + i q^{28} + 2 q^{29} + 4 q^{31} + 5 i q^{32} -2 q^{34} - q^{36} -2 i q^{37} + 8 i q^{38} + 6 q^{39} -6 q^{41} + i q^{42} -4 i q^{43} + 8 q^{46} + 8 i q^{47} + i q^{48} - q^{49} + 2 q^{51} + 6 i q^{52} -10 i q^{53} - q^{54} -3 q^{56} -8 i q^{57} + 2 i q^{58} -4 q^{59} -2 q^{61} + 4 i q^{62} -i q^{63} -7 q^{64} + 4 i q^{67} + 2 i q^{68} -8 q^{69} -12 q^{71} -3 i q^{72} + 2 i q^{73} + 2 q^{74} + 8 q^{76} + 6 i q^{78} -8 q^{79} + q^{81} -6 i q^{82} + 4 i q^{83} + q^{84} + 4 q^{86} -2 i q^{87} + 6 q^{89} -6 q^{91} -8 i q^{92} -4 i q^{93} -8 q^{94} + 5 q^{96} -18 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{6} - 2q^{9} - 2q^{14} - 2q^{16} + 16q^{19} + 2q^{21} + 6q^{24} - 12q^{26} + 4q^{29} + 8q^{31} - 4q^{34} - 2q^{36} + 12q^{39} - 12q^{41} + 16q^{46} - 2q^{49} + 4q^{51} - 2q^{54} - 6q^{56} - 8q^{59} - 4q^{61} - 14q^{64} - 16q^{69} - 24q^{71} + 4q^{74} + 16q^{76} - 16q^{79} + 2q^{81} + 2q^{84} + 8q^{86} + 12q^{89} - 12q^{91} - 16q^{94} + 10q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 1.00000i 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 1.00000 1.00000i 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 525.2.d.b 2
3.b odd 2 1 1575.2.d.b 2
5.b even 2 1 inner 525.2.d.b 2
5.c odd 4 1 105.2.a.a 1
5.c odd 4 1 525.2.a.a 1
15.d odd 2 1 1575.2.d.b 2
15.e even 4 1 315.2.a.a 1
15.e even 4 1 1575.2.a.h 1
20.e even 4 1 1680.2.a.f 1
20.e even 4 1 8400.2.a.co 1
35.f even 4 1 735.2.a.f 1
35.f even 4 1 3675.2.a.f 1
35.k even 12 2 735.2.i.b 2
35.l odd 12 2 735.2.i.a 2
40.i odd 4 1 6720.2.a.p 1
40.k even 4 1 6720.2.a.bk 1
60.l odd 4 1 5040.2.a.d 1
105.k odd 4 1 2205.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 5.c odd 4 1
315.2.a.a 1 15.e even 4 1
525.2.a.a 1 5.c odd 4 1
525.2.d.b 2 1.a even 1 1 trivial
525.2.d.b 2 5.b even 2 1 inner
735.2.a.f 1 35.f even 4 1
735.2.i.a 2 35.l odd 12 2
735.2.i.b 2 35.k even 12 2
1575.2.a.h 1 15.e even 4 1
1575.2.d.b 2 3.b odd 2 1
1575.2.d.b 2 15.d odd 2 1
1680.2.a.f 1 20.e even 4 1
2205.2.a.b 1 105.k odd 4 1
3675.2.a.f 1 35.f even 4 1
5040.2.a.d 1 60.l odd 4 1
6720.2.a.p 1 40.i odd 4 1
6720.2.a.bk 1 40.k even 4 1
8400.2.a.co 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 324 + T^{2} \)
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