Properties

Label 8100.2.d.h
Level $8100$
Weight $2$
Character orbit 8100.d
Analytic conductor $64.679$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + i q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{7} + 3 q^{11} - i q^{13} - 6 i q^{17} + 4 q^{19} - 3 i q^{23} - 3 q^{29} + 5 q^{31} - 2 i q^{37} + 3 q^{41} - i q^{43} + 9 i q^{47} + 6 q^{49} - 6 i q^{53} + 3 q^{59} - 13 q^{61} + 7 i q^{67} - 12 q^{71} - 10 i q^{73} + 3 i q^{77} - 11 q^{79} - 9 i q^{83} - 6 q^{89} + q^{91} - 11 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{11} + 8 q^{19} - 6 q^{29} + 10 q^{31} + 6 q^{41} + 12 q^{49} + 6 q^{59} - 26 q^{61} - 24 q^{71} - 22 q^{79} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
649.2 0 0 0 0 0 1.00000i 0 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 8100.2.d.h 2
3.b odd 2 1 8100.2.d.c 2
5.b even 2 1 inner 8100.2.d.h 2
5.c odd 4 1 324.2.a.c 1
5.c odd 4 1 8100.2.a.j 1
9.c even 3 2 900.2.s.b 4
9.d odd 6 2 2700.2.s.b 4
15.d odd 2 1 8100.2.d.c 2
15.e even 4 1 324.2.a.a 1
15.e even 4 1 8100.2.a.g 1
20.e even 4 1 1296.2.a.k 1
40.i odd 4 1 5184.2.a.e 1
40.k even 4 1 5184.2.a.f 1
45.h odd 6 2 2700.2.s.b 4
45.j even 6 2 900.2.s.b 4
45.k odd 12 2 36.2.e.a 2
45.k odd 12 2 900.2.i.b 2
45.l even 12 2 108.2.e.a 2
45.l even 12 2 2700.2.i.b 2
60.l odd 4 1 1296.2.a.b 1
120.q odd 4 1 5184.2.a.bb 1
120.w even 4 1 5184.2.a.ba 1
180.v odd 12 2 432.2.i.c 2
180.x even 12 2 144.2.i.a 2
315.bs even 12 2 1764.2.i.c 2
315.bt odd 12 2 1764.2.i.a 2
315.bu odd 12 2 5292.2.i.a 2
315.bv even 12 2 5292.2.i.c 2
315.bw odd 12 2 5292.2.l.c 2
315.bx even 12 2 5292.2.l.a 2
315.cb even 12 2 1764.2.j.b 2
315.cf odd 12 2 5292.2.j.a 2
315.cg even 12 2 1764.2.l.a 2
315.ch odd 12 2 1764.2.l.c 2
360.bo even 12 2 576.2.i.e 2
360.br even 12 2 1728.2.i.d 2
360.bt odd 12 2 1728.2.i.c 2
360.bu odd 12 2 576.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 45.k odd 12 2
108.2.e.a 2 45.l even 12 2
144.2.i.a 2 180.x even 12 2
324.2.a.a 1 15.e even 4 1
324.2.a.c 1 5.c odd 4 1
432.2.i.c 2 180.v odd 12 2
576.2.i.e 2 360.bo even 12 2
576.2.i.f 2 360.bu odd 12 2
900.2.i.b 2 45.k odd 12 2
900.2.s.b 4 9.c even 3 2
900.2.s.b 4 45.j even 6 2
1296.2.a.b 1 60.l odd 4 1
1296.2.a.k 1 20.e even 4 1
1728.2.i.c 2 360.bt odd 12 2
1728.2.i.d 2 360.br even 12 2
1764.2.i.a 2 315.bt odd 12 2
1764.2.i.c 2 315.bs even 12 2
1764.2.j.b 2 315.cb even 12 2
1764.2.l.a 2 315.cg even 12 2
1764.2.l.c 2 315.ch odd 12 2
2700.2.i.b 2 45.l even 12 2
2700.2.s.b 4 9.d odd 6 2
2700.2.s.b 4 45.h odd 6 2
5184.2.a.e 1 40.i odd 4 1
5184.2.a.f 1 40.k even 4 1
5184.2.a.ba 1 120.w even 4 1
5184.2.a.bb 1 120.q odd 4 1
5292.2.i.a 2 315.bu odd 12 2
5292.2.i.c 2 315.bv even 12 2
5292.2.j.a 2 315.cf odd 12 2
5292.2.l.a 2 315.bx even 12 2
5292.2.l.c 2 315.bw odd 12 2
8100.2.a.g 1 15.e even 4 1
8100.2.a.j 1 5.c odd 4 1
8100.2.d.c 2 3.b odd 2 1
8100.2.d.c 2 15.d odd 2 1
8100.2.d.h 2 1.a even 1 1 trivial
8100.2.d.h 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8100, [\chi])\):

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{29} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) 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
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 49 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 121 \) Copy content Toggle raw display
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