Properties

Label 8100.2.d.c
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: \(x^{2} + 1\)
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.

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

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

Hecke kernels

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

\( T_{7}^{2} + 1 \)
\( T_{11} + 3 \)
\( T_{29} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 9 + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 3 + T )^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 49 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 121 + T^{2} \)
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