Properties

Label 2205.2.d.b
Level $2205$
Weight $2$
Character orbit 2205.d
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -2 q^{4} + ( -2 + i ) q^{5} +O(q^{10})\) \( q + 2 i q^{2} -2 q^{4} + ( -2 + i ) q^{5} + ( -2 - 4 i ) q^{10} + 3 q^{11} -i q^{13} -4 q^{16} -7 i q^{17} + ( 4 - 2 i ) q^{20} + 6 i q^{22} -6 i q^{23} + ( 3 - 4 i ) q^{25} + 2 q^{26} -5 q^{29} -2 q^{31} -8 i q^{32} + 14 q^{34} -2 i q^{37} + 2 q^{41} -4 i q^{43} -6 q^{44} + 12 q^{46} + 3 i q^{47} + ( 8 + 6 i ) q^{50} + 2 i q^{52} -6 i q^{53} + ( -6 + 3 i ) q^{55} -10 i q^{58} -10 q^{59} + 8 q^{61} -4 i q^{62} + 8 q^{64} + ( 1 + 2 i ) q^{65} -2 i q^{67} + 14 i q^{68} + 8 q^{71} -6 i q^{73} + 4 q^{74} + 5 q^{79} + ( 8 - 4 i ) q^{80} + 4 i q^{82} -4 i q^{83} + ( 7 + 14 i ) q^{85} + 8 q^{86} + 12 i q^{92} -6 q^{94} + 7 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{5} + O(q^{10}) \) \( 2 q - 4 q^{4} - 4 q^{5} - 4 q^{10} + 6 q^{11} - 8 q^{16} + 8 q^{20} + 6 q^{25} + 4 q^{26} - 10 q^{29} - 4 q^{31} + 28 q^{34} + 4 q^{41} - 12 q^{44} + 24 q^{46} + 16 q^{50} - 12 q^{55} - 20 q^{59} + 16 q^{61} + 16 q^{64} + 2 q^{65} + 16 q^{71} + 8 q^{74} + 10 q^{79} + 16 q^{80} + 14 q^{85} + 16 q^{86} - 12 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
2.00000i 0 −2.00000 −2.00000 1.00000i 0 0 0 0 −2.00000 + 4.00000i
1324.2 2.00000i 0 −2.00000 −2.00000 + 1.00000i 0 0 0 0 −2.00000 4.00000i
\(n\): 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 2205.2.d.b 2
3.b odd 2 1 245.2.b.a 2
5.b even 2 1 inner 2205.2.d.b 2
7.b odd 2 1 315.2.d.a 2
15.d odd 2 1 245.2.b.a 2
15.e even 4 1 1225.2.a.a 1
15.e even 4 1 1225.2.a.i 1
21.c even 2 1 35.2.b.a 2
21.g even 6 2 245.2.j.e 4
21.h odd 6 2 245.2.j.d 4
28.d even 2 1 5040.2.t.p 2
35.c odd 2 1 315.2.d.a 2
35.f even 4 1 1575.2.a.a 1
35.f even 4 1 1575.2.a.k 1
84.h odd 2 1 560.2.g.b 2
105.g even 2 1 35.2.b.a 2
105.k odd 4 1 175.2.a.a 1
105.k odd 4 1 175.2.a.c 1
105.o odd 6 2 245.2.j.d 4
105.p even 6 2 245.2.j.e 4
140.c even 2 1 5040.2.t.p 2
168.e odd 2 1 2240.2.g.g 2
168.i even 2 1 2240.2.g.h 2
420.o odd 2 1 560.2.g.b 2
420.w even 4 1 2800.2.a.l 1
420.w even 4 1 2800.2.a.w 1
840.b odd 2 1 2240.2.g.g 2
840.u even 2 1 2240.2.g.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 21.c even 2 1
35.2.b.a 2 105.g even 2 1
175.2.a.a 1 105.k odd 4 1
175.2.a.c 1 105.k odd 4 1
245.2.b.a 2 3.b odd 2 1
245.2.b.a 2 15.d odd 2 1
245.2.j.d 4 21.h odd 6 2
245.2.j.d 4 105.o odd 6 2
245.2.j.e 4 21.g even 6 2
245.2.j.e 4 105.p even 6 2
315.2.d.a 2 7.b odd 2 1
315.2.d.a 2 35.c odd 2 1
560.2.g.b 2 84.h odd 2 1
560.2.g.b 2 420.o odd 2 1
1225.2.a.a 1 15.e even 4 1
1225.2.a.i 1 15.e even 4 1
1575.2.a.a 1 35.f even 4 1
1575.2.a.k 1 35.f even 4 1
2205.2.d.b 2 1.a even 1 1 trivial
2205.2.d.b 2 5.b even 2 1 inner
2240.2.g.g 2 168.e odd 2 1
2240.2.g.g 2 840.b odd 2 1
2240.2.g.h 2 168.i even 2 1
2240.2.g.h 2 840.u even 2 1
2800.2.a.l 1 420.w even 4 1
2800.2.a.w 1 420.w even 4 1
5040.2.t.p 2 28.d even 2 1
5040.2.t.p 2 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{2} + 4 \)
\( T_{11} - 3 \)
\( T_{13}^{2} + 1 \)
\( T_{19} \)
\( T_{29} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 49 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 49 + T^{2} \)
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