Properties

Label 900.2.k.c
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.18653618192\)
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 20)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 2) q^{8} + ( - i + 1) q^{13} - 4 q^{16} + (3 i + 3) q^{17} - 2 q^{26} - 4 i q^{29} + (4 i + 4) q^{32} - 6 i q^{34} + (7 i + 7) q^{37} + 8 q^{41} + 7 i q^{49} + (2 i + 2) q^{52} + ( - 9 i + 9) q^{53} + (4 i - 4) q^{58} + 12 q^{61} - 8 i q^{64} + (6 i - 6) q^{68} + ( - 11 i + 11) q^{73} - 14 i q^{74} + ( - 8 i - 8) q^{82} + 16 i q^{89} + ( - 13 i - 13) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{8} + 2 q^{13} - 8 q^{16} + 6 q^{17} - 4 q^{26} + 8 q^{32} + 14 q^{37} + 16 q^{41} + 4 q^{52} + 18 q^{53} - 8 q^{58} + 24 q^{61} - 12 q^{68} + 22 q^{73} - 16 q^{82} - 26 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 0 2.00000 2.00000i 0 0
343.1 −1.00000 + 1.00000i 0 2.00000i 0 0 0 2.00000 + 2.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.c 2
3.b odd 2 1 100.2.e.b 2
4.b odd 2 1 CM 900.2.k.c 2
5.b even 2 1 180.2.k.c 2
5.c odd 4 1 180.2.k.c 2
5.c odd 4 1 inner 900.2.k.c 2
12.b even 2 1 100.2.e.b 2
15.d odd 2 1 20.2.e.a 2
15.e even 4 1 20.2.e.a 2
15.e even 4 1 100.2.e.b 2
20.d odd 2 1 180.2.k.c 2
20.e even 4 1 180.2.k.c 2
20.e even 4 1 inner 900.2.k.c 2
24.f even 2 1 1600.2.n.h 2
24.h odd 2 1 1600.2.n.h 2
60.h even 2 1 20.2.e.a 2
60.l odd 4 1 20.2.e.a 2
60.l odd 4 1 100.2.e.b 2
105.g even 2 1 980.2.k.a 2
105.k odd 4 1 980.2.k.a 2
105.o odd 6 2 980.2.x.d 4
105.p even 6 2 980.2.x.c 4
105.w odd 12 2 980.2.x.c 4
105.x even 12 2 980.2.x.d 4
120.i odd 2 1 320.2.n.e 2
120.m even 2 1 320.2.n.e 2
120.q odd 4 1 320.2.n.e 2
120.q odd 4 1 1600.2.n.h 2
120.w even 4 1 320.2.n.e 2
120.w even 4 1 1600.2.n.h 2
240.t even 4 1 1280.2.o.g 2
240.t even 4 1 1280.2.o.j 2
240.z odd 4 1 1280.2.o.j 2
240.bb even 4 1 1280.2.o.j 2
240.bd odd 4 1 1280.2.o.g 2
240.bf even 4 1 1280.2.o.g 2
240.bm odd 4 1 1280.2.o.g 2
240.bm odd 4 1 1280.2.o.j 2
420.o odd 2 1 980.2.k.a 2
420.w even 4 1 980.2.k.a 2
420.ba even 6 2 980.2.x.d 4
420.be odd 6 2 980.2.x.c 4
420.bp odd 12 2 980.2.x.d 4
420.br even 12 2 980.2.x.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 15.d odd 2 1
20.2.e.a 2 15.e even 4 1
20.2.e.a 2 60.h even 2 1
20.2.e.a 2 60.l odd 4 1
100.2.e.b 2 3.b odd 2 1
100.2.e.b 2 12.b even 2 1
100.2.e.b 2 15.e even 4 1
100.2.e.b 2 60.l odd 4 1
180.2.k.c 2 5.b even 2 1
180.2.k.c 2 5.c odd 4 1
180.2.k.c 2 20.d odd 2 1
180.2.k.c 2 20.e even 4 1
320.2.n.e 2 120.i odd 2 1
320.2.n.e 2 120.m even 2 1
320.2.n.e 2 120.q odd 4 1
320.2.n.e 2 120.w even 4 1
900.2.k.c 2 1.a even 1 1 trivial
900.2.k.c 2 4.b odd 2 1 CM
900.2.k.c 2 5.c odd 4 1 inner
900.2.k.c 2 20.e even 4 1 inner
980.2.k.a 2 105.g even 2 1
980.2.k.a 2 105.k odd 4 1
980.2.k.a 2 420.o odd 2 1
980.2.k.a 2 420.w even 4 1
980.2.x.c 4 105.p even 6 2
980.2.x.c 4 105.w odd 12 2
980.2.x.c 4 420.be odd 6 2
980.2.x.c 4 420.br even 12 2
980.2.x.d 4 105.o odd 6 2
980.2.x.d 4 105.x even 12 2
980.2.x.d 4 420.ba even 6 2
980.2.x.d 4 420.bp odd 12 2
1280.2.o.g 2 240.t even 4 1
1280.2.o.g 2 240.bd odd 4 1
1280.2.o.g 2 240.bf even 4 1
1280.2.o.g 2 240.bm odd 4 1
1280.2.o.j 2 240.t even 4 1
1280.2.o.j 2 240.z odd 4 1
1280.2.o.j 2 240.bb even 4 1
1280.2.o.j 2 240.bm odd 4 1
1600.2.n.h 2 24.f even 2 1
1600.2.n.h 2 24.h odd 2 1
1600.2.n.h 2 120.q odd 4 1
1600.2.n.h 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 18 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 242 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} + 26T + 338 \) Copy content Toggle raw display
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