Properties

Label 1650.2.c.d
Level $1650$
Weight $2$
Character orbit 1650.c
Analytic conductor $13.175$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.1753163335\)
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 66)
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.

\(f(q)\) \(=\) \( q - i q^{2} - i q^{3} - q^{4} - q^{6} + 2 i q^{7} + i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - i q^{3} - q^{4} - q^{6} + 2 i q^{7} + i q^{8} - q^{9} - q^{11} + i q^{12} + 4 i q^{13} + 2 q^{14} + q^{16} - 6 i q^{17} + i q^{18} + 4 q^{19} + 2 q^{21} + i q^{22} - 6 i q^{23} + q^{24} + 4 q^{26} + i q^{27} - 2 i q^{28} - 6 q^{29} + 8 q^{31} - i q^{32} + i q^{33} - 6 q^{34} + q^{36} - 10 i q^{37} - 4 i q^{38} + 4 q^{39} + 6 q^{41} - 2 i q^{42} - 8 i q^{43} + q^{44} - 6 q^{46} - 6 i q^{47} - i q^{48} + 3 q^{49} - 6 q^{51} - 4 i q^{52} + q^{54} - 2 q^{56} - 4 i q^{57} + 6 i q^{58} + 8 q^{61} - 8 i q^{62} - 2 i q^{63} - q^{64} + q^{66} - 4 i q^{67} + 6 i q^{68} - 6 q^{69} + 6 q^{71} - i q^{72} - 2 i q^{73} - 10 q^{74} - 4 q^{76} - 2 i q^{77} - 4 i q^{78} - 14 q^{79} + q^{81} - 6 i q^{82} + 12 i q^{83} - 2 q^{84} - 8 q^{86} + 6 i q^{87} - i q^{88} + 6 q^{89} - 8 q^{91} + 6 i q^{92} - 8 i q^{93} - 6 q^{94} - q^{96} + 14 i q^{97} - 3 i q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} + 8 q^{26} - 12 q^{29} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} + 2 q^{44} - 12 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 4 q^{56} + 16 q^{61} - 2 q^{64} + 2 q^{66} - 12 q^{69} + 12 q^{71} - 20 q^{74} - 8 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 12 q^{89} - 16 q^{91} - 12 q^{94} - 2 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i −1.00000 0
199.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i −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 1650.2.c.d 2
3.b odd 2 1 4950.2.c.r 2
5.b even 2 1 inner 1650.2.c.d 2
5.c odd 4 1 66.2.a.a 1
5.c odd 4 1 1650.2.a.m 1
15.d odd 2 1 4950.2.c.r 2
15.e even 4 1 198.2.a.e 1
15.e even 4 1 4950.2.a.g 1
20.e even 4 1 528.2.a.d 1
35.f even 4 1 3234.2.a.d 1
40.i odd 4 1 2112.2.a.i 1
40.k even 4 1 2112.2.a.v 1
45.k odd 12 2 1782.2.e.s 2
45.l even 12 2 1782.2.e.f 2
55.e even 4 1 726.2.a.i 1
55.k odd 20 4 726.2.e.k 4
55.l even 20 4 726.2.e.b 4
60.l odd 4 1 1584.2.a.h 1
105.k odd 4 1 9702.2.a.bu 1
120.q odd 4 1 6336.2.a.bf 1
120.w even 4 1 6336.2.a.bj 1
165.l odd 4 1 2178.2.a.b 1
220.i odd 4 1 5808.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 5.c odd 4 1
198.2.a.e 1 15.e even 4 1
528.2.a.d 1 20.e even 4 1
726.2.a.i 1 55.e even 4 1
726.2.e.b 4 55.l even 20 4
726.2.e.k 4 55.k odd 20 4
1584.2.a.h 1 60.l odd 4 1
1650.2.a.m 1 5.c odd 4 1
1650.2.c.d 2 1.a even 1 1 trivial
1650.2.c.d 2 5.b even 2 1 inner
1782.2.e.f 2 45.l even 12 2
1782.2.e.s 2 45.k odd 12 2
2112.2.a.i 1 40.i odd 4 1
2112.2.a.v 1 40.k even 4 1
2178.2.a.b 1 165.l odd 4 1
3234.2.a.d 1 35.f even 4 1
4950.2.a.g 1 15.e even 4 1
4950.2.c.r 2 3.b odd 2 1
4950.2.c.r 2 15.d odd 2 1
5808.2.a.l 1 220.i odd 4 1
6336.2.a.bf 1 120.q odd 4 1
6336.2.a.bj 1 120.w even 4 1
9702.2.a.bu 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
show more
show less