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

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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: \(\Q(\sqrt{-1}) \)
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.

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

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 \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} + 36 \)
\( T_{19} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 58 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T^{2} + 9409 T^{4} \)
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