Properties

Label 4950.2.c.p
Level $4950$
Weight $2$
Character orbit 4950.c
Analytic conductor $39.526$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.5259490005\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + 4 i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + 4 i q^{7} -i q^{8} + q^{11} -6 i q^{13} -4 q^{14} + q^{16} + 2 i q^{17} -4 q^{19} + i q^{22} -4 i q^{23} + 6 q^{26} -4 i q^{28} + 6 q^{29} + i q^{32} -2 q^{34} -6 i q^{37} -4 i q^{38} + 6 q^{41} + 4 i q^{43} - q^{44} + 4 q^{46} -12 i q^{47} -9 q^{49} + 6 i q^{52} -2 i q^{53} + 4 q^{56} + 6 i q^{58} + 12 q^{59} -14 q^{61} - q^{64} -4 i q^{67} -2 i q^{68} + 12 q^{71} -6 i q^{73} + 6 q^{74} + 4 q^{76} + 4 i q^{77} + 4 q^{79} + 6 i q^{82} -4 i q^{83} -4 q^{86} -i q^{88} + 10 q^{89} + 24 q^{91} + 4 i q^{92} + 12 q^{94} + 14 i q^{97} -9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{11} - 8q^{14} + 2q^{16} - 8q^{19} + 12q^{26} + 12q^{29} - 4q^{34} + 12q^{41} - 2q^{44} + 8q^{46} - 18q^{49} + 8q^{56} + 24q^{59} - 28q^{61} - 2q^{64} + 24q^{71} + 12q^{74} + 8q^{76} + 8q^{79} - 8q^{86} + 20q^{89} + 48q^{91} + 24q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2377\) \(4501\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 0 −1.00000 0 0 4.00000i 1.00000i 0 0
199.2 1.00000i 0 −1.00000 0 0 4.00000i 1.00000i 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
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 4950.2.c.p 2
3.b odd 2 1 1650.2.c.e 2
5.b even 2 1 inner 4950.2.c.p 2
5.c odd 4 1 198.2.a.a 1
5.c odd 4 1 4950.2.a.bu 1
15.d odd 2 1 1650.2.c.e 2
15.e even 4 1 66.2.a.b 1
15.e even 4 1 1650.2.a.k 1
20.e even 4 1 1584.2.a.f 1
35.f even 4 1 9702.2.a.x 1
40.i odd 4 1 6336.2.a.bw 1
40.k even 4 1 6336.2.a.cj 1
45.k odd 12 2 1782.2.e.v 2
45.l even 12 2 1782.2.e.e 2
55.e even 4 1 2178.2.a.g 1
60.l odd 4 1 528.2.a.j 1
105.k odd 4 1 3234.2.a.t 1
120.q odd 4 1 2112.2.a.e 1
120.w even 4 1 2112.2.a.r 1
165.l odd 4 1 726.2.a.c 1
165.u odd 20 4 726.2.e.o 4
165.v even 20 4 726.2.e.g 4
660.q even 4 1 5808.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 15.e even 4 1
198.2.a.a 1 5.c odd 4 1
528.2.a.j 1 60.l odd 4 1
726.2.a.c 1 165.l odd 4 1
726.2.e.g 4 165.v even 20 4
726.2.e.o 4 165.u odd 20 4
1584.2.a.f 1 20.e even 4 1
1650.2.a.k 1 15.e even 4 1
1650.2.c.e 2 3.b odd 2 1
1650.2.c.e 2 15.d odd 2 1
1782.2.e.e 2 45.l even 12 2
1782.2.e.v 2 45.k odd 12 2
2112.2.a.e 1 120.q odd 4 1
2112.2.a.r 1 120.w even 4 1
2178.2.a.g 1 55.e even 4 1
3234.2.a.t 1 105.k odd 4 1
4950.2.a.bu 1 5.c odd 4 1
4950.2.c.p 2 1.a even 1 1 trivial
4950.2.c.p 2 5.b even 2 1 inner
5808.2.a.bc 1 660.q even 4 1
6336.2.a.bw 1 40.i odd 4 1
6336.2.a.cj 1 40.k even 4 1
9702.2.a.x 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4950, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 4 \)
\( T_{19} + 4 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 196 + T^{2} \)
show more
show less