Properties

Label 950.2.b.b
Level 950
Weight 2
Character orbit 950.b
Analytic conductor 7.586
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
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 38)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} -i q^{8} + 2 q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} -i q^{8} + 2 q^{9} -6 q^{11} -i q^{12} + 5 i q^{13} - q^{14} + q^{16} -3 i q^{17} + 2 i q^{18} - q^{19} - q^{21} -6 i q^{22} + 3 i q^{23} + q^{24} -5 q^{26} + 5 i q^{27} -i q^{28} -9 q^{29} -4 q^{31} + i q^{32} -6 i q^{33} + 3 q^{34} -2 q^{36} -2 i q^{37} -i q^{38} -5 q^{39} -i q^{42} + 8 i q^{43} + 6 q^{44} -3 q^{46} + i q^{48} + 6 q^{49} + 3 q^{51} -5 i q^{52} -3 i q^{53} -5 q^{54} + q^{56} -i q^{57} -9 i q^{58} -9 q^{59} -10 q^{61} -4 i q^{62} + 2 i q^{63} - q^{64} + 6 q^{66} -5 i q^{67} + 3 i q^{68} -3 q^{69} -6 q^{71} -2 i q^{72} -7 i q^{73} + 2 q^{74} + q^{76} -6 i q^{77} -5 i q^{78} + 10 q^{79} + q^{81} -6 i q^{83} + q^{84} -8 q^{86} -9 i q^{87} + 6 i q^{88} + 12 q^{89} -5 q^{91} -3 i q^{92} -4 i q^{93} - q^{96} + 10 i q^{97} + 6 i q^{98} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} - 12q^{11} - 2q^{14} + 2q^{16} - 2q^{19} - 2q^{21} + 2q^{24} - 10q^{26} - 18q^{29} - 8q^{31} + 6q^{34} - 4q^{36} - 10q^{39} + 12q^{44} - 6q^{46} + 12q^{49} + 6q^{51} - 10q^{54} + 2q^{56} - 18q^{59} - 20q^{61} - 2q^{64} + 12q^{66} - 6q^{69} - 12q^{71} + 4q^{74} + 2q^{76} + 20q^{79} + 2q^{81} + 2q^{84} - 16q^{86} + 24q^{89} - 10q^{91} - 2q^{96} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 2.00000 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 950.2.b.b 2
5.b even 2 1 inner 950.2.b.b 2
5.c odd 4 1 38.2.a.a 1
5.c odd 4 1 950.2.a.d 1
15.e even 4 1 342.2.a.e 1
15.e even 4 1 8550.2.a.m 1
20.e even 4 1 304.2.a.c 1
20.e even 4 1 7600.2.a.n 1
35.f even 4 1 1862.2.a.b 1
40.i odd 4 1 1216.2.a.e 1
40.k even 4 1 1216.2.a.m 1
55.e even 4 1 4598.2.a.p 1
60.l odd 4 1 2736.2.a.n 1
65.h odd 4 1 6422.2.a.h 1
95.g even 4 1 722.2.a.e 1
95.l even 12 2 722.2.c.c 2
95.m odd 12 2 722.2.c.e 2
95.q odd 36 6 722.2.e.f 6
95.r even 36 6 722.2.e.e 6
285.j odd 4 1 6498.2.a.f 1
380.j odd 4 1 5776.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 5.c odd 4 1
304.2.a.c 1 20.e even 4 1
342.2.a.e 1 15.e even 4 1
722.2.a.e 1 95.g even 4 1
722.2.c.c 2 95.l even 12 2
722.2.c.e 2 95.m odd 12 2
722.2.e.e 6 95.r even 36 6
722.2.e.f 6 95.q odd 36 6
950.2.a.d 1 5.c odd 4 1
950.2.b.b 2 1.a even 1 1 trivial
950.2.b.b 2 5.b even 2 1 inner
1216.2.a.e 1 40.i odd 4 1
1216.2.a.m 1 40.k even 4 1
1862.2.a.b 1 35.f even 4 1
2736.2.a.n 1 60.l odd 4 1
4598.2.a.p 1 55.e even 4 1
5776.2.a.m 1 380.j odd 4 1
6422.2.a.h 1 65.h odd 4 1
6498.2.a.f 1 285.j odd 4 1
7600.2.a.n 1 20.e even 4 1
8550.2.a.m 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 1 \)
\( T_{11} + 6 \)
\( T_{13}^{2} + 25 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 - 5 T^{2} + 9 T^{4} \)
$5$ 1
$7$ \( 1 - 13 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( 1 - T^{2} + 169 T^{4} \)
$17$ \( 1 - 25 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 1 - 37 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 9 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 97 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 9 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 109 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 97 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 12 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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