Properties

Label 950.2.b.c
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} -i q^{8} + 2 q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} -i q^{8} + 2 q^{9} + 2 q^{11} -i q^{12} + i q^{13} -3 q^{14} + q^{16} + 3 i q^{17} + 2 i q^{18} + q^{19} -3 q^{21} + 2 i q^{22} + i q^{23} + q^{24} - q^{26} + 5 i q^{27} -3 i q^{28} + 5 q^{29} -8 q^{31} + i q^{32} + 2 i q^{33} -3 q^{34} -2 q^{36} -2 i q^{37} + i q^{38} - q^{39} -8 q^{41} -3 i q^{42} -4 i q^{43} -2 q^{44} - q^{46} + 8 i q^{47} + i q^{48} -2 q^{49} -3 q^{51} -i q^{52} + i q^{53} -5 q^{54} + 3 q^{56} + i q^{57} + 5 i q^{58} -15 q^{59} + 2 q^{61} -8 i q^{62} + 6 i q^{63} - q^{64} -2 q^{66} + 3 i q^{67} -3 i q^{68} - q^{69} + 2 q^{71} -2 i q^{72} -9 i q^{73} + 2 q^{74} - q^{76} + 6 i q^{77} -i q^{78} + 10 q^{79} + q^{81} -8 i q^{82} + 6 i q^{83} + 3 q^{84} + 4 q^{86} + 5 i q^{87} -2 i q^{88} -3 q^{91} -i q^{92} -8 i q^{93} -8 q^{94} - q^{96} -2 i q^{97} -2 i q^{98} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 4q^{9} + 4q^{11} - 6q^{14} + 2q^{16} + 2q^{19} - 6q^{21} + 2q^{24} - 2q^{26} + 10q^{29} - 16q^{31} - 6q^{34} - 4q^{36} - 2q^{39} - 16q^{41} - 4q^{44} - 2q^{46} - 4q^{49} - 6q^{51} - 10q^{54} + 6q^{56} - 30q^{59} + 4q^{61} - 2q^{64} - 4q^{66} - 2q^{69} + 4q^{71} + 4q^{74} - 2q^{76} + 20q^{79} + 2q^{81} + 6q^{84} + 8q^{86} - 6q^{91} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100}) \)

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 3.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 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.c 2
5.b even 2 1 inner 950.2.b.c 2
5.c odd 4 1 38.2.a.b 1
5.c odd 4 1 950.2.a.b 1
15.e even 4 1 342.2.a.d 1
15.e even 4 1 8550.2.a.u 1
20.e even 4 1 304.2.a.d 1
20.e even 4 1 7600.2.a.h 1
35.f even 4 1 1862.2.a.f 1
40.i odd 4 1 1216.2.a.n 1
40.k even 4 1 1216.2.a.g 1
55.e even 4 1 4598.2.a.a 1
60.l odd 4 1 2736.2.a.w 1
65.h odd 4 1 6422.2.a.b 1
95.g even 4 1 722.2.a.b 1
95.l even 12 2 722.2.c.f 2
95.m odd 12 2 722.2.c.d 2
95.q odd 36 6 722.2.e.c 6
95.r even 36 6 722.2.e.d 6
285.j odd 4 1 6498.2.a.y 1
380.j odd 4 1 5776.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 5.c odd 4 1
304.2.a.d 1 20.e even 4 1
342.2.a.d 1 15.e even 4 1
722.2.a.b 1 95.g even 4 1
722.2.c.d 2 95.m odd 12 2
722.2.c.f 2 95.l even 12 2
722.2.e.c 6 95.q odd 36 6
722.2.e.d 6 95.r even 36 6
950.2.a.b 1 5.c odd 4 1
950.2.b.c 2 1.a even 1 1 trivial
950.2.b.c 2 5.b even 2 1 inner
1216.2.a.g 1 40.k even 4 1
1216.2.a.n 1 40.i odd 4 1
1862.2.a.f 1 35.f even 4 1
2736.2.a.w 1 60.l odd 4 1
4598.2.a.a 1 55.e even 4 1
5776.2.a.d 1 380.j odd 4 1
6422.2.a.b 1 65.h odd 4 1
6498.2.a.y 1 285.j odd 4 1
7600.2.a.h 1 20.e even 4 1
8550.2.a.u 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 9 \)
\( T_{11} - 2 \)
\( T_{13}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( ( 15 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 81 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 + T^{2} \)
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