Properties

Label 950.2.b.a
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 190)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 3 i q^{3} - q^{4} -3 q^{6} -5 i q^{7} -i q^{8} -6 q^{9} +O(q^{10})\) \( q + i q^{2} + 3 i q^{3} - q^{4} -3 q^{6} -5 i q^{7} -i q^{8} -6 q^{9} -4 q^{11} -3 i q^{12} + i q^{13} + 5 q^{14} + q^{16} -3 i q^{17} -6 i q^{18} - q^{19} + 15 q^{21} -4 i q^{22} -7 i q^{23} + 3 q^{24} - q^{26} -9 i q^{27} + 5 i q^{28} + 3 q^{29} -2 q^{31} + i q^{32} -12 i q^{33} + 3 q^{34} + 6 q^{36} -2 i q^{37} -i q^{38} -3 q^{39} -6 q^{41} + 15 i q^{42} -6 i q^{43} + 4 q^{44} + 7 q^{46} + 3 i q^{48} -18 q^{49} + 9 q^{51} -i q^{52} + 13 i q^{53} + 9 q^{54} -5 q^{56} -3 i q^{57} + 3 i q^{58} + 9 q^{59} -12 q^{61} -2 i q^{62} + 30 i q^{63} - q^{64} + 12 q^{66} -3 i q^{67} + 3 i q^{68} + 21 q^{69} + 6 i q^{72} -11 i q^{73} + 2 q^{74} + q^{76} + 20 i q^{77} -3 i q^{78} + 2 q^{79} + 9 q^{81} -6 i q^{82} + 10 i q^{83} -15 q^{84} + 6 q^{86} + 9 i q^{87} + 4 i q^{88} -2 q^{89} + 5 q^{91} + 7 i q^{92} -6 i q^{93} -3 q^{96} -2 i q^{97} -18 i q^{98} + 24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 6q^{6} - 12q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 6q^{6} - 12q^{9} - 8q^{11} + 10q^{14} + 2q^{16} - 2q^{19} + 30q^{21} + 6q^{24} - 2q^{26} + 6q^{29} - 4q^{31} + 6q^{34} + 12q^{36} - 6q^{39} - 12q^{41} + 8q^{44} + 14q^{46} - 36q^{49} + 18q^{51} + 18q^{54} - 10q^{56} + 18q^{59} - 24q^{61} - 2q^{64} + 24q^{66} + 42q^{69} + 4q^{74} + 2q^{76} + 4q^{79} + 18q^{81} - 30q^{84} + 12q^{86} - 4q^{89} + 10q^{91} - 6q^{96} + 48q^{99} + O(q^{100}) \)

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 3.00000i −1.00000 0 −3.00000 5.00000i 1.00000i −6.00000 0
799.2 1.00000i 3.00000i −1.00000 0 −3.00000 5.00000i 1.00000i −6.00000 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.a 2
5.b even 2 1 inner 950.2.b.a 2
5.c odd 4 1 190.2.a.b 1
5.c odd 4 1 950.2.a.c 1
15.e even 4 1 1710.2.a.g 1
15.e even 4 1 8550.2.a.bm 1
20.e even 4 1 1520.2.a.j 1
20.e even 4 1 7600.2.a.a 1
35.f even 4 1 9310.2.a.u 1
40.i odd 4 1 6080.2.a.x 1
40.k even 4 1 6080.2.a.b 1
95.g even 4 1 3610.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.b 1 5.c odd 4 1
950.2.a.c 1 5.c odd 4 1
950.2.b.a 2 1.a even 1 1 trivial
950.2.b.a 2 5.b even 2 1 inner
1520.2.a.j 1 20.e even 4 1
1710.2.a.g 1 15.e even 4 1
3610.2.a.e 1 95.g even 4 1
6080.2.a.b 1 40.k even 4 1
6080.2.a.x 1 40.i odd 4 1
7600.2.a.a 1 20.e even 4 1
8550.2.a.bm 1 15.e even 4 1
9310.2.a.u 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{7}^{2} + 25 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 49 + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 169 + T^{2} \)
$59$ \( ( -9 + T )^{2} \)
$61$ \( ( 12 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( -2 + T )^{2} \)
$83$ \( 100 + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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