Properties

Label 950.2.b.e
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.

\(f(q)\) \(=\) \( q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} + 2 q^{9} +O(q^{10})\) \( q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} + 2 q^{9} -i q^{12} -i q^{13} + q^{14} + q^{16} + 3 i q^{17} -2 i q^{18} - q^{19} - q^{21} + 3 i q^{23} - q^{24} - q^{26} + 5 i q^{27} -i q^{28} + 3 q^{29} + 2 q^{31} -i q^{32} + 3 q^{34} -2 q^{36} + 10 i q^{37} + i q^{38} + q^{39} + 6 q^{41} + i q^{42} + 2 i q^{43} + 3 q^{46} + i q^{48} + 6 q^{49} -3 q^{51} + i q^{52} + 3 i q^{53} + 5 q^{54} - q^{56} -i q^{57} -3 i q^{58} -3 q^{59} + 8 q^{61} -2 i q^{62} + 2 i q^{63} - q^{64} + 7 i q^{67} -3 i q^{68} -3 q^{69} + 12 q^{71} + 2 i q^{72} -13 i q^{73} + 10 q^{74} + q^{76} -i q^{78} -14 q^{79} + q^{81} -6 i q^{82} + 6 i q^{83} + q^{84} + 2 q^{86} + 3 i q^{87} -6 q^{89} + q^{91} -3 i q^{92} + 2 i q^{93} + q^{96} + 10 i q^{97} -6 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} + 4q^{9} + 2q^{14} + 2q^{16} - 2q^{19} - 2q^{21} - 2q^{24} - 2q^{26} + 6q^{29} + 4q^{31} + 6q^{34} - 4q^{36} + 2q^{39} + 12q^{41} + 6q^{46} + 12q^{49} - 6q^{51} + 10q^{54} - 2q^{56} - 6q^{59} + 16q^{61} - 2q^{64} - 6q^{69} + 24q^{71} + 20q^{74} + 2q^{76} - 28q^{79} + 2q^{81} + 2q^{84} + 4q^{86} - 12q^{89} + 2q^{91} + 2q^{96} + O(q^{100}) \)

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.e 2
5.b even 2 1 inner 950.2.b.e 2
5.c odd 4 1 190.2.a.c 1
5.c odd 4 1 950.2.a.a 1
15.e even 4 1 1710.2.a.d 1
15.e even 4 1 8550.2.a.bd 1
20.e even 4 1 1520.2.a.d 1
20.e even 4 1 7600.2.a.m 1
35.f even 4 1 9310.2.a.o 1
40.i odd 4 1 6080.2.a.h 1
40.k even 4 1 6080.2.a.p 1
95.g even 4 1 3610.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 5.c odd 4 1
950.2.a.a 1 5.c odd 4 1
950.2.b.e 2 1.a even 1 1 trivial
950.2.b.e 2 5.b even 2 1 inner
1520.2.a.d 1 20.e even 4 1
1710.2.a.d 1 15.e even 4 1
3610.2.a.b 1 95.g even 4 1
6080.2.a.h 1 40.i odd 4 1
6080.2.a.p 1 40.k even 4 1
7600.2.a.m 1 20.e even 4 1
8550.2.a.bd 1 15.e even 4 1
9310.2.a.o 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 9 + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 9 + T^{2} \)
$59$ \( ( 3 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 49 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 169 + T^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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