Properties

Label 950.3.d.a
Level $950$
Weight $3$
Character orbit 950.d
Analytic conductor $25.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 950.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8856251142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + 2 q^{4} -4 q^{6} + 5 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + 2 q^{4} -4 q^{6} + 5 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} + 5 q^{11} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{12} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{13} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{14} + 4 q^{16} -25 \zeta_{8}^{2} q^{17} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{18} -19 q^{19} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{21} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{22} + 10 \zeta_{8}^{2} q^{23} -8 q^{24} + 24 q^{26} + ( 20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{27} + 10 \zeta_{8}^{2} q^{28} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{29} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{33} + ( -25 \zeta_{8} - 25 \zeta_{8}^{3} ) q^{34} -2 q^{36} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{37} + ( -19 \zeta_{8} + 19 \zeta_{8}^{3} ) q^{38} -48 q^{39} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{41} -20 \zeta_{8}^{2} q^{42} -5 \zeta_{8}^{2} q^{43} + 10 q^{44} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{46} + 5 \zeta_{8}^{2} q^{47} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{48} + 24 q^{49} + ( 50 \zeta_{8} + 50 \zeta_{8}^{3} ) q^{51} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{52} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{53} + 40 q^{54} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{56} + ( 38 \zeta_{8} - 38 \zeta_{8}^{3} ) q^{57} + 60 \zeta_{8}^{2} q^{58} + ( 60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{59} + 95 q^{61} + 60 \zeta_{8}^{2} q^{62} -5 \zeta_{8}^{2} q^{63} + 8 q^{64} -20 q^{66} + ( 78 \zeta_{8} - 78 \zeta_{8}^{3} ) q^{67} -50 \zeta_{8}^{2} q^{68} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{69} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + 25 \zeta_{8}^{2} q^{73} -36 q^{74} -38 q^{76} + 25 \zeta_{8}^{2} q^{77} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{78} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{79} -71 q^{81} + 60 \zeta_{8}^{2} q^{82} + 130 \zeta_{8}^{2} q^{83} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{84} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{86} -120 \zeta_{8}^{2} q^{87} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{88} + ( -90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{89} + ( 60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{91} + 20 \zeta_{8}^{2} q^{92} -120 \zeta_{8}^{2} q^{93} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{94} -16 q^{96} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{97} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{98} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 16q^{6} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 16q^{6} - 4q^{9} + 20q^{11} + 16q^{16} - 76q^{19} - 32q^{24} + 96q^{26} - 8q^{36} - 192q^{39} + 40q^{44} + 96q^{49} + 160q^{54} + 380q^{61} + 32q^{64} - 80q^{66} - 144q^{74} - 152q^{76} - 284q^{81} - 64q^{96} - 20q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 2.82843 2.00000 0 −4.00000 5.00000i −2.82843 −1.00000 0
949.2 −1.41421 2.82843 2.00000 0 −4.00000 5.00000i −2.82843 −1.00000 0
949.3 1.41421 −2.82843 2.00000 0 −4.00000 5.00000i 2.82843 −1.00000 0
949.4 1.41421 −2.82843 2.00000 0 −4.00000 5.00000i 2.82843 −1.00000 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
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.3.d.a 4
5.b even 2 1 inner 950.3.d.a 4
5.c odd 4 1 38.3.b.a 2
5.c odd 4 1 950.3.c.a 2
15.e even 4 1 342.3.d.a 2
19.b odd 2 1 inner 950.3.d.a 4
20.e even 4 1 304.3.e.c 2
40.i odd 4 1 1216.3.e.j 2
40.k even 4 1 1216.3.e.i 2
60.l odd 4 1 2736.3.o.h 2
95.d odd 2 1 inner 950.3.d.a 4
95.g even 4 1 38.3.b.a 2
95.g even 4 1 950.3.c.a 2
285.j odd 4 1 342.3.d.a 2
380.j odd 4 1 304.3.e.c 2
760.t even 4 1 1216.3.e.j 2
760.y odd 4 1 1216.3.e.i 2
1140.w even 4 1 2736.3.o.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 5.c odd 4 1
38.3.b.a 2 95.g even 4 1
304.3.e.c 2 20.e even 4 1
304.3.e.c 2 380.j odd 4 1
342.3.d.a 2 15.e even 4 1
342.3.d.a 2 285.j odd 4 1
950.3.c.a 2 5.c odd 4 1
950.3.c.a 2 95.g even 4 1
950.3.d.a 4 1.a even 1 1 trivial
950.3.d.a 4 5.b even 2 1 inner
950.3.d.a 4 19.b odd 2 1 inner
950.3.d.a 4 95.d odd 2 1 inner
1216.3.e.i 2 40.k even 4 1
1216.3.e.i 2 760.y odd 4 1
1216.3.e.j 2 40.i odd 4 1
1216.3.e.j 2 760.t even 4 1
2736.3.o.h 2 60.l odd 4 1
2736.3.o.h 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( ( -8 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 25 + T^{2} )^{2} \)
$11$ \( ( -5 + T )^{4} \)
$13$ \( ( -288 + T^{2} )^{2} \)
$17$ \( ( 625 + T^{2} )^{2} \)
$19$ \( ( 19 + T )^{4} \)
$23$ \( ( 100 + T^{2} )^{2} \)
$29$ \( ( 1800 + T^{2} )^{2} \)
$31$ \( ( 1800 + T^{2} )^{2} \)
$37$ \( ( -648 + T^{2} )^{2} \)
$41$ \( ( 1800 + T^{2} )^{2} \)
$43$ \( ( 25 + T^{2} )^{2} \)
$47$ \( ( 25 + T^{2} )^{2} \)
$53$ \( ( -648 + T^{2} )^{2} \)
$59$ \( ( 7200 + T^{2} )^{2} \)
$61$ \( ( -95 + T )^{4} \)
$67$ \( ( -12168 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 625 + T^{2} )^{2} \)
$79$ \( ( 1800 + T^{2} )^{2} \)
$83$ \( ( 16900 + T^{2} )^{2} \)
$89$ \( ( 16200 + T^{2} )^{2} \)
$97$ \( ( -288 + T^{2} )^{2} \)
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