Properties

Label 950.6.a.d
Level $950$
Weight $6$
Character orbit 950.a
Self dual yes
Analytic conductor $152.365$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(152.364628822\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1441})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( -1 - \beta ) q^{3} + 16 q^{4} + ( -4 - 4 \beta ) q^{6} + ( -59 + 4 \beta ) q^{7} + 64 q^{8} + ( 118 + 3 \beta ) q^{9} +O(q^{10})\) \( q + 4 q^{2} + ( -1 - \beta ) q^{3} + 16 q^{4} + ( -4 - 4 \beta ) q^{6} + ( -59 + 4 \beta ) q^{7} + 64 q^{8} + ( 118 + 3 \beta ) q^{9} + ( 330 + \beta ) q^{11} + ( -16 - 16 \beta ) q^{12} + ( -809 + 5 \beta ) q^{13} + ( -236 + 16 \beta ) q^{14} + 256 q^{16} + ( -27 - 10 \beta ) q^{17} + ( 472 + 12 \beta ) q^{18} + 361 q^{19} + ( -1381 + 51 \beta ) q^{21} + ( 1320 + 4 \beta ) q^{22} + ( 1617 - 49 \beta ) q^{23} + ( -64 - 64 \beta ) q^{24} + ( -3236 + 20 \beta ) q^{26} + ( -955 + 119 \beta ) q^{27} + ( -944 + 64 \beta ) q^{28} + ( -1083 - 315 \beta ) q^{29} + ( -748 + 316 \beta ) q^{31} + 1024 q^{32} + ( -690 - 332 \beta ) q^{33} + ( -108 - 40 \beta ) q^{34} + ( 1888 + 48 \beta ) q^{36} + ( -5330 + 172 \beta ) q^{37} + 1444 q^{38} + ( -991 + 799 \beta ) q^{39} + ( 8616 - 602 \beta ) q^{41} + ( -5524 + 204 \beta ) q^{42} + ( -5792 + 281 \beta ) q^{43} + ( 5280 + 16 \beta ) q^{44} + ( 6468 - 196 \beta ) q^{46} + ( 5520 + 1115 \beta ) q^{47} + ( -256 - 256 \beta ) q^{48} + ( -7566 - 456 \beta ) q^{49} + ( 3627 + 47 \beta ) q^{51} + ( -12944 + 80 \beta ) q^{52} + ( -10593 + 601 \beta ) q^{53} + ( -3820 + 476 \beta ) q^{54} + ( -3776 + 256 \beta ) q^{56} + ( -361 - 361 \beta ) q^{57} + ( -4332 - 1260 \beta ) q^{58} + ( -39327 + 73 \beta ) q^{59} + ( 21398 + 825 \beta ) q^{61} + ( -2992 + 1264 \beta ) q^{62} + ( -2642 + 307 \beta ) q^{63} + 4096 q^{64} + ( -2760 - 1328 \beta ) q^{66} + ( -5453 + 3101 \beta ) q^{67} + ( -432 - 160 \beta ) q^{68} + ( 16023 - 1519 \beta ) q^{69} + ( -31878 + 1268 \beta ) q^{71} + ( 7552 + 192 \beta ) q^{72} + ( -6617 - 2984 \beta ) q^{73} + ( -21320 + 688 \beta ) q^{74} + 5776 q^{76} + ( -18030 + 1265 \beta ) q^{77} + ( -3964 + 3196 \beta ) q^{78} + ( 33494 + 134 \beta ) q^{79} + ( -70559 - 12 \beta ) q^{81} + ( 34464 - 2408 \beta ) q^{82} + ( 4134 + 2446 \beta ) q^{83} + ( -22096 + 816 \beta ) q^{84} + ( -23168 + 1124 \beta ) q^{86} + ( 114483 + 1713 \beta ) q^{87} + ( 21120 + 64 \beta ) q^{88} + ( 61956 + 4276 \beta ) q^{89} + ( 54931 - 3511 \beta ) q^{91} + ( 25872 - 784 \beta ) q^{92} + ( -113012 + 116 \beta ) q^{93} + ( 22080 + 4460 \beta ) q^{94} + ( -1024 - 1024 \beta ) q^{96} + ( -90590 + 2622 \beta ) q^{97} + ( -30264 - 1824 \beta ) q^{98} + ( 40020 + 1111 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + O(q^{10}) \) \( 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + 661q^{11} - 48q^{12} - 1613q^{13} - 456q^{14} + 512q^{16} - 64q^{17} + 956q^{18} + 722q^{19} - 2711q^{21} + 2644q^{22} + 3185q^{23} - 192q^{24} - 6452q^{26} - 1791q^{27} - 1824q^{28} - 2481q^{29} - 1180q^{31} + 2048q^{32} - 1712q^{33} - 256q^{34} + 3824q^{36} - 10488q^{37} + 2888q^{38} - 1183q^{39} + 16630q^{41} - 10844q^{42} - 11303q^{43} + 10576q^{44} + 12740q^{46} + 12155q^{47} - 768q^{48} - 15588q^{49} + 7301q^{51} - 25808q^{52} - 20585q^{53} - 7164q^{54} - 7296q^{56} - 1083q^{57} - 9924q^{58} - 78581q^{59} + 43621q^{61} - 4720q^{62} - 4977q^{63} + 8192q^{64} - 6848q^{66} - 7805q^{67} - 1024q^{68} + 30527q^{69} - 62488q^{71} + 15296q^{72} - 16218q^{73} - 41952q^{74} + 11552q^{76} - 34795q^{77} - 4732q^{78} + 67122q^{79} - 141130q^{81} + 66520q^{82} + 10714q^{83} - 43376q^{84} - 45212q^{86} + 230679q^{87} + 42304q^{88} + 128188q^{89} + 106351q^{91} + 50960q^{92} - 225908q^{93} + 48620q^{94} - 3072q^{96} - 178558q^{97} - 62352q^{98} + 81151q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
19.4803
−18.4803
4.00000 −20.4803 16.0000 0 −81.9210 18.9210 64.0000 176.441 0
1.2 4.00000 17.4803 16.0000 0 69.9210 −132.921 64.0000 62.5592 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.6.a.d 2
5.b even 2 1 38.6.a.c 2
15.d odd 2 1 342.6.a.i 2
20.d odd 2 1 304.6.a.f 2
95.d odd 2 1 722.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.c 2 5.b even 2 1
304.6.a.f 2 20.d odd 2 1
342.6.a.i 2 15.d odd 2 1
722.6.a.c 2 95.d odd 2 1
950.6.a.d 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 T_{3} - 358 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{2} \)
$3$ \( -358 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -2515 + 114 T + T^{2} \)
$11$ \( 108870 - 661 T + T^{2} \)
$13$ \( 641436 + 1613 T + T^{2} \)
$17$ \( -35001 + 64 T + T^{2} \)
$19$ \( ( -361 + T )^{2} \)
$23$ \( 1671096 - 3185 T + T^{2} \)
$29$ \( -34206966 + 2481 T + T^{2} \)
$31$ \( -35625024 + 1180 T + T^{2} \)
$37$ \( 16841900 + 10488 T + T^{2} \)
$41$ \( -61416816 - 16630 T + T^{2} \)
$43$ \( 3493752 + 11303 T + T^{2} \)
$47$ \( -410935800 - 12155 T + T^{2} \)
$53$ \( -24187104 + 20585 T + T^{2} \)
$59$ \( 1541823618 + 78581 T + T^{2} \)
$61$ \( 230502754 - 43621 T + T^{2} \)
$67$ \( -3449006904 + 7805 T + T^{2} \)
$71$ \( 396968940 + 62488 T + T^{2} \)
$73$ \( -3142002343 + 16218 T + T^{2} \)
$79$ \( 1119872072 - 67122 T + T^{2} \)
$83$ \( -2126648040 - 10714 T + T^{2} \)
$89$ \( -2478833568 - 128188 T + T^{2} \)
$97$ \( 5494062880 + 178558 T + T^{2} \)
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