Properties

Label 50.6.b.d
Level 50
Weight 6
Character orbit 50.b
Analytic conductor 8.019
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -26 i q^{3} -16 q^{4} + 104 q^{6} + 22 i q^{7} -64 i q^{8} -433 q^{9} +O(q^{10})\) \( q + 4 i q^{2} -26 i q^{3} -16 q^{4} + 104 q^{6} + 22 i q^{7} -64 i q^{8} -433 q^{9} -768 q^{11} + 416 i q^{12} -46 i q^{13} -88 q^{14} + 256 q^{16} -378 i q^{17} -1732 i q^{18} -1100 q^{19} + 572 q^{21} -3072 i q^{22} -1986 i q^{23} -1664 q^{24} + 184 q^{26} + 4940 i q^{27} -352 i q^{28} + 5610 q^{29} -3988 q^{31} + 1024 i q^{32} + 19968 i q^{33} + 1512 q^{34} + 6928 q^{36} + 142 i q^{37} -4400 i q^{38} -1196 q^{39} + 1542 q^{41} + 2288 i q^{42} -5026 i q^{43} + 12288 q^{44} + 7944 q^{46} -24738 i q^{47} -6656 i q^{48} + 16323 q^{49} -9828 q^{51} + 736 i q^{52} -14166 i q^{53} -19760 q^{54} + 1408 q^{56} + 28600 i q^{57} + 22440 i q^{58} -28380 q^{59} + 5522 q^{61} -15952 i q^{62} -9526 i q^{63} -4096 q^{64} -79872 q^{66} + 24742 i q^{67} + 6048 i q^{68} -51636 q^{69} + 42372 q^{71} + 27712 i q^{72} -52126 i q^{73} -568 q^{74} + 17600 q^{76} -16896 i q^{77} -4784 i q^{78} + 39640 q^{79} + 23221 q^{81} + 6168 i q^{82} -59826 i q^{83} -9152 q^{84} + 20104 q^{86} -145860 i q^{87} + 49152 i q^{88} -57690 q^{89} + 1012 q^{91} + 31776 i q^{92} + 103688 i q^{93} + 98952 q^{94} + 26624 q^{96} + 144382 i q^{97} + 65292 i q^{98} + 332544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 208q^{6} - 866q^{9} + O(q^{10}) \) \( 2q - 32q^{4} + 208q^{6} - 866q^{9} - 1536q^{11} - 176q^{14} + 512q^{16} - 2200q^{19} + 1144q^{21} - 3328q^{24} + 368q^{26} + 11220q^{29} - 7976q^{31} + 3024q^{34} + 13856q^{36} - 2392q^{39} + 3084q^{41} + 24576q^{44} + 15888q^{46} + 32646q^{49} - 19656q^{51} - 39520q^{54} + 2816q^{56} - 56760q^{59} + 11044q^{61} - 8192q^{64} - 159744q^{66} - 103272q^{69} + 84744q^{71} - 1136q^{74} + 35200q^{76} + 79280q^{79} + 46442q^{81} - 18304q^{84} + 40208q^{86} - 115380q^{89} + 2024q^{91} + 197904q^{94} + 53248q^{96} + 665088q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
4.00000i 26.0000i −16.0000 0 104.000 22.0000i 64.0000i −433.000 0
49.2 4.00000i 26.0000i −16.0000 0 104.000 22.0000i 64.0000i −433.000 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 50.6.b.d 2
3.b odd 2 1 450.6.c.o 2
4.b odd 2 1 400.6.c.a 2
5.b even 2 1 inner 50.6.b.d 2
5.c odd 4 1 10.6.a.a 1
5.c odd 4 1 50.6.a.g 1
15.d odd 2 1 450.6.c.o 2
15.e even 4 1 90.6.a.f 1
15.e even 4 1 450.6.a.h 1
20.d odd 2 1 400.6.c.a 2
20.e even 4 1 80.6.a.h 1
20.e even 4 1 400.6.a.a 1
35.f even 4 1 490.6.a.j 1
40.i odd 4 1 320.6.a.p 1
40.k even 4 1 320.6.a.a 1
60.l odd 4 1 720.6.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 5.c odd 4 1
50.6.a.g 1 5.c odd 4 1
50.6.b.d 2 1.a even 1 1 trivial
50.6.b.d 2 5.b even 2 1 inner
80.6.a.h 1 20.e even 4 1
90.6.a.f 1 15.e even 4 1
320.6.a.a 1 40.k even 4 1
320.6.a.p 1 40.i odd 4 1
400.6.a.a 1 20.e even 4 1
400.6.c.a 2 4.b odd 2 1
400.6.c.a 2 20.d odd 2 1
450.6.a.h 1 15.e even 4 1
450.6.c.o 2 3.b odd 2 1
450.6.c.o 2 15.d odd 2 1
490.6.a.j 1 35.f even 4 1
720.6.a.r 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 676 \) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T^{2} \)
$3$ \( 1 + 190 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 - 33130 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 768 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 740470 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 2696830 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1100 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 8928490 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 - 5610 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 3988 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 138667750 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 1542 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 268756210 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 + 153278630 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 635715430 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 28380 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 5522 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 2088083650 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 42372 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 1429023310 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 39640 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 4298931010 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 57690 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 3671481410 T^{2} + 73742412689492826049 T^{4} \)
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