Properties

Label 50.10.b.d
Level 50
Weight 10
Character orbit 50.b
Analytic conductor 25.752
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.7517918082\)
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.

\(f(q)\) \(=\) \( q -16 i q^{2} + 174 i q^{3} -256 q^{4} + 2784 q^{6} -4658 i q^{7} + 4096 i q^{8} -10593 q^{9} +O(q^{10})\) \( q -16 i q^{2} + 174 i q^{3} -256 q^{4} + 2784 q^{6} -4658 i q^{7} + 4096 i q^{8} -10593 q^{9} + 28992 q^{11} -44544 i q^{12} -164446 i q^{13} -74528 q^{14} + 65536 q^{16} + 594822 i q^{17} + 169488 i q^{18} + 295780 q^{19} + 810492 q^{21} -463872 i q^{22} + 2544534 i q^{23} -712704 q^{24} -2631136 q^{26} + 1581660 i q^{27} + 1192448 i q^{28} + 3722970 q^{29} + 2335772 q^{31} -1048576 i q^{32} + 5044608 i q^{33} + 9517152 q^{34} + 2711808 q^{36} -10840418 i q^{37} -4732480 i q^{38} + 28613604 q^{39} + 21593862 q^{41} -12967872 i q^{42} + 10832294 i q^{43} -7421952 q^{44} + 40712544 q^{46} -5172138 i q^{47} + 11403264 i q^{48} + 18656643 q^{49} -103499028 q^{51} + 42098176 i q^{52} + 98179674 i q^{53} + 25306560 q^{54} + 19079168 q^{56} + 51465720 i q^{57} -59567520 i q^{58} -16162860 q^{59} -43928158 q^{61} -37372352 i q^{62} + 49342194 i q^{63} -16777216 q^{64} + 80713728 q^{66} + 81557422 i q^{67} -152274432 i q^{68} -442748916 q^{69} + 161307732 q^{71} -43388928 i q^{72} -247147966 i q^{73} -173446688 q^{74} -75719680 q^{76} -135044736 i q^{77} -457817664 i q^{78} + 583345720 q^{79} -483710859 q^{81} -345501792 i q^{82} -14571786 i q^{83} -207485952 q^{84} + 173316704 q^{86} + 647796780 i q^{87} + 118751232 i q^{88} -470133690 q^{89} -765989468 q^{91} -651400704 i q^{92} + 406424328 i q^{93} -82754208 q^{94} + 182452224 q^{96} + 117838462 i q^{97} -298506288 i q^{98} -307112256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 512q^{4} + 5568q^{6} - 21186q^{9} + O(q^{10}) \) \( 2q - 512q^{4} + 5568q^{6} - 21186q^{9} + 57984q^{11} - 149056q^{14} + 131072q^{16} + 591560q^{19} + 1620984q^{21} - 1425408q^{24} - 5262272q^{26} + 7445940q^{29} + 4671544q^{31} + 19034304q^{34} + 5423616q^{36} + 57227208q^{39} + 43187724q^{41} - 14843904q^{44} + 81425088q^{46} + 37313286q^{49} - 206998056q^{51} + 50613120q^{54} + 38158336q^{56} - 32325720q^{59} - 87856316q^{61} - 33554432q^{64} + 161427456q^{66} - 885497832q^{69} + 322615464q^{71} - 346893376q^{74} - 151439360q^{76} + 1166691440q^{79} - 967421718q^{81} - 414971904q^{84} + 346633408q^{86} - 940267380q^{89} - 1531978936q^{91} - 165508416q^{94} + 364904448q^{96} - 614224512q^{99} + O(q^{100}) \)

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
16.0000i 174.000i −256.000 0 2784.00 4658.00i 4096.00i −10593.0 0
49.2 16.0000i 174.000i −256.000 0 2784.00 4658.00i 4096.00i −10593.0 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 50.10.b.d 2
4.b odd 2 1 400.10.c.c 2
5.b even 2 1 inner 50.10.b.d 2
5.c odd 4 1 10.10.a.c 1
5.c odd 4 1 50.10.a.a 1
15.e even 4 1 90.10.a.e 1
20.d odd 2 1 400.10.c.c 2
20.e even 4 1 80.10.a.a 1
20.e even 4 1 400.10.a.j 1
40.i odd 4 1 320.10.a.b 1
40.k even 4 1 320.10.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.c 1 5.c odd 4 1
50.10.a.a 1 5.c odd 4 1
50.10.b.d 2 1.a even 1 1 trivial
50.10.b.d 2 5.b even 2 1 inner
80.10.a.a 1 20.e even 4 1
90.10.a.e 1 15.e even 4 1
320.10.a.b 1 40.i odd 4 1
320.10.a.i 1 40.k even 4 1
400.10.a.j 1 20.e even 4 1
400.10.c.c 2 4.b odd 2 1
400.10.c.c 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 30276 \) acting on \(S_{10}^{\mathrm{new}}(50, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 256 T^{2} \)
$3$ \( 1 - 9090 T^{2} + 387420489 T^{4} \)
$5$ \( \)
$7$ \( 1 - 59010250 T^{2} + 1628413597910449 T^{4} \)
$11$ \( ( 1 - 28992 T + 2357947691 T^{2} )^{2} \)
$13$ \( 1 + 5833488170 T^{2} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 + 116637458690 T^{2} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( ( 1 - 295780 T + 322687697779 T^{2} )^{2} \)
$23$ \( 1 + 2872347954230 T^{2} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( ( 1 - 3722970 T + 14507145975869 T^{2} )^{2} \)
$31$ \( ( 1 - 2335772 T + 26439622160671 T^{2} )^{2} \)
$37$ \( 1 - 142408817175430 T^{2} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( ( 1 - 21593862 T + 327381934393961 T^{2} )^{2} \)
$43$ \( 1 - 887846630571250 T^{2} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 2211509934714490 T^{2} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + 3039721203142010 T^{2} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( ( 1 + 16162860 T + 8662995818654939 T^{2} )^{2} \)
$61$ \( ( 1 + 43928158 T + 11694146092834141 T^{2} )^{2} \)
$67$ \( 1 - 47761455709303810 T^{2} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( ( 1 - 161307732 T + 45848500718449031 T^{2} )^{2} \)
$73$ \( 1 - 56661056318598670 T^{2} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( ( 1 - 583345720 T + 119851595982618319 T^{2} )^{2} \)
$83$ \( 1 - 373668173587851010 T^{2} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( ( 1 + 470133690 T + 350356403707485209 T^{2} )^{2} \)
$97$ \( 1 - 1506576214182604990 T^{2} + \)\(57\!\cdots\!89\)\( T^{4} \)
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