Properties

Label 200.6.a.g
Level $200$
Weight $6$
Character orbit 200.a
Self dual yes
Analytic conductor $32.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Defining polynomial: \( x^{2} - x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 6) q^{3} + (3 \beta - 26) q^{7} + (12 \beta + 309) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 6) q^{3} + (3 \beta - 26) q^{7} + (12 \beta + 309) q^{9} + (6 \beta + 280) q^{11} + ( - 12 \beta - 694) q^{13} + (84 \beta - 74) q^{17} + (36 \beta - 500) q^{19} + ( - 8 \beta + 1392) q^{21} + ( - 123 \beta + 1226) q^{23} + (138 \beta + 6588) q^{27} + (312 \beta + 670) q^{29} + (54 \beta - 1124) q^{31} + (316 \beta + 4776) q^{33} + ( - 216 \beta + 2970) q^{37} + ( - 766 \beta - 10356) q^{39} + ( - 156 \beta + 11538) q^{41} + ( - 339 \beta - 8842) q^{43} + ( - 885 \beta + 1454) q^{47} + ( - 156 \beta - 11487) q^{49} + (430 \beta + 42900) q^{51} + (540 \beta + 2706) q^{53} + ( - 284 \beta + 15576) q^{57} + ( - 504 \beta + 31292) q^{59} + ( - 1104 \beta + 7054) q^{61} + (615 \beta + 10542) q^{63} + (543 \beta + 42706) q^{67} + (488 \beta - 56112) q^{69} + ( - 546 \beta + 23604) q^{71} + ( - 1308 \beta + 33726) q^{73} + (684 \beta + 2008) q^{77} + (2508 \beta - 32952) q^{79} + (4500 \beta + 35649) q^{81} + ( - 711 \beta - 54362) q^{83} + (2542 \beta + 165012) q^{87} + ( - 1464 \beta - 27510) q^{89} + ( - 1770 \beta - 532) q^{91} + ( - 800 \beta + 21120) q^{93} + (4620 \beta - 73834) q^{97} + (5214 \beta + 123672) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{3} - 52 q^{7} + 618 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{3} - 52 q^{7} + 618 q^{9} + 560 q^{11} - 1388 q^{13} - 148 q^{17} - 1000 q^{19} + 2784 q^{21} + 2452 q^{23} + 13176 q^{27} + 1340 q^{29} - 2248 q^{31} + 9552 q^{33} + 5940 q^{37} - 20712 q^{39} + 23076 q^{41} - 17684 q^{43} + 2908 q^{47} - 22974 q^{49} + 85800 q^{51} + 5412 q^{53} + 31152 q^{57} + 62584 q^{59} + 14108 q^{61} + 21084 q^{63} + 85412 q^{67} - 112224 q^{69} + 47208 q^{71} + 67452 q^{73} + 4016 q^{77} - 65904 q^{79} + 71298 q^{81} - 108724 q^{83} + 330024 q^{87} - 55020 q^{89} - 1064 q^{91} + 42240 q^{93} - 147668 q^{97} + 247344 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−5.17891
6.17891
0 −16.7156 0 0 0 −94.1469 0 36.4124 0
1.2 0 28.7156 0 0 0 42.1469 0 581.588 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 200.6.a.g 2
4.b odd 2 1 400.6.a.q 2
5.b even 2 1 40.6.a.d 2
5.c odd 4 2 200.6.c.e 4
15.d odd 2 1 360.6.a.l 2
20.d odd 2 1 80.6.a.i 2
20.e even 4 2 400.6.c.l 4
40.e odd 2 1 320.6.a.q 2
40.f even 2 1 320.6.a.w 2
60.h even 2 1 720.6.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.d 2 5.b even 2 1
80.6.a.i 2 20.d odd 2 1
200.6.a.g 2 1.a even 1 1 trivial
200.6.c.e 4 5.c odd 4 2
320.6.a.q 2 40.e odd 2 1
320.6.a.w 2 40.f even 2 1
360.6.a.l 2 15.d odd 2 1
400.6.a.q 2 4.b odd 2 1
400.6.c.l 4 20.e even 4 2
720.6.a.z 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 12T_{3} - 480 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 12T - 480 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 52T - 3968 \) Copy content Toggle raw display
$11$ \( T^{2} - 560T + 59824 \) Copy content Toggle raw display
$13$ \( T^{2} + 1388 T + 407332 \) Copy content Toggle raw display
$17$ \( T^{2} + 148 T - 3635420 \) Copy content Toggle raw display
$19$ \( T^{2} + 1000 T - 418736 \) Copy content Toggle raw display
$23$ \( T^{2} - 2452 T - 6303488 \) Copy content Toggle raw display
$29$ \( T^{2} - 1340 T - 49780604 \) Copy content Toggle raw display
$31$ \( T^{2} + 2248 T - 241280 \) Copy content Toggle raw display
$37$ \( T^{2} - 5940 T - 15253596 \) Copy content Toggle raw display
$41$ \( T^{2} - 23076 T + 120568068 \) Copy content Toggle raw display
$43$ \( T^{2} + 17684 T + 18881728 \) Copy content Toggle raw display
$47$ \( T^{2} - 2908 T - 402029984 \) Copy content Toggle raw display
$53$ \( T^{2} - 5412 T - 143143164 \) Copy content Toggle raw display
$59$ \( T^{2} - 62584 T + 848117008 \) Copy content Toggle raw display
$61$ \( T^{2} - 14108 T - 579150140 \) Copy content Toggle raw display
$67$ \( T^{2} - 85412 T + 1671660352 \) Copy content Toggle raw display
$71$ \( T^{2} - 47208 T + 403320960 \) Copy content Toggle raw display
$73$ \( T^{2} - 67452 T + 254637252 \) Copy content Toggle raw display
$79$ \( T^{2} + 65904 T - 2159838720 \) Copy content Toggle raw display
$83$ \( T^{2} + 108724 T + 2694378208 \) Copy content Toggle raw display
$89$ \( T^{2} + 55020 T - 349140636 \) Copy content Toggle raw display
$97$ \( T^{2} + 147668 T - 5562250844 \) Copy content Toggle raw display
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