Properties

Label 200.6.a.j
Level $200$
Weight $6$
Character orbit 200.a
Self dual yes
Analytic conductor $32.077$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1595208.1
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} + 33x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 - 37) q^{7} + ( - 2 \beta_{3} - \beta_{2} + 7 \beta_1 + 125) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 - 37) q^{7} + ( - 2 \beta_{3} - \beta_{2} + 7 \beta_1 + 125) q^{9} + (2 \beta_{2} + 18 \beta_1 - 92) q^{11} + ( - 3 \beta_{3} + 2 \beta_{2} - 110) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} + 44 \beta_1 + 168) q^{17} + (8 \beta_{3} - 2 \beta_{2} + 46 \beta_1 - 172) q^{19} + (6 \beta_{3} - 3 \beta_{2} + 149 \beta_1 + 248) q^{21} + (5 \beta_{3} - 8 \beta_{2} - 23 \beta_1 - 1123) q^{23} + (2 \beta_{3} + 16 \beta_{2} - 222 \beta_1 - 2038) q^{27} + (20 \beta_{2} - 12 \beta_1 - 734) q^{29} + ( - 24 \beta_{3} - 12 \beta_{2} + 84 \beta_1 + 528) q^{31} + (28 \beta_{3} + 8 \beta_{2} + 156 \beta_1 - 6716) q^{33} + ( - 17 \beta_{3} - 2 \beta_{2} + 428 \beta_1 - 2198) q^{37} + ( - 32 \beta_{3} - 4 \beta_{2} - 36 \beta_1 + 376) q^{39} + (2 \beta_{3} - 3 \beta_{2} - 491 \beta_1 + 2950) q^{41} + ( - 54 \beta_{3} - 189 \beta_1 - 12069) q^{43} + ( - 19 \beta_{3} - 56 \beta_{2} - 485 \beta_1 - 3681) q^{47} + (6 \beta_{3} - 69 \beta_{2} - 285 \beta_1 + 5625) q^{49} + (88 \beta_{3} + 68 \beta_{2} - 988 \beta_1 - 15600) q^{51} + ( - 53 \beta_{3} - 34 \beta_{2} - 64 \beta_1 - 21074) q^{53} + (164 \beta_{3} + 40 \beta_{2} + 572 \beta_1 - 17756) q^{57} + (88 \beta_{3} + 26 \beta_{2} - 598 \beta_1 - 11460) q^{59} + ( - 82 \beta_{3} + 61 \beta_{2} - 683 \beta_1 + 15482) q^{61} + (115 \beta_{3} + 152 \beta_{2} - 521 \beta_1 - 46573) q^{63} + (26 \beta_{3} + 32 \beta_{2} + 73 \beta_1 - 18175) q^{67} + (26 \beta_{3} + 7 \beta_{2} + 1103 \beta_1 + 9592) q^{69} + (24 \beta_{3} - 232 \beta_{2} + 1560 \beta_1 - 15704) q^{71} + ( - 48 \beta_{3} - 80 \beta_{2} + 1212 \beta_1 - 33268) q^{73} + ( - 440 \beta_{3} + 56 \beta_{2} - 1972 \beta_1 - 2860) q^{77} + ( - 328 \beta_{3} + 100 \beta_{2} + 196 \beta_1 - 5408) q^{79} + ( - 6 \beta_{3} - 63 \beta_{2} + 3257 \beta_1 + 51209) q^{81} + (166 \beta_{3} - 352 \beta_{2} + 1679 \beta_1 - 18665) q^{83} + ( - 104 \beta_{3} - 112 \beta_{2} + 2526 \beta_1 + 3118) q^{87} + (96 \beta_{3} + 300 \beta_{2} - 564 \beta_1 + 5238) q^{89} + ( - 440 \beta_{3} + 200 \beta_{2} + 2120 \beta_1 - 60952) q^{91} + (24 \beta_{3} + 192 \beta_{2} - 4608 \beta_1 - 26400) q^{93} + (402 \beta_{3} - 172 \beta_{2} - 1860 \beta_1 - 14864) q^{97} + (504 \beta_{3} - 426 \beta_{2} + 5062 \beta_1 - 33356) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 148 q^{7} + 500 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 148 q^{7} + 500 q^{9} - 368 q^{11} - 440 q^{13} + 672 q^{17} - 688 q^{19} + 992 q^{21} - 4492 q^{23} - 8152 q^{27} - 2936 q^{29} + 2112 q^{31} - 26864 q^{33} - 8792 q^{37} + 1504 q^{39} + 11800 q^{41} - 48276 q^{43} - 14724 q^{47} + 22500 q^{49} - 62400 q^{51} - 84296 q^{53} - 71024 q^{57} - 45840 q^{59} + 61928 q^{61} - 186292 q^{63} - 72700 q^{67} + 38368 q^{69} - 62816 q^{71} - 133072 q^{73} - 11440 q^{77} - 21632 q^{79} + 204836 q^{81} - 74660 q^{83} + 12472 q^{87} + 20952 q^{89} - 243808 q^{91} - 105600 q^{93} - 59456 q^{97} - 133424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 20x^{2} + 33x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{3} + 40\nu - 29 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\nu^{3} + 40\nu^{2} - 240\nu - 141 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{3} + 40\nu^{2} + 120\nu - 516 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 5\beta _1 + 20 ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} + \beta_{2} - 3\beta _1 + 820 ) / 80 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 3\beta _1 - 38 ) / 4 \) 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
1.64654
3.98753
−4.73066
0.0965878
0 −28.9338 0 0 0 −146.828 0 594.165 0
1.2 0 −4.69449 0 0 0 −10.2635 0 −220.962 0
1.3 0 5.49000 0 0 0 188.968 0 −212.860 0
1.4 0 24.1383 0 0 0 −179.876 0 339.657 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.j 4
4.b odd 2 1 400.6.a.ba 4
5.b even 2 1 200.6.a.k 4
5.c odd 4 2 40.6.c.a 8
15.e even 4 2 360.6.f.b 8
20.d odd 2 1 400.6.a.z 4
20.e even 4 2 80.6.c.d 8
40.i odd 4 2 320.6.c.j 8
40.k even 4 2 320.6.c.i 8
60.l odd 4 2 720.6.f.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 5.c odd 4 2
80.6.c.d 8 20.e even 4 2
200.6.a.j 4 1.a even 1 1 trivial
200.6.a.k 4 5.b even 2 1
320.6.c.i 8 40.k even 4 2
320.6.c.j 8 40.i odd 4 2
360.6.f.b 8 15.e even 4 2
400.6.a.z 4 20.d odd 2 1
400.6.a.ba 4 4.b odd 2 1
720.6.f.n 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} - 728T_{3}^{2} + 432T_{3} + 18000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - 728 T^{2} + \cdots + 18000 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 148 T^{3} + \cdots - 51222832 \) Copy content Toggle raw display
$11$ \( T^{4} + 368 T^{3} + \cdots + 37397137664 \) Copy content Toggle raw display
$13$ \( T^{4} + 440 T^{3} + \cdots + 10683053312 \) Copy content Toggle raw display
$17$ \( T^{4} - 672 T^{3} + \cdots + 22118400000 \) Copy content Toggle raw display
$19$ \( T^{4} + 688 T^{3} + \cdots + 1042985883904 \) Copy content Toggle raw display
$23$ \( T^{4} + 4492 T^{3} + \cdots - 5643370924592 \) Copy content Toggle raw display
$29$ \( T^{4} + 2936 T^{3} + \cdots - 97147517576176 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 244229603328000 \) Copy content Toggle raw display
$37$ \( T^{4} + 8792 T^{3} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 296811236945008 \) Copy content Toggle raw display
$43$ \( T^{4} + 48276 T^{3} + \cdots - 74\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{4} + 14724 T^{3} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + 84296 T^{3} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} + 45840 T^{3} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{4} - 61928 T^{3} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + 72700 T^{3} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{4} + 62816 T^{3} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + 133072 T^{3} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{4} + 21632 T^{3} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + 74660 T^{3} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{4} - 20952 T^{3} + \cdots - 93\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{4} + 59456 T^{3} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
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