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: \(x^{4} - x^{3} - 20 x^{2} + 33 x - 3\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -2 \nu^{3} + 40 \nu - 29 \)
\(\beta_{2}\)\(=\)\((\)\( 22 \nu^{3} + 40 \nu^{2} - 240 \nu - 141 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 40 \nu^{2} + 120 \nu - 516 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 5 \beta_{1} + 20\)\()/80\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{3} + \beta_{2} - 3 \beta_{1} + 820\)\()/80\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 3 \beta_{1} - 38\)\()/4\)

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T + 244 T^{2} + 3348 T^{3} + 18486 T^{4} + 813564 T^{5} + 14407956 T^{6} + 57395628 T^{7} + 3486784401 T^{8} \)
$5$ 1
$7$ \( 1 + 148 T + 33316 T^{2} + 2108948 T^{3} + 503710694 T^{4} + 35445089036 T^{5} + 9410945395684 T^{6} + 702639103471564 T^{7} + 79792266297612001 T^{8} \)
$11$ \( 1 + 368 T + 176204 T^{2} + 51158896 T^{3} + 42277949270 T^{4} + 8239191359696 T^{5} + 4570277964394604 T^{6} + 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( 1 + 440 T + 886036 T^{2} + 564452872 T^{3} + 392923998710 T^{4} + 209577400203496 T^{5} + 122147586683920564 T^{6} + 22521792926199933080 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} \)
$17$ \( 1 - 672 T + 3356228 T^{2} - 3540729312 T^{3} + 5520858237894 T^{4} - 5027329298748384 T^{5} + 6766135176516146372 T^{6} - \)\(19\!\cdots\!96\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 + 4492 T + 27426980 T^{2} + 79192657228 T^{3} + 264562508116390 T^{4} + 509711105000837204 T^{5} + \)\(11\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!44\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( 1 + 2936 T + 58625996 T^{2} + 77951973928 T^{3} + 1466411094282230 T^{4} + 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$31$ \( 1 - 2112 T + 80187004 T^{2} - 163080265536 T^{3} + 3196344720873606 T^{4} - 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} - \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$37$ \( 1 + 8792 T + 164536948 T^{2} + 653354012200 T^{3} + 11523689002604438 T^{4} + 45306152527774275400 T^{5} + \)\(79\!\cdots\!52\)\( T^{6} + \)\(29\!\cdots\!56\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$43$ \( 1 + 48276 T + 1306891924 T^{2} + 24102642735684 T^{3} + 333606713965182294 T^{4} + \)\(35\!\cdots\!12\)\( T^{5} + \)\(28\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!32\)\( T^{7} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 14724 T + 590074244 T^{2} + 7755349482468 T^{3} + 177012391425566214 T^{4} + \)\(17\!\cdots\!76\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 + 84296 T + 4144600628 T^{2} + 135070817073080 T^{3} + 3234464962764166486 T^{4} + \)\(56\!\cdots\!40\)\( T^{5} + \)\(72\!\cdots\!72\)\( T^{6} + \)\(61\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} + \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 + 72700 T + 7283604532 T^{2} + 314621051775212 T^{3} + 16093744308113184182 T^{4} + \)\(42\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( 1 + 62816 T + 3398787356 T^{2} + 184024084124896 T^{3} + 8353296562609817510 T^{4} + \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} \)
$73$ \( 1 + 133072 T + 13424418340 T^{2} + 846379227250928 T^{3} + 45561026057566462310 T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} + \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \)
$83$ \( 1 + 74660 T + 6167006708 T^{2} + 554766838355444 T^{3} + 42874466663031633142 T^{4} + \)\(21\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!92\)\( T^{6} + \)\(45\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( 1 - 20952 T + 16118164796 T^{2} - 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} - \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \)
$97$ \( 1 + 59456 T + 24399465604 T^{2} + 684112850575552 T^{3} + \)\(25\!\cdots\!74\)\( T^{4} + \)\(58\!\cdots\!64\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} + \)\(37\!\cdots\!08\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \)
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