Properties

Label 800.6.c.k
Level 800
Weight 6
Character orbit 800.c
Analytic conductor 128.307
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.6140289600.1
Defining polynomial: \(x^{6} - 2 x^{5} + 32 x^{4} + 116 x^{3} + 256 x^{2} + 2778 x + 7605\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \beta_{1} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( -157 - 6 \beta_{2} + 2 \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( -2 \beta_{1} + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( -157 - 6 \beta_{2} + 2 \beta_{4} ) q^{9} + ( 131 - 3 \beta_{4} ) q^{11} + ( -53 \beta_{1} - 20 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -211 \beta_{1} + 60 \beta_{3} + 6 \beta_{5} ) q^{17} + ( 1072 + 25 \beta_{2} - \beta_{4} ) q^{19} + ( -260 + 36 \beta_{2} + 4 \beta_{4} ) q^{21} + ( -1009 \beta_{1} - 43 \beta_{3} - 7 \beta_{5} ) q^{23} + ( 2368 \beta_{1} - 182 \beta_{3} + 20 \beta_{5} ) q^{27} + ( -118 + 20 \beta_{2} + 52 \beta_{4} ) q^{29} + ( -1061 + 135 \beta_{2} - 42 \beta_{4} ) q^{31} + ( -526 \beta_{1} + 380 \beta_{3} + 6 \beta_{5} ) q^{33} + ( 2019 \beta_{1} - 200 \beta_{3} + 76 \beta_{5} ) q^{37} + ( 7083 - 55 \beta_{2} - 44 \beta_{4} ) q^{39} + ( 4154 + 34 \beta_{2} - 22 \beta_{4} ) q^{41} + ( -4460 \beta_{1} + 221 \beta_{3} + 14 \beta_{5} ) q^{43} + ( 6093 \beta_{1} + 77 \beta_{3} - 25 \beta_{5} ) q^{47} + ( 1311 + 158 \beta_{2} - 74 \beta_{4} ) q^{49} + ( -23839 - 205 \beta_{2} + 132 \beta_{4} ) q^{51} + ( -3869 \beta_{1} + 1140 \beta_{3} + 114 \beta_{5} ) q^{53} + ( -12082 \beta_{1} + 1580 \beta_{3} - 98 \beta_{5} ) q^{57} + ( 11730 - 35 \beta_{2} + 185 \beta_{4} ) q^{59} + ( -7726 + 992 \beta_{2} - 32 \beta_{4} ) q^{61} + ( -13555 \beta_{1} + 263 \beta_{3} + 91 \beta_{5} ) q^{63} + ( -2404 \beta_{1} + 2345 \beta_{3} - 82 \beta_{5} ) q^{67} + ( 8604 - 1124 \beta_{2} - 100 \beta_{4} ) q^{69} + ( -29517 - 515 \beta_{2} + 56 \beta_{4} ) q^{71} + ( 10295 \beta_{1} + 2340 \beta_{3} - 150 \beta_{5} ) q^{73} + ( 22426 \beta_{1} + 860 \beta_{3} - 106 \beta_{5} ) q^{77} + ( -30922 + 350 \beta_{2} - 164 \beta_{4} ) q^{79} + ( 51337 + 2458 \beta_{2} + 162 \beta_{4} ) q^{81} + ( -4194 \beta_{1} - 2923 \beta_{3} - 376 \beta_{5} ) q^{83} + ( -3068 \beta_{1} - 4094 \beta_{3} - 184 \beta_{5} ) q^{87} + ( -57474 + 300 \beta_{2} - 36 \beta_{4} ) q^{89} + ( 35217 - 1055 \beta_{2} - 106 \beta_{4} ) q^{91} + ( -54764 \beta_{1} + 4720 \beta_{3} - 456 \beta_{5} ) q^{93} + ( -28155 \beta_{1} - 1420 \beta_{3} - 430 \beta_{5} ) q^{97} + ( -117731 - 1800 \beta_{2} + 43 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 934q^{9} + O(q^{10}) \) \( 6q - 934q^{9} + 792q^{11} + 6384q^{19} - 1640q^{21} - 852q^{29} - 6552q^{31} + 42696q^{39} + 24900q^{41} + 7698q^{49} - 142888q^{51} + 70080q^{59} - 48276q^{61} + 54072q^{69} - 176184q^{71} - 185904q^{79} + 302782q^{81} - 345372q^{89} + 213624q^{91} - 702872q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 32 x^{4} + 116 x^{3} + 256 x^{2} + 2778 x + 7605\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 7 \nu^{5} - 275 \nu^{4} + 2174 \nu^{3} - 5515 \nu^{2} + 8137 \nu + 106320 \)\()/58125\)
\(\beta_{2}\)\(=\)\((\)\( 8 \nu^{5} + 256 \nu^{3} + 1440 \nu^{2} - 72 \nu + 25205 \)\()/625\)
\(\beta_{3}\)\(=\)\((\)\( -173 \nu^{5} + 685 \nu^{4} - 6166 \nu^{3} + 2645 \nu^{2} - 27323 \nu - 212280 \)\()/11625\)
\(\beta_{4}\)\(=\)\((\)\( -16 \nu^{5} + 200 \nu^{4} - 112 \nu^{3} - 680 \nu^{2} + 20544 \nu + 12965 \)\()/625\)
\(\beta_{5}\)\(=\)\((\)\( 1061 \nu^{5} - 1825 \nu^{4} + 32802 \nu^{3} + 89655 \nu^{2} + 693051 \nu + 1899360 \)\()/19375\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} + 10\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{5} - 2 \beta_{4} + 22 \beta_{3} + 8 \beta_{2} + 79 \beta_{1} - 318\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 28 \beta_{2} + 395 \beta_{1} - 1408\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(-115 \beta_{5} + 106 \beta_{4} - 454 \beta_{3} + 204 \beta_{2} - 2959 \beta_{1} - 2030\)\()/32\)
\(\nu^{5}\)\(=\)\((\)\(-211 \beta_{5} + 104 \beta_{4} - 4070 \beta_{3} - 750 \beta_{2} - 39455 \beta_{1} + 46622\)\()/32\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−2.90341 + 0.978064i
3.05894 4.88658i
0.844467 + 4.86464i
0.844467 4.86464i
3.05894 + 4.88658i
−2.90341 0.978064i
0 29.2272i 0 0 0 44.5253i 0 −611.232 0
449.2 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.3 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.4 0 0.755735i 0 0 0 172.424i 0 242.429 0
449.5 0 18.4715i 0 0 0 121.899i 0 −98.1968 0
449.6 0 29.2272i 0 0 0 44.5253i 0 −611.232 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.6
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 800.6.c.k 6
4.b odd 2 1 800.6.c.j 6
5.b even 2 1 inner 800.6.c.k 6
5.c odd 4 1 160.6.a.f 3
5.c odd 4 1 800.6.a.o 3
20.d odd 2 1 800.6.c.j 6
20.e even 4 1 160.6.a.g yes 3
20.e even 4 1 800.6.a.n 3
40.i odd 4 1 320.6.a.y 3
40.k even 4 1 320.6.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 5.c odd 4 1
160.6.a.g yes 3 20.e even 4 1
320.6.a.x 3 40.k even 4 1
320.6.a.y 3 40.i odd 4 1
800.6.a.n 3 20.e even 4 1
800.6.a.o 3 5.c odd 4 1
800.6.c.j 6 4.b odd 2 1
800.6.c.j 6 20.d odd 2 1
800.6.c.k 6 1.a even 1 1 trivial
800.6.c.k 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{6} + 1196 T_{3}^{4} + 292144 T_{3}^{2} + 166464 \)
\( T_{11}^{3} - 396 T_{11}^{2} - 155280 T_{11} + 59934400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 262 T^{2} + 15367 T^{4} - 5058036 T^{6} + 907405983 T^{8} - 913537513062 T^{10} + 205891132094649 T^{12} \)
$5$ 1
$7$ \( 1 - 54270 T^{2} + 1636355967 T^{4} - 32963917489060 T^{6} + 462230059230960783 T^{8} - \)\(43\!\cdots\!70\)\( T^{10} + \)\(22\!\cdots\!49\)\( T^{12} \)
$11$ \( ( 1 - 396 T + 327873 T^{2} - 67617992 T^{3} + 52804274523 T^{4} - 10271220141996 T^{5} + 4177248169415651 T^{6} )^{2} \)
$13$ \( 1 - 1565442 T^{2} + 1135881445383 T^{4} - 513409425866197436 T^{6} + \)\(15\!\cdots\!67\)\( T^{8} - \)\(29\!\cdots\!42\)\( T^{10} + \)\(26\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 - 2487258 T^{2} + 7838169507183 T^{4} - 10425763805475099564 T^{6} + \)\(15\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!58\)\( T^{10} + \)\(81\!\cdots\!49\)\( T^{12} \)
$19$ \( ( 1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6} )^{2} \)
$23$ \( 1 - 21917310 T^{2} + 235143882878367 T^{4} - \)\(17\!\cdots\!80\)\( T^{6} + \)\(97\!\cdots\!83\)\( T^{8} - \)\(37\!\cdots\!10\)\( T^{10} + \)\(71\!\cdots\!49\)\( T^{12} \)
$29$ \( ( 1 + 426 T - 939693 T^{2} - 141773503364 T^{3} - 19274183137257 T^{4} + 179221281385885626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} )^{2} \)
$31$ \( ( 1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} )^{2} \)
$37$ \( 1 - 47567538 T^{2} + 8632350495280023 T^{4} - \)\(24\!\cdots\!84\)\( T^{6} + \)\(41\!\cdots\!27\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(11\!\cdots\!49\)\( T^{12} \)
$41$ \( ( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} )^{2} \)
$43$ \( 1 - 587098422 T^{2} + 171971726662843287 T^{4} - \)\(31\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!63\)\( T^{8} - \)\(27\!\cdots\!22\)\( T^{10} + \)\(10\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - 888472110 T^{2} + 408520605892907727 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!23\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{10} + \)\(14\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 - 343748754 T^{2} + 524992621000918647 T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(91\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(53\!\cdots\!49\)\( T^{12} \)
$59$ \( ( 1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} )^{2} \)
$61$ \( ( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} + \)\(17\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} )^{2} \)
$67$ \( 1 - 1227086118 T^{2} + 624926915335274247 T^{4} + \)\(11\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!03\)\( T^{8} - \)\(40\!\cdots\!18\)\( T^{10} + \)\(60\!\cdots\!49\)\( T^{12} \)
$71$ \( ( 1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} + \)\(28\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} )^{2} \)
$73$ \( 1 - 3294592458 T^{2} + 16216281639521153535 T^{4} - \)\(29\!\cdots\!40\)\( T^{6} + \)\(69\!\cdots\!15\)\( T^{8} - \)\(60\!\cdots\!58\)\( T^{10} + \)\(79\!\cdots\!49\)\( T^{12} \)
$79$ \( ( 1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 34353638858281213923 T^{4} + \)\(88\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} )^{2} \)
$83$ \( 1 - 7671858246 T^{2} + 26481453986477119527 T^{4} - \)\(57\!\cdots\!28\)\( T^{6} + \)\(41\!\cdots\!23\)\( T^{8} - \)\(18\!\cdots\!46\)\( T^{10} + \)\(37\!\cdots\!49\)\( T^{12} \)
$89$ \( ( 1 + 172686 T + 26445328791 T^{2} + 2103593815517412 T^{3} + \)\(14\!\cdots\!59\)\( T^{4} + \)\(53\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} )^{2} \)
$97$ \( 1 - 31357705242 T^{2} + \)\(46\!\cdots\!35\)\( T^{4} - \)\(45\!\cdots\!60\)\( T^{6} + \)\(34\!\cdots\!15\)\( T^{8} - \)\(17\!\cdots\!42\)\( T^{10} + \)\(40\!\cdots\!49\)\( T^{12} \)
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