# Properties

 Label 200.6.c.e Level 200 Weight 6 Character orbit 200.c Analytic conductor 32.077 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0767639626$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{129})$$ Defining polynomial: $$x^{4} + 65 x^{2} + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 \beta_{1} + \beta_{2} ) q^{3} + ( -13 \beta_{1} - 3 \beta_{2} ) q^{7} + ( -309 + 6 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -3 \beta_{1} + \beta_{2} ) q^{3} + ( -13 \beta_{1} - 3 \beta_{2} ) q^{7} + ( -309 + 6 \beta_{3} ) q^{9} + ( 280 - 3 \beta_{3} ) q^{11} + ( 347 \beta_{1} - 12 \beta_{2} ) q^{13} + ( -37 \beta_{1} - 84 \beta_{2} ) q^{17} + ( 500 + 18 \beta_{3} ) q^{19} + ( 1392 + 4 \beta_{3} ) q^{21} + ( -613 \beta_{1} - 123 \beta_{2} ) q^{23} + ( 3294 \beta_{1} - 138 \beta_{2} ) q^{27} + ( -670 + 156 \beta_{3} ) q^{29} + ( -1124 - 27 \beta_{3} ) q^{31} + ( -2388 \beta_{1} + 316 \beta_{2} ) q^{33} + ( 1485 \beta_{1} + 216 \beta_{2} ) q^{37} + ( 10356 - 383 \beta_{3} ) q^{39} + ( 11538 + 78 \beta_{3} ) q^{41} + ( 4421 \beta_{1} - 339 \beta_{2} ) q^{43} + ( 727 \beta_{1} + 885 \beta_{2} ) q^{47} + ( 11487 - 78 \beta_{3} ) q^{49} + ( 42900 - 215 \beta_{3} ) q^{51} + ( -1353 \beta_{1} + 540 \beta_{2} ) q^{53} + ( 7788 \beta_{1} + 284 \beta_{2} ) q^{57} + ( -31292 - 252 \beta_{3} ) q^{59} + ( 7054 + 552 \beta_{3} ) q^{61} + ( -5271 \beta_{1} + 615 \beta_{2} ) q^{63} + ( 21353 \beta_{1} - 543 \beta_{2} ) q^{67} + ( 56112 + 244 \beta_{3} ) q^{69} + ( 23604 + 273 \beta_{3} ) q^{71} + ( -16863 \beta_{1} - 1308 \beta_{2} ) q^{73} + ( 1004 \beta_{1} - 684 \beta_{2} ) q^{77} + ( 32952 + 1254 \beta_{3} ) q^{79} + ( 35649 - 2250 \beta_{3} ) q^{81} + ( 27181 \beta_{1} - 711 \beta_{2} ) q^{83} + ( 82506 \beta_{1} - 2542 \beta_{2} ) q^{87} + ( 27510 - 732 \beta_{3} ) q^{89} + ( -532 + 885 \beta_{3} ) q^{91} + ( -10560 \beta_{1} - 800 \beta_{2} ) q^{93} + ( -36917 \beta_{1} - 4620 \beta_{2} ) q^{97} + ( -123672 + 2607 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1236q^{9} + O(q^{10})$$ $$4q - 1236q^{9} + 1120q^{11} + 2000q^{19} + 5568q^{21} - 2680q^{29} - 4496q^{31} + 41424q^{39} + 46152q^{41} + 45948q^{49} + 171600q^{51} - 125168q^{59} + 28216q^{61} + 224448q^{69} + 94416q^{71} + 131808q^{79} + 142596q^{81} + 110040q^{89} - 2128q^{91} - 494688q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 65 x^{2} + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 33 \nu$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 97 \nu$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} + 260$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 260$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-33 \beta_{2} + 97 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 6.17891i − 5.17891i 5.17891i 6.17891i
0 28.7156i 0 0 0 42.1469i 0 −581.588 0
49.2 0 16.7156i 0 0 0 94.1469i 0 −36.4124 0
49.3 0 16.7156i 0 0 0 94.1469i 0 −36.4124 0
49.4 0 28.7156i 0 0 0 42.1469i 0 −581.588 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 200.6.c.e 4
4.b odd 2 1 400.6.c.l 4
5.b even 2 1 inner 200.6.c.e 4
5.c odd 4 1 40.6.a.d 2
5.c odd 4 1 200.6.a.g 2
15.e even 4 1 360.6.a.l 2
20.d odd 2 1 400.6.c.l 4
20.e even 4 1 80.6.a.i 2
20.e even 4 1 400.6.a.q 2
40.i odd 4 1 320.6.a.w 2
40.k even 4 1 320.6.a.q 2
60.l odd 4 1 720.6.a.z 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 1104 T_{3}^{2} + 230400$$ acting on $$S_{6}^{\mathrm{new}}(200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 132 T^{2} + 48150 T^{4} + 7794468 T^{6} + 3486784401 T^{8}$$
$5$ 1
$7$ $$1 - 56588 T^{2} + 1352943558 T^{4} - 15984709390412 T^{6} + 79792266297612001 T^{8}$$
$11$ $$( 1 - 560 T + 381926 T^{2} - 90188560 T^{3} + 25937424601 T^{4} )^{2}$$
$13$ $$1 - 373292 T^{2} + 167403787638 T^{4} - 51461472139296908 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$1 + 1613316 T^{2} + 4602934743878 T^{4} + 3252435215496778884 T^{6} +$$$$40\!\cdots\!01$$$$T^{8}$$
$19$ $$( 1 - 1000 T + 4533462 T^{2} - 2476099000 T^{3} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 - 7126092 T^{2} + 48612883261958 T^{4} -$$$$29\!\cdots\!08$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 + 1340 T - 8758306 T^{2} + 27484939660 T^{3} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 2248 T + 57017022 T^{2} + 64358331448 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 211585036 T^{2} + 19959790722550422 T^{4} -$$$$10\!\cdots\!64$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 313073372 T^{2} + 49182412950657078 T^{4} -$$$$67\!\cdots\!28$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 104863596 T^{2} + 104529727880710502 T^{4} -$$$$55\!\cdots\!04$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$1 - 1357205900 T^{2} + 805869805574922198 T^{4} -$$$$23\!\cdots\!00$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$( 1 + 62584 T + 2277965606 T^{2} + 44742822328616 T^{3} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$1 - 1448611388 T^{2} + 3060385933795078038 T^{4} -$$$$26\!\cdots\!12$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 47208 T + 4011779662 T^{2} - 85174059202008 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$1 - 4251788572 T^{2} + 9098112686809466598 T^{4} -$$$$18\!\cdots\!28$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8}$$
$79$ $$( 1 - 65904 T + 3994274078 T^{2} - 202790324919696 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 9324010812 T^{2} + 49682906641812448598 T^{4} -$$$$14\!\cdots\!88$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 - 55020 T + 10818978262 T^{2} - 307234950883980 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$1 - 1419021116 T^{2} - 92174956617487206138 T^{4} -$$$$10\!\cdots\!84$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$