LMFDB - Embedded newform 200.6.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,6,Mod(1,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 33x - 3 $ x^4 - x^3 - 20x^2 + 33x - 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:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.73066$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.49000 q^{3} -188.968 q^{7} -212.860 q^{9} -501.871 q^{11} +1061.48 q^{13} -29.5861 q^{17} +1578.33 q^{19} +1037.43 q^{21} -1295.86 q^{23} +2502.67 q^{27} -3586.63 q^{29} -3526.32 q^{31} +2755.27 q^{33} +8413.79 q^{37} -5827.54 q^{39} +7015.12 q^{41} +22694.2 q^{43} -3501.99 q^{47} +18901.8 q^{49} +162.428 q^{51} +27309.1 q^{53} -8665.04 q^{57} +7925.39 q^{59} -7020.54 q^{61} +40223.6 q^{63} +17631.2 q^{67} +7114.27 q^{69} +13432.9 q^{71} +39946.8 q^{73} +94837.3 q^{77} -93321.2 q^{79} +37985.3 q^{81} -58448.5 q^{83} +19690.6 q^{87} -13989.4 q^{89} -200586. q^{91} +19359.5 q^{93} -110640. q^{97} +106828. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 133424 q^{99}+O(q^{100}) $ 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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−5.49000	−0.352184	−0.176092	−	0.984374i	$-0.556346\pi$
$3$	−5.49000	−0.352184	−0.176092	+	0.984374i	$0.556346\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−188.968	−1.45761	−0.728807	−	0.684720i	$-0.759925\pi$
$7$	−188.968	−1.45761	−0.728807	+	0.684720i	$0.759925\pi$
$8$	0	0
$9$	−212.860	−0.875967
$10$	0	0
$11$	−501.871	−1.25058	−0.625288	−	0.780394i	$-0.715018\pi$
$11$	−501.871	−1.25058	−0.625288	+	0.780394i	$0.715018\pi$
$12$	0	0
$13$	1061.48	1.74203	0.871013	−	0.491259i	$-0.163463\pi$
$13$	1061.48	1.74203	0.871013	+	0.491259i	$0.163463\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−29.5861	−0.0248294	−0.0124147	−	0.999923i	$-0.503952\pi$
$17$	−29.5861	−0.0248294	−0.0124147	+	0.999923i	$0.503952\pi$
$18$	0	0
$19$	1578.33	1.00303	0.501515	−	0.865149i	$-0.332776\pi$
$19$	1578.33	1.00303	0.501515	+	0.865149i	$0.332776\pi$
$20$	0	0
$21$	1037.43	0.513347
$22$	0	0
$23$	−1295.86	−0.510786	−0.255393	−	0.966837i	$-0.582205\pi$
$23$	−1295.86	−0.510786	−0.255393	+	0.966837i	$0.582205\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2502.67	0.660685
$28$	0	0
$29$	−3586.63	−0.791938	−0.395969	−	0.918264i	$-0.629591\pi$
$29$	−3586.63	−0.791938	−0.395969	+	0.918264i	$0.629591\pi$
$30$	0	0
$31$	−3526.32	−0.659048	−0.329524	−	0.944147i	$-0.606888\pi$
$31$	−3526.32	−0.659048	−0.329524	+	0.944147i	$0.606888\pi$
$32$	0	0
$33$	2755.27	0.440432
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8413.79	1.01039	0.505193	−	0.863006i	$-0.331421\pi$
$37$	8413.79	1.01039	0.505193	+	0.863006i	$0.331421\pi$
$38$	0	0
$39$	−5827.54	−0.613513
$40$	0	0
$41$	7015.12	0.651742	0.325871	−	0.945414i	$-0.394343\pi$
$41$	7015.12	0.651742	0.325871	+	0.945414i	$0.394343\pi$
$42$	0	0
$43$	22694.2	1.87173	0.935865	−	0.352358i	$-0.114620\pi$
$43$	22694.2	1.87173	0.935865	+	0.352358i	$0.114620\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3501.99	−0.231244	−0.115622	−	0.993293i	$-0.536886\pi$
$47$	−3501.99	−0.231244	−0.115622	+	0.993293i	$0.536886\pi$
$48$	0	0
$49$	18901.8	1.12464
$50$	0	0
$51$	162.428	0.00874449
$52$	0	0
$53$	27309.1	1.33542	0.667710	−	0.744422i	$-0.267275\pi$
$53$	27309.1	1.33542	0.667710	+	0.744422i	$0.267275\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8665.04	−0.353251
$58$	0	0
$59$	7925.39	0.296409	0.148204	−	0.988957i	$-0.452651\pi$
$59$	7925.39	0.296409	0.148204	+	0.988957i	$0.452651\pi$
$60$	0	0
$61$	−7020.54	−0.241572	−0.120786	−	0.992679i	$-0.538541\pi$
$61$	−7020.54	−0.241572	−0.120786	+	0.992679i	$0.538541\pi$
$62$	0	0
$63$	40223.6	1.27682
$64$	0	0
$65$	0	0
$66$	0	0
$67$	17631.2	0.479837	0.239919	−	0.970793i	$-0.422879\pi$
$67$	17631.2	0.479837	0.239919	+	0.970793i	$0.422879\pi$
$68$	0	0
$69$	7114.27	0.179890
$70$	0	0
$71$	13432.9	0.316246	0.158123	−	0.987419i	$-0.449456\pi$
$71$	13432.9	0.316246	0.158123	+	0.987419i	$0.449456\pi$
$72$	0	0
$73$	39946.8	0.877353	0.438677	−	0.898645i	$-0.355447\pi$
$73$	39946.8	0.877353	0.438677	+	0.898645i	$0.355447\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	94837.3	1.82286
$78$	0	0
$79$	−93321.2	−1.68234	−0.841168	−	0.540774i	$-0.818131\pi$
$79$	−93321.2	−1.68234	−0.841168	+	0.540774i	$0.818131\pi$
$80$	0	0
$81$	37985.3	0.643284
$82$	0	0
$83$	−58448.5	−0.931275	−0.465638	−	0.884975i	$-0.654175\pi$
$83$	−58448.5	−0.931275	−0.465638	+	0.884975i	$0.654175\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19690.6	0.278908
$88$	0	0
$89$	−13989.4	−0.187208	−0.0936038	−	0.995610i	$-0.529839\pi$
$89$	−13989.4	−0.187208	−0.0936038	+	0.995610i	$0.529839\pi$
$90$	0	0
$91$	−200586.	−2.53920
$92$	0	0
$93$	19359.5	0.232106
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−110640.	−1.19394	−0.596969	−	0.802264i	$-0.703629\pi$
$97$	−110640.	−1.19394	−0.596969	+	0.802264i	$0.703629\pi$
$98$	0	0
$99$	106828.	1.09546
$100$	0	0
$101$	198468.	1.93592	0.967960	−	0.251103i	$-0.0807934\pi$
$101$	198468.	1.93592	0.967960	+	0.251103i	$0.0807934\pi$
$102$	0	0
$103$	134780.	1.25179	0.625897	−	0.779906i	$-0.284733\pi$
$103$	134780.	1.25179	0.625897	+	0.779906i	$0.284733\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	39785.5	0.335943	0.167971	−	0.985792i	$-0.446278\pi$
$107$	39785.5	0.335943	0.167971	+	0.985792i	$0.446278\pi$
$108$	0	0
$109$	92692.6	0.747272	0.373636	−	0.927575i	$-0.378111\pi$
$109$	92692.6	0.747272	0.373636	+	0.927575i	$0.378111\pi$
$110$	0	0
$111$	−46191.7	−0.355841
$112$	0	0
$113$	−66923.1	−0.493037	−0.246519	−	0.969138i	$-0.579287\pi$
$113$	−66923.1	−0.493037	−0.246519	+	0.969138i	$0.579287\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−225947.	−1.52596
$118$	0	0
$119$	5590.81	0.0361916
$120$	0	0
$121$	90823.1	0.563940
$122$	0	0
$123$	−38513.0	−0.229533
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−58998.9	−0.324589	−0.162295	−	0.986742i	$-0.551890\pi$
$127$	−58998.9	−0.324589	−0.162295	+	0.986742i	$0.551890\pi$
$128$	0	0
$129$	−124591.	−0.659193
$130$	0	0
$131$	−209380.	−1.06600	−0.533001	−	0.846115i	$-0.678936\pi$
$131$	−209380.	−1.06600	−0.533001	+	0.846115i	$0.678936\pi$
$132$	0	0
$133$	−298254.	−1.46203
$134$	0	0
$135$	0	0
$136$	0	0
$137$	326432.	1.48591	0.742953	−	0.669344i	$-0.233425\pi$
$137$	326432.	1.48591	0.742953	+	0.669344i	$0.233425\pi$
$138$	0	0
$139$	−233648.	−1.02571	−0.512856	−	0.858475i	$-0.671412\pi$
$139$	−233648.	−1.02571	−0.512856	+	0.858475i	$0.671412\pi$
$140$	0	0
$141$	19225.9	0.0814403
$142$	0	0
$143$	−532727.	−2.17854
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−103771.	−0.396078
$148$	0	0
$149$	145336.	0.536299	0.268149	−	0.963377i	$-0.413588\pi$
$149$	145336.	0.536299	0.268149	+	0.963377i	$0.413588\pi$
$150$	0	0
$151$	287343.	1.02555	0.512777	−	0.858522i	$-0.328617\pi$
$151$	287343.	1.02555	0.512777	+	0.858522i	$0.328617\pi$
$152$	0	0
$153$	6297.69	0.0217497
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−42626.2	−0.138015	−0.0690077	−	0.997616i	$-0.521983\pi$
$157$	−42626.2	−0.138015	−0.0690077	+	0.997616i	$0.521983\pi$
$158$	0	0
$159$	−149927.	−0.470313
$160$	0	0
$161$	244876.	0.744528
$162$	0	0
$163$	−221599.	−0.653279	−0.326639	−	0.945149i	$-0.605916\pi$
$163$	−221599.	−0.653279	−0.326639	+	0.945149i	$0.605916\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−334835.	−0.929051	−0.464525	−	0.885560i	$-0.653775\pi$
$167$	−334835.	−0.929051	−0.464525	+	0.885560i	$0.653775\pi$
$168$	0	0
$169$	755454.	2.03466
$170$	0	0
$171$	−335964.	−0.878622
$172$	0	0
$173$	−268475.	−0.682006	−0.341003	−	0.940062i	$-0.610767\pi$
$173$	−268475.	−0.682006	−0.341003	+	0.940062i	$0.610767\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−43510.4	−0.104390
$178$	0	0
$179$	−268052.	−0.625297	−0.312649	−	0.949869i	$-0.601216\pi$
$179$	−268052.	−0.625297	−0.312649	+	0.949869i	$0.601216\pi$
$180$	0	0
$181$	−90565.5	−0.205479	−0.102739	−	0.994708i	$-0.532761\pi$
$181$	−90565.5	−0.205479	−0.102739	+	0.994708i	$0.532761\pi$
$182$	0	0
$183$	38542.8	0.0850776
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14848.4	0.0310510
$188$	0	0
$189$	−472924.	−0.963023
$190$	0	0
$191$	−706725.	−1.40174	−0.700870	−	0.713290i	$-0.747204\pi$
$191$	−706725.	−1.40174	−0.700870	+	0.713290i	$0.747204\pi$
$192$	0	0
$193$	236999.	0.457988	0.228994	−	0.973428i	$-0.426456\pi$
$193$	236999.	0.457988	0.228994	+	0.973428i	$0.426456\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−607901.	−1.11601	−0.558004	−	0.829838i	$-0.688433\pi$
$197$	−607901.	−1.11601	−0.558004	+	0.829838i	$0.688433\pi$
$198$	0	0
$199$	621060.	1.11173	0.555867	−	0.831272i	$-0.312387\pi$
$199$	621060.	1.11173	0.555867	+	0.831272i	$0.312387\pi$
$200$	0	0
$201$	−96795.1	−0.168991
$202$	0	0
$203$	677756.	1.15434
$204$	0	0
$205$	0	0
$206$	0	0
$207$	275837.	0.447431
$208$	0	0
$209$	−792118.	−1.25437
$210$	0	0
$211$	1.06177e6	1.64181	0.820906	−	0.571063i	$-0.193469\pi$
$211$	1.06177e6	1.64181	0.820906	+	0.571063i	$0.193469\pi$
$212$	0	0
$213$	−73746.8	−0.111377
$214$	0	0
$215$	0	0
$216$	0	0
$217$	666360.	0.960638
$218$	0	0
$219$	−219308.	−0.308989
$220$	0	0
$221$	−31405.2	−0.0432534
$222$	0	0
$223$	240721.	0.324154	0.162077	−	0.986778i	$-0.448181\pi$
$223$	240721.	0.324154	0.162077	+	0.986778i	$0.448181\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−202494.	−0.260824	−0.130412	−	0.991460i	$-0.541630\pi$
$227$	−202494.	−0.260824	−0.130412	+	0.991460i	$0.541630\pi$
$228$	0	0
$229$	−284279.	−0.358225	−0.179112	−	0.983829i	$-0.557323\pi$
$229$	−284279.	−0.358225	−0.179112	+	0.983829i	$0.557323\pi$
$230$	0	0
$231$	−520657.	−0.641980
$232$	0	0
$233$	345405.	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$233$	345405.	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	512333.	0.592491
$238$	0	0
$239$	1.59920e6	1.81095	0.905476	−	0.424398i	$-0.139514\pi$
$239$	1.59920e6	1.81095	0.905476	+	0.424398i	$0.139514\pi$
$240$	0	0
$241$	−42246.4	−0.0468541	−0.0234270	−	0.999726i	$-0.507458\pi$
$241$	−42246.4	−0.0468541	−0.0234270	+	0.999726i	$0.507458\pi$
$242$	0	0
$243$	−816688.	−0.887239
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.67537e6	1.74731
$248$	0	0
$249$	320882.	0.327980
$250$	0	0
$251$	802841.	0.804351	0.402175	−	0.915563i	$-0.368254\pi$
$251$	802841.	0.804351	0.402175	+	0.915563i	$0.368254\pi$
$252$	0	0
$253$	650354.	0.638776
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−941153.	−0.888848	−0.444424	−	0.895817i	$-0.646592\pi$
$257$	−941153.	−0.888848	−0.444424	+	0.895817i	$0.646592\pi$
$258$	0	0
$259$	−1.58993e6	−1.47275
$260$	0	0
$261$	763449.	0.693711
$262$	0	0
$263$	1.53130e6	1.36512	0.682560	−	0.730830i	$-0.260867\pi$
$263$	1.53130e6	1.36512	0.682560	+	0.730830i	$0.260867\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	76801.7	0.0659315
$268$	0	0
$269$	1.25712e6	1.05924	0.529621	−	0.848234i	$-0.322334\pi$
$269$	1.25712e6	1.05924	0.529621	+	0.848234i	$0.322334\pi$
$270$	0	0
$271$	843976.	0.698083	0.349041	−	0.937107i	$-0.386507\pi$
$271$	843976.	0.698083	0.349041	+	0.937107i	$0.386507\pi$
$272$	0	0
$273$	1.10122e6	0.894265
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.52070e6	1.19082	0.595408	−	0.803423i	$-0.296990\pi$
$277$	1.52070e6	1.19082	0.595408	+	0.803423i	$0.296990\pi$
$278$	0	0
$279$	750612.	0.577305
$280$	0	0
$281$	−1.97975e6	−1.49570	−0.747848	−	0.663870i	$-0.768913\pi$
$281$	−1.97975e6	−1.49570	−0.747848	+	0.663870i	$0.768913\pi$
$282$	0	0
$283$	2.02901e6	1.50597	0.752986	−	0.658036i	$-0.228613\pi$
$283$	2.02901e6	1.50597	0.752986	+	0.658036i	$0.228613\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.32563e6	−0.949987
$288$	0	0
$289$	−1.41898e6	−0.999384
$290$	0	0
$291$	607412.	0.420485
$292$	0	0
$293$	2.23140e6	1.51848	0.759238	−	0.650813i	$-0.225572\pi$
$293$	2.23140e6	1.51848	0.759238	+	0.650813i	$0.225572\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.25602e6	−0.826236
$298$	0	0
$299$	−1.37553e6	−0.889802
$300$	0	0
$301$	−4.28847e6	−2.72826
$302$	0	0
$303$	−1.08959e6	−0.681799
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−334318.	−0.202448	−0.101224	−	0.994864i	$-0.532276\pi$
$307$	−334318.	−0.202448	−0.101224	+	0.994864i	$0.532276\pi$
$308$	0	0
$309$	−739943.	−0.440861
$310$	0	0
$311$	2.39090e6	1.40172	0.700858	−	0.713300i	$-0.252800\pi$
$311$	2.39090e6	1.40172	0.700858	+	0.713300i	$0.252800\pi$
$312$	0	0
$313$	−1.88563e6	−1.08791	−0.543957	−	0.839113i	$-0.683075\pi$
$313$	−1.88563e6	−1.08791	−0.543957	+	0.839113i	$0.683075\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	179636.	0.100403	0.0502014	−	0.998739i	$-0.484014\pi$
$317$	179636.	0.100403	0.0502014	+	0.998739i	$0.484014\pi$
$318$	0	0
$319$	1.80002e6	0.990378
$320$	0	0
$321$	−218422.	−0.118313
$322$	0	0
$323$	−46696.7	−0.0249046
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−508882.	−0.263177
$328$	0	0
$329$	661763.	0.337064
$330$	0	0
$331$	168620.	0.0845937	0.0422968	−	0.999105i	$-0.486532\pi$
$331$	168620.	0.0845937	0.0422968	+	0.999105i	$0.486532\pi$
$332$	0	0
$333$	−1.79096e6	−0.885064
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.74240e6	0.835744	0.417872	−	0.908506i	$-0.362776\pi$
$337$	1.74240e6	0.835744	0.417872	+	0.908506i	$0.362776\pi$
$338$	0	0
$339$	367407.	0.173640
$340$	0	0
$341$	1.76976e6	0.824190
$342$	0	0
$343$	−395842.	−0.181672
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.89447e6	1.29046	0.645232	−	0.763987i	$-0.276761\pi$
$347$	2.89447e6	1.29046	0.645232	+	0.763987i	$0.276761\pi$
$348$	0	0
$349$	63803.8	0.0280403	0.0140202	−	0.999902i	$-0.495537\pi$
$349$	63803.8	0.0280403	0.0140202	+	0.999902i	$0.495537\pi$
$350$	0	0
$351$	2.65654e6	1.15093
$352$	0	0
$353$	−2.79206e6	−1.19258	−0.596290	−	0.802769i	$-0.703359\pi$
$353$	−2.79206e6	−1.19258	−0.596290	+	0.802769i	$0.703359\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−30693.6	−0.0127461
$358$	0	0
$359$	1.48391e6	0.607674	0.303837	−	0.952724i	$-0.401732\pi$
$359$	1.48391e6	0.607674	0.303837	+	0.952724i	$0.401732\pi$
$360$	0	0
$361$	15032.0	0.00607083
$362$	0	0
$363$	−498619.	−0.198610
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.08188e6	0.419289	0.209644	−	0.977778i	$-0.432769\pi$
$367$	1.08188e6	0.419289	0.209644	+	0.977778i	$0.432769\pi$
$368$	0	0
$369$	−1.49324e6	−0.570904
$370$	0	0
$371$	−5.16053e6	−1.94652
$372$	0	0
$373$	−127750.	−0.0475434	−0.0237717	−	0.999717i	$-0.507567\pi$
$373$	−127750.	−0.0475434	−0.0237717	+	0.999717i	$0.507567\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.80714e6	−1.37958
$378$	0	0
$379$	−1.61295e6	−0.576798	−0.288399	−	0.957510i	$-0.593123\pi$
$379$	−1.61295e6	−0.576798	−0.288399	+	0.957510i	$0.593123\pi$
$380$	0	0
$381$	323904.	0.114315
$382$	0	0
$383$	−3.96656e6	−1.38171	−0.690855	−	0.722993i	$-0.742766\pi$
$383$	−3.96656e6	−1.38171	−0.690855	+	0.722993i	$0.742766\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.83068e6	−1.63957
$388$	0	0
$389$	−3.54852e6	−1.18898	−0.594488	−	0.804104i	$-0.702645\pi$
$389$	−3.54852e6	−1.18898	−0.594488	+	0.804104i	$0.702645\pi$
$390$	0	0
$391$	38339.5	0.0126825
$392$	0	0
$393$	1.14950e6	0.375428
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−580854.	−0.184965	−0.0924827	−	0.995714i	$-0.529480\pi$
$397$	−580854.	−0.184965	−0.0924827	+	0.995714i	$0.529480\pi$
$398$	0	0
$399$	1.63741e6	0.514903
$400$	0	0
$401$	2.10413e6	0.653449	0.326725	−	0.945120i	$-0.394055\pi$
$401$	2.10413e6	0.653449	0.326725	+	0.945120i	$0.394055\pi$
$402$	0	0
$403$	−3.74313e6	−1.14808
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.22263e6	−1.26356
$408$	0	0
$409$	3.23687e6	0.956790	0.478395	−	0.878145i	$-0.341219\pi$
$409$	3.23687e6	0.956790	0.478395	+	0.878145i	$0.341219\pi$
$410$	0	0
$411$	−1.79211e6	−0.523312
$412$	0	0
$413$	−1.49764e6	−0.432049
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.28273e6	0.361239
$418$	0	0
$419$	−2.03059e6	−0.565049	−0.282525	−	0.959260i	$-0.591172\pi$
$419$	−2.03059e6	−0.565049	−0.282525	+	0.959260i	$0.591172\pi$
$420$	0	0
$421$	2.15883e6	0.593625	0.296813	−	0.954936i	$-0.404076\pi$
$421$	2.15883e6	0.593625	0.296813	+	0.954936i	$0.404076\pi$
$422$	0	0
$423$	745434.	0.202562
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.32665e6	0.352118
$428$	0	0
$429$	2.92467e6	0.767245
$430$	0	0
$431$	5.49812e6	1.42568	0.712838	−	0.701328i	$-0.247409\pi$
$431$	5.49812e6	1.42568	0.712838	+	0.701328i	$0.247409\pi$
$432$	0	0
$433$	3.87201e6	0.992470	0.496235	−	0.868188i	$-0.334716\pi$
$433$	3.87201e6	0.992470	0.496235	+	0.868188i	$0.334716\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.04530e6	−0.512334
$438$	0	0
$439$	3.15465e6	0.781250	0.390625	−	0.920550i	$-0.372259\pi$
$439$	3.15465e6	0.781250	0.390625	+	0.920550i	$0.372259\pi$
$440$	0	0
$441$	−4.02343e6	−0.985144
$442$	0	0
$443$	−4.07021e6	−0.985389	−0.492694	−	0.870202i	$-0.663988\pi$
$443$	−4.07021e6	−0.985389	−0.492694	+	0.870202i	$0.663988\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−797893.	−0.188876
$448$	0	0
$449$	957718.	0.224193	0.112096	−	0.993697i	$-0.464243\pi$
$449$	957718.	0.224193	0.112096	+	0.993697i	$0.464243\pi$
$450$	0	0
$451$	−3.52068e6	−0.815052
$452$	0	0
$453$	−1.57751e6	−0.361183
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.38167e6	1.65335	0.826673	−	0.562682i	$-0.190231\pi$
$457$	7.38167e6	1.65335	0.826673	+	0.562682i	$0.190231\pi$
$458$	0	0
$459$	−74044.2	−0.0164044
$460$	0	0
$461$	−2.96862e6	−0.650582	−0.325291	−	0.945614i	$-0.605462\pi$
$461$	−2.96862e6	−0.650582	−0.325291	+	0.945614i	$0.605462\pi$
$462$	0	0
$463$	6.84593e6	1.48416	0.742079	−	0.670313i	$-0.233840\pi$
$463$	6.84593e6	1.48416	0.742079	+	0.670313i	$0.233840\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.20816e6	0.892895	0.446447	−	0.894810i	$-0.352689\pi$
$467$	4.20816e6	0.892895	0.446447	+	0.894810i	$0.352689\pi$
$468$	0	0
$469$	−3.33172e6	−0.699417
$470$	0	0
$471$	234018.	0.0486068
$472$	0	0
$473$	−1.13895e7	−2.34074
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.81301e6	−1.16978
$478$	0	0
$479$	−5.73077e6	−1.14123	−0.570617	−	0.821217i	$-0.693296\pi$
$479$	−5.73077e6	−1.14123	−0.570617	+	0.821217i	$0.693296\pi$
$480$	0	0
$481$	8.93110e6	1.76012
$482$	0	0
$483$	−1.34437e6	−0.262211
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.63249e6	1.26723	0.633613	−	0.773650i	$-0.281571\pi$
$487$	6.63249e6	1.26723	0.633613	+	0.773650i	$0.281571\pi$
$488$	0	0
$489$	1.21658e6	0.230074
$490$	0	0
$491$	−3.26463e6	−0.611125	−0.305563	−	0.952172i	$-0.598844\pi$
$491$	−3.26463e6	−0.611125	−0.305563	+	0.952172i	$0.598844\pi$
$492$	0	0
$493$	106114.	0.0196633
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.53839e6	−0.460965
$498$	0	0
$499$	−8.51527e6	−1.53090	−0.765450	−	0.643495i	$-0.777484\pi$
$499$	−8.51527e6	−1.53090	−0.765450	+	0.643495i	$0.777484\pi$
$500$	0	0
$501$	1.83824e6	0.327196
$502$	0	0
$503$	3.24174e6	0.571292	0.285646	−	0.958335i	$-0.407792\pi$
$503$	3.24174e6	0.571292	0.285646	+	0.958335i	$0.407792\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.14744e6	−0.716573
$508$	0	0
$509$	1.04569e7	1.78900	0.894499	−	0.447070i	$-0.147533\pi$
$509$	1.04569e7	1.78900	0.894499	+	0.447070i	$0.147533\pi$
$510$	0	0
$511$	−7.54865e6	−1.27884
$512$	0	0
$513$	3.95004e6	0.662687
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.75755e6	0.289188
$518$	0	0
$519$	1.47393e6	0.240191
$520$	0	0
$521$	−328142.	−0.0529625	−0.0264812	−	0.999649i	$-0.508430\pi$
$521$	−328142.	−0.0529625	−0.0264812	+	0.999649i	$0.508430\pi$
$522$	0	0
$523$	9.42102e6	1.50606	0.753032	−	0.657984i	$-0.228590\pi$
$523$	9.42102e6	1.50606	0.753032	+	0.657984i	$0.228590\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	104330.	0.0163637
$528$	0	0
$529$	−4.75709e6	−0.739098
$530$	0	0
$531$	−1.68700e6	−0.259644
$532$	0	0
$533$	7.44643e6	1.13535
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.47160e6	0.220219
$538$	0	0
$539$	−9.48624e6	−1.40644
$540$	0	0
$541$	6.08041e6	0.893182	0.446591	−	0.894738i	$-0.352638\pi$
$541$	6.08041e6	0.893182	0.446591	+	0.894738i	$0.352638\pi$
$542$	0	0
$543$	497205.	0.0723662
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.32182e6	0.617587	0.308794	−	0.951129i	$-0.400075\pi$
$547$	4.32182e6	0.617587	0.308794	+	0.951129i	$0.400075\pi$
$548$	0	0
$549$	1.49439e6	0.211609
$550$	0	0
$551$	−5.66089e6	−0.794338
$552$	0	0
$553$	1.76347e7	2.45220
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.41019e6	1.01203	0.506013	−	0.862526i	$-0.331119\pi$
$557$	7.41019e6	1.01203	0.506013	+	0.862526i	$0.331119\pi$
$558$	0	0
$559$	2.40895e7	3.26061
$560$	0	0
$561$	−81517.7	−0.0109356
$562$	0	0
$563$	1.08039e7	1.43652	0.718260	−	0.695775i	$-0.244939\pi$
$563$	1.08039e7	1.43652	0.718260	+	0.695775i	$0.244939\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.17799e6	−0.937660
$568$	0	0
$569$	1.77404e6	0.229712	0.114856	−	0.993382i	$-0.463359\pi$
$569$	1.77404e6	0.229712	0.114856	+	0.993382i	$0.463359\pi$
$570$	0	0
$571$	−70981.0	−0.00911070	−0.00455535	−	0.999990i	$-0.501450\pi$
$571$	−70981.0	−0.00911070	−0.00455535	+	0.999990i	$0.501450\pi$
$572$	0	0
$573$	3.87992e6	0.493669
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.04410e6	−0.505688	−0.252844	−	0.967507i	$-0.581366\pi$
$577$	−4.04410e6	−0.505688	−0.252844	+	0.967507i	$0.581366\pi$
$578$	0	0
$579$	−1.30113e6	−0.161296
$580$	0	0
$581$	1.10449e7	1.35744
$582$	0	0
$583$	−1.37056e7	−1.67004
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.86862e6	−0.343620	−0.171810	−	0.985130i	$-0.554962\pi$
$587$	−2.86862e6	−0.343620	−0.171810	+	0.985130i	$0.554962\pi$
$588$	0	0
$589$	−5.56570e6	−0.661046
$590$	0	0
$591$	3.33737e6	0.393040
$592$	0	0
$593$	−3.23131e6	−0.377347	−0.188674	−	0.982040i	$-0.560419\pi$
$593$	−3.23131e6	−0.377347	−0.188674	+	0.982040i	$0.560419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.40962e6	−0.391534
$598$	0	0
$599$	3.80078e6	0.432819	0.216409	−	0.976303i	$-0.430565\pi$
$599$	3.80078e6	0.432819	0.216409	+	0.976303i	$0.430565\pi$
$600$	0	0
$601$	5.45289e6	0.615801	0.307900	−	0.951419i	$-0.400374\pi$
$601$	5.45289e6	0.615801	0.307900	+	0.951419i	$0.400374\pi$
$602$	0	0
$603$	−3.75297e6	−0.420321
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.50483e7	1.65774	0.828870	−	0.559441i	$-0.188984\pi$
$607$	1.50483e7	1.65774	0.828870	+	0.559441i	$0.188984\pi$
$608$	0	0
$609$	−3.72088e6	−0.406539
$610$	0	0
$611$	−3.71731e6	−0.402833
$612$	0	0
$613$	−2.30782e6	−0.248056	−0.124028	−	0.992279i	$-0.539581\pi$
$613$	−2.30782e6	−0.248056	−0.124028	+	0.992279i	$0.539581\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	381761.	0.0403718	0.0201859	−	0.999796i	$-0.493574\pi$
$617$	381761.	0.0403718	0.0201859	+	0.999796i	$0.493574\pi$
$618$	0	0
$619$	−1.07208e7	−1.12460	−0.562302	−	0.826932i	$-0.690084\pi$
$619$	−1.07208e7	−1.12460	−0.562302	+	0.826932i	$0.690084\pi$
$620$	0	0
$621$	−3.24311e6	−0.337468
$622$	0	0
$623$	2.64354e6	0.272876
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.34873e6	0.441767
$628$	0	0
$629$	−248931.	−0.0250872
$630$	0	0
$631$	−4.56189e6	−0.456112	−0.228056	−	0.973648i	$-0.573237\pi$
$631$	−4.56189e6	−0.456112	−0.228056	+	0.973648i	$0.573237\pi$
$632$	0	0
$633$	−5.82910e6	−0.578219
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00639e7	1.95915
$638$	0	0
$639$	−2.85933e6	−0.277021
$640$	0	0
$641$	−2.70983e6	−0.260494	−0.130247	−	0.991482i	$-0.541577\pi$
$641$	−2.70983e6	−0.260494	−0.130247	+	0.991482i	$0.541577\pi$
$642$	0	0
$643$	6.10481e6	0.582297	0.291149	−	0.956678i	$-0.405963\pi$
$643$	6.10481e6	0.582297	0.291149	+	0.956678i	$0.405963\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.22081e7	−1.14654	−0.573269	−	0.819367i	$-0.694325\pi$
$647$	−1.22081e7	−1.14654	−0.573269	+	0.819367i	$0.694325\pi$
$648$	0	0
$649$	−3.97752e6	−0.370681
$650$	0	0
$651$	−3.65832e6	−0.338321
$652$	0	0
$653$	6.26514e6	0.574973	0.287487	−	0.957785i	$-0.407180\pi$
$653$	6.26514e6	0.574973	0.287487	+	0.957785i	$0.407180\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.50307e6	−0.768532
$658$	0	0
$659$	−1.85107e7	−1.66039	−0.830194	−	0.557474i	$-0.811771\pi$
$659$	−1.85107e7	−1.66039	−0.830194	+	0.557474i	$0.811771\pi$
$660$	0	0
$661$	−8.46041e6	−0.753161	−0.376581	−	0.926384i	$-0.622900\pi$
$661$	−8.46041e6	−0.753161	−0.376581	+	0.926384i	$0.622900\pi$
$662$	0	0
$663$	172414.	0.0152331
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.64777e6	0.404511
$668$	0	0
$669$	−1.32156e6	−0.114162
$670$	0	0
$671$	3.52340e6	0.302104
$672$	0	0
$673$	−7.25182e6	−0.617177	−0.308588	−	0.951196i	$-0.599857\pi$
$673$	−7.25182e6	−0.617177	−0.308588	+	0.951196i	$0.599857\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.74893e6	−0.398221	−0.199110	−	0.979977i	$-0.563805\pi$
$677$	−4.74893e6	−0.398221	−0.199110	+	0.979977i	$0.563805\pi$
$678$	0	0
$679$	2.09073e7	1.74030
$680$	0	0
$681$	1.11169e6	0.0918578
$682$	0	0
$683$	−1.37712e7	−1.12959	−0.564793	−	0.825233i	$-0.691044\pi$
$683$	−1.37712e7	−1.12959	−0.564793	+	0.825233i	$0.691044\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.56069e6	0.126161
$688$	0	0
$689$	2.89882e7	2.32634
$690$	0	0
$691$	1.64305e7	1.30905	0.654526	−	0.756040i	$-0.272868\pi$
$691$	1.64305e7	1.30905	0.654526	+	0.756040i	$0.272868\pi$
$692$	0	0
$693$	−2.01871e7	−1.59676
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−207550.	−0.0161823
$698$	0	0
$699$	−1.89628e6	−0.146794
$700$	0	0
$701$	−1.05170e7	−0.808342	−0.404171	−	0.914683i	$-0.632440\pi$
$701$	−1.05170e7	−0.808342	−0.404171	+	0.914683i	$0.632440\pi$
$702$	0	0
$703$	1.32797e7	1.01345
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.75041e7	−2.82182
$708$	0	0
$709$	−1.32691e6	−0.0991349	−0.0495674	−	0.998771i	$-0.515784\pi$
$709$	−1.32691e6	−0.0991349	−0.0495674	+	0.998771i	$0.515784\pi$
$710$	0	0
$711$	1.98643e7	1.47367
$712$	0	0
$713$	4.56962e6	0.336633
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.77958e6	−0.637787
$718$	0	0
$719$	4.65753e6	0.335995	0.167998	−	0.985787i	$-0.446270\pi$
$719$	4.65753e6	0.335995	0.167998	+	0.985787i	$0.446270\pi$
$720$	0	0
$721$	−2.54691e7	−1.82463
$722$	0	0
$723$	231933.	0.0165012
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.47884e6	0.103773	0.0518867	−	0.998653i	$-0.483477\pi$
$727$	1.47884e6	0.103773	0.0518867	+	0.998653i	$0.483477\pi$
$728$	0	0
$729$	−4.74681e6	−0.330814
$730$	0	0
$731$	−671432.	−0.0464739
$732$	0	0
$733$	−1.41118e7	−0.970115	−0.485057	−	0.874482i	$-0.661201\pi$
$733$	−1.41118e7	−0.970115	−0.485057	+	0.874482i	$0.661201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.84856e6	−0.600073
$738$	0	0
$739$	−5.94045e6	−0.400137	−0.200068	−	0.979782i	$-0.564116\pi$
$739$	−5.94045e6	−0.400137	−0.200068	+	0.979782i	$0.564116\pi$
$740$	0	0
$741$	−9.19780e6	−0.615373
$742$	0	0
$743$	−1.46589e7	−0.974156	−0.487078	−	0.873359i	$-0.661937\pi$
$743$	−1.46589e7	−0.974156	−0.487078	+	0.873359i	$0.661937\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.24413e7	0.815766
$748$	0	0
$749$	−7.51817e6	−0.489674
$750$	0	0
$751$	−6.08263e6	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$751$	−6.08263e6	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$752$	0	0
$753$	−4.40760e6	−0.283279
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.11757e6	−0.514856	−0.257428	−	0.966297i	$-0.582875\pi$
$757$	−8.11757e6	−0.514856	−0.257428	+	0.966297i	$0.582875\pi$
$758$	0	0
$759$	−3.57044e6	−0.224966
$760$	0	0
$761$	1.63567e7	1.02384	0.511921	−	0.859033i	$-0.328934\pi$
$761$	1.63567e7	1.02384	0.511921	+	0.859033i	$0.328934\pi$
$762$	0	0
$763$	−1.75159e7	−1.08923
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.41267e6	0.516352
$768$	0	0
$769$	−2.09586e7	−1.27805	−0.639024	−	0.769187i	$-0.720661\pi$
$769$	−2.09586e7	−1.27805	−0.639024	+	0.769187i	$0.720661\pi$
$770$	0	0
$771$	5.16693e6	0.313038
$772$	0	0
$773$	8.95651e6	0.539126	0.269563	−	0.962983i	$-0.413121\pi$
$773$	8.95651e6	0.539126	0.269563	+	0.962983i	$0.413121\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.72873e6	0.518679
$778$	0	0
$779$	1.10722e7	0.653717
$780$	0	0
$781$	−6.74160e6	−0.395490
$782$	0	0
$783$	−8.97614e6	−0.523221
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.42009e7	1.39282	0.696411	−	0.717643i	$-0.254779\pi$
$787$	2.42009e7	1.39282	0.696411	+	0.717643i	$0.254779\pi$
$788$	0	0
$789$	−8.40683e6	−0.480773
$790$	0	0
$791$	1.26463e7	0.718657
$792$	0	0
$793$	−7.45219e6	−0.420824
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.91659e6	−0.552989	−0.276494	−	0.961016i	$-0.589173\pi$
$797$	−9.91659e6	−0.552989	−0.276494	+	0.961016i	$0.589173\pi$
$798$	0	0
$799$	103610.	0.00574164
$800$	0	0
$801$	2.97778e6	0.163988
$802$	0	0
$803$	−2.00481e7	−1.09720
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.90158e6	−0.373048
$808$	0	0
$809$	−9.15657e6	−0.491883	−0.245941	−	0.969285i	$-0.579097\pi$
$809$	−9.15657e6	−0.491883	−0.245941	+	0.969285i	$0.579097\pi$
$810$	0	0
$811$	1.49145e7	0.796264	0.398132	−	0.917328i	$-0.369659\pi$
$811$	1.49145e7	0.796264	0.398132	+	0.917328i	$0.369659\pi$
$812$	0	0
$813$	−4.63343e6	−0.245853
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.58189e7	1.87740
$818$	0	0
$819$	4.26967e7	2.22426
$820$	0	0
$821$	9.39549e6	0.486476	0.243238	−	0.969967i	$-0.421790\pi$
$821$	9.39549e6	0.486476	0.243238	+	0.969967i	$0.421790\pi$
$822$	0	0
$823$	−1.59508e7	−0.820884	−0.410442	−	0.911887i	$-0.634626\pi$
$823$	−1.59508e7	−0.820884	−0.410442	+	0.911887i	$0.634626\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.68518e7	−0.856805	−0.428403	−	0.903588i	$-0.640923\pi$
$827$	−1.68518e7	−0.856805	−0.428403	+	0.903588i	$0.640923\pi$
$828$	0	0
$829$	−3.92620e6	−0.198420	−0.0992101	−	0.995067i	$-0.531632\pi$
$829$	−3.92620e6	−0.198420	−0.0992101	+	0.995067i	$0.531632\pi$
$830$	0	0
$831$	−8.34865e6	−0.419386
$832$	0	0
$833$	−559229.	−0.0279240
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.82521e6	−0.435423
$838$	0	0
$839$	−1.23626e7	−0.606322	−0.303161	−	0.952939i	$-0.598042\pi$
$839$	−1.23626e7	−0.606322	−0.303161	+	0.952939i	$0.598042\pi$
$840$	0	0
$841$	−7.64726e6	−0.372834
$842$	0	0
$843$	1.08688e7	0.526760
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.71626e7	−0.822006
$848$	0	0
$849$	−1.11392e7	−0.530379
$850$	0	0
$851$	−1.09031e7	−0.516091
$852$	0	0
$853$	3.17612e6	0.149460	0.0747299	−	0.997204i	$-0.476191\pi$
$853$	3.17612e6	0.149460	0.0747299	+	0.997204i	$0.476191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.97233e7	−0.917334	−0.458667	−	0.888608i	$-0.651673\pi$
$857$	−1.97233e7	−0.917334	−0.458667	+	0.888608i	$0.651673\pi$
$858$	0	0
$859$	4.18456e6	0.193494	0.0967470	−	0.995309i	$-0.469156\pi$
$859$	4.18456e6	0.193494	0.0967470	+	0.995309i	$0.469156\pi$
$860$	0	0
$861$	7.27771e6	0.334570
$862$	0	0
$863$	2.89684e7	1.32403	0.662015	−	0.749490i	$-0.269701\pi$
$863$	2.89684e7	1.32403	0.662015	+	0.749490i	$0.269701\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.79021e6	0.351966
$868$	0	0
$869$	4.68352e7	2.10389
$870$	0	0
$871$	1.87152e7	0.835889
$872$	0	0
$873$	2.35508e7	1.04585
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.78801e7	−1.22404	−0.612019	−	0.790843i	$-0.709642\pi$
$877$	−2.78801e7	−1.22404	−0.612019	+	0.790843i	$0.709642\pi$
$878$	0	0
$879$	−1.22504e7	−0.534783
$880$	0	0
$881$	−4.21471e7	−1.82948	−0.914741	−	0.404040i	$-0.867606\pi$
$881$	−4.21471e7	−1.82948	−0.914741	+	0.404040i	$0.867606\pi$
$882$	0	0
$883$	4.38615e7	1.89314	0.946568	−	0.322505i	$-0.104525\pi$
$883$	4.38615e7	1.89314	0.946568	+	0.322505i	$0.104525\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.32386e7	0.991747	0.495874	−	0.868395i	$-0.334848\pi$
$887$	2.32386e7	0.991747	0.495874	+	0.868395i	$0.334848\pi$
$888$	0	0
$889$	1.11489e7	0.473126
$890$	0	0
$891$	−1.90637e7	−0.804476
$892$	0	0
$893$	−5.52731e6	−0.231945
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.55168e6	0.313374
$898$	0	0
$899$	1.26476e7	0.521925
$900$	0	0
$901$	−807970.	−0.0331576
$902$	0	0
$903$	2.35437e7	0.960848
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.74916e6	0.312778	0.156389	−	0.987696i	$-0.450015\pi$
$907$	7.74916e6	0.312778	0.156389	+	0.987696i	$0.450015\pi$
$908$	0	0
$909$	−4.22459e7	−1.69580
$910$	0	0
$911$	1.54221e7	0.615668	0.307834	−	0.951440i	$-0.400396\pi$
$911$	1.54221e7	0.615668	0.307834	+	0.951440i	$0.400396\pi$
$912$	0	0
$913$	2.93336e7	1.16463
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.95661e7	1.55382
$918$	0	0
$919$	−2.40744e7	−0.940302	−0.470151	−	0.882586i	$-0.655801\pi$
$919$	−2.40744e7	−0.940302	−0.470151	+	0.882586i	$0.655801\pi$
$920$	0	0
$921$	1.83540e6	0.0712989
$922$	0	0
$923$	1.42588e7	0.550909
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.86893e7	−1.09653
$928$	0	0
$929$	−1.51619e7	−0.576387	−0.288193	−	0.957572i	$-0.593055\pi$
$929$	−1.51619e7	−0.576387	−0.288193	+	0.957572i	$0.593055\pi$
$930$	0	0
$931$	2.98333e7	1.12804
$932$	0	0
$933$	−1.31260e7	−0.493662
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.90390e6	−0.182470	−0.0912352	−	0.995829i	$-0.529082\pi$
$937$	−4.90390e6	−0.182470	−0.0912352	+	0.995829i	$0.529082\pi$
$938$	0	0
$939$	1.03521e7	0.383146
$940$	0	0
$941$	−1.62296e7	−0.597495	−0.298748	−	0.954332i	$-0.596569\pi$
$941$	−1.62296e7	−0.597495	−0.298748	+	0.954332i	$0.596569\pi$
$942$	0	0
$943$	−9.09062e6	−0.332900
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.16817e7	1.51033	0.755163	−	0.655537i	$-0.227558\pi$
$947$	4.16817e7	1.51033	0.755163	+	0.655537i	$0.227558\pi$
$948$	0	0
$949$	4.24028e7	1.52837
$950$	0	0
$951$	−986202.	−0.0353602
$952$	0	0
$953$	−1.61135e7	−0.574723	−0.287362	−	0.957822i	$-0.592778\pi$
$953$	−1.61135e7	−0.574723	−0.287362	+	0.957822i	$0.592778\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.88212e6	−0.348795
$958$	0	0
$959$	−6.16851e7	−2.16588
$960$	0	0
$961$	−1.61942e7	−0.565655
$962$	0	0
$963$	−8.46873e6	−0.294274
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.51707e7	−0.521722	−0.260861	−	0.965376i	$-0.584006\pi$
$967$	−1.51707e7	−0.521722	−0.260861	+	0.965376i	$0.584006\pi$
$968$	0	0
$969$	256365.	0.00877099
$970$	0	0
$971$	−2.82884e7	−0.962854	−0.481427	−	0.876486i	$-0.659881\pi$
$971$	−2.82884e7	−0.962854	−0.481427	+	0.876486i	$0.659881\pi$
$972$	0	0
$973$	4.41519e7	1.49509
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.88218e7	0.966017	0.483009	−	0.875616i	$-0.339544\pi$
$977$	2.88218e7	0.966017	0.483009	+	0.875616i	$0.339544\pi$
$978$	0	0
$979$	7.02086e6	0.234117
$980$	0	0
$981$	−1.97305e7	−0.654586
$982$	0	0
$983$	−2.26448e6	−0.0747453	−0.0373727	−	0.999301i	$-0.511899\pi$
$983$	−2.26448e6	−0.0747453	−0.0373727	+	0.999301i	$0.511899\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.63308e6	−0.118709
$988$	0	0
$989$	−2.94085e7	−0.956053
$990$	0	0
$991$	995592.	0.0322031	0.0161015	−	0.999870i	$-0.494875\pi$
$991$	995592.	0.0322031	0.0161015	+	0.999870i	$0.494875\pi$
$992$	0	0
$993$	−925721.	−0.0297925
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.57133e7	1.77509	0.887547	−	0.460718i	$-0.152408\pi$
$997$	5.57133e7	1.77509	0.887547	+	0.460718i	$0.152408\pi$
$998$	0	0
$999$	2.10569e7	0.667546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.a.k.1.2		4
4.3	odd	2		400.6.a.z.1.3		4
5.2	odd	4		40.6.c.a.9.6	yes	8
5.3	odd	4		40.6.c.a.9.3	✓	8
5.4	even	2		200.6.a.j.1.3		4
15.2	even	4		360.6.f.b.289.8		8
15.8	even	4		360.6.f.b.289.7		8
20.3	even	4		80.6.c.d.49.6		8
20.7	even	4		80.6.c.d.49.3		8
20.19	odd	2		400.6.a.ba.1.2		4
40.3	even	4		320.6.c.i.129.3		8
40.13	odd	4		320.6.c.j.129.6		8
40.27	even	4		320.6.c.i.129.6		8
40.37	odd	4		320.6.c.j.129.3		8
60.23	odd	4		720.6.f.n.289.7		8
60.47	odd	4		720.6.f.n.289.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.3	✓	8	5.3	odd	4
40.6.c.a.9.6	yes	8	5.2	odd	4
80.6.c.d.49.3		8	20.7	even	4
80.6.c.d.49.6		8	20.3	even	4
200.6.a.j.1.3		4	5.4	even	2
200.6.a.k.1.2		4	1.1	even	1	trivial
320.6.c.i.129.3		8	40.3	even	4
320.6.c.i.129.6		8	40.27	even	4
320.6.c.j.129.3		8	40.37	odd	4
320.6.c.j.129.6		8	40.13	odd	4
360.6.f.b.289.7		8	15.8	even	4
360.6.f.b.289.8		8	15.2	even	4
400.6.a.z.1.3		4	4.3	odd	2
400.6.a.ba.1.2		4	20.19	odd	2
720.6.f.n.289.7		8	60.23	odd	4
720.6.f.n.289.8		8	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-5.49000 q^{3} -188.968 q^{7} -212.860 q^{9} -501.871 q^{11} +1061.48 q^{13} -29.5861 q^{17} +1578.33 q^{19} +1037.43 q^{21} -1295.86 q^{23} +2502.67 q^{27} -3586.63 q^{29} -3526.32 q^{31} +2755.27 q^{33} +8413.79 q^{37} -5827.54 q^{39} +7015.12 q^{41} +22694.2 q^{43} -3501.99 q^{47} +18901.8 q^{49} +162.428 q^{51} +27309.1 q^{53} -8665.04 q^{57} +7925.39 q^{59} -7020.54 q^{61} +40223.6 q^{63} +17631.2 q^{67} +7114.27 q^{69} +13432.9 q^{71} +39946.8 q^{73} +94837.3 q^{77} -93321.2 q^{79} +37985.3 q^{81} -58448.5 q^{83} +19690.6 q^{87} -13989.4 q^{89} -200586. q^{91} +19359.5 q^{93} -110640. q^{97} +106828. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} - 368 q^{11} + 440 q^{13} - 672 q^{17} - 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} + 2112 q^{31} + 26864 q^{33} + 8792 q^{37} + 1504 q^{39}+ \cdots - 133424 q^{99}+O(q^{100}) \) 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

Introduction

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