LMFDB - Embedded newform 200.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(1,200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.47217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 38x - 24 $ x^3 - x^2 - 38*x - 24
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.30760$ 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)

$f(q)$	$=$	$q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} +O(q^{10})$	$q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} -18.2768 q^{11} -834.918 q^{13} +292.543 q^{17} -2287.27 q^{19} +4011.84 q^{21} -3788.91 q^{23} +2318.76 q^{27} +4554.90 q^{29} -1458.49 q^{31} +348.965 q^{33} -5509.69 q^{37} +15941.4 q^{39} -616.780 q^{41} -15866.5 q^{43} +29157.2 q^{47} +27342.2 q^{49} -5585.64 q^{51} +36944.9 q^{53} +43671.6 q^{57} +18713.3 q^{59} +33063.9 q^{61} -25541.2 q^{63} -5592.37 q^{67} +72343.0 q^{69} +31534.1 q^{71} -86147.7 q^{73} +3840.26 q^{77} -34527.0 q^{79} -73811.2 q^{81} -89370.3 q^{83} -86968.5 q^{87} +45489.4 q^{89} +175431. q^{91} +27847.6 q^{93} +133312. q^{97} -2221.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9} - 19 q^{11} - 196 q^{13} + 1223 q^{17} + 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} + 7442 q^{31} - 6969 q^{33} + 14766 q^{37} + 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} + 43165 q^{51} - 12878 q^{53} + 77791 q^{57} + 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} + 86676 q^{71} + 8305 q^{73} + 106822 q^{77} + 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} + 149224 q^{91} + 196470 q^{93} + 76342 q^{97} - 149130 q^{99}+O(q^{100}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9 - 19 * q^11 - 196 * q^13 + 1223 * q^17 + 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 + 7442 * q^31 - 6969 * q^33 + 14766 * q^37 + 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 + 43165 * q^51 - 12878 * q^53 + 77791 * q^57 + 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 + 86676 * q^71 + 8305 * q^73 + 106822 * q^77 + 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 + 149224 * q^91 + 196470 * q^93 + 76342 * q^97 - 149130 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−19.0934	−1.22484	−0.612420	−	0.790532i	$-0.709804\pi$
$3$	−19.0934	−1.22484	−0.612420	+	0.790532i	$0.709804\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−210.117	−1.62075	−0.810376	−	0.585911i	$-0.800737\pi$
$7$	−210.117	−1.62075	−0.810376	+	0.585911i	$0.800737\pi$
$8$	0	0
$9$	121.557	0.500234
$10$	0	0
$11$	−18.2768	−0.0455426	−0.0227713	−	0.999741i	$-0.507249\pi$
$11$	−18.2768	−0.0455426	−0.0227713	+	0.999741i	$0.507249\pi$
$12$	0	0
$13$	−834.918	−1.37021	−0.685103	−	0.728447i	$-0.740243\pi$
$13$	−834.918	−1.37021	−0.685103	+	0.728447i	$0.740243\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	292.543	0.245509	0.122755	−	0.992437i	$-0.460827\pi$
$17$	292.543	0.245509	0.122755	+	0.992437i	$0.460827\pi$
$18$	0	0
$19$	−2287.27	−1.45356	−0.726779	−	0.686871i	$-0.758984\pi$
$19$	−2287.27	−1.45356	−0.726779	+	0.686871i	$0.758984\pi$
$20$	0	0
$21$	4011.84	1.98516
$22$	0	0
$23$	−3788.91	−1.49346	−0.746731	−	0.665126i	$-0.768378\pi$
$23$	−3788.91	−1.49346	−0.746731	+	0.665126i	$0.768378\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2318.76	0.612133
$28$	0	0
$29$	4554.90	1.00574	0.502868	−	0.864363i	$-0.332278\pi$
$29$	4554.90	1.00574	0.502868	+	0.864363i	$0.332278\pi$
$30$	0	0
$31$	−1458.49	−0.272584	−0.136292	−	0.990669i	$-0.543519\pi$
$31$	−1458.49	−0.272584	−0.136292	+	0.990669i	$0.543519\pi$
$32$	0	0
$33$	348.965	0.0557824
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5509.69	−0.661642	−0.330821	−	0.943693i	$-0.607326\pi$
$37$	−5509.69	−0.661642	−0.330821	+	0.943693i	$0.607326\pi$
$38$	0	0
$39$	15941.4	1.67828
$40$	0	0
$41$	−616.780	−0.0573021	−0.0286511	−	0.999589i	$-0.509121\pi$
$41$	−616.780	−0.0573021	−0.0286511	+	0.999589i	$0.509121\pi$
$42$	0	0
$43$	−15866.5	−1.30861	−0.654303	−	0.756232i	$-0.727038\pi$
$43$	−15866.5	−1.30861	−0.654303	+	0.756232i	$0.727038\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	29157.2	1.92531	0.962655	−	0.270730i	$-0.0872650\pi$
$47$	29157.2	1.92531	0.962655	+	0.270730i	$0.0872650\pi$
$48$	0	0
$49$	27342.2	1.62683
$50$	0	0
$51$	−5585.64	−0.300710
$52$	0	0
$53$	36944.9	1.80661	0.903305	−	0.428998i	$-0.141133\pi$
$53$	36944.9	1.80661	0.903305	+	0.428998i	$0.141133\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	43671.6	1.78038
$58$	0	0
$59$	18713.3	0.699874	0.349937	−	0.936773i	$-0.386203\pi$
$59$	18713.3	0.699874	0.349937	+	0.936773i	$0.386203\pi$
$60$	0	0
$61$	33063.9	1.13770	0.568852	−	0.822440i	$-0.307388\pi$
$61$	33063.9	1.13770	0.568852	+	0.822440i	$0.307388\pi$
$62$	0	0
$63$	−25541.2	−0.810755
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5592.37	−0.152198	−0.0760990	−	0.997100i	$-0.524246\pi$
$67$	−5592.37	−0.152198	−0.0760990	+	0.997100i	$0.524246\pi$
$68$	0	0
$69$	72343.0	1.82925
$70$	0	0
$71$	31534.1	0.742394	0.371197	−	0.928554i	$-0.378947\pi$
$71$	31534.1	0.742394	0.371197	+	0.928554i	$0.378947\pi$
$72$	0	0
$73$	−86147.7	−1.89207	−0.946034	−	0.324067i	$-0.894950\pi$
$73$	−86147.7	−1.89207	−0.946034	+	0.324067i	$0.894950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3840.26	0.0738132
$78$	0	0
$79$	−34527.0	−0.622430	−0.311215	−	0.950340i	$-0.600736\pi$
$79$	−34527.0	−0.622430	−0.311215	+	0.950340i	$0.600736\pi$
$80$	0	0
$81$	−73811.2	−1.25000
$82$	0	0
$83$	−89370.3	−1.42396	−0.711980	−	0.702199i	$-0.752201\pi$
$83$	−89370.3	−1.42396	−0.711980	+	0.702199i	$0.752201\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−86968.5	−1.23187
$88$	0	0
$89$	45489.4	0.608745	0.304373	−	0.952553i	$-0.401553\pi$
$89$	45489.4	0.608745	0.304373	+	0.952553i	$0.401553\pi$
$90$	0	0
$91$	175431.	2.22076
$92$	0	0
$93$	27847.6	0.333872
$94$	0	0
$95$	0	0
$96$	0	0
$97$	133312.	1.43860	0.719302	−	0.694697i	$-0.244462\pi$
$97$	133312.	1.43860	0.719302	+	0.694697i	$0.244462\pi$
$98$	0	0
$99$	−2221.67	−0.0227820
$100$	0	0
$101$	−67742.9	−0.660785	−0.330393	−	0.943844i	$-0.607181\pi$
$101$	−67742.9	−0.660785	−0.330393	+	0.943844i	$0.607181\pi$
$102$	0	0
$103$	−62032.7	−0.576139	−0.288070	−	0.957609i	$-0.593013\pi$
$103$	−62032.7	−0.576139	−0.288070	+	0.957609i	$0.593013\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	22529.0	0.190231	0.0951156	−	0.995466i	$-0.469678\pi$
$107$	22529.0	0.190231	0.0951156	+	0.995466i	$0.469678\pi$
$108$	0	0
$109$	−212316.	−1.71166	−0.855829	−	0.517259i	$-0.826953\pi$
$109$	−212316.	−1.71166	−0.855829	+	0.517259i	$0.826953\pi$
$110$	0	0
$111$	105199.	0.810406
$112$	0	0
$113$	198742.	1.46417	0.732087	−	0.681211i	$-0.238546\pi$
$113$	198742.	1.46417	0.732087	+	0.681211i	$0.238546\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−101490.	−0.685423
$118$	0	0
$119$	−61468.4	−0.397910
$120$	0	0
$121$	−160717.	−0.997926
$122$	0	0
$123$	11776.4	0.0701860
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−84802.8	−0.466553	−0.233276	−	0.972410i	$-0.574945\pi$
$127$	−84802.8	−0.466553	−0.233276	+	0.972410i	$0.574945\pi$
$128$	0	0
$129$	302944.	1.60283
$130$	0	0
$131$	−220540.	−1.12282	−0.561410	−	0.827538i	$-0.689741\pi$
$131$	−220540.	−1.12282	−0.561410	+	0.827538i	$0.689741\pi$
$132$	0	0
$133$	480594.	2.35586
$134$	0	0
$135$	0	0
$136$	0	0
$137$	46832.2	0.213178	0.106589	−	0.994303i	$-0.466007\pi$
$137$	46832.2	0.213178	0.106589	+	0.994303i	$0.466007\pi$
$138$	0	0
$139$	−169151.	−0.742569	−0.371284	−	0.928519i	$-0.621083\pi$
$139$	−169151.	−0.742569	−0.371284	+	0.928519i	$0.621083\pi$
$140$	0	0
$141$	−556709.	−2.35820
$142$	0	0
$143$	15259.6	0.0624027
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−522055.	−1.99261
$148$	0	0
$149$	−280036.	−1.03335	−0.516675	−	0.856181i	$-0.672831\pi$
$149$	−280036.	−1.03335	−0.516675	+	0.856181i	$0.672831\pi$
$150$	0	0
$151$	98970.7	0.353235	0.176618	−	0.984280i	$-0.443484\pi$
$151$	98970.7	0.353235	0.176618	+	0.984280i	$0.443484\pi$
$152$	0	0
$153$	35560.7	0.122812
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−31033.4	−0.100480	−0.0502401	−	0.998737i	$-0.515999\pi$
$157$	−31033.4	−0.100480	−0.0502401	+	0.998737i	$0.515999\pi$
$158$	0	0
$159$	−705402.	−2.21281
$160$	0	0
$161$	796114.	2.42053
$162$	0	0
$163$	98784.1	0.291218	0.145609	−	0.989342i	$-0.453486\pi$
$163$	98784.1	0.291218	0.145609	+	0.989342i	$0.453486\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	418435.	1.16101	0.580506	−	0.814256i	$-0.302855\pi$
$167$	418435.	1.16101	0.580506	+	0.814256i	$0.302855\pi$
$168$	0	0
$169$	325796.	0.877462
$170$	0	0
$171$	−278033.	−0.727120
$172$	0	0
$173$	284193.	0.721934	0.360967	−	0.932579i	$-0.382447\pi$
$173$	284193.	0.721934	0.360967	+	0.932579i	$0.382447\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−357299.	−0.857233
$178$	0	0
$179$	−783123.	−1.82683	−0.913414	−	0.407032i	$-0.866564\pi$
$179$	−783123.	−1.82683	−0.913414	+	0.407032i	$0.866564\pi$
$180$	0	0
$181$	−64283.0	−0.145848	−0.0729238	−	0.997338i	$-0.523233\pi$
$181$	−64283.0	−0.145848	−0.0729238	+	0.997338i	$0.523233\pi$
$182$	0	0
$183$	−631302.	−1.39351
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5346.75	−0.0111811
$188$	0	0
$189$	−487211.	−0.992116
$190$	0	0
$191$	−112334.	−0.222806	−0.111403	−	0.993775i	$-0.535535\pi$
$191$	−112334.	−0.222806	−0.111403	+	0.993775i	$0.535535\pi$
$192$	0	0
$193$	161036.	0.311193	0.155596	−	0.987821i	$-0.450270\pi$
$193$	161036.	0.311193	0.155596	+	0.987821i	$0.450270\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−456071.	−0.837273	−0.418636	−	0.908154i	$-0.637492\pi$
$197$	−456071.	−0.837273	−0.418636	+	0.908154i	$0.637492\pi$
$198$	0	0
$199$	569030.	1.01860	0.509299	−	0.860590i	$-0.329905\pi$
$199$	569030.	1.01860	0.509299	+	0.860590i	$0.329905\pi$
$200$	0	0
$201$	106777.	0.186418
$202$	0	0
$203$	−957064.	−1.63005
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−460568.	−0.747081
$208$	0	0
$209$	41803.8	0.0661988
$210$	0	0
$211$	402226.	0.621963	0.310981	−	0.950416i	$-0.399342\pi$
$211$	402226.	0.621963	0.310981	+	0.950416i	$0.399342\pi$
$212$	0	0
$213$	−602092.	−0.909314
$214$	0	0
$215$	0	0
$216$	0	0
$217$	306455.	0.441791
$218$	0	0
$219$	1.64485e6	2.31748
$220$	0	0
$221$	−244250.	−0.336398
$222$	0	0
$223$	1.24567e6	1.67742	0.838709	−	0.544579i	$-0.183311\pi$
$223$	1.24567e6	1.67742	0.838709	+	0.544579i	$0.183311\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	475220.	0.612111	0.306055	−	0.952014i	$-0.400991\pi$
$227$	475220.	0.612111	0.306055	+	0.952014i	$0.400991\pi$
$228$	0	0
$229$	−399176.	−0.503009	−0.251505	−	0.967856i	$-0.580925\pi$
$229$	−399176.	−0.503009	−0.251505	+	0.967856i	$0.580925\pi$
$230$	0	0
$231$	−73323.5	−0.0904094
$232$	0	0
$233$	−280117.	−0.338026	−0.169013	−	0.985614i	$-0.554058\pi$
$233$	−280117.	−0.338026	−0.169013	+	0.985614i	$0.554058\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	659236.	0.762378
$238$	0	0
$239$	190860.	0.216133	0.108067	−	0.994144i	$-0.465534\pi$
$239$	190860.	0.216133	0.108067	+	0.994144i	$0.465534\pi$
$240$	0	0
$241$	−31402.5	−0.0348275	−0.0174137	−	0.999848i	$-0.505543\pi$
$241$	−31402.5	−0.0348275	−0.0174137	+	0.999848i	$0.505543\pi$
$242$	0	0
$243$	845847.	0.918917
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.90968e6	1.99167
$248$	0	0
$249$	1.70638e6	1.74412
$250$	0	0
$251$	47989.7	0.0480799	0.0240400	−	0.999711i	$-0.492347\pi$
$251$	47989.7	0.0480799	0.0240400	+	0.999711i	$0.492347\pi$
$252$	0	0
$253$	69248.9	0.0680161
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.79852e6	1.69857	0.849284	−	0.527937i	$-0.177034\pi$
$257$	1.79852e6	1.69857	0.849284	+	0.527937i	$0.177034\pi$
$258$	0	0
$259$	1.15768e6	1.07236
$260$	0	0
$261$	553680.	0.503104
$262$	0	0
$263$	967118.	0.862164	0.431082	−	0.902313i	$-0.358132\pi$
$263$	967118.	0.862164	0.431082	+	0.902313i	$0.358132\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−868547.	−0.745616
$268$	0	0
$269$	878552.	0.740264	0.370132	−	0.928979i	$-0.379312\pi$
$269$	878552.	0.740264	0.370132	+	0.928979i	$0.379312\pi$
$270$	0	0
$271$	1.19503e6	0.988455	0.494228	−	0.869333i	$-0.335451\pi$
$271$	1.19503e6	0.988455	0.494228	+	0.869333i	$0.335451\pi$
$272$	0	0
$273$	−3.34956e6	−2.72008
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−850774.	−0.666215	−0.333108	−	0.942889i	$-0.608097\pi$
$277$	−850774.	−0.666215	−0.333108	+	0.942889i	$0.608097\pi$
$278$	0	0
$279$	−177290.	−0.136356
$280$	0	0
$281$	−1.05422e6	−0.796462	−0.398231	−	0.917285i	$-0.630376\pi$
$281$	−1.05422e6	−0.796462	−0.398231	+	0.917285i	$0.630376\pi$
$282$	0	0
$283$	−226845.	−0.168369	−0.0841846	−	0.996450i	$-0.526829\pi$
$283$	−226845.	−0.168369	−0.0841846	+	0.996450i	$0.526829\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	129596.	0.0928725
$288$	0	0
$289$	−1.33428e6	−0.939725
$290$	0	0
$291$	−2.54538e6	−1.76206
$292$	0	0
$293$	−1.64442e6	−1.11904	−0.559519	−	0.828818i	$-0.689014\pi$
$293$	−1.64442e6	−1.11904	−0.559519	+	0.828818i	$0.689014\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−42379.4	−0.0278781
$298$	0	0
$299$	3.16343e6	2.04635
$300$	0	0
$301$	3.33382e6	2.12093
$302$	0	0
$303$	1.29344e6	0.809356
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.09665e6	−1.26964	−0.634820	−	0.772660i	$-0.718926\pi$
$307$	−2.09665e6	−1.26964	−0.634820	+	0.772660i	$0.718926\pi$
$308$	0	0
$309$	1.18441e6	0.705678
$310$	0	0
$311$	1.46148e6	0.856825	0.428413	−	0.903583i	$-0.359073\pi$
$311$	1.46148e6	0.856825	0.428413	+	0.903583i	$0.359073\pi$
$312$	0	0
$313$	−1.03377e6	−0.596437	−0.298219	−	0.954498i	$-0.596392\pi$
$313$	−1.03377e6	−0.596437	−0.298219	+	0.954498i	$0.596392\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−148727.	−0.0831272	−0.0415636	−	0.999136i	$-0.513234\pi$
$317$	−148727.	−0.0831272	−0.0415636	+	0.999136i	$0.513234\pi$
$318$	0	0
$319$	−83248.9	−0.0458038
$320$	0	0
$321$	−430154.	−0.233003
$322$	0	0
$323$	−669125.	−0.356862
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.05383e6	2.09651
$328$	0	0
$329$	−6.12642e6	−3.12045
$330$	0	0
$331$	1.54319e6	0.774194	0.387097	−	0.922039i	$-0.373478\pi$
$331$	1.54319e6	0.774194	0.387097	+	0.922039i	$0.373478\pi$
$332$	0	0
$333$	−669741.	−0.330976
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.22154e6	−1.54522	−0.772608	−	0.634883i	$-0.781048\pi$
$337$	−3.22154e6	−1.54522	−0.772608	+	0.634883i	$0.781048\pi$
$338$	0	0
$339$	−3.79465e6	−1.79338
$340$	0	0
$341$	26656.6	0.0124142
$342$	0	0
$343$	−2.21363e6	−1.01594
$344$	0	0
$345$	0	0
$346$	0	0
$347$	217036.	0.0967626	0.0483813	−	0.998829i	$-0.484594\pi$
$347$	217036.	0.0967626	0.0483813	+	0.998829i	$0.484594\pi$
$348$	0	0
$349$	−37149.9	−0.0163265	−0.00816326	−	0.999967i	$-0.502598\pi$
$349$	−37149.9	−0.0163265	−0.00816326	+	0.999967i	$0.502598\pi$
$350$	0	0
$351$	−1.93597e6	−0.838748
$352$	0	0
$353$	2.92411e6	1.24898	0.624492	−	0.781031i	$-0.285306\pi$
$353$	2.92411e6	1.24898	0.624492	+	0.781031i	$0.285306\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.17364e6	0.487376
$358$	0	0
$359$	1.55303e6	0.635980	0.317990	−	0.948094i	$-0.396992\pi$
$359$	1.55303e6	0.635980	0.317990	+	0.948094i	$0.396992\pi$
$360$	0	0
$361$	2.75549e6	1.11283
$362$	0	0
$363$	3.06863e6	1.22230
$364$	0	0
$365$	0	0
$366$	0	0
$367$	413541.	0.160271	0.0801353	−	0.996784i	$-0.474465\pi$
$367$	413541.	0.160271	0.0801353	+	0.996784i	$0.474465\pi$
$368$	0	0
$369$	−74973.9	−0.0286645
$370$	0	0
$371$	−7.76275e6	−2.92807
$372$	0	0
$373$	2.06546e6	0.768677	0.384338	−	0.923192i	$-0.374430\pi$
$373$	2.06546e6	0.768677	0.384338	+	0.923192i	$0.374430\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.80297e6	−1.37807
$378$	0	0
$379$	2.27742e6	0.814412	0.407206	−	0.913336i	$-0.366503\pi$
$379$	2.27742e6	0.814412	0.407206	+	0.913336i	$0.366503\pi$
$380$	0	0
$381$	1.61917e6	0.571453
$382$	0	0
$383$	2.88095e6	1.00355	0.501774	−	0.864999i	$-0.332681\pi$
$383$	2.88095e6	1.00355	0.501774	+	0.864999i	$0.332681\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.92868e6	−0.654610
$388$	0	0
$389$	634299.	0.212530	0.106265	−	0.994338i	$-0.466111\pi$
$389$	634299.	0.212530	0.106265	+	0.994338i	$0.466111\pi$
$390$	0	0
$391$	−1.10842e6	−0.366659
$392$	0	0
$393$	4.21086e6	1.37527
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.36800e6	−1.39093	−0.695466	−	0.718559i	$-0.744802\pi$
$397$	−4.36800e6	−1.39093	−0.695466	+	0.718559i	$0.744802\pi$
$398$	0	0
$399$	−9.17616e6	−2.88555
$400$	0	0
$401$	−5.50000e6	−1.70806	−0.854028	−	0.520228i	$-0.825847\pi$
$401$	−5.50000e6	−1.70806	−0.854028	+	0.520228i	$0.825847\pi$
$402$	0	0
$403$	1.21772e6	0.373496
$404$	0	0
$405$	0	0
$406$	0	0
$407$	100699.	0.0301329
$408$	0	0
$409$	3.21295e6	0.949721	0.474860	−	0.880061i	$-0.342499\pi$
$409$	3.21295e6	0.949721	0.474860	+	0.880061i	$0.342499\pi$
$410$	0	0
$411$	−894184.	−0.261109
$412$	0	0
$413$	−3.93198e6	−1.13432
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.22966e6	0.909528
$418$	0	0
$419$	890973.	0.247930	0.123965	−	0.992287i	$-0.460439\pi$
$419$	890973.	0.247930	0.123965	+	0.992287i	$0.460439\pi$
$420$	0	0
$421$	−2.87922e6	−0.791716	−0.395858	−	0.918312i	$-0.629553\pi$
$421$	−2.87922e6	−0.791716	−0.395858	+	0.918312i	$0.629553\pi$
$422$	0	0
$423$	3.54426e6	0.963106
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.94729e6	−1.84394
$428$	0	0
$429$	−291357.	−0.0764333
$430$	0	0
$431$	6.29450e6	1.63218	0.816090	−	0.577924i	$-0.196137\pi$
$431$	6.29450e6	1.63218	0.816090	+	0.577924i	$0.196137\pi$
$432$	0	0
$433$	5.96375e6	1.52862	0.764311	−	0.644848i	$-0.223079\pi$
$433$	5.96375e6	1.52862	0.764311	+	0.644848i	$0.223079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.66623e6	2.17084
$438$	0	0
$439$	−4.91648e6	−1.21757	−0.608784	−	0.793336i	$-0.708342\pi$
$439$	−4.91648e6	−1.21757	−0.608784	+	0.793336i	$0.708342\pi$
$440$	0	0
$441$	3.32363e6	0.813798
$442$	0	0
$443$	−5.53513e6	−1.34004	−0.670021	−	0.742342i	$-0.733715\pi$
$443$	−5.53513e6	−1.34004	−0.670021	+	0.742342i	$0.733715\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.34683e6	1.26569
$448$	0	0
$449$	−7.20215e6	−1.68596	−0.842978	−	0.537948i	$-0.819200\pi$
$449$	−7.20215e6	−1.68596	−0.842978	+	0.537948i	$0.819200\pi$
$450$	0	0
$451$	11272.7	0.00260969
$452$	0	0
$453$	−1.88968e6	−0.432657
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.49984e6	−1.23185	−0.615927	−	0.787803i	$-0.711218\pi$
$457$	−5.49984e6	−1.23185	−0.615927	+	0.787803i	$0.711218\pi$
$458$	0	0
$459$	678338.	0.150285
$460$	0	0
$461$	−3.79223e6	−0.831078	−0.415539	−	0.909575i	$-0.636407\pi$
$461$	−3.79223e6	−0.831078	−0.415539	+	0.909575i	$0.636407\pi$
$462$	0	0
$463$	4.78947e6	1.03833	0.519164	−	0.854674i	$-0.326243\pi$
$463$	4.78947e6	1.03833	0.519164	+	0.854674i	$0.326243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.62497e6	−1.19352	−0.596758	−	0.802422i	$-0.703545\pi$
$467$	−5.62497e6	−1.19352	−0.596758	+	0.802422i	$0.703545\pi$
$468$	0	0
$469$	1.17505e6	0.246675
$470$	0	0
$471$	592532.	0.123072
$472$	0	0
$473$	289988.	0.0595973
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.49090e6	0.903728
$478$	0	0
$479$	760544.	0.151456	0.0757278	−	0.997129i	$-0.475872\pi$
$479$	760544.	0.151456	0.0757278	+	0.997129i	$0.475872\pi$
$480$	0	0
$481$	4.60014e6	0.906586
$482$	0	0
$483$	−1.52005e7	−2.96476
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.52969e6	0.483332	0.241666	−	0.970360i	$-0.422306\pi$
$487$	2.52969e6	0.483332	0.241666	+	0.970360i	$0.422306\pi$
$488$	0	0
$489$	−1.88612e6	−0.356695
$490$	0	0
$491$	4.83619e6	0.905316	0.452658	−	0.891684i	$-0.350476\pi$
$491$	4.83619e6	0.905316	0.452658	+	0.891684i	$0.350476\pi$
$492$	0	0
$493$	1.33251e6	0.246918
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.62585e6	−1.20324
$498$	0	0
$499$	9.17357e6	1.64925	0.824626	−	0.565678i	$-0.191385\pi$
$499$	9.17357e6	1.64925	0.824626	+	0.565678i	$0.191385\pi$
$500$	0	0
$501$	−7.98933e6	−1.42205
$502$	0	0
$503$	−1.02119e7	−1.79964	−0.899820	−	0.436262i	$-0.856302\pi$
$503$	−1.02119e7	−1.79964	−0.899820	+	0.436262i	$0.856302\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.22054e6	−1.07475
$508$	0	0
$509$	6.61475e6	1.13167	0.565834	−	0.824519i	$-0.308554\pi$
$509$	6.61475e6	1.13167	0.565834	+	0.824519i	$0.308554\pi$
$510$	0	0
$511$	1.81011e7	3.06657
$512$	0	0
$513$	−5.30362e6	−0.889772
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−532899.	−0.0876836
$518$	0	0
$519$	−5.42620e6	−0.884254
$520$	0	0
$521$	−9.91327e6	−1.60001	−0.800005	−	0.599993i	$-0.795170\pi$
$521$	−9.91327e6	−1.60001	−0.800005	+	0.599993i	$0.795170\pi$
$522$	0	0
$523$	4.66698e6	0.746074	0.373037	−	0.927816i	$-0.378316\pi$
$523$	4.66698e6	0.746074	0.373037	+	0.927816i	$0.378316\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−426673.	−0.0669220
$528$	0	0
$529$	7.91946e6	1.23043
$530$	0	0
$531$	2.27473e6	0.350101
$532$	0	0
$533$	514961.	0.0785157
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.49525e7	2.23757
$538$	0	0
$539$	−499727.	−0.0740902
$540$	0	0
$541$	−1.47136e6	−0.216136	−0.108068	−	0.994144i	$-0.534466\pi$
$541$	−1.47136e6	−0.216136	−0.108068	+	0.994144i	$0.534466\pi$
$542$	0	0
$543$	1.22738e6	0.178640
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.55704e6	0.651200	0.325600	−	0.945508i	$-0.394434\pi$
$547$	4.55704e6	0.651200	0.325600	+	0.945508i	$0.394434\pi$
$548$	0	0
$549$	4.01915e6	0.569119
$550$	0	0
$551$	−1.04183e7	−1.46190
$552$	0	0
$553$	7.25470e6	1.00880
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.05967e7	1.44721	0.723605	−	0.690215i	$-0.242484\pi$
$557$	1.05967e7	1.44721	0.723605	+	0.690215i	$0.242484\pi$
$558$	0	0
$559$	1.32472e7	1.79306
$560$	0	0
$561$	102087.	0.0136951
$562$	0	0
$563$	−2.17451e6	−0.289129	−0.144564	−	0.989495i	$-0.546178\pi$
$563$	−2.17451e6	−0.289129	−0.144564	+	0.989495i	$0.546178\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.55090e7	2.02594
$568$	0	0
$569$	9.36030e6	1.21202	0.606009	−	0.795458i	$-0.292769\pi$
$569$	9.36030e6	1.21202	0.606009	+	0.795458i	$0.292769\pi$
$570$	0	0
$571$	1.64209e6	0.210770	0.105385	−	0.994432i	$-0.466393\pi$
$571$	1.64209e6	0.210770	0.105385	+	0.994432i	$0.466393\pi$
$572$	0	0
$573$	2.14484e6	0.272902
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.20742e6	0.651153	0.325576	−	0.945516i	$-0.394442\pi$
$577$	5.20742e6	0.651153	0.325576	+	0.945516i	$0.394442\pi$
$578$	0	0
$579$	−3.07472e6	−0.381161
$580$	0	0
$581$	1.87782e7	2.30789
$582$	0	0
$583$	−675233.	−0.0822777
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.06067e7	−1.27053	−0.635267	−	0.772292i	$-0.719110\pi$
$587$	−1.06067e7	−1.27053	−0.635267	+	0.772292i	$0.719110\pi$
$588$	0	0
$589$	3.33596e6	0.396217
$590$	0	0
$591$	8.70794e6	1.02553
$592$	0	0
$593$	−1.99180e6	−0.232600	−0.116300	−	0.993214i	$-0.537103\pi$
$593$	−1.99180e6	−0.232600	−0.116300	+	0.993214i	$0.537103\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.08647e7	−1.24762
$598$	0	0
$599$	−1.16108e7	−1.32219	−0.661095	−	0.750303i	$-0.729908\pi$
$599$	−1.16108e7	−1.32219	−0.661095	+	0.750303i	$0.729908\pi$
$600$	0	0
$601$	−628490.	−0.0709761	−0.0354881	−	0.999370i	$-0.511299\pi$
$601$	−628490.	−0.0709761	−0.0354881	+	0.999370i	$0.511299\pi$
$602$	0	0
$603$	−679791.	−0.0761346
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−707145.	−0.0778998	−0.0389499	−	0.999241i	$-0.512401\pi$
$607$	−707145.	−0.0778998	−0.0389499	+	0.999241i	$0.512401\pi$
$608$	0	0
$609$	1.82736e7	1.99655
$610$	0	0
$611$	−2.43439e7	−2.63807
$612$	0	0
$613$	1.37228e7	1.47500	0.737500	−	0.675347i	$-0.236006\pi$
$613$	1.37228e7	1.47500	0.737500	+	0.675347i	$0.236006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.47885e7	1.56390	0.781952	−	0.623339i	$-0.214224\pi$
$617$	1.47885e7	1.56390	0.781952	+	0.623339i	$0.214224\pi$
$618$	0	0
$619$	2.72872e6	0.286241	0.143121	−	0.989705i	$-0.454286\pi$
$619$	2.72872e6	0.286241	0.143121	+	0.989705i	$0.454286\pi$
$620$	0	0
$621$	−8.78556e6	−0.914198
$622$	0	0
$623$	−9.55811e6	−0.986624
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−798176.	−0.0810830
$628$	0	0
$629$	−1.61183e6	−0.162439
$630$	0	0
$631$	5.75522e6	0.575425	0.287713	−	0.957717i	$-0.407105\pi$
$631$	5.75522e6	0.575425	0.287713	+	0.957717i	$0.407105\pi$
$632$	0	0
$633$	−7.67986e6	−0.761805
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.28285e7	−2.22910
$638$	0	0
$639$	3.83319e6	0.371371
$640$	0	0
$641$	3.94835e6	0.379551	0.189776	−	0.981827i	$-0.439224\pi$
$641$	3.94835e6	0.379551	0.189776	+	0.981827i	$0.439224\pi$
$642$	0	0
$643$	−1.12977e7	−1.07761	−0.538804	−	0.842431i	$-0.681124\pi$
$643$	−1.12977e7	−1.07761	−0.538804	+	0.842431i	$0.681124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.60010e7	1.50275	0.751376	−	0.659875i	$-0.229391\pi$
$647$	1.60010e7	1.50275	0.751376	+	0.659875i	$0.229391\pi$
$648$	0	0
$649$	−342018.	−0.0318741
$650$	0	0
$651$	−5.85125e6	−0.541124
$652$	0	0
$653$	8.51457e6	0.781412	0.390706	−	0.920516i	$-0.372231\pi$
$653$	8.51457e6	0.781412	0.390706	+	0.920516i	$0.372231\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.04719e7	−0.946477
$658$	0	0
$659$	1.15442e7	1.03550	0.517751	−	0.855531i	$-0.326769\pi$
$659$	1.15442e7	1.03550	0.517751	+	0.855531i	$0.326769\pi$
$660$	0	0
$661$	1.51328e7	1.34715	0.673576	−	0.739118i	$-0.264757\pi$
$661$	1.51328e7	1.34715	0.673576	+	0.739118i	$0.264757\pi$
$662$	0	0
$663$	4.66355e6	0.412034
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.72581e7	−1.50203
$668$	0	0
$669$	−2.37841e7	−2.05457
$670$	0	0
$671$	−604301.	−0.0518140
$672$	0	0
$673$	−1.82845e7	−1.55613	−0.778063	−	0.628187i	$-0.783797\pi$
$673$	−1.82845e7	−1.55613	−0.778063	+	0.628187i	$0.783797\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.67653e6	−0.308295	−0.154147	−	0.988048i	$-0.549263\pi$
$677$	−3.67653e6	−0.308295	−0.154147	+	0.988048i	$0.549263\pi$
$678$	0	0
$679$	−2.80112e7	−2.33162
$680$	0	0
$681$	−9.07355e6	−0.749738
$682$	0	0
$683$	−1.19937e6	−0.0983784	−0.0491892	−	0.998789i	$-0.515664\pi$
$683$	−1.19937e6	−0.0983784	−0.0491892	+	0.998789i	$0.515664\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.62162e6	0.616106
$688$	0	0
$689$	−3.08460e7	−2.47543
$690$	0	0
$691$	1.32762e7	1.05774	0.528868	−	0.848704i	$-0.322617\pi$
$691$	1.32762e7	1.05774	0.528868	+	0.848704i	$0.322617\pi$
$692$	0	0
$693$	466810.	0.0369239
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−180435.	−0.0140682
$698$	0	0
$699$	5.34839e6	0.414028
$700$	0	0
$701$	9.25654e6	0.711466	0.355733	−	0.934588i	$-0.384231\pi$
$701$	9.25654e6	0.711466	0.355733	+	0.934588i	$0.384231\pi$
$702$	0	0
$703$	1.26021e7	0.961736
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.42339e7	1.07097
$708$	0	0
$709$	−9.58869e6	−0.716381	−0.358190	−	0.933649i	$-0.616606\pi$
$709$	−9.58869e6	−0.716381	−0.358190	+	0.933649i	$0.616606\pi$
$710$	0	0
$711$	−4.19699e6	−0.311361
$712$	0	0
$713$	5.52610e6	0.407094
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.64417e6	−0.264728
$718$	0	0
$719$	1.17477e6	0.0847483	0.0423741	−	0.999102i	$-0.486508\pi$
$719$	1.17477e6	0.0847483	0.0423741	+	0.999102i	$0.486508\pi$
$720$	0	0
$721$	1.30341e7	0.933778
$722$	0	0
$723$	599581.	0.0426581
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.68157e7	−1.17999	−0.589997	−	0.807405i	$-0.700871\pi$
$727$	−1.68157e7	−1.17999	−0.589997	+	0.807405i	$0.700871\pi$
$728$	0	0
$729$	1.78606e6	0.124473
$730$	0	0
$731$	−4.64163e6	−0.321275
$732$	0	0
$733$	−5.18574e6	−0.356493	−0.178247	−	0.983986i	$-0.557042\pi$
$733$	−5.18574e6	−0.356493	−0.178247	+	0.983986i	$0.557042\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	102210.	0.00693149
$738$	0	0
$739$	−1.07382e7	−0.723302	−0.361651	−	0.932313i	$-0.617787\pi$
$739$	−1.07382e7	−0.723302	−0.361651	+	0.932313i	$0.617787\pi$
$740$	0	0
$741$	−3.64622e7	−2.43948
$742$	0	0
$743$	1.88845e7	1.25497	0.627487	−	0.778627i	$-0.284084\pi$
$743$	1.88845e7	1.25497	0.627487	+	0.778627i	$0.284084\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.08636e7	−0.712314
$748$	0	0
$749$	−4.73372e6	−0.308318
$750$	0	0
$751$	7.40163e6	0.478881	0.239440	−	0.970911i	$-0.423036\pi$
$751$	7.40163e6	0.478881	0.239440	+	0.970911i	$0.423036\pi$
$752$	0	0
$753$	−916286.	−0.0588903
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.24822e6	−0.459718	−0.229859	−	0.973224i	$-0.573827\pi$
$757$	−7.24822e6	−0.459718	−0.229859	+	0.973224i	$0.573827\pi$
$758$	0	0
$759$	−1.32220e6	−0.0833089
$760$	0	0
$761$	5.50511e6	0.344591	0.172296	−	0.985045i	$-0.444882\pi$
$761$	5.50511e6	0.344591	0.172296	+	0.985045i	$0.444882\pi$
$762$	0	0
$763$	4.46113e7	2.77417
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.56241e7	−0.958970
$768$	0	0
$769$	1.81612e7	1.10746	0.553730	−	0.832696i	$-0.313204\pi$
$769$	1.81612e7	1.10746	0.553730	+	0.832696i	$0.313204\pi$
$770$	0	0
$771$	−3.43398e7	−2.08047
$772$	0	0
$773$	7.39969e6	0.445415	0.222707	−	0.974885i	$-0.428511\pi$
$773$	7.39969e6	0.445415	0.222707	+	0.974885i	$0.428511\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.21040e7	−1.31347
$778$	0	0
$779$	1.41074e6	0.0832920
$780$	0	0
$781$	−576341.	−0.0338105
$782$	0	0
$783$	1.05617e7	0.615645
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.85122e7	1.06542	0.532709	−	0.846298i	$-0.321174\pi$
$787$	1.85122e7	1.06542	0.532709	+	0.846298i	$0.321174\pi$
$788$	0	0
$789$	−1.84655e7	−1.05601
$790$	0	0
$791$	−4.17590e7	−2.37306
$792$	0	0
$793$	−2.76057e7	−1.55889
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.68656e7	0.940493	0.470247	−	0.882535i	$-0.344165\pi$
$797$	1.68656e7	0.940493	0.470247	+	0.882535i	$0.344165\pi$
$798$	0	0
$799$	8.52974e6	0.472682
$800$	0	0
$801$	5.52955e6	0.304515
$802$	0	0
$803$	1.57450e6	0.0861697
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.67745e7	−0.906706
$808$	0	0
$809$	−5.53056e6	−0.297097	−0.148548	−	0.988905i	$-0.547460\pi$
$809$	−5.53056e6	−0.297097	−0.148548	+	0.988905i	$0.547460\pi$
$810$	0	0
$811$	2.91422e7	1.55586	0.777929	−	0.628353i	$-0.216270\pi$
$811$	2.91422e7	1.55586	0.777929	+	0.628353i	$0.216270\pi$
$812$	0	0
$813$	−2.28172e7	−1.21070
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.62908e7	1.90214
$818$	0	0
$819$	2.13248e7	1.11090
$820$	0	0
$821$	−1.44892e7	−0.750216	−0.375108	−	0.926981i	$-0.622394\pi$
$821$	−1.44892e7	−0.750216	−0.375108	+	0.926981i	$0.622394\pi$
$822$	0	0
$823$	2.71404e7	1.39674	0.698371	−	0.715736i	$-0.253908\pi$
$823$	2.71404e7	1.39674	0.698371	+	0.715736i	$0.253908\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.01025e7	−1.53052	−0.765259	−	0.643722i	$-0.777389\pi$
$827$	−3.01025e7	−1.53052	−0.765259	+	0.643722i	$0.777389\pi$
$828$	0	0
$829$	−2.47022e7	−1.24839	−0.624194	−	0.781269i	$-0.714573\pi$
$829$	−2.47022e7	−1.24839	−0.624194	+	0.781269i	$0.714573\pi$
$830$	0	0
$831$	1.62441e7	0.816008
$832$	0	0
$833$	7.99878e6	0.399403
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.38190e6	−0.166858
$838$	0	0
$839$	1.23647e7	0.606428	0.303214	−	0.952922i	$-0.401940\pi$
$839$	1.23647e7	0.606428	0.303214	+	0.952922i	$0.401940\pi$
$840$	0	0
$841$	236010.	0.0115064
$842$	0	0
$843$	2.01286e7	0.975539
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.37694e7	1.61739
$848$	0	0
$849$	4.33123e6	0.206225
$850$	0	0
$851$	2.08757e7	0.988137
$852$	0	0
$853$	1.20993e7	0.569363	0.284681	−	0.958622i	$-0.408112\pi$
$853$	1.20993e7	0.569363	0.284681	+	0.958622i	$0.408112\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.86289e6	0.319194	0.159597	−	0.987182i	$-0.448980\pi$
$857$	6.86289e6	0.319194	0.159597	+	0.987182i	$0.448980\pi$
$858$	0	0
$859$	−2.52569e7	−1.16788	−0.583938	−	0.811798i	$-0.698489\pi$
$859$	−2.52569e7	−1.16788	−0.583938	+	0.811798i	$0.698489\pi$
$860$	0	0
$861$	−2.47443e6	−0.113754
$862$	0	0
$863$	2.45114e6	0.112032	0.0560159	−	0.998430i	$-0.482160\pi$
$863$	2.45114e6	0.112032	0.0560159	+	0.998430i	$0.482160\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.54758e7	1.15101
$868$	0	0
$869$	631041.	0.0283471
$870$	0	0
$871$	4.66917e6	0.208542
$872$	0	0
$873$	1.62050e7	0.719639
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.14652e7	−0.942401	−0.471201	−	0.882026i	$-0.656179\pi$
$877$	−2.14652e7	−0.942401	−0.471201	+	0.882026i	$0.656179\pi$
$878$	0	0
$879$	3.13976e7	1.37064
$880$	0	0
$881$	1.68815e7	0.732775	0.366388	−	0.930462i	$-0.380594\pi$
$881$	1.68815e7	0.732775	0.366388	+	0.930462i	$0.380594\pi$
$882$	0	0
$883$	−3.37649e7	−1.45735	−0.728676	−	0.684859i	$-0.759864\pi$
$883$	−3.37649e7	−1.45735	−0.728676	+	0.684859i	$0.759864\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.20696e7	1.36862	0.684312	−	0.729189i	$-0.260103\pi$
$887$	3.20696e7	1.36862	0.684312	+	0.729189i	$0.260103\pi$
$888$	0	0
$889$	1.78185e7	0.756166
$890$	0	0
$891$	1.34903e6	0.0569282
$892$	0	0
$893$	−6.66902e7	−2.79855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.04005e7	−2.50645
$898$	0	0
$899$	−6.64330e6	−0.274148
$900$	0	0
$901$	1.08080e7	0.443540
$902$	0	0
$903$	−6.36538e7	−2.59780
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.29290e7	0.521853	0.260926	−	0.965359i	$-0.415972\pi$
$907$	1.29290e7	0.521853	0.260926	+	0.965359i	$0.415972\pi$
$908$	0	0
$909$	−8.23461e6	−0.330547
$910$	0	0
$911$	4.00076e6	0.159715	0.0798577	−	0.996806i	$-0.474553\pi$
$911$	4.00076e6	0.159715	0.0798577	+	0.996806i	$0.474553\pi$
$912$	0	0
$913$	1.63340e6	0.0648508
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.63393e7	1.81981
$918$	0	0
$919$	2.55409e7	0.997581	0.498790	−	0.866723i	$-0.333778\pi$
$919$	2.55409e7	0.997581	0.498790	+	0.866723i	$0.333778\pi$
$920$	0	0
$921$	4.00321e7	1.55511
$922$	0	0
$923$	−2.63284e7	−1.01723
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.54050e6	−0.288204
$928$	0	0
$929$	−2.98080e7	−1.13316	−0.566582	−	0.824005i	$-0.691735\pi$
$929$	−2.98080e7	−1.13316	−0.566582	+	0.824005i	$0.691735\pi$
$930$	0	0
$931$	−6.25389e7	−2.36470
$932$	0	0
$933$	−2.79046e7	−1.04947
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.52937e7	−1.31325	−0.656627	−	0.754216i	$-0.728017\pi$
$937$	−3.52937e7	−1.31325	−0.656627	+	0.754216i	$0.728017\pi$
$938$	0	0
$939$	1.97382e7	0.730540
$940$	0	0
$941$	4.71557e7	1.73604	0.868020	−	0.496529i	$-0.165392\pi$
$941$	4.71557e7	1.73604	0.868020	+	0.496529i	$0.165392\pi$
$942$	0	0
$943$	2.33692e6	0.0855785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.56893e7	−1.65554	−0.827770	−	0.561067i	$-0.810391\pi$
$947$	−4.56893e7	−1.65554	−0.827770	+	0.561067i	$0.810391\pi$
$948$	0	0
$949$	7.19263e7	2.59252
$950$	0	0
$951$	2.83971e6	0.101818
$952$	0	0
$953$	2.47162e7	0.881556	0.440778	−	0.897616i	$-0.354703\pi$
$953$	2.47162e7	0.881556	0.440778	+	0.897616i	$0.354703\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.58950e6	0.0561024
$958$	0	0
$959$	−9.84024e6	−0.345509
$960$	0	0
$961$	−2.65019e7	−0.925698
$962$	0	0
$963$	2.73855e6	0.0951602
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.56789e6	−0.0883102	−0.0441551	−	0.999025i	$-0.514060\pi$
$967$	−2.56789e6	−0.0883102	−0.0441551	+	0.999025i	$0.514060\pi$
$968$	0	0
$969$	1.27758e7	0.437099
$970$	0	0
$971$	3.46122e7	1.17810	0.589049	−	0.808097i	$-0.299502\pi$
$971$	3.46122e7	1.17810	0.589049	+	0.808097i	$0.299502\pi$
$972$	0	0
$973$	3.55414e7	1.20352
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.51384e7	1.84807	0.924034	−	0.382309i	$-0.124871\pi$
$977$	5.51384e7	1.84807	0.924034	+	0.382309i	$0.124871\pi$
$978$	0	0
$979$	−831400.	−0.0277238
$980$	0	0
$981$	−2.58085e7	−0.856230
$982$	0	0
$983$	−3.37599e7	−1.11434	−0.557169	−	0.830399i	$-0.688113\pi$
$983$	−3.37599e7	−1.11434	−0.557169	+	0.830399i	$0.688113\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.16974e8	3.82205
$988$	0	0
$989$	6.01166e7	1.95435
$990$	0	0
$991$	−1.25256e7	−0.405150	−0.202575	−	0.979267i	$-0.564931\pi$
$991$	−1.25256e7	−0.405150	−0.202575	+	0.979267i	$0.564931\pi$
$992$	0	0
$993$	−2.94647e7	−0.948265
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.02324e7	0.963240	0.481620	−	0.876380i	$-0.340048\pi$
$997$	3.02324e7	0.963240	0.481620	+	0.876380i	$0.340048\pi$
$998$	0	0
$999$	−1.27757e7	−0.405013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.a.i.1.1	yes	3
4.3	odd	2		400.6.a.x.1.3		3
5.2	odd	4		200.6.c.g.49.5		6
5.3	odd	4		200.6.c.g.49.2		6
5.4	even	2		200.6.a.h.1.3	✓	3
20.3	even	4		400.6.c.p.49.5		6
20.7	even	4		400.6.c.p.49.2		6
20.19	odd	2		400.6.a.y.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.a.h.1.3	✓	3	5.4	even	2
200.6.a.i.1.1	yes	3	1.1	even	1	trivial
200.6.c.g.49.2		6	5.3	odd	4
200.6.c.g.49.5		6	5.2	odd	4
400.6.a.x.1.3		3	4.3	odd	2
400.6.a.y.1.1		3	20.19	odd	2
400.6.c.p.49.2		6	20.7	even	4
400.6.c.p.49.5		6	20.3	even	4

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Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} +O(q^{10})\)	\(q-19.0934 q^{3} -210.117 q^{7} +121.557 q^{9} -18.2768 q^{11} -834.918 q^{13} +292.543 q^{17} -2287.27 q^{19} +4011.84 q^{21} -3788.91 q^{23} +2318.76 q^{27} +4554.90 q^{29} -1458.49 q^{31} +348.965 q^{33} -5509.69 q^{37} +15941.4 q^{39} -616.780 q^{41} -15866.5 q^{43} +29157.2 q^{47} +27342.2 q^{49} -5585.64 q^{51} +36944.9 q^{53} +43671.6 q^{57} +18713.3 q^{59} +33063.9 q^{61} -25541.2 q^{63} -5592.37 q^{67} +72343.0 q^{69} +31534.1 q^{71} -86147.7 q^{73} +3840.26 q^{77} -34527.0 q^{79} -73811.2 q^{81} -89370.3 q^{83} -86968.5 q^{87} +45489.4 q^{89} +175431. q^{91} +27847.6 q^{93} +133312. q^{97} -2221.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9} - 19 q^{11} - 196 q^{13} + 1223 q^{17} + 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} + 7442 q^{31} - 6969 q^{33} + 14766 q^{37} + 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} + 43165 q^{51} - 12878 q^{53} + 77791 q^{57} + 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} + 86676 q^{71} + 8305 q^{73} + 106822 q^{77} + 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} + 149224 q^{91} + 196470 q^{93} + 76342 q^{97} - 149130 q^{99}+O(q^{100}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9 - 19 * q^11 - 196 * q^13 + 1223 * q^17 + 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 + 7442 * q^31 - 6969 * q^33 + 14766 * q^37 + 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 + 43165 * q^51 - 12878 * q^53 + 77791 * q^57 + 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 + 86676 * q^71 + 8305 * q^73 + 106822 * q^77 + 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 + 149224 * q^91 + 196470 * q^93 + 76342 * q^97 - 149130 * q^99

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