LMFDB - Embedded newform 600.6.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,6,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("600.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.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:	$96.2302918878$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.20073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 30x - 24 $ x^3 - x^2 - 30*x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.33429$ of defining polynomial
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -85.8595 q^{7} +81.0000 q^{9} -488.087 q^{11} -38.5971 q^{13} +692.837 q^{17} +2489.26 q^{19} +772.735 q^{21} +4124.41 q^{23} -729.000 q^{27} -1886.71 q^{29} -2299.60 q^{31} +4392.78 q^{33} -10628.6 q^{37} +347.373 q^{39} -15268.5 q^{41} +9468.20 q^{43} +14323.3 q^{47} -9435.15 q^{49} -6235.53 q^{51} +13918.1 q^{53} -22403.3 q^{57} +39692.5 q^{59} +38221.6 q^{61} -6954.62 q^{63} +1416.38 q^{67} -37119.6 q^{69} -11637.1 q^{71} +31082.1 q^{73} +41906.9 q^{77} -43476.3 q^{79} +6561.00 q^{81} -86699.9 q^{83} +16980.4 q^{87} -100834. q^{89} +3313.92 q^{91} +20696.4 q^{93} +39911.0 q^{97} -39535.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 27 q^{3} + 104 q^{7} + 243 q^{9} - 332 q^{11} + 88 q^{13} - 1116 q^{17} - 144 q^{19} - 936 q^{21} + 2868 q^{23} - 2187 q^{27} + 946 q^{29} - 5124 q^{31} + 2988 q^{33} - 7808 q^{37} - 792 q^{39} - 662 q^{41}+ \cdots - 26892 q^{99}+O(q^{100}) $ 3 * q - 27 * q^3 + 104 * q^7 + 243 * q^9 - 332 * q^11 + 88 * q^13 - 1116 * q^17 - 144 * q^19 - 936 * q^21 + 2868 * q^23 - 2187 * q^27 + 946 * q^29 - 5124 * q^31 + 2988 * q^33 - 7808 * q^37 - 792 * q^39 - 662 * q^41 + 25188 * q^43 + 25428 * q^47 - 14205 * q^49 + 10044 * q^51 + 36924 * q^53 + 1296 * q^57 + 5148 * q^59 - 26058 * q^61 + 8424 * q^63 + 30644 * q^67 - 25812 * q^69 - 14144 * q^71 + 37736 * q^73 + 46800 * q^77 - 110100 * q^79 + 19683 * q^81 + 7620 * q^83 - 8514 * q^87 - 175710 * q^89 - 8544 * q^91 + 46116 * q^93 - 6632 * q^97 - 26892 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−85.8595	−0.662282	−0.331141	−	0.943581i	$-0.607434\pi$
$7$	−85.8595	−0.662282	−0.331141	+	0.943581i	$0.607434\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−488.087	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$11$	−488.087	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$12$	0	0
$13$	−38.5971	−0.0633426	−0.0316713	−	0.999498i	$-0.510083\pi$
$13$	−38.5971	−0.0633426	−0.0316713	+	0.999498i	$0.510083\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	692.837	0.581445	0.290723	−	0.956807i	$-0.406104\pi$
$17$	692.837	0.581445	0.290723	+	0.956807i	$0.406104\pi$
$18$	0	0
$19$	2489.26	1.58192	0.790962	−	0.611865i	$-0.209581\pi$
$19$	2489.26	1.58192	0.790962	+	0.611865i	$0.209581\pi$
$20$	0	0
$21$	772.735	0.382369
$22$	0	0
$23$	4124.41	1.62571	0.812853	−	0.582470i	$-0.197913\pi$
$23$	4124.41	1.62571	0.812853	+	0.582470i	$0.197913\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−1886.71	−0.416592	−0.208296	−	0.978066i	$-0.566792\pi$
$29$	−1886.71	−0.416592	−0.208296	+	0.978066i	$0.566792\pi$
$30$	0	0
$31$	−2299.60	−0.429781	−0.214891	−	0.976638i	$-0.568940\pi$
$31$	−2299.60	−0.429781	−0.214891	+	0.976638i	$0.568940\pi$
$32$	0	0
$33$	4392.78	0.702190
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10628.6	−1.27636	−0.638178	−	0.769888i	$-0.720312\pi$
$37$	−10628.6	−1.27636	−0.638178	+	0.769888i	$0.720312\pi$
$38$	0	0
$39$	347.373	0.0365709
$40$	0	0
$41$	−15268.5	−1.41853	−0.709264	−	0.704943i	$-0.750973\pi$
$41$	−15268.5	−1.41853	−0.709264	+	0.704943i	$0.750973\pi$
$42$	0	0
$43$	9468.20	0.780901	0.390451	−	0.920624i	$-0.372319\pi$
$43$	9468.20	0.780901	0.390451	+	0.920624i	$0.372319\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	14323.3	0.945800	0.472900	−	0.881116i	$-0.343207\pi$
$47$	14323.3	0.945800	0.472900	+	0.881116i	$0.343207\pi$
$48$	0	0
$49$	−9435.15	−0.561382
$50$	0	0
$51$	−6235.53	−0.335698
$52$	0	0
$53$	13918.1	0.680596	0.340298	−	0.940318i	$-0.389472\pi$
$53$	13918.1	0.680596	0.340298	+	0.940318i	$0.389472\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−22403.3	−0.913324
$58$	0	0
$59$	39692.5	1.48449	0.742247	−	0.670126i	$-0.233760\pi$
$59$	39692.5	1.48449	0.742247	+	0.670126i	$0.233760\pi$
$60$	0	0
$61$	38221.6	1.31518	0.657588	−	0.753377i	$-0.271577\pi$
$61$	38221.6	1.31518	0.657588	+	0.753377i	$0.271577\pi$
$62$	0	0
$63$	−6954.62	−0.220761
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1416.38	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$67$	1416.38	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$68$	0	0
$69$	−37119.6	−0.938601
$70$	0	0
$71$	−11637.1	−0.273966	−0.136983	−	0.990573i	$-0.543741\pi$
$71$	−11637.1	−0.273966	−0.136983	+	0.990573i	$0.543741\pi$
$72$	0	0
$73$	31082.1	0.682657	0.341329	−	0.939944i	$-0.389123\pi$
$73$	31082.1	0.682657	0.341329	+	0.939944i	$0.389123\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	41906.9	0.805487
$78$	0	0
$79$	−43476.3	−0.783763	−0.391881	−	0.920016i	$-0.628176\pi$
$79$	−43476.3	−0.783763	−0.391881	+	0.920016i	$0.628176\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−86699.9	−1.38141	−0.690706	−	0.723135i	$-0.742700\pi$
$83$	−86699.9	−1.38141	−0.690706	+	0.723135i	$0.742700\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16980.4	0.240519
$88$	0	0
$89$	−100834.	−1.34937	−0.674686	−	0.738105i	$-0.735721\pi$
$89$	−100834.	−1.34937	−0.674686	+	0.738105i	$0.735721\pi$
$90$	0	0
$91$	3313.92	0.0419507
$92$	0	0
$93$	20696.4	0.248134
$94$	0	0
$95$	0	0
$96$	0	0
$97$	39911.0	0.430688	0.215344	−	0.976538i	$-0.430913\pi$
$97$	39911.0	0.430688	0.215344	+	0.976538i	$0.430913\pi$
$98$	0	0
$99$	−39535.0	−0.405410
$100$	0	0
$101$	−117147.	−1.14269	−0.571344	−	0.820711i	$-0.693578\pi$
$101$	−117147.	−1.14269	−0.571344	+	0.820711i	$0.693578\pi$
$102$	0	0
$103$	−266.000	−0.00247052	−0.00123526	−	0.999999i	$-0.500393\pi$
$103$	−266.000	−0.00247052	−0.00123526	+	0.999999i	$0.500393\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	120151.	1.01454	0.507270	−	0.861787i	$-0.330655\pi$
$107$	120151.	1.01454	0.507270	+	0.861787i	$0.330655\pi$
$108$	0	0
$109$	−135567.	−1.09292	−0.546458	−	0.837486i	$-0.684024\pi$
$109$	−135567.	−1.09292	−0.546458	+	0.837486i	$0.684024\pi$
$110$	0	0
$111$	95657.5	0.736905
$112$	0	0
$113$	−94060.5	−0.692965	−0.346482	−	0.938056i	$-0.612624\pi$
$113$	−94060.5	−0.692965	−0.346482	+	0.938056i	$0.612624\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3126.36	−0.0211142
$118$	0	0
$119$	−59486.6	−0.385081
$120$	0	0
$121$	77177.7	0.479213
$122$	0	0
$123$	137417.	0.818988
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−118472.	−0.651786	−0.325893	−	0.945407i	$-0.605665\pi$
$127$	−118472.	−0.651786	−0.325893	+	0.945407i	$0.605665\pi$
$128$	0	0
$129$	−85213.8	−0.450854
$130$	0	0
$131$	−232998.	−1.18624	−0.593122	−	0.805113i	$-0.702105\pi$
$131$	−232998.	−1.18624	−0.593122	+	0.805113i	$0.702105\pi$
$132$	0	0
$133$	−213726.	−1.04768
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−139571.	−0.635321	−0.317661	−	0.948204i	$-0.602897\pi$
$137$	−139571.	−0.635321	−0.317661	+	0.948204i	$0.602897\pi$
$138$	0	0
$139$	−263812.	−1.15813	−0.579066	−	0.815280i	$-0.696583\pi$
$139$	−263812.	−1.15813	−0.579066	+	0.815280i	$0.696583\pi$
$140$	0	0
$141$	−128910.	−0.546058
$142$	0	0
$143$	18838.7	0.0770391
$144$	0	0
$145$	0	0
$146$	0	0
$147$	84916.4	0.324114
$148$	0	0
$149$	344172.	1.27002	0.635008	−	0.772505i	$-0.280997\pi$
$149$	344172.	1.27002	0.635008	+	0.772505i	$0.280997\pi$
$150$	0	0
$151$	−12286.8	−0.0438527	−0.0219263	−	0.999760i	$-0.506980\pi$
$151$	−12286.8	−0.0438527	−0.0219263	+	0.999760i	$0.506980\pi$
$152$	0	0
$153$	56119.8	0.193815
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−240704.	−0.779354	−0.389677	−	0.920952i	$-0.627413\pi$
$157$	−240704.	−0.779354	−0.389677	+	0.920952i	$0.627413\pi$
$158$	0	0
$159$	−125263.	−0.392942
$160$	0	0
$161$	−354119.	−1.07668
$162$	0	0
$163$	565359.	1.66669	0.833346	−	0.552752i	$-0.186422\pi$
$163$	565359.	1.66669	0.833346	+	0.552752i	$0.186422\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−411038.	−1.14049	−0.570244	−	0.821475i	$-0.693151\pi$
$167$	−411038.	−1.14049	−0.570244	+	0.821475i	$0.693151\pi$
$168$	0	0
$169$	−369803.	−0.995988
$170$	0	0
$171$	201630.	0.527308
$172$	0	0
$173$	−188716.	−0.479395	−0.239698	−	0.970848i	$-0.577048\pi$
$173$	−188716.	−0.479395	−0.239698	+	0.970848i	$0.577048\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−357233.	−0.857073
$178$	0	0
$179$	−601974.	−1.40425	−0.702126	−	0.712053i	$-0.747766\pi$
$179$	−601974.	−1.40425	−0.702126	+	0.712053i	$0.747766\pi$
$180$	0	0
$181$	−836032.	−1.89682	−0.948410	−	0.317045i	$-0.897309\pi$
$181$	−836032.	−1.89682	−0.948410	+	0.317045i	$0.897309\pi$
$182$	0	0
$183$	−343994.	−0.759318
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−338165.	−0.707170
$188$	0	0
$189$	62591.6	0.127456
$190$	0	0
$191$	502698.	0.997066	0.498533	−	0.866871i	$-0.333872\pi$
$191$	502698.	0.997066	0.498533	+	0.866871i	$0.333872\pi$
$192$	0	0
$193$	−731154.	−1.41291	−0.706457	−	0.707756i	$-0.749707\pi$
$193$	−731154.	−1.41291	−0.706457	+	0.707756i	$0.749707\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	871888.	1.60065	0.800323	−	0.599569i	$-0.204661\pi$
$197$	871888.	1.60065	0.800323	+	0.599569i	$0.204661\pi$
$198$	0	0
$199$	−101311.	−0.181352	−0.0906760	−	0.995880i	$-0.528903\pi$
$199$	−101311.	−0.181352	−0.0906760	+	0.995880i	$0.528903\pi$
$200$	0	0
$201$	−12747.4	−0.0222553
$202$	0	0
$203$	161992.	0.275901
$204$	0	0
$205$	0	0
$206$	0	0
$207$	334077.	0.541902
$208$	0	0
$209$	−1.21497e6	−1.92398
$210$	0	0
$211$	−725153.	−1.12130	−0.560652	−	0.828051i	$-0.689450\pi$
$211$	−725153.	−1.12130	−0.560652	+	0.828051i	$0.689450\pi$
$212$	0	0
$213$	104734.	0.158175
$214$	0	0
$215$	0	0
$216$	0	0
$217$	197442.	0.284637
$218$	0	0
$219$	−279739.	−0.394132
$220$	0	0
$221$	−26741.5	−0.0368302
$222$	0	0
$223$	680623.	0.916526	0.458263	−	0.888817i	$-0.348472\pi$
$223$	680623.	0.916526	0.458263	+	0.888817i	$0.348472\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.01003e6	1.30098	0.650488	−	0.759516i	$-0.274564\pi$
$227$	1.01003e6	1.30098	0.650488	+	0.759516i	$0.274564\pi$
$228$	0	0
$229$	−1.22740e6	−1.54667	−0.773337	−	0.633996i	$-0.781414\pi$
$229$	−1.22740e6	−1.54667	−0.773337	+	0.633996i	$0.781414\pi$
$230$	0	0
$231$	−377162.	−0.465048
$232$	0	0
$233$	599483.	0.723415	0.361707	−	0.932292i	$-0.382194\pi$
$233$	599483.	0.723415	0.361707	+	0.932292i	$0.382194\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	391287.	0.452506
$238$	0	0
$239$	−1.37162e6	−1.55324	−0.776620	−	0.629970i	$-0.783067\pi$
$239$	−1.37162e6	−1.55324	−0.776620	+	0.629970i	$0.783067\pi$
$240$	0	0
$241$	1.36857e6	1.51784	0.758918	−	0.651186i	$-0.225728\pi$
$241$	1.36857e6	1.51784	0.758918	+	0.651186i	$0.225728\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−96078.0	−0.100203
$248$	0	0
$249$	780299.	0.797559
$250$	0	0
$251$	730293.	0.731666	0.365833	−	0.930681i	$-0.380784\pi$
$251$	730293.	0.731666	0.365833	+	0.930681i	$0.380784\pi$
$252$	0	0
$253$	−2.01307e6	−1.97723
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.36832e6	−1.29228	−0.646138	−	0.763221i	$-0.723617\pi$
$257$	−1.36832e6	−1.29228	−0.646138	+	0.763221i	$0.723617\pi$
$258$	0	0
$259$	912567.	0.845309
$260$	0	0
$261$	−152824.	−0.138864
$262$	0	0
$263$	1.26265e6	1.12562	0.562811	−	0.826585i	$-0.309720\pi$
$263$	1.26265e6	1.12562	0.562811	+	0.826585i	$0.309720\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	907505.	0.779060
$268$	0	0
$269$	652896.	0.550127	0.275064	−	0.961426i	$-0.411301\pi$
$269$	652896.	0.550127	0.275064	+	0.961426i	$0.411301\pi$
$270$	0	0
$271$	−649071.	−0.536870	−0.268435	−	0.963298i	$-0.586506\pi$
$271$	−649071.	−0.536870	−0.268435	+	0.963298i	$0.586506\pi$
$272$	0	0
$273$	−29825.3	−0.0242202
$274$	0	0
$275$	0	0
$276$	0	0
$277$	668500.	0.523482	0.261741	−	0.965138i	$-0.415703\pi$
$277$	668500.	0.523482	0.261741	+	0.965138i	$0.415703\pi$
$278$	0	0
$279$	−186267.	−0.143260
$280$	0	0
$281$	1.03323e6	0.780607	0.390304	−	0.920686i	$-0.372370\pi$
$281$	1.03323e6	0.780607	0.390304	+	0.920686i	$0.372370\pi$
$282$	0	0
$283$	−384887.	−0.285672	−0.142836	−	0.989746i	$-0.545622\pi$
$283$	−384887.	−0.285672	−0.142836	+	0.989746i	$0.545622\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.31095e6	0.939466
$288$	0	0
$289$	−939834.	−0.661922
$290$	0	0
$291$	−359199.	−0.248658
$292$	0	0
$293$	74753.1	0.0508698	0.0254349	−	0.999676i	$-0.491903\pi$
$293$	74753.1	0.0508698	0.0254349	+	0.999676i	$0.491903\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	355815.	0.234063
$298$	0	0
$299$	−159190.	−0.102976
$300$	0	0
$301$	−812934.	−0.517177
$302$	0	0
$303$	1.05432e6	0.659731
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.96989e6	−1.19288	−0.596438	−	0.802659i	$-0.703418\pi$
$307$	−1.96989e6	−1.19288	−0.596438	+	0.802659i	$0.703418\pi$
$308$	0	0
$309$	2394.00	0.00142635
$310$	0	0
$311$	−1.92570e6	−1.12899	−0.564493	−	0.825438i	$-0.690928\pi$
$311$	−1.92570e6	−1.12899	−0.564493	+	0.825438i	$0.690928\pi$
$312$	0	0
$313$	1.05127e6	0.606531	0.303265	−	0.952906i	$-0.401923\pi$
$313$	1.05127e6	0.606531	0.303265	+	0.952906i	$0.401923\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.68469e6	−1.50053	−0.750267	−	0.661134i	$-0.770075\pi$
$317$	−2.68469e6	−1.50053	−0.750267	+	0.661134i	$0.770075\pi$
$318$	0	0
$319$	920879.	0.506671
$320$	0	0
$321$	−1.08136e6	−0.585745
$322$	0	0
$323$	1.72465e6	0.919802
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.22010e6	0.630996
$328$	0	0
$329$	−1.22979e6	−0.626387
$330$	0	0
$331$	−1.50456e6	−0.754813	−0.377407	−	0.926048i	$-0.623184\pi$
$331$	−1.50456e6	−0.754813	−0.377407	+	0.926048i	$0.623184\pi$
$332$	0	0
$333$	−860917.	−0.425452
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.84030e6	−1.84200	−0.921001	−	0.389559i	$-0.872627\pi$
$337$	−3.84030e6	−1.84200	−0.921001	+	0.389559i	$0.872627\pi$
$338$	0	0
$339$	846544.	0.400083
$340$	0	0
$341$	1.12240e6	0.522713
$342$	0	0
$343$	2.25314e6	1.03408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−939488.	−0.418859	−0.209429	−	0.977824i	$-0.567161\pi$
$347$	−939488.	−0.418859	−0.209429	+	0.977824i	$0.567161\pi$
$348$	0	0
$349$	3.56398e6	1.56629	0.783144	−	0.621841i	$-0.213615\pi$
$349$	3.56398e6	1.56629	0.783144	+	0.621841i	$0.213615\pi$
$350$	0	0
$351$	28137.2	0.0121903
$352$	0	0
$353$	−2.24033e6	−0.956919	−0.478459	−	0.878110i	$-0.658805\pi$
$353$	−2.24033e6	−0.956919	−0.478459	+	0.878110i	$0.658805\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	535380.	0.222327
$358$	0	0
$359$	290038.	0.118773	0.0593867	−	0.998235i	$-0.481085\pi$
$359$	290038.	0.118773	0.0593867	+	0.998235i	$0.481085\pi$
$360$	0	0
$361$	3.72030e6	1.50248
$362$	0	0
$363$	−694599.	−0.276674
$364$	0	0
$365$	0	0
$366$	0	0
$367$	961035.	0.372455	0.186228	−	0.982507i	$-0.440374\pi$
$367$	961035.	0.372455	0.186228	+	0.982507i	$0.440374\pi$
$368$	0	0
$369$	−1.23675e6	−0.472843
$370$	0	0
$371$	−1.19500e6	−0.450746
$372$	0	0
$373$	−3.48605e6	−1.29736	−0.648681	−	0.761060i	$-0.724679\pi$
$373$	−3.48605e6	−1.29736	−0.648681	+	0.761060i	$0.724679\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	72821.5	0.0263880
$378$	0	0
$379$	3.08939e6	1.10478	0.552389	−	0.833587i	$-0.313716\pi$
$379$	3.08939e6	1.10478	0.552389	+	0.833587i	$0.313716\pi$
$380$	0	0
$381$	1.06624e6	0.376309
$382$	0	0
$383$	513758.	0.178962	0.0894812	−	0.995989i	$-0.471479\pi$
$383$	513758.	0.178962	0.0894812	+	0.995989i	$0.471479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	766924.	0.260300
$388$	0	0
$389$	5.32273e6	1.78345	0.891725	−	0.452578i	$-0.149496\pi$
$389$	5.32273e6	1.78345	0.891725	+	0.452578i	$0.149496\pi$
$390$	0	0
$391$	2.85754e6	0.945258
$392$	0	0
$393$	2.09698e6	0.684878
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.10400e6	−0.669990	−0.334995	−	0.942220i	$-0.608735\pi$
$397$	−2.10400e6	−0.669990	−0.334995	+	0.942220i	$0.608735\pi$
$398$	0	0
$399$	1.92354e6	0.604878
$400$	0	0
$401$	1.41705e6	0.440072	0.220036	−	0.975492i	$-0.429382\pi$
$401$	1.41705e6	0.440072	0.220036	+	0.975492i	$0.429382\pi$
$402$	0	0
$403$	88757.7	0.0272235
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.18768e6	1.55234
$408$	0	0
$409$	1.07044e6	0.316414	0.158207	−	0.987406i	$-0.449429\pi$
$409$	1.07044e6	0.316414	0.158207	+	0.987406i	$0.449429\pi$
$410$	0	0
$411$	1.25614e6	0.366803
$412$	0	0
$413$	−3.40798e6	−0.983154
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.37431e6	0.668648
$418$	0	0
$419$	2.75665e6	0.767090	0.383545	−	0.923522i	$-0.374703\pi$
$419$	2.75665e6	0.767090	0.383545	+	0.923522i	$0.374703\pi$
$420$	0	0
$421$	−3.32521e6	−0.914352	−0.457176	−	0.889376i	$-0.651139\pi$
$421$	−3.32521e6	−0.914352	−0.457176	+	0.889376i	$0.651139\pi$
$422$	0	0
$423$	1.16019e6	0.315267
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.28169e6	−0.871018
$428$	0	0
$429$	−169548.	−0.0444785
$430$	0	0
$431$	1.26909e6	0.329078	0.164539	−	0.986371i	$-0.447386\pi$
$431$	1.26909e6	0.329078	0.164539	+	0.986371i	$0.447386\pi$
$432$	0	0
$433$	−5.17143e6	−1.32553	−0.662767	−	0.748826i	$-0.730618\pi$
$433$	−5.17143e6	−1.32553	−0.662767	+	0.748826i	$0.730618\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.02667e7	2.57174
$438$	0	0
$439$	−6.78923e6	−1.68136	−0.840678	−	0.541536i	$-0.817843\pi$
$439$	−6.78923e6	−1.68136	−0.840678	+	0.541536i	$0.817843\pi$
$440$	0	0
$441$	−764247.	−0.187127
$442$	0	0
$443$	3.16608e6	0.766500	0.383250	−	0.923645i	$-0.374805\pi$
$443$	3.16608e6	0.766500	0.383250	+	0.923645i	$0.374805\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.09755e6	−0.733245
$448$	0	0
$449$	4.91937e6	1.15158	0.575790	−	0.817598i	$-0.304695\pi$
$449$	4.91937e6	1.15158	0.575790	+	0.817598i	$0.304695\pi$
$450$	0	0
$451$	7.45238e6	1.72526
$452$	0	0
$453$	110581.	0.0253184
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.30601e6	−1.63640	−0.818201	−	0.574932i	$-0.805028\pi$
$457$	−7.30601e6	−1.63640	−0.818201	+	0.574932i	$0.805028\pi$
$458$	0	0
$459$	−505078.	−0.111899
$460$	0	0
$461$	−2.52155e6	−0.552606	−0.276303	−	0.961071i	$-0.589109\pi$
$461$	−2.52155e6	−0.552606	−0.276303	+	0.961071i	$0.589109\pi$
$462$	0	0
$463$	−6.44859e6	−1.39802	−0.699008	−	0.715114i	$-0.746375\pi$
$463$	−6.44859e6	−1.39802	−0.699008	+	0.715114i	$0.746375\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.39798e6	1.14535	0.572676	−	0.819782i	$-0.305905\pi$
$467$	5.39798e6	1.14535	0.572676	+	0.819782i	$0.305905\pi$
$468$	0	0
$469$	−121610.	−0.0255291
$470$	0	0
$471$	2.16634e6	0.449960
$472$	0	0
$473$	−4.62130e6	−0.949755
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.12736e6	0.226865
$478$	0	0
$479$	5.25882e6	1.04725	0.523624	−	0.851950i	$-0.324580\pi$
$479$	5.25882e6	1.04725	0.523624	+	0.851950i	$0.324580\pi$
$480$	0	0
$481$	410233.	0.0808477
$482$	0	0
$483$	3.18707e6	0.621619
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.61141e6	−1.07214	−0.536068	−	0.844175i	$-0.680091\pi$
$487$	−5.61141e6	−1.07214	−0.536068	+	0.844175i	$0.680091\pi$
$488$	0	0
$489$	−5.08823e6	−0.962265
$490$	0	0
$491$	5.14176e6	0.962516	0.481258	−	0.876579i	$-0.340180\pi$
$491$	5.14176e6	0.962516	0.481258	+	0.876579i	$0.340180\pi$
$492$	0	0
$493$	−1.30718e6	−0.242225
$494$	0	0
$495$	0	0
$496$	0	0
$497$	999151.	0.181443
$498$	0	0
$499$	4.53158e6	0.814701	0.407350	−	0.913272i	$-0.366453\pi$
$499$	4.53158e6	0.814701	0.407350	+	0.913272i	$0.366453\pi$
$500$	0	0
$501$	3.69934e6	0.658461
$502$	0	0
$503$	3.93518e6	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$503$	3.93518e6	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.32823e6	0.575034
$508$	0	0
$509$	−1.37582e6	−0.235378	−0.117689	−	0.993050i	$-0.537549\pi$
$509$	−1.37582e6	−0.235378	−0.117689	+	0.993050i	$0.537549\pi$
$510$	0	0
$511$	−2.66869e6	−0.452112
$512$	0	0
$513$	−1.81467e6	−0.304441
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.99103e6	−1.15031
$518$	0	0
$519$	1.69844e6	0.276779
$520$	0	0
$521$	−7.67174e6	−1.23822	−0.619112	−	0.785302i	$-0.712507\pi$
$521$	−7.67174e6	−1.23822	−0.619112	+	0.785302i	$0.712507\pi$
$522$	0	0
$523$	1.16971e7	1.86993	0.934964	−	0.354743i	$-0.115432\pi$
$523$	1.16971e7	1.86993	0.934964	+	0.354743i	$0.115432\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.59325e6	−0.249894
$528$	0	0
$529$	1.05744e7	1.64292
$530$	0	0
$531$	3.21509e6	0.494831
$532$	0	0
$533$	589321.	0.0898533
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.41777e6	0.810745
$538$	0	0
$539$	4.60517e6	0.682769
$540$	0	0
$541$	−3.26111e6	−0.479041	−0.239521	−	0.970891i	$-0.576990\pi$
$541$	−3.26111e6	−0.479041	−0.239521	+	0.970891i	$0.576990\pi$
$542$	0	0
$543$	7.52429e6	1.09513
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−707108.	−0.101046	−0.0505228	−	0.998723i	$-0.516089\pi$
$547$	−707108.	−0.101046	−0.0505228	+	0.998723i	$0.516089\pi$
$548$	0	0
$549$	3.09595e6	0.438392
$550$	0	0
$551$	−4.69651e6	−0.659016
$552$	0	0
$553$	3.73285e6	0.519072
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.55292e6	−0.348657	−0.174329	−	0.984688i	$-0.555776\pi$
$557$	−2.55292e6	−0.348657	−0.174329	+	0.984688i	$0.555776\pi$
$558$	0	0
$559$	−365445.	−0.0494643
$560$	0	0
$561$	3.04348e6	0.408285
$562$	0	0
$563$	1.60880e6	0.213910	0.106955	−	0.994264i	$-0.465890\pi$
$563$	1.60880e6	0.213910	0.106955	+	0.994264i	$0.465890\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−563324.	−0.0735869
$568$	0	0
$569$	1.25953e7	1.63091	0.815453	−	0.578823i	$-0.196488\pi$
$569$	1.25953e7	1.63091	0.815453	+	0.578823i	$0.196488\pi$
$570$	0	0
$571$	2.22289e6	0.285317	0.142659	−	0.989772i	$-0.454435\pi$
$571$	2.22289e6	0.285317	0.142659	+	0.989772i	$0.454435\pi$
$572$	0	0
$573$	−4.52429e6	−0.575657
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.71842e6	−1.09018	−0.545090	−	0.838377i	$-0.683505\pi$
$577$	−8.71842e6	−1.09018	−0.545090	+	0.838377i	$0.683505\pi$
$578$	0	0
$579$	6.58039e6	0.815746
$580$	0	0
$581$	7.44401e6	0.914885
$582$	0	0
$583$	−6.79322e6	−0.827760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.00682e6	0.360174	0.180087	−	0.983651i	$-0.442362\pi$
$587$	3.00682e6	0.360174	0.180087	+	0.983651i	$0.442362\pi$
$588$	0	0
$589$	−5.72429e6	−0.679882
$590$	0	0
$591$	−7.84700e6	−0.924133
$592$	0	0
$593$	−224622.	−0.0262311	−0.0131155	−	0.999914i	$-0.504175\pi$
$593$	−224622.	−0.0262311	−0.0131155	+	0.999914i	$0.504175\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	911795.	0.104704
$598$	0	0
$599$	−1.14533e7	−1.30426	−0.652129	−	0.758108i	$-0.726124\pi$
$599$	−1.14533e7	−1.30426	−0.652129	+	0.758108i	$0.726124\pi$
$600$	0	0
$601$	−1.17677e7	−1.32894	−0.664470	−	0.747315i	$-0.731343\pi$
$601$	−1.17677e7	−1.32894	−0.664470	+	0.747315i	$0.731343\pi$
$602$	0	0
$603$	114727.	0.0128491
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.24589e6	−0.137248	−0.0686242	−	0.997643i	$-0.521861\pi$
$607$	−1.24589e6	−0.137248	−0.0686242	+	0.997643i	$0.521861\pi$
$608$	0	0
$609$	−1.45793e6	−0.159292
$610$	0	0
$611$	−552838.	−0.0599094
$612$	0	0
$613$	1.45412e7	1.56296	0.781480	−	0.623931i	$-0.214465\pi$
$613$	1.45412e7	1.56296	0.781480	+	0.623931i	$0.214465\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.06609e6	0.535747	0.267874	−	0.963454i	$-0.413679\pi$
$617$	5.06609e6	0.535747	0.267874	+	0.963454i	$0.413679\pi$
$618$	0	0
$619$	1.03229e7	1.08287	0.541435	−	0.840742i	$-0.317881\pi$
$619$	1.03229e7	1.08287	0.541435	+	0.840742i	$0.317881\pi$
$620$	0	0
$621$	−3.00669e6	−0.312867
$622$	0	0
$623$	8.65755e6	0.893665
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.09348e7	1.11081
$628$	0	0
$629$	−7.36389e6	−0.742132
$630$	0	0
$631$	1.27332e7	1.27310	0.636552	−	0.771234i	$-0.280360\pi$
$631$	1.27332e7	1.27310	0.636552	+	0.771234i	$0.280360\pi$
$632$	0	0
$633$	6.52638e6	0.647386
$634$	0	0
$635$	0	0
$636$	0	0
$637$	364169.	0.0355594
$638$	0	0
$639$	−942602.	−0.0913221
$640$	0	0
$641$	1.37535e7	1.32211	0.661057	−	0.750336i	$-0.270108\pi$
$641$	1.37535e7	1.32211	0.661057	+	0.750336i	$0.270108\pi$
$642$	0	0
$643$	−1.62140e7	−1.54654	−0.773272	−	0.634075i	$-0.781381\pi$
$643$	−1.62140e7	−1.54654	−0.773272	+	0.634075i	$0.781381\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.27009e7	−1.19282	−0.596410	−	0.802680i	$-0.703407\pi$
$647$	−1.27009e7	−1.19282	−0.596410	+	0.802680i	$0.703407\pi$
$648$	0	0
$649$	−1.93734e7	−1.80548
$650$	0	0
$651$	−1.77698e6	−0.164335
$652$	0	0
$653$	1.73886e7	1.59581	0.797907	−	0.602781i	$-0.205941\pi$
$653$	1.73886e7	1.59581	0.797907	+	0.602781i	$0.205941\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.51765e6	0.227552
$658$	0	0
$659$	−1.26691e7	−1.13640	−0.568200	−	0.822891i	$-0.692360\pi$
$659$	−1.26691e7	−1.13640	−0.568200	+	0.822891i	$0.692360\pi$
$660$	0	0
$661$	6.69054e6	0.595604	0.297802	−	0.954628i	$-0.403746\pi$
$661$	6.69054e6	0.595604	0.297802	+	0.954628i	$0.403746\pi$
$662$	0	0
$663$	240673.	0.0212639
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.78157e6	−0.677255
$668$	0	0
$669$	−6.12561e6	−0.529156
$670$	0	0
$671$	−1.86555e7	−1.59956
$672$	0	0
$673$	−1.31246e7	−1.11699	−0.558494	−	0.829509i	$-0.688621\pi$
$673$	−1.31246e7	−1.11699	−0.558494	+	0.829509i	$0.688621\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.74331e6	0.565460	0.282730	−	0.959200i	$-0.408760\pi$
$677$	6.74331e6	0.565460	0.282730	+	0.959200i	$0.408760\pi$
$678$	0	0
$679$	−3.42674e6	−0.285237
$680$	0	0
$681$	−9.09027e6	−0.751119
$682$	0	0
$683$	−1.57439e7	−1.29140	−0.645698	−	0.763593i	$-0.723434\pi$
$683$	−1.57439e7	−1.29140	−0.645698	+	0.763593i	$0.723434\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.10466e7	0.892972
$688$	0	0
$689$	−537196.	−0.0431107
$690$	0	0
$691$	−1.89262e7	−1.50789	−0.753943	−	0.656940i	$-0.771850\pi$
$691$	−1.89262e7	−1.50789	−0.753943	+	0.656940i	$0.771850\pi$
$692$	0	0
$693$	3.39446e6	0.268496
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.05786e7	−0.824797
$698$	0	0
$699$	−5.39535e6	−0.417664
$700$	0	0
$701$	1.21080e7	0.930628	0.465314	−	0.885146i	$-0.345941\pi$
$701$	1.21080e7	0.930628	0.465314	+	0.885146i	$0.345941\pi$
$702$	0	0
$703$	−2.64573e7	−2.01910
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.00582e7	0.756782
$708$	0	0
$709$	−1.59350e6	−0.119052	−0.0595259	−	0.998227i	$-0.518959\pi$
$709$	−1.59350e6	−0.119052	−0.0595259	+	0.998227i	$0.518959\pi$
$710$	0	0
$711$	−3.52158e6	−0.261254
$712$	0	0
$713$	−9.48447e6	−0.698698
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.23446e7	0.896763
$718$	0	0
$719$	−1.65457e7	−1.19361	−0.596806	−	0.802386i	$-0.703564\pi$
$719$	−1.65457e7	−1.19361	−0.596806	+	0.802386i	$0.703564\pi$
$720$	0	0
$721$	22838.6	0.00163618
$722$	0	0
$723$	−1.23171e7	−0.876323
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.71115e7	1.20075	0.600373	−	0.799720i	$-0.295019\pi$
$727$	1.71115e7	1.20075	0.600373	+	0.799720i	$0.295019\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	6.55992e6	0.454051
$732$	0	0
$733$	−1.21563e7	−0.835683	−0.417841	−	0.908520i	$-0.637213\pi$
$733$	−1.21563e7	−0.835683	−0.417841	+	0.908520i	$0.637213\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−691317.	−0.0468823
$738$	0	0
$739$	981165.	0.0660893	0.0330446	−	0.999454i	$-0.489480\pi$
$739$	981165.	0.0660893	0.0330446	+	0.999454i	$0.489480\pi$
$740$	0	0
$741$	864702.	0.0578523
$742$	0	0
$743$	2.75144e6	0.182847	0.0914237	−	0.995812i	$-0.470858\pi$
$743$	2.75144e6	0.182847	0.0914237	+	0.995812i	$0.470858\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.02269e6	−0.460471
$748$	0	0
$749$	−1.03161e7	−0.671912
$750$	0	0
$751$	2.03594e6	0.131724	0.0658622	−	0.997829i	$-0.479020\pi$
$751$	2.03594e6	0.131724	0.0658622	+	0.997829i	$0.479020\pi$
$752$	0	0
$753$	−6.57263e6	−0.422427
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.50989e7	−1.59189	−0.795947	−	0.605366i	$-0.793027\pi$
$757$	−2.50989e7	−1.59189	−0.795947	+	0.605366i	$0.793027\pi$
$758$	0	0
$759$	1.81176e7	1.14155
$760$	0	0
$761$	715007.	0.0447557	0.0223779	−	0.999750i	$-0.492876\pi$
$761$	715007.	0.0447557	0.0223779	+	0.999750i	$0.492876\pi$
$762$	0	0
$763$	1.16397e7	0.723819
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.53201e6	−0.0940317
$768$	0	0
$769$	−2.97361e6	−0.181329	−0.0906646	−	0.995881i	$-0.528899\pi$
$769$	−2.97361e6	−0.181329	−0.0906646	+	0.995881i	$0.528899\pi$
$770$	0	0
$771$	1.23149e7	0.746095
$772$	0	0
$773$	2.89819e7	1.74453	0.872264	−	0.489035i	$-0.162651\pi$
$773$	2.89819e7	1.74453	0.872264	+	0.489035i	$0.162651\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.21310e6	−0.488039
$778$	0	0
$779$	−3.80073e7	−2.24400
$780$	0	0
$781$	5.67989e6	0.333206
$782$	0	0
$783$	1.37541e6	0.0801731
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.25612e7	−1.87397	−0.936987	−	0.349364i	$-0.886398\pi$
$787$	−3.25612e7	−1.87397	−0.936987	+	0.349364i	$0.886398\pi$
$788$	0	0
$789$	−1.13638e7	−0.649879
$790$	0	0
$791$	8.07598e6	0.458938
$792$	0	0
$793$	−1.47524e6	−0.0833067
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.77148e7	0.987848	0.493924	−	0.869505i	$-0.335562\pi$
$797$	1.77148e7	0.987848	0.493924	+	0.869505i	$0.335562\pi$
$798$	0	0
$799$	9.92373e6	0.549931
$800$	0	0
$801$	−8.16755e6	−0.449791
$802$	0	0
$803$	−1.51707e7	−0.830268
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.87606e6	−0.317616
$808$	0	0
$809$	−2.42153e6	−0.130083	−0.0650413	−	0.997883i	$-0.520718\pi$
$809$	−2.42153e6	−0.130083	−0.0650413	+	0.997883i	$0.520718\pi$
$810$	0	0
$811$	−6.47458e6	−0.345668	−0.172834	−	0.984951i	$-0.555292\pi$
$811$	−6.47458e6	−0.345668	−0.172834	+	0.984951i	$0.555292\pi$
$812$	0	0
$813$	5.84164e6	0.309962
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.35688e7	1.23533
$818$	0	0
$819$	268428.	0.0139836
$820$	0	0
$821$	−5.71963e6	−0.296149	−0.148074	−	0.988976i	$-0.547308\pi$
$821$	−5.71963e6	−0.296149	−0.148074	+	0.988976i	$0.547308\pi$
$822$	0	0
$823$	1.45747e7	0.750066	0.375033	−	0.927011i	$-0.377631\pi$
$823$	1.45747e7	0.750066	0.375033	+	0.927011i	$0.377631\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.71684e6	−0.290664	−0.145332	−	0.989383i	$-0.546425\pi$
$827$	−5.71684e6	−0.290664	−0.145332	+	0.989383i	$0.546425\pi$
$828$	0	0
$829$	−3.16906e7	−1.60156	−0.800781	−	0.598958i	$-0.795582\pi$
$829$	−3.16906e7	−1.60156	−0.800781	+	0.598958i	$0.795582\pi$
$830$	0	0
$831$	−6.01650e6	−0.302233
$832$	0	0
$833$	−6.53702e6	−0.326413
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.67641e6	0.0827115
$838$	0	0
$839$	−1.30388e7	−0.639487	−0.319744	−	0.947504i	$-0.603597\pi$
$839$	−1.30388e7	−0.639487	−0.319744	+	0.947504i	$0.603597\pi$
$840$	0	0
$841$	−1.69515e7	−0.826451
$842$	0	0
$843$	−9.29910e6	−0.450684
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.62644e6	−0.317374
$848$	0	0
$849$	3.46399e6	0.164933
$850$	0	0
$851$	−4.38367e7	−2.07498
$852$	0	0
$853$	−1.19404e6	−0.0561883	−0.0280941	−	0.999605i	$-0.508944\pi$
$853$	−1.19404e6	−0.0561883	−0.0280941	+	0.999605i	$0.508944\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.25697e7	−0.584617	−0.292309	−	0.956324i	$-0.594423\pi$
$857$	−1.25697e7	−0.584617	−0.292309	+	0.956324i	$0.594423\pi$
$858$	0	0
$859$	−8.48043e6	−0.392135	−0.196067	−	0.980590i	$-0.562817\pi$
$859$	−8.48043e6	−0.392135	−0.196067	+	0.980590i	$0.562817\pi$
$860$	0	0
$861$	−1.17985e7	−0.542401
$862$	0	0
$863$	−1.53122e6	−0.0699859	−0.0349929	−	0.999388i	$-0.511141\pi$
$863$	−1.53122e6	−0.0699859	−0.0349929	+	0.999388i	$0.511141\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.45850e6	0.382161
$868$	0	0
$869$	2.12202e7	0.953235
$870$	0	0
$871$	−54668.1	−0.00244168
$872$	0	0
$873$	3.23279e6	0.143563
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.58663e7	1.57466	0.787331	−	0.616530i	$-0.211462\pi$
$877$	3.58663e7	1.57466	0.787331	+	0.616530i	$0.211462\pi$
$878$	0	0
$879$	−672777.	−0.0293697
$880$	0	0
$881$	−2.74419e7	−1.19117	−0.595586	−	0.803291i	$-0.703080\pi$
$881$	−2.74419e7	−1.19117	−0.595586	+	0.803291i	$0.703080\pi$
$882$	0	0
$883$	1.71743e7	0.741273	0.370637	−	0.928778i	$-0.379139\pi$
$883$	1.71743e7	0.741273	0.370637	+	0.928778i	$0.379139\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.60304e6	−0.409826	−0.204913	−	0.978780i	$-0.565691\pi$
$887$	−9.60304e6	−0.409826	−0.204913	+	0.978780i	$0.565691\pi$
$888$	0	0
$889$	1.01719e7	0.431666
$890$	0	0
$891$	−3.20234e6	−0.135137
$892$	0	0
$893$	3.56544e7	1.49618
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.43271e6	0.0594534
$898$	0	0
$899$	4.33868e6	0.179043
$900$	0	0
$901$	9.64295e6	0.395729
$902$	0	0
$903$	7.31641e6	0.298592
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.28904e7	−1.73118	−0.865590	−	0.500754i	$-0.833056\pi$
$907$	−4.28904e7	−1.73118	−0.865590	+	0.500754i	$0.833056\pi$
$908$	0	0
$909$	−9.48890e6	−0.380896
$910$	0	0
$911$	2.43996e7	0.974061	0.487031	−	0.873385i	$-0.338080\pi$
$911$	2.43996e7	0.974061	0.487031	+	0.873385i	$0.338080\pi$
$912$	0	0
$913$	4.23171e7	1.68011
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.00051e7	0.785628
$918$	0	0
$919$	−4.90096e7	−1.91422	−0.957112	−	0.289719i	$-0.906438\pi$
$919$	−4.90096e7	−1.91422	−0.957112	+	0.289719i	$0.906438\pi$
$920$	0	0
$921$	1.77290e7	0.688707
$922$	0	0
$923$	449156.	0.0173537
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−21546.0	−0.000823506 0
$928$	0	0
$929$	4.34428e7	1.65150	0.825750	−	0.564036i	$-0.190752\pi$
$929$	4.34428e7	1.65150	0.825750	+	0.564036i	$0.190752\pi$
$930$	0	0
$931$	−2.34865e7	−0.888064
$932$	0	0
$933$	1.73313e7	0.651820
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.45065e6	−0.0911868	−0.0455934	−	0.998960i	$-0.514518\pi$
$937$	−2.45065e6	−0.0911868	−0.0455934	+	0.998960i	$0.514518\pi$
$938$	0	0
$939$	−9.46141e6	−0.350181
$940$	0	0
$941$	9.14331e6	0.336612	0.168306	−	0.985735i	$-0.446170\pi$
$941$	9.14331e6	0.336612	0.168306	+	0.985735i	$0.446170\pi$
$942$	0	0
$943$	−6.29737e7	−2.30611
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.87759e7	0.680341	0.340171	−	0.940364i	$-0.389515\pi$
$947$	1.87759e7	0.680341	0.340171	+	0.940364i	$0.389515\pi$
$948$	0	0
$949$	−1.19968e6	−0.0432413
$950$	0	0
$951$	2.41622e7	0.866334
$952$	0	0
$953$	5.76978e6	0.205791	0.102896	−	0.994692i	$-0.467189\pi$
$953$	5.76978e6	0.205791	0.102896	+	0.994692i	$0.467189\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.28791e6	−0.292527
$958$	0	0
$959$	1.19835e7	0.420762
$960$	0	0
$961$	−2.33410e7	−0.815288
$962$	0	0
$963$	9.73227e6	0.338180
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.51533e6	0.0865024	0.0432512	−	0.999064i	$-0.486228\pi$
$967$	2.51533e6	0.0865024	0.0432512	+	0.999064i	$0.486228\pi$
$968$	0	0
$969$	−1.55218e7	−0.531048
$970$	0	0
$971$	3.98957e7	1.35793	0.678966	−	0.734170i	$-0.262428\pi$
$971$	3.98957e7	1.35793	0.678966	+	0.734170i	$0.262428\pi$
$972$	0	0
$973$	2.26508e7	0.767011
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.40009e7	−1.80994	−0.904971	−	0.425473i	$-0.860108\pi$
$977$	−5.40009e7	−1.80994	−0.904971	+	0.425473i	$0.860108\pi$
$978$	0	0
$979$	4.92157e7	1.64115
$980$	0	0
$981$	−1.09809e7	−0.364306
$982$	0	0
$983$	−3.78231e7	−1.24846	−0.624228	−	0.781242i	$-0.714586\pi$
$983$	−3.78231e7	−1.24846	−0.624228	+	0.781242i	$0.714586\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.10681e7	0.361644
$988$	0	0
$989$	3.90507e7	1.26952
$990$	0	0
$991$	5.61819e6	0.181724	0.0908620	−	0.995863i	$-0.471038\pi$
$991$	5.61819e6	0.181724	0.0908620	+	0.995863i	$0.471038\pi$
$992$	0	0
$993$	1.35410e7	0.435792
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.35983e7	1.38910	0.694548	−	0.719447i	$-0.255605\pi$
$997$	4.35983e7	1.38910	0.694548	+	0.719447i	$0.255605\pi$
$998$	0	0
$999$	7.74826e6	0.245635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.a.r.1.1		3
5.2	odd	4		120.6.f.a.49.5	yes	6
5.3	odd	4		120.6.f.a.49.2	✓	6
5.4	even	2		600.6.a.s.1.3		3
15.2	even	4		360.6.f.a.289.4		6
15.8	even	4		360.6.f.a.289.3		6
20.3	even	4		240.6.f.e.49.5		6
20.7	even	4		240.6.f.e.49.2		6
60.23	odd	4		720.6.f.l.289.3		6
60.47	odd	4		720.6.f.l.289.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.a.49.2	✓	6	5.3	odd	4
120.6.f.a.49.5	yes	6	5.2	odd	4
240.6.f.e.49.2		6	20.7	even	4
240.6.f.e.49.5		6	20.3	even	4
360.6.f.a.289.3		6	15.8	even	4
360.6.f.a.289.4		6	15.2	even	4
600.6.a.r.1.1		3	1.1	even	1	trivial
600.6.a.s.1.3		3	5.4	even	2
720.6.f.l.289.3		6	60.23	odd	4
720.6.f.l.289.4		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -85.8595 q^{7} +81.0000 q^{9} -488.087 q^{11} -38.5971 q^{13} +692.837 q^{17} +2489.26 q^{19} +772.735 q^{21} +4124.41 q^{23} -729.000 q^{27} -1886.71 q^{29} -2299.60 q^{31} +4392.78 q^{33} -10628.6 q^{37} +347.373 q^{39} -15268.5 q^{41} +9468.20 q^{43} +14323.3 q^{47} -9435.15 q^{49} -6235.53 q^{51} +13918.1 q^{53} -22403.3 q^{57} +39692.5 q^{59} +38221.6 q^{61} -6954.62 q^{63} +1416.38 q^{67} -37119.6 q^{69} -11637.1 q^{71} +31082.1 q^{73} +41906.9 q^{77} -43476.3 q^{79} +6561.00 q^{81} -86699.9 q^{83} +16980.4 q^{87} -100834. q^{89} +3313.92 q^{91} +20696.4 q^{93} +39911.0 q^{97} -39535.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 27 q^{3} + 104 q^{7} + 243 q^{9} - 332 q^{11} + 88 q^{13} - 1116 q^{17} - 144 q^{19} - 936 q^{21} + 2868 q^{23} - 2187 q^{27} + 946 q^{29} - 5124 q^{31} + 2988 q^{33} - 7808 q^{37} - 792 q^{39} - 662 q^{41}+ \cdots - 26892 q^{99}+O(q^{100}) \) 3 * q - 27 * q^3 + 104 * q^7 + 243 * q^9 - 332 * q^11 + 88 * q^13 - 1116 * q^17 - 144 * q^19 - 936 * q^21 + 2868 * q^23 - 2187 * q^27 + 946 * q^29 - 5124 * q^31 + 2988 * q^33 - 7808 * q^37 - 792 * q^39 - 662 * q^41 + 25188 * q^43 + 25428 * q^47 - 14205 * q^49 + 10044 * q^51 + 36924 * q^53 + 1296 * q^57 + 5148 * q^59 - 26058 * q^61 + 8424 * q^63 + 30644 * q^67 - 25812 * q^69 - 14144 * q^71 + 37736 * q^73 + 46800 * q^77 - 110100 * q^79 + 19683 * q^81 + 7620 * q^83 - 8514 * q^87 - 175710 * q^89 - 8544 * q^91 + 46116 * q^93 - 6632 * q^97 - 26892 * q^99

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