LMFDB - Embedded newform 700.6.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [700,6,Mod(1,700)] 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("700.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-22] 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:	$112.268673869$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 393x^{2} - 1002x + 16596 $ x^4 - 2x^3 - 393x^2 - 1002*x + 16596
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.45426$ of defining polynomial
Character	$\chi$	$=$	700.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.545745 q^{3} -49.0000 q^{7} -242.702 q^{9} +151.850 q^{11} -174.069 q^{13} -30.2562 q^{17} +1357.30 q^{19} +26.7415 q^{21} +4538.39 q^{23} +265.069 q^{27} -3112.43 q^{29} +8409.35 q^{31} -82.8715 q^{33} -9394.61 q^{37} +94.9973 q^{39} -9093.67 q^{41} -16911.3 q^{43} +27542.3 q^{47} +2401.00 q^{49} +16.5121 q^{51} +11364.1 q^{53} -740.740 q^{57} -18160.6 q^{59} -30788.8 q^{61} +11892.4 q^{63} -42283.7 q^{67} -2476.80 q^{69} -25775.3 q^{71} +14938.3 q^{73} -7440.67 q^{77} +53908.3 q^{79} +58832.0 q^{81} -18023.9 q^{83} +1698.59 q^{87} -132078. q^{89} +8529.38 q^{91} -4589.36 q^{93} -31392.1 q^{97} -36854.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{3} - 196 q^{7} - 62 q^{9} + 48 q^{11} + 430 q^{13} + 1286 q^{17} - 384 q^{19} + 1078 q^{21} - 1256 q^{23} + 1196 q^{27} - 4994 q^{29} + 4618 q^{31} + 15268 q^{33} + 16922 q^{37} + 3200 q^{39}+ \cdots - 170416 q^{99}+O(q^{100}) $ 4 * q - 22 * q^3 - 196 * q^7 - 62 * q^9 + 48 * q^11 + 430 * q^13 + 1286 * q^17 - 384 * q^19 + 1078 * q^21 - 1256 * q^23 + 1196 * q^27 - 4994 * q^29 + 4618 * q^31 + 15268 * q^33 + 16922 * q^37 + 3200 * q^39 - 9192 * q^41 + 12368 * q^43 - 4476 * q^47 + 9604 * q^49 - 30396 * q^51 + 16736 * q^53 - 29946 * q^57 - 14614 * q^59 - 7122 * q^61 + 3038 * q^63 - 21408 * q^67 + 11822 * q^69 - 39748 * q^71 + 89364 * q^73 - 2352 * q^77 - 37640 * q^79 - 125192 * q^81 + 83842 * q^83 - 90982 * q^87 - 12444 * q^89 - 21070 * q^91 + 137840 * q^93 - 57034 * q^97 - 170416 * 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
$3$	−0.545745	−0.0350096	−0.0175048	−	0.999847i	$-0.505572\pi$
$3$	−0.545745	−0.0350096	−0.0175048	+	0.999847i	$0.505572\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−242.702	−0.998774
$10$	0	0
$11$	151.850	0.378385	0.189193	−	0.981940i	$-0.439413\pi$
$11$	151.850	0.378385	0.189193	+	0.981940i	$0.439413\pi$
$12$	0	0
$13$	−174.069	−0.285669	−0.142834	−	0.989747i	$-0.545622\pi$
$13$	−174.069	−0.285669	−0.142834	+	0.989747i	$0.545622\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−30.2562	−0.0253917	−0.0126958	−	0.999919i	$-0.504041\pi$
$17$	−30.2562	−0.0253917	−0.0126958	+	0.999919i	$0.504041\pi$
$18$	0	0
$19$	1357.30	0.862565	0.431283	−	0.902217i	$-0.358061\pi$
$19$	1357.30	0.862565	0.431283	+	0.902217i	$0.358061\pi$
$20$	0	0
$21$	26.7415	0.0132324
$22$	0	0
$23$	4538.39	1.78888	0.894442	−	0.447184i	$-0.147573\pi$
$23$	4538.39	1.78888	0.894442	+	0.447184i	$0.147573\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	265.069	0.0699762
$28$	0	0
$29$	−3112.43	−0.687233	−0.343617	−	0.939110i	$-0.611652\pi$
$29$	−3112.43	−0.687233	−0.343617	+	0.939110i	$0.611652\pi$
$30$	0	0
$31$	8409.35	1.57166	0.785829	−	0.618444i	$-0.212237\pi$
$31$	8409.35	1.57166	0.785829	+	0.618444i	$0.212237\pi$
$32$	0	0
$33$	−82.8715	−0.0132471
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9394.61	−1.12817	−0.564085	−	0.825717i	$-0.690771\pi$
$37$	−9394.61	−1.12817	−0.564085	+	0.825717i	$0.690771\pi$
$38$	0	0
$39$	94.9973	0.0100011
$40$	0	0
$41$	−9093.67	−0.844849	−0.422425	−	0.906398i	$-0.638821\pi$
$41$	−9093.67	−0.844849	−0.422425	+	0.906398i	$0.638821\pi$
$42$	0	0
$43$	−16911.3	−1.39478	−0.697389	−	0.716693i	$-0.745655\pi$
$43$	−16911.3	−1.39478	−0.697389	+	0.716693i	$0.745655\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	27542.3	1.81868	0.909338	−	0.416058i	$-0.136589\pi$
$47$	27542.3	1.81868	0.909338	+	0.416058i	$0.136589\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	16.5121	0.000888951 0
$52$	0	0
$53$	11364.1	0.555704	0.277852	−	0.960624i	$-0.410377\pi$
$53$	11364.1	0.555704	0.277852	+	0.960624i	$0.410377\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−740.740	−0.0301980
$58$	0	0
$59$	−18160.6	−0.679202	−0.339601	−	0.940570i	$-0.610292\pi$
$59$	−18160.6	−0.679202	−0.339601	+	0.940570i	$0.610292\pi$
$60$	0	0
$61$	−30788.8	−1.05942	−0.529711	−	0.848178i	$-0.677699\pi$
$61$	−30788.8	−1.05942	−0.529711	+	0.848178i	$0.677699\pi$
$62$	0	0
$63$	11892.4	0.377501
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−42283.7	−1.15076	−0.575381	−	0.817885i	$-0.695146\pi$
$67$	−42283.7	−1.15076	−0.575381	+	0.817885i	$0.695146\pi$
$68$	0	0
$69$	−2476.80	−0.0626280
$70$	0	0
$71$	−25775.3	−0.606817	−0.303408	−	0.952861i	$-0.598125\pi$
$71$	−25775.3	−0.606817	−0.303408	+	0.952861i	$0.598125\pi$
$72$	0	0
$73$	14938.3	0.328091	0.164046	−	0.986453i	$-0.447546\pi$
$73$	14938.3	0.328091	0.164046	+	0.986453i	$0.447546\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7440.67	−0.143016
$78$	0	0
$79$	53908.3	0.971826	0.485913	−	0.874007i	$-0.338487\pi$
$79$	53908.3	0.971826	0.485913	+	0.874007i	$0.338487\pi$
$80$	0	0
$81$	58832.0	0.996324
$82$	0	0
$83$	−18023.9	−0.287180	−0.143590	−	0.989637i	$-0.545865\pi$
$83$	−18023.9	−0.287180	−0.143590	+	0.989637i	$0.545865\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1698.59	0.0240597
$88$	0	0
$89$	−132078.	−1.76749	−0.883743	−	0.467973i	$-0.844984\pi$
$89$	−132078.	−1.76749	−0.883743	+	0.467973i	$0.844984\pi$
$90$	0	0
$91$	8529.38	0.107973
$92$	0	0
$93$	−4589.36	−0.0550230
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−31392.1	−0.338759	−0.169380	−	0.985551i	$-0.554176\pi$
$97$	−31392.1	−0.338759	−0.169380	+	0.985551i	$0.554176\pi$
$98$	0	0
$99$	−36854.4	−0.377921
$100$	0	0
$101$	−50890.2	−0.496399	−0.248199	−	0.968709i	$-0.579839\pi$
$101$	−50890.2	−0.496399	−0.248199	+	0.968709i	$0.579839\pi$
$102$	0	0
$103$	56577.6	0.525474	0.262737	−	0.964868i	$-0.415375\pi$
$103$	56577.6	0.525474	0.262737	+	0.964868i	$0.415375\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−159424.	−1.34615	−0.673077	−	0.739572i	$-0.735028\pi$
$107$	−159424.	−1.34615	−0.673077	+	0.739572i	$0.735028\pi$
$108$	0	0
$109$	−1000.73	−0.00806771	−0.00403385	−	0.999992i	$-0.501284\pi$
$109$	−1000.73	−0.00806771	−0.00403385	+	0.999992i	$0.501284\pi$
$110$	0	0
$111$	5127.06	0.0394967
$112$	0	0
$113$	220151.	1.62190	0.810952	−	0.585113i	$-0.198950\pi$
$113$	220151.	1.62190	0.810952	+	0.585113i	$0.198950\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	42246.9	0.285319
$118$	0	0
$119$	1482.55	0.00959715
$120$	0	0
$121$	−137992.	−0.856825
$122$	0	0
$123$	4962.82	0.0295778
$124$	0	0
$125$	0	0
$126$	0	0
$127$	107554.	0.591723	0.295862	−	0.955231i	$-0.404393\pi$
$127$	107554.	0.591723	0.295862	+	0.955231i	$0.404393\pi$
$128$	0	0
$129$	9229.24	0.0488306
$130$	0	0
$131$	300017.	1.52745	0.763727	−	0.645539i	$-0.223367\pi$
$131$	300017.	1.52745	0.763727	+	0.645539i	$0.223367\pi$
$132$	0	0
$133$	−66507.7	−0.326019
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−83510.4	−0.380136	−0.190068	−	0.981771i	$-0.560871\pi$
$137$	−83510.4	−0.380136	−0.190068	+	0.981771i	$0.560871\pi$
$138$	0	0
$139$	−415785.	−1.82529	−0.912645	−	0.408754i	$-0.865963\pi$
$139$	−415785.	−1.82529	−0.912645	+	0.408754i	$0.865963\pi$
$140$	0	0
$141$	−15031.1	−0.0636710
$142$	0	0
$143$	−26432.4	−0.108093
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1310.33	−0.00500136
$148$	0	0
$149$	−298108.	−1.10004	−0.550020	−	0.835151i	$-0.685380\pi$
$149$	−298108.	−1.10004	−0.550020	+	0.835151i	$0.685380\pi$
$150$	0	0
$151$	−453998.	−1.62036	−0.810180	−	0.586181i	$-0.800631\pi$
$151$	−453998.	−1.62036	−0.810180	+	0.586181i	$0.800631\pi$
$152$	0	0
$153$	7343.23	0.0253606
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−390947.	−1.26581	−0.632905	−	0.774229i	$-0.718138\pi$
$157$	−390947.	−1.26581	−0.632905	+	0.774229i	$0.718138\pi$
$158$	0	0
$159$	−6201.88	−0.0194550
$160$	0	0
$161$	−222381.	−0.676135
$162$	0	0
$163$	161033.	0.474728	0.237364	−	0.971421i	$-0.423717\pi$
$163$	161033.	0.474728	0.237364	+	0.971421i	$0.423717\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−47355.4	−0.131395	−0.0656974	−	0.997840i	$-0.520927\pi$
$167$	−47355.4	−0.131395	−0.0656974	+	0.997840i	$0.520927\pi$
$168$	0	0
$169$	−340993.	−0.918393
$170$	0	0
$171$	−329420.	−0.861508
$172$	0	0
$173$	643520.	1.63473	0.817367	−	0.576118i	$-0.195433\pi$
$173$	643520.	1.63473	0.817367	+	0.576118i	$0.195433\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9911.03	0.0237786
$178$	0	0
$179$	−347904.	−0.811570	−0.405785	−	0.913968i	$-0.633002\pi$
$179$	−347904.	−0.811570	−0.405785	+	0.913968i	$0.633002\pi$
$180$	0	0
$181$	214485.	0.486632	0.243316	−	0.969947i	$-0.421765\pi$
$181$	214485.	0.486632	0.243316	+	0.969947i	$0.421765\pi$
$182$	0	0
$183$	16802.8	0.0370899
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4594.41	−0.00960783
$188$	0	0
$189$	−12988.4	−0.0264485
$190$	0	0
$191$	−401879.	−0.797098	−0.398549	−	0.917147i	$-0.630486\pi$
$191$	−401879.	−0.797098	−0.398549	+	0.917147i	$0.630486\pi$
$192$	0	0
$193$	−38300.7	−0.0740139	−0.0370070	−	0.999315i	$-0.511782\pi$
$193$	−38300.7	−0.0740139	−0.0370070	+	0.999315i	$0.511782\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−345600.	−0.634466	−0.317233	−	0.948348i	$-0.602754\pi$
$197$	−345600.	−0.634466	−0.317233	+	0.948348i	$0.602754\pi$
$198$	0	0
$199$	−733756.	−1.31347	−0.656733	−	0.754123i	$-0.728062\pi$
$199$	−733756.	−1.31347	−0.656733	+	0.754123i	$0.728062\pi$
$200$	0	0
$201$	23076.1	0.0402877
$202$	0	0
$203$	152509.	0.259750
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.10148e6	−1.78669
$208$	0	0
$209$	206106.	0.326382
$210$	0	0
$211$	537383.	0.830955	0.415477	−	0.909603i	$-0.363615\pi$
$211$	537383.	0.830955	0.415477	+	0.909603i	$0.363615\pi$
$212$	0	0
$213$	14066.7	0.0212444
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−412058.	−0.594031
$218$	0	0
$219$	−8152.51	−0.0114863
$220$	0	0
$221$	5266.66	0.00725361
$222$	0	0
$223$	913530.	1.23016	0.615078	−	0.788466i	$-0.289124\pi$
$223$	913530.	1.23016	0.615078	+	0.788466i	$0.289124\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−384996.	−0.495898	−0.247949	−	0.968773i	$-0.579756\pi$
$227$	−384996.	−0.495898	−0.247949	+	0.968773i	$0.579756\pi$
$228$	0	0
$229$	−1.30305e6	−1.64199	−0.820996	−	0.570934i	$-0.806581\pi$
$229$	−1.30305e6	−1.64199	−0.820996	+	0.570934i	$0.806581\pi$
$230$	0	0
$231$	4060.71	0.00500693
$232$	0	0
$233$	329721.	0.397884	0.198942	−	0.980011i	$-0.436249\pi$
$233$	329721.	0.397884	0.198942	+	0.980011i	$0.436249\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−29420.2	−0.0340232
$238$	0	0
$239$	−793186.	−0.898215	−0.449107	−	0.893478i	$-0.648258\pi$
$239$	−793186.	−0.898215	−0.449107	+	0.893478i	$0.648258\pi$
$240$	0	0
$241$	26163.1	0.0290166	0.0145083	−	0.999895i	$-0.495382\pi$
$241$	26163.1	0.0290166	0.0145083	+	0.999895i	$0.495382\pi$
$242$	0	0
$243$	−96519.1	−0.104857
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−236264.	−0.246408
$248$	0	0
$249$	9836.47	0.0100540
$250$	0	0
$251$	−909018.	−0.910727	−0.455363	−	0.890306i	$-0.650491\pi$
$251$	−909018.	−0.910727	−0.455363	+	0.890306i	$0.650491\pi$
$252$	0	0
$253$	689156.	0.676887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	268481.	0.253560	0.126780	−	0.991931i	$-0.459536\pi$
$257$	268481.	0.253560	0.126780	+	0.991931i	$0.459536\pi$
$258$	0	0
$259$	460336.	0.426408
$260$	0	0
$261$	755393.	0.686391
$262$	0	0
$263$	405970.	0.361914	0.180957	−	0.983491i	$-0.442081\pi$
$263$	405970.	0.361914	0.180957	+	0.983491i	$0.442081\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	72081.0	0.0618789
$268$	0	0
$269$	270513.	0.227933	0.113966	−	0.993485i	$-0.463644\pi$
$269$	270513.	0.227933	0.113966	+	0.993485i	$0.463644\pi$
$270$	0	0
$271$	−1.76528e6	−1.46013	−0.730063	−	0.683380i	$-0.760509\pi$
$271$	−1.76528e6	−1.46013	−0.730063	+	0.683380i	$0.760509\pi$
$272$	0	0
$273$	−4654.87	−0.00378008
$274$	0	0
$275$	0	0
$276$	0	0
$277$	509279.	0.398801	0.199400	−	0.979918i	$-0.436101\pi$
$277$	509279.	0.398801	0.199400	+	0.979918i	$0.436101\pi$
$278$	0	0
$279$	−2.04097e6	−1.56973
$280$	0	0
$281$	−1.93287e6	−1.46029	−0.730143	−	0.683295i	$-0.760546\pi$
$281$	−1.93287e6	−1.46029	−0.730143	+	0.683295i	$0.760546\pi$
$282$	0	0
$283$	840394.	0.623759	0.311880	−	0.950122i	$-0.399041\pi$
$283$	840394.	0.623759	0.311880	+	0.950122i	$0.399041\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	445590.	0.319323
$288$	0	0
$289$	−1.41894e6	−0.999355
$290$	0	0
$291$	17132.1	0.0118598
$292$	0	0
$293$	−388615.	−0.264454	−0.132227	−	0.991219i	$-0.542213\pi$
$293$	−388615.	−0.264454	−0.132227	+	0.991219i	$0.542213\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	40250.9	0.0264779
$298$	0	0
$299$	−789993.	−0.511029
$300$	0	0
$301$	828652.	0.527177
$302$	0	0
$303$	27773.1	0.0173787
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−921196.	−0.557836	−0.278918	−	0.960315i	$-0.589976\pi$
$307$	−921196.	−0.557836	−0.278918	+	0.960315i	$0.589976\pi$
$308$	0	0
$309$	−30876.9	−0.0183966
$310$	0	0
$311$	1.18654e6	0.695638	0.347819	−	0.937562i	$-0.386922\pi$
$311$	1.18654e6	0.695638	0.347819	+	0.937562i	$0.386922\pi$
$312$	0	0
$313$	−1.49842e6	−0.864516	−0.432258	−	0.901750i	$-0.642283\pi$
$313$	−1.49842e6	−0.864516	−0.432258	+	0.901750i	$0.642283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.78562e6	−1.55695	−0.778474	−	0.627676i	$-0.784006\pi$
$317$	−2.78562e6	−1.55695	−0.778474	+	0.627676i	$0.784006\pi$
$318$	0	0
$319$	−472623.	−0.260039
$320$	0	0
$321$	87005.0	0.0471283
$322$	0	0
$323$	−41066.7	−0.0219020
$324$	0	0
$325$	0	0
$326$	0	0
$327$	546.143	0.000282447 0
$328$	0	0
$329$	−1.34957e6	−0.687395
$330$	0	0
$331$	−576838.	−0.289390	−0.144695	−	0.989476i	$-0.546220\pi$
$331$	−576838.	−0.289390	−0.144695	+	0.989476i	$0.546220\pi$
$332$	0	0
$333$	2.28009e6	1.12679
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−261511.	−0.125434	−0.0627171	−	0.998031i	$-0.519977\pi$
$337$	−261511.	−0.125434	−0.0627171	+	0.998031i	$0.519977\pi$
$338$	0	0
$339$	−120146.	−0.0567821
$340$	0	0
$341$	1.27696e6	0.594692
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.10913e6	0.940330	0.470165	−	0.882578i	$-0.344194\pi$
$347$	2.10913e6	0.940330	0.470165	+	0.882578i	$0.344194\pi$
$348$	0	0
$349$	3.12887e6	1.37507	0.687534	−	0.726152i	$-0.258693\pi$
$349$	3.12887e6	1.37507	0.687534	+	0.726152i	$0.258693\pi$
$350$	0	0
$351$	−46140.4	−0.0199900
$352$	0	0
$353$	1.18006e6	0.504043	0.252021	−	0.967722i	$-0.418905\pi$
$353$	1.18006e6	0.504043	0.252021	+	0.967722i	$0.418905\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−809.095	−0.000335992 0
$358$	0	0
$359$	1.17557e6	0.481405	0.240703	−	0.970599i	$-0.422622\pi$
$359$	1.17557e6	0.481405	0.240703	+	0.970599i	$0.422622\pi$
$360$	0	0
$361$	−633835.	−0.255981
$362$	0	0
$363$	75308.7	0.0299970
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−932760.	−0.361497	−0.180748	−	0.983529i	$-0.557852\pi$
$367$	−932760.	−0.361497	−0.180748	+	0.983529i	$0.557852\pi$
$368$	0	0
$369$	2.20705e6	0.843814
$370$	0	0
$371$	−556839.	−0.210037
$372$	0	0
$373$	−1.66637e6	−0.620152	−0.310076	−	0.950712i	$-0.600355\pi$
$373$	−1.66637e6	−0.620152	−0.310076	+	0.950712i	$0.600355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	541777.	0.196321
$378$	0	0
$379$	2.39950e6	0.858071	0.429035	−	0.903288i	$-0.358854\pi$
$379$	2.39950e6	0.858071	0.429035	+	0.903288i	$0.358854\pi$
$380$	0	0
$381$	−58697.2	−0.0207160
$382$	0	0
$383$	314459.	0.109539	0.0547693	−	0.998499i	$-0.482558\pi$
$383$	314459.	0.109539	0.0547693	+	0.998499i	$0.482558\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.10440e6	1.39307
$388$	0	0
$389$	−786882.	−0.263655	−0.131827	−	0.991273i	$-0.542084\pi$
$389$	−786882.	−0.263655	−0.131827	+	0.991273i	$0.542084\pi$
$390$	0	0
$391$	−137314.	−0.0454228
$392$	0	0
$393$	−163733.	−0.0534755
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.13754e6	−1.63598	−0.817992	−	0.575229i	$-0.804913\pi$
$397$	−5.13754e6	−1.63598	−0.817992	+	0.575229i	$0.804913\pi$
$398$	0	0
$399$	36296.2	0.0114138
$400$	0	0
$401$	529007.	0.164286	0.0821429	−	0.996621i	$-0.473824\pi$
$401$	529007.	0.164286	0.0821429	+	0.996621i	$0.473824\pi$
$402$	0	0
$403$	−1.46381e6	−0.448974
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.42657e6	−0.426882
$408$	0	0
$409$	−4.02850e6	−1.19079	−0.595395	−	0.803433i	$-0.703004\pi$
$409$	−4.02850e6	−1.19079	−0.595395	+	0.803433i	$0.703004\pi$
$410$	0	0
$411$	45575.4	0.0133084
$412$	0	0
$413$	889867.	0.256714
$414$	0	0
$415$	0	0
$416$	0	0
$417$	226912.	0.0639026
$418$	0	0
$419$	2.13674e6	0.594589	0.297294	−	0.954786i	$-0.403916\pi$
$419$	2.13674e6	0.594589	0.297294	+	0.954786i	$0.403916\pi$
$420$	0	0
$421$	5.32374e6	1.46390	0.731950	−	0.681358i	$-0.238610\pi$
$421$	5.32374e6	1.46390	0.731950	+	0.681358i	$0.238610\pi$
$422$	0	0
$423$	−6.68457e6	−1.81645
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.50865e6	0.400424
$428$	0	0
$429$	14425.4	0.00378428
$430$	0	0
$431$	1.92677e6	0.499615	0.249808	−	0.968296i	$-0.419633\pi$
$431$	1.92677e6	0.499615	0.249808	+	0.968296i	$0.419633\pi$
$432$	0	0
$433$	2.79031e6	0.715209	0.357604	−	0.933873i	$-0.383594\pi$
$433$	2.79031e6	0.715209	0.357604	+	0.933873i	$0.383594\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.15996e6	1.54303
$438$	0	0
$439$	5.51339e6	1.36539	0.682695	−	0.730703i	$-0.260808\pi$
$439$	5.51339e6	1.36539	0.682695	+	0.730703i	$0.260808\pi$
$440$	0	0
$441$	−582728.	−0.142682
$442$	0	0
$443$	−1.28906e6	−0.312078	−0.156039	−	0.987751i	$-0.549873\pi$
$443$	−1.28906e6	−0.312078	−0.156039	+	0.987751i	$0.549873\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	162691.	0.0385119
$448$	0	0
$449$	2.66556e6	0.623983	0.311992	−	0.950085i	$-0.399004\pi$
$449$	2.66556e6	0.623983	0.311992	+	0.950085i	$0.399004\pi$
$450$	0	0
$451$	−1.38088e6	−0.319678
$452$	0	0
$453$	247767.	0.0567281
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.93520e6	1.10539	0.552694	−	0.833384i	$-0.313600\pi$
$457$	4.93520e6	1.10539	0.552694	+	0.833384i	$0.313600\pi$
$458$	0	0
$459$	−8019.98	−0.00177681
$460$	0	0
$461$	−2.29552e6	−0.503071	−0.251535	−	0.967848i	$-0.580936\pi$
$461$	−2.29552e6	−0.503071	−0.251535	+	0.967848i	$0.580936\pi$
$462$	0	0
$463$	−2.84800e6	−0.617430	−0.308715	−	0.951155i	$-0.599899\pi$
$463$	−2.84800e6	−0.617430	−0.308715	+	0.951155i	$0.599899\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	596639.	0.126596	0.0632979	−	0.997995i	$-0.479838\pi$
$467$	596639.	0.126596	0.0632979	+	0.997995i	$0.479838\pi$
$468$	0	0
$469$	2.07190e6	0.434947
$470$	0	0
$471$	213357.	0.0443154
$472$	0	0
$473$	−2.56798e6	−0.527763
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.75808e6	−0.555023
$478$	0	0
$479$	4.37534e6	0.871310	0.435655	−	0.900114i	$-0.356517\pi$
$479$	4.37534e6	0.871310	0.435655	+	0.900114i	$0.356517\pi$
$480$	0	0
$481$	1.63531e6	0.322283
$482$	0	0
$483$	121363.	0.0236712
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.10280e6	1.16602	0.583011	−	0.812464i	$-0.301875\pi$
$487$	6.10280e6	1.16602	0.583011	+	0.812464i	$0.301875\pi$
$488$	0	0
$489$	−87882.7	−0.0166200
$490$	0	0
$491$	−6.65964e6	−1.24666	−0.623329	−	0.781960i	$-0.714220\pi$
$491$	−6.65964e6	−1.24666	−0.623329	+	0.781960i	$0.714220\pi$
$492$	0	0
$493$	94170.1	0.0174500
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.26299e6	0.229355
$498$	0	0
$499$	−4.73748e6	−0.851719	−0.425859	−	0.904789i	$-0.640028\pi$
$499$	−4.73748e6	−0.851719	−0.425859	+	0.904789i	$0.640028\pi$
$500$	0	0
$501$	25843.9	0.00460007
$502$	0	0
$503$	−5.98106e6	−1.05404	−0.527021	−	0.849852i	$-0.676691\pi$
$503$	−5.98106e6	−1.05404	−0.527021	+	0.849852i	$0.676691\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	186095.	0.0321525
$508$	0	0
$509$	369411.	0.0631998	0.0315999	−	0.999501i	$-0.489940\pi$
$509$	369411.	0.0631998	0.0315999	+	0.999501i	$0.489940\pi$
$510$	0	0
$511$	−731977.	−0.124007
$512$	0	0
$513$	359779.	0.0603590
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.18230e6	0.688160
$518$	0	0
$519$	−351198.	−0.0572313
$520$	0	0
$521$	1.04526e7	1.68706	0.843531	−	0.537080i	$-0.180473\pi$
$521$	1.04526e7	1.68706	0.843531	+	0.537080i	$0.180473\pi$
$522$	0	0
$523$	−2.74533e6	−0.438875	−0.219437	−	0.975627i	$-0.570422\pi$
$523$	−2.74533e6	−0.438875	−0.219437	+	0.975627i	$0.570422\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−254434.	−0.0399070
$528$	0	0
$529$	1.41606e7	2.20011
$530$	0	0
$531$	4.40761e6	0.678370
$532$	0	0
$533$	1.58292e6	0.241347
$534$	0	0
$535$	0	0
$536$	0	0
$537$	189867.	0.0284127
$538$	0	0
$539$	364593.	0.0540550
$540$	0	0
$541$	1.15164e7	1.69171	0.845853	−	0.533416i	$-0.179092\pi$
$541$	1.15164e7	1.69171	0.845853	+	0.533416i	$0.179092\pi$
$542$	0	0
$543$	−117054.	−0.0170368
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.12929e7	1.61376	0.806878	−	0.590718i	$-0.201155\pi$
$547$	1.12929e7	1.61376	0.806878	+	0.590718i	$0.201155\pi$
$548$	0	0
$549$	7.47252e6	1.05812
$550$	0	0
$551$	−4.22450e6	−0.592784
$552$	0	0
$553$	−2.64151e6	−0.367316
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.58505e6	−0.762763	−0.381381	−	0.924418i	$-0.624551\pi$
$557$	−5.58505e6	−0.762763	−0.381381	+	0.924418i	$0.624551\pi$
$558$	0	0
$559$	2.94373e6	0.398445
$560$	0	0
$561$	2507.37	0.000336366 0
$562$	0	0
$563$	−8.31797e6	−1.10598	−0.552989	−	0.833189i	$-0.686513\pi$
$563$	−8.31797e6	−1.10598	−0.552989	+	0.833189i	$0.686513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.88277e6	−0.376575
$568$	0	0
$569$	1.12407e7	1.45550	0.727748	−	0.685844i	$-0.240567\pi$
$569$	1.12407e7	1.45550	0.727748	+	0.685844i	$0.240567\pi$
$570$	0	0
$571$	2.87087e6	0.368487	0.184244	−	0.982881i	$-0.441016\pi$
$571$	2.87087e6	0.368487	0.184244	+	0.982881i	$0.441016\pi$
$572$	0	0
$573$	219323.	0.0279061
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.24646e7	−1.55861	−0.779307	−	0.626642i	$-0.784429\pi$
$577$	−1.24646e7	−1.55861	−0.779307	+	0.626642i	$0.784429\pi$
$578$	0	0
$579$	20902.4	0.00259119
$580$	0	0
$581$	883173.	0.108544
$582$	0	0
$583$	1.72564e6	0.210270
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.89583e6	0.466665	0.233333	−	0.972397i	$-0.425037\pi$
$587$	3.89583e6	0.466665	0.233333	+	0.972397i	$0.425037\pi$
$588$	0	0
$589$	1.14140e7	1.35566
$590$	0	0
$591$	188609.	0.0222124
$592$	0	0
$593$	−1.26763e7	−1.48032	−0.740162	−	0.672429i	$-0.765251\pi$
$593$	−1.26763e7	−1.48032	−0.740162	+	0.672429i	$0.765251\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	400444.	0.0459839
$598$	0	0
$599$	−3.33879e6	−0.380208	−0.190104	−	0.981764i	$-0.560883\pi$
$599$	−3.33879e6	−0.380208	−0.190104	+	0.981764i	$0.560883\pi$
$600$	0	0
$601$	1.72769e6	0.195110	0.0975548	−	0.995230i	$-0.468898\pi$
$601$	1.72769e6	0.195110	0.0975548	+	0.995230i	$0.468898\pi$
$602$	0	0
$603$	1.02623e7	1.14935
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.39823e6	1.03532	0.517660	−	0.855587i	$-0.326803\pi$
$607$	9.39823e6	1.03532	0.517660	+	0.855587i	$0.326803\pi$
$608$	0	0
$609$	−83231.0	−0.00909372
$610$	0	0
$611$	−4.79426e6	−0.519539
$612$	0	0
$613$	1.54893e7	1.66487	0.832437	−	0.554119i	$-0.186945\pi$
$613$	1.54893e7	1.66487	0.832437	+	0.554119i	$0.186945\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.28160e6	0.981545	0.490772	−	0.871288i	$-0.336715\pi$
$617$	9.28160e6	0.981545	0.490772	+	0.871288i	$0.336715\pi$
$618$	0	0
$619$	−1.07700e7	−1.12977	−0.564884	−	0.825171i	$-0.691079\pi$
$619$	−1.07700e7	−1.12977	−0.564884	+	0.825171i	$0.691079\pi$
$620$	0	0
$621$	1.20299e6	0.125179
$622$	0	0
$623$	6.47183e6	0.668047
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−112482.	−0.0114265
$628$	0	0
$629$	284245.	0.0286461
$630$	0	0
$631$	−6.65740e6	−0.665628	−0.332814	−	0.942993i	$-0.607998\pi$
$631$	−6.65740e6	−0.665628	−0.332814	+	0.942993i	$0.607998\pi$
$632$	0	0
$633$	−293274.	−0.0290914
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−417940.	−0.0408098
$638$	0	0
$639$	6.25571e6	0.606073
$640$	0	0
$641$	−680.688	−6.54340e−5 0	−3.27170e−5	−	1.00000i	$-0.500010\pi$
$641$	−680.688	−6.54340e−5 0	−3.27170e−5		1.00000i	$0.500010\pi$
$642$	0	0
$643$	−1.68280e7	−1.60511	−0.802556	−	0.596576i	$-0.796527\pi$
$643$	−1.68280e7	−1.60511	−0.802556	+	0.596576i	$0.796527\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.14869e6	0.295712	0.147856	−	0.989009i	$-0.452763\pi$
$647$	3.14869e6	0.295712	0.147856	+	0.989009i	$0.452763\pi$
$648$	0	0
$649$	−2.75769e6	−0.257000
$650$	0	0
$651$	224879.	0.0207968
$652$	0	0
$653$	−1.78640e7	−1.63944	−0.819722	−	0.572761i	$-0.805872\pi$
$653$	−1.78640e7	−1.63944	−0.819722	+	0.572761i	$0.805872\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.62556e6	−0.327689
$658$	0	0
$659$	4.25078e6	0.381290	0.190645	−	0.981659i	$-0.438942\pi$
$659$	4.25078e6	0.381290	0.190645	+	0.981659i	$0.438942\pi$
$660$	0	0
$661$	−1.47601e6	−0.131397	−0.0656984	−	0.997840i	$-0.520928\pi$
$661$	−1.47601e6	−0.131397	−0.0656984	+	0.997840i	$0.520928\pi$
$662$	0	0
$663$	−2874.25	−0.000253946 0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.41254e7	−1.22938
$668$	0	0
$669$	−498554.	−0.0430672
$670$	0	0
$671$	−4.67529e6	−0.400869
$672$	0	0
$673$	−5.98979e6	−0.509769	−0.254885	−	0.966971i	$-0.582037\pi$
$673$	−5.98979e6	−0.509769	−0.254885	+	0.966971i	$0.582037\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.42251e7	1.19284	0.596421	−	0.802672i	$-0.296589\pi$
$677$	1.42251e7	1.19284	0.596421	+	0.802672i	$0.296589\pi$
$678$	0	0
$679$	1.53821e6	0.128039
$680$	0	0
$681$	210110.	0.0173612
$682$	0	0
$683$	−1.38904e7	−1.13937	−0.569683	−	0.821865i	$-0.692934\pi$
$683$	−1.38904e7	−1.13937	−0.569683	+	0.821865i	$0.692934\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	711131.	0.0574854
$688$	0	0
$689$	−1.97813e6	−0.158747
$690$	0	0
$691$	4.58886e6	0.365603	0.182801	−	0.983150i	$-0.441483\pi$
$691$	4.58886e6	0.365603	0.182801	+	0.983150i	$0.441483\pi$
$692$	0	0
$693$	1.80587e6	0.142841
$694$	0	0
$695$	0	0
$696$	0	0
$697$	275139.	0.0214521
$698$	0	0
$699$	−179943.	−0.0139297
$700$	0	0
$701$	1.22396e7	0.940749	0.470375	−	0.882467i	$-0.344119\pi$
$701$	1.22396e7	0.940749	0.470375	+	0.882467i	$0.344119\pi$
$702$	0	0
$703$	−1.27513e7	−0.973120
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.49362e6	0.187621
$708$	0	0
$709$	−9.27736e6	−0.693121	−0.346560	−	0.938028i	$-0.612650\pi$
$709$	−9.27736e6	−0.693121	−0.346560	+	0.938028i	$0.612650\pi$
$710$	0	0
$711$	−1.30837e7	−0.970634
$712$	0	0
$713$	3.81649e7	2.81151
$714$	0	0
$715$	0	0
$716$	0	0
$717$	432877.	0.0314461
$718$	0	0
$719$	−1.34945e7	−0.973494	−0.486747	−	0.873543i	$-0.661817\pi$
$719$	−1.34945e7	−0.973494	−0.486747	+	0.873543i	$0.661817\pi$
$720$	0	0
$721$	−2.77230e6	−0.198610
$722$	0	0
$723$	−14278.4	−0.00101586
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.24916e7	0.876563	0.438281	−	0.898838i	$-0.355587\pi$
$727$	1.24916e7	0.876563	0.438281	+	0.898838i	$0.355587\pi$
$728$	0	0
$729$	−1.42435e7	−0.992653
$730$	0	0
$731$	511670.	0.0354158
$732$	0	0
$733$	2.28106e7	1.56811	0.784056	−	0.620690i	$-0.213147\pi$
$733$	2.28106e7	1.56811	0.784056	+	0.620690i	$0.213147\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.42079e6	−0.435431
$738$	0	0
$739$	1.49101e7	1.00431	0.502156	−	0.864777i	$-0.332540\pi$
$739$	1.49101e7	1.00431	0.502156	+	0.864777i	$0.332540\pi$
$740$	0	0
$741$	128940.	0.00862664
$742$	0	0
$743$	1.88035e7	1.24959	0.624794	−	0.780790i	$-0.285183\pi$
$743$	1.88035e7	1.24959	0.624794	+	0.780790i	$0.285183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.37445e6	0.286828
$748$	0	0
$749$	7.81179e6	0.508799
$750$	0	0
$751$	−1.95078e7	−1.26214	−0.631070	−	0.775726i	$-0.717384\pi$
$751$	−1.95078e7	−1.26214	−0.631070	+	0.775726i	$0.717384\pi$
$752$	0	0
$753$	496092.	0.0318841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.92282e7	−1.21955	−0.609773	−	0.792576i	$-0.708739\pi$
$757$	−1.92282e7	−1.21955	−0.609773	+	0.792576i	$0.708739\pi$
$758$	0	0
$759$	−376103.	−0.0236975
$760$	0	0
$761$	2.52863e7	1.58279	0.791397	−	0.611303i	$-0.209354\pi$
$761$	2.52863e7	1.58279	0.791397	+	0.611303i	$0.209354\pi$
$762$	0	0
$763$	49035.7	0.00304931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.16119e6	0.194027
$768$	0	0
$769$	1.47943e7	0.902151	0.451075	−	0.892486i	$-0.351041\pi$
$769$	1.47943e7	0.902151	0.451075	+	0.892486i	$0.351041\pi$
$770$	0	0
$771$	−146522.	−0.00887701
$772$	0	0
$773$	−2.24891e7	−1.35370	−0.676852	−	0.736119i	$-0.736656\pi$
$773$	−2.24891e7	−1.35370	−0.676852	+	0.736119i	$0.736656\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−251226.	−0.0149284
$778$	0	0
$779$	−1.23428e7	−0.728738
$780$	0	0
$781$	−3.91398e6	−0.229610
$782$	0	0
$783$	−825010.	−0.0480900
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.68992e6	−0.442573	−0.221287	−	0.975209i	$-0.571026\pi$
$787$	−7.68992e6	−0.442573	−0.221287	+	0.975209i	$0.571026\pi$
$788$	0	0
$789$	−221556.	−0.0126704
$790$	0	0
$791$	−1.07874e7	−0.613022
$792$	0	0
$793$	5.35938e6	0.302644
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.14602e7	−1.19671	−0.598354	−	0.801232i	$-0.704178\pi$
$797$	−2.14602e7	−1.19671	−0.598354	+	0.801232i	$0.704178\pi$
$798$	0	0
$799$	−833324.	−0.0461792
$800$	0	0
$801$	3.20556e7	1.76532
$802$	0	0
$803$	2.26839e6	0.124145
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−147631.	−0.00797983
$808$	0	0
$809$	−1.18026e7	−0.634026	−0.317013	−	0.948421i	$-0.602680\pi$
$809$	−1.18026e7	−0.634026	−0.317013	+	0.948421i	$0.602680\pi$
$810$	0	0
$811$	−5.44831e6	−0.290877	−0.145439	−	0.989367i	$-0.546459\pi$
$811$	−5.44831e6	−0.290877	−0.145439	+	0.989367i	$0.546459\pi$
$812$	0	0
$813$	963393.	0.0511184
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.29537e7	−1.20309
$818$	0	0
$819$	−2.07010e6	−0.107840
$820$	0	0
$821$	7.53753e6	0.390275	0.195138	−	0.980776i	$-0.437485\pi$
$821$	7.53753e6	0.390275	0.195138	+	0.980776i	$0.437485\pi$
$822$	0	0
$823$	7.89571e6	0.406342	0.203171	−	0.979143i	$-0.434875\pi$
$823$	7.89571e6	0.406342	0.203171	+	0.979143i	$0.434875\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.11093e7	0.564836	0.282418	−	0.959291i	$-0.408864\pi$
$827$	1.11093e7	0.564836	0.282418	+	0.959291i	$0.408864\pi$
$828$	0	0
$829$	−3.51531e7	−1.77655	−0.888276	−	0.459311i	$-0.848096\pi$
$829$	−3.51531e7	−1.77655	−0.888276	+	0.459311i	$0.848096\pi$
$830$	0	0
$831$	−277936.	−0.0139618
$832$	0	0
$833$	−72645.0	−0.00362738
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.22906e6	0.109979
$838$	0	0
$839$	−2.39689e7	−1.17555	−0.587777	−	0.809023i	$-0.699997\pi$
$839$	−2.39689e7	−1.17555	−0.587777	+	0.809023i	$0.699997\pi$
$840$	0	0
$841$	−1.08239e7	−0.527710
$842$	0	0
$843$	1.05486e6	0.0511239
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.76163e6	0.323849
$848$	0	0
$849$	−458641.	−0.0218375
$850$	0	0
$851$	−4.26364e7	−2.01816
$852$	0	0
$853$	1.77178e7	0.833754	0.416877	−	0.908963i	$-0.363125\pi$
$853$	1.77178e7	0.833754	0.416877	+	0.908963i	$0.363125\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.13278e7	−0.991959	−0.495979	−	0.868334i	$-0.665191\pi$
$857$	−2.13278e7	−0.991959	−0.495979	+	0.868334i	$0.665191\pi$
$858$	0	0
$859$	−2.06255e7	−0.953723	−0.476861	−	0.878978i	$-0.658226\pi$
$859$	−2.06255e7	−0.953723	−0.476861	+	0.878978i	$0.658226\pi$
$860$	0	0
$861$	−243178.	−0.0111794
$862$	0	0
$863$	−2.66276e7	−1.21704	−0.608519	−	0.793539i	$-0.708236\pi$
$863$	−2.66276e7	−1.21704	−0.608519	+	0.793539i	$0.708236\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	774380.	0.0349870
$868$	0	0
$869$	8.18600e6	0.367724
$870$	0	0
$871$	7.36028e6	0.328737
$872$	0	0
$873$	7.61893e6	0.338344
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.67155e7	−0.733873	−0.366937	−	0.930246i	$-0.619593\pi$
$877$	−1.67155e7	−0.733873	−0.366937	+	0.930246i	$0.619593\pi$
$878$	0	0
$879$	212084.	0.00925842
$880$	0	0
$881$	−1.79274e7	−0.778174	−0.389087	−	0.921201i	$-0.627209\pi$
$881$	−1.79274e7	−0.778174	−0.389087	+	0.921201i	$0.627209\pi$
$882$	0	0
$883$	2.35062e7	1.01457	0.507284	−	0.861779i	$-0.330649\pi$
$883$	2.35062e7	1.01457	0.507284	+	0.861779i	$0.330649\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.03067e7	−0.866624	−0.433312	−	0.901244i	$-0.642655\pi$
$887$	−2.03067e7	−0.866624	−0.433312	+	0.901244i	$0.642655\pi$
$888$	0	0
$889$	−5.27016e6	−0.223650
$890$	0	0
$891$	8.93365e6	0.376994
$892$	0	0
$893$	3.73831e7	1.56873
$894$	0	0
$895$	0	0
$896$	0	0
$897$	431135.	0.0178909
$898$	0	0
$899$	−2.61735e7	−1.08010
$900$	0	0
$901$	−343833.	−0.0141103
$902$	0	0
$903$	−452233.	−0.0184562
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.51170e6	−0.262831	−0.131415	−	0.991327i	$-0.541952\pi$
$907$	−6.51170e6	−0.262831	−0.131415	+	0.991327i	$0.541952\pi$
$908$	0	0
$909$	1.23512e7	0.495790
$910$	0	0
$911$	−2.90005e7	−1.15774	−0.578869	−	0.815421i	$-0.696506\pi$
$911$	−2.90005e7	−1.15774	−0.578869	+	0.815421i	$0.696506\pi$
$912$	0	0
$913$	−2.73694e6	−0.108665
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.47009e7	−0.577324
$918$	0	0
$919$	−1.60353e7	−0.626309	−0.313154	−	0.949702i	$-0.601386\pi$
$919$	−1.60353e7	−0.626309	−0.313154	+	0.949702i	$0.601386\pi$
$920$	0	0
$921$	502738.	0.0195296
$922$	0	0
$923$	4.48667e6	0.173349
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.37315e7	−0.524830
$928$	0	0
$929$	−9.08140e6	−0.345234	−0.172617	−	0.984989i	$-0.555222\pi$
$929$	−9.08140e6	−0.345234	−0.172617	+	0.984989i	$0.555222\pi$
$930$	0	0
$931$	3.25888e6	0.123224
$932$	0	0
$933$	−647551.	−0.0243540
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.35342e7	0.503596	0.251798	−	0.967780i	$-0.418978\pi$
$937$	1.35342e7	0.503596	0.251798	+	0.967780i	$0.418978\pi$
$938$	0	0
$939$	817756.	0.0302663
$940$	0	0
$941$	−3.60708e7	−1.32795	−0.663975	−	0.747755i	$-0.731132\pi$
$941$	−3.60708e7	−1.32795	−0.663975	+	0.747755i	$0.731132\pi$
$942$	0	0
$943$	−4.12706e7	−1.51134
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.26133e7	1.18173	0.590867	−	0.806769i	$-0.298786\pi$
$947$	3.26133e7	1.18173	0.590867	+	0.806769i	$0.298786\pi$
$948$	0	0
$949$	−2.60030e6	−0.0937254
$950$	0	0
$951$	1.52024e6	0.0545081
$952$	0	0
$953$	3.44202e7	1.22767	0.613834	−	0.789435i	$-0.289627\pi$
$953$	3.44202e7	1.22767	0.613834	+	0.789435i	$0.289627\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	257932.	0.00910384
$958$	0	0
$959$	4.09201e6	0.143678
$960$	0	0
$961$	4.20879e7	1.47011
$962$	0	0
$963$	3.86926e7	1.34450
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.11368e7	0.382995	0.191498	−	0.981493i	$-0.438666\pi$
$967$	1.11368e7	0.382995	0.191498	+	0.981493i	$0.438666\pi$
$968$	0	0
$969$	22411.9	0.000766778 0
$970$	0	0
$971$	−4.43758e7	−1.51042	−0.755211	−	0.655481i	$-0.772466\pi$
$971$	−4.43758e7	−1.51042	−0.755211	+	0.655481i	$0.772466\pi$
$972$	0	0
$973$	2.03735e7	0.689894
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.66921e6	−0.190014	−0.0950070	−	0.995477i	$-0.530287\pi$
$977$	−5.66921e6	−0.190014	−0.0950070	+	0.995477i	$0.530287\pi$
$978$	0	0
$979$	−2.00561e7	−0.668790
$980$	0	0
$981$	242879.	0.00805782
$982$	0	0
$983$	−4.46329e7	−1.47323	−0.736616	−	0.676311i	$-0.763578\pi$
$983$	−4.46329e7	−1.47323	−0.736616	+	0.676311i	$0.763578\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	736522.	0.0240654
$988$	0	0
$989$	−7.67500e7	−2.49510
$990$	0	0
$991$	1.59365e7	0.515475	0.257738	−	0.966215i	$-0.417023\pi$
$991$	1.59365e7	0.515475	0.257738	+	0.966215i	$0.417023\pi$
$992$	0	0
$993$	314806.	0.0101314
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.48778e7	−0.792636	−0.396318	−	0.918113i	$-0.629712\pi$
$997$	−2.48778e7	−0.792636	−0.396318	+	0.918113i	$0.629712\pi$
$998$	0	0
$999$	−2.49022e6	−0.0789450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.a.k.1.3	✓	4
5.2	odd	4		700.6.e.j.449.5		8
5.3	odd	4		700.6.e.j.449.4		8
5.4	even	2		700.6.a.n.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.6.a.k.1.3	✓	4	1.1	even	1	trivial
700.6.a.n.1.2	yes	4	5.4	even	2
700.6.e.j.449.4		8	5.3	odd	4
700.6.e.j.449.5		8	5.2	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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

\(f(q)\)	\(=\)	\(q-0.545745 q^{3} -49.0000 q^{7} -242.702 q^{9} +151.850 q^{11} -174.069 q^{13} -30.2562 q^{17} +1357.30 q^{19} +26.7415 q^{21} +4538.39 q^{23} +265.069 q^{27} -3112.43 q^{29} +8409.35 q^{31} -82.8715 q^{33} -9394.61 q^{37} +94.9973 q^{39} -9093.67 q^{41} -16911.3 q^{43} +27542.3 q^{47} +2401.00 q^{49} +16.5121 q^{51} +11364.1 q^{53} -740.740 q^{57} -18160.6 q^{59} -30788.8 q^{61} +11892.4 q^{63} -42283.7 q^{67} -2476.80 q^{69} -25775.3 q^{71} +14938.3 q^{73} -7440.67 q^{77} +53908.3 q^{79} +58832.0 q^{81} -18023.9 q^{83} +1698.59 q^{87} -132078. q^{89} +8529.38 q^{91} -4589.36 q^{93} -31392.1 q^{97} -36854.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{3} - 196 q^{7} - 62 q^{9} + 48 q^{11} + 430 q^{13} + 1286 q^{17} - 384 q^{19} + 1078 q^{21} - 1256 q^{23} + 1196 q^{27} - 4994 q^{29} + 4618 q^{31} + 15268 q^{33} + 16922 q^{37} + 3200 q^{39}+ \cdots - 170416 q^{99}+O(q^{100}) \) 4 * q - 22 * q^3 - 196 * q^7 - 62 * q^9 + 48 * q^11 + 430 * q^13 + 1286 * q^17 - 384 * q^19 + 1078 * q^21 - 1256 * q^23 + 1196 * q^27 - 4994 * q^29 + 4618 * q^31 + 15268 * q^33 + 16922 * q^37 + 3200 * q^39 - 9192 * q^41 + 12368 * q^43 - 4476 * q^47 + 9604 * q^49 - 30396 * q^51 + 16736 * q^53 - 29946 * q^57 - 14614 * q^59 - 7122 * q^61 + 3038 * q^63 - 21408 * q^67 + 11822 * q^69 - 39748 * q^71 + 89364 * q^73 - 2352 * q^77 - 37640 * q^79 - 125192 * q^81 + 83842 * q^83 - 90982 * q^87 - 12444 * q^89 - 21070 * q^91 + 137840 * q^93 - 57034 * q^97 - 170416 * q^99

Introduction

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