LMFDB - Embedded newform 1764.4.a.bb.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(1,1764)] 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("1764.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{7}, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 67x^{2} + 68x + 393 $ x^4 - 2x^3 - 67x^2 + 68*x + 393
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.36590$ of defining polynomial
Character	$\chi$	$=$	1764.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-10.5830 q^{5} +55.2449 q^{11} +83.5225 q^{13} -95.2470 q^{17} +83.5225 q^{19} +165.735 q^{23} -13.0000 q^{25} -110.490 q^{29} -83.5225 q^{31} +78.0000 q^{37} -412.737 q^{41} +148.000 q^{43} +465.652 q^{47} -110.490 q^{53} -584.657 q^{55} -550.316 q^{59} -584.657 q^{61} -883.919 q^{65} -260.000 q^{67} +718.184 q^{71} +668.180 q^{73} +664.000 q^{79} -126.996 q^{83} +1008.00 q^{85} -878.389 q^{89} -883.919 q^{95} +1169.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 52 q^{25} + 312 q^{37} + 592 q^{43} - 1040 q^{67} + 2656 q^{79} + 4032 q^{85}+O(q^{100}) $ 4 * q - 52 * q^25 + 312 * q^37 + 592 * q^43 - 1040 * q^67 + 2656 * q^79 + 4032 * q^85

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	0
$3$	0	0
$4$	0	0
$5$	−10.5830	−0.946573	−0.473286	−	0.880909i	$-0.656932\pi$
$5$	−10.5830	−0.946573	−0.473286	+	0.880909i	$0.656932\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	55.2449	1.51427	0.757135	−	0.653259i	$-0.226599\pi$
$11$	55.2449	1.51427	0.757135	+	0.653259i	$0.226599\pi$
$12$	0	0
$13$	83.5225	1.78192	0.890960	−	0.454082i	$-0.150033\pi$
$13$	83.5225	1.78192	0.890960	+	0.454082i	$0.150033\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−95.2470	−1.35887	−0.679435	−	0.733735i	$-0.737775\pi$
$17$	−95.2470	−1.35887	−0.679435	+	0.733735i	$0.737775\pi$
$18$	0	0
$19$	83.5225	1.00849	0.504246	−	0.863560i	$-0.331770\pi$
$19$	83.5225	1.00849	0.504246	+	0.863560i	$0.331770\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	165.735	1.50253	0.751263	−	0.660003i	$-0.229445\pi$
$23$	165.735	1.50253	0.751263	+	0.660003i	$0.229445\pi$
$24$	0	0
$25$	−13.0000	−0.104000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−110.490	−0.707498	−0.353749	−	0.935340i	$-0.615093\pi$
$29$	−110.490	−0.707498	−0.353749	+	0.935340i	$0.615093\pi$
$30$	0	0
$31$	−83.5225	−0.483906	−0.241953	−	0.970288i	$-0.577788\pi$
$31$	−83.5225	−0.483906	−0.241953	+	0.970288i	$0.577788\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	78.0000	0.346571	0.173285	−	0.984872i	$-0.444562\pi$
$37$	78.0000	0.346571	0.173285	+	0.984872i	$0.444562\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−412.737	−1.57216	−0.786082	−	0.618122i	$-0.787894\pi$
$41$	−412.737	−1.57216	−0.786082	+	0.618122i	$0.787894\pi$
$42$	0	0
$43$	148.000	0.524879	0.262439	−	0.964948i	$-0.415473\pi$
$43$	148.000	0.524879	0.262439	+	0.964948i	$0.415473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	465.652	1.44516	0.722578	−	0.691289i	$-0.242957\pi$
$47$	465.652	1.44516	0.722578	+	0.691289i	$0.242957\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−110.490	−0.286357	−0.143179	−	0.989697i	$-0.545732\pi$
$53$	−110.490	−0.286357	−0.143179	+	0.989697i	$0.545732\pi$
$54$	0	0
$55$	−584.657	−1.43337
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−550.316	−1.21432	−0.607162	−	0.794578i	$-0.707692\pi$
$59$	−550.316	−1.21432	−0.607162	+	0.794578i	$0.707692\pi$
$60$	0	0
$61$	−584.657	−1.22718	−0.613588	−	0.789627i	$-0.710274\pi$
$61$	−584.657	−1.22718	−0.613588	+	0.789627i	$0.710274\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−883.919	−1.68672
$66$	0	0
$67$	−260.000	−0.474090	−0.237045	−	0.971499i	$-0.576179\pi$
$67$	−260.000	−0.474090	−0.237045	+	0.971499i	$0.576179\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	718.184	1.20046	0.600231	−	0.799827i	$-0.295075\pi$
$71$	718.184	1.20046	0.600231	+	0.799827i	$0.295075\pi$
$72$	0	0
$73$	668.180	1.07129	0.535647	−	0.844442i	$-0.320068\pi$
$73$	668.180	1.07129	0.535647	+	0.844442i	$0.320068\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	664.000	0.945644	0.472822	−	0.881158i	$-0.343235\pi$
$79$	664.000	0.945644	0.472822	+	0.881158i	$0.343235\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−126.996	−0.167947	−0.0839737	−	0.996468i	$-0.526761\pi$
$83$	−126.996	−0.167947	−0.0839737	+	0.996468i	$0.526761\pi$
$84$	0	0
$85$	1008.00	1.28627
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−878.389	−1.04617	−0.523085	−	0.852281i	$-0.675219\pi$
$89$	−878.389	−1.04617	−0.523085	+	0.852281i	$0.675219\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−883.919	−0.954612
$96$	0	0
$97$	1169.31	1.22398	0.611989	−	0.790866i	$-0.290370\pi$
$97$	1169.31	1.22398	0.611989	+	0.790866i	$0.290370\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−455.069	−0.448328	−0.224164	−	0.974551i	$-0.571965\pi$
$101$	−455.069	−0.448328	−0.224164	+	0.974551i	$0.571965\pi$
$102$	0	0
$103$	1085.79	1.03870	0.519351	−	0.854561i	$-0.326174\pi$
$103$	1085.79	1.03870	0.519351	+	0.854561i	$0.326174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	939.163	0.848527	0.424263	−	0.905539i	$-0.360533\pi$
$107$	939.163	0.848527	0.424263	+	0.905539i	$0.360533\pi$
$108$	0	0
$109$	−650.000	−0.571181	−0.285590	−	0.958352i	$-0.592190\pi$
$109$	−650.000	−0.571181	−0.285590	+	0.958352i	$0.592190\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1546.86	−1.28775	−0.643877	−	0.765129i	$-0.722675\pi$
$113$	−1546.86	−1.28775	−0.643877	+	0.765129i	$0.722675\pi$
$114$	0	0
$115$	−1753.97	−1.42225
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1721.00	1.29301
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1460.45	1.04502
$126$	0	0
$127$	2288.00	1.59864	0.799320	−	0.600906i	$-0.205193\pi$
$127$	2288.00	1.59864	0.799320	+	0.600906i	$0.205193\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	888.972	0.592900	0.296450	−	0.955048i	$-0.404197\pi$
$131$	888.972	0.592900	0.296450	+	0.955048i	$0.404197\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2872.74	1.79149	0.895746	−	0.444567i	$-0.146642\pi$
$137$	2872.74	1.79149	0.895746	+	0.444567i	$0.146642\pi$
$138$	0	0
$139$	501.135	0.305796	0.152898	−	0.988242i	$-0.451139\pi$
$139$	501.135	0.305796	0.152898	+	0.988242i	$0.451139\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4614.19	2.69831
$144$	0	0
$145$	1169.31	0.669698
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1878.33	1.03274	0.516371	−	0.856365i	$-0.327283\pi$
$149$	1878.33	1.03274	0.516371	+	0.856365i	$0.327283\pi$
$150$	0	0
$151$	−1976.00	−1.06493	−0.532466	−	0.846452i	$-0.678734\pi$
$151$	−1976.00	−1.06493	−0.532466	+	0.846452i	$0.678734\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	883.919	0.458052
$156$	0	0
$157$	584.657	0.297202	0.148601	−	0.988897i	$-0.452523\pi$
$157$	584.657	0.297202	0.148601	+	0.988897i	$0.452523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3348.00	1.60881	0.804404	−	0.594083i	$-0.202485\pi$
$163$	3348.00	1.60881	0.804404	+	0.594083i	$0.202485\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−84.6640	−0.0392305	−0.0196153	−	0.999808i	$-0.506244\pi$
$167$	−84.6640	−0.0392305	−0.0196153	+	0.999808i	$0.506244\pi$
$168$	0	0
$169$	4779.00	2.17524
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1513.37	−0.665083	−0.332542	−	0.943089i	$-0.607906\pi$
$173$	−1513.37	−0.665083	−0.332542	+	0.943089i	$0.607906\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	165.735	0.0692044	0.0346022	−	0.999401i	$-0.488984\pi$
$179$	165.735	0.0692044	0.0346022	+	0.999401i	$0.488984\pi$
$180$	0	0
$181$	−3591.47	−1.47487	−0.737435	−	0.675418i	$-0.763963\pi$
$181$	−3591.47	−1.47487	−0.737435	+	0.675418i	$0.763963\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−825.474	−0.328055
$186$	0	0
$187$	−5261.91	−2.05770
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2154.55	−0.816219	−0.408110	−	0.912933i	$-0.633812\pi$
$191$	−2154.55	−0.816219	−0.408110	+	0.912933i	$0.633812\pi$
$192$	0	0
$193$	−1378.00	−0.513941	−0.256970	−	0.966419i	$-0.582724\pi$
$193$	−1378.00	−0.513941	−0.256970	+	0.966419i	$0.582724\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	994.408	0.359638	0.179819	−	0.983700i	$-0.442449\pi$
$197$	994.408	0.359638	0.179819	+	0.983700i	$0.442449\pi$
$198$	0	0
$199$	4677.26	1.66614	0.833070	−	0.553167i	$-0.186581\pi$
$199$	4677.26	1.66614	0.833070	+	0.553167i	$0.186581\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4368.00	1.48817
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4614.19	1.52713
$210$	0	0
$211$	5108.00	1.66658	0.833292	−	0.552833i	$-0.186453\pi$
$211$	5108.00	1.66658	0.833292	+	0.552833i	$0.186453\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1566.28	−0.496836
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7955.27	−2.42140
$222$	0	0
$223$	1169.31	0.351135	0.175567	−	0.984467i	$-0.443824\pi$
$223$	1169.31	0.351135	0.175567	+	0.984467i	$0.443824\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1354.62	−0.396078	−0.198039	−	0.980194i	$-0.563457\pi$
$227$	−1354.62	−0.396078	−0.198039	+	0.980194i	$0.563457\pi$
$228$	0	0
$229$	1085.79	0.313324	0.156662	−	0.987652i	$-0.449927\pi$
$229$	1085.79	0.313324	0.156662	+	0.987652i	$0.449927\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3756.65	−1.05625	−0.528126	−	0.849166i	$-0.677105\pi$
$233$	−3756.65	−1.05625	−0.528126	+	0.849166i	$0.677105\pi$
$234$	0	0
$235$	−4928.00	−1.36795
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3148.96	−0.852256	−0.426128	−	0.904663i	$-0.640123\pi$
$239$	−3148.96	−0.852256	−0.426128	+	0.904663i	$0.640123\pi$
$240$	0	0
$241$	−5846.57	−1.56270	−0.781350	−	0.624093i	$-0.785469\pi$
$241$	−5846.57	−1.56270	−0.781350	+	0.624093i	$0.785469\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6976.00	1.79705
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6011.15	−1.51163	−0.755817	−	0.654783i	$-0.772760\pi$
$251$	−6011.15	−1.51163	−0.755817	+	0.654783i	$0.772760\pi$
$252$	0	0
$253$	9156.00	2.27523
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1005.39	−0.244024	−0.122012	−	0.992529i	$-0.538935\pi$
$257$	−1005.39	−0.244024	−0.122012	+	0.992529i	$0.538935\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3701.41	0.867828	0.433914	−	0.900954i	$-0.357132\pi$
$263$	3701.41	0.867828	0.433914	+	0.900954i	$0.357132\pi$
$264$	0	0
$265$	1169.31	0.271058
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4201.45	−0.952294	−0.476147	−	0.879366i	$-0.657967\pi$
$269$	−4201.45	−0.952294	−0.476147	+	0.879366i	$0.657967\pi$
$270$	0	0
$271$	7600.54	1.70369	0.851845	−	0.523794i	$-0.175484\pi$
$271$	7600.54	1.70369	0.851845	+	0.523794i	$0.175484\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−718.184	−0.157484
$276$	0	0
$277$	−1058.00	−0.229491	−0.114746	−	0.993395i	$-0.536605\pi$
$277$	−1058.00	−0.229491	−0.114746	+	0.993395i	$0.536605\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8618.21	1.82961	0.914803	−	0.403901i	$-0.132346\pi$
$281$	8618.21	1.82961	0.914803	+	0.403901i	$0.132346\pi$
$282$	0	0
$283$	−1085.79	−0.228069	−0.114035	−	0.993477i	$-0.536377\pi$
$283$	−1085.79	−0.228069	−0.114035	+	0.993477i	$0.536377\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4159.00	0.846530
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6000.56	−1.19644	−0.598220	−	0.801332i	$-0.704125\pi$
$293$	−6000.56	−1.19644	−0.598220	+	0.801332i	$0.704125\pi$
$294$	0	0
$295$	5824.00	1.14945
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13842.6	2.67738
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6187.43	1.16161
$306$	0	0
$307$	6932.36	1.28877	0.644383	−	0.764703i	$-0.277114\pi$
$307$	6932.36	1.28877	0.644383	+	0.764703i	$0.277114\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	550.316	0.100339	0.0501697	−	0.998741i	$-0.484024\pi$
$311$	550.316	0.100339	0.0501697	+	0.998741i	$0.484024\pi$
$312$	0	0
$313$	501.135	0.0904978	0.0452489	−	0.998976i	$-0.485592\pi$
$313$	501.135	0.0904978	0.0452489	+	0.998976i	$0.485592\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9612.61	1.70315	0.851574	−	0.524234i	$-0.175648\pi$
$317$	9612.61	1.70315	0.851574	+	0.524234i	$0.175648\pi$
$318$	0	0
$319$	−6104.00	−1.07134
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7955.27	−1.37041
$324$	0	0
$325$	−1085.79	−0.185320
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8324.00	1.38226	0.691131	−	0.722730i	$-0.257113\pi$
$331$	8324.00	1.38226	0.691131	+	0.722730i	$0.257113\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2751.58	0.448761
$336$	0	0
$337$	9794.00	1.58313	0.791563	−	0.611088i	$-0.209268\pi$
$337$	9794.00	1.58313	0.791563	+	0.611088i	$0.209268\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4614.19	−0.732764
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2154.55	0.333321	0.166660	−	0.986014i	$-0.446702\pi$
$347$	2154.55	0.333321	0.166660	+	0.986014i	$0.446702\pi$
$348$	0	0
$349$	−1753.97	−0.269020	−0.134510	−	0.990912i	$-0.542946\pi$
$349$	−1753.97	−0.269020	−0.134510	+	0.990912i	$0.542946\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6191.06	−0.933475	−0.466738	−	0.884396i	$-0.654571\pi$
$353$	−6191.06	−0.933475	−0.466738	+	0.884396i	$0.654571\pi$
$354$	0	0
$355$	−7600.54	−1.13632
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2927.98	0.430454	0.215227	−	0.976564i	$-0.430951\pi$
$359$	2927.98	0.430454	0.215227	+	0.976564i	$0.430951\pi$
$360$	0	0
$361$	117.000	0.0170579
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7071.35	−1.01406
$366$	0	0
$367$	8686.34	1.23549	0.617743	−	0.786380i	$-0.288047\pi$
$367$	8686.34	1.23549	0.617743	+	0.786380i	$0.288047\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4654.00	0.646046	0.323023	−	0.946391i	$-0.395301\pi$
$373$	4654.00	0.646046	0.323023	+	0.946391i	$0.395301\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9228.38	−1.26070
$378$	0	0
$379$	2388.00	0.323650	0.161825	−	0.986819i	$-0.448262\pi$
$379$	2388.00	0.323650	0.161825	+	0.986819i	$0.448262\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8254.74	−1.10130	−0.550650	−	0.834736i	$-0.685620\pi$
$383$	−8254.74	−1.10130	−0.550650	+	0.834736i	$0.685620\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4309.10	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$389$	−4309.10	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$390$	0	0
$391$	−15785.7	−2.04174
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7027.12	−0.895120
$396$	0	0
$397$	−10440.3	−1.31986	−0.659929	−	0.751328i	$-0.729414\pi$
$397$	−10440.3	−1.31986	−0.659929	+	0.751328i	$0.729414\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−662.939	−0.0825576	−0.0412788	−	0.999148i	$-0.513143\pi$
$401$	−662.939	−0.0825576	−0.0412788	+	0.999148i	$0.513143\pi$
$402$	0	0
$403$	−6976.00	−0.862281
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4309.10	0.524802
$408$	0	0
$409$	7517.02	0.908784	0.454392	−	0.890802i	$-0.349857\pi$
$409$	7517.02	0.908784	0.454392	+	0.890802i	$0.349857\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1344.00	0.158974
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13419.3	1.56461	0.782307	−	0.622893i	$-0.214043\pi$
$419$	13419.3	1.56461	0.782307	+	0.622893i	$0.214043\pi$
$420$	0	0
$421$	8238.00	0.953671	0.476836	−	0.878993i	$-0.341784\pi$
$421$	8238.00	0.953671	0.476836	+	0.878993i	$0.341784\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1238.21	0.141323
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8341.98	−0.932295	−0.466147	−	0.884707i	$-0.654358\pi$
$431$	−8341.98	−0.932295	−0.466147	+	0.884707i	$0.654358\pi$
$432$	0	0
$433$	1837.49	0.203936	0.101968	−	0.994788i	$-0.467486\pi$
$433$	1837.49	0.203936	0.101968	+	0.994788i	$0.467486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13842.6	1.51529
$438$	0	0
$439$	10523.8	1.14413	0.572067	−	0.820207i	$-0.306142\pi$
$439$	10523.8	1.14413	0.572067	+	0.820207i	$0.306142\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13756.0	−1.47532	−0.737660	−	0.675173i	$-0.764069\pi$
$443$	−13756.0	−1.47532	−0.737660	+	0.675173i	$0.764069\pi$
$444$	0	0
$445$	9296.00	0.990276
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9281.14	0.975511	0.487755	−	0.872980i	$-0.337816\pi$
$449$	9281.14	0.975511	0.487755	+	0.872980i	$0.337816\pi$
$450$	0	0
$451$	−22801.6	−2.38068
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2890.00	−0.295817	−0.147909	−	0.989001i	$-0.547254\pi$
$457$	−2890.00	−0.295817	−0.147909	+	0.989001i	$0.547254\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14191.8	1.43379	0.716896	−	0.697180i	$-0.245562\pi$
$461$	14191.8	1.43379	0.716896	+	0.697180i	$0.245562\pi$
$462$	0	0
$463$	−8632.00	−0.866443	−0.433221	−	0.901288i	$-0.642623\pi$
$463$	−8632.00	−0.866443	−0.433221	+	0.901288i	$0.642623\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4402.53	−0.436242	−0.218121	−	0.975922i	$-0.569993\pi$
$467$	−4402.53	−0.436242	−0.218121	+	0.975922i	$0.569993\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8176.25	0.794808
$474$	0	0
$475$	−1085.79	−0.104883
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9270.71	0.884321	0.442160	−	0.896936i	$-0.354212\pi$
$479$	9270.71	0.884321	0.442160	+	0.896936i	$0.354212\pi$
$480$	0	0
$481$	6514.75	0.617562
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12374.9	−1.15858
$486$	0	0
$487$	−12536.0	−1.16645	−0.583224	−	0.812311i	$-0.698209\pi$
$487$	−12536.0	−1.16645	−0.583224	+	0.812311i	$0.698209\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19391.0	1.78228	0.891142	−	0.453724i	$-0.149905\pi$
$491$	19391.0	1.78228	0.891142	+	0.453724i	$0.149905\pi$
$492$	0	0
$493$	10523.8	0.961398
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6756.00	0.606092	0.303046	−	0.952976i	$-0.401996\pi$
$499$	6756.00	0.606092	0.303046	+	0.952976i	$0.401996\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16721.1	−1.48222	−0.741112	−	0.671381i	$-0.765701\pi$
$503$	−16721.1	−1.48222	−0.741112	+	0.671381i	$0.765701\pi$
$504$	0	0
$505$	4816.00	0.424375
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10995.7	0.957520	0.478760	−	0.877946i	$-0.341086\pi$
$509$	10995.7	0.957520	0.478760	+	0.877946i	$0.341086\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11490.9	−0.983207
$516$	0	0
$517$	25724.9	2.18836
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20774.4	1.74692	0.873459	−	0.486897i	$-0.161871\pi$
$521$	20774.4	1.74692	0.873459	+	0.486897i	$0.161871\pi$
$522$	0	0
$523$	−1670.45	−0.139663	−0.0698314	−	0.997559i	$-0.522246\pi$
$523$	−1670.45	−0.139663	−0.0698314	+	0.997559i	$0.522246\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7955.27	0.657565
$528$	0	0
$529$	15301.0	1.25758
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−34472.8	−2.80147
$534$	0	0
$535$	−9939.17	−0.803192
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19370.0	−1.53934	−0.769669	−	0.638444i	$-0.779579\pi$
$541$	−19370.0	−1.53934	−0.769669	+	0.638444i	$0.779579\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6878.95	0.540664
$546$	0	0
$547$	−20396.0	−1.59428	−0.797139	−	0.603796i	$-0.793654\pi$
$547$	−20396.0	−1.59428	−0.797139	+	0.603796i	$0.793654\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9228.38	−0.713507
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4309.10	−0.327796	−0.163898	−	0.986477i	$-0.552407\pi$
$557$	−4309.10	−0.327796	−0.163898	+	0.986477i	$0.552407\pi$
$558$	0	0
$559$	12361.3	0.935292
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.3320	0.00316888	0.00158444	−	0.999999i	$-0.499496\pi$
$563$	42.3320	0.00316888	0.00158444	+	0.999999i	$0.499496\pi$
$564$	0	0
$565$	16370.4	1.21895
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6629.39	0.488433	0.244217	−	0.969721i	$-0.421469\pi$
$569$	6629.39	0.488433	0.244217	+	0.969721i	$0.421469\pi$
$570$	0	0
$571$	1308.00	0.0958636	0.0479318	−	0.998851i	$-0.484737\pi$
$571$	1308.00	0.0958636	0.0479318	+	0.998851i	$0.484737\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2154.55	−0.156263
$576$	0	0
$577$	8686.34	0.626719	0.313359	−	0.949635i	$-0.398546\pi$
$577$	8686.34	0.626719	0.313359	+	0.949635i	$0.398546\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6104.00	−0.433622
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25737.9	1.80974	0.904868	−	0.425691i	$-0.139969\pi$
$587$	25737.9	1.80974	0.904868	+	0.425691i	$0.139969\pi$
$588$	0	0
$589$	−6976.00	−0.488015
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−222.243	−0.0153903	−0.00769514	−	0.999970i	$-0.502449\pi$
$593$	−222.243	−0.0153903	−0.00769514	+	0.999970i	$0.502449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22374.2	1.52618	0.763092	−	0.646290i	$-0.223680\pi$
$599$	22374.2	1.52618	0.763092	+	0.646290i	$0.223680\pi$
$600$	0	0
$601$	−21715.8	−1.47389	−0.736945	−	0.675953i	$-0.763732\pi$
$601$	−21715.8	−1.47389	−0.736945	+	0.675953i	$0.763732\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18213.4	−1.22393
$606$	0	0
$607$	−13530.6	−0.904764	−0.452382	−	0.891824i	$-0.649426\pi$
$607$	−13530.6	−0.904764	−0.452382	+	0.891824i	$0.649426\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	38892.4	2.57515
$612$	0	0
$613$	12770.0	0.841396	0.420698	−	0.907201i	$-0.361785\pi$
$613$	12770.0	0.841396	0.420698	+	0.907201i	$0.361785\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11049.0	0.720932	0.360466	−	0.932772i	$-0.382618\pi$
$617$	11049.0	0.720932	0.360466	+	0.932772i	$0.382618\pi$
$618$	0	0
$619$	18207.9	1.18229	0.591145	−	0.806566i	$-0.298676\pi$
$619$	18207.9	1.18229	0.591145	+	0.806566i	$0.298676\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7429.27	−0.470945
$630$	0	0
$631$	−23952.0	−1.51112	−0.755558	−	0.655082i	$-0.772634\pi$
$631$	−23952.0	−1.51112	−0.755558	+	0.655082i	$0.772634\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−24213.9	−1.51323
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7734.29	0.476577	0.238289	−	0.971194i	$-0.423414\pi$
$641$	7734.29	0.476577	0.238289	+	0.971194i	$0.423414\pi$
$642$	0	0
$643$	−16286.9	−0.998899	−0.499449	−	0.866343i	$-0.666464\pi$
$643$	−16286.9	−0.998899	−0.499449	+	0.866343i	$0.666464\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13207.6	−0.802541	−0.401270	−	0.915960i	$-0.631431\pi$
$647$	−13207.6	−0.802541	−0.401270	+	0.915960i	$0.631431\pi$
$648$	0	0
$649$	−30402.2	−1.83881
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18672.8	−1.11902	−0.559512	−	0.828822i	$-0.689011\pi$
$653$	−18672.8	−1.11902	−0.559512	+	0.828822i	$0.689011\pi$
$654$	0	0
$655$	−9408.00	−0.561223
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6353.16	−0.375545	−0.187772	−	0.982213i	$-0.560127\pi$
$659$	−6353.16	−0.375545	−0.187772	+	0.982213i	$0.560127\pi$
$660$	0	0
$661$	−19961.9	−1.17462	−0.587312	−	0.809361i	$-0.699814\pi$
$661$	−19961.9	−1.17462	−0.587312	+	0.809361i	$0.699814\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18312.0	−1.06303
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32299.3	−1.85827
$672$	0	0
$673$	−12046.0	−0.689954	−0.344977	−	0.938611i	$-0.612113\pi$
$673$	−12046.0	−0.689954	−0.344977	+	0.938611i	$0.612113\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27653.4	1.56988	0.784938	−	0.619574i	$-0.212695\pi$
$677$	27653.4	1.56988	0.784938	+	0.619574i	$0.212695\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7900.02	0.442586	0.221293	−	0.975207i	$-0.428972\pi$
$683$	7900.02	0.442586	0.221293	+	0.975207i	$0.428972\pi$
$684$	0	0
$685$	−30402.2	−1.69578
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9228.38	−0.510266
$690$	0	0
$691$	7684.07	0.423033	0.211516	−	0.977374i	$-0.432160\pi$
$691$	7684.07	0.423033	0.211516	+	0.977374i	$0.432160\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5303.51	−0.289458
$696$	0	0
$697$	39312.0	2.13637
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12043.4	0.648891	0.324445	−	0.945904i	$-0.394822\pi$
$701$	12043.4	0.648891	0.324445	+	0.945904i	$0.394822\pi$
$702$	0	0
$703$	6514.75	0.349514
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25538.0	−1.35275	−0.676375	−	0.736557i	$-0.736450\pi$
$709$	−25538.0	−1.35275	−0.676375	+	0.736557i	$0.736450\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13842.6	−0.727080
$714$	0	0
$715$	−48832.0	−2.55414
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4952.85	−0.256898	−0.128449	−	0.991716i	$-0.541000\pi$
$719$	−4952.85	−0.256898	−0.128449	+	0.991716i	$0.541000\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1436.37	0.0735798
$726$	0	0
$727$	14115.3	0.720093	0.360046	−	0.932934i	$-0.382761\pi$
$727$	14115.3	0.720093	0.360046	+	0.932934i	$0.382761\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14096.6	−0.713242
$732$	0	0
$733$	12946.0	0.652347	0.326174	−	0.945310i	$-0.394241\pi$
$733$	12946.0	0.652347	0.326174	+	0.945310i	$0.394241\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14363.7	−0.717901
$738$	0	0
$739$	−14540.0	−0.723765	−0.361883	−	0.932224i	$-0.617866\pi$
$739$	−14540.0	−0.723765	−0.361883	+	0.932224i	$0.617866\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1160.14	0.0572833	0.0286417	−	0.999590i	$-0.490882\pi$
$743$	1160.14	0.0572833	0.0286417	+	0.999590i	$0.490882\pi$
$744$	0	0
$745$	−19878.3	−0.977565
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6768.00	−0.328852	−0.164426	−	0.986389i	$-0.552577\pi$
$751$	−6768.00	−0.328852	−0.164426	+	0.986389i	$0.552577\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20912.0	1.00803
$756$	0	0
$757$	−2354.00	−0.113022	−0.0565110	−	0.998402i	$-0.517998\pi$
$757$	−2354.00	−0.113022	−0.0565110	+	0.998402i	$0.517998\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33071.9	1.57537	0.787684	−	0.616079i	$-0.211280\pi$
$761$	33071.9	1.57537	0.787684	+	0.616079i	$0.211280\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45963.8	−2.16383
$768$	0	0
$769$	22885.2	1.07316	0.536580	−	0.843850i	$-0.319716\pi$
$769$	22885.2	1.07316	0.536580	+	0.843850i	$0.319716\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24351.5	1.13307	0.566535	−	0.824038i	$-0.308284\pi$
$773$	24351.5	1.13307	0.566535	+	0.824038i	$0.308284\pi$
$774$	0	0
$775$	1085.79	0.0503262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34472.8	−1.58552
$780$	0	0
$781$	39676.0	1.81782
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6187.43	−0.281323
$786$	0	0
$787$	36248.7	1.64184	0.820920	−	0.571043i	$-0.193461\pi$
$787$	36248.7	1.64184	0.820920	+	0.571043i	$0.193461\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−48832.0	−2.18673
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33431.7	−1.48584	−0.742918	−	0.669382i	$-0.766559\pi$
$797$	−33431.7	−1.48584	−0.742918	+	0.669382i	$0.766559\pi$
$798$	0	0
$799$	−44352.0	−1.96378
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36913.5	1.62223
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35798.7	−1.55577	−0.777883	−	0.628409i	$-0.783707\pi$
$809$	−35798.7	−1.55577	−0.777883	+	0.628409i	$0.783707\pi$
$810$	0	0
$811$	30402.2	1.31636	0.658178	−	0.752862i	$-0.271327\pi$
$811$	30402.2	1.31636	0.658178	+	0.752862i	$0.271327\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−35431.9	−1.52285
$816$	0	0
$817$	12361.3	0.529337
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33036.5	1.40436	0.702180	−	0.711999i	$-0.252210\pi$
$821$	33036.5	1.40436	0.702180	+	0.711999i	$0.252210\pi$
$822$	0	0
$823$	−10400.0	−0.440487	−0.220244	−	0.975445i	$-0.570685\pi$
$823$	−10400.0	−0.440487	−0.220244	+	0.975445i	$0.570685\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10109.8	−0.425094	−0.212547	−	0.977151i	$-0.568176\pi$
$827$	−10109.8	−0.425094	−0.212547	+	0.977151i	$0.568176\pi$
$828$	0	0
$829$	−14115.3	−0.591368	−0.295684	−	0.955286i	$-0.595548\pi$
$829$	−14115.3	−0.591368	−0.295684	+	0.955286i	$0.595548\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	896.000	0.0371346
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21843.3	−0.898826	−0.449413	−	0.893324i	$-0.648367\pi$
$839$	−21843.3	−0.898826	−0.449413	+	0.893324i	$0.648367\pi$
$840$	0	0
$841$	−12181.0	−0.499446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−50576.2	−2.05902
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12927.3	0.520731
$852$	0	0
$853$	25641.4	1.02924	0.514622	−	0.857417i	$-0.327932\pi$
$853$	25641.4	1.02924	0.514622	+	0.857417i	$0.327932\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	730.227	0.0291063	0.0145531	−	0.999894i	$-0.495367\pi$
$857$	730.227	0.0291063	0.0145531	+	0.999894i	$0.495367\pi$
$858$	0	0
$859$	22801.6	0.905683	0.452841	−	0.891591i	$-0.350410\pi$
$859$	22801.6	0.905683	0.452841	+	0.891591i	$0.350410\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32318.3	−1.27477	−0.637385	−	0.770545i	$-0.719984\pi$
$863$	−32318.3	−1.27477	−0.637385	+	0.770545i	$0.719984\pi$
$864$	0	0
$865$	16016.0	0.629550
$866$	0	0
$867$	0	0
$868$	0	0
$869$	36682.6	1.43196
$870$	0	0
$871$	−21715.8	−0.844791
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14166.0	−0.545441	−0.272721	−	0.962093i	$-0.587923\pi$
$877$	−14166.0	−0.545441	−0.272721	+	0.962093i	$0.587923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19673.8	0.752358	0.376179	−	0.926547i	$-0.377238\pi$
$881$	19673.8	0.752358	0.376179	+	0.926547i	$0.377238\pi$
$882$	0	0
$883$	−44876.0	−1.71030	−0.855152	−	0.518378i	$-0.826536\pi$
$883$	−44876.0	−1.71030	−0.855152	+	0.518378i	$0.826536\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15747.5	−0.596110	−0.298055	−	0.954549i	$-0.596338\pi$
$887$	−15747.5	−0.596110	−0.298055	+	0.954549i	$0.596338\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38892.4	1.45743
$894$	0	0
$895$	−1753.97	−0.0655070
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9228.38	0.342362
$900$	0	0
$901$	10523.8	0.389123
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38008.5	1.39607
$906$	0	0
$907$	−356.000	−0.0130328	−0.00651642	−	0.999979i	$-0.502074\pi$
$907$	−356.000	−0.0130328	−0.00651642	+	0.999979i	$0.502074\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4916.80	0.178815	0.0894077	−	0.995995i	$-0.471503\pi$
$911$	4916.80	0.178815	0.0894077	+	0.995995i	$0.471503\pi$
$912$	0	0
$913$	−7015.89	−0.254318
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−43056.0	−1.54547	−0.772735	−	0.634729i	$-0.781112\pi$
$919$	−43056.0	−1.54547	−0.772735	+	0.634729i	$0.781112\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59984.5	2.13913
$924$	0	0
$925$	−1014.00	−0.0360434
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43210.4	−1.52604	−0.763018	−	0.646377i	$-0.776283\pi$
$929$	−43210.4	−1.52604	−0.763018	+	0.646377i	$0.776283\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	55686.9	1.94776
$936$	0	0
$937$	−52118.0	−1.81710	−0.908549	−	0.417778i	$-0.862809\pi$
$937$	−52118.0	−1.81710	−0.908549	+	0.417778i	$0.862809\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4667.11	−0.161683	−0.0808413	−	0.996727i	$-0.525761\pi$
$941$	−4667.11	−0.161683	−0.0808413	+	0.996727i	$0.525761\pi$
$942$	0	0
$943$	−68404.9	−2.36222
$944$	0	0
$945$	0	0
$946$	0	0
$947$	718.184	0.0246440	0.0123220	−	0.999924i	$-0.496078\pi$
$947$	718.184	0.0246440	0.0123220	+	0.999924i	$0.496078\pi$
$948$	0	0
$949$	55808.0	1.90896
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36461.6	−1.23936	−0.619679	−	0.784855i	$-0.712737\pi$
$953$	−36461.6	−1.23936	−0.619679	+	0.784855i	$0.712737\pi$
$954$	0	0
$955$	22801.6	0.772611
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22815.0	−0.765835
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14583.4	0.486483
$966$	0	0
$967$	−39992.0	−1.32994	−0.664972	−	0.746868i	$-0.731557\pi$
$967$	−39992.0	−1.32994	−0.664972	+	0.746868i	$0.731557\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38183.5	1.26196	0.630982	−	0.775798i	$-0.282652\pi$
$971$	38183.5	1.26196	0.630982	+	0.775798i	$0.282652\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11049.0	−0.361810	−0.180905	−	0.983501i	$-0.557903\pi$
$977$	−11049.0	−0.361810	−0.180905	+	0.983501i	$0.557903\pi$
$978$	0	0
$979$	−48526.5	−1.58418
$980$	0	0
$981$	0	0
$982$	0	0
$983$	46311.2	1.50264	0.751322	−	0.659936i	$-0.229417\pi$
$983$	46311.2	1.50264	0.751322	+	0.659936i	$0.229417\pi$
$984$	0	0
$985$	−10523.8	−0.340423
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24528.7	0.788644
$990$	0	0
$991$	−51064.0	−1.63683	−0.818416	−	0.574626i	$-0.805148\pi$
$991$	−51064.0	−1.63683	−0.818416	+	0.574626i	$0.805148\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−49499.4	−1.57712
$996$	0	0
$997$	14115.3	0.448381	0.224191	−	0.974545i	$-0.428026\pi$
$997$	14115.3	0.448381	0.224191	+	0.974545i	$0.428026\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.a.bb.1.2	yes	4
3.2	odd	2	inner	1764.4.a.bb.1.3	yes	4
7.2	even	3		1764.4.k.bc.361.3		8
7.3	odd	6		1764.4.k.bc.1549.1		8
7.4	even	3		1764.4.k.bc.1549.3		8
7.5	odd	6		1764.4.k.bc.361.1		8
7.6	odd	2	inner	1764.4.a.bb.1.4	yes	4
21.2	odd	6		1764.4.k.bc.361.2		8
21.5	even	6		1764.4.k.bc.361.4		8
21.11	odd	6		1764.4.k.bc.1549.2		8
21.17	even	6		1764.4.k.bc.1549.4		8
21.20	even	2	inner	1764.4.a.bb.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.a.bb.1.1	✓	4	21.20	even	2	inner
1764.4.a.bb.1.2	yes	4	1.1	even	1	trivial
1764.4.a.bb.1.3	yes	4	3.2	odd	2	inner
1764.4.a.bb.1.4	yes	4	7.6	odd	2	inner
1764.4.k.bc.361.1		8	7.5	odd	6
1764.4.k.bc.361.2		8	21.2	odd	6
1764.4.k.bc.361.3		8	7.2	even	3
1764.4.k.bc.361.4		8	21.5	even	6
1764.4.k.bc.1549.1		8	7.3	odd	6
1764.4.k.bc.1549.2		8	21.11	odd	6
1764.4.k.bc.1549.3		8	7.4	even	3
1764.4.k.bc.1549.4		8	21.17	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

\(f(q)\)	\(=\)	\(q-10.5830 q^{5} +55.2449 q^{11} +83.5225 q^{13} -95.2470 q^{17} +83.5225 q^{19} +165.735 q^{23} -13.0000 q^{25} -110.490 q^{29} -83.5225 q^{31} +78.0000 q^{37} -412.737 q^{41} +148.000 q^{43} +465.652 q^{47} -110.490 q^{53} -584.657 q^{55} -550.316 q^{59} -584.657 q^{61} -883.919 q^{65} -260.000 q^{67} +718.184 q^{71} +668.180 q^{73} +664.000 q^{79} -126.996 q^{83} +1008.00 q^{85} -878.389 q^{89} -883.919 q^{95} +1169.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 52 q^{25} + 312 q^{37} + 592 q^{43} - 1040 q^{67} + 2656 q^{79} + 4032 q^{85}+O(q^{100}) \) 4 * q - 52 * q^25 + 312 * q^37 + 592 * q^43 - 1040 * q^67 + 2656 * q^79 + 4032 * q^85

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