LMFDB - Embedded newform 416.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(1,416)] 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("416.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0] 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.58874$ of defining polynomial
Character	$\chi$	$=$	416.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81616 q^{3} +2.70156 q^{5} -3.36131 q^{7} +0.298438 q^{9} +5.17748 q^{11} +1.00000 q^{13} +4.90647 q^{15} +6.70156 q^{17} -5.17748 q^{19} -6.10469 q^{21} +2.29844 q^{25} -4.90647 q^{27} -2.00000 q^{29} -8.80980 q^{31} +9.40312 q^{33} -9.08080 q^{35} +2.70156 q^{37} +1.81616 q^{39} +3.40312 q^{41} +8.53879 q^{43} +0.806248 q^{45} -3.36131 q^{47} +4.29844 q^{49} +12.1711 q^{51} -11.4031 q^{53} +13.9873 q^{55} -9.40312 q^{57} -2.08717 q^{59} -3.40312 q^{61} -1.00314 q^{63} +2.70156 q^{65} -12.4421 q^{67} +10.6260 q^{71} -6.00000 q^{73} +4.17433 q^{75} -17.4031 q^{77} +3.09031 q^{79} -9.80625 q^{81} -1.54515 q^{83} +18.1047 q^{85} -3.63232 q^{87} -6.00000 q^{89} -3.36131 q^{91} -16.0000 q^{93} -13.9873 q^{95} -16.8062 q^{97} +1.54515 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 14 q^{9} + 4 q^{13} + 14 q^{17} + 14 q^{21} + 22 q^{25} - 8 q^{29} + 12 q^{33} - 2 q^{37} - 12 q^{41} - 48 q^{45} + 30 q^{49} - 20 q^{53} - 12 q^{57} + 12 q^{61} - 2 q^{65} - 24 q^{73} - 44 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 14 * q^9 + 4 * q^13 + 14 * q^17 + 14 * q^21 + 22 * q^25 - 8 * q^29 + 12 * q^33 - 2 * q^37 - 12 * q^41 - 48 * q^45 + 30 * q^49 - 20 * q^53 - 12 * q^57 + 12 * q^61 - 2 * q^65 - 24 * q^73 - 44 * q^77 + 12 * q^81 + 34 * q^85 - 24 * q^89 - 64 * q^93 - 16 * q^97

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
$3$	1.81616	1.04856	0.524280	−	0.851546i	$-0.324334\pi$
$3$	1.81616	1.04856	0.524280	+	0.851546i	$0.324334\pi$
$4$	0	0
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	−3.36131	−1.27046	−0.635229	−	0.772324i	$-0.719094\pi$
$7$	−3.36131	−1.27046	−0.635229	+	0.772324i	$0.719094\pi$
$8$	0	0
$9$	0.298438	0.0994793
$10$	0	0
$11$	5.17748	1.56107	0.780534	−	0.625114i	$-0.214947\pi$
$11$	5.17748	1.56107	0.780534	+	0.625114i	$0.214947\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	4.90647	1.26685
$16$	0	0
$17$	6.70156	1.62537	0.812684	−	0.582705i	$-0.198006\pi$
$17$	6.70156	1.62537	0.812684	+	0.582705i	$0.198006\pi$
$18$	0	0
$19$	−5.17748	−1.18779	−0.593897	−	0.804541i	$-0.702411\pi$
$19$	−5.17748	−1.18779	−0.593897	+	0.804541i	$0.702411\pi$
$20$	0	0
$21$	−6.10469	−1.33215
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	−4.90647	−0.944251
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−8.80980	−1.58229	−0.791143	−	0.611631i	$-0.790514\pi$
$31$	−8.80980	−1.58229	−0.791143	+	0.611631i	$0.790514\pi$
$32$	0	0
$33$	9.40312	1.63687
$34$	0	0
$35$	−9.08080	−1.53494
$36$	0	0
$37$	2.70156	0.444134	0.222067	−	0.975031i	$-0.428720\pi$
$37$	2.70156	0.444134	0.222067	+	0.975031i	$0.428720\pi$
$38$	0	0
$39$	1.81616	0.290818
$40$	0	0
$41$	3.40312	0.531479	0.265739	−	0.964045i	$-0.414384\pi$
$41$	3.40312	0.531479	0.265739	+	0.964045i	$0.414384\pi$
$42$	0	0
$43$	8.53879	1.30215	0.651077	−	0.759012i	$-0.274318\pi$
$43$	8.53879	1.30215	0.651077	+	0.759012i	$0.274318\pi$
$44$	0	0
$45$	0.806248	0.120188
$46$	0	0
$47$	−3.36131	−0.490298	−0.245149	−	0.969485i	$-0.578837\pi$
$47$	−3.36131	−0.490298	−0.245149	+	0.969485i	$0.578837\pi$
$48$	0	0
$49$	4.29844	0.614063
$50$	0	0
$51$	12.1711	1.70430
$52$	0	0
$53$	−11.4031	−1.56634	−0.783170	−	0.621808i	$-0.786398\pi$
$53$	−11.4031	−1.56634	−0.783170	+	0.621808i	$0.786398\pi$
$54$	0	0
$55$	13.9873	1.88604
$56$	0	0
$57$	−9.40312	−1.24547
$58$	0	0
$59$	−2.08717	−0.271726	−0.135863	−	0.990728i	$-0.543381\pi$
$59$	−2.08717	−0.271726	−0.135863	+	0.990728i	$0.543381\pi$
$60$	0	0
$61$	−3.40312	−0.435725	−0.217863	−	0.975979i	$-0.569909\pi$
$61$	−3.40312	−0.435725	−0.217863	+	0.975979i	$0.569909\pi$
$62$	0	0
$63$	−1.00314	−0.126384
$64$	0	0
$65$	2.70156	0.335088
$66$	0	0
$67$	−12.4421	−1.52005	−0.760023	−	0.649896i	$-0.774812\pi$
$67$	−12.4421	−1.52005	−0.760023	+	0.649896i	$0.774812\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.6260	1.26107	0.630534	−	0.776161i	$-0.282836\pi$
$71$	10.6260	1.26107	0.630534	+	0.776161i	$0.282836\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	4.17433	0.482010
$76$	0	0
$77$	−17.4031	−1.98327
$78$	0	0
$79$	3.09031	0.347687	0.173843	−	0.984773i	$-0.444381\pi$
$79$	3.09031	0.347687	0.173843	+	0.984773i	$0.444381\pi$
$80$	0	0
$81$	−9.80625	−1.08958
$82$	0	0
$83$	−1.54515	−0.169603	−0.0848014	−	0.996398i	$-0.527026\pi$
$83$	−1.54515	−0.169603	−0.0848014	+	0.996398i	$0.527026\pi$
$84$	0	0
$85$	18.1047	1.96373
$86$	0	0
$87$	−3.63232	−0.389426
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−3.36131	−0.352362
$92$	0	0
$93$	−16.0000	−1.65912
$94$	0	0
$95$	−13.9873	−1.43506
$96$	0	0
$97$	−16.8062	−1.70642	−0.853208	−	0.521571i	$-0.825346\pi$
$97$	−16.8062	−1.70642	−0.853208	+	0.521571i	$0.825346\pi$
$98$	0	0
$99$	1.54515	0.155294
$100$	0	0
$101$	7.40312	0.736638	0.368319	−	0.929699i	$-0.379933\pi$
$101$	7.40312	0.736638	0.368319	+	0.929699i	$0.379933\pi$
$102$	0	0
$103$	7.26464	0.715806	0.357903	−	0.933759i	$-0.383492\pi$
$103$	7.26464	0.715806	0.357903	+	0.933759i	$0.383492\pi$
$104$	0	0
$105$	−16.4922	−1.60947
$106$	0	0
$107$	−3.09031	−0.298751	−0.149376	−	0.988781i	$-0.547726\pi$
$107$	−3.09031	−0.298751	−0.149376	+	0.988781i	$0.547726\pi$
$108$	0	0
$109$	10.7016	1.02502	0.512512	−	0.858680i	$-0.328715\pi$
$109$	10.7016	1.02502	0.512512	+	0.858680i	$0.328715\pi$
$110$	0	0
$111$	4.90647	0.465701
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.298438	0.0275906
$118$	0	0
$119$	−22.5261	−2.06496
$120$	0	0
$121$	15.8062	1.43693
$122$	0	0
$123$	6.18062	0.557288
$124$	0	0
$125$	−7.29844	−0.652792
$126$	0	0
$127$	13.4453	1.19307	0.596537	−	0.802586i	$-0.296543\pi$
$127$	13.4453	1.19307	0.596537	+	0.802586i	$0.296543\pi$
$128$	0	0
$129$	15.5078	1.36539
$130$	0	0
$131$	18.8937	1.65075	0.825377	−	0.564582i	$-0.190963\pi$
$131$	18.8937	1.65075	0.825377	+	0.564582i	$0.190963\pi$
$132$	0	0
$133$	17.4031	1.50904
$134$	0	0
$135$	−13.2551	−1.14082
$136$	0	0
$137$	−18.2094	−1.55573	−0.777866	−	0.628430i	$-0.783698\pi$
$137$	−18.2094	−1.55573	−0.777866	+	0.628430i	$0.783698\pi$
$138$	0	0
$139$	−9.08080	−0.770223	−0.385112	−	0.922870i	$-0.625837\pi$
$139$	−9.08080	−0.770223	−0.385112	+	0.922870i	$0.625837\pi$
$140$	0	0
$141$	−6.10469	−0.514107
$142$	0	0
$143$	5.17748	0.432962
$144$	0	0
$145$	−5.40312	−0.448705
$146$	0	0
$147$	7.80665	0.643882
$148$	0	0
$149$	−12.8062	−1.04913	−0.524564	−	0.851371i	$-0.675772\pi$
$149$	−12.8062	−1.04913	−0.524564	+	0.851371i	$0.675772\pi$
$150$	0	0
$151$	−6.99364	−0.569134	−0.284567	−	0.958656i	$-0.591850\pi$
$151$	−6.99364	−0.569134	−0.284567	+	0.958656i	$0.591850\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−23.8002	−1.91168
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−20.7099	−1.64240
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.9001	−0.932088	−0.466044	−	0.884762i	$-0.654321\pi$
$163$	−11.9001	−0.932088	−0.466044	+	0.884762i	$0.654321\pi$
$164$	0	0
$165$	25.4031	1.97763
$166$	0	0
$167$	22.2551	1.72215	0.861074	−	0.508480i	$-0.169792\pi$
$167$	22.2551	1.72215	0.861074	+	0.508480i	$0.169792\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.54515	−0.118161
$172$	0	0
$173$	−3.40312	−0.258735	−0.129367	−	0.991597i	$-0.541295\pi$
$173$	−3.40312	−0.258735	−0.129367	+	0.991597i	$0.541295\pi$
$174$	0	0
$175$	−7.72577	−0.584014
$176$	0	0
$177$	−3.79063	−0.284921
$178$	0	0
$179$	8.53879	0.638219	0.319110	−	0.947718i	$-0.396616\pi$
$179$	8.53879	0.638219	0.319110	+	0.947718i	$0.396616\pi$
$180$	0	0
$181$	15.4031	1.14491	0.572453	−	0.819938i	$-0.305992\pi$
$181$	15.4031	1.14491	0.572453	+	0.819938i	$0.305992\pi$
$182$	0	0
$183$	−6.18062	−0.456884
$184$	0	0
$185$	7.29844	0.536592
$186$	0	0
$187$	34.6972	2.53731
$188$	0	0
$189$	16.4922	1.19963
$190$	0	0
$191$	13.9873	1.01208	0.506042	−	0.862509i	$-0.331108\pi$
$191$	13.9873	1.01208	0.506042	+	0.862509i	$0.331108\pi$
$192$	0	0
$193$	8.59688	0.618817	0.309408	−	0.950929i	$-0.399869\pi$
$193$	8.59688	0.618817	0.309408	+	0.950929i	$0.399869\pi$
$194$	0	0
$195$	4.90647	0.351360
$196$	0	0
$197$	−22.7016	−1.61742	−0.808710	−	0.588208i	$-0.799834\pi$
$197$	−22.7016	−1.61742	−0.808710	+	0.588208i	$0.799834\pi$
$198$	0	0
$199$	−10.8970	−0.772465	−0.386233	−	0.922401i	$-0.626224\pi$
$199$	−10.8970	−0.772465	−0.386233	+	0.922401i	$0.626224\pi$
$200$	0	0
$201$	−22.5969	−1.59386
$202$	0	0
$203$	6.72263	0.471836
$204$	0	0
$205$	9.19375	0.642119
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−26.8062	−1.85423
$210$	0	0
$211$	12.7131	0.875207	0.437604	−	0.899168i	$-0.355827\pi$
$211$	12.7131	0.875207	0.437604	+	0.899168i	$0.355827\pi$
$212$	0	0
$213$	19.2984	1.32231
$214$	0	0
$215$	23.0681	1.57323
$216$	0	0
$217$	29.6125	2.01023
$218$	0	0
$219$	−10.8970	−0.736348
$220$	0	0
$221$	6.70156	0.450796
$222$	0	0
$223$	17.8906	1.19804	0.599021	−	0.800733i	$-0.295556\pi$
$223$	17.8906	1.19804	0.599021	+	0.800733i	$0.295556\pi$
$224$	0	0
$225$	0.685941	0.0457294
$226$	0	0
$227$	−8.80980	−0.584727	−0.292363	−	0.956307i	$-0.594442\pi$
$227$	−8.80980	−0.584727	−0.292363	+	0.956307i	$0.594442\pi$
$228$	0	0
$229$	28.1047	1.85721	0.928605	−	0.371070i	$-0.121009\pi$
$229$	28.1047	1.85721	0.928605	+	0.371070i	$0.121009\pi$
$230$	0	0
$231$	−31.6069	−2.07958
$232$	0	0
$233$	3.89531	0.255190	0.127595	−	0.991826i	$-0.459274\pi$
$233$	3.89531	0.255190	0.127595	+	0.991826i	$0.459274\pi$
$234$	0	0
$235$	−9.08080	−0.592366
$236$	0	0
$237$	5.61250	0.364571
$238$	0	0
$239$	14.2583	0.922291	0.461146	−	0.887324i	$-0.347439\pi$
$239$	14.2583	0.922291	0.461146	+	0.887324i	$0.347439\pi$
$240$	0	0
$241$	20.8062	1.34025	0.670124	−	0.742249i	$-0.266241\pi$
$241$	20.8062	1.34025	0.670124	+	0.742249i	$0.266241\pi$
$242$	0	0
$243$	−3.09031	−0.198243
$244$	0	0
$245$	11.6125	0.741895
$246$	0	0
$247$	−5.17748	−0.329435
$248$	0	0
$249$	−2.80625	−0.177839
$250$	0	0
$251$	−13.4453	−0.848657	−0.424329	−	0.905508i	$-0.639490\pi$
$251$	−13.4453	−0.848657	−0.424329	+	0.905508i	$0.639490\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	32.8810	2.05909
$256$	0	0
$257$	17.5078	1.09211	0.546054	−	0.837750i	$-0.316129\pi$
$257$	17.5078	1.09211	0.546054	+	0.837750i	$0.316129\pi$
$258$	0	0
$259$	−9.08080	−0.564254
$260$	0	0
$261$	−0.596876	−0.0369457
$262$	0	0
$263$	10.3550	0.638514	0.319257	−	0.947668i	$-0.396567\pi$
$263$	10.3550	0.638514	0.319257	+	0.947668i	$0.396567\pi$
$264$	0	0
$265$	−30.8062	−1.89241
$266$	0	0
$267$	−10.8970	−0.666883
$268$	0	0
$269$	26.2094	1.59801	0.799007	−	0.601322i	$-0.205359\pi$
$269$	26.2094	1.59801	0.799007	+	0.601322i	$0.205359\pi$
$270$	0	0
$271$	−3.36131	−0.204185	−0.102093	−	0.994775i	$-0.532554\pi$
$271$	−3.36131	−0.204185	−0.102093	+	0.994775i	$0.532554\pi$
$272$	0	0
$273$	−6.10469	−0.369472
$274$	0	0
$275$	11.9001	0.717603
$276$	0	0
$277$	−14.2094	−0.853758	−0.426879	−	0.904309i	$-0.640387\pi$
$277$	−14.2094	−0.853758	−0.426879	+	0.904309i	$0.640387\pi$
$278$	0	0
$279$	−2.62918	−0.157405
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	−25.4031	−1.50475
$286$	0	0
$287$	−11.4390	−0.675221
$288$	0	0
$289$	27.9109	1.64182
$290$	0	0
$291$	−30.5228	−1.78928
$292$	0	0
$293$	−0.104686	−0.00611584	−0.00305792	−	0.999995i	$-0.500973\pi$
$293$	−0.104686	−0.00611584	−0.00305792	+	0.999995i	$0.500973\pi$
$294$	0	0
$295$	−5.63861	−0.328292
$296$	0	0
$297$	−25.4031	−1.47404
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−28.7016	−1.65433
$302$	0	0
$303$	13.4453	0.772410
$304$	0	0
$305$	−9.19375	−0.526433
$306$	0	0
$307$	−8.26778	−0.471867	−0.235934	−	0.971769i	$-0.575815\pi$
$307$	−8.26778	−0.471867	−0.235934	+	0.971769i	$0.575815\pi$
$308$	0	0
$309$	13.1938	0.750566
$310$	0	0
$311$	−10.8970	−0.617910	−0.308955	−	0.951077i	$-0.599979\pi$
$311$	−10.8970	−0.617910	−0.308955	+	0.951077i	$0.599979\pi$
$312$	0	0
$313$	−29.5078	−1.66788	−0.833940	−	0.551855i	$-0.813920\pi$
$313$	−29.5078	−1.66788	−0.833940	+	0.551855i	$0.813920\pi$
$314$	0	0
$315$	−2.71006	−0.152694
$316$	0	0
$317$	0.806248	0.0452834	0.0226417	−	0.999744i	$-0.492792\pi$
$317$	0.806248	0.0452834	0.0226417	+	0.999744i	$0.492792\pi$
$318$	0	0
$319$	−10.3550	−0.579766
$320$	0	0
$321$	−5.61250	−0.313259
$322$	0	0
$323$	−34.6972	−1.93060
$324$	0	0
$325$	2.29844	0.127494
$326$	0	0
$327$	19.4358	1.07480
$328$	0	0
$329$	11.2984	0.622903
$330$	0	0
$331$	−14.9904	−0.823948	−0.411974	−	0.911196i	$-0.635160\pi$
$331$	−14.9904	−0.823948	−0.411974	+	0.911196i	$0.635160\pi$
$332$	0	0
$333$	0.806248	0.0441821
$334$	0	0
$335$	−33.6131	−1.83648
$336$	0	0
$337$	16.1047	0.877278	0.438639	−	0.898663i	$-0.355461\pi$
$337$	16.1047	0.877278	0.438639	+	0.898663i	$0.355461\pi$
$338$	0	0
$339$	18.1616	0.986403
$340$	0	0
$341$	−45.6125	−2.47006
$342$	0	0
$343$	9.08080	0.490317
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1584	1.40426	0.702128	−	0.712051i	$-0.252234\pi$
$347$	26.1584	1.40426	0.702128	+	0.712051i	$0.252234\pi$
$348$	0	0
$349$	−10.9109	−0.584049	−0.292024	−	0.956411i	$-0.594329\pi$
$349$	−10.9109	−0.584049	−0.292024	+	0.956411i	$0.594329\pi$
$350$	0	0
$351$	−4.90647	−0.261888
$352$	0	0
$353$	27.4031	1.45852	0.729261	−	0.684236i	$-0.239864\pi$
$353$	27.4031	1.45852	0.729261	+	0.684236i	$0.239864\pi$
$354$	0	0
$355$	28.7067	1.52359
$356$	0	0
$357$	−40.9109	−2.16524
$358$	0	0
$359$	−19.1647	−1.01148	−0.505738	−	0.862687i	$-0.668780\pi$
$359$	−19.1647	−1.01148	−0.505738	+	0.862687i	$0.668780\pi$
$360$	0	0
$361$	7.80625	0.410855
$362$	0	0
$363$	28.7067	1.50671
$364$	0	0
$365$	−16.2094	−0.848437
$366$	0	0
$367$	−28.5166	−1.48855	−0.744276	−	0.667872i	$-0.767205\pi$
$367$	−28.5166	−1.48855	−0.744276	+	0.667872i	$0.767205\pi$
$368$	0	0
$369$	1.01562	0.0528711
$370$	0	0
$371$	38.3295	1.98997
$372$	0	0
$373$	−22.2094	−1.14996	−0.574979	−	0.818168i	$-0.694990\pi$
$373$	−22.2094	−1.14996	−0.574979	+	0.818168i	$0.694990\pi$
$374$	0	0
$375$	−13.2551	−0.684492
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	33.6940	1.73075	0.865373	−	0.501128i	$-0.167082\pi$
$379$	33.6940	1.73075	0.865373	+	0.501128i	$0.167082\pi$
$380$	0	0
$381$	24.4187	1.25101
$382$	0	0
$383$	24.0712	1.22998	0.614991	−	0.788534i	$-0.289160\pi$
$383$	24.0712	1.22998	0.614991	+	0.788534i	$0.289160\pi$
$384$	0	0
$385$	−47.0156	−2.39614
$386$	0	0
$387$	2.54830	0.129537
$388$	0	0
$389$	−7.19375	−0.364738	−0.182369	−	0.983230i	$-0.558376\pi$
$389$	−7.19375	−0.364738	−0.182369	+	0.983230i	$0.558376\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	34.3141	1.73092
$394$	0	0
$395$	8.34866	0.420067
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	31.6069	1.58232
$400$	0	0
$401$	14.2094	0.709582	0.354791	−	0.934946i	$-0.384552\pi$
$401$	14.2094	0.709582	0.354791	+	0.934946i	$0.384552\pi$
$402$	0	0
$403$	−8.80980	−0.438847
$404$	0	0
$405$	−26.4922	−1.31641
$406$	0	0
$407$	13.9873	0.693323
$408$	0	0
$409$	3.40312	0.168274	0.0841368	−	0.996454i	$-0.473187\pi$
$409$	3.40312	0.168274	0.0841368	+	0.996454i	$0.473187\pi$
$410$	0	0
$411$	−33.0711	−1.63128
$412$	0	0
$413$	7.01562	0.345216
$414$	0	0
$415$	−4.17433	−0.204910
$416$	0	0
$417$	−16.4922	−0.807626
$418$	0	0
$419$	−4.36446	−0.213218	−0.106609	−	0.994301i	$-0.533999\pi$
$419$	−4.36446	−0.213218	−0.106609	+	0.994301i	$0.533999\pi$
$420$	0	0
$421$	−0.104686	−0.00510210	−0.00255105	−	0.999997i	$-0.500812\pi$
$421$	−0.104686	−0.00510210	−0.00255105	+	0.999997i	$0.500812\pi$
$422$	0	0
$423$	−1.00314	−0.0487745
$424$	0	0
$425$	15.4031	0.747161
$426$	0	0
$427$	11.4390	0.553571
$428$	0	0
$429$	9.40312	0.453987
$430$	0	0
$431$	−6.45162	−0.310764	−0.155382	−	0.987854i	$-0.549661\pi$
$431$	−6.45162	−0.310764	−0.155382	+	0.987854i	$0.549661\pi$
$432$	0	0
$433$	24.1047	1.15840	0.579199	−	0.815186i	$-0.303366\pi$
$433$	24.1047	1.15840	0.579199	+	0.815186i	$0.303366\pi$
$434$	0	0
$435$	−9.81294	−0.470494
$436$	0	0
$437$	0	0
$438$	0	0
$439$	30.5228	1.45678	0.728388	−	0.685165i	$-0.240270\pi$
$439$	30.5228	1.45678	0.728388	+	0.685165i	$0.240270\pi$
$440$	0	0
$441$	1.28282	0.0610865
$442$	0	0
$443$	−39.6036	−1.88163	−0.940813	−	0.338926i	$-0.889936\pi$
$443$	−39.6036	−1.88163	−0.940813	+	0.338926i	$0.889936\pi$
$444$	0	0
$445$	−16.2094	−0.768398
$446$	0	0
$447$	−23.2582	−1.10008
$448$	0	0
$449$	−0.806248	−0.0380492	−0.0190246	−	0.999819i	$-0.506056\pi$
$449$	−0.806248	−0.0380492	−0.0190246	+	0.999819i	$0.506056\pi$
$450$	0	0
$451$	17.6196	0.829674
$452$	0	0
$453$	−12.7016	−0.596771
$454$	0	0
$455$	−9.08080	−0.425715
$456$	0	0
$457$	−24.8062	−1.16039	−0.580194	−	0.814479i	$-0.697023\pi$
$457$	−24.8062	−1.16039	−0.580194	+	0.814479i	$0.697023\pi$
$458$	0	0
$459$	−32.8810	−1.53475
$460$	0	0
$461$	−5.29844	−0.246773	−0.123386	−	0.992359i	$-0.539375\pi$
$461$	−5.29844	−0.246773	−0.123386	+	0.992359i	$0.539375\pi$
$462$	0	0
$463$	−12.9841	−0.603424	−0.301712	−	0.953399i	$-0.597558\pi$
$463$	−12.9841	−0.603424	−0.301712	+	0.953399i	$0.597558\pi$
$464$	0	0
$465$	−43.2250	−2.00451
$466$	0	0
$467$	6.18062	0.286005	0.143002	−	0.989722i	$-0.454324\pi$
$467$	6.18062	0.286005	0.143002	+	0.989722i	$0.454324\pi$
$468$	0	0
$469$	41.8219	1.93115
$470$	0	0
$471$	−3.63232	−0.167368
$472$	0	0
$473$	44.2094	2.03275
$474$	0	0
$475$	−11.9001	−0.546014
$476$	0	0
$477$	−3.40312	−0.155818
$478$	0	0
$479$	2.81930	0.128817	0.0644086	−	0.997924i	$-0.479484\pi$
$479$	2.81930	0.128817	0.0644086	+	0.997924i	$0.479484\pi$
$480$	0	0
$481$	2.70156	0.123181
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−45.4031	−2.06165
$486$	0	0
$487$	19.1647	0.868438	0.434219	−	0.900807i	$-0.357024\pi$
$487$	19.1647	0.868438	0.434219	+	0.900807i	$0.357024\pi$
$488$	0	0
$489$	−21.6125	−0.977351
$490$	0	0
$491$	14.7194	0.664278	0.332139	−	0.943231i	$-0.392230\pi$
$491$	14.7194	0.664278	0.332139	+	0.943231i	$0.392230\pi$
$492$	0	0
$493$	−13.4031	−0.603646
$494$	0	0
$495$	4.17433	0.187622
$496$	0	0
$497$	−35.7172	−1.60213
$498$	0	0
$499$	18.0807	0.809404	0.404702	−	0.914449i	$-0.367375\pi$
$499$	18.0807	0.809404	0.404702	+	0.914449i	$0.367375\pi$
$500$	0	0
$501$	40.4187	1.80578
$502$	0	0
$503$	6.72263	0.299747	0.149874	−	0.988705i	$-0.452113\pi$
$503$	6.72263	0.299747	0.149874	+	0.988705i	$0.452113\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	1.81616	0.0806585
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	20.1679	0.892175
$512$	0	0
$513$	25.4031	1.12158
$514$	0	0
$515$	19.6259	0.864820
$516$	0	0
$517$	−17.4031	−0.765389
$518$	0	0
$519$	−6.18062	−0.271299
$520$	0	0
$521$	−13.5078	−0.591788	−0.295894	−	0.955221i	$-0.595618\pi$
$521$	−13.5078	−0.591788	−0.295894	+	0.955221i	$0.595618\pi$
$522$	0	0
$523$	−23.8002	−1.04071	−0.520355	−	0.853950i	$-0.674200\pi$
$523$	−23.8002	−1.04071	−0.520355	+	0.853950i	$0.674200\pi$
$524$	0	0
$525$	−14.0312	−0.612374
$526$	0	0
$527$	−59.0394	−2.57180
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−0.622889	−0.0270311
$532$	0	0
$533$	3.40312	0.147406
$534$	0	0
$535$	−8.34866	−0.360944
$536$	0	0
$537$	15.5078	0.669211
$538$	0	0
$539$	22.2551	0.958593
$540$	0	0
$541$	−14.7016	−0.632070	−0.316035	−	0.948748i	$-0.602352\pi$
$541$	−14.7016	−0.632070	−0.316035	+	0.948748i	$0.602352\pi$
$542$	0	0
$543$	27.9745	1.20050
$544$	0	0
$545$	28.9109	1.23841
$546$	0	0
$547$	−12.7131	−0.543574	−0.271787	−	0.962357i	$-0.587615\pi$
$547$	−12.7131	−0.543574	−0.271787	+	0.962357i	$0.587615\pi$
$548$	0	0
$549$	−1.01562	−0.0433457
$550$	0	0
$551$	10.3550	0.441136
$552$	0	0
$553$	−10.3875	−0.441722
$554$	0	0
$555$	13.2551	0.562649
$556$	0	0
$557$	1.29844	0.0550166	0.0275083	−	0.999622i	$-0.491243\pi$
$557$	1.29844	0.0550166	0.0275083	+	0.999622i	$0.491243\pi$
$558$	0	0
$559$	8.53879	0.361152
$560$	0	0
$561$	63.0156	2.66052
$562$	0	0
$563$	−40.1457	−1.69194	−0.845969	−	0.533232i	$-0.820977\pi$
$563$	−40.1457	−1.69194	−0.845969	+	0.533232i	$0.820977\pi$
$564$	0	0
$565$	27.0156	1.13656
$566$	0	0
$567$	32.9619	1.38427
$568$	0	0
$569$	29.2984	1.22825	0.614127	−	0.789207i	$-0.289508\pi$
$569$	29.2984	1.22825	0.614127	+	0.789207i	$0.289508\pi$
$570$	0	0
$571$	−8.53879	−0.357337	−0.178669	−	0.983909i	$-0.557179\pi$
$571$	−8.53879	−0.357337	−0.178669	+	0.983909i	$0.557179\pi$
$572$	0	0
$573$	25.4031	1.06123
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−32.8062	−1.36574	−0.682871	−	0.730539i	$-0.739269\pi$
$577$	−32.8062	−1.36574	−0.682871	+	0.730539i	$0.739269\pi$
$578$	0	0
$579$	15.6133	0.648867
$580$	0	0
$581$	5.19375	0.215473
$582$	0	0
$583$	−59.0394	−2.44516
$584$	0	0
$585$	0.806248	0.0333343
$586$	0	0
$587$	11.9001	0.491170	0.245585	−	0.969375i	$-0.421020\pi$
$587$	11.9001	0.491170	0.245585	+	0.969375i	$0.421020\pi$
$588$	0	0
$589$	45.6125	1.87943
$590$	0	0
$591$	−41.2297	−1.69596
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−60.8556	−2.49483
$596$	0	0
$597$	−19.7906	−0.809976
$598$	0	0
$599$	7.80665	0.318971	0.159486	−	0.987200i	$-0.449016\pi$
$599$	7.80665	0.318971	0.159486	+	0.987200i	$0.449016\pi$
$600$	0	0
$601$	30.7016	1.25234	0.626171	−	0.779685i	$-0.284621\pi$
$601$	30.7016	1.25234	0.626171	+	0.779685i	$0.284621\pi$
$602$	0	0
$603$	−3.71320	−0.151213
$604$	0	0
$605$	42.7016	1.73607
$606$	0	0
$607$	−41.9618	−1.70318	−0.851589	−	0.524211i	$-0.824360\pi$
$607$	−41.9618	−1.70318	−0.851589	+	0.524211i	$0.824360\pi$
$608$	0	0
$609$	12.2094	0.494749
$610$	0	0
$611$	−3.36131	−0.135984
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	16.6973	0.673301
$616$	0	0
$617$	19.4031	0.781140	0.390570	−	0.920573i	$-0.372278\pi$
$617$	19.4031	0.781140	0.390570	+	0.920573i	$0.372278\pi$
$618$	0	0
$619$	7.72577	0.310525	0.155263	−	0.987873i	$-0.450378\pi$
$619$	7.72577	0.310525	0.155263	+	0.987873i	$0.450378\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	20.1679	0.808009
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	−48.6844	−1.94427
$628$	0	0
$629$	18.1047	0.721881
$630$	0	0
$631$	41.6908	1.65968	0.829842	−	0.557998i	$-0.188430\pi$
$631$	41.6908	1.65968	0.829842	+	0.557998i	$0.188430\pi$
$632$	0	0
$633$	23.0891	0.917708
$634$	0	0
$635$	36.3232	1.44144
$636$	0	0
$637$	4.29844	0.170310
$638$	0	0
$639$	3.17119	0.125450
$640$	0	0
$641$	7.19375	0.284136	0.142068	−	0.989857i	$-0.454625\pi$
$641$	7.19375	0.284136	0.142068	+	0.989857i	$0.454625\pi$
$642$	0	0
$643$	28.9777	1.14277	0.571384	−	0.820683i	$-0.306407\pi$
$643$	28.9777	1.14277	0.571384	+	0.820683i	$0.306407\pi$
$644$	0	0
$645$	41.8953	1.64963
$646$	0	0
$647$	−10.8970	−0.428404	−0.214202	−	0.976789i	$-0.568715\pi$
$647$	−10.8970	−0.428404	−0.214202	+	0.976789i	$0.568715\pi$
$648$	0	0
$649$	−10.8062	−0.424182
$650$	0	0
$651$	53.7810	2.10785
$652$	0	0
$653$	−17.0156	−0.665873	−0.332936	−	0.942949i	$-0.608039\pi$
$653$	−17.0156	−0.665873	−0.332936	+	0.942949i	$0.608039\pi$
$654$	0	0
$655$	51.0426	1.99440
$656$	0	0
$657$	−1.79063	−0.0698590
$658$	0	0
$659$	−23.8002	−0.927125	−0.463562	−	0.886064i	$-0.653429\pi$
$659$	−23.8002	−0.927125	−0.463562	+	0.886064i	$0.653429\pi$
$660$	0	0
$661$	3.19375	0.124223	0.0621113	−	0.998069i	$-0.480217\pi$
$661$	3.19375	0.124223	0.0621113	+	0.998069i	$0.480217\pi$
$662$	0	0
$663$	12.1711	0.472687
$664$	0	0
$665$	47.0156	1.82319
$666$	0	0
$667$	0	0
$668$	0	0
$669$	32.4922	1.25622
$670$	0	0
$671$	−17.6196	−0.680197
$672$	0	0
$673$	28.3141	1.09143	0.545713	−	0.837972i	$-0.316259\pi$
$673$	28.3141	1.09143	0.545713	+	0.837972i	$0.316259\pi$
$674$	0	0
$675$	−11.2772	−0.434060
$676$	0	0
$677$	4.59688	0.176672	0.0883361	−	0.996091i	$-0.471845\pi$
$677$	4.59688	0.176672	0.0883361	+	0.996091i	$0.471845\pi$
$678$	0	0
$679$	56.4911	2.16793
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	0.461133	0.0176448	0.00882238	−	0.999961i	$-0.497192\pi$
$683$	0.461133	0.0176448	0.00882238	+	0.999961i	$0.497192\pi$
$684$	0	0
$685$	−49.1938	−1.87960
$686$	0	0
$687$	51.0426	1.94740
$688$	0	0
$689$	−11.4031	−0.434424
$690$	0	0
$691$	−35.7003	−1.35810	−0.679052	−	0.734090i	$-0.737609\pi$
$691$	−35.7003	−1.35810	−0.679052	+	0.734090i	$0.737609\pi$
$692$	0	0
$693$	−5.19375	−0.197294
$694$	0	0
$695$	−24.5323	−0.930565
$696$	0	0
$697$	22.8062	0.863848
$698$	0	0
$699$	7.07451	0.267583
$700$	0	0
$701$	4.59688	0.173622	0.0868108	−	0.996225i	$-0.472332\pi$
$701$	4.59688	0.173622	0.0868108	+	0.996225i	$0.472332\pi$
$702$	0	0
$703$	−13.9873	−0.527540
$704$	0	0
$705$	−16.4922	−0.621132
$706$	0	0
$707$	−24.8842	−0.935868
$708$	0	0
$709$	−28.8062	−1.08184	−0.540921	−	0.841074i	$-0.681924\pi$
$709$	−28.8062	−1.08184	−0.540921	+	0.841074i	$0.681924\pi$
$710$	0	0
$711$	0.922265	0.0345877
$712$	0	0
$713$	0	0
$714$	0	0
$715$	13.9873	0.523094
$716$	0	0
$717$	25.8953	0.967078
$718$	0	0
$719$	−18.1616	−0.677313	−0.338657	−	0.940910i	$-0.609973\pi$
$719$	−18.1616	−0.677313	−0.338657	+	0.940910i	$0.609973\pi$
$720$	0	0
$721$	−24.4187	−0.909402
$722$	0	0
$723$	37.7875	1.40533
$724$	0	0
$725$	−4.59688	−0.170724
$726$	0	0
$727$	−45.0521	−1.67089	−0.835445	−	0.549574i	$-0.814790\pi$
$727$	−45.0521	−1.67089	−0.835445	+	0.549574i	$0.814790\pi$
$728$	0	0
$729$	23.8062	0.881713
$730$	0	0
$731$	57.2232	2.11648
$732$	0	0
$733$	6.91093	0.255261	0.127631	−	0.991822i	$-0.459263\pi$
$733$	6.91093	0.255261	0.127631	+	0.991822i	$0.459263\pi$
$734$	0	0
$735$	21.0902	0.777922
$736$	0	0
$737$	−64.4187	−2.37289
$738$	0	0
$739$	16.0744	0.591308	0.295654	−	0.955295i	$-0.404463\pi$
$739$	16.0744	0.591308	0.295654	+	0.955295i	$0.404463\pi$
$740$	0	0
$741$	−9.40312	−0.345432
$742$	0	0
$743$	6.99364	0.256572	0.128286	−	0.991737i	$-0.459053\pi$
$743$	6.99364	0.256572	0.128286	+	0.991737i	$0.459053\pi$
$744$	0	0
$745$	−34.5969	−1.26753
$746$	0	0
$747$	−0.461133	−0.0168720
$748$	0	0
$749$	10.3875	0.379551
$750$	0	0
$751$	29.0586	1.06036	0.530181	−	0.847884i	$-0.322124\pi$
$751$	29.0586	1.06036	0.530181	+	0.847884i	$0.322124\pi$
$752$	0	0
$753$	−24.4187	−0.889869
$754$	0	0
$755$	−18.8937	−0.687614
$756$	0	0
$757$	26.2094	0.952596	0.476298	−	0.879284i	$-0.341978\pi$
$757$	26.2094	0.952596	0.476298	+	0.879284i	$0.341978\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.6125	−1.00095	−0.500476	−	0.865750i	$-0.666842\pi$
$761$	−27.6125	−1.00095	−0.500476	+	0.865750i	$0.666842\pi$
$762$	0	0
$763$	−35.9713	−1.30225
$764$	0	0
$765$	5.40312	0.195350
$766$	0	0
$767$	−2.08717	−0.0753632
$768$	0	0
$769$	−26.2094	−0.945134	−0.472567	−	0.881295i	$-0.656673\pi$
$769$	−26.2094	−0.945134	−0.472567	+	0.881295i	$0.656673\pi$
$770$	0	0
$771$	31.7970	1.14514
$772$	0	0
$773$	−16.1047	−0.579245	−0.289623	−	0.957141i	$-0.593530\pi$
$773$	−16.1047	−0.579245	−0.289623	+	0.957141i	$0.593530\pi$
$774$	0	0
$775$	−20.2488	−0.727357
$776$	0	0
$777$	−16.4922	−0.591654
$778$	0	0
$779$	−17.6196	−0.631287
$780$	0	0
$781$	55.0156	1.96861
$782$	0	0
$783$	9.81294	0.350686
$784$	0	0
$785$	−5.40312	−0.192846
$786$	0	0
$787$	29.5197	1.05226	0.526132	−	0.850403i	$-0.323642\pi$
$787$	29.5197	1.05226	0.526132	+	0.850403i	$0.323642\pi$
$788$	0	0
$789$	18.8062	0.669520
$790$	0	0
$791$	−33.6131	−1.19515
$792$	0	0
$793$	−3.40312	−0.120848
$794$	0	0
$795$	−55.9491	−1.98431
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−22.5261	−0.796915
$800$	0	0
$801$	−1.79063	−0.0632687
$802$	0	0
$803$	−31.0649	−1.09625
$804$	0	0
$805$	0	0
$806$	0	0
$807$	47.6004	1.67561
$808$	0	0
$809$	−7.89531	−0.277584	−0.138792	−	0.990322i	$-0.544322\pi$
$809$	−7.89531	−0.277584	−0.138792	+	0.990322i	$0.544322\pi$
$810$	0	0
$811$	−42.9650	−1.50870	−0.754352	−	0.656470i	$-0.772049\pi$
$811$	−42.9650	−1.50870	−0.754352	+	0.656470i	$0.772049\pi$
$812$	0	0
$813$	−6.10469	−0.214101
$814$	0	0
$815$	−32.1489	−1.12613
$816$	0	0
$817$	−44.2094	−1.54669
$818$	0	0
$819$	−1.00314	−0.0350527
$820$	0	0
$821$	−18.9109	−0.659996	−0.329998	−	0.943982i	$-0.607048\pi$
$821$	−18.9109	−0.659996	−0.329998	+	0.943982i	$0.607048\pi$
$822$	0	0
$823$	−51.2327	−1.78586	−0.892931	−	0.450194i	$-0.851355\pi$
$823$	−51.2327	−1.78586	−0.892931	+	0.450194i	$0.851355\pi$
$824$	0	0
$825$	21.6125	0.752451
$826$	0	0
$827$	12.4421	0.432655	0.216327	−	0.976321i	$-0.430592\pi$
$827$	12.4421	0.432655	0.216327	+	0.976321i	$0.430592\pi$
$828$	0	0
$829$	0.387503	0.0134585	0.00672927	−	0.999977i	$-0.497858\pi$
$829$	0.387503	0.0134585	0.00672927	+	0.999977i	$0.497858\pi$
$830$	0	0
$831$	−25.8065	−0.895217
$832$	0	0
$833$	28.8062	0.998077
$834$	0	0
$835$	60.1234	2.08066
$836$	0	0
$837$	43.2250	1.49407
$838$	0	0
$839$	−12.9841	−0.448262	−0.224131	−	0.974559i	$-0.571954\pi$
$839$	−12.9841	−0.448262	−0.224131	+	0.974559i	$0.571954\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	18.1616	0.625519
$844$	0	0
$845$	2.70156	0.0929366
$846$	0	0
$847$	−53.1298	−1.82556
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−3.89531	−0.133373	−0.0666865	−	0.997774i	$-0.521243\pi$
$853$	−3.89531	−0.133373	−0.0666865	+	0.997774i	$0.521243\pi$
$854$	0	0
$855$	−4.17433	−0.142759
$856$	0	0
$857$	12.8062	0.437453	0.218727	−	0.975786i	$-0.429810\pi$
$857$	12.8062	0.437453	0.218727	+	0.975786i	$0.429810\pi$
$858$	0	0
$859$	3.09031	0.105440	0.0527200	−	0.998609i	$-0.483211\pi$
$859$	3.09031	0.105440	0.0527200	+	0.998609i	$0.483211\pi$
$860$	0	0
$861$	−20.7750	−0.708010
$862$	0	0
$863$	13.1743	0.448457	0.224228	−	0.974537i	$-0.428014\pi$
$863$	13.1743	0.448457	0.224228	+	0.974537i	$0.428014\pi$
$864$	0	0
$865$	−9.19375	−0.312597
$866$	0	0
$867$	50.6907	1.72155
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	−12.4421	−0.421585
$872$	0	0
$873$	−5.01562	−0.169753
$874$	0	0
$875$	24.5323	0.829345
$876$	0	0
$877$	25.7172	0.868408	0.434204	−	0.900815i	$-0.357030\pi$
$877$	25.7172	0.868408	0.434204	+	0.900815i	$0.357030\pi$
$878$	0	0
$879$	−0.190127	−0.00641283
$880$	0	0
$881$	−49.2984	−1.66091	−0.830453	−	0.557088i	$-0.811918\pi$
$881$	−49.2984	−1.66091	−0.830453	+	0.557088i	$0.811918\pi$
$882$	0	0
$883$	11.0871	0.373110	0.186555	−	0.982444i	$-0.440268\pi$
$883$	11.0871	0.373110	0.186555	+	0.982444i	$0.440268\pi$
$884$	0	0
$885$	−10.2406	−0.344235
$886$	0	0
$887$	−23.8002	−0.799133	−0.399566	−	0.916704i	$-0.630839\pi$
$887$	−23.8002	−0.799133	−0.399566	+	0.916704i	$0.630839\pi$
$888$	0	0
$889$	−45.1938	−1.51575
$890$	0	0
$891$	−50.7716	−1.70091
$892$	0	0
$893$	17.4031	0.582373
$894$	0	0
$895$	23.0681	0.771081
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.6196	0.587646
$900$	0	0
$901$	−76.4187	−2.54588
$902$	0	0
$903$	−52.1266	−1.73467
$904$	0	0
$905$	41.6125	1.38325
$906$	0	0
$907$	1.27415	0.0423074	0.0211537	−	0.999776i	$-0.493266\pi$
$907$	1.27415	0.0423074	0.0211537	+	0.999776i	$0.493266\pi$
$908$	0	0
$909$	2.20937	0.0732803
$910$	0	0
$911$	20.1679	0.668192	0.334096	−	0.942539i	$-0.391569\pi$
$911$	20.1679	0.668192	0.334096	+	0.942539i	$0.391569\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−16.6973	−0.551997
$916$	0	0
$917$	−63.5078	−2.09721
$918$	0	0
$919$	3.09031	0.101940	0.0509700	−	0.998700i	$-0.483769\pi$
$919$	3.09031	0.101940	0.0509700	+	0.998700i	$0.483769\pi$
$920$	0	0
$921$	−15.0156	−0.494782
$922$	0	0
$923$	10.6260	0.349758
$924$	0	0
$925$	6.20937	0.204163
$926$	0	0
$927$	2.16804	0.0712079
$928$	0	0
$929$	−7.40312	−0.242889	−0.121444	−	0.992598i	$-0.538753\pi$
$929$	−7.40312	−0.242889	−0.121444	+	0.992598i	$0.538753\pi$
$930$	0	0
$931$	−22.2551	−0.729380
$932$	0	0
$933$	−19.7906	−0.647916
$934$	0	0
$935$	93.7366	3.06551
$936$	0	0
$937$	7.19375	0.235010	0.117505	−	0.993072i	$-0.462510\pi$
$937$	7.19375	0.235010	0.117505	+	0.993072i	$0.462510\pi$
$938$	0	0
$939$	−53.5909	−1.74887
$940$	0	0
$941$	36.1047	1.17698	0.588490	−	0.808505i	$-0.299723\pi$
$941$	36.1047	1.17698	0.588490	+	0.808505i	$0.299723\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	44.5547	1.44936
$946$	0	0
$947$	40.4167	1.31336	0.656682	−	0.754167i	$-0.271959\pi$
$947$	40.4167	1.31336	0.656682	+	0.754167i	$0.271959\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	1.46428	0.0474824
$952$	0	0
$953$	−8.31406	−0.269319	−0.134659	−	0.990892i	$-0.542994\pi$
$953$	−8.31406	−0.269319	−0.134659	+	0.990892i	$0.542994\pi$
$954$	0	0
$955$	37.7875	1.22277
$956$	0	0
$957$	−18.8062	−0.607920
$958$	0	0
$959$	61.2074	1.97649
$960$	0	0
$961$	46.6125	1.50363
$962$	0	0
$963$	−0.922265	−0.0297196
$964$	0	0
$965$	23.2250	0.747639
$966$	0	0
$967$	−17.3486	−0.557893	−0.278946	−	0.960307i	$-0.589985\pi$
$967$	−17.3486	−0.557893	−0.278946	+	0.960307i	$0.589985\pi$
$968$	0	0
$969$	−63.0156	−2.02435
$970$	0	0
$971$	−33.4230	−1.07260	−0.536298	−	0.844029i	$-0.680178\pi$
$971$	−33.4230	−1.07260	−0.536298	+	0.844029i	$0.680178\pi$
$972$	0	0
$973$	30.5234	0.978536
$974$	0	0
$975$	4.17433	0.133686
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−31.0649	−0.992837
$980$	0	0
$981$	3.19375	0.101969
$982$	0	0
$983$	−11.7100	−0.373490	−0.186745	−	0.982408i	$-0.559794\pi$
$983$	−11.7100	−0.373490	−0.186745	+	0.982408i	$0.559794\pi$
$984$	0	0
$985$	−61.3297	−1.95413
$986$	0	0
$987$	20.5198	0.653152
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0.542011	0.0172175	0.00860877	−	0.999963i	$-0.497260\pi$
$991$	0.542011	0.0172175	0.00860877	+	0.999963i	$0.497260\pi$
$992$	0	0
$993$	−27.2250	−0.863959
$994$	0	0
$995$	−29.4388	−0.933273
$996$	0	0
$997$	−20.8062	−0.658941	−0.329470	−	0.944166i	$-0.606870\pi$
$997$	−20.8062	−0.658941	−0.329470	+	0.944166i	$0.606870\pi$
$998$	0	0
$999$	−13.2551	−0.419374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.a.f.1.3	yes	4
3.2	odd	2		3744.2.a.be.1.1		4
4.3	odd	2	inner	416.2.a.f.1.2	✓	4
8.3	odd	2		832.2.a.p.1.3		4
8.5	even	2		832.2.a.p.1.2		4
12.11	even	2		3744.2.a.be.1.2		4
13.12	even	2		5408.2.a.bj.1.3		4
16.3	odd	4		3328.2.b.bb.1665.3		8
16.5	even	4		3328.2.b.bb.1665.4		8
16.11	odd	4		3328.2.b.bb.1665.6		8
16.13	even	4		3328.2.b.bb.1665.5		8
24.5	odd	2		7488.2.a.da.1.3		4
24.11	even	2		7488.2.a.da.1.4		4
52.51	odd	2		5408.2.a.bj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.f.1.2	✓	4	4.3	odd	2	inner
416.2.a.f.1.3	yes	4	1.1	even	1	trivial
832.2.a.p.1.2		4	8.5	even	2
832.2.a.p.1.3		4	8.3	odd	2
3328.2.b.bb.1665.3		8	16.3	odd	4
3328.2.b.bb.1665.4		8	16.5	even	4
3328.2.b.bb.1665.5		8	16.13	even	4
3328.2.b.bb.1665.6		8	16.11	odd	4
3744.2.a.be.1.1		4	3.2	odd	2
3744.2.a.be.1.2		4	12.11	even	2
5408.2.a.bj.1.2		4	52.51	odd	2
5408.2.a.bj.1.3		4	13.12	even	2
7488.2.a.da.1.3		4	24.5	odd	2
7488.2.a.da.1.4		4	24.11	even	2

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.81616 q^{3} +2.70156 q^{5} -3.36131 q^{7} +0.298438 q^{9} +5.17748 q^{11} +1.00000 q^{13} +4.90647 q^{15} +6.70156 q^{17} -5.17748 q^{19} -6.10469 q^{21} +2.29844 q^{25} -4.90647 q^{27} -2.00000 q^{29} -8.80980 q^{31} +9.40312 q^{33} -9.08080 q^{35} +2.70156 q^{37} +1.81616 q^{39} +3.40312 q^{41} +8.53879 q^{43} +0.806248 q^{45} -3.36131 q^{47} +4.29844 q^{49} +12.1711 q^{51} -11.4031 q^{53} +13.9873 q^{55} -9.40312 q^{57} -2.08717 q^{59} -3.40312 q^{61} -1.00314 q^{63} +2.70156 q^{65} -12.4421 q^{67} +10.6260 q^{71} -6.00000 q^{73} +4.17433 q^{75} -17.4031 q^{77} +3.09031 q^{79} -9.80625 q^{81} -1.54515 q^{83} +18.1047 q^{85} -3.63232 q^{87} -6.00000 q^{89} -3.36131 q^{91} -16.0000 q^{93} -13.9873 q^{95} -16.8062 q^{97} +1.54515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 14 q^{9} + 4 q^{13} + 14 q^{17} + 14 q^{21} + 22 q^{25} - 8 q^{29} + 12 q^{33} - 2 q^{37} - 12 q^{41} - 48 q^{45} + 30 q^{49} - 20 q^{53} - 12 q^{57} + 12 q^{61} - 2 q^{65} - 24 q^{73} - 44 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 14 * q^9 + 4 * q^13 + 14 * q^17 + 14 * q^21 + 22 * q^25 - 8 * q^29 + 12 * q^33 - 2 * q^37 - 12 * q^41 - 48 * q^45 + 30 * q^49 - 20 * q^53 - 12 * q^57 + 12 * q^61 - 2 * q^65 - 24 * q^73 - 44 * q^77 + 12 * q^81 + 34 * q^85 - 24 * q^89 - 64 * q^93 - 16 * q^97

Introduction

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