LMFDB - Embedded newform 1372.2.a.c.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1372,2,Mod(1,1372)] 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("1372.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1372, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1372 = 2^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1372.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,2,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.9554751573$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1229312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 24x^{2} - 8 $ x^6 - 10x^4 + 24x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.54832$ of defining polynomial
Character	$\chi$	$=$	1372.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54832 q^{3} +0.349282 q^{5} +3.49396 q^{9} +1.55496 q^{11} +0.784829 q^{13} +0.890084 q^{15} +5.72603 q^{17} -1.69432 q^{19} +4.35690 q^{23} -4.87800 q^{25} +1.25877 q^{27} -6.45473 q^{29} +6.70470 q^{31} +3.96254 q^{33} +9.65279 q^{37} +2.00000 q^{39} -5.41514 q^{41} +1.97823 q^{43} +1.22038 q^{45} -7.96347 q^{47} +14.5918 q^{51} +4.02177 q^{53} +0.543119 q^{55} -4.31767 q^{57} -6.29994 q^{59} +12.7279 q^{61} +0.274127 q^{65} +2.25906 q^{67} +11.1028 q^{69} +12.9119 q^{71} -14.7468 q^{73} -12.4307 q^{75} +2.02177 q^{79} -7.27413 q^{81} -13.2464 q^{83} +2.00000 q^{85} -16.4487 q^{87} +11.8568 q^{89} +17.0858 q^{93} -0.591794 q^{95} -8.80378 q^{97} +5.43296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} + 10 q^{11} + 4 q^{15} + 18 q^{23} + 10 q^{25} + 6 q^{29} + 22 q^{37} + 12 q^{39} + 18 q^{43} + 32 q^{51} + 18 q^{53} + 8 q^{57} - 20 q^{65} + 42 q^{67} + 70 q^{71} + 6 q^{79} - 22 q^{81}+ \cdots - 6 q^{99}+O(q^{100}) $ 6 * q + 2 * q^9 + 10 * q^11 + 4 * q^15 + 18 * q^23 + 10 * q^25 + 6 * q^29 + 22 * q^37 + 12 * q^39 + 18 * q^43 + 32 * q^51 + 18 * q^53 + 8 * q^57 - 20 * q^65 + 42 * q^67 + 70 * q^71 + 6 * q^79 - 22 * q^81 + 12 * q^85 + 28 * q^93 + 52 * q^95 - 6 * 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
$3$	2.54832	1.47128	0.735638	−	0.677375i	$-0.236882\pi$
$3$	2.54832	1.47128	0.735638	+	0.677375i	$0.236882\pi$
$4$	0	0
$5$	0.349282	0.156204	0.0781018	−	0.996945i	$-0.475114\pi$
$5$	0.349282	0.156204	0.0781018	+	0.996945i	$0.475114\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.49396	1.16465
$10$	0	0
$11$	1.55496	0.468838	0.234419	−	0.972136i	$-0.424681\pi$
$11$	1.55496	0.468838	0.234419	+	0.972136i	$0.424681\pi$
$12$	0	0
$13$	0.784829	0.217672	0.108836	−	0.994060i	$-0.465288\pi$
$13$	0.784829	0.217672	0.108836	+	0.994060i	$0.465288\pi$
$14$	0	0
$15$	0.890084	0.229819
$16$	0	0
$17$	5.72603	1.38877	0.694384	−	0.719605i	$-0.255677\pi$
$17$	5.72603	1.38877	0.694384	+	0.719605i	$0.255677\pi$
$18$	0	0
$19$	−1.69432	−0.388703	−0.194351	−	0.980932i	$-0.562260\pi$
$19$	−1.69432	−0.388703	−0.194351	+	0.980932i	$0.562260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.35690	0.908476	0.454238	−	0.890880i	$-0.349912\pi$
$23$	4.35690	0.908476	0.454238	+	0.890880i	$0.349912\pi$
$24$	0	0
$25$	−4.87800	−0.975600
$26$	0	0
$27$	1.25877	0.242250
$28$	0	0
$29$	−6.45473	−1.19861	−0.599307	−	0.800520i	$-0.704557\pi$
$29$	−6.45473	−1.19861	−0.599307	+	0.800520i	$0.704557\pi$
$30$	0	0
$31$	6.70470	1.20420	0.602100	−	0.798421i	$-0.294331\pi$
$31$	6.70470	1.20420	0.602100	+	0.798421i	$0.294331\pi$
$32$	0	0
$33$	3.96254	0.689789
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.65279	1.58691	0.793455	−	0.608629i	$-0.208280\pi$
$37$	9.65279	1.58691	0.793455	+	0.608629i	$0.208280\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−5.41514	−0.845703	−0.422852	−	0.906199i	$-0.638971\pi$
$41$	−5.41514	−0.845703	−0.422852	+	0.906199i	$0.638971\pi$
$42$	0	0
$43$	1.97823	0.301677	0.150839	−	0.988558i	$-0.451803\pi$
$43$	1.97823	0.301677	0.150839	+	0.988558i	$0.451803\pi$
$44$	0	0
$45$	1.22038	0.181923
$46$	0	0
$47$	−7.96347	−1.16159	−0.580796	−	0.814049i	$-0.697258\pi$
$47$	−7.96347	−1.16159	−0.580796	+	0.814049i	$0.697258\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	14.5918	2.04326
$52$	0	0
$53$	4.02177	0.552433	0.276216	−	0.961095i	$-0.410919\pi$
$53$	4.02177	0.552433	0.276216	+	0.961095i	$0.410919\pi$
$54$	0	0
$55$	0.543119	0.0732341
$56$	0	0
$57$	−4.31767	−0.571889
$58$	0	0
$59$	−6.29994	−0.820182	−0.410091	−	0.912045i	$-0.634503\pi$
$59$	−6.29994	−0.820182	−0.410091	+	0.912045i	$0.634503\pi$
$60$	0	0
$61$	12.7279	1.62964	0.814822	−	0.579712i	$-0.196835\pi$
$61$	12.7279	1.62964	0.814822	+	0.579712i	$0.196835\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.274127	0.0340012
$66$	0	0
$67$	2.25906	0.275988	0.137994	−	0.990433i	$-0.455934\pi$
$67$	2.25906	0.275988	0.137994	+	0.990433i	$0.455934\pi$
$68$	0	0
$69$	11.1028	1.33662
$70$	0	0
$71$	12.9119	1.53236	0.766178	−	0.642629i	$-0.222156\pi$
$71$	12.9119	1.53236	0.766178	+	0.642629i	$0.222156\pi$
$72$	0	0
$73$	−14.7468	−1.72599	−0.862993	−	0.505216i	$-0.831413\pi$
$73$	−14.7468	−1.72599	−0.862993	+	0.505216i	$0.831413\pi$
$74$	0	0
$75$	−12.4307	−1.43538
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.02177	0.227467	0.113733	−	0.993511i	$-0.463719\pi$
$79$	2.02177	0.227467	0.113733	+	0.993511i	$0.463719\pi$
$80$	0	0
$81$	−7.27413	−0.808236
$82$	0	0
$83$	−13.2464	−1.45398	−0.726988	−	0.686650i	$-0.759080\pi$
$83$	−13.2464	−1.45398	−0.726988	+	0.686650i	$0.759080\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−16.4487	−1.76349
$88$	0	0
$89$	11.8568	1.25682	0.628411	−	0.777882i	$-0.283706\pi$
$89$	11.8568	1.25682	0.628411	+	0.777882i	$0.283706\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.0858	1.77171
$94$	0	0
$95$	−0.591794	−0.0607168
$96$	0	0
$97$	−8.80378	−0.893888	−0.446944	−	0.894562i	$-0.647488\pi$
$97$	−8.80378	−0.893888	−0.446944	+	0.894562i	$0.647488\pi$
$98$	0	0
$99$	5.43296	0.546033
$100$	0	0
$101$	−8.94213	−0.889776	−0.444888	−	0.895586i	$-0.646756\pi$
$101$	−8.94213	−0.889776	−0.444888	+	0.895586i	$0.646756\pi$
$102$	0	0
$103$	−13.6127	−1.34130	−0.670651	−	0.741773i	$-0.733985\pi$
$103$	−13.6127	−1.34130	−0.670651	+	0.741773i	$0.733985\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.53319	0.438240	0.219120	−	0.975698i	$-0.429681\pi$
$107$	4.53319	0.438240	0.219120	+	0.975698i	$0.429681\pi$
$108$	0	0
$109$	7.99761	0.766032	0.383016	−	0.923742i	$-0.374885\pi$
$109$	7.99761	0.766032	0.383016	+	0.923742i	$0.374885\pi$
$110$	0	0
$111$	24.5985	2.33478
$112$	0	0
$113$	−8.26875	−0.777859	−0.388929	−	0.921268i	$-0.627155\pi$
$113$	−8.26875	−0.777859	−0.388929	+	0.921268i	$0.627155\pi$
$114$	0	0
$115$	1.52178	0.141907
$116$	0	0
$117$	2.74216	0.253513
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.58211	−0.780191
$122$	0	0
$123$	−13.7995	−1.24426
$124$	0	0
$125$	−3.45021	−0.308596
$126$	0	0
$127$	−2.88471	−0.255976	−0.127988	−	0.991776i	$-0.540852\pi$
$127$	−2.88471	−0.255976	−0.127988	+	0.991776i	$0.540852\pi$
$128$	0	0
$129$	5.04117	0.443850
$130$	0	0
$131$	6.67391	0.583102	0.291551	−	0.956555i	$-0.405829\pi$
$131$	6.67391	0.583102	0.291551	+	0.956555i	$0.405829\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.439665	0.0378403
$136$	0	0
$137$	19.0465	1.62725	0.813627	−	0.581387i	$-0.197490\pi$
$137$	19.0465	1.62725	0.813627	+	0.581387i	$0.197490\pi$
$138$	0	0
$139$	−22.8733	−1.94009	−0.970046	−	0.242921i	$-0.921894\pi$
$139$	−22.8733	−1.94009	−0.970046	+	0.242921i	$0.921894\pi$
$140$	0	0
$141$	−20.2935	−1.70902
$142$	0	0
$143$	1.22038	0.102053
$144$	0	0
$145$	−2.25452	−0.187228
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.29159	0.105811	0.0529054	−	0.998600i	$-0.483152\pi$
$149$	1.29159	0.105811	0.0529054	+	0.998600i	$0.483152\pi$
$150$	0	0
$151$	−21.3545	−1.73780	−0.868902	−	0.494983i	$-0.835174\pi$
$151$	−21.3545	−1.73780	−0.868902	+	0.494983i	$0.835174\pi$
$152$	0	0
$153$	20.0065	1.61743
$154$	0	0
$155$	2.34183	0.188100
$156$	0	0
$157$	−6.92932	−0.553020	−0.276510	−	0.961011i	$-0.589178\pi$
$157$	−6.92932	−0.553020	−0.276510	+	0.961011i	$0.589178\pi$
$158$	0	0
$159$	10.2488	0.812781
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.57002	−0.514604	−0.257302	−	0.966331i	$-0.582834\pi$
$163$	−6.57002	−0.514604	−0.257302	+	0.966331i	$0.582834\pi$
$164$	0	0
$165$	1.38404	0.107748
$166$	0	0
$167$	12.6033	0.975270	0.487635	−	0.873048i	$-0.337860\pi$
$167$	12.6033	0.975270	0.487635	+	0.873048i	$0.337860\pi$
$168$	0	0
$169$	−12.3840	−0.952619
$170$	0	0
$171$	−5.91987	−0.452704
$172$	0	0
$173$	−20.3729	−1.54892	−0.774461	−	0.632621i	$-0.781979\pi$
$173$	−20.3729	−1.54892	−0.774461	+	0.632621i	$0.781979\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.0543	−1.20671
$178$	0	0
$179$	17.8756	1.33609	0.668043	−	0.744123i	$-0.267132\pi$
$179$	17.8756	1.33609	0.668043	+	0.744123i	$0.267132\pi$
$180$	0	0
$181$	−16.9535	−1.26014	−0.630071	−	0.776538i	$-0.716974\pi$
$181$	−16.9535	−1.26014	−0.630071	+	0.776538i	$0.716974\pi$
$182$	0	0
$183$	32.4349	2.39766
$184$	0	0
$185$	3.37155	0.247881
$186$	0	0
$187$	8.90374	0.651106
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.96316	0.359122	0.179561	−	0.983747i	$-0.442532\pi$
$191$	4.96316	0.359122	0.179561	+	0.983747i	$0.442532\pi$
$192$	0	0
$193$	−12.3327	−0.887730	−0.443865	−	0.896094i	$-0.646393\pi$
$193$	−12.3327	−0.887730	−0.443865	+	0.896094i	$0.646393\pi$
$194$	0	0
$195$	0.698564	0.0500252
$196$	0	0
$197$	6.49827	0.462983	0.231491	−	0.972837i	$-0.425640\pi$
$197$	6.49827	0.462983	0.231491	+	0.972837i	$0.425640\pi$
$198$	0	0
$199$	16.1823	1.14714	0.573568	−	0.819158i	$-0.305559\pi$
$199$	16.1823	1.14714	0.573568	+	0.819158i	$0.305559\pi$
$200$	0	0
$201$	5.75682	0.406055
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.89141	−0.132102
$206$	0	0
$207$	15.2228	1.05806
$208$	0	0
$209$	−2.63459	−0.182238
$210$	0	0
$211$	7.44265	0.512373	0.256187	−	0.966627i	$-0.417534\pi$
$211$	7.44265	0.512373	0.256187	+	0.966627i	$0.417534\pi$
$212$	0	0
$213$	32.9036	2.25452
$214$	0	0
$215$	0.690960	0.0471231
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−37.5797	−2.53940
$220$	0	0
$221$	4.49396	0.302296
$222$	0	0
$223$	2.49623	0.167160	0.0835800	−	0.996501i	$-0.473365\pi$
$223$	2.49623	0.167160	0.0835800	+	0.996501i	$0.473365\pi$
$224$	0	0
$225$	−17.0435	−1.13624
$226$	0	0
$227$	24.4985	1.62602	0.813011	−	0.582249i	$-0.197827\pi$
$227$	24.4985	1.62602	0.813011	+	0.582249i	$0.197827\pi$
$228$	0	0
$229$	−1.29716	−0.0857188	−0.0428594	−	0.999081i	$-0.513647\pi$
$229$	−1.29716	−0.0857188	−0.0428594	+	0.999081i	$0.513647\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.2838	−0.870252	−0.435126	−	0.900370i	$-0.643296\pi$
$233$	−13.2838	−0.870252	−0.435126	+	0.900370i	$0.643296\pi$
$234$	0	0
$235$	−2.78150	−0.181445
$236$	0	0
$237$	5.15213	0.334667
$238$	0	0
$239$	−10.9487	−0.708212	−0.354106	−	0.935205i	$-0.615215\pi$
$239$	−10.9487	−0.708212	−0.354106	+	0.935205i	$0.615215\pi$
$240$	0	0
$241$	18.2506	1.17563	0.587813	−	0.808997i	$-0.299989\pi$
$241$	18.2506	1.17563	0.587813	+	0.808997i	$0.299989\pi$
$242$	0	0
$243$	−22.3131	−1.43139
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.32975	−0.0846099
$248$	0	0
$249$	−33.7560	−2.13920
$250$	0	0
$251$	−19.9450	−1.25891	−0.629457	−	0.777035i	$-0.716723\pi$
$251$	−19.9450	−1.25891	−0.629457	+	0.777035i	$0.716723\pi$
$252$	0	0
$253$	6.77479	0.425927
$254$	0	0
$255$	5.09665	0.319165
$256$	0	0
$257$	6.79097	0.423609	0.211804	−	0.977312i	$-0.432066\pi$
$257$	6.79097	0.423609	0.211804	+	0.977312i	$0.432066\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−22.5526	−1.39597
$262$	0	0
$263$	24.8713	1.53363	0.766815	−	0.641868i	$-0.221840\pi$
$263$	24.8713	1.53363	0.766815	+	0.641868i	$0.221840\pi$
$264$	0	0
$265$	1.40473	0.0862920
$266$	0	0
$267$	30.2150	1.84913
$268$	0	0
$269$	−5.67056	−0.345740	−0.172870	−	0.984945i	$-0.555304\pi$
$269$	−5.67056	−0.345740	−0.172870	+	0.984945i	$0.555304\pi$
$270$	0	0
$271$	20.4729	1.24364	0.621819	−	0.783161i	$-0.286394\pi$
$271$	20.4729	1.24364	0.621819	+	0.783161i	$0.286394\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.58509	−0.457398
$276$	0	0
$277$	−1.13946	−0.0684633	−0.0342316	−	0.999414i	$-0.510898\pi$
$277$	−1.13946	−0.0684633	−0.0342316	+	0.999414i	$0.510898\pi$
$278$	0	0
$279$	23.4259	1.40247
$280$	0	0
$281$	7.02608	0.419141	0.209570	−	0.977794i	$-0.432793\pi$
$281$	7.02608	0.419141	0.209570	+	0.977794i	$0.432793\pi$
$282$	0	0
$283$	−0.0137019	−0.000814491 0	−0.000407246	−	1.00000i	$-0.500130\pi$
$283$	−0.0137019	−0.000814491 0	−0.000407246		1.00000i	$0.500130\pi$
$284$	0	0
$285$	−1.50808	−0.0893311
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.7875	0.928674
$290$	0	0
$291$	−22.4349	−1.31516
$292$	0	0
$293$	−13.3539	−0.780144	−0.390072	−	0.920784i	$-0.627550\pi$
$293$	−13.3539	−0.780144	−0.390072	+	0.920784i	$0.627550\pi$
$294$	0	0
$295$	−2.20046	−0.128115
$296$	0	0
$297$	1.95733	0.113576
$298$	0	0
$299$	3.41942	0.197750
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−22.7875	−1.30911
$304$	0	0
$305$	4.44563	0.254556
$306$	0	0
$307$	−6.32463	−0.360966	−0.180483	−	0.983578i	$-0.557766\pi$
$307$	−6.32463	−0.360966	−0.180483	+	0.983578i	$0.557766\pi$
$308$	0	0
$309$	−34.6896	−1.97342
$310$	0	0
$311$	−8.08813	−0.458636	−0.229318	−	0.973352i	$-0.573649\pi$
$311$	−8.08813	−0.458636	−0.229318	+	0.973352i	$0.573649\pi$
$312$	0	0
$313$	−31.3169	−1.77013	−0.885067	−	0.465464i	$-0.845887\pi$
$313$	−31.3169	−1.77013	−0.885067	+	0.465464i	$0.845887\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.0441	−1.68745	−0.843723	−	0.536778i	$-0.819641\pi$
$317$	−30.0441	−1.68745	−0.843723	+	0.536778i	$0.819641\pi$
$318$	0	0
$319$	−10.0368	−0.561955
$320$	0	0
$321$	11.5520	0.644772
$322$	0	0
$323$	−9.70171	−0.539818
$324$	0	0
$325$	−3.82840	−0.212361
$326$	0	0
$327$	20.3805	1.12704
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.97152	−0.548085	−0.274042	−	0.961718i	$-0.588361\pi$
$331$	−9.97152	−0.548085	−0.274042	+	0.961718i	$0.588361\pi$
$332$	0	0
$333$	33.7265	1.84820
$334$	0	0
$335$	0.789049	0.0431104
$336$	0	0
$337$	−24.1903	−1.31773	−0.658865	−	0.752262i	$-0.728963\pi$
$337$	−24.1903	−1.31773	−0.658865	+	0.752262i	$0.728963\pi$
$338$	0	0
$339$	−21.0715	−1.14444
$340$	0	0
$341$	10.4255	0.564574
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.87800	0.208785
$346$	0	0
$347$	17.2078	0.923760	0.461880	−	0.886942i	$-0.347175\pi$
$347$	17.2078	0.923760	0.461880	+	0.886942i	$0.347175\pi$
$348$	0	0
$349$	3.94884	0.211376	0.105688	−	0.994399i	$-0.466295\pi$
$349$	3.94884	0.211376	0.105688	+	0.994399i	$0.466295\pi$
$350$	0	0
$351$	0.987918	0.0527312
$352$	0	0
$353$	−11.7663	−0.626259	−0.313130	−	0.949710i	$-0.601377\pi$
$353$	−11.7663	−0.626259	−0.313130	+	0.949710i	$0.601377\pi$
$354$	0	0
$355$	4.50988	0.239359
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.7888	−1.25552	−0.627762	−	0.778405i	$-0.716029\pi$
$359$	−23.7888	−1.25552	−0.627762	+	0.778405i	$0.716029\pi$
$360$	0	0
$361$	−16.1293	−0.848910
$362$	0	0
$363$	−21.8700	−1.14788
$364$	0	0
$365$	−5.15080	−0.269605
$366$	0	0
$367$	−0.761645	−0.0397576	−0.0198788	−	0.999802i	$-0.506328\pi$
$367$	−0.761645	−0.0397576	−0.0198788	+	0.999802i	$0.506328\pi$
$368$	0	0
$369$	−18.9203	−0.984951
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.6732	−1.12220	−0.561099	−	0.827749i	$-0.689621\pi$
$373$	−21.6732	−1.12220	−0.561099	+	0.827749i	$0.689621\pi$
$374$	0	0
$375$	−8.79225	−0.454030
$376$	0	0
$377$	−5.06586	−0.260905
$378$	0	0
$379$	18.5569	0.953203	0.476601	−	0.879120i	$-0.341869\pi$
$379$	18.5569	0.953203	0.476601	+	0.879120i	$0.341869\pi$
$380$	0	0
$381$	−7.35117	−0.376612
$382$	0	0
$383$	−14.9236	−0.762559	−0.381280	−	0.924460i	$-0.624517\pi$
$383$	−14.9236	−0.762559	−0.381280	+	0.924460i	$0.624517\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.91185	0.351349
$388$	0	0
$389$	19.1075	0.968790	0.484395	−	0.874849i	$-0.339040\pi$
$389$	19.1075	0.968790	0.484395	+	0.874849i	$0.339040\pi$
$390$	0	0
$391$	24.9477	1.26166
$392$	0	0
$393$	17.0073	0.857905
$394$	0	0
$395$	0.706168	0.0355312
$396$	0	0
$397$	4.42278	0.221973	0.110986	−	0.993822i	$-0.464599\pi$
$397$	4.42278	0.221973	0.110986	+	0.993822i	$0.464599\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.91053	0.395033	0.197516	−	0.980300i	$-0.436712\pi$
$401$	7.91053	0.395033	0.197516	+	0.980300i	$0.436712\pi$
$402$	0	0
$403$	5.26205	0.262121
$404$	0	0
$405$	−2.54072	−0.126249
$406$	0	0
$407$	15.0097	0.744003
$408$	0	0
$409$	−29.2979	−1.44869	−0.724345	−	0.689438i	$-0.757858\pi$
$409$	−29.2979	−1.44869	−0.724345	+	0.689438i	$0.757858\pi$
$410$	0	0
$411$	48.5367	2.39414
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.62671	−0.227116
$416$	0	0
$417$	−58.2887	−2.85441
$418$	0	0
$419$	−12.7184	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$419$	−12.7184	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$420$	0	0
$421$	−10.8334	−0.527987	−0.263994	−	0.964524i	$-0.585040\pi$
$421$	−10.8334	−0.527987	−0.263994	+	0.964524i	$0.585040\pi$
$422$	0	0
$423$	−27.8240	−1.35285
$424$	0	0
$425$	−27.9316	−1.35488
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.10992	0.150148
$430$	0	0
$431$	26.0519	1.25488	0.627438	−	0.778667i	$-0.284104\pi$
$431$	26.0519	1.25488	0.627438	+	0.778667i	$0.284104\pi$
$432$	0	0
$433$	22.1995	1.06684	0.533419	−	0.845851i	$-0.320907\pi$
$433$	22.1995	1.06684	0.533419	+	0.845851i	$0.320907\pi$
$434$	0	0
$435$	−5.74525	−0.275464
$436$	0	0
$437$	−7.38196	−0.353127
$438$	0	0
$439$	2.59432	0.123820	0.0619101	−	0.998082i	$-0.480281\pi$
$439$	2.59432	0.123820	0.0619101	+	0.998082i	$0.480281\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.0301	1.80687	0.903433	−	0.428729i	$-0.141039\pi$
$443$	38.0301	1.80687	0.903433	+	0.428729i	$0.141039\pi$
$444$	0	0
$445$	4.14138	0.196320
$446$	0	0
$447$	3.29138	0.155677
$448$	0	0
$449$	−4.11960	−0.194416	−0.0972081	−	0.995264i	$-0.530991\pi$
$449$	−4.11960	−0.194416	−0.0972081	+	0.995264i	$0.530991\pi$
$450$	0	0
$451$	−8.42032	−0.396497
$452$	0	0
$453$	−54.4182	−2.55679
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.5453	−0.540065	−0.270032	−	0.962851i	$-0.587034\pi$
$457$	−11.5453	−0.540065	−0.270032	+	0.962851i	$0.587034\pi$
$458$	0	0
$459$	7.20775	0.336429
$460$	0	0
$461$	0.798531	0.0371913	0.0185957	−	0.999827i	$-0.494080\pi$
$461$	0.798531	0.0371913	0.0185957	+	0.999827i	$0.494080\pi$
$462$	0	0
$463$	32.7385	1.52149	0.760745	−	0.649051i	$-0.224834\pi$
$463$	32.7385	1.52149	0.760745	+	0.649051i	$0.224834\pi$
$464$	0	0
$465$	5.96774	0.276748
$466$	0	0
$467$	41.0301	1.89865	0.949324	−	0.314300i	$-0.101770\pi$
$467$	41.0301	1.89865	0.949324	+	0.314300i	$0.101770\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−17.6582	−0.813646
$472$	0	0
$473$	3.07606	0.141438
$474$	0	0
$475$	8.26488	0.379219
$476$	0	0
$477$	14.0519	0.643392
$478$	0	0
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	7.57579	0.345427
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.07500	−0.139629
$486$	0	0
$487$	−24.4349	−1.10725	−0.553625	−	0.832766i	$-0.686756\pi$
$487$	−24.4349	−1.10725	−0.553625	+	0.832766i	$0.686756\pi$
$488$	0	0
$489$	−16.7426	−0.757124
$490$	0	0
$491$	33.9734	1.53320	0.766600	−	0.642125i	$-0.221947\pi$
$491$	33.9734	1.53320	0.766600	+	0.642125i	$0.221947\pi$
$492$	0	0
$493$	−36.9600	−1.66459
$494$	0	0
$495$	1.89763	0.0852923
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.32304	0.372591	0.186295	−	0.982494i	$-0.440352\pi$
$499$	8.32304	0.372591	0.186295	+	0.982494i	$0.440352\pi$
$500$	0	0
$501$	32.1172	1.43489
$502$	0	0
$503$	4.99668	0.222791	0.111396	−	0.993776i	$-0.464468\pi$
$503$	4.99668	0.222791	0.111396	+	0.993776i	$0.464468\pi$
$504$	0	0
$505$	−3.12333	−0.138986
$506$	0	0
$507$	−31.5586	−1.40157
$508$	0	0
$509$	−8.82358	−0.391098	−0.195549	−	0.980694i	$-0.562649\pi$
$509$	−8.82358	−0.391098	−0.195549	+	0.980694i	$0.562649\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.13275	−0.0941633
$514$	0	0
$515$	−4.75468	−0.209516
$516$	0	0
$517$	−12.3829	−0.544598
$518$	0	0
$519$	−51.9168	−2.27889
$520$	0	0
$521$	−37.6586	−1.64985	−0.824926	−	0.565241i	$-0.808783\pi$
$521$	−37.6586	−1.64985	−0.824926	+	0.565241i	$0.808783\pi$
$522$	0	0
$523$	−20.3634	−0.890430	−0.445215	−	0.895424i	$-0.646873\pi$
$523$	−20.3634	−0.890430	−0.445215	+	0.895424i	$0.646873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.3913	1.67235
$528$	0	0
$529$	−4.01746	−0.174672
$530$	0	0
$531$	−22.0117	−0.955228
$532$	0	0
$533$	−4.24996	−0.184086
$534$	0	0
$535$	1.58336	0.0684546
$536$	0	0
$537$	45.5529	1.96575
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.32544	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$541$	9.32544	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$542$	0	0
$543$	−43.2030	−1.85402
$544$	0	0
$545$	2.79342	0.119657
$546$	0	0
$547$	−35.0863	−1.50018	−0.750092	−	0.661334i	$-0.769991\pi$
$547$	−35.0863	−1.50018	−0.750092	+	0.661334i	$0.769991\pi$
$548$	0	0
$549$	44.4708	1.89797
$550$	0	0
$551$	10.9364	0.465904
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.59179	0.364701
$556$	0	0
$557$	31.4601	1.33301	0.666504	−	0.745502i	$-0.267790\pi$
$557$	31.4601	1.33301	0.666504	+	0.745502i	$0.267790\pi$
$558$	0	0
$559$	1.55257	0.0656668
$560$	0	0
$561$	22.6896	0.957957
$562$	0	0
$563$	36.9353	1.55664	0.778319	−	0.627869i	$-0.216073\pi$
$563$	36.9353	1.55664	0.778319	+	0.627869i	$0.216073\pi$
$564$	0	0
$565$	−2.88812	−0.121504
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.4620	0.815891	0.407945	−	0.913006i	$-0.366245\pi$
$569$	19.4620	0.815891	0.407945	+	0.913006i	$0.366245\pi$
$570$	0	0
$571$	−14.6920	−0.614842	−0.307421	−	0.951574i	$-0.599466\pi$
$571$	−14.6920	−0.614842	−0.307421	+	0.951574i	$0.599466\pi$
$572$	0	0
$573$	12.6478	0.528368
$574$	0	0
$575$	−21.2529	−0.886309
$576$	0	0
$577$	42.4785	1.76840	0.884202	−	0.467106i	$-0.154703\pi$
$577$	42.4785	1.76840	0.884202	+	0.467106i	$0.154703\pi$
$578$	0	0
$579$	−31.4278	−1.30610
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.25368	0.259001
$584$	0	0
$585$	0.957787	0.0395996
$586$	0	0
$587$	25.6539	1.05885	0.529425	−	0.848357i	$-0.322408\pi$
$587$	25.6539	1.05885	0.529425	+	0.848357i	$0.322408\pi$
$588$	0	0
$589$	−11.3599	−0.468076
$590$	0	0
$591$	16.5597	0.681175
$592$	0	0
$593$	−11.9644	−0.491319	−0.245659	−	0.969356i	$-0.579005\pi$
$593$	−11.9644	−0.491319	−0.245659	+	0.969356i	$0.579005\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.2379	1.68775
$598$	0	0
$599$	33.0944	1.35220	0.676100	−	0.736810i	$-0.263669\pi$
$599$	33.0944	1.35220	0.676100	+	0.736810i	$0.263669\pi$
$600$	0	0
$601$	−31.0785	−1.26772	−0.633860	−	0.773448i	$-0.718530\pi$
$601$	−31.0785	−1.26772	−0.633860	+	0.773448i	$0.718530\pi$
$602$	0	0
$603$	7.89307	0.321431
$604$	0	0
$605$	−2.99757	−0.121869
$606$	0	0
$607$	1.36822	0.0555342	0.0277671	−	0.999614i	$-0.491160\pi$
$607$	1.36822	0.0555342	0.0277671	+	0.999614i	$0.491160\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.24996	−0.252846
$612$	0	0
$613$	−17.7458	−0.716748	−0.358374	−	0.933578i	$-0.616669\pi$
$613$	−17.7458	−0.716748	−0.358374	+	0.933578i	$0.616669\pi$
$614$	0	0
$615$	−4.81993	−0.194358
$616$	0	0
$617$	−15.4252	−0.620995	−0.310497	−	0.950574i	$-0.600496\pi$
$617$	−15.4252	−0.620995	−0.310497	+	0.950574i	$0.600496\pi$
$618$	0	0
$619$	23.4985	0.944485	0.472242	−	0.881469i	$-0.343445\pi$
$619$	23.4985	0.944485	0.472242	+	0.881469i	$0.343445\pi$
$620$	0	0
$621$	5.48432	0.220078
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.1849	0.927397
$626$	0	0
$627$	−6.71379	−0.268123
$628$	0	0
$629$	55.2722	2.20385
$630$	0	0
$631$	25.4534	1.01328	0.506642	−	0.862157i	$-0.330887\pi$
$631$	25.4534	1.01328	0.506642	+	0.862157i	$0.330887\pi$
$632$	0	0
$633$	18.9663	0.753842
$634$	0	0
$635$	−1.00758	−0.0399844
$636$	0	0
$637$	0	0
$638$	0	0
$639$	45.1135	1.78466
$640$	0	0
$641$	28.9245	1.14245	0.571225	−	0.820793i	$-0.306468\pi$
$641$	28.9245	1.14245	0.571225	+	0.820793i	$0.306468\pi$
$642$	0	0
$643$	−26.9974	−1.06467	−0.532337	−	0.846532i	$-0.678686\pi$
$643$	−26.9974	−1.06467	−0.532337	+	0.846532i	$0.678686\pi$
$644$	0	0
$645$	1.76079	0.0693310
$646$	0	0
$647$	24.3841	0.958640	0.479320	−	0.877640i	$-0.340883\pi$
$647$	24.3841	0.958640	0.479320	+	0.877640i	$0.340883\pi$
$648$	0	0
$649$	−9.79614	−0.384532
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.2519	−0.988183	−0.494091	−	0.869410i	$-0.664499\pi$
$653$	−25.2519	−0.988183	−0.494091	+	0.869410i	$0.664499\pi$
$654$	0	0
$655$	2.33108	0.0910827
$656$	0	0
$657$	−51.5248	−2.01017
$658$	0	0
$659$	−27.3836	−1.06671	−0.533356	−	0.845891i	$-0.679069\pi$
$659$	−27.3836	−1.06671	−0.533356	+	0.845891i	$0.679069\pi$
$660$	0	0
$661$	45.7296	1.77868	0.889338	−	0.457251i	$-0.151166\pi$
$661$	45.7296	1.77868	0.889338	+	0.457251i	$0.151166\pi$
$662$	0	0
$663$	11.4521	0.444761
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.1226	−1.08891
$668$	0	0
$669$	6.36121	0.245938
$670$	0	0
$671$	19.7914	0.764038
$672$	0	0
$673$	38.4782	1.48322	0.741612	−	0.670829i	$-0.234062\pi$
$673$	38.4782	1.48322	0.741612	+	0.670829i	$0.234062\pi$
$674$	0	0
$675$	−6.14028	−0.236339
$676$	0	0
$677$	16.1961	0.622465	0.311232	−	0.950334i	$-0.399258\pi$
$677$	16.1961	0.622465	0.311232	+	0.950334i	$0.399258\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	62.4301	2.39233
$682$	0	0
$683$	8.30367	0.317731	0.158865	−	0.987300i	$-0.449216\pi$
$683$	8.30367	0.317731	0.158865	+	0.987300i	$0.449216\pi$
$684$	0	0
$685$	6.65261	0.254183
$686$	0	0
$687$	−3.30559	−0.126116
$688$	0	0
$689$	3.15640	0.120249
$690$	0	0
$691$	8.45111	0.321495	0.160748	−	0.986996i	$-0.448609\pi$
$691$	8.45111	0.321495	0.160748	+	0.986996i	$0.448609\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.98925	−0.303049
$696$	0	0
$697$	−31.0073	−1.17449
$698$	0	0
$699$	−33.8515	−1.28038
$700$	0	0
$701$	2.96316	0.111917	0.0559586	−	0.998433i	$-0.482179\pi$
$701$	2.96316	0.111917	0.0559586	+	0.998433i	$0.482179\pi$
$702$	0	0
$703$	−16.3549	−0.616836
$704$	0	0
$705$	−7.08815	−0.266955
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.5760	1.37364	0.686820	−	0.726827i	$-0.259006\pi$
$709$	36.5760	1.37364	0.686820	+	0.726827i	$0.259006\pi$
$710$	0	0
$711$	7.06398	0.264920
$712$	0	0
$713$	29.2117	1.09399
$714$	0	0
$715$	0.426256	0.0159411
$716$	0	0
$717$	−27.9008	−1.04198
$718$	0	0
$719$	−18.8279	−0.702163	−0.351082	−	0.936345i	$-0.614186\pi$
$719$	−18.8279	−0.702163	−0.351082	+	0.936345i	$0.614186\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	46.5086	1.72967
$724$	0	0
$725$	31.4862	1.16937
$726$	0	0
$727$	40.7979	1.51311	0.756555	−	0.653930i	$-0.226881\pi$
$727$	40.7979	1.51311	0.756555	+	0.653930i	$0.226881\pi$
$728$	0	0
$729$	−35.0388	−1.29773
$730$	0	0
$731$	11.3274	0.418959
$732$	0	0
$733$	14.9175	0.550990	0.275495	−	0.961303i	$-0.411158\pi$
$733$	14.9175	0.550990	0.275495	+	0.961303i	$0.411158\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.51275	0.129394
$738$	0	0
$739$	−31.9232	−1.17431	−0.587157	−	0.809473i	$-0.699753\pi$
$739$	−31.9232	−1.17431	−0.587157	+	0.809473i	$0.699753\pi$
$740$	0	0
$741$	−3.38863	−0.124484
$742$	0	0
$743$	29.3056	1.07512	0.537559	−	0.843226i	$-0.319347\pi$
$743$	29.3056	1.07512	0.537559	+	0.843226i	$0.319347\pi$
$744$	0	0
$745$	0.451127	0.0165280
$746$	0	0
$747$	−46.2822	−1.69338
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31.1105	1.13524	0.567619	−	0.823291i	$-0.307865\pi$
$751$	31.1105	1.13524	0.567619	+	0.823291i	$0.307865\pi$
$752$	0	0
$753$	−50.8262	−1.85221
$754$	0	0
$755$	−7.45874	−0.271451
$756$	0	0
$757$	35.2121	1.27980	0.639902	−	0.768456i	$-0.278975\pi$
$757$	35.2121	1.27980	0.639902	+	0.768456i	$0.278975\pi$
$758$	0	0
$759$	17.2644	0.626657
$760$	0	0
$761$	42.9600	1.55730	0.778650	−	0.627458i	$-0.215905\pi$
$761$	42.9600	1.55730	0.778650	+	0.627458i	$0.215905\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.98792	0.252649
$766$	0	0
$767$	−4.94438	−0.178531
$768$	0	0
$769$	7.97717	0.287664	0.143832	−	0.989602i	$-0.454057\pi$
$769$	7.97717	0.287664	0.143832	+	0.989602i	$0.454057\pi$
$770$	0	0
$771$	17.3056	0.623245
$772$	0	0
$773$	33.4775	1.20410	0.602051	−	0.798458i	$-0.294350\pi$
$773$	33.4775	1.20410	0.602051	+	0.798458i	$0.294350\pi$
$774$	0	0
$775$	−32.7055	−1.17482
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.17496	0.328727
$780$	0	0
$781$	20.0774	0.718426
$782$	0	0
$783$	−8.12501	−0.290364
$784$	0	0
$785$	−2.42029	−0.0863838
$786$	0	0
$787$	13.6743	0.487436	0.243718	−	0.969846i	$-0.421633\pi$
$787$	13.6743	0.487436	0.243718	+	0.969846i	$0.421633\pi$
$788$	0	0
$789$	63.3801	2.25639
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.98925	0.354729
$794$	0	0
$795$	3.57971	0.126959
$796$	0	0
$797$	−35.4321	−1.25507	−0.627535	−	0.778588i	$-0.715936\pi$
$797$	−35.4321	−1.25507	−0.627535	+	0.778588i	$0.715936\pi$
$798$	0	0
$799$	−45.5991	−1.61318
$800$	0	0
$801$	41.4273	1.46376
$802$	0	0
$803$	−22.9307	−0.809207
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.4504	−0.508679
$808$	0	0
$809$	−21.5851	−0.758891	−0.379446	−	0.925214i	$-0.623885\pi$
$809$	−21.5851	−0.758891	−0.379446	+	0.925214i	$0.623885\pi$
$810$	0	0
$811$	13.7682	0.483466	0.241733	−	0.970343i	$-0.422284\pi$
$811$	13.7682	0.483466	0.241733	+	0.970343i	$0.422284\pi$
$812$	0	0
$813$	52.1715	1.82973
$814$	0	0
$815$	−2.29479	−0.0803830
$816$	0	0
$817$	−3.35175	−0.117263
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.5846	−1.13721	−0.568605	−	0.822610i	$-0.692517\pi$
$821$	−32.5846	−1.13721	−0.568605	+	0.822610i	$0.692517\pi$
$822$	0	0
$823$	17.5144	0.610514	0.305257	−	0.952270i	$-0.401258\pi$
$823$	17.5144	0.610514	0.305257	+	0.952270i	$0.401258\pi$
$824$	0	0
$825$	−19.3293	−0.672959
$826$	0	0
$827$	−2.81163	−0.0977698	−0.0488849	−	0.998804i	$-0.515567\pi$
$827$	−2.81163	−0.0977698	−0.0488849	+	0.998804i	$0.515567\pi$
$828$	0	0
$829$	−46.9321	−1.63002	−0.815009	−	0.579448i	$-0.803268\pi$
$829$	−46.9321	−1.63002	−0.815009	+	0.579448i	$0.803268\pi$
$830$	0	0
$831$	−2.90370	−0.100728
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.40209	0.152341
$836$	0	0
$837$	8.43967	0.291717
$838$	0	0
$839$	−20.8145	−0.718598	−0.359299	−	0.933223i	$-0.616984\pi$
$839$	−20.8145	−0.718598	−0.359299	+	0.933223i	$0.616984\pi$
$840$	0	0
$841$	12.6635	0.436674
$842$	0	0
$843$	17.9047	0.616672
$844$	0	0
$845$	−4.32552	−0.148802
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.0349168	−0.00119834
$850$	0	0
$851$	42.0562	1.44167
$852$	0	0
$853$	29.2903	1.00288	0.501441	−	0.865192i	$-0.332803\pi$
$853$	29.2903	1.00288	0.501441	+	0.865192i	$0.332803\pi$
$854$	0	0
$855$	−2.06770	−0.0707140
$856$	0	0
$857$	−28.9103	−0.987556	−0.493778	−	0.869588i	$-0.664384\pi$
$857$	−28.9103	−0.987556	−0.493778	+	0.869588i	$0.664384\pi$
$858$	0	0
$859$	26.6975	0.910908	0.455454	−	0.890259i	$-0.349477\pi$
$859$	26.6975	0.910908	0.455454	+	0.890259i	$0.349477\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.4494	1.47903	0.739517	−	0.673138i	$-0.235054\pi$
$863$	43.4494	1.47903	0.739517	+	0.673138i	$0.235054\pi$
$864$	0	0
$865$	−7.11588	−0.241947
$866$	0	0
$867$	40.2316	1.36634
$868$	0	0
$869$	3.14377	0.106645
$870$	0	0
$871$	1.77298	0.0600750
$872$	0	0
$873$	−30.7600	−1.04107
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.4741	1.09657	0.548286	−	0.836291i	$-0.315280\pi$
$877$	32.4741	1.09657	0.548286	+	0.836291i	$0.315280\pi$
$878$	0	0
$879$	−34.0301	−1.14781
$880$	0	0
$881$	31.6353	1.06582	0.532911	−	0.846171i	$-0.321098\pi$
$881$	31.6353	1.06582	0.532911	+	0.846171i	$0.321098\pi$
$882$	0	0
$883$	−11.3002	−0.380282	−0.190141	−	0.981757i	$-0.560895\pi$
$883$	−11.3002	−0.380282	−0.190141	+	0.981757i	$0.560895\pi$
$884$	0	0
$885$	−5.60747	−0.188493
$886$	0	0
$887$	−27.5192	−0.924006	−0.462003	−	0.886878i	$-0.652869\pi$
$887$	−27.5192	−0.924006	−0.462003	+	0.886878i	$0.652869\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.3110	−0.378931
$892$	0	0
$893$	13.4926	0.451514
$894$	0	0
$895$	6.24363	0.208701
$896$	0	0
$897$	8.71379	0.290945
$898$	0	0
$899$	−43.2770	−1.44337
$900$	0	0
$901$	23.0288	0.767200
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.92154	−0.196839
$906$	0	0
$907$	37.5018	1.24523	0.622614	−	0.782529i	$-0.286071\pi$
$907$	37.5018	1.24523	0.622614	+	0.782529i	$0.286071\pi$
$908$	0	0
$909$	−31.2435	−1.03628
$910$	0	0
$911$	−33.8678	−1.12209	−0.561046	−	0.827785i	$-0.689601\pi$
$911$	−33.8678	−1.12209	−0.561046	+	0.827785i	$0.689601\pi$
$912$	0	0
$913$	−20.5975	−0.681678
$914$	0	0
$915$	11.3289	0.374522
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−33.5254	−1.10590	−0.552951	−	0.833214i	$-0.686498\pi$
$919$	−33.5254	−1.10590	−0.552951	+	0.833214i	$0.686498\pi$
$920$	0	0
$921$	−16.1172	−0.531080
$922$	0	0
$923$	10.1336	0.333552
$924$	0	0
$925$	−47.0863	−1.54819
$926$	0	0
$927$	−47.5623	−1.56215
$928$	0	0
$929$	33.8917	1.11195	0.555976	−	0.831198i	$-0.312345\pi$
$929$	33.8917	1.11195	0.555976	+	0.831198i	$0.312345\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20.6112	−0.674779
$934$	0	0
$935$	3.10992	0.101705
$936$	0	0
$937$	18.2412	0.595913	0.297956	−	0.954580i	$-0.403695\pi$
$937$	18.2412	0.595913	0.297956	+	0.954580i	$0.403695\pi$
$938$	0	0
$939$	−79.8055	−2.60435
$940$	0	0
$941$	−51.2960	−1.67220	−0.836101	−	0.548576i	$-0.815170\pi$
$941$	−51.2960	−1.67220	−0.836101	+	0.548576i	$0.815170\pi$
$942$	0	0
$943$	−23.5932	−0.768301
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6980	1.51748	0.758740	−	0.651393i	$-0.225815\pi$
$947$	46.6980	1.51748	0.758740	+	0.651393i	$0.225815\pi$
$948$	0	0
$949$	−11.5737	−0.375700
$950$	0	0
$951$	−76.5622	−2.48270
$952$	0	0
$953$	1.69633	0.0549496	0.0274748	−	0.999622i	$-0.491253\pi$
$953$	1.69633	0.0549496	0.0274748	+	0.999622i	$0.491253\pi$
$954$	0	0
$955$	1.73354	0.0560961
$956$	0	0
$957$	−25.5771	−0.826791
$958$	0	0
$959$	0	0
$960$	0	0
$961$	13.9530	0.450097
$962$	0	0
$963$	15.8388	0.510397
$964$	0	0
$965$	−4.30760	−0.138667
$966$	0	0
$967$	−21.8495	−0.702633	−0.351317	−	0.936257i	$-0.614266\pi$
$967$	−21.8495	−0.702633	−0.351317	+	0.936257i	$0.614266\pi$
$968$	0	0
$969$	−24.7231	−0.794221
$970$	0	0
$971$	−21.0894	−0.676790	−0.338395	−	0.941004i	$-0.609884\pi$
$971$	−21.0894	−0.676790	−0.338395	+	0.941004i	$0.609884\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−9.75600	−0.312442
$976$	0	0
$977$	−28.7385	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$977$	−28.7385	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$978$	0	0
$979$	18.4369	0.589245
$980$	0	0
$981$	27.9433	0.892161
$982$	0	0
$983$	−7.37248	−0.235145	−0.117573	−	0.993064i	$-0.537511\pi$
$983$	−7.37248	−0.235145	−0.117573	+	0.993064i	$0.537511\pi$
$984$	0	0
$985$	2.26973	0.0723196
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.61894	0.274066
$990$	0	0
$991$	20.9729	0.666225	0.333112	−	0.942887i	$-0.391901\pi$
$991$	20.9729	0.666225	0.333112	+	0.942887i	$0.391901\pi$
$992$	0	0
$993$	−25.4107	−0.806384
$994$	0	0
$995$	5.65220	0.179187
$996$	0	0
$997$	−2.35298	−0.0745197	−0.0372598	−	0.999306i	$-0.511863\pi$
$997$	−2.35298	−0.0745197	−0.0372598	+	0.999306i	$0.511863\pi$
$998$	0	0
$999$	12.1506	0.384429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1372.2.a.c.1.6	yes	6
4.3	odd	2		5488.2.a.j.1.1		6
7.2	even	3		1372.2.e.b.361.1		12
7.3	odd	6		1372.2.e.b.1353.6		12
7.4	even	3		1372.2.e.b.1353.1		12
7.5	odd	6		1372.2.e.b.361.6		12
7.6	odd	2	inner	1372.2.a.c.1.1	✓	6
28.27	even	2		5488.2.a.j.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1372.2.a.c.1.1	✓	6	7.6	odd	2	inner
1372.2.a.c.1.6	yes	6	1.1	even	1	trivial
1372.2.e.b.361.1		12	7.2	even	3
1372.2.e.b.361.6		12	7.5	odd	6
1372.2.e.b.1353.1		12	7.4	even	3
1372.2.e.b.1353.6		12	7.3	odd	6
5488.2.a.j.1.1		6	4.3	odd	2
5488.2.a.j.1.6		6	28.27	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:	\( N \)	\(=\)	\( 1372 = 2^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1372.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54832 q^{3} +0.349282 q^{5} +3.49396 q^{9} +1.55496 q^{11} +0.784829 q^{13} +0.890084 q^{15} +5.72603 q^{17} -1.69432 q^{19} +4.35690 q^{23} -4.87800 q^{25} +1.25877 q^{27} -6.45473 q^{29} +6.70470 q^{31} +3.96254 q^{33} +9.65279 q^{37} +2.00000 q^{39} -5.41514 q^{41} +1.97823 q^{43} +1.22038 q^{45} -7.96347 q^{47} +14.5918 q^{51} +4.02177 q^{53} +0.543119 q^{55} -4.31767 q^{57} -6.29994 q^{59} +12.7279 q^{61} +0.274127 q^{65} +2.25906 q^{67} +11.1028 q^{69} +12.9119 q^{71} -14.7468 q^{73} -12.4307 q^{75} +2.02177 q^{79} -7.27413 q^{81} -13.2464 q^{83} +2.00000 q^{85} -16.4487 q^{87} +11.8568 q^{89} +17.0858 q^{93} -0.591794 q^{95} -8.80378 q^{97} +5.43296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} + 10 q^{11} + 4 q^{15} + 18 q^{23} + 10 q^{25} + 6 q^{29} + 22 q^{37} + 12 q^{39} + 18 q^{43} + 32 q^{51} + 18 q^{53} + 8 q^{57} - 20 q^{65} + 42 q^{67} + 70 q^{71} + 6 q^{79} - 22 q^{81}+ \cdots - 6 q^{99}+O(q^{100}) \) 6 * q + 2 * q^9 + 10 * q^11 + 4 * q^15 + 18 * q^23 + 10 * q^25 + 6 * q^29 + 22 * q^37 + 12 * q^39 + 18 * q^43 + 32 * q^51 + 18 * q^53 + 8 * q^57 - 20 * q^65 + 42 * q^67 + 70 * q^71 + 6 * q^79 - 22 * q^81 + 12 * q^85 + 28 * q^93 + 52 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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