LMFDB - Embedded newform 2320.2.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(1,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,1,0,5,0,1,0,12,0,4] 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:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6083172.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 $ x^5 - x^4 - 13x^3 + 10x^2 + 40*x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.689426$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.689426 q^{3} +1.00000 q^{5} -4.55109 q^{7} -2.52469 q^{9} +0.973602 q^{11} +2.28418 q^{13} +0.689426 q^{15} +5.82707 q^{17} +0.302375 q^{19} -3.13764 q^{21} +1.41345 q^{23} +1.00000 q^{25} -3.80887 q^{27} -1.00000 q^{29} +2.55109 q^{31} +0.671227 q^{33} -4.55109 q^{35} +1.02640 q^{37} +1.57477 q^{39} +4.65483 q^{41} +3.92994 q^{43} -2.52469 q^{45} -1.07648 q^{47} +13.7124 q^{49} +4.01733 q^{51} -4.85075 q^{53} +0.973602 q^{55} +0.208465 q^{57} -1.66303 q^{59} +12.1969 q^{61} +11.4901 q^{63} +2.28418 q^{65} -0.352455 q^{67} +0.974469 q^{69} +6.94651 q^{71} +16.9793 q^{73} +0.689426 q^{75} -4.43095 q^{77} -9.96540 q^{79} +4.94814 q^{81} +13.9201 q^{83} +5.82707 q^{85} -0.689426 q^{87} +4.82690 q^{89} -10.3955 q^{91} +1.75879 q^{93} +0.302375 q^{95} +13.7388 q^{97} -2.45805 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 5 q^{5} + q^{7} + 12 q^{9} + 4 q^{11} + 13 q^{13} + q^{15} + 3 q^{17} + 8 q^{21} + 7 q^{23} + 5 q^{25} + 4 q^{27} - 5 q^{29} - 11 q^{31} + 4 q^{33} + q^{35} + 6 q^{37} - 21 q^{39} + 16 q^{41}+ \cdots - 10 q^{99}+O(q^{100}) $ 5 * q + q^3 + 5 * q^5 + q^7 + 12 * q^9 + 4 * q^11 + 13 * q^13 + q^15 + 3 * q^17 + 8 * q^21 + 7 * q^23 + 5 * q^25 + 4 * q^27 - 5 * q^29 - 11 * q^31 + 4 * q^33 + q^35 + 6 * q^37 - 21 * q^39 + 16 * q^41 - 9 * q^43 + 12 * q^45 - 2 * q^47 + 16 * q^49 - 2 * q^51 + 7 * q^53 + 4 * q^55 + 2 * q^57 - 5 * q^59 + 7 * q^61 + 28 * q^63 + 13 * q^65 + 4 * q^67 + 23 * q^69 - 12 * q^71 + 7 * q^73 + q^75 + 26 * q^77 - 45 * q^79 - 7 * q^81 + 22 * q^83 + 3 * q^85 - q^87 + 24 * q^89 + 20 * q^91 - 10 * q^93 + 17 * q^97 - 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.689426	0.398040	0.199020	−	0.979995i	$-0.436224\pi$
$3$	0.689426	0.398040	0.199020	+	0.979995i	$0.436224\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.55109	−1.72015	−0.860075	−	0.510168i	$-0.829583\pi$
$7$	−4.55109	−1.72015	−0.860075	+	0.510168i	$0.829583\pi$
$8$	0	0
$9$	−2.52469	−0.841564
$10$	0	0
$11$	0.973602	0.293552	0.146776	−	0.989170i	$-0.453110\pi$
$11$	0.973602	0.293552	0.146776	+	0.989170i	$0.453110\pi$
$12$	0	0
$13$	2.28418	0.633516	0.316758	−	0.948506i	$-0.397406\pi$
$13$	2.28418	0.633516	0.316758	+	0.948506i	$0.397406\pi$
$14$	0	0
$15$	0.689426	0.178009
$16$	0	0
$17$	5.82707	1.41327	0.706636	−	0.707578i	$-0.250212\pi$
$17$	5.82707	1.41327	0.706636	+	0.707578i	$0.250212\pi$
$18$	0	0
$19$	0.302375	0.0693697	0.0346848	−	0.999398i	$-0.488957\pi$
$19$	0.302375	0.0693697	0.0346848	+	0.999398i	$0.488957\pi$
$20$	0	0
$21$	−3.13764	−0.684689
$22$	0	0
$23$	1.41345	0.294724	0.147362	−	0.989083i	$-0.452922\pi$
$23$	1.41345	0.294724	0.147362	+	0.989083i	$0.452922\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.80887	−0.733017
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.55109	0.458189	0.229095	−	0.973404i	$-0.426423\pi$
$31$	2.55109	0.458189	0.229095	+	0.973404i	$0.426423\pi$
$32$	0	0
$33$	0.671227	0.116846
$34$	0	0
$35$	−4.55109	−0.769274
$36$	0	0
$37$	1.02640	0.168739	0.0843694	−	0.996435i	$-0.473112\pi$
$37$	1.02640	0.168739	0.0843694	+	0.996435i	$0.473112\pi$
$38$	0	0
$39$	1.57477	0.252165
$40$	0	0
$41$	4.65483	0.726962	0.363481	−	0.931602i	$-0.381588\pi$
$41$	4.65483	0.726962	0.363481	+	0.931602i	$0.381588\pi$
$42$	0	0
$43$	3.92994	0.599310	0.299655	−	0.954048i	$-0.403128\pi$
$43$	3.92994	0.599310	0.299655	+	0.954048i	$0.403128\pi$
$44$	0	0
$45$	−2.52469	−0.376359
$46$	0	0
$47$	−1.07648	−0.157020	−0.0785102	−	0.996913i	$-0.525016\pi$
$47$	−1.07648	−0.157020	−0.0785102	+	0.996913i	$0.525016\pi$
$48$	0	0
$49$	13.7124	1.95892
$50$	0	0
$51$	4.01733	0.562539
$52$	0	0
$53$	−4.85075	−0.666302	−0.333151	−	0.942874i	$-0.608112\pi$
$53$	−4.85075	−0.666302	−0.333151	+	0.942874i	$0.608112\pi$
$54$	0	0
$55$	0.973602	0.131281
$56$	0	0
$57$	0.208465	0.0276119
$58$	0	0
$59$	−1.66303	−0.216508	−0.108254	−	0.994123i	$-0.534526\pi$
$59$	−1.66303	−0.216508	−0.108254	+	0.994123i	$0.534526\pi$
$60$	0	0
$61$	12.1969	1.56165	0.780824	−	0.624752i	$-0.214800\pi$
$61$	12.1969	1.56165	0.780824	+	0.624752i	$0.214800\pi$
$62$	0	0
$63$	11.4901	1.44762
$64$	0	0
$65$	2.28418	0.283317
$66$	0	0
$67$	−0.352455	−0.0430592	−0.0215296	−	0.999768i	$-0.506854\pi$
$67$	−0.352455	−0.0430592	−0.0215296	+	0.999768i	$0.506854\pi$
$68$	0	0
$69$	0.974469	0.117312
$70$	0	0
$71$	6.94651	0.824399	0.412199	−	0.911094i	$-0.364761\pi$
$71$	6.94651	0.824399	0.412199	+	0.911094i	$0.364761\pi$
$72$	0	0
$73$	16.9793	1.98728	0.993640	−	0.112605i	$-0.0359195\pi$
$73$	16.9793	1.98728	0.993640	+	0.112605i	$0.0359195\pi$
$74$	0	0
$75$	0.689426	0.0796081
$76$	0	0
$77$	−4.43095	−0.504954
$78$	0	0
$79$	−9.96540	−1.12120	−0.560598	−	0.828088i	$-0.689429\pi$
$79$	−9.96540	−1.12120	−0.560598	+	0.828088i	$0.689429\pi$
$80$	0	0
$81$	4.94814	0.549793
$82$	0	0
$83$	13.9201	1.52793	0.763965	−	0.645257i	$-0.223250\pi$
$83$	13.9201	1.52793	0.763965	+	0.645257i	$0.223250\pi$
$84$	0	0
$85$	5.82707	0.632034
$86$	0	0
$87$	−0.689426	−0.0739143
$88$	0	0
$89$	4.82690	0.511650	0.255825	−	0.966723i	$-0.417653\pi$
$89$	4.82690	0.511650	0.255825	+	0.966723i	$0.417653\pi$
$90$	0	0
$91$	−10.3955	−1.08974
$92$	0	0
$93$	1.75879	0.182378
$94$	0	0
$95$	0.302375	0.0310231
$96$	0	0
$97$	13.7388	1.39496	0.697482	−	0.716602i	$-0.254304\pi$
$97$	13.7388	1.39496	0.697482	+	0.716602i	$0.254304\pi$
$98$	0	0
$99$	−2.45805	−0.247043
$100$	0	0
$101$	1.50736	0.149988	0.0749939	−	0.997184i	$-0.476106\pi$
$101$	1.50736	0.149988	0.0749939	+	0.997184i	$0.476106\pi$
$102$	0	0
$103$	3.64755	0.359403	0.179702	−	0.983721i	$-0.442487\pi$
$103$	3.64755	0.359403	0.179702	+	0.983721i	$0.442487\pi$
$104$	0	0
$105$	−3.13764	−0.306202
$106$	0	0
$107$	−8.78341	−0.849124	−0.424562	−	0.905399i	$-0.639572\pi$
$107$	−8.78341	−0.849124	−0.424562	+	0.905399i	$0.639572\pi$
$108$	0	0
$109$	3.34517	0.320409	0.160205	−	0.987084i	$-0.448785\pi$
$109$	3.34517	0.320409	0.160205	+	0.987084i	$0.448785\pi$
$110$	0	0
$111$	0.707626	0.0671648
$112$	0	0
$113$	7.79160	0.732972	0.366486	−	0.930423i	$-0.380561\pi$
$113$	7.79160	0.732972	0.366486	+	0.930423i	$0.380561\pi$
$114$	0	0
$115$	1.41345	0.131805
$116$	0	0
$117$	−5.76684	−0.533145
$118$	0	0
$119$	−26.5195	−2.43104
$120$	0	0
$121$	−10.0521	−0.913827
$122$	0	0
$123$	3.20916	0.289360
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.5231	1.11125	0.555624	−	0.831434i	$-0.312479\pi$
$127$	12.5231	1.11125	0.555624	+	0.831434i	$0.312479\pi$
$128$	0	0
$129$	2.70941	0.238550
$130$	0	0
$131$	−7.24617	−0.633101	−0.316550	−	0.948576i	$-0.602525\pi$
$131$	−7.24617	−0.633101	−0.316550	+	0.948576i	$0.602525\pi$
$132$	0	0
$133$	−1.37614	−0.119326
$134$	0	0
$135$	−3.80887	−0.327815
$136$	0	0
$137$	−16.1170	−1.37697	−0.688483	−	0.725253i	$-0.741723\pi$
$137$	−16.1170	−1.37697	−0.688483	+	0.725253i	$0.741723\pi$
$138$	0	0
$139$	5.36902	0.455394	0.227697	−	0.973732i	$-0.426880\pi$
$139$	5.36902	0.455394	0.227697	+	0.973732i	$0.426880\pi$
$140$	0	0
$141$	−0.742152	−0.0625005
$142$	0	0
$143$	2.22388	0.185970
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	9.45370	0.779728
$148$	0	0
$149$	−18.0824	−1.48137	−0.740683	−	0.671855i	$-0.765498\pi$
$149$	−18.0824	−1.48137	−0.740683	+	0.671855i	$0.765498\pi$
$150$	0	0
$151$	−10.3782	−0.844562	−0.422281	−	0.906465i	$-0.638771\pi$
$151$	−10.3782	−0.844562	−0.422281	+	0.906465i	$0.638771\pi$
$152$	0	0
$153$	−14.7115	−1.18936
$154$	0	0
$155$	2.55109	0.204908
$156$	0	0
$157$	0.923523	0.0737051	0.0368526	−	0.999321i	$-0.488267\pi$
$157$	0.923523	0.0737051	0.0368526	+	0.999321i	$0.488267\pi$
$158$	0	0
$159$	−3.34423	−0.265215
$160$	0	0
$161$	−6.43273	−0.506970
$162$	0	0
$163$	22.9695	1.79911	0.899555	−	0.436808i	$-0.143891\pi$
$163$	22.9695	1.79911	0.899555	+	0.436808i	$0.143891\pi$
$164$	0	0
$165$	0.671227	0.0522550
$166$	0	0
$167$	5.79067	0.448095	0.224048	−	0.974578i	$-0.428073\pi$
$167$	5.79067	0.448095	0.224048	+	0.974578i	$0.428073\pi$
$168$	0	0
$169$	−7.78254	−0.598657
$170$	0	0
$171$	−0.763404	−0.0583790
$172$	0	0
$173$	−1.14747	−0.0872407	−0.0436203	−	0.999048i	$-0.513889\pi$
$173$	−1.14747	−0.0872407	−0.0436203	+	0.999048i	$0.513889\pi$
$174$	0	0
$175$	−4.55109	−0.344030
$176$	0	0
$177$	−1.14654	−0.0861789
$178$	0	0
$179$	3.95882	0.295896	0.147948	−	0.988995i	$-0.452733\pi$
$179$	3.95882	0.295896	0.147948	+	0.988995i	$0.452733\pi$
$180$	0	0
$181$	−11.2068	−0.832994	−0.416497	−	0.909137i	$-0.636742\pi$
$181$	−11.2068	−0.832994	−0.416497	+	0.909137i	$0.636742\pi$
$182$	0	0
$183$	8.40883	0.621599
$184$	0	0
$185$	1.02640	0.0754623
$186$	0	0
$187$	5.67325	0.414869
$188$	0	0
$189$	17.3345	1.26090
$190$	0	0
$191$	23.3047	1.68627	0.843134	−	0.537704i	$-0.180708\pi$
$191$	23.3047	1.68627	0.843134	+	0.537704i	$0.180708\pi$
$192$	0	0
$193$	−14.8383	−1.06808	−0.534041	−	0.845459i	$-0.679327\pi$
$193$	−14.8383	−1.06808	−0.534041	+	0.845459i	$0.679327\pi$
$194$	0	0
$195$	1.57477	0.112772
$196$	0	0
$197$	−21.9556	−1.56427	−0.782137	−	0.623106i	$-0.785871\pi$
$197$	−21.9556	−1.56427	−0.782137	+	0.623106i	$0.785871\pi$
$198$	0	0
$199$	−24.8592	−1.76222	−0.881110	−	0.472910i	$-0.843203\pi$
$199$	−24.8592	−1.76222	−0.881110	+	0.472910i	$0.843203\pi$
$200$	0	0
$201$	−0.242992	−0.0171393
$202$	0	0
$203$	4.55109	0.319424
$204$	0	0
$205$	4.65483	0.325107
$206$	0	0
$207$	−3.56852	−0.248029
$208$	0	0
$209$	0.294393	0.0203636
$210$	0	0
$211$	0.368852	0.0253928	0.0126964	−	0.999919i	$-0.495959\pi$
$211$	0.368852	0.0253928	0.0126964	+	0.999919i	$0.495959\pi$
$212$	0	0
$213$	4.78911	0.328144
$214$	0	0
$215$	3.92994	0.268020
$216$	0	0
$217$	−11.6102	−0.788154
$218$	0	0
$219$	11.7060	0.791018
$220$	0	0
$221$	13.3100	0.895331
$222$	0	0
$223$	−16.6853	−1.11733	−0.558666	−	0.829393i	$-0.688687\pi$
$223$	−16.6853	−1.11733	−0.558666	+	0.829393i	$0.688687\pi$
$224$	0	0
$225$	−2.52469	−0.168313
$226$	0	0
$227$	12.6805	0.841636	0.420818	−	0.907145i	$-0.361743\pi$
$227$	12.6805	0.841636	0.420818	+	0.907145i	$0.361743\pi$
$228$	0	0
$229$	17.5274	1.15824	0.579122	−	0.815241i	$-0.303396\pi$
$229$	17.5274	1.15824	0.579122	+	0.815241i	$0.303396\pi$
$230$	0	0
$231$	−3.05481	−0.200992
$232$	0	0
$233$	20.0988	1.31671	0.658357	−	0.752706i	$-0.271252\pi$
$233$	20.0988	1.31671	0.658357	+	0.752706i	$0.271252\pi$
$234$	0	0
$235$	−1.07648	−0.0702216
$236$	0	0
$237$	−6.87041	−0.446281
$238$	0	0
$239$	3.42893	0.221799	0.110900	−	0.993832i	$-0.464627\pi$
$239$	3.42893	0.221799	0.110900	+	0.993832i	$0.464627\pi$
$240$	0	0
$241$	−27.2126	−1.75291	−0.876457	−	0.481479i	$-0.840100\pi$
$241$	−27.2126	−1.75291	−0.876457	+	0.481479i	$0.840100\pi$
$242$	0	0
$243$	14.8380	0.951857
$244$	0	0
$245$	13.7124	0.876054
$246$	0	0
$247$	0.690678	0.0439468
$248$	0	0
$249$	9.59689	0.608178
$250$	0	0
$251$	19.6604	1.24095	0.620476	−	0.784225i	$-0.286939\pi$
$251$	19.6604	1.24095	0.620476	+	0.784225i	$0.286939\pi$
$252$	0	0
$253$	1.37614	0.0865170
$254$	0	0
$255$	4.01733	0.251575
$256$	0	0
$257$	12.6006	0.786006	0.393003	−	0.919537i	$-0.371436\pi$
$257$	12.6006	0.786006	0.393003	+	0.919537i	$0.371436\pi$
$258$	0	0
$259$	−4.67123	−0.290256
$260$	0	0
$261$	2.52469	0.156274
$262$	0	0
$263$	−12.5905	−0.776363	−0.388182	−	0.921583i	$-0.626897\pi$
$263$	−12.5905	−0.776363	−0.388182	+	0.921583i	$0.626897\pi$
$264$	0	0
$265$	−4.85075	−0.297979
$266$	0	0
$267$	3.32779	0.203657
$268$	0	0
$269$	−1.31894	−0.0804173	−0.0402086	−	0.999191i	$-0.512802\pi$
$269$	−1.31894	−0.0804173	−0.0402086	+	0.999191i	$0.512802\pi$
$270$	0	0
$271$	9.43824	0.573332	0.286666	−	0.958031i	$-0.407453\pi$
$271$	9.43824	0.573332	0.286666	+	0.958031i	$0.407453\pi$
$272$	0	0
$273$	−7.16692	−0.433762
$274$	0	0
$275$	0.973602	0.0587104
$276$	0	0
$277$	9.15226	0.549906	0.274953	−	0.961458i	$-0.411338\pi$
$277$	9.15226	0.549906	0.274953	+	0.961458i	$0.411338\pi$
$278$	0	0
$279$	−6.44071	−0.385595
$280$	0	0
$281$	−13.8164	−0.824215	−0.412108	−	0.911135i	$-0.635207\pi$
$281$	−13.8164	−0.824215	−0.412108	+	0.911135i	$0.635207\pi$
$282$	0	0
$283$	28.5404	1.69655	0.848276	−	0.529555i	$-0.177641\pi$
$283$	28.5404	1.69655	0.848276	+	0.529555i	$0.177641\pi$
$284$	0	0
$285$	0.208465	0.0123484
$286$	0	0
$287$	−21.1845	−1.25048
$288$	0	0
$289$	16.9547	0.997336
$290$	0	0
$291$	9.47190	0.555252
$292$	0	0
$293$	−9.92011	−0.579539	−0.289770	−	0.957096i	$-0.593579\pi$
$293$	−9.92011	−0.579539	−0.289770	+	0.957096i	$0.593579\pi$
$294$	0	0
$295$	−1.66303	−0.0968253
$296$	0	0
$297$	−3.70832	−0.215179
$298$	0	0
$299$	3.22857	0.186713
$300$	0	0
$301$	−17.8855	−1.03090
$302$	0	0
$303$	1.03921	0.0597012
$304$	0	0
$305$	12.1969	0.698390
$306$	0	0
$307$	8.52313	0.486441	0.243220	−	0.969971i	$-0.421796\pi$
$307$	8.52313	0.486441	0.243220	+	0.969971i	$0.421796\pi$
$308$	0	0
$309$	2.51471	0.143057
$310$	0	0
$311$	−16.4437	−0.932438	−0.466219	−	0.884669i	$-0.654384\pi$
$311$	−16.4437	−0.932438	−0.466219	+	0.884669i	$0.654384\pi$
$312$	0	0
$313$	−5.68964	−0.321598	−0.160799	−	0.986987i	$-0.551407\pi$
$313$	−5.68964	−0.321598	−0.160799	+	0.986987i	$0.551407\pi$
$314$	0	0
$315$	11.4901	0.647394
$316$	0	0
$317$	−9.35447	−0.525400	−0.262700	−	0.964878i	$-0.584613\pi$
$317$	−9.35447	−0.525400	−0.262700	+	0.964878i	$0.584613\pi$
$318$	0	0
$319$	−0.973602	−0.0545113
$320$	0	0
$321$	−6.05551	−0.337986
$322$	0	0
$323$	1.76196	0.0980381
$324$	0	0
$325$	2.28418	0.126703
$326$	0	0
$327$	2.30625	0.127536
$328$	0	0
$329$	4.89914	0.270099
$330$	0	0
$331$	−34.1245	−1.87565	−0.937825	−	0.347108i	$-0.887164\pi$
$331$	−34.1245	−1.87565	−0.937825	+	0.347108i	$0.887164\pi$
$332$	0	0
$333$	−2.59134	−0.142004
$334$	0	0
$335$	−0.352455	−0.0192567
$336$	0	0
$337$	27.4240	1.49388	0.746939	−	0.664893i	$-0.231523\pi$
$337$	27.4240	1.49388	0.746939	+	0.664893i	$0.231523\pi$
$338$	0	0
$339$	5.37174	0.291753
$340$	0	0
$341$	2.48375	0.134502
$342$	0	0
$343$	−30.5488	−1.64948
$344$	0	0
$345$	0.974469	0.0524636
$346$	0	0
$347$	14.8362	0.796449	0.398225	−	0.917288i	$-0.369626\pi$
$347$	14.8362	0.796449	0.398225	+	0.917288i	$0.369626\pi$
$348$	0	0
$349$	9.61773	0.514826	0.257413	−	0.966302i	$-0.417130\pi$
$349$	9.61773	0.514826	0.257413	+	0.966302i	$0.417130\pi$
$350$	0	0
$351$	−8.70012	−0.464378
$352$	0	0
$353$	−20.6698	−1.10014	−0.550072	−	0.835117i	$-0.685400\pi$
$353$	−20.6698	−1.10014	−0.550072	+	0.835117i	$0.685400\pi$
$354$	0	0
$355$	6.94651	0.368682
$356$	0	0
$357$	−18.2832	−0.967652
$358$	0	0
$359$	16.8577	0.889717	0.444858	−	0.895601i	$-0.353254\pi$
$359$	16.8577	0.889717	0.444858	+	0.895601i	$0.353254\pi$
$360$	0	0
$361$	−18.9086	−0.995188
$362$	0	0
$363$	−6.93018	−0.363740
$364$	0	0
$365$	16.9793	0.888738
$366$	0	0
$367$	6.45533	0.336965	0.168483	−	0.985705i	$-0.446113\pi$
$367$	6.45533	0.336965	0.168483	+	0.985705i	$0.446113\pi$
$368$	0	0
$369$	−11.7520	−0.611785
$370$	0	0
$371$	22.0762	1.14614
$372$	0	0
$373$	−26.7447	−1.38479	−0.692395	−	0.721518i	$-0.743445\pi$
$373$	−26.7447	−1.38479	−0.692395	+	0.721518i	$0.743445\pi$
$374$	0	0
$375$	0.689426	0.0356018
$376$	0	0
$377$	−2.28418	−0.117641
$378$	0	0
$379$	12.6260	0.648554	0.324277	−	0.945962i	$-0.394879\pi$
$379$	12.6260	0.648554	0.324277	+	0.945962i	$0.394879\pi$
$380$	0	0
$381$	8.63377	0.442322
$382$	0	0
$383$	5.29596	0.270611	0.135305	−	0.990804i	$-0.456798\pi$
$383$	5.29596	0.270611	0.135305	+	0.990804i	$0.456798\pi$
$384$	0	0
$385$	−4.43095	−0.225822
$386$	0	0
$387$	−9.92189	−0.504358
$388$	0	0
$389$	−7.63685	−0.387204	−0.193602	−	0.981080i	$-0.562017\pi$
$389$	−7.63685	−0.387204	−0.193602	+	0.981080i	$0.562017\pi$
$390$	0	0
$391$	8.23626	0.416526
$392$	0	0
$393$	−4.99570	−0.252000
$394$	0	0
$395$	−9.96540	−0.501414
$396$	0	0
$397$	10.5090	0.527431	0.263716	−	0.964600i	$-0.415052\pi$
$397$	10.5090	0.527431	0.263716	+	0.964600i	$0.415052\pi$
$398$	0	0
$399$	−0.948745	−0.0474967
$400$	0	0
$401$	−31.3631	−1.56620	−0.783100	−	0.621896i	$-0.786363\pi$
$401$	−31.3631	−1.56620	−0.783100	+	0.621896i	$0.786363\pi$
$402$	0	0
$403$	5.82714	0.290270
$404$	0	0
$405$	4.94814	0.245875
$406$	0	0
$407$	0.999303	0.0495336
$408$	0	0
$409$	−4.79139	−0.236919	−0.118459	−	0.992959i	$-0.537796\pi$
$409$	−4.79139	−0.236919	−0.118459	+	0.992959i	$0.537796\pi$
$410$	0	0
$411$	−11.1115	−0.548088
$412$	0	0
$413$	7.56859	0.372426
$414$	0	0
$415$	13.9201	0.683311
$416$	0	0
$417$	3.70154	0.181265
$418$	0	0
$419$	2.03068	0.0992050	0.0496025	−	0.998769i	$-0.484205\pi$
$419$	2.03068	0.0992050	0.0496025	+	0.998769i	$0.484205\pi$
$420$	0	0
$421$	−8.13601	−0.396525	−0.198262	−	0.980149i	$-0.563530\pi$
$421$	−8.13601	−0.396525	−0.198262	+	0.980149i	$0.563530\pi$
$422$	0	0
$423$	2.71777	0.132143
$424$	0	0
$425$	5.82707	0.282654
$426$	0	0
$427$	−55.5090	−2.68627
$428$	0	0
$429$	1.53320	0.0740236
$430$	0	0
$431$	−30.7727	−1.48227	−0.741135	−	0.671356i	$-0.765712\pi$
$431$	−30.7727	−1.48227	−0.741135	+	0.671356i	$0.765712\pi$
$432$	0	0
$433$	10.1231	0.486487	0.243244	−	0.969965i	$-0.421789\pi$
$433$	10.1231	0.486487	0.243244	+	0.969965i	$0.421789\pi$
$434$	0	0
$435$	−0.689426	−0.0330555
$436$	0	0
$437$	0.427392	0.0204449
$438$	0	0
$439$	−22.9298	−1.09438	−0.547189	−	0.837009i	$-0.684302\pi$
$439$	−22.9298	−1.09438	−0.547189	+	0.837009i	$0.684302\pi$
$440$	0	0
$441$	−34.6196	−1.64855
$442$	0	0
$443$	−10.0833	−0.479071	−0.239536	−	0.970888i	$-0.576995\pi$
$443$	−10.0833	−0.479071	−0.239536	+	0.970888i	$0.576995\pi$
$444$	0	0
$445$	4.82690	0.228817
$446$	0	0
$447$	−12.4665	−0.589643
$448$	0	0
$449$	15.3911	0.726353	0.363176	−	0.931720i	$-0.381692\pi$
$449$	15.3911	0.726353	0.363176	+	0.931720i	$0.381692\pi$
$450$	0	0
$451$	4.53195	0.213401
$452$	0	0
$453$	−7.15497	−0.336170
$454$	0	0
$455$	−10.3955	−0.487348
$456$	0	0
$457$	−9.15226	−0.428125	−0.214062	−	0.976820i	$-0.568670\pi$
$457$	−9.15226	−0.428125	−0.214062	+	0.976820i	$0.568670\pi$
$458$	0	0
$459$	−22.1945	−1.03595
$460$	0	0
$461$	16.9227	0.788167	0.394083	−	0.919075i	$-0.371062\pi$
$461$	16.9227	0.788167	0.394083	+	0.919075i	$0.371062\pi$
$462$	0	0
$463$	32.1236	1.49291	0.746456	−	0.665435i	$-0.231754\pi$
$463$	32.1236	1.49291	0.746456	+	0.665435i	$0.231754\pi$
$464$	0	0
$465$	1.75879	0.0815618
$466$	0	0
$467$	−39.0981	−1.80924	−0.904622	−	0.426215i	$-0.859847\pi$
$467$	−39.0981	−1.80924	−0.904622	+	0.426215i	$0.859847\pi$
$468$	0	0
$469$	1.60405	0.0740683
$470$	0	0
$471$	0.636701	0.0293376
$472$	0	0
$473$	3.82620	0.175929
$474$	0	0
$475$	0.302375	0.0138739
$476$	0	0
$477$	12.2466	0.560735
$478$	0	0
$479$	13.4631	0.615143	0.307572	−	0.951525i	$-0.400484\pi$
$479$	13.4631	0.615143	0.307572	+	0.951525i	$0.400484\pi$
$480$	0	0
$481$	2.34447	0.106899
$482$	0	0
$483$	−4.43489	−0.201795
$484$	0	0
$485$	13.7388	0.623847
$486$	0	0
$487$	−35.1691	−1.59366	−0.796831	−	0.604202i	$-0.793492\pi$
$487$	−35.1691	−1.59366	−0.796831	+	0.604202i	$0.793492\pi$
$488$	0	0
$489$	15.8358	0.716118
$490$	0	0
$491$	20.3333	0.917631	0.458815	−	0.888532i	$-0.348274\pi$
$491$	20.3333	0.917631	0.458815	+	0.888532i	$0.348274\pi$
$492$	0	0
$493$	−5.82707	−0.262438
$494$	0	0
$495$	−2.45805	−0.110481
$496$	0	0
$497$	−31.6142	−1.41809
$498$	0	0
$499$	−17.2453	−0.772004	−0.386002	−	0.922498i	$-0.626144\pi$
$499$	−17.2453	−0.772004	−0.386002	+	0.922498i	$0.626144\pi$
$500$	0	0
$501$	3.99224	0.178360
$502$	0	0
$503$	18.5377	0.826555	0.413278	−	0.910605i	$-0.364384\pi$
$503$	18.5377	0.826555	0.413278	+	0.910605i	$0.364384\pi$
$504$	0	0
$505$	1.50736	0.0670766
$506$	0	0
$507$	−5.36549	−0.238290
$508$	0	0
$509$	0.549240	0.0243446	0.0121723	−	0.999926i	$-0.496125\pi$
$509$	0.549240	0.0243446	0.0121723	+	0.999926i	$0.496125\pi$
$510$	0	0
$511$	−77.2744	−3.41842
$512$	0	0
$513$	−1.15171	−0.0508491
$514$	0	0
$515$	3.64755	0.160730
$516$	0	0
$517$	−1.04806	−0.0460937
$518$	0	0
$519$	−0.791097	−0.0347253
$520$	0	0
$521$	22.4194	0.982211	0.491106	−	0.871100i	$-0.336593\pi$
$521$	22.4194	0.982211	0.491106	+	0.871100i	$0.336593\pi$
$522$	0	0
$523$	−26.6988	−1.16746	−0.583729	−	0.811949i	$-0.698407\pi$
$523$	−26.6988	−1.16746	−0.583729	+	0.811949i	$0.698407\pi$
$524$	0	0
$525$	−3.13764	−0.136938
$526$	0	0
$527$	14.8654	0.647546
$528$	0	0
$529$	−21.0022	−0.913138
$530$	0	0
$531$	4.19863	0.182205
$532$	0	0
$533$	10.6325	0.460543
$534$	0	0
$535$	−8.78341	−0.379740
$536$	0	0
$537$	2.72931	0.117778
$538$	0	0
$539$	13.3504	0.575044
$540$	0	0
$541$	−12.6310	−0.543048	−0.271524	−	0.962432i	$-0.587528\pi$
$541$	−12.6310	−0.543048	−0.271524	+	0.962432i	$0.587528\pi$
$542$	0	0
$543$	−7.72625	−0.331565
$544$	0	0
$545$	3.34517	0.143291
$546$	0	0
$547$	−23.4737	−1.00366	−0.501832	−	0.864965i	$-0.667341\pi$
$547$	−23.4737	−1.00366	−0.501832	+	0.864965i	$0.667341\pi$
$548$	0	0
$549$	−30.7933	−1.31423
$550$	0	0
$551$	−0.302375	−0.0128816
$552$	0	0
$553$	45.3534	1.92862
$554$	0	0
$555$	0.707626	0.0300370
$556$	0	0
$557$	−40.2073	−1.70364	−0.851819	−	0.523836i	$-0.824501\pi$
$557$	−40.2073	−1.70364	−0.851819	+	0.523836i	$0.824501\pi$
$558$	0	0
$559$	8.97668	0.379673
$560$	0	0
$561$	3.91128	0.165135
$562$	0	0
$563$	45.4917	1.91725	0.958623	−	0.284677i	$-0.0918864\pi$
$563$	45.4917	1.91725	0.958623	+	0.284677i	$0.0918864\pi$
$564$	0	0
$565$	7.79160	0.327795
$566$	0	0
$567$	−22.5194	−0.945727
$568$	0	0
$569$	−10.1338	−0.424833	−0.212416	−	0.977179i	$-0.568133\pi$
$569$	−10.1338	−0.424833	−0.212416	+	0.977179i	$0.568133\pi$
$570$	0	0
$571$	0.503946	0.0210895	0.0105447	−	0.999944i	$-0.496643\pi$
$571$	0.503946	0.0210895	0.0105447	+	0.999944i	$0.496643\pi$
$572$	0	0
$573$	16.0669	0.671203
$574$	0	0
$575$	1.41345	0.0589449
$576$	0	0
$577$	7.22914	0.300953	0.150477	−	0.988614i	$-0.451919\pi$
$577$	7.22914	0.300953	0.150477	+	0.988614i	$0.451919\pi$
$578$	0	0
$579$	−10.2299	−0.425140
$580$	0	0
$581$	−63.3517	−2.62827
$582$	0	0
$583$	−4.72270	−0.195594
$584$	0	0
$585$	−5.76684	−0.238429
$586$	0	0
$587$	46.6409	1.92508	0.962538	−	0.271146i	$-0.0874026\pi$
$587$	46.6409	1.92508	0.962538	+	0.271146i	$0.0874026\pi$
$588$	0	0
$589$	0.771386	0.0317844
$590$	0	0
$591$	−15.1368	−0.622645
$592$	0	0
$593$	38.0320	1.56179	0.780893	−	0.624665i	$-0.214765\pi$
$593$	38.0320	1.56179	0.780893	+	0.624665i	$0.214765\pi$
$594$	0	0
$595$	−26.5195	−1.08719
$596$	0	0
$597$	−17.1386	−0.701435
$598$	0	0
$599$	−4.53380	−0.185246	−0.0926231	−	0.995701i	$-0.529525\pi$
$599$	−4.53380	−0.185246	−0.0926231	+	0.995701i	$0.529525\pi$
$600$	0	0
$601$	−12.7897	−0.521701	−0.260850	−	0.965379i	$-0.584003\pi$
$601$	−12.7897	−0.521701	−0.260850	+	0.965379i	$0.584003\pi$
$602$	0	0
$603$	0.889840	0.0362371
$604$	0	0
$605$	−10.0521	−0.408676
$606$	0	0
$607$	18.5377	0.752422	0.376211	−	0.926534i	$-0.377227\pi$
$607$	18.5377	0.752422	0.376211	+	0.926534i	$0.377227\pi$
$608$	0	0
$609$	3.13764	0.127144
$610$	0	0
$611$	−2.45886	−0.0994750
$612$	0	0
$613$	−21.5184	−0.869121	−0.434560	−	0.900643i	$-0.643096\pi$
$613$	−21.5184	−0.869121	−0.434560	+	0.900643i	$0.643096\pi$
$614$	0	0
$615$	3.20916	0.129406
$616$	0	0
$617$	33.6646	1.35528	0.677642	−	0.735392i	$-0.263002\pi$
$617$	33.6646	1.35528	0.677642	+	0.735392i	$0.263002\pi$
$618$	0	0
$619$	35.9652	1.44556	0.722782	−	0.691076i	$-0.242863\pi$
$619$	35.9652	1.44556	0.722782	+	0.691076i	$0.242863\pi$
$620$	0	0
$621$	−5.38364	−0.216038
$622$	0	0
$623$	−21.9676	−0.880115
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.202962	0.00810554
$628$	0	0
$629$	5.98089	0.238474
$630$	0	0
$631$	−15.0130	−0.597657	−0.298829	−	0.954307i	$-0.596596\pi$
$631$	−15.0130	−0.597657	−0.298829	+	0.954307i	$0.596596\pi$
$632$	0	0
$633$	0.254296	0.0101074
$634$	0	0
$635$	12.5231	0.496965
$636$	0	0
$637$	31.3216	1.24101
$638$	0	0
$639$	−17.5378	−0.693784
$640$	0	0
$641$	21.2382	0.838858	0.419429	−	0.907788i	$-0.362230\pi$
$641$	21.2382	0.838858	0.419429	+	0.907788i	$0.362230\pi$
$642$	0	0
$643$	19.0214	0.750133	0.375066	−	0.926998i	$-0.377620\pi$
$643$	19.0214	0.750133	0.375066	+	0.926998i	$0.377620\pi$
$644$	0	0
$645$	2.70941	0.106683
$646$	0	0
$647$	20.1477	0.792087	0.396044	−	0.918232i	$-0.370383\pi$
$647$	20.1477	0.792087	0.396044	+	0.918232i	$0.370383\pi$
$648$	0	0
$649$	−1.61913	−0.0635564
$650$	0	0
$651$	−8.00440	−0.313717
$652$	0	0
$653$	29.0601	1.13721	0.568604	−	0.822611i	$-0.307484\pi$
$653$	29.0601	1.13721	0.568604	+	0.822611i	$0.307484\pi$
$654$	0	0
$655$	−7.24617	−0.283131
$656$	0	0
$657$	−42.8676	−1.67242
$658$	0	0
$659$	−13.0420	−0.508042	−0.254021	−	0.967199i	$-0.581753\pi$
$659$	−13.0420	−0.508042	−0.254021	+	0.967199i	$0.581753\pi$
$660$	0	0
$661$	51.0275	1.98474	0.992370	−	0.123299i	$-0.0393475\pi$
$661$	51.0275	1.98474	0.992370	+	0.123299i	$0.0393475\pi$
$662$	0	0
$663$	9.17630	0.356378
$664$	0	0
$665$	−1.37614	−0.0533643
$666$	0	0
$667$	−1.41345	−0.0547289
$668$	0	0
$669$	−11.5033	−0.444743
$670$	0	0
$671$	11.8749	0.458425
$672$	0	0
$673$	−5.96004	−0.229743	−0.114871	−	0.993380i	$-0.536646\pi$
$673$	−5.96004	−0.229743	−0.114871	+	0.993380i	$0.536646\pi$
$674$	0	0
$675$	−3.80887	−0.146603
$676$	0	0
$677$	−22.2465	−0.855001	−0.427500	−	0.904015i	$-0.640606\pi$
$677$	−22.2465	−0.855001	−0.427500	+	0.904015i	$0.640606\pi$
$678$	0	0
$679$	−62.5265	−2.39955
$680$	0	0
$681$	8.74229	0.335005
$682$	0	0
$683$	47.2308	1.80723	0.903617	−	0.428341i	$-0.140902\pi$
$683$	47.2308	1.80723	0.903617	+	0.428341i	$0.140902\pi$
$684$	0	0
$685$	−16.1170	−0.615798
$686$	0	0
$687$	12.0839	0.461028
$688$	0	0
$689$	−11.0800	−0.422113
$690$	0	0
$691$	−24.9859	−0.950509	−0.475254	−	0.879848i	$-0.657644\pi$
$691$	−24.9859	−0.950509	−0.475254	+	0.879848i	$0.657644\pi$
$692$	0	0
$693$	11.1868	0.424951
$694$	0	0
$695$	5.36902	0.203659
$696$	0	0
$697$	27.1240	1.02739
$698$	0	0
$699$	13.8566	0.524105
$700$	0	0
$701$	−15.5135	−0.585936	−0.292968	−	0.956122i	$-0.594643\pi$
$701$	−15.5135	−0.585936	−0.292968	+	0.956122i	$0.594643\pi$
$702$	0	0
$703$	0.310357	0.0117053
$704$	0	0
$705$	−0.742152	−0.0279511
$706$	0	0
$707$	−6.86012	−0.258001
$708$	0	0
$709$	44.7814	1.68180	0.840901	−	0.541189i	$-0.182026\pi$
$709$	44.7814	1.68180	0.840901	+	0.541189i	$0.182026\pi$
$710$	0	0
$711$	25.1596	0.943558
$712$	0	0
$713$	3.60583	0.135040
$714$	0	0
$715$	2.22388	0.0831684
$716$	0	0
$717$	2.36400	0.0882850
$718$	0	0
$719$	−30.9233	−1.15324	−0.576622	−	0.817011i	$-0.695629\pi$
$719$	−30.9233	−1.15324	−0.576622	+	0.817011i	$0.695629\pi$
$720$	0	0
$721$	−16.6003	−0.618228
$722$	0	0
$723$	−18.7611	−0.697731
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	44.0771	1.63473	0.817365	−	0.576121i	$-0.195434\pi$
$727$	44.0771	1.63473	0.817365	+	0.576121i	$0.195434\pi$
$728$	0	0
$729$	−4.61473	−0.170916
$730$	0	0
$731$	22.9000	0.846988
$732$	0	0
$733$	16.8154	0.621089	0.310545	−	0.950559i	$-0.399489\pi$
$733$	16.8154	0.621089	0.310545	+	0.950559i	$0.399489\pi$
$734$	0	0
$735$	9.45370	0.348705
$736$	0	0
$737$	−0.343151	−0.0126401
$738$	0	0
$739$	−32.7803	−1.20584	−0.602921	−	0.797801i	$-0.705997\pi$
$739$	−32.7803	−1.20584	−0.602921	+	0.797801i	$0.705997\pi$
$740$	0	0
$741$	0.476172	0.0174926
$742$	0	0
$743$	−1.90184	−0.0697718	−0.0348859	−	0.999391i	$-0.511107\pi$
$743$	−1.90184	−0.0697718	−0.0348859	+	0.999391i	$0.511107\pi$
$744$	0	0
$745$	−18.0824	−0.662487
$746$	0	0
$747$	−35.1440	−1.28585
$748$	0	0
$749$	39.9741	1.46062
$750$	0	0
$751$	−21.3282	−0.778277	−0.389138	−	0.921179i	$-0.627227\pi$
$751$	−21.3282	−0.778277	−0.389138	+	0.921179i	$0.627227\pi$
$752$	0	0
$753$	13.5544	0.493949
$754$	0	0
$755$	−10.3782	−0.377700
$756$	0	0
$757$	−14.5631	−0.529304	−0.264652	−	0.964344i	$-0.585257\pi$
$757$	−14.5631	−0.529304	−0.264652	+	0.964344i	$0.585257\pi$
$758$	0	0
$759$	0.948745	0.0344373
$760$	0	0
$761$	53.6193	1.94370	0.971850	−	0.235601i	$-0.0757058\pi$
$761$	53.6193	1.94370	0.971850	+	0.235601i	$0.0757058\pi$
$762$	0	0
$763$	−15.2242	−0.551152
$764$	0	0
$765$	−14.7115	−0.531897
$766$	0	0
$767$	−3.79865	−0.137161
$768$	0	0
$769$	−20.1352	−0.726093	−0.363046	−	0.931771i	$-0.618263\pi$
$769$	−20.1352	−0.726093	−0.363046	+	0.931771i	$0.618263\pi$
$770$	0	0
$771$	8.68721	0.312862
$772$	0	0
$773$	33.1762	1.19327	0.596633	−	0.802514i	$-0.296505\pi$
$773$	33.1762	1.19327	0.596633	+	0.802514i	$0.296505\pi$
$774$	0	0
$775$	2.55109	0.0916378
$776$	0	0
$777$	−3.22047	−0.115534
$778$	0	0
$779$	1.40751	0.0504291
$780$	0	0
$781$	6.76314	0.242004
$782$	0	0
$783$	3.80887	0.136118
$784$	0	0
$785$	0.923523	0.0329619
$786$	0	0
$787$	−19.4683	−0.693970	−0.346985	−	0.937871i	$-0.612795\pi$
$787$	−19.4683	−0.693970	−0.346985	+	0.937871i	$0.612795\pi$
$788$	0	0
$789$	−8.68022	−0.309024
$790$	0	0
$791$	−35.4603	−1.26082
$792$	0	0
$793$	27.8598	0.989329
$794$	0	0
$795$	−3.34423	−0.118608
$796$	0	0
$797$	−33.4032	−1.18320	−0.591600	−	0.806231i	$-0.701504\pi$
$797$	−33.4032	−1.18320	−0.591600	+	0.806231i	$0.701504\pi$
$798$	0	0
$799$	−6.27270	−0.221912
$800$	0	0
$801$	−12.1864	−0.430586
$802$	0	0
$803$	16.5311	0.583370
$804$	0	0
$805$	−6.43273	−0.226724
$806$	0	0
$807$	−0.909313	−0.0320093
$808$	0	0
$809$	−26.2208	−0.921874	−0.460937	−	0.887433i	$-0.652487\pi$
$809$	−26.2208	−0.921874	−0.460937	+	0.887433i	$0.652487\pi$
$810$	0	0
$811$	−22.5751	−0.792717	−0.396359	−	0.918096i	$-0.629726\pi$
$811$	−22.5751	−0.792717	−0.396359	+	0.918096i	$0.629726\pi$
$812$	0	0
$813$	6.50697	0.228209
$814$	0	0
$815$	22.9695	0.804586
$816$	0	0
$817$	1.18832	0.0415740
$818$	0	0
$819$	26.2454	0.917089
$820$	0	0
$821$	−6.46276	−0.225552	−0.112776	−	0.993620i	$-0.535974\pi$
$821$	−6.46276	−0.225552	−0.112776	+	0.993620i	$0.535974\pi$
$822$	0	0
$823$	−8.76701	−0.305599	−0.152799	−	0.988257i	$-0.548829\pi$
$823$	−8.76701	−0.305599	−0.152799	+	0.988257i	$0.548829\pi$
$824$	0	0
$825$	0.671227	0.0233691
$826$	0	0
$827$	27.1637	0.944574	0.472287	−	0.881445i	$-0.343429\pi$
$827$	27.1637	0.944574	0.472287	+	0.881445i	$0.343429\pi$
$828$	0	0
$829$	−28.3498	−0.984628	−0.492314	−	0.870418i	$-0.663849\pi$
$829$	−28.3498	−0.984628	−0.492314	+	0.870418i	$0.663849\pi$
$830$	0	0
$831$	6.30981	0.218885
$832$	0	0
$833$	79.9031	2.76848
$834$	0	0
$835$	5.79067	0.200394
$836$	0	0
$837$	−9.71676	−0.335860
$838$	0	0
$839$	−9.73165	−0.335974	−0.167987	−	0.985789i	$-0.553727\pi$
$839$	−9.73165	−0.335974	−0.167987	+	0.985789i	$0.553727\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−9.52537	−0.328071
$844$	0	0
$845$	−7.78254	−0.267728
$846$	0	0
$847$	45.7480	1.57192
$848$	0	0
$849$	19.6765	0.675296
$850$	0	0
$851$	1.45076	0.0497314
$852$	0	0
$853$	−21.9885	−0.752870	−0.376435	−	0.926443i	$-0.622850\pi$
$853$	−21.9885	−0.752870	−0.376435	+	0.926443i	$0.622850\pi$
$854$	0	0
$855$	−0.763404	−0.0261079
$856$	0	0
$857$	−14.1578	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$857$	−14.1578	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$858$	0	0
$859$	33.6722	1.14888	0.574440	−	0.818547i	$-0.305220\pi$
$859$	33.6722	1.14888	0.574440	+	0.818547i	$0.305220\pi$
$860$	0	0
$861$	−14.6052	−0.497743
$862$	0	0
$863$	−56.3189	−1.91712	−0.958560	−	0.284892i	$-0.908042\pi$
$863$	−56.3189	−1.91712	−0.958560	+	0.284892i	$0.908042\pi$
$864$	0	0
$865$	−1.14747	−0.0390152
$866$	0	0
$867$	11.6890	0.396980
$868$	0	0
$869$	−9.70234	−0.329129
$870$	0	0
$871$	−0.805069	−0.0272787
$872$	0	0
$873$	−34.6863	−1.17395
$874$	0	0
$875$	−4.55109	−0.153855
$876$	0	0
$877$	36.2864	1.22530	0.612652	−	0.790353i	$-0.290103\pi$
$877$	36.2864	1.22530	0.612652	+	0.790353i	$0.290103\pi$
$878$	0	0
$879$	−6.83918	−0.230680
$880$	0	0
$881$	21.5495	0.726022	0.363011	−	0.931785i	$-0.381749\pi$
$881$	21.5495	0.726022	0.363011	+	0.931785i	$0.381749\pi$
$882$	0	0
$883$	−9.49375	−0.319490	−0.159745	−	0.987158i	$-0.551067\pi$
$883$	−9.49375	−0.319490	−0.159745	+	0.987158i	$0.551067\pi$
$884$	0	0
$885$	−1.14654	−0.0385404
$886$	0	0
$887$	−10.5323	−0.353639	−0.176819	−	0.984243i	$-0.556581\pi$
$887$	−10.5323	−0.353639	−0.176819	+	0.984243i	$0.556581\pi$
$888$	0	0
$889$	−56.9939	−1.91151
$890$	0	0
$891$	4.81752	0.161393
$892$	0	0
$893$	−0.325500	−0.0108924
$894$	0	0
$895$	3.95882	0.132329
$896$	0	0
$897$	2.22586	0.0743192
$898$	0	0
$899$	−2.55109	−0.0850836
$900$	0	0
$901$	−28.2656	−0.941665
$902$	0	0
$903$	−12.3307	−0.410341
$904$	0	0
$905$	−11.2068	−0.372526
$906$	0	0
$907$	−37.9285	−1.25940	−0.629698	−	0.776840i	$-0.716822\pi$
$907$	−37.9285	−1.25940	−0.629698	+	0.776840i	$0.716822\pi$
$908$	0	0
$909$	−3.80561	−0.126224
$910$	0	0
$911$	−40.5953	−1.34498	−0.672491	−	0.740105i	$-0.734776\pi$
$911$	−40.5953	−1.34498	−0.672491	+	0.740105i	$0.734776\pi$
$912$	0	0
$913$	13.5527	0.448527
$914$	0	0
$915$	8.40883	0.277987
$916$	0	0
$917$	32.9780	1.08903
$918$	0	0
$919$	−6.56890	−0.216688	−0.108344	−	0.994113i	$-0.534555\pi$
$919$	−6.56890	−0.216688	−0.108344	+	0.994113i	$0.534555\pi$
$920$	0	0
$921$	5.87607	0.193623
$922$	0	0
$923$	15.8670	0.522270
$924$	0	0
$925$	1.02640	0.0337477
$926$	0	0
$927$	−9.20893	−0.302461
$928$	0	0
$929$	4.77110	0.156535	0.0782673	−	0.996932i	$-0.475061\pi$
$929$	4.77110	0.156535	0.0782673	+	0.996932i	$0.475061\pi$
$930$	0	0
$931$	4.14629	0.135889
$932$	0	0
$933$	−11.3367	−0.371148
$934$	0	0
$935$	5.67325	0.185535
$936$	0	0
$937$	−7.45076	−0.243406	−0.121703	−	0.992567i	$-0.538835\pi$
$937$	−7.45076	−0.243406	−0.121703	+	0.992567i	$0.538835\pi$
$938$	0	0
$939$	−3.92259	−0.128009
$940$	0	0
$941$	30.9656	1.00945	0.504725	−	0.863280i	$-0.331594\pi$
$941$	30.9656	1.00945	0.504725	+	0.863280i	$0.331594\pi$
$942$	0	0
$943$	6.57936	0.214254
$944$	0	0
$945$	17.3345	0.563891
$946$	0	0
$947$	21.3943	0.695223	0.347611	−	0.937639i	$-0.386993\pi$
$947$	21.3943	0.695223	0.347611	+	0.937639i	$0.386993\pi$
$948$	0	0
$949$	38.7838	1.25897
$950$	0	0
$951$	−6.44922	−0.209130
$952$	0	0
$953$	−55.9522	−1.81247	−0.906234	−	0.422776i	$-0.861056\pi$
$953$	−55.9522	−1.81247	−0.906234	+	0.422776i	$0.861056\pi$
$954$	0	0
$955$	23.3047	0.754122
$956$	0	0
$957$	−0.671227	−0.0216977
$958$	0	0
$959$	73.3497	2.36859
$960$	0	0
$961$	−24.4919	−0.790063
$962$	0	0
$963$	22.1754	0.714592
$964$	0	0
$965$	−14.8383	−0.477661
$966$	0	0
$967$	−9.93468	−0.319478	−0.159739	−	0.987159i	$-0.551065\pi$
$967$	−9.93468	−0.319478	−0.159739	+	0.987159i	$0.551065\pi$
$968$	0	0
$969$	1.21474	0.0390231
$970$	0	0
$971$	22.0481	0.707558	0.353779	−	0.935329i	$-0.384897\pi$
$971$	22.0481	0.707558	0.353779	+	0.935329i	$0.384897\pi$
$972$	0	0
$973$	−24.4349	−0.783347
$974$	0	0
$975$	1.57477	0.0504330
$976$	0	0
$977$	5.39441	0.172582	0.0862912	−	0.996270i	$-0.472498\pi$
$977$	5.39441	0.172582	0.0862912	+	0.996270i	$0.472498\pi$
$978$	0	0
$979$	4.69948	0.150196
$980$	0	0
$981$	−8.44552	−0.269645
$982$	0	0
$983$	62.0136	1.97793	0.988963	−	0.148162i	$-0.0473358\pi$
$983$	62.0136	1.97793	0.988963	+	0.148162i	$0.0473358\pi$
$984$	0	0
$985$	−21.9556	−0.699565
$986$	0	0
$987$	3.37760	0.107510
$988$	0	0
$989$	5.55477	0.176631
$990$	0	0
$991$	−35.5441	−1.12910	−0.564548	−	0.825400i	$-0.690949\pi$
$991$	−35.5441	−1.12910	−0.564548	+	0.825400i	$0.690949\pi$
$992$	0	0
$993$	−23.5263	−0.746585
$994$	0	0
$995$	−24.8592	−0.788089
$996$	0	0
$997$	−17.8032	−0.563833	−0.281917	−	0.959439i	$-0.590970\pi$
$997$	−17.8032	−0.563833	−0.281917	+	0.959439i	$0.590970\pi$
$998$	0	0
$999$	−3.90941	−0.123688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.w.1.3		5
4.3	odd	2		1160.2.a.i.1.3	✓	5
8.3	odd	2		9280.2.a.cj.1.3		5
8.5	even	2		9280.2.a.ch.1.3		5
20.19	odd	2		5800.2.a.v.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.i.1.3	✓	5	4.3	odd	2
2320.2.a.w.1.3		5	1.1	even	1	trivial
5800.2.a.v.1.3		5	20.19	odd	2
9280.2.a.ch.1.3		5	8.5	even	2
9280.2.a.cj.1.3		5	8.3	odd	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+0.689426 q^{3} +1.00000 q^{5} -4.55109 q^{7} -2.52469 q^{9} +0.973602 q^{11} +2.28418 q^{13} +0.689426 q^{15} +5.82707 q^{17} +0.302375 q^{19} -3.13764 q^{21} +1.41345 q^{23} +1.00000 q^{25} -3.80887 q^{27} -1.00000 q^{29} +2.55109 q^{31} +0.671227 q^{33} -4.55109 q^{35} +1.02640 q^{37} +1.57477 q^{39} +4.65483 q^{41} +3.92994 q^{43} -2.52469 q^{45} -1.07648 q^{47} +13.7124 q^{49} +4.01733 q^{51} -4.85075 q^{53} +0.973602 q^{55} +0.208465 q^{57} -1.66303 q^{59} +12.1969 q^{61} +11.4901 q^{63} +2.28418 q^{65} -0.352455 q^{67} +0.974469 q^{69} +6.94651 q^{71} +16.9793 q^{73} +0.689426 q^{75} -4.43095 q^{77} -9.96540 q^{79} +4.94814 q^{81} +13.9201 q^{83} +5.82707 q^{85} -0.689426 q^{87} +4.82690 q^{89} -10.3955 q^{91} +1.75879 q^{93} +0.302375 q^{95} +13.7388 q^{97} -2.45805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 5 q^{5} + q^{7} + 12 q^{9} + 4 q^{11} + 13 q^{13} + q^{15} + 3 q^{17} + 8 q^{21} + 7 q^{23} + 5 q^{25} + 4 q^{27} - 5 q^{29} - 11 q^{31} + 4 q^{33} + q^{35} + 6 q^{37} - 21 q^{39} + 16 q^{41}+ \cdots - 10 q^{99}+O(q^{100}) \) 5 * q + q^3 + 5 * q^5 + q^7 + 12 * q^9 + 4 * q^11 + 13 * q^13 + q^15 + 3 * q^17 + 8 * q^21 + 7 * q^23 + 5 * q^25 + 4 * q^27 - 5 * q^29 - 11 * q^31 + 4 * q^33 + q^35 + 6 * q^37 - 21 * q^39 + 16 * q^41 - 9 * q^43 + 12 * q^45 - 2 * q^47 + 16 * q^49 - 2 * q^51 + 7 * q^53 + 4 * q^55 + 2 * q^57 - 5 * q^59 + 7 * q^61 + 28 * q^63 + 13 * q^65 + 4 * q^67 + 23 * q^69 - 12 * q^71 + 7 * q^73 + q^75 + 26 * q^77 - 45 * q^79 - 7 * q^81 + 22 * q^83 + 3 * q^85 - q^87 + 24 * q^89 + 20 * q^91 - 10 * q^93 + 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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