LMFDB - Embedded newform 8004.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.434018$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.434018 q^{5} -2.25969 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.434018 q^{5} -2.25969 q^{7} +1.00000 q^{9} +3.49169 q^{11} +5.02170 q^{13} -0.434018 q^{15} -0.922556 q^{17} +5.87027 q^{19} -2.25969 q^{21} +1.00000 q^{23} -4.81163 q^{25} +1.00000 q^{27} -1.00000 q^{29} +9.57575 q^{31} +3.49169 q^{33} +0.980749 q^{35} +2.89227 q^{37} +5.02170 q^{39} -5.70503 q^{41} +4.57680 q^{43} -0.434018 q^{45} -1.28633 q^{47} -1.89378 q^{49} -0.922556 q^{51} -9.88390 q^{53} -1.51546 q^{55} +5.87027 q^{57} -0.268728 q^{59} -4.59157 q^{61} -2.25969 q^{63} -2.17951 q^{65} +12.9610 q^{67} +1.00000 q^{69} -1.25321 q^{71} -2.10581 q^{73} -4.81163 q^{75} -7.89016 q^{77} -9.21562 q^{79} +1.00000 q^{81} +3.24748 q^{83} +0.400406 q^{85} -1.00000 q^{87} +8.36608 q^{89} -11.3475 q^{91} +9.57575 q^{93} -2.54781 q^{95} +0.502057 q^{97} +3.49169 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.434018	−0.194099	−0.0970495	−	0.995280i	$-0.530941\pi$
$5$	−0.434018	−0.194099	−0.0970495	+	0.995280i	$0.530941\pi$
$6$	0	0
$7$	−2.25969	−0.854084	−0.427042	−	0.904232i	$-0.640444\pi$
$7$	−2.25969	−0.854084	−0.427042	+	0.904232i	$0.640444\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.49169	1.05278	0.526392	−	0.850242i	$-0.323544\pi$
$11$	3.49169	1.05278	0.526392	+	0.850242i	$0.323544\pi$
$12$	0	0
$13$	5.02170	1.39277	0.696384	−	0.717669i	$-0.254791\pi$
$13$	5.02170	1.39277	0.696384	+	0.717669i	$0.254791\pi$
$14$	0	0
$15$	−0.434018	−0.112063
$16$	0	0
$17$	−0.922556	−0.223753	−0.111876	−	0.993722i	$-0.535686\pi$
$17$	−0.922556	−0.223753	−0.111876	+	0.993722i	$0.535686\pi$
$18$	0	0
$19$	5.87027	1.34673	0.673367	−	0.739309i	$-0.264848\pi$
$19$	5.87027	1.34673	0.673367	+	0.739309i	$0.264848\pi$
$20$	0	0
$21$	−2.25969	−0.493106
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.81163	−0.962326
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.57575	1.71986	0.859928	−	0.510416i	$-0.170508\pi$
$31$	9.57575	1.71986	0.859928	+	0.510416i	$0.170508\pi$
$32$	0	0
$33$	3.49169	0.607826
$34$	0	0
$35$	0.980749	0.165777
$36$	0	0
$37$	2.89227	0.475487	0.237743	−	0.971328i	$-0.423592\pi$
$37$	2.89227	0.475487	0.237743	+	0.971328i	$0.423592\pi$
$38$	0	0
$39$	5.02170	0.804115
$40$	0	0
$41$	−5.70503	−0.890975	−0.445488	−	0.895288i	$-0.646970\pi$
$41$	−5.70503	−0.890975	−0.445488	+	0.895288i	$0.646970\pi$
$42$	0	0
$43$	4.57680	0.697956	0.348978	−	0.937131i	$-0.386529\pi$
$43$	4.57680	0.697956	0.348978	+	0.937131i	$0.386529\pi$
$44$	0	0
$45$	−0.434018	−0.0646996
$46$	0	0
$47$	−1.28633	−0.187631	−0.0938155	−	0.995590i	$-0.529906\pi$
$47$	−1.28633	−0.187631	−0.0938155	+	0.995590i	$0.529906\pi$
$48$	0	0
$49$	−1.89378	−0.270541
$50$	0	0
$51$	−0.922556	−0.129184
$52$	0	0
$53$	−9.88390	−1.35766	−0.678829	−	0.734296i	$-0.737512\pi$
$53$	−9.88390	−1.35766	−0.678829	+	0.734296i	$0.737512\pi$
$54$	0	0
$55$	−1.51546	−0.204344
$56$	0	0
$57$	5.87027	0.777537
$58$	0	0
$59$	−0.268728	−0.0349855	−0.0174927	−	0.999847i	$-0.505568\pi$
$59$	−0.268728	−0.0349855	−0.0174927	+	0.999847i	$0.505568\pi$
$60$	0	0
$61$	−4.59157	−0.587890	−0.293945	−	0.955822i	$-0.594968\pi$
$61$	−4.59157	−0.587890	−0.293945	+	0.955822i	$0.594968\pi$
$62$	0	0
$63$	−2.25969	−0.284695
$64$	0	0
$65$	−2.17951	−0.270335
$66$	0	0
$67$	12.9610	1.58344	0.791719	−	0.610886i	$-0.209187\pi$
$67$	12.9610	1.58344	0.791719	+	0.610886i	$0.209187\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.25321	−0.148728	−0.0743640	−	0.997231i	$-0.523693\pi$
$71$	−1.25321	−0.148728	−0.0743640	+	0.997231i	$0.523693\pi$
$72$	0	0
$73$	−2.10581	−0.246466	−0.123233	−	0.992378i	$-0.539326\pi$
$73$	−2.10581	−0.246466	−0.123233	+	0.992378i	$0.539326\pi$
$74$	0	0
$75$	−4.81163	−0.555599
$76$	0	0
$77$	−7.89016	−0.899167
$78$	0	0
$79$	−9.21562	−1.03684	−0.518419	−	0.855127i	$-0.673479\pi$
$79$	−9.21562	−1.03684	−0.518419	+	0.855127i	$0.673479\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.24748	0.356457	0.178228	−	0.983989i	$-0.442963\pi$
$83$	3.24748	0.356457	0.178228	+	0.983989i	$0.442963\pi$
$84$	0	0
$85$	0.400406	0.0434301
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.36608	0.886803	0.443401	−	0.896323i	$-0.353772\pi$
$89$	8.36608	0.886803	0.443401	+	0.896323i	$0.353772\pi$
$90$	0	0
$91$	−11.3475	−1.18954
$92$	0	0
$93$	9.57575	0.992959
$94$	0	0
$95$	−2.54781	−0.261399
$96$	0	0
$97$	0.502057	0.0509761	0.0254881	−	0.999675i	$-0.491886\pi$
$97$	0.502057	0.0509761	0.0254881	+	0.999675i	$0.491886\pi$
$98$	0	0
$99$	3.49169	0.350928
$100$	0	0
$101$	9.47938	0.943234	0.471617	−	0.881803i	$-0.343670\pi$
$101$	9.47938	0.943234	0.471617	+	0.881803i	$0.343670\pi$
$102$	0	0
$103$	7.20742	0.710168	0.355084	−	0.934834i	$-0.384452\pi$
$103$	7.20742	0.710168	0.355084	+	0.934834i	$0.384452\pi$
$104$	0	0
$105$	0.980749	0.0957113
$106$	0	0
$107$	8.72557	0.843532	0.421766	−	0.906705i	$-0.361410\pi$
$107$	8.72557	0.843532	0.421766	+	0.906705i	$0.361410\pi$
$108$	0	0
$109$	−12.9076	−1.23633	−0.618164	−	0.786049i	$-0.712123\pi$
$109$	−12.9076	−1.23633	−0.618164	+	0.786049i	$0.712123\pi$
$110$	0	0
$111$	2.89227	0.274522
$112$	0	0
$113$	18.2048	1.71256	0.856281	−	0.516510i	$-0.172769\pi$
$113$	18.2048	1.71256	0.856281	+	0.516510i	$0.172769\pi$
$114$	0	0
$115$	−0.434018	−0.0404724
$116$	0	0
$117$	5.02170	0.464256
$118$	0	0
$119$	2.08469	0.191104
$120$	0	0
$121$	1.19192	0.108356
$122$	0	0
$123$	−5.70503	−0.514405
$124$	0	0
$125$	4.25843	0.380885
$126$	0	0
$127$	−6.38726	−0.566777	−0.283389	−	0.959005i	$-0.591459\pi$
$127$	−6.38726	−0.566777	−0.283389	+	0.959005i	$0.591459\pi$
$128$	0	0
$129$	4.57680	0.402965
$130$	0	0
$131$	−3.83334	−0.334920	−0.167460	−	0.985879i	$-0.553557\pi$
$131$	−3.83334	−0.334920	−0.167460	+	0.985879i	$0.553557\pi$
$132$	0	0
$133$	−13.2650	−1.15022
$134$	0	0
$135$	−0.434018	−0.0373544
$136$	0	0
$137$	10.7808	0.921069	0.460535	−	0.887642i	$-0.347658\pi$
$137$	10.7808	0.921069	0.460535	+	0.887642i	$0.347658\pi$
$138$	0	0
$139$	7.97508	0.676438	0.338219	−	0.941067i	$-0.390176\pi$
$139$	7.97508	0.676438	0.338219	+	0.941067i	$0.390176\pi$
$140$	0	0
$141$	−1.28633	−0.108329
$142$	0	0
$143$	17.5342	1.46629
$144$	0	0
$145$	0.434018	0.0360433
$146$	0	0
$147$	−1.89378	−0.156197
$148$	0	0
$149$	−13.6244	−1.11616	−0.558078	−	0.829788i	$-0.688461\pi$
$149$	−13.6244	−1.11616	−0.558078	+	0.829788i	$0.688461\pi$
$150$	0	0
$151$	−0.216471	−0.0176161	−0.00880807	−	0.999961i	$-0.502804\pi$
$151$	−0.216471	−0.0176161	−0.00880807	+	0.999961i	$0.502804\pi$
$152$	0	0
$153$	−0.922556	−0.0745842
$154$	0	0
$155$	−4.15605	−0.333822
$156$	0	0
$157$	2.63225	0.210076	0.105038	−	0.994468i	$-0.466504\pi$
$157$	2.63225	0.210076	0.105038	+	0.994468i	$0.466504\pi$
$158$	0	0
$159$	−9.88390	−0.783844
$160$	0	0
$161$	−2.25969	−0.178089
$162$	0	0
$163$	−16.0811	−1.25957	−0.629786	−	0.776769i	$-0.716857\pi$
$163$	−16.0811	−1.25957	−0.629786	+	0.776769i	$0.716857\pi$
$164$	0	0
$165$	−1.51546	−0.117978
$166$	0	0
$167$	−2.56463	−0.198457	−0.0992285	−	0.995065i	$-0.531637\pi$
$167$	−2.56463	−0.198457	−0.0992285	+	0.995065i	$0.531637\pi$
$168$	0	0
$169$	12.2174	0.939804
$170$	0	0
$171$	5.87027	0.448911
$172$	0	0
$173$	−6.88517	−0.523470	−0.261735	−	0.965140i	$-0.584295\pi$
$173$	−6.88517	−0.523470	−0.261735	+	0.965140i	$0.584295\pi$
$174$	0	0
$175$	10.8728	0.821907
$176$	0	0
$177$	−0.268728	−0.0201989
$178$	0	0
$179$	−5.43773	−0.406435	−0.203217	−	0.979134i	$-0.565140\pi$
$179$	−5.43773	−0.406435	−0.203217	+	0.979134i	$0.565140\pi$
$180$	0	0
$181$	21.5095	1.59878	0.799392	−	0.600809i	$-0.205155\pi$
$181$	21.5095	1.59878	0.799392	+	0.600809i	$0.205155\pi$
$182$	0	0
$183$	−4.59157	−0.339418
$184$	0	0
$185$	−1.25530	−0.0922915
$186$	0	0
$187$	−3.22128	−0.235563
$188$	0	0
$189$	−2.25969	−0.164369
$190$	0	0
$191$	−22.6965	−1.64226	−0.821132	−	0.570738i	$-0.806657\pi$
$191$	−22.6965	−1.64226	−0.821132	+	0.570738i	$0.806657\pi$
$192$	0	0
$193$	22.6343	1.62925	0.814627	−	0.579985i	$-0.196942\pi$
$193$	22.6343	1.62925	0.814627	+	0.579985i	$0.196942\pi$
$194$	0	0
$195$	−2.17951	−0.156078
$196$	0	0
$197$	10.3349	0.736332	0.368166	−	0.929760i	$-0.379986\pi$
$197$	10.3349	0.736332	0.368166	+	0.929760i	$0.379986\pi$
$198$	0	0
$199$	1.80944	0.128268	0.0641339	−	0.997941i	$-0.479572\pi$
$199$	1.80944	0.128268	0.0641339	+	0.997941i	$0.479572\pi$
$200$	0	0
$201$	12.9610	0.914198
$202$	0	0
$203$	2.25969	0.158599
$204$	0	0
$205$	2.47609	0.172937
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	20.4972	1.41782
$210$	0	0
$211$	−9.37416	−0.645343	−0.322672	−	0.946511i	$-0.604581\pi$
$211$	−9.37416	−0.645343	−0.322672	+	0.946511i	$0.604581\pi$
$212$	0	0
$213$	−1.25321	−0.0858682
$214$	0	0
$215$	−1.98642	−0.135472
$216$	0	0
$217$	−21.6383	−1.46890
$218$	0	0
$219$	−2.10581	−0.142297
$220$	0	0
$221$	−4.63280	−0.311636
$222$	0	0
$223$	7.80672	0.522776	0.261388	−	0.965234i	$-0.415820\pi$
$223$	7.80672	0.522776	0.261388	+	0.965234i	$0.415820\pi$
$224$	0	0
$225$	−4.81163	−0.320775
$226$	0	0
$227$	1.27849	0.0848562	0.0424281	−	0.999100i	$-0.486491\pi$
$227$	1.27849	0.0848562	0.0424281	+	0.999100i	$0.486491\pi$
$228$	0	0
$229$	23.8626	1.57689	0.788443	−	0.615108i	$-0.210888\pi$
$229$	23.8626	1.57689	0.788443	+	0.615108i	$0.210888\pi$
$230$	0	0
$231$	−7.89016	−0.519134
$232$	0	0
$233$	10.8170	0.708648	0.354324	−	0.935123i	$-0.384711\pi$
$233$	10.8170	0.708648	0.354324	+	0.935123i	$0.384711\pi$
$234$	0	0
$235$	0.558292	0.0364190
$236$	0	0
$237$	−9.21562	−0.598619
$238$	0	0
$239$	25.8482	1.67198	0.835992	−	0.548742i	$-0.184893\pi$
$239$	25.8482	1.67198	0.835992	+	0.548742i	$0.184893\pi$
$240$	0	0
$241$	−3.80834	−0.245317	−0.122658	−	0.992449i	$-0.539142\pi$
$241$	−3.80834	−0.245317	−0.122658	+	0.992449i	$0.539142\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.821937	0.0525116
$246$	0	0
$247$	29.4787	1.87569
$248$	0	0
$249$	3.24748	0.205800
$250$	0	0
$251$	−27.6031	−1.74229	−0.871147	−	0.491023i	$-0.836623\pi$
$251$	−27.6031	−1.74229	−0.871147	+	0.491023i	$0.836623\pi$
$252$	0	0
$253$	3.49169	0.219521
$254$	0	0
$255$	0.400406	0.0250744
$256$	0	0
$257$	9.90359	0.617769	0.308885	−	0.951100i	$-0.400044\pi$
$257$	9.90359	0.617769	0.308885	+	0.951100i	$0.400044\pi$
$258$	0	0
$259$	−6.53565	−0.406106
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−22.5105	−1.38805	−0.694027	−	0.719949i	$-0.744165\pi$
$263$	−22.5105	−1.38805	−0.694027	+	0.719949i	$0.744165\pi$
$264$	0	0
$265$	4.28980	0.263520
$266$	0	0
$267$	8.36608	0.511996
$268$	0	0
$269$	20.7434	1.26475	0.632373	−	0.774664i	$-0.282081\pi$
$269$	20.7434	1.26475	0.632373	+	0.774664i	$0.282081\pi$
$270$	0	0
$271$	20.7623	1.26122	0.630611	−	0.776099i	$-0.282805\pi$
$271$	20.7623	1.26122	0.630611	+	0.776099i	$0.282805\pi$
$272$	0	0
$273$	−11.3475	−0.686782
$274$	0	0
$275$	−16.8007	−1.01312
$276$	0	0
$277$	22.1310	1.32973	0.664863	−	0.746966i	$-0.268490\pi$
$277$	22.1310	1.32973	0.664863	+	0.746966i	$0.268490\pi$
$278$	0	0
$279$	9.57575	0.573285
$280$	0	0
$281$	19.5397	1.16564	0.582821	−	0.812600i	$-0.301949\pi$
$281$	19.5397	1.16564	0.582821	+	0.812600i	$0.301949\pi$
$282$	0	0
$283$	−14.4356	−0.858109	−0.429054	−	0.903279i	$-0.641153\pi$
$283$	−14.4356	−0.858109	−0.429054	+	0.903279i	$0.641153\pi$
$284$	0	0
$285$	−2.54781	−0.150919
$286$	0	0
$287$	12.8916	0.760968
$288$	0	0
$289$	−16.1489	−0.949935
$290$	0	0
$291$	0.502057	0.0294311
$292$	0	0
$293$	−6.35213	−0.371095	−0.185548	−	0.982635i	$-0.559406\pi$
$293$	−6.35213	−0.371095	−0.185548	+	0.982635i	$0.559406\pi$
$294$	0	0
$295$	0.116633	0.00679064
$296$	0	0
$297$	3.49169	0.202609
$298$	0	0
$299$	5.02170	0.290412
$300$	0	0
$301$	−10.3422	−0.596113
$302$	0	0
$303$	9.47938	0.544576
$304$	0	0
$305$	1.99282	0.114109
$306$	0	0
$307$	13.6533	0.779232	0.389616	−	0.920977i	$-0.372608\pi$
$307$	13.6533	0.779232	0.389616	+	0.920977i	$0.372608\pi$
$308$	0	0
$309$	7.20742	0.410016
$310$	0	0
$311$	30.0516	1.70407	0.852034	−	0.523486i	$-0.175369\pi$
$311$	30.0516	1.70407	0.852034	+	0.523486i	$0.175369\pi$
$312$	0	0
$313$	18.8442	1.06513	0.532567	−	0.846388i	$-0.321227\pi$
$313$	18.8442	1.06513	0.532567	+	0.846388i	$0.321227\pi$
$314$	0	0
$315$	0.980749	0.0552589
$316$	0	0
$317$	13.1249	0.737166	0.368583	−	0.929595i	$-0.379843\pi$
$317$	13.1249	0.737166	0.368583	+	0.929595i	$0.379843\pi$
$318$	0	0
$319$	−3.49169	−0.195497
$320$	0	0
$321$	8.72557	0.487014
$322$	0	0
$323$	−5.41565	−0.301335
$324$	0	0
$325$	−24.1625	−1.34030
$326$	0	0
$327$	−12.9076	−0.713794
$328$	0	0
$329$	2.90672	0.160253
$330$	0	0
$331$	30.6848	1.68659	0.843293	−	0.537454i	$-0.180614\pi$
$331$	30.6848	1.68659	0.843293	+	0.537454i	$0.180614\pi$
$332$	0	0
$333$	2.89227	0.158496
$334$	0	0
$335$	−5.62531	−0.307343
$336$	0	0
$337$	−15.9484	−0.868767	−0.434384	−	0.900728i	$-0.643034\pi$
$337$	−15.9484	−0.868767	−0.434384	+	0.900728i	$0.643034\pi$
$338$	0	0
$339$	18.2048	0.988748
$340$	0	0
$341$	33.4356	1.81064
$342$	0	0
$343$	20.0972	1.08515
$344$	0	0
$345$	−0.434018	−0.0233668
$346$	0	0
$347$	−2.32704	−0.124922	−0.0624610	−	0.998047i	$-0.519895\pi$
$347$	−2.32704	−0.124922	−0.0624610	+	0.998047i	$0.519895\pi$
$348$	0	0
$349$	30.6041	1.63820	0.819101	−	0.573649i	$-0.194473\pi$
$349$	30.6041	1.63820	0.819101	+	0.573649i	$0.194473\pi$
$350$	0	0
$351$	5.02170	0.268038
$352$	0	0
$353$	11.1607	0.594025	0.297013	−	0.954874i	$-0.404010\pi$
$353$	11.1607	0.594025	0.297013	+	0.954874i	$0.404010\pi$
$354$	0	0
$355$	0.543914	0.0288680
$356$	0	0
$357$	2.08469	0.110334
$358$	0	0
$359$	19.9648	1.05370	0.526851	−	0.849957i	$-0.323372\pi$
$359$	19.9648	1.05370	0.526851	+	0.849957i	$0.323372\pi$
$360$	0	0
$361$	15.4601	0.813690
$362$	0	0
$363$	1.19192	0.0625595
$364$	0	0
$365$	0.913959	0.0478388
$366$	0	0
$367$	−31.2584	−1.63168	−0.815839	−	0.578280i	$-0.803724\pi$
$367$	−31.2584	−1.63168	−0.815839	+	0.578280i	$0.803724\pi$
$368$	0	0
$369$	−5.70503	−0.296992
$370$	0	0
$371$	22.3346	1.15955
$372$	0	0
$373$	−3.08153	−0.159556	−0.0797778	−	0.996813i	$-0.525421\pi$
$373$	−3.08153	−0.159556	−0.0797778	+	0.996813i	$0.525421\pi$
$374$	0	0
$375$	4.25843	0.219904
$376$	0	0
$377$	−5.02170	−0.258631
$378$	0	0
$379$	15.6639	0.804600	0.402300	−	0.915508i	$-0.368211\pi$
$379$	15.6639	0.804600	0.402300	+	0.915508i	$0.368211\pi$
$380$	0	0
$381$	−6.38726	−0.327229
$382$	0	0
$383$	−31.6604	−1.61777	−0.808886	−	0.587965i	$-0.799929\pi$
$383$	−31.6604	−1.61777	−0.808886	+	0.587965i	$0.799929\pi$
$384$	0	0
$385$	3.42447	0.174527
$386$	0	0
$387$	4.57680	0.232652
$388$	0	0
$389$	−14.6235	−0.741439	−0.370719	−	0.928745i	$-0.620889\pi$
$389$	−14.6235	−0.741439	−0.370719	+	0.928745i	$0.620889\pi$
$390$	0	0
$391$	−0.922556	−0.0466556
$392$	0	0
$393$	−3.83334	−0.193366
$394$	0	0
$395$	3.99975	0.201249
$396$	0	0
$397$	−10.6325	−0.533628	−0.266814	−	0.963748i	$-0.585971\pi$
$397$	−10.6325	−0.533628	−0.266814	+	0.963748i	$0.585971\pi$
$398$	0	0
$399$	−13.2650	−0.664082
$400$	0	0
$401$	−10.3394	−0.516326	−0.258163	−	0.966101i	$-0.583117\pi$
$401$	−10.3394	−0.516326	−0.258163	+	0.966101i	$0.583117\pi$
$402$	0	0
$403$	48.0865	2.39536
$404$	0	0
$405$	−0.434018	−0.0215665
$406$	0	0
$407$	10.0989	0.500585
$408$	0	0
$409$	31.1147	1.53852	0.769262	−	0.638933i	$-0.220624\pi$
$409$	31.1147	1.53852	0.769262	+	0.638933i	$0.220624\pi$
$410$	0	0
$411$	10.7808	0.531780
$412$	0	0
$413$	0.607244	0.0298805
$414$	0	0
$415$	−1.40946	−0.0691879
$416$	0	0
$417$	7.97508	0.390541
$418$	0	0
$419$	−1.74209	−0.0851067	−0.0425534	−	0.999094i	$-0.513549\pi$
$419$	−1.74209	−0.0851067	−0.0425534	+	0.999094i	$0.513549\pi$
$420$	0	0
$421$	−9.80160	−0.477701	−0.238850	−	0.971056i	$-0.576771\pi$
$421$	−9.80160	−0.477701	−0.238850	+	0.971056i	$0.576771\pi$
$422$	0	0
$423$	−1.28633	−0.0625437
$424$	0	0
$425$	4.43899	0.215323
$426$	0	0
$427$	10.3755	0.502107
$428$	0	0
$429$	17.5342	0.846560
$430$	0	0
$431$	2.75416	0.132663	0.0663316	−	0.997798i	$-0.478870\pi$
$431$	2.75416	0.132663	0.0663316	+	0.997798i	$0.478870\pi$
$432$	0	0
$433$	−7.90262	−0.379776	−0.189888	−	0.981806i	$-0.560812\pi$
$433$	−7.90262	−0.379776	−0.189888	+	0.981806i	$0.560812\pi$
$434$	0	0
$435$	0.434018	0.0208096
$436$	0	0
$437$	5.87027	0.280813
$438$	0	0
$439$	16.8389	0.803679	0.401839	−	0.915710i	$-0.368371\pi$
$439$	16.8389	0.803679	0.401839	+	0.915710i	$0.368371\pi$
$440$	0	0
$441$	−1.89378	−0.0901802
$442$	0	0
$443$	−26.6857	−1.26788	−0.633938	−	0.773384i	$-0.718562\pi$
$443$	−26.6857	−1.26788	−0.633938	+	0.773384i	$0.718562\pi$
$444$	0	0
$445$	−3.63103	−0.172127
$446$	0	0
$447$	−13.6244	−0.644413
$448$	0	0
$449$	−8.79784	−0.415196	−0.207598	−	0.978214i	$-0.566565\pi$
$449$	−8.79784	−0.415196	−0.207598	+	0.978214i	$0.566565\pi$
$450$	0	0
$451$	−19.9202	−0.938005
$452$	0	0
$453$	−0.216471	−0.0101707
$454$	0	0
$455$	4.92502	0.230889
$456$	0	0
$457$	35.9992	1.68397	0.841986	−	0.539500i	$-0.181387\pi$
$457$	35.9992	1.68397	0.841986	+	0.539500i	$0.181387\pi$
$458$	0	0
$459$	−0.922556	−0.0430612
$460$	0	0
$461$	−41.4402	−1.93006	−0.965031	−	0.262134i	$-0.915574\pi$
$461$	−41.4402	−1.93006	−0.965031	+	0.262134i	$0.915574\pi$
$462$	0	0
$463$	−0.188109	−0.00874218	−0.00437109	−	0.999990i	$-0.501391\pi$
$463$	−0.188109	−0.00874218	−0.00437109	+	0.999990i	$0.501391\pi$
$464$	0	0
$465$	−4.15605	−0.192732
$466$	0	0
$467$	−31.3961	−1.45284	−0.726420	−	0.687251i	$-0.758817\pi$
$467$	−31.3961	−1.45284	−0.726420	+	0.687251i	$0.758817\pi$
$468$	0	0
$469$	−29.2879	−1.35239
$470$	0	0
$471$	2.63225	0.121288
$472$	0	0
$473$	15.9808	0.734797
$474$	0	0
$475$	−28.2456	−1.29600
$476$	0	0
$477$	−9.88390	−0.452553
$478$	0	0
$479$	−36.6453	−1.67437	−0.837183	−	0.546922i	$-0.815799\pi$
$479$	−36.6453	−1.67437	−0.837183	+	0.546922i	$0.815799\pi$
$480$	0	0
$481$	14.5241	0.662243
$482$	0	0
$483$	−2.25969	−0.102820
$484$	0	0
$485$	−0.217902	−0.00989441
$486$	0	0
$487$	−22.4622	−1.01786	−0.508929	−	0.860808i	$-0.669958\pi$
$487$	−22.4622	−1.01786	−0.508929	+	0.860808i	$0.669958\pi$
$488$	0	0
$489$	−16.0811	−0.727214
$490$	0	0
$491$	−21.7683	−0.982388	−0.491194	−	0.871050i	$-0.663439\pi$
$491$	−21.7683	−0.982388	−0.491194	+	0.871050i	$0.663439\pi$
$492$	0	0
$493$	0.922556	0.0415498
$494$	0	0
$495$	−1.51546	−0.0681148
$496$	0	0
$497$	2.83186	0.127026
$498$	0	0
$499$	18.1913	0.814356	0.407178	−	0.913349i	$-0.366513\pi$
$499$	18.1913	0.814356	0.407178	+	0.913349i	$0.366513\pi$
$500$	0	0
$501$	−2.56463	−0.114579
$502$	0	0
$503$	19.5579	0.872044	0.436022	−	0.899936i	$-0.356387\pi$
$503$	19.5579	0.872044	0.436022	+	0.899936i	$0.356387\pi$
$504$	0	0
$505$	−4.11423	−0.183081
$506$	0	0
$507$	12.2174	0.542596
$508$	0	0
$509$	25.2850	1.12074	0.560369	−	0.828243i	$-0.310659\pi$
$509$	25.2850	1.12074	0.560369	+	0.828243i	$0.310659\pi$
$510$	0	0
$511$	4.75848	0.210503
$512$	0	0
$513$	5.87027	0.259179
$514$	0	0
$515$	−3.12815	−0.137843
$516$	0	0
$517$	−4.49148	−0.197535
$518$	0	0
$519$	−6.88517	−0.302225
$520$	0	0
$521$	−3.75838	−0.164658	−0.0823289	−	0.996605i	$-0.526236\pi$
$521$	−3.75838	−0.164658	−0.0823289	+	0.996605i	$0.526236\pi$
$522$	0	0
$523$	−37.4445	−1.63734	−0.818668	−	0.574268i	$-0.805287\pi$
$523$	−37.4445	−1.63734	−0.818668	+	0.574268i	$0.805287\pi$
$524$	0	0
$525$	10.8728	0.474528
$526$	0	0
$527$	−8.83416	−0.384822
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.268728	−0.0116618
$532$	0	0
$533$	−28.6489	−1.24092
$534$	0	0
$535$	−3.78706	−0.163729
$536$	0	0
$537$	−5.43773	−0.234655
$538$	0	0
$539$	−6.61251	−0.284821
$540$	0	0
$541$	−36.7346	−1.57934	−0.789671	−	0.613530i	$-0.789749\pi$
$541$	−36.7346	−1.57934	−0.789671	+	0.613530i	$0.789749\pi$
$542$	0	0
$543$	21.5095	0.923059
$544$	0	0
$545$	5.60215	0.239970
$546$	0	0
$547$	−6.53085	−0.279239	−0.139620	−	0.990205i	$-0.544588\pi$
$547$	−6.53085	−0.279239	−0.139620	+	0.990205i	$0.544588\pi$
$548$	0	0
$549$	−4.59157	−0.195963
$550$	0	0
$551$	−5.87027	−0.250082
$552$	0	0
$553$	20.8245	0.885547
$554$	0	0
$555$	−1.25530	−0.0532845
$556$	0	0
$557$	−27.6425	−1.17125	−0.585625	−	0.810582i	$-0.699151\pi$
$557$	−27.6425	−1.17125	−0.585625	+	0.810582i	$0.699151\pi$
$558$	0	0
$559$	22.9833	0.972090
$560$	0	0
$561$	−3.22128	−0.136003
$562$	0	0
$563$	−11.6355	−0.490376	−0.245188	−	0.969476i	$-0.578850\pi$
$563$	−11.6355	−0.490376	−0.245188	+	0.969476i	$0.578850\pi$
$564$	0	0
$565$	−7.90121	−0.332407
$566$	0	0
$567$	−2.25969	−0.0948982
$568$	0	0
$569$	26.2384	1.09997	0.549986	−	0.835174i	$-0.314633\pi$
$569$	26.2384	1.09997	0.549986	+	0.835174i	$0.314633\pi$
$570$	0	0
$571$	−8.60075	−0.359930	−0.179965	−	0.983673i	$-0.557598\pi$
$571$	−8.60075	−0.359930	−0.179965	+	0.983673i	$0.557598\pi$
$572$	0	0
$573$	−22.6965	−0.948162
$574$	0	0
$575$	−4.81163	−0.200659
$576$	0	0
$577$	7.96052	0.331401	0.165700	−	0.986176i	$-0.447012\pi$
$577$	7.96052	0.331401	0.165700	+	0.986176i	$0.447012\pi$
$578$	0	0
$579$	22.6343	0.940650
$580$	0	0
$581$	−7.33830	−0.304444
$582$	0	0
$583$	−34.5116	−1.42932
$584$	0	0
$585$	−2.17951	−0.0901116
$586$	0	0
$587$	22.4480	0.926527	0.463264	−	0.886221i	$-0.346678\pi$
$587$	22.4480	0.926527	0.463264	+	0.886221i	$0.346678\pi$
$588$	0	0
$589$	56.2123	2.31619
$590$	0	0
$591$	10.3349	0.425122
$592$	0	0
$593$	21.2147	0.871185	0.435593	−	0.900144i	$-0.356539\pi$
$593$	21.2147	0.871185	0.435593	+	0.900144i	$0.356539\pi$
$594$	0	0
$595$	−0.904795	−0.0370930
$596$	0	0
$597$	1.80944	0.0740554
$598$	0	0
$599$	−10.2582	−0.419139	−0.209570	−	0.977794i	$-0.567206\pi$
$599$	−10.2582	−0.419139	−0.209570	+	0.977794i	$0.567206\pi$
$600$	0	0
$601$	−26.8975	−1.09717	−0.548586	−	0.836094i	$-0.684833\pi$
$601$	−26.8975	−1.09717	−0.548586	+	0.836094i	$0.684833\pi$
$602$	0	0
$603$	12.9610	0.527812
$604$	0	0
$605$	−0.517315	−0.0210318
$606$	0	0
$607$	19.4719	0.790339	0.395170	−	0.918608i	$-0.370686\pi$
$607$	19.4719	0.790339	0.395170	+	0.918608i	$0.370686\pi$
$608$	0	0
$609$	2.25969	0.0915674
$610$	0	0
$611$	−6.45958	−0.261327
$612$	0	0
$613$	−35.4825	−1.43313	−0.716563	−	0.697523i	$-0.754286\pi$
$613$	−35.4825	−1.43313	−0.716563	+	0.697523i	$0.754286\pi$
$614$	0	0
$615$	2.47609	0.0998454
$616$	0	0
$617$	−19.6980	−0.793012	−0.396506	−	0.918032i	$-0.629777\pi$
$617$	−19.6980	−0.793012	−0.396506	+	0.918032i	$0.629777\pi$
$618$	0	0
$619$	32.3551	1.30046	0.650230	−	0.759737i	$-0.274672\pi$
$619$	32.3551	1.30046	0.650230	+	0.759737i	$0.274672\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−18.9048	−0.757404
$624$	0	0
$625$	22.2099	0.888396
$626$	0	0
$627$	20.4972	0.818579
$628$	0	0
$629$	−2.66828	−0.106391
$630$	0	0
$631$	44.7436	1.78121	0.890607	−	0.454773i	$-0.150280\pi$
$631$	44.7436	1.78121	0.890607	+	0.454773i	$0.150280\pi$
$632$	0	0
$633$	−9.37416	−0.372589
$634$	0	0
$635$	2.77219	0.110011
$636$	0	0
$637$	−9.51001	−0.376800
$638$	0	0
$639$	−1.25321	−0.0495760
$640$	0	0
$641$	4.62045	0.182497	0.0912483	−	0.995828i	$-0.470914\pi$
$641$	4.62045	0.182497	0.0912483	+	0.995828i	$0.470914\pi$
$642$	0	0
$643$	16.8433	0.664234	0.332117	−	0.943238i	$-0.392237\pi$
$643$	16.8433	0.664234	0.332117	+	0.943238i	$0.392237\pi$
$644$	0	0
$645$	−1.98642	−0.0782150
$646$	0	0
$647$	−26.5057	−1.04205	−0.521023	−	0.853542i	$-0.674450\pi$
$647$	−26.5057	−1.04205	−0.521023	+	0.853542i	$0.674450\pi$
$648$	0	0
$649$	−0.938317	−0.0368322
$650$	0	0
$651$	−21.6383	−0.848070
$652$	0	0
$653$	17.9743	0.703388	0.351694	−	0.936115i	$-0.385606\pi$
$653$	17.9743	0.703388	0.351694	+	0.936115i	$0.385606\pi$
$654$	0	0
$655$	1.66374	0.0650077
$656$	0	0
$657$	−2.10581	−0.0821554
$658$	0	0
$659$	−5.15635	−0.200863	−0.100431	−	0.994944i	$-0.532022\pi$
$659$	−5.15635	−0.200863	−0.100431	+	0.994944i	$0.532022\pi$
$660$	0	0
$661$	−1.14690	−0.0446092	−0.0223046	−	0.999751i	$-0.507100\pi$
$661$	−1.14690	−0.0446092	−0.0223046	+	0.999751i	$0.507100\pi$
$662$	0	0
$663$	−4.63280	−0.179923
$664$	0	0
$665$	5.75726	0.223257
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	7.80672	0.301825
$670$	0	0
$671$	−16.0323	−0.618922
$672$	0	0
$673$	−42.0563	−1.62115	−0.810576	−	0.585634i	$-0.800846\pi$
$673$	−42.0563	−1.62115	−0.810576	+	0.585634i	$0.800846\pi$
$674$	0	0
$675$	−4.81163	−0.185200
$676$	0	0
$677$	−18.5900	−0.714472	−0.357236	−	0.934014i	$-0.616281\pi$
$677$	−18.5900	−0.714472	−0.357236	+	0.934014i	$0.616281\pi$
$678$	0	0
$679$	−1.13449	−0.0435379
$680$	0	0
$681$	1.27849	0.0489918
$682$	0	0
$683$	−26.1459	−1.00044	−0.500222	−	0.865897i	$-0.666748\pi$
$683$	−26.1459	−1.00044	−0.500222	+	0.865897i	$0.666748\pi$
$684$	0	0
$685$	−4.67908	−0.178779
$686$	0	0
$687$	23.8626	0.910415
$688$	0	0
$689$	−49.6340	−1.89090
$690$	0	0
$691$	9.14832	0.348018	0.174009	−	0.984744i	$-0.444328\pi$
$691$	9.14832	0.348018	0.174009	+	0.984744i	$0.444328\pi$
$692$	0	0
$693$	−7.89016	−0.299722
$694$	0	0
$695$	−3.46133	−0.131296
$696$	0	0
$697$	5.26320	0.199358
$698$	0	0
$699$	10.8170	0.409138
$700$	0	0
$701$	−45.3203	−1.71172	−0.855862	−	0.517204i	$-0.826973\pi$
$701$	−45.3203	−1.71172	−0.855862	+	0.517204i	$0.826973\pi$
$702$	0	0
$703$	16.9784	0.640354
$704$	0	0
$705$	0.558292	0.0210265
$706$	0	0
$707$	−21.4205	−0.805601
$708$	0	0
$709$	21.9814	0.825530	0.412765	−	0.910837i	$-0.364563\pi$
$709$	21.9814	0.825530	0.412765	+	0.910837i	$0.364563\pi$
$710$	0	0
$711$	−9.21562	−0.345613
$712$	0	0
$713$	9.57575	0.358615
$714$	0	0
$715$	−7.61018	−0.284604
$716$	0	0
$717$	25.8482	0.965320
$718$	0	0
$719$	15.1433	0.564748	0.282374	−	0.959304i	$-0.408878\pi$
$719$	15.1433	0.564748	0.282374	+	0.959304i	$0.408878\pi$
$720$	0	0
$721$	−16.2866	−0.606543
$722$	0	0
$723$	−3.80834	−0.141634
$724$	0	0
$725$	4.81163	0.178699
$726$	0	0
$727$	15.3237	0.568324	0.284162	−	0.958776i	$-0.408285\pi$
$727$	15.3237	0.568324	0.284162	+	0.958776i	$0.408285\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.22235	−0.156169
$732$	0	0
$733$	−15.1025	−0.557825	−0.278912	−	0.960317i	$-0.589974\pi$
$733$	−15.1025	−0.557825	−0.278912	+	0.960317i	$0.589974\pi$
$734$	0	0
$735$	0.821937	0.0303176
$736$	0	0
$737$	45.2558	1.66702
$738$	0	0
$739$	43.2234	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$739$	43.2234	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$740$	0	0
$741$	29.4787	1.08293
$742$	0	0
$743$	−3.71900	−0.136437	−0.0682185	−	0.997670i	$-0.521732\pi$
$743$	−3.71900	−0.136437	−0.0682185	+	0.997670i	$0.521732\pi$
$744$	0	0
$745$	5.91325	0.216645
$746$	0	0
$747$	3.24748	0.118819
$748$	0	0
$749$	−19.7171	−0.720448
$750$	0	0
$751$	−7.61239	−0.277780	−0.138890	−	0.990308i	$-0.544353\pi$
$751$	−7.61239	−0.277780	−0.138890	+	0.990308i	$0.544353\pi$
$752$	0	0
$753$	−27.6031	−1.00591
$754$	0	0
$755$	0.0939523	0.00341927
$756$	0	0
$757$	27.1191	0.985660	0.492830	−	0.870126i	$-0.335962\pi$
$757$	27.1191	0.985660	0.492830	+	0.870126i	$0.335962\pi$
$758$	0	0
$759$	3.49169	0.126740
$760$	0	0
$761$	16.1490	0.585400	0.292700	−	0.956204i	$-0.405446\pi$
$761$	16.1490	0.585400	0.292700	+	0.956204i	$0.405446\pi$
$762$	0	0
$763$	29.1673	1.05593
$764$	0	0
$765$	0.400406	0.0144767
$766$	0	0
$767$	−1.34947	−0.0487266
$768$	0	0
$769$	−54.4394	−1.96314	−0.981568	−	0.191111i	$-0.938791\pi$
$769$	−54.4394	−1.96314	−0.981568	+	0.191111i	$0.938791\pi$
$770$	0	0
$771$	9.90359	0.356669
$772$	0	0
$773$	−35.7818	−1.28698	−0.643492	−	0.765453i	$-0.722515\pi$
$773$	−35.7818	−1.28698	−0.643492	+	0.765453i	$0.722515\pi$
$774$	0	0
$775$	−46.0750	−1.65506
$776$	0	0
$777$	−6.53565	−0.234465
$778$	0	0
$779$	−33.4901	−1.19991
$780$	0	0
$781$	−4.37581	−0.156579
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−1.14244	−0.0407756
$786$	0	0
$787$	45.1538	1.60956	0.804780	−	0.593573i	$-0.202283\pi$
$787$	45.1538	1.60956	0.804780	+	0.593573i	$0.202283\pi$
$788$	0	0
$789$	−22.5105	−0.801393
$790$	0	0
$791$	−41.1372	−1.46267
$792$	0	0
$793$	−23.0575	−0.818794
$794$	0	0
$795$	4.28980	0.152143
$796$	0	0
$797$	33.1561	1.17445	0.587225	−	0.809424i	$-0.300220\pi$
$797$	33.1561	1.17445	0.587225	+	0.809424i	$0.300220\pi$
$798$	0	0
$799$	1.18671	0.0419829
$800$	0	0
$801$	8.36608	0.295601
$802$	0	0
$803$	−7.35283	−0.259476
$804$	0	0
$805$	0.980749	0.0345668
$806$	0	0
$807$	20.7434	0.730201
$808$	0	0
$809$	52.4675	1.84466	0.922329	−	0.386405i	$-0.126283\pi$
$809$	52.4675	1.84466	0.922329	+	0.386405i	$0.126283\pi$
$810$	0	0
$811$	−7.74967	−0.272128	−0.136064	−	0.990700i	$-0.543445\pi$
$811$	−7.74967	−0.272128	−0.136064	+	0.990700i	$0.543445\pi$
$812$	0	0
$813$	20.7623	0.728166
$814$	0	0
$815$	6.97951	0.244482
$816$	0	0
$817$	26.8671	0.939960
$818$	0	0
$819$	−11.3475	−0.396514
$820$	0	0
$821$	−21.4881	−0.749940	−0.374970	−	0.927037i	$-0.622347\pi$
$821$	−21.4881	−0.749940	−0.374970	+	0.927037i	$0.622347\pi$
$822$	0	0
$823$	−36.5549	−1.27422	−0.637112	−	0.770772i	$-0.719871\pi$
$823$	−36.5549	−1.27422	−0.637112	+	0.770772i	$0.719871\pi$
$824$	0	0
$825$	−16.8007	−0.584926
$826$	0	0
$827$	−33.0011	−1.14756	−0.573781	−	0.819009i	$-0.694524\pi$
$827$	−33.0011	−1.14756	−0.573781	+	0.819009i	$0.694524\pi$
$828$	0	0
$829$	−4.66918	−0.162167	−0.0810837	−	0.996707i	$-0.525838\pi$
$829$	−4.66918	−0.162167	−0.0810837	+	0.996707i	$0.525838\pi$
$830$	0	0
$831$	22.1310	0.767717
$832$	0	0
$833$	1.74712	0.0605342
$834$	0	0
$835$	1.11310	0.0385203
$836$	0	0
$837$	9.57575	0.330986
$838$	0	0
$839$	36.5811	1.26292	0.631459	−	0.775409i	$-0.282456\pi$
$839$	36.5811	1.26292	0.631459	+	0.775409i	$0.282456\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	19.5397	0.672984
$844$	0	0
$845$	−5.30260	−0.182415
$846$	0	0
$847$	−2.69337	−0.0925453
$848$	0	0
$849$	−14.4356	−0.495429
$850$	0	0
$851$	2.89227	0.0991458
$852$	0	0
$853$	−54.0770	−1.85156	−0.925780	−	0.378062i	$-0.876591\pi$
$853$	−54.0770	−1.85156	−0.925780	+	0.378062i	$0.876591\pi$
$854$	0	0
$855$	−2.54781	−0.0871331
$856$	0	0
$857$	25.5237	0.871874	0.435937	−	0.899977i	$-0.356417\pi$
$857$	25.5237	0.871874	0.435937	+	0.899977i	$0.356417\pi$
$858$	0	0
$859$	33.6725	1.14889	0.574446	−	0.818542i	$-0.305218\pi$
$859$	33.6725	1.14889	0.574446	+	0.818542i	$0.305218\pi$
$860$	0	0
$861$	12.8916	0.439345
$862$	0	0
$863$	−50.3410	−1.71363	−0.856814	−	0.515626i	$-0.827559\pi$
$863$	−50.3410	−1.71363	−0.856814	+	0.515626i	$0.827559\pi$
$864$	0	0
$865$	2.98829	0.101605
$866$	0	0
$867$	−16.1489	−0.548445
$868$	0	0
$869$	−32.1781	−1.09157
$870$	0	0
$871$	65.0862	2.20536
$872$	0	0
$873$	0.502057	0.0169920
$874$	0	0
$875$	−9.62274	−0.325308
$876$	0	0
$877$	15.2179	0.513873	0.256936	−	0.966428i	$-0.417287\pi$
$877$	15.2179	0.513873	0.256936	+	0.966428i	$0.417287\pi$
$878$	0	0
$879$	−6.35213	−0.214252
$880$	0	0
$881$	39.4235	1.32821	0.664105	−	0.747639i	$-0.268813\pi$
$881$	39.4235	1.32821	0.664105	+	0.747639i	$0.268813\pi$
$882$	0	0
$883$	−30.1861	−1.01584	−0.507921	−	0.861404i	$-0.669586\pi$
$883$	−30.1861	−1.01584	−0.507921	+	0.861404i	$0.669586\pi$
$884$	0	0
$885$	0.116633	0.00392058
$886$	0	0
$887$	−19.3380	−0.649305	−0.324653	−	0.945833i	$-0.605247\pi$
$887$	−19.3380	−0.649305	−0.324653	+	0.945833i	$0.605247\pi$
$888$	0	0
$889$	14.4332	0.484075
$890$	0	0
$891$	3.49169	0.116976
$892$	0	0
$893$	−7.55113	−0.252689
$894$	0	0
$895$	2.36007	0.0788885
$896$	0	0
$897$	5.02170	0.167670
$898$	0	0
$899$	−9.57575	−0.319369
$900$	0	0
$901$	9.11845	0.303780
$902$	0	0
$903$	−10.3422	−0.344166
$904$	0	0
$905$	−9.33550	−0.310322
$906$	0	0
$907$	−36.4290	−1.20960	−0.604802	−	0.796376i	$-0.706748\pi$
$907$	−36.4290	−1.20960	−0.604802	+	0.796376i	$0.706748\pi$
$908$	0	0
$909$	9.47938	0.314411
$910$	0	0
$911$	−45.6790	−1.51341	−0.756707	−	0.653754i	$-0.773193\pi$
$911$	−45.6790	−1.51341	−0.756707	+	0.653754i	$0.773193\pi$
$912$	0	0
$913$	11.3392	0.375272
$914$	0	0
$915$	1.99282	0.0658807
$916$	0	0
$917$	8.66217	0.286050
$918$	0	0
$919$	29.1818	0.962617	0.481309	−	0.876551i	$-0.340162\pi$
$919$	29.1818	0.962617	0.481309	+	0.876551i	$0.340162\pi$
$920$	0	0
$921$	13.6533	0.449890
$922$	0	0
$923$	−6.29322	−0.207144
$924$	0	0
$925$	−13.9165	−0.457573
$926$	0	0
$927$	7.20742	0.236723
$928$	0	0
$929$	32.7941	1.07594	0.537971	−	0.842964i	$-0.319191\pi$
$929$	32.7941	1.07594	0.537971	+	0.842964i	$0.319191\pi$
$930$	0	0
$931$	−11.1170	−0.364346
$932$	0	0
$933$	30.0516	0.983844
$934$	0	0
$935$	1.39810	0.0457226
$936$	0	0
$937$	15.6579	0.511521	0.255760	−	0.966740i	$-0.417674\pi$
$937$	15.6579	0.511521	0.255760	+	0.966740i	$0.417674\pi$
$938$	0	0
$939$	18.8442	0.614956
$940$	0	0
$941$	−54.5724	−1.77901	−0.889504	−	0.456927i	$-0.848950\pi$
$941$	−54.5724	−1.77901	−0.889504	+	0.456927i	$0.848950\pi$
$942$	0	0
$943$	−5.70503	−0.185781
$944$	0	0
$945$	0.980749	0.0319038
$946$	0	0
$947$	−9.79570	−0.318318	−0.159159	−	0.987253i	$-0.550878\pi$
$947$	−9.79570	−0.318318	−0.159159	+	0.987253i	$0.550878\pi$
$948$	0	0
$949$	−10.5747	−0.343270
$950$	0	0
$951$	13.1249	0.425603
$952$	0	0
$953$	19.2918	0.624924	0.312462	−	0.949930i	$-0.398846\pi$
$953$	19.2918	0.624924	0.312462	+	0.949930i	$0.398846\pi$
$954$	0	0
$955$	9.85072	0.318762
$956$	0	0
$957$	−3.49169	−0.112870
$958$	0	0
$959$	−24.3614	−0.786670
$960$	0	0
$961$	60.6950	1.95790
$962$	0	0
$963$	8.72557	0.281177
$964$	0	0
$965$	−9.82371	−0.316236
$966$	0	0
$967$	12.0831	0.388565	0.194282	−	0.980946i	$-0.437762\pi$
$967$	12.0831	0.388565	0.194282	+	0.980946i	$0.437762\pi$
$968$	0	0
$969$	−5.41565	−0.173976
$970$	0	0
$971$	13.7988	0.442825	0.221412	−	0.975180i	$-0.428933\pi$
$971$	13.7988	0.442825	0.221412	+	0.975180i	$0.428933\pi$
$972$	0	0
$973$	−18.0212	−0.577734
$974$	0	0
$975$	−24.1625	−0.773821
$976$	0	0
$977$	−36.4835	−1.16721	−0.583605	−	0.812038i	$-0.698358\pi$
$977$	−36.4835	−1.16721	−0.583605	+	0.812038i	$0.698358\pi$
$978$	0	0
$979$	29.2118	0.933613
$980$	0	0
$981$	−12.9076	−0.412109
$982$	0	0
$983$	1.07110	0.0341627	0.0170814	−	0.999854i	$-0.494563\pi$
$983$	1.07110	0.0341627	0.0170814	+	0.999854i	$0.494563\pi$
$984$	0	0
$985$	−4.48554	−0.142921
$986$	0	0
$987$	2.90672	0.0925219
$988$	0	0
$989$	4.57680	0.145534
$990$	0	0
$991$	14.9592	0.475196	0.237598	−	0.971364i	$-0.423640\pi$
$991$	14.9592	0.475196	0.237598	+	0.971364i	$0.423640\pi$
$992$	0	0
$993$	30.6848	0.973751
$994$	0	0
$995$	−0.785330	−0.0248966
$996$	0	0
$997$	34.3969	1.08936	0.544680	−	0.838644i	$-0.316651\pi$
$997$	34.3969	1.08936	0.544680	+	0.838644i	$0.316651\pi$
$998$	0	0
$999$	2.89227	0.0915075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.8	✓	16	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.434018 q^{5} -2.25969 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.434018 q^{5} -2.25969 q^{7} +1.00000 q^{9} +3.49169 q^{11} +5.02170 q^{13} -0.434018 q^{15} -0.922556 q^{17} +5.87027 q^{19} -2.25969 q^{21} +1.00000 q^{23} -4.81163 q^{25} +1.00000 q^{27} -1.00000 q^{29} +9.57575 q^{31} +3.49169 q^{33} +0.980749 q^{35} +2.89227 q^{37} +5.02170 q^{39} -5.70503 q^{41} +4.57680 q^{43} -0.434018 q^{45} -1.28633 q^{47} -1.89378 q^{49} -0.922556 q^{51} -9.88390 q^{53} -1.51546 q^{55} +5.87027 q^{57} -0.268728 q^{59} -4.59157 q^{61} -2.25969 q^{63} -2.17951 q^{65} +12.9610 q^{67} +1.00000 q^{69} -1.25321 q^{71} -2.10581 q^{73} -4.81163 q^{75} -7.89016 q^{77} -9.21562 q^{79} +1.00000 q^{81} +3.24748 q^{83} +0.400406 q^{85} -1.00000 q^{87} +8.36608 q^{89} -11.3475 q^{91} +9.57575 q^{93} -2.54781 q^{95} +0.502057 q^{97} +3.49169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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