LMFDB - Embedded newform 6664.2.a.bd.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 14 x^{10} + 64 x^{9} + 59 x^{8} - 348 x^{7} - 74 x^{6} + 760 x^{5} + 27 x^{4} + \cdots - 14 $ x^12 - 4x^11 - 14x^10 + 64x^9 + 59x^8 - 348x^7 - 74x^6 + 760x^5 + 27x^4 - 616x^3 - 46x^2 + 108*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.291629$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.291629 q^{3} -0.918965 q^{5} -2.91495 q^{9} +4.23575 q^{11} +2.07576 q^{13} -0.267996 q^{15} -1.00000 q^{17} -6.92165 q^{19} +1.49967 q^{23} -4.15550 q^{25} -1.72497 q^{27} +10.0305 q^{29} -1.01989 q^{31} +1.23527 q^{33} -6.04815 q^{37} +0.605350 q^{39} +3.68863 q^{41} +0.609194 q^{43} +2.67874 q^{45} +5.20448 q^{47} -0.291629 q^{51} -1.85227 q^{53} -3.89251 q^{55} -2.01855 q^{57} -7.29351 q^{59} -7.07698 q^{61} -1.90755 q^{65} +0.512702 q^{67} +0.437347 q^{69} -6.48743 q^{71} -2.87363 q^{73} -1.21186 q^{75} -3.32001 q^{79} +8.24181 q^{81} +8.42303 q^{83} +0.918965 q^{85} +2.92519 q^{87} +8.79728 q^{89} -0.297430 q^{93} +6.36075 q^{95} -3.73505 q^{97} -12.3470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} - 8 q^{5} + 8 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} - 12 q^{17} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 4 q^{31} - 16 q^{33} + 12 q^{37} - 16 q^{39} - 24 q^{41} - 4 q^{43} - 28 q^{45} - 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 - 8 * q^5 + 8 * q^9 - 4 * q^11 - 12 * q^13 - 8 * q^15 - 12 * q^17 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 4 * q^31 - 16 * q^33 + 12 * q^37 - 16 * q^39 - 24 * q^41 - 4 * q^43 - 28 * q^45 - 28 * q^47 - 4 * q^51 + 4 * q^55 + 12 * q^57 - 36 * q^61 + 20 * q^65 - 4 * q^69 - 40 * q^73 - 20 * q^75 - 4 * q^79 - 4 * q^81 - 12 * q^83 + 8 * q^85 - 36 * q^89 - 16 * q^93 - 8 * q^95 - 12 * q^97 - 20 * 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
$3$	0.291629	0.168372	0.0841860	−	0.996450i	$-0.473171\pi$
$3$	0.291629	0.168372	0.0841860	+	0.996450i	$0.473171\pi$
$4$	0	0
$5$	−0.918965	−0.410973	−0.205487	−	0.978660i	$-0.565878\pi$
$5$	−0.918965	−0.410973	−0.205487	+	0.978660i	$0.565878\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.91495	−0.971651
$10$	0	0
$11$	4.23575	1.27713	0.638564	−	0.769569i	$-0.279529\pi$
$11$	4.23575	1.27713	0.638564	+	0.769569i	$0.279529\pi$
$12$	0	0
$13$	2.07576	0.575711	0.287856	−	0.957674i	$-0.407058\pi$
$13$	2.07576	0.575711	0.287856	+	0.957674i	$0.407058\pi$
$14$	0	0
$15$	−0.267996	−0.0691964
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.92165	−1.58793	−0.793967	−	0.607960i	$-0.791988\pi$
$19$	−6.92165	−1.58793	−0.793967	+	0.607960i	$0.791988\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.49967	0.312703	0.156351	−	0.987701i	$-0.450027\pi$
$23$	1.49967	0.312703	0.156351	+	0.987701i	$0.450027\pi$
$24$	0	0
$25$	−4.15550	−0.831101
$26$	0	0
$27$	−1.72497	−0.331971
$28$	0	0
$29$	10.0305	1.86262	0.931310	−	0.364226i	$-0.118667\pi$
$29$	10.0305	1.86262	0.931310	+	0.364226i	$0.118667\pi$
$30$	0	0
$31$	−1.01989	−0.183178	−0.0915891	−	0.995797i	$-0.529195\pi$
$31$	−1.01989	−0.183178	−0.0915891	+	0.995797i	$0.529195\pi$
$32$	0	0
$33$	1.23527	0.215032
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.04815	−0.994309	−0.497155	−	0.867662i	$-0.665622\pi$
$37$	−6.04815	−0.994309	−0.497155	+	0.867662i	$0.665622\pi$
$38$	0	0
$39$	0.605350	0.0969336
$40$	0	0
$41$	3.68863	0.576067	0.288034	−	0.957620i	$-0.406999\pi$
$41$	3.68863	0.576067	0.288034	+	0.957620i	$0.406999\pi$
$42$	0	0
$43$	0.609194	0.0929012	0.0464506	−	0.998921i	$-0.485209\pi$
$43$	0.609194	0.0929012	0.0464506	+	0.998921i	$0.485209\pi$
$44$	0	0
$45$	2.67874	0.399323
$46$	0	0
$47$	5.20448	0.759151	0.379575	−	0.925161i	$-0.376070\pi$
$47$	5.20448	0.759151	0.379575	+	0.925161i	$0.376070\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.291629	−0.0408362
$52$	0	0
$53$	−1.85227	−0.254429	−0.127214	−	0.991875i	$-0.540604\pi$
$53$	−1.85227	−0.254429	−0.127214	+	0.991875i	$0.540604\pi$
$54$	0	0
$55$	−3.89251	−0.524866
$56$	0	0
$57$	−2.01855	−0.267364
$58$	0	0
$59$	−7.29351	−0.949534	−0.474767	−	0.880111i	$-0.657468\pi$
$59$	−7.29351	−0.949534	−0.474767	+	0.880111i	$0.657468\pi$
$60$	0	0
$61$	−7.07698	−0.906114	−0.453057	−	0.891481i	$-0.649667\pi$
$61$	−7.07698	−0.906114	−0.453057	+	0.891481i	$0.649667\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.90755	−0.236602
$66$	0	0
$67$	0.512702	0.0626365	0.0313183	−	0.999509i	$-0.490029\pi$
$67$	0.512702	0.0626365	0.0313183	+	0.999509i	$0.490029\pi$
$68$	0	0
$69$	0.437347	0.0526504
$70$	0	0
$71$	−6.48743	−0.769917	−0.384958	−	0.922934i	$-0.625784\pi$
$71$	−6.48743	−0.769917	−0.384958	+	0.922934i	$0.625784\pi$
$72$	0	0
$73$	−2.87363	−0.336333	−0.168167	−	0.985759i	$-0.553785\pi$
$73$	−2.87363	−0.336333	−0.168167	+	0.985759i	$0.553785\pi$
$74$	0	0
$75$	−1.21186	−0.139934
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.32001	−0.373530	−0.186765	−	0.982405i	$-0.559800\pi$
$79$	−3.32001	−0.373530	−0.186765	+	0.982405i	$0.559800\pi$
$80$	0	0
$81$	8.24181	0.915756
$82$	0	0
$83$	8.42303	0.924547	0.462274	−	0.886737i	$-0.347034\pi$
$83$	8.42303	0.924547	0.462274	+	0.886737i	$0.347034\pi$
$84$	0	0
$85$	0.918965	0.0996757
$86$	0	0
$87$	2.92519	0.313613
$88$	0	0
$89$	8.79728	0.932509	0.466255	−	0.884651i	$-0.345603\pi$
$89$	8.79728	0.932509	0.466255	+	0.884651i	$0.345603\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.297430	−0.0308421
$94$	0	0
$95$	6.36075	0.652599
$96$	0	0
$97$	−3.73505	−0.379237	−0.189619	−	0.981858i	$-0.560725\pi$
$97$	−3.73505	−0.379237	−0.189619	+	0.981858i	$0.560725\pi$
$98$	0	0
$99$	−12.3470	−1.24092
$100$	0	0
$101$	−11.6767	−1.16188	−0.580938	−	0.813948i	$-0.697314\pi$
$101$	−11.6767	−1.16188	−0.580938	+	0.813948i	$0.697314\pi$
$102$	0	0
$103$	−9.83627	−0.969196	−0.484598	−	0.874737i	$-0.661034\pi$
$103$	−9.83627	−0.969196	−0.484598	+	0.874737i	$0.661034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.2043	−1.17983	−0.589916	−	0.807465i	$-0.700839\pi$
$107$	−12.2043	−1.17983	−0.589916	+	0.807465i	$0.700839\pi$
$108$	0	0
$109$	9.47222	0.907274	0.453637	−	0.891187i	$-0.350126\pi$
$109$	9.47222	0.907274	0.453637	+	0.891187i	$0.350126\pi$
$110$	0	0
$111$	−1.76381	−0.167414
$112$	0	0
$113$	−15.2706	−1.43653	−0.718267	−	0.695768i	$-0.755064\pi$
$113$	−15.2706	−1.43653	−0.718267	+	0.695768i	$0.755064\pi$
$114$	0	0
$115$	−1.37814	−0.128513
$116$	0	0
$117$	−6.05073	−0.559390
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.94160	0.631055
$122$	0	0
$123$	1.07571	0.0969935
$124$	0	0
$125$	8.41358	0.752534
$126$	0	0
$127$	−5.97703	−0.530375	−0.265188	−	0.964197i	$-0.585434\pi$
$127$	−5.97703	−0.530375	−0.265188	+	0.964197i	$0.585434\pi$
$128$	0	0
$129$	0.177658	0.0156419
$130$	0	0
$131$	−18.4392	−1.61104	−0.805520	−	0.592568i	$-0.798114\pi$
$131$	−18.4392	−1.61104	−0.805520	+	0.592568i	$0.798114\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.58519	0.136431
$136$	0	0
$137$	11.7293	1.00210	0.501050	−	0.865418i	$-0.332947\pi$
$137$	11.7293	1.00210	0.501050	+	0.865418i	$0.332947\pi$
$138$	0	0
$139$	−19.1369	−1.62317	−0.811585	−	0.584234i	$-0.801395\pi$
$139$	−19.1369	−1.62317	−0.811585	+	0.584234i	$0.801395\pi$
$140$	0	0
$141$	1.51777	0.127820
$142$	0	0
$143$	8.79239	0.735257
$144$	0	0
$145$	−9.21769	−0.765488
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.0090	1.72113	0.860563	−	0.509344i	$-0.170112\pi$
$149$	21.0090	1.72113	0.860563	+	0.509344i	$0.170112\pi$
$150$	0	0
$151$	17.7684	1.44597	0.722986	−	0.690863i	$-0.242769\pi$
$151$	17.7684	1.44597	0.722986	+	0.690863i	$0.242769\pi$
$152$	0	0
$153$	2.91495	0.235660
$154$	0	0
$155$	0.937245	0.0752813
$156$	0	0
$157$	−20.6530	−1.64828	−0.824142	−	0.566383i	$-0.808342\pi$
$157$	−20.6530	−1.64828	−0.824142	+	0.566383i	$0.808342\pi$
$158$	0	0
$159$	−0.540175	−0.0428386
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.7615	1.23454	0.617268	−	0.786753i	$-0.288239\pi$
$163$	15.7615	1.23454	0.617268	+	0.786753i	$0.288239\pi$
$164$	0	0
$165$	−1.13517	−0.0883726
$166$	0	0
$167$	23.0442	1.78322	0.891608	−	0.452808i	$-0.149578\pi$
$167$	23.0442	1.78322	0.891608	+	0.452808i	$0.149578\pi$
$168$	0	0
$169$	−8.69123	−0.668556
$170$	0	0
$171$	20.1763	1.54292
$172$	0	0
$173$	−8.79468	−0.668647	−0.334324	−	0.942458i	$-0.608508\pi$
$173$	−8.79468	−0.668647	−0.334324	+	0.942458i	$0.608508\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.12700	−0.159875
$178$	0	0
$179$	3.80147	0.284135	0.142068	−	0.989857i	$-0.454625\pi$
$179$	3.80147	0.284135	0.142068	+	0.989857i	$0.454625\pi$
$180$	0	0
$181$	−9.41070	−0.699491	−0.349746	−	0.936845i	$-0.613732\pi$
$181$	−9.41070	−0.699491	−0.349746	+	0.936845i	$0.613732\pi$
$182$	0	0
$183$	−2.06385	−0.152564
$184$	0	0
$185$	5.55803	0.408635
$186$	0	0
$187$	−4.23575	−0.309749
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.59522	−0.549571	−0.274786	−	0.961506i	$-0.588607\pi$
$191$	−7.59522	−0.549571	−0.274786	+	0.961506i	$0.588607\pi$
$192$	0	0
$193$	−4.09707	−0.294913	−0.147457	−	0.989069i	$-0.547109\pi$
$193$	−4.09707	−0.294913	−0.147457	+	0.989069i	$0.547109\pi$
$194$	0	0
$195$	−0.556296	−0.0398371
$196$	0	0
$197$	15.4115	1.09802	0.549011	−	0.835815i	$-0.315004\pi$
$197$	15.4115	1.09802	0.549011	+	0.835815i	$0.315004\pi$
$198$	0	0
$199$	−5.35308	−0.379470	−0.189735	−	0.981835i	$-0.560763\pi$
$199$	−5.35308	−0.379470	−0.189735	+	0.981835i	$0.560763\pi$
$200$	0	0
$201$	0.149519	0.0105462
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.38972	−0.236748
$206$	0	0
$207$	−4.37147	−0.303838
$208$	0	0
$209$	−29.3184	−2.02800
$210$	0	0
$211$	−18.6090	−1.28110	−0.640548	−	0.767918i	$-0.721293\pi$
$211$	−18.6090	−1.28110	−0.640548	+	0.767918i	$0.721293\pi$
$212$	0	0
$213$	−1.89192	−0.129632
$214$	0	0
$215$	−0.559827	−0.0381799
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.838034	−0.0566291
$220$	0	0
$221$	−2.07576	−0.139631
$222$	0	0
$223$	2.71170	0.181589	0.0907943	−	0.995870i	$-0.471059\pi$
$223$	2.71170	0.181589	0.0907943	+	0.995870i	$0.471059\pi$
$224$	0	0
$225$	12.1131	0.807540
$226$	0	0
$227$	−15.0620	−0.999700	−0.499850	−	0.866112i	$-0.666612\pi$
$227$	−15.0620	−0.999700	−0.499850	+	0.866112i	$0.666612\pi$
$228$	0	0
$229$	−9.88748	−0.653383	−0.326691	−	0.945131i	$-0.605934\pi$
$229$	−9.88748	−0.653383	−0.326691	+	0.945131i	$0.605934\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.8318	−0.840638	−0.420319	−	0.907376i	$-0.638082\pi$
$233$	−12.8318	−0.840638	−0.420319	+	0.907376i	$0.638082\pi$
$234$	0	0
$235$	−4.78273	−0.311991
$236$	0	0
$237$	−0.968210	−0.0628920
$238$	0	0
$239$	−19.3490	−1.25158	−0.625790	−	0.779992i	$-0.715223\pi$
$239$	−19.3490	−1.25158	−0.625790	+	0.779992i	$0.715223\pi$
$240$	0	0
$241$	−14.8004	−0.953376	−0.476688	−	0.879073i	$-0.658163\pi$
$241$	−14.8004	−0.953376	−0.476688	+	0.879073i	$0.658163\pi$
$242$	0	0
$243$	7.57846	0.486158
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.3677	−0.914192
$248$	0	0
$249$	2.45640	0.155668
$250$	0	0
$251$	5.28983	0.333891	0.166945	−	0.985966i	$-0.446610\pi$
$251$	5.28983	0.333891	0.166945	+	0.985966i	$0.446610\pi$
$252$	0	0
$253$	6.35223	0.399361
$254$	0	0
$255$	0.267996	0.0167826
$256$	0	0
$257$	−21.6667	−1.35153	−0.675765	−	0.737117i	$-0.736187\pi$
$257$	−21.6667	−1.35153	−0.675765	+	0.737117i	$0.736187\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−29.2385	−1.80982
$262$	0	0
$263$	12.6189	0.778117	0.389059	−	0.921213i	$-0.372800\pi$
$263$	12.6189	0.778117	0.389059	+	0.921213i	$0.372800\pi$
$264$	0	0
$265$	1.70217	0.104563
$266$	0	0
$267$	2.56554	0.157008
$268$	0	0
$269$	−13.3202	−0.812148	−0.406074	−	0.913840i	$-0.633102\pi$
$269$	−13.3202	−0.812148	−0.406074	+	0.913840i	$0.633102\pi$
$270$	0	0
$271$	5.54207	0.336657	0.168328	−	0.985731i	$-0.446163\pi$
$271$	5.54207	0.336657	0.168328	+	0.985731i	$0.446163\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17.6017	−1.06142
$276$	0	0
$277$	−12.2442	−0.735681	−0.367841	−	0.929889i	$-0.619903\pi$
$277$	−12.2442	−0.735681	−0.367841	+	0.929889i	$0.619903\pi$
$278$	0	0
$279$	2.97294	0.177985
$280$	0	0
$281$	9.03606	0.539046	0.269523	−	0.962994i	$-0.413134\pi$
$281$	9.03606	0.539046	0.269523	+	0.962994i	$0.413134\pi$
$282$	0	0
$283$	30.4360	1.80923	0.904615	−	0.426229i	$-0.140158\pi$
$283$	30.4360	1.80923	0.904615	+	0.426229i	$0.140158\pi$
$284$	0	0
$285$	1.85498	0.109879
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.08925	−0.0638529
$292$	0	0
$293$	−27.8001	−1.62410	−0.812050	−	0.583588i	$-0.801648\pi$
$293$	−27.8001	−1.62410	−0.812050	+	0.583588i	$0.801648\pi$
$294$	0	0
$295$	6.70248	0.390233
$296$	0	0
$297$	−7.30655	−0.423969
$298$	0	0
$299$	3.11295	0.180027
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.40526	−0.195627
$304$	0	0
$305$	6.50349	0.372389
$306$	0	0
$307$	1.22668	0.0700106	0.0350053	−	0.999387i	$-0.488855\pi$
$307$	1.22668	0.0700106	0.0350053	+	0.999387i	$0.488855\pi$
$308$	0	0
$309$	−2.86854	−0.163185
$310$	0	0
$311$	−25.1287	−1.42492	−0.712459	−	0.701713i	$-0.752419\pi$
$311$	−25.1287	−1.42492	−0.712459	+	0.701713i	$0.752419\pi$
$312$	0	0
$313$	−0.349772	−0.0197703	−0.00988513	−	0.999951i	$-0.503147\pi$
$313$	−0.349772	−0.0197703	−0.00988513	+	0.999951i	$0.503147\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.4311	−1.87768	−0.938839	−	0.344355i	$-0.888097\pi$
$317$	−33.4311	−1.87768	−0.938839	+	0.344355i	$0.888097\pi$
$318$	0	0
$319$	42.4868	2.37880
$320$	0	0
$321$	−3.55912	−0.198651
$322$	0	0
$323$	6.92165	0.385131
$324$	0	0
$325$	−8.62582	−0.478474
$326$	0	0
$327$	2.76237	0.152759
$328$	0	0
$329$	0	0
$330$	0	0
$331$	31.3869	1.72518	0.862589	−	0.505905i	$-0.168841\pi$
$331$	31.3869	1.72518	0.862589	+	0.505905i	$0.168841\pi$
$332$	0	0
$333$	17.6301	0.966122
$334$	0	0
$335$	−0.471155	−0.0257419
$336$	0	0
$337$	−18.5350	−1.00967	−0.504833	−	0.863217i	$-0.668446\pi$
$337$	−18.5350	−1.00967	−0.504833	+	0.863217i	$0.668446\pi$
$338$	0	0
$339$	−4.45334	−0.241872
$340$	0	0
$341$	−4.32001	−0.233942
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.401906	−0.0216379
$346$	0	0
$347$	−8.01743	−0.430398	−0.215199	−	0.976570i	$-0.569040\pi$
$347$	−8.01743	−0.430398	−0.215199	+	0.976570i	$0.569040\pi$
$348$	0	0
$349$	−13.6450	−0.730402	−0.365201	−	0.930929i	$-0.619000\pi$
$349$	−13.6450	−0.730402	−0.365201	+	0.930929i	$0.619000\pi$
$350$	0	0
$351$	−3.58062	−0.191119
$352$	0	0
$353$	−32.0572	−1.70623	−0.853115	−	0.521723i	$-0.825290\pi$
$353$	−32.0572	−1.70623	−0.853115	+	0.521723i	$0.825290\pi$
$354$	0	0
$355$	5.96172	0.316415
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.9406	−1.52743	−0.763714	−	0.645554i	$-0.776626\pi$
$359$	−28.9406	−1.52743	−0.763714	+	0.645554i	$0.776626\pi$
$360$	0	0
$361$	28.9092	1.52154
$362$	0	0
$363$	2.02437	0.106252
$364$	0	0
$365$	2.64077	0.138224
$366$	0	0
$367$	−16.4541	−0.858895	−0.429447	−	0.903092i	$-0.641292\pi$
$367$	−16.4541	−0.858895	−0.429447	+	0.903092i	$0.641292\pi$
$368$	0	0
$369$	−10.7522	−0.559736
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.59119	−0.237723	−0.118861	−	0.992911i	$-0.537924\pi$
$373$	−4.59119	−0.237723	−0.118861	+	0.992911i	$0.537924\pi$
$374$	0	0
$375$	2.45364	0.126706
$376$	0	0
$377$	20.8209	1.07233
$378$	0	0
$379$	−15.4301	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$379$	−15.4301	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$380$	0	0
$381$	−1.74307	−0.0893003
$382$	0	0
$383$	0.307511	0.0157131	0.00785655	−	0.999969i	$-0.497499\pi$
$383$	0.307511	0.0157131	0.00785655	+	0.999969i	$0.497499\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.77577	−0.0902675
$388$	0	0
$389$	−4.43047	−0.224634	−0.112317	−	0.993672i	$-0.535827\pi$
$389$	−4.43047	−0.224634	−0.112317	+	0.993672i	$0.535827\pi$
$390$	0	0
$391$	−1.49967	−0.0758416
$392$	0	0
$393$	−5.37740	−0.271254
$394$	0	0
$395$	3.05097	0.153511
$396$	0	0
$397$	−18.4721	−0.927089	−0.463545	−	0.886074i	$-0.653423\pi$
$397$	−18.4721	−0.927089	−0.463545	+	0.886074i	$0.653423\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.76660	−0.487721	−0.243860	−	0.969810i	$-0.578414\pi$
$401$	−9.76660	−0.487721	−0.243860	+	0.969810i	$0.578414\pi$
$402$	0	0
$403$	−2.11705	−0.105458
$404$	0	0
$405$	−7.57393	−0.376352
$406$	0	0
$407$	−25.6185	−1.26986
$408$	0	0
$409$	31.1755	1.54153	0.770763	−	0.637121i	$-0.219875\pi$
$409$	31.1755	1.54153	0.770763	+	0.637121i	$0.219875\pi$
$410$	0	0
$411$	3.42060	0.168726
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.74046	−0.379964
$416$	0	0
$417$	−5.58087	−0.273296
$418$	0	0
$419$	−5.20108	−0.254090	−0.127045	−	0.991897i	$-0.540549\pi$
$419$	−5.20108	−0.254090	−0.127045	+	0.991897i	$0.540549\pi$
$420$	0	0
$421$	−18.8562	−0.918994	−0.459497	−	0.888179i	$-0.651970\pi$
$421$	−18.8562	−0.918994	−0.459497	+	0.888179i	$0.651970\pi$
$422$	0	0
$423$	−15.1708	−0.737630
$424$	0	0
$425$	4.15550	0.201572
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.56411	0.123797
$430$	0	0
$431$	11.0183	0.530732	0.265366	−	0.964148i	$-0.414507\pi$
$431$	11.0183	0.530732	0.265366	+	0.964148i	$0.414507\pi$
$432$	0	0
$433$	4.52386	0.217403	0.108701	−	0.994074i	$-0.465331\pi$
$433$	4.52386	0.217403	0.108701	+	0.994074i	$0.465331\pi$
$434$	0	0
$435$	−2.68814	−0.128887
$436$	0	0
$437$	−10.3802	−0.496552
$438$	0	0
$439$	1.62534	0.0775732	0.0387866	−	0.999248i	$-0.487651\pi$
$439$	1.62534	0.0775732	0.0387866	+	0.999248i	$0.487651\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.1103	1.52561	0.762804	−	0.646630i	$-0.223822\pi$
$443$	32.1103	1.52561	0.762804	+	0.646630i	$0.223822\pi$
$444$	0	0
$445$	−8.08439	−0.383237
$446$	0	0
$447$	6.12683	0.289789
$448$	0	0
$449$	33.4297	1.57765	0.788824	−	0.614620i	$-0.210690\pi$
$449$	33.4297	1.57765	0.788824	+	0.614620i	$0.210690\pi$
$450$	0	0
$451$	15.6241	0.735711
$452$	0	0
$453$	5.18177	0.243461
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.5266	−0.773084	−0.386542	−	0.922272i	$-0.626331\pi$
$457$	−16.5266	−0.773084	−0.386542	+	0.922272i	$0.626331\pi$
$458$	0	0
$459$	1.72497	0.0805147
$460$	0	0
$461$	−6.73410	−0.313638	−0.156819	−	0.987627i	$-0.550124\pi$
$461$	−6.73410	−0.313638	−0.156819	+	0.987627i	$0.550124\pi$
$462$	0	0
$463$	18.7842	0.872975	0.436487	−	0.899710i	$-0.356222\pi$
$463$	18.7842	0.872975	0.436487	+	0.899710i	$0.356222\pi$
$464$	0	0
$465$	0.273328	0.0126753
$466$	0	0
$467$	−4.78208	−0.221288	−0.110644	−	0.993860i	$-0.535291\pi$
$467$	−4.78208	−0.221288	−0.110644	+	0.993860i	$0.535291\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.02299	−0.277525
$472$	0	0
$473$	2.58039	0.118647
$474$	0	0
$475$	28.7629	1.31973
$476$	0	0
$477$	5.39927	0.247216
$478$	0	0
$479$	18.9861	0.867496	0.433748	−	0.901034i	$-0.357191\pi$
$479$	18.9861	0.867496	0.433748	+	0.901034i	$0.357191\pi$
$480$	0	0
$481$	−12.5545	−0.572435
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.43238	0.155856
$486$	0	0
$487$	27.7269	1.25643	0.628213	−	0.778042i	$-0.283787\pi$
$487$	27.7269	1.25643	0.628213	+	0.778042i	$0.283787\pi$
$488$	0	0
$489$	4.59651	0.207861
$490$	0	0
$491$	7.45169	0.336290	0.168145	−	0.985762i	$-0.446222\pi$
$491$	7.45169	0.336290	0.168145	+	0.985762i	$0.446222\pi$
$492$	0	0
$493$	−10.0305	−0.451752
$494$	0	0
$495$	11.3465	0.509986
$496$	0	0
$497$	0	0
$498$	0	0
$499$	24.4689	1.09538	0.547689	−	0.836682i	$-0.315508\pi$
$499$	24.4689	1.09538	0.547689	+	0.836682i	$0.315508\pi$
$500$	0	0
$501$	6.72036	0.300244
$502$	0	0
$503$	−5.66361	−0.252528	−0.126264	−	0.991997i	$-0.540299\pi$
$503$	−5.66361	−0.252528	−0.126264	+	0.991997i	$0.540299\pi$
$504$	0	0
$505$	10.7305	0.477500
$506$	0	0
$507$	−2.53461	−0.112566
$508$	0	0
$509$	−20.3716	−0.902953	−0.451477	−	0.892283i	$-0.649103\pi$
$509$	−20.3716	−0.902953	−0.451477	+	0.892283i	$0.649103\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	11.9396	0.527148
$514$	0	0
$515$	9.03918	0.398314
$516$	0	0
$517$	22.0449	0.969533
$518$	0	0
$519$	−2.56478	−0.112581
$520$	0	0
$521$	14.6419	0.641474	0.320737	−	0.947168i	$-0.396069\pi$
$521$	14.6419	0.641474	0.320737	+	0.947168i	$0.396069\pi$
$522$	0	0
$523$	24.1198	1.05469	0.527343	−	0.849652i	$-0.323188\pi$
$523$	24.1198	1.05469	0.527343	+	0.849652i	$0.323188\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.01989	0.0444272
$528$	0	0
$529$	−20.7510	−0.902217
$530$	0	0
$531$	21.2602	0.922616
$532$	0	0
$533$	7.65670	0.331648
$534$	0	0
$535$	11.2153	0.484880
$536$	0	0
$537$	1.10862	0.0478404
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.8789	0.596702	0.298351	−	0.954456i	$-0.403563\pi$
$541$	13.8789	0.596702	0.298351	+	0.954456i	$0.403563\pi$
$542$	0	0
$543$	−2.74443	−0.117775
$544$	0	0
$545$	−8.70463	−0.372865
$546$	0	0
$547$	−9.36991	−0.400628	−0.200314	−	0.979732i	$-0.564196\pi$
$547$	−9.36991	−0.400628	−0.200314	+	0.979732i	$0.564196\pi$
$548$	0	0
$549$	20.6291	0.880427
$550$	0	0
$551$	−69.4277	−2.95772
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.62088	0.0688026
$556$	0	0
$557$	−3.66251	−0.155186	−0.0775929	−	0.996985i	$-0.524723\pi$
$557$	−3.66251	−0.155186	−0.0775929	+	0.996985i	$0.524723\pi$
$558$	0	0
$559$	1.26454	0.0534843
$560$	0	0
$561$	−1.23527	−0.0521530
$562$	0	0
$563$	26.3625	1.11105	0.555523	−	0.831501i	$-0.312518\pi$
$563$	26.3625	1.11105	0.555523	+	0.831501i	$0.312518\pi$
$564$	0	0
$565$	14.0331	0.590377
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.9091	−0.457333	−0.228666	−	0.973505i	$-0.573436\pi$
$569$	−10.9091	−0.457333	−0.228666	+	0.973505i	$0.573436\pi$
$570$	0	0
$571$	22.4156	0.938064	0.469032	−	0.883181i	$-0.344603\pi$
$571$	22.4156	0.938064	0.469032	+	0.883181i	$0.344603\pi$
$572$	0	0
$573$	−2.21499	−0.0925323
$574$	0	0
$575$	−6.23189	−0.259888
$576$	0	0
$577$	−25.7241	−1.07091	−0.535453	−	0.844565i	$-0.679859\pi$
$577$	−25.7241	−1.07091	−0.535453	+	0.844565i	$0.679859\pi$
$578$	0	0
$579$	−1.19482	−0.0496551
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.84575	−0.324938
$584$	0	0
$585$	5.56041	0.229895
$586$	0	0
$587$	−46.0724	−1.90161	−0.950805	−	0.309790i	$-0.899741\pi$
$587$	−46.0724	−1.90161	−0.950805	+	0.309790i	$0.899741\pi$
$588$	0	0
$589$	7.05934	0.290875
$590$	0	0
$591$	4.49443	0.184876
$592$	0	0
$593$	16.7710	0.688703	0.344351	−	0.938841i	$-0.388099\pi$
$593$	16.7710	0.688703	0.344351	+	0.938841i	$0.388099\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.56111	−0.0638921
$598$	0	0
$599$	11.0366	0.450942	0.225471	−	0.974250i	$-0.427608\pi$
$599$	11.0366	0.450942	0.225471	+	0.974250i	$0.427608\pi$
$600$	0	0
$601$	−13.1473	−0.536290	−0.268145	−	0.963379i	$-0.586411\pi$
$601$	−13.1473	−0.536290	−0.268145	+	0.963379i	$0.586411\pi$
$602$	0	0
$603$	−1.49450	−0.0608608
$604$	0	0
$605$	−6.37909	−0.259347
$606$	0	0
$607$	−5.44007	−0.220806	−0.110403	−	0.993887i	$-0.535214\pi$
$607$	−5.44007	−0.220806	−0.110403	+	0.993887i	$0.535214\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.8032	0.437052
$612$	0	0
$613$	25.6865	1.03747	0.518734	−	0.854936i	$-0.326403\pi$
$613$	25.6865	1.03747	0.518734	+	0.854936i	$0.326403\pi$
$614$	0	0
$615$	−0.988540	−0.0398618
$616$	0	0
$617$	30.6828	1.23524	0.617622	−	0.786475i	$-0.288096\pi$
$617$	30.6828	1.23524	0.617622	+	0.786475i	$0.288096\pi$
$618$	0	0
$619$	8.62156	0.346530	0.173265	−	0.984875i	$-0.444568\pi$
$619$	8.62156	0.346530	0.173265	+	0.984875i	$0.444568\pi$
$620$	0	0
$621$	−2.58689	−0.103808
$622$	0	0
$623$	0	0
$624$	0	0
$625$	13.0457	0.521829
$626$	0	0
$627$	−8.55009	−0.341458
$628$	0	0
$629$	6.04815	0.241155
$630$	0	0
$631$	−49.1275	−1.95573	−0.977867	−	0.209228i	$-0.932905\pi$
$631$	−49.1275	−1.95573	−0.977867	+	0.209228i	$0.932905\pi$
$632$	0	0
$633$	−5.42692	−0.215701
$634$	0	0
$635$	5.49268	0.217970
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.9106	0.748090
$640$	0	0
$641$	−14.7169	−0.581282	−0.290641	−	0.956832i	$-0.593868\pi$
$641$	−14.7169	−0.581282	−0.290641	+	0.956832i	$0.593868\pi$
$642$	0	0
$643$	40.6089	1.60146	0.800730	−	0.599026i	$-0.204445\pi$
$643$	40.6089	1.60146	0.800730	+	0.599026i	$0.204445\pi$
$644$	0	0
$645$	−0.163262	−0.00642843
$646$	0	0
$647$	−15.4655	−0.608012	−0.304006	−	0.952670i	$-0.598324\pi$
$647$	−15.4655	−0.608012	−0.304006	+	0.952670i	$0.598324\pi$
$648$	0	0
$649$	−30.8935	−1.21268
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.6585	−0.925829	−0.462914	−	0.886403i	$-0.653196\pi$
$653$	−23.6585	−0.925829	−0.462914	+	0.886403i	$0.653196\pi$
$654$	0	0
$655$	16.9450	0.662095
$656$	0	0
$657$	8.37651	0.326799
$658$	0	0
$659$	−20.0742	−0.781981	−0.390990	−	0.920395i	$-0.627867\pi$
$659$	−20.0742	−0.781981	−0.390990	+	0.920395i	$0.627867\pi$
$660$	0	0
$661$	−35.8354	−1.39384	−0.696918	−	0.717151i	$-0.745446\pi$
$661$	−35.8354	−1.39384	−0.696918	+	0.717151i	$0.745446\pi$
$662$	0	0
$663$	−0.605350	−0.0235099
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.0425	0.582447
$668$	0	0
$669$	0.790808	0.0305744
$670$	0	0
$671$	−29.9763	−1.15722
$672$	0	0
$673$	6.74520	0.260008	0.130004	−	0.991513i	$-0.458501\pi$
$673$	6.74520	0.260008	0.130004	+	0.991513i	$0.458501\pi$
$674$	0	0
$675$	7.16812	0.275901
$676$	0	0
$677$	20.9493	0.805146	0.402573	−	0.915388i	$-0.368116\pi$
$677$	20.9493	0.805146	0.402573	+	0.915388i	$0.368116\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.39251	−0.168321
$682$	0	0
$683$	−6.25012	−0.239154	−0.119577	−	0.992825i	$-0.538154\pi$
$683$	−6.25012	−0.239154	−0.119577	+	0.992825i	$0.538154\pi$
$684$	0	0
$685$	−10.7788	−0.411837
$686$	0	0
$687$	−2.88347	−0.110011
$688$	0	0
$689$	−3.84486	−0.146477
$690$	0	0
$691$	−12.4155	−0.472306	−0.236153	−	0.971716i	$-0.575887\pi$
$691$	−12.4155	−0.472306	−0.236153	+	0.971716i	$0.575887\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.5861	0.667080
$696$	0	0
$697$	−3.68863	−0.139717
$698$	0	0
$699$	−3.74212	−0.141540
$700$	0	0
$701$	20.1448	0.760860	0.380430	−	0.924810i	$-0.375776\pi$
$701$	20.1448	0.760860	0.380430	+	0.924810i	$0.375776\pi$
$702$	0	0
$703$	41.8632	1.57890
$704$	0	0
$705$	−1.39478	−0.0525305
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.3308	−0.575759	−0.287880	−	0.957667i	$-0.592950\pi$
$709$	−15.3308	−0.575759	−0.287880	+	0.957667i	$0.592950\pi$
$710$	0	0
$711$	9.67767	0.362941
$712$	0	0
$713$	−1.52950	−0.0572803
$714$	0	0
$715$	−8.07990	−0.302171
$716$	0	0
$717$	−5.64271	−0.210731
$718$	0	0
$719$	−6.31496	−0.235508	−0.117754	−	0.993043i	$-0.537569\pi$
$719$	−6.31496	−0.235508	−0.117754	+	0.993043i	$0.537569\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.31622	−0.160522
$724$	0	0
$725$	−41.6819	−1.54803
$726$	0	0
$727$	−5.24320	−0.194459	−0.0972297	−	0.995262i	$-0.530998\pi$
$727$	−5.24320	−0.194459	−0.0972297	+	0.995262i	$0.530998\pi$
$728$	0	0
$729$	−22.5153	−0.833901
$730$	0	0
$731$	−0.609194	−0.0225318
$732$	0	0
$733$	34.8235	1.28623	0.643117	−	0.765768i	$-0.277641\pi$
$733$	34.8235	1.28623	0.643117	+	0.765768i	$0.277641\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.17168	0.0799948
$738$	0	0
$739$	14.3046	0.526203	0.263102	−	0.964768i	$-0.415255\pi$
$739$	14.3046	0.526203	0.263102	+	0.964768i	$0.415255\pi$
$740$	0	0
$741$	−4.19002	−0.153924
$742$	0	0
$743$	−42.1623	−1.54679	−0.773393	−	0.633927i	$-0.781442\pi$
$743$	−42.1623	−1.54679	−0.773393	+	0.633927i	$0.781442\pi$
$744$	0	0
$745$	−19.3065	−0.707337
$746$	0	0
$747$	−24.5527	−0.898337
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.7831	1.34223	0.671116	−	0.741352i	$-0.265815\pi$
$751$	36.7831	1.34223	0.671116	+	0.741352i	$0.265815\pi$
$752$	0	0
$753$	1.54267	0.0562178
$754$	0	0
$755$	−16.3285	−0.594256
$756$	0	0
$757$	−35.7753	−1.30028	−0.650138	−	0.759816i	$-0.725289\pi$
$757$	−35.7753	−1.30028	−0.650138	+	0.759816i	$0.725289\pi$
$758$	0	0
$759$	1.85249	0.0672413
$760$	0	0
$761$	−7.42261	−0.269069	−0.134535	−	0.990909i	$-0.542954\pi$
$761$	−7.42261	−0.269069	−0.134535	+	0.990909i	$0.542954\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.67874	−0.0968500
$766$	0	0
$767$	−15.1396	−0.546658
$768$	0	0
$769$	−21.6273	−0.779899	−0.389950	−	0.920836i	$-0.627508\pi$
$769$	−21.6273	−0.779899	−0.389950	+	0.920836i	$0.627508\pi$
$770$	0	0
$771$	−6.31863	−0.227560
$772$	0	0
$773$	38.9887	1.40233	0.701164	−	0.713001i	$-0.252664\pi$
$773$	38.9887	1.40233	0.701164	+	0.713001i	$0.252664\pi$
$774$	0	0
$775$	4.23817	0.152239
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.5314	−0.914757
$780$	0	0
$781$	−27.4792	−0.983282
$782$	0	0
$783$	−17.3023	−0.618336
$784$	0	0
$785$	18.9793	0.677401
$786$	0	0
$787$	−1.10721	−0.0394677	−0.0197339	−	0.999805i	$-0.506282\pi$
$787$	−1.10721	−0.0394677	−0.0197339	+	0.999805i	$0.506282\pi$
$788$	0	0
$789$	3.68005	0.131013
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14.6901	−0.521660
$794$	0	0
$795$	0.496401	0.0176055
$796$	0	0
$797$	−28.9926	−1.02697	−0.513485	−	0.858099i	$-0.671646\pi$
$797$	−28.9926	−1.02697	−0.513485	+	0.858099i	$0.671646\pi$
$798$	0	0
$799$	−5.20448	−0.184121
$800$	0	0
$801$	−25.6436	−0.906074
$802$	0	0
$803$	−12.1720	−0.429541
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.88456	−0.136743
$808$	0	0
$809$	0.390358	0.0137242	0.00686212	−	0.999976i	$-0.497816\pi$
$809$	0.390358	0.0137242	0.00686212	+	0.999976i	$0.497816\pi$
$810$	0	0
$811$	33.3616	1.17148	0.585742	−	0.810498i	$-0.300803\pi$
$811$	33.3616	1.17148	0.585742	+	0.810498i	$0.300803\pi$
$812$	0	0
$813$	1.61623	0.0566835
$814$	0	0
$815$	−14.4843	−0.507362
$816$	0	0
$817$	−4.21662	−0.147521
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.55974	0.263837	0.131918	−	0.991261i	$-0.457886\pi$
$821$	7.55974	0.263837	0.131918	+	0.991261i	$0.457886\pi$
$822$	0	0
$823$	−42.0005	−1.46405	−0.732023	−	0.681280i	$-0.761424\pi$
$823$	−42.0005	−1.46405	−0.732023	+	0.681280i	$0.761424\pi$
$824$	0	0
$825$	−5.13316	−0.178714
$826$	0	0
$827$	44.5625	1.54959	0.774795	−	0.632212i	$-0.217853\pi$
$827$	44.5625	1.54959	0.774795	+	0.632212i	$0.217853\pi$
$828$	0	0
$829$	−0.417296	−0.0144933	−0.00724664	−	0.999974i	$-0.502307\pi$
$829$	−0.417296	−0.0144933	−0.00724664	+	0.999974i	$0.502307\pi$
$830$	0	0
$831$	−3.57075	−0.123868
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−21.1768	−0.732855
$836$	0	0
$837$	1.75928	0.0608098
$838$	0	0
$839$	28.9911	1.00088	0.500441	−	0.865771i	$-0.333171\pi$
$839$	28.9911	1.00088	0.500441	+	0.865771i	$0.333171\pi$
$840$	0	0
$841$	71.6113	2.46936
$842$	0	0
$843$	2.63517	0.0907602
$844$	0	0
$845$	7.98694	0.274759
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.87600	0.304624
$850$	0	0
$851$	−9.07023	−0.310923
$852$	0	0
$853$	40.1812	1.37578	0.687889	−	0.725816i	$-0.258538\pi$
$853$	40.1812	1.37578	0.687889	+	0.725816i	$0.258538\pi$
$854$	0	0
$855$	−18.5413	−0.634099
$856$	0	0
$857$	14.6607	0.500800	0.250400	−	0.968142i	$-0.419438\pi$
$857$	14.6607	0.500800	0.250400	+	0.968142i	$0.419438\pi$
$858$	0	0
$859$	57.4224	1.95923	0.979614	−	0.200890i	$-0.0643834\pi$
$859$	57.4224	1.95923	0.979614	+	0.200890i	$0.0643834\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.72184	−0.296895	−0.148448	−	0.988920i	$-0.547428\pi$
$863$	−8.72184	−0.296895	−0.148448	+	0.988920i	$0.547428\pi$
$864$	0	0
$865$	8.08200	0.274796
$866$	0	0
$867$	0.291629	0.00990423
$868$	0	0
$869$	−14.0627	−0.477046
$870$	0	0
$871$	1.06424	0.0360605
$872$	0	0
$873$	10.8875	0.368486
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.1000	0.847565	0.423783	−	0.905764i	$-0.360702\pi$
$877$	25.1000	0.847565	0.423783	+	0.905764i	$0.360702\pi$
$878$	0	0
$879$	−8.10731	−0.273453
$880$	0	0
$881$	−3.85588	−0.129908	−0.0649539	−	0.997888i	$-0.520690\pi$
$881$	−3.85588	−0.129908	−0.0649539	+	0.997888i	$0.520690\pi$
$882$	0	0
$883$	44.5288	1.49851	0.749257	−	0.662279i	$-0.230411\pi$
$883$	44.5288	1.49851	0.749257	+	0.662279i	$0.230411\pi$
$884$	0	0
$885$	1.95464	0.0657044
$886$	0	0
$887$	−28.1866	−0.946414	−0.473207	−	0.880951i	$-0.656904\pi$
$887$	−28.1866	−0.946414	−0.473207	+	0.880951i	$0.656904\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	34.9103	1.16954
$892$	0	0
$893$	−36.0236	−1.20548
$894$	0	0
$895$	−3.49342	−0.116772
$896$	0	0
$897$	0.907826	0.0303114
$898$	0	0
$899$	−10.2301	−0.341191
$900$	0	0
$901$	1.85227	0.0617080
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.64810	0.287472
$906$	0	0
$907$	−23.5411	−0.781668	−0.390834	−	0.920461i	$-0.627813\pi$
$907$	−23.5411	−0.781668	−0.390834	+	0.920461i	$0.627813\pi$
$908$	0	0
$909$	34.0371	1.12894
$910$	0	0
$911$	0.648280	0.0214785	0.0107392	−	0.999942i	$-0.496582\pi$
$911$	0.648280	0.0214785	0.0107392	+	0.999942i	$0.496582\pi$
$912$	0	0
$913$	35.6779	1.18076
$914$	0	0
$915$	1.89661	0.0626998
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−20.9924	−0.692476	−0.346238	−	0.938147i	$-0.612541\pi$
$919$	−20.9924	−0.692476	−0.346238	+	0.938147i	$0.612541\pi$
$920$	0	0
$921$	0.357737	0.0117878
$922$	0	0
$923$	−13.4663	−0.443250
$924$	0	0
$925$	25.1331	0.826371
$926$	0	0
$927$	28.6723	0.941720
$928$	0	0
$929$	19.2232	0.630693	0.315347	−	0.948977i	$-0.397879\pi$
$929$	19.2232	0.630693	0.315347	+	0.948977i	$0.397879\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.32826	−0.239916
$934$	0	0
$935$	3.89251	0.127299
$936$	0	0
$937$	−55.9691	−1.82843	−0.914215	−	0.405229i	$-0.867192\pi$
$937$	−55.9691	−1.82843	−0.914215	+	0.405229i	$0.867192\pi$
$938$	0	0
$939$	−0.102003	−0.00332876
$940$	0	0
$941$	−4.68016	−0.152569	−0.0762845	−	0.997086i	$-0.524306\pi$
$941$	−4.68016	−0.152569	−0.0762845	+	0.997086i	$0.524306\pi$
$942$	0	0
$943$	5.53173	0.180138
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.6864	0.509738	0.254869	−	0.966976i	$-0.417968\pi$
$947$	15.6864	0.509738	0.254869	+	0.966976i	$0.417968\pi$
$948$	0	0
$949$	−5.96497	−0.193631
$950$	0	0
$951$	−9.74947	−0.316148
$952$	0	0
$953$	46.8719	1.51833	0.759164	−	0.650899i	$-0.225608\pi$
$953$	46.8719	1.51833	0.759164	+	0.650899i	$0.225608\pi$
$954$	0	0
$955$	6.97974	0.225859
$956$	0	0
$957$	12.3904	0.400524
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.9598	−0.966446
$962$	0	0
$963$	35.5749	1.14639
$964$	0	0
$965$	3.76506	0.121202
$966$	0	0
$967$	32.8507	1.05641	0.528204	−	0.849118i	$-0.322866\pi$
$967$	32.8507	1.05641	0.528204	+	0.849118i	$0.322866\pi$
$968$	0	0
$969$	2.01855	0.0648452
$970$	0	0
$971$	46.8602	1.50382	0.751908	−	0.659268i	$-0.229134\pi$
$971$	46.8602	1.50382	0.751908	+	0.659268i	$0.229134\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.51554	−0.0805616
$976$	0	0
$977$	−33.8208	−1.08202	−0.541012	−	0.841015i	$-0.681959\pi$
$977$	−33.8208	−1.08202	−0.541012	+	0.841015i	$0.681959\pi$
$978$	0	0
$979$	37.2631	1.19093
$980$	0	0
$981$	−27.6111	−0.881553
$982$	0	0
$983$	−8.84124	−0.281992	−0.140996	−	0.990010i	$-0.545030\pi$
$983$	−8.84124	−0.281992	−0.140996	+	0.990010i	$0.545030\pi$
$984$	0	0
$985$	−14.1626	−0.451258
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.913590	0.0290505
$990$	0	0
$991$	−6.03473	−0.191700	−0.0958498	−	0.995396i	$-0.530557\pi$
$991$	−6.03473	−0.191700	−0.0958498	+	0.995396i	$0.530557\pi$
$992$	0	0
$993$	9.15331	0.290472
$994$	0	0
$995$	4.91929	0.155952
$996$	0	0
$997$	−30.3580	−0.961447	−0.480723	−	0.876872i	$-0.659626\pi$
$997$	−30.3580	−0.961447	−0.480723	+	0.876872i	$0.659626\pi$
$998$	0	0
$999$	10.4329	0.330082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.bd.1.7	yes	12
7.6	odd	2		6664.2.a.bc.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6664.2.a.bc.1.6	✓	12	7.6	odd	2
6664.2.a.bd.1.7	yes	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+0.291629 q^{3} -0.918965 q^{5} -2.91495 q^{9} +4.23575 q^{11} +2.07576 q^{13} -0.267996 q^{15} -1.00000 q^{17} -6.92165 q^{19} +1.49967 q^{23} -4.15550 q^{25} -1.72497 q^{27} +10.0305 q^{29} -1.01989 q^{31} +1.23527 q^{33} -6.04815 q^{37} +0.605350 q^{39} +3.68863 q^{41} +0.609194 q^{43} +2.67874 q^{45} +5.20448 q^{47} -0.291629 q^{51} -1.85227 q^{53} -3.89251 q^{55} -2.01855 q^{57} -7.29351 q^{59} -7.07698 q^{61} -1.90755 q^{65} +0.512702 q^{67} +0.437347 q^{69} -6.48743 q^{71} -2.87363 q^{73} -1.21186 q^{75} -3.32001 q^{79} +8.24181 q^{81} +8.42303 q^{83} +0.918965 q^{85} +2.92519 q^{87} +8.79728 q^{89} -0.297430 q^{93} +6.36075 q^{95} -3.73505 q^{97} -12.3470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} - 8 q^{5} + 8 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} - 12 q^{17} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 4 q^{31} - 16 q^{33} + 12 q^{37} - 16 q^{39} - 24 q^{41} - 4 q^{43} - 28 q^{45} - 28 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 - 8 * q^5 + 8 * q^9 - 4 * q^11 - 12 * q^13 - 8 * q^15 - 12 * q^17 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 4 * q^31 - 16 * q^33 + 12 * q^37 - 16 * q^39 - 24 * q^41 - 4 * q^43 - 28 * q^45 - 28 * q^47 - 4 * q^51 + 4 * q^55 + 12 * q^57 - 36 * q^61 + 20 * q^65 - 4 * q^69 - 40 * q^73 - 20 * q^75 - 4 * q^79 - 4 * q^81 - 12 * q^83 + 8 * q^85 - 36 * q^89 - 16 * q^93 - 8 * q^95 - 12 * q^97 - 20 * q^99

Introduction

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Modular forms

Varieties

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Database

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